LMFDB - Embedded newform 2001.2.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, 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("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.386616$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.386616 q^{2} +1.00000 q^{3} -1.85053 q^{4} +0.287925 q^{5} -0.386616 q^{6} -2.57107 q^{7} +1.48868 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.386616 q^{2} +1.00000 q^{3} -1.85053 q^{4} +0.287925 q^{5} -0.386616 q^{6} -2.57107 q^{7} +1.48868 q^{8} +1.00000 q^{9} -0.111317 q^{10} -4.70956 q^{11} -1.85053 q^{12} -0.906534 q^{13} +0.994018 q^{14} +0.287925 q^{15} +3.12551 q^{16} +4.50938 q^{17} -0.386616 q^{18} +3.83061 q^{19} -0.532813 q^{20} -2.57107 q^{21} +1.82080 q^{22} -1.00000 q^{23} +1.48868 q^{24} -4.91710 q^{25} +0.350481 q^{26} +1.00000 q^{27} +4.75784 q^{28} -1.00000 q^{29} -0.111317 q^{30} -0.585232 q^{31} -4.18573 q^{32} -4.70956 q^{33} -1.74340 q^{34} -0.740275 q^{35} -1.85053 q^{36} -3.38692 q^{37} -1.48098 q^{38} -0.906534 q^{39} +0.428627 q^{40} +11.5065 q^{41} +0.994018 q^{42} +5.57988 q^{43} +8.71518 q^{44} +0.287925 q^{45} +0.386616 q^{46} -0.820738 q^{47} +3.12551 q^{48} -0.389600 q^{49} +1.90103 q^{50} +4.50938 q^{51} +1.67757 q^{52} +6.67652 q^{53} -0.386616 q^{54} -1.35600 q^{55} -3.82749 q^{56} +3.83061 q^{57} +0.386616 q^{58} +10.9989 q^{59} -0.532813 q^{60} +8.33911 q^{61} +0.226260 q^{62} -2.57107 q^{63} -4.63275 q^{64} -0.261014 q^{65} +1.82080 q^{66} +15.2588 q^{67} -8.34473 q^{68} -1.00000 q^{69} +0.286203 q^{70} +13.9732 q^{71} +1.48868 q^{72} -8.09288 q^{73} +1.30944 q^{74} -4.91710 q^{75} -7.08865 q^{76} +12.1086 q^{77} +0.350481 q^{78} +8.56579 q^{79} +0.899912 q^{80} +1.00000 q^{81} -4.44859 q^{82} -9.72061 q^{83} +4.75784 q^{84} +1.29836 q^{85} -2.15728 q^{86} -1.00000 q^{87} -7.01102 q^{88} -15.0639 q^{89} -0.111317 q^{90} +2.33076 q^{91} +1.85053 q^{92} -0.585232 q^{93} +0.317311 q^{94} +1.10293 q^{95} -4.18573 q^{96} -2.21161 q^{97} +0.150626 q^{98} -4.70956 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9	\( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9 + 11 * q^10 + 8 * q^11 + 25 * q^12 + 19 * q^13 + 16 * q^14 + 3 * q^15 + 31 * q^16 - 4 * q^17 - q^18 + 19 * q^19 + 16 * q^20 + 13 * q^21 + 6 * q^22 - 16 * q^23 + 23 * q^25 - 15 * q^26 + 16 * q^27 + 18 * q^28 - 16 * q^29 + 11 * q^30 + 24 * q^31 - 21 * q^32 + 8 * q^33 - 9 * q^34 - 13 * q^35 + 25 * q^36 + 26 * q^37 + 19 * q^39 - 22 * q^40 - 15 * q^41 + 16 * q^42 + 33 * q^43 + 6 * q^44 + 3 * q^45 + q^46 + 13 * q^47 + 31 * q^48 + 41 * q^49 + 13 * q^50 - 4 * q^51 - 26 * q^52 + 5 * q^53 - q^54 + 9 * q^55 + 40 * q^56 + 19 * q^57 + q^58 + 2 * q^59 + 16 * q^60 + 29 * q^61 - 32 * q^62 + 13 * q^63 + 28 * q^64 + 18 * q^65 + 6 * q^66 + 32 * q^67 - 26 * q^68 - 16 * q^69 + 18 * q^70 + 29 * q^71 + 19 * q^73 - 16 * q^74 + 23 * q^75 + 64 * q^76 - 21 * q^77 - 15 * q^78 + 56 * q^79 + 16 * q^81 + 14 * q^82 + 5 * q^83 + 18 * q^84 + 16 * q^85 - 20 * q^86 - 16 * q^87 + q^88 + 7 * q^89 + 11 * q^90 - 6 * q^91 - 25 * q^92 + 24 * q^93 - 11 * q^94 + 39 * q^95 - 21 * q^96 + 35 * q^97 - 109 * q^98 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.386616	−0.273379	−0.136690	−	0.990614i	$-0.543646\pi$
$2$	−0.386616	−0.273379	−0.136690	+	0.990614i	$0.543646\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.85053	−0.925264
$5$	0.287925	0.128764	0.0643820	−	0.997925i	$-0.479492\pi$
$5$	0.287925	0.128764	0.0643820	+	0.997925i	$0.479492\pi$
$6$	−0.386616	−0.157836
$7$	−2.57107	−0.971773	−0.485887	−	0.874022i	$-0.661503\pi$
$7$	−2.57107	−0.971773	−0.485887	+	0.874022i	$0.661503\pi$
$8$	1.48868	0.526327
$9$	1.00000	0.333333
$10$	−0.111317	−0.0352014
$11$	−4.70956	−1.41999	−0.709994	−	0.704208i	$-0.751302\pi$
$11$	−4.70956	−1.41999	−0.709994	+	0.704208i	$0.751302\pi$
$12$	−1.85053	−0.534201
$13$	−0.906534	−0.251427	−0.125714	−	0.992067i	$-0.540122\pi$
$13$	−0.906534	−0.251427	−0.125714	+	0.992067i	$0.540122\pi$
$14$	0.994018	0.265662
$15$	0.287925	0.0743419
$16$	3.12551	0.781377
$17$	4.50938	1.09369	0.546843	−	0.837235i	$-0.315830\pi$
$17$	4.50938	1.09369	0.546843	+	0.837235i	$0.315830\pi$
$18$	−0.386616	−0.0911264
$19$	3.83061	0.878802	0.439401	−	0.898291i	$-0.355191\pi$
$19$	3.83061	0.878802	0.439401	+	0.898291i	$0.355191\pi$
$20$	−0.532813	−0.119141
$21$	−2.57107	−0.561053
$22$	1.82080	0.388195
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.48868	0.303875
$25$	−4.91710	−0.983420
$26$	0.350481	0.0687350
$27$	1.00000	0.192450
$28$	4.75784	0.899146
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−0.111317	−0.0203235
$31$	−0.585232	−0.105111	−0.0525554	−	0.998618i	$-0.516737\pi$
$31$	−0.585232	−0.105111	−0.0525554	+	0.998618i	$0.516737\pi$
$32$	−4.18573	−0.739939
$33$	−4.70956	−0.819830
$34$	−1.74340	−0.298991
$35$	−0.740275	−0.125129
$36$	−1.85053	−0.308421
$37$	−3.38692	−0.556806	−0.278403	−	0.960464i	$-0.589805\pi$
$37$	−3.38692	−0.556806	−0.278403	+	0.960464i	$0.589805\pi$
$38$	−1.48098	−0.240246
$39$	−0.906534	−0.145162
$40$	0.428627	0.0677720
$41$	11.5065	1.79701	0.898504	−	0.438965i	$-0.144655\pi$
$41$	11.5065	1.79701	0.898504	+	0.438965i	$0.144655\pi$
$42$	0.994018	0.153380
$43$	5.57988	0.850924	0.425462	−	0.904976i	$-0.360112\pi$
$43$	5.57988	0.850924	0.425462	+	0.904976i	$0.360112\pi$
$44$	8.71518	1.31386
$45$	0.287925	0.0429213
$46$	0.386616	0.0570035
$47$	−0.820738	−0.119717	−0.0598585	−	0.998207i	$-0.519065\pi$
$47$	−0.820738	−0.119717	−0.0598585	+	0.998207i	$0.519065\pi$
$48$	3.12551	0.451128
$49$	−0.389600	−0.0556572
$50$	1.90103	0.268846
$51$	4.50938	0.631439
$52$	1.67757	0.232637
$53$	6.67652	0.917091	0.458545	−	0.888671i	$-0.348371\pi$
$53$	6.67652	0.917091	0.458545	+	0.888671i	$0.348371\pi$
$54$	−0.386616	−0.0526118
$55$	−1.35600	−0.182843
$56$	−3.82749	−0.511470
$57$	3.83061	0.507377
$58$	0.386616	0.0507652
