LMFDB - Embedded newform 7381.2.a.r.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7381,2,Mod(1,7381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7381, 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("7381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7381 = 11^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7381.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:	$58.9375817319$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	7381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.822245 q^{2} -0.874235 q^{3} -1.32391 q^{4} +0.00515383 q^{5} -0.718836 q^{6} +3.88567 q^{7} -2.73307 q^{8} -2.23571 q^{9} +O(q^{10})$	\(q+0.822245 q^{2} -0.874235 q^{3} -1.32391 q^{4} +0.00515383 q^{5} -0.718836 q^{6} +3.88567 q^{7} -2.73307 q^{8} -2.23571 q^{9} +0.00423771 q^{10} +1.15741 q^{12} -2.53500 q^{13} +3.19497 q^{14} -0.00450566 q^{15} +0.400570 q^{16} -2.29459 q^{17} -1.83830 q^{18} +0.555985 q^{19} -0.00682322 q^{20} -3.39698 q^{21} -4.57637 q^{23} +2.38935 q^{24} -4.99997 q^{25} -2.08439 q^{26} +4.57724 q^{27} -5.14428 q^{28} -1.66746 q^{29} -0.00370475 q^{30} -6.23461 q^{31} +5.79551 q^{32} -1.88672 q^{34} +0.0200261 q^{35} +2.95989 q^{36} +9.46770 q^{37} +0.457156 q^{38} +2.21618 q^{39} -0.0140858 q^{40} -0.618681 q^{41} -2.79316 q^{42} +7.23804 q^{43} -0.0115225 q^{45} -3.76290 q^{46} +6.86683 q^{47} -0.350192 q^{48} +8.09840 q^{49} -4.11120 q^{50} +2.00601 q^{51} +3.35611 q^{52} -7.04806 q^{53} +3.76362 q^{54} -10.6198 q^{56} -0.486061 q^{57} -1.37107 q^{58} -4.74989 q^{59} +0.00596509 q^{60} +1.00000 q^{61} -5.12638 q^{62} -8.68724 q^{63} +3.96419 q^{64} -0.0130649 q^{65} +4.87867 q^{67} +3.03784 q^{68} +4.00082 q^{69} +0.0164663 q^{70} +12.7458 q^{71} +6.11037 q^{72} -0.350430 q^{73} +7.78477 q^{74} +4.37115 q^{75} -0.736075 q^{76} +1.82225 q^{78} -10.0701 q^{79} +0.00206447 q^{80} +2.70556 q^{81} -0.508708 q^{82} -13.5440 q^{83} +4.49731 q^{84} -0.0118259 q^{85} +5.95144 q^{86} +1.45776 q^{87} +14.2211 q^{89} -0.00947431 q^{90} -9.85015 q^{91} +6.05872 q^{92} +5.45051 q^{93} +5.64622 q^{94} +0.00286545 q^{95} -5.06664 q^{96} -17.8342 q^{97} +6.65887 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9} + 12 q^{10} - 14 q^{12} + 8 q^{13} - 2 q^{14} - 16 q^{15} + 42 q^{16} + 22 q^{17} + 32 q^{19} - 8 q^{20} + 24 q^{21} + 12 q^{23} - 8 q^{24} + 40 q^{25} + 10 q^{26} - 8 q^{27} + 72 q^{28} + 56 q^{29} + 24 q^{30} + 10 q^{31} + 70 q^{32} - 32 q^{34} + 70 q^{35} + 34 q^{36} - 8 q^{37} - 14 q^{38} + 96 q^{39} - 54 q^{40} + 56 q^{41} - 8 q^{42} + 44 q^{43} - 24 q^{45} - 4 q^{46} - 4 q^{47} - 28 q^{48} + 38 q^{49} + 120 q^{50} + 76 q^{51} + 24 q^{52} + 4 q^{53} + 48 q^{54} - 18 q^{56} + 8 q^{57} + 28 q^{58} + 12 q^{59} - 60 q^{60} + 50 q^{61} + 8 q^{62} + 30 q^{63} + 10 q^{64} + 64 q^{65} + 18 q^{67} - 22 q^{68} - 8 q^{69} + 34 q^{70} + 12 q^{71} + 104 q^{72} - 16 q^{73} + 84 q^{74} - 26 q^{75} + 64 q^{76} + 40 q^{78} + 78 q^{79} - 36 q^{80} + 34 q^{81} + 54 q^{82} + 68 q^{83} - 78 q^{84} - 4 q^{85} + 36 q^{86} + 48 q^{87} + 26 q^{89} - 20 q^{90} + 32 q^{92} + 22 q^{93} + 156 q^{94} + 100 q^{95} - 4 q^{96} - 14 q^{97} + 70 q^{98}+O(q^{100}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9 + 12 * q^10 - 14 * q^12 + 8 * q^13 - 2 * q^14 - 16 * q^15 + 42 * q^16 + 22 * q^17 + 32 * q^19 - 8 * q^20 + 24 * q^21 + 12 * q^23 - 8 * q^24 + 40 * q^25 + 10 * q^26 - 8 * q^27 + 72 * q^28 + 56 * q^29 + 24 * q^30 + 10 * q^31 + 70 * q^32 - 32 * q^34 + 70 * q^35 + 34 * q^36 - 8 * q^37 - 14 * q^38 + 96 * q^39 - 54 * q^40 + 56 * q^41 - 8 * q^42 + 44 * q^43 - 24 * q^45 - 4 * q^46 - 4 * q^47 - 28 * q^48 + 38 * q^49 + 120 * q^50 + 76 * q^51 + 24 * q^52 + 4 * q^53 + 48 * q^54 - 18 * q^56 + 8 * q^57 + 28 * q^58 + 12 * q^59 - 60 * q^60 + 50 * q^61 + 8 * q^62 + 30 * q^63 + 10 * q^64 + 64 * q^65 + 18 * q^67 - 22 * q^68 - 8 * q^69 + 34 * q^70 + 12 * q^71 + 104 * q^72 - 16 * q^73 + 84 * q^74 - 26 * q^75 + 64 * q^76 + 40 * q^78 + 78 * q^79 - 36 * q^80 + 34 * q^81 + 54 * q^82 + 68 * q^83 - 78 * q^84 - 4 * q^85 + 36 * q^86 + 48 * q^87 + 26 * q^89 - 20 * q^90 + 32 * q^92 + 22 * q^93 + 156 * q^94 + 100 * q^95 - 4 * q^96 - 14 * q^97 + 70 * 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 $

$2$	0.822245	0.581415	0.290708	−	0.956812i	$-0.406109\pi$
$2$	0.822245	0.581415	0.290708	+	0.956812i	$0.406109\pi$
$3$	−0.874235	−0.504740	−0.252370	−	0.967631i	$-0.581210\pi$
$3$	−0.874235	−0.504740	−0.252370	+	0.967631i	$0.581210\pi$
$4$	−1.32391	−0.661956
$5$	0.00515383	0.00230486	0.00115243	−	0.999999i	$-0.499633\pi$
$5$	0.00515383	0.00230486	0.00115243	+	0.999999i	$0.499633\pi$
$6$	−0.718836	−0.293463
$7$	3.88567	1.46864	0.734322	−	0.678801i	$-0.237500\pi$
$7$	3.88567	1.46864	0.734322	+	0.678801i	$0.237500\pi$
$8$	−2.73307	−0.966287
$9$	−2.23571	−0.745238
$10$	0.00423771	0.00134008
$11$	0	0
$11$	0	0
$12$	1.15741	0.334116
$13$	−2.53500	−0.703082	−0.351541	−	0.936173i	$-0.614342\pi$
$13$	−2.53500	−0.703082	−0.351541	+	0.936173i	$0.614342\pi$
$14$	3.19497	0.853892
$15$	−0.00450566	−0.00116336
$16$	0.400570	0.100142
$17$	−2.29459	−0.556520	−0.278260	−	0.960506i	$-0.589758\pi$
$17$	−2.29459	−0.556520	−0.278260	+	0.960506i	$0.589758\pi$
$18$	−1.83830	−0.433293
$19$	0.555985	0.127552	0.0637758	−	0.997964i	$-0.479686\pi$
$19$	0.555985	0.127552	0.0637758	+	0.997964i	$0.479686\pi$
$20$	−0.00682322	−0.00152572
$21$	−3.39698	−0.741283
$22$	0	0
$23$	−4.57637	−0.954239	−0.477120	−	0.878838i	$-0.658319\pi$
$23$	−4.57637	−0.954239	−0.477120	+	0.878838i	$0.658319\pi$
$24$	2.38935	0.487723
$25$	−4.99997	−0.999995
$26$	−2.08439	−0.408782
$27$	4.57724	0.880891
$28$	−5.14428	−0.972178
$29$	−1.66746	−0.309640	−0.154820	−	0.987943i	$-0.549480\pi$
$29$	−1.66746	−0.309640	−0.154820	+	0.987943i	$0.549480\pi$
$30$	−0.00370475	−0.000676393 0
$31$	−6.23461	−1.11977	−0.559885	−	0.828571i	$-0.689155\pi$
$31$	−6.23461	−1.11977	−0.559885	+	0.828571i	$0.689155\pi$
$32$	5.79551	1.02451
$33$	0	0
$34$	−1.88672	−0.323569
$35$	0.0200261	0.00338502
$36$	2.95989	0.493315
