LMFDB - Embedded newform 4730.2.a.bf.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4730,2,Mod(1,4730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4730, 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("4730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4730 = 2 \cdot 5 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4730.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:	$37.7692401561$
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} - 32 x^{11} - 5 x^{10} + 376 x^{9} + 100 x^{8} - 1985 x^{7} - 576 x^{6} + 4708 x^{5} + 889 x^{4} + \cdots - 128 $ x^13 - 32x^11 - 5x^10 + 376x^9 + 100x^8 - 1985x^7 - 576x^6 + 4708x^5 + 889x^4 - 4774x^3 - 296x^2 + 1648*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.0802958$ of defining polynomial
Character	$\chi$	$=$	4730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.0802958 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.0802958 q^{6} +2.53237 q^{7} +1.00000 q^{8} -2.99355 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.0802958 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.0802958 q^{6} +2.53237 q^{7} +1.00000 q^{8} -2.99355 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.0802958 q^{12} +5.53942 q^{13} +2.53237 q^{14} +0.0802958 q^{15} +1.00000 q^{16} -1.79237 q^{17} -2.99355 q^{18} -1.35154 q^{19} -1.00000 q^{20} -0.203338 q^{21} +1.00000 q^{22} +2.94296 q^{23} -0.0802958 q^{24} +1.00000 q^{25} +5.53942 q^{26} +0.481257 q^{27} +2.53237 q^{28} +5.51381 q^{29} +0.0802958 q^{30} -4.14071 q^{31} +1.00000 q^{32} -0.0802958 q^{33} -1.79237 q^{34} -2.53237 q^{35} -2.99355 q^{36} +11.9002 q^{37} -1.35154 q^{38} -0.444792 q^{39} -1.00000 q^{40} -8.42383 q^{41} -0.203338 q^{42} +1.00000 q^{43} +1.00000 q^{44} +2.99355 q^{45} +2.94296 q^{46} -5.26799 q^{47} -0.0802958 q^{48} -0.587126 q^{49} +1.00000 q^{50} +0.143919 q^{51} +5.53942 q^{52} +3.73022 q^{53} +0.481257 q^{54} -1.00000 q^{55} +2.53237 q^{56} +0.108523 q^{57} +5.51381 q^{58} -6.86311 q^{59} +0.0802958 q^{60} +3.71484 q^{61} -4.14071 q^{62} -7.58077 q^{63} +1.00000 q^{64} -5.53942 q^{65} -0.0802958 q^{66} +11.0507 q^{67} -1.79237 q^{68} -0.236307 q^{69} -2.53237 q^{70} -9.33354 q^{71} -2.99355 q^{72} +14.5726 q^{73} +11.9002 q^{74} -0.0802958 q^{75} -1.35154 q^{76} +2.53237 q^{77} -0.444792 q^{78} +13.5992 q^{79} -1.00000 q^{80} +8.94201 q^{81} -8.42383 q^{82} +2.07886 q^{83} -0.203338 q^{84} +1.79237 q^{85} +1.00000 q^{86} -0.442736 q^{87} +1.00000 q^{88} -4.41684 q^{89} +2.99355 q^{90} +14.0278 q^{91} +2.94296 q^{92} +0.332482 q^{93} -5.26799 q^{94} +1.35154 q^{95} -0.0802958 q^{96} -11.3450 q^{97} -0.587126 q^{98} -2.99355 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 13 q^{4} - 13 q^{5} + 7 q^{7} + 13 q^{8} + 25 q^{9}+O(q^{10}) $ 13 * q + 13 * q^2 + 13 * q^4 - 13 * q^5 + 7 * q^7 + 13 * q^8 + 25 * q^9	$ 13 q + 13 q^{2} + 13 q^{4} - 13 q^{5} + 7 q^{7} + 13 q^{8} + 25 q^{9} - 13 q^{10} + 13 q^{11} + 11 q^{13} + 7 q^{14} + 13 q^{16} - 2 q^{17} + 25 q^{18} + 7 q^{19} - 13 q^{20} + 12 q^{21} + 13 q^{22} + 12 q^{23} + 13 q^{25} + 11 q^{26} - 15 q^{27} + 7 q^{28} + 14 q^{29} + 20 q^{31} + 13 q^{32} - 2 q^{34} - 7 q^{35} + 25 q^{36} + 17 q^{37} + 7 q^{38} - 4 q^{39} - 13 q^{40} + 9 q^{41} + 12 q^{42} + 13 q^{43} + 13 q^{44} - 25 q^{45} + 12 q^{46} - 9 q^{47} + 30 q^{49} + 13 q^{50} - 3 q^{51} + 11 q^{52} + 22 q^{53} - 15 q^{54} - 13 q^{55} + 7 q^{56} + 17 q^{57} + 14 q^{58} + 19 q^{59} + 2 q^{61} + 20 q^{62} + 12 q^{63} + 13 q^{64} - 11 q^{65} + 9 q^{67} - 2 q^{68} - 6 q^{69} - 7 q^{70} + 6 q^{71} + 25 q^{72} + 7 q^{73} + 17 q^{74} + 7 q^{76} + 7 q^{77} - 4 q^{78} + 50 q^{79} - 13 q^{80} + 85 q^{81} + 9 q^{82} + q^{83} + 12 q^{84} + 2 q^{85} + 13 q^{86} - 21 q^{87} + 13 q^{88} + 5 q^{89} - 25 q^{90} + 5 q^{91} + 12 q^{92} + 3 q^{93} - 9 q^{94} - 7 q^{95} + 20 q^{97} + 30 q^{98} + 25 q^{99}+O(q^{100}) $ 13 * q + 13 * q^2 + 13 * q^4 - 13 * q^5 + 7 * q^7 + 13 * q^8 + 25 * q^9 - 13 * q^10 + 13 * q^11 + 11 * q^13 + 7 * q^14 + 13 * q^16 - 2 * q^17 + 25 * q^18 + 7 * q^19 - 13 * q^20 + 12 * q^21 + 13 * q^22 + 12 * q^23 + 13 * q^25 + 11 * q^26 - 15 * q^27 + 7 * q^28 + 14 * q^29 + 20 * q^31 + 13 * q^32 - 2 * q^34 - 7 * q^35 + 25 * q^36 + 17 * q^37 + 7 * q^38 - 4 * q^39 - 13 * q^40 + 9 * q^41 + 12 * q^42 + 13 * q^43 + 13 * q^44 - 25 * q^45 + 12 * q^46 - 9 * q^47 + 30 * q^49 + 13 * q^50 - 3 * q^51 + 11 * q^52 + 22 * q^53 - 15 * q^54 - 13 * q^55 + 7 * q^56 + 17 * q^57 + 14 * q^58 + 19 * q^59 + 2 * q^61 + 20 * q^62 + 12 * q^63 + 13 * q^64 - 11 * q^65 + 9 * q^67 - 2 * q^68 - 6 * q^69 - 7 * q^70 + 6 * q^71 + 25 * q^72 + 7 * q^73 + 17 * q^74 + 7 * q^76 + 7 * q^77 - 4 * q^78 + 50 * q^79 - 13 * q^80 + 85 * q^81 + 9 * q^82 + q^83 + 12 * q^84 + 2 * q^85 + 13 * q^86 - 21 * q^87 + 13 * q^88 + 5 * q^89 - 25 * q^90 + 5 * q^91 + 12 * q^92 + 3 * q^93 - 9 * q^94 - 7 * q^95 + 20 * q^97 + 30 * q^98 + 25 * 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
$3$	−0.0802958	−0.0463588	−0.0231794	−	0.999731i	$-0.507379\pi$
$3$	−0.0802958	−0.0463588	−0.0231794	+	0.999731i	$0.507379\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.0802958	−0.0327806
$7$	2.53237	0.957144	0.478572	−	0.878048i	$-0.341155\pi$
$7$	2.53237	0.957144	0.478572	+	0.878048i	$0.341155\pi$
$8$	1.00000	0.353553
$9$	−2.99355	−0.997851
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.0802958	−0.0231794
$13$	5.53942	1.53636	0.768179	−	0.640235i	$-0.221163\pi$
$13$	5.53942	1.53636	0.768179	+	0.640235i	$0.221163\pi$
$14$	2.53237	0.676803
$15$	0.0802958	0.0207323
$16$	1.00000	0.250000
$17$	−1.79237	−0.434713	−0.217356	−	0.976092i	$-0.569743\pi$
$17$	−1.79237	−0.434713	−0.217356	+	0.976092i	$0.569743\pi$
$18$	−2.99355	−0.705587
$19$	−1.35154	−0.310065	−0.155032	−	0.987909i	$-0.549548\pi$
$19$	−1.35154	−0.310065	−0.155032	+	0.987909i	$0.549548\pi$
$20$	−1.00000	−0.223607
$21$	−0.203338	−0.0443720
$22$	1.00000	0.213201
$23$	2.94296	0.613650	0.306825	−	0.951766i	$-0.400733\pi$
$23$	2.94296	0.613650	0.306825	+	0.951766i	$0.400733\pi$
$24$	−0.0802958	−0.0163903
$25$	1.00000	0.200000
$26$	5.53942	1.08637
$27$	0.481257	0.0926180
$28$	2.53237	0.478572
$29$	5.51381	1.02389	0.511944	−	0.859019i	$-0.328925\pi$
$29$	5.51381	1.02389	0.511944	+	0.859019i	$0.328925\pi$
