LMFDB - Embedded newform 1296.5.e.g.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,5,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (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:	$133.967472157$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 $ x^8 - 3x^7 + 6x^6 + 121x^5 + 1104x^4 - 1647x^3 + 6529x^2 + 85254*x + 440076
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-3.41053 - 2.74723i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.g.161.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-40.2664i q^{5} -14.7738 q^{7} +O(q^{10})$	$q-40.2664i q^{5} -14.7738 q^{7} -81.6517i q^{11} +278.107 q^{13} +10.8854i q^{17} -532.815 q^{19} +810.651i q^{23} -996.386 q^{25} +297.497i q^{29} -195.089 q^{31} +594.887i q^{35} -2097.18 q^{37} -1569.71i q^{41} +92.1727 q^{43} +2135.21i q^{47} -2182.74 q^{49} -2579.42i q^{53} -3287.82 q^{55} -1557.74i q^{59} +5371.95 q^{61} -11198.4i q^{65} -915.282 q^{67} +8215.93i q^{71} -3438.63 q^{73} +1206.30i q^{77} -4632.34 q^{79} -5999.27i q^{83} +438.317 q^{85} +8434.43i q^{89} -4108.69 q^{91} +21454.6i q^{95} -6031.15 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 26 q^{7}+O(q^{10}) $ 8 * q + 26 * q^7	$ 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) $ 8 * q + 26 * q^7 + 10 * q^13 - 562 * q^19 - 706 * q^25 + 374 * q^31 + 16 * q^37 - 136 * q^43 + 654 * q^49 + 1818 * q^55 + 3874 * q^61 + 308 * q^67 - 7802 * q^73 - 4390 * q^79 + 6084 * q^85 - 15830 * q^91 - 14564 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 40.2664i	− 1.61066i	−0.592829	−	0.805329i	$-0.701989\pi$
$5$	− 40.2664i	− 1.61066i	0.592829	−	0.805329i	$-0.298011\pi$
$6$	0	0
$7$	−14.7738	−0.301505	−0.150753	−	0.988572i	$-0.548170\pi$
$7$	−14.7738	−0.301505	−0.150753	+	0.988572i	$0.548170\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 81.6517i	− 0.674808i	−0.941360	−	0.337404i	$-0.890451\pi$
$11$	− 81.6517i	− 0.674808i	0.941360	−	0.337404i	$-0.109549\pi$
$12$	0	0
$13$	278.107	1.64560	0.822801	−	0.568329i	$-0.192410\pi$
$13$	278.107	1.64560	0.822801	+	0.568329i	$0.192410\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10.8854i	0.0376658i	0.999823	+	0.0188329i	$0.00599505\pi$
$17$	10.8854i	0.0376658i	−0.999823	+	0.0188329i	$0.994005\pi$
$18$	0	0
$19$	−532.815	−1.47594	−0.737971	−	0.674833i	$-0.764216\pi$
$19$	−532.815	−1.47594	−0.737971	+	0.674833i	$0.764216\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	810.651i	1.53242i	0.642589	+	0.766211i	$0.277860\pi$
$23$	810.651i	1.53242i	−0.642589	+	0.766211i	$0.722140\pi$
$24$	0	0
$25$	−996.386	−1.59422
$26$	0	0
$27$	0	0
$28$	0	0
$29$	297.497i	0.353742i	0.984234	+	0.176871i	$0.0565976\pi$
$29$	297.497i	0.353742i	−0.984234	+	0.176871i	$0.943402\pi$
$30$	0	0
$31$	−195.089	−0.203007	−0.101503	−	0.994835i	$-0.532365\pi$
$31$	−195.089	−0.203007	−0.101503	+	0.994835i	$0.532365\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	594.887i	0.485622i
$36$	0	0
$37$	−2097.18	−1.53191	−0.765954	−	0.642896i	$-0.777733\pi$
$37$	−2097.18	−1.53191	−0.765954	+	0.642896i	$0.777733\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1569.71i	− 0.933794i	−0.884312	−	0.466897i	$-0.845372\pi$
$41$	− 1569.71i	− 0.933794i	0.884312	−	0.466897i	$-0.154628\pi$
$42$	0	0
$43$	92.1727	0.0498500	0.0249250	−	0.999689i	$-0.492065\pi$
$43$	92.1727	0.0498500	0.0249250	+	0.999689i	$0.492065\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2135.21i	0.966597i	0.875456	+	0.483299i	$0.160561\pi$
$47$	2135.21i	0.966597i	−0.875456	+	0.483299i	$0.839439\pi$
$48$	0	0
$49$	−2182.74	−0.909094
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2579.42i	− 0.918271i	−0.888366	−	0.459135i	$-0.848159\pi$
$53$	− 2579.42i	− 0.918271i	0.888366	−	0.459135i	$-0.151841\pi$
$54$	0	0
$55$	−3287.82	−1.08688
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1557.74i	− 0.447497i	−0.974647	−	0.223748i	$-0.928171\pi$
$59$	− 1557.74i	− 0.447497i	0.974647	−	0.223748i	$-0.0718294\pi$
$60$	0	0
$61$	5371.95	1.44368	0.721842	−	0.692058i	$-0.243296\pi$
$61$	5371.95	1.44368	0.721842	+	0.692058i	$0.243296\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 11198.4i	− 2.65050i
$66$	0	0
$67$	−915.282	−0.203894	−0.101947	−	0.994790i	$-0.532507\pi$
$67$	−915.282	−0.203894	−0.101947	+	0.994790i	$0.532507\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8215.93i	1.62982i	0.579587	+	0.814910i	$0.303214\pi$
$71$	8215.93i	1.62982i	−0.579587	+	0.814910i	$0.696786\pi$
$72$	0	0
$73$	−3438.63	−0.645267	−0.322634	−	0.946524i	$-0.604568\pi$
$73$	−3438.63	−0.645267	−0.322634	+	0.946524i	$0.604568\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1206.30i	0.203458i
