LMFDB - Embedded newform 3640.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} -5.65685 q^{11} +1.00000 q^{13} +7.65685 q^{17} +4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} -9.65685 q^{31} -1.00000 q^{35} +6.00000 q^{37} -3.65685 q^{41} +5.65685 q^{43} -3.00000 q^{45} +1.00000 q^{49} +3.65685 q^{53} -5.65685 q^{55} +6.00000 q^{61} +3.00000 q^{63} +1.00000 q^{65} +4.00000 q^{67} +4.00000 q^{71} +10.0000 q^{73} +5.65685 q^{77} +11.3137 q^{79} +9.00000 q^{81} +4.00000 q^{83} +7.65685 q^{85} -3.65685 q^{89} -1.00000 q^{91} -9.31371 q^{97} +16.9706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^9	$ 2 q + 2 q^{5} - 2 q^{7} - 6 q^{9} + 2 q^{13} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} - 2 q^{35} + 12 q^{37} + 4 q^{41} - 6 q^{45} + 2 q^{49} - 4 q^{53} + 12 q^{61} + 6 q^{63} + 2 q^{65} + 8 q^{67} + 8 q^{71} + 20 q^{73} + 18 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{89} - 2 q^{91} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^9 + 2 * q^13 + 4 * q^17 + 8 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 - 2 * q^35 + 12 * q^37 + 4 * q^41 - 6 * q^45 + 2 * q^49 - 4 * q^53 + 12 * q^61 + 6 * q^63 + 2 * q^65 + 8 * q^67 + 8 * q^71 + 20 * q^73 + 18 * q^81 + 8 * q^83 + 4 * q^85 + 4 * q^89 - 2 * q^91 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−3.00000	−1.00000
$10$	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
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	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
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.65685	0.644658
$78$	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
$81$	9.00000	1.00000
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.31371	−0.945664	−0.472832	−	0.881153i	$-0.656768\pi$
$97$	−9.31371	−0.945664	−0.472832	+	0.881153i	$0.656768\pi$
$98$	0	0
$99$	16.9706	1.70561
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	1.65685	0.163255	0.0816274	−	0.996663i	$-0.473988\pi$
$103$	1.65685	0.163255	0.0816274	+	0.996663i	$0.473988\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	0	0
$112$	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
$115$	4.00000	0.373002
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	−7.65685	−0.701903
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.3137	1.13747	0.568733	−	0.822522i	$-0.307434\pi$
$137$	13.3137	1.13747	0.568733	+	0.822522i	$0.307434\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	0	0
$151$	0.686292	0.0558496	0.0279248	−	0.999610i	$-0.491110\pi$
$151$	0.686292	0.0558496	0.0279248	+	0.999610i	$0.491110\pi$
$152$	0	0
$153$	−22.9706	−1.85706
$154$	0	0
$155$	−9.65685	−0.775657
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.3137	−1.49454	−0.747270	−	0.664521i	$-0.768636\pi$
$167$	−19.3137	−1.49454	−0.747270	+	0.664521i	$0.768636\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	−43.3137	−3.16741
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	−17.3137	−1.24627	−0.623134	−	0.782115i	$-0.714141\pi$
$193$	−17.3137	−1.24627	−0.623134	+	0.782115i	$0.714141\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−19.3137	−1.36911	−0.684556	−	0.728960i	$-0.740004\pi$
$199$	−19.3137	−1.36911	−0.684556	+	0.728960i	$0.740004\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−3.65685	−0.255406
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	9.65685	0.655550
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.65685	0.515056
$222$	0	0
$223$	27.3137	1.82906	0.914531	−	0.404517i	$-0.132560\pi$
