LMFDB - Embedded newform 864.5.e.g.161.14

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			161.14
Root			$-1.80390 - 1.84121i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.5.e.g.161.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+26.0821i q^{5} +48.1712 q^{7} +40.7201i q^{11} +248.819 q^{13} -239.132i q^{17} +416.251 q^{19} -630.391i q^{23} -55.2781 q^{25} -1245.41i q^{29} -365.546 q^{31} +1256.41i q^{35} -911.684 q^{37} -427.438i q^{41} +2576.66 q^{43} +983.883i q^{47} -80.5328 q^{49} -4185.09i q^{53} -1062.07 q^{55} -3365.25i q^{59} +4341.68 q^{61} +6489.74i q^{65} -6089.47 q^{67} +899.338i q^{71} +7731.62 q^{73} +1961.54i q^{77} -8521.08 q^{79} -1878.93i q^{83} +6237.06 q^{85} -6120.92i q^{89} +11985.9 q^{91} +10856.7i q^{95} +1311.39 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	26.0821i	1.04329i	0.853164	+	0.521643i	$0.174681\pi$
$5$	26.0821i	1.04329i	−0.853164	+	0.521643i	$0.825319\pi$
$6$	0	0
$7$	48.1712	0.983086	0.491543	−	0.870853i	$-0.336433\pi$
$7$	48.1712	0.983086	0.491543	+	0.870853i	$0.336433\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	40.7201i	0.336530i	0.985742	+	0.168265i	$0.0538164\pi$
$11$	40.7201i	0.336530i	−0.985742	+	0.168265i	$0.946184\pi$
$12$	0	0
$13$	248.819	1.47230	0.736152	−	0.676816i	$-0.236641\pi$
$13$	248.819	1.47230	0.736152	+	0.676816i	$0.236641\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 239.132i	− 0.827445i	−0.910403	−	0.413722i	$-0.864228\pi$
$17$	− 239.132i	− 0.827445i	0.910403	−	0.413722i	$-0.135772\pi$
$18$	0	0
$19$	416.251	1.15305	0.576524	−	0.817080i	$-0.304409\pi$
$19$	416.251	1.15305	0.576524	+	0.817080i	$0.304409\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 630.391i	− 1.19166i	−0.803109	−	0.595832i	$-0.796822\pi$
$23$	− 630.391i	− 1.19166i	0.803109	−	0.595832i	$-0.203178\pi$
$24$	0	0
$25$	−55.2781	−0.0884449
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1245.41i	− 1.48086i	−0.672132	−	0.740431i	$-0.734621\pi$
$29$	− 1245.41i	− 1.48086i	0.672132	−	0.740431i	$-0.265379\pi$
$30$	0	0
$31$	−365.546	−0.380380	−0.190190	−	0.981747i	$-0.560910\pi$
$31$	−365.546	−0.380380	−0.190190	+	0.981747i	$0.560910\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1256.41i	1.02564i
$36$	0	0
$37$	−911.684	−0.665949	−0.332974	−	0.942936i	$-0.608052\pi$
$37$	−911.684	−0.665949	−0.332974	+	0.942936i	$0.608052\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 427.438i	− 0.254276i	−0.991885	−	0.127138i	$-0.959421\pi$
$41$	− 427.438i	− 0.254276i	0.991885	−	0.127138i	$-0.0405791\pi$
$42$	0	0
$43$	2576.66	1.39354	0.696772	−	0.717293i	$-0.254619\pi$
$43$	2576.66	1.39354	0.696772	+	0.717293i	$0.254619\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	983.883i	0.445398i	0.974887	+	0.222699i	$0.0714867\pi$
$47$	983.883i	0.445398i	−0.974887	+	0.222699i	$0.928513\pi$
$48$	0	0
$49$	−80.5328	−0.0335413
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4185.09i	− 1.48989i	−0.667127	−	0.744944i	$-0.732476\pi$
$53$	− 4185.09i	− 1.48989i	0.667127	−	0.744944i	$-0.267524\pi$
$54$	0	0
$55$	−1062.07	−0.351097
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3365.25i	− 0.966749i	−0.875414	−	0.483374i	$-0.839411\pi$
$59$	− 3365.25i	− 0.966749i	0.875414	−	0.483374i	$-0.160589\pi$
$60$	0	0
$61$	4341.68	1.16681	0.583403	−	0.812183i	$-0.301721\pi$
$61$	4341.68	1.16681	0.583403	+	0.812183i	$0.301721\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6489.74i	1.53603i
$66$	0	0
$67$	−6089.47	−1.35653	−0.678266	−	0.734817i	$-0.737268\pi$
$67$	−6089.47	−1.35653	−0.678266	+	0.734817i	$0.737268\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	899.338i	0.178405i	0.996014	+	0.0892023i	$0.0284318\pi$
$71$	899.338i	0.178405i	−0.996014	+	0.0892023i	$0.971568\pi$
$72$	0	0
$73$	7731.62	1.45086	0.725428	−	0.688298i	$-0.241642\pi$
$73$	7731.62	1.45086	0.725428	+	0.688298i	$0.241642\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1961.54i	0.330838i
$78$	0	0
$79$	−8521.08	−1.36534	−0.682669	−	0.730727i	$-0.739181\pi$
$79$	−8521.08	−1.36534	−0.682669	+	0.730727i	$0.739181\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1878.93i	− 0.272744i	−0.990658	−	0.136372i	$-0.956456\pi$
$83$	− 1878.93i	− 0.272744i	0.990658	−	0.136372i	$-0.0435442\pi$
$84$	0	0
$85$	6237.06	0.863261
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6120.92i	− 0.772745i	−0.922343	−	0.386373i	$-0.873728\pi$
$89$	− 6120.92i	− 0.772745i	0.922343	−	0.386373i	$-0.126272\pi$
