LMFDB - Embedded newform 1664.2.b.f.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)

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")

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);

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 = [4,0,0,0,0,0,0,0,-20] 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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.f.833.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{3} +2.00000i q^{5} +4.24264 q^{7} -5.00000 q^{9} -1.41421i q^{11} +1.00000i q^{13} -5.65685 q^{15} -4.00000 q^{17} +1.41421i q^{19} +12.0000i q^{21} -5.65685 q^{23} +1.00000 q^{25} -5.65685i q^{27} +2.00000i q^{29} -9.89949 q^{31} +4.00000 q^{33} +8.48528i q^{35} +2.00000i q^{37} -2.82843 q^{39} +6.00000 q^{41} +11.3137i q^{43} -10.0000i q^{45} -7.07107 q^{47} +11.0000 q^{49} -11.3137i q^{51} +12.0000i q^{53} +2.82843 q^{55} -4.00000 q^{57} -9.89949i q^{59} +12.0000i q^{61} -21.2132 q^{63} -2.00000 q^{65} -9.89949i q^{67} -16.0000i q^{69} +4.24264 q^{71} +14.0000 q^{73} +2.82843i q^{75} -6.00000i q^{77} +8.48528 q^{79} +1.00000 q^{81} -7.07107i q^{83} -8.00000i q^{85} -5.65685 q^{87} -14.0000 q^{89} +4.24264i q^{91} -28.0000i q^{93} -2.82843 q^{95} -2.00000 q^{97} +7.07107i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9} - 16 q^{17} + 4 q^{25} + 16 q^{33} + 24 q^{41} + 44 q^{49} - 16 q^{57} - 8 q^{65} + 56 q^{73} + 4 q^{81} - 56 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q - 20 * q^9 - 16 * q^17 + 4 * q^25 + 16 * q^33 + 24 * q^41 + 44 * q^49 - 16 * q^57 - 8 * q^65 + 56 * q^73 + 4 * q^81 - 56 * q^89 - 8 * 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
$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$	2.82843i	1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	4.24264	1.60357	0.801784	−	0.597614i	$-0.203885\pi$
$7$	4.24264	1.60357	0.801784	+	0.597614i	$0.203885\pi$
$8$	0	0
$9$	−5.00000	−1.66667
$10$	0	0
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−5.65685	−1.46059
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.41421i	0.324443i	0.986754	+	0.162221i	$0.0518659\pi$
$19$	1.41421i	0.324443i	−0.986754	+	0.162221i	$0.948134\pi$
$20$	0	0
$21$	12.0000i	2.61861i
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	−9.89949	−1.77800	−0.889001	−	0.457905i	$-0.848600\pi$
$31$	−9.89949	−1.77800	−0.889001	+	0.457905i	$0.848600\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	8.48528i	1.43427i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−2.82843	−0.452911
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	11.3137i	1.72532i	0.505781	+	0.862662i	$0.331205\pi$
$43$	11.3137i	1.72532i	−0.505781	+	0.862662i	$0.668795\pi$
$44$	0	0
$45$	− 10.0000i	− 1.49071i
$46$	0	0
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	− 11.3137i	− 1.58424i
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 9.89949i	− 1.28880i	−0.764687	−	0.644402i	$-0.777106\pi$
$59$	− 9.89949i	− 1.28880i	0.764687	−	0.644402i	$-0.222894\pi$
$60$	0	0
$61$	12.0000i	1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$	12.0000i	1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−21.2132	−2.67261
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	− 9.89949i	− 1.20942i	−0.796447	−	0.604708i	$-0.793290\pi$
$67$	− 9.89949i	− 1.20942i	0.796447	−	0.604708i	$-0.206710\pi$
$68$	0	0
$69$	− 16.0000i	− 1.92617i
$70$	0	0
$71$	4.24264	0.503509	0.251754	−	0.967791i	$-0.418992\pi$
$71$	4.24264	0.503509	0.251754	+	0.967791i	$0.418992\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	2.82843i	0.326599i
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 7.07107i	− 0.776151i	−0.921628	−	0.388075i	$-0.873140\pi$
$83$	− 7.07107i	− 0.776151i	0.921628	−	0.388075i	$-0.126860\pi$
$84$	0	0
$85$	− 8.00000i	− 0.867722i
