LMFDB - Embedded newform 1014.4.b.q.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1014.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8279367458$
Analytic rank:	$0$
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} + 845x^{10} + 287958x^{8} + 50362537x^{6} + 4731667920x^{4} + 224458698240x^{2} + 4178851762176 $ x^12 + 845x^10 + 287958x^8 + 50362537x^6 + 4731667920x^4 + 224458698240*x^2 + 4178851762176
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$12.9677i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.4.b.q.337.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +17.9285i q^{5} -6.00000i q^{6} -26.1605i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +17.9285i q^{5} -6.00000i q^{6} -26.1605i q^{7} +8.00000i q^{8} +9.00000 q^{9} +35.8570 q^{10} -12.4604i q^{11} -12.0000 q^{12} -52.3210 q^{14} +53.7855i q^{15} +16.0000 q^{16} +115.900 q^{17} -18.0000i q^{18} -92.5926i q^{19} -71.7140i q^{20} -78.4815i q^{21} -24.9208 q^{22} -203.684 q^{23} +24.0000i q^{24} -196.431 q^{25} +27.0000 q^{27} +104.642i q^{28} +165.582 q^{29} +107.571 q^{30} +223.081i q^{31} -32.0000i q^{32} -37.3812i q^{33} -231.799i q^{34} +469.019 q^{35} -36.0000 q^{36} +137.629i q^{37} -185.185 q^{38} -143.428 q^{40} -371.997i q^{41} -156.963 q^{42} -131.567 q^{43} +49.8416i q^{44} +161.357i q^{45} +407.369i q^{46} -82.8384i q^{47} +48.0000 q^{48} -341.373 q^{49} +392.863i q^{50} +347.699 q^{51} +433.489 q^{53} -54.0000i q^{54} +223.397 q^{55} +209.284 q^{56} -277.778i q^{57} -331.164i q^{58} -459.436i q^{59} -215.142i q^{60} -376.203 q^{61} +446.163 q^{62} -235.445i q^{63} -64.0000 q^{64} -74.7625 q^{66} -206.572i q^{67} -463.599 q^{68} -611.053 q^{69} -938.038i q^{70} -264.483i q^{71} +72.0000i q^{72} -843.011i q^{73} +275.258 q^{74} -589.294 q^{75} +370.370i q^{76} -325.971 q^{77} +869.486 q^{79} +286.856i q^{80} +81.0000 q^{81} -743.994 q^{82} -879.562i q^{83} +313.926i q^{84} +2077.91i q^{85} +263.133i q^{86} +496.746 q^{87} +99.6833 q^{88} -161.301i q^{89} +322.713 q^{90} +814.738 q^{92} +669.244i q^{93} -165.677 q^{94} +1660.05 q^{95} -96.0000i q^{96} -1660.23i q^{97} +682.745i q^{98} -112.144i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) $ 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9	$ 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) $ 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9 + 12 * q^10 - 144 * q^12 - 4 * q^14 + 192 * q^16 - 102 * q^17 + 256 * q^22 - 444 * q^23 - 370 * q^25 + 324 * q^27 + 658 * q^29 + 36 * q^30 + 1688 * q^35 - 432 * q^36 - 852 * q^38 - 48 * q^40 - 12 * q^42 - 982 * q^43 + 576 * q^48 - 2266 * q^49 - 306 * q^51 + 4604 * q^53 - 658 * q^55 + 16 * q^56 + 690 * q^61 - 1156 * q^62 - 768 * q^64 + 768 * q^66 + 408 * q^68 - 1332 * q^69 + 1880 * q^74 - 1110 * q^75 - 6582 * q^77 + 6200 * q^79 + 972 * q^81 + 1284 * q^82 + 1974 * q^87 - 1024 * q^88 + 108 * q^90 + 1776 * q^92 - 2564 * q^94 + 1330 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.00000	−0.500000
$5$	17.9285i	1.60357i	0.597610	+	0.801787i	$0.296117\pi$
$5$	17.9285i	1.60357i	−0.597610	+	0.801787i	$0.703883\pi$
$6$	− 6.00000i	− 0.408248i
$7$	− 26.1605i	− 1.41254i	−0.707945	−	0.706268i	$-0.750378\pi$
$7$	− 26.1605i	− 1.41254i	0.707945	−	0.706268i	$-0.249622\pi$
$8$	8.00000i	0.353553i
$9$	9.00000	0.333333
$10$	35.8570	1.13390
$11$	− 12.4604i	− 0.341541i	−0.985311	−	0.170771i	$-0.945374\pi$
$11$	− 12.4604i	− 0.341541i	0.985311	−	0.170771i	$-0.0546257\pi$
$12$	−12.0000	−0.288675
$13$	0	0
$13$	0	0
$14$	−52.3210	−0.998813
$15$	53.7855i	0.925824i
$16$	16.0000	0.250000
$17$	115.900	1.65352	0.826759	−	0.562556i	$-0.190182\pi$
$17$	115.900	1.65352	0.826759	+	0.562556i	$0.190182\pi$
$18$	− 18.0000i	− 0.235702i
$19$	− 92.5926i	− 1.11801i	−0.829164	−	0.559005i	$-0.811183\pi$
$19$	− 92.5926i	− 1.11801i	0.829164	−	0.559005i	$-0.188817\pi$
$20$	− 71.7140i	− 0.801787i
$21$	− 78.4815i	− 0.815527i
$22$	−24.9208	−0.241506
$23$	−203.684	−1.84657	−0.923286	−	0.384114i	$-0.874507\pi$
$23$	−203.684	−1.84657	−0.923286	+	0.384114i	$0.874507\pi$
$24$	24.0000i	0.204124i
$25$	−196.431	−1.57145
$26$	0	0
$27$	27.0000	0.192450
$28$	104.642i	0.706268i
$29$	165.582	1.06027	0.530134	−	0.847914i	$-0.322142\pi$
$29$	165.582	1.06027	0.530134	+	0.847914i	$0.322142\pi$
$30$	107.571	0.654657
$31$	223.081i	1.29247i	0.763138	+	0.646235i	$0.223657\pi$
$31$	223.081i	1.29247i	−0.763138	+	0.646235i	$0.776343\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 37.3812i	− 0.197189i
$34$	− 231.799i	− 1.16921i
$35$	469.019	2.26511
$36$	−36.0000	−0.166667
$37$	137.629i	0.611516i	0.952109	+	0.305758i	$0.0989098\pi$
$37$	137.629i	0.611516i	−0.952109	+	0.305758i	$0.901090\pi$
$38$	−185.185	−0.790552
$39$	0	0
$40$	−143.428	−0.566949
$41$	− 371.997i	− 1.41698i	−0.705721	−	0.708490i	$-0.749377\pi$
$41$	− 371.997i	− 1.41698i	0.705721	−	0.708490i	$-0.250623\pi$
$42$	−156.963	−0.576665
$43$	−131.567	−0.466598	−0.233299	−	0.972405i	$-0.574952\pi$
$43$	−131.567	−0.466598	−0.233299	+	0.972405i	$0.574952\pi$
$44$	49.8416i	0.170771i
$45$	161.357i	0.534525i
$46$	407.369i	1.30572i
$47$	− 82.8384i	− 0.257090i	−0.991704	−	0.128545i	$-0.958969\pi$
$47$	− 82.8384i	− 0.257090i	0.991704	−	0.128545i	$-0.0410306\pi$
$48$	48.0000	0.144338
$49$	−341.373	−0.995255
$50$	392.863i	1.11118i
$51$	347.699	0.954659
$52$	0	0
$53$	433.489	1.12348	0.561739	−	0.827315i	$-0.310133\pi$
$53$	433.489	1.12348	0.561739	+	0.827315i	$0.310133\pi$
$54$	− 54.0000i	− 0.136083i
$55$	223.397	0.547687
$56$	209.284	0.499407
$57$	− 277.778i	− 0.645483i
$58$	− 331.164i	− 0.749723i
$59$	− 459.436i	− 1.01379i	−0.862009	−	0.506894i	$-0.830794\pi$
$59$	− 459.436i	− 1.01379i	0.862009	−	0.506894i	$-0.169206\pi$
$60$	− 215.142i	− 0.462912i
$61$	−376.203	−0.789637	−0.394819	−	0.918759i	$-0.629193\pi$
