LMFDB - Embedded newform 2006.2.a.t.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2006,2,Mod(1,2006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2006 = 2 \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2006.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:	$16.0179906455$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 13x^{6} + 24x^{5} + 50x^{4} - 50x^{3} - 62x^{2} + 28x + 24 $ x^8 - 3x^7 - 13x^6 + 24x^5 + 50x^4 - 50x^3 - 62x^2 + 28*x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.19822$ of defining polynomial
Character	$\chi$	$=$	2006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.14832 q^{3} +1.00000 q^{4} +0.589028 q^{5} +3.14832 q^{6} -2.62587 q^{7} +1.00000 q^{8} +6.91190 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.14832 q^{3} +1.00000 q^{4} +0.589028 q^{5} +3.14832 q^{6} -2.62587 q^{7} +1.00000 q^{8} +6.91190 q^{9} +0.589028 q^{10} +3.58585 q^{11} +3.14832 q^{12} -4.90040 q^{13} -2.62587 q^{14} +1.85445 q^{15} +1.00000 q^{16} -1.00000 q^{17} +6.91190 q^{18} +8.15303 q^{19} +0.589028 q^{20} -8.26708 q^{21} +3.58585 q^{22} -1.78981 q^{23} +3.14832 q^{24} -4.65305 q^{25} -4.90040 q^{26} +12.3159 q^{27} -2.62587 q^{28} +5.42280 q^{29} +1.85445 q^{30} +5.43645 q^{31} +1.00000 q^{32} +11.2894 q^{33} -1.00000 q^{34} -1.54671 q^{35} +6.91190 q^{36} -3.32482 q^{37} +8.15303 q^{38} -15.4280 q^{39} +0.589028 q^{40} +9.74951 q^{41} -8.26708 q^{42} -9.25681 q^{43} +3.58585 q^{44} +4.07130 q^{45} -1.78981 q^{46} -9.60374 q^{47} +3.14832 q^{48} -0.104783 q^{49} -4.65305 q^{50} -3.14832 q^{51} -4.90040 q^{52} +10.9465 q^{53} +12.3159 q^{54} +2.11217 q^{55} -2.62587 q^{56} +25.6683 q^{57} +5.42280 q^{58} -1.00000 q^{59} +1.85445 q^{60} -3.90881 q^{61} +5.43645 q^{62} -18.1498 q^{63} +1.00000 q^{64} -2.88647 q^{65} +11.2894 q^{66} -9.90992 q^{67} -1.00000 q^{68} -5.63490 q^{69} -1.54671 q^{70} -9.05114 q^{71} +6.91190 q^{72} -4.75977 q^{73} -3.32482 q^{74} -14.6493 q^{75} +8.15303 q^{76} -9.41600 q^{77} -15.4280 q^{78} -1.62676 q^{79} +0.589028 q^{80} +18.0386 q^{81} +9.74951 q^{82} +1.59084 q^{83} -8.26708 q^{84} -0.589028 q^{85} -9.25681 q^{86} +17.0727 q^{87} +3.58585 q^{88} -11.6838 q^{89} +4.07130 q^{90} +12.8678 q^{91} -1.78981 q^{92} +17.1157 q^{93} -9.60374 q^{94} +4.80236 q^{95} +3.14832 q^{96} +4.19543 q^{97} -0.104783 q^{98} +24.7850 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 5 q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 8 q^{8} + 17 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 5 * q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 8 * q^8 + 17 * q^9	\( 8 q + 8 q^{2} + 5 q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 8 q^{8} + 17 q^{9} - 5 q^{10} - 5 q^{11} + 5 q^{12} + q^{13} + 9 q^{15} + 8 q^{16} - 8 q^{17} + 17 q^{18} + 8 q^{19} - 5 q^{20} + 13 q^{21} - 5 q^{22} - 8 q^{23} + 5 q^{24} + 29 q^{25} + q^{26} + 44 q^{27} + 20 q^{29} + 9 q^{30} + 18 q^{31} + 8 q^{32} - 23 q^{33} - 8 q^{34} + 9 q^{35} + 17 q^{36} - 7 q^{37} + 8 q^{38} + 6 q^{39} - 5 q^{40} + 12 q^{41} + 13 q^{42} - 8 q^{43} - 5 q^{44} + 5 q^{45} - 8 q^{46} + 30 q^{47} + 5 q^{48} + 28 q^{49} + 29 q^{50} - 5 q^{51} + q^{52} - 5 q^{53} + 44 q^{54} + 23 q^{55} + 13 q^{57} + 20 q^{58} - 8 q^{59} + 9 q^{60} + 12 q^{61} + 18 q^{62} + 21 q^{63} + 8 q^{64} - 52 q^{65} - 23 q^{66} - 4 q^{67} - 8 q^{68} + 24 q^{69} + 9 q^{70} + 4 q^{71} + 17 q^{72} - 18 q^{73} - 7 q^{74} - 49 q^{75} + 8 q^{76} - 13 q^{77} + 6 q^{78} - 17 q^{79} - 5 q^{80} + 12 q^{81} + 12 q^{82} + 9 q^{83} + 13 q^{84} + 5 q^{85} - 8 q^{86} + 48 q^{87} - 5 q^{88} - 12 q^{89} + 5 q^{90} + 31 q^{91} - 8 q^{92} - 3 q^{93} + 30 q^{94} - 7 q^{95} + 5 q^{96} - 11 q^{97} + 28 q^{98} - 27 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 5 * q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 8 * q^8 + 17 * q^9 - 5 * q^10 - 5 * q^11 + 5 * q^12 + q^13 + 9 * q^15 + 8 * q^16 - 8 * q^17 + 17 * q^18 + 8 * q^19 - 5 * q^20 + 13 * q^21 - 5 * q^22 - 8 * q^23 + 5 * q^24 + 29 * q^25 + q^26 + 44 * q^27 + 20 * q^29 + 9 * q^30 + 18 * q^31 + 8 * q^32 - 23 * q^33 - 8 * q^34 + 9 * q^35 + 17 * q^36 - 7 * q^37 + 8 * q^38 + 6 * q^39 - 5 * q^40 + 12 * q^41 + 13 * q^42 - 8 * q^43 - 5 * q^44 + 5 * q^45 - 8 * q^46 + 30 * q^47 + 5 * q^48 + 28 * q^49 + 29 * q^50 - 5 * q^51 + q^52 - 5 * q^53 + 44 * q^54 + 23 * q^55 + 13 * q^57 + 20 * q^58 - 8 * q^59 + 9 * q^60 + 12 * q^61 + 18 * q^62 + 21 * q^63 + 8 * q^64 - 52 * q^65 - 23 * q^66 - 4 * q^67 - 8 * q^68 + 24 * q^69 + 9 * q^70 + 4 * q^71 + 17 * q^72 - 18 * q^73 - 7 * q^74 - 49 * q^75 + 8 * q^76 - 13 * q^77 + 6 * q^78 - 17 * q^79 - 5 * q^80 + 12 * q^81 + 12 * q^82 + 9 * q^83 + 13 * q^84 + 5 * q^85 - 8 * q^86 + 48 * q^87 - 5 * q^88 - 12 * q^89 + 5 * q^90 + 31 * q^91 - 8 * q^92 - 3 * q^93 + 30 * q^94 - 7 * q^95 + 5 * q^96 - 11 * q^97 + 28 * q^98 - 27 * 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
$3$	3.14832	1.81768	0.908841	−	0.417143i	$-0.136969\pi$
$3$	3.14832	1.81768	0.908841	+	0.417143i	$0.136969\pi$
$4$	1.00000	0.500000
$5$	0.589028	0.263421	0.131711	−	0.991288i	$-0.457953\pi$
$5$	0.589028	0.263421	0.131711	+	0.991288i	$0.457953\pi$
$6$	3.14832	1.28529
$7$	−2.62587	−0.992487	−0.496244	−	0.868183i	$-0.665288\pi$
$7$	−2.62587	−0.992487	−0.496244	+	0.868183i	$0.665288\pi$
$8$	1.00000	0.353553
$9$	6.91190	2.30397
$10$	0.589028	0.186267
$11$	3.58585	1.08117	0.540587	−	0.841288i	$-0.318202\pi$
$11$	3.58585	1.08117	0.540587	+	0.841288i	$0.318202\pi$
$12$	3.14832	0.908841
$13$	−4.90040	−1.35913	−0.679563	−	0.733617i	$-0.737831\pi$
$13$	−4.90040	−1.35913	−0.679563	+	0.733617i	$0.737831\pi$
$14$	−2.62587	−0.701794
$15$	1.85445	0.478816
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	6.91190	1.62915
$19$	8.15303	1.87043	0.935217	−	0.354075i	$-0.115204\pi$
$19$	8.15303	1.87043	0.935217	+	0.354075i	$0.115204\pi$
$20$	0.589028	0.131711
$21$	−8.26708	−1.80403
$22$	3.58585	0.764506
$23$	−1.78981	−0.373202	−0.186601	−	0.982436i	$-0.559747\pi$
$23$	−1.78981	−0.373202	−0.186601	+	0.982436i	$0.559747\pi$
$24$	3.14832	0.642647
$25$	−4.65305	−0.930609
$26$	−4.90040	−0.961047
$27$	12.3159	2.37019
$28$	−2.62587	−0.496244
$29$	5.42280	1.00699	0.503494	−	0.863999i	$-0.332048\pi$
