LMFDB - Embedded newform 1600.2.c.n.449.3

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:	$4$
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:	$2^{5}$
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1600.449
Dual form			1600.2.c.n.449.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{3} -2.82843i q^{7} -5.00000 q^{9} -5.65685 q^{11} +2.00000i q^{13} -2.00000i q^{17} +8.00000 q^{21} -2.82843i q^{23} -5.65685i q^{27} +6.00000 q^{29} -5.65685 q^{31} -16.0000i q^{33} -10.0000i q^{37} -5.65685 q^{39} +2.00000 q^{41} -8.48528i q^{43} -2.82843i q^{47} -1.00000 q^{49} +5.65685 q^{51} -6.00000i q^{53} -11.3137 q^{59} +2.00000 q^{61} +14.1421i q^{63} +2.82843i q^{67} +8.00000 q^{69} -5.65685 q^{71} -6.00000i q^{73} +16.0000i q^{77} +11.3137 q^{79} +1.00000 q^{81} +2.82843i q^{83} +16.9706i q^{87} -10.0000 q^{89} +5.65685 q^{91} -16.0000i q^{93} -2.00000i q^{97} +28.2843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 20 q^{9} + 32 q^{21} + 24 q^{29} + 8 q^{41} - 4 q^{49} + 8 q^{61} + 32 q^{69} + 4 q^{81} - 40 q^{89}+O(q^{100})$ 4 * q - 20 * q^9 + 32 * q^21 + 24 * q^29 + 8 * q^41 - 4 * q^49 + 8 * q^61 + 32 * q^69 + 4 * q^81 - 40 * q^89

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
$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$	2.82843i	1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	−5.00000	−1.66667
$10$	0	0
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	0	0
$23$	− 2.82843i	− 0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$	− 2.82843i	− 0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	− 16.0000i	− 2.78524i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	−5.65685	−0.905822
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 8.48528i	− 1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	− 8.48528i	− 1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.82843i	− 0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$	− 2.82843i	− 0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	5.65685	0.792118
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	14.1421i	1.78174i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.82843i	0.345547i	0.984962	+	0.172774i	$0.0552729\pi$
$67$	2.82843i	0.345547i	−0.984962	+	0.172774i	$0.944727\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.0000i	1.82337i
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.82843i	0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$	2.82843i	0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.9706i	1.81944i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	5.65685	0.592999
$92$	0	0
$93$	− 16.0000i	− 1.65912i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	28.2843	2.84268
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	− 14.1421i	− 1.39347i	−0.717331	−	0.696733i	$-0.754636\pi$
$103$	− 14.1421i	− 1.39347i	0.717331	−	0.696733i	$-0.245364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	28.2843	2.68462
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 10.0000i	− 0.924500i
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	5.65685i	0.510061i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.82843i	− 0.250982i	−0.992095	−	0.125491i	$-0.959949\pi$
$127$	− 2.82843i	− 0.250982i	0.992095	−	0.125491i	$-0.0400507\pi$
$128$	0	0
$129$	24.0000	2.11308
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	− 11.3137i	− 0.946100i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.82843i	− 0.233285i
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	16.9706	1.38104	0.690522	−	0.723311i	$-0.257381\pi$
$151$	16.9706	1.38104	0.690522	+	0.723311i	$0.257381\pi$
$152$	0	0
$153$	10.0000i	0.808452i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	16.9706	1.34585
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	14.1421i	1.10770i	0.832617	+	0.553849i	$0.186841\pi$
$163$	14.1421i	1.10770i	−0.832617	+	0.553849i	$0.813159\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.1421i	− 1.09435i	−0.837018	−	0.547176i	$-0.815703\pi$