$61$	−376.203	−0.789637	−0.394819	+	0.918759i	$0.629193\pi$
$62$	446.163	0.913914
$63$	− 235.445i	− 0.470845i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−74.7625	−0.139434
$67$	− 206.572i	− 0.376669i	−0.982105	−	0.188334i	$-0.939691\pi$
$67$	− 206.572i	− 0.376669i	0.982105	−	0.188334i	$-0.0603088\pi$
$68$	−463.599	−0.826759
$69$	−611.053	−1.06612
$70$	− 938.038i	− 1.60167i
$71$	− 264.483i	− 0.442089i	−0.975264	−	0.221045i	$-0.929053\pi$
$71$	− 264.483i	− 0.442089i	0.975264	−	0.221045i	$-0.0709467\pi$
$72$	72.0000i	0.117851i
$73$	− 843.011i	− 1.35160i	−0.737084	−	0.675801i	$-0.763798\pi$
$73$	− 843.011i	− 1.35160i	0.737084	−	0.675801i	$-0.236202\pi$
$74$	275.258	0.432407
$75$	−589.294	−0.907278
$76$	370.370i	0.559005i
$77$	−325.971	−0.482439
$78$	0	0
$79$	869.486	1.23829	0.619144	−	0.785277i	$-0.287480\pi$
$79$	869.486	1.23829	0.619144	+	0.785277i	$0.287480\pi$
$80$	286.856i	0.400894i
$81$	81.0000	0.111111
$82$	−743.994	−1.00196
$83$	− 879.562i	− 1.16319i	−0.813480	−	0.581593i	$-0.802430\pi$
$83$	− 879.562i	− 1.16319i	0.813480	−	0.581593i	$-0.197570\pi$
$84$	313.926i	0.407764i
$85$	2077.91i	2.65154i
$86$	263.133i	0.329934i
$87$	496.746	0.612146
$88$	99.6833	0.120753
$89$	− 161.301i	− 0.192112i	−0.995376	−	0.0960558i	$-0.969377\pi$
$89$	− 161.301i	− 0.192112i	0.995376	−	0.0960558i	$-0.0306227\pi$
$90$	322.713	0.377966
$91$	0	0
$92$	814.738	0.923286
$93$	669.244i	0.746208i
$94$	−165.677	−0.181790
$95$	1660.05	1.79281
$96$	− 96.0000i	− 0.102062i
$97$	− 1660.23i	− 1.73784i	−0.494953	−	0.868920i	$-0.664815\pi$
$97$	− 1660.23i	− 1.73784i	0.494953	−	0.868920i	$-0.335185\pi$
$98$	682.745i	0.703752i
$99$	− 112.144i	− 0.113847i
$100$	785.726	0.785726
$101$	1085.66	1.06957	0.534786	−	0.844987i	$-0.320392\pi$
$101$	1085.66	1.06957	0.534786	+	0.844987i	$0.320392\pi$
$102$	− 695.398i	− 0.675046i
$103$	503.780	0.481931	0.240966	−	0.970534i	$-0.422536\pi$
$103$	503.780	0.481931	0.240966	+	0.970534i	$0.422536\pi$
$104$	0	0
$105$	1407.06	1.30776
$106$	− 866.978i	− 0.794419i
$107$	988.080	0.892722	0.446361	−	0.894853i	$-0.352720\pi$
$107$	988.080	0.892722	0.446361	+	0.894853i	$0.352720\pi$
$108$	−108.000	−0.0962250
$109$	− 359.841i	− 0.316207i	−0.987423	−	0.158103i	$-0.949462\pi$
$109$	− 359.841i	− 0.316207i	0.987423	−	0.158103i	$-0.0505379\pi$
$110$	− 446.793i	− 0.387273i
$111$	412.887i	0.353059i
$112$	− 418.568i	− 0.353134i
$113$	2014.23	1.67684	0.838421	−	0.545023i	$-0.183479\pi$
$113$	2014.23	1.67684	0.838421	+	0.545023i	$0.183479\pi$
$114$	−555.555	−0.456426
$115$	− 3651.76i	− 2.96112i
$116$	−662.327	−0.530134
$117$	0	0
$118$	−918.872	−0.716856
$119$	− 3032.00i	− 2.33565i
$120$	−430.284	−0.327328
$121$	1175.74	0.883350
$122$	752.406i	0.558358i
$123$	− 1115.99i	− 0.818093i
$124$	− 892.325i	− 0.646235i
$125$	− 1280.66i	− 0.916365i
$126$	−470.889	−0.332938
$127$	2121.43	1.48226	0.741129	−	0.671363i	$-0.234291\pi$
$127$	2121.43	1.48226	0.741129	+	0.671363i	$0.234291\pi$
$128$	128.000i	0.0883883i
$129$	−394.700	−0.269390
$130$	0	0
$131$	−738.025	−0.492225	−0.246113	−	0.969241i	$-0.579153\pi$
$131$	−738.025	−0.492225	−0.246113	+	0.969241i	$0.579153\pi$
$132$	149.525i	0.0985945i
$133$	−2422.27	−1.57923
$134$	−413.144	−0.266345
$135$	484.070i	0.308608i
$136$	927.198i	0.584607i
$137$	− 16.8617i	− 0.0105153i	−0.999986	−	0.00525763i	$-0.998326\pi$
$137$	− 16.8617i	− 0.0105153i	0.999986	−	0.00525763i	$-0.00167356\pi$
$138$	1222.11i	0.753860i
$139$	1176.86	0.718132	0.359066	−	0.933312i	$-0.383095\pi$
$139$	1176.86	0.718132	0.359066	+	0.933312i	$0.383095\pi$
$140$	−1876.08	−1.13255
$141$	− 248.515i	− 0.148431i
$142$	−528.966	−0.312604
$143$	0	0
$144$	144.000	0.0833333
$145$	2968.64i	1.70022i
$146$	−1686.02	−0.955727
$147$	−1024.12	−0.574611
$148$	− 550.516i	− 0.305758i
$149$	178.593i	0.0981940i	0.998794	+	0.0490970i	$0.0156343\pi$
$149$	178.593i	0.0981940i	−0.998794	+	0.0490970i	$0.984366\pi$
$150$	1178.59i	0.641542i
$151$	− 1509.69i	− 0.813621i	−0.913513	−	0.406811i	$-0.866641\pi$
$151$	− 1509.69i	− 0.813621i	0.913513	−	0.406811i	$-0.133359\pi$
$152$	740.740	0.395276
$153$	1043.10	0.551173
$154$	651.941i	0.341136i
$155$	−3999.51	−2.07257
$156$	0	0
$157$	−388.527	−0.197502	−0.0987511	−	0.995112i	$-0.531485\pi$
$157$	−388.527	−0.197502	−0.0987511	+	0.995112i	$0.531485\pi$
$158$	− 1738.97i	− 0.875602i
$159$	1300.47	0.648640
$160$	573.712	0.283475
$161$	5328.49i	2.60835i
$162$	− 162.000i	− 0.0785674i
$163$	1446.94i	0.695294i	0.937625	+	0.347647i	$0.113019\pi$
$163$	1446.94i	0.695294i	−0.937625	+	0.347647i	$0.886981\pi$
$164$	1487.99i	0.708490i
$165$	670.190	0.316207
$166$	−1759.12	−0.822497
$167$	− 1536.22i	− 0.711836i	−0.934517	−	0.355918i	$-0.884168\pi$
$167$	− 1536.22i	− 0.711836i	0.934517	−	0.355918i	$-0.115832\pi$
$168$	627.852	0.288333
$169$	0	0
$170$	4155.82	1.87492
$171$	− 833.333i	− 0.372670i
$172$	526.266	0.233299
$173$	765.636	0.336475	0.168238	−	0.985746i	$-0.446192\pi$
$173$	765.636	0.336475	0.168238	+	0.985746i	$0.446192\pi$
$174$	− 993.491i	− 0.432853i
$175$	5138.75i	2.21973i
$176$	− 199.367i	− 0.0853853i
$177$	− 1378.31i	− 0.585310i
$178$	−322.603	−0.135843
$179$	−4430.22	−1.84989	−0.924945	−	0.380102i	$-0.875889\pi$
$179$	−4430.22	−1.84989	−0.924945	+	0.380102i	$0.875889\pi$
$180$	− 645.426i	− 0.267262i
$181$	−4003.64	−1.64413	−0.822067	−	0.569391i	$-0.807179\pi$
$181$	−4003.64	−1.64413	−0.822067	+	0.569391i	$0.807179\pi$
$182$	0	0
$183$	−1128.61	−0.455897
$184$	− 1629.48i	− 0.652862i
$185$	−2467.48	−0.980611
$186$	1338.49	0.527649
$187$	− 1444.16i	− 0.564745i
$188$	331.354i	0.128545i
$189$	− 706.334i	− 0.271842i
$190$	− 3320.09i	− 1.26771i
$191$	−1820.19	−0.689550	−0.344775	−	0.938685i	$-0.612045\pi$
$191$	−1820.19	−0.689550	−0.344775	+	0.938685i	$0.612045\pi$
$192$	−192.000	−0.0721688
$193$	2157.08i	0.804506i	0.915528	+	0.402253i	$0.131773\pi$