$86$	0	0
$87$	−5.65685	−0.606478
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	4.24264i	0.444750i
$92$	0	0
$93$	− 28.0000i	− 2.90346i
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	7.07107i	0.710669i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−24.0000	−2.34216
$106$	0	0
$107$	− 2.82843i	− 0.273434i	−0.990610	−	0.136717i	$-0.956345\pi$
$107$	− 2.82843i	− 0.273434i	0.990610	−	0.136717i	$-0.0436552\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	−5.65685	−0.536925
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	− 11.3137i	− 1.05501i
$116$	0	0
$117$	− 5.00000i	− 0.462250i
$118$	0	0
$119$	−16.9706	−1.55569
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	16.9706i	1.53018i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	0	0
$129$	−32.0000	−2.81744
$130$	0	0
$131$	2.82843i	0.247121i	0.992337	+	0.123560i	$0.0394313\pi$
$131$	2.82843i	0.247121i	−0.992337	+	0.123560i	$0.960569\pi$
$132$	0	0
$133$	6.00000i	0.520266i
$134$	0	0
$135$	11.3137	0.973729
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	2.82843i	0.239904i	0.992780	+	0.119952i	$0.0382741\pi$
$139$	2.82843i	0.239904i	−0.992780	+	0.119952i	$0.961726\pi$
$140$	0	0
$141$	− 20.0000i	− 1.68430i
$142$	0	0
$143$	1.41421	0.118262
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	31.1127i	2.56613i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	−12.7279	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$151$	−12.7279	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$152$	0	0
$153$	20.0000	1.61690
$154$	0	0
$155$	− 19.7990i	− 1.59029i
$156$	0	0
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	0	0
$159$	−33.9411	−2.69171
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	1.41421i	0.110770i	0.998465	+	0.0553849i	$0.0176386\pi$
$163$	1.41421i	0.110770i	−0.998465	+	0.0553849i	$0.982361\pi$
$164$	0	0
$165$	8.00000i	0.622799i
$166$	0	0
$167$	18.3848	1.42266	0.711328	−	0.702860i	$-0.248094\pi$
$167$	18.3848	1.42266	0.711328	+	0.702860i	$0.248094\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 7.07107i	− 0.540738i
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	4.24264	0.320713
$176$	0	0
$177$	28.0000	2.10461
$178$	0	0
$179$	− 22.6274i	− 1.69125i	−0.533775	−	0.845626i	$-0.679227\pi$
$179$	− 22.6274i	− 1.69125i	0.533775	−	0.845626i	$-0.320773\pi$
$180$	0	0
$181$	26.0000i	1.93256i	0.257485	+	0.966282i	$0.417106\pi$
$181$	26.0000i	1.93256i	−0.257485	+	0.966282i	$0.582894\pi$
$182$	0	0
$183$	−33.9411	−2.50900
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	5.65685i	0.413670i
$188$	0	0
$189$	− 24.0000i	− 1.74574i
$190$	0	0
$191$	14.1421	1.02329	0.511645	−	0.859197i	$-0.329036\pi$
$191$	14.1421	1.02329	0.511645	+	0.859197i	$0.329036\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	− 5.65685i	− 0.405096i
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	28.0000	1.97497
$202$	0	0
$203$	8.48528i	0.595550i
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	28.2843	1.96589
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	14.1421i	0.973585i	0.873518	+	0.486792i	$0.161833\pi$
$211$	14.1421i	0.973585i	−0.873518	+	0.486792i	$0.838167\pi$
$212$	0	0
$213$	12.0000i	0.822226i
$214$	0	0
$215$	−22.6274	−1.54318
$216$	0	0
$217$	−42.0000	−2.85115
$218$	0	0
$219$	39.5980i	2.67578i
$220$	0	0
$221$	− 4.00000i	− 0.269069i
$222$	0	0
$223$	15.5563	1.04173	0.520865	−	0.853639i	$-0.325609\pi$
$223$	15.5563	1.04173	0.520865	+	0.853639i	$0.325609\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	18.3848i	1.22024i	0.792309	+	0.610120i	$0.208879\pi$
$227$	18.3848i	1.22024i	−0.792309	+	0.610120i	$0.791121\pi$
$228$	0	0
$229$	2.00000i	0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$	2.00000i	0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	0	0
