LMFDB - Embedded newform 864.5.e.g.161.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [864,5,Mod(161,864)] 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("864.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	864.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,-32] 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:	$89.3116481044$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 22 x^{14} - 60 x^{13} + 313 x^{12} + 1368 x^{11} + 1844 x^{10} - 4788 x^{9} - 11779 x^{8} + \cdots + 16848900 $ x^16 - 22x^14 - 60x^13 + 313x^12 + 1368x^11 + 1844x^10 - 4788x^9 - 11779x^8 - 45888x^7 + 8308x^6 + 466392x^5 + 2503364x^4 + 5337264x^3 + 10353832x^2 + 14664480x + 16848900
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{56}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.15
Root			$-3.06658 + 1.75061i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.5.e.g.161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+33.5878i q^{5} -14.6267 q^{7} +144.752i q^{11} +19.3503 q^{13} +166.163i q^{17} +173.755 q^{19} +220.826i q^{23} -503.139 q^{25} +1289.78i q^{29} +171.391 q^{31} -491.280i q^{35} +1264.16 q^{37} +1695.63i q^{41} +878.215 q^{43} -1324.53i q^{47} -2187.06 q^{49} +1670.43i q^{53} -4861.91 q^{55} -1733.62i q^{59} -4001.81 q^{61} +649.933i q^{65} +7996.28 q^{67} +2799.71i q^{71} +1218.51 q^{73} -2117.26i q^{77} -8781.35 q^{79} +5917.23i q^{83} -5581.05 q^{85} -12798.2i q^{89} -283.032 q^{91} +5836.05i q^{95} -6356.47 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{13} + 704 q^{25} - 2624 q^{37} + 1728 q^{49} + 3264 q^{61} - 5424 q^{73} + 4704 q^{85} - 80 q^{97}+O(q^{100}) $ 16 * q - 32 * q^13 + 704 * q^25 - 2624 * q^37 + 1728 * q^49 + 3264 * q^61 - 5424 * q^73 + 4704 * q^85 - 80 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	33.5878i	1.34351i	0.740773	+	0.671756i	$0.234460\pi$
$5$	33.5878i	1.34351i	−0.740773	+	0.671756i	$0.765540\pi$
$6$	0	0
$7$	−14.6267	−0.298505	−0.149253	−	0.988799i	$-0.547687\pi$
$7$	−14.6267	−0.298505	−0.149253	+	0.988799i	$0.547687\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	144.752i	1.19630i	0.801384	+	0.598150i	$0.204097\pi$
$11$	144.752i	1.19630i	−0.801384	+	0.598150i	$0.795903\pi$
$12$	0	0
$13$	19.3503	0.114499	0.0572494	−	0.998360i	$-0.481767\pi$
$13$	19.3503	0.114499	0.0572494	+	0.998360i	$0.481767\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	166.163i	0.574959i	0.957787	+	0.287479i	$0.0928173\pi$
$17$	166.163i	0.574959i	−0.957787	+	0.287479i	$0.907183\pi$
$18$	0	0
$19$	173.755	0.481316	0.240658	−	0.970610i	$-0.422637\pi$
$19$	173.755	0.481316	0.240658	+	0.970610i	$0.422637\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	220.826i	0.417441i	0.977975	+	0.208721i	$0.0669299\pi$
$23$	220.826i	0.417441i	−0.977975	+	0.208721i	$0.933070\pi$
$24$	0	0
$25$	−503.139	−0.805023
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1289.78i	1.53362i	0.641873	+	0.766811i	$0.278158\pi$
$29$	1289.78i	1.53362i	−0.641873	+	0.766811i	$0.721842\pi$
$30$	0	0
$31$	171.391	0.178347	0.0891734	−	0.996016i	$-0.471577\pi$
$31$	171.391	0.178347	0.0891734	+	0.996016i	$0.471577\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 491.280i	− 0.401045i
$36$	0	0
$37$	1264.16	0.923422	0.461711	−	0.887030i	$-0.347236\pi$
$37$	1264.16	0.923422	0.461711	+	0.887030i	$0.347236\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1695.63i	1.00871i	0.863498	+	0.504353i	$0.168269\pi$
$41$	1695.63i	1.00871i	−0.863498	+	0.504353i	$0.831731\pi$
$42$	0	0
$43$	878.215	0.474967	0.237484	−	0.971392i	$-0.423677\pi$
$43$	878.215	0.474967	0.237484	+	0.971392i	$0.423677\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1324.53i	− 0.599608i	−0.954001	−	0.299804i	$-0.903079\pi$
$47$	− 1324.53i	− 0.599608i	0.954001	−	0.299804i	$-0.0969213\pi$
$48$	0	0
$49$	−2187.06	−0.910895
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1670.43i	0.594670i	0.954773	+	0.297335i	$0.0960978\pi$
$53$	1670.43i	0.594670i	−0.954773	+	0.297335i	$0.903902\pi$
$54$	0	0
$55$	−4861.91	−1.60724
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1733.62i	− 0.498023i	−0.968501	−	0.249012i	$-0.919894\pi$
$59$	− 1733.62i	− 0.498023i	0.968501	−	0.249012i	$-0.0801057\pi$
$60$	0	0
$61$	−4001.81	−1.07547	−0.537733	−	0.843115i	$-0.680719\pi$
$61$	−4001.81	−1.07547	−0.537733	+	0.843115i	$0.680719\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	649.933i	0.153830i
$66$	0	0
$67$	7996.28	1.78130	0.890652	−	0.454685i	$-0.150248\pi$
$67$	7996.28	1.78130	0.890652	+	0.454685i	$0.150248\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2799.71i	0.555388i	0.960670	+	0.277694i	$0.0895702\pi$
