LMFDB - Embedded newform 23.16.a.a.1.10

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			1.10
Root			$98.0924$ 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)

$f(q)$	$=$	$q+174.185 q^{2} +1818.41 q^{3} -2427.63 q^{4} +106224. q^{5} +316739. q^{6} -648096. q^{7} -6.13055e6 q^{8} -1.10423e7 q^{9} +O(q^{10})$	\(q+174.185 q^{2} +1818.41 q^{3} -2427.63 q^{4} +106224. q^{5} +316739. q^{6} -648096. q^{7} -6.13055e6 q^{8} -1.10423e7 q^{9} +1.85026e7 q^{10} -4.87730e7 q^{11} -4.41442e6 q^{12} +1.24432e8 q^{13} -1.12888e8 q^{14} +1.93159e8 q^{15} -9.88300e8 q^{16} -1.77251e9 q^{17} -1.92340e9 q^{18} +1.60464e8 q^{19} -2.57873e8 q^{20} -1.17850e9 q^{21} -8.49551e9 q^{22} +3.40483e9 q^{23} -1.11478e10 q^{24} -1.92340e10 q^{25} +2.16742e10 q^{26} -4.61716e10 q^{27} +1.57334e9 q^{28} +2.15381e10 q^{29} +3.36453e10 q^{30} -1.28733e11 q^{31} +2.87388e10 q^{32} -8.86891e10 q^{33} -3.08745e11 q^{34} -6.88434e10 q^{35} +2.68066e10 q^{36} +2.84106e11 q^{37} +2.79504e10 q^{38} +2.26269e11 q^{39} -6.51212e11 q^{40} +1.28409e11 q^{41} -2.05277e11 q^{42} -2.41406e12 q^{43} +1.18403e11 q^{44} -1.17296e12 q^{45} +5.93069e11 q^{46} -6.33751e11 q^{47} -1.79713e12 q^{48} -4.32753e12 q^{49} -3.35027e12 q^{50} -3.22315e12 q^{51} -3.02076e11 q^{52} -8.90922e12 q^{53} -8.04239e12 q^{54} -5.18086e12 q^{55} +3.97318e12 q^{56} +2.91789e11 q^{57} +3.75160e12 q^{58} +1.74394e13 q^{59} -4.68918e11 q^{60} +7.96896e12 q^{61} -2.24234e13 q^{62} +7.15647e12 q^{63} +3.73905e13 q^{64} +1.32177e13 q^{65} -1.54483e13 q^{66} +9.50943e13 q^{67} +4.30301e12 q^{68} +6.19136e12 q^{69} -1.19915e13 q^{70} +1.93273e13 q^{71} +6.76953e13 q^{72} +7.24717e13 q^{73} +4.94869e13 q^{74} -3.49753e13 q^{75} -3.89547e11 q^{76} +3.16095e13 q^{77} +3.94126e13 q^{78} -8.70408e13 q^{79} -1.04981e14 q^{80} +7.44863e13 q^{81} +2.23668e13 q^{82} +6.31191e13 q^{83} +2.86096e12 q^{84} -1.88284e14 q^{85} -4.20493e14 q^{86} +3.91650e13 q^{87} +2.99005e14 q^{88} -2.08888e13 q^{89} -2.04312e14 q^{90} -8.06441e13 q^{91} -8.26565e12 q^{92} -2.34090e14 q^{93} -1.10390e14 q^{94} +1.70451e13 q^{95} +5.22589e13 q^{96} +7.66567e14 q^{97} -7.53791e14 q^{98} +5.38566e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	174.185	0.962245	0.481122	−	0.876653i	$-0.340229\pi$
$2$	174.185	0.962245	0.481122	+	0.876653i	$0.340229\pi$
$3$	1818.41	0.480045	0.240022	−	0.970767i	$-0.422845\pi$
$3$	1818.41	0.480045	0.240022	+	0.970767i	$0.422845\pi$
$4$	−2427.63	−0.0740854
$5$	106224.	0.608062	0.304031	−	0.952662i	$-0.401667\pi$
$5$	106224.	0.608062	0.304031	+	0.952662i	$0.401667\pi$
$6$	316739.	0.461920
$7$	−648096.	−0.297443	−0.148722	−	0.988879i	$-0.547516\pi$
$7$	−648096.	−0.297443	−0.148722	+	0.988879i	$0.547516\pi$
$8$	−6.13055e6	−1.03353
$9$	−1.10423e7	−0.769557
$10$	1.85026e7	0.585105
$11$	−4.87730e7	−0.754630	−0.377315	−	0.926085i	$-0.623153\pi$
$11$	−4.87730e7	−0.754630	−0.377315	+	0.926085i	$0.623153\pi$
$12$	−4.41442e6	−0.0355643
$13$	1.24432e8	0.549994	0.274997	−	0.961445i	$-0.411323\pi$
$13$	1.24432e8	0.549994	0.274997	+	0.961445i	$0.411323\pi$
$14$	−1.12888e8	−0.286213
$15$	1.93159e8	0.291897
$16$	−9.88300e8	−0.920426
$17$	−1.77251e9	−1.04767	−0.523833	−	0.851821i	$-0.675498\pi$
$17$	−1.77251e9	−1.04767	−0.523833	+	0.851821i	$0.675498\pi$
$18$	−1.92340e9	−0.740502
$19$	1.60464e8	0.0411837	0.0205918	−	0.999788i	$-0.493445\pi$
$19$	1.60464e8	0.0411837	0.0205918	+	0.999788i	$0.493445\pi$
$20$	−2.57873e8	−0.0450485
$21$	−1.17850e9	−0.142786
$22$	−8.49551e9	−0.726138
$23$	3.40483e9	0.208514
$23$	3.40483e9	0.208514
$24$	−1.11478e10	−0.496142
$25$	−1.92340e10	−0.630260
$26$	2.16742e10	0.529229
$27$	−4.61716e10	−0.849467
$28$	1.57334e9	0.0220362
$29$	2.15381e10	0.231858	0.115929	−	0.993258i	$-0.463016\pi$
$29$	2.15381e10	0.231858	0.115929	+	0.993258i	$0.463016\pi$
$30$	3.36453e10	0.280876
$31$	−1.28733e11	−0.840385	−0.420193	−	0.907435i	$-0.638038\pi$
$31$	−1.28733e11	−0.840385	−0.420193	+	0.907435i	$0.638038\pi$
$32$	2.87388e10	0.147858
$33$	−8.86891e10	−0.362256
$34$	−3.08745e11	−1.00811
$35$	−6.88434e10	−0.180864
$36$	2.68066e10	0.0570129
$37$	2.84106e11	0.492002	0.246001	−	0.969270i	$-0.420883\pi$
$37$	2.84106e11	0.492002	0.246001	+	0.969270i	$0.420883\pi$
$38$	2.79504e10	0.0396288
$39$	2.26269e11	0.264022
$40$	−6.51212e11	−0.628452
$41$	1.28409e11	0.102971	0.0514855	−	0.998674i	$-0.483604\pi$
$41$	1.28409e11	0.102971	0.0514855	+	0.998674i	$0.483604\pi$
$42$	−2.05277e11	−0.137395
$43$	−2.41406e12	−1.35436	−0.677182	−	0.735816i	$-0.736799\pi$
$43$	−2.41406e12	−1.35436	−0.677182	+	0.735816i	$0.736799\pi$
$44$	1.18403e11	0.0559070
$45$	−1.17296e12	−0.467939
$46$	5.93069e11	0.200642
$47$	−6.33751e11	−0.182467	−0.0912336	−	0.995830i	$-0.529081\pi$
$47$	−6.33751e11	−0.182467	−0.0912336	+	0.995830i	$0.529081\pi$
$48$	−1.79713e12	−0.441846
$49$	−4.32753e12	−0.911528
$50$	−3.35027e12	−0.606464
$51$	−3.22315e12	−0.502927
$52$	−3.02076e11	−0.0407465
$53$	−8.90922e12	−1.04177	−0.520884	−	0.853628i	$-0.674398\pi$
$53$	−8.90922e12	−1.04177	−0.520884	+	0.853628i	$0.674398\pi$
$54$	−8.04239e12	−0.817395
$55$	−5.18086e12	−0.458862
$56$	3.97318e12	0.307417
$57$	2.91789e11	0.0197700
$58$	3.75160e12	0.223104
$59$	1.74394e13	0.912309	0.456155	−	0.889901i	$-0.349226\pi$
$59$	1.74394e13	0.912309	0.456155	+	0.889901i	$0.349226\pi$
$60$	−4.68918e11	−0.0216253
$61$	7.96896e12	0.324659	0.162330	−	0.986737i	$-0.448099\pi$
$61$	7.96896e12	0.324659	0.162330	+	0.986737i	$0.448099\pi$
$62$	−2.24234e13	−0.808656
$63$	7.15647e12	0.228899
$64$	3.73905e13	1.06270
$65$	1.32177e13	0.334431
$66$	−1.54483e13	−0.348579
$67$	9.50943e13	1.91687	0.958437	−	0.285304i	$-0.0920944\pi$
$67$	9.50943e13	1.91687	0.958437	+	0.285304i	$0.0920944\pi$
$68$	4.30301e12	0.0776167
$69$	6.19136e12	0.100096
$70$	−1.19915e13	−0.174035
$71$	1.93273e13	0.252193	0.126097	−	0.992018i	$-0.459755\pi$
$71$	1.93273e13	0.252193	0.126097	+	0.992018i	$0.459755\pi$
$72$	6.76953e13	0.795362
$73$	7.24717e13	0.767798	0.383899	−	0.923375i	$-0.374581\pi$
$73$	7.24717e13	0.767798	0.383899	+	0.923375i	$0.374581\pi$
$74$	4.94869e13	0.473426