$29$	5.42280	1.00699	0.503494	+	0.863999i	$0.332048\pi$
$30$	1.85445	0.338574
$31$	5.43645	0.976415	0.488207	−	0.872728i	$-0.337651\pi$
$31$	5.43645	0.976415	0.488207	+	0.872728i	$0.337651\pi$
$32$	1.00000	0.176777
$33$	11.2894	1.96523
$34$	−1.00000	−0.171499
$35$	−1.54671	−0.261442
$36$	6.91190	1.15198
$37$	−3.32482	−0.546597	−0.273298	−	0.961929i	$-0.588115\pi$
$37$	−3.32482	−0.546597	−0.273298	+	0.961929i	$0.588115\pi$
$38$	8.15303	1.32260
$39$	−15.4280	−2.47046
$40$	0.589028	0.0931335
$41$	9.74951	1.52262	0.761309	−	0.648389i	$-0.224557\pi$
$41$	9.74951	1.52262	0.761309	+	0.648389i	$0.224557\pi$
$42$	−8.26708	−1.27564
$43$	−9.25681	−1.41165	−0.705825	−	0.708386i	$-0.749424\pi$
$43$	−9.25681	−1.41165	−0.705825	+	0.708386i	$0.749424\pi$
$44$	3.58585	0.540587
$45$	4.07130	0.606914
$46$	−1.78981	−0.263894
$47$	−9.60374	−1.40085	−0.700425	−	0.713726i	$-0.747006\pi$
$47$	−9.60374	−1.40085	−0.700425	+	0.713726i	$0.747006\pi$
$48$	3.14832	0.454420
$49$	−0.104783	−0.0149690
$50$	−4.65305	−0.658040
$51$	−3.14832	−0.440852
$52$	−4.90040	−0.679563
$53$	10.9465	1.50361	0.751807	−	0.659383i	$-0.229183\pi$
$53$	10.9465	1.50361	0.751807	+	0.659383i	$0.229183\pi$
$54$	12.3159	1.67598
$55$	2.11217	0.284804
$56$	−2.62587	−0.350897
$57$	25.6683	3.39985
$58$	5.42280	0.712048
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	1.85445	0.239408
$61$	−3.90881	−0.500472	−0.250236	−	0.968185i	$-0.580508\pi$
$61$	−3.90881	−0.500472	−0.250236	+	0.968185i	$0.580508\pi$
$62$	5.43645	0.690429
$63$	−18.1498	−2.28666
$64$	1.00000	0.125000
$65$	−2.88647	−0.358023
$66$	11.2894	1.38963
$67$	−9.90992	−1.21069	−0.605345	−	0.795963i	$-0.706965\pi$
$67$	−9.90992	−1.21069	−0.605345	+	0.795963i	$0.706965\pi$
$68$	−1.00000	−0.121268
$69$	−5.63490	−0.678363
$70$	−1.54671	−0.184868
$71$	−9.05114	−1.07417	−0.537086	−	0.843527i	$-0.680475\pi$
$71$	−9.05114	−1.07417	−0.537086	+	0.843527i	$0.680475\pi$
$72$	6.91190	0.814575
$73$	−4.75977	−0.557089	−0.278545	−	0.960423i	$-0.589852\pi$
$73$	−4.75977	−0.557089	−0.278545	+	0.960423i	$0.589852\pi$
$74$	−3.32482	−0.386502
$75$	−14.6493	−1.69155
$76$	8.15303	0.935217
$77$	−9.41600	−1.07305
$78$	−15.4280	−1.74688
$79$	−1.62676	−0.183025	−0.0915124	−	0.995804i	$-0.529170\pi$
$79$	−1.62676	−0.183025	−0.0915124	+	0.995804i	$0.529170\pi$
$80$	0.589028	0.0658553
$81$	18.0386	2.00429
$82$	9.74951	1.07665
$83$	1.59084	0.174618	0.0873088	−	0.996181i	$-0.472173\pi$
$83$	1.59084	0.174618	0.0873088	+	0.996181i	$0.472173\pi$
$84$	−8.26708	−0.902013
$85$	−0.589028	−0.0638890
$86$	−9.25681	−0.998188
$87$	17.0727	1.83038
$88$	3.58585	0.382253
$89$	−11.6838	−1.23848	−0.619239	−	0.785202i	$-0.712559\pi$
$89$	−11.6838	−1.23848	−0.619239	+	0.785202i	$0.712559\pi$
$90$	4.07130	0.429153
$91$	12.8678	1.34892
$92$	−1.78981	−0.186601
$93$	17.1157	1.77481
$94$	−9.60374	−0.990550
$95$	4.80236	0.492712
$96$	3.14832	0.321324
$97$	4.19543	0.425982	0.212991	−	0.977054i	$-0.431680\pi$
$97$	4.19543	0.425982	0.212991	+	0.977054i	$0.431680\pi$
$98$	−0.104783	−0.0105847
$99$	24.7850	2.49099
$100$	−4.65305	−0.465305
$101$	6.48405	0.645187	0.322594	−	0.946538i	$-0.395445\pi$
$101$	6.48405	0.645187	0.322594	+	0.946538i	$0.395445\pi$
$102$	−3.14832	−0.311730
$103$	−12.0123	−1.18361	−0.591805	−	0.806081i	$-0.701584\pi$
$103$	−12.0123	−1.18361	−0.591805	+	0.806081i	$0.701584\pi$
$104$	−4.90040	−0.480524
$105$	−4.86954	−0.475219
$106$	10.9465	1.06322
$107$	−0.472405	−0.0456691	−0.0228346	−	0.999739i	$-0.507269\pi$
$107$	−0.472405	−0.0456691	−0.0228346	+	0.999739i	$0.507269\pi$
$108$	12.3159	1.18510
$109$	0.620390	0.0594226	0.0297113	−	0.999559i	$-0.490541\pi$
$109$	0.620390	0.0594226	0.0297113	+	0.999559i	$0.490541\pi$
$110$	2.11217	0.201387
$111$	−10.4676	−0.993539
$112$	−2.62587	−0.248122
$113$	−1.76952	−0.166462	−0.0832311	−	0.996530i	$-0.526524\pi$
$113$	−1.76952	−0.166462	−0.0832311	+	0.996530i	$0.526524\pi$
$114$	25.6683	2.40406
$115$	−1.05425	−0.0983094
$116$	5.42280	0.503494
$117$	−33.8710	−3.13138
$118$	−1.00000	−0.0920575
$119$	2.62587	0.240714
$120$	1.85445	0.169287
$121$	1.85833	0.168939
$122$	−3.90881	−0.353887
$123$	30.6946	2.76764
$124$	5.43645	0.488207
$125$	−5.68591	−0.508564
$126$	−18.1498	−1.61691
$127$	0.878973	0.0779963	0.0389981	−	0.999239i	$-0.487583\pi$
$127$	0.878973	0.0779963	0.0389981	+	0.999239i	$0.487583\pi$
$128$	1.00000	0.0883883
$129$	−29.1434	−2.56593
$130$	−2.88647	−0.253160
$131$	4.95512	0.432931	0.216465	−	0.976290i	$-0.430547\pi$
$131$	4.95512	0.432931	0.216465	+	0.976290i	$0.430547\pi$
$132$	11.2894	0.982616
$133$	−21.4088	−1.85638
$134$	−9.90992	−0.856087
$135$	7.25440	0.624359
$136$	−1.00000	−0.0857493
$137$	12.0158	1.02658	0.513291	−	0.858214i	$-0.328426\pi$
$137$	12.0158	1.02658	0.513291	+	0.858214i	$0.328426\pi$
$138$	−5.63490	−0.479675
$139$	−11.7361	−0.995439	−0.497720	−	0.867338i	$-0.665829\pi$
$139$	−11.7361	−0.995439	−0.497720	+	0.867338i	$0.665829\pi$
$140$	−1.54671	−0.130721
$141$	−30.2356	−2.54630
$142$	−9.05114	−0.759555
$143$	−17.5721	−1.46945
$144$	6.91190	0.575991
$145$	3.19418	0.265262
$146$	−4.75977	−0.393922
$147$	−0.329890	−0.0272089
$148$	−3.32482	−0.273298
$149$	1.59813	0.130924	0.0654621	−	0.997855i	$-0.479148\pi$
$149$	1.59813	0.130924	0.0654621	+	0.997855i	$0.479148\pi$
$150$	−14.6493	−1.19611
$151$	−0.621394	−0.0505683	−0.0252842	−	0.999680i	$-0.508049\pi$
$151$	−0.621394	−0.0505683	−0.0252842	+	0.999680i	$0.508049\pi$
$152$	8.15303	0.661298
$153$	−6.91190	−0.558794
$154$	−9.41600	−0.758763
$155$	3.20222	0.257208
$156$	−15.4280	−1.23523
$157$	−0.0424167	−0.00338522	−0.00169261	−	0.999999i	$-0.500539\pi$
$157$	−0.0424167	−0.00338522	−0.00169261	+	0.999999i	$0.500539\pi$
$158$	−1.62676	−0.129418
$159$	34.4630	2.73309
$160$	0.589028	0.0465667
$161$	4.69983	0.370398
$162$	18.0386	1.41725
$163$	7.66429	0.600313	0.300157	−	0.953890i	$-0.402961\pi$
$163$	7.66429	0.600313	0.300157	+	0.953890i	$0.402961\pi$
$164$	9.74951	0.761309
$165$	6.64977	0.517684
$166$	1.59084	0.123473
$167$	−11.4214	−0.883814	−0.441907	−	0.897061i	$-0.645698\pi$
$167$	−11.4214	−0.883814	−0.441907	+	0.897061i	$0.645698\pi$
$168$	−8.26708	−0.637819
$169$	11.0139	0.847224
$170$	−0.589028	−0.0451764
