LMFDB - Embedded newform 1296.5.e.h.161.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,5,Mod(161,1296)] 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("1296.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage: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")

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:traces = [12,0,0,0,0,0,-96,0,0,0,0,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$133.967472157$
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:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$-4.92153i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.h.161.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.73487i q^{5} +24.6181 q^{7} +233.028i q^{11} -167.718 q^{13} +185.487i q^{17} +138.178 q^{19} +188.857i q^{23} +548.702 q^{25} +912.488i q^{29} -665.725 q^{31} +215.036i q^{35} +923.695 q^{37} +35.8773i q^{41} -1227.74 q^{43} -340.723i q^{47} -1794.95 q^{49} +1647.28i q^{53} -2035.47 q^{55} -2838.38i q^{59} -4837.60 q^{61} -1465.00i q^{65} +8204.36 q^{67} -5112.36i q^{71} -3679.12 q^{73} +5736.71i q^{77} -7800.33 q^{79} -6144.53i q^{83} -1620.20 q^{85} -5578.86i q^{89} -4128.91 q^{91} +1206.96i q^{95} +17069.6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

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

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
$5$	8.73487i	0.349395i	0.984622	+	0.174697i	$0.0558947\pi$
$5$	8.73487i	0.349395i	−0.984622	+	0.174697i	$0.944105\pi$
$6$	0	0
$7$	24.6181	0.502411	0.251205	−	0.967934i	$-0.419173\pi$
$7$	24.6181	0.502411	0.251205	+	0.967934i	$0.419173\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	233.028i	1.92585i	0.269770	+	0.962925i	$0.413052\pi$
$11$	233.028i	1.92585i	−0.269770	+	0.962925i	$0.586948\pi$
$12$	0	0
$13$	−167.718	−0.992417	−0.496208	−	0.868203i	$-0.665275\pi$
$13$	−167.718	−0.992417	−0.496208	+	0.868203i	$0.665275\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	185.487i	0.641824i	0.947109	+	0.320912i	$0.103989\pi$
$17$	185.487i	0.641824i	−0.947109	+	0.320912i	$0.896011\pi$
$18$	0	0
$19$	138.178	0.382764	0.191382	−	0.981516i	$-0.438703\pi$
$19$	138.178	0.382764	0.191382	+	0.981516i	$0.438703\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	188.857i	0.357008i	0.983939	+	0.178504i	$0.0571257\pi$
$23$	188.857i	0.357008i	−0.983939	+	0.178504i	$0.942874\pi$
$24$	0	0
$25$	548.702	0.877923
$26$	0	0
$27$	0	0
$28$	0	0
$29$	912.488i	1.08500i	0.840055	+	0.542502i	$0.182523\pi$
$29$	912.488i	1.08500i	−0.840055	+	0.542502i	$0.817477\pi$
$30$	0	0
$31$	−665.725	−0.692742	−0.346371	−	0.938098i	$-0.612586\pi$
$31$	−665.725	−0.692742	−0.346371	+	0.938098i	$0.612586\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	215.036i	0.175540i
$36$	0	0
$37$	923.695	0.674722	0.337361	−	0.941375i	$-0.390466\pi$
$37$	923.695	0.674722	0.337361	+	0.941375i	$0.390466\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	35.8773i	0.0213428i	0.999943	+	0.0106714i	$0.00339688\pi$
$41$	35.8773i	0.0213428i	−0.999943	+	0.0106714i	$0.996603\pi$
$42$	0	0
$43$	−1227.74	−0.664002	−0.332001	−	0.943279i	$-0.607724\pi$
$43$	−1227.74	−0.664002	−0.332001	+	0.943279i	$0.607724\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 340.723i	− 0.154243i	−0.997022	−	0.0771215i	$-0.975427\pi$
$47$	− 340.723i	− 0.154243i	0.997022	−	0.0771215i	$-0.0245729\pi$
$48$	0	0
$49$	−1794.95	−0.747583
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1647.28i	0.586429i	0.956047	+	0.293214i	$0.0947249\pi$
$53$	1647.28i	0.586429i	−0.956047	+	0.293214i	$0.905275\pi$
$54$	0	0
$55$	−2035.47	−0.672881
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2838.38i	− 0.815391i	−0.913118	−	0.407695i	$-0.866333\pi$
$59$	− 2838.38i	− 0.815391i	0.913118	−	0.407695i	$-0.133667\pi$
$60$	0	0
$61$	−4837.60	−1.30008	−0.650041	−	0.759899i	$-0.725248\pi$
$61$	−4837.60	−1.30008	−0.650041	+	0.759899i	$0.725248\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1465.00i	− 0.346745i
$66$	0	0
$67$	8204.36	1.82766	0.913829	−	0.406099i	$-0.133111\pi$
$67$	8204.36	1.82766	0.913829	+	0.406099i	$0.133111\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 5112.36i	− 1.01415i	−0.861900	−	0.507077i	$-0.830726\pi$
$71$	− 5112.36i	− 1.01415i	0.861900	−	0.507077i	$-0.169274\pi$
$72$	0	0
$73$	−3679.12	−0.690396	−0.345198	−	0.938530i	$-0.612188\pi$
$73$	−3679.12	−0.690396	−0.345198	+	0.938530i	$0.612188\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5736.71i	0.967567i
$78$	0	0
$79$	−7800.33	−1.24985	−0.624926	−	0.780684i	$-0.714871\pi$
