LMFDB - Embedded newform 5070.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
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			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.13706 q^{11} -1.00000 q^{12} +3.04892 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.69202 q^{17} +1.00000 q^{18} +1.55496 q^{19} +1.00000 q^{20} -3.04892 q^{21} +1.13706 q^{22} +5.40581 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.04892 q^{28} -6.15883 q^{29} -1.00000 q^{30} -0.246980 q^{31} +1.00000 q^{32} -1.13706 q^{33} +2.69202 q^{34} +3.04892 q^{35} +1.00000 q^{36} -0.554958 q^{37} +1.55496 q^{38} +1.00000 q^{40} +8.26875 q^{41} -3.04892 q^{42} +2.03923 q^{43} +1.13706 q^{44} +1.00000 q^{45} +5.40581 q^{46} -6.70171 q^{47} -1.00000 q^{48} +2.29590 q^{49} +1.00000 q^{50} -2.69202 q^{51} +5.77479 q^{53} -1.00000 q^{54} +1.13706 q^{55} +3.04892 q^{56} -1.55496 q^{57} -6.15883 q^{58} +1.36898 q^{59} -1.00000 q^{60} +10.9487 q^{61} -0.246980 q^{62} +3.04892 q^{63} +1.00000 q^{64} -1.13706 q^{66} -2.47219 q^{67} +2.69202 q^{68} -5.40581 q^{69} +3.04892 q^{70} -8.57002 q^{71} +1.00000 q^{72} -14.8334 q^{73} -0.554958 q^{74} -1.00000 q^{75} +1.55496 q^{76} +3.46681 q^{77} +5.33513 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.26875 q^{82} -5.45473 q^{83} -3.04892 q^{84} +2.69202 q^{85} +2.03923 q^{86} +6.15883 q^{87} +1.13706 q^{88} +16.8877 q^{89} +1.00000 q^{90} +5.40581 q^{92} +0.246980 q^{93} -6.70171 q^{94} +1.55496 q^{95} -1.00000 q^{96} +1.93900 q^{97} +2.29590 q^{98} +1.13706 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 5 q^{19} + 3 q^{20} - 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 3 q^{30} + 4 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 3 q^{36} - 2 q^{37} + 5 q^{38} + 3 q^{40} + 17 q^{41} + 19 q^{43} - 2 q^{44} + 3 q^{45} + 3 q^{46} + 7 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} - 2 q^{55} - 5 q^{57} - 10 q^{58} + 19 q^{59} - 3 q^{60} + q^{61} + 4 q^{62} + 3 q^{64} + 2 q^{66} - q^{67} + 3 q^{68} - 3 q^{69} - q^{71} + 3 q^{72} - 15 q^{73} - 2 q^{74} - 3 q^{75} + 5 q^{76} + 7 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} + 6 q^{83} + 3 q^{85} + 19 q^{86} + 10 q^{87} - 2 q^{88} + 9 q^{89} + 3 q^{90} + 3 q^{92} - 4 q^{93} + 7 q^{94} + 5 q^{95} - 3 q^{96} - 4 q^{97} - 7 q^{98} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 2 * q^11 - 3 * q^12 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 5 * q^19 + 3 * q^20 - 2 * q^22 + 3 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 3 * q^30 + 4 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 3 * q^36 - 2 * q^37 + 5 * q^38 + 3 * q^40 + 17 * q^41 + 19 * q^43 - 2 * q^44 + 3 * q^45 + 3 * q^46 + 7 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 - 2 * q^55 - 5 * q^57 - 10 * q^58 + 19 * q^59 - 3 * q^60 + q^61 + 4 * q^62 + 3 * q^64 + 2 * q^66 - q^67 + 3 * q^68 - 3 * q^69 - q^71 + 3 * q^72 - 15 * q^73 - 2 * q^74 - 3 * q^75 + 5 * q^76 + 7 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 + 6 * q^83 + 3 * q^85 + 19 * q^86 + 10 * q^87 - 2 * q^88 + 9 * q^89 + 3 * q^90 + 3 * q^92 - 4 * q^93 + 7 * q^94 + 5 * q^95 - 3 * q^96 - 4 * q^97 - 7 * q^98 - 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	3.04892	1.15238	0.576191	−	0.817315i	$-0.304538\pi$
$7$	3.04892	1.15238	0.576191	+	0.817315i	$0.304538\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.13706	0.342837	0.171419	−	0.985198i	$-0.445165\pi$
$11$	1.13706	0.342837	0.171419	+	0.985198i	$0.445165\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	3.04892	0.814857
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.69202	0.652911	0.326456	−	0.945213i	$-0.394146\pi$
$17$	2.69202	0.652911	0.326456	+	0.945213i	$0.394146\pi$
$18$	1.00000	0.235702
$19$	1.55496	0.356732	0.178366	−	0.983964i	$-0.442919\pi$
$19$	1.55496	0.356732	0.178366	+	0.983964i	$0.442919\pi$
$20$	1.00000	0.223607
$21$	−3.04892	−0.665328
$22$	1.13706	0.242423
$23$	5.40581	1.12719	0.563595	−	0.826051i	$-0.309418\pi$
$23$	5.40581	1.12719	0.563595	+	0.826051i	$0.309418\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.04892	0.576191
$29$	−6.15883	−1.14367	−0.571833	−	0.820370i	$-0.693768\pi$
$29$	−6.15883	−1.14367	−0.571833	+	0.820370i	$0.693768\pi$
$30$	−1.00000	−0.182574
$31$	−0.246980	−0.0443588	−0.0221794	−	0.999754i	$-0.507061\pi$
$31$	−0.246980	−0.0443588	−0.0221794	+	0.999754i	$0.507061\pi$
$32$	1.00000	0.176777
$33$	−1.13706	−0.197937
$34$	2.69202	0.461678
$35$	3.04892	0.515361
$36$	1.00000	0.166667
$37$	−0.554958	−0.0912346	−0.0456173	−	0.998959i	$-0.514525\pi$
$37$	−0.554958	−0.0912346	−0.0456173	+	0.998959i	$0.514525\pi$
$38$	1.55496	0.252248
$39$	0	0
$40$	1.00000	0.158114
$41$	8.26875	1.29136	0.645681	−	0.763607i	$-0.276574\pi$
$41$	8.26875	1.29136	0.645681	+	0.763607i	$0.276574\pi$
$42$	−3.04892	−0.470458
$43$	2.03923	0.310979	0.155490	−	0.987838i	$-0.450304\pi$
$43$	2.03923	0.310979	0.155490	+	0.987838i	$0.450304\pi$
$44$	1.13706	0.171419
$45$	1.00000	0.149071
$46$	5.40581	0.797044
$47$	−6.70171	−0.977545	−0.488772	−	0.872411i	$-0.662555\pi$
$47$	−6.70171	−0.977545	−0.488772	+	0.872411i	$0.662555\pi$
$48$	−1.00000	−0.144338
$49$	2.29590	0.327985
$50$	1.00000	0.141421
$51$	−2.69202	−0.376958
$52$	0	0
$53$	5.77479	0.793229	0.396614	−	0.917985i	$-0.370185\pi$
$53$	5.77479	0.793229	0.396614	+	0.917985i	$0.370185\pi$
$54$	−1.00000	−0.136083
$55$	1.13706	0.153322
$56$	3.04892	0.407429
$57$	−1.55496	−0.205959
$58$	−6.15883	−0.808694
$59$	1.36898	0.178226	0.0891128	−	0.996022i	$-0.471597\pi$
$59$	1.36898	0.178226	0.0891128	+	0.996022i	$0.471597\pi$
$60$	−1.00000	−0.129099
$61$	10.9487	1.40184	0.700918	−	0.713242i	$-0.252774\pi$
$61$	10.9487	1.40184	0.700918	+	0.713242i	$0.252774\pi$
