LMFDB - Embedded newform 1664.2.b.k.833.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.4
Root			$1.41127 + 0.0912546i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.k.833.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.86678i q^{3} +1.48486i q^{5} -2.55248 q^{7} -0.484862 q^{9} +1.59083i q^{11} -1.00000i q^{13} +2.77191 q^{15} +4.76491 q^{17} -3.51413i q^{19} +4.76491i q^{21} -3.73356 q^{23} +2.79518 q^{25} -4.69521i q^{27} -2.00000i q^{29} -0.685698 q^{31} +2.96972 q^{33} -3.79008i q^{35} -1.73463i q^{37} -1.86678 q^{39} +7.52982 q^{41} -4.69521i q^{43} -0.719953i q^{45} +5.38090 q^{47} -0.484862 q^{49} -8.89503i q^{51} -5.52982i q^{53} -2.36216 q^{55} -6.56009 q^{57} -10.9812i q^{59} -12.4995i q^{61} +1.23760 q^{63} +1.48486 q^{65} +2.96222i q^{67} +6.96972i q^{69} +11.3910 q^{71} -9.46927 q^{73} -5.21799i q^{75} -4.06055i q^{77} +6.91521 q^{79} -10.2195 q^{81} -4.06603i q^{83} +7.07523i q^{85} -3.73356 q^{87} +0.969724 q^{89} +2.55248i q^{91} +1.28005i q^{93} +5.21799 q^{95} +3.93945 q^{97} -0.771332i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 8 * q^17 - 28 * q^25 + 32 * q^33 - 40 * q^41 - 4 * q^49 + 48 * q^57 + 16 * q^65 + 24 * q^73 - 52 * q^81 + 8 * q^89 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.86678i	− 1.07779i	−0.842375	−	0.538893i	$-0.818843\pi$
$3$	− 1.86678i	− 1.07779i	0.842375	−	0.538893i	$-0.181157\pi$
$4$	0	0
$5$	1.48486i	0.664050i	0.943271	+	0.332025i	$0.107732\pi$
$5$	1.48486i	0.664050i	−0.943271	+	0.332025i	$0.892268\pi$
$6$	0	0
$7$	−2.55248	−0.964746	−0.482373	−	0.875966i	$-0.660225\pi$
$7$	−2.55248	−0.964746	−0.482373	+	0.875966i	$0.660225\pi$
$8$	0	0
$9$	−0.484862	−0.161621
$10$	0	0
$11$	1.59083i	0.479653i	0.970816	+	0.239826i	$0.0770905\pi$
$11$	1.59083i	0.479653i	−0.970816	+	0.239826i	$0.922909\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	2.77191	0.715704
$16$	0	0
$17$	4.76491	1.15566	0.577830	−	0.816157i	$-0.303900\pi$
$17$	4.76491	1.15566	0.577830	+	0.816157i	$0.303900\pi$
$18$	0	0
$19$	− 3.51413i	− 0.806196i	−0.915157	−	0.403098i	$-0.867933\pi$
$19$	− 3.51413i	− 0.806196i	0.915157	−	0.403098i	$-0.132067\pi$
$20$	0	0
$21$	4.76491i	1.03979i
$22$	0	0
$23$	−3.73356	−0.778500	−0.389250	−	0.921132i	$-0.627266\pi$
$23$	−3.73356	−0.778500	−0.389250	+	0.921132i	$0.627266\pi$
$24$	0	0
$25$	2.79518	0.559037
$26$	0	0
$27$	− 4.69521i	− 0.903593i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−0.685698	−0.123155	−0.0615775	−	0.998102i	$-0.519613\pi$
$31$	−0.685698	−0.123155	−0.0615775	+	0.998102i	$0.519613\pi$
$32$	0	0
$33$	2.96972	0.516963
$34$	0	0
$35$	− 3.79008i	− 0.640640i
$36$	0	0
$37$	− 1.73463i	− 0.285172i	−0.989782	−	0.142586i	$-0.954458\pi$
$37$	− 1.73463i	− 0.285172i	0.989782	−	0.142586i	$-0.0455417\pi$
$38$	0	0
$39$	−1.86678	−0.298924
$40$	0	0
$41$	7.52982	1.17596	0.587980	−	0.808875i	$-0.299923\pi$
$41$	7.52982	1.17596	0.587980	+	0.808875i	$0.299923\pi$
$42$	0	0
$43$	− 4.69521i	− 0.716012i	−0.933719	−	0.358006i	$-0.883457\pi$
$43$	− 4.69521i	− 0.716012i	0.933719	−	0.358006i	$-0.116543\pi$
$44$	0	0
$45$	− 0.719953i	− 0.107324i
$46$	0	0
$47$	5.38090	0.784886	0.392443	−	0.919776i	$-0.371630\pi$
$47$	5.38090	0.784886	0.392443	+	0.919776i	$0.371630\pi$
$48$	0	0
$49$	−0.484862	−0.0692660
$50$	0	0
$51$	− 8.89503i	− 1.24555i
$52$	0	0
$53$	− 5.52982i	− 0.759579i	−0.925073	−	0.379789i	$-0.875996\pi$
$53$	− 5.52982i	− 0.759579i	0.925073	−	0.379789i	$-0.124004\pi$
$54$	0	0
$55$	−2.36216	−0.318514
$56$	0	0
$57$	−6.56009	−0.868906
$58$	0	0
$59$	− 10.9812i	− 1.42964i	−0.699311	−	0.714818i	$-0.746510\pi$
$59$	− 10.9812i	− 1.42964i	0.699311	−	0.714818i	$-0.253490\pi$
$60$	0	0
$61$	− 12.4995i	− 1.60040i	−0.599732	−	0.800201i	$-0.704726\pi$
$61$	− 12.4995i	− 1.60040i	0.599732	−	0.800201i	$-0.295274\pi$
$62$	0	0
$63$	1.23760	0.155923
$64$	0	0
$65$	1.48486	0.184174
$66$	0	0
$67$	2.96222i	0.361893i	0.983493	+	0.180947i	$0.0579161\pi$
$67$	2.96222i	0.361893i	−0.983493	+	0.180947i	$0.942084\pi$
$68$	0	0
$69$	6.96972i	0.839056i
$70$	0	0
$71$	11.3910	1.35186	0.675931	−	0.736965i	$-0.263742\pi$
$71$	11.3910	1.35186	0.675931	+	0.736965i	$0.263742\pi$
$72$	0	0
$73$	−9.46927	−1.10829	−0.554147	−	0.832419i	$-0.686955\pi$
$73$	−9.46927	−1.10829	−0.554147	+	0.832419i	$0.686955\pi$
$74$	0	0
$75$	− 5.21799i	− 0.602522i
$76$	0	0
$77$	− 4.06055i	− 0.462743i
$78$	0	0
$79$	6.91521	0.778022	0.389011	−	0.921233i	$-0.372817\pi$
$79$	6.91521	0.778022	0.389011	+	0.921233i	$0.372817\pi$
$80$	0	0
$81$	−10.2195	−1.13550
$82$	0	0
$83$	− 4.06603i	− 0.446304i	−0.974784	−	0.223152i	$-0.928365\pi$
$83$	− 4.06603i	− 0.446304i	0.974784	−	0.223152i	$-0.0716347\pi$
$84$	0	0