$37$	9.46770	1.55648	0.778240	−	0.627967i	$-0.216113\pi$
$37$	9.46770	1.55648	0.778240	+	0.627967i	$0.216113\pi$
$38$	0.457156	0.0741604
$39$	2.21618	0.354873
$40$	−0.0140858	−0.00222716
$41$	−0.618681	−0.0966218	−0.0483109	−	0.998832i	$-0.515384\pi$
$41$	−0.618681	−0.0966218	−0.0483109	+	0.998832i	$0.515384\pi$
$42$	−2.79316	−0.430993
$43$	7.23804	1.10379	0.551895	−	0.833913i	$-0.313905\pi$
$43$	7.23804	1.10379	0.551895	+	0.833913i	$0.313905\pi$
$44$	0	0
$45$	−0.0115225	−0.00171767
$46$	−3.76290	−0.554809
$47$	6.86683	1.00163	0.500815	−	0.865554i	$-0.333034\pi$
$47$	6.86683	1.00163	0.500815	+	0.865554i	$0.333034\pi$
$48$	−0.350192	−0.0505459
$49$	8.09840	1.15691
$50$	−4.11120	−0.581412
$51$	2.00601	0.280898
$52$	3.35611	0.465409
$53$	−7.04806	−0.968125	−0.484063	−	0.875033i	$-0.660839\pi$
$53$	−7.04806	−0.968125	−0.484063	+	0.875033i	$0.660839\pi$
$54$	3.76362	0.512163
$55$	0	0
$56$	−10.6198	−1.41913
$57$	−0.486061	−0.0643804
$58$	−1.37107	−0.180030
$59$	−4.74989	−0.618383	−0.309192	−	0.951000i	$-0.600058\pi$
$59$	−4.74989	−0.618383	−0.309192	+	0.951000i	$0.600058\pi$
$60$	0.00596509	0.000770090 0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−5.12638	−0.651051
$63$	−8.68724	−1.09449
$64$	3.96419	0.495524
$65$	−0.0130649	−0.00162051
$66$	0	0
$67$	4.87867	0.596024	0.298012	−	0.954562i	$-0.403676\pi$
$67$	4.87867	0.596024	0.298012	+	0.954562i	$0.403676\pi$
$68$	3.03784	0.368392
$69$	4.00082	0.481642
$70$	0.0164663	0.00196810
$71$	12.7458	1.51264	0.756322	−	0.654200i	$-0.226995\pi$
$71$	12.7458	1.51264	0.756322	+	0.654200i	$0.226995\pi$
$72$	6.11037	0.720113
$73$	−0.350430	−0.0410147	−0.0205074	−	0.999790i	$-0.506528\pi$
$73$	−0.350430	−0.0410147	−0.0205074	+	0.999790i	$0.506528\pi$
$74$	7.78477	0.904961
$75$	4.37115	0.504737
$76$	−0.736075	−0.0844336
$77$	0	0
$78$	1.82225	0.206329
$79$	−10.0701	−1.13297	−0.566487	−	0.824071i	$-0.691698\pi$
$79$	−10.0701	−1.13297	−0.566487	+	0.824071i	$0.691698\pi$
$80$	0.00206447	0.000230814 0
$81$	2.70556	0.300617
$82$	−0.508708	−0.0561774
$83$	−13.5440	−1.48665	−0.743325	−	0.668930i	$-0.766753\pi$
$83$	−13.5440	−1.48665	−0.743325	+	0.668930i	$0.766753\pi$
$84$	4.49731	0.490697
$85$	−0.0118259	−0.00128270
$86$	5.95144	0.641761
$87$	1.45776	0.156288
$88$	0	0
$89$	14.2211	1.50743	0.753717	−	0.657199i	$-0.228259\pi$
$89$	14.2211	1.50743	0.753717	+	0.657199i	$0.228259\pi$
$90$	−0.00947431	−0.000998680 0
$91$	−9.85015	−1.03258
$92$	6.05872	0.631665
$93$	5.45051	0.565192
$94$	5.64622	0.582363
$95$	0.00286545	0.000293989 0
$96$	−5.06664	−0.517111
$97$	−17.8342	−1.81078	−0.905392	−	0.424576i	$-0.860423\pi$
$97$	−17.8342	−1.81078	−0.905392	+	0.424576i	$0.860423\pi$
$98$	6.65887	0.672648
$99$	0	0
$100$	6.61953	0.661953
$101$	5.48467	0.545745	0.272872	−	0.962050i	$-0.412026\pi$
$101$	5.48467	0.545745	0.272872	+	0.962050i	$0.412026\pi$
$102$	1.64943	0.163318
$103$	6.96059	0.685847	0.342924	−	0.939363i	$-0.388583\pi$
$103$	6.96059	0.685847	0.342924	+	0.939363i	$0.388583\pi$
$104$	6.92833	0.679379
$105$	−0.0175075	−0.00170855
$106$	−5.79523	−0.562883
$107$	7.47334	0.722475	0.361237	−	0.932474i	$-0.382354\pi$
$107$	7.47334	0.722475	0.361237	+	0.932474i	$0.382354\pi$
$108$	−6.05987	−0.583111
$109$	−15.4145	−1.47644	−0.738222	−	0.674558i	$-0.764334\pi$
$109$	−15.4145	−1.47644	−0.738222	+	0.674558i	$0.764334\pi$
$110$	0	0
$111$	−8.27699	−0.785617
$112$	1.55648	0.147074
$113$	19.8993	1.87197	0.935985	−	0.352041i	$-0.114512\pi$
$113$	19.8993	1.87197	0.935985	+	0.352041i	$0.114512\pi$
$114$	−0.399661	−0.0374317
$115$	−0.0235858	−0.00219939
$116$	2.20758	0.204968
$117$	5.66753	0.523963
$118$	−3.90558	−0.359537
$119$	−8.91601	−0.817330
$120$	0.0123143	0.00112413
$121$	0	0
$122$	0.822245	0.0744426
$123$	0.540873	0.0487688
$124$	8.25408	0.741238
$125$	−0.0515381	−0.00460971
$126$	−7.14304	−0.636353
$127$	17.3495	1.53952	0.769759	−	0.638334i	$-0.220376\pi$
$127$	17.3495	1.53952	0.769759	+	0.638334i	$0.220376\pi$
$128$	−8.33148	−0.736406
$129$	−6.32775	−0.557127
$130$	−0.0107426	−0.000942187 0
$131$	−5.92786	−0.517920	−0.258960	−	0.965888i	$-0.583380\pi$
$131$	−5.92786	−0.517920	−0.258960	+	0.965888i	$0.583380\pi$
$132$	0	0
$133$	2.16037	0.187328
$134$	4.01146	0.346538
$135$	0.0235903	0.00203033
$136$	6.27128	0.537758
$137$	13.9298	1.19010	0.595052	−	0.803687i	$-0.297131\pi$
$137$	13.9298	1.19010	0.595052	+	0.803687i	$0.297131\pi$
$138$	3.28966	0.280034
$139$	−21.7356	−1.84359	−0.921797	−	0.387673i	$-0.873279\pi$
$139$	−21.7356	−1.84359	−0.921797	+	0.387673i	$0.873279\pi$
$140$	−0.0265127	−0.00224074
$141$	−6.00322	−0.505562
$142$	10.4801	0.879474
$143$	0	0
$144$	−0.895559	−0.0746299
$145$	−0.00859383	−0.000713678 0
$146$	−0.288139	−0.0238466
$147$	−7.07991	−0.583941
$148$	−12.5344	−1.03032
$149$	15.5039	1.27013	0.635064	−	0.772459i	$-0.280974\pi$
$149$	15.5039	1.27013	0.635064	+	0.772459i	$0.280974\pi$
$150$	3.59416	0.293462
$151$	6.42358	0.522743	0.261372	−	0.965238i	$-0.415825\pi$
$151$	6.42358	0.522743	0.261372	+	0.965238i	$0.415825\pi$
$152$	−1.51955	−0.123251
$153$	5.13005	0.414740
$154$	0	0
$155$	−0.0321321	−0.00258091
$156$	−2.93403	−0.234911
$157$	13.4576	1.07404	0.537019	−	0.843570i	$-0.319551\pi$
$157$	13.4576	1.07404	0.537019	+	0.843570i	$0.319551\pi$
$158$	−8.28009	−0.658729
$159$	6.16166	0.488651
$160$	0.0298691	0.00236136
$161$	−17.7823	−1.40144
$162$	2.22463	0.174783
$163$	6.27802	0.491733	0.245866	−	0.969304i	$-0.420928\pi$
$163$	6.27802	0.491733	0.245866	+	0.969304i	$0.420928\pi$
$164$	0.819080	0.0639594
$165$	0	0
$166$	−11.1365	−0.864361
$167$	1.67109	0.129313	0.0646565	−	0.997908i	$-0.479405\pi$
$167$	1.67109	0.129313	0.0646565	+	0.997908i	$0.479405\pi$
$168$	9.28420	0.716292
$169$	−6.57379	−0.505676
$170$	−0.00972381	−0.000745782 0
$171$	−1.24302	−0.0950563
$172$	−9.58253	−0.730661
$173$	−3.47051	−0.263858	−0.131929	−	0.991259i	$-0.542117\pi$
$173$	−3.47051	−0.263858	−0.131929	+	0.991259i	$0.542117\pi$
$174$	1.19863	0.0908681
$175$	−19.4282	−1.46864
$176$	0	0
$177$	4.15252	0.312123
$178$	11.6932	0.876445
