LMFDB - Embedded newform 619.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [619,2,Mod(1,619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(619, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	619.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:	$4.94273988512$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61841 q^{2} +0.252107 q^{3} +4.85606 q^{4} +3.30541 q^{5} -0.660120 q^{6} -1.70947 q^{7} -7.47834 q^{8} -2.93644 q^{9} +O(q^{10})$	\(q-2.61841 q^{2} +0.252107 q^{3} +4.85606 q^{4} +3.30541 q^{5} -0.660120 q^{6} -1.70947 q^{7} -7.47834 q^{8} -2.93644 q^{9} -8.65491 q^{10} +5.06820 q^{11} +1.22425 q^{12} +5.40015 q^{13} +4.47610 q^{14} +0.833318 q^{15} +9.86923 q^{16} -0.784551 q^{17} +7.68880 q^{18} +1.17656 q^{19} +16.0513 q^{20} -0.430971 q^{21} -13.2706 q^{22} -1.79832 q^{23} -1.88535 q^{24} +5.92572 q^{25} -14.1398 q^{26} -1.49662 q^{27} -8.30131 q^{28} +7.10342 q^{29} -2.18197 q^{30} -9.66396 q^{31} -10.8850 q^{32} +1.27773 q^{33} +2.05428 q^{34} -5.65050 q^{35} -14.2595 q^{36} -3.34566 q^{37} -3.08072 q^{38} +1.36142 q^{39} -24.7190 q^{40} +8.63584 q^{41} +1.12846 q^{42} +1.57817 q^{43} +24.6115 q^{44} -9.70614 q^{45} +4.70874 q^{46} +4.16147 q^{47} +2.48811 q^{48} -4.07771 q^{49} -15.5160 q^{50} -0.197791 q^{51} +26.2235 q^{52} -1.42628 q^{53} +3.91877 q^{54} +16.7525 q^{55} +12.7840 q^{56} +0.296620 q^{57} -18.5997 q^{58} +9.64463 q^{59} +4.04665 q^{60} +13.8920 q^{61} +25.3042 q^{62} +5.01976 q^{63} +8.76290 q^{64} +17.8497 q^{65} -3.34562 q^{66} -10.7620 q^{67} -3.80983 q^{68} -0.453370 q^{69} +14.7953 q^{70} -0.285596 q^{71} +21.9597 q^{72} -16.4070 q^{73} +8.76030 q^{74} +1.49392 q^{75} +5.71346 q^{76} -8.66394 q^{77} -3.56475 q^{78} +7.56950 q^{79} +32.6218 q^{80} +8.43202 q^{81} -22.6122 q^{82} -2.48547 q^{83} -2.09282 q^{84} -2.59326 q^{85} -4.13228 q^{86} +1.79082 q^{87} -37.9017 q^{88} +10.8686 q^{89} +25.4146 q^{90} -9.23141 q^{91} -8.73276 q^{92} -2.43636 q^{93} -10.8964 q^{94} +3.88901 q^{95} -2.74419 q^{96} -0.0677252 q^{97} +10.6771 q^{98} -14.8825 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{2} + q^{3} + 33 q^{4} + 21 q^{5} + 6 q^{6} + 2 q^{7} + 27 q^{8} + 43 q^{9}+O(q^{10}) $ 30 * q + 9 * q^2 + q^3 + 33 * q^4 + 21 * q^5 + 6 * q^6 + 2 * q^7 + 27 * q^8 + 43 * q^9	\( 30 q + 9 q^{2} + q^{3} + 33 q^{4} + 21 q^{5} + 6 q^{6} + 2 q^{7} + 27 q^{8} + 43 q^{9} + 5 q^{10} + 23 q^{11} - 6 q^{12} + 9 q^{13} + 7 q^{14} - 2 q^{15} + 35 q^{16} + 4 q^{17} + 10 q^{18} - q^{19} + 29 q^{20} + 30 q^{21} + 4 q^{23} + 4 q^{24} + 35 q^{25} + q^{26} - 5 q^{27} - 13 q^{28} + 90 q^{29} - 31 q^{30} + 2 q^{31} + 43 q^{32} - 6 q^{33} - 9 q^{34} + 9 q^{35} + 33 q^{36} + 19 q^{37} + 5 q^{38} + 32 q^{39} - 12 q^{40} + 59 q^{41} - 25 q^{42} - 4 q^{43} + 52 q^{44} + 30 q^{45} - q^{46} + 4 q^{47} - 44 q^{48} + 30 q^{49} + 31 q^{50} - 12 q^{52} + 34 q^{53} - 28 q^{54} - 17 q^{55} + 2 q^{56} - 8 q^{57} + 6 q^{58} + 13 q^{59} - 64 q^{60} + 16 q^{61} + 28 q^{62} - 40 q^{63} + 37 q^{64} + 31 q^{65} - 59 q^{66} - 11 q^{67} - 52 q^{68} + 6 q^{69} - 40 q^{70} + 42 q^{71} + 6 q^{72} - 4 q^{73} + 16 q^{74} - 52 q^{75} - 42 q^{76} + 29 q^{77} - 56 q^{78} + 3 q^{79} + 21 q^{80} + 30 q^{81} - 43 q^{82} - 11 q^{83} - 36 q^{84} + 19 q^{85} - 11 q^{86} - 20 q^{87} - 47 q^{88} + 58 q^{89} - 33 q^{90} - 39 q^{91} - 7 q^{92} - 15 q^{93} - 46 q^{94} + 23 q^{95} - 70 q^{96} - 9 q^{97} - 8 q^{99}+O(q^{100}) \) 30 * q + 9 * q^2 + q^3 + 33 * q^4 + 21 * q^5 + 6 * q^6 + 2 * q^7 + 27 * q^8 + 43 * q^9 + 5 * q^10 + 23 * q^11 - 6 * q^12 + 9 * q^13 + 7 * q^14 - 2 * q^15 + 35 * q^16 + 4 * q^17 + 10 * q^18 - q^19 + 29 * q^20 + 30 * q^21 + 4 * q^23 + 4 * q^24 + 35 * q^25 + q^26 - 5 * q^27 - 13 * q^28 + 90 * q^29 - 31 * q^30 + 2 * q^31 + 43 * q^32 - 6 * q^33 - 9 * q^34 + 9 * q^35 + 33 * q^36 + 19 * q^37 + 5 * q^38 + 32 * q^39 - 12 * q^40 + 59 * q^41 - 25 * q^42 - 4 * q^43 + 52 * q^44 + 30 * q^45 - q^46 + 4 * q^47 - 44 * q^48 + 30 * q^49 + 31 * q^50 - 12 * q^52 + 34 * q^53 - 28 * q^54 - 17 * q^55 + 2 * q^56 - 8 * q^57 + 6 * q^58 + 13 * q^59 - 64 * q^60 + 16 * q^61 + 28 * q^62 - 40 * q^63 + 37 * q^64 + 31 * q^65 - 59 * q^66 - 11 * q^67 - 52 * q^68 + 6 * q^69 - 40 * q^70 + 42 * q^71 + 6 * q^72 - 4 * q^73 + 16 * q^74 - 52 * q^75 - 42 * q^76 + 29 * q^77 - 56 * q^78 + 3 * q^79 + 21 * q^80 + 30 * q^81 - 43 * q^82 - 11 * q^83 - 36 * q^84 + 19 * q^85 - 11 * q^86 - 20 * q^87 - 47 * q^88 + 58 * q^89 - 33 * q^90 - 39 * q^91 - 7 * q^92 - 15 * q^93 - 46 * q^94 + 23 * q^95 - 70 * q^96 - 9 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.61841	−1.85149	−0.925747	−	0.378143i	$-0.876563\pi$
$2$	−2.61841	−1.85149	−0.925747	+	0.378143i	$0.876563\pi$
$3$	0.252107	0.145554	0.0727772	−	0.997348i	$-0.476814\pi$
$3$	0.252107	0.145554	0.0727772	+	0.997348i	$0.476814\pi$
$4$	4.85606	2.42803
$5$	3.30541	1.47822	0.739112	−	0.673583i	$-0.235246\pi$
$5$	3.30541	1.47822	0.739112	+	0.673583i	$0.235246\pi$
$6$	−0.660120	−0.269493
$7$	−1.70947	−0.646120	−0.323060	−	0.946379i	$-0.604712\pi$
$7$	−1.70947	−0.646120	−0.323060	+	0.946379i	$0.604712\pi$
$8$	−7.47834	−2.64399
$9$	−2.93644	−0.978814
$10$	−8.65491	−2.73692
$11$	5.06820	1.52812	0.764059	−	0.645146i	$-0.223203\pi$
$11$	5.06820	1.52812	0.764059	+	0.645146i	$0.223203\pi$
$12$	1.22425	0.353411
$13$	5.40015	1.49773	0.748867	−	0.662721i	$-0.230598\pi$
$13$	5.40015	1.49773	0.748867	+	0.662721i	$0.230598\pi$
$14$	4.47610	1.19629
$15$	0.833318	0.215162
$16$	9.86923	2.46731
$17$	−0.784551	−0.190282	−0.0951408	−	0.995464i	$-0.530330\pi$
$17$	−0.784551	−0.190282	−0.0951408	+	0.995464i	$0.530330\pi$
$18$	7.68880	1.81227
$19$	1.17656	0.269922	0.134961	−	0.990851i	$-0.456909\pi$
$19$	1.17656	0.269922	0.134961	+	0.990851i	$0.456909\pi$
$20$	16.0513	3.58917
$21$	−0.430971	−0.0940455
$22$	−13.2706	−2.82930
$23$	−1.79832	−0.374976	−0.187488	−	0.982267i	$-0.560035\pi$
$23$	−1.79832	−0.374976	−0.187488	+	0.982267i	$0.560035\pi$
$24$	−1.88535	−0.384845
$25$	5.92572	1.18514
$26$	−14.1398	−2.77305
$27$	−1.49662	−0.288025
$28$	−8.30131	−1.56880
$29$	7.10342	1.31907	0.659536	−	0.751673i	$-0.270753\pi$
$29$	7.10342	1.31907	0.659536	+	0.751673i	$0.270753\pi$
$30$	−2.18197	−0.398371
