LMFDB - Embedded newform 637.2.a.l.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.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:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.746052.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 8x + 2 $ x^5 - x^4 - 7x^3 + 8x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.72525$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.72525 q^{2} -1.34642 q^{3} +5.42699 q^{4} +2.18716 q^{5} -3.66932 q^{6} +9.33940 q^{8} -1.18716 q^{9} +5.96057 q^{10} -1.04815 q^{11} -7.30699 q^{12} +1.00000 q^{13} -2.94483 q^{15} +14.5982 q^{16} -5.29125 q^{17} -3.23532 q^{18} +0.756906 q^{19} +11.8697 q^{20} -2.85648 q^{22} +0.653584 q^{23} -12.5747 q^{24} -0.216314 q^{25} +2.72525 q^{26} +5.63766 q^{27} -3.10408 q^{29} -8.02541 q^{30} +1.02791 q^{31} +21.1050 q^{32} +1.41125 q^{33} -14.4200 q^{34} -6.44273 q^{36} -10.8932 q^{37} +2.06276 q^{38} -1.34642 q^{39} +20.4268 q^{40} +7.32040 q^{41} +0.887771 q^{43} -5.68833 q^{44} -2.59652 q^{45} +1.78118 q^{46} +2.33751 q^{47} -19.6553 q^{48} -0.589510 q^{50} +7.12422 q^{51} +5.42699 q^{52} +4.88814 q^{53} +15.3640 q^{54} -2.29249 q^{55} -1.01911 q^{57} -8.45941 q^{58} -1.04815 q^{59} -15.9816 q^{60} -12.4998 q^{61} +2.80132 q^{62} +28.3200 q^{64} +2.18716 q^{65} +3.84602 q^{66} +4.47889 q^{67} -28.7155 q^{68} -0.879996 q^{69} -6.60274 q^{71} -11.0874 q^{72} -8.28347 q^{73} -29.6868 q^{74} +0.291249 q^{75} +4.10772 q^{76} -3.66932 q^{78} +2.14014 q^{79} +31.9287 q^{80} -4.02915 q^{81} +19.9499 q^{82} -6.66558 q^{83} -11.5728 q^{85} +2.41940 q^{86} +4.17939 q^{87} -9.78914 q^{88} -5.76777 q^{89} -7.07617 q^{90} +3.54699 q^{92} -1.38400 q^{93} +6.37030 q^{94} +1.65548 q^{95} -28.4162 q^{96} -2.88777 q^{97} +1.24433 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

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 $

$2$	2.72525	1.92704	0.963521	−	0.267631i	$-0.0862408\pi$
$2$	2.72525	1.92704	0.963521	+	0.267631i	$0.0862408\pi$
$3$	−1.34642	−0.777354	−0.388677	−	0.921374i	$-0.627068\pi$
$3$	−1.34642	−0.777354	−0.388677	+	0.921374i	$0.627068\pi$
$4$	5.42699	2.71349
$5$	2.18716	0.978129	0.489065	−	0.872247i	$-0.337338\pi$
$5$	2.18716	0.978129	0.489065	+	0.872247i	$0.337338\pi$
$6$	−3.66932	−1.49799
$7$	0	0
$7$	0	0
$8$	9.33940	3.30198
$9$	−1.18716	−0.395721
$10$	5.96057	1.88490
$11$	−1.04815	−0.316031	−0.158015	−	0.987437i	$-0.550510\pi$
$11$	−1.04815	−0.316031	−0.158015	+	0.987437i	$0.550510\pi$
$12$	−7.30699	−2.10934
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.94483	−0.760352
$16$	14.5982	3.64956
$17$	−5.29125	−1.28332	−0.641658	−	0.766991i	$-0.721753\pi$
$17$	−5.29125	−1.28332	−0.641658	+	0.766991i	$0.721753\pi$
$18$	−3.23532	−0.762572
$19$	0.756906	0.173646	0.0868231	−	0.996224i	$-0.472329\pi$
$19$	0.756906	0.173646	0.0868231	+	0.996224i	$0.472329\pi$
$20$	11.8697	2.65415
$21$	0	0
$22$	−2.85648	−0.609005
$23$	0.653584	0.136282	0.0681408	−	0.997676i	$-0.478293\pi$
$23$	0.653584	0.136282	0.0681408	+	0.997676i	$0.478293\pi$
$24$	−12.5747	−2.56680
$25$	−0.216314	−0.0432628
$26$	2.72525	0.534466
$27$	5.63766	1.08497
$28$	0	0
$29$	−3.10408	−0.576414	−0.288207	−	0.957568i	$-0.593059\pi$
$29$	−3.10408	−0.576414	−0.288207	+	0.957568i	$0.593059\pi$
$30$	−8.02541	−1.46523
$31$	1.02791	0.184618	0.0923092	−	0.995730i	$-0.470575\pi$
$31$	1.02791	0.184618	0.0923092	+	0.995730i	$0.470575\pi$
$32$	21.1050	3.73088
$33$	1.41125	0.245668
$34$	−14.4200	−2.47301
$35$	0	0
$36$	−6.44273	−1.07379
$37$	−10.8932	−1.79084	−0.895418	−	0.445227i	$-0.853123\pi$
$37$	−10.8932	−1.79084	−0.895418	+	0.445227i	$0.853123\pi$
$38$	2.06276	0.334624
$39$	−1.34642	−0.215599
$40$	20.4268	3.22976
$41$	7.32040	1.14325	0.571627	−	0.820514i	$-0.306312\pi$
$41$	7.32040	1.14325	0.571627	+	0.820514i	$0.306312\pi$
$42$	0	0
$43$	0.887771	0.135384	0.0676919	−	0.997706i	$-0.478437\pi$
$43$	0.887771	0.135384	0.0676919	+	0.997706i	$0.478437\pi$
$44$	−5.68833	−0.857547
$45$	−2.59652	−0.387067
$46$	1.78118	0.262621
$47$	2.33751	0.340961	0.170480	−	0.985361i	$-0.445468\pi$
$47$	2.33751	0.340961	0.170480	+	0.985361i	$0.445468\pi$
$48$	−19.6553	−2.83700
$49$	0	0
$50$	−0.589510	−0.0833692
$51$	7.12422	0.997591
$52$	5.42699	0.752588
$53$	4.88814	0.671438	0.335719	−	0.941962i	$-0.391021\pi$
$53$	4.88814	0.671438	0.335719	+	0.941962i	$0.391021\pi$
$54$	15.3640	2.09078
$55$	−2.29249	−0.309119
$56$	0	0
$57$	−1.01911	−0.134985
$58$	−8.45941	−1.11077
$59$	−1.04815	−0.136458	−0.0682291	−	0.997670i	$-0.521735\pi$
$59$	−1.04815	−0.136458	−0.0682291	+	0.997670i	$0.521735\pi$
$60$	−15.9816	−2.06321
$61$	−12.4998	−1.60043	−0.800217	−	0.599711i	$-0.795282\pi$
$61$	−12.4998	−1.60043	−0.800217	+	0.599711i	$0.795282\pi$
$62$	2.80132	0.355768
$63$	0	0
$64$	28.3200	3.54000
$65$	2.18716	0.271284
$66$	3.84602	0.473412
$67$	4.47889	0.547183	0.273592	−	0.961846i	$-0.411788\pi$
$67$	4.47889	0.547183	0.273592	+	0.961846i	$0.411788\pi$
$68$	−28.7155	−3.48227
$69$	−0.879996	−0.105939
$70$	0	0
$71$	−6.60274	−0.783601	−0.391801	−	0.920050i	$-0.628148\pi$
$71$	−6.60274	−0.783601	−0.391801	+	0.920050i	$0.628148\pi$
$72$	−11.0874	−1.30666
$73$	−8.28347	−0.969507	−0.484754	−	0.874651i	$-0.661091\pi$
$73$	−8.28347	−0.969507	−0.484754	+	0.874651i	$0.661091\pi$
$74$	−29.6868	−3.45102
$75$	0.291249	0.0336305
$76$	4.10772	0.471188