$193$	2157.08i	0.804506i	−0.915528	+	0.402253i	$0.868227\pi$
$194$	−3320.45	−1.22884
$195$	0	0
$196$	1365.49	0.497628
$197$	2033.93i	0.735593i	0.929906	+	0.367796i	$0.119888\pi$
$197$	2033.93i	0.735593i	−0.929906	+	0.367796i	$0.880112\pi$
$198$	−224.287	−0.0805021
$199$	4385.37	1.56216	0.781082	−	0.624428i	$-0.214668\pi$
$199$	4385.37	1.56216	0.781082	+	0.624428i	$0.214668\pi$
$200$	− 1571.45i	− 0.555592i
$201$	− 619.716i	− 0.217470i
$202$	− 2171.31i	− 0.756302i
$203$	− 4331.71i	− 1.49767i
$204$	−1390.80	−0.477330
$205$	6669.35	2.27223
$206$	− 1007.56i	− 0.340777i
$207$	−1833.16	−0.615524
$208$	0	0
$209$	−1153.74	−0.381847
$210$	− 2814.11i	− 0.924725i
$211$	−4254.83	−1.38822	−0.694110	−	0.719869i	$-0.744202\pi$
$211$	−4254.83	−1.38822	−0.694110	+	0.719869i	$0.744202\pi$
$212$	−1733.96	−0.561739
$213$	− 793.449i	− 0.255240i
$214$	− 1976.16i	− 0.631250i
$215$	− 2358.79i	− 0.748224i
$216$	216.000i	0.0680414i
$217$	5835.92	1.82566
$218$	−719.682	−0.223592
$219$	− 2529.03i	− 0.780348i
$220$	−893.586	−0.273844
$221$	0	0
$222$	825.775	0.249650
$223$	121.615i	0.0365199i	0.999833	+	0.0182599i	$0.00581264\pi$
$223$	121.615i	0.0365199i	−0.999833	+	0.0182599i	$0.994187\pi$
$224$	−837.136	−0.249703
$225$	−1767.88	−0.523817
$226$	− 4028.47i	− 1.18571i
$227$	− 1871.14i	− 0.547101i	−0.961858	−	0.273551i	$-0.911802\pi$
$227$	− 1871.14i	− 0.547101i	0.961858	−	0.273551i	$-0.0881981\pi$
$228$	1111.11i	0.322742i
$229$	2999.14i	0.865454i	0.901525	+	0.432727i	$0.142449\pi$
$229$	2999.14i	0.865454i	−0.901525	+	0.432727i	$0.857551\pi$
$230$	−7303.52	−2.09382
$231$	−977.912	−0.278536
$232$	1324.65i	0.374861i
$233$	−1743.02	−0.490081	−0.245040	−	0.969513i	$-0.578801\pi$
$233$	−1743.02	−0.490081	−0.245040	+	0.969513i	$0.578801\pi$
$234$	0	0
$235$	1485.17	0.412263
$236$	1837.74i	0.506894i
$237$	2608.46	0.714926
$238$	−6063.99	−1.65156
$239$	− 5265.86i	− 1.42519i	−0.701576	−	0.712595i	$-0.747520\pi$
$239$	− 5265.86i	− 1.42519i	0.701576	−	0.712595i	$-0.252480\pi$
$240$	860.568i	0.231456i
$241$	5318.69i	1.42161i	0.703392	+	0.710803i	$0.251668\pi$
$241$	5318.69i	1.42161i	−0.703392	+	0.710803i	$0.748332\pi$
$242$	− 2351.48i	− 0.624622i
$243$	243.000	0.0641500
$244$	1504.81	0.394819
$245$	− 6120.30i	− 1.59597i
$246$	−2231.98	−0.578479
$247$	0	0
$248$	−1784.65	−0.456957
$249$	− 2638.69i	− 0.671566i
$250$	−2561.32	−0.647968
$251$	−5683.28	−1.42919	−0.714593	−	0.699541i	$-0.753388\pi$
$251$	−5683.28	−1.42919	−0.714593	+	0.699541i	$0.753388\pi$
$252$	941.779i	0.235423i
$253$	2537.99i	0.630681i
$254$	− 4242.87i	− 1.04811i
$255$	6233.73i	1.53087i
$256$	256.000	0.0625000
$257$	−5246.94	−1.27352	−0.636761	−	0.771061i	$-0.719726\pi$
$257$	−5246.94	−1.27352	−0.636761	+	0.771061i	$0.719726\pi$
$258$	789.399i	0.190488i
$259$	3600.45	0.863788
$260$	0	0
$261$	1490.24	0.353423
$262$	1476.05i	0.348056i
$263$	5224.32	1.22489	0.612444	−	0.790514i	$-0.290187\pi$
$263$	5224.32	1.22489	0.612444	+	0.790514i	$0.290187\pi$
$264$	299.050	0.0697168
$265$	7771.81i	1.80158i
$266$	4844.54i	1.11668i
$267$	− 483.904i	− 0.110916i
$268$	826.288i	0.188334i
$269$	3002.32	0.680501	0.340251	−	0.940335i	$-0.389488\pi$
$269$	3002.32	0.680501	0.340251	+	0.940335i	$0.389488\pi$
$270$	968.139	0.218219
$271$	− 3025.56i	− 0.678191i	−0.940752	−	0.339096i	$-0.889879\pi$
$271$	− 3025.56i	− 0.678191i	0.940752	−	0.339096i	$-0.110121\pi$
$272$	1854.40	0.413380
$273$	0	0
$274$	−33.7234	−0.00743542
$275$	2447.62i	0.536716i
$276$	2444.21	0.533059
$277$	−5797.86	−1.25762	−0.628808	−	0.777561i	$-0.716457\pi$
$277$	−5797.86	−1.25762	−0.628808	+	0.777561i	$0.716457\pi$
$278$	− 2353.73i	− 0.507796i
$279$	2007.73i	0.430823i
$280$	3752.15i	0.800836i
$281$	− 2584.43i	− 0.548662i	−0.961635	−	0.274331i	$-0.911544\pi$
$281$	− 2584.43i	− 0.548662i	0.961635	−	0.274331i	$-0.0884564\pi$
$282$	−497.030	−0.104956
$283$	4491.70	0.943477	0.471738	−	0.881739i	$-0.343627\pi$
$283$	4491.70	0.943477	0.471738	+	0.881739i	$0.343627\pi$
$284$	1057.93i	0.221045i
$285$	4980.14	1.03508
$286$	0	0
$287$	−9731.63	−2.00153
$288$	− 288.000i	− 0.0589256i
$289$	8519.75	1.73412
$290$	5937.27	1.20224
$291$	− 4980.68i	− 1.00334i
$292$	3372.04i	0.675801i
$293$	3045.79i	0.607294i	0.952785	+	0.303647i	$0.0982043\pi$
$293$	3045.79i	0.607294i	−0.952785	+	0.303647i	$0.901796\pi$
$294$	2048.24i	0.406311i
$295$	8237.00	1.62568
$296$	−1101.03	−0.216204
$297$	− 336.431i	− 0.0657297i
$298$	357.186	0.0694336
$299$	0	0
$300$	2357.18	0.453639
$301$	3441.85i	0.659086i
$302$	−3019.38	−0.575317
$303$	3256.97	0.617518
$304$	− 1481.48i	− 0.279502i
$305$	− 6744.76i	− 1.26624i
$306$	− 2086.20i	− 0.389738i
$307$	4630.13i	0.860767i	0.902646	+	0.430384i	$0.141622\pi$
$307$	4630.13i	0.860767i	−0.902646	+	0.430384i	$0.858378\pi$
$308$	1303.88	0.241220
$309$	1511.34	0.278243
$310$	7999.03i	1.46553i
$311$	4333.04	0.790045	0.395023	−	0.918671i	$-0.370737\pi$
$311$	4333.04	0.790045	0.395023	+	0.918671i	$0.370737\pi$
$312$	0	0
$313$	−7590.94	−1.37082	−0.685408	−	0.728159i	$-0.740376\pi$
$313$	−7590.94	−1.37082	−0.685408	+	0.728159i	$0.740376\pi$
$314$	777.054i	0.139655i
$315$	4221.17	0.755035
$316$	−3477.94	−0.619144
$317$	6928.13i	1.22752i	0.789494	+	0.613758i	$0.210343\pi$
$317$	6928.13i	1.22752i	−0.789494	+	0.613758i	$0.789657\pi$
$318$	− 2600.93i	− 0.458658i
$319$	− 2063.22i	− 0.362125i
$320$	− 1147.42i	− 0.200447i
$321$	2964.24	0.515414
$322$	10657.0	1.84438
$323$	− 10731.5i	− 1.84865i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	2893.88	0.491647
$327$	− 1079.52i	− 0.182562i
$328$	2975.97	0.500978
$329$	−2167.10	−0.363148
$330$	− 1340.38i	− 0.223592i
$331$	− 2341.39i	− 0.388805i	−0.980922	−	0.194402i	$-0.937723\pi$
$331$	− 2341.39i	− 0.388805i	0.980922	−	0.194402i	$-0.0622767\pi$
$332$	3518.25i	0.581593i
$333$	1238.66i	0.203839i
$334$	−3072.45	−0.503344
$335$	3703.53	0.604016