$85$	7.07523i	0.767417i
$86$	0	0
$87$	−3.73356	−0.400279
$88$	0	0
$89$	0.969724	0.102791	0.0513953	−	0.998678i	$-0.483633\pi$
$89$	0.969724	0.102791	0.0513953	+	0.998678i	$0.483633\pi$
$90$	0	0
$91$	2.55248i	0.267572i
$92$	0	0
$93$	1.28005i	0.132735i
$94$	0	0
$95$	5.21799	0.535355
$96$	0	0
$97$	3.93945	0.399990	0.199995	−	0.979797i	$-0.435907\pi$
$97$	3.93945	0.399990	0.199995	+	0.979797i	$0.435907\pi$
$98$	0	0
$99$	− 0.771332i	− 0.0775218i
$100$	0	0
$101$	18.5601i	1.84680i	0.383841	+	0.923399i	$0.374601\pi$
$101$	18.5601i	1.84680i	−0.383841	+	0.923399i	$0.625399\pi$
$102$	0	0
$103$	−6.47635	−0.638134	−0.319067	−	0.947732i	$-0.603369\pi$
$103$	−6.47635	−0.638134	−0.319067	+	0.947732i	$0.603369\pi$
$104$	0	0
$105$	−7.07523	−0.690472
$106$	0	0
$107$	− 0.0856337i	− 0.00827852i	−0.999991	−	0.00413926i	$-0.998682\pi$
$107$	− 0.0856337i	− 0.00827852i	0.999991	−	0.00413926i	$-0.00131757\pi$
$108$	0	0
$109$	− 8.82546i	− 0.845326i	−0.906287	−	0.422663i	$-0.861095\pi$
$109$	− 8.82546i	− 0.845326i	0.906287	−	0.422663i	$-0.138905\pi$
$110$	0	0
$111$	−3.23818	−0.307354
$112$	0	0
$113$	−15.4693	−1.45523	−0.727613	−	0.685988i	$-0.759370\pi$
$113$	−15.4693	−1.45523	−0.727613	+	0.685988i	$0.759370\pi$
$114$	0	0
$115$	− 5.54382i	− 0.516964i
$116$	0	0
$117$	0.484862i	0.0448255i
$118$	0	0
$119$	−12.1623	−1.11492
$120$	0	0
$121$	8.46927	0.769933
$122$	0	0
$123$	− 14.0565i	− 1.26743i
$124$	0	0
$125$	11.5748i	1.03528i
$126$	0	0
$127$	−7.58015	−0.672630	−0.336315	−	0.941750i	$-0.609181\pi$
$127$	−7.58015	−0.672630	−0.336315	+	0.941750i	$0.609181\pi$
$128$	0	0
$129$	−8.76491	−0.771707
$130$	0	0
$131$	− 2.33305i	− 0.203839i	−0.994793	−	0.101920i	$-0.967502\pi$
$131$	− 2.33305i	− 0.203839i	0.994793	−	0.101920i	$-0.0324984\pi$
$132$	0	0
$133$	8.96972i	0.777774i
$134$	0	0
$135$	6.97173	0.600031
$136$	0	0
$137$	−0.0605522	−0.00517332	−0.00258666	−	0.999997i	$-0.500823\pi$
$137$	−0.0605522	−0.00517332	−0.00258666	+	0.999997i	$0.500823\pi$
$138$	0	0
$139$	− 6.41983i	− 0.544523i	−0.962223	−	0.272262i	$-0.912228\pi$
$139$	− 6.41983i	− 0.544523i	0.962223	−	0.272262i	$-0.0877716\pi$
$140$	0	0
$141$	− 10.0450i	− 0.845938i
$142$	0	0
$143$	1.59083	0.133032
$144$	0	0
$145$	2.96972	0.246622
$146$	0	0
$147$	0.905130i	0.0746539i
$148$	0	0
$149$	7.93945i	0.650425i	0.945641	+	0.325212i	$0.105436\pi$
$149$	7.93945i	0.650425i	−0.945641	+	0.325212i	$0.894564\pi$
$150$	0	0
$151$	2.63811	0.214686	0.107343	−	0.994222i	$-0.465766\pi$
$151$	2.63811	0.214686	0.107343	+	0.994222i	$0.465766\pi$
$152$	0	0
$153$	−2.31032	−0.186779
$154$	0	0
$155$	− 1.01817i	− 0.0817812i
$156$	0	0
$157$	− 11.5298i	− 0.920180i	−0.887872	−	0.460090i	$-0.847817\pi$
$157$	− 11.5298i	− 0.920180i	0.887872	−	0.460090i	$-0.152183\pi$
$158$	0	0
$159$	−10.3229	−0.818663
$160$	0	0
$161$	9.52982	0.751055
$162$	0	0
$163$	17.8965i	1.40176i	0.713280	+	0.700879i	$0.247209\pi$
$163$	17.8965i	1.40176i	−0.713280	+	0.700879i	$0.752791\pi$
$164$	0	0
$165$	4.40963i	0.343289i
$166$	0	0
$167$	19.9054	1.54032	0.770162	−	0.637848i	$-0.220175\pi$
$167$	19.9054	1.54032	0.770162	+	0.637848i	$0.220175\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	1.70387i	0.130298i
$172$	0	0
$173$	− 23.4693i	− 1.78434i	−0.451704	−	0.892168i	$-0.649184\pi$
$173$	− 23.4693i	− 1.78434i	0.451704	−	0.892168i	$-0.350816\pi$
$174$	0	0
$175$	−7.13464	−0.539328
$176$	0	0
$177$	−20.4995	−1.54084
$178$	0	0
$179$	− 24.6488i	− 1.84233i	−0.389167	−	0.921167i	$-0.627237\pi$
$179$	− 24.6488i	− 1.84233i	0.389167	−	0.921167i	$-0.372763\pi$
$180$	0	0
$181$	− 18.4995i	− 1.37506i	−0.726156	−	0.687530i	$-0.758695\pi$
$181$	− 18.4995i	− 1.37506i	0.726156	−	0.687530i	$-0.241305\pi$
$182$	0	0
$183$	−23.3339	−1.72489
$184$	0	0
$185$	2.57569	0.189369
$186$	0	0
$187$	7.58015i	0.554316i
$188$	0	0
$189$	11.9844i	0.871737i
$190$	0	0
$191$	−2.36216	−0.170920	−0.0854600	−	0.996342i	$-0.527236\pi$
$191$	−2.36216	−0.170920	−0.0854600	+	0.996342i	$0.527236\pi$
$192$	0	0
$193$	0.0605522	0.00435864	0.00217932	−	0.999998i	$-0.499306\pi$
$193$	0.0605522	0.00435864	0.00217932	+	0.999998i	$0.499306\pi$
$194$	0	0
$195$	− 2.77191i	− 0.198500i
$196$	0	0
$197$	15.7952i	1.12536i	0.826675	+	0.562680i	$0.190230\pi$
$197$	15.7952i	1.12536i	−0.826675	+	0.562680i	$0.809770\pi$
$198$	0	0
$199$	22.5144	1.59600	0.798001	−	0.602656i	$-0.205891\pi$
$199$	22.5144	1.59600	0.798001	+	0.602656i	$0.205891\pi$
$200$	0	0
$201$	5.52982	0.390043
$202$	0	0
$203$	5.10495i	0.358298i
$204$	0	0
$205$	11.1807i	0.780897i
$206$	0	0
$207$	1.81026	0.125822
$208$	0	0
$209$	5.59037	0.386694
$210$	0	0
$211$	24.8474i	1.71057i	0.518160	+	0.855284i	$0.326617\pi$
