LMFDB - Embedded newform 648.5.e.b.161.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,5,Mod(161,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	648.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$66.9837360783$
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} + 160x^{10} + 9733x^{8} + 278004x^{6} + 3678300x^{4} + 18632592x^{2} + 25765776 $ x^12 + 160x^10 + 9733x^8 + 278004x^6 + 3678300x^4 + 18632592*x^2 + 25765776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.6
Root			$-7.11605i$ of defining polynomial
Character	$\chi$	$=$	648.161
Dual form			648.5.e.b.161.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.08864i q^{5} -52.9803 q^{7} -23.8108i q^{11} +277.813 q^{13} +91.5027i q^{17} -332.045 q^{19} +401.662i q^{23} +574.751 q^{25} +365.240i q^{29} -351.065 q^{31} +375.558i q^{35} -42.6630 q^{37} -43.1905i q^{41} +43.2199 q^{43} -2808.21i q^{47} +405.914 q^{49} -4128.48i q^{53} -168.786 q^{55} -6356.33i q^{59} +1697.07 q^{61} -1969.32i q^{65} -2538.81 q^{67} -1550.91i q^{71} -4521.97 q^{73} +1261.50i q^{77} +7001.74 q^{79} +5729.75i q^{83} +648.630 q^{85} +6177.83i q^{89} -14718.6 q^{91} +2353.75i q^{95} +10727.6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 96 q^{7} - 72 q^{13} + 336 q^{19} + 84 q^{25} - 1536 q^{31} - 492 q^{37} + 10128 q^{43} - 6828 q^{49} - 13968 q^{55} + 26268 q^{61} - 20784 q^{67} - 9984 q^{73} + 44592 q^{79} - 45348 q^{85} - 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) $ 12 * q + 96 * q^7 - 72 * q^13 + 336 * q^19 + 84 * q^25 - 1536 * q^31 - 492 * q^37 + 10128 * q^43 - 6828 * q^49 - 13968 * q^55 + 26268 * q^61 - 20784 * q^67 - 9984 * q^73 + 44592 * q^79 - 45348 * q^85 - 11808 * q^91 + 76608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 7.08864i	− 0.283546i	−0.989899	−	0.141773i	$-0.954720\pi$
$5$	− 7.08864i	− 0.283546i	0.989899	−	0.141773i	$-0.0452802\pi$
$6$	0	0
$7$	−52.9803	−1.08123	−0.540615	−	0.841270i	$-0.681809\pi$
$7$	−52.9803	−1.08123	−0.540615	+	0.841270i	$0.681809\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 23.8108i	− 0.196784i	−0.995148	−	0.0983918i	$-0.968630\pi$
$11$	− 23.8108i	− 0.196784i	0.995148	−	0.0983918i	$-0.0313698\pi$
$12$	0	0
$13$	277.813	1.64387	0.821933	−	0.569585i	$-0.192896\pi$
$13$	277.813	1.64387	0.821933	+	0.569585i	$0.192896\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	91.5027i	0.316618i	0.987390	+	0.158309i	$0.0506043\pi$
$17$	91.5027i	0.316618i	−0.987390	+	0.158309i	$0.949396\pi$
$18$	0	0
$19$	−332.045	−0.919793	−0.459896	−	0.887973i	$-0.652113\pi$
$19$	−332.045	−0.919793	−0.459896	+	0.887973i	$0.652113\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	401.662i	0.759285i	0.925133	+	0.379642i	$0.123953\pi$
$23$	401.662i	0.759285i	−0.925133	+	0.379642i	$0.876047\pi$
$24$	0	0
$25$	574.751	0.919602
$26$	0	0
$27$	0	0
$28$	0	0
$29$	365.240i	0.434292i	0.976139	+	0.217146i	$0.0696748\pi$
$29$	365.240i	0.434292i	−0.976139	+	0.217146i	$0.930325\pi$
$30$	0	0
$31$	−351.065	−0.365312	−0.182656	−	0.983177i	$-0.558469\pi$
$31$	−351.065	−0.365312	−0.182656	+	0.983177i	$0.558469\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	375.558i	0.306578i
$36$	0	0
$37$	−42.6630	−0.0311636	−0.0155818	−	0.999879i	$-0.504960\pi$
$37$	−42.6630	−0.0311636	−0.0155818	+	0.999879i	$0.504960\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 43.1905i	− 0.0256934i	−0.999917	−	0.0128467i	$-0.995911\pi$
$41$	− 43.1905i	− 0.0256934i	0.999917	−	0.0128467i	$-0.00408934\pi$
$42$	0	0
$43$	43.2199	0.0233747	0.0116874	−	0.999932i	$-0.496280\pi$
$43$	43.2199	0.0233747	0.0116874	+	0.999932i	$0.496280\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2808.21i	− 1.27126i	−0.771994	−	0.635630i	$-0.780741\pi$
$47$	− 2808.21i	− 1.27126i	0.771994	−	0.635630i	$-0.219259\pi$
$48$	0	0
$49$	405.914	0.169060
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4128.48i	− 1.46973i	−0.678212	−	0.734866i	$-0.737245\pi$
$53$	− 4128.48i	− 1.46973i	0.678212	−	0.734866i	$-0.262755\pi$
$54$	0	0
$55$	−168.786	−0.0557971
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6356.33i	− 1.82601i	−0.407953	−	0.913003i	$-0.633757\pi$
$59$	− 6356.33i	− 1.82601i	0.407953	−	0.913003i	$-0.366243\pi$
$60$	0	0
$61$	1697.07	0.456079	0.228040	−	0.973652i	$-0.426768\pi$
$61$	1697.07	0.456079	0.228040	+	0.973652i	$0.426768\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1969.32i	− 0.466111i
$66$	0	0
$67$	−2538.81	−0.565562	−0.282781	−	0.959184i	$-0.591257\pi$
$67$	−2538.81	−0.565562	−0.282781	+	0.959184i	$0.591257\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 1550.91i	− 0.307659i	−0.988097	−	0.153829i	$-0.950839\pi$