$30$	0.0802958	0.0146599
$31$	−4.14071	−0.743693	−0.371847	−	0.928294i	$-0.621275\pi$
$31$	−4.14071	−0.743693	−0.371847	+	0.928294i	$0.621275\pi$
$32$	1.00000	0.176777
$33$	−0.0802958	−0.0139777
$34$	−1.79237	−0.307388
$35$	−2.53237	−0.428048
$36$	−2.99355	−0.498925
$37$	11.9002	1.95638	0.978188	−	0.207719i	$-0.0666040\pi$
$37$	11.9002	1.95638	0.978188	+	0.207719i	$0.0666040\pi$
$38$	−1.35154	−0.219249
$39$	−0.444792	−0.0712237
$40$	−1.00000	−0.158114
$41$	−8.42383	−1.31558	−0.657791	−	0.753201i	$-0.728509\pi$
$41$	−8.42383	−1.31558	−0.657791	+	0.753201i	$0.728509\pi$
$42$	−0.203338	−0.0313758
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	1.00000	0.150756
$45$	2.99355	0.446252
$46$	2.94296	0.433916
$47$	−5.26799	−0.768416	−0.384208	−	0.923247i	$-0.625525\pi$
$47$	−5.26799	−0.768416	−0.384208	+	0.923247i	$0.625525\pi$
$48$	−0.0802958	−0.0115897
$49$	−0.587126	−0.0838752
$50$	1.00000	0.141421
$51$	0.143919	0.0201528
$52$	5.53942	0.768179
$53$	3.73022	0.512385	0.256192	−	0.966626i	$-0.417532\pi$
$53$	3.73022	0.512385	0.256192	+	0.966626i	$0.417532\pi$
$54$	0.481257	0.0654908
$55$	−1.00000	−0.134840
$56$	2.53237	0.338402
$57$	0.108523	0.0143742
$58$	5.51381	0.723998
$59$	−6.86311	−0.893500	−0.446750	−	0.894659i	$-0.647419\pi$
$59$	−6.86311	−0.893500	−0.446750	+	0.894659i	$0.647419\pi$
$60$	0.0802958	0.0103661
$61$	3.71484	0.475637	0.237818	−	0.971310i	$-0.423568\pi$
$61$	3.71484	0.475637	0.237818	+	0.971310i	$0.423568\pi$
$62$	−4.14071	−0.525871
$63$	−7.58077	−0.955087
$64$	1.00000	0.125000
$65$	−5.53942	−0.687080
$66$	−0.0802958	−0.00988373
$67$	11.0507	1.35006	0.675031	−	0.737789i	$-0.264130\pi$
$67$	11.0507	1.35006	0.675031	+	0.737789i	$0.264130\pi$
$68$	−1.79237	−0.217356
$69$	−0.236307	−0.0284481
$70$	−2.53237	−0.302676
$71$	−9.33354	−1.10769	−0.553844	−	0.832621i	$-0.686839\pi$
$71$	−9.33354	−1.10769	−0.553844	+	0.832621i	$0.686839\pi$
$72$	−2.99355	−0.352794
$73$	14.5726	1.70559	0.852795	−	0.522247i	$-0.174906\pi$
$73$	14.5726	1.70559	0.852795	+	0.522247i	$0.174906\pi$
$74$	11.9002	1.38337
$75$	−0.0802958	−0.00927176
$76$	−1.35154	−0.155032
$77$	2.53237	0.288590
$78$	−0.444792	−0.0503628
$79$	13.5992	1.53003	0.765014	−	0.644014i	$-0.222732\pi$
$79$	13.5992	1.53003	0.765014	+	0.644014i	$0.222732\pi$
$80$	−1.00000	−0.111803
$81$	8.94201	0.993557
$82$	−8.42383	−0.930256
$83$	2.07886	0.228184	0.114092	−	0.993470i	$-0.463604\pi$
$83$	2.07886	0.228184	0.114092	+	0.993470i	$0.463604\pi$
$84$	−0.203338	−0.0221860
$85$	1.79237	0.194409
$86$	1.00000	0.107833
$87$	−0.442736	−0.0474662
$88$	1.00000	0.106600
$89$	−4.41684	−0.468184	−0.234092	−	0.972214i	$-0.575212\pi$
$89$	−4.41684	−0.468184	−0.234092	+	0.972214i	$0.575212\pi$
$90$	2.99355	0.315548
$91$	14.0278	1.47052
$92$	2.94296	0.306825
$93$	0.332482	0.0344767
$94$	−5.26799	−0.543352
$95$	1.35154	0.138665
$96$	−0.0802958	−0.00819515
$97$	−11.3450	−1.15191	−0.575955	−	0.817481i	$-0.695370\pi$
$97$	−11.3450	−1.15191	−0.575955	+	0.817481i	$0.695370\pi$
$98$	−0.587126	−0.0593087
$99$	−2.99355	−0.300863
$100$	1.00000	0.100000
$101$	−10.8959	−1.08418	−0.542089	−	0.840321i	$-0.682367\pi$
$101$	−10.8959	−1.08418	−0.542089	+	0.840321i	$0.682367\pi$
$102$	0.143919	0.0142502
$103$	6.15015	0.605992	0.302996	−	0.952992i	$-0.402013\pi$
$103$	6.15015	0.605992	0.302996	+	0.952992i	$0.402013\pi$
$104$	5.53942	0.543185
$105$	0.203338	0.0198438
$106$	3.73022	0.362311
$107$	15.8106	1.52847	0.764233	−	0.644941i	$-0.223118\pi$
$107$	15.8106	1.52847	0.764233	+	0.644941i	$0.223118\pi$
$108$	0.481257	0.0463090
$109$	7.13096	0.683022	0.341511	−	0.939878i	$-0.389061\pi$
$109$	7.13096	0.683022	0.341511	+	0.939878i	$0.389061\pi$
$110$	−1.00000	−0.0953463
$111$	−0.955534	−0.0906953
$112$	2.53237	0.239286
$113$	18.8176	1.77021	0.885105	−	0.465392i	$-0.154086\pi$
$113$	18.8176	1.77021	0.885105	+	0.465392i	$0.154086\pi$
$114$	0.108523	0.0101641
$115$	−2.94296	−0.274433
$116$	5.51381	0.511944
$117$	−16.5825	−1.53306
$118$	−6.86311	−0.631800
$119$	−4.53893	−0.416083
$120$	0.0802958	0.00732997
$121$	1.00000	0.0909091
$122$	3.71484	0.336326
$123$	0.676398	0.0609888
$124$	−4.14071	−0.371847
$125$	−1.00000	−0.0894427
$126$	−7.58077	−0.675349
$127$	−1.63100	−0.144728	−0.0723640	−	0.997378i	$-0.523054\pi$
$127$	−1.63100	−0.144728	−0.0723640	+	0.997378i	$0.523054\pi$
$128$	1.00000	0.0883883
$129$	−0.0802958	−0.00706965
$130$	−5.53942	−0.485839
$131$	15.3193	1.33846	0.669228	−	0.743057i	$-0.266625\pi$
$131$	15.3193	1.33846	0.669228	+	0.743057i	$0.266625\pi$
$132$	−0.0802958	−0.00698885
$133$	−3.42259	−0.296777
$134$	11.0507	0.954638
$135$	−0.481257	−0.0414200
$136$	−1.79237	−0.153694
$137$	1.72740	0.147581	0.0737907	−	0.997274i	$-0.476490\pi$
$137$	1.72740	0.147581	0.0737907	+	0.997274i	$0.476490\pi$
$138$	−0.236307	−0.0201158
$139$	−0.527955	−0.0447806	−0.0223903	−	0.999749i	$-0.507128\pi$
$139$	−0.527955	−0.0447806	−0.0223903	+	0.999749i	$0.507128\pi$
$140$	−2.53237	−0.214024
$141$	0.422998	0.0356228
$142$	−9.33354	−0.783253
$143$	5.53942	0.463229
$144$	−2.99355	−0.249463
$145$	−5.51381	−0.457897
$146$	14.5726	1.20603
$147$	0.0471438	0.00388835
$148$	11.9002	0.978188
$149$	0.606138	0.0496568	0.0248284	−	0.999692i	$-0.492096\pi$
$149$	0.606138	0.0496568	0.0248284	+	0.999692i	$0.492096\pi$
$150$	−0.0802958	−0.00655612
$151$	−14.0485	−1.14325	−0.571625	−	0.820515i	$-0.693687\pi$
$151$	−14.0485	−1.14325	−0.571625	+	0.820515i	$0.693687\pi$
$152$	−1.35154	−0.109624
$153$	5.36554	0.433779
$154$	2.53237	0.204064
$155$	4.14071	0.332590
$156$	−0.444792	−0.0356119
$157$	−12.7865	−1.02047	−0.510235	−	0.860035i	$-0.670442\pi$
$157$	−12.7865	−1.02047	−0.510235	+	0.860035i	$0.670442\pi$
$158$	13.5992	1.08189
$159$	−0.299521	−0.0237535
$160$	−1.00000	−0.0790569
$161$	7.45265	0.587351
$162$	8.94201	0.702551
$163$	1.14615	0.0897733	0.0448866	−	0.998992i	$-0.485707\pi$
$163$	1.14615	0.0897733	0.0448866	+	0.998992i	$0.485707\pi$
$164$	−8.42383	−0.657791
$165$	0.0802958	0.00625102
$166$	2.07886	0.161351
$167$	18.8684	1.46008	0.730041	−	0.683403i	$-0.239501\pi$
$167$	18.8684	1.46008	0.730041	+	0.683403i	$0.239501\pi$
$168$	−0.203338	−0.0156879
$169$	17.6852	1.36040
$170$	1.79237	0.137468
$171$	4.04591	0.309398
$172$	1.00000	0.0762493
