LMFDB - Embedded newform 325.6.b.f.274.10

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 326x^{10} + 40401x^{8} + 2365264x^{6} + 65636064x^{4} + 738923264x^{2} + 2250553600 $ x^12 + 326x^10 + 40401x^8 + 2365264x^6 + 65636064x^4 + 738923264*x^2 + 2250553600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.10
Root			$7.62628i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.f.274.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+7.62628i q^{2} +29.7832i q^{3} -26.1601 q^{4} -227.135 q^{6} +171.199i q^{7} +44.5366i q^{8} -644.042 q^{9} +452.811 q^{11} -779.133i q^{12} +169.000i q^{13} -1305.61 q^{14} -1176.77 q^{16} +1469.49i q^{17} -4911.64i q^{18} -2007.18 q^{19} -5098.87 q^{21} +3453.26i q^{22} +763.605i q^{23} -1326.44 q^{24} -1288.84 q^{26} -11944.3i q^{27} -4478.59i q^{28} +6493.90 q^{29} +5299.68 q^{31} -7549.22i q^{32} +13486.2i q^{33} -11206.7 q^{34} +16848.2 q^{36} -4537.59i q^{37} -15307.3i q^{38} -5033.37 q^{39} +13699.1 q^{41} -38885.4i q^{42} -3824.25i q^{43} -11845.6 q^{44} -5823.46 q^{46} +15653.3i q^{47} -35048.1i q^{48} -12502.1 q^{49} -43766.2 q^{51} -4421.06i q^{52} +16260.1i q^{53} +91090.7 q^{54} -7624.63 q^{56} -59780.3i q^{57} +49524.3i q^{58} -16817.3 q^{59} +9690.04 q^{61} +40416.8i q^{62} -110259. i q^{63} +19915.7 q^{64} -102849. q^{66} -5726.95i q^{67} -38442.0i q^{68} -22742.6 q^{69} +12847.1 q^{71} -28683.4i q^{72} +41868.8i q^{73} +34604.9 q^{74} +52508.0 q^{76} +77520.8i q^{77} -38385.9i q^{78} +33399.1 q^{79} +199239. q^{81} +104473. i q^{82} +58851.7i q^{83} +133387. q^{84} +29164.8 q^{86} +193409. i q^{87} +20166.7i q^{88} +94010.3 q^{89} -28932.7 q^{91} -19976.0i q^{92} +157842. i q^{93} -119377. q^{94} +224840. q^{96} -126692. i q^{97} -95344.8i q^{98} -291629. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{4} - 104 q^{6} - 2068 q^{9} + 1600 q^{11} - 4216 q^{14} + 644 q^{16} - 10608 q^{19} + 2144 q^{21} - 9512 q^{24} - 676 q^{26} + 7328 q^{29} + 33328 q^{31} - 2760 q^{34} + 3636 q^{36} + 6760 q^{39}+ \cdots - 602528 q^{99}+O(q^{100}) $ 12 * q - 268 * q^4 - 104 * q^6 - 2068 * q^9 + 1600 * q^11 - 4216 * q^14 + 644 * q^16 - 10608 * q^19 + 2144 * q^21 - 9512 * q^24 - 676 * q^26 + 7328 * q^29 + 33328 * q^31 - 2760 * q^34 + 3636 * q^36 + 6760 * q^39 + 24872 * q^41 + 70248 * q^44 - 9112 * q^46 + 55684 * q^49 - 55328 * q^51 + 265824 * q^54 + 88840 * q^56 + 23792 * q^59 + 114464 * q^61 + 79828 * q^64 + 19816 * q^66 - 144808 * q^69 + 428208 * q^71 + 364208 * q^74 + 733272 * q^76 - 72048 * q^79 + 701028 * q^81 - 12048 * q^84 + 58408 * q^86 + 104056 * q^89 - 58136 * q^91 - 315144 * q^94 + 1095816 * q^96 - 602528 * q^99

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 $

$2$	7.62628i	1.34815i	0.738664	+	0.674074i	$0.235457\pi$
$2$	7.62628i	1.34815i	−0.738664	+	0.674074i	$0.764543\pi$
$3$	29.7832i	1.91060i	0.295645	+	0.955298i	$0.404466\pi$
$3$	29.7832i	1.91060i	−0.295645	+	0.955298i	$0.595534\pi$
$4$	−26.1601	−0.817503
$5$	0	0
$5$	0	0
$6$	−227.135	−2.57577
$7$	171.199i	1.32055i	0.751022	+	0.660277i	$0.229561\pi$
$7$	171.199i	1.32055i	−0.751022	+	0.660277i	$0.770439\pi$
$8$	44.5366i	0.246032i
$9$	−644.042	−2.65038
$10$	0	0
$11$	452.811	1.12833	0.564163	−	0.825663i	$-0.309199\pi$
$11$	452.811	1.12833	0.564163	+	0.825663i	$0.309199\pi$
$12$	− 779.133i	− 1.56192i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−1305.61	−1.78030
$15$	0	0
$16$	−1176.77	−1.14919
$17$	1469.49i	1.23323i	0.787264	+	0.616615i	$0.211497\pi$
$17$	1469.49i	1.23323i	−0.787264	+	0.616615i	$0.788503\pi$
$18$	− 4911.64i	− 3.57310i
$19$	−2007.18	−1.27556	−0.637781	−	0.770218i	$-0.720148\pi$
$19$	−2007.18	−1.27556	−0.637781	+	0.770218i	$0.720148\pi$
$20$	0	0
$21$	−5098.87	−2.52305
$22$	3453.26i	1.52115i
$23$	763.605i	0.300988i	0.988611	+	0.150494i	$0.0480864\pi$
$23$	763.605i	0.300988i	−0.988611	+	0.150494i	$0.951914\pi$
$24$	−1326.44	−0.470068
$25$	0	0
$26$	−1288.84	−0.373909
$27$	− 11944.3i	− 3.15320i
$28$	− 4478.59i	− 1.07956i
$29$	6493.90	1.43387	0.716936	−	0.697139i	$-0.245544\pi$
$29$	6493.90	1.43387	0.716936	+	0.697139i	$0.245544\pi$
$30$	0	0
$31$	5299.68	0.990479	0.495239	−	0.868757i	$-0.335080\pi$
$31$	5299.68	0.990479	0.495239	+	0.868757i	$0.335080\pi$
$32$	− 7549.22i	− 1.30325i
$33$	13486.2i	2.15578i
$34$	−11206.7	−1.66258
$35$	0	0
$36$	16848.2	2.16669
$37$	− 4537.59i	− 0.544906i	−0.962169	−	0.272453i	$-0.912165\pi$
$37$	− 4537.59i	− 0.544906i	0.962169	−	0.272453i	$-0.0878349\pi$
$38$	− 15307.3i	− 1.71965i
$39$	−5033.37	−0.529904
$40$	0	0
$41$	13699.1	1.27272	0.636359	−	0.771393i	$-0.280440\pi$
$41$	13699.1	1.27272	0.636359	+	0.771393i	$0.280440\pi$
$42$	− 38885.4i	− 3.40144i
$43$	− 3824.25i	− 0.315409i	−0.987486	−	0.157705i	$-0.949591\pi$
$43$	− 3824.25i	− 0.315409i	0.987486	−	0.157705i	$-0.0504094\pi$
$44$	−11845.6	−0.922411
$45$	0	0
$46$	−5823.46	−0.405776
$47$	15653.3i	1.03362i	0.856099	+	0.516811i	$0.172881\pi$
$47$	15653.3i	1.03362i	−0.856099	+	0.516811i	$0.827119\pi$
$48$	− 35048.1i	− 2.19564i
$49$	−12502.1	−0.743865
$50$	0	0
$51$	−43766.2	−2.35621
$52$	− 4421.06i	− 0.226735i
$53$	16260.1i	0.795120i	0.917576	+	0.397560i	$0.130143\pi$
$53$	16260.1i	0.795120i	−0.917576	+	0.397560i	$0.869857\pi$
$54$	91090.7	4.25099
$55$	0	0
$56$	−7624.63	−0.324899
$57$	− 59780.3i	− 2.43708i
$58$	49524.3i	1.93307i
$59$	−16817.3	−0.628965	−0.314483	−	0.949263i	$-0.601831\pi$
$59$	−16817.3	−0.628965	−0.314483	+	0.949263i	$0.601831\pi$
$60$	0	0
$61$	9690.04	0.333427	0.166714	−	0.986005i	$-0.446684\pi$
$61$	9690.04	0.333427	0.166714	+	0.986005i	$0.446684\pi$
$62$	40416.8i	1.33531i
$63$	− 110259.i	− 3.49997i
$64$	19915.7	0.607780
$65$	0	0
$66$	−102849.	−2.90631
