LMFDB - Embedded newform 2368.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.820249$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82025 q^{3} -4.05632 q^{5} +0.493058 q^{7} +0.313307 q^{9} -0.908877 q^{11} +5.79933 q^{13} +7.38351 q^{15} -2.00000 q^{17} -5.01388 q^{19} -0.897489 q^{21} +8.81321 q^{23} +11.4537 q^{25} +4.89045 q^{27} -2.70058 q^{29} +3.58418 q^{31} +1.65438 q^{33} -2.00000 q^{35} -1.00000 q^{37} -10.5562 q^{39} -7.23856 q^{41} +11.3936 q^{43} -1.27087 q^{45} -0.965194 q^{47} -6.75689 q^{49} +3.64050 q^{51} -3.72913 q^{53} +3.68669 q^{55} +9.12652 q^{57} -10.1126 q^{59} +9.36508 q^{61} +0.154479 q^{63} -23.5239 q^{65} +1.87407 q^{67} -16.0422 q^{69} +11.6196 q^{71} -5.96769 q^{73} -20.8486 q^{75} -0.448129 q^{77} -7.46759 q^{79} -9.84176 q^{81} +5.89421 q^{83} +8.11263 q^{85} +4.91572 q^{87} -0.444368 q^{89} +2.85941 q^{91} -6.52410 q^{93} +20.3379 q^{95} -3.52786 q^{97} -0.284758 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + q^{9} - 3 q^{11} - 3 q^{13} + 8 q^{15} - 8 q^{17} - 12 q^{19} - 8 q^{21} + q^{23} + 3 q^{25} - 6 q^{27} - 3 q^{29} + 19 q^{31} - 10 q^{33} - 8 q^{35} - 4 q^{37} + 5 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + q^9 - 3 * q^11 - 3 * q^13 + 8 * q^15 - 8 * q^17 - 12 * q^19 - 8 * q^21 + q^23 + 3 * q^25 - 6 * q^27 - 3 * q^29 + 19 * q^31 - 10 * q^33 - 8 * q^35 - 4 * q^37 + 5 * q^39 - 17 * q^41 + 2 * q^43 - 10 * q^45 + 10 * q^47 - 6 * q^49 + 6 * q^51 - 10 * q^53 + 15 * q^55 + 2 * q^57 - 14 * q^59 - 9 * q^61 + 18 * q^63 - 30 * q^65 - 7 * q^67 + 9 * q^69 + 16 * q^71 - 7 * q^73 - 14 * q^75 - 14 * q^77 + 21 * q^79 - 8 * q^81 + 12 * q^83 + 6 * q^85 - 19 * q^87 - 14 * q^91 - 30 * q^93 + 2 * q^95 - 32 * q^97 + 2 * q^99

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
$3$	−1.82025	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$3$	−1.82025	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$4$	0	0
$5$	−4.05632	−1.81404	−0.907020	−	0.421087i	$-0.861649\pi$
$5$	−4.05632	−1.81404	−0.907020	+	0.421087i	$0.861649\pi$
$6$	0	0
$7$	0.493058	0.186358	0.0931792	−	0.995649i	$-0.470297\pi$
$7$	0.493058	0.186358	0.0931792	+	0.995649i	$0.470297\pi$
$8$	0	0
$9$	0.313307	0.104436
$10$	0	0
$11$	−0.908877	−0.274037	−0.137018	−	0.990569i	$-0.543752\pi$
$11$	−0.908877	−0.274037	−0.137018	+	0.990569i	$0.543752\pi$
$12$	0	0
$13$	5.79933	1.60844	0.804222	−	0.594329i	$-0.202582\pi$
$13$	5.79933	1.60844	0.804222	+	0.594329i	$0.202582\pi$
$14$	0	0
$15$	7.38351	1.90641
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−5.01388	−1.15026	−0.575132	−	0.818061i	$-0.695049\pi$
$19$	−5.01388	−1.15026	−0.575132	+	0.818061i	$0.695049\pi$
$20$	0	0
$21$	−0.897489	−0.195848
$22$	0	0
$23$	8.81321	1.83768	0.918841	−	0.394629i	$-0.129127\pi$
$23$	8.81321	1.83768	0.918841	+	0.394629i	$0.129127\pi$
$24$	0	0
$25$	11.4537	2.29074
$26$	0	0
$27$	4.89045	0.941168
$28$	0	0
$29$	−2.70058	−0.501484	−0.250742	−	0.968054i	$-0.580675\pi$
$29$	−2.70058	−0.501484	−0.250742	+	0.968054i	$0.580675\pi$
$30$	0	0
$31$	3.58418	0.643738	0.321869	−	0.946784i	$-0.395689\pi$
$31$	3.58418	0.643738	0.321869	+	0.946784i	$0.395689\pi$
$32$	0	0
$33$	1.65438	0.287991
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−10.5562	−1.69035
$40$	0	0
$41$	−7.23856	−1.13047	−0.565237	−	0.824929i	$-0.691215\pi$
$41$	−7.23856	−1.13047	−0.565237	+	0.824929i	$0.691215\pi$
$42$	0	0
$43$	11.3936	1.73751	0.868756	−	0.495240i	$-0.164920\pi$
$43$	11.3936	1.73751	0.868756	+	0.495240i	$0.164920\pi$
$44$	0	0
$45$	−1.27087	−0.189451
$46$	0	0
$47$	−0.965194	−0.140788	−0.0703940	−	0.997519i	$-0.522426\pi$
$47$	−0.965194	−0.140788	−0.0703940	+	0.997519i	$0.522426\pi$
$48$	0	0
$49$	−6.75689	−0.965271
$50$	0	0
$51$	3.64050	0.509772
$52$	0	0
$53$	−3.72913	−0.512235	−0.256117	−	0.966646i	$-0.582443\pi$
$53$	−3.72913	−0.512235	−0.256117	+	0.966646i	$0.582443\pi$
$54$	0	0
$55$	3.68669	0.497114
$56$	0	0
$57$	9.12652	1.20884
$58$	0	0
$59$	−10.1126	−1.31655	−0.658276	−	0.752776i	$-0.728714\pi$
$59$	−10.1126	−1.31655	−0.658276	+	0.752776i	$0.728714\pi$
$60$	0	0
$61$	9.36508	1.19908	0.599538	−	0.800346i	$-0.295351\pi$
$61$	9.36508	1.19908	0.599538	+	0.800346i	$0.295351\pi$
$62$	0	0
$63$	0.154479	0.0194625
$64$	0	0
$65$	−23.5239	−2.91778
$66$	0	0
$67$	1.87407	0.228954	0.114477	−	0.993426i	$-0.463481\pi$
$67$	1.87407	0.228954	0.114477	+	0.993426i	$0.463481\pi$
$68$	0	0
$69$	−16.0422	−1.93126
$70$	0	0
$71$	11.6196	1.37899	0.689495	−	0.724290i	$-0.257833\pi$
$71$	11.6196	1.37899	0.689495	+	0.724290i	$0.257833\pi$
$72$	0	0
$73$	−5.96769	−0.698465	−0.349233	−	0.937036i	$-0.613558\pi$