$78$	0	0
$79$	−4632.34	−0.742244	−0.371122	−	0.928584i	$-0.621027\pi$
$79$	−4632.34	−0.742244	−0.371122	+	0.928584i	$0.621027\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5999.27i	− 0.870847i	−0.900226	−	0.435424i	$-0.856599\pi$
$83$	− 5999.27i	− 0.870847i	0.900226	−	0.435424i	$-0.143401\pi$
$84$	0	0
$85$	438.317	0.0606667
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8434.43i	1.06482i	0.846487	+	0.532409i	$0.178713\pi$
$89$	8434.43i	1.06482i	−0.846487	+	0.532409i	$0.821287\pi$
$90$	0	0
$91$	−4108.69	−0.496158
$92$	0	0
$93$	0	0
$94$	0	0
$95$	21454.6i	2.37724i
$96$	0	0
$97$	−6031.15	−0.640998	−0.320499	−	0.947249i	$-0.603851\pi$
$97$	−6031.15	−0.640998	−0.320499	+	0.947249i	$0.603851\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5982.79i	− 0.586490i	−0.956037	−	0.293245i	$-0.905265\pi$
$101$	− 5982.79i	− 0.586490i	0.956037	−	0.293245i	$-0.0947352\pi$
$102$	0	0
$103$	1531.44	0.144353	0.0721763	−	0.997392i	$-0.477006\pi$
$103$	1531.44	0.144353	0.0721763	+	0.997392i	$0.477006\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20499.8i	1.79053i	0.445536	+	0.895264i	$0.353013\pi$
$107$	20499.8i	1.79053i	−0.445536	+	0.895264i	$0.646987\pi$
$108$	0	0
$109$	−9404.21	−0.791534	−0.395767	−	0.918351i	$-0.629521\pi$
$109$	−9404.21	−0.791534	−0.395767	+	0.918351i	$0.629521\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3000.59i	0.234990i	0.993073	+	0.117495i	$0.0374864\pi$
$113$	3000.59i	0.234990i	−0.993073	+	0.117495i	$0.962514\pi$
$114$	0	0
$115$	32642.0	2.46821
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 160.818i	− 0.0113564i
$120$	0	0
$121$	7974.00	0.544635
$122$	0	0
$123$	0	0
$124$	0	0
$125$	14954.4i	0.957081i
$126$	0	0
$127$	−9817.40	−0.608680	−0.304340	−	0.952563i	$-0.598436\pi$
$127$	−9817.40	−0.608680	−0.304340	+	0.952563i	$0.598436\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	638.774i	0.0372224i	0.999827	+	0.0186112i	$0.00592447\pi$
$131$	638.774i	0.0372224i	−0.999827	+	0.0186112i	$0.994076\pi$
$132$	0	0
$133$	7871.68	0.445004
$134$	0	0
$135$	0	0
$136$	0	0
$137$	32004.0i	1.70515i	0.522604	+	0.852575i	$0.324960\pi$
$137$	32004.0i	1.70515i	−0.522604	+	0.852575i	$0.675040\pi$
$138$	0	0
$139$	5584.26	0.289025	0.144513	−	0.989503i	$-0.453839\pi$
$139$	5584.26	0.289025	0.144513	+	0.989503i	$0.453839\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 22707.9i	− 1.11047i
$144$	0	0
$145$	11979.2	0.569758
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8220.03i	− 0.370255i	−0.982715	−	0.185127i	$-0.940730\pi$
$149$	− 8220.03i	− 0.370255i	0.982715	−	0.185127i	$-0.0592698\pi$
$150$	0	0
$151$	−20472.0	−0.897854	−0.448927	−	0.893568i	$-0.648194\pi$
$151$	−20472.0	−0.897854	−0.448927	+	0.893568i	$0.648194\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7855.56i	0.326974i
$156$	0	0
$157$	25793.9	1.04645	0.523224	−	0.852195i	$-0.324729\pi$
$157$	25793.9	1.04645	0.523224	+	0.852195i	$0.324729\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 11976.4i	− 0.462034i
$162$	0	0
$163$	16488.3	0.620584	0.310292	−	0.950641i	$-0.399573\pi$
$163$	16488.3	0.620584	0.310292	+	0.950641i	$0.399573\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 39563.6i	− 1.41861i	−0.704903	−	0.709304i	$-0.749009\pi$
$167$	− 39563.6i	− 1.41861i	0.704903	−	0.709304i	$-0.250991\pi$
$168$	0	0
$169$	48782.4	1.70801
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4854.61i	− 0.162204i	−0.996706	−	0.0811022i	$-0.974156\pi$
$173$	− 4854.61i	− 0.162204i	0.996706	−	0.0811022i	$-0.0258440\pi$
$174$	0	0
$175$	14720.4	0.480665
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4764.02i	− 0.148685i	−0.997233	−	0.0743425i	$-0.976314\pi$
$179$	− 4764.02i	− 0.148685i	0.997233	−	0.0743425i	$-0.0236858\pi$
$180$	0	0
$181$	3741.40	0.114203	0.0571014	−	0.998368i	$-0.481814\pi$
$181$	3741.40	0.114203	0.0571014	+	0.998368i	$0.481814\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	84446.0i	2.46738i
$186$	0	0
$187$	888.812	0.0254171
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10396.5i	0.284984i	0.989796	+	0.142492i	$0.0455115\pi$
$191$	10396.5i	0.284984i	−0.989796	+	0.142492i	$0.954489\pi$
$192$	0	0
$193$	−43212.2	−1.16009	−0.580045	−	0.814584i	$-0.696965\pi$
$193$	−43212.2	−1.16009	−0.580045	+	0.814584i	$0.696965\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	49899.7i	1.28578i	0.765960	+	0.642889i	$0.222264\pi$
$197$	49899.7i	1.28578i	−0.765960	+	0.642889i	$0.777736\pi$
$198$	0	0
$199$	7613.76	0.192262	0.0961310	−	0.995369i	$-0.469353\pi$
$199$	7613.76	0.192262	0.0961310	+	0.995369i	$0.469353\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4395.16i	− 0.106655i