$336$	− 1255.70i	− 0.203882i
$337$	−3112.36	−0.503089	−0.251545	−	0.967846i	$-0.580939\pi$
$337$	−3112.36	−0.503089	−0.251545	+	0.967846i	$0.580939\pi$
$338$	0	0
$339$	6042.70	0.968125
$340$	− 8311.64i	− 1.32577i
$341$	2779.68	0.441432
$342$	−1666.67	−0.263517
$343$	− 42.5744i	− 0.00670204i
$344$	− 1052.53i	− 0.164967i
$345$	− 10955.3i	− 1.70960i
$346$	− 1531.27i	− 0.237924i
$347$	−2913.56	−0.450744	−0.225372	−	0.974273i	$-0.572360\pi$
$347$	−2913.56	−0.450744	−0.225372	+	0.974273i	$0.572360\pi$
$348$	−1986.98	−0.306073
$349$	7300.80i	1.11978i	0.828567	+	0.559890i	$0.189156\pi$
$349$	7300.80i	1.11978i	−0.828567	+	0.559890i	$0.810844\pi$
$350$	10277.5	1.56959
$351$	0	0
$352$	−398.733	−0.0603766
$353$	5352.93i	0.807104i	0.914957	+	0.403552i	$0.132225\pi$
$353$	5352.93i	0.807104i	−0.914957	+	0.403552i	$0.867775\pi$
$354$	−2756.62	−0.413877
$355$	4741.78	0.708923
$356$	645.206i	0.0960558i
$357$	− 9095.99i	− 1.34849i
$358$	8860.44i	1.30807i
$359$	7301.69i	1.07345i	0.843757	+	0.536725i	$0.180339\pi$
$359$	7301.69i	1.07345i	−0.843757	+	0.536725i	$0.819661\pi$
$360$	−1290.85	−0.188983
$361$	−1714.38	−0.249946
$362$	8007.28i	1.16258i
$363$	3527.21	0.510002
$364$	0	0
$365$	15113.9	2.16740
$366$	2257.22i	0.322368i
$367$	−7323.28	−1.04161	−0.520807	−	0.853675i	$-0.674369\pi$
$367$	−7323.28	−1.04161	−0.520807	+	0.853675i	$0.674369\pi$
$368$	−3258.95	−0.461643
$369$	− 3347.97i	− 0.472326i
$370$	4934.97i	0.693397i
$371$	− 11340.3i	− 1.58695i
$372$	− 2676.98i	− 0.373104i
$373$	−8160.03	−1.13274	−0.566368	−	0.824153i	$-0.691652\pi$
$373$	−8160.03	−1.13274	−0.566368	+	0.824153i	$0.691652\pi$
$374$	−2888.32	−0.399335
$375$	− 3841.98i	− 0.529063i
$376$	662.707	0.0908950
$377$	0	0
$378$	−1412.67	−0.192222
$379$	− 3082.49i	− 0.417775i	−0.977940	−	0.208887i	$-0.933016\pi$
$379$	− 3082.49i	− 0.417775i	0.977940	−	0.208887i	$-0.0669842\pi$
$380$	−6640.19	−0.896406
$381$	6364.30	0.855782
$382$	3640.37i	0.487585i
$383$	− 2486.73i	− 0.331765i	−0.986146	−	0.165882i	$-0.946953\pi$
$383$	− 2486.73i	− 0.331765i	0.986146	−	0.165882i	$-0.0530472\pi$
$384$	384.000i	0.0510310i
$385$	− 5844.17i	− 0.773627i
$386$	4314.15	0.568872
$387$	−1184.10	−0.155533
$388$	6640.91i	0.868920i
$389$	−1280.65	−0.166919	−0.0834595	−	0.996511i	$-0.526597\pi$
$389$	−1280.65	−0.166919	−0.0834595	+	0.996511i	$0.526597\pi$
$390$	0	0
$391$	−23607.0	−3.05334
$392$	− 2730.98i	− 0.351876i
$393$	−2214.07	−0.284186
$394$	4067.87	0.520143
$395$	15588.6i	1.98569i
$396$	448.575i	0.0569236i
$397$	1229.41i	0.155421i	0.996976	+	0.0777107i	$0.0247611\pi$
$397$	1229.41i	0.155421i	−0.996976	+	0.0777107i	$0.975239\pi$
$398$	− 8770.74i	− 1.10462i
$399$	−7266.81	−0.911768
$400$	−3142.90	−0.392863
$401$	14066.3i	1.75171i	0.482570	+	0.875857i	$0.339703\pi$
$401$	14066.3i	1.75171i	−0.482570	+	0.875857i	$0.660297\pi$
$402$	−1239.43	−0.153774
$403$	0	0
$404$	−4342.63	−0.534786
$405$	1452.21i	0.178175i
$406$	−8663.41	−1.05901
$407$	1714.92	0.208858
$408$	2781.59i	0.337523i
$409$	− 2346.26i	− 0.283655i	−0.989891	−	0.141827i	$-0.954702\pi$
$409$	− 2346.26i	− 0.283655i	0.989891	−	0.141827i	$-0.0452978\pi$
$410$	− 13338.7i	− 1.60671i
$411$	− 50.5851i	− 0.00607099i
$412$	−2015.12	−0.240966
$413$	−12019.1	−1.43201
$414$	3666.32i	0.435241i
$415$	15769.2	1.86526
$416$	0	0
$417$	3530.59	0.414614
$418$	2307.48i	0.270006i
$419$	13040.4	1.52044	0.760218	−	0.649668i	$-0.225092\pi$
$419$	13040.4	1.52044	0.760218	+	0.649668i	$0.225092\pi$
$420$	−5628.23	−0.653880
$421$	− 3468.10i	− 0.401484i	−0.979644	−	0.200742i	$-0.935665\pi$
$421$	− 3468.10i	− 0.401484i	0.979644	−	0.200742i	$-0.0643354\pi$
$422$	8509.66i	0.981620i
$423$	− 745.546i	− 0.0856966i
$424$	3467.91i	0.397209i
$425$	−22766.4	−2.59842
$426$	−1586.90	−0.180482
$427$	9841.67i	1.11539i
$428$	−3952.32	−0.446361
$429$	0	0
$430$	−4717.58	−0.529075
$431$	3001.30i	0.335423i	0.985836	+	0.167712i	$0.0536377\pi$
$431$	3001.30i	0.335423i	−0.985836	+	0.167712i	$0.946362\pi$
$432$	432.000	0.0481125
$433$	9861.25	1.09446	0.547230	−	0.836982i	$-0.315682\pi$
$433$	9861.25	1.09446	0.547230	+	0.836982i	$0.315682\pi$
$434$	− 11671.8i	− 1.29094i
$435$	8905.91i	0.981622i
$436$	1439.36i	0.158103i
$437$	18859.7i	2.06449i
$438$	−5058.07	−0.551789
$439$	7325.65	0.796433	0.398216	−	0.917291i	$-0.369629\pi$
$439$	7325.65	0.796433	0.398216	+	0.917291i	$0.369629\pi$
$440$	1787.17i	0.193637i
$441$	−3072.35	−0.331752
$442$	0	0
$443$	5043.57	0.540919	0.270459	−	0.962731i	$-0.412824\pi$
$443$	5043.57	0.540919	0.270459	+	0.962731i	$0.412824\pi$
$444$	− 1651.55i	− 0.176529i
$445$	2891.90	0.308065
$446$	243.230	0.0258235
$447$	535.779i	0.0566923i
$448$	1674.27i	0.176567i
$449$	− 11135.2i	− 1.17038i	−0.810896	−	0.585191i	$-0.801020\pi$
$449$	− 11135.2i	− 1.17038i	0.810896	−	0.585191i	$-0.198980\pi$
$450$	3535.77i	0.370395i
$451$	−4635.23	−0.483957
$452$	−8056.93	−0.838421
$453$	− 4529.07i	− 0.469744i
$454$	−3742.28	−0.386859
$455$	0	0
$456$	2222.22	0.228213
$457$	9454.44i	0.967747i	0.875138	+	0.483873i	$0.160770\pi$
$457$	9454.44i	0.967747i	−0.875138	+	0.483873i	$0.839230\pi$
$458$	5998.29	0.611969
$459$	3129.29	0.318220
$460$	14607.0i	1.48056i
$461$	− 16071.8i	− 1.62372i	−0.583849	−	0.811862i	$-0.698454\pi$
$461$	− 16071.8i	− 1.62372i	0.583849	−	0.811862i	$-0.301546\pi$
$462$	1955.82i	0.196955i
$463$	− 14024.1i	− 1.40768i	−0.710360	−	0.703839i	$-0.751468\pi$
$463$	− 14024.1i	− 1.40768i	0.710360	−	0.703839i	$-0.248532\pi$
$464$	2649.31	0.265067
$465$	−11998.5	−1.19660
$466$	3486.03i	0.346539i
$467$	4312.46	0.427316	0.213658	−	0.976908i	$-0.431462\pi$
$467$	4312.46	0.427316	0.213658	+	0.976908i	$0.431462\pi$
$468$	0	0
$469$	−5404.03	−0.532058
$470$	− 2970.34i	− 0.291514i
$471$	−1165.58	−0.114028
$472$	3675.49	0.358428
$473$	1639.37i	0.159362i
$474$	− 5216.92i	− 0.505529i
$475$	18188.1i	1.75690i
$476$	12128.0i	1.16783i
