LMFDB - Embedded newform 6292.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.223824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 2x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 2x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.29200$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13260 q^{3} -1.36675 q^{5} -0.330724 q^{7} +1.54798 q^{9} +O(q^{10})$	$q+2.13260 q^{3} -1.36675 q^{5} -0.330724 q^{7} +1.54798 q^{9} +1.00000 q^{13} -2.91473 q^{15} -2.24546 q^{17} +5.48246 q^{19} -0.705303 q^{21} +7.30062 q^{23} -3.13199 q^{25} -3.09657 q^{27} +0.475928 q^{29} -7.05864 q^{31} +0.452018 q^{35} +4.44643 q^{37} +2.13260 q^{39} +4.89845 q^{41} +10.0778 q^{43} -2.11571 q^{45} +1.43880 q^{47} -6.89062 q^{49} -4.78866 q^{51} -5.54668 q^{53} +11.6919 q^{57} +9.54892 q^{59} +6.66369 q^{61} -0.511955 q^{63} -1.36675 q^{65} +1.04141 q^{67} +15.5693 q^{69} +9.19063 q^{71} -0.133208 q^{73} -6.67928 q^{75} +9.45011 q^{79} -11.2477 q^{81} -2.13199 q^{83} +3.06898 q^{85} +1.01496 q^{87} -2.25308 q^{89} -0.330724 q^{91} -15.0533 q^{93} -7.49316 q^{95} +7.37745 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q - q^3 + 2 * q^7 + 2 * q^9	$ 5 q - q^{3} + 2 q^{7} + 2 q^{9} + 5 q^{13} - 2 q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 7 q^{23} - q^{25} - 7 q^{27} + 3 q^{29} + 10 q^{31} + 8 q^{35} - q^{39} + 8 q^{41} - 5 q^{43} + 8 q^{45} - 6 q^{47} - 7 q^{49} + 3 q^{51} + 15 q^{53} + 20 q^{57} - 12 q^{59} + 9 q^{61} + 10 q^{67} - 7 q^{69} - 14 q^{71} + 8 q^{73} + 21 q^{75} + 13 q^{79} - 23 q^{81} + 4 q^{83} + 24 q^{85} - 7 q^{87} + 14 q^{89} + 2 q^{91} - 22 q^{93} + 12 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q - q^3 + 2 * q^7 + 2 * q^9 + 5 * q^13 - 2 * q^15 + 5 * q^17 + 2 * q^19 - 4 * q^21 + 7 * q^23 - q^25 - 7 * q^27 + 3 * q^29 + 10 * q^31 + 8 * q^35 - q^39 + 8 * q^41 - 5 * q^43 + 8 * q^45 - 6 * q^47 - 7 * q^49 + 3 * q^51 + 15 * q^53 + 20 * q^57 - 12 * q^59 + 9 * q^61 + 10 * q^67 - 7 * q^69 - 14 * q^71 + 8 * q^73 + 21 * q^75 + 13 * q^79 - 23 * q^81 + 4 * q^83 + 24 * q^85 - 7 * q^87 + 14 * q^89 + 2 * q^91 - 22 * q^93 + 12 * q^95 + 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.13260	1.23126	0.615629	−	0.788036i	$-0.288902\pi$
$3$	2.13260	1.23126	0.615629	+	0.788036i	$0.288902\pi$
$4$	0	0
$5$	−1.36675	−0.611230	−0.305615	−	0.952155i	$-0.598862\pi$
$5$	−1.36675	−0.611230	−0.305615	+	0.952155i	$0.598862\pi$
$6$	0	0
$7$	−0.330724	−0.125002	−0.0625010	−	0.998045i	$-0.519908\pi$
$7$	−0.330724	−0.125002	−0.0625010	+	0.998045i	$0.519908\pi$
$8$	0	0
$9$	1.54798	0.515994
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.91473	−0.752581
$16$	0	0
$17$	−2.24546	−0.544603	−0.272302	−	0.962212i	$-0.587785\pi$
$17$	−2.24546	−0.544603	−0.272302	+	0.962212i	$0.587785\pi$
$18$	0	0
$19$	5.48246	1.25776	0.628881	−	0.777502i	$-0.283513\pi$
$19$	5.48246	1.25776	0.628881	+	0.777502i	$0.283513\pi$
$20$	0	0
$21$	−0.705303	−0.153910
$22$	0	0
$23$	7.30062	1.52228	0.761142	−	0.648585i	$-0.224639\pi$
$23$	7.30062	1.52228	0.761142	+	0.648585i	$0.224639\pi$
$24$	0	0
$25$	−3.13199	−0.626398
$26$	0	0
$27$	−3.09657	−0.595936
$28$	0	0
$29$	0.475928	0.0883777	0.0441888	−	0.999023i	$-0.485930\pi$
$29$	0.475928	0.0883777	0.0441888	+	0.999023i	$0.485930\pi$
$30$	0	0
$31$	−7.05864	−1.26777	−0.633885	−	0.773428i	$-0.718541\pi$
$31$	−7.05864	−1.26777	−0.633885	+	0.773428i	$0.718541\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.452018	0.0764050
$36$	0	0
$37$	4.44643	0.730989	0.365494	−	0.930814i	$-0.380900\pi$
$37$	4.44643	0.730989	0.365494	+	0.930814i	$0.380900\pi$
$38$	0	0
$39$	2.13260	0.341489
$40$	0	0
$41$	4.89845	0.765009	0.382505	−	0.923954i	$-0.375062\pi$
$41$	4.89845	0.765009	0.382505	+	0.923954i	$0.375062\pi$
$42$	0	0
$43$	10.0778	1.53685	0.768423	−	0.639942i	$-0.221042\pi$
$43$	10.0778	1.53685	0.768423	+	0.639942i	$0.221042\pi$
$44$	0	0
$45$	−2.11571	−0.315391
$46$	0	0
$47$	1.43880	0.209871	0.104936	−	0.994479i	$-0.466536\pi$
$47$	1.43880	0.209871	0.104936	+	0.994479i	$0.466536\pi$
$48$	0	0
$49$	−6.89062	−0.984374
$50$	0	0
$51$	−4.78866	−0.670547
$52$	0	0
$53$	−5.54668	−0.761896	−0.380948	−	0.924597i	$-0.624402\pi$
$53$	−5.54668	−0.761896	−0.380948	+	0.924597i	$0.624402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.6919	1.54863
$58$	0	0
$59$	9.54892	1.24316	0.621582	−	0.783349i	$-0.286490\pi$
$59$	9.54892	1.24316	0.621582	+	0.783349i	$0.286490\pi$
$60$	0	0
$61$	6.66369	0.853198	0.426599	−	0.904441i	$-0.359712\pi$
$61$	6.66369	0.853198	0.426599	+	0.904441i	$0.359712\pi$
$62$	0	0
$63$	−0.511955	−0.0645003
$64$	0	0
$65$	−1.36675	−0.169525
$66$	0	0
$67$	1.04141	0.127229	0.0636144	−	0.997975i	$-0.479737\pi$