$211$	24.8474i	1.71057i	−0.518160	+	0.855284i	$0.673383\pi$
$212$	0	0
$213$	− 21.2645i	− 1.45702i
$214$	0	0
$215$	6.97173	0.475468
$216$	0	0
$217$	1.75023	0.118813
$218$	0	0
$219$	17.6770i	1.19450i
$220$	0	0
$221$	− 4.76491i	− 0.320522i
$222$	0	0
$223$	−15.3232	−1.02612	−0.513059	−	0.858353i	$-0.671488\pi$
$223$	−15.3232	−1.02612	−0.513059	+	0.858353i	$0.671488\pi$
$224$	0	0
$225$	−1.35528	−0.0903519
$226$	0	0
$227$	− 6.25692i	− 0.415286i	−0.978205	−	0.207643i	$-0.933421\pi$
$227$	− 6.25692i	− 0.415286i	0.978205	−	0.207643i	$-0.0665793\pi$
$228$	0	0
$229$	21.0752i	1.39269i	0.717707	+	0.696345i	$0.245192\pi$
$229$	21.0752i	1.39269i	−0.717707	+	0.696345i	$0.754808\pi$
$230$	0	0
$231$	−7.58015	−0.498737
$232$	0	0
$233$	−3.01468	−0.197498	−0.0987491	−	0.995112i	$-0.531484\pi$
$233$	−3.01468	−0.197498	−0.0987491	+	0.995112i	$0.531484\pi$
$234$	0	0
$235$	7.98990i	0.521204i
$236$	0	0
$237$	− 12.9092i	− 0.838541i
$238$	0	0
$239$	−18.3062	−1.18413	−0.592065	−	0.805890i	$-0.701687\pi$
$239$	−18.3062	−1.18413	−0.592065	+	0.805890i	$0.701687\pi$
$240$	0	0
$241$	28.0294	1.80553	0.902765	−	0.430134i	$-0.141534\pi$
$241$	28.0294	1.80553	0.902765	+	0.430134i	$0.141534\pi$
$242$	0	0
$243$	4.99192i	0.320232i
$244$	0	0
$245$	− 0.719953i	− 0.0459961i
$246$	0	0
$247$	−3.51413	−0.223598
$248$	0	0
$249$	−7.59037	−0.481020
$250$	0	0
$251$	9.85668i	0.622148i	0.950386	+	0.311074i	$0.100689\pi$
$251$	9.85668i	0.622148i	−0.950386	+	0.311074i	$0.899311\pi$
$252$	0	0
$253$	− 5.93945i	− 0.373410i
$254$	0	0
$255$	13.2079	0.827110
$256$	0	0
$257$	−0.204815	−0.0127760	−0.00638800	−	0.999980i	$-0.502033\pi$
$257$	−0.204815	−0.0127760	−0.00638800	+	0.999980i	$0.502033\pi$
$258$	0	0
$259$	4.42761i	0.275118i
$260$	0	0
$261$	0.969724i	0.0600244i
$262$	0	0
$263$	15.2018	0.937385	0.468692	−	0.883361i	$-0.344725\pi$
$263$	15.2018	0.937385	0.468692	+	0.883361i	$0.344725\pi$
$264$	0	0
$265$	8.21102	0.504399
$266$	0	0
$267$	− 1.81026i	− 0.110786i
$268$	0	0
$269$	− 9.09083i	− 0.554278i	−0.960830	−	0.277139i	$-0.910614\pi$
$269$	− 9.09083i	− 0.554278i	0.960830	−	0.277139i	$-0.0893862\pi$
$270$	0	0
$271$	4.67445	0.283952	0.141976	−	0.989870i	$-0.454654\pi$
$271$	4.67445	0.283952	0.141976	+	0.989870i	$0.454654\pi$
$272$	0	0
$273$	4.76491	0.288385
$274$	0	0
$275$	4.44666i	0.268144i
$276$	0	0
$277$	− 2.06055i	− 0.123807i	−0.998082	−	0.0619033i	$-0.980283\pi$
$277$	− 2.06055i	− 0.123807i	0.998082	−	0.0619033i	$-0.0197170\pi$
$278$	0	0
$279$	0.332469	0.0199044
$280$	0	0
$281$	17.5904	1.04935	0.524677	−	0.851302i	$-0.324186\pi$
$281$	17.5904	1.04935	0.524677	+	0.851302i	$0.324186\pi$
$282$	0	0
$283$	14.5810i	0.866750i	0.901214	+	0.433375i	$0.142678\pi$
$283$	14.5810i	0.866750i	−0.901214	+	0.433375i	$0.857322\pi$
$284$	0	0
$285$	− 9.74083i	− 0.576997i
$286$	0	0
$287$	−19.2197	−1.13450
$288$	0	0
$289$	5.70436	0.335550
$290$	0	0
$291$	− 7.35408i	− 0.431104i
$292$	0	0
$293$	14.8860i	0.869650i	0.900515	+	0.434825i	$0.143190\pi$
$293$	14.8860i	0.869650i	−0.900515	+	0.434825i	$0.856810\pi$
$294$	0	0
$295$	16.3056	0.949350
$296$	0	0
$297$	7.46927	0.433411
$298$	0	0
$299$	3.73356i	0.215917i
$300$	0	0
$301$	11.9844i	0.690770i
$302$	0	0
$303$	34.6476	1.99045
$304$	0	0
$305$	18.5601	1.06275
$306$	0	0
$307$	17.3030i	0.987536i	0.869594	+	0.493768i	$0.164381\pi$
$307$	17.3030i	0.987536i	−0.869594	+	0.493768i	$0.835619\pi$
$308$	0	0
$309$	12.0899i	0.687771i
$310$	0	0
$311$	4.55305	0.258180	0.129090	−	0.991633i	$-0.458794\pi$
$311$	4.55305	0.258180	0.129090	+	0.991633i	$0.458794\pi$
$312$	0	0
$313$	14.5445	0.822104	0.411052	−	0.911612i	$-0.365162\pi$
$313$	14.5445	0.822104	0.411052	+	0.911612i	$0.365162\pi$
$314$	0	0
$315$	1.83766i	0.103541i
$316$	0	0
$317$	− 21.0596i	− 1.18283i	−0.806368	−	0.591413i	$-0.798570\pi$
$317$	− 21.0596i	− 1.18283i	0.806368	−	0.591413i	$-0.201430\pi$
$318$	0	0
$319$	3.18166	0.178139
$320$	0	0
$321$	−0.159859	−0.00892247
$322$	0	0
$323$	− 16.7445i	− 0.931688i
$324$	0	0
$325$	− 2.79518i	− 0.155049i
$326$	0	0
$327$	−16.4752	−0.911080
$328$	0	0
$329$	−13.7346	−0.757215
$330$	0	0
$331$	17.1900i	0.944848i	0.881372	+	0.472424i	$0.156621\pi$
$331$	17.1900i	0.944848i	−0.881372	+	0.472424i	$0.843379\pi$
$332$	0	0
$333$	0.841057i	0.0460897i
$334$	0	0
$335$	−4.39850	−0.240315
$336$	0	0
$337$	−4.10551	−0.223641	−0.111821	−	0.993728i	$-0.535668\pi$
$337$	−4.10551	−0.223641	−0.111821	+	0.993728i	$0.535668\pi$
$338$	0	0
$339$	28.8777i	1.56842i
$340$	0	0
$341$	− 1.09083i	− 0.0590717i
$342$	0	0
$343$	19.1049	1.03157
$344$	0	0
$345$	−10.3491	−0.557176
$346$	0	0
$347$	3.79008i	0.203462i	0.994812	+	0.101731i	$0.0324381\pi$
$347$	3.79008i	0.203462i	−0.994812	+	0.101731i	$0.967562\pi$
