LMFDB - Embedded newform 23.16.a.a.1.1

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			1.1
Root			$-155.855$ 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)

$f(q)$	$=$	$q-333.711 q^{2} -2416.34 q^{3} +78595.0 q^{4} +165276. q^{5} +806359. q^{6} -2.92451e6 q^{7} -1.52930e7 q^{8} -8.51021e6 q^{9} +O(q^{10})$	\(q-333.711 q^{2} -2416.34 q^{3} +78595.0 q^{4} +165276. q^{5} +806359. q^{6} -2.92451e6 q^{7} -1.52930e7 q^{8} -8.51021e6 q^{9} -5.51544e7 q^{10} +4.75026e7 q^{11} -1.89912e8 q^{12} +2.36191e8 q^{13} +9.75940e8 q^{14} -3.99363e8 q^{15} +2.52803e9 q^{16} -3.07369e9 q^{17} +2.83995e9 q^{18} +5.39647e9 q^{19} +1.29899e10 q^{20} +7.06660e9 q^{21} -1.58521e10 q^{22} +3.40483e9 q^{23} +3.69530e10 q^{24} -3.20141e9 q^{25} -7.88195e10 q^{26} +5.52354e10 q^{27} -2.29852e11 q^{28} +1.80287e11 q^{29} +1.33272e11 q^{30} -6.34108e10 q^{31} -3.42512e11 q^{32} -1.14782e11 q^{33} +1.02572e12 q^{34} -4.83351e11 q^{35} -6.68860e11 q^{36} +1.83672e11 q^{37} -1.80086e12 q^{38} -5.70717e11 q^{39} -2.52756e12 q^{40} -1.28119e11 q^{41} -2.35820e12 q^{42} +3.51325e10 q^{43} +3.73347e12 q^{44} -1.40653e12 q^{45} -1.13623e12 q^{46} -2.32654e12 q^{47} -6.10858e12 q^{48} +3.80518e12 q^{49} +1.06835e12 q^{50} +7.42707e12 q^{51} +1.85634e13 q^{52} +5.76570e12 q^{53} -1.84327e13 q^{54} +7.85104e12 q^{55} +4.47244e13 q^{56} -1.30397e13 q^{57} -6.01639e13 q^{58} -5.62580e12 q^{59} -3.13879e13 q^{60} -2.88357e13 q^{61} +2.11609e13 q^{62} +2.48882e13 q^{63} +3.14614e13 q^{64} +3.90367e13 q^{65} +3.83041e13 q^{66} -6.66692e13 q^{67} -2.41577e14 q^{68} -8.22721e12 q^{69} +1.61300e14 q^{70} +5.30198e13 q^{71} +1.30146e14 q^{72} -6.67534e13 q^{73} -6.12934e13 q^{74} +7.73570e12 q^{75} +4.24136e14 q^{76} -1.38922e14 q^{77} +1.90455e14 q^{78} -1.45809e14 q^{79} +4.17823e14 q^{80} -1.13552e13 q^{81} +4.27546e13 q^{82} -4.90022e13 q^{83} +5.55400e14 q^{84} -5.08007e14 q^{85} -1.17241e13 q^{86} -4.35636e14 q^{87} -7.26456e14 q^{88} +4.81324e14 q^{89} +4.69376e14 q^{90} -6.90742e14 q^{91} +2.67602e14 q^{92} +1.53222e14 q^{93} +7.76392e14 q^{94} +8.91907e14 q^{95} +8.27625e14 q^{96} -7.04163e14 q^{97} -1.26983e15 q^{98} -4.04257e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−333.711	−1.84351	−0.921755	−	0.387773i	$-0.873245\pi$
$2$	−333.711	−1.84351	−0.921755	+	0.387773i	$0.873245\pi$
$3$	−2416.34	−0.637894	−0.318947	−	0.947773i	$-0.603329\pi$
$3$	−2416.34	−0.637894	−0.318947	+	0.947773i	$0.603329\pi$
$4$	78595.0	2.39853
$5$	165276.	0.946095	0.473048	−	0.881037i	$-0.343154\pi$
$5$	165276.	0.946095	0.473048	+	0.881037i	$0.343154\pi$
$6$	806359.	1.17596
$7$	−2.92451e6	−1.34220	−0.671100	−	0.741366i	$-0.734178\pi$
$7$	−2.92451e6	−1.34220	−0.671100	+	0.741366i	$0.734178\pi$
$8$	−1.52930e7	−2.57820
$9$	−8.51021e6	−0.593091
$10$	−5.51544e7	−1.74414
$11$	4.75026e7	0.734974	0.367487	−	0.930029i	$-0.380218\pi$
$11$	4.75026e7	0.734974	0.367487	+	0.930029i	$0.380218\pi$
$12$	−1.89912e8	−1.53001
$13$	2.36191e8	1.04397	0.521985	−	0.852955i	$-0.325192\pi$
$13$	2.36191e8	1.04397	0.521985	+	0.852955i	$0.325192\pi$
$14$	9.75940e8	2.47436
$15$	−3.99363e8	−0.603508
$16$	2.52803e9	2.35441
$17$	−3.07369e9	−1.81674	−0.908370	−	0.418167i	$-0.862673\pi$
$17$	−3.07369e9	−1.81674	−0.908370	+	0.418167i	$0.862673\pi$
$18$	2.83995e9	1.09337
$19$	5.39647e9	1.38503	0.692513	−	0.721406i	$-0.256504\pi$
$19$	5.39647e9	1.38503	0.692513	+	0.721406i	$0.256504\pi$
$20$	1.29899e10	2.26924
$21$	7.06660e9	0.856182
$22$	−1.58521e10	−1.35493
$23$	3.40483e9	0.208514
$23$	3.40483e9	0.208514
$24$	3.69530e10	1.64462
$25$	−3.20141e9	−0.104904
$26$	−7.88195e10	−1.92457
$27$	5.52354e10	1.01622
$28$	−2.29852e11	−3.21931
$29$	1.80287e11	1.94080	0.970399	−	0.241505i	$-0.0776411\pi$
$29$	1.80287e11	1.94080	0.970399	+	0.241505i	$0.0776411\pi$
$30$	1.33272e11	1.11257
$31$	−6.34108e10	−0.413953	−0.206976	−	0.978346i	$-0.566362\pi$
$31$	−6.34108e10	−0.413953	−0.206976	+	0.978346i	$0.566362\pi$
$32$	−3.42512e11	−1.76218
$33$	−1.14782e11	−0.468835
$34$	1.02572e12	3.34918
$35$	−4.83351e11	−1.26985
$36$	−6.68860e11	−1.42255
$37$	1.83672e11	0.318075	0.159038	−	0.987273i	$-0.449161\pi$
$37$	1.83672e11	0.318075	0.159038	+	0.987273i	$0.449161\pi$
$38$	−1.80086e12	−2.55331
$39$	−5.70717e11	−0.665942
$40$	−2.52756e12	−2.43923
$41$	−1.28119e11	−0.102739	−0.0513693	−	0.998680i	$-0.516359\pi$
$41$	−1.28119e11	−0.102739	−0.0513693	+	0.998680i	$0.516359\pi$
$42$	−2.35820e12	−1.57838
$43$	3.51325e10	0.0197104	0.00985520	−	0.999951i	$-0.496863\pi$
$43$	3.51325e10	0.0197104	0.00985520	+	0.999951i	$0.496863\pi$
$44$	3.73347e12	1.76286
$45$	−1.40653e12	−0.561121
$46$	−1.13623e12	−0.384398
$47$	−2.32654e12	−0.669848	−0.334924	−	0.942245i	$-0.608711\pi$
$47$	−2.32654e12	−0.669848	−0.334924	+	0.942245i	$0.608711\pi$
$48$	−6.10858e12	−1.50187
$49$	3.80518e12	0.801503
$50$	1.06835e12	0.193391
$51$	7.42707e12	1.15889
$52$	1.85634e13	2.50399
$53$	5.76570e12	0.674191	0.337096	−	0.941470i	$-0.390555\pi$
$53$	5.76570e12	0.674191	0.337096	+	0.941470i	$0.390555\pi$
$54$	−1.84327e13	−1.87342
$55$	7.85104e12	0.695355
$56$	4.47244e13	3.46047
$57$	−1.30397e13	−0.883499
$58$	−6.01639e13	−3.57788
$59$	−5.62580e12	−0.294303	−0.147151	−	0.989114i	$-0.547010\pi$
$59$	−5.62580e12	−0.294303	−0.147151	+	0.989114i	$0.547010\pi$
$60$	−3.13879e13	−1.44753
$61$	−2.88357e13	−1.17478	−0.587389	−	0.809304i	$-0.699844\pi$
$61$	−2.88357e13	−1.17478	−0.587389	+	0.809304i	$0.699844\pi$
$62$	2.11609e13	0.763126
$63$	2.48882e13	0.796048
$64$	3.14614e13	0.894188
$65$	3.90367e13	0.987695
$66$	3.83041e13	0.864303
$67$	−6.66692e13	−1.34389	−0.671946	−	0.740600i	$-0.734541\pi$
$67$	−6.66692e13	−1.34389	−0.671946	+	0.740600i	$0.734541\pi$
$68$	−2.41577e14	−4.35751
$69$	−8.22721e12	−0.133010
$70$	1.61300e14	2.34098
$71$	5.30198e13	0.691832	0.345916	−	0.938266i	$-0.387568\pi$
$71$	5.30198e13	0.691832	0.345916	+	0.938266i	$0.387568\pi$
$72$	1.30146e14	1.52911
$73$	−6.67534e13	−0.707216	−0.353608	−	0.935394i	$-0.615045\pi$
$73$	−6.67534e13	−0.707216	−0.353608	+	0.935394i	$0.615045\pi$
