LMFDB - Embedded newform 8034.2.a.bd.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$16$
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} - 5 x^{15} - 36 x^{14} + 196 x^{13} + 498 x^{12} - 3101 x^{11} - 3150 x^{10} + 25368 x^{9} + \cdots - 66432 $ x^16 - 5x^15 - 36x^14 + 196x^13 + 498x^12 - 3101x^11 - 3150x^10 + 25368x^9 + 6763x^8 - 113788x^7 + 19731x^6 + 270913x^5 - 122680x^4 - 296326x^3 + 185524x^2 + 94528*x - 66432
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			1.11
Root			$1.78856$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.78856 q^{5} +1.00000 q^{6} +0.327596 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.78856 q^{5} +1.00000 q^{6} +0.327596 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.78856 q^{10} -4.85229 q^{11} +1.00000 q^{12} -1.00000 q^{13} +0.327596 q^{14} +1.78856 q^{15} +1.00000 q^{16} +1.60542 q^{17} +1.00000 q^{18} +1.42429 q^{19} +1.78856 q^{20} +0.327596 q^{21} -4.85229 q^{22} +5.62903 q^{23} +1.00000 q^{24} -1.80104 q^{25} -1.00000 q^{26} +1.00000 q^{27} +0.327596 q^{28} +4.09609 q^{29} +1.78856 q^{30} +9.12528 q^{31} +1.00000 q^{32} -4.85229 q^{33} +1.60542 q^{34} +0.585926 q^{35} +1.00000 q^{36} +10.0665 q^{37} +1.42429 q^{38} -1.00000 q^{39} +1.78856 q^{40} +7.43278 q^{41} +0.327596 q^{42} +3.11586 q^{43} -4.85229 q^{44} +1.78856 q^{45} +5.62903 q^{46} -11.0796 q^{47} +1.00000 q^{48} -6.89268 q^{49} -1.80104 q^{50} +1.60542 q^{51} -1.00000 q^{52} +0.0340349 q^{53} +1.00000 q^{54} -8.67863 q^{55} +0.327596 q^{56} +1.42429 q^{57} +4.09609 q^{58} -6.56032 q^{59} +1.78856 q^{60} -6.82393 q^{61} +9.12528 q^{62} +0.327596 q^{63} +1.00000 q^{64} -1.78856 q^{65} -4.85229 q^{66} +2.98653 q^{67} +1.60542 q^{68} +5.62903 q^{69} +0.585926 q^{70} +14.2442 q^{71} +1.00000 q^{72} -13.9756 q^{73} +10.0665 q^{74} -1.80104 q^{75} +1.42429 q^{76} -1.58959 q^{77} -1.00000 q^{78} -6.52386 q^{79} +1.78856 q^{80} +1.00000 q^{81} +7.43278 q^{82} +5.30980 q^{83} +0.327596 q^{84} +2.87139 q^{85} +3.11586 q^{86} +4.09609 q^{87} -4.85229 q^{88} -13.1160 q^{89} +1.78856 q^{90} -0.327596 q^{91} +5.62903 q^{92} +9.12528 q^{93} -11.0796 q^{94} +2.54743 q^{95} +1.00000 q^{96} +8.44438 q^{97} -6.89268 q^{98} -4.85229 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9 + 5 * q^10 + 18 * q^11 + 16 * q^12 - 16 * q^13 + 4 * q^14 + 5 * q^15 + 16 * q^16 + 17 * q^17 + 16 * q^18 + 8 * q^19 + 5 * q^20 + 4 * q^21 + 18 * q^22 + 9 * q^23 + 16 * q^24 + 17 * q^25 - 16 * q^26 + 16 * q^27 + 4 * q^28 + 14 * q^29 + 5 * q^30 + 12 * q^31 + 16 * q^32 + 18 * q^33 + 17 * q^34 + 16 * q^35 + 16 * q^36 + 31 * q^37 + 8 * q^38 - 16 * q^39 + 5 * q^40 + 29 * q^41 + 4 * q^42 + 30 * q^43 + 18 * q^44 + 5 * q^45 + 9 * q^46 - q^47 + 16 * q^48 + 36 * q^49 + 17 * q^50 + 17 * q^51 - 16 * q^52 + 12 * q^53 + 16 * q^54 + 30 * q^55 + 4 * q^56 + 8 * q^57 + 14 * q^58 + 38 * q^59 + 5 * q^60 + 12 * q^62 + 4 * q^63 + 16 * q^64 - 5 * q^65 + 18 * q^66 + 28 * q^67 + 17 * q^68 + 9 * q^69 + 16 * q^70 + 32 * q^71 + 16 * q^72 + 20 * q^73 + 31 * q^74 + 17 * q^75 + 8 * q^76 + 26 * q^77 - 16 * q^78 + 13 * q^79 + 5 * q^80 + 16 * q^81 + 29 * q^82 + 39 * q^83 + 4 * q^84 + 31 * q^85 + 30 * q^86 + 14 * q^87 + 18 * q^88 + 9 * q^89 + 5 * q^90 - 4 * q^91 + 9 * q^92 + 12 * q^93 - q^94 - 20 * q^95 + 16 * q^96 + 35 * q^97 + 36 * q^98 + 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.78856	0.799869	0.399935	−	0.916544i	$-0.369033\pi$
$5$	1.78856	0.799869	0.399935	+	0.916544i	$0.369033\pi$
$6$	1.00000	0.408248
$7$	0.327596	0.123820	0.0619099	−	0.998082i	$-0.480281\pi$
$7$	0.327596	0.123820	0.0619099	+	0.998082i	$0.480281\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.78856	0.565593
$11$	−4.85229	−1.46302	−0.731511	−	0.681830i	$-0.761185\pi$
$11$	−4.85229	−1.46302	−0.731511	+	0.681830i	$0.761185\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.327596	0.0875538
$15$	1.78856	0.461805
$16$	1.00000	0.250000
$17$	1.60542	0.389371	0.194685	−	0.980866i	$-0.437631\pi$
$17$	1.60542	0.389371	0.194685	+	0.980866i	$0.437631\pi$
$18$	1.00000	0.235702
$19$	1.42429	0.326755	0.163377	−	0.986564i	$-0.447761\pi$
$19$	1.42429	0.326755	0.163377	+	0.986564i	$0.447761\pi$
$20$	1.78856	0.399935
$21$	0.327596	0.0714874
$22$	−4.85229	−1.03451
$23$	5.62903	1.17373	0.586867	−	0.809684i	$-0.300361\pi$
$23$	5.62903	1.17373	0.586867	+	0.809684i	$0.300361\pi$
$24$	1.00000	0.204124
$25$	−1.80104	−0.360209
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	0.327596	0.0619099
$29$	4.09609	0.760625	0.380313	−	0.924858i	$-0.375816\pi$
$29$	4.09609	0.760625	0.380313	+	0.924858i	$0.375816\pi$
$30$	1.78856	0.326545
$31$	9.12528	1.63895	0.819475	−	0.573115i	$-0.194265\pi$
$31$	9.12528	1.63895	0.819475	+	0.573115i	$0.194265\pi$
$32$	1.00000	0.176777
$33$	−4.85229	−0.844676
$34$	1.60542	0.275327
$35$	0.585926	0.0990396
$36$	1.00000	0.166667
$37$	10.0665	1.65492	0.827461	−	0.561523i	$-0.189784\pi$
$37$	10.0665	1.65492	0.827461	+	0.561523i	$0.189784\pi$
$38$	1.42429	0.231050
$39$	−1.00000	−0.160128
$40$	1.78856	0.282797
$41$	7.43278	1.16081	0.580403	−	0.814330i	$-0.302895\pi$
$41$	7.43278	1.16081	0.580403	+	0.814330i	$0.302895\pi$
$42$	0.327596	0.0505492
$43$	3.11586	0.475164	0.237582	−	0.971368i	$-0.423645\pi$
$43$	3.11586	0.475164	0.237582	+	0.971368i	$0.423645\pi$
$44$	−4.85229	−0.731511
$45$	1.78856	0.266623
$46$	5.62903	0.829955
$47$	−11.0796	−1.61612	−0.808062	−	0.589097i	$-0.799483\pi$
$47$	−11.0796	−1.61612	−0.808062	+	0.589097i	$0.799483\pi$
$48$	1.00000	0.144338
$49$	−6.89268	−0.984669
$50$	−1.80104	−0.254706
$51$	1.60542	0.224803
$52$	−1.00000	−0.138675
$53$	0.0340349	0.00467505	0.00233753	−	0.999997i	$-0.499256\pi$
$53$	0.0340349	0.00467505	0.00233753	+	0.999997i	$0.499256\pi$
$54$	1.00000	0.136083
$55$	−8.67863	−1.17023
$56$	0.327596	0.0437769
$57$	1.42429	0.188652