$59$	10.9989	1.43193	0.715965	−	0.698136i	$-0.245987\pi$
$59$	10.9989	1.43193	0.715965	+	0.698136i	$0.245987\pi$
$60$	−0.532813	−0.0687859
$61$	8.33911	1.06771	0.533857	−	0.845575i	$-0.320742\pi$
$61$	8.33911	1.06771	0.533857	+	0.845575i	$0.320742\pi$
$62$	0.226260	0.0287351
$63$	−2.57107	−0.323924
$64$	−4.63275	−0.579093
$65$	−0.261014	−0.0323748
$66$	1.82080	0.224124
$67$	15.2588	1.86416	0.932081	−	0.362250i	$-0.117991\pi$
$67$	15.2588	1.86416	0.932081	+	0.362250i	$0.117991\pi$
$68$	−8.34473	−1.01195
$69$	−1.00000	−0.120386
$70$	0.286203	0.0342078
$71$	13.9732	1.65831	0.829155	−	0.559018i	$-0.188822\pi$
$71$	13.9732	1.65831	0.829155	+	0.559018i	$0.188822\pi$
$72$	1.48868	0.175442
$73$	−8.09288	−0.947200	−0.473600	−	0.880740i	$-0.657046\pi$
$73$	−8.09288	−0.947200	−0.473600	+	0.880740i	$0.657046\pi$
$74$	1.30944	0.152219
$75$	−4.91710	−0.567778
$76$	−7.08865	−0.813124
$77$	12.1086	1.37991
$78$	0.350481	0.0396842
$79$	8.56579	0.963727	0.481864	−	0.876246i	$-0.339960\pi$
$79$	8.56579	0.963727	0.481864	+	0.876246i	$0.339960\pi$
$80$	0.899912	0.100613
$81$	1.00000	0.111111
$82$	−4.44859	−0.491265
$83$	−9.72061	−1.06698	−0.533488	−	0.845808i	$-0.679119\pi$
$83$	−9.72061	−1.06698	−0.533488	+	0.845808i	$0.679119\pi$
$84$	4.75784	0.519122
$85$	1.29836	0.140827
$86$	−2.15728	−0.232625
$87$	−1.00000	−0.107211
$88$	−7.01102	−0.747377
$89$	−15.0639	−1.59677	−0.798387	−	0.602145i	$-0.794313\pi$
$89$	−15.0639	−1.59677	−0.798387	+	0.602145i	$0.794313\pi$
$90$	−0.111317	−0.0117338
$91$	2.33076	0.244330
$92$	1.85053	0.192931
$93$	−0.585232	−0.0606857
$94$	0.317311	0.0327281
$95$	1.10293	0.113158
$96$	−4.18573	−0.427204
$97$	−2.21161	−0.224555	−0.112278	−	0.993677i	$-0.535815\pi$
$97$	−2.21161	−0.224555	−0.112278	+	0.993677i	$0.535815\pi$
$98$	0.150626	0.0152155
$99$	−4.70956	−0.473329
$100$	9.09923	0.909923
$101$	1.72771	0.171914	0.0859570	−	0.996299i	$-0.472605\pi$
$101$	1.72771	0.171914	0.0859570	+	0.996299i	$0.472605\pi$
$102$	−1.74340	−0.172622
$103$	14.5991	1.43850	0.719248	−	0.694754i	$-0.244487\pi$
$103$	14.5991	1.43850	0.719248	+	0.694754i	$0.244487\pi$
$104$	−1.34954	−0.132333
$105$	−0.740275	−0.0722435
$106$	−2.58125	−0.250713
$107$	−7.84440	−0.758347	−0.379173	−	0.925326i	$-0.623792\pi$
$107$	−7.84440	−0.758347	−0.379173	+	0.925326i	$0.623792\pi$
$108$	−1.85053	−0.178067
$109$	−5.62566	−0.538840	−0.269420	−	0.963023i	$-0.586832\pi$
$109$	−5.62566	−0.538840	−0.269420	+	0.963023i	$0.586832\pi$
$110$	0.524253	0.0499855
$111$	−3.38692	−0.321472
$112$	−8.03590	−0.759321
$113$	11.1189	1.04597	0.522987	−	0.852341i	$-0.324818\pi$
$113$	11.1189	1.04597	0.522987	+	0.852341i	$0.324818\pi$
$114$	−1.48098	−0.138706
$115$	−0.287925	−0.0268491
$116$	1.85053	0.171817
$117$	−0.906534	−0.0838091
$118$	−4.25234	−0.391460
$119$	−11.5939	−1.06281
$120$	0.428627	0.0391282
$121$	11.1800	1.01636
$122$	−3.22404	−0.291891
$123$	11.5065	1.03750
$124$	1.08299	0.0972551
$125$	−2.85538	−0.255393
$126$	0.994018	0.0885541
$127$	19.0377	1.68933	0.844663	−	0.535298i	$-0.179801\pi$
$127$	19.0377	1.68933	0.844663	+	0.535298i	$0.179801\pi$
$128$	10.1626	0.898251
$129$	5.57988	0.491281
$130$	0.100912	0.00885059
$131$	3.00962	0.262951	0.131476	−	0.991319i	$-0.458028\pi$
$131$	3.00962	0.262951	0.131476	+	0.991319i	$0.458028\pi$
$132$	8.71518	0.758559
$133$	−9.84876	−0.853996
$134$	−5.89931	−0.509623
$135$	0.287925	0.0247806
$136$	6.71301	0.575636
$137$	−21.3718	−1.82591	−0.912957	−	0.408057i	$-0.866207\pi$
$137$	−21.3718	−1.82591	−0.912957	+	0.408057i	$0.866207\pi$
$138$	0.386616	0.0329110
$139$	16.6957	1.41611	0.708057	−	0.706156i	$-0.249572\pi$
$139$	16.6957	1.41611	0.708057	+	0.706156i	$0.249572\pi$
$140$	1.36990	0.115778
$141$	−0.820738	−0.0691186
$142$	−5.40226	−0.453348
$143$	4.26938	0.357024
$144$	3.12551	0.260459
$145$	−0.287925	−0.0239109
$146$	3.12884	0.258945
$147$	−0.389600	−0.0321337
$148$	6.26758	0.515192
$149$	−0.179232	−0.0146832	−0.00734162	−	0.999973i	$-0.502337\pi$
$149$	−0.179232	−0.0146832	−0.00734162	+	0.999973i	$0.502337\pi$
$150$	1.90103	0.155219
$151$	−1.37683	−0.112045	−0.0560223	−	0.998430i	$-0.517842\pi$
$151$	−1.37683	−0.112045	−0.0560223	+	0.998430i	$0.517842\pi$
$152$	5.70254	0.462537
$153$	4.50938	0.364562
$154$	−4.68139	−0.377237
$155$	−0.168503	−0.0135345
$156$	1.67757	0.134313
$157$	17.6294	1.40698	0.703488	−	0.710707i	$-0.251625\pi$
$157$	17.6294	1.40698	0.703488	+	0.710707i	$0.251625\pi$
$158$	−3.31168	−0.263463
$159$	6.67652	0.529482
$160$	−1.20518	−0.0952775
$161$	2.57107	0.202629
$162$	−0.386616	−0.0303755
$163$	−1.94175	−0.152090	−0.0760449	−	0.997104i	$-0.524229\pi$
$163$	−1.94175	−0.152090	−0.0760449	+	0.997104i	$0.524229\pi$
$164$	−21.2930	−1.66271
$165$	−1.35600	−0.105565
$166$	3.75815	0.291689
$167$	7.84563	0.607113	0.303557	−	0.952813i	$-0.401826\pi$
$167$	7.84563	0.607113	0.303557	+	0.952813i	$0.401826\pi$
$168$	−3.82749	−0.295298
$169$	−12.1782	−0.936784
$170$	−0.501969	−0.0384992
$171$	3.83061	0.292934
$172$	−10.3257	−0.787330
$173$	3.11956	0.237176	0.118588	−	0.992944i	$-0.462163\pi$
$173$	3.11956	0.237176	0.118588	+	0.992944i	$0.462163\pi$
$174$	0.386616	0.0293093
$175$	12.6422	0.955661
$176$	−14.7198	−1.10955
$177$	10.9989	0.826725
$178$	5.82396	0.436524
$179$	−21.4735	−1.60500	−0.802502	−	0.596649i	$-0.796498\pi$
$179$	−21.4735	−1.60500	−0.802502	+	0.596649i	$0.796498\pi$
$180$	−0.532813	−0.0397136
$181$	14.0756	1.04623	0.523117	−	0.852261i	$-0.324769\pi$
$181$	14.0756	1.04623	0.523117	+	0.852261i	$0.324769\pi$
$182$	−0.901111	−0.0667948
$183$	8.33911	0.616445
$184$	−1.48868	−0.109747
$185$	−0.975178	−0.0716965
$186$	0.226260	0.0165902
$187$	−21.2372	−1.55302
$188$	1.51880	0.110770
$189$	−2.57107	−0.187018
$190$	−0.426410	−0.0309351
$191$	4.16450	0.301333	0.150666	−	0.988585i	$-0.451858\pi$
$191$	4.16450	0.301333	0.150666	+	0.988585i	$0.451858\pi$
$192$	−4.63275	−0.334340
