LMFDB - Embedded newform 91.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, 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("91.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	91.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:	$0.726638658394$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	91.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.470683 q^{2} +2.24914 q^{3} -1.77846 q^{4} +0.529317 q^{5} +1.05863 q^{6} -1.00000 q^{7} -1.77846 q^{8} +2.05863 q^{9} +O(q^{10})$	\(q+0.470683 q^{2} +2.24914 q^{3} -1.77846 q^{4} +0.529317 q^{5} +1.05863 q^{6} -1.00000 q^{7} -1.77846 q^{8} +2.05863 q^{9} +0.249141 q^{10} -2.24914 q^{11} -4.00000 q^{12} +1.00000 q^{13} -0.470683 q^{14} +1.19051 q^{15} +2.71982 q^{16} -1.30777 q^{17} +0.968964 q^{18} -1.47068 q^{19} -0.941367 q^{20} -2.24914 q^{21} -1.05863 q^{22} +5.83709 q^{23} -4.00000 q^{24} -4.71982 q^{25} +0.470683 q^{26} -2.11727 q^{27} +1.77846 q^{28} +5.22154 q^{29} +0.560352 q^{30} -7.02760 q^{31} +4.83709 q^{32} -5.05863 q^{33} -0.615547 q^{34} -0.529317 q^{35} -3.66119 q^{36} -2.36641 q^{37} -0.692226 q^{38} +2.24914 q^{39} -0.941367 q^{40} +6.49828 q^{41} -1.05863 q^{42} +11.3940 q^{43} +4.00000 q^{44} +1.08967 q^{45} +2.74742 q^{46} +8.58451 q^{47} +6.11727 q^{48} +1.00000 q^{49} -2.22154 q^{50} -2.94137 q^{51} -1.77846 q^{52} +11.2767 q^{53} -0.996562 q^{54} -1.19051 q^{55} +1.77846 q^{56} -3.30777 q^{57} +2.45769 q^{58} -12.1725 q^{59} -2.11727 q^{60} -2.00000 q^{61} -3.30777 q^{62} -2.05863 q^{63} -3.16291 q^{64} +0.529317 q^{65} -2.38101 q^{66} -15.9379 q^{67} +2.32582 q^{68} +13.1284 q^{69} -0.249141 q^{70} +1.19051 q^{71} -3.66119 q^{72} +7.64315 q^{73} -1.11383 q^{74} -10.6155 q^{75} +2.61555 q^{76} +2.24914 q^{77} +1.05863 q^{78} -1.33881 q^{79} +1.43965 q^{80} -10.9379 q^{81} +3.05863 q^{82} -16.3500 q^{83} +4.00000 q^{84} -0.692226 q^{85} +5.36297 q^{86} +11.7440 q^{87} +4.00000 q^{88} +6.91033 q^{89} +0.512889 q^{90} -1.00000 q^{91} -10.3810 q^{92} -15.8061 q^{93} +4.04059 q^{94} -0.778457 q^{95} +10.8793 q^{96} -3.47068 q^{97} +0.470683 q^{98} -4.63016 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9}+O(q^{10}) $ 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9	\( 3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9} - 8 q^{10} + 2 q^{11} - 12 q^{12} + 3 q^{13} - q^{14} - 6 q^{15} - q^{16} + 4 q^{17} - 15 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + q^{26} - 8 q^{27} - 3 q^{28} + 24 q^{29} + 20 q^{30} - 4 q^{31} + 7 q^{32} - 16 q^{33} + 14 q^{34} - 2 q^{35} - q^{36} - 10 q^{38} - 2 q^{39} - 2 q^{40} + 2 q^{41} - 4 q^{42} + 10 q^{43} + 12 q^{44} + 22 q^{45} - 18 q^{46} - 8 q^{47} + 20 q^{48} + 3 q^{49} - 15 q^{50} - 8 q^{51} + 3 q^{52} + 8 q^{53} + 32 q^{54} + 6 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 4 q^{59} - 8 q^{60} - 6 q^{61} - 2 q^{62} - 7 q^{63} - 17 q^{64} + 2 q^{65} + 12 q^{66} - 12 q^{67} + 22 q^{68} - 6 q^{69} + 8 q^{70} - 6 q^{71} - q^{72} - 10 q^{73} + 30 q^{74} - 16 q^{75} - 8 q^{76} - 2 q^{77} + 4 q^{78} - 14 q^{79} - 14 q^{80} + 3 q^{81} + 10 q^{82} - 12 q^{83} + 12 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 12 q^{88} + 2 q^{89} - 28 q^{90} - 3 q^{91} - 12 q^{92} - 22 q^{93} - 10 q^{94} + 6 q^{95} - 4 q^{96} - 10 q^{97} + q^{98} + 14 q^{99}+O(q^{100}) \) 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9 - 8 * q^10 + 2 * q^11 - 12 * q^12 + 3 * q^13 - q^14 - 6 * q^15 - q^16 + 4 * q^17 - 15 * q^18 - 4 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 + 10 * q^23 - 12 * q^24 - 5 * q^25 + q^26 - 8 * q^27 - 3 * q^28 + 24 * q^29 + 20 * q^30 - 4 * q^31 + 7 * q^32 - 16 * q^33 + 14 * q^34 - 2 * q^35 - q^36 - 10 * q^38 - 2 * q^39 - 2 * q^40 + 2 * q^41 - 4 * q^42 + 10 * q^43 + 12 * q^44 + 22 * q^45 - 18 * q^46 - 8 * q^47 + 20 * q^48 + 3 * q^49 - 15 * q^50 - 8 * q^51 + 3 * q^52 + 8 * q^53 + 32 * q^54 + 6 * q^55 - 3 * q^56 - 2 * q^57 + 12 * q^58 - 4 * q^59 - 8 * q^60 - 6 * q^61 - 2 * q^62 - 7 * q^63 - 17 * q^64 + 2 * q^65 + 12 * q^66 - 12 * q^67 + 22 * q^68 - 6 * q^69 + 8 * q^70 - 6 * q^71 - q^72 - 10 * q^73 + 30 * q^74 - 16 * q^75 - 8 * q^76 - 2 * q^77 + 4 * q^78 - 14 * q^79 - 14 * q^80 + 3 * q^81 + 10 * q^82 - 12 * q^83 + 12 * q^84 - 10 * q^85 - 26 * q^86 - 26 * q^87 + 12 * q^88 + 2 * q^89 - 28 * q^90 - 3 * q^91 - 12 * q^92 - 22 * q^93 - 10 * q^94 + 6 * q^95 - 4 * q^96 - 10 * q^97 + q^98 + 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.470683	0.332823	0.166412	−	0.986056i	$-0.446782\pi$
$2$	0.470683	0.332823	0.166412	+	0.986056i	$0.446782\pi$
$3$	2.24914	1.29854	0.649271	−	0.760557i	$-0.275074\pi$
$3$	2.24914	1.29854	0.649271	+	0.760557i	$0.275074\pi$
$4$	−1.77846	−0.889229
$5$	0.529317	0.236718	0.118359	−	0.992971i	$-0.462237\pi$
$5$	0.529317	0.236718	0.118359	+	0.992971i	$0.462237\pi$
$6$	1.05863	0.432185
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.77846	−0.628780
$9$	2.05863	0.686211
$10$	0.249141	0.0787852
$11$	−2.24914	−0.678141	−0.339071	−	0.940761i	$-0.610113\pi$
$11$	−2.24914	−0.678141	−0.339071	+	0.940761i	$0.610113\pi$
$12$	−4.00000	−1.15470
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.470683	−0.125795
$15$	1.19051	0.307388
$16$	2.71982	0.679956
$17$	−1.30777	−0.317182	−0.158591	−	0.987344i	$-0.550695\pi$
$17$	−1.30777	−0.317182	−0.158591	+	0.987344i	$0.550695\pi$
$18$	0.968964	0.228387
$19$	−1.47068	−0.337398	−0.168699	−	0.985668i	$-0.553957\pi$
$19$	−1.47068	−0.337398	−0.168699	+	0.985668i	$0.553957\pi$
$20$	−0.941367	−0.210496
$21$	−2.24914	−0.490803
$22$	−1.05863	−0.225701
$23$	5.83709	1.21712	0.608559	−	0.793509i	$-0.291748\pi$
$23$	5.83709	1.21712	0.608559	+	0.793509i	$0.291748\pi$
$24$	−4.00000	−0.816497
$25$	−4.71982	−0.943965
$26$	0.470683	0.0923086
$27$	−2.11727	−0.407468
$28$	1.77846	0.336097
$29$	5.22154	0.969616	0.484808	−	0.874621i	$-0.338889\pi$
$29$	5.22154	0.969616	0.484808	+	0.874621i	$0.338889\pi$
$30$	0.560352	0.102306
$31$	−7.02760	−1.26219	−0.631097	−	0.775704i	$-0.717395\pi$
$31$	−7.02760	−1.26219	−0.631097	+	0.775704i	$0.717395\pi$
$32$	4.83709	0.855085
$33$	−5.05863	−0.880595
$34$	−0.615547	−0.105566
$35$	−0.529317	−0.0894708
$36$	−3.66119	−0.610198
$37$	−2.36641	−0.389035	−0.194517	−	0.980899i	$-0.562314\pi$