$77$	0	0
$78$	−3.66932	−0.415469
$79$	2.14014	0.240785	0.120392	−	0.992726i	$-0.461585\pi$
$79$	2.14014	0.240785	0.120392	+	0.992726i	$0.461585\pi$
$80$	31.9287	3.56974
$81$	−4.02915	−0.447683
$82$	19.9499	2.20310
$83$	−6.66558	−0.731642	−0.365821	−	0.930685i	$-0.619212\pi$
$83$	−6.66558	−0.731642	−0.365821	+	0.930685i	$0.619212\pi$
$84$	0	0
$85$	−11.5728	−1.25525
$86$	2.41940	0.260890
$87$	4.17939	0.448078
$88$	−9.78914	−1.04353
$89$	−5.76777	−0.611382	−0.305691	−	0.952131i	$-0.598887\pi$
$89$	−5.76777	−0.611382	−0.305691	+	0.952131i	$0.598887\pi$
$90$	−7.07617	−0.745894
$91$	0	0
$92$	3.54699	0.369800
$93$	−1.38400	−0.143514
$94$	6.37030	0.657046
$95$	1.65548	0.169848
$96$	−28.4162	−2.90021
$97$	−2.88777	−0.293209	−0.146604	−	0.989195i	$-0.546834\pi$
$97$	−2.88777	−0.293209	−0.146604	+	0.989195i	$0.546834\pi$
$98$	0	0
$99$	1.24433	0.125060
$100$	−1.17393	−0.117393
$101$	−11.2543	−1.11985	−0.559924	−	0.828544i	$-0.689170\pi$
$101$	−11.2543	−1.11985	−0.559924	+	0.828544i	$0.689170\pi$
$102$	19.4153	1.92240
$103$	20.2334	1.99366	0.996828	−	0.0795900i	$-0.0253611\pi$
$103$	20.2334	1.99366	0.996828	+	0.0795900i	$0.0253611\pi$
$104$	9.33940	0.915804
$105$	0	0
$106$	13.3214	1.29389
$107$	9.05517	0.875396	0.437698	−	0.899122i	$-0.355794\pi$
$107$	9.05517	0.875396	0.437698	+	0.899122i	$0.355794\pi$
$108$	30.5955	2.94406
$109$	15.1014	1.44645	0.723226	−	0.690612i	$-0.242659\pi$
$109$	15.1014	1.44645	0.723226	+	0.690612i	$0.242659\pi$
$110$	−6.24760	−0.595685
$111$	14.6668	1.39211
$112$	0	0
$113$	3.10408	0.292008	0.146004	−	0.989284i	$-0.453359\pi$
$113$	3.10408	0.292008	0.146004	+	0.989284i	$0.453359\pi$
$114$	−2.77733	−0.260121
$115$	1.42950	0.133301
$116$	−16.8458	−1.56410
$117$	−1.18716	−0.109753
$118$	−2.85648	−0.262961
$119$	0	0
$120$	−27.5030	−2.51067
$121$	−9.90137	−0.900125
$122$	−34.0651	−3.08410
$123$	−9.85630	−0.888713
$124$	5.57847	0.500961
$125$	−11.4089	−1.02045
$126$	0	0
$127$	8.78914	0.779910	0.389955	−	0.920834i	$-0.372491\pi$
$127$	8.78914	0.779910	0.389955	+	0.920834i	$0.372491\pi$
$128$	34.9691	3.09086
$129$	−1.19531	−0.105241
$130$	5.96057	0.522776
$131$	−10.5145	−0.918653	−0.459327	−	0.888267i	$-0.651909\pi$
$131$	−10.5145	−0.918653	−0.459327	+	0.888267i	$0.651909\pi$
$132$	7.65885	0.666618
$133$	0	0
$134$	12.2061	1.05445
$135$	12.3305	1.06124
$136$	−49.4171	−4.23748
$137$	8.73165	0.745995	0.372998	−	0.927832i	$-0.378330\pi$
$137$	8.73165	0.745995	0.372998	+	0.927832i	$0.378330\pi$
$138$	−2.39821	−0.204149
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−3.14726	−0.265047
$142$	−17.9941	−1.51003
$143$	−1.04815	−0.0876511
$144$	−17.3305	−1.44421
$145$	−6.78914	−0.563808
$146$	−22.5745	−1.86828
$147$	0	0
$148$	−59.1174	−4.85942
$149$	15.3926	1.26101	0.630507	−	0.776183i	$-0.282847\pi$
$149$	15.3926	1.26101	0.630507	+	0.776183i	$0.282847\pi$
$150$	0.793725	0.0648074
$151$	−13.6757	−1.11291	−0.556457	−	0.830876i	$-0.687840\pi$
$151$	−13.6757	−1.11291	−0.556457	+	0.830876i	$0.687840\pi$
$152$	7.06905	0.573376
$153$	6.28158	0.507836
$154$	0	0
$155$	2.24821	0.180581
$156$	−7.30699	−0.585027
$157$	3.38756	0.270357	0.135178	−	0.990821i	$-0.456839\pi$
$157$	3.38756	0.270357	0.135178	+	0.990821i	$0.456839\pi$
$158$	5.83242	0.464002
$159$	−6.58147	−0.521945
$160$	46.1602	3.64928
$161$	0	0
$162$	−10.9804	−0.862705
$163$	−13.8100	−1.08169	−0.540843	−	0.841124i	$-0.681895\pi$
$163$	−13.8100	−1.08169	−0.540843	+	0.841124i	$0.681895\pi$
$164$	39.7277	3.10221
$165$	3.08664	0.240295
$166$	−18.1654	−1.40991
$167$	16.3783	1.26739	0.633695	−	0.773583i	$-0.281538\pi$
$167$	16.3783	1.26739	0.633695	+	0.773583i	$0.281538\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−31.5389	−2.41892
$171$	−0.898572	−0.0687155
$172$	4.81792	0.367363
$173$	4.12546	0.313653	0.156826	−	0.987626i	$-0.449874\pi$
$173$	4.12546	0.313653	0.156826	+	0.987626i	$0.449874\pi$
$174$	11.3899	0.863465
$175$	0	0
$176$	−15.3012	−1.15337
$177$	1.41125	0.106076
$178$	−15.7186	−1.17816
$179$	14.4136	1.07732	0.538661	−	0.842523i	$-0.318931\pi$
$179$	14.4136	1.07732	0.538661	+	0.842523i	$0.318931\pi$
$180$	−14.0913	−1.05030
$181$	18.1014	1.34547	0.672733	−	0.739885i	$-0.265120\pi$
$181$	18.1014	1.34547	0.672733	+	0.739885i	$0.265120\pi$
$182$	0	0
$183$	16.8299	1.24410
$184$	6.10408	0.449999
$185$	−23.8253	−1.75167
$186$	−3.77174	−0.276557
$187$	5.54605	0.405567
$188$	12.6856	0.925195
$189$	0	0
$190$	4.51159	0.327305
$191$	5.54135	0.400958	0.200479	−	0.979698i	$-0.435750\pi$
$191$	5.54135	0.400958	0.200479	+	0.979698i	$0.435750\pi$
$192$	−38.1305	−2.75184
$193$	−8.74088	−0.629182	−0.314591	−	0.949227i	$-0.601867\pi$
$193$	−8.74088	−0.629182	−0.314591	+	0.949227i	$0.601867\pi$
$194$	−7.86990	−0.565026
$195$	−2.94483	−0.210884
$196$	0	0
$197$	−5.46874	−0.389632	−0.194816	−	0.980840i	$-0.562411\pi$
$197$	−5.46874	−0.389632	−0.194816	+	0.980840i	$0.562411\pi$
$198$	3.39112	0.240996
$199$	19.5368	1.38493	0.692463	−	0.721454i	$-0.256526\pi$
$199$	19.5368	1.38493	0.692463	+	0.721454i	$0.256526\pi$
$200$	−2.02024	−0.142853
$201$	−6.03045	−0.425355
$202$	−30.6708	−2.15799
$203$	0	0
$204$	38.6631	2.70696
$205$	16.0109	1.11825
$206$	55.1411	3.84186
$207$	−0.775911	−0.0539296
$208$	14.5982	1.01221
$209$	−0.793355	−0.0548775
$210$	0	0
$211$	16.6905	1.14902	0.574511	−	0.818497i	$-0.305192\pi$
