LMFDB - Embedded newform 619.2.a.b.1.28

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			1.28
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)

$f(q)$	$=$	$q+2.58096 q^{2} -2.01571 q^{3} +4.66134 q^{4} -1.70302 q^{5} -5.20246 q^{6} +3.16609 q^{7} +6.86880 q^{8} +1.06308 q^{9} +O(q^{10})$	\(q+2.58096 q^{2} -2.01571 q^{3} +4.66134 q^{4} -1.70302 q^{5} -5.20246 q^{6} +3.16609 q^{7} +6.86880 q^{8} +1.06308 q^{9} -4.39543 q^{10} +3.07985 q^{11} -9.39590 q^{12} +1.76673 q^{13} +8.17153 q^{14} +3.43280 q^{15} +8.40540 q^{16} +3.12955 q^{17} +2.74378 q^{18} -1.60811 q^{19} -7.93837 q^{20} -6.38191 q^{21} +7.94897 q^{22} +5.10270 q^{23} -13.8455 q^{24} -2.09971 q^{25} +4.55984 q^{26} +3.90426 q^{27} +14.7582 q^{28} -0.870825 q^{29} +8.85992 q^{30} -5.41510 q^{31} +7.95637 q^{32} -6.20809 q^{33} +8.07724 q^{34} -5.39192 q^{35} +4.95540 q^{36} +1.53424 q^{37} -4.15045 q^{38} -3.56120 q^{39} -11.6977 q^{40} -9.86530 q^{41} -16.4714 q^{42} -1.06159 q^{43} +14.3562 q^{44} -1.81046 q^{45} +13.1698 q^{46} -6.84081 q^{47} -16.9428 q^{48} +3.02411 q^{49} -5.41926 q^{50} -6.30827 q^{51} +8.23530 q^{52} +8.24355 q^{53} +10.0767 q^{54} -5.24507 q^{55} +21.7472 q^{56} +3.24148 q^{57} -2.24756 q^{58} +3.71187 q^{59} +16.0015 q^{60} -6.66271 q^{61} -13.9761 q^{62} +3.36582 q^{63} +3.72426 q^{64} -3.00878 q^{65} -16.0228 q^{66} +4.94846 q^{67} +14.5879 q^{68} -10.2856 q^{69} -13.9163 q^{70} -15.3563 q^{71} +7.30212 q^{72} -9.17543 q^{73} +3.95981 q^{74} +4.23240 q^{75} -7.49593 q^{76} +9.75108 q^{77} -9.19132 q^{78} +0.733523 q^{79} -14.3146 q^{80} -11.0591 q^{81} -25.4619 q^{82} -1.05432 q^{83} -29.7483 q^{84} -5.32970 q^{85} -2.73992 q^{86} +1.75533 q^{87} +21.1549 q^{88} +1.86252 q^{89} -4.67272 q^{90} +5.59361 q^{91} +23.7854 q^{92} +10.9153 q^{93} -17.6558 q^{94} +2.73865 q^{95} -16.0377 q^{96} +5.89701 q^{97} +7.80509 q^{98} +3.27415 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.58096	1.82501	0.912506	−	0.409063i	$-0.134144\pi$
$2$	2.58096	1.82501	0.912506	+	0.409063i	$0.134144\pi$
$3$	−2.01571	−1.16377	−0.581885	−	0.813271i	$-0.697685\pi$
$3$	−2.01571	−1.16377	−0.581885	+	0.813271i	$0.697685\pi$
$4$	4.66134	2.33067
$5$	−1.70302	−0.761616	−0.380808	−	0.924654i	$-0.624354\pi$
$5$	−1.70302	−0.761616	−0.380808	+	0.924654i	$0.624354\pi$
$6$	−5.20246	−2.12390
$7$	3.16609	1.19667	0.598334	−	0.801247i	$-0.295830\pi$
$7$	3.16609	1.19667	0.598334	+	0.801247i	$0.295830\pi$
$8$	6.86880	2.42849
$9$	1.06308	0.354362
$10$	−4.39543	−1.38996
$11$	3.07985	0.928611	0.464305	−	0.885675i	$-0.346304\pi$
$11$	3.07985	0.928611	0.464305	+	0.885675i	$0.346304\pi$
$12$	−9.39590	−2.71236
$13$	1.76673	0.490001	0.245001	−	0.969523i	$-0.421212\pi$
$13$	1.76673	0.490001	0.245001	+	0.969523i	$0.421212\pi$
$14$	8.17153	2.18393
$15$	3.43280	0.886346
$16$	8.40540	2.10135
$17$	3.12955	0.759028	0.379514	−	0.925186i	$-0.376091\pi$
$17$	3.12955	0.759028	0.379514	+	0.925186i	$0.376091\pi$
$18$	2.74378	0.646714
$19$	−1.60811	−0.368925	−0.184463	−	0.982840i	$-0.559054\pi$
$19$	−1.60811	−0.368925	−0.184463	+	0.982840i	$0.559054\pi$
$20$	−7.93837	−1.77507
$21$	−6.38191	−1.39265
$22$	7.94897	1.69473
$23$	5.10270	1.06399	0.531993	−	0.846749i	$-0.321443\pi$
$23$	5.10270	1.06399	0.531993	+	0.846749i	$0.321443\pi$
$24$	−13.8455	−2.82620
$25$	−2.09971	−0.419942
$26$	4.55984	0.894259
$27$	3.90426	0.751375
$28$	14.7582	2.78904
$29$	−0.870825	−0.161708	−0.0808541	−	0.996726i	$-0.525765\pi$
$29$	−0.870825	−0.161708	−0.0808541	+	0.996726i	$0.525765\pi$
$30$	8.85992	1.61759
$31$	−5.41510	−0.972580	−0.486290	−	0.873797i	$-0.661650\pi$
$31$	−5.41510	−0.972580	−0.486290	+	0.873797i	$0.661650\pi$
$32$	7.95637	1.40650
$33$	−6.20809	−1.08069
$34$	8.07724	1.38524
$35$	−5.39192	−0.911401
$36$	4.95540	0.825900
$37$	1.53424	0.252228	0.126114	−	0.992016i	$-0.459750\pi$
$37$	1.53424	0.252228	0.126114	+	0.992016i	$0.459750\pi$
$38$	−4.15045	−0.673293
$39$	−3.56120	−0.570249
$40$	−11.6977	−1.84957
$41$	−9.86530	−1.54070	−0.770351	−	0.637620i	$-0.779919\pi$
$41$	−9.86530	−1.54070	−0.770351	+	0.637620i	$0.779919\pi$
$42$	−16.4714	−2.54160
$43$	−1.06159	−0.161891	−0.0809456	−	0.996719i	$-0.525794\pi$
$43$	−1.06159	−0.161891	−0.0809456	+	0.996719i	$0.525794\pi$
$44$	14.3562	2.16428
$45$	−1.81046	−0.269887
$46$	13.1698	1.94179
$47$	−6.84081	−0.997835	−0.498917	−	0.866650i	$-0.666269\pi$
$47$	−6.84081	−0.997835	−0.498917	+	0.866650i	$0.666269\pi$
$48$	−16.9428	−2.44549
$49$	3.02411	0.432015
$50$	−5.41926	−0.766398
$51$	−6.30827	−0.883334
$52$	8.23530	1.14203
$53$	8.24355	1.13234	0.566169	−	0.824289i	$-0.308425\pi$
$53$	8.24355	1.13234	0.566169	+	0.824289i	$0.308425\pi$
$54$	10.0767	1.37127
$55$	−5.24507	−0.707245
$56$	21.7472	2.90609
$57$	3.24148	0.429344
$58$	−2.24756	−0.295119
$59$	3.71187	0.483245	0.241622	−	0.970370i	$-0.422320\pi$
$59$	3.71187	0.483245	0.241622	+	0.970370i	$0.422320\pi$
$60$	16.0015	2.06578
$61$	−6.66271	−0.853072	−0.426536	−	0.904470i	$-0.640266\pi$
$61$	−6.66271	−0.853072	−0.426536	+	0.904470i	$0.640266\pi$
$62$	−13.9761	−1.77497
$63$	3.36582	0.424053
$64$	3.72426	0.465533
$65$	−3.00878	−0.373193
$66$	−16.0228	−1.97227
$67$	4.94846	0.604551	0.302276	−	0.953221i	$-0.402254\pi$
$67$	4.94846	0.604551	0.302276	+	0.953221i	$0.402254\pi$
$68$	14.5879	1.76904
$69$	−10.2856	−1.23824
$70$	−13.9163	−1.66332
$71$	−15.3563	−1.82246	−0.911231	−	0.411895i	$-0.864867\pi$
$71$	−15.3563	−1.82246	−0.911231	+	0.411895i	$0.864867\pi$
$72$	7.30212	0.860563
$73$	−9.17543	−1.07390	−0.536951	−	0.843613i	$-0.680424\pi$
