LMFDB - Embedded newform 6004.2.a.h.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12838 q^{3} -3.66649 q^{5} +0.118556 q^{7} -1.72677 q^{9} +O(q^{10})$	$q-1.12838 q^{3} -3.66649 q^{5} +0.118556 q^{7} -1.72677 q^{9} +0.229479 q^{11} +2.85074 q^{13} +4.13717 q^{15} +4.90976 q^{17} -1.00000 q^{19} -0.133775 q^{21} -2.13227 q^{23} +8.44312 q^{25} +5.33357 q^{27} -7.59361 q^{29} -1.48512 q^{31} -0.258939 q^{33} -0.434682 q^{35} -11.3920 q^{37} -3.21670 q^{39} +7.15039 q^{41} -10.4953 q^{43} +6.33117 q^{45} +3.54734 q^{47} -6.98594 q^{49} -5.54006 q^{51} -0.184954 q^{53} -0.841383 q^{55} +1.12838 q^{57} -11.7250 q^{59} -2.46608 q^{61} -0.204718 q^{63} -10.4522 q^{65} -6.54492 q^{67} +2.40600 q^{69} +0.940551 q^{71} +8.23800 q^{73} -9.52701 q^{75} +0.0272061 q^{77} -1.00000 q^{79} -0.837963 q^{81} +1.36710 q^{83} -18.0016 q^{85} +8.56844 q^{87} -13.8220 q^{89} +0.337971 q^{91} +1.67577 q^{93} +3.66649 q^{95} -2.19022 q^{97} -0.396258 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−1.12838	−0.651468	−0.325734	−	0.945461i	$-0.605611\pi$
$3$	−1.12838	−0.651468	−0.325734	+	0.945461i	$0.605611\pi$
$4$	0	0
$5$	−3.66649	−1.63970	−0.819851	−	0.572576i	$-0.805944\pi$
$5$	−3.66649	−1.63970	−0.819851	+	0.572576i	$0.805944\pi$
$6$	0	0
$7$	0.118556	0.0448098	0.0224049	−	0.999749i	$-0.492868\pi$
$7$	0.118556	0.0448098	0.0224049	+	0.999749i	$0.492868\pi$
$8$	0	0
$9$	−1.72677	−0.575590
$10$	0	0
$11$	0.229479	0.0691907	0.0345953	−	0.999401i	$-0.488986\pi$
$11$	0.229479	0.0691907	0.0345953	+	0.999401i	$0.488986\pi$
$12$	0	0
$13$	2.85074	0.790652	0.395326	−	0.918541i	$-0.370632\pi$
$13$	2.85074	0.790652	0.395326	+	0.918541i	$0.370632\pi$
$14$	0	0
$15$	4.13717	1.06821
$16$	0	0
$17$	4.90976	1.19079	0.595396	−	0.803432i	$-0.296995\pi$
$17$	4.90976	1.19079	0.595396	+	0.803432i	$0.296995\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.133775	−0.0291921
$22$	0	0
$23$	−2.13227	−0.444609	−0.222304	−	0.974977i	$-0.571358\pi$
$23$	−2.13227	−0.444609	−0.222304	+	0.974977i	$0.571358\pi$
$24$	0	0
$25$	8.44312	1.68862
$26$	0	0
$27$	5.33357	1.02645
$28$	0	0
$29$	−7.59361	−1.41010	−0.705049	−	0.709159i	$-0.749075\pi$
$29$	−7.59361	−1.41010	−0.705049	+	0.709159i	$0.749075\pi$
$30$	0	0
$31$	−1.48512	−0.266735	−0.133367	−	0.991067i	$-0.542579\pi$
$31$	−1.48512	−0.266735	−0.133367	+	0.991067i	$0.542579\pi$
$32$	0	0
$33$	−0.258939	−0.0450755
$34$	0	0
$35$	−0.434682	−0.0734747
$36$	0	0
$37$	−11.3920	−1.87284	−0.936418	−	0.350886i	$-0.885881\pi$
$37$	−11.3920	−1.87284	−0.936418	+	0.350886i	$0.885881\pi$
$38$	0	0
$39$	−3.21670	−0.515085
$40$	0	0
$41$	7.15039	1.11670	0.558351	−	0.829605i	$-0.311434\pi$
$41$	7.15039	1.11670	0.558351	+	0.829605i	$0.311434\pi$
$42$	0	0
$43$	−10.4953	−1.60051	−0.800256	−	0.599659i	$-0.795303\pi$
$43$	−10.4953	−1.60051	−0.800256	+	0.599659i	$0.795303\pi$
$44$	0	0
$45$	6.33117	0.943796
$46$	0	0
$47$	3.54734	0.517433	0.258717	−	0.965953i	$-0.416700\pi$
$47$	3.54734	0.517433	0.258717	+	0.965953i	$0.416700\pi$
$48$	0	0
$49$	−6.98594	−0.997992
$50$	0	0
$51$	−5.54006	−0.775763
$52$	0	0
$53$	−0.184954	−0.0254054	−0.0127027	−	0.999919i	$-0.504044\pi$
$53$	−0.184954	−0.0254054	−0.0127027	+	0.999919i	$0.504044\pi$
$54$	0	0
$55$	−0.841383	−0.113452
$56$	0	0
$57$	1.12838	0.149457
$58$	0	0
$59$	−11.7250	−1.52646	−0.763230	−	0.646127i	$-0.776387\pi$
$59$	−11.7250	−1.52646	−0.763230	+	0.646127i	$0.776387\pi$
$60$	0	0
$61$	−2.46608	−0.315750	−0.157875	−	0.987459i	$-0.550464\pi$
$61$	−2.46608	−0.315750	−0.157875	+	0.987459i	$0.550464\pi$
$62$	0	0
$63$	−0.204718	−0.0257920
$64$	0	0
$65$	−10.4522	−1.29643
$66$	0	0
$67$	−6.54492	−0.799589	−0.399794	−	0.916605i	$-0.630918\pi$
$67$	−6.54492	−0.799589	−0.399794	+	0.916605i	$0.630918\pi$
$68$	0	0
$69$	2.40600	0.289648
$70$	0	0
$71$	0.940551	0.111623	0.0558114	−	0.998441i	$-0.482225\pi$
$71$	0.940551	0.111623	0.0558114	+	0.998441i	$0.482225\pi$
$72$	0	0
$73$	8.23800	0.964185	0.482093	−	0.876120i	$-0.339877\pi$
$73$	8.23800	0.964185	0.482093	+	0.876120i	$0.339877\pi$
$74$	0	0
$75$	−9.52701	−1.10008
$76$	0	0
$77$	0.0272061	0.00310042
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−0.837963	−0.0931070
$82$	0	0
$83$	1.36710	0.150059	0.0750294	−	0.997181i	$-0.476095\pi$
$83$	1.36710	0.150059	0.0750294	+	0.997181i	$0.476095\pi$
$84$	0	0
$85$	−18.0016	−1.95255
$86$	0	0
$87$	8.56844	0.918633
$88$	0	0
$89$	−13.8220	−1.46513	−0.732563	−	0.680699i	$-0.761676\pi$