$67$	1.04141	0.127229	0.0636144	+	0.997975i	$0.479737\pi$
$68$	0	0
$69$	15.5693	1.87432
$70$	0	0
$71$	9.19063	1.09073	0.545364	−	0.838200i	$-0.316392\pi$
$71$	9.19063	1.09073	0.545364	+	0.838200i	$0.316392\pi$
$72$	0	0
$73$	−0.133208	−0.0155909	−0.00779544	−	0.999970i	$-0.502481\pi$
$73$	−0.133208	−0.0155909	−0.00779544	+	0.999970i	$0.502481\pi$
$74$	0	0
$75$	−6.67928	−0.771257
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.45011	1.06322	0.531610	−	0.846989i	$-0.321587\pi$
$79$	9.45011	1.06322	0.531610	+	0.846989i	$0.321587\pi$
$80$	0	0
$81$	−11.2477	−1.24974
$82$	0	0
$83$	−2.13199	−0.234016	−0.117008	−	0.993131i	$-0.537330\pi$
$83$	−2.13199	−0.234016	−0.117008	+	0.993131i	$0.537330\pi$
$84$	0	0
$85$	3.06898	0.332878
$86$	0	0
$87$	1.01496	0.108816
$88$	0	0
$89$	−2.25308	−0.238826	−0.119413	−	0.992845i	$-0.538101\pi$
$89$	−2.25308	−0.238826	−0.119413	+	0.992845i	$0.538101\pi$
$90$	0	0
$91$	−0.330724	−0.0346693
$92$	0	0
$93$	−15.0533	−1.56095
$94$	0	0
$95$	−7.49316	−0.768782
$96$	0	0
$97$	7.37745	0.749066	0.374533	−	0.927213i	$-0.377803\pi$
$97$	7.37745	0.749066	0.374533	+	0.927213i	$0.377803\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.48804	0.645585	0.322792	−	0.946470i	$-0.395378\pi$
$101$	6.48804	0.645585	0.322792	+	0.946470i	$0.395378\pi$
$102$	0	0
$103$	−2.91971	−0.287688	−0.143844	−	0.989600i	$-0.545946\pi$
$103$	−2.91971	−0.287688	−0.143844	+	0.989600i	$0.545946\pi$
$104$	0	0
$105$	0.963973	0.0940741
$106$	0	0
$107$	8.32575	0.804880	0.402440	−	0.915446i	$-0.368162\pi$
$107$	8.32575	0.804880	0.402440	+	0.915446i	$0.368162\pi$
$108$	0	0
$109$	−11.1025	−1.06343	−0.531713	−	0.846925i	$-0.678451\pi$
$109$	−11.1025	−1.06343	−0.531713	+	0.846925i	$0.678451\pi$
$110$	0	0
$111$	9.48246	0.900035
$112$	0	0
$113$	0.0611547	0.00575295	0.00287648	−	0.999996i	$-0.499084\pi$
$113$	0.0611547	0.00575295	0.00287648	+	0.999996i	$0.499084\pi$
$114$	0	0
$115$	−9.97813	−0.930465
$116$	0	0
$117$	1.54798	0.143111
$118$	0	0
$119$	0.742627	0.0680765
$120$	0	0
$121$	0	0
$122$	0	0
$123$	10.4464	0.941923
$124$	0	0
$125$	11.1144	0.994103
$126$	0	0
$127$	14.0010	1.24238	0.621192	−	0.783658i	$-0.286649\pi$
$127$	14.0010	1.24238	0.621192	+	0.783658i	$0.286649\pi$
$128$	0	0
$129$	21.4919	1.89225
$130$	0	0
$131$	−4.43391	−0.387392	−0.193696	−	0.981062i	$-0.562048\pi$
$131$	−4.43391	−0.387392	−0.193696	+	0.981062i	$0.562048\pi$
$132$	0	0
$133$	−1.81318	−0.157223
$134$	0	0
$135$	4.23224	0.364254
$136$	0	0
$137$	−10.7172	−0.915634	−0.457817	−	0.889047i	$-0.651368\pi$
$137$	−10.7172	−0.915634	−0.457817	+	0.889047i	$0.651368\pi$
$138$	0	0
$139$	12.9764	1.10065	0.550323	−	0.834952i	$-0.314505\pi$
$139$	12.9764	1.10065	0.550323	+	0.834952i	$0.314505\pi$
$140$	0	0
$141$	3.06840	0.258405
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.650476	−0.0540191
$146$	0	0
$147$	−14.6949	−1.21202
$148$	0	0
$149$	10.7555	0.881124	0.440562	−	0.897722i	$-0.354779\pi$
$149$	10.7555	0.881124	0.440562	+	0.897722i	$0.354779\pi$
$150$	0	0
$151$	−1.99878	−0.162659	−0.0813293	−	0.996687i	$-0.525917\pi$
$151$	−1.99878	−0.162659	−0.0813293	+	0.996687i	$0.525917\pi$
$152$	0	0
$153$	−3.47593	−0.281012
$154$	0	0
$155$	9.64740	0.774898
$156$	0	0
$157$	6.49791	0.518590	0.259295	−	0.965798i	$-0.416510\pi$
$157$	6.49791	0.518590	0.259295	+	0.965798i	$0.416510\pi$
$158$	0	0
$159$	−11.8289	−0.938089
$160$	0	0
$161$	−2.41449	−0.190289
$162$	0	0
$163$	15.5869	1.22086	0.610429	−	0.792071i	$-0.290997\pi$
$163$	15.5869	1.22086	0.610429	+	0.792071i	$0.290997\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.1649	0.786587	0.393294	−	0.919413i	$-0.371336\pi$
$167$	10.1649	0.786587	0.393294	+	0.919413i	$0.371336\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.48675	0.648998
$172$	0	0
$173$	15.4099	1.17160	0.585798	−	0.810457i	$-0.300781\pi$
$173$	15.4099	1.17160	0.585798	+	0.810457i	$0.300781\pi$
$174$	0	0
$175$	1.03583	0.0783011
$176$	0	0
$177$	20.3640	1.53065
$178$	0	0
$179$	−4.65667	−0.348056	−0.174028	−	0.984741i	$-0.555678\pi$
$179$	−4.65667	−0.348056	−0.174028	+	0.984741i	$0.555678\pi$
$180$	0	0
$181$	−7.37458	−0.548148	−0.274074	−	0.961709i	$-0.588371\pi$
$181$	−7.37458	−0.548148	−0.274074	+	0.961709i	$0.588371\pi$
$182$	0	0
$183$	14.2110	1.05051
$184$	0	0
$185$	−6.07716	−0.446802
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.02411	0.0744932
$190$	0	0
$191$	−16.3640	−1.18406	−0.592030	−	0.805916i	$-0.701673\pi$
$191$	−16.3640	−1.18406	−0.592030	+	0.805916i	$0.701673\pi$
$192$	0	0
$193$	−27.2263	−1.95979	−0.979896	−	0.199509i	$-0.936065\pi$
$193$	−27.2263	−1.95979	−0.979896	+	0.199509i	$0.936065\pi$
$194$	0	0
$195$	−2.91473	−0.208728
$196$	0	0