$167$	− 14.1421i	− 1.09435i	0.837018	−	0.547176i	$-0.184297\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 32.0000i	− 2.40527i
$178$	0	0
$179$	−11.3137	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$179$	−11.3137	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	5.65685i	0.418167i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.3137i	0.827340i
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−22.6274	−1.60402	−0.802008	−	0.597314i	$-0.796235\pi$
$199$	−22.6274	−1.60402	−0.802008	+	0.597314i	$0.796235\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	− 16.9706i	− 1.19110i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.1421i	0.982946i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.9706	−1.16830	−0.584151	−	0.811645i	$-0.698572\pi$
$211$	−16.9706	−1.16830	−0.584151	+	0.811645i	$0.698572\pi$
$212$	0	0
$213$	− 16.0000i	− 1.09630i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	16.9706	1.14676
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	8.48528i	0.568216i	0.958792	+	0.284108i	$0.0916975\pi$
$223$	8.48528i	0.568216i	−0.958792	+	0.284108i	$0.908302\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19.7990i	− 1.31411i	−0.753845	−	0.657053i	$-0.771803\pi$
$227$	− 19.7990i	− 1.31411i	0.753845	−	0.657053i	$-0.228197\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−45.2548	−2.97755
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.0000i	2.07862i
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	− 14.1421i	− 0.907218i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	0	0
$259$	−28.2843	−1.75750
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	19.7990i	1.22086i	0.792071	+	0.610429i	$0.209003\pi$
$263$	19.7990i	1.22086i	−0.792071	+	0.610429i	$0.790997\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 28.2843i	− 1.73097i
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	16.0000i	0.968364i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	28.2843	1.69334
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	− 8.48528i	− 0.504398i	−0.967675	−	0.252199i	$-0.918846\pi$
$283$	− 8.48528i	− 0.504398i	0.967675	−	0.252199i	$-0.0811537\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.65685i	− 0.333914i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	5.65685	0.331611
$292$	0	0
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	32.0000i	1.85683i
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	5.65685i	0.324978i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.82843i	0.161427i	0.996737	+	0.0807134i	$0.0257199\pi$
$307$	2.82843i	0.161427i	−0.996737	+	0.0807134i	$0.974280\pi$
$308$	0	0
$309$	40.0000	2.27552
$310$	0	0
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	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
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	0	0
$319$	−33.9411	−1.90034
$320$	0	0
$321$	−40.0000	−2.23258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 50.9117i	− 2.81542i
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	5.65685	0.310929	0.155464	−	0.987841i	$-0.450313\pi$
$331$	5.65685	0.310929	0.155464	+	0.987841i	$0.450313\pi$
$332$	0	0
$333$	50.0000i	2.73998i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.48528i	− 0.455514i	−0.973718	−	0.227757i	$-0.926861\pi$
$347$	− 8.48528i	− 0.455514i	0.973718	−	0.227757i	$-0.0731391\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	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
$357$	− 16.0000i	− 0.846810i
$358$	0	0
$359$	22.6274	1.19423	0.597115	−	0.802156i	$-0.296314\pi$
$359$	22.6274	1.19423	0.597115	+	0.802156i	$0.296314\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	59.3970i	3.11753i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.48528i	0.442928i	0.975169	+	0.221464i	$0.0710835\pi$
$367$	8.48528i	0.442928i	−0.975169	+	0.221464i	$0.928916\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−16.9706	−0.881068
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	22.6274	1.16229	0.581146	−	0.813799i	$-0.302604\pi$