$204$	0	0
$205$	−63206.5	−1.50402
$206$	0	0
$207$	0	0
$208$	0	0
$209$	43505.2i	0.995976i
$210$	0	0
$211$	56195.5	1.26222	0.631112	−	0.775692i	$-0.282599\pi$
$211$	56195.5	1.26222	0.631112	+	0.775692i	$0.282599\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 3711.47i	− 0.0802913i
$216$	0	0
$217$	2882.21	0.0612076
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3027.31i	0.0619829i
$222$	0	0
$223$	−37922.3	−0.762578	−0.381289	−	0.924456i	$-0.624520\pi$
$223$	−37922.3	−0.762578	−0.381289	+	0.924456i	$0.624520\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9770.20i	0.189606i	0.995496	+	0.0948029i	$0.0302221\pi$
$227$	9770.20i	0.189606i	−0.995496	+	0.0948029i	$0.969778\pi$
$228$	0	0
$229$	37930.0	0.723290	0.361645	−	0.932316i	$-0.382215\pi$
$229$	37930.0	0.723290	0.361645	+	0.932316i	$0.382215\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3372.35i	0.0621185i	0.999518	+	0.0310592i	$0.00988805\pi$
$233$	3372.35i	0.0621185i	−0.999518	+	0.0310592i	$0.990112\pi$
$234$	0	0
$235$	85977.4	1.55686
$236$	0	0
$237$	0	0
$238$	0	0
$239$	74615.5i	1.30627i	0.757241	+	0.653135i	$0.226547\pi$
$239$	74615.5i	1.30627i	−0.757241	+	0.653135i	$0.773453\pi$
$240$	0	0
$241$	−24254.2	−0.417592	−0.208796	−	0.977959i	$-0.566954\pi$
$241$	−24254.2	−0.417592	−0.208796	+	0.977959i	$0.566954\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	87891.0i	1.46424i
$246$	0	0
$247$	−148179.	−2.42881
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25543.7i	0.405449i	0.979236	+	0.202724i	$0.0649795\pi$
$251$	25543.7i	0.405449i	−0.979236	+	0.202724i	$0.935020\pi$
$252$	0	0
$253$	66191.1	1.03409
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21895.1i	0.331498i	0.986168	+	0.165749i	$0.0530042\pi$
$257$	21895.1i	0.331498i	−0.986168	+	0.165749i	$0.946996\pi$
$258$	0	0
$259$	30983.3	0.461878
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7550.84i	0.109165i	0.998509	+	0.0545825i	$0.0173828\pi$
$263$	7550.84i	0.109165i	−0.998509	+	0.0545825i	$0.982617\pi$
$264$	0	0
$265$	−103864.	−1.47902
$266$	0	0
$267$	0	0
$268$	0	0
$269$	106275.i	1.46868i	0.678780	+	0.734342i	$0.262509\pi$
$269$	106275.i	1.46868i	−0.678780	+	0.734342i	$0.737491\pi$
$270$	0	0
$271$	−82971.5	−1.12977	−0.564885	−	0.825170i	$-0.691080\pi$
$271$	−82971.5	−1.12977	−0.564885	+	0.825170i	$0.691080\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	81356.6i	1.07579i
$276$	0	0
$277$	−65233.7	−0.850183	−0.425091	−	0.905150i	$-0.639758\pi$
$277$	−65233.7	−0.850183	−0.425091	+	0.905150i	$0.639758\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	94850.2i	1.20123i	0.799539	+	0.600614i	$0.205077\pi$
$281$	94850.2i	1.20123i	−0.799539	+	0.600614i	$0.794923\pi$
$282$	0	0
$283$	−105668.	−1.31938	−0.659692	−	0.751536i	$-0.729313\pi$
$283$	−105668.	−1.31938	−0.659692	+	0.751536i	$0.729313\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23190.5i	0.281544i
$288$	0	0
$289$	83402.5	0.998581
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 70904.0i	− 0.825915i	−0.910750	−	0.412958i	$-0.864496\pi$
$293$	− 70904.0i	− 0.825915i	0.910750	−	0.412958i	$-0.135504\pi$
$294$	0	0
$295$	−62724.5	−0.720764
$296$	0	0
$297$	0	0
$298$	0	0
$299$	225448.i	2.52176i
$300$	0	0
$301$	−1361.74	−0.0150301
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 216309.i	− 2.32528i
$306$	0	0
$307$	−27339.7	−0.290079	−0.145040	−	0.989426i	$-0.546331\pi$
$307$	−27339.7	−0.290079	−0.145040	+	0.989426i	$0.546331\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 101749.i	− 1.05199i	−0.850488	−	0.525994i	$-0.823694\pi$
$311$	− 101749.i	− 1.05199i	0.850488	−	0.525994i	$-0.176306\pi$
$312$	0	0
$313$	−45621.6	−0.465674	−0.232837	−	0.972516i	$-0.574801\pi$
$313$	−45621.6	−0.465674	−0.232837	+	0.972516i	$0.574801\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23736.8i	− 0.236213i	−0.993001	−	0.118107i	$-0.962318\pi$
$317$	− 23736.8i	− 0.236213i	0.993001	−	0.118107i	$-0.0376824\pi$
$318$	0	0
$319$	24291.2	0.238708
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5799.91i	− 0.0555925i
$324$	0	0
$325$	−277102.	−2.62345
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 31545.1i	− 0.291434i
$330$	0	0
$331$	−127402.	−1.16284	−0.581422	−	0.813602i	$-0.697504\pi$
$331$	−127402.	−1.16284	−0.581422	+	0.813602i	$0.697504\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	36855.2i	0.328404i
$336$	0	0
$337$	−16707.6	−0.147114	−0.0735571	−	0.997291i	$-0.523435\pi$
$337$	−16707.6	−0.147114	−0.0735571	+	0.997291i	$0.523435\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15929.4i	0.136990i
$342$	0	0
$343$	67719.0	0.575602