$58$	4.09609	0.537843
$59$	−6.56032	−0.854080	−0.427040	−	0.904233i	$-0.640444\pi$
$59$	−6.56032	−0.854080	−0.427040	+	0.904233i	$0.640444\pi$
$60$	1.78856	0.230902
$61$	−6.82393	−0.873715	−0.436857	−	0.899531i	$-0.643909\pi$
$61$	−6.82393	−0.873715	−0.436857	+	0.899531i	$0.643909\pi$
$62$	9.12528	1.15891
$63$	0.327596	0.0412732
$64$	1.00000	0.125000
$65$	−1.78856	−0.221844
$66$	−4.85229	−0.597276
$67$	2.98653	0.364863	0.182432	−	0.983219i	$-0.441603\pi$
$67$	2.98653	0.364863	0.182432	+	0.983219i	$0.441603\pi$
$68$	1.60542	0.194685
$69$	5.62903	0.677655
$70$	0.585926	0.0700316
$71$	14.2442	1.69047	0.845236	−	0.534394i	$-0.179460\pi$
$71$	14.2442	1.69047	0.845236	+	0.534394i	$0.179460\pi$
$72$	1.00000	0.117851
$73$	−13.9756	−1.63572	−0.817860	−	0.575418i	$-0.804839\pi$
$73$	−13.9756	−1.63572	−0.817860	+	0.575418i	$0.804839\pi$
$74$	10.0665	1.17021
$75$	−1.80104	−0.207967
$76$	1.42429	0.163377
$77$	−1.58959	−0.181151
$78$	−1.00000	−0.113228
$79$	−6.52386	−0.733992	−0.366996	−	0.930223i	$-0.619614\pi$
$79$	−6.52386	−0.733992	−0.366996	+	0.930223i	$0.619614\pi$
$80$	1.78856	0.199967
$81$	1.00000	0.111111
$82$	7.43278	0.820813
$83$	5.30980	0.582826	0.291413	−	0.956597i	$-0.405875\pi$
$83$	5.30980	0.582826	0.291413	+	0.956597i	$0.405875\pi$
$84$	0.327596	0.0357437
$85$	2.87139	0.311446
$86$	3.11586	0.335992
$87$	4.09609	0.439147
$88$	−4.85229	−0.517256
$89$	−13.1160	−1.39029	−0.695147	−	0.718867i	$-0.744661\pi$
$89$	−13.1160	−1.39029	−0.695147	+	0.718867i	$0.744661\pi$
$90$	1.78856	0.188531
$91$	−0.327596	−0.0343414
$92$	5.62903	0.586867
$93$	9.12528	0.946248
$94$	−11.0796	−1.14277
$95$	2.54743	0.261361
$96$	1.00000	0.102062
$97$	8.44438	0.857397	0.428698	−	0.903448i	$-0.358972\pi$
$97$	8.44438	0.857397	0.428698	+	0.903448i	$0.358972\pi$
$98$	−6.89268	−0.696266
$99$	−4.85229	−0.487674
$100$	−1.80104	−0.180104
$101$	10.1762	1.01257	0.506283	−	0.862367i	$-0.331019\pi$
$101$	10.1762	1.01257	0.506283	+	0.862367i	$0.331019\pi$
$102$	1.60542	0.158960
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	0.585926	0.0571806
$106$	0.0340349	0.00330576
$107$	19.2490	1.86088	0.930438	−	0.366451i	$-0.119427\pi$
$107$	19.2490	1.86088	0.930438	+	0.366451i	$0.119427\pi$
$108$	1.00000	0.0962250
$109$	15.5044	1.48505	0.742524	−	0.669819i	$-0.233629\pi$
$109$	15.5044	1.48505	0.742524	+	0.669819i	$0.233629\pi$
$110$	−8.67863	−0.827475
$111$	10.0665	0.955470
$112$	0.327596	0.0309549
$113$	16.3227	1.53551	0.767757	−	0.640741i	$-0.221373\pi$
$113$	16.3227	1.53551	0.767757	+	0.640741i	$0.221373\pi$
$114$	1.42429	0.133397
$115$	10.0679	0.938834
$116$	4.09609	0.380313
$117$	−1.00000	−0.0924500
$118$	−6.56032	−0.603926
$119$	0.525929	0.0482118
$120$	1.78856	0.163273
$121$	12.5448	1.14043
$122$	−6.82393	−0.617810
$123$	7.43278	0.670191
$124$	9.12528	0.819475
$125$	−12.1641	−1.08799
$126$	0.327596	0.0291846
$127$	6.72756	0.596975	0.298487	−	0.954414i	$-0.403518\pi$
$127$	6.72756	0.596975	0.298487	+	0.954414i	$0.403518\pi$
$128$	1.00000	0.0883883
$129$	3.11586	0.274336
$130$	−1.78856	−0.156867
$131$	−10.2917	−0.899189	−0.449594	−	0.893233i	$-0.648431\pi$
$131$	−10.2917	−0.899189	−0.449594	+	0.893233i	$0.648431\pi$
$132$	−4.85229	−0.422338
$133$	0.466592	0.0404587
$134$	2.98653	0.257997
$135$	1.78856	0.153935
$136$	1.60542	0.137663
$137$	8.07567	0.689951	0.344975	−	0.938612i	$-0.387887\pi$
$137$	8.07567	0.689951	0.344975	+	0.938612i	$0.387887\pi$
$138$	5.62903	0.479175
$139$	−2.04231	−0.173226	−0.0866131	−	0.996242i	$-0.527604\pi$
$139$	−2.04231	−0.173226	−0.0866131	+	0.996242i	$0.527604\pi$
$140$	0.585926	0.0495198
$141$	−11.0796	−0.933070
$142$	14.2442	1.19534
$143$	4.85229	0.405769
$144$	1.00000	0.0833333
$145$	7.32612	0.608401
$146$	−13.9756	−1.15663
$147$	−6.89268	−0.568499
$148$	10.0665	0.827461
$149$	−2.43705	−0.199651	−0.0998255	−	0.995005i	$-0.531828\pi$
$149$	−2.43705	−0.199651	−0.0998255	+	0.995005i	$0.531828\pi$
$150$	−1.80104	−0.147055
$151$	9.18020	0.747074	0.373537	−	0.927615i	$-0.378145\pi$
$151$	9.18020	0.747074	0.373537	+	0.927615i	$0.378145\pi$
$152$	1.42429	0.115525
$153$	1.60542	0.129790
$154$	−1.58959	−0.128093
$155$	16.3211	1.31095
$156$	−1.00000	−0.0800641
$157$	21.0011	1.67607	0.838035	−	0.545616i	$-0.183704\pi$
$157$	21.0011	1.67607	0.838035	+	0.545616i	$0.183704\pi$
$158$	−6.52386	−0.519010
$159$	0.0340349	0.00269914
$160$	1.78856	0.141398
$161$	1.84405	0.145331
$162$	1.00000	0.0785674
$163$	−4.76711	−0.373389	−0.186694	−	0.982418i	$-0.559777\pi$
$163$	−4.76711	−0.373389	−0.186694	+	0.982418i	$0.559777\pi$
$164$	7.43278	0.580403
$165$	−8.67863	−0.675631
$166$	5.30980	0.412120
$167$	2.77768	0.214943	0.107471	−	0.994208i	$-0.465725\pi$
$167$	2.77768	0.214943	0.107471	+	0.994208i	$0.465725\pi$
$168$	0.327596	0.0252746
$169$	1.00000	0.0769231
$170$	2.87139	0.220226
$171$	1.42429	0.108918
$172$	3.11586	0.237582
$173$	7.70162	0.585543	0.292772	−	0.956182i	$-0.405422\pi$
$173$	7.70162	0.585543	0.292772	+	0.956182i	$0.405422\pi$
$174$	4.09609	0.310524
$175$	−0.590015	−0.0446010
$176$	−4.85229	−0.365755
$177$	−6.56032	−0.493104
$178$	−13.1160	−0.983087
$179$	−19.3613	−1.44713	−0.723565	−	0.690256i	$-0.757498\pi$
$179$	−19.3613	−1.44713	−0.723565	+	0.690256i	$0.757498\pi$
$180$	1.78856	0.133312
$181$	1.26309	0.0938846	0.0469423	−	0.998898i	$-0.485052\pi$
$181$	1.26309	0.0938846	0.0469423	+	0.998898i	$0.485052\pi$
$182$	−0.327596	−0.0242831
$183$	−6.82393	−0.504439
$184$	5.62903	0.414977
$185$	18.0046	1.32372
$186$	9.12528	0.669098
$187$	−7.78996	−0.569658
$188$	−11.0796	−0.808062
$189$	0.327596	0.0238291
$190$	2.54743	0.184810
$191$	16.0940	1.16452	0.582261	−	0.813002i	$-0.302168\pi$
$191$	16.0940	1.16452	0.582261	+	0.813002i	$0.302168\pi$
$192$	1.00000	0.0721688
$193$	11.0494	0.795352	0.397676	−	0.917526i	$-0.369817\pi$