$73$	−9.17543	−1.07390	−0.536951	+	0.843613i	$0.680424\pi$
$74$	3.95981	0.460319
$75$	4.23240	0.488716
$76$	−7.49593	−0.859842
$77$	9.75108	1.11124
$78$	−9.19132	−1.04071
$79$	0.733523	0.0825278	0.0412639	−	0.999148i	$-0.486862\pi$
$79$	0.733523	0.0825278	0.0412639	+	0.999148i	$0.486862\pi$
$80$	−14.3146	−1.60042
$81$	−11.0591	−1.22879
$82$	−25.4619	−2.81180
$83$	−1.05432	−0.115726	−0.0578632	−	0.998325i	$-0.518429\pi$
$83$	−1.05432	−0.115726	−0.0578632	+	0.998325i	$0.518429\pi$
$84$	−29.7483	−3.24580
$85$	−5.32970	−0.578088
$86$	−2.73992	−0.295453
$87$	1.75533	0.188191
$88$	21.1549	2.25512
$89$	1.86252	0.197427	0.0987136	−	0.995116i	$-0.468527\pi$
$89$	1.86252	0.197427	0.0987136	+	0.995116i	$0.468527\pi$
$90$	−4.67272	−0.492548
$91$	5.59361	0.586369
$92$	23.7854	2.47980
$93$	10.9153	1.13186
$94$	−17.6558	−1.82106
$95$	2.73865	0.280979
$96$	−16.0377	−1.63685
$97$	5.89701	0.598750	0.299375	−	0.954135i	$-0.403222\pi$
$97$	5.89701	0.598750	0.299375	+	0.954135i	$0.403222\pi$
$98$	7.80509	0.788433
$99$	3.27415	0.329064
$100$	−9.78745	−0.978745
$101$	7.69037	0.765220	0.382610	−	0.923910i	$-0.375025\pi$
$101$	7.69037	0.765220	0.382610	+	0.923910i	$0.375025\pi$
$102$	−16.2814	−1.61210
$103$	−16.1527	−1.59158	−0.795788	−	0.605576i	$-0.792943\pi$
$103$	−16.1527	−1.59158	−0.795788	+	0.605576i	$0.792943\pi$
$104$	12.1353	1.18996
$105$	10.8686	1.06066
$106$	21.2762	2.06653
$107$	−13.3167	−1.28737	−0.643685	−	0.765291i	$-0.722595\pi$
$107$	−13.3167	−1.28737	−0.643685	+	0.765291i	$0.722595\pi$
$108$	18.1991	1.75121
$109$	15.1458	1.45071	0.725353	−	0.688377i	$-0.241677\pi$
$109$	15.1458	1.45071	0.725353	+	0.688377i	$0.241677\pi$
$110$	−13.5373	−1.29073
$111$	−3.09258	−0.293535
$112$	26.6122	2.51462
$113$	20.0109	1.88247	0.941235	−	0.337752i	$-0.109666\pi$
$113$	20.0109	1.88247	0.941235	+	0.337752i	$0.109666\pi$
$114$	8.36611	0.783558
$115$	−8.69002	−0.810348
$116$	−4.05921	−0.376888
$117$	1.87818	0.173638
$118$	9.58019	0.881928
$119$	9.90844	0.908305
$120$	23.5792	2.15248
$121$	−1.51450	−0.137682
$122$	−17.1962	−1.55687
$123$	19.8856	1.79302
$124$	−25.2416	−2.26676
$125$	12.0910	1.08145
$126$	8.68703	0.773902
$127$	−18.4277	−1.63519	−0.817596	−	0.575793i	$-0.804693\pi$
$127$	−18.4277	−1.63519	−0.817596	+	0.575793i	$0.804693\pi$
$128$	−6.30059	−0.556899
$129$	2.13986	0.188404
$130$	−7.76552	−0.681081
$131$	19.9094	1.73949	0.869747	−	0.493499i	$-0.164282\pi$
$131$	19.9094	1.73949	0.869747	+	0.493499i	$0.164282\pi$
$132$	−28.9380	−2.51873
$133$	−5.09141	−0.441481
$134$	12.7718	1.10331
$135$	−6.64905	−0.572259
$136$	21.4963	1.84329
$137$	−14.6207	−1.24913	−0.624566	−	0.780972i	$-0.714724\pi$
$137$	−14.6207	−1.24913	−0.624566	+	0.780972i	$0.714724\pi$
$138$	−26.5466	−2.25979
$139$	−1.31190	−0.111274	−0.0556371	−	0.998451i	$-0.517719\pi$
$139$	−1.31190	−0.111274	−0.0556371	+	0.998451i	$0.517719\pi$
$140$	−25.1336	−2.12418
$141$	13.7891	1.16125
$142$	−39.6341	−3.32602
$143$	5.44126	0.455021
$144$	8.93565	0.744638
$145$	1.48304	0.123159
$146$	−23.6814	−1.95988
$147$	−6.09572	−0.502767
$148$	7.15162	0.587859
$149$	18.5569	1.52024	0.760119	−	0.649784i	$-0.225141\pi$
$149$	18.5569	1.52024	0.760119	+	0.649784i	$0.225141\pi$
$150$	10.9236	0.891912
$151$	−8.45448	−0.688015	−0.344008	−	0.938967i	$-0.611785\pi$
$151$	−8.45448	−0.688015	−0.344008	+	0.938967i	$0.611785\pi$
$152$	−11.0458	−0.895930
$153$	3.32698	0.268970
$154$	25.1671	2.02803
$155$	9.22204	0.740732
$156$	−16.6000	−1.32906
$157$	−13.4961	−1.07711	−0.538554	−	0.842591i	$-0.681029\pi$
$157$	−13.4961	−1.07711	−0.538554	+	0.842591i	$0.681029\pi$
$158$	1.89319	0.150614
$159$	−16.6166	−1.31778
$160$	−13.5499	−1.07121
$161$	16.1556	1.27324
$162$	−28.5431	−2.24256
$163$	−6.87027	−0.538121	−0.269060	−	0.963123i	$-0.586713\pi$
$163$	−6.87027	−0.538121	−0.269060	+	0.963123i	$0.586713\pi$
$164$	−45.9855	−3.59087
$165$	10.5725	0.823070
$166$	−2.72115	−0.211202
$167$	9.77794	0.756639	0.378320	−	0.925675i	$-0.376502\pi$
$167$	9.77794	0.756639	0.378320	+	0.925675i	$0.376502\pi$
$168$	−43.8361	−3.38203
$169$	−9.87868	−0.759899
$170$	−13.7557	−1.05502
$171$	−1.70955	−0.130733
$172$	−4.94844	−0.377315
$173$	6.43554	0.489285	0.244642	−	0.969613i	$-0.421329\pi$
$173$	6.43554	0.489285	0.244642	+	0.969613i	$0.421329\pi$
$174$	4.53043	0.343451
$175$	−6.64786	−0.502531
$176$	25.8874	1.95134
$177$	−7.48206	−0.562386
$178$	4.80710	0.360307
$179$	7.01587	0.524391	0.262195	−	0.965015i	$-0.415554\pi$
$179$	7.01587	0.524391	0.262195	+	0.965015i	$0.415554\pi$
$180$	−8.43916	−0.629018
$181$	1.05259	0.0782385	0.0391193	−	0.999235i	$-0.487545\pi$
$181$	1.05259	0.0782385	0.0391193	+	0.999235i	$0.487545\pi$
$182$	14.4369	1.07013
$183$	13.4301	0.992780
$184$	35.0494	2.58388
$185$	−2.61285	−0.192101
$186$	28.1718	2.06566
$187$	9.63856	0.704842
$188$	−31.8873	−2.32562
$189$	12.3612	0.899147
$190$	7.06833	0.512790
$191$	18.1174	1.31093	0.655464	−	0.755226i	$-0.272473\pi$
$191$	18.1174	1.31093	0.655464	+	0.755226i	$0.272473\pi$
$192$	−7.50703	−0.541773
$193$	17.5598	1.26398	0.631992	−	0.774975i	$-0.282238\pi$
$193$	17.5598	1.26398	0.631992	+	0.774975i	$0.282238\pi$
$194$	15.2199	1.09273
$195$	6.06482	0.434311
$196$	14.0964	1.00688
$197$	−15.6469	−1.11480	−0.557398	−	0.830245i	$-0.688200\pi$
$197$	−15.6469	−1.11480	−0.557398	+	0.830245i	$0.688200\pi$
$198$	8.45043	0.600546
$199$	25.2021	1.78653	0.893266	−	0.449529i	$-0.148408\pi$
$199$	25.2021	1.78653	0.893266	+	0.449529i	$0.148408\pi$
$200$	−14.4225	−1.01982
$201$	−9.97467	−0.703559
$202$	19.8485	1.39654
