LMFDB - Embedded newform 325.6.b.b.274.1

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

$q$-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	$-0.867902\pi$
$2$	− 4.56155i	− 0.806376i	0.915117	−	0.403188i	$-0.132098\pi$
$3$	1.63068i	0.104608i	0.998631	+	0.0523042i	$0.0166565\pi$
$3$	1.63068i	0.104608i	−0.998631	+	0.0523042i	$0.983343\pi$
$4$	11.1922	0.349757
$5$	0	0
$5$	0	0
$6$	7.43845	0.0843537
$7$	126.309i	0.974290i	0.873321	+	0.487145i	$0.161962\pi$
$7$	126.309i	0.974290i	−0.873321	+	0.487145i	$0.838038\pi$
$8$	− 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	$-0.854602\pi$
$17$	− 1051.12i	− 0.882126i	0.897476	−	0.441063i	$-0.145398\pi$
$18$	− 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	$0.313892\pi$
$23$	4231.16i	1.66778i	−0.551928	+	0.833892i	$0.686108\pi$
$24$	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	$-0.910214\pi$
$37$	− 4635.74i	− 0.556692i	0.960481	−	0.278346i	$-0.0897862\pi$
$38$	− 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	$0.114226\pi$
$43$	8516.41i	0.702401i	−0.936300	+	0.351201i	$0.885774\pi$
$44$	−165.976	−0.0129245
$45$	0	0
$46$	19300.7	1.34486
$47$	24921.2i	1.64560i	0.568330	+	0.822801i	$0.307590\pi$
$47$	24921.2i	1.64560i	−0.568330	+	0.822801i	$0.692410\pi$
$48$	− 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	$0.0611463\pi$
$53$	7808.46i	0.381835i	−0.981606	+	0.190917i	$0.938854\pi$
$54$	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	$-0.844153\pi$
$67$	− 34559.9i	− 0.940559i	0.882518	−	0.470279i	$-0.155847\pi$
$68$	− 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	$0.00369283\pi$
$73$	1056.42i	0.0232022i	−0.999933	+	0.0116011i	$0.996307\pi$
$74$	−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	$0.203203\pi$
$83$	74799.0i	1.19179i	−0.803061	+	0.595896i	$0.796797\pi$
$84$	−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	$-0.731528\pi$
$97$	− 138432.i	− 1.49385i	0.664906	−	0.746927i	$-0.268472\pi$
$98$	− 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	$-0.848754\pi$
$103$	− 98512.2i	− 0.914950i	0.889223	−	0.457475i	$-0.151246\pi$
$104$	33297.0	0.301871
$105$	0	0
$106$	35618.7	0.307902
$107$	− 26848.4i	− 0.226704i	−0.993555	−	0.113352i	$-0.963841\pi$
$107$	− 26848.4i	− 0.226704i	0.993555	−	0.113352i	$-0.0361587\pi$
$108$	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	$-0.861494\pi$
$113$	− 114434.i	− 0.843058i	0.906815	−	0.421529i	$-0.138506\pi$
$114$	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	$-0.759978\pi$
$127$	− 248871.i	− 1.36919i	0.728922	−	0.684596i	$-0.240022\pi$
$128$	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	$-0.973863\pi$
$137$	− 36037.4i	− 0.164041i	0.996631	−	0.0820204i	$-0.0261373\pi$
$138$	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	$0.0462667\pi$
$157$	89467.9i	0.289680i	−0.989455	+	0.144840i	$0.953733\pi$
$158$	− 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	$0.107939\pi$
$163$	225668.i	0.665275i	−0.943055	+	0.332637i	$0.892061\pi$
$164$	88912.8	0.258140
$165$	0	0
$166$	341200.	0.961033
$167$	209528.i	0.581367i	0.956819	+	0.290683i	$0.0938827\pi$
$167$	209528.i	0.581367i	−0.956819	+	0.290683i	$0.906117\pi$
$168$	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	$0.201209\pi$
$173$	465184.i	1.18171i	−0.806779	+	0.590853i	$0.798791\pi$
$174$	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	$0.221022\pi$
$193$	662265.i	1.27979i	−0.768463	+	0.639895i	$0.778978\pi$
$194$	−631467.	−1.20461
$195$	0	0
$196$	9548.24	0.0177535