$67$	− 5726.95i	− 0.155861i	−0.996959	−	0.0779303i	$-0.975169\pi$
$67$	− 5726.95i	− 0.155861i	0.996959	−	0.0779303i	$-0.0248311\pi$
$68$	− 38442.0i	− 1.00817i
$69$	−22742.6	−0.575067
$70$	0	0
$71$	12847.1	0.302455	0.151227	−	0.988499i	$-0.451677\pi$
$71$	12847.1	0.302455	0.151227	+	0.988499i	$0.451677\pi$
$72$	− 28683.4i	− 0.652079i
$73$	41868.8i	0.919566i	0.888031	+	0.459783i	$0.152073\pi$
$73$	41868.8i	0.919566i	−0.888031	+	0.459783i	$0.847927\pi$
$74$	34604.9	0.734614
$75$	0	0
$76$	52508.0	1.04278
$77$	77520.8i	1.49002i
$78$	− 38385.9i	− 0.714389i
$79$	33399.1	0.602097	0.301049	−	0.953609i	$-0.402663\pi$
$79$	33399.1	0.602097	0.301049	+	0.953609i	$0.402663\pi$
$80$	0	0
$81$	199239.	3.37412
$82$	104473.i	1.71581i
$83$	58851.7i	0.937700i	0.883278	+	0.468850i	$0.155331\pi$
$83$	58851.7i	0.937700i	−0.883278	+	0.468850i	$0.844669\pi$
$84$	133387.	2.06260
$85$	0	0
$86$	29164.8	0.425219
$87$	193409.i	2.73955i
$88$	20166.7i	0.277605i
$89$	94010.3	1.25806	0.629029	−	0.777382i	$-0.283453\pi$
$89$	94010.3	1.25806	0.629029	+	0.777382i	$0.283453\pi$
$90$	0	0
$91$	−28932.7	−0.366256
$92$	− 19976.0i	− 0.246059i
$93$	157842.i	1.89241i
$94$	−119377.	−1.39348
$95$	0	0
$96$	224840.	2.48998
$97$	− 126692.i	− 1.36716i	−0.729874	−	0.683581i	$-0.760421\pi$
$97$	− 126692.i	− 1.36716i	0.729874	−	0.683581i	$-0.239579\pi$
$98$	− 95344.8i	− 1.00284i
$99$	−291629.	−2.99049
$100$	0	0
$101$	35394.0	0.345245	0.172622	−	0.984988i	$-0.444776\pi$
$101$	35394.0	0.345245	0.172622	+	0.984988i	$0.444776\pi$
$102$	− 333773.i	− 3.17652i
$103$	33346.8i	0.309714i	0.987937	+	0.154857i	$0.0494917\pi$
$103$	33346.8i	0.309714i	−0.987937	+	0.154857i	$0.950508\pi$
$104$	−7526.69	−0.0682371
$105$	0	0
$106$	−124004.	−1.07194
$107$	− 1497.08i	− 0.0126411i	−0.999980	−	0.00632054i	$-0.997988\pi$
$107$	− 1497.08i	− 0.0126411i	0.999980	−	0.00632054i	$-0.00201190\pi$
$108$	312465.i	2.57776i
$109$	7731.80	0.0623325	0.0311662	−	0.999514i	$-0.490078\pi$
$109$	7731.80	0.0623325	0.0311662	+	0.999514i	$0.490078\pi$
$110$	0	0
$111$	135144.	1.04109
$112$	− 201462.i	− 1.51757i
$113$	149733.i	1.10312i	0.834136	+	0.551558i	$0.185967\pi$
$113$	149733.i	1.10312i	−0.834136	+	0.551558i	$0.814033\pi$
$114$	455901.	3.28555
$115$	0	0
$116$	−169881.	−1.17220
$117$	− 108843.i	− 0.735082i
$118$	− 128254.i	− 0.847938i
$119$	−251575.	−1.62855
$120$	0	0
$121$	43986.5	0.273122
$122$	73898.9i	0.449509i
$123$	408003.i	2.43165i
$124$	−138640.	−0.809720
$125$	0	0
$126$	840869.	4.71848
$127$	239284.i	1.31645i	0.752822	+	0.658224i	$0.228692\pi$
$127$	239284.i	1.31645i	−0.752822	+	0.658224i	$0.771308\pi$
$128$	− 89692.1i	− 0.483871i
$129$	113898.	0.602620
$130$	0	0
$131$	243815.	1.24132	0.620659	−	0.784081i	$-0.286865\pi$
$131$	243815.	1.24132	0.620659	+	0.784081i	$0.286865\pi$
$132$	− 352800.i	− 1.76236i
$133$	− 343627.i	− 1.68445i
$134$	43675.3	0.210123
$135$	0	0
$136$	−65446.1	−0.303415
$137$	− 164453.i	− 0.748583i	−0.927311	−	0.374292i	$-0.877886\pi$
$137$	− 164453.i	− 0.748583i	0.927311	−	0.374292i	$-0.122114\pi$
$138$	− 173442.i	− 0.775275i
$139$	−101444.	−0.445338	−0.222669	−	0.974894i	$-0.571477\pi$
$139$	−101444.	−0.445338	−0.222669	+	0.974894i	$0.571477\pi$
$140$	0	0
$141$	−466207.	−1.97484
$142$	97975.8i	0.407754i
$143$	76525.0i	0.312942i
$144$	757890.	3.04579
$145$	0	0
$146$	−319303.	−1.23971
$147$	− 372354.i	− 1.42123i
$148$	118704.i	0.445462i
$149$	56069.7	0.206901	0.103450	−	0.994635i	$-0.467012\pi$
$149$	56069.7	0.206901	0.103450	+	0.994635i	$0.467012\pi$
$150$	0	0
$151$	−103359.	−0.368897	−0.184449	−	0.982842i	$-0.559050\pi$
$151$	−103359.	−0.368897	−0.184449	+	0.982842i	$0.559050\pi$
$152$	− 89392.9i	− 0.313830i
$153$	− 946413.i	− 3.26853i
$154$	−591195.	−2.00876
$155$	0	0
$156$	131673.	0.433198
$157$	− 290816.i	− 0.941606i	−0.882238	−	0.470803i	$-0.843964\pi$
$157$	− 290816.i	− 0.941606i	0.882238	−	0.470803i	$-0.156036\pi$
$158$	254711.i	0.811716i
$159$	−484278.	−1.51915
$160$	0	0
$161$	−130729.	−0.397471
$162$	1.51945e6i	4.54882i
$163$	265789.i	0.783553i	0.920060	+	0.391777i	$0.128139\pi$
$163$	265789.i	0.783553i	−0.920060	+	0.391777i	$0.871861\pi$
$164$	−358370.	−1.04045
$165$	0	0
$166$	−448819.	−1.26416
$167$	− 279706.i	− 0.776087i	−0.921641	−	0.388043i	$-0.873151\pi$
$167$	− 279706.i	− 0.776087i	0.921641	−	0.388043i	$-0.126849\pi$
$168$	− 227086.i	− 0.620751i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	1.29271e6	3.38072
$172$	100043.i	0.257848i
$173$	657222.i	1.66954i	0.550599	+	0.834770i	$0.314399\pi$
$173$	657222.i	1.66954i	−0.550599	+	0.834770i	$0.685601\pi$
$174$	−1.47499e6	−3.69332
$175$	0	0
$176$	−532855.	−1.29666
$177$	− 500874.i	− 1.20170i
$178$	716949.i	1.69605i
$179$	140536.	0.327836	0.163918	−	0.986474i	$-0.447587\pi$
$179$	140536.	0.327836	0.163918	+	0.986474i	$0.447587\pi$
$180$	0	0
$181$	554170.	1.25732	0.628662	−	0.777679i	$-0.283603\pi$
$181$	554170.	1.25732	0.628662	+	0.777679i	$0.283603\pi$
$182$	− 220648.i	− 0.493767i
$183$	288601.i	0.637045i
$184$	−34008.4	−0.0740528
$185$	0	0
$186$	−1.20374e6	−2.55124
$187$	665401.i	1.39149i
$188$	− 409493.i	− 0.844990i
$189$	2.04486e6	4.16398
$190$	0	0
$191$	379250.	0.752215	0.376107	−	0.926576i	$-0.377262\pi$
$191$	379250.	0.752215	0.376107	+	0.926576i	$0.377262\pi$
$192$	593155.i	1.16122i
$193$	− 884530.i	− 1.70930i	−0.519201	−	0.854652i	$-0.673770\pi$
$193$	− 884530.i	− 1.70930i	0.519201	−	0.854652i	$-0.326230\pi$
$194$	966189.	1.84314
$195$	0	0
$196$	327057.	0.608112
$197$	− 529090.i	− 0.971323i	−0.874147	−	0.485661i	$-0.838579\pi$
$197$	− 529090.i	− 0.971323i	0.874147	−	0.485661i	$-0.161421\pi$
$198$	− 2.22404e6i	− 4.03163i
$199$	−8938.89	−0.0160011	−0.00800057	−	0.999968i	$-0.502547\pi$
$199$	−8938.89	−0.0160011	−0.00800057	+	0.999968i	$0.502547\pi$
