LMFDB - Embedded newform 4680.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.46410 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.46410 q^{7} +1.26795 q^{11} +1.00000 q^{13} +3.46410 q^{17} -6.73205 q^{19} -2.19615 q^{23} +1.00000 q^{25} -1.46410 q^{29} -1.26795 q^{31} -3.46410 q^{35} +10.9282 q^{37} +4.53590 q^{41} +0.732051 q^{43} +4.53590 q^{47} +5.00000 q^{49} -4.53590 q^{53} +1.26795 q^{55} +12.1962 q^{59} +1.46410 q^{61} +1.00000 q^{65} -11.8564 q^{67} +5.66025 q^{71} -4.39230 q^{77} +9.46410 q^{79} -4.53590 q^{83} +3.46410 q^{85} -8.92820 q^{89} -3.46410 q^{91} -6.73205 q^{95} +15.8564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 6 q^{11} + 2 q^{13} - 10 q^{19} + 6 q^{23} + 2 q^{25} + 4 q^{29} - 6 q^{31} + 8 q^{37} + 16 q^{41} - 2 q^{43} + 16 q^{47} + 10 q^{49} - 16 q^{53} + 6 q^{55} + 14 q^{59} - 4 q^{61} + 2 q^{65} + 4 q^{67} - 6 q^{71} + 12 q^{77} + 12 q^{79} - 16 q^{83} - 4 q^{89} - 10 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 6 * q^11 + 2 * q^13 - 10 * q^19 + 6 * q^23 + 2 * q^25 + 4 * q^29 - 6 * q^31 + 8 * q^37 + 16 * q^41 - 2 * q^43 + 16 * q^47 + 10 * q^49 - 16 * q^53 + 6 * q^55 + 14 * q^59 - 4 * q^61 + 2 * q^65 + 4 * q^67 - 6 * q^71 + 12 * q^77 + 12 * q^79 - 16 * q^83 - 4 * q^89 - 10 * q^95 + 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
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−6.73205	−1.54444	−0.772219	−	0.635356i	$-0.780853\pi$
$19$	−6.73205	−1.54444	−0.772219	+	0.635356i	$0.780853\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.46410	−0.271877	−0.135938	−	0.990717i	$-0.543405\pi$
$29$	−1.46410	−0.271877	−0.135938	+	0.990717i	$0.543405\pi$
$30$	0	0
$31$	−1.26795	−0.227730	−0.113865	−	0.993496i	$-0.536323\pi$
$31$	−1.26795	−0.227730	−0.113865	+	0.993496i	$0.536323\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	10.9282	1.79659	0.898293	−	0.439397i	$-0.144808\pi$
$37$	10.9282	1.79659	0.898293	+	0.439397i	$0.144808\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.53590	0.708388	0.354194	−	0.935172i	$-0.384755\pi$
$41$	4.53590	0.708388	0.354194	+	0.935172i	$0.384755\pi$
$42$	0	0
$43$	0.732051	0.111637	0.0558184	−	0.998441i	$-0.482223\pi$
$43$	0.732051	0.111637	0.0558184	+	0.998441i	$0.482223\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.53590	0.661629	0.330814	−	0.943696i	$-0.392677\pi$
$47$	4.53590	0.661629	0.330814	+	0.943696i	$0.392677\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.53590	−0.623054	−0.311527	−	0.950237i	$-0.600840\pi$
$53$	−4.53590	−0.623054	−0.311527	+	0.950237i	$0.600840\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.1962	1.58780	0.793902	−	0.608046i	$-0.208046\pi$
$59$	12.1962	1.58780	0.793902	+	0.608046i	$0.208046\pi$
$60$	0	0
$61$	1.46410	0.187459	0.0937295	−	0.995598i	$-0.470121\pi$
$61$	1.46410	0.187459	0.0937295	+	0.995598i	$0.470121\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−11.8564	−1.44849	−0.724245	−	0.689542i	$-0.757812\pi$
$67$	−11.8564	−1.44849	−0.724245	+	0.689542i	$0.757812\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.66025	0.671749	0.335874	−	0.941907i	$-0.390968\pi$
$71$	5.66025	0.671749	0.335874	+	0.941907i	$0.390968\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.39230	−0.500550
$78$	0	0
$79$	9.46410	1.06479	0.532397	−	0.846495i	$-0.321291\pi$
$79$	9.46410	1.06479	0.532397	+	0.846495i	$0.321291\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.53590	−0.497880	−0.248940	−	0.968519i	$-0.580082\pi$
$83$	−4.53590	−0.497880	−0.248940	+	0.968519i	$0.580082\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.73205	−0.690694
$96$	0	0
$97$	15.8564	1.60997	0.804987	−	0.593292i	$-0.202172\pi$
$97$	15.8564	1.60997	0.804987	+	0.593292i	$0.202172\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.9282	1.68442	0.842210	−	0.539150i	$-0.181255\pi$
$101$	16.9282	1.68442	0.842210	+	0.539150i	$0.181255\pi$
$102$	0	0
$103$	−3.26795	−0.322001	−0.161000	−	0.986954i	$-0.551472\pi$