$193$	11.0494	0.795352	0.397676	+	0.917526i	$0.369817\pi$
$194$	8.44438	0.606271
$195$	−1.78856	−0.128082
$196$	−6.89268	−0.492334
$197$	−17.1674	−1.22313	−0.611563	−	0.791196i	$-0.709459\pi$
$197$	−17.1674	−1.22313	−0.611563	+	0.791196i	$0.709459\pi$
$198$	−4.85229	−0.344838
$199$	−23.7518	−1.68372	−0.841861	−	0.539694i	$-0.818540\pi$
$199$	−23.7518	−1.68372	−0.841861	+	0.539694i	$0.818540\pi$
$200$	−1.80104	−0.127353
$201$	2.98653	0.210654
$202$	10.1762	0.715992
$203$	1.34186	0.0941805
$204$	1.60542	0.112402
$205$	13.2940	0.928493
$206$	1.00000	0.0696733
$207$	5.62903	0.391244
$208$	−1.00000	−0.0693375
$209$	−6.91108	−0.478049
$210$	0.585926	0.0404328
$211$	−20.5256	−1.41304	−0.706521	−	0.707692i	$-0.749736\pi$
$211$	−20.5256	−1.41304	−0.706521	+	0.707692i	$0.749736\pi$
$212$	0.0340349	0.00233753
$213$	14.2442	0.975994
$214$	19.2490	1.31584
$215$	5.57291	0.380069
$216$	1.00000	0.0680414
$217$	2.98941	0.202934
$218$	15.5044	1.05009
$219$	−13.9756	−0.944383
$220$	−8.67863	−0.585113
$221$	−1.60542	−0.107992
$222$	10.0665	0.675619
$223$	−23.3905	−1.56634	−0.783172	−	0.621805i	$-0.786400\pi$
$223$	−23.3905	−1.56634	−0.783172	+	0.621805i	$0.786400\pi$
$224$	0.327596	0.0218884
$225$	−1.80104	−0.120070
$226$	16.3227	1.08577
$227$	−16.0329	−1.06414	−0.532070	−	0.846700i	$-0.678586\pi$
$227$	−16.0329	−1.06414	−0.532070	+	0.846700i	$0.678586\pi$
$228$	1.42429	0.0943259
$229$	7.38355	0.487919	0.243959	−	0.969785i	$-0.421554\pi$
$229$	7.38355	0.487919	0.243959	+	0.969785i	$0.421554\pi$
$230$	10.0679	0.663856
$231$	−1.58959	−0.104588
$232$	4.09609	0.268922
$233$	−13.8005	−0.904099	−0.452050	−	0.891993i	$-0.649307\pi$
$233$	−13.8005	−0.904099	−0.452050	+	0.891993i	$0.649307\pi$
$234$	−1.00000	−0.0653720
$235$	−19.8165	−1.29269
$236$	−6.56032	−0.427040
$237$	−6.52386	−0.423770
$238$	0.525929	0.0340909
$239$	14.5338	0.940112	0.470056	−	0.882637i	$-0.344234\pi$
$239$	14.5338	0.940112	0.470056	+	0.882637i	$0.344234\pi$
$240$	1.78856	0.115451
$241$	−13.6312	−0.878064	−0.439032	−	0.898471i	$-0.644679\pi$
$241$	−13.6312	−0.878064	−0.439032	+	0.898471i	$0.644679\pi$
$242$	12.5448	0.806408
$243$	1.00000	0.0641500
$244$	−6.82393	−0.436857
$245$	−12.3280	−0.787606
$246$	7.43278	0.473897
$247$	−1.42429	−0.0906254
$248$	9.12528	0.579456
$249$	5.30980	0.336495
$250$	−12.1641	−0.769325
$251$	−1.87650	−0.118443	−0.0592217	−	0.998245i	$-0.518862\pi$
$251$	−1.87650	−0.118443	−0.0592217	+	0.998245i	$0.518862\pi$
$252$	0.327596	0.0206366
$253$	−27.3137	−1.71720
$254$	6.72756	0.422125
$255$	2.87139	0.179813
$256$	1.00000	0.0625000
$257$	−18.5169	−1.15505	−0.577527	−	0.816372i	$-0.695982\pi$
$257$	−18.5169	−1.15505	−0.577527	+	0.816372i	$0.695982\pi$
$258$	3.11586	0.193985
$259$	3.29775	0.204912
$260$	−1.78856	−0.110922
$261$	4.09609	0.253542
$262$	−10.2917	−0.635822
$263$	27.1806	1.67603	0.838015	−	0.545648i	$-0.183716\pi$
$263$	27.1806	1.67603	0.838015	+	0.545648i	$0.183716\pi$
$264$	−4.85229	−0.298638
$265$	0.0608736	0.00373943
$266$	0.466592	0.0286086
$267$	−13.1160	−0.802687
$268$	2.98653	0.182432
$269$	8.48305	0.517221	0.258610	−	0.965982i	$-0.416735\pi$
$269$	8.48305	0.517221	0.258610	+	0.965982i	$0.416735\pi$
$270$	1.78856	0.108848
$271$	13.7121	0.832951	0.416475	−	0.909147i	$-0.363265\pi$
$271$	13.7121	0.832951	0.416475	+	0.909147i	$0.363265\pi$
$272$	1.60542	0.0973427
$273$	−0.327596	−0.0198270
$274$	8.07567	0.487869
$275$	8.73920	0.526993
$276$	5.62903	0.338828
$277$	0.792420	0.0476119	0.0238059	−	0.999717i	$-0.492422\pi$
$277$	0.792420	0.0476119	0.0238059	+	0.999717i	$0.492422\pi$
$278$	−2.04231	−0.122489
$279$	9.12528	0.546316
$280$	0.585926	0.0350158
$281$	0.251176	0.0149839	0.00749195	−	0.999972i	$-0.497615\pi$
$281$	0.251176	0.0149839	0.00749195	+	0.999972i	$0.497615\pi$
$282$	−11.0796	−0.659780
$283$	−22.9126	−1.36201	−0.681006	−	0.732277i	$-0.738457\pi$
$283$	−22.9126	−1.36201	−0.681006	+	0.732277i	$0.738457\pi$
$284$	14.2442	0.845236
$285$	2.54743	0.150897
$286$	4.85229	0.286922
$287$	2.43495	0.143731
$288$	1.00000	0.0589256
$289$	−14.4226	−0.848390
$290$	7.32612	0.430205
$291$	8.44438	0.495018
$292$	−13.9756	−0.817860
$293$	−25.5969	−1.49539	−0.747693	−	0.664044i	$-0.768839\pi$
$293$	−25.5969	−1.49539	−0.747693	+	0.664044i	$0.768839\pi$
$294$	−6.89268	−0.401989
$295$	−11.7335	−0.683153
$296$	10.0665	0.585103
$297$	−4.85229	−0.281559
$298$	−2.43705	−0.141175
$299$	−5.62903	−0.325535
$300$	−1.80104	−0.103983
$301$	1.02074	0.0588347
$302$	9.18020	0.528261
$303$	10.1762	0.584605
$304$	1.42429	0.0816887
$305$	−12.2050	−0.698858
$306$	1.60542	0.0917756
$307$	−7.43248	−0.424194	−0.212097	−	0.977249i	$-0.568029\pi$
$307$	−7.43248	−0.424194	−0.212097	+	0.977249i	$0.568029\pi$
$308$	−1.58959	−0.0905755
$309$	1.00000	0.0568880
$310$	16.3211	0.926979
$311$	−13.7862	−0.781746	−0.390873	−	0.920445i	$-0.627827\pi$
$311$	−13.7862	−0.781746	−0.390873	+	0.920445i	$0.627827\pi$
$312$	−1.00000	−0.0566139
$313$	18.8499	1.06546	0.532730	−	0.846285i	$-0.321166\pi$
$313$	18.8499	1.06546	0.532730	+	0.846285i	$0.321166\pi$
$314$	21.0011	1.18516
$315$	0.585926	0.0330132
$316$	−6.52386	−0.366996
$317$	27.4394	1.54115	0.770576	−	0.637348i	$-0.219968\pi$
$317$	27.4394	1.54115	0.770576	+	0.637348i	$0.219968\pi$
$318$	0.0340349	0.00190858
$319$	−19.8755	−1.11281
$320$	1.78856	0.0999837
$321$	19.2490	1.07438
$322$	1.84405	0.102765
$323$	2.28658	0.127229
$324$	1.00000	0.0555556
$325$	1.80104	0.0999040
$326$	−4.76711	−0.264026
$327$	15.5044	0.857393
$328$	7.43278	0.410407
$329$	−3.62963	−0.200108
$330$	−8.67863	−0.477743
$331$	−6.35189	−0.349131	−0.174566	−	0.984646i	$-0.555852\pi$
$331$	−6.35189	−0.349131	−0.174566	+	0.984646i	$0.555852\pi$
$332$	5.30980	0.291413
$333$	10.0665	0.551641
$334$	2.77768	0.151988
$335$	5.34160	0.291843
$336$	0.327596	0.0178718