$193$	−4.67082	−0.336213	−0.168107	−	0.985769i	$-0.553765\pi$
$193$	−4.67082	−0.336213	−0.168107	+	0.985769i	$0.553765\pi$
$194$	0.855046	0.0613887
$195$	−0.261014	−0.0186916
$196$	0.720966	0.0514976
$197$	13.0423	0.929222	0.464611	−	0.885515i	$-0.346194\pi$
$197$	13.0423	0.929222	0.464611	+	0.885515i	$0.346194\pi$
$198$	1.82080	0.129398
$199$	23.2352	1.64710	0.823551	−	0.567242i	$-0.191990\pi$
$199$	23.2352	1.64710	0.823551	+	0.567242i	$0.191990\pi$
$200$	−7.31997	−0.517600
$201$	15.2588	1.07627
$202$	−0.667963	−0.0469977
$203$	2.57107	0.180454
$204$	−8.34473	−0.584248
$205$	3.31300	0.231390
$206$	−5.64426	−0.393255
$207$	−1.00000	−0.0695048
$208$	−2.83338	−0.196460
$209$	−18.0405	−1.24789
$210$	0.286203	0.0197499
$211$	−7.35145	−0.506095	−0.253047	−	0.967454i	$-0.581433\pi$
$211$	−7.35145	−0.506095	−0.253047	+	0.967454i	$0.581433\pi$
$212$	−12.3551	−0.848551
$213$	13.9732	0.957426
$214$	3.03277	0.207316
$215$	1.60659	0.109568
$216$	1.48868	0.101292
$217$	1.50467	0.102144
$218$	2.17497	0.147308
$219$	−8.09288	−0.546866
$220$	2.50932	0.169178
$221$	−4.08791	−0.274982
$222$	1.30944	0.0878837
$223$	17.2637	1.15606	0.578030	−	0.816016i	$-0.303822\pi$
$223$	17.2637	1.15606	0.578030	+	0.816016i	$0.303822\pi$
$224$	10.7618	0.719053
$225$	−4.91710	−0.327807
$226$	−4.29873	−0.285947
$227$	24.4444	1.62243	0.811215	−	0.584747i	$-0.198806\pi$
$227$	24.4444	1.62243	0.811215	+	0.584747i	$0.198806\pi$
$228$	−7.08865	−0.469457
$229$	6.54407	0.432444	0.216222	−	0.976344i	$-0.430626\pi$
$229$	6.54407	0.432444	0.216222	+	0.976344i	$0.430626\pi$
$230$	0.111317	0.00734000
$231$	12.1086	0.796689
$232$	−1.48868	−0.0977365
$233$	−26.1002	−1.70988	−0.854939	−	0.518728i	$-0.826406\pi$
$233$	−26.1002	−1.70988	−0.854939	+	0.518728i	$0.826406\pi$
$234$	0.350481	0.0229117
$235$	−0.236311	−0.0154152
$236$	−20.3537	−1.32491
$237$	8.56579	0.556408
$238$	4.48240	0.290551
$239$	−7.25068	−0.469008	−0.234504	−	0.972115i	$-0.575347\pi$
$239$	−7.25068	−0.469008	−0.234504	+	0.972115i	$0.575347\pi$
$240$	0.899912	0.0580891
$241$	−22.9939	−1.48117	−0.740583	−	0.671965i	$-0.765451\pi$
$241$	−22.9939	−1.48117	−0.740583	+	0.671965i	$0.765451\pi$
$242$	−4.32237	−0.277853
$243$	1.00000	0.0641500
$244$	−15.4318	−0.987917
$245$	−0.112176	−0.00716664
$246$	−4.44859	−0.283632
$247$	−3.47258	−0.220955
$248$	−0.871221	−0.0553226
$249$	−9.72061	−0.616019
$250$	1.10394	0.0698191
$251$	16.7032	1.05430	0.527148	−	0.849774i	$-0.323261\pi$
$251$	16.7032	1.05430	0.527148	+	0.849774i	$0.323261\pi$
$252$	4.75784	0.299715
$253$	4.70956	0.296088
$254$	−7.36031	−0.461827
$255$	1.29836	0.0813067
$256$	5.33648	0.333530
$257$	−31.4979	−1.96479	−0.982393	−	0.186826i	$-0.940180\pi$
$257$	−31.4979	−1.96479	−0.982393	+	0.186826i	$0.940180\pi$
$258$	−2.15728	−0.134306
$259$	8.70800	0.541089
$260$	0.483013	0.0299552
$261$	−1.00000	−0.0618984
$262$	−1.16357	−0.0718854
$263$	−28.5998	−1.76354	−0.881769	−	0.471681i	$-0.843647\pi$
$263$	−28.5998	−1.76354	−0.881769	+	0.471681i	$0.843647\pi$
$264$	−7.01102	−0.431499
$265$	1.92234	0.118088
$266$	3.80769	0.233465
$267$	−15.0639	−0.921897
$268$	−28.2369	−1.72484
$269$	−12.3966	−0.755837	−0.377919	−	0.925839i	$-0.623360\pi$
$269$	−12.3966	−0.755837	−0.377919	+	0.925839i	$0.623360\pi$
$270$	−0.111317	−0.00677451
$271$	9.61788	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$271$	9.61788	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$272$	14.0941	0.854580
$273$	2.33076	0.141064
$274$	8.26268	0.499167
$275$	23.1574	1.39644
$276$	1.85053	0.111389
$277$	−5.69568	−0.342220	−0.171110	−	0.985252i	$-0.554735\pi$
$277$	−5.69568	−0.342220	−0.171110	+	0.985252i	$0.554735\pi$
$278$	−6.45484	−0.387136
$279$	−0.585232	−0.0350369
$280$	−1.10203	−0.0658590
$281$	6.76690	0.403680	0.201840	−	0.979419i	$-0.435308\pi$
$281$	6.76690	0.403680	0.201840	+	0.979419i	$0.435308\pi$
$282$	0.317311	0.0188956
$283$	12.1255	0.720789	0.360394	−	0.932800i	$-0.382642\pi$
$283$	12.1255	0.720789	0.360394	+	0.932800i	$0.382642\pi$
$284$	−25.8577	−1.53438
$285$	1.10293	0.0653318
$286$	−1.65061	−0.0976028
$287$	−29.5839	−1.74628
$288$	−4.18573	−0.246646
$289$	3.33450	0.196147
$290$	0.111317	0.00653673
$291$	−2.21161	−0.129647
$292$	14.9761	0.876410
$293$	1.15709	0.0675979	0.0337989	−	0.999429i	$-0.489239\pi$
$293$	1.15709	0.0675979	0.0337989	+	0.999429i	$0.489239\pi$
$294$	0.150626	0.00878468
$295$	3.16685	0.184381
$296$	−5.04203	−0.293062
$297$	−4.70956	−0.273277
$298$	0.0692939	0.00401409
$299$	0.906534	0.0524262
$300$	9.09923	0.525344
$301$	−14.3463	−0.826905
$302$	0.532304	0.0306307
$303$	1.72771	0.0992546
$304$	11.9726	0.686676
$305$	2.40104	0.137483
$306$	−1.74340	−0.0996635
$307$	−2.29266	−0.130849	−0.0654246	−	0.997858i	$-0.520840\pi$
$307$	−2.29266	−0.130849	−0.0654246	+	0.997858i	$0.520840\pi$
$308$	−22.4073	−1.27678
$309$	14.5991	0.830516
$310$	0.0651460	0.00370004
$311$	−15.1271	−0.857782	−0.428891	−	0.903356i	$-0.641096\pi$
$311$	−15.1271	−0.857782	−0.428891	+	0.903356i	$0.641096\pi$
$312$	−1.34954	−0.0764025
$313$	6.96731	0.393816	0.196908	−	0.980422i	$-0.436910\pi$
$313$	6.96731	0.393816	0.196908	+	0.980422i	$0.436910\pi$
$314$	−6.81580	−0.384638
$315$	−0.740275	−0.0417098
$316$	−15.8512	−0.891702
$317$	−18.7165	−1.05122	−0.525611	−	0.850725i	$-0.676163\pi$
$317$	−18.7165	−1.05122	−0.525611	+	0.850725i	$0.676163\pi$
$318$	−2.58125	−0.144749
$319$	4.70956	0.263685
$320$	−1.33388	−0.0745663
$321$	−7.84440	−0.437832
$322$	−0.994018	−0.0553944
$323$	17.2737	0.961133
$324$	−1.85053	−0.102807
$325$	4.45752	0.247259
$326$	0.750714	0.0415782
$327$	−5.62566	−0.311100
$328$	17.1294	0.945814
$329$	2.11018	0.116338
$330$	0.524253	0.0288592
$331$	9.50628	0.522512	0.261256	−	0.965269i	$-0.415863\pi$
$331$	9.50628	0.522512	0.261256	+	0.965269i	$0.415863\pi$
$332$	17.9883	0.987234
$333$	−3.38692	−0.185602
$334$	−3.03325	−0.165972
$335$	4.39340	0.240037