$211$	16.6905	1.14902	0.574511	+	0.818497i	$0.305192\pi$
$212$	26.5279	1.82194
$213$	8.89004	0.609135
$214$	24.6776	1.68693
$215$	1.94170	0.132423
$216$	52.6524	3.58254
$217$	0	0
$218$	41.1551	2.78737
$219$	11.1530	0.753650
$220$	−12.4413	−0.838792
$221$	−5.29125	−0.355928
$222$	39.9707	2.68266
$223$	−5.34217	−0.357738	−0.178869	−	0.983873i	$-0.557244\pi$
$223$	−5.34217	−0.357738	−0.178869	+	0.983873i	$0.557244\pi$
$224$	0	0
$225$	0.256800	0.0171200
$226$	8.45941	0.562711
$227$	20.1215	1.33551	0.667757	−	0.744380i	$-0.267255\pi$
$227$	20.1215	1.33551	0.667757	+	0.744380i	$0.267255\pi$
$228$	−5.53070	−0.366280
$229$	25.2497	1.66855	0.834275	−	0.551349i	$-0.185887\pi$
$229$	25.2497	1.66855	0.834275	+	0.551349i	$0.185887\pi$
$230$	3.89573	0.256877
$231$	0	0
$232$	−28.9903	−1.90331
$233$	−0.793355	−0.0519744	−0.0259872	−	0.999662i	$-0.508273\pi$
$233$	−0.793355	−0.0519744	−0.0259872	+	0.999662i	$0.508273\pi$
$234$	−3.23532	−0.211499
$235$	5.11252	0.333504
$236$	−5.68833	−0.370278
$237$	−2.88152	−0.187175
$238$	0	0
$239$	20.0488	1.29685	0.648425	−	0.761279i	$-0.275428\pi$
$239$	20.0488	1.29685	0.648425	+	0.761279i	$0.275428\pi$
$240$	−42.9894	−2.77495
$241$	13.8120	0.889712	0.444856	−	0.895602i	$-0.353255\pi$
$241$	13.8120	0.889712	0.444856	+	0.895602i	$0.353255\pi$
$242$	−26.9837	−1.73458
$243$	−11.4881	−0.736961
$244$	−67.8362	−4.34277
$245$	0	0
$246$	−26.8609	−1.71259
$247$	0.756906	0.0481608
$248$	9.60008	0.609606
$249$	8.97464	0.568745
$250$	−31.0922	−1.96644
$251$	−26.1095	−1.64802	−0.824010	−	0.566576i	$-0.808268\pi$
$251$	−26.1095	−1.64802	−0.824010	+	0.566576i	$0.808268\pi$
$252$	0	0
$253$	−0.685057	−0.0430692
$254$	23.9526	1.50292
$255$	15.5818	0.975773
$256$	38.6595	2.41622
$257$	10.6198	0.662444	0.331222	−	0.943553i	$-0.392539\pi$
$257$	10.6198	0.662444	0.331222	+	0.943553i	$0.392539\pi$
$258$	−3.25752	−0.202804
$259$	0	0
$260$	11.8697	0.736128
$261$	3.68506	0.228099
$262$	−28.6546	−1.77028
$263$	10.3578	0.638686	0.319343	−	0.947639i	$-0.396538\pi$
$263$	10.3578	0.638686	0.319343	+	0.947639i	$0.396538\pi$
$264$	13.1803	0.811189
$265$	10.6912	0.656753
$266$	0	0
$267$	7.76581	0.475260
$268$	24.3069	1.48478
$269$	11.9701	0.729827	0.364914	−	0.931041i	$-0.381099\pi$
$269$	11.9701	0.729827	0.364914	+	0.931041i	$0.381099\pi$
$270$	33.6037	2.04506
$271$	−2.75691	−0.167470	−0.0837351	−	0.996488i	$-0.526685\pi$
$271$	−2.75691	−0.167470	−0.0837351	+	0.996488i	$0.526685\pi$
$272$	−77.2429	−4.68354
$273$	0	0
$274$	23.7959	1.43757
$275$	0.226731	0.0136724
$276$	−4.77573	−0.287465
$277$	−23.9275	−1.43766	−0.718831	−	0.695185i	$-0.755322\pi$
$277$	−23.9275	−1.43766	−0.718831	+	0.695185i	$0.755322\pi$
$278$	−10.9010	−0.653799
$279$	−1.22030	−0.0730574
$280$	0	0
$281$	−3.87870	−0.231384	−0.115692	−	0.993285i	$-0.536909\pi$
$281$	−3.87870	−0.231384	−0.115692	+	0.993285i	$0.536909\pi$
$282$	−8.57707	−0.510757
$283$	−6.20999	−0.369146	−0.184573	−	0.982819i	$-0.559090\pi$
$283$	−6.20999	−0.369146	−0.184573	+	0.982819i	$0.559090\pi$
$284$	−35.8330	−2.12630
$285$	−2.22896	−0.132032
$286$	−2.85648	−0.168907
$287$	0	0
$288$	−25.0551	−1.47639
$289$	10.9973	0.646901
$290$	−18.5021	−1.08648
$291$	3.88814	0.227927
$292$	−44.9543	−2.63075
$293$	16.5754	0.968347	0.484174	−	0.874972i	$-0.339120\pi$
$293$	16.5754	0.968347	0.484174	+	0.874972i	$0.339120\pi$
$294$	0	0
$295$	−2.29249	−0.133474
$296$	−101.736	−5.91330
$297$	−5.90915	−0.342883
$298$	41.9488	2.43003
$299$	0.653584	0.0377977
$300$	1.58060	0.0912561
$301$	0	0
$302$	−37.2698	−2.14463
$303$	15.1530	0.870517
$304$	11.0495	0.633732
$305$	−27.3391	−1.56543
$306$	17.1189	0.978621
$307$	−7.05788	−0.402815	−0.201407	−	0.979508i	$-0.564551\pi$
$307$	−7.05788	−0.402815	−0.201407	+	0.979508i	$0.564551\pi$
$308$	0	0
$309$	−27.2426	−1.54978
$310$	6.12694	0.347987
$311$	21.1102	1.19705	0.598525	−	0.801104i	$-0.295754\pi$
$311$	21.1102	1.19705	0.598525	+	0.801104i	$0.295754\pi$
$312$	−12.5747	−0.711903
$313$	1.98052	0.111946	0.0559728	−	0.998432i	$-0.482174\pi$
$313$	1.98052	0.111946	0.0559728	+	0.998432i	$0.482174\pi$
$314$	9.23194	0.520989
$315$	0	0
$316$	11.6145	0.653368
$317$	−18.0459	−1.01356	−0.506781	−	0.862075i	$-0.669165\pi$
$317$	−18.0459	−1.01356	−0.506781	+	0.862075i	$0.669165\pi$
$318$	−17.9362	−1.00581
$319$	3.25356	0.182164
$320$	61.9406	3.46258
$321$	−12.1920	−0.680492
$322$	0	0
$323$	−4.00498	−0.222843
$324$	−21.8662	−1.21479
$325$	−0.216314	−0.0119989
$326$	−37.6358	−2.08446
$327$	−20.3328	−1.12440
$328$	68.3682	3.77500
$329$	0	0
$330$	8.41187	0.463058
$331$	−14.6738	−0.806544	−0.403272	−	0.915080i	$-0.632127\pi$
$331$	−14.6738	−0.806544	−0.403272	+	0.915080i	$0.632127\pi$
$332$	−36.1740	−1.98531
$333$	12.9320	0.708672
$334$	44.6349	2.44231
$335$	9.79606	0.535216
$336$	0	0
$337$	12.8080	0.697698	0.348849	−	0.937179i	$-0.386573\pi$
$337$	12.8080	0.697698	0.348849	+	0.937179i	$0.386573\pi$
$338$	2.72525	0.148234
$339$	−4.17939	−0.226993
$340$	−62.8056	−3.40611
$341$	−1.07741	−0.0583451
$342$	−2.44883	−0.132418
$343$	0	0
$344$	8.29125	0.447034
$345$	−1.92470	−0.103622
$346$	11.2429	0.604423
$347$	20.2054	1.08468	0.542342	−	0.840158i	$-0.317538\pi$
$347$	20.2054	1.08468	0.542342	+	0.840158i	$0.317538\pi$
$348$	22.6815	1.21586
$349$	−18.4434	−0.987252	−0.493626	−	0.869674i	$-0.664329\pi$
