LMFDB - Embedded newform 409.2.a.b.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,2,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.26588144267$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$2.49897$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.49897 q^{2} -0.710812 q^{3} +4.24487 q^{4} +2.06301 q^{5} -1.77630 q^{6} +2.11869 q^{7} +5.60986 q^{8} -2.49475 q^{9} +5.15540 q^{10} -4.92148 q^{11} -3.01730 q^{12} -2.71950 q^{13} +5.29456 q^{14} -1.46641 q^{15} +5.52916 q^{16} -2.73270 q^{17} -6.23430 q^{18} +4.78990 q^{19} +8.75718 q^{20} -1.50599 q^{21} -12.2987 q^{22} +4.00349 q^{23} -3.98756 q^{24} -0.744008 q^{25} -6.79595 q^{26} +3.90573 q^{27} +8.99358 q^{28} +7.23994 q^{29} -3.66452 q^{30} -6.84951 q^{31} +2.59750 q^{32} +3.49825 q^{33} -6.82895 q^{34} +4.37088 q^{35} -10.5899 q^{36} +4.49991 q^{37} +11.9698 q^{38} +1.93305 q^{39} +11.5732 q^{40} +0.680218 q^{41} -3.76344 q^{42} -8.59956 q^{43} -20.8910 q^{44} -5.14668 q^{45} +10.0046 q^{46} -2.83227 q^{47} -3.93019 q^{48} -2.51113 q^{49} -1.85926 q^{50} +1.94244 q^{51} -11.5439 q^{52} -1.94045 q^{53} +9.76032 q^{54} -10.1530 q^{55} +11.8856 q^{56} -3.40472 q^{57} +18.0924 q^{58} +5.99606 q^{59} -6.22471 q^{60} -12.1551 q^{61} -17.1167 q^{62} -5.28561 q^{63} -4.56724 q^{64} -5.61034 q^{65} +8.74203 q^{66} +15.7509 q^{67} -11.6000 q^{68} -2.84573 q^{69} +10.9227 q^{70} +9.92348 q^{71} -13.9952 q^{72} -0.526315 q^{73} +11.2451 q^{74} +0.528850 q^{75} +20.3325 q^{76} -10.4271 q^{77} +4.83064 q^{78} +3.94118 q^{79} +11.4067 q^{80} +4.70800 q^{81} +1.69985 q^{82} -2.75033 q^{83} -6.39274 q^{84} -5.63758 q^{85} -21.4901 q^{86} -5.14624 q^{87} -27.6088 q^{88} -2.59517 q^{89} -12.8614 q^{90} -5.76179 q^{91} +16.9943 q^{92} +4.86871 q^{93} -7.07776 q^{94} +9.88159 q^{95} -1.84633 q^{96} -5.67590 q^{97} -6.27526 q^{98} +12.2779 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Coefficient data

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

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

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

$2$	2.49897	1.76704	0.883520	−	0.468393i	$-0.155167\pi$
$2$	2.49897	1.76704	0.883520	+	0.468393i	$0.155167\pi$
$3$	−0.710812	−0.410387	−0.205194	−	0.978721i	$-0.565782\pi$
$3$	−0.710812	−0.410387	−0.205194	+	0.978721i	$0.565782\pi$
$4$	4.24487	2.12243
$5$	2.06301	0.922604	0.461302	−	0.887243i	$-0.347382\pi$
$5$	2.06301	0.922604	0.461302	+	0.887243i	$0.347382\pi$
$6$	−1.77630	−0.725171
$7$	2.11869	0.800791	0.400396	−	0.916342i	$-0.368873\pi$
$7$	2.11869	0.800791	0.400396	+	0.916342i	$0.368873\pi$
$8$	5.60986	1.98339
$9$	−2.49475	−0.831582
$10$	5.15540	1.63028
$11$	−4.92148	−1.48388	−0.741942	−	0.670465i	$-0.766095\pi$
$11$	−4.92148	−1.48388	−0.741942	+	0.670465i	$0.766095\pi$
$12$	−3.01730	−0.871020
$13$	−2.71950	−0.754253	−0.377127	−	0.926162i	$-0.623088\pi$
$13$	−2.71950	−0.754253	−0.377127	+	0.926162i	$0.623088\pi$
$14$	5.29456	1.41503
$15$	−1.46641	−0.378625
$16$	5.52916	1.38229
$17$	−2.73270	−0.662778	−0.331389	−	0.943494i	$-0.607517\pi$
$17$	−2.73270	−0.662778	−0.331389	+	0.943494i	$0.607517\pi$
$18$	−6.23430	−1.46944
$19$	4.78990	1.09888	0.549439	−	0.835534i	$-0.314841\pi$
$19$	4.78990	1.09888	0.549439	+	0.835534i	$0.314841\pi$
$20$	8.75718	1.95817
$21$	−1.50599	−0.328635
$22$	−12.2987	−2.62208
$23$	4.00349	0.834785	0.417392	−	0.908726i	$-0.362944\pi$
$23$	4.00349	0.834785	0.417392	+	0.908726i	$0.362944\pi$
$24$	−3.98756	−0.813957
$25$	−0.744008	−0.148802
$26$	−6.79595	−1.33280
$27$	3.90573	0.751658
$28$	8.99358	1.69963
$29$	7.23994	1.34442	0.672212	−	0.740359i	$-0.265344\pi$
$29$	7.23994	1.34442	0.672212	+	0.740359i	$0.265344\pi$
$30$	−3.66452	−0.669046
$31$	−6.84951	−1.23021	−0.615104	−	0.788446i	$-0.710886\pi$
$31$	−6.84951	−1.23021	−0.615104	+	0.788446i	$0.710886\pi$
$32$	2.59750	0.459178
$33$	3.49825	0.608967
$34$	−6.82895	−1.17116
$35$	4.37088	0.738813
$36$	−10.5899	−1.76498
$37$	4.49991	0.739780	0.369890	−	0.929076i	$-0.379395\pi$
$37$	4.49991	0.739780	0.369890	+	0.929076i	$0.379395\pi$
$38$	11.9698	1.94176
$39$	1.93305	0.309536
$40$	11.5732	1.82988
$41$	0.680218	0.106232	0.0531161	−	0.998588i	$-0.483085\pi$
$41$	0.680218	0.106232	0.0531161	+	0.998588i	$0.483085\pi$
$42$	−3.76344	−0.580711
$43$	−8.59956	−1.31142	−0.655711	−	0.755012i	$-0.727631\pi$
$43$	−8.59956	−1.31142	−0.655711	+	0.755012i	$0.727631\pi$
$44$	−20.8910	−3.14944
$45$	−5.14668	−0.767221
$46$	10.0046	1.47510
$47$	−2.83227	−0.413129	−0.206564	−	0.978433i	$-0.566228\pi$
$47$	−2.83227	−0.413129	−0.206564	+	0.978433i	$0.566228\pi$
$48$	−3.93019	−0.567275
$49$	−2.51113	−0.358733
$50$	−1.85926	−0.262938
$51$	1.94244	0.271996
$52$	−11.5439	−1.60085
$53$	−1.94045	−0.266542	−0.133271	−	0.991080i	$-0.542548\pi$
$53$	−1.94045	−0.266542	−0.133271	+	0.991080i	$0.542548\pi$
$54$	9.76032	1.32821
$55$	−10.1530	−1.36904
$56$	11.8856	1.58828
$57$	−3.40472	−0.450966
$58$	18.0924	2.37565
$59$	5.99606	0.780620	0.390310	−	0.920683i	$-0.372368\pi$
$59$	5.99606	0.780620	0.390310	+	0.920683i	$0.372368\pi$
$60$	−6.22471	−0.803607
$61$	−12.1551	−1.55630	−0.778152	−	0.628076i	$-0.783843\pi$
$61$	−12.1551	−1.55630	−0.778152	+	0.628076i	$0.783843\pi$
$62$	−17.1167	−2.17383
$63$	−5.28561	−0.665924
$64$	−4.56724	−0.570905
$65$	−5.61034	−0.695877
$66$	8.74203	1.07607
$67$	15.7509	1.92428	0.962139	−	0.272559i	$-0.0878699\pi$
$67$	15.7509	1.92428	0.962139	+	0.272559i	$0.0878699\pi$
$68$	−11.6000	−1.40670
$69$	−2.84573	−0.342585
$70$	10.9227	1.30551
$71$	9.92348	1.17770	0.588850	−	0.808242i	$-0.299581\pi$