$171$	56.3529	4.30942
$172$	−9.25681	−0.705825
$173$	−9.80512	−0.745469	−0.372735	−	0.927938i	$-0.621580\pi$
$173$	−9.80512	−0.745469	−0.372735	+	0.927938i	$0.621580\pi$
$174$	17.0727	1.29428
$175$	12.2183	0.923618
$176$	3.58585	0.270294
$177$	−3.14832	−0.236642
$178$	−11.6838	−0.875737
$179$	−8.42530	−0.629736	−0.314868	−	0.949135i	$-0.601960\pi$
$179$	−8.42530	−0.629736	−0.314868	+	0.949135i	$0.601960\pi$
$180$	4.07130	0.303457
$181$	−8.07166	−0.599962	−0.299981	−	0.953945i	$-0.596980\pi$
$181$	−8.07166	−0.599962	−0.299981	+	0.953945i	$0.596980\pi$
$182$	12.8678	0.953827
$183$	−12.3062	−0.909699
$184$	−1.78981	−0.131947
$185$	−1.95841	−0.143985
$186$	17.1157	1.25498
$187$	−3.58585	−0.262223
$188$	−9.60374	−0.700425
$189$	−32.3400	−2.35239
$190$	4.80236	0.348400
$191$	14.7346	1.06616	0.533080	−	0.846065i	$-0.321035\pi$
$191$	14.7346	1.06616	0.533080	+	0.846065i	$0.321035\pi$
$192$	3.14832	0.227210
$193$	−25.2110	−1.81473	−0.907363	−	0.420349i	$-0.861908\pi$
$193$	−25.2110	−1.81473	−0.907363	+	0.420349i	$0.861908\pi$
$194$	4.19543	0.301215
$195$	−9.08753	−0.650771
$196$	−0.104783	−0.00748451
$197$	6.41372	0.456958	0.228479	−	0.973549i	$-0.426625\pi$
$197$	6.41372	0.456958	0.228479	+	0.973549i	$0.426625\pi$
$198$	24.7850	1.76140
$199$	−26.2951	−1.86401	−0.932005	−	0.362445i	$-0.881942\pi$
$199$	−26.2951	−1.86401	−0.932005	+	0.362445i	$0.881942\pi$
$200$	−4.65305	−0.329020
$201$	−31.1996	−2.20065
$202$	6.48405	0.456216
$203$	−14.2396	−0.999423
$204$	−3.14832	−0.220426
$205$	5.74274	0.401090
$206$	−12.0123	−0.836938
$207$	−12.3710	−0.859845
$208$	−4.90040	−0.339782
$209$	29.2356	2.02227
$210$	−4.86954	−0.336030
$211$	21.9976	1.51438	0.757190	−	0.653195i	$-0.226572\pi$
$211$	21.9976	1.51438	0.757190	+	0.653195i	$0.226572\pi$
$212$	10.9465	0.751807
$213$	−28.4958	−1.95250
$214$	−0.472405	−0.0322929
$215$	−5.45252	−0.371859
$216$	12.3159	0.837990
$217$	−14.2754	−0.969079
$218$	0.620390	0.0420181
$219$	−14.9853	−1.01261
$220$	2.11217	0.142402
$221$	4.90040	0.329637
$222$	−10.4676	−0.702538
$223$	−9.32882	−0.624704	−0.312352	−	0.949966i	$-0.601117\pi$
$223$	−9.32882	−0.624704	−0.312352	+	0.949966i	$0.601117\pi$
$224$	−2.62587	−0.175449
$225$	−32.1614	−2.14409
$226$	−1.76952	−0.117707
$227$	26.8160	1.77984	0.889922	−	0.456114i	$-0.150759\pi$
$227$	26.8160	1.77984	0.889922	+	0.456114i	$0.150759\pi$
$228$	25.6683	1.69993
$229$	−23.0821	−1.52531	−0.762653	−	0.646808i	$-0.776103\pi$
$229$	−23.0821	−1.52531	−0.762653	+	0.646808i	$0.776103\pi$
$230$	−1.05425	−0.0695152
$231$	−29.6445	−1.95047
$232$	5.42280	0.356024
$233$	9.38591	0.614891	0.307446	−	0.951566i	$-0.400526\pi$
$233$	9.38591	0.614891	0.307446	+	0.951566i	$0.400526\pi$
$234$	−33.8710	−2.21422
$235$	−5.65687	−0.369014
$236$	−1.00000	−0.0650945
$237$	−5.12155	−0.332681
$238$	2.62587	0.170210
$239$	−10.0692	−0.651321	−0.325660	−	0.945487i	$-0.605587\pi$
$239$	−10.0692	−0.651321	−0.325660	+	0.945487i	$0.605587\pi$
$240$	1.85445	0.119704
$241$	2.13124	0.137285	0.0686425	−	0.997641i	$-0.478133\pi$
$241$	2.13124	0.137285	0.0686425	+	0.997641i	$0.478133\pi$
$242$	1.85833	0.119458
$243$	19.8436	1.27297
$244$	−3.90881	−0.250236
$245$	−0.0617202	−0.00394316
$246$	30.6946	1.95701
$247$	−39.9531	−2.54216
$248$	5.43645	0.345215
$249$	5.00847	0.317399
$250$	−5.68591	−0.359609
$251$	−1.55051	−0.0978673	−0.0489337	−	0.998802i	$-0.515582\pi$
$251$	−1.55051	−0.0978673	−0.0489337	+	0.998802i	$0.515582\pi$
$252$	−18.1498	−1.14333
$253$	−6.41801	−0.403497
$254$	0.878973	0.0551517
$255$	−1.85445	−0.116130
$256$	1.00000	0.0625000
$257$	29.3848	1.83297	0.916485	−	0.400068i	$-0.131014\pi$
$257$	29.3848	1.83297	0.916485	+	0.400068i	$0.131014\pi$
$258$	−29.1434	−1.81439
$259$	8.73056	0.542490
$260$	−2.88647	−0.179011
$261$	37.4818	2.32007
$262$	4.95512	0.306128
$263$	10.0650	0.620637	0.310319	−	0.950633i	$-0.399564\pi$
$263$	10.0650	0.620637	0.310319	+	0.950633i	$0.399564\pi$
$264$	11.2894	0.694814
$265$	6.44778	0.396084
$266$	−21.4088	−1.31266
$267$	−36.7843	−2.25116
$268$	−9.90992	−0.605345
$269$	6.73694	0.410759	0.205379	−	0.978682i	$-0.434157\pi$
$269$	6.73694	0.410759	0.205379	+	0.978682i	$0.434157\pi$
$270$	7.25440	0.441489
$271$	19.6009	1.19067	0.595334	−	0.803479i	$-0.297020\pi$
$271$	19.6009	1.19067	0.595334	+	0.803479i	$0.297020\pi$
$272$	−1.00000	−0.0606339
$273$	40.5120	2.45190
$274$	12.0158	0.725904
$275$	−16.6851	−1.00615
$276$	−5.63490	−0.339181
$277$	22.6929	1.36349	0.681743	−	0.731592i	$-0.261222\pi$
$277$	22.6929	1.36349	0.681743	+	0.731592i	$0.261222\pi$
$278$	−11.7361	−0.703882
$279$	37.5762	2.24963
$280$	−1.54671	−0.0924338
$281$	2.38415	0.142226	0.0711132	−	0.997468i	$-0.477345\pi$
$281$	2.38415	0.142226	0.0711132	+	0.997468i	$0.477345\pi$
$282$	−30.2356	−1.80050
$283$	29.5839	1.75858	0.879290	−	0.476287i	$-0.158018\pi$
$283$	29.5839	1.75858	0.879290	+	0.476287i	$0.158018\pi$
$284$	−9.05114	−0.537086
$285$	15.1194	0.895594
$286$	−17.5721	−1.03906
$287$	−25.6010	−1.51118
$288$	6.91190	0.407287
$289$	1.00000	0.0588235
$290$	3.19418	0.187569
$291$	13.2086	0.774299
$292$	−4.75977	−0.278545
$293$	−8.70231	−0.508394	−0.254197	−	0.967152i	$-0.581811\pi$
$293$	−8.70231	−0.508394	−0.254197	+	0.967152i	$0.581811\pi$
$294$	−0.329890	−0.0192396
$295$	−0.589028	−0.0342945
$296$	−3.32482	−0.193251
$297$	44.1629	2.56259
$298$	1.59813	0.0925774
$299$	8.77081	0.507229
$300$	−14.6493	−0.845775
$301$	24.3072	1.40105
$302$	−0.621394	−0.0357572
$303$	20.4138	1.17274
$304$	8.15303	0.467609
$305$	−2.30240	−0.131835
$306$	−6.91190	−0.395127
$307$	26.3123	1.50172	0.750860	−	0.660461i	$-0.229639\pi$
$307$	26.3123	1.50172	0.750860	+	0.660461i	$0.229639\pi$
$308$	−9.41600	−0.536526
$309$	−37.8186	−2.15142
$310$	3.20222	0.181874
$311$	−1.11665	−0.0633196	−0.0316598	−	0.999499i	$-0.510079\pi$
$311$	−1.11665	−0.0633196	−0.0316598	+	0.999499i	$0.510079\pi$
$312$	−15.4280	−0.873439
$313$	2.15112	0.121588	0.0607942	−	0.998150i	$-0.480637\pi$
$313$	2.15112	0.121588	0.0607942	+	0.998150i	$0.480637\pi$
$314$	−0.0424167	−0.00239371
$315$	−10.6907	−0.602354
$316$	−1.62676	−0.0915124
$317$	−2.80446	−0.157514	−0.0787572	−	0.996894i	$-0.525095\pi$
$317$	−2.80446	−0.157514	−0.0787572	+	0.996894i	$0.525095\pi$
