LMFDB - Embedded newform 4680.2.a.bj.1.3

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:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.17741$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +5.09593 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +5.09593 q^{7} -1.25889 q^{11} -1.00000 q^{13} -7.09593 q^{17} +6.35482 q^{19} +5.61371 q^{23} +1.00000 q^{25} -4.35482 q^{29} -2.00000 q^{31} +5.09593 q^{35} +0.741113 q^{37} +0.741113 q^{41} +5.48223 q^{43} +5.48223 q^{47} +18.9685 q^{49} +6.74111 q^{53} -1.25889 q^{55} -4.00000 q^{59} +4.74111 q^{61} -1.00000 q^{65} +3.48223 q^{67} +11.4508 q^{71} -3.83705 q^{73} -6.41520 q^{77} -12.9330 q^{79} -4.00000 q^{83} -7.09593 q^{85} +15.9685 q^{89} -5.09593 q^{91} +6.35482 q^{95} +13.0959 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - q^7	$ 3 q + 3 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} - 5 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{25} + 2 q^{29} - 6 q^{31} - q^{35} + q^{37} + q^{41} + 14 q^{43} + 14 q^{47} + 28 q^{49} + 19 q^{53} - 5 q^{55} - 12 q^{59} + 13 q^{61} - 3 q^{65} + 8 q^{67} + 3 q^{71} + 6 q^{73} + 17 q^{77} - 5 q^{79} - 12 q^{83} - 5 q^{85} + 19 q^{89} + q^{91} + 4 q^{95} + 23 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - q^7 - 5 * q^11 - 3 * q^13 - 5 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^25 + 2 * q^29 - 6 * q^31 - q^35 + q^37 + q^41 + 14 * q^43 + 14 * q^47 + 28 * q^49 + 19 * q^53 - 5 * q^55 - 12 * q^59 + 13 * q^61 - 3 * q^65 + 8 * q^67 + 3 * q^71 + 6 * q^73 + 17 * q^77 - 5 * q^79 - 12 * q^83 - 5 * q^85 + 19 * q^89 + q^91 + 4 * q^95 + 23 * 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
$7$	5.09593	1.92608	0.963041	−	0.269356i	$-0.0868108\pi$
$7$	5.09593	1.92608	0.963041	+	0.269356i	$0.0868108\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.25889	−0.379569	−0.189784	−	0.981826i	$-0.560779\pi$
$11$	−1.25889	−0.379569	−0.189784	+	0.981826i	$0.560779\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.09593	−1.72102	−0.860508	−	0.509437i	$-0.829854\pi$
$17$	−7.09593	−1.72102	−0.860508	+	0.509437i	$0.829854\pi$
$18$	0	0
$19$	6.35482	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$19$	6.35482	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.61371	1.17054	0.585269	−	0.810839i	$-0.300989\pi$
$23$	5.61371	1.17054	0.585269	+	0.810839i	$0.300989\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.35482	−0.808670	−0.404335	−	0.914611i	$-0.632497\pi$
$29$	−4.35482	−0.808670	−0.404335	+	0.914611i	$0.632497\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.09593	0.861370
$36$	0	0
$37$	0.741113	0.121838	0.0609191	−	0.998143i	$-0.480597\pi$
$37$	0.741113	0.121838	0.0609191	+	0.998143i	$0.480597\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.741113	0.115742	0.0578712	−	0.998324i	$-0.481569\pi$
$41$	0.741113	0.115742	0.0578712	+	0.998324i	$0.481569\pi$
$42$	0	0
$43$	5.48223	0.836032	0.418016	−	0.908440i	$-0.362726\pi$
$43$	5.48223	0.836032	0.418016	+	0.908440i	$0.362726\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.48223	0.799665	0.399832	−	0.916588i	$-0.369068\pi$
$47$	5.48223	0.799665	0.399832	+	0.916588i	$0.369068\pi$
$48$	0	0
$49$	18.9685	2.70979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.74111	0.925963	0.462982	−	0.886368i	$-0.346780\pi$
$53$	6.74111	0.925963	0.462982	+	0.886368i	$0.346780\pi$
$54$	0	0
$55$	−1.25889	−0.169748
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	4.74111	0.607037	0.303519	−	0.952825i	$-0.401839\pi$
$61$	4.74111	0.607037	0.303519	+	0.952825i	$0.401839\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	3.48223	0.425422	0.212711	−	0.977115i	$-0.431771\pi$
$67$	3.48223	0.425422	0.212711	+	0.977115i	$0.431771\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.4508	1.35895	0.679477	−	0.733697i	$-0.262207\pi$
$71$	11.4508	1.35895	0.679477	+	0.733697i	$0.262207\pi$
$72$	0	0
$73$	−3.83705	−0.449092	−0.224546	−	0.974463i	$-0.572090\pi$
$73$	−3.83705	−0.449092	−0.224546	+	0.974463i	$0.572090\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.41520	−0.731080
$78$	0	0
$79$	−12.9330	−1.45507	−0.727537	−	0.686069i	$-0.759335\pi$
$79$	−12.9330	−1.45507	−0.727537	+	0.686069i	$0.759335\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−7.09593	−0.769662
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.9685	1.69266	0.846330	−	0.532659i	$-0.178807\pi$
