LMFDB - Embedded newform 8034.2.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 40 x^{11} + 34 x^{10} + 604 x^{9} - 381 x^{8} - 4352 x^{7} + 1474 x^{6} + 15809 x^{5} + \cdots + 8832 $ x^13 - x^12 - 40x^11 + 34x^10 + 604x^9 - 381x^8 - 4352x^7 + 1474x^6 + 15809x^5 - 368x^4 - 27369x^3 - 7621x^2 + 17300*x + 8832
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			1.6
Root			$-0.975286$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.975286 q^{5} -1.00000 q^{6} -3.46524 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.975286 q^{5} -1.00000 q^{6} -3.46524 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.975286 q^{10} -0.271160 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.46524 q^{14} -0.975286 q^{15} +1.00000 q^{16} +5.01313 q^{17} -1.00000 q^{18} +1.63890 q^{19} -0.975286 q^{20} -3.46524 q^{21} +0.271160 q^{22} +4.90131 q^{23} -1.00000 q^{24} -4.04882 q^{25} -1.00000 q^{26} +1.00000 q^{27} -3.46524 q^{28} -6.97587 q^{29} +0.975286 q^{30} +4.30230 q^{31} -1.00000 q^{32} -0.271160 q^{33} -5.01313 q^{34} +3.37960 q^{35} +1.00000 q^{36} -6.45496 q^{37} -1.63890 q^{38} +1.00000 q^{39} +0.975286 q^{40} -6.97295 q^{41} +3.46524 q^{42} +1.56065 q^{43} -0.271160 q^{44} -0.975286 q^{45} -4.90131 q^{46} +3.44629 q^{47} +1.00000 q^{48} +5.00792 q^{49} +4.04882 q^{50} +5.01313 q^{51} +1.00000 q^{52} +4.10931 q^{53} -1.00000 q^{54} +0.264458 q^{55} +3.46524 q^{56} +1.63890 q^{57} +6.97587 q^{58} -10.8986 q^{59} -0.975286 q^{60} +12.8926 q^{61} -4.30230 q^{62} -3.46524 q^{63} +1.00000 q^{64} -0.975286 q^{65} +0.271160 q^{66} +2.73443 q^{67} +5.01313 q^{68} +4.90131 q^{69} -3.37960 q^{70} -5.88972 q^{71} -1.00000 q^{72} +3.77261 q^{73} +6.45496 q^{74} -4.04882 q^{75} +1.63890 q^{76} +0.939634 q^{77} -1.00000 q^{78} +8.97116 q^{79} -0.975286 q^{80} +1.00000 q^{81} +6.97295 q^{82} -9.85969 q^{83} -3.46524 q^{84} -4.88924 q^{85} -1.56065 q^{86} -6.97587 q^{87} +0.271160 q^{88} +9.07183 q^{89} +0.975286 q^{90} -3.46524 q^{91} +4.90131 q^{92} +4.30230 q^{93} -3.44629 q^{94} -1.59840 q^{95} -1.00000 q^{96} -4.57937 q^{97} -5.00792 q^{98} -0.271160 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9	\( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9} - q^{10} + 6 q^{11} + 13 q^{12} + 13 q^{13} + q^{14} + q^{15} + 13 q^{16} + 18 q^{17} - 13 q^{18} - q^{19} + q^{20} - q^{21} - 6 q^{22} + 20 q^{23} - 13 q^{24} + 16 q^{25} - 13 q^{26} + 13 q^{27} - q^{28} + 25 q^{29} - q^{30} - 5 q^{31} - 13 q^{32} + 6 q^{33} - 18 q^{34} + 26 q^{35} + 13 q^{36} - 6 q^{37} + q^{38} + 13 q^{39} - q^{40} + 13 q^{41} + q^{42} - 2 q^{43} + 6 q^{44} + q^{45} - 20 q^{46} + 5 q^{47} + 13 q^{48} - 16 q^{50} + 18 q^{51} + 13 q^{52} + 21 q^{53} - 13 q^{54} + 2 q^{55} + q^{56} - q^{57} - 25 q^{58} + 22 q^{59} + q^{60} + 2 q^{61} + 5 q^{62} - q^{63} + 13 q^{64} + q^{65} - 6 q^{66} - q^{67} + 18 q^{68} + 20 q^{69} - 26 q^{70} + 32 q^{71} - 13 q^{72} - 13 q^{73} + 6 q^{74} + 16 q^{75} - q^{76} + 33 q^{77} - 13 q^{78} - 7 q^{79} + q^{80} + 13 q^{81} - 13 q^{82} + 17 q^{83} - q^{84} - 25 q^{85} + 2 q^{86} + 25 q^{87} - 6 q^{88} - 12 q^{89} - q^{90} - q^{91} + 20 q^{92} - 5 q^{93} - 5 q^{94} + 36 q^{95} - 13 q^{96} - 42 q^{97} + 6 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9 - q^10 + 6 * q^11 + 13 * q^12 + 13 * q^13 + q^14 + q^15 + 13 * q^16 + 18 * q^17 - 13 * q^18 - q^19 + q^20 - q^21 - 6 * q^22 + 20 * q^23 - 13 * q^24 + 16 * q^25 - 13 * q^26 + 13 * q^27 - q^28 + 25 * q^29 - q^30 - 5 * q^31 - 13 * q^32 + 6 * q^33 - 18 * q^34 + 26 * q^35 + 13 * q^36 - 6 * q^37 + q^38 + 13 * q^39 - q^40 + 13 * q^41 + q^42 - 2 * q^43 + 6 * q^44 + q^45 - 20 * q^46 + 5 * q^47 + 13 * q^48 - 16 * q^50 + 18 * q^51 + 13 * q^52 + 21 * q^53 - 13 * q^54 + 2 * q^55 + q^56 - q^57 - 25 * q^58 + 22 * q^59 + q^60 + 2 * q^61 + 5 * q^62 - q^63 + 13 * q^64 + q^65 - 6 * q^66 - q^67 + 18 * q^68 + 20 * q^69 - 26 * q^70 + 32 * q^71 - 13 * q^72 - 13 * q^73 + 6 * q^74 + 16 * q^75 - q^76 + 33 * q^77 - 13 * q^78 - 7 * q^79 + q^80 + 13 * q^81 - 13 * q^82 + 17 * q^83 - q^84 - 25 * q^85 + 2 * q^86 + 25 * q^87 - 6 * q^88 - 12 * q^89 - q^90 - q^91 + 20 * q^92 - 5 * q^93 - 5 * q^94 + 36 * q^95 - 13 * q^96 - 42 * q^97 + 6 * 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
$5$	−0.975286	−0.436161	−0.218081	−	0.975931i	$-0.569980\pi$
$5$	−0.975286	−0.436161	−0.218081	+	0.975931i	$0.569980\pi$
$6$	−1.00000	−0.408248
$7$	−3.46524	−1.30974	−0.654870	−	0.755742i	$-0.727277\pi$
$7$	−3.46524	−1.30974	−0.654870	+	0.755742i	$0.727277\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.975286	0.308412
$11$	−0.271160	−0.0817577	−0.0408788	−	0.999164i	$-0.513016\pi$
$11$	−0.271160	−0.0817577	−0.0408788	+	0.999164i	$0.513016\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.46524	0.926126
$15$	−0.975286	−0.251818
$16$	1.00000	0.250000
$17$	5.01313	1.21586	0.607932	−	0.793989i	$-0.291999\pi$
$17$	5.01313	1.21586	0.607932	+	0.793989i	$0.291999\pi$
$18$	−1.00000	−0.235702
$19$	1.63890	0.375990	0.187995	−	0.982170i	$-0.439801\pi$
$19$	1.63890	0.375990	0.187995	+	0.982170i	$0.439801\pi$
$20$	−0.975286	−0.218081
$21$	−3.46524	−0.756178
$22$	0.271160	0.0578114
$23$	4.90131	1.02199	0.510997	−	0.859583i	$-0.329276\pi$
$23$	4.90131	1.02199	0.510997	+	0.859583i	$0.329276\pi$
$24$	−1.00000	−0.204124
$25$	−4.04882	−0.809764
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−3.46524	−0.654870
$29$	−6.97587	−1.29539	−0.647693	−	0.761901i	$-0.724266\pi$
$29$	−6.97587	−1.29539	−0.647693	+	0.761901i	$0.724266\pi$
$30$	0.975286	0.178062
$31$	4.30230	0.772716	0.386358	−	0.922349i	$-0.373733\pi$
$31$	4.30230	0.772716	0.386358	+	0.922349i	$0.373733\pi$
$32$	−1.00000	−0.176777
$33$	−0.271160	−0.0472028
$34$	−5.01313	−0.859745
$35$	3.37960	0.571257
$36$	1.00000	0.166667
$37$	−6.45496	−1.06119	−0.530594	−	0.847626i	$-0.678031\pi$
$37$	−6.45496	−1.06119	−0.530594	+	0.847626i	$0.678031\pi$
$38$	−1.63890	−0.265865
$39$	1.00000	0.160128
$40$	0.975286	0.154206
$41$	−6.97295	−1.08899	−0.544496	−	0.838763i	$-0.683279\pi$
$41$	−6.97295	−1.08899	−0.544496	+	0.838763i	$0.683279\pi$