$31$	−9.66396	−1.73570	−0.867849	−	0.496828i	$-0.834498\pi$
$31$	−9.66396	−1.73570	−0.867849	+	0.496828i	$0.834498\pi$
$32$	−10.8850	−1.92421
$33$	1.27773	0.222424
$34$	2.05428	0.352305
$35$	−5.65050	−0.955109
$36$	−14.2595	−2.37659
$37$	−3.34566	−0.550023	−0.275011	−	0.961441i	$-0.588682\pi$
$37$	−3.34566	−0.550023	−0.275011	+	0.961441i	$0.588682\pi$
$38$	−3.08072	−0.499758
$39$	1.36142	0.218002
$40$	−24.7190	−3.90841
$41$	8.63584	1.34869	0.674346	−	0.738416i	$-0.264426\pi$
$41$	8.63584	1.34869	0.674346	+	0.738416i	$0.264426\pi$
$42$	1.12846	0.174125
$43$	1.57817	0.240668	0.120334	−	0.992733i	$-0.461603\pi$
$43$	1.57817	0.240668	0.120334	+	0.992733i	$0.461603\pi$
$44$	24.6115	3.71032
$45$	−9.70614	−1.44691
$46$	4.70874	0.694266
$47$	4.16147	0.607013	0.303506	−	0.952829i	$-0.401843\pi$
$47$	4.16147	0.607013	0.303506	+	0.952829i	$0.401843\pi$
$48$	2.48811	0.359127
$49$	−4.07771	−0.582529
$50$	−15.5160	−2.19429
$51$	−0.197791	−0.0276963
$52$	26.2235	3.63655
$53$	−1.42628	−0.195914	−0.0979572	−	0.995191i	$-0.531231\pi$
$53$	−1.42628	−0.195914	−0.0979572	+	0.995191i	$0.531231\pi$
$54$	3.91877	0.533277
$55$	16.7525	2.25890
$56$	12.7840	1.70834
$57$	0.296620	0.0392883
$58$	−18.5997	−2.44225
$59$	9.64463	1.25562	0.627812	−	0.778365i	$-0.283951\pi$
$59$	9.64463	1.25562	0.627812	+	0.778365i	$0.283951\pi$
$60$	4.04665	0.522420
$61$	13.8920	1.77869	0.889345	−	0.457237i	$-0.151161\pi$
$61$	13.8920	1.77869	0.889345	+	0.457237i	$0.151161\pi$
$62$	25.3042	3.21364
$63$	5.01976	0.632431
$64$	8.76290	1.09536
$65$	17.8497	2.21398
$66$	−3.34562	−0.411817
$67$	−10.7620	−1.31479	−0.657394	−	0.753547i	$-0.728341\pi$
$67$	−10.7620	−1.31479	−0.657394	+	0.753547i	$0.728341\pi$
$68$	−3.80983	−0.462010
$69$	−0.453370	−0.0545794
$70$	14.7953	1.76838
$71$	−0.285596	−0.0338940	−0.0169470	−	0.999856i	$-0.505395\pi$
$71$	−0.285596	−0.0338940	−0.0169470	+	0.999856i	$0.505395\pi$
$72$	21.9597	2.58798
$73$	−16.4070	−1.92030	−0.960149	−	0.279487i	$-0.909836\pi$
$73$	−16.4070	−1.92030	−0.960149	+	0.279487i	$0.909836\pi$
$74$	8.76030	1.01836
$75$	1.49392	0.172503
$76$	5.71346	0.655378
$77$	−8.66394	−0.987348
$78$	−3.56475	−0.403629
$79$	7.56950	0.851635	0.425818	−	0.904809i	$-0.359986\pi$
$79$	7.56950	0.851635	0.425818	+	0.904809i	$0.359986\pi$
$80$	32.6218	3.64723
$81$	8.43202	0.936891
$82$	−22.6122	−2.49710
$83$	−2.48547	−0.272816	−0.136408	−	0.990653i	$-0.543556\pi$
$83$	−2.48547	−0.272816	−0.136408	+	0.990653i	$0.543556\pi$
$84$	−2.09282	−0.228346
$85$	−2.59326	−0.281279
$86$	−4.13228	−0.445596
$87$	1.79082	0.191997
$88$	−37.9017	−4.04034
$89$	10.8686	1.15207	0.576033	−	0.817426i	$-0.304600\pi$
$89$	10.8686	1.15207	0.576033	+	0.817426i	$0.304600\pi$
$90$	25.4146	2.67894
$91$	−9.23141	−0.967715
$92$	−8.73276	−0.910453
$93$	−2.43636	−0.252638
$94$	−10.8964	−1.12388
$95$	3.88901	0.399004
$96$	−2.74419	−0.280078
$97$	−0.0677252	−0.00687645	−0.00343823	−	0.999994i	$-0.501094\pi$
$97$	−0.0677252	−0.00687645	−0.00343823	+	0.999994i	$0.501094\pi$
$98$	10.6771	1.07855
$99$	−14.8825	−1.49574
$100$	28.7757	2.87757
$101$	−10.5228	−1.04706	−0.523529	−	0.852008i	$-0.675385\pi$
$101$	−10.5228	−1.04706	−0.523529	+	0.852008i	$0.675385\pi$
$102$	0.517898	0.0512796
$103$	18.0538	1.77889	0.889445	−	0.457042i	$-0.151091\pi$
$103$	18.0538	1.77889	0.889445	+	0.457042i	$0.151091\pi$
$104$	−40.3842	−3.96000
$105$	−1.42453	−0.139020
$106$	3.73458	0.362734
$107$	5.71828	0.552807	0.276403	−	0.961042i	$-0.410857\pi$
$107$	5.71828	0.552807	0.276403	+	0.961042i	$0.410857\pi$
$108$	−7.26769	−0.699334
$109$	9.50553	0.910465	0.455233	−	0.890373i	$-0.349556\pi$
$109$	9.50553	0.910465	0.455233	+	0.890373i	$0.349556\pi$
$110$	−43.8648	−4.18234
$111$	−0.843465	−0.0800582
$112$	−16.8712	−1.59418
$113$	14.3872	1.35343	0.676715	−	0.736245i	$-0.263403\pi$
$113$	14.3872	1.35343	0.676715	+	0.736245i	$0.263403\pi$
$114$	−0.776672	−0.0727420
$115$	−5.94419	−0.554298
$116$	34.4947	3.20275
$117$	−15.8572	−1.46600
$118$	−25.2536	−2.32478
$119$	1.34117	0.122945
$120$	−6.23184	−0.568886
$121$	14.6866	1.33515
$122$	−36.3750	−3.29323
$123$	2.17716	0.196308
$124$	−46.9288	−4.21433
$125$	3.05989	0.273685
$126$	−13.1438	−1.17094
$127$	−9.34245	−0.829008	−0.414504	−	0.910047i	$-0.636045\pi$
$127$	−9.34245	−0.829008	−0.414504	+	0.910047i	$0.636045\pi$
$128$	−1.17487	−0.103844
$129$	0.397867	0.0350303
$130$	−46.7379	−4.09918
$131$	−0.740956	−0.0647377	−0.0323688	−	0.999476i	$-0.510305\pi$
$131$	−0.740956	−0.0647377	−0.0323688	+	0.999476i	$0.510305\pi$
$132$	6.20474	0.540053
$133$	−2.01130	−0.174402
$134$	28.1793	2.43432
$135$	−4.94694	−0.425765
$136$	5.86714	0.503103
$137$	−2.52142	−0.215419	−0.107710	−	0.994182i	$-0.534352\pi$
$137$	−2.52142	−0.215419	−0.107710	+	0.994182i	$0.534352\pi$
$138$	1.18711	0.101053
$139$	−0.597484	−0.0506780	−0.0253390	−	0.999679i	$-0.508067\pi$
$139$	−0.597484	−0.0506780	−0.0253390	+	0.999679i	$0.508067\pi$
$140$	−27.4392	−2.31904
$141$	1.04914	0.0883533
$142$	0.747807	0.0627546
$143$	27.3690	2.28871
$144$	−28.9804	−2.41504
$145$	23.4797	1.94988
$146$	42.9603	3.55542
$147$	−1.02802	−0.0847897
$148$	−16.2467	−1.33547
$149$	17.6301	1.44432	0.722159	−	0.691727i	$-0.243150\pi$
$149$	17.6301	1.44432	0.722159	+	0.691727i	$0.243150\pi$
$150$	−3.91169	−0.319388
$151$	−6.88670	−0.560432	−0.280216	−	0.959937i	$-0.590406\pi$
$151$	−6.88670	−0.560432	−0.280216	+	0.959937i	$0.590406\pi$
$152$	−8.79873	−0.713671
$153$	2.30379	0.186250
$154$	22.6857	1.82807
$155$	−31.9433	−2.56575
$156$	6.61114	0.529315
$157$	5.22375	0.416900	0.208450	−	0.978033i	$-0.433158\pi$
$157$	5.22375	0.416900	0.208450	+	0.978033i	$0.433158\pi$
$158$	−19.8200	−1.57680
$159$	−0.359575	−0.0285162
$160$	−35.9793	−2.84442
$161$	3.07418	0.242279
$162$	−22.0785	−1.73465
$163$	2.57402	0.201613	0.100806	−	0.994906i	$-0.467858\pi$
$163$	2.57402	0.201613	0.100806	+	0.994906i	$0.467858\pi$
$164$	41.9362	3.27467
$165$	4.22342	0.328793
$166$	6.50798	0.505117
$167$	−22.4337	−1.73597	−0.867985	−	0.496591i	$-0.834585\pi$
$167$	−22.4337	−1.73597	−0.867985	+	0.496591i	$0.834585\pi$
$168$	3.22295	0.248656
$169$	16.1617	1.24321
$170$	6.79022	0.520786
$171$	−3.45490	−0.264203
$172$	7.66368	0.584350