$71$	2799.71i	0.555388i	−0.960670	+	0.277694i	$0.910430\pi$
$72$	0	0
$73$	1218.51	0.228657	0.114329	−	0.993443i	$-0.463528\pi$
$73$	1218.51	0.228657	0.114329	+	0.993443i	$0.463528\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2117.26i	− 0.357102i
$78$	0	0
$79$	−8781.35	−1.40704	−0.703521	−	0.710674i	$-0.748390\pi$
$79$	−8781.35	−1.40704	−0.703521	+	0.710674i	$0.748390\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5917.23i	0.858938i	0.903082	+	0.429469i	$0.141299\pi$
$83$	5917.23i	0.858938i	−0.903082	+	0.429469i	$0.858701\pi$
$84$	0	0
$85$	−5581.05	−0.772464
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12798.2i	− 1.61573i	−0.589370	−	0.807863i	$-0.700624\pi$
$89$	− 12798.2i	− 1.61573i	0.589370	−	0.807863i	$-0.299376\pi$
$90$	0	0
$91$	−283.032	−0.0341785
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5836.05i	0.646653i
$96$	0	0
$97$	−6356.47	−0.675574	−0.337787	−	0.941223i	$-0.609678\pi$
$97$	−6356.47	−0.675574	−0.337787	+	0.941223i	$0.609678\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 12782.6i	− 1.25307i	−0.779392	−	0.626537i	$-0.784472\pi$
$101$	− 12782.6i	− 1.25307i	0.779392	−	0.626537i	$-0.215528\pi$
$102$	0	0
$103$	11151.4	1.05113	0.525563	−	0.850755i	$-0.323855\pi$
$103$	11151.4	1.05113	0.525563	+	0.850755i	$0.323855\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8160.18i	− 0.712741i	−0.934345	−	0.356371i	$-0.884014\pi$
$107$	− 8160.18i	− 0.712741i	0.934345	−	0.356371i	$-0.115986\pi$
$108$	0	0
$109$	−21304.4	−1.79315	−0.896575	−	0.442891i	$-0.853953\pi$
$109$	−21304.4	−1.79315	−0.896575	+	0.442891i	$0.853953\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4969.44i	− 0.389180i	−0.980885	−	0.194590i	$-0.937662\pi$
$113$	− 4969.44i	− 0.389180i	0.980885	−	0.194590i	$-0.0623377\pi$
$114$	0	0
$115$	−7417.07	−0.560837
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2430.43i	− 0.171628i
$120$	0	0
$121$	−6312.23	−0.431134
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4093.04i	0.261954i
$126$	0	0
$127$	−17154.5	−1.06358	−0.531792	−	0.846875i	$-0.678481\pi$
$127$	−17154.5	−1.06358	−0.531792	+	0.846875i	$0.678481\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 21193.8i	− 1.23500i	−0.786572	−	0.617499i	$-0.788146\pi$
$131$	− 21193.8i	− 1.23500i	0.786572	−	0.617499i	$-0.211854\pi$
$132$	0	0
$133$	−2541.47	−0.143675
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17726.8i	0.944470i	0.881473	+	0.472235i	$0.156553\pi$
$137$	17726.8i	0.944470i	−0.881473	+	0.472235i	$0.843447\pi$
$138$	0	0
$139$	32235.7	1.66843	0.834214	−	0.551441i	$-0.185922\pi$
$139$	32235.7	1.66843	0.834214	+	0.551441i	$0.185922\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2801.00i	0.136975i
$144$	0	0
$145$	−43320.7	−2.06044
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 6952.42i	− 0.313158i	−0.987665	−	0.156579i	$-0.949953\pi$
$149$	− 6952.42i	− 0.313158i	0.987665	−	0.156579i	$-0.0500466\pi$
$150$	0	0
$151$	19213.1	0.842643	0.421322	−	0.906911i	$-0.361566\pi$
$151$	19213.1	0.842643	0.421322	+	0.906911i	$0.361566\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5756.65i	0.239611i
$156$	0	0
$157$	−18752.7	−0.760788	−0.380394	−	0.924824i	$-0.624212\pi$
$157$	−18752.7	−0.760788	−0.380394	+	0.924824i	$0.624212\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3229.97i	− 0.124608i
$162$	0	0
$163$	−35433.3	−1.33363	−0.666816	−	0.745222i	$-0.732343\pi$
$163$	−35433.3	−1.33363	−0.666816	+	0.745222i	$0.732343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9710.35i	0.348179i	0.984730	+	0.174089i	$0.0556981\pi$
$167$	9710.35i	0.348179i	−0.984730	+	0.174089i	$0.944302\pi$
$168$	0	0
$169$	−28186.6	−0.986890
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3029.12i	0.101210i	0.998719	+	0.0506051i	$0.0161150\pi$
$173$	3029.12i	0.101210i	−0.998719	+	0.0506051i	$0.983885\pi$
$174$	0	0
$175$	7359.29	0.240303
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 6740.21i	− 0.210362i	−0.994453	−	0.105181i	$-0.966458\pi$
$179$	− 6740.21i	− 0.210362i	0.994453	−	0.105181i	$-0.0335422\pi$
$180$	0	0
$181$	−20114.9	−0.613990	−0.306995	−	0.951711i	$-0.599323\pi$
$181$	−20114.9	−0.613990	−0.306995	+	0.951711i	$0.599323\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	42460.5i	1.24063i
$186$	0	0
$187$	−24052.5	−0.687823
$188$	0	0
$189$	0	0
$190$	0	0
$191$	53663.0i	1.47098i	0.677533	+	0.735492i	$0.263049\pi$
$191$	53663.0i	1.47098i	−0.677533	+	0.735492i	$0.736951\pi$
$192$	0	0
$193$	−56279.7	−1.51090	−0.755452	−	0.655204i	$-0.772583\pi$