$197$	11.1295	0.792942	0.396471	−	0.918047i	$-0.370235\pi$
$197$	11.1295	0.792942	0.396471	+	0.918047i	$0.370235\pi$
$198$	0	0
$199$	0.711498	0.0504368	0.0252184	−	0.999682i	$-0.491972\pi$
$199$	0.711498	0.0504368	0.0252184	+	0.999682i	$0.491972\pi$
$200$	0	0
$201$	2.22092	0.156651
$202$	0	0
$203$	−0.157401	−0.0110474
$204$	0	0
$205$	−6.69496	−0.467596
$206$	0	0
$207$	11.3012	0.785490
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.01212	0.138520	0.0692599	−	0.997599i	$-0.477936\pi$
$211$	2.01212	0.138520	0.0692599	+	0.997599i	$0.477936\pi$
$212$	0	0
$213$	19.5999	1.34297
$214$	0	0
$215$	−13.7738	−0.939366
$216$	0	0
$217$	2.33446	0.158474
$218$	0	0
$219$	−0.284080	−0.0191964
$220$	0	0
$221$	−2.24546	−0.151046
$222$	0	0
$223$	−15.7112	−1.05210	−0.526049	−	0.850454i	$-0.676327\pi$
$223$	−15.7112	−1.05210	−0.526049	+	0.850454i	$0.676327\pi$
$224$	0	0
$225$	−4.84827	−0.323218
$226$	0	0
$227$	1.65921	0.110126	0.0550628	−	0.998483i	$-0.482464\pi$
$227$	1.65921	0.110126	0.0550628	+	0.998483i	$0.482464\pi$
$228$	0	0
$229$	9.47475	0.626109	0.313055	−	0.949735i	$-0.398648\pi$
$229$	9.47475	0.626109	0.313055	+	0.949735i	$0.398648\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.9461	−0.782616	−0.391308	−	0.920260i	$-0.627977\pi$
$233$	−11.9461	−0.782616	−0.391308	+	0.920260i	$0.627977\pi$
$234$	0	0
$235$	−1.96649	−0.128280
$236$	0	0
$237$	20.1533	1.30910
$238$	0	0
$239$	22.7577	1.47207	0.736036	−	0.676942i	$-0.236695\pi$
$239$	22.7577	1.47207	0.736036	+	0.676942i	$0.236695\pi$
$240$	0	0
$241$	−2.42661	−0.156312	−0.0781558	−	0.996941i	$-0.524903\pi$
$241$	−2.42661	−0.156312	−0.0781558	+	0.996941i	$0.524903\pi$
$242$	0	0
$243$	−14.6971	−0.942821
$244$	0	0
$245$	9.41776	0.601679
$246$	0	0
$247$	5.48246	0.348840
$248$	0	0
$249$	−4.54668	−0.288134
$250$	0	0
$251$	23.5700	1.48772	0.743862	−	0.668333i	$-0.232992\pi$
$251$	23.5700	1.48772	0.743862	+	0.668333i	$0.232992\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.54491	0.409858
$256$	0	0
$257$	18.6793	1.16518	0.582592	−	0.812765i	$-0.302039\pi$
$257$	18.6793	1.16518	0.582592	+	0.812765i	$0.302039\pi$
$258$	0	0
$259$	−1.47054	−0.0913751
$260$	0	0
$261$	0.736729	0.0456024
$262$	0	0
$263$	4.95920	0.305797	0.152899	−	0.988242i	$-0.451139\pi$
$263$	4.95920	0.305797	0.152899	+	0.988242i	$0.451139\pi$
$264$	0	0
$265$	7.58094	0.465693
$266$	0	0
$267$	−4.80493	−0.294057
$268$	0	0
$269$	−18.2170	−1.11071	−0.555354	−	0.831614i	$-0.687417\pi$
$269$	−18.2170	−1.11071	−0.555354	+	0.831614i	$0.687417\pi$
$270$	0	0
$271$	31.9390	1.94016	0.970079	−	0.242789i	$-0.0780621\pi$
$271$	31.9390	1.94016	0.970079	+	0.242789i	$0.0780621\pi$
$272$	0	0
$273$	−0.705303	−0.0426869
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.5818	0.996301	0.498151	−	0.867090i	$-0.334013\pi$
$277$	16.5818	0.996301	0.498151	+	0.867090i	$0.334013\pi$
$278$	0	0
$279$	−10.9266	−0.654161
$280$	0	0
$281$	27.2726	1.62694	0.813472	−	0.581604i	$-0.197575\pi$
$281$	27.2726	1.62694	0.813472	+	0.581604i	$0.197575\pi$
$282$	0	0
$283$	−12.6362	−0.751147	−0.375573	−	0.926793i	$-0.622554\pi$
$283$	−12.6362	−0.751147	−0.375573	+	0.926793i	$0.622554\pi$
$284$	0	0
$285$	−15.9799	−0.946568
$286$	0	0
$287$	−1.62004	−0.0956277
$288$	0	0
$289$	−11.9579	−0.703407
$290$	0	0
$291$	15.7331	0.922293
$292$	0	0
$293$	3.31751	0.193811	0.0969055	−	0.995294i	$-0.469106\pi$
$293$	3.31751	0.193811	0.0969055	+	0.995294i	$0.469106\pi$
$294$	0	0
$295$	−13.0510	−0.759859
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.30062	0.422206
$300$	0	0
$301$	−3.33296	−0.192109
$302$	0	0
$303$	13.8364	0.794881
$304$	0	0
$305$	−9.10760	−0.521500
$306$	0	0
$307$	−8.16495	−0.465998	−0.232999	−	0.972477i	$-0.574854\pi$
$307$	−8.16495	−0.465998	−0.232999	+	0.972477i	$0.574854\pi$
$308$	0	0
$309$	−6.22658	−0.354218
$310$	0	0
$311$	−27.9148	−1.58290	−0.791451	−	0.611232i	$-0.790674\pi$
$311$	−27.9148	−1.58290	−0.791451	+	0.611232i	$0.790674\pi$
$312$	0	0
$313$	−19.4488	−1.09931	−0.549655	−	0.835392i	$-0.685241\pi$
$313$	−19.4488	−1.09931	−0.549655	+	0.835392i	$0.685241\pi$
$314$	0	0
$315$	0.699716	0.0394245
$316$	0	0
$317$	22.7858	1.27978	0.639890	−	0.768466i	$-0.278980\pi$
$317$	22.7858	1.27978	0.639890	+	0.768466i	$0.278980\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	17.7555	0.991015
$322$	0	0
$323$	−12.3106	−0.684982
$324$	0	0
$325$	−3.13199	−0.173732
$326$	0	0
$327$	−23.6772	−1.30935
$328$	0	0
$329$	−0.475848	−0.0262343
$330$	0	0
$331$	24.9437	1.37103	0.685514	−	0.728059i	$-0.259577\pi$
$331$	24.9437	1.37103	0.685514	+	0.728059i	$0.259577\pi$
$332$	0	0
$333$	6.88300	0.377186
$334$	0	0
$335$	−1.42335	−0.0777660
$336$	0	0
$337$	−21.4790	−1.17004	−0.585019	−	0.811020i	$-0.698913\pi$