$173$	−5.91288	−0.449548	−0.224774	−	0.974411i	$-0.572164\pi$
$173$	−5.91288	−0.449548	−0.224774	+	0.974411i	$0.572164\pi$
$174$	−4.68911	−0.355481
$175$	−10.1299	−0.765745
$176$	50.0192	3.77034
$177$	2.43148	0.182761
$178$	−28.4584	−2.13305
$179$	−10.2097	−0.763111	−0.381556	−	0.924346i	$-0.624612\pi$
$179$	−10.2097	−0.763111	−0.381556	+	0.924346i	$0.624612\pi$
$180$	−47.1336	−3.51313
$181$	−23.9368	−1.77921	−0.889603	−	0.456735i	$-0.849019\pi$
$181$	−23.9368	−1.77921	−0.889603	+	0.456735i	$0.849019\pi$
$182$	24.1716	1.79172
$183$	3.50228	0.258896
$184$	13.4485	0.991434
$185$	−11.0588	−0.813057
$186$	6.37938	0.467759
$187$	−3.97626	−0.290773
$188$	20.2084	1.47385
$189$	2.55843	0.186099
$190$	−10.1830	−0.738755
$191$	−20.4041	−1.47639	−0.738195	−	0.674587i	$-0.764322\pi$
$191$	−20.4041	−1.47639	−0.738195	+	0.674587i	$0.764322\pi$
$192$	2.20919	0.159435
$193$	−6.29806	−0.453345	−0.226672	−	0.973971i	$-0.572785\pi$
$193$	−6.29806	−0.453345	−0.226672	+	0.973971i	$0.572785\pi$
$194$	0.177332	0.0127317
$195$	4.50005	0.322255
$196$	−19.8016	−1.41440
$197$	3.45995	0.246511	0.123256	−	0.992375i	$-0.460666\pi$
$197$	3.45995	0.246511	0.123256	+	0.992375i	$0.460666\pi$
$198$	38.9684	2.76936
$199$	−5.57896	−0.395482	−0.197741	−	0.980254i	$-0.563361\pi$
$199$	−5.57896	−0.395482	−0.197741	+	0.980254i	$0.563361\pi$
$200$	−44.3146	−3.13351
$201$	−2.71318	−0.191373
$202$	27.5530	1.93862
$203$	−12.1431	−0.852278
$204$	−0.960487	−0.0672475
$205$	28.5450	1.99367
$206$	−47.2721	−3.29361
$207$	5.28067	0.367032
$208$	53.2954	3.69537
$209$	5.96304	0.412472
$210$	3.73001	0.257395
$211$	−19.2958	−1.32838	−0.664190	−	0.747564i	$-0.731224\pi$
$211$	−19.2958	−1.32838	−0.664190	+	0.747564i	$0.731224\pi$
$212$	−6.92610	−0.475686
$213$	−0.0720009	−0.00493342
$214$	−14.9728	−1.02352
$215$	5.21648	0.355761
$216$	11.1922	0.761536
$217$	16.5203	1.12147
$218$	−24.8894	−1.68572
$219$	−4.13634	−0.279508
$220$	81.3510	5.48468
$221$	−4.23670	−0.284991
$222$	2.20854	0.148227
$223$	−19.5492	−1.30911	−0.654555	−	0.756014i	$-0.727144\pi$
$223$	−19.5492	−1.30911	−0.654555	+	0.756014i	$0.727144\pi$
$224$	18.6076	1.24327
$225$	−17.4005	−1.16004
$226$	−37.6715	−2.50587
$227$	−10.8304	−0.718840	−0.359420	−	0.933176i	$-0.617025\pi$
$227$	−10.8304	−0.718840	−0.359420	+	0.933176i	$0.617025\pi$
$228$	1.44040	0.0953931
$229$	−9.28215	−0.613382	−0.306691	−	0.951809i	$-0.599222\pi$
$229$	−9.28215	−0.613382	−0.306691	+	0.951809i	$0.599222\pi$
$230$	15.5643	1.02628
$231$	−2.18424	−0.143713
$232$	−53.1218	−3.48762
$233$	8.22304	0.538709	0.269355	−	0.963041i	$-0.413190\pi$
$233$	8.22304	0.538709	0.269355	+	0.963041i	$0.413190\pi$
$234$	41.5207	2.71430
$235$	13.7554	0.897300
$236$	46.8349	3.04869
$237$	1.90833	0.123959
$238$	−3.51173	−0.227631
$239$	−6.85298	−0.443282	−0.221641	−	0.975128i	$-0.571141\pi$
$239$	−6.85298	−0.443282	−0.221641	+	0.975128i	$0.571141\pi$
$240$	8.22421	0.530870
$241$	17.0291	1.09694	0.548472	−	0.836169i	$-0.315210\pi$
$241$	17.0291	1.09694	0.548472	+	0.836169i	$0.315210\pi$
$242$	−38.4556	−2.47202
$243$	6.61564	0.424393
$244$	67.4605	4.31872
$245$	−13.4785	−0.861109
$246$	−5.70069	−0.363463
$247$	6.35361	0.404271
$248$	72.2704	4.58918
$249$	−0.626606	−0.0397095
$250$	−8.01205	−0.506727
$251$	−22.9201	−1.44670	−0.723351	−	0.690480i	$-0.757399\pi$
$251$	−22.9201	−1.44670	−0.723351	+	0.690480i	$0.757399\pi$
$252$	24.3763	1.53556
$253$	−9.11425	−0.573008
$254$	24.4624	1.53490
$255$	−0.653781	−0.0409413
$256$	−14.4495	−0.903095
$257$	5.29694	0.330414	0.165207	−	0.986259i	$-0.447171\pi$
$257$	5.29694	0.330414	0.165207	+	0.986259i	$0.447171\pi$
$258$	−1.04178	−0.0648584
$259$	5.71931	0.355381
$260$	86.6794	5.37563
$261$	−20.8588	−1.29113
$262$	1.94013	0.119861
$263$	−17.9004	−1.10379	−0.551893	−	0.833915i	$-0.686094\pi$
$263$	−17.9004	−1.10379	−0.551893	+	0.833915i	$0.686094\pi$
$264$	−9.55531	−0.588088
$265$	−4.71443	−0.289605
$266$	5.26640	0.322904
$267$	2.74005	0.167688
$268$	−52.2609	−3.19235
$269$	−8.53581	−0.520437	−0.260219	−	0.965550i	$-0.583795\pi$
$269$	−8.53581	−0.520437	−0.260219	+	0.965550i	$0.583795\pi$
$270$	12.9531	0.788302
$271$	23.7832	1.44473	0.722365	−	0.691512i	$-0.243055\pi$
$271$	23.7832	1.44473	0.722365	+	0.691512i	$0.243055\pi$
$272$	−7.74292	−0.469483
$273$	−2.32731	−0.140855
$274$	6.60210	0.398848
$275$	30.0327	1.81104
$276$	−2.20159	−0.132520
$277$	−27.8210	−1.67160	−0.835801	−	0.549032i	$-0.814996\pi$
$277$	−27.8210	−1.67160	−0.835801	+	0.549032i	$0.814996\pi$
$278$	1.56446	0.0938300
$279$	28.3777	1.69893
$280$	42.2564	2.52530
$281$	−0.382276	−0.0228047	−0.0114023	−	0.999935i	$-0.503630\pi$
$281$	−0.382276	−0.0228047	−0.0114023	+	0.999935i	$0.503630\pi$
$282$	−2.74707	−0.163586
$283$	22.2189	1.32078	0.660389	−	0.750924i	$-0.270391\pi$
$283$	22.2189	1.32078	0.660389	+	0.750924i	$0.270391\pi$
$284$	−1.38687	−0.0822957
$285$	0.980450	0.0580768
$286$	−71.6634	−4.23754
$287$	−14.7627	−0.871416
$288$	31.9631	1.88345
$289$	−16.3845	−0.963793
$290$	−61.4794	−3.61020
$291$	−0.0170740	−0.00100090
$292$	−79.6736	−4.66255
$293$	32.2860	1.88617	0.943084	−	0.332553i	$-0.107910\pi$
$293$	32.2860	1.88617	0.943084	+	0.332553i	$0.107910\pi$
$294$	2.69178	0.156988
$295$	31.8794	1.85609
$296$	25.0200	1.45426
$297$	−7.58517	−0.440136
$298$	−46.1629	−2.67415
$299$	−9.71121	−0.561614
$300$	7.25457	0.418843
$301$	−2.69783	−0.155500
$302$	18.0322	1.03764
$303$	−2.65288	−0.152404
$304$	11.6118	0.665980
$305$	45.9188	2.62930
$306$	−6.03226	−0.344841
$307$	15.0228	0.857395	0.428698	−	0.903448i	$-0.358973\pi$
$307$	15.0228	0.857395	0.428698	+	0.903448i	$0.358973\pi$
$308$	−42.0726	−2.39731
$309$	4.55149	0.258925
$310$	83.6407	4.75047
$311$	−27.8289	−1.57803	−0.789016	−	0.614372i	$-0.789409\pi$
$311$	−27.8289	−1.57803	−0.789016	+	0.614372i	$0.789409\pi$
$312$	−10.1812	−0.576395
$313$	7.98394	0.451279	0.225639	−	0.974211i	$-0.427553\pi$
$313$	7.98394	0.451279	0.225639	+	0.974211i	$0.427553\pi$
$314$	−13.6779	−0.771889
$315$	16.5924	0.934874
$316$	36.7580	2.06780
$317$	−8.67321	−0.487136	−0.243568	−	0.969884i	$-0.578318\pi$
$317$	−8.67321	−0.487136	−0.243568	+	0.969884i	$0.578318\pi$
$318$	0.941515	0.0527976
$319$	36.0015	2.01570