$89$	−13.8220	−1.46513	−0.732563	+	0.680699i	$0.761676\pi$
$90$	0	0
$91$	0.337971	0.0354290
$92$	0	0
$93$	1.67577	0.173769
$94$	0	0
$95$	3.66649	0.376174
$96$	0	0
$97$	−2.19022	−0.222383	−0.111192	−	0.993799i	$-0.535467\pi$
$97$	−2.19022	−0.222383	−0.111192	+	0.993799i	$0.535467\pi$
$98$	0	0
$99$	−0.396258	−0.0398254
$100$	0	0
$101$	13.4955	1.34285	0.671426	−	0.741072i	$-0.265682\pi$
$101$	13.4955	1.34285	0.671426	+	0.741072i	$0.265682\pi$
$102$	0	0
$103$	9.88801	0.974295	0.487147	−	0.873320i	$-0.338037\pi$
$103$	9.88801	0.974295	0.487147	+	0.873320i	$0.338037\pi$
$104$	0	0
$105$	0.490485	0.0478664
$106$	0	0
$107$	16.3905	1.58453	0.792266	−	0.610176i	$-0.208901\pi$
$107$	16.3905	1.58453	0.792266	+	0.610176i	$0.208901\pi$
$108$	0	0
$109$	−2.36814	−0.226827	−0.113414	−	0.993548i	$-0.536178\pi$
$109$	−2.36814	−0.226827	−0.113414	+	0.993548i	$0.536178\pi$
$110$	0	0
$111$	12.8545	1.22009
$112$	0	0
$113$	−7.86349	−0.739735	−0.369867	−	0.929085i	$-0.620597\pi$
$113$	−7.86349	−0.739735	−0.369867	+	0.929085i	$0.620597\pi$
$114$	0	0
$115$	7.81794	0.729027
$116$	0	0
$117$	−4.92257	−0.455091
$118$	0	0
$119$	0.582080	0.0533592
$120$	0	0
$121$	−10.9473	−0.995213
$122$	0	0
$123$	−8.06832	−0.727496
$124$	0	0
$125$	−12.6242	−1.12914
$126$	0	0
$127$	7.37532	0.654454	0.327227	−	0.944946i	$-0.393886\pi$
$127$	7.37532	0.654454	0.327227	+	0.944946i	$0.393886\pi$
$128$	0	0
$129$	11.8426	1.04268
$130$	0	0
$131$	−14.0027	−1.22342	−0.611712	−	0.791080i	$-0.709519\pi$
$131$	−14.0027	−1.22342	−0.611712	+	0.791080i	$0.709519\pi$
$132$	0	0
$133$	−0.118556	−0.0102801
$134$	0	0
$135$	−19.5555	−1.68307
$136$	0	0
$137$	10.4782	0.895212	0.447606	−	0.894231i	$-0.352277\pi$
$137$	10.4782	0.895212	0.447606	+	0.894231i	$0.352277\pi$
$138$	0	0
$139$	0.866458	0.0734920	0.0367460	−	0.999325i	$-0.488301\pi$
$139$	0.866458	0.0734920	0.0367460	+	0.999325i	$0.488301\pi$
$140$	0	0
$141$	−4.00274	−0.337091
$142$	0	0
$143$	0.654186	0.0547058
$144$	0	0
$145$	27.8419	2.31214
$146$	0	0
$147$	7.88277	0.650160
$148$	0	0
$149$	−3.82175	−0.313090	−0.156545	−	0.987671i	$-0.550036\pi$
$149$	−3.82175	−0.313090	−0.156545	+	0.987671i	$0.550036\pi$
$150$	0	0
$151$	14.8413	1.20777	0.603884	−	0.797072i	$-0.293619\pi$
$151$	14.8413	1.20777	0.603884	+	0.797072i	$0.293619\pi$
$152$	0	0
$153$	−8.47803	−0.685408
$154$	0	0
$155$	5.44516	0.437366
$156$	0	0
$157$	13.0306	1.03995	0.519977	−	0.854180i	$-0.325941\pi$
$157$	13.0306	1.03995	0.519977	+	0.854180i	$0.325941\pi$
$158$	0	0
$159$	0.208698	0.0165508
$160$	0	0
$161$	−0.252792	−0.0199228
$162$	0	0
$163$	10.0007	0.783313	0.391657	−	0.920111i	$-0.371902\pi$
$163$	10.0007	0.783313	0.391657	+	0.920111i	$0.371902\pi$
$164$	0	0
$165$	0.949396	0.0739104
$166$	0	0
$167$	18.3039	1.41640	0.708198	−	0.706014i	$-0.249508\pi$
$167$	18.3039	1.41640	0.708198	+	0.706014i	$0.249508\pi$
$168$	0	0
$169$	−4.87329	−0.374869
$170$	0	0
$171$	1.72677	0.132049
$172$	0	0
$173$	23.5104	1.78746	0.893731	−	0.448604i	$-0.148079\pi$
$173$	23.5104	1.78746	0.893731	+	0.448604i	$0.148079\pi$
$174$	0	0
$175$	1.00098	0.0756669
$176$	0	0
$177$	13.2302	0.994439
$178$	0	0
$179$	5.13234	0.383609	0.191805	−	0.981433i	$-0.438566\pi$
$179$	5.13234	0.383609	0.191805	+	0.981433i	$0.438566\pi$
$180$	0	0
$181$	3.00212	0.223146	0.111573	−	0.993756i	$-0.464411\pi$
$181$	3.00212	0.223146	0.111573	+	0.993756i	$0.464411\pi$
$182$	0	0
$183$	2.78267	0.205701
$184$	0	0
$185$	41.7687	3.07090
$186$	0	0
$187$	1.12669	0.0823917
$188$	0	0
$189$	0.632324	0.0459948
$190$	0	0
$191$	0.527257	0.0381510	0.0190755	−	0.999818i	$-0.493928\pi$
$191$	0.527257	0.0381510	0.0190755	+	0.999818i	$0.493928\pi$
$192$	0	0
$193$	−26.7354	−1.92445	−0.962227	−	0.272247i	$-0.912233\pi$
$193$	−26.7354	−1.92445	−0.962227	+	0.272247i	$0.912233\pi$
$194$	0	0
$195$	11.7940	0.844586
$196$	0	0
$197$	11.9186	0.849165	0.424583	−	0.905389i	$-0.360421\pi$
$197$	11.9186	0.849165	0.424583	+	0.905389i	$0.360421\pi$
$198$	0	0
$199$	7.40181	0.524700	0.262350	−	0.964973i	$-0.415502\pi$
$199$	7.40181	0.524700	0.262350	+	0.964973i	$0.415502\pi$
$200$	0	0
$201$	7.38512	0.520906
$202$	0	0
$203$	−0.900264	−0.0631862
$204$	0	0
$205$	−26.2168	−1.83106
$206$	0	0
$207$	3.68194	0.255912
$208$	0	0
$209$	−0.229479	−0.0158734
$210$	0	0
$211$	−19.3155	−1.32973	−0.664866	−	0.746962i	$-0.731511\pi$
$211$	−19.3155	−1.32973	−0.664866	+	0.746962i	$0.731511\pi$
$212$	0	0
$213$	−1.06129	−0.0727187
$214$	0	0