$73$	−5.96769	−0.698465	−0.349233	+	0.937036i	$0.613558\pi$
$74$	0	0
$75$	−20.8486	−2.40739
$76$	0	0
$77$	−0.448129	−0.0510690
$78$	0	0
$79$	−7.46759	−0.840170	−0.420085	−	0.907485i	$-0.638000\pi$
$79$	−7.46759	−0.840170	−0.420085	+	0.907485i	$0.638000\pi$
$80$	0	0
$81$	−9.84176	−1.09353
$82$	0	0
$83$	5.89421	0.646974	0.323487	−	0.946233i	$-0.395145\pi$
$83$	5.89421	0.646974	0.323487	+	0.946233i	$0.395145\pi$
$84$	0	0
$85$	8.11263	0.879939
$86$	0	0
$87$	4.91572	0.527021
$88$	0	0
$89$	−0.444368	−0.0471029	−0.0235515	−	0.999723i	$-0.507497\pi$
$89$	−0.444368	−0.0471029	−0.0235515	+	0.999723i	$0.507497\pi$
$90$	0	0
$91$	2.85941	0.299747
$92$	0	0
$93$	−6.52410	−0.676518
$94$	0	0
$95$	20.3379	2.08662
$96$	0	0
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	0	0
$99$	−0.284758	−0.0286192
$100$	0	0
$101$	−7.18738	−0.715171	−0.357585	−	0.933880i	$-0.616400\pi$
$101$	−7.18738	−0.715171	−0.357585	+	0.933880i	$0.616400\pi$
$102$	0	0
$103$	−10.9778	−1.08168	−0.540838	−	0.841127i	$-0.681893\pi$
$103$	−10.9778	−1.08168	−0.540838	+	0.841127i	$0.681893\pi$
$104$	0	0
$105$	3.64050	0.355276
$106$	0	0
$107$	9.91196	0.958225	0.479113	−	0.877753i	$-0.340958\pi$
$107$	9.91196	0.958225	0.479113	+	0.877753i	$0.340958\pi$
$108$	0	0
$109$	−14.3226	−1.37186	−0.685930	−	0.727667i	$-0.740605\pi$
$109$	−14.3226	−1.37186	−0.685930	+	0.727667i	$0.740605\pi$
$110$	0	0
$111$	1.82025	0.172770
$112$	0	0
$113$	−15.2114	−1.43097	−0.715483	−	0.698630i	$-0.753793\pi$
$113$	−15.2114	−1.43097	−0.715483	+	0.698630i	$0.753793\pi$
$114$	0	0
$115$	−35.7492	−3.33363
$116$	0	0
$117$	1.81697	0.167979
$118$	0	0
$119$	−0.986116	−0.0903971
$120$	0	0
$121$	−10.1739	−0.924904
$122$	0	0
$123$	13.1760	1.18804
$124$	0	0
$125$	−26.1783	−2.34146
$126$	0	0
$127$	2.31081	0.205051	0.102526	−	0.994730i	$-0.467308\pi$
$127$	2.31081	0.205051	0.102526	+	0.994730i	$0.467308\pi$
$128$	0	0
$129$	−20.7392	−1.82599
$130$	0	0
$131$	−18.6125	−1.62619	−0.813093	−	0.582135i	$-0.802218\pi$
$131$	−18.6125	−1.62619	−0.813093	+	0.582135i	$0.802218\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	0	0
$135$	−19.8372	−1.70732
$136$	0	0
$137$	−6.62084	−0.565657	−0.282828	−	0.959171i	$-0.591273\pi$
$137$	−6.62084	−0.565657	−0.282828	+	0.959171i	$0.591273\pi$
$138$	0	0
$139$	3.25758	0.276304	0.138152	−	0.990411i	$-0.455884\pi$
$139$	3.25758	0.276304	0.138152	+	0.990411i	$0.455884\pi$
$140$	0	0
$141$	1.75689	0.147957
$142$	0	0
$143$	−5.27087	−0.440773
$144$	0	0
$145$	10.9544	0.909713
$146$	0	0
$147$	12.2992	1.01442
$148$	0	0
$149$	17.1505	1.40503	0.702513	−	0.711671i	$-0.252061\pi$
$149$	17.1505	1.40503	0.702513	+	0.711671i	$0.252061\pi$
$150$	0	0
$151$	10.1404	0.825214	0.412607	−	0.910909i	$-0.364618\pi$
$151$	10.1404	0.825214	0.412607	+	0.910909i	$0.364618\pi$
$152$	0	0
$153$	−0.626615	−0.0506588
$154$	0	0
$155$	−14.5386	−1.16777
$156$	0	0
$157$	−3.90638	−0.311763	−0.155882	−	0.987776i	$-0.549822\pi$
$157$	−3.90638	−0.311763	−0.155882	+	0.987776i	$0.549822\pi$
$158$	0	0
$159$	6.78794	0.538319
$160$	0	0
$161$	4.34542	0.342467
$162$	0	0
$163$	4.74677	0.371796	0.185898	−	0.982569i	$-0.440481\pi$
$163$	4.74677	0.371796	0.185898	+	0.982569i	$0.440481\pi$
$164$	0	0
$165$	−6.71070	−0.522427
$166$	0	0
$167$	−13.4499	−1.04079	−0.520394	−	0.853926i	$-0.674215\pi$
$167$	−13.4499	−1.04079	−0.520394	+	0.853926i	$0.674215\pi$
$168$	0	0
$169$	20.6322	1.58709
$170$	0	0
$171$	−1.57089	−0.120129
$172$	0	0
$173$	−13.0379	−0.991252	−0.495626	−	0.868536i	$-0.665061\pi$
$173$	−13.0379	−0.991252	−0.495626	+	0.868536i	$0.665061\pi$
$174$	0	0
$175$	5.64734	0.426899
$176$	0	0
$177$	18.4075	1.38359
$178$	0	0
$179$	6.41504	0.479482	0.239741	−	0.970837i	$-0.422937\pi$
$179$	6.41504	0.479482	0.239741	+	0.970837i	$0.422937\pi$
$180$	0	0
$181$	19.9266	1.48113	0.740567	−	0.671982i	$-0.234557\pi$
$181$	19.9266	1.48113	0.740567	+	0.671982i	$0.234557\pi$
$182$	0	0
$183$	−17.0468	−1.26013
$184$	0	0
$185$	4.05632	0.298226
$186$	0	0
$187$	1.81775	0.132927
$188$	0	0
$189$	2.41128	0.175395
$190$	0	0
$191$	−22.0716	−1.59704	−0.798521	−	0.601966i	$-0.794384\pi$
$191$	−22.0716	−1.59704	−0.798521	+	0.601966i	$0.794384\pi$
$192$	0	0
$193$	−4.67716	−0.336669	−0.168335	−	0.985730i	$-0.553839\pi$
$193$	−4.67716	−0.336669	−0.168335	+	0.985730i	$0.553839\pi$
$194$	0	0
$195$	42.8194	3.06636
$196$	0	0
$197$	1.39240	0.0992045	0.0496022	−	0.998769i	$-0.484205\pi$
$197$	1.39240	0.0992045	0.0496022	+	0.998769i	$0.484205\pi$
$198$	0	0
$199$	14.6974	1.04187	0.520936	−	0.853596i	$-0.325583\pi$
$199$	14.6974	1.04187	0.520936	+	0.853596i	$0.325583\pi$
$200$	0	0
$201$	−3.41128	−0.240613
$202$	0	0
