LMFDB - Embedded newform 325.6.b.i.274.8

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:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.8
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.78432i q^{2} -8.20788i q^{3} +17.6789 q^{4} -31.0612 q^{6} +88.9969i q^{7} -188.001i q^{8} +175.631 q^{9} +O(q^{10})$	\(q-3.78432i q^{2} -8.20788i q^{3} +17.6789 q^{4} -31.0612 q^{6} +88.9969i q^{7} -188.001i q^{8} +175.631 q^{9} -156.575 q^{11} -145.107i q^{12} -169.000i q^{13} +336.793 q^{14} -145.729 q^{16} +447.890i q^{17} -664.642i q^{18} +269.669 q^{19} +730.476 q^{21} +592.531i q^{22} -1371.05i q^{23} -1543.09 q^{24} -639.550 q^{26} -3436.07i q^{27} +1573.37i q^{28} +3698.11 q^{29} +797.974 q^{31} -5464.54i q^{32} +1285.15i q^{33} +1694.96 q^{34} +3104.97 q^{36} -4394.83i q^{37} -1020.51i q^{38} -1387.13 q^{39} +14320.9 q^{41} -2764.35i q^{42} -11337.1i q^{43} -2768.09 q^{44} -5188.50 q^{46} -9974.03i q^{47} +1196.13i q^{48} +8886.54 q^{49} +3676.23 q^{51} -2987.74i q^{52} -11587.1i q^{53} -13003.2 q^{54} +16731.5 q^{56} -2213.41i q^{57} -13994.8i q^{58} -7133.45 q^{59} +16629.8 q^{61} -3019.79i q^{62} +15630.6i q^{63} -25342.9 q^{64} +4863.42 q^{66} +4211.88i q^{67} +7918.22i q^{68} -11253.4 q^{69} +12871.4 q^{71} -33018.7i q^{72} +13097.1i q^{73} -16631.4 q^{74} +4767.46 q^{76} -13934.7i q^{77} +5249.35i q^{78} -74729.2 q^{79} +14475.4 q^{81} -54194.7i q^{82} -8547.83i q^{83} +12914.0 q^{84} -42903.0 q^{86} -30353.6i q^{87} +29436.3i q^{88} -59586.0 q^{89} +15040.5 q^{91} -24238.8i q^{92} -6549.68i q^{93} -37744.9 q^{94} -44852.3 q^{96} -17376.2i q^{97} -33629.5i q^{98} -27499.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * 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 $

$2$	− 3.78432i	− 0.668979i	−0.942399	−	0.334490i	$-0.891436\pi$
$2$	− 3.78432i	− 0.668979i	0.942399	−	0.334490i	$-0.108564\pi$
$3$	− 8.20788i	− 0.526536i	−0.964723	−	0.263268i	$-0.915200\pi$
$3$	− 8.20788i	− 0.526536i	0.964723	−	0.263268i	$-0.0848003\pi$
$4$	17.6789	0.552467
$5$	0	0
$5$	0	0
$6$	−31.0612	−0.352241
$7$	88.9969i	0.686483i	0.939247	+	0.343242i	$0.111525\pi$
$7$	88.9969i	0.686483i	−0.939247	+	0.343242i	$0.888475\pi$
$8$	− 188.001i	− 1.03857i
$9$	175.631	0.722760
$10$	0	0
$11$	−156.575	−0.390159	−0.195080	−	0.980787i	$-0.562497\pi$
$11$	−156.575	−0.390159	−0.195080	+	0.980787i	$0.562497\pi$
$12$	− 145.107i	− 0.290893i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	336.793	0.459243
$15$	0	0
$16$	−145.729	−0.142314
$17$	447.890i	0.375880i	0.982181	+	0.187940i	$0.0601810\pi$
$17$	447.890i	0.375880i	−0.982181	+	0.187940i	$0.939819\pi$
$18$	− 664.642i	− 0.483512i
$19$	269.669	0.171375	0.0856875	−	0.996322i	$-0.472691\pi$
$19$	269.669	0.171375	0.0856875	+	0.996322i	$0.472691\pi$
$20$	0	0
$21$	730.476	0.361458
$22$	592.531i	0.261008i
$23$	− 1371.05i	− 0.540424i	−0.962801	−	0.270212i	$-0.912906\pi$
$23$	− 1371.05i	− 0.540424i	0.962801	−	0.270212i	$-0.0870938\pi$
$24$	−1543.09	−0.546843
$25$	0	0
$26$	−639.550	−0.185541
$27$	− 3436.07i	− 0.907095i
$28$	1573.37i	0.379259i
$29$	3698.11	0.816554	0.408277	−	0.912858i	$-0.366130\pi$
$29$	3698.11	0.816554	0.408277	+	0.912858i	$0.366130\pi$
$30$	0	0
$31$	797.974	0.149137	0.0745684	−	0.997216i	$-0.476242\pi$
$31$	797.974	0.149137	0.0745684	+	0.997216i	$0.476242\pi$
$32$	− 5464.54i	− 0.943363i
$33$	1285.15i	0.205433i
$34$	1694.96	0.251456
$35$	0	0
$36$	3104.97	0.399301
$37$	− 4394.83i	− 0.527761i	−0.964555	−	0.263881i	$-0.914997\pi$
$37$	− 4394.83i	− 0.527761i	0.964555	−	0.263881i	$-0.0850025\pi$
$38$	− 1020.51i	− 0.114646i
$39$	−1387.13	−0.146035
$40$	0	0
$41$	14320.9	1.33048	0.665242	−	0.746628i	$-0.268328\pi$
$41$	14320.9	1.33048	0.665242	+	0.746628i	$0.268328\pi$
$42$	− 2764.35i	− 0.241808i
$43$	− 11337.1i	− 0.935037i	−0.883983	−	0.467519i	$-0.845148\pi$
$43$	− 11337.1i	− 0.935037i	0.883983	−	0.467519i	$-0.154852\pi$
$44$	−2768.09	−0.215550
$45$	0	0
$46$	−5188.50	−0.361532
$47$	− 9974.03i	− 0.658607i	−0.944224	−	0.329303i	$-0.893186\pi$
$47$	− 9974.03i	− 0.658607i	0.944224	−	0.329303i	$-0.106814\pi$
$48$	1196.13i	0.0749331i
$49$	8886.54	0.528741
$50$	0	0
$51$	3676.23	0.197914
$52$	− 2987.74i	− 0.153227i
$53$	− 11587.1i	− 0.566611i	−0.959030	−	0.283305i	$-0.908569\pi$
$53$	− 11587.1i	− 0.566611i	0.959030	−	0.283305i	$-0.0914310\pi$
$54$	−13003.2	−0.606827
$55$	0	0
$56$	16731.5	0.712960
$57$	− 2213.41i	− 0.0902350i
$58$	− 13994.8i	− 0.546258i
$59$	−7133.45	−0.266790	−0.133395	−	0.991063i	$-0.542588\pi$
$59$	−7133.45	−0.266790	−0.133395	+	0.991063i	$0.542588\pi$
$60$	0	0
$61$	16629.8	0.572220	0.286110	−	0.958197i	$-0.407638\pi$
$61$	16629.8	0.572220	0.286110	+	0.958197i	$0.407638\pi$
$62$	− 3019.79i	− 0.0997694i
$63$	15630.6i	0.496163i
$64$	−25342.9	−0.773404
$65$	0	0
$66$	4863.42	0.137430
$67$	4211.88i	0.114628i	0.998356	+	0.0573138i	$0.0182536\pi$
$67$	4211.88i	0.114628i	−0.998356	+	0.0573138i	$0.981746\pi$
$68$	7918.22i	0.207661i
$69$	−11253.4	−0.284552
$70$	0	0
$71$	12871.4	0.303026	0.151513	−	0.988455i	$-0.451585\pi$
$71$	12871.4	0.303026	0.151513	+	0.988455i	$0.451585\pi$
$72$	− 33018.7i	− 0.750636i
$73$	13097.1i	0.287653i	0.989603	+	0.143826i	$0.0459407\pi$
$73$	13097.1i	0.287653i	−0.989603	+	0.143826i	$0.954059\pi$
$74$	−16631.4	−0.353061
$75$	0	0
$76$	4767.46	0.0946790
$77$	− 13934.7i	− 0.267838i
$78$	5249.35i	0.0976942i
$79$	−74729.2	−1.34717	−0.673586	−	0.739109i	$-0.735247\pi$
$79$	−74729.2	−1.34717	−0.673586	+	0.739109i	$0.735247\pi$
$80$	0	0
$81$	14475.4	0.245143