$37$	−2.36641	−0.389035	−0.194517	+	0.980899i	$0.562314\pi$
$38$	−0.692226	−0.112294
$39$	2.24914	0.360151
$40$	−0.941367	−0.148843
$41$	6.49828	1.01486	0.507431	−	0.861693i	$-0.330595\pi$
$41$	6.49828	1.01486	0.507431	+	0.861693i	$0.330595\pi$
$42$	−1.05863	−0.163351
$43$	11.3940	1.73757	0.868785	−	0.495190i	$-0.164902\pi$
$43$	11.3940	1.73757	0.868785	+	0.495190i	$0.164902\pi$
$44$	4.00000	0.603023
$45$	1.08967	0.162438
$46$	2.74742	0.405085
$47$	8.58451	1.25218	0.626090	−	0.779751i	$-0.284654\pi$
$47$	8.58451	1.25218	0.626090	+	0.779751i	$0.284654\pi$
$48$	6.11727	0.882951
$49$	1.00000	0.142857
$50$	−2.22154	−0.314174
$51$	−2.94137	−0.411874
$52$	−1.77846	−0.246628
$53$	11.2767	1.54898	0.774490	−	0.632587i	$-0.218007\pi$
$53$	11.2767	1.54898	0.774490	+	0.632587i	$0.218007\pi$
$54$	−0.996562	−0.135615
$55$	−1.19051	−0.160528
$56$	1.77846	0.237656
$57$	−3.30777	−0.438125
$58$	2.45769	0.322711
$59$	−12.1725	−1.58472	−0.792360	−	0.610054i	$-0.791148\pi$
$59$	−12.1725	−1.58472	−0.792360	+	0.610054i	$0.791148\pi$
$60$	−2.11727	−0.273338
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−3.30777	−0.420088
$63$	−2.05863	−0.259363
$64$	−3.16291	−0.395364
$65$	0.529317	0.0656536
$66$	−2.38101	−0.293083
$67$	−15.9379	−1.94713	−0.973564	−	0.228415i	$-0.926646\pi$
$67$	−15.9379	−1.94713	−0.973564	+	0.228415i	$0.926646\pi$
$68$	2.32582	0.282047
$69$	13.1284	1.58048
$70$	−0.249141	−0.0297780
$71$	1.19051	0.141287	0.0706436	−	0.997502i	$-0.477495\pi$
$71$	1.19051	0.141287	0.0706436	+	0.997502i	$0.477495\pi$
$72$	−3.66119	−0.431475
$73$	7.64315	0.894562	0.447281	−	0.894393i	$-0.352392\pi$
$73$	7.64315	0.894562	0.447281	+	0.894393i	$0.352392\pi$
$74$	−1.11383	−0.129480
$75$	−10.6155	−1.22578
$76$	2.61555	0.300024
$77$	2.24914	0.256313
$78$	1.05863	0.119867
$79$	−1.33881	−0.150628	−0.0753139	−	0.997160i	$-0.523996\pi$
$79$	−1.33881	−0.150628	−0.0753139	+	0.997160i	$0.523996\pi$
$80$	1.43965	0.160958
$81$	−10.9379	−1.21533
$82$	3.05863	0.337770
$83$	−16.3500	−1.79464	−0.897322	−	0.441377i	$-0.854490\pi$
$83$	−16.3500	−1.79464	−0.897322	+	0.441377i	$0.854490\pi$
$84$	4.00000	0.436436
$85$	−0.692226	−0.0750825
$86$	5.36297	0.578304
$87$	11.7440	1.25909
$88$	4.00000	0.426401
$89$	6.91033	0.732494	0.366247	−	0.930518i	$-0.380643\pi$
$89$	6.91033	0.732494	0.366247	+	0.930518i	$0.380643\pi$
$90$	0.512889	0.0540632
$91$	−1.00000	−0.104828
$92$	−10.3810	−1.08230
$93$	−15.8061	−1.63901
$94$	4.04059	0.416755
$95$	−0.778457	−0.0798680
$96$	10.8793	1.11036
$97$	−3.47068	−0.352395	−0.176197	−	0.984355i	$-0.556380\pi$
$97$	−3.47068	−0.352395	−0.176197	+	0.984355i	$0.556380\pi$
$98$	0.470683	0.0475462
$99$	−4.63016	−0.465348
$100$	8.39400	0.839400
$101$	7.75086	0.771239	0.385620	−	0.922658i	$-0.373988\pi$
$101$	7.75086	0.771239	0.385620	+	0.922658i	$0.373988\pi$
$102$	−1.38445	−0.137081
$103$	−16.9966	−1.67472	−0.837361	−	0.546651i	$-0.815902\pi$
$103$	−16.9966	−1.67472	−0.837361	+	0.546651i	$0.815902\pi$
$104$	−1.77846	−0.174392
$105$	−1.19051	−0.116182
$106$	5.30777	0.515537
$107$	−5.55691	−0.537207	−0.268604	−	0.963251i	$-0.586562\pi$
$107$	−5.55691	−0.537207	−0.268604	+	0.963251i	$0.586562\pi$
$108$	3.76547	0.362332
$109$	7.92332	0.758917	0.379458	−	0.925209i	$-0.376110\pi$
$109$	7.92332	0.758917	0.379458	+	0.925209i	$0.376110\pi$
$110$	−0.560352	−0.0534275
$111$	−5.32238	−0.505178
$112$	−2.71982	−0.256999
$113$	−9.89229	−0.930588	−0.465294	−	0.885156i	$-0.654051\pi$
$113$	−9.89229	−0.930588	−0.465294	+	0.885156i	$0.654051\pi$
$114$	−1.55691	−0.145818
$115$	3.08967	0.288113
$116$	−9.28629	−0.862210
$117$	2.05863	0.190321
$118$	−5.72938	−0.527432
$119$	1.30777	0.119883
$120$	−2.11727	−0.193279
$121$	−5.94137	−0.540124
$122$	−0.941367	−0.0852273
$123$	14.6155	1.31784
$124$	12.4983	1.12238
$125$	−5.14486	−0.460171
$126$	−0.968964	−0.0863222
$127$	0.824101	0.0731271	0.0365635	−	0.999331i	$-0.488359\pi$
$127$	0.824101	0.0731271	0.0365635	+	0.999331i	$0.488359\pi$
$128$	−11.1629	−0.986671
$129$	25.6267	2.25631
$130$	0.249141	0.0218511
$131$	10.6155	0.927485	0.463742	−	0.885970i	$-0.346506\pi$
$131$	10.6155	0.927485	0.463742	+	0.885970i	$0.346506\pi$
$132$	8.99656	0.783050
$133$	1.47068	0.127524
$134$	−7.50172	−0.648050
$135$	−1.12070	−0.0964549
$136$	2.32582	0.199437
$137$	11.3630	0.970804	0.485402	−	0.874291i	$-0.338673\pi$
$137$	11.3630	0.970804	0.485402	+	0.874291i	$0.338673\pi$
$138$	6.17934	0.526020
$139$	−13.9233	−1.18096	−0.590480	−	0.807052i	$-0.701062\pi$
$139$	−13.9233	−1.18096	−0.590480	+	0.807052i	$0.701062\pi$
$140$	0.941367	0.0795600
$141$	19.3078	1.62601
$142$	0.560352	0.0470237
$143$	−2.24914	−0.188083
$144$	5.59912	0.466593
$145$	2.76385	0.229525
$146$	3.59750	0.297731
$147$	2.24914	0.185506
$148$	4.20855	0.345941
$149$	9.30777	0.762523	0.381261	−	0.924467i	$-0.375490\pi$
$149$	9.30777	0.762523	0.381261	+	0.924467i	$0.375490\pi$
$150$	−4.99656	−0.407968
$151$	7.07324	0.575612	0.287806	−	0.957689i	$-0.407074\pi$
$151$	7.07324	0.575612	0.287806	+	0.957689i	$0.407074\pi$
$152$	2.61555	0.212149
$153$	−2.69223	−0.217654
$154$	1.05863	0.0853071
$155$	−3.71982	−0.298783
$156$	−4.00000	−0.320256
$157$	−6.04059	−0.482091	−0.241046	−	0.970514i	$-0.577490\pi$
$157$	−6.04059	−0.482091	−0.241046	+	0.970514i	$0.577490\pi$
$158$	−0.630155	−0.0501325
$159$	25.3630	2.01141
$160$	2.56035	0.202414
$161$	−5.83709	−0.460027
$162$	−5.14830	−0.404489
$163$	6.38101	0.499800	0.249900	−	0.968272i	$-0.419602\pi$
$163$	6.38101	0.499800	0.249900	+	0.968272i	$0.419602\pi$
$164$	−11.5569	−0.902443
$165$	−2.67762	−0.208452
$166$	−7.69566	−0.597299
$167$	−16.5845	−1.28335	−0.641674	−	0.766977i	$-0.721760\pi$
$167$	−16.5845	−1.28335	−0.641674	+	0.766977i	$0.721760\pi$
$168$	4.00000	0.308607
$169$	1.00000	0.0769231
$170$	−0.325819	−0.0249892
$171$	−3.02760	−0.231526
$172$	−20.2637	−1.54510
$173$	−23.3009	−1.77153	−0.885767	−	0.464130i	$-0.846367\pi$
$173$	−23.3009	−1.77153	−0.885767	+	0.464130i	$0.846367\pi$
$174$	5.52770	0.419054
$175$	4.71982	0.356785
$176$	−6.11727	−0.461106
$177$	−27.3776	−2.05782
$178$	3.25258	0.243791
$179$	21.0422	1.57277	0.786384	−	0.617738i	$-0.211951\pi$
$179$	21.0422	1.57277	0.786384	+	0.617738i	$0.211951\pi$
$180$	−1.93793	−0.144445
$181$	16.7474	1.24483	0.622413	−	0.782689i	$-0.286152\pi$