$103$	−3.26795	−0.322001	−0.161000	+	0.986954i	$0.551472\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1962	1.37239	0.686197	−	0.727416i	$-0.259279\pi$
$107$	14.1962	1.37239	0.686197	+	0.727416i	$0.259279\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.3923	−1.35391	−0.676957	−	0.736022i	$-0.736702\pi$
$113$	−14.3923	−1.35391	−0.676957	+	0.736022i	$0.736702\pi$
$114$	0	0
$115$	−2.19615	−0.204792
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.1244	1.51954	0.759770	−	0.650191i	$-0.225311\pi$
$127$	17.1244	1.51954	0.759770	+	0.650191i	$0.225311\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.9282	−0.954802	−0.477401	−	0.878686i	$-0.658421\pi$
$131$	−10.9282	−0.954802	−0.477401	+	0.878686i	$0.658421\pi$
$132$	0	0
$133$	23.3205	2.02214
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.26795	0.106031
$144$	0	0
$145$	−1.46410	−0.121587
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.8564	1.95439	0.977196	−	0.212337i	$-0.0681075\pi$
$149$	23.8564	1.95439	0.977196	+	0.212337i	$0.0681075\pi$
$150$	0	0
$151$	−5.26795	−0.428700	−0.214350	−	0.976757i	$-0.568763\pi$
$151$	−5.26795	−0.428700	−0.214350	+	0.976757i	$0.568763\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.26795	−0.101844
$156$	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
$161$	7.60770	0.599570
$162$	0	0
$163$	11.8564	0.928665	0.464333	−	0.885661i	$-0.346294\pi$
$163$	11.8564	0.928665	0.464333	+	0.885661i	$0.346294\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.39230	0.494651	0.247326	−	0.968932i	$-0.420448\pi$
$167$	6.39230	0.494651	0.247326	+	0.968932i	$0.420448\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.60770	0.122231	0.0611154	−	0.998131i	$-0.480534\pi$
$173$	1.60770	0.122231	0.0611154	+	0.998131i	$0.480534\pi$
$174$	0	0
$175$	−3.46410	−0.261861
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	−23.3205	−1.73340	−0.866700	−	0.498830i	$-0.833763\pi$
$181$	−23.3205	−1.73340	−0.866700	+	0.498830i	$0.833763\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.9282	0.803457
$186$	0	0
$187$	4.39230	0.321197
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.9282	0.921096	0.460548	−	0.887635i	$-0.347653\pi$
$197$	12.9282	0.921096	0.460548	+	0.887635i	$0.347653\pi$
$198$	0	0
$199$	1.07180	0.0759777	0.0379888	−	0.999278i	$-0.487905\pi$
$199$	1.07180	0.0759777	0.0379888	+	0.999278i	$0.487905\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.07180	0.355970
$204$	0	0
$205$	4.53590	0.316801
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.53590	−0.590440
$210$	0	0
$211$	−16.7846	−1.15550	−0.577750	−	0.816214i	$-0.696069\pi$
$211$	−16.7846	−1.15550	−0.577750	+	0.816214i	$0.696069\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.732051	0.0499255
$216$	0	0
$217$	4.39230	0.298169
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.46410	0.233021
$222$	0	0
$223$	18.3923	1.23164	0.615820	−	0.787887i	$-0.288825\pi$
$223$	18.3923	1.23164	0.615820	+	0.787887i	$0.288825\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.0718	1.00035	0.500175	−	0.865924i	$-0.333269\pi$
$227$	15.0718	1.00035	0.500175	+	0.865924i	$0.333269\pi$
$228$	0	0
$229$	−7.46410	−0.493242	−0.246621	−	0.969112i	$-0.579320\pi$
$229$	−7.46410	−0.493242	−0.246621	+	0.969112i	$0.579320\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	4.53590	0.295889
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.5167	1.26243	0.631214	−	0.775609i	$-0.282557\pi$
$239$	19.5167	1.26243	0.631214	+	0.775609i	$0.282557\pi$
$240$	0	0
$241$	−18.3923	−1.18475	−0.592376	−	0.805661i	$-0.701810\pi$
$241$	−18.3923	−1.18475	−0.592376	+	0.805661i	$0.701810\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	−6.73205	−0.428350
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.53590	0.412542	0.206271	−	0.978495i	$-0.433867\pi$
$251$	6.53590	0.412542	0.206271	+	0.978495i	$0.433867\pi$
$252$	0	0
$253$	−2.78461	−0.175067
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.8564	−0.739582	−0.369791	−	0.929115i	$-0.620571\pi$
$257$	−11.8564	−0.739582	−0.369791	+	0.929115i	$0.620571\pi$
$258$	0	0
