LMFDB - Embedded newform 2368.2.a.y.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{3} -3.85410 q^{5} -3.23607 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} -3.85410 q^{5} -3.23607 q^{7} -0.381966 q^{9} +1.38197 q^{11} +2.85410 q^{13} -6.23607 q^{15} -4.47214 q^{17} -4.47214 q^{19} -5.23607 q^{21} +2.85410 q^{23} +9.85410 q^{25} -5.47214 q^{27} +9.32624 q^{29} +7.38197 q^{31} +2.23607 q^{33} +12.4721 q^{35} +1.00000 q^{37} +4.61803 q^{39} +9.61803 q^{41} +5.23607 q^{43} +1.47214 q^{45} -1.23607 q^{47} +3.47214 q^{49} -7.23607 q^{51} -0.472136 q^{53} -5.32624 q^{55} -7.23607 q^{57} +4.76393 q^{59} -10.6180 q^{61} +1.23607 q^{63} -11.0000 q^{65} -1.09017 q^{67} +4.61803 q^{69} +2.94427 q^{71} +7.09017 q^{73} +15.9443 q^{75} -4.47214 q^{77} -8.56231 q^{79} -7.70820 q^{81} +14.4721 q^{83} +17.2361 q^{85} +15.0902 q^{87} -1.52786 q^{89} -9.23607 q^{91} +11.9443 q^{93} +17.2361 q^{95} -0.472136 q^{97} -0.527864 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9	$ 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9} + 5 q^{11} - q^{13} - 8 q^{15} - 6 q^{21} - q^{23} + 13 q^{25} - 2 q^{27} + 3 q^{29} + 17 q^{31} + 16 q^{35} + 2 q^{37} + 7 q^{39} + 17 q^{41} + 6 q^{43} - 6 q^{45} + 2 q^{47} - 2 q^{49} - 10 q^{51} + 8 q^{53} + 5 q^{55} - 10 q^{57} + 14 q^{59} - 19 q^{61} - 2 q^{63} - 22 q^{65} + 9 q^{67} + 7 q^{69} - 12 q^{71} + 3 q^{73} + 14 q^{75} + 3 q^{79} - 2 q^{81} + 20 q^{83} + 30 q^{85} + 19 q^{87} - 12 q^{89} - 14 q^{91} + 6 q^{93} + 30 q^{95} + 8 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9 + 5 * q^11 - q^13 - 8 * q^15 - 6 * q^21 - q^23 + 13 * q^25 - 2 * q^27 + 3 * q^29 + 17 * q^31 + 16 * q^35 + 2 * q^37 + 7 * q^39 + 17 * q^41 + 6 * q^43 - 6 * q^45 + 2 * q^47 - 2 * q^49 - 10 * q^51 + 8 * q^53 + 5 * q^55 - 10 * q^57 + 14 * q^59 - 19 * q^61 - 2 * q^63 - 22 * q^65 + 9 * q^67 + 7 * q^69 - 12 * q^71 + 3 * q^73 + 14 * q^75 + 3 * q^79 - 2 * q^81 + 20 * q^83 + 30 * q^85 + 19 * q^87 - 12 * q^89 - 14 * q^91 + 6 * q^93 + 30 * q^95 + 8 * q^97 - 10 * q^99

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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	2.85410	0.791585	0.395793	−	0.918340i	$-0.370470\pi$
$13$	2.85410	0.791585	0.395793	+	0.918340i	$0.370470\pi$
$14$	0	0
$15$	−6.23607	−1.61015
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	−5.23607	−1.14260
$22$	0	0
$23$	2.85410	0.595121	0.297561	−	0.954703i	$-0.403827\pi$
$23$	2.85410	0.595121	0.297561	+	0.954703i	$0.403827\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	9.32624	1.73184	0.865919	−	0.500183i	$-0.166734\pi$
$29$	9.32624	1.73184	0.865919	+	0.500183i	$0.166734\pi$
$30$	0	0
$31$	7.38197	1.32584	0.662920	−	0.748690i	$-0.269317\pi$
$31$	7.38197	1.32584	0.662920	+	0.748690i	$0.269317\pi$
$32$	0	0
$33$	2.23607	0.389249
$34$	0	0
$35$	12.4721	2.10818
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	4.61803	0.739477
$40$	0	0
$41$	9.61803	1.50208	0.751042	−	0.660254i	$-0.229551\pi$
$41$	9.61803	1.50208	0.751042	+	0.660254i	$0.229551\pi$
$42$	0	0
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	0	0
$45$	1.47214	0.219453
$46$	0	0
$47$	−1.23607	−0.180299	−0.0901495	−	0.995928i	$-0.528734\pi$
$47$	−1.23607	−0.180299	−0.0901495	+	0.995928i	$0.528734\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	−7.23607	−1.01325
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	−5.32624	−0.718190
$56$	0	0
$57$	−7.23607	−0.958441
$58$	0	0
$59$	4.76393	0.620211	0.310106	−	0.950702i	$-0.399636\pi$
$59$	4.76393	0.620211	0.310106	+	0.950702i	$0.399636\pi$
$60$	0	0
$61$	−10.6180	−1.35950	−0.679750	−	0.733444i	$-0.737912\pi$
$61$	−10.6180	−1.35950	−0.679750	+	0.733444i	$0.737912\pi$
$62$	0	0
$63$	1.23607	0.155730
$64$	0	0
$65$	−11.0000	−1.36438
$66$	0	0
$67$	−1.09017	−0.133185	−0.0665927	−	0.997780i	$-0.521213\pi$
$67$	−1.09017	−0.133185	−0.0665927	+	0.997780i	$0.521213\pi$
$68$	0	0
$69$	4.61803	0.555946
$70$	0	0
$71$	2.94427	0.349421	0.174710	−	0.984620i	$-0.444101\pi$
$71$	2.94427	0.349421	0.174710	+	0.984620i	$0.444101\pi$
$72$	0	0
$73$	7.09017	0.829842	0.414921	−	0.909858i	$-0.363809\pi$
$73$	7.09017	0.829842	0.414921	+	0.909858i	$0.363809\pi$
$74$	0	0
$75$	15.9443	1.84109
$76$	0	0
$77$	−4.47214	−0.509647
$78$	0	0
$79$	−8.56231	−0.963335	−0.481667	−	0.876354i	$-0.659969\pi$
$79$	−8.56231	−0.963335	−0.481667	+	0.876354i	$0.659969\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	14.4721	1.58852	0.794262	−	0.607576i	$-0.207858\pi$
$83$	14.4721	1.58852	0.794262	+	0.607576i	$0.207858\pi$
$84$	0	0
$85$	17.2361	1.86951
$86$	0	0