$200$	0	0
$201$	170567.	0.297786
$202$	269925.i	0.465441i
$203$	1.11175e6i	1.89351i
$204$	1.14493e6	1.92621
$205$	0	0
$206$	−254312.	−0.417541
$207$	− 491794.i	− 0.797732i
$208$	− 198874.i	− 0.318728i
$209$	−908871.	−1.43925
$210$	0	0
$211$	−1.11242e6	−1.72013	−0.860066	−	0.510183i	$-0.829578\pi$
$211$	−1.11242e6	−1.72013	−0.860066	+	0.510183i	$0.829578\pi$
$212$	− 425365.i	− 0.650014i
$213$	382629.i	0.577869i
$214$	11417.1	0.0170421
$215$	0	0
$216$	531960.	0.775790
$217$	907300.i	1.30798i
$218$	58964.9i	0.0840334i
$219$	−1.24699e6	−1.75692
$220$	0	0
$221$	−248344.	−0.342037
$222$	1.03065e6i	1.40355i
$223$	− 1.27817e6i	− 1.72118i	−0.509301	−	0.860588i	$-0.670096\pi$
$223$	− 1.27817e6i	− 1.72118i	0.509301	−	0.860588i	$-0.329904\pi$
$224$	1.29242e6	1.72101
$225$	0	0
$226$	−1.14191e6	−1.48716
$227$	− 1.31606e6i	− 1.69516i	−0.530669	−	0.847579i	$-0.678059\pi$
$227$	− 1.31606e6i	− 1.69516i	0.530669	−	0.847579i	$-0.321941\pi$
$228$	1.56386e6i	1.99233i
$229$	10410.5	0.0131185	0.00655925	−	0.999978i	$-0.497912\pi$
$229$	10410.5	0.0131185	0.00655925	+	0.999978i	$0.497912\pi$
$230$	0	0
$231$	−2.30882e6	−2.84682
$232$	289216.i	0.352779i
$233$	− 1.42470e6i	− 1.71923i	−0.510941	−	0.859616i	$-0.670703\pi$
$233$	− 1.42470e6i	− 1.71923i	0.510941	−	0.859616i	$-0.329297\pi$
$234$	830067.	0.991000
$235$	0	0
$236$	439943.	0.514181
$237$	994733.i	1.15036i
$238$	− 1.91858e6i	− 2.19553i
$239$	336053.	0.380552	0.190276	−	0.981731i	$-0.439062\pi$
$239$	336053.	0.380552	0.190276	+	0.981731i	$0.439062\pi$
$240$	0	0
$241$	1.13898e6	1.26320	0.631599	−	0.775295i	$-0.282399\pi$
$241$	1.13898e6	1.26320	0.631599	+	0.775295i	$0.282399\pi$
$242$	335453.i	0.368208i
$243$	3.03150e6i	3.29338i
$244$	−253493.	−0.272578
$245$	0	0
$246$	−3.11155e6	−3.27822
$247$	− 339213.i	− 0.353777i
$248$	236030.i	0.243690i
$249$	−1.75279e6	−1.79157
$250$	0	0
$251$	−1.58387e6	−1.58684	−0.793421	−	0.608673i	$-0.791702\pi$
$251$	−1.58387e6	−1.58684	−0.793421	+	0.608673i	$0.791702\pi$
$252$	2.88440e6i	2.86124i
$253$	345769.i	0.339613i
$254$	−1.82484e6	−1.77477
$255$	0	0
$256$	1.32132e6	1.26011
$257$	− 1.60657e6i	− 1.51729i	−0.651507	−	0.758643i	$-0.725863\pi$
$257$	− 1.60657e6i	− 1.51729i	0.651507	−	0.758643i	$-0.274137\pi$
$258$	868621.i	0.812421i
$259$	776832.	0.719578
$260$	0	0
$261$	−4.18234e6	−3.80030
$262$	1.85940e6i	1.67348i
$263$	− 1.09488e6i	− 0.976064i	−0.872826	−	0.488032i	$-0.837715\pi$
$263$	− 1.09488e6i	− 0.976064i	0.872826	−	0.488032i	$-0.162285\pi$
$264$	−600628.	−0.530391
$265$	0	0
$266$	2.62059e6	2.27089
$267$	2.79993e6i	2.40364i
$268$	149818.i	0.127417i
$269$	−1.40894e6	−1.18716	−0.593582	−	0.804774i	$-0.702287\pi$
$269$	−1.40894e6	−1.18716	−0.593582	+	0.804774i	$0.702287\pi$
$270$	0	0
$271$	−416084.	−0.344158	−0.172079	−	0.985083i	$-0.555048\pi$
$271$	−416084.	−0.344158	−0.172079	+	0.985083i	$0.555048\pi$
$272$	− 1.72926e6i	− 1.41722i
$273$	− 861708.i	− 0.699767i
$274$	1.25416e6	1.00920
$275$	0	0
$276$	594950.	0.470119
$277$	1.07825e6i	0.844349i	0.906515	+	0.422174i	$0.138733\pi$
$277$	1.07825e6i	0.844349i	−0.906515	+	0.422174i	$0.861267\pi$
$278$	− 773641.i	− 0.600382i
$279$	−3.41321e6	−2.62514
$280$	0	0
$281$	−1.78529e6	−1.34879	−0.674393	−	0.738372i	$-0.735595\pi$
$281$	−1.78529e6	−1.34879	−0.674393	+	0.738372i	$0.735595\pi$
$282$	− 3.55542e6i	− 2.66237i
$283$	− 171905.i	− 0.127591i	−0.997963	−	0.0637957i	$-0.979679\pi$
$283$	− 171905.i	− 0.127591i	0.997963	−	0.0637957i	$-0.0203206\pi$
$284$	−336082.	−0.247258
$285$	0	0
$286$	−583601.	−0.421892
$287$	2.34527e6i	1.68069i
$288$	4.86201e6i	3.45410i
$289$	−739545.	−0.520859
$290$	0	0
$291$	3.77330e6	2.61210
$292$	− 1.09529e6i	− 0.751749i
$293$	− 976888.i	− 0.664776i	−0.943143	−	0.332388i	$-0.892146\pi$
$293$	− 976888.i	− 0.664776i	0.943143	−	0.332388i	$-0.107854\pi$
$294$	2.83968e6	1.91602
$295$	0	0
$296$	202089.	0.134064
$297$	− 5.40852e6i	− 3.55784i
$298$	427603.i	0.278933i
$299$	−129049.	−0.0834791
$300$	0	0
$301$	654708.	0.416515
$302$	− 788243.i	− 0.497328i
$303$	1.05415e6i	0.659623i
$304$	2.36199e6	1.46587
$305$	0	0
$306$	7.21761e6	4.40646
$307$	1.74098e6i	1.05426i	0.849784	+	0.527131i	$0.176732\pi$
$307$	1.74098e6i	1.05426i	−0.849784	+	0.527131i	$0.823268\pi$
$308$	− 2.02795e6i	− 1.21809i
$309$	−993176.	−0.591739
$310$	0	0
$311$	1.00483e6	0.589104	0.294552	−	0.955635i	$-0.404830\pi$
$311$	1.00483e6	0.589104	0.294552	+	0.955635i	$0.404830\pi$
$312$	− 224169.i	− 0.130374i
$313$	− 458407.i	− 0.264478i	−0.991218	−	0.132239i	$-0.957783\pi$
$313$	− 458407.i	− 0.264478i	0.991218	−	0.132239i	$-0.0422167\pi$
$314$	2.21784e6	1.26942
$315$	0	0
$316$	−873724.	−0.492217
$317$	383397.i	0.214289i	0.994243	+	0.107145i	$0.0341708\pi$
$317$	383397.i	0.214289i	−0.994243	+	0.107145i	$0.965829\pi$
$318$	− 3.69324e6i	− 2.04804i
$319$	2.94051e6	1.61788
$320$	0	0
$321$	44587.8	0.0241520
$322$	− 996972.i	− 0.535850i
$323$	− 2.94953e6i	− 1.57306i
$324$	−5.21210e6	−2.75836
$325$	0	0
$326$	−2.02698e6	−1.05635
$327$	230278.i	0.119092i
$328$	610111.i	0.313130i
$329$	−2.67984e6	−1.36496
$330$	0	0
$331$	622094.	0.312095	0.156047	−	0.987750i	$-0.450125\pi$
$331$	622094.	0.312095	0.156047	+	0.987750i	$0.450125\pi$
$332$	− 1.53957e6i	− 0.766573i
$333$	2.92240e6i	1.44421i
$334$	2.13311e6	1.04628
$335$	0	0
$336$	6.00020e6	2.89946
$337$	− 624562.i	− 0.299572i	−0.988718	−	0.149786i	$-0.952142\pi$
$337$	− 624562.i	− 0.299572i	0.988718	−	0.149786i	$-0.0478584\pi$
$338$	− 217814.i	− 0.103704i
$339$	−4.45953e6	−2.10761
$340$	0	0
$341$	2.39975e6	1.11758
$342$	9.85853e6i	4.55771i
$343$	736988.i	0.338240i
$344$	170319.	0.0776009
$345$	0	0
$346$	−5.01216e6	−2.25079
$347$	− 1.43412e6i	− 0.639382i	−0.947522	−	0.319691i	$-0.896421\pi$