$173$	−21.7578	−1.65421	−0.827107	−	0.562044i	$-0.810015\pi$
$173$	−21.7578	−1.65421	−0.827107	+	0.562044i	$0.810015\pi$
$174$	−0.442736	−0.0335637
$175$	2.53237	0.191429
$176$	1.00000	0.0753778
$177$	0.551078	0.0414216
$178$	−4.41684	−0.331056
$179$	10.6020	0.792428	0.396214	−	0.918158i	$-0.370324\pi$
$179$	10.6020	0.792428	0.396214	+	0.918158i	$0.370324\pi$
$180$	2.99355	0.223126
$181$	−21.1772	−1.57409	−0.787043	−	0.616898i	$-0.788389\pi$
$181$	−21.1772	−1.57409	−0.787043	+	0.616898i	$0.788389\pi$
$182$	14.0278	1.03981
$183$	−0.298286	−0.0220499
$184$	2.94296	0.216958
$185$	−11.9002	−0.874918
$186$	0.332482	0.0243787
$187$	−1.79237	−0.131071
$188$	−5.26799	−0.384208
$189$	1.21872	0.0886487
$190$	1.35154	0.0980511
$191$	−17.3295	−1.25392	−0.626958	−	0.779053i	$-0.715700\pi$
$191$	−17.3295	−1.25392	−0.626958	+	0.779053i	$0.715700\pi$
$192$	−0.0802958	−0.00579485
$193$	−18.5364	−1.33428	−0.667140	−	0.744932i	$-0.732482\pi$
$193$	−18.5364	−1.33428	−0.667140	+	0.744932i	$0.732482\pi$
$194$	−11.3450	−0.814523
$195$	0.444792	0.0318522
$196$	−0.587126	−0.0419376
$197$	16.1741	1.15236	0.576178	−	0.817324i	$-0.304543\pi$
$197$	16.1741	1.15236	0.576178	+	0.817324i	$0.304543\pi$
$198$	−2.99355	−0.212743
$199$	19.3164	1.36931	0.684653	−	0.728869i	$-0.259954\pi$
$199$	19.3164	1.36931	0.684653	+	0.728869i	$0.259954\pi$
$200$	1.00000	0.0707107
$201$	−0.887327	−0.0625872
$202$	−10.8959	−0.766630
$203$	13.9630	0.980009
$204$	0.143919	0.0100764
$205$	8.42383	0.588346
$206$	6.15015	0.428501
$207$	−8.80991	−0.612331
$208$	5.53942	0.384090
$209$	−1.35154	−0.0934880
$210$	0.203338	0.0140317
$211$	11.3192	0.779247	0.389624	−	0.920974i	$-0.372605\pi$
$211$	11.3192	0.779247	0.389624	+	0.920974i	$0.372605\pi$
$212$	3.73022	0.256192
$213$	0.749444	0.0513511
$214$	15.8106	1.08079
$215$	−1.00000	−0.0681994
$216$	0.481257	0.0327454
$217$	−10.4858	−0.711822
$218$	7.13096	0.482970
$219$	−1.17012	−0.0790691
$220$	−1.00000	−0.0674200
$221$	−9.92867	−0.667875
$222$	−0.955534	−0.0641312
$223$	8.13650	0.544860	0.272430	−	0.962176i	$-0.412173\pi$
$223$	8.13650	0.544860	0.272430	+	0.962176i	$0.412173\pi$
$224$	2.53237	0.169201
$225$	−2.99355	−0.199570
$226$	18.8176	1.25173
$227$	26.4976	1.75871	0.879353	−	0.476170i	$-0.157975\pi$
$227$	26.4976	1.75871	0.879353	+	0.476170i	$0.157975\pi$
$228$	0.108523	0.00718711
$229$	20.7862	1.37359	0.686794	−	0.726852i	$-0.259017\pi$
$229$	20.7862	1.37359	0.686794	+	0.726852i	$0.259017\pi$
$230$	−2.94296	−0.194053
$231$	−0.203338	−0.0133787
$232$	5.51381	0.361999
$233$	24.6597	1.61551	0.807754	−	0.589519i	$-0.200683\pi$
$233$	24.6597	1.61551	0.807754	+	0.589519i	$0.200683\pi$
$234$	−16.5825	−1.08403
$235$	5.26799	0.343646
$236$	−6.86311	−0.446750
$237$	−1.09196	−0.0709303
$238$	−4.53893	−0.294215
$239$	−25.4210	−1.64435	−0.822173	−	0.569238i	$-0.807238\pi$
$239$	−25.4210	−1.64435	−0.822173	+	0.569238i	$0.807238\pi$
$240$	0.0802958	0.00518307
$241$	0.339825	0.0218901	0.0109450	−	0.999940i	$-0.496516\pi$
$241$	0.339825	0.0218901	0.0109450	+	0.999940i	$0.496516\pi$
$242$	1.00000	0.0642824
$243$	−2.16178	−0.138678
$244$	3.71484	0.237818
$245$	0.587126	0.0375101
$246$	0.676398	0.0431256
$247$	−7.48675	−0.476370
$248$	−4.14071	−0.262935
$249$	−0.166923	−0.0105783
$250$	−1.00000	−0.0632456
$251$	−14.0703	−0.888112	−0.444056	−	0.895999i	$-0.646461\pi$
$251$	−14.0703	−0.888112	−0.444056	+	0.895999i	$0.646461\pi$
$252$	−7.58077	−0.477544
$253$	2.94296	0.185022
$254$	−1.63100	−0.102338
$255$	−0.143919	−0.00901259
$256$	1.00000	0.0625000
$257$	3.24329	0.202311	0.101156	−	0.994871i	$-0.467746\pi$
$257$	3.24329	0.202311	0.101156	+	0.994871i	$0.467746\pi$
$258$	−0.0802958	−0.00499900
$259$	30.1356	1.87253
$260$	−5.53942	−0.343540
$261$	−16.5059	−1.02169
$262$	15.3193	0.946432
$263$	−20.8739	−1.28714	−0.643569	−	0.765388i	$-0.722547\pi$
$263$	−20.8739	−1.28714	−0.643569	+	0.765388i	$0.722547\pi$
$264$	−0.0802958	−0.00494186
$265$	−3.73022	−0.229145
$266$	−3.42259	−0.209853
$267$	0.354654	0.0217045
$268$	11.0507	0.675031
$269$	7.57253	0.461706	0.230853	−	0.972989i	$-0.425848\pi$
$269$	7.57253	0.461706	0.230853	+	0.972989i	$0.425848\pi$
$270$	−0.481257	−0.0292884
$271$	−12.0129	−0.729731	−0.364865	−	0.931060i	$-0.618885\pi$
$271$	−12.0129	−0.729731	−0.364865	+	0.931060i	$0.618885\pi$
$272$	−1.79237	−0.108678
$273$	−1.12638	−0.0681714
$274$	1.72740	0.104356
$275$	1.00000	0.0603023
$276$	−0.236307	−0.0142240
$277$	26.4077	1.58669	0.793343	−	0.608775i	$-0.208339\pi$
$277$	26.4077	1.58669	0.793343	+	0.608775i	$0.208339\pi$
$278$	−0.527955	−0.0316647
$279$	12.3954	0.742095
$280$	−2.53237	−0.151338
$281$	4.35199	0.259618	0.129809	−	0.991539i	$-0.458564\pi$
$281$	4.35199	0.259618	0.129809	+	0.991539i	$0.458564\pi$
$282$	0.422998	0.0251891
$283$	30.9015	1.83691	0.918453	−	0.395530i	$-0.129439\pi$
$283$	30.9015	1.83691	0.918453	+	0.395530i	$0.129439\pi$
$284$	−9.33354	−0.553844
$285$	−0.108523	−0.00642835
$286$	5.53942	0.327553
$287$	−21.3322	−1.25920
$288$	−2.99355	−0.176397
$289$	−13.7874	−0.811025
$290$	−5.51381	−0.323782
$291$	0.910955	0.0534012
$292$	14.5726	0.852795
$293$	19.5789	1.14381	0.571905	−	0.820320i	$-0.306205\pi$
$293$	19.5789	1.14381	0.571905	+	0.820320i	$0.306205\pi$
$294$	0.0471438	0.00274948
$295$	6.86311	0.399585
$296$	11.9002	0.691684
$297$	0.481257	0.0279254
$298$	0.606138	0.0351126
$299$	16.3023	0.942786
$300$	−0.0802958	−0.00463588
$301$	2.53237	0.145963
$302$	−14.0485	−0.808399
$303$	0.874892	0.0502612
$304$	−1.35154	−0.0775162
$305$	−3.71484	−0.212711
$306$	5.36554	0.306728
$307$	23.5744	1.34546	0.672730	−	0.739888i	$-0.265122\pi$
$307$	23.5744	1.34546	0.672730	+	0.739888i	$0.265122\pi$
$308$	2.53237	0.144295
$309$	−0.493831	−0.0280931
$310$	4.14071	0.235176
$311$	−27.0824	−1.53570	−0.767852	−	0.640628i	$-0.778674\pi$
$311$	−27.0824	−1.53570	−0.767852	+	0.640628i	$0.778674\pi$
$312$	−0.444792	−0.0251814
$313$	−15.2582	−0.862447	−0.431223	−	0.902245i	$-0.641918\pi$
$313$	−15.2582	−0.862447	−0.431223	+	0.902245i	$0.641918\pi$
$314$	−12.7865	−0.721581
$315$	7.58077	0.427128
$316$	13.5992	0.765014
$317$	4.50710	0.253144	0.126572	−	0.991957i	$-0.459603\pi$
$317$	4.50710	0.253144	0.126572	+	0.991957i	$0.459603\pi$
$318$	−0.299521	−0.0167963
$319$	5.51381	0.308714