$231$	16.9706	1.11658
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	− 14.1421i	− 0.922531i
$236$	0	0
$237$	24.0000i	1.55897i
$238$	0	0
$239$	−1.41421	−0.0914779	−0.0457389	−	0.998953i	$-0.514564\pi$
$239$	−1.41421	−0.0914779	−0.0457389	+	0.998953i	$0.514564\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 14.1421i	− 0.907218i
$244$	0	0
$245$	22.0000i	1.40553i
$246$	0	0
$247$	−1.41421	−0.0899843
$248$	0	0
$249$	20.0000	1.26745
$250$	0	0
$251$	− 5.65685i	− 0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$	− 5.65685i	− 0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	0	0
$255$	22.6274	1.41698
$256$	0	0
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	8.48528i	0.527250i
$260$	0	0
$261$	− 10.0000i	− 0.618984i
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	− 39.5980i	− 2.42336i
$268$	0	0
$269$	14.0000i	0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$	14.0000i	0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$	0	0
$271$	26.8701	1.63224	0.816120	−	0.577883i	$-0.196121\pi$
$271$	26.8701	1.63224	0.816120	+	0.577883i	$0.196121\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	− 1.41421i	− 0.0852803i
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	49.4975	2.96334
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	25.4558	1.50261
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	− 5.65685i	− 0.331611i
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	19.7990	1.15274
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	− 5.65685i	− 0.327144i
$300$	0	0
$301$	48.0000i	2.76667i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	7.07107i	0.403567i	0.979430	+	0.201784i	$0.0646738\pi$
$307$	7.07107i	0.403567i	−0.979430	+	0.201784i	$0.935326\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.6274	−1.28308	−0.641542	−	0.767088i	$-0.721705\pi$
$311$	−22.6274	−1.28308	−0.641542	+	0.767088i	$0.721705\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	− 42.4264i	− 2.39046i
$316$	0	0
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	0	0
$319$	2.82843	0.158362
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	− 5.65685i	− 0.314756i
$324$	0	0
$325$	1.00000i	0.0554700i
$326$	0	0
$327$	5.65685	0.312825
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	− 9.89949i	− 0.544125i	−0.962280	−	0.272063i	$-0.912294\pi$
$331$	− 9.89949i	− 0.544125i	0.962280	−	0.272063i	$-0.0877058\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	19.7990	1.08173
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.0000i	0.758143i
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	32.0000	1.72282
$346$	0	0
$347$	− 2.82843i	− 0.151838i	−0.997114	−	0.0759190i	$-0.975811\pi$
$347$	− 2.82843i	− 0.151838i	0.997114	−	0.0759190i	$-0.0241890\pi$
$348$	0	0
$349$	22.0000i	1.17763i	0.808267	+	0.588817i	$0.200406\pi$
$349$	22.0000i	1.17763i	−0.808267	+	0.588817i	$0.799594\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	8.48528i	0.450352i
$356$	0	0
$357$	− 48.0000i	− 2.54043i
$358$	0	0
$359$	−18.3848	−0.970311	−0.485156	−	0.874428i	$-0.661237\pi$
$359$	−18.3848	−0.970311	−0.485156	+	0.874428i	$0.661237\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	25.4558i	1.33609i
$364$	0	0
$365$	28.0000i	1.46559i
$366$	0	0
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	50.9117i	2.64320i
$372$	0	0
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	−33.9411	−1.75271
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	− 9.89949i	− 0.508503i	−0.967138	−	0.254251i	$-0.918171\pi$
$379$	− 9.89949i	− 0.508503i	0.967138	−	0.254251i	$-0.0818291\pi$
$380$	0	0
$381$	48.0000i	2.45911i
$382$	0	0
$383$	−24.0416	−1.22847	−0.614235	−	0.789123i	$-0.710535\pi$
