LMFDB - Embedded newform 325.6.b.b.274.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.4
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.b.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.56155i q^{2} -1.63068i q^{3} +11.1922 q^{4} +7.43845 q^{6} -126.309i q^{7} +197.024i q^{8} +240.341 q^{9} +O(q^{10})$	\(q+4.56155i q^{2} -1.63068i q^{3} +11.1922 q^{4} +7.43845 q^{6} -126.309i q^{7} +197.024i q^{8} +240.341 q^{9} -14.8296 q^{11} -18.2510i q^{12} -169.000i q^{13} +576.164 q^{14} -540.582 q^{16} +1051.12i q^{17} +1096.33i q^{18} +213.723 q^{19} -205.969 q^{21} -67.6458i q^{22} -4231.16i q^{23} +321.283 q^{24} +770.902 q^{26} -788.176i q^{27} -1413.68i q^{28} +504.955 q^{29} +4783.58 q^{31} +3838.86i q^{32} +24.1823i q^{33} -4794.75 q^{34} +2689.95 q^{36} +4635.74i q^{37} +974.911i q^{38} -275.585 q^{39} +7944.15 q^{41} -939.541i q^{42} -8516.41i q^{43} -165.976 q^{44} +19300.7 q^{46} -24921.2i q^{47} +881.518i q^{48} +853.113 q^{49} +1714.05 q^{51} -1891.49i q^{52} -7808.46i q^{53} +3595.31 q^{54} +24885.8 q^{56} -348.515i q^{57} +2303.38i q^{58} +37337.5 q^{59} -18172.2 q^{61} +21820.5i q^{62} -30357.1i q^{63} -34809.8 q^{64} -110.309 q^{66} +34559.9i q^{67} +11764.4i q^{68} -6899.68 q^{69} +41255.7 q^{71} +47352.8i q^{72} -1056.42i q^{73} -21146.2 q^{74} +2392.04 q^{76} +1873.10i q^{77} -1257.10i q^{78} +47719.3 q^{79} +57117.6 q^{81} +36237.7i q^{82} -74799.0i q^{83} -2305.26 q^{84} +38848.0 q^{86} -823.421i q^{87} -2921.78i q^{88} -9799.26 q^{89} -21346.2 q^{91} -47356.1i q^{92} -7800.50i q^{93} +113679. q^{94} +6259.97 q^{96} +138432. i q^{97} +3891.52i q^{98} -3564.15 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9}+O(q^{10}) $ 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9	$ 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9} - 752 q^{11} + 1010 q^{14} + 930 q^{16} + 624 q^{19} + 8148 q^{21} + 2118 q^{24} + 1690 q^{26} + 1624 q^{29} + 15440 q^{31} - 10974 q^{34} - 23396 q^{36} - 9464 q^{39} + 15680 q^{41} - 23308 q^{44} + 37192 q^{46} - 17368 q^{49} + 86696 q^{51} + 2350 q^{54} + 40690 q^{56} + 77872 q^{59} + 3968 q^{61} - 6434 q^{64} - 8572 q^{66} + 70960 q^{69} + 134792 q^{71} - 53010 q^{74} + 11036 q^{76} + 110592 q^{79} + 185524 q^{81} + 267662 q^{84} + 68106 q^{86} - 233016 q^{89} + 12168 q^{91} + 214250 q^{94} + 82986 q^{96} + 319616 q^{99}+O(q^{100}) $ 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9 - 752 * q^11 + 1010 * q^14 + 930 * q^16 + 624 * q^19 + 8148 * q^21 + 2118 * q^24 + 1690 * q^26 + 1624 * q^29 + 15440 * q^31 - 10974 * q^34 - 23396 * q^36 - 9464 * q^39 + 15680 * q^41 - 23308 * q^44 + 37192 * q^46 - 17368 * q^49 + 86696 * q^51 + 2350 * q^54 + 40690 * q^56 + 77872 * q^59 + 3968 * q^61 - 6434 * q^64 - 8572 * q^66 + 70960 * q^69 + 134792 * q^71 - 53010 * q^74 + 11036 * q^76 + 110592 * q^79 + 185524 * q^81 + 267662 * q^84 + 68106 * q^86 - 233016 * q^89 + 12168 * q^91 + 214250 * q^94 + 82986 * q^96 + 319616 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.56155i	0.806376i	0.915117	+	0.403188i	$0.132098\pi$
$2$	4.56155i	0.806376i	−0.915117	+	0.403188i	$0.867902\pi$
$3$	− 1.63068i	− 0.104608i	−0.998631	−	0.0523042i	$-0.983343\pi$
$3$	− 1.63068i	− 0.104608i	0.998631	−	0.0523042i	$-0.0166565\pi$
$4$	11.1922	0.349757
$5$	0	0
$5$	0	0
$6$	7.43845	0.0843537
$7$	− 126.309i	− 0.974290i	−0.873321	−	0.487145i	$-0.838038\pi$
$7$	− 126.309i	− 0.974290i	0.873321	−	0.487145i	$-0.161962\pi$
$8$	197.024i	1.08841i
$9$	240.341	0.989057
$10$	0	0
$11$	−14.8296	−0.0369527	−0.0184764	−	0.999829i	$-0.505882\pi$
$11$	−14.8296	−0.0369527	−0.0184764	+	0.999829i	$0.505882\pi$
$12$	− 18.2510i	− 0.0365875i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	576.164	0.785644
$15$	0	0
$16$	−540.582	−0.527912
$17$	1051.12i	0.882126i	0.897476	+	0.441063i	$0.145398\pi$
$17$	1051.12i	0.882126i	−0.897476	+	0.441063i	$0.854602\pi$
$18$	1096.33i	0.797552i
$19$	213.723	0.135821	0.0679107	−	0.997691i	$-0.478367\pi$
$19$	213.723	0.135821	0.0679107	+	0.997691i	$0.478367\pi$
$20$	0	0
$21$	−205.969	−0.101919
$22$	− 67.6458i	− 0.0297978i
$23$	− 4231.16i	− 1.66778i	−0.551928	−	0.833892i	$-0.686108\pi$
$23$	− 4231.16i	− 1.66778i	0.551928	−	0.833892i	$-0.313892\pi$
$24$	321.283	0.113857
$25$	0	0
$26$	770.902	0.223649
$27$	− 788.176i	− 0.208072i
$28$	− 1413.68i	− 0.340765i
$29$	504.955	0.111495	0.0557477	−	0.998445i	$-0.482246\pi$
$29$	504.955	0.111495	0.0557477	+	0.998445i	$0.482246\pi$
$30$	0	0
$31$	4783.58	0.894022	0.447011	−	0.894528i	$-0.352488\pi$
$31$	4783.58	0.894022	0.447011	+	0.894528i	$0.352488\pi$
$32$	3838.86i	0.662716i
$33$	24.1823i	0.00386557i
$34$	−4794.75	−0.711325
$35$	0	0
$36$	2689.95	0.345930
$37$	4635.74i	0.556692i	0.960481	+	0.278346i	$0.0897862\pi$
$37$	4635.74i	0.556692i	−0.960481	+	0.278346i	$0.910214\pi$
$38$	974.911i	0.109523i
$39$	−275.585	−0.0290131
$40$	0	0
$41$	7944.15	0.738054	0.369027	−	0.929419i	$-0.379691\pi$
$41$	7944.15	0.738054	0.369027	+	0.929419i	$0.379691\pi$
$42$	− 939.541i	− 0.0821850i
$43$	− 8516.41i	− 0.702401i	−0.936300	−	0.351201i	$-0.885774\pi$
$43$	− 8516.41i	− 0.702401i	0.936300	−	0.351201i	$-0.114226\pi$
$44$	−165.976	−0.0129245
$45$	0	0
$46$	19300.7	1.34486
$47$	− 24921.2i	− 1.64560i	−0.568330	−	0.822801i	$-0.692410\pi$
$47$	− 24921.2i	− 1.64560i	0.568330	−	0.822801i	$-0.307590\pi$
$48$	881.518i	0.0552241i
$49$	853.113	0.0507594
$50$	0	0
$51$	1714.05	0.0922777
$52$	− 1891.49i	− 0.0970052i
$53$	− 7808.46i	− 0.381835i	−0.981606	−	0.190917i	$-0.938854\pi$
$53$	− 7808.46i	− 0.381835i	0.981606	−	0.190917i	$-0.0611463\pi$
$54$	3595.31	0.167784
$55$	0	0
$56$	24885.8	1.06043
$57$	− 348.515i	− 0.0142081i
$58$	2303.38i	0.0899073i
$59$	37337.5	1.39642	0.698209	−	0.715894i	$-0.253980\pi$
$59$	37337.5	1.39642	0.698209	+	0.715894i	$0.253980\pi$
$60$	0	0
$61$	−18172.2	−0.625292	−0.312646	−	0.949870i	$-0.601215\pi$