$259$	−37.8564	−2.35228
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.58846	0.406262	0.203131	−	0.979152i	$-0.434888\pi$
$263$	6.58846	0.406262	0.203131	+	0.979152i	$0.434888\pi$
$264$	0	0
$265$	−4.53590	−0.278638
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.8564	0.722898	0.361449	−	0.932392i	$-0.382282\pi$
$269$	11.8564	0.722898	0.361449	+	0.932392i	$0.382282\pi$
$270$	0	0
$271$	7.80385	0.474050	0.237025	−	0.971504i	$-0.423828\pi$
$271$	7.80385	0.474050	0.237025	+	0.971504i	$0.423828\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.26795	0.0764602
$276$	0	0
$277$	5.60770	0.336934	0.168467	−	0.985707i	$-0.446118\pi$
$277$	5.60770	0.336934	0.168467	+	0.985707i	$0.446118\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.60770	0.573147	0.286574	−	0.958058i	$-0.407484\pi$
$281$	9.60770	0.573147	0.286574	+	0.958058i	$0.407484\pi$
$282$	0	0
$283$	−26.1962	−1.55720	−0.778600	−	0.627521i	$-0.784070\pi$
$283$	−26.1962	−1.55720	−0.778600	+	0.627521i	$0.784070\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.7128	−0.927498
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.85641	−0.108452	−0.0542262	−	0.998529i	$-0.517269\pi$
$293$	−1.85641	−0.108452	−0.0542262	+	0.998529i	$0.517269\pi$
$294$	0	0
$295$	12.1962	0.710087
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.19615	−0.127007
$300$	0	0
$301$	−2.53590	−0.146167
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.46410	0.0838342
$306$	0	0
$307$	−15.4641	−0.882583	−0.441291	−	0.897364i	$-0.645479\pi$
$307$	−15.4641	−0.882583	−0.441291	+	0.897364i	$0.645479\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.53590	0.370617	0.185308	−	0.982680i	$-0.440672\pi$
$311$	6.53590	0.370617	0.185308	+	0.982680i	$0.440672\pi$
$312$	0	0
$313$	−28.2487	−1.59671	−0.798356	−	0.602186i	$-0.794297\pi$
$313$	−28.2487	−1.59671	−0.798356	+	0.602186i	$0.794297\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.7846	1.61670	0.808352	−	0.588699i	$-0.200360\pi$
$317$	28.7846	1.61670	0.808352	+	0.588699i	$0.200360\pi$
$318$	0	0
$319$	−1.85641	−0.103939
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.3205	−1.29759
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−15.7128	−0.866275
$330$	0	0
$331$	28.9808	1.59293	0.796463	−	0.604687i	$-0.206702\pi$
$331$	28.9808	1.59293	0.796463	+	0.604687i	$0.206702\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.8564	−0.647785
$336$	0	0
$337$	5.60770	0.305471	0.152735	−	0.988267i	$-0.451192\pi$
$337$	5.60770	0.305471	0.152735	+	0.988267i	$0.451192\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.60770	−0.0870616
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.3397	−0.662432	−0.331216	−	0.943555i	$-0.607459\pi$
$347$	−12.3397	−0.662432	−0.331216	+	0.943555i	$0.607459\pi$
$348$	0	0
$349$	29.3205	1.56949	0.784745	−	0.619818i	$-0.212794\pi$
$349$	29.3205	1.56949	0.784745	+	0.619818i	$0.212794\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.8564	1.37620	0.688099	−	0.725617i	$-0.258446\pi$
$353$	25.8564	1.37620	0.688099	+	0.725617i	$0.258446\pi$
$354$	0	0
$355$	5.66025	0.300415
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.1962	−1.91036	−0.955180	−	0.296026i	$-0.904339\pi$
$359$	−36.1962	−1.91036	−0.955180	+	0.296026i	$0.904339\pi$
$360$	0	0
$361$	26.3205	1.38529
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.05256	−0.420340	−0.210170	−	0.977665i	$-0.567402\pi$
$367$	−8.05256	−0.420340	−0.210170	+	0.977665i	$0.567402\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.7128	0.815769
$372$	0	0
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.46410	−0.0754051
$378$	0	0
$379$	−10.3397	−0.531117	−0.265559	−	0.964095i	$-0.585556\pi$
$379$	−10.3397	−0.531117	−0.265559	+	0.964095i	$0.585556\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.3923	1.55297	0.776487	−	0.630133i	$-0.217000\pi$
$383$	30.3923	1.55297	0.776487	+	0.630133i	$0.217000\pi$
$384$	0	0
$385$	−4.39230	−0.223853
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.7128	−1.30369	−0.651846	−	0.758352i	$-0.726005\pi$
