LMFDB - Embedded newform 23.16.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,16,Mod(1,23)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	23.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.8195061730$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 68942 x^{10} - 977032 x^{9} + 1644150380 x^{8} + 50352376602 x^{7} + \cdots + 34\!\cdots\!76 $ x^12 - 4x^11 - 68942x^10 - 977032x^9 + 1644150380x^8 + 50352376602x^7 - 15614385123802x^6 - 702635376772966x^5 + 50805039593451715x^4 + 3084648153192808082x^3 + 3701665188552036536x^2 - 1120833250638227525562*x + 3479819676773674797376
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	multiple of $ 2^{22}\cdot 3^{6}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$15.1401$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.28020 q^{2} -6881.62 q^{3} -32699.4 q^{4} -91711.3 q^{5} -56981.2 q^{6} +1.74802e6 q^{7} -542083. q^{8} +3.30077e7 q^{9} +O(q^{10})$	\(q+8.28020 q^{2} -6881.62 q^{3} -32699.4 q^{4} -91711.3 q^{5} -56981.2 q^{6} +1.74802e6 q^{7} -542083. q^{8} +3.30077e7 q^{9} -759388. q^{10} +1.56771e7 q^{11} +2.25025e8 q^{12} +2.14877e7 q^{13} +1.44740e7 q^{14} +6.31122e8 q^{15} +1.06701e9 q^{16} -1.10073e9 q^{17} +2.73311e8 q^{18} -1.03531e9 q^{19} +2.99891e9 q^{20} -1.20292e10 q^{21} +1.29809e8 q^{22} +3.40483e9 q^{23} +3.73041e9 q^{24} -2.21066e10 q^{25} +1.77923e8 q^{26} -1.28403e11 q^{27} -5.71594e10 q^{28} +1.26499e11 q^{29} +5.22582e9 q^{30} +1.97030e11 q^{31} +2.65980e10 q^{32} -1.07883e11 q^{33} -9.11427e9 q^{34} -1.60314e11 q^{35} -1.07933e12 q^{36} +6.58396e11 q^{37} -8.57261e9 q^{38} -1.47870e11 q^{39} +4.97152e10 q^{40} +2.90098e11 q^{41} -9.96044e10 q^{42} -3.25665e12 q^{43} -5.12631e11 q^{44} -3.02718e12 q^{45} +2.81926e10 q^{46} -3.48829e12 q^{47} -7.34273e12 q^{48} -1.69197e12 q^{49} -1.83047e11 q^{50} +7.57481e12 q^{51} -7.02637e11 q^{52} +5.53420e12 q^{53} -1.06320e12 q^{54} -1.43776e12 q^{55} -9.47575e11 q^{56} +7.12464e12 q^{57} +1.04744e12 q^{58} +2.14287e13 q^{59} -2.06373e13 q^{60} -1.84260e13 q^{61} +1.63145e12 q^{62} +5.76983e13 q^{63} -3.47434e13 q^{64} -1.97067e12 q^{65} -8.93297e11 q^{66} +3.99125e13 q^{67} +3.59933e13 q^{68} -2.34307e13 q^{69} -1.32743e12 q^{70} -1.07718e14 q^{71} -1.78930e13 q^{72} +1.14573e11 q^{73} +5.45165e12 q^{74} +1.52129e14 q^{75} +3.38542e13 q^{76} +2.74039e13 q^{77} -1.22440e12 q^{78} +2.01176e14 q^{79} -9.78566e13 q^{80} +4.09995e14 q^{81} +2.40207e12 q^{82} +3.66828e14 q^{83} +3.93349e14 q^{84} +1.00949e14 q^{85} -2.69657e13 q^{86} -8.70520e14 q^{87} -8.49827e12 q^{88} -4.48834e14 q^{89} -2.50657e13 q^{90} +3.75611e13 q^{91} -1.11336e14 q^{92} -1.35589e15 q^{93} -2.88837e13 q^{94} +9.49500e13 q^{95} -1.83037e14 q^{96} -6.09640e14 q^{97} -1.40099e13 q^{98} +5.17464e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9}+O(q^{10}) $ 12 * q - 256 * q^2 - 1745 * q^3 + 163840 * q^4 - 204476 * q^5 - 199430 * q^6 - 5717328 * q^7 + 10323168 * q^8 + 32660341 * q^9	\( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9} - 137846540 q^{10} - 87636002 q^{11} - 398208076 q^{12} - 292496079 q^{13} + 415954912 q^{14} + 548079030 q^{15} + 4273503168 q^{16} - 2462528162 q^{17} + 7261215718 q^{18} + 175321758 q^{19} + 2660811480 q^{20} + 205665472 q^{21} - 21718153768 q^{22} + 40857905364 q^{23} - 63413289624 q^{24} + 20443225284 q^{25} - 137268652810 q^{26} - 151915208903 q^{27} - 325638721712 q^{28} - 164667697193 q^{29} - 356944003956 q^{30} + 20222384151 q^{31} - 369109524032 q^{32} + 132365097022 q^{33} - 582887018988 q^{34} - 1578083373112 q^{35} - 1903913944516 q^{36} - 869669414912 q^{37} - 5525312078376 q^{38} - 5762413466499 q^{39} - 4733269274576 q^{40} - 7510147709883 q^{41} - 7436463221624 q^{42} - 5682603487020 q^{43} - 11849381658176 q^{44} - 10780493432442 q^{45} - 871635314432 q^{46} - 5828073094301 q^{47} - 29418911592496 q^{48} - 6518780198860 q^{49} - 16781003942456 q^{50} - 771327642584 q^{51} - 3841511618340 q^{52} + 1452974784324 q^{53} - 32167598069522 q^{54} - 14882020037092 q^{55} + 416192984288 q^{56} - 12135794354818 q^{57} - 60065613521022 q^{58} - 11503084624084 q^{59} - 6378557828664 q^{60} - 23587566667200 q^{61} + 49359974806402 q^{62} + 87886039196104 q^{63} + 80321007324160 q^{64} + 54548135308138 q^{65} + 316922278045948 q^{66} + 61525019345122 q^{67} + 45114528974104 q^{68} - 5941420405015 q^{69} + 374016699556320 q^{70} + 197895887067063 q^{71} + 439014895837656 q^{72} - 22888563242709 q^{73} + 694696716227036 q^{74} + 612085940395201 q^{75} + 301381886149904 q^{76} + 209007839834200 q^{77} + 350406148895766 q^{78} + 229938065096294 q^{79} + 555529032250016 q^{80} + 37596523177660 q^{81} - 414508112727306 q^{82} + 369402590629184 q^{83} + 559863541234208 q^{84} - 343366303925348 q^{85} + 12\!\cdots\!08 q^{86}+ \cdots - 32\!\cdots\!24 q^{99}+O(q^{100}) \) 12 * q - 256 * q^2 - 1745 * q^3 + 163840 * q^4 - 204476 * q^5 - 199430 * q^6 - 5717328 * q^7 + 10323168 * q^8 + 32660341 * q^9 - 137846540 * q^10 - 87636002 * q^11 - 398208076 * q^12 - 292496079 * q^13 + 415954912 * q^14 + 548079030 * q^15 + 4273503168 * q^16 - 2462528162 * q^17 + 7261215718 * q^18 + 175321758 * q^19 + 2660811480 * q^20 + 205665472 * q^21 - 21718153768 * q^22 + 40857905364 * q^23 - 63413289624 * q^24 + 20443225284 * q^25 - 137268652810 * q^26 - 151915208903 * q^27 - 325638721712 * q^28 - 164667697193 * q^29 - 356944003956 * q^30 + 20222384151 * q^31 - 369109524032 * q^32 + 132365097022 * q^33 - 582887018988 * q^34 - 1578083373112 * q^35 - 1903913944516 * q^36 - 869669414912 * q^37 - 5525312078376 * q^38 - 5762413466499 * q^39 - 4733269274576 * q^40 - 7510147709883 * q^41 - 7436463221624 * q^42 - 5682603487020 * q^43 - 11849381658176 * q^44 - 10780493432442 * q^45 - 871635314432 * q^46 - 5828073094301 * q^47 - 29418911592496 * q^48 - 6518780198860 * q^49 - 16781003942456 * q^50 - 771327642584 * q^51 - 3841511618340 * q^52 + 1452974784324 * q^53 - 32167598069522 * q^54 - 14882020037092 * q^55 + 416192984288 * q^56 - 12135794354818 * q^57 - 60065613521022 * q^58 - 11503084624084 * q^59 - 6378557828664 * q^60 - 23587566667200 * q^61 + 49359974806402 * q^62 + 87886039196104 * q^63 + 80321007324160 * q^64 + 54548135308138 * q^65 + 316922278045948 * q^66 + 61525019345122 * q^67 + 45114528974104 * q^68 - 5941420405015 * q^69 + 374016699556320 * q^70 + 197895887067063 * q^71 + 439014895837656 * q^72 - 22888563242709 * q^73 + 694696716227036 * q^74 + 612085940395201 * q^75 + 301381886149904 * q^76 + 209007839834200 * q^77 + 350406148895766 * q^78 + 229938065096294 * q^79 + 555529032250016 * q^80 + 37596523177660 * q^81 - 414508112727306 * q^82 + 369402590629184 * q^83 + 559863541234208 * q^84 - 343366303925348 * q^85 + 1273071253885508 * q^86 - 725794048127953 * q^87 - 2461430290350720 * q^88 - 85082918654478 * q^89 + 125194382337864 * q^90 - 180251669559688 * q^91 + 557846601236480 * q^92 + 190758447044005 * q^93 + 640973603901258 * q^94 + 462985554800484 * q^95 - 2815189465233952 * q^96 - 5046858838511686 * q^97 - 3547540222626416 * q^98 - 3251416762282424 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.28020	0.0457421	0.0228710	−	0.999738i	$-0.492719\pi$
$2$	8.28020	0.0457421	0.0228710	+	0.999738i	$0.492719\pi$
