LMFDB - Embedded newform 4010.2.a.k.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, 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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} + \cdots - 512 $ x^15 - 7x^14 - 7x^13 + 133x^12 - 99x^11 - 941x^10 + 1290x^9 + 3031x^8 - 5452x^7 - 4098x^6 + 9986x^5 + 850x^4 - 7216x^3 + 1688x^2 + 1441x - 512
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.95650$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.918908 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.918908 q^{6} -3.92602 q^{7} -1.00000 q^{8} -2.15561 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.918908 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.918908 q^{6} -3.92602 q^{7} -1.00000 q^{8} -2.15561 q^{9} +1.00000 q^{10} +1.54426 q^{11} -0.918908 q^{12} -0.562082 q^{13} +3.92602 q^{14} +0.918908 q^{15} +1.00000 q^{16} -6.59798 q^{17} +2.15561 q^{18} +6.11315 q^{19} -1.00000 q^{20} +3.60766 q^{21} -1.54426 q^{22} +4.55514 q^{23} +0.918908 q^{24} +1.00000 q^{25} +0.562082 q^{26} +4.73753 q^{27} -3.92602 q^{28} +4.69768 q^{29} -0.918908 q^{30} +5.10220 q^{31} -1.00000 q^{32} -1.41903 q^{33} +6.59798 q^{34} +3.92602 q^{35} -2.15561 q^{36} -2.86189 q^{37} -6.11315 q^{38} +0.516502 q^{39} +1.00000 q^{40} +6.29523 q^{41} -3.60766 q^{42} -3.87296 q^{43} +1.54426 q^{44} +2.15561 q^{45} -4.55514 q^{46} -2.99679 q^{47} -0.918908 q^{48} +8.41366 q^{49} -1.00000 q^{50} +6.06294 q^{51} -0.562082 q^{52} +8.81050 q^{53} -4.73753 q^{54} -1.54426 q^{55} +3.92602 q^{56} -5.61743 q^{57} -4.69768 q^{58} +8.63699 q^{59} +0.918908 q^{60} -5.01728 q^{61} -5.10220 q^{62} +8.46297 q^{63} +1.00000 q^{64} +0.562082 q^{65} +1.41903 q^{66} +1.74019 q^{67} -6.59798 q^{68} -4.18576 q^{69} -3.92602 q^{70} -7.67511 q^{71} +2.15561 q^{72} +3.93595 q^{73} +2.86189 q^{74} -0.918908 q^{75} +6.11315 q^{76} -6.06278 q^{77} -0.516502 q^{78} -14.4964 q^{79} -1.00000 q^{80} +2.11347 q^{81} -6.29523 q^{82} +4.99765 q^{83} +3.60766 q^{84} +6.59798 q^{85} +3.87296 q^{86} -4.31674 q^{87} -1.54426 q^{88} -14.6418 q^{89} -2.15561 q^{90} +2.20675 q^{91} +4.55514 q^{92} -4.68846 q^{93} +2.99679 q^{94} -6.11315 q^{95} +0.918908 q^{96} -4.83469 q^{97} -8.41366 q^{98} -3.32881 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^10 - 2 * q^11 - 6 * q^12 - 13 * q^13 + 5 * q^14 + 6 * q^15 + 15 * q^16 + 11 * q^17 - 19 * q^18 - 15 * q^19 - 15 * q^20 - 2 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 15 * q^25 + 13 * q^26 - 12 * q^27 - 5 * q^28 + 28 * q^29 - 6 * q^30 - 12 * q^31 - 15 * q^32 - 22 * q^33 - 11 * q^34 + 5 * q^35 + 19 * q^36 - 23 * q^37 + 15 * q^38 - 2 * q^39 + 15 * q^40 + 24 * q^41 + 2 * q^42 - 24 * q^43 - 2 * q^44 - 19 * q^45 + 3 * q^46 - 3 * q^47 - 6 * q^48 + 20 * q^49 - 15 * q^50 - 5 * q^51 - 13 * q^52 + 10 * q^53 + 12 * q^54 + 2 * q^55 + 5 * q^56 - 11 * q^57 - 28 * q^58 + 2 * q^59 + 6 * q^60 + 15 * q^61 + 12 * q^62 - 2 * q^63 + 15 * q^64 + 13 * q^65 + 22 * q^66 - 48 * q^67 + 11 * q^68 + 21 * q^69 - 5 * q^70 + 15 * q^71 - 19 * q^72 - 47 * q^73 + 23 * q^74 - 6 * q^75 - 15 * q^76 + 7 * q^77 + 2 * q^78 - 34 * q^79 - 15 * q^80 + 43 * q^81 - 24 * q^82 - 32 * q^83 - 2 * q^84 - 11 * q^85 + 24 * q^86 + 14 * q^87 + 2 * q^88 + 25 * q^89 + 19 * q^90 - 32 * q^91 - 3 * q^92 - 42 * q^93 + 3 * q^94 + 15 * q^95 + 6 * q^96 - 34 * q^97 - 20 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.918908	−0.530532	−0.265266	−	0.964175i	$-0.585460\pi$
$3$	−0.918908	−0.530532	−0.265266	+	0.964175i	$0.585460\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.918908	0.375143
$7$	−3.92602	−1.48390	−0.741949	−	0.670457i	$-0.766098\pi$
$7$	−3.92602	−1.48390	−0.741949	+	0.670457i	$0.766098\pi$
$8$	−1.00000	−0.353553
$9$	−2.15561	−0.718536
$10$	1.00000	0.316228
$11$	1.54426	0.465611	0.232805	−	0.972523i	$-0.425210\pi$
$11$	1.54426	0.465611	0.232805	+	0.972523i	$0.425210\pi$
$12$	−0.918908	−0.265266
$13$	−0.562082	−0.155894	−0.0779468	−	0.996958i	$-0.524836\pi$
$13$	−0.562082	−0.155894	−0.0779468	+	0.996958i	$0.524836\pi$
$14$	3.92602	1.04927
$15$	0.918908	0.237261
$16$	1.00000	0.250000
$17$	−6.59798	−1.60024	−0.800122	−	0.599837i	$-0.795232\pi$
$17$	−6.59798	−1.60024	−0.800122	+	0.599837i	$0.795232\pi$
$18$	2.15561	0.508082
$19$	6.11315	1.40245	0.701226	−	0.712939i	$-0.252636\pi$
$19$	6.11315	1.40245	0.701226	+	0.712939i	$0.252636\pi$
$20$	−1.00000	−0.223607
$21$	3.60766	0.787255
$22$	−1.54426	−0.329236
$23$	4.55514	0.949813	0.474907	−	0.880036i	$-0.342482\pi$
$23$	4.55514	0.949813	0.474907	+	0.880036i	$0.342482\pi$
$24$	0.918908	0.187571
$25$	1.00000	0.200000
$26$	0.562082	0.110233
$27$	4.73753	0.911738
$28$	−3.92602	−0.741949
$29$	4.69768	0.872338	0.436169	−	0.899865i	$-0.356335\pi$
$29$	4.69768	0.872338	0.436169	+	0.899865i	$0.356335\pi$
$30$	−0.918908	−0.167769
$31$	5.10220	0.916383	0.458191	−	0.888854i	$-0.348497\pi$
$31$	5.10220	0.916383	0.458191	+	0.888854i	$0.348497\pi$
$32$	−1.00000	−0.176777
$33$	−1.41903	−0.247021
$34$	6.59798	1.13154
$35$	3.92602	0.663619
$36$	−2.15561	−0.359268
$37$	−2.86189	−0.470492	−0.235246	−	0.971936i	$-0.575590\pi$
$37$	−2.86189	−0.470492	−0.235246	+	0.971936i	$0.575590\pi$
$38$	−6.11315	−0.991684
$39$	0.516502	0.0827065
$40$	1.00000	0.158114
$41$	6.29523	0.983150	0.491575	−	0.870835i	$-0.336421\pi$
$41$	6.29523	0.983150	0.491575	+	0.870835i	$0.336421\pi$
$42$	−3.60766	−0.556673
$43$	−3.87296	−0.590621	−0.295311	−	0.955401i	$-0.595423\pi$
$43$	−3.87296	−0.590621	−0.295311	+	0.955401i	$0.595423\pi$
$44$	1.54426	0.232805
$45$	2.15561	0.321339
$46$	−4.55514	−0.671619
$47$	−2.99679	−0.437127	−0.218564	−	0.975823i	$-0.570137\pi$
$47$	−2.99679	−0.437127	−0.218564	+	0.975823i	$0.570137\pi$
$48$	−0.918908	−0.132633
$49$	8.41366	1.20195
$50$	−1.00000	−0.141421
$51$	6.06294	0.848981
$52$	−0.562082	−0.0779468
$53$	8.81050	1.21022	0.605108	−	0.796144i	$-0.293130\pi$