$389$	−25.7128	−1.30369	−0.651846	+	0.758352i	$0.726005\pi$
$390$	0	0
$391$	−7.60770	−0.384738
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.46410	0.476191
$396$	0	0
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.7846	−1.93681	−0.968405	−	0.249381i	$-0.919773\pi$
$401$	−38.7846	−1.93681	−0.968405	+	0.249381i	$0.919773\pi$
$402$	0	0
$403$	−1.26795	−0.0631610
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.8564	0.686837
$408$	0	0
$409$	−37.3205	−1.84538	−0.922690	−	0.385542i	$-0.874014\pi$
$409$	−37.3205	−1.84538	−0.922690	+	0.385542i	$0.874014\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−42.2487	−2.07892
$414$	0	0
$415$	−4.53590	−0.222658
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.3205	1.13928	0.569641	−	0.821894i	$-0.307082\pi$
$419$	23.3205	1.13928	0.569641	+	0.821894i	$0.307082\pi$
$420$	0	0
$421$	18.7846	0.915506	0.457753	−	0.889079i	$-0.348654\pi$
$421$	18.7846	0.915506	0.457753	+	0.889079i	$0.348654\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−5.07180	−0.245441
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.4449	−1.27380	−0.636902	−	0.770945i	$-0.719784\pi$
$431$	−26.4449	−1.27380	−0.636902	+	0.770945i	$0.719784\pi$
$432$	0	0
$433$	26.7846	1.28719	0.643593	−	0.765368i	$-0.277443\pi$
$433$	26.7846	1.28719	0.643593	+	0.765368i	$0.277443\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.7846	0.707244
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.12436	0.243465	0.121733	−	0.992563i	$-0.461155\pi$
$443$	5.12436	0.243465	0.121733	+	0.992563i	$0.461155\pi$
$444$	0	0
$445$	−8.92820	−0.423237
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.32051	0.439862	0.219931	−	0.975515i	$-0.429417\pi$
$449$	9.32051	0.439862	0.219931	+	0.975515i	$0.429417\pi$
$450$	0	0
$451$	5.75129	0.270818
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.46410	−0.162400
$456$	0	0
$457$	18.7846	0.878707	0.439353	−	0.898314i	$-0.355208\pi$
$457$	18.7846	0.878707	0.439353	+	0.898314i	$0.355208\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.53590	−0.211258	−0.105629	−	0.994406i	$-0.533686\pi$
$461$	−4.53590	−0.211258	−0.105629	+	0.994406i	$0.533686\pi$
$462$	0	0
$463$	−30.7846	−1.43068	−0.715341	−	0.698775i	$-0.753729\pi$
$463$	−30.7846	−1.43068	−0.715341	+	0.698775i	$0.753729\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.12436	0.0520290	0.0260145	−	0.999662i	$-0.491718\pi$
$467$	1.12436	0.0520290	0.0260145	+	0.999662i	$0.491718\pi$
$468$	0	0
$469$	41.0718	1.89652
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.928203	0.0426788
$474$	0	0
$475$	−6.73205	−0.308888
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.8038	0.539332	0.269666	−	0.962954i	$-0.413087\pi$
$479$	11.8038	0.539332	0.269666	+	0.962954i	$0.413087\pi$
$480$	0	0
$481$	10.9282	0.498283
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.8564	0.720002
$486$	0	0
$487$	36.9282	1.67338	0.836688	−	0.547679i	$-0.184489\pi$
$487$	36.9282	1.67338	0.836688	+	0.547679i	$0.184489\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.53590	−0.294961	−0.147480	−	0.989065i	$-0.547116\pi$
$491$	−6.53590	−0.294961	−0.147480	+	0.989065i	$0.547116\pi$
$492$	0	0
$493$	−5.07180	−0.228422
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.6077	−0.879525
$498$	0	0
$499$	−33.6603	−1.50684	−0.753420	−	0.657540i	$-0.771597\pi$
$499$	−33.6603	−1.50684	−0.753420	+	0.657540i	$0.771597\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.33975	0.371851	0.185925	−	0.982564i	$-0.440472\pi$
$503$	8.33975	0.371851	0.185925	+	0.982564i	$0.440472\pi$
$504$	0	0
$505$	16.9282	0.753295
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.3923	0.637928	0.318964	−	0.947767i	$-0.396665\pi$
$509$	14.3923	0.637928	0.318964	+	0.947767i	$0.396665\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.26795	−0.144003
$516$	0	0
$517$	5.75129	0.252941
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.2487	−1.32522	−0.662610	−	0.748965i	$-0.730551\pi$
$521$	−30.2487	−1.32522	−0.662610	+	0.748965i	$0.730551\pi$
