LMFDB - Embedded newform 6480.2.h.c.2591.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(2591,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6480.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$51.7430605098$
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} - 12x^{14} + 99x^{12} - 432x^{10} + 1368x^{8} - 2214x^{6} + 2511x^{4} - 486x^{2} + 81 $ x^16 - 12x^14 + 99x^12 - 432x^10 + 1368x^8 - 2214x^6 + 2511x^4 - 486*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.14
Root			$0.384853 - 0.222195i$ of defining polynomial
Character	$\chi$	$=$	6480.2591
Dual form			6480.2.h.c.2591.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +1.68980i q^{7} +O(q^{10})$	$q+1.00000i q^{5} +1.68980i q^{7} +1.21410 q^{11} +5.65662 q^{13} -2.05158i q^{17} +0.325316i q^{19} +7.19521 q^{23} -1.00000 q^{25} +0.408882i q^{29} -9.55857i q^{31} -1.68980 q^{35} +5.55345 q^{37} -0.948415i q^{41} -6.71950i q^{43} -10.5977 q^{47} +4.14457 q^{49} -11.2101i q^{53} +1.21410i q^{55} +2.72897 q^{59} +3.64270 q^{61} +5.65662i q^{65} -3.31477i q^{67} +2.86199 q^{71} -2.81776 q^{73} +2.05158i q^{77} +13.5643i q^{79} -13.7621 q^{83} +2.05158 q^{85} +10.1171i q^{89} +9.55857i q^{91} -0.325316 q^{95} +8.39230 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{13} - 16 q^{25} + 8 q^{37} - 8 q^{49} + 56 q^{61} - 32 q^{73} + 24 q^{85} - 32 q^{97}+O(q^{100}) $ 16 * q - 8 * q^13 - 16 * q^25 + 8 * q^37 - 8 * q^49 + 56 * q^61 - 32 * q^73 + 24 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6480\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$6401$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	1.68980i	0.638685i	0.947639	+	0.319343i	$0.103462\pi$
$7$	1.68980i	0.638685i	−0.947639	+	0.319343i	$0.896538\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.21410	0.366064	0.183032	−	0.983107i	$-0.441409\pi$
$11$	1.21410	0.366064	0.183032	+	0.983107i	$0.441409\pi$
$12$	0	0
$13$	5.65662	1.56886	0.784432	−	0.620215i	$-0.212955\pi$
$13$	5.65662	1.56886	0.784432	+	0.620215i	$0.212955\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.05158i	− 0.497582i	−0.968557	−	0.248791i	$-0.919967\pi$
$17$	− 2.05158i	− 0.497582i	0.968557	−	0.248791i	$-0.0800333\pi$
$18$	0	0
$19$	0.325316i	0.0746327i	0.999304	+	0.0373164i	$0.0118809\pi$
$19$	0.325316i	0.0746327i	−0.999304	+	0.0373164i	$0.988119\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.19521	1.50030	0.750152	−	0.661265i	$-0.229980\pi$
$23$	7.19521	1.50030	0.750152	+	0.661265i	$0.229980\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.408882i	0.0759274i	0.999279	+	0.0379637i	$0.0120871\pi$
$29$	0.408882i	0.0759274i	−0.999279	+	0.0379637i	$0.987913\pi$
$30$	0	0
$31$	− 9.55857i	− 1.71677i	−0.513006	−	0.858385i	$-0.671468\pi$
$31$	− 9.55857i	− 1.71677i	0.513006	−	0.858385i	$-0.328532\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.68980	−0.285629
$36$	0	0
$37$	5.55345	0.912981	0.456491	−	0.889728i	$-0.349106\pi$
$37$	5.55345	0.912981	0.456491	+	0.889728i	$0.349106\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.948415i	− 0.148118i	−0.997254	−	0.0740588i	$-0.976405\pi$
$41$	− 0.948415i	− 0.148118i	0.997254	−	0.0740588i	$-0.0235952\pi$
$42$	0	0
$43$	− 6.71950i	− 1.02471i	−0.858772	−	0.512357i	$-0.828772\pi$
$43$	− 6.71950i	− 1.02471i	0.858772	−	0.512357i	$-0.171228\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.5977	−1.54584	−0.772920	−	0.634504i	$-0.781204\pi$
$47$	−10.5977	−1.54584	−0.772920	+	0.634504i	$0.781204\pi$
$48$	0	0
$49$	4.14457	0.592081
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.2101i	− 1.53982i	−0.638152	−	0.769911i	$-0.720301\pi$
$53$	− 11.2101i	− 1.53982i	0.638152	−	0.769911i	$-0.279699\pi$
$54$	0	0
$55$	1.21410i	0.163709i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.72897	0.355282	0.177641	−	0.984095i	$-0.443153\pi$
$59$	2.72897	0.355282	0.177641	+	0.984095i	$0.443153\pi$
$60$	0	0
$61$	3.64270	0.466400	0.233200	−	0.972429i	$-0.425080\pi$
$61$	3.64270	0.466400	0.233200	+	0.972429i	$0.425080\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.65662i	0.701617i
$66$	0	0
$67$	− 3.31477i	− 0.404964i	−0.979286	−	0.202482i	$-0.935099\pi$
$67$	− 3.31477i	− 0.404964i	0.979286	−	0.202482i	$-0.0649007\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.86199	0.339655	0.169828	−	0.985474i	$-0.445679\pi$
$71$	2.86199	0.339655	0.169828	+	0.985474i	$0.445679\pi$
$72$	0	0
$73$	−2.81776	−0.329794	−0.164897	−	0.986311i	$-0.552729\pi$
$73$	−2.81776	−0.329794	−0.164897	+	0.986311i	$0.552729\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.05158i	0.233800i
$78$	0	0
$79$	13.5643i	1.52610i	0.646340	+	0.763050i	$0.276299\pi$