$348$	0	0
$349$	35.3931i	1.89455i	0.320422	+	0.947275i	$0.396175\pi$
$349$	35.3931i	1.89455i	−0.320422	+	0.947275i	$0.603825\pi$
$350$	0	0
$351$	−4.69521	−0.250612
$352$	0	0
$353$	7.43991	0.395986	0.197993	−	0.980203i	$-0.436558\pi$
$353$	7.43991	0.395986	0.197993	+	0.980203i	$0.436558\pi$
$354$	0	0
$355$	16.9140i	0.897704i
$356$	0	0
$357$	22.7044i	1.20164i
$358$	0	0
$359$	5.63609	0.297462	0.148731	−	0.988878i	$-0.452481\pi$
$359$	5.63609	0.297462	0.148731	+	0.988878i	$0.452481\pi$
$360$	0	0
$361$	6.65092	0.350049
$362$	0	0
$363$	− 15.8102i	− 0.829823i
$364$	0	0
$365$	− 14.0606i	− 0.735963i
$366$	0	0
$367$	−34.9734	−1.82560	−0.912799	−	0.408410i	$-0.866083\pi$
$367$	−34.9734	−1.82560	−0.912799	+	0.408410i	$0.866083\pi$
$368$	0	0
$369$	−3.65092	−0.190059
$370$	0	0
$371$	14.1147i	0.732800i
$372$	0	0
$373$	− 21.5904i	− 1.11791i	−0.829199	−	0.558953i	$-0.811203\pi$
$373$	− 21.5904i	− 1.11791i	0.829199	−	0.558953i	$-0.188797\pi$
$374$	0	0
$375$	21.6075	1.11581
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	32.1242i	1.65011i	0.565052	+	0.825055i	$0.308856\pi$
$379$	32.1242i	1.65011i	−0.565052	+	0.825055i	$0.691144\pi$
$380$	0	0
$381$	14.1505i	0.724950i
$382$	0	0
$383$	3.62887	0.185427	0.0927134	−	0.995693i	$-0.470446\pi$
$383$	3.62887	0.185427	0.0927134	+	0.995693i	$0.470446\pi$
$384$	0	0
$385$	6.02936	0.307285
$386$	0	0
$387$	2.27653i	0.115722i
$388$	0	0
$389$	− 17.9688i	− 0.911055i	−0.890222	−	0.455527i	$-0.849451\pi$
$389$	− 17.9688i	− 0.911055i	0.890222	−	0.455527i	$-0.150549\pi$
$390$	0	0
$391$	−17.7901	−0.899682
$392$	0	0
$393$	−4.35528	−0.219695
$394$	0	0
$395$	10.2681i	0.516646i
$396$	0	0
$397$	30.9991i	1.55580i	0.628388	+	0.777900i	$0.283715\pi$
$397$	30.9991i	1.55580i	−0.628388	+	0.777900i	$0.716285\pi$
$398$	0	0
$399$	16.7445	0.838273
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	0.685698i	0.0341571i
$404$	0	0
$405$	− 15.1745i	− 0.754029i
$406$	0	0
$407$	2.75950	0.136783
$408$	0	0
$409$	−30.9991	−1.53281	−0.766403	−	0.642360i	$-0.777955\pi$
$409$	−30.9991	−1.53281	−0.766403	+	0.642360i	$0.777955\pi$
$410$	0	0
$411$	0.113038i	0.00557573i
$412$	0	0
$413$	28.0294i	1.37923i
$414$	0	0
$415$	6.03749	0.296369
$416$	0	0
$417$	−11.9844	−0.586879
$418$	0	0
$419$	18.7243i	0.914742i	0.889276	+	0.457371i	$0.151209\pi$
$419$	18.7243i	0.914742i	−0.889276	+	0.457371i	$0.848791\pi$
$420$	0	0
$421$	− 13.5748i	− 0.661594i	−0.943702	−	0.330797i	$-0.892682\pi$
$421$	− 13.5748i	− 0.661594i	0.943702	−	0.330797i	$-0.107318\pi$
$422$	0	0
$423$	−2.60900	−0.126854
$424$	0	0
$425$	13.3188	0.646057
$426$	0	0
$427$	31.9048i	1.54398i
$428$	0	0
$429$	− 2.96972i	− 0.143380i
$430$	0	0
$431$	−32.0951	−1.54597	−0.772983	−	0.634426i	$-0.781236\pi$
$431$	−32.0951	−1.54597	−0.772983	+	0.634426i	$0.781236\pi$
$432$	0	0
$433$	20.0450	0.963299	0.481650	−	0.876364i	$-0.340038\pi$
$433$	20.0450	0.963299	0.481650	+	0.876364i	$0.340038\pi$
$434$	0	0
$435$	− 5.54382i	− 0.265806i
$436$	0	0
$437$	13.1202i	0.627624i
$438$	0	0
$439$	−35.7514	−1.70632	−0.853160	−	0.521649i	$-0.825317\pi$
$439$	−35.7514	−1.70632	−0.853160	+	0.521649i	$0.825317\pi$
$440$	0	0
$441$	0.235091	0.0111948
$442$	0	0
$443$	− 40.6011i	− 1.92902i	−0.264049	−	0.964509i	$-0.585058\pi$
$443$	− 40.6011i	− 1.92902i	0.264049	−	0.964509i	$-0.414942\pi$
$444$	0	0
$445$	1.43991i	0.0682581i
$446$	0	0
$447$	14.8212	0.701018
$448$	0	0
$449$	−23.4986	−1.10897	−0.554484	−	0.832194i	$-0.687084\pi$
$449$	−23.4986	−1.10897	−0.554484	+	0.832194i	$0.687084\pi$
$450$	0	0
$451$	11.9786i	0.564052i
$452$	0	0
$453$	− 4.92477i	− 0.231386i
$454$	0	0
$455$	−3.79008	−0.177681
$456$	0	0
$457$	−32.0294	−1.49827	−0.749135	−	0.662417i	$-0.769531\pi$
$457$	−32.0294	−1.49827	−0.749135	+	0.662417i	$0.769531\pi$
$458$	0	0
$459$	− 22.3722i	− 1.04425i
$460$	0	0
$461$	− 25.7034i	− 1.19713i	−0.801075	−	0.598564i	$-0.795738\pi$
$461$	− 25.7034i	− 1.19713i	0.801075	−	0.598564i	$-0.204262\pi$
$462$	0	0
$463$	30.5542	1.41997	0.709986	−	0.704216i	$-0.248701\pi$
$463$	30.5542	1.41997	0.709986	+	0.704216i	$0.248701\pi$
$464$	0	0
$465$	−1.90069	−0.0881425
$466$	0	0
$467$	18.9628i	0.877493i	0.898611	+	0.438747i	$0.144577\pi$
$467$	18.9628i	0.877493i	−0.898611	+	0.438747i	$0.855423\pi$
$468$	0	0
$469$	− 7.56101i	− 0.349135i
$470$	0	0
$471$	−21.5236	−0.991756
$472$	0	0
$473$	7.46927	0.343437
$474$	0	0
$475$	− 9.82263i	− 0.450693i
$476$	0	0
$477$	2.68120i	0.122764i
$478$	0	0
$479$	33.0002	1.50782	0.753910	−	0.656978i	$-0.228166\pi$
$479$	33.0002	1.50782	0.753910	+	0.656978i	$0.228166\pi$
$480$	0	0
$481$	−1.73463	−0.0790924
$482$	0	0
$483$	− 17.7901i	− 0.809476i
$484$	0	0
$485$	5.84954i	0.265614i
$486$	0	0