$349$	−18.4434	−0.987252	−0.493626	+	0.869674i	$0.664329\pi$
$350$	0	0
$351$	5.63766	0.300916
$352$	−22.1213	−1.17907
$353$	−8.14436	−0.433480	−0.216740	−	0.976229i	$-0.569542\pi$
$353$	−8.14436	−0.433480	−0.216740	+	0.976229i	$0.569542\pi$
$354$	3.84602	0.204413
$355$	−14.4413	−0.766463
$356$	−31.3016	−1.65898
$357$	0	0
$358$	39.2806	2.07604
$359$	−32.6100	−1.72109	−0.860545	−	0.509375i	$-0.829877\pi$
$359$	−32.6100	−1.72109	−0.860545	+	0.509375i	$0.829877\pi$
$360$	−24.2500	−1.27809
$361$	−18.4271	−0.969847
$362$	49.3308	2.59277
$363$	13.3314	0.699715
$364$	0	0
$365$	−18.1173	−0.948304
$366$	45.8657	2.39744
$367$	−3.16012	−0.164957	−0.0824786	−	0.996593i	$-0.526284\pi$
$367$	−3.16012	−0.164957	−0.0824786	+	0.996593i	$0.526284\pi$
$368$	9.54117	0.497368
$369$	−8.69051	−0.452410
$370$	−64.9298	−3.37554
$371$	0	0
$372$	−7.51094	−0.389424
$373$	−1.47770	−0.0765123	−0.0382561	−	0.999268i	$-0.512180\pi$
$373$	−1.47770	−0.0765123	−0.0382561	+	0.999268i	$0.512180\pi$
$374$	15.1144	0.781545
$375$	15.3612	0.793247
$376$	21.8309	1.12584
$377$	−3.10408	−0.159868
$378$	0	0
$379$	10.7254	0.550927	0.275463	−	0.961312i	$-0.411169\pi$
$379$	10.7254	0.550927	0.275463	+	0.961312i	$0.411169\pi$
$380$	8.98426	0.460883
$381$	−11.8338	−0.606266
$382$	15.1016	0.772664
$383$	−21.4109	−1.09405	−0.547023	−	0.837118i	$-0.684239\pi$
$383$	−21.4109	−1.09405	−0.547023	+	0.837118i	$0.684239\pi$
$384$	−47.0830	−2.40269
$385$	0	0
$386$	−23.8211	−1.21246
$387$	−1.05393	−0.0535742
$388$	−15.6719	−0.795620
$389$	34.7819	1.76351	0.881755	−	0.471707i	$-0.156362\pi$
$389$	34.7819	1.76351	0.881755	+	0.471707i	$0.156362\pi$
$390$	−8.02541	−0.406382
$391$	−3.45828	−0.174893
$392$	0	0
$393$	14.1568	0.714119
$394$	−14.9037	−0.750837
$395$	4.68084	0.235519
$396$	6.75297	0.339350
$397$	4.45211	0.223445	0.111722	−	0.993739i	$-0.464363\pi$
$397$	4.45211	0.223445	0.111722	+	0.993739i	$0.464363\pi$
$398$	53.2426	2.66881
$399$	0	0
$400$	−3.15780	−0.157890
$401$	−13.7537	−0.686829	−0.343415	−	0.939184i	$-0.611584\pi$
$401$	−13.7537	−0.686829	−0.343415	+	0.939184i	$0.611584\pi$
$402$	−16.4345	−0.819677
$403$	1.02791	0.0512039
$404$	−61.0771	−3.03870
$405$	−8.81241	−0.437892
$406$	0	0
$407$	11.4178	0.565959
$408$	66.5360	3.29402
$409$	−3.49207	−0.172672	−0.0863358	−	0.996266i	$-0.527516\pi$
$409$	−3.49207	−0.172672	−0.0863358	+	0.996266i	$0.527516\pi$
$410$	43.6337	2.15492
$411$	−11.7564	−0.579902
$412$	109.806	5.40977
$413$	0	0
$414$	−2.11455	−0.103925
$415$	−14.5787	−0.715641
$416$	21.1050	1.03476
$417$	5.38566	0.263737
$418$	−2.16209	−0.105751
$419$	−3.56737	−0.174278	−0.0871388	−	0.996196i	$-0.527772\pi$
$419$	−3.56737	−0.174278	−0.0871388	+	0.996196i	$0.527772\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	45.4858	2.21422
$423$	−2.77501	−0.134925
$424$	45.6523	2.21707
$425$	1.14457	0.0555198
$426$	24.2276	1.17383
$427$	0	0
$428$	49.1423	2.37538
$429$	1.41125	0.0681359
$430$	5.29162	0.255185
$431$	−11.3642	−0.547396	−0.273698	−	0.961816i	$-0.588247\pi$
$431$	−11.3642	−0.547396	−0.273698	+	0.961816i	$0.588247\pi$
$432$	82.2999	3.95966
$433$	−21.2136	−1.01946	−0.509731	−	0.860334i	$-0.670255\pi$
$433$	−21.2136	−1.01946	−0.509731	+	0.860334i	$0.670255\pi$
$434$	0	0
$435$	9.14101	0.438278
$436$	81.9551	3.92494
$437$	0.494702	0.0236648
$438$	30.3947	1.45232
$439$	−24.5007	−1.16935	−0.584676	−	0.811267i	$-0.698778\pi$
$439$	−24.5007	−1.16935	−0.584676	+	0.811267i	$0.698778\pi$
$440$	−21.4105	−1.02070
$441$	0	0
$442$	−14.4200	−0.685888
$443$	40.4688	1.92273	0.961366	−	0.275274i	$-0.0887686\pi$
$443$	40.4688	1.92273	0.961366	+	0.275274i	$0.0887686\pi$
$444$	79.5966	3.77749
$445$	−12.6151	−0.598011
$446$	−14.5587	−0.689376
$447$	−20.7249	−0.980254
$448$	0	0
$449$	−27.7638	−1.31025	−0.655127	−	0.755519i	$-0.727385\pi$
$449$	−27.7638	−1.31025	−0.655127	+	0.755519i	$0.727385\pi$
$450$	0.699845	0.0329910
$451$	−7.67291	−0.361303
$452$	16.8458	0.792361
$453$	18.4132	0.865128
$454$	54.8362	2.57359
$455$	0	0
$456$	−9.51789	−0.445716
$457$	−11.1939	−0.523629	−0.261815	−	0.965118i	$-0.584321\pi$
$457$	−11.1939	−0.523629	−0.261815	+	0.965118i	$0.584321\pi$
$458$	68.8119	3.21537
$459$	−29.8303	−1.39236
$460$	7.75786	0.361712
$461$	−9.29773	−0.433038	−0.216519	−	0.976278i	$-0.569470\pi$
$461$	−9.29773	−0.433038	−0.216519	+	0.976278i	$0.569470\pi$
$462$	0	0
$463$	28.2439	1.31260	0.656302	−	0.754499i	$-0.272120\pi$
$463$	28.2439	1.31260	0.656302	+	0.754499i	$0.272120\pi$
$464$	−45.3142	−2.10366
$465$	−3.02703	−0.140375
$466$	−2.16209	−0.100157
$467$	22.2606	1.03010	0.515050	−	0.857160i	$-0.327773\pi$
$467$	22.2606	1.03010	0.515050	+	0.857160i	$0.327773\pi$
$468$	−6.44273	−0.297815
$469$	0	0
$470$	13.9329	0.642676
$471$	−4.56106	−0.210163
$472$	−9.78914	−0.450582
$473$	−0.930521	−0.0427854
$474$	−7.85286	−0.360694
$475$	−0.163729	−0.00751242
$476$	0	0
$477$	−5.80302	−0.265702
$478$	54.6380	2.49909
$479$	−32.8764	−1.50216	−0.751081	−	0.660210i	$-0.770467\pi$
$479$	−32.8764	−1.50216	−0.751081	+	0.660210i	$0.770467\pi$
$480$	−62.1508	−2.83678
$481$	−10.8932	−0.496688
$482$	37.6413	1.71451
$483$	0	0
$484$	−53.7346	−2.44248
$485$	−6.31603	−0.286796
$486$	−31.3079	−1.42016
$487$	−27.8924	−1.26392	−0.631962	−	0.774999i	$-0.717750\pi$
$487$	−27.8924	−1.26392	−0.631962	+	0.774999i	$0.717750\pi$
$488$	−116.741	−5.28460
$489$	18.5941	0.840852