$74$	−6.12934e13	−0.586375
$75$	7.73570e12	0.0669176
$76$	4.24136e14	3.32202
$77$	−1.38922e14	−0.986483
$78$	1.90455e14	1.22767
$79$	−1.45809e14	−0.854242	−0.427121	−	0.904194i	$-0.640472\pi$
$79$	−1.45809e14	−0.854242	−0.427121	+	0.904194i	$0.640472\pi$
$80$	4.17823e14	2.22750
$81$	−1.13552e13	−0.0551514
$82$	4.27546e13	0.189400
$83$	−4.90022e13	−0.198212	−0.0991060	−	0.995077i	$-0.531598\pi$
$83$	−4.90022e13	−0.198212	−0.0991060	+	0.995077i	$0.531598\pi$
$84$	5.55400e14	2.05358
$85$	−5.08007e14	−1.71881
$86$	−1.17241e13	−0.0363363
$87$	−4.35636e14	−1.23802
$88$	−7.26456e14	−1.89491
$89$	4.81324e14	1.15349	0.576743	−	0.816925i	$-0.304323\pi$
$89$	4.81324e14	1.15349	0.576743	+	0.816925i	$0.304323\pi$
$90$	4.69376e14	1.03443
$91$	−6.90742e14	−1.40122
$92$	2.67602e14	0.500128
$93$	1.53222e14	0.264058
$94$	7.76392e14	1.23487
$95$	8.91907e14	1.31037
$96$	8.27625e14	1.12409
$97$	−7.04163e14	−0.884882	−0.442441	−	0.896798i	$-0.645887\pi$
$97$	−7.04163e14	−0.884882	−0.442441	+	0.896798i	$0.645887\pi$
$98$	−1.26983e15	−1.47758
$99$	−4.04257e14	−0.435907
$100$	−2.51615e14	−0.251615
$101$	−1.87791e15	−1.74287	−0.871433	−	0.490515i	$-0.836809\pi$
$101$	−1.87791e15	−1.74287	−0.871433	+	0.490515i	$0.836809\pi$
$102$	−2.47850e15	−2.13642
$103$	−1.02113e15	−0.818092	−0.409046	−	0.912514i	$-0.634138\pi$
$103$	−1.02113e15	−0.818092	−0.409046	+	0.912514i	$0.634138\pi$
$104$	−3.61206e15	−2.69157
$105$	1.16794e15	0.810029
$106$	−1.92408e15	−1.24288
$107$	2.69746e15	1.62396	0.811982	−	0.583683i	$-0.198389\pi$
$107$	2.69746e15	1.62396	0.811982	+	0.583683i	$0.198389\pi$
$108$	4.34123e15	2.43744
$109$	−3.31863e15	−1.73884	−0.869420	−	0.494074i	$-0.835507\pi$
$109$	−3.31863e15	−1.73884	−0.869420	+	0.494074i	$0.835507\pi$
$110$	−2.61998e15	−1.28189
$111$	−4.43814e14	−0.202898
$112$	−7.39325e15	−3.16010
$113$	−2.31538e14	−0.0925834	−0.0462917	−	0.998928i	$-0.514740\pi$
$113$	−2.31538e14	−0.0925834	−0.0462917	+	0.998928i	$0.514740\pi$
$114$	4.35149e15	1.62874
$115$	5.62736e14	0.197274
$116$	1.41697e16	4.65506
$117$	−2.01003e15	−0.619170
$118$	1.87739e15	0.542550
$119$	8.98903e15	2.43843
$120$	6.10745e15	1.55597
$121$	−1.92075e15	−0.459813
$122$	9.62278e15	2.16572
$123$	3.09578e14	0.0655363
$124$	−4.98378e15	−0.992878
$125$	−5.57294e15	−1.04534
$126$	−8.30546e15	−1.46752
$127$	6.02553e15	1.00338	0.501692	−	0.865046i	$-0.332711\pi$
$127$	6.02553e15	1.00338	0.501692	+	0.865046i	$0.332711\pi$
$128$	7.24408e14	0.113739
$129$	−8.48920e13	−0.0125732
$130$	−1.30270e16	−1.82083
$131$	−5.52214e15	−0.728740	−0.364370	−	0.931254i	$-0.618716\pi$
$131$	−5.52214e15	−0.728740	−0.364370	+	0.931254i	$0.618716\pi$
$132$	−9.02132e15	−1.12452
$133$	−1.57820e16	−1.85898
$134$	2.22482e16	2.47748
$135$	9.12908e15	0.961444
$136$	4.70058e16	4.68393
$137$	−1.06537e16	−1.00484	−0.502422	−	0.864623i	$-0.667557\pi$
$137$	−1.06537e16	−1.00484	−0.502422	+	0.864623i	$0.667557\pi$
$138$	2.74551e15	0.245205
$139$	−1.04573e16	−0.884725	−0.442363	−	0.896836i	$-0.645860\pi$
$139$	−1.04573e16	−0.884725	−0.442363	+	0.896836i	$0.645860\pi$
$140$	−3.79890e16	−3.04577
$141$	5.62171e15	0.427292
$142$	−1.76933e16	−1.27540
$143$	1.12197e16	0.767291
$144$	−2.15141e16	−1.39638
$145$	2.97972e16	1.83618
$146$	2.22763e16	1.30376
$147$	−9.19462e15	−0.511274
$148$	1.44357e16	0.762913
$149$	1.93141e16	0.970461	0.485230	−	0.874386i	$-0.338736\pi$
$149$	1.93141e16	0.970461	0.485230	+	0.874386i	$0.338736\pi$
$150$	−2.58149e15	−0.123363
$151$	6.08560e15	0.276679	0.138340	−	0.990385i	$-0.455823\pi$
$151$	6.08560e15	0.276679	0.138340	+	0.990385i	$0.455823\pi$
$152$	−8.25281e16	−3.57088
$153$	2.61577e16	1.07749
$154$	4.63597e16	1.81859
$155$	−1.04803e16	−0.391639
$156$	−4.48555e16	−1.59728
$157$	1.35350e16	0.459421	0.229710	−	0.973259i	$-0.426222\pi$
$157$	1.35350e16	0.459421	0.229710	+	0.973259i	$0.426222\pi$
$158$	4.86580e16	1.57480
$159$	−1.39319e16	−0.430063
$160$	−5.66090e16	−1.66719
$161$	−9.95744e15	−0.279868
$162$	3.78935e15	0.101672
$163$	1.73774e16	0.445222	0.222611	−	0.974907i	$-0.428542\pi$
$163$	1.73774e16	0.445222	0.222611	+	0.974907i	$0.428542\pi$
$164$	−1.00695e16	−0.246422
$165$	−1.89708e16	−0.443563
$166$	1.63526e16	0.365406
$167$	−7.27150e16	−1.55328	−0.776641	−	0.629943i	$-0.783078\pi$
$167$	−7.27150e16	−1.55328	−0.776641	+	0.629943i	$0.783078\pi$
$168$	−1.08069e17	−2.20741
$169$	4.60026e15	0.0898737
$170$	1.69528e17	3.16864
$171$	−4.59251e16	−0.821447
$172$	2.76124e15	0.0472760
$173$	−1.02587e17	−1.68169	−0.840846	−	0.541274i	$-0.817942\pi$
$173$	−1.02587e17	−1.68169	−0.840846	+	0.541274i	$0.817942\pi$
$174$	1.45376e17	2.28231
$175$	9.36256e15	0.140802
$176$	1.20088e17	1.73043
$177$	1.35938e16	0.187734
$178$	−1.60623e17	−2.12646
$179$	1.38915e17	1.76340	0.881702	−	0.471806i	$-0.156398\pi$
$179$	1.38915e17	1.76340	0.881702	+	0.471806i	$0.156398\pi$
$180$	−1.10547e17	−1.34586
$181$	−2.36548e16	−0.276268	−0.138134	−	0.990414i	$-0.544110\pi$
$181$	−2.36548e16	−0.276268	−0.138134	+	0.990414i	$0.544110\pi$
$182$	2.30508e17	2.58316
$183$	6.96767e16	0.749384
$184$	−5.20699e16	−0.537592
$185$	3.03566e16	0.300930
$186$	−5.11319e16	−0.486794
$187$	−1.46008e17	−1.33526
$188$	−1.82854e17	−1.60665
$189$	−1.61536e17	−1.36398
$190$	−2.97639e17	−2.41567
$191$	1.17152e17	0.914116	0.457058	−	0.889437i	$-0.348903\pi$
$191$	1.17152e17	0.914116	0.457058	+	0.889437i	$0.348903\pi$
$192$	−7.60215e16	−0.570397
$193$	−2.51807e17	−1.81714	−0.908568	−	0.417737i	$-0.862823\pi$
$193$	−2.51807e17	−1.81714	−0.908568	+	0.417737i	$0.862823\pi$
$194$	2.34987e17	1.63129
$195$	−9.43259e16	−0.630045
$196$	2.99069e17	1.92243
$197$	−2.44314e17	−1.51165	−0.755826	−	0.654772i	$-0.772765\pi$
$197$	−2.44314e17	−1.51165	−0.755826	+	0.654772i	$0.772765\pi$
$198$	1.34905e17	0.803598
$199$	−1.21144e17	−0.694869	−0.347435	−	0.937704i	$-0.612947\pi$
$199$	−1.21144e17	−0.694869	−0.347435	+	0.937704i	$0.612947\pi$
$200$	4.89591e16	0.270464
$201$	1.61095e17	0.857260
$202$	6.26678e17	3.21299
$203$	−5.27252e17	−2.60494