$336$	−8.03590	−0.438394
$337$	20.8700	1.13686	0.568430	−	0.822732i	$-0.307551\pi$
$337$	20.8700	1.13686	0.568430	+	0.822732i	$0.307551\pi$
$338$	4.70829	0.256097
$339$	11.1189	0.603893
$340$	−2.40266	−0.130302
$341$	2.75619	0.149256
$342$	−1.48098	−0.0800820
$343$	18.9992	1.02586
$344$	8.30665	0.447864
$345$	−0.287925	−0.0155014
$346$	−1.20607	−0.0648389
$347$	−6.79461	−0.364754	−0.182377	−	0.983229i	$-0.558379\pi$
$347$	−6.79461	−0.364754	−0.182377	+	0.983229i	$0.558379\pi$
$348$	1.85053	0.0991987
$349$	24.4527	1.30892	0.654462	−	0.756095i	$-0.272895\pi$
$349$	24.4527	1.30892	0.654462	+	0.756095i	$0.272895\pi$
$350$	−4.88768	−0.261258
$351$	−0.906534	−0.0483872
$352$	19.7130	1.05070
$353$	31.6663	1.68543	0.842714	−	0.538362i	$-0.180957\pi$
$353$	31.6663	1.68543	0.842714	+	0.538362i	$0.180957\pi$
$354$	−4.25234	−0.226009
$355$	4.02323	0.213531
$356$	27.8762	1.47744
$357$	−11.5939	−0.613616
$358$	8.30200	0.438775
$359$	24.1599	1.27511	0.637555	−	0.770404i	$-0.279946\pi$
$359$	24.1599	1.27511	0.637555	+	0.770404i	$0.279946\pi$
$360$	0.428627	0.0225907
$361$	−4.32643	−0.227707
$362$	−5.44187	−0.286019
$363$	11.1800	0.586798
$364$	−4.31314	−0.226070
$365$	−2.33014	−0.121965
$366$	−3.22404	−0.168523
$367$	28.3996	1.48245	0.741224	−	0.671258i	$-0.234246\pi$
$367$	28.3996	1.48245	0.741224	+	0.671258i	$0.234246\pi$
$368$	−3.12551	−0.162928
$369$	11.5065	0.599003
$370$	0.377020	0.0196003
$371$	−17.1658	−0.891204
$372$	1.08299	0.0561503
$373$	−23.7148	−1.22791	−0.613953	−	0.789343i	$-0.710422\pi$
$373$	−23.7148	−1.22791	−0.613953	+	0.789343i	$0.710422\pi$
$374$	8.21066	0.424563
$375$	−2.85538	−0.147451
$376$	−1.22181	−0.0630103
$377$	0.906534	0.0466889
$378$	0.994018	0.0511268
$379$	−34.7595	−1.78548	−0.892738	−	0.450575i	$-0.851219\pi$
$379$	−34.7595	−1.78548	−0.892738	+	0.450575i	$0.851219\pi$
$380$	−2.04100	−0.104701
$381$	19.0377	0.975333
$382$	−1.61007	−0.0823781
$383$	−22.5166	−1.15054	−0.575271	−	0.817963i	$-0.695104\pi$
$383$	−22.5166	−1.15054	−0.575271	+	0.817963i	$0.695104\pi$
$384$	10.1626	0.518606
$385$	3.48637	0.177682
$386$	1.80582	0.0919136
$387$	5.57988	0.283641
$388$	4.09265	0.207773
$389$	−2.98565	−0.151378	−0.0756892	−	0.997131i	$-0.524116\pi$
$389$	−2.98565	−0.151378	−0.0756892	+	0.997131i	$0.524116\pi$
$390$	0.100912	0.00510989
$391$	−4.50938	−0.228049
$392$	−0.579989	−0.0292939
$393$	3.00962	0.151815
$394$	−5.04235	−0.254030
$395$	2.46631	0.124093
$396$	8.71518	0.437954
$397$	−29.6332	−1.48725	−0.743625	−	0.668597i	$-0.766895\pi$
$397$	−29.6332	−1.48725	−0.743625	+	0.668597i	$0.766895\pi$
$398$	−8.98312	−0.450283
$399$	−9.84876	−0.493055
$400$	−15.3684	−0.768422
$401$	13.3800	0.668167	0.334083	−	0.942544i	$-0.391573\pi$
$401$	13.3800	0.668167	0.334083	+	0.942544i	$0.391573\pi$
$402$	−5.89931	−0.294231
$403$	0.530533	0.0264277
$404$	−3.19718	−0.159066
$405$	0.287925	0.0143071
$406$	−0.994018	−0.0493323
$407$	15.9509	0.790657
$408$	6.71301	0.332344
$409$	−15.2597	−0.754544	−0.377272	−	0.926102i	$-0.623138\pi$
$409$	−15.2597	−0.754544	−0.377272	+	0.926102i	$0.623138\pi$
$410$	−1.28086	−0.0632572
$411$	−21.3718	−1.05419
$412$	−27.0161	−1.33099
$413$	−28.2789	−1.39151
$414$	0.386616	0.0190012
$415$	−2.79881	−0.137388
$416$	3.79451	0.186041
$417$	16.6957	0.817594
$418$	6.97476	0.341146
$419$	−1.51590	−0.0740567	−0.0370283	−	0.999314i	$-0.511789\pi$
$419$	−1.51590	−0.0740567	−0.0370283	+	0.999314i	$0.511789\pi$
$420$	1.36990	0.0668443
$421$	−30.9888	−1.51030	−0.755151	−	0.655551i	$-0.772437\pi$
$421$	−30.9888	−1.51030	−0.755151	+	0.655551i	$0.772437\pi$
$422$	2.84219	0.138356
$423$	−0.820738	−0.0399057
$424$	9.93918	0.482689
$425$	−22.1731	−1.07555
$426$	−5.40226	−0.261740
$427$	−21.4404	−1.03758
$428$	14.5163	0.701671
$429$	4.26938	0.206128
$430$	−0.621134	−0.0299537
$431$	4.42544	0.213166	0.106583	−	0.994304i	$-0.466009\pi$
$431$	4.42544	0.213166	0.106583	+	0.994304i	$0.466009\pi$
$432$	3.12551	0.150376
$433$	−1.75648	−0.0844108	−0.0422054	−	0.999109i	$-0.513438\pi$
$433$	−1.75648	−0.0844108	−0.0422054	+	0.999109i	$0.513438\pi$
$434$	−0.581731	−0.0279240
$435$	−0.287925	−0.0138049
$436$	10.4104	0.498569
$437$	−3.83061	−0.183243
$438$	3.12884	0.149502
$439$	10.6827	0.509857	0.254929	−	0.966960i	$-0.417948\pi$
$439$	10.6827	0.509857	0.254929	+	0.966960i	$0.417948\pi$
$440$	−2.01865	−0.0962353
$441$	−0.389600	−0.0185524
$442$	1.58045	0.0751744
$443$	−18.5116	−0.879512	−0.439756	−	0.898117i	$-0.644935\pi$
$443$	−18.5116	−0.879512	−0.439756	+	0.898117i	$0.644935\pi$
$444$	6.26758	0.297446
$445$	−4.33728	−0.205607
$446$	−6.67441	−0.316043
$447$	−0.179232	−0.00847737
$448$	11.9111	0.562747
$449$	6.30134	0.297379	0.148689	−	0.988884i	$-0.452495\pi$
$449$	6.30134	0.297379	0.148689	+	0.988884i	$0.452495\pi$
$450$	1.90103	0.0896155
$451$	−54.1905	−2.55173
$452$	−20.5758	−0.967802
$453$	−1.37683	−0.0646890
$454$	−9.45060	−0.443539
$455$	0.671085	0.0314609
$456$	5.70254	0.267046
$457$	21.1799	0.990756	0.495378	−	0.868677i	$-0.335030\pi$
$457$	21.1799	0.990756	0.495378	+	0.868677i	$0.335030\pi$
$458$	−2.53005	−0.118221
$459$	4.50938	0.210480
$460$	0.532813	0.0248425
$461$	1.47287	0.0685984	0.0342992	−	0.999412i	$-0.489080\pi$
$461$	1.47287	0.0685984	0.0342992	+	0.999412i	$0.489080\pi$
$462$	−4.68139	−0.217798
$463$	27.3343	1.27033	0.635167	−	0.772375i	$-0.280931\pi$
$463$	27.3343	1.27033	0.635167	+	0.772375i	$0.280931\pi$
$464$	−3.12551	−0.145098
$465$	−0.168503	−0.00781413
$466$	10.0908	0.467445
$467$	−41.9669	−1.94200	−0.970998	−	0.239086i	$-0.923152\pi$
$467$	−41.9669	−1.94200	−0.970998	+	0.239086i	$0.923152\pi$
$468$	1.67757	0.0775456
$469$	−39.2315	−1.81154
$470$	0.0913618	0.00421420
$471$	17.6294	0.812318
$472$	16.3738	0.753664
$473$	−26.2788	−1.20830
$474$	−3.31168	−0.152110
$475$	−18.8355	−0.864231
$476$	21.4549	0.983383
$477$	6.67652	0.305697
$478$	2.80323	0.128217