$71$	− 1550.91i	− 0.307659i	0.988097	−	0.153829i	$-0.0491606\pi$
$72$	0	0
$73$	−4521.97	−0.848558	−0.424279	−	0.905532i	$-0.639472\pi$
$73$	−4521.97	−0.848558	−0.424279	+	0.905532i	$0.639472\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1261.50i	0.212769i
$78$	0	0
$79$	7001.74	1.12189	0.560947	−	0.827852i	$-0.310437\pi$
$79$	7001.74	1.12189	0.560947	+	0.827852i	$0.310437\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5729.75i	0.831724i	0.909428	+	0.415862i	$0.136520\pi$
$83$	5729.75i	0.831724i	−0.909428	+	0.415862i	$0.863480\pi$
$84$	0	0
$85$	648.630	0.0897758
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6177.83i	0.779930i	0.920830	+	0.389965i	$0.127513\pi$
$89$	6177.83i	0.779930i	−0.920830	+	0.389965i	$0.872487\pi$
$90$	0	0
$91$	−14718.6	−1.77740
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2353.75i	0.260803i
$96$	0	0
$97$	10727.6	1.14014	0.570071	−	0.821595i	$-0.306916\pi$
$97$	10727.6	1.14014	0.570071	+	0.821595i	$0.306916\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 12388.9i	− 1.21448i	−0.794520	−	0.607238i	$-0.792277\pi$
$101$	− 12388.9i	− 1.21448i	0.794520	−	0.607238i	$-0.207723\pi$
$102$	0	0
$103$	6922.98	0.652558	0.326279	−	0.945274i	$-0.394205\pi$
$103$	6922.98	0.652558	0.326279	+	0.945274i	$0.394205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18280.7i	− 1.59671i	−0.602190	−	0.798353i	$-0.705705\pi$
$107$	− 18280.7i	− 1.59671i	0.602190	−	0.798353i	$-0.294295\pi$
$108$	0	0
$109$	−16716.1	−1.40696	−0.703479	−	0.710716i	$-0.748371\pi$
$109$	−16716.1	−1.40696	−0.703479	+	0.710716i	$0.748371\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 16195.9i	− 1.26837i	−0.773180	−	0.634187i	$-0.781335\pi$
$113$	− 16195.9i	− 1.26837i	0.773180	−	0.634187i	$-0.218665\pi$
$114$	0	0
$115$	2847.24	0.215292
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4847.84i	− 0.342338i
$120$	0	0
$121$	14074.0	0.961276
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8504.60i	− 0.544295i
$126$	0	0
$127$	−14891.2	−0.923254	−0.461627	−	0.887074i	$-0.652734\pi$
$127$	−14891.2	−0.923254	−0.461627	+	0.887074i	$0.652734\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 19437.2i	− 1.13264i	−0.824185	−	0.566320i	$-0.808366\pi$
$131$	− 19437.2i	− 1.13264i	0.824185	−	0.566320i	$-0.191634\pi$
$132$	0	0
$133$	17591.9	0.994509
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 26654.4i	− 1.42013i	−0.704136	−	0.710065i	$-0.748666\pi$
$137$	− 26654.4i	− 1.42013i	0.704136	−	0.710065i	$-0.251334\pi$
$138$	0	0
$139$	−3868.41	−0.200218	−0.100109	−	0.994976i	$-0.531919\pi$
$139$	−3868.41	−0.200218	−0.100109	+	0.994976i	$0.531919\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6614.96i	− 0.323486i
$144$	0	0
$145$	2589.05	0.123142
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22727.7i	1.02372i	0.859068	+	0.511862i	$0.171044\pi$
$149$	22727.7i	1.02372i	−0.859068	+	0.511862i	$0.828956\pi$
$150$	0	0
$151$	26485.5	1.16159	0.580797	−	0.814049i	$-0.302741\pi$
$151$	26485.5	1.16159	0.580797	+	0.814049i	$0.302741\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2488.57i	0.103583i
$156$	0	0
$157$	−25193.4	−1.02208	−0.511042	−	0.859556i	$-0.670740\pi$
$157$	−25193.4	−1.02208	−0.511042	+	0.859556i	$0.670740\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 21280.2i	− 0.820962i
$162$	0	0
$163$	−20189.7	−0.759897	−0.379948	−	0.925008i	$-0.624058\pi$
$163$	−20189.7	−0.759897	−0.379948	+	0.925008i	$0.624058\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 50099.7i	− 1.79640i	−0.439590	−	0.898198i	$-0.644876\pi$
$167$	− 50099.7i	− 1.79640i	0.439590	−	0.898198i	$-0.355124\pi$
$168$	0	0
$169$	48619.2	1.70229
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 43788.9i	− 1.46309i	−0.681792	−	0.731546i	$-0.738799\pi$
$173$	− 43788.9i	− 1.46309i	0.681792	−	0.731546i	$-0.261201\pi$
$174$	0	0
$175$	−30450.5	−0.994302
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9037.18i	0.282050i	0.990006	+	0.141025i	$0.0450399\pi$
$179$	9037.18i	0.282050i	−0.990006	+	0.141025i	$0.954960\pi$
$180$	0	0
$181$	−13305.4	−0.406136	−0.203068	−	0.979165i	$-0.565091\pi$
$181$	−13305.4	−0.406136	−0.203068	+	0.979165i	$0.565091\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	302.422i	0.00883630i
$186$	0	0
$187$	2178.75	0.0623053
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21093.5i	0.578206i	0.957298	+	0.289103i	$0.0933569\pi$
$191$	21093.5i	0.578206i	−0.957298	+	0.289103i	$0.906643\pi$
$192$	0	0
$193$	43627.8	1.17125	0.585624	−	0.810583i	$-0.300850\pi$
$193$	43627.8	1.17125	0.585624	+	0.810583i	$0.300850\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 51851.1i	− 1.33606i	−0.744135	−	0.668029i	$-0.767138\pi$