$79$	13.5643i	1.52610i	−0.646340	+	0.763050i	$0.723701\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.7621	−1.51059	−0.755294	−	0.655386i	$-0.772506\pi$
$83$	−13.7621	−1.51059	−0.755294	+	0.655386i	$0.772506\pi$
$84$	0	0
$85$	2.05158	0.222526
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.1171i	1.07241i	0.844088	+	0.536204i	$0.180142\pi$
$89$	10.1171i	1.07241i	−0.844088	+	0.536204i	$0.819858\pi$
$90$	0	0
$91$	9.55857i	1.00201i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.325316	−0.0333768
$96$	0	0
$97$	8.39230	0.852109	0.426055	−	0.904697i	$-0.359903\pi$
$97$	8.39230	0.852109	0.426055	+	0.904697i	$0.359903\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.9183i	− 1.48442i	−0.670165	−	0.742212i	$-0.733777\pi$
$101$	− 14.9183i	− 1.48442i	0.670165	−	0.742212i	$-0.266223\pi$
$102$	0	0
$103$	− 12.3976i	− 1.22158i	−0.791795	−	0.610788i	$-0.790853\pi$
$103$	− 12.3976i	− 1.22158i	0.791795	−	0.610788i	$-0.209147\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.70439	−0.454790	−0.227395	−	0.973803i	$-0.573021\pi$
$107$	−4.70439	−0.454790	−0.227395	+	0.973803i	$0.573021\pi$
$108$	0	0
$109$	5.24774	0.502642	0.251321	−	0.967904i	$-0.419135\pi$
$109$	5.24774	0.502642	0.251321	+	0.967904i	$0.419135\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 13.4164i	− 1.26211i	−0.775738	−	0.631055i	$-0.782622\pi$
$113$	− 13.4164i	− 1.26211i	0.775738	−	0.631055i	$-0.217378\pi$
$114$	0	0
$115$	7.19521i	0.670957i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.46677	0.317799
$120$	0	0
$121$	−9.52597	−0.865997
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	6.56912i	0.582915i	0.956584	+	0.291457i	$0.0941402\pi$
$127$	6.56912i	0.582915i	−0.956584	+	0.291457i	$0.905860\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.4665	1.61343	0.806713	−	0.590943i	$-0.201244\pi$
$131$	18.4665	1.61343	0.806713	+	0.590943i	$0.201244\pi$
$132$	0	0
$133$	−0.549721	−0.0476668
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.817763i	− 0.0698662i	−0.999390	−	0.0349331i	$-0.988878\pi$
$137$	− 0.817763i	− 0.0698662i	0.999390	−	0.0349331i	$-0.0111218\pi$
$138$	0	0
$139$	− 10.9650i	− 0.930036i	−0.885301	−	0.465018i	$-0.846048\pi$
$139$	− 10.9650i	− 0.930036i	0.885301	−	0.465018i	$-0.153952\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.86769	0.574305
$144$	0	0
$145$	−0.408882	−0.0339558
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.47438i	0.448479i	0.974534	+	0.224239i	$0.0719898\pi$
$149$	5.47438i	0.448479i	−0.974534	+	0.224239i	$0.928010\pi$
$150$	0	0
$151$	− 3.88044i	− 0.315786i	−0.987456	−	0.157893i	$-0.949530\pi$
$151$	− 3.88044i	− 0.315786i	0.987456	−	0.157893i	$-0.0504701\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.55857	0.767763
$156$	0	0
$157$	−2.92093	−0.233116	−0.116558	−	0.993184i	$-0.537186\pi$
$157$	−2.92093	−0.233116	−0.116558	+	0.993184i	$0.537186\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.1585i	0.958223i
$162$	0	0
$163$	18.5521i	1.45311i	0.687109	+	0.726555i	$0.258880\pi$
$163$	18.5521i	1.45311i	−0.687109	+	0.726555i	$0.741120\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.01248	0.697406	0.348703	−	0.937233i	$-0.386622\pi$
$167$	9.01248	0.697406	0.348703	+	0.937233i	$0.386622\pi$
$168$	0	0
$169$	18.9973	1.46133
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.31324i	0.251901i	0.992037	+	0.125950i	$0.0401980\pi$
$173$	3.31324i	0.251901i	−0.992037	+	0.125950i	$0.959802\pi$
$174$	0	0
$175$	− 1.68980i	− 0.127737i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.7298	−1.17570	−0.587849	−	0.808970i	$-0.700025\pi$
$179$	−15.7298	−1.17570	−0.587849	+	0.808970i	$0.700025\pi$
$180$	0	0
$181$	15.9183	1.18320	0.591598	−	0.806233i	$-0.298497\pi$
$181$	15.9183	1.18320	0.591598	+	0.806233i	$0.298497\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.55345i	0.408298i
$186$	0	0
$187$	− 2.49082i	− 0.182147i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.5862	−1.48956	−0.744781	−	0.667309i	$-0.767446\pi$
$191$	−20.5862	−1.48956	−0.744781	+	0.667309i	$0.767446\pi$
$192$	0	0
$193$	26.0489	1.87504	0.937521	−	0.347928i	$-0.113115\pi$
$193$	26.0489	1.87504	0.937521	+	0.347928i	$0.113115\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.7055i	1.83144i	0.401812	+	0.915722i	$0.368380\pi$
$197$	25.7055i	1.83144i	−0.401812	+	0.915722i	$0.631620\pi$
$198$	0	0
$199$	− 21.8216i	− 1.54689i	−0.633863	−	0.773445i	$-0.718532\pi$
$199$	− 21.8216i	− 1.54689i	0.633863	−	0.773445i	$-0.281468\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.690929	−0.0484937
$204$	0	0
$205$	0.948415	0.0662402
