LMFDB - Embedded newform 6042.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + \cdots + 1496 $ x^12 - 3x^11 - 36x^10 + 105x^9 + 465x^8 - 1287x^7 - 2668x^6 + 6443x^5 + 7140x^4 - 10858x^3 - 10086x^2 + 2072*x + 1496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.73947$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73947 q^{5} +1.00000 q^{6} +3.23526 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73947 q^{5} +1.00000 q^{6} +3.23526 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.73947 q^{10} +4.83000 q^{11} -1.00000 q^{12} +2.97959 q^{13} -3.23526 q^{14} +2.73947 q^{15} +1.00000 q^{16} -4.98645 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.73947 q^{20} -3.23526 q^{21} -4.83000 q^{22} -4.19849 q^{23} +1.00000 q^{24} +2.50467 q^{25} -2.97959 q^{26} -1.00000 q^{27} +3.23526 q^{28} +3.90827 q^{29} -2.73947 q^{30} +0.453059 q^{31} -1.00000 q^{32} -4.83000 q^{33} +4.98645 q^{34} -8.86290 q^{35} +1.00000 q^{36} -4.36263 q^{37} +1.00000 q^{38} -2.97959 q^{39} +2.73947 q^{40} -9.85613 q^{41} +3.23526 q^{42} +9.43566 q^{43} +4.83000 q^{44} -2.73947 q^{45} +4.19849 q^{46} +1.30429 q^{47} -1.00000 q^{48} +3.46693 q^{49} -2.50467 q^{50} +4.98645 q^{51} +2.97959 q^{52} -1.00000 q^{53} +1.00000 q^{54} -13.2316 q^{55} -3.23526 q^{56} +1.00000 q^{57} -3.90827 q^{58} -11.0725 q^{59} +2.73947 q^{60} -13.2395 q^{61} -0.453059 q^{62} +3.23526 q^{63} +1.00000 q^{64} -8.16248 q^{65} +4.83000 q^{66} +7.76940 q^{67} -4.98645 q^{68} +4.19849 q^{69} +8.86290 q^{70} +3.85139 q^{71} -1.00000 q^{72} -8.13755 q^{73} +4.36263 q^{74} -2.50467 q^{75} -1.00000 q^{76} +15.6263 q^{77} +2.97959 q^{78} -6.64258 q^{79} -2.73947 q^{80} +1.00000 q^{81} +9.85613 q^{82} -14.2520 q^{83} -3.23526 q^{84} +13.6602 q^{85} -9.43566 q^{86} -3.90827 q^{87} -4.83000 q^{88} -0.669080 q^{89} +2.73947 q^{90} +9.63976 q^{91} -4.19849 q^{92} -0.453059 q^{93} -1.30429 q^{94} +2.73947 q^{95} +1.00000 q^{96} -6.36404 q^{97} -3.46693 q^{98} +4.83000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 7 * q^11 - 12 * q^12 + 4 * q^13 + q^14 + 3 * q^15 + 12 * q^16 - 4 * q^17 - 12 * q^18 - 12 * q^19 - 3 * q^20 + q^21 + 7 * q^22 - 18 * q^23 + 12 * q^24 + 21 * q^25 - 4 * q^26 - 12 * q^27 - q^28 + 10 * q^29 - 3 * q^30 + 14 * q^31 - 12 * q^32 + 7 * q^33 + 4 * q^34 - 19 * q^35 + 12 * q^36 + 14 * q^37 + 12 * q^38 - 4 * q^39 + 3 * q^40 - 9 * q^41 - q^42 - 4 * q^43 - 7 * q^44 - 3 * q^45 + 18 * q^46 - 14 * q^47 - 12 * q^48 + 37 * q^49 - 21 * q^50 + 4 * q^51 + 4 * q^52 - 12 * q^53 + 12 * q^54 - 4 * q^55 + q^56 + 12 * q^57 - 10 * q^58 - 18 * q^59 + 3 * q^60 - 7 * q^61 - 14 * q^62 - q^63 + 12 * q^64 + 3 * q^65 - 7 * q^66 + 8 * q^67 - 4 * q^68 + 18 * q^69 + 19 * q^70 - q^71 - 12 * q^72 - 31 * q^73 - 14 * q^74 - 21 * q^75 - 12 * q^76 - 25 * q^77 + 4 * q^78 - 3 * q^80 + 12 * q^81 + 9 * q^82 - 48 * q^83 + q^84 + 4 * q^85 + 4 * q^86 - 10 * q^87 + 7 * q^88 + 13 * q^89 + 3 * q^90 + 9 * q^91 - 18 * q^92 - 14 * q^93 + 14 * q^94 + 3 * q^95 + 12 * q^96 + 25 * q^97 - 37 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.73947	−1.22513	−0.612563	−	0.790422i	$-0.709862\pi$
$5$	−2.73947	−1.22513	−0.612563	+	0.790422i	$0.709862\pi$
$6$	1.00000	0.408248
$7$	3.23526	1.22281	0.611407	−	0.791316i	$-0.290604\pi$
$7$	3.23526	1.22281	0.611407	+	0.791316i	$0.290604\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.73947	0.866295
$11$	4.83000	1.45630	0.728149	−	0.685419i	$-0.240381\pi$
$11$	4.83000	1.45630	0.728149	+	0.685419i	$0.240381\pi$
$12$	−1.00000	−0.288675
$13$	2.97959	0.826389	0.413195	−	0.910643i	$-0.364413\pi$
$13$	2.97959	0.826389	0.413195	+	0.910643i	$0.364413\pi$
$14$	−3.23526	−0.864661
$15$	2.73947	0.707327
$16$	1.00000	0.250000
$17$	−4.98645	−1.20939	−0.604696	−	0.796456i	$-0.706705\pi$
$17$	−4.98645	−1.20939	−0.604696	+	0.796456i	$0.706705\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.73947	−0.612563
$21$	−3.23526	−0.705993
$22$	−4.83000	−1.02976
$23$	−4.19849	−0.875445	−0.437723	−	0.899110i	$-0.644215\pi$
$23$	−4.19849	−0.875445	−0.437723	+	0.899110i	$0.644215\pi$
$24$	1.00000	0.204124
$25$	2.50467	0.500935
$26$	−2.97959	−0.584345
$27$	−1.00000	−0.192450
$28$	3.23526	0.611407
$29$	3.90827	0.725748	0.362874	−	0.931838i	$-0.381795\pi$
$29$	3.90827	0.725748	0.362874	+	0.931838i	$0.381795\pi$
$30$	−2.73947	−0.500156
$31$	0.453059	0.0813717	0.0406859	−	0.999172i	$-0.487046\pi$
$31$	0.453059	0.0813717	0.0406859	+	0.999172i	$0.487046\pi$
$32$	−1.00000	−0.176777
$33$	−4.83000	−0.840794
$34$	4.98645	0.855169
$35$	−8.86290	−1.49810
$36$	1.00000	0.166667
$37$	−4.36263	−0.717212	−0.358606	−	0.933489i	$-0.616748\pi$
$37$	−4.36263	−0.717212	−0.358606	+	0.933489i	$0.616748\pi$
$38$	1.00000	0.162221
$39$	−2.97959	−0.477116
$40$	2.73947	0.433148
$41$	−9.85613	−1.53927	−0.769634	−	0.638485i	$-0.779561\pi$
$41$	−9.85613	−1.53927	−0.769634	+	0.638485i	$0.779561\pi$
$42$	3.23526	0.499212
$43$	9.43566	1.43893	0.719463	−	0.694531i	$-0.244388\pi$
$43$	9.43566	1.43893	0.719463	+	0.694531i	$0.244388\pi$
$44$	4.83000	0.728149
$45$	−2.73947	−0.408375
$46$	4.19849	0.619033
$47$	1.30429	0.190250	0.0951250	−	0.995465i	$-0.469675\pi$
$47$	1.30429	0.190250	0.0951250	+	0.995465i	$0.469675\pi$
$48$	−1.00000	−0.144338
$49$	3.46693	0.495276
$50$	−2.50467	−0.354214
$51$	4.98645	0.698243
$52$	2.97959	0.413195
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	−13.2316	−1.78415
$56$	−3.23526	−0.432330
$57$	1.00000	0.132453
$58$	−3.90827	−0.513182
$59$	−11.0725	−1.44151	−0.720757	−	0.693188i	$-0.756205\pi$
$59$	−11.0725	−1.44151	−0.720757	+	0.693188i	$0.756205\pi$
$60$	2.73947	0.353664
$61$	−13.2395	−1.69515	−0.847573	−	0.530679i	$-0.821937\pi$
$61$	−13.2395	−1.69515	−0.847573	+	0.530679i	$0.821937\pi$
$62$	−0.453059	−0.0575385
