LMFDB - Embedded newform 4338.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4338,2,Mod(1,4338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4338 = 2 \cdot 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4338.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:	$34.6391043968$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 20x^{5} + 26x^{4} + 95x^{3} - 121x^{2} - 126x + 158 $ x^7 - x^6 - 20x^5 + 26x^4 + 95x^3 - 121x^2 - 126*x + 158
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.36253$ of defining polynomial
Character	$\chi$	$=$	4338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.18526 q^{5} +2.78499 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.18526 q^{5} +2.78499 q^{7} +1.00000 q^{8} +1.18526 q^{10} +5.75385 q^{11} +4.78099 q^{13} +2.78499 q^{14} +1.00000 q^{16} -6.74428 q^{17} +0.00400639 q^{19} +1.18526 q^{20} +5.75385 q^{22} +8.32980 q^{23} -3.59516 q^{25} +4.78099 q^{26} +2.78499 q^{28} -4.96886 q^{29} -1.28541 q^{31} +1.00000 q^{32} -6.74428 q^{34} +3.30094 q^{35} +8.74428 q^{37} +0.00400639 q^{38} +1.18526 q^{40} -1.30094 q^{41} +2.29136 q^{43} +5.75385 q^{44} +8.32980 q^{46} +3.72423 q^{47} +0.756197 q^{49} -3.59516 q^{50} +4.78099 q^{52} -10.7493 q^{53} +6.81981 q^{55} +2.78499 q^{56} -4.96886 q^{58} -0.960614 q^{59} -12.3595 q^{61} -1.28541 q^{62} +1.00000 q^{64} +5.66671 q^{65} +10.7827 q^{67} -6.74428 q^{68} +3.30094 q^{70} -4.36651 q^{71} -13.8591 q^{73} +8.74428 q^{74} +0.00400639 q^{76} +16.0245 q^{77} +7.52558 q^{79} +1.18526 q^{80} -1.30094 q^{82} -17.1260 q^{83} -7.99371 q^{85} +2.29136 q^{86} +5.75385 q^{88} -8.03882 q^{89} +13.3150 q^{91} +8.32980 q^{92} +3.72423 q^{94} +0.00474861 q^{95} +6.08372 q^{97} +0.756197 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.18526	0.530064	0.265032	−	0.964240i	$-0.414618\pi$
$5$	1.18526	0.530064	0.265032	+	0.964240i	$0.414618\pi$
$6$	0	0
$7$	2.78499	1.05263	0.526315	−	0.850290i	$-0.323573\pi$
$7$	2.78499	1.05263	0.526315	+	0.850290i	$0.323573\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.18526	0.374812
$11$	5.75385	1.73485	0.867426	−	0.497566i	$-0.165773\pi$
$11$	5.75385	1.73485	0.867426	+	0.497566i	$0.165773\pi$
$12$	0	0
$13$	4.78099	1.32601	0.663004	−	0.748616i	$-0.269281\pi$
$13$	4.78099	1.32601	0.663004	+	0.748616i	$0.269281\pi$
$14$	2.78499	0.744321
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.74428	−1.63573	−0.817863	−	0.575412i	$-0.804842\pi$
$17$	−6.74428	−1.63573	−0.817863	+	0.575412i	$0.804842\pi$
$18$	0	0
$19$	0.00400639	0.000919128 0	0.000459564	−	1.00000i	$-0.499854\pi$
$19$	0.00400639	0.000919128 0	0.000459564		1.00000i	$0.499854\pi$
$20$	1.18526	0.265032
$21$	0	0
$22$	5.75385	1.22673
$23$	8.32980	1.73688	0.868441	−	0.495792i	$-0.165122\pi$
$23$	8.32980	1.73688	0.868441	+	0.495792i	$0.165122\pi$
$24$	0	0
$25$	−3.59516	−0.719032
$26$	4.78099	0.937629
$27$	0	0
$28$	2.78499	0.526315
$29$	−4.96886	−0.922694	−0.461347	−	0.887220i	$-0.652634\pi$
$29$	−4.96886	−0.922694	−0.461347	+	0.887220i	$0.652634\pi$
$30$	0	0
$31$	−1.28541	−0.230866	−0.115433	−	0.993315i	$-0.536826\pi$
$31$	−1.28541	−0.230866	−0.115433	+	0.993315i	$0.536826\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.74428	−1.15663
$35$	3.30094	0.557961
$36$	0	0
$37$	8.74428	1.43755	0.718775	−	0.695243i	$-0.244703\pi$
$37$	8.74428	1.43755	0.718775	+	0.695243i	$0.244703\pi$
$38$	0.00400639	0.000649922 0
$39$	0	0
$40$	1.18526	0.187406
$41$	−1.30094	−0.203173	−0.101586	−	0.994827i	$-0.532392\pi$
$41$	−1.30094	−0.203173	−0.101586	+	0.994827i	$0.532392\pi$
$42$	0	0
$43$	2.29136	0.349429	0.174715	−	0.984619i	$-0.444100\pi$
$43$	2.29136	0.349429	0.174715	+	0.984619i	$0.444100\pi$
$44$	5.75385	0.867426
$45$	0	0
$46$	8.32980	1.22816
$47$	3.72423	0.543235	0.271618	−	0.962405i	$-0.412441\pi$
$47$	3.72423	0.543235	0.271618	+	0.962405i	$0.412441\pi$
$48$	0	0
$49$	0.756197	0.108028
$50$	−3.59516	−0.508433
$51$	0	0
$52$	4.78099	0.663004
$53$	−10.7493	−1.47653	−0.738264	−	0.674512i	$-0.764354\pi$
$53$	−10.7493	−1.47653	−0.738264	+	0.674512i	$0.764354\pi$
$54$	0	0
$55$	6.81981	0.919582
$56$	2.78499	0.372161
$57$	0	0
$58$	−4.96886	−0.652443
$59$	−0.960614	−0.125061	−0.0625307	−	0.998043i	$-0.519917\pi$
$59$	−0.960614	−0.125061	−0.0625307	+	0.998043i	$0.519917\pi$
$60$	0	0
$61$	−12.3595	−1.58247	−0.791235	−	0.611513i	$-0.790561\pi$
$61$	−12.3595	−1.58247	−0.791235	+	0.611513i	$0.790561\pi$
$62$	−1.28541	−0.163247
$63$	0	0
$64$	1.00000	0.125000
$65$	5.66671	0.702869
$66$	0	0
$67$	10.7827	1.31731	0.658655	−	0.752445i	$-0.271126\pi$
$67$	10.7827	1.31731	0.658655	+	0.752445i	$0.271126\pi$
$68$	−6.74428	−0.817863
$69$	0	0
$70$	3.30094	0.394538
$71$	−4.36651	−0.518210	−0.259105	−	0.965849i	$-0.583427\pi$
$71$	−4.36651	−0.518210	−0.259105	+	0.965849i	$0.583427\pi$
$72$	0	0
$73$	−13.8591	−1.62208	−0.811041	−	0.584990i	$-0.801098\pi$