$82$	− 54194.7i	− 0.890066i
$83$	− 8547.83i	− 0.136195i	−0.997679	−	0.0680974i	$-0.978307\pi$
$83$	− 8547.83i	− 0.136195i	0.997679	−	0.0680974i	$-0.0216929\pi$
$84$	12914.0	0.199694
$85$	0	0
$86$	−42903.0	−0.625521
$87$	− 30353.6i	− 0.429945i
$88$	29436.3i	0.405207i
$89$	−59586.0	−0.797387	−0.398694	−	0.917084i	$-0.630536\pi$
$89$	−59586.0	−0.797387	−0.398694	+	0.917084i	$0.630536\pi$
$90$	0	0
$91$	15040.5	0.190396
$92$	− 24238.8i	− 0.298566i
$93$	− 6549.68i	− 0.0785258i
$94$	−37744.9	−0.440594
$95$	0	0
$96$	−44852.3	−0.496714
$97$	− 17376.2i	− 0.187510i	−0.995595	−	0.0937552i	$-0.970113\pi$
$97$	− 17376.2i	− 0.187510i	0.995595	−	0.0937552i	$-0.0298871\pi$
$98$	− 33629.5i	− 0.353716i
$99$	−27499.5	−0.281992
$100$	0	0
$101$	68669.7	0.669825	0.334913	−	0.942249i	$-0.391293\pi$
$101$	68669.7	0.669825	0.334913	+	0.942249i	$0.391293\pi$
$102$	− 13912.0i	− 0.132400i
$103$	− 191203.i	− 1.77584i	−0.460002	−	0.887918i	$-0.652151\pi$
$103$	− 191203.i	− 1.77584i	0.460002	−	0.887918i	$-0.347849\pi$
$104$	−31772.2	−0.288047
$105$	0	0
$106$	−43849.2	−0.379051
$107$	− 46117.6i	− 0.389410i	−0.980862	−	0.194705i	$-0.937625\pi$
$107$	− 46117.6i	− 0.389410i	0.980862	−	0.194705i	$-0.0623750\pi$
$108$	− 60746.1i	− 0.501140i
$109$	−187271.	−1.50975	−0.754873	−	0.655871i	$-0.772302\pi$
$109$	−187271.	−1.50975	−0.754873	+	0.655871i	$0.772302\pi$
$110$	0	0
$111$	−36072.2	−0.277885
$112$	− 12969.4i	− 0.0976959i
$113$	− 189262.i	− 1.39434i	−0.716907	−	0.697169i	$-0.754443\pi$
$113$	− 189262.i	− 1.39434i	0.716907	−	0.697169i	$-0.245557\pi$
$114$	−8376.25	−0.0603653
$115$	0	0
$116$	65378.7	0.451119
$117$	− 29681.6i	− 0.200458i
$118$	26995.2i	0.178477i
$119$	−39860.9	−0.258035
$120$	0	0
$121$	−136535.	−0.847776
$122$	− 62932.5i	− 0.382803i
$123$	− 117544.i	− 0.700547i
$124$	14107.3	0.0823931
$125$	0	0
$126$	59151.2	0.331923
$127$	183029.i	1.00695i	0.864009	+	0.503477i	$0.167946\pi$
$127$	183029.i	1.00695i	−0.864009	+	0.503477i	$0.832054\pi$
$128$	− 78959.8i	− 0.425972i
$129$	−93053.1	−0.492331
$130$	0	0
$131$	221035.	1.12534	0.562668	−	0.826683i	$-0.309775\pi$
$131$	221035.	1.12534	0.562668	+	0.826683i	$0.309775\pi$
$132$	22720.1i	0.113495i
$133$	23999.7i	0.117646i
$134$	15939.1	0.0766835
$135$	0	0
$136$	84203.7	0.390377
$137$	353821.i	1.61058i	0.592881	+	0.805290i	$0.297991\pi$
$137$	353821.i	1.61058i	−0.592881	+	0.805290i	$0.702009\pi$
$138$	42586.6i	0.190360i
$139$	−282625.	−1.24072	−0.620359	−	0.784318i	$-0.713013\pi$
$139$	−282625.	−1.24072	−0.620359	+	0.784318i	$0.713013\pi$
$140$	0	0
$141$	−81865.7	−0.346780
$142$	− 48709.4i	− 0.202718i
$143$	26461.2i	0.108211i
$144$	−25594.5	−0.102859
$145$	0	0
$146$	49563.7	0.192434
$147$	− 72939.7i	− 0.278401i
$148$	− 77695.9i	− 0.291571i
$149$	77579.6	0.286274	0.143137	−	0.989703i	$-0.454281\pi$
$149$	77579.6	0.286274	0.143137	+	0.989703i	$0.454281\pi$
$150$	0	0
$151$	−209693.	−0.748412	−0.374206	−	0.927346i	$-0.622085\pi$
$151$	−209693.	−0.748412	−0.374206	+	0.927346i	$0.622085\pi$
$152$	− 50698.0i	− 0.177985i
$153$	78663.3i	0.271671i
$154$	−52733.5	−0.179178
$155$	0	0
$156$	−24523.0	−0.0806793
$157$	127745.i	0.413612i	0.978382	+	0.206806i	$0.0663070\pi$
$157$	127745.i	0.413612i	−0.978382	+	0.206806i	$0.933693\pi$
$158$	282799.i	0.901230i
$159$	−95105.4	−0.298341
$160$	0	0
$161$	122019.	0.370992
$162$	− 54779.6i	− 0.163995i
$163$	− 33444.3i	− 0.0985947i	−0.998784	−	0.0492973i	$-0.984302\pi$
$163$	− 33444.3i	− 0.0985947i	0.998784	−	0.0492973i	$-0.0156982\pi$
$164$	253178.	0.735048
$165$	0	0
$166$	−32347.7	−0.0911115
$167$	− 317479.i	− 0.880893i	−0.897779	−	0.440447i	$-0.854820\pi$
$167$	− 317479.i	− 0.880893i	0.897779	−	0.440447i	$-0.145180\pi$
$168$	− 137330.i	− 0.375399i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	47362.2	0.123863
$172$	− 200427.i	− 0.516577i
$173$	− 84505.1i	− 0.214668i	−0.994223	−	0.107334i	$-0.965769\pi$
$173$	− 84505.1i	− 0.214668i	0.994223	−	0.107334i	$-0.0342314\pi$
$174$	−114868.	−0.287624
$175$	0	0
$176$	22817.6	0.0555249
$177$	58550.5i	0.140474i
$178$	225492.i	0.533435i
$179$	734183.	1.71266	0.856331	−	0.516428i	$-0.172739\pi$
$179$	734183.	1.71266	0.856331	+	0.516428i	$0.172739\pi$
$180$	0	0
$181$	363356.	0.824396	0.412198	−	0.911094i	$-0.364761\pi$
$181$	363356.	0.824396	0.412198	+	0.911094i	$0.364761\pi$
$182$	− 56918.0i	− 0.127371i
$183$	− 136496.i	− 0.301294i
$184$	−257759.	−0.561267
$185$	0	0
$186$	−24786.1	−0.0525321
$187$	− 70128.6i	− 0.146653i
$188$	− 176330.i	− 0.363858i
$189$	305800.	0.622705
$190$	0	0
$191$	54142.3	0.107387	0.0536937	−	0.998557i	$-0.482901\pi$
$191$	54142.3	0.107387	0.0536937	+	0.998557i	$0.482901\pi$
$192$	208011.i	0.407225i
$193$	208911.i	0.403709i	0.979416	+	0.201855i	$0.0646968\pi$
$193$	208911.i	0.403709i	−0.979416	+	0.201855i	$0.935303\pi$
$194$	−65757.1	−0.125441
$195$	0	0
$196$	157105.	0.292112
$197$	581675.i	1.06786i	0.845528	+	0.533931i	$0.179286\pi$
$197$	581675.i	1.06786i	−0.845528	+	0.533931i	$0.820714\pi$
$198$	104067.i	0.188646i
$199$	624788.	1.11841	0.559204	−	0.829030i	$-0.311107\pi$
$199$	624788.	1.11841	0.559204	+	0.829030i	$0.311107\pi$
$200$	0	0
$201$	34570.6	0.0603555
$202$	− 259868.i	− 0.448099i
$203$	329121.i	0.560551i
$204$	64991.8	0.109341
$205$	0	0
$206$	−723575.	−1.18800
$207$	− 240799.i	− 0.390597i
$208$	24628.2i	0.0394707i
$209$	−42223.6	−0.0668635
$210$	0	0
$211$	628250.	0.971464	0.485732	−	0.874108i	$-0.338553\pi$
$211$	628250.	0.971464	0.485732	+	0.874108i	$0.338553\pi$
$212$	− 204847.i	− 0.313034i
