LMFDB - Embedded newform 320.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,2,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$320 = 2^{6} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	320.a (trivial)

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:	yes
Analytic conductor:	$2.55521286468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{3} -1.00000 q^{5} +2.82843 q^{7} +5.00000 q^{9} -5.65685 q^{11} +2.00000 q^{13} -2.82843 q^{15} +2.00000 q^{17} +8.00000 q^{21} -2.82843 q^{23} +1.00000 q^{25} +5.65685 q^{27} -6.00000 q^{29} -5.65685 q^{31} -16.0000 q^{33} -2.82843 q^{35} +10.0000 q^{37} +5.65685 q^{39} +2.00000 q^{41} -8.48528 q^{43} -5.00000 q^{45} +2.82843 q^{47} +1.00000 q^{49} +5.65685 q^{51} -6.00000 q^{53} +5.65685 q^{55} +11.3137 q^{59} +2.00000 q^{61} +14.1421 q^{63} -2.00000 q^{65} -2.82843 q^{67} -8.00000 q^{69} -5.65685 q^{71} -6.00000 q^{73} +2.82843 q^{75} -16.0000 q^{77} -11.3137 q^{79} +1.00000 q^{81} +2.82843 q^{83} -2.00000 q^{85} -16.9706 q^{87} +10.0000 q^{89} +5.65685 q^{91} -16.0000 q^{93} +2.00000 q^{97} -28.2843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} + 10 q^{9} + 4 q^{13} + 4 q^{17} + 16 q^{21} + 2 q^{25} - 12 q^{29} - 32 q^{33} + 20 q^{37} + 4 q^{41} - 10 q^{45} + 2 q^{49} - 12 q^{53} + 4 q^{61} - 4 q^{65} - 16 q^{69} - 12 q^{73} - 32 q^{77}+ \cdots + 4 q^{97}+O(q^{100})$ 2 * q - 2 * q^5 + 10 * q^9 + 4 * q^13 + 4 * q^17 + 16 * q^21 + 2 * q^25 - 12 * q^29 - 32 * q^33 + 20 * q^37 + 4 * q^41 - 10 * q^45 + 2 * q^49 - 12 * q^53 + 4 * q^61 - 4 * q^65 - 16 * q^69 - 12 * q^73 - 32 * q^77 + 2 * q^81 - 4 * q^85 + 20 * q^89 - 32 * q^93 + 4 * q^97

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
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	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
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$33$	−16.0000	−2.78524
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	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
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	−5.00000	−0.745356
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.65685	0.792118
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	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
$63$	14.1421	1.78174
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−2.82843	−0.345547	−0.172774	−	0.984962i	$-0.555273\pi$
$67$	−2.82843	−0.345547	−0.172774	+	0.984962i	$0.555273\pi$
$68$	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
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	2.82843	0.326599
$76$	0	0
$77$	−16.0000	−1.82337
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−16.9706	−1.81944
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	5.65685	0.592999
$92$	0	0
$93$	−16.0000	−1.65912
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	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
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	−14.1421	−1.36717	−0.683586	−	0.729870i	$-0.739581\pi$
$107$	−14.1421	−1.36717	−0.683586	+	0.729870i	$0.739581\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	28.2843	2.68462
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	0	0
$117$	10.0000	0.924500
$118$	0	0
$119$	5.65685	0.518563
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	5.65685	0.510061
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.82843	0.250982	0.125491	−	0.992095i	$-0.459949\pi$
$127$	2.82843	0.250982	0.125491	+	0.992095i	$0.459949\pi$
$128$	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
$135$	−5.65685	−0.486864
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	11.3137	0.959616	0.479808	−	0.877373i	$-0.340706\pi$
$139$	11.3137	0.959616	0.479808	+	0.877373i	$0.340706\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−11.3137	−0.946100
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	2.82843	0.233285
$148$	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
$151$	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
$153$	10.0000	0.808452
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−16.9706	−1.34585
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	14.1421	1.10770	0.553849	−	0.832617i	$-0.313159\pi$
$163$	14.1421	1.10770	0.553849	+	0.832617i	$0.313159\pi$
$164$	0	0
$165$	16.0000	1.24560
$166$	0	0
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	32.0000	2.40527
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	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
$183$	5.65685	0.418167