$320$	−1.00000	−0.0559017
$321$	−1.26952	−0.0708578
$322$	7.45265	0.415320
$323$	2.42246	0.134789
$324$	8.94201	0.496779
$325$	5.53942	0.307272
$326$	1.14615	0.0634793
$327$	−0.572586	−0.0316641
$328$	−8.42383	−0.465128
$329$	−13.3405	−0.735484
$330$	0.0802958	0.00442014
$331$	−15.3588	−0.844198	−0.422099	−	0.906550i	$-0.638707\pi$
$331$	−15.3588	−0.844198	−0.422099	+	0.906550i	$0.638707\pi$
$332$	2.07886	0.114092
$333$	−35.6238	−1.95217
$334$	18.8684	1.03243
$335$	−11.0507	−0.603766
$336$	−0.203338	−0.0110930
$337$	−20.9110	−1.13910	−0.569549	−	0.821958i	$-0.692882\pi$
$337$	−20.9110	−1.13910	−0.569549	+	0.821958i	$0.692882\pi$
$338$	17.6852	0.961946
$339$	−1.51097	−0.0820648
$340$	1.79237	0.0972047
$341$	−4.14071	−0.224232
$342$	4.04591	0.218778
$343$	−19.2134	−1.03742
$344$	1.00000	0.0539164
$345$	0.236307	0.0127224
$346$	−21.7578	−1.16971
$347$	−15.2696	−0.819715	−0.409858	−	0.912150i	$-0.634422\pi$
$347$	−15.2696	−0.819715	−0.409858	+	0.912150i	$0.634422\pi$
$348$	−0.442736	−0.0237331
$349$	−7.74515	−0.414588	−0.207294	−	0.978279i	$-0.566466\pi$
$349$	−7.74515	−0.414588	−0.207294	+	0.978279i	$0.566466\pi$
$350$	2.53237	0.135361
$351$	2.66588	0.142294
$352$	1.00000	0.0533002
$353$	−27.2898	−1.45249	−0.726246	−	0.687435i	$-0.758737\pi$
$353$	−27.2898	−1.45249	−0.726246	+	0.687435i	$0.758737\pi$
$354$	0.551078	0.0292895
$355$	9.33354	0.495373
$356$	−4.41684	−0.234092
$357$	0.364457	0.0192891
$358$	10.6020	0.560331
$359$	−22.5389	−1.18956	−0.594778	−	0.803890i	$-0.702760\pi$
$359$	−22.5389	−1.18956	−0.594778	+	0.803890i	$0.702760\pi$
$360$	2.99355	0.157774
$361$	−17.1733	−0.903860
$362$	−21.1772	−1.11305
$363$	−0.0802958	−0.00421444
$364$	14.0278	0.735258
$365$	−14.5726	−0.762763
$366$	−0.298286	−0.0155917
$367$	−3.64335	−0.190181	−0.0950907	−	0.995469i	$-0.530314\pi$
$367$	−3.64335	−0.190181	−0.0950907	+	0.995469i	$0.530314\pi$
$368$	2.94296	0.153412
$369$	25.2172	1.31275
$370$	−11.9002	−0.618661
$371$	9.44627	0.490426
$372$	0.332482	0.0172384
$373$	−22.3092	−1.15513	−0.577563	−	0.816346i	$-0.695996\pi$
$373$	−22.3092	−1.15513	−0.577563	+	0.816346i	$0.695996\pi$
$374$	−1.79237	−0.0926811
$375$	0.0802958	0.00414646
$376$	−5.26799	−0.271676
$377$	30.5433	1.57306
$378$	1.21872	0.0626841
$379$	−4.46027	−0.229108	−0.114554	−	0.993417i	$-0.536544\pi$
$379$	−4.46027	−0.229108	−0.114554	+	0.993417i	$0.536544\pi$
$380$	1.35154	0.0693326
$381$	0.130963	0.00670942
$382$	−17.3295	−0.886652
$383$	−20.1980	−1.03207	−0.516034	−	0.856568i	$-0.672592\pi$
$383$	−20.1980	−1.03207	−0.516034	+	0.856568i	$0.672592\pi$
$384$	−0.0802958	−0.00409758
$385$	−2.53237	−0.129061
$386$	−18.5364	−0.943479
$387$	−2.99355	−0.152171
$388$	−11.3450	−0.575955
$389$	19.0027	0.963474	0.481737	−	0.876316i	$-0.340006\pi$
$389$	19.0027	0.963474	0.481737	+	0.876316i	$0.340006\pi$
$390$	0.444792	0.0225229
$391$	−5.27487	−0.266761
$392$	−0.587126	−0.0296543
$393$	−1.23008	−0.0620492
$394$	16.1741	0.814839
$395$	−13.5992	−0.684249
$396$	−2.99355	−0.150432
$397$	−1.53517	−0.0770480	−0.0385240	−	0.999258i	$-0.512266\pi$
$397$	−1.53517	−0.0770480	−0.0385240	+	0.999258i	$0.512266\pi$
$398$	19.3164	0.968245
$399$	0.274820	0.0137582
$400$	1.00000	0.0500000
$401$	9.82030	0.490403	0.245201	−	0.969472i	$-0.421146\pi$
$401$	9.82030	0.490403	0.245201	+	0.969472i	$0.421146\pi$
$402$	−0.887327	−0.0442559
$403$	−22.9371	−1.14258
$404$	−10.8959	−0.542089
$405$	−8.94201	−0.444332
$406$	13.9630	0.692971
$407$	11.9002	0.589870
$408$	0.143919	0.00712508
$409$	20.0718	0.992485	0.496242	−	0.868184i	$-0.334713\pi$
$409$	20.0718	0.992485	0.496242	+	0.868184i	$0.334713\pi$
$410$	8.42383	0.416023
$411$	−0.138703	−0.00684170
$412$	6.15015	0.302996
$413$	−17.3799	−0.855208
$414$	−8.80991	−0.432984
$415$	−2.07886	−0.102047
$416$	5.53942	0.271592
$417$	0.0423926	0.00207597
$418$	−1.35154	−0.0661060
$419$	−34.7946	−1.69983	−0.849915	−	0.526920i	$-0.823347\pi$
$419$	−34.7946	−1.69983	−0.849915	+	0.526920i	$0.823347\pi$
$420$	0.203338	0.00992189
$421$	−10.3465	−0.504259	−0.252130	−	0.967693i	$-0.581131\pi$
$421$	−10.3465	−0.504259	−0.252130	+	0.967693i	$0.581131\pi$
$422$	11.3192	0.551011
$423$	15.7700	0.766764
$424$	3.73022	0.181155
$425$	−1.79237	−0.0869426
$426$	0.749444	0.0363107
$427$	9.40734	0.455253
$428$	15.8106	0.764233
$429$	−0.444792	−0.0214748
$430$	−1.00000	−0.0482243
$431$	−10.2756	−0.494960	−0.247480	−	0.968893i	$-0.579602\pi$
$431$	−10.2756	−0.494960	−0.247480	+	0.968893i	$0.579602\pi$
$432$	0.481257	0.0231545
$433$	24.6166	1.18300	0.591499	−	0.806305i	$-0.298536\pi$
$433$	24.6166	1.18300	0.591499	+	0.806305i	$0.298536\pi$
$434$	−10.4858	−0.503334
$435$	0.442736	0.0212275
$436$	7.13096	0.341511
$437$	−3.97753	−0.190271
$438$	−1.17012	−0.0559103
$439$	2.26151	0.107936	0.0539680	−	0.998543i	$-0.482813\pi$
$439$	2.26151	0.107936	0.0539680	+	0.998543i	$0.482813\pi$
$440$	−1.00000	−0.0476731
$441$	1.75759	0.0836949
$442$	−9.92867	−0.472259
$443$	−28.6143	−1.35950	−0.679752	−	0.733442i	$-0.737913\pi$
$443$	−28.6143	−1.35950	−0.679752	+	0.733442i	$0.737913\pi$
$444$	−0.955534	−0.0453476
$445$	4.41684	0.209378
$446$	8.13650	0.385274
$447$	−0.0486703	−0.00230203
$448$	2.53237	0.119643
$449$	10.5148	0.496222	0.248111	−	0.968732i	$-0.420190\pi$
$449$	10.5148	0.496222	0.248111	+	0.968732i	$0.420190\pi$
$450$	−2.99355	−0.141117
$451$	−8.42383	−0.396663
$452$	18.8176	0.885105
$453$	1.12803	0.0529997
$454$	26.4976	1.24359
$455$	−14.0278	−0.657635
$456$	0.108523	0.00508206
$457$	34.8469	1.63007	0.815035	−	0.579412i	$-0.196718\pi$
$457$	34.8469	1.63007	0.815035	+	0.579412i	$0.196718\pi$
$458$	20.7862	0.971273
$459$	−0.862589	−0.0402622
$460$	−2.94296	−0.137216
$461$	−20.8632	−0.971695	−0.485847	−	0.874044i	$-0.661489\pi$
$461$	−20.8632	−0.971695	−0.485847	+	0.874044i	$0.661489\pi$
$462$	−0.203338	−0.00946015
$463$	7.25956	0.337380	0.168690	−	0.985669i	$-0.446046\pi$
$463$	7.25956	0.337380	0.168690	+	0.985669i	$0.446046\pi$
$464$	5.51381	0.255972
$465$	−0.332482	−0.0154185
$466$	24.6597	1.14234
$467$	−14.0704	−0.651100	−0.325550	−	0.945525i	$-0.605549\pi$
$467$	−14.0704	−0.651100	−0.325550	+	0.945525i	$0.605549\pi$
$468$	−16.5825	−0.766528
$469$	27.9845	1.29220
$470$	5.26799	0.242994
$471$	1.02670	0.0473078
$472$	−6.86311	−0.315900