$320$	28.9650	1.61919
$321$	1.44162	0.0804634
$322$	−8.04946	−0.448579
$323$	−0.923072	−0.0513611
$324$	40.9464	2.27480
$325$	31.9998	1.77503
$326$	−6.73984	−0.373285
$327$	2.39642	0.132522
$328$	−64.5818	−3.56593
$329$	−7.11392	−0.392203
$330$	−11.0586	−0.608758
$331$	0.549307	0.0301926	0.0150963	−	0.999886i	$-0.495195\pi$
$331$	0.549307	0.0301926	0.0150963	+	0.999886i	$0.495195\pi$
$332$	−12.0696	−0.662405
$333$	9.82433	0.538370
$334$	58.7405	3.21414
$335$	−35.5728	−1.94355
$336$	−4.25335	−0.232039
$337$	23.2445	1.26621	0.633103	−	0.774067i	$-0.281781\pi$
$337$	23.2445	1.26621	0.633103	+	0.774067i	$0.281781\pi$
$338$	−42.3179	−2.30179
$339$	3.62711	0.196998
$340$	−12.5930	−0.682954
$341$	−48.9789	−2.65235
$342$	9.04635	0.489170
$343$	18.9370	1.02250
$344$	−11.8021	−0.636325
$345$	−1.49857	−0.0806805
$346$	15.4823	0.832336
$347$	−33.9095	−1.82036	−0.910179	−	0.414216i	$-0.864056\pi$
$347$	−33.9095	−1.82036	−0.910179	+	0.414216i	$0.864056\pi$
$348$	8.69636	0.466174
$349$	−9.16422	−0.490549	−0.245275	−	0.969454i	$-0.578878\pi$
$349$	−9.16422	−0.490549	−0.245275	+	0.969454i	$0.578878\pi$
$350$	26.5241	1.41777
$351$	−8.08199	−0.431385
$352$	−55.1673	−2.94043
$353$	−24.5982	−1.30923	−0.654615	−	0.755962i	$-0.727169\pi$
$353$	−24.5982	−1.30923	−0.654615	+	0.755962i	$0.727169\pi$
$354$	−6.36662	−0.338382
$355$	−0.944011	−0.0501029
$356$	52.7785	2.79725
$357$	0.338119	0.0178951
$358$	26.7333	1.41290
$359$	0.901238	0.0475655	0.0237828	−	0.999717i	$-0.492429\pi$
$359$	0.901238	0.0475655	0.0237828	+	0.999717i	$0.492429\pi$
$360$	72.5858	3.82561
$361$	−17.6157	−0.927142
$362$	62.6762	3.29419
$363$	3.70261	0.194336
$364$	−44.8283	−2.34964
$365$	−54.2320	−2.83863
$366$	−9.17040	−0.479345
$367$	−19.2141	−1.00297	−0.501483	−	0.865167i	$-0.667212\pi$
$367$	−19.2141	−1.00297	−0.501483	+	0.865167i	$0.667212\pi$
$368$	−17.7480	−0.925181
$369$	−25.3586	−1.32012
$370$	28.9564	1.50537
$371$	2.43818	0.126584
$372$	−11.8311	−0.613414
$373$	−26.7886	−1.38706	−0.693530	−	0.720428i	$-0.743946\pi$
$373$	−26.7886	−1.38706	−0.693530	+	0.720428i	$0.743946\pi$
$374$	10.4115	0.538365
$375$	0.771422	0.0398361
$376$	−31.1209	−1.60494
$377$	38.3596	1.97562
$378$	−6.69902	−0.344560
$379$	18.8658	0.969071	0.484535	−	0.874772i	$-0.338989\pi$
$379$	18.8658	0.969071	0.484535	+	0.874772i	$0.338989\pi$
$380$	18.8853	0.968796
$381$	−2.35530	−0.120666
$382$	53.4263	2.73353
$383$	23.4746	1.19949	0.599747	−	0.800190i	$-0.295268\pi$
$383$	23.4746	1.19949	0.599747	+	0.800190i	$0.295268\pi$
$384$	−0.296192	−0.0151150
$385$	−28.6379	−1.45952
$386$	16.4909	0.839365
$387$	−4.63419	−0.235569
$388$	−0.328878	−0.0166962
$389$	20.9012	1.05974	0.529868	−	0.848080i	$-0.322242\pi$
$389$	20.9012	1.05974	0.529868	+	0.848080i	$0.322242\pi$
$390$	−11.7830	−0.596653
$391$	1.41088	0.0713510
$392$	30.4945	1.54020
$393$	−0.186801	−0.00942285
$394$	−9.05956	−0.456414
$395$	25.0203	1.25891
$396$	−72.2702	−3.63171
$397$	15.7384	0.789888	0.394944	−	0.918705i	$-0.370764\pi$
$397$	15.7384	0.789888	0.394944	+	0.918705i	$0.370764\pi$
$398$	14.6080	0.732233
$399$	−0.507063	−0.0253849
$400$	58.4823	2.92412
$401$	−18.6777	−0.932719	−0.466359	−	0.884595i	$-0.654435\pi$
$401$	−18.6777	−0.932719	−0.466359	+	0.884595i	$0.654435\pi$
$402$	7.10421	0.354326
$403$	−52.1869	−2.59961
$404$	−51.0994	−2.54229
$405$	27.8713	1.38493
$406$	31.7956	1.57799
$407$	−16.9565	−0.840500
$408$	1.47915	0.0732289
$409$	−8.27923	−0.409382	−0.204691	−	0.978827i	$-0.565619\pi$
$409$	−8.27923	−0.409382	−0.204691	+	0.978827i	$0.565619\pi$
$410$	−74.7424	−3.69127
$411$	−0.635668	−0.0313552
$412$	87.6703	4.31920
$413$	−16.4872	−0.811283
$414$	−13.8269	−0.679557
$415$	−8.21549	−0.403283
$416$	−58.7806	−2.88196
$417$	−0.150630	−0.00737640
$418$	−15.6137	−0.763690
$419$	−27.2743	−1.33244	−0.666218	−	0.745757i	$-0.732088\pi$
$419$	−27.2743	−1.33244	−0.666218	+	0.745757i	$0.732088\pi$
$420$	−6.91763	−0.337546
$421$	20.6546	1.00664	0.503321	−	0.864100i	$-0.332111\pi$
$421$	20.6546	1.00664	0.503321	+	0.864100i	$0.332111\pi$
$422$	50.5244	2.45949
$423$	−12.2199	−0.594153
$424$	10.6662	0.517996
$425$	−4.64903	−0.225511
$426$	0.188528	0.00913420
$427$	−23.7480	−1.14925
$428$	27.7683	1.34223
$429$	6.89994	0.333132
$430$	−13.6589	−0.658690
$431$	5.21636	0.251263	0.125632	−	0.992077i	$-0.459904\pi$
$431$	5.21636	0.251263	0.125632	+	0.992077i	$0.459904\pi$
$432$	−14.7705	−0.710646
$433$	−8.58335	−0.412490	−0.206245	−	0.978500i	$-0.566124\pi$
$433$	−8.58335	−0.412490	−0.206245	+	0.978500i	$0.566124\pi$
$434$	−43.2568	−2.07639
$435$	5.91941	0.283814
$436$	46.1595	2.21064
$437$	−2.11583	−0.101214
$438$	10.8306	0.517507
$439$	16.4167	0.783529	0.391764	−	0.920066i	$-0.371865\pi$
$439$	16.4167	0.783529	0.391764	+	0.920066i	$0.371865\pi$
$440$	−125.281	−5.97252
$441$	11.9739	0.570188
$442$	11.0934	0.527660
$443$	−12.8410	−0.610092	−0.305046	−	0.952338i	$-0.598672\pi$
$443$	−12.8410	−0.610092	−0.305046	+	0.952338i	$0.598672\pi$
$444$	−4.09592	−0.194384
$445$	35.9251	1.70301
$446$	51.1877	2.42381
$447$	4.44469	0.210227
$448$	−14.9799	−0.707735
$449$	31.8372	1.50249	0.751246	−	0.660023i	$-0.229453\pi$
$449$	31.8372	1.50249	0.751246	+	0.660023i	$0.229453\pi$
$450$	45.5617	2.14780
$451$	43.7681	2.06096
$452$	69.8650	3.28617
$453$	−1.73619	−0.0815733
$454$	28.3585	1.33093
$455$	−30.5136	−1.43050
$456$	−2.21822	−0.103878
$457$	−2.29825	−0.107508	−0.0537538	−	0.998554i	$-0.517119\pi$
$457$	−2.29825	−0.107508	−0.0537538	+	0.998554i	$0.517119\pi$
$458$	24.3045	1.13567
$459$	1.17418	0.0548059
$460$	−28.8653	−1.34585
$461$	−18.8355	−0.877257	−0.438629	−	0.898668i	$-0.644536\pi$
$461$	−18.8355	−0.877257	−0.438629	+	0.898668i	$0.644536\pi$
$462$	5.71924	0.266083
$463$	−24.5974	−1.14314	−0.571569	−	0.820554i	$-0.693665\pi$
$463$	−24.5974	−1.14314	−0.571569	+	0.820554i	$0.693665\pi$
$464$	70.1053	3.25456
$465$	−8.05315	−0.373456
$466$	−21.5313	−0.997417
$467$	−27.4443	−1.26997	−0.634986	−	0.772523i	$-0.718994\pi$
$467$	−27.4443	−1.26997	−0.634986	+	0.772523i	$0.718994\pi$
$468$	−77.0038	−3.55950
$469$	18.3973	0.849510
$470$	−36.0171	−1.66135
$471$	1.31695	0.0606817
$472$	−72.1258	−3.31986
$473$	7.99846	0.367769