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.394966i	0.0273204i
$210$	0	0
$211$	22.9976i	1.58322i	0.611028	+	0.791609i	$0.290756\pi$
$211$	22.9976i	1.58322i	−0.611028	+	0.791609i	$0.709244\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.71950	0.458266
$216$	0	0
$217$	16.1521	1.09648
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 11.6050i	− 0.780639i
$222$	0	0
$223$	1.17380i	0.0786035i	0.999227	+	0.0393018i	$0.0125134\pi$
$223$	1.17380i	0.0786035i	−0.999227	+	0.0393018i	$0.987487\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.97443	0.462909	0.231455	−	0.972846i	$-0.425651\pi$
$227$	6.97443	0.462909	0.231455	+	0.972846i	$0.425651\pi$
$228$	0	0
$229$	4.70820	0.311127	0.155563	−	0.987826i	$-0.450281\pi$
$229$	4.70820	0.311127	0.155563	+	0.987826i	$0.450281\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.1585i	− 0.993065i	−0.868018	−	0.496533i	$-0.834606\pi$
$233$	− 15.1585i	− 0.993065i	0.868018	−	0.496533i	$-0.165394\pi$
$234$	0	0
$235$	− 10.5977i	− 0.691320i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.3990	−1.77229	−0.886146	−	0.463405i	$-0.846627\pi$
$239$	−27.3990	−1.77229	−0.886146	+	0.463405i	$0.846627\pi$
$240$	0	0
$241$	24.9533	1.60738	0.803691	−	0.595047i	$-0.202866\pi$
$241$	24.9533	1.60738	0.803691	+	0.595047i	$0.202866\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.14457i	0.264787i
$246$	0	0
$247$	1.84019i	0.117089i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.37126	0.402151	0.201075	−	0.979576i	$-0.435556\pi$
$251$	6.37126	0.402151	0.201075	+	0.979576i	$0.435556\pi$
$252$	0	0
$253$	8.73569	0.549208
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.7846i	0.922239i	0.887338	+	0.461119i	$0.152552\pi$
$257$	14.7846i	0.922239i	−0.887338	+	0.461119i	$0.847448\pi$
$258$	0	0
$259$	9.38423i	0.583108i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.6174	−1.08634	−0.543169	−	0.839624i	$-0.682776\pi$
$263$	−17.6174	−1.08634	−0.543169	+	0.839624i	$0.682776\pi$
$264$	0	0
$265$	11.2101	0.688629
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 19.1521i	− 1.16772i	−0.811853	−	0.583862i	$-0.801541\pi$
$269$	− 19.1521i	− 1.16772i	0.811853	−	0.583862i	$-0.198459\pi$
$270$	0	0
$271$	− 3.40414i	− 0.206787i	−0.994641	−	0.103394i	$-0.967030\pi$
$271$	− 3.40414i	− 0.206787i	0.994641	−	0.103394i	$-0.0329701\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.21410	−0.0732128
$276$	0	0
$277$	15.7598	0.946914	0.473457	−	0.880817i	$-0.343006\pi$
$277$	15.7598	0.946914	0.473457	+	0.880817i	$0.343006\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.553449i	0.0330160i	0.999864	+	0.0165080i	$0.00525490\pi$
$281$	0.553449i	0.0330160i	−0.999864	+	0.0165080i	$0.994745\pi$
$282$	0	0
$283$	18.2703i	1.08605i	0.839715	+	0.543027i	$0.182722\pi$
$283$	18.2703i	1.08605i	−0.839715	+	0.543027i	$0.817278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.60263	0.0946005
$288$	0	0
$289$	12.7910	0.752412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0516i	0.820902i	0.911883	+	0.410451i	$0.134629\pi$
$293$	14.0516i	0.820902i	−0.911883	+	0.410451i	$0.865371\pi$
$294$	0	0
$295$	2.72897i	0.158887i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	40.7006	2.35377
$300$	0	0
$301$	11.3546	0.654470
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.64270i	0.208581i
$306$	0	0
$307$	12.1126i	0.691303i	0.938363	+	0.345652i	$0.112342\pi$
$307$	12.1126i	0.691303i	−0.938363	+	0.345652i	$0.887658\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.16492	0.292876	0.146438	−	0.989220i	$-0.453219\pi$
$311$	5.16492	0.292876	0.146438	+	0.989220i	$0.453219\pi$
$312$	0	0
$313$	8.83886	0.499602	0.249801	−	0.968297i	$-0.419635\pi$
$313$	8.83886	0.499602	0.249801	+	0.968297i	$0.419635\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0553i	0.957922i	0.877836	+	0.478961i	$0.158986\pi$
$317$	17.0553i	0.957922i	−0.877836	+	0.478961i	$0.841014\pi$
$318$	0	0
$319$	0.496422i	0.0277943i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.667414	0.0371359
$324$	0	0
$325$	−5.65662	−0.313773
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 17.9081i	− 0.987305i
$330$	0	0
$331$	− 15.2122i	− 0.836136i	−0.908416	−	0.418068i	$-0.862707\pi$
$331$	− 15.2122i	− 0.836136i	0.908416	−	0.418068i	$-0.137293\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.31477	0.181105
$336$	0	0
$337$	28.9156	1.57513	0.787567	−	0.616229i	$-0.211341\pi$
$337$	28.9156	1.57513	0.787567	+	0.616229i	$0.211341\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 11.6050i	− 0.628448i
$342$	0	0
$343$	18.8321i	1.01684i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.5219	−1.10167	−0.550837	−	0.834613i	$-0.685691\pi$