$71$	9.92348	1.17770	0.588850	+	0.808242i	$0.299581\pi$
$72$	−13.9952	−1.64935
$73$	−0.526315	−0.0616005	−0.0308002	−	0.999526i	$-0.509806\pi$
$73$	−0.526315	−0.0616005	−0.0308002	+	0.999526i	$0.509806\pi$
$74$	11.2451	1.30722
$75$	0.528850	0.0610663
$76$	20.3325	2.33230
$77$	−10.4271	−1.18828
$78$	4.83064	0.546963
$79$	3.94118	0.443418	0.221709	−	0.975113i	$-0.428837\pi$
$79$	3.94118	0.443418	0.221709	+	0.975113i	$0.428837\pi$
$80$	11.4067	1.27531
$81$	4.70800	0.523111
$82$	1.69985	0.187717
$83$	−2.75033	−0.301888	−0.150944	−	0.988542i	$-0.548231\pi$
$83$	−2.75033	−0.301888	−0.150944	+	0.988542i	$0.548231\pi$
$84$	−6.39274	−0.697505
$85$	−5.63758	−0.611481
$86$	−21.4901	−2.31733
$87$	−5.14624	−0.551735
$88$	−27.6088	−2.94311
$89$	−2.59517	−0.275087	−0.137544	−	0.990496i	$-0.543921\pi$
$89$	−2.59517	−0.275087	−0.137544	+	0.990496i	$0.543921\pi$
$90$	−12.8614	−1.35571
$91$	−5.76179	−0.603999
$92$	16.9943	1.77177
$93$	4.86871	0.504862
$94$	−7.07776	−0.730015
$95$	9.88159	1.01383
$96$	−1.84633	−0.188441
$97$	−5.67590	−0.576300	−0.288150	−	0.957585i	$-0.593040\pi$
$97$	−5.67590	−0.576300	−0.288150	+	0.957585i	$0.593040\pi$
$98$	−6.27526	−0.633897
$99$	12.2779	1.23397
$100$	−3.15821	−0.315821
$101$	−5.23407	−0.520810	−0.260405	−	0.965500i	$-0.583856\pi$
$101$	−5.23407	−0.520810	−0.260405	+	0.965500i	$0.583856\pi$
$102$	4.85410	0.480627
$103$	14.3813	1.41703	0.708513	−	0.705697i	$-0.249366\pi$
$103$	14.3813	1.41703	0.708513	+	0.705697i	$0.249366\pi$
$104$	−15.2560	−1.49597
$105$	−3.10687	−0.303200
$106$	−4.84914	−0.470990
$107$	3.27840	0.316935	0.158467	−	0.987364i	$-0.449345\pi$
$107$	3.27840	0.316935	0.158467	+	0.987364i	$0.449345\pi$
$108$	16.5793	1.59534
$109$	−13.5773	−1.30047	−0.650235	−	0.759733i	$-0.725330\pi$
$109$	−13.5773	−1.30047	−0.650235	+	0.759733i	$0.725330\pi$
$110$	−25.3722	−2.41914
$111$	−3.19859	−0.303596
$112$	11.7146	1.10693
$113$	17.3173	1.62908	0.814539	−	0.580109i	$-0.196990\pi$
$113$	17.3173	1.62908	0.814539	+	0.580109i	$0.196990\pi$
$114$	−8.50830	−0.796876
$115$	8.25922	0.770176
$116$	30.7326	2.85345
$117$	6.78446	0.627223
$118$	14.9840	1.37939
$119$	−5.78976	−0.530747
$120$	−8.22635	−0.750960
$121$	13.2210	1.20191
$122$	−30.3753	−2.75005
$123$	−0.483507	−0.0435963
$124$	−29.0753	−2.61104
$125$	−11.8499	−1.05989
$126$	−13.2086	−1.17671
$127$	18.4811	1.63993	0.819964	−	0.572415i	$-0.193993\pi$
$127$	18.4811	1.63993	0.819964	+	0.572415i	$0.193993\pi$
$128$	−16.6084	−1.46799
$129$	6.11267	0.538191
$130$	−14.0201	−1.22964
$131$	19.2072	1.67814	0.839072	−	0.544020i	$-0.183099\pi$
$131$	19.2072	1.67814	0.839072	+	0.544020i	$0.183099\pi$
$132$	14.8496	1.29249
$133$	10.1483	0.879973
$134$	39.3611	3.40028
$135$	8.05755	0.693483
$136$	−15.3301	−1.31454
$137$	11.5802	0.989367	0.494684	−	0.869073i	$-0.335284\pi$
$137$	11.5802	0.989367	0.494684	+	0.869073i	$0.335284\pi$
$138$	−7.11139	−0.605362
$139$	−8.60393	−0.729776	−0.364888	−	0.931051i	$-0.618893\pi$
$139$	−8.60393	−0.729776	−0.364888	+	0.931051i	$0.618893\pi$
$140$	18.5538	1.56808
$141$	2.01321	0.169543
$142$	24.7985	2.08105
$143$	13.3840	1.11922
$144$	−13.7939	−1.14949
$145$	14.9360	1.24037
$146$	−1.31525	−0.108851
$147$	1.78494	0.147220
$148$	19.1015	1.57013
$149$	3.49380	0.286223	0.143112	−	0.989707i	$-0.454289\pi$
$149$	3.49380	0.286223	0.143112	+	0.989707i	$0.454289\pi$
$150$	1.32158	0.107907
$151$	0.161151	0.0131142	0.00655712	−	0.999979i	$-0.497913\pi$
$151$	0.161151	0.0131142	0.00655712	+	0.999979i	$0.497913\pi$
$152$	26.8707	2.17950
$153$	6.81740	0.551154
$154$	−26.0571	−2.09974
$155$	−14.1306	−1.13500
$156$	8.20555	0.656970
$157$	−14.8623	−1.18614	−0.593072	−	0.805149i	$-0.702085\pi$
$157$	−14.8623	−1.18614	−0.593072	+	0.805149i	$0.702085\pi$
$158$	9.84891	0.783537
$159$	1.37930	0.109385
$160$	5.35866	0.423639
$161$	8.48217	0.668488
$162$	11.7652	0.924358
$163$	15.7403	1.23287	0.616437	−	0.787404i	$-0.288575\pi$
$163$	15.7403	1.23287	0.616437	+	0.787404i	$0.288575\pi$
$164$	2.88743	0.225471
$165$	7.21691	0.561836
$166$	−6.87299	−0.533448
$167$	−18.3107	−1.41692	−0.708461	−	0.705750i	$-0.750610\pi$
$167$	−18.3107	−1.41692	−0.708461	+	0.705750i	$0.750610\pi$
$168$	−8.44842	−0.651809
$169$	−5.60433	−0.431102
$170$	−14.0882	−1.08051
$171$	−11.9496	−0.913808
$172$	−36.5040	−2.78340
$173$	18.5721	1.41201	0.706005	−	0.708207i	$-0.250496\pi$
$173$	18.5721	1.41201	0.706005	+	0.708207i	$0.250496\pi$
$174$	−12.8603	−0.974938
$175$	−1.57633	−0.119159
$176$	−27.2117	−2.05116
$177$	−4.26207	−0.320357
$178$	−6.48525	−0.486090
$179$	−17.9310	−1.34023	−0.670114	−	0.742258i	$-0.733755\pi$
$179$	−17.9310	−1.34023	−0.670114	+	0.742258i	$0.733755\pi$
$180$	−21.8470	−1.62838
$181$	−13.5683	−1.00853	−0.504264	−	0.863550i	$-0.668236\pi$
$181$	−13.5683	−1.00853	−0.504264	+	0.863550i	$0.668236\pi$
$182$	−14.3985	−1.06729
$183$	8.64001	0.638688
$184$	22.4590	1.65570
$185$	9.28333	0.682524
$186$	12.1668	0.892112
$187$	13.4489	0.983484
$188$	−12.0226	−0.876838
$189$	8.27505	0.601921
$190$	24.6938	1.79148
$191$	−6.91490	−0.500344	−0.250172	−	0.968201i	$-0.580487\pi$
$191$	−6.91490	−0.500344	−0.250172	+	0.968201i	$0.580487\pi$
$192$	3.24645	0.234292
$193$	−7.74170	−0.557260	−0.278630	−	0.960399i	$-0.589880\pi$
$193$	−7.74170	−0.557260	−0.278630	+	0.960399i	$0.589880\pi$
$194$	−14.1839	−1.01835
$195$	3.98790	0.285579
$196$	−10.6594	−0.761388
$197$	−21.5203	−1.53326	−0.766630	−	0.642089i	$-0.778068\pi$
$197$	−21.5203	−1.53326	−0.766630	+	0.642089i	$0.778068\pi$
$198$	30.6820	2.18048