$3$	−6881.62	−1.81669	−0.908346	−	0.418220i	$-0.862654\pi$
$3$	−6881.62	−1.81669	−0.908346	+	0.418220i	$0.862654\pi$
$4$	−32699.4	−0.997908
$5$	−91711.3	−0.524986	−0.262493	−	0.964934i	$-0.584545\pi$
$5$	−91711.3	−0.524986	−0.262493	+	0.964934i	$0.584545\pi$
$6$	−56981.2	−0.0830992
$7$	1.74802e6	0.802254	0.401127	−	0.916022i	$-0.368619\pi$
$7$	1.74802e6	0.802254	0.401127	+	0.916022i	$0.368619\pi$
$8$	−542083.	−0.0913884
$9$	3.30077e7	2.30037
$10$	−759388.	−0.0240140
$11$	1.56771e7	0.242560	0.121280	−	0.992618i	$-0.461300\pi$
$11$	1.56771e7	0.242560	0.121280	+	0.992618i	$0.461300\pi$
$12$	2.25025e8	1.81289
$13$	2.14877e7	0.0949763	0.0474882	−	0.998872i	$-0.484878\pi$
$13$	2.14877e7	0.0949763	0.0474882	+	0.998872i	$0.484878\pi$
$14$	1.44740e7	0.0366968
$15$	6.31122e8	0.953738
$16$	1.06701e9	0.993727
$17$	−1.10073e9	−0.650600	−0.325300	−	0.945611i	$-0.605465\pi$
$17$	−1.10073e9	−0.650600	−0.325300	+	0.945611i	$0.605465\pi$
$18$	2.73311e8	0.105224
$19$	−1.03531e9	−0.265717	−0.132859	−	0.991135i	$-0.542416\pi$
$19$	−1.03531e9	−0.265717	−0.132859	+	0.991135i	$0.542416\pi$
$20$	2.99891e9	0.523888
$21$	−1.20292e10	−1.45745
$22$	1.29809e8	0.0110952
$23$	3.40483e9	0.208514
$23$	3.40483e9	0.208514
$24$	3.73041e9	0.166025
$25$	−2.21066e10	−0.724390
$26$	1.77923e8	0.00434441
$27$	−1.28403e11	−2.36236
$28$	−5.71594e10	−0.800576
$29$	1.26499e11	1.36177	0.680884	−	0.732391i	$-0.261596\pi$
$29$	1.26499e11	1.36177	0.680884	+	0.732391i	$0.261596\pi$
$30$	5.22582e9	0.0436259
$31$	1.97030e11	1.28623	0.643117	−	0.765768i	$-0.277641\pi$
$31$	1.97030e11	1.28623	0.643117	+	0.765768i	$0.277641\pi$
$32$	2.65980e10	0.136844
$33$	−1.07883e11	−0.440657
$34$	−9.11427e9	−0.0297598
$35$	−1.60314e11	−0.421172
$36$	−1.07933e12	−2.29555
$37$	6.58396e11	1.14018	0.570091	−	0.821581i	$-0.306908\pi$
$37$	6.58396e11	1.14018	0.570091	+	0.821581i	$0.306908\pi$
$38$	−8.57261e9	−0.0121545
$39$	−1.47870e11	−0.172543
$40$	4.97152e10	0.0479777
$41$	2.90098e11	0.232630	0.116315	−	0.993212i	$-0.462892\pi$
$41$	2.90098e11	0.232630	0.116315	+	0.993212i	$0.462892\pi$
$42$	−9.96044e10	−0.0666667
$43$	−3.25665e12	−1.82708	−0.913540	−	0.406750i	$-0.866662\pi$
$43$	−3.25665e12	−1.82708	−0.913540	+	0.406750i	$0.866662\pi$
$44$	−5.12631e11	−0.242053
$45$	−3.02718e12	−1.20766
$46$	2.81926e10	0.00953788
$47$	−3.48829e12	−1.00433	−0.502167	−	0.864771i	$-0.667464\pi$
$47$	−3.48829e12	−1.00433	−0.502167	+	0.864771i	$0.667464\pi$
$48$	−7.34273e12	−1.80530
$49$	−1.69197e12	−0.356388
$50$	−1.83047e11	−0.0331351
$51$	7.57481e12	1.18194
$52$	−7.02637e11	−0.0947776
$53$	5.53420e12	0.647122	0.323561	−	0.946207i	$-0.395120\pi$
$53$	5.53420e12	0.647122	0.323561	+	0.946207i	$0.395120\pi$
$54$	−1.06320e12	−0.108059
$55$	−1.43776e12	−0.127341
$56$	−9.47575e11	−0.0733168
$57$	7.12464e12	0.482727
$58$	1.04744e12	0.0622901
$59$	2.14287e13	1.12100	0.560499	−	0.828155i	$-0.310609\pi$
$59$	2.14287e13	1.12100	0.560499	+	0.828155i	$0.310609\pi$
$60$	−2.06373e13	−0.951742
$61$	−1.84260e13	−0.750685	−0.375343	−	0.926886i	$-0.622475\pi$
$61$	−1.84260e13	−0.750685	−0.375343	+	0.926886i	$0.622475\pi$
$62$	1.63145e12	0.0588351
$63$	5.76983e13	1.84548
$64$	−3.47434e13	−0.987468
$65$	−1.97067e12	−0.0498613
$66$	−8.93297e11	−0.0201566
$67$	3.99125e13	0.804541	0.402270	−	0.915521i	$-0.368221\pi$
$67$	3.99125e13	0.804541	0.402270	+	0.915521i	$0.368221\pi$
$68$	3.59933e13	0.649239
$69$	−2.34307e13	−0.378806
$70$	−1.32743e12	−0.0192653
$71$	−1.07718e14	−1.40557	−0.702784	−	0.711404i	$-0.748060\pi$
$71$	−1.07718e14	−1.40557	−0.702784	+	0.711404i	$0.748060\pi$
$72$	−1.78930e13	−0.210227
$73$	1.14573e11	0.00121383	0.000606917	−	1.00000i	$-0.499807\pi$
$73$	1.14573e11	0.00121383	0.000606917		1.00000i	$0.499807\pi$
$74$	5.45165e12	0.0521543
$75$	1.52129e14	1.31599
$76$	3.38542e13	0.265161
$77$	2.74039e13	0.194595
$78$	−1.22440e12	−0.00789246
$79$	2.01176e14	1.17862	0.589308	−	0.807908i	$-0.299400\pi$
$79$	2.01176e14	1.17862	0.589308	+	0.807908i	$0.299400\pi$
$80$	−9.78566e13	−0.521693
$81$	4.09995e14	1.99132
$82$	2.40207e12	0.0106410
$83$	3.66828e14	1.48381	0.741903	−	0.670507i	$-0.233923\pi$
$83$	3.66828e14	1.48381	0.741903	+	0.670507i	$0.233923\pi$
$84$	3.93349e14	1.45440
$85$	1.00949e14	0.341556
$86$	−2.69657e13	−0.0835744
$87$	−8.70520e14	−2.47391
$88$	−8.49827e12	−0.0221672
$89$	−4.48834e14	−1.07562	−0.537812	−	0.843065i	$-0.680749\pi$
$89$	−4.48834e14	−1.07562	−0.537812	+	0.843065i	$0.680749\pi$
$90$	−2.50657e13	−0.0552409
$91$	3.75611e13	0.0761952
$92$	−1.11336e14	−0.208078
$93$	−1.35589e15	−2.33669
$94$	−2.88837e13	−0.0459403
$95$	9.49500e13	0.139498
$96$	−1.83037e14	−0.248603
$97$	−6.09640e14	−0.766100	−0.383050	−	0.923728i	$-0.625126\pi$
$97$	−6.09640e14	−0.766100	−0.383050	+	0.923728i	$0.625126\pi$
$98$	−1.40099e13	−0.0163019
$99$	5.17464e14	0.557977
$100$	7.22874e14	0.722874
$101$	−2.06151e15	−1.91326	−0.956630	−	0.291305i	$-0.905911\pi$
$101$	−2.06151e15	−1.91326	−0.956630	+	0.291305i	$0.905911\pi$
$102$	6.27209e13	0.0540644
$103$	−1.77326e15	−1.42067	−0.710333	−	0.703866i	$-0.751456\pi$
$103$	−1.77326e15	−1.42067	−0.710333	+	0.703866i	$0.751456\pi$
$104$	−1.16481e13	−0.00867974
$105$	1.10322e15	0.765140
$106$	4.58243e13	0.0296007
$107$	−3.86090e14	−0.232440	−0.116220	−	0.993224i	$-0.537078\pi$
$107$	−3.86090e14	−0.232440	−0.116220	+	0.993224i	$0.537078\pi$
$108$	4.19871e15	2.35742
$109$	3.36923e15	1.76536	0.882678	−	0.469978i	$-0.155738\pi$
$109$	3.36923e15	1.76536	0.882678	+	0.469978i	$0.155738\pi$
$110$	−1.19050e13	−0.00582483
$111$	−4.53083e15	−2.07136
$112$	1.86515e15	0.797222
$113$	6.92039e14	0.276721	0.138361	−	0.990382i	$-0.455817\pi$
$113$	6.92039e14	0.276721	0.138361	+	0.990382i	$0.455817\pi$
$114$	5.89934e13	0.0220809
$115$	−3.12261e14	−0.109467
$116$	−4.13646e15	−1.35892
$117$	7.09262e14	0.218480
$118$	1.77434e14	0.0512768
$119$	−1.92410e15	−0.521947
$120$	−3.42121e14	−0.0871606
$121$	−3.93148e15	−0.941165
$122$	−1.52571e14	−0.0343379
$123$	−1.99634e15	−0.422616
$124$	−6.44278e15	−1.28354
$125$	4.82623e15	0.905281
$126$	4.77754e14	0.0844160
$127$	3.82547e14	0.0637026	0.0318513	−	0.999493i	$-0.489860\pi$
$127$	3.82547e14	0.0637026	0.0318513	+	0.999493i	$0.489860\pi$
$128$	−1.15925e15	−0.182012
$129$	2.24110e16	3.31924
$130$	−1.63175e13	−0.00228076
$131$	−7.92615e13	−0.0104599	−0.00522995	−	0.999986i	$-0.501665\pi$
$131$	−7.92615e13	−0.0104599	−0.00522995	+	0.999986i	$0.501665\pi$
$132$	3.52773e15	0.439735
$133$	−1.80975e15	−0.213173
$134$	3.30484e14	0.0368014
$135$	1.17760e16	1.24021
$136$	5.96688e14	0.0594574
$137$	−1.72413e16	−1.62617	−0.813085	−	0.582145i	$-0.802214\pi$
$137$	−1.72413e16	−1.62617	−0.813085	+	0.582145i	$0.802214\pi$
$138$	−1.94011e14	−0.0173274
$139$	2.35516e15	0.199255	0.0996277	−	0.995025i	$-0.468235\pi$
$139$	2.35516e15	0.199255	0.0996277	+	0.995025i	$0.468235\pi$
$140$	5.24216e15	0.420291
$141$	2.40051e16	1.82456
$142$	−8.91929e14	−0.0642936
$143$	3.36864e14	0.0230375
$144$	3.52195e16	2.28594
$145$	−1.16014e16	−0.714909
$146$	9.48685e11	5.55233e−5 0
$147$	1.16435e16	0.647447
$148$	−2.15292e16	−1.13780
$149$	−1.48002e16	−0.743651	−0.371825	−	0.928303i	$-0.621268\pi$
$149$	−1.48002e16	−0.743651	−0.371825	+	0.928303i	$0.621268\pi$