$87$	15.0902	1.61784
$88$	0	0
$89$	−1.52786	−0.161953	−0.0809766	−	0.996716i	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	0	0
$91$	−9.23607	−0.968203
$92$	0	0
$93$	11.9443	1.23856
$94$	0	0
$95$	17.2361	1.76838
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	0	0
$99$	−0.527864	−0.0530523
$100$	0	0
$101$	−3.52786	−0.351036	−0.175518	−	0.984476i	$-0.556160\pi$
$101$	−3.52786	−0.351036	−0.175518	+	0.984476i	$0.556160\pi$
$102$	0	0
$103$	17.2705	1.70171	0.850857	−	0.525397i	$-0.176083\pi$
$103$	17.2705	1.70171	0.850857	+	0.525397i	$0.176083\pi$
$104$	0	0
$105$	20.1803	1.96940
$106$	0	0
$107$	7.32624	0.708254	0.354127	−	0.935197i	$-0.384778\pi$
$107$	7.32624	0.708254	0.354127	+	0.935197i	$0.384778\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	1.61803	0.153577
$112$	0	0
$113$	−6.94427	−0.653262	−0.326631	−	0.945152i	$-0.605913\pi$
$113$	−6.94427	−0.653262	−0.326631	+	0.945152i	$0.605913\pi$
$114$	0	0
$115$	−11.0000	−1.02576
$116$	0	0
$117$	−1.09017	−0.100786
$118$	0	0
$119$	14.4721	1.32666
$120$	0	0
$121$	−9.09017	−0.826379
$122$	0	0
$123$	15.5623	1.40321
$124$	0	0
$125$	−18.7082	−1.67331
$126$	0	0
$127$	−8.47214	−0.751780	−0.375890	−	0.926664i	$-0.622663\pi$
$127$	−8.47214	−0.751780	−0.375890	+	0.926664i	$0.622663\pi$
$128$	0	0
$129$	8.47214	0.745930
$130$	0	0
$131$	22.6525	1.97916	0.989578	−	0.143998i	$-0.0459958\pi$
$131$	22.6525	1.97916	0.989578	+	0.143998i	$0.0459958\pi$
$132$	0	0
$133$	14.4721	1.25489
$134$	0	0
$135$	21.0902	1.81515
$136$	0	0
$137$	3.67376	0.313871	0.156935	−	0.987609i	$-0.449839\pi$
$137$	3.67376	0.313871	0.156935	+	0.987609i	$0.449839\pi$
$138$	0	0
$139$	−4.85410	−0.411720	−0.205860	−	0.978581i	$-0.565999\pi$
$139$	−4.85410	−0.411720	−0.205860	+	0.978581i	$0.565999\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	3.94427	0.329837
$144$	0	0
$145$	−35.9443	−2.98501
$146$	0	0
$147$	5.61803	0.463368
$148$	0	0
$149$	−16.1803	−1.32555	−0.662773	−	0.748821i	$-0.730620\pi$
$149$	−16.1803	−1.32555	−0.662773	+	0.748821i	$0.730620\pi$
$150$	0	0
$151$	4.29180	0.349261	0.174631	−	0.984634i	$-0.444127\pi$
$151$	4.29180	0.349261	0.174631	+	0.984634i	$0.444127\pi$
$152$	0	0
$153$	1.70820	0.138100
$154$	0	0
$155$	−28.4508	−2.28523
$156$	0	0
$157$	16.4721	1.31462	0.657310	−	0.753620i	$-0.271694\pi$
$157$	16.4721	1.31462	0.657310	+	0.753620i	$0.271694\pi$
$158$	0	0
$159$	−0.763932	−0.0605838
$160$	0	0
$161$	−9.23607	−0.727904
$162$	0	0
$163$	3.52786	0.276324	0.138162	−	0.990410i	$-0.455881\pi$
$163$	3.52786	0.276324	0.138162	+	0.990410i	$0.455881\pi$
$164$	0	0
$165$	−8.61803	−0.670913
$166$	0	0
$167$	13.8541	1.07206	0.536031	−	0.844198i	$-0.319923\pi$
$167$	13.8541	1.07206	0.536031	+	0.844198i	$0.319923\pi$
$168$	0	0
$169$	−4.85410	−0.373392
$170$	0	0
$171$	1.70820	0.130630
$172$	0	0
$173$	0.472136	0.0358958	0.0179479	−	0.999839i	$-0.494287\pi$
$173$	0.472136	0.0358958	0.0179479	+	0.999839i	$0.494287\pi$
$174$	0	0
$175$	−31.8885	−2.41055
$176$	0	0
$177$	7.70820	0.579384
$178$	0	0
$179$	12.6525	0.945690	0.472845	−	0.881146i	$-0.343227\pi$
$179$	12.6525	0.945690	0.472845	+	0.881146i	$0.343227\pi$
$180$	0	0
$181$	−14.4721	−1.07571	−0.537853	−	0.843039i	$-0.680764\pi$
$181$	−14.4721	−1.07571	−0.537853	+	0.843039i	$0.680764\pi$
$182$	0	0
$183$	−17.1803	−1.27001
$184$	0	0
$185$	−3.85410	−0.283359
$186$	0	0
$187$	−6.18034	−0.451951
$188$	0	0
$189$	17.7082	1.28808
$190$	0	0
$191$	−7.09017	−0.513027	−0.256513	−	0.966541i	$-0.582574\pi$
$191$	−7.09017	−0.513027	−0.256513	+	0.966541i	$0.582574\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	−17.7984	−1.27457
$196$	0	0
$197$	7.52786	0.536338	0.268169	−	0.963372i	$-0.413581\pi$
$197$	7.52786	0.536338	0.268169	+	0.963372i	$0.413581\pi$
$198$	0	0
$199$	3.05573	0.216615	0.108307	−	0.994117i	$-0.465457\pi$
$199$	3.05573	0.216615	0.108307	+	0.994117i	$0.465457\pi$
$200$	0	0
$201$	−1.76393	−0.124418
$202$	0	0
$203$	−30.1803	−2.11824
$204$	0	0
$205$	−37.0689	−2.58900
$206$	0	0
$207$	−1.09017	−0.0757720
$208$	0	0
$209$	−6.18034	−0.427503
$210$	0	0
$211$	−11.2705	−0.775894	−0.387947	−	0.921682i	$-0.626816\pi$
$211$	−11.2705	−0.775894	−0.387947	+	0.921682i	$0.626816\pi$
$212$	0	0
$213$	4.76393	0.326419
$214$	0	0
$215$	−20.1803	−1.37629
$216$	0	0
$217$	−23.8885	−1.62166
$218$	0	0
$219$	11.4721	0.775215
$220$	0	0
$221$	−12.7639	−0.858595
$222$	0	0
$223$	14.1803	0.949586	0.474793	−	0.880098i	$-0.342523\pi$
$223$	14.1803	0.949586	0.474793	+	0.880098i	$0.342523\pi$
$224$	0	0
$225$	−3.76393	−0.250929
$226$	0	0
$227$	4.29180	0.284857	0.142428	−	0.989805i	$-0.454509\pi$