$479$	−17.5408	−0.801460	−0.400730	−	0.916196i	$-0.631243\pi$
$479$	−17.5408	−0.801460	−0.400730	+	0.916196i	$0.631243\pi$
$480$	−1.20518	−0.0550085
$481$	3.07036	0.139996
$482$	8.88981	0.404920
$483$	2.57107	0.116988
$484$	−20.6889	−0.940405
$485$	−0.636779	−0.0289146
$486$	−0.386616	−0.0175373
$487$	−14.5887	−0.661079	−0.330539	−	0.943792i	$-0.607231\pi$
$487$	−14.5887	−0.661079	−0.330539	+	0.943792i	$0.607231\pi$
$488$	12.4142	0.561967
$489$	−1.94175	−0.0878091
$490$	0.0433690	0.00195921
$491$	5.38388	0.242971	0.121486	−	0.992593i	$-0.461234\pi$
$491$	5.38388	0.242971	0.121486	+	0.992593i	$0.461234\pi$
$492$	−21.2930	−0.959964
$493$	−4.50938	−0.203092
$494$	1.34256	0.0604045
$495$	−1.35600	−0.0609477
$496$	−1.82915	−0.0821311
$497$	−35.9260	−1.61150
$498$	3.75815	0.168407
$499$	4.25174	0.190334	0.0951669	−	0.995461i	$-0.469662\pi$
$499$	4.25174	0.190334	0.0951669	+	0.995461i	$0.469662\pi$
$500$	5.28396	0.236306
$501$	7.84563	0.350517
$502$	−6.45772	−0.288222
$503$	27.5946	1.23038	0.615190	−	0.788379i	$-0.289079\pi$
$503$	27.5946	1.23038	0.615190	+	0.788379i	$0.289079\pi$
$504$	−3.82749	−0.170490
$505$	0.497452	0.0221363
$506$	−1.82080	−0.0809442
$507$	−12.1782	−0.540853
$508$	−35.2299	−1.56307
$509$	−13.8528	−0.614013	−0.307007	−	0.951707i	$-0.599327\pi$
$509$	−13.8528	−0.614013	−0.307007	+	0.951707i	$0.599327\pi$
$510$	−0.501969	−0.0222275
$511$	20.8074	0.920463
$512$	−22.3883	−0.989431
$513$	3.83061	0.169126
$514$	12.1776	0.537131
$515$	4.20346	0.185226
$516$	−10.3257	−0.454565
$517$	3.86532	0.169997
$518$	−3.36666	−0.147922
$519$	3.11956	0.136933
$520$	−0.388565	−0.0170397
$521$	26.3988	1.15655	0.578277	−	0.815841i	$-0.303725\pi$
$521$	26.3988	1.15655	0.578277	+	0.815841i	$0.303725\pi$
$522$	0.386616	0.0169217
$523$	39.6615	1.73428	0.867139	−	0.498067i	$-0.165957\pi$
$523$	39.6615	1.73428	0.867139	+	0.498067i	$0.165957\pi$
$524$	−5.56938	−0.243299
$525$	12.6422	0.551751
$526$	11.0571	0.482115
$527$	−2.63903	−0.114958
$528$	−14.7198	−0.640596
$529$	1.00000	0.0434783
$530$	−0.743207	−0.0322829
$531$	10.9989	0.477310
$532$	18.2254	0.790172
$533$	−10.4310	−0.451817
$534$	5.82396	0.252028
$535$	−2.25860	−0.0976478
$536$	22.7155	0.981158
$537$	−21.4735	−0.926649
$538$	4.79275	0.206630
$539$	1.83485	0.0790325
$540$	−0.532813	−0.0229286
$541$	−21.8378	−0.938880	−0.469440	−	0.882964i	$-0.655544\pi$
$541$	−21.8378	−0.938880	−0.469440	+	0.882964i	$0.655544\pi$
$542$	−3.71843	−0.159720
$543$	14.0756	0.604043
$544$	−18.8750	−0.809260
$545$	−1.61977	−0.0693832
$546$	−0.901111	−0.0385640
$547$	−33.5249	−1.43342	−0.716712	−	0.697370i	$-0.754354\pi$
$547$	−33.5249	−1.43342	−0.716712	+	0.697370i	$0.754354\pi$
$548$	39.5490	1.68945
$549$	8.33911	0.355905
$550$	−8.95303	−0.381759
$551$	−3.83061	−0.163189
$552$	−1.48868	−0.0633623
$553$	−22.0233	−0.936524
$554$	2.20204	0.0935558
$555$	−0.975178	−0.0413940
$556$	−30.8959	−1.31028
$557$	−12.8124	−0.542878	−0.271439	−	0.962456i	$-0.587500\pi$
$557$	−12.8124	−0.542878	−0.271439	+	0.962456i	$0.587500\pi$
$558$	0.226260	0.00957836
$559$	−5.05836	−0.213946
$560$	−2.31374	−0.0977732
$561$	−21.2372	−0.896636
$562$	−2.61620	−0.110358
$563$	3.39408	0.143043	0.0715217	−	0.997439i	$-0.477214\pi$
$563$	3.39408	0.143043	0.0715217	+	0.997439i	$0.477214\pi$
$564$	1.51880	0.0639530
$565$	3.20140	0.134684
$566$	−4.68793	−0.197049
$567$	−2.57107	−0.107975
$568$	20.8016	0.872814
$569$	23.4806	0.984358	0.492179	−	0.870494i	$-0.336201\pi$
$569$	23.4806	0.984358	0.492179	+	0.870494i	$0.336201\pi$
$570$	−0.426410	−0.0178604
$571$	−40.4939	−1.69462	−0.847309	−	0.531101i	$-0.821779\pi$
$571$	−40.4939	−1.69462	−0.847309	+	0.531101i	$0.821779\pi$
$572$	−7.90061	−0.330341
$573$	4.16450	0.173975
$574$	11.4376	0.477398
$575$	4.91710	0.205057
$576$	−4.63275	−0.193031
$577$	5.68234	0.236559	0.118279	−	0.992980i	$-0.462262\pi$
$577$	5.68234	0.236559	0.118279	+	0.992980i	$0.462262\pi$
$578$	−1.28917	−0.0536225
$579$	−4.67082	−0.194113
$580$	0.532813	0.0221239
$581$	24.9924	1.03686
$582$	0.855046	0.0354428
$583$	−31.4435	−1.30226
$584$	−12.0477	−0.498537
$585$	−0.261014	−0.0107916
$586$	−0.447350	−0.0184798
$587$	−14.5024	−0.598580	−0.299290	−	0.954162i	$-0.596750\pi$
$587$	−14.5024	−0.598580	−0.299290	+	0.954162i	$0.596750\pi$
$588$	0.720966	0.0297321
$589$	−2.24179	−0.0923715
$590$	−1.22436	−0.0504059
$591$	13.0423	0.536487
$592$	−10.5858	−0.435075
$593$	7.79222	0.319988	0.159994	−	0.987118i	$-0.448852\pi$
$593$	7.79222	0.319988	0.159994	+	0.987118i	$0.448852\pi$
$594$	1.82080	0.0747081
$595$	−3.33818	−0.136852
$596$	0.331673	0.0135859
$597$	23.2352	0.950955
$598$	−0.350481	−0.0143322
$599$	−19.8445	−0.810823	−0.405412	−	0.914134i	$-0.632872\pi$
$599$	−19.8445	−0.810823	−0.405412	+	0.914134i	$0.632872\pi$
$600$	−7.31997	−0.298837
$601$	44.8062	1.82768	0.913841	−	0.406072i	$-0.133102\pi$
$601$	44.8062	1.82768	0.913841	+	0.406072i	$0.133102\pi$
$602$	5.54650	0.226059
$603$	15.2588	0.621387
$604$	2.54786	0.103671
$605$	3.21900	0.130871
$606$	−0.667963	−0.0271341
$607$	−24.1809	−0.981472	−0.490736	−	0.871308i	$-0.663272\pi$
$607$	−24.1809	−0.981472	−0.490736	+	0.871308i	$0.663272\pi$
$608$	−16.0339	−0.650260
$609$	2.57107	0.104185
$610$	−0.928281	−0.0375850
$611$	0.744027	0.0301001
$612$	−8.34473	−0.337316
$613$	35.9050	1.45019	0.725095	−	0.688648i	$-0.241796\pi$
$613$	35.9050	1.45019	0.725095	+	0.688648i	$0.241796\pi$
$614$	0.886381	0.0357714
$615$	3.31300	0.133593
$616$	18.0258	0.726281
$617$	−25.5271	−1.02768	−0.513841	−	0.857885i	$-0.671778\pi$
$617$	−25.5271	−1.02768	−0.513841	+	0.857885i	$0.671778\pi$
$618$	−5.64426	−0.227046
$619$	1.27108	0.0510891	0.0255446	−	0.999674i	$-0.491868\pi$
$619$	1.27108	0.0510891	0.0255446	+	0.999674i	$0.491868\pi$
$620$	0.311819	0.0125230
$621$	−1.00000	−0.0401286
$622$	5.84840	0.234500
$623$	38.7304	1.55170
$624$	−2.83338	−0.113426
$625$	23.7634	0.950534