$215$	38.4807	2.62436
$216$	0	0
$217$	−0.176069	−0.0119523
$218$	0	0
$219$	−9.29556	−0.628136
$220$	0	0
$221$	13.9964	0.941503
$222$	0	0
$223$	−9.69265	−0.649068	−0.324534	−	0.945874i	$-0.605207\pi$
$223$	−9.69265	−0.649068	−0.324534	+	0.945874i	$0.605207\pi$
$224$	0	0
$225$	−14.5793	−0.971955
$226$	0	0
$227$	−10.8192	−0.718096	−0.359048	−	0.933319i	$-0.616899\pi$
$227$	−10.8192	−0.718096	−0.359048	+	0.933319i	$0.616899\pi$
$228$	0	0
$229$	8.70622	0.575323	0.287662	−	0.957732i	$-0.407122\pi$
$229$	8.70622	0.575323	0.287662	+	0.957732i	$0.407122\pi$
$230$	0	0
$231$	−0.0306987	−0.00201982
$232$	0	0
$233$	23.6198	1.54738	0.773691	−	0.633563i	$-0.218408\pi$
$233$	23.6198	1.54738	0.773691	+	0.633563i	$0.218408\pi$
$234$	0	0
$235$	−13.0063	−0.848437
$236$	0	0
$237$	1.12838	0.0732959
$238$	0	0
$239$	14.9311	0.965813	0.482907	−	0.875672i	$-0.339581\pi$
$239$	14.9311	0.965813	0.482907	+	0.875672i	$0.339581\pi$
$240$	0	0
$241$	11.4616	0.738310	0.369155	−	0.929368i	$-0.379647\pi$
$241$	11.4616	0.738310	0.369155	+	0.929368i	$0.379647\pi$
$242$	0	0
$243$	−15.0552	−0.965790
$244$	0	0
$245$	25.6139	1.63641
$246$	0	0
$247$	−2.85074	−0.181388
$248$	0	0
$249$	−1.54260	−0.0977585
$250$	0	0
$251$	4.10271	0.258961	0.129481	−	0.991582i	$-0.458669\pi$
$251$	4.10271	0.258961	0.129481	+	0.991582i	$0.458669\pi$
$252$	0	0
$253$	−0.489312	−0.0307628
$254$	0	0
$255$	20.3125	1.27202
$256$	0	0
$257$	−21.3525	−1.33193	−0.665967	−	0.745981i	$-0.731981\pi$
$257$	−21.3525	−1.33193	−0.665967	+	0.745981i	$0.731981\pi$
$258$	0	0
$259$	−1.35059	−0.0839214
$260$	0	0
$261$	13.1124	0.811637
$262$	0	0
$263$	−24.7410	−1.52559	−0.762796	−	0.646639i	$-0.776174\pi$
$263$	−24.7410	−1.52559	−0.762796	+	0.646639i	$0.776174\pi$
$264$	0	0
$265$	0.678132	0.0416573
$266$	0	0
$267$	15.5964	0.954483
$268$	0	0
$269$	15.3422	0.935433	0.467717	−	0.883878i	$-0.345077\pi$
$269$	15.3422	0.935433	0.467717	+	0.883878i	$0.345077\pi$
$270$	0	0
$271$	−14.2660	−0.866599	−0.433299	−	0.901250i	$-0.642651\pi$
$271$	−14.2660	−0.866599	−0.433299	+	0.901250i	$0.642651\pi$
$272$	0	0
$273$	−0.381358	−0.0230808
$274$	0	0
$275$	1.93752	0.116837
$276$	0	0
$277$	7.60922	0.457194	0.228597	−	0.973521i	$-0.426586\pi$
$277$	7.60922	0.457194	0.228597	+	0.973521i	$0.426586\pi$
$278$	0	0
$279$	2.56445	0.153530
$280$	0	0
$281$	29.1324	1.73789	0.868945	−	0.494908i	$-0.164798\pi$
$281$	29.1324	1.73789	0.868945	+	0.494908i	$0.164798\pi$
$282$	0	0
$283$	17.6625	1.04993	0.524963	−	0.851125i	$-0.324079\pi$
$283$	17.6625	1.04993	0.524963	+	0.851125i	$0.324079\pi$
$284$	0	0
$285$	−4.13717	−0.245065
$286$	0	0
$287$	0.847718	0.0500392
$288$	0	0
$289$	7.10578	0.417987
$290$	0	0
$291$	2.47139	0.144875
$292$	0	0
$293$	4.19304	0.244960	0.122480	−	0.992471i	$-0.460915\pi$
$293$	4.19304	0.244960	0.122480	+	0.992471i	$0.460915\pi$
$294$	0	0
$295$	42.9894	2.50294
$296$	0	0
$297$	1.22394	0.0710205
$298$	0	0
$299$	−6.07854	−0.351531
$300$	0	0
$301$	−1.24427	−0.0717186
$302$	0	0
$303$	−15.2280	−0.874825
$304$	0	0
$305$	9.04186	0.517735
$306$	0	0
$307$	−25.0870	−1.43179	−0.715896	−	0.698207i	$-0.753981\pi$
$307$	−25.0870	−1.43179	−0.715896	+	0.698207i	$0.753981\pi$
$308$	0	0
$309$	−11.1574	−0.634722
$310$	0	0
$311$	8.76941	0.497268	0.248634	−	0.968598i	$-0.420018\pi$
$311$	8.76941	0.497268	0.248634	+	0.968598i	$0.420018\pi$
$312$	0	0
$313$	−19.9322	−1.12663	−0.563317	−	0.826241i	$-0.690475\pi$
$313$	−19.9322	−1.12663	−0.563317	+	0.826241i	$0.690475\pi$
$314$	0	0
$315$	0.750596	0.0422913
$316$	0	0
$317$	−1.36219	−0.0765081	−0.0382540	−	0.999268i	$-0.512180\pi$
$317$	−1.36219	−0.0765081	−0.0382540	+	0.999268i	$0.512180\pi$
$318$	0	0
$319$	−1.74258	−0.0975656
$320$	0	0
$321$	−18.4947	−1.03227
$322$	0	0
$323$	−4.90976	−0.273187
$324$	0	0
$325$	24.0691	1.33512
$326$	0	0
$327$	2.67216	0.147771
$328$	0	0
$329$	0.420557	0.0231861
$330$	0	0
$331$	2.84638	0.156451	0.0782256	−	0.996936i	$-0.475075\pi$
$331$	2.84638	0.156451	0.0782256	+	0.996936i	$0.475075\pi$
$332$	0	0
$333$	19.6714	1.07799
$334$	0	0
$335$	23.9968	1.31109
$336$	0	0
$337$	−20.8964	−1.13830	−0.569150	−	0.822234i	$-0.692728\pi$
$337$	−20.8964	−1.13830	−0.569150	+	0.822234i	$0.692728\pi$
$338$	0	0
$339$	8.87297	0.481914
$340$	0	0
$341$	−0.340804	−0.0184556
$342$	0	0
$343$	−1.65811	−0.0895296
$344$	0	0
$345$	−8.82157	−0.474937
$346$	0	0
$347$	5.53511	0.297140	0.148570	−	0.988902i	$-0.452533\pi$
$347$	5.53511	0.297140	0.148570	+	0.988902i	$0.452533\pi$