$193$	−56279.7	−1.51090	−0.755452	+	0.655204i	$0.772583\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	73594.0i	1.89631i	0.317803	+	0.948157i	$0.397055\pi$
$197$	73594.0i	1.89631i	−0.317803	+	0.948157i	$0.602945\pi$
$198$	0	0
$199$	41991.5	1.06036	0.530182	−	0.847884i	$-0.322124\pi$
$199$	41991.5	1.06036	0.530182	+	0.847884i	$0.322124\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 18865.2i	− 0.457794i
$204$	0	0
$205$	−56952.6	−1.35521
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25151.4i	0.575798i
$210$	0	0
$211$	−71125.2	−1.59756	−0.798782	−	0.601620i	$-0.794522\pi$
$211$	−71125.2	−1.59756	−0.798782	+	0.601620i	$0.794522\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	29497.3i	0.638124i
$216$	0	0
$217$	−2506.90	−0.0532374
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3215.30i	0.0658320i
$222$	0	0
$223$	10162.0	0.204347	0.102173	−	0.994767i	$-0.467420\pi$
$223$	10162.0	0.204347	0.102173	+	0.994767i	$0.467420\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 32258.8i	− 0.626032i	−0.949748	−	0.313016i	$-0.898661\pi$
$227$	− 32258.8i	− 0.626032i	0.949748	−	0.313016i	$-0.101339\pi$
$228$	0	0
$229$	−27216.5	−0.518993	−0.259497	−	0.965744i	$-0.583557\pi$
$229$	−27216.5	−0.518993	−0.259497	+	0.965744i	$0.583557\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 50535.6i	− 0.930863i	−0.885084	−	0.465431i	$-0.845899\pi$
$233$	− 50535.6i	− 0.930863i	0.885084	−	0.465431i	$-0.154101\pi$
$234$	0	0
$235$	44488.2	0.805580
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 19945.9i	− 0.349186i	−0.984641	−	0.174593i	$-0.944139\pi$
$239$	− 19945.9i	− 0.349186i	0.984641	−	0.174593i	$-0.0558610\pi$
$240$	0	0
$241$	−24031.9	−0.413765	−0.206882	−	0.978366i	$-0.566332\pi$
$241$	−24031.9	−0.413765	−0.206882	+	0.978366i	$0.566332\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 73458.4i	− 1.22380i
$246$	0	0
$247$	3362.21	0.0551101
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 109770.i	− 1.74235i	−0.490974	−	0.871174i	$-0.663359\pi$
$251$	− 109770.i	− 1.74235i	0.490974	−	0.871174i	$-0.336641\pi$
$252$	0	0
$253$	−31965.1	−0.499385
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 40324.2i	− 0.610520i	−0.952269	−	0.305260i	$-0.901257\pi$
$257$	− 40324.2i	− 0.610520i	0.952269	−	0.305260i	$-0.0987433\pi$
$258$	0	0
$259$	−18490.6	−0.275646
$260$	0	0
$261$	0	0
$262$	0	0
$263$	131526.i	1.90152i	0.309929	+	0.950760i	$0.399695\pi$
$263$	131526.i	1.90152i	−0.309929	+	0.950760i	$0.600305\pi$
$264$	0	0
$265$	−56106.0	−0.798946
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30889.1i	0.426876i	0.976957	+	0.213438i	$0.0684661\pi$
$269$	30889.1i	0.426876i	−0.976957	+	0.213438i	$0.931534\pi$
$270$	0	0
$271$	10337.2	0.140755	0.0703774	−	0.997520i	$-0.477580\pi$
$271$	10337.2	0.140755	0.0703774	+	0.997520i	$0.477580\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 72830.5i	− 0.963049i
$276$	0	0
$277$	−105073.	−1.36940	−0.684701	−	0.728825i	$-0.740067\pi$
$277$	−105073.	−1.36940	−0.684701	+	0.728825i	$0.740067\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3501.73i	− 0.0443476i	−0.999754	−	0.0221738i	$-0.992941\pi$
$281$	− 3501.73i	− 0.0443476i	0.999754	−	0.0221738i	$-0.00705872\pi$
$282$	0	0
$283$	−20809.2	−0.259826	−0.129913	−	0.991525i	$-0.541470\pi$
$283$	−20809.2	−0.259826	−0.129913	+	0.991525i	$0.541470\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24801.6i	− 0.301104i
$288$	0	0
$289$	55910.8	0.669422
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 84883.3i	− 0.988752i	−0.869248	−	0.494376i	$-0.835397\pi$
$293$	− 84883.3i	− 0.988752i	0.869248	−	0.494376i	$-0.164603\pi$
$294$	0	0
$295$	58228.4	0.669100
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4273.05i	0.0477965i
$300$	0	0
$301$	−12845.4	−0.141780
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 134412.i	− 1.44490i
$306$	0	0
$307$	28347.7	0.300774	0.150387	−	0.988627i	$-0.451948\pi$
$307$	28347.7	0.300774	0.150387	+	0.988627i	$0.451948\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 20485.9i	− 0.211804i	−0.994377	−	0.105902i	$-0.966227\pi$
$311$	− 20485.9i	− 0.211804i	0.994377	−	0.105902i	$-0.0337730\pi$
$312$	0	0
$313$	−18831.2	−0.192216	−0.0961078	−	0.995371i	$-0.530639\pi$
$313$	−18831.2	−0.192216	−0.0961078	+	0.995371i	$0.530639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 69100.0i	− 0.687638i	−0.939036	−	0.343819i	$-0.888279\pi$
$317$	− 69100.0i	− 0.687638i	0.939036	−	0.343819i	$-0.111721\pi$
$318$	0	0
$319$	−186698.	−1.83467
$320$	0	0
$321$	0	0
$322$	0	0
$323$	28871.7i	0.276737i
$324$	0	0
$325$	−9735.89	−0.0921741