$42$	3.46524	0.534699
$43$	1.56065	0.237997	0.118998	−	0.992894i	$-0.462032\pi$
$43$	1.56065	0.237997	0.118998	+	0.992894i	$0.462032\pi$
$44$	−0.271160	−0.0408788
$45$	−0.975286	−0.145387
$46$	−4.90131	−0.722659
$47$	3.44629	0.502693	0.251347	−	0.967897i	$-0.419127\pi$
$47$	3.44629	0.502693	0.251347	+	0.967897i	$0.419127\pi$
$48$	1.00000	0.144338
$49$	5.00792	0.715417
$50$	4.04882	0.572589
$51$	5.01313	0.701979
$52$	1.00000	0.138675
$53$	4.10931	0.564457	0.282228	−	0.959347i	$-0.408926\pi$
$53$	4.10931	0.564457	0.282228	+	0.959347i	$0.408926\pi$
$54$	−1.00000	−0.136083
$55$	0.264458	0.0356595
$56$	3.46524	0.463063
$57$	1.63890	0.217078
$58$	6.97587	0.915977
$59$	−10.8986	−1.41888	−0.709440	−	0.704765i	$-0.751052\pi$
$59$	−10.8986	−1.41888	−0.709440	+	0.704765i	$0.751052\pi$
$60$	−0.975286	−0.125909
$61$	12.8926	1.65073	0.825364	−	0.564602i	$-0.190970\pi$
$61$	12.8926	1.65073	0.825364	+	0.564602i	$0.190970\pi$
$62$	−4.30230	−0.546393
$63$	−3.46524	−0.436580
$64$	1.00000	0.125000
$65$	−0.975286	−0.120969
$66$	0.271160	0.0333774
$67$	2.73443	0.334064	0.167032	−	0.985951i	$-0.446582\pi$
$67$	2.73443	0.334064	0.167032	+	0.985951i	$0.446582\pi$
$68$	5.01313	0.607932
$69$	4.90131	0.590048
$70$	−3.37960	−0.403940
$71$	−5.88972	−0.698982	−0.349491	−	0.936940i	$-0.613645\pi$
$71$	−5.88972	−0.698982	−0.349491	+	0.936940i	$0.613645\pi$
$72$	−1.00000	−0.117851
$73$	3.77261	0.441551	0.220775	−	0.975325i	$-0.429141\pi$
$73$	3.77261	0.441551	0.220775	+	0.975325i	$0.429141\pi$
$74$	6.45496	0.750374
$75$	−4.04882	−0.467517
$76$	1.63890	0.187995
$77$	0.939634	0.107081
$78$	−1.00000	−0.113228
$79$	8.97116	1.00933	0.504667	−	0.863314i	$-0.331615\pi$
$79$	8.97116	1.00933	0.504667	+	0.863314i	$0.331615\pi$
$80$	−0.975286	−0.109040
$81$	1.00000	0.111111
$82$	6.97295	0.770034
$83$	−9.85969	−1.08224	−0.541121	−	0.840945i	$-0.682000\pi$
$83$	−9.85969	−1.08224	−0.541121	+	0.840945i	$0.682000\pi$
$84$	−3.46524	−0.378089
$85$	−4.88924	−0.530312
$86$	−1.56065	−0.168289
$87$	−6.97587	−0.747892
$88$	0.271160	0.0289057
$89$	9.07183	0.961612	0.480806	−	0.876827i	$-0.340344\pi$
$89$	9.07183	0.961612	0.480806	+	0.876827i	$0.340344\pi$
$90$	0.975286	0.102804
$91$	−3.46524	−0.363256
$92$	4.90131	0.510997
$93$	4.30230	0.446128
$94$	−3.44629	−0.355458
$95$	−1.59840	−0.163992
$96$	−1.00000	−0.102062
$97$	−4.57937	−0.464964	−0.232482	−	0.972601i	$-0.574685\pi$
$97$	−4.57937	−0.464964	−0.232482	+	0.972601i	$0.574685\pi$
$98$	−5.00792	−0.505876
$99$	−0.271160	−0.0272526
$100$	−4.04882	−0.404882
$101$	−5.15362	−0.512804	−0.256402	−	0.966570i	$-0.582537\pi$
$101$	−5.15362	−0.512804	−0.256402	+	0.966570i	$0.582537\pi$
$102$	−5.01313	−0.496374
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	3.37960	0.329816
$106$	−4.10931	−0.399131
$107$	−1.46461	−0.141589	−0.0707944	−	0.997491i	$-0.522553\pi$
$107$	−1.46461	−0.141589	−0.0707944	+	0.997491i	$0.522553\pi$
$108$	1.00000	0.0962250
$109$	−7.46462	−0.714981	−0.357490	−	0.933917i	$-0.616367\pi$
$109$	−7.46462	−0.714981	−0.357490	+	0.933917i	$0.616367\pi$
$110$	−0.264458	−0.0252151
$111$	−6.45496	−0.612678
$112$	−3.46524	−0.327435
$113$	17.6138	1.65697	0.828486	−	0.560010i	$-0.189203\pi$
$113$	17.6138	1.65697	0.828486	+	0.560010i	$0.189203\pi$
$114$	−1.63890	−0.153497
$115$	−4.78018	−0.445754
$116$	−6.97587	−0.647693
$117$	1.00000	0.0924500
$118$	10.8986	1.00330
$119$	−17.3717	−1.59246
$120$	0.975286	0.0890310
$121$	−10.9265	−0.993316
$122$	−12.8926	−1.16724
$123$	−6.97295	−0.628730
$124$	4.30230	0.386358
$125$	8.82518	0.789348
$126$	3.46524	0.308709
$127$	−6.59691	−0.585381	−0.292690	−	0.956207i	$-0.594551\pi$
$127$	−6.59691	−0.585381	−0.292690	+	0.956207i	$0.594551\pi$
$128$	−1.00000	−0.0883883
$129$	1.56065	0.137408
$130$	0.975286	0.0855382
$131$	7.06615	0.617373	0.308686	−	0.951164i	$-0.400111\pi$
$131$	7.06615	0.617373	0.308686	+	0.951164i	$0.400111\pi$
$132$	−0.271160	−0.0236014
$133$	−5.67921	−0.492450
$134$	−2.73443	−0.236219
$135$	−0.975286	−0.0839392
$136$	−5.01313	−0.429873
$137$	15.5474	1.32830	0.664150	−	0.747599i	$-0.268793\pi$
$137$	15.5474	1.32830	0.664150	+	0.747599i	$0.268793\pi$
$138$	−4.90131	−0.417227
$139$	−4.72308	−0.400606	−0.200303	−	0.979734i	$-0.564193\pi$
$139$	−4.72308	−0.400606	−0.200303	+	0.979734i	$0.564193\pi$
$140$	3.37960	0.285629
$141$	3.44629	0.290230
$142$	5.88972	0.494255
$143$	−0.271160	−0.0226755
$144$	1.00000	0.0833333
$145$	6.80347	0.564997
$146$	−3.77261	−0.312224
$147$	5.00792	0.413046
$148$	−6.45496	−0.530594
$149$	19.5888	1.60478	0.802389	−	0.596802i	$-0.203562\pi$
$149$	19.5888	1.60478	0.802389	+	0.596802i	$0.203562\pi$
$150$	4.04882	0.330585
$151$	−13.1420	−1.06948	−0.534739	−	0.845017i	$-0.679590\pi$
$151$	−13.1420	−1.06948	−0.534739	+	0.845017i	$0.679590\pi$
$152$	−1.63890	−0.132933
$153$	5.01313	0.405288
$154$	−0.939634	−0.0757179
$155$	−4.19597	−0.337029
$156$	1.00000	0.0800641
$157$	−4.35584	−0.347634	−0.173817	−	0.984778i	$-0.555610\pi$
$157$	−4.35584	−0.347634	−0.173817	+	0.984778i	$0.555610\pi$
$158$	−8.97116	−0.713707
$159$	4.10931	0.325889
$160$	0.975286	0.0771031
$161$	−16.9842	−1.33855
$162$	−1.00000	−0.0785674
$163$	−21.4431	−1.67955	−0.839776	−	0.542932i	$-0.817314\pi$
$163$	−21.4431	−1.67955	−0.839776	+	0.542932i	$0.817314\pi$
$164$	−6.97295	−0.544496
$165$	0.264458	0.0205880
$166$	9.85969	0.765260
$167$	11.1361	0.861734	0.430867	−	0.902415i	$-0.358208\pi$
$167$	11.1361	0.861734	0.430867	+	0.902415i	$0.358208\pi$
$168$	3.46524	0.267349
$169$	1.00000	0.0769231
$170$	4.88924	0.374988
$171$	1.63890	0.125330
$172$	1.56065	0.118998
$173$	−1.50329	−0.114293	−0.0571464	−	0.998366i	$-0.518200\pi$
$173$	−1.50329	−0.114293	−0.0571464	+	0.998366i	$0.518200\pi$
$174$	6.97587	0.528839
$175$	14.0301	1.06058
$176$	−0.271160	−0.0204394
$177$	−10.8986	−0.819191
$178$	−9.07183	−0.679962
$179$	1.82523	0.136424	0.0682121	−	0.997671i	$-0.478271\pi$
$179$	1.82523	0.136424	0.0682121	+	0.997671i	$0.478271\pi$
$180$	−0.975286	−0.0726935
$181$	9.51084	0.706935	0.353468	−	0.935447i	$-0.385002\pi$