$347$	−20.5219	−1.10167	−0.550837	+	0.834613i	$0.685691\pi$
$348$	0	0
$349$	−19.0278	−1.01854	−0.509268	−	0.860608i	$-0.670084\pi$
$349$	−19.0278	−1.01854	−0.509268	+	0.860608i	$0.670084\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.9209i	0.581263i	0.956835	+	0.290631i	$0.0938653\pi$
$353$	10.9209i	0.581263i	−0.956835	+	0.290631i	$0.906135\pi$
$354$	0	0
$355$	2.86199i	0.151899i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.6487	−0.773128	−0.386564	−	0.922262i	$-0.626338\pi$
$359$	−14.6487	−0.773128	−0.386564	+	0.922262i	$0.626338\pi$
$360$	0	0
$361$	18.8942	0.994430
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.81776i	− 0.147488i
$366$	0	0
$367$	8.02077i	0.418681i	0.977843	+	0.209340i	$0.0671317\pi$
$367$	8.02077i	0.418681i	−0.977843	+	0.209340i	$0.932868\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.9428	0.983461
$372$	0	0
$373$	0.103170	0.00534193	0.00267096	−	0.999996i	$-0.499150\pi$
$373$	0.103170	0.00534193	0.00267096	+	0.999996i	$0.499150\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.31289i	0.119120i
$378$	0	0
$379$	− 28.3962i	− 1.45862i	−0.684185	−	0.729308i	$-0.739842\pi$
$379$	− 28.3962i	− 1.45862i	0.684185	−	0.729308i	$-0.260158\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.8723	1.11762	0.558810	−	0.829296i	$-0.311258\pi$
$383$	21.8723	1.11762	0.558810	+	0.829296i	$0.311258\pi$
$384$	0	0
$385$	−2.05158	−0.104558
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 19.2376i	− 0.975383i	−0.873016	−	0.487691i	$-0.837839\pi$
$389$	− 19.2376i	− 0.975383i	0.873016	−	0.487691i	$-0.162161\pi$
$390$	0	0
$391$	− 14.7616i	− 0.746525i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.5643	−0.682492
$396$	0	0
$397$	−24.5233	−1.23079	−0.615395	−	0.788219i	$-0.711003\pi$
$397$	−24.5233	−1.23079	−0.615395	+	0.788219i	$0.711003\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.0214i	1.54914i	0.632490	+	0.774568i	$0.282033\pi$
$401$	31.0214i	1.54914i	−0.632490	+	0.774568i	$0.717967\pi$
$402$	0	0
$403$	− 54.0692i	− 2.69338i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.74243	0.334210
$408$	0	0
$409$	2.36447	0.116916	0.0584578	−	0.998290i	$-0.481382\pi$
$409$	2.36447	0.116916	0.0584578	+	0.998290i	$0.481382\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.61142i	0.226913i
$414$	0	0
$415$	− 13.7621i	− 0.675556i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.5186	−1.44208	−0.721040	−	0.692893i	$-0.756336\pi$
$419$	−29.5186	−1.44208	−0.721040	+	0.692893i	$0.756336\pi$
$420$	0	0
$421$	−15.4164	−0.751350	−0.375675	−	0.926752i	$-0.622589\pi$
$421$	−15.4164	−0.751350	−0.375675	+	0.926752i	$0.622589\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.05158i	0.0995165i
$426$	0	0
$427$	6.15545i	0.297883i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.1393	1.83711	0.918554	−	0.395296i	$-0.129358\pi$
$431$	38.1393	1.83711	0.918554	+	0.395296i	$0.129358\pi$
$432$	0	0
$433$	15.9458	0.766304	0.383152	−	0.923685i	$-0.374839\pi$
$433$	15.9458	0.766304	0.383152	+	0.923685i	$0.374839\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.34072i	0.111972i
$438$	0	0
$439$	24.7048i	1.17909i	0.807734	+	0.589547i	$0.200694\pi$
$439$	24.7048i	1.17909i	−0.807734	+	0.589547i	$0.799306\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.75804	−0.463618	−0.231809	−	0.972761i	$-0.574464\pi$
$443$	−9.75804	−0.463618	−0.231809	+	0.972761i	$0.574464\pi$
$444$	0	0
$445$	−10.1171	−0.479596
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 37.0214i	− 1.74715i	−0.486690	−	0.873575i	$-0.661796\pi$
$449$	− 37.0214i	− 1.74715i	0.486690	−	0.873575i	$-0.338204\pi$
$450$	0	0
$451$	− 1.15147i	− 0.0542205i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.55857	−0.448113
$456$	0	0
$457$	−23.8908	−1.11756	−0.558782	−	0.829315i	$-0.688731\pi$
$457$	−23.8908	−1.11756	−0.558782	+	0.829315i	$0.688731\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.61522i	0.401251i	0.979668	+	0.200625i	$0.0642974\pi$
$461$	8.61522i	0.401251i	−0.979668	+	0.200625i	$0.935703\pi$
$462$	0	0
$463$	36.1678i	1.68086i	0.541921	+	0.840429i	$0.317697\pi$
$463$	36.1678i	1.68086i	−0.541921	+	0.840429i	$0.682303\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.43551	0.297800	0.148900	−	0.988852i	$-0.452427\pi$
$467$	6.43551	0.297800	0.148900	+	0.988852i	$0.452427\pi$
$468$	0	0
$469$	5.60131	0.258644
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8.15813i	− 0.375111i
$474$	0	0
$475$	− 0.325316i	− 0.0149265i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.3487	1.24959	0.624797	−	0.780787i	$-0.285182\pi$
$479$	27.3487	1.24959	0.624797	+	0.780787i	$0.285182\pi$