$61$	−18172.2	−0.625292	−0.312646	+	0.949870i	$0.601215\pi$
$62$	21820.5i	0.720918i
$63$	− 30357.1i	− 0.963628i
$64$	−34809.8	−1.06231
$65$	0	0
$66$	−110.309	−0.00311710
$67$	34559.9i	0.940559i	0.882518	+	0.470279i	$0.155847\pi$
$67$	34559.9i	0.940559i	−0.882518	+	0.470279i	$0.844153\pi$
$68$	11764.4i	0.308530i
$69$	−6899.68	−0.174464
$70$	0	0
$71$	41255.7	0.971265	0.485632	−	0.874163i	$-0.338589\pi$
$71$	41255.7	0.971265	0.485632	+	0.874163i	$0.338589\pi$
$72$	47352.8i	1.07650i
$73$	− 1056.42i	− 0.0232022i	−0.999933	−	0.0116011i	$-0.996307\pi$
$73$	− 1056.42i	− 0.0232022i	0.999933	−	0.0116011i	$-0.00369283\pi$
$74$	−21146.2	−0.448903
$75$	0	0
$76$	2392.04	0.0475045
$77$	1873.10i	0.0360027i
$78$	− 1257.10i	− 0.0233955i
$79$	47719.3	0.860253	0.430126	−	0.902769i	$-0.358469\pi$
$79$	47719.3	0.860253	0.430126	+	0.902769i	$0.358469\pi$
$80$	0	0
$81$	57117.6	0.967291
$82$	36237.7i	0.595149i
$83$	− 74799.0i	− 1.19179i	−0.803061	−	0.595896i	$-0.796797\pi$
$83$	− 74799.0i	− 1.19179i	0.803061	−	0.595896i	$-0.203203\pi$
$84$	−2305.26	−0.0356469
$85$	0	0
$86$	38848.0	0.566400
$87$	− 823.421i	− 0.0116634i
$88$	− 2921.78i	− 0.0402198i
$89$	−9799.26	−0.131135	−0.0655675	−	0.997848i	$-0.520886\pi$
$89$	−9799.26	−0.131135	−0.0655675	+	0.997848i	$0.520886\pi$
$90$	0	0
$91$	−21346.2	−0.270219
$92$	− 47356.1i	− 0.583320i
$93$	− 7800.50i	− 0.0935222i
$94$	113679.	1.32697
$95$	0	0
$96$	6259.97	0.0693257
$97$	138432.i	1.49385i	0.664906	+	0.746927i	$0.268472\pi$
$97$	138432.i	1.49385i	−0.664906	+	0.746927i	$0.731528\pi$
$98$	3891.52i	0.0409312i
$99$	−3564.15	−0.0365484
$100$	0	0
$101$	139151.	1.35733	0.678663	−	0.734450i	$-0.262560\pi$
$101$	139151.	1.35733	0.678663	+	0.734450i	$0.262560\pi$
$102$	7818.71i	0.0744106i
$103$	98512.2i	0.914950i	0.889223	+	0.457475i	$0.151246\pi$
$103$	98512.2i	0.914950i	−0.889223	+	0.457475i	$0.848754\pi$
$104$	33297.0	0.301871
$105$	0	0
$106$	35618.7	0.307902
$107$	26848.4i	0.226704i	0.993555	+	0.113352i	$0.0361587\pi$
$107$	26848.4i	0.226704i	−0.993555	+	0.113352i	$0.963841\pi$
$108$	− 8821.45i	− 0.0727747i
$109$	63220.9	0.509676	0.254838	−	0.966984i	$-0.417978\pi$
$109$	63220.9	0.509676	0.254838	+	0.966984i	$0.417978\pi$
$110$	0	0
$111$	7559.43	0.0582346
$112$	68280.2i	0.514340i
$113$	114434.i	0.843058i	0.906815	+	0.421529i	$0.138506\pi$
$113$	114434.i	0.843058i	−0.906815	+	0.421529i	$0.861494\pi$
$114$	1589.77	0.0114570
$115$	0	0
$116$	5651.57	0.0389964
$117$	− 40617.6i	− 0.274315i
$118$	170317.i	1.12604i
$119$	132766.	0.859446
$120$	0	0
$121$	−160831.	−0.998634
$122$	− 82893.4i	− 0.504221i
$123$	− 12954.4i	− 0.0772066i
$124$	53538.9	0.312691
$125$	0	0
$126$	138476.	0.777047
$127$	248871.i	1.36919i	0.728922	+	0.684596i	$0.240022\pi$
$127$	248871.i	1.36919i	−0.728922	+	0.684596i	$0.759978\pi$
$128$	− 35943.2i	− 0.193906i
$129$	−13887.6	−0.0734770
$130$	0	0
$131$	102963.	0.524205	0.262102	−	0.965040i	$-0.415584\pi$
$131$	102963.	0.524205	0.262102	+	0.965040i	$0.415584\pi$
$132$	270.654i	0.00135201i
$133$	− 26995.1i	− 0.132329i
$134$	−157647.	−0.758444
$135$	0	0
$136$	−207096.	−0.960117
$137$	36037.4i	0.164041i	0.996631	+	0.0820204i	$0.0261373\pi$
$137$	36037.4i	0.164041i	−0.996631	+	0.0820204i	$0.973863\pi$
$138$	− 31473.3i	− 0.140684i
$139$	−152655.	−0.670151	−0.335076	−	0.942191i	$-0.608762\pi$
$139$	−152655.	−0.670151	−0.335076	+	0.942191i	$0.608762\pi$
$140$	0	0
$141$	−40638.6	−0.172144
$142$	188190.i	0.783205i
$143$	2506.20i	0.0102488i
$144$	−129924.	−0.522136
$145$	0	0
$146$	4818.92	0.0187097
$147$	− 1391.16i	− 0.00530986i
$148$	51884.3i	0.194707i
$149$	−72547.1	−0.267704	−0.133852	−	0.991001i	$-0.542735\pi$
$149$	−72547.1	−0.267704	−0.133852	+	0.991001i	$0.542735\pi$
$150$	0	0
$151$	489021.	1.74536	0.872681	−	0.488291i	$-0.162379\pi$
$151$	489021.	1.74536	0.872681	+	0.488291i	$0.162379\pi$
$152$	42108.6i	0.147830i
$153$	252627.i	0.872473i
$154$	−8544.26	−0.0290317
$155$	0	0
$156$	−3084.42	−0.0101476
$157$	− 89467.9i	− 0.289680i	−0.989455	−	0.144840i	$-0.953733\pi$
$157$	− 89467.9i	− 0.289680i	0.989455	−	0.144840i	$-0.0462667\pi$
$158$	217674.i	0.693687i
$159$	−12733.1	−0.0399431
$160$	0	0
$161$	−534432.	−1.62490
$162$	260545.i	0.780000i
$163$	− 225668.i	− 0.665275i	−0.943055	−	0.332637i	$-0.892061\pi$
$163$	− 225668.i	− 0.665275i	0.943055	−	0.332637i	$-0.107939\pi$
$164$	88912.8	0.258140
$165$	0	0
$166$	341200.	0.961033
$167$	− 209528.i	− 0.581367i	−0.956819	−	0.290683i	$-0.906117\pi$
$167$	− 209528.i	− 0.581367i	0.956819	−	0.290683i	$-0.0938827\pi$
$168$	− 40580.9i	− 0.110930i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	51366.5	0.134335
$172$	− 95317.6i	− 0.245670i
$173$	− 465184.i	− 1.18171i	−0.806779	−	0.590853i	$-0.798791\pi$
$173$	− 465184.i	− 1.18171i	0.806779	−	0.590853i	$-0.201209\pi$
$174$	3756.08	0.00940506
$175$	0	0
$176$	8016.60	0.0195078
$177$	− 60885.7i	− 0.146077i
$178$	− 44699.9i	− 0.105744i
$179$	472573.	1.10239	0.551197	−	0.834375i	$-0.314171\pi$
$179$	472573.	1.10239	0.551197	+	0.834375i	$0.314171\pi$
$180$	0	0
$181$	−74099.4	−0.168120	−0.0840598	−	0.996461i	$-0.526789\pi$
$181$	−74099.4	−0.168120	−0.0840598	+	0.996461i	$0.526789\pi$
$182$	− 97371.7i	− 0.217898i
$183$	29633.1i	0.0654108i
$184$	833638.	1.81524
$185$	0	0
$186$	35582.4	0.0754141
$187$	− 15587.7i	− 0.0325970i
$188$	− 278924.i	− 0.575561i
$189$	−99553.5	−0.202722
$190$	0	0
$191$	−224128.	−0.444543	−0.222271	−	0.974985i	$-0.571347\pi$
$191$	−224128.	−0.444543	−0.222271	+	0.974985i	$0.571347\pi$
$192$	56763.8i	0.111127i
$193$	− 662265.i	− 1.27979i	−0.768463	−	0.639895i	$-0.778978\pi$
$193$	− 662265.i	− 1.27979i	0.768463	−	0.639895i	$-0.221022\pi$
$194$	−631467.	−1.20461
$195$	0	0
$196$	9548.24	0.0177535
$197$	− 666464.i	− 1.22352i	−0.791043	−	0.611760i	$-0.790462\pi$
$197$	− 666464.i	− 1.22352i	0.791043	−	0.611760i	$-0.209538\pi$
$198$	− 16258.1i	− 0.0294717i