$62$	−0.246980	−0.0313664
$63$	3.04892	0.384127
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.13706	−0.139963
$67$	−2.47219	−0.302026	−0.151013	−	0.988532i	$-0.548254\pi$
$67$	−2.47219	−0.302026	−0.151013	+	0.988532i	$0.548254\pi$
$68$	2.69202	0.326456
$69$	−5.40581	−0.650783
$70$	3.04892	0.364415
$71$	−8.57002	−1.01707	−0.508537	−	0.861040i	$-0.669814\pi$
$71$	−8.57002	−1.01707	−0.508537	+	0.861040i	$0.669814\pi$
$72$	1.00000	0.117851
$73$	−14.8334	−1.73612	−0.868059	−	0.496461i	$-0.834632\pi$
$73$	−14.8334	−1.73612	−0.868059	+	0.496461i	$0.834632\pi$
$74$	−0.554958	−0.0645126
$75$	−1.00000	−0.115470
$76$	1.55496	0.178366
$77$	3.46681	0.395080
$78$	0	0
$79$	5.33513	0.600249	0.300124	−	0.953900i	$-0.402972\pi$
$79$	5.33513	0.600249	0.300124	+	0.953900i	$0.402972\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	8.26875	0.913131
$83$	−5.45473	−0.598734	−0.299367	−	0.954138i	$-0.596776\pi$
$83$	−5.45473	−0.598734	−0.299367	+	0.954138i	$0.596776\pi$
$84$	−3.04892	−0.332664
$85$	2.69202	0.291991
$86$	2.03923	0.219896
$87$	6.15883	0.660296
$88$	1.13706	0.121211
$89$	16.8877	1.79009	0.895046	−	0.445974i	$-0.147143\pi$
$89$	16.8877	1.79009	0.895046	+	0.445974i	$0.147143\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	5.40581	0.563595
$93$	0.246980	0.0256106
$94$	−6.70171	−0.691229
$95$	1.55496	0.159535
$96$	−1.00000	−0.102062
$97$	1.93900	0.196876	0.0984379	−	0.995143i	$-0.468615\pi$
$97$	1.93900	0.196876	0.0984379	+	0.995143i	$0.468615\pi$
$98$	2.29590	0.231921
$99$	1.13706	0.114279
$100$	1.00000	0.100000
$101$	−14.4940	−1.44220	−0.721101	−	0.692830i	$-0.756364\pi$
$101$	−14.4940	−1.44220	−0.721101	+	0.692830i	$0.756364\pi$
$102$	−2.69202	−0.266550
$103$	−16.1836	−1.59462	−0.797308	−	0.603572i	$-0.793743\pi$
$103$	−16.1836	−1.59462	−0.797308	+	0.603572i	$0.793743\pi$
$104$	0	0
$105$	−3.04892	−0.297544
$106$	5.77479	0.560897
$107$	4.92931	0.476535	0.238267	−	0.971200i	$-0.423421\pi$
$107$	4.92931	0.476535	0.238267	+	0.971200i	$0.423421\pi$
$108$	−1.00000	−0.0962250
$109$	0.801938	0.0768117	0.0384059	−	0.999262i	$-0.487772\pi$
$109$	0.801938	0.0768117	0.0384059	+	0.999262i	$0.487772\pi$
$110$	1.13706	0.108415
$111$	0.554958	0.0526743
$112$	3.04892	0.288096
$113$	−11.5211	−1.08381	−0.541907	−	0.840438i	$-0.682298\pi$
$113$	−11.5211	−1.08381	−0.541907	+	0.840438i	$0.682298\pi$
$114$	−1.55496	−0.145635
$115$	5.40581	0.504095
$116$	−6.15883	−0.571833
$117$	0	0
$118$	1.36898	0.126025
$119$	8.20775	0.752403
$120$	−1.00000	−0.0912871
$121$	−9.70709	−0.882462
$122$	10.9487	0.991248
$123$	−8.26875	−0.745568
$124$	−0.246980	−0.0221794
$125$	1.00000	0.0894427
$126$	3.04892	0.271619
$127$	6.22521	0.552398	0.276199	−	0.961100i	$-0.410925\pi$
$127$	6.22521	0.552398	0.276199	+	0.961100i	$0.410925\pi$
$128$	1.00000	0.0883883
$129$	−2.03923	−0.179544
$130$	0	0
$131$	3.34481	0.292238	0.146119	−	0.989267i	$-0.453322\pi$
$131$	3.34481	0.292238	0.146119	+	0.989267i	$0.453322\pi$
$132$	−1.13706	−0.0989687
$133$	4.74094	0.411092
$134$	−2.47219	−0.213565
$135$	−1.00000	−0.0860663
$136$	2.69202	0.230839
$137$	−9.53079	−0.814271	−0.407135	−	0.913368i	$-0.633472\pi$
$137$	−9.53079	−0.814271	−0.407135	+	0.913368i	$0.633472\pi$
$138$	−5.40581	−0.460173
$139$	18.7017	1.58626	0.793129	−	0.609053i	$-0.208451\pi$
$139$	18.7017	1.58626	0.793129	+	0.609053i	$0.208451\pi$
$140$	3.04892	0.257681
$141$	6.70171	0.564386
$142$	−8.57002	−0.719180
$143$	0	0
$144$	1.00000	0.0833333
$145$	−6.15883	−0.511463
$146$	−14.8334	−1.22762
$147$	−2.29590	−0.189362
$148$	−0.554958	−0.0456173
$149$	16.4209	1.34525	0.672625	−	0.739983i	$-0.265167\pi$
$149$	16.4209	1.34525	0.672625	+	0.739983i	$0.265167\pi$
$150$	−1.00000	−0.0816497
$151$	−9.26636	−0.754085	−0.377043	−	0.926196i	$-0.623059\pi$
$151$	−9.26636	−0.754085	−0.377043	+	0.926196i	$0.623059\pi$
$152$	1.55496	0.126124
$153$	2.69202	0.217637
$154$	3.46681	0.279364
$155$	−0.246980	−0.0198379
$156$	0	0
$157$	17.0761	1.36282	0.681409	−	0.731903i	$-0.261367\pi$
$157$	17.0761	1.36282	0.681409	+	0.731903i	$0.261367\pi$
$158$	5.33513	0.424440
$159$	−5.77479	−0.457971
$160$	1.00000	0.0790569
$161$	16.4819	1.29895
$162$	1.00000	0.0785674
$163$	−9.61894	−0.753414	−0.376707	−	0.926333i	$-0.622944\pi$
$163$	−9.61894	−0.753414	−0.376707	+	0.926333i	$0.622944\pi$
$164$	8.26875	0.645681
$165$	−1.13706	−0.0885203
$166$	−5.45473	−0.423369
$167$	21.8509	1.69087	0.845435	−	0.534078i	$-0.179341\pi$
$167$	21.8509	1.69087	0.845435	+	0.534078i	$0.179341\pi$
$168$	−3.04892	−0.235229
$169$	0	0
$170$	2.69202	0.206469
$171$	1.55496	0.118911
$172$	2.03923	0.155490
$173$	16.6853	1.26856	0.634281	−	0.773103i	$-0.281296\pi$
$173$	16.6853	1.26856	0.634281	+	0.773103i	$0.281296\pi$
$174$	6.15883	0.466900
$175$	3.04892	0.230476
$176$	1.13706	0.0857094
$177$	−1.36898	−0.102899
$178$	16.8877	1.26579
$179$	−3.94438	−0.294817	−0.147408	−	0.989076i	$-0.547093\pi$
$179$	−3.94438	−0.294817	−0.147408	+	0.989076i	$0.547093\pi$
$180$	1.00000	0.0745356
$181$	−20.5308	−1.52604	−0.763021	−	0.646374i	$-0.776285\pi$
$181$	−20.5308	−1.52604	−0.763021	+	0.646374i	$0.776285\pi$
$182$	0	0
$183$	−10.9487	−0.809350
$184$	5.40581	0.398522
$185$	−0.554958	−0.0408013
$186$	0.246980	0.0181094
$187$	3.06100	0.223842
$188$	−6.70171	−0.488772
$189$	−3.04892	−0.221776
$190$	1.55496	0.112809
$191$	−6.66487	−0.482253	−0.241127	−	0.970494i	$-0.577517\pi$
$191$	−6.66487	−0.482253	−0.241127	+	0.970494i	$0.577517\pi$
$192$	−1.00000	−0.0721688
$193$	16.6558	1.19891	0.599455	−	0.800409i	$-0.295384\pi$
$193$	16.6558	1.19891	0.599455	+	0.800409i	$0.295384\pi$
$194$	1.93900	0.139212
$195$	0	0
$196$	2.29590	0.163993
$197$	19.4494	1.38571	0.692855	−	0.721077i	$-0.256353\pi$