$348$	0	0
$349$	−34.2380	−1.83272	−0.916359	−	0.400358i	$-0.868886\pi$
$349$	−34.2380	−1.83272	−0.916359	+	0.400358i	$0.868886\pi$
$350$	0	0
$351$	15.2046	0.811562
$352$	0	0
$353$	−2.01101	−0.107035	−0.0535177	−	0.998567i	$-0.517043\pi$
$353$	−2.01101	−0.107035	−0.0535177	+	0.998567i	$0.517043\pi$
$354$	0	0
$355$	−3.44852	−0.183028
$356$	0	0
$357$	−0.656805	−0.0347618
$358$	0	0
$359$	1.69275	0.0893398	0.0446699	−	0.999002i	$-0.485776\pi$
$359$	1.69275	0.0893398	0.0446699	+	0.999002i	$0.485776\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	12.3527	0.648349
$364$	0	0
$365$	−30.2045	−1.58098
$366$	0	0
$367$	23.2567	1.21399	0.606995	−	0.794706i	$-0.292375\pi$
$367$	23.2567	1.21399	0.606995	+	0.794706i	$0.292375\pi$
$368$	0	0
$369$	−12.3471	−0.642762
$370$	0	0
$371$	−0.0219273	−0.00113841
$372$	0	0
$373$	29.6610	1.53579	0.767895	−	0.640576i	$-0.221304\pi$
$373$	29.6610	1.53579	0.767895	+	0.640576i	$0.221304\pi$
$374$	0	0
$375$	14.2448	0.735598
$376$	0	0
$377$	−21.6474	−1.11490
$378$	0	0
$379$	19.5285	1.00311	0.501555	−	0.865126i	$-0.332762\pi$
$379$	19.5285	1.00311	0.501555	+	0.865126i	$0.332762\pi$
$380$	0	0
$381$	−8.32213	−0.426355
$382$	0	0
$383$	7.62636	0.389689	0.194844	−	0.980834i	$-0.437580\pi$
$383$	7.62636	0.389689	0.194844	+	0.980834i	$0.437580\pi$
$384$	0	0
$385$	−0.0997507	−0.00508377
$386$	0	0
$387$	18.1229	0.921238
$388$	0	0
$389$	−7.67379	−0.389076	−0.194538	−	0.980895i	$-0.562321\pi$
$389$	−7.67379	−0.389076	−0.194538	+	0.980895i	$0.562321\pi$
$390$	0	0
$391$	−10.4689	−0.529437
$392$	0	0
$393$	15.8003	0.797022
$394$	0	0
$395$	3.66649	0.184481
$396$	0	0
$397$	7.58560	0.380710	0.190355	−	0.981715i	$-0.439036\pi$
$397$	7.58560	0.380710	0.190355	+	0.981715i	$0.439036\pi$
$398$	0	0
$399$	0.133775	0.00669714
$400$	0	0
$401$	29.9076	1.49351	0.746757	−	0.665096i	$-0.231610\pi$
$401$	29.9076	1.49351	0.746757	+	0.665096i	$0.231610\pi$
$402$	0	0
$403$	−4.23368	−0.210895
$404$	0	0
$405$	3.07238	0.152668
$406$	0	0
$407$	−2.61423	−0.129583
$408$	0	0
$409$	10.1542	0.502094	0.251047	−	0.967975i	$-0.419225\pi$
$409$	10.1542	0.502094	0.251047	+	0.967975i	$0.419225\pi$
$410$	0	0
$411$	−11.8233	−0.583202
$412$	0	0
$413$	−1.39006	−0.0684003
$414$	0	0
$415$	−5.01246	−0.246052
$416$	0	0
$417$	−0.977690	−0.0478777
$418$	0	0
$419$	−39.5071	−1.93005	−0.965025	−	0.262159i	$-0.915565\pi$
$419$	−39.5071	−1.93005	−0.965025	+	0.262159i	$0.915565\pi$
$420$	0	0
$421$	−6.14272	−0.299378	−0.149689	−	0.988733i	$-0.547827\pi$
$421$	−6.14272	−0.299378	−0.149689	+	0.988733i	$0.547827\pi$
$422$	0	0
$423$	−6.12544	−0.297829
$424$	0	0
$425$	41.4537	2.01080
$426$	0	0
$427$	−0.292368	−0.0141487
$428$	0	0
$429$	−0.738167	−0.0356391
$430$	0	0
$431$	36.5829	1.76214	0.881068	−	0.472990i	$-0.156825\pi$
$431$	36.5829	1.76214	0.881068	+	0.472990i	$0.156825\pi$
$432$	0	0
$433$	25.2810	1.21493	0.607463	−	0.794348i	$-0.292187\pi$
$433$	25.2810	1.21493	0.607463	+	0.794348i	$0.292187\pi$
$434$	0	0
$435$	−31.4161	−1.50629
$436$	0	0
$437$	2.13227	0.102000
$438$	0	0
$439$	−35.7527	−1.70638	−0.853192	−	0.521596i	$-0.825337\pi$
$439$	−35.7527	−1.70638	−0.853192	+	0.521596i	$0.825337\pi$
$440$	0	0
$441$	12.0631	0.574434
$442$	0	0
$443$	−7.82837	−0.371937	−0.185968	−	0.982556i	$-0.559542\pi$
$443$	−7.82837	−0.371937	−0.185968	+	0.982556i	$0.559542\pi$
$444$	0	0
$445$	50.6781	2.40237
$446$	0	0
$447$	4.31236	0.203968
$448$	0	0
$449$	−5.81107	−0.274242	−0.137121	−	0.990554i	$-0.543785\pi$
$449$	−5.81107	−0.274242	−0.137121	+	0.990554i	$0.543785\pi$
$450$	0	0
$451$	1.64087	0.0772654
$452$	0	0
$453$	−16.7466	−0.786822
$454$	0	0
$455$	−1.23917	−0.0580930
$456$	0	0
$457$	0.824579	0.0385722	0.0192861	−	0.999814i	$-0.493861\pi$
$457$	0.824579	0.0385722	0.0192861	+	0.999814i	$0.493861\pi$
$458$	0	0
$459$	26.1866	1.22228
$460$	0	0
$461$	−4.14882	−0.193230	−0.0966149	−	0.995322i	$-0.530802\pi$
$461$	−4.14882	−0.193230	−0.0966149	+	0.995322i	$0.530802\pi$
$462$	0	0
$463$	34.9834	1.62582	0.812909	−	0.582390i	$-0.197882\pi$
$463$	34.9834	1.62582	0.812909	+	0.582390i	$0.197882\pi$
$464$	0	0
$465$	−6.14419	−0.284930
$466$	0	0
$467$	−24.1720	−1.11855	−0.559274	−	0.828983i	$-0.688920\pi$
$467$	−24.1720	−1.11855	−0.559274	+	0.828983i	$0.688920\pi$
$468$	0	0
$469$	−0.775936	−0.0358294
$470$	0	0
$471$	−14.7034	−0.677496
$472$	0	0
$473$	−2.40845	−0.110740
$474$	0	0
$475$	−8.44312	−0.387397
$476$	0	0
$477$	0.319373	0.0146231
