LMFDB - Embedded newform 6080.2.a.ce.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.78292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 8x + 18 $ x^4 - x^3 - 10x^2 + 8x + 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.09502$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.09502 q^{3} +1.00000 q^{5} +2.80094 q^{7} -1.80094 q^{9} +O(q^{10})$	$q+1.09502 q^{3} +1.00000 q^{5} +2.80094 q^{7} -1.80094 q^{9} +2.45617 q^{11} -3.55118 q^{13} +1.09502 q^{15} -4.34477 q^{17} -1.00000 q^{19} +3.06707 q^{21} -1.19906 q^{23} +1.00000 q^{25} -5.25711 q^{27} -7.44714 q^{29} -2.91234 q^{31} +2.68954 q^{33} +2.80094 q^{35} -7.89596 q^{37} -3.88860 q^{39} -10.5142 q^{41} +2.95568 q^{43} -1.80094 q^{45} -13.1457 q^{47} +0.845261 q^{49} -4.75760 q^{51} -3.74122 q^{53} +2.45617 q^{55} -1.09502 q^{57} +12.3595 q^{59} +2.45617 q^{61} -5.04432 q^{63} -3.55118 q^{65} -9.59453 q^{67} -1.31299 q^{69} +7.60188 q^{71} +2.75760 q^{73} +1.09502 q^{75} +6.87958 q^{77} -9.79191 q^{79} -0.353800 q^{81} +6.95568 q^{83} -4.34477 q^{85} -8.15474 q^{87} +8.49951 q^{89} -9.94665 q^{91} -3.18905 q^{93} -1.00000 q^{95} +2.01540 q^{97} -4.42341 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9} + 6 q^{11} - 5 q^{13} - q^{15} - 5 q^{17} - 4 q^{19} + 3 q^{21} - 21 q^{23} + 4 q^{25} - q^{27} + q^{29} - 4 q^{31} - 14 q^{33} - 5 q^{35} - 10 q^{37} - 7 q^{39} - 2 q^{41} - 6 q^{43} + 9 q^{45} - 24 q^{47} + 5 q^{49} - 13 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 11 q^{59} + 6 q^{61} - 38 q^{63} - 5 q^{65} - 19 q^{67} + 7 q^{69} - 2 q^{71} + 5 q^{73} - q^{75} - 8 q^{77} + 4 q^{79} - 16 q^{81} + 10 q^{83} - 5 q^{85} - 31 q^{87} + 20 q^{89} + 5 q^{91} + 26 q^{93} - 4 q^{95} - 6 q^{97} + 14 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9 + 6 * q^11 - 5 * q^13 - q^15 - 5 * q^17 - 4 * q^19 + 3 * q^21 - 21 * q^23 + 4 * q^25 - q^27 + q^29 - 4 * q^31 - 14 * q^33 - 5 * q^35 - 10 * q^37 - 7 * q^39 - 2 * q^41 - 6 * q^43 + 9 * q^45 - 24 * q^47 + 5 * q^49 - 13 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 11 * q^59 + 6 * q^61 - 38 * q^63 - 5 * q^65 - 19 * q^67 + 7 * q^69 - 2 * q^71 + 5 * q^73 - q^75 - 8 * q^77 + 4 * q^79 - 16 * q^81 + 10 * q^83 - 5 * q^85 - 31 * q^87 + 20 * q^89 + 5 * q^91 + 26 * q^93 - 4 * q^95 - 6 * q^97 + 14 * 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
$3$	1.09502	0.632208	0.316104	−	0.948725i	$-0.397625\pi$
$3$	1.09502	0.632208	0.316104	+	0.948725i	$0.397625\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.80094	1.05866	0.529328	−	0.848417i	$-0.322444\pi$
$7$	2.80094	1.05866	0.529328	+	0.848417i	$0.322444\pi$
$8$	0	0
$9$	−1.80094	−0.600313
$10$	0	0
$11$	2.45617	0.740562	0.370281	−	0.928920i	$-0.379261\pi$
$11$	2.45617	0.740562	0.370281	+	0.928920i	$0.379261\pi$
$12$	0	0
$13$	−3.55118	−0.984921	−0.492461	−	0.870335i	$-0.663902\pi$
$13$	−3.55118	−0.984921	−0.492461	+	0.870335i	$0.663902\pi$
$14$	0	0
$15$	1.09502	0.282732
$16$	0	0
$17$	−4.34477	−1.05376	−0.526881	−	0.849939i	$-0.676639\pi$
$17$	−4.34477	−1.05376	−0.526881	+	0.849939i	$0.676639\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.06707	0.669290
$22$	0	0
$23$	−1.19906	−0.250021	−0.125011	−	0.992155i	$-0.539897\pi$
$23$	−1.19906	−0.250021	−0.125011	+	0.992155i	$0.539897\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.25711	−1.01173
$28$	0	0
$29$	−7.44714	−1.38290	−0.691450	−	0.722425i	$-0.743028\pi$
$29$	−7.44714	−1.38290	−0.691450	+	0.722425i	$0.743028\pi$
$30$	0	0
$31$	−2.91234	−0.523071	−0.261535	−	0.965194i	$-0.584229\pi$
$31$	−2.91234	−0.523071	−0.261535	+	0.965194i	$0.584229\pi$
$32$	0	0
$33$	2.68954	0.468189
$34$	0	0
$35$	2.80094	0.473445
$36$	0	0
$37$	−7.89596	−1.29809	−0.649044	−	0.760751i	$-0.724831\pi$
$37$	−7.89596	−1.29809	−0.649044	+	0.760751i	$0.724831\pi$
$38$	0	0
$39$	−3.88860	−0.622675
$40$	0	0
$41$	−10.5142	−1.64204	−0.821022	−	0.570896i	$-0.806596\pi$
$41$	−10.5142	−1.64204	−0.821022	+	0.570896i	$0.806596\pi$
$42$	0	0
$43$	2.95568	0.450737	0.225368	−	0.974274i	$-0.427641\pi$
$43$	2.95568	0.450737	0.225368	+	0.974274i	$0.427641\pi$
$44$	0	0
$45$	−1.80094	−0.268468
$46$	0	0
$47$	−13.1457	−1.91750	−0.958750	−	0.284252i	$-0.908255\pi$
$47$	−13.1457	−1.91750	−0.958750	+	0.284252i	$0.908255\pi$
$48$	0	0
$49$	0.845261	0.120752
$50$	0	0
$51$	−4.75760	−0.666197
$52$	0	0
$53$	−3.74122	−0.513896	−0.256948	−	0.966425i	$-0.582717\pi$
$53$	−3.74122	−0.513896	−0.256948	+	0.966425i	$0.582717\pi$
$54$	0	0
$55$	2.45617	0.331190
$56$	0	0
$57$	−1.09502	−0.145038
$58$	0	0
$59$	12.3595	1.60907	0.804533	−	0.593908i	$-0.202416\pi$
$59$	12.3595	1.60907	0.804533	+	0.593908i	$0.202416\pi$
$60$	0	0
$61$	2.45617	0.314480	0.157240	−	0.987560i	$-0.449740\pi$
$61$	2.45617	0.314480	0.157240	+	0.987560i	$0.449740\pi$