$89$	15.9685	1.69266	0.846330	+	0.532659i	$0.178807\pi$
$90$	0	0
$91$	−5.09593	−0.534199
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.35482	0.651991
$96$	0	0
$97$	13.0959	1.32969	0.664845	−	0.746981i	$-0.268497\pi$
$97$	13.0959	1.32969	0.664845	+	0.746981i	$0.268497\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.87259	0.683849	0.341924	−	0.939727i	$-0.388921\pi$
$101$	6.87259	0.683849	0.341924	+	0.939727i	$0.388921\pi$
$102$	0	0
$103$	−2.19186	−0.215971	−0.107985	−	0.994152i	$-0.534440\pi$
$103$	−2.19186	−0.215971	−0.107985	+	0.994152i	$0.534440\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.9330	−1.25028	−0.625139	−	0.780513i	$-0.714958\pi$
$107$	−12.9330	−1.25028	−0.625139	+	0.780513i	$0.714958\pi$
$108$	0	0
$109$	11.8370	1.13378	0.566892	−	0.823792i	$-0.308146\pi$
$109$	11.8370	1.13378	0.566892	+	0.823792i	$0.308146\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.35482	−0.785955	−0.392978	−	0.919548i	$-0.628555\pi$
$113$	−8.35482	−0.785955	−0.392978	+	0.919548i	$0.628555\pi$
$114$	0	0
$115$	5.61371	0.523481
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−36.1604	−3.31482
$120$	0	0
$121$	−9.41520	−0.855928
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.5467	1.27095	0.635475	−	0.772122i	$-0.280804\pi$
$131$	14.5467	1.27095	0.635475	+	0.772122i	$0.280804\pi$
$132$	0	0
$133$	32.3837	2.80803
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.4822	0.980993	0.490496	−	0.871443i	$-0.336815\pi$
$137$	11.4822	0.980993	0.490496	+	0.871443i	$0.336815\pi$
$138$	0	0
$139$	0.549248	0.0465866	0.0232933	−	0.999729i	$-0.492585\pi$
$139$	0.549248	0.0465866	0.0232933	+	0.999729i	$0.492585\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.25889	0.105273
$144$	0	0
$145$	−4.35482	−0.361648
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.4152	−1.01709	−0.508547	−	0.861035i	$-0.669817\pi$
$149$	−12.4152	−1.01709	−0.508547	+	0.861035i	$0.669817\pi$
$150$	0	0
$151$	12.5178	1.01868	0.509341	−	0.860565i	$-0.329889\pi$
$151$	12.5178	1.01868	0.509341	+	0.860565i	$0.329889\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−12.9015	−1.02965	−0.514826	−	0.857295i	$-0.672143\pi$
$157$	−12.9015	−1.02965	−0.514826	+	0.857295i	$0.672143\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	28.6071	2.25455
$162$	0	0
$163$	0.294435	0.0230620	0.0115310	−	0.999934i	$-0.496329\pi$
$163$	0.294435	0.0230620	0.0115310	+	0.999934i	$0.496329\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.19186	−0.479141	−0.239570	−	0.970879i	$-0.577007\pi$
$167$	−6.19186	−0.479141	−0.239570	+	0.970879i	$0.577007\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.2274	−1.46183	−0.730917	−	0.682467i	$-0.760907\pi$
$173$	−19.2274	−1.46183	−0.730917	+	0.682467i	$0.760907\pi$
$174$	0	0
$175$	5.09593	0.385216
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.64518	−0.571428	−0.285714	−	0.958315i	$-0.592231\pi$
$179$	−7.64518	−0.571428	−0.285714	+	0.958315i	$0.592231\pi$
$180$	0	0
$181$	−24.6782	−1.83431	−0.917157	−	0.398527i	$-0.869522\pi$
$181$	−24.6782	−1.83431	−0.917157	+	0.398527i	$0.869522\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.741113	0.0544877
$186$	0	0
$187$	8.93298	0.653244
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.5178	−1.05047	−0.525235	−	0.850957i	$-0.676022\pi$
$191$	−14.5178	−1.05047	−0.525235	+	0.850957i	$0.676022\pi$
$192$	0	0
$193$	−5.09593	−0.366813	−0.183407	−	0.983037i	$-0.558712\pi$
$193$	−5.09593	−0.366813	−0.183407	+	0.983037i	$0.558712\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.22741	−0.372438	−0.186219	−	0.982508i	$-0.559623\pi$
$197$	−5.22741	−0.372438	−0.186219	+	0.982508i	$0.559623\pi$
$198$	0	0
$199$	−25.4193	−1.80192	−0.900962	−	0.433897i	$-0.857138\pi$
$199$	−25.4193	−1.80192	−0.900962	+	0.433897i	$0.857138\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−22.1919	−1.55756
$204$	0	0
$205$	0.741113	0.0517616
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	24.7096	1.70108	0.850541	−	0.525909i	$-0.176275\pi$
$211$	24.7096	1.70108	0.850541	+	0.525909i	$0.176275\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.48223	0.373885
$216$	0	0
$217$	−10.1919	−0.691869
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.09593	0.477324
$222$	0	0
$223$	7.39037	0.494896	0.247448	−	0.968901i	$-0.420408\pi$