$63$	3.23526	0.407605
$64$	1.00000	0.125000
$65$	−8.16248	−1.01243
$66$	4.83000	0.594531
$67$	7.76940	0.949184	0.474592	−	0.880206i	$-0.342596\pi$
$67$	7.76940	0.949184	0.474592	+	0.880206i	$0.342596\pi$
$68$	−4.98645	−0.604696
$69$	4.19849	0.505439
$70$	8.86290	1.05932
$71$	3.85139	0.457076	0.228538	−	0.973535i	$-0.426606\pi$
$71$	3.85139	0.457076	0.228538	+	0.973535i	$0.426606\pi$
$72$	−1.00000	−0.117851
$73$	−8.13755	−0.952428	−0.476214	−	0.879329i	$-0.657991\pi$
$73$	−8.13755	−0.952428	−0.476214	+	0.879329i	$0.657991\pi$
$74$	4.36263	0.507146
$75$	−2.50467	−0.289215
$76$	−1.00000	−0.114708
$77$	15.6263	1.78078
$78$	2.97959	0.337372
$79$	−6.64258	−0.747349	−0.373674	−	0.927560i	$-0.621902\pi$
$79$	−6.64258	−0.747349	−0.373674	+	0.927560i	$0.621902\pi$
$80$	−2.73947	−0.306282
$81$	1.00000	0.111111
$82$	9.85613	1.08843
$83$	−14.2520	−1.56436	−0.782181	−	0.623051i	$-0.785893\pi$
$83$	−14.2520	−1.56436	−0.782181	+	0.623051i	$0.785893\pi$
$84$	−3.23526	−0.352996
$85$	13.6602	1.48166
$86$	−9.43566	−1.01747
$87$	−3.90827	−0.419011
$88$	−4.83000	−0.514879
$89$	−0.669080	−0.0709223	−0.0354612	−	0.999371i	$-0.511290\pi$
$89$	−0.669080	−0.0709223	−0.0354612	+	0.999371i	$0.511290\pi$
$90$	2.73947	0.288765
$91$	9.63976	1.01052
$92$	−4.19849	−0.437723
$93$	−0.453059	−0.0469800
$94$	−1.30429	−0.134527
$95$	2.73947	0.281063
$96$	1.00000	0.102062
$97$	−6.36404	−0.646170	−0.323085	−	0.946370i	$-0.604720\pi$
$97$	−6.36404	−0.646170	−0.323085	+	0.946370i	$0.604720\pi$
$98$	−3.46693	−0.350213
$99$	4.83000	0.485433
$100$	2.50467	0.250467
$101$	−14.3374	−1.42663	−0.713313	−	0.700846i	$-0.752806\pi$
$101$	−14.3374	−1.42663	−0.713313	+	0.700846i	$0.752806\pi$
$102$	−4.98645	−0.493732
$103$	−7.41212	−0.730338	−0.365169	−	0.930941i	$-0.618989\pi$
$103$	−7.41212	−0.730338	−0.365169	+	0.930941i	$0.618989\pi$
$104$	−2.97959	−0.292173
$105$	8.86290	0.864930
$106$	1.00000	0.0971286
$107$	12.2910	1.18822	0.594109	−	0.804385i	$-0.297505\pi$
$107$	12.2910	1.18822	0.594109	+	0.804385i	$0.297505\pi$
$108$	−1.00000	−0.0962250
$109$	15.9702	1.52967	0.764834	−	0.644227i	$-0.222821\pi$
$109$	15.9702	1.52967	0.764834	+	0.644227i	$0.222821\pi$
$110$	13.2316	1.26158
$111$	4.36263	0.414083
$112$	3.23526	0.305704
$113$	4.12317	0.387876	0.193938	−	0.981014i	$-0.437874\pi$
$113$	4.12317	0.387876	0.193938	+	0.981014i	$0.437874\pi$
$114$	−1.00000	−0.0936586
$115$	11.5016	1.07253
$116$	3.90827	0.362874
$117$	2.97959	0.275463
$118$	11.0725	1.01930
$119$	−16.1325	−1.47886
$120$	−2.73947	−0.250078
$121$	12.3289	1.12080
$122$	13.2395	1.19865
$123$	9.85613	0.888697
$124$	0.453059	0.0406859
$125$	6.83586	0.611418
$126$	−3.23526	−0.288220
$127$	1.57229	0.139518	0.0697592	−	0.997564i	$-0.477777\pi$
$127$	1.57229	0.139518	0.0697592	+	0.997564i	$0.477777\pi$
$128$	−1.00000	−0.0883883
$129$	−9.43566	−0.830764
$130$	8.16248	0.715897
$131$	18.4843	1.61498	0.807491	−	0.589880i	$-0.200825\pi$
$131$	18.4843	1.61498	0.807491	+	0.589880i	$0.200825\pi$
$132$	−4.83000	−0.420397
$133$	−3.23526	−0.280533
$134$	−7.76940	−0.671174
$135$	2.73947	0.235776
$136$	4.98645	0.427585
$137$	−1.25735	−0.107423	−0.0537113	−	0.998557i	$-0.517105\pi$
$137$	−1.25735	−0.107423	−0.0537113	+	0.998557i	$0.517105\pi$
$138$	−4.19849	−0.357399
$139$	13.5380	1.14828	0.574139	−	0.818758i	$-0.305337\pi$
$139$	13.5380	1.14828	0.574139	+	0.818758i	$0.305337\pi$
$140$	−8.86290	−0.749051
$141$	−1.30429	−0.109841
$142$	−3.85139	−0.323201
$143$	14.3914	1.20347
$144$	1.00000	0.0833333
$145$	−10.7066	−0.889133
$146$	8.13755	0.673468
$147$	−3.46693	−0.285948
$148$	−4.36263	−0.358606
$149$	−8.21091	−0.672664	−0.336332	−	0.941743i	$-0.609186\pi$
$149$	−8.21091	−0.672664	−0.336332	+	0.941743i	$0.609186\pi$
$150$	2.50467	0.204506
$151$	21.2332	1.72793	0.863966	−	0.503551i	$-0.167973\pi$
$151$	21.2332	1.72793	0.863966	+	0.503551i	$0.167973\pi$
$152$	1.00000	0.0811107
$153$	−4.98645	−0.403131
$154$	−15.6263	−1.25920
$155$	−1.24114	−0.0996906
$156$	−2.97959	−0.238558
$157$	−20.3747	−1.62608	−0.813039	−	0.582210i	$-0.802188\pi$
$157$	−20.3747	−1.62608	−0.813039	+	0.582210i	$0.802188\pi$
$158$	6.64258	0.528455
$159$	1.00000	0.0793052
$160$	2.73947	0.216574
$161$	−13.5832	−1.07051
$162$	−1.00000	−0.0785674
$163$	−1.07494	−0.0841957	−0.0420979	−	0.999113i	$-0.513404\pi$
$163$	−1.07494	−0.0841957	−0.0420979	+	0.999113i	$0.513404\pi$
$164$	−9.85613	−0.769634
$165$	13.2316	1.03008
$166$	14.2520	1.10617
$167$	16.6424	1.28783	0.643914	−	0.765098i	$-0.277310\pi$
$167$	16.6424	1.28783	0.643914	+	0.765098i	$0.277310\pi$
$168$	3.23526	0.249606
$169$	−4.12205	−0.317081
$170$	−13.6602	−1.04769
$171$	−1.00000	−0.0764719
$172$	9.43566	0.719463
$173$	−19.2757	−1.46550	−0.732752	−	0.680496i	$-0.761764\pi$
$173$	−19.2757	−1.46550	−0.732752	+	0.680496i	$0.761764\pi$
$174$	3.90827	0.296286
$175$	8.10328	0.612550
$176$	4.83000	0.364075
$177$	11.0725	0.832258
$178$	0.669080	0.0501496
$179$	12.4357	0.929486	0.464743	−	0.885446i	$-0.346147\pi$
$179$	12.4357	0.929486	0.464743	+	0.885446i	$0.346147\pi$
$180$	−2.73947	−0.204188
$181$	16.0228	1.19097	0.595484	−	0.803367i	$-0.296960\pi$
$181$	16.0228	1.19097	0.595484	+	0.803367i	$0.296960\pi$
$182$	−9.63976	−0.714546
$183$	13.2395	0.978693
$184$	4.19849	0.309517
$185$	11.9513	0.878676
$186$	0.453059	0.0332199
$187$	−24.0845	−1.76124
$188$	1.30429	0.0951250
$189$	−3.23526	−0.235331
$190$	−2.73947	−0.198742
$191$	−7.67553	−0.555382	−0.277691	−	0.960671i	$-0.589569\pi$
$191$	−7.67553	−0.555382	−0.277691	+	0.960671i	$0.589569\pi$
$192$	−1.00000	−0.0721688
$193$	−4.15995	−0.299440	−0.149720	−	0.988728i	$-0.547837\pi$
$193$	−4.15995	−0.299440	−0.149720	+	0.988728i	$0.547837\pi$
$194$	6.36404	0.456911
$195$	8.16248	0.584527
$196$	3.46693	0.247638