$179$	−9.68200	−0.723667	−0.361833	−	0.932243i	$-0.617849\pi$
$179$	−9.68200	−0.723667	−0.361833	+	0.932243i	$0.617849\pi$
$180$	0.0152548	0.00113702
$181$	1.73973	0.129313	0.0646564	−	0.997908i	$-0.479405\pi$
$181$	1.73973	0.129313	0.0646564	+	0.997908i	$0.479405\pi$
$182$	−8.09924	−0.600356
$183$	−0.874235	−0.0646253
$184$	12.5075	0.922069
$185$	0.0487949	0.00358747
$186$	4.48166	0.328611
$187$	0	0
$188$	−9.09108	−0.663035
$189$	17.7856	1.29371
$190$	0.00235610	0.000170930 0
$191$	−4.07147	−0.294601	−0.147301	−	0.989092i	$-0.547058\pi$
$191$	−4.07147	−0.294601	−0.147301	+	0.989092i	$0.547058\pi$
$192$	−3.46563	−0.250111
$193$	10.4247	0.750387	0.375193	−	0.926947i	$-0.377576\pi$
$193$	10.4247	0.750387	0.375193	+	0.926947i	$0.377576\pi$
$194$	−14.6641	−1.05282
$195$	0.0114218	0.000817934 0
$196$	−10.7216	−0.765827
$197$	−9.08962	−0.647608	−0.323804	−	0.946124i	$-0.604962\pi$
$197$	−9.08962	−0.647608	−0.323804	+	0.946124i	$0.604962\pi$
$198$	0	0
$199$	−2.99857	−0.212563	−0.106281	−	0.994336i	$-0.533894\pi$
$199$	−2.99857	−0.212563	−0.106281	+	0.994336i	$0.533894\pi$
$200$	13.6653	0.966282
$201$	−4.26510	−0.300837
$202$	4.50974	0.317304
$203$	−6.47921	−0.454751
$204$	−2.65578	−0.185942
$205$	−0.00318858	−0.000222700 0
$206$	5.72331	0.398762
$207$	10.2315	0.711135
$208$	−1.01544	−0.0704083
$209$	0	0
$210$	−0.0143954	−0.000993380 0
$211$	2.61293	0.179881	0.0899406	−	0.995947i	$-0.471332\pi$
$211$	2.61293	0.179881	0.0899406	+	0.995947i	$0.471332\pi$
$212$	9.33101	0.640857
$213$	−11.1428	−0.763491
$214$	6.14492	0.420058
$215$	0.0373036	0.00254409
$216$	−12.5099	−0.851193
$217$	−24.2256	−1.64454
$218$	−12.6745	−0.858427
$219$	0.306358	0.0207018
$220$	0	0
$221$	5.81678	0.391279
$222$	−6.80572	−0.456770
$223$	−26.3911	−1.76728	−0.883638	−	0.468171i	$-0.844913\pi$
$223$	−26.3911	−1.76728	−0.883638	+	0.468171i	$0.844913\pi$
$224$	22.5194	1.50464
$225$	11.1785	0.745234
$226$	16.3621	1.08839
$227$	23.4169	1.55423	0.777116	−	0.629357i	$-0.216682\pi$
$227$	23.4169	1.55423	0.777116	+	0.629357i	$0.216682\pi$
$228$	0.643502	0.0426170
$229$	15.4099	1.01831	0.509157	−	0.860674i	$-0.329957\pi$
$229$	15.4099	1.01831	0.509157	+	0.860674i	$0.329957\pi$
$230$	−0.0193933	−0.00127876
$231$	0	0
$232$	4.55730	0.299201
$233$	5.81544	0.380982	0.190491	−	0.981689i	$-0.438992\pi$
$233$	5.81544	0.380982	0.190491	+	0.981689i	$0.438992\pi$
$234$	4.66010	0.304640
$235$	0.0353904	0.00230862
$236$	6.28844	0.409343
$237$	8.80363	0.571857
$238$	−7.33115	−0.475208
$239$	13.4929	0.872782	0.436391	−	0.899757i	$-0.356256\pi$
$239$	13.4929	0.872782	0.436391	+	0.899757i	$0.356256\pi$
$240$	−0.00180483	−0.000116501 0
$241$	18.2848	1.17782	0.588912	−	0.808197i	$-0.299556\pi$
$241$	18.2848	1.17782	0.588912	+	0.808197i	$0.299556\pi$
$242$	0	0
$243$	−16.0970	−1.03262
$244$	−1.32391	−0.0847548
$245$	0.0417378	0.00266653
$246$	0.444730	0.0283550
$247$	−1.40942	−0.0896792
$248$	17.0396	1.08202
$249$	11.8407	0.750372
$250$	−0.0423770	−0.00268016
$251$	−12.2025	−0.770217	−0.385108	−	0.922871i	$-0.625836\pi$
$251$	−12.2025	−0.770217	−0.385108	+	0.922871i	$0.625836\pi$
$252$	11.5011	0.724504
$253$	0	0
$254$	14.2655	0.895100
$255$	0.0103386	0.000647430 0
$256$	−14.7789	−0.923682
$257$	19.1450	1.19423	0.597116	−	0.802155i	$-0.296313\pi$
$257$	19.1450	1.19423	0.597116	+	0.802155i	$0.296313\pi$
$258$	−5.20296	−0.323922
$259$	36.7883	2.28591
$260$	0.0172968	0.00107270
$261$	3.72797	0.230756
$262$	−4.87416	−0.301126
$263$	6.21510	0.383239	0.191620	−	0.981469i	$-0.438626\pi$
$263$	6.21510	0.383239	0.191620	+	0.981469i	$0.438626\pi$
$264$	0	0
$265$	−0.0363245	−0.00223140
$266$	1.77635	0.108915
$267$	−12.4326	−0.760862
$268$	−6.45893	−0.394542
$269$	−0.147851	−0.00901462	−0.00450731	−	0.999990i	$-0.501435\pi$
$269$	−0.147851	−0.00901462	−0.00450731	+	0.999990i	$0.501435\pi$
$270$	0.0193970	0.00118047
$271$	5.48781	0.333361	0.166680	−	0.986011i	$-0.446695\pi$
$271$	5.48781	0.333361	0.166680	+	0.986011i	$0.446695\pi$
$272$	−0.919144	−0.0557313
$273$	8.61135	0.521182
$274$	11.4537	0.691945
$275$	0	0
$276$	−5.29674	−0.318826
$277$	23.4332	1.40797	0.703983	−	0.710216i	$-0.251403\pi$
$277$	23.4332	1.40797	0.703983	+	0.710216i	$0.251403\pi$
$278$	−17.8720	−1.07189
$279$	13.9388	0.834494
$280$	−0.0547326	−0.00327090
$281$	2.45697	0.146571	0.0732853	−	0.997311i	$-0.476652\pi$
$281$	2.45697	0.146571	0.0732853	+	0.997311i	$0.476652\pi$
$282$	−4.93612	−0.293942
$283$	25.7322	1.52962	0.764811	−	0.644255i	$-0.222832\pi$
$283$	25.7322	1.52962	0.764811	+	0.644255i	$0.222832\pi$
$284$	−16.8743	−1.00130
$285$	−0.00250507	−0.000148388 0
$286$	0	0
$287$	−2.40399	−0.141903
$288$	−12.9571	−0.763504
$289$	−11.7349	−0.690285
$290$	−0.00706623	−0.000414943 0
$291$	15.5912	0.913975
$292$	0.463939	0.0271500
$293$	2.89474	0.169112	0.0845561	−	0.996419i	$-0.473053\pi$
$293$	2.89474	0.169112	0.0845561	+	0.996419i	$0.473053\pi$
$294$	−5.82142	−0.339512
$295$	−0.0244801	−0.00142529
$296$	−25.8759	−1.50401
$297$	0	0
$298$	12.7480	0.738472
$299$	11.6011	0.670908
$300$	−5.78702	−0.334114
$301$	28.1246	1.62108
$302$	5.28176	0.303931
$303$	−4.79489	−0.275459
$304$	0.222711	0.0127733
$305$	0.00515383	0.000295107 0
$306$	4.21816	0.241136
$307$	22.6498	1.29269	0.646347	−	0.763044i	$-0.276296\pi$
$307$	22.6498	1.29269	0.646347	+	0.763044i	$0.276296\pi$
$308$	0	0
$309$	−6.08519	−0.346174
$310$	−0.0264205	−0.00150058
$311$	20.5865	1.16735	0.583676	−	0.811987i	$-0.301614\pi$
$311$	20.5865	1.16735	0.583676	+	0.811987i	$0.301614\pi$
$312$	−6.05699	−0.342909
$313$	10.9388	0.618295	0.309147	−	0.951014i	$-0.399956\pi$
$313$	10.9388	0.618295	0.309147	+	0.951014i	$0.399956\pi$
$314$	11.0655	0.624462
$315$	−0.0447725	−0.00252265
$316$	13.3319	0.749980
$317$	25.8950	1.45441	0.727203	−	0.686422i	$-0.240820\pi$
$317$	25.8950	1.45441	0.727203	+	0.686422i	$0.240820\pi$
$318$	5.06640	0.284109
$319$	0	0
$320$	0.0204308	0.00114211
$321$	−6.53345	−0.364662
$322$	−14.6214	−0.814817
$323$	−1.27576	−0.0709850
$324$	−3.58192	−0.198995
$325$	12.6749	0.703078
$326$	5.16207	0.285901