$473$	1.00000	0.0459800
$474$	−1.09196	−0.0501553
$475$	−1.35154	−0.0620129
$476$	−4.53893	−0.208041
$477$	−11.1666	−0.511284
$478$	−25.4210	−1.16273
$479$	37.2717	1.70299	0.851495	−	0.524363i	$-0.175697\pi$
$479$	37.2717	1.70299	0.851495	+	0.524363i	$0.175697\pi$
$480$	0.0802958	0.00366498
$481$	65.9201	3.00570
$482$	0.339825	0.0154786
$483$	−0.598417	−0.0272289
$484$	1.00000	0.0454545
$485$	11.3450	0.515150
$486$	−2.16178	−0.0980602
$487$	−27.3833	−1.24086	−0.620428	−	0.784263i	$-0.713041\pi$
$487$	−27.3833	−1.24086	−0.620428	+	0.784263i	$0.713041\pi$
$488$	3.71484	0.168163
$489$	−0.0920309	−0.00416178
$490$	0.587126	0.0265237
$491$	18.8903	0.852508	0.426254	−	0.904603i	$-0.359833\pi$
$491$	18.8903	0.852508	0.426254	+	0.904603i	$0.359833\pi$
$492$	0.676398	0.0304944
$493$	−9.88277	−0.445097
$494$	−7.48675	−0.336845
$495$	2.99355	0.134550
$496$	−4.14071	−0.185923
$497$	−23.6359	−1.06022
$498$	−0.166923	−0.00748002
$499$	7.35085	0.329069	0.164535	−	0.986371i	$-0.447388\pi$
$499$	7.35085	0.329069	0.164535	+	0.986371i	$0.447388\pi$
$500$	−1.00000	−0.0447214
$501$	−1.51505	−0.0676876
$502$	−14.0703	−0.627990
$503$	23.0497	1.02773	0.513867	−	0.857870i	$-0.328212\pi$
$503$	23.0497	1.02773	0.513867	+	0.857870i	$0.328212\pi$
$504$	−7.58077	−0.337674
$505$	10.8959	0.484859
$506$	2.94296	0.130831
$507$	−1.42004	−0.0630664
$508$	−1.63100	−0.0723640
$509$	−5.43536	−0.240918	−0.120459	−	0.992718i	$-0.538437\pi$
$509$	−5.43536	−0.240918	−0.120459	+	0.992718i	$0.538437\pi$
$510$	−0.143919	−0.00637286
$511$	36.9030	1.63249
$512$	1.00000	0.0441942
$513$	−0.650438	−0.0287176
$514$	3.24329	0.143056
$515$	−6.15015	−0.271008
$516$	−0.0802958	−0.00353483
$517$	−5.26799	−0.231686
$518$	30.1356	1.32408
$519$	1.74706	0.0766874
$520$	−5.53942	−0.242920
$521$	−14.0472	−0.615418	−0.307709	−	0.951481i	$-0.599562\pi$
$521$	−14.0472	−0.615418	−0.307709	+	0.951481i	$0.599562\pi$
$522$	−16.5059	−0.722442
$523$	−12.4419	−0.544047	−0.272024	−	0.962291i	$-0.587693\pi$
$523$	−12.4419	−0.544047	−0.272024	+	0.962291i	$0.587693\pi$
$524$	15.3193	0.669228
$525$	−0.203338	−0.00887441
$526$	−20.8739	−0.910145
$527$	7.42167	0.323293
$528$	−0.0802958	−0.00349443
$529$	−14.3390	−0.623434
$530$	−3.73022	−0.162030
$531$	20.5451	0.891580
$532$	−3.42259	−0.148388
$533$	−46.6631	−2.02120
$534$	0.354654	0.0153474
$535$	−15.8106	−0.683551
$536$	11.0507	0.477319
$537$	−0.851293	−0.0367360
$538$	7.57253	0.326475
$539$	−0.587126	−0.0252893
$540$	−0.481257	−0.0207100
$541$	−22.6114	−0.972141	−0.486071	−	0.873919i	$-0.661570\pi$
$541$	−22.6114	−0.972141	−0.486071	+	0.873919i	$0.661570\pi$
$542$	−12.0129	−0.515998
$543$	1.70044	0.0729727
$544$	−1.79237	−0.0768471
$545$	−7.13096	−0.305457
$546$	−1.12638	−0.0482044
$547$	−18.6541	−0.797593	−0.398796	−	0.917039i	$-0.630572\pi$
$547$	−18.6541	−0.797593	−0.398796	+	0.917039i	$0.630572\pi$
$548$	1.72740	0.0737907
$549$	−11.1206	−0.474615
$550$	1.00000	0.0426401
$551$	−7.45214	−0.317472
$552$	−0.236307	−0.0100579
$553$	34.4381	1.46446
$554$	26.4077	1.12196
$555$	0.955534	0.0405602
$556$	−0.527955	−0.0223903
$557$	15.6159	0.661668	0.330834	−	0.943689i	$-0.392670\pi$
$557$	15.6159	0.661668	0.330834	+	0.943689i	$0.392670\pi$
$558$	12.3954	0.524740
$559$	5.53942	0.234292
$560$	−2.53237	−0.107012
$561$	0.143919	0.00607629
$562$	4.35199	0.183577
$563$	−7.18692	−0.302892	−0.151446	−	0.988466i	$-0.548393\pi$
$563$	−7.18692	−0.302892	−0.151446	+	0.988466i	$0.548393\pi$
$564$	0.422998	0.0178114
$565$	−18.8176	−0.791662
$566$	30.9015	1.29889
$567$	22.6444	0.950977
$568$	−9.33354	−0.391627
$569$	−5.13595	−0.215310	−0.107655	−	0.994188i	$-0.534334\pi$
$569$	−5.13595	−0.215310	−0.107655	+	0.994188i	$0.534334\pi$
$570$	−0.108523	−0.00454553
$571$	−18.9978	−0.795034	−0.397517	−	0.917595i	$-0.630128\pi$
$571$	−18.9978	−0.795034	−0.397517	+	0.917595i	$0.630128\pi$
$572$	5.53942	0.231615
$573$	1.39148	0.0581300
$574$	−21.3322	−0.890390
$575$	2.94296	0.122730
$576$	−2.99355	−0.124731
$577$	−39.3920	−1.63991	−0.819954	−	0.572429i	$-0.806001\pi$
$577$	−39.3920	−1.63991	−0.819954	+	0.572429i	$0.806001\pi$
$578$	−13.7874	−0.573481
$579$	1.48840	0.0618557
$580$	−5.51381	−0.228948
$581$	5.26443	0.218405
$582$	0.910955	0.0377603
$583$	3.73022	0.154490
$584$	14.5726	0.603017
$585$	16.5825	0.685604
$586$	19.5789	0.808796
$587$	−28.1015	−1.15987	−0.579936	−	0.814662i	$-0.696922\pi$
$587$	−28.1015	−1.15987	−0.579936	+	0.814662i	$0.696922\pi$
$588$	0.0471438	0.00194418
$589$	5.59634	0.230593
$590$	6.86311	0.282550
$591$	−1.29871	−0.0534219
$592$	11.9002	0.489094
$593$	24.7093	1.01469	0.507345	−	0.861743i	$-0.330627\pi$
$593$	24.7093	1.01469	0.507345	+	0.861743i	$0.330627\pi$
$594$	0.481257	0.0197462
$595$	4.53893	0.186078
$596$	0.606138	0.0248284
$597$	−1.55103	−0.0634794
$598$	16.3023	0.666651
$599$	32.9405	1.34591	0.672956	−	0.739683i	$-0.265024\pi$
$599$	32.9405	1.34591	0.672956	+	0.739683i	$0.265024\pi$
$600$	−0.0802958	−0.00327806
$601$	−31.7248	−1.29408	−0.647042	−	0.762455i	$-0.723994\pi$
$601$	−31.7248	−1.29408	−0.647042	+	0.762455i	$0.723994\pi$
$602$	2.53237	0.103212
$603$	−33.0809	−1.34716
$604$	−14.0485	−0.571625
$605$	−1.00000	−0.0406558
$606$	0.874892	0.0355400
$607$	36.6188	1.48631	0.743156	−	0.669118i	$-0.233328\pi$
$607$	36.6188	1.48631	0.743156	+	0.669118i	$0.233328\pi$
$608$	−1.35154	−0.0548122
$609$	−1.12117	−0.0454320
$610$	−3.71484	−0.150410
$611$	−29.1816	−1.18056
$612$	5.36554	0.216889
$613$	44.1965	1.78508	0.892539	−	0.450969i	$-0.148922\pi$
$613$	44.1965	1.78508	0.892539	+	0.450969i	$0.148922\pi$
$614$	23.5744	0.951384
$615$	−0.676398	−0.0272750
$616$	2.53237	0.102032
$617$	−12.6211	−0.508105	−0.254053	−	0.967190i	$-0.581764\pi$
$617$	−12.6211	−0.508105	−0.254053	+	0.967190i	$0.581764\pi$
$618$	−0.493831	−0.0198648
$619$	12.0355	0.483746	0.241873	−	0.970308i	$-0.422238\pi$
$619$	12.0355	0.483746	0.241873	+	0.970308i	$0.422238\pi$
$620$	4.14071	0.166295
$621$	1.41632	0.0568350
$622$	−27.0824	−1.08591
$623$	−11.1851	−0.448120
$624$	−0.444792	−0.0178059
$625$	1.00000	0.0400000
$626$	−15.2582	−0.609842
$627$	0.108523	0.00433399
$628$	−12.7865	−0.510235
$629$	−21.3295	−0.850462
$630$	7.58077	0.302025
$631$	10.4654	0.416619	0.208310	−	0.978063i	$-0.433204\pi$