$337$	−18.1461	−0.988483	−0.494241	−	0.869325i	$-0.664554\pi$
$337$	−18.1461	−0.988483	−0.494241	+	0.869325i	$0.664554\pi$
$338$	1.00000	0.0543928
$339$	16.3227	0.886530
$340$	2.87139	0.155723
$341$	−44.2786	−2.39782
$342$	1.42429	0.0770168
$343$	−4.55119	−0.245741
$344$	3.11586	0.167996
$345$	10.0679	0.542036
$346$	7.70162	0.414042
$347$	22.6085	1.21369	0.606845	−	0.794820i	$-0.292435\pi$
$347$	22.6085	1.21369	0.606845	+	0.794820i	$0.292435\pi$
$348$	4.09609	0.219574
$349$	−0.414156	−0.0221693	−0.0110846	−	0.999939i	$-0.503528\pi$
$349$	−0.414156	−0.0221693	−0.0110846	+	0.999939i	$0.503528\pi$
$350$	−0.590015	−0.0315376
$351$	−1.00000	−0.0533761
$352$	−4.85229	−0.258628
$353$	25.3197	1.34763	0.673815	−	0.738900i	$-0.264654\pi$
$353$	25.3197	1.34763	0.673815	+	0.738900i	$0.264654\pi$
$354$	−6.56032	−0.348677
$355$	25.4766	1.35216
$356$	−13.1160	−0.695147
$357$	0.525929	0.0278351
$358$	−19.3613	−1.02328
$359$	25.4768	1.34461	0.672307	−	0.740272i	$-0.265303\pi$
$359$	25.4768	1.34461	0.672307	+	0.740272i	$0.265303\pi$
$360$	1.78856	0.0942655
$361$	−16.9714	−0.893231
$362$	1.26309	0.0663864
$363$	12.5448	0.658429
$364$	−0.327596	−0.0171707
$365$	−24.9962	−1.30836
$366$	−6.82393	−0.356693
$367$	10.0623	0.525248	0.262624	−	0.964898i	$-0.415412\pi$
$367$	10.0623	0.525248	0.262624	+	0.964898i	$0.415412\pi$
$368$	5.62903	0.293433
$369$	7.43278	0.386935
$370$	18.0046	0.936013
$371$	0.0111497	0.000578864 0
$372$	9.12528	0.473124
$373$	1.66576	0.0862496	0.0431248	−	0.999070i	$-0.486269\pi$
$373$	1.66576	0.0862496	0.0431248	+	0.999070i	$0.486269\pi$
$374$	−7.78996	−0.402809
$375$	−12.1641	−0.628151
$376$	−11.0796	−0.571386
$377$	−4.09609	−0.210960
$378$	0.327596	0.0168497
$379$	2.67524	0.137418	0.0687089	−	0.997637i	$-0.478112\pi$
$379$	2.67524	0.137418	0.0687089	+	0.997637i	$0.478112\pi$
$380$	2.54743	0.130681
$381$	6.72756	0.344663
$382$	16.0940	0.823442
$383$	−13.6728	−0.698648	−0.349324	−	0.937002i	$-0.613589\pi$
$383$	−13.6728	−0.698648	−0.349324	+	0.937002i	$0.613589\pi$
$384$	1.00000	0.0510310
$385$	−2.84309	−0.144897
$386$	11.0494	0.562399
$387$	3.11586	0.158388
$388$	8.44438	0.428698
$389$	19.5623	0.991850	0.495925	−	0.868365i	$-0.334829\pi$
$389$	19.5623	0.991850	0.495925	+	0.868365i	$0.334829\pi$
$390$	−1.78856	−0.0905674
$391$	9.03694	0.457018
$392$	−6.89268	−0.348133
$393$	−10.2917	−0.519147
$394$	−17.1674	−0.864880
$395$	−11.6683	−0.587097
$396$	−4.85229	−0.243837
$397$	−5.18513	−0.260234	−0.130117	−	0.991499i	$-0.541535\pi$
$397$	−5.18513	−0.260234	−0.130117	+	0.991499i	$0.541535\pi$
$398$	−23.7518	−1.19057
$399$	0.466592	0.0233588
$400$	−1.80104	−0.0900522
$401$	15.2041	0.759255	0.379628	−	0.925139i	$-0.376052\pi$
$401$	15.2041	0.759255	0.379628	+	0.925139i	$0.376052\pi$
$402$	2.98653	0.148955
$403$	−9.12528	−0.454563
$404$	10.1762	0.506283
$405$	1.78856	0.0888744
$406$	1.34186	0.0665956
$407$	−48.8456	−2.42119
$408$	1.60542	0.0794800
$409$	23.7697	1.17533	0.587667	−	0.809103i	$-0.300047\pi$
$409$	23.7697	1.17533	0.587667	+	0.809103i	$0.300047\pi$
$410$	13.2940	0.656544
$411$	8.07567	0.398343
$412$	1.00000	0.0492665
$413$	−2.14913	−0.105752
$414$	5.62903	0.276652
$415$	9.49691	0.466185
$416$	−1.00000	−0.0490290
$417$	−2.04231	−0.100012
$418$	−6.91108	−0.338032
$419$	−31.9619	−1.56144	−0.780720	−	0.624881i	$-0.785147\pi$
$419$	−31.9619	−1.56144	−0.780720	+	0.624881i	$0.785147\pi$
$420$	0.585926	0.0285903
$421$	8.75934	0.426904	0.213452	−	0.976954i	$-0.431529\pi$
$421$	8.75934	0.426904	0.213452	+	0.976954i	$0.431529\pi$
$422$	−20.5256	−0.999171
$423$	−11.0796	−0.538708
$424$	0.0340349	0.00165288
$425$	−2.89143	−0.140255
$426$	14.2442	0.690132
$427$	−2.23549	−0.108183
$428$	19.2490	0.930438
$429$	4.85229	0.234271
$430$	5.57291	0.268749
$431$	−31.1691	−1.50136	−0.750680	−	0.660666i	$-0.770274\pi$
$431$	−31.1691	−1.50136	−0.750680	+	0.660666i	$0.770274\pi$
$432$	1.00000	0.0481125
$433$	9.47459	0.455320	0.227660	−	0.973741i	$-0.426893\pi$
$433$	9.47459	0.455320	0.227660	+	0.973741i	$0.426893\pi$
$434$	2.98941	0.143496
$435$	7.32612	0.351261
$436$	15.5044	0.742524
$437$	8.01737	0.383523
$438$	−13.9756	−0.667780
$439$	−8.73654	−0.416972	−0.208486	−	0.978025i	$-0.566854\pi$
$439$	−8.73654	−0.416972	−0.208486	+	0.978025i	$0.566854\pi$
$440$	−8.67863	−0.413738
$441$	−6.89268	−0.328223
$442$	−1.60542	−0.0763619
$443$	−30.5265	−1.45036	−0.725178	−	0.688562i	$-0.758242\pi$
$443$	−30.5265	−1.45036	−0.725178	+	0.688562i	$0.758242\pi$
$444$	10.0665	0.477735
$445$	−23.4588	−1.11205
$446$	−23.3905	−1.10757
$447$	−2.43705	−0.115269
$448$	0.327596	0.0154775
$449$	−27.7482	−1.30952	−0.654758	−	0.755838i	$-0.727230\pi$
$449$	−27.7482	−1.30952	−0.654758	+	0.755838i	$0.727230\pi$
$450$	−1.80104	−0.0849020
$451$	−36.0660	−1.69828
$452$	16.3227	0.767757
$453$	9.18020	0.431324
$454$	−16.0329	−0.752461
$455$	−0.585926	−0.0274687
$456$	1.42429	0.0666985
$457$	−28.1737	−1.31791	−0.658954	−	0.752183i	$-0.729001\pi$
$457$	−28.1737	−1.31791	−0.658954	+	0.752183i	$0.729001\pi$
$458$	7.38355	0.345011
$459$	1.60542	0.0749345
$460$	10.0679	0.469417
$461$	20.6173	0.960243	0.480122	−	0.877202i	$-0.340593\pi$
$461$	20.6173	0.960243	0.480122	+	0.877202i	$0.340593\pi$
$462$	−1.58959	−0.0739546
$463$	−21.8807	−1.01688	−0.508442	−	0.861096i	$-0.669778\pi$
$463$	−21.8807	−1.01688	−0.508442	+	0.861096i	$0.669778\pi$
$464$	4.09609	0.190156
$465$	16.3211	0.756875
$466$	−13.8005	−0.639295
$467$	−34.1345	−1.57955	−0.789777	−	0.613394i	$-0.789804\pi$
$467$	−34.1345	−1.57955	−0.789777	+	0.613394i	$0.789804\pi$
$468$	−1.00000	−0.0462250
$469$	0.978377	0.0451773
$470$	−19.8165	−0.914069
$471$	21.0011	0.967680
$472$	−6.56032	−0.301963
$473$	−15.1191	−0.695175
$474$	−6.52386	−0.299651
$475$	−2.56521	−0.117700
$476$	0.525929	0.0241059
$477$	0.0340349	0.00155835
$478$	14.5338	0.664759