$347$	− 1.43412e6i	− 0.639382i	0.947522	−	0.319691i	$-0.103579\pi$
$348$	− 5.05961e6i	− 2.23959i
$349$	1.80725e6	0.794247	0.397124	−	0.917765i	$-0.370008\pi$
$349$	1.80725e6	0.794247	0.397124	+	0.917765i	$0.370008\pi$
$350$	0	0
$351$	2.01859e6	0.874541
$352$	− 3.41837e6i	− 1.47049i
$353$	2.71033e6i	1.15767i	0.815444	+	0.578835i	$0.196493\pi$
$353$	2.71033e6i	1.15767i	−0.815444	+	0.578835i	$0.803507\pi$
$354$	3.81981e6	1.62007
$355$	0	0
$356$	−2.45932e6	−1.02847
$357$	− 7.49273e6i	− 3.11150i
$358$	1.07177e6i	0.441971i
$359$	1.91002e6	0.782171	0.391085	−	0.920354i	$-0.372100\pi$
$359$	1.91002e6	0.782171	0.391085	+	0.920354i	$0.372100\pi$
$360$	0	0
$361$	1.55266e6	0.627060
$362$	4.22626e6i	1.69506i
$363$	1.31006e6i	0.521825i
$364$	756881.	0.299416
$365$	0	0
$366$	−2.20095e6	−0.858831
$367$	2.09294e6i	0.811131i	0.914066	+	0.405566i	$0.132925\pi$
$367$	2.09294e6i	0.811131i	−0.914066	+	0.405566i	$0.867075\pi$
$368$	− 898589.i	− 0.345893i
$369$	−8.82278e6	−3.37318
$370$	0	0
$371$	−2.78371e6	−1.05000
$372$	− 4.12915e6i	− 1.54705i
$373$	− 1.20939e6i	− 0.450083i	−0.974349	−	0.225041i	$-0.927748\pi$
$373$	− 1.20939e6i	− 0.450083i	0.974349	−	0.225041i	$-0.0722518\pi$
$374$	−5.07453e6	−1.87593
$375$	0	0
$376$	−697146.	−0.254305
$377$	1.09747e6i	0.397685i
$378$	1.55947e7i	5.61366i
$379$	594902.	0.212739	0.106370	−	0.994327i	$-0.466077\pi$
$379$	594902.	0.212739	0.106370	+	0.994327i	$0.466077\pi$
$380$	0	0
$381$	−7.12664e6	−2.51520
$382$	2.89226e6i	1.01410i
$383$	2.34526e6i	0.816946i	0.912770	+	0.408473i	$0.133939\pi$
$383$	2.34526e6i	0.816946i	−0.912770	+	0.408473i	$0.866061\pi$
$384$	2.67132e6	0.924481
$385$	0	0
$386$	6.74567e6	2.30440
$387$	2.46297e6i	0.835954i
$388$	3.31428e6i	1.11766i
$389$	4.27642e6	1.43287	0.716434	−	0.697655i	$-0.245773\pi$
$389$	4.27642e6	1.43287	0.716434	+	0.697655i	$0.245773\pi$
$390$	0	0
$391$	−1.12211e6	−0.371188
$392$	− 556803.i	− 0.183015i
$393$	7.26161e6i	2.37166i
$394$	4.03498e6	1.30949
$395$	0	0
$396$	7.62905e6	2.44474
$397$	4.05375e6i	1.29087i	0.763817	+	0.645433i	$0.223323\pi$
$397$	4.05375e6i	1.29087i	−0.763817	+	0.645433i	$0.776677\pi$
$398$	− 68170.5i	− 0.0215719i
$399$	1.02343e7	3.21830
$400$	0	0
$401$	−2.70195e6	−0.839105	−0.419553	−	0.907731i	$-0.637813\pi$
$401$	−2.70195e6	−0.839105	−0.419553	+	0.907731i	$0.637813\pi$
$402$	1.30079e6i	0.401460i
$403$	895646.i	0.274709i
$404$	−925912.	−0.282239
$405$	0	0
$406$	−8.47851e6	−2.55273
$407$	− 2.05467e6i	− 0.614832i
$408$	− 1.94920e6i	− 0.579703i
$409$	1.61265e6	0.476686	0.238343	−	0.971181i	$-0.423396\pi$
$409$	1.61265e6	0.476686	0.238343	+	0.971181i	$0.423396\pi$
$410$	0	0
$411$	4.89794e6	1.43024
$412$	− 872356.i	− 0.253192i
$413$	− 2.87911e6i	− 0.830583i
$414$	3.75055e6	1.07546
$415$	0	0
$416$	1.27582e6	0.361456
$417$	− 3.02133e6i	− 0.850861i
$418$	− 6.93131e6i	− 1.94032i
$419$	1.08547e6	0.302054	0.151027	−	0.988530i	$-0.451742\pi$
$419$	1.08547e6	0.302054	0.151027	+	0.988530i	$0.451742\pi$
$420$	0	0
$421$	−5.35894e6	−1.47358	−0.736790	−	0.676121i	$-0.763659\pi$
$421$	−5.35894e6	−1.47358	−0.736790	+	0.676121i	$0.763659\pi$
$422$	− 8.48361e6i	− 2.31899i
$423$	− 1.00814e7i	− 2.73949i
$424$	−724169.	−0.195625
$425$	0	0
$426$	−2.91804e6	−0.779053
$427$	1.65893e6i	0.440309i
$428$	39163.7i	0.0103341i
$429$	−2.27916e6	−0.597905
$430$	0	0
$431$	−2.11538e6	−0.548524	−0.274262	−	0.961655i	$-0.588434\pi$
$431$	−2.11538e6	−0.548524	−0.274262	+	0.961655i	$0.588434\pi$
$432$	1.40557e7i	3.62364i
$433$	− 5.31501e6i	− 1.36234i	−0.732128	−	0.681168i	$-0.761473\pi$
$433$	− 5.31501e6i	− 1.36234i	0.732128	−	0.681168i	$-0.238527\pi$
$434$	−6.91932e6	−1.76335
$435$	0	0
$436$	−202265.	−0.0509570
$437$	− 1.53269e6i	− 0.383929i
$438$	− 9.50987e6i	− 2.36859i
$439$	−3.32599e6	−0.823682	−0.411841	−	0.911256i	$-0.635114\pi$
$439$	−3.32599e6	−0.823682	−0.411841	+	0.911256i	$0.635114\pi$
$440$	0	0
$441$	8.05190e6	1.97152
$442$	− 1.89394e6i	− 0.461116i
$443$	− 5.41394e6i	− 1.31070i	−0.755324	−	0.655352i	$-0.772520\pi$
$443$	− 5.41394e6i	− 1.31070i	0.755324	−	0.655352i	$-0.227480\pi$
$444$	−3.53539e6	−0.851098
$445$	0	0
$446$	9.74766e6	2.32040
$447$	1.66994e6i	0.395304i
$448$	3.40956e6i	0.802607i
$449$	−692641.	−0.162141	−0.0810704	−	0.996708i	$-0.525834\pi$
$449$	−692641.	−0.162141	−0.0810704	+	0.996708i	$0.525834\pi$
$450$	0	0
$451$	6.20309e6	1.43604
$452$	− 3.91703e6i	− 0.901801i
$453$	− 3.07836e6i	− 0.704813i
$454$	1.00366e7	2.28532
$455$	0	0
$456$	2.66241e6	0.599602
$457$	3.90454e6i	0.874539i	0.899330	+	0.437270i	$0.144054\pi$
$457$	3.90454e6i	0.874539i	−0.899330	+	0.437270i	$0.855946\pi$
$458$	79393.6i	0.0176857i
$459$	1.75521e7	3.88863
$460$	0	0
$461$	−8.82344e6	−1.93368	−0.966842	−	0.255375i	$-0.917801\pi$
$461$	−8.82344e6	−1.93368	−0.966842	+	0.255375i	$0.917801\pi$
$462$	− 1.76077e7i	− 3.83794i
$463$	5.52375e6i	1.19752i	0.800930	+	0.598758i	$0.204339\pi$
$463$	5.52375e6i	1.19752i	−0.800930	+	0.598758i	$0.795661\pi$
$464$	−7.64184e6	−1.64779
$465$	0	0
$466$	1.08652e7	2.31778
$467$	− 5.29356e6i	− 1.12320i	−0.827410	−	0.561598i	$-0.810187\pi$
$467$	− 5.29356e6i	− 1.12320i	0.827410	−	0.561598i	$-0.189813\pi$
$468$	2.84735e6i	0.600932i
$469$	980448.	0.205822
$470$	0	0
$471$	8.66145e6	1.79903
$472$	− 748986.i	− 0.154746i
$473$	− 1.73166e6i	− 0.355885i
$474$	−7.58611e6	−1.55086
$475$	0	0
$476$	6.58124e6	1.33134
$477$	− 1.04722e7i	− 2.10737i
$478$	2.56284e6i	0.513040i
$479$	4.55616e6	0.907319	0.453659	−	0.891175i	$-0.350118\pi$
$479$	4.55616e6	0.907319	0.453659	+	0.891175i	$0.350118\pi$
$480$	0	0
$481$	766853.	0.151130
$482$	8.68614e6i	1.70298i
$483$	− 3.89352e6i	− 0.759407i
$484$	−1.15069e6	−0.223278
$485$	0	0
$486$	−2.31191e7	−4.43996