$204$	5.83731e17	2.77963
$205$	−2.11749e16	−0.0972005
$206$	3.40763e17	1.50816
$207$	−2.89758e16	−0.123668
$208$	5.97098e17	2.45794
$209$	2.56346e17	1.01796
$210$	−3.89754e17	−1.49330
$211$	1.42100e17	0.525383	0.262691	−	0.964880i	$-0.415390\pi$
$211$	1.42100e17	0.525383	0.262691	+	0.964880i	$0.415390\pi$
$212$	4.53156e17	1.61707
$213$	−1.28114e17	−0.441315
$214$	−9.00171e17	−2.99379
$215$	5.80656e15	0.0186479
$216$	−8.44713e17	−2.62003
$217$	1.85446e17	0.555608
$218$	1.10746e18	3.20557
$219$	1.61299e17	0.451129
$220$	6.17052e17	1.66783
$221$	−7.25977e17	−1.89662
$222$	1.48106e17	0.374045
$223$	1.07429e17	0.262323	0.131161	−	0.991361i	$-0.458129\pi$
$223$	1.07429e17	0.262323	0.131161	+	0.991361i	$0.458129\pi$
$224$	1.00168e18	2.36520
$225$	2.72447e16	0.0622176
$226$	7.72667e16	0.170679
$227$	5.43501e17	1.16147	0.580733	−	0.814094i	$-0.302766\pi$
$227$	5.43501e17	1.16147	0.580733	+	0.814094i	$0.302766\pi$
$228$	−1.02486e18	−2.11910
$229$	2.68290e17	0.536832	0.268416	−	0.963303i	$-0.413500\pi$
$229$	2.68290e17	0.536832	0.268416	+	0.963303i	$0.413500\pi$
$230$	−1.87791e17	−0.363678
$231$	3.35682e17	0.629271
$232$	−2.75713e18	−5.00377
$233$	−3.36778e17	−0.591799	−0.295900	−	0.955219i	$-0.595619\pi$
$233$	−3.36778e17	−0.591799	−0.295900	+	0.955219i	$0.595619\pi$
$234$	6.70771e17	1.14145
$235$	−3.84521e17	−0.633740
$236$	−4.42160e17	−0.705894
$237$	3.52324e17	0.544916
$238$	−2.99974e18	−4.49527
$239$	−3.13483e17	−0.455229	−0.227615	−	0.973751i	$-0.573093\pi$
$239$	−3.13483e17	−0.455229	−0.227615	+	0.973751i	$0.573093\pi$
$240$	−1.00960e18	−1.42091
$241$	−2.78447e17	−0.379852	−0.189926	−	0.981798i	$-0.560825\pi$
$241$	−2.78447e17	−0.379852	−0.189926	+	0.981798i	$0.560825\pi$
$242$	6.40977e17	0.847670
$243$	−7.65129e17	−0.981043
$244$	−2.26634e18	−2.81774
$245$	6.28906e17	0.758298
$246$	−1.03310e17	−0.120817
$247$	1.27460e18	1.44593
$248$	9.69741e17	1.06725
$249$	1.18406e17	0.126438
$250$	1.85975e18	1.92710
$251$	1.06679e18	1.07282	0.536410	−	0.843958i	$-0.319780\pi$
$251$	1.06679e18	1.07282	0.536410	+	0.843958i	$0.319780\pi$
$252$	1.95609e18	1.90934
$253$	1.61738e17	0.153253
$254$	−2.01079e18	−1.84975
$255$	1.22752e18	1.09642
$256$	−1.27267e18	−1.10387
$257$	−1.01037e18	−0.851104	−0.425552	−	0.904934i	$-0.639920\pi$
$257$	−1.01037e18	−0.851104	−0.425552	+	0.904934i	$0.639920\pi$
$258$	2.83294e16	0.0231787
$259$	−5.37150e17	−0.426921
$260$	3.06809e18	2.36902
$261$	−1.53428e18	−1.15107
$262$	1.84280e18	1.34344
$263$	6.35795e17	0.450453	0.225226	−	0.974306i	$-0.427688\pi$
$263$	6.35795e17	0.450453	0.225226	+	0.974306i	$0.427688\pi$
$264$	1.75536e18	1.20875
$265$	9.52933e17	0.637849
$266$	5.26663e18	3.42705
$267$	−1.16304e18	−0.735802
$268$	−5.23987e18	−3.22336
$269$	2.74844e18	1.64416	0.822081	−	0.569370i	$-0.192813\pi$
$269$	2.74844e18	1.64416	0.822081	+	0.569370i	$0.192813\pi$
$270$	−3.04648e18	−1.77243
$271$	−1.17563e18	−0.665276	−0.332638	−	0.943055i	$-0.607939\pi$
$271$	−1.17563e18	−0.665276	−0.332638	+	0.943055i	$0.607939\pi$
$272$	−7.77039e18	−4.27736
$273$	1.66907e18	0.893828
$274$	3.55527e18	1.85244
$275$	−1.52075e17	−0.0771016
$276$	−6.46618e17	−0.319029
$277$	−4.30831e17	−0.206875	−0.103438	−	0.994636i	$-0.532984\pi$
$277$	−4.30831e17	−0.206875	−0.103438	+	0.994636i	$0.532984\pi$
$278$	3.48972e18	1.63100
$279$	5.39640e17	0.245512
$280$	7.39188e18	3.27393
$281$	−3.03795e18	−1.31004	−0.655019	−	0.755613i	$-0.727339\pi$
$281$	−3.03795e18	−1.31004	−0.655019	+	0.755613i	$0.727339\pi$
$282$	−1.87603e18	−0.787717
$283$	4.48768e18	1.83495	0.917474	−	0.397797i	$-0.130225\pi$
$283$	4.48768e18	1.83495	0.917474	+	0.397797i	$0.130225\pi$
$284$	4.16709e18	1.65938
$285$	−2.15515e18	−0.835875
$286$	−3.74413e18	−1.41451
$287$	3.74684e17	0.137896
$288$	2.91485e18	1.04514
$289$	6.58514e18	2.30055
$290$	−9.94365e18	−3.38502
$291$	1.70150e18	0.564461
$292$	−5.24648e18	−1.69628
$293$	2.68530e17	0.0846226	0.0423113	−	0.999104i	$-0.486528\pi$
$293$	2.68530e17	0.0846226	0.0423113	+	0.999104i	$0.486528\pi$
$294$	3.06834e18	0.942539
$295$	−9.29810e17	−0.278438
$296$	−2.80889e18	−0.820063
$297$	2.62382e18	0.746898
$298$	−6.44533e18	−1.78905
$299$	8.04189e17	0.217683
$300$	6.07987e17	0.160504
$301$	−1.02745e17	−0.0264553
$302$	−2.03083e18	−0.510061
$303$	4.53766e18	1.11176
$304$	1.36425e19	3.26092
$305$	−4.76584e18	−1.11145
$306$	−8.72913e18	−1.98637
$307$	−3.63599e18	−0.807393	−0.403696	−	0.914893i	$-0.632275\pi$
$307$	−3.63599e18	−0.807393	−0.403696	+	0.914893i	$0.632275\pi$
$308$	−1.09185e19	−2.36611
$309$	2.46740e18	0.521856
$310$	3.49739e18	0.721990
$311$	−6.98465e18	−1.40748	−0.703739	−	0.710459i	$-0.748488\pi$
$311$	−6.98465e18	−1.40748	−0.703739	+	0.710459i	$0.748488\pi$
$312$	8.72797e18	1.71693
$313$	−1.79489e18	−0.344712	−0.172356	−	0.985035i	$-0.555138\pi$
$313$	−1.79489e18	−0.344712	−0.172356	+	0.985035i	$0.555138\pi$
$314$	−4.51678e18	−0.846947
$315$	4.11342e18	0.753137
$316$	−1.14599e19	−2.04892
$317$	7.17704e18	1.25314	0.626572	−	0.779364i	$-0.284458\pi$
$317$	7.17704e18	1.25314	0.626572	+	0.779364i	$0.284458\pi$
$318$	4.64923e18	0.792825
$319$	8.56412e18	1.42644
$320$	5.19982e18	0.845987
$321$	−6.51797e18	−1.03592
$322$	3.32291e18	0.515940
$323$	−1.65871e19	−2.51623
$324$	−8.92460e17	−0.132282
$325$	−7.56145e17	−0.109517
$326$	−5.79902e18	−0.820772
$327$	8.01893e18	1.10920
$328$	1.95932e18	0.264881
$329$	6.80398e18	0.899071
$330$	6.33075e18	0.817713
$331$	8.93091e17	0.112768	0.0563840	−	0.998409i	$-0.482043\pi$
$331$	8.93091e17	0.112768	0.0563840	+	0.998409i	$0.482043\pi$
$332$	−3.85133e18	−0.475417
$333$	−1.56309e18	−0.188648
$334$	2.42658e19	2.86349
$335$	−1.10188e19	−1.27145
$336$	1.78646e19	2.01581
$337$	−9.11048e18	−1.00535	−0.502675	−	0.864476i	$-0.667651\pi$
$337$	−9.11048e18	−1.00535	−0.502675	+	0.864476i	$0.667651\pi$
$338$	−1.53516e18	−0.165683
$339$	5.59474e17	0.0590584
$340$	−3.99268e19	−4.12262
$341$	−3.01218e18	−0.304245
$342$	1.53257e19	1.51435
$343$	2.75599e18	0.266423
$344$	−5.37280e17	−0.0508174