$337$	−21.4790	−1.17004	−0.585019	+	0.811020i	$0.698913\pi$
$338$	0	0
$339$	0.130419	0.00708336
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.59397	0.248051
$344$	0	0
$345$	−21.2794	−1.14564
$346$	0	0
$347$	−17.2539	−0.926240	−0.463120	−	0.886296i	$-0.653270\pi$
$347$	−17.2539	−0.926240	−0.463120	+	0.886296i	$0.653270\pi$
$348$	0	0
$349$	−15.1765	−0.812380	−0.406190	−	0.913789i	$-0.633143\pi$
$349$	−15.1765	−0.812380	−0.406190	+	0.913789i	$0.633143\pi$
$350$	0	0
$351$	−3.09657	−0.165283
$352$	0	0
$353$	−0.932235	−0.0496179	−0.0248089	−	0.999692i	$-0.507898\pi$
$353$	−0.932235	−0.0496179	−0.0248089	+	0.999692i	$0.507898\pi$
$354$	0	0
$355$	−12.5613	−0.666685
$356$	0	0
$357$	1.58373	0.0838197
$358$	0	0
$359$	15.3062	0.807832	0.403916	−	0.914796i	$-0.367649\pi$
$359$	15.3062	0.807832	0.403916	+	0.914796i	$0.367649\pi$
$360$	0	0
$361$	11.0573	0.581965
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.182063	0.00952961
$366$	0	0
$367$	−17.5601	−0.916630	−0.458315	−	0.888790i	$-0.651547\pi$
$367$	−17.5601	−0.916630	−0.458315	+	0.888790i	$0.651547\pi$
$368$	0	0
$369$	7.58271	0.394740
$370$	0	0
$371$	1.83442	0.0952385
$372$	0	0
$373$	−23.5668	−1.22024	−0.610121	−	0.792308i	$-0.708879\pi$
$373$	−23.5668	−1.22024	−0.610121	+	0.792308i	$0.708879\pi$
$374$	0	0
$375$	23.7026	1.22400
$376$	0	0
$377$	0.475928	0.0245116
$378$	0	0
$379$	17.8586	0.917335	0.458668	−	0.888608i	$-0.348327\pi$
$379$	17.8586	0.917335	0.458668	+	0.888608i	$0.348327\pi$
$380$	0	0
$381$	29.8585	1.52969
$382$	0	0
$383$	29.7649	1.52091	0.760457	−	0.649388i	$-0.224975\pi$
$383$	29.7649	1.52091	0.760457	+	0.649388i	$0.224975\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.6002	0.793003
$388$	0	0
$389$	−0.607920	−0.0308228	−0.0154114	−	0.999881i	$-0.504906\pi$
$389$	−0.607920	−0.0308228	−0.0154114	+	0.999881i	$0.504906\pi$
$390$	0	0
$391$	−16.3932	−0.829041
$392$	0	0
$393$	−9.45575	−0.476979
$394$	0	0
$395$	−12.9160	−0.649872
$396$	0	0
$397$	6.53477	0.327971	0.163985	−	0.986463i	$-0.447565\pi$
$397$	6.53477	0.327971	0.163985	+	0.986463i	$0.447565\pi$
$398$	0	0
$399$	−3.86679	−0.193582
$400$	0	0
$401$	−30.7429	−1.53523	−0.767614	−	0.640912i	$-0.778556\pi$
$401$	−30.7429	−1.53523	−0.767614	+	0.640912i	$0.778556\pi$
$402$	0	0
$403$	−7.05864	−0.351616
$404$	0	0
$405$	15.3728	0.763881
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.537164	0.0265610	0.0132805	−	0.999912i	$-0.495773\pi$
$409$	0.537164	0.0265610	0.0132805	+	0.999912i	$0.495773\pi$
$410$	0	0
$411$	−22.8555	−1.12738
$412$	0	0
$413$	−3.15806	−0.155398
$414$	0	0
$415$	2.91390	0.143038
$416$	0	0
$417$	27.6735	1.35518
$418$	0	0
$419$	4.36816	0.213399	0.106699	−	0.994291i	$-0.465972\pi$
$419$	4.36816	0.213399	0.106699	+	0.994291i	$0.465972\pi$
$420$	0	0
$421$	−13.5501	−0.660390	−0.330195	−	0.943913i	$-0.607114\pi$
$421$	−13.5501	−0.660390	−0.330195	+	0.943913i	$0.607114\pi$
$422$	0	0
$423$	2.22724	0.108292
$424$	0	0
$425$	7.03275	0.341139
$426$	0	0
$427$	−2.20384	−0.106651
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.5075	1.51766	0.758831	−	0.651288i	$-0.225771\pi$
$431$	31.5075	1.51766	0.758831	+	0.651288i	$0.225771\pi$
$432$	0	0
$433$	−6.34040	−0.304700	−0.152350	−	0.988327i	$-0.548684\pi$
$433$	−6.34040	−0.304700	−0.152350	+	0.988327i	$0.548684\pi$
$434$	0	0
$435$	−1.38720	−0.0665114
$436$	0	0
$437$	40.0253	1.91467
$438$	0	0
$439$	6.62863	0.316367	0.158184	−	0.987410i	$-0.449436\pi$
$439$	6.62863	0.316367	0.158184	+	0.987410i	$0.449436\pi$
$440$	0	0
$441$	−10.6666	−0.507931
$442$	0	0
$443$	−27.2913	−1.29665	−0.648324	−	0.761364i	$-0.724530\pi$
$443$	−27.2913	−1.29665	−0.648324	+	0.761364i	$0.724530\pi$
$444$	0	0
$445$	3.07940	0.145978
$446$	0	0
$447$	22.9371	1.08489
$448$	0	0
$449$	10.5556	0.498150	0.249075	−	0.968484i	$-0.419873\pi$
$449$	10.5556	0.498150	0.249075	+	0.968484i	$0.419873\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−4.26260	−0.200275
$454$	0	0
$455$	0.452018	0.0211909
$456$	0	0
$457$	−14.5290	−0.679637	−0.339819	−	0.940491i	$-0.610366\pi$
$457$	−14.5290	−0.679637	−0.339819	+	0.940491i	$0.610366\pi$
$458$	0	0
$459$	6.95322	0.324549
$460$	0	0
$461$	−17.3853	−0.809714	−0.404857	−	0.914380i	$-0.632679\pi$
$461$	−17.3853	−0.809714	−0.404857	+	0.914380i	$0.632679\pi$
$462$	0	0
$463$	19.2834	0.896175	0.448087	−	0.893990i	$-0.352105\pi$
$463$	19.2834	0.896175	0.448087	+	0.893990i	$0.352105\pi$
$464$	0	0
$465$	20.5741	0.954099
$466$	0	0
$467$	13.8880	0.642660	0.321330	−	0.946967i	$-0.395870\pi$
$467$	13.8880	0.642660	0.321330	+	0.946967i	$0.395870\pi$
$468$	0	0
$469$	−0.344420	−0.0159039
$470$	0	0
$471$	13.8574	0.638517
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−17.1710	−0.787860
