LMFDB - Embedded newform 619.2.a.b.1.11

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			1.11
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)

$f(q)$	$=$	$q-0.394547 q^{2} +1.27337 q^{3} -1.84433 q^{4} +1.92856 q^{5} -0.502406 q^{6} +3.31459 q^{7} +1.51677 q^{8} -1.37852 q^{9} +O(q^{10})$	\(q-0.394547 q^{2} +1.27337 q^{3} -1.84433 q^{4} +1.92856 q^{5} -0.502406 q^{6} +3.31459 q^{7} +1.51677 q^{8} -1.37852 q^{9} -0.760909 q^{10} +2.52548 q^{11} -2.34852 q^{12} -4.15823 q^{13} -1.30776 q^{14} +2.45578 q^{15} +3.09023 q^{16} +2.64063 q^{17} +0.543891 q^{18} +6.63734 q^{19} -3.55691 q^{20} +4.22072 q^{21} -0.996422 q^{22} -4.34053 q^{23} +1.93142 q^{24} -1.28064 q^{25} +1.64062 q^{26} -5.57549 q^{27} -6.11321 q^{28} +0.725269 q^{29} -0.968922 q^{30} +6.55612 q^{31} -4.25278 q^{32} +3.21588 q^{33} -1.04185 q^{34} +6.39241 q^{35} +2.54245 q^{36} +6.31493 q^{37} -2.61874 q^{38} -5.29499 q^{39} +2.92519 q^{40} -0.0915471 q^{41} -1.66527 q^{42} +0.725844 q^{43} -4.65783 q^{44} -2.65856 q^{45} +1.71254 q^{46} +10.3201 q^{47} +3.93501 q^{48} +3.98653 q^{49} +0.505274 q^{50} +3.36251 q^{51} +7.66917 q^{52} -4.81831 q^{53} +2.19979 q^{54} +4.87055 q^{55} +5.02748 q^{56} +8.45182 q^{57} -0.286153 q^{58} +12.8732 q^{59} -4.52928 q^{60} -3.71827 q^{61} -2.58670 q^{62} -4.56923 q^{63} -4.50253 q^{64} -8.01942 q^{65} -1.26882 q^{66} -7.85813 q^{67} -4.87020 q^{68} -5.52711 q^{69} -2.52211 q^{70} -8.16026 q^{71} -2.09090 q^{72} -12.1531 q^{73} -2.49154 q^{74} -1.63074 q^{75} -12.2415 q^{76} +8.37095 q^{77} +2.08912 q^{78} +14.4595 q^{79} +5.95970 q^{80} -2.96412 q^{81} +0.0361197 q^{82} -2.55883 q^{83} -7.78440 q^{84} +5.09262 q^{85} -0.286380 q^{86} +0.923538 q^{87} +3.83058 q^{88} -9.14859 q^{89} +1.04893 q^{90} -13.7829 q^{91} +8.00538 q^{92} +8.34839 q^{93} -4.07177 q^{94} +12.8005 q^{95} -5.41538 q^{96} +0.730351 q^{97} -1.57288 q^{98} -3.48143 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.394547	−0.278987	−0.139493	−	0.990223i	$-0.544547\pi$
$2$	−0.394547	−0.278987	−0.139493	+	0.990223i	$0.544547\pi$
$3$	1.27337	0.735183	0.367591	−	0.929987i	$-0.380183\pi$
$3$	1.27337	0.735183	0.367591	+	0.929987i	$0.380183\pi$
$4$	−1.84433	−0.922166
$5$	1.92856	0.862480	0.431240	−	0.902237i	$-0.358076\pi$
$5$	1.92856	0.862480	0.431240	+	0.902237i	$0.358076\pi$
$6$	−0.502406	−0.205106
$7$	3.31459	1.25280	0.626399	−	0.779502i	$-0.284528\pi$
$7$	3.31459	1.25280	0.626399	+	0.779502i	$0.284528\pi$
$8$	1.51677	0.536259
$9$	−1.37852	−0.459507
$10$	−0.760909	−0.240621
$11$	2.52548	0.761461	0.380731	−	0.924686i	$-0.375672\pi$
$11$	2.52548	0.761461	0.380731	+	0.924686i	$0.375672\pi$
$12$	−2.34852	−0.677961
$13$	−4.15823	−1.15329	−0.576643	−	0.816996i	$-0.695638\pi$
$13$	−4.15823	−1.15329	−0.576643	+	0.816996i	$0.695638\pi$
$14$	−1.30776	−0.349515
$15$	2.45578	0.634080
$16$	3.09023	0.772557
$17$	2.64063	0.640447	0.320223	−	0.947342i	$-0.396242\pi$
$17$	2.64063	0.640447	0.320223	+	0.947342i	$0.396242\pi$
$18$	0.543891	0.128196
$19$	6.63734	1.52271	0.761355	−	0.648335i	$-0.224534\pi$
$19$	6.63734	1.52271	0.761355	+	0.648335i	$0.224534\pi$
$20$	−3.55691	−0.795350
$21$	4.22072	0.921036
$22$	−0.996422	−0.212438
$23$	−4.34053	−0.905063	−0.452531	−	0.891749i	$-0.649479\pi$
$23$	−4.34053	−0.905063	−0.452531	+	0.891749i	$0.649479\pi$
$24$	1.93142	0.394249
$25$	−1.28064	−0.256128
$26$	1.64062	0.321752
$27$	−5.57549	−1.07300
$28$	−6.11321	−1.15529
$29$	0.725269	0.134679	0.0673395	−	0.997730i	$-0.478549\pi$
$29$	0.725269	0.134679	0.0673395	+	0.997730i	$0.478549\pi$
$30$	−0.968922	−0.176900
$31$	6.55612	1.17751	0.588757	−	0.808310i	$-0.299617\pi$
$31$	6.55612	1.17751	0.588757	+	0.808310i	$0.299617\pi$
$32$	−4.25278	−0.751793
$33$	3.21588	0.559813
$34$	−1.04185	−0.178676
$35$	6.39241	1.08051
$36$	2.54245	0.423741
$37$	6.31493	1.03817	0.519084	−	0.854723i	$-0.326273\pi$
$37$	6.31493	1.03817	0.519084	+	0.854723i	$0.326273\pi$
$38$	−2.61874	−0.424816
$39$	−5.29499	−0.847876
$40$	2.92519	0.462513
$41$	−0.0915471	−0.0142973	−0.00714863	−	0.999974i	$-0.502275\pi$
$41$	−0.0915471	−0.0142973	−0.00714863	+	0.999974i	$0.502275\pi$
$42$	−1.66527	−0.256957
$43$	0.725844	0.110690	0.0553451	−	0.998467i	$-0.482374\pi$
$43$	0.725844	0.110690	0.0553451	+	0.998467i	$0.482374\pi$
$44$	−4.65783	−0.702194
$45$	−2.65856	−0.396315
$46$	1.71254	0.252501
$47$	10.3201	1.50534	0.752671	−	0.658397i	$-0.228765\pi$
$47$	10.3201	1.50534	0.752671	+	0.658397i	$0.228765\pi$
$48$	3.93501	0.567970
$49$	3.98653	0.569505
$50$	0.505274	0.0714565
$51$	3.36251	0.470845
$52$	7.66917	1.06352
$53$	−4.81831	−0.661845	−0.330923	−	0.943658i	$-0.607360\pi$
$53$	−4.81831	−0.661845	−0.330923	+	0.943658i	$0.607360\pi$
$54$	2.19979	0.299354
$55$	4.87055	0.656745
$56$	5.02748	0.671825
$57$	8.45182	1.11947
$58$	−0.286153	−0.0375737
$59$	12.8732	1.67595	0.837974	−	0.545710i	$-0.183740\pi$
$59$	12.8732	1.67595	0.837974	+	0.545710i	$0.183740\pi$
$60$	−4.52928	−0.584727
$61$	−3.71827	−0.476076	−0.238038	−	0.971256i	$-0.576504\pi$
$61$	−3.71827	−0.476076	−0.238038	+	0.971256i	$0.576504\pi$
$62$	−2.58670	−0.328511
$63$	−4.56923	−0.575669
$64$	−4.50253	−0.562817
$65$	−8.01942	−0.994687
$66$	−1.26882	−0.156181
$67$	−7.85813	−0.960023	−0.480012	−	0.877262i	$-0.659368\pi$
$67$	−7.85813	−0.960023	−0.480012	+	0.877262i	$0.659368\pi$
$68$	−4.87020	−0.590598
$69$	−5.52711	−0.665386
$70$	−2.52211	−0.301449
$71$	−8.16026	−0.968445	−0.484223	−	0.874945i	$-0.660898\pi$
$71$	−8.16026	−0.968445	−0.484223	+	0.874945i	$0.660898\pi$
$72$	−2.09090	−0.246415
$73$	−12.1531	−1.42241	−0.711205	−	0.702984i	$-0.751850\pi$