$327$	13.4759	0.745219
$328$	1.69090	0.0933643
$329$	26.6822	1.47104
$330$	0	0
$331$	−1.47244	−0.0809325	−0.0404662	−	0.999181i	$-0.512884\pi$
$331$	−1.47244	−0.0809325	−0.0404662	+	0.999181i	$0.512884\pi$
$332$	17.9311	0.984098
$333$	−21.1671	−1.15995
$334$	1.37405	0.0751846
$335$	0.0251438	0.00137375
$336$	−1.36073	−0.0742339
$337$	23.5396	1.28229	0.641143	−	0.767421i	$-0.278461\pi$
$337$	23.5396	1.28229	0.641143	+	0.767421i	$0.278461\pi$
$338$	−5.40527	−0.294008
$339$	−17.3967	−0.944857
$340$	0.0156565	0.000849092 0
$341$	0	0
$342$	−1.02207	−0.0552672
$343$	4.26803	0.230452
$344$	−19.7821	−1.06658
$345$	0.0206196	0.00111012
$346$	−2.85361	−0.153411
$347$	−16.1793	−0.868552	−0.434276	−	0.900780i	$-0.642996\pi$
$347$	−16.1793	−0.868552	−0.434276	+	0.900780i	$0.642996\pi$
$348$	−1.92994	−0.103456
$349$	7.05207	0.377489	0.188744	−	0.982026i	$-0.439558\pi$
$349$	7.05207	0.377489	0.188744	+	0.982026i	$0.439558\pi$
$350$	−15.9748	−0.853887
$351$	−11.6033	−0.619338
$352$	0	0
$353$	22.8995	1.21882	0.609410	−	0.792856i	$-0.291407\pi$
$353$	22.8995	1.21882	0.609410	+	0.792856i	$0.291407\pi$
$354$	3.41439	0.181473
$355$	0.0656894	0.00348643
$356$	−18.8275	−0.997856
$357$	7.79469	0.412539
$358$	−7.96098	−0.420751
$359$	−9.33694	−0.492785	−0.246392	−	0.969170i	$-0.579245\pi$
$359$	−9.33694	−0.492785	−0.246392	+	0.969170i	$0.579245\pi$
$360$	0.0314918	0.00165976
$361$	−18.6909	−0.983731
$362$	1.43048	0.0751844
$363$	0	0
$364$	13.0407	0.683521
$365$	−0.00180606	−9.45333e−5 0
$366$	−0.718836	−0.0375741
$367$	−9.53738	−0.497847	−0.248924	−	0.968523i	$-0.580077\pi$
$367$	−9.53738	−0.497847	−0.248924	+	0.968523i	$0.580077\pi$
$368$	−1.83316	−0.0955599
$369$	1.38319	0.0720062
$370$	0.0401214	0.00208581
$371$	−27.3864	−1.42183
$372$	−7.21600	−0.374132
$373$	−16.9911	−0.879767	−0.439883	−	0.898055i	$-0.644980\pi$
$373$	−16.9911	−0.879767	−0.439883	+	0.898055i	$0.644980\pi$
$374$	0	0
$375$	0.0450564	0.00232670
$376$	−18.7675	−0.967862
$377$	4.22702	0.217703
$378$	14.6242	0.752186
$379$	23.2934	1.19650	0.598252	−	0.801308i	$-0.295862\pi$
$379$	23.2934	1.19650	0.598252	+	0.801308i	$0.295862\pi$
$380$	−0.00379360	−0.000194608 0
$381$	−15.1675	−0.777056
$382$	−3.34775	−0.171286
$383$	−20.6889	−1.05715	−0.528577	−	0.848886i	$-0.677274\pi$
$383$	−20.6889	−1.05715	−0.528577	+	0.848886i	$0.677274\pi$
$384$	7.28367	0.371693
$385$	0	0
$386$	8.57166	0.436286
$387$	−16.1822	−0.822587
$388$	23.6109	1.19866
$389$	34.2978	1.73897	0.869484	−	0.493962i	$-0.164452\pi$
$389$	34.2978	1.73897	0.869484	+	0.493962i	$0.164452\pi$
$390$	0.00939154	0.000475559 0
$391$	10.5009	0.531053
$392$	−22.1335	−1.11791
$393$	5.18234	0.261415
$394$	−7.47390	−0.376529
$395$	−0.0518995	−0.00261135
$396$	0	0
$397$	0.681150	0.0341860	0.0170930	−	0.999854i	$-0.494559\pi$
$397$	0.681150	0.0341860	0.0170930	+	0.999854i	$0.494559\pi$
$398$	−2.46556	−0.123587
$399$	−1.88867	−0.0945518
$400$	−2.00284	−0.100142
$401$	31.6303	1.57954	0.789771	−	0.613402i	$-0.210199\pi$
$401$	31.6303	1.57954	0.789771	+	0.613402i	$0.210199\pi$
$402$	−3.50696	−0.174911
$403$	15.8047	0.787289
$404$	−7.26122	−0.361259
$405$	0.0139440	0.000692881 0
$406$	−5.32750	−0.264399
$407$	0	0
$408$	−5.48257	−0.271428
$409$	−20.4148	−1.00945	−0.504724	−	0.863281i	$-0.668406\pi$
$409$	−20.4148	−1.00945	−0.504724	+	0.863281i	$0.668406\pi$
$410$	−0.00262179	−0.000129481 0
$411$	−12.1779	−0.600693
$412$	−9.21521	−0.454001
$413$	−18.4565	−0.908185
$414$	8.41277	0.413465
$415$	−0.0698036	−0.00342652
$416$	−14.6916	−0.720315
$417$	19.0021	0.930535
$418$	0	0
$419$	24.1643	1.18050	0.590251	−	0.807219i	$-0.299029\pi$
$419$	24.1643	1.18050	0.590251	+	0.807219i	$0.299029\pi$
$420$	0.0231784	0.00113099
$421$	2.05896	0.100348	0.0501738	−	0.998741i	$-0.484022\pi$
$421$	2.05896	0.100348	0.0501738	+	0.998741i	$0.484022\pi$
$422$	2.14847	0.104586
$423$	−15.3523	−0.746452
$424$	19.2629	0.935487
$425$	11.4729	0.556517
$426$	−9.16210	−0.443905
$427$	3.88567	0.188041
$428$	−9.89404	−0.478247
$429$	0	0
$430$	0.0306727	0.00147917
$431$	30.6911	1.47834	0.739168	−	0.673521i	$-0.235219\pi$
$431$	30.6911	1.47834	0.739168	+	0.673521i	$0.235219\pi$
$432$	1.83351	0.0882146
$433$	−16.3076	−0.783694	−0.391847	−	0.920030i	$-0.628164\pi$
$433$	−16.3076	−0.783694	−0.391847	+	0.920030i	$0.628164\pi$
$434$	−19.9194	−0.956162
$435$	0.00751302	0.000360222 0
$436$	20.4075	0.977341
$437$	−2.54439	−0.121715
$438$	0.251902	0.0120363
$439$	28.8835	1.37853	0.689267	−	0.724507i	$-0.257933\pi$
$439$	28.8835	1.37853	0.689267	+	0.724507i	$0.257933\pi$
$440$	0	0
$441$	−18.1057	−0.862177
$442$	4.78282	0.227496
$443$	−0.160622	−0.00763137	−0.00381569	−	0.999993i	$-0.501215\pi$
$443$	−0.160622	−0.00763137	−0.00381569	+	0.999993i	$0.501215\pi$
$444$	10.9580	0.520044
$445$	0.0732931	0.00347443
$446$	−21.6999	−1.02752
$447$	−13.5540	−0.641084
$448$	15.4035	0.727748
$449$	−23.3731	−1.10304	−0.551521	−	0.834161i	$-0.685952\pi$
$449$	−23.3731	−1.10304	−0.551521	+	0.834161i	$0.685952\pi$
$450$	9.19148	0.433290
$451$	0	0
$452$	−26.3449	−1.23916
$453$	−5.61572	−0.263849
$454$	19.2544	0.903655
$455$	−0.0507660	−0.00237995
$456$	1.32844	0.0622099
$457$	−25.2761	−1.18236	−0.591182	−	0.806538i	$-0.701339\pi$
$457$	−25.2761	−1.18236	−0.591182	+	0.806538i	$0.701339\pi$
$458$	12.6707	0.592063
$459$	−10.5029	−0.490233
$460$	0.0312256	0.00145590
$461$	−16.8755	−0.785973	−0.392986	−	0.919544i	$-0.628558\pi$
$461$	−16.8755	−0.785973	−0.392986	+	0.919544i	$0.628558\pi$
$462$	0	0
$463$	−36.5403	−1.69817	−0.849085	−	0.528256i	$-0.822846\pi$
$463$	−36.5403	−1.69817	−0.849085	+	0.528256i	$0.822846\pi$
$464$	−0.667936	−0.0310081
$465$	0.0280910	0.00130269
$466$	4.78172	0.221509
$467$	15.5083	0.717639	0.358820	−	0.933407i	$-0.383179\pi$
$467$	15.5083	0.717639	0.358820	+	0.933407i	$0.383179\pi$
$468$	−7.50331	−0.346841
$469$	18.9569	0.875347
$470$	0.0290996	0.00134227
$471$	−11.7651	−0.542109
$472$	12.9818	0.597536
$473$	0	0
$474$	7.23874	0.332487
$475$	−2.77991	−0.127551
$476$	11.8040	0.541037
$477$	15.7574	0.721484