$197$	666464.i	1.22352i	0.791043	+	0.611760i	$0.209538\pi$
$197$	666464.i	1.22352i	−0.791043	+	0.611760i	$0.790462\pi$
$198$	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	$0.0125071\pi$
$223$	58342.9i	0.0785644i	−0.999228	+	0.0392822i	$0.987493\pi$
$224$	484882.	0.645678
$225$	0	0
$226$	−521995.	−0.679822
$227$	111768.i	0.143964i	0.997406	+	0.0719820i	$0.0229324\pi$
$227$	111768.i	0.143964i	−0.997406	+	0.0719820i	$0.977068\pi$
$228$	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	$-0.875785\pi$
$233$	− 630470.i	− 0.760807i	0.924821	−	0.380404i	$-0.124215\pi$
$234$	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	$0.0389158\pi$
$257$	258260.i	0.243907i	−0.992536	+	0.121953i	$0.961084\pi$
$258$	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	$0.178058\pi$
$263$	1.19053e6i	1.06133i	−0.847582	+	0.530665i	$0.821942\pi$
$264$	−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	$0.128128\pi$
$277$	1.00054e6i	0.783490i	−0.920074	+	0.391745i	$0.871872\pi$
$278$	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	$0.156706\pi$
$283$	1.27363e6i	0.945319i	−0.881245	+	0.472660i	$0.843294\pi$
$284$	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	$0.000221344\pi$
$293$	2043.70i	0.00139075i	−1.00000		0.000695374i	$0.999779\pi$
$294$	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	$-0.961227\pi$
$307$	− 401308.i	− 0.243014i	0.992591	−	0.121507i	$-0.0387728\pi$
$308$	− 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	$-0.842593\pi$
$313$	− 1.64519e6i	− 0.949196i	0.880203	−	0.474598i	$-0.157407\pi$
$314$	408113.	0.233591
$315$	0	0
$316$	534085.	0.300880
$317$	− 1.99476e6i	− 1.11492i	−0.830205	−	0.557459i	$-0.811777\pi$
$317$	− 1.99476e6i	− 1.11492i	0.830205	−	0.557459i	$-0.188223\pi$
$318$	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	$-0.828848\pi$
$337$	− 2.13552e6i	− 1.02430i	0.858895	−	0.512152i	$-0.171152\pi$
$338$	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	$-0.805596\pi$
$347$	− 2.57257e6i	− 1.14695i	0.819225	−	0.573473i	$-0.194404\pi$
$348$	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	$-0.855898\pi$
$353$	− 2.04810e6i	− 0.874810i	0.899265	−	0.437405i	$-0.144102\pi$
$354$	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	$-0.730661\pi$
$367$	− 3.86389e6i	− 1.49747i	0.662867	−	0.748737i	$-0.269339\pi$
$368$	− 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	$0.0941841\pi$
$373$	1.56702e6i	0.583179i	−0.956544	+	0.291589i	$0.905816\pi$
$374$	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	$-0.977782\pi$
$383$	− 400432.i	− 0.139486i	0.997565	−	0.0697432i	$-0.0222180\pi$
$384$	−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	$-0.994783\pi$
$397$	− 102926.i	− 0.0327753i	0.999866	−	0.0163877i	$-0.00521659\pi$
$398$	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	$0.00137617\pi$
$433$	33734.3i	0.00864673i	−0.999991	+	0.00432337i	$0.998624\pi$
$434$	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	$0.129974\pi$
$443$	3.28028e6i	0.794148i	−0.917786	+	0.397074i	$0.870026\pi$
$444$	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	$-0.877513\pi$
$457$	− 3.35187e6i	− 0.750753i	0.926872	−	0.375377i	$-0.122487\pi$
$458$	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	$-0.743907\pi$
$463$	− 6.64697e6i	− 1.44102i	0.693442	−	0.720512i	$-0.256093\pi$
$464$	−272969.	−0.0588599
$465$	0	0
$466$	−2.87592e6	−0.613497
$467$	− 3.14141e6i	− 0.666549i	−0.942830	−	0.333275i	$-0.891846\pi$
$467$	− 3.14141e6i	− 0.666549i	0.942830	−	0.333275i	$-0.108154\pi$