$73$	−13.8591	−1.62208	−0.811041	+	0.584990i	$0.801098\pi$
$74$	8.74428	1.01650
$75$	0	0
$76$	0.00400639	0.000459564 0
$77$	16.0245	1.82616
$78$	0	0
$79$	7.52558	0.846694	0.423347	−	0.905968i	$-0.360855\pi$
$79$	7.52558	0.846694	0.423347	+	0.905968i	$0.360855\pi$
$80$	1.18526	0.132516
$81$	0	0
$82$	−1.30094	−0.143665
$83$	−17.1260	−1.87982	−0.939911	−	0.341419i	$-0.889093\pi$
$83$	−17.1260	−1.87982	−0.939911	+	0.341419i	$0.889093\pi$
$84$	0	0
$85$	−7.99371	−0.867040
$86$	2.29136	0.247084
$87$	0	0
$88$	5.75385	0.613363
$89$	−8.03882	−0.852113	−0.426056	−	0.904697i	$-0.640098\pi$
$89$	−8.03882	−0.852113	−0.426056	+	0.904697i	$0.640098\pi$
$90$	0	0
$91$	13.3150	1.39579
$92$	8.32980	0.868441
$93$	0	0
$94$	3.72423	0.384125
$95$	0.00474861	0.000487197 0
$96$	0	0
$97$	6.08372	0.617708	0.308854	−	0.951109i	$-0.400055\pi$
$97$	6.08372	0.617708	0.308854	+	0.951109i	$0.400055\pi$
$98$	0.756197	0.0763874
$99$	0	0
$100$	−3.59516	−0.359516
$101$	−8.20030	−0.815960	−0.407980	−	0.912991i	$-0.633767\pi$
$101$	−8.20030	−0.815960	−0.407980	+	0.912991i	$0.633767\pi$
$102$	0	0
$103$	−15.7843	−1.55527	−0.777636	−	0.628715i	$-0.783581\pi$
$103$	−15.7843	−1.55527	−0.777636	+	0.628715i	$0.783581\pi$
$104$	4.78099	0.468815
$105$	0	0
$106$	−10.7493	−1.04406
$107$	−20.0887	−1.94204	−0.971022	−	0.238991i	$-0.923183\pi$
$107$	−20.0887	−1.94204	−0.971022	+	0.238991i	$0.923183\pi$
$108$	0	0
$109$	8.88693	0.851214	0.425607	−	0.904908i	$-0.360061\pi$
$109$	8.88693	0.851214	0.425607	+	0.904908i	$0.360061\pi$
$110$	6.81981	0.650243
$111$	0	0
$112$	2.78499	0.263157
$113$	3.96968	0.373436	0.186718	−	0.982414i	$-0.440215\pi$
$113$	3.96968	0.373436	0.186718	+	0.982414i	$0.440215\pi$
$114$	0	0
$115$	9.87297	0.920659
$116$	−4.96886	−0.461347
$117$	0	0
$118$	−0.960614	−0.0884317
$119$	−18.7828	−1.72181
$120$	0	0
$121$	22.1068	2.00971
$122$	−12.3595	−1.11897
$123$	0	0
$124$	−1.28541	−0.115433
$125$	−10.1875	−0.911197
$126$	0	0
$127$	13.4218	1.19099	0.595495	−	0.803359i	$-0.296956\pi$
$127$	13.4218	1.19099	0.595495	+	0.803359i	$0.296956\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.66671	0.497003
$131$	8.60563	0.751877	0.375938	−	0.926645i	$-0.377320\pi$
$131$	8.60563	0.751877	0.375938	+	0.926645i	$0.377320\pi$
$132$	0	0
$133$	0.0111578	0.000967501 0
$134$	10.7827	0.931479
$135$	0	0
$136$	−6.74428	−0.578317
$137$	−17.8706	−1.52679	−0.763394	−	0.645933i	$-0.776468\pi$
$137$	−17.8706	−1.52679	−0.763394	+	0.645933i	$0.776468\pi$
$138$	0	0
$139$	13.8358	1.17354	0.586768	−	0.809755i	$-0.300400\pi$
$139$	13.8358	1.17354	0.586768	+	0.809755i	$0.300400\pi$
$140$	3.30094	0.278980
$141$	0	0
$142$	−4.36651	−0.366429
$143$	27.5091	2.30043
$144$	0	0
$145$	−5.88938	−0.489087
$146$	−13.8591	−1.14698
$147$	0	0
$148$	8.74428	0.718775
$149$	−7.37935	−0.604540	−0.302270	−	0.953222i	$-0.597744\pi$
$149$	−7.37935	−0.604540	−0.302270	+	0.953222i	$0.597744\pi$
$150$	0	0
$151$	6.50866	0.529667	0.264834	−	0.964294i	$-0.414683\pi$
$151$	6.50866	0.529667	0.264834	+	0.964294i	$0.414683\pi$
$152$	0.00400639	0.000324961 0
$153$	0	0
$154$	16.0245	1.29129
$155$	−1.52354	−0.122374
$156$	0	0
$157$	15.9153	1.27018	0.635090	−	0.772438i	$-0.280963\pi$
$157$	15.9153	1.27018	0.635090	+	0.772438i	$0.280963\pi$
$158$	7.52558	0.598703
$159$	0	0
$160$	1.18526	0.0937029
$161$	23.1984	1.82829
$162$	0	0
$163$	−8.34896	−0.653941	−0.326970	−	0.945035i	$-0.606028\pi$
$163$	−8.34896	−0.653941	−0.326970	+	0.945035i	$0.606028\pi$
$164$	−1.30094	−0.101586
$165$	0	0
$166$	−17.1260	−1.32924
$167$	0.536383	0.0415066	0.0207533	−	0.999785i	$-0.493394\pi$
$167$	0.536383	0.0415066	0.0207533	+	0.999785i	$0.493394\pi$
$168$	0	0
$169$	9.85785	0.758296
$170$	−7.99371	−0.613090
$171$	0	0
$172$	2.29136	0.174715
$173$	−7.46927	−0.567878	−0.283939	−	0.958842i	$-0.591641\pi$
$173$	−7.46927	−0.567878	−0.283939	+	0.958842i	$0.591641\pi$
$174$	0	0
$175$	−10.0125	−0.756874
$176$	5.75385	0.433713
$177$	0	0
$178$	−8.03882	−0.602535
$179$	−3.88654	−0.290493	−0.145247	−	0.989395i	$-0.546398\pi$
$179$	−3.88654	−0.290493	−0.145247	+	0.989395i	$0.546398\pi$
$180$	0	0
$181$	0.131783	0.00979539	0.00489769	−	0.999988i	$-0.498441\pi$
$181$	0.131783	0.00979539	0.00489769	+	0.999988i	$0.498441\pi$
$182$	13.3150	0.986976
$183$	0	0
$184$	8.32980	0.614081
$185$	10.3642	0.761993
$186$	0	0
$187$	−38.8056	−2.83774
$188$	3.72423	0.271618
$189$	0	0
$190$	0.00474861	0.000344500 0
$191$	25.7048	1.85994	0.929968	−	0.367641i	$-0.119834\pi$
$191$	25.7048	1.85994	0.929968	+	0.367641i	$0.119834\pi$
$192$	0	0
$193$	−6.35960	−0.457774	−0.228887	−	0.973453i	$-0.573509\pi$
$193$	−6.35960	−0.457774	−0.228887	+	0.973453i	$0.573509\pi$
$194$	6.08372	0.436785
$195$	0	0
$196$	0.756197	0.0540141
$197$	27.1869	1.93699	0.968493	−	0.249041i	$-0.0801153\pi$
$197$	27.1869	1.93699	0.968493	+	0.249041i	$0.0801153\pi$
$198$	0	0
$199$	14.4146	1.02182	0.510911	−	0.859633i	$-0.329308\pi$