$203$	−2.75711	−0.193511
$204$	−29.4050	−2.05876
$205$	16.8009	1.17342
$206$	−41.6895	−2.90464
$207$	5.42460	0.377036
$208$	14.8500	1.02966
$209$	−4.95273	−0.342588
$210$	28.0513	1.93572
$211$	1.86294	0.128250	0.0641251	−	0.997942i	$-0.479574\pi$
$211$	1.86294	0.128250	0.0641251	+	0.997942i	$0.479574\pi$
$212$	38.4260	2.63911
$213$	30.9539	2.12093
$214$	−34.3697	−2.34946
$215$	1.80792	0.123299
$216$	26.8176	1.82470
$217$	−17.1447	−1.16386
$218$	39.0907	2.64756
$219$	18.4950	1.24978
$220$	−24.4490	−1.64835
$221$	5.52906	0.371925
$222$	−7.98183	−0.535705
$223$	−4.91903	−0.329403	−0.164701	−	0.986343i	$-0.552666\pi$
$223$	−4.91903	−0.329403	−0.164701	+	0.986343i	$0.552666\pi$
$224$	25.1906	1.68312
$225$	−2.23217	−0.148811
$226$	51.6474	3.43553
$227$	−9.50921	−0.631149	−0.315574	−	0.948901i	$-0.602197\pi$
$227$	−9.50921	−0.631149	−0.315574	+	0.948901i	$0.602197\pi$
$228$	15.1096	1.00066
$229$	−11.4060	−0.753727	−0.376864	−	0.926269i	$-0.622997\pi$
$229$	−11.4060	−0.753727	−0.376864	+	0.926269i	$0.622997\pi$
$230$	−22.4286	−1.47890
$231$	−19.6554	−1.29323
$232$	−5.98152	−0.392706
$233$	28.3769	1.85903	0.929515	−	0.368785i	$-0.120226\pi$
$233$	28.3769	1.85903	0.929515	+	0.368785i	$0.120226\pi$
$234$	4.84750	0.316891
$235$	11.6501	0.759967
$236$	17.3023	1.12628
$237$	−1.47857	−0.0960434
$238$	25.5732	1.65767
$239$	−29.6601	−1.91855	−0.959276	−	0.282472i	$-0.908846\pi$
$239$	−29.6601	−1.91855	−0.959276	+	0.282472i	$0.908846\pi$
$240$	28.8541	1.86252
$241$	5.82270	0.375073	0.187536	−	0.982258i	$-0.439950\pi$
$241$	5.82270	0.375073	0.187536	+	0.982258i	$0.439950\pi$
$242$	−3.90886	−0.251271
$243$	10.5792	0.678654
$244$	−31.0571	−1.98823
$245$	−5.15013	−0.329030
$246$	51.3238	3.27229
$247$	−2.84108	−0.180774
$248$	−37.1952	−2.36190
$249$	2.12520	0.134679
$250$	31.2063	1.97366
$251$	−22.9779	−1.45035	−0.725177	−	0.688562i	$-0.758242\pi$
$251$	−22.9779	−1.45035	−0.725177	+	0.688562i	$0.758242\pi$
$252$	15.6892	0.988328
$253$	15.7156	0.988029
$254$	−47.5610	−2.98424
$255$	10.7431	0.672761
$256$	−23.7101	−1.48188
$257$	−7.71742	−0.481399	−0.240700	−	0.970600i	$-0.577377\pi$
$257$	−7.71742	−0.481399	−0.240700	+	0.970600i	$0.577377\pi$
$258$	5.52289	0.343840
$259$	4.85754	0.301833
$260$	−14.0249	−0.869789
$261$	−0.925761	−0.0573031
$262$	51.3853	3.17460
$263$	−14.9336	−0.920843	−0.460421	−	0.887700i	$-0.652302\pi$
$263$	−14.9336	−0.920843	−0.460421	+	0.887700i	$0.652302\pi$
$264$	−42.6421	−2.62444
$265$	−14.0390	−0.862407
$266$	−13.1407	−0.805708
$267$	−3.75431	−0.229760
$268$	23.0665	1.40901
$269$	23.1253	1.40997	0.704986	−	0.709221i	$-0.250953\pi$
$269$	23.1253	1.40997	0.704986	+	0.709221i	$0.250953\pi$
$270$	−17.1609	−1.04438
$271$	14.2167	0.863604	0.431802	−	0.901968i	$-0.357878\pi$
$271$	14.2167	0.863604	0.431802	+	0.901968i	$0.357878\pi$
$272$	26.3051	1.59498
$273$	−11.2751	−0.682399
$274$	−37.7354	−2.27968
$275$	−6.46679	−0.389962
$276$	−47.9444	−2.88592
$277$	1.23241	0.0740484	0.0370242	−	0.999314i	$-0.488212\pi$
$277$	1.23241	0.0740484	0.0370242	+	0.999314i	$0.488212\pi$
$278$	−3.38597	−0.203077
$279$	−5.75671	−0.344645
$280$	−37.0360	−2.21333
$281$	24.3436	1.45222	0.726109	−	0.687579i	$-0.241327\pi$
$281$	24.3436	1.45222	0.726109	+	0.687579i	$0.241327\pi$
$282$	35.5890	2.11930
$283$	−8.99433	−0.534657	−0.267329	−	0.963605i	$-0.586141\pi$
$283$	−8.99433	−0.534657	−0.267329	+	0.963605i	$0.586141\pi$
$284$	−71.5811	−4.24756
$285$	−5.52031	−0.326995
$286$	14.0436	0.830418
$287$	−31.2344	−1.84371
$288$	8.45830	0.498410
$289$	−7.20590	−0.423877
$290$	3.82765	0.224767
$291$	−11.8867	−0.696808
$292$	−42.7698	−2.50291
$293$	9.45893	0.552596	0.276298	−	0.961072i	$-0.410892\pi$
$293$	9.45893	0.552596	0.276298	+	0.961072i	$0.410892\pi$
$294$	−15.7328	−0.917555
$295$	−6.32141	−0.368047
$296$	10.5384	0.612532
$297$	12.0245	0.697735
$298$	47.8945	2.77445
$299$	9.01506	0.521355
$300$	19.7287	1.13903
$301$	−3.36109	−0.193730
$302$	−21.8206	−1.25564
$303$	−15.5015	−0.890541
$304$	−13.5168	−0.775241
$305$	11.3468	0.649713
$306$	8.58679	0.490874
$307$	16.7495	0.955947	0.477973	−	0.878374i	$-0.341372\pi$
$307$	16.7495	0.955947	0.477973	+	0.878374i	$0.341372\pi$
$308$	45.4531	2.58993
$309$	32.5592	1.85223
$310$	23.8017	1.35185
$311$	−15.6140	−0.885387	−0.442693	−	0.896673i	$-0.645977\pi$
$311$	−15.6140	−0.885387	−0.442693	+	0.896673i	$0.645977\pi$
$312$	−24.4612	−1.38484
$313$	15.0080	0.848302	0.424151	−	0.905592i	$-0.360573\pi$
$313$	15.0080	0.848302	0.424151	+	0.905592i	$0.360573\pi$
$314$	−34.8329	−1.96574
$315$	−5.73207	−0.322966
$316$	3.41920	0.192345
$317$	12.8209	0.720096	0.360048	−	0.932934i	$-0.382760\pi$
$317$	12.8209	0.720096	0.360048	+	0.932934i	$0.382760\pi$
$318$	−42.8867	−2.40497
$319$	−2.68201	−0.150164
$320$	−6.34251	−0.354557
$321$	26.8425	1.49820
$322$	41.6969	2.32367
$323$	−5.03265	−0.280024
$324$	−51.5502	−2.86390
$325$	−3.70961	−0.205772
$326$	−17.7319	−0.982077
$327$	−30.5296	−1.68829
$328$	−67.7628	−3.74158
$329$	−21.6586	−1.19408
$330$	27.2872	1.50211
$331$	−23.1410	−1.27194	−0.635971	−	0.771713i	$-0.719400\pi$
$331$	−23.1410	−1.27194	−0.635971	+	0.771713i	$0.719400\pi$
$332$	−4.91453	−0.269720
$333$	1.63103	0.0893798
$334$	25.2364	1.38088
$335$	−8.42736	−0.460436
$336$	−53.6425	−2.92644
$337$	−11.7253	−0.638720	−0.319360	−	0.947633i	$-0.603468\pi$
$337$	−11.7253	−0.638720	−0.319360	+	0.947633i	$0.603468\pi$
$338$	−25.4965	−1.38682
$339$	−40.3362	−2.19076
$340$	−24.8436	−1.34733
$341$	−16.6777	−0.903148
$342$	−4.41228	−0.238589