$199$	4.48559	0.317975	0.158988	−	0.987281i	$-0.449177\pi$
$199$	4.48559	0.317975	0.158988	+	0.987281i	$0.449177\pi$
$200$	−4.17378	−0.295131
$201$	−11.1959	−0.789700
$202$	−13.0798	−0.920292
$203$	15.3392	1.07660
$204$	8.24539	0.577293
$205$	1.40329	0.0980102
$206$	35.9384	2.50394
$207$	−9.98768	−0.694192
$208$	−15.0365	−1.04260
$209$	−23.5734	−1.63061
$210$	−7.76399	−0.535766
$211$	14.3883	0.990532	0.495266	−	0.868741i	$-0.335071\pi$
$211$	14.3883	0.990532	0.495266	+	0.868741i	$0.335071\pi$
$212$	−8.23696	−0.565717
$213$	−7.05373	−0.483314
$214$	8.19263	0.560037
$215$	−17.7409	−1.20992
$216$	21.9106	1.49083
$217$	−14.5120	−0.985140
$218$	−33.9293	−2.29798
$219$	0.374111	0.0252801
$220$	−43.0983	−2.90569
$221$	7.43158	0.499902
$222$	−7.99318	−0.536467
$223$	−1.02481	−0.0686266	−0.0343133	−	0.999411i	$-0.510924\pi$
$223$	−1.02481	−0.0686266	−0.0343133	+	0.999411i	$0.510924\pi$
$224$	5.50331	0.367705
$225$	1.85611	0.123741
$226$	43.2755	2.87865
$227$	−17.3253	−1.14992	−0.574960	−	0.818182i	$-0.694982\pi$
$227$	−17.3253	−1.14992	−0.574960	+	0.818182i	$0.694982\pi$
$228$	−14.4526	−0.957146
$229$	−5.72855	−0.378553	−0.189277	−	0.981924i	$-0.560614\pi$
$229$	−5.72855	−0.378553	−0.189277	+	0.981924i	$0.560614\pi$
$230$	20.6396	1.36093
$231$	7.41172	0.487656
$232$	40.6151	2.66651
$233$	10.2748	0.673122	0.336561	−	0.941662i	$-0.390736\pi$
$233$	10.2748	0.673122	0.336561	+	0.941662i	$0.390736\pi$
$234$	16.9542	1.10833
$235$	−5.84298	−0.381154
$236$	25.4525	1.65681
$237$	−2.80144	−0.181973
$238$	−14.4685	−0.937851
$239$	23.8763	1.54443	0.772215	−	0.635362i	$-0.219149\pi$
$239$	23.8763	1.54443	0.772215	+	0.635362i	$0.219149\pi$
$240$	−8.10801	−0.523370
$241$	7.85890	0.506236	0.253118	−	0.967435i	$-0.418544\pi$
$241$	7.85890	0.506236	0.253118	+	0.967435i	$0.418544\pi$
$242$	33.0389	2.12382
$243$	−15.0637	−0.966337
$244$	−51.5969	−3.30315
$245$	−5.18048	−0.330969
$246$	−1.20827	−0.0770365
$247$	−13.0261	−0.828833
$248$	−38.4248	−2.43998
$249$	1.95497	0.123891
$250$	−29.6126	−1.87287
$251$	13.5071	0.852561	0.426281	−	0.904591i	$-0.359824\pi$
$251$	13.5071	0.852561	0.426281	+	0.904591i	$0.359824\pi$
$252$	−22.4367	−1.41338
$253$	−19.7031	−1.23872
$254$	46.1836	2.89782
$255$	4.00726	0.250944
$256$	−32.3695	−2.02309
$257$	−5.91517	−0.368978	−0.184489	−	0.982835i	$-0.559063\pi$
$257$	−5.91517	−0.368978	−0.184489	+	0.982835i	$0.559063\pi$
$258$	15.2754	0.951005
$259$	9.53392	0.592409
$260$	−23.8151	−1.47695
$261$	−18.0618	−1.11800
$262$	47.9984	2.96535
$263$	−8.37072	−0.516161	−0.258080	−	0.966123i	$-0.583090\pi$
$263$	−8.37072	−0.516161	−0.258080	+	0.966123i	$0.583090\pi$
$264$	19.6247	1.20782
$265$	−4.00316	−0.245912
$266$	25.3604	1.55495
$267$	1.84468	0.112892
$268$	66.8605	4.08415
$269$	6.51391	0.397160	0.198580	−	0.980085i	$-0.436367\pi$
$269$	6.51391	0.397160	0.198580	+	0.980085i	$0.436367\pi$
$270$	20.1356	1.22541
$271$	22.0590	1.33999	0.669993	−	0.742367i	$-0.266297\pi$
$271$	22.0590	1.33999	0.669993	+	0.742367i	$0.266297\pi$
$272$	−15.1095	−0.916151
$273$	4.09555	0.247874
$274$	28.9387	1.74825
$275$	3.66162	0.220804
$276$	−12.0797	−0.727114
$277$	−5.88264	−0.353454	−0.176727	−	0.984260i	$-0.556551\pi$
$277$	−5.88264	−0.353454	−0.176727	+	0.984260i	$0.556551\pi$
$278$	−21.5010	−1.28954
$279$	17.0878	1.02302
$280$	24.5200	1.46535
$281$	24.4698	1.45975	0.729874	−	0.683582i	$-0.239579\pi$
$281$	24.4698	1.45975	0.729874	+	0.683582i	$0.239579\pi$
$282$	5.03096	0.299589
$283$	−30.5409	−1.81547	−0.907734	−	0.419547i	$-0.862189\pi$
$283$	−30.5409	−1.81547	−0.907734	+	0.419547i	$0.862189\pi$
$284$	42.1239	2.49959
$285$	−7.02396	−0.416063
$286$	33.4462	1.97771
$287$	1.44117	0.0850698
$288$	−6.48010	−0.381844
$289$	−9.53234	−0.560726
$290$	37.3248	2.19179
$291$	4.03450	0.236506
$292$	−2.23414	−0.130743
$293$	13.3748	0.781366	0.390683	−	0.920525i	$-0.372239\pi$
$293$	13.3748	0.781366	0.390683	+	0.920525i	$0.372239\pi$
$294$	4.46053	0.260143
$295$	12.3699	0.720204
$296$	25.2438	1.46727
$297$	−19.2220	−1.11537
$298$	8.73091	0.505768
$299$	−10.8875	−0.629639
$300$	2.24490	0.129609
$301$	−18.2198	−1.05017
$302$	0.402711	0.0231734
$303$	3.72044	0.213734
$304$	26.4841	1.51897
$305$	−25.0761	−1.43585
$306$	17.0365	0.973912
$307$	17.8977	1.02148	0.510739	−	0.859736i	$-0.329372\pi$
$307$	17.8977	1.02148	0.510739	+	0.859736i	$0.329372\pi$
$308$	−44.2617	−2.52205
$309$	−10.2224	−0.581530
$310$	−35.3119	−2.00558
$311$	−5.74437	−0.325733	−0.162867	−	0.986648i	$-0.552074\pi$
$311$	−5.74437	−0.325733	−0.162867	+	0.986648i	$0.552074\pi$
$312$	10.8442	0.613929
$313$	26.2773	1.48528	0.742639	−	0.669692i	$-0.233574\pi$
$313$	26.2773	1.48528	0.742639	+	0.669692i	$0.233574\pi$
$314$	−37.1406	−2.09596
$315$	−10.9042	−0.614384
$316$	16.7298	0.941125
$317$	0.763861	0.0429027	0.0214514	−	0.999770i	$-0.493171\pi$
$317$	0.763861	0.0429027	0.0214514	+	0.999770i	$0.493171\pi$
$318$	3.44682	0.193288
$319$	−35.6313	−1.99497
$320$	−9.42224	−0.526719
$321$	−2.33033	−0.130066
$322$	21.1967	1.18125
$323$	−13.0894	−0.728312
$324$	19.9848	1.11027
$325$	2.02333	0.112234
$326$	39.3346	2.17854
$327$	9.65092	0.533697
$328$	3.81593	0.210699
$329$	−6.00071	−0.330830
$330$	18.0349	0.992786
$331$	15.6234	0.858738	0.429369	−	0.903129i	$-0.358736\pi$
$331$	15.6234	0.858738	0.429369	+	0.903129i	$0.358736\pi$
$332$	−11.6748	−0.640736
$333$	−11.2261	−0.615188
$334$	−45.7578	−2.50376
$335$	32.4942	1.77535
$336$	−8.32688	−0.454269
$337$	7.07528	0.385415	0.192708	−	0.981256i	$-0.438273\pi$