$626$	−2.69368	−0.107661
$627$	−18.0405	−0.720468
$628$	−32.6236	−1.30182
$629$	−15.2729	−0.608970
$630$	0.286203	0.0114026
$631$	−25.7750	−1.02609	−0.513044	−	0.858362i	$-0.671482\pi$
$631$	−25.7750	−1.02609	−0.513044	+	0.858362i	$0.671482\pi$
$632$	12.7517	0.507235
$633$	−7.35145	−0.292194
$634$	7.23610	0.287382
$635$	5.48144	0.217524
$636$	−12.3551	−0.489911
$637$	0.353186	0.0139937
$638$	−1.82080	−0.0720860
$639$	13.9732	0.552770
$640$	2.92605	0.115662
$641$	13.5772	0.536267	0.268133	−	0.963382i	$-0.413593\pi$
$641$	13.5772	0.536267	0.268133	+	0.963382i	$0.413593\pi$
$642$	3.03277	0.119694
$643$	3.72566	0.146926	0.0734629	−	0.997298i	$-0.476595\pi$
$643$	3.72566	0.146926	0.0734629	+	0.997298i	$0.476595\pi$
$644$	−4.75784	−0.187485
$645$	1.60659	0.0632594
$646$	−6.67829	−0.262754
$647$	−20.2185	−0.794869	−0.397435	−	0.917630i	$-0.630099\pi$
$647$	−20.2185	−0.794869	−0.397435	+	0.917630i	$0.630099\pi$
$648$	1.48868	0.0584808
$649$	−51.7999	−2.03332
$650$	−1.72335	−0.0675954
$651$	1.50467	0.0589727
$652$	3.59327	0.140723
$653$	41.9179	1.64037	0.820187	−	0.572096i	$-0.193869\pi$
$653$	41.9179	1.64037	0.820187	+	0.572096i	$0.193869\pi$
$654$	2.17497	0.0850481
$655$	0.866544	0.0338587
$656$	35.9636	1.40414
$657$	−8.09288	−0.315733
$658$	−0.815829	−0.0318043
$659$	−20.1209	−0.783800	−0.391900	−	0.920008i	$-0.628182\pi$
$659$	−20.1209	−0.783800	−0.391900	+	0.920008i	$0.628182\pi$
$660$	2.50932	0.0976751
$661$	47.1610	1.83435	0.917176	−	0.398483i	$-0.130463\pi$
$661$	47.1610	1.83435	0.917176	+	0.398483i	$0.130463\pi$
$662$	−3.67528	−0.142844
$663$	−4.08791	−0.158761
$664$	−14.4709	−0.561578
$665$	−2.83571	−0.109964
$666$	1.30944	0.0507397
$667$	1.00000	0.0387202
$668$	−14.5186	−0.561740
$669$	17.2637	0.667451
$670$	−1.69856	−0.0656211
$671$	−39.2736	−1.51614
$672$	10.7618	0.415145
$673$	44.6166	1.71984	0.859922	−	0.510425i	$-0.170512\pi$
$673$	44.6166	1.71984	0.859922	+	0.510425i	$0.170512\pi$
$674$	−8.06867	−0.310794
$675$	−4.91710	−0.189259
$676$	22.5361	0.866773
$677$	15.4385	0.593349	0.296675	−	0.954979i	$-0.404122\pi$
$677$	15.4385	0.593349	0.296675	+	0.954979i	$0.404122\pi$
$678$	−4.29873	−0.165092
$679$	5.68621	0.218217
$680$	1.93284	0.0741212
$681$	24.4444	0.936711
$682$	−1.06559	−0.0408034
$683$	42.7476	1.63569	0.817846	−	0.575437i	$-0.195168\pi$
$683$	42.7476	1.63569	0.817846	+	0.575437i	$0.195168\pi$
$684$	−7.08865	−0.271041
$685$	−6.15347	−0.235112
$686$	−7.34539	−0.280448
$687$	6.54407	0.249672
$688$	17.4400	0.664893
$689$	−6.05249	−0.230582
$690$	0.111317	0.00423775
$691$	8.76609	0.333478	0.166739	−	0.986001i	$-0.446676\pi$
$691$	8.76609	0.333478	0.166739	+	0.986001i	$0.446676\pi$
$692$	−5.77283	−0.219450
$693$	12.1086	0.459968
$694$	2.62691	0.0997160
$695$	4.80712	0.182344
$696$	−1.48868	−0.0564282
$697$	51.8870	1.96536
$698$	−9.45382	−0.357833
$699$	−26.1002	−0.987199
$700$	−23.3948	−0.884238
$701$	22.6787	0.856564	0.428282	−	0.903645i	$-0.359119\pi$
$701$	22.6787	0.856564	0.428282	+	0.903645i	$0.359119\pi$
$702$	0.350481	0.0132281
$703$	−12.9740	−0.489322
$704$	21.8182	0.822305
$705$	−0.236311	−0.00889999
$706$	−12.2427	−0.460761
$707$	−4.44208	−0.167061
$708$	−20.3537	−0.764939
$709$	−19.5032	−0.732460	−0.366230	−	0.930524i	$-0.619352\pi$
$709$	−19.5032	−0.732460	−0.366230	+	0.930524i	$0.619352\pi$
$710$	−1.55545	−0.0583748
$711$	8.56579	0.321242
$712$	−22.4253	−0.840425
$713$	0.585232	0.0219171
$714$	4.48240	0.167750
$715$	1.22926	0.0459718
$716$	39.7373	1.48505
$717$	−7.25068	−0.270782
$718$	−9.34062	−0.348589
$719$	11.9661	0.446261	0.223130	−	0.974789i	$-0.428372\pi$
$719$	11.9661	0.446261	0.223130	+	0.974789i	$0.428372\pi$
$720$	0.899912	0.0335377
$721$	−37.5354	−1.39789
$722$	1.67267	0.0622503
$723$	−22.9939	−0.855152
$724$	−26.0474	−0.968042
$725$	4.91710	0.182616
$726$	−4.32237	−0.160418
$727$	−23.6984	−0.878923	−0.439462	−	0.898261i	$-0.644831\pi$
$727$	−23.6984	−0.878923	−0.439462	+	0.898261i	$0.644831\pi$
$728$	3.46975	0.128598
$729$	1.00000	0.0370370
$730$	0.900871	0.0333427
$731$	25.1618	0.930643
$732$	−15.4318	−0.570374
$733$	51.4839	1.90160	0.950801	−	0.309802i	$-0.100263\pi$
$733$	51.4839	1.90160	0.950801	+	0.309802i	$0.100263\pi$
$734$	−10.9798	−0.405270
$735$	−0.112176	−0.00413766
$736$	4.18573	0.154288
$737$	−71.8624	−2.64709
$738$	−4.44859	−0.163755
$739$	−47.9688	−1.76456	−0.882280	−	0.470725i	$-0.843992\pi$
$739$	−47.9688	−1.76456	−0.882280	+	0.470725i	$0.843992\pi$
$740$	1.80459	0.0663382
$741$	−3.47258	−0.127568
$742$	6.63658	0.243637
$743$	−11.4140	−0.418740	−0.209370	−	0.977837i	$-0.567141\pi$
$743$	−11.4140	−0.418740	−0.209370	+	0.977837i	$0.567141\pi$
$744$	−0.871221	−0.0319405
$745$	−0.0516053	−0.00189067
$746$	9.16853	0.335684
$747$	−9.72061	−0.355659
$748$	39.3000	1.43695
$749$	20.1685	0.736941
$750$	1.10394	0.0403101
$751$	38.4086	1.40155	0.700776	−	0.713382i	$-0.252837\pi$
$751$	38.4086	1.40155	0.700776	+	0.713382i	$0.252837\pi$
$752$	−2.56522	−0.0935441
$753$	16.7032	0.608698
$754$	−0.350481	−0.0127638
$755$	−0.396423	−0.0144273
$756$	4.75784	0.173041
$757$	−3.25821	−0.118421	−0.0592107	−	0.998246i	$-0.518858\pi$
$757$	−3.25821	−0.118421	−0.0592107	+	0.998246i	$0.518858\pi$
$758$	13.4386	0.488112
$759$	4.70956	0.170946
$760$	1.64190	0.0595581
$761$	9.73174	0.352775	0.176388	−	0.984321i	$-0.443559\pi$
$761$	9.73174	0.352775	0.176388	+	0.984321i	$0.443559\pi$
$762$	−7.36031	−0.266636
$763$	14.4640	0.523630
$764$	−7.70653	−0.278812
$765$	1.29836	0.0469424
$766$	8.70527	0.314534
$767$	−9.97085	−0.360026
$768$	5.33648	0.192564
$769$	30.7353	1.10834	0.554171	−	0.832403i	$-0.313035\pi$
$769$	30.7353	1.10834	0.554171	+	0.832403i	$0.313035\pi$
$770$	−1.34789	−0.0485746
$771$	−31.4979	−1.13437
$772$	8.64348	0.311086
$773$	9.05798	0.325793	0.162896	−	0.986643i	$-0.447916\pi$
$773$	9.05798	0.325793	0.162896	+	0.986643i	$0.447916\pi$