$490$	0	0
$491$	10.6571	0.480948	0.240474	−	0.970656i	$-0.422697\pi$
$491$	10.6571	0.480948	0.240474	+	0.970656i	$0.422697\pi$
$492$	−53.4900	−2.41152
$493$	16.4245	0.739722
$494$	2.06276	0.0928079
$495$	2.72156	0.122325
$496$	15.0057	0.673776
$497$	0	0
$498$	24.4581	1.09600
$499$	−24.5114	−1.09728	−0.548641	−	0.836058i	$-0.684854\pi$
$499$	−24.5114	−1.09728	−0.548641	+	0.836058i	$0.684854\pi$
$500$	−61.9162	−2.76897
$501$	−22.0520	−0.985210
$502$	−71.1550	−3.17580
$503$	−38.0054	−1.69458	−0.847288	−	0.531134i	$-0.821766\pi$
$503$	−38.0054	−1.69458	−0.847288	+	0.531134i	$0.821766\pi$
$504$	0	0
$505$	−24.6151	−1.09536
$506$	−1.86695	−0.0829962
$507$	−1.34642	−0.0597964
$508$	47.6986	2.11628
$509$	39.8501	1.76632	0.883161	−	0.469070i	$-0.155411\pi$
$509$	39.8501	1.76632	0.883161	+	0.469070i	$0.155411\pi$
$510$	42.4644	1.88036
$511$	0	0
$512$	35.4186	1.56530
$513$	4.26718	0.188401
$514$	28.9416	1.27656
$515$	44.2537	1.95005
$516$	−6.48693	−0.285571
$517$	−2.45007	−0.107754
$518$	0	0
$519$	−5.55459	−0.243819
$520$	20.4268	0.895775
$521$	−19.6334	−0.860155	−0.430077	−	0.902792i	$-0.641514\pi$
$521$	−19.6334	−0.860155	−0.430077	+	0.902792i	$0.641514\pi$
$522$	10.0427	0.439557
$523$	22.8324	0.998391	0.499195	−	0.866489i	$-0.333629\pi$
$523$	22.8324	0.998391	0.499195	+	0.866489i	$0.333629\pi$
$524$	−57.0619	−2.49276
$525$	0	0
$526$	28.2275	1.23078
$527$	−5.43894	−0.236924
$528$	20.6018	0.896578
$529$	−22.5728	−0.981427
$530$	29.1361	1.26559
$531$	1.24433	0.0539994
$532$	0	0
$533$	7.32040	0.317082
$534$	21.1638	0.915847
$535$	19.8051	0.856251
$536$	41.8301	1.80679
$537$	−19.4067	−0.837460
$538$	32.6214	1.40641
$539$	0	0
$540$	66.9175	2.87967
$541$	−9.64668	−0.414743	−0.207372	−	0.978262i	$-0.566491\pi$
$541$	−9.64668	−0.414743	−0.207372	+	0.978262i	$0.566491\pi$
$542$	−7.51326	−0.322722
$543$	−24.3720	−1.04590
$544$	−111.672	−4.78790
$545$	33.0292	1.41482
$546$	0	0
$547$	−43.8570	−1.87519	−0.937596	−	0.347728i	$-0.886953\pi$
$547$	−43.8570	−1.87519	−0.937596	+	0.347728i	$0.886953\pi$
$548$	47.3866	2.02425
$549$	14.8393	0.633326
$550$	0.617897	0.0263472
$551$	−2.34950	−0.100092
$552$	−8.21864	−0.349808
$553$	0	0
$554$	−65.2083	−2.77044
$555$	32.0787	1.36167
$556$	−21.7080	−0.920622
$557$	14.9195	0.632161	0.316080	−	0.948732i	$-0.397633\pi$
$557$	14.9195	0.632161	0.316080	+	0.948732i	$0.397633\pi$
$558$	−3.32562	−0.140785
$559$	0.887771	0.0375487
$560$	0	0
$561$	−7.46729	−0.315269
$562$	−10.5704	−0.445886
$563$	−17.2697	−0.727832	−0.363916	−	0.931432i	$-0.618560\pi$
$563$	−17.2697	−0.727832	−0.363916	+	0.931432i	$0.618560\pi$
$564$	−17.0801	−0.719204
$565$	6.78914	0.285621
$566$	−16.9238	−0.711359
$567$	0	0
$568$	−61.6657	−2.58743
$569$	26.5324	1.11230	0.556148	−	0.831083i	$-0.312279\pi$
$569$	26.5324	1.11230	0.556148	+	0.831083i	$0.312279\pi$
$570$	−6.07448	−0.254432
$571$	−1.98569	−0.0830985	−0.0415492	−	0.999136i	$-0.513229\pi$
$571$	−1.98569	−0.0830985	−0.0415492	+	0.999136i	$0.513229\pi$
$572$	−5.68833	−0.237841
$573$	−7.46097	−0.311686
$574$	0	0
$575$	−0.141379	−0.00589593
$576$	−33.6205	−1.40086
$577$	11.8983	0.495333	0.247666	−	0.968845i	$-0.420336\pi$
$577$	11.8983	0.495333	0.247666	+	0.968845i	$0.420336\pi$
$578$	29.9704	1.24661
$579$	11.7689	0.489097
$580$	−36.8446	−1.52989
$581$	0	0
$582$	10.5962	0.439225
$583$	−5.12353	−0.212195
$584$	−77.3627	−3.20129
$585$	−2.59652	−0.107353
$586$	45.1722	1.86605
$587$	33.5122	1.38320	0.691598	−	0.722283i	$-0.256907\pi$
$587$	33.5122	1.38320	0.691598	+	0.722283i	$0.256907\pi$
$588$	0	0
$589$	0.778033	0.0320583
$590$	−6.24760	−0.257210
$591$	7.36320	0.302882
$592$	−159.022	−6.53576
$593$	35.2815	1.44884	0.724419	−	0.689360i	$-0.242108\pi$
$593$	35.2815	1.44884	0.724419	+	0.689360i	$0.242108\pi$
$594$	−16.1039	−0.660751
$595$	0	0
$596$	83.5357	3.42176
$597$	−26.3046	−1.07658
$598$	1.78118	0.0728379
$599$	−25.0068	−1.02175	−0.510876	−	0.859655i	$-0.670679\pi$
$599$	−25.0068	−1.02175	−0.510876	+	0.859655i	$0.670679\pi$
$600$	2.72009	0.111047
$601$	−28.4688	−1.16127	−0.580634	−	0.814165i	$-0.697195\pi$
$601$	−28.4688	−1.16127	−0.580634	+	0.814165i	$0.697195\pi$
$602$	0	0
$603$	−5.31717	−0.216532
$604$	−74.2180	−3.01989
$605$	−21.6559	−0.880438
$606$	41.2957	1.67752
$607$	36.0469	1.46310	0.731549	−	0.681789i	$-0.238798\pi$
$607$	36.0469	1.46310	0.731549	+	0.681789i	$0.238798\pi$
$608$	15.9745	0.647853
$609$	0	0
$610$	−74.5058	−3.01665
$611$	2.33751	0.0945655
$612$	34.0901	1.37801
$613$	18.3253	0.740151	0.370075	−	0.929002i	$-0.379332\pi$
$613$	18.3253	0.740151	0.370075	+	0.929002i	$0.379332\pi$
$614$	−19.2345	−0.776241
$615$	−21.5573	−0.869276
$616$	0	0
$617$	44.3782	1.78660	0.893299	−	0.449463i	$-0.148385\pi$
$617$	44.3782	1.78660	0.893299	+	0.449463i	$0.148385\pi$
$618$	−74.2428	−2.98648
$619$	−25.0085	−1.00518	−0.502588	−	0.864526i	$-0.667619\pi$
$619$	−25.0085	−1.00518	−0.502588	+	0.864526i	$0.667619\pi$
$620$	12.2010	0.490005
$621$	3.68469	0.147861
$622$	57.5306	2.30677
$623$	0	0
$624$	−19.6553	−0.786842
$625$	−23.8716	−0.954866
$626$	5.39741	0.215724
$627$	1.06819	0.0426592
$628$	18.3842	0.733611
$629$	57.6388	2.29821
$630$	0	0
$631$	−18.4638	−0.735032	−0.367516	−	0.930017i	$-0.619792\pi$
$631$	−18.4638	−0.735032	−0.367516	+	0.930017i	$0.619792\pi$
$632$	19.9876	0.795066
$633$	−22.4724	−0.893197