$75$	−3.49753e13	−0.302553
$76$	−3.89547e11	−0.00305111
$77$	3.16095e13	0.224459
$78$	3.94126e13	0.254054
$79$	−8.70408e13	−0.509941	−0.254970	−	0.966949i	$-0.582066\pi$
$79$	−8.70408e13	−0.509941	−0.254970	+	0.966949i	$0.582066\pi$
$80$	−1.04981e14	−0.559676
$81$	7.44863e13	0.361775
$82$	2.23668e13	0.0990833
$83$	6.31191e13	0.255314	0.127657	−	0.991818i	$-0.459254\pi$
$83$	6.31191e13	0.255314	0.127657	+	0.991818i	$0.459254\pi$
$84$	2.86096e12	0.0105783
$85$	−1.88284e14	−0.637046
$86$	−4.20493e14	−1.30323
$87$	3.91650e13	0.111302
$88$	2.99005e14	0.779935
$89$	−2.08888e13	−0.0500598	−0.0250299	−	0.999687i	$-0.507968\pi$
$89$	−2.08888e13	−0.0500598	−0.0250299	+	0.999687i	$0.507968\pi$
$90$	−2.04312e14	−0.450271
$91$	−8.06441e13	−0.163592
$92$	−8.26565e12	−0.0154479
$93$	−2.34090e14	−0.403423
$94$	−1.10390e14	−0.175578
$95$	1.70451e13	0.0250423
$96$	5.22589e13	0.0709784
$97$	7.66567e14	0.963302	0.481651	−	0.876363i	$-0.340037\pi$
$97$	7.66567e14	0.963302	0.481651	+	0.876363i	$0.340037\pi$
$98$	−7.53791e14	−0.877113
$99$	5.38566e14	0.580731
$100$	4.66931e13	0.0466931
$101$	1.53308e15	1.42283	0.711416	−	0.702771i	$-0.248054\pi$
$101$	1.53308e15	1.42283	0.711416	+	0.702771i	$0.248054\pi$
$102$	−5.61425e14	−0.483938
$103$	7.98029e13	0.0639351	0.0319675	−	0.999489i	$-0.489823\pi$
$103$	7.98029e13	0.0639351	0.0319675	+	0.999489i	$0.489823\pi$
$104$	−7.62838e14	−0.568437
$105$	−1.25185e14	−0.0868228
$106$	−1.55185e15	−1.00244
$107$	−1.97363e15	−1.18819	−0.594096	−	0.804394i	$-0.702490\pi$
$107$	−1.97363e15	−1.18819	−0.594096	+	0.804394i	$0.702490\pi$
$108$	1.12087e14	0.0629330
$109$	−2.72004e15	−1.42520	−0.712601	−	0.701569i	$-0.752483\pi$
$109$	−2.72004e15	−1.42520	−0.712601	+	0.701569i	$0.752483\pi$
$110$	−9.02428e14	−0.441537
$111$	5.16620e14	0.236183
$112$	6.40513e14	0.273774
$113$	1.69811e15	0.679012	0.339506	−	0.940604i	$-0.389740\pi$
$113$	1.69811e15	0.679012	0.339506	+	0.940604i	$0.389740\pi$
$114$	5.08252e13	0.0190236
$115$	3.61675e14	0.126790
$116$	−5.22864e13	−0.0171773
$117$	−1.37402e15	−0.423252
$118$	3.03768e15	0.877865
$119$	1.14876e15	0.311621
$120$	−1.18417e15	−0.301685
$121$	−1.79845e15	−0.430534
$122$	1.38807e15	0.312402
$123$	2.33499e14	0.0494307
$124$	3.12517e14	0.0622602
$125$	−5.28482e15	−0.991300
$126$	1.24655e15	0.220257
$127$	−2.16779e14	−0.0360985	−0.0180492	−	0.999837i	$-0.505746\pi$
$127$	−2.16779e14	−0.0360985	−0.0180492	+	0.999837i	$0.505746\pi$
$128$	5.57114e15	0.874721
$129$	−4.38975e15	−0.650155
$130$	2.30233e15	0.321804
$131$	5.84372e15	0.771178	0.385589	−	0.922671i	$-0.373998\pi$
$131$	5.84372e15	0.771178	0.385589	+	0.922671i	$0.373998\pi$
$132$	2.15304e14	0.0268379
$133$	−1.03996e14	−0.0122498
$134$	1.65640e16	1.84450
$135$	−4.90453e15	−0.516529
$136$	1.08665e16	1.08280
$137$	−1.06591e16	−1.00534	−0.502672	−	0.864477i	$-0.667650\pi$
$137$	−1.06591e16	−1.00534	−0.502672	+	0.864477i	$0.667650\pi$
$138$	1.07844e15	0.0963171
$139$	1.73311e16	1.46627	0.733136	−	0.680082i	$-0.238056\pi$
$139$	1.73311e16	1.46627	0.733136	+	0.680082i	$0.238056\pi$
$140$	1.67126e14	0.0133994
$141$	−1.15242e15	−0.0875924
$142$	3.36652e15	0.242672
$143$	−6.06893e15	−0.415042
$144$	1.09131e16	0.708320
$145$	2.28786e15	0.140984
$146$	1.26235e16	0.738809
$147$	−7.86922e15	−0.437574
$148$	−6.89703e14	−0.0364501
$149$	−1.68072e16	−0.844500	−0.422250	−	0.906479i	$-0.638760\pi$
$149$	−1.68072e16	−0.844500	−0.422250	+	0.906479i	$0.638760\pi$
$150$	−6.09216e15	−0.291130
$151$	−7.60055e14	−0.0345556	−0.0172778	−	0.999851i	$-0.505500\pi$
$151$	−7.60055e14	−0.0345556	−0.0172778	+	0.999851i	$0.505500\pi$
$152$	−9.83731e14	−0.0425647
$153$	1.95726e16	0.806239
$154$	5.50590e15	0.215985
$155$	−1.36746e16	−0.511007
$156$	−5.49296e14	−0.0195602
$157$	−3.12117e16	−1.05942	−0.529712	−	0.848178i	$-0.677700\pi$
$157$	−3.12117e16	−1.05942	−0.529712	+	0.848178i	$0.677700\pi$
$158$	−1.51612e16	−0.490688
$159$	−1.62006e16	−0.500095
$160$	3.05276e15	0.0899068
$161$	−2.20665e15	−0.0620212
$162$	1.29744e16	0.348116
$163$	−3.17675e16	−0.813908	−0.406954	−	0.913449i	$-0.633409\pi$
$163$	−3.17675e16	−0.813908	−0.406954	+	0.913449i	$0.633409\pi$
$164$	−3.11728e14	−0.00762865
$165$	−9.42092e15	−0.220274
$166$	1.09944e16	0.245675
$167$	−7.18995e16	−1.53586	−0.767932	−	0.640532i	$-0.778714\pi$
$167$	−7.18995e16	−1.53586	−0.767932	+	0.640532i	$0.778714\pi$
$168$	7.22486e15	0.147574
$169$	−3.57025e16	−0.697506
$170$	−3.27962e16	−0.612994
$171$	−1.77189e15	−0.0316932
$172$	5.86045e15	0.100339
$173$	−2.75194e16	−0.451122	−0.225561	−	0.974229i	$-0.572421\pi$
$173$	−2.75194e16	−0.451122	−0.225561	+	0.974229i	$0.572421\pi$
$174$	6.82194e15	0.107100
$175$	1.24655e16	0.187467
$176$	4.82023e16	0.694581
$177$	3.17120e16	0.437949
$178$	−3.63852e15	−0.0481698
$179$	−8.70692e16	−1.10527	−0.552633	−	0.833425i	$-0.686377\pi$
$179$	−8.70692e16	−1.10527	−0.552633	+	0.833425i	$0.686377\pi$
$180$	2.84751e15	0.0346674
$181$	5.60663e16	0.654806	0.327403	−	0.944885i	$-0.393827\pi$
$181$	5.60663e16	0.654806	0.327403	+	0.944885i	$0.393827\pi$
$182$	−1.40470e16	−0.157415
$183$	1.44908e16	0.155851
$184$	−2.08734e16	−0.215506
$185$	3.01789e16	0.299168
$186$	−4.07749e16	−0.388191
$187$	8.64508e16	0.790600
$188$	1.53851e15	0.0135181
$189$	2.99236e16	0.252668
$190$	2.96900e15	0.0240968
$191$	−4.15713e16	−0.324372	−0.162186	−	0.986760i	$-0.551854\pi$
$191$	−4.15713e16	−0.324372	−0.162186	+	0.986760i	$0.551854\pi$
$192$	6.79911e16	0.510144
$193$	7.60300e16	0.548663	0.274331	−	0.961635i	$-0.411544\pi$
$193$	7.60300e16	0.548663	0.274331	+	0.961635i	$0.411544\pi$
$194$	1.33524e17	0.926932
$195$	2.40352e16	0.160542
$196$	1.05056e16	0.0675308
$197$	−1.90838e17	−1.18078	−0.590390	−	0.807118i	$-0.701026\pi$
$197$	−1.90838e17	−1.18078	−0.590390	+	0.807118i	$0.701026\pi$
$198$	9.38100e16	0.558805
$199$	1.15451e17	0.662213	0.331107	−	0.943593i	$-0.392578\pi$
$199$	1.15451e17	0.662213	0.331107	+	0.943593i	$0.392578\pi$
$200$	1.17915e17	0.651395
$201$	1.72920e17	0.920186
$202$	2.67039e17	1.36911
$203$	−1.39587e16	−0.0689645
$204$	7.82462e15	0.0372595