$150$	1.25966e15	0.0601962
$151$	1.45661e15	0.0662241	0.0331121	−	0.999452i	$-0.489458\pi$
$151$	1.45661e15	0.0662241	0.0331121	+	0.999452i	$0.489458\pi$
$152$	5.61227e14	0.0242835
$153$	−3.63327e16	−1.49662
$154$	2.26909e14	0.00890117
$155$	−1.80699e16	−0.675256
$156$	4.83528e15	0.172182
$157$	−2.21204e16	−0.750838	−0.375419	−	0.926855i	$-0.622501\pi$
$157$	−2.21204e16	−0.750838	−0.375419	+	0.926855i	$0.622501\pi$
$158$	1.66578e15	0.0539124
$159$	−3.80843e16	−1.17562
$160$	−2.43934e15	−0.0718410
$161$	5.95172e15	0.167282
$162$	3.39484e15	0.0910871
$163$	−8.75351e15	−0.224272	−0.112136	−	0.993693i	$-0.535769\pi$
$163$	−8.75351e15	−0.224272	−0.112136	+	0.993693i	$0.535769\pi$
$164$	−9.48603e15	−0.232143
$165$	9.89414e15	0.231339
$166$	3.03741e15	0.0678724
$167$	−2.48015e16	−0.529791	−0.264895	−	0.964277i	$-0.585337\pi$
$167$	−2.48015e16	−0.529791	−0.264895	+	0.964277i	$0.585337\pi$
$168$	6.52085e15	0.133194
$169$	−5.07242e16	−0.990979
$170$	8.35882e14	0.0156235
$171$	−3.41734e16	−0.611248
$172$	1.06491e17	1.82326
$173$	3.58696e16	0.588005	0.294003	−	0.955805i	$-0.405013\pi$
$173$	3.58696e16	0.588005	0.294003	+	0.955805i	$0.405013\pi$
$174$	−7.20808e15	−0.113162
$175$	−3.86429e16	−0.581145
$176$	1.67275e16	0.241039
$177$	−1.47464e17	−2.03651
$178$	−3.71644e15	−0.0492013
$179$	9.09287e16	1.15426	0.577129	−	0.816653i	$-0.304173\pi$
$179$	9.09287e16	1.15426	0.577129	+	0.816653i	$0.304173\pi$
$180$	9.89872e16	1.20513
$181$	−1.39324e17	−1.62719	−0.813594	−	0.581433i	$-0.802492\pi$
$181$	−1.39324e17	−1.62719	−0.813594	+	0.581433i	$0.802492\pi$
$182$	3.11013e14	0.00348533
$183$	1.26801e17	1.36376
$184$	−1.84570e15	−0.0190558
$185$	−6.03824e16	−0.598580
$186$	−1.12270e16	−0.106885
$187$	−1.72562e16	−0.157810
$188$	1.14065e17	1.00223
$189$	−2.24452e17	−1.89522
$190$	7.86205e14	0.00638093
$191$	1.48656e17	1.15993	0.579965	−	0.814642i	$-0.303066\pi$
$191$	1.48656e17	1.15993	0.579965	+	0.814642i	$0.303066\pi$
$192$	2.39091e17	1.79392
$193$	1.92845e17	1.39164	0.695821	−	0.718215i	$-0.255041\pi$
$193$	1.92845e17	1.39164	0.695821	+	0.718215i	$0.255041\pi$
$194$	−5.04794e15	−0.0350430
$195$	1.35614e16	0.0905825
$196$	5.53266e16	0.355642
$197$	−2.73609e17	−1.69291	−0.846454	−	0.532461i	$-0.821267\pi$
$197$	−2.73609e17	−1.69291	−0.846454	+	0.532461i	$0.821267\pi$
$198$	4.28471e15	0.0255230
$199$	−6.31784e15	−0.0362386	−0.0181193	−	0.999836i	$-0.505768\pi$
$199$	−6.31784e15	−0.0362386	−0.0181193	+	0.999836i	$0.505768\pi$
$200$	1.19836e16	0.0662008
$201$	−2.74663e17	−1.46160
$202$	−1.70697e16	−0.0875165
$203$	2.21124e17	1.09248
$204$	−2.47692e17	−1.17947
$205$	−2.66052e16	−0.122127
$206$	−1.46829e16	−0.0649842
$207$	1.12386e17	0.479660
$208$	2.29276e16	0.0943806
$209$	−1.62307e16	−0.0644525
$210$	9.13485e15	0.0349991
$211$	1.95258e17	0.721923	0.360961	−	0.932581i	$-0.382449\pi$
$211$	1.95258e17	0.721923	0.360961	+	0.932581i	$0.382449\pi$
$212$	−1.80965e17	−0.645768
$213$	7.41276e17	2.55348
$214$	−3.19690e15	−0.0106323
$215$	2.98671e17	0.959191
$216$	6.96051e16	0.215893
$217$	3.44414e17	1.03189
$218$	2.78979e16	0.0807511
$219$	−7.88445e14	−0.00220516
$220$	4.70141e16	0.127074
$221$	−2.36522e16	−0.0617917
$222$	−3.75162e16	−0.0947483
$223$	−3.38368e17	−0.826234	−0.413117	−	0.910678i	$-0.635560\pi$
$223$	−3.38368e17	−0.826234	−0.413117	+	0.910678i	$0.635560\pi$
$224$	4.64940e16	0.109783
$225$	−7.29690e17	−1.66636
$226$	5.73022e15	0.0126578
$227$	3.59850e17	0.769002	0.384501	−	0.923125i	$-0.374374\pi$
$227$	3.59850e17	0.769002	0.384501	+	0.923125i	$0.374374\pi$
$228$	−2.32972e17	−0.481717
$229$	−3.64184e17	−0.728709	−0.364355	−	0.931260i	$-0.618710\pi$
$229$	−3.64184e17	−0.728709	−0.364355	+	0.931260i	$0.618710\pi$
$230$	−2.58558e15	−0.00500726
$231$	−1.88583e17	−0.353519
$232$	−6.85732e16	−0.124450
$233$	5.10415e17	0.896921	0.448461	−	0.893803i	$-0.351972\pi$
$233$	5.10415e17	0.896921	0.448461	+	0.893803i	$0.351972\pi$
$234$	5.87283e15	0.00999375
$235$	3.19915e17	0.527262
$236$	−7.00705e17	−1.11865
$237$	−1.38441e18	−2.14118
$238$	−1.59320e16	−0.0238749
$239$	−6.62751e17	−0.962424	−0.481212	−	0.876604i	$-0.659803\pi$
$239$	−6.62751e17	−0.962424	−0.481212	+	0.876604i	$0.659803\pi$
$240$	6.73412e17	0.947755
$241$	5.03613e17	0.687019	0.343510	−	0.939149i	$-0.388384\pi$
$241$	5.03613e17	0.687019	0.343510	+	0.939149i	$0.388384\pi$
$242$	−3.25534e16	−0.0430508
$243$	−9.78987e17	−1.25525
$244$	6.02521e17	0.749114
$245$	1.55173e17	0.187099
$246$	−1.65301e16	−0.0193314
$247$	−2.22466e16	−0.0252369
$248$	−1.06807e17	−0.117547
$249$	−2.52437e18	−2.69562
$250$	3.99622e16	0.0414094
$251$	7.36382e17	0.740543	0.370271	−	0.928924i	$-0.379265\pi$
$251$	7.36382e17	0.740543	0.370271	+	0.928924i	$0.379265\pi$
$252$	−1.88670e18	−1.84162
$253$	5.33776e16	0.0505773
$254$	3.16757e15	0.00291389
$255$	−6.94696e17	−0.620502
$256$	1.12887e18	0.979142
$257$	−8.61538e17	−0.725731	−0.362866	−	0.931841i	$-0.618202\pi$
$257$	−8.61538e17	−0.725731	−0.362866	+	0.931841i	$0.618202\pi$
$258$	1.85568e17	0.151829
$259$	1.15089e18	0.914716
$260$	6.44397e16	0.0497569
$261$	4.17546e18	3.13257
$262$	−6.56301e14	−0.000478457 0
$263$	1.31039e18	0.928398	0.464199	−	0.885731i	$-0.346342\pi$
$263$	1.31039e18	0.928398	0.464199	+	0.885731i	$0.346342\pi$
$264$	5.84819e16	0.0402709
$265$	−5.07549e17	−0.339730
$266$	−1.49851e16	−0.00975097
$267$	3.08870e18	1.95408
$268$	−1.30512e18	−0.802857
$269$	1.24148e18	0.742670	0.371335	−	0.928499i	$-0.378900\pi$
$269$	1.24148e18	0.742670	0.371335	+	0.928499i	$0.378900\pi$
$270$	9.75077e16	0.0567297
$271$	−1.83477e18	−1.03827	−0.519136	−	0.854692i	$-0.673746\pi$
$271$	−1.83477e18	−1.03827	−0.519136	+	0.854692i	$0.673746\pi$
$272$	−1.17449e18	−0.646519
$273$	−2.58481e17	−0.138423
$274$	−1.42761e17	−0.0743844
$275$	−3.46567e17	−0.175708
$276$	7.66171e17	0.378014
$277$	−1.43366e18	−0.688413	−0.344206	−	0.938894i	$-0.611852\pi$
$277$	−1.43366e18	−0.688413	−0.344206	+	0.938894i	$0.611852\pi$
$278$	1.95012e16	0.00911435
$279$	6.50353e18	2.95881
$280$	8.69033e16	0.0384903
$281$	−3.42250e18	−1.47586	−0.737931	−	0.674876i	$-0.764197\pi$
$281$	−3.42250e18	−1.47586	−0.737931	+	0.674876i	$0.764197\pi$
$282$	1.98767e17	0.0834594
$283$	−3.87150e18	−1.58300	−0.791499	−	0.611170i	$-0.790699\pi$
$283$	−3.87150e18	−1.58300	−0.791499	+	0.611170i	$0.790699\pi$
$284$	3.52233e18	1.40263
$285$	−6.53410e17	−0.253425
$286$	2.78930e15	0.00105378
$287$	5.07098e17	0.186628
$288$	8.77941e17	0.314790
$289$	−1.65081e18	−0.576719
$290$	−9.60621e16	−0.0327014
$291$	4.19531e18	1.39177
$292$	−3.74646e15	−0.00121130
$293$	−1.57787e18	−0.497237	−0.248619	−	0.968601i	$-0.579977\pi$
$293$	−1.57787e18	−0.497237	−0.248619	+	0.968601i	$0.579977\pi$
$294$	9.64107e16	0.0296156
$295$	−1.96525e18	−0.588509
$296$	−3.56906e17	−0.104200
$297$	−2.01298e18	−0.573015
$298$	−1.22548e17	−0.0340161
$299$	7.31620e16	0.0198039
$300$	−4.97454e18	−1.31324
$301$	−5.69270e18	−1.46578
$302$	1.20610e16	0.00302923
$303$	1.41865e19	3.47580
$304$	−1.10469e18	−0.264051
$305$	1.68987e18	0.394099
$306$	−3.00842e17	−0.0684585
$307$	7.16921e17	0.159197	0.0795983	−	0.996827i	$-0.474636\pi$
$307$	7.16921e17	0.159197	0.0795983	+	0.996827i	$0.474636\pi$
$308$	−8.96091e17	−0.194188
$309$	1.22029e19	2.58091