$197$	− 51851.1i	− 1.33606i	0.744135	−	0.668029i	$-0.232862\pi$
$198$	0	0
$199$	−77783.6	−1.96418	−0.982092	−	0.188404i	$-0.939668\pi$
$199$	−77783.6	−1.96418	−0.982092	+	0.188404i	$0.939668\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 19350.5i	− 0.469570i
$204$	0	0
$205$	−306.162	−0.00728524
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7906.27i	0.181000i
$210$	0	0
$211$	29081.4	0.653207	0.326604	−	0.945161i	$-0.394096\pi$
$211$	29081.4	0.653207	0.326604	+	0.945161i	$0.394096\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 306.370i	− 0.00662780i
$216$	0	0
$217$	18599.5	0.394986
$218$	0	0
$219$	0	0
$220$	0	0
$221$	25420.7i	0.520478i
$222$	0	0
$223$	−67639.0	−1.36015	−0.680076	−	0.733142i	$-0.738053\pi$
$223$	−67639.0	−1.36015	−0.680076	+	0.733142i	$0.738053\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19307.6i	− 0.374694i	−0.982294	−	0.187347i	$-0.940011\pi$
$227$	− 19307.6i	− 0.374694i	0.982294	−	0.187347i	$-0.0599889\pi$
$228$	0	0
$229$	80173.7	1.52884	0.764418	−	0.644721i	$-0.223026\pi$
$229$	80173.7	1.52884	0.764418	+	0.644721i	$0.223026\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	64412.4i	1.18647i	0.805029	+	0.593236i	$0.202150\pi$
$233$	64412.4i	1.18647i	−0.805029	+	0.593236i	$0.797850\pi$
$234$	0	0
$235$	−19906.4	−0.360460
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 25352.0i	− 0.443830i	−0.975066	−	0.221915i	$-0.928769\pi$
$239$	− 25352.0i	− 0.443830i	0.975066	−	0.221915i	$-0.0712308\pi$
$240$	0	0
$241$	54732.7	0.942351	0.471176	−	0.882039i	$-0.343830\pi$
$241$	54732.7	0.942351	0.471176	+	0.882039i	$0.343830\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2877.38i	− 0.0479364i
$246$	0	0
$247$	−92246.6	−1.51202
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 35904.7i	− 0.569908i	−0.958541	−	0.284954i	$-0.908022\pi$
$251$	− 35904.7i	− 0.569908i	0.958541	−	0.284954i	$-0.0919782\pi$
$252$	0	0
$253$	9563.89	0.149415
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26218.9i	0.396962i	0.980105	+	0.198481i	$0.0636008\pi$
$257$	26218.9i	0.396962i	−0.980105	+	0.198481i	$0.936399\pi$
$258$	0	0
$259$	2260.30	0.0336951
$260$	0	0
$261$	0	0
$262$	0	0
$263$	130253.i	1.88311i	0.336860	+	0.941555i	$0.390635\pi$
$263$	130253.i	1.88311i	−0.336860	+	0.941555i	$0.609365\pi$
$264$	0	0
$265$	−29265.3	−0.416736
$266$	0	0
$267$	0	0
$268$	0	0
$269$	56557.5i	0.781602i	0.920475	+	0.390801i	$0.127802\pi$
$269$	56557.5i	0.781602i	−0.920475	+	0.390801i	$0.872198\pi$
$270$	0	0
$271$	−8171.92	−0.111272	−0.0556359	−	0.998451i	$-0.517719\pi$
$271$	−8171.92	−0.111272	−0.0556359	+	0.998451i	$0.517719\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 13685.3i	− 0.180963i
$276$	0	0
$277$	48440.5	0.631319	0.315660	−	0.948872i	$-0.397774\pi$
$277$	48440.5	0.631319	0.315660	+	0.948872i	$0.397774\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 136493.i	− 1.72862i	−0.502962	−	0.864309i	$-0.667756\pi$
$281$	− 136493.i	− 1.72862i	0.502962	−	0.864309i	$-0.332244\pi$
$282$	0	0
$283$	−33657.6	−0.420252	−0.210126	−	0.977674i	$-0.567387\pi$
$283$	−33657.6	−0.420252	−0.210126	+	0.977674i	$0.567387\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2288.25i	0.0277805i
$288$	0	0
$289$	75148.3	0.899753
$290$	0	0
$291$	0	0
$292$	0	0
$293$	113888.i	1.32661i	0.748349	+	0.663305i	$0.230847\pi$
$293$	113888.i	1.32661i	−0.748349	+	0.663305i	$0.769153\pi$
$294$	0	0
$295$	−45057.7	−0.517756
$296$	0	0
$297$	0	0
$298$	0	0
$299$	111587.i	1.24816i
$300$	0	0
$301$	−2289.80	−0.0252735
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12029.9i	− 0.129319i
$306$	0	0
$307$	144481.	1.53297	0.766487	−	0.642260i	$-0.222003\pi$
$307$	144481.	1.53297	0.766487	+	0.642260i	$0.222003\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 102367.i	− 1.05838i	−0.848504	−	0.529189i	$-0.822496\pi$
$311$	− 102367.i	− 1.05838i	0.848504	−	0.529189i	$-0.177504\pi$
$312$	0	0
$313$	−142406.	−1.45358	−0.726789	−	0.686861i	$-0.758988\pi$
$313$	−142406.	−1.45358	−0.726789	+	0.686861i	$0.758988\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 97384.2i	− 0.969103i	−0.874763	−	0.484552i	$-0.838983\pi$
$317$	− 97384.2i	− 0.969103i	0.874763	−	0.484552i	$-0.161017\pi$
$318$	0	0
$319$	8696.66	0.0854616
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 30383.0i	− 0.291223i
$324$	0	0
$325$	159673.	1.51170
$326$	0	0
$327$	0	0
$328$	0	0
$329$	148780.i	1.37452i
$330$	0	0
$331$	−128696.	−1.17465	−0.587324	−	0.809352i	$-0.699819\pi$
$331$	−128696.	−1.17465	−0.587324	+	0.809352i	$0.699819\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	17996.7i	0.160363i
$336$	0	0
$337$	15645.3	0.137760	0.0688799	−	0.997625i	$-0.478057\pi$