$343$	−12.5880	−0.679689
$344$	−7.29186	−0.393151
$345$	17.5165	0.943059
$346$	16.6098	0.892951
$347$	−14.4163	−0.773906	−0.386953	−	0.922099i	$-0.626472\pi$
$347$	−14.4163	−0.773906	−0.386953	+	0.922099i	$0.626472\pi$
$348$	8.18219	0.438611
$349$	−19.6116	−1.04979	−0.524893	−	0.851168i	$-0.675895\pi$
$349$	−19.6116	−1.04979	−0.524893	+	0.851168i	$0.675895\pi$
$350$	−17.1578	−0.917125
$351$	6.89775	0.368175
$352$	24.5045	1.30609
$353$	−20.9387	−1.11445	−0.557227	−	0.830360i	$-0.688135\pi$
$353$	−20.9387	−1.11445	−0.557227	+	0.830360i	$0.688135\pi$
$354$	−19.3109	−1.02636
$355$	26.1522	1.38802
$356$	8.68186	0.460137
$357$	−19.9725	−1.05706
$358$	18.1077	0.957019
$359$	37.6285	1.98596	0.992978	−	0.118302i	$-0.0377451\pi$
$359$	37.6285	1.98596	0.992978	+	0.118302i	$0.0377451\pi$
$360$	−12.4357	−0.655418
$361$	−16.4140	−0.863894
$362$	2.71669	0.142786
$363$	3.05279	0.160230
$364$	26.0737	1.36663
$365$	15.6260	0.817901
$366$	34.6625	1.81184
$367$	9.43449	0.492476	0.246238	−	0.969209i	$-0.420805\pi$
$367$	9.43449	0.492476	0.246238	+	0.969209i	$0.420805\pi$
$368$	42.8902	2.23581
$369$	−10.4877	−0.545966
$370$	−6.74365	−0.350586
$371$	26.0998	1.35503
$372$	50.8797	2.63799
$373$	−30.0391	−1.55537	−0.777684	−	0.628656i	$-0.783605\pi$
$373$	−30.0391	−1.55537	−0.777684	+	0.628656i	$0.783605\pi$
$374$	24.8767	1.28634
$375$	−24.3719	−1.25856
$376$	−46.9882	−2.42323
$377$	−1.53851	−0.0792372
$378$	31.9038	1.64095
$379$	3.41140	0.175232	0.0876158	−	0.996154i	$-0.472075\pi$
$379$	3.41140	0.175232	0.0876158	+	0.996154i	$0.472075\pi$
$380$	12.7658	0.654869
$381$	37.1448	1.90299
$382$	46.7602	2.39246
$383$	−17.3443	−0.886253	−0.443126	−	0.896459i	$-0.646131\pi$
$383$	−17.3443	−0.886253	−0.443126	+	0.896459i	$0.646131\pi$
$384$	12.7002	0.648103
$385$	−16.6063	−0.846337
$386$	45.3211	2.30679
$387$	−1.12856	−0.0573680
$388$	27.4880	1.39549
$389$	−6.43887	−0.326464	−0.163232	−	0.986588i	$-0.552192\pi$
$389$	−6.43887	−0.326464	−0.163232	+	0.986588i	$0.552192\pi$
$390$	15.6530	0.792622
$391$	15.9692	0.807595
$392$	20.7720	1.04914
$393$	−40.1316	−2.02437
$394$	−40.3840	−2.03452
$395$	−1.24921	−0.0628545
$396$	15.2619	0.766939
$397$	24.7960	1.24448	0.622238	−	0.782828i	$-0.286224\pi$
$397$	24.7960	1.24448	0.622238	+	0.782828i	$0.286224\pi$
$398$	65.0456	3.26044
$399$	10.2628	0.513782
$400$	−17.6489	−0.882444
$401$	18.4118	0.919441	0.459720	−	0.888064i	$-0.347950\pi$
$401$	18.4118	0.919441	0.459720	+	0.888064i	$0.347950\pi$
$402$	−25.7442	−1.28400
$403$	−9.56699	−0.476566
$404$	35.8474	1.78348
$405$	18.8339	0.935865
$406$	−7.11598	−0.353160
$407$	4.72524	0.234221
$408$	−43.3302	−2.14517
$409$	22.8498	1.12985	0.564924	−	0.825143i	$-0.308905\pi$
$409$	22.8498	1.12985	0.564924	+	0.825143i	$0.308905\pi$
$410$	43.3623	2.14151
$411$	29.4711	1.45370
$412$	−75.2933	−3.70944
$413$	11.7521	0.578284
$414$	14.0007	0.688095
$415$	1.79553	0.0881391
$416$	14.0567	0.689188
$417$	2.64442	0.129498
$418$	−12.7828	−0.625227
$419$	−32.0411	−1.56531	−0.782656	−	0.622455i	$-0.786136\pi$
$419$	−32.0411	−1.56531	−0.782656	+	0.622455i	$0.786136\pi$
$420$	50.6620	2.47205
$421$	−28.0886	−1.36895	−0.684477	−	0.729034i	$-0.739970\pi$
$421$	−28.0886	−1.36895	−0.684477	+	0.729034i	$0.739970\pi$
$422$	4.80817	0.234058
$423$	−7.27236	−0.353594
$424$	56.6233	2.74987
$425$	−6.57115	−0.318747
$426$	79.8908	3.87072
$427$	−21.0947	−1.02084
$428$	−62.0734	−3.00043
$429$	−10.9680	−0.529540
$430$	4.66615	0.225022
$431$	6.01265	0.289619	0.144809	−	0.989460i	$-0.453743\pi$
$431$	6.01265	0.289619	0.144809	+	0.989460i	$0.453743\pi$
$432$	32.8169	1.57890
$433$	40.2174	1.93273	0.966363	−	0.257183i	$-0.0827943\pi$
$433$	40.2174	1.93273	0.966363	+	0.257183i	$0.0827943\pi$
$434$	−44.2496	−2.12405
$435$	−2.98937	−0.143329
$436$	70.5998	3.38112
$437$	−8.20568	−0.392531
$438$	47.7348	2.28086
$439$	−17.6477	−0.842280	−0.421140	−	0.906996i	$-0.638370\pi$
$439$	−17.6477	−0.842280	−0.421140	+	0.906996i	$0.638370\pi$
$440$	−36.0273	−1.71753
$441$	3.21488	0.153090
$442$	14.2703	0.678767
$443$	8.35591	0.397001	0.198501	−	0.980101i	$-0.436393\pi$
$443$	8.35591	0.397001	0.198501	+	0.980101i	$0.436393\pi$
$444$	−14.4156	−0.684133
$445$	−3.17192	−0.150364
$446$	−12.6958	−0.601164
$447$	−37.4053	−1.76921
$448$	11.7913	0.557088
$449$	21.8549	1.03139	0.515697	−	0.856771i	$-0.327533\pi$
$449$	21.8549	1.03139	0.515697	+	0.856771i	$0.327533\pi$
$450$	−5.76113	−0.271582
$451$	−30.3837	−1.43071
$452$	93.2777	4.38742
$453$	17.0418	0.800692
$454$	−24.5429	−1.15185
$455$	−9.52605	−0.446588
$456$	22.2651	1.04266
$457$	23.3971	1.09447	0.547236	−	0.836978i	$-0.315680\pi$
$457$	23.3971	1.09447	0.547236	+	0.836978i	$0.315680\pi$
$458$	−29.4383	−1.37556
$459$	12.2186	0.570315
$460$	−40.5071	−1.88865
$461$	8.62821	0.401856	0.200928	−	0.979606i	$-0.435604\pi$
$461$	8.62821	0.401856	0.200928	+	0.979606i	$0.435604\pi$
$462$	−50.7296	−2.36016
$463$	8.21241	0.381663	0.190832	−	0.981623i	$-0.438882\pi$
$463$	8.21241	0.381663	0.190832	+	0.981623i	$0.438882\pi$
$464$	−7.31963	−0.339805
$465$	−18.5890	−0.862042
$466$	73.2394	3.39275
$467$	−0.192039	−0.00888652	−0.00444326	−	0.999990i	$-0.501414\pi$
$467$	−0.192039	−0.00888652	−0.00444326	+	0.999990i	$0.501414\pi$
$468$	8.75483	0.404692
$469$	15.6673	0.723447
$470$	30.0683	1.38695
$471$	27.2043	1.25351
$472$	25.4961	1.17355
$473$	−3.26955	−0.150334
$474$	−3.81612	−0.175280
$475$	3.37655	0.154927
$476$	46.1866	2.11696
$477$	8.76359	0.401257
$478$	−76.5514	−3.50138
$479$	35.7226	1.63221	0.816104	−	0.577905i	$-0.196130\pi$