$223$	7.39037	0.494896	0.247448	+	0.968901i	$0.420408\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.70964	0.312590	0.156295	−	0.987710i	$-0.450045\pi$
$227$	4.70964	0.312590	0.156295	+	0.987710i	$0.450045\pi$
$228$	0	0
$229$	−3.06446	−0.202505	−0.101253	−	0.994861i	$-0.532285\pi$
$229$	−3.06446	−0.202505	−0.101253	+	0.994861i	$0.532285\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.8056	1.29751	0.648753	−	0.760999i	$-0.275291\pi$
$233$	19.8056	1.29751	0.648753	+	0.760999i	$0.275291\pi$
$234$	0	0
$235$	5.48223	0.357621
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.29444	−0.148415	−0.0742073	−	0.997243i	$-0.523643\pi$
$239$	−2.29444	−0.148415	−0.0742073	+	0.997243i	$0.523643\pi$
$240$	0	0
$241$	29.3482	1.89048	0.945241	−	0.326372i	$-0.105826\pi$
$241$	29.3482	1.89048	0.945241	+	0.326372i	$0.105826\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	18.9685	1.21185
$246$	0	0
$247$	−6.35482	−0.404347
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−26.5467	−1.67561	−0.837806	−	0.545968i	$-0.816162\pi$
$251$	−26.5467	−1.67561	−0.837806	+	0.545968i	$0.816162\pi$
$252$	0	0
$253$	−7.06702	−0.444300
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.6452	0.975920	0.487960	−	0.872866i	$-0.337741\pi$
$257$	15.6452	0.975920	0.487960	+	0.872866i	$0.337741\pi$
$258$	0	0
$259$	3.77666	0.234670
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.87259	0.423782	0.211891	−	0.977293i	$-0.432038\pi$
$263$	6.87259	0.423782	0.211891	+	0.977293i	$0.432038\pi$
$264$	0	0
$265$	6.74111	0.414103
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.31927	0.446264	0.223132	−	0.974788i	$-0.428372\pi$
$269$	7.31927	0.446264	0.223132	+	0.974788i	$0.428372\pi$
$270$	0	0
$271$	28.9015	1.75564	0.877821	−	0.478989i	$-0.158997\pi$
$271$	28.9015	1.75564	0.877821	+	0.478989i	$0.158997\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.25889	−0.0759137
$276$	0	0
$277$	−3.92890	−0.236065	−0.118032	−	0.993010i	$-0.537659\pi$
$277$	−3.92890	−0.236065	−0.118032	+	0.993010i	$0.537659\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.3837	0.619441	0.309721	−	0.950828i	$-0.399764\pi$
$281$	10.3837	0.619441	0.309721	+	0.950828i	$0.399764\pi$
$282$	0	0
$283$	−13.4193	−0.797693	−0.398847	−	0.917018i	$-0.630589\pi$
$283$	−13.4193	−0.797693	−0.398847	+	0.917018i	$0.630589\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.77666	0.222929
$288$	0	0
$289$	33.3523	1.96190
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.70964	−0.391981	−0.195991	−	0.980606i	$-0.562792\pi$
$293$	−6.70964	−0.391981	−0.195991	+	0.980606i	$0.562792\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.61371	−0.324649
$300$	0	0
$301$	27.9371	1.61026
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.74111	0.271475
$306$	0	0
$307$	31.9685	1.82454	0.912270	−	0.409589i	$-0.134328\pi$
$307$	31.9685	1.82454	0.912270	+	0.409589i	$0.134328\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.55332	0.201490	0.100745	−	0.994912i	$-0.467877\pi$
$311$	3.55332	0.201490	0.100745	+	0.994912i	$0.467877\pi$
$312$	0	0
$313$	−4.19186	−0.236938	−0.118469	−	0.992958i	$-0.537799\pi$
$313$	−4.19186	−0.236938	−0.118469	+	0.992958i	$0.537799\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.03555	0.170493	0.0852467	−	0.996360i	$-0.472832\pi$
$317$	3.03555	0.170493	0.0852467	+	0.996360i	$0.472832\pi$
$318$	0	0
$319$	5.48223	0.306946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−45.0934	−2.50906
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.9371	1.54022
$330$	0	0
$331$	8.28372	0.455315	0.227657	−	0.973741i	$-0.426893\pi$
$331$	8.28372	0.455315	0.227657	+	0.973741i	$0.426893\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.48223	0.190254
$336$	0	0
$337$	−14.3837	−0.783532	−0.391766	−	0.920065i	$-0.628136\pi$
$337$	−14.3837	−0.783532	−0.391766	+	0.920065i	$0.628136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.51777	0.136345
$342$	0	0
$343$	60.9908	3.29319
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.74111	0.147151	0.0735753	−	0.997290i	$-0.476559\pi$
$347$	2.74111	0.147151	0.0735753	+	0.997290i	$0.476559\pi$
$348$	0	0
$349$	8.28372	0.443418	0.221709	−	0.975113i	$-0.428837\pi$
$349$	8.28372	0.443418	0.221709	+	0.975113i	$0.428837\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.8030	1.90560	0.952801	−	0.303596i	$-0.0981873\pi$
$353$	35.8030	1.90560	0.952801	+	0.303596i	$0.0981873\pi$
$354$	0	0
$355$	11.4508	0.607743