$53$	8.81050	1.21022	0.605108	+	0.796144i	$0.293130\pi$
$54$	−4.73753	−0.644696
$55$	−1.54426	−0.208227
$56$	3.92602	0.524637
$57$	−5.61743	−0.744046
$58$	−4.69768	−0.616836
$59$	8.63699	1.12444	0.562220	−	0.826988i	$-0.309947\pi$
$59$	8.63699	1.12444	0.562220	+	0.826988i	$0.309947\pi$
$60$	0.918908	0.118631
$61$	−5.01728	−0.642396	−0.321198	−	0.947012i	$-0.604086\pi$
$61$	−5.01728	−0.642396	−0.321198	+	0.947012i	$0.604086\pi$
$62$	−5.10220	−0.647980
$63$	8.46297	1.06623
$64$	1.00000	0.125000
$65$	0.562082	0.0697177
$66$	1.41903	0.174670
$67$	1.74019	0.212598	0.106299	−	0.994334i	$-0.466100\pi$
$67$	1.74019	0.212598	0.106299	+	0.994334i	$0.466100\pi$
$68$	−6.59798	−0.800122
$69$	−4.18576	−0.503906
$70$	−3.92602	−0.469250
$71$	−7.67511	−0.910868	−0.455434	−	0.890270i	$-0.650516\pi$
$71$	−7.67511	−0.910868	−0.455434	+	0.890270i	$0.650516\pi$
$72$	2.15561	0.254041
$73$	3.93595	0.460668	0.230334	−	0.973112i	$-0.426018\pi$
$73$	3.93595	0.460668	0.230334	+	0.973112i	$0.426018\pi$
$74$	2.86189	0.332688
$75$	−0.918908	−0.106106
$76$	6.11315	0.701226
$77$	−6.06278	−0.690918
$78$	−0.516502	−0.0584823
$79$	−14.4964	−1.63097	−0.815486	−	0.578777i	$-0.803530\pi$
$79$	−14.4964	−1.63097	−0.815486	+	0.578777i	$0.803530\pi$
$80$	−1.00000	−0.111803
$81$	2.11347	0.234830
$82$	−6.29523	−0.695192
$83$	4.99765	0.548564	0.274282	−	0.961649i	$-0.411560\pi$
$83$	4.99765	0.548564	0.274282	+	0.961649i	$0.411560\pi$
$84$	3.60766	0.393628
$85$	6.59798	0.715651
$86$	3.87296	0.417632
$87$	−4.31674	−0.462803
$88$	−1.54426	−0.164618
$89$	−14.6418	−1.55202	−0.776012	−	0.630719i	$-0.782760\pi$
$89$	−14.6418	−1.55202	−0.776012	+	0.630719i	$0.782760\pi$
$90$	−2.15561	−0.227221
$91$	2.20675	0.231330
$92$	4.55514	0.474907
$93$	−4.68846	−0.486170
$94$	2.99679	0.309096
$95$	−6.11315	−0.627196
$96$	0.918908	0.0937857
$97$	−4.83469	−0.490888	−0.245444	−	0.969411i	$-0.578934\pi$
$97$	−4.83469	−0.490888	−0.245444	+	0.969411i	$0.578934\pi$
$98$	−8.41366	−0.849908
$99$	−3.32881	−0.334558
$100$	1.00000	0.100000
$101$	−8.29206	−0.825090	−0.412545	−	0.910937i	$-0.635360\pi$
$101$	−8.29206	−0.825090	−0.412545	+	0.910937i	$0.635360\pi$
$102$	−6.06294	−0.600320
$103$	−0.0164182	−0.00161774	−0.000808868	−	1.00000i	$-0.500257\pi$
$103$	−0.0164182	−0.00161774	−0.000808868		1.00000i	$0.500257\pi$
$104$	0.562082	0.0551167
$105$	−3.60766	−0.352071
$106$	−8.81050	−0.855752
$107$	−8.49762	−0.821496	−0.410748	−	0.911749i	$-0.634732\pi$
$107$	−8.49762	−0.821496	−0.410748	+	0.911749i	$0.634732\pi$
$108$	4.73753	0.455869
$109$	8.07761	0.773694	0.386847	−	0.922144i	$-0.373564\pi$
$109$	8.07761	0.773694	0.386847	+	0.922144i	$0.373564\pi$
$110$	1.54426	0.147239
$111$	2.62982	0.249611
$112$	−3.92602	−0.370974
$113$	−9.68966	−0.911527	−0.455763	−	0.890101i	$-0.650634\pi$
$113$	−9.68966	−0.911527	−0.455763	+	0.890101i	$0.650634\pi$
$114$	5.61743	0.526120
$115$	−4.55514	−0.424769
$116$	4.69768	0.436169
$117$	1.21163	0.112015
$118$	−8.63699	−0.795100
$119$	25.9038	2.37460
$120$	−0.918908	−0.0838845
$121$	−8.61527	−0.783207
$122$	5.01728	0.454243
$123$	−5.78474	−0.521592
$124$	5.10220	0.458191
$125$	−1.00000	−0.0894427
$126$	−8.46297	−0.753941
$127$	16.5317	1.46695	0.733477	−	0.679715i	$-0.237896\pi$
$127$	16.5317	1.46695	0.733477	+	0.679715i	$0.237896\pi$
$128$	−1.00000	−0.0883883
$129$	3.55890	0.313343
$130$	−0.562082	−0.0492979
$131$	−10.2662	−0.896966	−0.448483	−	0.893791i	$-0.648036\pi$
$131$	−10.2662	−0.896966	−0.448483	+	0.893791i	$0.648036\pi$
$132$	−1.41903	−0.123511
$133$	−24.0004	−2.08110
$134$	−1.74019	−0.150329
$135$	−4.73753	−0.407742
$136$	6.59798	0.565772
$137$	−2.55263	−0.218086	−0.109043	−	0.994037i	$-0.534779\pi$
$137$	−2.55263	−0.218086	−0.109043	+	0.994037i	$0.534779\pi$
$138$	4.18576	0.356316
$139$	1.40272	0.118977	0.0594885	−	0.998229i	$-0.481053\pi$
$139$	1.40272	0.118977	0.0594885	+	0.998229i	$0.481053\pi$
$140$	3.92602	0.331810
$141$	2.75378	0.231910
$142$	7.67511	0.644081
$143$	−0.867999	−0.0725857
$144$	−2.15561	−0.179634
$145$	−4.69768	−0.390121
$146$	−3.93595	−0.325741
$147$	−7.73139	−0.637674
$148$	−2.86189	−0.235246
$149$	23.7432	1.94512	0.972558	−	0.232659i	$-0.0747424\pi$
$149$	23.7432	1.94512	0.972558	+	0.232659i	$0.0747424\pi$
$150$	0.918908	0.0750286
$151$	−18.6292	−1.51602	−0.758012	−	0.652241i	$-0.773829\pi$
$151$	−18.6292	−1.51602	−0.758012	+	0.652241i	$0.773829\pi$
$152$	−6.11315	−0.495842
$153$	14.2226	1.14983
$154$	6.06278	0.488553
$155$	−5.10220	−0.409819
$156$	0.516502	0.0413533
$157$	−6.43230	−0.513354	−0.256677	−	0.966497i	$-0.582628\pi$
$157$	−6.43230	−0.513354	−0.256677	+	0.966497i	$0.582628\pi$
$158$	14.4964	1.15327
$159$	−8.09604	−0.642058
$160$	1.00000	0.0790569
$161$	−17.8836	−1.40943
$162$	−2.11347	−0.166050
$163$	16.4413	1.28778	0.643892	−	0.765117i	$-0.277319\pi$
$163$	16.4413	1.28778	0.643892	+	0.765117i	$0.277319\pi$
$164$	6.29523	0.491575
$165$	1.41903	0.110471
$166$	−4.99765	−0.387893
$167$	3.26012	0.252275	0.126138	−	0.992013i	$-0.459742\pi$
$167$	3.26012	0.252275	0.126138	+	0.992013i	$0.459742\pi$
$168$	−3.60766	−0.278337
$169$	−12.6841	−0.975697
$170$	−6.59798	−0.506042
$171$	−13.1776	−1.00771
$172$	−3.87296	−0.295311
$173$	17.5469	1.33407	0.667035	−	0.745026i	$-0.267563\pi$
$173$	17.5469	1.33407	0.667035	+	0.745026i	$0.267563\pi$
$174$	4.31674	0.327251
$175$	−3.92602	−0.296780
$176$	1.54426	0.116403
$177$	−7.93660	−0.596552
$178$	14.6418	1.09745
$179$	−1.40640	−0.105119	−0.0525597	−	0.998618i	$-0.516738\pi$
$179$	−1.40640	−0.105119	−0.0525597	+	0.998618i	$0.516738\pi$
$180$	2.15561	0.160669
$181$	8.51792	0.633132	0.316566	−	0.948571i	$-0.397470\pi$
$181$	8.51792	0.633132	0.316566	+	0.948571i	$0.397470\pi$
$182$	−2.20675	−0.163575
$183$	4.61042	0.340812
$184$	−4.55514	−0.335810
$185$	2.86189	0.210410
$186$	4.68846	0.343774
$187$	−10.1890	−0.745091
$188$	−2.99679	−0.218564