$197$	2.45480	0.174897	0.0874486	−	0.996169i	$-0.472129\pi$
$197$	2.45480	0.174897	0.0874486	+	0.996169i	$0.472129\pi$
$198$	−4.83000	−0.343253
$199$	3.02649	0.214542	0.107271	−	0.994230i	$-0.465789\pi$
$199$	3.02649	0.214542	0.107271	+	0.994230i	$0.465789\pi$
$200$	−2.50467	−0.177107
$201$	−7.76940	−0.548011
$202$	14.3374	1.00878
$203$	12.6443	0.887456
$204$	4.98645	0.349121
$205$	27.0005	1.88580
$206$	7.41212	0.516427
$207$	−4.19849	−0.291815
$208$	2.97959	0.206597
$209$	−4.83000	−0.334098
$210$	−8.86290	−0.611598
$211$	−10.9141	−0.751359	−0.375680	−	0.926750i	$-0.622591\pi$
$211$	−10.9141	−0.751359	−0.375680	+	0.926750i	$0.622591\pi$
$212$	−1.00000	−0.0686803
$213$	−3.85139	−0.263893
$214$	−12.2910	−0.840197
$215$	−25.8487	−1.76287
$216$	1.00000	0.0680414
$217$	1.46576	0.0995025
$218$	−15.9702	−1.08164
$219$	8.13755	0.549884
$220$	−13.2316	−0.892075
$221$	−14.8576	−0.999428
$222$	−4.36263	−0.292801
$223$	4.05519	0.271556	0.135778	−	0.990739i	$-0.456647\pi$
$223$	4.05519	0.271556	0.135778	+	0.990739i	$0.456647\pi$
$224$	−3.23526	−0.216165
$225$	2.50467	0.166978
$226$	−4.12317	−0.274269
$227$	−28.4295	−1.88693	−0.943466	−	0.331469i	$-0.892456\pi$
$227$	−28.4295	−1.88693	−0.943466	+	0.331469i	$0.892456\pi$
$228$	1.00000	0.0662266
$229$	−5.62659	−0.371815	−0.185908	−	0.982567i	$-0.559523\pi$
$229$	−5.62659	−0.371815	−0.185908	+	0.982567i	$0.559523\pi$
$230$	−11.5016	−0.758394
$231$	−15.6263	−1.02814
$232$	−3.90827	−0.256591
$233$	0.751998	0.0492650	0.0246325	−	0.999697i	$-0.492158\pi$
$233$	0.751998	0.0492650	0.0246325	+	0.999697i	$0.492158\pi$
$234$	−2.97959	−0.194782
$235$	−3.57305	−0.233080
$236$	−11.0725	−0.720757
$237$	6.64258	0.431482
$238$	16.1325	1.04571
$239$	25.0652	1.62133	0.810666	−	0.585509i	$-0.199105\pi$
$239$	25.0652	1.62133	0.810666	+	0.585509i	$0.199105\pi$
$240$	2.73947	0.176832
$241$	1.99807	0.128707	0.0643534	−	0.997927i	$-0.479502\pi$
$241$	1.99807	0.128707	0.0643534	+	0.997927i	$0.479502\pi$
$242$	−12.3289	−0.792529
$243$	−1.00000	−0.0641500
$244$	−13.2395	−0.847573
$245$	−9.49755	−0.606776
$246$	−9.85613	−0.628404
$247$	−2.97959	−0.189587
$248$	−0.453059	−0.0287692
$249$	14.2520	0.903185
$250$	−6.83586	−0.432338
$251$	0.357960	0.0225942	0.0112971	−	0.999936i	$-0.496404\pi$
$251$	0.357960	0.0225942	0.0112971	+	0.999936i	$0.496404\pi$
$252$	3.23526	0.203802
$253$	−20.2787	−1.27491
$254$	−1.57229	−0.0986544
$255$	−13.6602	−0.855435
$256$	1.00000	0.0625000
$257$	−13.6812	−0.853408	−0.426704	−	0.904391i	$-0.640325\pi$
$257$	−13.6812	−0.853408	−0.426704	+	0.904391i	$0.640325\pi$
$258$	9.43566	0.587439
$259$	−14.1143	−0.877018
$260$	−8.16248	−0.506216
$261$	3.90827	0.241916
$262$	−18.4843	−1.14196
$263$	−20.8869	−1.28794	−0.643970	−	0.765050i	$-0.722714\pi$
$263$	−20.8869	−1.28794	−0.643970	+	0.765050i	$0.722714\pi$
$264$	4.83000	0.297266
$265$	2.73947	0.168284
$266$	3.23526	0.198367
$267$	0.669080	0.0409470
$268$	7.76940	0.474592
$269$	−3.09618	−0.188777	−0.0943886	−	0.995535i	$-0.530090\pi$
$269$	−3.09618	−0.188777	−0.0943886	+	0.995535i	$0.530090\pi$
$270$	−2.73947	−0.166719
$271$	6.39364	0.388386	0.194193	−	0.980963i	$-0.437791\pi$
$271$	6.39364	0.388386	0.194193	+	0.980963i	$0.437791\pi$
$272$	−4.98645	−0.302348
$273$	−9.63976	−0.583425
$274$	1.25735	0.0759592
$275$	12.0976	0.729510
$276$	4.19849	0.252719
$277$	−22.5081	−1.35238	−0.676191	−	0.736726i	$-0.736371\pi$
$277$	−22.5081	−1.35238	−0.676191	+	0.736726i	$0.736371\pi$
$278$	−13.5380	−0.811955
$279$	0.453059	0.0271239
$280$	8.86290	0.529659
$281$	−19.0729	−1.13780	−0.568898	−	0.822408i	$-0.692630\pi$
$281$	−19.0729	−1.13780	−0.568898	+	0.822408i	$0.692630\pi$
$282$	1.30429	0.0776692
$283$	13.4804	0.801328	0.400664	−	0.916225i	$-0.368780\pi$
$283$	13.4804	0.801328	0.400664	+	0.916225i	$0.368780\pi$
$284$	3.85139	0.228538
$285$	−2.73947	−0.162272
$286$	−14.3914	−0.850981
$287$	−31.8872	−1.88224
$288$	−1.00000	−0.0589256
$289$	7.86468	0.462628
$290$	10.7066	0.628712
$291$	6.36404	0.373067
$292$	−8.13755	−0.476214
$293$	−19.9454	−1.16522	−0.582610	−	0.812752i	$-0.697969\pi$
$293$	−19.9454	−1.16522	−0.582610	+	0.812752i	$0.697969\pi$
$294$	3.46693	0.202196
$295$	30.3327	1.76604
$296$	4.36263	0.253573
$297$	−4.83000	−0.280265
$298$	8.21091	0.475645
$299$	−12.5098	−0.723459
$300$	−2.50467	−0.144607
$301$	30.5269	1.75954
$302$	−21.2332	−1.22183
$303$	14.3374	0.823662
$304$	−1.00000	−0.0573539
$305$	36.2692	2.07677
$306$	4.98645	0.285056
$307$	−13.3207	−0.760251	−0.380125	−	0.924935i	$-0.624119\pi$
$307$	−13.3207	−0.760251	−0.380125	+	0.924935i	$0.624119\pi$
$308$	15.6263	0.890392
$309$	7.41212	0.421661
$310$	1.24114	0.0704919
$311$	−25.3383	−1.43680	−0.718402	−	0.695629i	$-0.755126\pi$
$311$	−25.3383	−1.43680	−0.718402	+	0.695629i	$0.755126\pi$
$312$	2.97959	0.168686
$313$	13.2581	0.749393	0.374697	−	0.927148i	$-0.377747\pi$
$313$	13.2581	0.749393	0.374697	+	0.927148i	$0.377747\pi$
$314$	20.3747	1.14981
$315$	−8.86290	−0.499368
$316$	−6.64258	−0.373674
$317$	−2.36254	−0.132693	−0.0663467	−	0.997797i	$-0.521134\pi$
$317$	−2.36254	−0.132693	−0.0663467	+	0.997797i	$0.521134\pi$
$318$	−1.00000	−0.0560772
$319$	18.8769	1.05691
$320$	−2.73947	−0.153141
$321$	−12.2910	−0.686018
$322$	13.5832	0.756963
$323$	4.98645	0.277453
$324$	1.00000	0.0555556
$325$	7.46290	0.413967
$326$	1.07494	0.0595354
$327$	−15.9702	−0.883155
$328$	9.85613	0.544214
$329$	4.21972	0.232641
$330$	−13.2316	−0.728376
$331$	−30.7261	−1.68886	−0.844430	−	0.535666i	$-0.820061\pi$
$331$	−30.7261	−1.68886	−0.844430	+	0.535666i	$0.820061\pi$
$332$	−14.2520	−0.782181
$333$	−4.36263	−0.239071
$334$	−16.6424	−0.910631
$335$	−21.2840	−1.16287
$336$	−3.23526	−0.176498
$337$	−23.0972	−1.25819	−0.629094	−	0.777329i	$-0.716574\pi$
$337$	−23.0972	−1.25819	−0.629094	+	0.777329i	$0.716574\pi$