$203$	−1.33154	−0.0934559
$204$	0	0
$205$	29.3619	2.05072
$206$	0	0
$207$	2.76124	0.191920
$208$	0	0
$209$	4.55700	0.315214
$210$	0	0
$211$	−13.7405	−0.945936	−0.472968	−	0.881080i	$-0.656817\pi$
$211$	−13.7405	−0.945936	−0.472968	+	0.881080i	$0.656817\pi$
$212$	0	0
$213$	−21.1505	−1.44921
$214$	0	0
$215$	−46.2162	−3.15192
$216$	0	0
$217$	1.76721	0.119966
$218$	0	0
$219$	10.8627	0.734032
$220$	0	0
$221$	−11.5987	−0.780210
$222$	0	0
$223$	28.2043	1.88870	0.944351	−	0.328938i	$-0.106691\pi$
$223$	28.2043	1.88870	0.944351	+	0.328938i	$0.106691\pi$
$224$	0	0
$225$	3.58853	0.239235
$226$	0	0
$227$	−8.47713	−0.562647	−0.281323	−	0.959613i	$-0.590773\pi$
$227$	−8.47713	−0.562647	−0.281323	+	0.959613i	$0.590773\pi$
$228$	0	0
$229$	−16.4038	−1.08399	−0.541995	−	0.840382i	$-0.682331\pi$
$229$	−16.4038	−1.08399	−0.541995	+	0.840382i	$0.682331\pi$
$230$	0	0
$231$	0.815707	0.0536696
$232$	0	0
$233$	13.7347	0.899791	0.449895	−	0.893081i	$-0.351461\pi$
$233$	13.7347	0.899791	0.449895	+	0.893081i	$0.351461\pi$
$234$	0	0
$235$	3.91513	0.255395
$236$	0	0
$237$	13.5929	0.882953
$238$	0	0
$239$	−18.8880	−1.22176	−0.610880	−	0.791723i	$-0.709184\pi$
$239$	−18.8880	−1.22176	−0.610880	+	0.791723i	$0.709184\pi$
$240$	0	0
$241$	23.4719	1.51196	0.755980	−	0.654594i	$-0.227161\pi$
$241$	23.4719	1.51196	0.755980	+	0.654594i	$0.227161\pi$
$242$	0	0
$243$	3.24311	0.208045
$244$	0	0
$245$	27.4081	1.75104
$246$	0	0
$247$	−29.0772	−1.85013
$248$	0	0
$249$	−10.7289	−0.679919
$250$	0	0
$251$	5.12153	0.323268	0.161634	−	0.986851i	$-0.448324\pi$
$251$	5.12153	0.323268	0.161634	+	0.986851i	$0.448324\pi$
$252$	0	0
$253$	−8.01012	−0.503592
$254$	0	0
$255$	−14.7670	−0.924746
$256$	0	0
$257$	15.0897	0.941267	0.470634	−	0.882329i	$-0.344025\pi$
$257$	15.0897	0.941267	0.470634	+	0.882329i	$0.344025\pi$
$258$	0	0
$259$	−0.493058	−0.0306371
$260$	0	0
$261$	−0.846110	−0.0523729
$262$	0	0
$263$	−23.0485	−1.42123	−0.710616	−	0.703580i	$-0.751583\pi$
$263$	−23.0485	−1.42123	−0.710616	+	0.703580i	$0.751583\pi$
$264$	0	0
$265$	15.1265	0.929215
$266$	0	0
$267$	0.808861	0.0495015
$268$	0	0
$269$	3.10374	0.189238	0.0946192	−	0.995514i	$-0.469837\pi$
$269$	3.10374	0.189238	0.0946192	+	0.995514i	$0.469837\pi$
$270$	0	0
$271$	−29.6257	−1.79964	−0.899818	−	0.436266i	$-0.856301\pi$
$271$	−29.6257	−1.79964	−0.899818	+	0.436266i	$0.856301\pi$
$272$	0	0
$273$	−5.20483	−0.315011
$274$	0	0
$275$	−10.4100	−0.627747
$276$	0	0
$277$	−7.84474	−0.471345	−0.235672	−	0.971833i	$-0.575729\pi$
$277$	−7.84474	−0.471345	−0.235672	+	0.971833i	$0.575729\pi$
$278$	0	0
$279$	1.12295	0.0672293
$280$	0	0
$281$	0.719003	0.0428921	0.0214461	−	0.999770i	$-0.493173\pi$
$281$	0.719003	0.0428921	0.0214461	+	0.999770i	$0.493173\pi$
$282$	0	0
$283$	2.26212	0.134469	0.0672346	−	0.997737i	$-0.478582\pi$
$283$	2.26212	0.134469	0.0672346	+	0.997737i	$0.478582\pi$
$284$	0	0
$285$	−37.0201	−2.19288
$286$	0	0
$287$	−3.56903	−0.210673
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.42159	0.376440
$292$	0	0
$293$	7.62642	0.445540	0.222770	−	0.974871i	$-0.428490\pi$
$293$	7.62642	0.445540	0.222770	+	0.974871i	$0.428490\pi$
$294$	0	0
$295$	41.0201	2.38828
$296$	0	0
$297$	−4.44482	−0.257914
$298$	0	0
$299$	51.1107	2.95581
$300$	0	0
$301$	5.61772	0.323800
$302$	0	0
$303$	13.0828	0.751588
$304$	0	0
$305$	−37.9877	−2.17517
$306$	0	0
$307$	−16.4328	−0.937869	−0.468934	−	0.883233i	$-0.655362\pi$
$307$	−16.4328	−0.937869	−0.468934	+	0.883233i	$0.655362\pi$
$308$	0	0
$309$	19.9824	1.13676
$310$	0	0
$311$	13.2157	0.749396	0.374698	−	0.927147i	$-0.377746\pi$
$311$	13.2157	0.749396	0.374698	+	0.927147i	$0.377746\pi$
$312$	0	0
$313$	−22.1606	−1.25259	−0.626297	−	0.779585i	$-0.715430\pi$
$313$	−22.1606	−1.25259	−0.626297	+	0.779585i	$0.715430\pi$
$314$	0	0
$315$	−0.626615	−0.0353057
$316$	0	0
$317$	−5.79498	−0.325478	−0.162739	−	0.986669i	$-0.552033\pi$
$317$	−5.79498	−0.325478	−0.162739	+	0.986669i	$0.552033\pi$
$318$	0	0
$319$	2.45449	0.137425
$320$	0	0
$321$	−18.0422	−1.00702
$322$	0	0
$323$	10.0278	0.557960
$324$	0	0
$325$	66.4238	3.68453
$326$	0	0
$327$	26.0708	1.44172
$328$	0	0
$329$	−0.475897	−0.0262370
$330$	0	0
$331$	17.6314	0.969110	0.484555	−	0.874761i	$-0.338982\pi$
$331$	17.6314	0.969110	0.484555	+	0.874761i	$0.338982\pi$
$332$	0	0
$333$	−0.313307	−0.0171691
$334$	0	0
$335$	−7.60183	−0.415332
$336$	0	0
$337$	6.97658	0.380039	0.190019	−	0.981780i	$-0.439145\pi$
$337$	6.97658	0.380039	0.190019	+	0.981780i	$0.439145\pi$
$338$	0	0
$339$	27.6885	1.50383
$340$	0	0
$341$	−3.25758	−0.176408
$342$	0	0
$343$	−6.78295	−0.366245
$344$	0	0
$345$	65.0724	3.50338