$199$	−645720.	−1.15588	−0.577938	−	0.816081i	$-0.696142\pi$
$199$	−645720.	−1.15588	−0.577938	+	0.816081i	$0.696142\pi$
$200$	0	0
$201$	56356.3	0.0983903
$202$	634746.i	1.09451i
$203$	− 63780.1i	− 0.108629i
$204$	19184.0	0.0322748
$205$	0	0
$206$	−449369.	−0.737794
$207$	− 1.01692e6i	− 1.64953i
$208$	91358.4i	0.146417i
$209$	−3169.43	−0.00501897
$210$	0	0
$211$	−868021.	−1.34222	−0.671110	−	0.741357i	$-0.734182\pi$
$211$	−868021.	−1.34222	−0.671110	+	0.741357i	$0.734182\pi$
$212$	− 87394.1i	− 0.133550i
$213$	− 67274.9i	− 0.101602i
$214$	−122470.	−0.182808
$215$	0	0
$216$	155289.	0.226468
$217$	− 604207.i	− 0.871037i
$218$	288385.i	0.410991i
$219$	−1722.69	−0.00242715
$220$	0	0
$221$	177639.	0.244658
$222$	34482.7i	0.0469590i
$223$	− 58342.9i	− 0.0785644i	−0.999228	−	0.0392822i	$-0.987493\pi$
$223$	− 58342.9i	− 0.0785644i	0.999228	−	0.0392822i	$-0.0125071\pi$
$224$	484882.	0.645678
$225$	0	0
$226$	−521995.	−0.679822
$227$	− 111768.i	− 0.143964i	−0.997406	−	0.0719820i	$-0.977068\pi$
$227$	− 111768.i	− 0.143964i	0.997406	−	0.0719820i	$-0.0229324\pi$
$228$	− 3900.67i	− 0.00496937i
$229$	−1.14984e6	−1.44893	−0.724467	−	0.689309i	$-0.757914\pi$
$229$	−1.14984e6	−1.44893	−0.724467	+	0.689309i	$0.757914\pi$
$230$	0	0
$231$	3054.44	0.00376618
$232$	99488.0i	0.121353i
$233$	630470.i	0.760807i	0.924821	+	0.380404i	$0.124215\pi$
$233$	630470.i	0.760807i	−0.924821	+	0.380404i	$0.875785\pi$
$234$	185279.	0.221201
$235$	0	0
$236$	417891.	0.488408
$237$	− 77815.0i	− 0.0899897i
$238$	605618.i	0.693037i
$239$	165628.	0.187559	0.0937797	−	0.995593i	$-0.470105\pi$
$239$	165628.	0.187559	0.0937797	+	0.995593i	$0.470105\pi$
$240$	0	0
$241$	690968.	0.766329	0.383165	−	0.923680i	$-0.374834\pi$
$241$	690968.	0.766329	0.383165	+	0.923680i	$0.374834\pi$
$242$	− 733639.i	− 0.805275i
$243$	− 284667.i	− 0.309259i
$244$	−203387.	−0.218700
$245$	0	0
$246$	59092.1	0.0622575
$247$	− 36119.3i	− 0.0376701i
$248$	942478.i	0.973065i
$249$	−121974.	−0.124671
$250$	0	0
$251$	−386887.	−0.387615	−0.193807	−	0.981040i	$-0.562084\pi$
$251$	−386887.	−0.387615	−0.193807	+	0.981040i	$0.562084\pi$
$252$	− 339764.i	− 0.337036i
$253$	62746.2i	0.0616292i
$254$	−1.13524e6	−1.10408
$255$	0	0
$256$	−949957.	−0.905950
$257$	− 258260.i	− 0.243907i	−0.992536	−	0.121953i	$-0.961084\pi$
$257$	− 258260.i	− 0.243907i	0.992536	−	0.121953i	$-0.0389158\pi$
$258$	− 63348.8i	− 0.0592501i
$259$	585535.	0.542379
$260$	0	0
$261$	121361.	0.110275
$262$	469669.i	0.422706i
$263$	− 1.19053e6i	− 1.06133i	−0.847582	−	0.530665i	$-0.821942\pi$
$263$	− 1.19053e6i	− 1.06133i	0.847582	−	0.530665i	$-0.178058\pi$
$264$	−4764.49	−0.00420733
$265$	0	0
$266$	123140.	0.106707
$267$	15979.5i	0.0137178i
$268$	386803.i	0.328967i
$269$	−850968.	−0.717022	−0.358511	−	0.933525i	$-0.616715\pi$
$269$	−850968.	−0.717022	−0.358511	+	0.933525i	$0.616715\pi$
$270$	0	0
$271$	−40926.1	−0.0338514	−0.0169257	−	0.999857i	$-0.505388\pi$
$271$	−40926.1	−0.0338514	−0.0169257	+	0.999857i	$0.505388\pi$
$272$	− 568218.i	− 0.465685i
$273$	34808.8i	0.0282672i
$274$	−164387.	−0.132279
$275$	0	0
$276$	−77222.8	−0.0610201
$277$	− 1.00054e6i	− 0.783490i	−0.920074	−	0.391745i	$-0.871872\pi$
$277$	− 1.00054e6i	− 0.783490i	0.920074	−	0.391745i	$-0.128128\pi$
$278$	− 696342.i	− 0.540394i
$279$	1.14969e6	0.884239
$280$	0	0
$281$	−1.73596e6	−1.31151	−0.655757	−	0.754972i	$-0.727650\pi$
$281$	−1.73596e6	−1.31151	−0.655757	+	0.754972i	$0.727650\pi$
$282$	− 185375.i	− 0.138813i
$283$	− 1.27363e6i	− 0.945319i	−0.881245	−	0.472660i	$-0.843294\pi$
$283$	− 1.27363e6i	− 0.945319i	0.881245	−	0.472660i	$-0.156706\pi$
$284$	461743.	0.339707
$285$	0	0
$286$	−11432.1	−0.00826443
$287$	− 1.00342e6i	− 0.719078i
$288$	922636.i	0.655464i
$289$	315001.	0.221854
$290$	0	0
$291$	225739.	0.156270
$292$	− 11823.7i	− 0.00811515i
$293$	− 2043.70i	− 0.00139075i	−1.00000		0.000695374i	$-0.999779\pi$
$293$	− 2043.70i	− 0.00139075i	1.00000		0.000695374i	$-0.000221344\pi$
$294$	6345.84	0.00428174
$295$	0	0
$296$	−913351.	−0.605910
$297$	11688.3i	0.00768883i
$298$	− 330927.i	− 0.215870i
$299$	−715066.	−0.462560
$300$	0	0
$301$	−1.07570e6	−0.684342
$302$	2.23070e6i	1.40742i
$303$	− 226912.i	− 0.141988i
$304$	−115535.	−0.0717018
$305$	0	0
$306$	−1.15237e6	−0.703541
$307$	401308.i	0.243014i	0.992591	+	0.121507i	$0.0387728\pi$
$307$	401308.i	0.243014i	−0.992591	+	0.121507i	$0.961227\pi$
$308$	20964.2i	0.0125922i
$309$	160642.	0.0957114
$310$	0	0
$311$	−1.92628e6	−1.12933	−0.564663	−	0.825322i	$-0.690994\pi$
$311$	−1.92628e6	−1.12933	−0.564663	+	0.825322i	$0.690994\pi$
$312$	− 54296.9i	− 0.0315783i
$313$	1.64519e6i	0.949196i	0.880203	+	0.474598i	$0.157407\pi$
$313$	1.64519e6i	0.949196i	−0.880203	+	0.474598i	$0.842593\pi$
$314$	408113.	0.233591
$315$	0	0
$316$	534085.	0.300880
$317$	1.99476e6i	1.11492i	0.830205	+	0.557459i	$0.188223\pi$
$317$	1.99476e6i	1.11492i	−0.830205	+	0.557459i	$0.811777\pi$
$318$	− 58082.8i	− 0.0322092i
$319$	−7488.26	−0.00412006
$320$	0	0
$321$	43781.2	0.0237151
$322$	− 2.43784e6i	− 1.31028i
$323$	224649.i	0.119812i
$324$	639273.	0.338317
$325$	0	0
$326$	1.02940e6	0.536462
$327$	− 103093.i	− 0.0533164i
$328$	1.56519e6i	0.803306i
$329$	−3.14777e6	−1.60329
$330$	0	0
$331$	−675924.	−0.339100	−0.169550	−	0.985522i	$-0.554231\pi$
$331$	−675924.	−0.339100	−0.169550	+	0.985522i	$0.554231\pi$
$332$	− 837168.i	− 0.416838i
$333$	1.11416e6i	0.550600i
$334$	955772.	0.468800
$335$	0	0
$336$	111343.	0.0538042
$337$	2.13552e6i	1.02430i	0.858895	+	0.512152i	$0.171152\pi$
$337$	2.13552e6i	1.02430i	−0.858895	+	0.512152i	$0.828848\pi$
$338$	− 130283.i	− 0.0620289i
$339$	186605.	0.0881909
$340$	0	0
$341$	−70938.3	−0.0330366
$342$	234311.i	0.108325i
$343$	− 2.23063e6i	− 1.02374i
$344$	1.67793e6	0.764502
$345$	0	0
$346$	2.12196e6	0.952899
$347$	2.57257e6i	1.14695i	0.819225	+	0.573473i	$0.194404\pi$
$347$	2.57257e6i	1.14695i	−0.819225	+	0.573473i	$0.805596\pi$