$310$	−1.49622e17	−0.0308876
$311$	9.27806e18	1.86962	0.934812	−	0.355143i	$-0.115568\pi$
$311$	9.27806e18	1.86962	0.934812	+	0.355143i	$0.115568\pi$
$312$	8.01581e16	0.0157684
$313$	−9.41803e18	−1.80875	−0.904373	−	0.426744i	$-0.859661\pi$
$313$	−9.41803e18	−1.80875	−0.904373	+	0.426744i	$0.859661\pi$
$314$	−1.83162e17	−0.0343449
$315$	−5.29159e18	−0.968851
$316$	−6.57833e18	−1.17615
$317$	6.37821e18	1.11367	0.556833	−	0.830625i	$-0.312017\pi$
$317$	6.37821e18	1.11367	0.556833	+	0.830625i	$0.312017\pi$
$318$	−3.15345e17	−0.0537753
$319$	1.98314e18	0.330311
$320$	3.18637e18	0.518407
$321$	2.65692e18	0.422271
$322$	4.92814e16	0.00765181
$323$	1.13960e18	0.172876
$324$	−1.34066e19	−1.98715
$325$	−4.75021e17	−0.0687999
$326$	−7.24808e16	−0.0102587
$327$	−2.31858e19	−3.20711
$328$	−1.57257e17	−0.0212597
$329$	−6.09761e18	−0.805731
$330$	8.19254e16	0.0105819
$331$	5.74380e18	0.725252	0.362626	−	0.931935i	$-0.381880\pi$
$331$	5.74380e18	0.725252	0.362626	+	0.931935i	$0.381880\pi$
$332$	−1.19951e19	−1.48070
$333$	2.17322e19	2.62284
$334$	−2.05361e17	−0.0242337
$335$	−3.66043e18	−0.422373
$336$	−1.28353e19	−1.44831
$337$	−1.46241e19	−1.61378	−0.806889	−	0.590704i	$-0.798850\pi$
$337$	−1.46241e19	−1.61378	−0.806889	+	0.590704i	$0.798850\pi$
$338$	−4.20006e17	−0.0453295
$339$	−4.76235e18	−0.502717
$340$	−3.30099e18	−0.340842
$341$	3.08885e18	0.311989
$342$	−2.82962e17	−0.0279597
$343$	−1.12565e19	−1.08817
$344$	1.76537e18	0.166974
$345$	2.14886e18	0.198868
$346$	2.97008e17	0.0268966
$347$	−2.06957e19	−1.83404	−0.917019	−	0.398844i	$-0.869411\pi$
$347$	−2.06957e19	−1.83404	−0.917019	+	0.398844i	$0.869411\pi$
$348$	2.84655e19	2.46874
$349$	1.20776e19	1.02515	0.512577	−	0.858641i	$-0.328691\pi$
$349$	1.20776e19	1.02515	0.512577	+	0.858641i	$0.328691\pi$
$350$	−3.19971e17	−0.0265828
$351$	−2.75909e18	−0.224369
$352$	4.16979e17	0.0331928
$353$	2.00220e19	1.56026	0.780131	−	0.625616i	$-0.215152\pi$
$353$	2.00220e19	1.56026	0.780131	+	0.625616i	$0.215152\pi$
$354$	−1.22103e18	−0.0931541
$355$	9.87898e18	0.737903
$356$	1.46766e19	1.07337
$357$	1.32409e19	0.948216
$358$	7.52908e17	0.0527982
$359$	−3.39598e18	−0.233215	−0.116608	−	0.993178i	$-0.537202\pi$
$359$	−3.39598e18	−0.233215	−0.116608	+	0.993178i	$0.537202\pi$
$360$	1.64099e18	0.110366
$361$	−1.41093e19	−0.929394
$362$	−1.15363e18	−0.0744309
$363$	2.70549e19	1.70981
$364$	−1.22823e18	−0.0760358
$365$	−1.05076e16	−0.000637247 0
$366$	1.04994e18	0.0623813
$367$	−1.77572e19	−1.03366	−0.516832	−	0.856087i	$-0.672889\pi$
$367$	−1.77572e19	−1.03366	−0.516832	+	0.856087i	$0.672889\pi$
$368$	3.63297e18	0.207206
$369$	9.57547e18	0.535134
$370$	−4.99978e17	−0.0273803
$371$	9.67392e18	0.519156
$372$	4.43367e19	2.33180
$373$	6.81600e18	0.351328	0.175664	−	0.984450i	$-0.443793\pi$
$373$	6.81600e18	0.351328	0.175664	+	0.984450i	$0.443793\pi$
$374$	−1.42885e17	−0.00721854
$375$	−3.32123e19	−1.64462
$376$	1.89094e18	0.0917845
$377$	2.71818e18	0.129336
$378$	−1.85850e18	−0.0866912
$379$	2.66512e19	1.21877	0.609386	−	0.792874i	$-0.291416\pi$
$379$	2.66512e19	1.21877	0.609386	+	0.792874i	$0.291416\pi$
$380$	−3.10481e18	−0.139206
$381$	−2.63254e18	−0.115728
$382$	1.23090e18	0.0530576
$383$	−8.13577e18	−0.343881	−0.171940	−	0.985107i	$-0.555004\pi$
$383$	−8.13577e18	−0.343881	−0.171940	+	0.985107i	$0.555004\pi$
$384$	7.97749e18	0.330660
$385$	−2.51324e18	−0.102160
$386$	1.59679e18	0.0636566
$387$	−1.07495e20	−4.20295
$388$	1.99349e19	0.764497
$389$	6.49082e18	0.244162	0.122081	−	0.992520i	$-0.461043\pi$
$389$	6.49082e18	0.244162	0.122081	+	0.992520i	$0.461043\pi$
$390$	1.12291e17	0.00414343
$391$	−3.74780e18	−0.135660
$392$	9.17191e17	0.0325698
$393$	5.45447e17	0.0190024
$394$	−2.26554e18	−0.0774372
$395$	−1.84501e19	−0.618757
$396$	−1.69208e19	−0.556810
$397$	−2.55775e19	−0.825904	−0.412952	−	0.910753i	$-0.635502\pi$
$397$	−2.55775e19	−0.825904	−0.412952	+	0.910753i	$0.635502\pi$
$398$	−5.23130e16	−0.00165763
$399$	1.24540e19	0.387269
$400$	−2.35879e19	−0.719846
$401$	4.34572e19	1.30160	0.650802	−	0.759248i	$-0.274433\pi$
$401$	4.34572e19	1.30160	0.650802	+	0.759248i	$0.274433\pi$
$402$	−2.27426e18	−0.0668567
$403$	4.23373e18	0.122162
$404$	6.74101e19	1.90926
$405$	−3.76012e19	−1.04542
$406$	1.83095e18	0.0499725
$407$	1.03217e19	0.276563
$408$	−4.10618e18	−0.108016
$409$	1.90840e18	0.0492883	0.0246442	−	0.999696i	$-0.492155\pi$
$409$	1.90840e18	0.0492883	0.0246442	+	0.999696i	$0.492155\pi$
$410$	−2.20297e17	−0.00558636
$411$	1.18648e20	2.95425
$412$	5.79844e19	1.41769
$413$	3.74578e19	0.899326
$414$	9.30575e17	0.0219406
$415$	−3.36423e19	−0.778978
$416$	5.71531e17	0.0129969
$417$	−1.62073e19	−0.361985
$418$	−1.34393e17	−0.00294819
$419$	−2.06181e19	−0.444266	−0.222133	−	0.975016i	$-0.571302\pi$
$419$	−2.06181e19	−0.444266	−0.222133	+	0.975016i	$0.571302\pi$
$420$	−3.60746e19	−0.763539
$421$	−7.93926e19	−1.65069	−0.825343	−	0.564632i	$-0.809018\pi$
$421$	−7.93926e19	−1.65069	−0.825343	+	0.564632i	$0.809018\pi$
$422$	1.61678e18	0.0330223
$423$	−1.15140e20	−2.31034
$424$	−3.00000e18	−0.0591394
$425$	2.43334e19	0.471288
$426$	6.13791e18	0.116802
$427$	−3.22091e19	−0.602240
$428$	1.26249e19	0.231953
$429$	−2.31817e18	−0.0418520
$430$	2.47306e18	0.0438754
$431$	2.61100e19	0.455226	0.227613	−	0.973752i	$-0.426908\pi$
$431$	2.61100e19	0.455226	0.227613	+	0.973752i	$0.426908\pi$
$432$	−1.37007e20	−2.34755
$433$	−4.45387e19	−0.750029	−0.375014	−	0.927019i	$-0.622362\pi$
$433$	−4.45387e19	−0.750029	−0.375014	+	0.927019i	$0.622362\pi$
$434$	2.85181e18	0.0472007
$435$	7.98365e19	1.29877
$436$	−1.10172e20	−1.76166
$437$	−3.52506e18	−0.0554059
$438$	−6.52848e15	−0.000100869 0
$439$	−1.79466e19	−0.272582	−0.136291	−	0.990669i	$-0.543518\pi$
$439$	−1.79466e19	−0.272582	−0.136291	+	0.990669i	$0.543518\pi$
$440$	7.79388e17	0.0116375
$441$	−5.58483e19	−0.819823
$442$	−1.95845e17	−0.00282648
$443$	8.91318e17	0.0126475	0.00632375	−	0.999980i	$-0.497987\pi$
$443$	8.91318e17	0.0126475	0.00632375	+	0.999980i	$0.497987\pi$
$444$	1.48156e20	2.06703
$445$	4.11632e19	0.564688
$446$	−2.80176e18	−0.0377937
$447$	1.01849e20	1.35098
$448$	−6.07324e19	−0.792200
$449$	3.29553e19	0.422744	0.211372	−	0.977406i	$-0.432207\pi$
$449$	3.29553e19	0.422744	0.211372	+	0.977406i	$0.432207\pi$
$450$	−6.04197e18	−0.0762228
$451$	4.54788e18	0.0564267
$452$	−2.26293e19	−0.276142
$453$	−1.00238e19	−0.120309
$454$	2.97963e18	0.0351757
$455$	−3.44477e18	−0.0400014
$456$	−3.86215e18	−0.0441156
$457$	−1.57945e20	−1.77473	−0.887367	−	0.461064i	$-0.847468\pi$
$457$	−1.57945e20	−1.77473	−0.887367	+	0.461064i	$0.847468\pi$
$458$	−3.01551e18	−0.0333327
$459$	1.41337e20	1.53696
$460$	1.02108e19	0.109238
$461$	−1.69843e20	−1.78768	−0.893842	−	0.448382i	$-0.852000\pi$
$461$	−1.69843e20	−1.78768	−0.893842	+	0.448382i	$0.852000\pi$
$462$	−1.56150e18	−0.0161707
$463$	−9.47884e19	−0.965823	−0.482912	−	0.875669i	$-0.660421\pi$
$463$	−9.47884e19	−0.965823	−0.482912	+	0.875669i	$0.660421\pi$
$464$	1.34976e20	1.35323
$465$	1.24350e20	1.22673
$466$	4.22634e18	0.0410270
$467$	5.35197e19	0.511254	0.255627	−	0.966775i	$-0.417718\pi$
$467$	5.35197e19	0.511254	0.255627	+	0.966775i	$0.417718\pi$
$468$	−2.31925e19	−0.218023
$469$	6.97680e19	0.645446
$470$	2.64896e18	0.0241180