$522$	0	0
$523$	27.6603	1.20950	0.604749	−	0.796416i	$-0.293273\pi$
$523$	27.6603	1.20950	0.604749	+	0.796416i	$0.293273\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.39230	−0.191332
$528$	0	0
$529$	−18.1769	−0.790301
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.53590	0.196472
$534$	0	0
$535$	14.1962	0.613753
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.33975	0.273072
$540$	0	0
$541$	11.4641	0.492880	0.246440	−	0.969158i	$-0.420739\pi$
$541$	11.4641	0.492880	0.246440	+	0.969158i	$0.420739\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	23.6603	1.01164	0.505820	−	0.862639i	$-0.331190\pi$
$547$	23.6603	1.01164	0.505820	+	0.862639i	$0.331190\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.85641	0.419897
$552$	0	0
$553$	−32.7846	−1.39414
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.0718	−0.892841	−0.446420	−	0.894823i	$-0.647301\pi$
$557$	−21.0718	−0.892841	−0.446420	+	0.894823i	$0.647301\pi$
$558$	0	0
$559$	0.732051	0.0309625
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.2679	0.643467	0.321734	−	0.946830i	$-0.395734\pi$
$563$	15.2679	0.643467	0.321734	+	0.946830i	$0.395734\pi$
$564$	0	0
$565$	−14.3923	−0.605489
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.39230	−0.184135	−0.0920675	−	0.995753i	$-0.529348\pi$
$569$	−4.39230	−0.184135	−0.0920675	+	0.995753i	$0.529348\pi$
$570$	0	0
$571$	22.2487	0.931080	0.465540	−	0.885027i	$-0.345860\pi$
$571$	22.2487	0.931080	0.465540	+	0.885027i	$0.345860\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.19615	−0.0915859
$576$	0	0
$577$	16.0000	0.666089	0.333044	−	0.942911i	$-0.391924\pi$
$577$	16.0000	0.666089	0.333044	+	0.942911i	$0.391924\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.7128	0.651877
$582$	0	0
$583$	−5.75129	−0.238194
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.7128	−0.731086	−0.365543	−	0.930794i	$-0.619117\pi$
$587$	−17.7128	−0.731086	−0.365543	+	0.930794i	$0.619117\pi$
$588$	0	0
$589$	8.53590	0.351716
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.92820	0.366637	0.183319	−	0.983054i	$-0.441316\pi$
$593$	8.92820	0.366637	0.183319	+	0.983054i	$0.441316\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.3923	−0.833207	−0.416603	−	0.909088i	$-0.636780\pi$
$599$	−20.3923	−0.833207	−0.416603	+	0.909088i	$0.636780\pi$
$600$	0	0
$601$	39.8564	1.62578	0.812888	−	0.582420i	$-0.197894\pi$
$601$	39.8564	1.62578	0.812888	+	0.582420i	$0.197894\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.39230	−0.381851
$606$	0	0
$607$	3.26795	0.132642	0.0663210	−	0.997798i	$-0.478874\pi$
$607$	3.26795	0.132642	0.0663210	+	0.997798i	$0.478874\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.53590	0.183503
$612$	0	0
$613$	−16.1436	−0.652034	−0.326017	−	0.945364i	$-0.605707\pi$
$613$	−16.1436	−0.652034	−0.326017	+	0.945364i	$0.605707\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−36.1962	−1.45485	−0.727423	−	0.686189i	$-0.759282\pi$
$619$	−36.1962	−1.45485	−0.727423	+	0.686189i	$0.759282\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	30.9282	1.23911
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37.8564	1.50943
$630$	0	0
$631$	31.1244	1.23904	0.619521	−	0.784980i	$-0.287327\pi$
$631$	31.1244	1.23904	0.619521	+	0.784980i	$0.287327\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17.1244	0.679559
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.0718	0.437310	0.218655	−	0.975802i	$-0.429833\pi$
$641$	11.0718	0.437310	0.218655	+	0.975802i	$0.429833\pi$
$642$	0	0
$643$	−16.5359	−0.652112	−0.326056	−	0.945350i	$-0.605720\pi$
$643$	−16.5359	−0.652112	−0.326056	+	0.945350i	$0.605720\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.5885	0.730788	0.365394	−	0.930853i	$-0.380934\pi$
$647$	18.5885	0.730788	0.365394	+	0.930853i	$0.380934\pi$
$648$	0	0
$649$	15.4641	0.607019
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.14359	−0.162151	−0.0810757	−	0.996708i	$-0.525836\pi$
$653$	−4.14359	−0.162151	−0.0810757	+	0.996708i	$0.525836\pi$
$654$	0	0