$348$	− 9215.92i	− 0.00407935i
$349$	−2.02363e6	−0.889339	−0.444670	−	0.895695i	$-0.646679\pi$
$349$	−2.02363e6	−0.889339	−0.444670	+	0.895695i	$0.646679\pi$
$350$	0	0
$351$	−133202.	−0.0577088
$352$	− 56928.7i	− 0.0244892i
$353$	2.04810e6i	0.874810i	0.899265	+	0.437405i	$0.144102\pi$
$353$	2.04810e6i	0.874810i	−0.899265	+	0.437405i	$0.855898\pi$
$354$	277733.	0.117793
$355$	0	0
$356$	−109676.	−0.0458654
$357$	− 216499.i	− 0.0899053i
$358$	2.15567e6i	0.888944i
$359$	1.59901e6	0.654808	0.327404	−	0.944885i	$-0.393826\pi$
$359$	1.59901e6	0.654808	0.327404	+	0.944885i	$0.393826\pi$
$360$	0	0
$361$	−2.43042e6	−0.981553
$362$	− 338008.i	− 0.135568i
$363$	262265.i	0.104466i
$364$	−238911.	−0.0945112
$365$	0	0
$366$	−135173.	−0.0527457
$367$	3.86389e6i	1.49747i	0.662867	+	0.748737i	$0.269339\pi$
$367$	3.86389e6i	1.49747i	−0.662867	+	0.748737i	$0.730661\pi$
$368$	2.28729e6i	0.880444i
$369$	1.90930e6	0.729977
$370$	0	0
$371$	−986276.	−0.372018
$372$	− 87305.0i	− 0.0327101i
$373$	− 1.56702e6i	− 0.583179i	−0.956544	−	0.291589i	$-0.905816\pi$
$373$	− 1.56702e6i	− 0.583179i	0.956544	−	0.291589i	$-0.0941841\pi$
$374$	71104.0	0.0262854
$375$	0	0
$376$	4.91007e6	1.79109
$377$	− 85337.3i	− 0.0309233i
$378$	− 454118.i	− 0.163471i
$379$	−3.19239e6	−1.14161	−0.570805	−	0.821086i	$-0.693369\pi$
$379$	−3.19239e6	−1.14161	−0.570805	+	0.821086i	$0.693369\pi$
$380$	0	0
$381$	405829.	0.143229
$382$	− 1.02237e6i	− 0.358469i
$383$	400432.i	0.139486i	0.997565	+	0.0697432i	$0.0222180\pi$
$383$	400432.i	0.139486i	−0.997565	+	0.0697432i	$0.977782\pi$
$384$	−58611.9	−0.0202842
$385$	0	0
$386$	3.02096e6	1.03199
$387$	− 2.04684e6i	− 0.694715i
$388$	1.54937e6i	0.522487i
$389$	−413440.	−0.138528	−0.0692642	−	0.997598i	$-0.522065\pi$
$389$	−413440.	−0.138528	−0.0692642	+	0.997598i	$0.522065\pi$
$390$	0	0
$391$	4.44746e6	1.47120
$392$	168083.i	0.0552471i
$393$	− 167899.i	− 0.0548362i
$394$	3.04011e6	0.986618
$395$	0	0
$396$	−39890.8	−0.0127831
$397$	102926.i	0.0327753i	0.999866	+	0.0163877i	$0.00521659\pi$
$397$	102926.i	0.0327753i	−0.999866	+	0.0163877i	$0.994783\pi$
$398$	− 2.94548e6i	− 0.932071i
$399$	−44020.5	−0.0138428
$400$	0	0
$401$	2.23365e6	0.693671	0.346836	−	0.937926i	$-0.387256\pi$
$401$	2.23365e6	0.693671	0.346836	+	0.937926i	$0.387256\pi$
$402$	257072.i	0.0793396i
$403$	− 808424.i	− 0.247957i
$404$	1.55741e6	0.474734
$405$	0	0
$406$	290937.	0.0875958
$407$	− 68746.0i	− 0.0205713i
$408$	337708.i	0.100436i
$409$	4.46150e6	1.31878	0.659390	−	0.751801i	$-0.270815\pi$
$409$	4.46150e6	1.31878	0.659390	+	0.751801i	$0.270815\pi$
$410$	0	0
$411$	58765.6	0.0171600
$412$	1.10257e6i	0.320010i
$413$	− 4.71606e6i	− 1.36052i
$414$	4.63874e6	1.33014
$415$	0	0
$416$	648768.	0.183804
$417$	248931.i	0.0701034i
$418$	− 14457.5i	− 0.00404718i
$419$	−4.22792e6	−1.17650	−0.588250	−	0.808679i	$-0.700183\pi$
$419$	−4.22792e6	−1.17650	−0.588250	+	0.808679i	$0.700183\pi$
$420$	0	0
$421$	4.11791e6	1.13233	0.566163	−	0.824293i	$-0.308427\pi$
$421$	4.11791e6	1.13233	0.566163	+	0.824293i	$0.308427\pi$
$422$	− 3.95952e6i	− 1.08233i
$423$	− 5.98959e6i	− 1.62759i
$424$	1.53845e6	0.415594
$425$	0	0
$426$	306878.	0.0819298
$427$	2.29531e6i	0.609216i
$428$	300493.i	0.0792912i
$429$	4086.81	0.00107212
$430$	0	0
$431$	1.15324e6	0.299038	0.149519	−	0.988759i	$-0.452227\pi$
$431$	1.15324e6	0.299038	0.149519	+	0.988759i	$0.452227\pi$
$432$	426074.i	0.109844i
$433$	− 33734.3i	− 0.00864673i	−0.999991	−	0.00432337i	$-0.998624\pi$
$433$	− 33734.3i	− 0.00864673i	0.999991	−	0.00432337i	$-0.00137617\pi$
$434$	2.75612e6	0.702383
$435$	0	0
$436$	707583.	0.178263
$437$	− 904298.i	− 0.226521i
$438$	− 7858.13i	− 0.00195719i
$439$	−7.48363e6	−1.85332	−0.926661	−	0.375898i	$-0.877334\pi$
$439$	−7.48363e6	−1.85332	−0.926661	+	0.375898i	$0.877334\pi$
$440$	0	0
$441$	205038.	0.0502039
$442$	810312.i	0.197286i
$443$	− 3.28028e6i	− 0.794148i	−0.917786	−	0.397074i	$-0.870026\pi$
$443$	− 3.28028e6i	− 0.794148i	0.917786	−	0.397074i	$-0.129974\pi$
$444$	84606.9	0.0203680
$445$	0	0
$446$	266134.	0.0633525
$447$	118301.i	0.0280040i
$448$	4.39678e6i	1.03500i
$449$	−7.95356e6	−1.86185	−0.930927	−	0.365205i	$-0.880999\pi$
$449$	−7.95356e6	−1.86185	−0.930927	+	0.365205i	$0.880999\pi$
$450$	0	0
$451$	−117808.	−0.0272731
$452$	1.28077e6i	0.294866i
$453$	− 797439.i	− 0.182579i
$454$	509837.	0.116089
$455$	0	0
$456$	68665.8	0.0154642
$457$	3.35187e6i	0.750753i	0.926872	+	0.375377i	$0.122487\pi$
$457$	3.35187e6i	0.750753i	−0.926872	+	0.375377i	$0.877513\pi$
$458$	− 5.24506e6i	− 1.16839i
$459$	828468.	0.183546
$460$	0	0
$461$	4.68627e6	1.02701	0.513505	−	0.858086i	$-0.328347\pi$
$461$	4.68627e6	1.02701	0.513505	+	0.858086i	$0.328347\pi$
$462$	13933.0i	0.00303696i
$463$	6.64697e6i	1.44102i	0.693442	+	0.720512i	$0.256093\pi$
$463$	6.64697e6i	1.44102i	−0.693442	+	0.720512i	$0.743907\pi$
$464$	−272969.	−0.0588599
$465$	0	0
$466$	−2.87592e6	−0.613497
$467$	3.14141e6i	0.666549i	0.942830	+	0.333275i	$0.108154\pi$
$467$	3.14141e6i	0.666549i	−0.942830	+	0.333275i	$0.891846\pi$
$468$	− 454602.i	− 0.0959437i
$469$	4.36522e6	0.916377
$470$	0	0
$471$	−145894.	−0.0303029
$472$	7.35638e6i	1.51988i
$473$	126295.i	0.0259556i
$474$	354957.	0.0725655
$475$	0	0
$476$	1.48595e6	0.300598
$477$	− 1.87669e6i	− 0.377656i
$478$	755521.i	0.151243i
$479$	6.68286e6	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$479$	6.68286e6	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$480$	0	0
$481$	783441.	0.154399
$482$	3.15189e6i	0.617950i
$483$	871489.i	0.169979i
$484$	−1.80006e6	−0.349280
$485$	0	0
$486$	1.29853e6	0.249379
$487$	4.06478e6i	0.776631i	0.921526	+	0.388316i	$0.126943\pi$
$487$	4.06478e6i	0.776631i	−0.921526	+	0.388316i	$0.873057\pi$
$488$	− 3.58035e6i	− 0.680575i
$489$	−367993.	−0.0695933
$490$	0	0
$491$	−2.10434e6	−0.393923	−0.196962	−	0.980411i	$-0.563107\pi$