$227$	4.29180	0.284857	0.142428	+	0.989805i	$0.454509\pi$
$228$	0	0
$229$	23.1246	1.52812	0.764059	−	0.645147i	$-0.223204\pi$
$229$	23.1246	1.52812	0.764059	+	0.645147i	$0.223204\pi$
$230$	0	0
$231$	−7.23607	−0.476098
$232$	0	0
$233$	−6.56231	−0.429911	−0.214955	−	0.976624i	$-0.568961\pi$
$233$	−6.56231	−0.429911	−0.214955	+	0.976624i	$0.568961\pi$
$234$	0	0
$235$	4.76393	0.310765
$236$	0	0
$237$	−13.8541	−0.899921
$238$	0	0
$239$	9.85410	0.637409	0.318704	−	0.947854i	$-0.396752\pi$
$239$	9.85410	0.637409	0.318704	+	0.947854i	$0.396752\pi$
$240$	0	0
$241$	−1.52786	−0.0984184	−0.0492092	−	0.998788i	$-0.515670\pi$
$241$	−1.52786	−0.0984184	−0.0492092	+	0.998788i	$0.515670\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	−13.3820	−0.854942
$246$	0	0
$247$	−12.7639	−0.812150
$248$	0	0
$249$	23.4164	1.48395
$250$	0	0
$251$	20.9443	1.32199	0.660995	−	0.750390i	$-0.270134\pi$
$251$	20.9443	1.32199	0.660995	+	0.750390i	$0.270134\pi$
$252$	0	0
$253$	3.94427	0.247974
$254$	0	0
$255$	27.8885	1.74645
$256$	0	0
$257$	−1.05573	−0.0658545	−0.0329273	−	0.999458i	$-0.510483\pi$
$257$	−1.05573	−0.0658545	−0.0329273	+	0.999458i	$0.510483\pi$
$258$	0	0
$259$	−3.23607	−0.201079
$260$	0	0
$261$	−3.56231	−0.220501
$262$	0	0
$263$	−13.2361	−0.816171	−0.408085	−	0.912944i	$-0.633803\pi$
$263$	−13.2361	−0.816171	−0.408085	+	0.912944i	$0.633803\pi$
$264$	0	0
$265$	1.81966	0.111781
$266$	0	0
$267$	−2.47214	−0.151292
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	12.9443	0.786309	0.393154	−	0.919473i	$-0.371384\pi$
$271$	12.9443	0.786309	0.393154	+	0.919473i	$0.371384\pi$
$272$	0	0
$273$	−14.9443	−0.904468
$274$	0	0
$275$	13.6180	0.821198
$276$	0	0
$277$	−16.7984	−1.00932	−0.504658	−	0.863319i	$-0.668381\pi$
$277$	−16.7984	−1.00932	−0.504658	+	0.863319i	$0.668381\pi$
$278$	0	0
$279$	−2.81966	−0.168809
$280$	0	0
$281$	29.8885	1.78300	0.891501	−	0.453020i	$-0.149653\pi$
$281$	29.8885	1.78300	0.891501	+	0.453020i	$0.149653\pi$
$282$	0	0
$283$	−6.76393	−0.402074	−0.201037	−	0.979584i	$-0.564431\pi$
$283$	−6.76393	−0.402074	−0.201037	+	0.979584i	$0.564431\pi$
$284$	0	0
$285$	27.8885	1.65197
$286$	0	0
$287$	−31.1246	−1.83723
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−0.763932	−0.0447825
$292$	0	0
$293$	12.6525	0.739166	0.369583	−	0.929198i	$-0.379501\pi$
$293$	12.6525	0.739166	0.369583	+	0.929198i	$0.379501\pi$
$294$	0	0
$295$	−18.3607	−1.06900
$296$	0	0
$297$	−7.56231	−0.438809
$298$	0	0
$299$	8.14590	0.471089
$300$	0	0
$301$	−16.9443	−0.976652
$302$	0	0
$303$	−5.70820	−0.327928
$304$	0	0
$305$	40.9230	2.34324
$306$	0	0
$307$	12.8541	0.733622	0.366811	−	0.930295i	$-0.380450\pi$
$307$	12.8541	0.733622	0.366811	+	0.930295i	$0.380450\pi$
$308$	0	0
$309$	27.9443	1.58969
$310$	0	0
$311$	−27.0344	−1.53298	−0.766491	−	0.642255i	$-0.777999\pi$
$311$	−27.0344	−1.53298	−0.766491	+	0.642255i	$0.777999\pi$
$312$	0	0
$313$	28.1803	1.59285	0.796423	−	0.604739i	$-0.206723\pi$
$313$	28.1803	1.59285	0.796423	+	0.604739i	$0.206723\pi$
$314$	0	0
$315$	−4.76393	−0.268417
$316$	0	0
$317$	−20.9443	−1.17635	−0.588174	−	0.808735i	$-0.700153\pi$
$317$	−20.9443	−1.17635	−0.588174	+	0.808735i	$0.700153\pi$
$318$	0	0
$319$	12.8885	0.721620
$320$	0	0
$321$	11.8541	0.661631
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	28.1246	1.56007
$326$	0	0
$327$	−4.76393	−0.263446
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−0.381966	−0.0209316
$334$	0	0
$335$	4.20163	0.229559
$336$	0	0
$337$	12.0344	0.655558	0.327779	−	0.944754i	$-0.393700\pi$
$337$	12.0344	0.655558	0.327779	+	0.944754i	$0.393700\pi$
$338$	0	0
$339$	−11.2361	−0.610259
$340$	0	0
$341$	10.2016	0.552449
$342$	0	0
$343$	11.4164	0.616428
$344$	0	0
$345$	−17.7984	−0.958232
$346$	0	0
$347$	−17.2361	−0.925281	−0.462640	−	0.886546i	$-0.653098\pi$
$347$	−17.2361	−0.925281	−0.462640	+	0.886546i	$0.653098\pi$
$348$	0	0
$349$	10.1803	0.544941	0.272471	−	0.962164i	$-0.412159\pi$
$349$	10.1803	0.544941	0.272471	+	0.962164i	$0.412159\pi$
$350$	0	0
$351$	−15.6180	−0.833629
$352$	0	0
$353$	16.2918	0.867125	0.433562	−	0.901124i	$-0.357256\pi$
$353$	16.2918	0.867125	0.433562	+	0.901124i	$0.357256\pi$
$354$	0	0
$355$	−11.3475	−0.602264
$356$	0	0
$357$	23.4164	1.23933
$358$	0	0
$359$	−4.47214	−0.236030	−0.118015	−	0.993012i	$-0.537653\pi$
$359$	−4.47214	−0.236030	−0.118015	+	0.993012i	$0.537653\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−14.7082	−0.771980
$364$	0	0
$365$	−27.3262	−1.43032
$366$	0	0
$367$	13.1246	0.685099	0.342550	−	0.939500i	$-0.388710\pi$
$367$	13.1246	0.685099	0.342550	+	0.939500i	$0.388710\pi$