$487$	13.3708	0.605889	0.302944	−	0.953008i	$-0.402030\pi$
$487$	13.3708	0.605889	0.302944	+	0.953008i	$0.402030\pi$
$488$	0	0
$489$	33.4087	1.51079
$490$	0	0
$491$	33.5729i	1.51512i	0.652763	+	0.757562i	$0.273610\pi$
$491$	33.5729i	1.51512i	−0.652763	+	0.757562i	$0.726390\pi$
$492$	0	0
$493$	− 9.52982i	− 0.429201i
$494$	0	0
$495$	1.14532	0.0514784
$496$	0	0
$497$	−29.0752	−1.30420
$498$	0	0
$499$	22.5625i	1.01004i	0.863108	+	0.505019i	$0.168515\pi$
$499$	22.5625i	1.01004i	−0.863108	+	0.505019i	$0.831485\pi$
$500$	0	0
$501$	− 37.1589i	− 1.66014i
$502$	0	0
$503$	−34.9152	−1.55679	−0.778395	−	0.627774i	$-0.783966\pi$
$503$	−34.9152	−1.55679	−0.778395	+	0.627774i	$0.783966\pi$
$504$	0	0
$505$	−27.5592	−1.22637
$506$	0	0
$507$	1.86678i	0.0829065i
$508$	0	0
$509$	− 16.9385i	− 0.750787i	−0.926865	−	0.375394i	$-0.877508\pi$
$509$	− 16.9385i	− 0.750787i	0.926865	−	0.375394i	$-0.122492\pi$
$510$	0	0
$511$	24.1701	1.06922
$512$	0	0
$513$	−16.4995	−0.728473
$514$	0	0
$515$	− 9.61649i	− 0.423753i
$516$	0	0
$517$	8.56009i	0.376472i
$518$	0	0
$519$	−43.8119	−1.92313
$520$	0	0
$521$	8.10551	0.355109	0.177554	−	0.984111i	$-0.443181\pi$
$521$	8.10551	0.355109	0.177554	+	0.984111i	$0.443181\pi$
$522$	0	0
$523$	− 41.8030i	− 1.82792i	−0.405808	−	0.913959i	$-0.633010\pi$
$523$	− 41.8030i	− 1.82792i	0.405808	−	0.913959i	$-0.366990\pi$
$524$	0	0
$525$	13.3188i	0.581280i
$526$	0	0
$527$	−3.26729	−0.142325
$528$	0	0
$529$	−9.06055	−0.393937
$530$	0	0
$531$	5.32439i	0.231059i
$532$	0	0
$533$	− 7.52982i	− 0.326153i
$534$	0	0
$535$	0.127154	0.00549736
$536$	0	0
$537$	−46.0138	−1.98564
$538$	0	0
$539$	− 0.771332i	− 0.0332236i
$540$	0	0
$541$	22.2654i	0.957263i	0.878016	+	0.478631i	$0.158867\pi$
$541$	22.2654i	0.957263i	−0.878016	+	0.478631i	$0.841133\pi$
$542$	0	0
$543$	−34.5345	−1.48202
$544$	0	0
$545$	13.1046	0.561339
$546$	0	0
$547$	8.74047i	0.373716i	0.982387	+	0.186858i	$0.0598304\pi$
$547$	8.74047i	0.373716i	−0.982387	+	0.186858i	$0.940170\pi$
$548$	0	0
$549$	6.06055i	0.258658i
$550$	0	0
$551$	−7.02825	−0.299414
$552$	0	0
$553$	−17.6509	−0.750594
$554$	0	0
$555$	− 4.80824i	− 0.204099i
$556$	0	0
$557$	− 15.1745i	− 0.642966i	−0.946915	−	0.321483i	$-0.895819\pi$
$557$	− 15.1745i	− 0.642966i	0.946915	−	0.321483i	$-0.104181\pi$
$558$	0	0
$559$	−4.69521	−0.198586
$560$	0	0
$561$	14.1505	0.597433
$562$	0	0
$563$	− 2.41868i	− 0.101935i	−0.998700	−	0.0509676i	$-0.983769\pi$
$563$	− 2.41868i	− 0.101935i	0.998700	−	0.0509676i	$-0.0162305\pi$
$564$	0	0
$565$	− 22.9697i	− 0.966344i
$566$	0	0
$567$	26.0850	1.09547
$568$	0	0
$569$	5.61353	0.235331	0.117666	−	0.993053i	$-0.462459\pi$
$569$	5.61353	0.235331	0.117666	+	0.993053i	$0.462459\pi$
$570$	0	0
$571$	23.8293i	0.997223i	0.866825	+	0.498612i	$0.166157\pi$
$571$	23.8293i	0.997223i	−0.866825	+	0.498612i	$0.833843\pi$
$572$	0	0
$573$	4.40963i	0.184215i
$574$	0	0
$575$	−10.4360	−0.435211
$576$	0	0
$577$	−34.4683	−1.43494	−0.717468	−	0.696591i	$-0.754699\pi$
$577$	−34.4683	−1.43494	−0.717468	+	0.696591i	$0.754699\pi$
$578$	0	0
$579$	− 0.113038i	− 0.00469768i
$580$	0	0
$581$	10.3784i	0.430570i
$582$	0	0
$583$	8.79699	0.364334
$584$	0	0
$585$	−0.719953	−0.0297664
$586$	0	0
$587$	7.51528i	0.310189i	0.987900	+	0.155094i	$0.0495682\pi$
$587$	7.51528i	0.310189i	−0.987900	+	0.155094i	$0.950432\pi$
$588$	0	0
$589$	2.40963i	0.0992871i
$590$	0	0
$591$	29.4861	1.21290
$592$	0	0
$593$	−24.6812	−1.01354	−0.506768	−	0.862083i	$-0.669160\pi$
$593$	−24.6812	−1.01354	−0.506768	+	0.862083i	$0.669160\pi$
$594$	0	0
$595$	− 18.0594i	− 0.740362i
$596$	0	0
$597$	− 42.0294i	− 1.72015i
$598$	0	0
$599$	5.69837	0.232829	0.116415	−	0.993201i	$-0.462860\pi$
$599$	5.69837	0.232829	0.116415	+	0.993201i	$0.462860\pi$
$600$	0	0
$601$	46.6732	1.90384	0.951919	−	0.306350i	$-0.0991077\pi$
$601$	46.6732	1.90384	0.951919	+	0.306350i	$0.0991077\pi$
$602$	0	0
$603$	− 1.43627i	− 0.0584894i
$604$	0	0
$605$	12.5757i	0.511275i
$606$	0	0
$607$	−13.3916	−0.543547	−0.271773	−	0.962361i	$-0.587610\pi$
$607$	−13.3916	−0.543547	−0.271773	+	0.962361i	$0.587610\pi$
$608$	0	0
$609$	9.52982	0.386168
$610$	0	0
$611$	− 5.38090i	− 0.217688i
$612$	0	0
$613$	28.9385i	1.16882i	0.811460	+	0.584408i	$0.198673\pi$
$613$	28.9385i	1.16882i	−0.811460	+	0.584408i	$0.801327\pi$
$614$	0	0
$615$	20.8720	0.841639
$616$	0	0
$617$	−7.65092	−0.308015	−0.154007	−	0.988070i	$-0.549218\pi$
$617$	−7.65092	−0.308015	−0.154007	+	0.988070i	$0.549218\pi$
$618$	0	0
$619$	30.4685i	1.22463i	0.790613	+	0.612317i	$0.209762\pi$
$619$	30.4685i	1.22463i	−0.790613	+	0.612317i	$0.790238\pi$
$620$	0	0
$621$	17.5298i	0.703447i
$622$	0	0
$623$	−2.47520	−0.0991667
$624$	0	0