$344$	0	0
$345$	0	0
$346$	0	0
$347$	176749.i	1.46790i	0.679202	+	0.733951i	$0.262326\pi$
$347$	176749.i	1.46790i	−0.679202	+	0.733951i	$0.737674\pi$
$348$	0	0
$349$	98439.7	0.808201	0.404101	−	0.914715i	$-0.367585\pi$
$349$	98439.7	0.808201	0.404101	+	0.914715i	$0.367585\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 125037.i	− 1.00343i	−0.865032	−	0.501716i	$-0.832702\pi$
$353$	− 125037.i	− 1.00343i	0.865032	−	0.501716i	$-0.167298\pi$
$354$	0	0
$355$	330826.	2.62508
$356$	0	0
$357$	0	0
$358$	0	0
$359$	130777.i	1.01471i	0.861737	+	0.507356i	$0.169377\pi$
$359$	130777.i	1.01471i	−0.861737	+	0.507356i	$0.830623\pi$
$360$	0	0
$361$	153571.	1.17840
$362$	0	0
$363$	0	0
$364$	0	0
$365$	138461.i	1.03930i
$366$	0	0
$367$	128239.	0.952109	0.476054	−	0.879416i	$-0.342067\pi$
$367$	128239.	0.952109	0.476054	+	0.879416i	$0.342067\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	38107.8i	0.276864i
$372$	0	0
$373$	63208.3	0.454315	0.227157	−	0.973858i	$-0.427057\pi$
$373$	63208.3	0.454315	0.227157	+	0.973858i	$0.427057\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	82736.1i	0.582120i
$378$	0	0
$379$	101202.	0.704547	0.352273	−	0.935897i	$-0.385409\pi$
$379$	101202.	0.704547	0.352273	+	0.935897i	$0.385409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19489.7i	0.132864i	0.997791	+	0.0664319i	$0.0211615\pi$
$383$	19489.7i	0.132864i	−0.997791	+	0.0664319i	$0.978838\pi$
$384$	0	0
$385$	48573.5	0.327701
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 57179.7i	− 0.377870i	−0.981990	−	0.188935i	$-0.939496\pi$
$389$	− 57179.7i	− 0.377870i	0.981990	−	0.188935i	$-0.0605036\pi$
$390$	0	0
$391$	−8824.27	−0.0577199
$392$	0	0
$393$	0	0
$394$	0	0
$395$	186528.i	1.19550i
$396$	0	0
$397$	−75878.7	−0.481436	−0.240718	−	0.970595i	$-0.577383\pi$
$397$	−75878.7	−0.481436	−0.240718	+	0.970595i	$0.577383\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	242637.i	1.50893i	0.656341	+	0.754464i	$0.272103\pi$
$401$	242637.i	1.50893i	−0.656341	+	0.754464i	$0.727897\pi$
$402$	0	0
$403$	−54255.7	−0.334068
$404$	0	0
$405$	0	0
$406$	0	0
$407$	171238.i	1.03374i
$408$	0	0
$409$	121832.	0.728306	0.364153	−	0.931339i	$-0.381359\pi$
$409$	121832.	0.728306	0.364153	+	0.931339i	$0.381359\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23013.6i	0.134923i
$414$	0	0
$415$	−241569.	−1.40264
$416$	0	0
$417$	0	0
$418$	0	0
$419$	124571.i	0.709561i	0.934950	+	0.354780i	$0.115444\pi$
$419$	124571.i	0.709561i	−0.934950	+	0.354780i	$0.884556\pi$
$420$	0	0
$421$	25295.8	0.142720	0.0713600	−	0.997451i	$-0.477266\pi$
$421$	25295.8	0.142720	0.0713600	+	0.997451i	$0.477266\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 10846.1i	− 0.0600474i
$426$	0	0
$427$	−79363.9	−0.435279
$428$	0	0
$429$	0	0
$430$	0	0
$431$	153800.i	0.827944i	0.910290	+	0.413972i	$0.135859\pi$
$431$	153800.i	0.827944i	−0.910290	+	0.413972i	$0.864141\pi$
$432$	0	0
$433$	112975.	0.602569	0.301285	−	0.953534i	$-0.402585\pi$
$433$	112975.	0.602569	0.301285	+	0.953534i	$0.402585\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 431927.i	− 2.26176i
$438$	0	0
$439$	178167.	0.924479	0.462239	−	0.886755i	$-0.347046\pi$
$439$	178167.	0.924479	0.462239	+	0.886755i	$0.347046\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 298570.i	− 1.52138i	−0.649114	−	0.760691i	$-0.724860\pi$
$443$	− 298570.i	− 1.52138i	0.649114	−	0.760691i	$-0.275140\pi$
$444$	0	0
$445$	339624.	1.71506
$446$	0	0
$447$	0	0
$448$	0	0
$449$	201956.i	1.00176i	0.865516	+	0.500881i	$0.166991\pi$
$449$	201956.i	1.00176i	−0.865516	+	0.500881i	$0.833009\pi$
$450$	0	0
$451$	−128169.	−0.630131
$452$	0	0
$453$	0	0
$454$	0	0
$455$	165442.i	0.799141i
$456$	0	0
$457$	295317.	1.41402	0.707011	−	0.707202i	$-0.250043\pi$
$457$	295317.	1.41402	0.707011	+	0.707202i	$0.250043\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	197593.i	0.929757i	0.885375	+	0.464878i	$0.153902\pi$
$461$	197593.i	0.929757i	−0.885375	+	0.464878i	$0.846098\pi$
$462$	0	0
$463$	−86462.6	−0.403335	−0.201668	−	0.979454i	$-0.564636\pi$
$463$	−86462.6	−0.403335	−0.201668	+	0.979454i	$0.564636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 249159.i	− 1.14247i	−0.820788	−	0.571233i	$-0.806465\pi$
$467$	− 249159.i	− 1.14247i	0.820788	−	0.571233i	$-0.193535\pi$
$468$	0	0
$469$	13522.2	0.0614753
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 7526.06i	− 0.0336392i
$474$	0	0
$475$	530889.	2.35297
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 420214.i	− 1.83147i	−0.401785	−	0.915734i	$-0.631610\pi$
$479$	− 420214.i	− 1.83147i	0.401785	−	0.915734i	$-0.368390\pi$
$480$	0	0
$481$	−583241.	−2.52091