$337$	15645.3	0.137760	0.0688799	+	0.997625i	$0.478057\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8359.14i	0.0718874i
$342$	0	0
$343$	105700.	0.898438
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 49635.9i	− 0.412228i	−0.978528	−	0.206114i	$-0.933918\pi$
$347$	− 49635.9i	− 0.412228i	0.978528	−	0.206114i	$-0.0660817\pi$
$348$	0	0
$349$	−85292.6	−0.700262	−0.350131	−	0.936701i	$-0.613863\pi$
$349$	−85292.6	−0.700262	−0.350131	+	0.936701i	$0.613863\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17347.9i	0.139218i	0.997574	+	0.0696091i	$0.0221752\pi$
$353$	17347.9i	0.139218i	−0.997574	+	0.0696091i	$0.977825\pi$
$354$	0	0
$355$	−10993.8	−0.0872353
$356$	0	0
$357$	0	0
$358$	0	0
$359$	184846.i	1.43424i	0.696952	+	0.717118i	$0.254539\pi$
$359$	184846.i	1.43424i	−0.696952	+	0.717118i	$0.745461\pi$
$360$	0	0
$361$	−20067.0	−0.153981
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32054.6i	0.240605i
$366$	0	0
$367$	−51775.7	−0.384409	−0.192205	−	0.981355i	$-0.561564\pi$
$367$	−51775.7	−0.384409	−0.192205	+	0.981355i	$0.561564\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	218728.i	1.58912i
$372$	0	0
$373$	55047.1	0.395655	0.197828	−	0.980237i	$-0.436611\pi$
$373$	55047.1	0.395655	0.197828	+	0.980237i	$0.436611\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	101468.i	0.713918i
$378$	0	0
$379$	−119817.	−0.834143	−0.417072	−	0.908874i	$-0.636944\pi$
$379$	−119817.	−0.834143	−0.417072	+	0.908874i	$0.636944\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	155736.i	1.06167i	0.847474	+	0.530837i	$0.178123\pi$
$383$	155736.i	1.06167i	−0.847474	+	0.530837i	$0.821877\pi$
$384$	0	0
$385$	8942.35	0.0603296
$386$	0	0
$387$	0	0
$388$	0	0
$389$	154849.i	1.02332i	0.859189	+	0.511658i	$0.170968\pi$
$389$	154849.i	1.02332i	−0.859189	+	0.511658i	$0.829032\pi$
$390$	0	0
$391$	−36753.1	−0.240404
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 49632.8i	− 0.318108i
$396$	0	0
$397$	148940.	0.944997	0.472498	−	0.881332i	$-0.343352\pi$
$397$	148940.	0.944997	0.472498	+	0.881332i	$0.343352\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	213407.i	1.32715i	0.748109	+	0.663576i	$0.230962\pi$
$401$	213407.i	1.32715i	−0.748109	+	0.663576i	$0.769038\pi$
$402$	0	0
$403$	−97530.4	−0.600524
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1015.84i	0.00613249i
$408$	0	0
$409$	49642.8	0.296763	0.148381	−	0.988930i	$-0.452594\pi$
$409$	49642.8	0.296763	0.148381	+	0.988930i	$0.452594\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	336760.i	1.97433i
$414$	0	0
$415$	40616.1	0.235832
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22487.4i	0.128089i	0.997947	+	0.0640444i	$0.0203999\pi$
$419$	22487.4i	0.128089i	−0.997947	+	0.0640444i	$0.979600\pi$
$420$	0	0
$421$	103433.	0.583570	0.291785	−	0.956484i	$-0.405751\pi$
$421$	103433.	0.583570	0.291785	+	0.956484i	$0.405751\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	52591.3i	0.291163i
$426$	0	0
$427$	−89911.4	−0.493127
$428$	0	0
$429$	0	0
$430$	0	0
$431$	116100.i	0.624996i	0.949918	+	0.312498i	$0.101166\pi$
$431$	116100.i	0.624996i	−0.949918	+	0.312498i	$0.898834\pi$
$432$	0	0
$433$	296628.	1.58211	0.791054	−	0.611747i	$-0.209533\pi$
$433$	296628.	1.58211	0.791054	+	0.611747i	$0.209533\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 133370.i	− 0.698385i
$438$	0	0
$439$	−324820.	−1.68544	−0.842722	−	0.538350i	$-0.819048\pi$
$439$	−324820.	−1.68544	−0.842722	+	0.538350i	$0.819048\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	309783.i	1.57852i	0.614058	+	0.789261i	$0.289536\pi$
$443$	309783.i	1.57852i	−0.614058	+	0.789261i	$0.710464\pi$
$444$	0	0
$445$	43792.4	0.221146
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 367550.i	− 1.82316i	−0.411127	−	0.911578i	$-0.634865\pi$
$449$	− 367550.i	− 1.82316i	0.411127	−	0.911578i	$-0.365135\pi$
$450$	0	0
$451$	−1028.40	−0.00505603
$452$	0	0
$453$	0	0
$454$	0	0
$455$	104335.i	0.503973i
$456$	0	0
$457$	−175614.	−0.840864	−0.420432	−	0.907324i	$-0.638121\pi$
$457$	−175614.	−0.840864	−0.420432	+	0.907324i	$0.638121\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 104038.i	− 0.489541i	−0.969581	−	0.244770i	$-0.921287\pi$
$461$	− 104038.i	− 0.489541i	0.969581	−	0.244770i	$-0.0787125\pi$
$462$	0	0
$463$	−44023.3	−0.205362	−0.102681	−	0.994714i	$-0.532742\pi$
$463$	−44023.3	−0.205362	−0.102681	+	0.994714i	$0.532742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	85970.6i	0.394199i	0.980383	+	0.197100i	$0.0631523\pi$
$467$	85970.6i	0.394199i	−0.980383	+	0.197100i	$0.936848\pi$
$468$	0	0
$469$	134507.	0.611503
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1029.10i	− 0.00459976i