$62$	0	0
$63$	−5.04432	−0.635525
$64$	0	0
$65$	−3.55118	−0.440470
$66$	0	0
$67$	−9.59453	−1.17216	−0.586079	−	0.810254i	$-0.699329\pi$
$67$	−9.59453	−1.17216	−0.586079	+	0.810254i	$0.699329\pi$
$68$	0	0
$69$	−1.31299	−0.158066
$70$	0	0
$71$	7.60188	0.902177	0.451089	−	0.892479i	$-0.351036\pi$
$71$	7.60188	0.902177	0.451089	+	0.892479i	$0.351036\pi$
$72$	0	0
$73$	2.75760	0.322752	0.161376	−	0.986893i	$-0.448407\pi$
$73$	2.75760	0.322752	0.161376	+	0.986893i	$0.448407\pi$
$74$	0	0
$75$	1.09502	0.126442
$76$	0	0
$77$	6.87958	0.784000
$78$	0	0
$79$	−9.79191	−1.10168	−0.550838	−	0.834612i	$-0.685692\pi$
$79$	−9.79191	−1.10168	−0.550838	+	0.834612i	$0.685692\pi$
$80$	0	0
$81$	−0.353800	−0.0393111
$82$	0	0
$83$	6.95568	0.763485	0.381742	−	0.924269i	$-0.375324\pi$
$83$	6.95568	0.763485	0.381742	+	0.924269i	$0.375324\pi$
$84$	0	0
$85$	−4.34477	−0.471257
$86$	0	0
$87$	−8.15474	−0.874280
$88$	0	0
$89$	8.49951	0.900946	0.450473	−	0.892790i	$-0.351255\pi$
$89$	8.49951	0.900946	0.450473	+	0.892790i	$0.351255\pi$
$90$	0	0
$91$	−9.94665	−1.04269
$92$	0	0
$93$	−3.18905	−0.330690
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.01540	0.204633	0.102316	−	0.994752i	$-0.467375\pi$
$97$	2.01540	0.204633	0.102316	+	0.994752i	$0.467375\pi$
$98$	0	0
$99$	−4.42341	−0.444569
$100$	0	0
$101$	−4.95568	−0.493108	−0.246554	−	0.969129i	$-0.579298\pi$
$101$	−4.95568	−0.493108	−0.246554	+	0.969129i	$0.579298\pi$
$102$	0	0
$103$	−3.83791	−0.378160	−0.189080	−	0.981962i	$-0.560551\pi$
$103$	−3.83791	−0.378160	−0.189080	+	0.981962i	$0.560551\pi$
$104$	0	0
$105$	3.06707	0.299316
$106$	0	0
$107$	9.59453	0.927538	0.463769	−	0.885956i	$-0.346497\pi$
$107$	9.59453	0.927538	0.463769	+	0.885956i	$0.346497\pi$
$108$	0	0
$109$	−0.534804	−0.0512250	−0.0256125	−	0.999672i	$-0.508154\pi$
$109$	−0.534804	−0.0512250	−0.0256125	+	0.999672i	$0.508154\pi$
$110$	0	0
$111$	−8.64620	−0.820661
$112$	0	0
$113$	13.6879	1.28765	0.643823	−	0.765174i	$-0.277347\pi$
$113$	13.6879	1.28765	0.643823	+	0.765174i	$0.277347\pi$
$114$	0	0
$115$	−1.19906	−0.111813
$116$	0	0
$117$	6.39547	0.591261
$118$	0	0
$119$	−12.1694	−1.11557
$120$	0	0
$121$	−4.96724	−0.451567
$122$	0	0
$123$	−11.5132	−1.03811
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.5739	−1.55943	−0.779717	−	0.626132i	$-0.784637\pi$
$127$	−17.5739	−1.55943	−0.779717	+	0.626132i	$0.784637\pi$
$128$	0	0
$129$	3.23652	0.284959
$130$	0	0
$131$	−4.32418	−0.377805	−0.188903	−	0.981996i	$-0.560493\pi$
$131$	−4.32418	−0.377805	−0.188903	+	0.981996i	$0.560493\pi$
$132$	0	0
$133$	−2.80094	−0.242872
$134$	0	0
$135$	−5.25711	−0.452460
$136$	0	0
$137$	−5.63717	−0.481616	−0.240808	−	0.970573i	$-0.577412\pi$
$137$	−5.63717	−0.481616	−0.240808	+	0.970573i	$0.577412\pi$
$138$	0	0
$139$	20.7476	1.75979	0.879894	−	0.475170i	$-0.157614\pi$
$139$	20.7476	1.75979	0.879894	+	0.475170i	$0.157614\pi$
$140$	0	0
$141$	−14.3948	−1.21226
$142$	0	0
$143$	−8.72230	−0.729396
$144$	0	0
$145$	−7.44714	−0.618451
$146$	0	0
$147$	0.925574	0.0763401
$148$	0	0
$149$	17.2799	1.41562	0.707811	−	0.706402i	$-0.249683\pi$
$149$	17.2799	1.41562	0.707811	+	0.706402i	$0.249683\pi$
$150$	0	0
$151$	−8.10139	−0.659282	−0.329641	−	0.944106i	$-0.606928\pi$
$151$	−8.10139	−0.659282	−0.329641	+	0.944106i	$0.606928\pi$
$152$	0	0
$153$	7.82467	0.632587
$154$	0	0
$155$	−2.91234	−0.233924
$156$	0	0
$157$	6.76013	0.539517	0.269759	−	0.962928i	$-0.413056\pi$
$157$	6.76013	0.539517	0.269759	+	0.962928i	$0.413056\pi$
$158$	0	0
$159$	−4.09669	−0.324889
$160$	0	0
$161$	−3.35850	−0.264687
$162$	0	0
$163$	3.86801	0.302966	0.151483	−	0.988460i	$-0.451595\pi$
$163$	3.86801	0.302966	0.151483	+	0.988460i	$0.451595\pi$
$164$	0	0
$165$	2.68954	0.209381
$166$	0	0
$167$	−23.4545	−1.81496	−0.907481	−	0.420092i	$-0.861998\pi$
$167$	−23.4545	−1.81496	−0.907481	+	0.420092i	$0.861998\pi$
$168$	0	0
$169$	−0.389093	−0.0299303
$170$	0	0
$171$	1.80094	0.137721
$172$	0	0
$173$	20.7902	1.58065	0.790326	−	0.612686i	$-0.209911\pi$
$173$	20.7902	1.58065	0.790326	+	0.612686i	$0.209911\pi$
$174$	0	0
$175$	2.80094	0.211731
$176$	0	0
$177$	13.5338	1.01726
$178$	0	0
$179$	7.27770	0.543961	0.271980	−	0.962303i	$-0.412321\pi$
$179$	7.27770	0.543961	0.271980	+	0.962303i	$0.412321\pi$
$180$	0	0
$181$	−8.82369	−0.655860	−0.327930	−	0.944702i	$-0.606351\pi$
$181$	−8.82369	−0.655860	−0.327930	+	0.944702i	$0.606351\pi$
$182$	0	0
$183$	2.68954	0.198817
$184$	0	0
$185$	−7.89596	−0.580522
$186$	0	0
$187$	−10.6715	−0.780376
$188$	0	0
$189$	−14.7248	−1.07107
$190$	0	0
$191$	−6.03529	−0.436699	−0.218349	−	0.975871i	$-0.570067\pi$
$191$	−6.03529	−0.436699	−0.218349	+	0.975871i	$0.570067\pi$
$192$	0	0
$193$	−4.42823	−0.318751	−0.159375	−	0.987218i	$-0.550948\pi$