$205$	1.36401e16	0.0626128
$206$	1.39005e16	0.0615212
$207$	−3.75971e16	−0.160464
$208$	−1.22976e17	−0.506229
$209$	−7.82630e15	−0.0310784
$210$	−2.18054e16	−0.0835447
$211$	−3.40205e17	−1.25783	−0.628915	−	0.777474i	$-0.716501\pi$
$211$	−3.40205e17	−1.25783	−0.628915	+	0.777474i	$0.716501\pi$
$212$	2.16283e16	0.0771797
$213$	3.51449e16	0.121064
$214$	−3.43776e17	−1.14333
$215$	−2.56432e17	−0.823537
$216$	2.83057e17	0.877952
$217$	8.34315e16	0.249967
$218$	−4.73790e17	−1.37139
$219$	1.31783e17	0.368577
$220$	1.25772e16	0.0339950
$221$	−2.20558e17	−0.576210
$222$	8.99873e16	0.227266
$223$	−1.22291e17	−0.298612	−0.149306	−	0.988791i	$-0.547704\pi$
$223$	−1.22291e17	−0.298612	−0.149306	+	0.988791i	$0.547704\pi$
$224$	−1.86255e16	−0.0439793
$225$	2.12388e17	0.485021
$226$	2.95785e17	0.653375
$227$	2.34394e17	0.500901	0.250450	−	0.968129i	$-0.419421\pi$
$227$	2.34394e17	0.500901	0.250450	+	0.968129i	$0.419421\pi$
$228$	−7.08355e14	−0.00146467
$229$	2.96005e17	0.592288	0.296144	−	0.955143i	$-0.404299\pi$
$229$	2.96005e17	0.592288	0.296144	+	0.955143i	$0.404299\pi$
$230$	6.29982e16	0.122003
$231$	5.74790e16	0.107751
$232$	−1.32040e17	−0.239633
$233$	−1.52151e17	−0.267366	−0.133683	−	0.991024i	$-0.542680\pi$
$233$	−1.52151e17	−0.267366	−0.133683	+	0.991024i	$0.542680\pi$
$234$	−2.39333e17	−0.407272
$235$	−6.73196e16	−0.110951
$236$	−4.23365e16	−0.0675888
$237$	−1.58276e17	−0.244794
$238$	2.00096e17	0.299856
$239$	−5.13069e17	−0.745060	−0.372530	−	0.928020i	$-0.621510\pi$
$239$	−5.13069e17	−0.745060	−0.372530	+	0.928020i	$0.621510\pi$
$240$	−1.90899e17	−0.268670
$241$	−7.82625e16	−0.106764	−0.0533821	−	0.998574i	$-0.517000\pi$
$241$	−7.82625e16	−0.106764	−0.0533821	+	0.998574i	$0.517000\pi$
$242$	−3.13262e17	−0.414279
$243$	7.97958e17	1.02313
$244$	−1.93457e16	−0.0240525
$245$	−4.59688e17	−0.554266
$246$	4.06720e16	0.0475644
$247$	1.99669e16	0.0226508
$248$	7.89206e17	0.868566
$249$	1.14776e17	0.122562
$250$	−9.20536e17	−0.953873
$251$	−6.15106e17	−0.618582	−0.309291	−	0.950967i	$-0.600092\pi$
$251$	−6.15106e17	−0.618582	−0.309291	+	0.950967i	$0.600092\pi$
$252$	−1.73732e16	−0.0169581
$253$	−1.66063e17	−0.157351
$254$	−3.77596e16	−0.0347356
$255$	−3.42377e17	−0.305811
$256$	−2.54803e17	−0.221006
$257$	1.03801e18	0.874383	0.437191	−	0.899369i	$-0.355973\pi$
$257$	1.03801e18	0.874383	0.437191	+	0.899369i	$0.355973\pi$
$258$	−7.64628e17	−0.625608
$259$	−1.84128e17	−0.146343
$260$	−3.20877e16	−0.0247764
$261$	−2.37830e17	−0.178428
$262$	1.01789e18	0.742062
$263$	−9.81321e17	−0.695253	−0.347627	−	0.937633i	$-0.613012\pi$
$263$	−9.81321e17	−0.695253	−0.347627	+	0.937633i	$0.613012\pi$
$264$	5.43713e17	0.374404
$265$	−9.46374e17	−0.633460
$266$	−1.81145e16	−0.0117873
$267$	−3.79844e16	−0.0240309
$268$	−2.30854e17	−0.142012
$269$	−7.01436e15	−0.00419610	−0.00209805	−	0.999998i	$-0.500668\pi$
$269$	−7.01436e15	−0.00419610	−0.00209805	+	0.999998i	$0.500668\pi$
$270$	−8.54295e17	−0.497027
$271$	−1.67153e18	−0.945898	−0.472949	−	0.881090i	$-0.656811\pi$
$271$	−1.67153e18	−0.945898	−0.472949	+	0.881090i	$0.656811\pi$
$272$	1.75178e18	0.964299
$273$	−1.46644e17	−0.0785315
$274$	−1.85665e18	−0.967388
$275$	9.38100e17	0.475613
$276$	−1.50303e16	−0.00741567
$277$	1.06789e18	0.512776	0.256388	−	0.966574i	$-0.417468\pi$
$277$	1.06789e18	0.512776	0.256388	+	0.966574i	$0.417468\pi$
$278$	3.01881e18	1.41091
$279$	1.42151e18	0.646724
$280$	4.22048e17	0.186929
$281$	2.90388e18	1.25222	0.626111	−	0.779734i	$-0.284646\pi$
$281$	2.90388e18	1.25222	0.626111	+	0.779734i	$0.284646\pi$
$282$	−2.00734e17	−0.0842853
$283$	2.06679e18	0.845079	0.422540	−	0.906344i	$-0.361139\pi$
$283$	2.06679e18	0.845079	0.422540	+	0.906344i	$0.361139\pi$
$284$	−4.69195e16	−0.0186838
$285$	3.09950e16	0.0120214
$286$	−1.05712e18	−0.399372
$287$	−8.32210e16	−0.0306280
$288$	−3.17343e17	−0.113785
$289$	2.79385e17	0.0976043
$290$	3.98511e17	0.135661
$291$	1.39393e18	0.462428
$292$	−1.75934e17	−0.0568826
$293$	3.94914e18	1.24450	0.622251	−	0.782818i	$-0.286218\pi$
$293$	3.94914e18	1.24450	0.622251	+	0.782818i	$0.286218\pi$
$294$	−1.37070e18	−0.421053
$295$	1.85249e18	0.554741
$296$	−1.74172e18	−0.508500
$297$	2.25192e18	0.641033
$298$	−2.92757e18	−0.812615
$299$	4.23670e17	0.114682
$300$	8.49070e16	0.0224148
$301$	1.56454e18	0.402846
$302$	−1.32390e17	−0.0332509
$303$	2.78776e18	0.683023
$304$	−1.58586e17	−0.0379065
$305$	8.46496e17	0.197413
$306$	3.40926e18	0.775799
$307$	−5.31072e18	−1.17928	−0.589638	−	0.807668i	$-0.700730\pi$
$307$	−5.31072e18	−1.17928	−0.589638	+	0.807668i	$0.700730\pi$
$308$	−7.67362e16	−0.0166292
$309$	1.45114e17	0.0306917
$310$	−2.38191e18	−0.491713
$311$	3.35276e17	0.0675615	0.0337808	−	0.999429i	$-0.489245\pi$
$311$	3.35276e17	0.0675615	0.0337808	+	0.999429i	$0.489245\pi$
$312$	−1.38715e18	−0.272875
$313$	6.32925e17	0.121554	0.0607771	−	0.998151i	$-0.480642\pi$
$313$	6.32925e17	0.121554	0.0607771	+	0.998151i	$0.480642\pi$
$314$	−5.43660e18	−1.01942
$315$	7.60189e17	0.139185
$316$	2.11303e17	0.0377791
$317$	−5.93963e18	−1.03709	−0.518543	−	0.855051i	$-0.673525\pi$
$317$	−5.93963e18	−1.03709	−0.518543	+	0.855051i	$0.673525\pi$
$318$	−2.82190e18	−0.481214
$319$	−1.05048e18	−0.174967
$320$	3.97177e18	0.646189
$321$	−3.58886e18	−0.570385
$322$	−3.84365e17	−0.0596795
$323$	−2.84425e17	−0.0431468
$324$	−1.80825e17	−0.0268022
$325$	−2.39333e18	−0.346640
$326$	−5.53341e18	−0.783179
$327$	−4.94614e18	−0.684161
$328$	−7.87215e17	−0.106424
$329$	4.10731e17	0.0542736
$330$	−1.64098e18	−0.211958
$331$	−7.09914e18	−0.896387	−0.448193	−	0.893937i	$-0.647932\pi$
$331$	−7.09914e18	−0.896387	−0.448193	+	0.893937i	$0.647932\pi$
$332$	−1.53230e17	−0.0189150
$333$	−3.13718e18	−0.378623
$334$	−1.25238e19	−1.47788
$335$	1.01013e19	1.16558
$336$	1.16471e18	0.131424
$337$	−8.23709e17	−0.0908970	−0.0454485	−	0.998967i	$-0.514472\pi$
$337$	−8.23709e17	−0.0908970	−0.0454485	+	0.998967i	$0.514472\pi$
$338$	−6.21883e18	−0.671172
$339$	3.08785e18	0.325956
$340$	4.57083e17	0.0471958
$341$	6.27871e18	0.634180
$342$	−3.08637e17	−0.0304966
$343$	5.88153e18	0.568571