$383$	−24.0416	−1.22847	−0.614235	+	0.789123i	$0.710535\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	− 56.5685i	− 2.87554i
$388$	0	0
$389$	10.0000i	0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$	10.0000i	0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$	0	0
$391$	22.6274	1.14432
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	16.9706i	0.853882i
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	−16.9706	−0.849591
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	− 9.89949i	− 0.493129i
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	2.82843	0.140200
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	50.9117i	2.51129i
$412$	0	0
$413$	− 42.0000i	− 2.06668i
$414$	0	0
$415$	14.1421	0.694210
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	− 14.1421i	− 0.690889i	−0.938439	−	0.345444i	$-0.887728\pi$
$419$	− 14.1421i	− 0.690889i	0.938439	−	0.345444i	$-0.112272\pi$
$420$	0	0
$421$	− 18.0000i	− 0.877266i	−0.898666	−	0.438633i	$-0.855463\pi$
$421$	− 18.0000i	− 0.877266i	0.898666	−	0.438633i	$-0.144537\pi$
$422$	0	0
$423$	35.3553	1.71904
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	50.9117i	2.46379i
$428$	0	0
$429$	4.00000i	0.193122i
$430$	0	0
$431$	−15.5563	−0.749323	−0.374661	−	0.927162i	$-0.622241\pi$
$431$	−15.5563	−0.749323	−0.374661	+	0.927162i	$0.622241\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	− 11.3137i	− 0.542451i
$436$	0	0
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	−55.0000	−2.61905
$442$	0	0
$443$	− 19.7990i	− 0.940678i	−0.882486	−	0.470339i	$-0.844132\pi$
$443$	− 19.7990i	− 0.940678i	0.882486	−	0.470339i	$-0.155868\pi$
$444$	0	0
$445$	− 28.0000i	− 1.32733i
$446$	0	0
$447$	−16.9706	−0.802680
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	− 8.48528i	− 0.399556i
$452$	0	0
$453$	− 36.0000i	− 1.69143i
$454$	0	0
$455$	−8.48528	−0.397796
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	22.6274i	1.05616i
$460$	0	0
$461$	34.0000i	1.58354i	0.610821	+	0.791769i	$0.290840\pi$
$461$	34.0000i	1.58354i	−0.610821	+	0.791769i	$0.709160\pi$
$462$	0	0
$463$	9.89949	0.460069	0.230034	−	0.973183i	$-0.426116\pi$
$463$	9.89949	0.460069	0.230034	+	0.973183i	$0.426116\pi$
$464$	0	0
$465$	56.0000	2.59694
$466$	0	0
$467$	16.9706i	0.785304i	0.919687	+	0.392652i	$0.128442\pi$
$467$	16.9706i	0.785304i	−0.919687	+	0.392652i	$0.871558\pi$
$468$	0	0
$469$	− 42.0000i	− 1.93938i
$470$	0	0
$471$	62.2254	2.86719
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	1.41421i	0.0648886i
$476$	0	0
$477$	− 60.0000i	− 2.74721i
$478$	0	0
$479$	−21.2132	−0.969256	−0.484628	−	0.874720i	$-0.661045\pi$
$479$	−21.2132	−0.969256	−0.484628	+	0.874720i	$0.661045\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	− 67.8823i	− 3.08875i
$484$	0	0
$485$	− 4.00000i	− 0.181631i
$486$	0	0
$487$	−15.5563	−0.704925	−0.352463	−	0.935826i	$-0.614656\pi$
$487$	−15.5563	−0.704925	−0.352463	+	0.935826i	$0.614656\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	11.3137i	0.510581i	0.966864	+	0.255290i	$0.0821710\pi$
$491$	11.3137i	0.510581i	−0.966864	+	0.255290i	$0.917829\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	−14.1421	−0.635642
$496$	0	0
$497$	18.0000	0.807410
$498$	0	0
$499$	− 4.24264i	− 0.189927i	−0.995481	−	0.0949633i	$-0.969727\pi$
$499$	− 4.24264i	− 0.189927i	0.995481	−	0.0949633i	$-0.0302734\pi$
$500$	0	0
$501$	52.0000i	2.32319i
$502$	0	0
$503$	−2.82843	−0.126113	−0.0630567	−	0.998010i	$-0.520085\pi$
$503$	−2.82843	−0.126113	−0.0630567	+	0.998010i	$0.520085\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.82843i	− 0.125615i
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	59.3970	2.62757
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.0000i	0.439799i
$518$	0	0
$519$	45.2548	1.98647