$474$	0	0
$475$	−190843.	−0.845843
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 380507.i	− 1.65841i	−0.558946	−	0.829204i	$-0.688794\pi$
$479$	− 380507.i	− 1.65841i	0.558946	−	0.829204i	$-0.311206\pi$
$480$	0	0
$481$	−11852.3	−0.0512288
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 76044.1i	− 0.323282i
$486$	0	0
$487$	−103685.	−0.437179	−0.218590	−	0.975817i	$-0.570146\pi$
$487$	−103685.	−0.437179	−0.218590	+	0.975817i	$0.570146\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 266664.i	− 1.10612i	−0.833142	−	0.553060i	$-0.813460\pi$
$491$	− 266664.i	− 1.10612i	0.833142	−	0.553060i	$-0.186540\pi$
$492$	0	0
$493$	−33420.4	−0.137505
$494$	0	0
$495$	0	0
$496$	0	0
$497$	82167.6i	0.332650i
$498$	0	0
$499$	19875.5	0.0798209	0.0399104	−	0.999203i	$-0.487293\pi$
$499$	19875.5	0.0798209	0.0399104	+	0.999203i	$0.487293\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 12736.5i	− 0.0503402i	−0.999683	−	0.0251701i	$-0.991987\pi$
$503$	− 12736.5i	− 0.0503402i	0.999683	−	0.0251701i	$-0.00801273\pi$
$504$	0	0
$505$	−87820.3	−0.344360
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32695.7i	0.126199i	0.998007	+	0.0630995i	$0.0200985\pi$
$509$	32695.7i	0.126199i	−0.998007	+	0.0630995i	$0.979901\pi$
$510$	0	0
$511$	239575.	0.917487
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 49074.6i	− 0.185030i
$516$	0	0
$517$	−66865.8	−0.250163
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1464.29i	− 0.00539451i	−0.999996	−	0.00269725i	$-0.999141\pi$
$521$	− 1464.29i	− 0.00539451i	0.999996	−	0.00269725i	$-0.000858564\pi$
$522$	0	0
$523$	443548.	1.62158	0.810788	−	0.585339i	$-0.199039\pi$
$523$	443548.	1.62158	0.810788	+	0.585339i	$0.199039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 32123.4i	− 0.115664i
$528$	0	0
$529$	118509.	0.423487
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 11998.9i	− 0.0422364i
$534$	0	0
$535$	−129585.	−0.452739
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 9665.15i	− 0.0332683i
$540$	0	0
$541$	−74322.5	−0.253937	−0.126968	−	0.991907i	$-0.540525\pi$
$541$	−74322.5	−0.253937	−0.126968	+	0.991907i	$0.540525\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	118494.i	0.398937i
$546$	0	0
$547$	592495.	1.98021	0.990103	−	0.140345i	$-0.0448212\pi$
$547$	592495.	1.98021	0.990103	+	0.140345i	$0.0448212\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 121276.i	− 0.399459i
$552$	0	0
$553$	−370954.	−1.21303
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 101727.i	− 0.327888i	−0.986470	−	0.163944i	$-0.947578\pi$
$557$	− 101727.i	− 0.327888i	0.986470	−	0.163944i	$-0.0524216\pi$
$558$	0	0
$559$	12007.1	0.0384249
$560$	0	0
$561$	0	0
$562$	0	0
$563$	433330.i	1.36710i	0.729902	+	0.683552i	$0.239566\pi$
$563$	433330.i	1.36710i	−0.729902	+	0.683552i	$0.760434\pi$
$564$	0	0
$565$	−114807.	−0.359642
$566$	0	0
$567$	0	0
$568$	0	0
$569$	76489.1i	0.236252i	0.992999	+	0.118126i	$0.0376887\pi$
$569$	76489.1i	0.236252i	−0.992999	+	0.118126i	$0.962311\pi$
$570$	0	0
$571$	304738.	0.934661	0.467330	−	0.884083i	$-0.345216\pi$
$571$	304738.	0.934661	0.467330	+	0.884083i	$0.345216\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	230856.i	0.698240i
$576$	0	0
$577$	−583342.	−1.75215	−0.876075	−	0.482174i	$-0.839847\pi$
$577$	−583342.	−1.75215	−0.876075	+	0.482174i	$0.839847\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 303564.i	− 0.899286i
$582$	0	0
$583$	−98302.5	−0.289219
$584$	0	0
$585$	0	0
$586$	0	0
$587$	258531.i	0.750303i	0.926963	+	0.375152i	$0.122409\pi$
$587$	258531.i	0.750303i	−0.926963	+	0.375152i	$0.877591\pi$
$588$	0	0
$589$	116569.	0.336011
$590$	0	0
$591$	0	0
$592$	0	0
$593$	348330.i	0.990563i	0.868733	+	0.495281i	$0.164935\pi$
$593$	348330.i	0.990563i	−0.868733	+	0.495281i	$0.835065\pi$
$594$	0	0
$595$	−34364.6	−0.0970683
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 223519.i	− 0.622961i	−0.950252	−	0.311481i	$-0.899175\pi$
$599$	− 223519.i	− 0.622961i	0.950252	−	0.311481i	$-0.100825\pi$
$600$	0	0
$601$	35005.4	0.0969139	0.0484570	−	0.998825i	$-0.484570\pi$
$601$	35005.4	0.0969139	0.0484570	+	0.998825i	$0.484570\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 99765.8i	− 0.272566i
$606$	0	0
$607$	412760.	1.12026	0.560132	−	0.828403i	$-0.310750\pi$
$607$	412760.	1.12026	0.560132	+	0.828403i	$0.310750\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 780158.i	− 2.08978i
$612$	0	0
$613$	−431167.	−1.14743	−0.573713	−	0.819056i	$-0.694497\pi$
$613$	−431167.	−1.14743	−0.573713	+	0.819056i	$0.694497\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 152955.i	− 0.401785i	−0.979613	−	0.200893i	$-0.935616\pi$