$189$	−18.5997	−1.35293
$190$	6.11315	0.443495
$191$	0.207405	0.0150073	0.00750365	−	0.999972i	$-0.497611\pi$
$191$	0.207405	0.0150073	0.00750365	+	0.999972i	$0.497611\pi$
$192$	−0.918908	−0.0663165
$193$	−18.1703	−1.30793	−0.653964	−	0.756525i	$-0.726895\pi$
$193$	−18.1703	−1.30793	−0.653964	+	0.756525i	$0.726895\pi$
$194$	4.83469	0.347110
$195$	−0.516502	−0.0369875
$196$	8.41366	0.600976
$197$	12.9568	0.923132	0.461566	−	0.887106i	$-0.347288\pi$
$197$	12.9568	0.923132	0.461566	+	0.887106i	$0.347288\pi$
$198$	3.32881	0.236568
$199$	−12.2094	−0.865503	−0.432751	−	0.901513i	$-0.642457\pi$
$199$	−12.2094	−0.865503	−0.432751	+	0.901513i	$0.642457\pi$
$200$	−1.00000	−0.0707107
$201$	−1.59907	−0.112790
$202$	8.29206	0.583427
$203$	−18.4432	−1.29446
$204$	6.06294	0.424490
$205$	−6.29523	−0.439678
$206$	0.0164182	0.00114391
$207$	−9.81910	−0.682475
$208$	−0.562082	−0.0389734
$209$	9.44027	0.652997
$210$	3.60766	0.248952
$211$	12.0336	0.828430	0.414215	−	0.910179i	$-0.364056\pi$
$211$	12.0336	0.828430	0.414215	+	0.910179i	$0.364056\pi$
$212$	8.81050	0.605108
$213$	7.05272	0.483245
$214$	8.49762	0.580886
$215$	3.87296	0.264134
$216$	−4.73753	−0.322348
$217$	−20.0314	−1.35982
$218$	−8.07761	−0.547085
$219$	−3.61677	−0.244399
$220$	−1.54426	−0.104114
$221$	3.70861	0.249468
$222$	−2.62982	−0.176502
$223$	−17.7792	−1.19058	−0.595292	−	0.803510i	$-0.702964\pi$
$223$	−17.7792	−1.19058	−0.595292	+	0.803510i	$0.702964\pi$
$224$	3.92602	0.262319
$225$	−2.15561	−0.143707
$226$	9.68966	0.644547
$227$	−10.8774	−0.721956	−0.360978	−	0.932574i	$-0.617557\pi$
$227$	−10.8774	−0.721956	−0.360978	+	0.932574i	$0.617557\pi$
$228$	−5.61743	−0.372023
$229$	−10.1215	−0.668845	−0.334422	−	0.942423i	$-0.608541\pi$
$229$	−10.1215	−0.668845	−0.334422	+	0.942423i	$0.608541\pi$
$230$	4.55514	0.300357
$231$	5.57114	0.366554
$232$	−4.69768	−0.308418
$233$	−7.83644	−0.513382	−0.256691	−	0.966493i	$-0.582632\pi$
$233$	−7.83644	−0.513382	−0.256691	+	0.966493i	$0.582632\pi$
$234$	−1.21163	−0.0792066
$235$	2.99679	0.195489
$236$	8.63699	0.562220
$237$	13.3209	0.865283
$238$	−25.9038	−1.67910
$239$	7.90410	0.511274	0.255637	−	0.966773i	$-0.417715\pi$
$239$	7.90410	0.511274	0.255637	+	0.966773i	$0.417715\pi$
$240$	0.918908	0.0593153
$241$	15.6685	1.00930	0.504650	−	0.863324i	$-0.331622\pi$
$241$	15.6685	1.00930	0.504650	+	0.863324i	$0.331622\pi$
$242$	8.61527	0.553811
$243$	−16.1547	−1.03632
$244$	−5.01728	−0.321198
$245$	−8.41366	−0.537529
$246$	5.78474	0.368822
$247$	−3.43609	−0.218633
$248$	−5.10220	−0.323990
$249$	−4.59239	−0.291031
$250$	1.00000	0.0632456
$251$	5.67050	0.357919	0.178959	−	0.983856i	$-0.442727\pi$
$251$	5.67050	0.357919	0.178959	+	0.983856i	$0.442727\pi$
$252$	8.46297	0.533117
$253$	7.03431	0.442243
$254$	−16.5317	−1.03729
$255$	−6.06294	−0.379676
$256$	1.00000	0.0625000
$257$	−6.28655	−0.392144	−0.196072	−	0.980590i	$-0.562819\pi$
$257$	−6.28655	−0.392144	−0.196072	+	0.980590i	$0.562819\pi$
$258$	−3.55890	−0.221567
$259$	11.2359	0.698162
$260$	0.562082	0.0348589
$261$	−10.1264	−0.626806
$262$	10.2662	0.634251
$263$	−28.9758	−1.78673	−0.893363	−	0.449336i	$-0.851660\pi$
$263$	−28.9758	−1.78673	−0.893363	+	0.449336i	$0.851660\pi$
$264$	1.41903	0.0873352
$265$	−8.81050	−0.541225
$266$	24.0004	1.47156
$267$	13.4544	0.823398
$268$	1.74019	0.106299
$269$	22.5756	1.37646	0.688228	−	0.725495i	$-0.258389\pi$
$269$	22.5756	1.37646	0.688228	+	0.725495i	$0.258389\pi$
$270$	4.73753	0.288317
$271$	−5.19284	−0.315443	−0.157721	−	0.987484i	$-0.550415\pi$
$271$	−5.19284	−0.315443	−0.157721	+	0.987484i	$0.550415\pi$
$272$	−6.59798	−0.400061
$273$	−2.02780	−0.122728
$274$	2.55263	0.154210
$275$	1.54426	0.0931221
$276$	−4.18576	−0.251953
$277$	−4.42858	−0.266088	−0.133044	−	0.991110i	$-0.542475\pi$
$277$	−4.42858	−0.266088	−0.133044	+	0.991110i	$0.542475\pi$
$278$	−1.40272	−0.0841295
$279$	−10.9983	−0.658454
$280$	−3.92602	−0.234625
$281$	15.4096	0.919258	0.459629	−	0.888111i	$-0.347982\pi$
$281$	15.4096	0.919258	0.459629	+	0.888111i	$0.347982\pi$
$282$	−2.75378	−0.163985
$283$	3.11244	0.185015	0.0925077	−	0.995712i	$-0.470512\pi$
$283$	3.11244	0.185015	0.0925077	+	0.995712i	$0.470512\pi$
$284$	−7.67511	−0.455434
$285$	5.61743	0.332748
$286$	0.867999	0.0513258
$287$	−24.7152	−1.45889
$288$	2.15561	0.127020
$289$	26.5333	1.56078
$290$	4.69768	0.275857
$291$	4.44264	0.260432
$292$	3.93595	0.230334
$293$	22.9346	1.33985	0.669926	−	0.742428i	$-0.266326\pi$
$293$	22.9346	1.33985	0.669926	+	0.742428i	$0.266326\pi$
$294$	7.73139	0.450904
$295$	−8.63699	−0.502865
$296$	2.86189	0.166344
$297$	7.31596	0.424515
$298$	−23.7432	−1.37541
$299$	−2.56037	−0.148070
$300$	−0.918908	−0.0530532
$301$	15.2053	0.876422
$302$	18.6292	1.07199
$303$	7.61964	0.437737
$304$	6.11315	0.350613
$305$	5.01728	0.287288
$306$	−14.2226	−0.813055
$307$	9.29961	0.530757	0.265378	−	0.964144i	$-0.414503\pi$
$307$	9.29961	0.530757	0.265378	+	0.964144i	$0.414503\pi$
$308$	−6.06278	−0.345459
$309$	0.0150869	0.000858261 0
$310$	5.10220	0.289786
$311$	−16.6788	−0.945771	−0.472885	−	0.881124i	$-0.656788\pi$
$311$	−16.6788	−0.945771	−0.472885	+	0.881124i	$0.656788\pi$
$312$	−0.516502	−0.0292412
$313$	7.07243	0.399757	0.199879	−	0.979821i	$-0.435945\pi$
$313$	7.07243	0.399757	0.199879	+	0.979821i	$0.435945\pi$
$314$	6.43230	0.362996
$315$	−8.46297	−0.476834
$316$	−14.4964	−0.815486
$317$	10.9145	0.613021	0.306511	−	0.951867i	$-0.400839\pi$
$317$	10.9145	0.613021	0.306511	+	0.951867i	$0.400839\pi$
$318$	8.09604	0.454004
$319$	7.25442	0.406170
$320$	−1.00000	−0.0559017
$321$	7.80854	0.435830
$322$	17.8836	0.996614
$323$	−40.3344	−2.24427
$324$	2.11347	0.117415
$325$	−0.562082	−0.0311787
$326$	−16.4413	−0.910600
$327$	−7.42258	−0.410470
$328$	−6.29523	−0.347596
$329$	11.7655	0.648652
$330$	−1.41903	−0.0781150
$331$	−8.13592	−0.447191	−0.223595	−	0.974682i	$-0.571779\pi$