$479$	21.3228	0.974262	0.487131	−	0.873329i	$-0.338043\pi$
$479$	21.3228	0.974262	0.487131	+	0.873329i	$0.338043\pi$
$480$	1.78856	0.0816363
$481$	−10.0665	−0.458993
$482$	−13.6312	−0.620885
$483$	1.84405	0.0839071
$484$	12.5448	0.570216
$485$	15.1033	0.685805
$486$	1.00000	0.0453609
$487$	−22.7417	−1.03052	−0.515262	−	0.857033i	$-0.672305\pi$
$487$	−22.7417	−1.03052	−0.515262	+	0.857033i	$0.672305\pi$
$488$	−6.82393	−0.308905
$489$	−4.76711	−0.215576
$490$	−12.3280	−0.556922
$491$	−15.9175	−0.718346	−0.359173	−	0.933271i	$-0.616941\pi$
$491$	−15.9175	−0.718346	−0.359173	+	0.933271i	$0.616941\pi$
$492$	7.43278	0.335096
$493$	6.57594	0.296165
$494$	−1.42429	−0.0640819
$495$	−8.67863	−0.390076
$496$	9.12528	0.409737
$497$	4.66633	0.209314
$498$	5.30980	0.237938
$499$	41.3480	1.85099	0.925496	−	0.378756i	$-0.123648\pi$
$499$	41.3480	1.85099	0.925496	+	0.378756i	$0.123648\pi$
$500$	−12.1641	−0.543995
$501$	2.77768	0.124097
$502$	−1.87650	−0.0837522
$503$	2.17512	0.0969837	0.0484919	−	0.998824i	$-0.484559\pi$
$503$	2.17512	0.0969837	0.0484919	+	0.998824i	$0.484559\pi$
$504$	0.327596	0.0145923
$505$	18.2007	0.809921
$506$	−27.3137	−1.21424
$507$	1.00000	0.0444116
$508$	6.72756	0.298487
$509$	−11.7048	−0.518806	−0.259403	−	0.965769i	$-0.583526\pi$
$509$	−11.7048	−0.518806	−0.259403	+	0.965769i	$0.583526\pi$
$510$	2.87139	0.127147
$511$	−4.57835	−0.202534
$512$	1.00000	0.0441942
$513$	1.42429	0.0628840
$514$	−18.5169	−0.816746
$515$	1.78856	0.0788135
$516$	3.11586	0.137168
$517$	53.7614	2.36442
$518$	3.29775	0.144895
$519$	7.70162	0.338064
$520$	−1.78856	−0.0784337
$521$	−19.5333	−0.855772	−0.427886	−	0.903833i	$-0.640741\pi$
$521$	−19.5333	−0.855772	−0.427886	+	0.903833i	$0.640741\pi$
$522$	4.09609	0.179281
$523$	−23.5001	−1.02759	−0.513794	−	0.857914i	$-0.671760\pi$
$523$	−23.5001	−1.02759	−0.513794	+	0.857914i	$0.671760\pi$
$524$	−10.2917	−0.449594
$525$	−0.590015	−0.0257504
$526$	27.1806	1.18513
$527$	14.6499	0.638159
$528$	−4.85229	−0.211169
$529$	8.68595	0.377650
$530$	0.0608736	0.00264418
$531$	−6.56032	−0.284693
$532$	0.466592	0.0202293
$533$	−7.43278	−0.321949
$534$	−13.1160	−0.567585
$535$	34.4281	1.48846
$536$	2.98653	0.128999
$537$	−19.3613	−0.835501
$538$	8.48305	0.365730
$539$	33.4453	1.44059
$540$	1.78856	0.0769675
$541$	12.5265	0.538558	0.269279	−	0.963062i	$-0.413215\pi$
$541$	12.5265	0.538558	0.269279	+	0.963062i	$0.413215\pi$
$542$	13.7121	0.588985
$543$	1.26309	0.0542043
$544$	1.60542	0.0688317
$545$	27.7305	1.18784
$546$	−0.327596	−0.0140198
$547$	−22.9069	−0.979427	−0.489714	−	0.871883i	$-0.662899\pi$
$547$	−22.9069	−0.979427	−0.489714	+	0.871883i	$0.662899\pi$
$548$	8.07567	0.344975
$549$	−6.82393	−0.291238
$550$	8.73920	0.372641
$551$	5.83403	0.248538
$552$	5.62903	0.239587
$553$	−2.13719	−0.0908826
$554$	0.792420	0.0336667
$555$	18.0046	0.764251
$556$	−2.04231	−0.0866131
$557$	2.89110	0.122500	0.0612498	−	0.998122i	$-0.480491\pi$
$557$	2.89110	0.122500	0.0612498	+	0.998122i	$0.480491\pi$
$558$	9.12528	0.386304
$559$	−3.11586	−0.131787
$560$	0.585926	0.0247599
$561$	−7.78996	−0.328892
$562$	0.251176	0.0105952
$563$	10.5501	0.444634	0.222317	−	0.974974i	$-0.428638\pi$
$563$	10.5501	0.444634	0.222317	+	0.974974i	$0.428638\pi$
$564$	−11.0796	−0.466535
$565$	29.1942	1.22821
$566$	−22.9126	−0.963089
$567$	0.327596	0.0137577
$568$	14.2442	0.597672
$569$	19.0627	0.799148	0.399574	−	0.916701i	$-0.369158\pi$
$569$	19.0627	0.799148	0.399574	+	0.916701i	$0.369158\pi$
$570$	2.54743	0.106700
$571$	8.32391	0.348345	0.174172	−	0.984715i	$-0.444275\pi$
$571$	8.32391	0.348345	0.174172	+	0.984715i	$0.444275\pi$
$572$	4.85229	0.202885
$573$	16.0940	0.672337
$574$	2.43495	0.101633
$575$	−10.1381	−0.422789
$576$	1.00000	0.0416667
$577$	0.685674	0.0285450	0.0142725	−	0.999898i	$-0.495457\pi$
$577$	0.685674	0.0285450	0.0142725	+	0.999898i	$0.495457\pi$
$578$	−14.4226	−0.599902
$579$	11.0494	0.459197
$580$	7.32612	0.304201
$581$	1.73947	0.0721654
$582$	8.44438	0.350031
$583$	−0.165147	−0.00683971
$584$	−13.9756	−0.578314
$585$	−1.78856	−0.0739480
$586$	−25.5969	−1.05740
$587$	−17.9424	−0.740563	−0.370281	−	0.928920i	$-0.620739\pi$
$587$	−17.9424	−0.740563	−0.370281	+	0.928920i	$0.620739\pi$
$588$	−6.89268	−0.284249
$589$	12.9971	0.535534
$590$	−11.7335	−0.483062
$591$	−17.1674	−0.706172
$592$	10.0665	0.413731
$593$	5.69264	0.233769	0.116884	−	0.993146i	$-0.462709\pi$
$593$	5.69264	0.233769	0.116884	+	0.993146i	$0.462709\pi$
$594$	−4.85229	−0.199092
$595$	0.940657	0.0385632
$596$	−2.43705	−0.0998255
$597$	−23.7518	−0.972098
$598$	−5.62903	−0.230188
$599$	−7.49483	−0.306231	−0.153115	−	0.988208i	$-0.548931\pi$
$599$	−7.49483	−0.306231	−0.153115	+	0.988208i	$0.548931\pi$
$600$	−1.80104	−0.0735273
$601$	−20.7656	−0.847046	−0.423523	−	0.905885i	$-0.639207\pi$
$601$	−20.7656	−0.847046	−0.423523	+	0.905885i	$0.639207\pi$
$602$	1.02074	0.0416024
$603$	2.98653	0.121621
$604$	9.18020	0.373537
$605$	22.4371	0.912198
$606$	10.1762	0.413378
$607$	−12.9123	−0.524093	−0.262046	−	0.965055i	$-0.584397\pi$
$607$	−12.9123	−0.524093	−0.262046	+	0.965055i	$0.584397\pi$
$608$	1.42429	0.0577626
$609$	1.34186	0.0543751
$610$	−12.2050	−0.494167
$611$	11.0796	0.448232
$612$	1.60542	0.0648952
$613$	24.6570	0.995885	0.497943	−	0.867210i	$-0.334089\pi$
$613$	24.6570	0.995885	0.497943	+	0.867210i	$0.334089\pi$
$614$	−7.43248	−0.299950
$615$	13.2940	0.536066
$616$	−1.58959	−0.0640465
$617$	−29.0629	−1.17003	−0.585013	−	0.811024i	$-0.698911\pi$
$617$	−29.0629	−1.17003	−0.585013	+	0.811024i	$0.698911\pi$
$618$	1.00000	0.0402259
$619$	−27.7258	−1.11440	−0.557198	−	0.830380i	$-0.688124\pi$
$619$	−27.7258	−1.11440	−0.557198	+	0.830380i	$0.688124\pi$
$620$	16.3211	0.655473
$621$	5.62903	0.225885
$622$	−13.7862	−0.552778
$623$	−4.29676	−0.172146