$213$	− 105647.i	− 0.159554i
$214$	−174524.	−0.260507
$215$	0	0
$216$	−645984.	−0.942079
$217$	71017.3i	0.102380i
$218$	708693.i	1.00999i
$219$	107500.	0.151459
$220$	0	0
$221$	75693.4	0.104250
$222$	136509.i	0.185899i
$223$	1.09757e6i	1.47798i	0.673715	+	0.738992i	$0.264698\pi$
$223$	1.09757e6i	1.47798i	−0.673715	+	0.738992i	$0.735302\pi$
$224$	486328.	0.647603
$225$	0	0
$226$	−716229.	−0.932783
$227$	161937.i	0.208584i	0.994547	+	0.104292i	$0.0332577\pi$
$227$	161937.i	0.208584i	−0.994547	+	0.104292i	$0.966742\pi$
$228$	− 39130.8i	− 0.0498518i
$229$	645210.	0.813040	0.406520	−	0.913642i	$-0.366742\pi$
$229$	645210.	0.813040	0.406520	+	0.913642i	$0.366742\pi$
$230$	0	0
$231$	−114375.	−0.141026
$232$	− 695248.i	− 0.848047i
$233$	839907.i	1.01354i	0.862081	+	0.506770i	$0.169161\pi$
$233$	839907.i	1.01354i	−0.862081	+	0.506770i	$0.830839\pi$
$234$	−112325.	−0.134102
$235$	0	0
$236$	−126112.	−0.147393
$237$	613369.i	0.709334i
$238$	150846.i	0.172620i
$239$	532168.	0.602635	0.301317	−	0.953524i	$-0.402574\pi$
$239$	532168.	0.602635	0.301317	+	0.953524i	$0.402574\pi$
$240$	0	0
$241$	−11282.2	−0.0125127	−0.00625633	−	0.999980i	$-0.501991\pi$
$241$	−11282.2	−0.0125127	−0.00625633	+	0.999980i	$0.501991\pi$
$242$	516692.i	0.567144i
$243$	− 953778.i	− 1.03617i
$244$	293998.	0.316133
$245$	0	0
$246$	−444824.	−0.468652
$247$	− 45574.1i	− 0.0475308i
$248$	− 150020.i	− 0.154889i
$249$	−70159.5	−0.0717114
$250$	0	0
$251$	−592652.	−0.593766	−0.296883	−	0.954914i	$-0.595947\pi$
$251$	−592652.	−0.593766	−0.296883	+	0.954914i	$0.595947\pi$
$252$	276332.i	0.274114i
$253$	214673.i	0.210851i
$254$	692639.	0.673632
$255$	0	0
$256$	−1.10978e6	−1.05837
$257$	− 161530.i	− 0.152552i	−0.997087	−	0.0762762i	$-0.975697\pi$
$257$	− 161530.i	− 0.152552i	0.997087	−	0.0762762i	$-0.0243031\pi$
$258$	352143.i	0.329359i
$259$	391126.	0.362299
$260$	0	0
$261$	649502.	0.590173
$262$	− 836466.i	− 0.752826i
$263$	377791.i	0.336793i	0.985719	+	0.168396i	$0.0538588\pi$
$263$	377791.i	0.336793i	−0.985719	+	0.168396i	$0.946141\pi$
$264$	241610.	0.213356
$265$	0	0
$266$	90822.6	0.0787027
$267$	489075.i	0.419853i
$268$	74461.6i	0.0633280i
$269$	689.090	0.000580624 0	0.000290312	−	1.00000i	$-0.499908\pi$
$269$	689.090	0.000580624 0	0.000290312		1.00000i	$0.499908\pi$
$270$	0	0
$271$	625466.	0.517345	0.258673	−	0.965965i	$-0.416715\pi$
$271$	625466.	0.517345	0.258673	+	0.965965i	$0.416715\pi$
$272$	− 65270.6i	− 0.0534928i
$273$	− 123450.i	− 0.100250i
$274$	1.33897e6	1.07744
$275$	0	0
$276$	−198949.	−0.157206
$277$	744000.i	0.582604i	0.956631	+	0.291302i	$0.0940885\pi$
$277$	744000.i	0.582604i	−0.956631	+	0.291302i	$0.905912\pi$
$278$	1.06954e6i	0.830015i
$279$	140149.	0.107790
$280$	0	0
$281$	1.43786e6	1.08631	0.543153	−	0.839634i	$-0.317230\pi$
$281$	1.43786e6	1.08631	0.543153	+	0.839634i	$0.317230\pi$
$282$	309806.i	0.231989i
$283$	1.24662e6i	0.925267i	0.886550	+	0.462634i	$0.153095\pi$
$283$	1.24662e6i	0.925267i	−0.886550	+	0.462634i	$0.846905\pi$
$284$	227553.	0.167412
$285$	0	0
$286$	100138.	0.0723907
$287$	1.27451e6i	0.913355i
$288$	− 959742.i	− 0.681825i
$289$	1.21925e6	0.858714
$290$	0	0
$291$	−142622.	−0.0987309
$292$	231543.i	0.158919i
$293$	− 1.74658e6i	− 1.18855i	−0.804261	−	0.594276i	$-0.797439\pi$
$293$	− 1.74658e6i	− 1.18855i	0.804261	−	0.594276i	$-0.202561\pi$
$294$	−276027.	−0.186244
$295$	0	0
$296$	−826231.	−0.548116
$297$	538004.i	0.353911i
$298$	− 293586.i	− 0.191511i
$299$	−231708.	−0.149887
$300$	0	0
$301$	1.00896e6	0.641888
$302$	793543.i	0.500672i
$303$	− 563632.i	− 0.352687i
$304$	−39298.6	−0.0243890
$305$	0	0
$306$	297687.	0.181742
$307$	− 239330.i	− 0.144928i	−0.997371	−	0.0724638i	$-0.976914\pi$
$307$	− 239330.i	− 0.144928i	0.997371	−	0.0724638i	$-0.0230862\pi$
$308$	− 246351.i	− 0.147972i
$309$	−1.56937e6	−0.935041
$310$	0	0
$311$	1.10338e6	0.646883	0.323441	−	0.946248i	$-0.395160\pi$
$311$	1.10338e6	0.646883	0.323441	+	0.946248i	$0.395160\pi$
$312$	260782.i	0.151667i
$313$	934831.i	0.539352i	0.962951	+	0.269676i	$0.0869165\pi$
$313$	934831.i	0.539352i	−0.962951	+	0.269676i	$0.913083\pi$
$314$	483426.	0.276698
$315$	0	0
$316$	−1.32113e6	−0.744268
$317$	2.65437e6i	1.48359i	0.670628	+	0.741794i	$0.266025\pi$
$317$	2.65437e6i	1.48359i	−0.670628	+	0.741794i	$0.733975\pi$
$318$	359909.i	0.199584i
$319$	−579033.	−0.318586
$320$	0	0
$321$	−378528.	−0.205038
$322$	− 461761.i	− 0.248186i
$323$	120782.i	0.0644164i
$324$	255910.	0.135433
$325$	0	0
$326$	−126564.	−0.0659578
$327$	1.53710e6i	0.794935i
$328$	− 2.69234e6i	− 1.38180i
$329$	887659.	0.452123
$330$	0	0
$331$	648335.	0.325259	0.162630	−	0.986687i	$-0.448002\pi$
$331$	648335.	0.325259	0.162630	+	0.986687i	$0.448002\pi$
$332$	− 151117.i	− 0.0752431i
$333$	− 771867.i	− 0.381445i
$334$	−1.20144e6	−0.589299
$335$	0	0
$336$	−106452.	−0.0514404
$337$	− 761016.i	− 0.365022i	−0.983204	−	0.182511i	$-0.941577\pi$
$337$	− 761016.i	− 0.365022i	0.983204	−	0.182511i	$-0.0584226\pi$
$338$	108084.i	0.0514599i
$339$	−1.55344e6	−0.734169
$340$	0	0
$341$	−124943.	−0.0581871
$342$	− 179234.i	− 0.0828617i
$343$	2.28665e6i	1.04945i
$344$	−2.13138e6	−0.971100
$345$	0	0
$346$	−319794.	−0.143609
$347$	− 1.06891e6i	− 0.476561i	−0.971196	−	0.238281i	$-0.923416\pi$
$347$	− 1.06891e6i	− 0.476561i	0.971196	−	0.238281i	$-0.0765838\pi$
$348$	− 536620.i	− 0.237530i
$349$	−1.57765e6	−0.693342	−0.346671	−	0.937987i	$-0.612688\pi$
$349$	−1.57765e6	−0.693342	−0.346671	+	0.937987i	$0.612688\pi$
$350$	0	0
$351$	−580696.	−0.251583