$480$	0	0
$481$	31.4137	1.43234
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.39230i	0.381075i
$486$	0	0
$487$	− 33.6602i	− 1.52529i	−0.646819	−	0.762644i	$-0.723901\pi$
$487$	− 33.6602i	− 1.52529i	0.646819	−	0.762644i	$-0.276099\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−31.3846	−1.41637	−0.708183	−	0.706029i	$-0.750485\pi$
$491$	−31.3846	−1.41637	−0.708183	+	0.706029i	$0.750485\pi$
$492$	0	0
$493$	0.838856	0.0377802
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.83619i	0.216933i
$498$	0	0
$499$	15.2612i	0.683187i	0.939848	+	0.341593i	$0.110967\pi$
$499$	15.2612i	0.683187i	−0.939848	+	0.341593i	$0.889033\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.3218	0.995281	0.497640	−	0.867384i	$-0.334200\pi$
$503$	22.3218	0.995281	0.497640	+	0.867384i	$0.334200\pi$
$504$	0	0
$505$	14.9183	0.663854
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 21.9258i	− 0.971844i	−0.874002	−	0.485922i	$-0.838484\pi$
$509$	− 21.9258i	− 0.971844i	0.874002	−	0.485922i	$-0.161516\pi$
$510$	0	0
$511$	− 4.76146i	− 0.210635i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.3976	0.546305
$516$	0	0
$517$	−12.8667	−0.565876
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.5121i	0.460541i	0.973127	+	0.230271i	$0.0739612\pi$
$521$	10.5121i	0.460541i	−0.973127	+	0.230271i	$0.926039\pi$
$522$	0	0
$523$	− 0.159762i	− 0.00698589i	−0.999994	−	0.00349295i	$-0.998888\pi$
$523$	− 0.159762i	− 0.00698589i	0.999994	−	0.00349295i	$-0.00111184\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.6102	−0.854234
$528$	0	0
$529$	28.7710	1.25091
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.36482i	− 0.232376i
$534$	0	0
$535$	− 4.70439i	− 0.203388i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.03191	0.216740
$540$	0	0
$541$	−34.5474	−1.48531	−0.742655	−	0.669675i	$-0.766434\pi$
$541$	−34.5474	−1.48531	−0.742655	+	0.669675i	$0.766434\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.24774i	0.224788i
$546$	0	0
$547$	− 40.1164i	− 1.71525i	−0.514273	−	0.857626i	$-0.671938\pi$
$547$	− 40.1164i	− 1.71525i	0.514273	−	0.857626i	$-0.328062\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.133016	−0.00566667
$552$	0	0
$553$	−22.9209	−0.974697
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.8365i	1.00999i	0.863123	+	0.504993i	$0.168505\pi$
$557$	23.8365i	1.00999i	−0.863123	+	0.504993i	$0.831495\pi$
$558$	0	0
$559$	− 38.0097i	− 1.60764i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.47345	0.104244	0.0521218	−	0.998641i	$-0.483402\pi$
$563$	2.47345	0.104244	0.0521218	+	0.998641i	$0.483402\pi$
$564$	0	0
$565$	13.4164	0.564433
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 42.4074i	− 1.77781i	−0.458093	−	0.888904i	$-0.651467\pi$
$569$	− 42.4074i	− 1.77781i	0.458093	−	0.888904i	$-0.348533\pi$
$570$	0	0
$571$	20.3235i	0.850511i	0.905073	+	0.425256i	$0.139816\pi$
$571$	20.3235i	0.850511i	−0.905073	+	0.425256i	$0.860184\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.19521	−0.300061
$576$	0	0
$577$	21.6777	0.902455	0.451227	−	0.892409i	$-0.350986\pi$
$577$	21.6777	0.902455	0.451227	+	0.892409i	$0.350986\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 23.2553i	− 0.964791i
$582$	0	0
$583$	− 13.6101i	− 0.563673i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.0468	−1.11634	−0.558171	−	0.829726i	$-0.688497\pi$
$587$	−27.0468	−1.11634	−0.558171	+	0.829726i	$0.688497\pi$
$588$	0	0
$589$	3.10956	0.128127
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.3136i	1.16270i	0.813654	+	0.581350i	$0.197475\pi$
$593$	28.3136i	1.16270i	−0.813654	+	0.581350i	$0.802525\pi$
$594$	0	0
$595$	3.46677i	0.142124i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.20634	0.0492897	0.0246449	−	0.999696i	$-0.492155\pi$
$599$	1.20634	0.0492897	0.0246449	+	0.999696i	$0.492155\pi$
$600$	0	0
$601$	7.69049	0.313702	0.156851	−	0.987622i	$-0.449866\pi$
$601$	7.69049	0.313702	0.156851	+	0.987622i	$0.449866\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 9.52597i	− 0.387286i
$606$	0	0
$607$	21.4314i	0.869874i	0.900461	+	0.434937i	$0.143229\pi$
$607$	21.4314i	0.869874i	−0.900461	+	0.434937i	$0.856771\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−59.9474	−2.42521
$612$	0	0
$613$	−13.8148	−0.557973	−0.278986	−	0.960295i	$-0.589998\pi$
$613$	−13.8148	−0.557973	−0.278986	+	0.960295i	$0.589998\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.4992i	1.67070i	0.549722	+	0.835348i	$0.314734\pi$
$617$	41.4992i	1.67070i	−0.549722	+	0.835348i	$0.685266\pi$
$618$	0	0
$619$	30.8379i	1.23948i	0.784808	+	0.619739i	$0.212762\pi$