$617$	− 152955.i	− 0.401785i	0.979613	−	0.200893i	$-0.0643843\pi$
$618$	0	0
$619$	396066.	1.03368	0.516840	−	0.856082i	$-0.327108\pi$
$619$	396066.	1.03368	0.516840	+	0.856082i	$0.327108\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 327303.i	− 0.843285i
$624$	0	0
$625$	298933.	0.765270
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3903.78i	− 0.00986697i
$630$	0	0
$631$	−699103.	−1.75583	−0.877915	−	0.478817i	$-0.841066\pi$
$631$	−699103.	−1.75583	−0.877915	+	0.478817i	$0.841066\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	105558.i	0.261785i
$636$	0	0
$637$	112768.	0.277913
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 105349.i	− 0.256399i	−0.991748	−	0.128199i	$-0.959080\pi$
$641$	− 105349.i	− 0.256399i	0.991748	−	0.128199i	$-0.0409197\pi$
$642$	0	0
$643$	360149.	0.871084	0.435542	−	0.900168i	$-0.356557\pi$
$643$	360149.	0.871084	0.435542	+	0.900168i	$0.356557\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 109548.i	− 0.261694i	−0.991403	−	0.130847i	$-0.958230\pi$
$647$	− 109548.i	− 0.261694i	0.991403	−	0.130847i	$-0.0417697\pi$
$648$	0	0
$649$	−151349.	−0.359328
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 302941.i	− 0.710447i	−0.934781	−	0.355223i	$-0.884405\pi$
$653$	− 302941.i	− 0.710447i	0.934781	−	0.355223i	$-0.115595\pi$
$654$	0	0
$655$	−137784.	−0.321155
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 863699.i	− 1.98880i	−0.105671	−	0.994401i	$-0.533699\pi$
$659$	− 863699.i	− 1.98880i	0.105671	−	0.994401i	$-0.466301\pi$
$660$	0	0
$661$	−110418.	−0.252718	−0.126359	−	0.991985i	$-0.540329\pi$
$661$	−110418.	−0.252718	−0.126359	+	0.991985i	$0.540329\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 124702.i	− 0.281989i
$666$	0	0
$667$	−146703.	−0.329751
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 40408.7i	− 0.0897490i
$672$	0	0
$673$	56355.6	0.124425	0.0622124	−	0.998063i	$-0.480184\pi$
$673$	56355.6	0.124425	0.0622124	+	0.998063i	$0.480184\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	346467.i	0.755934i	0.925819	+	0.377967i	$0.123377\pi$
$677$	346467.i	0.755934i	−0.925819	+	0.377967i	$0.876623\pi$
$678$	0	0
$679$	−568352.	−1.23276
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 421086.i	− 0.902670i	−0.892355	−	0.451335i	$-0.850948\pi$
$683$	− 421086.i	− 0.902670i	0.892355	−	0.451335i	$-0.149052\pi$
$684$	0	0
$685$	−188944.	−0.402672
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.14695e6i	− 2.41604i
$690$	0	0
$691$	778744.	1.63094	0.815471	−	0.578797i	$-0.196478\pi$
$691$	778744.	1.63094	0.815471	+	0.578797i	$0.196478\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27421.7i	0.0567709i
$696$	0	0
$697$	3952.05	0.00813499
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 388023.i	− 0.789626i	−0.918761	−	0.394813i	$-0.870809\pi$
$701$	− 388023.i	− 0.789626i	0.918761	−	0.394813i	$-0.129191\pi$
$702$	0	0
$703$	14166.0	0.0286641
$704$	0	0
$705$	0	0
$706$	0	0
$707$	656367.i	1.31313i
$708$	0	0
$709$	324628.	0.645793	0.322896	−	0.946434i	$-0.395344\pi$
$709$	324628.	0.645793	0.322896	+	0.946434i	$0.395344\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 141009.i	− 0.277376i
$714$	0	0
$715$	−46891.1	−0.0917230
$716$	0	0
$717$	0	0
$718$	0	0
$719$	284337.i	0.550017i	0.961442	+	0.275008i	$0.0886807\pi$
$719$	284337.i	0.550017i	−0.961442	+	0.275008i	$0.911319\pi$
$720$	0	0
$721$	−366782.	−0.705566
$722$	0	0
$723$	0	0
$724$	0	0
$725$	209922.i	0.399376i
$726$	0	0
$727$	149352.	0.282580	0.141290	−	0.989968i	$-0.454875\pi$
$727$	149352.	0.282580	0.141290	+	0.989968i	$0.454875\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3954.74i	0.00740087i
$732$	0	0
$733$	−578133.	−1.07602	−0.538009	−	0.842939i	$-0.680824\pi$
$733$	−578133.	−1.07602	−0.538009	+	0.842939i	$0.680824\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60451.1i	0.111293i
$738$	0	0
$739$	499172.	0.914031	0.457016	−	0.889459i	$-0.348918\pi$
$739$	499172.	0.914031	0.457016	+	0.889459i	$0.348918\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 542955.i	− 0.983526i	−0.870729	−	0.491763i	$-0.836353\pi$
$743$	− 542955.i	− 0.983526i	0.870729	−	0.491763i	$-0.163647\pi$
$744$	0	0
$745$	161109.	0.290273
$746$	0	0
$747$	0	0
$748$	0	0
$749$	968516.i	1.72641i
$750$	0	0
$751$	−281745.	−0.499546	−0.249773	−	0.968304i	$-0.580356\pi$
$751$	−281745.	−0.499546	−0.249773	+	0.968304i	$0.580356\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 187746.i	− 0.329365i
$756$	0	0
$757$	−1.00873e6	−1.76029	−0.880143	−	0.474708i	$-0.842554\pi$
$757$	−1.00873e6	−1.76029	−0.880143	+	0.474708i	$0.842554\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 654039.i	− 1.12936i	−0.825308	−	0.564682i	$-0.808999\pi$