$346$	0	0
$347$	−33.4276	−1.79449	−0.897243	−	0.441537i	$-0.854433\pi$
$347$	−33.4276	−1.79449	−0.897243	+	0.441537i	$0.854433\pi$
$348$	0	0
$349$	−20.8797	−1.11766	−0.558831	−	0.829282i	$-0.688750\pi$
$349$	−20.8797	−1.11766	−0.558831	+	0.829282i	$0.688750\pi$
$350$	0	0
$351$	28.3613	1.51382
$352$	0	0
$353$	5.23416	0.278586	0.139293	−	0.990251i	$-0.455517\pi$
$353$	5.23416	0.278586	0.139293	+	0.990251i	$0.455517\pi$
$354$	0	0
$355$	−47.1327	−2.50154
$356$	0	0
$357$	1.79498	0.0950003
$358$	0	0
$359$	28.6194	1.51047	0.755237	−	0.655452i	$-0.227522\pi$
$359$	28.6194	1.51047	0.755237	+	0.655452i	$0.227522\pi$
$360$	0	0
$361$	6.13903	0.323107
$362$	0	0
$363$	18.5191	0.972001
$364$	0	0
$365$	24.2068	1.26704
$366$	0	0
$367$	9.87984	0.515724	0.257862	−	0.966182i	$-0.416982\pi$
$367$	9.87984	0.515724	0.257862	+	0.966182i	$0.416982\pi$
$368$	0	0
$369$	−2.26790	−0.118062
$370$	0	0
$371$	−1.83868	−0.0954593
$372$	0	0
$373$	26.2063	1.35691	0.678454	−	0.734643i	$-0.262650\pi$
$373$	26.2063	1.35691	0.678454	+	0.734643i	$0.262650\pi$
$374$	0	0
$375$	47.6510	2.46069
$376$	0	0
$377$	−15.6615	−0.806610
$378$	0	0
$379$	−27.3493	−1.40484	−0.702419	−	0.711763i	$-0.747897\pi$
$379$	−27.3493	−1.40484	−0.702419	+	0.711763i	$0.747897\pi$
$380$	0	0
$381$	−4.20625	−0.215493
$382$	0	0
$383$	−7.46577	−0.381483	−0.190742	−	0.981640i	$-0.561089\pi$
$383$	−7.46577	−0.381483	−0.190742	+	0.981640i	$0.561089\pi$
$384$	0	0
$385$	1.81775	0.0926413
$386$	0	0
$387$	3.56971	0.181458
$388$	0	0
$389$	−1.70694	−0.0865452	−0.0432726	−	0.999063i	$-0.513778\pi$
$389$	−1.70694	−0.0865452	−0.0432726	+	0.999063i	$0.513778\pi$
$390$	0	0
$391$	−17.6264	−0.891406
$392$	0	0
$393$	33.8795	1.70899
$394$	0	0
$395$	30.2909	1.52410
$396$	0	0
$397$	−30.7960	−1.54561	−0.772804	−	0.634645i	$-0.781146\pi$
$397$	−30.7960	−1.54561	−0.772804	+	0.634645i	$0.781146\pi$
$398$	0	0
$399$	4.49990	0.225277
$400$	0	0
$401$	11.9076	0.594638	0.297319	−	0.954778i	$-0.403908\pi$
$401$	11.9076	0.594638	0.297319	+	0.954778i	$0.403908\pi$
$402$	0	0
$403$	20.7858	1.03542
$404$	0	0
$405$	39.9213	1.98371
$406$	0	0
$407$	0.908877	0.0450514
$408$	0	0
$409$	−12.6050	−0.623278	−0.311639	−	0.950201i	$-0.600878\pi$
$409$	−12.6050	−0.623278	−0.311639	+	0.950201i	$0.600878\pi$
$410$	0	0
$411$	12.0516	0.594461
$412$	0	0
$413$	−4.98612	−0.245351
$414$	0	0
$415$	−23.9088	−1.17364
$416$	0	0
$417$	−5.92961	−0.290374
$418$	0	0
$419$	6.85691	0.334982	0.167491	−	0.985874i	$-0.446433\pi$
$419$	6.85691	0.334982	0.167491	+	0.985874i	$0.446433\pi$
$420$	0	0
$421$	17.4473	0.850332	0.425166	−	0.905115i	$-0.360216\pi$
$421$	17.4473	0.850332	0.425166	+	0.905115i	$0.360216\pi$
$422$	0	0
$423$	−0.302402	−0.0147033
$424$	0	0
$425$	−22.9074	−1.11117
$426$	0	0
$427$	4.61753	0.223458
$428$	0	0
$429$	9.59430	0.463217
$430$	0	0
$431$	−2.19750	−0.105850	−0.0529250	−	0.998598i	$-0.516854\pi$
$431$	−2.19750	−0.105850	−0.0529250	+	0.998598i	$0.516854\pi$
$432$	0	0
$433$	−14.2399	−0.684328	−0.342164	−	0.939640i	$-0.611160\pi$
$433$	−14.2399	−0.684328	−0.342164	+	0.939640i	$0.611160\pi$
$434$	0	0
$435$	−19.9397	−0.956037
$436$	0	0
$437$	−44.1884	−2.11382
$438$	0	0
$439$	24.6778	1.17781	0.588904	−	0.808203i	$-0.299560\pi$
$439$	24.6778	1.17781	0.588904	+	0.808203i	$0.299560\pi$
$440$	0	0
$441$	−2.11698	−0.100809
$442$	0	0
$443$	11.4709	0.544998	0.272499	−	0.962156i	$-0.412150\pi$
$443$	11.4709	0.544998	0.272499	+	0.962156i	$0.412150\pi$
$444$	0	0
$445$	1.80250	0.0854466
$446$	0	0
$447$	−31.2182	−1.47657
$448$	0	0
$449$	−36.2112	−1.70891	−0.854456	−	0.519524i	$-0.826109\pi$
$449$	−36.2112	−1.70891	−0.854456	+	0.519524i	$0.826109\pi$
$450$	0	0
$451$	6.57896	0.309791
$452$	0	0
$453$	−18.4581	−0.867235
$454$	0	0
$455$	−11.5987	−0.543753
$456$	0	0
$457$	13.3593	0.624922	0.312461	−	0.949931i	$-0.398847\pi$
$457$	13.3593	0.624922	0.312461	+	0.949931i	$0.398847\pi$
$458$	0	0
$459$	−9.78090	−0.456533
$460$	0	0
$461$	20.6100	0.959904	0.479952	−	0.877295i	$-0.340654\pi$
$461$	20.6100	0.959904	0.479952	+	0.877295i	$0.340654\pi$
$462$	0	0
$463$	12.0462	0.559834	0.279917	−	0.960024i	$-0.409693\pi$
$463$	12.0462	0.559834	0.279917	+	0.960024i	$0.409693\pi$
$464$	0	0
$465$	26.4638	1.22723
$466$	0	0
$467$	2.64939	0.122599	0.0612996	−	0.998119i	$-0.480475\pi$
$467$	2.64939	0.122599	0.0612996	+	0.998119i	$0.480475\pi$
$468$	0	0
$469$	0.924026	0.0426675
$470$	0	0
$471$	7.11059	0.327639
$472$	0	0
$473$	−10.3554	−0.476142
$474$	0	0
$475$	−57.4276	−2.63496
$476$	0	0
$477$	−1.16836	−0.0534956
$478$	0	0
$479$	−32.4412	−1.48228	−0.741138	−	0.671353i	$-0.765714\pi$
$479$	−32.4412	−1.48228	−0.741138	+	0.671353i	$0.765714\pi$