$379$	22.6274	1.16229	0.581146	+	0.813799i	$0.302604\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 36.7696i	− 1.87884i	−0.342773	−	0.939418i	$-0.611366\pi$
$383$	− 36.7696i	− 1.87884i	0.342773	−	0.939418i	$-0.388634\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	42.4264i	2.15666i
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	16.0000i	0.807093i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	0	0
$400$	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
$403$	− 11.3137i	− 0.563576i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	56.5685i	2.80400i
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	0	0
$413$	32.0000i	1.57462i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 32.0000i	− 1.56705i
$418$	0	0
$419$	11.3137	0.552711	0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137	0.552711	0.276355	+	0.961056i	$0.410873\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	0	0
$423$	14.1421i	0.687614i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5.65685i	− 0.273754i
$428$	0	0
$429$	32.0000	1.54497
$430$	0	0
$431$	5.65685	0.272481	0.136241	−	0.990676i	$-0.456498\pi$
$431$	5.65685	0.272481	0.136241	+	0.990676i	$0.456498\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	2.82843i	0.134383i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$	2.82843i	0.134383i	−0.997740	+	0.0671913i	$0.978596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 28.2843i	− 1.33780i
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	0	0
$453$	48.0000i	2.25524i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	0	0
$459$	−11.3137	−0.528079
$460$	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
$463$	8.48528i	0.394344i	0.980369	+	0.197172i	$0.0631758\pi$
$463$	8.48528i	0.394344i	−0.980369	+	0.197172i	$0.936824\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.1421i	0.654420i	0.944952	+	0.327210i	$0.106108\pi$
$467$	14.1421i	0.654420i	−0.944952	+	0.327210i	$0.893892\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	50.9117	2.34589
$472$	0	0
$473$	48.0000i	2.20704i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.0000i	1.37361i
$478$	0	0
$479$	−33.9411	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$479$	−33.9411	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	− 22.6274i	− 1.02958i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.1127i	1.40985i	0.709281	+	0.704925i	$0.249020\pi$
$487$	31.1127i	1.40985i	−0.709281	+	0.704925i	$0.750980\pi$
$488$	0	0
$489$	−40.0000	−1.80886
$490$	0	0
$491$	39.5980	1.78703	0.893516	−	0.449032i	$-0.148231\pi$
$491$	39.5980	1.78703	0.893516	+	0.449032i	$0.148231\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.0000i	0.717698i
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	40.0000	1.78707
$502$	0	0
$503$	8.48528i	0.378340i	0.981944	+	0.189170i	$0.0605797\pi$
$503$	8.48528i	0.378340i	−0.981944	+	0.189170i	$0.939420\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	25.4558i	1.13053i
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−16.9706	−0.750733
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−5.65685	−0.248308
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	− 8.48528i	− 0.371035i	−0.982641	−	0.185518i	$-0.940604\pi$
$523$	− 8.48528i	− 0.371035i	0.982641	−	0.185518i	$-0.0593962\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.3137i	0.492833i
$528$	0	0
$529$	15.0000	0.652174
$530$	0	0
$531$	56.5685	2.45487
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 32.0000i	− 1.38090i
$538$	0	0
$539$	5.65685	0.243658
$540$	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
$543$	− 39.5980i	− 1.69931i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 42.4264i	− 1.81402i	−0.421107	−	0.907011i	$-0.638358\pi$
$547$	− 42.4264i	− 1.81402i	0.421107	−	0.907011i	$-0.361642\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	0	0
$561$	−32.0000	−1.35104
$562$	0	0
$563$	− 19.7990i	− 0.834428i	−0.908808	−	0.417214i	$-0.863007\pi$
$563$	− 19.7990i	− 0.834428i	0.908808	−	0.417214i	$-0.136993\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.82843i	− 0.118783i