$761$	− 654039.i	− 1.12936i	0.825308	−	0.564682i	$-0.191001\pi$
$762$	0	0
$763$	885623.	1.52125
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.76587e6i	− 3.00171i
$768$	0	0
$769$	−510663.	−0.863538	−0.431769	−	0.901984i	$-0.642111\pi$
$769$	−510663.	−0.863538	−0.431769	+	0.901984i	$0.642111\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 795361.i	− 1.33108i	−0.746360	−	0.665542i	$-0.768200\pi$
$773$	− 795361.i	− 1.33108i	0.746360	−	0.665542i	$-0.231800\pi$
$774$	0	0
$775$	−201775.	−0.335941
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14341.2i	0.0236326i
$780$	0	0
$781$	−36928.4	−0.0605422
$782$	0	0
$783$	0	0
$784$	0	0
$785$	178587.i	0.289807i
$786$	0	0
$787$	1.03663e6	1.67369	0.836843	−	0.547443i	$-0.184399\pi$
$787$	1.03663e6	1.67369	0.836843	+	0.547443i	$0.184399\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	858062.i	1.37140i
$792$	0	0
$793$	471469.	0.749733
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 166408.i	− 0.261973i	−0.991384	−	0.130986i	$-0.958186\pi$
$797$	− 166408.i	− 0.261973i	0.991384	−	0.130986i	$-0.0418144\pi$
$798$	0	0
$799$	256959.	0.402504
$800$	0	0
$801$	0	0
$802$	0	0
$803$	107672.i	0.166982i
$804$	0	0
$805$	−150847.	−0.232780
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 751726.i	− 1.14858i	−0.818651	−	0.574292i	$-0.805278\pi$
$809$	− 751726.i	− 1.14858i	0.818651	−	0.574292i	$-0.194722\pi$
$810$	0	0
$811$	−25761.8	−0.0391682	−0.0195841	−	0.999808i	$-0.506234\pi$
$811$	−25761.8	−0.0391682	−0.0195841	+	0.999808i	$0.506234\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	143117.i	0.215465i
$816$	0	0
$817$	−14351.0	−0.0214999
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.20407e6i	− 1.78635i	−0.449711	−	0.893174i	$-0.648473\pi$
$821$	− 1.20407e6i	− 1.78635i	0.449711	−	0.893174i	$-0.351527\pi$
$822$	0	0
$823$	400642.	0.591503	0.295751	−	0.955265i	$-0.404430\pi$
$823$	400642.	0.591503	0.295751	+	0.955265i	$0.404430\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 326642.i	− 0.477596i	−0.971069	−	0.238798i	$-0.923247\pi$
$827$	− 326642.i	− 0.477596i	0.971069	−	0.238798i	$-0.0767534\pi$
$828$	0	0
$829$	−296231.	−0.431043	−0.215522	−	0.976499i	$-0.569145\pi$
$829$	−296231.	−0.431043	−0.215522	+	0.976499i	$0.569145\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	37142.3i	0.0535277i
$834$	0	0
$835$	−355139.	−0.509360
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 668472.i	− 0.949641i	−0.880083	−	0.474821i	$-0.842513\pi$
$839$	− 668472.i	− 0.949641i	0.880083	−	0.474821i	$-0.157487\pi$
$840$	0	0
$841$	573881.	0.811390
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 344644.i	− 0.482678i
$846$	0	0
$847$	−745647.	−1.03936
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 17136.1i	− 0.0236621i
$852$	0	0
$853$	−906815.	−1.24629	−0.623147	−	0.782104i	$-0.714146\pi$
$853$	−906815.	−1.24629	−0.623147	+	0.782104i	$0.714146\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.41075e6i	1.92083i	0.278567	+	0.960417i	$0.410140\pi$
$857$	1.41075e6i	1.92083i	−0.278567	+	0.960417i	$0.589860\pi$
$858$	0	0
$859$	364568.	0.494074	0.247037	−	0.969006i	$-0.420543\pi$
$859$	364568.	0.494074	0.247037	+	0.969006i	$0.420543\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	625950.i	0.840462i	0.907417	+	0.420231i	$0.138051\pi$
$863$	625950.i	0.840462i	−0.907417	+	0.420231i	$0.861949\pi$
$864$	0	0
$865$	−310404.	−0.414853
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 166717.i	− 0.220770i
$870$	0	0
$871$	−705315.	−0.929708
$872$	0	0
$873$	0	0
$874$	0	0
$875$	450577.i	0.588508i
$876$	0	0
$877$	248442.	0.323017	0.161509	−	0.986871i	$-0.448364\pi$
$877$	248442.	0.323017	0.161509	+	0.986871i	$0.448364\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	616016.i	0.793671i	0.917890	+	0.396835i	$0.129892\pi$
$881$	616016.i	0.793671i	−0.917890	+	0.396835i	$0.870108\pi$
$882$	0	0
$883$	498838.	0.639791	0.319895	−	0.947453i	$-0.396352\pi$
$883$	498838.	0.639791	0.319895	+	0.947453i	$0.396352\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.55587e6i	1.97754i	0.149449	+	0.988769i	$0.452250\pi$
$887$	1.55587e6i	1.97754i	−0.149449	+	0.988769i	$0.547750\pi$
$888$	0	0
$889$	788938.	0.998250
$890$	0	0
$891$	0	0
$892$	0	0
$893$	932453.i	1.16929i
$894$	0	0
$895$	64061.3	0.0799741
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 128223.i	− 0.158652i
$900$	0	0
$901$	377767.	0.465344
$902$	0	0
$903$	0	0
$904$	0	0
$905$	94317.3i	0.115158i
$906$	0	0
$907$	729390.	0.886635	0.443318	−	0.896365i	$-0.353801\pi$
$907$	729390.	0.886635	0.443318	+	0.896365i	$0.353801\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 275628.i	− 0.332114i	−0.986116	−	0.166057i	$-0.946896\pi$