$468$	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	$-0.873057\pi$
$487$	− 4.06478e6i	− 0.776631i	0.921526	−	0.388316i	$-0.126943\pi$
$488$	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	$0.344085\pi$
$503$	1.00144e7i	1.76483i	−0.470467	+	0.882417i	$0.655915\pi$
$504$	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	$-0.770030\pi$
$523$	− 8.27263e6i	− 1.32248i	0.750174	−	0.661240i	$-0.229970\pi$
$524$	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	$-0.985884\pi$
$547$	− 620452.i	− 0.0886624i	0.999017	−	0.0443312i	$-0.0141157\pi$
$548$	− 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	$0.0857471\pi$
$557$	3.89737e6i	0.532272i	−0.963935	+	0.266136i	$0.914253\pi$
$558$	− 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	$-0.989190\pi$
$563$	− 510725.i	− 0.0679073i	0.999423	−	0.0339536i	$-0.0108099\pi$
$564$	−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	$-0.976810\pi$
$577$	− 1.16423e6i	− 0.145579i	0.997347	−	0.0727896i	$-0.0231902\pi$
$578$	− 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	$0.128950\pi$
$587$	6.58038e6i	0.788234i	−0.919060	+	0.394117i	$0.871050\pi$
$588$	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	$0.0356522\pi$
$593$	1.91423e6i	0.223541i	−0.993734	+	0.111771i	$0.964348\pi$
$594$	−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	$-0.997891\pi$
$607$	− 120274.i	− 0.0132495i	0.999978	−	0.00662474i	$-0.00210874\pi$
$608$	− 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	$-0.742650\pi$
$613$	− 1.34576e7i	− 1.44649i	0.690592	−	0.723245i	$-0.257350\pi$
$614$	−1.83059e6	−0.195961
$615$	0	0
$616$	−369046.	−0.0391858
$617$	− 6.84879e6i	− 0.724270i	−0.932126	−	0.362135i	$-0.882048\pi$
$617$	− 6.84879e6i	− 0.724270i	0.932126	−	0.362135i	$-0.117952\pi$
$618$	− 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	$-0.730111\pi$
$643$	− 1.57235e7i	− 1.49976i	0.661574	−	0.749880i	$-0.269889\pi$
$644$	−5.98149e6	−0.568322
$645$	0	0
$646$	−1.02475e6	−0.0966132
$647$	− 1.58173e7i	− 1.48549i	−0.669572	−	0.742747i	$-0.733523\pi$
$647$	− 1.58173e7i	− 1.48549i	0.669572	−	0.742747i	$-0.266477\pi$
$648$	− 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	$-0.921618\pi$
$653$	− 5.31229e6i	− 0.487527i	0.969835	−	0.243763i	$-0.0783820\pi$
$654$	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	$0.190678\pi$
$673$	1.32503e7i	1.12769i	−0.825881	+	0.563844i	$0.809322\pi$
$674$	−9.74129e6	−0.825975
$675$	0	0
$676$	−319661.	−0.0269044
$677$	2.23310e7i	1.87257i	0.351245	+	0.936284i	$0.385758\pi$
$677$	2.23310e7i	1.87257i	−0.351245	+	0.936284i	$0.614242\pi$
$678$	− 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	$0.194757\pi$
$683$	1.40049e7i	1.14876i	−0.818588	+	0.574380i	$0.805243\pi$
$684$	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	$-0.945771\pi$
$727$	− 4.83218e6i	− 0.339084i	0.985523	−	0.169542i	$-0.0542288\pi$
$728$	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	$0.324513\pi$
$733$	2.47827e7i	1.70368i	−0.523803	+	0.851840i	$0.675487\pi$
$734$	−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	$-0.775469\pi$
$743$	− 1.95117e7i	− 1.29665i	0.761362	−	0.648327i	$-0.224531\pi$
$744$	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	$0.0124094\pi$
$757$	1.22902e6i	0.0779508i	−0.999240	+	0.0389754i	$0.987591\pi$
$758$	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	$-0.859956\pi$
$773$	− 1.41511e7i	− 0.851807i	0.904769	−	0.425903i	$-0.140044\pi$
$774$	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	$-0.990555\pi$