$193$	−4.42823	−0.318751	−0.159375	+	0.987218i	$0.550948\pi$
$194$	0	0
$195$	−3.88860	−0.278469
$196$	0	0
$197$	16.9957	1.21089	0.605446	−	0.795887i	$-0.292995\pi$
$197$	16.9957	1.21089	0.605446	+	0.795887i	$0.292995\pi$
$198$	0	0
$199$	22.4609	1.59221	0.796104	−	0.605160i	$-0.206891\pi$
$199$	22.4609	1.59221	0.796104	+	0.605160i	$0.206891\pi$
$200$	0	0
$201$	−10.5062	−0.741048
$202$	0	0
$203$	−20.8590	−1.46401
$204$	0	0
$205$	−10.5142	−0.734345
$206$	0	0
$207$	2.15944	0.150091
$208$	0	0
$209$	−2.45617	−0.169897
$210$	0	0
$211$	4.22533	0.290883	0.145442	−	0.989367i	$-0.453540\pi$
$211$	4.22533	0.290883	0.145442	+	0.989367i	$0.453540\pi$
$212$	0	0
$213$	8.32418	0.570363
$214$	0	0
$215$	2.95568	0.201576
$216$	0	0
$217$	−8.15727	−0.553752
$218$	0	0
$219$	3.01961	0.204047
$220$	0	0
$221$	15.4291	1.03787
$222$	0	0
$223$	13.8673	0.928624	0.464312	−	0.885672i	$-0.346302\pi$
$223$	13.8673	0.928624	0.464312	+	0.885672i	$0.346302\pi$
$224$	0	0
$225$	−1.80094	−0.120063
$226$	0	0
$227$	25.0211	1.66071	0.830354	−	0.557237i	$-0.188138\pi$
$227$	25.0211	1.66071	0.830354	+	0.557237i	$0.188138\pi$
$228$	0	0
$229$	19.4699	1.28661	0.643303	−	0.765611i	$-0.277563\pi$
$229$	19.4699	1.28661	0.643303	+	0.765611i	$0.277563\pi$
$230$	0	0
$231$	7.53325	0.495651
$232$	0	0
$233$	−2.58815	−0.169556	−0.0847778	−	0.996400i	$-0.527018\pi$
$233$	−2.58815	−0.169556	−0.0847778	+	0.996400i	$0.527018\pi$
$234$	0	0
$235$	−13.1457	−0.857532
$236$	0	0
$237$	−10.7223	−0.696488
$238$	0	0
$239$	10.0500	0.650080	0.325040	−	0.945700i	$-0.394622\pi$
$239$	10.0500	0.650080	0.325040	+	0.945700i	$0.394622\pi$
$240$	0	0
$241$	11.4677	0.738701	0.369351	−	0.929290i	$-0.379580\pi$
$241$	11.4677	0.738701	0.369351	+	0.929290i	$0.379580\pi$
$242$	0	0
$243$	15.3839	0.986878
$244$	0	0
$245$	0.845261	0.0540017
$246$	0	0
$247$	3.55118	0.225956
$248$	0	0
$249$	7.61658	0.482681
$250$	0	0
$251$	−17.6725	−1.11548	−0.557738	−	0.830017i	$-0.688331\pi$
$251$	−17.6725	−1.11548	−0.557738	+	0.830017i	$0.688331\pi$
$252$	0	0
$253$	−2.94509	−0.185156
$254$	0	0
$255$	−4.75760	−0.297932
$256$	0	0
$257$	−23.6879	−1.47761	−0.738804	−	0.673920i	$-0.764609\pi$
$257$	−23.6879	−1.47761	−0.738804	+	0.673920i	$0.764609\pi$
$258$	0	0
$259$	−22.1161	−1.37423
$260$	0	0
$261$	13.4118	0.830172
$262$	0	0
$263$	−10.4381	−0.643642	−0.321821	−	0.946801i	$-0.604295\pi$
$263$	−10.4381	−0.643642	−0.321821	+	0.946801i	$0.604295\pi$
$264$	0	0
$265$	−3.74122	−0.229821
$266$	0	0
$267$	9.30710	0.569585
$268$	0	0
$269$	−21.0137	−1.28123	−0.640615	−	0.767862i	$-0.721320\pi$
$269$	−21.0137	−1.28123	−0.640615	+	0.767862i	$0.721320\pi$
$270$	0	0
$271$	−14.4028	−0.874909	−0.437454	−	0.899241i	$-0.644120\pi$
$271$	−14.4028	−0.874909	−0.437454	+	0.899241i	$0.644120\pi$
$272$	0	0
$273$	−10.8917	−0.659198
$274$	0	0
$275$	2.45617	0.148112
$276$	0	0
$277$	13.6905	0.822584	0.411292	−	0.911504i	$-0.365078\pi$
$277$	13.6905	0.822584	0.411292	+	0.911504i	$0.365078\pi$
$278$	0	0
$279$	5.24494	0.314006
$280$	0	0
$281$	−3.31046	−0.197485	−0.0987426	−	0.995113i	$-0.531482\pi$
$281$	−3.31046	−0.197485	−0.0987426	+	0.995113i	$0.531482\pi$
$282$	0	0
$283$	4.62814	0.275115	0.137557	−	0.990494i	$-0.456075\pi$
$283$	4.62814	0.275115	0.137557	+	0.990494i	$0.456075\pi$
$284$	0	0
$285$	−1.09502	−0.0648632
$286$	0	0
$287$	−29.4497	−1.73836
$288$	0	0
$289$	1.87704	0.110414
$290$	0	0
$291$	2.20690	0.129371
$292$	0	0
$293$	−27.7867	−1.62332	−0.811659	−	0.584132i	$-0.801435\pi$
$293$	−27.7867	−1.62332	−0.811659	+	0.584132i	$0.801435\pi$
$294$	0	0
$295$	12.3595	0.719596
$296$	0	0
$297$	−12.9123	−0.749250
$298$	0	0
$299$	4.25809	0.246251
$300$	0	0
$301$	8.27868	0.477175
$302$	0	0
$303$	−5.42655	−0.311747
$304$	0	0
$305$	2.45617	0.140640
$306$	0	0
$307$	20.1260	1.14865	0.574325	−	0.818627i	$-0.305265\pi$
$307$	20.1260	1.14865	0.574325	+	0.818627i	$0.305265\pi$
$308$	0	0
$309$	−4.20257	−0.239076
$310$	0	0
$311$	21.5611	1.22262	0.611308	−	0.791393i	$-0.290644\pi$
$311$	21.5611	1.22262	0.611308	+	0.791393i	$0.290644\pi$
$312$	0	0
$313$	8.02059	0.453351	0.226675	−	0.973970i	$-0.427214\pi$
$313$	8.02059	0.453351	0.226675	+	0.973970i	$0.427214\pi$
$314$	0	0
$315$	−5.04432	−0.284215
$316$	0	0
$317$	29.0156	1.62968	0.814838	−	0.579689i	$-0.196826\pi$
$317$	29.0156	1.62968	0.814838	+	0.579689i	$0.196826\pi$
$318$	0	0
$319$	−18.2914	−1.02412
$320$	0	0
$321$	10.5062	0.586397
$322$	0	0
$323$	4.34477	0.241750
$324$	0	0
$325$	−3.55118	−0.196984
$326$	0	0
$327$	−0.585620	−0.0323848
$328$	0	0
$329$	−36.8203	−2.02997
$330$	0	0
$331$	−18.5529	−1.01976	−0.509879	−	0.860246i	$-0.670310\pi$
$331$	−18.5529	−1.01976	−0.509879	+	0.860246i	$0.670310\pi$
$332$	0	0
$333$	14.2201	0.779259
$334$	0	0
$335$	−9.59453	−0.524205
$336$	0	0
$337$	13.1144	0.714388	0.357194	−	0.934030i	$-0.383734\pi$