$368$	0	0
$369$	−3.67376	−0.191248
$370$	0	0
$371$	1.52786	0.0793227
$372$	0	0
$373$	−27.7082	−1.43468	−0.717338	−	0.696725i	$-0.754640\pi$
$373$	−27.7082	−1.43468	−0.717338	+	0.696725i	$0.754640\pi$
$374$	0	0
$375$	−30.2705	−1.56316
$376$	0	0
$377$	26.6180	1.37090
$378$	0	0
$379$	−28.0902	−1.44290	−0.721448	−	0.692469i	$-0.756523\pi$
$379$	−28.0902	−1.44290	−0.721448	+	0.692469i	$0.756523\pi$
$380$	0	0
$381$	−13.7082	−0.702293
$382$	0	0
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0	0
$385$	17.2361	0.878431
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	6.85410	0.347517	0.173758	−	0.984788i	$-0.444409\pi$
$389$	6.85410	0.347517	0.173758	+	0.984788i	$0.444409\pi$
$390$	0	0
$391$	−12.7639	−0.645500
$392$	0	0
$393$	36.6525	1.84887
$394$	0	0
$395$	33.0000	1.66041
$396$	0	0
$397$	−20.6525	−1.03652	−0.518259	−	0.855224i	$-0.673420\pi$
$397$	−20.6525	−1.03652	−0.518259	+	0.855224i	$0.673420\pi$
$398$	0	0
$399$	23.4164	1.17229
$400$	0	0
$401$	−4.76393	−0.237899	−0.118950	−	0.992900i	$-0.537953\pi$
$401$	−4.76393	−0.237899	−0.118950	+	0.992900i	$0.537953\pi$
$402$	0	0
$403$	21.0689	1.04952
$404$	0	0
$405$	29.7082	1.47621
$406$	0	0
$407$	1.38197	0.0685015
$408$	0	0
$409$	−26.1803	−1.29453	−0.647267	−	0.762263i	$-0.724088\pi$
$409$	−26.1803	−1.29453	−0.647267	+	0.762263i	$0.724088\pi$
$410$	0	0
$411$	5.94427	0.293209
$412$	0	0
$413$	−15.4164	−0.758592
$414$	0	0
$415$	−55.7771	−2.73799
$416$	0	0
$417$	−7.85410	−0.384617
$418$	0	0
$419$	10.5623	0.516002	0.258001	−	0.966145i	$-0.416936\pi$
$419$	10.5623	0.516002	0.258001	+	0.966145i	$0.416936\pi$
$420$	0	0
$421$	31.0344	1.51253	0.756263	−	0.654268i	$-0.227023\pi$
$421$	31.0344	1.51253	0.756263	+	0.654268i	$0.227023\pi$
$422$	0	0
$423$	0.472136	0.0229560
$424$	0	0
$425$	−44.0689	−2.13765
$426$	0	0
$427$	34.3607	1.66283
$428$	0	0
$429$	6.38197	0.308124
$430$	0	0
$431$	12.3607	0.595393	0.297696	−	0.954661i	$-0.403782\pi$
$431$	12.3607	0.595393	0.297696	+	0.954661i	$0.403782\pi$
$432$	0	0
$433$	−20.6738	−0.993518	−0.496759	−	0.867889i	$-0.665477\pi$
$433$	−20.6738	−0.993518	−0.496759	+	0.867889i	$0.665477\pi$
$434$	0	0
$435$	−58.1591	−2.78851
$436$	0	0
$437$	−12.7639	−0.610582
$438$	0	0
$439$	7.79837	0.372196	0.186098	−	0.982531i	$-0.440416\pi$
$439$	7.79837	0.372196	0.186098	+	0.982531i	$0.440416\pi$
$440$	0	0
$441$	−1.32624	−0.0631542
$442$	0	0
$443$	−15.2705	−0.725524	−0.362762	−	0.931882i	$-0.618166\pi$
$443$	−15.2705	−0.725524	−0.362762	+	0.931882i	$0.618166\pi$
$444$	0	0
$445$	5.88854	0.279144
$446$	0	0
$447$	−26.1803	−1.23829
$448$	0	0
$449$	−28.4721	−1.34368	−0.671842	−	0.740695i	$-0.734496\pi$
$449$	−28.4721	−1.34368	−0.671842	+	0.740695i	$0.734496\pi$
$450$	0	0
$451$	13.2918	0.625886
$452$	0	0
$453$	6.94427	0.326270
$454$	0	0
$455$	35.5967	1.66880
$456$	0	0
$457$	−24.7639	−1.15841	−0.579204	−	0.815183i	$-0.696637\pi$
$457$	−24.7639	−1.15841	−0.579204	+	0.815183i	$0.696637\pi$
$458$	0	0
$459$	24.4721	1.14226
$460$	0	0
$461$	38.9443	1.81382	0.906908	−	0.421329i	$-0.138436\pi$
$461$	38.9443	1.81382	0.906908	+	0.421329i	$0.138436\pi$
$462$	0	0
$463$	−4.56231	−0.212028	−0.106014	−	0.994365i	$-0.533809\pi$
$463$	−4.56231	−0.212028	−0.106014	+	0.994365i	$0.533809\pi$
$464$	0	0
$465$	−46.0344	−2.13480
$466$	0	0
$467$	24.3607	1.12728	0.563639	−	0.826021i	$-0.309401\pi$
$467$	24.3607	1.12728	0.563639	+	0.826021i	$0.309401\pi$
$468$	0	0
$469$	3.52786	0.162902
$470$	0	0
$471$	26.6525	1.22808
$472$	0	0
$473$	7.23607	0.332715
$474$	0	0
$475$	−44.0689	−2.02202
$476$	0	0
$477$	0.180340	0.00825720
$478$	0	0
$479$	36.5623	1.67057	0.835287	−	0.549814i	$-0.185301\pi$
$479$	36.5623	1.67057	0.835287	+	0.549814i	$0.185301\pi$
$480$	0	0
$481$	2.85410	0.130136
$482$	0	0
$483$	−14.9443	−0.679988
$484$	0	0
$485$	1.81966	0.0826265
$486$	0	0
$487$	37.3050	1.69045	0.845224	−	0.534412i	$-0.179467\pi$
$487$	37.3050	1.69045	0.845224	+	0.534412i	$0.179467\pi$
$488$	0	0
$489$	5.70820	0.258134
$490$	0	0
$491$	28.4508	1.28397	0.641984	−	0.766718i	$-0.278111\pi$
$491$	28.4508	1.28397	0.641984	+	0.766718i	$0.278111\pi$
$492$	0	0
$493$	−41.7082	−1.87844
$494$	0	0
$495$	2.03444	0.0914414
$496$	0	0
$497$	−9.52786	−0.427383
$498$	0	0
$499$	−10.2918	−0.460724	−0.230362	−	0.973105i	$-0.573991\pi$
$499$	−10.2918	−0.460724	−0.230362	+	0.973105i	$0.573991\pi$
$500$	0	0
$501$	22.4164	1.00149
$502$	0	0
$503$	19.0902	0.851189	0.425594	−	0.904914i	$-0.360065\pi$
$503$	19.0902	0.851189	0.425594	+	0.904914i	$0.360065\pi$
$504$	0	0
$505$	13.5967	0.605047
$506$	0	0
$507$	−7.85410	−0.348813
$508$	0	0