$478$	0	0
$479$	−9.44176	−0.431405	−0.215702	−	0.976459i	$-0.569204\pi$
$479$	−9.44176	−0.431405	−0.215702	+	0.976459i	$0.569204\pi$
$480$	0	0
$481$	−32.4757	−1.48076
$482$	0	0
$483$	0.285245	0.0129791
$484$	0	0
$485$	8.03041	0.364642
$486$	0	0
$487$	−30.6803	−1.39026	−0.695129	−	0.718885i	$-0.744653\pi$
$487$	−30.6803	−1.39026	−0.695129	+	0.718885i	$0.744653\pi$
$488$	0	0
$489$	−11.2845	−0.510304
$490$	0	0
$491$	−17.1772	−0.775196	−0.387598	−	0.921828i	$-0.626695\pi$
$491$	−17.1772	−0.775196	−0.387598	+	0.921828i	$0.626695\pi$
$492$	0	0
$493$	−37.2828	−1.67913
$494$	0	0
$495$	1.45287	0.0653019
$496$	0	0
$497$	0.111508	0.00500180
$498$	0	0
$499$	28.8487	1.29144	0.645722	−	0.763572i	$-0.276556\pi$
$499$	28.8487	1.29144	0.645722	+	0.763572i	$0.276556\pi$
$500$	0	0
$501$	−20.6536	−0.922736
$502$	0	0
$503$	−19.5226	−0.870468	−0.435234	−	0.900317i	$-0.643334\pi$
$503$	−19.5226	−0.870468	−0.435234	+	0.900317i	$0.643334\pi$
$504$	0	0
$505$	−49.4811	−2.20188
$506$	0	0
$507$	5.49890	0.244215
$508$	0	0
$509$	42.5265	1.88496	0.942478	−	0.334268i	$-0.108489\pi$
$509$	42.5265	1.88496	0.942478	+	0.334268i	$0.108489\pi$
$510$	0	0
$511$	0.976661	0.0432049
$512$	0	0
$513$	−5.33357	−0.235483
$514$	0	0
$515$	−36.2543	−1.59755
$516$	0	0
$517$	0.814043	0.0358016
$518$	0	0
$519$	−26.5285	−1.16447
$520$	0	0
$521$	42.6083	1.86670	0.933351	−	0.358966i	$-0.116871\pi$
$521$	42.6083	1.86670	0.933351	+	0.358966i	$0.116871\pi$
$522$	0	0
$523$	20.9534	0.916228	0.458114	−	0.888893i	$-0.348525\pi$
$523$	20.9534	0.916228	0.458114	+	0.888893i	$0.348525\pi$
$524$	0	0
$525$	−1.12948	−0.0492946
$526$	0	0
$527$	−7.29157	−0.317626
$528$	0	0
$529$	−18.4534	−0.802323
$530$	0	0
$531$	20.2463	0.878614
$532$	0	0
$533$	20.3839	0.882924
$534$	0	0
$535$	−60.0957	−2.59816
$536$	0	0
$537$	−5.79121	−0.249909
$538$	0	0
$539$	−1.60313	−0.0690517
$540$	0	0
$541$	7.55773	0.324932	0.162466	−	0.986714i	$-0.448055\pi$
$541$	7.55773	0.324932	0.162466	+	0.986714i	$0.448055\pi$
$542$	0	0
$543$	−3.38752	−0.145372
$544$	0	0
$545$	8.68277	0.371929
$546$	0	0
$547$	13.3736	0.571815	0.285907	−	0.958257i	$-0.407705\pi$
$547$	13.3736	0.571815	0.285907	+	0.958257i	$0.407705\pi$
$548$	0	0
$549$	4.25836	0.181742
$550$	0	0
$551$	7.59361	0.323499
$552$	0	0
$553$	−0.118556	−0.00504150
$554$	0	0
$555$	−47.1308	−2.00059
$556$	0	0
$557$	8.06772	0.341840	0.170920	−	0.985285i	$-0.445326\pi$
$557$	8.06772	0.341840	0.170920	+	0.985285i	$0.445326\pi$
$558$	0	0
$559$	−29.9192	−1.26545
$560$	0	0
$561$	−1.27133	−0.0536756
$562$	0	0
$563$	21.1932	0.893185	0.446592	−	0.894738i	$-0.352637\pi$
$563$	21.1932	0.893185	0.446592	+	0.894738i	$0.352637\pi$
$564$	0	0
$565$	28.8314	1.21295
$566$	0	0
$567$	−0.0993452	−0.00417211
$568$	0	0
$569$	4.37273	0.183314	0.0916572	−	0.995791i	$-0.470784\pi$
$569$	4.37273	0.183314	0.0916572	+	0.995791i	$0.470784\pi$
$570$	0	0
$571$	11.0245	0.461362	0.230681	−	0.973029i	$-0.425905\pi$
$571$	11.0245	0.461362	0.230681	+	0.973029i	$0.425905\pi$
$572$	0	0
$573$	−0.594944	−0.0248542
$574$	0	0
$575$	−18.0030	−0.750778
$576$	0	0
$577$	41.5721	1.73067	0.865334	−	0.501196i	$-0.167106\pi$
$577$	41.5721	1.73067	0.865334	+	0.501196i	$0.167106\pi$
$578$	0	0
$579$	30.1676	1.25372
$580$	0	0
$581$	0.162077	0.00672410
$582$	0	0
$583$	−0.0424432	−0.00175782
$584$	0	0
$585$	18.0485	0.746214
$586$	0	0
$587$	41.8144	1.72587	0.862933	−	0.505319i	$-0.168625\pi$
$587$	41.8144	1.72587	0.862933	+	0.505319i	$0.168625\pi$
$588$	0	0
$589$	1.48512	0.0611932
$590$	0	0
$591$	−13.4487	−0.553204
$592$	0	0
$593$	42.5331	1.74662	0.873312	−	0.487161i	$-0.161967\pi$
$593$	42.5331	1.74662	0.873312	+	0.487161i	$0.161967\pi$
$594$	0	0
$595$	−2.13419	−0.0874932
$596$	0	0
$597$	−8.35202	−0.341825
$598$	0	0
$599$	5.06548	0.206970	0.103485	−	0.994631i	$-0.467001\pi$
$599$	5.06548	0.206970	0.103485	+	0.994631i	$0.467001\pi$
$600$	0	0
$601$	31.4695	1.28367	0.641835	−	0.766843i	$-0.278174\pi$
$601$	31.4695	1.28367	0.641835	+	0.766843i	$0.278174\pi$
$602$	0	0
$603$	11.3016	0.460235
$604$	0	0
$605$	40.1383	1.63185
$606$	0	0
$607$	−30.7548	−1.24830	−0.624149	−	0.781305i	$-0.714554\pi$
$607$	−30.7548	−1.24830	−0.624149	+	0.781305i	$0.714554\pi$
$608$	0	0
$609$	1.01584	0.0411638
$610$	0	0
$611$	10.1125	0.409110
$612$	0	0
$613$	44.2637	1.78779	0.893897	−	0.448272i	$-0.147961\pi$
$613$	44.2637	1.78779	0.893897	+	0.448272i	$0.147961\pi$
$614$	0	0
$615$	29.5824	1.19288
$616$	0	0