$619$	30.8379i	1.23948i	−0.784808	+	0.619739i	$0.787238\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.0959	−0.684932
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 11.3934i	− 0.454284i
$630$	0	0
$631$	38.9655i	1.55119i	0.631230	+	0.775596i	$0.282551\pi$
$631$	38.9655i	1.55119i	−0.631230	+	0.775596i	$0.717449\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.56912	−0.260687
$636$	0	0
$637$	23.4442	0.928895
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.1799i	1.46852i	0.678869	+	0.734259i	$0.262470\pi$
$641$	37.1799i	1.46852i	−0.678869	+	0.734259i	$0.737530\pi$
$642$	0	0
$643$	− 26.7003i	− 1.05296i	−0.850189	−	0.526478i	$-0.823512\pi$
$643$	− 26.7003i	− 1.05296i	0.850189	−	0.526478i	$-0.176488\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.0210	−0.865736	−0.432868	−	0.901457i	$-0.642498\pi$
$647$	−22.0210	−0.865736	−0.432868	+	0.901457i	$0.642498\pi$
$648$	0	0
$649$	3.31324	0.130056
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.39265i	0.211031i	0.994418	+	0.105515i	$0.0336492\pi$
$653$	5.39265i	0.211031i	−0.994418	+	0.105515i	$0.966351\pi$
$654$	0	0
$655$	18.4665i	0.721546i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.6436	1.73907	0.869534	−	0.493872i	$-0.164419\pi$
$659$	44.6436	1.73907	0.869534	+	0.493872i	$0.164419\pi$
$660$	0	0
$661$	−20.2010	−0.785729	−0.392864	−	0.919596i	$-0.628516\pi$
$661$	−20.2010	−0.785729	−0.392864	+	0.919596i	$0.628516\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.549721i	− 0.0213172i
$666$	0	0
$667$	2.94199i	0.113914i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.42260	0.170732
$672$	0	0
$673$	28.2553	1.08916	0.544580	−	0.838709i	$-0.316689\pi$
$673$	28.2553	1.08916	0.544580	+	0.838709i	$0.316689\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.3317i	1.66537i	0.553746	+	0.832686i	$0.313198\pi$
$677$	43.3317i	1.66537i	−0.553746	+	0.832686i	$0.686802\pi$
$678$	0	0
$679$	14.1813i	0.544230i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.247521	−0.00947114	−0.00473557	−	0.999989i	$-0.501507\pi$
$683$	−0.247521	−0.00947114	−0.00473557	+	0.999989i	$0.501507\pi$
$684$	0	0
$685$	0.817763	0.0312451
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 63.4111i	− 2.41577i
$690$	0	0
$691$	− 12.4832i	− 0.474883i	−0.971402	−	0.237441i	$-0.923691\pi$
$691$	− 12.4832i	− 0.474883i	0.971402	−	0.237441i	$-0.0763088\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.9650	0.415925
$696$	0	0
$697$	−1.94575	−0.0737007
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 27.7609i	− 1.04851i	−0.851560	−	0.524257i	$-0.824343\pi$
$701$	− 27.7609i	− 1.04851i	0.851560	−	0.524257i	$-0.175657\pi$
$702$	0	0
$703$	1.80663i	0.0681383i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.2089	0.948080
$708$	0	0
$709$	−10.3859	−0.390051	−0.195026	−	0.980798i	$-0.562479\pi$
$709$	−10.3859	−0.390051	−0.195026	+	0.980798i	$0.562479\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 68.7759i	− 2.57568i
$714$	0	0
$715$	6.86769i	0.256837i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.60371	0.0971021	0.0485511	−	0.998821i	$-0.484540\pi$
$719$	2.60371	0.0971021	0.0485511	+	0.998821i	$0.484540\pi$
$720$	0	0
$721$	20.9496	0.780202
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 0.408882i	− 0.0151855i
$726$	0	0
$727$	3.61555i	0.134093i	0.997750	+	0.0670466i	$0.0213576\pi$
$727$	3.61555i	0.134093i	−0.997750	+	0.0670466i	$0.978642\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.7856	−0.509880
$732$	0	0
$733$	−22.0151	−0.813145	−0.406572	−	0.913619i	$-0.633276\pi$
$733$	−22.0151	−0.813145	−0.406572	+	0.913619i	$0.633276\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.02445i	− 0.148243i
$738$	0	0
$739$	12.0113i	0.441843i	0.975292	+	0.220921i	$0.0709064\pi$
$739$	12.0113i	0.441843i	−0.975292	+	0.220921i	$0.929094\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.7190	−0.466615	−0.233308	−	0.972403i	$-0.574955\pi$
$743$	−12.7190	−0.466615	−0.233308	+	0.972403i	$0.574955\pi$
$744$	0	0
$745$	−5.47438	−0.200566
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 7.94948i	− 0.290468i
$750$	0	0
$751$	− 12.2875i	− 0.448379i	−0.974546	−	0.224189i	$-0.928027\pi$
$751$	− 12.2875i	− 0.448379i	0.974546	−	0.224189i	$-0.0719734\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.88044	0.141224
$756$	0	0
$757$	2.39904	0.0871948	0.0435974	−	0.999049i	$-0.486118\pi$
$757$	2.39904	0.0871948	0.0435974	+	0.999049i	$0.486118\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.90063i	− 0.213898i	−0.994265	−	0.106949i	$-0.965892\pi$
$761$	− 5.90063i	− 0.213898i	0.994265	−	0.106949i	$-0.0341081\pi$
$762$	0	0