$480$	0	0
$481$	−5.79933	−0.264427
$482$	0	0
$483$	−7.90976	−0.359906
$484$	0	0
$485$	14.3101	0.649790
$486$	0	0
$487$	2.60384	0.117991	0.0589956	−	0.998258i	$-0.481210\pi$
$487$	2.60384	0.117991	0.0589956	+	0.998258i	$0.481210\pi$
$488$	0	0
$489$	−8.64031	−0.390728
$490$	0	0
$491$	8.15630	0.368089	0.184044	−	0.982918i	$-0.441081\pi$
$491$	8.15630	0.368089	0.184044	+	0.982918i	$0.441081\pi$
$492$	0	0
$493$	5.40115	0.243256
$494$	0	0
$495$	1.15507	0.0519164
$496$	0	0
$497$	5.72913	0.256986
$498$	0	0
$499$	30.3859	1.36026	0.680130	−	0.733091i	$-0.261923\pi$
$499$	30.3859	1.36026	0.680130	+	0.733091i	$0.261923\pi$
$500$	0	0
$501$	24.4823	1.09379
$502$	0	0
$503$	−19.5904	−0.873491	−0.436745	−	0.899585i	$-0.643869\pi$
$503$	−19.5904	−0.873491	−0.436745	+	0.899585i	$0.643869\pi$
$504$	0	0
$505$	29.1543	1.29735
$506$	0	0
$507$	−37.5557	−1.66791
$508$	0	0
$509$	−32.0115	−1.41888	−0.709442	−	0.704764i	$-0.751053\pi$
$509$	−32.0115	−1.41888	−0.709442	+	0.704764i	$0.751053\pi$
$510$	0	0
$511$	−2.94242	−0.130165
$512$	0	0
$513$	−24.5201	−1.08259
$514$	0	0
$515$	44.5295	1.96220
$516$	0	0
$517$	0.877242	0.0385811
$518$	0	0
$519$	23.7322	1.04173
$520$	0	0
$521$	−43.8252	−1.92001	−0.960007	−	0.279975i	$-0.909674\pi$
$521$	−43.8252	−1.92001	−0.960007	+	0.279975i	$0.909674\pi$
$522$	0	0
$523$	−24.1404	−1.05559	−0.527793	−	0.849373i	$-0.676980\pi$
$523$	−24.1404	−1.05559	−0.527793	+	0.849373i	$0.676980\pi$
$524$	0	0
$525$	−10.2796	−0.448637
$526$	0	0
$527$	−7.16836	−0.312259
$528$	0	0
$529$	54.6727	2.37707
$530$	0	0
$531$	−3.16836	−0.137495
$532$	0	0
$533$	−41.9788	−1.81830
$534$	0	0
$535$	−40.2061	−1.73826
$536$	0	0
$537$	−11.6770	−0.503898
$538$	0	0
$539$	6.14118	0.264520
$540$	0	0
$541$	−40.5471	−1.74326	−0.871629	−	0.490167i	$-0.836936\pi$
$541$	−40.5471	−1.74326	−0.871629	+	0.490167i	$0.836936\pi$
$542$	0	0
$543$	−36.2714	−1.55656
$544$	0	0
$545$	58.0972	2.48861
$546$	0	0
$547$	−23.0291	−0.984655	−0.492327	−	0.870410i	$-0.663854\pi$
$547$	−23.0291	−0.984655	−0.492327	+	0.870410i	$0.663854\pi$
$548$	0	0
$549$	2.93415	0.125226
$550$	0	0
$551$	13.5404	0.576839
$552$	0	0
$553$	−3.68196	−0.156573
$554$	0	0
$555$	−7.38351	−0.313412
$556$	0	0
$557$	−46.0181	−1.94985	−0.974925	−	0.222534i	$-0.928567\pi$
$557$	−46.0181	−1.94985	−0.974925	+	0.222534i	$0.928567\pi$
$558$	0	0
$559$	66.0754	2.79469
$560$	0	0
$561$	−3.30876	−0.139696
$562$	0	0
$563$	16.7392	0.705475	0.352738	−	0.935722i	$-0.385251\pi$
$563$	16.7392	0.705475	0.352738	+	0.935722i	$0.385251\pi$
$564$	0	0
$565$	61.7022	2.59583
$566$	0	0
$567$	−4.85256	−0.203788
$568$	0	0
$569$	32.1379	1.34729	0.673645	−	0.739055i	$-0.264728\pi$
$569$	32.1379	1.34729	0.673645	+	0.739055i	$0.264728\pi$
$570$	0	0
$571$	−8.25435	−0.345434	−0.172717	−	0.984971i	$-0.555255\pi$
$571$	−8.25435	−0.345434	−0.172717	+	0.984971i	$0.555255\pi$
$572$	0	0
$573$	40.1758	1.67837
$574$	0	0
$575$	100.944	4.20965
$576$	0	0
$577$	−18.2556	−0.759989	−0.379995	−	0.924989i	$-0.624074\pi$
$577$	−18.2556	−0.759989	−0.379995	+	0.924989i	$0.624074\pi$
$578$	0	0
$579$	8.51359	0.353813
$580$	0	0
$581$	2.90619	0.120569
$582$	0	0
$583$	3.38932	0.140371
$584$	0	0
$585$	−7.37021	−0.304721
$586$	0	0
$587$	−6.63161	−0.273716	−0.136858	−	0.990591i	$-0.543700\pi$
$587$	−6.63161	−0.273716	−0.136858	+	0.990591i	$0.543700\pi$
$588$	0	0
$589$	−17.9707	−0.740468
$590$	0	0
$591$	−2.53452	−0.104256
$592$	0	0
$593$	26.3686	1.08283	0.541415	−	0.840755i	$-0.317889\pi$
$593$	26.3686	1.08283	0.541415	+	0.840755i	$0.317889\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	−26.7529	−1.09492
$598$	0	0
$599$	0.832313	0.0340074	0.0170037	−	0.999855i	$-0.494587\pi$
$599$	0.832313	0.0340074	0.0170037	+	0.999855i	$0.494587\pi$
$600$	0	0
$601$	12.1259	0.494627	0.247313	−	0.968936i	$-0.420452\pi$
$601$	12.1259	0.494627	0.247313	+	0.968936i	$0.420452\pi$
$602$	0	0
$603$	0.587160	0.0239110
$604$	0	0
$605$	41.2687	1.67781
$606$	0	0
$607$	−24.9168	−1.01134	−0.505670	−	0.862727i	$-0.668755\pi$
$607$	−24.9168	−1.01134	−0.505670	+	0.862727i	$0.668755\pi$
$608$	0	0
$609$	2.42374	0.0982148
$610$	0	0
$611$	−5.59748	−0.226450
$612$	0	0
$613$	0.596053	0.0240743	0.0120372	−	0.999928i	$-0.496168\pi$
$613$	0.596053	0.0240743	0.0120372	+	0.999928i	$0.496168\pi$
$614$	0	0
$615$	−53.4460	−2.15515
$616$	0	0
$617$	22.1182	0.890446	0.445223	−	0.895420i	$-0.353124\pi$
$617$	22.1182	0.890446	0.445223	+	0.895420i	$0.353124\pi$
$618$	0	0
$619$	−45.2757	−1.81979	−0.909893	−	0.414844i	$-0.863836\pi$
$619$	−45.2757	−1.81979	−0.909893	+	0.414844i	$0.863836\pi$
$620$	0	0
$621$	43.1006	1.72957
$622$	0	0