$181$	16.7474	1.24483	0.622413	+	0.782689i	$0.286152\pi$
$182$	−0.470683	−0.0348894
$183$	−4.49828	−0.332523
$184$	−10.3810	−0.765299
$185$	−1.25258	−0.0920914
$186$	−7.43965	−0.545501
$187$	2.94137	0.215094
$188$	−15.2672	−1.11347
$189$	2.11727	0.154008
$190$	−0.366407	−0.0265819
$191$	−7.43965	−0.538314	−0.269157	−	0.963096i	$-0.586745\pi$
$191$	−7.43965	−0.538314	−0.269157	+	0.963096i	$0.586745\pi$
$192$	−7.11383	−0.513396
$193$	−1.50172	−0.108096	−0.0540480	−	0.998538i	$-0.517212\pi$
$193$	−1.50172	−0.108096	−0.0540480	+	0.998538i	$0.517212\pi$
$194$	−1.63359	−0.117285
$195$	1.19051	0.0852540
$196$	−1.77846	−0.127033
$197$	23.9931	1.70944	0.854720	−	0.519090i	$-0.173729\pi$
$197$	23.9931	1.70944	0.854720	+	0.519090i	$0.173729\pi$
$198$	−2.17934	−0.154879
$199$	−2.01461	−0.142812	−0.0714059	−	0.997447i	$-0.522749\pi$
$199$	−2.01461	−0.142812	−0.0714059	+	0.997447i	$0.522749\pi$
$200$	8.39400	0.593546
$201$	−35.8466	−2.52843
$202$	3.64820	0.256687
$203$	−5.22154	−0.366480
$204$	5.23109	0.366250
$205$	3.43965	0.240235
$206$	−8.00000	−0.557386
$207$	12.0164	0.835199
$208$	2.71982	0.188586
$209$	3.30777	0.228803
$210$	−0.560352	−0.0386680
$211$	10.1008	0.695370	0.347685	−	0.937611i	$-0.386968\pi$
$211$	10.1008	0.695370	0.347685	+	0.937611i	$0.386968\pi$
$212$	−20.0552	−1.37740
$213$	2.67762	0.183467
$214$	−2.61555	−0.178795
$215$	6.03104	0.411313
$216$	3.76547	0.256208
$217$	7.02760	0.477064
$218$	3.72938	0.252585
$219$	17.1905	1.16163
$220$	2.11727	0.142746
$221$	−1.30777	−0.0879704
$222$	−2.50516	−0.168135
$223$	10.1414	0.679120	0.339560	−	0.940584i	$-0.389722\pi$
$223$	10.1414	0.679120	0.339560	+	0.940584i	$0.389722\pi$
$224$	−4.83709	−0.323192
$225$	−9.71639	−0.647759
$226$	−4.65613	−0.309721
$227$	−5.38445	−0.357379	−0.178689	−	0.983906i	$-0.557186\pi$
$227$	−5.38445	−0.357379	−0.178689	+	0.983906i	$0.557186\pi$
$228$	5.88273	0.389594
$229$	3.32238	0.219549	0.109775	−	0.993957i	$-0.464987\pi$
$229$	3.32238	0.219549	0.109775	+	0.993957i	$0.464987\pi$
$230$	1.45426	0.0958908
$231$	5.05863	0.332834
$232$	−9.28629	−0.609675
$233$	13.7198	0.898816	0.449408	−	0.893327i	$-0.351635\pi$
$233$	13.7198	0.898816	0.449408	+	0.893327i	$0.351635\pi$
$234$	0.968964	0.0633432
$235$	4.54392	0.296413
$236$	21.6482	1.40918
$237$	−3.01117	−0.195597
$238$	0.615547	0.0399000
$239$	−3.50172	−0.226507	−0.113254	−	0.993566i	$-0.536127\pi$
$239$	−3.50172	−0.226507	−0.113254	+	0.993566i	$0.536127\pi$
$240$	3.23797	0.209010
$241$	1.58795	0.102289	0.0511444	−	0.998691i	$-0.483713\pi$
$241$	1.58795	0.102289	0.0511444	+	0.998691i	$0.483713\pi$
$242$	−2.79650	−0.179766
$243$	−18.2491	−1.17068
$244$	3.55691	0.227708
$245$	0.529317	0.0338168
$246$	6.87930	0.438608
$247$	−1.47068	−0.0935773
$248$	12.4983	0.793642
$249$	−36.7734	−2.33042
$250$	−2.42160	−0.153156
$251$	−4.92676	−0.310974	−0.155487	−	0.987838i	$-0.549695\pi$
$251$	−4.92676	−0.310974	−0.155487	+	0.987838i	$0.549695\pi$
$252$	3.66119	0.230633
$253$	−13.1284	−0.825378
$254$	0.387890	0.0243384
$255$	−1.55691	−0.0974978
$256$	1.07162	0.0669764
$257$	−8.01461	−0.499938	−0.249969	−	0.968254i	$-0.580420\pi$
$257$	−8.01461	−0.499938	−0.249969	+	0.968254i	$0.580420\pi$
$258$	12.0621	0.750952
$259$	2.36641	0.147041
$260$	−0.941367	−0.0583811
$261$	10.7492	0.665361
$262$	4.99656	0.308689
$263$	1.60256	0.0988179	0.0494090	−	0.998779i	$-0.484266\pi$
$263$	1.60256	0.0988179	0.0494090	+	0.998779i	$0.484266\pi$
$264$	8.99656	0.553700
$265$	5.96896	0.366671
$266$	0.692226	0.0424431
$267$	15.5423	0.951174
$268$	28.3449	1.73144
$269$	−11.8207	−0.720719	−0.360359	−	0.932814i	$-0.617346\pi$
$269$	−11.8207	−0.720719	−0.360359	+	0.932814i	$0.617346\pi$
$270$	−0.527497	−0.0321024
$271$	21.8827	1.32928	0.664641	−	0.747163i	$-0.268585\pi$
$271$	21.8827	1.32928	0.664641	+	0.747163i	$0.268585\pi$
$272$	−3.55691	−0.215670
$273$	−2.24914	−0.136124
$274$	5.34836	0.323106
$275$	10.6155	0.640142
$276$	−23.3484	−1.40541
$277$	−10.2181	−0.613946	−0.306973	−	0.951718i	$-0.599316\pi$
$277$	−10.2181	−0.613946	−0.306973	+	0.951718i	$0.599316\pi$
$278$	−6.55348	−0.393051
$279$	−14.4672	−0.866131
$280$	0.941367	0.0562574
$281$	1.54231	0.0920063	0.0460031	−	0.998941i	$-0.485352\pi$
$281$	1.54231	0.0920063	0.0460031	+	0.998941i	$0.485352\pi$
$282$	9.08785	0.541174
$283$	15.8466	0.941985	0.470993	−	0.882137i	$-0.343896\pi$
$283$	15.8466	0.941985	0.470993	+	0.882137i	$0.343896\pi$
$284$	−2.11727	−0.125637
$285$	−1.75086	−0.103712
$286$	−1.05863	−0.0625983
$287$	−6.49828	−0.383581
$288$	9.95779	0.586769
$289$	−15.2897	−0.899396
$290$	1.30090	0.0763914
$291$	−7.80605	−0.457599
$292$	−13.5930	−0.795470
$293$	11.0828	0.647464	0.323732	−	0.946149i	$-0.395062\pi$
$293$	11.0828	0.647464	0.323732	+	0.946149i	$0.395062\pi$
$294$	1.05863	0.0617407
$295$	−6.44309	−0.375131
$296$	4.20855	0.244617
$297$	4.76203	0.276321
$298$	4.38101	0.253785
$299$	5.83709	0.337568
$300$	18.8793	1.09000
$301$	−11.3940	−0.656740
$302$	3.32926	0.191577
$303$	17.4328	1.00149
$304$	−4.00000	−0.229416
$305$	−1.05863	−0.0606172
$306$	−1.26719	−0.0724402
$307$	20.4121	1.16498	0.582489	−	0.812839i	$-0.302079\pi$
$307$	20.4121	1.16498	0.582489	+	0.812839i	$0.302079\pi$
$308$	−4.00000	−0.227921
$309$	−38.2277	−2.17470
$310$	−1.75086	−0.0994421
$311$	−1.92332	−0.109062	−0.0545308	−	0.998512i	$-0.517366\pi$
$311$	−1.92332	−0.109062	−0.0545308	+	0.998512i	$0.517366\pi$
$312$	−4.00000	−0.226455
$313$	−14.3664	−0.812037	−0.406019	−	0.913865i	$-0.633083\pi$
$313$	−14.3664	−0.812037	−0.406019	+	0.913865i	$0.633083\pi$
$314$	−2.84320	−0.160451
$315$	−1.08967	−0.0613959
$316$	2.38101	0.133943
$317$	15.5569	0.873763	0.436882	−	0.899519i	$-0.356083\pi$
$317$	15.5569	0.873763	0.436882	+	0.899519i	$0.356083\pi$
$318$	11.9379	0.669446
$319$	−11.7440	−0.657537
$320$	−1.67418	−0.0935895
$321$	−12.4983	−0.697586
$322$	−2.74742	−0.153108
$323$	1.92332	0.107016
$324$	19.4526	1.08070
$325$	−4.71982	−0.261809
$326$	3.00344	0.166345
$327$	17.8207	0.985485
$328$	−11.5569	−0.638124
$329$	−8.58451	−0.473279
$330$	−1.26031	−0.0693778
$331$	31.5500	1.73415	0.867073	−	0.498180i	$-0.165998\pi$
$331$	31.5500	1.73415	0.867073	+	0.498180i	$0.165998\pi$
$332$	29.0777	1.59585
$333$	−4.87156	−0.266960
$334$	−7.80605	−0.427128