$79$	−7800.33	−1.24985	−0.624926	+	0.780684i	$0.714871\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6144.53i	− 0.891933i	−0.895050	−	0.445967i	$-0.852860\pi$
$83$	− 6144.53i	− 0.891933i	0.895050	−	0.445967i	$-0.147140\pi$
$84$	0	0
$85$	−1620.20	−0.224250
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 5578.86i	− 0.704313i	−0.935941	−	0.352156i	$-0.885449\pi$
$89$	− 5578.86i	− 0.704313i	0.935941	−	0.352156i	$-0.114551\pi$
$90$	0	0
$91$	−4128.91	−0.498601
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1206.96i	0.133736i
$96$	0	0
$97$	17069.6	1.81418	0.907090	−	0.420938i	$-0.138299\pi$
$97$	17069.6	1.81418	0.907090	+	0.420938i	$0.138299\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18653.8i	1.82863i	0.405005	+	0.914315i	$0.367270\pi$
$101$	18653.8i	1.82863i	−0.405005	+	0.914315i	$0.632730\pi$
$102$	0	0
$103$	6446.05	0.607602	0.303801	−	0.952736i	$-0.401744\pi$
$103$	6446.05	0.607602	0.303801	+	0.952736i	$0.401744\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6316.31i	− 0.551691i	−0.961202	−	0.275845i	$-0.911042\pi$
$107$	− 6316.31i	− 0.551691i	0.961202	−	0.275845i	$-0.0889578\pi$
$108$	0	0
$109$	17568.8	1.47873	0.739366	−	0.673304i	$-0.235125\pi$
$109$	17568.8	1.47873	0.739366	+	0.673304i	$0.235125\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7902.70i	0.618897i	0.950916	+	0.309449i	$0.100144\pi$
$113$	7902.70i	0.618897i	−0.950916	+	0.309449i	$0.899856\pi$
$114$	0	0
$115$	−1649.64	−0.124737
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4566.34i	0.322459i
$120$	0	0
$121$	−39660.9	−2.70890
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10252.1i	0.656136i
$126$	0	0
$127$	−3647.38	−0.226138	−0.113069	−	0.993587i	$-0.536068\pi$
$127$	−3647.38	−0.226138	−0.113069	+	0.993587i	$0.536068\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1023.33i	0.0596311i	0.999555	+	0.0298156i	$0.00949200\pi$
$131$	1023.33i	0.0596311i	−0.999555	+	0.0298156i	$0.990508\pi$
$132$	0	0
$133$	3401.68	0.192305
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3650.80i	− 0.194512i	−0.995259	−	0.0972560i	$-0.968993\pi$
$137$	− 3650.80i	− 0.194512i	0.995259	−	0.0972560i	$-0.0310066\pi$
$138$	0	0
$139$	3752.13	0.194199	0.0970997	−	0.995275i	$-0.469043\pi$
$139$	3752.13	0.194199	0.0970997	+	0.995275i	$0.469043\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 39083.1i	− 1.91125i
$144$	0	0
$145$	−7970.46	−0.379094
$146$	0	0
$147$	0	0
$148$	0	0
$149$	35016.9i	1.57727i	0.614864	+	0.788633i	$0.289211\pi$
$149$	35016.9i	1.57727i	−0.614864	+	0.788633i	$0.710789\pi$
$150$	0	0
$151$	−16719.8	−0.733292	−0.366646	−	0.930360i	$-0.619494\pi$
$151$	−16719.8	−0.733292	−0.366646	+	0.930360i	$0.619494\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 5815.02i	− 0.242040i
$156$	0	0
$157$	−25908.1	−1.05108	−0.525541	−	0.850768i	$-0.676137\pi$
$157$	−25908.1	−1.05108	−0.525541	+	0.850768i	$0.676137\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4649.31i	0.179365i
$162$	0	0
$163$	−36433.8	−1.37129	−0.685645	−	0.727936i	$-0.740480\pi$
$163$	−36433.8	−1.37129	−0.685645	+	0.727936i	$0.740480\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16846.7i	− 0.604063i	−0.953298	−	0.302031i	$-0.902335\pi$
$167$	− 16846.7i	− 0.604063i	0.953298	−	0.302031i	$-0.0976647\pi$
$168$	0	0
$169$	−431.524	−0.0151088
$170$	0	0
$171$	0	0
$172$	0	0
$173$	31153.0i	1.04090i	0.853893	+	0.520448i	$0.174235\pi$
$173$	31153.0i	1.04090i	−0.853893	+	0.520448i	$0.825765\pi$
$174$	0	0
$175$	13508.0	0.441078
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 32225.9i	− 1.00577i	−0.864353	−	0.502885i	$-0.832272\pi$
$179$	− 32225.9i	− 1.00577i	0.864353	−	0.502885i	$-0.167728\pi$
$180$	0	0
$181$	−14660.2	−0.447488	−0.223744	−	0.974648i	$-0.571828\pi$
$181$	−14660.2	−0.447488	−0.223744	+	0.974648i	$0.571828\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8068.35i	0.235744i
$186$	0	0
$187$	−43223.6	−1.23606
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 47655.1i	− 1.30630i	−0.757229	−	0.653149i	$-0.773447\pi$
$191$	− 47655.1i	− 1.30630i	0.757229	−	0.653149i	$-0.226553\pi$
$192$	0	0
$193$	−13190.1	−0.354107	−0.177054	−	0.984201i	$-0.556657\pi$
$193$	−13190.1	−0.354107	−0.177054	+	0.984201i	$0.556657\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 64005.6i	− 1.64925i	−0.565683	−	0.824623i	$-0.691387\pi$
$197$	− 64005.6i	− 1.64925i	0.565683	−	0.824623i	$-0.308613\pi$
$198$	0	0
$199$	−51481.8	−1.30001	−0.650006	−	0.759929i	$-0.725234\pi$
$199$	−51481.8	−1.30001	−0.650006	+	0.759929i	$0.725234\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	22463.7i	0.545117i