$625$	−3.21102	−0.128441
$626$	0	0
$627$	− 10.4360i	− 0.416773i
$628$	0	0
$629$	− 8.26537i	− 0.329562i
$630$	0	0
$631$	−2.25748	−0.0898688	−0.0449344	−	0.998990i	$-0.514308\pi$
$631$	−2.25748	−0.0898688	−0.0449344	+	0.998990i	$0.514308\pi$
$632$	0	0
$633$	46.3846	1.84362
$634$	0	0
$635$	− 11.2555i	− 0.446660i
$636$	0	0
$637$	0.484862i	0.0192109i
$638$	0	0
$639$	−5.52306	−0.218489
$640$	0	0
$641$	11.7115	0.462575	0.231288	−	0.972885i	$-0.425706\pi$
$641$	11.7115	0.462575	0.231288	+	0.972885i	$0.425706\pi$
$642$	0	0
$643$	22.0106i	0.868015i	0.900909	+	0.434008i	$0.142901\pi$
$643$	22.0106i	0.868015i	−0.900909	+	0.434008i	$0.857099\pi$
$644$	0	0
$645$	− 13.0147i	− 0.512453i
$646$	0	0
$647$	−15.1603	−0.596013	−0.298007	−	0.954564i	$-0.596322\pi$
$647$	−15.1603	−0.596013	−0.298007	+	0.954564i	$0.596322\pi$
$648$	0	0
$649$	17.4693	0.685729
$650$	0	0
$651$	− 3.26729i	− 0.128055i
$652$	0	0
$653$	4.15046i	0.162420i	0.996697	+	0.0812101i	$0.0258785\pi$
$653$	4.15046i	0.162420i	−0.996697	+	0.0812101i	$0.974122\pi$
$654$	0	0
$655$	3.46425	0.135359
$656$	0	0
$657$	4.59129	0.179123
$658$	0	0
$659$	− 17.7627i	− 0.691935i	−0.938247	−	0.345967i	$-0.887551\pi$
$659$	− 17.7627i	− 0.691935i	0.938247	−	0.345967i	$-0.112449\pi$
$660$	0	0
$661$	− 23.9394i	− 0.931137i	−0.885012	−	0.465568i	$-0.845850\pi$
$661$	− 23.9394i	− 0.931137i	0.885012	−	0.465568i	$-0.154150\pi$
$662$	0	0
$663$	−8.89503	−0.345454
$664$	0	0
$665$	−13.3188	−0.516481
$666$	0	0
$667$	7.46711i	0.289128i
$668$	0	0
$669$	28.6050i	1.10594i
$670$	0	0
$671$	19.8846	0.767637
$672$	0	0
$673$	−25.6741	−0.989663	−0.494832	−	0.868989i	$-0.664770\pi$
$673$	−25.6741	−0.989663	−0.494832	+	0.868989i	$0.664770\pi$
$674$	0	0
$675$	− 13.1240i	− 0.505142i
$676$	0	0
$677$	13.0303i	0.500794i	0.968143	+	0.250397i	$0.0805612\pi$
$677$	13.0303i	0.500794i	−0.968143	+	0.250397i	$0.919439\pi$
$678$	0	0
$679$	−10.0553	−0.385889
$680$	0	0
$681$	−11.6803	−0.447589
$682$	0	0
$683$	− 48.2171i	− 1.84497i	−0.386028	−	0.922487i	$-0.626153\pi$
$683$	− 48.2171i	− 1.84497i	0.386028	−	0.922487i	$-0.373847\pi$
$684$	0	0
$685$	− 0.0899116i	− 0.00343535i
$686$	0	0
$687$	39.3428	1.50102
$688$	0	0
$689$	−5.52982	−0.210669
$690$	0	0
$691$	− 37.7644i	− 1.43662i	−0.695721	−	0.718312i	$-0.744915\pi$
$691$	− 37.7644i	− 1.43662i	0.695721	−	0.718312i	$-0.255085\pi$
$692$	0	0
$693$	1.96881i	0.0747888i
$694$	0	0
$695$	9.53256	0.361591
$696$	0	0
$697$	35.8789	1.35901
$698$	0	0
$699$	5.62774i	0.212861i
$700$	0	0
$701$	41.4381i	1.56509i	0.622591	+	0.782547i	$0.286080\pi$
$701$	41.4381i	1.56509i	−0.622591	+	0.782547i	$0.713920\pi$
$702$	0	0
$703$	−6.09572	−0.229904
$704$	0	0
$705$	14.9154	0.561745
$706$	0	0
$707$	− 47.3742i	− 1.78169i
$708$	0	0
$709$	5.05964i	0.190019i	0.995476	+	0.0950093i	$0.0302881\pi$
$709$	5.05964i	0.190019i	−0.995476	+	0.0950093i	$0.969712\pi$
$710$	0	0
$711$	−3.35292	−0.125744
$712$	0	0
$713$	2.56009	0.0958763
$714$	0	0
$715$	2.36216i	0.0883398i
$716$	0	0
$717$	34.1736i	1.27624i
$718$	0	0
$719$	50.5726	1.88604	0.943019	−	0.332738i	$-0.107972\pi$
$719$	50.5726	1.88604	0.943019	+	0.332738i	$0.107972\pi$
$720$	0	0
$721$	16.5307	0.615637
$722$	0	0
$723$	− 52.3246i	− 1.94597i
$724$	0	0
$725$	− 5.59037i	− 0.207621i
$726$	0	0
$727$	48.3650	1.79376	0.896879	−	0.442276i	$-0.145829\pi$
$727$	48.3650	1.79376	0.896879	+	0.442276i	$0.145829\pi$
$728$	0	0
$729$	−21.3397	−0.790359
$730$	0	0
$731$	− 22.3722i	− 0.827467i
$732$	0	0
$733$	18.1055i	0.668742i	0.942441	+	0.334371i	$0.108524\pi$
$733$	18.1055i	0.668742i	−0.942441	+	0.334371i	$0.891476\pi$
$734$	0	0
$735$	−1.34399	−0.0495739
$736$	0	0
$737$	−4.71239	−0.173583
$738$	0	0
$739$	− 20.9236i	− 0.769685i	−0.922982	−	0.384843i	$-0.874256\pi$
$739$	− 20.9236i	− 0.769685i	0.922982	−	0.384843i	$-0.125744\pi$
$740$	0	0
$741$	6.56009i	0.240991i
$742$	0	0
$743$	−23.8941	−0.876591	−0.438295	−	0.898831i	$-0.644418\pi$
$743$	−23.8941	−0.876591	−0.438295	+	0.898831i	$0.644418\pi$
$744$	0	0
$745$	−11.7890	−0.431915
$746$	0	0
$747$	1.97146i	0.0721320i
$748$	0	0
$749$	0.218578i	0.00798667i
$750$	0	0
$751$	−48.8754	−1.78349	−0.891743	−	0.452541i	$-0.850518\pi$
$751$	−48.8754	−1.78349	−0.891743	+	0.452541i	$0.850518\pi$
$752$	0	0
$753$	18.4002	0.670542
$754$	0	0
$755$	3.91723i	0.142563i
$756$	0	0
$757$	− 26.1505i	− 0.950455i	−0.879863	−	0.475227i	$-0.842366\pi$
$757$	− 26.1505i	− 0.950455i	0.879863	−	0.475227i	$-0.157634\pi$
$758$	0	0
$759$	−11.0876	−0.402456
$760$	0	0
$761$	−35.9083	−1.30167	−0.650837	−	0.759218i	$-0.725582\pi$
$761$	−35.9083	−1.30167	−0.650837	+	0.759218i	$0.725582\pi$
$762$	0	0
$763$	22.5268i	0.815524i
$764$	0	0
$765$	− 3.43051i	− 0.124030i