$73$	−12.1531	−1.42241	−0.711205	+	0.702984i	$0.751850\pi$
$74$	−2.49154	−0.289635
$75$	−1.63074	−0.188301
$76$	−12.2415	−1.40419
$77$	8.37095	0.953958
$78$	2.08912	0.236546
$79$	14.4595	1.62683	0.813413	−	0.581686i	$-0.197607\pi$
$79$	14.4595	1.62683	0.813413	+	0.581686i	$0.197607\pi$
$80$	5.95970	0.666315
$81$	−2.96412	−0.329347
$82$	0.0361197	0.00398875
$83$	−2.55883	−0.280868	−0.140434	−	0.990090i	$-0.544850\pi$
$83$	−2.55883	−0.280868	−0.140434	+	0.990090i	$0.544850\pi$
$84$	−7.78440	−0.849348
$85$	5.09262	0.552372
$86$	−0.286380	−0.0308811
$87$	0.923538	0.0990137
$88$	3.83058	0.408341
$89$	−9.14859	−0.969748	−0.484874	−	0.874584i	$-0.661135\pi$
$89$	−9.14859	−0.969748	−0.484874	+	0.874584i	$0.661135\pi$
$90$	1.04893	0.110567
$91$	−13.7829	−1.44484
$92$	8.00538	0.834618
$93$	8.34839	0.865688
$94$	−4.07177	−0.419971
$95$	12.8005	1.31331
$96$	−5.41538	−0.552705
$97$	0.730351	0.0741559	0.0370780	−	0.999312i	$-0.488195\pi$
$97$	0.730351	0.0741559	0.0370780	+	0.999312i	$0.488195\pi$
$98$	−1.57288	−0.158884
$99$	−3.48143	−0.349897
$100$	2.36193	0.236193
$101$	−3.22161	−0.320563	−0.160281	−	0.987071i	$-0.551240\pi$
$101$	−3.22161	−0.320563	−0.160281	+	0.987071i	$0.551240\pi$
$102$	−1.32667	−0.131360
$103$	3.68117	0.362716	0.181358	−	0.983417i	$-0.441951\pi$
$103$	3.68117	0.362716	0.181358	+	0.983417i	$0.441951\pi$
$104$	−6.30709	−0.618461
$105$	8.13992	0.794375
$106$	1.90105	0.184646
$107$	−9.76971	−0.944474	−0.472237	−	0.881472i	$-0.656553\pi$
$107$	−9.76971	−0.944474	−0.472237	+	0.881472i	$0.656553\pi$
$108$	10.2831	0.989488
$109$	−18.7013	−1.79126	−0.895631	−	0.444798i	$-0.853275\pi$
$109$	−18.7013	−1.79126	−0.895631	+	0.444798i	$0.853275\pi$
$110$	−1.92166	−0.183223
$111$	8.04126	0.763243
$112$	10.2429	0.967858
$113$	7.61333	0.716202	0.358101	−	0.933683i	$-0.383424\pi$
$113$	7.61333	0.716202	0.358101	+	0.933683i	$0.383424\pi$
$114$	−3.33464	−0.312318
$115$	−8.37098	−0.780598
$116$	−1.33764	−0.124196
$117$	5.73221	0.529943
$118$	−5.07908	−0.467568
$119$	8.75261	0.802351
$120$	3.72486	0.340031
$121$	−4.62194	−0.420176
$122$	1.46703	0.132819
$123$	−0.116574	−0.0105111
$124$	−12.0917	−1.08586
$125$	−12.1126	−1.08339
$126$	1.80278	0.160604
$127$	−15.5583	−1.38057	−0.690287	−	0.723535i	$-0.742516\pi$
$127$	−15.5583	−1.38057	−0.690287	+	0.723535i	$0.742516\pi$
$128$	10.2820	0.908811
$129$	0.924270	0.0813774
$130$	3.16404	0.277505
$131$	−1.28101	−0.111923	−0.0559614	−	0.998433i	$-0.517822\pi$
$131$	−1.28101	−0.111923	−0.0559614	+	0.998433i	$0.517822\pi$
$132$	−5.93116	−0.516241
$133$	22.0001	1.90765
$134$	3.10040	0.267834
$135$	−10.7527	−0.925444
$136$	4.00523	0.343446
$137$	0.214753	0.0183476	0.00917378	−	0.999958i	$-0.497080\pi$
$137$	0.214753	0.0183476	0.00917378	+	0.999958i	$0.497080\pi$
$138$	2.18071	0.185634
$139$	−4.63534	−0.393165	−0.196582	−	0.980487i	$-0.562984\pi$
$139$	−4.63534	−0.393165	−0.196582	+	0.980487i	$0.562984\pi$
$140$	−11.7897	−0.996413
$141$	13.1414	1.10670
$142$	3.21961	0.270184
$143$	−10.5015	−0.878183
$144$	−4.25994	−0.354995
$145$	1.39873	0.116158
$146$	4.79496	0.396834
$147$	5.07635	0.418690
$148$	−11.6468	−0.957363
$149$	2.25407	0.184660	0.0923302	−	0.995728i	$-0.470568\pi$
$149$	2.25407	0.184660	0.0923302	+	0.995728i	$0.470568\pi$
$150$	0.643402	0.0525336
$151$	−22.3859	−1.82174	−0.910870	−	0.412694i	$-0.864588\pi$
$151$	−22.3859	−1.82174	−0.910870	+	0.412694i	$0.864588\pi$
$152$	10.0673	0.816568
$153$	−3.64016	−0.294289
$154$	−3.30273	−0.266142
$155$	12.6439	1.01558
$156$	9.76571	0.781883
$157$	−7.69459	−0.614095	−0.307047	−	0.951694i	$-0.599341\pi$
$157$	−7.69459	−0.614095	−0.307047	+	0.951694i	$0.599341\pi$
$158$	−5.70497	−0.453863
$159$	−6.13550	−0.486577
$160$	−8.20176	−0.648406
$161$	−14.3871	−1.13386
$162$	1.16949	0.0918836
$163$	−20.7421	−1.62465	−0.812325	−	0.583205i	$-0.801798\pi$
$163$	−20.7421	−1.62465	−0.812325	+	0.583205i	$0.801798\pi$
$164$	0.168843	0.0131844
$165$	6.20203	0.482828
$166$	1.00958	0.0783584
$167$	9.68268	0.749268	0.374634	−	0.927173i	$-0.377768\pi$
$167$	9.68268	0.749268	0.374634	+	0.927173i	$0.377768\pi$
$168$	6.40186	0.493914
$169$	4.29091	0.330070
$170$	−2.00928	−0.154105
$171$	−9.14971	−0.699696
$172$	−1.33870	−0.102075
$173$	−13.6349	−1.03664	−0.518322	−	0.855186i	$-0.673443\pi$
$173$	−13.6349	−1.03664	−0.518322	+	0.855186i	$0.673443\pi$
$174$	−0.364379	−0.0276235
$175$	−4.24481	−0.320877
$176$	7.80431	0.588272
$177$	16.3924	1.23213
$178$	3.60955	0.270547
$179$	21.4754	1.60515	0.802575	−	0.596551i	$-0.203463\pi$
$179$	21.4754	1.60515	0.802575	+	0.596551i	$0.203463\pi$
$180$	4.90327	0.365469
$181$	11.5422	0.857922	0.428961	−	0.903323i	$-0.358880\pi$
$181$	11.5422	0.857922	0.428961	+	0.903323i	$0.358880\pi$
$182$	5.43799	0.403090
$183$	−4.73475	−0.350003
$184$	−6.58358	−0.485348
$185$	12.1787	0.895399
$186$	−3.29384	−0.241516
$187$	6.66886	0.487676
$188$	−19.0337	−1.38818
$189$	−18.4805	−1.34426
$190$	−5.05042	−0.366396
$191$	−13.8904	−1.00507	−0.502536	−	0.864556i	$-0.667599\pi$
$191$	−13.8904	−1.00507	−0.502536	+	0.864556i	$0.667599\pi$
$192$	−5.73341	−0.413773
$193$	16.8713	1.21442	0.607210	−	0.794541i	$-0.292289\pi$
$193$	16.8713	1.21442	0.607210	+	0.794541i	$0.292289\pi$
$194$	−0.288158	−0.0206885
$195$	−10.2117	−0.731276
$196$	−7.35249	−0.525178
$197$	26.7475	1.90568	0.952840	−	0.303473i	$-0.0981461\pi$
$197$	26.7475	1.90568	0.952840	+	0.303473i	$0.0981461\pi$
$198$	1.37359	0.0976166
$199$	−16.1990	−1.14832	−0.574159	−	0.818744i	$-0.694671\pi$
$199$	−16.1990	−1.14832	−0.574159	+	0.818744i	$0.694671\pi$
$200$	−1.94244	−0.137351
$201$	−10.0063	−0.705792