$655$	−10.9282	−0.427000
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.4641	−1.61521	−0.807606	−	0.589722i	$-0.799237\pi$
$659$	−41.4641	−1.61521	−0.807606	+	0.589722i	$0.799237\pi$
$660$	0	0
$661$	10.7846	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$661$	10.7846	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.3205	0.904331
$666$	0	0
$667$	3.21539	0.124500
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.85641	0.0716658
$672$	0	0
$673$	−8.53590	−0.329035	−0.164517	−	0.986374i	$-0.552607\pi$
$673$	−8.53590	−0.329035	−0.164517	+	0.986374i	$0.552607\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.6077	0.830451	0.415226	−	0.909718i	$-0.363703\pi$
$677$	21.6077	0.830451	0.415226	+	0.909718i	$0.363703\pi$
$678$	0	0
$679$	−54.9282	−2.10795
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.7128	−0.830818	−0.415409	−	0.909635i	$-0.636361\pi$
$683$	−21.7128	−0.830818	−0.415409	+	0.909635i	$0.636361\pi$
$684$	0	0
$685$	12.9282	0.493961
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.53590	−0.172804
$690$	0	0
$691$	−33.6603	−1.28050	−0.640248	−	0.768168i	$-0.721169\pi$
$691$	−33.6603	−1.28050	−0.640248	+	0.768168i	$0.721169\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.39230	0.318338
$696$	0	0
$697$	15.7128	0.595165
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−73.5692	−2.77472
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−58.6410	−2.20542
$708$	0	0
$709$	−25.6077	−0.961717	−0.480859	−	0.876798i	$-0.659675\pi$
$709$	−25.6077	−0.961717	−0.480859	+	0.876798i	$0.659675\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.78461	0.104284
$714$	0	0
$715$	1.26795	0.0474186
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−38.9282	−1.45178	−0.725889	−	0.687812i	$-0.758571\pi$
$719$	−38.9282	−1.45178	−0.725889	+	0.687812i	$0.758571\pi$
$720$	0	0
$721$	11.3205	0.421598
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.46410	−0.0543754
$726$	0	0
$727$	−37.9090	−1.40597	−0.702983	−	0.711207i	$-0.748149\pi$
$727$	−37.9090	−1.40597	−0.702983	+	0.711207i	$0.748149\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.53590	0.0937936
$732$	0	0
$733$	0.143594	0.00530375	0.00265187	−	0.999996i	$-0.499156\pi$
$733$	0.143594	0.00530375	0.00265187	+	0.999996i	$0.499156\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.0333	−0.553759
$738$	0	0
$739$	−24.5885	−0.904501	−0.452251	−	0.891891i	$-0.649379\pi$
$739$	−24.5885	−0.904501	−0.452251	+	0.891891i	$0.649379\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.46410	0.127086	0.0635428	−	0.997979i	$-0.479760\pi$
$743$	3.46410	0.127086	0.0635428	+	0.997979i	$0.479760\pi$
$744$	0	0
$745$	23.8564	0.874031
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−49.1769	−1.79689
$750$	0	0
$751$	−10.5359	−0.384460	−0.192230	−	0.981350i	$-0.561572\pi$
$751$	−10.5359	−0.384460	−0.192230	+	0.981350i	$0.561572\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.26795	−0.191720
$756$	0	0
$757$	19.4641	0.707435	0.353717	−	0.935352i	$-0.384917\pi$
$757$	19.4641	0.707435	0.353717	+	0.935352i	$0.384917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.71281	0.352089	0.176045	−	0.984382i	$-0.443670\pi$
$761$	9.71281	0.352089	0.176045	+	0.984382i	$0.443670\pi$
$762$	0	0
$763$	20.7846	0.752453
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.1962	0.440378
$768$	0	0
$769$	26.7846	0.965878	0.482939	−	0.875654i	$-0.339569\pi$
$769$	26.7846	0.965878	0.482939	+	0.875654i	$0.339569\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−37.8564	−1.36160	−0.680800	−	0.732469i	$-0.738368\pi$
$773$	−37.8564	−1.36160	−0.680800	+	0.732469i	$0.738368\pi$
$774$	0	0
$775$	−1.26795	−0.0455461
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.5359	−1.09406
$780$	0	0
$781$	7.17691	0.256810
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−12.5359	−0.446857	−0.223428	−	0.974720i	$-0.571725\pi$
$787$	−12.5359	−0.446857	−0.223428	+	0.974720i	$0.571725\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	49.8564	1.77269
$792$	0	0