$766$	0	0
$767$	−10.9812	−0.396510
$768$	0	0
$769$	17.7115	0.638692	0.319346	−	0.947638i	$-0.396537\pi$
$769$	17.7115	0.638692	0.319346	+	0.947638i	$0.396537\pi$
$770$	0	0
$771$	0.382344i	0.0137698i
$772$	0	0
$773$	− 1.07523i	− 0.0386734i	−0.999813	−	0.0193367i	$-0.993845\pi$
$773$	− 1.07523i	− 0.0386734i	0.999813	−	0.0193367i	$-0.00615545\pi$
$774$	0	0
$775$	−1.91665	−0.0688482
$776$	0	0
$777$	8.26537	0.296518
$778$	0	0
$779$	− 26.4607i	− 0.948054i
$780$	0	0
$781$	18.1211i	0.648424i
$782$	0	0
$783$	−9.39041	−0.335586
$784$	0	0
$785$	17.1202	0.611046
$786$	0	0
$787$	25.6479i	0.914248i	0.889403	+	0.457124i	$0.151120\pi$
$787$	25.6479i	0.914248i	−0.889403	+	0.457124i	$0.848880\pi$
$788$	0	0
$789$	− 28.3784i	− 1.01030i
$790$	0	0
$791$	39.4849	1.40392
$792$	0	0
$793$	−12.4995	−0.443872
$794$	0	0
$795$	− 15.3281i	− 0.543633i
$796$	0	0
$797$	− 12.0294i	− 0.426102i	−0.977041	−	0.213051i	$-0.931660\pi$
$797$	− 12.0294i	− 0.426102i	0.977041	−	0.213051i	$-0.0683400\pi$
$798$	0	0
$799$	25.6395	0.907061
$800$	0	0
$801$	−0.470182	−0.0166131
$802$	0	0
$803$	− 15.0640i	− 0.531596i
$804$	0	0
$805$	14.1505i	0.498738i
$806$	0	0
$807$	−16.9706	−0.597392
$808$	0	0
$809$	24.1055	0.847505	0.423752	−	0.905778i	$-0.360713\pi$
$809$	24.1055	0.847505	0.423752	+	0.905778i	$0.360713\pi$
$810$	0	0
$811$	35.4189i	1.24373i	0.783126	+	0.621863i	$0.213624\pi$
$811$	35.4189i	1.24373i	−0.783126	+	0.621863i	$0.786376\pi$
$812$	0	0
$813$	− 8.72615i	− 0.306040i
$814$	0	0
$815$	−26.5738	−0.930838
$816$	0	0
$817$	−16.4995	−0.577246
$818$	0	0
$819$	− 1.23760i	− 0.0432452i
$820$	0	0
$821$	− 26.8936i	− 0.938592i	−0.883041	−	0.469296i	$-0.844508\pi$
$821$	− 26.8936i	− 0.938592i	0.883041	−	0.469296i	$-0.155492\pi$
$822$	0	0
$823$	31.1983	1.08751	0.543753	−	0.839245i	$-0.317003\pi$
$823$	31.1983	1.08751	0.543753	+	0.839245i	$0.317003\pi$
$824$	0	0
$825$	8.30093	0.289001
$826$	0	0
$827$	− 39.8007i	− 1.38401i	−0.721895	−	0.692003i	$-0.756728\pi$
$827$	− 39.8007i	− 1.38401i	0.721895	−	0.692003i	$-0.243272\pi$
$828$	0	0
$829$	− 10.0294i	− 0.348334i	−0.984716	−	0.174167i	$-0.944277\pi$
$829$	− 10.0294i	− 0.348334i	0.984716	−	0.174167i	$-0.0557233\pi$
$830$	0	0
$831$	−3.84659	−0.133437
$832$	0	0
$833$	−2.31032	−0.0800479
$834$	0	0
$835$	29.5567i	1.02285i
$836$	0	0
$837$	3.21949i	0.111282i
$838$	0	0
$839$	−43.7745	−1.51126	−0.755631	−	0.654998i	$-0.772670\pi$
$839$	−43.7745	−1.51126	−0.755631	+	0.654998i	$0.772670\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 32.8373i	− 1.13098i
$844$	0	0
$845$	− 1.48486i	− 0.0510808i
$846$	0	0
$847$	−21.6176	−0.742790
$848$	0	0
$849$	27.2195	0.934171
$850$	0	0
$851$	6.47635i	0.222006i
$852$	0	0
$853$	− 38.6050i	− 1.32181i	−0.750469	−	0.660906i	$-0.770172\pi$
$853$	− 38.6050i	− 1.32181i	0.750469	−	0.660906i	$-0.229828\pi$
$854$	0	0
$855$	−2.53001	−0.0865244
$856$	0	0
$857$	3.93945	0.134569	0.0672845	−	0.997734i	$-0.478566\pi$
$857$	3.93945	0.134569	0.0672845	+	0.997734i	$0.478566\pi$
$858$	0	0
$859$	− 8.65655i	− 0.295358i	−0.989035	−	0.147679i	$-0.952820\pi$
$859$	− 8.65655i	− 0.295358i	0.989035	−	0.147679i	$-0.0471802\pi$
$860$	0	0
$861$	35.8789i	1.22275i
$862$	0	0
$863$	−0.361584	−0.0123085	−0.00615423	−	0.999981i	$-0.501959\pi$
$863$	−0.361584	−0.0123085	−0.00615423	+	0.999981i	$0.501959\pi$
$864$	0	0
$865$	34.8486	1.18489
$866$	0	0
$867$	− 10.6488i	− 0.361651i
$868$	0	0
$869$	11.0009i	0.373181i
$870$	0	0
$871$	2.96222	0.100371
$872$	0	0
$873$	−1.91009	−0.0646467
$874$	0	0
$875$	− 29.5443i	− 0.998781i
$876$	0	0
$877$	− 0.0761486i	− 0.00257135i	−0.999999	−	0.00128568i	$-0.999591\pi$
$877$	− 0.0761486i	− 0.00257135i	0.999999	−	0.00128568i	$-0.000409244\pi$
$878$	0	0
$879$	27.7889	0.937296
$880$	0	0
$881$	47.3544	1.59541	0.797704	−	0.603049i	$-0.206047\pi$
$881$	47.3544	1.59541	0.797704	+	0.603049i	$0.206047\pi$
$882$	0	0
$883$	3.47837i	0.117056i	0.998286	+	0.0585282i	$0.0186408\pi$
$883$	3.47837i	0.117056i	−0.998286	+	0.0585282i	$0.981359\pi$
$884$	0	0
$885$	− 30.4390i	− 1.02320i
$886$	0	0
$887$	−28.5519	−0.958678	−0.479339	−	0.877630i	$-0.659124\pi$
$887$	−28.5519	−0.958678	−0.479339	+	0.877630i	$0.659124\pi$
$888$	0	0
$889$	19.3482	0.648916
$890$	0	0
$891$	− 16.2575i	− 0.544645i
$892$	0	0
$893$	− 18.9092i	− 0.632771i
$894$	0	0
$895$	36.6000	1.22340
$896$	0	0
$897$	6.96972	0.232712
$898$	0	0
$899$	1.37140i	0.0457386i
$900$	0	0
$901$	− 26.3491i	− 0.877815i
$902$	0	0
$903$	22.3722	0.744501
$904$	0	0
$905$	27.4693	0.913109
$906$	0	0
$907$	16.8284i	0.558778i	0.960178	+	0.279389i	$0.0901319\pi$
$907$	16.8284i	0.558778i	−0.960178	+	0.279389i	$0.909868\pi$
$908$	0	0
$909$	− 8.99908i	− 0.298481i
$910$	0	0
