LMFDB - Embedded newform 1600.2.c.f.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(449,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1600.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.c (of order $2$, degree $1$, not 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.449
Dual form			1600.2.c.f.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} -2.00000i q^{17} +8.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +4.00000i q^{27} -2.00000 q^{29} +4.00000 q^{31} +8.00000i q^{33} +2.00000i q^{37} -12.0000 q^{39} -10.0000 q^{41} +2.00000i q^{43} +2.00000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -2.00000i q^{53} +16.0000i q^{57} -2.00000 q^{61} -2.00000i q^{63} -6.00000i q^{67} +12.0000 q^{69} -12.0000 q^{71} +10.0000i q^{73} +8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +10.0000i q^{83} -4.00000i q^{87} +6.00000 q^{89} -12.0000 q^{91} +8.00000i q^{93} -10.0000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 8 q^{11} + 16 q^{19} - 8 q^{21} - 4 q^{29} + 8 q^{31} - 24 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} - 4 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} - 22 q^{81} + 12 q^{89} - 24 q^{91} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 8 * q^11 + 16 * q^19 - 8 * q^21 - 4 * q^29 + 8 * q^31 - 24 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^51 - 4 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 - 22 * q^81 + 12 * q^89 - 24 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	8.00000i	1.39262i
$34$	0	0
$35$	0	0
$36$	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
$39$	−12.0000	−1.92154
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.0000i	2.11925i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	10.0000i	1.09764i	0.835940	+	0.548821i	$0.184923\pi$
$83$	10.0000i	1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.00000i	− 0.428845i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 20.0000i	− 1.80334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	24.0000i	2.00698i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000i	0.494872i
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−8.00000	−0.611775
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000i	0.417029i
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00000i	0.543075i
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	0	0
$233$	2.00000i	0.131024i	0.997852	+	0.0655122i	$0.0208681\pi$
$233$	2.00000i	0.131024i	−0.997852	+	0.0655122i	$0.979132\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	48.0000i	3.05417i
$248$	0	0
$249$	−20.0000	−1.26745
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	− 24.0000i	− 1.50887i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.0000i	− 1.62184i	−0.585160	−	0.810918i	$-0.698968\pi$
$257$	− 26.0000i	− 1.62184i	0.585160	−	0.810918i	$-0.301032\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	2.00000i	0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$	2.00000i	0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.0000i	0.734388i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	− 24.0000i	− 1.45255i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 20.0000i	− 1.18056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.0000i	0.928414i
$298$	0	0
$299$	36.0000	2.08193
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	− 28.0000i	− 1.60856i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 28.0000i	− 1.54840i
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.0000i	0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$	18.0000i	0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	− 14.0000i	− 0.715367i	−0.933843	−	0.357683i	$-0.883567\pi$
$383$	− 14.0000i	− 0.715367i	0.933843	−	0.357683i	$-0.116433\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.00000i	− 0.101666i
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	− 8.00000i	− 0.403547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	0	0
$399$	−32.0000	−1.60200
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	24.0000i	1.19553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	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
$411$	4.00000	0.197305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 32.0000i	− 1.56705i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	− 2.00000i	− 0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	−48.0000	−2.31746
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 48.0000i	− 2.29615i
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	18.0000i	0.855206i	0.903967	+	0.427603i	$0.140642\pi$
$443$	18.0000i	0.855206i	−0.903967	+	0.427603i	$0.859358\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.0000i	0.945968i
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	0	0
$453$	40.0000i	1.87936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	− 38.0000i	− 1.76601i	−0.469364	−	0.883005i	$-0.655517\pi$
$463$	− 38.0000i	− 1.76601i	0.469364	−	0.883005i	$-0.344483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	24.0000i	1.09204i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	28.0000	1.25095
$502$	0	0
$503$	− 14.0000i	− 0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$	− 14.0000i	− 0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 46.0000i	− 2.04293i