$344$	1.47995e19	1.39978
$345$	6.57672e17	0.0608648
$346$	−4.79347e18	−0.434089
$347$	1.06864e19	0.947019	0.473510	−	0.880789i	$-0.342987\pi$
$347$	1.06864e19	0.947019	0.473510	+	0.880789i	$0.342987\pi$
$348$	−9.50780e16	−0.00824586
$349$	−2.07174e19	−1.75851	−0.879254	−	0.476353i	$-0.841958\pi$
$349$	−2.07174e19	−1.75851	−0.879254	+	0.476353i	$0.841958\pi$
$350$	2.17130e18	0.180389
$351$	−5.74524e18	−0.467202
$352$	−1.40168e18	−0.111578
$353$	1.06603e19	0.830730	0.415365	−	0.909655i	$-0.363654\pi$
$353$	1.06603e19	0.830730	0.415365	+	0.909655i	$0.363654\pi$
$354$	5.52375e18	0.421414
$355$	2.05302e18	0.153349
$356$	5.07103e16	0.00370870
$357$	2.08891e18	0.149592
$358$	−1.51661e19	−1.06354
$359$	2.60663e19	1.79007	0.895037	−	0.445993i	$-0.147149\pi$
$359$	2.60663e19	1.79007	0.895037	+	0.445993i	$0.147149\pi$
$360$	7.19088e18	0.483630
$361$	−1.51554e19	−0.998304
$362$	9.76590e18	0.630083
$363$	−3.27031e18	−0.206675
$364$	1.95774e17	0.0121198
$365$	7.69824e18	0.466869
$366$	2.52408e18	0.149967
$367$	1.64267e19	0.956211	0.478106	−	0.878302i	$-0.341324\pi$
$367$	1.64267e19	0.956211	0.478106	+	0.878302i	$0.341324\pi$
$368$	−3.36499e18	−0.191922
$369$	−1.41793e18	−0.0792421
$370$	5.25670e18	0.287873
$371$	5.77403e18	0.309866
$372$	5.68283e17	0.0298877
$373$	2.47800e18	0.127728	0.0638639	−	0.997959i	$-0.479658\pi$
$373$	2.47800e18	0.127728	0.0638639	+	0.997959i	$0.479658\pi$
$374$	1.50584e19	0.760751
$375$	−9.60995e18	−0.475868
$376$	3.88524e18	0.188586
$377$	2.68003e18	0.127520
$378$	5.21224e18	0.243128
$379$	−4.92785e18	−0.225353	−0.112677	−	0.993632i	$-0.535942\pi$
$379$	−4.92785e18	−0.225353	−0.112677	+	0.993632i	$0.535942\pi$
$380$	−4.13793e16	−0.00185526
$381$	−3.94192e17	−0.0173289
$382$	−7.24109e18	−0.312125
$383$	−2.15872e19	−0.912442	−0.456221	−	0.889866i	$-0.650797\pi$
$383$	−2.15872e19	−0.912442	−0.456221	+	0.889866i	$0.650797\pi$
$384$	1.01306e19	0.419905
$385$	3.35770e18	0.136485
$386$	1.32433e19	0.527948
$387$	2.66568e19	1.04226
$388$	−1.86094e18	−0.0713666
$389$	−4.11041e18	−0.154619	−0.0773095	−	0.997007i	$-0.524633\pi$
$389$	−4.11041e18	−0.154619	−0.0773095	+	0.997007i	$0.524633\pi$
$390$	4.18657e18	0.154480
$391$	−6.03510e18	−0.218453
$392$	2.65301e19	0.942094
$393$	1.06263e19	0.370200
$394$	−3.32412e19	−1.13620
$395$	−9.24583e18	−0.310076
$396$	−1.30744e18	−0.0430236
$397$	−5.84675e19	−1.88793	−0.943964	−	0.330048i	$-0.892935\pi$
$397$	−5.84675e19	−1.88793	−0.943964	+	0.330048i	$0.892935\pi$
$398$	2.01097e19	0.637211
$399$	−1.89107e17	−0.00588045
$400$	1.90090e19	0.580108
$401$	−3.43033e19	−1.02743	−0.513715	−	0.857961i	$-0.671731\pi$
$401$	−3.43033e19	−1.02743	−0.513715	+	0.857961i	$0.671731\pi$
$402$	3.01201e19	0.885444
$403$	−1.60186e19	−0.462207
$404$	−3.72174e18	−0.105411
$405$	7.91224e18	0.219982
$406$	−2.43140e18	−0.0663607
$407$	−1.38567e19	−0.371279
$408$	1.97597e19	0.519791
$409$	−3.90851e19	−1.00945	−0.504727	−	0.863279i	$-0.668407\pi$
$409$	−3.90851e19	−1.00945	−0.504727	+	0.863279i	$0.668407\pi$
$410$	2.37590e18	0.0602488
$411$	−1.93825e19	−0.482611
$412$	−1.93732e17	−0.00473665
$413$	−1.13024e19	−0.271360
$414$	−6.54885e18	−0.154405
$415$	6.70477e18	0.155247
$416$	3.57604e18	0.0813210
$417$	3.15149e19	0.703876
$418$	−1.36322e18	−0.0299051
$419$	−7.36900e19	−1.58783	−0.793914	−	0.608030i	$-0.791960\pi$
$419$	−7.36900e19	−1.58783	−0.793914	+	0.608030i	$0.791960\pi$
$420$	3.03903e17	0.00643230
$421$	−6.34719e19	−1.31967	−0.659835	−	0.751410i	$-0.729374\pi$
$421$	−6.34719e19	−1.31967	−0.659835	+	0.751410i	$0.729374\pi$
$422$	−5.92585e19	−1.21034
$423$	6.99807e18	0.140419
$424$	5.46184e19	1.07670
$425$	3.40926e19	0.660302
$426$	6.12171e18	0.116493
$427$	−5.16465e18	−0.0965677
$428$	4.79123e18	0.0880276
$429$	−1.10358e19	−0.199239
$430$	−4.46665e19	−0.792444
$431$	2.05401e19	0.358115	0.179058	−	0.983839i	$-0.442695\pi$
$431$	2.05401e19	0.358115	0.179058	+	0.983839i	$0.442695\pi$
$432$	4.56313e19	0.781871
$433$	9.80159e19	1.65058	0.825291	−	0.564708i	$-0.191011\pi$
$433$	9.80159e19	1.65058	0.825291	+	0.564708i	$0.191011\pi$
$434$	1.45325e19	0.240529
$435$	4.16026e18	0.0676786
$436$	6.60325e18	0.105587
$437$	5.46352e17	0.00858739
$438$	2.29546e19	0.354662
$439$	−5.11706e19	−0.777207	−0.388604	−	0.921405i	$-0.627042\pi$
$439$	−5.11706e19	−0.777207	−0.388604	+	0.921405i	$0.627042\pi$
$440$	3.17615e19	0.474249
$441$	4.77859e19	0.701472
$442$	−3.84179e19	−0.554455
$443$	−4.16617e19	−0.591165	−0.295583	−	0.955317i	$-0.595514\pi$
$443$	−4.16617e19	−0.591165	−0.295583	+	0.955317i	$0.595514\pi$
$444$	−1.25416e18	−0.0174977
$445$	−2.21890e18	−0.0304395
$446$	−2.13012e19	−0.287337
$447$	−3.05624e19	−0.405398
$448$	−2.42326e19	−0.316093
$449$	−1.51402e20	−1.94215	−0.971076	−	0.238769i	$-0.923256\pi$
$449$	−1.51402e20	−1.94215	−0.971076	+	0.238769i	$0.923256\pi$
$450$	3.69947e19	0.466709
$451$	−6.26287e18	−0.0777050
$452$	−4.12238e18	−0.0503048
$453$	−1.38209e18	−0.0165882
$454$	4.08278e19	0.481989
$455$	−8.56634e18	−0.0994741
$456$	−1.78882e18	−0.0204330
$457$	−8.54780e19	−0.960468	−0.480234	−	0.877140i	$-0.659448\pi$
$457$	−8.54780e19	−0.960468	−0.480234	+	0.877140i	$0.659448\pi$
$458$	5.15596e19	0.569926
$459$	8.18398e19	0.889957
$460$	−8.78012e17	−0.00939326
$461$	2.36983e19	0.249437	0.124719	−	0.992192i	$-0.460197\pi$
$461$	2.36983e19	0.249437	0.124719	+	0.992192i	$0.460197\pi$
$462$	1.00120e19	0.103682
$463$	1.43300e20	1.46013	0.730063	−	0.683380i	$-0.239491\pi$
$463$	1.43300e20	1.46013	0.730063	+	0.683380i	$0.239491\pi$
$464$	−2.12861e19	−0.213408
$465$	−2.48660e19	−0.245306
$466$	−2.65024e19	−0.257271
$467$	6.81009e19	0.650544	0.325272	−	0.945621i	$-0.394544\pi$
$467$	6.81009e19	0.650544	0.325272	+	0.945621i	$0.394544\pi$
$468$	3.33561e18	0.0313568
$469$	−6.16302e19	−0.570161
$470$	−1.17261e19	−0.106762
$471$	−5.67555e19	−0.508571
$472$	−1.06913e20	−0.942902
$473$	1.17741e20	1.02204
$474$	−2.75692e19	−0.235552
$475$	−3.08637e18	−0.0259564
$476$	−2.78876e18	−0.0230866
$477$	9.83783e19	0.801700
$478$	−8.93688e19	−0.716930
$479$	9.88223e19	0.780438	0.390219	−	0.920722i	$-0.372399\pi$
