LMFDB - Embedded newform 578.2.b.f.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [578,2,Mod(577,578)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("578.577"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 578 = 2 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	578.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,6,0,6,0,0,0,6,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$4.61535323683$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.419904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 9x^{2} + 1 $ x^6 + 6x^4 + 9x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.87939i$ of defining polynomial
Character	$\chi$	$=$	578.577
Dual form			578.2.b.f.577.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -3.22668i q^{3} +1.00000 q^{4} -1.18479i q^{5} -3.22668i q^{6} +1.12061i q^{7} +1.00000 q^{8} -7.41147 q^{9} -1.18479i q^{10} -3.41147i q^{11} -3.22668i q^{12} +0.347296 q^{13} +1.12061i q^{14} -3.82295 q^{15} +1.00000 q^{16} -7.41147 q^{18} -0.347296 q^{19} -1.18479i q^{20} +3.61587 q^{21} -3.41147i q^{22} -0.411474i q^{23} -3.22668i q^{24} +3.59627 q^{25} +0.347296 q^{26} +14.2344i q^{27} +1.12061i q^{28} +8.78106i q^{29} -3.82295 q^{30} -8.71688i q^{31} +1.00000 q^{32} -11.0077 q^{33} +1.32770 q^{35} -7.41147 q^{36} +0.475652i q^{37} -0.347296 q^{38} -1.12061i q^{39} -1.18479i q^{40} +2.63816i q^{41} +3.61587 q^{42} +9.33275 q^{43} -3.41147i q^{44} +8.78106i q^{45} -0.411474i q^{46} -7.86484 q^{47} -3.22668i q^{48} +5.74422 q^{49} +3.59627 q^{50} +0.347296 q^{52} -8.41921 q^{53} +14.2344i q^{54} -4.04189 q^{55} +1.12061i q^{56} +1.12061i q^{57} +8.78106i q^{58} +6.41147 q^{59} -3.82295 q^{60} -5.70233i q^{61} -8.71688i q^{62} -8.30541i q^{63} +1.00000 q^{64} -0.411474i q^{65} -11.0077 q^{66} +7.31315 q^{67} -1.32770 q^{69} +1.32770 q^{70} -7.59627i q^{71} -7.41147 q^{72} -9.04189i q^{73} +0.475652i q^{74} -11.6040i q^{75} -0.347296 q^{76} +3.82295 q^{77} -1.12061i q^{78} +13.2763i q^{79} -1.18479i q^{80} +23.6955 q^{81} +2.63816i q^{82} +7.73917 q^{83} +3.61587 q^{84} +9.33275 q^{86} +28.3337 q^{87} -3.41147i q^{88} +7.18479 q^{89} +8.78106i q^{90} +0.389185i q^{91} -0.411474i q^{92} -28.1266 q^{93} -7.86484 q^{94} +0.411474i q^{95} -3.22668i q^{96} -0.221629i q^{97} +5.74422 q^{98} +25.2841i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} - 24 q^{9} + 18 q^{15} + 6 q^{16} - 24 q^{18} - 6 q^{25} + 18 q^{30} + 6 q^{32} - 18 q^{33} - 24 q^{36} + 18 q^{43} - 24 q^{49} - 6 q^{50} + 18 q^{53} - 18 q^{55} + 18 q^{59}+ \cdots - 24 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 - 24 * q^9 + 18 * q^15 + 6 * q^16 - 24 * q^18 - 6 * q^25 + 18 * q^30 + 6 * q^32 - 18 * q^33 - 24 * q^36 + 18 * q^43 - 24 * q^49 - 6 * q^50 + 18 * q^53 - 18 * q^55 + 18 * q^59 + 18 * q^60 + 6 * q^64 - 18 * q^66 - 24 * q^72 - 18 * q^77 + 6 * q^81 + 18 * q^83 + 18 * q^86 + 36 * q^89 - 18 * q^93 - 24 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/578\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	− 3.22668i	− 1.86293i	−0.363837	−	0.931463i	$-0.618533\pi$
$3$	− 3.22668i	− 1.86293i	0.363837	−	0.931463i	$-0.381467\pi$
$4$	1.00000	0.500000
$5$	− 1.18479i	− 0.529855i	−0.964268	−	0.264928i	$-0.914652\pi$
$5$	− 1.18479i	− 0.529855i	0.964268	−	0.264928i	$-0.0853481\pi$
$6$	− 3.22668i	− 1.31729i
$7$	1.12061i	0.423553i	0.977318	+	0.211776i	$0.0679248\pi$
$7$	1.12061i	0.423553i	−0.977318	+	0.211776i	$0.932075\pi$
$8$	1.00000	0.353553
$9$	−7.41147	−2.47049
$10$	− 1.18479i	− 0.374664i
$11$	− 3.41147i	− 1.02860i	−0.857611	−	0.514299i	$-0.828052\pi$
$11$	− 3.41147i	− 1.02860i	0.857611	−	0.514299i	$-0.171948\pi$
$12$	− 3.22668i	− 0.931463i
$13$	0.347296	0.0963227	0.0481613	−	0.998840i	$-0.484664\pi$
$13$	0.347296	0.0963227	0.0481613	+	0.998840i	$0.484664\pi$
$14$	1.12061i	0.299497i
$15$	−3.82295	−0.987081
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−7.41147	−1.74690
$19$	−0.347296	−0.0796752	−0.0398376	−	0.999206i	$-0.512684\pi$
$19$	−0.347296	−0.0796752	−0.0398376	+	0.999206i	$0.512684\pi$
$20$	− 1.18479i	− 0.264928i
$21$	3.61587	0.789047
$22$	− 3.41147i	− 0.727329i
$23$	− 0.411474i	− 0.0857983i	−0.999079	−	0.0428991i	$-0.986341\pi$
$23$	− 0.411474i	− 0.0857983i	0.999079	−	0.0428991i	$-0.0136594\pi$
$24$	− 3.22668i	− 0.658644i
$25$	3.59627	0.719253
$26$	0.347296	0.0681104
$27$	14.2344i	2.73942i
$28$	1.12061i	0.211776i
$29$	8.78106i	1.63060i	0.579038	+	0.815301i	$0.303428\pi$
$29$	8.78106i	1.63060i	−0.579038	+	0.815301i	$0.696572\pi$
$30$	−3.82295	−0.697972
$31$	− 8.71688i	− 1.56560i	−0.622275	−	0.782799i	$-0.713791\pi$
$31$	− 8.71688i	− 1.56560i	0.622275	−	0.782799i	$-0.286209\pi$
$32$	1.00000	0.176777
$33$	−11.0077	−1.91620
$34$	0	0
$35$	1.32770	0.224422
$36$	−7.41147	−1.23525
$37$	0.475652i	0.0781967i	0.999235	+	0.0390983i	$0.0124486\pi$
$37$	0.475652i	0.0781967i	−0.999235	+	0.0390983i	$0.987551\pi$
$38$	−0.347296	−0.0563389
$39$	− 1.12061i	− 0.179442i
$40$	− 1.18479i	− 0.187332i
$41$	2.63816i	0.412011i	0.978551	+	0.206005i	$0.0660464\pi$
$41$	2.63816i	0.412011i	−0.978551	+	0.206005i	$0.933954\pi$
$42$	3.61587	0.557940
$43$	9.33275	1.42323	0.711615	−	0.702569i	$-0.247964\pi$
$43$	9.33275	1.42323	0.711615	+	0.702569i	$0.247964\pi$