$491$	−2.10434e6	−0.393923	−0.196962	+	0.980411i	$0.563107\pi$
$492$	− 144989.i	− 0.0270036i
$493$	530768.i	0.0983530i
$494$	164760.	0.0303763
$495$	0	0
$496$	−2.58592e6	−0.471966
$497$	− 5.21095e6i	− 0.946293i
$498$	− 556389.i	− 0.100532i
$499$	−5.96715e6	−1.07279	−0.536396	−	0.843966i	$-0.680215\pi$
$499$	−5.96715e6	−1.07279	−0.536396	+	0.843966i	$0.680215\pi$
$500$	0	0
$501$	−341673.	−0.0608158
$502$	− 1.76481e6i	− 0.312563i
$503$	− 1.00144e7i	− 1.76483i	−0.470467	−	0.882417i	$-0.655915\pi$
$503$	− 1.00144e7i	− 1.76483i	0.470467	−	0.882417i	$-0.344085\pi$
$504$	5.98108e6	1.04882
$505$	0	0
$506$	−286220.	−0.0496963
$507$	46573.9i	0.00804680i
$508$	2.78542e6i	0.478885i
$509$	8.47321e6	1.44962	0.724809	−	0.688950i	$-0.241928\pi$
$509$	8.47321e6	1.44962	0.724809	+	0.688950i	$0.241928\pi$
$510$	0	0
$511$	−133435.	−0.0226057
$512$	− 5.48346e6i	− 0.924442i
$513$	− 168452.i	− 0.0282606i
$514$	1.17807e6	0.196681
$515$	0	0
$516$	−155433.	−0.0256991
$517$	369571.i	0.0608095i
$518$	2.67095e6i	0.437362i
$519$	−758567.	−0.123616
$520$	0	0
$521$	−197614.	−0.0318951	−0.0159476	−	0.999873i	$-0.505076\pi$
$521$	−197614.	−0.0318951	−0.0159476	+	0.999873i	$0.505076\pi$
$522$	553596.i	0.0889235i
$523$	8.27263e6i	1.32248i	0.750174	+	0.661240i	$0.229970\pi$
$523$	8.27263e6i	1.32248i	−0.750174	+	0.661240i	$0.770030\pi$
$524$	1.15238e6	0.183345
$525$	0	0
$526$	5.43066e6	0.855831
$527$	5.02812e6i	0.788640i
$528$	− 13072.5i	− 0.00204068i
$529$	−1.14664e7	−1.78150
$530$	0	0
$531$	8.97374e6	1.38114
$532$	− 302136.i	− 0.0462832i
$533$	− 1.34256e6i	− 0.204699i
$534$	−72891.3	−0.0110617
$535$	0	0
$536$	−6.80913e6	−1.02372
$537$	− 770618.i	− 0.115320i
$538$	− 3.88174e6i	− 0.578190i
$539$	−12651.3	−0.00187570
$540$	0	0
$541$	363216.	0.0533546	0.0266773	−	0.999644i	$-0.491507\pi$
$541$	363216.	0.0533546	0.0266773	+	0.999644i	$0.491507\pi$
$542$	− 186686.i	− 0.0272970i
$543$	120833.i	0.0175867i
$544$	−4.03511e6	−0.584599
$545$	0	0
$546$	−158782.	−0.0227940
$547$	620452.i	0.0886624i	0.999017	+	0.0443312i	$0.0141157\pi$
$547$	620452.i	0.0886624i	−0.999017	+	0.0443312i	$0.985884\pi$
$548$	403339.i	0.0573745i
$549$	−4.36752e6	−0.618449
$550$	0	0
$551$	107921.	0.0151435
$552$	− 1.35940e6i	− 0.189889i
$553$	− 6.02736e6i	− 0.838136i
$554$	4.56400e6	0.631788
$555$	0	0
$556$	−1.70855e6	−0.234390
$557$	− 3.89737e6i	− 0.532272i	−0.963935	−	0.266136i	$-0.914253\pi$
$557$	− 3.89737e6i	− 0.532272i	0.963935	−	0.266136i	$-0.0857471\pi$
$558$	5.24437e6i	0.713029i
$559$	−1.43927e6	−0.194811
$560$	0	0
$561$	−25418.5	−0.00340992
$562$	− 7.91865e6i	− 1.05757i
$563$	510725.i	0.0679073i	0.999423	+	0.0339536i	$0.0108099\pi$
$563$	510725.i	0.0679073i	−0.999423	+	0.0339536i	$0.989190\pi$
$564$	−454837.	−0.0602085
$565$	0	0
$566$	5.80975e6	0.762283
$567$	− 7.21445e6i	− 0.942422i
$568$	8.12834e6i	1.05714i
$569$	−9.75625e6	−1.26329	−0.631644	−	0.775259i	$-0.717619\pi$
$569$	−9.75625e6	−1.26329	−0.631644	+	0.775259i	$0.717619\pi$
$570$	0	0
$571$	−1.41952e7	−1.82201	−0.911006	−	0.412393i	$-0.864693\pi$
$571$	−1.41952e7	−1.82201	−0.911006	+	0.412393i	$0.864693\pi$
$572$	28049.9i	0.00358461i
$573$	365482.i	0.0465029i
$574$	4.57713e6	0.579847
$575$	0	0
$576$	−8.36622e6	−1.05069
$577$	1.16423e6i	0.145579i	0.997347	+	0.0727896i	$0.0231902\pi$
$577$	1.16423e6i	0.145579i	−0.997347	+	0.0727896i	$0.976810\pi$
$578$	1.43689e6i	0.178898i
$579$	−1.07994e6	−0.133877
$580$	0	0
$581$	−9.44777e6	−1.16115
$582$	1.02972e6i	0.126012i
$583$	115796.i	0.0141098i
$584$	208140.	0.0252536
$585$	0	0
$586$	9322.45	0.00112147
$587$	− 6.58038e6i	− 0.788234i	−0.919060	−	0.394117i	$-0.871050\pi$
$587$	− 6.58038e6i	− 0.788234i	0.919060	−	0.394117i	$-0.128950\pi$
$588$	− 15570.2i	− 0.00185716i
$589$	1.02236e6	0.121427
$590$	0	0
$591$	−1.08679e6	−0.127991
$592$	− 2.50600e6i	− 0.293885i
$593$	− 1.91423e6i	− 0.223541i	−0.993734	−	0.111771i	$-0.964348\pi$
$593$	− 1.91423e6i	− 0.223541i	0.993734	−	0.111771i	$-0.0356522\pi$
$594$	−53316.8	−0.00620009
$595$	0	0
$596$	−811964.	−0.0936313
$597$	1.05296e6i	0.120914i
$598$	− 3.26181e6i	− 0.372997i
$599$	2.33678e6	0.266104	0.133052	−	0.991109i	$-0.457522\pi$
$599$	2.33678e6	0.266104	0.133052	+	0.991109i	$0.457522\pi$
$600$	0	0
$601$	1.04273e7	1.17757	0.588786	−	0.808289i	$-0.299606\pi$
$601$	1.04273e7	1.17757	0.588786	+	0.808289i	$0.299606\pi$
$602$	− 4.90684e6i	− 0.551837i
$603$	8.30617e6i	0.930266i
$604$	5.47324e6	0.610453
$605$	0	0
$606$	1.03507e6	0.114495
$607$	120274.i	0.0132495i	0.999978	+	0.00662474i	$0.00210874\pi$
$607$	120274.i	0.0132495i	−0.999978	+	0.00662474i	$0.997891\pi$
$608$	820455.i	0.0900111i
$609$	−104005.	−0.0113635
$610$	0	0
$611$	−4.21169e6	−0.456408
$612$	2.82747e6i	0.305154i
$613$	1.34576e7i	1.44649i	0.690592	+	0.723245i	$0.257350\pi$
$613$	1.34576e7i	1.44649i	−0.690592	+	0.723245i	$0.742650\pi$
$614$	−1.83059e6	−0.195961
$615$	0	0
$616$	−369046.	−0.0391858
$617$	6.84879e6i	0.724270i	0.932126	+	0.362135i	$0.117952\pi$
$617$	6.84879e6i	0.724270i	−0.932126	+	0.362135i	$0.882048\pi$
$618$	732778.i	0.0771794i
$619$	5.40663e6	0.567153	0.283577	−	0.958950i	$-0.408479\pi$
$619$	5.40663e6	0.567153	0.283577	+	0.958950i	$0.408479\pi$
$620$	0	0
$621$	−3.33490e6	−0.347019
$622$	− 8.78684e6i	− 0.910661i
$623$	1.23773e6i	0.127763i
$624$	148977.	0.0153164
$625$	0	0
$626$	−7.50463e6	−0.765409
$627$	5168.33i	0 0.000525027i
$628$	− 1.00135e6i	− 0.101318i
$629$	−4.87273e6	−0.491072
$630$	0	0
$631$	−9.62552e6	−0.962390	−0.481195	−	0.876614i	$-0.659797\pi$
$631$	−9.62552e6	−0.962390	−0.481195	+	0.876614i	$0.659797\pi$
$632$	9.40183e6i	0.936310i
$633$	1.41547e6i	0.140408i
$634$	−9.09920e6	−0.899043
$635$	0	0
$636$	−142512.	−0.0139704
$637$	− 144176.i	− 0.0140781i
$638$	− 34158.1i	− 0.00332232i
$639$	9.91542e6	0.960636
$640$	0	0
$641$	−1.58752e7	−1.52607	−0.763037	−	0.646355i	$-0.776293\pi$
$641$	−1.58752e7	−1.52607	−0.763037	+	0.646355i	$0.776293\pi$