$202$	1.27108	0.0894328
$203$	2.40397	0.168726
$204$	−6.20158	−0.434198
$205$	−0.176554	−0.0123311
$206$	−1.45239	−0.101193
$207$	5.98350	0.415882
$208$	−12.8499	−0.890980
$209$	16.7625	1.15949
$210$	−3.21158	−0.221620
$211$	−17.5817	−1.21038	−0.605188	−	0.796082i	$-0.706902\pi$
$211$	−17.5817	−1.21038	−0.605188	+	0.796082i	$0.706902\pi$
$212$	8.88656	0.610331
$213$	−10.3911	−0.711984
$214$	3.85461	0.263496
$215$	1.39984	0.0954680
$216$	−8.45674	−0.575408
$217$	21.7309	1.47519
$218$	7.37855	0.499739
$219$	−15.4754	−1.04573
$220$	−8.98292	−0.605628
$221$	−10.9804	−0.738619
$222$	−3.17266	−0.212935
$223$	2.98314	0.199766	0.0998830	−	0.994999i	$-0.468153\pi$
$223$	2.98314	0.199766	0.0998830	+	0.994999i	$0.468153\pi$
$224$	−14.0962	−0.941845
$225$	1.76539	0.117693
$226$	−3.00382	−0.199811
$227$	−0.00326963	−0.000217013 0	−0.000108507	−	1.00000i	$-0.500035\pi$
$227$	−0.00326963	−0.000217013 0	−0.000108507		1.00000i	$0.500035\pi$
$228$	−15.5880	−1.03234
$229$	−6.94976	−0.459253	−0.229627	−	0.973279i	$-0.573750\pi$
$229$	−6.94976	−0.459253	−0.229627	+	0.973279i	$0.573750\pi$
$230$	3.30275	0.217777
$231$	10.6593	0.701333
$232$	1.10007	0.0722229
$233$	1.93016	0.126449	0.0632246	−	0.997999i	$-0.479862\pi$
$233$	1.93016	0.126449	0.0632246	+	0.997999i	$0.479862\pi$
$234$	−2.26163	−0.147847
$235$	19.9030	1.29833
$236$	−23.7425	−1.54550
$237$	18.4124	1.19601
$238$	−3.45332	−0.223845
$239$	−8.21315	−0.531265	−0.265632	−	0.964074i	$-0.585581\pi$
$239$	−8.21315	−0.531265	−0.265632	+	0.964074i	$0.585581\pi$
$240$	7.58893	0.489863
$241$	13.0765	0.842328	0.421164	−	0.906984i	$-0.361622\pi$
$241$	13.0765	0.842328	0.421164	+	0.906984i	$0.361622\pi$
$242$	1.82357	0.117224
$243$	12.9520	0.830874
$244$	6.85773	0.439021
$245$	7.68828	0.491186
$246$	0.0459938	0.00293246
$247$	−27.5996	−1.75612
$248$	9.94413	0.631453
$249$	−3.25834	−0.206489
$250$	4.77900	0.302250
$251$	5.46727	0.345091	0.172545	−	0.985002i	$-0.444801\pi$
$251$	5.46727	0.345091	0.172545	+	0.985002i	$0.444801\pi$
$252$	8.42719	0.530863
$253$	−10.9619	−0.689170
$254$	6.13848	0.385162
$255$	6.48481	0.406095
$256$	4.94832	0.309270
$257$	−12.6379	−0.788330	−0.394165	−	0.919040i	$-0.628966\pi$
$257$	−12.6379	−0.788330	−0.394165	+	0.919040i	$0.628966\pi$
$258$	−0.364668	−0.0227032
$259$	20.9314	1.30061
$260$	14.7905	0.917266
$261$	−0.999798	−0.0618859
$262$	0.505421	0.0312250
$263$	−15.9455	−0.983244	−0.491622	−	0.870809i	$-0.663596\pi$
$263$	−15.9455	−0.983244	−0.491622	+	0.870809i	$0.663596\pi$
$264$	4.87775	0.300205
$265$	−9.29241	−0.570828
$266$	−8.68008	−0.532210
$267$	−11.6496	−0.712942
$268$	14.4930	0.885301
$269$	27.9611	1.70482	0.852409	−	0.522876i	$-0.175141\pi$
$269$	27.9611	1.70482	0.852409	+	0.522876i	$0.175141\pi$
$270$	4.24244	0.258187
$271$	9.71775	0.590311	0.295156	−	0.955449i	$-0.404629\pi$
$271$	9.71775	0.590311	0.295156	+	0.955449i	$0.404629\pi$
$272$	8.16015	0.494782
$273$	−17.5507	−1.06222
$274$	−0.0847301	−0.00511873
$275$	−3.23424	−0.195032
$276$	10.1938	0.613597
$277$	4.10155	0.246438	0.123219	−	0.992379i	$-0.460678\pi$
$277$	4.10155	0.246438	0.123219	+	0.992379i	$0.460678\pi$
$278$	1.82886	0.109688
$279$	−9.03775	−0.541076
$280$	9.69581	0.579436
$281$	14.7833	0.881899	0.440950	−	0.897532i	$-0.354642\pi$
$281$	14.7833	0.881899	0.440950	+	0.897532i	$0.354642\pi$
$282$	−5.18488	−0.308755
$283$	−2.82855	−0.168140	−0.0840699	−	0.996460i	$-0.526792\pi$
$283$	−2.82855	−0.168140	−0.0840699	+	0.996460i	$0.526792\pi$
$284$	15.0502	0.893067
$285$	16.2999	0.965521
$286$	4.14335	0.245002
$287$	−0.303442	−0.0179116
$288$	5.86254	0.345454
$289$	−10.0271	−0.589828
$290$	−0.551864	−0.0324066
$291$	0.930010	0.0545182
$292$	22.4143	1.31170
$293$	−3.10899	−0.181629	−0.0908147	−	0.995868i	$-0.528947\pi$
$293$	−3.10899	−0.181629	−0.0908147	+	0.995868i	$0.528947\pi$
$294$	−2.00286	−0.116809
$295$	24.8268	1.44547
$296$	9.57829	0.556727
$297$	−14.0808	−0.817051
$298$	−0.889336	−0.0515179
$299$	18.0489	1.04380
$300$	3.00762	0.173645
$301$	2.40588	0.138672
$302$	8.83230	0.508242
$303$	−4.10232	−0.235672
$304$	20.5109	1.17638
$305$	−7.17092	−0.410606
$306$	1.43621	0.0821029
$307$	−6.67202	−0.380792	−0.190396	−	0.981707i	$-0.560977\pi$
$307$	−6.67202	−0.380792	−0.190396	+	0.981707i	$0.560977\pi$
$308$	−15.4388	−0.879708
$309$	4.68750	0.266663
$310$	−4.98862	−0.283334
$311$	34.7399	1.96992	0.984960	−	0.172782i	$-0.0552755\pi$
$311$	34.7399	1.96992	0.984960	+	0.172782i	$0.0552755\pi$
$312$	−8.03128	−0.454682
$313$	14.6252	0.826667	0.413334	−	0.910580i	$-0.364364\pi$
$313$	14.6252	0.826667	0.413334	+	0.910580i	$0.364364\pi$
$314$	3.03588	0.171324
$315$	−8.81206	−0.496503
$316$	−26.6682	−1.50020
$317$	−22.0932	−1.24088	−0.620438	−	0.784255i	$-0.713045\pi$
$317$	−22.0932	−1.24088	−0.620438	+	0.784255i	$0.713045\pi$
$318$	2.42075	0.135749
$319$	1.83165	0.102553
$320$	−8.68342	−0.485418
$321$	−12.4405	−0.694361
$322$	5.67638	0.316333
$323$	17.5268	0.975215
$324$	5.46683	0.303713
$325$	5.32521	0.295389
$326$	8.18375	0.453256
$327$	−23.8138	−1.31690
$328$	−0.138856	−0.00766704
$329$	34.2070	1.88589
$330$	−2.44699	−0.134703
$331$	−22.8034	−1.25339	−0.626694	−	0.779266i	$-0.715592\pi$
$331$	−22.8034	−1.25339	−0.626694	+	0.779266i	$0.715592\pi$
$332$	4.71933	0.259007
$333$	−8.70525	−0.477045
$334$	−3.82027	−0.209036
$335$	−15.1549	−0.828001
$336$	13.0430	0.711553
$337$	−14.0107	−0.763214	−0.381607	−	0.924325i	$-0.624629\pi$
$337$	−14.0107	−0.763214	−0.381607	+	0.924325i	$0.624629\pi$
$338$	−1.69297	−0.0920853
$339$	9.69461	0.526539
$340$	−9.39249	−0.509379
$341$	16.5574	0.896632
$342$	3.60999	0.195206