$318$	34.4630	1.93259
$319$	19.4453	1.08873
$320$	0.589028	0.0329277
$321$	−1.48728	−0.0830119
$322$	4.69983	0.261911
$323$	−8.15303	−0.453647
$324$	18.0386	1.00215
$325$	22.8018	1.26482
$326$	7.66429	0.424486
$327$	1.95319	0.108011
$328$	9.74951	0.538327
$329$	25.2182	1.39033
$330$	6.64977	0.366058
$331$	9.32074	0.512314	0.256157	−	0.966635i	$-0.417544\pi$
$331$	9.32074	0.512314	0.256157	+	0.966635i	$0.417544\pi$
$332$	1.59084	0.0873088
$333$	−22.9808	−1.25934
$334$	−11.4214	−0.624951
$335$	−5.83722	−0.318921
$336$	−8.26708	−0.451006
$337$	−36.1661	−1.97009	−0.985047	−	0.172287i	$-0.944884\pi$
$337$	−36.1661	−1.97009	−0.985047	+	0.172287i	$0.944884\pi$
$338$	11.0139	0.599078
$339$	−5.57100	−0.302575
$340$	−0.589028	−0.0319445
$341$	19.4943	1.05568
$342$	56.3529	3.04722
$343$	18.6563	1.00734
$344$	−9.25681	−0.499094
$345$	−3.31911	−0.178695
$346$	−9.80512	−0.527126
$347$	−33.3397	−1.78977	−0.894886	−	0.446296i	$-0.852743\pi$
$347$	−33.3397	−1.78977	−0.894886	+	0.446296i	$0.852743\pi$
$348$	17.0727	0.915192
$349$	30.2492	1.61920	0.809600	−	0.586982i	$-0.199684\pi$
$349$	30.2492	1.61920	0.809600	+	0.586982i	$0.199684\pi$
$350$	12.2183	0.653096
$351$	−60.3528	−3.22139
$352$	3.58585	0.191127
$353$	−8.70630	−0.463389	−0.231695	−	0.972789i	$-0.574427\pi$
$353$	−8.70630	−0.463389	−0.231695	+	0.972789i	$0.574427\pi$
$354$	−3.14832	−0.167331
$355$	−5.33137	−0.282960
$356$	−11.6838	−0.619239
$357$	8.26708	0.437540
$358$	−8.42530	−0.445291
$359$	20.6559	1.09017	0.545087	−	0.838379i	$-0.316497\pi$
$359$	20.6559	1.09017	0.545087	+	0.838379i	$0.316497\pi$
$360$	4.07130	0.214576
$361$	47.4720	2.49852
$362$	−8.07166	−0.424237
$363$	5.85061	0.307078
$364$	12.8678	0.674458
$365$	−2.80364	−0.146749
$366$	−12.3062	−0.643254
$367$	−32.7200	−1.70797	−0.853985	−	0.520298i	$-0.825821\pi$
$367$	−32.7200	−1.70797	−0.853985	+	0.520298i	$0.825821\pi$
$368$	−1.78981	−0.0933005
$369$	67.3876	3.50806
$370$	−1.95841	−0.101813
$371$	−28.7441	−1.49232
$372$	17.1157	0.887405
$373$	5.93052	0.307070	0.153535	−	0.988143i	$-0.450934\pi$
$373$	5.93052	0.307070	0.153535	+	0.988143i	$0.450934\pi$
$374$	−3.58585	−0.185420
$375$	−17.9011	−0.924407
$376$	−9.60374	−0.495275
$377$	−26.5739	−1.36862
$378$	−32.3400	−1.66339
$379$	34.6780	1.78129	0.890645	−	0.454699i	$-0.150253\pi$
$379$	34.6780	1.78129	0.890645	+	0.454699i	$0.150253\pi$
$380$	4.80236	0.246356
$381$	2.76729	0.141772
$382$	14.7346	0.753889
$383$	32.2031	1.64550	0.822751	−	0.568402i	$-0.192438\pi$
$383$	32.2031	1.64550	0.822751	+	0.568402i	$0.192438\pi$
$384$	3.14832	0.160662
$385$	−5.54628	−0.282665
$386$	−25.2110	−1.28320
$387$	−63.9821	−3.25239
$388$	4.19543	0.212991
$389$	4.87596	0.247221	0.123611	−	0.992331i	$-0.460553\pi$
$389$	4.87596	0.247221	0.123611	+	0.992331i	$0.460553\pi$
$390$	−9.08753	−0.460165
$391$	1.78981	0.0905148
$392$	−0.104783	−0.00529234
$393$	15.6003	0.786930
$394$	6.41372	0.323118
$395$	−0.958207	−0.0482126
$396$	24.7850	1.24549
$397$	8.60978	0.432112	0.216056	−	0.976381i	$-0.430681\pi$
$397$	8.60978	0.432112	0.216056	+	0.976381i	$0.430681\pi$
$398$	−26.2951	−1.31805
$399$	−67.4018	−3.37431
$400$	−4.65305	−0.232652
$401$	18.7279	0.935228	0.467614	−	0.883933i	$-0.345114\pi$
$401$	18.7279	0.935228	0.467614	+	0.883933i	$0.345114\pi$
$402$	−31.1996	−1.55609
$403$	−26.6408	−1.32707
$404$	6.48405	0.322594
$405$	10.6252	0.527973
$406$	−14.2396	−0.706699
$407$	−11.9223	−0.590967
$408$	−3.14832	−0.155865
$409$	−12.9012	−0.637921	−0.318961	−	0.947768i	$-0.603334\pi$
$409$	−12.9012	−0.637921	−0.318961	+	0.947768i	$0.603334\pi$
$410$	5.74274	0.283614
$411$	37.8297	1.86600
$412$	−12.0123	−0.591805
$413$	2.62587	0.129211
$414$	−12.3710	−0.608002
$415$	0.937050	0.0459980
$416$	−4.90040	−0.240262
$417$	−36.9488	−1.80939
$418$	29.2356	1.42996
$419$	13.2122	0.645458	0.322729	−	0.946491i	$-0.395400\pi$
$419$	13.2122	0.645458	0.322729	+	0.946491i	$0.395400\pi$
$420$	−4.86954	−0.237609
$421$	7.23797	0.352757	0.176379	−	0.984322i	$-0.443562\pi$
$421$	7.23797	0.352757	0.176379	+	0.984322i	$0.443562\pi$
$422$	21.9976	1.07083
$423$	−66.3800	−3.22751
$424$	10.9465	0.531608
$425$	4.65305	0.225706
$426$	−28.4958	−1.38063
$427$	10.2641	0.496712
$428$	−0.472405	−0.0228346
$429$	−55.3225	−2.67100
$430$	−5.45252	−0.262944
$431$	−30.9120	−1.48898	−0.744490	−	0.667633i	$-0.767307\pi$
$431$	−30.9120	−1.48898	−0.744490	+	0.667633i	$0.767307\pi$
$432$	12.3159	0.592548
$433$	−29.4556	−1.41554	−0.707772	−	0.706441i	$-0.750300\pi$
$433$	−29.4556	−1.41554	−0.707772	+	0.706441i	$0.750300\pi$
$434$	−14.2754	−0.685242
$435$	10.0563	0.482162
$436$	0.620390	0.0297113
$437$	−14.5924	−0.698050
$438$	−14.9853	−0.716024
$439$	5.57890	0.266266	0.133133	−	0.991098i	$-0.457496\pi$
$439$	5.57890	0.266266	0.133133	+	0.991098i	$0.457496\pi$
$440$	2.11217	0.100694
$441$	−0.724250	−0.0344881
$442$	4.90040	0.233088
$443$	15.9821	0.759333	0.379667	−	0.925123i	$-0.376039\pi$
$443$	15.9821	0.759333	0.379667	+	0.925123i	$0.376039\pi$
$444$	−10.4676	−0.496769
$445$	−6.88208	−0.326242
$446$	−9.32882	−0.441733
$447$	5.03143	0.237979
$448$	−2.62587	−0.124061
$449$	−13.9942	−0.660429	−0.330215	−	0.943906i	$-0.607121\pi$
$449$	−13.9942	−0.660429	−0.330215	+	0.943906i	$0.607121\pi$
$450$	−32.1614	−1.51610
$451$	34.9603	1.64622
$452$	−1.76952	−0.0832311
$453$	−1.95635	−0.0919171
$454$	26.8160	1.25854
$455$	7.57951	0.355333
$456$	25.6683	1.20203
$457$	−21.1888	−0.991169	−0.495585	−	0.868560i	$-0.665046\pi$
$457$	−21.1888	−0.991169	−0.495585	+	0.868560i	$0.665046\pi$
$458$	−23.0821	−1.07855
$459$	−12.3159	−0.574856
$460$	−1.05425	−0.0491547
$461$	−20.9031	−0.973552	−0.486776	−	0.873527i	$-0.661827\pi$
$461$	−20.9031	−0.973552	−0.486776	+	0.873527i	$0.661827\pi$
$462$	−29.6445	−1.37919
$463$	40.6488	1.88911	0.944554	−	0.328356i	$-0.106495\pi$
$463$	40.6488	1.88911	0.944554	+	0.328356i	$0.106495\pi$
$464$	5.42280	0.251747
$465$	10.0816	0.467523
$466$	9.38591	0.434794
$467$	−16.6371	−0.769871	−0.384936	−	0.922943i	$-0.625776\pi$
$467$	−16.6371	−0.769871	−0.384936	+	0.922943i	$0.625776\pi$
$468$	−33.8710	−1.56569
$469$	26.0222	1.20159
$470$	−5.65687	−0.260932
$471$	−0.133541	−0.00615325
$472$	−1.00000	−0.0460287
$473$	−33.1936	−1.52624