$335$	−8.43621	−0.460919
$336$	−6.11727	−0.333724
$337$	−8.42666	−0.459029	−0.229515	−	0.973305i	$-0.573714\pi$
$337$	−8.42666	−0.459029	−0.229515	+	0.973305i	$0.573714\pi$
$338$	0.470683	0.0256018
$339$	−22.2491	−1.20841
$340$	1.23109	0.0667655
$341$	15.8061	0.855946
$342$	−1.42504	−0.0770573
$343$	−1.00000	−0.0539949
$344$	−20.2637	−1.09255
$345$	6.94910	0.374127
$346$	−10.9673	−0.589608
$347$	19.3484	1.03867	0.519337	−	0.854569i	$-0.326179\pi$
$347$	19.3484	1.03867	0.519337	+	0.854569i	$0.326179\pi$
$348$	−20.8862	−1.11962
$349$	27.2553	1.45894	0.729470	−	0.684013i	$-0.239767\pi$
$349$	27.2553	1.45894	0.729470	+	0.684013i	$0.239767\pi$
$350$	2.22154	0.118746
$351$	−2.11727	−0.113011
$352$	−10.8793	−0.579868
$353$	−25.6742	−1.36650	−0.683249	−	0.730185i	$-0.739434\pi$
$353$	−25.6742	−1.36650	−0.683249	+	0.730185i	$0.739434\pi$
$354$	−12.8862	−0.684892
$355$	0.630155	0.0334452
$356$	−12.2897	−0.651354
$357$	2.94137	0.155674
$358$	9.90422	0.523454
$359$	−23.4182	−1.23596	−0.617982	−	0.786192i	$-0.712049\pi$
$359$	−23.4182	−1.23596	−0.617982	+	0.786192i	$0.712049\pi$
$360$	−1.93793	−0.102138
$361$	−16.8371	−0.886163
$362$	7.88273	0.414307
$363$	−13.3630	−0.701374
$364$	1.77846	0.0932165
$365$	4.04564	0.211759
$366$	−2.11727	−0.110671
$367$	−14.6854	−0.766569	−0.383285	−	0.923630i	$-0.625207\pi$
$367$	−14.6854	−0.766569	−0.383285	+	0.923630i	$0.625207\pi$
$368$	15.8759	0.827586
$369$	13.3776	0.696409
$370$	−0.589568	−0.0306502
$371$	−11.2767	−0.585459
$372$	28.1104	1.45746
$373$	−23.6673	−1.22545	−0.612723	−	0.790298i	$-0.709926\pi$
$373$	−23.6673	−1.22545	−0.612723	+	0.790298i	$0.709926\pi$
$374$	1.38445	0.0715883
$375$	−11.5715	−0.597551
$376$	−15.2672	−0.787345
$377$	5.22154	0.268923
$378$	0.996562	0.0512576
$379$	−32.7405	−1.68177	−0.840884	−	0.541215i	$-0.817965\pi$
$379$	−32.7405	−1.68177	−0.840884	+	0.541215i	$0.817965\pi$
$380$	1.38445	0.0710209
$381$	1.85352	0.0949586
$382$	−3.50172	−0.179164
$383$	−22.6155	−1.15560	−0.577800	−	0.816178i	$-0.696089\pi$
$383$	−22.6155	−1.15560	−0.577800	+	0.816178i	$0.696089\pi$
$384$	−25.1070	−1.28123
$385$	1.19051	0.0606739
$386$	−0.706834	−0.0359769
$387$	23.4561	1.19234
$388$	6.17246	0.313359
$389$	38.0483	1.92913	0.964563	−	0.263852i	$-0.0849930\pi$
$389$	38.0483	1.92913	0.964563	+	0.263852i	$0.0849930\pi$
$390$	0.560352	0.0283745
$391$	−7.63359	−0.386047
$392$	−1.77846	−0.0898256
$393$	23.8759	1.20438
$394$	11.2932	0.568941
$395$	−0.708654	−0.0356562
$396$	8.23453	0.413801
$397$	−11.7052	−0.587468	−0.293734	−	0.955887i	$-0.594898\pi$
$397$	−11.7052	−0.587468	−0.293734	+	0.955887i	$0.594898\pi$
$398$	−0.948243	−0.0475311
$399$	3.30777	0.165596
$400$	−12.8371	−0.641855
$401$	3.55691	0.177624	0.0888119	−	0.996048i	$-0.471693\pi$
$401$	3.55691	0.177624	0.0888119	+	0.996048i	$0.471693\pi$
$402$	−16.8724	−0.841520
$403$	−7.02760	−0.350070
$404$	−13.7846	−0.685808
$405$	−5.78963	−0.287689
$406$	−2.45769	−0.121973
$407$	5.32238	0.263821
$408$	5.23109	0.258978
$409$	5.26213	0.260196	0.130098	−	0.991501i	$-0.458471\pi$
$409$	5.26213	0.260196	0.130098	+	0.991501i	$0.458471\pi$
$410$	1.61899	0.0799560
$411$	25.5569	1.26063
$412$	30.2277	1.48921
$413$	12.1725	0.598968
$414$	5.65593	0.277974
$415$	−8.65432	−0.424824
$416$	4.83709	0.237158
$417$	−31.3155	−1.53353
$418$	1.55691	0.0761512
$419$	26.0337	1.27183	0.635915	−	0.771759i	$-0.280623\pi$
$419$	26.0337	1.27183	0.635915	+	0.771759i	$0.280623\pi$
$420$	2.11727	0.103312
$421$	22.2423	1.08402	0.542011	−	0.840372i	$-0.317663\pi$
$421$	22.2423	1.08402	0.542011	+	0.840372i	$0.317663\pi$
$422$	4.75430	0.231436
$423$	17.6724	0.859260
$424$	−20.0552	−0.973966
$425$	6.17246	0.299408
$426$	1.26031	0.0610622
$427$	2.00000	0.0967868
$428$	9.88273	0.477700
$429$	−5.05863	−0.244233
$430$	2.83871	0.136895
$431$	27.6742	1.33302	0.666509	−	0.745497i	$-0.267788\pi$
$431$	27.6742	1.33302	0.666509	+	0.745497i	$0.267788\pi$
$432$	−5.75859	−0.277060
$433$	−12.7880	−0.614552	−0.307276	−	0.951620i	$-0.599418\pi$
$433$	−12.7880	−0.614552	−0.307276	+	0.951620i	$0.599418\pi$
$434$	3.30777	0.158778
$435$	6.21629	0.298048
$436$	−14.0913	−0.674850
$437$	−8.58451	−0.410653
$438$	8.09129	0.386617
$439$	−18.1656	−0.866996	−0.433498	−	0.901155i	$-0.642721\pi$
$439$	−18.1656	−0.866996	−0.433498	+	0.901155i	$0.642721\pi$
$440$	2.11727	0.100937
$441$	2.05863	0.0980302
$442$	−0.615547	−0.0292786
$443$	0.107714	0.00511767	0.00255883	−	0.999997i	$-0.499185\pi$
$443$	0.107714	0.00511767	0.00255883	+	0.999997i	$0.499185\pi$
$444$	9.46563	0.449219
$445$	3.65775	0.173394
$446$	4.77340	0.226027
$447$	20.9345	0.990167
$448$	3.16291	0.149433
$449$	−22.1725	−1.04638	−0.523192	−	0.852215i	$-0.675259\pi$
$449$	−22.1725	−1.04638	−0.523192	+	0.852215i	$0.675259\pi$
$450$	−4.57334	−0.215589
$451$	−14.6155	−0.688219
$452$	17.5930	0.827505
$453$	15.9087	0.747457
$454$	−2.53437	−0.118944
$455$	−0.529317	−0.0248147
$456$	5.88273	0.275484
$457$	4.35953	0.203930	0.101965	−	0.994788i	$-0.467487\pi$
$457$	4.35953	0.203930	0.101965	+	0.994788i	$0.467487\pi$
$458$	1.56379	0.0730711
$459$	2.76891	0.129241
$460$	−5.49484	−0.256198
$461$	32.3810	1.50813	0.754067	−	0.656797i	$-0.228089\pi$
$461$	32.3810	1.50813	0.754067	+	0.656797i	$0.228089\pi$
$462$	2.38101	0.110775
$463$	8.36641	0.388820	0.194410	−	0.980920i	$-0.437721\pi$
$463$	8.36641	0.388820	0.194410	+	0.980920i	$0.437721\pi$
$464$	14.2017	0.659296
$465$	−8.36641	−0.387983
$466$	6.45769	0.299147
$467$	−19.5423	−0.904310	−0.452155	−	0.891939i	$-0.649345\pi$
$467$	−19.5423	−0.904310	−0.452155	+	0.891939i	$0.649345\pi$
$468$	−3.66119	−0.169239
$469$	15.9379	0.735945
$470$	2.13875	0.0986532
$471$	−13.5861	−0.626016
$472$	21.6482	0.996439
$473$	−25.6267	−1.17832
$474$	−1.41731	−0.0650991
$475$	6.94137	0.318492
$476$	−2.32582	−0.106604
$477$	23.2147	1.06293
$478$	−1.64820	−0.0753870
$479$	−28.5224	−1.30322	−0.651612	−	0.758553i	$-0.725907\pi$
$479$	−28.5224	−1.30322	−0.651612	+	0.758553i	$0.725907\pi$
$480$	5.75859	0.262843
$481$	−2.36641	−0.107899
$482$	0.747422	0.0340441
$483$	−13.1284	−0.597365
$484$	10.5665	0.480294
$485$	−1.83709	−0.0834180
$486$	−8.58957	−0.389631
$487$	24.8241	1.12489	0.562444	−	0.826836i	$-0.309861\pi$
$487$	24.8241	1.12489	0.562444	+	0.826836i	$0.309861\pi$