$476$	0	0
$477$	−8.58617	−0.393134
$478$	0	0
$479$	−8.06536	−0.368516	−0.184258	−	0.982878i	$-0.558988\pi$
$479$	−8.06536	−0.368516	−0.184258	+	0.982878i	$0.558988\pi$
$480$	0	0
$481$	4.44643	0.202740
$482$	0	0
$483$	−5.14914	−0.234294
$484$	0	0
$485$	−10.0831	−0.457852
$486$	0	0
$487$	11.8214	0.535681	0.267840	−	0.963463i	$-0.413690\pi$
$487$	11.8214	0.535681	0.267840	+	0.963463i	$0.413690\pi$
$488$	0	0
$489$	33.2406	1.50319
$490$	0	0
$491$	41.9391	1.89268	0.946342	−	0.323166i	$-0.104747\pi$
$491$	41.9391	1.89268	0.946342	+	0.323166i	$0.104747\pi$
$492$	0	0
$493$	−1.06868	−0.0481308
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.03956	−0.136343
$498$	0	0
$499$	−6.31641	−0.282762	−0.141381	−	0.989955i	$-0.545154\pi$
$499$	−6.31641	−0.282762	−0.141381	+	0.989955i	$0.545154\pi$
$500$	0	0
$501$	21.6778	0.968491
$502$	0	0
$503$	−34.0809	−1.51959	−0.759796	−	0.650162i	$-0.774701\pi$
$503$	−34.0809	−1.51959	−0.759796	+	0.650162i	$0.774701\pi$
$504$	0	0
$505$	−8.86754	−0.394600
$506$	0	0
$507$	2.13260	0.0947121
$508$	0	0
$509$	6.75128	0.299245	0.149623	−	0.988743i	$-0.452194\pi$
$509$	6.75128	0.299245	0.149623	+	0.988743i	$0.452194\pi$
$510$	0	0
$511$	0.0440553	0.00194889
$512$	0	0
$513$	−16.9768	−0.749545
$514$	0	0
$515$	3.99052	0.175843
$516$	0	0
$517$	0	0
$518$	0	0
$519$	32.8632	1.44254
$520$	0	0
$521$	−30.1705	−1.32180	−0.660898	−	0.750476i	$-0.729824\pi$
$521$	−30.1705	−1.32180	−0.660898	+	0.750476i	$0.729824\pi$
$522$	0	0
$523$	−7.63433	−0.333826	−0.166913	−	0.985972i	$-0.553380\pi$
$523$	−7.63433	−0.333826	−0.166913	+	0.985972i	$0.553380\pi$
$524$	0	0
$525$	2.20900	0.0964087
$526$	0	0
$527$	15.8499	0.690431
$528$	0	0
$529$	30.2990	1.31735
$530$	0	0
$531$	14.7816	0.641465
$532$	0	0
$533$	4.89845	0.212175
$534$	0	0
$535$	−11.3792	−0.491967
$536$	0	0
$537$	−9.93082	−0.428546
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−33.2855	−1.43105	−0.715527	−	0.698585i	$-0.753813\pi$
$541$	−33.2855	−1.43105	−0.715527	+	0.698585i	$0.753813\pi$
$542$	0	0
$543$	−15.7270	−0.674911
$544$	0	0
$545$	15.1743	0.649998
$546$	0	0
$547$	−30.2932	−1.29524	−0.647622	−	0.761962i	$-0.724236\pi$
$547$	−30.2932	−1.29524	−0.647622	+	0.761962i	$0.724236\pi$
$548$	0	0
$549$	10.3153	0.440245
$550$	0	0
$551$	2.60926	0.111158
$552$	0	0
$553$	−3.12538	−0.132905
$554$	0	0
$555$	−12.9602	−0.550128
$556$	0	0
$557$	29.5140	1.25055	0.625275	−	0.780405i	$-0.284987\pi$
$557$	29.5140	1.25055	0.625275	+	0.780405i	$0.284987\pi$
$558$	0	0
$559$	10.0778	0.426244
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.9785	−1.60060	−0.800302	−	0.599597i	$-0.795328\pi$
$563$	−37.9785	−1.60060	−0.800302	+	0.599597i	$0.795328\pi$
$564$	0	0
$565$	−0.0835833	−0.00351638
$566$	0	0
$567$	3.71989	0.156221
$568$	0	0
$569$	−32.4507	−1.36040	−0.680202	−	0.733024i	$-0.738108\pi$
$569$	−32.4507	−1.36040	−0.680202	+	0.733024i	$0.738108\pi$
$570$	0	0
$571$	−23.0745	−0.965638	−0.482819	−	0.875720i	$-0.660387\pi$
$571$	−23.0745	−0.965638	−0.482819	+	0.875720i	$0.660387\pi$
$572$	0	0
$573$	−34.8979	−1.45788
$574$	0	0
$575$	−22.8655	−0.953556
$576$	0	0
$577$	−13.5922	−0.565850	−0.282925	−	0.959142i	$-0.591305\pi$
$577$	−13.5922	−0.565850	−0.282925	+	0.959142i	$0.591305\pi$
$578$	0	0
$579$	−58.0628	−2.41301
$580$	0	0
$581$	0.705101	0.0292525
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.11571	−0.0874737
$586$	0	0
$587$	42.5799	1.75746	0.878731	−	0.477318i	$-0.158391\pi$
$587$	42.5799	1.75746	0.878731	+	0.477318i	$0.158391\pi$
$588$	0	0
$589$	−38.6987	−1.59455
$590$	0	0
$591$	23.7347	0.976316
$592$	0	0
$593$	−45.3469	−1.86217	−0.931087	−	0.364796i	$-0.881139\pi$
$593$	−45.3469	−1.86217	−0.931087	+	0.364796i	$0.881139\pi$
$594$	0	0
$595$	−1.01499	−0.0416104
$596$	0	0
$597$	1.51734	0.0621006
$598$	0	0
$599$	−45.4710	−1.85790	−0.928948	−	0.370209i	$-0.879286\pi$
$599$	−45.4710	−1.85790	−0.928948	+	0.370209i	$0.879286\pi$
$600$	0	0
$601$	−33.4720	−1.36535	−0.682676	−	0.730721i	$-0.739184\pi$
$601$	−33.4720	−1.36535	−0.682676	+	0.730721i	$0.739184\pi$
$602$	0	0
$603$	1.61209	0.0656493
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−25.0294	−1.01591	−0.507955	−	0.861384i	$-0.669598\pi$
$607$	−25.0294	−1.01591	−0.507955	+	0.861384i	$0.669598\pi$
$608$	0	0
$609$	−0.335674	−0.0136022
$610$	0	0
$611$	1.43880	0.0582078
$612$	0	0
$613$	6.34456	0.256254	0.128127	−	0.991758i	$-0.459103\pi$
$613$	6.34456	0.256254	0.128127	+	0.991758i	$0.459103\pi$
$614$	0	0
$615$	−14.2777	−0.575731
$616$	0	0
$617$	2.76591	0.111351	0.0556757	−	0.998449i	$-0.482269\pi$
$617$	2.76591	0.111351	0.0556757	+	0.998449i	$0.482269\pi$
$618$	0	0
$619$	−30.9883	−1.24552	−0.622762	−	0.782411i	$-0.713989\pi$