$337$	7.07528	0.385415	0.192708	+	0.981256i	$0.438273\pi$
$338$	−14.0051	−0.761775
$339$	−12.3094	−0.668553
$340$	−23.9308	−1.29783
$341$	33.7098	1.82549
$342$	−29.8617	−1.61474
$343$	−20.1512	−1.08806
$344$	−48.2424	−2.60105
$345$	−5.87075	−0.316071
$346$	46.4111	2.49508
$347$	0.0161358	0.000866217 0	0.000433109	−	1.00000i	$-0.499862\pi$
$347$	0.0161358	0.000866217 0	0.000433109		1.00000i	$0.499862\pi$
$348$	−21.8451	−1.17102
$349$	−9.51072	−0.509097	−0.254549	−	0.967060i	$-0.581927\pi$
$349$	−9.51072	−0.509097	−0.254549	+	0.967060i	$0.581927\pi$
$350$	−3.93919	−0.210559
$351$	−10.6216	−0.566941
$352$	−12.7836	−0.681366
$353$	7.61742	0.405434	0.202717	−	0.979237i	$-0.435023\pi$
$353$	7.61742	0.405434	0.202717	+	0.979237i	$0.435023\pi$
$354$	−10.6508	−0.566084
$355$	20.4722	1.08655
$356$	−11.0161	−0.583854
$357$	4.11543	0.217812
$358$	−44.8092	−2.36824
$359$	28.6776	1.51354	0.756772	−	0.653679i	$-0.226775\pi$
$359$	28.6776	1.51354	0.756772	+	0.653679i	$0.226775\pi$
$360$	−28.8721	−1.52170
$361$	3.94316	0.207535
$362$	−33.9069	−1.78211
$363$	−9.39765	−0.493248
$364$	−24.4580	−1.28195
$365$	−1.08579	−0.0568329
$366$	21.5911	1.12859
$367$	−28.8671	−1.50685	−0.753426	−	0.657533i	$-0.771600\pi$
$367$	−28.8671	−1.50685	−0.753426	+	0.657533i	$0.771600\pi$
$368$	22.1359	1.15391
$369$	−1.69697	−0.0883407
$370$	23.1988	1.20605
$371$	−4.11122	−0.213444
$372$	20.6670	1.07154
$373$	−32.9576	−1.70648	−0.853241	−	0.521518i	$-0.825366\pi$
$373$	−32.9576	−1.70648	−0.853241	+	0.521518i	$0.825366\pi$
$374$	33.6086	1.73786
$375$	8.42307	0.434965
$376$	−15.8886	−0.819393
$377$	−19.6890	−1.01404
$378$	20.6791	1.06362
$379$	3.02041	0.155148	0.0775741	−	0.996987i	$-0.475283\pi$
$379$	3.02041	0.155148	0.0775741	+	0.996987i	$0.475283\pi$
$380$	41.9461	2.15179
$381$	−13.1366	−0.673006
$382$	−17.2801	−0.884129
$383$	−4.77435	−0.243958	−0.121979	−	0.992533i	$-0.538924\pi$
$383$	−4.77435	−0.243958	−0.121979	+	0.992533i	$0.538924\pi$
$384$	11.8055	0.602445
$385$	−21.5112	−1.09631
$386$	−19.3463	−0.984701
$387$	21.4537	1.09055
$388$	−24.0934	−1.22316
$389$	32.9920	1.67276	0.836381	−	0.548149i	$-0.184667\pi$
$389$	32.9920	1.67276	0.836381	+	0.548149i	$0.184667\pi$
$390$	9.96565	0.504630
$391$	−10.9403	−0.553277
$392$	−14.0871	−0.711507
$393$	−13.6527	−0.688689
$394$	−53.7788	−2.70933
$395$	8.13069	0.409099
$396$	52.1178	2.61902
$397$	−14.5793	−0.731716	−0.365858	−	0.930671i	$-0.619224\pi$
$397$	−14.5793	−0.731716	−0.365858	+	0.930671i	$0.619224\pi$
$398$	11.2094	0.561875
$399$	−7.21356	−0.361130
$400$	−4.11374	−0.205687
$401$	−11.2575	−0.562173	−0.281086	−	0.959682i	$-0.590695\pi$
$401$	−11.2575	−0.562173	−0.281086	+	0.959682i	$0.590695\pi$
$402$	−27.9783	−1.39543
$403$	18.6272	0.927888
$404$	−22.2179	−1.10538
$405$	9.71263	0.482624
$406$	38.3323	1.90240
$407$	−22.1462	−1.09775
$408$	10.8968	0.539472
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	3.50679	0.173188
$411$	−8.23138	−0.406024
$412$	61.0465	3.00754
$413$	12.7038	0.625114
$414$	−24.9590	−1.22667
$415$	−5.67394	−0.278523
$416$	−7.06390	−0.346336
$417$	6.11578	0.299491
$418$	−58.9094	−2.88135
$419$	−16.5481	−0.808426	−0.404213	−	0.914665i	$-0.632455\pi$
$419$	−16.5481	−0.808426	−0.404213	+	0.914665i	$0.632455\pi$
$420$	−13.1883	−0.643521
$421$	−29.1207	−1.41925	−0.709627	−	0.704578i	$-0.751136\pi$
$421$	−29.1207	−1.41925	−0.709627	+	0.704578i	$0.751136\pi$
$422$	35.9560	1.75031
$423$	7.06579	0.343550
$424$	−10.8857	−0.528655
$425$	2.03315	0.0986223
$426$	−17.6271	−0.854035
$427$	−25.7530	−1.24627
$428$	13.9164	0.672673
$429$	−9.51348	−0.459315
$430$	−44.3342	−2.13798
$431$	17.7069	0.852913	0.426456	−	0.904508i	$-0.359762\pi$
$431$	17.7069	0.852913	0.426456	+	0.904508i	$0.359762\pi$
$432$	21.5954	1.03901
$433$	2.62036	0.125927	0.0629633	−	0.998016i	$-0.479945\pi$
$433$	2.62036	0.125927	0.0629633	+	0.998016i	$0.479945\pi$
$434$	−36.2651	−1.74078
$435$	−10.6167	−0.509033
$436$	−57.6339	−2.76016
$437$	19.1763	0.917327
$438$	0.934893	0.0446709
$439$	2.62470	0.125270	0.0626351	−	0.998036i	$-0.480050\pi$
$439$	2.62470	0.125270	0.0626351	+	0.998036i	$0.480050\pi$
$440$	−56.9572	−2.71533
$441$	6.26464	0.298316
$442$	18.5713	0.883347
$443$	−10.0144	−0.475797	−0.237899	−	0.971290i	$-0.576459\pi$
$443$	−10.0144	−0.475797	−0.237899	+	0.971290i	$0.576459\pi$
$444$	−13.5776	−0.644363
$445$	−5.35384	−0.253797
$446$	−2.56098	−0.121266
$447$	−2.48343	−0.117462
$448$	−9.67658	−0.457176
$449$	39.6980	1.87346	0.936732	−	0.350048i	$-0.113835\pi$
$449$	39.6980	1.87346	0.936732	+	0.350048i	$0.113835\pi$
$450$	4.63837	0.218655
$451$	−3.34768	−0.157636
$452$	73.5098	3.45761
$453$	−0.114548	−0.00538192
$454$	−43.2954	−2.03195
$455$	−11.8866	−0.557252
$456$	−19.1000	−0.894440
$457$	−25.4737	−1.19161	−0.595806	−	0.803129i	$-0.703167\pi$
$457$	−25.4737	−1.19161	−0.595806	+	0.803129i	$0.703167\pi$
$458$	−14.3155	−0.668919
$459$	−10.6732	−0.498182
$460$	35.0593	1.63465
$461$	9.00933	0.419606	0.209803	−	0.977744i	$-0.432718\pi$
$461$	9.00933	0.419606	0.209803	+	0.977744i	$0.432718\pi$
$462$	18.5217	0.861707
$463$	−39.9032	−1.85446	−0.927230	−	0.374492i	$-0.877817\pi$
$463$	−39.9032	−1.85446	−0.927230	+	0.374492i	$0.877817\pi$
$464$	40.0308	1.85838
$465$	10.0442	0.465788
$466$	25.6764	1.18943
$467$	−9.22161	−0.426725	−0.213363	−	0.976973i	$-0.568442\pi$
$467$	−9.22161	−0.426725	−0.213363	+	0.976973i	$0.568442\pi$
$468$	28.7991	1.33124
$469$	33.3713	1.54095
$470$	−14.6015	−0.673515
$471$	10.5643	0.486779
$472$	33.6371	1.54827
$473$	42.3226	1.94600
$474$	−7.00073	−0.321554