$471$	1.52224e20	1.36404
$472$	−1.16161e19	−0.102446
$473$	−5.10546e19	−0.443177
$474$	−1.14632e19	−0.0979421
$475$	2.28873e19	0.192483
$476$	6.29171e19	0.520855
$477$	1.82672e20	1.48862
$478$	−5.48771e18	−0.0440233
$479$	1.94113e20	1.53298	0.766492	−	0.642254i	$-0.222000\pi$
$479$	1.94113e20	1.53298	0.766492	+	0.642254i	$0.222000\pi$
$480$	1.67866e19	0.130513
$481$	1.41474e19	0.108290
$482$	4.17002e18	0.0314257
$483$	−4.09574e19	−0.303899
$484$	1.28557e20	0.939195
$485$	5.59109e19	0.402192
$486$	−8.10621e18	−0.0574177
$487$	−2.11125e20	−1.47255	−0.736277	−	0.676680i	$-0.763418\pi$
$487$	−2.11125e20	−1.47255	−0.736277	+	0.676680i	$0.763418\pi$
$488$	9.98844e18	0.0686039
$489$	6.02383e19	0.407433
$490$	1.28487e18	0.00855829
$491$	−1.34404e20	−0.881662	−0.440831	−	0.897590i	$-0.645316\pi$
$491$	−1.34404e20	−0.881662	−0.440831	+	0.897590i	$0.645316\pi$
$492$	6.52792e19	0.421732
$493$	−1.39242e20	−0.885967
$494$	−1.84206e17	−0.00115439
$495$	−4.74573e19	−0.292930
$496$	2.10233e20	1.27817
$497$	−1.88294e20	−1.12762
$498$	−2.09023e19	−0.123303
$499$	−1.96324e20	−1.14083	−0.570413	−	0.821358i	$-0.693217\pi$
$499$	−1.96324e20	−1.14083	−0.570413	+	0.821358i	$0.693217\pi$
$500$	−1.57815e20	−0.903387
$501$	1.70674e20	0.962466
$502$	6.09739e18	0.0338740
$503$	4.21374e19	0.230626	0.115313	−	0.993329i	$-0.463213\pi$
$503$	4.21374e19	0.230626	0.115313	+	0.993329i	$0.463213\pi$
$504$	−3.12773e19	−0.168655
$505$	1.89063e20	1.00444
$506$	4.41977e17	0.00231351
$507$	3.49064e20	1.80030
$508$	−1.25091e19	−0.0635693
$509$	−3.40694e20	−1.70601	−0.853004	−	0.521904i	$-0.825222\pi$
$509$	−3.40694e20	−1.70601	−0.853004	+	0.521904i	$0.825222\pi$
$510$	−5.75222e18	−0.0283831
$511$	2.00276e17	0.000973804 0
$512$	4.73335e19	0.226800
$513$	1.32937e20	0.627722
$514$	−7.13370e18	−0.0331965
$515$	1.62628e20	0.745830
$516$	−7.32827e20	−3.31229
$517$	−5.46861e19	−0.243611
$518$	9.52962e18	0.0418410
$519$	−2.46841e20	−1.06822
$520$	1.06827e18	0.00455674
$521$	−3.68421e19	−0.154904	−0.0774518	−	0.996996i	$-0.524678\pi$
$521$	−3.68421e19	−0.154904	−0.0774518	+	0.996996i	$0.524678\pi$
$522$	3.45736e19	0.143290
$523$	−3.27314e20	−1.33722	−0.668609	−	0.743615i	$-0.733110\pi$
$523$	−3.27314e20	−1.33722	−0.668609	+	0.743615i	$0.733110\pi$
$524$	2.59181e18	0.0104380
$525$	2.65926e20	1.05576
$526$	1.08503e19	0.0424668
$527$	−2.16877e20	−0.836825
$528$	−1.15112e20	−0.437893
$529$	1.15928e19	0.0434783
$530$	−4.20261e18	−0.0155400
$531$	7.07312e20	2.57871
$532$	5.91779e19	0.212727
$533$	6.23354e18	0.0220943
$534$	2.55751e19	0.0893836
$535$	3.54088e19	0.122028
$536$	−2.16359e19	−0.0735257
$537$	−6.25736e20	−2.09693
$538$	1.02797e19	0.0339713
$539$	−2.65252e19	−0.0864455
$540$	−3.85069e20	−1.23761
$541$	−4.62262e20	−1.46524	−0.732620	−	0.680638i	$-0.761703\pi$
$541$	−4.62262e20	−1.46524	−0.732620	+	0.680638i	$0.761703\pi$
$542$	−1.51922e19	−0.0474927
$543$	9.58777e20	2.95610
$544$	−2.92773e19	−0.0890305
$545$	−3.08997e20	−0.926788
$546$	−2.14027e18	−0.00633176
$547$	4.30612e20	1.25655	0.628277	−	0.777990i	$-0.283760\pi$
$547$	4.30612e20	1.25655	0.628277	+	0.777990i	$0.283760\pi$
$548$	5.63781e20	1.62277
$549$	−6.08201e20	−1.72685
$550$	−2.86964e18	−0.00803725
$551$	−1.30967e20	−0.361846
$552$	1.27014e19	0.0346185
$553$	3.51660e20	0.945550
$554$	−1.18710e19	−0.0314894
$555$	4.15528e20	1.08744
$556$	−7.70125e19	−0.198838
$557$	−3.20330e19	−0.0815988	−0.0407994	−	0.999167i	$-0.512990\pi$
$557$	−3.20330e19	−0.0815988	−0.0407994	+	0.999167i	$0.512990\pi$
$558$	5.38505e19	0.135342
$559$	−6.99780e19	−0.173529
$560$	−1.71056e20	−0.418531
$561$	1.18751e20	0.286691
$562$	−2.83390e19	−0.0675090
$563$	−4.99917e20	−1.17513	−0.587564	−	0.809178i	$-0.699913\pi$
$563$	−4.99917e20	−1.17513	−0.587564	+	0.809178i	$0.699913\pi$
$564$	−7.84952e20	−1.82075
$565$	−6.34678e19	−0.145275
$566$	−3.20568e19	−0.0724096
$567$	7.16681e20	1.59755
$568$	5.83923e19	0.128453
$569$	5.36683e20	1.16513	0.582567	−	0.812783i	$-0.302048\pi$
$569$	5.36683e20	1.16513	0.582567	+	0.812783i	$0.302048\pi$
$570$	−5.41036e18	−0.0115922
$571$	7.40303e20	1.56545	0.782724	−	0.622369i	$-0.213830\pi$
$571$	7.40303e20	1.56545	0.782724	+	0.622369i	$0.213830\pi$
$572$	−1.10153e19	−0.0229893
$573$	−1.02299e21	−2.10723
$574$	4.19887e18	0.00853676
$575$	−7.52692e19	−0.151046
$576$	−1.14680e21	−2.27154
$577$	5.67065e20	1.10870	0.554351	−	0.832283i	$-0.312967\pi$
$577$	5.67065e20	1.10870	0.554351	+	0.832283i	$0.312967\pi$
$578$	−1.36691e19	−0.0263803
$579$	−1.32708e21	−2.52818
$580$	3.79360e20	0.713414
$581$	6.41225e20	1.19039
$582$	3.47380e19	0.0636623
$583$	8.67600e19	0.156966
$584$	−6.21080e16	−0.000110930 0
$585$	−6.50473e19	−0.114699
$586$	−1.30651e19	−0.0227447
$587$	4.87900e20	0.838582	0.419291	−	0.907852i	$-0.362279\pi$
$587$	4.87900e20	0.838582	0.419291	+	0.907852i	$0.362279\pi$
$588$	−3.80737e20	−0.646092
$589$	−2.03988e20	−0.341775
$590$	−1.62727e19	−0.0269196
$591$	1.88287e21	3.07549
$592$	7.02513e20	1.13303
$593$	−1.65312e20	−0.263266	−0.131633	−	0.991299i	$-0.542022\pi$
$593$	−1.65312e20	−0.263266	−0.131633	+	0.991299i	$0.542022\pi$
$594$	−1.66679e19	−0.0262109
$595$	1.76462e20	0.274015
$596$	4.83957e20	0.742095
$597$	4.34770e19	0.0658343
$598$	6.05796e17	0.000905873 0
$599$	1.61949e20	0.239154	0.119577	−	0.992825i	$-0.461846\pi$
$599$	1.61949e20	0.239154	0.119577	+	0.992825i	$0.461846\pi$
$600$	−8.24667e19	−0.120266
$601$	−4.49753e20	−0.647761	−0.323881	−	0.946098i	$-0.604988\pi$
$601$	−4.49753e20	−0.647761	−0.323881	+	0.946098i	$0.604988\pi$
$602$	−4.71367e19	−0.0670479
$603$	1.31742e21	1.85074
$604$	−4.76303e19	−0.0660855
$605$	3.60561e20	0.494098
$606$	1.17467e20	0.158990
$607$	−2.94437e20	−0.393620	−0.196810	−	0.980442i	$-0.563058\pi$
$607$	−2.94437e20	−0.393620	−0.196810	+	0.980442i	$0.563058\pi$
$608$	−2.75373e19	−0.0363617
$609$	−1.52169e21	−1.98471
$610$	1.39925e19	0.0180269
$611$	−7.49554e19	−0.0953880
$612$	1.18806e21	1.49349
$613$	1.73681e20	0.215675	0.107837	−	0.994169i	$-0.465607\pi$
$613$	1.73681e20	0.215675	0.107837	+	0.994169i	$0.465607\pi$
$614$	5.93625e18	0.00728198
$615$	1.83087e20	0.221868
$616$	−1.48552e19	−0.0177837
$617$	2.25992e20	0.267272	0.133636	−	0.991030i	$-0.457335\pi$
$617$	2.25992e20	0.267272	0.133636	+	0.991030i	$0.457335\pi$
$618$	1.01042e20	0.118056
$619$	9.89990e20	1.14275	0.571375	−	0.820689i	$-0.306410\pi$
$619$	9.89990e20	1.14275	0.571375	+	0.820689i	$0.306410\pi$
$620$	5.90876e20	0.673843
$621$	−4.37190e20	−0.492587
$622$	7.68242e19	0.0855205
$623$	−7.84573e20	−0.862924
$624$	−1.57779e20	−0.171460
$625$	2.32020e20	0.249130
$626$	−7.79831e19	−0.0827357
$627$	1.11693e20	0.117090
$628$	7.23325e20	0.749267
$629$	−7.24717e20	−0.741803
$630$	−4.38154e19	−0.0443173
$631$	1.35530e20	0.135461	0.0677306	−	0.997704i	$-0.478424\pi$
$631$	1.35530e20	0.135461	0.0677306	+	0.997704i	$0.478424\pi$
$632$	−1.09054e20	−0.107712
$633$	−1.34369e21	−1.31151
$634$	5.28129e19	0.0509413
$635$	−3.50839e19	−0.0334430
$636$	1.24533e21	1.17316
$637$	−3.63567e19	−0.0338484
$638$	1.64208e19	0.0151091
$639$	−3.55554e21	−3.23332
$640$	1.06316e20	0.0955540
$641$	−2.16529e21	−1.92345	−0.961724	−	0.274019i	$-0.911647\pi$
$641$	−2.16529e21	−1.92345	−0.961724	+	0.274019i	$0.911647\pi$