$197$	19.4494	1.38571	0.692855	+	0.721077i	$0.256353\pi$
$198$	1.13706	0.0808076
$199$	−21.6558	−1.53514	−0.767569	−	0.640967i	$-0.778534\pi$
$199$	−21.6558	−1.53514	−0.767569	+	0.640967i	$0.778534\pi$
$200$	1.00000	0.0707107
$201$	2.47219	0.174375
$202$	−14.4940	−1.01979
$203$	−18.7778	−1.31794
$204$	−2.69202	−0.188479
$205$	8.26875	0.577515
$206$	−16.1836	−1.12756
$207$	5.40581	0.375730
$208$	0	0
$209$	1.76809	0.122301
$210$	−3.04892	−0.210395
$211$	12.1521	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$211$	12.1521	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$212$	5.77479	0.396614
$213$	8.57002	0.587208
$214$	4.92931	0.336961
$215$	2.03923	0.139074
$216$	−1.00000	−0.0680414
$217$	−0.753020	−0.0511184
$218$	0.801938	0.0543141
$219$	14.8334	1.00235
$220$	1.13706	0.0766608
$221$	0	0
$222$	0.554958	0.0372464
$223$	1.11960	0.0749743	0.0374871	−	0.999297i	$-0.488065\pi$
$223$	1.11960	0.0749743	0.0374871	+	0.999297i	$0.488065\pi$
$224$	3.04892	0.203714
$225$	1.00000	0.0666667
$226$	−11.5211	−0.766373
$227$	4.06638	0.269895	0.134947	−	0.990853i	$-0.456913\pi$
$227$	4.06638	0.269895	0.134947	+	0.990853i	$0.456913\pi$
$228$	−1.55496	−0.102980
$229$	−16.5187	−1.09159	−0.545794	−	0.837920i	$-0.683772\pi$
$229$	−16.5187	−1.09159	−0.545794	+	0.837920i	$0.683772\pi$
$230$	5.40581	0.356449
$231$	−3.46681	−0.228099
$232$	−6.15883	−0.404347
$233$	−6.16421	−0.403831	−0.201915	−	0.979403i	$-0.564717\pi$
$233$	−6.16421	−0.403831	−0.201915	+	0.979403i	$0.564717\pi$
$234$	0	0
$235$	−6.70171	−0.437171
$236$	1.36898	0.0891128
$237$	−5.33513	−0.346554
$238$	8.20775	0.532029
$239$	18.1793	1.17592	0.587960	−	0.808890i	$-0.299931\pi$
$239$	18.1793	1.17592	0.587960	+	0.808890i	$0.299931\pi$
$240$	−1.00000	−0.0645497
$241$	−9.21744	−0.593747	−0.296874	−	0.954917i	$-0.595944\pi$
$241$	−9.21744	−0.593747	−0.296874	+	0.954917i	$0.595944\pi$
$242$	−9.70709	−0.623995
$243$	−1.00000	−0.0641500
$244$	10.9487	0.700918
$245$	2.29590	0.146679
$246$	−8.26875	−0.527196
$247$	0	0
$248$	−0.246980	−0.0156832
$249$	5.45473	0.345680
$250$	1.00000	0.0632456
$251$	17.2054	1.08599	0.542996	−	0.839735i	$-0.317290\pi$
$251$	17.2054	1.08599	0.542996	+	0.839735i	$0.317290\pi$
$252$	3.04892	0.192064
$253$	6.14675	0.386443
$254$	6.22521	0.390604
$255$	−2.69202	−0.168581
$256$	1.00000	0.0625000
$257$	−19.5157	−1.21736	−0.608679	−	0.793417i	$-0.708300\pi$
$257$	−19.5157	−1.21736	−0.608679	+	0.793417i	$0.708300\pi$
$258$	−2.03923	−0.126957
$259$	−1.69202	−0.105137
$260$	0	0
$261$	−6.15883	−0.381222
$262$	3.34481	0.206643
$263$	12.8834	0.794423	0.397212	−	0.917727i	$-0.369978\pi$
$263$	12.8834	0.794423	0.397212	+	0.917727i	$0.369978\pi$
$264$	−1.13706	−0.0699814
$265$	5.77479	0.354743
$266$	4.74094	0.290686
$267$	−16.8877	−1.03351
$268$	−2.47219	−0.151013
$269$	−30.9138	−1.88485	−0.942423	−	0.334423i	$-0.891459\pi$
$269$	−30.9138	−1.88485	−0.942423	+	0.334423i	$0.891459\pi$
$270$	−1.00000	−0.0608581
$271$	7.24027	0.439815	0.219908	−	0.975521i	$-0.429424\pi$
$271$	7.24027	0.439815	0.219908	+	0.975521i	$0.429424\pi$
$272$	2.69202	0.163228
$273$	0	0
$274$	−9.53079	−0.575776
$275$	1.13706	0.0685675
$276$	−5.40581	−0.325392
$277$	12.2446	0.735706	0.367853	−	0.929884i	$-0.380093\pi$
$277$	12.2446	0.735706	0.367853	+	0.929884i	$0.380093\pi$
$278$	18.7017	1.12165
$279$	−0.246980	−0.0147863
$280$	3.04892	0.182208
$281$	−2.74632	−0.163831	−0.0819157	−	0.996639i	$-0.526104\pi$
$281$	−2.74632	−0.163831	−0.0819157	+	0.996639i	$0.526104\pi$
$282$	6.70171	0.399081
$283$	14.9312	0.887570	0.443785	−	0.896133i	$-0.353635\pi$
$283$	14.9312	0.887570	0.443785	+	0.896133i	$0.353635\pi$
$284$	−8.57002	−0.508537
$285$	−1.55496	−0.0921078
$286$	0	0
$287$	25.2107	1.48814
$288$	1.00000	0.0589256
$289$	−9.75302	−0.573707
$290$	−6.15883	−0.361659
$291$	−1.93900	−0.113666
$292$	−14.8334	−0.868059
$293$	18.5810	1.08552	0.542758	−	0.839889i	$-0.317380\pi$
$293$	18.5810	1.08552	0.542758	+	0.839889i	$0.317380\pi$
$294$	−2.29590	−0.133899
$295$	1.36898	0.0797049
$296$	−0.554958	−0.0322563
$297$	−1.13706	−0.0659791
$298$	16.4209	0.951236
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	6.21744	0.358367
$302$	−9.26636	−0.533219
$303$	14.4940	0.832656
$304$	1.55496	0.0891830
$305$	10.9487	0.626920
$306$	2.69202	0.153893
$307$	32.4359	1.85122	0.925609	−	0.378482i	$-0.123554\pi$
$307$	32.4359	1.85122	0.925609	+	0.378482i	$0.123554\pi$
$308$	3.46681	0.197540
$309$	16.1836	0.920652
$310$	−0.246980	−0.0140275
$311$	−21.2771	−1.20651	−0.603257	−	0.797547i	$-0.706131\pi$
$311$	−21.2771	−1.20651	−0.603257	+	0.797547i	$0.706131\pi$
$312$	0	0
$313$	−18.7060	−1.05733	−0.528663	−	0.848832i	$-0.677307\pi$
$313$	−18.7060	−1.05733	−0.528663	+	0.848832i	$0.677307\pi$
$314$	17.0761	0.963658
$315$	3.04892	0.171787
$316$	5.33513	0.300124
$317$	17.4776	0.981638	0.490819	−	0.871262i	$-0.336698\pi$
$317$	17.4776	0.981638	0.490819	+	0.871262i	$0.336698\pi$
$318$	−5.77479	−0.323834
$319$	−7.00298	−0.392092
$320$	1.00000	0.0559017
$321$	−4.92931	−0.275127
$322$	16.4819	0.918499
$323$	4.18598	0.232914
$324$	1.00000	0.0555556
$325$	0	0
$326$	−9.61894	−0.532744
$327$	−0.801938	−0.0443473
$328$	8.26875	0.456565
$329$	−20.4330	−1.12651
$330$	−1.13706	−0.0625933
$331$	−23.1890	−1.27458	−0.637290	−	0.770624i	$-0.719945\pi$
$331$	−23.1890	−1.27458	−0.637290	+	0.770624i	$0.719945\pi$
$332$	−5.45473	−0.299367
$333$	−0.554958	−0.0304115
$334$	21.8509	1.19563
$335$	−2.47219	−0.135070
$336$	−3.04892	−0.166332
$337$	−26.1400	−1.42394	−0.711970	−	0.702210i	$-0.752197\pi$
$337$	−26.1400	−1.42394	−0.711970	+	0.702210i	$0.752197\pi$
$338$	0	0
$339$	11.5211	0.625741
$340$	2.69202	0.145995
$341$	−0.280831	−0.0152079
$342$	1.55496	0.0840825
$343$	−14.3424	−0.774418