$204$	0	0
$205$	−313.383	−0.00745707
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32199.3i	0.737146i
$210$	0	0
$211$	32713.7	0.734793	0.367396	−	0.930064i	$-0.380249\pi$
$211$	32713.7	0.734793	0.367396	+	0.930064i	$0.380249\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 10724.1i	− 0.231999i
$216$	0	0
$217$	−16388.9	−0.348041
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 31109.6i	− 0.636957i
$222$	0	0
$223$	−50779.5	−1.02112	−0.510562	−	0.859841i	$-0.670563\pi$
$223$	−50779.5	−1.02112	−0.510562	+	0.859841i	$0.670563\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	74220.5i	1.44036i	0.693785	+	0.720182i	$0.255942\pi$
$227$	74220.5i	1.44036i	−0.693785	+	0.720182i	$0.744058\pi$
$228$	0	0
$229$	−41014.5	−0.782107	−0.391054	−	0.920368i	$-0.627889\pi$
$229$	−41014.5	−0.782107	−0.391054	+	0.920368i	$0.627889\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 28741.8i	− 0.529423i	−0.964328	−	0.264711i	$-0.914723\pi$
$233$	− 28741.8i	− 0.529423i	0.964328	−	0.264711i	$-0.0852767\pi$
$234$	0	0
$235$	2976.17	0.0538917
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 41248.4i	− 0.722123i	−0.932542	−	0.361062i	$-0.882414\pi$
$239$	− 41248.4i	− 0.722123i	0.932542	−	0.361062i	$-0.117586\pi$
$240$	0	0
$241$	−91441.7	−1.57438	−0.787191	−	0.616709i	$-0.788466\pi$
$241$	−91441.7	−1.57438	−0.787191	+	0.616709i	$0.788466\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 15678.6i	− 0.261202i
$246$	0	0
$247$	−23175.0	−0.379861
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 107498.i	− 1.70629i	−0.521676	−	0.853144i	$-0.674693\pi$
$251$	− 107498.i	− 1.70629i	0.521676	−	0.853144i	$-0.325307\pi$
$252$	0	0
$253$	−44009.0	−0.687543
$254$	0	0
$255$	0	0
$256$	0	0
$257$	126064.i	1.90864i	0.298788	+	0.954320i	$0.403418\pi$
$257$	126064.i	1.90864i	−0.298788	+	0.954320i	$0.596582\pi$
$258$	0	0
$259$	22739.6	0.338988
$260$	0	0
$261$	0	0
$262$	0	0
$263$	86393.7i	1.24902i	0.781015	+	0.624512i	$0.214702\pi$
$263$	86393.7i	1.24902i	−0.781015	+	0.624512i	$0.785298\pi$
$264$	0	0
$265$	−14388.8	−0.204895
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 53675.3i	− 0.741771i	−0.928679	−	0.370885i	$-0.879054\pi$
$269$	− 53675.3i	− 0.741771i	0.928679	−	0.370885i	$-0.120946\pi$
$270$	0	0
$271$	96148.9	1.30920	0.654600	−	0.755976i	$-0.272837\pi$
$271$	96148.9	1.30920	0.654600	+	0.755976i	$0.272837\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	127863.i	1.69075i
$276$	0	0
$277$	68959.1	0.898735	0.449368	−	0.893347i	$-0.351649\pi$
$277$	68959.1	0.898735	0.449368	+	0.893347i	$0.351649\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	78514.0i	0.994339i	0.867653	+	0.497170i	$0.165627\pi$
$281$	78514.0i	0.994339i	−0.867653	+	0.497170i	$0.834373\pi$
$282$	0	0
$283$	−136308.	−1.70196	−0.850978	−	0.525201i	$-0.823990\pi$
$283$	−136308.	−1.70196	−0.850978	+	0.525201i	$0.823990\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	883.231i	0.0107229i
$288$	0	0
$289$	49115.6	0.588062
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20945.9i	0.243985i	0.992531	+	0.121993i	$0.0389284\pi$
$293$	20945.9i	0.243985i	−0.992531	+	0.121993i	$0.961072\pi$
$294$	0	0
$295$	24792.8	0.284893
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 31674.8i	− 0.354301i
$300$	0	0
$301$	−30224.6	−0.333602
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 42255.8i	− 0.454242i
$306$	0	0
$307$	−123270.	−1.30792	−0.653958	−	0.756531i	$-0.726893\pi$
$307$	−123270.	−1.30792	−0.653958	+	0.756531i	$0.726893\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	188522.i	1.94913i	0.224100	+	0.974566i	$0.428056\pi$
$311$	188522.i	1.94913i	−0.224100	+	0.974566i	$0.571944\pi$
$312$	0	0
$313$	−135319.	−1.38125	−0.690623	−	0.723215i	$-0.742664\pi$
$313$	−135319.	−1.38125	−0.690623	+	0.723215i	$0.742664\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2171.98i	0.0216141i	0.999942	+	0.0108071i	$0.00344007\pi$
$317$	2171.98i	0.0216141i	−0.999942	+	0.0108071i	$0.996560\pi$
$318$	0	0
$319$	−212635.	−2.08955
$320$	0	0
$321$	0	0
$322$	0	0
$323$	25630.2i	0.245667i
$324$	0	0
$325$	−92027.5	−0.871266
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8387.96i	− 0.0774933i
$330$	0	0
$331$	−17030.3	−0.155442	−0.0777208	−	0.996975i	$-0.524764\pi$
$331$	−17030.3	−0.155442	−0.0777208	+	0.996975i	$0.524764\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	71664.0i	0.638574i
$336$	0	0
$337$	145849.	1.28424	0.642118	−	0.766606i	$-0.278056\pi$
$337$	145849.	1.28424	0.642118	+	0.766606i	$0.278056\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 155132.i	− 1.33412i