$474$	−4.99678	−0.229510
$475$	6.97197	0.319896
$476$	6.51280	0.298514
$477$	4.18818	0.191764
$478$	17.9439	0.820734
$479$	8.97630	0.410137	0.205069	−	0.978748i	$-0.434258\pi$
$479$	8.97630	0.410137	0.205069	+	0.978748i	$0.434258\pi$
$480$	−9.07066	−0.414017
$481$	−18.0671	−0.823788
$482$	−44.5893	−2.03099
$483$	0.775024	0.0352648
$484$	71.3192	3.24178
$485$	−0.223859	−0.0101649
$486$	−17.3224	−0.785762
$487$	−32.2972	−1.46352	−0.731762	−	0.681560i	$-0.761302\pi$
$487$	−32.2972	−1.46352	−0.731762	+	0.681560i	$0.761302\pi$
$488$	−103.889	−4.70284
$489$	0.648930	0.0293456
$490$	35.2922	1.59434
$491$	8.41411	0.379724	0.189862	−	0.981811i	$-0.439196\pi$
$491$	8.41411	0.379724	0.189862	+	0.981811i	$0.439196\pi$
$492$	10.5724	0.476642
$493$	−5.57300	−0.250995
$494$	−16.6364	−0.748505
$495$	−49.1926	−2.21104
$496$	−95.3759	−4.28250
$497$	0.488218	0.0218996
$498$	1.64071	0.0735219
$499$	−41.0243	−1.83650	−0.918250	−	0.396001i	$-0.870398\pi$
$499$	−41.0243	−1.83650	−0.918250	+	0.396001i	$0.870398\pi$
$500$	14.8590	0.664516
$501$	−5.65570	−0.252678
$502$	60.0141	2.67856
$503$	26.9003	1.19942	0.599712	−	0.800216i	$-0.295282\pi$
$503$	26.9003	1.19942	0.599712	+	0.800216i	$0.295282\pi$
$504$	−37.5395	−1.67214
$505$	−34.7822	−1.54779
$506$	23.8648	1.06092
$507$	4.07448	0.180954
$508$	−45.3675	−2.01286
$509$	9.93633	0.440420	0.220210	−	0.975453i	$-0.429326\pi$
$509$	9.93633	0.440420	0.220210	+	0.975453i	$0.429326\pi$
$510$	1.71187	0.0758027
$511$	28.0474	1.24074
$512$	40.1845	1.77592
$513$	−1.76087	−0.0777441
$514$	−13.8696	−0.611760
$515$	59.6751	2.62960
$516$	1.93207	0.0850546
$517$	21.0911	0.927588
$518$	−14.9755	−0.657985
$519$	−1.49068	−0.0654337
$520$	−133.486	−5.85376
$521$	−0.0576909	−0.00252749	−0.00126374	−	0.999999i	$-0.500402\pi$
$521$	−0.0576909	−0.00252749	−0.00126374	+	0.999999i	$0.500402\pi$
$522$	54.6168	2.39051
$523$	−2.63561	−0.115247	−0.0576236	−	0.998338i	$-0.518352\pi$
$523$	−2.63561	−0.115247	−0.0576236	+	0.998338i	$0.518352\pi$
$524$	−3.59813	−0.157185
$525$	−2.55381	−0.111458
$526$	46.8705	2.04365
$527$	7.58187	0.330272
$528$	12.6102	0.548789
$529$	−19.7660	−0.859393
$530$	12.3443	0.536202
$531$	−28.3209	−1.22902
$532$	−9.76699	−0.423453
$533$	46.6349	2.01998
$534$	−7.17457	−0.310474
$535$	18.9012	0.817172
$536$	80.4819	3.47629
$537$	−2.57395	−0.111074
$538$	22.3502	0.963587
$539$	−20.6666	−0.890174
$540$	−24.0227	−1.03377
$541$	−17.6990	−0.760940	−0.380470	−	0.924793i	$-0.624238\pi$
$541$	−17.6990	−0.760940	−0.380470	+	0.924793i	$0.624238\pi$
$542$	−62.2743	−2.67491
$543$	−6.03464	−0.258971
$544$	8.53984	0.366142
$545$	31.4197	1.34587
$546$	6.09384	0.260792
$547$	8.98778	0.384290	0.192145	−	0.981367i	$-0.438456\pi$
$547$	8.98778	0.384290	0.192145	+	0.981367i	$0.438456\pi$
$548$	−12.2442	−0.523045
$549$	−40.7931	−1.74101
$550$	−78.6380	−3.35313
$551$	8.35760	0.356046
$552$	3.39046	0.144307
$553$	−12.9398	−0.550258
$554$	72.8468	3.09496
$555$	−2.78800	−0.118344
$556$	−2.90142	−0.123048
$557$	2.75405	0.116693	0.0583464	−	0.998296i	$-0.481417\pi$
$557$	2.75405	0.116693	0.0583464	+	0.998296i	$0.481417\pi$
$558$	−74.3043	−3.14555
$559$	8.52234	0.360457
$560$	−55.7661	−2.35655
$561$	−1.00245	−0.0423233
$562$	1.00096	0.0422228
$563$	28.7946	1.21355	0.606774	−	0.794874i	$-0.292463\pi$
$563$	28.7946	1.21355	0.606774	+	0.794874i	$0.292463\pi$
$564$	5.09468	0.214525
$565$	47.5554	2.00067
$566$	−58.1782	−2.44541
$567$	−14.4143	−0.605343
$568$	2.13578	0.0896155
$569$	−36.3249	−1.52282	−0.761409	−	0.648272i	$-0.775492\pi$
$569$	−36.3249	−1.52282	−0.761409	+	0.648272i	$0.775492\pi$
$570$	−2.56722	−0.107529
$571$	−33.9472	−1.42065	−0.710324	−	0.703875i	$-0.751452\pi$
$571$	−33.9472	−1.42065	−0.710324	+	0.703875i	$0.751452\pi$
$572$	132.906	5.55707
$573$	−5.14403	−0.214895
$574$	38.6549	1.61342
$575$	−10.6564	−0.444401
$576$	−25.7317	−1.07216
$577$	26.8651	1.11841	0.559205	−	0.829029i	$-0.311107\pi$
$577$	26.8651	1.11841	0.559205	+	0.829029i	$0.311107\pi$
$578$	42.9013	1.78446
$579$	−1.58779	−0.0659863
$580$	114.019	4.73438
$581$	4.24884	0.176272
$582$	0.0447068	0.00185316
$583$	−7.22866	−0.299380
$584$	122.697	5.07726
$585$	−52.4147	−2.16708
$586$	−84.5380	−3.49223
$587$	6.68080	0.275746	0.137873	−	0.990450i	$-0.455973\pi$
$587$	6.68080	0.275746	0.137873	+	0.990450i	$0.455973\pi$
$588$	−4.99213	−0.205872
$589$	−11.3702	−0.468503
$590$	−83.4734	−3.43654
$591$	0.872279	0.0358808
$592$	−33.0191	−1.35708
$593$	−0.897450	−0.0368539	−0.0184269	−	0.999830i	$-0.505866\pi$
$593$	−0.897450	−0.0368539	−0.0184269	+	0.999830i	$0.505866\pi$
$594$	19.8611	0.814910
$595$	4.43311	0.181740
$596$	85.6131	3.50685
$597$	−1.40650	−0.0575642
$598$	25.4279	1.03983
$599$	−10.2892	−0.420405	−0.210202	−	0.977658i	$-0.567412\pi$
$599$	−10.2892	−0.420405	−0.210202	+	0.977658i	$0.567412\pi$
$600$	−11.1720	−0.456097
$601$	−13.8082	−0.563247	−0.281624	−	0.959525i	$-0.590873\pi$
$601$	−13.8082	−0.563247	−0.281624	+	0.959525i	$0.590873\pi$
$602$	7.06402	0.287908
$603$	31.6020	1.28693
$604$	−33.4423	−1.36075
$605$	48.5453	1.97365
$606$	6.94632	0.282175
$607$	7.56231	0.306945	0.153472	−	0.988153i	$-0.450954\pi$
$607$	7.56231	0.306945	0.153472	+	0.988153i	$0.450954\pi$
$608$	−12.8069	−0.519387
$609$	−3.06136	−0.124053
$610$	−120.234	−4.86814
$611$	22.4726	0.909143
$612$	11.1873	0.452222
$613$	−40.2607	−1.62612	−0.813058	−	0.582183i	$-0.802199\pi$
$613$	−40.2607	−1.62612	−0.813058	+	0.582183i	$0.802199\pi$
$614$	−39.3358	−1.58746
$615$	7.19640	0.290187
$616$	64.7919	2.61054
$617$	−40.0364	−1.61181	−0.805903	−	0.592048i	$-0.798320\pi$
$617$	−40.0364	−1.61181	−0.805903	+	0.592048i	$0.798320\pi$
$618$	−11.9177	−0.479399
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−155.119	−6.22972
$621$	2.69141	0.108002
$622$	72.8675	2.92172
$623$	−18.5795	−0.744373
$624$	13.4362	0.537877
$625$	−19.5144	−0.780577
$626$	−20.9052	−0.835541
$627$	1.50333	0.0600371
$628$	25.3669	1.01225
$629$	2.62484	0.104659
$630$	−43.4456	−1.73091
$631$	27.7851	1.10611	0.553053	−	0.833146i	$-0.313463\pi$
$631$	27.7851	1.10611	0.553053	+	0.833146i	$0.313463\pi$
$632$	−56.6073	−2.25172
$633$	−4.86463	−0.193352
$634$	22.7100	0.901929
$635$	−30.8806	−1.22546