$44$	− 3.41147i	− 0.514299i
$45$	8.78106i	1.30900i
$46$	− 0.411474i	− 0.0606686i
$47$	−7.86484	−1.14720	−0.573602	−	0.819134i	$-0.694454\pi$
$47$	−7.86484	−1.14720	−0.573602	+	0.819134i	$0.694454\pi$
$48$	− 3.22668i	− 0.465731i
$49$	5.74422	0.820603
$50$	3.59627	0.508589
$51$	0	0
$52$	0.347296	0.0481613
$53$	−8.41921	−1.15647	−0.578234	−	0.815871i	$-0.696258\pi$
$53$	−8.41921	−1.15647	−0.578234	+	0.815871i	$0.696258\pi$
$54$	14.2344i	1.93706i
$55$	−4.04189	−0.545008
$56$	1.12061i	0.149748i
$57$	1.12061i	0.148429i
$58$	8.78106i	1.15301i
$59$	6.41147	0.834703	0.417351	−	0.908745i	$-0.362958\pi$
$59$	6.41147	0.834703	0.417351	+	0.908745i	$0.362958\pi$
$60$	−3.82295	−0.493540
$61$	− 5.70233i	− 0.730109i	−0.930986	−	0.365054i	$-0.881050\pi$
$61$	− 5.70233i	− 0.730109i	0.930986	−	0.365054i	$-0.118950\pi$
$62$	− 8.71688i	− 1.10705i
$63$	− 8.30541i	− 1.04638i
$64$	1.00000	0.125000
$65$	− 0.411474i	− 0.0510371i
$66$	−11.0077	−1.35496
$67$	7.31315	0.893443	0.446722	−	0.894673i	$-0.352591\pi$
$67$	7.31315	0.893443	0.446722	+	0.894673i	$0.352591\pi$
$68$	0	0
$69$	−1.32770	−0.159836
$70$	1.32770	0.158690
$71$	− 7.59627i	− 0.901511i	−0.892647	−	0.450755i	$-0.851155\pi$
$71$	− 7.59627i	− 0.901511i	0.892647	−	0.450755i	$-0.148845\pi$
$72$	−7.41147	−0.873451
$73$	− 9.04189i	− 1.05827i	−0.848537	−	0.529137i	$-0.822516\pi$
$73$	− 9.04189i	− 1.05827i	0.848537	−	0.529137i	$-0.177484\pi$
$74$	0.475652i	0.0552934i
$75$	− 11.6040i	− 1.33992i
$76$	−0.347296	−0.0398376
$77$	3.82295	0.435665
$78$	− 1.12061i	− 0.126885i
$79$	13.2763i	1.49370i	0.664992	+	0.746851i	$0.268435\pi$
$79$	13.2763i	1.49370i	−0.664992	+	0.746851i	$0.731565\pi$
$80$	− 1.18479i	− 0.132464i
$81$	23.6955	2.63284
$82$	2.63816i	0.291336i
$83$	7.73917	0.849484	0.424742	−	0.905314i	$-0.360365\pi$
$83$	7.73917	0.849484	0.424742	+	0.905314i	$0.360365\pi$
$84$	3.61587	0.394523
$85$	0	0
$86$	9.33275	1.00638
$87$	28.3337	3.03769
$88$	− 3.41147i	− 0.363664i
$89$	7.18479	0.761586	0.380793	−	0.924660i	$-0.375651\pi$
$89$	7.18479	0.761586	0.380793	+	0.924660i	$0.375651\pi$
$90$	8.78106i	0.925605i
$91$	0.389185i	0.0407977i
$92$	− 0.411474i	− 0.0428991i
$93$	−28.1266	−2.91659
$94$	−7.86484	−0.811196
$95$	0.411474i	0.0422164i
$96$	− 3.22668i	− 0.329322i
$97$	− 0.221629i	− 0.0225030i	−0.999937	−	0.0112515i	$-0.996418\pi$
$97$	− 0.221629i	− 0.0225030i	0.999937	−	0.0112515i	$-0.00358154\pi$
$98$	5.74422	0.580254
$99$	25.2841i	2.54114i
$100$	3.59627	0.359627
$101$	−4.86484	−0.484069	−0.242035	−	0.970268i	$-0.577815\pi$
$101$	−4.86484	−0.484069	−0.242035	+	0.970268i	$0.577815\pi$
$102$	0	0