$331$	−8.13592	−0.447191	−0.223595	+	0.974682i	$0.571779\pi$
$332$	4.99765	0.274282
$333$	6.16911	0.338065
$334$	−3.26012	−0.178386
$335$	−1.74019	−0.0950765
$336$	3.60766	0.196814
$337$	−26.1203	−1.42286	−0.711432	−	0.702755i	$-0.751953\pi$
$337$	−26.1203	−1.42286	−0.711432	+	0.702755i	$0.751953\pi$
$338$	12.6841	0.689922
$339$	8.90391	0.483594
$340$	6.59798	0.357826
$341$	7.87910	0.426677
$342$	13.1776	0.712560
$343$	−5.55007	−0.299676
$344$	3.87296	0.208816
$345$	4.18576	0.225354
$346$	−17.5469	−0.943330
$347$	4.10689	0.220470	0.110235	−	0.993906i	$-0.464840\pi$
$347$	4.10689	0.220470	0.110235	+	0.993906i	$0.464840\pi$
$348$	−4.31674	−0.231402
$349$	−25.9138	−1.38713	−0.693566	−	0.720393i	$-0.743962\pi$
$349$	−25.9138	−1.38713	−0.693566	+	0.720393i	$0.743962\pi$
$350$	3.92602	0.209855
$351$	−2.66288	−0.142134
$352$	−1.54426	−0.0823091
$353$	−11.8873	−0.632697	−0.316349	−	0.948643i	$-0.602457\pi$
$353$	−11.8873	−0.632697	−0.316349	+	0.948643i	$0.602457\pi$
$354$	7.93660	0.421826
$355$	7.67511	0.407353
$356$	−14.6418	−0.776012
$357$	−23.8032	−1.25980
$358$	1.40640	0.0743306
$359$	−12.0315	−0.635000	−0.317500	−	0.948258i	$-0.602843\pi$
$359$	−12.0315	−0.635000	−0.317500	+	0.948258i	$0.602843\pi$
$360$	−2.15561	−0.113610
$361$	18.3706	0.966874
$362$	−8.51792	−0.447692
$363$	7.91665	0.415516
$364$	2.20675	0.115665
$365$	−3.93595	−0.206017
$366$	−4.61042	−0.240990
$367$	−10.8051	−0.564022	−0.282011	−	0.959411i	$-0.591001\pi$
$367$	−10.8051	−0.564022	−0.282011	+	0.959411i	$0.591001\pi$
$368$	4.55514	0.237453
$369$	−13.5700	−0.706428
$370$	−2.86189	−0.148783
$371$	−34.5902	−1.79584
$372$	−4.68846	−0.243085
$373$	29.1390	1.50876	0.754380	−	0.656438i	$-0.227938\pi$
$373$	29.1390	1.50876	0.754380	+	0.656438i	$0.227938\pi$
$374$	10.1890	0.526859
$375$	0.918908	0.0474522
$376$	2.99679	0.154548
$377$	−2.64048	−0.135992
$378$	18.5997	0.956663
$379$	−27.4798	−1.41154	−0.705771	−	0.708440i	$-0.749399\pi$
$379$	−27.4798	−1.41154	−0.705771	+	0.708440i	$0.749399\pi$
$380$	−6.11315	−0.313598
$381$	−15.1911	−0.778266
$382$	−0.207405	−0.0106118
$383$	−6.59847	−0.337166	−0.168583	−	0.985687i	$-0.553919\pi$
$383$	−6.59847	−0.337166	−0.168583	+	0.985687i	$0.553919\pi$
$384$	0.918908	0.0468928
$385$	6.06278	0.308988
$386$	18.1703	0.924845
$387$	8.34859	0.424383
$388$	−4.83469	−0.245444
$389$	−13.2124	−0.669894	−0.334947	−	0.942237i	$-0.608718\pi$
$389$	−13.2124	−0.669894	−0.334947	+	0.942237i	$0.608718\pi$
$390$	0.516502	0.0261541
$391$	−30.0547	−1.51993
$392$	−8.41366	−0.424954
$393$	9.43374	0.475869
$394$	−12.9568	−0.652753
$395$	14.4964	0.729393
$396$	−3.32881	−0.167279
$397$	16.6594	0.836112	0.418056	−	0.908421i	$-0.362712\pi$
$397$	16.6594	0.836112	0.418056	+	0.908421i	$0.362712\pi$
$398$	12.2094	0.612003
$399$	22.0541	1.10409
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	1.59907	0.0797544
$403$	−2.86786	−0.142858
$404$	−8.29206	−0.412545
$405$	−2.11347	−0.105019
$406$	18.4432	0.915321
$407$	−4.41949	−0.219066
$408$	−6.06294	−0.300160
$409$	27.9156	1.38034	0.690168	−	0.723649i	$-0.257537\pi$
$409$	27.9156	1.38034	0.690168	+	0.723649i	$0.257537\pi$
$410$	6.29523	0.310899
$411$	2.34564	0.115702
$412$	−0.0164182	−0.000808868 0
$413$	−33.9090	−1.66855
$414$	9.81910	0.482583
$415$	−4.99765	−0.245325
$416$	0.562082	0.0275583
$417$	−1.28897	−0.0631211
$418$	−9.44027	−0.461739
$419$	4.77625	0.233335	0.116668	−	0.993171i	$-0.462779\pi$
$419$	4.77625	0.233335	0.116668	+	0.993171i	$0.462779\pi$
$420$	−3.60766	−0.176036
$421$	−29.3901	−1.43238	−0.716192	−	0.697903i	$-0.754117\pi$
$421$	−29.3901	−1.43238	−0.716192	+	0.697903i	$0.754117\pi$
$422$	−12.0336	−0.585789
$423$	6.45991	0.314092
$424$	−8.81050	−0.427876
$425$	−6.59798	−0.320049
$426$	−7.05272	−0.341705
$427$	19.6979	0.953250
$428$	−8.49762	−0.410748
$429$	0.797611	0.0385090
$430$	−3.87296	−0.186771
$431$	21.6388	1.04230	0.521151	−	0.853464i	$-0.325503\pi$
$431$	21.6388	1.04230	0.521151	+	0.853464i	$0.325503\pi$
$432$	4.73753	0.227935
$433$	−30.3104	−1.45662	−0.728312	−	0.685246i	$-0.759695\pi$
$433$	−30.3104	−1.45662	−0.728312	+	0.685246i	$0.759695\pi$
$434$	20.0314	0.961537
$435$	4.31674	0.206972
$436$	8.07761	0.386847
$437$	27.8463	1.33207
$438$	3.61677	0.172816
$439$	−14.8601	−0.709235	−0.354617	−	0.935012i	$-0.615389\pi$
$439$	−14.8601	−0.709235	−0.354617	+	0.935012i	$0.615389\pi$
$440$	1.54426	0.0736195
$441$	−18.1366	−0.863645
$442$	−3.70861	−0.176400
$443$	−24.8300	−1.17971	−0.589853	−	0.807510i	$-0.700814\pi$
$443$	−24.8300	−1.17971	−0.589853	+	0.807510i	$0.700814\pi$
$444$	2.62982	0.124806
$445$	14.6418	0.694086
$446$	17.7792	0.841870
$447$	−21.8178	−1.03195
$448$	−3.92602	−0.185487
$449$	−1.04233	−0.0491905	−0.0245953	−	0.999697i	$-0.507830\pi$
$449$	−1.04233	−0.0491905	−0.0245953	+	0.999697i	$0.507830\pi$
$450$	2.15561	0.101616
$451$	9.72144	0.457765
$452$	−9.68966	−0.455763
$453$	17.1185	0.804299
$454$	10.8774	0.510500
$455$	−2.20675	−0.103454
$456$	5.61743	0.263060
$457$	15.2698	0.714292	0.357146	−	0.934049i	$-0.383750\pi$
$457$	15.2698	0.714292	0.357146	+	0.934049i	$0.383750\pi$
$458$	10.1215	0.472945
$459$	−31.2581	−1.45900
$460$	−4.55514	−0.212385
$461$	−7.72789	−0.359924	−0.179962	−	0.983674i	$-0.557597\pi$
$461$	−7.72789	−0.359924	−0.179962	+	0.983674i	$0.557597\pi$
$462$	−5.57114	−0.259193
$463$	−24.4397	−1.13581	−0.567906	−	0.823094i	$-0.692246\pi$
$463$	−24.4397	−1.13581	−0.567906	+	0.823094i	$0.692246\pi$
$464$	4.69768	0.218084
$465$	4.68846	0.217422
$466$	7.83644	0.363016
$467$	−30.9905	−1.43407	−0.717034	−	0.697038i	$-0.754501\pi$
$467$	−30.9905	−1.43407	−0.717034	+	0.697038i	$0.754501\pi$
$468$	1.21163	0.0560076
$469$	−6.83201	−0.315473
$470$	−2.99679	−0.138232
$471$	5.91070	0.272351
$472$	−8.63699	−0.397550
$473$	−5.98084	−0.275000
$474$	−13.3209	−0.611847
$475$	6.11315	0.280491
$476$	25.9038	1.18730