$474$	−5.12155	−0.235241
$475$	−37.9364	−1.74064
$476$	2.62587	0.120357
$477$	75.6609	3.46427
$478$	−10.0692	−0.460553
$479$	30.6239	1.39924	0.699620	−	0.714515i	$-0.253352\pi$
$479$	30.6239	1.39924	0.699620	+	0.714515i	$0.253352\pi$
$480$	1.85445	0.0846435
$481$	16.2929	0.742894
$482$	2.13124	0.0970752
$483$	14.7965	0.673266
$484$	1.85833	0.0844696
$485$	2.47123	0.112213
$486$	19.8436	0.900124
$487$	−37.4899	−1.69883	−0.849416	−	0.527724i	$-0.823045\pi$
$487$	−37.4899	−1.69883	−0.849416	+	0.527724i	$0.823045\pi$
$488$	−3.90881	−0.176944
$489$	24.1296	1.09118
$490$	−0.0617202	−0.00278823
$491$	−25.1759	−1.13617	−0.568087	−	0.822969i	$-0.692316\pi$
$491$	−25.1759	−1.13617	−0.568087	+	0.822969i	$0.692316\pi$
$492$	30.6946	1.38382
$493$	−5.42280	−0.244230
$494$	−39.9531	−1.79758
$495$	14.5991	0.656180
$496$	5.43645	0.244104
$497$	23.7672	1.06610
$498$	5.00847	0.224435
$499$	−13.5773	−0.607806	−0.303903	−	0.952703i	$-0.598290\pi$
$499$	−13.5773	−0.607806	−0.303903	+	0.952703i	$0.598290\pi$
$500$	−5.68591	−0.254282
$501$	−35.9582	−1.60649
$502$	−1.55051	−0.0692026
$503$	6.89724	0.307533	0.153766	−	0.988107i	$-0.450860\pi$
$503$	6.89724	0.307533	0.153766	+	0.988107i	$0.450860\pi$
$504$	−18.1498	−0.808455
$505$	3.81929	0.169956
$506$	−6.41801	−0.285315
$507$	34.6753	1.53998
$508$	0.878973	0.0389981
$509$	−11.9721	−0.530655	−0.265327	−	0.964158i	$-0.585480\pi$
$509$	−11.9721	−0.530655	−0.265327	+	0.964158i	$0.585480\pi$
$510$	−1.85445	−0.0821163
$511$	12.4986	0.552904
$512$	1.00000	0.0441942
$513$	100.412	4.43329
$514$	29.3848	1.29611
$515$	−7.07559	−0.311788
$516$	−29.1434	−1.28297
$517$	−34.4376	−1.51456
$518$	8.73056	0.383599
$519$	−30.8696	−1.35503
$520$	−2.88647	−0.126580
$521$	−13.8530	−0.606909	−0.303455	−	0.952846i	$-0.598140\pi$
$521$	−13.8530	−0.606909	−0.303455	+	0.952846i	$0.598140\pi$
$522$	37.4818	1.64053
$523$	26.0093	1.13731	0.568653	−	0.822578i	$-0.307465\pi$
$523$	26.0093	1.13731	0.568653	+	0.822578i	$0.307465\pi$
$524$	4.95512	0.216465
$525$	38.4671	1.67884
$526$	10.0650	0.438857
$527$	−5.43645	−0.236815
$528$	11.2894	0.491308
$529$	−19.7966	−0.860720
$530$	6.44778	0.280074
$531$	−6.91190	−0.299951
$532$	−21.4088	−0.928191
$533$	−47.7765	−2.06943
$534$	−36.7843	−1.59181
$535$	−0.278260	−0.0120302
$536$	−9.90992	−0.428043
$537$	−26.5255	−1.14466
$538$	6.73694	0.290450
$539$	−0.375737	−0.0161841
$540$	7.25440	0.312180
$541$	46.2395	1.98799	0.993996	−	0.109419i	$-0.0348990\pi$
$541$	46.2395	1.98799	0.993996	+	0.109419i	$0.0348990\pi$
$542$	19.6009	0.841929
$543$	−25.4121	−1.09054
$544$	−1.00000	−0.0428746
$545$	0.365427	0.0156532
$546$	40.5120	1.73375
$547$	0.147916	0.00632442	0.00316221	−	0.999995i	$-0.498993\pi$
$547$	0.147916	0.00632442	0.00316221	+	0.999995i	$0.498993\pi$
$548$	12.0158	0.513291
$549$	−27.0173	−1.15307
$550$	−16.6851	−0.711456
$551$	44.2122	1.88350
$552$	−5.63490	−0.239837
$553$	4.27167	0.181650
$554$	22.6929	0.964130
$555$	−6.16570	−0.261719
$556$	−11.7361	−0.497720
$557$	−44.2063	−1.87308	−0.936541	−	0.350558i	$-0.885992\pi$
$557$	−44.2063	−1.87308	−0.936541	+	0.350558i	$0.885992\pi$
$558$	37.5762	1.59073
$559$	45.3621	1.91861
$560$	−1.54671	−0.0653606
$561$	−11.2894	−0.476639
$562$	2.38415	0.100569
$563$	5.11645	0.215633	0.107816	−	0.994171i	$-0.465614\pi$
$563$	5.11645	0.215633	0.107816	+	0.994171i	$0.465614\pi$
$564$	−30.2356	−1.27315
$565$	−1.04230	−0.0438497
$566$	29.5839	1.24350
$567$	−47.3671	−1.98923
$568$	−9.05114	−0.379777
$569$	44.3822	1.86060	0.930299	−	0.366803i	$-0.119548\pi$
$569$	44.3822	1.86060	0.930299	+	0.366803i	$0.119548\pi$
$570$	15.1194	0.633280
$571$	11.9347	0.499452	0.249726	−	0.968316i	$-0.419659\pi$
$571$	11.9347	0.499452	0.249726	+	0.968316i	$0.419659\pi$
$572$	−17.5721	−0.734727
$573$	46.3892	1.93794
$574$	−25.6010	−1.06857
$575$	8.32809	0.347305
$576$	6.91190	0.287996
$577$	−31.9722	−1.33102	−0.665510	−	0.746389i	$-0.731786\pi$
$577$	−31.9722	−1.33102	−0.665510	+	0.746389i	$0.731786\pi$
$578$	1.00000	0.0415945
$579$	−79.3721	−3.29859
$580$	3.19418	0.132631
$581$	−4.17735	−0.173306
$582$	13.2086	0.547512
$583$	39.2524	1.62567
$584$	−4.75977	−0.196961
$585$	−19.9510	−0.824872
$586$	−8.70231	−0.359489
$587$	−6.58269	−0.271697	−0.135848	−	0.990730i	$-0.543376\pi$
$587$	−6.58269	−0.271697	−0.135848	+	0.990730i	$0.543376\pi$
$588$	−0.329890	−0.0136044
$589$	44.3235	1.82632
$590$	−0.589028	−0.0242499
$591$	20.1924	0.830605
$592$	−3.32482	−0.136649
$593$	−5.61928	−0.230756	−0.115378	−	0.993322i	$-0.536808\pi$
$593$	−5.61928	−0.230756	−0.115378	+	0.993322i	$0.536808\pi$
$594$	44.1629	1.81203
$595$	1.54671	0.0634091
$596$	1.59813	0.0654621
$597$	−82.7853	−3.38818
$598$	8.77081	0.358665
$599$	−20.7347	−0.847196	−0.423598	−	0.905850i	$-0.639233\pi$
$599$	−20.7347	−0.847196	−0.423598	+	0.905850i	$0.639233\pi$
$600$	−14.6493	−0.598054
$601$	27.7369	1.13141	0.565706	−	0.824607i	$-0.308604\pi$
$601$	27.7369	1.13141	0.565706	+	0.824607i	$0.308604\pi$
$602$	24.3072	0.990689
$603$	−68.4964	−2.78939
$604$	−0.621394	−0.0252842
$605$	1.09461	0.0445022
$606$	20.4138	0.829256
$607$	22.2879	0.904636	0.452318	−	0.891857i	$-0.350597\pi$
$607$	22.2879	0.904636	0.452318	+	0.891857i	$0.350597\pi$
$608$	8.15303	0.330649
$609$	−44.8307	−1.81663
$610$	−2.30240	−0.0932214
$611$	47.0622	1.90393
$612$	−6.91190	−0.279397
$613$	−14.5885	−0.589224	−0.294612	−	0.955617i	$-0.595190\pi$
$613$	−14.5885	−0.589224	−0.294612	+	0.955617i	$0.595190\pi$
$614$	26.3123	1.06188
$615$	18.0800	0.729054
$616$	−9.41600	−0.379381
$617$	−39.1794	−1.57731	−0.788653	−	0.614839i	$-0.789221\pi$
$617$	−39.1794	−1.57731	−0.788653	+	0.614839i	$0.789221\pi$
$618$	−37.8186	−1.52129
$619$	0.0685856	0.00275669	0.00137834	−	0.999999i	$-0.499561\pi$
$619$	0.0685856	0.00275669	0.00137834	+	0.999999i	$0.499561\pi$
$620$	3.20222	0.128604
$621$	−22.0432	−0.884561
$622$	−1.11665	−0.0447737
$623$	30.6802	1.22917
$624$	−15.4280	−0.617615
$625$	19.9161	0.796643
$626$	2.15112	0.0859760
$627$	92.0428	3.67584
$628$	−0.0424167	−0.00169261
$629$	3.32482	0.132569
$630$	−10.6907	−0.425929
$631$	−3.07945	−0.122591	−0.0612954	−	0.998120i	$-0.519523\pi$
$631$	−3.07945	−0.122591	−0.0612954	+	0.998120i	$0.519523\pi$