$617$	21.8812	0.880904	0.440452	−	0.897776i	$-0.354818\pi$
$617$	21.8812	0.880904	0.440452	+	0.897776i	$0.354818\pi$
$618$	0	0
$619$	−47.4805	−1.90840	−0.954201	−	0.299167i	$-0.903291\pi$
$619$	−47.4805	−1.90840	−0.954201	+	0.299167i	$0.903291\pi$
$620$	0	0
$621$	−11.3726	−0.456367
$622$	0	0
$623$	−1.63867	−0.0656520
$624$	0	0
$625$	4.07073	0.162829
$626$	0	0
$627$	0.258939	0.0103410
$628$	0	0
$629$	−55.9321	−2.23016
$630$	0	0
$631$	−32.4858	−1.29324	−0.646621	−	0.762812i	$-0.723818\pi$
$631$	−32.4858	−1.29324	−0.646621	+	0.762812i	$0.723818\pi$
$632$	0	0
$633$	21.7951	0.866278
$634$	0	0
$635$	−27.0415	−1.07311
$636$	0	0
$637$	−19.9151	−0.789065
$638$	0	0
$639$	−1.62411	−0.0642489
$640$	0	0
$641$	11.2877	0.445836	0.222918	−	0.974837i	$-0.428442\pi$
$641$	11.2877	0.445836	0.222918	+	0.974837i	$0.428442\pi$
$642$	0	0
$643$	−30.7702	−1.21346	−0.606729	−	0.794909i	$-0.707519\pi$
$643$	−30.7702	−1.21346	−0.606729	+	0.794909i	$0.707519\pi$
$644$	0	0
$645$	−43.4207	−1.70969
$646$	0	0
$647$	15.5751	0.612319	0.306159	−	0.951980i	$-0.400956\pi$
$647$	15.5751	0.612319	0.306159	+	0.951980i	$0.400956\pi$
$648$	0	0
$649$	−2.69064	−0.105617
$650$	0	0
$651$	0.198672	0.00778656
$652$	0	0
$653$	−38.1943	−1.49466	−0.747330	−	0.664453i	$-0.768664\pi$
$653$	−38.1943	−1.49466	−0.747330	+	0.664453i	$0.768664\pi$
$654$	0	0
$655$	51.3408	2.00605
$656$	0	0
$657$	−14.2251	−0.554975
$658$	0	0
$659$	23.1959	0.903583	0.451792	−	0.892123i	$-0.350785\pi$
$659$	23.1959	0.903583	0.451792	+	0.892123i	$0.350785\pi$
$660$	0	0
$661$	−5.55038	−0.215885	−0.107942	−	0.994157i	$-0.534426\pi$
$661$	−5.55038	−0.215885	−0.107942	+	0.994157i	$0.534426\pi$
$662$	0	0
$663$	−15.7932	−0.613359
$664$	0	0
$665$	0.434682	0.0168563
$666$	0	0
$667$	16.1916	0.626942
$668$	0	0
$669$	10.9369	0.422847
$670$	0	0
$671$	−0.565915	−0.0218469
$672$	0	0
$673$	18.4694	0.711945	0.355972	−	0.934496i	$-0.384150\pi$
$673$	18.4694	0.711945	0.355972	+	0.934496i	$0.384150\pi$
$674$	0	0
$675$	45.0320	1.73328
$676$	0	0
$677$	7.51393	0.288784	0.144392	−	0.989521i	$-0.453877\pi$
$677$	7.51393	0.288784	0.144392	+	0.989521i	$0.453877\pi$
$678$	0	0
$679$	−0.259663	−0.00996494
$680$	0	0
$681$	12.2081	0.467816
$682$	0	0
$683$	−18.5589	−0.710136	−0.355068	−	0.934840i	$-0.615542\pi$
$683$	−18.5589	−0.710136	−0.355068	+	0.934840i	$0.615542\pi$
$684$	0	0
$685$	−38.4181	−1.46788
$686$	0	0
$687$	−9.82388	−0.374804
$688$	0	0
$689$	−0.527256	−0.0200868
$690$	0	0
$691$	−39.0100	−1.48401	−0.742006	−	0.670393i	$-0.766125\pi$
$691$	−39.0100	−1.48401	−0.742006	+	0.670393i	$0.766125\pi$
$692$	0	0
$693$	−0.0469786	−0.00178457
$694$	0	0
$695$	−3.17686	−0.120505
$696$	0	0
$697$	35.1067	1.32976
$698$	0	0
$699$	−26.6520	−1.00807
$700$	0	0
$701$	20.0498	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$701$	20.0498	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$702$	0	0
$703$	11.3920	0.429658
$704$	0	0
$705$	14.6760	0.552729
$706$	0	0
$707$	1.59997	0.0601729
$708$	0	0
$709$	9.51908	0.357497	0.178748	−	0.983895i	$-0.442795\pi$
$709$	9.51908	0.357497	0.178748	+	0.983895i	$0.442795\pi$
$710$	0	0
$711$	1.72677	0.0647589
$712$	0	0
$713$	3.16667	0.118593
$714$	0	0
$715$	−2.39856	−0.0897012
$716$	0	0
$717$	−16.8479	−0.629196
$718$	0	0
$719$	45.6447	1.70226	0.851130	−	0.524955i	$-0.175918\pi$
$719$	45.6447	1.70226	0.851130	+	0.524955i	$0.175918\pi$
$720$	0	0
$721$	1.17228	0.0436579
$722$	0	0
$723$	−12.9330	−0.480985
$724$	0	0
$725$	−64.1138	−2.38113
$726$	0	0
$727$	15.7689	0.584837	0.292419	−	0.956290i	$-0.405540\pi$
$727$	15.7689	0.584837	0.292419	+	0.956290i	$0.405540\pi$
$728$	0	0
$729$	19.5018	0.722288
$730$	0	0
$731$	−51.5292	−1.90588
$732$	0	0
$733$	9.33141	0.344663	0.172332	−	0.985039i	$-0.444870\pi$
$733$	9.33141	0.344663	0.172332	+	0.985039i	$0.444870\pi$
$734$	0	0
$735$	−28.9021	−1.06607
$736$	0	0
$737$	−1.50192	−0.0553241
$738$	0	0
$739$	−4.36731	−0.160654	−0.0803271	−	0.996769i	$-0.525596\pi$
$739$	−4.36731	−0.160654	−0.0803271	+	0.996769i	$0.525596\pi$
$740$	0	0
$741$	3.21670	0.118169
$742$	0	0
$743$	−13.9671	−0.512403	−0.256201	−	0.966623i	$-0.582471\pi$
$743$	−13.9671	−0.512403	−0.256201	+	0.966623i	$0.582471\pi$
$744$	0	0
$745$	14.0124	0.513374
$746$	0	0
$747$	−2.36067	−0.0863723
$748$	0	0
$749$	1.94319	0.0710026
$750$	0	0
$751$	−25.7677	−0.940278	−0.470139	−	0.882592i	$-0.655796\pi$
$751$	−25.7677	−0.940278	−0.470139	+	0.882592i	$0.655796\pi$
$752$	0	0
$753$	−4.62940	−0.168705