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	28.2843	1.18366	0.591830	−	0.806063i	$-0.298406\pi$
$571$	28.2843	1.18366	0.591830	+	0.806063i	$0.298406\pi$
$572$	0	0
$573$	− 48.0000i	− 2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0000i	0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$	14.0000i	0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$	0	0
$579$	−50.9117	−2.11582
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	33.9411i	1.40570i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.4558i	1.05068i	0.850894	+	0.525338i	$0.176061\pi$
$587$	25.4558i	1.05068i	−0.850894	+	0.525338i	$0.823939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−16.9706	−0.698076
$592$	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
$597$	− 64.0000i	− 2.61935i
$598$	0	0
$599$	11.3137	0.462266	0.231133	−	0.972922i	$-0.425757\pi$
$599$	11.3137	0.462266	0.231133	+	0.972922i	$0.425757\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	− 14.1421i	− 0.575912i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.82843i	− 0.114802i	−0.998351	−	0.0574012i	$-0.981719\pi$
$607$	− 2.82843i	− 0.114802i	0.998351	−	0.0574012i	$-0.0182814\pi$
$608$	0	0
$609$	48.0000	1.94506
$610$	0	0
$611$	5.65685	0.228852
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.0000i	− 0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$	− 10.0000i	− 0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$	0	0
$619$	45.2548	1.81895	0.909473	−	0.415764i	$-0.136486\pi$
$619$	45.2548	1.81895	0.909473	+	0.415764i	$0.136486\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	28.2843i	1.13319i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	16.9706	0.675587	0.337794	−	0.941220i	$-0.390319\pi$
$631$	16.9706	0.675587	0.337794	+	0.941220i	$0.390319\pi$
$632$	0	0
$633$	− 48.0000i	− 1.90783i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	28.2843	1.11891
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	− 31.1127i	− 1.22697i	−0.789708	−	0.613483i	$-0.789768\pi$
$643$	− 31.1127i	− 1.22697i	0.789708	−	0.613483i	$-0.210232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 14.1421i	− 0.555985i	−0.960583	−	0.277992i	$-0.910331\pi$
$647$	− 14.1421i	− 0.555985i	0.960583	−	0.277992i	$-0.0896690\pi$
$648$	0	0
$649$	64.0000	2.51222
$650$	0	0
$651$	−45.2548	−1.77368
$652$	0	0
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000i	1.17041i
$658$	0	0
$659$	33.9411	1.32216	0.661079	−	0.750316i	$-0.270099\pi$
$659$	33.9411	1.32216	0.661079	+	0.750316i	$0.270099\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	11.3137i	0.439388i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.9706i	− 0.657103i
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	−5.65685	−0.217090
$680$	0	0
$681$	56.0000	2.14592
$682$	0	0
$683$	14.1421i	0.541134i	0.962701	+	0.270567i	$0.0872111\pi$
$683$	14.1421i	0.541134i	−0.962701	+	0.270567i	$0.912789\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	39.5980i	1.51076i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.2843	−1.07598	−0.537992	−	0.842950i	$-0.680817\pi$
$691$	−28.2843	−1.07598	−0.537992	+	0.842950i	$0.680817\pi$
$692$	0	0
$693$	− 80.0000i	− 3.03895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	−28.2843	−1.06981
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.65685i	− 0.212748i
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−56.5685	−2.12149
$712$	0	0
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 32.0000i	− 1.19506i
$718$	0	0
$719$	−11.3137	−0.421930	−0.210965	−	0.977494i	$-0.567661\pi$
$719$	−11.3137	−0.421930	−0.210965	+	0.977494i	$0.567661\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	73.5391i	2.73495i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 25.4558i	− 0.944105i	−0.881570	−	0.472052i	$-0.843513\pi$
$727$	− 25.4558i	− 0.944105i	0.881570	−	0.472052i	$-0.156487\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	−16.9706	−0.627679
$732$	0	0
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	11.3137	0.416181	0.208091	−	0.978110i	$-0.433275\pi$