$479$	9.88223e19	0.780438	0.390219	+	0.920722i	$0.372399\pi$
$480$	5.55116e18	0.0431593
$481$	3.53519e19	0.270598
$482$	−1.36321e19	−0.102733
$483$	−4.01259e18	−0.0297729
$484$	4.36596e18	0.0318962
$485$	8.14279e19	0.585748
$486$	1.38992e20	0.984506
$487$	1.53748e20	1.07236	0.536181	−	0.844103i	$-0.319867\pi$
$487$	1.53748e20	1.07236	0.536181	+	0.844103i	$0.319867\pi$
$488$	−4.88541e19	−0.335546
$489$	−5.77662e19	−0.390712
$490$	−8.00708e19	−0.533339
$491$	7.76823e19	0.509578	0.254789	−	0.966997i	$-0.417994\pi$
$491$	7.76823e19	0.509578	0.254789	+	0.966997i	$0.417994\pi$
$492$	−5.66849e17	−0.00366209
$493$	−3.81765e19	−0.242910
$494$	3.47793e18	0.0217956
$495$	5.72087e19	0.353120
$496$	1.27227e20	0.773512
$497$	−1.25259e19	−0.0750131
$498$	1.99923e19	0.117935
$499$	−4.80573e19	−0.279258	−0.139629	−	0.990204i	$-0.544591\pi$
$499$	−4.80573e19	−0.279258	−0.139629	+	0.990204i	$0.544591\pi$
$500$	1.28296e19	0.0734408
$501$	−1.30743e20	−0.737283
$502$	−1.07142e20	−0.595227
$503$	2.19817e20	1.20310	0.601550	−	0.798835i	$-0.294550\pi$
$503$	2.19817e20	1.20310	0.601550	+	0.798835i	$0.294550\pi$
$504$	−4.38731e19	−0.236575
$505$	1.62850e20	0.865170
$506$	−2.89257e19	−0.151410
$507$	−6.49217e19	−0.334834
$508$	5.26259e17	0.00267437
$509$	3.18443e20	1.59459	0.797294	−	0.603591i	$-0.206264\pi$
$509$	3.18443e20	1.59459	0.797294	+	0.603591i	$0.206264\pi$
$510$	−5.96368e19	−0.294265
$511$	−4.69686e19	−0.228376
$512$	−2.26938e20	−1.08738
$513$	−7.40887e18	−0.0349842
$514$	1.80805e20	0.841370
$515$	8.47699e18	0.0388765
$516$	1.06567e19	0.0481670
$517$	3.09099e19	0.137695
$518$	−3.20722e19	−0.140817
$519$	−5.00415e19	−0.216559
$520$	−8.10318e19	−0.345645
$521$	−4.08364e20	−1.71698	−0.858488	−	0.512834i	$-0.828596\pi$
$521$	−4.08364e20	−1.71698	−0.858488	+	0.512834i	$0.828596\pi$
$522$	−4.14264e19	−0.171691
$523$	3.11801e20	1.27384	0.636921	−	0.770929i	$-0.280208\pi$
$523$	3.11801e20	1.27384	0.636921	+	0.770929i	$0.280208\pi$
$524$	−1.41864e19	−0.0571330
$525$	2.26673e19	0.0899923
$526$	−1.70931e20	−0.669004
$527$	2.28182e20	0.880443
$528$	8.76514e19	0.333430
$529$	1.15928e19	0.0434783
$530$	−1.64844e20	−0.609543
$531$	−1.92571e20	−0.702074
$532$	2.52464e17	0.000907531 0
$533$	1.59782e19	0.0566335
$534$	−6.61631e18	−0.0231236
$535$	−2.09647e20	−0.722495
$536$	−5.82980e20	−1.98115
$537$	−1.58327e20	−0.530577
$538$	−1.22180e18	−0.00403768
$539$	2.11067e20	0.687866
$540$	1.19064e19	0.0382672
$541$	−4.88575e20	−1.54864	−0.774322	−	0.632792i	$-0.781909\pi$
$541$	−4.88575e20	−1.54864	−0.774322	+	0.632792i	$0.781909\pi$
$542$	−2.91155e20	−0.910185
$543$	1.01951e20	0.314336
$544$	−5.09400e19	−0.154906
$545$	−2.88934e20	−0.866612
$546$	−2.55431e19	−0.0755665
$547$	−4.65982e20	−1.35977	−0.679883	−	0.733321i	$-0.737969\pi$
$547$	−4.65982e20	−1.35977	−0.679883	+	0.733321i	$0.737969\pi$
$548$	2.58763e19	0.0744813
$549$	−8.79957e19	−0.249844
$550$	1.63403e20	0.457656
$551$	3.45608e18	0.00954876
$552$	−3.79564e19	−0.103453
$553$	5.64108e19	0.151678
$554$	1.86010e20	0.493416
$555$	5.48774e19	0.143614
$556$	−4.20734e19	−0.108629
$557$	−4.80399e20	−1.22374	−0.611869	−	0.790959i	$-0.709582\pi$
$557$	−4.80399e20	−1.22374	−0.611869	+	0.790959i	$0.709582\pi$
$558$	2.47606e20	0.622307
$559$	−3.00388e20	−0.744892
$560$	6.80379e19	0.166472
$561$	1.57203e20	0.379523
$562$	5.05812e20	1.20494
$563$	6.76496e20	1.59020	0.795101	−	0.606476i	$-0.207418\pi$
$563$	6.76496e20	1.59020	0.795101	+	0.606476i	$0.207418\pi$
$564$	2.79764e18	0.00648931
$565$	1.80380e20	0.412881
$566$	3.60003e20	0.813173
$567$	−4.82742e19	−0.107607
$568$	−1.18487e20	−0.260650
$569$	5.32079e20	1.15514	0.577569	−	0.816342i	$-0.304001\pi$
$569$	5.32079e20	1.15514	0.577569	+	0.816342i	$0.304001\pi$
$570$	5.39886e18	0.0115675
$571$	5.90195e20	1.24803	0.624014	−	0.781413i	$-0.285501\pi$
$571$	5.90195e20	1.24803	0.624014	+	0.781413i	$0.285501\pi$
$572$	1.47331e19	0.0307485
$573$	−7.55935e19	−0.155713
$574$	−1.44958e19	−0.0294716
$575$	−6.54885e19	−0.131418
$576$	−4.12877e20	−0.817809
$577$	1.67313e20	0.327123	0.163561	−	0.986533i	$-0.447702\pi$
$577$	1.67313e20	0.327123	0.163561	+	0.986533i	$0.447702\pi$
$578$	4.86646e19	0.0939192
$579$	1.38254e20	0.263383
$580$	−5.55408e18	−0.0104448
$581$	−4.09072e19	−0.0759414
$582$	2.42802e20	0.444969
$583$	4.34529e20	0.786149
$584$	−4.44291e20	−0.793544
$585$	−1.45954e20	−0.257364
$586$	6.87881e20	1.19752
$587$	−2.35939e20	−0.405522	−0.202761	−	0.979228i	$-0.564991\pi$
$587$	−2.35939e20	−0.405522	−0.202761	+	0.979228i	$0.564991\pi$
$588$	1.91035e19	0.0324178
$589$	−2.06571e19	−0.0346102
$590$	3.22675e20	0.533796
$591$	−3.47022e20	−0.566827
$592$	−2.80781e20	−0.452851
$593$	8.86048e20	1.41106	0.705532	−	0.708678i	$-0.250708\pi$
$593$	8.86048e20	1.41106	0.705532	+	0.708678i	$0.250708\pi$
$594$	3.92251e20	0.616830
$595$	1.22026e20	0.189485
$596$	4.08018e19	0.0625651
$597$	2.09936e20	0.317892
$598$	7.37970e19	0.110352
$599$	7.35434e20	1.08603	0.543016	−	0.839722i	$-0.317282\pi$
$599$	7.35434e20	1.08603	0.543016	+	0.839722i	$0.317282\pi$
$600$	2.14418e20	0.312699
$601$	−1.69623e20	−0.244301	−0.122151	−	0.992512i	$-0.538979\pi$
$601$	−1.69623e20	−0.244301	−0.122151	+	0.992512i	$0.538979\pi$
$602$	2.72520e20	0.387636
$603$	−1.05006e21	−1.47514
$604$	1.84513e18	0.00256006
$605$	−1.91038e20	−0.261791
$606$	4.85585e20	0.657235
$607$	−2.16302e20	−0.289164	−0.144582	−	0.989493i	$-0.546184\pi$
$607$	−2.16302e20	−0.289164	−0.144582	+	0.989493i	$0.546184\pi$
$608$	4.61155e18	0.00608933
$609$	−2.53826e19	−0.0331060
$610$	1.47447e20	0.189960
$611$	−7.88591e19	−0.100356
$612$	−4.75151e19	−0.0597305
$613$	−2.96244e20	−0.367872	−0.183936	−	0.982938i	$-0.558884\pi$
$613$	−2.96244e20	−0.367872	−0.183936	+	0.982938i	$0.558884\pi$
$614$	−9.25047e20	−1.13475
$615$	2.48032e19	0.0300570
$616$	−1.93784e20	−0.231986
$617$	−9.21169e20	−1.08943	−0.544717	−	0.838620i	$-0.683363\pi$
$617$	−9.21169e20	−1.08943	−0.544717	+	0.838620i	$0.683363\pi$
$618$	2.52767e19	0.0295329
$619$	9.19614e20	1.06151	0.530757	−	0.847524i	$-0.321908\pi$
$619$	9.19614e20	1.06151	0.530757	+	0.847524i	$0.321908\pi$