$345$	−1.35976e18	−0.125840
$346$	3.42344e19	3.10022
$347$	−8.47584e18	−0.751124	−0.375562	−	0.926797i	$-0.622550\pi$
$347$	−8.47584e18	−0.751124	−0.375562	+	0.926797i	$0.622550\pi$
$348$	−3.42388e19	−2.96944
$349$	−7.16428e18	−0.608110	−0.304055	−	0.952655i	$-0.598341\pi$
$349$	−7.16428e18	−0.608110	−0.304055	+	0.952655i	$0.598341\pi$
$350$	−3.12439e18	−0.259570
$351$	1.30461e19	1.06091
$352$	−1.62702e19	−1.29516
$353$	4.02776e18	0.313872	0.156936	−	0.987609i	$-0.449838\pi$
$353$	4.02776e18	0.313872	0.156936	+	0.987609i	$0.449838\pi$
$354$	−4.53641e18	−0.346089
$355$	8.76290e18	0.654539
$356$	3.78297e19	2.76667
$357$	−2.17205e19	−1.55546
$358$	−4.63575e19	−3.25085
$359$	8.53211e18	0.585933	0.292967	−	0.956123i	$-0.405358\pi$
$359$	8.53211e18	0.585933	0.292967	+	0.956123i	$0.405358\pi$
$360$	2.15101e19	1.44668
$361$	1.39408e19	0.918296
$362$	7.89386e18	0.509302
$363$	4.64119e18	0.293312
$364$	−5.42889e19	−3.36086
$365$	−1.10327e19	−0.669093
$366$	−2.32519e19	−1.38150
$367$	−1.00228e19	−0.583437	−0.291719	−	0.956504i	$-0.594227\pi$
$367$	−1.00228e19	−0.583437	−0.291719	+	0.956504i	$0.594227\pi$
$368$	8.60751e18	0.490929
$369$	1.09032e18	0.0609334
$370$	−1.01303e19	−0.554767
$371$	−1.68618e19	−0.904900
$372$	1.20425e19	0.633351
$373$	−1.14465e19	−0.590007	−0.295004	−	0.955496i	$-0.595321\pi$
$373$	−1.14465e19	−0.590007	−0.295004	+	0.955496i	$0.595321\pi$
$374$	4.87245e19	2.46156
$375$	1.34661e19	0.666819
$376$	3.55797e19	1.72700
$377$	4.25823e19	2.02614
$378$	5.39064e19	2.51450
$379$	−2.65452e19	−1.21392	−0.606962	−	0.794731i	$-0.707612\pi$
$379$	−2.65452e19	−1.21392	−0.606962	+	0.794731i	$0.707612\pi$
$380$	7.00995e19	3.14295
$381$	−1.45597e19	−0.640053
$382$	−3.90951e19	−1.68518
$383$	1.67988e19	0.710047	0.355023	−	0.934857i	$-0.384473\pi$
$383$	1.67988e19	0.710047	0.355023	+	0.934857i	$0.384473\pi$
$384$	−1.75041e18	−0.0725532
$385$	−2.29604e19	−0.933307
$386$	8.40306e19	3.34991
$387$	−2.98985e17	−0.0116901
$388$	−5.53437e19	−2.12242
$389$	−8.69040e18	−0.326902	−0.163451	−	0.986551i	$-0.552263\pi$
$389$	−8.69040e18	−0.326902	−0.163451	+	0.986551i	$0.552263\pi$
$390$	3.14776e19	1.16149
$391$	−1.04654e19	−0.378817
$392$	−5.81926e19	−2.06644
$393$	1.33434e19	0.464859
$394$	8.15303e19	2.78675
$395$	−2.40987e19	−0.808194
$396$	−3.17726e19	−1.04554
$397$	4.52576e19	1.46138	0.730690	−	0.682709i	$-0.239198\pi$
$397$	4.52576e19	1.46138	0.730690	+	0.682709i	$0.239198\pi$
$398$	4.04270e19	1.28100
$399$	3.81347e19	1.18583
$400$	−8.09328e18	−0.246987
$401$	−1.37310e19	−0.411261	−0.205631	−	0.978630i	$-0.565925\pi$
$401$	−1.37310e19	−0.411261	−0.205631	+	0.978630i	$0.565925\pi$
$402$	−5.37593e19	−1.58037
$403$	−1.49771e19	−0.432154
$404$	−1.47594e20	−4.18031
$405$	−1.87674e18	−0.0521784
$406$	1.75950e20	4.80224
$407$	8.72490e18	0.233777
$408$	−1.13582e20	−2.98785
$409$	−1.63941e19	−0.423413	−0.211706	−	0.977333i	$-0.567902\pi$
$409$	−1.63941e19	−0.423413	−0.211706	+	0.977333i	$0.567902\pi$
$410$	7.06631e18	0.179190
$411$	2.57431e19	0.640983
$412$	−8.02558e19	−1.96222
$413$	1.64527e19	0.395013
$414$	9.66954e18	0.227983
$415$	−8.09889e18	−0.187527
$416$	−8.08982e19	−1.83967
$417$	2.52684e19	0.564361
$418$	−8.55456e19	−1.87662
$419$	−4.70909e19	−1.01469	−0.507344	−	0.861744i	$-0.669373\pi$
$419$	−4.70909e19	−1.01469	−0.507344	+	0.861744i	$0.669373\pi$
$420$	9.17943e19	1.94288
$421$	−4.47556e19	−0.930533	−0.465267	−	0.885171i	$-0.654042\pi$
$421$	−4.47556e19	−0.930533	−0.465267	+	0.885171i	$0.654042\pi$
$422$	−4.74203e19	−0.968548
$423$	1.97993e19	0.397281
$424$	−8.81748e19	−1.73820
$425$	9.84014e18	0.190583
$426$	4.27530e19	0.813569
$427$	8.43301e19	1.57679
$428$	2.12007e20	3.89513
$429$	−2.71105e19	−0.489450
$430$	−1.93771e18	−0.0343776
$431$	−2.71125e19	−0.472705	−0.236352	−	0.971667i	$-0.575952\pi$
$431$	−2.71125e19	−0.472705	−0.236352	+	0.971667i	$0.575952\pi$
$432$	1.39637e20	2.39261
$433$	−3.54388e19	−0.596787	−0.298393	−	0.954443i	$-0.596451\pi$
$433$	−3.54388e19	−0.596787	−0.298393	+	0.954443i	$0.596451\pi$
$434$	−6.18852e19	−1.02427
$435$	−7.20001e19	−1.17129
$436$	−2.60827e20	−4.17066
$437$	1.83740e19	0.288798
$438$	−5.38272e19	−0.831660
$439$	−2.23647e19	−0.339688	−0.169844	−	0.985471i	$-0.554326\pi$
$439$	−2.23647e19	−0.339688	−0.169844	+	0.985471i	$0.554326\pi$
$440$	−1.20066e20	−1.79277
$441$	−3.23829e19	−0.475364
$442$	2.42267e20	3.49644
$443$	8.65664e19	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$443$	8.65664e19	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$444$	−3.48816e19	−0.486658
$445$	7.95513e19	1.09131
$446$	−3.58503e19	−0.483594
$447$	−4.66695e19	−0.619051
$448$	−9.20092e19	−1.20018
$449$	1.15790e20	1.48533	0.742665	−	0.669664i	$-0.233562\pi$
$449$	1.15790e20	1.48533	0.742665	+	0.669664i	$0.233562\pi$
$450$	−9.09186e18	−0.114699
$451$	−6.08597e18	−0.0755102
$452$	−1.81977e19	−0.222064
$453$	−1.47049e19	−0.176492
$454$	−1.81372e20	−2.14117
$455$	−1.14163e20	−1.32569
$456$	1.99416e20	2.27784
$457$	−4.46987e19	−0.502254	−0.251127	−	0.967954i	$-0.580801\pi$
$457$	−4.46987e19	−0.502254	−0.251127	+	0.967954i	$0.580801\pi$
$458$	−8.95313e19	−0.989655
$459$	−1.69776e20	−1.84621
$460$	4.42282e19	0.473169
$461$	2.12672e19	0.223848	0.111924	−	0.993717i	$-0.464299\pi$
$461$	2.12672e19	0.223848	0.111924	+	0.993717i	$0.464299\pi$
$462$	−1.12021e20	−1.16007
$463$	6.40979e19	0.653110	0.326555	−	0.945178i	$-0.394112\pi$
$463$	6.40979e19	0.653110	0.326555	+	0.945178i	$0.394112\pi$
$464$	4.55773e20	4.56944
$465$	2.53239e19	0.249824
$466$	1.12387e20	1.09099
$467$	2.73164e19	0.260943	0.130472	−	0.991452i	$-0.458351\pi$
$467$	2.73164e19	0.260943	0.130472	+	0.991452i	$0.458351\pi$
$468$	−1.57979e20	−1.48510
$469$	1.94975e20	1.80377
$470$	1.28319e20	1.16831
$471$	−3.27051e19	−0.293062
$472$	8.60352e19	0.758772
$473$	1.66888e18	0.0144866
$474$	−1.17574e20	−1.00456
$475$	−1.72763e19	−0.145295
$476$	7.06493e20	5.84865
$477$	−4.90674e19	−0.399857
$478$	1.04613e20	0.839220
$479$	−6.77309e19	−0.534898	−0.267449	−	0.963572i	$-0.586181\pi$