$739$	11.3137	0.416181	0.208091	+	0.978110i	$0.433275\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 14.1421i	− 0.518825i	−0.965767	−	0.259412i	$-0.916471\pi$
$743$	− 14.1421i	− 0.518825i	0.965767	−	0.259412i	$-0.0835289\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 14.1421i	− 0.517434i
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	16.9706	0.619265	0.309632	−	0.950856i	$-0.399794\pi$
$751$	16.9706	0.619265	0.309632	+	0.950856i	$0.399794\pi$
$752$	0	0
$753$	− 16.0000i	− 0.583072i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	−45.2548	−1.64265
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	50.9117i	1.84313i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 22.6274i	− 0.817029i
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	−39.5980	−1.42609
$772$	0	0
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 80.0000i	− 2.86998i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	− 33.9411i	− 1.21296i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.48528i	− 0.302468i	−0.988498	−	0.151234i	$-0.951675\pi$
$787$	− 8.48528i	− 0.302468i	0.988498	−	0.151234i	$-0.0483246\pi$
$788$	0	0
$789$	−56.0000	−1.99365
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	4.00000i	0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	−5.65685	−0.200125
$800$	0	0
$801$	50.0000	1.76666
$802$	0	0
$803$	33.9411i	1.19776i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 50.9117i	− 1.79218i
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	− 48.0000i	− 1.68343i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−28.2843	−0.988332
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 25.4558i	− 0.887335i	−0.896191	−	0.443667i	$-0.853677\pi$
$823$	− 25.4558i	− 0.887335i	0.896191	−	0.443667i	$-0.146323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 31.1127i	− 1.08189i	−0.841057	−	0.540947i	$-0.818066\pi$
$827$	− 31.1127i	− 1.08189i	0.841057	−	0.540947i	$-0.181934\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	28.2843	0.981170
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	32.0000i	1.10608i
$838$	0	0
$839$	11.3137	0.390593	0.195296	−	0.980744i	$-0.437433\pi$
$839$	11.3137	0.390593	0.195296	+	0.980744i	$0.437433\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 84.8528i	− 2.92249i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 59.3970i	− 2.04090i
$848$	0	0
$849$	24.0000	0.823678
$850$	0	0
$851$	−28.2843	−0.969572
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	−33.9411	−1.15806	−0.579028	−	0.815308i	$-0.696568\pi$
$859$	−33.9411	−1.15806	−0.579028	+	0.815308i	$0.696568\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	42.4264i	1.44421i	0.691783	+	0.722106i	$0.256826\pi$
$863$	42.4264i	1.44421i	−0.691783	+	0.722106i	$0.743174\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	36.7696i	1.24876i
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	−28.2843	−0.954005
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	2.82843i	0.0951842i	0.998867	+	0.0475921i	$0.0151548\pi$
$883$	2.82843i	0.0951842i	−0.998867	+	0.0475921i	$0.984845\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.82843i	− 0.0949693i	−0.998872	−	0.0474846i	$-0.984879\pi$
$887$	− 2.82843i	− 0.0949693i	0.998872	−	0.0474846i	$-0.0151205\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−5.65685	−0.189512
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000i	0.534224i
$898$	0	0
$899$	−33.9411	−1.13200
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	− 67.8823i	− 2.25898i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.4558i	0.845247i	0.906305	+	0.422624i	$0.138891\pi$
$907$	25.4558i	0.845247i	−0.906305	+	0.422624i	$0.861109\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	33.9411	1.11961	0.559807	−	0.828623i	$-0.310875\pi$
$919$	33.9411	1.11961	0.559807	+	0.828623i	$0.310875\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	− 11.3137i	− 0.372395i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	70.7107i	2.32244i
$928$	0	0