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	32.0000i	1.41283i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.00000i	− 0.348485i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 60.0000i	− 2.59889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	36.0000i	1.54491i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14.0000i	− 0.598597i	−0.954160	−	0.299298i	$-0.903247\pi$
$547$	− 14.0000i	− 0.598597i	0.954160	−	0.299298i	$-0.0967526\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	16.0000i	0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 46.0000i	− 1.94908i	−0.224208	−	0.974541i	$-0.571980\pi$
$557$	− 46.0000i	− 1.94908i	0.224208	−	0.974541i	$-0.428020\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	− 30.0000i	− 1.26435i	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	− 30.0000i	− 1.26435i	0.774826	−	0.632175i	$-0.217837\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 22.0000i	− 0.923913i
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	− 8.00000i	− 0.334205i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−20.0000	−0.829740
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.0000i	0.412744i	0.978474	+	0.206372i	$0.0661657\pi$
$587$	10.0000i	0.412744i	−0.978474	+	0.206372i	$0.933834\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	44.0000	1.80992
$592$	0	0
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	32.0000i	1.30967i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	6.00000i	0.244339i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.00000i	0.0811775i	0.999176	+	0.0405887i	$0.0129233\pi$
$607$	2.00000i	0.0811775i	−0.999176	+	0.0405887i	$0.987077\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	0	0
$623$	12.0000i	0.480770i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	64.0000i	2.55591i
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	36.0000	1.43314	0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000	1.43314	0.716569	+	0.697517i	$0.245712\pi$
$632$	0	0
$633$	− 8.00000i	− 0.317971i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 30.0000i	− 1.18308i	−0.806274	−	0.591542i	$-0.798519\pi$
$643$	− 30.0000i	− 1.18308i	0.806274	−	0.591542i	$-0.201481\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	24.0000i	0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.0000i	0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$	26.0000i	0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	34.0000i	1.30097i	0.759517	+	0.650487i	$0.225435\pi$
$683$	34.0000i	1.30097i	−0.759517	+	0.650487i	$0.774565\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 20.0000i	− 0.763048i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.0000i	0.757554i
$698$	0	0
$699$	−4.00000	−0.151294
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	16.0000i	0.603451i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 28.0000i	− 1.05305i
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	− 24.0000i	− 0.898807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	− 36.0000i	− 1.33885i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.00000i	0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$	2.00000i	0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	−96.0000	−3.52665
$742$	0	0
$743$	42.0000i	1.54083i	0.637542	+	0.770415i	$0.279951\pi$
$743$	42.0000i	1.54083i	−0.637542	+	0.770415i	$0.720049\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.0000i	− 0.365881i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	40.0000i	1.45768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	48.0000	1.74229
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	− 28.0000i	− 1.01367i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	52.0000	1.87273
$772$	0	0
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	−80.0000	−2.86630
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	− 8.00000i	− 0.285897i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 14.0000i	− 0.499046i	−0.968369	−	0.249523i	$-0.919726\pi$
$787$	− 14.0000i	− 0.499046i	0.968369	−	0.249523i	$-0.0802738\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	40.0000i	1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	36.0000i	1.26726i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	56.0000i	1.96401i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	− 6.00000i	− 0.209147i	−0.994517	−	0.104573i	$-0.966652\pi$
$823$	− 6.00000i	− 0.209147i	0.994517	−	0.104573i	$-0.0333477\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.0000i	1.73867i	0.494223	+	0.869335i	$0.335453\pi$
$827$	50.0000i	1.73867i	−0.494223	+	0.869335i	$0.664547\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	− 20.0000i	− 0.688837i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	0	0
$849$	−52.0000	−1.78464
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	40.0000	1.36320
$862$	0	0
$863$	− 46.0000i	− 1.56586i	−0.622111	−	0.782929i	$-0.713725\pi$
$863$	− 46.0000i	− 1.56586i	0.622111	−	0.782929i	$-0.286275\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.0000i	0.883006i
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	36.0000	1.21981
$872$	0	0