$338$	4.12205	0.224210
$339$	−4.12317	−0.223940
$340$	13.6602	0.740829
$341$	2.18827	0.118501
$342$	1.00000	0.0540738
$343$	−11.4304	−0.617184
$344$	−9.43566	−0.508737
$345$	−11.5016	−0.619226
$346$	19.2757	1.03627
$347$	−2.03583	−0.109289	−0.0546447	−	0.998506i	$-0.517403\pi$
$347$	−2.03583	−0.109289	−0.0546447	+	0.998506i	$0.517403\pi$
$348$	−3.90827	−0.209506
$349$	−27.5033	−1.47222	−0.736108	−	0.676864i	$-0.763339\pi$
$349$	−27.5033	−1.47222	−0.736108	+	0.676864i	$0.763339\pi$
$350$	−8.10328	−0.433139
$351$	−2.97959	−0.159039
$352$	−4.83000	−0.257440
$353$	20.6001	1.09643	0.548216	−	0.836337i	$-0.315307\pi$
$353$	20.6001	1.09643	0.548216	+	0.836337i	$0.315307\pi$
$354$	−11.0725	−0.588495
$355$	−10.5507	−0.559976
$356$	−0.669080	−0.0354612
$357$	16.1325	0.853821
$358$	−12.4357	−0.657246
$359$	−36.4576	−1.92416	−0.962079	−	0.272771i	$-0.912060\pi$
$359$	−36.4576	−1.92416	−0.962079	+	0.272771i	$0.912060\pi$
$360$	2.73947	0.144383
$361$	1.00000	0.0526316
$362$	−16.0228	−0.842142
$363$	−12.3289	−0.647097
$364$	9.63976	0.505260
$365$	22.2925	1.16684
$366$	−13.2395	−0.692040
$367$	−12.4924	−0.652098	−0.326049	−	0.945353i	$-0.605717\pi$
$367$	−12.4924	−0.652098	−0.326049	+	0.945353i	$0.605717\pi$
$368$	−4.19849	−0.218861
$369$	−9.85613	−0.513090
$370$	−11.9513	−0.621317
$371$	−3.23526	−0.167967
$372$	−0.453059	−0.0234900
$373$	−29.0502	−1.50416	−0.752081	−	0.659071i	$-0.770950\pi$
$373$	−29.0502	−1.50416	−0.752081	+	0.659071i	$0.770950\pi$
$374$	24.0845	1.24538
$375$	−6.83586	−0.353002
$376$	−1.30429	−0.0672635
$377$	11.6450	0.599751
$378$	3.23526	0.166404
$379$	22.7932	1.17081	0.585404	−	0.810742i	$-0.300936\pi$
$379$	22.7932	1.17081	0.585404	+	0.810742i	$0.300936\pi$
$380$	2.73947	0.140532
$381$	−1.57229	−0.0805510
$382$	7.67553	0.392714
$383$	17.9430	0.916847	0.458423	−	0.888734i	$-0.348414\pi$
$383$	17.9430	0.916847	0.458423	+	0.888734i	$0.348414\pi$
$384$	1.00000	0.0510310
$385$	−42.8077	−2.18168
$386$	4.15995	0.211736
$387$	9.43566	0.479642
$388$	−6.36404	−0.323085
$389$	−22.9838	−1.16532	−0.582661	−	0.812715i	$-0.697989\pi$
$389$	−22.9838	−1.16532	−0.582661	+	0.812715i	$0.697989\pi$
$390$	−8.16248	−0.413323
$391$	20.9356	1.05876
$392$	−3.46693	−0.175107
$393$	−18.4843	−0.932410
$394$	−2.45480	−0.123671
$395$	18.1971	0.915596
$396$	4.83000	0.242716
$397$	−4.46757	−0.224221	−0.112110	−	0.993696i	$-0.535761\pi$
$397$	−4.46757	−0.224221	−0.112110	+	0.993696i	$0.535761\pi$
$398$	−3.02649	−0.151704
$399$	3.23526	0.161966
$400$	2.50467	0.125234
$401$	34.8245	1.73905	0.869527	−	0.493885i	$-0.164424\pi$
$401$	34.8245	1.73905	0.869527	+	0.493885i	$0.164424\pi$
$402$	7.76940	0.387503
$403$	1.34993	0.0672447
$404$	−14.3374	−0.713313
$405$	−2.73947	−0.136125
$406$	−12.6443	−0.627526
$407$	−21.0715	−1.04447
$408$	−4.98645	−0.246866
$409$	−25.8053	−1.27599	−0.637994	−	0.770041i	$-0.720235\pi$
$409$	−25.8053	−1.27599	−0.637994	+	0.770041i	$0.720235\pi$
$410$	−27.0005	−1.33346
$411$	1.25735	0.0620204
$412$	−7.41212	−0.365169
$413$	−35.8224	−1.76270
$414$	4.19849	0.206344
$415$	39.0429	1.91654
$416$	−2.97959	−0.146086
$417$	−13.5380	−0.662959
$418$	4.83000	0.236243
$419$	−0.343947	−0.0168029	−0.00840144	−	0.999965i	$-0.502674\pi$
$419$	−0.343947	−0.0168029	−0.00840144	+	0.999965i	$0.502674\pi$
$420$	8.86290	0.432465
$421$	−10.0368	−0.489162	−0.244581	−	0.969629i	$-0.578650\pi$
$421$	−10.0368	−0.489162	−0.244581	+	0.969629i	$0.578650\pi$
$422$	10.9141	0.531291
$423$	1.30429	0.0634167
$424$	1.00000	0.0485643
$425$	−12.4894	−0.605826
$426$	3.85139	0.186600
$427$	−42.8333	−2.07285
$428$	12.2910	0.594109
$429$	−14.3914	−0.694823
$430$	25.8487	1.24653
$431$	−7.66750	−0.369331	−0.184665	−	0.982801i	$-0.559120\pi$
$431$	−7.66750	−0.369331	−0.184665	+	0.982801i	$0.559120\pi$
$432$	−1.00000	−0.0481125
$433$	−1.12411	−0.0540214	−0.0270107	−	0.999635i	$-0.508599\pi$
$433$	−1.12411	−0.0540214	−0.0270107	+	0.999635i	$0.508599\pi$
$434$	−1.46576	−0.0703589
$435$	10.7066	0.513341
$436$	15.9702	0.764834
$437$	4.19849	0.200841
$438$	−8.13755	−0.388827
$439$	27.7423	1.32407	0.662035	−	0.749473i	$-0.269693\pi$
$439$	27.7423	1.32407	0.662035	+	0.749473i	$0.269693\pi$
$440$	13.2316	0.630792
$441$	3.46693	0.165092
$442$	14.8576	0.706702
$443$	−21.0876	−1.00190	−0.500951	−	0.865476i	$-0.667016\pi$
$443$	−21.0876	−1.00190	−0.500951	+	0.865476i	$0.667016\pi$
$444$	4.36263	0.207041
$445$	1.83292	0.0868888
$446$	−4.05519	−0.192019
$447$	8.21091	0.388363
$448$	3.23526	0.152852
$449$	4.13304	0.195050	0.0975252	−	0.995233i	$-0.468907\pi$
$449$	4.13304	0.195050	0.0975252	+	0.995233i	$0.468907\pi$
$450$	−2.50467	−0.118071
$451$	−47.6051	−2.24163
$452$	4.12317	0.193938
$453$	−21.2332	−0.997622
$454$	28.4295	1.33426
$455$	−26.4078	−1.23802
$456$	−1.00000	−0.0468293
$457$	−10.2743	−0.480612	−0.240306	−	0.970697i	$-0.577248\pi$
$457$	−10.2743	−0.480612	−0.240306	+	0.970697i	$0.577248\pi$
$458$	5.62659	0.262913
$459$	4.98645	0.232748
$460$	11.5016	0.536266
$461$	13.3545	0.621980	0.310990	−	0.950413i	$-0.399339\pi$
$461$	13.3545	0.621980	0.310990	+	0.950413i	$0.399339\pi$
$462$	15.6263	0.727002
$463$	30.2503	1.40585	0.702925	−	0.711264i	$-0.251877\pi$
$463$	30.2503	1.40585	0.702925	+	0.711264i	$0.251877\pi$
$464$	3.90827	0.181437
$465$	1.24114	0.0575564
$466$	−0.751998	−0.0348356
$467$	−27.9883	−1.29514	−0.647571	−	0.762005i	$-0.724215\pi$
$467$	−27.9883	−1.29514	−0.647571	+	0.762005i	$0.724215\pi$
$468$	2.97959	0.137732
$469$	25.1361	1.16068
$470$	3.57305	0.164813
$471$	20.3747	0.938816
$472$	11.0725	0.509652
$473$	45.5742	2.09550
$474$	−6.64258	−0.305104
$475$	−2.50467	−0.114922
$476$	−16.1325	−0.739431
$477$	−1.00000	−0.0457869
$478$	−25.0652	−1.14645
$479$	40.3402	1.84319	0.921594	−	0.388154i	$-0.126887\pi$