$343$	−9.98842	−0.539324
$344$	1.10094	0.0593586
$345$	−10.6594	−0.573882
$346$	5.37962	0.289210
$347$	17.8681	0.959208	0.479604	−	0.877485i	$-0.340780\pi$
$347$	17.8681	0.959208	0.479604	+	0.877485i	$0.340780\pi$
$348$	−1.70331	−0.0913071
$349$	30.5664	1.63618	0.818090	−	0.575090i	$-0.195033\pi$
$349$	30.5664	1.63618	0.818090	+	0.575090i	$0.195033\pi$
$350$	1.67478	0.0895206
$351$	23.1842	1.23748
$352$	−10.7403	−0.572461
$353$	15.6993	0.835591	0.417796	−	0.908541i	$-0.362803\pi$
$353$	15.6993	0.835591	0.417796	+	0.908541i	$0.362803\pi$
$354$	−6.46757	−0.343748
$355$	−15.7376	−0.835265
$356$	16.8730	0.894269
$357$	11.1453	0.589874
$358$	−8.47307	−0.447816
$359$	10.8785	0.574147	0.287073	−	0.957909i	$-0.407318\pi$
$359$	10.8785	0.574147	0.287073	+	0.957909i	$0.407318\pi$
$360$	−4.03243	−0.212528
$361$	25.0543	1.31865
$362$	−4.55393	−0.239349
$363$	−5.88546	−0.308906
$364$	25.4202	1.33238
$365$	−23.4380	−1.22680
$366$	1.86808	0.0976462
$367$	34.1890	1.78465	0.892324	−	0.451395i	$-0.149073\pi$
$367$	34.1890	1.78465	0.892324	+	0.451395i	$0.149073\pi$
$368$	−13.4132	−0.699212
$369$	0.126200	0.00656968
$370$	−4.80509	−0.249805
$371$	−15.9707	−0.829159
$372$	−15.3972	−0.798308
$373$	−11.4843	−0.594635	−0.297317	−	0.954779i	$-0.596092\pi$
$373$	−11.4843	−0.594635	−0.297317	+	0.954779i	$0.596092\pi$
$374$	−2.63118	−0.136055
$375$	−15.4239	−0.796486
$376$	15.6532	0.807254
$377$	−3.01584	−0.155324
$378$	7.29142	0.375030
$379$	−29.6409	−1.52255	−0.761276	−	0.648428i	$-0.775427\pi$
$379$	−29.6409	−1.52255	−0.761276	+	0.648428i	$0.775427\pi$
$380$	−23.6085	−1.21109
$381$	−19.8115	−1.01497
$382$	5.48040	0.280402
$383$	1.97866	0.101105	0.0505525	−	0.998721i	$-0.483902\pi$
$383$	1.97866	0.101105	0.0505525	+	0.998721i	$0.483902\pi$
$384$	13.0929	0.668142
$385$	16.1439	0.822770
$386$	−6.65651	−0.338807
$387$	−1.00059	−0.0508628
$388$	−1.34701	−0.0683841
$389$	30.4944	1.54613	0.773065	−	0.634327i	$-0.218723\pi$
$389$	30.4944	1.54613	0.773065	+	0.634327i	$0.218723\pi$
$390$	4.02900	0.204017
$391$	−11.4617	−0.579644
$392$	6.04666	0.305402
$393$	−1.63121	−0.0822837
$394$	−10.5531	−0.531660
$395$	27.8862	1.40311
$396$	6.42091	0.322663
$397$	−3.31961	−0.166607	−0.0833033	−	0.996524i	$-0.526547\pi$
$397$	−3.31961	−0.166607	−0.0833033	+	0.996524i	$0.526547\pi$
$398$	6.39128	0.320366
$399$	28.0143	1.40247
$400$	−3.95747	−0.197874
$401$	0.0340859	0.00170217	0.000851085	−	1.00000i	$-0.499729\pi$
$401$	0.0340859	0.00170217	0.000851085		1.00000i	$0.499729\pi$
$402$	3.94797	0.196907
$403$	−27.2619	−1.35801
$404$	5.94173	0.295612
$405$	−5.71650	−0.284055
$406$	−0.948480	−0.0470723
$407$	15.9482	0.790525
$408$	5.10015	0.252495
$409$	−24.4981	−1.21135	−0.605677	−	0.795711i	$-0.707098\pi$
$409$	−24.4981	−1.21135	−0.605677	+	0.795711i	$0.707098\pi$
$410$	0.0696591	0.00344022
$411$	0.273460	0.0134888
$412$	−6.78930	−0.334485
$413$	42.6694	2.09963
$414$	−2.36077	−0.116026
$415$	−4.93486	−0.242243
$416$	17.6841	0.867032
$417$	−5.90252	−0.289048
$418$	−6.61359	−0.323481
$419$	27.6511	1.35085	0.675423	−	0.737431i	$-0.263961\pi$
$419$	27.6511	1.35085	0.675423	+	0.737431i	$0.263961\pi$
$420$	−15.0127	−0.732546
$421$	10.5638	0.514846	0.257423	−	0.966299i	$-0.417127\pi$
$421$	10.5638	0.514846	0.257423	+	0.966299i	$0.417127\pi$
$422$	6.93683	0.337679
$423$	−14.2265	−0.691715
$424$	−7.30826	−0.354921
$425$	−3.38170	−0.164037
$426$	4.09976	0.198634
$427$	−12.3246	−0.596427
$428$	18.0186	0.870962
$429$	−13.3724	−0.645625
$430$	−0.552301	−0.0266343
$431$	29.3274	1.41265	0.706327	−	0.707886i	$-0.250351\pi$
$431$	29.3274	1.41265	0.706327	+	0.707886i	$0.250351\pi$
$432$	−17.2295	−0.828957
$433$	−8.94627	−0.429930	−0.214965	−	0.976622i	$-0.568964\pi$
$433$	−8.94627	−0.429930	−0.214965	+	0.976622i	$0.568964\pi$
$434$	−8.57386	−0.411558
$435$	1.78110	0.0853973
$436$	34.4915	1.65184
$437$	−28.8096	−1.37815
$438$	6.10578	0.291745
$439$	18.8444	0.899395	0.449698	−	0.893181i	$-0.351532\pi$
$439$	18.8444	0.899395	0.449698	+	0.893181i	$0.351532\pi$
$440$	7.38751	0.352186
$441$	−5.49551	−0.261691
$442$	4.33227	0.206065
$443$	−23.5362	−1.11824	−0.559119	−	0.829087i	$-0.688861\pi$
$443$	−23.5362	−1.11824	−0.559119	+	0.829087i	$0.688861\pi$
$444$	−14.8308	−0.703837
$445$	−17.6436	−0.836389
$446$	−1.17699	−0.0557321
$447$	2.87027	0.135759
$448$	−14.9241	−0.705096
$449$	10.1071	0.476982	0.238491	−	0.971145i	$-0.423347\pi$
$449$	10.1071	0.476982	0.238491	+	0.971145i	$0.423347\pi$
$450$	−0.696530	−0.0328347
$451$	−0.231201	−0.0108868
$452$	−14.0415	−0.660457
$453$	−28.5056	−1.33931
$454$	0.00129002	6.05438e−5 0
$455$	−26.5811	−1.24614
$456$	12.8195	0.600326
$457$	−4.98983	−0.233415	−0.116707	−	0.993166i	$-0.537234\pi$
$457$	−4.98983	−0.233415	−0.116707	+	0.993166i	$0.537234\pi$
$458$	2.74201	0.128126
$459$	−14.7228	−0.687202
$460$	15.4389	0.719841
$461$	−15.0962	−0.703099	−0.351549	−	0.936169i	$-0.614345\pi$
$461$	−15.0962	−0.703099	−0.351549	+	0.936169i	$0.614345\pi$
$462$	−4.20561	−0.195663
$463$	−1.29951	−0.0603933	−0.0301967	−	0.999544i	$-0.509613\pi$
$463$	−1.29951	−0.0603933	−0.0301967	+	0.999544i	$0.509613\pi$
$464$	2.24125	0.104047
$465$	16.1004	0.746639
$466$	−0.761541	−0.0352777
$467$	24.1905	1.11940	0.559702	−	0.828694i	$-0.310916\pi$
$467$	24.1905	1.11940	0.559702	+	0.828694i	$0.310916\pi$
$468$	−10.5721	−0.488695
$469$	−26.0465	−1.20272
$470$	−7.85267	−0.362216
$471$	−9.79809	−0.451472
$472$	19.5257	0.898743
$473$	1.83311	0.0842863
$474$	−7.26456	−0.333672
$475$	−8.50006	−0.390009
$476$	−16.1427	−0.739901
$477$	6.64213	0.304122
$478$	3.24048	0.148216