$479$	−6.77309e19	−0.534898	−0.267449	+	0.963572i	$0.586181\pi$
$480$	1.36787e20	1.06349
$481$	4.33817e19	0.332061
$482$	9.29207e19	0.700261
$483$	2.40605e19	0.178526
$484$	−1.50962e20	−1.10288
$485$	−1.16381e20	−0.837183
$486$	2.55332e20	1.80856
$487$	1.25560e20	0.875756	0.437878	−	0.899034i	$-0.355730\pi$
$487$	1.25560e20	0.875756	0.437878	+	0.899034i	$0.355730\pi$
$488$	4.40983e20	3.02882
$489$	−4.19896e19	−0.284005
$490$	−2.09873e20	−1.39793
$491$	7.89449e19	0.517861	0.258930	−	0.965896i	$-0.416630\pi$
$491$	7.89449e19	0.517861	0.258930	+	0.965896i	$0.416630\pi$
$492$	2.43313e19	0.157191
$493$	−5.54148e20	−3.52593
$494$	−4.25347e20	−2.66558
$495$	−6.68140e19	−0.412409
$496$	−1.60305e20	−0.974616
$497$	−1.55057e20	−0.928577
$498$	−3.95134e19	−0.233090
$499$	6.31597e19	0.367017	0.183508	−	0.983018i	$-0.441255\pi$
$499$	6.31597e19	0.367017	0.183508	+	0.983018i	$0.441255\pi$
$500$	−4.38005e20	−2.50729
$501$	1.75704e20	0.990830
$502$	−3.56000e20	−1.97775
$503$	−1.01075e20	−0.553202	−0.276601	−	0.960985i	$-0.589208\pi$
$503$	−1.01075e20	−0.553202	−0.276601	+	0.960985i	$0.589208\pi$
$504$	−3.80614e20	−2.05237
$505$	−3.10373e20	−1.64892
$506$	−5.39737e19	−0.282523
$507$	−1.11158e19	−0.0573299
$508$	4.73577e20	2.40665
$509$	−2.90069e20	−1.45251	−0.726253	−	0.687427i	$-0.758740\pi$
$509$	−2.90069e20	−1.45251	−0.726253	+	0.687427i	$0.758740\pi$
$510$	−4.09636e20	−2.02126
$511$	1.95221e20	0.949226
$512$	4.00967e20	1.92125
$513$	2.98076e20	1.40750
$514$	3.37172e20	1.56902
$515$	−1.68768e20	−0.773993
$516$	−6.67209e18	−0.0301571
$517$	−1.10517e20	−0.492321
$518$	1.79253e20	0.787033
$519$	2.47885e20	1.07274
$520$	−5.96987e20	−2.54648
$521$	4.35313e20	1.83029	0.915143	−	0.403129i	$-0.132077\pi$
$521$	4.35313e20	1.83029	0.915143	+	0.403129i	$0.132077\pi$
$522$	5.12008e20	2.12201
$523$	4.34399e20	1.77471	0.887353	−	0.461090i	$-0.152541\pi$
$523$	4.34399e20	1.77471	0.887353	+	0.461090i	$0.152541\pi$
$524$	−4.34012e20	−1.74790
$525$	−2.26231e19	−0.0898168
$526$	−2.12172e20	−0.830414
$527$	1.94905e20	0.752045
$528$	−2.90173e20	−1.10383
$529$	1.15928e19	0.0434783
$530$	−3.18004e20	−1.17588
$531$	4.78768e19	0.174548
$532$	−1.24039e21	−4.45882
$533$	−3.02605e19	−0.107256
$534$	3.88120e20	1.35646
$535$	4.45825e20	1.53642
$536$	1.01957e21	3.46483
$537$	−3.35666e20	−1.12487
$538$	−9.17186e20	−3.03103
$539$	1.80756e20	0.589084
$540$	7.17501e20	2.30605
$541$	5.46135e19	0.173109	0.0865547	−	0.996247i	$-0.472414\pi$
$541$	5.46135e19	0.173109	0.0865547	+	0.996247i	$0.472414\pi$
$542$	3.92321e20	1.22644
$543$	5.71580e19	0.176229
$544$	1.05278e21	3.20143
$545$	−5.48489e20	−1.64511
$546$	−5.56986e20	−1.64778
$547$	4.22380e20	1.23253	0.616266	−	0.787538i	$-0.288645\pi$
$547$	4.22380e20	1.23253	0.616266	+	0.787538i	$0.288645\pi$
$548$	−8.37332e20	−2.41015
$549$	2.45398e20	0.696751
$550$	5.07492e19	0.142138
$551$	9.72916e20	2.68806
$552$	1.25819e20	0.342927
$553$	4.26419e20	1.14656
$554$	1.43773e20	0.381376
$555$	−7.33518e19	−0.191961
$556$	−8.21891e20	−2.12204
$557$	−5.57690e20	−1.42062	−0.710312	−	0.703887i	$-0.751446\pi$
$557$	−5.57690e20	−1.42062	−0.710312	+	0.703887i	$0.751446\pi$
$558$	−1.80084e20	−0.452604
$559$	8.29798e18	0.0205771
$560$	−1.22193e21	−2.98975
$561$	3.52805e20	0.851753
$562$	1.01380e21	2.41507
$563$	−3.28340e20	−0.771810	−0.385905	−	0.922539i	$-0.626111\pi$
$563$	−3.28340e20	−0.771810	−0.385905	+	0.922539i	$0.626111\pi$
$564$	4.41838e20	1.02487
$565$	−3.82676e19	−0.0875928
$566$	−1.49759e21	−3.38274
$567$	3.32083e19	0.0740242
$568$	−8.10831e20	−1.78368
$569$	4.42847e19	0.0961417	0.0480709	−	0.998844i	$-0.484693\pi$
$569$	4.42847e19	0.0961417	0.0480709	+	0.998844i	$0.484693\pi$
$570$	7.19197e20	1.54094
$571$	−5.10554e20	−1.07962	−0.539810	−	0.841787i	$-0.681504\pi$
$571$	−5.10554e20	−1.07962	−0.539810	+	0.841787i	$0.681504\pi$
$572$	8.81811e20	1.84037
$573$	−2.83080e20	−0.583109
$574$	−1.25036e20	−0.254212
$575$	−1.09003e19	−0.0218740
$576$	−2.67743e20	−0.530335
$577$	−5.08061e20	−0.993338	−0.496669	−	0.867940i	$-0.665444\pi$
$577$	−5.08061e20	−0.993338	−0.496669	+	0.867940i	$0.665444\pi$
$578$	−2.19753e21	−4.24108
$579$	6.08450e20	1.15914
$580$	2.34191e21	4.40413
$581$	1.43307e20	0.266040
$582$	−5.67808e20	−1.04059
$583$	2.73886e20	0.495513
$584$	1.02086e21	1.82335
$585$	−3.32211e20	−0.585793
$586$	−8.96115e19	−0.156003
$587$	2.51404e20	0.432102	0.216051	−	0.976382i	$-0.430682\pi$
$587$	2.51404e20	0.432102	0.216051	+	0.976382i	$0.430682\pi$
$588$	−7.22651e20	−1.22631
$589$	−3.42195e20	−0.573335
$590$	3.10288e20	0.513304
$591$	5.90346e20	0.964274
$592$	4.64329e20	0.748881
$593$	1.28228e19	0.0204209	0.0102104	−	0.999948i	$-0.496750\pi$
$593$	1.28228e19	0.0204209	0.0102104	+	0.999948i	$0.496750\pi$
$594$	−8.75599e20	−1.37691
$595$	1.48567e21	2.30699
$596$	1.51799e21	2.32768
$597$	2.92724e20	0.443253
$598$	−2.68367e20	−0.401300
$599$	−4.64879e20	−0.686498	−0.343249	−	0.939245i	$-0.611527\pi$
$599$	−4.64879e20	−0.686498	−0.343249	+	0.939245i	$0.611527\pi$
$600$	−1.18302e20	−0.172527
$601$	−9.04912e19	−0.130331	−0.0651655	−	0.997874i	$-0.520758\pi$
$601$	−9.04912e19	−0.130331	−0.0651655	+	0.997874i	$0.520758\pi$
$602$	3.42872e19	0.0487707
$603$	5.67369e20	0.797051
$604$	4.78298e20	0.663623
$605$	−3.17455e20	−0.435027
$606$	−1.51427e21	−2.04955
$607$	−1.04211e21	−1.39315	−0.696575	−	0.717484i	$-0.745294\pi$
$607$	−1.04211e21	−1.39315	−0.696575	+	0.717484i	$0.745294\pi$
$608$	−1.84836e21	−2.44067
$609$	1.27402e21	1.66168
$610$	1.59041e21	2.04897
$611$	−5.49507e20	−0.699302
$612$	2.05587e21	2.58440
$613$	−7.69650e20	−0.955740	−0.477870	−	0.878431i	$-0.658591\pi$
$613$	−7.69650e20	−0.955740	−0.477870	+	0.878431i	$0.658591\pi$
$614$	1.21337e21	1.48844
$615$	5.11659e19	0.0620036
$616$	2.12453e21	2.54335
$617$	2.10666e20	0.249147	0.124574	−	0.992210i	$-0.460244\pi$
$617$	2.10666e20	0.249147	0.124574	+	0.992210i	$0.460244\pi$
$618$	−8.23398e20	−0.962047
$619$	−2.41791e20	−0.279100	−0.139550	−	0.990215i	$-0.544566\pi$