$477$	−18.9920	−0.869583
$478$	−7.90410	−0.361525
$479$	−22.0256	−1.00638	−0.503188	−	0.864177i	$-0.667840\pi$
$479$	−22.0256	−1.00638	−0.503188	+	0.864177i	$0.667840\pi$
$480$	−0.918908	−0.0419422
$481$	1.60862	0.0733467
$482$	−15.6685	−0.713683
$483$	16.4334	0.747745
$484$	−8.61527	−0.391603
$485$	4.83469	0.219532
$486$	16.1547	0.732791
$487$	−21.0036	−0.951766	−0.475883	−	0.879509i	$-0.657871\pi$
$487$	−21.0036	−0.951766	−0.475883	+	0.879509i	$0.657871\pi$
$488$	5.01728	0.227121
$489$	−15.1081	−0.683210
$490$	8.41366	0.380091
$491$	0.318350	0.0143669	0.00718346	−	0.999974i	$-0.497713\pi$
$491$	0.318350	0.0143669	0.00718346	+	0.999974i	$0.497713\pi$
$492$	−5.78474	−0.260796
$493$	−30.9952	−1.39595
$494$	3.43609	0.154597
$495$	3.32881	0.149619
$496$	5.10220	0.229096
$497$	30.1327	1.35163
$498$	4.59239	0.205790
$499$	0.281372	0.0125959	0.00629796	−	0.999980i	$-0.497995\pi$
$499$	0.281372	0.0125959	0.00629796	+	0.999980i	$0.497995\pi$
$500$	−1.00000	−0.0447214
$501$	−2.99575	−0.133840
$502$	−5.67050	−0.253087
$503$	−9.01237	−0.401842	−0.200921	−	0.979607i	$-0.564393\pi$
$503$	−9.01237	−0.401842	−0.200921	+	0.979607i	$0.564393\pi$
$504$	−8.46297	−0.376970
$505$	8.29206	0.368992
$506$	−7.03431	−0.312713
$507$	11.6555	0.517639
$508$	16.5317	0.733477
$509$	34.6649	1.53650	0.768248	−	0.640153i	$-0.221129\pi$
$509$	34.6649	1.53650	0.768248	+	0.640153i	$0.221129\pi$
$510$	6.06294	0.268471
$511$	−15.4526	−0.683584
$512$	−1.00000	−0.0441942
$513$	28.9612	1.27867
$514$	6.28655	0.277288
$515$	0.0164182	0.000723474 0
$516$	3.55890	0.156672
$517$	−4.62782	−0.203531
$518$	−11.2359	−0.493675
$519$	−16.1240	−0.707767
$520$	−0.562082	−0.0246489
$521$	−7.84551	−0.343718	−0.171859	−	0.985122i	$-0.554977\pi$
$521$	−7.84551	−0.343718	−0.171859	+	0.985122i	$0.554977\pi$
$522$	10.1264	0.443219
$523$	−20.5593	−0.898995	−0.449498	−	0.893282i	$-0.648397\pi$
$523$	−20.5593	−0.898995	−0.449498	+	0.893282i	$0.648397\pi$
$524$	−10.2662	−0.448483
$525$	3.60766	0.157451
$526$	28.9758	1.26341
$527$	−33.6642	−1.46644
$528$	−1.41903	−0.0617553
$529$	−2.25065	−0.0978545
$530$	8.81050	0.382704
$531$	−18.6180	−0.807951
$532$	−24.0004	−1.04055
$533$	−3.53844	−0.153267
$534$	−13.4544	−0.582230
$535$	8.49762	0.367384
$536$	−1.74019	−0.0751646
$537$	1.29235	0.0557692
$538$	−22.5756	−0.973301
$539$	12.9928	0.559641
$540$	−4.73753	−0.203871
$541$	−45.8817	−1.97261	−0.986303	−	0.164942i	$-0.947256\pi$
$541$	−45.8817	−1.97261	−0.986303	+	0.164942i	$0.947256\pi$
$542$	5.19284	0.223052
$543$	−7.82719	−0.335897
$544$	6.59798	0.282886
$545$	−8.07761	−0.346007
$546$	2.02780	0.0867818
$547$	3.60148	0.153988	0.0769941	−	0.997032i	$-0.475468\pi$
$547$	3.60148	0.153988	0.0769941	+	0.997032i	$0.475468\pi$
$548$	−2.55263	−0.109043
$549$	10.8153	0.461585
$550$	−1.54426	−0.0658473
$551$	28.7176	1.22341
$552$	4.18576	0.178158
$553$	56.9132	2.42019
$554$	4.42858	0.188152
$555$	−2.62982	−0.111629
$556$	1.40272	0.0594885
$557$	4.64386	0.196767	0.0983833	−	0.995149i	$-0.468633\pi$
$557$	4.64386	0.196767	0.0983833	+	0.995149i	$0.468633\pi$
$558$	10.9983	0.465597
$559$	2.17692	0.0920741
$560$	3.92602	0.165905
$561$	9.36272	0.395294
$562$	−15.4096	−0.650014
$563$	40.3282	1.69963	0.849815	−	0.527081i	$-0.176714\pi$
$563$	40.3282	1.69963	0.849815	+	0.527081i	$0.176714\pi$
$564$	2.75378	0.115955
$565$	9.68966	0.407647
$566$	−3.11244	−0.130826
$567$	−8.29752	−0.348463
$568$	7.67511	0.322040
$569$	13.7368	0.575876	0.287938	−	0.957649i	$-0.407030\pi$
$569$	13.7368	0.575876	0.287938	+	0.957649i	$0.407030\pi$
$570$	−5.61743	−0.235288
$571$	−22.4222	−0.938340	−0.469170	−	0.883108i	$-0.655447\pi$
$571$	−22.4222	−0.938340	−0.469170	+	0.883108i	$0.655447\pi$
$572$	−0.867999	−0.0362928
$573$	−0.190586	−0.00796186
$574$	24.7152	1.03159
$575$	4.55514	0.189963
$576$	−2.15561	−0.0898170
$577$	−31.6058	−1.31577	−0.657883	−	0.753120i	$-0.728548\pi$
$577$	−31.6058	−1.31577	−0.657883	+	0.753120i	$0.728548\pi$
$578$	−26.5333	−1.10364
$579$	16.6969	0.693898
$580$	−4.69768	−0.195061
$581$	−19.6209	−0.814012
$582$	−4.44264	−0.184153
$583$	13.6057	0.563489
$584$	−3.93595	−0.162871
$585$	−1.21163	−0.0500947
$586$	−22.9346	−0.947418
$587$	−3.40326	−0.140468	−0.0702339	−	0.997531i	$-0.522375\pi$
$587$	−3.40326	−0.140468	−0.0702339	+	0.997531i	$0.522375\pi$
$588$	−7.73139	−0.318837
$589$	31.1905	1.28518
$590$	8.63699	0.355579
$591$	−11.9061	−0.489751
$592$	−2.86189	−0.117623
$593$	33.9699	1.39498	0.697489	−	0.716596i	$-0.254301\pi$
$593$	33.9699	1.39498	0.697489	+	0.716596i	$0.254301\pi$
$594$	−7.31596	−0.300177
$595$	−25.9038	−1.06195
$596$	23.7432	0.972558
$597$	11.2193	0.459177
$598$	2.56037	0.104701
$599$	0.760709	0.0310817	0.0155409	−	0.999879i	$-0.495053\pi$
$599$	0.760709	0.0310817	0.0155409	+	0.999879i	$0.495053\pi$
$600$	0.918908	0.0375143
$601$	−43.2346	−1.76357	−0.881787	−	0.471648i	$-0.843659\pi$
$601$	−43.2346	−1.76357	−0.881787	+	0.471648i	$0.843659\pi$
$602$	−15.2053	−0.619724
$603$	−3.75116	−0.152759
$604$	−18.6292	−0.758012
$605$	8.61527	0.350261
$606$	−7.61964	−0.309527
$607$	−35.2370	−1.43022	−0.715112	−	0.699010i	$-0.753624\pi$
$607$	−35.2370	−1.43022	−0.715112	+	0.699010i	$0.753624\pi$
$608$	−6.11315	−0.247921
$609$	16.9476	0.686752
$610$	−5.01728	−0.203144
$611$	1.68445	0.0681454
$612$	14.2226	0.574917
$613$	−15.5644	−0.628639	−0.314319	−	0.949317i	$-0.601776\pi$
$613$	−15.5644	−0.628639	−0.314319	+	0.949317i	$0.601776\pi$
$614$	−9.29961	−0.375302
$615$	5.78474	0.233263
$616$	6.06278	0.244277
$617$	−47.6815	−1.91958	−0.959792	−	0.280713i	$-0.909429\pi$
$617$	−47.6815	−1.91958	−0.959792	+	0.280713i	$0.909429\pi$
$618$	−0.0150869	−0.000606882 0
$619$	−12.4244	−0.499380	−0.249690	−	0.968326i	$-0.580329\pi$
$619$	−12.4244	−0.499380	−0.249690	+	0.968326i	$0.580329\pi$
$620$	−5.10220	−0.204909
$621$	21.5801	0.865981
$622$	16.6788	0.668761