$344$	2.03923	0.109948
$345$	−5.40581	−0.291039
$346$	16.6853	0.897008
$347$	15.2959	0.821127	0.410563	−	0.911832i	$-0.365332\pi$
$347$	15.2959	0.821127	0.410563	+	0.911832i	$0.365332\pi$
$348$	6.15883	0.330148
$349$	−19.2107	−1.02833	−0.514164	−	0.857692i	$-0.671898\pi$
$349$	−19.2107	−1.02833	−0.514164	+	0.857692i	$0.671898\pi$
$350$	3.04892	0.162971
$351$	0	0
$352$	1.13706	0.0606057
$353$	−5.35450	−0.284991	−0.142496	−	0.989795i	$-0.545513\pi$
$353$	−5.35450	−0.284991	−0.142496	+	0.989795i	$0.545513\pi$
$354$	−1.36898	−0.0727603
$355$	−8.57002	−0.454850
$356$	16.8877	0.895046
$357$	−8.20775	−0.434400
$358$	−3.94438	−0.208467
$359$	8.49934	0.448578	0.224289	−	0.974523i	$-0.427994\pi$
$359$	8.49934	0.448578	0.224289	+	0.974523i	$0.427994\pi$
$360$	1.00000	0.0527046
$361$	−16.5821	−0.872742
$362$	−20.5308	−1.07907
$363$	9.70709	0.509490
$364$	0	0
$365$	−14.8334	−0.776415
$366$	−10.9487	−0.572297
$367$	4.76032	0.248486	0.124243	−	0.992252i	$-0.460350\pi$
$367$	4.76032	0.248486	0.124243	+	0.992252i	$0.460350\pi$
$368$	5.40581	0.281797
$369$	8.26875	0.430454
$370$	−0.554958	−0.0288509
$371$	17.6069	0.914103
$372$	0.246980	0.0128053
$373$	−20.2064	−1.04625	−0.523124	−	0.852256i	$-0.675234\pi$
$373$	−20.2064	−1.04625	−0.523124	+	0.852256i	$0.675234\pi$
$374$	3.06100	0.158280
$375$	−1.00000	−0.0516398
$376$	−6.70171	−0.345614
$377$	0	0
$378$	−3.04892	−0.156819
$379$	24.4088	1.25380	0.626898	−	0.779101i	$-0.284324\pi$
$379$	24.4088	1.25380	0.626898	+	0.779101i	$0.284324\pi$
$380$	1.55496	0.0797677
$381$	−6.22521	−0.318927
$382$	−6.66487	−0.341005
$383$	32.0237	1.63633	0.818167	−	0.574981i	$-0.194990\pi$
$383$	32.0237	1.63633	0.818167	+	0.574981i	$0.194990\pi$
$384$	−1.00000	−0.0510310
$385$	3.46681	0.176685
$386$	16.6558	0.847757
$387$	2.03923	0.103660
$388$	1.93900	0.0984379
$389$	21.1879	1.07427	0.537135	−	0.843497i	$-0.319507\pi$
$389$	21.1879	1.07427	0.537135	+	0.843497i	$0.319507\pi$
$390$	0	0
$391$	14.5526	0.735955
$392$	2.29590	0.115960
$393$	−3.34481	−0.168724
$394$	19.4494	0.979844
$395$	5.33513	0.268439
$396$	1.13706	0.0571396
$397$	10.9608	0.550105	0.275053	−	0.961429i	$-0.411305\pi$
$397$	10.9608	0.550105	0.275053	+	0.961429i	$0.411305\pi$
$398$	−21.6558	−1.08551
$399$	−4.74094	−0.237344
$400$	1.00000	0.0500000
$401$	−29.2301	−1.45968	−0.729841	−	0.683617i	$-0.760406\pi$
$401$	−29.2301	−1.45968	−0.729841	+	0.683617i	$0.760406\pi$
$402$	2.47219	0.123302
$403$	0	0
$404$	−14.4940	−0.721101
$405$	1.00000	0.0496904
$406$	−18.7778	−0.931925
$407$	−0.631023	−0.0312786
$408$	−2.69202	−0.133275
$409$	36.3163	1.79573	0.897864	−	0.440274i	$-0.145119\pi$
$409$	36.3163	1.79573	0.897864	+	0.440274i	$0.145119\pi$
$410$	8.26875	0.408364
$411$	9.53079	0.470119
$412$	−16.1836	−0.797308
$413$	4.17390	0.205384
$414$	5.40581	0.265681
$415$	−5.45473	−0.267762
$416$	0	0
$417$	−18.7017	−0.915827
$418$	1.76809	0.0864799
$419$	11.0382	0.539250	0.269625	−	0.962965i	$-0.413100\pi$
$419$	11.0382	0.539250	0.269625	+	0.962965i	$0.413100\pi$
$420$	−3.04892	−0.148772
$421$	4.58642	0.223528	0.111764	−	0.993735i	$-0.464350\pi$
$421$	4.58642	0.223528	0.111764	+	0.993735i	$0.464350\pi$
$422$	12.1521	0.591556
$423$	−6.70171	−0.325848
$424$	5.77479	0.280449
$425$	2.69202	0.130582
$426$	8.57002	0.415219
$427$	33.3817	1.61545
$428$	4.92931	0.238267
$429$	0	0
$430$	2.03923	0.0983403
$431$	19.3642	0.932740	0.466370	−	0.884590i	$-0.345561\pi$
$431$	19.3642	0.932740	0.466370	+	0.884590i	$0.345561\pi$
$432$	−1.00000	−0.0481125
$433$	4.41119	0.211988	0.105994	−	0.994367i	$-0.466198\pi$
$433$	4.41119	0.211988	0.105994	+	0.994367i	$0.466198\pi$
$434$	−0.753020	−0.0361461
$435$	6.15883	0.295293
$436$	0.801938	0.0384059
$437$	8.40581	0.402105
$438$	14.8334	0.708767
$439$	39.5260	1.88647	0.943237	−	0.332121i	$-0.107764\pi$
$439$	39.5260	1.88647	0.943237	+	0.332121i	$0.107764\pi$
$440$	1.13706	0.0542074
$441$	2.29590	0.109328
$442$	0	0
$443$	−7.16182	−0.340268	−0.170134	−	0.985421i	$-0.554420\pi$
$443$	−7.16182	−0.340268	−0.170134	+	0.985421i	$0.554420\pi$
$444$	0.554958	0.0263371
$445$	16.8877	0.800553
$446$	1.11960	0.0530148
$447$	−16.4209	−0.776681
$448$	3.04892	0.144048
$449$	−4.46011	−0.210485	−0.105243	−	0.994447i	$-0.533562\pi$
$449$	−4.46011	−0.210485	−0.105243	+	0.994447i	$0.533562\pi$
$450$	1.00000	0.0471405
$451$	9.40209	0.442727
$452$	−11.5211	−0.541907
$453$	9.26636	0.435371
$454$	4.06638	0.190844
$455$	0	0
$456$	−1.55496	−0.0728176
$457$	−8.04593	−0.376373	−0.188186	−	0.982133i	$-0.560261\pi$
$457$	−8.04593	−0.376373	−0.188186	+	0.982133i	$0.560261\pi$
$458$	−16.5187	−0.771869
$459$	−2.69202	−0.125653
$460$	5.40581	0.252047
$461$	4.00538	0.186549	0.0932745	−	0.995640i	$-0.470267\pi$
$461$	4.00538	0.186549	0.0932745	+	0.995640i	$0.470267\pi$
$462$	−3.46681	−0.161291
$463$	−37.6142	−1.74808	−0.874039	−	0.485856i	$-0.838508\pi$
$463$	−37.6142	−1.74808	−0.874039	+	0.485856i	$0.838508\pi$
$464$	−6.15883	−0.285917
$465$	0.246980	0.0114534
$466$	−6.16421	−0.285552
$467$	−32.8079	−1.51817	−0.759084	−	0.650992i	$-0.774353\pi$
$467$	−32.8079	−1.51817	−0.759084	+	0.650992i	$0.774353\pi$
$468$	0	0
$469$	−7.53750	−0.348049
$470$	−6.70171	−0.309127
$471$	−17.0761	−0.786824
$472$	1.36898	0.0630123
$473$	2.31873	0.106615
$474$	−5.33513	−0.245050
$475$	1.55496	0.0713464
$476$	8.20775	0.376202
$477$	5.77479	0.264410
$478$	18.1793	0.831501
$479$	−9.88530	−0.451671	−0.225835	−	0.974165i	$-0.572511\pi$
$479$	−9.88530	−0.451671	−0.225835	+	0.974165i	$0.572511\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−9.21744	−0.419843
$483$	−16.4819	−0.749951
$484$	−9.70709	−0.441231
$485$	1.93900	0.0880455
$486$	−1.00000	−0.0453609
$487$	20.2610	0.918113	0.459056	−	0.888407i	$-0.348188\pi$