$619$	−2.41791e20	−0.279100	−0.139550	+	0.990215i	$0.544566\pi$
$620$	−8.23699e20	−0.939357
$621$	1.88067e20	0.211897
$622$	2.33085e21	2.59470
$623$	−1.40764e21	−1.54821
$624$	−1.44279e21	−1.56790
$625$	−8.23374e20	−0.884091
$626$	5.98975e20	0.635479
$627$	−6.19419e20	−0.649349
$628$	1.06378e21	1.10193
$629$	−5.64551e20	−0.577861
$630$	−1.37269e21	−1.38842
$631$	−2.08179e20	−0.208073	−0.104037	−	0.994573i	$-0.533176\pi$
$631$	−2.08179e20	−0.208073	−0.104037	+	0.994573i	$0.533176\pi$
$632$	2.22985e21	2.20241
$633$	−3.43362e20	−0.335138
$634$	−2.39506e21	−2.31018
$635$	9.95876e20	0.949298
$636$	−1.09498e21	−1.03152
$637$	8.98750e20	0.836745
$638$	−2.85794e21	−2.62965
$639$	−4.51210e20	−0.410319
$640$	1.19727e20	0.107608
$641$	1.62512e21	1.44361	0.721807	−	0.692094i	$-0.243312\pi$
$641$	1.62512e21	1.44361	0.721807	+	0.692094i	$0.243312\pi$
$642$	2.17512e21	1.90972
$643$	−1.36015e20	−0.118033	−0.0590167	−	0.998257i	$-0.518797\pi$
$643$	−1.36015e20	−0.118033	−0.0590167	+	0.998257i	$0.518797\pi$
$644$	−7.82605e20	−0.671272
$645$	−1.40306e19	−0.0118954
$646$	5.53529e21	4.63870
$647$	−2.19502e21	−1.81826	−0.909131	−	0.416511i	$-0.863253\pi$
$647$	−2.19502e21	−1.81826	−0.909131	+	0.416511i	$0.863253\pi$
$648$	1.73654e20	0.142191
$649$	−2.67240e20	−0.216305
$650$	2.52334e20	0.201895
$651$	−4.48099e20	−0.354419
$652$	1.36577e21	1.06788
$653$	−5.17615e20	−0.400090	−0.200045	−	0.979787i	$-0.564109\pi$
$653$	−5.17615e20	−0.400090	−0.200045	+	0.979787i	$0.564109\pi$
$654$	−2.67600e21	−2.04481
$655$	−9.12677e20	−0.689457
$656$	−3.23888e20	−0.241889
$657$	5.68086e20	0.419444
$658$	−2.27056e21	−1.65745
$659$	−8.27419e20	−0.597152	−0.298576	−	0.954386i	$-0.596512\pi$
$659$	−8.27419e20	−0.597152	−0.298576	+	0.954386i	$0.596512\pi$
$660$	−1.49101e21	−1.06390
$661$	−1.22424e21	−0.863686	−0.431843	−	0.901949i	$-0.642136\pi$
$661$	−1.22424e21	−0.863686	−0.431843	+	0.901949i	$0.642136\pi$
$662$	−2.98034e20	−0.207889
$663$	1.75421e21	1.20984
$664$	7.49389e20	0.511031
$665$	−2.60839e21	−1.75877
$666$	5.21620e20	0.347774
$667$	6.13847e20	0.404685
$668$	−5.71504e21	−3.72559
$669$	−2.59585e20	−0.167334
$670$	3.67710e21	2.34393
$671$	−1.36977e21	−0.863432
$672$	−2.42040e21	−1.50875
$673$	−2.91031e21	−1.79401	−0.897007	−	0.442016i	$-0.854263\pi$
$673$	−2.91031e21	−1.79401	−0.897007	+	0.442016i	$0.854263\pi$
$674$	3.04027e21	1.85337
$675$	−1.76831e20	−0.106606
$676$	3.61558e20	0.215565
$677$	−2.33155e21	−1.37477	−0.687385	−	0.726294i	$-0.741241\pi$
$677$	−2.33155e21	−1.37477	−0.687385	+	0.726294i	$0.741241\pi$
$678$	−1.86702e20	−0.108875
$679$	2.05933e21	1.18769
$680$	7.76894e21	4.43144
$681$	−1.31328e21	−0.740892
$682$	1.00520e21	0.560878
$683$	3.05745e21	1.68735	0.843674	−	0.536856i	$-0.180388\pi$
$683$	3.05745e21	1.68735	0.843674	+	0.536856i	$0.180388\pi$
$684$	−3.60948e21	−1.97026
$685$	−1.76081e21	−0.950677
$686$	−9.19704e20	−0.491153
$687$	−6.48280e20	−0.342442
$688$	8.88161e19	0.0464065
$689$	1.36181e21	0.703836
$690$	4.53767e20	0.231988
$691$	−2.06387e21	−1.04375	−0.521877	−	0.853021i	$-0.674768\pi$
$691$	−2.06387e21	−1.04375	−0.521877	+	0.853021i	$0.674768\pi$
$692$	−8.06283e21	−4.03359
$693$	1.18225e21	0.585074
$694$	2.82848e21	1.38470
$695$	−1.72834e21	−0.837034
$696$	6.66217e21	3.19188
$697$	3.93797e20	0.186649
$698$	2.39080e21	1.12106
$699$	8.13770e20	0.377505
$700$	7.35850e20	0.337718
$701$	−1.64369e21	−0.746335	−0.373168	−	0.927764i	$-0.621728\pi$
$701$	−1.64369e21	−0.746335	−0.373168	+	0.927764i	$0.621728\pi$
$702$	−4.35363e21	−1.95579
$703$	9.91181e20	0.440543
$704$	1.49450e21	0.657205
$705$	9.29133e20	0.404259
$706$	−1.34411e21	−0.578627
$707$	5.49195e21	2.33928
$708$	1.06841e21	0.450285
$709$	1.43632e21	0.598969	0.299485	−	0.954101i	$-0.403185\pi$
$709$	1.43632e21	0.598969	0.299485	+	0.954101i	$0.403185\pi$
$710$	−2.92428e21	−1.20665
$711$	1.24086e21	0.506644
$712$	−7.36088e21	−2.97392
$713$	−2.15903e20	−0.0863151
$714$	7.24838e21	2.86751
$715$	1.85434e21	0.725930
$716$	1.09180e22	4.22958
$717$	7.57482e20	0.290388
$718$	−2.84726e21	−1.08017
$719$	2.19520e20	0.0824154	0.0412077	−	0.999151i	$-0.486879\pi$
$719$	2.19520e20	0.0824154	0.0412077	+	0.999151i	$0.486879\pi$
$720$	−3.55576e21	−1.32111
$721$	2.98631e21	1.09804
$722$	−4.65219e21	−1.69289
$723$	6.72822e20	0.242305
$724$	−1.85915e21	−0.662636
$725$	−5.77175e20	−0.203597
$726$	−1.54882e21	−0.540724
$727$	4.62737e21	1.59892	0.799460	−	0.600720i	$-0.205119\pi$
$727$	4.62737e21	1.59892	0.799460	+	0.600720i	$0.205119\pi$
$728$	1.05635e22	3.61262
$729$	2.01175e21	0.680952
$730$	3.68175e21	1.23348
$731$	−1.07986e20	−0.0358087
$732$	5.47624e21	1.79742
$733$	−3.08625e20	−0.100266	−0.0501328	−	0.998743i	$-0.515964\pi$
$733$	−3.08625e20	−0.100266	−0.0501328	+	0.998743i	$0.515964\pi$
$734$	3.34472e21	1.07557
$735$	−1.51965e21	−0.483714
$736$	−1.16619e21	−0.367440
$737$	−3.16696e21	−0.987726
$738$	−3.63851e20	−0.112331
$739$	−7.11539e20	−0.217453	−0.108726	−	0.994072i	$-0.534677\pi$
$739$	−7.11539e20	−0.217453	−0.108726	+	0.994072i	$0.534677\pi$
$740$	2.38588e21	0.721789
$741$	−3.07986e21	−0.922347
$742$	5.62698e21	1.66819
$743$	−2.43145e21	−0.713590	−0.356795	−	0.934183i	$-0.616131\pi$
$743$	−2.43145e21	−0.713590	−0.356795	+	0.934183i	$0.616131\pi$
$744$	−2.34322e21	−0.680795
$745$	3.19216e21	0.918148
$746$	3.81983e21	1.08768
$747$	4.17019e20	0.117558
$748$	−1.14755e22	−3.20265
$749$	−7.88874e21	−2.17969
$750$	−4.49379e21	−1.22929
$751$	−6.74224e21	−1.82602	−0.913008	−	0.407942i	$-0.866246\pi$
$751$	−6.74224e21	−1.82602	−0.913008	+	0.407942i	$0.866246\pi$
$752$	−5.88157e21	−1.57710
$753$	−2.57773e21	−0.684345
$754$	−1.42102e22	−3.73520
$755$	1.00580e21	0.261765
$756$	−1.26959e22	−3.27154
$757$	2.48329e20	0.0633590	0.0316795	−	0.999498i	$-0.489914\pi$
$757$	2.48329e20	0.0633590	0.0316795	+	0.999498i	$0.489914\pi$
$758$	8.85842e21	2.23788
$759$	−3.90814e20	−0.0977590
$760$	−1.36399e22	−3.37839
$761$	7.42069e20	0.181995	0.0909975	−	0.995851i	$-0.470994\pi$
$761$	7.42069e20	0.181995	0.0909975	+	0.995851i	$0.470994\pi$