$90$	0	0
$91$	11985.9	1.44740
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10856.7i	1.20296i
$96$	0	0
$97$	1311.39	0.139376	0.0696881	−	0.997569i	$-0.477800\pi$
$97$	1311.39	0.139376	0.0696881	+	0.997569i	$0.477800\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11536.8i	1.13095i	0.824766	+	0.565475i	$0.191307\pi$
$101$	11536.8i	1.13095i	−0.824766	+	0.565475i	$0.808693\pi$
$102$	0	0
$103$	−9606.38	−0.905493	−0.452747	−	0.891639i	$-0.649556\pi$
$103$	−9606.38	−0.905493	−0.452747	+	0.891639i	$0.649556\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10583.3i	− 0.924387i	−0.886779	−	0.462194i	$-0.847063\pi$
$107$	− 10583.3i	− 0.924387i	0.886779	−	0.462194i	$-0.152937\pi$
$108$	0	0
$109$	−1154.69	−0.0971882	−0.0485941	−	0.998819i	$-0.515474\pi$
$109$	−1154.69	−0.0971882	−0.0485941	+	0.998819i	$0.515474\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11484.3i	0.899392i	0.893182	+	0.449696i	$0.148468\pi$
$113$	11484.3i	0.899392i	−0.893182	+	0.449696i	$0.851532\pi$
$114$	0	0
$115$	16441.9	1.24325
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 11519.3i	− 0.813450i
$120$	0	0
$121$	12982.9	0.886748
$122$	0	0
$123$	0	0
$124$	0	0
$125$	14859.6i	0.951012i
$126$	0	0
$127$	18047.0	1.11891	0.559457	−	0.828860i	$-0.311010\pi$
$127$	18047.0	1.11891	0.559457	+	0.828860i	$0.311010\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17139.8i	0.998764i	0.866382	+	0.499382i	$0.166440\pi$
$131$	17139.8i	0.998764i	−0.866382	+	0.499382i	$0.833560\pi$
$132$	0	0
$133$	20051.3	1.13355
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5308.06i	− 0.282810i	−0.989952	−	0.141405i	$-0.954838\pi$
$137$	− 5308.06i	− 0.282810i	0.989952	−	0.141405i	$-0.0451620\pi$
$138$	0	0
$139$	−12796.4	−0.662307	−0.331153	−	0.943577i	$-0.607438\pi$
$139$	−12796.4	−0.662307	−0.331153	+	0.943577i	$0.607438\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10132.0i	0.495474i
$144$	0	0
$145$	32482.8	1.54496
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3105.45i	− 0.139879i	−0.997551	−	0.0699395i	$-0.977719\pi$
$149$	− 3105.45i	− 0.139879i	0.997551	−	0.0699395i	$-0.0222806\pi$
$150$	0	0
$151$	16915.4	0.741873	0.370937	−	0.928658i	$-0.379037\pi$
$151$	16915.4	0.741873	0.370937	+	0.928658i	$0.379037\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9534.21i	− 0.396845i
$156$	0	0
$157$	34031.7	1.38065	0.690327	−	0.723497i	$-0.257467\pi$
$157$	34031.7	1.38065	0.690327	+	0.723497i	$0.257467\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 30366.7i	− 1.17151i
$162$	0	0
$163$	−29232.3	−1.10024	−0.550121	−	0.835085i	$-0.685418\pi$
$163$	−29232.3	−1.10024	−0.550121	+	0.835085i	$0.685418\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3995.34i	0.143259i	0.997431	+	0.0716294i	$0.0228199\pi$
$167$	3995.34i	0.143259i	−0.997431	+	0.0716294i	$0.977180\pi$
$168$	0	0
$169$	33350.1	1.16768
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 56731.5i	− 1.89554i	−0.318959	−	0.947769i	$-0.603333\pi$
$173$	− 56731.5i	− 1.89554i	0.318959	−	0.947769i	$-0.396667\pi$
$174$	0	0
$175$	−2662.81	−0.0869490
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9534.66i	− 0.297577i	−0.988869	−	0.148788i	$-0.952463\pi$
$179$	− 9534.66i	− 0.297577i	0.988869	−	0.148788i	$-0.0475373\pi$
$180$	0	0
$181$	64512.2	1.96918	0.984588	−	0.174890i	$-0.0559571\pi$
$181$	64512.2	1.96918	0.984588	+	0.174890i	$0.0559571\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 23778.7i	− 0.694775i
$186$	0	0
$187$	9737.46	0.278460
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14948.1i	0.409749i	0.978788	+	0.204875i	$0.0656787\pi$
$191$	14948.1i	0.409749i	−0.978788	+	0.204875i	$0.934321\pi$
$192$	0	0
$193$	16198.3	0.434864	0.217432	−	0.976075i	$-0.430232\pi$
$193$	16198.3	0.434864	0.217432	+	0.976075i	$0.430232\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	66093.1i	1.70303i	0.524327	+	0.851517i	$0.324317\pi$
$197$	66093.1i	1.70303i	−0.524327	+	0.851517i	$0.675683\pi$
$198$	0	0
$199$	34440.5	0.869688	0.434844	−	0.900506i	$-0.356803\pi$
$199$	34440.5	0.869688	0.434844	+	0.900506i	$0.356803\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 59992.7i	− 1.45582i
$204$	0	0
$205$	11148.5	0.265282
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16949.8i	0.388035i
$210$	0	0
$211$	−52835.8	−1.18676	−0.593381	−	0.804922i	$-0.702207\pi$
$211$	−52835.8	−1.18676	−0.593381	+	0.804922i	$0.702207\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	67204.9i	1.45386i