$181$	9.51084	0.706935	0.353468	+	0.935447i	$0.385002\pi$
$182$	3.46524	0.256861
$183$	12.8926	0.953048
$184$	−4.90131	−0.361329
$185$	6.29543	0.462849
$186$	−4.30230	−0.315460
$187$	−1.35936	−0.0994062
$188$	3.44629	0.251347
$189$	−3.46524	−0.252059
$190$	1.59840	0.115960
$191$	23.7141	1.71589	0.857947	−	0.513738i	$-0.171740\pi$
$191$	23.7141	1.71589	0.857947	+	0.513738i	$0.171740\pi$
$192$	1.00000	0.0721688
$193$	−0.846035	−0.0608989	−0.0304495	−	0.999536i	$-0.509694\pi$
$193$	−0.846035	−0.0608989	−0.0304495	+	0.999536i	$0.509694\pi$
$194$	4.57937	0.328780
$195$	−0.975286	−0.0698417
$196$	5.00792	0.357709
$197$	22.3654	1.59347	0.796733	−	0.604332i	$-0.206560\pi$
$197$	22.3654	1.59347	0.796733	+	0.604332i	$0.206560\pi$
$198$	0.271160	0.0192705
$199$	−0.874237	−0.0619730	−0.0309865	−	0.999520i	$-0.509865\pi$
$199$	−0.874237	−0.0619730	−0.0309865	+	0.999520i	$0.509865\pi$
$200$	4.04882	0.286295
$201$	2.73443	0.192872
$202$	5.15362	0.362608
$203$	24.1731	1.69662
$204$	5.01313	0.350990
$205$	6.80062	0.474976
$206$	−1.00000	−0.0696733
$207$	4.90131	0.340665
$208$	1.00000	0.0693375
$209$	−0.444405	−0.0307401
$210$	−3.37960	−0.233215
$211$	−13.7130	−0.944039	−0.472019	−	0.881588i	$-0.656475\pi$
$211$	−13.7130	−0.944039	−0.472019	+	0.881588i	$0.656475\pi$
$212$	4.10931	0.282228
$213$	−5.88972	−0.403557
$214$	1.46461	0.100118
$215$	−1.52208	−0.103805
$216$	−1.00000	−0.0680414
$217$	−14.9085	−1.01206
$218$	7.46462	0.505568
$219$	3.77261	0.254929
$220$	0.264458	0.0178298
$221$	5.01313	0.337220
$222$	6.45496	0.433229
$223$	−5.69166	−0.381142	−0.190571	−	0.981673i	$-0.561034\pi$
$223$	−5.69166	−0.381142	−0.190571	+	0.981673i	$0.561034\pi$
$224$	3.46524	0.231531
$225$	−4.04882	−0.269921
$226$	−17.6138	−1.17166
$227$	−1.11936	−0.0742944	−0.0371472	−	0.999310i	$-0.511827\pi$
$227$	−1.11936	−0.0742944	−0.0371472	+	0.999310i	$0.511827\pi$
$228$	1.63890	0.108539
$229$	7.06465	0.466845	0.233423	−	0.972375i	$-0.425007\pi$
$229$	7.06465	0.466845	0.233423	+	0.972375i	$0.425007\pi$
$230$	4.78018	0.315196
$231$	0.939634	0.0618234
$232$	6.97587	0.457988
$233$	−1.27464	−0.0835044	−0.0417522	−	0.999128i	$-0.513294\pi$
$233$	−1.27464	−0.0835044	−0.0417522	+	0.999128i	$0.513294\pi$
$234$	−1.00000	−0.0653720
$235$	−3.36112	−0.219255
$236$	−10.8986	−0.709440
$237$	8.97116	0.582739
$238$	17.3717	1.12604
$239$	24.5955	1.59095	0.795475	−	0.605987i	$-0.207222\pi$
$239$	24.5955	1.59095	0.795475	+	0.605987i	$0.207222\pi$
$240$	−0.975286	−0.0629544
$241$	11.4375	0.736757	0.368378	−	0.929676i	$-0.379913\pi$
$241$	11.4375	0.736757	0.368378	+	0.929676i	$0.379913\pi$
$242$	10.9265	0.702380
$243$	1.00000	0.0641500
$244$	12.8926	0.825364
$245$	−4.88415	−0.312037
$246$	6.97295	0.444579
$247$	1.63890	0.104281
$248$	−4.30230	−0.273196
$249$	−9.85969	−0.624832
$250$	−8.82518	−0.558154
$251$	4.73336	0.298767	0.149384	−	0.988779i	$-0.452271\pi$
$251$	4.73336	0.298767	0.149384	+	0.988779i	$0.452271\pi$
$252$	−3.46524	−0.218290
$253$	−1.32904	−0.0835558
$254$	6.59691	0.413927
$255$	−4.88924	−0.306176
$256$	1.00000	0.0625000
$257$	−3.74522	−0.233620	−0.116810	−	0.993154i	$-0.537267\pi$
$257$	−3.74522	−0.233620	−0.116810	+	0.993154i	$0.537267\pi$
$258$	−1.56065	−0.0971619
$259$	22.3680	1.38988
$260$	−0.975286	−0.0604847
$261$	−6.97587	−0.431795
$262$	−7.06615	−0.436548
$263$	7.62061	0.469907	0.234953	−	0.972007i	$-0.424506\pi$
$263$	7.62061	0.469907	0.234953	+	0.972007i	$0.424506\pi$
$264$	0.271160	0.0166887
$265$	−4.00775	−0.246194
$266$	5.67921	0.348214
$267$	9.07183	0.555187
$268$	2.73443	0.167032
$269$	0.350569	0.0213745	0.0106873	−	0.999943i	$-0.496598\pi$
$269$	0.350569	0.0213745	0.0106873	+	0.999943i	$0.496598\pi$
$270$	0.975286	0.0593540
$271$	18.3518	1.11479	0.557397	−	0.830246i	$-0.311800\pi$
$271$	18.3518	1.11479	0.557397	+	0.830246i	$0.311800\pi$
$272$	5.01313	0.303966
$273$	−3.46524	−0.209726
$274$	−15.5474	−0.939250
$275$	1.09788	0.0662044
$276$	4.90131	0.295024
$277$	6.71964	0.403744	0.201872	−	0.979412i	$-0.435298\pi$
$277$	6.71964	0.403744	0.201872	+	0.979412i	$0.435298\pi$
$278$	4.72308	0.283271
$279$	4.30230	0.257572
$280$	−3.37960	−0.201970
$281$	31.4305	1.87499	0.937493	−	0.348004i	$-0.113141\pi$
$281$	31.4305	1.87499	0.937493	+	0.348004i	$0.113141\pi$
$282$	−3.44629	−0.205224
$283$	−1.25736	−0.0747421	−0.0373710	−	0.999301i	$-0.511898\pi$
$283$	−1.25736	−0.0747421	−0.0373710	+	0.999301i	$0.511898\pi$
$284$	−5.88972	−0.349491
$285$	−1.59840	−0.0946811
$286$	0.271160	0.0160340
$287$	24.1630	1.42630
$288$	−1.00000	−0.0589256
$289$	8.13152	0.478325
$290$	−6.80347	−0.399513
$291$	−4.57937	−0.268447
$292$	3.77261	0.220775
$293$	4.33316	0.253146	0.126573	−	0.991957i	$-0.459602\pi$
$293$	4.33316	0.253146	0.126573	+	0.991957i	$0.459602\pi$
$294$	−5.00792	−0.292068
$295$	10.6293	0.618861
$296$	6.45496	0.375187
$297$	−0.271160	−0.0157343
$298$	−19.5888	−1.13475
$299$	4.90131	0.283450
$300$	−4.04882	−0.233759
$301$	−5.40804	−0.311714
$302$	13.1420	0.756235
$303$	−5.15362	−0.296068
$304$	1.63890	0.0939976
$305$	−12.5740	−0.719983
$306$	−5.01313	−0.286582
$307$	16.8300	0.960541	0.480270	−	0.877120i	$-0.340539\pi$
$307$	16.8300	0.960541	0.480270	+	0.877120i	$0.340539\pi$
$308$	0.939634	0.0535406
$309$	1.00000	0.0568880
$310$	4.19597	0.238315
$311$	16.1061	0.913291	0.456645	−	0.889649i	$-0.349051\pi$
$311$	16.1061	0.913291	0.456645	+	0.889649i	$0.349051\pi$
$312$	−1.00000	−0.0566139
$313$	−1.45026	−0.0819735	−0.0409867	−	0.999160i	$-0.513050\pi$
$313$	−1.45026	−0.0819735	−0.0409867	+	0.999160i	$0.513050\pi$
$314$	4.35584	0.245814
$315$	3.37960	0.190419
$316$	8.97116	0.504667
$317$	−6.27555	−0.352470	−0.176235	−	0.984348i	$-0.556392\pi$
$317$	−6.27555	−0.352470	−0.176235	+	0.984348i	$0.556392\pi$
$318$	−4.10931	−0.230439
$319$	1.89157	0.105908
$320$	−0.975286	−0.0545201
$321$	−1.46461	−0.0817464
$322$	16.9842	0.946494
$323$	8.21605	0.457153
$324$	1.00000	0.0555556
$325$	−4.04882	−0.224588
$326$	21.4431	1.18762
$327$	−7.46462	−0.412794
$328$	6.97295	0.385017
$329$	−11.9422	−0.658397
$330$	−0.264458	−0.0145579