$479$	40.3402	1.84319	0.921594	+	0.388154i	$0.126887\pi$
$480$	−2.73947	−0.125039
$481$	−12.9988	−0.592696
$482$	−1.99807	−0.0910094
$483$	13.5832	0.618058
$484$	12.3289	0.560402
$485$	17.4341	0.791640
$486$	1.00000	0.0453609
$487$	36.7511	1.66535	0.832677	−	0.553759i	$-0.186807\pi$
$487$	36.7511	1.66535	0.832677	+	0.553759i	$0.186807\pi$
$488$	13.2395	0.599324
$489$	1.07494	0.0486104
$490$	9.49755	0.429056
$491$	28.1976	1.27254	0.636271	−	0.771466i	$-0.280476\pi$
$491$	28.1976	1.27254	0.636271	+	0.771466i	$0.280476\pi$
$492$	9.85613	0.444349
$493$	−19.4884	−0.877714
$494$	2.97959	0.134058
$495$	−13.2316	−0.594717
$496$	0.453059	0.0203429
$497$	12.4603	0.558919
$498$	−14.2520	−0.638648
$499$	−6.01554	−0.269293	−0.134646	−	0.990894i	$-0.542990\pi$
$499$	−6.01554	−0.269293	−0.134646	+	0.990894i	$0.542990\pi$
$500$	6.83586	0.305709
$501$	−16.6424	−0.743527
$502$	−0.357960	−0.0159765
$503$	−5.34487	−0.238316	−0.119158	−	0.992875i	$-0.538019\pi$
$503$	−5.34487	−0.238316	−0.119158	+	0.992875i	$0.538019\pi$
$504$	−3.23526	−0.144110
$505$	39.2768	1.74780
$506$	20.2787	0.901497
$507$	4.12205	0.183067
$508$	1.57229	0.0697592
$509$	17.3495	0.769003	0.384501	−	0.923124i	$-0.374373\pi$
$509$	17.3495	0.769003	0.384501	+	0.923124i	$0.374373\pi$
$510$	13.6602	0.604884
$511$	−26.3271	−1.16464
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	13.6812	0.603450
$515$	20.3053	0.894756
$516$	−9.43566	−0.415382
$517$	6.29971	0.277061
$518$	14.1143	0.620145
$519$	19.2757	0.846109
$520$	8.16248	0.357948
$521$	−17.6919	−0.775094	−0.387547	−	0.921850i	$-0.626678\pi$
$521$	−17.6919	−0.775094	−0.387547	+	0.921850i	$0.626678\pi$
$522$	−3.90827	−0.171061
$523$	14.7189	0.643612	0.321806	−	0.946806i	$-0.395710\pi$
$523$	14.7189	0.643612	0.321806	+	0.946806i	$0.395710\pi$
$524$	18.4843	0.807491
$525$	−8.10328	−0.353656
$526$	20.8869	0.910712
$527$	−2.25915	−0.0984103
$528$	−4.83000	−0.210199
$529$	−5.37269	−0.233595
$530$	−2.73947	−0.118995
$531$	−11.0725	−0.480504
$532$	−3.23526	−0.140266
$533$	−29.3672	−1.27203
$534$	−0.669080	−0.0289539
$535$	−33.6708	−1.45572
$536$	−7.76940	−0.335587
$537$	−12.4357	−0.536639
$538$	3.09618	0.133486
$539$	16.7453	0.721270
$540$	2.73947	0.117888
$541$	−39.6695	−1.70553	−0.852763	−	0.522298i	$-0.825075\pi$
$541$	−39.6695	−1.70553	−0.852763	+	0.522298i	$0.825075\pi$
$542$	−6.39364	−0.274630
$543$	−16.0228	−0.687606
$544$	4.98645	0.213792
$545$	−43.7499	−1.87404
$546$	9.63976	0.412543
$547$	6.51671	0.278634	0.139317	−	0.990248i	$-0.455509\pi$
$547$	6.51671	0.278634	0.139317	+	0.990248i	$0.455509\pi$
$548$	−1.25735	−0.0537113
$549$	−13.2395	−0.565049
$550$	−12.0976	−0.515842
$551$	−3.90827	−0.166498
$552$	−4.19849	−0.178700
$553$	−21.4905	−0.913869
$554$	22.5081	0.956278
$555$	−11.9513	−0.507304
$556$	13.5380	0.574139
$557$	5.63616	0.238812	0.119406	−	0.992846i	$-0.461901\pi$
$557$	5.63616	0.238812	0.119406	+	0.992846i	$0.461901\pi$
$558$	−0.453059	−0.0191795
$559$	28.1144	1.18911
$560$	−8.86290	−0.374526
$561$	24.0845	1.01685
$562$	19.0729	0.804543
$563$	16.0257	0.675405	0.337702	−	0.941253i	$-0.390350\pi$
$563$	16.0257	0.675405	0.337702	+	0.941253i	$0.390350\pi$
$564$	−1.30429	−0.0549205
$565$	−11.2953	−0.475197
$566$	−13.4804	−0.566624
$567$	3.23526	0.135868
$568$	−3.85139	−0.161601
$569$	27.6548	1.15935	0.579676	−	0.814847i	$-0.303179\pi$
$569$	27.6548	1.15935	0.579676	+	0.814847i	$0.303179\pi$
$570$	2.73947	0.114744
$571$	−19.2149	−0.804119	−0.402060	−	0.915613i	$-0.631706\pi$
$571$	−19.2149	−0.804119	−0.402060	+	0.915613i	$0.631706\pi$
$572$	14.3914	0.601735
$573$	7.67553	0.320650
$574$	31.8872	1.33095
$575$	−10.5158	−0.438541
$576$	1.00000	0.0416667
$577$	16.2859	0.677991	0.338995	−	0.940788i	$-0.389913\pi$
$577$	16.2859	0.677991	0.338995	+	0.940788i	$0.389913\pi$
$578$	−7.86468	−0.327128
$579$	4.15995	0.172882
$580$	−10.7066	−0.444567
$581$	−46.1090	−1.91293
$582$	−6.36404	−0.263798
$583$	−4.83000	−0.200038
$584$	8.13755	0.336734
$585$	−8.16248	−0.337477
$586$	19.9454	0.823935
$587$	−31.5738	−1.30319	−0.651595	−	0.758567i	$-0.725900\pi$
$587$	−31.5738	−1.30319	−0.651595	+	0.758567i	$0.725900\pi$
$588$	−3.46693	−0.142974
$589$	−0.453059	−0.0186680
$590$	−30.3327	−1.24878
$591$	−2.45480	−0.100977
$592$	−4.36263	−0.179303
$593$	−24.1287	−0.990845	−0.495422	−	0.868652i	$-0.664987\pi$
$593$	−24.1287	−0.990845	−0.495422	+	0.868652i	$0.664987\pi$
$594$	4.83000	0.198177
$595$	44.1944	1.81179
$596$	−8.21091	−0.336332
$597$	−3.02649	−0.123866
$598$	12.5098	0.511563
$599$	18.8733	0.771144	0.385572	−	0.922678i	$-0.374004\pi$
$599$	18.8733	0.771144	0.385572	+	0.922678i	$0.374004\pi$
$600$	2.50467	0.102253
$601$	32.8228	1.33887	0.669435	−	0.742870i	$-0.266536\pi$
$601$	32.8228	1.33887	0.669435	+	0.742870i	$0.266536\pi$
$602$	−30.5269	−1.24418
$603$	7.76940	0.316395
$604$	21.2332	0.863966
$605$	−33.7745	−1.37313
$606$	−14.3374	−0.582417
$607$	11.5870	0.470301	0.235150	−	0.971959i	$-0.424442\pi$
$607$	11.5870	0.470301	0.235150	+	0.971959i	$0.424442\pi$
$608$	1.00000	0.0405554
$609$	−12.6443	−0.512373
$610$	−36.2692	−1.46850
$611$	3.88624	0.157221
$612$	−4.98645	−0.201565
$613$	−9.26799	−0.374331	−0.187165	−	0.982328i	$-0.559930\pi$
$613$	−9.26799	−0.374331	−0.187165	+	0.982328i	$0.559930\pi$
$614$	13.3207	0.537579
$615$	−27.0005	−1.08877
$616$	−15.6263	−0.629602
$617$	14.1040	0.567805	0.283902	−	0.958853i	$-0.408371\pi$
$617$	14.1040	0.567805	0.283902	+	0.958853i	$0.408371\pi$
$618$	−7.41212	−0.298159
$619$	−38.2805	−1.53862	−0.769312	−	0.638873i	$-0.779401\pi$
$619$	−38.2805	−1.53862	−0.769312	+	0.638873i	$0.779401\pi$
$620$	−1.24114	−0.0498453
$621$	4.19849	0.168480
$622$	25.3383	1.01597
$623$	−2.16465	−0.0867248
$624$	−2.97959	−0.119279
$625$	−31.2500	−1.25000