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.96445	−0.156458	−0.0782289	−	0.996935i	$-0.524926\pi$
$359$	−2.96445	−0.156458	−0.0782289	+	0.996935i	$0.524926\pi$
$360$	0	0
$361$	21.3837	1.12546
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.83705	−0.200840
$366$	0	0
$367$	23.3482	1.21877	0.609383	−	0.792876i	$-0.291417\pi$
$367$	23.3482	1.21877	0.609383	+	0.792876i	$0.291417\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	34.3523	1.78348
$372$	0	0
$373$	31.4193	1.62683	0.813414	−	0.581685i	$-0.197606\pi$
$373$	31.4193	1.62683	0.813414	+	0.581685i	$0.197606\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.35482	0.224285
$378$	0	0
$379$	−23.8370	−1.22443	−0.612213	−	0.790693i	$-0.709721\pi$
$379$	−23.8370	−1.22443	−0.612213	+	0.790693i	$0.709721\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.32591	−0.221044	−0.110522	−	0.993874i	$-0.535252\pi$
$383$	−4.32591	−0.221044	−0.110522	+	0.993874i	$0.535252\pi$
$384$	0	0
$385$	−6.41520	−0.326949
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.4837	−1.74839	−0.874197	−	0.485571i	$-0.838612\pi$
$389$	−34.4837	−1.74839	−0.874197	+	0.485571i	$0.838612\pi$
$390$	0	0
$391$	−39.8345	−2.01452
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.9330	−0.650729
$396$	0	0
$397$	−10.2233	−0.513095	−0.256547	−	0.966532i	$-0.582585\pi$
$397$	−10.2233	−0.513095	−0.256547	+	0.966532i	$0.582585\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.29036	0.0644376	0.0322188	−	0.999481i	$-0.489743\pi$
$401$	1.29036	0.0644376	0.0322188	+	0.999481i	$0.489743\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.932977	−0.0462460
$408$	0	0
$409$	4.19186	0.207274	0.103637	−	0.994615i	$-0.466952\pi$
$409$	4.19186	0.207274	0.103637	+	0.994615i	$0.466952\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−20.3837	−1.00302
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−35.3193	−1.72546	−0.862730	−	0.505665i	$-0.831247\pi$
$419$	−35.3193	−1.72546	−0.862730	+	0.505665i	$0.831247\pi$
$420$	0	0
$421$	4.10001	0.199822	0.0999110	−	0.994996i	$-0.468144\pi$
$421$	4.10001	0.199822	0.0999110	+	0.994996i	$0.468144\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.09593	−0.344203
$426$	0	0
$427$	24.1604	1.16920
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.0934	−1.01603	−0.508016	−	0.861348i	$-0.669621\pi$
$431$	−21.0934	−1.01603	−0.508016	+	0.861348i	$0.669621\pi$
$432$	0	0
$433$	−29.9371	−1.43868	−0.719341	−	0.694657i	$-0.755556\pi$
$433$	−29.9371	−1.43868	−0.719341	+	0.694657i	$0.755556\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	35.6741	1.70652
$438$	0	0
$439$	1.96853	0.0939526	0.0469763	−	0.998896i	$-0.485041\pi$
$439$	1.96853	0.0939526	0.0469763	+	0.998896i	$0.485041\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.6426	−1.40836	−0.704182	−	0.710020i	$-0.748686\pi$
$443$	−29.6426	−1.40836	−0.704182	+	0.710020i	$0.748686\pi$
$444$	0	0
$445$	15.9685	0.756981
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.45075	0.257237	0.128618	−	0.991694i	$-0.458946\pi$
$449$	5.45075	0.257237	0.128618	+	0.991694i	$0.458946\pi$
$450$	0	0
$451$	−0.932977	−0.0439322
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.09593	−0.238901
$456$	0	0
$457$	−17.3589	−0.812015	−0.406007	−	0.913870i	$-0.633079\pi$
$457$	−17.3589	−0.812015	−0.406007	+	0.913870i	$0.633079\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.6426	−0.542251	−0.271125	−	0.962544i	$-0.587396\pi$
$461$	−11.6426	−0.542251	−0.271125	+	0.962544i	$0.587396\pi$
$462$	0	0
$463$	−41.4167	−1.92480	−0.962399	−	0.271640i	$-0.912434\pi$
$463$	−41.4167	−1.92480	−0.962399	+	0.271640i	$0.912434\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.4152	1.03725	0.518626	−	0.855001i	$-0.326444\pi$
$467$	22.4152	1.03725	0.518626	+	0.855001i	$0.326444\pi$
$468$	0	0
$469$	17.7452	0.819397
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.90150	−0.317331
$474$	0	0
$475$	6.35482	0.291579
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.7137	1.26627	0.633136	−	0.774041i	$-0.281767\pi$
$479$	27.7137	1.26627	0.633136	+	0.774041i	$0.281767\pi$
$480$	0	0
$481$	−0.741113	−0.0337918
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.0959	0.594656
$486$	0	0
$487$	30.0685	1.36254	0.681268	−	0.732034i	$-0.261429\pi$
$487$	30.0685	1.36254	0.681268	+	0.732034i	$0.261429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.3904	−0.784816	−0.392408	−	0.919791i	$-0.628358\pi$