$337$	13.1144	0.714388	0.357194	+	0.934030i	$0.383734\pi$
$338$	0	0
$339$	14.9884	0.814060
$340$	0	0
$341$	−7.15318	−0.387367
$342$	0	0
$343$	−17.2391	−0.930821
$344$	0	0
$345$	−1.31299	−0.0706891
$346$	0	0
$347$	−23.1277	−1.24156	−0.620779	−	0.783986i	$-0.713183\pi$
$347$	−23.1277	−1.24156	−0.620779	+	0.783986i	$0.713183\pi$
$348$	0	0
$349$	16.9957	0.909757	0.454879	−	0.890553i	$-0.349683\pi$
$349$	16.9957	0.909757	0.454879	+	0.890553i	$0.349683\pi$
$350$	0	0
$351$	18.6690	0.996475
$352$	0	0
$353$	28.7542	1.53043	0.765217	−	0.643772i	$-0.222632\pi$
$353$	28.7542	1.53043	0.765217	+	0.643772i	$0.222632\pi$
$354$	0	0
$355$	7.60188	0.403466
$356$	0	0
$357$	−13.3257	−0.705273
$358$	0	0
$359$	−1.97725	−0.104355	−0.0521776	−	0.998638i	$-0.516616\pi$
$359$	−1.97725	−0.104355	−0.0521776	+	0.998638i	$0.516616\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−5.43921	−0.285484
$364$	0	0
$365$	2.75760	0.144339
$366$	0	0
$367$	−7.90919	−0.412857	−0.206428	−	0.978462i	$-0.566184\pi$
$367$	−7.90919	−0.412857	−0.206428	+	0.978462i	$0.566184\pi$
$368$	0	0
$369$	18.9355	0.985741
$370$	0	0
$371$	−10.4789	−0.544038
$372$	0	0
$373$	−8.96303	−0.464088	−0.232044	−	0.972705i	$-0.574541\pi$
$373$	−8.96303	−0.464088	−0.232044	+	0.972705i	$0.574541\pi$
$374$	0	0
$375$	1.09502	0.0565464
$376$	0	0
$377$	26.4462	1.36205
$378$	0	0
$379$	−36.9751	−1.89928	−0.949641	−	0.313340i	$-0.898552\pi$
$379$	−36.9751	−1.89928	−0.949641	+	0.313340i	$0.898552\pi$
$380$	0	0
$381$	−19.2437	−0.985887
$382$	0	0
$383$	−30.7469	−1.57109	−0.785546	−	0.618803i	$-0.787618\pi$
$383$	−30.7469	−1.57109	−0.785546	+	0.618803i	$0.787618\pi$
$384$	0	0
$385$	6.87958	0.350616
$386$	0	0
$387$	−5.32300	−0.270583
$388$	0	0
$389$	−32.6862	−1.65726	−0.828628	−	0.559800i	$-0.810878\pi$
$389$	−32.6862	−1.65726	−0.828628	+	0.559800i	$0.810878\pi$
$390$	0	0
$391$	5.20965	0.263463
$392$	0	0
$393$	−4.73505	−0.238852
$394$	0	0
$395$	−9.79191	−0.492685
$396$	0	0
$397$	2.27868	0.114363	0.0571817	−	0.998364i	$-0.481789\pi$
$397$	2.27868	0.114363	0.0571817	+	0.998364i	$0.481789\pi$
$398$	0	0
$399$	−3.06707	−0.153546
$400$	0	0
$401$	13.4971	0.674015	0.337007	−	0.941502i	$-0.390585\pi$
$401$	13.4971	0.674015	0.337007	+	0.941502i	$0.390585\pi$
$402$	0	0
$403$	10.3422	0.515184
$404$	0	0
$405$	−0.353800	−0.0175805
$406$	0	0
$407$	−19.3938	−0.961314
$408$	0	0
$409$	19.5691	0.967631	0.483815	−	0.875170i	$-0.339251\pi$
$409$	19.5691	0.967631	0.483815	+	0.875170i	$0.339251\pi$
$410$	0	0
$411$	−6.17280	−0.304482
$412$	0	0
$413$	34.6181	1.70345
$414$	0	0
$415$	6.95568	0.341441
$416$	0	0
$417$	22.7189	1.11255
$418$	0	0
$419$	−15.7507	−0.769473	−0.384737	−	0.923026i	$-0.625708\pi$
$419$	−15.7507	−0.769473	−0.384737	+	0.923026i	$0.625708\pi$
$420$	0	0
$421$	−9.16846	−0.446844	−0.223422	−	0.974722i	$-0.571723\pi$
$421$	−9.16846	−0.446844	−0.223422	+	0.974722i	$0.571723\pi$
$422$	0	0
$423$	23.6746	1.15110
$424$	0	0
$425$	−4.34477	−0.210752
$426$	0	0
$427$	6.87958	0.332926
$428$	0	0
$429$	−9.55106	−0.461130
$430$	0	0
$431$	10.2606	0.494237	0.247118	−	0.968985i	$-0.420516\pi$
$431$	10.2606	0.494237	0.247118	+	0.968985i	$0.420516\pi$
$432$	0	0
$433$	−26.1643	−1.25737	−0.628687	−	0.777659i	$-0.716407\pi$
$433$	−26.1643	−1.25737	−0.628687	+	0.777659i	$0.716407\pi$
$434$	0	0
$435$	−8.15474	−0.390990
$436$	0	0
$437$	1.19906	0.0573589
$438$	0	0
$439$	−14.0853	−0.672254	−0.336127	−	0.941817i	$-0.609117\pi$
$439$	−14.0853	−0.672254	−0.336127	+	0.941817i	$0.609117\pi$
$440$	0	0
$441$	−1.52226	−0.0724887
$442$	0	0
$443$	17.1751	0.816014	0.408007	−	0.912979i	$-0.366224\pi$
$443$	17.1751	0.816014	0.408007	+	0.912979i	$0.366224\pi$
$444$	0	0
$445$	8.49951	0.402915
$446$	0	0
$447$	18.9217	0.894967
$448$	0	0
$449$	−15.1024	−0.712725	−0.356362	−	0.934348i	$-0.615983\pi$
$449$	−15.1024	−0.712725	−0.356362	+	0.934348i	$0.615983\pi$
$450$	0	0
$451$	−25.8247	−1.21604
$452$	0	0
$453$	−8.87115	−0.416803
$454$	0	0
$455$	−9.94665	−0.466306
$456$	0	0
$457$	−25.5960	−1.19733	−0.598665	−	0.801000i	$-0.704302\pi$
$457$	−25.5960	−1.19733	−0.598665	+	0.801000i	$0.704302\pi$
$458$	0	0
$459$	22.8409	1.06612
$460$	0	0
$461$	−32.8531	−1.53012	−0.765061	−	0.643958i	$-0.777291\pi$
$461$	−32.8531	−1.53012	−0.765061	+	0.643958i	$0.777291\pi$
$462$	0	0
$463$	26.8182	1.24635	0.623173	−	0.782084i	$-0.285843\pi$
$463$	26.8182	1.24635	0.623173	+	0.782084i	$0.285843\pi$
$464$	0	0
$465$	−3.18905	−0.147889
$466$	0	0
$467$	−13.7074	−0.634302	−0.317151	−	0.948375i	$-0.602726\pi$
$467$	−13.7074	−0.634302	−0.317151	+	0.948375i	$0.602726\pi$
$468$	0	0
$469$	−26.8737	−1.24091
$470$	0	0
$471$	7.40245	0.341087
$472$	0	0
$473$	7.25964	0.333799
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	6.73770	0.308498