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 42.4264i	− 1.85518i	−0.373603	−	0.927589i	$-0.621878\pi$
$523$	− 42.4264i	− 1.85518i	0.373603	−	0.927589i	$-0.378122\pi$
$524$	0	0
$525$	12.0000i	0.523723i
$526$	0	0
$527$	39.5980	1.72492
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	49.4975i	2.14801i
$532$	0	0
$533$	6.00000i	0.259889i
$534$	0	0
$535$	5.65685	0.244567
$536$	0	0
$537$	64.0000	2.76180
$538$	0	0
$539$	− 15.5563i	− 0.670059i
$540$	0	0
$541$	− 6.00000i	− 0.257960i	−0.991647	−	0.128980i	$-0.958830\pi$
$541$	− 6.00000i	− 0.257960i	0.991647	−	0.128980i	$-0.0411703\pi$
$542$	0	0
$543$	−73.5391	−3.15587
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	25.4558i	1.08841i	0.838951	+	0.544207i	$0.183169\pi$
$547$	25.4558i	1.08841i	−0.838951	+	0.544207i	$0.816831\pi$
$548$	0	0
$549$	− 60.0000i	− 2.56074i
$550$	0	0
$551$	−2.82843	−0.120495
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	− 11.3137i	− 0.480240i
$556$	0	0
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	0	0
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	16.9706i	0.715224i	0.933870	+	0.357612i	$0.116409\pi$
$563$	16.9706i	0.715224i	−0.933870	+	0.357612i	$0.883591\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.24264	0.178174
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	5.65685i	0.236732i	0.992970	+	0.118366i	$0.0377656\pi$
$571$	5.65685i	0.236732i	−0.992970	+	0.118366i	$0.962234\pi$
$572$	0	0
$573$	40.0000i	1.67102i
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	62.2254i	2.58600i
$580$	0	0
$581$	− 30.0000i	− 1.24461i
$582$	0	0
$583$	16.9706	0.702849
$584$	0	0
$585$	10.0000	0.413449
$586$	0	0
$587$	35.3553i	1.45927i	0.683836	+	0.729636i	$0.260310\pi$
$587$	35.3553i	1.45927i	−0.683836	+	0.729636i	$0.739690\pi$
$588$	0	0
$589$	− 14.0000i	− 0.576860i
$590$	0	0
$591$	−5.65685	−0.232692
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	− 33.9411i	− 1.39145i
$596$	0	0
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	31.1127	1.27123	0.635615	−	0.772006i	$-0.280747\pi$
$599$	31.1127	1.27123	0.635615	+	0.772006i	$0.280747\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	49.4975i	2.01569i
$604$	0	0
$605$	18.0000i	0.731804i
$606$	0	0
$607$	14.1421	0.574012	0.287006	−	0.957929i	$-0.407340\pi$
$607$	14.1421	0.574012	0.287006	+	0.957929i	$0.407340\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	− 7.07107i	− 0.286065i
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	0	0
$615$	−33.9411	−1.36864
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	− 24.0416i	− 0.966315i	−0.875534	−	0.483157i	$-0.839490\pi$
$619$	− 24.0416i	− 0.966315i	0.875534	−	0.483157i	$-0.160510\pi$
$620$	0	0
$621$	32.0000i	1.28412i
$622$	0	0
$623$	−59.3970	−2.37969
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	5.65685i	0.225913i
$628$	0	0
$629$	− 8.00000i	− 0.318981i
$630$	0	0
$631$	9.89949	0.394093	0.197046	−	0.980394i	$-0.436865\pi$
$631$	9.89949	0.394093	0.197046	+	0.980394i	$0.436865\pi$
$632$	0	0
$633$	−40.0000	−1.58986
$634$	0	0
$635$	33.9411i	1.34691i
$636$	0	0
$637$	11.0000i	0.435836i
$638$	0	0
$639$	−21.2132	−0.839181
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	− 12.7279i	− 0.501940i	−0.967995	−	0.250970i	$-0.919250\pi$
$643$	− 12.7279i	− 0.501940i	0.967995	−	0.250970i	$-0.0807496\pi$
$644$	0	0
$645$	− 64.0000i	− 2.52000i
$646$	0	0
$647$	−33.9411	−1.33436	−0.667182	−	0.744895i	$-0.732500\pi$
$647$	−33.9411	−1.33436	−0.667182	+	0.744895i	$0.732500\pi$
$648$	0	0
$649$	−14.0000	−0.549548
$650$	0	0
$651$	− 118.794i	− 4.65590i
$652$	0	0
$653$	− 22.0000i	− 0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$	− 22.0000i	− 0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	−70.0000	−2.73096
$658$	0	0
$659$	2.82843i	0.110180i	0.998481	+	0.0550899i	$0.0175446\pi$