$642$	2.19999e19	0.0193156
$643$	−2.73604e20	−0.237433	−0.118716	−	0.992928i	$-0.537878\pi$
$643$	−2.73604e20	−0.237433	−0.118716	+	0.992928i	$0.537878\pi$
$644$	−1.94618e20	−0.166932
$645$	−2.05534e21	−1.74255
$646$	9.43614e18	0.00790770
$647$	6.17385e20	0.511415	0.255708	−	0.966754i	$-0.417692\pi$
$647$	6.17385e20	0.511415	0.255708	+	0.966754i	$0.417692\pi$
$648$	−2.22252e20	−0.181984
$649$	3.35938e20	0.271909
$650$	−3.93327e18	−0.00314705
$651$	−2.37012e21	−1.87462
$652$	2.86235e20	0.223803
$653$	−5.75944e20	−0.445176	−0.222588	−	0.974913i	$-0.571450\pi$
$653$	−5.75944e20	−0.445176	−0.222588	+	0.974913i	$0.571450\pi$
$654$	−1.91983e20	−0.146700
$655$	7.26917e18	0.00549130
$656$	3.09536e20	0.231171
$657$	3.78179e18	0.00279227
$658$	−5.04894e19	−0.0368558
$659$	1.37926e21	0.995418	0.497709	−	0.867344i	$-0.334175\pi$
$659$	1.37926e21	0.995418	0.497709	+	0.867344i	$0.334175\pi$
$660$	−3.23533e20	−0.230855
$661$	−8.22356e20	−0.580161	−0.290081	−	0.957002i	$-0.593682\pi$
$661$	−8.22356e20	−0.580161	−0.290081	+	0.957002i	$0.593682\pi$
$662$	4.75598e19	0.0331745
$663$	1.62766e20	0.112256
$664$	−1.98852e20	−0.135603
$665$	1.65975e20	0.111913
$666$	1.79947e20	0.119974
$667$	4.30708e20	0.283948
$668$	8.10995e20	0.528682
$669$	2.32852e21	1.50101
$670$	−3.03091e19	−0.0193202
$671$	−2.88866e20	−0.182086
$672$	−3.19954e20	−0.199442
$673$	2.04935e21	1.26329	0.631645	−	0.775258i	$-0.282380\pi$
$673$	2.04935e21	1.26329	0.631645	+	0.775258i	$0.282380\pi$
$674$	−1.21090e20	−0.0738175
$675$	2.83856e21	1.71127
$676$	1.65865e21	0.988906
$677$	−1.27188e21	−0.749947	−0.374973	−	0.927036i	$-0.622348\pi$
$677$	−1.27188e21	−0.749947	−0.374973	+	0.927036i	$0.622348\pi$
$678$	−3.94332e19	−0.0229953
$679$	−1.06566e21	−0.614607
$680$	−5.47231e19	−0.0312143
$681$	−2.47635e21	−1.39704
$682$	2.55763e19	0.0142710
$683$	−1.69418e21	−0.934983	−0.467492	−	0.883998i	$-0.654842\pi$
$683$	−1.69418e21	−0.934983	−0.467492	+	0.883998i	$0.654842\pi$
$684$	1.11745e21	0.609969
$685$	1.58122e21	0.853717
$686$	−9.32057e19	−0.0497751
$687$	2.50617e21	1.32384
$688$	−3.47487e21	−1.81562
$689$	1.18917e20	0.0614613
$690$	1.77930e19	0.00909664
$691$	1.55598e21	0.786901	0.393450	−	0.919346i	$-0.371281\pi$
$691$	1.55598e21	0.786901	0.393450	+	0.919346i	$0.371281\pi$
$692$	−1.17292e21	−0.586775
$693$	9.04540e20	0.447640
$694$	−1.71364e20	−0.0838927
$695$	−2.15995e20	−0.104606
$696$	4.71894e20	0.226087
$697$	−3.19320e20	−0.151349
$698$	1.00005e20	0.0468926
$699$	−3.51248e21	−1.62943
$700$	1.26360e21	0.579929
$701$	3.84342e21	1.74515	0.872574	−	0.488481i	$-0.162449\pi$
$701$	3.84342e21	1.74515	0.872574	+	0.488481i	$0.162449\pi$
$702$	−2.28458e19	−0.0102631
$703$	−6.81647e20	−0.302966
$704$	−5.44675e20	−0.239520
$705$	−2.20153e21	−0.957871
$706$	1.65786e20	0.0713696
$707$	−3.60356e21	−1.53492
$708$	4.82198e21	2.03225
$709$	1.45359e21	0.606170	0.303085	−	0.952964i	$-0.401983\pi$
$709$	1.45359e21	0.606170	0.303085	+	0.952964i	$0.401983\pi$
$710$	8.17999e19	0.0337532
$711$	6.64036e21	2.71125
$712$	2.43305e20	0.0982997
$713$	6.70854e20	0.268199
$714$	1.09638e20	0.0433734
$715$	−3.08943e19	−0.0120944
$716$	−2.97332e21	−1.15184
$717$	4.56080e21	1.74843
$718$	−2.81194e19	−0.0106678
$719$	−4.40634e21	−1.65429	−0.827144	−	0.561990i	$-0.810036\pi$
$719$	−4.40634e21	−1.65429	−0.827144	+	0.561990i	$0.810036\pi$
$720$	−3.23003e21	−1.20009
$721$	−3.09969e21	−1.13974
$722$	−1.16827e20	−0.0425124
$723$	−3.46567e21	−1.24810
$724$	4.55583e21	1.62378
$725$	−2.79647e21	−0.986451
$726$	2.24020e20	0.0782100
$727$	−4.47925e21	−1.54774	−0.773868	−	0.633347i	$-0.781681\pi$
$727$	−4.47925e21	−1.54774	−0.773868	+	0.633347i	$0.781681\pi$
$728$	−2.03612e19	−0.00696336
$729$	8.54034e20	0.289080
$730$	−8.70051e16	−2.91490e−5 0
$731$	3.58469e21	1.18870
$732$	−4.14632e21	−1.36091
$733$	2.12901e21	0.691669	0.345835	−	0.938295i	$-0.387596\pi$
$733$	2.12901e21	0.691669	0.345835	+	0.938295i	$0.387596\pi$
$734$	−1.47033e20	−0.0472819
$735$	−1.06784e21	−0.339901
$736$	9.05616e19	0.0285339
$737$	6.25711e20	0.195149
$738$	7.92868e19	0.0244781
$739$	4.55962e21	1.39346	0.696731	−	0.717333i	$-0.254637\pi$
$739$	4.55962e21	1.39346	0.696731	+	0.717333i	$0.254637\pi$
$740$	1.97447e21	0.597328
$741$	1.53092e20	0.0458476
$742$	8.01020e19	0.0237473
$743$	3.81675e21	1.12015	0.560076	−	0.828441i	$-0.310772\pi$
$743$	3.81675e21	1.12015	0.560076	+	0.828441i	$0.310772\pi$
$744$	7.35004e20	0.213547
$745$	1.35734e21	0.390406
$746$	5.64378e19	0.0160705
$747$	1.21082e22	3.41330
$748$	5.64269e20	0.157480
$749$	−6.74895e20	−0.186476
$750$	−2.75004e20	−0.0752281
$751$	−1.02399e21	−0.277330	−0.138665	−	0.990339i	$-0.544281\pi$
$751$	−1.02399e21	−0.277330	−0.138665	+	0.990339i	$0.544281\pi$
$752$	−3.72202e21	−0.998034
$753$	−5.06750e21	−1.34534
$754$	2.25071e19	0.00591609
$755$	−1.33588e20	−0.0347667
$756$	7.33944e21	1.89125
$757$	4.82500e21	1.23106	0.615528	−	0.788115i	$-0.288943\pi$
$757$	4.82500e21	1.23106	0.615528	+	0.788115i	$0.288943\pi$
$758$	2.20677e20	0.0557492
$759$	−3.67324e20	−0.0918833
$760$	−5.14708e19	−0.0127485
$761$	−3.23366e21	−0.793066	−0.396533	−	0.918020i	$-0.629787\pi$
$761$	−3.23366e21	−0.793066	−0.396533	+	0.918020i	$0.629787\pi$
$762$	−2.17980e19	−0.00529363
$763$	5.88950e21	1.41626
$764$	−4.86096e21	−1.15750
$765$	3.33212e21	0.785705
$766$	−6.73658e19	−0.0157298
$767$	4.60453e20	0.106468
$768$	−7.76848e21	−1.77880
$769$	−1.14299e21	−0.259176	−0.129588	−	0.991568i	$-0.541366\pi$
$769$	−1.14299e21	−0.259176	−0.129588	+	0.991568i	$0.541366\pi$
$770$	−2.08102e19	−0.00467299
$771$	5.92877e21	1.31843
$772$	−6.30591e21	−1.38873
$773$	−4.29024e21	−0.935698	−0.467849	−	0.883808i	$-0.654971\pi$
$773$	−4.29024e21	−0.935698	−0.467849	+	0.883808i	$0.654971\pi$
$774$	−8.90077e20	−0.192252
$775$	−4.35567e21	−0.931735
$776$	3.30476e20	0.0700126
$777$	−7.92000e21	−1.66176
$778$	5.37453e19	0.0111685
$779$	−3.00342e20	−0.0618138
$780$	−4.43450e20	−0.0903930
$781$	−1.68870e21	−0.340935
$782$	−3.10325e19	−0.00620535
$783$	−1.62429e22	−3.21699
$784$	−1.80535e21	−0.354153
$785$	2.02869e21	0.394179
$786$	4.51641e18	0.000869209 0
$787$	−4.49897e21	−0.857635	−0.428818	−	0.903391i	$-0.641070\pi$
$787$	−4.49897e21	−0.857635	−0.428818	+	0.903391i	$0.641070\pi$
$788$	8.94686e21	1.68937
$789$	−9.01764e21	−1.68661
$790$	−1.52770e20	−0.0283032
$791$	1.20970e21	0.222001
$792$	−2.80509e20	−0.0509927
$793$	−3.95933e20	−0.0712973
$794$	−2.11787e20	−0.0377785
$795$	3.49276e21	0.617184
$796$	2.06590e20	0.0361627
$797$	−3.30363e21	−0.572868	−0.286434	−	0.958100i	$-0.592470\pi$
$797$	−3.30363e21	−0.572868	−0.286434	+	0.958100i	$0.592470\pi$
$798$	1.03122e20	0.0177145
$799$	3.83967e21	0.653420
$800$	−5.87992e20	−0.0991281
$801$	−1.48150e22	−2.47433
$802$	3.59834e20	0.0595380
$803$	1.79616e18	0.000294428 0
$804$	8.98131e21	1.45854
$805$	−5.45840e20	−0.0878205
$806$	3.50562e19	0.00558794
$807$	−8.54336e21	−1.34920
$808$	1.11751e21	0.174850
$809$	5.57568e21	0.864338	0.432169	−	0.901793i	$-0.357748\pi$
$809$	5.57568e21	0.864338	0.432169	+	0.901793i	$0.357748\pi$
$810$	−3.11345e20	−0.0478195
$811$	1.25642e20	0.0191195	0.00955977	−	0.999954i	$-0.496957\pi$
$811$	1.25642e20	0.0191195	0.00955977	+	0.999954i	$0.496957\pi$
$812$	−7.23062e21	−1.09020