$478$	0	0
$479$	−15.3685	−0.702205	−0.351102	−	0.936337i	$-0.614193\pi$
$479$	−15.3685	−0.702205	−0.351102	+	0.936337i	$0.614193\pi$
$480$	0	0
$481$	28.0400	1.27851
$482$	0	0
$483$	−3.67761	−0.167337
$484$	0	0
$485$	2.01540	0.0915147
$486$	0	0
$487$	−11.6445	−0.527664	−0.263832	−	0.964569i	$-0.584986\pi$
$487$	−11.6445	−0.527664	−0.263832	+	0.964569i	$0.584986\pi$
$488$	0	0
$489$	4.23554	0.191538
$490$	0	0
$491$	−30.7421	−1.38737	−0.693685	−	0.720278i	$-0.744014\pi$
$491$	−30.7421	−1.38737	−0.693685	+	0.720278i	$0.744014\pi$
$492$	0	0
$493$	32.3561	1.45725
$494$	0	0
$495$	−4.42341	−0.198817
$496$	0	0
$497$	21.2924	0.955095
$498$	0	0
$499$	−4.43811	−0.198677	−0.0993386	−	0.995054i	$-0.531673\pi$
$499$	−4.43811	−0.198677	−0.0993386	+	0.995054i	$0.531673\pi$
$500$	0	0
$501$	−25.6830	−1.14743
$502$	0	0
$503$	23.2104	1.03490	0.517451	−	0.855713i	$-0.326881\pi$
$503$	23.2104	1.03490	0.517451	+	0.855713i	$0.326881\pi$
$504$	0	0
$505$	−4.95568	−0.220525
$506$	0	0
$507$	−0.426064	−0.0189221
$508$	0	0
$509$	−14.4403	−0.640054	−0.320027	−	0.947408i	$-0.603692\pi$
$509$	−14.4403	−0.640054	−0.320027	+	0.947408i	$0.603692\pi$
$510$	0	0
$511$	7.72386	0.341683
$512$	0	0
$513$	5.25711	0.232107
$514$	0	0
$515$	−3.83791	−0.169118
$516$	0	0
$517$	−32.2881	−1.42003
$518$	0	0
$519$	22.7656	0.999301
$520$	0	0
$521$	33.6293	1.47333	0.736664	−	0.676259i	$-0.236400\pi$
$521$	33.6293	1.47333	0.736664	+	0.676259i	$0.236400\pi$
$522$	0	0
$523$	−18.8542	−0.824435	−0.412218	−	0.911085i	$-0.635246\pi$
$523$	−18.8542	−0.824435	−0.412218	+	0.911085i	$0.635246\pi$
$524$	0	0
$525$	3.06707	0.133858
$526$	0	0
$527$	12.6534	0.551192
$528$	0	0
$529$	−21.5623	−0.937489
$530$	0	0
$531$	−22.2587	−0.965944
$532$	0	0
$533$	37.3379	1.61728
$534$	0	0
$535$	9.59453	0.414808
$536$	0	0
$537$	7.96920	0.343896
$538$	0	0
$539$	2.07610	0.0894241
$540$	0	0
$541$	21.3651	0.918560	0.459280	−	0.888292i	$-0.348108\pi$
$541$	21.3651	0.918560	0.459280	+	0.888292i	$0.348108\pi$
$542$	0	0
$543$	−9.66209	−0.414640
$544$	0	0
$545$	−0.534804	−0.0229085
$546$	0	0
$547$	−31.6118	−1.35162	−0.675811	−	0.737075i	$-0.736206\pi$
$547$	−31.6118	−1.35162	−0.675811	+	0.737075i	$0.736206\pi$
$548$	0	0
$549$	−4.42341	−0.188786
$550$	0	0
$551$	7.44714	0.317259
$552$	0	0
$553$	−27.4265	−1.16630
$554$	0	0
$555$	−8.64620	−0.367011
$556$	0	0
$557$	−11.4858	−0.486668	−0.243334	−	0.969943i	$-0.578241\pi$
$557$	−11.4858	−0.486668	−0.243334	+	0.969943i	$0.578241\pi$
$558$	0	0
$559$	−10.4962	−0.443940
$560$	0	0
$561$	−11.6855	−0.493360
$562$	0	0
$563$	17.4725	0.736380	0.368190	−	0.929751i	$-0.379978\pi$
$563$	17.4725	0.736380	0.368190	+	0.929751i	$0.379978\pi$
$564$	0	0
$565$	13.6879	0.575853
$566$	0	0
$567$	−0.990972	−0.0416169
$568$	0	0
$569$	−16.6748	−0.699046	−0.349523	−	0.936928i	$-0.613656\pi$
$569$	−16.6748	−0.699046	−0.349523	+	0.936928i	$0.613656\pi$
$570$	0	0
$571$	−9.23435	−0.386446	−0.193223	−	0.981155i	$-0.561894\pi$
$571$	−9.23435	−0.386446	−0.193223	+	0.981155i	$0.561894\pi$
$572$	0	0
$573$	−6.60874	−0.276084
$574$	0	0
$575$	−1.19906	−0.0500043
$576$	0	0
$577$	−30.4281	−1.26674	−0.633369	−	0.773850i	$-0.718329\pi$
$577$	−30.4281	−1.26674	−0.633369	+	0.773850i	$0.718329\pi$
$578$	0	0
$579$	−4.84898	−0.201517
$580$	0	0
$581$	19.4824	0.808268
$582$	0	0
$583$	−9.18905	−0.380572
$584$	0	0
$585$	6.39547	0.264420
$586$	0	0
$587$	−15.6040	−0.644048	−0.322024	−	0.946732i	$-0.604363\pi$
$587$	−15.6040	−0.644048	−0.322024	+	0.946732i	$0.604363\pi$
$588$	0	0
$589$	2.91234	0.120001
$590$	0	0
$591$	18.6105	0.765535
$592$	0	0
$593$	13.9639	0.573428	0.286714	−	0.958016i	$-0.407437\pi$
$593$	13.9639	0.573428	0.286714	+	0.958016i	$0.407437\pi$
$594$	0	0
$595$	−12.1694	−0.498898
$596$	0	0
$597$	24.5950	1.00661
$598$	0	0
$599$	23.7033	0.968489	0.484245	−	0.874933i	$-0.339094\pi$
$599$	23.7033	0.968489	0.484245	+	0.874933i	$0.339094\pi$
$600$	0	0
$601$	−46.5648	−1.89942	−0.949709	−	0.313135i	$-0.898621\pi$
$601$	−46.5648	−1.89942	−0.949709	+	0.313135i	$0.898621\pi$
$602$	0	0
$603$	17.2792	0.703662
$604$	0	0
$605$	−4.96724	−0.201947
$606$	0	0
$607$	−32.7957	−1.33114	−0.665569	−	0.746336i	$-0.731811\pi$
$607$	−32.7957	−1.33114	−0.665569	+	0.746336i	$0.731811\pi$
$608$	0	0
$609$	−22.8409	−0.925561
$610$	0	0
$611$	46.6828	1.88859
$612$	0	0
$613$	−2.98432	−0.120536	−0.0602678	−	0.998182i	$-0.519195\pi$
$613$	−2.98432	−0.120536	−0.0602678	+	0.998182i	$0.519195\pi$
$614$	0	0
$615$	−11.5132	−0.464258
$616$	0	0
$617$	28.0833	1.13059	0.565296	−	0.824888i	$-0.308762\pi$
$617$	28.0833	1.13059	0.565296	+	0.824888i	$0.308762\pi$
$618$	0	0
$619$	31.1551	1.25223	0.626115	−	0.779731i	$-0.284644\pi$
$619$	31.1551	1.25223	0.626115	+	0.779731i	$0.284644\pi$
$620$	0	0
$621$	6.30359	0.252954
$622$	0	0