$479$	35.7226	1.63221	0.816104	+	0.577905i	$0.196130\pi$
$480$	27.3127	1.24665
$481$	2.71058	0.123592
$482$	15.0281	0.684512
$483$	−32.5650	−1.48176
$484$	−7.05960	−0.320891
$485$	−10.0427	−0.456018
$486$	27.3044	1.23855
$487$	−21.6508	−0.981090	−0.490545	−	0.871416i	$-0.663202\pi$
$487$	−21.6508	−0.981090	−0.490545	+	0.871416i	$0.663202\pi$
$488$	−45.7648	−2.07168
$489$	13.8485	0.626249
$490$	−13.2923	−0.600483
$491$	−9.91269	−0.447353	−0.223677	−	0.974663i	$-0.571806\pi$
$491$	−9.91269	−0.447353	−0.223677	+	0.974663i	$0.571806\pi$
$492$	92.6935	4.17894
$493$	−2.72529	−0.122741
$494$	−7.33271	−0.329914
$495$	−5.57595	−0.250620
$496$	−45.5161	−2.04373
$497$	−48.6195	−2.18088
$498$	5.48505	0.245791
$499$	−9.57800	−0.428770	−0.214385	−	0.976749i	$-0.568775\pi$
$499$	−9.57800	−0.428770	−0.214385	+	0.976749i	$0.568775\pi$
$500$	56.3601	2.52050
$501$	−19.7095	−0.880555
$502$	−59.3051	−2.64692
$503$	−4.55582	−0.203134	−0.101567	−	0.994829i	$-0.532386\pi$
$503$	−4.55582	−0.203134	−0.101567	+	0.994829i	$0.532386\pi$
$504$	23.1191	1.02981
$505$	−13.0969	−0.582804
$506$	40.5612	1.80316
$507$	19.9126	0.884348
$508$	−85.8976	−3.81109
$509$	27.4516	1.21677	0.608385	−	0.793642i	$-0.291817\pi$
$509$	27.4516	1.21677	0.608385	+	0.793642i	$0.291817\pi$
$510$	27.7276	1.22780
$511$	−29.0502	−1.28510
$512$	−48.5935	−2.14755
$513$	−6.27847	−0.277201
$514$	−19.9183	−0.878560
$515$	27.5085	1.21217
$516$	9.97461	0.439108
$517$	−21.0687	−0.926600
$518$	12.5371	0.550849
$519$	−12.9722	−0.569415
$520$	−20.6667	−0.906294
$521$	−2.88161	−0.126246	−0.0631229	−	0.998006i	$-0.520106\pi$
$521$	−2.88161	−0.126246	−0.0631229	+	0.998006i	$0.520106\pi$
$522$	−2.38935	−0.104579
$523$	3.09008	0.135120	0.0675598	−	0.997715i	$-0.478479\pi$
$523$	3.09008	0.135120	0.0675598	+	0.997715i	$0.478479\pi$
$524$	92.8045	4.05418
$525$	13.4002	0.584831
$526$	−38.5429	−1.68055
$527$	−16.9468	−0.738215
$528$	−52.1815	−2.27091
$529$	3.03751	0.132066
$530$	−36.2340	−1.57390
$531$	3.94604	0.171243
$532$	−23.7328	−1.02895
$533$	−17.4293	−0.754946
$534$	−9.68971	−0.419315
$535$	22.6786	0.980481
$536$	33.9900	1.46814
$537$	−14.1420	−0.610270
$538$	59.6853	2.57322
$539$	9.31381	0.401174
$540$	−30.9935	−1.33375
$541$	29.9618	1.28816	0.644078	−	0.764959i	$-0.277241\pi$
$541$	29.9618	1.28816	0.644078	+	0.764959i	$0.277241\pi$
$542$	36.6927	1.57609
$543$	−2.12172	−0.0910517
$544$	24.8999	1.06757
$545$	−25.7937	−1.10488
$546$	−29.1005	−1.24539
$547$	−9.81005	−0.419448	−0.209724	−	0.977761i	$-0.567256\pi$
$547$	−9.81005	−0.419448	−0.209724	+	0.977761i	$0.567256\pi$
$548$	−68.1521	−2.91131
$549$	−7.08302	−0.302296
$550$	−16.6905	−0.711686
$551$	1.40038	0.0596582
$552$	−70.6494	−3.00704
$553$	2.32240	0.0987584
$554$	3.18080	0.135139
$555$	5.26675	0.223561
$556$	−6.11523	−0.259343
$557$	−15.1051	−0.640022	−0.320011	−	0.947414i	$-0.603687\pi$
$557$	−15.1051	−0.640022	−0.320011	+	0.947414i	$0.603687\pi$
$558$	−14.8578	−0.628981
$559$	−1.87554	−0.0793269
$560$	−45.3213	−1.91517
$561$	−19.4285	−0.820274
$562$	62.8298	2.65032
$563$	12.7689	0.538145	0.269073	−	0.963120i	$-0.413283\pi$
$563$	12.7689	0.538145	0.269073	+	0.963120i	$0.413283\pi$
$564$	64.2756	2.70649
$565$	−34.0791	−1.43372
$566$	−23.2140	−0.975756
$567$	−35.0141	−1.47045
$568$	−105.480	−4.42583
$569$	34.5924	1.45019	0.725095	−	0.688649i	$-0.241796\pi$
$569$	34.5924	1.45019	0.725095	+	0.688649i	$0.241796\pi$
$570$	−14.2477	−0.596770
$571$	2.27833	0.0953453	0.0476726	−	0.998863i	$-0.484820\pi$
$571$	2.27833	0.0953453	0.0476726	+	0.998863i	$0.484820\pi$
$572$	25.3635	1.06050
$573$	−36.5194	−1.52562
$574$	−80.6147	−3.36479
$575$	−10.7142	−0.446812
$576$	3.95920	0.164967
$577$	7.19988	0.299735	0.149867	−	0.988706i	$-0.452115\pi$
$577$	7.19988	0.299735	0.149867	+	0.988706i	$0.452115\pi$
$578$	−18.5981	−0.773580
$579$	−35.3955	−1.47099
$580$	6.91293	0.287044
$581$	−3.33806	−0.138486
$582$	−30.6789	−1.27168
$583$	25.3889	1.05150
$584$	−63.0242	−2.60796
$585$	−3.19858	−0.132245
$586$	24.4131	1.00850
$587$	42.8151	1.76717	0.883584	−	0.468272i	$-0.155123\pi$
$587$	42.8151	1.76717	0.883584	+	0.468272i	$0.155123\pi$
$588$	−28.4142	−1.17178
$589$	8.70805	0.358809
$590$	−16.3153	−0.671690
$591$	31.5396	1.29737
$592$	12.8959	0.530019
$593$	31.1437	1.27892	0.639460	−	0.768824i	$-0.279158\pi$
$593$	31.1437	1.27892	0.639460	+	0.768824i	$0.279158\pi$
$594$	31.0348	1.27337
$595$	−16.8743	−0.691779
$596$	86.4998	3.54317
$597$	−50.8002	−2.07911
$598$	23.2675	0.951478
$599$	−10.5301	−0.430248	−0.215124	−	0.976587i	$-0.569016\pi$
$599$	−10.5301	−0.430248	−0.215124	+	0.976587i	$0.569016\pi$
$600$	29.0715	1.18684
$601$	−27.2158	−1.11016	−0.555079	−	0.831798i	$-0.687312\pi$
$601$	−27.2158	−1.11016	−0.555079	+	0.831798i	$0.687312\pi$
$602$	−8.67483	−0.353560
$603$	5.26064	0.214230
$604$	−39.4092	−1.60354
$605$	2.57923	0.104861
$606$	−40.0088	−1.62525
$607$	−9.65556	−0.391907	−0.195953	−	0.980613i	$-0.562780\pi$
$607$	−9.65556	−0.391907	−0.195953	+	0.980613i	$0.562780\pi$
$608$	−12.7947	−0.518894
$609$	5.55753	0.225202
$610$	29.2855	1.18573
$611$	−12.0858	−0.488941
$612$	15.5082	0.626881
$613$	7.86977	0.317857	0.158928	−	0.987290i	$-0.449196\pi$
$613$	7.86977	0.317857	0.158928	+	0.987290i	$0.449196\pi$
$614$	43.2298	1.74461
$615$	−33.8656	−1.36559
$616$	66.9783	2.69863
$617$	−23.1681	−0.932714	−0.466357	−	0.884597i	$-0.654434\pi$
$617$	−23.1681	−0.932714	−0.466357	+	0.884597i	$0.654434\pi$
$618$	84.0339	3.38034
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	42.9871	1.72640