$199$	14.4146	1.02182	0.510911	+	0.859633i	$0.329308\pi$
$200$	−3.59516	−0.254216
$201$	0	0
$202$	−8.20030	−0.576971
$203$	−13.8382	−0.971255
$204$	0	0
$205$	−1.54195	−0.107694
$206$	−15.7843	−1.09974
$207$	0	0
$208$	4.78099	0.331502
$209$	0.0230522	0.00159455
$210$	0	0
$211$	7.76063	0.534263	0.267132	−	0.963660i	$-0.413924\pi$
$211$	7.76063	0.534263	0.267132	+	0.963660i	$0.413924\pi$
$212$	−10.7493	−0.738264
$213$	0	0
$214$	−20.0887	−1.37323
$215$	2.71585	0.185220
$216$	0	0
$217$	−3.57985	−0.243016
$218$	8.88693	0.601899
$219$	0	0
$220$	6.81981	0.459791
$221$	−32.2443	−2.16899
$222$	0	0
$223$	−1.73919	−0.116465	−0.0582325	−	0.998303i	$-0.518546\pi$
$223$	−1.73919	−0.116465	−0.0582325	+	0.998303i	$0.518546\pi$
$224$	2.78499	0.186080
$225$	0	0
$226$	3.96968	0.264059
$227$	−3.15551	−0.209439	−0.104719	−	0.994502i	$-0.533394\pi$
$227$	−3.15551	−0.209439	−0.104719	+	0.994502i	$0.533394\pi$
$228$	0	0
$229$	19.7834	1.30733	0.653663	−	0.756785i	$-0.273231\pi$
$229$	19.7834	1.30733	0.653663	+	0.756785i	$0.273231\pi$
$230$	9.87297	0.651004
$231$	0	0
$232$	−4.96886	−0.326222
$233$	23.9094	1.56636	0.783179	−	0.621797i	$-0.213597\pi$
$233$	23.9094	1.56636	0.783179	+	0.621797i	$0.213597\pi$
$234$	0	0
$235$	4.41418	0.287949
$236$	−0.960614	−0.0625307
$237$	0	0
$238$	−18.7828	−1.21751
$239$	24.8903	1.61002	0.805010	−	0.593261i	$-0.202160\pi$
$239$	24.8903	1.61002	0.805010	+	0.593261i	$0.202160\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	22.1068	1.42108
$243$	0	0
$244$	−12.3595	−0.791235
$245$	0.896289	0.0572618
$246$	0	0
$247$	0.0191545	0.00121877
$248$	−1.28541	−0.0816234
$249$	0	0
$250$	−10.1875	−0.644313
$251$	−15.0319	−0.948804	−0.474402	−	0.880308i	$-0.657336\pi$
$251$	−15.0319	−0.948804	−0.474402	+	0.880308i	$0.657336\pi$
$252$	0	0
$253$	47.9284	3.01324
$254$	13.4218	0.842157
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.44834	0.0903452	0.0451726	−	0.998979i	$-0.485616\pi$
$257$	1.44834	0.0903452	0.0451726	+	0.998979i	$0.485616\pi$
$258$	0	0
$259$	24.3528	1.51321
$260$	5.66671	0.351434
$261$	0	0
$262$	8.60563	0.531657
$263$	−2.45490	−0.151376	−0.0756878	−	0.997132i	$-0.524115\pi$
$263$	−2.45490	−0.151376	−0.0756878	+	0.997132i	$0.524115\pi$
$264$	0	0
$265$	−12.7407	−0.782654
$266$	0.0111578	0.000684127 0
$267$	0	0
$268$	10.7827	0.658655
$269$	9.65041	0.588396	0.294198	−	0.955745i	$-0.404948\pi$
$269$	9.65041	0.588396	0.294198	+	0.955745i	$0.404948\pi$
$270$	0	0
$271$	11.0972	0.674108	0.337054	−	0.941485i	$-0.390569\pi$
$271$	11.0972	0.674108	0.337054	+	0.941485i	$0.390569\pi$
$272$	−6.74428	−0.408932
$273$	0	0
$274$	−17.8706	−1.07960
$275$	−20.6860	−1.24741
$276$	0	0
$277$	−28.9706	−1.74068	−0.870338	−	0.492455i	$-0.836100\pi$
$277$	−28.9706	−1.74068	−0.870338	+	0.492455i	$0.836100\pi$
$278$	13.8358	0.829815
$279$	0	0
$280$	3.30094	0.197269
$281$	−25.9134	−1.54587	−0.772933	−	0.634487i	$-0.781211\pi$
$281$	−25.9134	−1.54587	−0.772933	+	0.634487i	$0.781211\pi$
$282$	0	0
$283$	−15.0528	−0.894797	−0.447398	−	0.894335i	$-0.647649\pi$
$283$	−15.0528	−0.894797	−0.447398	+	0.894335i	$0.647649\pi$
$284$	−4.36651	−0.259105
$285$	0	0
$286$	27.5091	1.62665
$287$	−3.62311	−0.213865
$288$	0	0
$289$	28.4852	1.67560
$290$	−5.88938	−0.345837
$291$	0	0
$292$	−13.8591	−0.811041
$293$	19.5077	1.13965	0.569826	−	0.821765i	$-0.307010\pi$
$293$	19.5077	1.13965	0.569826	+	0.821765i	$0.307010\pi$
$294$	0	0
$295$	−1.13858	−0.0662905
$296$	8.74428	0.508251
$297$	0	0
$298$	−7.37935	−0.427475
$299$	39.8247	2.30312
$300$	0	0
$301$	6.38143	0.367819
$302$	6.50866	0.374531
$303$	0	0
$304$	0.00400639	0.000229782 0
$305$	−14.6492	−0.838810
$306$	0	0
$307$	4.34421	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$307$	4.34421	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$308$	16.0245	0.913078
$309$	0	0
$310$	−1.52354	−0.0865313
$311$	4.03157	0.228609	0.114305	−	0.993446i	$-0.463536\pi$
$311$	4.03157	0.228609	0.114305	+	0.993446i	$0.463536\pi$
$312$	0	0
$313$	12.0214	0.679492	0.339746	−	0.940517i	$-0.389659\pi$
$313$	12.0214	0.679492	0.339746	+	0.940517i	$0.389659\pi$
$314$	15.9153	0.898153
$315$	0	0
$316$	7.52558	0.423347
$317$	−0.649585	−0.0364844	−0.0182422	−	0.999834i	$-0.505807\pi$
$317$	−0.649585	−0.0364844	−0.0182422	+	0.999834i	$0.505807\pi$
$318$	0	0
$319$	−28.5901	−1.60074
$320$	1.18526	0.0662580
$321$	0	0
$322$	23.1984	1.29280
$323$	−0.0270202	−0.00150344
$324$	0	0
$325$	−17.1884	−0.953442
$326$	−8.34896	−0.462406
$327$	0	0
$328$	−1.30094	−0.0718324
$329$	10.3720	0.571825
$330$	0	0
$331$	−33.9683	−1.86706	−0.933532	−	0.358493i	$-0.883291\pi$
$331$	−33.9683	−1.86706	−0.933532	+	0.358493i	$0.883291\pi$
$332$	−17.1260	−0.939911
$333$	0	0
$334$	0.536383	0.0293496
$335$	12.7802	0.698259
$336$	0	0
$337$	−17.8331	−0.971429	−0.485715	−	0.874117i	$-0.661441\pi$
$337$	−17.8331	−0.971429	−0.485715	+	0.874117i	$0.661441\pi$