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−11.3137	−0.827340
$188$	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
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	−5.65685	−0.405096
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	22.6274	1.60402	0.802008	−	0.597314i	$-0.203765\pi$
$199$	22.6274	1.60402	0.802008	+	0.597314i	$0.203765\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−16.9706	−1.19110
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−14.1421	−0.982946
$208$	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
$213$	−16.0000	−1.09630
$214$	0	0
$215$	8.48528	0.578691
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	−16.9706	−1.14676
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	8.48528	0.568216	0.284108	−	0.958792i	$-0.408302\pi$
$223$	8.48528	0.568216	0.284108	+	0.958792i	$0.408302\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	19.7990	1.31411	0.657053	−	0.753845i	$-0.271803\pi$
$227$	19.7990	1.31411	0.657053	+	0.753845i	$0.271803\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−45.2548	−2.97755
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−2.82843	−0.184506
$236$	0	0
$237$	−32.0000	−2.07862
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	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
$243$	−14.1421	−0.907218
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	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
$253$	16.0000	1.00591
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	28.2843	1.75750
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	19.7990	1.22086	0.610429	−	0.792071i	$-0.290997\pi$
$263$	19.7990	1.22086	0.610429	+	0.792071i	$0.290997\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	28.2843	1.73097
$268$	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
$271$	−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
$273$	16.0000	0.968364
$274$	0	0
$275$	−5.65685	−0.341121
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	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
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	5.65685	0.331611
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−11.3137	−0.658710
$296$	0	0
$297$	−32.0000	−1.85683
$298$	0	0
$299$	−5.65685	−0.327144
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	5.65685	0.324978
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−2.82843	−0.161427	−0.0807134	−	0.996737i	$-0.525720\pi$
$307$	−2.82843	−0.161427	−0.0807134	+	0.996737i	$0.525720\pi$
$308$	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
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	−14.1421	−0.796819
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	33.9411	1.90034
$320$	0	0
$321$	−40.0000	−2.23258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	50.9117	2.81542
$328$	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
$333$	50.0000	2.73998
$334$	0	0
$335$	2.82843	0.154533
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	5.65685	0.307238
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	8.48528	0.455514	0.227757	−	0.973718i	$-0.426861\pi$
$347$	8.48528	0.455514	0.227757	+	0.973718i	$0.426861\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	16.0000	0.846810
$358$	0	0
$359$	−22.6274	−1.19423	−0.597115	−	0.802156i	$-0.703686\pi$
$359$	−22.6274	−1.19423	−0.597115	+	0.802156i	$0.703686\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	59.3970	3.11753
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−16.9706	−0.881068
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−2.82843	−0.146059
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−22.6274	−1.16229	−0.581146	−	0.813799i	$-0.697396\pi$
$379$	−22.6274	−1.16229	−0.581146	+	0.813799i	$0.697396\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	−36.7696	−1.87884	−0.939418	−	0.342773i	$-0.888634\pi$
$383$	−36.7696	−1.87884	−0.939418	+	0.342773i	$0.888634\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	−42.4264	−2.15666
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	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
$403$	−11.3137	−0.563576
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−56.5685	−2.80400
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	0	0
$413$	32.0000	1.57462
$414$	0	0
$415$	−2.82843	−0.138842
$416$	0	0
$417$	32.0000	1.56705
$418$	0	0
$419$	−11.3137	−0.552711	−0.276355	−	0.961056i	$-0.589127\pi$
$419$	−11.3137	−0.552711	−0.276355	+	0.961056i	$0.589127\pi$
$420$	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
$423$	14.1421	0.687614