$623$	−0.219099	−0.00877803
$624$	0	0
$625$	48.9189	1.95676
$626$	0	0
$627$	−8.29488	−0.331266
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	40.1422	1.59804	0.799018	−	0.601307i	$-0.205353\pi$
$631$	40.1422	1.59804	0.799018	+	0.601307i	$0.205353\pi$
$632$	0	0
$633$	25.0112	0.994104
$634$	0	0
$635$	−9.37339	−0.371971
$636$	0	0
$637$	−39.1854	−1.55258
$638$	0	0
$639$	3.64050	0.144016
$640$	0	0
$641$	2.43469	0.0961646	0.0480823	−	0.998843i	$-0.484689\pi$
$641$	2.43469	0.0961646	0.0480823	+	0.998843i	$0.484689\pi$
$642$	0	0
$643$	32.4631	1.28022	0.640109	−	0.768284i	$-0.278889\pi$
$643$	32.4631	1.28022	0.640109	+	0.768284i	$0.278889\pi$
$644$	0	0
$645$	84.1250	3.31242
$646$	0	0
$647$	35.6373	1.40105	0.700524	−	0.713629i	$-0.252950\pi$
$647$	35.6373	1.40105	0.700524	+	0.713629i	$0.252950\pi$
$648$	0	0
$649$	9.19114	0.360784
$650$	0	0
$651$	−3.21676	−0.126075
$652$	0	0
$653$	28.2677	1.10620	0.553099	−	0.833115i	$-0.313445\pi$
$653$	28.2677	1.10620	0.553099	+	0.833115i	$0.313445\pi$
$654$	0	0
$655$	75.4984	2.94996
$656$	0	0
$657$	−1.86972	−0.0729448
$658$	0	0
$659$	0.553329	0.0215546	0.0107773	−	0.999942i	$-0.496569\pi$
$659$	0.553329	0.0215546	0.0107773	+	0.999942i	$0.496569\pi$
$660$	0	0
$661$	19.5893	0.761936	0.380968	−	0.924588i	$-0.375591\pi$
$661$	19.5893	0.761936	0.380968	+	0.924588i	$0.375591\pi$
$662$	0	0
$663$	21.1124	0.819939
$664$	0	0
$665$	10.0278	0.388860
$666$	0	0
$667$	−23.8008	−0.921569
$668$	0	0
$669$	−51.3389	−1.98488
$670$	0	0
$671$	−8.51171	−0.328591
$672$	0	0
$673$	−29.5471	−1.13896	−0.569479	−	0.822006i	$-0.692855\pi$
$673$	−29.5471	−1.13896	−0.569479	+	0.822006i	$0.692855\pi$
$674$	0	0
$675$	56.0138	2.15597
$676$	0	0
$677$	−9.28975	−0.357034	−0.178517	−	0.983937i	$-0.557130\pi$
$677$	−9.28975	−0.357034	−0.178517	+	0.983937i	$0.557130\pi$
$678$	0	0
$679$	−1.73944	−0.0667537
$680$	0	0
$681$	15.4305	0.591297
$682$	0	0
$683$	−1.58957	−0.0608232	−0.0304116	−	0.999537i	$-0.509682\pi$
$683$	−1.58957	−0.0608232	−0.0304116	+	0.999537i	$0.509682\pi$
$684$	0	0
$685$	26.8562	1.02612
$686$	0	0
$687$	29.8589	1.13919
$688$	0	0
$689$	−21.6264	−0.823901
$690$	0	0
$691$	−49.1795	−1.87088	−0.935439	−	0.353489i	$-0.884995\pi$
$691$	−49.1795	−1.87088	−0.935439	+	0.353489i	$0.884995\pi$
$692$	0	0
$693$	−0.140402	−0.00533344
$694$	0	0
$695$	−13.2138	−0.501227
$696$	0	0
$697$	14.4771	0.548360
$698$	0	0
$699$	−25.0006	−0.945609
$700$	0	0
$701$	13.9159	0.525597	0.262798	−	0.964851i	$-0.415355\pi$
$701$	13.9159	0.525597	0.262798	+	0.964851i	$0.415355\pi$
$702$	0	0
$703$	5.01388	0.189102
$704$	0	0
$705$	−7.12652	−0.268400
$706$	0	0
$707$	−3.54379	−0.133278
$708$	0	0
$709$	−39.1167	−1.46906	−0.734528	−	0.678578i	$-0.762597\pi$
$709$	−39.1167	−1.46906	−0.734528	+	0.678578i	$0.762597\pi$
$710$	0	0
$711$	−2.33965	−0.0877438
$712$	0	0
$713$	31.5881	1.18299
$714$	0	0
$715$	21.3803	0.799579
$716$	0	0
$717$	34.3808	1.28397
$718$	0	0
$719$	−48.6801	−1.81546	−0.907731	−	0.419552i	$-0.862187\pi$
$719$	−48.6801	−1.81546	−0.907731	+	0.419552i	$0.862187\pi$
$720$	0	0
$721$	−5.41270	−0.201579
$722$	0	0
$723$	−42.7248	−1.58895
$724$	0	0
$725$	−30.9316	−1.14877
$726$	0	0
$727$	3.78421	0.140349	0.0701743	−	0.997535i	$-0.477644\pi$
$727$	3.78421	0.140349	0.0701743	+	0.997535i	$0.477644\pi$
$728$	0	0
$729$	23.6220	0.874890
$730$	0	0
$731$	−22.7873	−0.842817
$732$	0	0
$733$	−26.3847	−0.974541	−0.487270	−	0.873251i	$-0.662007\pi$
$733$	−26.3847	−0.974541	−0.487270	+	0.873251i	$0.662007\pi$
$734$	0	0
$735$	−49.8896	−1.84020
$736$	0	0
$737$	−1.70330	−0.0627418
$738$	0	0
$739$	−45.5997	−1.67741	−0.838707	−	0.544584i	$-0.816688\pi$
$739$	−45.5997	−1.67741	−0.838707	+	0.544584i	$0.816688\pi$
$740$	0	0
$741$	52.9277	1.94435
$742$	0	0
$743$	−37.6979	−1.38300	−0.691501	−	0.722376i	$-0.743050\pi$
$743$	−37.6979	−1.38300	−0.691501	+	0.722376i	$0.743050\pi$
$744$	0	0
$745$	−69.5680	−2.54877
$746$	0	0
$747$	1.84670	0.0675672
$748$	0	0
$749$	4.88717	0.178573
$750$	0	0
$751$	3.56404	0.130054	0.0650269	−	0.997884i	$-0.479287\pi$
$751$	3.56404	0.130054	0.0650269	+	0.997884i	$0.479287\pi$
$752$	0	0
$753$	−9.32246	−0.339729
$754$	0	0
$755$	−41.1327	−1.49697
$756$	0	0
$757$	42.6648	1.55068	0.775339	−	0.631546i	$-0.217579\pi$
$757$	42.6648	1.55068	0.775339	+	0.631546i	$0.217579\pi$
$758$	0	0
$759$	14.5804	0.529236
$760$	0	0
$761$	−24.6877	−0.894930	−0.447465	−	0.894302i	$-0.647673\pi$
$761$	−24.6877	−0.894930	−0.447465	+	0.894302i	$0.647673\pi$
$762$	0	0
$763$	−7.06190	−0.255658
$764$	0	0
$765$	2.54175	0.0918971
$766$	0	0
$767$	−58.6465	−2.11760
$768$	0	0
$769$	12.3404	0.445007	0.222504	−	0.974932i	$-0.428577\pi$