$482$	0	0
$483$	0	0
$484$	0	0
$485$	242853.i	1.03243i
$486$	0	0
$487$	394498.	1.66336	0.831680	−	0.555255i	$-0.187379\pi$
$487$	394498.	1.66336	0.831680	+	0.555255i	$0.187379\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 337091.i	− 1.39825i	−0.715001	−	0.699123i	$-0.753574\pi$
$491$	− 337091.i	− 1.39825i	0.715001	−	0.699123i	$-0.246426\pi$
$492$	0	0
$493$	−3238.38	−0.0133240
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 121380.i	− 0.491400i
$498$	0	0
$499$	−455900.	−1.83092	−0.915458	−	0.402415i	$-0.868171\pi$
$499$	−455900.	−1.83092	−0.915458	+	0.402415i	$0.868171\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33893.3i	0.133961i	0.997754	+	0.0669804i	$0.0213365\pi$
$503$	33893.3i	0.133961i	−0.997754	+	0.0669804i	$0.978663\pi$
$504$	0	0
$505$	−240905.	−0.944635
$506$	0	0
$507$	0	0
$508$	0	0
$509$	214842.i	0.829245i	0.909994	+	0.414622i	$0.136086\pi$
$509$	214842.i	0.829245i	−0.909994	+	0.414622i	$0.863914\pi$
$510$	0	0
$511$	50801.5	0.194552
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 61665.5i	− 0.232502i
$516$	0	0
$517$	174344.	0.652267
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 256109.i	− 0.943516i	−0.881728	−	0.471758i	$-0.843620\pi$
$521$	− 256109.i	− 0.943516i	0.881728	−	0.471758i	$-0.156380\pi$
$522$	0	0
$523$	−372105.	−1.36038	−0.680192	−	0.733034i	$-0.738104\pi$
$523$	−372105.	−1.36038	−0.680192	+	0.733034i	$0.738104\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2123.63i	− 0.00764640i
$528$	0	0
$529$	−377314.	−1.34832
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 436546.i	− 1.53665i
$534$	0	0
$535$	825452.	2.88393
$536$	0	0
$537$	0	0
$538$	0	0
$539$	178224.i	0.613464i
$540$	0	0
$541$	38292.3	0.130833	0.0654164	−	0.997858i	$-0.479162\pi$
$541$	38292.3	0.130833	0.0654164	+	0.997858i	$0.479162\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	378674.i	1.27489i
$546$	0	0
$547$	−502206.	−1.67845	−0.839223	−	0.543788i	$-0.816990\pi$
$547$	−502206.	−1.67845	−0.839223	+	0.543788i	$0.816990\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 158511.i	− 0.522103i
$552$	0	0
$553$	68437.1	0.223790
$554$	0	0
$555$	0	0
$556$	0	0
$557$	601940.i	1.94018i	0.242737	+	0.970092i	$0.421955\pi$
$557$	601940.i	1.94018i	−0.242737	+	0.970092i	$0.578045\pi$
$558$	0	0
$559$	25633.9	0.0820333
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 157005.i	− 0.495332i	−0.968845	−	0.247666i	$-0.920336\pi$
$563$	− 157005.i	− 0.495332i	0.968845	−	0.247666i	$-0.0796636\pi$
$564$	0	0
$565$	120823.	0.378489
$566$	0	0
$567$	0	0
$568$	0	0
$569$	488107.i	1.50762i	0.657094	+	0.753808i	$0.271785\pi$
$569$	488107.i	1.50762i	−0.657094	+	0.753808i	$0.728215\pi$
$570$	0	0
$571$	171208.	0.525112	0.262556	−	0.964917i	$-0.415435\pi$
$571$	171208.	0.525112	0.262556	+	0.964917i	$0.415435\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 807721.i	− 2.44301i
$576$	0	0
$577$	−120517.	−0.361990	−0.180995	−	0.983484i	$-0.557932\pi$
$577$	−120517.	−0.361990	−0.180995	+	0.983484i	$0.557932\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	88631.7i	0.262565i
$582$	0	0
$583$	−210614.	−0.619656
$584$	0	0
$585$	0	0
$586$	0	0
$587$	46375.0i	0.134588i	0.997733	+	0.0672942i	$0.0214366\pi$
$587$	46375.0i	0.134588i	−0.997733	+	0.0672942i	$0.978563\pi$
$588$	0	0
$589$	103947.	0.299626
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 511267.i	− 1.45391i	−0.686683	−	0.726957i	$-0.740934\pi$
$593$	− 511267.i	− 1.45391i	0.686683	−	0.726957i	$-0.259066\pi$
$594$	0	0
$595$	−6475.59	−0.0182913
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 644972.i	− 1.79757i	−0.438385	−	0.898787i	$-0.644449\pi$
$599$	− 644972.i	− 1.79757i	0.438385	−	0.898787i	$-0.355551\pi$
$600$	0	0
$601$	593210.	1.64233	0.821163	−	0.570694i	$-0.193326\pi$
$601$	593210.	1.64233	0.821163	+	0.570694i	$0.193326\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 321085.i	− 0.877220i
$606$	0	0
$607$	−441020.	−1.19696	−0.598482	−	0.801136i	$-0.704229\pi$
$607$	−441020.	−1.19696	−0.598482	+	0.801136i	$0.704229\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	593817.i	1.59063i
$612$	0	0
$613$	−397447.	−1.05769	−0.528845	−	0.848718i	$-0.677375\pi$
$613$	−397447.	−1.05769	−0.528845	+	0.848718i	$0.677375\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1617.91i	− 0.00424994i	−0.999998	−	0.00212497i	$-0.999324\pi$
$617$	− 1617.91i	− 0.00424994i	0.999998	−	0.00212497i	$-0.000676400\pi$
$618$	0	0
$619$	−548970.	−1.43274	−0.716370	−	0.697720i	$-0.754198\pi$
$619$	−548970.	−1.43274	−0.716370	+	0.697720i	$0.754198\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 124608.i	− 0.321049i
$624$	0	0