$331$	−6.14581	−0.337804	−0.168902	−	0.985633i	$-0.554022\pi$
$331$	−6.14581	−0.337804	−0.168902	+	0.985633i	$0.554022\pi$
$332$	−9.85969	−0.541121
$333$	−6.45496	−0.353730
$334$	−11.1361	−0.609338
$335$	−2.66685	−0.145706
$336$	−3.46524	−0.189045
$337$	9.59614	0.522735	0.261368	−	0.965239i	$-0.415827\pi$
$337$	9.59614	0.522735	0.261368	+	0.965239i	$0.415827\pi$
$338$	−1.00000	−0.0543928
$339$	17.6138	0.956653
$340$	−4.88924	−0.265156
$341$	−1.16661	−0.0631755
$342$	−1.63890	−0.0886218
$343$	6.90304	0.372729
$344$	−1.56065	−0.0841446
$345$	−4.78018	−0.257356
$346$	1.50329	0.0808172
$347$	17.2096	0.923858	0.461929	−	0.886917i	$-0.347158\pi$
$347$	17.2096	0.923858	0.461929	+	0.886917i	$0.347158\pi$
$348$	−6.97587	−0.373946
$349$	10.6917	0.572313	0.286156	−	0.958183i	$-0.407622\pi$
$349$	10.6917	0.572313	0.286156	+	0.958183i	$0.407622\pi$
$350$	−14.0301	−0.749943
$351$	1.00000	0.0533761
$352$	0.271160	0.0144529
$353$	−3.31384	−0.176378	−0.0881890	−	0.996104i	$-0.528108\pi$
$353$	−3.31384	−0.176378	−0.0881890	+	0.996104i	$0.528108\pi$
$354$	10.8986	0.579256
$355$	5.74416	0.304869
$356$	9.07183	0.480806
$357$	−17.3717	−0.919410
$358$	−1.82523	−0.0964665
$359$	−0.667444	−0.0352264	−0.0176132	−	0.999845i	$-0.505607\pi$
$359$	−0.667444	−0.0352264	−0.0176132	+	0.999845i	$0.505607\pi$
$360$	0.975286	0.0514021
$361$	−16.3140	−0.858631
$362$	−9.51084	−0.499879
$363$	−10.9265	−0.573491
$364$	−3.46524	−0.181628
$365$	−3.67937	−0.192587
$366$	−12.8926	−0.673906
$367$	3.06087	0.159776	0.0798879	−	0.996804i	$-0.474544\pi$
$367$	3.06087	0.159776	0.0798879	+	0.996804i	$0.474544\pi$
$368$	4.90131	0.255498
$369$	−6.97295	−0.362997
$370$	−6.29543	−0.327284
$371$	−14.2398	−0.739291
$372$	4.30230	0.223064
$373$	12.4637	0.645345	0.322673	−	0.946511i	$-0.395419\pi$
$373$	12.4637	0.645345	0.322673	+	0.946511i	$0.395419\pi$
$374$	1.35936	0.0702908
$375$	8.82518	0.455731
$376$	−3.44629	−0.177729
$377$	−6.97587	−0.359276
$378$	3.46524	0.178233
$379$	−0.473301	−0.0243118	−0.0121559	−	0.999926i	$-0.503869\pi$
$379$	−0.473301	−0.0243118	−0.0121559	+	0.999926i	$0.503869\pi$
$380$	−1.59840	−0.0819962
$381$	−6.59691	−0.337970
$382$	−23.7141	−1.21332
$383$	32.7415	1.67301	0.836507	−	0.547956i	$-0.184594\pi$
$383$	32.7415	1.67301	0.836507	+	0.547956i	$0.184594\pi$
$384$	−1.00000	−0.0510310
$385$	−0.916412	−0.0467047
$386$	0.846035	0.0430620
$387$	1.56065	0.0793323
$388$	−4.57937	−0.232482
$389$	−26.2423	−1.33054	−0.665268	−	0.746604i	$-0.731683\pi$
$389$	−26.2423	−1.33054	−0.665268	+	0.746604i	$0.731683\pi$
$390$	0.975286	0.0493855
$391$	24.5709	1.24260
$392$	−5.00792	−0.252938
$393$	7.06615	0.356440
$394$	−22.3654	−1.12675
$395$	−8.74944	−0.440232
$396$	−0.271160	−0.0136263
$397$	−17.7221	−0.889446	−0.444723	−	0.895668i	$-0.646698\pi$
$397$	−17.7221	−0.889446	−0.444723	+	0.895668i	$0.646698\pi$
$398$	0.874237	0.0438215
$399$	−5.67921	−0.284316
$400$	−4.04882	−0.202441
$401$	−0.408256	−0.0203873	−0.0101937	−	0.999948i	$-0.503245\pi$
$401$	−0.408256	−0.0203873	−0.0101937	+	0.999948i	$0.503245\pi$
$402$	−2.73443	−0.136381
$403$	4.30230	0.214313
$404$	−5.15362	−0.256402
$405$	−0.975286	−0.0484623
$406$	−24.1731	−1.19969
$407$	1.75032	0.0867603
$408$	−5.01313	−0.248187
$409$	28.6172	1.41503	0.707515	−	0.706698i	$-0.249816\pi$
$409$	28.6172	1.41503	0.707515	+	0.706698i	$0.249816\pi$
$410$	−6.80062	−0.335859
$411$	15.5474	0.766895
$412$	1.00000	0.0492665
$413$	37.7664	1.85836
$414$	−4.90131	−0.240886
$415$	9.61602	0.472032
$416$	−1.00000	−0.0490290
$417$	−4.72308	−0.231290
$418$	0.444405	0.0217365
$419$	31.0898	1.51884	0.759419	−	0.650602i	$-0.225483\pi$
$419$	31.0898	1.51884	0.759419	+	0.650602i	$0.225483\pi$
$420$	3.37960	0.164908
$421$	3.59868	0.175389	0.0876944	−	0.996147i	$-0.472050\pi$
$421$	3.59868	0.175389	0.0876944	+	0.996147i	$0.472050\pi$
$422$	13.7130	0.667536
$423$	3.44629	0.167564
$424$	−4.10931	−0.199566
$425$	−20.2973	−0.984562
$426$	5.88972	0.285358
$427$	−44.6760	−2.16202
$428$	−1.46461	−0.0707944
$429$	−0.271160	−0.0130917
$430$	1.52208	0.0734012
$431$	30.1410	1.45184	0.725920	−	0.687779i	$-0.241414\pi$
$431$	30.1410	1.45184	0.725920	+	0.687779i	$0.241414\pi$
$432$	1.00000	0.0481125
$433$	36.4406	1.75122	0.875611	−	0.483017i	$-0.160459\pi$
$433$	36.4406	1.75122	0.875611	+	0.483017i	$0.160459\pi$
$434$	14.9085	0.715632
$435$	6.80347	0.326201
$436$	−7.46462	−0.357490
$437$	8.03278	0.384260
$438$	−3.77261	−0.180262
$439$	23.5840	1.12560	0.562802	−	0.826592i	$-0.309724\pi$
$439$	23.5840	1.12560	0.562802	+	0.826592i	$0.309724\pi$
$440$	−0.264458	−0.0126075
$441$	5.00792	0.238472
$442$	−5.01313	−0.238450
$443$	8.66283	0.411584	0.205792	−	0.978596i	$-0.434023\pi$
$443$	8.66283	0.411584	0.205792	+	0.978596i	$0.434023\pi$
$444$	−6.45496	−0.306339
$445$	−8.84763	−0.419418
$446$	5.69166	0.269508
$447$	19.5888	0.926519
$448$	−3.46524	−0.163717
$449$	−33.1353	−1.56375	−0.781876	−	0.623434i	$-0.785737\pi$
$449$	−33.1353	−1.56375	−0.781876	+	0.623434i	$0.785737\pi$
$450$	4.04882	0.190863
$451$	1.89078	0.0890335
$452$	17.6138	0.828486
$453$	−13.1420	−0.617463
$454$	1.11936	0.0525341
$455$	3.37960	0.158438
$456$	−1.63890	−0.0767487
$457$	13.2406	0.619371	0.309686	−	0.950839i	$-0.399776\pi$
$457$	13.2406	0.619371	0.309686	+	0.950839i	$0.399776\pi$
$458$	−7.06465	−0.330110
$459$	5.01313	0.233993
$460$	−4.78018	−0.222877
$461$	−14.6699	−0.683244	−0.341622	−	0.939837i	$-0.610976\pi$
$461$	−14.6699	−0.683244	−0.341622	+	0.939837i	$0.610976\pi$
$462$	−0.939634	−0.0437157
$463$	−14.3763	−0.668121	−0.334061	−	0.942552i	$-0.608419\pi$
$463$	−14.3763	−0.668121	−0.334061	+	0.942552i	$0.608419\pi$
$464$	−6.97587	−0.323847
$465$	−4.19597	−0.194584
$466$	1.27464	0.0590465
$467$	−3.22317	−0.149150	−0.0745751	−	0.997215i	$-0.523760\pi$
$467$	−3.22317	−0.149150	−0.0745751	+	0.997215i	$0.523760\pi$
$468$	1.00000	0.0462250
$469$	−9.47547	−0.437536
$470$	3.36112	0.155037
$471$	−4.35584	−0.200707
$472$	10.8986	0.501650
$473$	−0.423185	−0.0194581
$474$	−8.97116	−0.412059
$475$	−6.63563	−0.304463
$476$	−17.3717	−0.796232
$477$	4.10931	0.188152
$478$	−24.5955	−1.12497