$487$	8.00881e6i	1.53019i	0.643916	+	0.765096i	$0.277308\pi$
$487$	8.00881e6i	1.53019i	−0.643916	+	0.765096i	$0.722692\pi$
$488$	431562.i	0.0820339i
$489$	−7.91607e6	−1.49705
$490$	0	0
$491$	−5.83743e6	−1.09274	−0.546372	−	0.837543i	$-0.683991\pi$
$491$	−5.83743e6	−1.09274	−0.546372	+	0.837543i	$0.683991\pi$
$492$	− 1.06734e7i	− 1.98788i
$493$	9.54272e6i	1.76830i
$494$	2.58693e6	0.476944
$495$	0	0
$496$	−6.23651e6	−1.13825
$497$	2.19942e6i	0.399408i
$498$	− 1.33673e7i	− 2.41530i
$499$	2.89323e6	0.520153	0.260077	−	0.965588i	$-0.416252\pi$
$499$	2.89323e6	0.520153	0.260077	+	0.965588i	$0.416252\pi$
$500$	0	0
$501$	8.33055e6	1.48279
$502$	− 1.20790e7i	− 2.13930i
$503$	− 1.02497e7i	− 1.80631i	−0.429312	−	0.903156i	$-0.641244\pi$
$503$	− 1.02497e7i	− 1.80631i	0.429312	−	0.903156i	$-0.358756\pi$
$504$	4.91058e6	0.861106
$505$	0	0
$506$	−2.63693e6	−0.457849
$507$	− 850639.i	− 0.146969i
$508$	− 6.25969e6i	− 1.07620i
$509$	−6.87884e6	−1.17685	−0.588424	−	0.808552i	$-0.700251\pi$
$509$	−6.87884e6	−1.17685	−0.588424	+	0.808552i	$0.700251\pi$
$510$	0	0
$511$	−7.16790e6	−1.21434
$512$	7.20661e6i	1.21494i
$513$	2.39744e7i	4.02211i
$514$	1.22522e7	2.04553
$515$	0	0
$516$	−2.97960e6	−0.492644
$517$	7.08799e6i	1.16626i
$518$	5.92434e6i	0.970097i
$519$	−1.95742e7	−3.18982
$520$	0	0
$521$	1.11738e6	0.180346	0.0901728	−	0.995926i	$-0.471258\pi$
$521$	1.11738e6	0.180346	0.0901728	+	0.995926i	$0.471258\pi$
$522$	− 3.18957e7i	− 5.12337i
$523$	− 2.65174e6i	− 0.423912i	−0.977279	−	0.211956i	$-0.932017\pi$
$523$	− 2.65174e6i	− 0.423912i	0.977279	−	0.211956i	$-0.0679834\pi$
$524$	−6.37824e6	−1.01478
$525$	0	0
$526$	8.34988e6	1.31588
$527$	7.78783e6i	1.22149i
$528$	− 1.58702e7i	− 2.47740i
$529$	5.85325e6	0.909406
$530$	0	0
$531$	1.08311e7	1.66700
$532$	8.98932e6i	1.37704i
$533$	2.31515e6i	0.352988i
$534$	−2.13531e7	−3.24046
$535$	0	0
$536$	255059.	0.0383467
$537$	4.18563e6i	0.626362i
$538$	− 1.07449e7i	− 1.60047i
$539$	−5.66110e6	−0.839323
$540$	0	0
$541$	9.63494e6	1.41532	0.707662	−	0.706551i	$-0.249750\pi$
$541$	9.63494e6	1.41532	0.707662	+	0.706551i	$0.249750\pi$
$542$	− 3.17317e6i	− 0.463976i
$543$	1.65050e7i	2.40224i
$544$	1.10935e7	1.60721
$545$	0	0
$546$	6.57163e6	0.943390
$547$	1.83795e6i	0.262642i	0.991340	+	0.131321i	$0.0419219\pi$
$547$	1.83795e6i	0.262642i	−0.991340	+	0.131321i	$0.958078\pi$
$548$	4.30211e6i	0.611969i
$549$	−6.24079e6	−0.883708
$550$	0	0
$551$	−1.30344e7	−1.82899
$552$	− 1.01288e6i	− 0.141485i
$553$	5.71789e6i	0.795103i
$554$	−8.22307e6	−1.13831
$555$	0	0
$556$	2.65379e6	0.364065
$557$	− 5.72815e6i	− 0.782305i	−0.920326	−	0.391152i	$-0.872077\pi$
$557$	− 5.72815e6i	− 0.782305i	0.920326	−	0.391152i	$-0.127923\pi$
$558$	− 2.60301e7i	− 3.53908i
$559$	646298.	0.0874788
$560$	0	0
$561$	−1.98178e7	−2.65857
$562$	− 1.36151e7i	− 1.81836i
$563$	8.32198e6i	1.10651i	0.833012	+	0.553255i	$0.186615\pi$
$563$	8.32198e6i	1.10651i	−0.833012	+	0.553255i	$0.813385\pi$
$564$	1.21960e7	1.61443
$565$	0	0
$566$	1.31099e6	0.172012
$567$	3.41095e7i	4.45571i
$568$	572168.i	0.0744137i
$569$	1.13277e7	1.46676	0.733382	−	0.679817i	$-0.237941\pi$
$569$	1.13277e7	1.46676	0.733382	+	0.679817i	$0.237941\pi$
$570$	0	0
$571$	−1.31340e7	−1.68581	−0.842904	−	0.538064i	$-0.819156\pi$
$571$	−1.31340e7	−1.68581	−0.842904	+	0.538064i	$0.819156\pi$
$572$	− 2.00190e6i	− 0.255831i
$573$	1.12953e7i	1.43718i
$574$	−1.78857e7	−2.26582
$575$	0	0
$576$	−1.28266e7	−1.61085
$577$	− 5.24787e6i	− 0.656211i	−0.944641	−	0.328105i	$-0.893590\pi$
$577$	− 5.24787e6i	− 0.656211i	0.944641	−	0.328105i	$-0.106410\pi$
$578$	− 5.63997e6i	− 0.702195i
$579$	2.63442e7	3.26579
$580$	0	0
$581$	−1.00754e7	−1.23828
$582$	2.87762e7i	3.52149i
$583$	7.36274e6i	0.897156i
$584$	−1.86469e6	−0.226243
$585$	0	0
$586$	7.45002e6	0.896217
$587$	4.50054e6i	0.539100i	0.962986	+	0.269550i	$0.0868749\pi$
$587$	4.50054e6i	0.539100i	−0.962986	+	0.269550i	$0.913125\pi$
$588$	9.74083e6i	1.16186i
$589$	−1.06374e7	−1.26342
$590$	0	0
$591$	1.57580e7	1.85581
$592$	5.33971e6i	0.626201i
$593$	1.02931e7i	1.20201i	0.799245	+	0.601005i	$0.205233\pi$
$593$	1.02931e7i	1.20201i	−0.799245	+	0.601005i	$0.794767\pi$
$594$	4.12468e7	4.79650
$595$	0	0
$596$	−1.46679e6	−0.169142
$597$	− 266229.i	− 0.0305717i
$598$	− 984166.i	− 0.112542i
$599$	−1.05149e7	−1.19740	−0.598700	−	0.800973i	$-0.704316\pi$
$599$	−1.05149e7	−1.19740	−0.598700	+	0.800973i	$0.704316\pi$
$600$	0	0
$601$	1.66880e7	1.88460	0.942298	−	0.334774i	$-0.108660\pi$
$601$	1.66880e7	1.88460	0.942298	+	0.334774i	$0.108660\pi$
$602$	4.99298e6i	0.561525i
$603$	3.68839e6i	0.413089i
$604$	2.70388e6	0.301575
$605$	0	0
$606$	−8.03924e6	−0.889269
$607$	− 5.97807e6i	− 0.658551i	−0.944234	−	0.329275i	$-0.893196\pi$
$607$	− 5.97807e6i	− 0.658551i	0.944234	−	0.329275i	$-0.106804\pi$
$608$	1.51526e7i	1.66237i
$609$	−3.31115e7	−3.61773
$610$	0	0
$611$	−2.64541e6	−0.286675
$612$	2.47583e7i	2.67203i
$613$	449741.i	0.0483405i	0.999708	+	0.0241702i	$0.00769438\pi$
$613$	449741.i	0.0483405i	−0.999708	+	0.0241702i	$0.992306\pi$
$614$	−1.32772e7	−1.42130
$615$	0	0
$616$	−3.45251e6	−0.366593
$617$	3.68155e6i	0.389330i	0.980870	+	0.194665i	$0.0623620\pi$
$617$	3.68155e6i	0.389330i	−0.980870	+	0.194665i	$0.937638\pi$
$618$	− 7.57423e6i	− 0.797751i
$619$	1.14389e6	0.119993	0.0599967	−	0.998199i	$-0.480891\pi$
$619$	1.14389e6	0.119993	0.0599967	+	0.998199i	$0.480891\pi$
$620$	0	0
$621$	9.12075e6	0.949077
$622$	7.66312e6i	0.794199i
$623$	1.60945e7i	1.66133i
$624$	5.92313e6	0.608961
$625$	0	0
$626$	3.49594e6	0.356556
$627$	− 2.70691e7i	− 2.74983i
$628$	7.60778e6i	0.769766i
$629$	6.66795e6	0.671995
$630$	0	0
$631$	−1.75675e6	−0.175646	−0.0878228	−	0.996136i	$-0.527991\pi$