$491$	−17.3904	−0.784816	−0.392408	+	0.919791i	$0.628358\pi$
$492$	0	0
$493$	30.9015	1.39173
$494$	0	0
$495$	0	0
$496$	0	0
$497$	58.3523	2.61746
$498$	0	0
$499$	−16.8726	−0.755321	−0.377661	−	0.925944i	$-0.623271\pi$
$499$	−16.8726	−0.755321	−0.377661	+	0.925944i	$0.623271\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.6178	−0.919301	−0.459651	−	0.888100i	$-0.652025\pi$
$503$	−20.6178	−0.919301	−0.459651	+	0.888100i	$0.652025\pi$
$504$	0	0
$505$	6.87259	0.305826
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.6426	0.693347	0.346673	−	0.937986i	$-0.387311\pi$
$509$	15.6426	0.693347	0.346673	+	0.937986i	$0.387311\pi$
$510$	0	0
$511$	−19.5533	−0.864988
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.19186	−0.0965851
$516$	0	0
$517$	−6.90150	−0.303528
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.19186	−0.0960273	−0.0480137	−	0.998847i	$-0.515289\pi$
$521$	−2.19186	−0.0960273	−0.0480137	+	0.998847i	$0.515289\pi$
$522$	0	0
$523$	−29.4193	−1.28642	−0.643208	−	0.765692i	$-0.722397\pi$
$523$	−29.4193	−1.28642	−0.643208	+	0.765692i	$0.722397\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.1919	0.618207
$528$	0	0
$529$	8.51370	0.370161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.741113	−0.0321012
$534$	0	0
$535$	−12.9330	−0.559141
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23.8792	−1.02855
$540$	0	0
$541$	−22.7385	−0.977607	−0.488803	−	0.872394i	$-0.662566\pi$
$541$	−22.7385	−0.977607	−0.488803	+	0.872394i	$0.662566\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.8370	0.507043
$546$	0	0
$547$	−26.9015	−1.15023	−0.575113	−	0.818074i	$-0.695042\pi$
$547$	−26.9015	−1.15023	−0.575113	+	0.818074i	$0.695042\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.6741	−1.17896
$552$	0	0
$553$	−65.9056	−2.80259
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.3837	1.62637	0.813185	−	0.582005i	$-0.197731\pi$
$557$	38.3837	1.62637	0.813185	+	0.582005i	$0.197731\pi$
$558$	0	0
$559$	−5.48223	−0.231873
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.5441	0.865831	0.432916	−	0.901434i	$-0.357485\pi$
$563$	20.5441	0.865831	0.432916	+	0.901434i	$0.357485\pi$
$564$	0	0
$565$	−8.35482	−0.351490
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.5756	−1.95255	−0.976275	−	0.216533i	$-0.930525\pi$
$569$	−46.5756	−1.95255	−0.976275	+	0.216533i	$0.930525\pi$
$570$	0	0
$571$	3.06702	0.128351	0.0641754	−	0.997939i	$-0.479558\pi$
$571$	3.06702	0.128351	0.0641754	+	0.997939i	$0.479558\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.61371	0.234108
$576$	0	0
$577$	6.84112	0.284800	0.142400	−	0.989809i	$-0.454518\pi$
$577$	6.84112	0.284800	0.142400	+	0.989809i	$0.454518\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−20.3837	−0.845660
$582$	0	0
$583$	−8.48630	−0.351467
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.67409	−0.316744	−0.158372	−	0.987380i	$-0.550624\pi$
$587$	−7.67409	−0.316744	−0.158372	+	0.987380i	$0.550624\pi$
$588$	0	0
$589$	−12.7096	−0.523692
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.4548	1.00424	0.502120	−	0.864798i	$-0.332554\pi$
$593$	24.4548	1.00424	0.502120	+	0.864798i	$0.332554\pi$
$594$	0	0
$595$	−36.1604	−1.48243
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.8660	1.05685	0.528427	−	0.848979i	$-0.322782\pi$
$599$	25.8660	1.05685	0.528427	+	0.848979i	$0.322782\pi$
$600$	0	0
$601$	10.9330	0.445965	0.222983	−	0.974822i	$-0.428421\pi$
$601$	10.9330	0.445965	0.222983	+	0.974822i	$0.428421\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.41520	−0.382782
$606$	0	0
$607$	21.0355	0.853807	0.426903	−	0.904297i	$-0.359604\pi$
$607$	21.0355	0.853807	0.426903	+	0.904297i	$0.359604\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.48223	−0.221787
$612$	0	0
$613$	−33.7767	−1.36423	−0.682113	−	0.731247i	$-0.738939\pi$
$613$	−33.7767	−1.36423	−0.682113	+	0.731247i	$0.738939\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.1289	−0.810360	−0.405180	−	0.914237i	$-0.632791\pi$
$617$	−20.1289	−0.810360	−0.405180	+	0.914237i	$0.632791\pi$
$618$	0	0
$619$	−27.7741	−1.11634	−0.558168	−	0.829728i	$-0.688495\pi$
$619$	−27.7741	−1.11634	−0.558168	+	0.829728i	$0.688495\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	81.3745	3.26020
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.25889	−0.209686
$630$	0	0