$634$	−49.1797	−1.95318
$635$	19.2233	0.762853
$636$	−35.7176	−1.41629
$637$	0	0
$638$	8.86677	0.351039
$639$	7.83853	0.310088
$640$	76.4832	3.02326
$641$	−21.2567	−0.839589	−0.419795	−	0.907619i	$-0.637898\pi$
$641$	−21.2567	−0.839589	−0.419795	+	0.907619i	$0.637898\pi$
$642$	−33.2263	−1.31134
$643$	−36.0554	−1.42188	−0.710942	−	0.703251i	$-0.751731\pi$
$643$	−36.0554	−1.42188	−0.710942	+	0.703251i	$0.751731\pi$
$644$	0	0
$645$	−2.61434	−0.102939
$646$	−10.9146	−0.429428
$647$	39.8234	1.56562	0.782809	−	0.622262i	$-0.213786\pi$
$647$	39.8234	1.56562	0.782809	+	0.622262i	$0.213786\pi$
$648$	−37.6299	−1.47824
$649$	1.09863	0.0431250
$650$	−0.589510	−0.0231225
$651$	0	0
$652$	−74.9469	−2.93515
$653$	−32.4669	−1.27053	−0.635265	−	0.772295i	$-0.719109\pi$
$653$	−32.4669	−1.27053	−0.635265	+	0.772295i	$0.719109\pi$
$654$	−55.4119	−2.16678
$655$	−22.9969	−0.898562
$656$	106.865	4.17237
$657$	9.83384	0.383655
$658$	0	0
$659$	23.5230	0.916327	0.458164	−	0.888868i	$-0.348507\pi$
$659$	23.5230	0.916327	0.458164	+	0.888868i	$0.348507\pi$
$660$	16.7512	0.652038
$661$	14.0389	0.546049	0.273025	−	0.962007i	$-0.411976\pi$
$661$	14.0389	0.546049	0.273025	+	0.962007i	$0.411976\pi$
$662$	−39.9897	−1.55424
$663$	7.12422	0.276682
$664$	−62.2525	−2.41587
$665$	0	0
$666$	35.2431	1.36564
$667$	−2.02878	−0.0785547
$668$	88.8848	3.43905
$669$	7.19278	0.278089
$670$	26.6967	1.03138
$671$	13.1017	0.505786
$672$	0	0
$673$	−47.1937	−1.81918	−0.909592	−	0.415502i	$-0.863606\pi$
$673$	−47.1937	−1.81918	−0.909592	+	0.415502i	$0.863606\pi$
$674$	34.9051	1.34449
$675$	−1.21951	−0.0469388
$676$	5.42699	0.208730
$677$	−9.58876	−0.368526	−0.184263	−	0.982877i	$-0.558990\pi$
$677$	−9.58876	−0.368526	−0.184263	+	0.982877i	$0.558990\pi$
$678$	−11.3899	−0.437426
$679$	0	0
$680$	−108.083	−4.14481
$681$	−27.0920	−1.03817
$682$	−2.93621	−0.112433
$683$	47.3161	1.81050	0.905250	−	0.424879i	$-0.139683\pi$
$683$	47.3161	1.81050	0.905250	+	0.424879i	$0.139683\pi$
$684$	−4.87654	−0.186459
$685$	19.0976	0.729680
$686$	0	0
$687$	−33.9967	−1.29705
$688$	12.9599	0.494091
$689$	4.88814	0.186223
$690$	−5.24528	−0.199684
$691$	−27.1119	−1.03138	−0.515692	−	0.856774i	$-0.672465\pi$
$691$	−27.1119	−1.03138	−0.515692	+	0.856774i	$0.672465\pi$
$692$	22.3888	0.851095
$693$	0	0
$694$	55.0648	2.09023
$695$	−8.74866	−0.331855
$696$	39.0330	1.47954
$697$	−38.7340	−1.46716
$698$	−50.2628	−1.90248
$699$	1.06819	0.0404025
$700$	0	0
$701$	−1.79821	−0.0679176	−0.0339588	−	0.999423i	$-0.510811\pi$
$701$	−1.79821	−0.0679176	−0.0339588	+	0.999423i	$0.510811\pi$
$702$	15.3640	0.579879
$703$	−8.24515	−0.310972
$704$	−29.6838	−1.11875
$705$	−6.88357	−0.259250
$706$	−22.1954	−0.835336
$707$	0	0
$708$	7.65885	0.287837
$709$	−28.3230	−1.06369	−0.531846	−	0.846841i	$-0.678502\pi$
$709$	−28.3230	−1.06369	−0.531846	+	0.846841i	$0.678502\pi$
$710$	−39.3561	−1.47701
$711$	−2.54070	−0.0952836
$712$	−53.8675	−2.01877
$713$	0.671827	0.0251601
$714$	0	0
$715$	−2.29249	−0.0857341
$716$	78.2223	2.92331
$717$	−26.9940	−1.00811
$718$	−88.8704	−3.31661
$719$	−41.8971	−1.56250	−0.781249	−	0.624220i	$-0.785417\pi$
$719$	−41.8971	−1.56250	−0.781249	+	0.624220i	$0.785417\pi$
$720$	−37.9046	−1.41262
$721$	0	0
$722$	−50.2184	−1.86894
$723$	−18.5968	−0.691621
$724$	98.2361	3.65092
$725$	0.671457	0.0249373
$726$	36.3313	1.34838
$727$	19.5123	0.723670	0.361835	−	0.932242i	$-0.382150\pi$
$727$	19.5123	0.723670	0.361835	+	0.932242i	$0.382150\pi$
$728$	0	0
$729$	27.5552	1.02056
$730$	−49.3742	−1.82742
$731$	−4.69742	−0.173740
$732$	91.3358	3.37587
$733$	17.7540	0.655757	0.327879	−	0.944720i	$-0.393666\pi$
$733$	17.7540	0.655757	0.327879	+	0.944720i	$0.393666\pi$
$734$	−8.61213	−0.317880
$735$	0	0
$736$	13.7939	0.508450
$737$	−4.69457	−0.172927
$738$	−23.6838	−0.871814
$739$	44.3142	1.63012	0.815061	−	0.579375i	$-0.196703\pi$
$739$	44.3142	1.63012	0.815061	+	0.579375i	$0.196703\pi$
$740$	−129.299	−4.75314
$741$	−1.01911	−0.0374380
$742$	0	0
$743$	−7.16727	−0.262941	−0.131471	−	0.991320i	$-0.541970\pi$
$743$	−7.16727	−0.262941	−0.131471	+	0.991320i	$0.541970\pi$
$744$	−12.9257	−0.473879
$745$	33.6662	1.23344
$746$	−4.02710	−0.147442
$747$	7.91313	0.289526
$748$	30.0983	1.10050
$749$	0	0
$750$	41.8630	1.52862
$751$	−33.9065	−1.23726	−0.618632	−	0.785681i	$-0.712313\pi$
$751$	−33.9065	−1.23726	−0.618632	+	0.785681i	$0.712313\pi$
$752$	34.1235	1.24436
$753$	35.1543	1.28109
$754$	−8.45941	−0.308073
$755$	−29.9110	−1.08857
$756$	0	0
$757$	−0.906670	−0.0329535	−0.0164767	−	0.999864i	$-0.505245\pi$
$757$	−0.906670	−0.0329535	−0.0164767	+	0.999864i	$0.505245\pi$
$758$	29.2294	1.06166
$759$	0.922372	0.0334800
$760$	15.4612	0.560836
$761$	20.2494	0.734039	0.367020	−	0.930213i	$-0.380378\pi$
$761$	20.2494	0.734039	0.367020	+	0.930213i	$0.380378\pi$
$762$	−32.2502	−1.16830
$763$	0	0
$764$	30.0729	1.08800
$765$	13.7388	0.496729
$766$	−58.3500	−2.10827
$767$	−1.04815	−0.0378467
$768$	−52.0518	−1.87826
$769$	−36.9094	−1.33099	−0.665494	−	0.746403i	$-0.731779\pi$
$769$	−36.9094	−1.33099	−0.665494	+	0.746403i	$0.731779\pi$
$770$	0	0
$771$	−14.2987	−0.514954
$772$	−47.4367	−1.70728
$773$	−9.88037	−0.355372	−0.177686	−	0.984087i	$-0.556861\pi$
$773$	−9.88037	−0.355372	−0.177686	+	0.984087i	$0.556861\pi$
$774$	−2.87222	−0.103240
$775$	−0.222352	−0.00798711