$626$	−13.2581	−0.529901
$627$	4.83000	0.192891
$628$	−20.3747	−0.813039
$629$	21.7540	0.867390
$630$	8.86290	0.353106
$631$	−23.9281	−0.952562	−0.476281	−	0.879293i	$-0.658015\pi$
$631$	−23.9281	−0.952562	−0.476281	+	0.879293i	$0.658015\pi$
$632$	6.64258	0.264228
$633$	10.9141	0.433797
$634$	2.36254	0.0938284
$635$	−4.30724	−0.170928
$636$	1.00000	0.0396526
$637$	10.3300	0.409291
$638$	−18.8769	−0.747346
$639$	3.85139	0.152359
$640$	2.73947	0.108287
$641$	12.3535	0.487936	0.243968	−	0.969783i	$-0.421551\pi$
$641$	12.3535	0.487936	0.243968	+	0.969783i	$0.421551\pi$
$642$	12.2910	0.485088
$643$	−38.2671	−1.50911	−0.754553	−	0.656239i	$-0.772146\pi$
$643$	−38.2671	−1.50911	−0.754553	+	0.656239i	$0.772146\pi$
$644$	−13.5832	−0.535254
$645$	25.8487	1.01779
$646$	−4.98645	−0.196189
$647$	7.29864	0.286939	0.143470	−	0.989655i	$-0.454174\pi$
$647$	7.29864	0.286939	0.143470	+	0.989655i	$0.454174\pi$
$648$	−1.00000	−0.0392837
$649$	−53.4800	−2.09927
$650$	−7.46290	−0.292719
$651$	−1.46576	−0.0574478
$652$	−1.07494	−0.0420979
$653$	−38.5874	−1.51004	−0.755021	−	0.655700i	$-0.772373\pi$
$653$	−38.5874	−1.51004	−0.755021	+	0.655700i	$0.772373\pi$
$654$	15.9702	0.624485
$655$	−50.6372	−1.97856
$656$	−9.85613	−0.384817
$657$	−8.13755	−0.317476
$658$	−4.21972	−0.164502
$659$	−14.7441	−0.574350	−0.287175	−	0.957878i	$-0.592716\pi$
$659$	−14.7441	−0.574350	−0.287175	+	0.957878i	$0.592716\pi$
$660$	13.2316	0.515040
$661$	15.7136	0.611190	0.305595	−	0.952162i	$-0.401145\pi$
$661$	15.7136	0.611190	0.305595	+	0.952162i	$0.401145\pi$
$662$	30.7261	1.19420
$663$	14.8576	0.577020
$664$	14.2520	0.553086
$665$	8.86290	0.343688
$666$	4.36263	0.169049
$667$	−16.4088	−0.635353
$668$	16.6424	0.643914
$669$	−4.05519	−0.156783
$670$	21.2840	0.822273
$671$	−63.9468	−2.46864
$672$	3.23526	0.124803
$673$	6.14818	0.236995	0.118497	−	0.992954i	$-0.462192\pi$
$673$	6.14818	0.236995	0.118497	+	0.992954i	$0.462192\pi$
$674$	23.0972	0.889673
$675$	−2.50467	−0.0964049
$676$	−4.12205	−0.158541
$677$	−10.2309	−0.393206	−0.196603	−	0.980483i	$-0.562991\pi$
$677$	−10.2309	−0.393206	−0.196603	+	0.980483i	$0.562991\pi$
$678$	4.12317	0.158350
$679$	−20.5893	−0.790147
$680$	−13.6602	−0.523845
$681$	28.4295	1.08942
$682$	−2.18827	−0.0837932
$683$	−50.6531	−1.93819	−0.969093	−	0.246696i	$-0.920655\pi$
$683$	−50.6531	−1.93819	−0.969093	+	0.246696i	$0.920655\pi$
$684$	−1.00000	−0.0382360
$685$	3.44446	0.131606
$686$	11.4304	0.436415
$687$	5.62659	0.214668
$688$	9.43566	0.359731
$689$	−2.97959	−0.113513
$690$	11.5016	0.437859
$691$	19.7684	0.752024	0.376012	−	0.926615i	$-0.377295\pi$
$691$	19.7684	0.752024	0.376012	+	0.926615i	$0.377295\pi$
$692$	−19.2757	−0.732752
$693$	15.6263	0.593594
$694$	2.03583	0.0772792
$695$	−37.0869	−1.40679
$696$	3.90827	0.148143
$697$	49.1471	1.86158
$698$	27.5033	1.04101
$699$	−0.751998	−0.0284432
$700$	8.10328	0.306275
$701$	7.54714	0.285051	0.142526	−	0.989791i	$-0.454478\pi$
$701$	7.54714	0.285051	0.142526	+	0.989791i	$0.454478\pi$
$702$	2.97959	0.112457
$703$	4.36263	0.164540
$704$	4.83000	0.182037
$705$	3.57305	0.134569
$706$	−20.6001	−0.775295
$707$	−46.3853	−1.74450
$708$	11.0725	0.416129
$709$	−24.6979	−0.927549	−0.463775	−	0.885953i	$-0.653505\pi$
$709$	−24.6979	−0.927549	−0.463775	+	0.885953i	$0.653505\pi$
$710$	10.5507	0.395963
$711$	−6.64258	−0.249116
$712$	0.669080	0.0250748
$713$	−1.90216	−0.0712365
$714$	−16.1325	−0.603743
$715$	−39.4247	−1.47440
$716$	12.4357	0.464743
$717$	−25.0652	−0.936076
$718$	36.4576	1.36058
$719$	32.3276	1.20562	0.602808	−	0.797887i	$-0.294049\pi$
$719$	32.3276	1.20562	0.602808	+	0.797887i	$0.294049\pi$
$720$	−2.73947	−0.102094
$721$	−23.9802	−0.893068
$722$	−1.00000	−0.0372161
$723$	−1.99807	−0.0743089
$724$	16.0228	0.595484
$725$	9.78895	0.363553
$726$	12.3289	0.457567
$727$	19.2252	0.713021	0.356511	−	0.934291i	$-0.383966\pi$
$727$	19.2252	0.713021	0.356511	+	0.934291i	$0.383966\pi$
$728$	−9.63976	−0.357273
$729$	1.00000	0.0370370
$730$	−22.2925	−0.825084
$731$	−47.0505	−1.74022
$732$	13.2395	0.489346
$733$	−18.9632	−0.700423	−0.350212	−	0.936671i	$-0.613890\pi$
$733$	−18.9632	−0.700423	−0.350212	+	0.936671i	$0.613890\pi$
$734$	12.4924	0.461103
$735$	9.49755	0.350322
$736$	4.19849	0.154758
$737$	37.5262	1.38229
$738$	9.85613	0.362809
$739$	34.6201	1.27352	0.636761	−	0.771061i	$-0.280274\pi$
$739$	34.6201	1.27352	0.636761	+	0.771061i	$0.280274\pi$
$740$	11.9513	0.439338
$741$	2.97959	0.109458
$742$	3.23526	0.118770
$743$	−12.0258	−0.441185	−0.220592	−	0.975366i	$-0.570799\pi$
$743$	−12.0258	−0.441185	−0.220592	+	0.975366i	$0.570799\pi$
$744$	0.453059	0.0166099
$745$	22.4935	0.824099
$746$	29.0502	1.06360
$747$	−14.2520	−0.521454
$748$	−24.0845	−0.880618
$749$	39.7647	1.45297
$750$	6.83586	0.249610
$751$	−50.8353	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$751$	−50.8353	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$752$	1.30429	0.0475625
$753$	−0.357960	−0.0130448
$754$	−11.6450	−0.424088
$755$	−58.1676	−2.11693
$756$	−3.23526	−0.117665
$757$	38.2013	1.38845	0.694225	−	0.719758i	$-0.255747\pi$
$757$	38.2013	1.38845	0.694225	+	0.719758i	$0.255747\pi$
$758$	−22.7932	−0.827886
$759$	20.2787	0.736070
$760$	−2.73947	−0.0993709
$761$	45.9810	1.66681	0.833405	−	0.552663i	$-0.186388\pi$
$761$	45.9810	1.66681	0.833405	+	0.552663i	$0.186388\pi$
$762$	1.57229	0.0569582
$763$	51.6679	1.87050
$764$	−7.67553	−0.277691
$765$	13.6602	0.493886
$766$	−17.9430	−0.648309
$767$	−32.9914	−1.19125
$768$	−1.00000	−0.0360844
$769$	−29.3080	−1.05687	−0.528437	−	0.848973i	$-0.677222\pi$
$769$	−29.3080	−1.05687	−0.528437	+	0.848973i	$0.677222\pi$
$770$	42.8077	1.54268
$771$	13.6812	0.492715
$772$	−4.15995	−0.149720
$773$	5.95829	0.214305	0.107152	−	0.994243i	$-0.465827\pi$