$763$	8.86764i	0.321030i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.4368	0.557389
$768$	0	0
$769$	14.2074	0.512332	0.256166	−	0.966633i	$-0.417541\pi$
$769$	14.2074	0.512332	0.256166	+	0.966633i	$0.417541\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.6265i	0.669948i	0.942227	+	0.334974i	$0.108728\pi$
$773$	18.6265i	0.669948i	−0.942227	+	0.334974i	$0.891272\pi$
$774$	0	0
$775$	9.55857i	0.343354i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.308535	0.0110544
$780$	0	0
$781$	3.47473	0.124336
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.92093i	− 0.104253i
$786$	0	0
$787$	− 15.4274i	− 0.549927i	−0.961455	−	0.274963i	$-0.911334\pi$
$787$	− 15.4274i	− 0.549927i	0.961455	−	0.274963i	$-0.0886657\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.6711	0.806091
$792$	0	0
$793$	20.6054	0.731719
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.416058i	0.0147375i	0.999973	+	0.00736876i	$0.00234557\pi$
$797$	0.416058i	0.0147375i	−0.999973	+	0.00736876i	$0.997654\pi$
$798$	0	0
$799$	21.7422i	0.769182i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.42104	−0.120726
$804$	0	0
$805$	−12.1585	−0.428530
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.34000i	0.0822701i	0.999154	+	0.0411350i	$0.0130974\pi$
$809$	2.34000i	0.0822701i	−0.999154	+	0.0411350i	$0.986903\pi$
$810$	0	0
$811$	− 27.1285i	− 0.952612i	−0.879280	−	0.476306i	$-0.841976\pi$
$811$	− 27.1285i	− 0.952612i	0.879280	−	0.476306i	$-0.158024\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.5521	−0.649850
$816$	0	0
$817$	2.18597	0.0764772
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 34.2338i	− 1.19477i	−0.801955	−	0.597384i	$-0.796207\pi$
$821$	− 34.2338i	− 1.19477i	0.801955	−	0.597384i	$-0.203793\pi$
$822$	0	0
$823$	− 30.1425i	− 1.05070i	−0.850885	−	0.525351i	$-0.823934\pi$
$823$	− 30.1425i	− 1.05070i	0.850885	−	0.525351i	$-0.176066\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.5636	1.54963	0.774815	−	0.632189i	$-0.217843\pi$
$827$	44.5636	1.54963	0.774815	+	0.632189i	$0.217843\pi$
$828$	0	0
$829$	−37.2729	−1.29454	−0.647271	−	0.762260i	$-0.724090\pi$
$829$	−37.2729	−1.29454	−0.647271	+	0.762260i	$0.724090\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 8.50293i	− 0.294609i
$834$	0	0
$835$	9.01248i	0.311890i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−11.0489	−0.381451	−0.190725	−	0.981643i	$-0.561084\pi$
$839$	−11.0489	−0.381451	−0.190725	+	0.981643i	$0.561084\pi$
$840$	0	0
$841$	28.8328	0.994235
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.9973i	0.653528i
$846$	0	0
$847$	− 16.0970i	− 0.553100i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39.9582	1.36975
$852$	0	0
$853$	−1.02482	−0.0350892	−0.0175446	−	0.999846i	$-0.505585\pi$
$853$	−1.02482	−0.0350892	−0.0175446	+	0.999846i	$0.505585\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	55.6690i	1.90162i	0.309779	+	0.950809i	$0.399745\pi$
$857$	55.6690i	1.90162i	−0.309779	+	0.950809i	$0.600255\pi$
$858$	0	0
$859$	15.7968i	0.538980i	0.963003	+	0.269490i	$0.0868552\pi$
$859$	15.7968i	0.538980i	−0.963003	+	0.269490i	$0.913145\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.3402	0.590266	0.295133	−	0.955456i	$-0.404636\pi$
$863$	17.3402	0.590266	0.295133	+	0.955456i	$0.404636\pi$
$864$	0	0
$865$	−3.31324	−0.112653
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.4683i	0.558650i
$870$	0	0
$871$	− 18.7504i	− 0.635333i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.68980	0.0571258
$876$	0	0
$877$	8.88706	0.300095	0.150047	−	0.988679i	$-0.452057\pi$
$877$	8.88706	0.300095	0.150047	+	0.988679i	$0.452057\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 20.8452i	− 0.702294i	−0.936320	−	0.351147i	$-0.885792\pi$
$881$	− 20.8452i	− 0.702294i	0.936320	−	0.351147i	$-0.114208\pi$
$882$	0	0
$883$	20.0663i	0.675286i	0.941274	+	0.337643i	$0.109630\pi$
$883$	20.0663i	0.675286i	−0.941274	+	0.337643i	$0.890370\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.9701	1.47637	0.738185	−	0.674598i	$-0.235683\pi$
$887$	43.9701	1.47637	0.738185	+	0.674598i	$0.235683\pi$
$888$	0	0
$889$	−11.1005	−0.372299
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.44762i	− 0.115370i
$894$	0	0
$895$	− 15.7298i	− 0.525788i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.90832	0.130350
$900$	0	0
$901$	−22.9984	−0.766188
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.9183i	0.529141i
$906$	0	0
$907$	11.1829i	0.371324i	0.982614	+	0.185662i	$0.0594429\pi$
$907$	11.1829i	0.371324i	−0.982614	+	0.185662i	$0.940557\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.4886	−1.37458	−0.687290	−	0.726383i	$-0.741200\pi$