$475$	−3.56372	−0.163515
$476$	−24.5768	−1.12647
$477$	4.84093	0.221651
$478$	59.6662	2.72907
$479$	−24.8144	−1.13380	−0.566900	−	0.823786i	$-0.691858\pi$
$479$	−24.8144	−1.13380	−0.566900	+	0.823786i	$0.691858\pi$
$480$	−3.80900	−0.173856
$481$	−12.2375	−0.557981
$482$	19.6392	0.894541
$483$	−6.02922	−0.274339
$484$	56.1214	2.55097
$485$	−11.7094	−0.531697
$486$	−37.6438	−1.70756
$487$	−24.7309	−1.12066	−0.560331	−	0.828269i	$-0.689326\pi$
$487$	−24.7309	−1.12066	−0.560331	+	0.828269i	$0.689326\pi$
$488$	−68.1886	−3.08675
$489$	−11.1884	−0.505956
$490$	−12.9459	−0.584836
$491$	−14.8359	−0.669536	−0.334768	−	0.942301i	$-0.608658\pi$
$491$	−14.8359	−0.669536	−0.334768	+	0.942301i	$0.608658\pi$
$492$	−2.05242	−0.0925303
$493$	−19.7846	−0.891054
$494$	−32.5519	−1.46458
$495$	25.3293	1.13847
$496$	−37.8720	−1.70050
$497$	21.0248	0.943093
$498$	4.88541	0.218920
$499$	−11.7754	−0.527137	−0.263569	−	0.964641i	$-0.584900\pi$
$499$	−11.7754	−0.527137	−0.263569	+	0.964641i	$0.584900\pi$
$500$	−50.3013	−2.24954
$501$	13.0154	0.581487
$502$	33.7539	1.50651
$503$	1.95767	0.0872880	0.0436440	−	0.999047i	$-0.486103\pi$
$503$	1.95767	0.0872880	0.0436440	+	0.999047i	$0.486103\pi$
$504$	−29.6515	−1.32078
$505$	−10.7979	−0.480501
$506$	−49.2375	−2.18887
$507$	3.98363	0.176919
$508$	78.4496	3.48064
$509$	−20.6369	−0.914715	−0.457357	−	0.889283i	$-0.651204\pi$
$509$	−20.6369	−0.914715	−0.457357	+	0.889283i	$0.651204\pi$
$510$	10.0140	0.443429
$511$	−1.11510	−0.0493291
$512$	−47.6737	−2.10690
$513$	18.7081	0.825982
$514$	−14.7818	−0.651999
$515$	29.6686	1.30735
$516$	25.9475	1.14227
$517$	13.9390	0.613035
$518$	23.8250	1.04681
$519$	−13.2013	−0.579471
$520$	−31.4732	−1.38019
$521$	−42.7343	−1.87222	−0.936111	−	0.351705i	$-0.885602\pi$
$521$	−42.7343	−1.87222	−0.936111	+	0.351705i	$0.885602\pi$
$522$	−45.1360	−1.97555
$523$	−27.9890	−1.22387	−0.611936	−	0.790907i	$-0.709609\pi$
$523$	−27.9890	−1.22387	−0.611936	+	0.790907i	$0.709609\pi$
$524$	81.5322	3.56175
$525$	1.12047	0.0489014
$526$	−20.9182	−0.912077
$527$	18.7177	0.815355
$528$	19.3424	0.841769
$529$	−6.97209	−0.303135
$530$	−10.0038	−0.434537
$531$	−14.9586	−0.649150
$532$	43.0783	1.86768
$533$	−1.84985	−0.0801259
$534$	4.60980	0.199485
$535$	6.76336	0.292405
$536$	88.3604	3.81659
$537$	12.7456	0.550013
$538$	16.2781	0.701798
$539$	12.3585	0.532318
$540$	34.2032	1.47187
$541$	−10.9422	−0.470440	−0.235220	−	0.971942i	$-0.575581\pi$
$541$	−10.9422	−0.470440	−0.235220	+	0.971942i	$0.575581\pi$
$542$	55.1248	2.36781
$543$	9.64454	0.413887
$544$	−7.09819	−0.304333
$545$	−28.0101	−1.19982
$546$	10.2347	0.438003
$547$	−25.8979	−1.10731	−0.553657	−	0.832745i	$-0.686768\pi$
$547$	−25.8979	−1.10731	−0.553657	+	0.832745i	$0.686768\pi$
$548$	49.1566	2.09987
$549$	30.3239	1.29419
$550$	9.15029	0.390170
$551$	34.6786	1.47736
$552$	−15.9641	−0.679479
$553$	8.35017	0.355085
$554$	−14.7006	−0.624567
$555$	−6.59870	−0.280099
$556$	−36.5225	−1.54890
$557$	24.3755	1.03282	0.516412	−	0.856340i	$-0.327267\pi$
$557$	24.3755	1.03282	0.516412	+	0.856340i	$0.327267\pi$
$558$	42.7019	1.80772
$559$	23.3865	0.989143
$560$	24.1673	1.02125
$561$	−9.55967	−0.403610
$562$	61.1494	2.57943
$563$	16.1815	0.681969	0.340985	−	0.940069i	$-0.389240\pi$
$563$	16.1815	0.681969	0.340985	+	0.940069i	$0.389240\pi$
$564$	8.54581	0.359843
$565$	35.7258	1.50299
$566$	−76.3209	−3.20801
$567$	9.97481	0.418903
$568$	55.6694	2.33584
$569$	−7.77787	−0.326065	−0.163033	−	0.986621i	$-0.552128\pi$
$569$	−7.77787	−0.326065	−0.163033	+	0.986621i	$0.552128\pi$
$570$	−17.5527	−0.735201
$571$	29.6872	1.24237	0.621185	−	0.783664i	$-0.286652\pi$
$571$	29.6872	1.24237	0.621185	+	0.783664i	$0.286652\pi$
$572$	56.8131	2.37548
$573$	4.91519	0.205335
$574$	3.60145	0.150322
$575$	−2.97863	−0.124217
$576$	11.3941	0.474754
$577$	−11.4841	−0.478089	−0.239044	−	0.971009i	$-0.576834\pi$
$577$	−11.4841	−0.478089	−0.239044	+	0.971009i	$0.576834\pi$
$578$	−23.8211	−0.990825
$579$	5.50289	0.228692
$580$	63.4015	2.63260
$581$	−5.82710	−0.241749
$582$	10.0821	0.417916
$583$	9.54990	0.395516
$584$	−2.95255	−0.122178
$585$	13.9964	0.578679
$586$	33.4233	1.38071
$587$	−35.1932	−1.45258	−0.726289	−	0.687390i	$-0.758756\pi$
$587$	−35.1932	−1.45258	−0.726289	+	0.687390i	$0.758756\pi$
$588$	7.57685	0.312464
$589$	−32.8085	−1.35185
$590$	30.9121	1.27263
$591$	15.2969	0.629231
$592$	24.8807	1.02259
$593$	2.69312	0.110593	0.0552967	−	0.998470i	$-0.482390\pi$
$593$	2.69312	0.110593	0.0552967	+	0.998470i	$0.482390\pi$
$594$	−48.0352	−1.97091
$595$	−11.9443	−0.489669
$596$	14.8307	0.607489
$597$	−3.18841	−0.130493
$598$	−27.2075	−1.11260
$599$	41.2094	1.68377	0.841884	−	0.539658i	$-0.181446\pi$
$599$	41.2094	1.68377	0.841884	+	0.539658i	$0.181446\pi$
$600$	2.96677	0.121118
$601$	−36.8677	−1.50386	−0.751932	−	0.659241i	$-0.770878\pi$
$601$	−36.8677	−1.50386	−0.751932	+	0.659241i	$0.770878\pi$
$602$	−45.5309	−1.85570
$603$	−39.2945	−1.60020
$604$	0.684063	0.0278341
$605$	27.2750	1.10889
$606$	9.29728	0.377676
$607$	−29.8278	−1.21067	−0.605337	−	0.795970i	$-0.706961\pi$
$607$	−29.8278	−1.21067	−0.605337	+	0.795970i	$0.706961\pi$
$608$	12.4418	0.504580
$609$	−10.9033	−0.441824
$610$	−62.6645	−2.53721
$611$	7.70235	0.311604
$612$	28.9389	1.16979
$613$	−33.4737	−1.35199	−0.675994	−	0.736907i	$-0.736286\pi$
$613$	−33.4737	−1.35199	−0.675994	+	0.736907i	$0.736286\pi$
$614$	44.7260	1.80499
$615$	−0.997477	−0.0402222
$616$	−58.4947	−2.35682
$617$	3.29905	0.132815	0.0664074	−	0.997793i	$-0.478846\pi$