$488$	3.55691	0.161014
$489$	14.3518	0.649011
$490$	0.249141	0.0112550
$491$	29.1690	1.31638	0.658190	−	0.752852i	$-0.271322\pi$
$491$	29.1690	1.31638	0.658190	+	0.752852i	$0.271322\pi$
$492$	−25.9931	−1.17186
$493$	−6.82860	−0.307545
$494$	−0.692226	−0.0311447
$495$	−2.45082	−0.110156
$496$	−19.1138	−0.858236
$497$	−1.19051	−0.0534016
$498$	−17.3086	−0.775618
$499$	−33.3009	−1.49075	−0.745376	−	0.666644i	$-0.767730\pi$
$499$	−33.3009	−1.49075	−0.745376	+	0.666644i	$0.767730\pi$
$500$	9.14992	0.409197
$501$	−37.3009	−1.66648
$502$	−2.31894	−0.103500
$503$	−12.3258	−0.549581	−0.274791	−	0.961504i	$-0.588609\pi$
$503$	−12.3258	−0.549581	−0.274791	+	0.961504i	$0.588609\pi$
$504$	3.66119	0.163082
$505$	4.10266	0.182566
$506$	−6.17934	−0.274705
$507$	2.24914	0.0998878
$508$	−1.46563	−0.0650267
$509$	23.7052	1.05072	0.525358	−	0.850882i	$-0.323932\pi$
$509$	23.7052	1.05072	0.525358	+	0.850882i	$0.323932\pi$
$510$	−0.732814	−0.0324495
$511$	−7.64315	−0.338113
$512$	22.8302	1.00896
$513$	3.11383	0.137479
$514$	−3.77234	−0.166391
$515$	−8.99656	−0.396436
$516$	−45.5760	−2.00637
$517$	−19.3078	−0.849155
$518$	1.11383	0.0489388
$519$	−52.4070	−2.30041
$520$	−0.941367	−0.0412817
$521$	43.9018	1.92337	0.961687	−	0.274149i	$-0.0883962\pi$
$521$	43.9018	1.92337	0.961687	+	0.274149i	$0.0883962\pi$
$522$	5.05949	0.221448
$523$	37.4328	1.63682	0.818410	−	0.574634i	$-0.194856\pi$
$523$	37.4328	1.63682	0.818410	+	0.574634i	$0.194856\pi$
$524$	−18.8793	−0.824746
$525$	10.6155	0.463300
$526$	0.754297	0.0328889
$527$	9.19051	0.400345
$528$	−13.7586	−0.598766
$529$	11.0716	0.481375
$530$	2.80949	0.122037
$531$	−25.0586	−1.08745
$532$	−2.61555	−0.113398
$533$	6.49828	0.281472
$534$	7.31551	0.316573
$535$	−2.94137	−0.127166
$536$	28.3449	1.22431
$537$	47.3269	2.04231
$538$	−5.56379	−0.239872
$539$	−2.24914	−0.0968773
$540$	1.99312	0.0857704
$541$	−34.9751	−1.50370	−0.751848	−	0.659336i	$-0.770837\pi$
$541$	−34.9751	−1.50370	−0.751848	+	0.659336i	$0.770837\pi$
$542$	10.2998	0.442416
$543$	37.6673	1.61646
$544$	−6.32582	−0.271217
$545$	4.19395	0.179649
$546$	−1.05863	−0.0453053
$547$	6.50783	0.278255	0.139127	−	0.990274i	$-0.455570\pi$
$547$	6.50783	0.278255	0.139127	+	0.990274i	$0.455570\pi$
$548$	−20.2086	−0.863267
$549$	−4.11727	−0.175721
$550$	4.99656	0.213054
$551$	−7.67924	−0.327146
$552$	−23.3484	−0.993772
$553$	1.33881	0.0569320
$554$	−4.80949	−0.204336
$555$	−2.81722	−0.119585
$556$	24.7620	1.05014
$557$	−43.4328	−1.84031	−0.920153	−	0.391559i	$-0.871936\pi$
$557$	−43.4328	−1.84031	−0.920153	+	0.391559i	$0.871936\pi$
$558$	−6.80949	−0.288269
$559$	11.3940	0.481915
$560$	−1.43965	−0.0608362
$561$	6.61555	0.279309
$562$	0.725938	0.0306218
$563$	−33.8827	−1.42799	−0.713993	−	0.700152i	$-0.753115\pi$
$563$	−33.8827	−1.42799	−0.713993	+	0.700152i	$0.753115\pi$
$564$	−34.3380	−1.44589
$565$	−5.23615	−0.220287
$566$	7.45875	0.313515
$567$	10.9379	0.459350
$568$	−2.11727	−0.0888385
$569$	19.2147	0.805521	0.402760	−	0.915305i	$-0.368051\pi$
$569$	19.2147	0.805521	0.402760	+	0.915305i	$0.368051\pi$
$570$	−0.824101	−0.0345178
$571$	20.8268	0.871573	0.435787	−	0.900050i	$-0.356470\pi$
$571$	20.8268	0.871573	0.435787	+	0.900050i	$0.356470\pi$
$572$	4.00000	0.167248
$573$	−16.7328	−0.699023
$574$	−3.05863	−0.127665
$575$	−27.5500	−1.14892
$576$	−6.51127	−0.271303
$577$	28.6448	1.19250	0.596249	−	0.802800i	$-0.296657\pi$
$577$	28.6448	1.19250	0.596249	+	0.802800i	$0.296657\pi$
$578$	−7.19662	−0.299340
$579$	−3.37758	−0.140367
$580$	−4.91539	−0.204100
$581$	16.3500	0.678311
$582$	−3.67418	−0.152300
$583$	−25.3630	−1.05043
$584$	−13.5930	−0.562483
$585$	1.08967	0.0450523
$586$	5.21649	0.215491
$587$	−4.32076	−0.178337	−0.0891685	−	0.996017i	$-0.528421\pi$
$587$	−4.32076	−0.178337	−0.0891685	+	0.996017i	$0.528421\pi$
$588$	−4.00000	−0.164957
$589$	10.3354	0.425862
$590$	−3.03265	−0.124852
$591$	53.9639	2.21978
$592$	−6.43621	−0.264527
$593$	−15.9690	−0.655767	−0.327883	−	0.944718i	$-0.606335\pi$
$593$	−15.9690	−0.655767	−0.327883	+	0.944718i	$0.606335\pi$
$594$	2.24141	0.0919661
$595$	0.692226	0.0283785
$596$	−16.5535	−0.678057
$597$	−4.53114	−0.185447
$598$	2.74742	0.112350
$599$	−16.8697	−0.689279	−0.344640	−	0.938735i	$-0.611999\pi$
$599$	−16.8697	−0.689279	−0.344640	+	0.938735i	$0.611999\pi$
$600$	18.8793	0.770744
$601$	−15.3415	−0.625792	−0.312896	−	0.949787i	$-0.601299\pi$
$601$	−15.3415	−0.625792	−0.312896	+	0.949787i	$0.601299\pi$
$602$	−5.36297	−0.218578
$603$	−32.8103	−1.33614
$604$	−12.5795	−0.511851
$605$	−3.14486	−0.127857
$606$	8.20532	0.333318
$607$	35.8353	1.45451	0.727254	−	0.686368i	$-0.240796\pi$
$607$	35.8353	1.45451	0.727254	+	0.686368i	$0.240796\pi$
$608$	−7.11383	−0.288504
$609$	−11.7440	−0.475890
$610$	−0.498281	−0.0201748
$611$	8.58451	0.347292
$612$	4.78801	0.193544
$613$	19.6673	0.794355	0.397177	−	0.917742i	$-0.369990\pi$
$613$	19.6673	0.794355	0.397177	+	0.917742i	$0.369990\pi$
$614$	9.60761	0.387732
$615$	7.73625	0.311956
$616$	−4.00000	−0.161165
$617$	−41.4588	−1.66907	−0.834533	−	0.550958i	$-0.814263\pi$
$617$	−41.4588	−1.66907	−0.834533	+	0.550958i	$0.814263\pi$
$618$	−17.9931	−0.723790
$619$	10.8793	0.437276	0.218638	−	0.975806i	$-0.429839\pi$
$619$	10.8793	0.437276	0.218638	+	0.975806i	$0.429839\pi$
$620$	6.61555	0.265687
$621$	−12.3587	−0.495937
$622$	−0.905275	−0.0362982
$623$	−6.91033	−0.276857
$624$	6.11727	0.244887
$625$	20.8759	0.835034
$626$	−6.76203	−0.270265
$627$	7.43965	0.297111
$628$	10.7429	0.428689
$629$	3.09472	0.123395
$630$	−0.512889	−0.0204340
$631$	−31.4396	−1.25159	−0.625796	−	0.779987i	$-0.715226\pi$
$631$	−31.4396	−1.25159	−0.625796	+	0.779987i	$0.715226\pi$
$632$	2.38101	0.0947117
$633$	22.7182	0.902968
$634$	7.32238	0.290809
$635$	0.436210	0.0173105
$636$	−45.1070	−1.78861
$637$	1.00000	0.0396214
$638$	−5.52770	−0.218844
$639$	2.45082	0.0969529
$640$	−5.90871	−0.233562
$641$	3.04221	0.120160	0.0600799	−	0.998194i	$-0.480864\pi$
$641$	3.04221	0.120160	0.0600799	+	0.998194i	$0.480864\pi$
$642$	−5.88273	−0.232173
$643$	8.02922	0.316641	0.158321	−	0.987388i	$-0.449392\pi$
$643$	8.02922	0.316641	0.158321	+	0.987388i	$0.449392\pi$
$644$	10.3810	0.409069
$645$	13.5646	0.534107
$646$	0.905275	0.0356176