$509$	17.7082	0.784902	0.392451	−	0.919773i	$-0.371627\pi$
$509$	17.7082	0.784902	0.392451	+	0.919773i	$0.371627\pi$
$510$	0	0
$511$	−22.9443	−1.01499
$512$	0	0
$513$	24.4721	1.08047
$514$	0	0
$515$	−66.5623	−2.93309
$516$	0	0
$517$	−1.70820	−0.0751267
$518$	0	0
$519$	0.763932	0.0335329
$520$	0	0
$521$	−1.41641	−0.0620540	−0.0310270	−	0.999519i	$-0.509878\pi$
$521$	−1.41641	−0.0620540	−0.0310270	+	0.999519i	$0.509878\pi$
$522$	0	0
$523$	−11.8197	−0.516838	−0.258419	−	0.966033i	$-0.583201\pi$
$523$	−11.8197	−0.516838	−0.258419	+	0.966033i	$0.583201\pi$
$524$	0	0
$525$	−51.5967	−2.25187
$526$	0	0
$527$	−33.0132	−1.43808
$528$	0	0
$529$	−14.8541	−0.645831
$530$	0	0
$531$	−1.81966	−0.0789665
$532$	0	0
$533$	27.4508	1.18903
$534$	0	0
$535$	−28.2361	−1.22075
$536$	0	0
$537$	20.4721	0.883438
$538$	0	0
$539$	4.79837	0.206681
$540$	0	0
$541$	10.3262	0.443960	0.221980	−	0.975051i	$-0.428748\pi$
$541$	10.3262	0.443960	0.221980	+	0.975051i	$0.428748\pi$
$542$	0	0
$543$	−23.4164	−1.00489
$544$	0	0
$545$	11.3475	0.486075
$546$	0	0
$547$	16.0689	0.687056	0.343528	−	0.939142i	$-0.388378\pi$
$547$	16.0689	0.687056	0.343528	+	0.939142i	$0.388378\pi$
$548$	0	0
$549$	4.05573	0.173094
$550$	0	0
$551$	−41.7082	−1.77683
$552$	0	0
$553$	27.7082	1.17827
$554$	0	0
$555$	−6.23607	−0.264706
$556$	0	0
$557$	−19.5623	−0.828882	−0.414441	−	0.910076i	$-0.636023\pi$
$557$	−19.5623	−0.828882	−0.414441	+	0.910076i	$0.636023\pi$
$558$	0	0
$559$	14.9443	0.632075
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	−7.88854	−0.332462	−0.166231	−	0.986087i	$-0.553160\pi$
$563$	−7.88854	−0.332462	−0.166231	+	0.986087i	$0.553160\pi$
$564$	0	0
$565$	26.7639	1.12597
$566$	0	0
$567$	24.9443	1.04756
$568$	0	0
$569$	13.8885	0.582238	0.291119	−	0.956687i	$-0.405972\pi$
$569$	13.8885	0.582238	0.291119	+	0.956687i	$0.405972\pi$
$570$	0	0
$571$	10.5623	0.442019	0.221009	−	0.975272i	$-0.429065\pi$
$571$	10.5623	0.442019	0.221009	+	0.975272i	$0.429065\pi$
$572$	0	0
$573$	−11.4721	−0.479255
$574$	0	0
$575$	28.1246	1.17288
$576$	0	0
$577$	10.6525	0.443468	0.221734	−	0.975107i	$-0.428828\pi$
$577$	10.6525	0.443468	0.221734	+	0.975107i	$0.428828\pi$
$578$	0	0
$579$	6.47214	0.268973
$580$	0	0
$581$	−46.8328	−1.94295
$582$	0	0
$583$	−0.652476	−0.0270228
$584$	0	0
$585$	4.20163	0.173716
$586$	0	0
$587$	13.0557	0.538868	0.269434	−	0.963019i	$-0.413163\pi$
$587$	13.0557	0.538868	0.269434	+	0.963019i	$0.413163\pi$
$588$	0	0
$589$	−33.0132	−1.36028
$590$	0	0
$591$	12.1803	0.501032
$592$	0	0
$593$	17.5623	0.721197	0.360599	−	0.932721i	$-0.382572\pi$
$593$	17.5623	0.721197	0.360599	+	0.932721i	$0.382572\pi$
$594$	0	0
$595$	−55.7771	−2.28664
$596$	0	0
$597$	4.94427	0.202356
$598$	0	0
$599$	−6.36068	−0.259890	−0.129945	−	0.991521i	$-0.541480\pi$
$599$	−6.36068	−0.259890	−0.129945	+	0.991521i	$0.541480\pi$
$600$	0	0
$601$	−24.6869	−1.00700	−0.503500	−	0.863995i	$-0.667955\pi$
$601$	−24.6869	−1.00700	−0.503500	+	0.863995i	$0.667955\pi$
$602$	0	0
$603$	0.416408	0.0169574
$604$	0	0
$605$	35.0344	1.42435
$606$	0	0
$607$	35.0344	1.42200	0.711002	−	0.703190i	$-0.248242\pi$
$607$	35.0344	1.42200	0.711002	+	0.703190i	$0.248242\pi$
$608$	0	0
$609$	−48.8328	−1.97881
$610$	0	0
$611$	−3.52786	−0.142722
$612$	0	0
$613$	13.8197	0.558171	0.279085	−	0.960266i	$-0.409969\pi$
$613$	13.8197	0.558171	0.279085	+	0.960266i	$0.409969\pi$
$614$	0	0
$615$	−59.9787	−2.41858
$616$	0	0
$617$	0.0901699	0.00363011	0.00181505	−	0.999998i	$-0.499422\pi$
$617$	0.0901699	0.00363011	0.00181505	+	0.999998i	$0.499422\pi$
$618$	0	0
$619$	15.2705	0.613774	0.306887	−	0.951746i	$-0.400713\pi$
$619$	15.2705	0.613774	0.306887	+	0.951746i	$0.400713\pi$
$620$	0	0
$621$	−15.6180	−0.626730
$622$	0	0
$623$	4.94427	0.198088
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	−10.0000	−0.399362
$628$	0	0
$629$	−4.47214	−0.178316
$630$	0	0
$631$	−47.3951	−1.88677	−0.943385	−	0.331700i	$-0.892378\pi$
$631$	−47.3951	−1.88677	−0.943385	+	0.331700i	$0.892378\pi$
$632$	0	0
$633$	−18.2361	−0.724819
$634$	0	0
$635$	32.6525	1.29577
$636$	0	0
$637$	9.90983	0.392642
$638$	0	0
$639$	−1.12461	−0.0444890
$640$	0	0
$641$	15.5066	0.612473	0.306236	−	0.951955i	$-0.400930\pi$
$641$	15.5066	0.612473	0.306236	+	0.951955i	$0.400930\pi$
$642$	0	0
$643$	28.7639	1.13434	0.567169	−	0.823601i	$-0.308038\pi$
$643$	28.7639	1.13434	0.567169	+	0.823601i	$0.308038\pi$
$644$	0	0
$645$	−32.6525	−1.28569
$646$	0	0
$647$	30.0902	1.18297	0.591483	−	0.806317i	$-0.298543\pi$
$647$	30.0902	1.18297	0.591483	+	0.806317i	$0.298543\pi$
$648$	0	0
$649$	6.58359	0.258429
$650$	0	0