$623$	23.8066	0.953792
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.68954	−0.107410
$628$	0	0
$629$	34.3061	1.36787
$630$	0	0
$631$	25.3665	1.00983	0.504913	−	0.863170i	$-0.331525\pi$
$631$	25.3665	1.00983	0.504913	+	0.863170i	$0.331525\pi$
$632$	0	0
$633$	4.62680	0.183899
$634$	0	0
$635$	−17.5739	−0.697400
$636$	0	0
$637$	−3.00168	−0.118931
$638$	0	0
$639$	−13.6905	−0.541589
$640$	0	0
$641$	30.4942	1.20445	0.602224	−	0.798327i	$-0.294281\pi$
$641$	30.4942	1.20445	0.602224	+	0.798327i	$0.294281\pi$
$642$	0	0
$643$	−23.6599	−0.933056	−0.466528	−	0.884506i	$-0.654495\pi$
$643$	−23.6599	−0.933056	−0.466528	+	0.884506i	$0.654495\pi$
$644$	0	0
$645$	3.23652	0.127438
$646$	0	0
$647$	−48.9298	−1.92363	−0.961814	−	0.273704i	$-0.911751\pi$
$647$	−48.9298	−1.92363	−0.961814	+	0.273704i	$0.911751\pi$
$648$	0	0
$649$	30.3569	1.19161
$650$	0	0
$651$	−8.93235	−0.350086
$652$	0	0
$653$	−9.98530	−0.390755	−0.195377	−	0.980728i	$-0.562593\pi$
$653$	−9.98530	−0.390755	−0.195377	+	0.980728i	$0.562593\pi$
$654$	0	0
$655$	−4.32418	−0.168960
$656$	0	0
$657$	−4.96626	−0.193752
$658$	0	0
$659$	28.2889	1.10198	0.550989	−	0.834512i	$-0.314251\pi$
$659$	28.2889	1.10198	0.550989	+	0.834512i	$0.314251\pi$
$660$	0	0
$661$	32.3722	1.25913	0.629567	−	0.776946i	$-0.283232\pi$
$661$	32.3722	1.25913	0.629567	+	0.776946i	$0.283232\pi$
$662$	0	0
$663$	16.8951	0.656151
$664$	0	0
$665$	−2.80094	−0.108616
$666$	0	0
$667$	8.92957	0.345754
$668$	0	0
$669$	15.1849	0.587084
$670$	0	0
$671$	6.03276	0.232892
$672$	0	0
$673$	6.15993	0.237448	0.118724	−	0.992927i	$-0.462120\pi$
$673$	6.15993	0.237448	0.118724	+	0.992927i	$0.462120\pi$
$674$	0	0
$675$	−5.25711	−0.202346
$676$	0	0
$677$	−28.1962	−1.08367	−0.541834	−	0.840486i	$-0.682270\pi$
$677$	−28.1962	−1.08367	−0.541834	+	0.840486i	$0.682270\pi$
$678$	0	0
$679$	5.64502	0.216636
$680$	0	0
$681$	27.3985	1.04991
$682$	0	0
$683$	6.85163	0.262170	0.131085	−	0.991371i	$-0.458154\pi$
$683$	6.85163	0.262170	0.131085	+	0.991371i	$0.458154\pi$
$684$	0	0
$685$	−5.63717	−0.215385
$686$	0	0
$687$	21.3198	0.813403
$688$	0	0
$689$	13.2857	0.506147
$690$	0	0
$691$	−14.2334	−0.541463	−0.270732	−	0.962655i	$-0.587266\pi$
$691$	−14.2334	−0.541463	−0.270732	+	0.962655i	$0.587266\pi$
$692$	0	0
$693$	−12.3897	−0.470646
$694$	0	0
$695$	20.7476	0.787001
$696$	0	0
$697$	45.6819	1.73032
$698$	0	0
$699$	−2.83407	−0.107194
$700$	0	0
$701$	38.1561	1.44114	0.720568	−	0.693385i	$-0.243881\pi$
$701$	38.1561	1.44114	0.720568	+	0.693385i	$0.243881\pi$
$702$	0	0
$703$	7.89596	0.297802
$704$	0	0
$705$	−14.3948	−0.542138
$706$	0	0
$707$	−13.8806	−0.522032
$708$	0	0
$709$	−24.3767	−0.915487	−0.457743	−	0.889084i	$-0.651342\pi$
$709$	−24.3767	−0.915487	−0.457743	+	0.889084i	$0.651342\pi$
$710$	0	0
$711$	17.6346	0.661351
$712$	0	0
$713$	3.49207	0.130779
$714$	0	0
$715$	−8.72230	−0.326196
$716$	0	0
$717$	11.0049	0.410986
$718$	0	0
$719$	2.24554	0.0837447	0.0418723	−	0.999123i	$-0.486668\pi$
$719$	2.24554	0.0837447	0.0418723	+	0.999123i	$0.486668\pi$
$720$	0	0
$721$	−10.7498	−0.400342
$722$	0	0
$723$	12.5574	0.467013
$724$	0	0
$725$	−7.44714	−0.276580
$726$	0	0
$727$	−4.61091	−0.171009	−0.0855045	−	0.996338i	$-0.527250\pi$
$727$	−4.61091	−0.171009	−0.0855045	+	0.996338i	$0.527250\pi$
$728$	0	0
$729$	17.9070	0.663223
$730$	0	0
$731$	−12.8417	−0.474969
$732$	0	0
$733$	−35.2820	−1.30317	−0.651586	−	0.758575i	$-0.725896\pi$
$733$	−35.2820	−1.30317	−0.651586	+	0.758575i	$0.725896\pi$
$734$	0	0
$735$	0.925574	0.0341403
$736$	0	0
$737$	−23.5658	−0.868056
$738$	0	0
$739$	51.5065	1.89470	0.947349	−	0.320202i	$-0.103751\pi$
$739$	51.5065	1.89470	0.947349	+	0.320202i	$0.103751\pi$
$740$	0	0
$741$	3.88860	0.142851
$742$	0	0
$743$	−12.1587	−0.446061	−0.223030	−	0.974811i	$-0.571595\pi$
$743$	−12.1587	−0.446061	−0.223030	+	0.974811i	$0.571595\pi$
$744$	0	0
$745$	17.2799	0.633085
$746$	0	0
$747$	−12.5268	−0.458330
$748$	0	0
$749$	26.8737	0.981943
$750$	0	0
$751$	−21.7739	−0.794539	−0.397270	−	0.917702i	$-0.630042\pi$
$751$	−21.7739	−0.794539	−0.397270	+	0.917702i	$0.630042\pi$
$752$	0	0
$753$	−19.3516	−0.705213
$754$	0	0
$755$	−8.10139	−0.294840
$756$	0	0
$757$	−28.3473	−1.03030	−0.515150	−	0.857100i	$-0.672264\pi$
$757$	−28.3473	−1.03030	−0.515150	+	0.857100i	$0.672264\pi$
$758$	0	0
$759$	−3.22493	−0.117057
$760$	0	0
$761$	−54.1535	−1.96306	−0.981532	−	0.191297i	$-0.938731\pi$
$761$	−54.1535	−1.96306	−0.981532	+	0.191297i	$0.938731\pi$
$762$	0	0
$763$	−1.49795	−0.0542296
$764$	0	0
$765$	7.82467	0.282902
$766$	0	0
$767$	−43.8908	−1.58480
$768$	0	0
$769$	−44.7795	−1.61479	−0.807396	−	0.590011i	$-0.799124\pi$
$769$	−44.7795	−1.61479	−0.807396	+	0.590011i	$0.799124\pi$
$770$	0	0
$771$	−25.9386	−0.934156
$772$	0	0