$631$	−33.6741	−1.34054	−0.670272	−	0.742115i	$-0.733823\pi$
$631$	−33.6741	−1.34054	−0.670272	+	0.742115i	$0.733823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−18.9685	−0.751560
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−36.3837	−1.43707	−0.718535	−	0.695490i	$-0.755187\pi$
$641$	−36.3837	−1.43707	−0.718535	+	0.695490i	$0.755187\pi$
$642$	0	0
$643$	25.0041	0.986064	0.493032	−	0.870011i	$-0.335888\pi$
$643$	25.0041	0.986064	0.493032	+	0.870011i	$0.335888\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.7700	1.36695	0.683475	−	0.729974i	$-0.260468\pi$
$647$	34.7700	1.36695	0.683475	+	0.729974i	$0.260468\pi$
$648$	0	0
$649$	5.03555	0.197663
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.3208	1.57787	0.788937	−	0.614474i	$-0.210632\pi$
$653$	40.3208	1.57787	0.788937	+	0.614474i	$0.210632\pi$
$654$	0	0
$655$	14.5467	0.568386
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.4837	0.875842	0.437921	−	0.899013i	$-0.355715\pi$
$659$	22.4837	0.875842	0.437921	+	0.899013i	$0.355715\pi$
$660$	0	0
$661$	−49.6401	−1.93077	−0.965387	−	0.260821i	$-0.916007\pi$
$661$	−49.6401	−1.93077	−0.965387	+	0.260821i	$0.916007\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	32.3837	1.25579
$666$	0	0
$667$	−24.4467	−0.946579
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.96853	−0.230412
$672$	0	0
$673$	−19.6030	−0.755640	−0.377820	−	0.925879i	$-0.623326\pi$
$673$	−19.6030	−0.755640	−0.377820	+	0.925879i	$0.623326\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.7989	1.18370	0.591850	−	0.806048i	$-0.298398\pi$
$677$	30.7989	1.18370	0.591850	+	0.806048i	$0.298398\pi$
$678$	0	0
$679$	66.7360	2.56109
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.3126	−1.15988	−0.579940	−	0.814659i	$-0.696924\pi$
$683$	−30.3126	−1.15988	−0.579940	+	0.814659i	$0.696924\pi$
$684$	0	0
$685$	11.4822	0.438713
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.74111	−0.256816
$690$	0	0
$691$	−38.4126	−1.46129	−0.730643	−	0.682760i	$-0.760780\pi$
$691$	−38.4126	−1.46129	−0.730643	+	0.682760i	$0.760780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.549248	0.0208342
$696$	0	0
$697$	−5.25889	−0.199195
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−52.3497	−1.97722	−0.988610	−	0.150498i	$-0.951912\pi$
$701$	−52.3497	−1.97722	−0.988610	+	0.150498i	$0.951912\pi$
$702$	0	0
$703$	4.70964	0.177627
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.0223	1.31715
$708$	0	0
$709$	−4.60963	−0.173118	−0.0865592	−	0.996247i	$-0.527587\pi$
$709$	−4.60963	−0.173118	−0.0865592	+	0.996247i	$0.527587\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.2274	−0.420470
$714$	0	0
$715$	1.25889	0.0470797
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.7452	−1.25848	−0.629242	−	0.777210i	$-0.716634\pi$
$719$	−33.7452	−1.25848	−0.629242	+	0.777210i	$0.716634\pi$
$720$	0	0
$721$	−11.1696	−0.415977
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.35482	−0.161734
$726$	0	0
$727$	15.5533	0.576841	0.288420	−	0.957504i	$-0.406870\pi$
$727$	15.5533	0.576841	0.288420	+	0.957504i	$0.406870\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38.9015	−1.43882
$732$	0	0
$733$	34.5441	1.27592	0.637958	−	0.770071i	$-0.279779\pi$
$733$	34.5441	1.27592	0.637958	+	0.770071i	$0.279779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.38373	−0.161477
$738$	0	0
$739$	−39.8370	−1.46543	−0.732715	−	0.680536i	$-0.761747\pi$
$739$	−39.8370	−1.46543	−0.732715	+	0.680536i	$0.761747\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	45.8030	1.68035	0.840175	−	0.542316i	$-0.182452\pi$
$743$	45.8030	1.68035	0.840175	+	0.542316i	$0.182452\pi$
$744$	0	0
$745$	−12.4152	−0.454858
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−65.9056	−2.40814
$750$	0	0
$751$	15.8974	0.580105	0.290053	−	0.957011i	$-0.406327\pi$
$751$	15.8974	0.580105	0.290053	+	0.957011i	$0.406327\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.5178	0.455568
$756$	0	0
$757$	−13.4904	−0.490316	−0.245158	−	0.969483i	$-0.578840\pi$
$757$	−13.4904	−0.490316	−0.245158	+	0.969483i	$0.578840\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−44.1289	−1.59967	−0.799836	−	0.600219i	$-0.795080\pi$
$761$	−44.1289	−1.59967	−0.799836	+	0.600219i	$0.795080\pi$
$762$	0	0
$763$	60.3208	2.18376
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	12.9645	0.467510	0.233755	−	0.972296i	$-0.424899\pi$