$619$	−30.9883	−1.24552	−0.622762	+	0.782411i	$0.713989\pi$
$620$	0	0
$621$	−22.6069	−0.907184
$622$	0	0
$623$	0.745149	0.0298538
$624$	0	0
$625$	0.469325	0.0187730
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.98427	−0.398099
$630$	0	0
$631$	−32.9868	−1.31318	−0.656592	−	0.754246i	$-0.728002\pi$
$631$	−32.9868	−1.31318	−0.656592	+	0.754246i	$0.728002\pi$
$632$	0	0
$633$	4.29104	0.170553
$634$	0	0
$635$	−19.1358	−0.759382
$636$	0	0
$637$	−6.89062	−0.273016
$638$	0	0
$639$	14.2269	0.562809
$640$	0	0
$641$	36.2553	1.43200	0.716000	−	0.698100i	$-0.245971\pi$
$641$	36.2553	1.43200	0.716000	+	0.698100i	$0.245971\pi$
$642$	0	0
$643$	36.1562	1.42586	0.712931	−	0.701234i	$-0.247367\pi$
$643$	36.1562	1.42586	0.712931	+	0.701234i	$0.247367\pi$
$644$	0	0
$645$	−29.3740	−1.15660
$646$	0	0
$647$	3.68951	0.145049	0.0725247	−	0.997367i	$-0.476894\pi$
$647$	3.68951	0.145049	0.0725247	+	0.997367i	$0.476894\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.97848	0.195122
$652$	0	0
$653$	23.3344	0.913145	0.456572	−	0.889686i	$-0.349077\pi$
$653$	23.3344	0.913145	0.456572	+	0.889686i	$0.349077\pi$
$654$	0	0
$655$	6.06005	0.236786
$656$	0	0
$657$	−0.206204	−0.00804480
$658$	0	0
$659$	29.4583	1.14753	0.573767	−	0.819019i	$-0.305482\pi$
$659$	29.4583	1.14753	0.573767	+	0.819019i	$0.305482\pi$
$660$	0	0
$661$	−2.10532	−0.0818875	−0.0409437	−	0.999161i	$-0.513036\pi$
$661$	−2.10532	−0.0818875	−0.0409437	+	0.999161i	$0.513036\pi$
$662$	0	0
$663$	−4.78866	−0.185976
$664$	0	0
$665$	2.47817	0.0960993
$666$	0	0
$667$	3.47457	0.134536
$668$	0	0
$669$	−33.5056	−1.29540
$670$	0	0
$671$	0	0
$672$	0	0
$673$	31.1368	1.20024	0.600118	−	0.799911i	$-0.295120\pi$
$673$	31.1368	1.20024	0.600118	+	0.799911i	$0.295120\pi$
$674$	0	0
$675$	9.69844	0.373293
$676$	0	0
$677$	−30.5938	−1.17582	−0.587908	−	0.808928i	$-0.700048\pi$
$677$	−30.5938	−1.17582	−0.587908	+	0.808928i	$0.700048\pi$
$678$	0	0
$679$	−2.43990	−0.0936348
$680$	0	0
$681$	3.53843	0.135593
$682$	0	0
$683$	−33.4336	−1.27930	−0.639651	−	0.768665i	$-0.720921\pi$
$683$	−33.4336	−1.27930	−0.639651	+	0.768665i	$0.720921\pi$
$684$	0	0
$685$	14.6478	0.559663
$686$	0	0
$687$	20.2059	0.770901
$688$	0	0
$689$	−5.54668	−0.211312
$690$	0	0
$691$	7.32623	0.278703	0.139352	−	0.990243i	$-0.455498\pi$
$691$	7.32623	0.278703	0.139352	+	0.990243i	$0.455498\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.7355	−0.672747
$696$	0	0
$697$	−10.9993	−0.416627
$698$	0	0
$699$	−25.4763	−0.963602
$700$	0	0
$701$	1.58003	0.0596771	0.0298385	−	0.999555i	$-0.490501\pi$
$701$	1.58003	0.0596771	0.0298385	+	0.999555i	$0.490501\pi$
$702$	0	0
$703$	24.3774	0.919410
$704$	0	0
$705$	−4.19373	−0.157945
$706$	0	0
$707$	−2.14575	−0.0806994
$708$	0	0
$709$	−13.9716	−0.524714	−0.262357	−	0.964971i	$-0.584500\pi$
$709$	−13.9716	−0.524714	−0.262357	+	0.964971i	$0.584500\pi$
$710$	0	0
$711$	14.6286	0.548616
$712$	0	0
$713$	−51.5324	−1.92990
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.5330	1.81250
$718$	0	0
$719$	15.1610	0.565410	0.282705	−	0.959207i	$-0.408768\pi$
$719$	15.1610	0.565410	0.282705	+	0.959207i	$0.408768\pi$
$720$	0	0
$721$	0.965620	0.0359616
$722$	0	0
$723$	−5.17498	−0.192460
$724$	0	0
$725$	−1.49060	−0.0553596
$726$	0	0
$727$	23.0015	0.853078	0.426539	−	0.904469i	$-0.359733\pi$
$727$	23.0015	0.853078	0.426539	+	0.904469i	$0.359733\pi$
$728$	0	0
$729$	2.40002	0.0888895
$730$	0	0
$731$	−22.6292	−0.836972
$732$	0	0
$733$	−42.6029	−1.57357	−0.786787	−	0.617224i	$-0.788257\pi$
$733$	−42.6029	−1.57357	−0.786787	+	0.617224i	$0.788257\pi$
$734$	0	0
$735$	20.0843	0.740821
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−0.690681	−0.0254071	−0.0127036	−	0.999919i	$-0.504044\pi$
$739$	−0.690681	−0.0254071	−0.0127036	+	0.999919i	$0.504044\pi$
$740$	0	0
$741$	11.6919	0.429512
$742$	0	0
$743$	5.94301	0.218028	0.109014	−	0.994040i	$-0.465231\pi$
$743$	5.94301	0.218028	0.109014	+	0.994040i	$0.465231\pi$
$744$	0	0
$745$	−14.7001	−0.538569
$746$	0	0
$747$	−3.30028	−0.120751
$748$	0	0
$749$	−2.75353	−0.100612
$750$	0	0
$751$	−18.2583	−0.666253	−0.333127	−	0.942882i	$-0.608104\pi$
$751$	−18.2583	−0.666253	−0.333127	+	0.942882i	$0.608104\pi$
$752$	0	0
$753$	50.2653	1.83177
$754$	0	0
$755$	2.73184	0.0994218
$756$	0	0
$757$	18.5572	0.674474	0.337237	−	0.941420i	$-0.390508\pi$
$757$	18.5572	0.674474	0.337237	+	0.941420i	$0.390508\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.0424	0.726535	0.363267	−	0.931685i	$-0.381661\pi$
$761$	20.0424	0.726535	0.363267	+	0.931685i	$0.381661\pi$
$762$	0	0
$763$	3.67186	0.132930
$764$	0	0
$765$	4.75073	0.171763
$766$	0	0
$767$	9.54892	0.344792
$768$	0	0
$769$	−50.1683	−1.80912	−0.904558	−	0.426350i	$-0.859799\pi$