$623$	57.4839	2.30304
$624$	0.516502	0.0206766
$625$	1.00000	0.0400000
$626$	−7.07243	−0.282671
$627$	−8.67474	−0.346436
$628$	−6.43230	−0.256677
$629$	18.8827	0.752902
$630$	8.46297	0.337173
$631$	0.0738847	0.00294130	0.00147065	−	0.999999i	$-0.499532\pi$
$631$	0.0738847	0.00294130	0.00147065	+	0.999999i	$0.499532\pi$
$632$	14.4964	0.576636
$633$	−11.0578	−0.439509
$634$	−10.9145	−0.433471
$635$	−16.5317	−0.656042
$636$	−8.09604	−0.321029
$637$	−4.72917	−0.187377
$638$	−7.25442	−0.287205
$639$	16.5445	0.654491
$640$	1.00000	0.0395285
$641$	22.3087	0.881139	0.440569	−	0.897719i	$-0.354777\pi$
$641$	22.3087	0.881139	0.440569	+	0.897719i	$0.354777\pi$
$642$	−7.80854	−0.308178
$643$	−31.3979	−1.23821	−0.619105	−	0.785308i	$-0.712505\pi$
$643$	−31.3979	−1.23821	−0.619105	+	0.785308i	$0.712505\pi$
$644$	−17.8836	−0.704713
$645$	−3.55890	−0.140131
$646$	40.3344	1.58694
$647$	44.2051	1.73788	0.868941	−	0.494916i	$-0.164801\pi$
$647$	44.2051	1.73788	0.868941	+	0.494916i	$0.164801\pi$
$648$	−2.11347	−0.0830248
$649$	13.3377	0.523551
$650$	0.562082	0.0220467
$651$	18.4070	0.721427
$652$	16.4413	0.643892
$653$	−17.4459	−0.682711	−0.341355	−	0.939934i	$-0.610886\pi$
$653$	−17.4459	−0.682711	−0.341355	+	0.939934i	$0.610886\pi$
$654$	7.42258	0.290246
$655$	10.2662	0.401135
$656$	6.29523	0.245787
$657$	−8.48435	−0.331006
$658$	−11.7655	−0.458666
$659$	9.85922	0.384061	0.192030	−	0.981389i	$-0.438493\pi$
$659$	9.85922	0.384061	0.192030	+	0.981389i	$0.438493\pi$
$660$	1.41903	0.0552356
$661$	7.32314	0.284837	0.142419	−	0.989807i	$-0.454512\pi$
$661$	7.32314	0.284837	0.142419	+	0.989807i	$0.454512\pi$
$662$	8.13592	0.316212
$663$	−3.40787	−0.132351
$664$	−4.99765	−0.193947
$665$	24.0004	0.930695
$666$	−6.16911	−0.239048
$667$	21.3986	0.828558
$668$	3.26012	0.126138
$669$	16.3375	0.631643
$670$	1.74019	0.0672293
$671$	−7.74795	−0.299106
$672$	−3.60766	−0.139168
$673$	28.9692	1.11668	0.558341	−	0.829611i	$-0.311438\pi$
$673$	28.9692	1.11668	0.558341	+	0.829611i	$0.311438\pi$
$674$	26.1203	1.00612
$675$	4.73753	0.182348
$676$	−12.6841	−0.487849
$677$	−5.04167	−0.193767	−0.0968835	−	0.995296i	$-0.530887\pi$
$677$	−5.04167	−0.193767	−0.0968835	+	0.995296i	$0.530887\pi$
$678$	−8.90391	−0.341953
$679$	18.9811	0.728428
$680$	−6.59798	−0.253021
$681$	9.99530	0.383021
$682$	−7.87910	−0.301707
$683$	8.39087	0.321068	0.160534	−	0.987030i	$-0.448678\pi$
$683$	8.39087	0.321068	0.160534	+	0.987030i	$0.448678\pi$
$684$	−13.1776	−0.503856
$685$	2.55263	0.0975311
$686$	5.55007	0.211903
$687$	9.30069	0.354844
$688$	−3.87296	−0.147655
$689$	−4.95223	−0.188665
$690$	−4.18576	−0.159349
$691$	−18.7907	−0.714831	−0.357415	−	0.933946i	$-0.616342\pi$
$691$	−18.7907	−0.714831	−0.357415	+	0.933946i	$0.616342\pi$
$692$	17.5469	0.667035
$693$	13.0690	0.496450
$694$	−4.10689	−0.155896
$695$	−1.40272	−0.0532081
$696$	4.31674	0.163626
$697$	−41.5358	−1.57328
$698$	25.9138	0.980851
$699$	7.20097	0.272366
$700$	−3.92602	−0.148390
$701$	24.0207	0.907251	0.453625	−	0.891192i	$-0.350130\pi$
$701$	24.0207	0.907251	0.453625	+	0.891192i	$0.350130\pi$
$702$	2.66288	0.100504
$703$	−17.4952	−0.659843
$704$	1.54426	0.0582013
$705$	−2.75378	−0.103713
$706$	11.8873	0.447384
$707$	32.5548	1.22435
$708$	−7.93660	−0.298276
$709$	40.3710	1.51616	0.758082	−	0.652159i	$-0.226137\pi$
$709$	40.3710	1.51616	0.758082	+	0.652159i	$0.226137\pi$
$710$	−7.67511	−0.288042
$711$	31.2485	1.17191
$712$	14.6418	0.548723
$713$	23.2413	0.870393
$714$	23.8032	0.890814
$715$	0.867999	0.0324613
$716$	−1.40640	−0.0525597
$717$	−7.26314	−0.271247
$718$	12.0315	0.449013
$719$	1.59930	0.0596438	0.0298219	−	0.999555i	$-0.490506\pi$
$719$	1.59930	0.0596438	0.0298219	+	0.999555i	$0.490506\pi$
$720$	2.15561	0.0803347
$721$	0.0644584	0.00240056
$722$	−18.3706	−0.683683
$723$	−14.3980	−0.535466
$724$	8.51792	0.316566
$725$	4.69768	0.174468
$726$	−7.91665	−0.293814
$727$	39.5408	1.46649	0.733243	−	0.679966i	$-0.238006\pi$
$727$	39.5408	1.46649	0.733243	+	0.679966i	$0.238006\pi$
$728$	−2.20675	−0.0817875
$729$	8.50427	0.314973
$730$	3.93595	0.145676
$731$	25.5537	0.945139
$732$	4.61042	0.170406
$733$	−26.7411	−0.987703	−0.493852	−	0.869546i	$-0.664411\pi$
$733$	−26.7411	−0.987703	−0.493852	+	0.869546i	$0.664411\pi$
$734$	10.8051	0.398824
$735$	7.73139	0.285176
$736$	−4.55514	−0.167905
$737$	2.68729	0.0989877
$738$	13.5700	0.499520
$739$	23.5912	0.867816	0.433908	−	0.900957i	$-0.357134\pi$
$739$	23.5912	0.867816	0.433908	+	0.900957i	$0.357134\pi$
$740$	2.86189	0.105205
$741$	3.15746	0.115992
$742$	34.5902	1.26985
$743$	−17.6125	−0.646139	−0.323070	−	0.946375i	$-0.604715\pi$
$743$	−17.6125	−0.646139	−0.323070	+	0.946375i	$0.604715\pi$
$744$	4.68846	0.171887
$745$	−23.7432	−0.869883
$746$	−29.1390	−1.06685
$747$	−10.7730	−0.394163
$748$	−10.1890	−0.372545
$749$	33.3619	1.21902
$750$	−0.918908	−0.0335538
$751$	−22.8931	−0.835380	−0.417690	−	0.908590i	$-0.637160\pi$
$751$	−22.8931	−0.835380	−0.417690	+	0.908590i	$0.637160\pi$
$752$	−2.99679	−0.109282
$753$	−5.21067	−0.189887
$754$	2.64048	0.0961607
$755$	18.6292	0.677986
$756$	−18.5997	−0.676463
$757$	11.8800	0.431784	0.215892	−	0.976417i	$-0.430734\pi$
$757$	11.8800	0.431784	0.215892	+	0.976417i	$0.430734\pi$
$758$	27.4798	0.998111
$759$	−6.46388	−0.234624
$760$	6.11315	0.221747
$761$	4.84736	0.175717	0.0878584	−	0.996133i	$-0.471998\pi$
$761$	4.84736	0.175717	0.0878584	+	0.996133i	$0.471998\pi$
$762$	15.1911	0.550317
$763$	−31.7129	−1.14808
$764$	0.207405	0.00750365
$765$	−14.2226	−0.514221
$766$	6.59847	0.238412
$767$	−4.85470	−0.175293
$768$	−0.918908	−0.0331582
$769$	−44.5829	−1.60770	−0.803849	−	0.594833i	$-0.797218\pi$
$769$	−44.5829	−1.60770	−0.803849	+	0.594833i	$0.797218\pi$
$770$	−6.06278	−0.218488
$771$	5.77676	0.208045
$772$	−18.1703	−0.653964
$773$	−2.80103	−0.100746	−0.0503730	−	0.998730i	$-0.516041\pi$