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	80.0000i	2.61908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	0	0
$939$	16.9706	0.553813
$940$	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
$943$	− 5.65685i	− 0.184213i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.4264i	− 1.37867i	−0.724441	−	0.689336i	$-0.757902\pi$
$947$	− 42.4264i	− 1.37867i	0.724441	−	0.689336i	$-0.242098\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	−84.8528	−2.75154
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 96.0000i	− 3.10324i
$958$	0	0
$959$	16.9706	0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 70.7107i	− 2.27862i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	42.4264i	1.36434i	0.731193	+	0.682171i	$0.238964\pi$
$967$	42.4264i	1.36434i	−0.731193	+	0.682171i	$0.761036\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.2843	0.907685	0.453843	−	0.891082i	$-0.350053\pi$
$971$	28.2843	0.907685	0.453843	+	0.891082i	$0.350053\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	56.5685	1.80794
$980$	0	0
$981$	90.0000	2.87348
$982$	0	0
$983$	− 2.82843i	− 0.0902128i	−0.998982	−	0.0451064i	$-0.985637\pi$
$983$	− 2.82843i	− 0.0902128i	0.998982	−	0.0451064i	$-0.0143627\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 22.6274i	− 0.720239i
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	16.9706	0.539088	0.269544	−	0.962988i	$-0.413127\pi$
$991$	16.9706	0.539088	0.269544	+	0.962988i	$0.413127\pi$
$992$	0	0
$993$	16.0000i	0.507745i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	−56.5685	−1.78975

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a_p

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p

up to: 50 250 1000 (See

a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.c.n.449.3		4
4.3	odd	2	inner	1600.2.c.n.449.2		4
5.2	odd	4		320.2.a.g.1.2		2
5.3	odd	4		1600.2.a.bc.1.1		2
5.4	even	2	inner	1600.2.c.n.449.1		4
8.3	odd	2		800.2.c.f.449.3		4
8.5	even	2		800.2.c.f.449.2		4
15.2	even	4		2880.2.a.bk.1.2		2
20.3	even	4		1600.2.a.bc.1.2		2
20.7	even	4		320.2.a.g.1.1		2
20.19	odd	2	inner	1600.2.c.n.449.4		4
24.5	odd	2		7200.2.f.bh.6049.1		4
24.11	even	2		7200.2.f.bh.6049.4		4
40.3	even	4		800.2.a.m.1.1		2
40.13	odd	4		800.2.a.m.1.2		2
40.19	odd	2		800.2.c.f.449.1		4
40.27	even	4		160.2.a.c.1.2	yes	2
40.29	even	2		800.2.c.f.449.4		4
40.37	odd	4		160.2.a.c.1.1	✓	2
60.47	odd	4		2880.2.a.bk.1.1		2
80.27	even	4		1280.2.d.l.641.3		4
80.37	odd	4		1280.2.d.l.641.1		4
80.67	even	4		1280.2.d.l.641.2		4
80.77	odd	4		1280.2.d.l.641.4		4
120.29	odd	2		7200.2.f.bh.6049.3		4
120.53	even	4		7200.2.a.cm.1.1		2
120.59	even	2		7200.2.f.bh.6049.2		4
120.77	even	4		1440.2.a.o.1.2		2
120.83	odd	4		7200.2.a.cm.1.2		2
120.107	odd	4		1440.2.a.o.1.1		2
280.27	odd	4		7840.2.a.bf.1.1		2
280.237	even	4		7840.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.c.1.1	✓	2	40.37	odd	4
160.2.a.c.1.2	yes	2	40.27	even	4
320.2.a.g.1.1		2	20.7	even	4
320.2.a.g.1.2		2	5.2	odd	4
800.2.a.m.1.1		2	40.3	even	4
800.2.a.m.1.2		2	40.13	odd	4
800.2.c.f.449.1		4	40.19	odd	2
800.2.c.f.449.2		4	8.5	even	2
800.2.c.f.449.3		4	8.3	odd	2
800.2.c.f.449.4		4	40.29	even	2
1280.2.d.l.641.1		4	80.37	odd	4
1280.2.d.l.641.2		4	80.67	even	4
1280.2.d.l.641.3		4	80.27	even	4
1280.2.d.l.641.4		4	80.77	odd	4
1440.2.a.o.1.1		2	120.107	odd	4
1440.2.a.o.1.2		2	120.77	even	4
1600.2.a.bc.1.1		2	5.3	odd	4
1600.2.a.bc.1.2		2	20.3	even	4
1600.2.c.n.449.1		4	5.4	even	2	inner
1600.2.c.n.449.2		4	4.3	odd	2	inner
1600.2.c.n.449.3		4	1.1	even	1	trivial
1600.2.c.n.449.4		4	20.19	odd	2	inner
2880.2.a.bk.1.1		2	60.47	odd	4
2880.2.a.bk.1.2		2	15.2	even	4
7200.2.a.cm.1.1		2	120.53	even	4
7200.2.a.cm.1.2		2	120.83	odd	4
7200.2.f.bh.6049.1		4	24.5	odd	2
7200.2.f.bh.6049.2		4	120.59	even	2
7200.2.f.bh.6049.3		4	120.29	odd	2
7200.2.f.bh.6049.4		4	24.11	even	2
7840.2.a.bf.1.1		2	280.27	odd	4
7840.2.a.bf.1.2		2	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1600.2.c.n.449.3

Level	$1600$
Weight	$2$
Character	1600.449
Analytic conductor	$12.776$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$