$631$	−1.75675e6	−0.175646	−0.0878228	+	0.996136i	$0.527991\pi$
$632$	1.48748e6i	0.148135i
$633$	− 3.31314e7i	− 3.28648i
$634$	−2.92389e6	−0.288894
$635$	0	0
$636$	1.26688e7	1.24191
$637$	− 2.11286e6i	− 0.206311i
$638$	2.24251e7i	2.18114i
$639$	−8.27409e6	−0.801619
$640$	0	0
$641$	−9.00917e6	−0.866043	−0.433022	−	0.901384i	$-0.642553\pi$
$641$	−9.00917e6	−0.866043	−0.433022	+	0.901384i	$0.642553\pi$
$642$	340039.i	0.0325605i
$643$	1.98773e7i	1.89596i	0.318327	+	0.947981i	$0.396879\pi$
$643$	1.98773e7i	1.89596i	−0.318327	+	0.947981i	$0.603121\pi$
$644$	3.41987e6	0.324934
$645$	0	0
$646$	2.24939e7	2.12072
$647$	907817.i	0.0852585i	0.999091	+	0.0426293i	$0.0135734\pi$
$647$	907817.i	0.0852585i	−0.999091	+	0.0426293i	$0.986427\pi$
$648$	8.87341e6i	0.830143i
$649$	−7.61506e6	−0.709679
$650$	0	0
$651$	−2.70223e7	−2.49902
$652$	− 6.95308e6i	− 0.640557i
$653$	4.96671e6i	0.455812i	0.973683	+	0.227906i	$0.0731880\pi$
$653$	4.96671e6i	0.455812i	−0.973683	+	0.227906i	$0.926812\pi$
$654$	−1.75616e6	−0.160554
$655$	0	0
$656$	−1.61207e7	−1.46260
$657$	− 2.69652e7i	− 2.43720i
$658$	− 2.04372e7i	− 1.84016i
$659$	−1.50643e7	−1.35125	−0.675625	−	0.737245i	$-0.736126\pi$
$659$	−1.50643e7	−1.35125	−0.675625	+	0.737245i	$0.736126\pi$
$660$	0	0
$661$	2.41958e6	0.215395	0.107698	−	0.994184i	$-0.465652\pi$
$661$	2.41958e6	0.215395	0.107698	+	0.994184i	$0.465652\pi$
$662$	4.74426e6i	0.420750i
$663$	− 7.39649e6i	− 0.653494i
$664$	−2.62106e6	−0.230705
$665$	0	0
$666$	−2.22870e7	−1.94700
$667$	4.95877e6i	0.431578i
$668$	7.31714e6i	0.634454i
$669$	3.80680e7	3.28847
$670$	0	0
$671$	4.38775e6	0.376215
$672$	3.84925e7i	3.28816i
$673$	− 1.65638e7i	− 1.40969i	−0.709362	−	0.704844i	$-0.751017\pi$
$673$	− 1.65638e7i	− 1.40969i	0.709362	−	0.704844i	$-0.248983\pi$
$674$	4.76308e6	0.403867
$675$	0	0
$676$	747159.	0.0628849
$677$	5.75548e6i	0.482625i	0.970447	+	0.241313i	$0.0775779\pi$
$677$	5.75548e6i	0.482625i	−0.970447	+	0.241313i	$0.922422\pi$
$678$	− 3.40096e7i	− 2.84137i
$679$	2.16896e7	1.80541
$680$	0	0
$681$	3.91965e7	3.23876
$682$	1.83012e7i	1.50667i
$683$	5.72848e6i	0.469881i	0.972010	+	0.234940i	$0.0754895\pi$
$683$	5.72848e6i	0.469881i	−0.972010	+	0.234940i	$0.924511\pi$
$684$	−3.38173e7	−2.76375
$685$	0	0
$686$	−5.62048e6	−0.455998
$687$	310060.i	0.0250642i
$688$	4.50027e6i	0.362466i
$689$	−2.74795e6	−0.220527
$690$	0	0
$691$	1.03053e7	0.821041	0.410520	−	0.911851i	$-0.365347\pi$
$691$	1.03053e7	0.821041	0.410520	+	0.911851i	$0.365347\pi$
$692$	− 1.71930e7i	− 1.36485i
$693$	− 4.99266e7i	− 3.94911i
$694$	1.09370e7	0.861982
$695$	0	0
$696$	−8.61380e6	−0.674018
$697$	2.01307e7i	1.56955i
$698$	1.37826e7i	1.07076i
$699$	4.24323e7	3.28476
$700$	0	0
$701$	1.51125e7	1.16156	0.580781	−	0.814060i	$-0.302747\pi$
$701$	1.51125e7	1.16156	0.580781	+	0.814060i	$0.302747\pi$
$702$	1.53943e7i	1.17901i
$703$	9.10776e6i	0.695061i
$704$	9.01806e6	0.685775
$705$	0	0
$706$	−2.06697e7	−1.56071
$707$	6.05943e6i	0.455914i
$708$	1.31029e7i	0.982393i
$709$	−2.17724e7	−1.62664	−0.813318	−	0.581820i	$-0.802341\pi$
$709$	−2.17724e7	−1.62664	−0.813318	+	0.581820i	$0.802341\pi$
$710$	0	0
$711$	−2.15104e7	−1.59579
$712$	4.18690e6i	0.309523i
$713$	4.04686e6i	0.298122i
$714$	5.71417e7	4.19476
$715$	0	0
$716$	−3.67645e6	−0.268007
$717$	1.00088e7i	0.727080i
$718$	1.45663e7i	1.05448i
$719$	−7.54447e6	−0.544260	−0.272130	−	0.962260i	$-0.587728\pi$
$719$	−7.54447e6	−0.544260	−0.272130	+	0.962260i	$0.587728\pi$
$720$	0	0
$721$	−5.70894e6	−0.408995
$722$	1.18410e7i	0.845369i
$723$	3.39224e7i	2.41346i
$724$	−1.44972e7	−1.02787
$725$	0	0
$726$	−9.99089e6	−0.703498
$727$	2.06007e7i	1.44559i	0.691063	+	0.722795i	$0.257143\pi$
$727$	2.06007e7i	1.44559i	−0.691063	+	0.722795i	$0.742857\pi$
$728$	− 1.28856e6i	− 0.0901108i
$729$	−4.18729e7	−2.91820
$730$	0	0
$731$	5.61969e6	0.388973
$732$	− 7.54983e6i	− 0.520786i
$733$	4.42100e6i	0.303921i	0.988387	+	0.151960i	$0.0485586\pi$
$733$	4.42100e6i	0.303921i	−0.988387	+	0.151960i	$0.951441\pi$
$734$	−1.59613e7	−1.09353
$735$	0	0
$736$	5.76462e6	0.392262
$737$	− 2.59322e6i	− 0.175862i
$738$	− 6.72850e7i	− 4.54755i
$739$	−847676.	−0.0570977	−0.0285489	−	0.999592i	$-0.509089\pi$
$739$	−847676.	−0.0570977	−0.0285489	+	0.999592i	$0.509089\pi$
$740$	0	0
$741$	1.01029e7	0.675926
$742$	− 2.12294e7i	− 1.41556i
$743$	− 4.18581e6i	− 0.278168i	−0.990281	−	0.139084i	$-0.955584\pi$
$743$	− 4.18581e6i	− 0.278168i	0.990281	−	0.139084i	$-0.0444158\pi$
$744$	−7.02973e6	−0.465593
$745$	0	0
$746$	9.22311e6	0.606778
$747$	− 3.79030e7i	− 2.48526i
$748$	− 1.74070e7i	− 1.13755i
$749$	256298.	0.0166932
$750$	0	0
$751$	−2.60568e7	−1.68586	−0.842930	−	0.538023i	$-0.819171\pi$
$751$	−2.60568e7	−1.68586	−0.842930	+	0.538023i	$0.819171\pi$
$752$	− 1.84204e7i	− 1.18783i
$753$	− 4.71726e7i	− 3.03182i
$754$	−8.36960e6	−0.536138
$755$	0	0
$756$	−5.34937e7	−3.40407
$757$	− 1.34084e7i	− 0.850430i	−0.905092	−	0.425215i	$-0.860199\pi$
$757$	− 1.34084e7i	− 0.850430i	0.905092	−	0.425215i	$-0.139801\pi$
$758$	4.53689e6i	0.286804i
$759$	−1.02981e7	−0.648863
$760$	0	0
$761$	1.74233e7	1.09061	0.545303	−	0.838239i	$-0.316415\pi$
$761$	1.74233e7	1.09061	0.545303	+	0.838239i	$0.316415\pi$
$762$	− 5.43498e7i	− 3.39086i
$763$	1.32368e6i	0.0823135i
$764$	−9.92122e6	−0.614938
$765$	0	0
$766$	−1.78856e7	−1.10136
$767$	− 2.84213e6i	− 0.174444i
$768$	3.93532e7i	2.40756i
$769$	−1.67671e6	−0.102245	−0.0511227	−	0.998692i	$-0.516280\pi$
$769$	−1.67671e6	−0.102245	−0.0511227	+	0.998692i	$0.516280\pi$
$770$	0	0
$771$	4.78489e7	2.89892
$772$	2.31394e7i	1.39736i
$773$	− 2.08953e7i	− 1.25776i	−0.777501	−	0.628882i	$-0.783513\pi$
$773$	− 2.08953e7i	− 1.25776i	0.777501	−	0.628882i	$-0.216487\pi$
$774$	−1.87833e7	−1.12699