$338$	9.85785	0.536196
$339$	0	0
$340$	−7.99371	−0.433520
$341$	−7.39605	−0.400518
$342$	0	0
$343$	−17.3890	−0.938916
$344$	2.29136	0.123542
$345$	0	0
$346$	−7.46927	−0.401551
$347$	0.789138	0.0423631	0.0211816	−	0.999776i	$-0.493257\pi$
$347$	0.789138	0.0423631	0.0211816	+	0.999776i	$0.493257\pi$
$348$	0	0
$349$	−5.72752	−0.306587	−0.153294	−	0.988181i	$-0.548988\pi$
$349$	−5.72752	−0.306587	−0.153294	+	0.988181i	$0.548988\pi$
$350$	−10.0125	−0.535191
$351$	0	0
$352$	5.75385	0.306681
$353$	−20.8391	−1.10915	−0.554577	−	0.832133i	$-0.687120\pi$
$353$	−20.8391	−1.10915	−0.554577	+	0.832133i	$0.687120\pi$
$354$	0	0
$355$	−5.17545	−0.274684
$356$	−8.03882	−0.426056
$357$	0	0
$358$	−3.88654	−0.205410
$359$	22.4299	1.18380	0.591902	−	0.806010i	$-0.298377\pi$
$359$	22.4299	1.18380	0.591902	+	0.806010i	$0.298377\pi$
$360$	0	0
$361$	−19.0000	−0.999999
$362$	0.131783	0.00692638
$363$	0	0
$364$	13.3150	0.697897
$365$	−16.4266	−0.859807
$366$	0	0
$367$	−24.1401	−1.26010	−0.630052	−	0.776553i	$-0.716966\pi$
$367$	−24.1401	−1.26010	−0.630052	+	0.776553i	$0.716966\pi$
$368$	8.32980	0.434221
$369$	0	0
$370$	10.3642	0.538811
$371$	−29.9367	−1.55424
$372$	0	0
$373$	23.3172	1.20732	0.603660	−	0.797242i	$-0.293709\pi$
$373$	23.3172	1.20732	0.603660	+	0.797242i	$0.293709\pi$
$374$	−38.8056	−2.00659
$375$	0	0
$376$	3.72423	0.192063
$377$	−23.7561	−1.22350
$378$	0	0
$379$	2.14819	0.110345	0.0551726	−	0.998477i	$-0.482429\pi$
$379$	2.14819	0.110345	0.0551726	+	0.998477i	$0.482429\pi$
$380$	0.00474861	0.000243598 0
$381$	0	0
$382$	25.7048	1.31517
$383$	5.14072	0.262679	0.131339	−	0.991337i	$-0.458072\pi$
$383$	5.14072	0.262679	0.131339	+	0.991337i	$0.458072\pi$
$384$	0	0
$385$	18.9931	0.967979
$386$	−6.35960	−0.323695
$387$	0	0
$388$	6.08372	0.308854
$389$	−12.3447	−0.625900	−0.312950	−	0.949770i	$-0.601317\pi$
$389$	−12.3447	−0.625900	−0.312950	+	0.949770i	$0.601317\pi$
$390$	0	0
$391$	−56.1784	−2.84107
$392$	0.756197	0.0381937
$393$	0	0
$394$	27.1869	1.36966
$395$	8.91976	0.448802
$396$	0	0
$397$	−20.4753	−1.02763	−0.513814	−	0.857902i	$-0.671768\pi$
$397$	−20.4753	−1.02763	−0.513814	+	0.857902i	$0.671768\pi$
$398$	14.4146	0.722538
$399$	0	0
$400$	−3.59516	−0.179758
$401$	4.53148	0.226291	0.113146	−	0.993578i	$-0.463907\pi$
$401$	4.53148	0.226291	0.113146	+	0.993578i	$0.463907\pi$
$402$	0	0
$403$	−6.14552	−0.306130
$404$	−8.20030	−0.407980
$405$	0	0
$406$	−13.8382	−0.686781
$407$	50.3133	2.49394
$408$	0	0
$409$	−23.3866	−1.15639	−0.578196	−	0.815898i	$-0.696243\pi$
$409$	−23.3866	−1.15639	−0.578196	+	0.815898i	$0.696243\pi$
$410$	−1.54195	−0.0761515
$411$	0	0
$412$	−15.7843	−0.777636
$413$	−2.67531	−0.131643
$414$	0	0
$415$	−20.2987	−0.996426
$416$	4.78099	0.234407
$417$	0	0
$418$	0.0230522	0.00112752
$419$	−18.3339	−0.895671	−0.447836	−	0.894116i	$-0.647805\pi$
$419$	−18.3339	−0.895671	−0.447836	+	0.894116i	$0.647805\pi$
$420$	0	0
$421$	19.2156	0.936511	0.468256	−	0.883593i	$-0.344883\pi$
$421$	19.2156	0.936511	0.468256	+	0.883593i	$0.344883\pi$
$422$	7.76063	0.377781
$423$	0	0
$424$	−10.7493	−0.522031
$425$	24.2468	1.17614
$426$	0	0
$427$	−34.4211	−1.66575
$428$	−20.0887	−0.971022
$429$	0	0
$430$	2.71585	0.130970
$431$	−30.0514	−1.44753	−0.723763	−	0.690048i	$-0.757589\pi$
$431$	−30.0514	−1.44753	−0.723763	+	0.690048i	$0.757589\pi$
$432$	0	0
$433$	−19.6712	−0.945336	−0.472668	−	0.881241i	$-0.656709\pi$
$433$	−19.6712	−0.945336	−0.472668	+	0.881241i	$0.656709\pi$
$434$	−3.57985	−0.171838
$435$	0	0
$436$	8.88693	0.425607
$437$	0.0333724	0.00159642
$438$	0	0
$439$	28.2196	1.34685	0.673424	−	0.739257i	$-0.264823\pi$
$439$	28.2196	1.34685	0.673424	+	0.739257i	$0.264823\pi$
$440$	6.81981	0.325121
$441$	0	0
$442$	−32.2443	−1.53371
$443$	−16.6190	−0.789591	−0.394796	−	0.918769i	$-0.629185\pi$
$443$	−16.6190	−0.789591	−0.394796	+	0.918769i	$0.629185\pi$
$444$	0	0
$445$	−9.52808	−0.451674
$446$	−1.73919	−0.0823532
$447$	0	0
$448$	2.78499	0.131579
$449$	−6.93205	−0.327144	−0.163572	−	0.986531i	$-0.552302\pi$
$449$	−6.93205	−0.327144	−0.163572	+	0.986531i	$0.552302\pi$
$450$	0	0
$451$	−7.48542	−0.352475
$452$	3.96968	0.186718
$453$	0	0
$454$	−3.15551	−0.148096
$455$	15.7818	0.739860
$456$	0	0
$457$	−15.1772	−0.709958	−0.354979	−	0.934874i	$-0.615512\pi$
$457$	−15.1772	−0.709958	−0.354979	+	0.934874i	$0.615512\pi$
$458$	19.7834	0.924420
$459$	0	0
$460$	9.87297	0.460329
$461$	31.2933	1.45748	0.728738	−	0.684792i	$-0.240107\pi$
$461$	31.2933	1.45748	0.728738	+	0.684792i	$0.240107\pi$
$462$	0	0
$463$	−3.46957	−0.161245	−0.0806223	−	0.996745i	$-0.525691\pi$
$463$	−3.46957	−0.161245	−0.0806223	+	0.996745i	$0.525691\pi$
$464$	−4.96886	−0.230674
$465$	0	0
$466$	23.9094	1.10758
$467$	−30.5122	−1.41194	−0.705968	−	0.708244i	$-0.749488\pi$
$467$	−30.5122	−1.41194	−0.705968	+	0.708244i	$0.749488\pi$
$468$	0	0
$469$	30.0296	1.38664
$470$	4.41418	0.203611