$479$	14.0636	0.642583	0.321291	−	0.946980i	$-0.395883\pi$
$479$	14.0636	0.642583	0.321291	+	0.946980i	$0.395883\pi$
$480$	0.975286	0.0445155
$481$	−6.45496	−0.294321
$482$	−11.4375	−0.520966
$483$	−16.9842	−0.772809
$484$	−10.9265	−0.496658
$485$	4.46619	0.202799
$486$	−1.00000	−0.0453609
$487$	−27.9367	−1.26593	−0.632967	−	0.774178i	$-0.718163\pi$
$487$	−27.9367	−1.26593	−0.632967	+	0.774178i	$0.718163\pi$
$488$	−12.8926	−0.583620
$489$	−21.4431	−0.969690
$490$	4.88415	0.220644
$491$	10.2925	0.464493	0.232246	−	0.972657i	$-0.425392\pi$
$491$	10.2925	0.464493	0.232246	+	0.972657i	$0.425392\pi$
$492$	−6.97295	−0.314365
$493$	−34.9710	−1.57501
$494$	−1.63890	−0.0737378
$495$	0.264458	0.0118865
$496$	4.30230	0.193179
$497$	20.4093	0.915484
$498$	9.85969	0.441823
$499$	18.8157	0.842305	0.421153	−	0.906990i	$-0.361626\pi$
$499$	18.8157	0.842305	0.421153	+	0.906990i	$0.361626\pi$
$500$	8.82518	0.394674
$501$	11.1361	0.497522
$502$	−4.73336	−0.211260
$503$	24.9791	1.11377	0.556883	−	0.830591i	$-0.311997\pi$
$503$	24.9791	1.11377	0.556883	+	0.830591i	$0.311997\pi$
$504$	3.46524	0.154354
$505$	5.02625	0.223665
$506$	1.32904	0.0590829
$507$	1.00000	0.0444116
$508$	−6.59691	−0.292690
$509$	22.9593	1.01765	0.508826	−	0.860869i	$-0.330080\pi$
$509$	22.9593	1.01765	0.508826	+	0.860869i	$0.330080\pi$
$510$	4.88924	0.216499
$511$	−13.0730	−0.578316
$512$	−1.00000	−0.0441942
$513$	1.63890	0.0723594
$514$	3.74522	0.165195
$515$	−0.975286	−0.0429762
$516$	1.56065	0.0687038
$517$	−0.934495	−0.0410990
$518$	−22.3680	−0.982794
$519$	−1.50329	−0.0659870
$520$	0.975286	0.0427691
$521$	−27.3308	−1.19738	−0.598691	−	0.800980i	$-0.704312\pi$
$521$	−27.3308	−1.19738	−0.598691	+	0.800980i	$0.704312\pi$
$522$	6.97587	0.305326
$523$	−11.9198	−0.521217	−0.260608	−	0.965445i	$-0.583923\pi$
$523$	−11.9198	−0.521217	−0.260608	+	0.965445i	$0.583923\pi$
$524$	7.06615	0.308686
$525$	14.0301	0.612326
$526$	−7.62061	−0.332274
$527$	21.5680	0.939517
$528$	−0.271160	−0.0118007
$529$	1.02283	0.0444708
$530$	4.00775	0.174085
$531$	−10.8986	−0.472960
$532$	−5.67921	−0.246225
$533$	−6.97295	−0.302032
$534$	−9.07183	−0.392576
$535$	1.42841	0.0617556
$536$	−2.73443	−0.118109
$537$	1.82523	0.0787646
$538$	−0.350569	−0.0151141
$539$	−1.35795	−0.0584909
$540$	−0.975286	−0.0419696
$541$	−8.67884	−0.373133	−0.186566	−	0.982442i	$-0.559736\pi$
$541$	−8.67884	−0.373133	−0.186566	+	0.982442i	$0.559736\pi$
$542$	−18.3518	−0.788279
$543$	9.51084	0.408149
$544$	−5.01313	−0.214936
$545$	7.28013	0.311847
$546$	3.46524	0.148299
$547$	−9.12897	−0.390327	−0.195163	−	0.980771i	$-0.562524\pi$
$547$	−9.12897	−0.390327	−0.195163	+	0.980771i	$0.562524\pi$
$548$	15.5474	0.664150
$549$	12.8926	0.550242
$550$	−1.09788	−0.0468136
$551$	−11.4328	−0.487053
$552$	−4.90131	−0.208614
$553$	−31.0873	−1.32196
$554$	−6.71964	−0.285490
$555$	6.29543	0.267226
$556$	−4.72308	−0.200303
$557$	−40.7183	−1.72529	−0.862645	−	0.505809i	$-0.831194\pi$
$557$	−40.7183	−1.72529	−0.862645	+	0.505809i	$0.831194\pi$
$558$	−4.30230	−0.182131
$559$	1.56065	0.0660085
$560$	3.37960	0.142814
$561$	−1.35936	−0.0573922
$562$	−31.4305	−1.32582
$563$	3.15958	0.133160	0.0665801	−	0.997781i	$-0.478791\pi$
$563$	3.15958	0.133160	0.0665801	+	0.997781i	$0.478791\pi$
$564$	3.44629	0.145115
$565$	−17.1785	−0.722706
$566$	1.25736	0.0528506
$567$	−3.46524	−0.145527
$568$	5.88972	0.247127
$569$	−11.3769	−0.476944	−0.238472	−	0.971149i	$-0.576647\pi$
$569$	−11.3769	−0.476944	−0.238472	+	0.971149i	$0.576647\pi$
$570$	1.59840	0.0669496
$571$	40.8460	1.70935	0.854676	−	0.519163i	$-0.173756\pi$
$571$	40.8460	1.70935	0.854676	+	0.519163i	$0.173756\pi$
$572$	−0.271160	−0.0113378
$573$	23.7141	0.990672
$574$	−24.1630	−1.00854
$575$	−19.8445	−0.827573
$576$	1.00000	0.0416667
$577$	−29.7550	−1.23872	−0.619359	−	0.785108i	$-0.712607\pi$
$577$	−29.7550	−1.23872	−0.619359	+	0.785108i	$0.712607\pi$
$578$	−8.13152	−0.338227
$579$	−0.846035	−0.0351600
$580$	6.80347	0.282499
$581$	34.1662	1.41745
$582$	4.57937	0.189821
$583$	−1.11428	−0.0461487
$584$	−3.77261	−0.156112
$585$	−0.975286	−0.0403231
$586$	−4.33316	−0.179001
$587$	44.3019	1.82854	0.914268	−	0.405110i	$-0.132767\pi$
$587$	44.3019	1.82854	0.914268	+	0.405110i	$0.132767\pi$
$588$	5.00792	0.206523
$589$	7.05106	0.290534
$590$	−10.6293	−0.437600
$591$	22.3654	0.919988
$592$	−6.45496	−0.265297
$593$	−19.3307	−0.793816	−0.396908	−	0.917858i	$-0.629917\pi$
$593$	−19.3307	−0.793816	−0.396908	+	0.917858i	$0.629917\pi$
$594$	0.271160	0.0111258
$595$	16.9424	0.694571
$596$	19.5888	0.802389
$597$	−0.874237	−0.0357801
$598$	−4.90131	−0.200429
$599$	8.63585	0.352851	0.176426	−	0.984314i	$-0.443546\pi$
$599$	8.63585	0.352851	0.176426	+	0.984314i	$0.443546\pi$
$600$	4.04882	0.165292
$601$	0.0313753	0.00127983	0.000639913	−	1.00000i	$-0.499796\pi$
$601$	0.0313753	0.00127983	0.000639913		1.00000i	$0.499796\pi$
$602$	5.40804	0.220415
$603$	2.73443	0.111355
$604$	−13.1420	−0.534739
$605$	10.6564	0.433246
$606$	5.15362	0.209352
$607$	9.56415	0.388197	0.194099	−	0.980982i	$-0.437822\pi$
$607$	9.56415	0.388197	0.194099	+	0.980982i	$0.437822\pi$
$608$	−1.63890	−0.0664664
$609$	24.1731	0.979543
$610$	12.5740	0.509105
$611$	3.44629	0.139422
$612$	5.01313	0.202644
$613$	44.5149	1.79794	0.898970	−	0.438010i	$-0.144317\pi$
$613$	44.5149	1.79794	0.898970	+	0.438010i	$0.144317\pi$
$614$	−16.8300	−0.679205
$615$	6.80062	0.274227
$616$	−0.939634	−0.0378589
$617$	−23.0467	−0.927827	−0.463914	−	0.885880i	$-0.653555\pi$
$617$	−23.0467	−0.927827	−0.463914	+	0.885880i	$0.653555\pi$
$618$	−1.00000	−0.0402259
$619$	−18.6412	−0.749252	−0.374626	−	0.927176i	$-0.622229\pi$
$619$	−18.6412	−0.749252	−0.374626	+	0.927176i	$0.622229\pi$
$620$	−4.19597	−0.168514
$621$	4.90131	0.196683
$622$	−16.1061	−0.645794
$623$	−31.4361	−1.25946
$624$	1.00000	0.0400320
$625$	11.6370	0.465480
$626$	1.45026	0.0579640
$627$	−0.444405	−0.0177478
$628$	−4.35584	−0.173817
$629$	−32.3596	−1.29026
$630$	−3.37960	−0.134647
$631$	−28.1848	−1.12202	−0.561009	−	0.827810i	$-0.689587\pi$