$659$	2.82843i	0.110180i	−0.998481	+	0.0550899i	$0.982455\pi$
$660$	0	0
$661$	50.0000i	1.94477i	0.233373	+	0.972387i	$0.425024\pi$
$661$	50.0000i	1.94477i	−0.233373	+	0.972387i	$0.574976\pi$
$662$	0	0
$663$	11.3137	0.439388
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	− 11.3137i	− 0.438069i
$668$	0	0
$669$	44.0000i	1.70114i
$670$	0	0
$671$	16.9706	0.655141
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	− 5.65685i	− 0.217732i
$676$	0	0
$677$	20.0000i	0.768662i	0.923195	+	0.384331i	$0.125568\pi$
$677$	20.0000i	0.768662i	−0.923195	+	0.384331i	$0.874432\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	−52.0000	−1.99264
$682$	0	0
$683$	− 12.7279i	− 0.487020i	−0.969898	−	0.243510i	$-0.921701\pi$
$683$	− 12.7279i	− 0.487020i	0.969898	−	0.243510i	$-0.0782989\pi$
$684$	0	0
$685$	36.0000i	1.37549i
$686$	0	0
$687$	−5.65685	−0.215822
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	− 38.1838i	− 1.45258i	−0.687389	−	0.726289i	$-0.741243\pi$
$691$	− 38.1838i	− 1.45258i	0.687389	−	0.726289i	$-0.258757\pi$
$692$	0	0
$693$	30.0000i	1.13961i
$694$	0	0
$695$	−5.65685	−0.214577
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	− 16.9706i	− 0.641886i
$700$	0	0
$701$	− 18.0000i	− 0.679851i	−0.940452	−	0.339925i	$-0.889598\pi$
$701$	− 18.0000i	− 0.679851i	0.940452	−	0.339925i	$-0.110402\pi$
$702$	0	0
$703$	−2.82843	−0.106676
$704$	0	0
$705$	40.0000	1.50649
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	−42.4264	−1.59111
$712$	0	0
$713$	56.0000	2.09722
$714$	0	0
$715$	2.82843i	0.105777i
$716$	0	0
$717$	− 4.00000i	− 0.149383i
$718$	0	0
$719$	−33.9411	−1.26579	−0.632895	−	0.774237i	$-0.718134\pi$
$719$	−33.9411	−1.26579	−0.632895	+	0.774237i	$0.718134\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	28.2843i	1.05190i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−22.6274	−0.839204	−0.419602	−	0.907708i	$-0.637830\pi$
$727$	−22.6274	−0.839204	−0.419602	+	0.907708i	$0.637830\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	− 45.2548i	− 1.67381i
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	0	0
$735$	−62.2254	−2.29522
$736$	0	0
$737$	−14.0000	−0.515697
$738$	0	0
$739$	4.24264i	0.156068i	0.996951	+	0.0780340i	$0.0248643\pi$
$739$	4.24264i	0.156068i	−0.996951	+	0.0780340i	$0.975136\pi$
$740$	0	0
$741$	− 4.00000i	− 0.146944i
$742$	0	0
$743$	9.89949	0.363177	0.181589	−	0.983375i	$-0.441876\pi$
$743$	9.89949	0.363177	0.181589	+	0.983375i	$0.441876\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	35.3553i	1.29358i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	−22.6274	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$751$	−22.6274	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$752$	0	0
$753$	16.0000	0.583072
$754$	0	0
$755$	− 25.4558i	− 0.926433i
$756$	0	0
$757$	− 44.0000i	− 1.59921i	−0.600528	−	0.799604i	$-0.705043\pi$
$757$	− 44.0000i	− 1.59921i	0.600528	−	0.799604i	$-0.294957\pi$
$758$	0	0
$759$	−22.6274	−0.821323
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	− 8.48528i	− 0.307188i
$764$	0	0
$765$	40.0000i	1.44620i
$766$	0	0
$767$	9.89949	0.357450
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	84.8528i	3.05590i
$772$	0	0
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	−9.89949	−0.355600
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	8.48528i	0.304017i
$780$	0	0
$781$	− 6.00000i	− 0.214697i
$782$	0	0
$783$	11.3137	0.404319
$784$	0	0
$785$	44.0000	1.57043
$786$	0	0
$787$	− 21.2132i	− 0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$	− 21.2132i	− 0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$	0	0
$789$	− 24.0000i	− 0.854423i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	− 67.8823i	− 2.40754i
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	28.2843	1.00063