$754$	0	0
$755$	−54.4155	−1.98038
$756$	0	0
$757$	−44.3109	−1.61051	−0.805253	−	0.592932i	$-0.797970\pi$
$757$	−44.3109	−1.61051	−0.805253	+	0.592932i	$0.797970\pi$
$758$	0	0
$759$	0.552128	0.0200410
$760$	0	0
$761$	14.1067	0.511366	0.255683	−	0.966761i	$-0.417700\pi$
$761$	14.1067	0.511366	0.255683	+	0.966761i	$0.417700\pi$
$762$	0	0
$763$	−0.280757	−0.0101641
$764$	0	0
$765$	31.0846	1.12386
$766$	0	0
$767$	−33.4248	−1.20690
$768$	0	0
$769$	50.1576	1.80873	0.904365	−	0.426759i	$-0.140345\pi$
$769$	50.1576	1.80873	0.904365	+	0.426759i	$0.140345\pi$
$770$	0	0
$771$	24.0937	0.867713
$772$	0	0
$773$	24.0212	0.863983	0.431991	−	0.901878i	$-0.357811\pi$
$773$	24.0212	0.863983	0.431991	+	0.901878i	$0.357811\pi$
$774$	0	0
$775$	−12.5390	−0.450415
$776$	0	0
$777$	1.52397	0.0546721
$778$	0	0
$779$	−7.15039	−0.256189
$780$	0	0
$781$	0.215837	0.00772326
$782$	0	0
$783$	−40.5010	−1.44739
$784$	0	0
$785$	−47.7764	−1.70521
$786$	0	0
$787$	31.4023	1.11937	0.559686	−	0.828705i	$-0.310922\pi$
$787$	31.4023	1.11937	0.559686	+	0.828705i	$0.310922\pi$
$788$	0	0
$789$	27.9171	0.993875
$790$	0	0
$791$	−0.932261	−0.0331474
$792$	0	0
$793$	−7.03016	−0.249648
$794$	0	0
$795$	−0.765187	−0.0271384
$796$	0	0
$797$	17.1084	0.606009	0.303004	−	0.952989i	$-0.402010\pi$
$797$	17.1084	0.606009	0.303004	+	0.952989i	$0.402010\pi$
$798$	0	0
$799$	17.4166	0.616156
$800$	0	0
$801$	23.8674	0.843312
$802$	0	0
$803$	1.89045	0.0667126
$804$	0	0
$805$	0.926860	0.0326675
$806$	0	0
$807$	−17.3118	−0.609405
$808$	0	0
$809$	−42.1743	−1.48277	−0.741384	−	0.671081i	$-0.765830\pi$
$809$	−42.1743	−1.48277	−0.741384	+	0.671081i	$0.765830\pi$
$810$	0	0
$811$	2.17575	0.0764010	0.0382005	−	0.999270i	$-0.487837\pi$
$811$	2.17575	0.0764010	0.0382005	+	0.999270i	$0.487837\pi$
$812$	0	0
$813$	16.0974	0.564561
$814$	0	0
$815$	−36.6673	−1.28440
$816$	0	0
$817$	10.4953	0.367183
$818$	0	0
$819$	−0.583597	−0.0203925
$820$	0	0
$821$	25.1802	0.878794	0.439397	−	0.898293i	$-0.355192\pi$
$821$	25.1802	0.878794	0.439397	+	0.898293i	$0.355192\pi$
$822$	0	0
$823$	36.3609	1.26746	0.633730	−	0.773554i	$-0.281523\pi$
$823$	36.3609	1.26746	0.633730	+	0.773554i	$0.281523\pi$
$824$	0	0
$825$	−2.18625	−0.0761156
$826$	0	0
$827$	−21.5698	−0.750054	−0.375027	−	0.927014i	$-0.622367\pi$
$827$	−21.5698	−0.750054	−0.375027	+	0.927014i	$0.622367\pi$
$828$	0	0
$829$	−40.2900	−1.39933	−0.699665	−	0.714471i	$-0.746668\pi$
$829$	−40.2900	−1.39933	−0.699665	+	0.714471i	$0.746668\pi$
$830$	0	0
$831$	−8.58605	−0.297847
$832$	0	0
$833$	−34.2993	−1.18840
$834$	0	0
$835$	−67.1109	−2.32247
$836$	0	0
$837$	−7.92097	−0.273789
$838$	0	0
$839$	−25.4781	−0.879602	−0.439801	−	0.898095i	$-0.644951\pi$
$839$	−25.4781	−0.879602	−0.439801	+	0.898095i	$0.644951\pi$
$840$	0	0
$841$	28.6629	0.988375
$842$	0	0
$843$	−32.8722	−1.13218
$844$	0	0
$845$	17.8679	0.614673
$846$	0	0
$847$	−1.29787	−0.0445953
$848$	0	0
$849$	−19.9299	−0.683992
$850$	0	0
$851$	24.2909	0.832680
$852$	0	0
$853$	3.64561	0.124823	0.0624117	−	0.998050i	$-0.480121\pi$
$853$	3.64561	0.124823	0.0624117	+	0.998050i	$0.480121\pi$
$854$	0	0
$855$	−6.33117	−0.216522
$856$	0	0
$857$	37.9926	1.29780	0.648901	−	0.760873i	$-0.275229\pi$
$857$	37.9926	1.29780	0.648901	+	0.760873i	$0.275229\pi$
$858$	0	0
$859$	42.3555	1.44515	0.722575	−	0.691293i	$-0.242958\pi$
$859$	42.3555	1.44515	0.722575	+	0.691293i	$0.242958\pi$
$860$	0	0
$861$	−0.956544	−0.0325989
$862$	0	0
$863$	−23.3485	−0.794793	−0.397396	−	0.917647i	$-0.630086\pi$
$863$	−23.3485	−0.794793	−0.397396	+	0.917647i	$0.630086\pi$
$864$	0	0
$865$	−86.2005	−2.93090
$866$	0	0
$867$	−8.01798	−0.272305
$868$	0	0
$869$	−0.229479	−0.00778456
$870$	0	0
$871$	−18.6578	−0.632197
$872$	0	0
$873$	3.78200	0.128001
$874$	0	0
$875$	−1.49667	−0.0505965
$876$	0	0
$877$	44.2937	1.49569	0.747847	−	0.663872i	$-0.231088\pi$
$877$	44.2937	1.49569	0.747847	+	0.663872i	$0.231088\pi$
$878$	0	0
$879$	−4.73132	−0.159583
$880$	0	0
$881$	−6.10088	−0.205544	−0.102772	−	0.994705i	$-0.532771\pi$
$881$	−6.10088	−0.205544	−0.102772	+	0.994705i	$0.532771\pi$
$882$	0	0
$883$	−17.3927	−0.585309	−0.292655	−	0.956218i	$-0.594539\pi$
$883$	−17.3927	−0.585309	−0.292655	+	0.956218i	$0.594539\pi$
$884$	0	0
$885$	−48.5082	−1.63058
$886$	0	0
$887$	−49.6547	−1.66724	−0.833621	−	0.552337i	$-0.813736\pi$
$887$	−49.6547	−1.66724	−0.833621	+	0.552337i	$0.813736\pi$
$888$	0	0
$889$	0.874385	0.0293259
$890$	0	0