$487$	20.2610	0.918113	0.459056	+	0.888407i	$0.348188\pi$
$488$	10.9487	0.495624
$489$	9.61894	0.434984
$490$	2.29590	0.103718
$491$	−22.4940	−1.01514	−0.507569	−	0.861611i	$-0.669456\pi$
$491$	−22.4940	−1.01514	−0.507569	+	0.861611i	$0.669456\pi$
$492$	−8.26875	−0.372784
$493$	−16.5797	−0.746713
$494$	0	0
$495$	1.13706	0.0511072
$496$	−0.246980	−0.0110897
$497$	−26.1293	−1.17206
$498$	5.45473	0.244432
$499$	−15.2728	−0.683704	−0.341852	−	0.939754i	$-0.611054\pi$
$499$	−15.2728	−0.683704	−0.341852	+	0.939754i	$0.611054\pi$
$500$	1.00000	0.0447214
$501$	−21.8509	−0.976224
$502$	17.2054	0.767913
$503$	−2.95838	−0.131908	−0.0659538	−	0.997823i	$-0.521009\pi$
$503$	−2.95838	−0.131908	−0.0659538	+	0.997823i	$0.521009\pi$
$504$	3.04892	0.135810
$505$	−14.4940	−0.644973
$506$	6.14675	0.273256
$507$	0	0
$508$	6.22521	0.276199
$509$	−4.34614	−0.192639	−0.0963197	−	0.995350i	$-0.530707\pi$
$509$	−4.34614	−0.192639	−0.0963197	+	0.995350i	$0.530707\pi$
$510$	−2.69202	−0.119205
$511$	−45.2258	−2.00067
$512$	1.00000	0.0441942
$513$	−1.55496	−0.0686531
$514$	−19.5157	−0.860802
$515$	−16.1836	−0.713134
$516$	−2.03923	−0.0897720
$517$	−7.62027	−0.335139
$518$	−1.69202	−0.0743432
$519$	−16.6853	−0.732404
$520$	0	0
$521$	9.13228	0.400092	0.200046	−	0.979786i	$-0.435891\pi$
$521$	9.13228	0.400092	0.200046	+	0.979786i	$0.435891\pi$
$522$	−6.15883	−0.269565
$523$	10.2828	0.449633	0.224817	−	0.974401i	$-0.427822\pi$
$523$	10.2828	0.449633	0.224817	+	0.974401i	$0.427822\pi$
$524$	3.34481	0.146119
$525$	−3.04892	−0.133066
$526$	12.8834	0.561742
$527$	−0.664874	−0.0289624
$528$	−1.13706	−0.0494843
$529$	6.22282	0.270557
$530$	5.77479	0.250841
$531$	1.36898	0.0594086
$532$	4.74094	0.205546
$533$	0	0
$534$	−16.8877	−0.730802
$535$	4.92931	0.213113
$536$	−2.47219	−0.106782
$537$	3.94438	0.170212
$538$	−30.9138	−1.33279
$539$	2.61058	0.112446
$540$	−1.00000	−0.0430331
$541$	1.47517	0.0634226	0.0317113	−	0.999497i	$-0.489904\pi$
$541$	1.47517	0.0634226	0.0317113	+	0.999497i	$0.489904\pi$
$542$	7.24027	0.310996
$543$	20.5308	0.881061
$544$	2.69202	0.115419
$545$	0.801938	0.0343512
$546$	0	0
$547$	−4.82908	−0.206477	−0.103238	−	0.994657i	$-0.532920\pi$
$547$	−4.82908	−0.206477	−0.103238	+	0.994657i	$0.532920\pi$
$548$	−9.53079	−0.407135
$549$	10.9487	0.467279
$550$	1.13706	0.0484845
$551$	−9.57673	−0.407982
$552$	−5.40581	−0.230087
$553$	16.2664	0.691716
$554$	12.2446	0.520223
$555$	0.554958	0.0235567
$556$	18.7017	0.793129
$557$	3.26145	0.138192	0.0690961	−	0.997610i	$-0.477988\pi$
$557$	3.26145	0.138192	0.0690961	+	0.997610i	$0.477988\pi$
$558$	−0.246980	−0.0104555
$559$	0	0
$560$	3.04892	0.128840
$561$	−3.06100	−0.129235
$562$	−2.74632	−0.115846
$563$	18.1551	0.765147	0.382573	−	0.923925i	$-0.375038\pi$
$563$	18.1551	0.765147	0.382573	+	0.923925i	$0.375038\pi$
$564$	6.70171	0.282193
$565$	−11.5211	−0.484697
$566$	14.9312	0.627606
$567$	3.04892	0.128042
$568$	−8.57002	−0.359590
$569$	−39.2669	−1.64616	−0.823078	−	0.567928i	$-0.807745\pi$
$569$	−39.2669	−1.64616	−0.823078	+	0.567928i	$0.807745\pi$
$570$	−1.55496	−0.0651300
$571$	−36.8649	−1.54275	−0.771373	−	0.636383i	$-0.780430\pi$
$571$	−36.8649	−1.54275	−0.771373	+	0.636383i	$0.780430\pi$
$572$	0	0
$573$	6.66487	0.278429
$574$	25.2107	1.05228
$575$	5.40581	0.225438
$576$	1.00000	0.0416667
$577$	−20.3043	−0.845277	−0.422639	−	0.906298i	$-0.638896\pi$
$577$	−20.3043	−0.845277	−0.422639	+	0.906298i	$0.638896\pi$
$578$	−9.75302	−0.405672
$579$	−16.6558	−0.692190
$580$	−6.15883	−0.255732
$581$	−16.6310	−0.689971
$582$	−1.93900	−0.0803742
$583$	6.56630	0.271948
$584$	−14.8334	−0.613810
$585$	0	0
$586$	18.5810	0.767576
$587$	24.5472	1.01317	0.506585	−	0.862190i	$-0.330907\pi$
$587$	24.5472	1.01317	0.506585	+	0.862190i	$0.330907\pi$
$588$	−2.29590	−0.0946812
$589$	−0.384043	−0.0158242
$590$	1.36898	0.0563599
$591$	−19.4494	−0.800040
$592$	−0.554958	−0.0228086
$593$	−24.0844	−0.989029	−0.494514	−	0.869169i	$-0.664654\pi$
$593$	−24.0844	−0.989029	−0.494514	+	0.869169i	$0.664654\pi$
$594$	−1.13706	−0.0466543
$595$	8.20775	0.336485
$596$	16.4209	0.672625
$597$	21.6558	0.886312
$598$	0	0
$599$	1.84415	0.0753499	0.0376750	−	0.999290i	$-0.488005\pi$
$599$	1.84415	0.0753499	0.0376750	+	0.999290i	$0.488005\pi$
$600$	−1.00000	−0.0408248
$601$	−15.6770	−0.639476	−0.319738	−	0.947506i	$-0.603595\pi$
$601$	−15.6770	−0.639476	−0.319738	+	0.947506i	$0.603595\pi$
$602$	6.21744	0.253404
$603$	−2.47219	−0.100675
$604$	−9.26636	−0.377043
$605$	−9.70709	−0.394649
$606$	14.4940	0.588777
$607$	−29.6866	−1.20494	−0.602472	−	0.798140i	$-0.705817\pi$
$607$	−29.6866	−1.20494	−0.602472	+	0.798140i	$0.705817\pi$
$608$	1.55496	0.0630619
$609$	18.7778	0.760914
$610$	10.9487	0.443299
$611$	0	0
$612$	2.69202	0.108819
$613$	−42.5163	−1.71722	−0.858609	−	0.512631i	$-0.828671\pi$
$613$	−42.5163	−1.71722	−0.858609	+	0.512631i	$0.828671\pi$
$614$	32.4359	1.30901
$615$	−8.26875	−0.333428
$616$	3.46681	0.139682
$617$	27.3666	1.10174	0.550869	−	0.834592i	$-0.314297\pi$
$617$	27.3666	1.10174	0.550869	+	0.834592i	$0.314297\pi$
$618$	16.1836	0.650999
$619$	27.9420	1.12308	0.561542	−	0.827449i	$-0.310208\pi$
$619$	27.9420	1.12308	0.561542	+	0.827449i	$0.310208\pi$
$620$	−0.246980	−0.00991894
$621$	−5.40581	−0.216928
$622$	−21.2771	−0.853134
$623$	51.4892	2.06287
$624$	0	0
$625$	1.00000	0.0400000
$626$	−18.7060	−0.747643
$627$	−1.76809	−0.0706105
$628$	17.0761	0.681409
$629$	−1.49396	−0.0595681
$630$	3.04892	0.121472
$631$	7.43594	0.296020	0.148010	−	0.988986i	$-0.452713\pi$
$631$	7.43594	0.296020	0.148010	+	0.988986i	$0.452713\pi$
$632$	5.33513	0.212220
$633$	−12.1521	−0.483004
$634$	17.4776	0.694123
$635$	6.22521	0.247040
$636$	−5.77479	−0.228985
$637$	0	0