$769$	12.9645	0.467510	0.233755	+	0.972296i	$0.424899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.5756	−0.883923	−0.441961	−	0.897034i	$-0.645717\pi$
$773$	−24.5756	−0.883923	−0.441961	+	0.897034i	$0.645717\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.70964	0.168740
$780$	0	0
$781$	−14.4152	−0.515817
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.9015	−0.460474
$786$	0	0
$787$	−25.2274	−0.899260	−0.449630	−	0.893215i	$-0.648444\pi$
$787$	−25.2274	−0.899260	−0.449630	+	0.893215i	$0.648444\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−42.5756	−1.51381
$792$	0	0
$793$	−4.74111	−0.168362
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.6071	1.15500	0.577501	−	0.816390i	$-0.304028\pi$
$797$	32.6071	1.15500	0.577501	+	0.816390i	$0.304028\pi$
$798$	0	0
$799$	−38.9015	−1.37624
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.83041	0.170461
$804$	0	0
$805$	28.6071	1.00827
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27.3482	0.961511	0.480755	−	0.876855i	$-0.340362\pi$
$809$	27.3482	0.961511	0.480755	+	0.876855i	$0.340362\pi$
$810$	0	0
$811$	−24.5467	−0.861951	−0.430975	−	0.902364i	$-0.641830\pi$
$811$	−24.5467	−0.861951	−0.430975	+	0.902364i	$0.641830\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.294435	0.0103136
$816$	0	0
$817$	34.8386	1.21885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.10257	0.212981	0.106491	−	0.994314i	$-0.466039\pi$
$821$	6.10257	0.212981	0.106491	+	0.994314i	$0.466039\pi$
$822$	0	0
$823$	−18.6385	−0.649699	−0.324849	−	0.945766i	$-0.605314\pi$
$823$	−18.6385	−0.649699	−0.324849	+	0.945766i	$0.605314\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.44668	−0.154626	−0.0773131	−	0.997007i	$-0.524634\pi$
$827$	−4.44668	−0.154626	−0.0773131	+	0.997007i	$0.524634\pi$
$828$	0	0
$829$	−14.3837	−0.499568	−0.249784	−	0.968302i	$-0.580360\pi$
$829$	−14.3837	−0.499568	−0.249784	+	0.968302i	$0.580360\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−134.599	−4.66359
$834$	0	0
$835$	−6.19186	−0.214278
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46.0263	−1.58901	−0.794503	−	0.607260i	$-0.792269\pi$
$839$	−46.0263	−1.58901	−0.794503	+	0.607260i	$0.792269\pi$
$840$	0	0
$841$	−10.0355	−0.346053
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−47.9792	−1.64859
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.16039	0.142616
$852$	0	0
$853$	−17.1248	−0.586343	−0.293172	−	0.956060i	$-0.594711\pi$
$853$	−17.1248	−0.586343	−0.293172	+	0.956060i	$0.594711\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.1670	0.723052	0.361526	−	0.932362i	$-0.382256\pi$
$857$	21.1670	0.723052	0.361526	+	0.932362i	$0.382256\pi$
$858$	0	0
$859$	1.64262	0.0560453	0.0280227	−	0.999607i	$-0.491079\pi$
$859$	1.64262	0.0560453	0.0280227	+	0.999607i	$0.491079\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−48.5045	−1.65111	−0.825556	−	0.564320i	$-0.809138\pi$
$863$	−48.5045	−1.65111	−0.825556	+	0.564320i	$0.809138\pi$
$864$	0	0
$865$	−19.2274	−0.653752
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.2812	0.552300
$870$	0	0
$871$	−3.48223	−0.117991
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.09593	0.172274
$876$	0	0
$877$	7.41928	0.250531	0.125266	−	0.992123i	$-0.460022\pi$
$877$	7.41928	0.250531	0.125266	+	0.992123i	$0.460022\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21.1563	−0.712774	−0.356387	−	0.934338i	$-0.615992\pi$
$881$	−21.1563	−0.712774	−0.356387	+	0.934338i	$0.615992\pi$
$882$	0	0
$883$	24.8304	0.835610	0.417805	−	0.908537i	$-0.362800\pi$
$883$	24.8304	0.835610	0.417805	+	0.908537i	$0.362800\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.7071	−0.896736	−0.448368	−	0.893849i	$-0.647995\pi$
$887$	−26.7071	−0.896736	−0.448368	+	0.893849i	$0.647995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	34.8386	1.16583
$894$	0	0
$895$	−7.64518	−0.255550
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.70964	0.290483
$900$	0	0
$901$	−47.8345	−1.59360
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.6782	−0.820330
$906$	0	0
$907$	16.8304	0.558844	0.279422	−	0.960168i	$-0.409857\pi$
$907$	16.8304	0.558844	0.279422	+	0.960168i	$0.409857\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26.6385	0.882574	0.441287	−	0.897366i	$-0.354522\pi$
$911$	26.6385	0.882574	0.441287	+	0.897366i	$0.354522\pi$
$912$	0	0
$913$	5.03555	0.166652