$326$	0	0
$327$	0	0
$328$	0	0
$329$	19373.6i	0.178986i
$330$	0	0
$331$	185395.	1.69216	0.846082	−	0.533054i	$-0.178956\pi$
$331$	185395.	1.69216	0.846082	+	0.533054i	$0.178956\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	268577.i	2.39320i
$336$	0	0
$337$	57550.1	0.506741	0.253370	−	0.967369i	$-0.418461\pi$
$337$	57550.1	0.506741	0.253370	+	0.967369i	$0.418461\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24809.3i	0.213356i
$342$	0	0
$343$	67108.4	0.570412
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 154889.i	− 1.28636i	−0.765715	−	0.643180i	$-0.777615\pi$
$347$	− 154889.i	− 1.28636i	0.765715	−	0.643180i	$-0.222385\pi$
$348$	0	0
$349$	179372.	1.47267	0.736334	−	0.676618i	$-0.236555\pi$
$349$	179372.	1.47267	0.736334	+	0.676618i	$0.236555\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	107698.i	0.864286i	0.901805	+	0.432143i	$0.142242\pi$
$353$	107698.i	0.864286i	−0.901805	+	0.432143i	$0.857758\pi$
$354$	0	0
$355$	−94036.1	−0.746170
$356$	0	0
$357$	0	0
$358$	0	0
$359$	145895.i	1.13201i	0.824402	+	0.566005i	$0.191512\pi$
$359$	145895.i	1.13201i	−0.824402	+	0.566005i	$0.808488\pi$
$360$	0	0
$361$	−100130.	−0.768335
$362$	0	0
$363$	0	0
$364$	0	0
$365$	40927.2i	0.307204i
$366$	0	0
$367$	−167014.	−1.23999	−0.619997	−	0.784604i	$-0.712866\pi$
$367$	−167014.	−1.23999	−0.619997	+	0.784604i	$0.712866\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 24432.9i	− 0.177512i
$372$	0	0
$373$	92493.7	0.664806	0.332403	−	0.943138i	$-0.392141\pi$
$373$	92493.7	0.664806	0.332403	+	0.943138i	$0.392141\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24957.5i	0.175598i
$378$	0	0
$379$	223591.	1.55660	0.778299	−	0.627894i	$-0.216083\pi$
$379$	223591.	1.55660	0.778299	+	0.627894i	$0.216083\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24915.6i	0.169853i	0.996387	+	0.0849265i	$0.0270655\pi$
$383$	24915.6i	0.169853i	−0.996387	+	0.0849265i	$0.972934\pi$
$384$	0	0
$385$	71113.9	0.479770
$386$	0	0
$387$	0	0
$388$	0	0
$389$	156539.i	1.03448i	0.855839	+	0.517242i	$0.173041\pi$
$389$	156539.i	1.03448i	−0.855839	+	0.517242i	$0.826959\pi$
$390$	0	0
$391$	−36693.2	−0.240011
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 294946.i	− 1.89038i
$396$	0	0
$397$	3880.64	0.0246220	0.0123110	−	0.999924i	$-0.496081\pi$
$397$	3880.64	0.0246220	0.0123110	+	0.999924i	$0.496081\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	113022.i	0.702868i	0.936213	+	0.351434i	$0.114306\pi$
$401$	113022.i	0.702868i	−0.936213	+	0.351434i	$0.885694\pi$
$402$	0	0
$403$	3316.47	0.0204205
$404$	0	0
$405$	0	0
$406$	0	0
$407$	182991.i	1.10469i
$408$	0	0
$409$	−56305.9	−0.336594	−0.168297	−	0.985736i	$-0.553827\pi$
$409$	−56305.9	−0.336594	−0.168297	+	0.985736i	$0.553827\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	25357.2i	0.148662i
$414$	0	0
$415$	−198746.	−1.15399
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 302463.i	− 1.72284i	−0.507895	−	0.861419i	$-0.669576\pi$
$419$	− 302463.i	− 1.72284i	0.507895	−	0.861419i	$-0.330424\pi$
$420$	0	0
$421$	154089.	0.869374	0.434687	−	0.900582i	$-0.356859\pi$
$421$	154089.	0.869374	0.434687	+	0.900582i	$0.356859\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 83603.1i	− 0.462855i
$426$	0	0
$427$	58533.5	0.321032
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 198819.i	− 1.07030i	−0.844758	−	0.535148i	$-0.820256\pi$
$431$	− 198819.i	− 1.07030i	0.844758	−	0.535148i	$-0.179744\pi$
$432$	0	0
$433$	211411.	1.12759	0.563795	−	0.825915i	$-0.309341\pi$
$433$	211411.	1.12759	0.563795	+	0.825915i	$0.309341\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	38369.7i	0.200921i
$438$	0	0
$439$	46748.0	0.242568	0.121284	−	0.992618i	$-0.461299\pi$
$439$	46748.0	0.242568	0.121284	+	0.992618i	$0.461299\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	306327.i	1.56091i	0.625211	+	0.780456i	$0.285013\pi$
$443$	306327.i	1.56091i	−0.625211	+	0.780456i	$0.714987\pi$
$444$	0	0
$445$	429862.	2.17075
$446$	0	0
$447$	0	0
$448$	0	0
$449$	248443.i	1.23235i	0.787610	+	0.616175i	$0.211318\pi$
$449$	248443.i	1.23235i	−0.787610	+	0.616175i	$0.788682\pi$
$450$	0	0
$451$	−245447.	−1.20671
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 9506.41i	− 0.0459191i
$456$	0	0
$457$	311234.	1.49023	0.745117	−	0.666934i	$-0.232394\pi$
$457$	311234.	1.49023	0.745117	+	0.666934i	$0.232394\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	74685.1i	0.351424i	0.984442	+	0.175712i	$0.0562228\pi$
$461$	74685.1i	0.351424i	−0.984442	+	0.175712i	$0.943777\pi$
$462$	0	0
$463$	164457.	0.767169	0.383584	−	0.923506i	$-0.374690\pi$