$631$	−28.1848	−1.12202	−0.561009	+	0.827810i	$0.689587\pi$
$632$	−8.97116	−0.356853
$633$	−13.7130	−0.545041
$634$	6.27555	0.249234
$635$	6.43387	0.255320
$636$	4.10931	0.162945
$637$	5.00792	0.198421
$638$	−1.89157	−0.0748881
$639$	−5.88972	−0.232994
$640$	0.975286	0.0385516
$641$	19.6176	0.774849	0.387425	−	0.921901i	$-0.373365\pi$
$641$	19.6176	0.774849	0.387425	+	0.921901i	$0.373365\pi$
$642$	1.46461	0.0578034
$643$	−0.537263	−0.0211876	−0.0105938	−	0.999944i	$-0.503372\pi$
$643$	−0.537263	−0.0211876	−0.0105938	+	0.999944i	$0.503372\pi$
$644$	−16.9842	−0.669273
$645$	−1.52208	−0.0599318
$646$	−8.21605	−0.323256
$647$	−21.8809	−0.860228	−0.430114	−	0.902775i	$-0.641527\pi$
$647$	−21.8809	−0.860228	−0.430114	+	0.902775i	$0.641527\pi$
$648$	−1.00000	−0.0392837
$649$	2.95527	0.116004
$650$	4.04882	0.158808
$651$	−14.9085	−0.584311
$652$	−21.4431	−0.839776
$653$	−5.25653	−0.205704	−0.102852	−	0.994697i	$-0.532797\pi$
$653$	−5.25653	−0.205704	−0.102852	+	0.994697i	$0.532797\pi$
$654$	7.46462	0.291890
$655$	−6.89152	−0.269274
$656$	−6.97295	−0.272248
$657$	3.77261	0.147184
$658$	11.9422	0.465557
$659$	−8.63450	−0.336352	−0.168176	−	0.985757i	$-0.553788\pi$
$659$	−8.63450	−0.336352	−0.168176	+	0.985757i	$0.553788\pi$
$660$	0.264458	0.0102940
$661$	−18.8556	−0.733399	−0.366700	−	0.930339i	$-0.619512\pi$
$661$	−18.8556	−0.733399	−0.366700	+	0.930339i	$0.619512\pi$
$662$	6.14581	0.238864
$663$	5.01313	0.194694
$664$	9.85969	0.382630
$665$	5.53885	0.214787
$666$	6.45496	0.250125
$667$	−34.1909	−1.32388
$668$	11.1361	0.430867
$669$	−5.69166	−0.220052
$670$	2.66685	0.103029
$671$	−3.49595	−0.134960
$672$	3.46524	0.133675
$673$	−15.3417	−0.591379	−0.295690	−	0.955284i	$-0.595549\pi$
$673$	−15.3417	−0.591379	−0.295690	+	0.955284i	$0.595549\pi$
$674$	−9.59614	−0.369630
$675$	−4.04882	−0.155839
$676$	1.00000	0.0384615
$677$	15.9865	0.614413	0.307206	−	0.951643i	$-0.400606\pi$
$677$	15.9865	0.614413	0.307206	+	0.951643i	$0.400606\pi$
$678$	−17.6138	−0.676456
$679$	15.8686	0.608982
$680$	4.88924	0.187494
$681$	−1.11936	−0.0428939
$682$	1.16661	0.0446718
$683$	42.0202	1.60786	0.803929	−	0.594725i	$-0.202739\pi$
$683$	42.0202	1.60786	0.803929	+	0.594725i	$0.202739\pi$
$684$	1.63890	0.0626651
$685$	−15.1631	−0.579353
$686$	−6.90304	−0.263559
$687$	7.06465	0.269533
$688$	1.56065	0.0594992
$689$	4.10931	0.156552
$690$	4.78018	0.181978
$691$	−30.4374	−1.15789	−0.578947	−	0.815365i	$-0.696536\pi$
$691$	−30.4374	−1.15789	−0.578947	+	0.815365i	$0.696536\pi$
$692$	−1.50329	−0.0571464
$693$	0.939634	0.0356938
$694$	−17.2096	−0.653266
$695$	4.60635	0.174729
$696$	6.97587	0.264420
$697$	−34.9563	−1.32407
$698$	−10.6917	−0.404686
$699$	−1.27464	−0.0482113
$700$	14.0301	0.530290
$701$	−43.8052	−1.65450	−0.827250	−	0.561834i	$-0.810096\pi$
$701$	−43.8052	−1.65450	−0.827250	+	0.561834i	$0.810096\pi$
$702$	−1.00000	−0.0377426
$703$	−10.5791	−0.398997
$704$	−0.271160	−0.0102197
$705$	−3.36112	−0.126587
$706$	3.31384	0.124718
$707$	17.8586	0.671640
$708$	−10.8986	−0.409596
$709$	9.09349	0.341513	0.170757	−	0.985313i	$-0.445379\pi$
$709$	9.09349	0.341513	0.170757	+	0.985313i	$0.445379\pi$
$710$	−5.74416	−0.215575
$711$	8.97116	0.336445
$712$	−9.07183	−0.339981
$713$	21.0869	0.789711
$714$	17.3717	0.650121
$715$	0.264458	0.00989017
$716$	1.82523	0.0682121
$717$	24.5955	0.918535
$718$	0.667444	0.0249088
$719$	37.0126	1.38034	0.690169	−	0.723648i	$-0.257536\pi$
$719$	37.0126	1.38034	0.690169	+	0.723648i	$0.257536\pi$
$720$	−0.975286	−0.0363468
$721$	−3.46524	−0.129052
$722$	16.3140	0.607144
$723$	11.4375	0.425367
$724$	9.51084	0.353468
$725$	28.2440	1.04896
$726$	10.9265	0.405519
$727$	−2.16074	−0.0801374	−0.0400687	−	0.999197i	$-0.512758\pi$
$727$	−2.16074	−0.0801374	−0.0400687	+	0.999197i	$0.512758\pi$
$728$	3.46524	0.128431
$729$	1.00000	0.0370370
$730$	3.67937	0.136180
$731$	7.82375	0.289372
$732$	12.8926	0.476524
$733$	10.4755	0.386920	0.193460	−	0.981108i	$-0.438029\pi$
$733$	10.4755	0.386920	0.193460	+	0.981108i	$0.438029\pi$
$734$	−3.06087	−0.112979
$735$	−4.88415	−0.180155
$736$	−4.90131	−0.180665
$737$	−0.741467	−0.0273123
$738$	6.97295	0.256678
$739$	−0.517591	−0.0190399	−0.00951995	−	0.999955i	$-0.503030\pi$
$739$	−0.517591	−0.0190399	−0.00951995	+	0.999955i	$0.503030\pi$
$740$	6.29543	0.231425
$741$	1.63890	0.0602067
$742$	14.2398	0.522758
$743$	22.2248	0.815347	0.407674	−	0.913128i	$-0.366340\pi$
$743$	22.2248	0.815347	0.407674	+	0.913128i	$0.366340\pi$
$744$	−4.30230	−0.157730
$745$	−19.1047	−0.699942
$746$	−12.4637	−0.456328
$747$	−9.85969	−0.360747
$748$	−1.35936	−0.0497031
$749$	5.07522	0.185445
$750$	−8.82518	−0.322250
$751$	32.5202	1.18668	0.593340	−	0.804952i	$-0.297809\pi$
$751$	32.5202	1.18668	0.593340	+	0.804952i	$0.297809\pi$
$752$	3.44629	0.125673
$753$	4.73336	0.172493
$754$	6.97587	0.254046
$755$	12.8172	0.466465
$756$	−3.46524	−0.126030
$757$	−28.8415	−1.04826	−0.524130	−	0.851638i	$-0.675609\pi$
$757$	−28.8415	−1.04826	−0.524130	+	0.851638i	$0.675609\pi$
$758$	0.473301	0.0171911
$759$	−1.32904	−0.0482410
$760$	1.59840	0.0579801
$761$	−35.1562	−1.27441	−0.637206	−	0.770693i	$-0.719910\pi$
$761$	−35.1562	−1.27441	−0.637206	+	0.770693i	$0.719910\pi$
$762$	6.59691	0.238981
$763$	25.8667	0.936438
$764$	23.7141	0.857947
$765$	−4.88924	−0.176771
$766$	−32.7415	−1.18300
$767$	−10.8986	−0.393527
$768$	1.00000	0.0360844
$769$	37.9719	1.36930	0.684651	−	0.728871i	$-0.259955\pi$
$769$	37.9719	1.36930	0.684651	+	0.728871i	$0.259955\pi$
$770$	0.916412	0.0330252
$771$	−3.74522	−0.134881
$772$	−0.846035	−0.0304495
$773$	−39.4598	−1.41927	−0.709635	−	0.704570i	$-0.751140\pi$
$773$	−39.4598	−1.41927	−0.709635	+	0.704570i	$0.751140\pi$
$774$	−1.56065	−0.0560964
$775$	−17.4192	−0.625717
$776$	4.57937	0.164390
$777$	22.3680	0.802448
$778$	26.2423	0.940832
$779$	−11.4280	−0.409451
$780$	−0.975286	−0.0349208
$781$	1.59705	0.0571471
$782$	−24.5709	−0.878654
$783$	−6.97587	−0.249297
$784$	5.00792	0.178854
$785$	4.24819	0.151624
$786$	−7.06615	−0.252041
$787$	26.3522	0.939355	0.469678	−	0.882838i	$-0.344370\pi$