$477$	3901.40	0.374493
$478$	−10531.7	−1.00776
$479$	7401.52i	0.706021i	0.935619	+	0.353011i	$0.114842\pi$
$479$	7401.52i	0.706021i	−0.935619	+	0.353011i	$0.885158\pi$
$480$	1721.14	0.163664
$481$	0	0
$482$	10637.4	1.00523
$483$	15985.5i	1.50593i
$484$	−4702.95	−0.441675
$485$	29765.4	2.78676
$486$	− 486.000i	− 0.0453609i
$487$	2486.76i	0.231388i	0.993285	+	0.115694i	$0.0369091\pi$
$487$	2486.76i	0.231388i	−0.993285	+	0.115694i	$0.963091\pi$
$488$	− 3009.63i	− 0.279179i
$489$	4340.82i	0.401428i
$490$	−12240.6	−1.12852
$491$	4043.19	0.371622	0.185811	−	0.982585i	$-0.440509\pi$
$491$	4043.19	0.371622	0.185811	+	0.982585i	$0.440509\pi$
$492$	4463.96i	0.409047i
$493$	19190.9	1.75317
$494$	0	0
$495$	2010.57	0.182562
$496$	3569.30i	0.323118i
$497$	−6919.01	−0.624467
$498$	−5277.37	−0.474869
$499$	15860.0i	1.42283i	0.702775	+	0.711413i	$0.251944\pi$
$499$	15860.0i	1.42283i	−0.702775	+	0.711413i	$0.748056\pi$
$500$	5122.63i	0.458182i
$501$	− 4608.67i	− 0.410979i
$502$	11366.6i	1.01059i
$503$	10757.5	0.953584	0.476792	−	0.879016i	$-0.341800\pi$
$503$	10757.5	0.953584	0.476792	+	0.879016i	$0.341800\pi$
$504$	1883.56	0.166469
$505$	19464.2i	1.71514i
$506$	5075.98	0.445958
$507$	0	0
$508$	−8485.73	−0.741129
$509$	− 12830.6i	− 1.11730i	−0.829403	−	0.558650i	$-0.811319\pi$
$509$	− 12830.6i	− 1.11730i	0.829403	−	0.558650i	$-0.188681\pi$
$510$	12467.5	1.08249
$511$	−22053.6	−1.90919
$512$	− 512.000i	− 0.0441942i
$513$	− 2500.00i	− 0.215161i
$514$	10493.9i	0.900516i
$515$	9032.03i	0.772813i
$516$	1578.80	0.134695
$517$	−1032.20	−0.0878068
$518$	− 7200.90i	− 0.610790i
$519$	2296.91	0.194264
$520$	0	0
$521$	−12212.1	−1.02692	−0.513458	−	0.858115i	$-0.671636\pi$
$521$	−12212.1	−1.02692	−0.513458	+	0.858115i	$0.671636\pi$
$522$	− 2980.47i	− 0.249908i
$523$	−11486.4	−0.960354	−0.480177	−	0.877171i	$-0.659428\pi$
$523$	−11486.4	−0.960354	−0.480177	+	0.877171i	$0.659428\pi$
$524$	2952.10	0.246113
$525$	15416.2i	1.28156i
$526$	− 10448.6i	− 0.866126i
$527$	25855.1i	2.13712i
$528$	− 598.100i	− 0.0492972i
$529$	29320.4	2.40983
$530$	15543.6	1.27391
$531$	− 4134.92i	− 0.337929i
$532$	9689.08	0.789614
$533$	0	0
$534$	−967.809	−0.0784292
$535$	17714.8i	1.43155i
$536$	1652.58	0.133172
$537$	−13290.7	−1.06803
$538$	− 6004.65i	− 0.481187i
$539$	4253.64i	0.339921i
$540$	− 1936.28i	− 0.154304i
$541$	− 15390.3i	− 1.22307i	−0.791217	−	0.611535i	$-0.790552\pi$
$541$	− 15390.3i	− 1.22307i	0.791217	−	0.611535i	$-0.209448\pi$
$542$	−6051.12	−0.479554
$543$	−12010.9	−0.949241
$544$	− 3708.79i	− 0.292304i
$545$	6451.42	0.507061
$546$	0	0
$547$	−1031.47	−0.0806260	−0.0403130	−	0.999187i	$-0.512836\pi$
$547$	−1031.47	−0.0806260	−0.0403130	+	0.999187i	$0.512836\pi$
$548$	67.4468i	0.00525763i
$549$	−3385.83	−0.263212
$550$	4895.23	0.379515
$551$	− 15331.6i	− 1.18539i
$552$	− 4888.43i	− 0.376930i
$553$	− 22746.2i	− 1.74913i
$554$	11595.7i	0.889269i
$555$	−7402.45	−0.566156
$556$	−4707.46	−0.359066
$557$	− 18196.9i	− 1.38425i	−0.721777	−	0.692126i	$-0.756674\pi$
$557$	− 18196.9i	− 1.38425i	0.721777	−	0.692126i	$-0.243326\pi$
$558$	4015.46	0.304638
$559$	0	0
$560$	7504.30	0.566276
$561$	− 4332.47i	− 0.326056i
$562$	−5168.85	−0.387963
$563$	−15461.2	−1.15740	−0.578698	−	0.815542i	$-0.696439\pi$
$563$	−15461.2	−1.15740	−0.578698	+	0.815542i	$0.696439\pi$
$564$	994.061i	0.0742154i
$565$	36112.2i	2.68894i
$566$	− 8983.40i	− 0.667139i
$567$	− 2119.00i	− 0.156948i
$568$	2115.86	0.156302
$569$	15190.7	1.11920	0.559601	−	0.828762i	$-0.310954\pi$
$569$	15190.7	1.11920	0.559601	+	0.828762i	$0.310954\pi$
$570$	− 9960.28i	− 0.731913i
$571$	−8757.00	−0.641802	−0.320901	−	0.947113i	$-0.603986\pi$
$571$	−8757.00	−0.641802	−0.320901	+	0.947113i	$0.603986\pi$
$572$	0	0
$573$	−5460.56	−0.398112
$574$	19463.3i	1.41530i
$575$	40010.0	2.90180
$576$	−576.000	−0.0416667
$577$	− 13076.4i	− 0.943464i	−0.881742	−	0.471732i	$-0.843629\pi$
$577$	− 13076.4i	− 0.943464i	0.881742	−	0.471732i	$-0.156371\pi$
$578$	− 17039.5i	− 1.22621i
$579$	6471.23i	0.464482i
$580$	− 11874.5i	− 0.850109i
$581$	−23009.8	−1.64304
$582$	−9961.36	−0.709470
$583$	− 5401.45i	− 0.383714i
$584$	6744.09	0.477864
$585$	0	0
$586$	6091.59	0.429422
$587$	− 10601.6i	− 0.745445i	−0.927943	−	0.372722i	$-0.878424\pi$
$587$	− 10601.6i	− 0.745445i	0.927943	−	0.372722i	$-0.121576\pi$
$588$	4096.47	0.287305
$589$	20655.7	1.44499
$590$	− 16474.0i	− 1.14953i
$591$	6101.80i	0.424695i
$592$	2202.07i	0.152879i
$593$	13992.0i	0.968941i	0.874807	+	0.484471i	$0.160988\pi$
$593$	13992.0i	0.968941i	−0.874807	+	0.484471i	$0.839012\pi$
$594$	−672.862	−0.0464779
$595$	54359.2	3.74539
$596$	− 714.372i	− 0.0490970i
$597$	13156.1	0.901916
$598$	0	0
$599$	−5575.15	−0.380291	−0.190146	−	0.981756i	$-0.560896\pi$
$599$	−5575.15	−0.380291	−0.190146	+	0.981756i	$0.560896\pi$
$600$	− 4714.35i	− 0.320771i
$601$	26024.1	1.76630	0.883149	−	0.469092i	$-0.155419\pi$
$601$	26024.1	1.76630	0.883149	+	0.469092i	$0.155419\pi$
$602$	6883.70	0.466044
$603$	− 1859.15i	− 0.125556i
$604$	6038.76i	0.406811i
$605$	21079.2i	1.41652i
$606$	− 6513.94i	− 0.436651i
$607$	−1387.68	−0.0927913	−0.0463956	−	0.998923i	$-0.514773\pi$
$607$	−1387.68	−0.0927913	−0.0463956	+	0.998923i	$0.514773\pi$
$608$	−2962.96	−0.197638
$609$	− 12995.1i	− 0.864678i
$610$	−13489.5	−0.895369
$611$	0	0
$612$	−4172.39	−0.275586
$613$	23095.8i	1.52174i	0.648902	+	0.760872i	$0.275229\pi$
$613$	23095.8i	1.52174i	−0.648902	+	0.760872i	$0.724771\pi$
$614$	9260.26	0.608654
$615$	20008.0	1.31187
$616$	− 2607.77i	− 0.170568i
$617$	6824.78i	0.445309i	0.974897	+	0.222654i	$0.0714721\pi$
$617$	6824.78i	0.445309i	−0.974897	+	0.222654i	$0.928528\pi$
$618$	− 3022.68i	− 0.196748i
$619$	28173.0i	1.82935i	0.404185	+	0.914677i	$0.367555\pi$
$619$	28173.0i	1.82935i	−0.404185	+	0.914677i	$0.632445\pi$
$620$	15998.1	1.03629
$621$	−5499.48	−0.355373