$479$	1.78150	0.0813990	0.0406995	−	0.999171i	$-0.487041\pi$
$479$	1.78150	0.0813990	0.0406995	+	0.999171i	$0.487041\pi$
$480$	−10.4439	−0.476697
$481$	−26.2589	−1.19730
$482$	−5.15928	−0.234999
$483$	−18.3201	−0.833595
$484$	8.52440	0.387473
$485$	1.40853	0.0639580
$486$	−5.11019	−0.231803
$487$	−36.1287	−1.63715	−0.818574	−	0.574401i	$-0.805235\pi$
$487$	−36.1287	−1.63715	−0.818574	+	0.574401i	$0.805235\pi$
$488$	−5.63976	−0.255300
$489$	−26.4125	−1.19441
$490$	−3.03339	−0.137035
$491$	−20.4601	−0.923352	−0.461676	−	0.887049i	$-0.652752\pi$
$491$	−20.4601	−0.923352	−0.461676	+	0.887049i	$0.652752\pi$
$492$	0.215001	0.00969298
$493$	1.91517	0.0862548
$494$	10.8894	0.489935
$495$	−6.71415	−0.301779
$496$	20.2599	0.909697
$497$	−27.0480	−1.21327
$498$	1.28557	0.0576078
$499$	−26.8942	−1.20395	−0.601974	−	0.798516i	$-0.705619\pi$
$499$	−26.8942	−1.20395	−0.601974	+	0.798516i	$0.705619\pi$
$500$	22.3397	0.999062
$501$	12.3297	0.550849
$502$	−2.15709	−0.0962759
$503$	−39.3954	−1.75655	−0.878276	−	0.478154i	$-0.841306\pi$
$503$	−39.3954	−1.75655	−0.878276	+	0.478154i	$0.841306\pi$
$504$	−6.93048	−0.308708
$505$	−6.21309	−0.276479
$506$	4.32500	0.192270
$507$	5.46393	0.242662
$508$	28.6947	1.27312
$509$	−13.0536	−0.578591	−0.289296	−	0.957240i	$-0.593421\pi$
$509$	−13.0536	−0.578591	−0.289296	+	0.957240i	$0.593421\pi$
$510$	−2.55856	−0.113295
$511$	−40.2825	−1.78199
$512$	−22.5164	−0.995093
$513$	−37.0064	−1.63387
$514$	4.98624	0.219934
$515$	7.09937	0.312835
$516$	−1.70466	−0.0750435
$517$	26.0632	1.14626
$518$	−8.25843	−0.362855
$519$	−17.3623	−0.762122
$520$	−12.1636	−0.533410
$521$	−11.3612	−0.497744	−0.248872	−	0.968536i	$-0.580060\pi$
$521$	−11.3612	−0.497744	−0.248872	+	0.968536i	$0.580060\pi$
$522$	0.394467	0.0172654
$523$	−12.1481	−0.531201	−0.265600	−	0.964083i	$-0.585570\pi$
$523$	−12.1481	−0.531201	−0.265600	+	0.964083i	$0.585570\pi$
$524$	2.36262	0.103211
$525$	−5.40523	−0.235903
$526$	6.29126	0.274312
$527$	17.3123	0.754135
$528$	9.93781	0.432488
$529$	−4.15982	−0.180862
$530$	3.66629	0.159254
$531$	−17.7460	−0.770109
$532$	−40.5755	−1.75917
$533$	0.380674	0.0164888
$534$	4.59630	0.198902
$535$	−18.8415	−0.814590
$536$	−11.9190	−0.514821
$537$	27.3463	1.18008
$538$	−11.0320	−0.475622
$539$	10.0679	0.433656
$540$	19.8315	0.853413
$541$	39.5763	1.70152	0.850760	−	0.525554i	$-0.176142\pi$
$541$	39.5763	1.70152	0.850760	+	0.525554i	$0.176142\pi$
$542$	−3.83411	−0.164689
$543$	14.6975	0.630729
$544$	−11.2300	−0.481483
$545$	−36.0667	−1.54493
$546$	6.92459	0.296345
$547$	27.1079	1.15905	0.579524	−	0.814955i	$-0.303238\pi$
$547$	27.1079	1.15905	0.579524	+	0.814955i	$0.303238\pi$
$548$	−0.396075	−0.0169195
$549$	5.12571	0.218760
$550$	1.27606	0.0544114
$551$	4.81386	0.205077
$552$	−8.38336	−0.356820
$553$	47.9275	2.03809
$554$	−1.61826	−0.0687531
$555$	15.5081	0.658281
$556$	8.54911	0.362563
$557$	20.0432	0.849256	0.424628	−	0.905368i	$-0.360405\pi$
$557$	20.0432	0.849256	0.424628	+	0.905368i	$0.360405\pi$
$558$	3.56582	0.150953
$559$	−3.01823	−0.127657
$560$	19.7540	0.834758
$561$	8.49195	0.358531
$562$	−5.83272	−0.246038
$563$	30.9096	1.30269	0.651343	−	0.758784i	$-0.274206\pi$
$563$	30.9096	1.30269	0.651343	+	0.758784i	$0.274206\pi$
$564$	−24.2370	−1.02056
$565$	14.6828	0.617710
$566$	1.11600	0.0469088
$567$	−9.82487	−0.412606
$568$	−12.3772	−0.519338
$569$	5.05123	0.211759	0.105879	−	0.994379i	$-0.466234\pi$
$569$	5.05123	0.211759	0.105879	+	0.994379i	$0.466234\pi$
$570$	−6.43107	−0.269368
$571$	−13.6965	−0.573182	−0.286591	−	0.958053i	$-0.592522\pi$
$571$	−13.6965	−0.573182	−0.286591	+	0.958053i	$0.592522\pi$
$572$	19.3683	0.809831
$573$	−17.6876	−0.738911
$574$	0.119722	0.00499710
$575$	5.55866	0.231812
$576$	6.20683	0.258618
$577$	−8.28905	−0.345078	−0.172539	−	0.985003i	$-0.555197\pi$
$577$	−8.28905	−0.345078	−0.172539	+	0.985003i	$0.555197\pi$
$578$	3.95615	0.164554
$579$	21.4834	0.892821
$580$	−2.57972	−0.107117
$581$	−8.48147	−0.351871
$582$	−0.366933	−0.0152099
$583$	−12.1685	−0.503970
$584$	−18.4334	−0.762781
$585$	11.0549	0.457065
$586$	1.22664	0.0506722
$587$	35.6447	1.47122	0.735608	−	0.677408i	$-0.236897\pi$
$587$	35.6447	1.47122	0.735608	+	0.677408i	$0.236897\pi$
$588$	−9.36247	−0.386102
$589$	43.5152	1.79301
$590$	−9.79534	−0.403268
$591$	34.0596	1.40102
$592$	19.5146	0.802044
$593$	3.95476	0.162403	0.0812013	−	0.996698i	$-0.474124\pi$
$593$	3.95476	0.162403	0.0812013	+	0.996698i	$0.474124\pi$
$594$	5.55554	0.227947
$595$	16.8800	0.692012
$596$	−4.15725	−0.170288
$597$	−20.6274	−0.844224
$598$	−7.12115	−0.291206
$599$	−24.0090	−0.980983	−0.490492	−	0.871446i	$-0.663183\pi$
$599$	−24.0090	−0.980983	−0.490492	+	0.871446i	$0.663183\pi$
$600$	−2.47345	−0.100978
$601$	37.6063	1.53399	0.766996	−	0.641652i	$-0.221750\pi$
$601$	37.6063	1.53399	0.766996	+	0.641652i	$0.221750\pi$
$602$	−0.949232	−0.0386878
$603$	10.8326	0.441137
$604$	41.2871	1.67995
$605$	−8.91371	−0.362394
$606$	1.61856	0.0657494
$607$	28.3565	1.15095	0.575476	−	0.817818i	$-0.304817\pi$
$607$	28.3565	1.15095	0.575476	+	0.817818i	$0.304817\pi$
$608$	−28.2272	−1.14476
$609$	3.06115	0.124044
$610$	2.82927	0.114554
$611$	−42.9134	−1.73609
$612$	6.71367	0.271384
$613$	−14.7062	−0.593977	−0.296988	−	0.954881i	$-0.595982\pi$
$613$	−14.7062	−0.593977	−0.296988	+	0.954881i	$0.595982\pi$
$614$	2.63243	0.106236
$615$	−0.224820	−0.00906561
$616$	12.6968	0.511569
$617$	−43.4416	−1.74889	−0.874447	−	0.485121i	$-0.838776\pi$
$617$	−43.4416	−1.74889	−0.874447	+	0.485121i	$0.838776\pi$
$618$	−1.84944	−0.0743954
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−23.3196	−0.936536