$651$	−38.6525	−1.51491
$652$	0	0
$653$	38.2705	1.49764	0.748820	−	0.662773i	$-0.230621\pi$
$653$	38.2705	1.49764	0.748820	+	0.662773i	$0.230621\pi$
$654$	0	0
$655$	−87.3050	−3.41129
$656$	0	0
$657$	−2.70820	−0.105657
$658$	0	0
$659$	−40.4508	−1.57574	−0.787871	−	0.615841i	$-0.788816\pi$
$659$	−40.4508	−1.57574	−0.787871	+	0.615841i	$0.788816\pi$
$660$	0	0
$661$	−17.3262	−0.673913	−0.336956	−	0.941520i	$-0.609397\pi$
$661$	−17.3262	−0.673913	−0.336956	+	0.941520i	$0.609397\pi$
$662$	0	0
$663$	−20.6525	−0.802076
$664$	0	0
$665$	−55.7771	−2.16294
$666$	0	0
$667$	26.6180	1.03065
$668$	0	0
$669$	22.9443	0.887077
$670$	0	0
$671$	−14.6738	−0.566474
$672$	0	0
$673$	11.1459	0.429643	0.214821	−	0.976653i	$-0.431083\pi$
$673$	11.1459	0.429643	0.214821	+	0.976653i	$0.431083\pi$
$674$	0	0
$675$	−53.9230	−2.07550
$676$	0	0
$677$	−34.6525	−1.33180	−0.665901	−	0.746040i	$-0.731953\pi$
$677$	−34.6525	−1.33180	−0.665901	+	0.746040i	$0.731953\pi$
$678$	0	0
$679$	1.52786	0.0586340
$680$	0	0
$681$	6.94427	0.266105
$682$	0	0
$683$	8.58359	0.328442	0.164221	−	0.986424i	$-0.447489\pi$
$683$	8.58359	0.328442	0.164221	+	0.986424i	$0.447489\pi$
$684$	0	0
$685$	−14.1591	−0.540990
$686$	0	0
$687$	37.4164	1.42752
$688$	0	0
$689$	−1.34752	−0.0513366
$690$	0	0
$691$	−35.7771	−1.36102	−0.680512	−	0.732737i	$-0.738243\pi$
$691$	−35.7771	−1.36102	−0.680512	+	0.732737i	$0.738243\pi$
$692$	0	0
$693$	1.70820	0.0648893
$694$	0	0
$695$	18.7082	0.709643
$696$	0	0
$697$	−43.0132	−1.62924
$698$	0	0
$699$	−10.6180	−0.401611
$700$	0	0
$701$	42.9787	1.62328	0.811642	−	0.584155i	$-0.198574\pi$
$701$	42.9787	1.62328	0.811642	+	0.584155i	$0.198574\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	0	0
$705$	7.70820	0.290308
$706$	0	0
$707$	11.4164	0.429358
$708$	0	0
$709$	8.21478	0.308513	0.154256	−	0.988031i	$-0.450702\pi$
$709$	8.21478	0.308513	0.154256	+	0.988031i	$0.450702\pi$
$710$	0	0
$711$	3.27051	0.122654
$712$	0	0
$713$	21.0689	0.789036
$714$	0	0
$715$	−15.2016	−0.568509
$716$	0	0
$717$	15.9443	0.595450
$718$	0	0
$719$	27.4164	1.02246	0.511230	−	0.859444i	$-0.329190\pi$
$719$	27.4164	1.02246	0.511230	+	0.859444i	$0.329190\pi$
$720$	0	0
$721$	−55.8885	−2.08140
$722$	0	0
$723$	−2.47214	−0.0919397
$724$	0	0
$725$	91.9017	3.41314
$726$	0	0
$727$	−29.8541	−1.10723	−0.553614	−	0.832774i	$-0.686752\pi$
$727$	−29.8541	−1.10723	−0.553614	+	0.832774i	$0.686752\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−23.4164	−0.866087
$732$	0	0
$733$	−27.5279	−1.01676	−0.508382	−	0.861131i	$-0.669756\pi$
$733$	−27.5279	−1.01676	−0.508382	+	0.861131i	$0.669756\pi$
$734$	0	0
$735$	−21.6525	−0.798664
$736$	0	0
$737$	−1.50658	−0.0554955
$738$	0	0
$739$	10.9098	0.401325	0.200662	−	0.979660i	$-0.435691\pi$
$739$	10.9098	0.401325	0.200662	+	0.979660i	$0.435691\pi$
$740$	0	0
$741$	−20.6525	−0.758688
$742$	0	0
$743$	−48.0689	−1.76348	−0.881738	−	0.471739i	$-0.843626\pi$
$743$	−48.0689	−1.76348	−0.881738	+	0.471739i	$0.843626\pi$
$744$	0	0
$745$	62.3607	2.28472
$746$	0	0
$747$	−5.52786	−0.202254
$748$	0	0
$749$	−23.7082	−0.866279
$750$	0	0
$751$	−1.05573	−0.0385241	−0.0192620	−	0.999814i	$-0.506132\pi$
$751$	−1.05573	−0.0385241	−0.0192620	+	0.999814i	$0.506132\pi$
$752$	0	0
$753$	33.8885	1.23497
$754$	0	0
$755$	−16.5410	−0.601989
$756$	0	0
$757$	4.14590	0.150685	0.0753426	−	0.997158i	$-0.475995\pi$
$757$	4.14590	0.150685	0.0753426	+	0.997158i	$0.475995\pi$
$758$	0	0
$759$	6.38197	0.231651
$760$	0	0
$761$	−19.1459	−0.694038	−0.347019	−	0.937858i	$-0.612806\pi$
$761$	−19.1459	−0.694038	−0.347019	+	0.937858i	$0.612806\pi$
$762$	0	0
$763$	9.52786	0.344932
$764$	0	0
$765$	−6.58359	−0.238030
$766$	0	0
$767$	13.5967	0.490950
$768$	0	0
$769$	23.8885	0.861443	0.430721	−	0.902485i	$-0.358259\pi$
$769$	23.8885	0.861443	0.430721	+	0.902485i	$0.358259\pi$
$770$	0	0
$771$	−1.70820	−0.0615195
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	72.7426	2.61299
$776$	0	0
$777$	−5.23607	−0.187843
$778$	0	0
$779$	−43.0132	−1.54111
$780$	0	0
$781$	4.06888	0.145596
$782$	0	0
$783$	−51.0344	−1.82382
$784$	0	0
$785$	−63.4853	−2.26589
$786$	0	0
$787$	25.5279	0.909970	0.454985	−	0.890499i	$-0.349645\pi$
$787$	25.5279	0.909970	0.454985	+	0.890499i	$0.349645\pi$
$788$	0	0
$789$	−21.4164	−0.762444
$790$	0	0
$791$	22.4721	0.799017
$792$	0	0
$793$	−30.3050	−1.07616
$794$	0	0
$795$	2.94427	0.104423
$796$	0	0
$797$	−46.2705	−1.63899	−0.819493	−	0.573090i	$-0.805745\pi$
$797$	−46.2705	−1.63899	−0.819493	+	0.573090i	$0.805745\pi$
$798$	0	0
$799$	5.52786	0.195562
$800$	0	0
$801$	0.583592	0.0206202
$802$	0	0