$463$	164457.	0.767169	0.383584	+	0.923506i	$0.374690\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 170365.i	− 0.781170i	−0.920567	−	0.390585i	$-0.872273\pi$
$467$	− 170365.i	− 0.781170i	0.920567	−	0.390585i	$-0.127727\pi$
$468$	0	0
$469$	−116960.	−0.531728
$470$	0	0
$471$	0	0
$472$	0	0
$473$	127124.i	0.568203i
$474$	0	0
$475$	−87423.0	−0.387470
$476$	0	0
$477$	0	0
$478$	0	0
$479$	325738.i	1.41970i	0.704351	+	0.709852i	$0.251238\pi$
$479$	325738.i	1.41970i	−0.704351	+	0.709852i	$0.748762\pi$
$480$	0	0
$481$	24461.9	0.105731
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 213500.i	− 0.907641i
$486$	0	0
$487$	134748.	0.568151	0.284076	−	0.958802i	$-0.408313\pi$
$487$	134748.	0.568151	0.284076	+	0.958802i	$0.408313\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 359856.i	− 1.49267i	−0.665568	−	0.746337i	$-0.731811\pi$
$491$	− 359856.i	− 1.49267i	0.665568	−	0.746337i	$-0.268189\pi$
$492$	0	0
$493$	−214313.	−0.881770
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 40950.7i	− 0.165786i
$498$	0	0
$499$	342749.	1.37650	0.688248	−	0.725475i	$-0.258380\pi$
$499$	342749.	1.37650	0.688248	+	0.725475i	$0.258380\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	373142.i	1.47482i	0.675448	+	0.737408i	$0.263950\pi$
$503$	373142.i	1.47482i	−0.675448	+	0.737408i	$0.736050\pi$
$504$	0	0
$505$	429339.	1.68352
$506$	0	0
$507$	0	0
$508$	0	0
$509$	118958.i	0.459156i	0.973290	+	0.229578i	$0.0737345\pi$
$509$	118958.i	0.459156i	−0.973290	+	0.229578i	$0.926265\pi$
$510$	0	0
$511$	−17822.9	−0.0682554
$512$	0	0
$513$	0	0
$514$	0	0
$515$	374551.i	1.41220i
$516$	0	0
$517$	191729.	0.717311
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37178.7i	0.136968i	0.997652	+	0.0684840i	$0.0218162\pi$
$521$	37178.7i	0.136968i	−0.997652	+	0.0684840i	$0.978184\pi$
$522$	0	0
$523$	−337828.	−1.23507	−0.617536	−	0.786542i	$-0.711869\pi$
$523$	−337828.	−1.23507	−0.617536	+	0.786542i	$0.711869\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28478.9i	0.102542i
$528$	0	0
$529$	231077.	0.825743
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32811.0i	0.115495i
$534$	0	0
$535$	274082.	0.957576
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 316582.i	− 1.08970i
$540$	0	0
$541$	359312.	1.22766	0.613830	−	0.789439i	$-0.289628\pi$
$541$	359312.	1.22766	0.613830	+	0.789439i	$0.289628\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 715568.i	− 2.40912i
$546$	0	0
$547$	−102745.	−0.343388	−0.171694	−	0.985150i	$-0.554924\pi$
$547$	−102745.	−0.343388	−0.171694	+	0.985150i	$0.554924\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	224105.i	0.738157i
$552$	0	0
$553$	128443.	0.420009
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 510008.i	− 1.64387i	−0.569583	−	0.821934i	$-0.692895\pi$
$557$	− 510008.i	− 1.64387i	0.569583	−	0.821934i	$-0.307105\pi$
$558$	0	0
$559$	16993.7	0.0543832
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 184068.i	− 0.580712i	−0.956919	−	0.290356i	$-0.906226\pi$
$563$	− 184068.i	− 0.580712i	0.956919	−	0.290356i	$-0.0937737\pi$
$564$	0	0
$565$	166913.	0.522868
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 372582.i	− 1.15079i	−0.817875	−	0.575397i	$-0.804848\pi$
$569$	− 372582.i	− 1.15079i	0.817875	−	0.575397i	$-0.195152\pi$
$570$	0	0
$571$	48251.5	0.147992	0.0739961	−	0.997259i	$-0.476425\pi$
$571$	48251.5	0.147992	0.0739961	+	0.997259i	$0.476425\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 111106.i	− 0.336049i
$576$	0	0
$577$	321409.	0.965398	0.482699	−	0.875786i	$-0.339657\pi$
$577$	321409.	0.965398	0.482699	+	0.875786i	$0.339657\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 86549.8i	− 0.256397i
$582$	0	0
$583$	−241798.	−0.711403
$584$	0	0
$585$	0	0
$586$	0	0
$587$	185894.i	0.539496i	0.962931	+	0.269748i	$0.0869404\pi$
$587$	185894.i	0.539496i	−0.962931	+	0.269748i	$0.913060\pi$
$588$	0	0
$589$	29780.1	0.0858412
$590$	0	0
$591$	0	0
$592$	0	0
$593$	291755.i	0.829676i	0.909895	+	0.414838i	$0.136162\pi$
$593$	291755.i	0.829676i	−0.909895	+	0.414838i	$0.863838\pi$
$594$	0	0
$595$	81632.6	0.230584
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 256098.i	− 0.713761i	−0.934150	−	0.356881i	$-0.883840\pi$
$599$	− 256098.i	− 0.713761i	0.934150	−	0.356881i	$-0.116160\pi$
$600$	0	0
$601$	1472.07	0.00407549	0.00203775	−	0.999998i	$-0.499351\pi$
$601$	1472.07	0.00407549	0.00203775	+	0.999998i	$0.499351\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 212014.i	− 0.579233i
$606$	0	0
$607$	533495.	1.44795	0.723974	−	0.689827i	$-0.242313\pi$