$632$	−1.62676	−0.0647090
$633$	69.2555	2.75266
$634$	−2.80446	−0.111379
$635$	0.517740	0.0205459
$636$	34.4630	1.36655
$637$	0.513479	0.0203448
$638$	19.4453	0.769848
$639$	−62.5605	−2.47486
$640$	0.589028	0.0232834
$641$	−41.4532	−1.63730	−0.818651	−	0.574292i	$-0.805277\pi$
$641$	−41.4532	−1.63730	−0.818651	+	0.574292i	$0.805277\pi$
$642$	−1.48728	−0.0586983
$643$	22.6981	0.895128	0.447564	−	0.894252i	$-0.352292\pi$
$643$	22.6981	0.895128	0.447564	+	0.894252i	$0.352292\pi$
$644$	4.69983	0.185199
$645$	−17.1663	−0.675921
$646$	−8.15303	−0.320777
$647$	41.2330	1.62104	0.810519	−	0.585713i	$-0.199185\pi$
$647$	41.2330	1.62104	0.810519	+	0.585713i	$0.199185\pi$
$648$	18.0386	0.708624
$649$	−3.58585	−0.140757
$650$	22.8018	0.894360
$651$	−44.9436	−1.76148
$652$	7.66429	0.300157
$653$	−15.7894	−0.617888	−0.308944	−	0.951080i	$-0.599976\pi$
$653$	−15.7894	−0.617888	−0.308944	+	0.951080i	$0.599976\pi$
$654$	1.95319	0.0763756
$655$	2.91870	0.114043
$656$	9.74951	0.380655
$657$	−32.8991	−1.28351
$658$	25.2182	0.983108
$659$	−46.1195	−1.79656	−0.898280	−	0.439424i	$-0.855183\pi$
$659$	−46.1195	−1.79656	−0.898280	+	0.439424i	$0.855183\pi$
$660$	6.64977	0.258842
$661$	11.9859	0.466198	0.233099	−	0.972453i	$-0.425113\pi$
$661$	11.9859	0.466198	0.233099	+	0.972453i	$0.425113\pi$
$662$	9.32074	0.362261
$663$	15.4280	0.599174
$664$	1.59084	0.0617366
$665$	−12.6104	−0.489011
$666$	−22.9808	−0.890488
$667$	−9.70580	−0.375810
$668$	−11.4214	−0.441907
$669$	−29.3701	−1.13551
$670$	−5.83722	−0.225512
$671$	−14.0164	−0.541098
$672$	−8.26708	−0.318910
$673$	36.7056	1.41490	0.707448	−	0.706765i	$-0.249846\pi$
$673$	36.7056	1.41490	0.707448	+	0.706765i	$0.249846\pi$
$674$	−36.1661	−1.39307
$675$	−57.3064	−2.20572
$676$	11.0139	0.423612
$677$	7.22593	0.277715	0.138858	−	0.990312i	$-0.455657\pi$
$677$	7.22593	0.277715	0.138858	+	0.990312i	$0.455657\pi$
$678$	−5.57100	−0.213953
$679$	−11.0167	−0.422781
$680$	−0.589028	−0.0225882
$681$	84.4254	3.23519
$682$	19.4943	0.746475
$683$	44.6159	1.70718	0.853590	−	0.520945i	$-0.174420\pi$
$683$	44.6159	1.70718	0.853590	+	0.520945i	$0.174420\pi$
$684$	56.3529	2.15471
$685$	7.07767	0.270424
$686$	18.6563	0.712300
$687$	−72.6696	−2.77252
$688$	−9.25681	−0.352913
$689$	−53.6421	−2.04360
$690$	−3.31911	−0.126357
$691$	12.6916	0.482813	0.241407	−	0.970424i	$-0.422391\pi$
$691$	12.6916	0.482813	0.241407	+	0.970424i	$0.422391\pi$
$692$	−9.80512	−0.372735
$693$	−65.0824	−2.47228
$694$	−33.3397	−1.26556
$695$	−6.91286	−0.262220
$696$	17.0727	0.647138
$697$	−9.74951	−0.369289
$698$	30.2492	1.14495
$699$	29.5498	1.11768
$700$	12.2183	0.461809
$701$	19.8428	0.749453	0.374726	−	0.927135i	$-0.377737\pi$
$701$	19.8428	0.749453	0.374726	+	0.927135i	$0.377737\pi$
$702$	−60.3528	−2.27787
$703$	−27.1074	−1.02237
$704$	3.58585	0.135147
$705$	−17.8096	−0.670749
$706$	−8.70630	−0.327666
$707$	−17.0263	−0.640340
$708$	−3.14832	−0.118321
$709$	−34.6184	−1.30012	−0.650061	−	0.759882i	$-0.725257\pi$
$709$	−34.6184	−1.30012	−0.650061	+	0.759882i	$0.725257\pi$
$710$	−5.33137	−0.200083
$711$	−11.2440	−0.421683
$712$	−11.6838	−0.437868
$713$	−9.73023	−0.364400
$714$	8.26708	0.309388
$715$	−10.3505	−0.387085
$716$	−8.42530	−0.314868
$717$	−31.7009	−1.18389
$718$	20.6559	0.770870
$719$	22.8201	0.851046	0.425523	−	0.904948i	$-0.360090\pi$
$719$	22.8201	0.851046	0.425523	+	0.904948i	$0.360090\pi$
$720$	4.07130	0.151728
$721$	31.5428	1.17472
$722$	47.4720	1.76672
$723$	6.70981	0.249540
$724$	−8.07166	−0.299981
$725$	−25.2325	−0.937112
$726$	5.85061	0.217137
$727$	−16.8020	−0.623150	−0.311575	−	0.950221i	$-0.600857\pi$
$727$	−16.8020	−0.623150	−0.311575	+	0.950221i	$0.600857\pi$
$728$	12.8678	0.476914
$729$	8.35812	0.309560
$730$	−2.80364	−0.103767
$731$	9.25681	0.342376
$732$	−12.3062	−0.454849
$733$	−15.7741	−0.582629	−0.291314	−	0.956627i	$-0.594093\pi$
$733$	−15.7741	−0.582629	−0.291314	+	0.956627i	$0.594093\pi$
$734$	−32.7200	−1.20772
$735$	−0.194315	−0.00716740
$736$	−1.78981	−0.0659734
$737$	−35.5355	−1.30897
$738$	67.3876	2.48057
$739$	0.450468	0.0165707	0.00828537	−	0.999966i	$-0.497363\pi$
$739$	0.450468	0.0165707	0.00828537	+	0.999966i	$0.497363\pi$
$740$	−1.95841	−0.0719926
$741$	−125.785	−4.62083
$742$	−28.7441	−1.05523
$743$	6.69559	0.245637	0.122819	−	0.992429i	$-0.460807\pi$
$743$	6.69559	0.245637	0.122819	+	0.992429i	$0.460807\pi$
$744$	17.1157	0.627490
$745$	0.941346	0.0344882
$746$	5.93052	0.217132
$747$	10.9957	0.402313
$748$	−3.58585	−0.131112
$749$	1.24048	0.0453260
$750$	−17.9011	−0.653654
$751$	−20.3601	−0.742952	−0.371476	−	0.928443i	$-0.621148\pi$
$751$	−20.3601	−0.742952	−0.371476	+	0.928443i	$0.621148\pi$
$752$	−9.60374	−0.350212
$753$	−4.88150	−0.177892
$754$	−26.5739	−0.967763
$755$	−0.366019	−0.0133208
$756$	−32.3400	−1.17619
$757$	37.5495	1.36476	0.682380	−	0.730998i	$-0.260945\pi$
$757$	37.5495	1.36476	0.682380	+	0.730998i	$0.260945\pi$
$758$	34.6780	1.25956
$759$	−20.2059	−0.733429
$760$	4.80236	0.174200
$761$	−45.8112	−1.66066	−0.830328	−	0.557275i	$-0.811847\pi$
$761$	−45.8112	−1.66066	−0.830328	+	0.557275i	$0.811847\pi$
$762$	2.76729	0.100248
$763$	−1.62907	−0.0589762
$764$	14.7346	0.533080
$765$	−4.07130	−0.147198
$766$	32.2031	1.16355
$767$	4.90040	0.176943
$768$	3.14832	0.113605
$769$	−9.61553	−0.346745	−0.173372	−	0.984856i	$-0.555466\pi$
$769$	−9.61553	−0.346745	−0.173372	+	0.984856i	$0.555466\pi$
$770$	−5.54628	−0.199874
$771$	92.5125	3.33176
$772$	−25.2110	−0.907363
$773$	32.2908	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$773$	32.2908	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$774$	−63.9821	−2.29979
$775$	−25.2960	−0.908661
$776$	4.19543	0.150607
$777$	27.4866	0.986075
$778$	4.87596	0.174812
$779$	79.4881	2.84796
$780$	−9.08753	−0.325386
$781$	−32.4560	−1.16137
$782$	1.78981	0.0640036
$783$	66.7865	2.38676
$784$	−0.104783	−0.00374225
$785$	−0.0249846	−0.000891738 0
$786$	15.6003	0.556443
$787$	23.2881	0.830130	0.415065	−	0.909792i	$-0.363759\pi$
$787$	23.2881	0.830130	0.415065	+	0.909792i	$0.363759\pi$
$788$	6.41372	0.228479
$789$	31.6879	1.12812
$790$	−0.958207	−0.0340915
$791$	4.64653	0.165212
$792$	24.7850	0.880698
$793$	19.1547	0.680205
$794$	8.60978	0.305550