$342$	0	0
$343$	−103296.	−0.878005
$344$	0	0
$345$	0	0
$346$	0	0
$347$	44839.5i	0.372394i	0.982512	+	0.186197i	$0.0596162\pi$
$347$	44839.5i	0.372394i	−0.982512	+	0.186197i	$0.940384\pi$
$348$	0	0
$349$	−128119.	−1.05187	−0.525935	−	0.850525i	$-0.676284\pi$
$349$	−128119.	−1.05187	−0.525935	+	0.850525i	$0.676284\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 90919.9i	− 0.729641i	−0.931078	−	0.364821i	$-0.881130\pi$
$353$	− 90919.9i	− 0.729641i	0.931078	−	0.364821i	$-0.118870\pi$
$354$	0	0
$355$	44655.7	0.354340
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 76614.4i	− 0.594459i	−0.954806	−	0.297229i	$-0.903937\pi$
$359$	− 76614.4i	− 0.594459i	0.954806	−	0.297229i	$-0.0960626\pi$
$360$	0	0
$361$	−111228.	−0.853492
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 32136.6i	− 0.241221i
$366$	0	0
$367$	−66590.1	−0.494399	−0.247200	−	0.968965i	$-0.579510\pi$
$367$	−66590.1	−0.494399	−0.247200	+	0.968965i	$0.579510\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	40552.9i	0.294628i
$372$	0	0
$373$	118275.	0.850108	0.425054	−	0.905168i	$-0.360255\pi$
$373$	118275.	0.850108	0.425054	+	0.905168i	$0.360255\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 153041.i	− 1.07678i
$378$	0	0
$379$	199613.	1.38967	0.694833	−	0.719171i	$-0.255478\pi$
$379$	199613.	1.38967	0.694833	+	0.719171i	$0.255478\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 103259.i	− 0.703935i	−0.936012	−	0.351967i	$-0.885513\pi$
$383$	− 103259.i	− 0.703935i	0.936012	−	0.351967i	$-0.114487\pi$
$384$	0	0
$385$	−50109.4	−0.338063
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 208338.i	− 1.37680i	−0.725333	−	0.688399i	$-0.758314\pi$
$389$	− 208338.i	− 1.37680i	0.725333	−	0.688399i	$-0.241686\pi$
$390$	0	0
$391$	−35030.5	−0.229136
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 68134.8i	− 0.436692i
$396$	0	0
$397$	−204654.	−1.29849	−0.649245	−	0.760579i	$-0.724915\pi$
$397$	−204654.	−1.29849	−0.649245	+	0.760579i	$0.724915\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 111937.i	− 0.696124i	−0.937472	−	0.348062i	$-0.886840\pi$
$401$	− 111937.i	− 0.696124i	0.937472	−	0.348062i	$-0.113160\pi$
$402$	0	0
$403$	111654.	0.687489
$404$	0	0
$405$	0	0
$406$	0	0
$407$	215247.i	1.29941i
$408$	0	0
$409$	75918.4	0.453838	0.226919	−	0.973914i	$-0.427135\pi$
$409$	75918.4	0.453838	0.226919	+	0.973914i	$0.427135\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 69875.5i	− 0.409661i
$414$	0	0
$415$	53671.6	0.311637
$416$	0	0
$417$	0	0
$418$	0	0
$419$	136165.i	0.775601i	0.921743	+	0.387800i	$0.126765\pi$
$419$	136165.i	0.775601i	−0.921743	+	0.387800i	$0.873235\pi$
$420$	0	0
$421$	−199051.	−1.12305	−0.561527	−	0.827458i	$-0.689786\pi$
$421$	−199051.	−1.12305	−0.561527	+	0.827458i	$0.689786\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	101777.i	0.563472i
$426$	0	0
$427$	−119093.	−0.653175
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 94505.9i	− 0.508750i	−0.967106	−	0.254375i	$-0.918130\pi$
$431$	− 94505.9i	− 0.508750i	0.967106	−	0.254375i	$-0.0818697\pi$
$432$	0	0
$433$	−57190.1	−0.305032	−0.152516	−	0.988301i	$-0.548738\pi$
$433$	−57190.1	−0.305032	−0.152516	+	0.988301i	$0.548738\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26095.9i	0.136650i
$438$	0	0
$439$	74649.2	0.387344	0.193672	−	0.981066i	$-0.437960\pi$
$439$	74649.2	0.387344	0.193672	+	0.981066i	$0.437960\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 32815.8i	− 0.167215i	−0.996499	−	0.0836076i	$-0.973356\pi$
$443$	− 32815.8i	− 0.167215i	0.996499	−	0.0836076i	$-0.0266442\pi$
$444$	0	0
$445$	48730.6	0.246083
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 354122.i	− 1.75655i	−0.478157	−	0.878275i	$-0.658695\pi$
$449$	− 354122.i	− 1.75655i	0.478157	−	0.878275i	$-0.341305\pi$
$450$	0	0
$451$	−8360.40	−0.0411030
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 36065.5i	− 0.174208i
$456$	0	0
$457$	207905.	0.995481	0.497740	−	0.867326i	$-0.334163\pi$
$457$	207905.	0.995481	0.497740	+	0.867326i	$0.334163\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	240931.i	1.13368i	0.823827	+	0.566841i	$0.191835\pi$
$461$	240931.i	1.13368i	−0.823827	+	0.566841i	$0.808165\pi$
$462$	0	0
$463$	−360289.	−1.68070	−0.840348	−	0.542047i	$-0.817650\pi$
$463$	−360289.	−1.68070	−0.840348	+	0.542047i	$0.817650\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	267462.i	1.22639i	0.789932	+	0.613194i	$0.210116\pi$
$467$	267462.i	1.22639i	−0.789932	+	0.613194i	$0.789884\pi$
$468$	0	0
$469$	201976.	0.918235
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 286097.i	− 1.27877i