$478$	11.0945	0.507449
$479$	7.93828	0.362709	0.181355	−	0.983418i	$-0.441952\pi$
$479$	7.93828	0.362709	0.181355	+	0.983418i	$0.441952\pi$
$480$	−0.0261126	−0.00119187
$481$	−24.0006	−1.09433
$482$	15.0346	0.684805
$483$	15.5459	0.707361
$484$	0	0
$485$	−0.0919142	−0.00417361
$486$	−13.2357	−0.600384
$487$	−13.5924	−0.615929	−0.307964	−	0.951398i	$-0.599648\pi$
$487$	−13.5924	−0.615929	−0.307964	+	0.951398i	$0.599648\pi$
$488$	−2.73307	−0.123720
$489$	−5.48847	−0.248197
$490$	0.0343187	0.00155036
$491$	−0.750846	−0.0338852	−0.0169426	−	0.999856i	$-0.505393\pi$
$491$	−0.750846	−0.0338852	−0.0169426	+	0.999856i	$0.505393\pi$
$492$	−0.716068	−0.0322828
$493$	3.82615	0.172321
$494$	−1.15889	−0.0521409
$495$	0	0
$496$	−2.49740	−0.112136
$497$	49.5258	2.22153
$498$	9.73593	0.436277
$499$	−2.90960	−0.130252	−0.0651259	−	0.997877i	$-0.520745\pi$
$499$	−2.90960	−0.130252	−0.0651259	+	0.997877i	$0.520745\pi$
$500$	0.0682320	0.00305143
$501$	−1.46093	−0.0652694
$502$	−10.0335	−0.447816
$503$	33.3366	1.48640	0.743202	−	0.669067i	$-0.233306\pi$
$503$	33.3366	1.48640	0.743202	+	0.669067i	$0.233306\pi$
$504$	23.7428	1.05759
$505$	0.0282670	0.00125787
$506$	0	0
$507$	5.74704	0.255235
$508$	−22.9692	−1.01909
$509$	−13.3592	−0.592136	−0.296068	−	0.955167i	$-0.595676\pi$
$509$	−13.3592	−0.592136	−0.296068	+	0.955167i	$0.595676\pi$
$510$	0.00850090	0.000376426 0
$511$	−1.36165	−0.0602360
$512$	4.51108	0.199363
$513$	2.54488	0.112359
$514$	15.7419	0.694345
$515$	0.0358737	0.00158078
$516$	8.37738	0.368794
$517$	0	0
$518$	30.2490	1.32907
$519$	3.03404	0.133179
$520$	0.0357074	0.00156587
$521$	28.4519	1.24650	0.623250	−	0.782023i	$-0.285812\pi$
$521$	28.4519	1.24650	0.623250	+	0.782023i	$0.285812\pi$
$522$	3.06531	0.134165
$523$	−18.2972	−0.800080	−0.400040	−	0.916498i	$-0.631004\pi$
$523$	−18.2972	−0.800080	−0.400040	+	0.916498i	$0.631004\pi$
$524$	7.84797	0.342840
$525$	16.9848	0.741279
$526$	5.11033	0.222821
$527$	14.3059	0.623174
$528$	0	0
$529$	−2.05683	−0.0894273
$530$	−0.0298676	−0.00129737
$531$	10.6194	0.460843
$532$	−2.86014	−0.124003
$533$	1.56836	0.0679330
$534$	−10.2226	−0.442377
$535$	0.0385163	0.00166520
$536$	−13.3337	−0.575930
$537$	8.46434	0.365263
$538$	−0.121570	−0.00524124
$539$	0	0
$540$	−0.0312315	−0.00134399
$541$	−0.739219	−0.0317815	−0.0158907	−	0.999874i	$-0.505058\pi$
$541$	−0.739219	−0.0317815	−0.0158907	+	0.999874i	$0.505058\pi$
$542$	4.51233	0.193821
$543$	−1.52093	−0.0652693
$544$	−13.2983	−0.570161
$545$	−0.0794438	−0.00340300
$546$	7.08064	0.303023
$547$	10.4380	0.446296	0.223148	−	0.974785i	$-0.428367\pi$
$547$	10.4380	0.446296	0.223148	+	0.974785i	$0.428367\pi$
$548$	−18.4419	−0.787797
$549$	−2.23571	−0.0954179
$550$	0	0
$551$	−0.927085	−0.0394951
$552$	−10.9345	−0.465405
$553$	−39.1290	−1.66394
$554$	19.2679	0.818613
$555$	−0.0426582	−0.00181074
$556$	28.7761	1.22038
$557$	14.6662	0.621425	0.310713	−	0.950504i	$-0.399432\pi$
$557$	14.6662	0.621425	0.310713	+	0.950504i	$0.399432\pi$
$558$	11.4611	0.485188
$559$	−18.3484	−0.776055
$560$	0.00802183	0.000338984 0
$561$	0	0
$562$	2.02023	0.0852184
$563$	16.3171	0.687686	0.343843	−	0.939027i	$-0.388271\pi$
$563$	16.3171	0.687686	0.343843	+	0.939027i	$0.388271\pi$
$564$	7.94774	0.334660
$565$	0.102558	0.00431463
$566$	21.1582	0.889345
$567$	10.5129	0.441500
$568$	−34.8351	−1.46165
$569$	26.4031	1.10688	0.553438	−	0.832890i	$-0.313316\pi$
$569$	26.4031	1.10688	0.553438	+	0.832890i	$0.313316\pi$
$570$	−0.00205979	−8.62749e−5 0
$571$	−18.6535	−0.780626	−0.390313	−	0.920682i	$-0.627633\pi$
$571$	−18.6535	−0.780626	−0.390313	+	0.920682i	$0.627633\pi$
$572$	0	0
$573$	3.55942	0.148697
$574$	−1.97667	−0.0825046
$575$	22.8817	0.954234
$576$	−8.86280	−0.369283
$577$	3.68475	0.153398	0.0766990	−	0.997054i	$-0.475562\pi$
$577$	3.68475	0.153398	0.0766990	+	0.997054i	$0.475562\pi$
$578$	−9.64893	−0.401343
$579$	−9.11364	−0.378750
$580$	0.0113775	0.000472424 0
$581$	−52.6276	−2.18336
$582$	12.8198	0.531399
$583$	0	0
$584$	0.957750	0.0396320
$585$	0.0292095	0.00120766
$586$	2.38018	0.0983244
$587$	10.8093	0.446147	0.223073	−	0.974802i	$-0.428391\pi$
$587$	10.8093	0.446147	0.223073	+	0.974802i	$0.428391\pi$
$588$	9.37318	0.386543
$589$	−3.46635	−0.142828
$590$	−0.0201287	−0.000828684 0
$591$	7.94646	0.326874
$592$	3.79247	0.155870
$593$	30.4621	1.25093	0.625464	−	0.780253i	$-0.284910\pi$
$593$	30.4621	1.25093	0.625464	+	0.780253i	$0.284910\pi$
$594$	0	0
$595$	−0.0459516	−0.00188383
$596$	−20.5258	−0.840769
$597$	2.62145	0.107289
$598$	9.53894	0.390076
$599$	−26.8887	−1.09864	−0.549320	−	0.835612i	$-0.685113\pi$
$599$	−26.8887	−1.09864	−0.549320	+	0.835612i	$0.685113\pi$
$600$	−11.9467	−0.487721
$601$	4.72714	0.192824	0.0964121	−	0.995342i	$-0.469263\pi$
$601$	4.72714	0.192824	0.0964121	+	0.995342i	$0.469263\pi$
$602$	23.1253	0.942518
$603$	−10.9073	−0.444180
$604$	−8.50426	−0.346033
$605$	0	0
$606$	−3.94257	−0.160156
$607$	−32.8916	−1.33503	−0.667514	−	0.744597i	$-0.732642\pi$
$607$	−32.8916	−1.33503	−0.667514	+	0.744597i	$0.732642\pi$
$608$	3.22221	0.130678
$609$	5.66435	0.229531
$610$	0.00423771	0.000171580 0
$611$	−17.4074	−0.704228
$612$	−6.79173	−0.274540
$613$	−22.3712	−0.903563	−0.451782	−	0.892129i	$-0.649211\pi$
$613$	−22.3712	−0.903563	−0.451782	+	0.892129i	$0.649211\pi$
$614$	18.6237	0.751592
$615$	0.00278756	0.000112405 0
$616$	0	0
$617$	−16.5003	−0.664278	−0.332139	−	0.943230i	$-0.607770\pi$
$617$	−16.5003	−0.664278	−0.332139	+	0.943230i	$0.607770\pi$
$618$	−5.00352	−0.201271
$619$	−23.9640	−0.963194	−0.481597	−	0.876393i	$-0.659943\pi$
$619$	−23.9640	−0.963194	−0.481597	+	0.876393i	$0.659943\pi$
$620$	0.0425401	0.00170845
$621$	−20.9472	−0.840581
$622$	16.9271	0.678716
$623$	55.2585	2.21388
$624$	0.887736	0.0355379
$625$	24.9996	0.999984
$626$	8.99434	0.359486
$627$	0	0
$628$	−17.8167	−0.710966
$629$	−21.7245	−0.866212
$630$	−0.0368140	−0.00146670
$631$	−25.6753	−1.02212	−0.511060	−	0.859545i	$-0.670747\pi$
$631$	−25.6753	−1.02212	−0.511060	+	0.859545i	$0.670747\pi$