$621$	24.2006	0.971136
$622$	−13.7065	−0.549582
$623$	−30.3239	−1.21490
$624$	−16.3627	−0.655033
$625$	−16.9567	−0.678270
$626$	−5.77035	−0.230629
$627$	21.3449	0.852434
$628$	14.1914	0.566298
$629$	16.6754	0.664891
$630$	3.47677	0.138518
$631$	5.19347	0.206749	0.103374	−	0.994643i	$-0.467036\pi$
$631$	5.19347	0.206749	0.103374	+	0.994643i	$0.467036\pi$
$632$	21.9318	0.872401
$633$	−22.3881	−0.889848
$634$	8.71680	0.346188
$635$	−30.0051	−1.19072
$636$	11.3159	0.448705
$637$	−16.5769	−0.656802
$638$	−0.722674	−0.0286109
$639$	11.2491	0.445007
$640$	19.8295	0.783831
$641$	−4.63923	−0.183239	−0.0916193	−	0.995794i	$-0.529204\pi$
$641$	−4.63923	−0.183239	−0.0916193	+	0.995794i	$0.529204\pi$
$642$	4.90836	0.193718
$643$	19.9911	0.788372	0.394186	−	0.919031i	$-0.371027\pi$
$643$	19.9911	0.788372	0.394186	+	0.919031i	$0.371027\pi$
$644$	26.5346	1.04561
$645$	1.78251	0.0701864
$646$	−6.91513	−0.272072
$647$	−41.1854	−1.61916	−0.809582	−	0.587007i	$-0.800306\pi$
$647$	−41.1854	−1.61916	−0.809582	+	0.587007i	$0.800306\pi$
$648$	−4.49590	−0.176615
$649$	32.5110	1.27617
$650$	−2.10105	−0.0824098
$651$	27.6715	1.08453
$652$	38.2554	1.49820
$653$	−29.6784	−1.16141	−0.580703	−	0.814115i	$-0.697222\pi$
$653$	−29.6784	−1.16141	−0.580703	+	0.814115i	$0.697222\pi$
$654$	9.39565	0.367399
$655$	−2.47052	−0.0965311
$656$	−0.282901	−0.0110454
$657$	16.7533	0.653607
$658$	−13.4963	−0.526139
$659$	−22.7688	−0.886947	−0.443473	−	0.896288i	$-0.646254\pi$
$659$	−22.7688	−0.886947	−0.443473	+	0.896288i	$0.646254\pi$
$660$	−11.4386	−0.445247
$661$	11.6073	0.451473	0.225736	−	0.974188i	$-0.427521\pi$
$661$	11.6073	0.451473	0.225736	+	0.974188i	$0.427521\pi$
$662$	8.99701	0.349679
$663$	−13.9821	−0.543020
$664$	−3.88115	−0.150618
$665$	42.4286	1.64531
$666$	3.43463	0.133089
$667$	−3.14805	−0.121893
$668$	−17.8581	−0.690950
$669$	3.79865	0.146864
$670$	5.97932	0.231001
$671$	−9.39043	−0.362513
$672$	−17.9498	−0.692428
$673$	8.17196	0.315006	0.157503	−	0.987519i	$-0.449656\pi$
$673$	8.17196	0.315006	0.157503	+	0.987519i	$0.449656\pi$
$674$	5.52790	0.212927
$675$	7.14021	0.274827
$676$	−7.91387	−0.304380
$677$	8.58307	0.329874	0.164937	−	0.986304i	$-0.447258\pi$
$677$	8.58307	0.329874	0.164937	+	0.986304i	$0.447258\pi$
$678$	−3.82498	−0.146898
$679$	2.42082	0.0929025
$680$	7.72434	0.296215
$681$	−0.00416346	−0.000159544 0
$682$	−6.53266	−0.250149
$683$	−24.2687	−0.928614	−0.464307	−	0.885674i	$-0.653697\pi$
$683$	−24.2687	−0.928614	−0.464307	+	0.885674i	$0.653697\pi$
$684$	16.8751	0.645236
$685$	0.414164	0.0158244
$686$	3.94090	0.150464
$687$	−8.84964	−0.337635
$688$	2.24302	0.0855144
$689$	20.0356	0.763297
$690$	4.20563	0.160106
$691$	−3.30802	−0.125843	−0.0629216	−	0.998018i	$-0.520042\pi$
$691$	−3.30802	−0.125843	−0.0629216	+	0.998018i	$0.520042\pi$
$692$	25.1473	0.955957
$693$	−11.5395	−0.438350
$694$	−7.04979	−0.267607
$695$	−8.93956	−0.339097
$696$	1.40080	0.0530970
$697$	−0.241742	−0.00915663
$698$	−12.0599	−0.456473
$699$	2.45782	0.0929633
$700$	7.82884	0.295902
$701$	25.8447	0.976142	0.488071	−	0.872804i	$-0.337701\pi$
$701$	25.8447	0.976142	0.488071	+	0.872804i	$0.337701\pi$
$702$	−9.14726	−0.345241
$703$	41.9143	1.58083
$704$	−11.3711	−0.428563
$705$	25.3439	0.954508
$706$	−6.19413	−0.233119
$707$	−10.6783	−0.401600
$708$	−30.2330	−1.13623
$709$	−35.5194	−1.33396	−0.666980	−	0.745076i	$-0.732413\pi$
$709$	−35.5194	−1.33396	−0.666980	+	0.745076i	$0.732413\pi$
$710$	6.20922	0.233028
$711$	−19.9328	−0.747537
$712$	−13.8763	−0.520037
$713$	−28.4570	−1.06572
$714$	−4.39737	−0.164567
$715$	−20.2529	−0.757415
$716$	−39.6079	−1.48022
$717$	−10.4584	−0.390577
$718$	−4.29209	−0.160180
$719$	−39.7181	−1.48123	−0.740617	−	0.671927i	$-0.765467\pi$
$719$	−39.7181	−1.48123	−0.740617	+	0.671927i	$0.765467\pi$
$720$	−8.21557	−0.306176
$721$	12.2016	0.454410
$722$	−9.88511	−0.367886
$723$	16.6512	0.619265
$724$	−21.2876	−0.791147
$725$	−0.928810	−0.0344951
$726$	2.32209	0.0861809
$727$	−35.0458	−1.29978	−0.649888	−	0.760030i	$-0.725184\pi$
$727$	−35.0458	−1.29978	−0.649888	+	0.760030i	$0.725184\pi$
$728$	−20.9054	−0.774807
$729$	25.3852	0.940191
$730$	9.24739	0.342261
$731$	1.91668	0.0708911
$732$	8.73245	0.322761
$733$	−17.5734	−0.649087	−0.324543	−	0.945871i	$-0.605211\pi$
$733$	−17.5734	−0.649087	−0.324543	+	0.945871i	$0.605211\pi$
$734$	−13.4892	−0.497894
$735$	9.79006	0.361112
$736$	18.4593	0.680419
$737$	−19.8456	−0.731021
$738$	−0.0497917	−0.00183286
$739$	−16.2100	−0.596294	−0.298147	−	0.954520i	$-0.596369\pi$
$739$	−16.2100	−0.596294	−0.298147	+	0.954520i	$0.596369\pi$
$740$	−22.4616	−0.825706
$741$	−35.1446	−1.29107
$742$	6.30121	0.231325
$743$	45.8562	1.68230	0.841151	−	0.540801i	$-0.181879\pi$
$743$	45.8562	1.68230	0.841151	+	0.540801i	$0.181879\pi$
$744$	12.6626	0.464233
$745$	4.34712	0.159266
$746$	4.53110	0.165895
$747$	3.52739	0.129061
$748$	−12.2996	−0.449718
$749$	−32.3826	−1.18324
$750$	6.08545	0.222209
$751$	40.1237	1.46414	0.732068	−	0.681232i	$-0.238555\pi$
$751$	40.1237	1.46414	0.732068	+	0.681232i	$0.238555\pi$
$752$	31.8915	1.16296
$753$	6.96187	0.253705
$754$	1.18989	0.0433333
$755$	−43.1727	−1.57121
$756$	34.0842	1.23963
$757$	26.2866	0.955404	0.477702	−	0.878522i	$-0.341470\pi$
$757$	26.2866	0.955404	0.477702	+	0.878522i	$0.341470\pi$
$758$	11.6947	0.424772
$759$	−13.9586	−0.506666
$760$	19.4155	0.704273
$761$	39.3968	1.42813	0.714066	−	0.700079i	$-0.246852\pi$
$761$	39.3968	1.42813	0.714066	+	0.700079i	$0.246852\pi$
$762$	7.81658	0.283165
$763$	−61.9873	−2.24409
$764$	25.6185	0.926843