$911$	− 275628.i	− 0.332114i	0.986116	−	0.166057i	$-0.0531035\pi$
$912$	0	0
$913$	136430.	0.163670
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.02979e6i	1.22465i
$918$	0	0
$919$	146750.	0.173759	0.0868794	−	0.996219i	$-0.472311\pi$
$919$	146750.	0.173759	0.0868794	+	0.996219i	$0.472311\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 430863.i	− 0.505750i
$924$	0	0
$925$	−24520.6	−0.0286581
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 398351.i	− 0.461567i	−0.973005	−	0.230783i	$-0.925871\pi$
$929$	− 398351.i	− 0.461567i	0.973005	−	0.230783i	$-0.0741289\pi$
$930$	0	0
$931$	−134782.	−0.155501
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 15444.4i	− 0.0176664i
$936$	0	0
$937$	100126.	0.114043	0.0570214	−	0.998373i	$-0.481840\pi$
$937$	100126.	0.114043	0.0570214	+	0.998373i	$0.481840\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	450064.i	0.508271i	0.967169	+	0.254136i	$0.0817909\pi$
$941$	450064.i	0.508271i	−0.967169	+	0.254136i	$0.918209\pi$
$942$	0	0
$943$	17348.0	0.0195086
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.39068e6i	1.55069i	0.631536	+	0.775346i	$0.282425\pi$
$947$	1.39068e6i	1.55069i	−0.631536	+	0.775346i	$0.717575\pi$
$948$	0	0
$949$	−1.25626e6	−1.39492
$950$	0	0
$951$	0	0
$952$	0	0
$953$	965440.i	1.06301i	0.847054	+	0.531507i	$0.178374\pi$
$953$	965440.i	1.06301i	−0.847054	+	0.531507i	$0.821626\pi$
$954$	0	0
$955$	149524.	0.163948
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.41216e6i	1.53549i
$960$	0	0
$961$	−800275.	−0.866547
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 309262.i	− 0.332102i
$966$	0	0
$967$	−885546.	−0.947018	−0.473509	−	0.880789i	$-0.657013\pi$
$967$	−885546.	−0.947018	−0.473509	+	0.880789i	$0.657013\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.57335e6i	− 1.66873i	−0.551210	−	0.834366i	$-0.685834\pi$
$971$	− 1.57335e6i	− 1.66873i	0.551210	−	0.834366i	$-0.314166\pi$
$972$	0	0
$973$	204949.	0.216482
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.52131e6i	− 1.59378i	−0.604125	−	0.796889i	$-0.706477\pi$
$977$	− 1.52131e6i	− 1.59378i	0.604125	−	0.796889i	$-0.293523\pi$
$978$	0	0
$979$	147099.	0.153477
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.57338e6i	− 1.62827i	−0.580678	−	0.814133i	$-0.697213\pi$
$983$	− 1.57338e6i	− 1.62827i	0.580678	−	0.814133i	$-0.302787\pi$
$984$	0	0
$985$	−367553.	−0.378833
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17359.8i	0.0177481i
$990$	0	0
$991$	1.27581e6	1.29908	0.649542	−	0.760326i	$-0.274961\pi$
$991$	1.27581e6	1.29908	0.649542	+	0.760326i	$0.274961\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	551380.i	0.556936i
$996$	0	0
$997$	−688039.	−0.692186	−0.346093	−	0.938200i	$-0.612492\pi$
$997$	−688039.	−0.692186	−0.346093	+	0.938200i	$0.612492\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.5.e.b.161.6	✓	12
3.2	odd	2	inner	648.5.e.b.161.7	yes	12
4.3	odd	2		1296.5.e.h.161.6		12
9.2	odd	6		648.5.m.g.377.6		24
9.4	even	3		648.5.m.g.593.6		24
9.5	odd	6		648.5.m.g.593.7		24
9.7	even	3		648.5.m.g.377.7		24
12.11	even	2		1296.5.e.h.161.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.5.e.b.161.6	✓	12	1.1	even	1	trivial
648.5.e.b.161.7	yes	12	3.2	odd	2	inner
648.5.m.g.377.6		24	9.2	odd	6
648.5.m.g.377.7		24	9.7	even	3
648.5.m.g.593.6		24	9.4	even	3
648.5.m.g.593.7		24	9.5	odd	6
1296.5.e.h.161.6		12	4.3	odd	2
1296.5.e.h.161.7		12	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	648.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.08864i q^{5} -52.9803 q^{7} -23.8108i q^{11} +277.813 q^{13} +91.5027i q^{17} -332.045 q^{19} +401.662i q^{23} +574.751 q^{25} +365.240i q^{29} -351.065 q^{31} +375.558i q^{35} -42.6630 q^{37} -43.1905i q^{41} +43.2199 q^{43} -2808.21i q^{47} +405.914 q^{49} -4128.48i q^{53} -168.786 q^{55} -6356.33i q^{59} +1697.07 q^{61} -1969.32i q^{65} -2538.81 q^{67} -1550.91i q^{71} -4521.97 q^{73} +1261.50i q^{77} +7001.74 q^{79} +5729.75i q^{83} +648.630 q^{85} +6177.83i q^{89} -14718.6 q^{91} +2353.75i q^{95} +10727.6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 96 q^{7} - 72 q^{13} + 336 q^{19} + 84 q^{25} - 1536 q^{31} - 492 q^{37} + 10128 q^{43} - 6828 q^{49} - 13968 q^{55} + 26268 q^{61} - 20784 q^{67} - 9984 q^{73} + 44592 q^{79} - 45348 q^{85} - 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) \) 12 * q + 96 * q^7 - 72 * q^13 + 336 * q^19 + 84 * q^25 - 1536 * q^31 - 492 * q^37 + 10128 * q^43 - 6828 * q^49 - 13968 * q^55 + 26268 * q^61 - 20784 * q^67 - 9984 * q^73 + 44592 * q^79 - 45348 * q^85 - 11808 * q^91 + 76608 * q^97

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