$787$	− 1.03095e6i	− 0.0593338i	0.999560	−	0.0296669i	$-0.00944465\pi$
$788$	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	$-0.869131\pi$
$797$	− 1.43337e7i	− 0.799303i	0.916667	−	0.399651i	$-0.130869\pi$
$798$	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	$-0.974682\pi$
$823$	− 3.08787e6i	− 0.158913i	0.996838	−	0.0794564i	$-0.0253185\pi$
$824$	−1.94092e7	−0.995842
$825$	0	0
$826$	2.15125e7	1.09709
$827$	6.89555e6i	0.350595i	0.984516	+	0.175297i	$0.0560886\pi$
$827$	6.89555e6i	0.350595i	−0.984516	+	0.175297i	$0.943911\pi$
$828$	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	$0.101195\pi$
$853$	1.32853e7i	0.625170i	−0.949890	+	0.312585i	$0.898805\pi$
$854$	−1.04702e7	−0.491257
$855$	0	0
$856$	−5.28976e6	−0.246747
$857$	− 8.34473e6i	− 0.388115i	−0.980990	−	0.194057i	$-0.937835\pi$
$857$	− 8.34473e6i	− 0.388115i	0.980990	−	0.194057i	$-0.0621648\pi$
$858$	− 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	$0.130055\pi$
$863$	1.73853e7i	0.794614i	−0.917686	+	0.397307i	$0.869945\pi$
$864$	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	$0.0180569\pi$
$877$	2.58279e6i	0.113394i	−0.998391	+	0.0566971i	$0.981943\pi$
$878$	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	$-0.829595\pi$
$883$	− 2.36384e7i	− 1.02027i	0.860093	−	0.510137i	$-0.170405\pi$
$884$	1.98818e6	0.0855708
$885$	0	0
$886$	1.49632e7	0.640382
$887$	− 5.25660e6i	− 0.224334i	−0.993689	−	0.112167i	$-0.964221\pi$
$887$	− 5.25660e6i	− 0.224334i	0.993689	−	0.112167i	$-0.0357792\pi$
$888$	− 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	$0.225833\pi$
$907$	3.22789e7i	1.30287i	−0.758705	+	0.651435i	$0.774167\pi$
$908$	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	$0.189212\pi$
$937$	3.01018e7i	1.12007i	−0.828470	+	0.560033i	$0.810788\pi$
$938$	− 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	$-0.855718\pi$
$947$	− 2.41709e7i	− 0.875826i	0.899017	−	0.437913i	$-0.144282\pi$
$948$	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	$-0.724430\pi$
$953$	− 4.27043e7i	− 1.52314i	0.648084	−	0.761569i	$-0.275570\pi$
$954$	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	$-0.724605\pi$
$967$	− 4.42692e7i	− 1.52242i	0.648504	−	0.761211i	$-0.275395\pi$
$968$	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	$-0.849576\pi$
$977$	− 2.71611e7i	− 0.910354i	0.890401	−	0.455177i	$-0.150424\pi$
$978$	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	$0.105989\pi$
$983$	1.98048e7i	0.653714i	−0.945074	+	0.326857i	$0.894011\pi$
$984$	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	$0.168293\pi$
$997$	3.16635e7i	1.00884i	−0.863459	+	0.504418i	$0.831707\pi$
$998$	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.274.1		4
5.2	odd	4		325.6.a.b.1.2		2
5.3	odd	4		13.6.a.a.1.1	✓	2
5.4	even	2	inner	325.6.b.b.274.4		4
15.8	even	4		117.6.a.c.1.2		2
20.3	even	4		208.6.a.h.1.1		2
35.13	even	4		637.6.a.a.1.1		2
40.3	even	4		832.6.a.i.1.2		2
40.13	odd	4		832.6.a.p.1.1		2
65.8	even	4		169.6.b.a.168.1		4
65.18	even	4		169.6.b.a.168.4		4
65.38	odd	4		169.6.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	5.3	odd	4
117.6.a.c.1.2		2	15.8	even	4
169.6.a.a.1.2		2	65.38	odd	4
169.6.b.a.168.1		4	65.8	even	4
169.6.b.a.168.4		4	65.18	even	4
208.6.a.h.1.1		2	20.3	even	4
325.6.a.b.1.2		2	5.2	odd	4
325.6.b.b.274.1		4	1.1	even	1	trivial
325.6.b.b.274.4		4	5.4	even	2	inner
637.6.a.a.1.1		2	35.13	even	4
832.6.a.i.1.2		2	40.3	even	4
832.6.a.p.1.1		2	40.13	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