$352$	855613.i	0.368062i
$353$	− 109374.i	− 0.0467173i	−0.999727	−	0.0233587i	$-0.992564\pi$
$353$	− 109374.i	− 0.0467173i	0.999727	−	0.0233587i	$-0.00743597\pi$
$354$	221574.	0.0939745
$355$	0	0
$356$	−1.05342e6	−0.440530
$357$	327173.i	0.135865i
$358$	− 2.77838e6i	− 1.14573i
$359$	−3.75166e6	−1.53634	−0.768169	−	0.640247i	$-0.778832\pi$
$359$	−3.75166e6	−1.53634	−0.768169	+	0.640247i	$0.778832\pi$
$360$	0	0
$361$	−2.40338e6	−0.970631
$362$	− 1.37506e6i	− 0.551504i
$363$	1.12066e6i	0.446384i
$364$	265900.	0.105188
$365$	0	0
$366$	−516543.	−0.201560
$367$	− 2.12587e6i	− 0.823895i	−0.911208	−	0.411947i	$-0.864849\pi$
$367$	− 2.12587e6i	− 0.823895i	0.911208	−	0.411947i	$-0.135151\pi$
$368$	199802.i	0.0769096i
$369$	2.51518e6	0.961621
$370$	0	0
$371$	1.03122e6	0.388969
$372$	− 115791.i	− 0.0433829i
$373$	725713.i	0.270080i	0.990840	+	0.135040i	$0.0431163\pi$
$373$	725713.i	0.270080i	−0.990840	+	0.135040i	$0.956884\pi$
$374$	−265389.	−0.0981078
$375$	0	0
$376$	−1.87513e6	−0.684008
$377$	− 624981.i	− 0.226471i
$378$	− 1.15724e6i	− 0.416577i
$379$	−1.40410e6	−0.502111	−0.251056	−	0.967973i	$-0.580778\pi$
$379$	−1.40410e6	−0.502111	−0.251056	+	0.967973i	$0.580778\pi$
$380$	0	0
$381$	1.50228e6	0.530197
$382$	− 204892.i	− 0.0718400i
$383$	2.82085e6i	0.982615i	0.870986	+	0.491307i	$0.163481\pi$
$383$	2.82085e6i	0.982615i	−0.870986	+	0.491307i	$0.836519\pi$
$384$	−648093.	−0.224290
$385$	0	0
$386$	790587.	0.270073
$387$	− 1.99113e6i	− 0.675808i
$388$	− 307193.i	− 0.103593i
$389$	1.71293e6	0.573940	0.286970	−	0.957940i	$-0.407352\pi$
$389$	1.71293e6	0.573940	0.286970	+	0.957940i	$0.407352\pi$
$390$	0	0
$391$	614081.	0.203135
$392$	− 1.67068e6i	− 0.549133i
$393$	− 1.81423e6i	− 0.592530i
$394$	2.20124e6	0.714377
$395$	0	0
$396$	−486161.	−0.155791
$397$	− 3.80080e6i	− 1.21032i	−0.796106	−	0.605158i	$-0.793110\pi$
$397$	− 3.80080e6i	− 1.21032i	0.796106	−	0.605158i	$-0.206890\pi$
$398$	− 2.36440e6i	− 0.748191i
$399$	196987.	0.0619448
$400$	0	0
$401$	1.09440e6	0.339871	0.169936	−	0.985455i	$-0.445644\pi$
$401$	1.09440e6	0.339871	0.169936	+	0.985455i	$0.445644\pi$
$402$	− 130826.i	− 0.0403766i
$403$	− 134858.i	− 0.0413631i
$404$	1.21401e6	0.370056
$405$	0	0
$406$	1.24550e6	0.374997
$407$	688122.i	0.205911i
$408$	− 691134.i	− 0.205547i
$409$	−6.63162e6	−1.96025	−0.980125	−	0.198380i	$-0.936432\pi$
$409$	−6.63162e6	−1.96025	−0.980125	+	0.198380i	$0.936432\pi$
$410$	0	0
$411$	2.90412e6	0.848028
$412$	− 3.38027e6i	− 0.981090i
$413$	− 634855.i	− 0.183147i
$414$	−911260.	−0.261301
$415$	0	0
$416$	−923508.	−0.261642
$417$	2.31975e6i	0.653283i
$418$	159787.i	0.0447303i
$419$	1.50034e6	0.417499	0.208749	−	0.977969i	$-0.433061\pi$
$419$	1.50034e6	0.417499	0.208749	+	0.977969i	$0.433061\pi$
$420$	0	0
$421$	−1.91392e6	−0.526282	−0.263141	−	0.964757i	$-0.584758\pi$
$421$	−1.91392e6	−0.526282	−0.263141	+	0.964757i	$0.584758\pi$
$422$	− 2.37750e6i	− 0.649889i
$423$	− 1.75175e6i	− 0.476015i
$424$	−2.17838e6	−0.588464
$425$	0	0
$426$	−399801.	−0.106738
$427$	1.48000e6i	0.392820i
$428$	− 815311.i	− 0.215136i
$429$	217191.	0.0569768
$430$	0	0
$431$	6.04255e6	1.56685	0.783424	−	0.621488i	$-0.213471\pi$
$431$	6.04255e6	1.56685	0.783424	+	0.621488i	$0.213471\pi$
$432$	500735.i	0.129092i
$433$	1.81284e6i	0.464666i	0.972636	+	0.232333i	$0.0746359\pi$
$433$	1.81284e6i	0.464666i	−0.972636	+	0.232333i	$0.925364\pi$
$434$	268752.	0.0684900
$435$	0	0
$436$	−3.31075e6	−0.834085
$437$	− 369731.i	− 0.0926151i
$438$	− 406813.i	− 0.101323i
$439$	−3.45414e6	−0.855418	−0.427709	−	0.903916i	$-0.640679\pi$
$439$	−3.45414e6	−0.855418	−0.427709	+	0.903916i	$0.640679\pi$
$440$	0	0
$441$	1.56075e6	0.382153
$442$	− 286448.i	− 0.0697413i
$443$	1.64299e6i	0.397764i	0.980023	+	0.198882i	$0.0637311\pi$
$443$	1.64299e6i	0.397764i	−0.980023	+	0.198882i	$0.936269\pi$
$444$	−637718.	−0.153522
$445$	0	0
$446$	4.15355e6	0.988740
$447$	− 636764.i	− 0.150733i
$448$	− 2.25544e6i	− 0.530929i
$449$	−3.90173e6	−0.913359	−0.456680	−	0.889631i	$-0.650961\pi$
$449$	−3.90173e6	−0.913359	−0.456680	+	0.889631i	$0.650961\pi$
$450$	0	0
$451$	−2.24230e6	−0.519101
$452$	− 3.34596e6i	− 0.770325i
$453$	1.72113e6i	0.394066i
$454$	612821.	0.139539
$455$	0	0
$456$	−416123.	−0.0937152
$457$	− 6.09815e6i	− 1.36586i	−0.730482	−	0.682932i	$-0.760704\pi$
$457$	− 6.09815e6i	− 1.36586i	0.730482	−	0.682932i	$-0.239296\pi$
$458$	− 2.44168e6i	− 0.543907i
$459$	1.53898e6	0.340959
$460$	0	0
$461$	8.52589e6	1.86848	0.934238	−	0.356651i	$-0.116081\pi$
$461$	8.52589e6	1.86848	0.934238	+	0.356651i	$0.116081\pi$
$462$	432830.i	0.0943436i
$463$	2.70655e6i	0.586765i	0.955995	+	0.293382i	$0.0947809\pi$
$463$	2.70655e6i	0.586765i	−0.955995	+	0.293382i	$0.905219\pi$
$464$	−538922.	−0.116207
$465$	0	0
$466$	3.17847e6	0.678038
$467$	− 350172.i	− 0.0743001i	−0.999310	−	0.0371501i	$-0.988172\pi$
$467$	− 350172.i	− 0.0743001i	0.999310	−	0.0371501i	$-0.0118280\pi$
$468$	− 524739.i	− 0.110746i
$469$	−374845.	−0.0786900
$470$	0	0
$471$	1.04851e6	0.217782
$472$	1.34110e6i	0.277080i
$473$	1.77510e6i	0.364813i
$474$	2.32118e6	0.474529
$475$	0	0
$476$	−704698.	−0.142556
$477$	− 2.03505e6i	− 0.409524i
$478$	− 2.01389e6i	− 0.403150i
$479$	−8.49125e6	−1.69096	−0.845479	−	0.534009i	$-0.820685\pi$
$479$	−8.49125e6	−1.69096	−0.845479	+	0.534009i	$0.820685\pi$
$480$	0	0
$481$	−742726.	−0.146375
$482$	42695.3i	0.00837071i
$483$	− 1.00152e6i	− 0.195341i
$484$	−2.41380e6	−0.468368
$485$	0	0
$486$	−3.60940e6	−0.693177
$487$	3034.65i	0 0.000579811i	1.00000		0.000289905i	$9.22797e-5\pi$