$474$	0	0
$475$	75818.4	0.336037
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 49009.4i	− 0.213603i	−0.994280	−	0.106802i	$-0.965939\pi$
$479$	− 49009.4i	− 0.213603i	0.994280	−	0.106802i	$-0.0340610\pi$
$480$	0	0
$481$	−154921.	−0.669606
$482$	0	0
$483$	0	0
$484$	0	0
$485$	149101.i	0.633864i
$486$	0	0
$487$	−445429.	−1.87811	−0.939053	−	0.343772i	$-0.888295\pi$
$487$	−445429.	−1.87811	−0.939053	+	0.343772i	$0.888295\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	152664.i	0.633248i	0.948551	+	0.316624i	$0.102549\pi$
$491$	152664.i	0.633248i	−0.948551	+	0.316624i	$0.897451\pi$
$492$	0	0
$493$	−169255.	−0.696381
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 125857.i	− 0.509522i
$498$	0	0
$499$	−17606.6	−0.0707088	−0.0353544	−	0.999375i	$-0.511256\pi$
$499$	−17606.6	−0.0707088	−0.0353544	+	0.999375i	$0.511256\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	296477.i	1.17181i	0.810381	+	0.585903i	$0.199260\pi$
$503$	296477.i	1.17181i	−0.810381	+	0.585903i	$0.800740\pi$
$504$	0	0
$505$	−162939.	−0.638913
$506$	0	0
$507$	0	0
$508$	0	0
$509$	310574.i	1.19875i	0.800468	+	0.599376i	$0.204585\pi$
$509$	310574.i	1.19875i	−0.800468	+	0.599376i	$0.795415\pi$
$510$	0	0
$511$	−90573.0	−0.346862
$512$	0	0
$513$	0	0
$514$	0	0
$515$	56305.4i	0.212293i
$516$	0	0
$517$	79397.9	0.297049
$518$	0	0
$519$	0	0
$520$	0	0
$521$	435336.i	1.60380i	0.597461	+	0.801898i	$0.296176\pi$
$521$	435336.i	1.60380i	−0.597461	+	0.801898i	$0.703824\pi$
$522$	0	0
$523$	417724.	1.52717	0.763583	−	0.645710i	$-0.223438\pi$
$523$	417724.	1.52717	0.763583	+	0.645710i	$0.223438\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 123483.i	− 0.444618i
$528$	0	0
$529$	244174.	0.872545
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6017.28i	− 0.0211810i
$534$	0	0
$535$	55172.1	0.192758
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 418273.i	− 1.43973i
$540$	0	0
$541$	215875.	0.737578	0.368789	−	0.929513i	$-0.379773\pi$
$541$	215875.	0.737578	0.368789	+	0.929513i	$0.379773\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	153461.i	0.516661i
$546$	0	0
$547$	409002.	1.36695	0.683473	−	0.729976i	$-0.260469\pi$
$547$	409002.	1.36695	0.683473	+	0.729976i	$0.260469\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	126086.i	0.415300i
$552$	0	0
$553$	−192029.	−0.627939
$554$	0	0
$555$	0	0
$556$	0	0
$557$	381833.i	1.23073i	0.788242	+	0.615366i	$0.210992\pi$
$557$	381833.i	1.23073i	−0.788242	+	0.615366i	$0.789008\pi$
$558$	0	0
$559$	205914.	0.658966
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18580.6i	0.0586197i	0.999570	+	0.0293099i	$0.00933096\pi$
$563$	18580.6i	0.0586197i	−0.999570	+	0.0293099i	$0.990669\pi$
$564$	0	0
$565$	−69029.0	−0.216239
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 152379.i	− 0.470652i	−0.971917	−	0.235326i	$-0.924384\pi$
$569$	− 152379.i	− 0.470652i	0.971917	−	0.235326i	$-0.0756158\pi$
$570$	0	0
$571$	152843.	0.468785	0.234393	−	0.972142i	$-0.424690\pi$
$571$	152843.	0.468785	0.234393	+	0.972142i	$0.424690\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	103626.i	0.313425i
$576$	0	0
$577$	−62842.7	−0.188757	−0.0943786	−	0.995536i	$-0.530086\pi$
$577$	−62842.7	−0.188757	−0.0943786	+	0.995536i	$0.530086\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 151267.i	− 0.448117i
$582$	0	0
$583$	−383862.	−1.12937
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 429236.i	− 1.24572i	−0.782334	−	0.622859i	$-0.785971\pi$
$587$	− 429236.i	− 1.24572i	0.782334	−	0.622859i	$-0.214029\pi$
$588$	0	0
$589$	−91988.4	−0.265157
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 184842.i	− 0.525642i	−0.964845	−	0.262821i	$-0.915347\pi$
$593$	− 184842.i	− 0.525642i	0.964845	−	0.262821i	$-0.0846529\pi$
$594$	0	0
$595$	−39886.4	−0.112665
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 169801.i	− 0.473245i	−0.971602	−	0.236623i	$-0.923959\pi$
$599$	− 169801.i	− 0.473245i	0.971602	−	0.236623i	$-0.0760405\pi$
$600$	0	0
$601$	241540.	0.668713	0.334356	−	0.942447i	$-0.391481\pi$
$601$	241540.	0.668713	0.334356	+	0.942447i	$0.391481\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 346433.i	− 0.946474i
$606$	0	0
$607$	678570.	1.84169	0.920847	−	0.389924i	$-0.127499\pi$
$607$	678570.	1.84169	0.920847	+	0.389924i	$0.127499\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	57145.5i	0.153073i
$612$	0	0
$613$	238616.	0.635006	0.317503	−	0.948257i	$-0.397156\pi$
$613$	238616.	0.635006	0.317503	+	0.948257i	$0.397156\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	104832.i	0.275375i	0.990476	+	0.137688i	$0.0439670\pi$