$103$	12.3969	1.22151	0.610753	−	0.791821i	$-0.290867\pi$
$103$	12.3969	1.22151	0.610753	+	0.791821i	$0.290867\pi$
$104$	0.347296	0.0340552
$105$	− 4.28405i	− 0.418081i
$106$	−8.41921	−0.817746
$107$	8.85710i	0.856248i	0.903720	+	0.428124i	$0.140825\pi$
$107$	8.85710i	0.856248i	−0.903720	+	0.428124i	$0.859175\pi$
$108$	14.2344i	1.36971i
$109$	11.8726i	1.13719i	0.822619	+	0.568593i	$0.192512\pi$
$109$	11.8726i	1.13719i	−0.822619	+	0.568593i	$0.807488\pi$
$110$	−4.04189	−0.385379
$111$	1.53478	0.145675
$112$	1.12061i	0.105888i
$113$	− 10.8648i	− 1.02208i	−0.859558	−	0.511039i	$-0.829261\pi$
$113$	− 10.8648i	− 1.02208i	0.859558	−	0.511039i	$-0.170739\pi$
$114$	1.12061i	0.104955i
$115$	−0.487511	−0.0454607
$116$	8.78106i	0.815301i
$117$	−2.57398	−0.237964
$118$	6.41147	0.590224
$119$	0	0
$120$	−3.82295	−0.348986
$121$	−0.638156	−0.0580142
$122$	− 5.70233i	− 0.516265i
$123$	8.51249	0.767545
$124$	− 8.71688i	− 0.782799i
$125$	− 10.1848i	− 0.910956i
$126$	− 8.30541i	− 0.739904i
$127$	6.55169	0.581368	0.290684	−	0.956819i	$-0.406117\pi$
$127$	6.55169	0.581368	0.290684	+	0.956819i	$0.406117\pi$
$128$	1.00000	0.0883883
$129$	− 30.1138i	− 2.65137i
$130$	− 0.411474i	− 0.0360887i
$131$	− 0.0496299i	− 0.00433618i	−0.999998	−	0.00216809i	$-0.999310\pi$
$131$	− 0.0496299i	− 0.00433618i	0.999998	−	0.00216809i	$-0.000690125\pi$
$132$	−11.0077	−0.958101
$133$	− 0.389185i	− 0.0337467i
$134$	7.31315	0.631760
$135$	16.8648	1.45149
$136$	0	0
$137$	−8.13341	−0.694884	−0.347442	−	0.937701i	$-0.612950\pi$
$137$	−8.13341	−0.694884	−0.347442	+	0.937701i	$0.612950\pi$
$138$	−1.32770	−0.113021
$139$	15.3824i	1.30472i	0.757911	+	0.652358i	$0.226220\pi$
$139$	15.3824i	1.30472i	−0.757911	+	0.652358i	$0.773780\pi$
$140$	1.32770	0.112211
$141$	25.3773i	2.13716i
$142$	− 7.59627i	− 0.637465i
$143$	− 1.18479i	− 0.0990773i
$144$	−7.41147	−0.617623
$145$	10.4037	0.863983
$146$	− 9.04189i	− 0.748312i
$147$	− 18.5348i	− 1.52872i
$148$	0.475652i	0.0390983i
$149$	−16.1088	−1.31968	−0.659840	−	0.751406i	$-0.729376\pi$
$149$	−16.1088	−1.31968	−0.659840	+	0.751406i	$0.729376\pi$
$150$	− 11.6040i	− 0.947463i
$151$	2.70233	0.219913	0.109956	−	0.993936i	$-0.464929\pi$
$151$	2.70233	0.219913	0.109956	+	0.993936i	$0.464929\pi$
$152$	−0.347296	−0.0281695
$153$	0	0
$154$	3.82295	0.308062
$155$	−10.3277	−0.829541
$156$	− 1.12061i	− 0.0897210i
$157$	−23.5594	−1.88025	−0.940124	−	0.340834i	$-0.889291\pi$
$157$	−23.5594	−1.88025	−0.940124	+	0.340834i	$0.889291\pi$
$158$	13.2763i	1.05621i
$159$	27.1661i	2.15441i
$160$	− 1.18479i	− 0.0936661i
$161$	0.461104	0.0363401
$162$	23.6955	1.86170