$636$	−1.74612	−0.0692382
$637$	−22.0202	−0.872474
$638$	−94.2667	−3.73205
$639$	0.838636	0.0331759
$640$	−3.88341	−0.153505
$641$	15.6623	0.618625	0.309312	−	0.950960i	$-0.399901\pi$
$641$	15.6623	0.618625	0.309312	+	0.950960i	$0.399901\pi$
$642$	−3.77475	−0.148978
$643$	41.9228	1.65328	0.826638	−	0.562735i	$-0.190251\pi$
$643$	41.9228	1.65328	0.826638	+	0.562735i	$0.190251\pi$
$644$	14.9284	0.588262
$645$	1.31511	0.0517826
$646$	2.41698	0.0950948
$647$	14.5442	0.571792	0.285896	−	0.958261i	$-0.407709\pi$
$647$	14.5442	0.571792	0.285896	+	0.958261i	$0.407709\pi$
$648$	−63.0575	−2.47713
$649$	48.8809	1.91874
$650$	−83.7886	−3.28646
$651$	4.16488	0.163235
$652$	12.4996	0.489522
$653$	45.4250	1.77762	0.888809	−	0.458279i	$-0.151534\pi$
$653$	45.4250	1.77762	0.888809	+	0.458279i	$0.151534\pi$
$654$	−6.27480	−0.245364
$655$	−2.44916	−0.0956967
$656$	85.2291	3.32764
$657$	48.1783	1.87962
$658$	18.6271	0.726161
$659$	−3.90677	−0.152186	−0.0760931	−	0.997101i	$-0.524245\pi$
$659$	−3.90677	−0.152186	−0.0760931	+	0.997101i	$0.524245\pi$
$660$	20.5092	0.798319
$661$	14.2711	0.555080	0.277540	−	0.960714i	$-0.410481\pi$
$661$	14.2711	0.555080	0.277540	+	0.960714i	$0.410481\pi$
$662$	−1.43831	−0.0559015
$663$	−1.06810	−0.0414817
$664$	18.5872	0.721323
$665$	−6.64816	−0.257805
$666$	−25.7241	−0.996789
$667$	−12.7742	−0.494620
$668$	−108.939	−4.21499
$669$	−4.92849	−0.190547
$670$	93.1441	3.59847
$671$	70.4074	2.71805
$672$	4.69111	0.180964
$673$	31.6147	1.21866	0.609328	−	0.792918i	$-0.291439\pi$
$673$	31.6147	1.21866	0.609328	+	0.792918i	$0.291439\pi$
$674$	−60.8635	−2.34437
$675$	−8.86856	−0.341351
$676$	78.4821	3.01854
$677$	18.8231	0.723432	0.361716	−	0.932288i	$-0.382191\pi$
$677$	18.8231	0.723432	0.361716	+	0.932288i	$0.382191\pi$
$678$	−9.49726	−0.364740
$679$	0.115774	0.00444301
$680$	19.3933	0.743699
$681$	−2.73043	−0.104630
$682$	128.247	4.91082
$683$	7.18185	0.274806	0.137403	−	0.990515i	$-0.456125\pi$
$683$	7.18185	0.274806	0.137403	+	0.990515i	$0.456125\pi$
$684$	−16.7772	−0.641493
$685$	−8.33432	−0.318438
$686$	−49.5849	−1.89316
$687$	−2.34010	−0.0892804
$688$	15.5753	0.593802
$689$	−7.70212	−0.293427
$690$	3.92388	0.149379
$691$	−21.2098	−0.806859	−0.403429	−	0.915011i	$-0.632182\pi$
$691$	−21.2098	−0.806859	−0.403429	+	0.915011i	$0.632182\pi$
$692$	−28.7133	−1.09152
$693$	25.4412	0.966430
$694$	88.7889	3.37038
$695$	−1.97493	−0.0749134
$696$	−13.3924	−0.507638
$697$	−6.77526	−0.256631
$698$	23.9957	0.908249
$699$	2.07309	0.0784115
$700$	−49.1912	−1.85925
$701$	17.9212	0.676876	0.338438	−	0.940989i	$-0.390101\pi$
$701$	17.9212	0.676876	0.338438	+	0.940989i	$0.390101\pi$
$702$	21.1619	0.798706
$703$	−3.93637	−0.148463
$704$	44.4121	1.67384
$705$	3.46783	0.130606
$706$	64.4081	2.42403
$707$	17.9884	0.676525
$708$	11.8074	0.443751
$709$	37.0614	1.39187	0.695934	−	0.718106i	$-0.254991\pi$
$709$	37.0614	1.39187	0.695934	+	0.718106i	$0.254991\pi$
$710$	2.47181	0.0927653
$711$	−22.2274	−0.833593
$712$	−81.2789	−3.04606
$713$	17.3789	0.650845
$714$	−0.885333	−0.0331327
$715$	90.4659	3.38323
$716$	−49.5791	−1.85286
$717$	−1.72769	−0.0645216
$718$	−2.35981	−0.0880673
$719$	15.2676	0.569386	0.284693	−	0.958619i	$-0.408108\pi$
$719$	15.2676	0.569386	0.284693	+	0.958619i	$0.408108\pi$
$720$	−95.7921	−3.56996
$721$	−30.8624	−1.14938
$722$	46.1251	1.71660
$723$	4.29317	0.159665
$724$	−116.238	−4.31997
$725$	42.0929	1.56329
$726$	−9.69494	−0.359813
$727$	6.85952	0.254405	0.127203	−	0.991877i	$-0.459400\pi$
$727$	6.85952	0.254405	0.127203	+	0.991877i	$0.459400\pi$
$728$	69.0357	2.55863
$729$	−23.6282	−0.875118
$730$	142.001	5.25571
$731$	−1.23815	−0.0457947
$732$	17.0073	0.628608
$733$	−3.81613	−0.140952	−0.0704760	−	0.997513i	$-0.522452\pi$
$733$	−3.81613	−0.140952	−0.0704760	+	0.997513i	$0.522452\pi$
$734$	50.3103	1.85699
$735$	−3.39803	−0.125338
$736$	19.5747	0.721534
$737$	−54.5439	−2.00915
$738$	66.3993	2.44419
$739$	−5.79329	−0.213109	−0.106555	−	0.994307i	$-0.533982\pi$
$739$	−5.79329	−0.213109	−0.106555	+	0.994307i	$0.533982\pi$
$740$	−53.7021	−1.97413
$741$	1.60179	0.0588433
$742$	−6.38416	−0.234370
$743$	−33.3268	−1.22264	−0.611320	−	0.791383i	$-0.709361\pi$
$743$	−33.3268	−1.22264	−0.611320	+	0.791383i	$0.709361\pi$
$744$	18.2199	0.667974
$745$	58.2748	2.13502
$746$	70.1435	2.56813
$747$	7.29844	0.267036
$748$	−19.3090	−0.706006
$749$	−9.77524	−0.357179
$750$	−2.01990	−0.0737563
$751$	9.43349	0.344233	0.172117	−	0.985077i	$-0.444939\pi$
$751$	9.43349	0.344233	0.172117	+	0.985077i	$0.444939\pi$
$752$	41.0705	1.49769
$753$	−5.77832	−0.210574
$754$	−100.441	−3.65785
$755$	−22.7634	−0.828443
$756$	12.4239	0.451853
$757$	−39.1745	−1.42382	−0.711910	−	0.702270i	$-0.752170\pi$
$757$	−39.1745	−1.42382	−0.711910	+	0.702270i	$0.752170\pi$
$758$	−49.3984	−1.79423
$759$	−2.29777	−0.0834037
$760$	−29.0834	−1.05497
$761$	−27.9002	−1.01138	−0.505691	−	0.862714i	$-0.668763\pi$
$761$	−27.9002	−1.01138	−0.505691	+	0.862714i	$0.668763\pi$
$762$	6.16714	0.223412
$763$	−16.2494	−0.588269
$764$	−99.0837	−3.58472
$765$	7.61496	0.275320
$766$	−61.4660	−2.22086
$767$	52.0825	1.88059
$768$	−3.64283	−0.131449
$769$	11.2033	0.404000	0.202000	−	0.979385i	$-0.435256\pi$
$769$	11.2033	0.404000	0.202000	+	0.979385i	$0.435256\pi$
$770$	74.9856	2.70229
$771$	1.33540	0.0480932
$772$	−30.5838	−1.10074
$773$	30.9853	1.11446	0.557232	−	0.830357i	$-0.311863\pi$
$773$	30.9853	1.11446	0.557232	+	0.830357i	$0.311863\pi$
$774$	12.1342	0.436155
$775$	−57.2660	−2.05705
$776$	0.506472	0.0181813
$777$	1.44188	0.0517272
$778$	−54.7280	−1.96209
$779$	10.1606	0.364041
$780$	21.8525	0.782446
$781$	−1.44746	−0.0517941
$782$	−3.69425	−0.132106
$783$	−10.6311	−0.379925
$784$	−40.2438	−1.43728
$785$	17.2666	0.616272
$786$	0.489120	0.0174463
$787$	7.77104	0.277008	0.138504	−	0.990362i	$-0.455771\pi$
$787$	7.77104	0.277008	0.138504	+	0.990362i	$0.455771\pi$
$788$	16.8017	0.598537
$789$	−4.51282	−0.160661
$790$	−65.5133	−2.33086
$791$	−24.5944	−0.874478
$792$	111.296	3.95474
$793$	75.0190	2.66400
$794$	−41.2096	−1.46247
$795$	−1.18854	−0.0421533
$796$	−27.0918	−0.960244
$797$	35.2344	1.24807	0.624033	−	0.781398i	$-0.285493\pi$