$487$	3034.65i	0 0.000579811i	−1.00000		0.000289905i	$0.999908\pi$
$488$	− 3.12642e6i	− 0.594289i
$489$	−274507.	−0.0519136
$490$	0	0
$491$	−3.43114e6	−0.642295	−0.321147	−	0.947029i	$-0.604068\pi$
$491$	−3.43114e6	−0.642295	−0.321147	+	0.947029i	$0.604068\pi$
$492$	− 2.07805e6i	− 0.387029i
$493$	1.65635e6i	0.306926i
$494$	−172467.	−0.0317971
$495$	0	0
$496$	−116288.	−0.0212242
$497$	1.14551e6i	0.208022i
$498$	265506.i	0.0479735i
$499$	1.11566e6	0.200577	0.100289	−	0.994958i	$-0.468023\pi$
$499$	1.11566e6	0.200577	0.100289	+	0.994958i	$0.468023\pi$
$500$	0	0
$501$	−2.60583e6	−0.463822
$502$	2.24278e6i	0.397217i
$503$	− 4.99165e6i	− 0.879679i	−0.898076	−	0.439840i	$-0.855035\pi$
$503$	− 4.99165e6i	− 0.879679i	0.898076	−	0.439840i	$-0.144965\pi$
$504$	2.93857e6	0.515299
$505$	0	0
$506$	812391.	0.141055
$507$	234425.i	0.0405027i
$508$	3.23575e6i	0.556309i
$509$	398811.	0.0682296	0.0341148	−	0.999418i	$-0.489139\pi$
$509$	398811.	0.0682296	0.0341148	+	0.999418i	$0.489139\pi$
$510$	0	0
$511$	−1.16560e6	−0.197469
$512$	1.67305e6i	0.282056i
$513$	− 926602.i	− 0.155453i
$514$	−611279.	−0.102054
$515$	0	0
$516$	−1.64508e6	−0.271996
$517$	1.56169e6i	0.256962i
$518$	− 1.48015e6i	− 0.242371i
$519$	−693607.	−0.113030
$520$	0	0
$521$	9.33110e6	1.50605	0.753023	−	0.657994i	$-0.228595\pi$
$521$	9.33110e6	1.50605	0.753023	+	0.657994i	$0.228595\pi$
$522$	− 2.45792e6i	− 0.394813i
$523$	4.93297e6i	0.788596i	0.918983	+	0.394298i	$0.129012\pi$
$523$	4.93297e6i	0.788596i	−0.918983	+	0.394298i	$0.870988\pi$
$524$	3.90766e6	0.621711
$525$	0	0
$526$	1.42968e6	0.225307
$527$	357405.i	0.0560575i
$528$	− 187284.i	− 0.0292359i
$529$	4.55656e6	0.707942
$530$	0	0
$531$	−1.25285e6	−0.192825
$532$	424290.i	0.0649955i
$533$	− 2.42023e6i	− 0.369010i
$534$	1.85081e6	0.280873
$535$	0	0
$536$	791838.	0.119049
$537$	− 6.02608e6i	− 0.901777i
$538$	− 2607.73i	0 0.000388425i
$539$	−1.39141e6	−0.206293
$540$	0	0
$541$	2.72340e6	0.400054	0.200027	−	0.979790i	$-0.435897\pi$
$541$	2.72340e6	0.400054	0.200027	+	0.979790i	$0.435897\pi$
$542$	− 2.36696e6i	− 0.346093i
$543$	− 2.98238e6i	− 0.434074i
$544$	2.44751e6	0.354591
$545$	0	0
$546$	−467176.	−0.0670654
$547$	1.28669e7i	1.83867i	0.393472	+	0.919337i	$0.371274\pi$
$547$	1.28669e7i	1.83867i	−0.393472	+	0.919337i	$0.628726\pi$
$548$	6.25518e6i	0.889792i
$549$	2.92071e6	0.413578
$550$	0	0
$551$	997266.	0.139937
$552$	2.11566e6i	0.295527i
$553$	− 6.65067e6i	− 0.924811i
$554$	2.81553e6	0.389750
$555$	0	0
$556$	−4.99651e6	−0.685456
$557$	1.03791e7i	1.41750i	0.705459	+	0.708751i	$0.250741\pi$
$557$	1.03791e7i	1.41750i	−0.705459	+	0.708751i	$0.749259\pi$
$558$	− 530368.i	− 0.0721094i
$559$	−1.91596e6	−0.259333
$560$	0	0
$561$	−575607.	−0.0772180
$562$	− 5.44133e6i	− 0.726716i
$563$	3.37237e6i	0.448399i	0.974543	+	0.224199i	$0.0719767\pi$
$563$	3.37237e6i	0.448399i	−0.974543	+	0.224199i	$0.928023\pi$
$564$	−1.44730e6	−0.191584
$565$	0	0
$566$	4.71760e6	0.618984
$567$	1.28827e6i	0.168286i
$568$	− 2.41983e6i	− 0.314713i
$569$	9.65718e6	1.25046	0.625230	−	0.780441i	$-0.285005\pi$
$569$	9.65718e6	1.25046	0.625230	+	0.780441i	$0.285005\pi$
$570$	0	0
$571$	−3.19460e6	−0.410040	−0.205020	−	0.978758i	$-0.565726\pi$
$571$	−3.19460e6	−0.410040	−0.205020	+	0.978758i	$0.565726\pi$
$572$	467807.i	0.0597828i
$573$	− 444394.i	− 0.0565433i
$574$	4.82316e6	0.611016
$575$	0	0
$576$	−4.45099e6	−0.558986
$577$	1.01384e7i	1.26774i	0.773438	+	0.633872i	$0.218535\pi$
$577$	1.01384e7i	1.26774i	−0.773438	+	0.633872i	$0.781465\pi$
$578$	− 4.61403e6i	− 0.574462i
$579$	1.71472e6	0.212567
$580$	0	0
$581$	760731.	0.0934955
$582$	539726.i	0.0660489i
$583$	1.81425e6i	0.221068i
$584$	2.46227e6	0.298747
$585$	0	0
$586$	−6.60960e6	−0.795117
$587$	777799.i	0.0931691i	0.998914	+	0.0465846i	$0.0148337\pi$
$587$	777799.i	0.0931691i	−0.998914	+	0.0465846i	$0.985166\pi$
$588$	− 1.28950e6i	− 0.153807i
$589$	215189.	0.0255583
$590$	0	0
$591$	4.77432e6	0.562267
$592$	640454.i	0.0751076i
$593$	8.28738e6i	0.967789i	0.875126	+	0.483894i	$0.160778\pi$
$593$	8.28738e6i	0.967789i	−0.875126	+	0.483894i	$0.839222\pi$
$594$	2.03598e6	0.236759
$595$	0	0
$596$	1.37153e6	0.158157
$597$	− 5.12818e6i	− 0.588881i
$598$	876856.i	0.100271i
$599$	−1.39700e7	−1.59085	−0.795423	−	0.606054i	$-0.792751\pi$
$599$	−1.39700e7	−1.59085	−0.795423	+	0.606054i	$0.792751\pi$
$600$	0	0
$601$	−1.27397e7	−1.43871	−0.719355	−	0.694643i	$-0.755562\pi$
$601$	−1.27397e7	−1.43871	−0.719355	+	0.694643i	$0.755562\pi$
$602$	− 3.81824e6i	− 0.429409i
$603$	739736.i	0.0828483i
$604$	−3.70714e6	−0.413473
$605$	0	0
$606$	−2.13296e6	−0.235940
$607$	− 6.25976e6i	− 0.689583i	−0.938679	−	0.344791i	$-0.887950\pi$
$607$	− 6.25976e6i	− 0.689583i	0.938679	−	0.344791i	$-0.112050\pi$
$608$	− 1.47362e6i	− 0.161669i
$609$	2.70138e6	0.295150
$610$	0	0
$611$	−1.68561e6	−0.182665
$612$	1.39068e6i	0.150089i
$613$	1.17405e7i	1.26193i	0.775812	+	0.630964i	$0.217340\pi$
$613$	1.17405e7i	1.26193i	−0.775812	+	0.630964i	$0.782660\pi$
$614$	−905700.	−0.0969535
$615$	0	0
$616$	−2.61974e6	−0.278168
$617$	1.57982e7i	1.67069i	0.549729	+	0.835343i	$0.314731\pi$
$617$	1.57982e7i	1.67069i	−0.549729	+	0.835343i	$0.685269\pi$
$618$	5.93901e6i	0.625523i
$619$	−3.16596e6	−0.332108	−0.166054	−	0.986117i	$-0.553103\pi$
$619$	−3.16596e6	−0.332108	−0.166054	+	0.986117i	$0.553103\pi$
$620$	0	0
$621$	−4.71103e6	−0.490216
$622$	− 4.17556e6i	− 0.432751i
$623$	− 5.30297e6i	− 0.547393i
$624$	202145.	0.0207827
$625$	0	0