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	5.65685	0.273754
$428$	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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	16.9706	0.813676
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	2.82843	0.134383	0.0671913	−	0.997740i	$-0.478596\pi$
$443$	2.82843	0.134383	0.0671913	+	0.997740i	$0.478596\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	28.2843	1.33780
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	0	0
$453$	48.0000	2.25524
$454$	0	0
$455$	−5.65685	−0.265197
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	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
$463$	8.48528	0.394344	0.197172	−	0.980369i	$-0.436824\pi$
$463$	8.48528	0.394344	0.197172	+	0.980369i	$0.436824\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	−14.1421	−0.654420	−0.327210	−	0.944952i	$-0.606108\pi$
$467$	−14.1421	−0.654420	−0.327210	+	0.944952i	$0.606108\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	50.9117	2.34589
$472$	0	0
$473$	48.0000	2.20704
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.0000	−1.37361
$478$	0	0
$479$	33.9411	1.55081	0.775405	−	0.631464i	$-0.217546\pi$
$479$	33.9411	1.55081	0.775405	+	0.631464i	$0.217546\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	−22.6274	−1.02958
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−31.1127	−1.40985	−0.704925	−	0.709281i	$-0.749020\pi$
$487$	−31.1127	−1.40985	−0.704925	+	0.709281i	$0.749020\pi$
$488$	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
$493$	−12.0000	−0.540453
$494$	0	0
$495$	28.2843	1.27128
$496$	0	0
$497$	−16.0000	−0.717698
$498$	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
$503$	8.48528	0.378340	0.189170	−	0.981944i	$-0.439420\pi$
$503$	8.48528	0.378340	0.189170	+	0.981944i	$0.439420\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	−25.4558	−1.13053
$508$	0	0
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−16.9706	−0.750733
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.1421	0.623177
$516$	0	0
$517$	−16.0000	−0.703679
$518$	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
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	0	0
$525$	8.00000	0.349149
$526$	0	0
$527$	−11.3137	−0.492833
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	56.5685	2.45487
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	14.1421	0.611418
$536$	0	0
$537$	32.0000	1.38090
$538$	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
$543$	−39.5980	−1.69931
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	42.4264	1.81402	0.907011	−	0.421107i	$-0.138358\pi$
$547$	42.4264	1.81402	0.907011	+	0.421107i	$0.138358\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	−28.2843	−1.20060
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	−16.9706	−0.717778
$560$	0	0
$561$	−32.0000	−1.35104
$562$	0	0
$563$	−19.7990	−0.834428	−0.417214	−	0.908808i	$-0.636993\pi$
$563$	−19.7990	−0.834428	−0.417214	+	0.908808i	$0.636993\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	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
$573$	−48.0000	−2.00523
$574$	0	0
$575$	−2.82843	−0.117954
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	50.9117	2.11582
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	33.9411	1.40570
$584$	0	0
$585$	−10.0000	−0.413449
$586$	0	0
$587$	−25.4558	−1.05068	−0.525338	−	0.850894i	$-0.676061\pi$
$587$	−25.4558	−1.05068	−0.525338	+	0.850894i	$0.676061\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−16.9706	−0.698076
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−5.65685	−0.231908
$596$	0	0
$597$	64.0000	2.61935
$598$	0	0
$599$	−11.3137	−0.462266	−0.231133	−	0.972922i	$-0.574243\pi$
$599$	−11.3137	−0.462266	−0.231133	+	0.972922i	$0.574243\pi$
$600$	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
$603$	−14.1421	−0.575912
$604$	0	0
$605$	−21.0000	−0.853771
$606$	0	0
$607$	2.82843	0.114802	0.0574012	−	0.998351i	$-0.481719\pi$
$607$	2.82843	0.114802	0.0574012	+	0.998351i	$0.481719\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	5.65685	0.228852
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	−5.65685	−0.228106
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	−45.2548	−1.81895	−0.909473	−	0.415764i	$-0.863514\pi$
$619$	−45.2548	−1.81895	−0.909473	+	0.415764i	$0.863514\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	28.2843	1.13319
$624$	0	0
$625$	1.00000	0.0400000
$626$	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
$633$	−48.0000	−1.90783
$634$	0	0