$607$	533495.	1.44795	0.723974	+	0.689827i	$0.242313\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 25630.1i	− 0.0686543i
$612$	0	0
$613$	261329.	0.695451	0.347725	−	0.937596i	$-0.386954\pi$
$613$	261329.	0.695451	0.347725	+	0.937596i	$0.386954\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6677.95i	0.0175417i	0.999962	+	0.00877087i	$0.00279189\pi$
$617$	6677.95i	0.0175417i	−0.999962	+	0.00877087i	$0.997208\pi$
$618$	0	0
$619$	−467024.	−1.21887	−0.609436	−	0.792835i	$-0.708604\pi$
$619$	−467024.	−1.21887	−0.609436	+	0.792835i	$0.708604\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	187196.i	0.482302i
$624$	0	0
$625$	−451938.	−1.15696
$626$	0	0
$627$	0	0
$628$	0	0
$629$	210057.i	0.530929i
$630$	0	0
$631$	−355070.	−0.891775	−0.445887	−	0.895089i	$-0.647112\pi$
$631$	−355070.	−0.891775	−0.445887	+	0.895089i	$0.647112\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 576183.i	− 1.42894i
$636$	0	0
$637$	−42320.2	−0.104296
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 9656.48i	− 0.0235019i	−0.999931	−	0.0117509i	$-0.996259\pi$
$641$	− 9656.48i	− 0.0235019i	0.999931	−	0.0117509i	$-0.00374053\pi$
$642$	0	0
$643$	−357560.	−0.864823	−0.432411	−	0.901676i	$-0.642337\pi$
$643$	−357560.	−0.864823	−0.432411	+	0.901676i	$0.642337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 343333.i	− 0.820176i	−0.912046	−	0.410088i	$-0.865498\pi$
$647$	− 343333.i	− 0.820176i	0.912046	−	0.410088i	$-0.134502\pi$
$648$	0	0
$649$	250945.	0.595785
$650$	0	0
$651$	0	0
$652$	0	0
$653$	476404.i	1.11725i	0.829422	+	0.558623i	$0.188670\pi$
$653$	476404.i	1.11725i	−0.829422	+	0.558623i	$0.811330\pi$
$654$	0	0
$655$	711853.	1.65923
$656$	0	0
$657$	0	0
$658$	0	0
$659$	601236.i	1.38444i	0.721686	+	0.692220i	$0.243367\pi$
$659$	601236.i	1.38444i	−0.721686	+	0.692220i	$0.756633\pi$
$660$	0	0
$661$	679484.	1.55516	0.777582	−	0.628781i	$-0.216446\pi$
$661$	679484.	1.55516	0.777582	+	0.628781i	$0.216446\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 85362.4i	− 0.193029i
$666$	0	0
$667$	−284817.	−0.640197
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 579271.i	− 1.28658i
$672$	0	0
$673$	−75784.5	−0.167321	−0.0836605	−	0.996494i	$-0.526661\pi$
$673$	−75784.5	−0.167321	−0.0836605	+	0.996494i	$0.526661\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	612672.i	1.33675i	0.743824	+	0.668376i	$0.233010\pi$
$677$	612672.i	1.33675i	−0.743824	+	0.668376i	$0.766990\pi$
$678$	0	0
$679$	92974.6	0.201662
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 274139.i	− 0.587665i	−0.955857	−	0.293832i	$-0.905069\pi$
$683$	− 274139.i	− 0.587665i	0.955857	−	0.293832i	$-0.0949308\pi$
$684$	0	0
$685$	−595402.	−1.26891
$686$	0	0
$687$	0	0
$688$	0	0
$689$	32323.2i	0.0680889i
$690$	0	0
$691$	390853.	0.818573	0.409286	−	0.912406i	$-0.365778\pi$
$691$	390853.	0.818573	0.409286	+	0.912406i	$0.365778\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.08273e6i	2.24155i
$696$	0	0
$697$	−281752.	−0.579964
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 297715.i	− 0.605849i	−0.953015	−	0.302924i	$-0.902037\pi$
$701$	− 297715.i	− 0.605849i	0.953015	−	0.302924i	$-0.0979630\pi$
$702$	0	0
$703$	219655.	0.444458
$704$	0	0
$705$	0	0
$706$	0	0
$707$	186968.i	0.374049i
$708$	0	0
$709$	173670.	0.345487	0.172744	−	0.984967i	$-0.444737\pi$
$709$	173670.	0.345487	0.172744	+	0.984967i	$0.444737\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37847.7i	0.0744493i
$714$	0	0
$715$	−94079.3	−0.184027
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 456933.i	− 0.883882i	−0.897044	−	0.441941i	$-0.854290\pi$
$719$	− 456933.i	− 0.883882i	0.897044	−	0.441941i	$-0.145710\pi$
$720$	0	0
$721$	−163109.	−0.313766
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 648937.i	− 1.23460i
$726$	0	0
$727$	−607565.	−1.14954	−0.574770	−	0.818315i	$-0.694908\pi$
$727$	−607565.	−1.14954	−0.574770	+	0.818315i	$0.694908\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	145927.i	0.273087i
$732$	0	0
$733$	433466.	0.806765	0.403383	−	0.915031i	$-0.367834\pi$
$733$	433466.	0.806765	0.403383	+	0.915031i	$0.367834\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.15748e6i	2.13097i
$738$	0	0
$739$	−85028.7	−0.155696	−0.0778478	−	0.996965i	$-0.524805\pi$
$739$	−85028.7	−0.155696	−0.0778478	+	0.996965i	$0.524805\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	100283.i	0.181656i	0.995867	+	0.0908278i	$0.0289513\pi$
$743$	100283.i	0.181656i	−0.995867	+	0.0908278i	$0.971049\pi$
$744$	0	0
$745$	233516.	0.420731
$746$	0	0
$747$	0	0
$748$	0	0
$749$	119357.i	0.212757i
$750$	0	0