$795$	20.2996	0.719954
$796$	−26.2951	−0.932005
$797$	54.0863	1.91583	0.957917	−	0.287044i	$-0.0926727\pi$
$797$	54.0863	1.91583	0.957917	+	0.287044i	$0.0926727\pi$
$798$	−67.4018	−2.38600
$799$	9.60374	0.339756
$800$	−4.65305	−0.164510
$801$	−80.7571	−2.85341
$802$	18.7279	0.661306
$803$	−17.0678	−0.602311
$804$	−31.1996	−1.10032
$805$	2.76833	0.0975708
$806$	−26.6408	−0.938381
$807$	21.2100	0.746628
$808$	6.48405	0.228108
$809$	33.6627	1.18352	0.591758	−	0.806116i	$-0.298434\pi$
$809$	33.6627	1.18352	0.591758	+	0.806116i	$0.298434\pi$
$810$	10.6252	0.373333
$811$	−3.20530	−0.112553	−0.0562766	−	0.998415i	$-0.517923\pi$
$811$	−3.20530	−0.112553	−0.0562766	+	0.998415i	$0.517923\pi$
$812$	−14.2396	−0.499711
$813$	61.7097	2.16425
$814$	−11.9223	−0.417877
$815$	4.51448	0.158135
$816$	−3.14832	−0.110213
$817$	−75.4711	−2.64040
$818$	−12.9012	−0.451078
$819$	88.9411	3.10785
$820$	5.74274	0.200545
$821$	10.6513	0.371733	0.185866	−	0.982575i	$-0.440491\pi$
$821$	10.6513	0.371733	0.185866	+	0.982575i	$0.440491\pi$
$822$	37.8297	1.31946
$823$	−23.2965	−0.812066	−0.406033	−	0.913858i	$-0.633088\pi$
$823$	−23.2965	−0.812066	−0.406033	+	0.913858i	$0.633088\pi$
$824$	−12.0123	−0.418469
$825$	−52.5301	−1.82886
$826$	2.62587	0.0913659
$827$	−38.3685	−1.33420	−0.667101	−	0.744967i	$-0.732465\pi$
$827$	−38.3685	−1.33420	−0.667101	+	0.744967i	$0.732465\pi$
$828$	−12.3710	−0.429922
$829$	43.1479	1.49859	0.749294	−	0.662237i	$-0.230393\pi$
$829$	43.1479	1.49859	0.749294	+	0.662237i	$0.230393\pi$
$830$	0.937050	0.0325255
$831$	71.4445	2.47838
$832$	−4.90040	−0.169891
$833$	0.104783	0.00363052
$834$	−36.9488	−1.27943
$835$	−6.72752	−0.232815
$836$	29.2356	1.01113
$837$	66.9547	2.31429
$838$	13.2122	0.456408
$839$	−3.05301	−0.105402	−0.0527008	−	0.998610i	$-0.516783\pi$
$839$	−3.05301	−0.105402	−0.0527008	+	0.998610i	$0.516783\pi$
$840$	−4.86954	−0.168015
$841$	0.406722	0.0140249
$842$	7.23797	0.249437
$843$	7.50605	0.258522
$844$	21.9976	0.757190
$845$	6.48750	0.223177
$846$	−66.3800	−2.28219
$847$	−4.87974	−0.167670
$848$	10.9465	0.375903
$849$	93.1394	3.19654
$850$	4.65305	0.159598
$851$	5.95081	0.203991
$852$	−28.4958	−0.976252
$853$	19.7483	0.676167	0.338084	−	0.941116i	$-0.390221\pi$
$853$	19.7483	0.676167	0.338084	+	0.941116i	$0.390221\pi$
$854$	10.2641	0.351229
$855$	33.1934	1.13519
$856$	−0.472405	−0.0161465
$857$	−22.1097	−0.755253	−0.377626	−	0.925958i	$-0.623260\pi$
$857$	−22.1097	−0.755253	−0.377626	+	0.925958i	$0.623260\pi$
$858$	−55.3225	−1.88868
$859$	−14.6412	−0.499551	−0.249775	−	0.968304i	$-0.580357\pi$
$859$	−14.6412	−0.499551	−0.249775	+	0.968304i	$0.580357\pi$
$860$	−5.45252	−0.185929
$861$	−80.6001	−2.74684
$862$	−30.9120	−1.05287
$863$	−6.48574	−0.220777	−0.110389	−	0.993889i	$-0.535210\pi$
$863$	−6.48574	−0.220777	−0.110389	+	0.993889i	$0.535210\pi$
$864$	12.3159	0.418995
$865$	−5.77549	−0.196372
$866$	−29.4556	−1.00094
$867$	3.14832	0.106922
$868$	−14.2754	−0.484540
$869$	−5.83332	−0.197882
$870$	10.0563	0.340940
$871$	48.5626	1.64548
$872$	0.620390	0.0210091
$873$	28.9984	0.981447
$874$	−14.5924	−0.493596
$875$	14.9305	0.504743
$876$	−14.9853	−0.506305
$877$	−14.4030	−0.486356	−0.243178	−	0.969982i	$-0.578190\pi$
$877$	−14.4030	−0.486356	−0.243178	+	0.969982i	$0.578190\pi$
$878$	5.57890	0.188279
$879$	−27.3976	−0.924099
$880$	2.11217	0.0712011
$881$	−39.5663	−1.33302	−0.666512	−	0.745495i	$-0.732213\pi$
$881$	−39.5663	−1.33302	−0.666512	+	0.745495i	$0.732213\pi$
$882$	−0.724250	−0.0243868
$883$	3.34992	0.112734	0.0563670	−	0.998410i	$-0.482048\pi$
$883$	3.34992	0.112734	0.0563670	+	0.998410i	$0.482048\pi$
$884$	4.90040	0.164818
$885$	−1.85445	−0.0623365
$886$	15.9821	0.536930
$887$	−54.5587	−1.83190	−0.915951	−	0.401290i	$-0.868562\pi$
$887$	−54.5587	−1.83190	−0.915951	+	0.401290i	$0.868562\pi$
$888$	−10.4676	−0.351269
$889$	−2.30807	−0.0774103
$890$	−6.88208	−0.230688
$891$	64.6838	2.16699
$892$	−9.32882	−0.312352
$893$	−78.2996	−2.62020
$894$	5.03143	0.168276
$895$	−4.96274	−0.165886
$896$	−2.62587	−0.0877243
$897$	27.6133	0.921980
$898$	−13.9942	−0.466994
$899$	29.4807	0.983238
$900$	−32.1614	−1.07205
$901$	−10.9465	−0.364680
$902$	34.9603	1.16405
$903$	76.5268	2.54665
$904$	−1.76952	−0.0588533
$905$	−4.75443	−0.158043
$906$	−1.95635	−0.0649952
$907$	0.825674	0.0274160	0.0137080	−	0.999906i	$-0.495636\pi$
$907$	0.825674	0.0274160	0.0137080	+	0.999906i	$0.495636\pi$
$908$	26.8160	0.889922
$909$	44.8171	1.48649
$910$	7.57951	0.251258
$911$	−22.3106	−0.739183	−0.369591	−	0.929194i	$-0.620502\pi$
$911$	−22.3106	−0.739183	−0.369591	+	0.929194i	$0.620502\pi$
$912$	25.6683	0.849963
$913$	5.70452	0.188792
$914$	−21.1888	−0.700863
$915$	−7.24868	−0.239634
$916$	−23.0821	−0.762653
$917$	−13.0115	−0.429678
$918$	−12.3159	−0.406485
$919$	−1.38141	−0.0455684	−0.0227842	−	0.999740i	$-0.507253\pi$
$919$	−1.38141	−0.0455684	−0.0227842	+	0.999740i	$0.507253\pi$
$920$	−1.05425	−0.0347576
$921$	82.8394	2.72965
$922$	−20.9031	−0.688406
$923$	44.3542	1.45994
$924$	−29.6445	−0.975234
$925$	15.4705	0.508668
$926$	40.6488	1.33580
$927$	−83.0279	−2.72699
$928$	5.42280	0.178012
$929$	−2.61999	−0.0859590	−0.0429795	−	0.999076i	$-0.513685\pi$
$929$	−2.61999	−0.0859590	−0.0429795	+	0.999076i	$0.513685\pi$
$930$	10.0816	0.330589
$931$	−0.854300	−0.0279985
$932$	9.38591	0.307446
$933$	−3.51558	−0.115095
$934$	−16.6371	−0.544381
$935$	−2.11217	−0.0690752
$936$	−33.8710	−1.10711
$937$	13.0633	0.426759	0.213380	−	0.976969i	$-0.431553\pi$
$937$	13.0633	0.426759	0.213380	+	0.976969i	$0.431553\pi$
$938$	26.0222	0.849655
$939$	6.77240	0.221009
$940$	−5.65687	−0.184507
$941$	33.5340	1.09318	0.546589	−	0.837401i	$-0.315926\pi$
$941$	33.5340	1.09318	0.546589	+	0.837401i	$0.315926\pi$
$942$	−0.133541	−0.00435100
$943$	−17.4498	−0.568244
$944$	−1.00000	−0.0325472
$945$	−19.0491	−0.619669
$946$	−33.1936	−1.07922
$947$	40.5897	1.31899	0.659494	−	0.751710i	$-0.270770\pi$
$947$	40.5897	1.31899	0.659494	+	0.751710i	$0.270770\pi$
$948$	−5.12155	−0.166340
$949$	23.3248	0.757155
$950$	−37.9364	−1.23082
$951$	−8.82934	−0.286311
$952$	2.62587	0.0851051
$953$	30.2004	0.978286	0.489143	−	0.872204i	$-0.337310\pi$