$773$	−2.80103	−0.100746	−0.0503730	+	0.998730i	$0.516041\pi$
$774$	−8.34859	−0.300084
$775$	5.10220	0.183277
$776$	4.83469	0.173555
$777$	−10.3247	−0.370397
$778$	13.2124	0.473686
$779$	38.4837	1.37882
$780$	−0.516502	−0.0184937
$781$	−11.8523	−0.424110
$782$	30.0547	1.07476
$783$	22.2554	0.795344
$784$	8.41366	0.300488
$785$	6.43230	0.229579
$786$	−9.43374	−0.336490
$787$	54.9251	1.95787	0.978934	−	0.204175i	$-0.0654510\pi$
$787$	54.9251	1.95787	0.978934	+	0.204175i	$0.0654510\pi$
$788$	12.9568	0.461566
$789$	26.6261	0.947915
$790$	−14.4964	−0.515759
$791$	38.0419	1.35261
$792$	3.32881	0.118284
$793$	2.82012	0.100145
$794$	−16.6594	−0.591220
$795$	8.09604	0.287137
$796$	−12.2094	−0.432751
$797$	34.6507	1.22739	0.613696	−	0.789542i	$-0.289682\pi$
$797$	34.6507	1.22739	0.613696	+	0.789542i	$0.289682\pi$
$798$	−22.0541	−0.780708
$799$	19.7728	0.699511
$800$	−1.00000	−0.0353553
$801$	31.5619	1.11518
$802$	1.00000	0.0353112
$803$	6.07811	0.214492
$804$	−1.59907	−0.0563949
$805$	17.8836	0.630314
$806$	2.86786	0.101016
$807$	−20.7449	−0.730254
$808$	8.29206	0.291714
$809$	11.0131	0.387201	0.193600	−	0.981080i	$-0.437984\pi$
$809$	11.0131	0.387201	0.193600	+	0.981080i	$0.437984\pi$
$810$	2.11347	0.0742596
$811$	40.8479	1.43436	0.717182	−	0.696886i	$-0.245432\pi$
$811$	40.8479	1.43436	0.717182	+	0.696886i	$0.245432\pi$
$812$	−18.4432	−0.647230
$813$	4.77175	0.167352
$814$	4.41949	0.154903
$815$	−16.4413	−0.575914
$816$	6.06294	0.212245
$817$	−23.6760	−0.828319
$818$	−27.9156	−0.976045
$819$	−4.75688	−0.166219
$820$	−6.29523	−0.219839
$821$	−29.8030	−1.04013	−0.520066	−	0.854126i	$-0.674093\pi$
$821$	−29.8030	−1.04013	−0.520066	+	0.854126i	$0.674093\pi$
$822$	−2.34564	−0.0818135
$823$	−12.8083	−0.446470	−0.223235	−	0.974765i	$-0.571662\pi$
$823$	−12.8083	−0.446470	−0.223235	+	0.974765i	$0.571662\pi$
$824$	0.0164182	0.000571956 0
$825$	−1.41903	−0.0494043
$826$	33.9090	1.17985
$827$	−23.1998	−0.806736	−0.403368	−	0.915038i	$-0.632160\pi$
$827$	−23.1998	−0.806736	−0.403368	+	0.915038i	$0.632160\pi$
$828$	−9.81910	−0.341237
$829$	25.7076	0.892861	0.446430	−	0.894818i	$-0.352695\pi$
$829$	25.7076	0.892861	0.446430	+	0.894818i	$0.352695\pi$
$830$	4.99765	0.173471
$831$	4.06946	0.141168
$832$	−0.562082	−0.0194867
$833$	−55.5132	−1.92342
$834$	1.28897	0.0446334
$835$	−3.26012	−0.112821
$836$	9.44027	0.326498
$837$	24.1718	0.835501
$838$	−4.77625	−0.164993
$839$	30.1805	1.04195	0.520973	−	0.853573i	$-0.325569\pi$
$839$	30.1805	1.04195	0.520973	+	0.853573i	$0.325569\pi$
$840$	3.60766	0.124476
$841$	−6.93178	−0.239027
$842$	29.3901	1.01285
$843$	−14.1600	−0.487696
$844$	12.0336	0.414215
$845$	12.6841	0.436345
$846$	−6.45991	−0.222096
$847$	33.8238	1.16220
$848$	8.81050	0.302554
$849$	−2.86005	−0.0981566
$850$	6.59798	0.226309
$851$	−13.0363	−0.446880
$852$	7.05272	0.241622
$853$	37.1996	1.27369	0.636845	−	0.770992i	$-0.280239\pi$
$853$	37.1996	1.27369	0.636845	+	0.770992i	$0.280239\pi$
$854$	−19.6979	−0.674050
$855$	13.1776	0.450663
$856$	8.49762	0.290443
$857$	36.9242	1.26131	0.630653	−	0.776065i	$-0.282787\pi$
$857$	36.9242	1.26131	0.630653	+	0.776065i	$0.282787\pi$
$858$	−0.797611	−0.0272300
$859$	−40.7939	−1.39187	−0.695935	−	0.718105i	$-0.745010\pi$
$859$	−40.7939	−1.39187	−0.695935	+	0.718105i	$0.745010\pi$
$860$	3.87296	0.132067
$861$	22.7110	0.773990
$862$	−21.6388	−0.737019
$863$	−16.6230	−0.565855	−0.282927	−	0.959141i	$-0.591306\pi$
$863$	−16.6230	−0.565855	−0.282927	+	0.959141i	$0.591306\pi$
$864$	−4.73753	−0.161174
$865$	−17.5469	−0.596614
$866$	30.3104	1.02999
$867$	−24.3817	−0.828045
$868$	−20.0314	−0.679909
$869$	−22.3861	−0.759398
$870$	−4.31674	−0.146351
$871$	−0.978128	−0.0331426
$872$	−8.07761	−0.273542
$873$	10.4217	0.352721
$874$	−27.8463	−0.941915
$875$	3.92602	0.132724
$876$	−3.61677	−0.122199
$877$	49.9734	1.68748	0.843741	−	0.536751i	$-0.180349\pi$
$877$	49.9734	1.68748	0.843741	+	0.536751i	$0.180349\pi$
$878$	14.8601	0.501505
$879$	−21.0748	−0.710834
$880$	−1.54426	−0.0520568
$881$	−12.2411	−0.412413	−0.206207	−	0.978508i	$-0.566112\pi$
$881$	−12.2411	−0.412413	−0.206207	+	0.978508i	$0.566112\pi$
$882$	18.1366	0.610690
$883$	37.7681	1.27100	0.635498	−	0.772102i	$-0.280795\pi$
$883$	37.7681	1.27100	0.635498	+	0.772102i	$0.280795\pi$
$884$	3.70861	0.124734
$885$	7.93660	0.266786
$886$	24.8300	0.834179
$887$	52.1678	1.75162	0.875811	−	0.482654i	$-0.160327\pi$
$887$	52.1678	1.75162	0.875811	+	0.482654i	$0.160327\pi$
$888$	−2.62982	−0.0882508
$889$	−64.9039	−2.17681
$890$	−14.6418	−0.490793
$891$	3.26373	0.109339
$892$	−17.7792	−0.595292
$893$	−18.3199	−0.613051
$894$	21.8178	0.729697
$895$	1.40640	0.0470108
$896$	3.92602	0.131159
$897$	2.35274	0.0785558
$898$	1.04233	0.0347830
$899$	23.9685	0.799395
$900$	−2.15561	−0.0718536
$901$	−58.1315	−1.93664
$902$	−9.72144	−0.323689
$903$	−13.9723	−0.464970
$904$	9.68966	0.322273
$905$	−8.51792	−0.283145
$906$	−17.1185	−0.568725
$907$	−59.4904	−1.97535	−0.987674	−	0.156527i	$-0.949970\pi$
$907$	−59.4904	−1.97535	−0.987674	+	0.156527i	$0.949970\pi$
$908$	−10.8774	−0.360978
$909$	17.8744	0.592857
$910$	2.20675	0.0731530
$911$	7.66753	0.254037	0.127018	−	0.991900i	$-0.459459\pi$
$911$	7.66753	0.254037	0.127018	+	0.991900i	$0.459459\pi$
$912$	−5.61743	−0.186012
$913$	7.71765	0.255417
$914$	−15.2698	−0.505081
$915$	−4.61042	−0.152416
$916$	−10.1215	−0.334422
$917$	40.3055	1.33101
$918$	31.2581	1.03167
$919$	17.8523	0.588894	0.294447	−	0.955668i	$-0.404865\pi$
$919$	17.8523	0.588894	0.294447	+	0.955668i	$0.404865\pi$
$920$	4.55514	0.150179
$921$	−8.54549	−0.281584
$922$	7.72789	0.254505
$923$	4.31404	0.141998
$924$	5.57114	0.183277
$925$	−2.86189	−0.0940984
$926$	24.4397	0.803140
$927$	0.0353913	0.00116240
$928$	−4.69768	−0.154209
$929$	15.2945	0.501796	0.250898	−	0.968014i	$-0.419274\pi$
$929$	15.2945	0.501796	0.250898	+	0.968014i	$0.419274\pi$