$773$	−17.5331	−0.630623	−0.315311	−	0.948988i	$-0.602109\pi$
$773$	−17.5331	−0.630623	−0.315311	+	0.948988i	$0.602109\pi$
$774$	0	0
$775$	−2.91234	−0.104614
$776$	0	0
$777$	−24.2175	−0.868797
$778$	0	0
$779$	10.5142	0.376711
$780$	0	0
$781$	18.6715	0.668118
$782$	0	0
$783$	39.1504	1.39912
$784$	0	0
$785$	6.76013	0.241279
$786$	0	0
$787$	−6.34763	−0.226269	−0.113134	−	0.993580i	$-0.536089\pi$
$787$	−6.34763	−0.226269	−0.113134	+	0.993580i	$0.536089\pi$
$788$	0	0
$789$	−11.4299	−0.406915
$790$	0	0
$791$	38.3389	1.36317
$792$	0	0
$793$	−8.72230	−0.309738
$794$	0	0
$795$	−4.09669	−0.145295
$796$	0	0
$797$	−3.56393	−0.126241	−0.0631204	−	0.998006i	$-0.520105\pi$
$797$	−3.56393	−0.126241	−0.0631204	+	0.998006i	$0.520105\pi$
$798$	0	0
$799$	57.1151	2.02059
$800$	0	0
$801$	−15.3071	−0.540850
$802$	0	0
$803$	6.77312	0.239018
$804$	0	0
$805$	−3.35850	−0.118371
$806$	0	0
$807$	−23.0104	−0.810003
$808$	0	0
$809$	0.330069	0.0116046	0.00580230	−	0.999983i	$-0.498153\pi$
$809$	0.330069	0.0116046	0.00580230	+	0.999983i	$0.498153\pi$
$810$	0	0
$811$	−29.6827	−1.04230	−0.521150	−	0.853465i	$-0.674497\pi$
$811$	−29.6827	−1.04230	−0.521150	+	0.853465i	$0.674497\pi$
$812$	0	0
$813$	−15.7713	−0.553124
$814$	0	0
$815$	3.86801	0.135491
$816$	0	0
$817$	−2.95568	−0.103406
$818$	0	0
$819$	17.9133	0.625942
$820$	0	0
$821$	20.4179	0.712589	0.356295	−	0.934374i	$-0.384040\pi$
$821$	20.4179	0.712589	0.356295	+	0.934374i	$0.384040\pi$
$822$	0	0
$823$	−8.14922	−0.284064	−0.142032	−	0.989862i	$-0.545364\pi$
$823$	−8.14922	−0.284064	−0.142032	+	0.989862i	$0.545364\pi$
$824$	0	0
$825$	2.68954	0.0936379
$826$	0	0
$827$	25.7993	0.897128	0.448564	−	0.893751i	$-0.351936\pi$
$827$	25.7993	0.897128	0.448564	+	0.893751i	$0.351936\pi$
$828$	0	0
$829$	3.39968	0.118076	0.0590378	−	0.998256i	$-0.481197\pi$
$829$	3.39968	0.118076	0.0590378	+	0.998256i	$0.481197\pi$
$830$	0	0
$831$	14.9913	0.520044
$832$	0	0
$833$	−3.67247	−0.127243
$834$	0	0
$835$	−23.4545	−0.811676
$836$	0	0
$837$	15.3105	0.529207
$838$	0	0
$839$	12.8384	0.443231	0.221615	−	0.975134i	$-0.428867\pi$
$839$	12.8384	0.443231	0.221615	+	0.975134i	$0.428867\pi$
$840$	0	0
$841$	26.4599	0.912410
$842$	0	0
$843$	−3.62500	−0.124852
$844$	0	0
$845$	−0.389093	−0.0133852
$846$	0	0
$847$	−13.9129	−0.478054
$848$	0	0
$849$	5.06789	0.173930
$850$	0	0
$851$	9.46773	0.324550
$852$	0	0
$853$	−1.29436	−0.0443179	−0.0221590	−	0.999754i	$-0.507054\pi$
$853$	−1.29436	−0.0443179	−0.0221590	+	0.999754i	$0.507054\pi$
$854$	0	0
$855$	1.80094	0.0615908
$856$	0	0
$857$	−1.20977	−0.0413248	−0.0206624	−	0.999787i	$-0.506578\pi$
$857$	−1.20977	−0.0413248	−0.0206624	+	0.999787i	$0.506578\pi$
$858$	0	0
$859$	−54.0261	−1.84335	−0.921673	−	0.387969i	$-0.873177\pi$
$859$	−54.0261	−1.84335	−0.921673	+	0.387969i	$0.873177\pi$
$860$	0	0
$861$	−32.2479	−1.09900
$862$	0	0
$863$	−27.9654	−0.951952	−0.475976	−	0.879458i	$-0.657905\pi$
$863$	−27.9654	−0.951952	−0.475976	+	0.879458i	$0.657905\pi$
$864$	0	0
$865$	20.7902	0.706889
$866$	0	0
$867$	2.05539	0.0698047
$868$	0	0
$869$	−24.0506	−0.815860
$870$	0	0
$871$	34.0719	1.15448
$872$	0	0
$873$	−3.62962	−0.122844
$874$	0	0
$875$	2.80094	0.0946890
$876$	0	0
$877$	−6.75829	−0.228211	−0.114106	−	0.993469i	$-0.536400\pi$
$877$	−6.75829	−0.228211	−0.114106	+	0.993469i	$0.536400\pi$
$878$	0	0
$879$	−30.4269	−1.02627
$880$	0	0
$881$	49.1665	1.65646	0.828230	−	0.560388i	$-0.189348\pi$
$881$	49.1665	1.65646	0.828230	+	0.560388i	$0.189348\pi$
$882$	0	0
$883$	−16.8001	−0.565369	−0.282685	−	0.959213i	$-0.591225\pi$
$883$	−16.8001	−0.565369	−0.282685	+	0.959213i	$0.591225\pi$
$884$	0	0
$885$	13.5338	0.454935
$886$	0	0
$887$	−9.42509	−0.316463	−0.158232	−	0.987402i	$-0.550579\pi$
$887$	−9.42509	−0.316463	−0.158232	+	0.987402i	$0.550579\pi$
$888$	0	0
$889$	−49.2235	−1.65090
$890$	0	0
$891$	−0.868992	−0.0291123
$892$	0	0
$893$	13.1457	0.439904
$894$	0	0
$895$	7.27770	0.243267
$896$	0	0
$897$	4.66267	0.155682
$898$	0	0
$899$	21.6886	0.723354
$900$	0	0
$901$	16.2547	0.541524
$902$	0	0
$903$	9.06529	0.301674
$904$	0	0
$905$	−8.82369	−0.293309
$906$	0	0
$907$	−23.4045	−0.777133	−0.388567	−	0.921421i	$-0.627030\pi$
$907$	−23.4045	−0.777133	−0.388567	+	0.921421i	$0.627030\pi$
$908$	0	0
$909$	8.92488	0.296019
$910$	0	0
$911$	15.2596	0.505574	0.252787	−	0.967522i	$-0.418653\pi$
$911$	15.2596	0.505574	0.252787	+	0.967522i	$0.418653\pi$
$912$	0	0
$913$	17.0843	0.565408
$914$	0	0
$915$	2.68954	0.0889136
$916$	0	0
$917$	−12.1118	−0.399966
$918$	0	0
$919$	57.2588	1.88879	0.944397	−	0.328807i	$-0.106647\pi$
$919$	57.2588	1.88879	0.944397	+	0.328807i	$0.106647\pi$
$920$	0	0
$921$	22.0383	0.726186
$922$	0	0
$923$	−26.9957	−0.888573
$924$	0	0
$925$	−7.89596	−0.259617
$926$	0	0
$927$	6.91184	0.227015
$928$	0	0