$617$	104832.i	0.275375i	−0.990476	+	0.137688i	$0.956033\pi$
$618$	0	0
$619$	−420591.	−1.09769	−0.548843	−	0.835925i	$-0.684932\pi$
$619$	−420591.	−1.09769	−0.548843	+	0.835925i	$0.684932\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 137341.i	− 0.353854i
$624$	0	0
$625$	253388.	0.648673
$626$	0	0
$627$	0	0
$628$	0	0
$629$	171333.i	0.433053i
$630$	0	0
$631$	−240041.	−0.602874	−0.301437	−	0.953486i	$-0.597466\pi$
$631$	−240041.	−0.602874	−0.301437	+	0.953486i	$0.597466\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 31859.3i	− 0.0790113i
$636$	0	0
$637$	301046.	0.741914
$638$	0	0
$639$	0	0
$640$	0	0
$641$	257769.i	0.627357i	0.949529	+	0.313678i	$0.101561\pi$
$641$	257769.i	0.627357i	−0.949529	+	0.313678i	$0.898439\pi$
$642$	0	0
$643$	745688.	1.80358	0.901790	−	0.432176i	$-0.142254\pi$
$643$	745688.	1.80358	0.901790	+	0.432176i	$0.142254\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	221439.i	0.528989i	0.964387	+	0.264494i	$0.0852051\pi$
$647$	221439.i	0.528989i	−0.964387	+	0.264494i	$0.914795\pi$
$648$	0	0
$649$	661420.	1.57032
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 343306.i	− 0.805109i	−0.915396	−	0.402555i	$-0.868122\pi$
$653$	− 343306.i	− 0.805109i	0.915396	−	0.402555i	$-0.131878\pi$
$654$	0	0
$655$	−8938.65	−0.0208348
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 633646.i	− 1.45907i	−0.683944	−	0.729534i	$-0.739737\pi$
$659$	− 633646.i	− 1.45907i	0.683944	−	0.729534i	$-0.260263\pi$
$660$	0	0
$661$	148335.	0.339501	0.169750	−	0.985487i	$-0.445704\pi$
$661$	148335.	0.339501	0.169750	+	0.985487i	$0.445704\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29713.2i	0.0671902i
$666$	0	0
$667$	−172330.	−0.387355
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.12730e6i	− 2.50376i
$672$	0	0
$673$	83366.3	0.184060	0.0920302	−	0.995756i	$-0.470664\pi$
$673$	83366.3	0.184060	0.0920302	+	0.995756i	$0.470664\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	744037.i	1.62337i	0.584097	+	0.811684i	$0.301449\pi$
$677$	744037.i	1.62337i	−0.584097	+	0.811684i	$0.698551\pi$
$678$	0	0
$679$	420222.	0.911463
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 388064.i	− 0.831882i	−0.909392	−	0.415941i	$-0.863452\pi$
$683$	− 388064.i	− 0.831882i	0.909392	−	0.415941i	$-0.136548\pi$
$684$	0	0
$685$	31889.2	0.0679615
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 276279.i	− 0.581982i
$690$	0	0
$691$	−95380.4	−0.199757	−0.0998787	−	0.995000i	$-0.531845\pi$
$691$	−95380.4	−0.199757	−0.0998787	+	0.995000i	$0.531845\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32774.3i	0.0678523i
$696$	0	0
$697$	−6654.77	−0.0136983
$698$	0	0
$699$	0	0
$700$	0	0
$701$	95898.4i	0.195153i	0.995228	+	0.0975765i	$0.0311091\pi$
$701$	95898.4i	0.195153i	−0.995228	+	0.0975765i	$0.968891\pi$
$702$	0	0
$703$	127634.	0.258259
$704$	0	0
$705$	0	0
$706$	0	0
$707$	459223.i	0.918723i
$708$	0	0
$709$	514240.	1.02300	0.511498	−	0.859285i	$-0.329091\pi$
$709$	514240.	1.02300	0.511498	+	0.859285i	$0.329091\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 125727.i	− 0.247314i
$714$	0	0
$715$	341385.	0.667779
$716$	0	0
$717$	0	0
$718$	0	0
$719$	926662.i	1.79252i	0.443532	+	0.896259i	$0.353725\pi$
$719$	926662.i	1.79252i	−0.443532	+	0.896259i	$0.646275\pi$
$720$	0	0
$721$	158690.	0.305266
$722$	0	0
$723$	0	0
$724$	0	0
$725$	500684.i	0.952550i
$726$	0	0
$727$	396812.	0.750786	0.375393	−	0.926866i	$-0.377508\pi$
$727$	396812.	0.750786	0.375393	+	0.926866i	$0.377508\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 227730.i	− 0.426172i
$732$	0	0
$733$	−833114.	−1.55059	−0.775294	−	0.631600i	$-0.782398\pi$
$733$	−833114.	−1.55059	−0.775294	+	0.631600i	$0.782398\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.91184e6i	3.51979i
$738$	0	0
$739$	743893.	1.36214	0.681070	−	0.732219i	$-0.261515\pi$
$739$	743893.	1.36214	0.681070	+	0.732219i	$0.261515\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 849514.i	− 1.53884i	−0.638744	−	0.769419i	$-0.720546\pi$
$743$	− 849514.i	− 1.53884i	0.638744	−	0.769419i	$-0.279454\pi$
$744$	0	0
$745$	−305868.	−0.551088
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 155496.i	− 0.277175i
$750$	0	0
$751$	−83814.1	−0.148606	−0.0743032	−	0.997236i	$-0.523673\pi$
$751$	−83814.1	−0.148606	−0.0743032	+	0.997236i	$0.523673\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 146045.i	− 0.256208i
$756$	0	0
$757$	−271456.	−0.473704	−0.236852	−	0.971546i	$-0.576116\pi$