$769$	12.3404	0.445007	0.222504	+	0.974932i	$0.428577\pi$
$770$	0	0
$771$	−27.4670	−0.989198
$772$	0	0
$773$	−32.3922	−1.16507	−0.582533	−	0.812807i	$-0.697938\pi$
$773$	−32.3922	−1.16507	−0.582533	+	0.812807i	$0.697938\pi$
$774$	0	0
$775$	41.0522	1.47464
$776$	0	0
$777$	0.897489	0.0321972
$778$	0	0
$779$	36.2933	1.30034
$780$	0	0
$781$	−10.5608	−0.377894
$782$	0	0
$783$	−13.2070	−0.471981
$784$	0	0
$785$	15.8455	0.565551
$786$	0	0
$787$	−48.9623	−1.74532	−0.872658	−	0.488331i	$-0.837606\pi$
$787$	−48.9623	−1.74532	−0.872658	+	0.488331i	$0.837606\pi$
$788$	0	0
$789$	41.9540	1.49360
$790$	0	0
$791$	−7.50010	−0.266673
$792$	0	0
$793$	54.3112	1.92865
$794$	0	0
$795$	−27.5340	−0.976532
$796$	0	0
$797$	46.7891	1.65735	0.828677	−	0.559727i	$-0.189094\pi$
$797$	46.7891	1.65735	0.828677	+	0.559727i	$0.189094\pi$
$798$	0	0
$799$	1.93039	0.0682922
$800$	0	0
$801$	−0.139224	−0.00491923
$802$	0	0
$803$	5.42389	0.191405
$804$	0	0
$805$	−17.6264	−0.621250
$806$	0	0
$807$	−5.64958	−0.198875
$808$	0	0
$809$	47.2085	1.65976	0.829881	−	0.557941i	$-0.188408\pi$
$809$	47.2085	1.65976	0.829881	+	0.557941i	$0.188408\pi$
$810$	0	0
$811$	−25.0199	−0.878569	−0.439285	−	0.898348i	$-0.644768\pi$
$811$	−25.0199	−0.878569	−0.439285	+	0.898348i	$0.644768\pi$
$812$	0	0
$813$	53.9262	1.89128
$814$	0	0
$815$	−19.2544	−0.674452
$816$	0	0
$817$	−57.1263	−1.99860
$818$	0	0
$819$	0.895873	0.0313043
$820$	0	0
$821$	−6.59009	−0.229996	−0.114998	−	0.993366i	$-0.536686\pi$
$821$	−6.59009	−0.229996	−0.114998	+	0.993366i	$0.536686\pi$
$822$	0	0
$823$	−29.6633	−1.03400	−0.516998	−	0.855986i	$-0.672951\pi$
$823$	−29.6633	−1.03400	−0.516998	+	0.855986i	$0.672951\pi$
$824$	0	0
$825$	18.9488	0.659713
$826$	0	0
$827$	−8.39999	−0.292096	−0.146048	−	0.989277i	$-0.546655\pi$
$827$	−8.39999	−0.292096	−0.146048	+	0.989277i	$0.546655\pi$
$828$	0	0
$829$	6.52332	0.226564	0.113282	−	0.993563i	$-0.463864\pi$
$829$	6.52332	0.226564	0.113282	+	0.993563i	$0.463864\pi$
$830$	0	0
$831$	14.2794	0.495346
$832$	0	0
$833$	13.5138	0.468225
$834$	0	0
$835$	54.5573	1.88803
$836$	0	0
$837$	17.5283	0.605865
$838$	0	0
$839$	29.9556	1.03418	0.517092	−	0.855930i	$-0.327015\pi$
$839$	29.9556	1.03418	0.517092	+	0.855930i	$0.327015\pi$
$840$	0	0
$841$	−21.7069	−0.748513
$842$	0	0
$843$	−1.30876	−0.0450762
$844$	0	0
$845$	−83.6907	−2.87905
$846$	0	0
$847$	−5.01634	−0.172364
$848$	0	0
$849$	−4.11763	−0.141317
$850$	0	0
$851$	−8.81321	−0.302113
$852$	0	0
$853$	3.72458	0.127527	0.0637637	−	0.997965i	$-0.479690\pi$
$853$	3.72458	0.127527	0.0637637	+	0.997965i	$0.479690\pi$
$854$	0	0
$855$	6.37201	0.217918
$856$	0	0
$857$	−13.8758	−0.473989	−0.236994	−	0.971511i	$-0.576162\pi$
$857$	−13.8758	−0.473989	−0.236994	+	0.971511i	$0.576162\pi$
$858$	0	0
$859$	34.4567	1.17565	0.587824	−	0.808989i	$-0.299985\pi$
$859$	34.4567	1.17565	0.587824	+	0.808989i	$0.299985\pi$
$860$	0	0
$861$	6.49653	0.221401
$862$	0	0
$863$	32.3668	1.10178	0.550890	−	0.834578i	$-0.314288\pi$
$863$	32.3668	1.10178	0.550890	+	0.834578i	$0.314288\pi$
$864$	0	0
$865$	52.8858	1.79817
$866$	0	0
$867$	23.6632	0.803646
$868$	0	0
$869$	6.78712	0.230237
$870$	0	0
$871$	10.8683	0.368260
$872$	0	0
$873$	−1.10531	−0.0374089
$874$	0	0
$875$	−12.9074	−0.436350
$876$	0	0
$877$	−38.0305	−1.28420	−0.642100	−	0.766621i	$-0.721936\pi$
$877$	−38.0305	−1.28420	−0.642100	+	0.766621i	$0.721936\pi$
$878$	0	0
$879$	−13.8820	−0.468228
$880$	0	0
$881$	−37.9308	−1.27792	−0.638960	−	0.769240i	$-0.720635\pi$
$881$	−37.9308	−1.27792	−0.638960	+	0.769240i	$0.720635\pi$
$882$	0	0
$883$	55.3009	1.86102	0.930511	−	0.366264i	$-0.119363\pi$
$883$	55.3009	1.86102	0.930511	+	0.366264i	$0.119363\pi$
$884$	0	0
$885$	−74.6667	−2.50989
$886$	0	0
$887$	26.0118	0.873392	0.436696	−	0.899609i	$-0.356149\pi$
$887$	26.0118	0.873392	0.436696	+	0.899609i	$0.356149\pi$
$888$	0	0
$889$	1.13936	0.0382131
$890$	0	0
$891$	8.94495	0.299667
$892$	0	0
$893$	4.83937	0.161943
$894$	0	0
$895$	−26.0214	−0.869800
$896$	0	0
$897$	−93.0342	−3.10632
$898$	0	0
$899$	−9.67936	−0.322825
$900$	0	0
$901$	7.45825	0.248470
$902$	0	0
$903$	−10.2257	−0.340288
$904$	0	0
$905$	−80.8287	−2.68684
$906$	0	0
$907$	−30.6353	−1.01723	−0.508614	−	0.860994i	$-0.669842\pi$
$907$	−30.6353	−1.01723	−0.508614	+	0.860994i	$0.669842\pi$
$908$	0	0
$909$	−2.25186	−0.0746894
$910$	0	0
$911$	−9.02796	−0.299110	−0.149555	−	0.988753i	$-0.547784\pi$
$911$	−9.02796	−0.299110	−0.149555	+	0.988753i	$0.547784\pi$
$912$	0	0
$913$	−5.35711	−0.177295
$914$	0	0
$915$	69.1472	2.28593
$916$	0	0
$917$	−9.17706	−0.303053
$918$	0	0
$919$	−9.18704	−0.303053	−0.151526	−	0.988453i	$-0.548419\pi$