$774$	−2.15728	−0.0775417
$775$	2.87764	0.103368
$776$	−3.29238	−0.118189
$777$	8.70800	0.312398
$778$	1.15430	0.0413837
$779$	44.0768	1.57921
$780$	0.483013	0.0172947
$781$	−65.8076	−2.35478
$782$	1.74340	0.0623439
$783$	−1.00000	−0.0357371
$784$	−1.21770	−0.0434893
$785$	5.07593	0.181168
$786$	−1.16357	−0.0415031
$787$	9.41379	0.335565	0.167783	−	0.985824i	$-0.446339\pi$
$787$	9.41379	0.335565	0.167783	+	0.985824i	$0.446339\pi$
$788$	−24.1351	−0.859776
$789$	−28.5998	−1.01818
$790$	−0.953515	−0.0339245
$791$	−28.5874	−1.01645
$792$	−7.01102	−0.249126
$793$	−7.55969	−0.268452
$794$	11.4567	0.406583
$795$	1.92234	0.0681783
$796$	−42.9974	−1.52400
$797$	−43.4980	−1.54078	−0.770389	−	0.637574i	$-0.779938\pi$
$797$	−43.4980	−1.54078	−0.770389	+	0.637574i	$0.779938\pi$
$798$	3.80769	0.134791
$799$	−3.70102	−0.130933
$800$	20.5816	0.727671
$801$	−15.0639	−0.532258
$802$	−5.17294	−0.182663
$803$	38.1139	1.34501
$804$	−28.2369	−0.995838
$805$	0.740275	0.0260913
$806$	−0.205113	−0.00722478
$807$	−12.3966	−0.436383
$808$	2.57201	0.0904830
$809$	−11.9829	−0.421296	−0.210648	−	0.977562i	$-0.567557\pi$
$809$	−11.9829	−0.421296	−0.210648	+	0.977562i	$0.567557\pi$
$810$	−0.111317	−0.00391127
$811$	30.1902	1.06012	0.530061	−	0.847960i	$-0.322169\pi$
$811$	30.1902	1.06012	0.530061	+	0.847960i	$0.322169\pi$
$812$	−4.75784	−0.166967
$813$	9.61788	0.337314
$814$	−6.16688	−0.216149
$815$	−0.559079	−0.0195837
$816$	14.0941	0.493392
$817$	21.3744	0.747794
$818$	5.89965	0.206277
$819$	2.33076	0.0814434
$820$	−6.13080	−0.214097
$821$	−31.9957	−1.11666	−0.558328	−	0.829620i	$-0.688557\pi$
$821$	−31.9957	−1.11666	−0.558328	+	0.829620i	$0.688557\pi$
$822$	8.26268	0.288194
$823$	−28.1400	−0.980898	−0.490449	−	0.871470i	$-0.663167\pi$
$823$	−28.1400	−0.980898	−0.490449	+	0.871470i	$0.663167\pi$
$824$	21.7334	0.757119
$825$	23.1574	0.806237
$826$	10.9331	0.380410
$827$	12.4134	0.431656	0.215828	−	0.976431i	$-0.430755\pi$
$827$	12.4134	0.431656	0.215828	+	0.976431i	$0.430755\pi$
$828$	1.85053	0.0643103
$829$	16.7653	0.582284	0.291142	−	0.956680i	$-0.405965\pi$
$829$	16.7653	0.582284	0.291142	+	0.956680i	$0.405965\pi$
$830$	1.08207	0.0375590
$831$	−5.69568	−0.197581
$832$	4.19974	0.145600
$833$	−1.75686	−0.0608714
$834$	−6.45484	−0.223513
$835$	2.25895	0.0781743
$836$	33.3845	1.15463
$837$	−0.585232	−0.0202286
$838$	0.586073	0.0202455
$839$	8.12999	0.280679	0.140339	−	0.990103i	$-0.455181\pi$
$839$	8.12999	0.280679	0.140339	+	0.990103i	$0.455181\pi$
$840$	−1.10203	−0.0380237
$841$	1.00000	0.0344828
$842$	11.9808	0.412885
$843$	6.76690	0.233064
$844$	13.6041	0.468271
$845$	−3.50641	−0.120624
$846$	0.317311	0.0109094
$847$	−28.7446	−0.987675
$848$	20.8675	0.716594
$849$	12.1255	0.416147
$850$	8.57247	0.294033
$851$	3.38692	0.116102
$852$	−25.8577	−0.885872
$853$	1.26296	0.0432428	0.0216214	−	0.999766i	$-0.493117\pi$
$853$	1.26296	0.0432428	0.0216214	+	0.999766i	$0.493117\pi$
$854$	8.28923	0.283651
$855$	1.10293	0.0377194
$856$	−11.6778	−0.399138
$857$	−0.254072	−0.00867895	−0.00433947	−	0.999991i	$-0.501381\pi$
$857$	−0.254072	−0.00867895	−0.00433947	+	0.999991i	$0.501381\pi$
$858$	−1.65061	−0.0563510
$859$	41.0688	1.40125	0.700625	−	0.713530i	$-0.252905\pi$
$859$	41.0688	1.40125	0.700625	+	0.713530i	$0.252905\pi$
$860$	−2.97304	−0.101380
$861$	−29.5839	−1.00822
$862$	−1.71095	−0.0582751
$863$	−14.8984	−0.507147	−0.253574	−	0.967316i	$-0.581606\pi$
$863$	−14.8984	−0.507147	−0.253574	+	0.967316i	$0.581606\pi$
$864$	−4.18573	−0.142401
$865$	0.898199	0.0305397
$866$	0.679082	0.0230762
$867$	3.33450	0.113246
$868$	−2.78444	−0.0945099
$869$	−40.3412	−1.36848
$870$	0.111317	0.00377398
$871$	−13.8326	−0.468701
$872$	−8.37479	−0.283606
$873$	−2.21161	−0.0748518
$874$	1.48098	0.0500948
$875$	7.34138	0.248184
$876$	14.9761	0.505995
$877$	−4.72399	−0.159518	−0.0797589	−	0.996814i	$-0.525415\pi$
$877$	−4.72399	−0.159518	−0.0797589	+	0.996814i	$0.525415\pi$
$878$	−4.13011	−0.139384
$879$	1.15709	0.0390276
$880$	−4.23819	−0.142869
$881$	4.39995	0.148238	0.0741190	−	0.997249i	$-0.476386\pi$
$881$	4.39995	0.148238	0.0741190	+	0.997249i	$0.476386\pi$
$882$	0.150626	0.00507184
$883$	7.42849	0.249988	0.124994	−	0.992157i	$-0.460109\pi$
$883$	7.42849	0.249988	0.124994	+	0.992157i	$0.460109\pi$
$884$	7.56478	0.254431
$885$	3.16685	0.106452
$886$	7.15688	0.240440
$887$	8.17408	0.274459	0.137229	−	0.990539i	$-0.456180\pi$
$887$	8.17408	0.274459	0.137229	+	0.990539i	$0.456180\pi$
$888$	−5.04203	−0.169199
$889$	−48.9474	−1.64164
$890$	1.67686	0.0562086
$891$	−4.70956	−0.157776
$892$	−31.9469	−1.06966
$893$	−3.14393	−0.105208
$894$	0.0692939	0.00231754
$895$	−6.18276	−0.206667
$896$	−26.1286	−0.872896
$897$	0.906534	0.0302683
$898$	−2.43620	−0.0812971
$899$	0.585232	0.0195186
$900$	9.09923	0.303308
$901$	30.1070	1.00301
$902$	20.9509	0.697589
$903$	−14.3463	−0.477414
$904$	16.5524	0.550524
$905$	4.05273	0.134717
$906$	0.532304	0.0176846
$907$	8.71889	0.289506	0.144753	−	0.989468i	$-0.453761\pi$
$907$	8.71889	0.289506	0.144753	+	0.989468i	$0.453761\pi$
$908$	−45.2350	−1.50118
$909$	1.72771	0.0573047
$910$	−0.259452	−0.00860077
$911$	29.8552	0.989146	0.494573	−	0.869136i	$-0.335324\pi$
$911$	29.8552	0.989146	0.494573	+	0.869136i	$0.335324\pi$
$912$	11.9726	0.396452
$913$	45.7799	1.51509
$914$	−8.18852	−0.270852
$915$	2.40104	0.0793759
$916$	−12.1100	−0.400125
$917$	−7.73793	−0.255529
$918$	−1.74340	−0.0575408
$919$	49.5220	1.63358	0.816791	−	0.576934i	$-0.195751\pi$
$919$	49.5220	1.63358	0.816791	+	0.576934i	$0.195751\pi$
$920$	−0.428627	−0.0141314
$921$	−2.29266	−0.0755458
$922$	−0.569436	−0.0187534
$923$	−12.6672	−0.416945
$924$	−22.4073	−0.737147
$925$	16.6538	0.547574
$926$	−10.5679	−0.347283
$927$	14.5991	0.479498
$928$	4.18573	0.137403
$929$	27.6463	0.907045	0.453523	−	0.891245i	$-0.350167\pi$
$929$	27.6463	0.907045	0.453523	+	0.891245i	$0.350167\pi$