$223$	27.3137	1.82906	0.914531	+	0.404517i	$0.132560\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	23.3137	1.54739	0.773693	−	0.633561i	$-0.218408\pi$
$227$	23.3137	1.54739	0.773693	+	0.633561i	$0.218408\pi$
$228$	0	0
$229$	−5.31371	−0.351140	−0.175570	−	0.984467i	$-0.556177\pi$
$229$	−5.31371	−0.351140	−0.175570	+	0.984467i	$0.556177\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.3137	1.39631	0.698154	−	0.715948i	$-0.254005\pi$
$233$	21.3137	1.39631	0.698154	+	0.715948i	$0.254005\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	15.6569	1.00855	0.504273	−	0.863544i	$-0.331760\pi$
$241$	15.6569	1.00855	0.504273	+	0.863544i	$0.331760\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.31371	0.461637	0.230819	−	0.972997i	$-0.425860\pi$
$251$	7.31371	0.461637	0.230819	+	0.972997i	$0.425860\pi$
$252$	0	0
$253$	−22.6274	−1.42257
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.65685	0.477621	0.238811	−	0.971066i	$-0.423242\pi$
$257$	7.65685	0.477621	0.238811	+	0.971066i	$0.423242\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−31.3137	−1.93089	−0.965443	−	0.260614i	$-0.916075\pi$
$263$	−31.3137	−1.93089	−0.965443	+	0.260614i	$0.916075\pi$
$264$	0	0
$265$	3.65685	0.224639
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	25.6569	1.55854	0.779271	−	0.626687i	$-0.215589\pi$
$271$	25.6569	1.55854	0.779271	+	0.626687i	$0.215589\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.65685	−0.341121
$276$	0	0
$277$	14.9706	0.899494	0.449747	−	0.893156i	$-0.351514\pi$
$277$	14.9706	0.899494	0.449747	+	0.893156i	$0.351514\pi$
$278$	0	0
$279$	28.9706	1.73442
$280$	0	0
$281$	21.3137	1.27147	0.635735	−	0.771908i	$-0.280697\pi$
$281$	21.3137	1.27147	0.635735	+	0.771908i	$0.280697\pi$
$282$	0	0
$283$	−11.3137	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$283$	−11.3137	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.65685	0.215857
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.6274	1.67243	0.836216	−	0.548401i	$-0.184763\pi$
$293$	28.6274	1.67243	0.836216	+	0.548401i	$0.184763\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−5.65685	−0.326056
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	−23.3137	−1.33058	−0.665292	−	0.746583i	$-0.731693\pi$
$307$	−23.3137	−1.33058	−0.665292	+	0.746583i	$0.731693\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	−22.9706	−1.29837	−0.649186	−	0.760629i	$-0.724891\pi$
$313$	−22.9706	−1.29837	−0.649186	+	0.760629i	$0.724891\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	2.68629	0.150877	0.0754386	−	0.997150i	$-0.475964\pi$
$317$	2.68629	0.150877	0.0754386	+	0.997150i	$0.475964\pi$
$318$	0	0
$319$	11.3137	0.633446
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.34315	−0.128791	−0.0643955	−	0.997924i	$-0.520512\pi$
$331$	−2.34315	−0.128791	−0.0643955	+	0.997924i	$0.520512\pi$
$332$	0	0
$333$	−18.0000	−0.986394
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−25.3137	−1.37893	−0.689463	−	0.724321i	$-0.742153\pi$
$337$	−25.3137	−1.37893	−0.689463	+	0.724321i	$0.742153\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	54.6274	2.95824
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.3431	0.555249	0.277625	−	0.960690i	$-0.410453\pi$
$347$	10.3431	0.555249	0.277625	+	0.960690i	$0.410453\pi$
$348$	0	0
$349$	−6.68629	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$349$	−6.68629	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.3137	1.56021	0.780106	−	0.625648i	$-0.215165\pi$
$353$	29.3137	1.56021	0.780106	+	0.625648i	$0.215165\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	4.97056	0.259461	0.129731	−	0.991549i	$-0.458589\pi$