$751$	−833818.	−1.47840	−0.739199	−	0.673487i	$-0.764796\pi$
$751$	−833818.	−1.47840	−0.739199	+	0.673487i	$0.764796\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	645326.i	1.13210i
$756$	0	0
$757$	−864671.	−1.50889	−0.754447	−	0.656360i	$-0.772095\pi$
$757$	−864671.	−1.50889	−0.754447	+	0.656360i	$0.772095\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.02371e6i	1.76770i	0.467775	+	0.883848i	$0.345056\pi$
$761$	1.02371e6i	1.76770i	−0.467775	+	0.883848i	$0.654944\pi$
$762$	0	0
$763$	311614.	0.535265
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 33546.0i	− 0.0570230i
$768$	0	0
$769$	−572475.	−0.968064	−0.484032	−	0.875050i	$-0.660828\pi$
$769$	−572475.	−0.968064	−0.484032	+	0.875050i	$0.660828\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	479910.i	0.803158i	0.915824	+	0.401579i	$0.131538\pi$
$773$	479910.i	0.803158i	−0.915824	+	0.401579i	$0.868462\pi$
$774$	0	0
$775$	−86233.7	−0.143573
$776$	0	0
$777$	0	0
$778$	0	0
$779$	294625.i	0.485506i
$780$	0	0
$781$	−405265.	−0.664411
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 629861.i	− 1.02213i
$786$	0	0
$787$	1.03953e6	1.67837	0.839186	−	0.543844i	$-0.183032\pi$
$787$	1.03953e6	1.67837	0.839186	+	0.543844i	$0.183032\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	72686.8i	0.116172i
$792$	0	0
$793$	−77436.2	−0.123140
$794$	0	0
$795$	0	0
$796$	0	0
$797$	889741.i	1.40071i	0.713797	+	0.700353i	$0.246974\pi$
$797$	889741.i	1.40071i	−0.713797	+	0.700353i	$0.753026\pi$
$798$	0	0
$799$	220089.	0.344750
$800$	0	0
$801$	0	0
$802$	0	0
$803$	176383.i	0.273543i
$804$	0	0
$805$	108488.	0.167413
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 637100.i	− 0.973443i	−0.873557	−	0.486722i	$-0.838193\pi$
$809$	− 637100.i	− 0.973443i	0.873557	−	0.486722i	$-0.161807\pi$
$810$	0	0
$811$	1.29389e6	1.96723	0.983615	−	0.180282i	$-0.0577012\pi$
$811$	1.29389e6	1.96723	0.983615	+	0.180282i	$0.0577012\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.19013e6i	− 1.79175i
$816$	0	0
$817$	152594.	0.228609
$818$	0	0
$819$	0	0
$820$	0	0
$821$	415943.i	0.617089i	0.951210	+	0.308544i	$0.0998418\pi$
$821$	415943.i	0.617089i	−0.951210	+	0.308544i	$0.900158\pi$
$822$	0	0
$823$	−1.07471e6	−1.58669	−0.793344	−	0.608774i	$-0.791662\pi$
$823$	−1.07471e6	−1.58669	−0.793344	+	0.608774i	$0.791662\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23323.7i	− 0.0341025i	−0.999855	−	0.0170513i	$-0.994572\pi$
$827$	− 23323.7i	− 0.0341025i	0.999855	−	0.0170513i	$-0.00542785\pi$
$828$	0	0
$829$	−1.10602e6	−1.60937	−0.804683	−	0.593705i	$-0.797665\pi$
$829$	−1.10602e6	−1.60937	−0.804683	+	0.593705i	$0.797665\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 363408.i	− 0.523727i
$834$	0	0
$835$	−326149.	−0.467782
$836$	0	0
$837$	0	0
$838$	0	0
$839$	221608.i	0.314819i	0.987533	+	0.157409i	$0.0503143\pi$
$839$	221608.i	0.314819i	−0.987533	+	0.157409i	$0.949686\pi$
$840$	0	0
$841$	−956243.	−1.35200
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 946724.i	− 1.32590i
$846$	0	0
$847$	92327.4	0.128696
$848$	0	0
$849$	0	0
$850$	0	0
$851$	279161.i	0.385474i
$852$	0	0
$853$	−1.12832e6	−1.55072	−0.775362	−	0.631517i	$-0.782433\pi$
$853$	−1.12832e6	−1.55072	−0.775362	+	0.631517i	$0.782433\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	449136.i	0.611528i	0.952107	+	0.305764i	$0.0989118\pi$
$857$	449136.i	0.611528i	−0.952107	+	0.305764i	$0.901088\pi$
$858$	0	0
$859$	−1.24856e6	−1.69210	−0.846048	−	0.533107i	$-0.821024\pi$
$859$	−1.24856e6	−1.69210	−0.846048	+	0.533107i	$0.821024\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 826246.i	− 1.10940i	−0.832051	−	0.554699i	$-0.812833\pi$
$863$	− 826246.i	− 1.10940i	0.832051	−	0.554699i	$-0.187167\pi$
$864$	0	0
$865$	−101741.	−0.135977
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.27112e6i	− 1.68324i
$870$	0	0
$871$	154730.	0.203957
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 59867.8i	− 0.0781947i
$876$	0	0
$877$	−78630.5	−0.102233	−0.0511166	−	0.998693i	$-0.516278\pi$
$877$	−78630.5	−0.102233	−0.0511166	+	0.998693i	$0.516278\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 186139.i	− 0.239820i	−0.992785	−	0.119910i	$-0.961739\pi$
$881$	− 186139.i	− 0.239820i	0.992785	−	0.119910i	$-0.0382606\pi$
$882$	0	0
$883$	1.39533e6	1.78960	0.894802	−	0.446464i	$-0.147317\pi$
$883$	1.39533e6	1.78960	0.894802	+	0.446464i	$0.147317\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 874855.i	− 1.11196i	−0.831196	−	0.555980i	$-0.812343\pi$
$887$	− 874855.i	− 1.11196i	0.831196	−	0.555980i	$-0.187657\pi$
$888$	0	0
$889$	250915.	0.317485