$919$	−9.18704	−0.303053	−0.151526	+	0.988453i	$0.548419\pi$
$920$	0	0
$921$	29.9118	0.985626
$922$	0	0
$923$	67.3857	2.21803
$924$	0	0
$925$	−11.4537	−0.376596
$926$	0	0
$927$	−3.43943	−0.112966
$928$	0	0
$929$	−37.6500	−1.23526	−0.617628	−	0.786470i	$-0.711906\pi$
$929$	−37.6500	−1.23526	−0.617628	+	0.786470i	$0.711906\pi$
$930$	0	0
$931$	33.8783	1.11032
$932$	0	0
$933$	−24.0559	−0.787556
$934$	0	0
$935$	−7.37339	−0.241135
$936$	0	0
$937$	29.6145	0.967463	0.483732	−	0.875216i	$-0.339281\pi$
$937$	29.6145	0.967463	0.483732	+	0.875216i	$0.339281\pi$
$938$	0	0
$939$	40.3379	1.31638
$940$	0	0
$941$	−0.709918	−0.0231427	−0.0115713	−	0.999933i	$-0.503683\pi$
$941$	−0.709918	−0.0231427	−0.0115713	+	0.999933i	$0.503683\pi$
$942$	0	0
$943$	−63.7950	−2.07745
$944$	0	0
$945$	−9.78090	−0.318173
$946$	0	0
$947$	0.274635	0.00892443	0.00446221	−	0.999990i	$-0.498580\pi$
$947$	0.274635	0.00892443	0.00446221	+	0.999990i	$0.498580\pi$
$948$	0	0
$949$	−34.6086	−1.12344
$950$	0	0
$951$	10.5483	0.342052
$952$	0	0
$953$	−38.7902	−1.25654	−0.628270	−	0.777996i	$-0.716237\pi$
$953$	−38.7902	−1.25654	−0.628270	+	0.777996i	$0.716237\pi$
$954$	0	0
$955$	89.5293	2.89710
$956$	0	0
$957$	−4.46779	−0.144423
$958$	0	0
$959$	−3.26446	−0.105415
$960$	0	0
$961$	−18.1536	−0.585601
$962$	0	0
$963$	3.10549	0.100073
$964$	0	0
$965$	18.9720	0.610732
$966$	0	0
$967$	−17.9742	−0.578011	−0.289006	−	0.957327i	$-0.593325\pi$
$967$	−17.9742	−0.578011	−0.289006	+	0.957327i	$0.593325\pi$
$968$	0	0
$969$	−18.2530	−0.586372
$970$	0	0
$971$	−14.1830	−0.455155	−0.227578	−	0.973760i	$-0.573081\pi$
$971$	−14.1830	−0.455155	−0.227578	+	0.973760i	$0.573081\pi$
$972$	0	0
$973$	1.60618	0.0514916
$974$	0	0
$975$	−120.908	−3.87215
$976$	0	0
$977$	−14.6833	−0.469761	−0.234881	−	0.972024i	$-0.575470\pi$
$977$	−14.6833	−0.469761	−0.234881	+	0.972024i	$0.575470\pi$
$978$	0	0
$979$	0.403876	0.0129079
$980$	0	0
$981$	−4.48739	−0.143271
$982$	0	0
$983$	−8.93996	−0.285140	−0.142570	−	0.989785i	$-0.545537\pi$
$983$	−8.93996	−0.285140	−0.142570	+	0.989785i	$0.545537\pi$
$984$	0	0
$985$	−5.64802	−0.179961
$986$	0	0
$987$	0.866251	0.0275731
$988$	0	0
$989$	100.414	3.19299
$990$	0	0
$991$	27.8751	0.885482	0.442741	−	0.896650i	$-0.354006\pi$
$991$	27.8751	0.885482	0.442741	+	0.896650i	$0.354006\pi$
$992$	0	0
$993$	−32.0936	−1.01846
$994$	0	0
$995$	−59.6173	−1.89000
$996$	0	0
$997$	−48.1557	−1.52510	−0.762552	−	0.646926i	$-0.776054\pi$
$997$	−48.1557	−1.52510	−0.762552	+	0.646926i	$0.776054\pi$
$998$	0	0
$999$	−4.89045	−0.154727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bf.1.2		4
4.3	odd	2		2368.2.a.bi.1.3		4
8.3	odd	2		1184.2.a.n.1.2	✓	4
8.5	even	2		1184.2.a.o.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.n.1.2	✓	4	8.3	odd	2
1184.2.a.o.1.3	yes	4	8.5	even	2
2368.2.a.bf.1.2		4	1.1	even	1	trivial
2368.2.a.bi.1.3		4	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82025 q^{3} -4.05632 q^{5} +0.493058 q^{7} +0.313307 q^{9} -0.908877 q^{11} +5.79933 q^{13} +7.38351 q^{15} -2.00000 q^{17} -5.01388 q^{19} -0.897489 q^{21} +8.81321 q^{23} +11.4537 q^{25} +4.89045 q^{27} -2.70058 q^{29} +3.58418 q^{31} +1.65438 q^{33} -2.00000 q^{35} -1.00000 q^{37} -10.5562 q^{39} -7.23856 q^{41} +11.3936 q^{43} -1.27087 q^{45} -0.965194 q^{47} -6.75689 q^{49} +3.64050 q^{51} -3.72913 q^{53} +3.68669 q^{55} +9.12652 q^{57} -10.1126 q^{59} +9.36508 q^{61} +0.154479 q^{63} -23.5239 q^{65} +1.87407 q^{67} -16.0422 q^{69} +11.6196 q^{71} -5.96769 q^{73} -20.8486 q^{75} -0.448129 q^{77} -7.46759 q^{79} -9.84176 q^{81} +5.89421 q^{83} +8.11263 q^{85} +4.91572 q^{87} -0.444368 q^{89} +2.85941 q^{91} -6.52410 q^{93} +20.3379 q^{95} -3.52786 q^{97} -0.284758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + q^{9} - 3 q^{11} - 3 q^{13} + 8 q^{15} - 8 q^{17} - 12 q^{19} - 8 q^{21} + q^{23} + 3 q^{25} - 6 q^{27} - 3 q^{29} + 19 q^{31} - 10 q^{33} - 8 q^{35} - 4 q^{37} + 5 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + q^9 - 3 * q^11 - 3 * q^13 + 8 * q^15 - 8 * q^17 - 12 * q^19 - 8 * q^21 + q^23 + 3 * q^25 - 6 * q^27 - 3 * q^29 + 19 * q^31 - 10 * q^33 - 8 * q^35 - 4 * q^37 + 5 * q^39 - 17 * q^41 + 2 * q^43 - 10 * q^45 + 10 * q^47 - 6 * q^49 + 6 * q^51 - 10 * q^53 + 15 * q^55 + 2 * q^57 - 14 * q^59 - 9 * q^61 + 18 * q^63 - 30 * q^65 - 7 * q^67 + 9 * q^69 + 16 * q^71 - 7 * q^73 - 14 * q^75 - 14 * q^77 + 21 * q^79 - 8 * q^81 + 12 * q^83 + 6 * q^85 - 19 * q^87 - 14 * q^91 - 30 * q^93 + 2 * q^95 - 32 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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