$765$	−7.02028	−0.253819
$766$	−0.780676	−0.0282070
$767$	−53.5298	−1.93285
$768$	6.30106	0.227370
$769$	22.1652	0.799297	0.399648	−	0.916668i	$-0.369132\pi$
$769$	22.1652	0.799297	0.399648	+	0.916668i	$0.369132\pi$
$770$	−6.36953	−0.229542
$771$	−16.0928	−0.579566
$772$	−31.1162	−1.11990
$773$	7.91637	0.284732	0.142366	−	0.989814i	$-0.454529\pi$
$773$	7.91637	0.284732	0.142366	+	0.989814i	$0.454529\pi$
$774$	0.394780	0.0141901
$775$	−8.39605	−0.301595
$776$	1.10778	0.0397668
$777$	26.6535	0.956189
$778$	−12.0315	−0.431350
$779$	−0.607630	−0.0217706
$780$	18.8338	0.674358
$781$	−20.6086	−0.737434
$782$	4.52219	0.161713
$783$	−4.04373	−0.144511
$784$	12.3193	0.439975
$785$	−14.8395	−0.529645
$786$	0.643589	0.0229561
$787$	−17.5604	−0.625961	−0.312980	−	0.949760i	$-0.601327\pi$
$787$	−17.5604	−0.625961	−0.312980	+	0.949760i	$0.601327\pi$
$788$	−49.3313	−1.75735
$789$	−20.3046	−0.722864
$790$	−11.0024	−0.391448
$791$	25.2351	0.897257
$792$	−5.28052	−0.187635
$793$	15.4614	0.549052
$794$	1.30974	0.0464811
$795$	−11.8327	−0.419663
$796$	29.8764	1.05894
$797$	49.6381	1.75827	0.879136	−	0.476571i	$-0.158120\pi$
$797$	49.6381	1.75827	0.879136	+	0.476571i	$0.158120\pi$
$798$	−11.0530	−0.391271
$799$	27.2516	0.964092
$800$	5.44629	0.192555
$801$	12.6115	0.445606
$802$	−0.0134485	−0.000474883 0
$803$	−30.6924	−1.08311
$804$	18.4550	0.650858
$805$	−27.7464	−0.977933
$806$	10.7561	0.378868
$807$	35.6049	1.25335
$808$	−4.88645	−0.171905
$809$	1.41762	0.0498409	0.0249204	−	0.999689i	$-0.492067\pi$
$809$	1.41762	0.0498409	0.0249204	+	0.999689i	$0.492067\pi$
$810$	2.25543	0.0792477
$811$	31.2800	1.09839	0.549194	−	0.835695i	$-0.314935\pi$
$811$	31.2800	1.09839	0.549194	+	0.835695i	$0.314935\pi$
$812$	−4.43372	−0.155593
$813$	12.3743	0.433986
$814$	−6.29233	−0.220546
$815$	−40.0025	−1.40123
$816$	10.3909	0.363755
$817$	4.81767	0.168549
$818$	9.66566	0.337952
$819$	18.9999	0.663912
$820$	0.325625	0.0113713
$821$	−20.8653	−0.728205	−0.364102	−	0.931359i	$-0.618624\pi$
$821$	−20.8653	−0.728205	−0.364102	+	0.931359i	$0.618624\pi$
$822$	−0.107893	−0.00376320
$823$	38.3051	1.33523	0.667617	−	0.744505i	$-0.267315\pi$
$823$	38.3051	1.33523	0.667617	+	0.744505i	$0.267315\pi$
$824$	5.58349	0.194510
$825$	−4.11839	−0.143384
$826$	−16.8351	−0.585768
$827$	−55.7253	−1.93776	−0.968880	−	0.247533i	$-0.920380\pi$
$827$	−55.7253	−1.93776	−0.968880	+	0.247533i	$0.920380\pi$
$828$	−11.0356	−0.383513
$829$	−7.93394	−0.275557	−0.137778	−	0.990463i	$-0.543996\pi$
$829$	−7.93394	−0.275557	−0.137778	+	0.990463i	$0.543996\pi$
$830$	1.94704	0.0675826
$831$	5.22281	0.181177
$832$	18.7226	0.649089
$833$	10.5270	0.364737
$834$	2.32882	0.0806406
$835$	18.6737	0.646229
$836$	−30.9156	−1.06924
$837$	−36.5536	−1.26348
$838$	−10.9097	−0.376868
$839$	27.8168	0.960342	0.480171	−	0.877175i	$-0.340575\pi$
$839$	27.8168	0.960342	0.480171	+	0.877175i	$0.340575\pi$
$840$	12.3464	0.425991
$841$	−28.4740	−0.981862
$842$	−4.16790	−0.143635
$843$	18.8247	0.648357
$844$	32.4266	1.11617
$845$	8.27530	0.284679
$846$	5.61301	0.192979
$847$	−15.3199	−0.526397
$848$	−14.8897	−0.511313
$849$	−3.60180	−0.123613
$850$	1.33424	0.0457641
$851$	−27.4101	−0.939607
$852$	19.1646	0.656568
$853$	−3.26689	−0.111856	−0.0559281	−	0.998435i	$-0.517812\pi$
$853$	−3.26689	−0.111856	−0.0559281	+	0.998435i	$0.517812\pi$
$854$	4.86262	0.166395
$855$	−17.6458	−0.603473
$856$	−14.8184	−0.506483
$857$	24.4385	0.834805	0.417402	−	0.908722i	$-0.362941\pi$
$857$	24.4385	0.834805	0.417402	+	0.908722i	$0.362941\pi$
$858$	5.27604	0.180121
$859$	−25.9275	−0.884634	−0.442317	−	0.896859i	$-0.645843\pi$
$859$	−25.9275	−0.884634	−0.442317	+	0.896859i	$0.645843\pi$
$860$	−2.58176	−0.0880374
$861$	−0.386394	−0.0131683
$862$	−11.5711	−0.394112
$863$	−23.9799	−0.816286	−0.408143	−	0.912918i	$-0.633824\pi$
$863$	−23.9799	−0.816286	−0.408143	+	0.912918i	$0.633824\pi$
$864$	23.7113	0.806676
$865$	−26.2958	−0.894084
$866$	3.52972	0.119945
$867$	−12.7682	−0.433631
$868$	−40.0790	−1.36037
$869$	36.5173	1.23877
$870$	−0.702729	−0.0238247
$871$	32.6759	1.10718
$872$	−28.3656	−0.960581
$873$	−1.00680	−0.0340751
$874$	11.3667	0.384486
$875$	−40.1484	−1.35726
$876$	28.5418	0.964338
$877$	−8.83073	−0.298193	−0.149096	−	0.988823i	$-0.547636\pi$
$877$	−8.83073	−0.298193	−0.149096	+	0.988823i	$0.547636\pi$
$878$	−7.43501	−0.250920
$879$	−3.95891	−0.133531
$880$	15.0511	0.507373
$881$	−38.2539	−1.28881	−0.644403	−	0.764686i	$-0.722894\pi$
$881$	−38.2539	−1.28881	−0.644403	+	0.764686i	$0.722894\pi$
$882$	2.16824	0.0730084
$883$	53.5201	1.80110	0.900548	−	0.434757i	$-0.143166\pi$
$883$	53.5201	1.80110	0.900548	+	0.434757i	$0.143166\pi$
$884$	20.2514	0.681129
$885$	31.6138	1.06269
$886$	9.28614	0.311974
$887$	41.5586	1.39540	0.697701	−	0.716389i	$-0.254206\pi$
$887$	41.5586	1.39540	0.697701	+	0.716389i	$0.254206\pi$
$888$	12.1967	0.409296
$889$	−51.5694	−1.72958
$890$	6.96125	0.233341
$891$	−7.48584	−0.250785
$892$	−5.50190	−0.184217
$893$	68.4981	2.29220
$894$	−1.13246	−0.0378750
$895$	41.4168	1.38441
$896$	34.0807	1.13856
$897$	22.9830	0.767381
$898$	−3.98772	−0.133072
$899$	4.75495	0.158587
$900$	−3.25597	−0.108532
$901$	−12.7234	−0.423877
$902$	0.0912195	0.00303728
$903$	3.06358	0.101950
$904$	11.5477	0.384070
$905$	22.2598	0.739940
$906$	11.2468	0.373650
$907$	37.3221	1.23926	0.619630	−	0.784894i	$-0.287283\pi$
$907$	37.3221	1.23926	0.619630	+	0.784894i	$0.287283\pi$
$908$	0.00603029	0.000200122 0
$909$	4.44106	0.147301
$910$	10.4875	0.347657
$911$	49.5208	1.64070	0.820349	−	0.571863i	$-0.193779\pi$
$911$	49.5208	1.64070	0.820349	+	0.571863i	$0.193779\pi$