$757$	−271456.	−0.473704	−0.236852	+	0.971546i	$0.576116\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	819191.i	1.41454i	0.706942	+	0.707271i	$0.250074\pi$
$761$	819191.i	1.41454i	−0.706942	+	0.707271i	$0.749926\pi$
$762$	0	0
$763$	432511.	0.742931
$764$	0	0
$765$	0	0
$766$	0	0
$767$	476048.i	0.809207i
$768$	0	0
$769$	−599098.	−1.01308	−0.506541	−	0.862216i	$-0.669076\pi$
$769$	−599098.	−1.01308	−0.506541	+	0.862216i	$0.669076\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	334807.i	0.560319i	0.959954	+	0.280159i	$0.0903873\pi$
$773$	334807.i	0.560319i	−0.959954	+	0.280159i	$0.909613\pi$
$774$	0	0
$775$	−365285.	−0.608174
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4957.44i	0.00816926i
$780$	0	0
$781$	1.19132e6	1.95311
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 226304.i	− 0.367243i
$786$	0	0
$787$	637051.	1.02855	0.514275	−	0.857626i	$-0.328061\pi$
$787$	637051.	1.02855	0.514275	+	0.857626i	$0.328061\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	194550.i	0.310941i
$792$	0	0
$793$	811355.	1.29022
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 563207.i	− 0.886648i	−0.896361	−	0.443324i	$-0.853799\pi$
$797$	− 563207.i	− 0.886648i	0.896361	−	0.443324i	$-0.146201\pi$
$798$	0	0
$799$	63199.7	0.0989968
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 857337.i	− 1.32960i
$804$	0	0
$805$	−40611.1	−0.0626690
$806$	0	0
$807$	0	0
$808$	0	0
$809$	430042.i	0.657073i	0.944491	+	0.328536i	$0.106555\pi$
$809$	430042.i	0.657073i	−0.944491	+	0.328536i	$0.893445\pi$
$810$	0	0
$811$	181645.	0.276174	0.138087	−	0.990420i	$-0.455905\pi$
$811$	181645.	0.276174	0.138087	+	0.990420i	$0.455905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 318245.i	− 0.479122i
$816$	0	0
$817$	−169646.	−0.254156
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 641220.i	− 0.951307i	−0.879633	−	0.475654i	$-0.842212\pi$
$821$	− 641220.i	− 0.951307i	0.879633	−	0.475654i	$-0.157788\pi$
$822$	0	0
$823$	−1.20525e6	−1.77942	−0.889708	−	0.456529i	$-0.849092\pi$
$823$	−1.20525e6	−1.77942	−0.889708	+	0.456529i	$0.849092\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	726419.i	1.06213i	0.847332	+	0.531063i	$0.178207\pi$
$827$	726419.i	1.06213i	−0.847332	+	0.531063i	$0.821793\pi$
$828$	0	0
$829$	639915.	0.931136	0.465568	−	0.885012i	$-0.345850\pi$
$829$	639915.	0.931136	0.465568	+	0.885012i	$0.345850\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 332940.i	− 0.479817i
$834$	0	0
$835$	147154.	0.211056
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 508629.i	− 0.722566i	−0.932456	−	0.361283i	$-0.882339\pi$
$839$	− 508629.i	− 0.722566i	0.932456	−	0.361283i	$-0.117661\pi$
$840$	0	0
$841$	−125353.	−0.177232
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 3769.30i	− 0.00527895i
$846$	0	0
$847$	−976378.	−1.36098
$848$	0	0
$849$	0	0
$850$	0	0
$851$	174446.i	0.240881i
$852$	0	0
$853$	487596.	0.670135	0.335068	−	0.942194i	$-0.391241\pi$
$853$	487596.	0.670135	0.335068	+	0.942194i	$0.391241\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	880176.i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$857$	880176.i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$858$	0	0
$859$	850084.	1.15206	0.576031	−	0.817428i	$-0.304601\pi$
$859$	850084.	1.15206	0.576031	+	0.817428i	$0.304601\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	633599.i	0.850733i	0.905021	+	0.425366i	$0.139855\pi$
$863$	633599.i	0.850733i	−0.905021	+	0.425366i	$0.860145\pi$
$864$	0	0
$865$	−272117.	−0.363684
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.81769e6i	− 2.40703i
$870$	0	0
$871$	−1.37602e6	−1.81380
$872$	0	0
$873$	0	0
$874$	0	0
$875$	252388.i	0.329650i
$876$	0	0
$877$	474893.	0.617443	0.308721	−	0.951153i	$-0.400099\pi$
$877$	474893.	0.617443	0.308721	+	0.951153i	$0.400099\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 543957.i	− 0.700830i	−0.936595	−	0.350415i	$-0.886041\pi$
$881$	− 543957.i	− 0.700830i	0.936595	−	0.350415i	$-0.113959\pi$
$882$	0	0
$883$	532781.	0.683325	0.341662	−	0.939823i	$-0.389010\pi$
$883$	532781.	0.683325	0.341662	+	0.939823i	$0.389010\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 883957.i	− 1.12353i	−0.827297	−	0.561764i	$-0.810123\pi$
$887$	− 883957.i	− 1.12353i	0.827297	−	0.561764i	$-0.189877\pi$
$888$	0	0
$889$	−89791.6	−0.113614
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 47080.3i	− 0.0590387i
$894$	0	0
$895$	281489.	0.351411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 607466.i	− 0.751627i
$900$	0	0
$901$	−305549.	−0.376384
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 128055.i	− 0.156350i