$930$	0.0651460	0.00213622
$931$	−1.49241	−0.0489117
$932$	48.2991	1.58209
$933$	−15.1271	−0.495241
$934$	16.2251	0.530901
$935$	−6.11472	−0.199973
$936$	−1.34954	−0.0441110
$937$	58.7856	1.92044	0.960220	−	0.279244i	$-0.0900838\pi$
$937$	58.7856	1.92044	0.960220	+	0.279244i	$0.0900838\pi$
$938$	15.1675	0.495238
$939$	6.96731	0.227370
$940$	0.437300	0.0142632
$941$	−37.0088	−1.20645	−0.603226	−	0.797570i	$-0.706118\pi$
$941$	−37.0088	−1.20645	−0.603226	+	0.797570i	$0.706118\pi$
$942$	−6.81580	−0.222071
$943$	−11.5065	−0.374702
$944$	34.3770	1.11888
$945$	−0.740275	−0.0240812
$946$	10.1598	0.330324
$947$	−38.6118	−1.25472	−0.627358	−	0.778731i	$-0.715864\pi$
$947$	−38.6118	−1.25472	−0.627358	+	0.778731i	$0.715864\pi$
$948$	−15.8512	−0.514824
$949$	7.33647	0.238152
$950$	7.28211	0.236263
$951$	−18.7165	−0.606923
$952$	−17.2596	−0.559387
$953$	−24.2350	−0.785049	−0.392524	−	0.919742i	$-0.628398\pi$
$953$	−24.2350	−0.785049	−0.392524	+	0.919742i	$0.628398\pi$
$954$	−2.58125	−0.0835711
$955$	1.19907	0.0388008
$956$	13.4176	0.433956
$957$	4.70956	0.152239
$958$	6.78157	0.219102
$959$	54.9483	1.77437
$960$	−1.33388	−0.0430509
$961$	−30.6575	−0.988952
$962$	−1.18705	−0.0382720
$963$	−7.84440	−0.252782
$964$	42.5508	1.37047
$965$	−1.34485	−0.0432921
$966$	−0.994018	−0.0319820
$967$	−20.4310	−0.657018	−0.328509	−	0.944501i	$-0.606546\pi$
$967$	−20.4310	−0.657018	−0.328509	+	0.944501i	$0.606546\pi$
$968$	16.6434	0.534940
$969$	17.2737	0.554910
$970$	0.246189	0.00790466
$971$	4.62601	0.148456	0.0742278	−	0.997241i	$-0.476351\pi$
$971$	4.62601	0.148456	0.0742278	+	0.997241i	$0.476351\pi$
$972$	−1.85053	−0.0593557
$973$	−42.9259	−1.37614
$974$	5.64025	0.180725
$975$	4.45752	0.142755
$976$	26.0640	0.834287
$977$	−27.8443	−0.890819	−0.445409	−	0.895327i	$-0.646942\pi$
$977$	−27.8443	−0.890819	−0.445409	+	0.895327i	$0.646942\pi$
$978$	0.750714	0.0240052
$979$	70.9445	2.26740
$980$	0.207584	0.00663103
$981$	−5.62566	−0.179613
$982$	−2.08150	−0.0664232
$983$	33.0220	1.05324	0.526619	−	0.850102i	$-0.323460\pi$
$983$	33.0220	1.05324	0.526619	+	0.850102i	$0.323460\pi$
$984$	17.1294	0.546066
$985$	3.75519	0.119650
$986$	1.74340	0.0555212
$987$	2.11018	0.0671676
$988$	6.42610	0.204442
$989$	−5.57988	−0.177430
$990$	0.524253	0.0166618
$991$	−32.3931	−1.02900	−0.514500	−	0.857490i	$-0.672022\pi$
$991$	−32.3931	−1.02900	−0.514500	+	0.857490i	$0.672022\pi$
$992$	2.44962	0.0777755
$993$	9.50628	0.301673
$994$	13.8896	0.440551
$995$	6.69000	0.212087
$996$	17.9883	0.569980
$997$	−61.5979	−1.95083	−0.975413	−	0.220386i	$-0.929268\pi$
$997$	−61.5979	−1.95083	−0.975413	+	0.220386i	$0.929268\pi$
$998$	−1.64379	−0.0520333
$999$	−3.38692	−0.107157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.n.1.8	✓	16
3.2	odd	2		6003.2.a.r.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.8	✓	16	1.1	even	1	trivial
6003.2.a.r.1.9		16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.386616 q^{2} +1.00000 q^{3} -1.85053 q^{4} +0.287925 q^{5} -0.386616 q^{6} -2.57107 q^{7} +1.48868 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.386616 q^{2} +1.00000 q^{3} -1.85053 q^{4} +0.287925 q^{5} -0.386616 q^{6} -2.57107 q^{7} +1.48868 q^{8} +1.00000 q^{9} -0.111317 q^{10} -4.70956 q^{11} -1.85053 q^{12} -0.906534 q^{13} +0.994018 q^{14} +0.287925 q^{15} +3.12551 q^{16} +4.50938 q^{17} -0.386616 q^{18} +3.83061 q^{19} -0.532813 q^{20} -2.57107 q^{21} +1.82080 q^{22} -1.00000 q^{23} +1.48868 q^{24} -4.91710 q^{25} +0.350481 q^{26} +1.00000 q^{27} +4.75784 q^{28} -1.00000 q^{29} -0.111317 q^{30} -0.585232 q^{31} -4.18573 q^{32} -4.70956 q^{33} -1.74340 q^{34} -0.740275 q^{35} -1.85053 q^{36} -3.38692 q^{37} -1.48098 q^{38} -0.906534 q^{39} +0.428627 q^{40} +11.5065 q^{41} +0.994018 q^{42} +5.57988 q^{43} +8.71518 q^{44} +0.287925 q^{45} +0.386616 q^{46} -0.820738 q^{47} +3.12551 q^{48} -0.389600 q^{49} +1.90103 q^{50} +4.50938 q^{51} +1.67757 q^{52} +6.67652 q^{53} -0.386616 q^{54} -1.35600 q^{55} -3.82749 q^{56} +3.83061 q^{57} +0.386616 q^{58} +10.9989 q^{59} -0.532813 q^{60} +8.33911 q^{61} +0.226260 q^{62} -2.57107 q^{63} -4.63275 q^{64} -0.261014 q^{65} +1.82080 q^{66} +15.2588 q^{67} -8.34473 q^{68} -1.00000 q^{69} +0.286203 q^{70} +13.9732 q^{71} +1.48868 q^{72} -8.09288 q^{73} +1.30944 q^{74} -4.91710 q^{75} -7.08865 q^{76} +12.1086 q^{77} +0.350481 q^{78} +8.56579 q^{79} +0.899912 q^{80} +1.00000 q^{81} -4.44859 q^{82} -9.72061 q^{83} +4.75784 q^{84} +1.29836 q^{85} -2.15728 q^{86} -1.00000 q^{87} -7.01102 q^{88} -15.0639 q^{89} -0.111317 q^{90} +2.33076 q^{91} +1.85053 q^{92} -0.585232 q^{93} +0.317311 q^{94} +1.10293 q^{95} -4.18573 q^{96} -2.21161 q^{97} +0.150626 q^{98} -4.70956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9	\( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9 + 11 * q^10 + 8 * q^11 + 25 * q^12 + 19 * q^13 + 16 * q^14 + 3 * q^15 + 31 * q^16 - 4 * q^17 - q^18 + 19 * q^19 + 16 * q^20 + 13 * q^21 + 6 * q^22 - 16 * q^23 + 23 * q^25 - 15 * q^26 + 16 * q^27 + 18 * q^28 - 16 * q^29 + 11 * q^30 + 24 * q^31 - 21 * q^32 + 8 * q^33 - 9 * q^34 - 13 * q^35 + 25 * q^36 + 26 * q^37 + 19 * q^39 - 22 * q^40 - 15 * q^41 + 16 * q^42 + 33 * q^43 + 6 * q^44 + 3 * q^45 + q^46 + 13 * q^47 + 31 * q^48 + 41 * q^49 + 13 * q^50 - 4 * q^51 - 26 * q^52 + 5 * q^53 - q^54 + 9 * q^55 + 40 * q^56 + 19 * q^57 + q^58 + 2 * q^59 + 16 * q^60 + 29 * q^61 - 32 * q^62 + 13 * q^63 + 28 * q^64 + 18 * q^65 + 6 * q^66 + 32 * q^67 - 26 * q^68 - 16 * q^69 + 18 * q^70 + 29 * q^71 + 19 * q^73 - 16 * q^74 + 23 * q^75 + 64 * q^76 - 21 * q^77 - 15 * q^78 + 56 * q^79 + 16 * q^81 + 14 * q^82 + 5 * q^83 + 18 * q^84 + 16 * q^85 - 20 * q^86 - 16 * q^87 + q^88 + 7 * q^89 + 11 * q^90 - 6 * q^91 - 25 * q^92 + 24 * q^93 - 11 * q^94 + 39 * q^95 - 21 * q^96 + 35 * q^97 - 109 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more