$647$	7.07324	0.278078	0.139039	−	0.990287i	$-0.455599\pi$
$647$	7.07324	0.278078	0.139039	+	0.990287i	$0.455599\pi$
$648$	19.4526	0.764172
$649$	27.3776	1.07466
$650$	−2.22154	−0.0871361
$651$	15.8061	0.619488
$652$	−11.3484	−0.444436
$653$	−15.7586	−0.616681	−0.308341	−	0.951276i	$-0.599774\pi$
$653$	−15.7586	−0.616681	−0.308341	+	0.951276i	$0.599774\pi$
$654$	8.38789	0.327992
$655$	5.61899	0.219552
$656$	17.6742	0.690061
$657$	15.7344	0.613859
$658$	−4.04059	−0.157518
$659$	−12.2181	−0.475950	−0.237975	−	0.971271i	$-0.576484\pi$
$659$	−12.2181	−0.475950	−0.237975	+	0.971271i	$0.576484\pi$
$660$	4.76203	0.185362
$661$	3.73443	0.145253	0.0726263	−	0.997359i	$-0.476862\pi$
$661$	3.73443	0.145253	0.0726263	+	0.997359i	$0.476862\pi$
$662$	14.8501	0.577165
$663$	−2.94137	−0.114233
$664$	29.0777	1.12844
$665$	0.778457	0.0301873
$666$	−2.29296	−0.0888506
$667$	30.4786	1.18014
$668$	29.4948	1.14119
$669$	22.8095	0.881866
$670$	−3.97078	−0.153405
$671$	4.49828	0.173654
$672$	−10.8793	−0.419678
$673$	−5.65775	−0.218090	−0.109045	−	0.994037i	$-0.534779\pi$
$673$	−5.65775	−0.218090	−0.109045	+	0.994037i	$0.534779\pi$
$674$	−3.96629	−0.152776
$675$	9.99312	0.384636
$676$	−1.77846	−0.0684022
$677$	9.39906	0.361235	0.180618	−	0.983553i	$-0.442190\pi$
$677$	9.39906	0.361235	0.180618	+	0.983553i	$0.442190\pi$
$678$	−10.4723	−0.402186
$679$	3.47068	0.133193
$680$	1.23109	0.0472103
$681$	−12.1104	−0.464071
$682$	7.43965	0.284879
$683$	−20.7328	−0.793319	−0.396660	−	0.917966i	$-0.629831\pi$
$683$	−20.7328	−0.793319	−0.396660	+	0.917966i	$0.629831\pi$
$684$	5.38445	0.205880
$685$	6.01461	0.229806
$686$	−0.470683	−0.0179708
$687$	7.47250	0.285094
$688$	30.9897	1.18147
$689$	11.2767	0.429610
$690$	3.27083	0.124518
$691$	16.0862	0.611949	0.305975	−	0.952040i	$-0.401018\pi$
$691$	16.0862	0.611949	0.305975	+	0.952040i	$0.401018\pi$
$692$	41.4396	1.57530
$693$	4.63016	0.175885
$694$	9.10695	0.345695
$695$	−7.36984	−0.279554
$696$	−20.8862	−0.791688
$697$	−8.49828	−0.321895
$698$	12.8286	0.485570
$699$	30.8578	1.16715
$700$	−8.39400	−0.317264
$701$	−6.98013	−0.263636	−0.131818	−	0.991274i	$-0.542081\pi$
$701$	−6.98013	−0.263636	−0.131818	+	0.991274i	$0.542081\pi$
$702$	−0.996562	−0.0376128
$703$	3.48024	0.131260
$704$	7.11383	0.268112
$705$	10.2199	0.384905
$706$	−12.0844	−0.454803
$707$	−7.75086	−0.291501
$708$	48.6898	1.82988
$709$	8.39239	0.315183	0.157591	−	0.987504i	$-0.449627\pi$
$709$	8.39239	0.315183	0.157591	+	0.987504i	$0.449627\pi$
$710$	0.296604	0.0111313
$711$	−2.75612	−0.103362
$712$	−12.2897	−0.460577
$713$	−41.0207	−1.53624
$714$	1.38445	0.0518118
$715$	−1.19051	−0.0445225
$716$	−37.4227	−1.39855
$717$	−7.87586	−0.294129
$718$	−11.0225	−0.411358
$719$	−5.16129	−0.192484	−0.0962418	−	0.995358i	$-0.530682\pi$
$719$	−5.16129	−0.192484	−0.0962418	+	0.995358i	$0.530682\pi$
$720$	2.96371	0.110451
$721$	16.9966	0.632985
$722$	−7.92494	−0.294936
$723$	3.57152	0.132826
$724$	−29.7846	−1.10693
$725$	−24.6448	−0.915284
$726$	−6.28973	−0.233434
$727$	−40.4362	−1.49970	−0.749848	−	0.661610i	$-0.769873\pi$
$727$	−40.4362	−1.49970	−0.749848	+	0.661610i	$0.769873\pi$
$728$	1.77846	0.0659140
$729$	−8.23109	−0.304855
$730$	1.90422	0.0704782
$731$	−14.9008	−0.551125
$732$	8.00000	0.295689
$733$	−39.1311	−1.44534	−0.722670	−	0.691193i	$-0.757085\pi$
$733$	−39.1311	−1.44534	−0.722670	+	0.691193i	$0.757085\pi$
$734$	−6.91215	−0.255132
$735$	1.19051	0.0439125
$736$	28.2345	1.04074
$737$	35.8466	1.32043
$738$	6.29660	0.231781
$739$	−7.13531	−0.262477	−0.131238	−	0.991351i	$-0.541895\pi$
$739$	−7.13531	−0.262477	−0.131238	+	0.991351i	$0.541895\pi$
$740$	2.22766	0.0818903
$741$	−3.30777	−0.121514
$742$	−5.30777	−0.194855
$743$	13.8827	0.509308	0.254654	−	0.967032i	$-0.418038\pi$
$743$	13.8827	0.509308	0.254654	+	0.967032i	$0.418038\pi$
$744$	28.1104	1.03058
$745$	4.92676	0.180502
$746$	−11.1398	−0.407857
$747$	−33.6586	−1.23150
$748$	−5.23109	−0.191268
$749$	5.55691	0.203045
$750$	−5.44652	−0.198879
$751$	−37.3251	−1.36201	−0.681005	−	0.732278i	$-0.738457\pi$
$751$	−37.3251	−1.36201	−0.681005	+	0.732278i	$0.738457\pi$
$752$	23.3484	0.851427
$753$	−11.0810	−0.403813
$754$	2.45769	0.0895039
$755$	3.74398	0.136258
$756$	−3.76547	−0.136949
$757$	−7.10428	−0.258209	−0.129105	−	0.991631i	$-0.541210\pi$
$757$	−7.10428	−0.258209	−0.129105	+	0.991631i	$0.541210\pi$
$758$	−15.4104	−0.559732
$759$	−29.5277	−1.07179
$760$	1.38445	0.0502194
$761$	−25.9621	−0.941125	−0.470562	−	0.882367i	$-0.655949\pi$
$761$	−25.9621	−0.941125	−0.470562	+	0.882367i	$0.655949\pi$
$762$	0.872420	0.0316044
$763$	−7.92332	−0.286843
$764$	13.2311	0.478684
$765$	−1.42504	−0.0515224
$766$	−10.6448	−0.384611
$767$	−12.1725	−0.439522
$768$	2.41023	0.0869717
$769$	−21.4638	−0.774005	−0.387002	−	0.922079i	$-0.626489\pi$
$769$	−21.4638	−0.774005	−0.387002	+	0.922079i	$0.626489\pi$
$770$	0.560352	0.0201937
$771$	−18.0260	−0.649190
$772$	2.67074	0.0961221
$773$	−40.4914	−1.45637	−0.728187	−	0.685378i	$-0.759637\pi$
$773$	−40.4914	−1.45637	−0.728187	+	0.685378i	$0.759637\pi$
$774$	11.0404	0.396838
$775$	33.1690	1.19147
$776$	6.17246	0.221578
$777$	5.32238	0.190939
$778$	17.9087	0.642058
$779$	−9.55691	−0.342412
$780$	−2.11727	−0.0758103
$781$	−2.67762	−0.0958127
$782$	−3.59301	−0.128486
$783$	−11.0554	−0.395088
$784$	2.71982	0.0971366
$785$	−3.19738	−0.114119
$786$	11.2380	0.400845
$787$	−5.02072	−0.178969	−0.0894847	−	0.995988i	$-0.528522\pi$
$787$	−5.02072	−0.178969	−0.0894847	+	0.995988i	$0.528522\pi$
$788$	−42.6707	−1.52008
$789$	3.60438	0.128319
$790$	−0.333552	−0.0118672
$791$	9.89229	0.351729
$792$	8.23453	0.292601
$793$	−2.00000	−0.0710221
$794$	−5.50945	−0.195523
$795$	13.4250	0.476137
$796$	3.58289	0.126992
$797$	19.7002	0.697815	0.348908	−	0.937157i	$-0.386553\pi$
$797$	19.7002	0.697815	0.348908	+	0.937157i	$0.386553\pi$
$798$	1.55691	0.0551142
$799$	−11.2266	−0.397169
$800$	−22.8302	−0.807170
$801$	14.2258	0.502645
$802$	1.67418	0.0591174
$803$	−17.1905	−0.606640
$804$	63.7517	2.24835
$805$	−3.08967	−0.108897
$806$	−3.30777	−0.116511
$807$	−26.5863	−0.935883
$808$	−13.7846	−0.484940
$809$	−2.57678	−0.0905947	−0.0452974	−	0.998974i	$-0.514424\pi$
$809$	−2.57678	−0.0905947	−0.0452974	+	0.998974i	$0.514424\pi$