$625$	−20581.4	−0.0526883
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 22828.7i	− 0.0577005i
$630$	0	0
$631$	−378413.	−0.950403	−0.475202	−	0.879877i	$-0.657625\pi$
$631$	−378413.	−0.950403	−0.475202	+	0.879877i	$0.657625\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	395312.i	0.980375i
$636$	0	0
$637$	−607034.	−1.49601
$638$	0	0
$639$	0	0
$640$	0	0
$641$	71696.2i	0.174494i	0.996187	+	0.0872469i	$0.0278069\pi$
$641$	71696.2i	0.174494i	−0.996187	+	0.0872469i	$0.972193\pi$
$642$	0	0
$643$	−167165.	−0.404318	−0.202159	−	0.979353i	$-0.564796\pi$
$643$	−167165.	−0.404318	−0.202159	+	0.979353i	$0.564796\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 678022.i	− 1.61970i	−0.586635	−	0.809852i	$-0.699548\pi$
$647$	− 678022.i	− 1.61970i	0.586635	−	0.809852i	$-0.300452\pi$
$648$	0	0
$649$	−127192.	−0.301974
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 296149.i	− 0.694520i	−0.937769	−	0.347260i	$-0.887112\pi$
$653$	− 296149.i	− 0.694520i	0.937769	−	0.347260i	$-0.112888\pi$
$654$	0	0
$655$	25721.1	0.0599526
$656$	0	0
$657$	0	0
$658$	0	0
$659$	789890.i	1.81885i	0.415872	+	0.909423i	$0.363477\pi$
$659$	789890.i	1.81885i	−0.415872	+	0.909423i	$0.636523\pi$
$660$	0	0
$661$	−425488.	−0.973834	−0.486917	−	0.873448i	$-0.661879\pi$
$661$	−425488.	−0.973834	−0.486917	+	0.873448i	$0.661879\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 316964.i	− 0.716749i
$666$	0	0
$667$	−241167.	−0.542083
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 438629.i	− 0.974209i
$672$	0	0
$673$	−362806.	−0.801021	−0.400511	−	0.916292i	$-0.631167\pi$
$673$	−362806.	−0.801021	−0.400511	+	0.916292i	$0.631167\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	863547.i	1.88412i	0.335443	+	0.942061i	$0.391114\pi$
$677$	863547.i	1.88412i	−0.335443	+	0.942061i	$0.608886\pi$
$678$	0	0
$679$	89102.8	0.193264
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 323967.i	− 0.694479i	−0.937776	−	0.347239i	$-0.887119\pi$
$683$	− 323967.i	− 0.694479i	0.937776	−	0.347239i	$-0.112881\pi$
$684$	0	0
$685$	1.28869e6	2.74641
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 717355.i	− 1.51111i
$690$	0	0
$691$	657257.	1.37651	0.688255	−	0.725469i	$-0.258377\pi$
$691$	657257.	1.37651	0.688255	+	0.725469i	$0.258377\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 224858.i	− 0.465521i
$696$	0	0
$697$	17086.9	0.0351721
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 695995.i	− 1.41635i	−0.706038	−	0.708174i	$-0.749519\pi$
$701$	− 695995.i	− 1.41635i	0.706038	−	0.708174i	$-0.250481\pi$
$702$	0	0
$703$	1.11741e6	2.26101
$704$	0	0
$705$	0	0
$706$	0	0
$707$	88388.3i	0.176830i
$708$	0	0
$709$	−430484.	−0.856375	−0.428188	−	0.903690i	$-0.640848\pi$
$709$	−430484.	−0.856375	−0.428188	+	0.903690i	$0.640848\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 158149.i	− 0.311092i
$714$	0	0
$715$	−914366.	−1.78858
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 435311.i	− 0.842059i	−0.907047	−	0.421029i	$-0.861669\pi$
$719$	− 435311.i	− 0.842059i	0.907047	−	0.421029i	$-0.138331\pi$
$720$	0	0
$721$	−22625.1	−0.0435231
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 296422.i	− 0.563942i
$726$	0	0
$727$	6107.95	0.0115565	0.00577826	−	0.999983i	$-0.498161\pi$
$727$	6107.95	0.0115565	0.00577826	+	0.999983i	$0.498161\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1003.34i	0.00187764i
$732$	0	0
$733$	−369756.	−0.688189	−0.344094	−	0.938935i	$-0.611814\pi$
$733$	−369756.	−0.688189	−0.344094	+	0.938935i	$0.611814\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	74734.4i	0.137590i
$738$	0	0
$739$	381087.	0.697807	0.348904	−	0.937159i	$-0.386554\pi$
$739$	381087.	0.697807	0.348904	+	0.937159i	$0.386554\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	427947.i	0.775198i	0.921828	+	0.387599i	$0.126696\pi$
$743$	427947.i	0.775198i	−0.921828	+	0.387599i	$0.873304\pi$
$744$	0	0
$745$	−330991.	−0.596354
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 302859.i	− 0.539854i
$750$	0	0
$751$	423580.	0.751027	0.375514	−	0.926817i	$-0.377466\pi$
$751$	423580.	0.751027	0.375514	+	0.926817i	$0.377466\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	824333.i	1.44614i
$756$	0	0
$757$	−910525.	−1.58891	−0.794457	−	0.607321i	$-0.792244\pi$
$757$	−910525.	−1.58891	−0.794457	+	0.607321i	$0.792244\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 706744.i	− 1.22037i	−0.792257	−	0.610187i	$-0.791094\pi$
$761$	− 706744.i	− 1.22037i	0.792257	−	0.610187i	$-0.208906\pi$
$762$	0	0
$763$	138936.	0.238652
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 433217.i	− 0.736402i
$768$	0	0