$216$	0	0
$217$	−17608.8	−0.373947
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 59500.6i	− 1.21825i
$222$	0	0
$223$	−74220.3	−1.49249	−0.746247	−	0.665669i	$-0.768146\pi$
$223$	−74220.3	−1.49249	−0.746247	+	0.665669i	$0.768146\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 97835.5i	− 1.89865i	−0.314297	−	0.949325i	$-0.601769\pi$
$227$	− 97835.5i	− 1.89865i	0.314297	−	0.949325i	$-0.398231\pi$
$228$	0	0
$229$	500.763	0.00954907	0.00477453	−	0.999989i	$-0.498480\pi$
$229$	500.763	0.00954907	0.00477453	+	0.999989i	$0.498480\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21170.2i	− 0.389953i	−0.980808	−	0.194976i	$-0.937537\pi$
$233$	− 21170.2i	− 0.389953i	0.980808	−	0.194976i	$-0.0624630\pi$
$234$	0	0
$235$	−25661.8	−0.464677
$236$	0	0
$237$	0	0
$238$	0	0
$239$	73002.8i	1.27804i	0.769191	+	0.639019i	$0.220659\pi$
$239$	73002.8i	1.27804i	−0.769191	+	0.639019i	$0.779341\pi$
$240$	0	0
$241$	91442.8	1.57440	0.787201	−	0.616697i	$-0.211529\pi$
$241$	91442.8	1.57440	0.787201	+	0.616697i	$0.211529\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2100.47i	− 0.0349932i
$246$	0	0
$247$	103571.	1.69764
$248$	0	0
$249$	0	0
$250$	0	0
$251$	34216.9i	0.543116i	0.962422	+	0.271558i	$0.0875389\pi$
$251$	34216.9i	0.543116i	−0.962422	+	0.271558i	$0.912461\pi$
$252$	0	0
$253$	25669.6	0.401031
$254$	0	0
$255$	0	0
$256$	0	0
$257$	64000.8i	0.968990i	0.874794	+	0.484495i	$0.160997\pi$
$257$	64000.8i	0.968990i	−0.874794	+	0.484495i	$0.839003\pi$
$258$	0	0
$259$	−43916.9	−0.654685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	45688.9i	0.660540i	0.943886	+	0.330270i	$0.107140\pi$
$263$	45688.9i	0.660540i	−0.943886	+	0.330270i	$0.892860\pi$
$264$	0	0
$265$	109156.	1.55438
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30659.8i	0.423706i	0.977302	+	0.211853i	$0.0679498\pi$
$269$	30659.8i	0.423706i	−0.977302	+	0.211853i	$0.932050\pi$
$270$	0	0
$271$	−78049.4	−1.06275	−0.531375	−	0.847137i	$-0.678324\pi$
$271$	−78049.4	−1.06275	−0.531375	+	0.847137i	$0.678324\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2250.93i	− 0.0297643i
$276$	0	0
$277$	14251.2	0.185735	0.0928673	−	0.995678i	$-0.470397\pi$
$277$	14251.2	0.185735	0.0928673	+	0.995678i	$0.470397\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	63538.5i	0.804682i	0.915490	+	0.402341i	$0.131803\pi$
$281$	63538.5i	0.804682i	−0.915490	+	0.402341i	$0.868197\pi$
$282$	0	0
$283$	146475.	1.82890	0.914448	−	0.404702i	$-0.132625\pi$
$283$	146475.	1.82890	0.914448	+	0.404702i	$0.132625\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 20590.2i	− 0.249975i
$288$	0	0
$289$	26337.1	0.315335
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3680.34i	− 0.0428700i	−0.999770	−	0.0214350i	$-0.993177\pi$
$293$	− 3680.34i	− 0.0428700i	0.999770	−	0.0214350i	$-0.00682349\pi$
$294$	0	0
$295$	87773.0	1.00860
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 156853.i	− 1.75449i
$300$	0	0
$301$	124121.	1.36997
$302$	0	0
$303$	0	0
$304$	0	0
$305$	113240.i	1.21731i
$306$	0	0
$307$	42873.8	0.454899	0.227449	−	0.973790i	$-0.426961\pi$
$307$	42873.8	0.454899	0.227449	+	0.973790i	$0.426961\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	143339.i	1.48198i	0.671515	+	0.740991i	$0.265644\pi$
$311$	143339.i	1.48198i	−0.671515	+	0.740991i	$0.734356\pi$
$312$	0	0
$313$	−32423.5	−0.330956	−0.165478	−	0.986213i	$-0.552917\pi$
$313$	−32423.5	−0.330956	−0.165478	+	0.986213i	$0.552917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 60745.1i	− 0.604495i	−0.953230	−	0.302247i	$-0.902263\pi$
$317$	− 60745.1i	− 0.604495i	0.953230	−	0.302247i	$-0.0977368\pi$
$318$	0	0
$319$	50713.0	0.498354
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 99538.7i	− 0.954084i
$324$	0	0
$325$	−13754.3	−0.130218
$326$	0	0
$327$	0	0
$328$	0	0
$329$	47394.9i	0.437864i
$330$	0	0
$331$	40918.4	0.373476	0.186738	−	0.982410i	$-0.440208\pi$
$331$	40918.4	0.373476	0.186738	+	0.982410i	$0.440208\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 158826.i	− 1.41525i
$336$	0	0
$337$	−211024.	−1.85811	−0.929057	−	0.369936i	$-0.879380\pi$
$337$	−211024.	−1.85811	−0.929057	+	0.369936i	$0.879380\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 14885.1i	− 0.128009i
$342$	0	0
$343$	−119538.	−1.01606
$344$	0	0
$345$	0	0
$346$	0	0
$347$	197212.i	1.63785i	0.573903	+	0.818924i	$0.305429\pi$
$347$	197212.i	1.63785i	−0.573903	+	0.818924i	$0.694571\pi$