$617$	3.29905	0.132815	0.0664074	+	0.997793i	$0.478846\pi$
$618$	−25.5454	−1.02759
$619$	42.2688	1.69893	0.849464	−	0.527647i	$-0.176926\pi$
$619$	42.2688	1.69893	0.849464	+	0.527647i	$0.176926\pi$
$620$	−59.9824	−2.40895
$621$	15.6365	0.627473
$622$	−14.3550	−0.575584
$623$	−5.49837	−0.220287
$624$	10.6882	0.427869
$625$	−20.7264	−0.829057
$626$	65.6662	2.62455
$627$	16.7563	0.669181
$628$	−63.0887	−2.51751
$629$	−12.2969	−0.490310
$630$	−27.2494	−1.08564
$631$	−6.04440	−0.240624	−0.120312	−	0.992736i	$-0.538389\pi$
$631$	−6.04440	−0.240624	−0.120312	+	0.992736i	$0.538389\pi$
$632$	22.1095	0.879469
$633$	−10.2274	−0.406502
$634$	1.90887	0.0758109
$635$	38.1265	1.51300
$636$	5.85493	0.232163
$637$	6.82902	0.270576
$638$	−89.0416	−3.52519
$639$	−24.7566	−0.979355
$640$	−34.2632	−1.35437
$641$	−41.6202	−1.64390	−0.821950	−	0.569560i	$-0.807114\pi$
$641$	−41.6202	−1.64390	−0.821950	+	0.569560i	$0.807114\pi$
$642$	−5.82342	−0.229832
$643$	−41.8895	−1.65196	−0.825980	−	0.563699i	$-0.809378\pi$
$643$	−41.8895	−1.65196	−0.825980	+	0.563699i	$0.809378\pi$
$644$	36.0057	1.41882
$645$	12.6105	0.496537
$646$	−32.7100	−1.28696
$647$	35.6764	1.40258	0.701292	−	0.712874i	$-0.252607\pi$
$647$	35.6764	1.40258	0.701292	+	0.712874i	$0.252607\pi$
$648$	26.4112	1.03753
$649$	−29.5095	−1.15835
$650$	5.05624	0.198322
$651$	10.3153	0.404289
$652$	66.8154	2.61669
$653$	−4.72277	−0.184816	−0.0924082	−	0.995721i	$-0.529456\pi$
$653$	−4.72277	−0.184816	−0.0924082	+	0.995721i	$0.529456\pi$
$654$	24.1174	0.943064
$655$	39.6246	1.54826
$656$	3.76103	0.146844
$657$	1.31302	0.0512259
$658$	−14.9956	−0.584590
$659$	6.02891	0.234853	0.117427	−	0.993082i	$-0.462536\pi$
$659$	6.02891	0.234853	0.117427	+	0.993082i	$0.462536\pi$
$660$	30.6348	1.19246
$661$	46.0816	1.79237	0.896183	−	0.443684i	$-0.146329\pi$
$661$	46.0816	1.79237	0.896183	+	0.443684i	$0.146329\pi$
$662$	39.0424	1.51742
$663$	−5.28245	−0.205154
$664$	−15.4290	−0.598760
$665$	20.9361	0.811866
$666$	−28.0538	−1.08706
$667$	28.9850	1.12230
$668$	−77.7263	−3.00732
$669$	0.728450	0.0281635
$670$	81.2021	3.13711
$671$	59.8212	2.30937
$672$	−3.91182	−0.150902
$673$	40.0552	1.54402	0.772008	−	0.635613i	$-0.219253\pi$
$673$	40.0552	1.54402	0.772008	+	0.635613i	$0.219253\pi$
$674$	17.6809	0.681044
$675$	−2.90589	−0.111848
$676$	−23.7896	−0.914986
$677$	17.2019	0.661121	0.330561	−	0.943785i	$-0.392762\pi$
$677$	17.2019	0.661121	0.330561	+	0.943785i	$0.392762\pi$
$678$	−30.7608	−1.18136
$679$	−12.0255	−0.461496
$680$	−31.6260	−1.21280
$681$	12.3150	0.471913
$682$	84.2398	3.22571
$683$	−32.3563	−1.23808	−0.619040	−	0.785360i	$-0.712478\pi$
$683$	−32.3563	−1.23808	−0.619040	+	0.785360i	$0.712478\pi$
$684$	−50.7244	−1.93950
$685$	23.8901	0.912794
$686$	−50.3573	−1.92265
$687$	4.07192	0.155354
$688$	−47.5484	−1.81276
$689$	5.27705	0.201040
$690$	−14.6708	−0.558510
$691$	−29.0938	−1.10678	−0.553390	−	0.832922i	$-0.686666\pi$
$691$	−29.0938	−1.10678	−0.553390	+	0.832922i	$0.686666\pi$
$692$	78.8360	2.99690
$693$	26.0130	0.988153
$694$	0.0403230	0.00153064
$695$	−17.7500	−0.673294
$696$	−28.8697	−1.09430
$697$	−1.85883	−0.0704083
$698$	−23.7670	−0.899596
$699$	−7.30343	−0.276241
$700$	−6.69129	−0.252907
$701$	3.52492	0.133134	0.0665672	−	0.997782i	$-0.478795\pi$
$701$	3.52492	0.133134	0.0665672	+	0.997782i	$0.478795\pi$
$702$	−26.5432	−1.00181
$703$	21.5541	0.812928
$704$	22.4776	0.847156
$705$	4.15326	0.156421
$706$	19.0357	0.716419
$707$	−11.0894	−0.417060
$708$	−18.0919	−0.679936
$709$	0.241988	0.00908807	0.00454403	−	0.999990i	$-0.498554\pi$
$709$	0.241988	0.00908807	0.00454403	+	0.999990i	$0.498554\pi$
$710$	51.1595	1.91998
$711$	−9.83225	−0.368738
$712$	−14.5585	−0.545604
$713$	−27.4219	−1.02696
$714$	10.2844	0.384882
$715$	27.6112	1.03260
$716$	−76.1149	−2.84455
$717$	−16.9716	−0.633815
$718$	71.6644	2.67449
$719$	−3.07731	−0.114764	−0.0573821	−	0.998352i	$-0.518275\pi$
$719$	−3.07731	−0.114764	−0.0573821	+	0.998352i	$0.518275\pi$
$720$	−28.4568	−1.06052
$721$	30.4695	1.13474
$722$	9.85385	0.366722
$723$	−5.58620	−0.207753
$724$	−57.5958	−2.14053
$725$	−5.38657	−0.200052
$726$	−23.4845	−0.871590
$727$	51.0793	1.89443	0.947214	−	0.320602i	$-0.103885\pi$
$727$	51.0793	1.89443	0.947214	+	0.320602i	$0.103885\pi$
$728$	−32.3228	−1.19796
$729$	−3.41654	−0.126538
$730$	−2.71336	−0.100426
$731$	23.5000	0.869181
$732$	36.6757	1.35557
$733$	32.9294	1.21628	0.608138	−	0.793831i	$-0.291917\pi$
$733$	32.9294	1.21628	0.608138	+	0.793831i	$0.291917\pi$
$734$	−72.1382	−2.66267
$735$	3.68235	0.135826
$736$	10.3991	0.383314
$737$	−77.5178	−2.85540
$738$	−4.24068	−0.156102
$739$	9.28037	0.341384	0.170692	−	0.985324i	$-0.445400\pi$
$739$	9.28037	0.341384	0.170692	+	0.985324i	$0.445400\pi$
$740$	39.4065	1.44861
$741$	9.25913	0.340143
$742$	−10.2738	−0.377164
$743$	−23.4136	−0.858962	−0.429481	−	0.903076i	$-0.641303\pi$
$743$	−23.4136	−0.858962	−0.429481	+	0.903076i	$0.641303\pi$
$744$	27.3128	1.00134
$745$	7.20772	0.264071
$746$	−82.3602	−3.01542
$747$	6.86137	0.251044
$748$	57.0890	2.08738
$749$	6.94593	0.253799
$750$	21.0490	0.768601
$751$	28.3115	1.03310	0.516551	−	0.856257i	$-0.327216\pi$
$751$	28.3115	1.03310	0.516551	+	0.856257i	$0.327216\pi$
$752$	−15.6601	−0.571064
$753$	−9.60102	−0.349881
$754$	−49.2023	−1.79184
$755$	0.332455	0.0120993
$756$	35.1265	1.27754
$757$	−1.49938	−0.0544959	−0.0272480	−	0.999629i	$-0.508674\pi$
$757$	−1.49938	−0.0544959	−0.0272480	+	0.999629i	$0.508674\pi$
$758$	7.54793	0.274153
$759$	14.0052	0.508356
$760$	55.4344	2.01082