$810$	−2.72508	−0.0957496
$811$	41.2311	1.44782	0.723910	−	0.689895i	$-0.242343\pi$
$811$	41.2311	1.44782	0.723910	+	0.689895i	$0.242343\pi$
$812$	9.28629	0.325885
$813$	49.2173	1.72613
$814$	2.50516	0.0878057
$815$	3.37758	0.118311
$816$	−8.00000	−0.280056
$817$	−16.7570	−0.586252
$818$	2.47680	0.0865992
$819$	−2.05863	−0.0719345
$820$	−6.11727	−0.213624
$821$	5.26719	0.183826	0.0919130	−	0.995767i	$-0.470702\pi$
$821$	5.26719	0.183826	0.0919130	+	0.995767i	$0.470702\pi$
$822$	12.0292	0.419567
$823$	36.0191	1.25555	0.627774	−	0.778396i	$-0.283966\pi$
$823$	36.0191	1.25555	0.627774	+	0.778396i	$0.283966\pi$
$824$	30.2277	1.05303
$825$	23.8759	0.831251
$826$	5.72938	0.199350
$827$	−16.3157	−0.567353	−0.283676	−	0.958920i	$-0.591554\pi$
$827$	−16.3157	−0.567353	−0.283676	+	0.958920i	$0.591554\pi$
$828$	−21.3707	−0.742683
$829$	−30.3956	−1.05568	−0.527842	−	0.849343i	$-0.676999\pi$
$829$	−30.3956	−1.05568	−0.527842	+	0.849343i	$0.676999\pi$
$830$	−4.07344	−0.141391
$831$	−22.9820	−0.797235
$832$	−3.16291	−0.109654
$833$	−1.30777	−0.0453117
$834$	−14.7397	−0.510394
$835$	−8.77846	−0.303791
$836$	−5.88273	−0.203459
$837$	14.8793	0.514304
$838$	12.2536	0.423295
$839$	29.8398	1.03018	0.515092	−	0.857135i	$-0.327758\pi$
$839$	29.8398	1.03018	0.515092	+	0.857135i	$0.327758\pi$
$840$	2.11727	0.0730526
$841$	−1.73549	−0.0598445
$842$	10.4691	0.360788
$843$	3.46886	0.119474
$844$	−17.9639	−0.618343
$845$	0.529317	0.0182090
$846$	8.31809	0.285982
$847$	5.94137	0.204148
$848$	30.6707	1.05324
$849$	35.6413	1.22321
$850$	2.90528	0.0996501
$851$	−13.8129	−0.473501
$852$	−4.76203	−0.163144
$853$	−0.203497	−0.00696761	−0.00348380	−	0.999994i	$-0.501109\pi$
$853$	−0.203497	−0.00696761	−0.00348380	+	0.999994i	$0.501109\pi$
$854$	0.941367	0.0322129
$855$	−1.60256	−0.0548063
$856$	9.88273	0.337785
$857$	−12.6155	−0.430939	−0.215469	−	0.976511i	$-0.569128\pi$
$857$	−12.6155	−0.430939	−0.215469	+	0.976511i	$0.569128\pi$
$858$	−2.38101	−0.0812865
$859$	−27.9671	−0.954227	−0.477113	−	0.878842i	$-0.658317\pi$
$859$	−27.9671	−0.954227	−0.477113	+	0.878842i	$0.658317\pi$
$860$	−10.7259	−0.365751
$861$	−14.6155	−0.498097
$862$	13.0258	0.443660
$863$	−2.76891	−0.0942546	−0.0471273	−	0.998889i	$-0.515007\pi$
$863$	−2.76891	−0.0942546	−0.0471273	+	0.998889i	$0.515007\pi$
$864$	−10.2414	−0.348420
$865$	−12.3336	−0.419353
$866$	−6.01910	−0.204537
$867$	−34.3887	−1.16790
$868$	−12.4983	−0.424219
$869$	3.01117	0.102147
$870$	2.92590	0.0991974
$871$	−15.9379	−0.540036
$872$	−14.0913	−0.477191
$873$	−7.14486	−0.241817
$874$	−4.04059	−0.136675
$875$	5.14486	0.173928
$876$	−30.5726	−1.03295
$877$	6.71133	0.226626	0.113313	−	0.993559i	$-0.463854\pi$
$877$	6.71133	0.226626	0.113313	+	0.993559i	$0.463854\pi$
$878$	−8.55024	−0.288557
$879$	24.9268	0.840759
$880$	−3.23797	−0.109152
$881$	−18.3741	−0.619040	−0.309520	−	0.950893i	$-0.600168\pi$
$881$	−18.3741	−0.619040	−0.309520	+	0.950893i	$0.600168\pi$
$882$	0.968964	0.0326267
$883$	9.93105	0.334207	0.167103	−	0.985939i	$-0.446559\pi$
$883$	9.93105	0.334207	0.167103	+	0.985939i	$0.446559\pi$
$884$	2.32582	0.0782258
$885$	−14.4914	−0.487123
$886$	0.0506994	0.00170328
$887$	−51.3776	−1.72509	−0.862545	−	0.505980i	$-0.831131\pi$
$887$	−51.3776	−1.72509	−0.862545	+	0.505980i	$0.831131\pi$
$888$	9.46563	0.317646
$889$	−0.824101	−0.0276394
$890$	1.72164	0.0577096
$891$	24.6009	0.824162
$892$	−18.0361	−0.603893
$893$	−12.6251	−0.422483
$894$	9.85352	0.329551
$895$	11.1380	0.372302
$896$	11.1629	0.372927
$897$	13.1284	0.438346
$898$	−10.4362	−0.348261
$899$	−36.6949	−1.22384
$900$	17.2802	0.576006
$901$	−14.7474	−0.491308
$902$	−6.87930	−0.229055
$903$	−25.6267	−0.852804
$904$	17.5930	0.585135
$905$	8.86469	0.294672
$906$	7.48797	0.248771
$907$	4.34225	0.144182	0.0720910	−	0.997398i	$-0.477033\pi$
$907$	4.34225	0.144182	0.0720910	+	0.997398i	$0.477033\pi$
$908$	9.57602	0.317791
$909$	15.9562	0.529233
$910$	−0.249141	−0.00825893
$911$	31.4853	1.04315	0.521577	−	0.853204i	$-0.325344\pi$
$911$	31.4853	1.04315	0.521577	+	0.853204i	$0.325344\pi$
$912$	−8.99656	−0.297906
$913$	36.7734	1.21702
$914$	2.05196	0.0678728
$915$	−2.38101	−0.0787139
$916$	−5.90871	−0.195229
$917$	−10.6155	−0.350556
$918$	1.30328	0.0430146
$919$	43.4588	1.43357	0.716786	−	0.697293i	$-0.245612\pi$
$919$	43.4588	1.43357	0.716786	+	0.697293i	$0.245612\pi$
$920$	−5.49484	−0.181160
$921$	45.9096	1.51277
$922$	15.2412	0.501942
$923$	1.19051	0.0391860
$924$	−8.99656	−0.295965
$925$	11.1690	0.367235
$926$	3.93793	0.129408
$927$	−34.9897	−1.14921
$928$	25.2571	0.829104
$929$	4.40517	0.144529	0.0722645	−	0.997385i	$-0.476977\pi$
$929$	4.40517	0.144529	0.0722645	+	0.997385i	$0.476977\pi$
$930$	−3.93793	−0.129130
$931$	−1.47068	−0.0481997
$932$	−24.4001	−0.799252
$933$	−4.32582	−0.141621
$934$	−9.19824	−0.300976
$935$	1.55691	0.0509165
$936$	−3.66119	−0.119670
$937$	−34.0990	−1.11397	−0.556983	−	0.830524i	$-0.688041\pi$
$937$	−34.0990	−1.11397	−0.556983	+	0.830524i	$0.688041\pi$
$938$	7.50172	0.244940
$939$	−32.3121	−1.05446
$940$	−8.08117	−0.263579
$941$	44.4672	1.44959	0.724795	−	0.688964i	$-0.241934\pi$
$941$	44.4672	1.44959	0.724795	+	0.688964i	$0.241934\pi$
$942$	−6.39477	−0.208353
$943$	37.9311	1.23521
$944$	−33.1070	−1.07754
$945$	1.12070	0.0364565
$946$	−12.0621	−0.392172
$947$	57.9311	1.88251	0.941253	−	0.337702i	$-0.109650\pi$
$947$	57.9311	1.88251	0.941253	+	0.337702i	$0.109650\pi$
$948$	5.35524	0.173930
$949$	7.64315	0.248107
$950$	3.26719	0.106002
$951$	34.9897	1.13462
$952$	−2.32582	−0.0753802
$953$	35.3060	1.14367	0.571836	−	0.820368i	$-0.306231\pi$
$953$	35.3060	1.14367	0.571836	+	0.820368i	$0.306231\pi$
$954$	10.9268	0.353767
$955$	−3.93793	−0.127428
$956$	6.22766	0.201417
$957$	−26.4139	−0.853839
$958$	−13.4250	−0.433743
$959$	−11.3630	−0.366929
$960$	−3.76547	−0.121530
$961$	18.3871	0.593133
$962$	−1.11383	−0.0359113
$963$	−11.4396	−0.368638
$964$	−2.82410	−0.0909582
$965$	−0.794885	−0.0255882
$966$	−6.17934	−0.198817
$967$	23.7148	0.762616	0.381308	−	0.924448i	$-0.375474\pi$
$967$	23.7148	0.762616	0.381308	+	0.924448i	$0.375474\pi$
$968$	10.5665	0.339619
$969$	4.32582	0.138965
$970$	−0.864688	−0.0277635
$971$	−23.9379	−0.768205	−0.384102	−	0.923291i	$-0.625489\pi$