$471$	0	0
$472$	−0.960614	−0.0442159
$473$	13.1842	0.606208
$474$	0	0
$475$	−0.0144036	−0.000660883 0
$476$	−18.7828	−0.860907
$477$	0	0
$478$	24.8903	1.13846
$479$	8.85977	0.404813	0.202407	−	0.979302i	$-0.435124\pi$
$479$	8.85977	0.404813	0.202407	+	0.979302i	$0.435124\pi$
$480$	0	0
$481$	41.8063	1.90620
$482$	1.00000	0.0455488
$483$	0	0
$484$	22.1068	1.00486
$485$	7.21078	0.327425
$486$	0	0
$487$	22.4128	1.01562	0.507811	−	0.861468i	$-0.330455\pi$
$487$	22.4128	1.01562	0.507811	+	0.861468i	$0.330455\pi$
$488$	−12.3595	−0.559487
$489$	0	0
$490$	0.896289	0.0404902
$491$	−3.14038	−0.141723	−0.0708616	−	0.997486i	$-0.522575\pi$
$491$	−3.14038	−0.141723	−0.0708616	+	0.997486i	$0.522575\pi$
$492$	0	0
$493$	33.5114	1.50928
$494$	0.0191545	0.000861801 0
$495$	0	0
$496$	−1.28541	−0.0577165
$497$	−12.1607	−0.545482
$498$	0	0
$499$	−8.09558	−0.362408	−0.181204	−	0.983446i	$-0.557999\pi$
$499$	−8.09558	−0.362408	−0.181204	+	0.983446i	$0.557999\pi$
$500$	−10.1875	−0.455598
$501$	0	0
$502$	−15.0319	−0.670906
$503$	17.5742	0.783596	0.391798	−	0.920051i	$-0.371853\pi$
$503$	17.5742	0.783596	0.391798	+	0.920051i	$0.371853\pi$
$504$	0	0
$505$	−9.71947	−0.432511
$506$	47.9284	2.13068
$507$	0	0
$508$	13.4218	0.595495
$509$	9.21149	0.408292	0.204146	−	0.978940i	$-0.434558\pi$
$509$	9.21149	0.408292	0.204146	+	0.978940i	$0.434558\pi$
$510$	0	0
$511$	−38.5974	−1.70745
$512$	1.00000	0.0441942
$513$	0	0
$514$	1.44834	0.0638837
$515$	−18.7085	−0.824393
$516$	0	0
$517$	21.4287	0.942433
$518$	24.3528	1.07000
$519$	0	0
$520$	5.66671	0.248502
$521$	−40.8037	−1.78764	−0.893822	−	0.448423i	$-0.851986\pi$
$521$	−40.8037	−1.78764	−0.893822	+	0.448423i	$0.851986\pi$
$522$	0	0
$523$	−13.3010	−0.581614	−0.290807	−	0.956782i	$-0.593924\pi$
$523$	−13.3010	−0.581614	−0.290807	+	0.956782i	$0.593924\pi$
$524$	8.60563	0.375938
$525$	0	0
$526$	−2.45490	−0.107039
$527$	8.66914	0.377634
$528$	0	0
$529$	46.3855	2.01676
$530$	−12.7407	−0.553420
$531$	0	0
$532$	0.0111578	0.000483751 0
$533$	−6.21978	−0.269408
$534$	0	0
$535$	−23.8103	−1.02941
$536$	10.7827	0.465740
$537$	0	0
$538$	9.65041	0.416059
$539$	4.35105	0.187413
$540$	0	0
$541$	−9.90207	−0.425723	−0.212862	−	0.977082i	$-0.568278\pi$
$541$	−9.90207	−0.425723	−0.212862	+	0.977082i	$0.568278\pi$
$542$	11.0972	0.476667
$543$	0	0
$544$	−6.74428	−0.289158
$545$	10.5333	0.451198
$546$	0	0
$547$	17.7496	0.758918	0.379459	−	0.925209i	$-0.376110\pi$
$547$	17.7496	0.758918	0.379459	+	0.925209i	$0.376110\pi$
$548$	−17.8706	−0.763394
$549$	0	0
$550$	−20.6860	−0.882056
$551$	−0.0199072	−0.000848074 0
$552$	0	0
$553$	20.9587	0.891255
$554$	−28.9706	−1.23084
$555$	0	0
$556$	13.8358	0.586768
$557$	23.2016	0.983084	0.491542	−	0.870854i	$-0.336433\pi$
$557$	23.2016	0.983084	0.491542	+	0.870854i	$0.336433\pi$
$558$	0	0
$559$	10.9550	0.463346
$560$	3.30094	0.139490
$561$	0	0
$562$	−25.9134	−1.09309
$563$	−10.8545	−0.457462	−0.228731	−	0.973490i	$-0.573458\pi$
$563$	−10.8545	−0.457462	−0.228731	+	0.973490i	$0.573458\pi$
$564$	0	0
$565$	4.70510	0.197945
$566$	−15.0528	−0.632717
$567$	0	0
$568$	−4.36651	−0.183215
$569$	15.1996	0.637200	0.318600	−	0.947889i	$-0.396787\pi$
$569$	15.1996	0.637200	0.318600	+	0.947889i	$0.396787\pi$
$570$	0	0
$571$	8.71559	0.364736	0.182368	−	0.983230i	$-0.441624\pi$
$571$	8.71559	0.364736	0.182368	+	0.983230i	$0.441624\pi$
$572$	27.5091	1.15021
$573$	0	0
$574$	−3.62311	−0.151226
$575$	−29.9470	−1.24887
$576$	0	0
$577$	15.2068	0.633068	0.316534	−	0.948581i	$-0.397481\pi$
$577$	15.2068	0.633068	0.316534	+	0.948581i	$0.397481\pi$
$578$	28.4852	1.18483
$579$	0	0
$580$	−5.88938	−0.244543
$581$	−47.6958	−1.97876
$582$	0	0
$583$	−61.8498	−2.56156
$584$	−13.8591	−0.573492
$585$	0	0
$586$	19.5077	0.805856
$587$	22.7712	0.939867	0.469933	−	0.882702i	$-0.344278\pi$
$587$	22.7712	0.939867	0.469933	+	0.882702i	$0.344278\pi$
$588$	0	0
$589$	−0.00514984	−0.000212195 0
$590$	−1.13858	−0.0468745
$591$	0	0
$592$	8.74428	0.359387
$593$	7.13949	0.293184	0.146592	−	0.989197i	$-0.453170\pi$
$593$	7.13949	0.293184	0.146592	+	0.989197i	$0.453170\pi$
$594$	0	0
$595$	−22.2624	−0.912671
$596$	−7.37935	−0.302270
$597$	0	0
$598$	39.8247	1.62855
$599$	−47.0975	−1.92435	−0.962176	−	0.272428i	$-0.912173\pi$
$599$	−47.0975	−1.92435	−0.962176	+	0.272428i	$0.912173\pi$
$600$	0	0
$601$	35.4153	1.44462	0.722311	−	0.691568i	$-0.243080\pi$
$601$	35.4153	1.44462	0.722311	+	0.691568i	$0.243080\pi$
$602$	6.38143	0.260088
$603$	0	0
$604$	6.50866	0.264834
$605$	26.2023	1.06528
$606$	0	0
$607$	19.3317	0.784650	0.392325	−	0.919827i	$-0.371671\pi$
$607$	19.3317	0.784650	0.392325	+	0.919827i	$0.371671\pi$
$608$	0.00400639	0.000162480 0
$609$	0	0
$610$	−14.6492	−0.593128
$611$	17.8055	0.720334
$612$	0	0
$613$	−20.2213	−0.816731	−0.408366	−	0.912818i	$-0.633901\pi$
$613$	−20.2213	−0.816731	−0.408366	+	0.912818i	$0.633901\pi$
$614$	4.34421	0.175318