$953$	30.2004	0.978286	0.489143	+	0.872204i	$0.337310\pi$
$954$	75.6609	2.44961
$955$	8.67910	0.280849
$956$	−10.0692	−0.325660
$957$	61.2201	1.97896
$958$	30.6239	0.989413
$959$	−31.5521	−1.01887
$960$	1.85445	0.0598520
$961$	−1.44504	−0.0466143
$962$	16.2929	0.525305
$963$	−3.26521	−0.105220
$964$	2.13124	0.0686425
$965$	−14.8500	−0.478037
$966$	14.7965	0.476071
$967$	31.7672	1.02157	0.510783	−	0.859710i	$-0.329356\pi$
$967$	31.7672	1.02157	0.510783	+	0.859710i	$0.329356\pi$
$968$	1.85833	0.0597290
$969$	−25.6683	−0.824585
$970$	2.47123	0.0793463
$971$	44.8009	1.43773	0.718865	−	0.695150i	$-0.244662\pi$
$971$	44.8009	1.43773	0.718865	+	0.695150i	$0.244662\pi$
$972$	19.8436	0.636484
$973$	30.8174	0.987961
$974$	−37.4899	−1.20126
$975$	71.7872	2.29903
$976$	−3.90881	−0.125118
$977$	−28.5384	−0.913023	−0.456511	−	0.889718i	$-0.650901\pi$
$977$	−28.5384	−0.913023	−0.456511	+	0.889718i	$0.650901\pi$
$978$	24.1296	0.771580
$979$	−41.8963	−1.33901
$980$	−0.0617202	−0.00197158
$981$	4.28807	0.136908
$982$	−25.1759	−0.803396
$983$	3.29883	0.105216	0.0526082	−	0.998615i	$-0.483247\pi$
$983$	3.29883	0.105216	0.0526082	+	0.998615i	$0.483247\pi$
$984$	30.6946	0.978507
$985$	3.77786	0.120373
$986$	−5.42280	−0.172697
$987$	79.3949	2.52717
$988$	−39.9531	−1.27108
$989$	16.5680	0.526831
$990$	14.5991	0.463989
$991$	−34.7089	−1.10257	−0.551283	−	0.834319i	$-0.685861\pi$
$991$	−34.7089	−1.10257	−0.551283	+	0.834319i	$0.685861\pi$
$992$	5.43645	0.172607
$993$	29.3447	0.931224
$994$	23.7672	0.753848
$995$	−15.4885	−0.491020
$996$	5.00847	0.158700
$997$	53.7657	1.70278	0.851389	−	0.524535i	$-0.175761\pi$
$997$	53.7657	1.70278	0.851389	+	0.524535i	$0.175761\pi$
$998$	−13.5773	−0.429783
$999$	−40.9481	−1.29554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2006.2.a.t.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2006.2.a.t.1.7	✓	8	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:	\( N \)	\(=\)	\( 2006 = 2 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.14832 q^{3} +1.00000 q^{4} +0.589028 q^{5} +3.14832 q^{6} -2.62587 q^{7} +1.00000 q^{8} +6.91190 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.14832 q^{3} +1.00000 q^{4} +0.589028 q^{5} +3.14832 q^{6} -2.62587 q^{7} +1.00000 q^{8} +6.91190 q^{9} +0.589028 q^{10} +3.58585 q^{11} +3.14832 q^{12} -4.90040 q^{13} -2.62587 q^{14} +1.85445 q^{15} +1.00000 q^{16} -1.00000 q^{17} +6.91190 q^{18} +8.15303 q^{19} +0.589028 q^{20} -8.26708 q^{21} +3.58585 q^{22} -1.78981 q^{23} +3.14832 q^{24} -4.65305 q^{25} -4.90040 q^{26} +12.3159 q^{27} -2.62587 q^{28} +5.42280 q^{29} +1.85445 q^{30} +5.43645 q^{31} +1.00000 q^{32} +11.2894 q^{33} -1.00000 q^{34} -1.54671 q^{35} +6.91190 q^{36} -3.32482 q^{37} +8.15303 q^{38} -15.4280 q^{39} +0.589028 q^{40} +9.74951 q^{41} -8.26708 q^{42} -9.25681 q^{43} +3.58585 q^{44} +4.07130 q^{45} -1.78981 q^{46} -9.60374 q^{47} +3.14832 q^{48} -0.104783 q^{49} -4.65305 q^{50} -3.14832 q^{51} -4.90040 q^{52} +10.9465 q^{53} +12.3159 q^{54} +2.11217 q^{55} -2.62587 q^{56} +25.6683 q^{57} +5.42280 q^{58} -1.00000 q^{59} +1.85445 q^{60} -3.90881 q^{61} +5.43645 q^{62} -18.1498 q^{63} +1.00000 q^{64} -2.88647 q^{65} +11.2894 q^{66} -9.90992 q^{67} -1.00000 q^{68} -5.63490 q^{69} -1.54671 q^{70} -9.05114 q^{71} +6.91190 q^{72} -4.75977 q^{73} -3.32482 q^{74} -14.6493 q^{75} +8.15303 q^{76} -9.41600 q^{77} -15.4280 q^{78} -1.62676 q^{79} +0.589028 q^{80} +18.0386 q^{81} +9.74951 q^{82} +1.59084 q^{83} -8.26708 q^{84} -0.589028 q^{85} -9.25681 q^{86} +17.0727 q^{87} +3.58585 q^{88} -11.6838 q^{89} +4.07130 q^{90} +12.8678 q^{91} -1.78981 q^{92} +17.1157 q^{93} -9.60374 q^{94} +4.80236 q^{95} +3.14832 q^{96} +4.19543 q^{97} -0.104783 q^{98} +24.7850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 5 q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 8 q^{8} + 17 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 5 * q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 8 * q^8 + 17 * q^9	\( 8 q + 8 q^{2} + 5 q^{3} + 8 q^{4} - 5 q^{5} + 5 q^{6} + 8 q^{8} + 17 q^{9} - 5 q^{10} - 5 q^{11} + 5 q^{12} + q^{13} + 9 q^{15} + 8 q^{16} - 8 q^{17} + 17 q^{18} + 8 q^{19} - 5 q^{20} + 13 q^{21} - 5 q^{22} - 8 q^{23} + 5 q^{24} + 29 q^{25} + q^{26} + 44 q^{27} + 20 q^{29} + 9 q^{30} + 18 q^{31} + 8 q^{32} - 23 q^{33} - 8 q^{34} + 9 q^{35} + 17 q^{36} - 7 q^{37} + 8 q^{38} + 6 q^{39} - 5 q^{40} + 12 q^{41} + 13 q^{42} - 8 q^{43} - 5 q^{44} + 5 q^{45} - 8 q^{46} + 30 q^{47} + 5 q^{48} + 28 q^{49} + 29 q^{50} - 5 q^{51} + q^{52} - 5 q^{53} + 44 q^{54} + 23 q^{55} + 13 q^{57} + 20 q^{58} - 8 q^{59} + 9 q^{60} + 12 q^{61} + 18 q^{62} + 21 q^{63} + 8 q^{64} - 52 q^{65} - 23 q^{66} - 4 q^{67} - 8 q^{68} + 24 q^{69} + 9 q^{70} + 4 q^{71} + 17 q^{72} - 18 q^{73} - 7 q^{74} - 49 q^{75} + 8 q^{76} - 13 q^{77} + 6 q^{78} - 17 q^{79} - 5 q^{80} + 12 q^{81} + 12 q^{82} + 9 q^{83} + 13 q^{84} + 5 q^{85} - 8 q^{86} + 48 q^{87} - 5 q^{88} - 12 q^{89} + 5 q^{90} + 31 q^{91} - 8 q^{92} - 3 q^{93} + 30 q^{94} - 7 q^{95} + 5 q^{96} - 11 q^{97} + 28 q^{98} - 27 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 5 * q^3 + 8 * q^4 - 5 * q^5 + 5 * q^6 + 8 * q^8 + 17 * q^9 - 5 * q^10 - 5 * q^11 + 5 * q^12 + q^13 + 9 * q^15 + 8 * q^16 - 8 * q^17 + 17 * q^18 + 8 * q^19 - 5 * q^20 + 13 * q^21 - 5 * q^22 - 8 * q^23 + 5 * q^24 + 29 * q^25 + q^26 + 44 * q^27 + 20 * q^29 + 9 * q^30 + 18 * q^31 + 8 * q^32 - 23 * q^33 - 8 * q^34 + 9 * q^35 + 17 * q^36 - 7 * q^37 + 8 * q^38 + 6 * q^39 - 5 * q^40 + 12 * q^41 + 13 * q^42 - 8 * q^43 - 5 * q^44 + 5 * q^45 - 8 * q^46 + 30 * q^47 + 5 * q^48 + 28 * q^49 + 29 * q^50 - 5 * q^51 + q^52 - 5 * q^53 + 44 * q^54 + 23 * q^55 + 13 * q^57 + 20 * q^58 - 8 * q^59 + 9 * q^60 + 12 * q^61 + 18 * q^62 + 21 * q^63 + 8 * q^64 - 52 * q^65 - 23 * q^66 - 4 * q^67 - 8 * q^68 + 24 * q^69 + 9 * q^70 + 4 * q^71 + 17 * q^72 - 18 * q^73 - 7 * q^74 - 49 * q^75 + 8 * q^76 - 13 * q^77 + 6 * q^78 - 17 * q^79 - 5 * q^80 + 12 * q^81 + 12 * q^82 + 9 * q^83 + 13 * q^84 + 5 * q^85 - 8 * q^86 + 48 * q^87 - 5 * q^88 - 12 * q^89 + 5 * q^90 + 31 * q^91 - 8 * q^92 - 3 * q^93 + 30 * q^94 - 7 * q^95 + 5 * q^96 - 11 * q^97 + 28 * q^98 - 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more