$621$	19.9222	0.799452
$622$	−40.2990	−1.61584
$623$	5.89691	0.236255
$624$	−29.9334	−1.19829
$625$	−10.0927	−0.403707
$626$	38.7350	1.54816
$627$	9.98327	0.398694
$628$	−62.9100	−2.51038
$629$	4.80149	0.191448
$630$	−14.7942	−0.589416
$631$	5.02978	0.200232	0.100116	−	0.994976i	$-0.468079\pi$
$631$	5.02978	0.200232	0.100116	+	0.994976i	$0.468079\pi$
$632$	5.03842	0.200418
$633$	−3.75515	−0.149254
$634$	33.0903	1.31418
$635$	31.3828	1.24539
$636$	−77.4556	−3.07131
$637$	5.34277	0.211688
$638$	−6.92216	−0.274051
$639$	−16.3251	−0.645811
$640$	10.7301	0.424143
$641$	−2.12276	−0.0838439	−0.0419220	−	0.999121i	$-0.513348\pi$
$641$	−2.12276	−0.0838439	−0.0419220	+	0.999121i	$0.513348\pi$
$642$	69.2793	2.73424
$643$	16.8092	0.662889	0.331445	−	0.943475i	$-0.392464\pi$
$643$	16.8092	0.662889	0.331445	+	0.943475i	$0.392464\pi$
$644$	75.3066	2.96750
$645$	−3.64423	−0.143492
$646$	−12.9891	−0.511048
$647$	0.404862	0.0159168	0.00795839	−	0.999968i	$-0.497467\pi$
$647$	0.404862	0.0159168	0.00795839	+	0.999968i	$0.497467\pi$
$648$	−75.9628	−2.98410
$649$	11.4320	0.448746
$650$	−9.57434	−0.375536
$651$	34.5587	1.35446
$652$	−32.0246	−1.25418
$653$	−31.3137	−1.22540	−0.612701	−	0.790315i	$-0.709917\pi$
$653$	−31.3137	−1.22540	−0.612701	+	0.790315i	$0.709917\pi$
$654$	−78.7955	−3.08115
$655$	−33.9062	−1.32483
$656$	−82.9218	−3.23755
$657$	−9.75425	−0.380550
$658$	−55.8999	−2.17921
$659$	13.2047	0.514381	0.257191	−	0.966361i	$-0.417203\pi$
$659$	13.2047	0.514381	0.257191	+	0.966361i	$0.417203\pi$
$660$	49.2821	1.91830
$661$	29.8567	1.16129	0.580646	−	0.814156i	$-0.302800\pi$
$661$	29.8567	1.16129	0.580646	+	0.814156i	$0.302800\pi$
$662$	−59.7258	−2.32131
$663$	−11.1450	−0.432835
$664$	−7.24190	−0.281040
$665$	8.67079	0.336239
$666$	4.20961	0.163119
$667$	−4.44355	−0.172055
$668$	45.5783	1.76348
$669$	9.91535	0.383349
$670$	−21.7506	−0.840301
$671$	−20.5202	−0.792172
$672$	−50.7769	−1.95876
$673$	−14.1579	−0.545748	−0.272874	−	0.962050i	$-0.587974\pi$
$673$	−14.1579	−0.545748	−0.272874	+	0.962050i	$0.587974\pi$
$674$	−30.2626	−1.16567
$675$	−8.19780	−0.315534
$676$	−46.0479	−1.77107
$677$	12.0505	0.463138	0.231569	−	0.972819i	$-0.425614\pi$
$677$	12.0505	0.463138	0.231569	+	0.972819i	$0.425614\pi$
$678$	−104.106	−3.99817
$679$	18.6704	0.716506
$680$	−36.6087	−1.40388
$681$	19.1678	0.734512
$682$	−43.0444	−1.64826
$683$	10.4975	0.401674	0.200837	−	0.979625i	$-0.435634\pi$
$683$	10.4975	0.401674	0.200837	+	0.979625i	$0.435634\pi$
$684$	−7.96881	−0.304695
$685$	24.8994	0.951359
$686$	−32.4891	−1.24044
$687$	22.9911	0.877165
$688$	−8.92310	−0.340190
$689$	14.5641	0.554847
$690$	45.2095	1.72109
$691$	12.2267	0.465126	0.232563	−	0.972581i	$-0.425289\pi$
$691$	12.2267	0.465126	0.232563	+	0.972581i	$0.425289\pi$
$692$	29.9982	1.14036
$693$	10.3662	0.393780
$694$	−37.2078	−1.41239
$695$	2.23420	0.0847482
$696$	12.0570	0.457020
$697$	−30.8740	−1.16944
$698$	−50.6168	−1.91587
$699$	−57.1995	−2.16348
$700$	−30.9879	−1.17123
$701$	−51.7007	−1.95271	−0.976355	−	0.216173i	$-0.930642\pi$
$701$	−51.7007	−1.95271	−0.976355	+	0.216173i	$0.930642\pi$
$702$	17.8028	0.671923
$703$	−2.46722	−0.0930531
$704$	11.4702	0.432299
$705$	−23.4832	−0.884427
$706$	−54.0419	−2.03389
$707$	24.3484	0.915715
$708$	−34.8764	−1.31074
$709$	40.1300	1.50711	0.753557	−	0.657382i	$-0.228336\pi$
$709$	40.1300	1.50711	0.753557	+	0.657382i	$0.228336\pi$
$710$	67.4978	2.53315
$711$	0.779797	0.0292447
$712$	12.7933	0.479449
$713$	−27.6316	−1.03481
$714$	−51.5482	−1.92914
$715$	−9.26659	−0.346551
$716$	32.7033	1.22218
$717$	59.7861	2.23275
$718$	97.1175	3.62439
$719$	35.8890	1.33843	0.669216	−	0.743068i	$-0.266630\pi$
$719$	35.8890	1.33843	0.669216	+	0.743068i	$0.266630\pi$
$720$	−15.2176	−0.567128
$721$	−51.1409	−1.90459
$722$	−42.3638	−1.57662
$723$	−11.7369	−0.436499
$724$	4.90649	0.182348
$725$	1.82848	0.0679080
$726$	7.87913	0.292422
$727$	24.6485	0.914164	0.457082	−	0.889425i	$-0.348895\pi$
$727$	24.6485	0.914164	0.457082	+	0.889425i	$0.348895\pi$
$728$	38.4214	1.42399
$729$	11.8528	0.438992
$730$	40.3300	1.49268
$731$	−3.32231	−0.122880
$732$	62.6022	2.31384
$733$	−51.8727	−1.91596	−0.957980	−	0.286834i	$-0.907397\pi$
$733$	−51.8727	−1.91596	−0.957980	+	0.286834i	$0.907397\pi$
$734$	24.3500	0.898776
$735$	10.3812	0.382915
$736$	40.5990	1.49650
$737$	15.2405	0.561393
$738$	−27.0682	−0.996394
$739$	−13.6739	−0.503004	−0.251502	−	0.967857i	$-0.580925\pi$
$739$	−13.6739	−0.503004	−0.251502	+	0.967857i	$0.580925\pi$
$740$	−12.1794	−0.447723
$741$	5.72680	0.210379
$742$	67.3624	2.47295
$743$	12.7077	0.466201	0.233101	−	0.972453i	$-0.425113\pi$
$743$	12.7077	0.466201	0.233101	+	0.972453i	$0.425113\pi$
$744$	74.9747	2.74871
$745$	−31.6028	−1.15784
$746$	−77.5297	−2.83856
$747$	−1.12083	−0.0410090
$748$	44.9286	1.64275
$749$	−42.1617	−1.54055
$750$	−62.9028	−2.29689
$751$	−20.1202	−0.734196	−0.367098	−	0.930182i	$-0.619649\pi$
$751$	−20.1202	−0.734196	−0.367098	+	0.930182i	$0.619649\pi$
$752$	−57.4998	−2.09680
$753$	46.3169	1.68788
$754$	−3.97082	−0.144609
$755$	14.3982	0.524003
$756$	57.6198	2.09561
$757$	−11.6413	−0.423110	−0.211555	−	0.977366i	$-0.567853\pi$
$757$	−11.6413	−0.423110	−0.211555	+	0.977366i	$0.567853\pi$
$758$	8.80466	0.319800
$759$	−31.6780	−1.14984
$760$	18.8112	0.682354
$761$	26.1485	0.947884	0.473942	−	0.880556i	$-0.342831\pi$
$761$	26.1485	0.947884	0.473942	+	0.880556i	$0.342831\pi$
$762$	95.8692	3.47297
$763$	47.9530	1.73601
$764$	84.4513	3.05534
$765$	−5.66593	−0.204852
$766$	−44.7649	−1.61742