$367$	4.97056	0.259461	0.129731	+	0.991549i	$0.458589\pi$
$368$	0	0
$369$	10.9706	0.571105
$370$	0	0
$371$	−3.65685	−0.189854
$372$	0	0
$373$	26.2843	1.36095	0.680474	−	0.732772i	$-0.261774\pi$
$373$	26.2843	1.36095	0.680474	+	0.732772i	$0.261774\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−32.9706	−1.69358	−0.846792	−	0.531924i	$-0.821469\pi$
$379$	−32.9706	−1.69358	−0.846792	+	0.531924i	$0.821469\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.3137	1.39567	0.697833	−	0.716261i	$-0.254148\pi$
$383$	27.3137	1.39567	0.697833	+	0.716261i	$0.254148\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	0	0
$387$	−16.9706	−0.862662
$388$	0	0
$389$	−32.6274	−1.65428	−0.827138	−	0.561999i	$-0.810032\pi$
$389$	−32.6274	−1.65428	−0.827138	+	0.561999i	$0.810032\pi$
$390$	0	0
$391$	30.6274	1.54890
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	−8.62742	−0.432998	−0.216499	−	0.976283i	$-0.569464\pi$
$397$	−8.62742	−0.432998	−0.216499	+	0.976283i	$0.569464\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	−9.65685	−0.481042
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−33.9411	−1.68240
$408$	0	0
$409$	12.3431	0.610329	0.305165	−	0.952300i	$-0.401289\pi$
$409$	12.3431	0.610329	0.305165	+	0.952300i	$0.401289\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	−22.2843	−1.08607	−0.543034	−	0.839710i	$-0.682725\pi$
$421$	−22.2843	−1.08607	−0.543034	+	0.839710i	$0.682725\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.65685	0.371412
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	39.3137	1.89367	0.946837	−	0.321713i	$-0.104259\pi$
$431$	39.3137	1.89367	0.946837	+	0.321713i	$0.104259\pi$
$432$	0	0
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−3.31371	−0.158155	−0.0790773	−	0.996868i	$-0.525197\pi$
$439$	−3.31371	−0.158155	−0.0790773	+	0.996868i	$0.525197\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−36.2843	−1.72392	−0.861959	−	0.506978i	$-0.830762\pi$
$443$	−36.2843	−1.72392	−0.861959	+	0.506978i	$0.830762\pi$
$444$	0	0
$445$	−3.65685	−0.173352
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.68629	−0.126774	−0.0633870	−	0.997989i	$-0.520190\pi$
$449$	−2.68629	−0.126774	−0.0633870	+	0.997989i	$0.520190\pi$
$450$	0	0
$451$	20.6863	0.974079
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−9.31371	−0.435677	−0.217838	−	0.975985i	$-0.569901\pi$
$457$	−9.31371	−0.435677	−0.217838	+	0.975985i	$0.569901\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.3137	1.17898	0.589488	−	0.807777i	$-0.299329\pi$
$461$	25.3137	1.17898	0.589488	+	0.807777i	$0.299329\pi$
$462$	0	0
$463$	3.31371	0.154001	0.0770005	−	0.997031i	$-0.475466\pi$
$463$	3.31371	0.154001	0.0770005	+	0.997031i	$0.475466\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.62742	0.306680	0.153340	−	0.988173i	$-0.450997\pi$
$467$	6.62742	0.306680	0.153340	+	0.988173i	$0.450997\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−32.0000	−1.47136
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.9706	−0.502308
$478$	0	0
$479$	−4.97056	−0.227111	−0.113555	−	0.993532i	$-0.536224\pi$
$479$	−4.97056	−0.227111	−0.113555	+	0.993532i	$0.536224\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.31371	−0.422914
$486$	0	0
$487$	−4.68629	−0.212356	−0.106178	−	0.994347i	$-0.533861\pi$
$487$	−4.68629	−0.212356	−0.106178	+	0.994347i	$0.533861\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.6274	0.840644	0.420322	−	0.907375i	$-0.361917\pi$
$491$	18.6274	0.840644	0.420322	+	0.907375i	$0.361917\pi$
$492$	0	0
$493$	−15.3137	−0.689695
$494$	0	0