$775$	0	0
$776$	5.64244e6	0.336366
$777$	2.31366e7i	1.37482i
$778$	3.26132e7i	1.93172i
$779$	−2.74965e7	−1.62343
$780$	0	0
$781$	5.81732e6	0.341268
$782$	− 8.55753e6i	− 0.500416i
$783$	− 7.75652e7i	− 4.52129i
$784$	1.47122e7	0.854843
$785$	0	0
$786$	−5.53791e7	−3.19734
$787$	2.62205e7i	1.50905i	0.656269	+	0.754527i	$0.272133\pi$
$787$	2.62205e7i	1.50905i	−0.656269	+	0.754527i	$0.727867\pi$
$788$	1.38410e7i	0.794060i
$789$	3.26092e7	1.86486
$790$	0	0
$791$	−2.56342e7	−1.45673
$792$	− 1.29882e7i	− 0.735758i
$793$	1.63762e6i	0.0924761i
$794$	−3.09150e7	−1.74028
$795$	0	0
$796$	233842.	0.0130810
$797$	− 2.73769e6i	− 0.152665i	−0.997082	−	0.0763324i	$-0.975679\pi$
$797$	− 2.73769e6i	− 0.152665i	0.997082	−	0.0763324i	$-0.0243210\pi$
$798$	7.80498e7i	4.33875i
$799$	−2.30024e7	−1.27470
$800$	0	0
$801$	−6.05465e7	−3.33433
$802$	− 2.06058e7i	− 1.13124i
$803$	1.89586e7i	1.03757i
$804$	−4.46205e6	−0.243441
$805$	0	0
$806$	−6.83044e6	−0.370349
$807$	− 4.19627e7i	− 2.26819i
$808$	1.57633e6i	0.0849413i
$809$	2.09569e7	1.12578	0.562892	−	0.826530i	$-0.309689\pi$
$809$	2.09569e7	1.12578	0.562892	+	0.826530i	$0.309689\pi$
$810$	0	0
$811$	2.72798e6	0.145643	0.0728213	−	0.997345i	$-0.476800\pi$
$811$	2.72798e6	0.145643	0.0728213	+	0.997345i	$0.476800\pi$
$812$	− 2.90835e7i	− 1.54795i
$813$	− 1.23923e7i	− 0.657546i
$814$	1.56695e7	0.828884
$815$	0	0
$816$	5.15028e7	2.70773
$817$	7.67594e6i	0.402324i
$818$	1.22985e7i	0.642644i
$819$	1.86338e7	0.970717
$820$	0	0
$821$	1.56207e7	0.808801	0.404400	−	0.914582i	$-0.367480\pi$
$821$	1.56207e7	0.808801	0.404400	+	0.914582i	$0.367480\pi$
$822$	3.73531e7i	1.92818i
$823$	− 7.04631e6i	− 0.362629i	−0.983425	−	0.181314i	$-0.941965\pi$
$823$	− 7.04631e6i	− 0.362629i	0.983425	−	0.181314i	$-0.0580352\pi$
$824$	−1.48515e6	−0.0761997
$825$	0	0
$826$	2.19569e7	1.11975
$827$	− 3.60941e7i	− 1.83515i	−0.397559	−	0.917577i	$-0.630143\pi$
$827$	− 3.60941e7i	− 1.83515i	0.397559	−	0.917577i	$-0.369857\pi$
$828$	1.28654e7i	0.652149i
$829$	2.15501e7	1.08909	0.544543	−	0.838733i	$-0.316703\pi$
$829$	2.15501e7	1.08909	0.544543	+	0.838733i	$0.316703\pi$
$830$	0	0
$831$	−3.21139e7	−1.61321
$832$	3.36576e6i	0.168568i
$833$	− 1.83718e7i	− 0.917357i
$834$	2.30415e7	1.14709
$835$	0	0
$836$	2.37762e7	1.17659
$837$	− 6.33011e7i	− 3.12318i
$838$	8.27812e6i	0.407213i
$839$	−1.21355e7	−0.595187	−0.297593	−	0.954693i	$-0.596184\pi$
$839$	−1.21355e7	−0.595187	−0.297593	+	0.954693i	$0.596184\pi$
$840$	0	0
$841$	2.16596e7	1.05599
$842$	− 4.08688e7i	− 1.98660i
$843$	− 5.31717e7i	− 2.57699i
$844$	2.91010e7	1.40621
$845$	0	0
$846$	7.68835e7	3.69324
$847$	7.53045e6i	0.360672i
$848$	− 1.91344e7i	− 0.913746i
$849$	5.11988e6	0.243776
$850$	0	0
$851$	3.46493e6	0.164010
$852$	− 1.00096e7i	− 0.472410i
$853$	− 3.47231e7i	− 1.63398i	−0.576653	−	0.816990i	$-0.695641\pi$
$853$	− 3.47231e7i	− 1.63398i	0.576653	−	0.816990i	$-0.304359\pi$
$854$	−1.26514e7	−0.593602
$855$	0	0
$856$	66674.7	0.00311012
$857$	− 7.91521e6i	− 0.368138i	−0.982913	−	0.184069i	$-0.941073\pi$
$857$	− 7.91521e6i	− 0.368138i	0.982913	−	0.184069i	$-0.0589270\pi$
$858$	− 1.73815e7i	− 0.806064i
$859$	2.19414e7	1.01457	0.507284	−	0.861779i	$-0.330649\pi$
$859$	2.19414e7	1.01457	0.507284	+	0.861779i	$0.330649\pi$
$860$	0	0
$861$	−6.98498e7	−3.21113
$862$	− 1.61325e7i	− 0.739491i
$863$	2.34790e7i	1.07313i	0.843859	+	0.536566i	$0.180279\pi$
$863$	2.34790e7i	1.07313i	−0.843859	+	0.536566i	$0.819721\pi$
$864$	−9.01703e7	−4.10941
$865$	0	0
$866$	4.05337e7	1.83663
$867$	− 2.20260e7i	− 0.995150i
$868$	− 2.37351e7i	− 1.06928i
$869$	1.51235e7	0.679363
$870$	0	0
$871$	967854.	0.0432279
$872$	344348.i	0.0153358i
$873$	8.15950e7i	3.62350i
$874$	1.16887e7	0.517593
$875$	0	0
$876$	3.26213e7	1.43629
$877$	2.71398e7i	1.19154i	0.803156	+	0.595768i	$0.203152\pi$
$877$	2.71398e7i	1.19154i	−0.803156	+	0.595768i	$0.796848\pi$
$878$	− 2.53649e7i	− 1.11045i
$879$	2.90949e7	1.27012
$880$	0	0
$881$	4.57122e6	0.198423	0.0992115	−	0.995066i	$-0.468368\pi$
$881$	4.57122e6	0.198423	0.0992115	+	0.995066i	$0.468368\pi$
$882$	6.14060e7i	2.65791i
$883$	− 3.03713e7i	− 1.31087i	−0.755250	−	0.655437i	$-0.772484\pi$
$883$	− 3.03713e7i	− 1.31087i	0.755250	−	0.655437i	$-0.227516\pi$
$884$	6.49670e6	0.279616
$885$	0	0
$886$	4.12882e7	1.76702
$887$	3.39368e7i	1.44831i	0.689636	+	0.724156i	$0.257770\pi$
$887$	3.39368e7i	1.44831i	−0.689636	+	0.724156i	$0.742230\pi$
$888$	6.01887e6i	0.256143i
$889$	−4.09652e7	−1.73844
$890$	0	0
$891$	9.02173e7	3.80711
$892$	3.34370e7i	1.40707i
$893$	− 3.14190e7i	− 1.31845i
$894$	−1.27354e7	−0.532929
$895$	0	0
$896$	1.53552e7	0.638978
$897$	− 3.84351e6i	− 0.159495i
$898$	− 5.28227e6i	− 0.218590i
$899$	3.44156e7	1.42022
$900$	0	0
$901$	−2.38940e7	−0.980567
$902$	4.73065e7i	1.93600i
$903$	1.94993e7i	0.795793i
$904$	−6.66860e6	−0.271402
$905$	0	0
$906$	2.34764e7	0.950193
$907$	− 2.55351e7i	− 1.03067i	−0.856989	−	0.515334i	$-0.827668\pi$
$907$	− 2.55351e7i	− 1.03067i	0.856989	−	0.515334i	$-0.172332\pi$
$908$	3.44282e7i	1.38580i
$909$	−2.27952e7	−0.915028
$910$	0	0
$911$	−2.19240e6	−0.0875233	−0.0437616	−	0.999042i	$-0.513934\pi$
$911$	−2.19240e6	−0.0875233	−0.0437616	+	0.999042i	$0.513934\pi$
$912$	7.03477e7i	2.80068i
$913$	2.66487e7i	1.05803i
$914$	−2.97771e7	−1.17901
$915$	0	0
$916$	−272341.	−0.0107244
$917$	4.17410e7i	1.63923i
$918$	1.33857e8i	5.24245i
$919$	1.95519e6	0.0763659	0.0381829	−	0.999271i	$-0.487843\pi$
$919$	1.95519e6	0.0763659	0.0381829	+	0.999271i	$0.487843\pi$
$920$	0	0
$921$	−5.18521e7	−2.01427
$922$	− 6.72900e7i	− 2.60689i
$923$	2.17117e6i	0.0838858i
$924$	6.03990e7	2.32729
$925$	0	0
$926$	−4.21256e7	−1.61443
$927$	− 2.14767e7i	− 0.820859i
$928$	− 4.90239e7i	− 1.86869i