$911$	−41.4886	−1.37458	−0.687290	+	0.726383i	$0.741200\pi$
$912$	0	0
$913$	−16.7086	−0.552972
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31.2047i	1.03047i
$918$	0	0
$919$	35.7144i	1.17811i	0.808093	+	0.589054i	$0.200500\pi$
$919$	35.7144i	1.17811i	−0.808093	+	0.589054i	$0.799500\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.1892	0.532873
$924$	0	0
$925$	−5.55345	−0.182596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 30.8878i	− 1.01340i	−0.862124	−	0.506698i	$-0.830866\pi$
$929$	− 30.8878i	− 1.01340i	0.862124	−	0.506698i	$-0.169134\pi$
$930$	0	0
$931$	1.34830i	0.0441886i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.49082	0.0814587
$936$	0	0
$937$	−8.44051	−0.275739	−0.137870	−	0.990450i	$-0.544026\pi$
$937$	−8.44051	−0.275739	−0.137870	+	0.990450i	$0.544026\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 5.42766i	− 0.176937i	−0.996079	−	0.0884684i	$-0.971803\pi$
$941$	− 5.42766i	− 0.176937i	0.996079	−	0.0884684i	$-0.0281972\pi$
$942$	0	0
$943$	− 6.82405i	− 0.222222i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.797677	0.0259210	0.0129605	−	0.999916i	$-0.495874\pi$
$947$	0.797677	0.0259210	0.0129605	+	0.999916i	$0.495874\pi$
$948$	0	0
$949$	−15.9390	−0.517402
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 5.96347i	− 0.193176i	−0.995324	−	0.0965878i	$-0.969207\pi$
$953$	− 5.96347i	− 0.193176i	0.995324	−	0.0965878i	$-0.0307929\pi$
$954$	0	0
$955$	− 20.5862i	− 0.666153i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.38186	0.0446225
$960$	0	0
$961$	−60.3662	−1.94730
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26.0489i	0.838545i
$966$	0	0
$967$	− 9.98843i	− 0.321206i	−0.987019	−	0.160603i	$-0.948656\pi$
$967$	− 9.98843i	− 0.321206i	0.987019	−	0.160603i	$-0.0513439\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.4559	−0.752736	−0.376368	−	0.926470i	$-0.622827\pi$
$971$	−23.4559	−0.752736	−0.376368	+	0.926470i	$0.622827\pi$
$972$	0	0
$973$	18.5286	0.594001
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.4386i	1.22976i	0.788621	+	0.614879i	$0.210795\pi$
$977$	38.4386i	1.22976i	−0.788621	+	0.614879i	$0.789205\pi$
$978$	0	0
$979$	12.2831i	0.392570i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.0716	−1.05482	−0.527409	−	0.849611i	$-0.676837\pi$
$983$	−33.0716	−1.05482	−0.527409	+	0.849611i	$0.676837\pi$
$984$	0	0
$985$	−25.7055	−0.819047
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 48.3482i	− 1.53738i
$990$	0	0
$991$	30.4287i	0.966600i	0.875455	+	0.483300i	$0.160562\pi$
$991$	30.4287i	0.966600i	−0.875455	+	0.483300i	$0.839438\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.8216	0.691790
$996$	0	0
$997$	−55.4111	−1.75489	−0.877443	−	0.479680i	$-0.840753\pi$
$997$	−55.4111	−1.75489	−0.877443	+	0.479680i	$0.840753\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.h.c.2591.14		16
3.2	odd	2	inner	6480.2.h.c.2591.6		16
4.3	odd	2	inner	6480.2.h.c.2591.11		16
9.2	odd	6		720.2.bw.b.671.6	yes	16
9.4	even	3		720.2.bw.b.191.3	✓	16
9.5	odd	6		2160.2.bw.b.1871.1		16
9.7	even	3		2160.2.bw.b.1151.4		16
12.11	even	2	inner	6480.2.h.c.2591.3		16
36.7	odd	6		2160.2.bw.b.1151.1		16
36.11	even	6		720.2.bw.b.671.3	yes	16
36.23	even	6		2160.2.bw.b.1871.4		16
36.31	odd	6		720.2.bw.b.191.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bw.b.191.3	✓	16	9.4	even	3
720.2.bw.b.191.6	yes	16	36.31	odd	6
720.2.bw.b.671.3	yes	16	36.11	even	6
720.2.bw.b.671.6	yes	16	9.2	odd	6
2160.2.bw.b.1151.1		16	36.7	odd	6
2160.2.bw.b.1151.4		16	9.7	even	3
2160.2.bw.b.1871.1		16	9.5	odd	6
2160.2.bw.b.1871.4		16	36.23	even	6
6480.2.h.c.2591.3		16	12.11	even	2	inner
6480.2.h.c.2591.6		16	3.2	odd	2	inner
6480.2.h.c.2591.11		16	4.3	odd	2	inner
6480.2.h.c.2591.14		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +1.68980i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +1.68980i q^{7} +1.21410 q^{11} +5.65662 q^{13} -2.05158i q^{17} +0.325316i q^{19} +7.19521 q^{23} -1.00000 q^{25} +0.408882i q^{29} -9.55857i q^{31} -1.68980 q^{35} +5.55345 q^{37} -0.948415i q^{41} -6.71950i q^{43} -10.5977 q^{47} +4.14457 q^{49} -11.2101i q^{53} +1.21410i q^{55} +2.72897 q^{59} +3.64270 q^{61} +5.65662i q^{65} -3.31477i q^{67} +2.86199 q^{71} -2.81776 q^{73} +2.05158i q^{77} +13.5643i q^{79} -13.7621 q^{83} +2.05158 q^{85} +10.1171i q^{89} +9.55857i q^{91} -0.325316 q^{95} +8.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{13} - 16 q^{25} + 8 q^{37} - 8 q^{49} + 56 q^{61} - 32 q^{73} + 24 q^{85} - 32 q^{97}+O(q^{100}) \) 16 * q - 8 * q^13 - 16 * q^25 + 8 * q^37 - 8 * q^49 + 56 * q^61 - 32 * q^73 + 24 * q^85 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more