$906$	0	0
$907$	1.19043e6	1.44707	0.723536	−	0.690286i	$-0.242515\pi$
$907$	1.19043e6	1.44707	0.723536	+	0.690286i	$0.242515\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	81438.5i	0.0981280i	0.998796	+	0.0490640i	$0.0156238\pi$
$911$	81438.5i	0.0981280i	−0.998796	+	0.0490640i	$0.984376\pi$
$912$	0	0
$913$	1.43185e6	1.71773
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25192.5i	0.0299593i
$918$	0	0
$919$	954730.	1.13045	0.565223	−	0.824938i	$-0.308790\pi$
$919$	954730.	1.13045	0.565223	+	0.824938i	$0.308790\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	857436.i	1.00646i
$924$	0	0
$925$	506833.	0.592355
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.01164e6i	− 1.17218i	−0.810246	−	0.586089i	$-0.800667\pi$
$929$	− 1.01164e6i	− 1.17218i	0.810246	−	0.586089i	$-0.199333\pi$
$930$	0	0
$931$	−248022.	−0.286148
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 377553.i	− 0.431871i
$936$	0	0
$937$	−389712.	−0.443879	−0.221939	−	0.975060i	$-0.571239\pi$
$937$	−389712.	−0.443879	−0.221939	+	0.975060i	$0.571239\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.34464e6i	1.51854i	0.650775	+	0.759271i	$0.274444\pi$
$941$	1.34464e6i	1.51854i	−0.650775	+	0.759271i	$0.725556\pi$
$942$	0	0
$943$	−6775.68	−0.00761955
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 732017.i	− 0.816246i	−0.912927	−	0.408123i	$-0.866183\pi$
$947$	− 732017.i	− 0.816246i	0.912927	−	0.408123i	$-0.133817\pi$
$948$	0	0
$949$	617056.	0.685160
$950$	0	0
$951$	0	0
$952$	0	0
$953$	178149.i	0.196154i	0.995179	+	0.0980768i	$0.0312691\pi$
$953$	178149.i	0.196154i	−0.995179	+	0.0980768i	$0.968731\pi$
$954$	0	0
$955$	416261.	0.456414
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 89875.8i	− 0.0977250i
$960$	0	0
$961$	−480331.	−0.520109
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 115214.i	− 0.123723i
$966$	0	0
$967$	496360.	0.530816	0.265408	−	0.964136i	$-0.414493\pi$
$967$	496360.	0.530816	0.265408	+	0.964136i	$0.414493\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	388051.i	0.411576i	0.978597	+	0.205788i	$0.0659757\pi$
$971$	388051.i	0.411576i	−0.978597	+	0.205788i	$0.934024\pi$
$972$	0	0
$973$	92370.4	0.0975679
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 667564.i	− 0.699365i	−0.936868	−	0.349682i	$-0.886290\pi$
$977$	− 667564.i	− 0.699365i	0.936868	−	0.349682i	$-0.113710\pi$
$978$	0	0
$979$	1.30003e6	1.35640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.88486e6i	1.95062i	0.220837	+	0.975311i	$0.429121\pi$
$983$	1.88486e6i	1.95062i	−0.220837	+	0.975311i	$0.570879\pi$
$984$	0	0
$985$	559080.	0.576238
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 231867.i	− 0.237054i
$990$	0	0
$991$	−896441.	−0.912797	−0.456399	−	0.889775i	$-0.650861\pi$
$991$	−896441.	−0.912797	−0.456399	+	0.889775i	$0.650861\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 449687.i	− 0.454217i
$996$	0	0
$997$	−1.60644e6	−1.61613	−0.808063	−	0.589096i	$-0.799484\pi$
$997$	−1.60644e6	−1.61613	−0.808063	+	0.589096i	$0.799484\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.h.161.8		12
3.2	odd	2	inner	1296.5.e.h.161.5		12
4.3	odd	2		648.5.e.b.161.8	yes	12
12.11	even	2		648.5.e.b.161.5	✓	12
36.7	odd	6		648.5.m.g.377.5		24
36.11	even	6		648.5.m.g.377.8		24
36.23	even	6		648.5.m.g.593.5		24
36.31	odd	6		648.5.m.g.593.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.5.e.b.161.5	✓	12	12.11	even	2
648.5.e.b.161.8	yes	12	4.3	odd	2
648.5.m.g.377.5		24	36.7	odd	6
648.5.m.g.377.8		24	36.11	even	6
648.5.m.g.593.5		24	36.23	even	6
648.5.m.g.593.8		24	36.31	odd	6
1296.5.e.h.161.5		12	3.2	odd	2	inner
1296.5.e.h.161.8		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.73487i q^{5} +24.6181 q^{7} +233.028i q^{11} -167.718 q^{13} +185.487i q^{17} +138.178 q^{19} +188.857i q^{23} +548.702 q^{25} +912.488i q^{29} -665.725 q^{31} +215.036i q^{35} +923.695 q^{37} +35.8773i q^{41} -1227.74 q^{43} -340.723i q^{47} -1794.95 q^{49} +1647.28i q^{53} -2035.47 q^{55} -2838.38i q^{59} -4837.60 q^{61} -1465.00i q^{65} +8204.36 q^{67} -5112.36i q^{71} -3679.12 q^{73} +5736.71i q^{77} -7800.33 q^{79} -6144.53i q^{83} -1620.20 q^{85} -5578.86i q^{89} -4128.91 q^{91} +1206.96i q^{95} +17069.6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 96 q^{7} - 72 q^{13} - 336 q^{19} + 84 q^{25} + 1536 q^{31} - 492 q^{37} - 10128 q^{43} - 6828 q^{49} + 13968 q^{55} + 26268 q^{61} + 20784 q^{67} - 9984 q^{73} - 44592 q^{79} - 45348 q^{85} + 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

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