$631$	10.4654	0.416619	0.208310	+	0.978063i	$0.433204\pi$
$632$	13.5992	0.540947
$633$	−0.908886	−0.0361250
$634$	4.50710	0.179000
$635$	1.63100	0.0647243
$636$	−0.299521	−0.0118768
$637$	−3.25234	−0.128862
$638$	5.51381	0.218294
$639$	27.9404	1.10531
$640$	−1.00000	−0.0395285
$641$	25.7943	1.01881	0.509406	−	0.860526i	$-0.329865\pi$
$641$	25.7943	1.01881	0.509406	+	0.860526i	$0.329865\pi$
$642$	−1.26952	−0.0501040
$643$	−20.7620	−0.818772	−0.409386	−	0.912361i	$-0.634257\pi$
$643$	−20.7620	−0.818772	−0.409386	+	0.912361i	$0.634257\pi$
$644$	7.45265	0.293676
$645$	0.0802958	0.00316164
$646$	2.42246	0.0953103
$647$	−44.7180	−1.75805	−0.879024	−	0.476778i	$-0.841805\pi$
$647$	−44.7180	−1.75805	−0.879024	+	0.476778i	$0.841805\pi$
$648$	8.94201	0.351276
$649$	−6.86311	−0.269400
$650$	5.53942	0.217274
$651$	0.841965	0.0329992
$652$	1.14615	0.0448866
$653$	−12.5626	−0.491612	−0.245806	−	0.969319i	$-0.579053\pi$
$653$	−12.5626	−0.491612	−0.245806	+	0.969319i	$0.579053\pi$
$654$	−0.572586	−0.0223899
$655$	−15.3193	−0.598576
$656$	−8.42383	−0.328895
$657$	−43.6237	−1.70192
$658$	−13.3405	−0.520066
$659$	18.5262	0.721679	0.360839	−	0.932628i	$-0.382490\pi$
$659$	18.5262	0.721679	0.360839	+	0.932628i	$0.382490\pi$
$660$	0.0802958	0.00312551
$661$	34.3543	1.33623	0.668114	−	0.744059i	$-0.267102\pi$
$661$	34.3543	1.33623	0.668114	+	0.744059i	$0.267102\pi$
$662$	−15.3588	−0.596938
$663$	0.797230	0.0309619
$664$	2.07886	0.0806753
$665$	3.42259	0.132723
$666$	−35.6238	−1.38039
$667$	16.2269	0.628309
$668$	18.8684	0.730041
$669$	−0.653327	−0.0252591
$670$	−11.0507	−0.426927
$671$	3.71484	0.143410
$672$	−0.203338	−0.00784394
$673$	−28.8322	−1.11140	−0.555699	−	0.831384i	$-0.687549\pi$
$673$	−28.8322	−1.11140	−0.555699	+	0.831384i	$0.687549\pi$
$674$	−20.9110	−0.805463
$675$	0.481257	0.0185236
$676$	17.6852	0.680199
$677$	5.20811	0.200164	0.100082	−	0.994979i	$-0.468090\pi$
$677$	5.20811	0.200164	0.100082	+	0.994979i	$0.468090\pi$
$678$	−1.51097	−0.0580286
$679$	−28.7297	−1.10254
$680$	1.79237	0.0687341
$681$	−2.12764	−0.0815315
$682$	−4.14071	−0.158556
$683$	−30.0087	−1.14825	−0.574125	−	0.818768i	$-0.694658\pi$
$683$	−30.0087	−1.14825	−0.574125	+	0.818768i	$0.694658\pi$
$684$	4.04591	0.154699
$685$	−1.72740	−0.0660004
$686$	−19.2134	−0.733570
$687$	−1.66904	−0.0636779
$688$	1.00000	0.0381246
$689$	20.6632	0.787207
$690$	0.236307	0.00899607
$691$	20.1555	0.766750	0.383375	−	0.923593i	$-0.374762\pi$
$691$	20.1555	0.766750	0.383375	+	0.923593i	$0.374762\pi$
$692$	−21.7578	−0.827107
$693$	−7.58077	−0.287970
$694$	−15.2696	−0.579626
$695$	0.527955	0.0200265
$696$	−0.442736	−0.0167818
$697$	15.0986	0.571900
$698$	−7.74515	−0.293158
$699$	−1.98007	−0.0748931
$700$	2.53237	0.0957144
$701$	−38.1038	−1.43916	−0.719580	−	0.694409i	$-0.755666\pi$
$701$	−38.1038	−1.43916	−0.719580	+	0.694409i	$0.755666\pi$
$702$	2.66588	0.100617
$703$	−16.0836	−0.606603
$704$	1.00000	0.0376889
$705$	−0.422998	−0.0159310
$706$	−27.2898	−1.02707
$707$	−27.5923	−1.03772
$708$	0.551078	0.0207108
$709$	4.17569	0.156821	0.0784107	−	0.996921i	$-0.475015\pi$
$709$	4.17569	0.156821	0.0784107	+	0.996921i	$0.475015\pi$
$710$	9.33354	0.350281
$711$	−40.7099	−1.52674
$712$	−4.41684	−0.165528
$713$	−12.1859	−0.456367
$714$	0.364457	0.0136395
$715$	−5.53942	−0.207163
$716$	10.6020	0.396214
$717$	2.04120	0.0762299
$718$	−22.5389	−0.841143
$719$	−39.6845	−1.47998	−0.739990	−	0.672617i	$-0.765170\pi$
$719$	−39.6845	−1.47998	−0.739990	+	0.672617i	$0.765170\pi$
$720$	2.99355	0.111563
$721$	15.5744	0.580022
$722$	−17.1733	−0.639125
$723$	−0.0272865	−0.00101480
$724$	−21.1772	−0.787043
$725$	5.51381	0.204778
$726$	−0.0802958	−0.00298006
$727$	−30.1268	−1.11734	−0.558671	−	0.829390i	$-0.688688\pi$
$727$	−30.1268	−1.11734	−0.558671	+	0.829390i	$0.688688\pi$
$728$	14.0278	0.519906
$729$	−26.6525	−0.987128
$730$	−14.5726	−0.539355
$731$	−1.79237	−0.0662931
$732$	−0.298286	−0.0110250
$733$	40.0045	1.47760	0.738799	−	0.673925i	$-0.235393\pi$
$733$	40.0045	1.47760	0.738799	+	0.673925i	$0.235393\pi$
$734$	−3.64335	−0.134479
$735$	−0.0471438	−0.00173892
$736$	2.94296	0.108479
$737$	11.0507	0.407059
$738$	25.2172	0.928257
$739$	−27.5323	−1.01279	−0.506396	−	0.862301i	$-0.669022\pi$
$739$	−27.5323	−1.01279	−0.506396	+	0.862301i	$0.669022\pi$
$740$	−11.9002	−0.437459
$741$	0.601155	0.0220840
$742$	9.44627	0.346784
$743$	−29.9996	−1.10058	−0.550290	−	0.834974i	$-0.685483\pi$
$743$	−29.9996	−1.10058	−0.550290	+	0.834974i	$0.685483\pi$
$744$	0.332482	0.0121894
$745$	−0.606138	−0.0222072
$746$	−22.3092	−0.816797
$747$	−6.22317	−0.227694
$748$	−1.79237	−0.0655354
$749$	40.0381	1.46296
$750$	0.0802958	0.00293199
$751$	9.62663	0.351281	0.175640	−	0.984454i	$-0.443800\pi$
$751$	9.62663	0.351281	0.175640	+	0.984454i	$0.443800\pi$
$752$	−5.26799	−0.192104
$753$	1.12979	0.0411718
$754$	30.5433	1.11232
$755$	14.0485	0.511277
$756$	1.21872	0.0443244
$757$	−29.3151	−1.06548	−0.532738	−	0.846280i	$-0.678837\pi$
$757$	−29.3151	−1.06548	−0.532738	+	0.846280i	$0.678837\pi$
$758$	−4.46027	−0.162004
$759$	−0.236307	−0.00857742
$760$	1.35154	0.0490255
$761$	−16.6133	−0.602231	−0.301115	−	0.953588i	$-0.597359\pi$
$761$	−16.6133	−0.602231	−0.301115	+	0.953588i	$0.597359\pi$
$762$	0.130963	0.00474427
$763$	18.0582	0.653751
$764$	−17.3295	−0.626958
$765$	−5.36554	−0.193992
$766$	−20.1980	−0.729782
$767$	−38.0176	−1.37274
$768$	−0.0802958	−0.00289742
$769$	−18.2072	−0.656569	−0.328285	−	0.944579i	$-0.606470\pi$
$769$	−18.2072	−0.656569	−0.328285	+	0.944579i	$0.606470\pi$
$770$	−2.53237	−0.0912601
$771$	−0.260423	−0.00937890
$772$	−18.5364	−0.667140
$773$	14.2373	0.512080	0.256040	−	0.966666i	$-0.417582\pi$
$773$	14.2373	0.512080	0.256040	+	0.966666i	$0.417582\pi$
$774$	−2.99355	−0.107601
$775$	−4.14071	−0.148739
$776$	−11.3450	−0.407262
$777$	−2.41976	−0.0868085
$778$	19.0027	0.681279
$779$	11.3851	0.407915
$780$	0.444792	0.0159261
$781$	−9.33354	−0.333980
$782$	−5.27487	−0.188629
$783$	2.65356	0.0948305
$784$	−0.587126	−0.0209688
$785$	12.7865	0.456368
$786$	−1.23008	−0.0438754
$787$	−26.1616	−0.932562	−0.466281	−	0.884637i	$-0.654406\pi$
$787$	−26.1616	−0.932562	−0.466281	+	0.884637i	$0.654406\pi$
$788$	16.1741	0.576178
$789$	1.67608	0.0596702
$790$	−13.5992	−0.483837