$624$	−1.00000	−0.0400320
$625$	−12.7510	−0.510041
$626$	18.8499	0.753394
$627$	−6.91108	−0.276002
$628$	21.0011	0.838035
$629$	16.1609	0.644379
$630$	0.585926	0.0233439
$631$	−17.6055	−0.700865	−0.350433	−	0.936588i	$-0.613965\pi$
$631$	−17.6055	−0.700865	−0.350433	+	0.936588i	$0.613965\pi$
$632$	−6.52386	−0.259505
$633$	−20.5256	−0.815820
$634$	27.4394	1.08976
$635$	12.0327	0.477502
$636$	0.0340349	0.00134957
$637$	6.89268	0.273098
$638$	−19.8755	−0.786877
$639$	14.2442	0.563490
$640$	1.78856	0.0706991
$641$	5.26338	0.207891	0.103946	−	0.994583i	$-0.466853\pi$
$641$	5.26338	0.207891	0.103946	+	0.994583i	$0.466853\pi$
$642$	19.2490	0.759699
$643$	5.21867	0.205804	0.102902	−	0.994691i	$-0.467187\pi$
$643$	5.21867	0.205804	0.102902	+	0.994691i	$0.467187\pi$
$644$	1.84405	0.0726657
$645$	5.57291	0.219433
$646$	2.28658	0.0899643
$647$	3.63624	0.142955	0.0714777	−	0.997442i	$-0.477229\pi$
$647$	3.63624	0.142955	0.0714777	+	0.997442i	$0.477229\pi$
$648$	1.00000	0.0392837
$649$	31.8326	1.24954
$650$	1.80104	0.0706428
$651$	2.98941	0.117164
$652$	−4.76711	−0.186694
$653$	10.0214	0.392168	0.196084	−	0.980587i	$-0.437178\pi$
$653$	10.0214	0.392168	0.196084	+	0.980587i	$0.437178\pi$
$654$	15.5044	0.606268
$655$	−18.4073	−0.719233
$656$	7.43278	0.290201
$657$	−13.9756	−0.545240
$658$	−3.62963	−0.141498
$659$	−33.5987	−1.30882	−0.654411	−	0.756139i	$-0.727083\pi$
$659$	−33.5987	−1.30882	−0.654411	+	0.756139i	$0.727083\pi$
$660$	−8.67863	−0.337815
$661$	−5.22774	−0.203335	−0.101668	−	0.994818i	$-0.532418\pi$
$661$	−5.22774	−0.203335	−0.101668	+	0.994818i	$0.532418\pi$
$662$	−6.35189	−0.246873
$663$	−1.60542	−0.0623493
$664$	5.30980	0.206060
$665$	0.834529	0.0323617
$666$	10.0665	0.390069
$667$	23.0570	0.892772
$668$	2.77768	0.107471
$669$	−23.3905	−0.904329
$670$	5.34160	0.206364
$671$	33.1117	1.27826
$672$	0.327596	0.0126373
$673$	−10.8626	−0.418723	−0.209361	−	0.977838i	$-0.567139\pi$
$673$	−10.8626	−0.418723	−0.209361	+	0.977838i	$0.567139\pi$
$674$	−18.1461	−0.698963
$675$	−1.80104	−0.0693222
$676$	1.00000	0.0384615
$677$	21.1669	0.813508	0.406754	−	0.913538i	$-0.366661\pi$
$677$	21.1669	0.813508	0.406754	+	0.913538i	$0.366661\pi$
$678$	16.3227	0.626871
$679$	2.76635	0.106163
$680$	2.87139	0.110113
$681$	−16.0329	−0.614382
$682$	−44.2786	−1.69551
$683$	−1.07262	−0.0410426	−0.0205213	−	0.999789i	$-0.506533\pi$
$683$	−1.07262	−0.0410426	−0.0205213	+	0.999789i	$0.506533\pi$
$684$	1.42429	0.0544591
$685$	14.4438	0.551871
$686$	−4.55119	−0.173765
$687$	7.38355	0.281700
$688$	3.11586	0.118791
$689$	−0.0340349	−0.00129663
$690$	10.0679	0.383277
$691$	15.6179	0.594133	0.297066	−	0.954857i	$-0.403992\pi$
$691$	15.6179	0.594133	0.297066	+	0.954857i	$0.403992\pi$
$692$	7.70162	0.292772
$693$	−1.58959	−0.0603837
$694$	22.6085	0.858208
$695$	−3.65279	−0.138558
$696$	4.09609	0.155262
$697$	11.9327	0.451984
$698$	−0.414156	−0.0156760
$699$	−13.8005	−0.521982
$700$	−0.590015	−0.0223005
$701$	−8.50085	−0.321073	−0.160536	−	0.987030i	$-0.551322\pi$
$701$	−8.50085	−0.321073	−0.160536	+	0.987030i	$0.551322\pi$
$702$	−1.00000	−0.0377426
$703$	14.3376	0.540754
$704$	−4.85229	−0.182878
$705$	−19.8165	−0.746334
$706$	25.3197	0.952918
$707$	3.33367	0.125376
$708$	−6.56032	−0.246552
$709$	31.5413	1.18456	0.592278	−	0.805733i	$-0.298229\pi$
$709$	31.5413	1.18456	0.592278	+	0.805733i	$0.298229\pi$
$710$	25.4766	0.956119
$711$	−6.52386	−0.244664
$712$	−13.1160	−0.491543
$713$	51.3665	1.92369
$714$	0.525929	0.0196824
$715$	8.67863	0.324562
$716$	−19.3613	−0.723565
$717$	14.5338	0.542774
$718$	25.4768	0.950786
$719$	−30.1144	−1.12308	−0.561538	−	0.827451i	$-0.689790\pi$
$719$	−30.1144	−1.12308	−0.561538	+	0.827451i	$0.689790\pi$
$720$	1.78856	0.0666558
$721$	0.327596	0.0122003
$722$	−16.9714	−0.631610
$723$	−13.6312	−0.506951
$724$	1.26309	0.0469423
$725$	−7.37725	−0.273984
$726$	12.5448	0.465580
$727$	−7.19118	−0.266706	−0.133353	−	0.991069i	$-0.542574\pi$
$727$	−7.19118	−0.266706	−0.133353	+	0.991069i	$0.542574\pi$
$728$	−0.327596	−0.0121415
$729$	1.00000	0.0370370
$730$	−24.9962	−0.925152
$731$	5.00225	0.185015
$732$	−6.82393	−0.252220
$733$	−21.9397	−0.810360	−0.405180	−	0.914237i	$-0.632791\pi$
$733$	−21.9397	−0.810360	−0.405180	+	0.914237i	$0.632791\pi$
$734$	10.0623	0.371407
$735$	−12.3280	−0.454725
$736$	5.62903	0.207489
$737$	−14.4915	−0.533803
$738$	7.43278	0.273604
$739$	−20.0471	−0.737445	−0.368723	−	0.929539i	$-0.620205\pi$
$739$	−20.0471	−0.737445	−0.368723	+	0.929539i	$0.620205\pi$
$740$	18.0046	0.661861
$741$	−1.42429	−0.0523226
$742$	0.0111497	0.000409319 0
$743$	7.48156	0.274472	0.137236	−	0.990538i	$-0.456178\pi$
$743$	7.48156	0.274472	0.137236	+	0.990538i	$0.456178\pi$
$744$	9.12528	0.334549
$745$	−4.35882	−0.159695
$746$	1.66576	0.0609877
$747$	5.30980	0.194275
$748$	−7.78996	−0.284829
$749$	6.30591	0.230413
$750$	−12.1641	−0.444170
$751$	12.7212	0.464201	0.232101	−	0.972692i	$-0.425440\pi$
$751$	12.7212	0.464201	0.232101	+	0.972692i	$0.425440\pi$
$752$	−11.0796	−0.404031
$753$	−1.87650	−0.0683834
$754$	−4.09609	−0.149171
$755$	16.4194	0.597562
$756$	0.327596	0.0119146
$757$	15.1473	0.550540	0.275270	−	0.961367i	$-0.411233\pi$
$757$	15.1473	0.550540	0.275270	+	0.961367i	$0.411233\pi$
$758$	2.67524	0.0971690
$759$	−27.3137	−0.991425
$760$	2.54743	0.0924051
$761$	−34.8605	−1.26369	−0.631846	−	0.775094i	$-0.717703\pi$
$761$	−34.8605	−1.26369	−0.631846	+	0.775094i	$0.717703\pi$
$762$	6.72756	0.243714
$763$	5.07917	0.183878
$764$	16.0940	0.582261
$765$	2.87139	0.103815
$766$	−13.6728	−0.494019
$767$	6.56032	0.236879
$768$	1.00000	0.0360844
$769$	22.0048	0.793514	0.396757	−	0.917924i	$-0.370136\pi$
$769$	22.0048	0.793514	0.396757	+	0.917924i	$0.370136\pi$
$770$	−2.84309	−0.102458
$771$	−18.5169	−0.666870
$772$	11.0494	0.397676