$632$	27.5223	1.09478
$633$	−2.28431	−0.0907932
$634$	21.2920	0.845614
$635$	0.0894163	0.00354838
$636$	−8.15750	−0.323466
$637$	−20.5294	−0.813406
$638$	0	0
$639$	−28.4959	−1.12728
$640$	−0.0429390	−0.00169731
$641$	−43.4420	−1.71586	−0.857928	−	0.513770i	$-0.828249\pi$
$641$	−43.4420	−1.71586	−0.857928	+	0.513770i	$0.828249\pi$
$642$	−5.37210	−0.212020
$643$	8.02245	0.316374	0.158187	−	0.987409i	$-0.449435\pi$
$643$	8.02245	0.316374	0.158187	+	0.987409i	$0.449435\pi$
$644$	23.5421	0.927691
$645$	−0.0326121	−0.00128410
$646$	−1.04899	−0.0412718
$647$	19.2377	0.756310	0.378155	−	0.925742i	$-0.376559\pi$
$647$	19.2377	0.756310	0.378155	+	0.925742i	$0.376559\pi$
$648$	−7.39448	−0.290482
$649$	0	0
$650$	10.4219	0.408780
$651$	21.1789	0.830066
$652$	−8.31155	−0.325505
$653$	16.0732	0.628992	0.314496	−	0.949259i	$-0.398164\pi$
$653$	16.0732	0.628992	0.314496	+	0.949259i	$0.398164\pi$
$654$	11.0805	0.433282
$655$	−0.0305512	−0.00119373
$656$	−0.247825	−0.00967594
$657$	0.783461	0.0305657
$658$	21.9393	0.855284
$659$	3.29079	0.128191	0.0640956	−	0.997944i	$-0.479584\pi$
$659$	3.29079	0.128191	0.0640956	+	0.997944i	$0.479584\pi$
$660$	0	0
$661$	−19.5966	−0.762219	−0.381110	−	0.924530i	$-0.624458\pi$
$661$	−19.5966	−0.762219	−0.381110	+	0.924530i	$0.624458\pi$
$662$	−1.21070	−0.0470554
$663$	−5.08523	−0.197494
$664$	37.0168	1.43653
$665$	0.0111342	0.000431765 0
$666$	−17.4045	−0.674411
$667$	7.63094	0.295471
$668$	−2.21238	−0.0855996
$669$	23.0720	0.892014
$670$	0.0206744	0.000798721 0
$671$	0	0
$672$	−19.6873	−0.759453
$673$	25.1986	0.971334	0.485667	−	0.874144i	$-0.338577\pi$
$673$	25.1986	0.971334	0.485667	+	0.874144i	$0.338577\pi$
$674$	19.3554	0.745541
$675$	−22.8861	−0.880886
$676$	8.70312	0.334735
$677$	−7.07541	−0.271930	−0.135965	−	0.990714i	$-0.543413\pi$
$677$	−7.07541	−0.271930	−0.135965	+	0.990714i	$0.543413\pi$
$678$	−14.3043	−0.549354
$679$	−69.2976	−2.65940
$680$	0.0323211	0.00123946
$681$	−20.4718	−0.784483
$682$	0	0
$683$	2.78875	0.106709	0.0533543	−	0.998576i	$-0.483009\pi$
$683$	2.78875	0.106709	0.0533543	+	0.998576i	$0.483009\pi$
$684$	1.64565	0.0629231
$685$	0.0717919	0.00274303
$686$	3.50937	0.133988
$687$	−13.4719	−0.513983
$688$	2.89934	0.110536
$689$	17.8668	0.680671
$690$	0.0169543	0.000645440 0
$691$	−8.82809	−0.335836	−0.167918	−	0.985801i	$-0.553704\pi$
$691$	−8.82809	−0.335836	−0.167918	+	0.985801i	$0.553704\pi$
$692$	4.59465	0.174662
$693$	0	0
$694$	−13.3034	−0.504989
$695$	−0.112022	−0.00424923
$696$	−3.98415	−0.151019
$697$	1.41962	0.0537719
$698$	5.79853	0.219478
$699$	−5.08406	−0.192297
$700$	25.7213	0.972173
$701$	−9.26834	−0.350060	−0.175030	−	0.984563i	$-0.556002\pi$
$701$	−9.26834	−0.350060	−0.175030	+	0.984563i	$0.556002\pi$
$702$	−9.54076	−0.360093
$703$	5.26389	0.198531
$704$	0	0
$705$	−0.0309396	−0.00116525
$706$	18.8290	0.708640
$707$	21.3116	0.801504
$708$	−5.49758	−0.206612
$709$	−13.5033	−0.507127	−0.253563	−	0.967319i	$-0.581603\pi$
$709$	−13.5033	−0.507127	−0.253563	+	0.967319i	$0.581603\pi$
$710$	0.0540128	0.00202707
$711$	22.5139	0.844335
$712$	−38.8673	−1.45661
$713$	28.5319	1.06853
$714$	6.40915	0.239856
$715$	0	0
$716$	12.8181	0.479036
$717$	−11.7959	−0.440528
$718$	−7.67725	−0.286513
$719$	−10.8591	−0.404975	−0.202487	−	0.979285i	$-0.564903\pi$
$719$	−10.8591	−0.404975	−0.202487	+	0.979285i	$0.564903\pi$
$720$	−0.00461556	−0.000172012 0
$721$	27.0465	1.00727
$722$	−15.3685	−0.571956
$723$	−15.9852	−0.594495
$724$	−2.30324	−0.0855994
$725$	8.33728	0.309639
$726$	0	0
$727$	−2.40568	−0.0892218	−0.0446109	−	0.999004i	$-0.514205\pi$
$727$	−2.40568	−0.0892218	−0.0446109	+	0.999004i	$0.514205\pi$
$728$	26.9212	0.997765
$729$	5.95591	0.220589
$730$	−0.00148502	−5.49631e−5 0
$731$	−16.6083	−0.614282
$732$	1.15741	0.0427791
$733$	41.6939	1.54000	0.769999	−	0.638045i	$-0.220257\pi$
$733$	41.6939	1.54000	0.769999	+	0.638045i	$0.220257\pi$
$734$	−7.84207	−0.289456
$735$	−0.0364886	−0.00134590
$736$	−26.5224	−0.977629
$737$	0	0
$738$	1.13732	0.0418655
$739$	3.06557	0.112769	0.0563845	−	0.998409i	$-0.482043\pi$
$739$	3.06557	0.112769	0.0563845	+	0.998409i	$0.482043\pi$
$740$	−0.0646001	−0.00237475
$741$	1.23216	0.0452647
$742$	−22.5183	−0.826675
$743$	23.2651	0.853515	0.426757	−	0.904366i	$-0.359656\pi$
$743$	23.2651	0.853515	0.426757	+	0.904366i	$0.359656\pi$
$744$	−14.8966	−0.546138
$745$	0.0799044	0.00292747
$746$	−13.9709	−0.511510
$747$	30.2806	1.10791
$748$	0	0
$749$	29.0389	1.06106
$750$	0.0370474	0.00135278
$751$	16.3540	0.596766	0.298383	−	0.954446i	$-0.403553\pi$
$751$	16.3540	0.596766	0.298383	+	0.954446i	$0.403553\pi$
$752$	2.75064	0.100306
$753$	10.6679	0.388759
$754$	3.47565	0.126576
$755$	0.0331060	0.00120485
$756$	−23.5466	−0.856383
$757$	8.35181	0.303552	0.151776	−	0.988415i	$-0.451501\pi$
$757$	8.35181	0.303552	0.151776	+	0.988415i	$0.451501\pi$
$758$	19.1529	0.695666
$759$	0	0
$760$	−0.00783148	−0.000284077 0
$761$	46.8220	1.69730	0.848648	−	0.528958i	$-0.177417\pi$
$761$	46.8220	1.69730	0.848648	+	0.528958i	$0.177417\pi$
$762$	−12.4714	−0.451792
$763$	−59.8957	−2.16837
$764$	5.39027	0.195013
$765$	0.0264394	0.000955918 0
$766$	−17.0113	−0.614645
$767$	12.0410	0.434774
$768$	12.9202	0.466219
$769$	−20.1836	−0.727839	−0.363919	−	0.931430i	$-0.618562\pi$
$769$	−20.1836	−0.727839	−0.363919	+	0.931430i	$0.618562\pi$
$770$	0	0
$771$	−16.7372	−0.602777
$772$	−13.8014	−0.496723
$773$	−49.8900	−1.79442	−0.897209	−	0.441606i	$-0.854409\pi$
$773$	−49.8900	−1.79442	−0.897209	+	0.441606i	$0.854409\pi$
$774$	−13.3057	−0.478264
$775$	31.1729	1.11976
$776$	48.7420	1.74974
$777$	−32.1616	−1.15379
$778$	28.2012	1.01106
$779$	−0.343977	−0.0123243
$780$	−0.0151215	−0.000541436 0
$781$	0	0
$782$	8.63432	0.308762
$783$	−7.63239	−0.272759
$784$	3.24398	0.115856
$785$	0.0693584	0.00247551
$786$	4.26116	0.151990
$787$	−12.4118	−0.442432	−0.221216	−	0.975225i	$-0.571003\pi$
$787$	−12.4118	−0.442432	−0.221216	+	0.975225i	$0.571003\pi$
$788$	12.0339	0.428688
$789$	−5.43345	−0.193436
$790$	−0.0426742	−0.00151828