$914$	0	0
$915$	0	0
$916$	0	0
$917$	74.1289	2.44795
$918$	0	0
$919$	−31.8974	−1.05220	−0.526100	−	0.850423i	$-0.676346\pi$
$919$	−31.8974	−1.05220	−0.526100	+	0.850423i	$0.676346\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.4508	−0.376906
$924$	0	0
$925$	0.741113	0.0243676
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.4508	−0.441305	−0.220652	−	0.975353i	$-0.570819\pi$
$929$	−13.4508	−0.441305	−0.220652	+	0.975353i	$0.570819\pi$
$930$	0	0
$931$	120.542	3.95059
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.93298	0.292140
$936$	0	0
$937$	−19.8081	−0.647104	−0.323552	−	0.946210i	$-0.604877\pi$
$937$	−19.8081	−0.647104	−0.323552	+	0.946210i	$0.604877\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	58.0974	1.89392	0.946961	−	0.321348i	$-0.104136\pi$
$941$	58.0974	1.89392	0.946961	+	0.321348i	$0.104136\pi$
$942$	0	0
$943$	4.16039	0.135481
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.8660	0.970513	0.485257	−	0.874372i	$-0.338726\pi$
$947$	29.8660	0.970513	0.485257	+	0.874372i	$0.338726\pi$
$948$	0	0
$949$	3.83705	0.124556
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.6715	1.47945	0.739723	−	0.672912i	$-0.234957\pi$
$953$	45.6715	1.47945	0.739723	+	0.672912i	$0.234957\pi$
$954$	0	0
$955$	−14.5178	−0.469784
$956$	0	0
$957$	0	0
$958$	0	0
$959$	58.5126	1.88947
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.09593	−0.164044
$966$	0	0
$967$	−23.7741	−0.764523	−0.382262	−	0.924054i	$-0.624855\pi$
$967$	−23.7741	−0.764523	−0.382262	+	0.924054i	$0.624855\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.12741	−0.292912	−0.146456	−	0.989217i	$-0.546787\pi$
$971$	−9.12741	−0.292912	−0.146456	+	0.989217i	$0.546787\pi$
$972$	0	0
$973$	2.79893	0.0897297
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.4548	1.03832	0.519161	−	0.854677i	$-0.326245\pi$
$977$	32.4548	1.03832	0.519161	+	0.854677i	$0.326245\pi$
$978$	0	0
$979$	−20.1026	−0.642481
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.8030	−0.695408	−0.347704	−	0.937604i	$-0.613039\pi$
$983$	−21.8030	−0.695408	−0.347704	+	0.937604i	$0.613039\pi$
$984$	0	0
$985$	−5.22741	−0.166559
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.7756	0.978607
$990$	0	0
$991$	24.0396	0.763644	0.381822	−	0.924236i	$-0.375297\pi$
$991$	24.0396	0.763644	0.381822	+	0.924236i	$0.375297\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25.4193	−0.805845
$996$	0	0
$997$	42.3837	1.34231	0.671153	−	0.741319i	$-0.265799\pi$
$997$	42.3837	1.34231	0.671153	+	0.741319i	$0.265799\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bj.1.3	yes	3
3.2	odd	2		4680.2.a.bg.1.3	✓	3
4.3	odd	2		9360.2.a.de.1.1		3
12.11	even	2		9360.2.a.cz.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4680.2.a.bg.1.3	✓	3	3.2	odd	2
4680.2.a.bj.1.3	yes	3	1.1	even	1	trivial
9360.2.a.cz.1.1		3	12.11	even	2
9360.2.a.de.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +5.09593 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +5.09593 q^{7} -1.25889 q^{11} -1.00000 q^{13} -7.09593 q^{17} +6.35482 q^{19} +5.61371 q^{23} +1.00000 q^{25} -4.35482 q^{29} -2.00000 q^{31} +5.09593 q^{35} +0.741113 q^{37} +0.741113 q^{41} +5.48223 q^{43} +5.48223 q^{47} +18.9685 q^{49} +6.74111 q^{53} -1.25889 q^{55} -4.00000 q^{59} +4.74111 q^{61} -1.00000 q^{65} +3.48223 q^{67} +11.4508 q^{71} -3.83705 q^{73} -6.41520 q^{77} -12.9330 q^{79} -4.00000 q^{83} -7.09593 q^{85} +15.9685 q^{89} -5.09593 q^{91} +6.35482 q^{95} +13.0959 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - q^7	\( 3 q + 3 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} - 5 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{25} + 2 q^{29} - 6 q^{31} - q^{35} + q^{37} + q^{41} + 14 q^{43} + 14 q^{47} + 28 q^{49} + 19 q^{53} - 5 q^{55} - 12 q^{59} + 13 q^{61} - 3 q^{65} + 8 q^{67} + 3 q^{71} + 6 q^{73} + 17 q^{77} - 5 q^{79} - 12 q^{83} - 5 q^{85} + 19 q^{89} + q^{91} + 4 q^{95} + 23 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - q^7 - 5 * q^11 - 3 * q^13 - 5 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^25 + 2 * q^29 - 6 * q^31 - q^35 + q^37 + q^41 + 14 * q^43 + 14 * q^47 + 28 * q^49 + 19 * q^53 - 5 * q^55 - 12 * q^59 + 13 * q^61 - 3 * q^65 + 8 * q^67 + 3 * q^71 + 6 * q^73 + 17 * q^77 - 5 * q^79 - 12 * q^83 - 5 * q^85 + 19 * q^89 + q^91 + 4 * q^95 + 23 * q^97

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