$791$	47.6530	1.69435
$792$	−2.99355	−0.106371
$793$	20.5781	0.730749
$794$	−1.53517	−0.0544811
$795$	0.299521	0.0106229
$796$	19.3164	0.684653
$797$	−41.3388	−1.46430	−0.732148	−	0.681146i	$-0.761482\pi$
$797$	−41.3388	−1.46430	−0.732148	+	0.681146i	$0.761482\pi$
$798$	0.274820	0.00972852
$799$	9.44217	0.334040
$800$	1.00000	0.0353553
$801$	13.2221	0.467178
$802$	9.82030	0.346767
$803$	14.5726	0.514254
$804$	−0.887327	−0.0312936
$805$	−7.45265	−0.262672
$806$	−22.9371	−0.807926
$807$	−0.608043	−0.0214041
$808$	−10.8959	−0.383315
$809$	31.9418	1.12301	0.561507	−	0.827472i	$-0.310222\pi$
$809$	31.9418	1.12301	0.561507	+	0.827472i	$0.310222\pi$
$810$	−8.94201	−0.314190
$811$	−30.1792	−1.05974	−0.529868	−	0.848080i	$-0.677759\pi$
$811$	−30.1792	−1.05974	−0.529868	+	0.848080i	$0.677759\pi$
$812$	13.9630	0.490004
$813$	0.964584	0.0338294
$814$	11.9002	0.417101
$815$	−1.14615	−0.0401478
$816$	0.143919	0.00503819
$817$	−1.35154	−0.0472844
$818$	20.0718	0.701793
$819$	−41.9931	−1.46736
$820$	8.42383	0.294173
$821$	42.0871	1.46885	0.734425	−	0.678690i	$-0.237452\pi$
$821$	42.0871	1.46885	0.734425	+	0.678690i	$0.237452\pi$
$822$	−0.138703	−0.00483781
$823$	−15.3269	−0.534263	−0.267131	−	0.963660i	$-0.586076\pi$
$823$	−15.3269	−0.534263	−0.267131	+	0.963660i	$0.586076\pi$
$824$	6.15015	0.214250
$825$	−0.0802958	−0.00279554
$826$	−17.3799	−0.604724
$827$	−43.1924	−1.50195	−0.750973	−	0.660333i	$-0.770415\pi$
$827$	−43.1924	−1.50195	−0.750973	+	0.660333i	$0.770415\pi$
$828$	−8.80991	−0.306166
$829$	−45.4509	−1.57857	−0.789287	−	0.614025i	$-0.789550\pi$
$829$	−45.4509	−1.57857	−0.789287	+	0.614025i	$0.789550\pi$
$830$	−2.07886	−0.0721582
$831$	−2.12043	−0.0735569
$832$	5.53942	0.192045
$833$	1.05235	0.0364616
$834$	0.0423926	0.00146794
$835$	−18.8684	−0.652969
$836$	−1.35154	−0.0467440
$837$	−1.99275	−0.0688794
$838$	−34.7946	−1.20196
$839$	−37.8768	−1.30765	−0.653826	−	0.756645i	$-0.726837\pi$
$839$	−37.8768	−1.30765	−0.653826	+	0.756645i	$0.726837\pi$
$840$	0.203338	0.00701584
$841$	1.40208	0.0483475
$842$	−10.3465	−0.356565
$843$	−0.349446	−0.0120356
$844$	11.3192	0.389624
$845$	−17.6852	−0.608388
$846$	15.7700	0.542184
$847$	2.53237	0.0870131
$848$	3.73022	0.128096
$849$	−2.48126	−0.0851568
$850$	−1.79237	−0.0614777
$851$	35.0218	1.20053
$852$	0.749444	0.0256755
$853$	−22.8492	−0.782341	−0.391171	−	0.920318i	$-0.627930\pi$
$853$	−22.8492	−0.782341	−0.391171	+	0.920318i	$0.627930\pi$
$854$	9.40734	0.321912
$855$	−4.04591	−0.138367
$856$	15.8106	0.540394
$857$	−23.5090	−0.803052	−0.401526	−	0.915848i	$-0.631520\pi$
$857$	−23.5090	−0.803052	−0.401526	+	0.915848i	$0.631520\pi$
$858$	−0.444792	−0.0151849
$859$	3.44951	0.117696	0.0588479	−	0.998267i	$-0.481257\pi$
$859$	3.44951	0.117696	0.0588479	+	0.998267i	$0.481257\pi$
$860$	−1.00000	−0.0340997
$861$	1.71289	0.0583750
$862$	−10.2756	−0.349990
$863$	−42.4244	−1.44414	−0.722071	−	0.691819i	$-0.756810\pi$
$863$	−42.4244	−1.44414	−0.722071	+	0.691819i	$0.756810\pi$
$864$	0.481257	0.0163727
$865$	21.7578	0.739787
$866$	24.6166	0.836506
$867$	1.10707	0.0375981
$868$	−10.4858	−0.355911
$869$	13.5992	0.461321
$870$	0.442736	0.0150101
$871$	61.2146	2.07418
$872$	7.13096	0.241485
$873$	33.9618	1.14943
$874$	−3.97753	−0.134542
$875$	−2.53237	−0.0856096
$876$	−1.17012	−0.0395345
$877$	23.4047	0.790320	0.395160	−	0.918612i	$-0.370689\pi$
$877$	23.4047	0.790320	0.395160	+	0.918612i	$0.370689\pi$
$878$	2.26151	0.0763223
$879$	−1.57210	−0.0530256
$880$	−1.00000	−0.0337100
$881$	−31.0051	−1.04459	−0.522295	−	0.852765i	$-0.674924\pi$
$881$	−31.0051	−1.04459	−0.522295	+	0.852765i	$0.674924\pi$
$882$	1.75759	0.0591812
$883$	−19.9759	−0.672243	−0.336122	−	0.941819i	$-0.609115\pi$
$883$	−19.9759	−0.672243	−0.336122	+	0.941819i	$0.609115\pi$
$884$	−9.92867	−0.333937
$885$	−0.551078	−0.0185243
$886$	−28.6143	−0.961315
$887$	42.7824	1.43649	0.718246	−	0.695790i	$-0.244945\pi$
$887$	42.7824	1.43649	0.718246	+	0.695790i	$0.244945\pi$
$888$	−0.955534	−0.0320656
$889$	−4.13029	−0.138526
$890$	4.41684	0.148053
$891$	8.94201	0.299569
$892$	8.13650	0.272430
$893$	7.11990	0.238259
$894$	−0.0486703	−0.00162778
$895$	−10.6020	−0.354385
$896$	2.53237	0.0846004
$897$	−1.30901	−0.0437064
$898$	10.5148	0.350882
$899$	−22.8311	−0.761459
$900$	−2.99355	−0.0997851
$901$	−6.68592	−0.222740
$902$	−8.42383	−0.280483
$903$	−0.203338	−0.00676667
$904$	18.8176	0.625864
$905$	21.1772	0.703953
$906$	1.12803	0.0374764
$907$	−6.88502	−0.228613	−0.114307	−	0.993446i	$-0.536465\pi$
$907$	−6.88502	−0.228613	−0.114307	+	0.993446i	$0.536465\pi$
$908$	26.4976	0.879353
$909$	32.6173	1.08185
$910$	−14.0278	−0.465018
$911$	−14.2895	−0.473433	−0.236716	−	0.971579i	$-0.576071\pi$
$911$	−14.2895	−0.473433	−0.236716	+	0.971579i	$0.576071\pi$
$912$	0.108523	0.00359356
$913$	2.07886	0.0688001
$914$	34.8469	1.15263
$915$	0.298286	0.00986104
$916$	20.7862	0.686794
$917$	38.7942	1.28110
$918$	−0.862589	−0.0284697
$919$	21.1108	0.696380	0.348190	−	0.937424i	$-0.386796\pi$
$919$	21.1108	0.696380	0.348190	+	0.937424i	$0.386796\pi$
$920$	−2.94296	−0.0970266
$921$	−1.89292	−0.0623739
$922$	−20.8632	−0.687092
$923$	−51.7024	−1.70180
$924$	−0.203338	−0.00668934
$925$	11.9002	0.391275
$926$	7.25956	0.238564
$927$	−18.4108	−0.604690
$928$	5.51381	0.181000
$929$	15.0036	0.492252	0.246126	−	0.969238i	$-0.420842\pi$
$929$	15.0036	0.492252	0.246126	+	0.969238i	$0.420842\pi$
$930$	−0.332482	−0.0109025
$931$	0.793525	0.0260067
$932$	24.6597	0.807754
$933$	2.17460	0.0711934
$934$	−14.0704	−0.460397
$935$	1.79237	0.0586167
$936$	−16.5825	−0.542017
$937$	−2.66952	−0.0872096	−0.0436048	−	0.999049i	$-0.513884\pi$
$937$	−2.66952	−0.0872096	−0.0436048	+	0.999049i	$0.513884\pi$
$938$	27.9845	0.913726
$939$	1.22517	0.0399820
$940$	5.26799	0.171823
$941$	10.9184	0.355931	0.177965	−	0.984037i	$-0.443048\pi$
$941$	10.9184	0.355931	0.177965	+	0.984037i	$0.443048\pi$
$942$	1.02670	0.0334516
$943$	−24.7910	−0.807306
$944$	−6.86311	−0.223375
$945$	−1.21872	−0.0396449
$946$	1.00000	0.0325128
$947$	60.2218	1.95695	0.978474	−	0.206372i	$-0.0661658\pi$
$947$	60.2218	1.95695	0.978474	+	0.206372i	$0.0661658\pi$
$948$	−1.09196	−0.0354651
$949$	80.7235	2.62040
$950$	−1.35154	−0.0438498
$951$	−0.361901	−0.0117354