$626$	3.53770e6	0.360815
$627$	346566.i	0.0352060i
$628$	2.25839e6i	0.228507i
$629$	1.96840e6	0.198375
$630$	0	0
$631$	−1.95795e7	−1.95762	−0.978811	−	0.204764i	$-0.934357\pi$
$631$	−1.95795e7	−1.95762	−0.978811	+	0.204764i	$0.934357\pi$
$632$	1.40492e7i	1.39913i
$633$	− 5.15660e6i	− 0.511510i
$634$	1.00450e7	0.992489
$635$	0	0
$636$	−1.68136e6	−0.164823
$637$	− 1.50183e6i	− 0.146646i
$638$	2.19125e6i	0.213127i
$639$	2.26061e6	0.219015
$640$	0	0
$641$	−7.23060e6	−0.695072	−0.347536	−	0.937667i	$-0.612981\pi$
$641$	−7.23060e6	−0.695072	−0.347536	+	0.937667i	$0.612981\pi$
$642$	1.43247e6i	0.137166i
$643$	1.17438e7i	1.12016i	0.828438	+	0.560081i	$0.189230\pi$
$643$	1.17438e7i	1.12016i	−0.828438	+	0.560081i	$0.810770\pi$
$644$	2.15718e6	0.204961
$645$	0	0
$646$	457078.	0.0430932
$647$	9.24080e6i	0.867858i	0.900947	+	0.433929i	$0.142873\pi$
$647$	9.24080e6i	0.867858i	−0.900947	+	0.433929i	$0.857127\pi$
$648$	− 2.72139e6i	− 0.254597i
$649$	1.11692e6	0.104091
$650$	0	0
$651$	582901.	0.0539067
$652$	− 591260.i	− 0.0544703i
$653$	1.11349e7i	1.02188i	0.859615	+	0.510942i	$0.170703\pi$
$653$	1.11349e7i	1.02188i	−0.859615	+	0.510942i	$0.829297\pi$
$654$	5.81686e6	0.531795
$655$	0	0
$656$	−2.08697e6	−0.189346
$657$	2.30026e6i	0.207904i
$658$	− 3.35918e6i	− 0.302461i
$659$	−8.65776e6	−0.776590	−0.388295	−	0.921535i	$-0.626936\pi$
$659$	−8.65776e6	−0.776590	−0.388295	+	0.921535i	$0.626936\pi$
$660$	0	0
$661$	−1.23594e7	−1.10025	−0.550127	−	0.835081i	$-0.685421\pi$
$661$	−1.23594e7	−1.10025	−0.550127	+	0.835081i	$0.685421\pi$
$662$	− 2.45351e6i	− 0.217592i
$663$	− 621282.i	− 0.0548915i
$664$	−1.60700e6	−0.141448
$665$	0	0
$666$	−2.92099e6	−0.255179
$667$	− 5.07030e6i	− 0.441285i
$668$	− 5.61268e6i	− 0.486664i
$669$	9.00871e6	0.778211
$670$	0	0
$671$	−2.60382e6	−0.223257
$672$	− 3.99172e6i	− 0.340986i
$673$	6.11057e6i	0.520048i	0.965602	+	0.260024i	$0.0837305\pi$
$673$	6.11057e6i	0.520048i	−0.965602	+	0.260024i	$0.916269\pi$
$674$	−2.87993e6	−0.244192
$675$	0	0
$676$	−504928.	−0.0424974
$677$	1.48518e7i	1.24540i	0.782462	+	0.622698i	$0.213964\pi$
$677$	1.48518e7i	1.24540i	−0.782462	+	0.622698i	$0.786036\pi$
$678$	5.87872e6i	0.491144i
$679$	1.54643e6	0.128723
$680$	0	0
$681$	1.32916e6	0.109827
$682$	472825.i	0.0389260i
$683$	6.13145e6i	0.502934i	0.967866	+	0.251467i	$0.0809130\pi$
$683$	6.13145e6i	0.502934i	−0.967866	+	0.251467i	$0.919087\pi$
$684$	837313.	0.0684302
$685$	0	0
$686$	8.65340e6	0.702064
$687$	− 5.29580e6i	− 0.428095i
$688$	1.65214e6i	0.133068i
$689$	−1.95822e6	−0.157150
$690$	0	0
$691$	2.33358e7	1.85920	0.929602	−	0.368566i	$-0.120151\pi$
$691$	2.33358e7	1.85920	0.929602	+	0.368566i	$0.120151\pi$
$692$	− 1.49396e6i	− 0.118597i
$693$	− 2.44737e6i	− 0.193582i
$694$	−4.04511e6	−0.318809
$695$	0	0
$696$	−5.70651e6	−0.446527
$697$	6.41417e6i	0.500102i
$698$	5.97033e6i	0.463831i
$699$	6.89385e6	0.533665
$700$	0	0
$701$	2.53633e7	1.94944	0.974721	−	0.223426i	$-0.0717242\pi$
$701$	2.53633e7	1.94944	0.974721	+	0.223426i	$0.0717242\pi$
$702$	2.19754e6i	0.168304i
$703$	− 1.18515e6i	− 0.0904450i
$704$	3.96807e6	0.301751
$705$	0	0
$706$	−413907.	−0.0312529
$707$	6.11139e6i	0.459824i
$708$	1.03511e6i	0.0776075i
$709$	1.64964e7	1.23246	0.616232	−	0.787565i	$-0.288658\pi$
$709$	1.64964e7	1.23246	0.616232	+	0.787565i	$0.288658\pi$
$710$	0	0
$711$	−1.31248e7	−0.973682
$712$	1.12022e7i	0.828141i
$713$	− 1.09407e6i	− 0.0805971i
$714$	1.23813e6	0.0908907
$715$	0	0
$716$	1.29796e7	0.946189
$717$	− 4.36797e6i	− 0.317309i
$718$	1.41975e7i	1.02778i
$719$	9.96268e6	0.718710	0.359355	−	0.933201i	$-0.382997\pi$
$719$	9.96268e6	0.718710	0.359355	+	0.933201i	$0.382997\pi$
$720$	0	0
$721$	1.70165e7	1.21908
$722$	9.09514e6i	0.649332i
$723$	92602.6i	0.00658836i
$724$	6.42375e6	0.455452
$725$	0	0
$726$	4.24095e6	0.298622
$727$	− 2.53565e7i	− 1.77932i	−0.456627	−	0.889658i	$-0.650943\pi$
$727$	− 2.53565e7i	− 1.77932i	0.456627	−	0.889658i	$-0.349057\pi$
$728$	− 2.82762e6i	− 0.197739i
$729$	−4.31096e6	−0.300438
$730$	0	0
$731$	5.07775e6	0.351462
$732$	− 2.41310e6i	− 0.166455i
$733$	− 1.70653e7i	− 1.17315i	−0.809894	−	0.586576i	$-0.800476\pi$
$733$	− 1.70653e7i	− 1.17315i	0.809894	−	0.586576i	$-0.199524\pi$
$734$	−8.04497e6	−0.551168
$735$	0	0
$736$	−7.49218e6	−0.509816
$737$	− 659478.i	− 0.0447230i
$738$	− 9.51826e6i	− 0.643304i
$739$	1.40560e7	0.946781	0.473391	−	0.880853i	$-0.343030\pi$
$739$	1.40560e7	0.946781	0.473391	+	0.880853i	$0.343030\pi$
$740$	0	0
$741$	−374067.	−0.0250267
$742$	− 3.90245e6i	− 0.260212i
$743$	1.55718e7i	1.03483i	0.855736	+	0.517413i	$0.173105\pi$
$743$	1.55718e7i	1.03483i	−0.855736	+	0.517413i	$0.826895\pi$
$744$	−1.23135e6	−0.0815544
$745$	0	0
$746$	2.74633e6	0.180678
$747$	− 1.50126e6i	− 0.0984362i
$748$	− 1.23980e6i	− 0.0810209i
$749$	4.10433e6	0.267324
$750$	0	0
$751$	−5.89761e6	−0.381572	−0.190786	−	0.981632i	$-0.561104\pi$
$751$	−5.89761e6	−0.381572	−0.190786	+	0.981632i	$0.561104\pi$
$752$	1.45351e6i	0.0937287i
$753$	4.86441e6i	0.312639i
$754$	−2.36513e6	−0.151505
$755$	0	0
$756$	5.40622e6	0.344024
$757$	− 2.57205e6i	− 0.163132i	−0.996668	−	0.0815660i	$-0.974008\pi$
$757$	− 2.57205e6i	− 0.163132i	0.996668	−	0.0815660i	$-0.0259921\pi$
$758$	5.31356e6i	0.335902i
$759$	1.76201e6	0.111021
$760$	0	0
$761$	−2.56012e7	−1.60250	−0.801252	−	0.598328i	$-0.795832\pi$
$761$	−2.56012e7	−1.60250	−0.801252	+	0.598328i	$0.795832\pi$
$762$	− 5.68509e6i	− 0.354691i
$763$	− 1.66665e7i	− 1.03642i
$764$	957179.	0.0593280
$765$	0	0
$766$	1.06750e7	0.657349