$769$	−50.1683	−1.80912	−0.904558	+	0.426350i	$0.859799\pi$
$770$	0	0
$771$	39.8355	1.43464
$772$	0	0
$773$	−34.8506	−1.25349	−0.626745	−	0.779225i	$-0.715613\pi$
$773$	−34.8506	−1.25349	−0.626745	+	0.779225i	$0.715613\pi$
$774$	0	0
$775$	22.1076	0.794128
$776$	0	0
$777$	−3.13608	−0.112506
$778$	0	0
$779$	26.8555	0.962199
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.47375	−0.0526674
$784$	0	0
$785$	−8.88103	−0.316977
$786$	0	0
$787$	17.4381	0.621602	0.310801	−	0.950475i	$-0.399403\pi$
$787$	17.4381	0.621602	0.310801	+	0.950475i	$0.399403\pi$
$788$	0	0
$789$	10.5760	0.376515
$790$	0	0
$791$	−0.0202254	−0.000719131 0
$792$	0	0
$793$	6.66369	0.236635
$794$	0	0
$795$	16.1671	0.573388
$796$	0	0
$797$	−7.34362	−0.260124	−0.130062	−	0.991506i	$-0.541518\pi$
$797$	−7.34362	−0.260124	−0.130062	+	0.991506i	$0.541518\pi$
$798$	0	0
$799$	−3.23078	−0.114297
$800$	0	0
$801$	−3.48773	−0.123233
$802$	0	0
$803$	0	0
$804$	0	0
$805$	3.30001	0.116310
$806$	0	0
$807$	−38.8495	−1.36757
$808$	0	0
$809$	11.6797	0.410637	0.205318	−	0.978695i	$-0.434177\pi$
$809$	11.6797	0.410637	0.205318	+	0.978695i	$0.434177\pi$
$810$	0	0
$811$	−33.3433	−1.17084	−0.585420	−	0.810730i	$-0.699071\pi$
$811$	−33.3433	−1.17084	−0.585420	+	0.810730i	$0.699071\pi$
$812$	0	0
$813$	68.1132	2.38883
$814$	0	0
$815$	−21.3034	−0.746225
$816$	0	0
$817$	55.2510	1.93299
$818$	0	0
$819$	−0.511955	−0.0178892
$820$	0	0
$821$	39.9074	1.39278	0.696390	−	0.717664i	$-0.254789\pi$
$821$	39.9074	1.39278	0.696390	+	0.717664i	$0.254789\pi$
$822$	0	0
$823$	−22.5533	−0.786159	−0.393079	−	0.919505i	$-0.628590\pi$
$823$	−22.5533	−0.786159	−0.393079	+	0.919505i	$0.628590\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.5452	−0.540561	−0.270280	−	0.962782i	$-0.587116\pi$
$827$	−15.5452	−0.540561	−0.270280	+	0.962782i	$0.587116\pi$
$828$	0	0
$829$	33.0353	1.14736	0.573682	−	0.819078i	$-0.305515\pi$
$829$	33.0353	1.14736	0.573682	+	0.819078i	$0.305515\pi$
$830$	0	0
$831$	35.3623	1.22670
$832$	0	0
$833$	15.4726	0.536094
$834$	0	0
$835$	−13.8930	−0.480785
$836$	0	0
$837$	21.8576	0.755509
$838$	0	0
$839$	−21.3652	−0.737608	−0.368804	−	0.929507i	$-0.620233\pi$
$839$	−21.3652	−0.737608	−0.368804	+	0.929507i	$0.620233\pi$
$840$	0	0
$841$	−28.7735	−0.992189
$842$	0	0
$843$	58.1614	2.00319
$844$	0	0
$845$	−1.36675	−0.0470177
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−26.9480	−0.924854
$850$	0	0
$851$	32.4617	1.11277
$852$	0	0
$853$	21.9413	0.751256	0.375628	−	0.926770i	$-0.377427\pi$
$853$	21.9413	0.751256	0.375628	+	0.926770i	$0.377427\pi$
$854$	0	0
$855$	−11.5993	−0.396687
$856$	0	0
$857$	8.58531	0.293269	0.146634	−	0.989191i	$-0.453156\pi$
$857$	8.58531	0.293269	0.146634	+	0.989191i	$0.453156\pi$
$858$	0	0
$859$	−14.3088	−0.488211	−0.244105	−	0.969749i	$-0.578494\pi$
$859$	−14.3088	−0.488211	−0.244105	+	0.969749i	$0.578494\pi$
$860$	0	0
$861$	−3.45489	−0.117742
$862$	0	0
$863$	52.6979	1.79386	0.896928	−	0.442177i	$-0.145794\pi$
$863$	52.6979	1.79386	0.896928	+	0.442177i	$0.145794\pi$
$864$	0	0
$865$	−21.0616	−0.716114
$866$	0	0
$867$	−25.5015	−0.866075
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.04141	0.0352869
$872$	0	0
$873$	11.4202	0.386514
$874$	0	0
$875$	−3.67580	−0.124265
$876$	0	0
$877$	45.3122	1.53009	0.765043	−	0.643980i	$-0.222718\pi$
$877$	45.3122	1.53009	0.765043	+	0.643980i	$0.222718\pi$
$878$	0	0
$879$	7.07492	0.238631
$880$	0	0
$881$	−9.40015	−0.316699	−0.158350	−	0.987383i	$-0.550617\pi$
$881$	−9.40015	−0.316699	−0.158350	+	0.987383i	$0.550617\pi$
$882$	0	0
$883$	47.1135	1.58549	0.792747	−	0.609551i	$-0.208650\pi$
$883$	47.1135	1.58549	0.792747	+	0.609551i	$0.208650\pi$
$884$	0	0
$885$	−27.8326	−0.935582
$886$	0	0
$887$	−15.2687	−0.512673	−0.256336	−	0.966588i	$-0.582515\pi$
$887$	−15.2687	−0.512673	−0.256336	+	0.966588i	$0.582515\pi$
$888$	0	0
$889$	−4.63046	−0.155301
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.88819	0.263968
$894$	0	0
$895$	6.36451	0.212742
$896$	0	0
$897$	15.5693	0.519844
$898$	0	0
$899$	−3.35941	−0.112043
$900$	0	0
$901$	12.4548	0.414931
$902$	0	0
$903$	−7.10788	−0.236535
$904$	0	0
$905$	10.0792	0.335044
$906$	0	0
$907$	−13.7704	−0.457237	−0.228619	−	0.973516i	$-0.573421\pi$
$907$	−13.7704	−0.457237	−0.228619	+	0.973516i	$0.573421\pi$
$908$	0	0
$909$	10.0434	0.333118
$910$	0	0
$911$	−24.4324	−0.809483	−0.404741	−	0.914431i	$-0.632638\pi$
$911$	−24.4324	−0.809483	−0.404741	+	0.914431i	$0.632638\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−19.4229	−0.642100
$916$	0	0
$917$	1.46640	0.0484248
$918$	0	0
$919$	−40.5103	−1.33631	−0.668155	−	0.744022i	$-0.732916\pi$
$919$	−40.5103	−1.33631	−0.668155	+	0.744022i	$0.732916\pi$