$930$	−4.68846	−0.153741
$931$	51.4340	1.68568
$932$	−7.83644	−0.256691
$933$	15.3263	0.501762
$934$	30.9905	1.01404
$935$	10.1890	0.333215
$936$	−1.21163	−0.0396033
$937$	−11.8289	−0.386433	−0.193217	−	0.981156i	$-0.561892\pi$
$937$	−11.8289	−0.386433	−0.193217	+	0.981156i	$0.561892\pi$
$938$	6.83201	0.223073
$939$	−6.49892	−0.212084
$940$	2.99679	0.0977447
$941$	−47.0046	−1.53231	−0.766153	−	0.642658i	$-0.777832\pi$
$941$	−47.0046	−1.53231	−0.766153	+	0.642658i	$0.777832\pi$
$942$	−5.91070	−0.192581
$943$	28.6757	0.933809
$944$	8.63699	0.281110
$945$	18.5997	0.605047
$946$	5.98084	0.194454
$947$	−58.9092	−1.91429	−0.957146	−	0.289607i	$-0.906475\pi$
$947$	−58.9092	−1.91429	−0.957146	+	0.289607i	$0.906475\pi$
$948$	13.3209	0.432641
$949$	−2.21233	−0.0718151
$950$	−6.11315	−0.198337
$951$	−10.0295	−0.325227
$952$	−25.9038	−0.839548
$953$	13.5554	0.439103	0.219551	−	0.975601i	$-0.429541\pi$
$953$	13.5554	0.439103	0.219551	+	0.975601i	$0.429541\pi$
$954$	18.9920	0.614888
$955$	−0.207405	−0.00671147
$956$	7.90410	0.255637
$957$	−6.66615	−0.215486
$958$	22.0256	0.711616
$959$	10.0217	0.323618
$960$	0.918908	0.0296576
$961$	−4.96752	−0.160243
$962$	−1.60862	−0.0518639
$963$	18.3175	0.590274
$964$	15.6685	0.504650
$965$	18.1703	0.584923
$966$	−16.4334	−0.528736
$967$	−23.0039	−0.739756	−0.369878	−	0.929080i	$-0.620600\pi$
$967$	−23.0039	−0.739756	−0.369878	+	0.929080i	$0.620600\pi$
$968$	8.61527	0.276905
$969$	37.0636	1.19066
$970$	−4.83469	−0.155232
$971$	23.0033	0.738212	0.369106	−	0.929387i	$-0.379664\pi$
$971$	23.0033	0.738212	0.369106	+	0.929387i	$0.379664\pi$
$972$	−16.1547	−0.518161
$973$	−5.50711	−0.176550
$974$	21.0036	0.673000
$975$	0.516502	0.0165413
$976$	−5.01728	−0.160599
$977$	38.8437	1.24272	0.621361	−	0.783525i	$-0.286580\pi$
$977$	38.8437	1.24272	0.621361	+	0.783525i	$0.286580\pi$
$978$	15.1081	0.483103
$979$	−22.6106	−0.722638
$980$	−8.41366	−0.268765
$981$	−17.4122	−0.555927
$982$	−0.318350	−0.0101589
$983$	12.6053	0.402047	0.201024	−	0.979586i	$-0.435573\pi$
$983$	12.6053	0.402047	0.201024	+	0.979586i	$0.435573\pi$
$984$	5.78474	0.184411
$985$	−12.9568	−0.412837
$986$	30.9952	0.987088
$987$	−10.8114	−0.344131
$988$	−3.43609	−0.109317
$989$	−17.6419	−0.560980
$990$	−3.32881	−0.105796
$991$	−54.7272	−1.73847	−0.869234	−	0.494401i	$-0.835388\pi$
$991$	−54.7272	−1.73847	−0.869234	+	0.494401i	$0.835388\pi$
$992$	−5.10220	−0.161995
$993$	7.47617	0.237249
$994$	−30.1327	−0.955750
$995$	12.2094	0.387065
$996$	−4.59239	−0.145515
$997$	−4.45414	−0.141064	−0.0705321	−	0.997510i	$-0.522470\pi$
$997$	−4.45414	−0.141064	−0.0705321	+	0.997510i	$0.522470\pi$
$998$	−0.281372	−0.00890666
$999$	−13.5583	−0.428966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.k.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.k.1.7	✓	15	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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.918908 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.918908 q^{6} -3.92602 q^{7} -1.00000 q^{8} -2.15561 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.918908 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.918908 q^{6} -3.92602 q^{7} -1.00000 q^{8} -2.15561 q^{9} +1.00000 q^{10} +1.54426 q^{11} -0.918908 q^{12} -0.562082 q^{13} +3.92602 q^{14} +0.918908 q^{15} +1.00000 q^{16} -6.59798 q^{17} +2.15561 q^{18} +6.11315 q^{19} -1.00000 q^{20} +3.60766 q^{21} -1.54426 q^{22} +4.55514 q^{23} +0.918908 q^{24} +1.00000 q^{25} +0.562082 q^{26} +4.73753 q^{27} -3.92602 q^{28} +4.69768 q^{29} -0.918908 q^{30} +5.10220 q^{31} -1.00000 q^{32} -1.41903 q^{33} +6.59798 q^{34} +3.92602 q^{35} -2.15561 q^{36} -2.86189 q^{37} -6.11315 q^{38} +0.516502 q^{39} +1.00000 q^{40} +6.29523 q^{41} -3.60766 q^{42} -3.87296 q^{43} +1.54426 q^{44} +2.15561 q^{45} -4.55514 q^{46} -2.99679 q^{47} -0.918908 q^{48} +8.41366 q^{49} -1.00000 q^{50} +6.06294 q^{51} -0.562082 q^{52} +8.81050 q^{53} -4.73753 q^{54} -1.54426 q^{55} +3.92602 q^{56} -5.61743 q^{57} -4.69768 q^{58} +8.63699 q^{59} +0.918908 q^{60} -5.01728 q^{61} -5.10220 q^{62} +8.46297 q^{63} +1.00000 q^{64} +0.562082 q^{65} +1.41903 q^{66} +1.74019 q^{67} -6.59798 q^{68} -4.18576 q^{69} -3.92602 q^{70} -7.67511 q^{71} +2.15561 q^{72} +3.93595 q^{73} +2.86189 q^{74} -0.918908 q^{75} +6.11315 q^{76} -6.06278 q^{77} -0.516502 q^{78} -14.4964 q^{79} -1.00000 q^{80} +2.11347 q^{81} -6.29523 q^{82} +4.99765 q^{83} +3.60766 q^{84} +6.59798 q^{85} +3.87296 q^{86} -4.31674 q^{87} -1.54426 q^{88} -14.6418 q^{89} -2.15561 q^{90} +2.20675 q^{91} +4.55514 q^{92} -4.68846 q^{93} +2.99679 q^{94} -6.11315 q^{95} +0.918908 q^{96} -4.83469 q^{97} -8.41366 q^{98} -3.32881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^10 - 2 * q^11 - 6 * q^12 - 13 * q^13 + 5 * q^14 + 6 * q^15 + 15 * q^16 + 11 * q^17 - 19 * q^18 - 15 * q^19 - 15 * q^20 - 2 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 15 * q^25 + 13 * q^26 - 12 * q^27 - 5 * q^28 + 28 * q^29 - 6 * q^30 - 12 * q^31 - 15 * q^32 - 22 * q^33 - 11 * q^34 + 5 * q^35 + 19 * q^36 - 23 * q^37 + 15 * q^38 - 2 * q^39 + 15 * q^40 + 24 * q^41 + 2 * q^42 - 24 * q^43 - 2 * q^44 - 19 * q^45 + 3 * q^46 - 3 * q^47 - 6 * q^48 + 20 * q^49 - 15 * q^50 - 5 * q^51 - 13 * q^52 + 10 * q^53 + 12 * q^54 + 2 * q^55 + 5 * q^56 - 11 * q^57 - 28 * q^58 + 2 * q^59 + 6 * q^60 + 15 * q^61 + 12 * q^62 - 2 * q^63 + 15 * q^64 + 13 * q^65 + 22 * q^66 - 48 * q^67 + 11 * q^68 + 21 * q^69 - 5 * q^70 + 15 * q^71 - 19 * q^72 - 47 * q^73 + 23 * q^74 - 6 * q^75 - 15 * q^76 + 7 * q^77 + 2 * q^78 - 34 * q^79 - 15 * q^80 + 43 * q^81 - 24 * q^82 - 32 * q^83 - 2 * q^84 - 11 * q^85 + 24 * q^86 + 14 * q^87 + 2 * q^88 + 25 * q^89 + 19 * q^90 - 32 * q^91 - 3 * q^92 - 42 * q^93 + 3 * q^94 + 15 * q^95 + 6 * q^96 - 34 * q^97 - 20 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more