$793$	1.46410	0.0519918
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.92820	−0.316253	−0.158127	−	0.987419i	$-0.550545\pi$
$797$	−8.92820	−0.316253	−0.158127	+	0.987419i	$0.550545\pi$
$798$	0	0
$799$	15.7128	0.555879
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	7.60770	0.268136
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.0333	1.79424	0.897118	−	0.441791i	$-0.145657\pi$
$809$	51.0333	1.79424	0.897118	+	0.441791i	$0.145657\pi$
$810$	0	0
$811$	28.1962	0.990101	0.495050	−	0.868864i	$-0.335150\pi$
$811$	28.1962	0.990101	0.495050	+	0.868864i	$0.335150\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.8564	0.415312
$816$	0	0
$817$	−4.92820	−0.172416
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.9282	0.451197	0.225599	−	0.974220i	$-0.427566\pi$
$821$	12.9282	0.451197	0.225599	+	0.974220i	$0.427566\pi$
$822$	0	0
$823$	−24.3397	−0.848430	−0.424215	−	0.905561i	$-0.639450\pi$
$823$	−24.3397	−0.848430	−0.424215	+	0.905561i	$0.639450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.3923	−1.75231	−0.876156	−	0.482028i	$-0.839901\pi$
$827$	−50.3923	−1.75231	−0.876156	+	0.482028i	$0.839901\pi$
$828$	0	0
$829$	−11.6077	−0.403152	−0.201576	−	0.979473i	$-0.564606\pi$
$829$	−11.6077	−0.403152	−0.201576	+	0.979473i	$0.564606\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	17.3205	0.600120
$834$	0	0
$835$	6.39230	0.221215
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.6603	−1.02399	−0.511993	−	0.858990i	$-0.671093\pi$
$839$	−29.6603	−1.02399	−0.511993	+	0.858990i	$0.671093\pi$
$840$	0	0
$841$	−26.8564	−0.926083
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	32.5359	1.11795
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−50.6410	−1.73392	−0.866958	−	0.498382i	$-0.833928\pi$
$853$	−50.6410	−1.73392	−0.866958	+	0.498382i	$0.833928\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	7.32051	0.249773	0.124886	−	0.992171i	$-0.460143\pi$
$859$	7.32051	0.249773	0.124886	+	0.992171i	$0.460143\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.1769	−0.652790	−0.326395	−	0.945234i	$-0.605834\pi$
$863$	−19.1769	−0.652790	−0.326395	+	0.945234i	$0.605834\pi$
$864$	0	0
$865$	1.60770	0.0546633
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−11.8564	−0.401739
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.46410	−0.117108
$876$	0	0
$877$	−41.7128	−1.40854	−0.704271	−	0.709931i	$-0.748726\pi$
$877$	−41.7128	−1.40854	−0.704271	+	0.709931i	$0.748726\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.392305	0.0132171	0.00660854	−	0.999978i	$-0.497896\pi$
$881$	0.392305	0.0132171	0.00660854	+	0.999978i	$0.497896\pi$
$882$	0	0
$883$	−1.41154	−0.0475022	−0.0237511	−	0.999718i	$-0.507561\pi$
$883$	−1.41154	−0.0475022	−0.0237511	+	0.999718i	$0.507561\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.0526	0.538992	0.269496	−	0.963001i	$-0.413143\pi$
$887$	16.0526	0.538992	0.269496	+	0.963001i	$0.413143\pi$
$888$	0	0
$889$	−59.3205	−1.98955
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30.5359	−1.02184
$894$	0	0
$895$	13.8564	0.463169
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.85641	0.0619146
$900$	0	0
$901$	−15.7128	−0.523470
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.3205	−0.775200
$906$	0	0
$907$	−17.1244	−0.568605	−0.284302	−	0.958735i	$-0.591762\pi$
$907$	−17.1244	−0.568605	−0.284302	+	0.958735i	$0.591762\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.7846	0.953677	0.476838	−	0.878991i	$-0.341783\pi$
$911$	28.7846	0.953677	0.476838	+	0.878991i	$0.341783\pi$
$912$	0	0
$913$	−5.75129	−0.190340
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.8564	1.25013
$918$	0	0
$919$	29.1769	0.962458	0.481229	−	0.876595i	$-0.340191\pi$
$919$	29.1769	0.962458	0.481229	+	0.876595i	$0.340191\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.66025	0.186310
$924$	0	0
$925$	10.9282	0.359317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.248711	−0.00815995	−0.00407998	−	0.999992i	$-0.501299\pi$
$929$	−0.248711	−0.00815995	−0.00407998	+	0.999992i	$0.501299\pi$