$635$	−2.82843	−0.112243
$636$	0	0
$637$	2.00000	0.0792429
$638$	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
$643$	−31.1127	−1.22697	−0.613483	−	0.789708i	$-0.710232\pi$
$643$	−31.1127	−1.22697	−0.613483	+	0.789708i	$0.710232\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	14.1421	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$648$	0	0
$649$	−64.0000	−2.51222
$650$	0	0
$651$	−45.2548	−1.77368
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	−30.0000	−1.17041
$658$	0	0
$659$	−33.9411	−1.32216	−0.661079	−	0.750316i	$-0.729901\pi$
$659$	−33.9411	−1.32216	−0.661079	+	0.750316i	$0.729901\pi$
$660$	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
$663$	11.3137	0.439388
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.9706	0.657103
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	5.65685	0.217090
$680$	0	0
$681$	56.0000	2.14592
$682$	0	0
$683$	14.1421	0.541134	0.270567	−	0.962701i	$-0.412789\pi$
$683$	14.1421	0.541134	0.270567	+	0.962701i	$0.412789\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	−39.5980	−1.51076
$688$	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
$693$	−80.0000	−3.03895
$694$	0	0
$695$	−11.3137	−0.429153
$696$	0	0
$697$	4.00000	0.151511
$698$	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
$705$	−8.00000	−0.301297
$706$	0	0
$707$	5.65685	0.212748
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−56.5685	−2.12149
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	11.3137	0.423109
$716$	0	0
$717$	32.0000	1.19506
$718$	0	0
$719$	11.3137	0.421930	0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137	0.421930	0.210965	+	0.977494i	$0.432339\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	73.5391	2.73495
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	25.4558	0.944105	0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558	0.944105	0.472052	+	0.881570i	$0.343513\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−16.9706	−0.627679
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	−2.82843	−0.104328
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−11.3137	−0.416181	−0.208091	−	0.978110i	$-0.566725\pi$
$739$	−11.3137	−0.416181	−0.208091	+	0.978110i	$0.566725\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.1421	−0.518825	−0.259412	−	0.965767i	$-0.583529\pi$
$743$	−14.1421	−0.518825	−0.259412	+	0.965767i	$0.583529\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	14.1421	0.517434
$748$	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
$753$	−16.0000	−0.583072
$754$	0	0
$755$	−16.9706	−0.617622
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	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
$763$	50.9117	1.84313
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	22.6274	0.817029
$768$	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
$771$	−39.5980	−1.42609
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	80.0000	2.86998
$778$	0	0
$779$	0	0
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	−33.9411	−1.21296
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	8.48528	0.302468	0.151234	−	0.988498i	$-0.451675\pi$
$787$	8.48528	0.302468	0.151234	+	0.988498i	$0.451675\pi$
$788$	0	0
$789$	56.0000	1.99365
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	16.9706	0.601884
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	5.65685	0.200125
$800$	0	0
$801$	50.0000	1.76666
$802$	0	0
$803$	33.9411	1.19776
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	50.9117	1.79218
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	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
$813$	−48.0000	−1.68343
$814$	0	0
$815$	−14.1421	−0.495377
$816$	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
$823$	−25.4558	−0.887335	−0.443667	−	0.896191i	$-0.646323\pi$
$823$	−25.4558	−0.887335	−0.443667	+	0.896191i	$0.646323\pi$
$824$	0	0
$825$	−16.0000	−0.557048
$826$	0	0
$827$	31.1127	1.08189	0.540947	−	0.841057i	$-0.318066\pi$
$827$	31.1127	1.08189	0.540947	+	0.841057i	$0.318066\pi$
$828$	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
$831$	28.2843	0.981170
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−14.1421	−0.489409
$836$	0	0
$837$	−32.0000	−1.10608
$838$	0	0
$839$	−11.3137	−0.390593	−0.195296	−	0.980744i	$-0.562567\pi$
$839$	−11.3137	−0.390593	−0.195296	+	0.980744i	$0.562567\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−84.8528	−2.92249
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	59.3970	2.04090
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	−28.2843	−0.969572