$348$	0	0
$349$	−59585.4	−0.489203	−0.244602	−	0.969624i	$-0.578657\pi$
$349$	−59585.4	−0.489203	−0.244602	+	0.969624i	$0.578657\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 17784.6i	− 0.142723i	−0.997451	−	0.0713617i	$-0.977266\pi$
$353$	− 17784.6i	− 0.142723i	0.997451	−	0.0713617i	$-0.0227344\pi$
$354$	0	0
$355$	−23456.6	−0.186127
$356$	0	0
$357$	0	0
$358$	0	0
$359$	222312.i	1.72494i	0.506108	+	0.862470i	$0.331084\pi$
$359$	222312.i	1.72494i	−0.506108	+	0.862470i	$0.668916\pi$
$360$	0	0
$361$	42943.6	0.329521
$362$	0	0
$363$	0	0
$364$	0	0
$365$	201657.i	1.51366i
$366$	0	0
$367$	−131037.	−0.972888	−0.486444	−	0.873712i	$-0.661706\pi$
$367$	−131037.	−0.972888	−0.486444	+	0.873712i	$0.661706\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 201601.i	− 1.46469i
$372$	0	0
$373$	−139772.	−1.00462	−0.502311	−	0.864687i	$-0.667517\pi$
$373$	−139772.	−1.00462	−0.502311	+	0.864687i	$0.667517\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 309881.i	− 2.18028i
$378$	0	0
$379$	−20704.5	−0.144141	−0.0720703	−	0.997400i	$-0.522961\pi$
$379$	−20704.5	−0.144141	−0.0720703	+	0.997400i	$0.522961\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	188237.i	1.28324i	0.767024	+	0.641619i	$0.221737\pi$
$383$	188237.i	1.28324i	−0.767024	+	0.641619i	$0.778263\pi$
$384$	0	0
$385$	−51161.1	−0.345158
$386$	0	0
$387$	0	0
$388$	0	0
$389$	94390.4i	0.623776i	0.950119	+	0.311888i	$0.100961\pi$
$389$	94390.4i	0.623776i	−0.950119	+	0.311888i	$0.899039\pi$
$390$	0	0
$391$	−150746.	−0.986037
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 222248.i	− 1.42444i
$396$	0	0
$397$	−219505.	−1.39272	−0.696358	−	0.717694i	$-0.745198\pi$
$397$	−219505.	−1.39272	−0.696358	+	0.717694i	$0.745198\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 217971.i	− 1.35553i	−0.735277	−	0.677766i	$-0.762948\pi$
$401$	− 217971.i	− 1.35553i	0.735277	−	0.677766i	$-0.237052\pi$
$402$	0	0
$403$	−90954.9	−0.560036
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 37123.8i	− 0.224112i
$408$	0	0
$409$	−129752.	−0.775653	−0.387827	−	0.921732i	$-0.626774\pi$
$409$	−129752.	−0.775653	−0.387827	+	0.921732i	$0.626774\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 162108.i	− 0.950397i
$414$	0	0
$415$	49006.5	0.284550
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 144142.i	− 0.821036i	−0.911853	−	0.410518i	$-0.865348\pi$
$419$	− 144142.i	− 0.821036i	0.911853	−	0.410518i	$-0.134652\pi$
$420$	0	0
$421$	31049.1	0.175180	0.0875901	−	0.996157i	$-0.472083\pi$
$421$	31049.1	0.175180	0.0875901	+	0.996157i	$0.472083\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13218.7i	0.0731833i
$426$	0	0
$427$	209144.	1.14707
$428$	0	0
$429$	0	0
$430$	0	0
$431$	302768.i	1.62988i	0.579547	+	0.814939i	$0.303230\pi$
$431$	302768.i	1.62988i	−0.579547	+	0.814939i	$0.696770\pi$
$432$	0	0
$433$	169560.	0.904371	0.452185	−	0.891924i	$-0.350645\pi$
$433$	169560.	0.904371	0.452185	+	0.891924i	$0.350645\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 262400.i	− 1.37405i
$438$	0	0
$439$	127352.	0.660809	0.330405	−	0.943839i	$-0.392815\pi$
$439$	127352.	0.660809	0.330405	+	0.943839i	$0.392815\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 377780.i	− 1.92500i	−0.271277	−	0.962501i	$-0.587446\pi$
$443$	− 377780.i	− 1.92500i	0.271277	−	0.962501i	$-0.412554\pi$
$444$	0	0
$445$	159647.	0.806194
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 80369.1i	− 0.398654i	−0.979933	−	0.199327i	$-0.936124\pi$
$449$	− 80369.1i	− 0.398654i	0.979933	−	0.199327i	$-0.0638756\pi$
$450$	0	0
$451$	17405.3	0.0855714
$452$	0	0
$453$	0	0
$454$	0	0
$455$	312619.i	1.51005i
$456$	0	0
$457$	−287088.	−1.37462	−0.687311	−	0.726363i	$-0.741209\pi$
$457$	−287088.	−1.37462	−0.687311	+	0.726363i	$0.741209\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	135413.i	0.637175i	0.947893	+	0.318588i	$0.103208\pi$
$461$	135413.i	0.637175i	−0.947893	+	0.318588i	$0.896792\pi$
$462$	0	0
$463$	−368189.	−1.71755	−0.858774	−	0.512354i	$-0.828773\pi$
$463$	−368189.	−1.71755	−0.858774	+	0.512354i	$0.828773\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	311796.i	1.42967i	0.699291	+	0.714837i	$0.253499\pi$
$467$	311796.i	1.42967i	−0.699291	+	0.714837i	$0.746501\pi$
$468$	0	0
$469$	−293337.	−1.33359
$470$	0	0
$471$	0	0
$472$	0	0
$473$	104922.i	0.468969i
$474$	0	0
$475$	−23009.5	−0.101981
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 22235.0i	− 0.0969093i	−0.998825	−	0.0484547i	$-0.984570\pi$