$767$	6.55786	0.236791
$768$	47.7926	1.72457
$769$	−44.3241	−1.59837	−0.799184	−	0.601086i	$-0.794735\pi$
$769$	−44.3241	−1.59837	−0.799184	+	0.601086i	$0.794735\pi$
$770$	−42.8602	−1.54458
$771$	15.5561	0.560238
$772$	81.8523	2.94593
$773$	3.90690	0.140521	0.0702607	−	0.997529i	$-0.477617\pi$
$773$	3.90690	0.140521	0.0702607	+	0.997529i	$0.477617\pi$
$774$	−2.91277	−0.104697
$775$	11.3701	0.408427
$776$	40.5054	1.45406
$777$	−9.79139	−0.351264
$778$	−16.6184	−0.595800
$779$	15.8645	0.568403
$780$	28.2702	1.01223
$781$	−47.2953	−1.69236
$782$	41.2157	1.47387
$783$	−3.39993	−0.121503
$784$	25.4188	0.907815
$785$	22.9842	0.820343
$786$	−103.578	−3.69450
$787$	−21.5688	−0.768844	−0.384422	−	0.923158i	$-0.625599\pi$
$787$	−21.5688	−0.768844	−0.384422	+	0.923158i	$0.625599\pi$
$788$	−72.9355	−2.59822
$789$	30.1017	1.07165
$790$	−3.22415	−0.114710
$791$	63.3564	2.25269
$792$	22.4895	0.799128
$793$	−11.7712	−0.418007
$794$	63.9974	2.27118
$795$	28.2985	1.00364
$796$	117.476	4.16381
$797$	4.91379	0.174055	0.0870277	−	0.996206i	$-0.472263\pi$
$797$	4.91379	0.174055	0.0870277	+	0.996206i	$0.472263\pi$
$798$	26.4878	0.937659
$799$	−21.4087	−0.757385
$800$	−16.7061	−0.590649
$801$	1.98002	0.0699606
$802$	47.5200	1.67799
$803$	−28.2590	−0.997237
$804$	−46.4953	−1.63976
$805$	−27.5133	−0.969718
$806$	−24.6920	−0.869738
$807$	−46.6138	−1.64088
$808$	52.8236	1.85833
$809$	9.58417	0.336961	0.168481	−	0.985705i	$-0.446114\pi$
$809$	9.58417	0.336961	0.168481	+	0.985705i	$0.446114\pi$
$810$	48.6096	1.70797
$811$	30.3837	1.06692	0.533458	−	0.845827i	$-0.320892\pi$
$811$	30.3837	1.06692	0.533458	+	0.845827i	$0.320892\pi$
$812$	−12.8518	−0.451010
$813$	−28.6568	−1.00504
$814$	12.1956	0.427457
$815$	11.7002	0.409841
$816$	−53.0235	−1.85619
$817$	1.70715	0.0597257
$818$	58.9743	2.06199
$819$	5.94648	0.207787
$820$	78.3145	2.73486
$821$	−29.6974	−1.03645	−0.518223	−	0.855245i	$-0.673406\pi$
$821$	−29.6974	−1.03645	−0.518223	+	0.855245i	$0.673406\pi$
$822$	76.0637	2.65303
$823$	22.7377	0.792586	0.396293	−	0.918124i	$-0.370296\pi$
$823$	22.7377	0.792586	0.396293	+	0.918124i	$0.370296\pi$
$824$	−110.950	−3.86512
$825$	13.0352	0.453827
$826$	30.3317	1.05538
$827$	−19.4280	−0.675578	−0.337789	−	0.941222i	$-0.609679\pi$
$827$	−19.4280	−0.675578	−0.337789	+	0.941222i	$0.609679\pi$
$828$	25.2859	0.878745
$829$	26.5541	0.922260	0.461130	−	0.887332i	$-0.347444\pi$
$829$	26.5541	0.922260	0.461130	+	0.887332i	$0.347444\pi$
$830$	4.63418	0.160855
$831$	−2.48418	−0.0861754
$832$	6.57975	0.228112
$833$	9.46410	0.327912
$834$	6.82513	0.236335
$835$	−16.6521	−0.576268
$836$	−23.0864	−0.798459
$837$	−21.1419	−0.730772
$838$	−82.6968	−2.85671
$839$	48.8188	1.68541	0.842706	−	0.538374i	$-0.180961\pi$
$839$	48.8188	1.68541	0.842706	+	0.538374i	$0.180961\pi$
$840$	74.6539	2.57580
$841$	−28.2417	−0.973850
$842$	−72.4955	−2.49836
$843$	−49.0697	−1.69005
$844$	8.68380	0.298909
$845$	16.8236	0.578751
$846$	−18.7697	−0.645314
$847$	−4.79504	−0.164760
$848$	69.2903	2.37944
$849$	18.1300	0.622218
$850$	−16.9598	−0.581718
$851$	7.82877	0.268367
$852$	144.287	4.94318
$853$	−20.5013	−0.701950	−0.350975	−	0.936385i	$-0.614150\pi$
$853$	−20.5013	−0.701950	−0.350975	+	0.936385i	$0.614150\pi$
$854$	−54.4446	−1.86305
$855$	2.91141	0.0995682
$856$	−91.4694	−3.12636
$857$	−17.3353	−0.592162	−0.296081	−	0.955163i	$-0.595680\pi$
$857$	−17.3353	−0.592162	−0.296081	+	0.955163i	$0.595680\pi$
$858$	−28.3079	−0.966416
$859$	−44.0404	−1.50264	−0.751320	−	0.659938i	$-0.770582\pi$
$859$	−44.0404	−1.50264	−0.751320	+	0.659938i	$0.770582\pi$
$860$	8.42731	0.287369
$861$	62.9595	2.14565
$862$	15.5184	0.528558
$863$	−39.7354	−1.35261	−0.676304	−	0.736623i	$-0.736419\pi$
$863$	−39.7354	−1.35261	−0.676304	+	0.736623i	$0.736419\pi$
$864$	31.0637	1.05681
$865$	−10.9599	−0.372647
$866$	103.799	3.52725
$867$	14.5250	0.493295
$868$	−79.9171	−2.71256
$869$	2.25914	0.0766362
$870$	−7.71543	−0.261578
$871$	8.74258	0.296231
$872$	104.034	3.52302
$873$	6.26902	0.212174
$874$	−21.1785	−0.716374
$875$	38.2811	1.29414
$876$	86.2114	2.91281
$877$	39.2455	1.32523	0.662613	−	0.748962i	$-0.269447\pi$
$877$	39.2455	1.32523	0.662613	+	0.748962i	$0.269447\pi$
$878$	−45.5480	−1.53717
$879$	−19.0664	−0.643095
$880$	−44.0869	−1.48617
$881$	−13.3068	−0.448317	−0.224158	−	0.974553i	$-0.571963\pi$
$881$	−13.3068	−0.448317	−0.224158	+	0.974553i	$0.571963\pi$
$882$	8.29747	0.279390
$883$	−23.2186	−0.781367	−0.390683	−	0.920525i	$-0.627761\pi$
$883$	−23.2186	−0.781367	−0.390683	+	0.920525i	$0.627761\pi$
$884$	25.7728	0.866834
$885$	12.7421	0.428322
$886$	21.5662	0.724532
$887$	46.1980	1.55118	0.775589	−	0.631238i	$-0.217453\pi$
$887$	46.1980	1.55118	0.775589	+	0.631238i	$0.217453\pi$
$888$	−21.2423	−0.712846
$889$	−58.3436	−1.95678
$890$	−8.18660	−0.274415
$891$	−34.0604	−1.14107
$892$	−22.9293	−0.767729
$893$	11.0008	0.368126
$894$	−96.5413	−3.22883
$895$	−11.9482	−0.399384
$896$	−19.9482	−0.666423
$897$	−18.1717	−0.606737
$898$	56.4064	1.88231
$899$	4.71560	0.157274
$900$	−10.4049	−0.346830
$901$	25.7986	0.859476
$902$	−78.4190	−2.61107
$903$	6.77498	0.225457
$904$	137.451	4.57156
$905$	−1.79259	−0.0595877
$906$	43.9841	1.46127
$907$	32.9747	1.09491	0.547454	−	0.836836i	$-0.315597\pi$
$907$	32.9747	1.09491	0.547454	+	0.836836i	$0.315597\pi$
$908$	−44.3257	−1.47100
$909$	8.17551	0.271165
$910$	−24.5863	−0.815029
$911$	−19.0345	−0.630641	−0.315320	−	0.948985i	$-0.602112\pi$
$911$	−19.0345	−0.630641	−0.315320	+	0.948985i	$0.602112\pi$
$912$	27.2459	0.902202