$773$	53.6063	1.92808	0.964042	−	0.265751i	$-0.0856199\pi$
$773$	53.6063	1.92808	0.964042	+	0.265751i	$0.0856199\pi$
$774$	3.11586	0.111997
$775$	−16.4350	−0.590364
$776$	8.44438	0.303135
$777$	3.29775	0.118306
$778$	19.5623	0.701344
$779$	10.5864	0.379299
$780$	−1.78856	−0.0640408
$781$	−69.1169	−2.47320
$782$	9.03694	0.323160
$783$	4.09609	0.146382
$784$	−6.89268	−0.246167
$785$	37.5618	1.34064
$786$	−10.2917	−0.367092
$787$	−17.9484	−0.639793	−0.319897	−	0.947452i	$-0.603648\pi$
$787$	−17.9484	−0.639793	−0.319897	+	0.947452i	$0.603648\pi$
$788$	−17.1674	−0.611563
$789$	27.1806	0.967656
$790$	−11.6683	−0.415141
$791$	5.34727	0.190127
$792$	−4.85229	−0.172419
$793$	6.82393	0.242325
$794$	−5.18513	−0.184013
$795$	0.0608736	0.00215896
$796$	−23.7518	−0.841861
$797$	8.30234	0.294084	0.147042	−	0.989130i	$-0.453025\pi$
$797$	8.30234	0.294084	0.147042	+	0.989130i	$0.453025\pi$
$798$	0.466592	0.0165172
$799$	−17.7874	−0.629272
$800$	−1.80104	−0.0636765
$801$	−13.1160	−0.463432
$802$	15.2041	0.536875
$803$	67.8137	2.39309
$804$	2.98653	0.105327
$805$	3.29820	0.116246
$806$	−9.12528	−0.321424
$807$	8.48305	0.298617
$808$	10.1762	0.357996
$809$	−40.2389	−1.41473	−0.707363	−	0.706851i	$-0.750115\pi$
$809$	−40.2389	−1.41473	−0.707363	+	0.706851i	$0.750115\pi$
$810$	1.78856	0.0628437
$811$	−35.0201	−1.22972	−0.614860	−	0.788636i	$-0.710788\pi$
$811$	−35.0201	−1.22972	−0.614860	+	0.788636i	$0.710788\pi$
$812$	1.34186	0.0470902
$813$	13.7121	0.480904
$814$	−48.8456	−1.71204
$815$	−8.52627	−0.298662
$816$	1.60542	0.0562009
$817$	4.43789	0.155262
$818$	23.7697	0.831086
$819$	−0.327596	−0.0114471
$820$	13.2940	0.464246
$821$	−24.7122	−0.862461	−0.431231	−	0.902242i	$-0.641921\pi$
$821$	−24.7122	−0.862461	−0.431231	+	0.902242i	$0.641921\pi$
$822$	8.07567	0.281671
$823$	−49.5925	−1.72869	−0.864343	−	0.502903i	$-0.832265\pi$
$823$	−49.5925	−1.72869	−0.864343	+	0.502903i	$0.832265\pi$
$824$	1.00000	0.0348367
$825$	8.73920	0.304260
$826$	−2.14913	−0.0747780
$827$	−10.3137	−0.358642	−0.179321	−	0.983791i	$-0.557390\pi$
$827$	−10.3137	−0.358642	−0.179321	+	0.983791i	$0.557390\pi$
$828$	5.62903	0.195622
$829$	15.3441	0.532924	0.266462	−	0.963845i	$-0.414145\pi$
$829$	15.3441	0.532924	0.266462	+	0.963845i	$0.414145\pi$
$830$	9.49691	0.329642
$831$	0.792420	0.0274887
$832$	−1.00000	−0.0346688
$833$	−11.0656	−0.383401
$834$	−2.04231	−0.0707193
$835$	4.96805	0.171926
$836$	−6.91108	−0.239025
$837$	9.12528	0.315416
$838$	−31.9619	−1.10410
$839$	−44.0433	−1.52055	−0.760273	−	0.649604i	$-0.774935\pi$
$839$	−44.0433	−1.52055	−0.760273	+	0.649604i	$0.774935\pi$
$840$	0.585926	0.0202164
$841$	−12.2220	−0.421449
$842$	8.75934	0.301867
$843$	0.251176	0.00865096
$844$	−20.5256	−0.706521
$845$	1.78856	0.0615284
$846$	−11.0796	−0.380924
$847$	4.10962	0.141208
$848$	0.0340349	0.00116876
$849$	−22.9126	−0.786359
$850$	−2.89143	−0.0991752
$851$	56.6646	1.94244
$852$	14.2442	0.487997
$853$	33.4504	1.14532	0.572660	−	0.819793i	$-0.305912\pi$
$853$	33.4504	1.14532	0.572660	+	0.819793i	$0.305912\pi$
$854$	−2.23549	−0.0764970
$855$	2.54743	0.0871204
$856$	19.2490	0.657919
$857$	36.9757	1.26307	0.631533	−	0.775349i	$-0.282426\pi$
$857$	36.9757	1.26307	0.631533	+	0.775349i	$0.282426\pi$
$858$	4.85229	0.165655
$859$	6.73030	0.229635	0.114817	−	0.993387i	$-0.463372\pi$
$859$	6.73030	0.229635	0.114817	+	0.993387i	$0.463372\pi$
$860$	5.57291	0.190034
$861$	2.43495	0.0829829
$862$	−31.1691	−1.06162
$863$	−28.5718	−0.972597	−0.486298	−	0.873793i	$-0.661653\pi$
$863$	−28.5718	−0.972597	−0.486298	+	0.873793i	$0.661653\pi$
$864$	1.00000	0.0340207
$865$	13.7748	0.468358
$866$	9.47459	0.321960
$867$	−14.4226	−0.489818
$868$	2.98941	0.101467
$869$	31.6557	1.07385
$870$	7.32612	0.248379
$871$	−2.98653	−0.101195
$872$	15.5044	0.525044
$873$	8.44438	0.285799
$874$	8.01737	0.271192
$875$	−3.98491	−0.134715
$876$	−13.9756	−0.472191
$877$	−5.06396	−0.170998	−0.0854988	−	0.996338i	$-0.527248\pi$
$877$	−5.06396	−0.170998	−0.0854988	+	0.996338i	$0.527248\pi$
$878$	−8.73654	−0.294844
$879$	−25.5969	−0.863362
$880$	−8.67863	−0.292557
$881$	14.3344	0.482938	0.241469	−	0.970409i	$-0.422371\pi$
$881$	14.3344	0.482938	0.241469	+	0.970409i	$0.422371\pi$
$882$	−6.89268	−0.232089
$883$	7.36235	0.247763	0.123881	−	0.992297i	$-0.460466\pi$
$883$	7.36235	0.247763	0.123881	+	0.992297i	$0.460466\pi$
$884$	−1.60542	−0.0539960
$885$	−11.7335	−0.394418
$886$	−30.5265	−1.02556
$887$	12.4898	0.419366	0.209683	−	0.977769i	$-0.432757\pi$
$887$	12.4898	0.419366	0.209683	+	0.977769i	$0.432757\pi$
$888$	10.0665	0.337810
$889$	2.20392	0.0739172
$890$	−23.4588	−0.786341
$891$	−4.85229	−0.162558
$892$	−23.3905	−0.783172
$893$	−15.7806	−0.528076
$894$	−2.43705	−0.0815071
$895$	−34.6289	−1.15751
$896$	0.327596	0.0109442
$897$	−5.62903	−0.187948
$898$	−27.7482	−0.925968
$899$	37.3780	1.24663
$900$	−1.80104	−0.0600348
$901$	0.0546402	0.00182033
$902$	−36.0660	−1.20087
$903$	1.02074	0.0339682
$904$	16.3227	0.542886
$905$	2.25911	0.0750954
$906$	9.18020	0.304992
$907$	−22.6565	−0.752298	−0.376149	−	0.926559i	$-0.622752\pi$
$907$	−22.6565	−0.752298	−0.376149	+	0.926559i	$0.622752\pi$
$908$	−16.0329	−0.532070
$909$	10.1762	0.337522
$910$	−0.585926	−0.0194233
$911$	54.1381	1.79368	0.896838	−	0.442358i	$-0.145858\pi$
$911$	54.1381	1.79368	0.896838	+	0.442358i	$0.145858\pi$
$912$	1.42429	0.0471630
$913$	−25.7647	−0.852687
$914$	−28.1737	−0.931902
$915$	−12.2050	−0.403486
$916$	7.38355	0.243959
$917$	−3.37152	−0.111337
$918$	1.60542	0.0529867
$919$	−17.4812	−0.576650	−0.288325	−	0.957533i	$-0.593098\pi$
$919$	−17.4812	−0.576650	−0.288325	+	0.957533i	$0.593098\pi$
$920$	10.0679	0.331928
$921$	−7.43248	−0.244909
$922$	20.6173	0.678994
$923$	−14.2442	−0.468852
$924$	−1.58959	−0.0522938
$925$	−18.1302	−0.596118
$926$	−21.8807	−0.719045