$813$	1.26262e22	1.88622
$814$	8.54659e19	0.0126506
$815$	8.02796e20	0.117740
$816$	8.08237e21	1.17453
$817$	3.37165e21	0.485487
$818$	1.58019e19	0.00225455
$819$	1.23981e21	0.175277
$820$	8.69976e20	0.121872
$821$	1.14027e22	1.58283	0.791417	−	0.611277i	$-0.209344\pi$
$821$	1.14027e22	1.58283	0.791417	+	0.611277i	$0.209344\pi$
$822$	9.82430e20	0.135133
$823$	−2.47724e21	−0.337652	−0.168826	−	0.985646i	$-0.553998\pi$
$823$	−2.47724e21	−0.337652	−0.168826	+	0.985646i	$0.553998\pi$
$824$	9.61252e20	0.129832
$825$	2.38494e21	0.319207
$826$	3.10158e20	0.0411370
$827$	3.67731e21	0.483325	0.241662	−	0.970360i	$-0.422307\pi$
$827$	3.67731e21	0.483325	0.241662	+	0.970360i	$0.422307\pi$
$828$	−3.67495e21	−0.478656
$829$	−5.05700e21	−0.652731	−0.326366	−	0.945244i	$-0.605824\pi$
$829$	−5.05700e21	−0.652731	−0.326366	+	0.945244i	$0.605824\pi$
$830$	−2.78565e20	−0.0356321
$831$	9.86592e21	1.25063
$832$	−7.46558e20	−0.0937861
$833$	1.86241e21	0.231866
$834$	−1.34200e20	−0.0165580
$835$	2.27458e21	0.278133
$836$	5.30734e20	0.0643176
$837$	−2.52993e22	−3.03856
$838$	−1.70722e20	−0.0203217
$839$	1.13239e22	1.33592	0.667958	−	0.744199i	$-0.267168\pi$
$839$	1.13239e22	1.33592	0.667958	+	0.744199i	$0.267168\pi$
$840$	−5.98035e20	−0.0699250
$841$	7.37289e21	0.854413
$842$	−6.57387e20	−0.0755058
$843$	2.35523e22	2.68118
$844$	−6.38483e21	−0.720412
$845$	4.65198e21	0.520251
$846$	−9.53386e20	−0.105680
$847$	−6.87232e21	−0.755053
$848$	5.90503e21	0.643063
$849$	2.66422e22	2.87582
$850$	2.01486e20	0.0215577
$851$	2.24172e21	0.237745
$852$	−2.42393e22	−2.54814
$853$	−4.65004e21	−0.484550	−0.242275	−	0.970208i	$-0.577894\pi$
$853$	−4.65004e21	−0.484550	−0.242275	+	0.970208i	$0.577894\pi$
$854$	−2.66698e20	−0.0275477
$855$	3.13409e21	0.320897
$856$	2.09293e20	0.0212423
$857$	2.87175e19	0.00288928	0.00144464	−	0.999999i	$-0.499540\pi$
$857$	2.87175e19	0.00288928	0.00144464	+	0.999999i	$0.499540\pi$
$858$	−1.91949e19	−0.00191440
$859$	−2.32260e20	−0.0229629	−0.0114814	−	0.999934i	$-0.503655\pi$
$859$	−2.32260e20	−0.0229629	−0.0114814	+	0.999934i	$0.503655\pi$
$860$	−9.76639e21	−0.957184
$861$	−3.48965e21	−0.339046
$862$	2.16196e20	0.0208230
$863$	2.35064e21	0.224442	0.112221	−	0.993683i	$-0.464203\pi$
$863$	2.35064e21	0.224442	0.112221	+	0.993683i	$0.464203\pi$
$864$	−3.41527e21	−0.323274
$865$	−3.28965e21	−0.308695
$866$	−3.68789e20	−0.0343079
$867$	1.13603e22	1.04772
$868$	−1.12621e22	−1.02973
$869$	3.15384e21	0.285885
$870$	6.61062e20	0.0594084
$871$	8.57629e20	0.0764123
$872$	−1.82641e21	−0.161333
$873$	−2.01228e22	−1.76231
$874$	−2.91882e19	−0.00253438
$875$	8.43637e21	0.726265
$876$	2.57817e19	0.00220055
$877$	−6.11329e21	−0.517343	−0.258671	−	0.965965i	$-0.583285\pi$
$877$	−6.11329e21	−0.517343	−0.258671	+	0.965965i	$0.583285\pi$
$878$	−1.48601e20	−0.0124685
$879$	1.08583e22	0.903327
$880$	−1.53410e21	−0.126542
$881$	−7.09220e21	−0.580045	−0.290022	−	0.957020i	$-0.593663\pi$
$881$	−7.09220e21	−0.580045	−0.290022	+	0.957020i	$0.593663\pi$
$882$	−4.62435e20	−0.0375004
$883$	−1.40078e21	−0.112633	−0.0563165	−	0.998413i	$-0.517936\pi$
$883$	−1.40078e21	−0.112633	−0.0563165	+	0.998413i	$0.517936\pi$
$884$	7.73414e20	0.0616624
$885$	1.35241e22	1.06914
$886$	7.38029e18	0.000578523 0
$887$	−1.23384e22	−0.959030	−0.479515	−	0.877534i	$-0.659187\pi$
$887$	−1.23384e22	−0.959030	−0.479515	+	0.877534i	$0.659187\pi$
$888$	2.45609e21	0.189298
$889$	6.68701e20	0.0511057
$890$	3.40839e20	0.0258300
$891$	6.42752e21	0.483015
$892$	1.10645e22	0.824505
$893$	3.61147e21	0.266869
$894$	8.43330e20	0.0617968
$895$	−8.33919e21	−0.605970
$896$	−2.02639e21	−0.146020
$897$	−5.03473e20	−0.0359776
$898$	2.72877e20	0.0193372
$899$	2.49242e22	1.75155
$900$	2.38604e22	1.66287
$901$	−6.09167e21	−0.421018
$902$	3.76573e19	0.00258107
$903$	3.91750e22	2.66287
$904$	−3.75143e20	−0.0252891
$905$	1.27776e22	0.854251
$906$	−8.29993e19	−0.00550317
$907$	−1.63183e22	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$907$	−1.63183e22	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$908$	−1.17669e22	−0.767393
$909$	−6.80457e22	−4.40120
$910$	−2.85234e19	−0.00182975
$911$	−1.00282e22	−0.638022	−0.319011	−	0.947751i	$-0.603351\pi$
$911$	−1.00282e22	−0.638022	−0.319011	+	0.947751i	$0.603351\pi$
$912$	7.60203e21	0.479699
$913$	5.75079e21	0.359912
$914$	−1.30781e21	−0.0811800
$915$	−1.16291e22	−0.715957
$916$	1.19086e22	0.727185
$917$	−1.38551e20	−0.00839150
$918$	1.17030e21	0.0703035
$919$	1.86307e22	1.11010	0.555050	−	0.831817i	$-0.312699\pi$
$919$	1.86307e22	1.11010	0.555050	+	0.831817i	$0.312699\pi$
$920$	1.69272e20	0.0100040
$921$	−4.93358e21	−0.289211
$922$	−1.40633e21	−0.0817724
$923$	−2.31462e21	−0.133496
$924$	6.16656e21	0.352779
$925$	−1.45549e22	−0.825936
$926$	−7.84867e20	−0.0441788
$927$	−5.85312e22	−3.26805
$928$	3.36463e21	0.186349
$929$	2.62532e22	1.44233	0.721165	−	0.692763i	$-0.243607\pi$
$929$	2.62532e22	1.44233	0.721165	+	0.692763i	$0.243607\pi$
$930$	1.02964e21	0.0561132
$931$	1.75172e21	0.0946985
$932$	−1.66903e22	−0.895045
$933$	−6.38481e22	−3.39653
$934$	4.43154e20	0.0233858
$935$	1.58259e21	0.0828479
$936$	−3.84479e20	−0.0199666
$937$	−2.13339e22	−1.09907	−0.549533	−	0.835472i	$-0.685194\pi$
$937$	−2.13339e22	−1.09907	−0.549533	+	0.835472i	$0.685194\pi$
$938$	5.77693e20	0.0295240
$939$	6.48112e22	3.28593
$940$	−1.04611e22	−0.526158
$941$	−1.57289e22	−0.784829	−0.392415	−	0.919788i	$-0.628360\pi$
$941$	−1.57289e22	−0.784829	−0.392415	+	0.919788i	$0.628360\pi$
$942$	1.26045e21	0.0623940
$943$	9.87732e20	0.0485067
$944$	2.28645e22	1.11397
$945$	2.05847e22	0.994963
$946$	−4.22743e20	−0.0202718
$947$	1.31121e22	0.623801	0.311901	−	0.950115i	$-0.399034\pi$
$947$	1.31121e22	0.623801	0.311901	+	0.950115i	$0.399034\pi$
$948$	4.52696e22	2.13670
$949$	2.46191e18	0.000115286 0
$950$	1.89511e20	0.00880457
$951$	−4.38924e22	−2.02319
$952$	1.04303e21	0.0476999
$953$	−3.72497e22	−1.69015	−0.845076	−	0.534647i	$-0.820445\pi$
$953$	−3.72497e22	−1.69015	−0.845076	+	0.534647i	$0.820445\pi$
$954$	1.51256e21	0.0680924
$955$	−1.36334e22	−0.608947
$956$	2.16716e22	0.960410
$957$	−1.36472e22	−0.600072
$958$	1.60729e21	0.0701219
$959$	−3.01382e22	−1.30460
$960$	−2.19274e22	−0.941785
$961$	1.53557e22	0.654400
$962$	1.17144e20	0.00495343
$963$	−1.27440e22	−0.534697
$964$	−1.64679e22	−0.685582
$965$	−1.76860e22	−0.730593
$966$	−3.39136e20	−0.0139010
$967$	−2.33940e22	−0.951495	−0.475747	−	0.879582i	$-0.657822\pi$
$967$	−2.33940e22	−0.951495	−0.475747	+	0.879582i	$0.657822\pi$
$968$	2.13119e21	0.0860116
$969$	−7.84231e21	−0.314062
$970$	4.62953e20	0.0183971
$971$	3.20061e22	1.26209	0.631043	−	0.775748i	$-0.282627\pi$
$971$	3.20061e22	1.26209	0.631043	+	0.775748i	$0.282627\pi$
$972$	3.20123e22	1.25262
$973$	4.11688e21	0.159853
$974$	−1.74815e21	−0.0673577
$975$	3.26891e21	0.124988
$976$	−1.96607e22	−0.745976
$977$	2.48586e22	0.935982	0.467991	−	0.883733i	$-0.344978\pi$
$977$	2.48586e22	0.935982	0.467991	+	0.883733i	$0.344978\pi$
$978$	4.98785e20	0.0186368
$979$	−7.03640e21	−0.260904
$980$	−5.07408e21	−0.186707
$981$	1.11211e23	4.06097
$982$	−1.11289e21	−0.0403290
$983$	1.39390e21	0.0501280	0.0250640	−	0.999686i	$-0.492021\pi$
$983$	1.39390e21	0.0501280	0.0250640	+	0.999686i	$0.492021\pi$