$797$	35.2344	1.24807	0.624033	+	0.781398i	$0.285493\pi$
$798$	1.32770	0.0470000
$799$	−3.26489	−0.115503
$800$	−64.5015	−2.28047
$801$	−31.9149	−1.12766
$802$	48.9058	1.72692
$803$	−83.1541	−2.93444
$804$	−13.1754	−0.464660
$805$	10.1614	0.358143
$806$	136.647	4.81317
$807$	−2.15194	−0.0757519
$808$	78.6932	2.76842
$809$	−35.0943	−1.23385	−0.616925	−	0.787022i	$-0.711622\pi$
$809$	−35.0943	−1.23385	−0.616925	+	0.787022i	$0.711622\pi$
$810$	−72.9783	−2.56420
$811$	1.49321	0.0524337	0.0262168	−	0.999656i	$-0.491654\pi$
$811$	1.49321	0.0524337	0.0262168	+	0.999656i	$0.491654\pi$
$812$	−58.9676	−2.06936
$813$	5.99593	0.210287
$814$	44.3989	1.55618
$815$	8.50819	0.298029
$816$	−1.95205	−0.0683353
$817$	1.85681	0.0649615
$818$	21.6784	0.757968
$819$	27.1075	0.947213
$820$	138.616	4.84069
$821$	−7.80078	−0.272249	−0.136125	−	0.990692i	$-0.543465\pi$
$821$	−7.80078	−0.272249	−0.136125	+	0.990692i	$0.543465\pi$
$822$	1.66444	0.0580540
$823$	4.44031	0.154779	0.0773897	−	0.997001i	$-0.475341\pi$
$823$	4.44031	0.154779	0.0773897	+	0.997001i	$0.475341\pi$
$824$	−135.012	−4.70338
$825$	7.57148	0.263605
$826$	43.1703	1.50209
$827$	39.0009	1.35619	0.678097	−	0.734973i	$-0.262805\pi$
$827$	39.0009	1.35619	0.678097	+	0.734973i	$0.262805\pi$
$828$	25.6433	0.891165
$829$	−37.2181	−1.29264	−0.646320	−	0.763067i	$-0.723693\pi$
$829$	−37.2181	−1.29264	−0.646320	+	0.763067i	$0.723693\pi$
$830$	21.5115	0.746676
$831$	−7.01388	−0.243309
$832$	47.3210	1.64056
$833$	3.19917	0.110845
$834$	0.394412	0.0136574
$835$	−74.1524	−2.56615
$836$	28.9569	1.00150
$837$	14.4633	0.499924
$838$	71.4152	2.46700
$839$	−31.1291	−1.07469	−0.537347	−	0.843361i	$-0.680574\pi$
$839$	−31.1291	−1.07469	−0.537347	+	0.843361i	$0.680574\pi$
$840$	10.6532	0.367569
$841$	21.4585	0.739950
$842$	−54.0821	−1.86379
$843$	−0.0963747	−0.00331932
$844$	−93.7018	−3.22535
$845$	53.4209	1.83774
$846$	31.9967	1.10007
$847$	−25.1064	−0.862665
$848$	−14.0763	−0.483381
$849$	5.60156	0.192245
$850$	12.1731	0.417533
$851$	6.01657	0.206245
$852$	−0.349641	−0.0119785
$853$	11.4891	0.393379	0.196690	−	0.980466i	$-0.436981\pi$
$853$	11.4891	0.393379	0.196690	+	0.980466i	$0.436981\pi$
$854$	62.1820	2.12782
$855$	−11.4199	−0.390551
$856$	−42.7632	−1.46162
$857$	−9.67575	−0.330517	−0.165259	−	0.986250i	$-0.552846\pi$
$857$	−9.67575	−0.330517	−0.165259	+	0.986250i	$0.552846\pi$
$858$	−18.0669	−0.616793
$859$	1.37280	0.0468394	0.0234197	−	0.999726i	$-0.492545\pi$
$859$	1.37280	0.0468394	0.0234197	+	0.999726i	$0.492545\pi$
$860$	25.3316	0.863800
$861$	−3.72179	−0.126838
$862$	−13.6586	−0.465212
$863$	50.5277	1.71998	0.859992	−	0.510308i	$-0.170469\pi$
$863$	50.5277	1.71998	0.859992	+	0.510308i	$0.170469\pi$
$864$	16.2907	0.554221
$865$	−19.5445	−0.664532
$866$	22.4747	0.763722
$867$	−4.13065	−0.140284
$868$	80.2235	2.72296
$869$	38.3637	1.30140
$870$	−15.4994	−0.525480
$871$	−58.1165	−1.96920
$872$	−71.0856	−2.40726
$873$	0.198871	0.00673077
$874$	5.54012	0.187397
$875$	−5.23080	−0.176833
$876$	−20.0863	−0.678654
$877$	27.6349	0.933163	0.466581	−	0.884478i	$-0.345485\pi$
$877$	27.6349	0.933163	0.466581	+	0.884478i	$0.345485\pi$
$878$	−42.9858	−1.45070
$879$	8.13954	0.274540
$880$	165.334	5.57340
$881$	5.69198	0.191768	0.0958838	−	0.995393i	$-0.469432\pi$
$881$	5.69198	0.191768	0.0958838	+	0.995393i	$0.469432\pi$
$882$	−31.3527	−1.05570
$883$	−45.7074	−1.53818	−0.769089	−	0.639142i	$-0.779290\pi$
$883$	−45.7074	−1.53818	−0.769089	+	0.639142i	$0.779290\pi$
$884$	−20.5737	−0.691968
$885$	8.03704	0.270162
$886$	33.6229	1.12958
$887$	4.92217	0.165270	0.0826352	−	0.996580i	$-0.473666\pi$
$887$	4.92217	0.165270	0.0826352	+	0.996580i	$0.473666\pi$
$888$	6.30772	0.211673
$889$	15.9707	0.535639
$890$	−94.0665	−3.15312
$891$	42.7351	1.43168
$892$	−94.9321	−3.17856
$893$	4.89622	0.163846
$894$	−11.6380	−0.389233
$895$	−33.7473	−1.12805
$896$	2.00840	0.0670959
$897$	−2.44827	−0.0817453
$898$	−83.3629	−2.78185
$899$	−68.6471	−2.28951
$900$	−84.4981	−2.81660
$901$	1.11899	0.0372789
$902$	−114.603	−3.81586
$903$	−0.680143	−0.0226337
$904$	−107.592	−3.57846
$905$	−79.1208	−2.63006
$906$	4.54605	0.151032
$907$	2.84636	0.0945118	0.0472559	−	0.998883i	$-0.484952\pi$
$907$	2.84636	0.0945118	0.0472559	+	0.998883i	$0.484952\pi$
$908$	−52.5932	−1.74537
$909$	30.8996	1.02488
$910$	79.8970	2.64856
$911$	47.0407	1.55853	0.779263	−	0.626697i	$-0.215593\pi$
$911$	47.0407	1.55853	0.779263	+	0.626697i	$0.215593\pi$
$912$	2.92741	0.0969362
$913$	−12.5969	−0.416895
$914$	6.01776	0.199050
$915$	11.5765	0.382706
$916$	−45.0747	−1.48931
$917$	1.26664	0.0418283
$918$	−3.07447	−0.101473
$919$	−0.826432	−0.0272615	−0.0136307	−	0.999907i	$-0.504339\pi$
$919$	−0.826432	−0.0272615	−0.0136307	+	0.999907i	$0.504339\pi$
$920$	44.4527	1.46556
$921$	3.78735	0.124798
$922$	49.3191	1.62424
$923$	−1.54226	−0.0507642
$924$	−10.6068	−0.348939
$925$	−19.8254	−0.651857
$926$	64.4060	2.11651
$927$	−53.0138	−1.74120
$928$	−77.3207	−2.53817
$929$	17.9236	0.588056	0.294028	−	0.955797i	$-0.405004\pi$
$929$	17.9236	0.588056	0.294028	+	0.955797i	$0.405004\pi$
$930$	21.0864	0.691452
$931$	−4.79767	−0.157237
$932$	39.9316	1.30800
$933$	−7.01588	−0.229690
$934$	71.8605	2.35135
$935$	−13.1432	−0.429827
$936$	118.586	3.87610
$937$	−16.6746	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$937$	−16.6746	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$938$	−48.1717	−1.57286
$939$	2.01281	0.0656856
$940$	66.7969	2.17867
$941$	−49.9379	−1.62793	−0.813964	−	0.580915i	$-0.802695\pi$
$941$	−49.9379	−1.62793	−0.813964	+	0.580915i	$0.802695\pi$
$942$	−3.44830	−0.112352
$943$	−15.5300	−0.505727
$944$	95.1851	3.09801
$945$	8.45666	0.275095
$946$	−20.9432	−0.680923
$947$	47.1294	1.53150	0.765750	−	0.643138i	$-0.222368\pi$
$947$	47.1294	1.53150	0.765750	+	0.643138i	$0.222368\pi$
$948$	9.26696	0.300977
$949$	−88.6006	−2.87610
$950$	−18.2555	−0.592286
$951$	−2.18658	−0.0709047
$952$	−10.0297	−0.325065
$953$	−51.7129	−1.67514	−0.837572	−	0.546327i	$-0.816026\pi$
$953$	−51.7129	−1.67514	−0.837572	+	0.546327i	$0.816026\pi$
$954$	−10.9664	−0.355049
$955$	−67.4439	−2.18243
$956$	−33.2785	−1.07630
$957$	9.07625	0.293394
$958$	−23.5036	−0.759367