$642$	199710.i	0.0191233i
$643$	1.57235e7i	1.49976i	0.661574	+	0.749880i	$0.269889\pi$
$643$	1.57235e7i	1.49976i	−0.661574	+	0.749880i	$0.730111\pi$
$644$	−5.98149e6	−0.568322
$645$	0	0
$646$	−1.02475e6	−0.0966132
$647$	1.58173e7i	1.48549i	0.669572	+	0.742747i	$0.266477\pi$
$647$	1.58173e7i	1.48549i	−0.669572	+	0.742747i	$0.733523\pi$
$648$	1.12535e7i	1.05281i
$649$	−553699.	−0.0516015
$650$	0	0
$651$	−985270.	−0.0911178
$652$	− 2.52573e6i	− 0.232685i
$653$	5.31229e6i	0.487527i	0.969835	+	0.243763i	$0.0783820\pi$
$653$	5.31229e6i	0.487527i	−0.969835	+	0.243763i	$0.921618\pi$
$654$	470265.	0.0429931
$655$	0	0
$656$	−4.29447e6	−0.389628
$657$	− 253901.i	− 0.0229483i
$658$	− 1.43587e7i	− 1.29286i
$659$	−9.90554e6	−0.888514	−0.444257	−	0.895899i	$-0.646532\pi$
$659$	−9.90554e6	−0.888514	−0.444257	+	0.895899i	$0.646532\pi$
$660$	0	0
$661$	−1.29988e7	−1.15717	−0.578587	−	0.815621i	$-0.696396\pi$
$661$	−1.29988e7	−1.15717	−0.578587	+	0.815621i	$0.696396\pi$
$662$	− 3.08326e6i	− 0.273442i
$663$	− 289674.i	− 0.0255932i
$664$	1.47372e7	1.29716
$665$	0	0
$666$	−5.08229e6	−0.443991
$667$	− 2.13654e6i	− 0.185950i
$668$	− 2.34508e6i	− 0.203337i
$669$	−95138.8	−0.00821850
$670$	0	0
$671$	269486.	0.0231062
$672$	− 790688.i	− 0.0675433i
$673$	− 1.32503e7i	− 1.12769i	−0.825881	−	0.563844i	$-0.809322\pi$
$673$	− 1.32503e7i	− 1.12769i	0.825881	−	0.563844i	$-0.190678\pi$
$674$	−9.74129e6	−0.825975
$675$	0	0
$676$	−319661.	−0.0269044
$677$	− 2.23310e7i	− 1.87257i	−0.351245	−	0.936284i	$-0.614242\pi$
$677$	− 2.23310e7i	− 1.87257i	0.351245	−	0.936284i	$-0.385758\pi$
$678$	851208.i	0.0711151i
$679$	1.74852e7	1.45545
$680$	0	0
$681$	−182259.	−0.0150598
$682$	− 323589.i	− 0.0266399i
$683$	− 1.40049e7i	− 1.14876i	−0.818588	−	0.574380i	$-0.805243\pi$
$683$	− 1.40049e7i	− 1.14876i	0.818588	−	0.574380i	$-0.194757\pi$
$684$	574906.	0.0469847
$685$	0	0
$686$	1.01751e7	0.825523
$687$	1.87502e6i	0.151571i
$688$	4.60382e6i	0.370806i
$689$	−1.31963e6	−0.105902
$690$	0	0
$691$	−5.24817e6	−0.418132	−0.209066	−	0.977902i	$-0.567042\pi$
$691$	−5.24817e6	−0.418132	−0.209066	+	0.977902i	$0.567042\pi$
$692$	− 5.20645e6i	− 0.413310i
$693$	450183.i	0.0356087i
$694$	−1.17349e7	−0.924870
$695$	0	0
$696$	162233.	0.0126945
$697$	8.35027e6i	0.651056i
$698$	− 9.23090e6i	− 0.717142i
$699$	1.02810e6	0.0795868
$700$	0	0
$701$	−2.13994e7	−1.64477	−0.822386	−	0.568930i	$-0.807358\pi$
$701$	−2.13994e7	−1.64477	−0.822386	+	0.568930i	$0.807358\pi$
$702$	− 607607.i	− 0.0465350i
$703$	990767.i	0.0756107i
$704$	516214.	0.0392553
$705$	0	0
$706$	−9.34250e6	−0.705426
$707$	− 1.75760e7i	− 1.32243i
$708$	− 681447.i	− 0.0510915i
$709$	−4.35333e6	−0.325242	−0.162621	−	0.986689i	$-0.551995\pi$
$709$	−4.35333e6	−0.325242	−0.162621	+	0.986689i	$0.551995\pi$
$710$	0	0
$711$	1.14689e7	0.850839
$712$	− 1.93069e6i	− 0.142729i
$713$	− 2.02401e7i	− 1.49104i
$714$	987571.	0.0724975
$715$	0	0
$716$	5.28915e6	0.385570
$717$	− 270087.i	− 0.0196203i
$718$	7.29395e6i	0.528021i
$719$	−2.21389e7	−1.59710	−0.798552	−	0.601926i	$-0.794400\pi$
$719$	−2.21389e7	−1.59710	−0.798552	+	0.601926i	$0.794400\pi$
$720$	0	0
$721$	1.24430e7	0.891426
$722$	− 1.10865e7i	− 0.791501i
$723$	− 1.12675e6i	− 0.0801645i
$724$	−829338.	−0.0588010
$725$	0	0
$726$	−1.19633e6	−0.0842385
$727$	4.83218e6i	0.339084i	0.985523	+	0.169542i	$0.0542288\pi$
$727$	4.83218e6i	0.339084i	−0.985523	+	0.169542i	$0.945771\pi$
$728$	− 4.20570e6i	− 0.294110i
$729$	1.34154e7	0.934940
$730$	0	0
$731$	8.95178e6	0.619606
$732$	331661.i	0.0228779i
$733$	− 2.47827e7i	− 1.70368i	−0.523803	−	0.851840i	$-0.675487\pi$
$733$	− 2.47827e7i	− 1.70368i	0.523803	−	0.851840i	$-0.324513\pi$
$734$	−1.76253e7	−1.20753
$735$	0	0
$736$	1.62428e7	1.10527
$737$	− 512509.i	− 0.0347562i
$738$	8.70939e6i	0.588636i
$739$	−7.09289e6	−0.477762	−0.238881	−	0.971049i	$-0.576781\pi$
$739$	−7.09289e6	−0.477762	−0.238881	+	0.971049i	$0.576781\pi$
$740$	0	0
$741$	−58899.1	−0.00394061
$742$	− 4.49895e6i	− 0.299986i
$743$	1.95117e7i	1.29665i	0.761362	+	0.648327i	$0.224531\pi$
$743$	1.95117e7i	1.29665i	−0.761362	+	0.648327i	$0.775469\pi$
$744$	1.53688e6	0.101791
$745$	0	0
$746$	7.14803e6	0.470262
$747$	− 1.79773e7i	− 1.17875i
$748$	− 174461.i	− 0.0114010i
$749$	3.39118e6	0.220875
$750$	0	0
$751$	1.66103e7	1.07468	0.537339	−	0.843366i	$-0.319430\pi$
$751$	1.66103e7	1.07468	0.537339	+	0.843366i	$0.319430\pi$
$752$	1.34720e7i	0.868733i
$753$	630891.i	0.0405477i
$754$	389271.	0.0249358
$755$	0	0
$756$	−1.11423e6	−0.0709037
$757$	− 1.22902e6i	− 0.0779508i	−0.999240	−	0.0389754i	$-0.987591\pi$
$757$	− 1.22902e6i	− 0.0779508i	0.999240	−	0.0389754i	$-0.0124094\pi$
$758$	− 1.45622e7i	− 0.920567i
$759$	102319.	0.00644693
$760$	0	0
$761$	1.37482e7	0.860569	0.430284	−	0.902693i	$-0.358413\pi$
$761$	1.37482e7	0.860569	0.430284	+	0.902693i	$0.358413\pi$
$762$	1.85121e6i	0.115496i
$763$	− 7.98535e6i	− 0.496572i
$764$	−2.50850e6	−0.155482
$765$	0	0
$766$	−1.82659e6	−0.112478
$767$	− 6.31004e6i	− 0.387297i
$768$	1.54908e6i	0.0947699i
$769$	1.01549e7	0.619240	0.309620	−	0.950860i	$-0.399798\pi$
$769$	1.01549e7	0.619240	0.309620	+	0.950860i	$0.399798\pi$
$770$	0	0
$771$	−421140.	−0.0255147
$772$	− 7.41222e6i	− 0.447616i
$773$	1.41511e7i	0.851807i	0.904769	+	0.425903i	$0.140044\pi$
$773$	1.41511e7i	0.851807i	−0.904769	+	0.425903i	$0.859956\pi$
$774$	9.33677e6	0.560202
$775$	0	0
$776$	−2.72745e7	−1.62593
$777$	− 954821.i	− 0.0567374i
$778$	− 1.88593e6i	− 0.111706i
$779$	1.69785e6	0.100243
$780$	0	0
$781$	−611803.	−0.0358909
$782$	2.02873e7i	1.18634i
$783$	− 397993.i	− 0.0231991i
$784$	−461178.	−0.0267965
$785$	0	0
$786$	765882.	0.0442186
$787$	1.03095e6i	0.0593338i	0.999560	+	0.0296669i	$0.00944465\pi$
$787$	1.03095e6i	0.0593338i	−0.999560	+	0.0296669i	$0.990555\pi$
$788$	− 7.45923e6i	− 0.427935i
$789$	−1.94137e6	−0.111024
$790$	0	0
$791$	1.44540e7	0.821383