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	2.00000i	0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$	2.00000i	0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.00000i	0.0671534i	0.999436	+	0.0335767i	$0.0106898\pi$
$887$	2.00000i	0.0671534i	−0.999436	+	0.0335767i	$0.989310\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	−44.0000	−1.47406
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	72.0000i	2.40401i
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.00000i	0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$	2.00000i	0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	−28.0000	−0.927681	−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000	−0.927681	−0.463841	+	0.885919i	$0.653529\pi$
$912$	0	0
$913$	40.0000i	1.32381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	44.0000	1.44985
$922$	0	0
$923$	− 72.0000i	− 2.36991i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.00000i	− 0.0656886i
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	0	0
$933$	56.0000i	1.83336i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 34.0000i	− 1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	− 34.0000i	− 1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	60.0000i	1.95387i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.0000i	− 1.23483i	−0.786636	−	0.617417i	$-0.788179\pi$
$947$	− 38.0000i	− 1.23483i	0.786636	−	0.617417i	$-0.211821\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	−4.00000	−0.129709
$952$	0	0
$953$	58.0000i	1.87880i	0.342817	+	0.939402i	$0.388619\pi$
$953$	58.0000i	1.87880i	−0.342817	+	0.939402i	$0.611381\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 16.0000i	− 0.517207i
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	32.0000	1.02799
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	− 46.0000i	− 1.46717i	−0.679597	−	0.733586i	$-0.737845\pi$
$983$	− 46.0000i	− 1.46717i	0.679597	−	0.733586i	$-0.262155\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.c.f.449.2		2
4.3	odd	2		1600.2.c.c.449.1		2
5.2	odd	4		320.2.a.e.1.1		1
5.3	odd	4		1600.2.a.e.1.1		1
5.4	even	2	inner	1600.2.c.f.449.1		2
8.3	odd	2		800.2.c.b.449.2		2
8.5	even	2		800.2.c.a.449.1		2
15.2	even	4		2880.2.a.d.1.1		1
20.3	even	4		1600.2.a.t.1.1		1
20.7	even	4		320.2.a.b.1.1		1
20.19	odd	2		1600.2.c.c.449.2		2
24.5	odd	2		7200.2.f.w.6049.2		2
24.11	even	2		7200.2.f.g.6049.1		2
40.3	even	4		800.2.a.a.1.1		1
40.13	odd	4		800.2.a.i.1.1		1
40.19	odd	2		800.2.c.b.449.1		2
40.27	even	4		160.2.a.b.1.1	yes	1
40.29	even	2		800.2.c.a.449.2		2
40.37	odd	4		160.2.a.a.1.1	✓	1
60.47	odd	4		2880.2.a.o.1.1		1
80.27	even	4		1280.2.d.b.641.2		2
80.37	odd	4		1280.2.d.h.641.1		2
80.67	even	4		1280.2.d.b.641.1		2
80.77	odd	4		1280.2.d.h.641.2		2
120.29	odd	2		7200.2.f.w.6049.1		2
120.53	even	4		7200.2.a.bp.1.1		1
120.59	even	2		7200.2.f.g.6049.2		2
120.77	even	4		1440.2.a.i.1.1		1
120.83	odd	4		7200.2.a.l.1.1		1
120.107	odd	4		1440.2.a.l.1.1		1
280.27	odd	4		7840.2.a.e.1.1		1
280.237	even	4		7840.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.a.1.1	✓	1	40.37	odd	4
160.2.a.b.1.1	yes	1	40.27	even	4
320.2.a.b.1.1		1	20.7	even	4
320.2.a.e.1.1		1	5.2	odd	4
800.2.a.a.1.1		1	40.3	even	4
800.2.a.i.1.1		1	40.13	odd	4
800.2.c.a.449.1		2	8.5	even	2
800.2.c.a.449.2		2	40.29	even	2
800.2.c.b.449.1		2	40.19	odd	2
800.2.c.b.449.2		2	8.3	odd	2
1280.2.d.b.641.1		2	80.67	even	4
1280.2.d.b.641.2		2	80.27	even	4
1280.2.d.h.641.1		2	80.37	odd	4
1280.2.d.h.641.2		2	80.77	odd	4
1440.2.a.i.1.1		1	120.77	even	4
1440.2.a.l.1.1		1	120.107	odd	4
1600.2.a.e.1.1		1	5.3	odd	4
1600.2.a.t.1.1		1	20.3	even	4
1600.2.c.c.449.1		2	4.3	odd	2
1600.2.c.c.449.2		2	20.19	odd	2
1600.2.c.f.449.1		2	5.4	even	2	inner
1600.2.c.f.449.2		2	1.1	even	1	trivial
2880.2.a.d.1.1		1	15.2	even	4
2880.2.a.o.1.1		1	60.47	odd	4
7200.2.a.l.1.1		1	120.83	odd	4
7200.2.a.bp.1.1		1	120.53	even	4
7200.2.f.g.6049.1		2	24.11	even	2
7200.2.f.g.6049.2		2	120.59	even	2
7200.2.f.w.6049.1		2	120.29	odd	2
7200.2.f.w.6049.2		2	24.5	odd	2
7840.2.a.e.1.1		1	280.27	odd	4
7840.2.a.w.1.1		1	280.237	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} -2.00000i q^{17} +8.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +4.00000i q^{27} -2.00000 q^{29} +4.00000 q^{31} +8.00000i q^{33} +2.00000i q^{37} -12.0000 q^{39} -10.0000 q^{41} +2.00000i q^{43} +2.00000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -2.00000i q^{53} +16.0000i q^{57} -2.00000 q^{61} -2.00000i q^{63} -6.00000i q^{67} +12.0000 q^{69} -12.0000 q^{71} +10.0000i q^{73} +8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +10.0000i q^{83} -4.00000i q^{87} +6.00000 q^{89} -12.0000 q^{91} +8.00000i q^{93} -10.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 8 q^{11} + 16 q^{19} - 8 q^{21} - 4 q^{29} + 8 q^{31} - 24 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} - 4 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} - 22 q^{81} + 12 q^{89} - 24 q^{91} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 8 * q^11 + 16 * q^19 - 8 * q^21 - 4 * q^29 + 8 * q^31 - 24 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^51 - 4 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 - 22 * q^81 + 12 * q^89 - 24 * q^91 - 8 * q^99

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