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	33.9411	1.15806	0.579028	−	0.815308i	$-0.303432\pi$
$859$	33.9411	1.15806	0.579028	+	0.815308i	$0.303432\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	42.4264	1.44421	0.722106	−	0.691783i	$-0.243174\pi$
$863$	42.4264	1.44421	0.722106	+	0.691783i	$0.243174\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−36.7696	−1.24876
$868$	0	0
$869$	64.0000	2.17105
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	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
$883$	2.82843	0.0951842	0.0475921	−	0.998867i	$-0.484845\pi$
$883$	2.82843	0.0951842	0.0475921	+	0.998867i	$0.484845\pi$
$884$	0	0
$885$	−32.0000	−1.07567
$886$	0	0
$887$	2.82843	0.0949693	0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843	0.0949693	0.0474846	+	0.998872i	$0.484879\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−5.65685	−0.189512
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	33.9411	1.13200
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−67.8823	−2.25898
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	−25.4558	−0.845247	−0.422624	−	0.906305i	$-0.638891\pi$
$907$	−25.4558	−0.845247	−0.422624	+	0.906305i	$0.638891\pi$
$908$	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
$913$	−16.0000	−0.529523
$914$	0	0
$915$	−5.65685	−0.187010
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	−33.9411	−1.11961	−0.559807	−	0.828623i	$-0.689125\pi$
$919$	−33.9411	−1.11961	−0.559807	+	0.828623i	$0.689125\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	−11.3137	−0.372395
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	−70.7107	−2.32244
$928$	0	0
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	80.0000	2.61908
$934$	0	0
$935$	11.3137	0.369998
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	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
$943$	−5.65685	−0.184213
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	42.4264	1.37867	0.689336	−	0.724441i	$-0.257902\pi$
$947$	42.4264	1.37867	0.689336	+	0.724441i	$0.257902\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−84.8528	−2.75154
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	16.9706	0.549155
$956$	0	0
$957$	96.0000	3.10324
$958$	0	0
$959$	−16.9706	−0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−70.7107	−2.27862
$964$	0	0
$965$	−18.0000	−0.579441
$966$	0	0
$967$	−42.4264	−1.36434	−0.682171	−	0.731193i	$-0.738964\pi$
$967$	−42.4264	−1.36434	−0.682171	+	0.731193i	$0.738964\pi$
$968$	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
$973$	32.0000	1.02587
$974$	0	0
$975$	5.65685	0.181164
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−56.5685	−1.80794
$980$	0	0
$981$	90.0000	2.87348
$982$	0	0
$983$	−2.82843	−0.0902128	−0.0451064	−	0.998982i	$-0.514363\pi$
$983$	−2.82843	−0.0902128	−0.0451064	+	0.998982i	$0.514363\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	22.6274	0.720239
$988$	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
$993$	16.0000	0.507745
$994$	0	0
$995$	−22.6274	−0.717337
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	56.5685	1.78975

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

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.c.1.1	✓	2	8.5	even	2
160.2.a.c.1.2	yes	2	8.3	odd	2
320.2.a.g.1.1		2	4.3	odd	2	inner
320.2.a.g.1.2		2	1.1	even	1	trivial
800.2.a.m.1.1		2	40.19	odd	2
800.2.a.m.1.2		2	40.29	even	2
800.2.c.f.449.1		4	40.27	even	4
800.2.c.f.449.2		4	40.13	odd	4
800.2.c.f.449.3		4	40.3	even	4
800.2.c.f.449.4		4	40.37	odd	4
1280.2.d.l.641.1		4	16.5	even	4
1280.2.d.l.641.2		4	16.3	odd	4
1280.2.d.l.641.3		4	16.11	odd	4
1280.2.d.l.641.4		4	16.13	even	4
1440.2.a.o.1.1		2	24.11	even	2
1440.2.a.o.1.2		2	24.5	odd	2
1600.2.a.bc.1.1		2	5.4	even	2
1600.2.a.bc.1.2		2	20.19	odd	2
1600.2.c.n.449.1		4	5.2	odd	4
1600.2.c.n.449.2		4	20.3	even	4
1600.2.c.n.449.3		4	5.3	odd	4
1600.2.c.n.449.4		4	20.7	even	4
2880.2.a.bk.1.1		2	12.11	even	2
2880.2.a.bk.1.2		2	3.2	odd	2
7200.2.a.cm.1.1		2	120.29	odd	2
7200.2.a.cm.1.2		2	120.59	even	2
7200.2.f.bh.6049.1		4	120.53	even	4
7200.2.f.bh.6049.2		4	120.107	odd	4
7200.2.f.bh.6049.3		4	120.77	even	4
7200.2.f.bh.6049.4		4	120.83	odd	4
7840.2.a.bf.1.1		2	56.27	even	2
7840.2.a.bf.1.2		2	56.13	odd	2

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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	320.2.a.g.1.2

Level	$320$
Weight	$2$
Character	320.1
Self dual	yes
Analytic conductor	$2.555$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$