$761$	44.9404	1.62909	0.814543	−	0.580103i	$-0.196988\pi$
$761$	44.9404	1.62909	0.814543	+	0.580103i	$0.196988\pi$
$762$	−32.8279	−1.18923
$763$	−28.7662	−1.04141
$764$	−29.3528	−1.06195
$765$	14.0643	0.508497
$766$	−11.9310	−0.431083
$767$	−16.3063	−0.588785
$768$	23.0086	0.830252
$769$	21.4226	0.772518	0.386259	−	0.922390i	$-0.373767\pi$
$769$	21.4226	0.772518	0.386259	+	0.922390i	$0.373767\pi$
$770$	−53.7559	−1.93723
$771$	4.20457	0.151424
$772$	−32.8625	−1.18275
$773$	−27.9492	−1.00526	−0.502632	−	0.864500i	$-0.667635\pi$
$773$	−27.9492	−1.00526	−0.502632	+	0.864500i	$0.667635\pi$
$774$	53.6123	1.92705
$775$	5.09609	0.183057
$776$	−31.8410	−1.14303
$777$	−6.77683	−0.243117
$778$	82.4461	2.95584
$779$	3.25818	0.116736
$780$	16.9281	0.606123
$781$	−48.8383	−1.74757
$782$	−27.3396	−0.977662
$783$	28.2773	1.01055
$784$	−13.8845	−0.495874
$785$	−30.6611	−1.09434
$786$	−34.1178	−1.21694
$787$	−14.3627	−0.511976	−0.255988	−	0.966680i	$-0.582401\pi$
$787$	−14.3627	−0.511976	−0.255988	+	0.966680i	$0.582401\pi$
$788$	−91.3510	−3.25424
$789$	5.95001	0.211826
$790$	20.3184	0.722895
$791$	36.6901	1.30455
$792$	68.8771	2.44744
$793$	33.0558	1.17385
$794$	−36.4334	−1.29297
$795$	2.84550	0.100919
$796$	19.0407	0.674881
$797$	−44.7540	−1.58527	−0.792635	−	0.609697i	$-0.791291\pi$
$797$	−44.7540	−1.58527	−0.792635	+	0.609697i	$0.791291\pi$
$798$	−18.0265	−0.638131
$799$	7.73974	0.273812
$800$	−1.93256	−0.0683263
$801$	6.47428	0.228758
$802$	−28.1322	−0.993382
$803$	2.59025	0.0914079
$804$	−47.5252	−1.67609
$805$	17.4988	0.616750
$806$	46.5489	1.63962
$807$	−4.63016	−0.162990
$808$	−29.3624	−1.03297
$809$	−37.1363	−1.30564	−0.652821	−	0.757512i	$-0.726414\pi$
$809$	−37.1363	−1.30564	−0.652821	+	0.757512i	$0.726414\pi$
$810$	24.2716	0.852817
$811$	−23.0157	−0.808190	−0.404095	−	0.914717i	$-0.632413\pi$
$811$	−23.0157	−0.808190	−0.404095	+	0.914717i	$0.632413\pi$
$812$	65.1130	2.28502
$813$	−15.6798	−0.549914
$814$	−55.3428	−1.93976
$815$	32.4723	1.13746
$816$	10.7400	0.375977
$817$	−41.1911	−1.44109
$818$	2.49897	0.0873745
$819$	14.3742	0.502275
$820$	5.95679	0.208020
$821$	−39.9835	−1.39543	−0.697716	−	0.716374i	$-0.745800\pi$
$821$	−39.9835	−1.39543	−0.697716	+	0.716374i	$0.745800\pi$
$822$	−20.5700	−0.717461
$823$	17.9450	0.625525	0.312762	−	0.949831i	$-0.398746\pi$
$823$	17.9450	0.625525	0.312762	+	0.949831i	$0.398746\pi$
$824$	80.6768	2.81051
$825$	−2.60272	−0.0906152
$826$	31.7465	1.10460
$827$	−22.3948	−0.778745	−0.389372	−	0.921080i	$-0.627308\pi$
$827$	−22.3948	−0.778745	−0.389372	+	0.921080i	$0.627308\pi$
$828$	−42.3964	−1.47338
$829$	46.7725	1.62448	0.812238	−	0.583326i	$-0.198249\pi$
$829$	46.7725	1.62448	0.812238	+	0.583326i	$0.198249\pi$
$830$	−14.1790	−0.492161
$831$	4.18145	0.145053
$832$	12.4206	0.430607
$833$	6.86218	0.237760
$834$	15.2832	0.529213
$835$	−37.7750	−1.30726
$836$	−100.066	−3.46086
$837$	−26.7523	−0.924696
$838$	−41.3532	−1.42852
$839$	−24.1605	−0.834112	−0.417056	−	0.908881i	$-0.636938\pi$
$839$	−24.1605	−0.834112	−0.417056	+	0.908881i	$0.636938\pi$
$840$	−17.4291	−0.601362
$841$	23.4168	0.807475
$842$	−72.7717	−2.50788
$843$	−17.3934	−0.599062
$844$	61.0765	2.10234
$845$	−11.5618	−0.397737
$846$	17.6572	0.607068
$847$	28.0113	0.962478
$848$	−10.7291	−0.368438
$849$	21.7088	0.745045
$850$	5.08079	0.174270
$851$	18.0153	0.617557
$852$	−29.9422	−1.02580
$853$	49.3895	1.69106	0.845532	−	0.533925i	$-0.179284\pi$
$853$	49.3895	1.69106	0.845532	+	0.533925i	$0.179284\pi$
$854$	−64.3560	−2.20222
$855$	−24.6521	−0.843083
$856$	18.3914	0.628604
$857$	36.0098	1.23007	0.615036	−	0.788499i	$-0.289141\pi$
$857$	36.0098	1.23007	0.615036	+	0.788499i	$0.289141\pi$
$858$	−23.7739	−0.811629
$859$	48.1611	1.64324	0.821618	−	0.570039i	$-0.193072\pi$
$859$	48.1611	1.64324	0.821618	+	0.570039i	$0.193072\pi$
$860$	−75.3080	−2.56798
$861$	−1.02440	−0.0349116
$862$	44.2491	1.50713
$863$	28.9005	0.983783	0.491892	−	0.870656i	$-0.336306\pi$
$863$	28.9005	0.983783	0.491892	+	0.870656i	$0.336306\pi$
$864$	10.1451	0.345145
$865$	38.3143	1.30273
$866$	6.54822	0.222518
$867$	6.77570	0.230115
$868$	−61.6016	−2.09089
$869$	−19.3965	−0.657980
$870$	−26.5309	−0.899482
$871$	−42.8345	−1.45139
$872$	−76.1668	−2.57933
$873$	14.1599	0.479241
$874$	47.9211	1.62095
$875$	−25.1064	−0.848750
$876$	1.58805	0.0536553
$877$	−47.0058	−1.58727	−0.793636	−	0.608393i	$-0.791815\pi$
$877$	−47.0058	−1.58727	−0.793636	+	0.608393i	$0.791815\pi$
$878$	6.55906	0.221358
$879$	−9.50699	−0.320663
$880$	−56.1378	−1.89241
$881$	36.0302	1.21389	0.606944	−	0.794745i	$-0.292395\pi$
$881$	36.0302	1.21389	0.606944	+	0.794745i	$0.292395\pi$
$882$	15.6552	0.527137
$883$	14.4723	0.487033	0.243516	−	0.969897i	$-0.421699\pi$
$883$	14.4723	0.487033	0.243516	+	0.969897i	$0.421699\pi$
$884$	31.5461	1.06101
$885$	−8.79267	−0.295563
$886$	−25.0257	−0.840753
$887$	0.133140	0.00447039	0.00223520	−	0.999998i	$-0.499289\pi$
$887$	0.133140	0.00447039	0.00223520	+	0.999998i	$0.499289\pi$
$888$	−17.9436	−0.602149
$889$	39.1557	1.31324
$890$	−13.3791	−0.448469
$891$	−23.1703	−0.776235
$892$	−4.35020	−0.145655
$893$	−13.5663	−0.453978
$894$	−6.20603	−0.207561
$895$	−36.9918	−1.23650
$896$	−35.1881	−1.17555
$897$	7.73895	0.258396
$898$	99.2042	3.31049
$899$	−49.5901	−1.65392
$900$	7.87894	0.262631
$901$	5.30268	0.176658
$902$	−8.36576	−0.278549
$903$	12.9509	0.430978
$904$	97.1478	3.23109
$905$	−27.9916	−0.930471
$906$	−0.286252	−0.00951008
$907$	58.4083	1.93942	0.969708	−	0.244266i	$-0.0785469\pi$