$769$	495864.	0.838512	0.419256	−	0.907868i	$-0.362291\pi$
$769$	495864.	0.838512	0.419256	+	0.907868i	$0.362291\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 834415.i	− 1.39644i	−0.715882	−	0.698221i	$-0.753975\pi$
$773$	− 834415.i	− 1.39644i	0.715882	−	0.698221i	$-0.246025\pi$
$774$	0	0
$775$	194384.	0.323637
$776$	0	0
$777$	0	0
$778$	0	0
$779$	836363.i	1.37822i
$780$	0	0
$781$	670844.	1.09982
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.03863e6i	− 1.68547i
$786$	0	0
$787$	−129636.	−0.209304	−0.104652	−	0.994509i	$-0.533373\pi$
$787$	−129636.	−0.209304	−0.104652	+	0.994509i	$0.533373\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 44330.0i	− 0.0708508i
$792$	0	0
$793$	1.49398e6	2.37573
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 169450.i	− 0.266763i	−0.991065	−	0.133382i	$-0.957416\pi$
$797$	− 169450.i	− 0.266763i	0.991065	−	0.133382i	$-0.0425836\pi$
$798$	0	0
$799$	−23242.7	−0.0364076
$800$	0	0
$801$	0	0
$802$	0	0
$803$	280770.i	0.435431i
$804$	0	0
$805$	−482246.	−0.744178
$806$	0	0
$807$	0	0
$808$	0	0
$809$	622158.i	0.950613i	0.879820	+	0.475307i	$0.157663\pi$
$809$	622158.i	0.950613i	−0.879820	+	0.475307i	$0.842337\pi$
$810$	0	0
$811$	−543177.	−0.825846	−0.412923	−	0.910766i	$-0.635492\pi$
$811$	−543177.	−0.825846	−0.412923	+	0.910766i	$0.635492\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 663925.i	− 0.999548i
$816$	0	0
$817$	−49111.0	−0.0735757
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 246412.i	− 0.365574i	−0.983152	−	0.182787i	$-0.941488\pi$
$821$	− 246412.i	− 0.365574i	0.983152	−	0.182787i	$-0.0585119\pi$
$822$	0	0
$823$	−17172.3	−0.0253529	−0.0126765	−	0.999920i	$-0.504035\pi$
$823$	−17172.3	−0.0253529	−0.0126765	+	0.999920i	$0.504035\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 471086.i	− 0.688794i	−0.938824	−	0.344397i	$-0.888083\pi$
$827$	− 471086.i	− 0.688794i	0.938824	−	0.344397i	$-0.111917\pi$
$828$	0	0
$829$	197808.	0.287829	0.143915	−	0.989590i	$-0.454031\pi$
$829$	197808.	0.287829	0.143915	+	0.989590i	$0.454031\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 23760.0i	− 0.0342417i
$834$	0	0
$835$	−1.59308e6	−2.28489
$836$	0	0
$837$	0	0
$838$	0	0
$839$	249833.i	0.354917i	0.984128	+	0.177458i	$0.0567875\pi$
$839$	249833.i	0.354917i	−0.984128	+	0.177458i	$0.943213\pi$
$840$	0	0
$841$	618776.	0.874866
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.96429e6i	− 2.75102i
$846$	0	0
$847$	−117806.	−0.164210
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.70008e6i	− 2.34753i
$852$	0	0
$853$	193342.	0.265722	0.132861	−	0.991135i	$-0.457584\pi$
$853$	193342.	0.265722	0.132861	+	0.991135i	$0.457584\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 993867.i	− 1.35321i	−0.736344	−	0.676607i	$-0.763450\pi$
$857$	− 993867.i	− 1.35321i	0.736344	−	0.676607i	$-0.236550\pi$
$858$	0	0
$859$	1.01690e6	1.37814	0.689070	−	0.724695i	$-0.258019\pi$
$859$	1.01690e6	1.37814	0.689070	+	0.724695i	$0.258019\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17982.7i	0.0241453i	0.999927	+	0.0120727i	$0.00384294\pi$
$863$	17982.7i	0.0241453i	−0.999927	+	0.0120727i	$0.996157\pi$
$864$	0	0
$865$	−195478.	−0.261256
$866$	0	0
$867$	0	0
$868$	0	0
$869$	378239.i	0.500872i
$870$	0	0
$871$	−254546.	−0.335529
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 220933.i	− 0.288565i
$876$	0	0
$877$	535360.	0.696060	0.348030	−	0.937483i	$-0.386851\pi$
$877$	535360.	0.696060	0.348030	+	0.937483i	$0.386851\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	348052.i	0.448427i	0.974540	+	0.224214i	$0.0719813\pi$
$881$	348052.i	0.448427i	−0.974540	+	0.224214i	$0.928019\pi$
$882$	0	0
$883$	1.28216e6	1.64445	0.822227	−	0.569160i	$-0.192732\pi$
$883$	1.28216e6	1.64445	0.822227	+	0.569160i	$0.192732\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 621896.i	− 0.790443i	−0.918586	−	0.395222i	$-0.870668\pi$
$887$	− 621896.i	− 0.790443i	0.918586	−	0.395222i	$-0.129332\pi$
$888$	0	0
$889$	145040.	0.183520
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.13767e6i	− 1.42664i
$894$	0	0
$895$	−191830.	−0.239481
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 58038.6i	− 0.0718121i
$900$	0	0
$901$	28078.1	0.0345874
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 150653.i	− 0.183942i
$906$	0	0
$907$	1.25369e6	1.52397	0.761985	−	0.647595i	$-0.224225\pi$
$907$	1.25369e6	1.52397	0.761985	+	0.647595i	$0.224225\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 526452.i	− 0.634340i	−0.948369	−	0.317170i	$-0.897267\pi$
$911$	− 526452.i	− 0.634340i	0.948369	−	0.317170i	$-0.102733\pi$
$912$	0	0
$913$	−489850.	−0.587654