$638$	−7.00298	−0.277251
$639$	−8.57002	−0.339025
$640$	1.00000	0.0395285
$641$	−24.9444	−0.985244	−0.492622	−	0.870243i	$-0.663961\pi$
$641$	−24.9444	−0.985244	−0.492622	+	0.870243i	$0.663961\pi$
$642$	−4.92931	−0.194544
$643$	30.1661	1.18964	0.594818	−	0.803860i	$-0.297224\pi$
$643$	30.1661	1.18964	0.594818	+	0.803860i	$0.297224\pi$
$644$	16.4819	0.649477
$645$	−2.03923	−0.0802946
$646$	4.18598	0.164695
$647$	−44.4053	−1.74575	−0.872877	−	0.487940i	$-0.837748\pi$
$647$	−44.4053	−1.74575	−0.872877	+	0.487940i	$0.837748\pi$
$648$	1.00000	0.0392837
$649$	1.55661	0.0611024
$650$	0	0
$651$	0.753020	0.0295132
$652$	−9.61894	−0.376707
$653$	−29.2573	−1.14493	−0.572463	−	0.819931i	$-0.694012\pi$
$653$	−29.2573	−1.14493	−0.572463	+	0.819931i	$0.694012\pi$
$654$	−0.801938	−0.0313582
$655$	3.34481	0.130693
$656$	8.26875	0.322840
$657$	−14.8334	−0.578706
$658$	−20.4330	−0.796560
$659$	−11.4808	−0.447229	−0.223614	−	0.974678i	$-0.571786\pi$
$659$	−11.4808	−0.447229	−0.223614	+	0.974678i	$0.571786\pi$
$660$	−1.13706	−0.0442601
$661$	26.1129	1.01567	0.507837	−	0.861453i	$-0.330445\pi$
$661$	26.1129	1.01567	0.507837	+	0.861453i	$0.330445\pi$
$662$	−23.1890	−0.901265
$663$	0	0
$664$	−5.45473	−0.211685
$665$	4.74094	0.183846
$666$	−0.554958	−0.0215042
$667$	−33.2935	−1.28913
$668$	21.8509	0.845435
$669$	−1.11960	−0.0432864
$670$	−2.47219	−0.0955090
$671$	12.4494	0.480602
$672$	−3.04892	−0.117615
$673$	−12.4190	−0.478716	−0.239358	−	0.970931i	$-0.576937\pi$
$673$	−12.4190	−0.478716	−0.239358	+	0.970931i	$0.576937\pi$
$674$	−26.1400	−1.00688
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	12.7646	0.490585	0.245292	−	0.969449i	$-0.421116\pi$
$677$	12.7646	0.490585	0.245292	+	0.969449i	$0.421116\pi$
$678$	11.5211	0.442465
$679$	5.91185	0.226876
$680$	2.69202	0.103234
$681$	−4.06638	−0.155824
$682$	−0.280831	−0.0107536
$683$	31.6142	1.20968	0.604841	−	0.796346i	$-0.293237\pi$
$683$	31.6142	1.20968	0.604841	+	0.796346i	$0.293237\pi$
$684$	1.55496	0.0594553
$685$	−9.53079	−0.364153
$686$	−14.3424	−0.547596
$687$	16.5187	0.630228
$688$	2.03923	0.0777449
$689$	0	0
$690$	−5.40581	−0.205796
$691$	2.06962	0.0787322	0.0393661	−	0.999225i	$-0.487466\pi$
$691$	2.06962	0.0787322	0.0393661	+	0.999225i	$0.487466\pi$
$692$	16.6853	0.634281
$693$	3.46681	0.131693
$694$	15.2959	0.580624
$695$	18.7017	0.709396
$696$	6.15883	0.233450
$697$	22.2597	0.843144
$698$	−19.2107	−0.727137
$699$	6.16421	0.233152
$700$	3.04892	0.115238
$701$	−23.0030	−0.868811	−0.434405	−	0.900717i	$-0.643041\pi$
$701$	−23.0030	−0.868811	−0.434405	+	0.900717i	$0.643041\pi$
$702$	0	0
$703$	−0.862937	−0.0325463
$704$	1.13706	0.0428547
$705$	6.70171	0.252401
$706$	−5.35450	−0.201519
$707$	−44.1909	−1.66197
$708$	−1.36898	−0.0514493
$709$	−32.4373	−1.21821	−0.609104	−	0.793091i	$-0.708471\pi$
$709$	−32.4373	−1.21821	−0.609104	+	0.793091i	$0.708471\pi$
$710$	−8.57002	−0.321627
$711$	5.33513	0.200083
$712$	16.8877	0.632893
$713$	−1.33513	−0.0500008
$714$	−8.20775	−0.307167
$715$	0	0
$716$	−3.94438	−0.147408
$717$	−18.1793	−0.678917
$718$	8.49934	0.317192
$719$	−13.7845	−0.514074	−0.257037	−	0.966402i	$-0.582746\pi$
$719$	−13.7845	−0.514074	−0.257037	+	0.966402i	$0.582746\pi$
$720$	1.00000	0.0372678
$721$	−49.3424	−1.83761
$722$	−16.5821	−0.617122
$723$	9.21744	0.342800
$724$	−20.5308	−0.763021
$725$	−6.15883	−0.228733
$726$	9.70709	0.360264
$727$	4.64848	0.172403	0.0862013	−	0.996278i	$-0.472527\pi$
$727$	4.64848	0.172403	0.0862013	+	0.996278i	$0.472527\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−14.8334	−0.549009
$731$	5.48965	0.203042
$732$	−10.9487	−0.404675
$733$	−27.4717	−1.01469	−0.507345	−	0.861743i	$-0.669373\pi$
$733$	−27.4717	−1.01469	−0.507345	+	0.861743i	$0.669373\pi$
$734$	4.76032	0.175706
$735$	−2.29590	−0.0846854
$736$	5.40581	0.199261
$737$	−2.81104	−0.103546
$738$	8.26875	0.304377
$739$	−48.0170	−1.76633	−0.883167	−	0.469059i	$-0.844593\pi$
$739$	−48.0170	−1.76633	−0.883167	+	0.469059i	$0.844593\pi$
$740$	−0.554958	−0.0204007
$741$	0	0
$742$	17.6069	0.646368
$743$	−23.6517	−0.867698	−0.433849	−	0.900986i	$-0.642845\pi$
$743$	−23.6517	−0.867698	−0.433849	+	0.900986i	$0.642845\pi$
$744$	0.246980	0.00905471
$745$	16.4209	0.601614
$746$	−20.2064	−0.739810
$747$	−5.45473	−0.199578
$748$	3.06100	0.111921
$749$	15.0291	0.549150
$750$	−1.00000	−0.0365148
$751$	−39.5133	−1.44186	−0.720931	−	0.693007i	$-0.756286\pi$
$751$	−39.5133	−1.44186	−0.720931	+	0.693007i	$0.756286\pi$
$752$	−6.70171	−0.244386
$753$	−17.2054	−0.626998
$754$	0	0
$755$	−9.26636	−0.337237
$756$	−3.04892	−0.110888
$757$	−11.2239	−0.407939	−0.203969	−	0.978977i	$-0.565384\pi$
$757$	−11.2239	−0.407939	−0.203969	+	0.978977i	$0.565384\pi$
$758$	24.4088	0.886567
$759$	−6.14675	−0.223113
$760$	1.55496	0.0564043
$761$	−5.80864	−0.210563	−0.105282	−	0.994442i	$-0.533574\pi$
$761$	−5.80864	−0.210563	−0.105282	+	0.994442i	$0.533574\pi$
$762$	−6.22521	−0.225516
$763$	2.44504	0.0885165
$764$	−6.66487	−0.241127
$765$	2.69202	0.0973302
$766$	32.0237	1.15706
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−31.8316	−1.14788	−0.573938	−	0.818899i	$-0.694585\pi$
$769$	−31.8316	−1.14788	−0.573938	+	0.818899i	$0.694585\pi$
$770$	3.46681	0.124935
$771$	19.5157	0.702842
$772$	16.6558	0.599455
$773$	39.7434	1.42947	0.714736	−	0.699394i	$-0.246547\pi$
$773$	39.7434	1.42947	0.714736	+	0.699394i	$0.246547\pi$
$774$	2.03923	0.0732986
$775$	−0.246980	−0.00887177
$776$	1.93900	0.0696061
$777$	1.69202	0.0607009
$778$	21.1879	0.759623
$779$	12.8576	0.460670
$780$	0	0
$781$	−9.74466	−0.348691
$782$	14.5526	0.520399
$783$	6.15883	0.220099
$784$	2.29590	0.0819963
$785$	17.0761	0.609471
$786$	−3.34481	−0.119306