$479$	− 22235.0i	− 0.0969093i	0.998825	−	0.0484547i	$-0.0154296\pi$
$480$	0	0
$481$	−226845.	−0.980479
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34203.9i	0.145409i
$486$	0	0
$487$	−61386.5	−0.258830	−0.129415	−	0.991591i	$-0.541310\pi$
$487$	−61386.5	−0.258830	−0.129415	+	0.991591i	$0.541310\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 256257.i	− 1.06295i	−0.847074	−	0.531475i	$-0.821638\pi$
$491$	− 256257.i	− 1.06295i	0.847074	−	0.531475i	$-0.178362\pi$
$492$	0	0
$493$	−297816.	−1.22533
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43322.2i	0.175387i
$498$	0	0
$499$	65633.4	0.263587	0.131794	−	0.991277i	$-0.457926\pi$
$499$	65633.4	0.263587	0.131794	+	0.991277i	$0.457926\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	82370.7i	0.325564i	0.986662	+	0.162782i	$0.0520468\pi$
$503$	82370.7i	0.325564i	−0.986662	+	0.162782i	$0.947953\pi$
$504$	0	0
$505$	−300905.	−1.17990
$506$	0	0
$507$	0	0
$508$	0	0
$509$	260352.i	1.00491i	0.864604	+	0.502454i	$0.167569\pi$
$509$	260352.i	1.00491i	−0.864604	+	0.502454i	$0.832431\pi$
$510$	0	0
$511$	372441.	1.42632
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 250555.i	− 0.944688i
$516$	0	0
$517$	−40063.8	−0.149890
$518$	0	0
$519$	0	0
$520$	0	0
$521$	402531.i	1.48294i	0.670985	+	0.741471i	$0.265871\pi$
$521$	402531.i	1.48294i	−0.670985	+	0.741471i	$0.734129\pi$
$522$	0	0
$523$	137088.	0.501183	0.250592	−	0.968093i	$-0.419375\pi$
$523$	137088.	0.501183	0.250592	+	0.968093i	$0.419375\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	87413.5i	0.314744i
$528$	0	0
$529$	−117551.	−0.420065
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 106355.i	− 0.374371i
$534$	0	0
$535$	276035.	0.964400
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3279.30i	− 0.0112877i
$540$	0	0
$541$	−434083.	−1.48313	−0.741563	−	0.670883i	$-0.765915\pi$
$541$	−434083.	−1.48313	−0.741563	+	0.670883i	$0.765915\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 30116.9i	− 0.101395i
$546$	0	0
$547$	−439181.	−1.46781	−0.733904	−	0.679254i	$-0.762304\pi$
$547$	−439181.	−1.46781	−0.733904	+	0.679254i	$0.762304\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 518401.i	− 1.70751i
$552$	0	0
$553$	−410471.	−1.34225
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 169278.i	− 0.545621i	−0.962068	−	0.272810i	$-0.912047\pi$
$557$	− 169278.i	− 0.545621i	0.962068	−	0.272810i	$-0.0879532\pi$
$558$	0	0
$559$	641124.	2.05172
$560$	0	0
$561$	0	0
$562$	0	0
$563$	469385.i	1.48086i	0.672136	+	0.740428i	$0.265377\pi$
$563$	469385.i	1.48086i	−0.672136	+	0.740428i	$0.734623\pi$
$564$	0	0
$565$	−299536.	−0.938323
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 320777.i	− 0.990783i	−0.868670	−	0.495391i	$-0.835025\pi$
$569$	− 320777.i	− 0.990783i	0.868670	−	0.495391i	$-0.164975\pi$
$570$	0	0
$571$	−588961.	−1.80640	−0.903201	−	0.429217i	$-0.858789\pi$
$571$	−588961.	−1.80640	−0.903201	+	0.429217i	$0.858789\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	34846.8i	0.105397i
$576$	0	0
$577$	11156.6	0.0335106	0.0167553	−	0.999860i	$-0.494666\pi$
$577$	11156.6	0.0335106	0.0167553	+	0.999860i	$0.494666\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 90510.4i	− 0.268131i
$582$	0	0
$583$	170417.	0.501391
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 495563.i	− 1.43821i	−0.694901	−	0.719106i	$-0.744552\pi$
$587$	− 495563.i	− 1.43821i	0.694901	−	0.719106i	$-0.255448\pi$
$588$	0	0
$589$	−152159.	−0.438597
$590$	0	0
$591$	0	0
$592$	0	0
$593$	519925.i	1.47853i	0.673412	+	0.739267i	$0.264828\pi$
$593$	519925.i	1.47853i	−0.673412	+	0.739267i	$0.735172\pi$
$594$	0	0
$595$	300447.	0.848660
$596$	0	0
$597$	0	0
$598$	0	0
$599$	562456.i	1.56760i	0.621013	+	0.783800i	$0.286721\pi$
$599$	562456.i	1.56760i	−0.621013	+	0.783800i	$0.713279\pi$
$600$	0	0
$601$	507756.	1.40574	0.702872	−	0.711317i	$-0.251901\pi$
$601$	507756.	1.40574	0.702872	+	0.711317i	$0.251901\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	338621.i	0.925131i
$606$	0	0
$607$	−277348.	−0.752744	−0.376372	−	0.926469i	$-0.622829\pi$
$607$	−277348.	−0.752744	−0.376372	+	0.926469i	$0.622829\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	244809.i	0.655761i
$612$	0	0
$613$	107427.	0.285887	0.142943	−	0.989731i	$-0.454343\pi$
$613$	107427.	0.285887	0.142943	+	0.989731i	$0.454343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	643658.i	1.69077i	0.534157	+	0.845386i	$0.320629\pi$