$803$	9.79837	0.345777
$804$	0	0
$805$	35.5967	1.25462
$806$	0	0
$807$	−6.47214	−0.227830
$808$	0	0
$809$	−13.1246	−0.461437	−0.230718	−	0.973021i	$-0.574108\pi$
$809$	−13.1246	−0.461437	−0.230718	+	0.973021i	$0.574108\pi$
$810$	0	0
$811$	34.1033	1.19753	0.598765	−	0.800925i	$-0.295658\pi$
$811$	34.1033	1.19753	0.598765	+	0.800925i	$0.295658\pi$
$812$	0	0
$813$	20.9443	0.734548
$814$	0	0
$815$	−13.5967	−0.476273
$816$	0	0
$817$	−23.4164	−0.819236
$818$	0	0
$819$	3.52786	0.123274
$820$	0	0
$821$	5.41641	0.189034	0.0945170	−	0.995523i	$-0.469869\pi$
$821$	5.41641	0.189034	0.0945170	+	0.995523i	$0.469869\pi$
$822$	0	0
$823$	1.88854	0.0658305	0.0329152	−	0.999458i	$-0.489521\pi$
$823$	1.88854	0.0658305	0.0329152	+	0.999458i	$0.489521\pi$
$824$	0	0
$825$	22.0344	0.767141
$826$	0	0
$827$	2.06888	0.0719421	0.0359711	−	0.999353i	$-0.488548\pi$
$827$	2.06888	0.0719421	0.0359711	+	0.999353i	$0.488548\pi$
$828$	0	0
$829$	−55.7984	−1.93796	−0.968979	−	0.247144i	$-0.920508\pi$
$829$	−55.7984	−1.93796	−0.968979	+	0.247144i	$0.920508\pi$
$830$	0	0
$831$	−27.1803	−0.942876
$832$	0	0
$833$	−15.5279	−0.538009
$834$	0	0
$835$	−53.3951	−1.84781
$836$	0	0
$837$	−40.3951	−1.39626
$838$	0	0
$839$	−36.6525	−1.26538	−0.632692	−	0.774404i	$-0.718050\pi$
$839$	−36.6525	−1.26538	−0.632692	+	0.774404i	$0.718050\pi$
$840$	0	0
$841$	57.9787	1.99927
$842$	0	0
$843$	48.3607	1.66563
$844$	0	0
$845$	18.7082	0.643582
$846$	0	0
$847$	29.4164	1.01076
$848$	0	0
$849$	−10.9443	−0.375606
$850$	0	0
$851$	2.85410	0.0978374
$852$	0	0
$853$	−29.7426	−1.01837	−0.509184	−	0.860657i	$-0.670053\pi$
$853$	−29.7426	−1.01837	−0.509184	+	0.860657i	$0.670053\pi$
$854$	0	0
$855$	−6.58359	−0.225154
$856$	0	0
$857$	−9.05573	−0.309338	−0.154669	−	0.987966i	$-0.549431\pi$
$857$	−9.05573	−0.309338	−0.154669	+	0.987966i	$0.549431\pi$
$858$	0	0
$859$	−53.4164	−1.82254	−0.911272	−	0.411805i	$-0.864899\pi$
$859$	−53.4164	−1.82254	−0.911272	+	0.411805i	$0.864899\pi$
$860$	0	0
$861$	−50.3607	−1.71629
$862$	0	0
$863$	19.4164	0.660942	0.330471	−	0.943816i	$-0.392792\pi$
$863$	19.4164	0.660942	0.330471	+	0.943816i	$0.392792\pi$
$864$	0	0
$865$	−1.81966	−0.0618703
$866$	0	0
$867$	4.85410	0.164854
$868$	0	0
$869$	−11.8328	−0.401401
$870$	0	0
$871$	−3.11146	−0.105428
$872$	0	0
$873$	0.180340	0.00610358
$874$	0	0
$875$	60.5410	2.04666
$876$	0	0
$877$	16.8328	0.568404	0.284202	−	0.958764i	$-0.408271\pi$
$877$	16.8328	0.568404	0.284202	+	0.958764i	$0.408271\pi$
$878$	0	0
$879$	20.4721	0.690508
$880$	0	0
$881$	19.7426	0.665147	0.332573	−	0.943077i	$-0.392083\pi$
$881$	19.7426	0.665147	0.332573	+	0.943077i	$0.392083\pi$
$882$	0	0
$883$	33.3050	1.12080	0.560400	−	0.828222i	$-0.310647\pi$
$883$	33.3050	1.12080	0.560400	+	0.828222i	$0.310647\pi$
$884$	0	0
$885$	−29.7082	−0.998630
$886$	0	0
$887$	27.8885	0.936406	0.468203	−	0.883621i	$-0.344902\pi$
$887$	27.8885	0.936406	0.468203	+	0.883621i	$0.344902\pi$
$888$	0	0
$889$	27.4164	0.919517
$890$	0	0
$891$	−10.6525	−0.356871
$892$	0	0
$893$	5.52786	0.184983
$894$	0	0
$895$	−48.7639	−1.63000
$896$	0	0
$897$	13.1803	0.440079
$898$	0	0
$899$	68.8460	2.29614
$900$	0	0
$901$	2.11146	0.0703428
$902$	0	0
$903$	−27.4164	−0.912361
$904$	0	0
$905$	55.7771	1.85409
$906$	0	0
$907$	55.8885	1.85575	0.927874	−	0.372893i	$-0.121634\pi$
$907$	55.8885	1.85575	0.927874	+	0.372893i	$0.121634\pi$
$908$	0	0
$909$	1.34752	0.0446946
$910$	0	0
$911$	13.8885	0.460148	0.230074	−	0.973173i	$-0.426103\pi$
$911$	13.8885	0.460148	0.230074	+	0.973173i	$0.426103\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	66.2148	2.18899
$916$	0	0
$917$	−73.3050	−2.42074
$918$	0	0
$919$	−5.88854	−0.194245	−0.0971226	−	0.995272i	$-0.530964\pi$
$919$	−5.88854	−0.194245	−0.0971226	+	0.995272i	$0.530964\pi$
$920$	0	0
$921$	20.7984	0.685330
$922$	0	0
$923$	8.40325	0.276596
$924$	0	0
$925$	9.85410	0.324001
$926$	0	0
$927$	−6.59675	−0.216666
$928$	0	0
$929$	−27.4508	−0.900633	−0.450317	−	0.892869i	$-0.648689\pi$
$929$	−27.4508	−0.900633	−0.450317	+	0.892869i	$0.648689\pi$
$930$	0	0
$931$	−15.5279	−0.508905
$932$	0	0
$933$	−43.7426	−1.43207
$934$	0	0
$935$	23.8197	0.778986
$936$	0	0
$937$	−48.0476	−1.56965	−0.784823	−	0.619720i	$-0.787246\pi$
$937$	−48.0476	−1.56965	−0.784823	+	0.619720i	$0.787246\pi$
$938$	0	0
$939$	45.5967	1.48799
$940$	0	0
$941$	26.1803	0.853455	0.426727	−	0.904380i	$-0.359666\pi$
$941$	26.1803	0.853455	0.426727	+	0.904380i	$0.359666\pi$
$942$	0	0
$943$	27.4508	0.893923
$944$	0	0
$945$	−68.2492	−2.22015
$946$	0	0
$947$	−18.8328	−0.611984	−0.305992	−	0.952034i	$-0.598988\pi$