$792$	− 702222.i	− 0.0397797i
$793$	3.07110e6i	0.173425i
$794$	−469500.	−0.0264292
$795$	0	0
$796$	−7.22705e6	−0.404276
$797$	1.43337e7i	0.799303i	0.916667	+	0.399651i	$0.130869\pi$
$797$	1.43337e7i	0.799303i	−0.916667	+	0.399651i	$0.869131\pi$
$798$	− 200802.i	− 0.0111625i
$799$	2.61952e7	1.45163
$800$	0	0
$801$	−2.35516e6	−0.129700
$802$	1.01889e7i	0.559360i
$803$	15666.3i	0 0.000857386i
$804$	630753.	0.0344127
$805$	0	0
$806$	3.68767e6	0.199947
$807$	1.38766e6i	0.0750065i
$808$	2.74161e7i	1.47733i
$809$	−1.55020e7	−0.832751	−0.416376	−	0.909193i	$-0.636700\pi$
$809$	−1.55020e7	−0.832751	−0.416376	+	0.909193i	$0.636700\pi$
$810$	0	0
$811$	2.45861e7	1.31261	0.656307	−	0.754494i	$-0.272118\pi$
$811$	2.45861e7	1.31261	0.656307	+	0.754494i	$0.272118\pi$
$812$	− 713842.i	− 0.0379938i
$813$	66737.5i	0.00354114i
$814$	313589.	0.0165882
$815$	0	0
$816$	−926583.	−0.0487146
$817$	− 1.82016e6i	− 0.0954011i
$818$	2.03513e7i	1.06343i
$819$	−5.13036e6	−0.267262
$820$	0	0
$821$	−3.33396e6	−0.172624	−0.0863122	−	0.996268i	$-0.527508\pi$
$821$	−3.33396e6	−0.172624	−0.0863122	+	0.996268i	$0.527508\pi$
$822$	268062.i	0.0138375i
$823$	3.08787e6i	0.158913i	0.996838	+	0.0794564i	$0.0253185\pi$
$823$	3.08787e6i	0.158913i	−0.996838	+	0.0794564i	$0.974682\pi$
$824$	−1.94092e7	−0.995842
$825$	0	0
$826$	2.15125e7	1.09709
$827$	− 6.89555e6i	− 0.350595i	−0.984516	−	0.175297i	$-0.943911\pi$
$827$	− 6.89555e6i	− 0.350595i	0.984516	−	0.175297i	$-0.0560886\pi$
$828$	− 1.13816e7i	− 0.576936i
$829$	2.00007e7	1.01078	0.505391	−	0.862890i	$-0.331348\pi$
$829$	2.00007e7	1.01078	0.505391	+	0.862890i	$0.331348\pi$
$830$	0	0
$831$	−1.63156e6	−0.0819596
$832$	5.88286e6i	0.294632i
$833$	896725.i	0.0447762i
$834$	−1.13551e6	−0.0565297
$835$	0	0
$836$	−35473.0	−0.00175542
$837$	− 3.77030e6i	− 0.186021i
$838$	− 1.92859e7i	− 0.948701i
$839$	−5.23988e6	−0.256990	−0.128495	−	0.991710i	$-0.541015\pi$
$839$	−5.23988e6	−0.256990	−0.128495	+	0.991710i	$0.541015\pi$
$840$	0	0
$841$	−2.02562e7	−0.987569
$842$	1.87841e7i	0.913081i
$843$	2.83079e6i	0.137195i
$844$	−9.71509e6	−0.469452
$845$	0	0
$846$	2.73218e7	1.31245
$847$	2.03144e7i	0.972959i
$848$	4.22111e6i	0.201575i
$849$	−2.07689e6	−0.0988883
$850$	0	0
$851$	1.96146e7	0.928442
$852$	− 752957.i	− 0.0355362i
$853$	− 1.32853e7i	− 0.625170i	−0.949890	−	0.312585i	$-0.898805\pi$
$853$	− 1.32853e7i	− 0.625170i	0.949890	−	0.312585i	$-0.101195\pi$
$854$	−1.04702e7	−0.491257
$855$	0	0
$856$	−5.28976e6	−0.246747
$857$	8.34473e6i	0.388115i	0.980990	+	0.194057i	$0.0621648\pi$
$857$	8.34473e6i	0.388115i	−0.980990	+	0.194057i	$0.937835\pi$
$858$	18642.2i	0 0.000864528i
$859$	4.07521e7	1.88437	0.942187	−	0.335088i	$-0.108766\pi$
$859$	4.07521e7	1.88437	0.942187	+	0.335088i	$0.108766\pi$
$860$	0	0
$861$	−1.63625e6	−0.0752216
$862$	5.26056e6i	0.241137i
$863$	− 1.73853e7i	− 0.794614i	−0.917686	−	0.397307i	$-0.869945\pi$
$863$	− 1.73853e7i	− 0.794614i	0.917686	−	0.397307i	$-0.130055\pi$
$864$	3.02570e6	0.137893
$865$	0	0
$866$	153881.	0.00697252
$867$	− 513667.i	− 0.0232078i
$868$	− 6.76243e6i	− 0.304652i
$869$	−707656.	−0.0317887
$870$	0	0
$871$	5.84063e6	0.260864
$872$	1.24560e7i	0.554738i
$873$	3.32710e7i	1.47751i
$874$	4.12500e6	0.182661
$875$	0	0
$876$	−19280.7	−0.000848912 0
$877$	− 2.58279e6i	− 0.113394i	−0.998391	−	0.0566971i	$-0.981943\pi$
$877$	− 2.58279e6i	− 0.113394i	0.998391	−	0.0566971i	$-0.0180569\pi$
$878$	− 3.41370e7i	− 1.49447i
$879$	−3332.63	−0.000145484 0
$880$	0	0
$881$	−1.66814e7	−0.724090	−0.362045	−	0.932161i	$-0.617921\pi$
$881$	−1.66814e7	−0.724090	−0.362045	+	0.932161i	$0.617921\pi$
$882$	935291.i	0.0404833i
$883$	2.36384e7i	1.02027i	0.860093	+	0.510137i	$0.170405\pi$
$883$	2.36384e7i	1.02027i	−0.860093	+	0.510137i	$0.829595\pi$
$884$	1.98818e6	0.0855708
$885$	0	0
$886$	1.49632e7	0.640382
$887$	5.25660e6i	0.224334i	0.993689	+	0.112167i	$0.0357792\pi$
$887$	5.25660e6i	0.224334i	−0.993689	+	0.112167i	$0.964221\pi$
$888$	1.48939e6i	0.0633833i
$889$	3.14345e7	1.33399
$890$	0	0
$891$	−847029.	−0.0357441
$892$	− 652988.i	− 0.0274785i
$893$	− 5.32625e6i	− 0.223508i
$894$	−539638.	−0.0225818
$895$	0	0
$896$	−4.53994e6	−0.188921
$897$	1.16605e6i	0.0483876i
$898$	− 3.62806e7i	− 1.50135i
$899$	2.41549e6	0.0996795
$900$	0	0
$901$	8.20763e6	0.336826
$902$	− 537389.i	− 0.0219924i
$903$	1.75412e6i	0.0715879i
$904$	−2.25461e7	−0.917595
$905$	0	0
$906$	3.63756e6	0.147228
$907$	− 3.22789e7i	− 1.30287i	−0.758705	−	0.651435i	$-0.774167\pi$
$907$	− 3.22789e7i	− 1.30287i	0.758705	−	0.651435i	$-0.225833\pi$
$908$	− 1.25094e6i	− 0.0503525i
$909$	3.34438e7	1.34247
$910$	0	0
$911$	4.20975e7	1.68058	0.840292	−	0.542134i	$-0.182383\pi$
$911$	4.20975e7	1.68058	0.840292	+	0.542134i	$0.182383\pi$
$912$	188401.i	0.00750061i
$913$	1.10924e6i	0.0440400i
$914$	−1.52897e7	−0.605389
$915$	0	0
$916$	−1.28693e7	−0.506775
$917$	− 1.30051e7i	− 0.510728i
$918$	3.77910e6i	0.148007i
$919$	2.19460e7	0.857168	0.428584	−	0.903502i	$-0.359013\pi$
$919$	2.19460e7	0.857168	0.428584	+	0.903502i	$0.359013\pi$
$920$	0	0
$921$	654407.	0.0254213
$922$	2.13767e7i	0.828157i
$923$	− 6.97221e6i	− 0.269380i
$924$	34186.0	0.00131725
$925$	0	0
$926$	−3.03205e7	−1.16201
$927$	2.36765e7i	0.904937i
$928$	1.93845e6i	0.0738899i
$929$	1.35921e7	0.516710	0.258355	−	0.966050i	$-0.416820\pi$
$929$	1.35921e7	0.516710	0.258355	+	0.966050i	$0.416820\pi$
$930$	0	0
$931$	182330.	0.00689421
$932$	7.05637e6i	0.266098i
$933$	3.14116e6i	0.118137i
$934$	−1.43297e7	−0.537489
$935$	0	0
$936$	8.00263e6	0.298568
$937$	− 3.01018e7i	− 1.12007i	−0.828470	−	0.560033i	$-0.810788\pi$
$937$	− 3.01018e7i	− 1.12007i	0.828470	−	0.560033i	$-0.189212\pi$
$938$	1.99122e7i	0.738944i
$939$	2.68279e6	0.0992938
$940$	0	0
$941$	−1.06275e7	−0.391252	−0.195626	−	0.980679i	$-0.562674\pi$
$941$	−1.06275e7	−0.391252	−0.195626	+	0.980679i	$0.562674\pi$
$942$	− 665503.i	− 0.0244356i