$767$	1.20555e6i	0.0739942i
$768$	9.10895e6i	0.557270i
$769$	1.15095e7	0.701842	0.350921	−	0.936405i	$-0.385869\pi$
$769$	1.15095e7	0.701842	0.350921	+	0.936405i	$0.385869\pi$
$770$	0	0
$771$	−1.32582e6	−0.0803243
$772$	3.69333e6i	0.223036i
$773$	5.18479e6i	0.312092i	0.987750	+	0.156046i	$0.0498748\pi$
$773$	5.18479e6i	0.312092i	−0.987750	+	0.156046i	$0.950125\pi$
$774$	−7.53509e6	−0.452101
$775$	0	0
$776$	−3.26674e6	−0.194742
$777$	− 3.21032e6i	− 0.190764i
$778$	− 6.48229e6i	− 0.383954i
$779$	3.86190e6	0.228012
$780$	0	0
$781$	−2.01534e6	−0.118228
$782$	− 2.32388e6i	− 0.135893i
$783$	− 1.27070e7i	− 0.740692i
$784$	−1.29503e6	−0.0752469
$785$	0	0
$786$	−6.86561e6	−0.396390
$787$	− 1.28463e7i	− 0.739333i	−0.929165	−	0.369667i	$-0.879472\pi$
$787$	− 1.28463e7i	− 0.739333i	0.929165	−	0.369667i	$-0.120528\pi$
$788$	1.02834e7i	0.589958i
$789$	3.10087e6	0.177333
$790$	0	0
$791$	1.68438e7	0.957190
$792$	5.16992e6i	0.292867i
$793$	− 2.81044e6i	− 0.158705i
$794$	−1.43834e7	−0.809676
$795$	0	0
$796$	1.10456e7	0.617883
$797$	− 2.11768e7i	− 1.18090i	−0.807073	−	0.590452i	$-0.798950\pi$
$797$	− 2.11768e7i	− 1.18090i	0.807073	−	0.590452i	$-0.201050\pi$
$798$	− 745461.i	− 0.0414398i
$799$	4.46727e6	0.247557
$800$	0	0
$801$	−1.04651e7	−0.576320
$802$	− 4.14155e6i	− 0.227367i
$803$	− 2.05069e6i	− 0.112230i
$804$	611172.	0.0333444
$805$	0	0
$806$	−510344.	−0.0276711
$807$	− 5655.97i	0 0.000305719i
$808$	− 1.29100e7i	− 0.695659i
$809$	−1.58537e6	−0.0851648	−0.0425824	−	0.999093i	$-0.513558\pi$
$809$	−1.58537e6	−0.0851648	−0.0425824	+	0.999093i	$0.513558\pi$
$810$	0	0
$811$	−2.18155e7	−1.16470	−0.582350	−	0.812938i	$-0.697867\pi$
$811$	−2.18155e7	−1.16470	−0.582350	+	0.812938i	$0.697867\pi$
$812$	5.81850e6i	0.309686i
$813$	− 5.13375e6i	− 0.272401i
$814$	2.60407e6	0.137750
$815$	0	0
$816$	−535733.	−0.0281659
$817$	− 3.05725e6i	− 0.160242i
$818$	2.50962e7i	1.31137i
$819$	2.64157e6	0.137611
$820$	0	0
$821$	3.09541e7	1.60273	0.801365	−	0.598176i	$-0.204108\pi$
$821$	3.09541e7	1.60273	0.801365	+	0.598176i	$0.204108\pi$
$822$	− 1.09901e7i	− 0.567313i
$823$	− 2.68746e7i	− 1.38307i	−0.722345	−	0.691533i	$-0.756936\pi$
$823$	− 2.68746e7i	− 1.38307i	0.722345	−	0.691533i	$-0.243064\pi$
$824$	−3.59464e7	−1.84433
$825$	0	0
$826$	−2.40249e6	−0.122521
$827$	9.94152e6i	0.505462i	0.967537	+	0.252731i	$0.0813288\pi$
$827$	9.94152e6i	0.505462i	−0.967537	+	0.252731i	$0.918671\pi$
$828$	− 4.25707e6i	− 0.215792i
$829$	−7.20887e6	−0.364318	−0.182159	−	0.983269i	$-0.558309\pi$
$829$	−7.20887e6	−0.364318	−0.182159	+	0.983269i	$0.558309\pi$
$830$	0	0
$831$	6.10666e6	0.306762
$832$	4.28295e6i	0.214504i
$833$	3.98020e6i	0.198743i
$834$	8.77867e6	0.437032
$835$	0	0
$836$	−746468.	−0.0369399
$837$	− 2.74190e6i	− 0.135281i
$838$	− 5.67777e6i	− 0.279298i
$839$	−1.96670e7	−0.964569	−0.482285	−	0.876015i	$-0.660193\pi$
$839$	−1.96670e7	−0.964569	−0.482285	+	0.876015i	$0.660193\pi$
$840$	0	0
$841$	−6.83513e6	−0.333240
$842$	7.24288e6i	0.352072i
$843$	− 1.18018e7i	− 0.571979i
$844$	1.11068e7	0.536702
$845$	0	0
$846$	−6.62917e6	−0.318444
$847$	− 1.21512e7i	− 0.581984i
$848$	1.68858e6i	0.0806364i
$849$	1.02321e7	0.487186
$850$	0	0
$851$	−6.02554e6	−0.285215
$852$	− 1.86772e6i	− 0.0881482i
$853$	− 2.60044e7i	− 1.22370i	−0.790975	−	0.611848i	$-0.790426\pi$
$853$	− 2.60044e7i	− 1.22370i	0.790975	−	0.611848i	$-0.209574\pi$
$854$	5.60080e6	0.262788
$855$	0	0
$856$	−8.67015e6	−0.404429
$857$	− 2.74925e7i	− 1.27868i	−0.768923	−	0.639341i	$-0.779207\pi$
$857$	− 2.74925e7i	− 1.27868i	0.768923	−	0.639341i	$-0.220793\pi$
$858$	− 821919.i	− 0.0381163i
$859$	−3.86634e7	−1.78779	−0.893896	−	0.448274i	$-0.852039\pi$
$859$	−3.86634e7	−1.78779	−0.893896	+	0.448274i	$0.852039\pi$
$860$	0	0
$861$	1.04611e7	0.480914
$862$	− 2.28669e7i	− 1.04819i
$863$	− 8.41848e6i	− 0.384775i	−0.981319	−	0.192387i	$-0.938377\pi$
$863$	− 8.41848e6i	− 0.384775i	0.981319	−	0.192387i	$-0.0616230\pi$
$864$	−1.87766e7	−0.855720
$865$	0	0
$866$	6.86038e6	0.310852
$867$	− 1.00075e7i	− 0.452144i
$868$	1.25551e6i	0.0565615i
$869$	1.17008e7	0.525611
$870$	0	0
$871$	711808.	0.0317920
$872$	3.52071e7i	1.56797i
$873$	− 3.05179e6i	− 0.135525i
$874$	−1.39918e6	−0.0619576
$875$	0	0
$876$	1.90048e6	0.0836763
$877$	1.24226e7i	0.545399i	0.962099	+	0.272699i	$0.0879164\pi$
$877$	1.24226e7i	0.545399i	−0.962099	+	0.272699i	$0.912084\pi$
$878$	1.30716e7i	0.572257i
$879$	−1.43357e7	−0.625815
$880$	0	0
$881$	7.24684e6	0.314564	0.157282	−	0.987554i	$-0.449727\pi$
$881$	7.24684e6	0.314564	0.157282	+	0.987554i	$0.449727\pi$
$882$	− 5.90637e6i	− 0.255652i
$883$	− 1.58232e7i	− 0.682958i	−0.939889	−	0.341479i	$-0.889072\pi$
$883$	− 1.58232e7i	− 0.682958i	0.939889	−	0.341479i	$-0.110928\pi$
$884$	1.33818e6	0.0575949
$885$	0	0
$886$	6.21760e6	0.266096
$887$	− 1.85905e7i	− 0.793382i	−0.917952	−	0.396691i	$-0.870158\pi$
$887$	− 1.85905e7i	− 0.793382i	0.917952	−	0.396691i	$-0.129842\pi$
$888$	6.78161e6i	0.288603i
$889$	−1.62890e7	−0.691257
$890$	0	0
$891$	−2.26650e6	−0.0956446
$892$	1.94038e7i	0.816537i
$893$	− 2.68969e6i	− 0.112869i
$894$	−2.40972e6	−0.100838
$895$	0	0
$896$	7.02718e6	0.292423
$897$	1.90183e6i	0.0789206i
$898$	1.47654e7i	0.611018i
$899$	2.95100e6	0.121778
$900$	0	0
$901$	5.18974e6	0.212978
$902$	8.48556e6i	0.347267i
$903$	− 8.28144e6i	− 0.337977i
$904$	−3.55815e7	−1.44811
$905$	0	0
$906$	6.51331e6	0.263622
$907$	− 4.30877e7i	− 1.73914i	−0.493808	−	0.869571i	$-0.664395\pi$
$907$	− 4.30877e7i	− 1.73914i	0.493808	−	0.869571i	$-0.335605\pi$