$971$	−23.9379	−0.768205	−0.384102	+	0.923291i	$0.625489\pi$
$972$	32.4553	1.04100
$973$	13.9233	0.446361
$974$	11.6843	0.374389
$975$	−10.6155	−0.339970
$976$	−5.43965	−0.174119
$977$	−16.1871	−0.517870	−0.258935	−	0.965895i	$-0.583372\pi$
$977$	−16.1871	−0.517870	−0.258935	+	0.965895i	$0.583372\pi$
$978$	6.75515	0.216006
$979$	−15.5423	−0.496734
$980$	−0.941367	−0.0300709
$981$	16.3112	0.520777
$982$	13.7294	0.438122
$983$	45.7243	1.45838	0.729190	−	0.684312i	$-0.239897\pi$
$983$	45.7243	1.45838	0.729190	+	0.684312i	$0.239897\pi$
$984$	−25.9931	−0.828631
$985$	12.7000	0.404654
$986$	−3.21411	−0.102358
$987$	−19.3078	−0.614573
$988$	2.61555	0.0832116
$989$	66.5078	2.11483
$990$	−1.15356	−0.0366625
$991$	−5.40356	−0.171650	−0.0858248	−	0.996310i	$-0.527353\pi$
$991$	−5.40356	−0.171650	−0.0858248	+	0.996310i	$0.527353\pi$
$992$	−33.9931	−1.07928
$993$	70.9605	2.25186
$994$	−0.560352	−0.0177733
$995$	−1.06637	−0.0338061
$996$	65.3999	2.07228
$997$	6.04832	0.191552	0.0957761	−	0.995403i	$-0.469467\pi$
$997$	6.04832	0.191552	0.0957761	+	0.995403i	$0.469467\pi$
$998$	−15.6742	−0.496158
$999$	5.01031	0.158519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.2.a.d.1.2	✓	3
3.2	odd	2		819.2.a.i.1.2		3
4.3	odd	2		1456.2.a.t.1.1		3
5.4	even	2		2275.2.a.m.1.2		3
7.2	even	3		637.2.e.j.508.2		6
7.3	odd	6		637.2.e.i.79.2		6
7.4	even	3		637.2.e.j.79.2		6
7.5	odd	6		637.2.e.i.508.2		6
7.6	odd	2		637.2.a.j.1.2		3
8.3	odd	2		5824.2.a.bs.1.3		3
8.5	even	2		5824.2.a.by.1.1		3
13.5	odd	4		1183.2.c.f.337.3		6
13.8	odd	4		1183.2.c.f.337.4		6
13.12	even	2		1183.2.a.i.1.2		3
21.20	even	2		5733.2.a.x.1.2		3
91.90	odd	2		8281.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.2	✓	3	1.1	even	1	trivial
637.2.a.j.1.2		3	7.6	odd	2
637.2.e.i.79.2		6	7.3	odd	6
637.2.e.i.508.2		6	7.5	odd	6
637.2.e.j.79.2		6	7.4	even	3
637.2.e.j.508.2		6	7.2	even	3
819.2.a.i.1.2		3	3.2	odd	2
1183.2.a.i.1.2		3	13.12	even	2
1183.2.c.f.337.3		6	13.5	odd	4
1183.2.c.f.337.4		6	13.8	odd	4
1456.2.a.t.1.1		3	4.3	odd	2
2275.2.a.m.1.2		3	5.4	even	2
5733.2.a.x.1.2		3	21.20	even	2
5824.2.a.bs.1.3		3	8.3	odd	2
5824.2.a.by.1.1		3	8.5	even	2
8281.2.a.bg.1.2		3	91.90	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.a (trivial)

\(f(q)\)	\(=\)	\(q+0.470683 q^{2} +2.24914 q^{3} -1.77846 q^{4} +0.529317 q^{5} +1.05863 q^{6} -1.00000 q^{7} -1.77846 q^{8} +2.05863 q^{9} +O(q^{10})\)	\(q+0.470683 q^{2} +2.24914 q^{3} -1.77846 q^{4} +0.529317 q^{5} +1.05863 q^{6} -1.00000 q^{7} -1.77846 q^{8} +2.05863 q^{9} +0.249141 q^{10} -2.24914 q^{11} -4.00000 q^{12} +1.00000 q^{13} -0.470683 q^{14} +1.19051 q^{15} +2.71982 q^{16} -1.30777 q^{17} +0.968964 q^{18} -1.47068 q^{19} -0.941367 q^{20} -2.24914 q^{21} -1.05863 q^{22} +5.83709 q^{23} -4.00000 q^{24} -4.71982 q^{25} +0.470683 q^{26} -2.11727 q^{27} +1.77846 q^{28} +5.22154 q^{29} +0.560352 q^{30} -7.02760 q^{31} +4.83709 q^{32} -5.05863 q^{33} -0.615547 q^{34} -0.529317 q^{35} -3.66119 q^{36} -2.36641 q^{37} -0.692226 q^{38} +2.24914 q^{39} -0.941367 q^{40} +6.49828 q^{41} -1.05863 q^{42} +11.3940 q^{43} +4.00000 q^{44} +1.08967 q^{45} +2.74742 q^{46} +8.58451 q^{47} +6.11727 q^{48} +1.00000 q^{49} -2.22154 q^{50} -2.94137 q^{51} -1.77846 q^{52} +11.2767 q^{53} -0.996562 q^{54} -1.19051 q^{55} +1.77846 q^{56} -3.30777 q^{57} +2.45769 q^{58} -12.1725 q^{59} -2.11727 q^{60} -2.00000 q^{61} -3.30777 q^{62} -2.05863 q^{63} -3.16291 q^{64} +0.529317 q^{65} -2.38101 q^{66} -15.9379 q^{67} +2.32582 q^{68} +13.1284 q^{69} -0.249141 q^{70} +1.19051 q^{71} -3.66119 q^{72} +7.64315 q^{73} -1.11383 q^{74} -10.6155 q^{75} +2.61555 q^{76} +2.24914 q^{77} +1.05863 q^{78} -1.33881 q^{79} +1.43965 q^{80} -10.9379 q^{81} +3.05863 q^{82} -16.3500 q^{83} +4.00000 q^{84} -0.692226 q^{85} +5.36297 q^{86} +11.7440 q^{87} +4.00000 q^{88} +6.91033 q^{89} +0.512889 q^{90} -1.00000 q^{91} -10.3810 q^{92} -15.8061 q^{93} +4.04059 q^{94} -0.778457 q^{95} +10.8793 q^{96} -3.47068 q^{97} +0.470683 q^{98} -4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9	\( 3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9} - 8 q^{10} + 2 q^{11} - 12 q^{12} + 3 q^{13} - q^{14} - 6 q^{15} - q^{16} + 4 q^{17} - 15 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + q^{26} - 8 q^{27} - 3 q^{28} + 24 q^{29} + 20 q^{30} - 4 q^{31} + 7 q^{32} - 16 q^{33} + 14 q^{34} - 2 q^{35} - q^{36} - 10 q^{38} - 2 q^{39} - 2 q^{40} + 2 q^{41} - 4 q^{42} + 10 q^{43} + 12 q^{44} + 22 q^{45} - 18 q^{46} - 8 q^{47} + 20 q^{48} + 3 q^{49} - 15 q^{50} - 8 q^{51} + 3 q^{52} + 8 q^{53} + 32 q^{54} + 6 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 4 q^{59} - 8 q^{60} - 6 q^{61} - 2 q^{62} - 7 q^{63} - 17 q^{64} + 2 q^{65} + 12 q^{66} - 12 q^{67} + 22 q^{68} - 6 q^{69} + 8 q^{70} - 6 q^{71} - q^{72} - 10 q^{73} + 30 q^{74} - 16 q^{75} - 8 q^{76} - 2 q^{77} + 4 q^{78} - 14 q^{79} - 14 q^{80} + 3 q^{81} + 10 q^{82} - 12 q^{83} + 12 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 12 q^{88} + 2 q^{89} - 28 q^{90} - 3 q^{91} - 12 q^{92} - 22 q^{93} - 10 q^{94} + 6 q^{95} - 4 q^{96} - 10 q^{97} + q^{98} + 14 q^{99}+O(q^{100}) \) 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9 - 8 * q^10 + 2 * q^11 - 12 * q^12 + 3 * q^13 - q^14 - 6 * q^15 - q^16 + 4 * q^17 - 15 * q^18 - 4 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 + 10 * q^23 - 12 * q^24 - 5 * q^25 + q^26 - 8 * q^27 - 3 * q^28 + 24 * q^29 + 20 * q^30 - 4 * q^31 + 7 * q^32 - 16 * q^33 + 14 * q^34 - 2 * q^35 - q^36 - 10 * q^38 - 2 * q^39 - 2 * q^40 + 2 * q^41 - 4 * q^42 + 10 * q^43 + 12 * q^44 + 22 * q^45 - 18 * q^46 - 8 * q^47 + 20 * q^48 + 3 * q^49 - 15 * q^50 - 8 * q^51 + 3 * q^52 + 8 * q^53 + 32 * q^54 + 6 * q^55 - 3 * q^56 - 2 * q^57 + 12 * q^58 - 4 * q^59 - 8 * q^60 - 6 * q^61 - 2 * q^62 - 7 * q^63 - 17 * q^64 + 2 * q^65 + 12 * q^66 - 12 * q^67 + 22 * q^68 - 6 * q^69 + 8 * q^70 - 6 * q^71 - q^72 - 10 * q^73 + 30 * q^74 - 16 * q^75 - 8 * q^76 - 2 * q^77 + 4 * q^78 - 14 * q^79 - 14 * q^80 + 3 * q^81 + 10 * q^82 - 12 * q^83 + 12 * q^84 - 10 * q^85 - 26 * q^86 - 26 * q^87 + 12 * q^88 + 2 * q^89 - 28 * q^90 - 3 * q^91 - 12 * q^92 - 22 * q^93 - 10 * q^94 + 6 * q^95 - 4 * q^96 - 10 * q^97 + q^98 + 14 * q^99

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