$620$	3.31968e19	0.0378581
$621$	−1.57206e20	−0.177126
$622$	5.84000e19	0.0650107
$623$	1.35380e19	0.0148899
$624$	−2.23621e20	−0.243013
$625$	2.56004e19	0.0274882
$626$	1.10246e20	0.116965
$627$	−1.42314e19	−0.0149190
$628$	7.57703e19	0.0784878
$629$	−5.03581e20	−0.515454
$630$	1.32414e20	0.133930
$631$	−4.62648e20	−0.462413	−0.231207	−	0.972905i	$-0.574267\pi$
$631$	−4.62648e20	−0.462413	−0.231207	+	0.972905i	$0.574267\pi$
$632$	5.33608e20	0.527041
$633$	−6.18630e20	−0.603814
$634$	−1.03459e21	−0.997931
$635$	−2.30271e19	−0.0219501
$636$	3.93290e19	0.0370497
$637$	−5.38485e20	−0.501335
$638$	−1.82977e20	−0.168361
$639$	−2.13418e20	−0.194077
$640$	5.91790e20	0.531885
$641$	1.66213e20	0.147648	0.0738242	−	0.997271i	$-0.476480\pi$
$641$	1.66213e20	0.147648	0.0738242	+	0.997271i	$0.476480\pi$
$642$	−6.25125e20	−0.548850
$643$	1.64811e21	1.43022	0.715110	−	0.699012i	$-0.246377\pi$
$643$	1.64811e21	1.43022	0.715110	+	0.699012i	$0.246377\pi$
$644$	5.35693e18	0.00459486
$645$	−4.66297e20	−0.395335
$646$	−4.95425e19	−0.0415177
$647$	1.28262e21	1.06247	0.531234	−	0.847225i	$-0.321729\pi$
$647$	1.28262e21	1.06247	0.531234	+	0.847225i	$0.321729\pi$
$648$	−4.56642e20	−0.373906
$649$	−8.50572e20	−0.688456
$650$	−4.16883e20	−0.333552
$651$	1.51713e20	0.119995
$652$	7.71196e19	0.0602987
$653$	−4.17795e20	−0.322935	−0.161467	−	0.986878i	$-0.551623\pi$
$653$	−4.17795e20	−0.322935	−0.161467	+	0.986878i	$0.551623\pi$
$654$	−8.61543e20	−0.658330
$655$	6.20744e20	0.468924
$656$	−1.26906e20	−0.0947772
$657$	−8.00254e20	−0.590864
$658$	7.15432e19	0.0522245
$659$	−4.68778e20	−0.338319	−0.169160	−	0.985589i	$-0.554105\pi$
$659$	−4.68778e20	−0.338319	−0.169160	+	0.985589i	$0.554105\pi$
$660$	2.28705e19	0.0163191
$661$	−1.41215e20	−0.0996254	−0.0498127	−	0.998759i	$-0.515862\pi$
$661$	−1.41215e20	−0.0996254	−0.0498127	+	0.998759i	$0.515862\pi$
$662$	−1.23656e21	−0.862543
$663$	−4.01065e20	−0.276607
$664$	−3.86955e20	−0.263876
$665$	−1.10469e19	−0.00744864
$666$	−5.46449e20	−0.364328
$667$	7.33333e19	0.0483457
$668$	1.74545e20	0.113785
$669$	−2.22374e20	−0.143347
$670$	1.75950e21	1.12157
$671$	−3.88670e20	−0.244998
$672$	−3.38688e19	−0.0211120
$673$	1.69210e21	1.04307	0.521534	−	0.853230i	$-0.325360\pi$
$673$	1.69210e21	1.04307	0.521534	+	0.853230i	$0.325360\pi$
$674$	−1.43478e20	−0.0874652
$675$	8.88064e20	0.535385
$676$	8.66724e19	0.0516750
$677$	−2.59785e21	−1.53179	−0.765895	−	0.642965i	$-0.777704\pi$
$677$	−2.59785e21	−1.53179	−0.765895	+	0.642965i	$0.777704\pi$
$678$	5.37857e20	0.313649
$679$	−4.96809e20	−0.286527
$680$	1.15428e21	0.658408
$681$	4.26223e20	0.240455
$682$	1.09366e21	0.610236
$683$	2.01017e21	1.10938	0.554688	−	0.832059i	$-0.312838\pi$
$683$	2.01017e21	1.10938	0.554688	+	0.832059i	$0.312838\pi$
$684$	4.30149e18	0.00234800
$685$	−1.13225e21	−0.611312
$686$	1.02447e21	0.547104
$687$	5.38258e20	0.284325
$688$	2.38582e21	1.24659
$689$	−1.10860e21	−0.572966
$690$	1.14556e20	0.0585668
$691$	−2.16923e21	−1.09703	−0.548517	−	0.836139i	$-0.684808\pi$
$691$	−2.16923e21	−1.09703	−0.548517	+	0.836139i	$0.684808\pi$
$692$	6.68069e19	0.0334215
$693$	−3.49042e20	−0.172734
$694$	1.86140e21	0.911264
$695$	1.84098e21	0.891585
$696$	−2.40103e20	−0.115034
$697$	−2.27606e20	−0.107879
$698$	−3.60866e21	−1.69212
$699$	−2.76673e20	−0.128348
$700$	−3.02616e19	−0.0138885
$701$	−1.24887e21	−0.567065	−0.283533	−	0.958963i	$-0.591506\pi$
$701$	−1.24887e21	−0.567065	−0.283533	+	0.958963i	$0.591506\pi$
$702$	−1.00073e21	−0.449562
$703$	4.55887e19	0.0202625
$704$	−1.82364e21	−0.801946
$705$	−1.22415e20	−0.0532616
$706$	1.85687e21	0.799365
$707$	−9.93580e20	−0.423211
$708$	−7.69849e19	−0.0324456
$709$	1.25832e21	0.524739	0.262369	−	0.964968i	$-0.415496\pi$
$709$	1.25832e21	0.524739	0.262369	+	0.964968i	$0.415496\pi$
$710$	3.57606e20	0.147559
$711$	9.61131e20	0.392428
$712$	1.28060e20	0.0517385
$713$	−4.38315e20	−0.175232
$714$	3.63857e20	0.143944
$715$	−6.44667e20	−0.252371
$716$	2.11372e20	0.0818841
$717$	−9.32968e20	−0.357662
$718$	4.54035e21	1.72249
$719$	1.90065e21	0.713570	0.356785	−	0.934187i	$-0.383873\pi$
$719$	1.90065e21	0.713570	0.356785	+	0.934187i	$0.383873\pi$
$720$	1.15924e21	0.430703
$721$	−5.17199e19	−0.0190170
$722$	−2.63984e21	−0.960613
$723$	−1.42313e20	−0.0512516
$724$	−1.36108e20	−0.0485115
$725$	−4.14263e20	−0.146131
$726$	−5.69638e20	−0.198872
$727$	2.13472e21	0.737620	0.368810	−	0.929505i	$-0.379765\pi$
$727$	2.13472e21	0.737620	0.368810	+	0.929505i	$0.379765\pi$
$728$	4.94392e20	0.169078
$729$	3.82216e20	0.129375
$730$	1.34092e21	0.449242
$731$	4.27896e21	1.41892
$732$	−3.51783e19	−0.0115463
$733$	1.51700e21	0.492839	0.246419	−	0.969163i	$-0.420746\pi$
$733$	1.51700e21	0.492839	0.246419	+	0.969163i	$0.420746\pi$
$734$	2.86128e21	0.920109
$735$	−8.35901e20	−0.266072
$736$	9.78507e19	0.0308305
$737$	−4.63803e21	−1.44653
$738$	−2.46981e20	−0.0762503
$739$	−2.30866e21	−0.705549	−0.352774	−	0.935708i	$-0.614762\pi$
$739$	−2.30866e21	−0.705549	−0.352774	+	0.935708i	$0.614762\pi$
$740$	−7.32631e19	−0.0221639
$741$	3.63080e19	0.0108734
$742$	1.00575e21	0.298167
$743$	−1.58219e21	−0.464348	−0.232174	−	0.972674i	$-0.574584\pi$
$743$	−1.58219e21	−0.464348	−0.232174	+	0.972674i	$0.574584\pi$
$744$	1.43510e21	0.416950
$745$	−1.78533e21	−0.513509
$746$	4.31630e20	0.122905
$747$	−6.96980e20	−0.196479
$748$	−2.09870e20	−0.0585719
$749$	1.27910e21	0.353419
$750$	−1.67391e21	−0.457902
$751$	5.39670e21	1.46160	0.730801	−	0.682591i	$-0.239147\pi$
$751$	5.39670e21	1.46160	0.730801	+	0.682591i	$0.239147\pi$
$752$	6.26336e20	0.167948
$753$	−1.11851e21	−0.296947
$754$	4.66821e20	0.122706
$755$	−8.07362e19	−0.0210119
$756$	−7.26434e19	−0.0187190
$757$	−6.17467e21	−1.57541	−0.787707	−	0.616050i	$-0.788732\pi$
$757$	−6.17467e21	−1.57541	−0.787707	+	0.616050i	$0.788732\pi$
$758$	−8.58357e20	−0.216845
$759$	−3.01971e20	−0.0755356
$760$	−1.04496e20	−0.0258820
$761$	2.50634e21	0.614689	0.307344	−	0.951598i	$-0.400560\pi$
$761$	2.50634e21	0.614689	0.307344	+	0.951598i	$0.400560\pi$
$762$	−6.86623e19	−0.0166746