$930$	0	0
$931$	−33.6603	−1.10317
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.39230	0.143644
$936$	0	0
$937$	−22.7846	−0.744341	−0.372170	−	0.928164i	$-0.621386\pi$
$937$	−22.7846	−0.744341	−0.372170	+	0.928164i	$0.621386\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17.3205	−0.564632	−0.282316	−	0.959321i	$-0.591103\pi$
$941$	−17.3205	−0.564632	−0.282316	+	0.959321i	$0.591103\pi$
$942$	0	0
$943$	−9.96152	−0.324392
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.32051	−0.172893	−0.0864466	−	0.996256i	$-0.527551\pi$
$947$	−5.32051	−0.172893	−0.0864466	+	0.996256i	$0.527551\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.6410	0.798201	0.399100	−	0.916907i	$-0.369322\pi$
$953$	24.6410	0.798201	0.399100	+	0.916907i	$0.369322\pi$
$954$	0	0
$955$	−18.9282	−0.612502
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−44.7846	−1.44617
$960$	0	0
$961$	−29.3923	−0.948139
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	48.6410	1.56419	0.782095	−	0.623159i	$-0.214151\pi$
$967$	48.6410	1.56419	0.782095	+	0.623159i	$0.214151\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	−29.0718	−0.931999
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.4256	−1.77322	−0.886611	−	0.462515i	$-0.846947\pi$
$977$	−55.4256	−1.77322	−0.886611	+	0.462515i	$0.846947\pi$
$978$	0	0
$979$	−11.3205	−0.361805
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56.6410	−1.80657	−0.903284	−	0.429043i	$-0.858851\pi$
$983$	−56.6410	−1.80657	−0.903284	+	0.429043i	$0.858851\pi$
$984$	0	0
$985$	12.9282	0.411927
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.60770	−0.0511217
$990$	0	0
$991$	11.7128	0.372070	0.186035	−	0.982543i	$-0.440436\pi$
$991$	11.7128	0.372070	0.186035	+	0.982543i	$0.440436\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.07180	0.0339782
$996$	0	0
$997$	−30.3923	−0.962534	−0.481267	−	0.876574i	$-0.659823\pi$
$997$	−30.3923	−0.962534	−0.481267	+	0.876574i	$0.659823\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.be.1.1		2
3.2	odd	2		520.2.a.d.1.2	✓	2
4.3	odd	2		9360.2.a.cq.1.2		2
12.11	even	2		1040.2.a.m.1.1		2
15.2	even	4		2600.2.d.l.1249.2		4
15.8	even	4		2600.2.d.l.1249.3		4
15.14	odd	2		2600.2.a.u.1.1		2
24.5	odd	2		4160.2.a.bl.1.1		2
24.11	even	2		4160.2.a.v.1.2		2
39.38	odd	2		6760.2.a.p.1.2		2
60.59	even	2		5200.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.d.1.2	✓	2	3.2	odd	2
1040.2.a.m.1.1		2	12.11	even	2
2600.2.a.u.1.1		2	15.14	odd	2
2600.2.d.l.1249.2		4	15.2	even	4
2600.2.d.l.1249.3		4	15.8	even	4
4160.2.a.v.1.2		2	24.11	even	2
4160.2.a.bl.1.1		2	24.5	odd	2
4680.2.a.be.1.1		2	1.1	even	1	trivial
5200.2.a.bo.1.2		2	60.59	even	2
6760.2.a.p.1.2		2	39.38	odd	2
9360.2.a.cq.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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.46410 q^{7} +1.26795 q^{11} +1.00000 q^{13} +3.46410 q^{17} -6.73205 q^{19} -2.19615 q^{23} +1.00000 q^{25} -1.46410 q^{29} -1.26795 q^{31} -3.46410 q^{35} +10.9282 q^{37} +4.53590 q^{41} +0.732051 q^{43} +4.53590 q^{47} +5.00000 q^{49} -4.53590 q^{53} +1.26795 q^{55} +12.1962 q^{59} +1.46410 q^{61} +1.00000 q^{65} -11.8564 q^{67} +5.66025 q^{71} -4.39230 q^{77} +9.46410 q^{79} -4.53590 q^{83} +3.46410 q^{85} -8.92820 q^{89} -3.46410 q^{91} -6.73205 q^{95} +15.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 6 q^{11} + 2 q^{13} - 10 q^{19} + 6 q^{23} + 2 q^{25} + 4 q^{29} - 6 q^{31} + 8 q^{37} + 16 q^{41} - 2 q^{43} + 16 q^{47} + 10 q^{49} - 16 q^{53} + 6 q^{55} + 14 q^{59} - 4 q^{61} + 2 q^{65} + 4 q^{67} - 6 q^{71} + 12 q^{77} + 12 q^{79} - 16 q^{83} - 4 q^{89} - 10 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 6 * q^11 + 2 * q^13 - 10 * q^19 + 6 * q^23 + 2 * q^25 + 4 * q^29 - 6 * q^31 + 8 * q^37 + 16 * q^41 - 2 * q^43 + 16 * q^47 + 10 * q^49 - 16 * q^53 + 6 * q^55 + 14 * q^59 - 4 * q^61 + 2 * q^65 + 4 * q^67 - 6 * q^71 + 12 * q^77 + 12 * q^79 - 16 * q^83 - 4 * q^89 - 10 * q^95 + 4 * q^97

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