$495$	16.9706	0.762770
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	40.9706	1.83409	0.917047	−	0.398779i	$-0.130566\pi$
$499$	40.9706	1.83409	0.917047	+	0.398779i	$0.130566\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.9706	0.935031	0.467516	−	0.883985i	$-0.345149\pi$
$503$	20.9706	0.935031	0.467516	+	0.883985i	$0.345149\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.9411	−1.59306	−0.796531	−	0.604597i	$-0.793334\pi$
$509$	−35.9411	−1.59306	−0.796531	+	0.604597i	$0.793334\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.65685	0.0730097
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.9411	−1.74985	−0.874926	−	0.484256i	$-0.839090\pi$
$521$	−39.9411	−1.74985	−0.874926	+	0.484256i	$0.839090\pi$
$522$	0	0
$523$	−30.6274	−1.33924	−0.669622	−	0.742702i	$-0.733544\pi$
$523$	−30.6274	−1.33924	−0.669622	+	0.742702i	$0.733544\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−73.9411	−3.22093
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.65685	−0.158396
$534$	0	0
$535$	5.65685	0.244567
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.65685	−0.243658
$540$	0	0
$541$	24.3431	1.04659	0.523297	−	0.852150i	$-0.324702\pi$
$541$	24.3431	1.04659	0.523297	+	0.852150i	$0.324702\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.65685	−0.327984
$546$	0	0
$547$	−29.6569	−1.26804	−0.634018	−	0.773318i	$-0.718595\pi$
$547$	−29.6569	−1.26804	−0.634018	+	0.773318i	$0.718595\pi$
$548$	0	0
$549$	−18.0000	−0.768221
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.3137	−0.481108
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.6274	0.874012	0.437006	−	0.899459i	$-0.356039\pi$
$557$	20.6274	0.874012	0.437006	+	0.899459i	$0.356039\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	35.3137	1.48830	0.744148	−	0.668015i	$-0.232856\pi$
$563$	35.3137	1.48830	0.744148	+	0.668015i	$0.232856\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−20.6863	−0.856739
$584$	0	0
$585$	−3.00000	−0.124035
$586$	0	0
$587$	−16.6863	−0.688717	−0.344358	−	0.938838i	$-0.611904\pi$
$587$	−16.6863	−0.688717	−0.344358	+	0.938838i	$0.611904\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.31371	−0.382468	−0.191234	−	0.981544i	$-0.561249\pi$
$593$	−9.31371	−0.382468	−0.191234	+	0.981544i	$0.561249\pi$
$594$	0	0
$595$	−7.65685	−0.313900
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.6863	−0.518348	−0.259174	−	0.965831i	$-0.583450\pi$
$599$	−12.6863	−0.518348	−0.259174	+	0.965831i	$0.583450\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	21.0000	0.853771
$606$	0	0
$607$	9.65685	0.391960	0.195980	−	0.980608i	$-0.437211\pi$
$607$	9.65685	0.391960	0.195980	+	0.980608i	$0.437211\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	23.9411	0.966973	0.483486	−	0.875352i	$-0.339370\pi$
$613$	23.9411	0.966973	0.483486	+	0.875352i	$0.339370\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.3137	0.535990	0.267995	−	0.963420i	$-0.413639\pi$
$617$	13.3137	0.535990	0.267995	+	0.963420i	$0.413639\pi$
$618$	0	0
$619$	−11.3137	−0.454736	−0.227368	−	0.973809i	$-0.573012\pi$
$619$	−11.3137	−0.454736	−0.227368	+	0.973809i	$0.573012\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.65685	0.146509
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	45.9411	1.83179
$630$	0	0
$631$	−47.3137	−1.88353	−0.941764	−	0.336273i	$-0.890833\pi$
$631$	−47.3137	−1.88353	−0.941764	+	0.336273i	$0.890833\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.2843	1.26923	0.634613	−	0.772830i	$-0.281160\pi$