$787$	19.3183	0.688622	0.344311	−	0.938856i	$-0.388113\pi$
$787$	19.3183	0.688622	0.344311	+	0.938856i	$0.388113\pi$
$788$	19.4494	0.692855
$789$	−12.8834	−0.458660
$790$	5.33513	0.189815
$791$	−35.1269	−1.24897
$792$	1.13706	0.0404038
$793$	0	0
$794$	10.9608	0.388983
$795$	−5.77479	−0.204811
$796$	−21.6558	−0.767569
$797$	29.4946	1.04475	0.522375	−	0.852716i	$-0.325046\pi$
$797$	29.4946	1.04475	0.522375	+	0.852716i	$0.325046\pi$
$798$	−4.74094	−0.167827
$799$	−18.0411	−0.638250
$800$	1.00000	0.0353553
$801$	16.8877	0.596697
$802$	−29.2301	−1.03215
$803$	−16.8665	−0.595206
$804$	2.47219	0.0871874
$805$	16.4819	0.580910
$806$	0	0
$807$	30.9138	1.08822
$808$	−14.4940	−0.509896
$809$	14.0901	0.495380	0.247690	−	0.968839i	$-0.420329\pi$
$809$	14.0901	0.495380	0.247690	+	0.968839i	$0.420329\pi$
$810$	1.00000	0.0351364
$811$	−45.5060	−1.59793	−0.798967	−	0.601375i	$-0.794620\pi$
$811$	−45.5060	−1.59793	−0.798967	+	0.601375i	$0.794620\pi$
$812$	−18.7778	−0.658971
$813$	−7.24027	−0.253928
$814$	−0.631023	−0.0221173
$815$	−9.61894	−0.336937
$816$	−2.69202	−0.0942396
$817$	3.17092	0.110936
$818$	36.3163	1.26977
$819$	0	0
$820$	8.26875	0.288757
$821$	−14.6853	−0.512521	−0.256261	−	0.966608i	$-0.582491\pi$
$821$	−14.6853	−0.512521	−0.256261	+	0.966608i	$0.582491\pi$
$822$	9.53079	0.332425
$823$	−32.0551	−1.11737	−0.558686	−	0.829379i	$-0.688694\pi$
$823$	−32.0551	−1.11737	−0.558686	+	0.829379i	$0.688694\pi$
$824$	−16.1836	−0.563782
$825$	−1.13706	−0.0395875
$826$	4.17390	0.145229
$827$	8.09352	0.281439	0.140720	−	0.990049i	$-0.455058\pi$
$827$	8.09352	0.281439	0.140720	+	0.990049i	$0.455058\pi$
$828$	5.40581	0.187865
$829$	−23.8708	−0.829068	−0.414534	−	0.910034i	$-0.636055\pi$
$829$	−23.8708	−0.829068	−0.414534	+	0.910034i	$0.636055\pi$
$830$	−5.45473	−0.189336
$831$	−12.2446	−0.424760
$832$	0	0
$833$	6.18060	0.214145
$834$	−18.7017	−0.647587
$835$	21.8509	0.756180
$836$	1.76809	0.0611505
$837$	0.246980	0.00853686
$838$	11.0382	0.381307
$839$	−17.5569	−0.606131	−0.303065	−	0.952970i	$-0.598010\pi$
$839$	−17.5569	−0.606131	−0.303065	+	0.952970i	$0.598010\pi$
$840$	−3.04892	−0.105198
$841$	8.93123	0.307973
$842$	4.58642	0.158058
$843$	2.74632	0.0945881
$844$	12.1521	0.418294
$845$	0	0
$846$	−6.70171	−0.230410
$847$	−29.5961	−1.01693
$848$	5.77479	0.198307
$849$	−14.9312	−0.512439
$850$	2.69202	0.0923356
$851$	−3.00000	−0.102839
$852$	8.57002	0.293604
$853$	−40.2892	−1.37948	−0.689738	−	0.724059i	$-0.742274\pi$
$853$	−40.2892	−1.37948	−0.689738	+	0.724059i	$0.742274\pi$
$854$	33.3817	1.14230
$855$	1.55496	0.0531784
$856$	4.92931	0.168480
$857$	44.5913	1.52321	0.761605	−	0.648041i	$-0.224412\pi$
$857$	44.5913	1.52321	0.761605	+	0.648041i	$0.224412\pi$
$858$	0	0
$859$	−40.6467	−1.38685	−0.693423	−	0.720530i	$-0.743898\pi$
$859$	−40.6467	−1.38685	−0.693423	+	0.720530i	$0.743898\pi$
$860$	2.03923	0.0695371
$861$	−25.2107	−0.859180
$862$	19.3642	0.659547
$863$	−31.9748	−1.08843	−0.544217	−	0.838945i	$-0.683173\pi$
$863$	−31.9748	−1.08843	−0.544217	+	0.838945i	$0.683173\pi$
$864$	−1.00000	−0.0340207
$865$	16.6853	0.567318
$866$	4.41119	0.149898
$867$	9.75302	0.331230
$868$	−0.753020	−0.0255592
$869$	6.06638	0.205788
$870$	6.15883	0.208804
$871$	0	0
$872$	0.801938	0.0271570
$873$	1.93900	0.0656252
$874$	8.40581	0.284331
$875$	3.04892	0.103072
$876$	14.8334	0.501174
$877$	−39.2476	−1.32530	−0.662648	−	0.748931i	$-0.730567\pi$
$877$	−39.2476	−1.32530	−0.662648	+	0.748931i	$0.730567\pi$
$878$	39.5260	1.33394
$879$	−18.5810	−0.626723
$880$	1.13706	0.0383304
$881$	−24.3153	−0.819202	−0.409601	−	0.912265i	$-0.634332\pi$
$881$	−24.3153	−0.819202	−0.409601	+	0.912265i	$0.634332\pi$
$882$	2.29590	0.0773069
$883$	51.8756	1.74575	0.872877	−	0.487941i	$-0.162252\pi$
$883$	51.8756	1.74575	0.872877	+	0.487941i	$0.162252\pi$
$884$	0	0
$885$	−1.36898	−0.0460177
$886$	−7.16182	−0.240606
$887$	−13.4383	−0.451215	−0.225608	−	0.974218i	$-0.572437\pi$
$887$	−13.4383	−0.451215	−0.225608	+	0.974218i	$0.572437\pi$
$888$	0.554958	0.0186232
$889$	18.9801	0.636574
$890$	16.8877	0.566077
$891$	1.13706	0.0380931
$892$	1.11960	0.0374871
$893$	−10.4209	−0.348721
$894$	−16.4209	−0.549196
$895$	−3.94438	−0.131846
$896$	3.04892	0.101857
$897$	0	0
$898$	−4.46011	−0.148836
$899$	1.52111	0.0507317
$900$	1.00000	0.0333333
$901$	15.5459	0.517908
$902$	9.40209	0.313055
$903$	−6.21744	−0.206903
$904$	−11.5211	−0.383186
$905$	−20.5308	−0.682467
$906$	9.26636	0.307854
$907$	−6.58881	−0.218778	−0.109389	−	0.993999i	$-0.534889\pi$
$907$	−6.58881	−0.218778	−0.109389	+	0.993999i	$0.534889\pi$
$908$	4.06638	0.134947
$909$	−14.4940	−0.480734
$910$	0	0
$911$	51.2646	1.69847	0.849235	−	0.528015i	$-0.177063\pi$
$911$	51.2646	1.69847	0.849235	+	0.528015i	$0.177063\pi$
$912$	−1.55496	−0.0514898
$913$	−6.20237	−0.205269
$914$	−8.04593	−0.266136
$915$	−10.9487	−0.361953
$916$	−16.5187	−0.545794
$917$	10.1981	0.336770
$918$	−2.69202	−0.0888499
$919$	50.7066	1.67266	0.836328	−	0.548229i	$-0.184698\pi$
$919$	50.7066	1.67266	0.836328	+	0.548229i	$0.184698\pi$
$920$	5.40581	0.178224
$921$	−32.4359	−1.06880
$922$	4.00538	0.131910
$923$	0	0
$924$	−3.46681	−0.114050
$925$	−0.554958	−0.0182469
$926$	−37.6142	−1.23608
$927$	−16.1836	−0.531539
$928$	−6.15883	−0.202174
$929$	−18.2903	−0.600084	−0.300042	−	0.953926i	$-0.597001\pi$
$929$	−18.2903	−0.600084	−0.300042	+	0.953926i	$0.597001\pi$
$930$	0.246980	0.00809878
$931$	3.57002	0.117003
$932$	−6.16421	−0.201915
$933$	21.2771	0.696581
$934$	−32.8079	−1.07351
$935$	3.06100	0.100105
$936$	0	0
$937$	−59.6679	−1.94926	−0.974632	−	0.223814i	$-0.928149\pi$
$937$	−59.6679	−1.94926	−0.974632	+	0.223814i	$0.928149\pi$
$938$	−7.53750	−0.246108
$939$	18.7060	0.610448
$940$	−6.70171	−0.218586