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 230144.i	− 0.288601i
$894$	0	0
$895$	226389.	0.282624
$896$	0	0
$897$	0	0
$898$	0	0
$899$	221056.i	0.273517i
$900$	0	0
$901$	−277563.	−0.341911
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 675616.i	− 0.824902i
$906$	0	0
$907$	120934.	0.147006	0.0735028	−	0.997295i	$-0.476582\pi$
$907$	120934.	0.147006	0.0735028	+	0.997295i	$0.476582\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 287471.i	− 0.346383i	−0.984888	−	0.173192i	$-0.944592\pi$
$911$	− 287471.i	− 0.346383i	0.984888	−	0.173192i	$-0.0554080\pi$
$912$	0	0
$913$	−856532.	−1.02755
$914$	0	0
$915$	0	0
$916$	0	0
$917$	309996.i	0.368653i
$918$	0	0
$919$	539676.	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$919$	539676.	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	54175.2i	0.0635912i
$924$	0	0
$925$	−636051.	−0.743375
$926$	0	0
$927$	0	0
$928$	0	0
$929$	658350.i	0.762826i	0.924405	+	0.381413i	$0.124562\pi$
$929$	658350.i	0.762826i	−0.924405	+	0.381413i	$0.875438\pi$
$930$	0	0
$931$	−380012.	−0.438428
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 807870.i	− 0.924098i
$936$	0	0
$937$	255274.	0.290755	0.145378	−	0.989376i	$-0.453560\pi$
$937$	255274.	0.290755	0.145378	+	0.989376i	$0.453560\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	654533.i	0.739183i	0.929194	+	0.369592i	$0.120502\pi$
$941$	654533.i	0.739183i	−0.929194	+	0.369592i	$0.879498\pi$
$942$	0	0
$943$	−374441.	−0.421075
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.59897e6i	1.78295i	0.453066	+	0.891477i	$0.350330\pi$
$947$	1.59897e6i	1.78295i	−0.453066	+	0.891477i	$0.649670\pi$
$948$	0	0
$949$	23578.6	0.0261810
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.25008e6i	− 1.37642i	−0.725510	−	0.688212i	$-0.758396\pi$
$953$	− 1.25008e6i	− 1.37642i	0.725510	−	0.688212i	$-0.241604\pi$
$954$	0	0
$955$	−1.80242e6	−1.97628
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 259285.i	− 0.281929i
$960$	0	0
$961$	−894146.	−0.968192
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.89031e6i	− 2.02992i
$966$	0	0
$967$	−1.70048e6	−1.81852	−0.909260	−	0.416229i	$-0.863352\pi$
$967$	−1.70048e6	−1.81852	−0.909260	+	0.416229i	$0.863352\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	761568.i	0.807737i	0.914817	+	0.403869i	$0.132335\pi$
$971$	761568.i	0.807737i	−0.914817	+	0.403869i	$0.867665\pi$
$972$	0	0
$973$	−471503.	−0.498034
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 905620.i	− 0.948761i	−0.880320	−	0.474380i	$-0.842672\pi$
$977$	− 905620.i	− 0.948761i	0.880320	−	0.474380i	$-0.157328\pi$
$978$	0	0
$979$	1.85256e6	1.93289
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 291784.i	− 0.301963i	−0.988537	−	0.150982i	$-0.951757\pi$
$983$	− 291784.i	− 0.301963i	0.988537	−	0.150982i	$-0.0482434\pi$
$984$	0	0
$985$	−2.47186e6	−2.54772
$986$	0	0
$987$	0	0
$988$	0	0
$989$	193933.i	0.198271i
$990$	0	0
$991$	120352.	0.122547	0.0612737	−	0.998121i	$-0.480484\pi$
$991$	120352.	0.122547	0.0612737	+	0.998121i	$0.480484\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.41040e6i	1.42461i
$996$	0	0
$997$	−1.33758e6	−1.34564	−0.672819	−	0.739807i	$-0.734917\pi$
$997$	−1.33758e6	−1.34564	−0.672819	+	0.739807i	$0.734917\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.5.e.g.161.15	yes	16
3.2	odd	2	inner	864.5.e.g.161.1	✓	16
4.3	odd	2	inner	864.5.e.g.161.16	yes	16
12.11	even	2	inner	864.5.e.g.161.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.5.e.g.161.1	✓	16	3.2	odd	2	inner
864.5.e.g.161.2	yes	16	12.11	even	2	inner
864.5.e.g.161.15	yes	16	1.1	even	1	trivial
864.5.e.g.161.16	yes	16	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	864.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+33.5878i q^{5} -14.6267 q^{7} +144.752i q^{11} +19.3503 q^{13} +166.163i q^{17} +173.755 q^{19} +220.826i q^{23} -503.139 q^{25} +1289.78i q^{29} +171.391 q^{31} -491.280i q^{35} +1264.16 q^{37} +1695.63i q^{41} +878.215 q^{43} -1324.53i q^{47} -2187.06 q^{49} +1670.43i q^{53} -4861.91 q^{55} -1733.62i q^{59} -4001.81 q^{61} +649.933i q^{65} +7996.28 q^{67} +2799.71i q^{71} +1218.51 q^{73} -2117.26i q^{77} -8781.35 q^{79} +5917.23i q^{83} -5581.05 q^{85} -12798.2i q^{89} -283.032 q^{91} +5836.05i q^{95} -6356.47 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{13} + 704 q^{25} - 2624 q^{37} + 1728 q^{49} + 3264 q^{61} - 5424 q^{73} + 4704 q^{85} - 80 q^{97}+O(q^{100}) \) 16 * q - 32 * q^13 + 704 * q^25 - 2624 * q^37 + 1728 * q^49 + 3264 * q^61 - 5424 * q^73 + 4704 * q^85 - 80 * q^97

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