$984$	1.08218e21	0.0386223
$985$	2.50930e22	0.888754
$986$	−1.15295e21	−0.0405260
$987$	4.19614e22	1.46376
$988$	7.27450e20	0.0251841
$989$	−1.10883e22	−0.380972
$990$	−3.92956e20	−0.0133992
$991$	1.67530e22	0.566942	0.283471	−	0.958981i	$-0.408514\pi$
$991$	1.67530e22	0.566942	0.283471	+	0.958981i	$0.408514\pi$
$992$	5.24062e21	0.176013
$993$	−3.95266e22	−1.31756
$994$	−1.55911e21	−0.0515798
$995$	5.79418e20	0.0190247
$996$	8.25456e22	2.68998
$997$	−8.34210e21	−0.269812	−0.134906	−	0.990858i	$-0.543073\pi$
$997$	−8.34210e21	−0.269812	−0.134906	+	0.990858i	$0.543073\pi$
$998$	−1.62560e21	−0.0521838
$999$	−8.45401e22	−2.69353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.16.a.a.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.16.a.a.1.8	✓	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q+8.28020 q^{2} -6881.62 q^{3} -32699.4 q^{4} -91711.3 q^{5} -56981.2 q^{6} +1.74802e6 q^{7} -542083. q^{8} +3.30077e7 q^{9} +O(q^{10})\)	\(q+8.28020 q^{2} -6881.62 q^{3} -32699.4 q^{4} -91711.3 q^{5} -56981.2 q^{6} +1.74802e6 q^{7} -542083. q^{8} +3.30077e7 q^{9} -759388. q^{10} +1.56771e7 q^{11} +2.25025e8 q^{12} +2.14877e7 q^{13} +1.44740e7 q^{14} +6.31122e8 q^{15} +1.06701e9 q^{16} -1.10073e9 q^{17} +2.73311e8 q^{18} -1.03531e9 q^{19} +2.99891e9 q^{20} -1.20292e10 q^{21} +1.29809e8 q^{22} +3.40483e9 q^{23} +3.73041e9 q^{24} -2.21066e10 q^{25} +1.77923e8 q^{26} -1.28403e11 q^{27} -5.71594e10 q^{28} +1.26499e11 q^{29} +5.22582e9 q^{30} +1.97030e11 q^{31} +2.65980e10 q^{32} -1.07883e11 q^{33} -9.11427e9 q^{34} -1.60314e11 q^{35} -1.07933e12 q^{36} +6.58396e11 q^{37} -8.57261e9 q^{38} -1.47870e11 q^{39} +4.97152e10 q^{40} +2.90098e11 q^{41} -9.96044e10 q^{42} -3.25665e12 q^{43} -5.12631e11 q^{44} -3.02718e12 q^{45} +2.81926e10 q^{46} -3.48829e12 q^{47} -7.34273e12 q^{48} -1.69197e12 q^{49} -1.83047e11 q^{50} +7.57481e12 q^{51} -7.02637e11 q^{52} +5.53420e12 q^{53} -1.06320e12 q^{54} -1.43776e12 q^{55} -9.47575e11 q^{56} +7.12464e12 q^{57} +1.04744e12 q^{58} +2.14287e13 q^{59} -2.06373e13 q^{60} -1.84260e13 q^{61} +1.63145e12 q^{62} +5.76983e13 q^{63} -3.47434e13 q^{64} -1.97067e12 q^{65} -8.93297e11 q^{66} +3.99125e13 q^{67} +3.59933e13 q^{68} -2.34307e13 q^{69} -1.32743e12 q^{70} -1.07718e14 q^{71} -1.78930e13 q^{72} +1.14573e11 q^{73} +5.45165e12 q^{74} +1.52129e14 q^{75} +3.38542e13 q^{76} +2.74039e13 q^{77} -1.22440e12 q^{78} +2.01176e14 q^{79} -9.78566e13 q^{80} +4.09995e14 q^{81} +2.40207e12 q^{82} +3.66828e14 q^{83} +3.93349e14 q^{84} +1.00949e14 q^{85} -2.69657e13 q^{86} -8.70520e14 q^{87} -8.49827e12 q^{88} -4.48834e14 q^{89} -2.50657e13 q^{90} +3.75611e13 q^{91} -1.11336e14 q^{92} -1.35589e15 q^{93} -2.88837e13 q^{94} +9.49500e13 q^{95} -1.83037e14 q^{96} -6.09640e14 q^{97} -1.40099e13 q^{98} +5.17464e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9}+O(q^{10}) \) 12 * q - 256 * q^2 - 1745 * q^3 + 163840 * q^4 - 204476 * q^5 - 199430 * q^6 - 5717328 * q^7 + 10323168 * q^8 + 32660341 * q^9	\( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9} - 137846540 q^{10} - 87636002 q^{11} - 398208076 q^{12} - 292496079 q^{13} + 415954912 q^{14} + 548079030 q^{15} + 4273503168 q^{16} - 2462528162 q^{17} + 7261215718 q^{18} + 175321758 q^{19} + 2660811480 q^{20} + 205665472 q^{21} - 21718153768 q^{22} + 40857905364 q^{23} - 63413289624 q^{24} + 20443225284 q^{25} - 137268652810 q^{26} - 151915208903 q^{27} - 325638721712 q^{28} - 164667697193 q^{29} - 356944003956 q^{30} + 20222384151 q^{31} - 369109524032 q^{32} + 132365097022 q^{33} - 582887018988 q^{34} - 1578083373112 q^{35} - 1903913944516 q^{36} - 869669414912 q^{37} - 5525312078376 q^{38} - 5762413466499 q^{39} - 4733269274576 q^{40} - 7510147709883 q^{41} - 7436463221624 q^{42} - 5682603487020 q^{43} - 11849381658176 q^{44} - 10780493432442 q^{45} - 871635314432 q^{46} - 5828073094301 q^{47} - 29418911592496 q^{48} - 6518780198860 q^{49} - 16781003942456 q^{50} - 771327642584 q^{51} - 3841511618340 q^{52} + 1452974784324 q^{53} - 32167598069522 q^{54} - 14882020037092 q^{55} + 416192984288 q^{56} - 12135794354818 q^{57} - 60065613521022 q^{58} - 11503084624084 q^{59} - 6378557828664 q^{60} - 23587566667200 q^{61} + 49359974806402 q^{62} + 87886039196104 q^{63} + 80321007324160 q^{64} + 54548135308138 q^{65} + 316922278045948 q^{66} + 61525019345122 q^{67} + 45114528974104 q^{68} - 5941420405015 q^{69} + 374016699556320 q^{70} + 197895887067063 q^{71} + 439014895837656 q^{72} - 22888563242709 q^{73} + 694696716227036 q^{74} + 612085940395201 q^{75} + 301381886149904 q^{76} + 209007839834200 q^{77} + 350406148895766 q^{78} + 229938065096294 q^{79} + 555529032250016 q^{80} + 37596523177660 q^{81} - 414508112727306 q^{82} + 369402590629184 q^{83} + 559863541234208 q^{84} - 343366303925348 q^{85} + 12\!\cdots\!08 q^{86}+ \cdots - 32\!\cdots\!24 q^{99}+O(q^{100}) \) 12 * q - 256 * q^2 - 1745 * q^3 + 163840 * q^4 - 204476 * q^5 - 199430 * q^6 - 5717328 * q^7 + 10323168 * q^8 + 32660341 * q^9 - 137846540 * q^10 - 87636002 * q^11 - 398208076 * q^12 - 292496079 * q^13 + 415954912 * q^14 + 548079030 * q^15 + 4273503168 * q^16 - 2462528162 * q^17 + 7261215718 * q^18 + 175321758 * q^19 + 2660811480 * q^20 + 205665472 * q^21 - 21718153768 * q^22 + 40857905364 * q^23 - 63413289624 * q^24 + 20443225284 * q^25 - 137268652810 * q^26 - 151915208903 * q^27 - 325638721712 * q^28 - 164667697193 * q^29 - 356944003956 * q^30 + 20222384151 * q^31 - 369109524032 * q^32 + 132365097022 * q^33 - 582887018988 * q^34 - 1578083373112 * q^35 - 1903913944516 * q^36 - 869669414912 * q^37 - 5525312078376 * q^38 - 5762413466499 * q^39 - 4733269274576 * q^40 - 7510147709883 * q^41 - 7436463221624 * q^42 - 5682603487020 * q^43 - 11849381658176 * q^44 - 10780493432442 * q^45 - 871635314432 * q^46 - 5828073094301 * q^47 - 29418911592496 * q^48 - 6518780198860 * q^49 - 16781003942456 * q^50 - 771327642584 * q^51 - 3841511618340 * q^52 + 1452974784324 * q^53 - 32167598069522 * q^54 - 14882020037092 * q^55 + 416192984288 * q^56 - 12135794354818 * q^57 - 60065613521022 * q^58 - 11503084624084 * q^59 - 6378557828664 * q^60 - 23587566667200 * q^61 + 49359974806402 * q^62 + 87886039196104 * q^63 + 80321007324160 * q^64 + 54548135308138 * q^65 + 316922278045948 * q^66 + 61525019345122 * q^67 + 45114528974104 * q^68 - 5941420405015 * q^69 + 374016699556320 * q^70 + 197895887067063 * q^71 + 439014895837656 * q^72 - 22888563242709 * q^73 + 694696716227036 * q^74 + 612085940395201 * q^75 + 301381886149904 * q^76 + 209007839834200 * q^77 + 350406148895766 * q^78 + 229938065096294 * q^79 + 555529032250016 * q^80 + 37596523177660 * q^81 - 414508112727306 * q^82 + 369402590629184 * q^83 + 559863541234208 * q^84 - 343366303925348 * q^85 + 1273071253885508 * q^86 - 725794048127953 * q^87 - 2461430290350720 * q^88 - 85082918654478 * q^89 + 125194382337864 * q^90 - 180251669559688 * q^91 + 557846601236480 * q^92 + 190758447044005 * q^93 + 640973603901258 * q^94 + 462985554800484 * q^95 - 2815189465233952 * q^96 - 5046858838511686 * q^97 - 3547540222626416 * q^98 - 3251416762282424 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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