$929$	−2.43874e7	−0.927099	−0.463550	−	0.886071i	$-0.653424\pi$
$929$	−2.43874e7	−0.927099	−0.463550	+	0.886071i	$0.653424\pi$
$930$	0	0
$931$	2.50940e7	0.948846
$932$	3.72704e7i	1.40548i
$933$	2.99271e7i	1.12554i
$934$	4.03702e7	1.51424
$935$	0	0
$936$	4.84750e6	0.180854
$937$	− 4.49798e7i	− 1.67367i	−0.547458	−	0.836833i	$-0.684404\pi$
$937$	− 4.49798e7i	− 1.67367i	0.547458	−	0.836833i	$-0.315596\pi$
$938$	7.47717e6i	0.277479i
$939$	1.36528e7	0.505311
$940$	0	0
$941$	2.36994e7	0.872496	0.436248	−	0.899826i	$-0.356307\pi$
$941$	2.36994e7	0.872496	0.436248	+	0.899826i	$0.356307\pi$
$942$	6.60546e7i	2.42536i
$943$	1.04607e7i	0.383073i
$944$	1.97902e7	0.722802
$945$	0	0
$946$	1.32061e7	0.479786
$947$	2.24198e7i	0.812376i	0.913789	+	0.406188i	$0.133142\pi$
$947$	2.24198e7i	0.812376i	−0.913789	+	0.406188i	$0.866858\pi$
$948$	− 2.60223e7i	− 0.940427i
$949$	−7.07582e6	−0.255042
$950$	0	0
$951$	−1.14188e7	−0.409420
$952$	− 1.12043e7i	− 0.400676i
$953$	− 1.65747e7i	− 0.591171i	−0.955316	−	0.295585i	$-0.904485\pi$
$953$	− 1.65747e7i	− 0.591171i	0.955316	−	0.295585i	$-0.0955147\pi$
$954$	7.98636e7	2.84105
$955$	0	0
$956$	−8.79119e6	−0.311102
$957$	8.75778e7i	3.09111i
$958$	3.47465e7i	1.22320i
$959$	2.81542e7	0.988545
$960$	0	0
$961$	−542563.	−0.0189514
$962$	5.84824e6i	0.203745i
$963$	964179.i	0.0335036i
$964$	−2.97957e7	−1.03267
$965$	0	0
$966$	2.96931e7	1.02379
$967$	1.57319e7i	0.541021i	0.962717	+	0.270510i	$0.0871924\pi$
$967$	1.57319e7i	0.541021i	−0.962717	+	0.270510i	$0.912808\pi$
$968$	1.95901e6i	0.0671968i
$969$	8.78465e7	3.00549
$970$	0	0
$971$	−8.18745e6	−0.278677	−0.139338	−	0.990245i	$-0.544498\pi$
$971$	−8.18745e6	−0.278677	−0.139338	+	0.990245i	$0.544498\pi$
$972$	− 7.93044e7i	− 2.69235i
$973$	− 1.73671e7i	− 0.588093i
$974$	−6.10774e7	−2.06292
$975$	0	0
$976$	−1.14030e7	−0.383172
$977$	4.04123e7i	1.35450i	0.735755	+	0.677248i	$0.236827\pi$
$977$	4.04123e7i	1.35450i	−0.735755	+	0.677248i	$0.763173\pi$
$978$	− 6.03701e7i	− 2.01825i
$979$	4.25689e7	1.41950
$980$	0	0
$981$	−4.97960e6	−0.165205
$982$	− 4.45179e7i	− 1.47318i
$983$	3.44605e7i	1.13746i	0.822523	+	0.568732i	$0.192566\pi$
$983$	3.44605e7i	1.13746i	−0.822523	+	0.568732i	$0.807434\pi$
$984$	−1.81711e7	−0.598264
$985$	0	0
$986$	−7.27754e7	−2.38392
$987$	− 7.98142e7i	− 2.60788i
$988$	8.87385e6i	0.289214i
$989$	2.92021e6	0.0949345
$990$	0	0
$991$	1.80218e7	0.582927	0.291464	−	0.956582i	$-0.405858\pi$
$991$	1.80218e7	0.582927	0.291464	+	0.956582i	$0.405858\pi$
$992$	− 4.00084e7i	− 1.29084i
$993$	1.85280e7i	0.596287i
$994$	−1.67734e7	−0.538461
$995$	0	0
$996$	4.58533e7	1.46461
$997$	3.07170e7i	0.978679i	0.872093	+	0.489340i	$0.162762\pi$
$997$	3.07170e7i	0.978679i	−0.872093	+	0.489340i	$0.837238\pi$
$998$	2.20646e7i	0.701244i
$999$	−5.41985e7	−1.71820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.f.274.10		12
5.2	odd	4		325.6.a.f.1.2		6
5.3	odd	4		65.6.a.e.1.5	✓	6
5.4	even	2	inner	325.6.b.f.274.3		12
15.8	even	4		585.6.a.k.1.2		6
20.3	even	4		1040.6.a.r.1.6		6
65.38	odd	4		845.6.a.g.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.e.1.5	✓	6	5.3	odd	4
325.6.a.f.1.2		6	5.2	odd	4
325.6.b.f.274.3		12	5.4	even	2	inner
325.6.b.f.274.10		12	1.1	even	1	trivial
585.6.a.k.1.2		6	15.8	even	4
845.6.a.g.1.2		6	65.38	odd	4
1040.6.a.r.1.6		6	20.3	even	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)

\(f(q)\)	\(=\)	\(q+7.62628i q^{2} +29.7832i q^{3} -26.1601 q^{4} -227.135 q^{6} +171.199i q^{7} +44.5366i q^{8} -644.042 q^{9} +452.811 q^{11} -779.133i q^{12} +169.000i q^{13} -1305.61 q^{14} -1176.77 q^{16} +1469.49i q^{17} -4911.64i q^{18} -2007.18 q^{19} -5098.87 q^{21} +3453.26i q^{22} +763.605i q^{23} -1326.44 q^{24} -1288.84 q^{26} -11944.3i q^{27} -4478.59i q^{28} +6493.90 q^{29} +5299.68 q^{31} -7549.22i q^{32} +13486.2i q^{33} -11206.7 q^{34} +16848.2 q^{36} -4537.59i q^{37} -15307.3i q^{38} -5033.37 q^{39} +13699.1 q^{41} -38885.4i q^{42} -3824.25i q^{43} -11845.6 q^{44} -5823.46 q^{46} +15653.3i q^{47} -35048.1i q^{48} -12502.1 q^{49} -43766.2 q^{51} -4421.06i q^{52} +16260.1i q^{53} +91090.7 q^{54} -7624.63 q^{56} -59780.3i q^{57} +49524.3i q^{58} -16817.3 q^{59} +9690.04 q^{61} +40416.8i q^{62} -110259. i q^{63} +19915.7 q^{64} -102849. q^{66} -5726.95i q^{67} -38442.0i q^{68} -22742.6 q^{69} +12847.1 q^{71} -28683.4i q^{72} +41868.8i q^{73} +34604.9 q^{74} +52508.0 q^{76} +77520.8i q^{77} -38385.9i q^{78} +33399.1 q^{79} +199239. q^{81} +104473. i q^{82} +58851.7i q^{83} +133387. q^{84} +29164.8 q^{86} +193409. i q^{87} +20166.7i q^{88} +94010.3 q^{89} -28932.7 q^{91} -19976.0i q^{92} +157842. i q^{93} -119377. q^{94} +224840. q^{96} -126692. i q^{97} -95344.8i q^{98} -291629. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{4} - 104 q^{6} - 2068 q^{9} + 1600 q^{11} - 4216 q^{14} + 644 q^{16} - 10608 q^{19} + 2144 q^{21} - 9512 q^{24} - 676 q^{26} + 7328 q^{29} + 33328 q^{31} - 2760 q^{34} + 3636 q^{36} + 6760 q^{39}+ \cdots - 602528 q^{99}+O(q^{100}) \) 12 * q - 268 * q^4 - 104 * q^6 - 2068 * q^9 + 1600 * q^11 - 4216 * q^14 + 644 * q^16 - 10608 * q^19 + 2144 * q^21 - 9512 * q^24 - 676 * q^26 + 7328 * q^29 + 33328 * q^31 - 2760 * q^34 + 3636 * q^36 + 6760 * q^39 + 24872 * q^41 + 70248 * q^44 - 9112 * q^46 + 55684 * q^49 - 55328 * q^51 + 265824 * q^54 + 88840 * q^56 + 23792 * q^59 + 114464 * q^61 + 79828 * q^64 + 19816 * q^66 - 144808 * q^69 + 428208 * q^71 + 364208 * q^74 + 733272 * q^76 - 72048 * q^79 + 701028 * q^81 - 12048 * q^84 + 58408 * q^86 + 104056 * q^89 - 58136 * q^91 - 315144 * q^94 + 1095816 * q^96 - 602528 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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