$787$	26.3522	0.939355	0.469678	+	0.882838i	$0.344370\pi$
$788$	22.3654	0.796733
$789$	7.62061	0.271301
$790$	8.74944	0.311291
$791$	−61.0363	−2.17020
$792$	0.271160	0.00963524
$793$	12.8926	0.457829
$794$	17.7221	0.628934
$795$	−4.00775	−0.142140
$796$	−0.874237	−0.0309865
$797$	37.6646	1.33415	0.667075	−	0.744991i	$-0.267546\pi$
$797$	37.6646	1.33415	0.667075	+	0.744991i	$0.267546\pi$
$798$	5.67921	0.201042
$799$	17.2767	0.611207
$800$	4.04882	0.143147
$801$	9.07183	0.320537
$802$	0.408256	0.0144160
$803$	−1.02298	−0.0361002
$804$	2.73443	0.0964359
$805$	16.5645	0.583821
$806$	−4.30230	−0.151542
$807$	0.350569	0.0123406
$808$	5.15362	0.181304
$809$	27.4033	0.963447	0.481723	−	0.876323i	$-0.340011\pi$
$809$	27.4033	0.963447	0.481723	+	0.876323i	$0.340011\pi$
$810$	0.975286	0.0342680
$811$	−7.96951	−0.279847	−0.139924	−	0.990162i	$-0.544686\pi$
$811$	−7.96951	−0.279847	−0.139924	+	0.990162i	$0.544686\pi$
$812$	24.1731	0.848309
$813$	18.3518	0.643627
$814$	−1.75032	−0.0613488
$815$	20.9131	0.732556
$816$	5.01313	0.175495
$817$	2.55776	0.0894846
$818$	−28.6172	−1.00058
$819$	−3.46524	−0.121085
$820$	6.80062	0.237488
$821$	−0.629114	−0.0219562	−0.0109781	−	0.999940i	$-0.503495\pi$
$821$	−0.629114	−0.0219562	−0.0109781	+	0.999940i	$0.503495\pi$
$822$	−15.5474	−0.542276
$823$	0.403790	0.0140752	0.00703762	−	0.999975i	$-0.497760\pi$
$823$	0.403790	0.0140752	0.00703762	+	0.999975i	$0.497760\pi$
$824$	−1.00000	−0.0348367
$825$	1.09788	0.0382231
$826$	−37.7664	−1.31406
$827$	−37.6630	−1.30967	−0.654835	−	0.755772i	$-0.727262\pi$
$827$	−37.6630	−1.30967	−0.654835	+	0.755772i	$0.727262\pi$
$828$	4.90131	0.170332
$829$	30.2189	1.04955	0.524773	−	0.851242i	$-0.324150\pi$
$829$	30.2189	1.04955	0.524773	+	0.851242i	$0.324150\pi$
$830$	−9.61602	−0.333777
$831$	6.71964	0.233102
$832$	1.00000	0.0346688
$833$	25.1054	0.869850
$834$	4.72308	0.163547
$835$	−10.8608	−0.375855
$836$	−0.444405	−0.0153701
$837$	4.30230	0.148709
$838$	−31.0898	−1.07398
$839$	4.61853	0.159449	0.0797246	−	0.996817i	$-0.474596\pi$
$839$	4.61853	0.159449	0.0797246	+	0.996817i	$0.474596\pi$
$840$	−3.37960	−0.116607
$841$	19.6628	0.678026
$842$	−3.59868	−0.124019
$843$	31.4305	1.08252
$844$	−13.7130	−0.472019
$845$	−0.975286	−0.0335509
$846$	−3.44629	−0.118486
$847$	37.8629	1.30098
$848$	4.10931	0.141114
$849$	−1.25736	−0.0431524
$850$	20.2973	0.696191
$851$	−31.6378	−1.08453
$852$	−5.88972	−0.201779
$853$	−20.9811	−0.718378	−0.359189	−	0.933265i	$-0.616947\pi$
$853$	−20.9811	−0.718378	−0.359189	+	0.933265i	$0.616947\pi$
$854$	44.6760	1.52878
$855$	−1.59840	−0.0546641
$856$	1.46461	0.0500592
$857$	−33.8503	−1.15630	−0.578151	−	0.815930i	$-0.696226\pi$
$857$	−33.8503	−1.15630	−0.578151	+	0.815930i	$0.696226\pi$
$858$	0.271160	0.00925723
$859$	−6.16233	−0.210256	−0.105128	−	0.994459i	$-0.533525\pi$
$859$	−6.16233	−0.210256	−0.105128	+	0.994459i	$0.533525\pi$
$860$	−1.52208	−0.0519025
$861$	24.1630	0.823472
$862$	−30.1410	−1.02661
$863$	4.73172	0.161070	0.0805349	−	0.996752i	$-0.474337\pi$
$863$	4.73172	0.161070	0.0805349	+	0.996752i	$0.474337\pi$
$864$	−1.00000	−0.0340207
$865$	1.46613	0.0498501
$866$	−36.4406	−1.23830
$867$	8.13152	0.276161
$868$	−14.9085	−0.506028
$869$	−2.43261	−0.0825208
$870$	−6.80347	−0.230659
$871$	2.73443	0.0926526
$872$	7.46462	0.252784
$873$	−4.57937	−0.154988
$874$	−8.03278	−0.271713
$875$	−30.5814	−1.03384
$876$	3.77261	0.127465
$877$	−12.8861	−0.435134	−0.217567	−	0.976045i	$-0.569812\pi$
$877$	−12.8861	−0.435134	−0.217567	+	0.976045i	$0.569812\pi$
$878$	−23.5840	−0.795922
$879$	4.33316	0.146154
$880$	0.264458	0.00891488
$881$	20.0006	0.673838	0.336919	−	0.941534i	$-0.390615\pi$
$881$	20.0006	0.673838	0.336919	+	0.941534i	$0.390615\pi$
$882$	−5.00792	−0.168625
$883$	−44.9586	−1.51298	−0.756489	−	0.654006i	$-0.773087\pi$
$883$	−44.9586	−1.51298	−0.756489	+	0.654006i	$0.773087\pi$
$884$	5.01313	0.168610
$885$	10.6293	0.357299
$886$	−8.66283	−0.291034
$887$	12.9594	0.435133	0.217566	−	0.976046i	$-0.430188\pi$
$887$	12.9594	0.435133	0.217566	+	0.976046i	$0.430188\pi$
$888$	6.45496	0.216614
$889$	22.8599	0.766696
$890$	8.84763	0.296573
$891$	−0.271160	−0.00908419
$892$	−5.69166	−0.190571
$893$	5.64814	0.189008
$894$	−19.5888	−0.655148
$895$	−1.78012	−0.0595029
$896$	3.46524	0.115766
$897$	4.90131	0.163650
$898$	33.1353	1.10574
$899$	−30.0123	−1.00097
$900$	−4.04882	−0.134961
$901$	20.6005	0.686303
$902$	−1.89078	−0.0629562
$903$	−5.40804	−0.179968
$904$	−17.6138	−0.585828
$905$	−9.27579	−0.308338
$906$	13.1420	0.436612
$907$	21.3356	0.708438	0.354219	−	0.935162i	$-0.384747\pi$
$907$	21.3356	0.708438	0.354219	+	0.935162i	$0.384747\pi$
$908$	−1.11936	−0.0371472
$909$	−5.15362	−0.170935
$910$	−3.37960	−0.112033
$911$	17.6212	0.583817	0.291909	−	0.956446i	$-0.405710\pi$
$911$	17.6212	0.583817	0.291909	+	0.956446i	$0.405710\pi$
$912$	1.63890	0.0542696
$913$	2.67355	0.0884816
$914$	−13.2406	−0.437962
$915$	−12.5740	−0.415682
$916$	7.06465	0.233423
$917$	−24.4859	−0.808597
$918$	−5.01313	−0.165458
$919$	13.2955	0.438577	0.219288	−	0.975660i	$-0.429626\pi$
$919$	13.2955	0.438577	0.219288	+	0.975660i	$0.429626\pi$
$920$	4.78018	0.157598
$921$	16.8300	0.554569
$922$	14.6699	0.483126
$923$	−5.88972	−0.193863
$924$	0.939634	0.0309117
$925$	26.1350	0.859312
$926$	14.3763	0.472433
$927$	1.00000	0.0328443
$928$	6.97587	0.228994
$929$	50.0616	1.64247	0.821233	−	0.570593i	$-0.193286\pi$
$929$	50.0616	1.64247	0.821233	+	0.570593i	$0.193286\pi$
$930$	4.19597	0.137591
$931$	8.20750	0.268990
$932$	−1.27464	−0.0417522
$933$	16.1061	0.527289
$934$	3.22317	0.105465
$935$	1.32576	0.0433571
$936$	−1.00000	−0.0326860
$937$	−28.7112	−0.937955	−0.468977	−	0.883210i	$-0.655377\pi$
$937$	−28.7112	−0.937955	−0.468977	+	0.883210i	$0.655377\pi$
$938$	9.47547	0.309385
$939$	−1.45026	−0.0473274
$940$	−3.36112	−0.109628
$941$	13.8720	0.452214	0.226107	−	0.974102i	$-0.427400\pi$
$941$	13.8720	0.452214	0.226107	+	0.974102i	$0.427400\pi$
$942$	4.35584	0.141921
$943$	−34.1766	−1.11294
$944$	−10.8986	−0.354720
$945$	3.37960	0.109939