$776$	−26.9701	−0.968169
$777$	0	0
$778$	94.7893	3.39836
$779$	5.54086	0.198522
$780$	−15.9816	−0.572232
$781$	6.92069	0.247642
$782$	−9.42467	−0.337025
$783$	−17.4998	−0.625391
$784$	0	0
$785$	7.40915	0.264444
$786$	38.5810	1.37614
$787$	37.6821	1.34322	0.671611	−	0.740904i	$-0.265603\pi$
$787$	37.6821	1.34322	0.671611	+	0.740904i	$0.265603\pi$
$788$	−29.6788	−1.05726
$789$	−13.9458	−0.496485
$790$	12.7565	0.453854
$791$	0	0
$792$	11.6213	0.412945
$793$	−12.4998	−0.443880
$794$	12.1331	0.430588
$795$	−14.3948	−0.510529
$796$	106.026	3.75799
$797$	28.3837	1.00540	0.502701	−	0.864460i	$-0.332340\pi$
$797$	28.3837	1.00540	0.502701	+	0.864460i	$0.332340\pi$
$798$	0	0
$799$	−12.3683	−0.437560
$800$	−4.56531	−0.161408
$801$	6.84728	0.241937
$802$	−37.4824	−1.32355
$803$	8.68236	0.306394
$804$	−32.7272	−1.15420
$805$	0	0
$806$	2.80132	0.0986722
$807$	−16.1167	−0.567334
$808$	−105.109	−3.69771
$809$	−11.7465	−0.412987	−0.206493	−	0.978448i	$-0.566205\pi$
$809$	−11.7465	−0.412987	−0.206493	+	0.978448i	$0.566205\pi$
$810$	−24.0160	−0.843837
$811$	2.01940	0.0709108	0.0354554	−	0.999371i	$-0.488712\pi$
$811$	2.01940	0.0709108	0.0354554	+	0.999371i	$0.488712\pi$
$812$	0	0
$813$	3.71194	0.130184
$814$	31.1163	1.09063
$815$	−30.2048	−1.05803
$816$	104.001	3.64077
$817$	0.671959	0.0235089
$818$	−9.51676	−0.332746
$819$	0	0
$820$	86.8910	3.03437
$821$	−15.0842	−0.526441	−0.263220	−	0.964736i	$-0.584785\pi$
$821$	−15.0842	−0.526441	−0.263220	+	0.964736i	$0.584785\pi$
$822$	−32.0392	−1.11750
$823$	−14.7766	−0.515079	−0.257539	−	0.966268i	$-0.582912\pi$
$823$	−14.7766	−0.515079	−0.257539	+	0.966268i	$0.582912\pi$
$824$	188.968	6.58301
$825$	−0.305274	−0.0106283
$826$	0	0
$827$	13.0407	0.453471	0.226736	−	0.973956i	$-0.427195\pi$
$827$	13.0407	0.453471	0.226736	+	0.973956i	$0.427195\pi$
$828$	−4.21086	−0.146338
$829$	−25.4581	−0.884198	−0.442099	−	0.896966i	$-0.645766\pi$
$829$	−25.4581	−0.884198	−0.442099	+	0.896966i	$0.645766\pi$
$830$	−39.7306	−1.37907
$831$	32.2163	1.11757
$832$	28.3200	0.981820
$833$	0	0
$834$	14.6773	0.508233
$835$	35.8220	1.23967
$836$	−4.30553	−0.148910
$837$	5.79502	0.200305
$838$	−9.72198	−0.335840
$839$	32.1703	1.11064	0.555321	−	0.831636i	$-0.312596\pi$
$839$	32.1703	1.11064	0.555321	+	0.831636i	$0.312596\pi$
$840$	0	0
$841$	−19.3647	−0.667747
$842$	−27.2525	−0.939183
$843$	5.22234	0.179867
$844$	90.5792	3.11787
$845$	2.18716	0.0752407
$846$	−7.56259	−0.260007
$847$	0	0
$848$	71.3582	2.45045
$849$	8.36123	0.286957
$850$	3.11924	0.106989
$851$	−7.11964	−0.244058
$852$	48.2461	1.65288
$853$	−19.3910	−0.663934	−0.331967	−	0.943291i	$-0.607712\pi$
$853$	−19.3910	−0.663934	−0.331967	+	0.943291i	$0.607712\pi$
$854$	0	0
$855$	−1.96532	−0.0672127
$856$	84.5699	2.89054
$857$	17.4242	0.595199	0.297600	−	0.954691i	$-0.403814\pi$
$857$	17.4242	0.595199	0.297600	+	0.954691i	$0.403814\pi$
$858$	3.84602	0.131301
$859$	−35.4917	−1.21096	−0.605481	−	0.795860i	$-0.707019\pi$
$859$	−35.4917	−1.21096	−0.605481	+	0.795860i	$0.707019\pi$
$860$	10.5376	0.359329
$861$	0	0
$862$	−30.9704	−1.05486
$863$	56.0019	1.90633	0.953164	−	0.302455i	$-0.0978062\pi$
$863$	56.0019	1.90633	0.953164	+	0.302455i	$0.0978062\pi$
$864$	118.983	4.04789
$865$	9.02306	0.306793
$866$	−57.8124	−1.96455
$867$	−14.8070	−0.502871
$868$	0	0
$869$	−2.24320	−0.0760953
$870$	24.9115	0.844580
$871$	4.47889	0.151761
$872$	141.038	4.77615
$873$	3.42826	0.116029
$874$	1.34819	0.0456031
$875$	0	0
$876$	60.5272	2.04503
$877$	25.2062	0.851153	0.425577	−	0.904922i	$-0.360071\pi$
$877$	25.2062	0.851153	0.425577	+	0.904922i	$0.360071\pi$
$878$	−66.7704	−2.25339
$879$	−22.3174	−0.752748
$880$	−33.4663	−1.12815
$881$	−18.6082	−0.626925	−0.313463	−	0.949601i	$-0.601489\pi$
$881$	−18.6082	−0.626925	−0.313463	+	0.949601i	$0.601489\pi$
$882$	0	0
$883$	−11.2552	−0.378768	−0.189384	−	0.981903i	$-0.560649\pi$
$883$	−11.2552	−0.378768	−0.189384	+	0.981903i	$0.560649\pi$
$884$	−28.7155	−0.965808
$885$	3.08664	0.103756
$886$	110.288	3.70519
$887$	−39.2112	−1.31658	−0.658292	−	0.752762i	$-0.728721\pi$
$887$	−39.2112	−1.31658	−0.658292	+	0.752762i	$0.728721\pi$
$888$	136.979	4.59672
$889$	0	0
$890$	−34.3792	−1.15239
$891$	4.22317	0.141482
$892$	−28.9919	−0.970720
$893$	1.76928	0.0592066
$894$	−56.4805	−1.88899
$895$	31.5249	1.05376
$896$	0	0
$897$	−0.879996	−0.0293822
$898$	−75.6633	−2.52492
$899$	−3.19073	−0.106417
$900$	1.39365	0.0464550
$901$	−25.8644	−0.861667
$902$	−20.9106	−0.696247
$903$	0	0
$904$	28.9903	0.964203
$905$	39.5907	1.31604
$906$	50.1806	1.66714
$907$	21.5970	0.717116	0.358558	−	0.933508i	$-0.383269\pi$
$907$	21.5970	0.717116	0.358558	+	0.933508i	$0.383269\pi$
$908$	109.199	3.62391
$909$	13.3607	0.443147
$910$	0	0
$911$	−32.4434	−1.07490	−0.537449	−	0.843297i	$-0.680612\pi$
$911$	−32.4434	−1.07490	−0.537449	+	0.843297i	$0.680612\pi$
$912$	−14.8772	−0.492634
$913$	6.98656	0.231221
$914$	−30.5062	−1.00906
$915$	36.8098	1.21689
$916$	137.030	4.52760
$917$	0	0
$918$	−81.2950	−2.68313
$919$	35.7372	1.17886	0.589430	−	0.807819i	$-0.299352\pi$
$919$	35.7372	1.17886	0.589430	+	0.807819i	$0.299352\pi$
$920$	13.3506	0.440157
$921$	9.50285	0.313129
$922$	−25.3386	−0.834483
$923$	−6.60274	−0.217332
$924$	0	0
$925$	2.35636	0.0774765
$926$	76.9716	2.52944
$927$	−24.0204	−0.788932
$928$	−65.5118	−2.15053