$952$	−4.53893	−0.147107
$953$	−20.6439	−0.668721	−0.334361	−	0.942445i	$-0.608520\pi$
$953$	−20.6439	−0.668721	−0.334361	+	0.942445i	$0.608520\pi$
$954$	−11.1666	−0.361532
$955$	17.3295	0.560768
$956$	−25.4210	−0.822173
$957$	−0.442736	−0.0143116
$958$	37.2717	1.20420
$959$	4.37440	0.141257
$960$	0.0802958	0.00259154
$961$	−13.8545	−0.446920
$962$	65.9201	2.12535
$963$	−47.3298	−1.52518
$964$	0.339825	0.0109450
$965$	18.5364	0.596709
$966$	−0.598417	−0.0192537
$967$	49.1054	1.57912	0.789561	−	0.613672i	$-0.210308\pi$
$967$	49.1054	1.57912	0.789561	+	0.613672i	$0.210308\pi$
$968$	1.00000	0.0321412
$969$	−0.194513	−0.00624866
$970$	11.3450	0.364266
$971$	3.95375	0.126882	0.0634410	−	0.997986i	$-0.479793\pi$
$971$	3.95375	0.126882	0.0634410	+	0.997986i	$0.479793\pi$
$972$	−2.16178	−0.0693390
$973$	−1.33698	−0.0428615
$974$	−27.3833	−0.877418
$975$	−0.444792	−0.0142447
$976$	3.71484	0.118909
$977$	38.7865	1.24089	0.620445	−	0.784250i	$-0.286952\pi$
$977$	38.7865	1.24089	0.620445	+	0.784250i	$0.286952\pi$
$978$	−0.0920309	−0.00294282
$979$	−4.41684	−0.141163
$980$	0.587126	0.0187551
$981$	−21.3469	−0.681554
$982$	18.8903	0.602814
$983$	43.7599	1.39573	0.697863	−	0.716232i	$-0.254135\pi$
$983$	43.7599	1.39573	0.697863	+	0.716232i	$0.254135\pi$
$984$	0.676398	0.0215628
$985$	−16.1741	−0.515350
$986$	−9.88277	−0.314731
$987$	1.07118	0.0340962
$988$	−7.48675	−0.238185
$989$	2.94296	0.0935807
$990$	2.99355	0.0951413
$991$	21.9423	0.697019	0.348509	−	0.937305i	$-0.386688\pi$
$991$	21.9423	0.697019	0.348509	+	0.937305i	$0.386688\pi$
$992$	−4.14071	−0.131468
$993$	1.23325	0.0391360
$994$	−23.6359	−0.749686
$995$	−19.3164	−0.612372
$996$	−0.166923	−0.00528917
$997$	−51.3411	−1.62599	−0.812994	−	0.582272i	$-0.802164\pi$
$997$	−51.3411	−1.62599	−0.812994	+	0.582272i	$0.802164\pi$
$998$	7.35085	0.232687
$999$	5.72704	0.181196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.bf.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.bf.1.7	✓	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:	\( N \)	\(=\)	\( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4730.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.0802958 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.0802958 q^{6} +2.53237 q^{7} +1.00000 q^{8} -2.99355 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.0802958 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.0802958 q^{6} +2.53237 q^{7} +1.00000 q^{8} -2.99355 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.0802958 q^{12} +5.53942 q^{13} +2.53237 q^{14} +0.0802958 q^{15} +1.00000 q^{16} -1.79237 q^{17} -2.99355 q^{18} -1.35154 q^{19} -1.00000 q^{20} -0.203338 q^{21} +1.00000 q^{22} +2.94296 q^{23} -0.0802958 q^{24} +1.00000 q^{25} +5.53942 q^{26} +0.481257 q^{27} +2.53237 q^{28} +5.51381 q^{29} +0.0802958 q^{30} -4.14071 q^{31} +1.00000 q^{32} -0.0802958 q^{33} -1.79237 q^{34} -2.53237 q^{35} -2.99355 q^{36} +11.9002 q^{37} -1.35154 q^{38} -0.444792 q^{39} -1.00000 q^{40} -8.42383 q^{41} -0.203338 q^{42} +1.00000 q^{43} +1.00000 q^{44} +2.99355 q^{45} +2.94296 q^{46} -5.26799 q^{47} -0.0802958 q^{48} -0.587126 q^{49} +1.00000 q^{50} +0.143919 q^{51} +5.53942 q^{52} +3.73022 q^{53} +0.481257 q^{54} -1.00000 q^{55} +2.53237 q^{56} +0.108523 q^{57} +5.51381 q^{58} -6.86311 q^{59} +0.0802958 q^{60} +3.71484 q^{61} -4.14071 q^{62} -7.58077 q^{63} +1.00000 q^{64} -5.53942 q^{65} -0.0802958 q^{66} +11.0507 q^{67} -1.79237 q^{68} -0.236307 q^{69} -2.53237 q^{70} -9.33354 q^{71} -2.99355 q^{72} +14.5726 q^{73} +11.9002 q^{74} -0.0802958 q^{75} -1.35154 q^{76} +2.53237 q^{77} -0.444792 q^{78} +13.5992 q^{79} -1.00000 q^{80} +8.94201 q^{81} -8.42383 q^{82} +2.07886 q^{83} -0.203338 q^{84} +1.79237 q^{85} +1.00000 q^{86} -0.442736 q^{87} +1.00000 q^{88} -4.41684 q^{89} +2.99355 q^{90} +14.0278 q^{91} +2.94296 q^{92} +0.332482 q^{93} -5.26799 q^{94} +1.35154 q^{95} -0.0802958 q^{96} -11.3450 q^{97} -0.587126 q^{98} -2.99355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 13 q^{4} - 13 q^{5} + 7 q^{7} + 13 q^{8} + 25 q^{9}+O(q^{10}) \) 13 * q + 13 * q^2 + 13 * q^4 - 13 * q^5 + 7 * q^7 + 13 * q^8 + 25 * q^9	\( 13 q + 13 q^{2} + 13 q^{4} - 13 q^{5} + 7 q^{7} + 13 q^{8} + 25 q^{9} - 13 q^{10} + 13 q^{11} + 11 q^{13} + 7 q^{14} + 13 q^{16} - 2 q^{17} + 25 q^{18} + 7 q^{19} - 13 q^{20} + 12 q^{21} + 13 q^{22} + 12 q^{23} + 13 q^{25} + 11 q^{26} - 15 q^{27} + 7 q^{28} + 14 q^{29} + 20 q^{31} + 13 q^{32} - 2 q^{34} - 7 q^{35} + 25 q^{36} + 17 q^{37} + 7 q^{38} - 4 q^{39} - 13 q^{40} + 9 q^{41} + 12 q^{42} + 13 q^{43} + 13 q^{44} - 25 q^{45} + 12 q^{46} - 9 q^{47} + 30 q^{49} + 13 q^{50} - 3 q^{51} + 11 q^{52} + 22 q^{53} - 15 q^{54} - 13 q^{55} + 7 q^{56} + 17 q^{57} + 14 q^{58} + 19 q^{59} + 2 q^{61} + 20 q^{62} + 12 q^{63} + 13 q^{64} - 11 q^{65} + 9 q^{67} - 2 q^{68} - 6 q^{69} - 7 q^{70} + 6 q^{71} + 25 q^{72} + 7 q^{73} + 17 q^{74} + 7 q^{76} + 7 q^{77} - 4 q^{78} + 50 q^{79} - 13 q^{80} + 85 q^{81} + 9 q^{82} + q^{83} + 12 q^{84} + 2 q^{85} + 13 q^{86} - 21 q^{87} + 13 q^{88} + 5 q^{89} - 25 q^{90} + 5 q^{91} + 12 q^{92} + 3 q^{93} - 9 q^{94} - 7 q^{95} + 20 q^{97} + 30 q^{98} + 25 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 13 * q^4 - 13 * q^5 + 7 * q^7 + 13 * q^8 + 25 * q^9 - 13 * q^10 + 13 * q^11 + 11 * q^13 + 7 * q^14 + 13 * q^16 - 2 * q^17 + 25 * q^18 + 7 * q^19 - 13 * q^20 + 12 * q^21 + 13 * q^22 + 12 * q^23 + 13 * q^25 + 11 * q^26 - 15 * q^27 + 7 * q^28 + 14 * q^29 + 20 * q^31 + 13 * q^32 - 2 * q^34 - 7 * q^35 + 25 * q^36 + 17 * q^37 + 7 * q^38 - 4 * q^39 - 13 * q^40 + 9 * q^41 + 12 * q^42 + 13 * q^43 + 13 * q^44 - 25 * q^45 + 12 * q^46 - 9 * q^47 + 30 * q^49 + 13 * q^50 - 3 * q^51 + 11 * q^52 + 22 * q^53 - 15 * q^54 - 13 * q^55 + 7 * q^56 + 17 * q^57 + 14 * q^58 + 19 * q^59 + 2 * q^61 + 20 * q^62 + 12 * q^63 + 13 * q^64 - 11 * q^65 + 9 * q^67 - 2 * q^68 - 6 * q^69 - 7 * q^70 + 6 * q^71 + 25 * q^72 + 7 * q^73 + 17 * q^74 + 7 * q^76 + 7 * q^77 - 4 * q^78 + 50 * q^79 - 13 * q^80 + 85 * q^81 + 9 * q^82 + q^83 + 12 * q^84 + 2 * q^85 + 13 * q^86 - 21 * q^87 + 13 * q^88 + 5 * q^89 - 25 * q^90 + 5 * q^91 + 12 * q^92 + 3 * q^93 - 9 * q^94 - 7 * q^95 + 20 * q^97 + 30 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more