$800$	0	0
$801$	70.0000	2.47333
$802$	0	0
$803$	− 19.7990i	− 0.698691i
$804$	0	0
$805$	− 48.0000i	− 1.69178i
$806$	0	0
$807$	−39.5980	−1.39391
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	− 18.3848i	− 0.645577i	−0.946471	−	0.322788i	$-0.895380\pi$
$811$	− 18.3848i	− 0.645577i	0.946471	−	0.322788i	$-0.104620\pi$
$812$	0	0
$813$	76.0000i	2.66544i
$814$	0	0
$815$	−2.82843	−0.0990755
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	− 21.2132i	− 0.741249i
$820$	0	0
$821$	− 6.00000i	− 0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$	− 6.00000i	− 0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	0	0
$823$	5.65685	0.197186	0.0985928	−	0.995128i	$-0.468566\pi$
$823$	5.65685	0.197186	0.0985928	+	0.995128i	$0.468566\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	35.3553i	1.22943i	0.788751	+	0.614713i	$0.210728\pi$
$827$	35.3553i	1.22943i	−0.788751	+	0.614713i	$0.789272\pi$
$828$	0	0
$829$	40.0000i	1.38926i	0.719368	+	0.694629i	$0.244431\pi$
$829$	40.0000i	1.38926i	−0.719368	+	0.694629i	$0.755569\pi$
$830$	0	0
$831$	−22.6274	−0.784936
$832$	0	0
$833$	−44.0000	−1.52451
$834$	0	0
$835$	36.7696i	1.27246i
$836$	0	0
$837$	56.0000i	1.93564i
$838$	0	0
$839$	−12.7279	−0.439417	−0.219708	−	0.975566i	$-0.570511\pi$
$839$	−12.7279	−0.439417	−0.219708	+	0.975566i	$0.570511\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	62.2254i	2.14316i
$844$	0	0
$845$	− 2.00000i	− 0.0688021i
$846$	0	0
$847$	38.1838	1.31201
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 11.3137i	− 0.387829i
$852$	0	0
$853$	− 2.00000i	− 0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$	− 2.00000i	− 0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	0	0
$855$	14.1421	0.483651
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	36.7696i	1.25456i	0.778793	+	0.627280i	$0.215832\pi$
$859$	36.7696i	1.25456i	−0.778793	+	0.627280i	$0.784168\pi$
$860$	0	0
$861$	72.0000i	2.45375i
$862$	0	0
$863$	7.07107	0.240702	0.120351	−	0.992731i	$-0.461598\pi$
$863$	7.07107	0.240702	0.120351	+	0.992731i	$0.461598\pi$
$864$	0	0
$865$	32.0000	1.08803
$866$	0	0
$867$	− 2.82843i	− 0.0960584i
$868$	0	0
$869$	− 12.0000i	− 0.407072i
$870$	0	0
$871$	9.89949	0.335432
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	50.9117i	1.72113i
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	0	0
$879$	−39.5980	−1.33561
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	33.9411i	1.14221i	0.820877	+	0.571105i	$0.193485\pi$
$883$	33.9411i	1.14221i	−0.820877	+	0.571105i	$0.806515\pi$
$884$	0	0
$885$	56.0000i	1.88242i
$886$	0	0
$887$	−31.1127	−1.04466	−0.522331	−	0.852743i	$-0.674937\pi$
$887$	−31.1127	−1.04466	−0.522331	+	0.852743i	$0.674937\pi$
$888$	0	0
$889$	72.0000	2.41480
$890$	0	0
$891$	− 1.41421i	− 0.0473779i
$892$	0	0
$893$	− 10.0000i	− 0.334637i
$894$	0	0
$895$	45.2548	1.51270
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	− 19.7990i	− 0.660333i
$900$	0	0
$901$	− 48.0000i	− 1.59911i
$902$	0	0
$903$	−135.765	−4.51796
$904$	0	0
$905$	−52.0000	−1.72854
$906$	0	0
$907$	− 28.2843i	− 0.939164i	−0.882889	−	0.469582i	$-0.844405\pi$
$907$	− 28.2843i	− 0.939164i	0.882889	−	0.469582i	$-0.155595\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0833	−1.59307	−0.796535	−	0.604593i	$-0.793336\pi$
$911$	−48.0833	−1.59307	−0.796535	+	0.604593i	$0.793336\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	0	0
$915$	− 67.8823i	− 2.24412i
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	36.7696	1.21292	0.606458	−	0.795116i	$-0.292590\pi$
$919$	36.7696	1.21292	0.606458	+	0.795116i	$0.292590\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	4.24264i	0.139648i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	0	0
$931$	15.5563i	0.509839i