$908$	2.86288e6i	0.115236i
$909$	1.20605e7	0.484123
$910$	0	0
$911$	870987.	0.0347709	0.0173854	−	0.999849i	$-0.494466\pi$
$911$	870987.	0.0347709	0.0173854	+	0.999849i	$0.494466\pi$
$912$	322558.i	0.0128417i
$913$	1.33838e6i	0.0531377i
$914$	−2.30773e7	−0.913735
$915$	0	0
$916$	1.14066e7	0.449178
$917$	1.96714e7i	0.772524i
$918$	− 5.82400e6i	− 0.228094i
$919$	7.81477e6	0.305230	0.152615	−	0.988286i	$-0.451231\pi$
$919$	7.81477e6	0.305230	0.152615	+	0.988286i	$0.451231\pi$
$920$	0	0
$921$	−1.96439e6	−0.0763095
$922$	− 3.22647e7i	− 1.24997i
$923$	− 2.17526e6i	− 0.0840442i
$924$	−2.02202e6	−0.0779123
$925$	0	0
$926$	1.02425e7	0.392533
$927$	− 3.35812e7i	− 1.28350i
$928$	− 2.02085e7i	− 0.770307i
$929$	−6.93323e6	−0.263570	−0.131785	−	0.991278i	$-0.542071\pi$
$929$	−6.93323e6	−0.263570	−0.131785	+	0.991278i	$0.542071\pi$
$930$	0	0
$931$	2.39643e6	0.0906129
$932$	1.48487e7i	0.559948i
$933$	− 9.05644e6i	− 0.340607i
$934$	−1.32516e6	−0.0497052
$935$	0	0
$936$	−5.58017e6	−0.208189
$937$	3.31684e7i	1.23417i	0.786896	+	0.617085i	$0.211687\pi$
$937$	3.31684e7i	1.23417i	−0.786896	+	0.617085i	$0.788313\pi$
$938$	1.41853e6i	0.0526420i
$939$	7.67298e6	0.283988
$940$	0	0
$941$	−5.00778e6	−0.184362	−0.0921809	−	0.995742i	$-0.529384\pi$
$941$	−5.00778e6	−0.184362	−0.0921809	+	0.995742i	$0.529384\pi$
$942$	− 3.96790e6i	− 0.145691i
$943$	− 1.96347e7i	− 0.719025i
$944$	1.03955e6	0.0379678
$945$	0	0
$946$	6.71756e6	0.244053
$947$	7.37028e6i	0.267060i	0.991045	+	0.133530i	$0.0426313\pi$
$947$	7.37028e6i	0.267060i	−0.991045	+	0.133530i	$0.957369\pi$
$948$	1.08437e7i	0.391883i
$949$	2.21341e6	0.0797805
$950$	0	0
$951$	2.17867e7	0.781162
$952$	7.49388e6i	0.267987i
$953$	7.45227e6i	0.265801i	0.991129	+	0.132900i	$0.0424291\pi$
$953$	7.45227e6i	0.265801i	−0.991129	+	0.132900i	$0.957571\pi$
$954$	−7.70127e6	−0.273963
$955$	0	0
$956$	9.40817e6	0.332936
$957$	4.75263e6i	0.167747i
$958$	3.21336e7i	1.13122i
$959$	−3.14890e7	−1.10564
$960$	0	0
$961$	−2.79924e7	−0.977758
$962$	2.81071e6i	0.0979216i
$963$	− 8.09967e6i	− 0.281450i
$964$	−199457.	−0.00691283
$965$	0	0
$966$	−3.79007e6	−0.130679
$967$	4.78544e7i	1.64572i	0.568244	+	0.822860i	$0.307623\pi$
$967$	4.78544e7i	1.64572i	−0.568244	+	0.822860i	$0.692377\pi$
$968$	2.56687e7i	0.880473i
$969$	991365.	0.0339175
$970$	0	0
$971$	−2.54187e7	−0.865177	−0.432588	−	0.901592i	$-0.642400\pi$
$971$	−2.54187e7	−0.865177	−0.432588	+	0.901592i	$0.642400\pi$
$972$	− 1.68618e7i	− 0.572450i
$973$	− 2.51527e7i	− 0.851733i
$974$	11484.1	0.000387881 0
$975$	0	0
$976$	−2.42345e6	−0.0814347
$977$	− 3.10699e7i	− 1.04137i	−0.853750	−	0.520683i	$-0.825677\pi$
$977$	− 3.10699e7i	− 1.04137i	0.853750	−	0.520683i	$-0.174323\pi$
$978$	1.03882e6i	0.0347291i
$979$	9.32970e6	0.311108
$980$	0	0
$981$	−3.28905e7	−1.09118
$982$	1.29845e7i	0.429682i
$983$	1.28575e6i	0.0424398i	0.999775	+	0.0212199i	$0.00675501\pi$
$983$	1.28575e6i	0.0424398i	−0.999775	+	0.0212199i	$0.993245\pi$
$984$	−2.20984e7	−0.727566
$985$	0	0
$986$	6.26814e6	0.205327
$987$	− 7.28579e6i	− 0.238059i
$988$	− 805701.i	− 0.0262592i
$989$	−1.55437e7	−0.505317
$990$	0	0
$991$	−4.43766e6	−0.143539	−0.0717695	−	0.997421i	$-0.522865\pi$
$991$	−4.43766e6	−0.143539	−0.0717695	+	0.997421i	$0.522865\pi$
$992$	− 4.36057e6i	− 0.140690i
$993$	− 5.32146e6i	− 0.171261i
$994$	4.33499e6	0.139162
$995$	0	0
$996$	−1.24035e6	−0.0396182
$997$	2.72549e7i	0.868375i	0.900823	+	0.434187i	$0.142964\pi$
$997$	2.72549e7i	0.868375i	−0.900823	+	0.434187i	$0.857036\pi$
$998$	− 4.22202e6i	− 0.134182i
$999$	−1.51009e7	−0.478729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.8		22
5.2	odd	4		325.6.a.j.1.8	✓	11
5.3	odd	4		325.6.a.k.1.4	yes	11
5.4	even	2	inner	325.6.b.i.274.15		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.8	✓	11	5.2	odd	4
325.6.a.k.1.4	yes	11	5.3	odd	4
325.6.b.i.274.8		22	1.1	even	1	trivial
325.6.b.i.274.15		22	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.78432i q^{2} -8.20788i q^{3} +17.6789 q^{4} -31.0612 q^{6} +88.9969i q^{7} -188.001i q^{8} +175.631 q^{9} +O(q^{10})\)	\(q-3.78432i q^{2} -8.20788i q^{3} +17.6789 q^{4} -31.0612 q^{6} +88.9969i q^{7} -188.001i q^{8} +175.631 q^{9} -156.575 q^{11} -145.107i q^{12} -169.000i q^{13} +336.793 q^{14} -145.729 q^{16} +447.890i q^{17} -664.642i q^{18} +269.669 q^{19} +730.476 q^{21} +592.531i q^{22} -1371.05i q^{23} -1543.09 q^{24} -639.550 q^{26} -3436.07i q^{27} +1573.37i q^{28} +3698.11 q^{29} +797.974 q^{31} -5464.54i q^{32} +1285.15i q^{33} +1694.96 q^{34} +3104.97 q^{36} -4394.83i q^{37} -1020.51i q^{38} -1387.13 q^{39} +14320.9 q^{41} -2764.35i q^{42} -11337.1i q^{43} -2768.09 q^{44} -5188.50 q^{46} -9974.03i q^{47} +1196.13i q^{48} +8886.54 q^{49} +3676.23 q^{51} -2987.74i q^{52} -11587.1i q^{53} -13003.2 q^{54} +16731.5 q^{56} -2213.41i q^{57} -13994.8i q^{58} -7133.45 q^{59} +16629.8 q^{61} -3019.79i q^{62} +15630.6i q^{63} -25342.9 q^{64} +4863.42 q^{66} +4211.88i q^{67} +7918.22i q^{68} -11253.4 q^{69} +12871.4 q^{71} -33018.7i q^{72} +13097.1i q^{73} -16631.4 q^{74} +4767.46 q^{76} -13934.7i q^{77} +5249.35i q^{78} -74729.2 q^{79} +14475.4 q^{81} -54194.7i q^{82} -8547.83i q^{83} +12914.0 q^{84} -42903.0 q^{86} -30353.6i q^{87} +29436.3i q^{88} -59586.0 q^{89} +15040.5 q^{91} -24238.8i q^{92} -6549.68i q^{93} -37744.9 q^{94} -44852.3 q^{96} -17376.2i q^{97} -33629.5i q^{98} -27499.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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