$947$	−18.8328	−0.611984	−0.305992	+	0.952034i	$0.598988\pi$
$948$	0	0
$949$	20.2361	0.656891
$950$	0	0
$951$	−33.8885	−1.09891
$952$	0	0
$953$	11.4508	0.370929	0.185465	−	0.982651i	$-0.440621\pi$
$953$	11.4508	0.370929	0.185465	+	0.982651i	$0.440621\pi$
$954$	0	0
$955$	27.3262	0.884256
$956$	0	0
$957$	20.8541	0.674117
$958$	0	0
$959$	−11.8885	−0.383901
$960$	0	0
$961$	23.4934	0.757852
$962$	0	0
$963$	−2.79837	−0.0901763
$964$	0	0
$965$	−15.4164	−0.496272
$966$	0	0
$967$	45.2705	1.45580	0.727901	−	0.685683i	$-0.240496\pi$
$967$	45.2705	1.45580	0.727901	+	0.685683i	$0.240496\pi$
$968$	0	0
$969$	32.3607	1.03957
$970$	0	0
$971$	−42.3262	−1.35831	−0.679157	−	0.733993i	$-0.737654\pi$
$971$	−42.3262	−1.35831	−0.679157	+	0.733993i	$0.737654\pi$
$972$	0	0
$973$	15.7082	0.503582
$974$	0	0
$975$	45.5066	1.45738
$976$	0	0
$977$	43.5279	1.39258	0.696290	−	0.717761i	$-0.254833\pi$
$977$	43.5279	1.39258	0.696290	+	0.717761i	$0.254833\pi$
$978$	0	0
$979$	−2.11146	−0.0674824
$980$	0	0
$981$	1.12461	0.0359061
$982$	0	0
$983$	−31.7771	−1.01353	−0.506766	−	0.862084i	$-0.669159\pi$
$983$	−31.7771	−1.01353	−0.506766	+	0.862084i	$0.669159\pi$
$984$	0	0
$985$	−29.0132	−0.924436
$986$	0	0
$987$	6.47214	0.206010
$988$	0	0
$989$	14.9443	0.475200
$990$	0	0
$991$	33.1033	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$991$	33.1033	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$992$	0	0
$993$	−45.3050	−1.43771
$994$	0	0
$995$	−11.7771	−0.373359
$996$	0	0
$997$	17.7771	0.563006	0.281503	−	0.959560i	$-0.409167\pi$
$997$	17.7771	0.563006	0.281503	+	0.959560i	$0.409167\pi$
$998$	0	0
$999$	−5.47214	−0.173131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.y.1.2		2
4.3	odd	2		2368.2.a.u.1.1		2
8.3	odd	2		592.2.a.g.1.2		2
8.5	even	2		74.2.a.b.1.1	✓	2
24.5	odd	2		666.2.a.i.1.1		2
24.11	even	2		5328.2.a.bc.1.1		2
40.13	odd	4		1850.2.b.j.149.1		4
40.29	even	2		1850.2.a.t.1.2		2
40.37	odd	4		1850.2.b.j.149.4		4
56.13	odd	2		3626.2.a.s.1.2		2
88.21	odd	2		8954.2.a.j.1.1		2
296.221	even	2		2738.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.b.1.1	✓	2	8.5	even	2
592.2.a.g.1.2		2	8.3	odd	2
666.2.a.i.1.1		2	24.5	odd	2
1850.2.a.t.1.2		2	40.29	even	2
1850.2.b.j.149.1		4	40.13	odd	4
1850.2.b.j.149.4		4	40.37	odd	4
2368.2.a.u.1.1		2	4.3	odd	2
2368.2.a.y.1.2		2	1.1	even	1	trivial
2738.2.a.g.1.1		2	296.221	even	2
3626.2.a.s.1.2		2	56.13	odd	2
5328.2.a.bc.1.1		2	24.11	even	2
8954.2.a.j.1.1		2	88.21	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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -3.85410 q^{5} -3.23607 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} -3.85410 q^{5} -3.23607 q^{7} -0.381966 q^{9} +1.38197 q^{11} +2.85410 q^{13} -6.23607 q^{15} -4.47214 q^{17} -4.47214 q^{19} -5.23607 q^{21} +2.85410 q^{23} +9.85410 q^{25} -5.47214 q^{27} +9.32624 q^{29} +7.38197 q^{31} +2.23607 q^{33} +12.4721 q^{35} +1.00000 q^{37} +4.61803 q^{39} +9.61803 q^{41} +5.23607 q^{43} +1.47214 q^{45} -1.23607 q^{47} +3.47214 q^{49} -7.23607 q^{51} -0.472136 q^{53} -5.32624 q^{55} -7.23607 q^{57} +4.76393 q^{59} -10.6180 q^{61} +1.23607 q^{63} -11.0000 q^{65} -1.09017 q^{67} +4.61803 q^{69} +2.94427 q^{71} +7.09017 q^{73} +15.9443 q^{75} -4.47214 q^{77} -8.56231 q^{79} -7.70820 q^{81} +14.4721 q^{83} +17.2361 q^{85} +15.0902 q^{87} -1.52786 q^{89} -9.23607 q^{91} +11.9443 q^{93} +17.2361 q^{95} -0.472136 q^{97} -0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9	\( 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9} + 5 q^{11} - q^{13} - 8 q^{15} - 6 q^{21} - q^{23} + 13 q^{25} - 2 q^{27} + 3 q^{29} + 17 q^{31} + 16 q^{35} + 2 q^{37} + 7 q^{39} + 17 q^{41} + 6 q^{43} - 6 q^{45} + 2 q^{47} - 2 q^{49} - 10 q^{51} + 8 q^{53} + 5 q^{55} - 10 q^{57} + 14 q^{59} - 19 q^{61} - 2 q^{63} - 22 q^{65} + 9 q^{67} + 7 q^{69} - 12 q^{71} + 3 q^{73} + 14 q^{75} + 3 q^{79} - 2 q^{81} + 20 q^{83} + 30 q^{85} + 19 q^{87} - 12 q^{89} - 14 q^{91} + 6 q^{93} + 30 q^{95} + 8 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9 + 5 * q^11 - q^13 - 8 * q^15 - 6 * q^21 - q^23 + 13 * q^25 - 2 * q^27 + 3 * q^29 + 17 * q^31 + 16 * q^35 + 2 * q^37 + 7 * q^39 + 17 * q^41 + 6 * q^43 - 6 * q^45 + 2 * q^47 - 2 * q^49 - 10 * q^51 + 8 * q^53 + 5 * q^55 - 10 * q^57 + 14 * q^59 - 19 * q^61 - 2 * q^63 - 22 * q^65 + 9 * q^67 + 7 * q^69 - 12 * q^71 + 3 * q^73 + 14 * q^75 + 3 * q^79 - 2 * q^81 + 20 * q^83 + 30 * q^85 + 19 * q^87 - 12 * q^89 - 14 * q^91 + 6 * q^93 + 30 * q^95 + 8 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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