$932$	0	0
$933$	− 64.0000i	− 2.09527i
$934$	0	0
$935$	−11.3137	−0.369998
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	− 45.2548i	− 1.47684i
$940$	0	0
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	0	0
$943$	−33.9411	−1.10528
$944$	0	0
$945$	48.0000	1.56144
$946$	0	0
$947$	− 41.0122i	− 1.33272i	−0.745631	−	0.666359i	$-0.767852\pi$
$947$	− 41.0122i	− 1.33272i	0.745631	−	0.666359i	$-0.232148\pi$
$948$	0	0
$949$	14.0000i	0.454459i
$950$	0	0
$951$	62.2254	2.01780
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	28.2843i	0.915258i
$956$	0	0
$957$	8.00000i	0.258603i
$958$	0	0
$959$	76.3675	2.46604
$960$	0	0
$961$	67.0000	2.16129
$962$	0	0
$963$	14.1421i	0.455724i
$964$	0	0
$965$	44.0000i	1.41641i
$966$	0	0
$967$	9.89949	0.318346	0.159173	−	0.987251i	$-0.449117\pi$
$967$	9.89949	0.318346	0.159173	+	0.987251i	$0.449117\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	48.0833i	1.54307i	0.636190	+	0.771533i	$0.280510\pi$
$971$	48.0833i	1.54307i	−0.636190	+	0.771533i	$0.719490\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	−2.82843	−0.0905822
$976$	0	0
$977$	−58.0000	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$977$	−58.0000	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$978$	0	0
$979$	19.7990i	0.632778i
$980$	0	0
$981$	10.0000i	0.319275i
$982$	0	0
$983$	35.3553	1.12766	0.563830	−	0.825891i	$-0.309327\pi$
$983$	35.3553	1.12766	0.563830	+	0.825891i	$0.309327\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	0	0
$987$	− 84.8528i	− 2.70089i
$988$	0	0
$989$	− 64.0000i	− 2.03508i
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	− 11.3137i	− 0.358669i
$996$	0	0
$997$	− 52.0000i	− 1.64686i	−0.567420	−	0.823428i	$-0.692059\pi$
$997$	− 52.0000i	− 1.64686i	0.567420	−	0.823428i	$-0.307941\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.f.833.4	yes	4
4.3	odd	2	inner	1664.2.b.f.833.2	yes	4
8.3	odd	2	inner	1664.2.b.f.833.3	yes	4
8.5	even	2	inner	1664.2.b.f.833.1	✓	4
16.3	odd	4		3328.2.a.x.1.2		2
16.5	even	4		3328.2.a.t.1.2		2
16.11	odd	4		3328.2.a.t.1.1		2
16.13	even	4		3328.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.f.833.1	✓	4	8.5	even	2	inner
1664.2.b.f.833.2	yes	4	4.3	odd	2	inner
1664.2.b.f.833.3	yes	4	8.3	odd	2	inner
1664.2.b.f.833.4	yes	4	1.1	even	1	trivial
3328.2.a.t.1.1		2	16.11	odd	4
3328.2.a.t.1.2		2	16.5	even	4
3328.2.a.x.1.1		2	16.13	even	4
3328.2.a.x.1.2		2	16.3	odd	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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{3} +2.00000i q^{5} +4.24264 q^{7} -5.00000 q^{9} -1.41421i q^{11} +1.00000i q^{13} -5.65685 q^{15} -4.00000 q^{17} +1.41421i q^{19} +12.0000i q^{21} -5.65685 q^{23} +1.00000 q^{25} -5.65685i q^{27} +2.00000i q^{29} -9.89949 q^{31} +4.00000 q^{33} +8.48528i q^{35} +2.00000i q^{37} -2.82843 q^{39} +6.00000 q^{41} +11.3137i q^{43} -10.0000i q^{45} -7.07107 q^{47} +11.0000 q^{49} -11.3137i q^{51} +12.0000i q^{53} +2.82843 q^{55} -4.00000 q^{57} -9.89949i q^{59} +12.0000i q^{61} -21.2132 q^{63} -2.00000 q^{65} -9.89949i q^{67} -16.0000i q^{69} +4.24264 q^{71} +14.0000 q^{73} +2.82843i q^{75} -6.00000i q^{77} +8.48528 q^{79} +1.00000 q^{81} -7.07107i q^{83} -8.00000i q^{85} -5.65685 q^{87} -14.0000 q^{89} +4.24264i q^{91} -28.0000i q^{93} -2.82843 q^{95} -2.00000 q^{97} +7.07107i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9} - 16 q^{17} + 4 q^{25} + 16 q^{33} + 24 q^{41} + 44 q^{49} - 16 q^{57} - 8 q^{65} + 56 q^{73} + 4 q^{81} - 56 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q - 20 * q^9 - 16 * q^17 + 4 * q^25 + 16 * q^33 + 24 * q^41 + 44 * q^49 - 16 * q^57 - 8 * q^65 + 56 * q^73 + 4 * q^81 - 56 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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