$615$	0	0
$616$	16.0245	0.645644
$617$	21.7820	0.876910	0.438455	−	0.898753i	$-0.355526\pi$
$617$	21.7820	0.876910	0.438455	+	0.898753i	$0.355526\pi$
$618$	0	0
$619$	−44.2449	−1.77835	−0.889177	−	0.457564i	$-0.848722\pi$
$619$	−44.2449	−1.77835	−0.889177	+	0.457564i	$0.848722\pi$
$620$	−1.52354	−0.0611868
$621$	0	0
$622$	4.03157	0.161651
$623$	−22.3881	−0.896959
$624$	0	0
$625$	5.90100	0.236040
$626$	12.0214	0.480473
$627$	0	0
$628$	15.9153	0.635090
$629$	−58.9738	−2.35144
$630$	0	0
$631$	41.9386	1.66955	0.834774	−	0.550593i	$-0.185598\pi$
$631$	41.9386	1.66955	0.834774	+	0.550593i	$0.185598\pi$
$632$	7.52558	0.299352
$633$	0	0
$634$	−0.649585	−0.0257983
$635$	15.9083	0.631301
$636$	0	0
$637$	3.61537	0.143246
$638$	−28.5901	−1.13189
$639$	0	0
$640$	1.18526	0.0468515
$641$	15.0844	0.595797	0.297899	−	0.954598i	$-0.403714\pi$
$641$	15.0844	0.595797	0.297899	+	0.954598i	$0.403714\pi$
$642$	0	0
$643$	−5.22472	−0.206043	−0.103021	−	0.994679i	$-0.532851\pi$
$643$	−5.22472	−0.206043	−0.103021	+	0.994679i	$0.532851\pi$
$644$	23.1984	0.914147
$645$	0	0
$646$	−0.0270202	−0.00106309
$647$	44.3317	1.74286	0.871430	−	0.490520i	$-0.163193\pi$
$647$	44.3317	1.74286	0.871430	+	0.490520i	$0.163193\pi$
$648$	0	0
$649$	−5.52724	−0.216963
$650$	−17.1884	−0.674186
$651$	0	0
$652$	−8.34896	−0.326970
$653$	−9.54080	−0.373360	−0.186680	−	0.982421i	$-0.559773\pi$
$653$	−9.54080	−0.373360	−0.186680	+	0.982421i	$0.559773\pi$
$654$	0	0
$655$	10.1999	0.398543
$656$	−1.30094	−0.0507932
$657$	0	0
$658$	10.3720	0.404341
$659$	10.5227	0.409907	0.204953	−	0.978772i	$-0.434296\pi$
$659$	10.5227	0.409907	0.204953	+	0.978772i	$0.434296\pi$
$660$	0	0
$661$	−2.32235	−0.0903289	−0.0451645	−	0.998980i	$-0.514381\pi$
$661$	−2.32235	−0.0903289	−0.0451645	+	0.998980i	$0.514381\pi$
$662$	−33.9683	−1.32021
$663$	0	0
$664$	−17.1260	−0.664618
$665$	0.0132248	0.000512837 0
$666$	0	0
$667$	−41.3896	−1.60261
$668$	0.536383	0.0207533
$669$	0	0
$670$	12.7802	0.493743
$671$	−71.1146	−2.74535
$672$	0	0
$673$	−0.755644	−0.0291279	−0.0145640	−	0.999894i	$-0.504636\pi$
$673$	−0.755644	−0.0291279	−0.0145640	+	0.999894i	$0.504636\pi$
$674$	−17.8331	−0.686904
$675$	0	0
$676$	9.85785	0.379148
$677$	−11.8251	−0.454475	−0.227238	−	0.973839i	$-0.572969\pi$
$677$	−11.8251	−0.454475	−0.227238	+	0.973839i	$0.572969\pi$
$678$	0	0
$679$	16.9431	0.650217
$680$	−7.99371	−0.306545
$681$	0	0
$682$	−7.39605	−0.283209
$683$	9.30137	0.355907	0.177953	−	0.984039i	$-0.443052\pi$
$683$	9.30137	0.355907	0.177953	+	0.984039i	$0.443052\pi$
$684$	0	0
$685$	−21.1813	−0.809295
$686$	−17.3890	−0.663914
$687$	0	0
$688$	2.29136	0.0873573
$689$	−51.3922	−1.95789
$690$	0	0
$691$	0.0708326	0.00269460	0.00134730	−	0.999999i	$-0.499571\pi$
$691$	0.0708326	0.00269460	0.00134730	+	0.999999i	$0.499571\pi$
$692$	−7.46927	−0.283939
$693$	0	0
$694$	0.789138	0.0299553
$695$	16.3990	0.622049
$696$	0	0
$697$	8.77389	0.332335
$698$	−5.72752	−0.216790
$699$	0	0
$700$	−10.0125	−0.378437
$701$	−41.8498	−1.58065	−0.790323	−	0.612691i	$-0.790087\pi$
$701$	−41.8498	−1.58065	−0.790323	+	0.612691i	$0.790087\pi$
$702$	0	0
$703$	0.0350330	0.00132129
$704$	5.75385	0.216857
$705$	0	0
$706$	−20.8391	−0.784290
$707$	−22.8378	−0.858903
$708$	0	0
$709$	−20.0130	−0.751604	−0.375802	−	0.926700i	$-0.622633\pi$
$709$	−20.0130	−0.751604	−0.375802	+	0.926700i	$0.622633\pi$
$710$	−5.17545	−0.194231
$711$	0	0
$712$	−8.03882	−0.301267
$713$	−10.7072	−0.400987
$714$	0	0
$715$	32.6054	1.21937
$716$	−3.88654	−0.145247
$717$	0	0
$718$	22.4299	0.837076
$719$	−45.1857	−1.68514	−0.842571	−	0.538585i	$-0.818959\pi$
$719$	−45.1857	−1.68514	−0.842571	+	0.538585i	$0.818959\pi$
$720$	0	0
$721$	−43.9591	−1.63712
$722$	−19.0000	−0.707106
$723$	0	0
$724$	0.131783	0.00489769
$725$	17.8639	0.663447
$726$	0	0
$727$	47.5991	1.76535	0.882677	−	0.469979i	$-0.155739\pi$
$727$	47.5991	1.76535	0.882677	+	0.469979i	$0.155739\pi$
$728$	13.3150	0.493488
$729$	0	0
$730$	−16.4266	−0.607975
$731$	−15.4536	−0.571571
$732$	0	0
$733$	−21.6161	−0.798410	−0.399205	−	0.916862i	$-0.630714\pi$
$733$	−21.6161	−0.798410	−0.399205	+	0.916862i	$0.630714\pi$
$734$	−24.1401	−0.891028
$735$	0	0
$736$	8.32980	0.307040
$737$	62.0418	2.28534
$738$	0	0
$739$	−19.0881	−0.702166	−0.351083	−	0.936344i	$-0.614187\pi$
$739$	−19.0881	−0.702166	−0.351083	+	0.936344i	$0.614187\pi$
$740$	10.3642	0.380997
$741$	0	0
$742$	−29.9367	−1.09901
$743$	−1.60105	−0.0587370	−0.0293685	−	0.999569i	$-0.509350\pi$
$743$	−1.60105	−0.0587370	−0.0293685	+	0.999569i	$0.509350\pi$
$744$	0	0
$745$	−8.74644	−0.320445
$746$	23.3172	0.853704
$747$	0	0
$748$	−38.8056	−1.41887
$749$	−55.9468	−2.04425
$750$	0	0
$751$	−1.72512	−0.0629504	−0.0314752	−	0.999505i	$-0.510021\pi$
$751$	−1.72512	−0.0629504	−0.0314752	+	0.999505i	$0.510021\pi$
$752$	3.72423	0.135809
$753$	0	0
$754$	−23.7561	−0.865145
$755$	7.71444	0.280757