$622$	− 8666.08i	− 0.558646i
$623$	−4219.73	−0.271364
$624$	0	0
$625$	−1593.63	−0.101992
$626$	15181.9i	0.969313i
$627$	−3461.22	−0.220459
$628$	1554.11	0.0987511
$629$	15951.2i	1.01115i
$630$	− 8442.34i	− 0.533890i
$631$	− 2134.38i	− 0.134657i	−0.997731	−	0.0673284i	$-0.978552\pi$
$631$	− 2134.38i	− 0.134657i	0.997731	−	0.0673284i	$-0.0214475\pi$
$632$	6955.89i	0.437801i
$633$	−12764.5	−0.801489
$634$	13856.3	0.867985
$635$	38034.1i	2.37691i
$636$	−5201.87	−0.324320
$637$	0	0
$638$	−4126.44	−0.256061
$639$	− 2380.35i	− 0.147363i
$640$	−2294.85	−0.141737
$641$	−22227.7	−1.36965	−0.684823	−	0.728709i	$-0.740120\pi$
$641$	−22227.7	−1.36965	−0.684823	+	0.728709i	$0.740120\pi$
$642$	− 5928.48i	− 0.364452i
$643$	− 7837.12i	− 0.480662i	−0.970691	−	0.240331i	$-0.922744\pi$
$643$	− 7837.12i	− 0.480662i	0.970691	−	0.240331i	$-0.0772561\pi$
$644$	− 21314.0i	− 1.30417i
$645$	− 7076.37i	− 0.431988i
$646$	−21462.9	−1.30719
$647$	10032.5	0.609609	0.304805	−	0.952415i	$-0.401409\pi$
$647$	10032.5	0.609609	0.304805	+	0.952415i	$0.401409\pi$
$648$	648.000i	0.0392837i
$649$	−5724.76	−0.346250
$650$	0	0
$651$	17507.8	1.05405
$652$	− 5787.75i	− 0.347647i
$653$	−14044.9	−0.841681	−0.420841	−	0.907135i	$-0.638265\pi$
$653$	−14044.9	−0.841681	−0.420841	+	0.907135i	$0.638265\pi$
$654$	−2159.05	−0.129091
$655$	− 13231.7i	− 0.789320i
$656$	− 5951.95i	− 0.354245i
$657$	− 7587.10i	− 0.450534i
$658$	4334.19i	0.256785i
$659$	10782.8	0.637388	0.318694	−	0.947858i	$-0.396756\pi$
$659$	10782.8	0.637388	0.318694	+	0.947858i	$0.396756\pi$
$660$	−2680.76	−0.158104
$661$	17732.3i	1.04343i	0.853120	+	0.521714i	$0.174707\pi$
$661$	17732.3i	1.04343i	−0.853120	+	0.521714i	$0.825293\pi$
$662$	−4682.78	−0.274926
$663$	0	0
$664$	7036.50	0.411249
$665$	− 43427.7i	− 2.53241i
$666$	2477.32	0.144136
$667$	−33726.4	−1.95786
$668$	6144.90i	0.355918i
$669$	364.845i	0.0210848i
$670$	− 7407.06i	− 0.427104i
$671$	4687.65i	0.269694i
$672$	−2511.41	−0.144166
$673$	−20509.1	−1.17469	−0.587347	−	0.809335i	$-0.699828\pi$
$673$	−20509.1	−1.17469	−0.587347	+	0.809335i	$0.699828\pi$
$674$	6224.72i	0.355738i
$675$	−5303.65	−0.302426
$676$	0	0
$677$	−8840.45	−0.501870	−0.250935	−	0.968004i	$-0.580738\pi$
$677$	−8840.45	−0.501870	−0.250935	+	0.968004i	$0.580738\pi$
$678$	− 12085.4i	− 0.684568i
$679$	−43432.4	−2.45476
$680$	−16623.3	−0.937461
$681$	− 5613.42i	− 0.315869i
$682$	− 5559.37i	− 0.312140i
$683$	10339.3i	0.579241i	0.957142	+	0.289620i	$0.0935290\pi$
$683$	10339.3i	0.579241i	−0.957142	+	0.289620i	$0.906471\pi$
$684$	3333.33i	0.186335i
$685$	302.305	0.0168620
$686$	−85.1488	−0.00473906
$687$	8997.43i	0.499670i
$688$	−2105.06	−0.116649
$689$	0	0
$690$	−21910.6	−1.20887
$691$	− 3555.81i	− 0.195759i	−0.995198	−	0.0978795i	$-0.968794\pi$
$691$	− 3555.81i	− 0.195759i	0.995198	−	0.0978795i	$-0.0312060\pi$
$692$	−3062.54	−0.168238
$693$	−2933.74	−0.160813
$694$	5827.12i	0.318724i
$695$	21099.4i	1.15158i
$696$	3973.96i	0.216426i
$697$	− 43114.3i	− 2.34300i
$698$	14601.6	0.791803
$699$	−5229.05	−0.282948
$700$	− 20555.0i	− 1.10987i
$701$	7279.02	0.392190	0.196095	−	0.980585i	$-0.437174\pi$
$701$	7279.02	0.392190	0.196095	+	0.980585i	$0.437174\pi$
$702$	0	0
$703$	12743.4	0.683681
$704$	797.466i	0.0426927i
$705$	4455.51	0.238020
$706$	10705.9	0.570709
$707$	− 28401.3i	− 1.51081i
$708$	5513.23i	0.292655i
$709$	− 16199.9i	− 0.858112i	−0.903278	−	0.429056i	$-0.858846\pi$
$709$	− 16199.9i	− 0.858112i	0.903278	−	0.429056i	$-0.141154\pi$
$710$	− 9483.57i	− 0.501285i
$711$	7825.37	0.412763
$712$	1290.41	0.0679217
$713$	− 45438.2i	− 2.38664i
$714$	−18192.0	−0.953526
$715$	0	0
$716$	17720.9	0.924945
$717$	− 15797.6i	− 0.822834i
$718$	14603.4	0.759043
$719$	−21366.6	−1.10826	−0.554131	−	0.832430i	$-0.686949\pi$
$719$	−21366.6	−1.10826	−0.554131	+	0.832430i	$0.686949\pi$
$720$	2581.71i	0.133631i
$721$	− 13179.2i	− 0.680745i
$722$	3428.76i	0.176739i
$723$	15956.1i	0.820764i
$724$	16014.6	0.822067
$725$	−32525.5	−1.66616
$726$	− 7054.43i	− 0.360626i
$727$	4997.32	0.254939	0.127469	−	0.991843i	$-0.459315\pi$
$727$	4997.32	0.254939	0.127469	+	0.991843i	$0.459315\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 30227.9i	− 1.53258i
$731$	−15248.5	−0.771528
$732$	4514.44	0.227949
$733$	25015.4i	1.26052i	0.776382	+	0.630262i	$0.217053\pi$
$733$	25015.4i	1.26052i	−0.776382	+	0.630262i	$0.782947\pi$
$734$	14646.6i	0.736532i
$735$	− 18360.9i	− 0.921431i
$736$	6517.90i	0.326431i
$737$	−2573.97	−0.128648
$738$	−6695.94	−0.333985
$739$	24979.9i	1.24344i	0.783240	+	0.621719i	$0.213566\pi$
$739$	24979.9i	1.24344i	−0.783240	+	0.621719i	$0.786434\pi$
$740$	9869.94	0.490306
$741$	0	0
$742$	−22680.6	−1.12214
$743$	− 18100.6i	− 0.893737i	−0.894600	−	0.446868i	$-0.852539\pi$
$743$	− 18100.6i	− 0.893737i	0.894600	−	0.446868i	$-0.147461\pi$
$744$	−5353.95	−0.263824
$745$	−3201.91	−0.157461
$746$	16320.1i	0.800965i
$747$	− 7916.06i	− 0.387729i
$748$	5776.63i	0.282372i
$749$	− 25848.7i	− 1.26100i
$750$	−7683.95	−0.374104
$751$	−8778.40	−0.426536	−0.213268	−	0.976994i	$-0.568411\pi$
$751$	−8778.40	−0.426536	−0.213268	+	0.976994i	$0.568411\pi$
$752$	− 1325.41i	− 0.0642725i
$753$	−17049.8	−0.825141
$754$	0	0
$755$	27066.5	1.30470
$756$	2825.34i	0.135921i
$757$	−37514.1	−1.80115	−0.900576	−	0.434699i	$-0.856855\pi$
$757$	−37514.1	−1.80115	−0.900576	+	0.434699i	$0.856855\pi$
$758$	−6164.97	−0.295411
$759$	7613.98i	0.364124i
$760$	13280.4i	0.633855i
$761$	− 1797.41i	− 0.0856190i	−0.999083	−	0.0428095i	$-0.986369\pi$
$761$	− 1797.41i	− 0.0856190i	0.999083	−	0.0428095i	$-0.0136309\pi$
$762$	− 12728.6i	− 0.605129i
$763$	−9413.63	−0.446653
$764$	7280.74	0.344775
$765$	18701.2i	0.883847i
$766$	−4973.45	−0.234593
$767$	0	0
$768$	768.000	0.0360844
$769$	− 3966.45i	− 0.186000i	−0.995666	−	0.0930000i	$-0.970354\pi$