$920$	0	0
$921$	−17.4126	−0.573763
$922$	0	0
$923$	9.19063	0.302513
$924$	0	0
$925$	−13.9262	−0.457890
$926$	0	0
$927$	−4.51966	−0.148445
$928$	0	0
$929$	−28.5240	−0.935840	−0.467920	−	0.883771i	$-0.654997\pi$
$929$	−28.5240	−0.935840	−0.467920	+	0.883771i	$0.654997\pi$
$930$	0	0
$931$	−37.7775	−1.23811
$932$	0	0
$933$	−59.5311	−1.94896
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.11720	0.0691659	0.0345829	−	0.999402i	$-0.488990\pi$
$937$	2.11720	0.0691659	0.0345829	+	0.999402i	$0.488990\pi$
$938$	0	0
$939$	−41.4765	−1.35353
$940$	0	0
$941$	31.2163	1.01762	0.508811	−	0.860878i	$-0.330085\pi$
$941$	31.2163	1.01762	0.508811	+	0.860878i	$0.330085\pi$
$942$	0	0
$943$	35.7617	1.16456
$944$	0	0
$945$	−1.39971	−0.0455324
$946$	0	0
$947$	−54.3615	−1.76651	−0.883256	−	0.468890i	$-0.844654\pi$
$947$	−54.3615	−1.76651	−0.883256	+	0.468890i	$0.844654\pi$
$948$	0	0
$949$	−0.133208	−0.00432413
$950$	0	0
$951$	48.5931	1.57574
$952$	0	0
$953$	−22.4286	−0.726532	−0.363266	−	0.931685i	$-0.618338\pi$
$953$	−22.4286	−0.726532	−0.363266	+	0.931685i	$0.618338\pi$
$954$	0	0
$955$	22.3656	0.723733
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.54444	0.114456
$960$	0	0
$961$	18.8244	0.607238
$962$	0	0
$963$	12.8881	0.415313
$964$	0	0
$965$	37.2116	1.19788
$966$	0	0
$967$	−20.0980	−0.646309	−0.323155	−	0.946346i	$-0.604743\pi$
$967$	−20.0980	−0.646309	−0.323155	+	0.946346i	$0.604743\pi$
$968$	0	0
$969$	−26.2536	−0.843388
$970$	0	0
$971$	0.608448	0.0195260	0.00976301	−	0.999952i	$-0.496892\pi$
$971$	0.608448	0.0195260	0.00976301	+	0.999952i	$0.496892\pi$
$972$	0	0
$973$	−4.29162	−0.137583
$974$	0	0
$975$	−6.67928	−0.213908
$976$	0	0
$977$	−54.9086	−1.75668	−0.878341	−	0.478034i	$-0.841350\pi$
$977$	−54.9086	−1.75668	−0.878341	+	0.478034i	$0.841350\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−17.1865	−0.548722
$982$	0	0
$983$	59.5893	1.90060	0.950302	−	0.311328i	$-0.100774\pi$
$983$	59.5893	1.90060	0.950302	+	0.311328i	$0.100774\pi$
$984$	0	0
$985$	−15.2112	−0.484670
$986$	0	0
$987$	−1.01479	−0.0323012
$988$	0	0
$989$	73.5740	2.33952
$990$	0	0
$991$	−13.8682	−0.440537	−0.220268	−	0.975439i	$-0.570693\pi$
$991$	−13.8682	−0.440537	−0.220268	+	0.975439i	$0.570693\pi$
$992$	0	0
$993$	53.1949	1.68809
$994$	0	0
$995$	−0.972441	−0.0308285
$996$	0	0
$997$	47.2485	1.49637	0.748187	−	0.663488i	$-0.230925\pi$
$997$	47.2485	1.49637	0.748187	+	0.663488i	$0.230925\pi$
$998$	0	0
$999$	−13.7687	−0.435622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.t.1.5	yes	5
11.10	odd	2		6292.2.a.s.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.s.1.5	✓	5	11.10	odd	2
6292.2.a.t.1.5	yes	5	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:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13260 q^{3} -1.36675 q^{5} -0.330724 q^{7} +1.54798 q^{9} +O(q^{10})\)	\(q+2.13260 q^{3} -1.36675 q^{5} -0.330724 q^{7} +1.54798 q^{9} +1.00000 q^{13} -2.91473 q^{15} -2.24546 q^{17} +5.48246 q^{19} -0.705303 q^{21} +7.30062 q^{23} -3.13199 q^{25} -3.09657 q^{27} +0.475928 q^{29} -7.05864 q^{31} +0.452018 q^{35} +4.44643 q^{37} +2.13260 q^{39} +4.89845 q^{41} +10.0778 q^{43} -2.11571 q^{45} +1.43880 q^{47} -6.89062 q^{49} -4.78866 q^{51} -5.54668 q^{53} +11.6919 q^{57} +9.54892 q^{59} +6.66369 q^{61} -0.511955 q^{63} -1.36675 q^{65} +1.04141 q^{67} +15.5693 q^{69} +9.19063 q^{71} -0.133208 q^{73} -6.67928 q^{75} +9.45011 q^{79} -11.2477 q^{81} -2.13199 q^{83} +3.06898 q^{85} +1.01496 q^{87} -2.25308 q^{89} -0.330724 q^{91} -15.0533 q^{93} -7.49316 q^{95} +7.37745 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q - q^3 + 2 * q^7 + 2 * q^9	\( 5 q - q^{3} + 2 q^{7} + 2 q^{9} + 5 q^{13} - 2 q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 7 q^{23} - q^{25} - 7 q^{27} + 3 q^{29} + 10 q^{31} + 8 q^{35} - q^{39} + 8 q^{41} - 5 q^{43} + 8 q^{45} - 6 q^{47} - 7 q^{49} + 3 q^{51} + 15 q^{53} + 20 q^{57} - 12 q^{59} + 9 q^{61} + 10 q^{67} - 7 q^{69} - 14 q^{71} + 8 q^{73} + 21 q^{75} + 13 q^{79} - 23 q^{81} + 4 q^{83} + 24 q^{85} - 7 q^{87} + 14 q^{89} + 2 q^{91} - 22 q^{93} + 12 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q - q^3 + 2 * q^7 + 2 * q^9 + 5 * q^13 - 2 * q^15 + 5 * q^17 + 2 * q^19 - 4 * q^21 + 7 * q^23 - q^25 - 7 * q^27 + 3 * q^29 + 10 * q^31 + 8 * q^35 - q^39 + 8 * q^41 - 5 * q^43 + 8 * q^45 - 6 * q^47 - 7 * q^49 + 3 * q^51 + 15 * q^53 + 20 * q^57 - 12 * q^59 + 9 * q^61 + 10 * q^67 - 7 * q^69 - 14 * q^71 + 8 * q^73 + 21 * q^75 + 13 * q^79 - 23 * q^81 + 4 * q^83 + 24 * q^85 - 7 * q^87 + 14 * q^89 + 2 * q^91 - 22 * q^93 + 12 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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