$762$	4.85874e21	1.17994
$763$	9.70535e21	2.33387
$764$	9.20760e21	2.19253
$765$	4.32325e21	1.01941
$766$	−5.60594e21	−1.30898
$767$	−1.32876e21	−0.307243
$768$	3.07520e21	0.704150
$769$	−1.37070e21	−0.310810	−0.155405	−	0.987851i	$-0.549668\pi$
$769$	−1.37070e21	−0.310810	−0.155405	+	0.987851i	$0.549668\pi$
$770$	7.66214e21	1.72056
$771$	2.44140e21	0.542914
$772$	−1.97907e22	−4.35845
$773$	6.42687e21	1.40170	0.700848	−	0.713311i	$-0.252805\pi$
$773$	6.42687e21	1.40170	0.700848	+	0.713311i	$0.252805\pi$
$774$	9.97746e19	0.0215508
$775$	2.03004e20	0.0434253
$776$	1.07688e22	2.28141
$777$	1.29794e21	0.272330
$778$	2.90008e21	0.602647
$779$	−6.91389e20	−0.142296
$780$	−7.41355e21	−1.51118
$781$	2.51858e21	0.508478
$782$	3.49241e21	0.698352
$783$	9.95825e21	1.97229
$784$	9.61963e21	1.88707
$785$	2.23701e21	0.434656
$786$	−4.45282e21	−0.856972
$787$	2.18489e21	0.416503	0.208252	−	0.978075i	$-0.433223\pi$
$787$	2.18489e21	0.416503	0.208252	+	0.978075i	$0.433223\pi$
$788$	−1.92019e22	−3.62574
$789$	−1.53630e21	−0.287341
$790$	8.04201e21	1.48991
$791$	6.77134e20	0.124266
$792$	6.18229e21	1.12386
$793$	−6.81072e21	−1.22643
$794$	−1.51030e22	−2.69407
$795$	−2.30261e21	−0.406880
$796$	−9.52130e21	−1.66666
$797$	−5.64641e20	−0.0979117	−0.0489559	−	0.998801i	$-0.515589\pi$
$797$	−5.64641e20	−0.0979117	−0.0489559	+	0.998801i	$0.515589\pi$
$798$	−1.27260e22	−2.18610
$799$	7.15106e21	1.21694
$800$	1.09652e21	0.184860
$801$	−4.09617e21	−0.684123
$802$	4.58217e21	0.758164
$803$	−3.17096e21	−0.519785
$804$	1.26613e22	2.05616
$805$	−1.64573e21	−0.264782
$806$	4.99801e21	0.796681
$807$	−6.64117e21	−1.04880
$808$	2.87188e22	4.49346
$809$	−1.04377e22	−1.61805	−0.809025	−	0.587774i	$-0.800005\pi$
$809$	−1.04377e22	−1.61805	−0.809025	+	0.587774i	$0.800005\pi$
$810$	6.26288e20	0.0961915
$811$	1.11601e22	1.69829	0.849144	−	0.528161i	$-0.177118\pi$
$811$	1.11601e22	1.69829	0.849144	+	0.528161i	$0.177118\pi$
$812$	−4.14394e22	−6.24803
$813$	2.84073e21	0.424375
$814$	−2.91159e21	−0.430971
$815$	2.87206e21	0.421223
$816$	1.87759e22	2.72850
$817$	1.89591e20	0.0272994
$818$	5.47090e21	0.780565
$819$	5.87836e21	0.831050
$820$	−1.66425e21	−0.233138
$821$	−5.97010e21	−0.828720	−0.414360	−	0.910113i	$-0.635995\pi$
$821$	−5.97010e21	−0.828720	−0.414360	+	0.910113i	$0.635995\pi$
$822$	−8.59075e21	−1.18166
$823$	1.17920e22	1.60726	0.803632	−	0.595126i	$-0.202898\pi$
$823$	1.17920e22	1.60726	0.803632	+	0.595126i	$0.202898\pi$
$824$	1.56161e22	2.10921
$825$	3.67466e20	0.0491827
$826$	−5.49045e21	−0.728211
$827$	4.68835e21	0.616210	0.308105	−	0.951352i	$-0.400305\pi$
$827$	4.68835e21	0.616210	0.308105	+	0.951352i	$0.400305\pi$
$828$	−2.27735e21	−0.296622
$829$	−1.01230e22	−1.30662	−0.653312	−	0.757089i	$-0.726621\pi$
$829$	−1.01230e22	−1.30662	−0.653312	+	0.757089i	$0.726621\pi$
$830$	2.70269e21	0.345709
$831$	1.04103e21	0.131964
$832$	7.43090e21	0.933505
$833$	−1.16960e22	−1.45612
$834$	−8.43234e21	−1.04041
$835$	−1.20180e22	−1.46955
$836$	2.01475e22	2.44160
$837$	−3.50252e21	−0.420669
$838$	1.57148e22	1.87059
$839$	−2.54799e20	−0.0300596	−0.0150298	−	0.999887i	$-0.504784\pi$
$839$	−2.54799e20	−0.0300596	−0.0150298	+	0.999887i	$0.504784\pi$
$840$	−1.78613e22	−2.08842
$841$	2.38744e22	2.76670
$842$	1.49354e22	1.71545
$843$	7.34072e21	0.835665
$844$	1.11684e22	1.26015
$845$	7.60313e20	0.0850291
$846$	−6.60726e21	−0.732392
$847$	5.61726e21	0.617162
$848$	1.45759e22	1.58733
$849$	−1.08438e22	−1.17050
$850$	−3.28376e21	−0.351342
$851$	6.25371e20	0.0663233
$852$	−1.00691e22	−1.05851
$853$	1.32096e22	1.37649	0.688245	−	0.725479i	$-0.258382\pi$
$853$	1.32096e22	1.37649	0.688245	+	0.725479i	$0.258382\pi$
$854$	−2.81419e22	−2.90683
$855$	−7.59032e21	−0.777167
$856$	−4.12522e22	−4.18691
$857$	−8.91620e20	−0.0897064	−0.0448532	−	0.998994i	$-0.514282\pi$
$857$	−8.91620e20	−0.0897064	−0.0448532	+	0.998994i	$0.514282\pi$
$858$	9.04709e21	0.902306
$859$	−3.97231e21	−0.392731	−0.196365	−	0.980531i	$-0.562914\pi$
$859$	−3.97231e21	−0.392731	−0.196365	+	0.980531i	$0.562914\pi$
$860$	4.56367e20	0.0447276
$861$	−9.05364e20	−0.0879629
$862$	9.04774e21	0.871436
$863$	−4.67984e21	−0.446838	−0.223419	−	0.974723i	$-0.571722\pi$
$863$	−4.67984e21	−0.446838	−0.223419	+	0.974723i	$0.571722\pi$
$864$	−1.89188e22	−1.79077
$865$	−1.69552e22	−1.59104
$866$	1.18263e22	1.10018
$867$	−1.59119e22	−1.46750
$868$	1.45751e22	1.33264
$869$	−6.92630e21	−0.627846
$870$	2.40272e22	2.15928
$871$	−1.57467e22	−1.40298
$872$	5.07517e22	4.48308
$873$	5.99258e21	0.524816
$874$	−6.13162e21	−0.532402
$875$	1.62981e22	1.40306
$876$	1.26773e22	1.08205
$877$	1.49622e22	1.26619	0.633093	−	0.774076i	$-0.281785\pi$
$877$	1.49622e22	1.26619	0.633093	+	0.774076i	$0.281785\pi$
$878$	7.46336e21	0.626218
$879$	−6.48860e20	−0.0539802
$880$	1.98477e22	1.63715
$881$	2.00861e22	1.64277	0.821384	−	0.570376i	$-0.193203\pi$
$881$	2.00861e22	1.64277	0.821384	+	0.570376i	$0.193203\pi$
$882$	1.08065e22	0.876339
$883$	−1.37853e22	−1.10844	−0.554219	−	0.832371i	$-0.686983\pi$
$883$	−1.37853e22	−1.10844	−0.554219	+	0.832371i	$0.686983\pi$
$884$	−5.70582e22	−4.54911
$885$	2.24674e21	0.177614
$886$	−2.88882e22	−2.26447
$887$	−1.14043e22	−0.886422	−0.443211	−	0.896417i	$-0.646161\pi$
$887$	−1.14043e22	−0.886422	−0.443211	+	0.896417i	$0.646161\pi$
$888$	6.78724e21	0.523113
$889$	−1.76217e22	−1.34674
$890$	−2.65472e22	−2.01184
$891$	−5.39400e20	−0.0405348
$892$	8.44340e21	0.629188
$893$	−1.25551e22	−0.927757
$894$	1.55741e22	1.14123
$895$	2.29594e22	1.66835
$896$	−2.11854e21	−0.152660
$897$	−1.94319e21	−0.138859
$898$	−3.86403e22	−2.73822
$899$	−1.14322e22	−0.803399
$900$	2.14130e21	0.149231
$901$	−1.77220e22	−1.22483
$902$	2.03095e21	0.139204
$903$	2.48267e20	0.0168757
$904$	3.54090e21	0.238699
$905$	−3.90957e21	−0.261375
$906$	4.90718e21	0.325365
$907$	−1.06170e22	−0.698147	−0.349074	−	0.937095i	$-0.613504\pi$
$907$	−1.06170e22	−0.698147	−0.349074	+	0.937095i	$0.613504\pi$
$908$	4.27165e22	2.78581
$909$	1.59814e22	1.03368