$617$	643658.i	1.69077i	−0.534157	+	0.845386i	$0.679371\pi$
$618$	0	0
$619$	489273.	1.27694	0.638469	−	0.769648i	$-0.279568\pi$
$619$	489273.	1.27694	0.638469	+	0.769648i	$0.279568\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 294852.i	− 0.759675i
$624$	0	0
$625$	−422118.	−1.08062
$626$	0	0
$627$	0	0
$628$	0	0
$629$	218012.i	0.551036i
$630$	0	0
$631$	−216159.	−0.542894	−0.271447	−	0.962453i	$-0.587502\pi$
$631$	−216159.	−0.542894	−0.271447	+	0.962453i	$0.587502\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	470703.i	1.16735i
$636$	0	0
$637$	−20038.1	−0.0493831
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 535626.i	− 1.30360i	−0.758390	−	0.651802i	$-0.774013\pi$
$641$	− 535626.i	− 1.30360i	0.758390	−	0.651802i	$-0.225987\pi$
$642$	0	0
$643$	−439218.	−1.06233	−0.531163	−	0.847270i	$-0.678245\pi$
$643$	−439218.	−1.06233	−0.531163	+	0.847270i	$0.678245\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	152707.i	0.364797i	0.983225	+	0.182398i	$0.0583860\pi$
$647$	152707.i	0.364797i	−0.983225	+	0.182398i	$0.941614\pi$
$648$	0	0
$649$	137033.	0.325340
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 259997.i	− 0.609736i	−0.952395	−	0.304868i	$-0.901388\pi$
$653$	− 259997.i	− 0.609736i	0.952395	−	0.304868i	$-0.0986123\pi$
$654$	0	0
$655$	−447043.	−1.04200
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 353537.i	− 0.814074i	−0.913412	−	0.407037i	$-0.866562\pi$
$659$	− 353537.i	− 0.814074i	0.913412	−	0.407037i	$-0.133438\pi$
$660$	0	0
$661$	501941.	1.14881	0.574407	−	0.818570i	$-0.305233\pi$
$661$	501941.	1.14881	0.574407	+	0.818570i	$0.305233\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	522981.i	1.18261i
$666$	0	0
$667$	−785092.	−1.76469
$668$	0	0
$669$	0	0
$670$	0	0
$671$	176794.i	0.392665i
$672$	0	0
$673$	−157532.	−0.347808	−0.173904	−	0.984763i	$-0.555638\pi$
$673$	−157532.	−0.347808	−0.173904	+	0.984763i	$0.555638\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 153643.i	− 0.335224i	−0.985853	−	0.167612i	$-0.946394\pi$
$677$	− 153643.i	− 0.335224i	0.985853	−	0.167612i	$-0.0536056\pi$
$678$	0	0
$679$	63171.3	0.137019
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 558703.i	− 1.19768i	−0.800870	−	0.598838i	$-0.795629\pi$
$683$	− 558703.i	− 1.19768i	0.800870	−	0.598838i	$-0.204371\pi$
$684$	0	0
$685$	138446.	0.295052
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.04133e6i	− 2.19357i
$690$	0	0
$691$	432660.	0.906129	0.453065	−	0.891478i	$-0.350331\pi$
$691$	432660.	0.906129	0.453065	+	0.891478i	$0.350331\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 333758.i	− 0.690975i
$696$	0	0
$697$	−102214.	−0.210399
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 414368.i	− 0.843239i	−0.906773	−	0.421619i	$-0.861462\pi$
$701$	− 414368.i	− 0.843239i	0.906773	−	0.421619i	$-0.138538\pi$
$702$	0	0
$703$	−379489.	−0.767871
$704$	0	0
$705$	0	0
$706$	0	0
$707$	555743.i	1.11182i
$708$	0	0
$709$	−490238.	−0.975246	−0.487623	−	0.873054i	$-0.662136\pi$
$709$	−490238.	−0.975246	−0.487623	+	0.873054i	$0.662136\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	230437.i	0.453286i
$714$	0	0
$715$	−264263.	−0.516921
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3533.72i	0.00683557i	0.999994	+	0.00341778i	$0.00108792\pi$
$719$	3533.72i	0.00683557i	−0.999994	+	0.00341778i	$0.998912\pi$
$720$	0	0
$721$	−462751.	−0.890178
$722$	0	0
$723$	0	0
$724$	0	0
$725$	68843.6i	0.130975i
$726$	0	0
$727$	−363437.	−0.687639	−0.343819	−	0.939036i	$-0.611721\pi$
$727$	−363437.	−0.687639	−0.343819	+	0.939036i	$0.611721\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 616161.i	− 1.15308i
$732$	0	0
$733$	380607.	0.708385	0.354192	−	0.935173i	$-0.384756\pi$
$733$	380607.	0.708385	0.354192	+	0.935173i	$0.384756\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 247964.i	− 0.456513i
$738$	0	0
$739$	801943.	1.46843	0.734217	−	0.678915i	$-0.237549\pi$
$739$	801943.	1.46843	0.734217	+	0.678915i	$0.237549\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	399832.i	0.724269i	0.932126	+	0.362135i	$0.117952\pi$
$743$	399832.i	0.724269i	−0.932126	+	0.362135i	$0.882048\pi$
$744$	0	0
$745$	80996.9	0.145934
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 509811.i	− 0.908752i
$750$	0	0
$751$	983568.	1.74391	0.871956	−	0.489584i	$-0.162851\pi$
$751$	983568.	1.74391	0.871956	+	0.489584i	$0.162851\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	441191.i	0.773986i