$941$	−36.2513	−1.18176	−0.590879	−	0.806760i	$-0.701219\pi$
$941$	−36.2513	−1.18176	−0.590879	+	0.806760i	$0.701219\pi$
$942$	−17.0761	−0.556368
$943$	44.6993	1.45561
$944$	1.36898	0.0445564
$945$	−3.04892	−0.0991813
$946$	2.31873	0.0753885
$947$	7.02475	0.228274	0.114137	−	0.993465i	$-0.463590\pi$
$947$	7.02475	0.228274	0.114137	+	0.993465i	$0.463590\pi$
$948$	−5.33513	−0.173277
$949$	0	0
$950$	1.55496	0.0504495
$951$	−17.4776	−0.566749
$952$	8.20775	0.266015
$953$	12.5730	0.407280	0.203640	−	0.979046i	$-0.434723\pi$
$953$	12.5730	0.407280	0.203640	+	0.979046i	$0.434723\pi$
$954$	5.77479	0.186966
$955$	−6.66487	−0.215670
$956$	18.1793	0.587960
$957$	7.00298	0.226374
$958$	−9.88530	−0.319379
$959$	−29.0586	−0.938351
$960$	−1.00000	−0.0322749
$961$	−30.9390	−0.998032
$962$	0	0
$963$	4.92931	0.158845
$964$	−9.21744	−0.296874
$965$	16.6558	0.536168
$966$	−16.4819	−0.530296
$967$	−23.1371	−0.744038	−0.372019	−	0.928225i	$-0.621334\pi$
$967$	−23.1371	−0.744038	−0.372019	+	0.928225i	$0.621334\pi$
$968$	−9.70709	−0.311998
$969$	−4.18598	−0.134473
$970$	1.93900	0.0622576
$971$	−26.5394	−0.851690	−0.425845	−	0.904796i	$-0.640023\pi$
$971$	−26.5394	−0.851690	−0.425845	+	0.904796i	$0.640023\pi$
$972$	−1.00000	−0.0320750
$973$	57.0200	1.82798
$974$	20.2610	0.649204
$975$	0	0
$976$	10.9487	0.350459
$977$	19.5670	0.626005	0.313003	−	0.949752i	$-0.398665\pi$
$977$	19.5670	0.626005	0.313003	+	0.949752i	$0.398665\pi$
$978$	9.61894	0.307580
$979$	19.2024	0.613711
$980$	2.29590	0.0733397
$981$	0.801938	0.0256039
$982$	−22.4940	−0.717811
$983$	−8.82610	−0.281509	−0.140754	−	0.990045i	$-0.544953\pi$
$983$	−8.82610	−0.281509	−0.140754	+	0.990045i	$0.544953\pi$
$984$	−8.26875	−0.263598
$985$	19.4494	0.619708
$986$	−16.5797	−0.528006
$987$	20.4330	0.650388
$988$	0	0
$989$	11.0237	0.350533
$990$	1.13706	0.0361382
$991$	0.601483	0.0191067	0.00955336	−	0.999954i	$-0.496959\pi$
$991$	0.601483	0.0191067	0.00955336	+	0.999954i	$0.496959\pi$
$992$	−0.246980	−0.00784161
$993$	23.1890	0.735880
$994$	−26.1293	−0.828771
$995$	−21.6558	−0.686534
$996$	5.45473	0.172840
$997$	13.4101	0.424703	0.212351	−	0.977193i	$-0.431888\pi$
$997$	13.4101	0.424703	0.212351	+	0.977193i	$0.431888\pi$
$998$	−15.2728	−0.483452
$999$	0.554958	0.0175581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bu.1.3	yes	3
13.5	odd	4		5070.2.b.t.1351.3		6
13.8	odd	4		5070.2.b.t.1351.4		6
13.12	even	2		5070.2.a.bj.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bj.1.1	✓	3	13.12	even	2
5070.2.a.bu.1.3	yes	3	1.1	even	1	trivial
5070.2.b.t.1351.3		6	13.5	odd	4
5070.2.b.t.1351.4		6	13.8	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.13706 q^{11} -1.00000 q^{12} +3.04892 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.69202 q^{17} +1.00000 q^{18} +1.55496 q^{19} +1.00000 q^{20} -3.04892 q^{21} +1.13706 q^{22} +5.40581 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.04892 q^{28} -6.15883 q^{29} -1.00000 q^{30} -0.246980 q^{31} +1.00000 q^{32} -1.13706 q^{33} +2.69202 q^{34} +3.04892 q^{35} +1.00000 q^{36} -0.554958 q^{37} +1.55496 q^{38} +1.00000 q^{40} +8.26875 q^{41} -3.04892 q^{42} +2.03923 q^{43} +1.13706 q^{44} +1.00000 q^{45} +5.40581 q^{46} -6.70171 q^{47} -1.00000 q^{48} +2.29590 q^{49} +1.00000 q^{50} -2.69202 q^{51} +5.77479 q^{53} -1.00000 q^{54} +1.13706 q^{55} +3.04892 q^{56} -1.55496 q^{57} -6.15883 q^{58} +1.36898 q^{59} -1.00000 q^{60} +10.9487 q^{61} -0.246980 q^{62} +3.04892 q^{63} +1.00000 q^{64} -1.13706 q^{66} -2.47219 q^{67} +2.69202 q^{68} -5.40581 q^{69} +3.04892 q^{70} -8.57002 q^{71} +1.00000 q^{72} -14.8334 q^{73} -0.554958 q^{74} -1.00000 q^{75} +1.55496 q^{76} +3.46681 q^{77} +5.33513 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.26875 q^{82} -5.45473 q^{83} -3.04892 q^{84} +2.69202 q^{85} +2.03923 q^{86} +6.15883 q^{87} +1.13706 q^{88} +16.8877 q^{89} +1.00000 q^{90} +5.40581 q^{92} +0.246980 q^{93} -6.70171 q^{94} +1.55496 q^{95} -1.00000 q^{96} +1.93900 q^{97} +2.29590 q^{98} +1.13706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 5 q^{19} + 3 q^{20} - 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 3 q^{30} + 4 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 3 q^{36} - 2 q^{37} + 5 q^{38} + 3 q^{40} + 17 q^{41} + 19 q^{43} - 2 q^{44} + 3 q^{45} + 3 q^{46} + 7 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} - 2 q^{55} - 5 q^{57} - 10 q^{58} + 19 q^{59} - 3 q^{60} + q^{61} + 4 q^{62} + 3 q^{64} + 2 q^{66} - q^{67} + 3 q^{68} - 3 q^{69} - q^{71} + 3 q^{72} - 15 q^{73} - 2 q^{74} - 3 q^{75} + 5 q^{76} + 7 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} + 6 q^{83} + 3 q^{85} + 19 q^{86} + 10 q^{87} - 2 q^{88} + 9 q^{89} + 3 q^{90} + 3 q^{92} - 4 q^{93} + 7 q^{94} + 5 q^{95} - 3 q^{96} - 4 q^{97} - 7 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 2 * q^11 - 3 * q^12 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 5 * q^19 + 3 * q^20 - 2 * q^22 + 3 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 3 * q^30 + 4 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 3 * q^36 - 2 * q^37 + 5 * q^38 + 3 * q^40 + 17 * q^41 + 19 * q^43 - 2 * q^44 + 3 * q^45 + 3 * q^46 + 7 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 - 2 * q^55 - 5 * q^57 - 10 * q^58 + 19 * q^59 - 3 * q^60 + q^61 + 4 * q^62 + 3 * q^64 + 2 * q^66 - q^67 + 3 * q^68 - 3 * q^69 - q^71 + 3 * q^72 - 15 * q^73 - 2 * q^74 - 3 * q^75 + 5 * q^76 + 7 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 + 6 * q^83 + 3 * q^85 + 19 * q^86 + 10 * q^87 - 2 * q^88 + 9 * q^89 + 3 * q^90 + 3 * q^92 - 4 * q^93 + 7 * q^94 + 5 * q^95 - 3 * q^96 - 4 * q^97 - 7 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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