$647$	32.2843	1.26923	0.634613	+	0.772830i	$0.281160\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.3431	0.639557	0.319778	−	0.947492i	$-0.396392\pi$
$653$	16.3431	0.639557	0.319778	+	0.947492i	$0.396392\pi$
$654$	0	0
$655$	7.31371	0.285770
$656$	0	0
$657$	−30.0000	−1.17041
$658$	0	0
$659$	−23.3137	−0.908173	−0.454087	−	0.890958i	$-0.650034\pi$
$659$	−23.3137	−0.908173	−0.454087	+	0.890958i	$0.650034\pi$
$660$	0	0
$661$	28.6274	1.11348	0.556739	−	0.830688i	$-0.312052\pi$
$661$	28.6274	1.11348	0.556739	+	0.830688i	$0.312052\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.9411	−1.31028
$672$	0	0
$673$	47.2548	1.82154	0.910770	−	0.412914i	$-0.135489\pi$
$673$	47.2548	1.82154	0.910770	+	0.412914i	$0.135489\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	9.31371	0.357427
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.68629	−0.332372	−0.166186	−	0.986094i	$-0.553145\pi$
$683$	−8.68629	−0.332372	−0.166186	+	0.986094i	$0.553145\pi$
$684$	0	0
$685$	13.3137	0.508691
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.65685	0.139315
$690$	0	0
$691$	1.37258	0.0522155	0.0261078	−	0.999659i	$-0.491689\pi$
$691$	1.37258	0.0522155	0.0261078	+	0.999659i	$0.491689\pi$
$692$	0	0
$693$	−16.9706	−0.644658
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.3137	1.56040	0.780199	−	0.625532i	$-0.215118\pi$
$701$	41.3137	1.56040	0.780199	+	0.625532i	$0.215118\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.0000	−0.526524
$708$	0	0
$709$	6.97056	0.261785	0.130892	−	0.991397i	$-0.458216\pi$
$709$	6.97056	0.261785	0.130892	+	0.991397i	$0.458216\pi$
$710$	0	0
$711$	−33.9411	−1.27289
$712$	0	0
$713$	−38.6274	−1.44661
$714$	0	0
$715$	−5.65685	−0.211554
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−1.65685	−0.0617045
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−38.3431	−1.42207	−0.711034	−	0.703157i	$-0.751773\pi$
$727$	−38.3431	−1.42207	−0.711034	+	0.703157i	$0.751773\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	43.3137	1.60202
$732$	0	0
$733$	20.6274	0.761891	0.380946	−	0.924597i	$-0.375599\pi$
$733$	20.6274	0.761891	0.380946	+	0.924597i	$0.375599\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.6274	−0.833492
$738$	0	0
$739$	28.2843	1.04045	0.520227	−	0.854028i	$-0.325847\pi$
$739$	28.2843	1.04045	0.520227	+	0.854028i	$0.325847\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.9411	0.951688	0.475844	−	0.879530i	$-0.342143\pi$
$743$	25.9411	0.951688	0.475844	+	0.879530i	$0.342143\pi$
$744$	0	0
$745$	0.343146	0.0125719
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−5.65685	−0.206697
$750$	0	0
$751$	49.9411	1.82238	0.911189	−	0.411989i	$-0.135166\pi$
$751$	49.9411	1.82238	0.911189	+	0.411989i	$0.135166\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.686292	0.0249767
$756$	0	0
$757$	46.9706	1.70717	0.853587	−	0.520950i	$-0.174422\pi$
$757$	46.9706	1.70717	0.853587	+	0.520950i	$0.174422\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.9706	−1.70268	−0.851341	−	0.524613i	$-0.824210\pi$
$761$	−46.9706	−1.70268	−0.851341	+	0.524613i	$0.824210\pi$
$762$	0	0
$763$	7.65685	0.277197
$764$	0	0
$765$	−22.9706	−0.830502
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−5.02944	−0.181366	−0.0906831	−	0.995880i	$-0.528905\pi$
$769$	−5.02944	−0.181366	−0.0906831	+	0.995880i	$0.528905\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	0	0
$775$	−9.65685	−0.346884
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−22.6274	−0.809673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	21.9411	0.782117	0.391058	−	0.920366i	$-0.372109\pi$