$943$	− 3.36130e7i	− 1.23091i
$944$	−2.01840e7	−0.737187
$945$	0	0
$946$	−576099.	−0.0209300
$947$	2.41709e7i	0.875826i	0.899017	+	0.437913i	$0.144282\pi$
$947$	2.41709e7i	0.875826i	−0.899017	+	0.437913i	$0.855718\pi$
$948$	− 870924.i	− 0.0314745i
$949$	−178535.	−0.00643514
$950$	0	0
$951$	3.25282e6	0.116630
$952$	2.61580e7i	0.935432i
$953$	4.27043e7i	1.52314i	0.648084	+	0.761569i	$0.275570\pi$
$953$	4.27043e7i	1.52314i	−0.648084	+	0.761569i	$0.724430\pi$
$954$	8.56063e6	0.304533
$955$	0	0
$956$	1.85375e6	0.0656003
$957$	12211.0i	0 0.000430993i
$958$	3.04842e7i	1.07315i
$959$	4.55184e6	0.159823
$960$	0	0
$961$	−5.74655e6	−0.200724
$962$	3.57371e6i	0.124503i
$963$	6.45276e6i	0.224223i
$964$	7.73348e6	0.268029
$965$	0	0
$966$	−3.97535e6	−0.137067
$967$	4.42692e7i	1.52242i	0.648504	+	0.761211i	$0.275395\pi$
$967$	4.42692e7i	1.52242i	−0.648504	+	0.761211i	$0.724605\pi$
$968$	− 3.16875e7i	− 1.08693i
$969$	366332.	0.0125333
$970$	0	0
$971$	3.88962e7	1.32391	0.661956	−	0.749542i	$-0.269726\pi$
$971$	3.88962e7	1.32391	0.661956	+	0.749542i	$0.269726\pi$
$972$	− 3.18606e6i	− 0.108166i
$973$	1.92816e7i	0.652922i
$974$	−1.85417e7	−0.626257
$975$	0	0
$976$	9.82357e6	0.330099
$977$	2.71611e7i	0.910354i	0.890401	+	0.455177i	$0.150424\pi$
$977$	2.71611e7i	0.910354i	−0.890401	+	0.455177i	$0.849576\pi$
$978$	− 1.67862e6i	− 0.0561184i
$979$	145319.	0.00484580
$980$	0	0
$981$	1.51946e7	0.504099
$982$	− 9.59904e6i	− 0.317650i
$983$	− 1.98048e7i	− 0.653714i	−0.945074	−	0.326857i	$-0.894011\pi$
$983$	− 1.98048e7i	− 0.653714i	0.945074	−	0.326857i	$-0.105989\pi$
$984$	2.55232e6	0.0840326
$985$	0	0
$986$	−2.42113e6	−0.0793096
$987$	5.13301e6i	0.167718i
$988$	− 404255.i	− 0.0131754i
$989$	−3.60343e7	−1.17145
$990$	0	0
$991$	−1.44104e7	−0.466115	−0.233057	−	0.972463i	$-0.574873\pi$
$991$	−1.44104e7	−0.466115	−0.233057	+	0.972463i	$0.574873\pi$
$992$	1.83635e7i	0.592483i
$993$	1.10222e6i	0.0354727i
$994$	2.37700e7	0.763068
$995$	0	0
$996$	−1.36516e6	−0.0436048
$997$	− 3.16635e7i	− 1.00884i	−0.863459	−	0.504418i	$-0.831707\pi$
$997$	− 3.16635e7i	− 1.00884i	0.863459	−	0.504418i	$-0.168293\pi$
$998$	− 2.72195e7i	− 0.865074i
$999$	3.65378e6	0.115832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.b.274.4		4
5.2	odd	4		13.6.a.a.1.1	✓	2
5.3	odd	4		325.6.a.b.1.2		2
5.4	even	2	inner	325.6.b.b.274.1		4
15.2	even	4		117.6.a.c.1.2		2
20.7	even	4		208.6.a.h.1.1		2
35.27	even	4		637.6.a.a.1.1		2
40.27	even	4		832.6.a.i.1.2		2
40.37	odd	4		832.6.a.p.1.1		2
65.12	odd	4		169.6.a.a.1.2		2
65.47	even	4		169.6.b.a.168.1		4
65.57	even	4		169.6.b.a.168.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	5.2	odd	4
117.6.a.c.1.2		2	15.2	even	4
169.6.a.a.1.2		2	65.12	odd	4
169.6.b.a.168.1		4	65.47	even	4
169.6.b.a.168.4		4	65.57	even	4
208.6.a.h.1.1		2	20.7	even	4
325.6.a.b.1.2		2	5.3	odd	4
325.6.b.b.274.1		4	5.4	even	2	inner
325.6.b.b.274.4		4	1.1	even	1	trivial
637.6.a.a.1.1		2	35.27	even	4
832.6.a.i.1.2		2	40.27	even	4
832.6.a.p.1.1		2	40.37	odd	4

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.56155i q^{2} -1.63068i q^{3} +11.1922 q^{4} +7.43845 q^{6} -126.309i q^{7} +197.024i q^{8} +240.341 q^{9} +O(q^{10})\)	\(q+4.56155i q^{2} -1.63068i q^{3} +11.1922 q^{4} +7.43845 q^{6} -126.309i q^{7} +197.024i q^{8} +240.341 q^{9} -14.8296 q^{11} -18.2510i q^{12} -169.000i q^{13} +576.164 q^{14} -540.582 q^{16} +1051.12i q^{17} +1096.33i q^{18} +213.723 q^{19} -205.969 q^{21} -67.6458i q^{22} -4231.16i q^{23} +321.283 q^{24} +770.902 q^{26} -788.176i q^{27} -1413.68i q^{28} +504.955 q^{29} +4783.58 q^{31} +3838.86i q^{32} +24.1823i q^{33} -4794.75 q^{34} +2689.95 q^{36} +4635.74i q^{37} +974.911i q^{38} -275.585 q^{39} +7944.15 q^{41} -939.541i q^{42} -8516.41i q^{43} -165.976 q^{44} +19300.7 q^{46} -24921.2i q^{47} +881.518i q^{48} +853.113 q^{49} +1714.05 q^{51} -1891.49i q^{52} -7808.46i q^{53} +3595.31 q^{54} +24885.8 q^{56} -348.515i q^{57} +2303.38i q^{58} +37337.5 q^{59} -18172.2 q^{61} +21820.5i q^{62} -30357.1i q^{63} -34809.8 q^{64} -110.309 q^{66} +34559.9i q^{67} +11764.4i q^{68} -6899.68 q^{69} +41255.7 q^{71} +47352.8i q^{72} -1056.42i q^{73} -21146.2 q^{74} +2392.04 q^{76} +1873.10i q^{77} -1257.10i q^{78} +47719.3 q^{79} +57117.6 q^{81} +36237.7i q^{82} -74799.0i q^{83} -2305.26 q^{84} +38848.0 q^{86} -823.421i q^{87} -2921.78i q^{88} -9799.26 q^{89} -21346.2 q^{91} -47356.1i q^{92} -7800.50i q^{93} +113679. q^{94} +6259.97 q^{96} +138432. i q^{97} +3891.52i q^{98} -3564.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9}+O(q^{10}) \) 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9	\( 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9} - 752 q^{11} + 1010 q^{14} + 930 q^{16} + 624 q^{19} + 8148 q^{21} + 2118 q^{24} + 1690 q^{26} + 1624 q^{29} + 15440 q^{31} - 10974 q^{34} - 23396 q^{36} - 9464 q^{39} + 15680 q^{41} - 23308 q^{44} + 37192 q^{46} - 17368 q^{49} + 86696 q^{51} + 2350 q^{54} + 40690 q^{56} + 77872 q^{59} + 3968 q^{61} - 6434 q^{64} - 8572 q^{66} + 70960 q^{69} + 134792 q^{71} - 53010 q^{74} + 11036 q^{76} + 110592 q^{79} + 185524 q^{81} + 267662 q^{84} + 68106 q^{86} - 233016 q^{89} + 12168 q^{91} + 214250 q^{94} + 82986 q^{96} + 319616 q^{99}+O(q^{100}) \) 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9 - 752 * q^11 + 1010 * q^14 + 930 * q^16 + 624 * q^19 + 8148 * q^21 + 2118 * q^24 + 1690 * q^26 + 1624 * q^29 + 15440 * q^31 - 10974 * q^34 - 23396 * q^36 - 9464 * q^39 + 15680 * q^41 - 23308 * q^44 + 37192 * q^46 - 17368 * q^49 + 86696 * q^51 + 2350 * q^54 + 40690 * q^56 + 77872 * q^59 + 3968 * q^61 - 6434 * q^64 - 8572 * q^66 + 70960 * q^69 + 134792 * q^71 - 53010 * q^74 + 11036 * q^76 + 110592 * q^79 + 185524 * q^81 + 267662 * q^84 + 68106 * q^86 - 233016 * q^89 + 12168 * q^91 + 214250 * q^94 + 82986 * q^96 + 319616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more