$907$	58.4083	1.93942	0.969708	+	0.244266i	$0.0785469\pi$
$908$	−73.5435	−2.44063
$909$	13.0577	0.433096
$910$	−29.7043	−0.984687
$911$	35.0365	1.16081	0.580406	−	0.814327i	$-0.302894\pi$
$911$	35.0365	1.16081	0.580406	+	0.814327i	$0.302894\pi$
$912$	−18.8252	−0.623366
$913$	13.5357	0.447966
$914$	−63.6582	−2.10563
$915$	17.8244	0.589256
$916$	−24.3169	−0.803454
$917$	40.6943	1.34384
$918$	−26.6720	−0.880309
$919$	3.17828	0.104842	0.0524209	−	0.998625i	$-0.483306\pi$
$919$	3.17828	0.104842	0.0524209	+	0.998625i	$0.483306\pi$
$920$	46.3331	1.52756
$921$	−12.7219	−0.419202
$922$	22.5141	0.741461
$923$	−26.9869	−0.888284
$924$	31.4618	1.03502
$925$	−3.34796	−0.110080
$926$	−99.7171	−3.27691
$927$	−35.8776	−1.17837
$928$	18.8058	0.617329
$929$	−4.34275	−0.142481	−0.0712405	−	0.997459i	$-0.522696\pi$
$929$	−4.34275	−0.142481	−0.0712405	+	0.997459i	$0.522696\pi$
$930$	25.1001	0.823066
$931$	−12.0281	−0.394205
$932$	43.6150	1.42866
$933$	4.08317	0.133677
$934$	−23.0445	−0.754041
$935$	27.7453	0.907367
$936$	38.0599	1.24403
$937$	−40.8706	−1.33518	−0.667592	−	0.744527i	$-0.732675\pi$
$937$	−40.8706	−1.33518	−0.667592	+	0.744527i	$0.732675\pi$
$938$	83.3941	2.72291
$939$	−18.6782	−0.609540
$940$	−24.8027	−0.808974
$941$	17.2424	0.562087	0.281043	−	0.959695i	$-0.409320\pi$
$941$	17.2424	0.562087	0.281043	+	0.959695i	$0.409320\pi$
$942$	26.4000	0.860158
$943$	2.72324	0.0886810
$944$	33.1532	1.07904
$945$	17.0715	0.555335
$946$	105.763	3.43865
$947$	−7.99358	−0.259757	−0.129878	−	0.991530i	$-0.541459\pi$
$947$	−7.99358	−0.259757	−0.129878	+	0.991530i	$0.541459\pi$
$948$	−11.8917	−0.386226
$949$	1.43131	0.0464624
$950$	−8.90565	−0.288937
$951$	−0.542962	−0.0176067
$952$	−32.4798	−1.05268
$953$	−25.2541	−0.818059	−0.409030	−	0.912521i	$-0.634133\pi$
$953$	−25.2541	−0.818059	−0.409030	+	0.912521i	$0.634133\pi$
$954$	12.0974	0.391667
$955$	−14.2655	−0.461620
$956$	101.352	3.27795
$957$	25.3271	0.818710
$958$	−62.0106	−2.00347
$959$	24.5350	0.792277
$960$	6.69744	0.216159
$961$	15.9158	0.513412
$962$	−30.5811	−0.985976
$963$	−8.17877	−0.263557
$964$	33.3600	1.07445
$965$	−15.9712	−0.514130
$966$	−15.0669	−0.484769
$967$	22.8680	0.735385	0.367692	−	0.929947i	$-0.380148\pi$
$967$	22.8680	0.735385	0.367692	+	0.929947i	$0.380148\pi$
$968$	74.1680	2.38385
$969$	9.30408	0.298890
$970$	−29.2615	−0.939530
$971$	−33.4596	−1.07377	−0.536885	−	0.843655i	$-0.680399\pi$
$971$	−33.4596	−1.07377	−0.536885	+	0.843655i	$0.680399\pi$
$972$	−63.9434	−2.05099
$973$	−18.2291	−0.584398
$974$	−61.8018	−1.98026
$975$	−1.43821	−0.0460594
$976$	−67.2076	−2.15126
$977$	40.1301	1.28388	0.641938	−	0.766756i	$-0.278130\pi$
$977$	40.1301	1.28388	0.641938	+	0.766756i	$0.278130\pi$
$978$	−27.9595	−0.894046
$979$	12.7721	0.408197
$980$	−21.9905	−0.702459
$981$	33.8719	1.08145
$982$	−37.0746	−1.18310
$983$	36.2849	1.15731	0.578655	−	0.815573i	$-0.303578\pi$
$983$	36.2849	1.15731	0.578655	+	0.815573i	$0.303578\pi$
$984$	−2.71241	−0.0864684
$985$	−44.3966	−1.41459
$986$	−49.4412	−1.57453
$987$	4.26538	0.135768
$988$	−55.2942	−1.75914
$989$	−34.4282	−1.09475
$990$	63.2972	2.01172
$991$	−4.82195	−0.153174	−0.0765871	−	0.997063i	$-0.524402\pi$
$991$	−4.82195	−0.153174	−0.0765871	+	0.997063i	$0.524402\pi$
$992$	−17.7916	−0.564884
$993$	−11.1053	−0.352415
$994$	52.5405	1.66648
$995$	9.25380	0.293365
$996$	8.29857	0.262950
$997$	−16.2191	−0.513664	−0.256832	−	0.966456i	$-0.582679\pi$
$997$	−16.2191	−0.513664	−0.256832	+	0.966456i	$0.582679\pi$
$998$	−29.4263	−0.931473
$999$	17.5754	0.556062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.18	✓	20
3.2	odd	2		3681.2.a.i.1.3		20
4.3	odd	2		6544.2.a.i.1.12		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.18	✓	20	1.1	even	1	trivial
3681.2.a.i.1.3		20	3.2	odd	2
6544.2.a.i.1.12		20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49897 q^{2} -0.710812 q^{3} +4.24487 q^{4} +2.06301 q^{5} -1.77630 q^{6} +2.11869 q^{7} +5.60986 q^{8} -2.49475 q^{9} +5.15540 q^{10} -4.92148 q^{11} -3.01730 q^{12} -2.71950 q^{13} +5.29456 q^{14} -1.46641 q^{15} +5.52916 q^{16} -2.73270 q^{17} -6.23430 q^{18} +4.78990 q^{19} +8.75718 q^{20} -1.50599 q^{21} -12.2987 q^{22} +4.00349 q^{23} -3.98756 q^{24} -0.744008 q^{25} -6.79595 q^{26} +3.90573 q^{27} +8.99358 q^{28} +7.23994 q^{29} -3.66452 q^{30} -6.84951 q^{31} +2.59750 q^{32} +3.49825 q^{33} -6.82895 q^{34} +4.37088 q^{35} -10.5899 q^{36} +4.49991 q^{37} +11.9698 q^{38} +1.93305 q^{39} +11.5732 q^{40} +0.680218 q^{41} -3.76344 q^{42} -8.59956 q^{43} -20.8910 q^{44} -5.14668 q^{45} +10.0046 q^{46} -2.83227 q^{47} -3.93019 q^{48} -2.51113 q^{49} -1.85926 q^{50} +1.94244 q^{51} -11.5439 q^{52} -1.94045 q^{53} +9.76032 q^{54} -10.1530 q^{55} +11.8856 q^{56} -3.40472 q^{57} +18.0924 q^{58} +5.99606 q^{59} -6.22471 q^{60} -12.1551 q^{61} -17.1167 q^{62} -5.28561 q^{63} -4.56724 q^{64} -5.61034 q^{65} +8.74203 q^{66} +15.7509 q^{67} -11.6000 q^{68} -2.84573 q^{69} +10.9227 q^{70} +9.92348 q^{71} -13.9952 q^{72} -0.526315 q^{73} +11.2451 q^{74} +0.528850 q^{75} +20.3325 q^{76} -10.4271 q^{77} +4.83064 q^{78} +3.94118 q^{79} +11.4067 q^{80} +4.70800 q^{81} +1.69985 q^{82} -2.75033 q^{83} -6.39274 q^{84} -5.63758 q^{85} -21.4901 q^{86} -5.14624 q^{87} -27.6088 q^{88} -2.59517 q^{89} -12.8614 q^{90} -5.76179 q^{91} +16.9943 q^{92} +4.86871 q^{93} -7.07776 q^{94} +9.88159 q^{95} -1.84633 q^{96} -5.67590 q^{97} -6.27526 q^{98} +12.2779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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