$959$	4.31029	0.139187
$960$	7.30228	0.235680
$961$	62.3921	2.01265
$962$	47.3070	1.52524
$963$	−16.7914	−0.541095
$964$	82.6946	2.66341
$965$	−20.8177	−0.670145
$966$	−2.02933	−0.0652926
$967$	0.962194	0.0309421	0.0154710	−	0.999880i	$-0.495075\pi$
$967$	0.962194	0.0309421	0.0154710	+	0.999880i	$0.495075\pi$
$968$	−109.832	−3.53012
$969$	−0.232713	−0.00747583
$970$	0.586156	0.0188203
$971$	−15.1556	−0.486367	−0.243183	−	0.969980i	$-0.578192\pi$
$971$	−15.1556	−0.486367	−0.243183	+	0.969980i	$0.578192\pi$
$972$	32.1260	1.03044
$973$	1.02138	0.0327440
$974$	84.5672	2.70971
$975$	8.06739	0.258363
$976$	137.103	4.38858
$977$	30.9833	0.991245	0.495623	−	0.868538i	$-0.334940\pi$
$977$	30.9833	0.991245	0.495623	+	0.868538i	$0.334940\pi$
$978$	−1.69916	−0.0543333
$979$	55.0841	1.76049
$980$	−65.4524	−2.09080
$981$	−27.9124	−0.891176
$982$	−22.0316	−0.703056
$983$	−4.51938	−0.144146	−0.0720730	−	0.997399i	$-0.522961\pi$
$983$	−4.51938	−0.144146	−0.0720730	+	0.997399i	$0.522961\pi$
$984$	−16.2815	−0.519037
$985$	11.4365	0.364399
$986$	14.5924	0.464716
$987$	−1.79347	−0.0570868
$988$	30.8535	0.981582
$989$	−2.83805	−0.0902447
$990$	128.806	4.09374
$991$	54.6240	1.73519	0.867595	−	0.497272i	$-0.165665\pi$
$991$	54.6240	1.73519	0.867595	+	0.497272i	$0.165665\pi$
$992$	105.192	3.33985
$993$	0.138484	0.00439467
$994$	−1.27836	−0.0405470
$995$	−18.4408	−0.584611
$996$	−3.04284	−0.0964160
$997$	16.7792	0.531403	0.265701	−	0.964055i	$-0.414396\pi$
$997$	16.7792	0.531403	0.265701	+	0.964055i	$0.414396\pi$
$998$	107.418	3.40027
$999$	5.00718	0.158420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	619.2.a.b.1.1	✓	30
3.2	odd	2		5571.2.a.g.1.30		30
4.3	odd	2		9904.2.a.n.1.15		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.1	✓	30	1.1	even	1	trivial
5571.2.a.g.1.30		30	3.2	odd	2
9904.2.a.n.1.15		30	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	619.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61841 q^{2} +0.252107 q^{3} +4.85606 q^{4} +3.30541 q^{5} -0.660120 q^{6} -1.70947 q^{7} -7.47834 q^{8} -2.93644 q^{9} +O(q^{10})\)	\(q-2.61841 q^{2} +0.252107 q^{3} +4.85606 q^{4} +3.30541 q^{5} -0.660120 q^{6} -1.70947 q^{7} -7.47834 q^{8} -2.93644 q^{9} -8.65491 q^{10} +5.06820 q^{11} +1.22425 q^{12} +5.40015 q^{13} +4.47610 q^{14} +0.833318 q^{15} +9.86923 q^{16} -0.784551 q^{17} +7.68880 q^{18} +1.17656 q^{19} +16.0513 q^{20} -0.430971 q^{21} -13.2706 q^{22} -1.79832 q^{23} -1.88535 q^{24} +5.92572 q^{25} -14.1398 q^{26} -1.49662 q^{27} -8.30131 q^{28} +7.10342 q^{29} -2.18197 q^{30} -9.66396 q^{31} -10.8850 q^{32} +1.27773 q^{33} +2.05428 q^{34} -5.65050 q^{35} -14.2595 q^{36} -3.34566 q^{37} -3.08072 q^{38} +1.36142 q^{39} -24.7190 q^{40} +8.63584 q^{41} +1.12846 q^{42} +1.57817 q^{43} +24.6115 q^{44} -9.70614 q^{45} +4.70874 q^{46} +4.16147 q^{47} +2.48811 q^{48} -4.07771 q^{49} -15.5160 q^{50} -0.197791 q^{51} +26.2235 q^{52} -1.42628 q^{53} +3.91877 q^{54} +16.7525 q^{55} +12.7840 q^{56} +0.296620 q^{57} -18.5997 q^{58} +9.64463 q^{59} +4.04665 q^{60} +13.8920 q^{61} +25.3042 q^{62} +5.01976 q^{63} +8.76290 q^{64} +17.8497 q^{65} -3.34562 q^{66} -10.7620 q^{67} -3.80983 q^{68} -0.453370 q^{69} +14.7953 q^{70} -0.285596 q^{71} +21.9597 q^{72} -16.4070 q^{73} +8.76030 q^{74} +1.49392 q^{75} +5.71346 q^{76} -8.66394 q^{77} -3.56475 q^{78} +7.56950 q^{79} +32.6218 q^{80} +8.43202 q^{81} -22.6122 q^{82} -2.48547 q^{83} -2.09282 q^{84} -2.59326 q^{85} -4.13228 q^{86} +1.79082 q^{87} -37.9017 q^{88} +10.8686 q^{89} +25.4146 q^{90} -9.23141 q^{91} -8.73276 q^{92} -2.43636 q^{93} -10.8964 q^{94} +3.88901 q^{95} -2.74419 q^{96} -0.0677252 q^{97} +10.6771 q^{98} -14.8825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{2} + q^{3} + 33 q^{4} + 21 q^{5} + 6 q^{6} + 2 q^{7} + 27 q^{8} + 43 q^{9}+O(q^{10}) \) 30 * q + 9 * q^2 + q^3 + 33 * q^4 + 21 * q^5 + 6 * q^6 + 2 * q^7 + 27 * q^8 + 43 * q^9	\( 30 q + 9 q^{2} + q^{3} + 33 q^{4} + 21 q^{5} + 6 q^{6} + 2 q^{7} + 27 q^{8} + 43 q^{9} + 5 q^{10} + 23 q^{11} - 6 q^{12} + 9 q^{13} + 7 q^{14} - 2 q^{15} + 35 q^{16} + 4 q^{17} + 10 q^{18} - q^{19} + 29 q^{20} + 30 q^{21} + 4 q^{23} + 4 q^{24} + 35 q^{25} + q^{26} - 5 q^{27} - 13 q^{28} + 90 q^{29} - 31 q^{30} + 2 q^{31} + 43 q^{32} - 6 q^{33} - 9 q^{34} + 9 q^{35} + 33 q^{36} + 19 q^{37} + 5 q^{38} + 32 q^{39} - 12 q^{40} + 59 q^{41} - 25 q^{42} - 4 q^{43} + 52 q^{44} + 30 q^{45} - q^{46} + 4 q^{47} - 44 q^{48} + 30 q^{49} + 31 q^{50} - 12 q^{52} + 34 q^{53} - 28 q^{54} - 17 q^{55} + 2 q^{56} - 8 q^{57} + 6 q^{58} + 13 q^{59} - 64 q^{60} + 16 q^{61} + 28 q^{62} - 40 q^{63} + 37 q^{64} + 31 q^{65} - 59 q^{66} - 11 q^{67} - 52 q^{68} + 6 q^{69} - 40 q^{70} + 42 q^{71} + 6 q^{72} - 4 q^{73} + 16 q^{74} - 52 q^{75} - 42 q^{76} + 29 q^{77} - 56 q^{78} + 3 q^{79} + 21 q^{80} + 30 q^{81} - 43 q^{82} - 11 q^{83} - 36 q^{84} + 19 q^{85} - 11 q^{86} - 20 q^{87} - 47 q^{88} + 58 q^{89} - 33 q^{90} - 39 q^{91} - 7 q^{92} - 15 q^{93} - 46 q^{94} + 23 q^{95} - 70 q^{96} - 9 q^{97} - 8 q^{99}+O(q^{100}) \) 30 * q + 9 * q^2 + q^3 + 33 * q^4 + 21 * q^5 + 6 * q^6 + 2 * q^7 + 27 * q^8 + 43 * q^9 + 5 * q^10 + 23 * q^11 - 6 * q^12 + 9 * q^13 + 7 * q^14 - 2 * q^15 + 35 * q^16 + 4 * q^17 + 10 * q^18 - q^19 + 29 * q^20 + 30 * q^21 + 4 * q^23 + 4 * q^24 + 35 * q^25 + q^26 - 5 * q^27 - 13 * q^28 + 90 * q^29 - 31 * q^30 + 2 * q^31 + 43 * q^32 - 6 * q^33 - 9 * q^34 + 9 * q^35 + 33 * q^36 + 19 * q^37 + 5 * q^38 + 32 * q^39 - 12 * q^40 + 59 * q^41 - 25 * q^42 - 4 * q^43 + 52 * q^44 + 30 * q^45 - q^46 + 4 * q^47 - 44 * q^48 + 30 * q^49 + 31 * q^50 - 12 * q^52 + 34 * q^53 - 28 * q^54 - 17 * q^55 + 2 * q^56 - 8 * q^57 + 6 * q^58 + 13 * q^59 - 64 * q^60 + 16 * q^61 + 28 * q^62 - 40 * q^63 + 37 * q^64 + 31 * q^65 - 59 * q^66 - 11 * q^67 - 52 * q^68 + 6 * q^69 - 40 * q^70 + 42 * q^71 + 6 * q^72 - 4 * q^73 + 16 * q^74 - 52 * q^75 - 42 * q^76 + 29 * q^77 - 56 * q^78 + 3 * q^79 + 21 * q^80 + 30 * q^81 - 43 * q^82 - 11 * q^83 - 36 * q^84 + 19 * q^85 - 11 * q^86 - 20 * q^87 - 47 * q^88 + 58 * q^89 - 33 * q^90 - 39 * q^91 - 7 * q^92 - 15 * q^93 - 46 * q^94 + 23 * q^95 - 70 * q^96 - 9 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more