$756$	0	0
$757$	−348508.	−0.608164	−0.304082	−	0.952646i	$-0.598350\pi$
$757$	−348508.	−0.608164	−0.304082	+	0.952646i	$0.598350\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 684017.i	− 1.18113i	−0.806990	−	0.590565i	$-0.798905\pi$
$761$	− 684017.i	− 1.18113i	0.806990	−	0.590565i	$-0.201095\pi$
$762$	0	0
$763$	−55623.0	−0.0955444
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 837340.i	− 1.42335i
$768$	0	0
$769$	196926.	0.333004	0.166502	−	0.986041i	$-0.446753\pi$
$769$	196926.	0.333004	0.166502	+	0.986041i	$0.446753\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 875710.i	− 1.46555i	−0.680469	−	0.732776i	$-0.738224\pi$
$773$	− 875710.i	− 1.46555i	0.680469	−	0.732776i	$-0.261776\pi$
$774$	0	0
$775$	20206.7	0.0336427
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 177921.i	− 0.293192i
$780$	0	0
$781$	−36621.1	−0.0600385
$782$	0	0
$783$	0	0
$784$	0	0
$785$	887621.i	1.44042i
$786$	0	0
$787$	1.00359e6	1.62035	0.810174	−	0.586189i	$-0.199372\pi$
$787$	1.00359e6	1.62035	0.810174	+	0.586189i	$0.199372\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	553215.i	0.884180i
$792$	0	0
$793$	1.08030e6	1.71789
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.14346e6i	1.80013i	0.435755	+	0.900066i	$0.356482\pi$
$797$	1.14346e6i	1.80013i	−0.435755	+	0.900066i	$0.643518\pi$
$798$	0	0
$799$	235278.	0.368542
$800$	0	0
$801$	0	0
$802$	0	0
$803$	314832.i	0.488257i
$804$	0	0
$805$	792028.	1.22222
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 839438.i	− 1.28260i	−0.767290	−	0.641301i	$-0.778395\pi$
$809$	− 839438.i	− 1.28260i	0.767290	−	0.641301i	$-0.221605\pi$
$810$	0	0
$811$	−838658.	−1.27510	−0.637549	−	0.770410i	$-0.720052\pi$
$811$	−838658.	−1.27510	−0.637549	+	0.770410i	$0.720052\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 762441.i	− 1.14787i
$816$	0	0
$817$	1.07254e6	1.60682
$818$	0	0
$819$	0	0
$820$	0	0
$821$	671694.i	0.996518i	0.867028	+	0.498259i	$0.166027\pi$
$821$	671694.i	0.996518i	−0.867028	+	0.498259i	$0.833973\pi$
$822$	0	0
$823$	−440508.	−0.650360	−0.325180	−	0.945652i	$-0.605425\pi$
$823$	−440508.	−0.650360	−0.325180	+	0.945652i	$0.605425\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.08223e6i	1.58237i	0.611580	+	0.791183i	$0.290534\pi$
$827$	1.08223e6i	1.58237i	−0.611580	+	0.791183i	$0.709466\pi$
$828$	0	0
$829$	600354.	0.873571	0.436786	−	0.899566i	$-0.356117\pi$
$829$	600354.	0.873571	0.436786	+	0.899566i	$0.356117\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	19257.9i	0.0277536i
$834$	0	0
$835$	−104207.	−0.149460
$836$	0	0
$837$	0	0
$838$	0	0
$839$	67217.9i	0.0954907i	0.998860	+	0.0477453i	$0.0152036\pi$
$839$	67217.9i	0.0954907i	−0.998860	+	0.0477453i	$0.984796\pi$
$840$	0	0
$841$	−843754.	−1.19295
$842$	0	0
$843$	0	0
$844$	0	0
$845$	869842.i	1.21822i
$846$	0	0
$847$	625401.	0.871750
$848$	0	0
$849$	0	0
$850$	0	0
$851$	574717.i	0.793587i
$852$	0	0
$853$	785261.	1.07923	0.539617	−	0.841910i	$-0.318569\pi$
$853$	785261.	1.07923	0.539617	+	0.841910i	$0.318569\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.43820e6i	− 1.95820i	−0.203372	−	0.979102i	$-0.565190\pi$
$857$	− 1.43820e6i	− 1.95820i	0.203372	−	0.979102i	$-0.434810\pi$
$858$	0	0
$859$	1.06162e6	1.43874	0.719372	−	0.694625i	$-0.244430\pi$
$859$	1.06162e6	1.43874	0.719372	+	0.694625i	$0.244430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.12954e6i	− 1.51663i	−0.651889	−	0.758314i	$-0.726023\pi$
$863$	− 1.12954e6i	− 1.51663i	0.651889	−	0.758314i	$-0.273977\pi$
$864$	0	0
$865$	1.47968e6	1.97759
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 346979.i	− 0.459477i
$870$	0	0
$871$	−1.51518e6	−1.99723
$872$	0	0
$873$	0	0
$874$	0	0
$875$	715804.i	0.934927i
$876$	0	0
$877$	973912.	1.26625	0.633126	−	0.774048i	$-0.281771\pi$
$877$	973912.	1.26625	0.633126	+	0.774048i	$0.281771\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	246060.i	0.317022i	0.987357	+	0.158511i	$0.0506694\pi$
$881$	246060.i	0.317022i	−0.987357	+	0.158511i	$0.949331\pi$
$882$	0	0
$883$	455874.	0.584687	0.292344	−	0.956313i	$-0.405565\pi$
$883$	455874.	0.584687	0.292344	+	0.956313i	$0.405565\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 883124.i	− 1.12247i	−0.827657	−	0.561234i	$-0.810327\pi$
$887$	− 883124.i	− 1.12247i	0.827657	−	0.561234i	$-0.189673\pi$
$888$	0	0
$889$	869344.	1.09999
$890$	0	0
$891$	0	0
$892$	0	0
$893$	409542.i	0.513565i