$769$	− 3966.45i	− 0.186000i	0.995666	−	0.0930000i	$-0.0296457\pi$
$770$	−11688.3	−0.547037
$771$	−15740.8	−0.735269
$772$	− 8628.30i	− 0.402253i
$773$	− 6714.61i	− 0.312429i	−0.987723	−	0.156215i	$-0.950071\pi$
$773$	− 6714.61i	− 0.312429i	0.987723	−	0.156215i	$-0.0499292\pi$
$774$	2368.20i	0.109978i
$775$	− 43820.2i	− 2.03105i
$776$	13281.8	0.614419
$777$	10801.3	0.498708
$778$	2561.30i	0.118030i
$779$	−34444.1	−1.58420
$780$	0	0
$781$	−3295.57	−0.150992
$782$	47214.0i	2.15904i
$783$	4470.71	0.204049
$784$	−5461.96	−0.248814
$785$	− 6965.71i	− 0.316710i
$786$	4428.15i	0.200950i
$787$	− 31945.0i	− 1.44691i	−0.690373	−	0.723454i	$-0.742553\pi$
$787$	− 31945.0i	− 1.44691i	0.690373	−	0.723454i	$-0.257447\pi$
$788$	− 8135.73i	− 0.367796i
$789$	15673.0	0.707189
$790$	31177.2	1.40409
$791$	− 52693.4i	− 2.36860i
$792$	897.149	0.0402510
$793$	0	0
$794$	2458.82	0.109900
$795$	23315.4i	1.04014i
$796$	−17541.5	−0.781082
$797$	20388.6	0.906152	0.453076	−	0.891472i	$-0.350327\pi$
$797$	20388.6	0.906152	0.453076	+	0.891472i	$0.350327\pi$
$798$	14533.6i	0.644717i
$799$	− 9600.95i	− 0.425103i
$800$	6285.81i	0.277796i
$801$	− 1451.71i	− 0.0640372i
$802$	28132.6	1.23865
$803$	−10504.3	−0.461628
$804$	2478.87i	0.108735i
$805$	−95531.9	−4.18268
$806$	0	0
$807$	9006.97	0.392888
$808$	8685.25i	0.378151i
$809$	4142.56	0.180031	0.0900153	−	0.995940i	$-0.471308\pi$
$809$	4142.56	0.180031	0.0900153	+	0.995940i	$0.471308\pi$
$810$	2904.42	0.125989
$811$	− 2126.48i	− 0.0920727i	−0.998940	−	0.0460364i	$-0.985341\pi$
$811$	− 2126.48i	− 0.0920727i	0.998940	−	0.0460364i	$-0.0146590\pi$
$812$	17326.8i	0.748833i
$813$	− 9076.69i	− 0.391554i
$814$	− 3429.83i	− 0.147685i
$815$	−25941.4	−1.11496
$816$	5563.19	0.238665
$817$	12182.1i	0.521661i
$818$	−4692.51	−0.200574
$819$	0	0
$820$	−26677.4	−1.13612
$821$	4536.13i	0.192828i	0.995341	+	0.0964142i	$0.0307373\pi$
$821$	4536.13i	0.192828i	−0.995341	+	0.0964142i	$0.969263\pi$
$822$	−101.170	−0.00429284
$823$	17618.0	0.746203	0.373101	−	0.927791i	$-0.378294\pi$
$823$	17618.0	0.746203	0.373101	+	0.927791i	$0.378294\pi$
$824$	4030.24i	0.170389i
$825$	7342.85i	0.309873i
$826$	24038.2i	1.01258i
$827$	3071.45i	0.129147i	0.997913	+	0.0645737i	$0.0205687\pi$
$827$	3071.45i	0.129147i	−0.997913	+	0.0645737i	$0.979431\pi$
$828$	7332.64	0.307762
$829$	−25730.4	−1.07799	−0.538996	−	0.842308i	$-0.681196\pi$
$829$	−25730.4	−1.07799	−0.538996	+	0.842308i	$0.681196\pi$
$830$	− 31538.5i	− 1.31894i
$831$	−17393.6	−0.726085
$832$	0	0
$833$	−39565.0	−1.64567
$834$	− 7061.19i	− 0.293176i
$835$	27542.2	1.14148
$836$	4614.96	0.190923
$837$	6023.19i	0.248736i
$838$	− 26080.7i	− 1.07511i
$839$	− 1662.37i	− 0.0684047i	−0.999415	−	0.0342023i	$-0.989111\pi$
$839$	− 1662.37i	− 0.0684047i	0.999415	−	0.0342023i	$-0.0108891\pi$
$840$	11256.5i	0.462363i
$841$	3028.35	0.124169
$842$	−6936.20	−0.283892
$843$	− 7753.28i	− 0.316770i
$844$	17019.3	0.694110
$845$	0	0
$846$	−1491.09	−0.0605967
$847$	− 30757.9i	− 1.24776i
$848$	6935.83	0.280869
$849$	13475.1	0.544717
$850$	45532.7i	1.83736i
$851$	− 28032.9i	− 1.12921i
$852$	3173.80i	0.127620i
$853$	35489.6i	1.42455i	0.701901	+	0.712275i	$0.252335\pi$
$853$	35489.6i	1.42455i	−0.701901	+	0.712275i	$0.747665\pi$
$854$	19683.3	0.788700
$855$	14940.4	0.597604
$856$	7904.64i	0.315625i
$857$	6817.57	0.271743	0.135871	−	0.990726i	$-0.456617\pi$
$857$	6817.57	0.271743	0.135871	+	0.990726i	$0.456617\pi$
$858$	0	0
$859$	34112.6	1.35496	0.677478	−	0.735543i	$-0.263073\pi$
$859$	34112.6	1.35496	0.677478	+	0.735543i	$0.263073\pi$
$860$	9435.17i	0.374112i
$861$	−29194.9	−1.15559
$862$	6002.59	0.237180
$863$	− 36591.4i	− 1.44332i	−0.692247	−	0.721660i	$-0.743379\pi$
$863$	− 36591.4i	− 1.44332i	0.692247	−	0.721660i	$-0.256621\pi$
$864$	− 864.000i	− 0.0340207i
$865$	13726.7i	0.539563i
$866$	− 19722.5i	− 0.773900i
$867$	25559.3	1.00120
$868$	−23343.7	−0.912830
$869$	− 10834.2i	− 0.422927i
$870$	17811.8	0.694111
$871$	0	0
$872$	2878.73	0.111796
$873$	− 14942.0i	− 0.579280i
$874$	37719.3	1.45981
$875$	−33502.7	−1.29440
$876$	10116.1i	0.390174i
$877$	46296.7i	1.78258i	0.453429	+	0.891292i	$0.350201\pi$
$877$	46296.7i	1.78258i	−0.453429	+	0.891292i	$0.649799\pi$
$878$	− 14651.3i	− 0.563163i
$879$	9137.38i	0.350621i
$880$	3574.34	0.136922
$881$	−1470.73	−0.0562432	−0.0281216	−	0.999605i	$-0.508953\pi$
$881$	−1470.73	−0.0562432	−0.0281216	+	0.999605i	$0.508953\pi$
$882$	6144.71i	0.234584i
$883$	28008.7	1.06746	0.533731	−	0.845654i	$-0.320789\pi$
$883$	28008.7	1.06746	0.533731	+	0.845654i	$0.320789\pi$
$884$	0	0
$885$	24711.0	0.938589
$886$	− 10087.1i	− 0.382487i
$887$	16063.6	0.608076	0.304038	−	0.952660i	$-0.401665\pi$
$887$	16063.6	0.608076	0.304038	+	0.952660i	$0.401665\pi$
$888$	−3303.10	−0.124825
$889$	− 55497.8i	− 2.09374i
$890$	− 5783.79i	− 0.217835i
$891$	− 1009.29i	− 0.0379490i
$892$	− 486.459i	− 0.0182599i
$893$	−7670.22	−0.287429
$894$	1071.56	0.0400875
$895$	− 79427.2i	− 2.96643i
$896$	3348.55	0.124852
$897$	0	0
$898$	−22270.3	−0.827585
$899$	36938.2i	1.37037i
$900$	7071.53	0.261909
$901$	50241.3	1.85769
$902$	9270.47i	0.342209i
$903$	10325.5i	0.380523i
$904$	16113.9i	0.592853i
$905$	− 71779.3i	− 2.63649i
$906$	−9058.14	−0.332159
$907$	25743.6	0.942450	0.471225	−	0.882013i	$-0.343812\pi$
$907$	25743.6	0.942450	0.471225	+	0.882013i	$0.343812\pi$
$908$	7484.57i	0.273551i
$909$	9770.91	0.356524
$910$	0	0
$911$	28512.1	1.03694	0.518468	−	0.855097i	$-0.326502\pi$
$911$	28512.1	1.03694	0.518468	+	0.855097i	$0.326502\pi$
$912$	− 4444.44i	− 0.161371i
$913$	−10959.7	−0.397276
$914$	18908.9	0.684300
$915$	− 20234.3i	− 0.731065i
$916$	− 11996.6i	− 0.432727i
$917$	19307.1i	0.695286i
$918$	− 6258.59i	− 0.225015i
$919$	−342.842	−0.0123061	−0.00615306	−	0.999981i	$-0.501959\pi$
$919$	−342.842	−0.0123061	−0.00615306	+	0.999981i	$0.501959\pi$
$920$	29214.1	1.04691