$929$	2.63112	0.0863244	0.0431622	−	0.999068i	$-0.486257\pi$
$929$	2.63112	0.0863244	0.0431622	+	0.999068i	$0.486257\pi$
$930$	0	0
$931$	−0.845261	−0.0277023
$932$	0	0
$933$	23.6097	0.772948
$934$	0	0
$935$	−10.6715	−0.348995
$936$	0	0
$937$	56.8145	1.85605	0.928024	−	0.372521i	$-0.121507\pi$
$937$	56.8145	1.85605	0.928024	+	0.372521i	$0.121507\pi$
$938$	0	0
$939$	8.78268	0.286612
$940$	0	0
$941$	−26.8590	−0.875578	−0.437789	−	0.899078i	$-0.644238\pi$
$941$	−26.8590	−0.875578	−0.437789	+	0.899078i	$0.644238\pi$
$942$	0	0
$943$	12.6072	0.410546
$944$	0	0
$945$	−14.7248	−0.478999
$946$	0	0
$947$	3.09318	0.100515	0.0502574	−	0.998736i	$-0.483996\pi$
$947$	3.09318	0.100515	0.0502574	+	0.998736i	$0.483996\pi$
$948$	0	0
$949$	−9.79273	−0.317885
$950$	0	0
$951$	31.7725	1.03029
$952$	0	0
$953$	27.3225	0.885063	0.442531	−	0.896753i	$-0.354081\pi$
$953$	27.3225	0.885063	0.442531	+	0.896753i	$0.354081\pi$
$954$	0	0
$955$	−6.03529	−0.195298
$956$	0	0
$957$	−20.0294	−0.647459
$958$	0	0
$959$	−15.7894	−0.509866
$960$	0	0
$961$	−22.5183	−0.726397
$962$	0	0
$963$	−17.2792	−0.556813
$964$	0	0
$965$	−4.42823	−0.142550
$966$	0	0
$967$	−2.08452	−0.0670338	−0.0335169	−	0.999438i	$-0.510671\pi$
$967$	−2.08452	−0.0670338	−0.0335169	+	0.999438i	$0.510671\pi$
$968$	0	0
$969$	4.75760	0.152836
$970$	0	0
$971$	51.0629	1.63869	0.819343	−	0.573303i	$-0.194338\pi$
$971$	51.0629	1.63869	0.819343	+	0.573303i	$0.194338\pi$
$972$	0	0
$973$	58.1127	1.86301
$974$	0	0
$975$	−3.88860	−0.124535
$976$	0	0
$977$	0.710255	0.0227231	0.0113615	−	0.999935i	$-0.496383\pi$
$977$	0.710255	0.0227231	0.0113615	+	0.999935i	$0.496383\pi$
$978$	0	0
$979$	20.8762	0.667207
$980$	0	0
$981$	0.963150	0.0307510
$982$	0	0
$983$	3.84730	0.122710	0.0613549	−	0.998116i	$-0.480458\pi$
$983$	3.84730	0.122710	0.0613549	+	0.998116i	$0.480458\pi$
$984$	0	0
$985$	16.9957	0.541527
$986$	0	0
$987$	−40.3189	−1.28336
$988$	0	0
$989$	−3.54404	−0.112694
$990$	0	0
$991$	11.3413	0.360267	0.180133	−	0.983642i	$-0.442347\pi$
$991$	11.3413	0.360267	0.180133	+	0.983642i	$0.442347\pi$
$992$	0	0
$993$	−20.3157	−0.644699
$994$	0	0
$995$	22.4609	0.712057
$996$	0	0
$997$	23.5313	0.745243	0.372622	−	0.927983i	$-0.378459\pi$
$997$	23.5313	0.745243	0.372622	+	0.927983i	$0.378459\pi$
$998$	0	0
$999$	41.5099	1.31331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.ce.1.3		4
4.3	odd	2		6080.2.a.cg.1.2		4
8.3	odd	2		3040.2.a.q.1.3	✓	4
8.5	even	2		3040.2.a.s.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.q.1.3	✓	4	8.3	odd	2
3040.2.a.s.1.2	yes	4	8.5	even	2
6080.2.a.ce.1.3		4	1.1	even	1	trivial
6080.2.a.cg.1.2		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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09502 q^{3} +1.00000 q^{5} +2.80094 q^{7} -1.80094 q^{9} +O(q^{10})\)	\(q+1.09502 q^{3} +1.00000 q^{5} +2.80094 q^{7} -1.80094 q^{9} +2.45617 q^{11} -3.55118 q^{13} +1.09502 q^{15} -4.34477 q^{17} -1.00000 q^{19} +3.06707 q^{21} -1.19906 q^{23} +1.00000 q^{25} -5.25711 q^{27} -7.44714 q^{29} -2.91234 q^{31} +2.68954 q^{33} +2.80094 q^{35} -7.89596 q^{37} -3.88860 q^{39} -10.5142 q^{41} +2.95568 q^{43} -1.80094 q^{45} -13.1457 q^{47} +0.845261 q^{49} -4.75760 q^{51} -3.74122 q^{53} +2.45617 q^{55} -1.09502 q^{57} +12.3595 q^{59} +2.45617 q^{61} -5.04432 q^{63} -3.55118 q^{65} -9.59453 q^{67} -1.31299 q^{69} +7.60188 q^{71} +2.75760 q^{73} +1.09502 q^{75} +6.87958 q^{77} -9.79191 q^{79} -0.353800 q^{81} +6.95568 q^{83} -4.34477 q^{85} -8.15474 q^{87} +8.49951 q^{89} -9.94665 q^{91} -3.18905 q^{93} -1.00000 q^{95} +2.01540 q^{97} -4.42341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9} + 6 q^{11} - 5 q^{13} - q^{15} - 5 q^{17} - 4 q^{19} + 3 q^{21} - 21 q^{23} + 4 q^{25} - q^{27} + q^{29} - 4 q^{31} - 14 q^{33} - 5 q^{35} - 10 q^{37} - 7 q^{39} - 2 q^{41} - 6 q^{43} + 9 q^{45} - 24 q^{47} + 5 q^{49} - 13 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 11 q^{59} + 6 q^{61} - 38 q^{63} - 5 q^{65} - 19 q^{67} + 7 q^{69} - 2 q^{71} + 5 q^{73} - q^{75} - 8 q^{77} + 4 q^{79} - 16 q^{81} + 10 q^{83} - 5 q^{85} - 31 q^{87} + 20 q^{89} + 5 q^{91} + 26 q^{93} - 4 q^{95} - 6 q^{97} + 14 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9 + 6 * q^11 - 5 * q^13 - q^15 - 5 * q^17 - 4 * q^19 + 3 * q^21 - 21 * q^23 + 4 * q^25 - q^27 + q^29 - 4 * q^31 - 14 * q^33 - 5 * q^35 - 10 * q^37 - 7 * q^39 - 2 * q^41 - 6 * q^43 + 9 * q^45 - 24 * q^47 + 5 * q^49 - 13 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 11 * q^59 + 6 * q^61 - 38 * q^63 - 5 * q^65 - 19 * q^67 + 7 * q^69 - 2 * q^71 + 5 * q^73 - q^75 - 8 * q^77 + 4 * q^79 - 16 * q^81 + 10 * q^83 - 5 * q^85 - 31 * q^87 + 20 * q^89 + 5 * q^91 + 26 * q^93 - 4 * q^95 - 6 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more