$911$	−48.1522	−1.59535	−0.797677	−	0.603086i	$-0.793938\pi$
$911$	−48.1522	−1.59535	−0.797677	+	0.603086i	$0.793938\pi$
$912$	0	0
$913$	6.46835	0.214071
$914$	0	0
$915$	− 34.6476i	− 1.14541i
$916$	0	0
$917$	5.95504i	0.196653i
$918$	0	0
$919$	37.8292	1.24787	0.623936	−	0.781476i	$-0.285533\pi$
$919$	37.8292	1.24787	0.623936	+	0.781476i	$0.285533\pi$
$920$	0	0
$921$	32.3009	1.06435
$922$	0	0
$923$	− 11.3910i	− 0.374939i
$924$	0	0
$925$	− 4.84862i	− 0.159422i
$926$	0	0
$927$	3.14014	0.103136
$928$	0	0
$929$	−21.9688	−0.720773	−0.360387	−	0.932803i	$-0.617355\pi$
$929$	−21.9688	−0.720773	−0.360387	+	0.932803i	$0.617355\pi$
$930$	0	0
$931$	1.70387i	0.0558419i
$932$	0	0
$933$	− 8.49954i	− 0.278263i
$934$	0	0
$935$	−11.2555	−0.368094
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	− 27.1514i	− 0.886051i
$940$	0	0
$941$	0.0449558i	0.00146552i	1.00000		0.000732759i	$0.000233244\pi$
$941$	0.0449558i	0.00146552i	−1.00000		0.000732759i	$0.999767\pi$
$942$	0	0
$943$	−28.1130	−0.915485
$944$	0	0
$945$	−17.7952	−0.578877
$946$	0	0
$947$	− 37.7811i	− 1.22772i	−0.789415	−	0.613860i	$-0.789616\pi$
$947$	− 37.7811i	− 1.22772i	0.789415	−	0.613860i	$-0.210384\pi$
$948$	0	0
$949$	9.46927i	0.307385i
$950$	0	0
$951$	−39.3137	−1.27483
$952$	0	0
$953$	−22.8860	−0.741351	−0.370675	−	0.928763i	$-0.620874\pi$
$953$	−22.8860	−0.741351	−0.370675	+	0.928763i	$0.620874\pi$
$954$	0	0
$955$	− 3.50748i	− 0.113499i
$956$	0	0
$957$	− 5.93945i	− 0.191995i
$958$	0	0
$959$	0.154558	0.00499094
$960$	0	0
$961$	−30.5298	−0.984833
$962$	0	0
$963$	0.0415205i	0.00133798i
$964$	0	0
$965$	0.0899116i	0.00289436i
$966$	0	0
$967$	−0.457912	−0.0147255	−0.00736273	−	0.999973i	$-0.502344\pi$
$967$	−0.457912	−0.0147255	−0.00736273	+	0.999973i	$0.502344\pi$
$968$	0	0
$969$	−31.2582	−1.00416
$970$	0	0
$971$	3.15254i	0.101170i	0.998720	+	0.0505849i	$0.0161086\pi$
$971$	3.15254i	0.101170i	−0.998720	+	0.0505849i	$0.983891\pi$
$972$	0	0
$973$	16.3865i	0.525326i
$974$	0	0
$975$	−5.21799	−0.167109
$976$	0	0
$977$	9.87890	0.316054	0.158027	−	0.987435i	$-0.449487\pi$
$977$	9.87890	0.316054	0.158027	+	0.987435i	$0.449487\pi$
$978$	0	0
$979$	1.54266i	0.0493038i
$980$	0	0
$981$	4.27913i	0.136622i
$982$	0	0
$983$	44.7802	1.42827	0.714133	−	0.700010i	$-0.246821\pi$
$983$	44.7802	1.42827	0.714133	+	0.700010i	$0.246821\pi$
$984$	0	0
$985$	−23.4537	−0.747296
$986$	0	0
$987$	25.6395i	0.816115i
$988$	0	0
$989$	17.5298i	0.557416i
$990$	0	0
$991$	12.2878	0.390334	0.195167	−	0.980770i	$-0.437475\pi$
$991$	12.2878	0.390334	0.195167	+	0.980770i	$0.437475\pi$
$992$	0	0
$993$	32.0899	1.01834
$994$	0	0
$995$	33.4307i	1.05983i
$996$	0	0
$997$	23.0908i	0.731294i	0.930754	+	0.365647i	$0.119152\pi$
$997$	23.0908i	0.731294i	−0.930754	+	0.365647i	$0.880848\pi$
$998$	0	0
$999$	−8.14446	−0.257679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.k.833.4	yes	12
4.3	odd	2	inner	1664.2.b.k.833.10	yes	12
8.3	odd	2	inner	1664.2.b.k.833.3	✓	12
8.5	even	2	inner	1664.2.b.k.833.9	yes	12
16.3	odd	4		3328.2.a.bp.1.2		6
16.5	even	4		3328.2.a.bo.1.2		6
16.11	odd	4		3328.2.a.bo.1.5		6
16.13	even	4		3328.2.a.bp.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.k.833.3	✓	12	8.3	odd	2	inner
1664.2.b.k.833.4	yes	12	1.1	even	1	trivial
1664.2.b.k.833.9	yes	12	8.5	even	2	inner
1664.2.b.k.833.10	yes	12	4.3	odd	2	inner
3328.2.a.bo.1.2		6	16.5	even	4
3328.2.a.bo.1.5		6	16.11	odd	4
3328.2.a.bp.1.2		6	16.3	odd	4
3328.2.a.bp.1.5		6	16.13	even	4

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:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.86678i q^{3} +1.48486i q^{5} -2.55248 q^{7} -0.484862 q^{9} +1.59083i q^{11} -1.00000i q^{13} +2.77191 q^{15} +4.76491 q^{17} -3.51413i q^{19} +4.76491i q^{21} -3.73356 q^{23} +2.79518 q^{25} -4.69521i q^{27} -2.00000i q^{29} -0.685698 q^{31} +2.96972 q^{33} -3.79008i q^{35} -1.73463i q^{37} -1.86678 q^{39} +7.52982 q^{41} -4.69521i q^{43} -0.719953i q^{45} +5.38090 q^{47} -0.484862 q^{49} -8.89503i q^{51} -5.52982i q^{53} -2.36216 q^{55} -6.56009 q^{57} -10.9812i q^{59} -12.4995i q^{61} +1.23760 q^{63} +1.48486 q^{65} +2.96222i q^{67} +6.96972i q^{69} +11.3910 q^{71} -9.46927 q^{73} -5.21799i q^{75} -4.06055i q^{77} +6.91521 q^{79} -10.2195 q^{81} -4.06603i q^{83} +7.07523i q^{85} -3.73356 q^{87} +0.969724 q^{89} +2.55248i q^{91} +1.28005i q^{93} +5.21799 q^{95} +3.93945 q^{97} -0.771332i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 8 * q^17 - 28 * q^25 + 32 * q^33 - 40 * q^41 - 4 * q^49 + 48 * q^57 + 16 * q^65 + 24 * q^73 - 52 * q^81 + 8 * q^89 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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