$756$	0	0
$757$	−30.1973	−1.09754	−0.548771	−	0.835973i	$-0.684904\pi$
$757$	−30.1973	−1.09754	−0.548771	+	0.835973i	$0.684904\pi$
$758$	2.14819	0.0780258
$759$	0	0
$760$	0.00474861	0.000172250 0
$761$	27.0903	0.982023	0.491011	−	0.871153i	$-0.336627\pi$
$761$	27.0903	0.982023	0.491011	+	0.871153i	$0.336627\pi$
$762$	0	0
$763$	24.7501	0.896012
$764$	25.7048	0.929968
$765$	0	0
$766$	5.14072	0.185742
$767$	−4.59269	−0.165832
$768$	0	0
$769$	2.73157	0.0985029	0.0492514	−	0.998786i	$-0.484316\pi$
$769$	2.73157	0.0985029	0.0492514	+	0.998786i	$0.484316\pi$
$770$	18.9931	0.684465
$771$	0	0
$772$	−6.35960	−0.228887
$773$	−44.1805	−1.58906	−0.794531	−	0.607224i	$-0.792283\pi$
$773$	−44.1805	−1.58906	−0.794531	+	0.607224i	$0.792283\pi$
$774$	0	0
$775$	4.62125	0.166000
$776$	6.08372	0.218393
$777$	0	0
$778$	−12.3447	−0.442578
$779$	−0.00521207	−0.000186742 0
$780$	0	0
$781$	−25.1243	−0.899017
$782$	−56.1784	−2.00894
$783$	0	0
$784$	0.756197	0.0270070
$785$	18.8638	0.673277
$786$	0	0
$787$	4.59324	0.163731	0.0818657	−	0.996643i	$-0.473912\pi$
$787$	4.59324	0.163731	0.0818657	+	0.996643i	$0.473912\pi$
$788$	27.1869	0.968493
$789$	0	0
$790$	8.91976	0.317351
$791$	11.0555	0.393090
$792$	0	0
$793$	−59.0905	−2.09837
$794$	−20.4753	−0.726642
$795$	0	0
$796$	14.4146	0.510911
$797$	9.23257	0.327034	0.163517	−	0.986540i	$-0.447716\pi$
$797$	9.23257	0.327034	0.163517	+	0.986540i	$0.447716\pi$
$798$	0	0
$799$	−25.1172	−0.888584
$800$	−3.59516	−0.127108
$801$	0	0
$802$	4.53148	0.160012
$803$	−79.7431	−2.81407
$804$	0	0
$805$	27.4962	0.969112
$806$	−6.14552	−0.216467
$807$	0	0
$808$	−8.20030	−0.288485
$809$	−51.3454	−1.80521	−0.902603	−	0.430473i	$-0.858347\pi$
$809$	−51.3454	−1.80521	−0.902603	+	0.430473i	$0.858347\pi$
$810$	0	0
$811$	−34.9332	−1.22667	−0.613336	−	0.789822i	$-0.710173\pi$
$811$	−34.9332	−1.22667	−0.613336	+	0.789822i	$0.710173\pi$
$812$	−13.8382	−0.485627
$813$	0	0
$814$	50.3133	1.76348
$815$	−9.89567	−0.346630
$816$	0	0
$817$	0.00918008	0.000321170 0
$818$	−23.3866	−0.817692
$819$	0	0
$820$	−1.54195	−0.0538472
$821$	38.2123	1.33362	0.666809	−	0.745228i	$-0.267659\pi$
$821$	38.2123	1.33362	0.666809	+	0.745228i	$0.267659\pi$
$822$	0	0
$823$	5.42918	0.189249	0.0946246	−	0.995513i	$-0.469835\pi$
$823$	5.42918	0.189249	0.0946246	+	0.995513i	$0.469835\pi$
$824$	−15.7843	−0.549871
$825$	0	0
$826$	−2.67531	−0.0930858
$827$	48.1757	1.67523	0.837617	−	0.546258i	$-0.183948\pi$
$827$	48.1757	1.67523	0.837617	+	0.546258i	$0.183948\pi$
$828$	0	0
$829$	19.2947	0.670132	0.335066	−	0.942195i	$-0.391241\pi$
$829$	19.2947	0.670132	0.335066	+	0.942195i	$0.391241\pi$
$830$	−20.2987	−0.704580
$831$	0	0
$832$	4.78099	0.165751
$833$	−5.10000	−0.176705
$834$	0	0
$835$	0.635753	0.0220011
$836$	0.0230522	0.000797276 0
$837$	0	0
$838$	−18.3339	−0.633335
$839$	17.7783	0.613773	0.306887	−	0.951746i	$-0.400713\pi$
$839$	17.7783	0.613773	0.306887	+	0.951746i	$0.400713\pi$
$840$	0	0
$841$	−4.31044	−0.148636
$842$	19.2156	0.662214
$843$	0	0
$844$	7.76063	0.267132
$845$	11.6841	0.401945
$846$	0	0
$847$	61.5674	2.11548
$848$	−10.7493	−0.369132
$849$	0	0
$850$	24.2468	0.831657
$851$	72.8380	2.49686
$852$	0	0
$853$	−6.46572	−0.221382	−0.110691	−	0.993855i	$-0.535306\pi$
$853$	−6.46572	−0.221382	−0.110691	+	0.993855i	$0.535306\pi$
$854$	−34.4211	−1.17787
$855$	0	0
$856$	−20.0887	−0.686616
$857$	33.4867	1.14388	0.571941	−	0.820295i	$-0.306191\pi$
$857$	33.4867	1.14388	0.571941	+	0.820295i	$0.306191\pi$
$858$	0	0
$859$	−34.2680	−1.16921	−0.584604	−	0.811319i	$-0.698750\pi$
$859$	−34.2680	−1.16921	−0.584604	+	0.811319i	$0.698750\pi$
$860$	2.71585	0.0926099
$861$	0	0
$862$	−30.0514	−1.02356
$863$	−30.2458	−1.02958	−0.514790	−	0.857316i	$-0.672130\pi$
$863$	−30.2458	−1.02958	−0.514790	+	0.857316i	$0.672130\pi$
$864$	0	0
$865$	−8.85302	−0.301012
$866$	−19.6712	−0.668454
$867$	0	0
$868$	−3.57985	−0.121508
$869$	43.3011	1.46889
$870$	0	0
$871$	51.5517	1.74676
$872$	8.88693	0.300949
$873$	0	0
$874$	0.0333724	0.00112884
$875$	−28.3721	−0.959152
$876$	0	0
$877$	47.6057	1.60753	0.803766	−	0.594946i	$-0.202827\pi$
$877$	47.6057	1.60753	0.803766	+	0.594946i	$0.202827\pi$
$878$	28.2196	0.952365
$879$	0	0
$880$	6.81981	0.229896
$881$	−30.0993	−1.01407	−0.507035	−	0.861925i	$-0.669259\pi$
$881$	−30.0993	−1.01407	−0.507035	+	0.861925i	$0.669259\pi$
$882$	0	0
$883$	−15.1646	−0.510328	−0.255164	−	0.966898i	$-0.582129\pi$
$883$	−15.1646	−0.510328	−0.255164	+	0.966898i	$0.582129\pi$
$884$	−32.2443	−1.08449
$885$	0	0
$886$	−16.6190	−0.558325
$887$	54.5507	1.83163	0.915817	−	0.401595i	$-0.131544\pi$
$887$	54.5507	1.83163	0.915817	+	0.401595i	$0.131544\pi$
$888$	0	0
$889$	37.3796	1.25367
$890$	−9.52808	−0.319382
$891$	0	0
$892$	−1.73919	−0.0582325
$893$	0.0149207	0.000499303 0
$894$	0	0
$895$	−4.60655	−0.153980
$896$	2.78499	0.0930402
$897$	0	0
$898$	−6.93205	−0.231325