$891$	−0.192295	−0.00644214
$892$	0	0
$893$	−3.54734	−0.118707
$894$	0	0
$895$	−18.8177	−0.629005
$896$	0	0
$897$	6.85888	0.229011
$898$	0	0
$899$	11.2774	0.376122
$900$	0	0
$901$	−0.908081	−0.0302526
$902$	0	0
$903$	1.40401	0.0467224
$904$	0	0
$905$	−11.0072	−0.365893
$906$	0	0
$907$	−15.5628	−0.516753	−0.258377	−	0.966044i	$-0.583188\pi$
$907$	−15.5628	−0.516753	−0.258377	+	0.966044i	$0.583188\pi$
$908$	0	0
$909$	−23.3036	−0.772932
$910$	0	0
$911$	11.0564	0.366315	0.183158	−	0.983084i	$-0.441368\pi$
$911$	11.0564	0.366315	0.183158	+	0.983084i	$0.441368\pi$
$912$	0	0
$913$	0.313721	0.0103827
$914$	0	0
$915$	−10.2026	−0.337288
$916$	0	0
$917$	−1.66010	−0.0548214
$918$	0	0
$919$	40.1868	1.32564	0.662821	−	0.748778i	$-0.269359\pi$
$919$	40.1868	1.32564	0.662821	+	0.748778i	$0.269359\pi$
$920$	0	0
$921$	28.3076	0.932766
$922$	0	0
$923$	2.68126	0.0882549
$924$	0	0
$925$	−96.1842	−3.16252
$926$	0	0
$927$	−17.0743	−0.560794
$928$	0	0
$929$	27.3269	0.896568	0.448284	−	0.893891i	$-0.352035\pi$
$929$	27.3269	0.896568	0.448284	+	0.893891i	$0.352035\pi$
$930$	0	0
$931$	6.98594	0.228955
$932$	0	0
$933$	−9.89519	−0.323954
$934$	0	0
$935$	−4.13099	−0.135098
$936$	0	0
$937$	−14.0882	−0.460240	−0.230120	−	0.973162i	$-0.573912\pi$
$937$	−14.0882	−0.460240	−0.230120	+	0.973162i	$0.573912\pi$
$938$	0	0
$939$	22.4910	0.733966
$940$	0	0
$941$	22.3662	0.729117	0.364558	−	0.931180i	$-0.381220\pi$
$941$	22.3662	0.729117	0.364558	+	0.931180i	$0.381220\pi$
$942$	0	0
$943$	−15.2466	−0.496496
$944$	0	0
$945$	−2.31841	−0.0754178
$946$	0	0
$947$	3.87067	0.125780	0.0628899	−	0.998020i	$-0.479968\pi$
$947$	3.87067	0.125780	0.0628899	+	0.998020i	$0.479968\pi$
$948$	0	0
$949$	23.4844	0.762336
$950$	0	0
$951$	1.53706	0.0498425
$952$	0	0
$953$	22.6276	0.732979	0.366490	−	0.930422i	$-0.380560\pi$
$953$	22.6276	0.732979	0.366490	+	0.930422i	$0.380560\pi$
$954$	0	0
$955$	−1.93318	−0.0625563
$956$	0	0
$957$	1.96628	0.0635608
$958$	0	0
$959$	1.24225	0.0401143
$960$	0	0
$961$	−28.7944	−0.928853
$962$	0	0
$963$	−28.3027	−0.912040
$964$	0	0
$965$	98.0249	3.15553
$966$	0	0
$967$	23.3594	0.751186	0.375593	−	0.926785i	$-0.377439\pi$
$967$	23.3594	0.751186	0.375593	+	0.926785i	$0.377439\pi$
$968$	0	0
$969$	5.54006	0.177972
$970$	0	0
$971$	−40.4569	−1.29832	−0.649161	−	0.760651i	$-0.724880\pi$
$971$	−40.4569	−1.29832	−0.649161	+	0.760651i	$0.724880\pi$
$972$	0	0
$973$	0.102723	0.00329316
$974$	0	0
$975$	−27.1590	−0.869785
$976$	0	0
$977$	−3.10330	−0.0992834	−0.0496417	−	0.998767i	$-0.515808\pi$
$977$	−3.10330	−0.0992834	−0.0496417	+	0.998767i	$0.515808\pi$
$978$	0	0
$979$	−3.17186	−0.101373
$980$	0	0
$981$	4.08924	0.130559
$982$	0	0
$983$	−19.5943	−0.624960	−0.312480	−	0.949924i	$-0.601160\pi$
$983$	−19.5943	−0.624960	−0.312480	+	0.949924i	$0.601160\pi$
$984$	0	0
$985$	−43.6994	−1.39238
$986$	0	0
$987$	−0.474547	−0.0151050
$988$	0	0
$989$	22.3787	0.711602
$990$	0	0
$991$	10.8770	0.345519	0.172760	−	0.984964i	$-0.444732\pi$
$991$	10.8770	0.345519	0.172760	+	0.984964i	$0.444732\pi$
$992$	0	0
$993$	−3.21179	−0.101923
$994$	0	0
$995$	−27.1386	−0.860353
$996$	0	0
$997$	−28.8875	−0.914877	−0.457439	−	0.889241i	$-0.651233\pi$
$997$	−28.8875	−0.914877	−0.457439	+	0.889241i	$0.651233\pi$
$998$	0	0
$999$	−60.7601	−1.92237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.13	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.13	✓	31	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.12838 q^{3} -3.66649 q^{5} +0.118556 q^{7} -1.72677 q^{9} +O(q^{10})\)	\(q-1.12838 q^{3} -3.66649 q^{5} +0.118556 q^{7} -1.72677 q^{9} +0.229479 q^{11} +2.85074 q^{13} +4.13717 q^{15} +4.90976 q^{17} -1.00000 q^{19} -0.133775 q^{21} -2.13227 q^{23} +8.44312 q^{25} +5.33357 q^{27} -7.59361 q^{29} -1.48512 q^{31} -0.258939 q^{33} -0.434682 q^{35} -11.3920 q^{37} -3.21670 q^{39} +7.15039 q^{41} -10.4953 q^{43} +6.33117 q^{45} +3.54734 q^{47} -6.98594 q^{49} -5.54006 q^{51} -0.184954 q^{53} -0.841383 q^{55} +1.12838 q^{57} -11.7250 q^{59} -2.46608 q^{61} -0.204718 q^{63} -10.4522 q^{65} -6.54492 q^{67} +2.40600 q^{69} +0.940551 q^{71} +8.23800 q^{73} -9.52701 q^{75} +0.0272061 q^{77} -1.00000 q^{79} -0.837963 q^{81} +1.36710 q^{83} -18.0016 q^{85} +8.56844 q^{87} -13.8220 q^{89} +0.337971 q^{91} +1.67577 q^{93} +3.66649 q^{95} -2.19022 q^{97} -0.396258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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