$913$	−3.24715	−0.107465
$914$	60.3870	1.99742
$915$	−22.8718	−0.756117
$916$	−53.1670	−1.75669
$917$	63.0349	2.08160
$918$	31.5356	1.04083
$919$	22.6059	0.745699	0.372850	−	0.927892i	$-0.378381\pi$
$919$	22.6059	0.745699	0.372850	+	0.927892i	$0.378381\pi$
$920$	−59.6900	−1.96792
$921$	−33.7622	−1.11250
$922$	22.2691	0.733392
$923$	−27.1304	−0.893009
$924$	−91.6203	−3.01409
$925$	−3.22146	−0.105921
$926$	21.1959	0.696540
$927$	−17.1717	−0.563993
$928$	−6.92861	−0.227443
$929$	−15.1631	−0.497486	−0.248743	−	0.968569i	$-0.580018\pi$
$929$	−15.1631	−0.497486	−0.248743	+	0.968569i	$0.580018\pi$
$930$	−47.9773	−1.57324
$931$	−4.86309	−0.159381
$932$	132.274	4.33278
$933$	31.4732	1.03039
$934$	−0.495645	−0.0162180
$935$	−16.4147	−0.536818
$936$	12.9008	0.421677
$937$	−11.3706	−0.371462	−0.185731	−	0.982601i	$-0.559465\pi$
$937$	−11.3706	−0.371462	−0.185731	+	0.982601i	$0.559465\pi$
$938$	40.4365	1.32030
$939$	−30.2517	−0.987228
$940$	54.3049	1.77123
$941$	26.9663	0.879077	0.439538	−	0.898224i	$-0.355142\pi$
$941$	26.9663	0.879077	0.439538	+	0.898224i	$0.355142\pi$
$942$	70.2131	2.28767
$943$	−50.3397	−1.63928
$944$	31.1998	1.01547
$945$	−21.0515	−0.684804
$946$	−8.43856	−0.274361
$947$	−6.59044	−0.214161	−0.107080	−	0.994250i	$-0.534150\pi$
$947$	−6.59044	−0.214161	−0.107080	+	0.994250i	$0.534150\pi$
$948$	−6.89211	−0.223845
$949$	−16.2105	−0.526214
$950$	8.71474	0.282744
$951$	−25.8433	−0.838027
$952$	68.0591	2.20581
$953$	−21.0244	−0.681046	−0.340523	−	0.940236i	$-0.610604\pi$
$953$	−21.0244	−0.681046	−0.340523	+	0.940236i	$0.610604\pi$
$954$	22.6184	0.732299
$955$	−30.8544	−0.998424
$956$	−138.256	−4.47151
$957$	5.40616	0.174756
$958$	92.1985	2.97880
$959$	−46.2905	−1.49480
$960$	12.7847	0.412623
$961$	−1.67673	−0.0540881
$962$	6.99590	0.225557
$963$	−14.1567	−0.456194
$964$	27.1416	0.874171
$965$	−29.9048	−0.962670
$966$	−84.0487	−2.70422
$967$	41.7081	1.34124	0.670620	−	0.741801i	$-0.266028\pi$
$967$	41.7081	1.34124	0.670620	+	0.741801i	$0.266028\pi$
$968$	−10.4028	−0.334359
$969$	10.1444	0.325884
$970$	−25.9199	−0.832238
$971$	36.8797	1.18353	0.591763	−	0.806112i	$-0.298432\pi$
$971$	36.8797	1.18353	0.591763	+	0.806112i	$0.298432\pi$
$972$	49.3131	1.58172
$973$	−4.15360	−0.133158
$974$	−55.8797	−1.79050
$975$	7.47749	0.239471
$976$	−56.0027	−1.79260
$977$	13.0561	0.417701	0.208850	−	0.977948i	$-0.433028\pi$
$977$	13.0561	0.417701	0.208850	+	0.977948i	$0.433028\pi$
$978$	35.7423	1.14291
$979$	5.73630	0.183333
$980$	−24.0065	−0.766859
$981$	16.1013	0.514074
$982$	−25.5842	−0.816425
$983$	−59.7269	−1.90499	−0.952496	−	0.304552i	$-0.901493\pi$
$983$	−59.7269	−1.90499	−0.952496	+	0.304552i	$0.901493\pi$
$984$	136.590	4.35433
$985$	26.6471	0.849046
$986$	−7.03386	−0.224004
$987$	43.6575	1.38963
$988$	−13.2433	−0.421324
$989$	−5.41698	−0.172250
$990$	−14.3913	−0.457385
$991$	−8.39032	−0.266527	−0.133264	−	0.991081i	$-0.542546\pi$
$991$	−8.39032	−0.266527	−0.133264	+	0.991081i	$0.542546\pi$
$992$	−43.0845	−1.36794
$993$	46.6455	1.48025
$994$	−125.485	−3.98014
$995$	−42.9198	−1.36065
$996$	9.90627	0.313892
$997$	23.7258	0.751403	0.375702	−	0.926741i	$-0.377402\pi$
$997$	23.7258	0.751403	0.375702	+	0.926741i	$0.377402\pi$
$998$	−24.7204	−0.782511
$999$	5.99007	0.189518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	619.2.a.b.1.28	✓	30
3.2	odd	2		5571.2.a.g.1.3		30
4.3	odd	2		9904.2.a.n.1.24		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.28	✓	30	1.1	even	1	trivial
5571.2.a.g.1.3		30	3.2	odd	2
9904.2.a.n.1.24		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)

\(f(q)\)	\(=\)	\(q+2.58096 q^{2} -2.01571 q^{3} +4.66134 q^{4} -1.70302 q^{5} -5.20246 q^{6} +3.16609 q^{7} +6.86880 q^{8} +1.06308 q^{9} +O(q^{10})\)	\(q+2.58096 q^{2} -2.01571 q^{3} +4.66134 q^{4} -1.70302 q^{5} -5.20246 q^{6} +3.16609 q^{7} +6.86880 q^{8} +1.06308 q^{9} -4.39543 q^{10} +3.07985 q^{11} -9.39590 q^{12} +1.76673 q^{13} +8.17153 q^{14} +3.43280 q^{15} +8.40540 q^{16} +3.12955 q^{17} +2.74378 q^{18} -1.60811 q^{19} -7.93837 q^{20} -6.38191 q^{21} +7.94897 q^{22} +5.10270 q^{23} -13.8455 q^{24} -2.09971 q^{25} +4.55984 q^{26} +3.90426 q^{27} +14.7582 q^{28} -0.870825 q^{29} +8.85992 q^{30} -5.41510 q^{31} +7.95637 q^{32} -6.20809 q^{33} +8.07724 q^{34} -5.39192 q^{35} +4.95540 q^{36} +1.53424 q^{37} -4.15045 q^{38} -3.56120 q^{39} -11.6977 q^{40} -9.86530 q^{41} -16.4714 q^{42} -1.06159 q^{43} +14.3562 q^{44} -1.81046 q^{45} +13.1698 q^{46} -6.84081 q^{47} -16.9428 q^{48} +3.02411 q^{49} -5.41926 q^{50} -6.30827 q^{51} +8.23530 q^{52} +8.24355 q^{53} +10.0767 q^{54} -5.24507 q^{55} +21.7472 q^{56} +3.24148 q^{57} -2.24756 q^{58} +3.71187 q^{59} +16.0015 q^{60} -6.66271 q^{61} -13.9761 q^{62} +3.36582 q^{63} +3.72426 q^{64} -3.00878 q^{65} -16.0228 q^{66} +4.94846 q^{67} +14.5879 q^{68} -10.2856 q^{69} -13.9163 q^{70} -15.3563 q^{71} +7.30212 q^{72} -9.17543 q^{73} +3.95981 q^{74} +4.23240 q^{75} -7.49593 q^{76} +9.75108 q^{77} -9.19132 q^{78} +0.733523 q^{79} -14.3146 q^{80} -11.0591 q^{81} -25.4619 q^{82} -1.05432 q^{83} -29.7483 q^{84} -5.32970 q^{85} -2.73992 q^{86} +1.75533 q^{87} +21.1549 q^{88} +1.86252 q^{89} -4.67272 q^{90} +5.59361 q^{91} +23.7854 q^{92} +10.9153 q^{93} -17.6558 q^{94} +2.73865 q^{95} -16.0377 q^{96} +5.89701 q^{97} +7.80509 q^{98} +3.27415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{2} + q^{3} + 33 q^{4} + 21 q^{5} + 6 q^{6} + 2 q^{7} + 27 q^{8} + 43 q^{9}+O(q^{10}) \) 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