$791$	77.3221	2.74926
$792$	0	0
$793$	−2.53500	−0.0900204
$794$	0.560073	0.0198762
$795$	0.0317561	0.00112627
$796$	3.96984	0.140707
$797$	16.4175	0.581538	0.290769	−	0.956793i	$-0.406089\pi$
$797$	16.4175	0.581538	0.290769	+	0.956793i	$0.406089\pi$
$798$	−1.55295	−0.0549739
$799$	−15.7566	−0.557427
$800$	−28.9774	−1.02451
$801$	−31.7943	−1.12340
$802$	26.0079	0.918370
$803$	0	0
$804$	5.64662	0.199141
$805$	−0.0916467	−0.00323012
$806$	12.9954	0.457742
$807$	0.129256	0.00455004
$808$	−14.9900	−0.527346
$809$	−0.951220	−0.0334431	−0.0167216	−	0.999860i	$-0.505323\pi$
$809$	−0.951220	−0.0334431	−0.0167216	+	0.999860i	$0.505323\pi$
$810$	0.0114654	0.000402852 0
$811$	18.4221	0.646886	0.323443	−	0.946248i	$-0.395160\pi$
$811$	18.4221	0.646886	0.323443	+	0.946248i	$0.395160\pi$
$812$	8.57791	0.301026
$813$	−4.79764	−0.168260
$814$	0	0
$815$	0.0323558	0.00113338
$816$	0.803547	0.0281298
$817$	4.02424	0.140790
$818$	−16.7860	−0.586908
$819$	22.0221	0.769515
$820$	0.00422140	0.000147418 0
$821$	4.49365	0.156829	0.0784147	−	0.996921i	$-0.475014\pi$
$821$	4.49365	0.156829	0.0784147	+	0.996921i	$0.475014\pi$
$822$	−10.0133	−0.349252
$823$	21.8096	0.760237	0.380118	−	0.924938i	$-0.375883\pi$
$823$	21.8096	0.760237	0.380118	+	0.924938i	$0.375883\pi$
$824$	−19.0238	−0.662725
$825$	0	0
$826$	−15.1758	−0.528033
$827$	−17.2762	−0.600753	−0.300376	−	0.953821i	$-0.597112\pi$
$827$	−17.2762	−0.600753	−0.300376	+	0.953821i	$0.597112\pi$
$828$	−13.5456	−0.470740
$829$	−2.91490	−0.101239	−0.0506194	−	0.998718i	$-0.516120\pi$
$829$	−2.91490	−0.101239	−0.0506194	+	0.998718i	$0.516120\pi$
$830$	−0.0573957	−0.00199223
$831$	−20.4862	−0.710657
$832$	−10.0492	−0.348394
$833$	−18.5825	−0.643846
$834$	15.6244	0.541027
$835$	0.00861253	0.000298049 0
$836$	0	0
$837$	−28.5373	−0.986394
$838$	19.8690	0.686362
$839$	−34.1630	−1.17944	−0.589718	−	0.807609i	$-0.700761\pi$
$839$	−34.1630	−1.17944	−0.589718	+	0.807609i	$0.700761\pi$
$840$	0.0478492	0.00165095
$841$	−26.2196	−0.904123
$842$	1.69297	0.0583436
$843$	−2.14797	−0.0739800
$844$	−3.45929	−0.119074
$845$	−0.0338802	−0.00116551
$846$	−12.6233	−0.433999
$847$	0	0
$848$	−2.82324	−0.0969504
$849$	−22.4960	−0.772061
$850$	9.43353	0.323568
$851$	−43.3277	−1.48525
$852$	14.7521	0.505398
$853$	32.3920	1.10908	0.554540	−	0.832157i	$-0.312894\pi$
$853$	32.3920	1.10908	0.554540	+	0.832157i	$0.312894\pi$
$854$	3.19497	0.109330
$855$	−0.00640632	−0.000219092 0
$856$	−20.4252	−0.698118
$857$	−31.7484	−1.08450	−0.542252	−	0.840216i	$-0.682428\pi$
$857$	−31.7484	−1.08450	−0.542252	+	0.840216i	$0.682428\pi$
$858$	0	0
$859$	31.4991	1.07474	0.537368	−	0.843348i	$-0.319419\pi$
$859$	31.4991	1.07474	0.537368	+	0.843348i	$0.319419\pi$
$860$	−0.0493867	−0.00168407
$861$	2.10165	0.0716241
$862$	25.2356	0.859528
$863$	−2.01345	−0.0685387	−0.0342694	−	0.999413i	$-0.510910\pi$
$863$	−2.01345	−0.0685387	−0.0342694	+	0.999413i	$0.510910\pi$
$864$	26.5275	0.902482
$865$	−0.0178864	−0.000608155 0
$866$	−13.4089	−0.455651
$867$	10.2590	0.348414
$868$	32.0726	1.08862
$869$	0	0
$870$	0.00617755	0.000209438 0
$871$	−12.3674	−0.419054
$872$	42.1290	1.42667
$873$	39.8721	1.34946
$874$	−2.09211	−0.0707668
$875$	−0.200260	−0.00677002
$876$	−0.405591	−0.0137037
$877$	12.3368	0.416583	0.208291	−	0.978067i	$-0.433210\pi$
$877$	12.3368	0.416583	0.208291	+	0.978067i	$0.433210\pi$
$878$	23.7493	0.801501
$879$	−2.53068	−0.0853577
$880$	0	0
$881$	−41.2161	−1.38861	−0.694303	−	0.719682i	$-0.744287\pi$
$881$	−41.2161	−1.38861	−0.694303	+	0.719682i	$0.744287\pi$
$882$	−14.8873	−0.501283
$883$	36.9722	1.24421	0.622107	−	0.782932i	$-0.286277\pi$
$883$	36.9722	1.24421	0.622107	+	0.782932i	$0.286277\pi$
$884$	−7.70091	−0.259010
$885$	0.0214014	0.000719400 0
$886$	−0.132071	−0.00443700
$887$	−55.1163	−1.85062	−0.925312	−	0.379208i	$-0.876197\pi$
$887$	−55.1163	−1.85062	−0.925312	+	0.379208i	$0.876197\pi$
$888$	22.6216	0.759131
$889$	67.4144	2.26100
$890$	0.0602649	0.00202009
$891$	0	0
$892$	34.9395	1.16986
$893$	3.81785	0.127759
$894$	−11.1447	−0.372736
$895$	−0.0498994	−0.00166795
$896$	−32.3734	−1.08152
$897$	−10.1421	−0.338634
$898$	−19.2184	−0.641326
$899$	10.3960	0.346726
$900$	−14.7994	−0.493312
$901$	16.1724	0.538781
$902$	0	0
$903$	−24.5875	−0.818221
$904$	−54.3862	−1.80886
$905$	0.00896624	0.000298048 0
$906$	−4.61750	−0.153406
$907$	28.0680	0.931982	0.465991	−	0.884789i	$-0.345698\pi$
$907$	28.0680	0.931982	0.465991	+	0.884789i	$0.345698\pi$
$908$	−31.0019	−1.02883
$909$	−12.2621	−0.406710
$910$	−0.0417421	−0.00138374
$911$	−16.8050	−0.556774	−0.278387	−	0.960469i	$-0.589800\pi$
$911$	−16.8050	−0.556774	−0.278387	+	0.960469i	$0.589800\pi$
$912$	−0.194701	−0.00644721
$913$	0	0
$914$	−20.7831	−0.687445
$915$	−0.00450566	−0.000148952 0
$916$	−20.4013	−0.674079
$917$	−23.0337	−0.760640
$918$	−8.63596	−0.285029
$919$	0.883904	0.0291573	0.0145786	−	0.999894i	$-0.495359\pi$
$919$	0.883904	0.0291573	0.0145786	+	0.999894i	$0.495359\pi$
$920$	0.0644618	0.00212524
$921$	−19.8013	−0.652474
$922$	−13.8758	−0.456976
$923$	−32.3105	−1.06351
$924$	0	0
$925$	−47.3382	−1.55647
$926$	−30.0451	−0.987342
$927$	−15.5619	−0.511119
$928$	−9.66381	−0.317230
$929$	−51.3018	−1.68316	−0.841578	−	0.540135i	$-0.818373\pi$
$929$	−51.3018	−1.68316	−0.841578	+	0.540135i	$0.818373\pi$
$930$	0.0230977	0.000757404 0
$931$	4.50259	0.147566
$932$	−7.69913	−0.252193
$933$	−17.9974	−0.589209
$934$	12.7516	0.417246
$935$	0	0
$936$	−15.4898	−0.506299
$937$	19.7965	0.646722	0.323361	−	0.946276i	$-0.395187\pi$
$937$	19.7965	0.646722	0.323361	+	0.946276i	$0.395187\pi$
$938$	15.5872	0.508940
$939$	−9.56304	−0.312078
$940$	−0.0468539	−0.00152820
$941$	8.32360	0.271342	0.135671	−	0.990754i	$-0.456681\pi$
$941$	8.32360	0.271342	0.135671	+	0.990754i	$0.456681\pi$
$942$	−9.67383	−0.315191
$943$	2.83131	0.0922003
$944$	−1.90266	−0.0619264
$945$	0.0916641	0.00298183
$946$	0	0
$947$	19.3318	0.628198	0.314099	−	0.949390i	$-0.398298\pi$
$947$	19.3318	0.628198	0.314099	+	0.949390i	$0.398298\pi$
$948$	−11.6552	−0.378544