$910$	3.80975e22	2.44391
$911$	−9.25426e21	−0.588781	−0.294391	−	0.955685i	$-0.595117\pi$
$911$	−9.25426e21	−0.588781	−0.294391	+	0.955685i	$0.595117\pi$
$912$	−3.29648e22	−2.08012
$913$	−2.32773e21	−0.145681
$914$	1.49165e22	0.925911
$915$	1.15159e22	0.708989
$916$	2.10863e22	1.28761
$917$	1.61495e22	0.978115
$918$	5.66562e22	3.40351
$919$	1.35306e22	0.806212	0.403106	−	0.915153i	$-0.367931\pi$
$919$	1.35306e22	0.806212	0.403106	+	0.915153i	$0.367931\pi$
$920$	−8.60591e21	−0.508614
$921$	8.78578e21	0.515031
$922$	−7.09710e21	−0.412666
$923$	1.25228e22	0.722252
$924$	2.63829e22	1.50933
$925$	−5.88010e20	−0.0333673
$926$	−2.13902e22	−1.20401
$927$	8.69004e21	0.485203
$928$	−6.17506e22	−3.42004
$929$	1.63537e22	0.898460	0.449230	−	0.893416i	$-0.351698\pi$
$929$	1.63537e22	0.898460	0.449230	+	0.893416i	$0.351698\pi$
$930$	−8.45088e21	−0.460553
$931$	2.05346e22	1.11010
$932$	−2.64691e22	−1.41945
$933$	1.68773e22	0.897822
$934$	−9.11577e21	−0.481052
$935$	−2.41316e22	−1.26328
$936$	3.07394e22	1.59635
$937$	1.63567e22	0.842654	0.421327	−	0.906909i	$-0.361565\pi$
$937$	1.63567e22	0.842654	0.421327	+	0.906909i	$0.361565\pi$
$938$	−6.50652e22	−3.32527
$939$	4.33707e21	0.219889
$940$	−3.02214e22	−1.52004
$941$	7.30630e21	0.364565	0.182283	−	0.983246i	$-0.441651\pi$
$941$	7.30630e21	0.364565	0.182283	+	0.983246i	$0.441651\pi$
$942$	1.09141e22	0.540262
$943$	−4.36222e20	−0.0214225
$944$	−1.42222e22	−0.692910
$945$	−2.66981e22	−1.29045
$946$	−5.56925e20	−0.0267063
$947$	1.63180e22	0.776322	0.388161	−	0.921592i	$-0.373110\pi$
$947$	1.63180e22	0.776322	0.388161	+	0.921592i	$0.373110\pi$
$948$	2.76909e22	1.30700
$949$	−1.57665e22	−0.738312
$950$	5.76530e21	0.267852
$951$	−1.73422e22	−0.799373
$952$	−1.37469e23	−6.28677
$953$	−8.63869e21	−0.391969	−0.195984	−	0.980607i	$-0.562790\pi$
$953$	−8.63869e21	−0.391969	−0.195984	+	0.980607i	$0.562790\pi$
$954$	1.63743e22	0.737141
$955$	1.93625e22	0.864840
$956$	−2.46382e22	−1.09188
$957$	−2.06938e22	−0.909915
$958$	2.26025e22	0.986089
$959$	3.11570e22	1.34870
$960$	−1.25645e22	−0.539650
$961$	−1.94443e22	−0.828643
$962$	−1.44769e22	−0.612158
$963$	−2.29559e22	−0.963159
$964$	−2.18845e22	−0.911086
$965$	−4.16176e22	−1.71918
$966$	−8.02927e21	−0.329115
$967$	1.46121e22	0.594312	0.297156	−	0.954829i	$-0.403962\pi$
$967$	1.46121e22	0.594312	0.297156	+	0.954829i	$0.403962\pi$
$968$	2.93740e22	1.18549
$969$	4.00800e22	1.60509
$970$	3.88377e22	1.54335
$971$	−3.09015e22	−1.21853	−0.609263	−	0.792968i	$-0.708535\pi$
$971$	−3.09015e22	−1.21853	−0.609263	+	0.792968i	$0.708535\pi$
$972$	−6.01354e22	−2.35306
$973$	3.05824e22	1.18748
$974$	−4.19007e22	−1.61447
$975$	1.82710e21	0.0698599
$976$	−7.28975e22	−2.76592
$977$	−7.33455e21	−0.276162	−0.138081	−	0.990421i	$-0.544093\pi$
$977$	−7.33455e21	−0.276162	−0.138081	+	0.990421i	$0.544093\pi$
$978$	1.40124e22	0.523566
$979$	2.28641e22	0.847783
$980$	4.94289e22	1.81880
$981$	2.82422e22	1.03129
$982$	−2.63448e22	−0.954681
$983$	−4.24790e22	−1.52765	−0.763824	−	0.645425i	$-0.776680\pi$
$983$	−4.24790e22	−1.52765	−0.763824	+	0.645425i	$0.776680\pi$
$984$	−4.73437e21	−0.168966
$985$	−4.03793e22	−1.43017
$986$	1.84925e23	6.50008
$987$	−1.64407e22	−0.573512
$988$	1.00177e23	3.46809
$989$	1.19620e20	0.00410990
$990$	2.22966e22	0.760281
$991$	2.02363e22	0.684824	0.342412	−	0.939550i	$-0.388756\pi$
$991$	2.02363e22	0.684824	0.342412	+	0.939550i	$0.388756\pi$
$992$	2.17190e22	0.729461
$993$	−2.15801e21	−0.0719340
$994$	5.17442e22	1.71184
$995$	−2.00222e22	−0.657412
$996$	9.30612e21	0.303266
$997$	−3.57106e22	−1.15500	−0.577501	−	0.816390i	$-0.695972\pi$
$997$	−3.57106e22	−1.15500	−0.577501	+	0.816390i	$0.695972\pi$
$998$	−2.10771e22	−0.676599
$999$	1.01452e22	0.323236

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.16.a.a.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-333.711 q^{2} -2416.34 q^{3} +78595.0 q^{4} +165276. q^{5} +806359. q^{6} -2.92451e6 q^{7} -1.52930e7 q^{8} -8.51021e6 q^{9} +O(q^{10})\)	\(q-333.711 q^{2} -2416.34 q^{3} +78595.0 q^{4} +165276. q^{5} +806359. q^{6} -2.92451e6 q^{7} -1.52930e7 q^{8} -8.51021e6 q^{9} -5.51544e7 q^{10} +4.75026e7 q^{11} -1.89912e8 q^{12} +2.36191e8 q^{13} +9.75940e8 q^{14} -3.99363e8 q^{15} +2.52803e9 q^{16} -3.07369e9 q^{17} +2.83995e9 q^{18} +5.39647e9 q^{19} +1.29899e10 q^{20} +7.06660e9 q^{21} -1.58521e10 q^{22} +3.40483e9 q^{23} +3.69530e10 q^{24} -3.20141e9 q^{25} -7.88195e10 q^{26} +5.52354e10 q^{27} -2.29852e11 q^{28} +1.80287e11 q^{29} +1.33272e11 q^{30} -6.34108e10 q^{31} -3.42512e11 q^{32} -1.14782e11 q^{33} +1.02572e12 q^{34} -4.83351e11 q^{35} -6.68860e11 q^{36} +1.83672e11 q^{37} -1.80086e12 q^{38} -5.70717e11 q^{39} -2.52756e12 q^{40} -1.28119e11 q^{41} -2.35820e12 q^{42} +3.51325e10 q^{43} +3.73347e12 q^{44} -1.40653e12 q^{45} -1.13623e12 q^{46} -2.32654e12 q^{47} -6.10858e12 q^{48} +3.80518e12 q^{49} +1.06835e12 q^{50} +7.42707e12 q^{51} +1.85634e13 q^{52} +5.76570e12 q^{53} -1.84327e13 q^{54} +7.85104e12 q^{55} +4.47244e13 q^{56} -1.30397e13 q^{57} -6.01639e13 q^{58} -5.62580e12 q^{59} -3.13879e13 q^{60} -2.88357e13 q^{61} +2.11609e13 q^{62} +2.48882e13 q^{63} +3.14614e13 q^{64} +3.90367e13 q^{65} +3.83041e13 q^{66} -6.66692e13 q^{67} -2.41577e14 q^{68} -8.22721e12 q^{69} +1.61300e14 q^{70} +5.30198e13 q^{71} +1.30146e14 q^{72} -6.67534e13 q^{73} -6.12934e13 q^{74} +7.73570e12 q^{75} +4.24136e14 q^{76} -1.38922e14 q^{77} +1.90455e14 q^{78} -1.45809e14 q^{79} +4.17823e14 q^{80} -1.13552e13 q^{81} +4.27546e13 q^{82} -4.90022e13 q^{83} +5.55400e14 q^{84} -5.08007e14 q^{85} -1.17241e13 q^{86} -4.35636e14 q^{87} -7.26456e14 q^{88} +4.81324e14 q^{89} +4.69376e14 q^{90} -6.90742e14 q^{91} +2.67602e14 q^{92} +1.53222e14 q^{93} +7.76392e14 q^{94} +8.91907e14 q^{95} +8.27625e14 q^{96} -7.04163e14 q^{97} -1.26983e15 q^{98} -4.04257e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9}+O(q^{10}) \) 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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