$773$	5.95829	0.214305	0.107152	+	0.994243i	$0.465827\pi$
$774$	−9.43566	−0.339158
$775$	1.13476	0.0407619
$776$	6.36404	0.228456
$777$	14.1143	0.506346
$778$	22.9838	0.824008
$779$	9.85613	0.353132
$780$	8.16248	0.292264
$781$	18.6022	0.665639
$782$	−20.9356	−0.748654
$783$	−3.90827	−0.139670
$784$	3.46693	0.123819
$785$	55.8158	1.99215
$786$	18.4843	0.659314
$787$	−10.8812	−0.387873	−0.193937	−	0.981014i	$-0.562126\pi$
$787$	−10.8812	−0.387873	−0.193937	+	0.981014i	$0.562126\pi$
$788$	2.45480	0.0874486
$789$	20.8869	0.743593
$790$	−18.1971	−0.647424
$791$	13.3396	0.474300
$792$	−4.83000	−0.171626
$793$	−39.4483	−1.40085
$794$	4.46757	0.158548
$795$	−2.73947	−0.0971588
$796$	3.02649	0.107271
$797$	−2.63294	−0.0932637	−0.0466318	−	0.998912i	$-0.514849\pi$
$797$	−2.63294	−0.0932637	−0.0466318	+	0.998912i	$0.514849\pi$
$798$	−3.23526	−0.114527
$799$	−6.50377	−0.230087
$800$	−2.50467	−0.0885536
$801$	−0.669080	−0.0236408
$802$	−34.8245	−1.22970
$803$	−39.3043	−1.38702
$804$	−7.76940	−0.274006
$805$	37.2108	1.31151
$806$	−1.34993	−0.0475492
$807$	3.09618	0.108991
$808$	14.3374	0.504388
$809$	3.19121	0.112197	0.0560985	−	0.998425i	$-0.482134\pi$
$809$	3.19121	0.112197	0.0560985	+	0.998425i	$0.482134\pi$
$810$	2.73947	0.0962550
$811$	16.5313	0.580491	0.290245	−	0.956952i	$-0.406263\pi$
$811$	16.5313	0.580491	0.290245	+	0.956952i	$0.406263\pi$
$812$	12.6443	0.443728
$813$	−6.39364	−0.224235
$814$	21.0715	0.738555
$815$	2.94476	0.103150
$816$	4.98645	0.174561
$817$	−9.43566	−0.330112
$818$	25.8053	0.902260
$819$	9.63976	0.336840
$820$	27.0005	0.942899
$821$	−23.9283	−0.835104	−0.417552	−	0.908653i	$-0.637112\pi$
$821$	−23.9283	−0.835104	−0.417552	+	0.908653i	$0.637112\pi$
$822$	−1.25735	−0.0438551
$823$	−19.1168	−0.666370	−0.333185	−	0.942862i	$-0.608123\pi$
$823$	−19.1168	−0.666370	−0.333185	+	0.942862i	$0.608123\pi$
$824$	7.41212	0.258213
$825$	−12.0976	−0.421183
$826$	35.8224	1.24642
$827$	12.9177	0.449191	0.224595	−	0.974452i	$-0.427894\pi$
$827$	12.9177	0.449191	0.224595	+	0.974452i	$0.427894\pi$
$828$	−4.19849	−0.145908
$829$	31.9293	1.10895	0.554476	−	0.832200i	$-0.312919\pi$
$829$	31.9293	1.10895	0.554476	+	0.832200i	$0.312919\pi$
$830$	−39.0429	−1.35520
$831$	22.5081	0.780798
$832$	2.97959	0.103299
$833$	−17.2877	−0.598983
$834$	13.5380	0.468783
$835$	−45.5913	−1.57775
$836$	−4.83000	−0.167049
$837$	−0.453059	−0.0156600
$838$	0.343947	0.0118814
$839$	−29.7960	−1.02867	−0.514337	−	0.857588i	$-0.671962\pi$
$839$	−29.7960	−1.02867	−0.514337	+	0.857588i	$0.671962\pi$
$840$	−8.86290	−0.305799
$841$	−13.7254	−0.473289
$842$	10.0368	0.345890
$843$	19.0729	0.656907
$844$	−10.9141	−0.375680
$845$	11.2922	0.388464
$846$	−1.30429	−0.0448424
$847$	39.8871	1.37054
$848$	−1.00000	−0.0343401
$849$	−13.4804	−0.462647
$850$	12.4894	0.428384
$851$	18.3165	0.627880
$852$	−3.85139	−0.131946
$853$	−10.2476	−0.350870	−0.175435	−	0.984491i	$-0.556133\pi$
$853$	−10.2476	−0.350870	−0.175435	+	0.984491i	$0.556133\pi$
$854$	42.8333	1.46573
$855$	2.73947	0.0936878
$856$	−12.2910	−0.420098
$857$	37.5884	1.28400	0.641998	−	0.766707i	$-0.278106\pi$
$857$	37.5884	1.28400	0.641998	+	0.766707i	$0.278106\pi$
$858$	14.3914	0.491314
$859$	15.8639	0.541270	0.270635	−	0.962682i	$-0.412766\pi$
$859$	15.8639	0.541270	0.270635	+	0.962682i	$0.412766\pi$
$860$	−25.8487	−0.881433
$861$	31.8872	1.08671
$862$	7.66750	0.261156
$863$	51.2597	1.74490	0.872451	−	0.488702i	$-0.162529\pi$
$863$	51.2597	1.74490	0.872451	+	0.488702i	$0.162529\pi$
$864$	1.00000	0.0340207
$865$	52.8051	1.79543
$866$	1.12411	0.0381989
$867$	−7.86468	−0.267099
$868$	1.46576	0.0497513
$869$	−32.0836	−1.08836
$870$	−10.7066	−0.362987
$871$	23.1496	0.784395
$872$	−15.9702	−0.540820
$873$	−6.36404	−0.215390
$874$	−4.19849	−0.142016
$875$	22.1158	0.747651
$876$	8.13755	0.274942
$877$	−43.3715	−1.46455	−0.732276	−	0.681008i	$-0.761542\pi$
$877$	−43.3715	−1.46455	−0.732276	+	0.681008i	$0.761542\pi$
$878$	−27.7423	−0.936259
$879$	19.9454	0.672740
$880$	−13.2316	−0.446037
$881$	6.91893	0.233105	0.116552	−	0.993185i	$-0.462816\pi$
$881$	6.91893	0.233105	0.116552	+	0.993185i	$0.462816\pi$
$882$	−3.46693	−0.116738
$883$	44.2612	1.48951	0.744754	−	0.667339i	$-0.232567\pi$
$883$	44.2612	1.48951	0.744754	+	0.667339i	$0.232567\pi$
$884$	−14.8576	−0.499714
$885$	−30.3327	−1.01962
$886$	21.0876	0.708452
$887$	16.3177	0.547896	0.273948	−	0.961744i	$-0.411670\pi$
$887$	16.3177	0.547896	0.273948	+	0.961744i	$0.411670\pi$
$888$	−4.36263	−0.146400
$889$	5.08678	0.170605
$890$	−1.83292	−0.0614396
$891$	4.83000	0.161811
$892$	4.05519	0.135778
$893$	−1.30429	−0.0436463
$894$	−8.21091	−0.274614
$895$	−34.0671	−1.13874
$896$	−3.23526	−0.108083
$897$	12.5098	0.417689
$898$	−4.13304	−0.137921
$899$	1.77068	0.0590554
$900$	2.50467	0.0834891
$901$	4.98645	0.166123
$902$	47.6051	1.58507
$903$	−30.5269	−1.01587
$904$	−4.12317	−0.137135
$905$	−43.8940	−1.45909
$906$	21.2332	0.705425
$907$	−11.7917	−0.391536	−0.195768	−	0.980650i	$-0.562720\pi$
$907$	−11.7917	−0.391536	−0.195768	+	0.980650i	$0.562720\pi$
$908$	−28.4295	−0.943466
$909$	−14.3374	−0.475542
$910$	26.4078	0.875409
$911$	20.0430	0.664054	0.332027	−	0.943270i	$-0.392267\pi$
$911$	20.0430	0.664054	0.332027	+	0.943270i	$0.392267\pi$
$912$	1.00000	0.0331133
$913$	−68.8372	−2.27818
$914$	10.2743	0.339844
$915$	−36.2692	−1.19902
$916$	−5.62659	−0.185908
$917$	59.8016	1.97482
$918$	−4.98645	−0.164577
$919$	−52.5487	−1.73342	−0.866710	−	0.498812i	$-0.833770\pi$
$919$	−52.5487	−1.73342	−0.866710	+	0.498812i	$0.833770\pi$
$920$	−11.5016	−0.379197
$921$	13.3207	0.438931
$922$	−13.3545	−0.439807
$923$	11.4756	0.377722
$924$	−15.6263	−0.514068
$925$	−10.9270	−0.359276
$926$	−30.2503	−0.994086
$927$	−7.41212	−0.243446