$946$	0.423185	0.0137589
$947$	−30.1789	−0.980683	−0.490341	−	0.871530i	$-0.663128\pi$
$947$	−30.1789	−0.980683	−0.490341	+	0.871530i	$0.663128\pi$
$948$	8.97116	0.291370
$949$	3.77261	0.122464
$950$	6.63563	0.215288
$951$	−6.27555	−0.203499
$952$	17.3717	0.563021
$953$	19.4760	0.630889	0.315444	−	0.948944i	$-0.397846\pi$
$953$	19.4760	0.630889	0.315444	+	0.948944i	$0.397846\pi$
$954$	−4.10931	−0.133044
$955$	−23.1281	−0.748406
$956$	24.5955	0.795475
$957$	1.89157	0.0611459
$958$	−14.0636	−0.454375
$959$	−53.8754	−1.73973
$960$	−0.975286	−0.0314772
$961$	−12.4902	−0.402910
$962$	6.45496	0.208116
$963$	−1.46461	−0.0471963
$964$	11.4375	0.368378
$965$	0.825126	0.0265617
$966$	16.9842	0.546459
$967$	−1.60807	−0.0517121	−0.0258560	−	0.999666i	$-0.508231\pi$
$967$	−1.60807	−0.0517121	−0.0258560	+	0.999666i	$0.508231\pi$
$968$	10.9265	0.351190
$969$	8.21605	0.263938
$970$	−4.46619	−0.143401
$971$	−10.6912	−0.343096	−0.171548	−	0.985176i	$-0.554877\pi$
$971$	−10.6912	−0.343096	−0.171548	+	0.985176i	$0.554877\pi$
$972$	1.00000	0.0320750
$973$	16.3666	0.524690
$974$	27.9367	0.895151
$975$	−4.04882	−0.129666
$976$	12.8926	0.412682
$977$	−24.3861	−0.780181	−0.390090	−	0.920777i	$-0.627556\pi$
$977$	−24.3861	−0.780181	−0.390090	+	0.920777i	$0.627556\pi$
$978$	21.4431	0.685675
$979$	−2.45991	−0.0786192
$980$	−4.88415	−0.156019
$981$	−7.46462	−0.238327
$982$	−10.2925	−0.328446
$983$	12.8836	0.410923	0.205462	−	0.978665i	$-0.434130\pi$
$983$	12.8836	0.410923	0.205462	+	0.978665i	$0.434130\pi$
$984$	6.97295	0.222290
$985$	−21.8126	−0.695008
$986$	34.9710	1.11370
$987$	−11.9422	−0.380126
$988$	1.63890	0.0521405
$989$	7.64923	0.243231
$990$	−0.264458	−0.00840503
$991$	33.2564	1.05642	0.528212	−	0.849113i	$-0.322863\pi$
$991$	33.2564	1.05642	0.528212	+	0.849113i	$0.322863\pi$
$992$	−4.30230	−0.136598
$993$	−6.14581	−0.195031
$994$	−20.4093	−0.647345
$995$	0.852631	0.0270302
$996$	−9.85969	−0.312416
$997$	−32.9668	−1.04407	−0.522035	−	0.852924i	$-0.674827\pi$
$997$	−32.9668	−1.04407	−0.522035	+	0.852924i	$0.674827\pi$
$998$	−18.8157	−0.595600
$999$	−6.45496	−0.204226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.x.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.x.1.6	✓	13	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.975286 q^{5} -1.00000 q^{6} -3.46524 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.975286 q^{5} -1.00000 q^{6} -3.46524 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.975286 q^{10} -0.271160 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.46524 q^{14} -0.975286 q^{15} +1.00000 q^{16} +5.01313 q^{17} -1.00000 q^{18} +1.63890 q^{19} -0.975286 q^{20} -3.46524 q^{21} +0.271160 q^{22} +4.90131 q^{23} -1.00000 q^{24} -4.04882 q^{25} -1.00000 q^{26} +1.00000 q^{27} -3.46524 q^{28} -6.97587 q^{29} +0.975286 q^{30} +4.30230 q^{31} -1.00000 q^{32} -0.271160 q^{33} -5.01313 q^{34} +3.37960 q^{35} +1.00000 q^{36} -6.45496 q^{37} -1.63890 q^{38} +1.00000 q^{39} +0.975286 q^{40} -6.97295 q^{41} +3.46524 q^{42} +1.56065 q^{43} -0.271160 q^{44} -0.975286 q^{45} -4.90131 q^{46} +3.44629 q^{47} +1.00000 q^{48} +5.00792 q^{49} +4.04882 q^{50} +5.01313 q^{51} +1.00000 q^{52} +4.10931 q^{53} -1.00000 q^{54} +0.264458 q^{55} +3.46524 q^{56} +1.63890 q^{57} +6.97587 q^{58} -10.8986 q^{59} -0.975286 q^{60} +12.8926 q^{61} -4.30230 q^{62} -3.46524 q^{63} +1.00000 q^{64} -0.975286 q^{65} +0.271160 q^{66} +2.73443 q^{67} +5.01313 q^{68} +4.90131 q^{69} -3.37960 q^{70} -5.88972 q^{71} -1.00000 q^{72} +3.77261 q^{73} +6.45496 q^{74} -4.04882 q^{75} +1.63890 q^{76} +0.939634 q^{77} -1.00000 q^{78} +8.97116 q^{79} -0.975286 q^{80} +1.00000 q^{81} +6.97295 q^{82} -9.85969 q^{83} -3.46524 q^{84} -4.88924 q^{85} -1.56065 q^{86} -6.97587 q^{87} +0.271160 q^{88} +9.07183 q^{89} +0.975286 q^{90} -3.46524 q^{91} +4.90131 q^{92} +4.30230 q^{93} -3.44629 q^{94} -1.59840 q^{95} -1.00000 q^{96} -4.57937 q^{97} -5.00792 q^{98} -0.271160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9	\( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9} - q^{10} + 6 q^{11} + 13 q^{12} + 13 q^{13} + q^{14} + q^{15} + 13 q^{16} + 18 q^{17} - 13 q^{18} - q^{19} + q^{20} - q^{21} - 6 q^{22} + 20 q^{23} - 13 q^{24} + 16 q^{25} - 13 q^{26} + 13 q^{27} - q^{28} + 25 q^{29} - q^{30} - 5 q^{31} - 13 q^{32} + 6 q^{33} - 18 q^{34} + 26 q^{35} + 13 q^{36} - 6 q^{37} + q^{38} + 13 q^{39} - q^{40} + 13 q^{41} + q^{42} - 2 q^{43} + 6 q^{44} + q^{45} - 20 q^{46} + 5 q^{47} + 13 q^{48} - 16 q^{50} + 18 q^{51} + 13 q^{52} + 21 q^{53} - 13 q^{54} + 2 q^{55} + q^{56} - q^{57} - 25 q^{58} + 22 q^{59} + q^{60} + 2 q^{61} + 5 q^{62} - q^{63} + 13 q^{64} + q^{65} - 6 q^{66} - q^{67} + 18 q^{68} + 20 q^{69} - 26 q^{70} + 32 q^{71} - 13 q^{72} - 13 q^{73} + 6 q^{74} + 16 q^{75} - q^{76} + 33 q^{77} - 13 q^{78} - 7 q^{79} + q^{80} + 13 q^{81} - 13 q^{82} + 17 q^{83} - q^{84} - 25 q^{85} + 2 q^{86} + 25 q^{87} - 6 q^{88} - 12 q^{89} - q^{90} - q^{91} + 20 q^{92} - 5 q^{93} - 5 q^{94} + 36 q^{95} - 13 q^{96} - 42 q^{97} + 6 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9 - q^10 + 6 * q^11 + 13 * q^12 + 13 * q^13 + q^14 + q^15 + 13 * q^16 + 18 * q^17 - 13 * q^18 - q^19 + q^20 - q^21 - 6 * q^22 + 20 * q^23 - 13 * q^24 + 16 * q^25 - 13 * q^26 + 13 * q^27 - q^28 + 25 * q^29 - q^30 - 5 * q^31 - 13 * q^32 + 6 * q^33 - 18 * q^34 + 26 * q^35 + 13 * q^36 - 6 * q^37 + q^38 + 13 * q^39 - q^40 + 13 * q^41 + q^42 - 2 * q^43 + 6 * q^44 + q^45 - 20 * q^46 + 5 * q^47 + 13 * q^48 - 16 * q^50 + 18 * q^51 + 13 * q^52 + 21 * q^53 - 13 * q^54 + 2 * q^55 + q^56 - q^57 - 25 * q^58 + 22 * q^59 + q^60 + 2 * q^61 + 5 * q^62 - q^63 + 13 * q^64 + q^65 - 6 * q^66 - q^67 + 18 * q^68 + 20 * q^69 - 26 * q^70 + 32 * q^71 - 13 * q^72 - 13 * q^73 + 6 * q^74 + 16 * q^75 - q^76 + 33 * q^77 - 13 * q^78 - 7 * q^79 + q^80 + 13 * q^81 - 13 * q^82 + 17 * q^83 - q^84 - 25 * q^85 + 2 * q^86 + 25 * q^87 - 6 * q^88 - 12 * q^89 - q^90 - q^91 + 20 * q^92 - 5 * q^93 - 5 * q^94 + 36 * q^95 - 13 * q^96 - 42 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more