$763$	1.76285e21	0.423917
$764$	1.00920e20	0.0240312
$765$	2.07909e21	0.490243
$766$	−3.76016e21	−0.877993
$767$	2.17003e21	0.501765
$768$	−4.63335e20	−0.106093
$769$	−2.24695e21	−0.509502	−0.254751	−	0.967007i	$-0.581993\pi$
$769$	−2.24695e21	−0.509502	−0.254751	+	0.967007i	$0.581993\pi$
$770$	5.84860e20	0.131332
$771$	1.88752e21	0.419743
$772$	−1.84573e20	−0.0406479
$773$	−7.13821e21	−1.55684	−0.778419	−	0.627745i	$-0.783978\pi$
$773$	−7.13821e21	−1.55684	−0.778419	+	0.627745i	$0.783978\pi$
$774$	4.64321e21	1.00291
$775$	2.47606e21	0.529661
$776$	−4.69948e21	−0.995604
$777$	−3.34819e20	−0.0702510
$778$	−7.15971e20	−0.148781
$779$	2.06049e19	0.00424073
$780$	−5.83485e19	−0.0118938
$781$	−9.42649e20	−0.190313
$782$	−1.05122e21	−0.210206
$783$	−9.94446e20	−0.196955
$784$	4.27690e21	0.838994
$785$	−3.31543e21	−0.644195
$786$	1.85093e21	0.356223
$787$	6.10968e21	1.16468	0.582342	−	0.812944i	$-0.302137\pi$
$787$	6.10968e21	1.16468	0.582342	+	0.812944i	$0.302137\pi$
$788$	4.63285e20	0.0874785
$789$	−1.78444e21	−0.333753
$790$	−1.61048e21	−0.298369
$791$	−1.10054e21	−0.201967
$792$	−3.30170e21	−0.600204
$793$	9.91597e20	0.178561
$794$	−1.01842e22	−1.81665
$795$	−1.72089e21	−0.304089
$796$	−2.80271e20	−0.0490603
$797$	5.70048e21	0.988494	0.494247	−	0.869321i	$-0.335444\pi$
$797$	5.70048e21	0.988494	0.494247	+	0.869321i	$0.335444\pi$
$798$	−3.29396e19	−0.00565844
$799$	1.12333e21	0.191165
$800$	−5.52763e20	−0.0931889
$801$	2.30661e20	0.0385239
$802$	−5.97511e21	−0.988640
$803$	−3.53466e21	−0.579403
$804$	−4.19786e20	−0.0681723
$805$	−2.34400e20	−0.0377127
$806$	−2.79020e21	−0.444756
$807$	−1.27550e19	−0.00201432
$808$	−9.39860e21	−1.47054
$809$	−7.14853e21	−1.10816	−0.554080	−	0.832463i	$-0.686930\pi$
$809$	−7.14853e21	−1.10816	−0.554080	+	0.832463i	$0.686930\pi$
$810$	1.37819e21	0.211676
$811$	1.02556e22	1.56064	0.780322	−	0.625378i	$-0.215055\pi$
$811$	1.02556e22	1.56064	0.780322	+	0.625378i	$0.215055\pi$
$812$	3.38866e19	0.00510926
$813$	−3.03952e21	−0.454073
$814$	−2.41362e21	−0.357261
$815$	−3.37447e21	−0.494907
$816$	3.18544e21	0.462907
$817$	−3.87370e20	−0.0557777
$818$	−6.80803e21	−0.971341
$819$	8.90496e20	0.125893
$820$	−3.31131e19	−0.00463869
$821$	8.09448e21	1.12361	0.561804	−	0.827270i	$-0.310107\pi$
$821$	8.09448e21	1.12361	0.561804	+	0.827270i	$0.310107\pi$
$822$	−3.37614e21	−0.464389
$823$	9.34241e21	1.27339	0.636693	−	0.771117i	$-0.280302\pi$
$823$	9.34241e21	1.27339	0.636693	+	0.771117i	$0.280302\pi$
$824$	−4.89235e20	−0.0660790
$825$	1.70585e21	0.228316
$826$	−1.96871e21	−0.261115
$827$	7.23524e21	0.950959	0.475480	−	0.879727i	$-0.342275\pi$
$827$	7.23524e21	0.950959	0.475480	+	0.879727i	$0.342275\pi$
$828$	9.12718e19	0.0118880
$829$	−4.47449e21	−0.577544	−0.288772	−	0.957398i	$-0.593247\pi$
$829$	−4.47449e21	−0.577544	−0.288772	+	0.957398i	$0.593247\pi$
$830$	1.16787e21	0.149386
$831$	1.94186e21	0.246155
$832$	4.65259e21	0.584480
$833$	7.67062e21	0.954977
$834$	5.48943e21	0.677301
$835$	−7.63746e21	−0.933901
$836$	1.89994e19	0.00230246
$837$	5.94382e21	0.713879
$838$	−1.28357e22	−1.52788
$839$	1.07816e22	1.27194	0.635970	−	0.771714i	$-0.280600\pi$
$839$	1.07816e22	1.27194	0.635970	+	0.771714i	$0.280600\pi$
$840$	7.67454e20	0.0897342
$841$	−8.16530e21	−0.946242
$842$	−1.10558e22	−1.26985
$843$	5.28044e21	0.601123
$844$	8.25890e20	0.0931867
$845$	−3.79246e21	−0.424127
$846$	1.21896e21	0.135117
$847$	1.16557e21	0.128059
$848$	8.80498e21	0.958870
$849$	3.75826e21	0.405676
$850$	5.93841e21	0.635372
$851$	9.67330e20	0.102589
$852$	−8.53187e19	−0.00896907
$853$	−3.87914e21	−0.404220	−0.202110	−	0.979363i	$-0.564780\pi$
$853$	−3.87914e21	−0.404220	−0.202110	+	0.979363i	$0.564780\pi$
$854$	−8.99604e20	−0.0929217
$855$	−1.88218e20	−0.0192714
$856$	1.20994e22	1.22804
$857$	1.84405e22	1.85531	0.927656	−	0.373437i	$-0.121821\pi$
$857$	1.84405e22	1.85531	0.927656	+	0.373437i	$0.121821\pi$
$858$	−1.92227e21	−0.191716
$859$	−1.52763e22	−1.51033	−0.755163	−	0.655537i	$-0.772442\pi$
$859$	−1.52763e22	−1.51033	−0.755163	+	0.655537i	$0.772442\pi$
$860$	6.22521e20	0.0610121
$861$	−1.51330e20	−0.0147028
$862$	3.57777e21	0.344594
$863$	1.29168e22	1.23332	0.616659	−	0.787231i	$-0.288486\pi$
$863$	1.29168e22	1.23332	0.616659	+	0.787231i	$0.288486\pi$
$864$	−1.32692e21	−0.125600
$865$	−2.92323e21	−0.274310
$866$	1.70729e22	1.58826
$867$	5.08035e20	0.0468544
$868$	−2.02541e20	−0.0185189
$869$	4.24524e21	0.384817
$870$	7.24655e20	0.0651234
$871$	1.18328e22	1.05427
$872$	1.66753e22	1.47299
$873$	−8.46467e21	−0.741316
$874$	9.51662e19	0.00826317
$875$	3.42507e21	0.294855
$876$	−3.19920e20	−0.0273062
$877$	−2.85565e21	−0.241662	−0.120831	−	0.992673i	$-0.538556\pi$
$877$	−2.85565e21	−0.241662	−0.120831	+	0.992673i	$0.538556\pi$
$878$	−8.91315e21	−0.747863
$879$	7.18115e21	0.597417
$880$	5.12025e21	0.422348
$881$	1.87739e22	1.53545	0.767726	−	0.640778i	$-0.221388\pi$
$881$	1.87739e22	1.53545	0.767726	+	0.640778i	$0.221388\pi$
$882$	8.32359e21	0.674988
$883$	−2.34852e22	−1.88838	−0.944191	−	0.329399i	$-0.893154\pi$
$883$	−2.34852e22	−1.88838	−0.944191	+	0.329399i	$0.893154\pi$
$884$	5.35433e20	0.0426887
$885$	3.36858e21	0.266300
$886$	−7.25683e21	−0.568845
$887$	−1.76570e22	−1.37242	−0.686212	−	0.727401i	$-0.740728\pi$
$887$	−1.76570e22	−1.37242	−0.686212	+	0.727401i	$0.740728\pi$
$888$	−3.16716e21	−0.244103
$889$	1.40493e20	0.0107372
$890$	−3.86498e20	−0.0292902
$891$	−3.63292e21	−0.273006
$892$	2.96876e20	0.0221227
$893$	−1.01694e20	−0.00751467
$894$	−5.32351e21	−0.390092
$895$	−9.24885e21	−0.672071
$896$	−3.61063e21	−0.260180
$897$	7.70405e20	0.0550524
$898$	−2.63719e22	−1.86883
$899$	−2.77267e21	−0.194850
$900$	−5.15599e20	−0.0359330
$901$	1.57917e22	1.09142
$902$	−1.09090e21	−0.0747712
$903$	2.84498e21	0.193384
$904$	−1.04103e22	−0.701781
$905$	5.95559e21	0.398163
$906$	−2.40739e20	−0.0159619
$907$	−2.73703e22	−1.79980	−0.899902	−	0.436092i	$-0.856362\pi$
$907$	−2.73703e22	−1.79980	−0.899902	+	0.436092i	$0.856362\pi$
$908$	−5.69021e20	−0.0371094
$909$	−1.69287e22	−1.09495
$910$	−1.49213e21	−0.0957184