$928$	−3.90827	−0.128295
$929$	30.1004	0.987563	0.493781	−	0.869586i	$-0.335614\pi$
$929$	30.1004	0.987563	0.493781	+	0.869586i	$0.335614\pi$
$930$	−1.24114	−0.0406985
$931$	−3.46693	−0.113624
$932$	0.751998	0.0246325
$933$	25.3383	0.829539
$934$	27.9883	0.915804
$935$	65.9787	2.15774
$936$	−2.97959	−0.0973909
$937$	23.8216	0.778217	0.389108	−	0.921192i	$-0.372783\pi$
$937$	23.8216	0.778217	0.389108	+	0.921192i	$0.372783\pi$
$938$	−25.1361	−0.820722
$939$	−13.2581	−0.432662
$940$	−3.57305	−0.116540
$941$	50.8379	1.65727	0.828634	−	0.559790i	$-0.189118\pi$
$941$	50.8379	1.65727	0.828634	+	0.559790i	$0.189118\pi$
$942$	−20.3747	−0.663843
$943$	41.3808	1.34755
$944$	−11.0725	−0.360378
$945$	8.86290	0.288310
$946$	−45.5742	−1.48175
$947$	−56.2667	−1.82842	−0.914212	−	0.405237i	$-0.867189\pi$
$947$	−56.2667	−1.82842	−0.914212	+	0.405237i	$0.867189\pi$
$948$	6.64258	0.215741
$949$	−24.2465	−0.787076
$950$	2.50467	0.0812623
$951$	2.36254	0.0766106
$952$	16.1325	0.522857
$953$	−30.8264	−0.998566	−0.499283	−	0.866439i	$-0.666403\pi$
$953$	−30.8264	−0.998566	−0.499283	+	0.866439i	$0.666403\pi$
$954$	1.00000	0.0323762
$955$	21.0268	0.680413
$956$	25.0652	0.810666
$957$	−18.8769	−0.610205
$958$	−40.3402	−1.30333
$959$	−4.06785	−0.131358
$960$	2.73947	0.0884159
$961$	−30.7947	−0.993379
$962$	12.9988	0.419100
$963$	12.2910	0.396073
$964$	1.99807	0.0643534
$965$	11.3960	0.366852
$966$	−13.5832	−0.437033
$967$	−25.3163	−0.814117	−0.407059	−	0.913402i	$-0.633446\pi$
$967$	−25.3163	−0.814117	−0.407059	+	0.913402i	$0.633446\pi$
$968$	−12.3289	−0.396264
$969$	−4.98645	−0.160188
$970$	−17.4341	−0.559774
$971$	11.3487	0.364198	0.182099	−	0.983280i	$-0.441711\pi$
$971$	11.3487	0.364198	0.182099	+	0.983280i	$0.441711\pi$
$972$	−1.00000	−0.0320750
$973$	43.7990	1.40413
$974$	−36.7511	−1.17758
$975$	−7.46290	−0.239004
$976$	−13.2395	−0.423786
$977$	−37.6312	−1.20393	−0.601964	−	0.798523i	$-0.705615\pi$
$977$	−37.6312	−1.20393	−0.601964	+	0.798523i	$0.705615\pi$
$978$	−1.07494	−0.0343728
$979$	−3.23165	−0.103284
$980$	−9.49755	−0.303388
$981$	15.9702	0.509890
$982$	−28.1976	−0.899823
$983$	−42.7658	−1.36402	−0.682008	−	0.731345i	$-0.738893\pi$
$983$	−42.7658	−1.36402	−0.682008	+	0.731345i	$0.738893\pi$
$984$	−9.85613	−0.314202
$985$	−6.72484	−0.214271
$986$	19.4884	0.620638
$987$	−4.21972	−0.134315
$988$	−2.97959	−0.0947933
$989$	−39.6155	−1.25970
$990$	13.2316	0.420528
$991$	−25.2934	−0.803471	−0.401735	−	0.915756i	$-0.631593\pi$
$991$	−25.2934	−0.803471	−0.401735	+	0.915756i	$0.631593\pi$
$992$	−0.453059	−0.0143846
$993$	30.7261	0.975064
$994$	−12.4603	−0.395215
$995$	−8.29096	−0.262841
$996$	14.2520	0.451593
$997$	32.9029	1.04204	0.521022	−	0.853543i	$-0.325551\pi$
$997$	32.9029	1.04204	0.521022	+	0.853543i	$0.325551\pi$
$998$	6.01554	0.190419
$999$	4.36263	0.138028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.be.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.be.1.4	✓	12	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73947 q^{5} +1.00000 q^{6} +3.23526 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73947 q^{5} +1.00000 q^{6} +3.23526 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.73947 q^{10} +4.83000 q^{11} -1.00000 q^{12} +2.97959 q^{13} -3.23526 q^{14} +2.73947 q^{15} +1.00000 q^{16} -4.98645 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.73947 q^{20} -3.23526 q^{21} -4.83000 q^{22} -4.19849 q^{23} +1.00000 q^{24} +2.50467 q^{25} -2.97959 q^{26} -1.00000 q^{27} +3.23526 q^{28} +3.90827 q^{29} -2.73947 q^{30} +0.453059 q^{31} -1.00000 q^{32} -4.83000 q^{33} +4.98645 q^{34} -8.86290 q^{35} +1.00000 q^{36} -4.36263 q^{37} +1.00000 q^{38} -2.97959 q^{39} +2.73947 q^{40} -9.85613 q^{41} +3.23526 q^{42} +9.43566 q^{43} +4.83000 q^{44} -2.73947 q^{45} +4.19849 q^{46} +1.30429 q^{47} -1.00000 q^{48} +3.46693 q^{49} -2.50467 q^{50} +4.98645 q^{51} +2.97959 q^{52} -1.00000 q^{53} +1.00000 q^{54} -13.2316 q^{55} -3.23526 q^{56} +1.00000 q^{57} -3.90827 q^{58} -11.0725 q^{59} +2.73947 q^{60} -13.2395 q^{61} -0.453059 q^{62} +3.23526 q^{63} +1.00000 q^{64} -8.16248 q^{65} +4.83000 q^{66} +7.76940 q^{67} -4.98645 q^{68} +4.19849 q^{69} +8.86290 q^{70} +3.85139 q^{71} -1.00000 q^{72} -8.13755 q^{73} +4.36263 q^{74} -2.50467 q^{75} -1.00000 q^{76} +15.6263 q^{77} +2.97959 q^{78} -6.64258 q^{79} -2.73947 q^{80} +1.00000 q^{81} +9.85613 q^{82} -14.2520 q^{83} -3.23526 q^{84} +13.6602 q^{85} -9.43566 q^{86} -3.90827 q^{87} -4.83000 q^{88} -0.669080 q^{89} +2.73947 q^{90} +9.63976 q^{91} -4.19849 q^{92} -0.453059 q^{93} -1.30429 q^{94} +2.73947 q^{95} +1.00000 q^{96} -6.36404 q^{97} -3.46693 q^{98} +4.83000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 7 * q^11 - 12 * q^12 + 4 * q^13 + q^14 + 3 * q^15 + 12 * q^16 - 4 * q^17 - 12 * q^18 - 12 * q^19 - 3 * q^20 + q^21 + 7 * q^22 - 18 * q^23 + 12 * q^24 + 21 * q^25 - 4 * q^26 - 12 * q^27 - q^28 + 10 * q^29 - 3 * q^30 + 14 * q^31 - 12 * q^32 + 7 * q^33 + 4 * q^34 - 19 * q^35 + 12 * q^36 + 14 * q^37 + 12 * q^38 - 4 * q^39 + 3 * q^40 - 9 * q^41 - q^42 - 4 * q^43 - 7 * q^44 - 3 * q^45 + 18 * q^46 - 14 * q^47 - 12 * q^48 + 37 * q^49 - 21 * q^50 + 4 * q^51 + 4 * q^52 - 12 * q^53 + 12 * q^54 - 4 * q^55 + q^56 + 12 * q^57 - 10 * q^58 - 18 * q^59 + 3 * q^60 - 7 * q^61 - 14 * q^62 - q^63 + 12 * q^64 + 3 * q^65 - 7 * q^66 + 8 * q^67 - 4 * q^68 + 18 * q^69 + 19 * q^70 - q^71 - 12 * q^72 - 31 * q^73 - 14 * q^74 - 21 * q^75 - 12 * q^76 - 25 * q^77 + 4 * q^78 - 3 * q^80 + 12 * q^81 + 9 * q^82 - 48 * q^83 + q^84 + 4 * q^85 + 4 * q^86 - 10 * q^87 + 7 * q^88 + 13 * q^89 + 3 * q^90 + 9 * q^91 - 18 * q^92 - 14 * q^93 + 14 * q^94 + 3 * q^95 + 12 * q^96 + 25 * q^97 - 37 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more