$899$	6.38701	0.213019
$900$	0	0
$901$	72.4961	2.41520
$902$	−7.48542	−0.249237
$903$	0	0
$904$	3.96968	0.132030
$905$	0.156197	0.00519218
$906$	0	0
$907$	−33.5054	−1.11253	−0.556263	−	0.831006i	$-0.687765\pi$
$907$	−33.5054	−1.11253	−0.556263	+	0.831006i	$0.687765\pi$
$908$	−3.15551	−0.104719
$909$	0	0
$910$	15.7818	0.523160
$911$	−24.5771	−0.814274	−0.407137	−	0.913367i	$-0.633473\pi$
$911$	−24.5771	−0.814274	−0.407137	+	0.913367i	$0.633473\pi$
$912$	0	0
$913$	−98.5405	−3.26121
$914$	−15.1772	−0.502016
$915$	0	0
$916$	19.7834	0.653663
$917$	23.9666	0.791448
$918$	0	0
$919$	13.4312	0.443054	0.221527	−	0.975154i	$-0.428896\pi$
$919$	13.4312	0.443054	0.221527	+	0.975154i	$0.428896\pi$
$920$	9.87297	0.325502
$921$	0	0
$922$	31.2933	1.03059
$923$	−20.8762	−0.687150
$924$	0	0
$925$	−31.4371	−1.03364
$926$	−3.46957	−0.114017
$927$	0	0
$928$	−4.96886	−0.163111
$929$	13.4983	0.442865	0.221433	−	0.975176i	$-0.428927\pi$
$929$	13.4983	0.442865	0.221433	+	0.975176i	$0.428927\pi$
$930$	0	0
$931$	0.00302962	9.92917e−5 0
$932$	23.9094	0.783179
$933$	0	0
$934$	−30.5122	−0.998389
$935$	−45.9946	−1.50419
$936$	0	0
$937$	29.0554	0.949197	0.474599	−	0.880202i	$-0.342593\pi$
$937$	29.0554	0.949197	0.474599	+	0.880202i	$0.342593\pi$
$938$	30.0296	0.980502
$939$	0	0
$940$	4.41418	0.143975
$941$	10.9140	0.355788	0.177894	−	0.984050i	$-0.443072\pi$
$941$	10.9140	0.355788	0.177894	+	0.984050i	$0.443072\pi$
$942$	0	0
$943$	−10.8366	−0.352887
$944$	−0.960614	−0.0312653
$945$	0	0
$946$	13.1842	0.428654
$947$	−47.8337	−1.55439	−0.777194	−	0.629261i	$-0.783358\pi$
$947$	−47.8337	−1.55439	−0.777194	+	0.629261i	$0.783358\pi$
$948$	0	0
$949$	−66.2600	−2.15089
$950$	−0.0144036	−0.000467315 0
$951$	0	0
$952$	−18.7828	−0.608753
$953$	29.5561	0.957417	0.478709	−	0.877974i	$-0.341105\pi$
$953$	29.5561	0.957417	0.478709	+	0.877974i	$0.341105\pi$
$954$	0	0
$955$	30.4669	0.985885
$956$	24.8903	0.805010
$957$	0	0
$958$	8.85977	0.286246
$959$	−49.7695	−1.60714
$960$	0	0
$961$	−29.3477	−0.946701
$962$	41.8063	1.34789
$963$	0	0
$964$	1.00000	0.0322078
$965$	−7.53778	−0.242650
$966$	0	0
$967$	2.93525	0.0943911	0.0471956	−	0.998886i	$-0.484972\pi$
$967$	2.93525	0.0943911	0.0471956	+	0.998886i	$0.484972\pi$
$968$	22.1068	0.710541
$969$	0	0
$970$	7.21078	0.231524
$971$	5.09762	0.163590	0.0817952	−	0.996649i	$-0.473935\pi$
$971$	5.09762	0.163590	0.0817952	+	0.996649i	$0.473935\pi$
$972$	0	0
$973$	38.5326	1.23530
$974$	22.4128	0.718154
$975$	0	0
$976$	−12.3595	−0.395617
$977$	15.2914	0.489216	0.244608	−	0.969622i	$-0.421341\pi$
$977$	15.2914	0.489216	0.244608	+	0.969622i	$0.421341\pi$
$978$	0	0
$979$	−46.2542	−1.47829
$980$	0.896289	0.0286309
$981$	0	0
$982$	−3.14038	−0.100213
$983$	26.7241	0.852367	0.426183	−	0.904637i	$-0.359858\pi$
$983$	26.7241	0.852367	0.426183	+	0.904637i	$0.359858\pi$
$984$	0	0
$985$	32.2235	1.02673
$986$	33.5114	1.06722
$987$	0	0
$988$	0.0191545	0.000609386 0
$989$	19.0866	0.606918
$990$	0	0
$991$	−10.1414	−0.322152	−0.161076	−	0.986942i	$-0.551496\pi$
$991$	−10.1414	−0.322152	−0.161076	+	0.986942i	$0.551496\pi$
$992$	−1.28541	−0.0408117
$993$	0	0
$994$	−12.1607	−0.385714
$995$	17.0850	0.541631
$996$	0	0
$997$	−15.0660	−0.477144	−0.238572	−	0.971125i	$-0.576679\pi$
$997$	−15.0660	−0.477144	−0.238572	+	0.971125i	$0.576679\pi$
$998$	−8.09558	−0.256261
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4338.2.a.w.1.5		7
3.2	odd	2		1446.2.a.n.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1446.2.a.n.1.3	✓	7	3.2	odd	2
4338.2.a.w.1.5		7	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:	\( N \)	\(=\)	\( 4338 = 2 \cdot 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.18526 q^{5} +2.78499 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.18526 q^{5} +2.78499 q^{7} +1.00000 q^{8} +1.18526 q^{10} +5.75385 q^{11} +4.78099 q^{13} +2.78499 q^{14} +1.00000 q^{16} -6.74428 q^{17} +0.00400639 q^{19} +1.18526 q^{20} +5.75385 q^{22} +8.32980 q^{23} -3.59516 q^{25} +4.78099 q^{26} +2.78499 q^{28} -4.96886 q^{29} -1.28541 q^{31} +1.00000 q^{32} -6.74428 q^{34} +3.30094 q^{35} +8.74428 q^{37} +0.00400639 q^{38} +1.18526 q^{40} -1.30094 q^{41} +2.29136 q^{43} +5.75385 q^{44} +8.32980 q^{46} +3.72423 q^{47} +0.756197 q^{49} -3.59516 q^{50} +4.78099 q^{52} -10.7493 q^{53} +6.81981 q^{55} +2.78499 q^{56} -4.96886 q^{58} -0.960614 q^{59} -12.3595 q^{61} -1.28541 q^{62} +1.00000 q^{64} +5.66671 q^{65} +10.7827 q^{67} -6.74428 q^{68} +3.30094 q^{70} -4.36651 q^{71} -13.8591 q^{73} +8.74428 q^{74} +0.00400639 q^{76} +16.0245 q^{77} +7.52558 q^{79} +1.18526 q^{80} -1.30094 q^{82} -17.1260 q^{83} -7.99371 q^{85} +2.29136 q^{86} +5.75385 q^{88} -8.03882 q^{89} +13.3150 q^{91} +8.32980 q^{92} +3.72423 q^{94} +0.00474861 q^{95} +6.08372 q^{97} +0.756197 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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