$927$	1.00000	0.0328443
$928$	4.09609	0.134461
$929$	4.21637	0.138335	0.0691674	−	0.997605i	$-0.477966\pi$
$929$	4.21637	0.138335	0.0691674	+	0.997605i	$0.477966\pi$
$930$	16.3211	0.535191
$931$	−9.81718	−0.321745
$932$	−13.8005	−0.452050
$933$	−13.7862	−0.451341
$934$	−34.1345	−1.11691
$935$	−13.9328	−0.455652
$936$	−1.00000	−0.0326860
$937$	−46.6625	−1.52440	−0.762198	−	0.647344i	$-0.775880\pi$
$937$	−46.6625	−1.52440	−0.762198	+	0.647344i	$0.775880\pi$
$938$	0.978377	0.0319452
$939$	18.8499	0.615143
$940$	−19.8165	−0.646344
$941$	12.1937	0.397504	0.198752	−	0.980050i	$-0.436311\pi$
$941$	12.1937	0.397504	0.198752	+	0.980050i	$0.436311\pi$
$942$	21.0011	0.684253
$943$	41.8393	1.36248
$944$	−6.56032	−0.213520
$945$	0.585926	0.0190602
$946$	−15.1191	−0.491563
$947$	−24.6049	−0.799551	−0.399775	−	0.916613i	$-0.630912\pi$
$947$	−24.6049	−0.799551	−0.399775	+	0.916613i	$0.630912\pi$
$948$	−6.52386	−0.211885
$949$	13.9756	0.453667
$950$	−2.56521	−0.0832264
$951$	27.4394	0.889785
$952$	0.525929	0.0170455
$953$	23.1241	0.749063	0.374532	−	0.927214i	$-0.377804\pi$
$953$	23.1241	0.749063	0.374532	+	0.927214i	$0.377804\pi$
$954$	0.0340349	0.00110192
$955$	28.7852	0.931466
$956$	14.5338	0.470056
$957$	−19.8755	−0.642482
$958$	21.3228	0.688907
$959$	2.64556	0.0854296
$960$	1.78856	0.0577256
$961$	52.2708	1.68616
$962$	−10.0665	−0.324557
$963$	19.2490	0.620292
$964$	−13.6312	−0.439032
$965$	19.7625	0.636178
$966$	1.84405	0.0593313
$967$	12.0996	0.389097	0.194549	−	0.980893i	$-0.437676\pi$
$967$	12.0996	0.389097	0.194549	+	0.980893i	$0.437676\pi$
$968$	12.5448	0.403204
$969$	2.28658	0.0734556
$970$	15.1033	0.484938
$971$	−19.4022	−0.622647	−0.311323	−	0.950304i	$-0.600772\pi$
$971$	−19.4022	−0.622647	−0.311323	+	0.950304i	$0.600772\pi$
$972$	1.00000	0.0320750
$973$	−0.669052	−0.0214488
$974$	−22.7417	−0.728690
$975$	1.80104	0.0576796
$976$	−6.82393	−0.218429
$977$	18.6746	0.597452	0.298726	−	0.954339i	$-0.403438\pi$
$977$	18.6746	0.597452	0.298726	+	0.954339i	$0.403438\pi$
$978$	−4.76711	−0.152435
$979$	63.6428	2.03403
$980$	−12.3280	−0.393803
$981$	15.5044	0.495016
$982$	−15.9175	−0.507948
$983$	36.0600	1.15013	0.575067	−	0.818106i	$-0.304976\pi$
$983$	36.0600	1.15013	0.575067	+	0.818106i	$0.304976\pi$
$984$	7.43278	0.236948
$985$	−30.7049	−0.978341
$986$	6.57594	0.209421
$987$	−3.62963	−0.115532
$988$	−1.42429	−0.0453127
$989$	17.5392	0.557716
$990$	−8.67863	−0.275825
$991$	−20.7148	−0.658027	−0.329013	−	0.944325i	$-0.606716\pi$
$991$	−20.7148	−0.658027	−0.329013	+	0.944325i	$0.606716\pi$
$992$	9.12528	0.289728
$993$	−6.35189	−0.201571
$994$	4.66633	0.148007
$995$	−42.4816	−1.34676
$996$	5.30980	0.168247
$997$	−8.62451	−0.273141	−0.136570	−	0.990630i	$-0.543608\pi$
$997$	−8.62451	−0.273141	−0.136570	+	0.990630i	$0.543608\pi$
$998$	41.3480	1.30885
$999$	10.0665	0.318490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bd.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bd.1.11	✓	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.78856 q^{5} +1.00000 q^{6} +0.327596 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.78856 q^{5} +1.00000 q^{6} +0.327596 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.78856 q^{10} -4.85229 q^{11} +1.00000 q^{12} -1.00000 q^{13} +0.327596 q^{14} +1.78856 q^{15} +1.00000 q^{16} +1.60542 q^{17} +1.00000 q^{18} +1.42429 q^{19} +1.78856 q^{20} +0.327596 q^{21} -4.85229 q^{22} +5.62903 q^{23} +1.00000 q^{24} -1.80104 q^{25} -1.00000 q^{26} +1.00000 q^{27} +0.327596 q^{28} +4.09609 q^{29} +1.78856 q^{30} +9.12528 q^{31} +1.00000 q^{32} -4.85229 q^{33} +1.60542 q^{34} +0.585926 q^{35} +1.00000 q^{36} +10.0665 q^{37} +1.42429 q^{38} -1.00000 q^{39} +1.78856 q^{40} +7.43278 q^{41} +0.327596 q^{42} +3.11586 q^{43} -4.85229 q^{44} +1.78856 q^{45} +5.62903 q^{46} -11.0796 q^{47} +1.00000 q^{48} -6.89268 q^{49} -1.80104 q^{50} +1.60542 q^{51} -1.00000 q^{52} +0.0340349 q^{53} +1.00000 q^{54} -8.67863 q^{55} +0.327596 q^{56} +1.42429 q^{57} +4.09609 q^{58} -6.56032 q^{59} +1.78856 q^{60} -6.82393 q^{61} +9.12528 q^{62} +0.327596 q^{63} +1.00000 q^{64} -1.78856 q^{65} -4.85229 q^{66} +2.98653 q^{67} +1.60542 q^{68} +5.62903 q^{69} +0.585926 q^{70} +14.2442 q^{71} +1.00000 q^{72} -13.9756 q^{73} +10.0665 q^{74} -1.80104 q^{75} +1.42429 q^{76} -1.58959 q^{77} -1.00000 q^{78} -6.52386 q^{79} +1.78856 q^{80} +1.00000 q^{81} +7.43278 q^{82} +5.30980 q^{83} +0.327596 q^{84} +2.87139 q^{85} +3.11586 q^{86} +4.09609 q^{87} -4.85229 q^{88} -13.1160 q^{89} +1.78856 q^{90} -0.327596 q^{91} +5.62903 q^{92} +9.12528 q^{93} -11.0796 q^{94} +2.54743 q^{95} +1.00000 q^{96} +8.44438 q^{97} -6.89268 q^{98} -4.85229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9 + 5 * q^10 + 18 * q^11 + 16 * q^12 - 16 * q^13 + 4 * q^14 + 5 * q^15 + 16 * q^16 + 17 * q^17 + 16 * q^18 + 8 * q^19 + 5 * q^20 + 4 * q^21 + 18 * q^22 + 9 * q^23 + 16 * q^24 + 17 * q^25 - 16 * q^26 + 16 * q^27 + 4 * q^28 + 14 * q^29 + 5 * q^30 + 12 * q^31 + 16 * q^32 + 18 * q^33 + 17 * q^34 + 16 * q^35 + 16 * q^36 + 31 * q^37 + 8 * q^38 - 16 * q^39 + 5 * q^40 + 29 * q^41 + 4 * q^42 + 30 * q^43 + 18 * q^44 + 5 * q^45 + 9 * q^46 - q^47 + 16 * q^48 + 36 * q^49 + 17 * q^50 + 17 * q^51 - 16 * q^52 + 12 * q^53 + 16 * q^54 + 30 * q^55 + 4 * q^56 + 8 * q^57 + 14 * q^58 + 38 * q^59 + 5 * q^60 + 12 * q^62 + 4 * q^63 + 16 * q^64 - 5 * q^65 + 18 * q^66 + 28 * q^67 + 17 * q^68 + 9 * q^69 + 16 * q^70 + 32 * q^71 + 16 * q^72 + 20 * q^73 + 31 * q^74 + 17 * q^75 + 8 * q^76 + 26 * q^77 - 16 * q^78 + 13 * q^79 + 5 * q^80 + 16 * q^81 + 29 * q^82 + 39 * q^83 + 4 * q^84 + 31 * q^85 + 30 * q^86 + 14 * q^87 + 18 * q^88 + 9 * q^89 + 5 * q^90 - 4 * q^91 + 9 * q^92 + 12 * q^93 - q^94 - 20 * q^95 + 16 * q^96 + 35 * q^97 + 36 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more