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9437.09i	− 0.0112228i
$918$	0	0
$919$	−317984.	−0.376508	−0.188254	−	0.982120i	$-0.560283\pi$
$919$	−317984.	−0.376508	−0.188254	+	0.982120i	$0.560283\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.28491e6i	2.68204i
$924$	0	0
$925$	2.08960e6	2.44219
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.46830e6i	1.70130i	0.525729	+	0.850652i	$0.323793\pi$
$929$	1.46830e6i	1.70130i	−0.525729	+	0.850652i	$0.676207\pi$
$930$	0	0
$931$	1.16299e6	1.34177
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 35789.3i	− 0.0409383i
$936$	0	0
$937$	−672991.	−0.766532	−0.383266	−	0.923638i	$-0.625201\pi$
$937$	−672991.	−0.766532	−0.383266	+	0.923638i	$0.625201\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 87597.2i	− 0.0989261i	−0.998776	−	0.0494631i	$-0.984249\pi$
$941$	− 87597.2i	− 0.0989261i	0.998776	−	0.0494631i	$-0.0157510\pi$
$942$	0	0
$943$	1.27249e6	1.43097
$944$	0	0
$945$	0	0
$946$	0	0
$947$	908007.i	1.01249i	0.862391	+	0.506243i	$0.168966\pi$
$947$	908007.i	1.01249i	−0.862391	+	0.506243i	$0.831034\pi$
$948$	0	0
$949$	−956306.	−1.06185
$950$	0	0
$951$	0	0
$952$	0	0
$953$	747370.i	0.822905i	0.911431	+	0.411452i	$0.134978\pi$
$953$	747370.i	0.822905i	−0.911431	+	0.411452i	$0.865022\pi$
$954$	0	0
$955$	418630.	0.459011
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 472819.i	− 0.514112i
$960$	0	0
$961$	−885461.	−0.958788
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.74000e6i	1.86851i
$966$	0	0
$967$	−324035.	−0.346528	−0.173264	−	0.984875i	$-0.555431\pi$
$967$	−324035.	−0.346528	−0.173264	+	0.984875i	$0.555431\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 873822.i	− 0.926797i	−0.886150	−	0.463398i	$-0.846630\pi$
$971$	− 873822.i	− 0.926797i	0.886150	−	0.463398i	$-0.153370\pi$
$972$	0	0
$973$	−82500.5	−0.0871427
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 284631.i	− 0.298190i	−0.988823	−	0.149095i	$-0.952364\pi$
$977$	− 284631.i	− 0.298190i	0.988823	−	0.149095i	$-0.0476361\pi$
$978$	0	0
$979$	688685.	0.718548
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 353.458i	0 0.000365789i	−1.00000		0.000182895i	$-0.999942\pi$
$983$	− 353.458i	0 0.000365789i	1.00000		0.000182895i	$-5.82171e-5\pi$
$984$	0	0
$985$	2.00928e6	2.07095
$986$	0	0
$987$	0	0
$988$	0	0
$989$	74719.9i	0.0763913i
$990$	0	0
$991$	−1.22572e6	−1.24808	−0.624041	−	0.781392i	$-0.714510\pi$
$991$	−1.22572e6	−1.24808	−0.624041	+	0.781392i	$0.714510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 306579.i	− 0.309668i
$996$	0	0
$997$	−1.32697e6	−1.33496	−0.667482	−	0.744626i	$-0.732628\pi$
$997$	−1.32697e6	−1.33496	−0.667482	+	0.744626i	$0.732628\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.g.161.1		8
3.2	odd	2	inner	1296.5.e.g.161.8		8
4.3	odd	2		324.5.c.a.161.1		8
9.2	odd	6		144.5.q.c.113.3		8
9.4	even	3		144.5.q.c.65.3		8
9.5	odd	6		432.5.q.c.305.4		8
9.7	even	3		432.5.q.c.17.4		8
12.11	even	2		324.5.c.a.161.8		8
36.7	odd	6		108.5.g.a.17.4		8
36.11	even	6		36.5.g.a.5.2	✓	8
36.23	even	6		108.5.g.a.89.4		8
36.31	odd	6		36.5.g.a.29.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.5.g.a.5.2	✓	8	36.11	even	6
36.5.g.a.29.2	yes	8	36.31	odd	6
108.5.g.a.17.4		8	36.7	odd	6
108.5.g.a.89.4		8	36.23	even	6
144.5.q.c.65.3		8	9.4	even	3
144.5.q.c.113.3		8	9.2	odd	6
324.5.c.a.161.1		8	4.3	odd	2
324.5.c.a.161.8		8	12.11	even	2
432.5.q.c.17.4		8	9.7	even	3
432.5.q.c.305.4		8	9.5	odd	6
1296.5.e.g.161.1		8	1.1	even	1	trivial
1296.5.e.g.161.8		8	3.2	odd	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-40.2664i q^{5} -14.7738 q^{7} +O(q^{10})\)	\(q-40.2664i q^{5} -14.7738 q^{7} -81.6517i q^{11} +278.107 q^{13} +10.8854i q^{17} -532.815 q^{19} +810.651i q^{23} -996.386 q^{25} +297.497i q^{29} -195.089 q^{31} +594.887i q^{35} -2097.18 q^{37} -1569.71i q^{41} +92.1727 q^{43} +2135.21i q^{47} -2182.74 q^{49} -2579.42i q^{53} -3287.82 q^{55} -1557.74i q^{59} +5371.95 q^{61} -11198.4i q^{65} -915.282 q^{67} +8215.93i q^{71} -3438.63 q^{73} +1206.30i q^{77} -4632.34 q^{79} -5999.27i q^{83} +438.317 q^{85} +8434.43i q^{89} -4108.69 q^{91} +21454.6i q^{95} -6031.15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 26 q^{7}+O(q^{10}) \) 8 * q + 26 * q^7	\( 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) \) 8 * q + 26 * q^7 + 10 * q^13 - 562 * q^19 - 706 * q^25 + 374 * q^31 + 16 * q^37 - 136 * q^43 + 654 * q^49 + 1818 * q^55 + 3874 * q^61 + 308 * q^67 - 7802 * q^73 - 4390 * q^79 + 6084 * q^85 - 15830 * q^91 - 14564 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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