$787$	21.9411	0.782117	0.391058	+	0.920366i	$0.372109\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.6274	1.29741	0.648705	−	0.761040i	$-0.275311\pi$
$797$	36.6274	1.29741	0.648705	+	0.761040i	$0.275311\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	10.9706	0.387626
$802$	0	0
$803$	−56.5685	−1.99626
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−51.2548	−1.80202	−0.901012	−	0.433794i	$-0.857175\pi$
$809$	−51.2548	−1.80202	−0.901012	+	0.433794i	$0.857175\pi$
$810$	0	0
$811$	11.3137	0.397278	0.198639	−	0.980073i	$-0.436348\pi$
$811$	11.3137	0.397278	0.198639	+	0.980073i	$0.436348\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.00000	0.104828
$820$	0	0
$821$	−38.2843	−1.33613	−0.668065	−	0.744103i	$-0.732877\pi$
$821$	−38.2843	−1.33613	−0.668065	+	0.744103i	$0.732877\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.9411	0.762968	0.381484	−	0.924376i	$-0.375413\pi$
$827$	21.9411	0.762968	0.381484	+	0.924376i	$0.375413\pi$
$828$	0	0
$829$	51.2548	1.78015	0.890077	−	0.455810i	$-0.150650\pi$
$829$	51.2548	1.78015	0.890077	+	0.455810i	$0.150650\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.65685	0.265294
$834$	0	0
$835$	−19.3137	−0.668378
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.2843	−0.838386	−0.419193	−	0.907897i	$-0.637687\pi$
$839$	−24.2843	−0.838386	−0.419193	+	0.907897i	$0.637687\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−21.0000	−0.721569
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	−55.2548	−1.89189	−0.945945	−	0.324328i	$-0.894862\pi$
$853$	−55.2548	−1.89189	−0.945945	+	0.324328i	$0.894862\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.9706	−0.784659	−0.392330	−	0.919825i	$-0.628331\pi$
$857$	−22.9706	−0.784659	−0.392330	+	0.919825i	$0.628331\pi$
$858$	0	0
$859$	−15.3137	−0.522497	−0.261248	−	0.965272i	$-0.584134\pi$
$859$	−15.3137	−0.522497	−0.261248	+	0.965272i	$0.584134\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.6274	1.31489	0.657446	−	0.753501i	$-0.271637\pi$
$863$	38.6274	1.31489	0.657446	+	0.753501i	$0.271637\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	27.9411	0.945664
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−24.6274	−0.831609	−0.415804	−	0.909454i	$-0.636500\pi$
$877$	−24.6274	−0.831609	−0.415804	+	0.909454i	$0.636500\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36.6274	−1.23401	−0.617005	−	0.786960i	$-0.711654\pi$
$881$	−36.6274	−1.23401	−0.617005	+	0.786960i	$0.711654\pi$
$882$	0	0
$883$	−52.2843	−1.75951	−0.879753	−	0.475431i	$-0.842292\pi$
$883$	−52.2843	−1.75951	−0.879753	+	0.475431i	$0.842292\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.9706	0.435509	0.217754	−	0.976004i	$-0.430127\pi$
$887$	12.9706	0.435509	0.217754	+	0.976004i	$0.430127\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	−50.9117	−1.70561
$892$	0	0
$893$	0	0
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.3137	0.644148
$900$	0	0
$901$	28.0000	0.932815
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−18.3431	−0.609074	−0.304537	−	0.952500i	$-0.598502\pi$
$907$	−18.3431	−0.609074	−0.304537	+	0.952500i	$0.598502\pi$
$908$	0	0
$909$	−42.0000	−1.39305
$910$	0	0
$911$	−14.6274	−0.484628	−0.242314	−	0.970198i	$-0.577906\pi$
$911$	−14.6274	−0.484628	−0.242314	+	0.970198i	$0.577906\pi$
$912$	0	0
$913$	−22.6274	−0.748858
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.31371	−0.241520
$918$	0	0
$919$	14.6274	0.482514	0.241257	−	0.970461i	$-0.422440\pi$
$919$	14.6274	0.482514	0.241257	+	0.970461i	$0.422440\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−4.97056	−0.163255
$928$	0	0