$929$	11.7769	0.386389	0.193194	−	0.981161i	$-0.438115\pi$
$929$	11.7769	0.386389	0.193194	+	0.981161i	$0.438115\pi$
$930$	−8.24941	−0.270509
$931$	0	0
$932$	−4.30553	−0.141032
$933$	−28.4231	−0.930531
$934$	60.6658	1.98505
$935$	12.1301	0.396697
$936$	−11.0874	−0.362403
$937$	18.9937	0.620497	0.310248	−	0.950655i	$-0.399588\pi$
$937$	18.9937	0.620497	0.310248	+	0.950655i	$0.399588\pi$
$938$	0	0
$939$	−2.66660	−0.0870213
$940$	27.7456	0.904961
$941$	6.81864	0.222281	0.111141	−	0.993805i	$-0.464550\pi$
$941$	6.81864	0.222281	0.111141	+	0.993805i	$0.464550\pi$
$942$	−12.4300	−0.404993
$943$	4.78450	0.155805
$944$	−15.3012	−0.498012
$945$	0	0
$946$	−2.53590	−0.0824493
$947$	1.05992	0.0344426	0.0172213	−	0.999852i	$-0.494518\pi$
$947$	1.05992	0.0344426	0.0172213	+	0.999852i	$0.494518\pi$
$948$	−15.6380	−0.507898
$949$	−8.28347	−0.268893
$950$	−0.446204	−0.0144768
$951$	24.2973	0.787895
$952$	0	0
$953$	−40.4127	−1.30910	−0.654548	−	0.756020i	$-0.727141\pi$
$953$	−40.4127	−1.30910	−0.654548	+	0.756020i	$0.727141\pi$
$954$	−15.8147	−0.512020
$955$	12.1199	0.392189
$956$	108.805	3.51899
$957$	−4.38065	−0.141606
$958$	−89.5964	−2.89473
$959$	0	0
$960$	−83.3978	−2.69165
$961$	−29.9434	−0.965916
$962$	−29.6868	−0.957140
$963$	−10.7500	−0.346413
$964$	74.9578	2.41423
$965$	−19.1177	−0.615422
$966$	0	0
$967$	−36.2949	−1.16717	−0.583583	−	0.812053i	$-0.698350\pi$
$967$	−36.2949	−1.16717	−0.583583	+	0.812053i	$0.698350\pi$
$968$	−92.4729	−2.97219
$969$	5.39237	0.173228
$970$	−17.2128	−0.552668
$971$	−21.4437	−0.688160	−0.344080	−	0.938940i	$-0.611809\pi$
$971$	−21.4437	−0.688160	−0.344080	+	0.938940i	$0.611809\pi$
$972$	−62.3457	−1.99974
$973$	0	0
$974$	−76.0137	−2.43564
$975$	0.291249	0.00932742
$976$	−182.475	−5.84088
$977$	−39.8277	−1.27420	−0.637100	−	0.770781i	$-0.719866\pi$
$977$	−39.8277	−1.27420	−0.637100	+	0.770781i	$0.719866\pi$
$978$	50.6735	1.62036
$979$	6.04551	0.193215
$980$	0	0
$981$	−17.9278	−0.572392
$982$	29.0432	0.926807
$983$	15.8814	0.506538	0.253269	−	0.967396i	$-0.418494\pi$
$983$	15.8814	0.506538	0.253269	+	0.967396i	$0.418494\pi$
$984$	−92.0520	−2.93451
$985$	−11.9610	−0.381110
$986$	44.7608	1.42548
$987$	0	0
$988$	4.10772	0.130684
$989$	0.580233	0.0184503
$990$	7.41693	0.235725
$991$	−17.6687	−0.561265	−0.280633	−	0.959815i	$-0.590544\pi$
$991$	−17.6687	−0.561265	−0.280633	+	0.959815i	$0.590544\pi$
$992$	21.6941	0.688789
$993$	19.7570	0.626970
$994$	0	0
$995$	42.7301	1.35464
$996$	48.7053	1.54329
$997$	−24.8608	−0.787350	−0.393675	−	0.919250i	$-0.628797\pi$
$997$	−24.8608	−0.787350	−0.393675	+	0.919250i	$0.628797\pi$
$998$	−66.7997	−2.11451
$999$	−61.4124	−1.94300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.l.1.5		5
3.2	odd	2		5733.2.a.bl.1.1		5
7.2	even	3		91.2.e.c.53.1	✓	10
7.3	odd	6		637.2.e.m.79.1		10
7.4	even	3		91.2.e.c.79.1	yes	10
7.5	odd	6		637.2.e.m.508.1		10
7.6	odd	2		637.2.a.k.1.5		5
13.12	even	2		8281.2.a.bw.1.1		5
21.2	odd	6		819.2.j.h.235.5		10
21.11	odd	6		819.2.j.h.352.5		10
21.20	even	2		5733.2.a.bm.1.1		5
28.11	odd	6		1456.2.r.p.625.2		10
28.23	odd	6		1456.2.r.p.417.2		10
91.25	even	6		1183.2.e.f.170.5		10
91.51	even	6		1183.2.e.f.508.5		10
91.90	odd	2		8281.2.a.bx.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.1	✓	10	7.2	even	3
91.2.e.c.79.1	yes	10	7.4	even	3
637.2.a.k.1.5		5	7.6	odd	2
637.2.a.l.1.5		5	1.1	even	1	trivial
637.2.e.m.79.1		10	7.3	odd	6
637.2.e.m.508.1		10	7.5	odd	6
819.2.j.h.235.5		10	21.2	odd	6
819.2.j.h.352.5		10	21.11	odd	6
1183.2.e.f.170.5		10	91.25	even	6
1183.2.e.f.508.5		10	91.51	even	6
1456.2.r.p.417.2		10	28.23	odd	6
1456.2.r.p.625.2		10	28.11	odd	6
5733.2.a.bl.1.1		5	3.2	odd	2
5733.2.a.bm.1.1		5	21.20	even	2
8281.2.a.bw.1.1		5	13.12	even	2
8281.2.a.bx.1.1		5	91.90	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:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q+2.72525 q^{2} -1.34642 q^{3} +5.42699 q^{4} +2.18716 q^{5} -3.66932 q^{6} +9.33940 q^{8} -1.18716 q^{9} +5.96057 q^{10} -1.04815 q^{11} -7.30699 q^{12} +1.00000 q^{13} -2.94483 q^{15} +14.5982 q^{16} -5.29125 q^{17} -3.23532 q^{18} +0.756906 q^{19} +11.8697 q^{20} -2.85648 q^{22} +0.653584 q^{23} -12.5747 q^{24} -0.216314 q^{25} +2.72525 q^{26} +5.63766 q^{27} -3.10408 q^{29} -8.02541 q^{30} +1.02791 q^{31} +21.1050 q^{32} +1.41125 q^{33} -14.4200 q^{34} -6.44273 q^{36} -10.8932 q^{37} +2.06276 q^{38} -1.34642 q^{39} +20.4268 q^{40} +7.32040 q^{41} +0.887771 q^{43} -5.68833 q^{44} -2.59652 q^{45} +1.78118 q^{46} +2.33751 q^{47} -19.6553 q^{48} -0.589510 q^{50} +7.12422 q^{51} +5.42699 q^{52} +4.88814 q^{53} +15.3640 q^{54} -2.29249 q^{55} -1.01911 q^{57} -8.45941 q^{58} -1.04815 q^{59} -15.9816 q^{60} -12.4998 q^{61} +2.80132 q^{62} +28.3200 q^{64} +2.18716 q^{65} +3.84602 q^{66} +4.47889 q^{67} -28.7155 q^{68} -0.879996 q^{69} -6.60274 q^{71} -11.0874 q^{72} -8.28347 q^{73} -29.6868 q^{74} +0.291249 q^{75} +4.10772 q^{76} -3.66932 q^{78} +2.14014 q^{79} +31.9287 q^{80} -4.02915 q^{81} +19.9499 q^{82} -6.66558 q^{83} -11.5728 q^{85} +2.41940 q^{86} +4.17939 q^{87} -9.78914 q^{88} -5.76777 q^{89} -7.07617 q^{90} +3.54699 q^{92} -1.38400 q^{93} +6.37030 q^{94} +1.65548 q^{95} -28.4162 q^{96} -2.88777 q^{97} +1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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