$894$	0	0
$895$	248684.	0.310458
$896$	0	0
$897$	0	0
$898$	0	0
$899$	455252.i	0.563291i
$900$	0	0
$901$	−1.00079e6	−1.23280
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.68262e6i	2.05441i
$906$	0	0
$907$	400867.	0.487288	0.243644	−	0.969865i	$-0.421657\pi$
$907$	400867.	0.487288	0.243644	+	0.969865i	$0.421657\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.46297e6i	− 1.76278i	−0.472385	−	0.881392i	$-0.656607\pi$
$911$	− 1.46297e6i	− 1.76278i	0.472385	−	0.881392i	$-0.343393\pi$
$912$	0	0
$913$	76510.3	0.0917864
$914$	0	0
$915$	0	0
$916$	0	0
$917$	825645.i	0.981872i
$918$	0	0
$919$	−1.16531e6	−1.37978	−0.689889	−	0.723915i	$-0.742341\pi$
$919$	−1.16531e6	−1.37978	−0.689889	+	0.723915i	$0.742341\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	223773.i	0.262666i
$924$	0	0
$925$	50396.1	0.0588997
$926$	0	0
$927$	0	0
$928$	0	0
$929$	986016.i	1.14249i	0.820780	+	0.571245i	$0.193539\pi$
$929$	986016.i	1.14249i	−0.820780	+	0.571245i	$0.806461\pi$
$930$	0	0
$931$	−33521.8	−0.0386748
$932$	0	0
$933$	0	0
$934$	0	0
$935$	253974.i	0.290513i
$936$	0	0
$937$	1.46816e6	1.67223	0.836114	−	0.548555i	$-0.184822\pi$
$937$	1.46816e6	1.67223	0.836114	+	0.548555i	$0.184822\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 229399.i	− 0.259067i	−0.991575	−	0.129534i	$-0.958652\pi$
$941$	− 229399.i	− 0.259067i	0.991575	−	0.129534i	$-0.0413480\pi$
$942$	0	0
$943$	−269453.	−0.303012
$944$	0	0
$945$	0	0
$946$	0	0
$947$	403740.i	0.450196i	0.974336	+	0.225098i	$0.0722703\pi$
$947$	403740.i	0.450196i	−0.974336	+	0.225098i	$0.927730\pi$
$948$	0	0
$949$	1.92378e6	2.13610
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 807857.i	− 0.889506i	−0.895653	−	0.444753i	$-0.853291\pi$
$953$	− 807857.i	− 0.889506i	0.895653	−	0.444753i	$-0.146709\pi$
$954$	0	0
$955$	−389877.	−0.427485
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 255696.i	− 0.278027i
$960$	0	0
$961$	−789897.	−0.855311
$962$	0	0
$963$	0	0
$964$	0	0
$965$	422485.i	0.453688i
$966$	0	0
$967$	778378.	0.832410	0.416205	−	0.909271i	$-0.363360\pi$
$967$	778378.	0.832410	0.416205	+	0.909271i	$0.363360\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	943638.i	1.00085i	0.865781	+	0.500423i	$0.166822\pi$
$971$	943638.i	1.00085i	−0.865781	+	0.500423i	$0.833178\pi$
$972$	0	0
$973$	−616420.	−0.651105
$974$	0	0
$975$	0	0
$976$	0	0
$977$	573080.i	0.600380i	0.953879	+	0.300190i	$0.0970501\pi$
$977$	573080.i	0.600380i	−0.953879	+	0.300190i	$0.902950\pi$
$978$	0	0
$979$	249244.	0.260052
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.43403e6i	1.48406i	0.670366	+	0.742030i	$0.266137\pi$
$983$	1.43403e6i	1.48406i	−0.670366	+	0.742030i	$0.733863\pi$
$984$	0	0
$985$	−1.72385e6	−1.77675
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.62430e6i	− 1.66064i
$990$	0	0
$991$	−309420.	−0.315066	−0.157533	−	0.987514i	$-0.550354\pi$
$991$	−309420.	−0.315066	−0.157533	+	0.987514i	$0.550354\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	898282.i	0.907333i
$996$	0	0
$997$	824998.	0.829970	0.414985	−	0.909828i	$-0.363787\pi$
$997$	824998.	0.829970	0.414985	+	0.909828i	$0.363787\pi$
$998$	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.161.14	yes	16
3.2	odd	2	inner	864.5.e.g.161.4	yes	16
4.3	odd	2	inner	864.5.e.g.161.13	yes	16
12.11	even	2	inner	864.5.e.g.161.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.5.e.g.161.3	✓	16	12.11	even	2	inner
864.5.e.g.161.4	yes	16	3.2	odd	2	inner
864.5.e.g.161.13	yes	16	4.3	odd	2	inner
864.5.e.g.161.14	yes	16	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	864.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+26.0821i q^{5} +48.1712 q^{7} +40.7201i q^{11} +248.819 q^{13} -239.132i q^{17} +416.251 q^{19} -630.391i q^{23} -55.2781 q^{25} -1245.41i q^{29} -365.546 q^{31} +1256.41i q^{35} -911.684 q^{37} -427.438i q^{41} +2576.66 q^{43} +983.883i q^{47} -80.5328 q^{49} -4185.09i q^{53} -1062.07 q^{55} -3365.25i q^{59} +4341.68 q^{61} +6489.74i q^{65} -6089.47 q^{67} +899.338i q^{71} +7731.62 q^{73} +1961.54i q^{77} -8521.08 q^{79} -1878.93i q^{83} +6237.06 q^{85} -6120.92i q^{89} +11985.9 q^{91} +10856.7i q^{95} +1311.39 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{13} + 704 q^{25} - 2624 q^{37} + 1728 q^{49} + 3264 q^{61} - 5424 q^{73} + 4704 q^{85} - 80 q^{97}+O(q^{100}) \) 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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