$929$	31.6569	1.03863	0.519314	−	0.854584i	$-0.326188\pi$
$929$	31.6569	1.03863	0.519314	+	0.854584i	$0.326188\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−43.3137	−1.41651
$936$	0	0
$937$	−43.6569	−1.42621	−0.713104	−	0.701059i	$-0.752711\pi$
$937$	−43.6569	−1.42621	−0.713104	+	0.701059i	$0.752711\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.62742	−0.281246	−0.140623	−	0.990063i	$-0.544911\pi$
$941$	−8.62742	−0.281246	−0.140623	+	0.990063i	$0.544911\pi$
$942$	0	0
$943$	−14.6274	−0.476334
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.9411	0.972956	0.486478	−	0.873693i	$-0.338281\pi$
$947$	29.9411	0.972956	0.486478	+	0.873693i	$0.338281\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.3137	−0.560846	−0.280423	−	0.959877i	$-0.590475\pi$
$953$	−17.3137	−0.560846	−0.280423	+	0.959877i	$0.590475\pi$
$954$	0	0
$955$	−3.31371	−0.107229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.3137	−0.429922
$960$	0	0
$961$	62.2548	2.00822
$962$	0	0
$963$	−16.9706	−0.546869
$964$	0	0
$965$	−17.3137	−0.557348
$966$	0	0
$967$	−27.3137	−0.878350	−0.439175	−	0.898402i	$-0.644729\pi$
$967$	−27.3137	−0.878350	−0.439175	+	0.898402i	$0.644729\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	58.6274	1.88144	0.940722	−	0.339180i	$-0.110149\pi$
$971$	58.6274	1.88144	0.940722	+	0.339180i	$0.110149\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.9411	−1.27783	−0.638915	−	0.769277i	$-0.720616\pi$
$977$	−39.9411	−1.27783	−0.638915	+	0.769277i	$0.720616\pi$
$978$	0	0
$979$	20.6863	0.661137
$980$	0	0
$981$	22.9706	0.733394
$982$	0	0
$983$	11.3137	0.360851	0.180426	−	0.983589i	$-0.442252\pi$
$983$	11.3137	0.360851	0.180426	+	0.983589i	$0.442252\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.6274	0.719510
$990$	0	0
$991$	46.6274	1.48117	0.740584	−	0.671963i	$-0.234549\pi$
$991$	46.6274	1.48117	0.740584	+	0.671963i	$0.234549\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−19.3137	−0.612286
$996$	0	0
$997$	49.3137	1.56178	0.780890	−	0.624668i	$-0.214766\pi$
$997$	49.3137	1.56178	0.780890	+	0.624668i	$0.214766\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.m.1.1	✓	2
4.3	odd	2		7280.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.m.1.1	✓	2	1.1	even	1	trivial
7280.2.a.bd.1.2		2	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} -5.65685 q^{11} +1.00000 q^{13} +7.65685 q^{17} +4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} -9.65685 q^{31} -1.00000 q^{35} +6.00000 q^{37} -3.65685 q^{41} +5.65685 q^{43} -3.00000 q^{45} +1.00000 q^{49} +3.65685 q^{53} -5.65685 q^{55} +6.00000 q^{61} +3.00000 q^{63} +1.00000 q^{65} +4.00000 q^{67} +4.00000 q^{71} +10.0000 q^{73} +5.65685 q^{77} +11.3137 q^{79} +9.00000 q^{81} +4.00000 q^{83} +7.65685 q^{85} -3.65685 q^{89} -1.00000 q^{91} -9.31371 q^{97} +16.9706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^9	\( 2 q + 2 q^{5} - 2 q^{7} - 6 q^{9} + 2 q^{13} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} - 2 q^{35} + 12 q^{37} + 4 q^{41} - 6 q^{45} + 2 q^{49} - 4 q^{53} + 12 q^{61} + 6 q^{63} + 2 q^{65} + 8 q^{67} + 8 q^{71} + 20 q^{73} + 18 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{89} - 2 q^{91} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^9 + 2 * q^13 + 4 * q^17 + 8 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 - 2 * q^35 + 12 * q^37 + 4 * q^41 - 6 * q^45 + 2 * q^49 - 4 * q^53 + 12 * q^61 + 6 * q^63 + 2 * q^65 + 8 * q^67 + 8 * q^71 + 20 * q^73 + 18 * q^81 + 8 * q^83 + 4 * q^85 + 4 * q^89 - 2 * q^91 + 4 * q^97

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