$921$	13890.4i	0.496964i
$922$	−32143.5	−1.14815
$923$	0	0
$924$	3911.65	0.139268
$925$	− 27034.7i	− 0.960967i
$926$	−28048.2	−0.995378
$927$	4534.02	0.160644
$928$	− 5298.62i	− 0.187431i
$929$	− 28384.4i	− 1.00244i	−0.865321	−	0.501218i	$-0.832885\pi$
$929$	− 28384.4i	− 1.00244i	0.865321	−	0.501218i	$-0.167115\pi$
$930$	23997.1i	0.846124i
$931$	31608.6i	1.11271i
$932$	6972.06	0.245040
$933$	12999.1	0.456133
$934$	− 8624.91i	− 0.302158i
$935$	25891.6	0.905611
$936$	0	0
$937$	−3889.74	−0.135616	−0.0678080	−	0.997698i	$-0.521601\pi$
$937$	−3889.74	−0.135616	−0.0678080	+	0.997698i	$0.521601\pi$
$938$	10808.1i	0.376222i
$939$	−22772.8	−0.791441
$940$	−5940.68	−0.206131
$941$	− 5939.11i	− 0.205749i	−0.994694	−	0.102874i	$-0.967196\pi$
$941$	− 5939.11i	− 0.205749i	0.994694	−	0.102874i	$-0.0328040\pi$
$942$	2331.16i	0.0806299i
$943$	75770.0i	2.61655i
$944$	− 7350.97i	− 0.253447i
$945$	12663.5	0.435920
$946$	3278.75	0.112686
$947$	20529.3i	0.704447i	0.935916	+	0.352224i	$0.114574\pi$
$947$	20529.3i	0.704447i	−0.935916	+	0.352224i	$0.885426\pi$
$948$	−10433.8	−0.357463
$949$	0	0
$950$	36376.2	1.24231
$951$	20784.4i	0.708707i
$952$	24256.0	0.825778
$953$	−24682.2	−0.838966	−0.419483	−	0.907763i	$-0.637789\pi$
$953$	−24682.2	−0.838966	−0.419483	+	0.907763i	$0.637789\pi$
$954$	− 7802.80i	− 0.264806i
$955$	− 32633.2i	− 1.10574i
$956$	21063.5i	0.712595i
$957$	− 6189.65i	− 0.209073i
$958$	14803.0	0.499232
$959$	−441.111	−0.0148532
$960$	− 3442.27i	− 0.115728i
$961$	−19974.2	−0.670479
$962$	0	0
$963$	8892.72	0.297574
$964$	− 21274.8i	− 0.710803i
$965$	−38673.2	−1.29009
$966$	31970.9	1.06485
$967$	4212.27i	0.140080i	0.997544	+	0.0700400i	$0.0223127\pi$
$967$	4212.27i	0.140080i	−0.997544	+	0.0700400i	$0.977687\pi$
$968$	9405.91i	0.312311i
$969$	− 32194.4i	− 1.06732i
$970$	− 59530.8i	− 1.97053i
$971$	−33834.4	−1.11823	−0.559114	−	0.829091i	$-0.688858\pi$
$971$	−33834.4	−1.11823	−0.559114	+	0.829091i	$0.688858\pi$
$972$	−972.000	−0.0320750
$973$	− 30787.4i	− 1.01439i
$974$	4973.52	0.163616
$975$	0	0
$976$	−6019.25	−0.197409
$977$	14119.2i	0.462346i	0.972913	+	0.231173i	$0.0742563\pi$
$977$	14119.2i	0.462346i	−0.972913	+	0.231173i	$0.925744\pi$
$978$	8681.63	0.283853
$979$	−2009.88	−0.0656140
$980$	24481.2i	0.797983i
$981$	− 3238.57i	− 0.105402i
$982$	− 8086.38i	− 0.262777i
$983$	29090.2i	0.943881i	0.881630	+	0.471940i	$0.156446\pi$
$983$	29090.2i	0.943881i	−0.881630	+	0.471940i	$0.843554\pi$
$984$	8927.92	0.289240
$985$	−36465.4	−1.17958
$986$	− 38381.8i	− 1.23968i
$987$	−6501.29	−0.209664
$988$	0	0
$989$	26798.1	0.861606
$990$	− 4021.14i	− 0.129091i
$991$	16640.1	0.533392	0.266696	−	0.963781i	$-0.414068\pi$
$991$	16640.1	0.533392	0.266696	+	0.963781i	$0.414068\pi$
$992$	7138.60	0.228479
$993$	− 7024.16i	− 0.224476i
$994$	13838.0i	0.441565i
$995$	78623.2i	2.50505i
$996$	10554.7i	0.335783i
$997$	998.778	0.0317268	0.0158634	−	0.999874i	$-0.494950\pi$
$997$	998.778	0.0317268	0.0158634	+	0.999874i	$0.494950\pi$
$998$	31719.9	1.00609
$999$	3715.99i	0.117686i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.4.b.q.337.6		12
13.5	odd	4		1014.4.a.bc.1.6	✓	6
13.8	odd	4		1014.4.a.be.1.1	yes	6
13.12	even	2	inner	1014.4.b.q.337.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.4.a.bc.1.6	✓	6	13.5	odd	4
1014.4.a.be.1.1	yes	6	13.8	odd	4
1014.4.b.q.337.6		12	1.1	even	1	trivial
1014.4.b.q.337.7		12	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +17.9285i q^{5} -6.00000i q^{6} -26.1605i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +17.9285i q^{5} -6.00000i q^{6} -26.1605i q^{7} +8.00000i q^{8} +9.00000 q^{9} +35.8570 q^{10} -12.4604i q^{11} -12.0000 q^{12} -52.3210 q^{14} +53.7855i q^{15} +16.0000 q^{16} +115.900 q^{17} -18.0000i q^{18} -92.5926i q^{19} -71.7140i q^{20} -78.4815i q^{21} -24.9208 q^{22} -203.684 q^{23} +24.0000i q^{24} -196.431 q^{25} +27.0000 q^{27} +104.642i q^{28} +165.582 q^{29} +107.571 q^{30} +223.081i q^{31} -32.0000i q^{32} -37.3812i q^{33} -231.799i q^{34} +469.019 q^{35} -36.0000 q^{36} +137.629i q^{37} -185.185 q^{38} -143.428 q^{40} -371.997i q^{41} -156.963 q^{42} -131.567 q^{43} +49.8416i q^{44} +161.357i q^{45} +407.369i q^{46} -82.8384i q^{47} +48.0000 q^{48} -341.373 q^{49} +392.863i q^{50} +347.699 q^{51} +433.489 q^{53} -54.0000i q^{54} +223.397 q^{55} +209.284 q^{56} -277.778i q^{57} -331.164i q^{58} -459.436i q^{59} -215.142i q^{60} -376.203 q^{61} +446.163 q^{62} -235.445i q^{63} -64.0000 q^{64} -74.7625 q^{66} -206.572i q^{67} -463.599 q^{68} -611.053 q^{69} -938.038i q^{70} -264.483i q^{71} +72.0000i q^{72} -843.011i q^{73} +275.258 q^{74} -589.294 q^{75} +370.370i q^{76} -325.971 q^{77} +869.486 q^{79} +286.856i q^{80} +81.0000 q^{81} -743.994 q^{82} -879.562i q^{83} +313.926i q^{84} +2077.91i q^{85} +263.133i q^{86} +496.746 q^{87} +99.6833 q^{88} -161.301i q^{89} +322.713 q^{90} +814.738 q^{92} +669.244i q^{93} -165.677 q^{94} +1660.05 q^{95} -96.0000i q^{96} -1660.23i q^{97} +682.745i q^{98} -112.144i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) \) 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9	\( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) \) 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9 + 12 * q^10 - 144 * q^12 - 4 * q^14 + 192 * q^16 - 102 * q^17 + 256 * q^22 - 444 * q^23 - 370 * q^25 + 324 * q^27 + 658 * q^29 + 36 * q^30 + 1688 * q^35 - 432 * q^36 - 852 * q^38 - 48 * q^40 - 12 * q^42 - 982 * q^43 + 576 * q^48 - 2266 * q^49 - 306 * q^51 + 4604 * q^53 - 658 * q^55 + 16 * q^56 + 690 * q^61 - 1156 * q^62 - 768 * q^64 + 768 * q^66 + 408 * q^68 - 1332 * q^69 + 1880 * q^74 - 1110 * q^75 - 6582 * q^77 + 6200 * q^79 + 972 * q^81 + 1284 * q^82 + 1974 * q^87 - 1024 * q^88 + 108 * q^90 + 1776 * q^92 - 2564 * q^94 + 1330 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more