$912$	26.1180	0.864855
$913$	−6.46227	−0.213870
$914$	1.96872	0.0651196
$915$	−9.13126	−0.301870
$916$	12.8177	0.423508
$917$	−4.24604	−0.140217
$918$	5.80884	0.191720
$919$	2.37680	0.0784034	0.0392017	−	0.999231i	$-0.487519\pi$
$919$	2.37680	0.0784034	0.0392017	+	0.999231i	$0.487519\pi$
$920$	−12.6969	−0.418603
$921$	−8.49597	−0.279952
$922$	5.95615	0.196155
$923$	33.9323	1.11689
$924$	−19.6594	−0.646746
$925$	−8.08716	−0.265904
$926$	0.512718	0.0168489
$927$	−5.07456	−0.166670
$928$	−3.08441	−0.101251
$929$	−43.2093	−1.41765	−0.708825	−	0.705384i	$-0.750774\pi$
$929$	−43.2093	−1.41765	−0.708825	+	0.705384i	$0.750774\pi$
$930$	−6.35237	−0.208302
$931$	26.4600	0.867191
$932$	−3.55986	−0.116607
$933$	44.2369	1.44825
$934$	−9.54430	−0.312299
$935$	12.8613	0.420610
$936$	8.69444	0.284187
$937$	−2.29674	−0.0750312	−0.0375156	−	0.999296i	$-0.511944\pi$
$937$	−2.29674	−0.0750312	−0.0375156	+	0.999296i	$0.511944\pi$
$938$	10.2766	0.335542
$939$	18.6234	0.607752
$940$	−36.7077	−1.19727
$941$	49.7980	1.62337	0.811684	−	0.584097i	$-0.198551\pi$
$941$	49.7980	1.62337	0.811684	+	0.584097i	$0.198551\pi$
$942$	3.86581	0.125955
$943$	0.397363	0.0129399
$944$	39.7811	1.29477
$945$	−35.6408	−1.15940
$946$	−0.723246	−0.0235148
$947$	−55.2779	−1.79629	−0.898145	−	0.439699i	$-0.855085\pi$
$947$	−55.2779	−1.79629	−0.898145	+	0.439699i	$0.855085\pi$
$948$	−33.9586	−1.10292
$949$	50.5354	1.64045
$950$	3.35367	0.108808
$951$	−28.1329	−0.912271
$952$	13.2757	0.430268
$953$	−9.80332	−0.317561	−0.158780	−	0.987314i	$-0.550756\pi$
$953$	−9.80332	−0.317561	−0.158780	+	0.987314i	$0.550756\pi$
$954$	−2.62063	−0.0848461
$955$	−26.7885	−0.866854
$956$	15.1478	0.489914
$957$	2.33238	0.0753951
$958$	−0.702887	−0.0227093
$959$	0.711818	0.0229858
$960$	−11.0572	−0.356871
$961$	11.9828	0.386541
$962$	10.3604	0.334032
$963$	13.4677	0.433992
$964$	−24.1173	−0.776767
$965$	32.5373	1.04741
$966$	7.22816	0.232562
$967$	−6.09145	−0.195888	−0.0979440	−	0.995192i	$-0.531227\pi$
$967$	−6.09145	−0.195888	−0.0979440	+	0.995192i	$0.531227\pi$
$968$	−7.01042	−0.225324
$969$	22.3181	0.716961
$970$	−0.555731	−0.0178434
$971$	30.5845	0.981502	0.490751	−	0.871300i	$-0.336723\pi$
$971$	30.5845	0.981502	0.490751	+	0.871300i	$0.336723\pi$
$972$	−23.8879	−0.766204
$973$	−15.3643	−0.492556
$974$	14.2545	0.456743
$975$	6.78098	0.217165
$976$	−11.4903	−0.367796
$977$	23.2970	0.745337	0.372668	−	0.927965i	$-0.378443\pi$
$977$	23.2970	0.745337	0.372668	+	0.927965i	$0.378443\pi$
$978$	10.4210	0.333226
$979$	−23.1046	−0.738426
$980$	−14.1798	−0.452956
$981$	25.7801	0.823096
$982$	8.07248	0.257603
$983$	41.4405	1.32175	0.660873	−	0.750498i	$-0.270186\pi$
$983$	41.4405	1.32175	0.660873	+	0.750498i	$0.270186\pi$
$984$	−0.176816	−0.00563667
$985$	51.5843	1.64361
$986$	−0.755624	−0.0240640
$987$	43.5582	1.38647
$988$	50.9029	1.61944
$989$	−3.15054	−0.100181
$990$	2.64905	0.0841923
$991$	44.3684	1.40941	0.704704	−	0.709502i	$-0.251080\pi$
$991$	44.3684	1.40941	0.704704	+	0.709502i	$0.251080\pi$
$992$	−27.8818	−0.885247
$993$	−29.0372	−0.921468
$994$	10.6717	0.338486
$995$	−31.2409	−0.990402
$996$	6.00947	0.190417
$997$	16.0732	0.509044	0.254522	−	0.967067i	$-0.418082\pi$
$997$	16.0732	0.509044	0.254522	+	0.967067i	$0.418082\pi$
$998$	10.6110	0.335886
$999$	−35.2088	−1.11396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	619.2.a.b.1.11	✓	30
3.2	odd	2		5571.2.a.g.1.20		30
4.3	odd	2		9904.2.a.n.1.11		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.11	✓	30	1.1	even	1	trivial
5571.2.a.g.1.20		30	3.2	odd	2
9904.2.a.n.1.11		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)

\(f(q)\)	\(=\)	\(q-0.394547 q^{2} +1.27337 q^{3} -1.84433 q^{4} +1.92856 q^{5} -0.502406 q^{6} +3.31459 q^{7} +1.51677 q^{8} -1.37852 q^{9} +O(q^{10})\)	\(q-0.394547 q^{2} +1.27337 q^{3} -1.84433 q^{4} +1.92856 q^{5} -0.502406 q^{6} +3.31459 q^{7} +1.51677 q^{8} -1.37852 q^{9} -0.760909 q^{10} +2.52548 q^{11} -2.34852 q^{12} -4.15823 q^{13} -1.30776 q^{14} +2.45578 q^{15} +3.09023 q^{16} +2.64063 q^{17} +0.543891 q^{18} +6.63734 q^{19} -3.55691 q^{20} +4.22072 q^{21} -0.996422 q^{22} -4.34053 q^{23} +1.93142 q^{24} -1.28064 q^{25} +1.64062 q^{26} -5.57549 q^{27} -6.11321 q^{28} +0.725269 q^{29} -0.968922 q^{30} +6.55612 q^{31} -4.25278 q^{32} +3.21588 q^{33} -1.04185 q^{34} +6.39241 q^{35} +2.54245 q^{36} +6.31493 q^{37} -2.61874 q^{38} -5.29499 q^{39} +2.92519 q^{40} -0.0915471 q^{41} -1.66527 q^{42} +0.725844 q^{43} -4.65783 q^{44} -2.65856 q^{45} +1.71254 q^{46} +10.3201 q^{47} +3.93501 q^{48} +3.98653 q^{49} +0.505274 q^{50} +3.36251 q^{51} +7.66917 q^{52} -4.81831 q^{53} +2.19979 q^{54} +4.87055 q^{55} +5.02748 q^{56} +8.45182 q^{57} -0.286153 q^{58} +12.8732 q^{59} -4.52928 q^{60} -3.71827 q^{61} -2.58670 q^{62} -4.56923 q^{63} -4.50253 q^{64} -8.01942 q^{65} -1.26882 q^{66} -7.85813 q^{67} -4.87020 q^{68} -5.52711 q^{69} -2.52211 q^{70} -8.16026 q^{71} -2.09090 q^{72} -12.1531 q^{73} -2.49154 q^{74} -1.63074 q^{75} -12.2415 q^{76} +8.37095 q^{77} +2.08912 q^{78} +14.4595 q^{79} +5.95970 q^{80} -2.96412 q^{81} +0.0361197 q^{82} -2.55883 q^{83} -7.78440 q^{84} +5.09262 q^{85} -0.286380 q^{86} +0.923538 q^{87} +3.83058 q^{88} -9.14859 q^{89} +1.04893 q^{90} -13.7829 q^{91} +8.00538 q^{92} +8.34839 q^{93} -4.07177 q^{94} +12.8005 q^{95} -5.41538 q^{96} +0.730351 q^{97} -1.57288 q^{98} -3.48143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{2} + q^{3} + 33 q^{4} + 21 q^{5} + 6 q^{6} + 2 q^{7} + 27 q^{8} + 43 q^{9}+O(q^{10}) \) 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