$949$	0.888339	0.0288367
$950$	−2.28577	−0.0741601
$951$	−22.6383	−0.734097
$952$	24.3681	0.789775
$953$	32.3978	1.04947	0.524734	−	0.851266i	$-0.324165\pi$
$953$	32.3978	1.04947	0.524734	+	0.851266i	$0.324165\pi$
$954$	12.9565	0.419482
$955$	−0.0209836	−0.000679015 0
$956$	−17.8634	−0.577743
$957$	0	0
$958$	6.52722	0.210885
$959$	54.1266	1.74784
$960$	−0.0178613	−0.000576470 0
$961$	7.87038	0.253883
$962$	−19.7344	−0.636261
$963$	−16.7082	−0.538416
$964$	−24.2074	−0.779668
$965$	0.0537271	0.00172954
$966$	12.7825	0.411271
$967$	−33.6628	−1.08252	−0.541262	−	0.840854i	$-0.682053\pi$
$967$	−33.6628	−1.08252	−0.541262	+	0.840854i	$0.682053\pi$
$968$	0	0
$969$	1.11531	0.0358290
$970$	−0.0755760	−0.00242660
$971$	−27.8036	−0.892260	−0.446130	−	0.894968i	$-0.647198\pi$
$971$	−27.8036	−0.892260	−0.446130	+	0.894968i	$0.647198\pi$
$972$	21.3110	0.683552
$973$	−84.4575	−2.70758
$974$	−11.1763	−0.358110
$975$	−11.0809	−0.354871
$976$	0.400570	0.0128219
$977$	46.7126	1.49447	0.747235	−	0.664560i	$-0.231381\pi$
$977$	46.7126	1.49447	0.747235	+	0.664560i	$0.231381\pi$
$978$	−4.51286	−0.144306
$979$	0	0
$980$	−0.0552572	−0.00176513
$981$	34.4624	1.10030
$982$	−0.617379	−0.0197014
$983$	−44.5187	−1.41993	−0.709963	−	0.704239i	$-0.751288\pi$
$983$	−44.5187	−1.41993	−0.709963	+	0.704239i	$0.751288\pi$
$984$	−1.47824	−0.0471247
$985$	−0.0468463	−0.00149265
$986$	3.14603	0.100190
$987$	−23.3265	−0.742491
$988$	1.86595	0.0593637
$989$	−33.1240	−1.05328
$990$	0	0
$991$	49.8180	1.58252	0.791261	−	0.611478i	$-0.209425\pi$
$991$	49.8180	1.58252	0.791261	+	0.611478i	$0.209425\pi$
$992$	−36.1328	−1.14722
$993$	1.28726	0.0408498
$994$	40.7223	1.29163
$995$	−0.0154541	−0.000489928 0
$996$	−15.6760	−0.496713
$997$	−48.7178	−1.54291	−0.771455	−	0.636284i	$-0.780471\pi$
$997$	−48.7178	−1.54291	−0.771455	+	0.636284i	$0.780471\pi$
$998$	−2.39241	−0.0757303
$999$	43.3359	1.37109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.r.1.30	yes	50
11.10	odd	2		7381.2.a.q.1.21	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7381.2.a.q.1.21	✓	50	11.10	odd	2
7381.2.a.r.1.30	yes	50	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:	\( N \)	\(=\)	\( 7381 = 11^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7381.a (trivial)

\(f(q)\)	\(=\)	\(q+0.822245 q^{2} -0.874235 q^{3} -1.32391 q^{4} +0.00515383 q^{5} -0.718836 q^{6} +3.88567 q^{7} -2.73307 q^{8} -2.23571 q^{9} +O(q^{10})\)	\(q+0.822245 q^{2} -0.874235 q^{3} -1.32391 q^{4} +0.00515383 q^{5} -0.718836 q^{6} +3.88567 q^{7} -2.73307 q^{8} -2.23571 q^{9} +0.00423771 q^{10} +1.15741 q^{12} -2.53500 q^{13} +3.19497 q^{14} -0.00450566 q^{15} +0.400570 q^{16} -2.29459 q^{17} -1.83830 q^{18} +0.555985 q^{19} -0.00682322 q^{20} -3.39698 q^{21} -4.57637 q^{23} +2.38935 q^{24} -4.99997 q^{25} -2.08439 q^{26} +4.57724 q^{27} -5.14428 q^{28} -1.66746 q^{29} -0.00370475 q^{30} -6.23461 q^{31} +5.79551 q^{32} -1.88672 q^{34} +0.0200261 q^{35} +2.95989 q^{36} +9.46770 q^{37} +0.457156 q^{38} +2.21618 q^{39} -0.0140858 q^{40} -0.618681 q^{41} -2.79316 q^{42} +7.23804 q^{43} -0.0115225 q^{45} -3.76290 q^{46} +6.86683 q^{47} -0.350192 q^{48} +8.09840 q^{49} -4.11120 q^{50} +2.00601 q^{51} +3.35611 q^{52} -7.04806 q^{53} +3.76362 q^{54} -10.6198 q^{56} -0.486061 q^{57} -1.37107 q^{58} -4.74989 q^{59} +0.00596509 q^{60} +1.00000 q^{61} -5.12638 q^{62} -8.68724 q^{63} +3.96419 q^{64} -0.0130649 q^{65} +4.87867 q^{67} +3.03784 q^{68} +4.00082 q^{69} +0.0164663 q^{70} +12.7458 q^{71} +6.11037 q^{72} -0.350430 q^{73} +7.78477 q^{74} +4.37115 q^{75} -0.736075 q^{76} +1.82225 q^{78} -10.0701 q^{79} +0.00206447 q^{80} +2.70556 q^{81} -0.508708 q^{82} -13.5440 q^{83} +4.49731 q^{84} -0.0118259 q^{85} +5.95144 q^{86} +1.45776 q^{87} +14.2211 q^{89} -0.00947431 q^{90} -9.85015 q^{91} +6.05872 q^{92} +5.45051 q^{93} +5.64622 q^{94} +0.00286545 q^{95} -5.06664 q^{96} -17.8342 q^{97} +6.65887 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9} + 12 q^{10} - 14 q^{12} + 8 q^{13} - 2 q^{14} - 16 q^{15} + 42 q^{16} + 22 q^{17} + 32 q^{19} - 8 q^{20} + 24 q^{21} + 12 q^{23} - 8 q^{24} + 40 q^{25} + 10 q^{26} - 8 q^{27} + 72 q^{28} + 56 q^{29} + 24 q^{30} + 10 q^{31} + 70 q^{32} - 32 q^{34} + 70 q^{35} + 34 q^{36} - 8 q^{37} - 14 q^{38} + 96 q^{39} - 54 q^{40} + 56 q^{41} - 8 q^{42} + 44 q^{43} - 24 q^{45} - 4 q^{46} - 4 q^{47} - 28 q^{48} + 38 q^{49} + 120 q^{50} + 76 q^{51} + 24 q^{52} + 4 q^{53} + 48 q^{54} - 18 q^{56} + 8 q^{57} + 28 q^{58} + 12 q^{59} - 60 q^{60} + 50 q^{61} + 8 q^{62} + 30 q^{63} + 10 q^{64} + 64 q^{65} + 18 q^{67} - 22 q^{68} - 8 q^{69} + 34 q^{70} + 12 q^{71} + 104 q^{72} - 16 q^{73} + 84 q^{74} - 26 q^{75} + 64 q^{76} + 40 q^{78} + 78 q^{79} - 36 q^{80} + 34 q^{81} + 54 q^{82} + 68 q^{83} - 78 q^{84} - 4 q^{85} + 36 q^{86} + 48 q^{87} + 26 q^{89} - 20 q^{90} + 32 q^{92} + 22 q^{93} + 156 q^{94} + 100 q^{95} - 4 q^{96} - 14 q^{97} + 70 q^{98}+O(q^{100}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9 + 12 * q^10 - 14 * q^12 + 8 * q^13 - 2 * q^14 - 16 * q^15 + 42 * q^16 + 22 * q^17 + 32 * q^19 - 8 * q^20 + 24 * q^21 + 12 * q^23 - 8 * q^24 + 40 * q^25 + 10 * q^26 - 8 * q^27 + 72 * q^28 + 56 * q^29 + 24 * q^30 + 10 * q^31 + 70 * q^32 - 32 * q^34 + 70 * q^35 + 34 * q^36 - 8 * q^37 - 14 * q^38 + 96 * q^39 - 54 * q^40 + 56 * q^41 - 8 * q^42 + 44 * q^43 - 24 * q^45 - 4 * q^46 - 4 * q^47 - 28 * q^48 + 38 * q^49 + 120 * q^50 + 76 * q^51 + 24 * q^52 + 4 * q^53 + 48 * q^54 - 18 * q^56 + 8 * q^57 + 28 * q^58 + 12 * q^59 - 60 * q^60 + 50 * q^61 + 8 * q^62 + 30 * q^63 + 10 * q^64 + 64 * q^65 + 18 * q^67 - 22 * q^68 - 8 * q^69 + 34 * q^70 + 12 * q^71 + 104 * q^72 - 16 * q^73 + 84 * q^74 - 26 * q^75 + 64 * q^76 + 40 * q^78 + 78 * q^79 - 36 * q^80 + 34 * q^81 + 54 * q^82 + 68 * q^83 - 78 * q^84 - 4 * q^85 + 36 * q^86 + 48 * q^87 + 26 * q^89 - 20 * q^90 + 32 * q^92 + 22 * q^93 + 156 * q^94 + 100 * q^95 - 4 * q^96 - 14 * q^97 + 70 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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