$911$	4.01270e21	0.255299	0.127650	−	0.991819i	$-0.459257\pi$
$911$	4.01270e21	0.255299	0.127650	+	0.991819i	$0.459257\pi$
$912$	−2.88375e20	−0.0181968
$913$	−3.07851e21	−0.192668
$914$	−1.48890e22	−0.924205
$915$	1.53927e21	0.0947671
$916$	−7.18591e20	−0.0438799
$917$	−3.78729e21	−0.229381
$918$	1.42552e22	0.856357
$919$	−5.14135e21	−0.306345	−0.153172	−	0.988199i	$-0.548949\pi$
$919$	−5.14135e21	−0.306345	−0.153172	+	0.988199i	$0.548949\pi$
$920$	−2.21726e21	−0.131041
$921$	−9.65705e21	−0.566105
$922$	4.12789e21	0.240019
$923$	2.40494e21	0.138705
$924$	−1.39538e20	−0.00798274
$925$	−5.46449e21	−0.310089
$926$	2.49608e22	1.40500
$927$	−8.81207e20	−0.0492017
$928$	6.18979e20	0.0342820
$929$	−3.08347e22	−1.69403	−0.847016	−	0.531568i	$-0.821603\pi$
$929$	−3.08347e22	−1.69403	−0.847016	+	0.531568i	$0.821603\pi$
$930$	−4.33128e21	−0.236044
$931$	−6.94413e20	−0.0375401
$932$	3.69367e20	0.0198079
$933$	6.09668e20	0.0324326
$934$	1.18622e22	0.625982
$935$	9.18316e21	0.480734
$936$	8.42349e21	0.437445
$937$	−3.20922e22	−1.65330	−0.826650	−	0.562716i	$-0.809757\pi$
$937$	−3.20922e22	−1.65330	−0.826650	+	0.562716i	$0.809757\pi$
$938$	−1.07351e22	−0.548634
$939$	1.15092e21	0.0583514
$940$	1.63427e20	0.00821987
$941$	2.74096e22	1.36767	0.683834	−	0.729638i	$-0.260311\pi$
$941$	2.74096e22	1.36767	0.683834	+	0.729638i	$0.260311\pi$
$942$	−9.88595e21	−0.489369
$943$	4.37209e20	0.0214709
$944$	−1.72354e22	−0.839713
$945$	3.17861e21	0.153638
$946$	2.05087e22	0.983456
$947$	7.52518e21	0.358007	0.179004	−	0.983848i	$-0.442713\pi$
$947$	7.52518e21	0.358007	0.179004	+	0.983848i	$0.442713\pi$
$948$	3.84234e20	0.0181357
$949$	9.01782e21	0.422284
$950$	−5.37598e20	−0.0249765
$951$	−1.08007e22	−0.497848
$952$	−7.04252e21	−0.322071
$953$	−3.32784e22	−1.50996	−0.754980	−	0.655748i	$-0.772353\pi$
$953$	−3.32784e22	−1.50996	−0.754980	+	0.655748i	$0.772353\pi$
$954$	1.71360e22	0.771431
$955$	−4.41587e21	−0.197238
$956$	1.24554e21	0.0551980
$957$	−1.91019e21	−0.0839919
$958$	1.72133e22	0.750973
$959$	6.90810e21	0.299033
$960$	7.22230e21	0.310200
$961$	−6.89298e21	−0.293753
$962$	6.15777e21	0.260382
$963$	2.17934e22	0.914381
$964$	1.89992e20	0.00790966
$965$	8.07622e21	0.333621
$966$	−6.98933e20	−0.0286488
$967$	−4.31838e22	−1.75639	−0.878197	−	0.478299i	$-0.841253\pi$
$967$	−4.31838e22	−1.75639	−0.878197	+	0.478299i	$0.841253\pi$
$968$	1.10255e22	0.444971
$969$	−5.17200e20	−0.0207124
$970$	1.41835e22	0.563632
$971$	2.77508e22	1.09429	0.547144	−	0.837038i	$-0.315715\pi$
$971$	2.77508e22	1.09429	0.547144	+	0.837038i	$0.315715\pi$
$972$	−1.93715e21	−0.0757993
$973$	−1.12322e22	−0.436132
$974$	2.67805e22	1.03187
$975$	−4.35206e21	−0.166402
$976$	−7.87572e21	−0.298825
$977$	4.89425e22	1.84280	0.921398	−	0.388620i	$-0.127048\pi$
$977$	4.89425e22	1.84280	0.921398	+	0.388620i	$0.127048\pi$
$978$	−1.00620e22	−0.375961
$979$	1.01881e21	0.0377766
$980$	1.11595e21	0.0410630
$981$	3.00355e22	1.09677
$982$	1.35311e22	0.490339
$983$	−3.14332e22	−1.13041	−0.565206	−	0.824950i	$-0.691203\pi$
$983$	−3.14332e22	−1.13041	−0.565206	+	0.824950i	$0.691203\pi$
$984$	−1.43148e21	−0.0510883
$985$	−2.02716e22	−0.717988
$986$	−6.64977e21	−0.233738
$987$	7.46877e20	0.0260537
$988$	−4.84722e19	−0.00167809
$989$	−8.21946e21	−0.282404
$990$	9.96489e21	0.339788
$991$	1.77608e22	0.601050	0.300525	−	0.953774i	$-0.402838\pi$
$991$	1.77608e22	0.601050	0.300525	+	0.953774i	$0.402838\pi$
$992$	−3.69965e21	−0.124258
$993$	−1.29091e22	−0.430306
$994$	−2.18183e21	−0.0721810
$995$	1.22636e22	0.402667
$996$	−2.78634e20	−0.00908007
$997$	8.82604e21	0.285464	0.142732	−	0.989761i	$-0.454411\pi$
$997$	8.82604e21	0.285464	0.142732	+	0.989761i	$0.454411\pi$
$998$	−8.37085e21	−0.268714
$999$	−1.31176e22	−0.417939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.16.a.a.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.16.a.a.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+174.185 q^{2} +1818.41 q^{3} -2427.63 q^{4} +106224. q^{5} +316739. q^{6} -648096. q^{7} -6.13055e6 q^{8} -1.10423e7 q^{9} +O(q^{10})\)	\(q+174.185 q^{2} +1818.41 q^{3} -2427.63 q^{4} +106224. q^{5} +316739. q^{6} -648096. q^{7} -6.13055e6 q^{8} -1.10423e7 q^{9} +1.85026e7 q^{10} -4.87730e7 q^{11} -4.41442e6 q^{12} +1.24432e8 q^{13} -1.12888e8 q^{14} +1.93159e8 q^{15} -9.88300e8 q^{16} -1.77251e9 q^{17} -1.92340e9 q^{18} +1.60464e8 q^{19} -2.57873e8 q^{20} -1.17850e9 q^{21} -8.49551e9 q^{22} +3.40483e9 q^{23} -1.11478e10 q^{24} -1.92340e10 q^{25} +2.16742e10 q^{26} -4.61716e10 q^{27} +1.57334e9 q^{28} +2.15381e10 q^{29} +3.36453e10 q^{30} -1.28733e11 q^{31} +2.87388e10 q^{32} -8.86891e10 q^{33} -3.08745e11 q^{34} -6.88434e10 q^{35} +2.68066e10 q^{36} +2.84106e11 q^{37} +2.79504e10 q^{38} +2.26269e11 q^{39} -6.51212e11 q^{40} +1.28409e11 q^{41} -2.05277e11 q^{42} -2.41406e12 q^{43} +1.18403e11 q^{44} -1.17296e12 q^{45} +5.93069e11 q^{46} -6.33751e11 q^{47} -1.79713e12 q^{48} -4.32753e12 q^{49} -3.35027e12 q^{50} -3.22315e12 q^{51} -3.02076e11 q^{52} -8.90922e12 q^{53} -8.04239e12 q^{54} -5.18086e12 q^{55} +3.97318e12 q^{56} +2.91789e11 q^{57} +3.75160e12 q^{58} +1.74394e13 q^{59} -4.68918e11 q^{60} +7.96896e12 q^{61} -2.24234e13 q^{62} +7.15647e12 q^{63} +3.73905e13 q^{64} +1.32177e13 q^{65} -1.54483e13 q^{66} +9.50943e13 q^{67} +4.30301e12 q^{68} +6.19136e12 q^{69} -1.19915e13 q^{70} +1.93273e13 q^{71} +6.76953e13 q^{72} +7.24717e13 q^{73} +4.94869e13 q^{74} -3.49753e13 q^{75} -3.89547e11 q^{76} +3.16095e13 q^{77} +3.94126e13 q^{78} -8.70408e13 q^{79} -1.04981e14 q^{80} +7.44863e13 q^{81} +2.23668e13 q^{82} +6.31191e13 q^{83} +2.86096e12 q^{84} -1.88284e14 q^{85} -4.20493e14 q^{86} +3.91650e13 q^{87} +2.99005e14 q^{88} -2.08888e13 q^{89} -2.04312e14 q^{90} -8.06441e13 q^{91} -8.26565e12 q^{92} -2.34090e14 q^{93} -1.10390e14 q^{94} +1.70451e13 q^{95} +5.22589e13 q^{96} +7.66567e14 q^{97} -7.53791e14 q^{98} +5.38566e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9}+O(q^{10}) \) 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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