$163$	1.87433i	0.146809i	0.997302	+	0.0734045i	$0.0233864\pi$
$163$	1.87433i	0.146809i	−0.997302	+	0.0734045i	$0.976614\pi$
$164$	2.63816i	0.206005i
$165$	13.0419i	1.01531i
$166$	7.73917	0.600676
$167$	− 8.85710i	− 0.685383i	−0.939448	−	0.342691i	$-0.888661\pi$
$167$	− 8.85710i	− 0.685383i	0.939448	−	0.342691i	$-0.111339\pi$
$168$	3.61587	0.278970
$169$	−12.8794	−0.990722
$170$	0	0
$171$	2.57398	0.196837
$172$	9.33275	0.711615
$173$	6.41147i	0.487455i	0.969844	+	0.243728i	$0.0783703\pi$
$173$	6.41147i	0.487455i	−0.969844	+	0.243728i	$0.921630\pi$
$174$	28.3337	2.14797
$175$	4.03003i	0.304642i
$176$	− 3.41147i	− 0.257150i
$177$	− 20.6878i	− 1.55499i
$178$	7.18479	0.538523
$179$	1.26083	0.0942388	0.0471194	−	0.998889i	$-0.484996\pi$
$179$	1.26083	0.0942388	0.0471194	+	0.998889i	$0.484996\pi$
$180$	8.78106i	0.654502i
$181$	15.3209i	1.13879i	0.822063	+	0.569396i	$0.192823\pi$
$181$	15.3209i	1.13879i	−0.822063	+	0.569396i	$0.807177\pi$
$182$	0.389185i	0.0288483i
$183$	−18.3996	−1.36014
$184$	− 0.411474i	− 0.0303343i
$185$	0.563549	0.0414329
$186$	−28.1266	−2.06234
$187$	0	0
$188$	−7.86484	−0.573602
$189$	−15.9513	−1.16029
$190$	0.411474i	0.0298515i
$191$	10.3601	0.749630	0.374815	−	0.927100i	$-0.377706\pi$
$191$	10.3601	0.749630	0.374815	+	0.927100i	$0.377706\pi$
$192$	− 3.22668i	− 0.232866i
$193$	15.2540i	1.09801i	0.835820	+	0.549004i	$0.184993\pi$
$193$	15.2540i	1.09801i	−0.835820	+	0.549004i	$0.815007\pi$
$194$	− 0.221629i	− 0.0159120i
$195$	−1.32770	−0.0950783
$196$	5.74422	0.410302
$197$	21.2422i	1.51344i	0.653738	+	0.756721i	$0.273200\pi$
$197$	21.2422i	1.51344i	−0.653738	+	0.756721i	$0.726800\pi$
$198$	25.2841i	1.79686i
$199$	− 13.0838i	− 0.927484i	−0.885970	−	0.463742i	$-0.846506\pi$
$199$	− 13.0838i	− 0.927484i	0.885970	−	0.463742i	$-0.153494\pi$
$200$	3.59627	0.254294
$201$	− 23.5972i	− 1.66442i
$202$	−4.86484	−0.342289
$203$	−9.84018	−0.690646
$204$	0	0
$205$	3.12567	0.218306
$206$	12.3969	0.863735
$207$	3.04963i	0.211964i
$208$	0.347296	0.0240807
$209$	1.18479i	0.0819538i
$210$	− 4.28405i	− 0.295628i
$211$	9.95130i	0.685076i	0.939504	+	0.342538i	$0.111287\pi$
$211$	9.95130i	0.685076i	−0.939504	+	0.342538i	$0.888713\pi$
$212$	−8.41921	−0.578234
$213$	−24.5107	−1.67945
$214$	8.85710i	0.605459i
$215$	− 11.0574i	− 0.754106i
$216$	14.2344i	0.968530i
$217$	9.76827	0.663113
$218$	11.8726i	0.804112i
$219$	−29.1753	−1.97148
$220$	−4.04189	−0.272504
$221$	0	0
$222$	1.53478	0.103008
$223$	−19.7638	−1.32348	−0.661742	−	0.749732i	$-0.730182\pi$
$223$	−19.7638	−1.32348	−0.661742	+	0.749732i	$0.730182\pi$
$224$	1.12061i	0.0748742i
$225$	−26.6536	−1.77691
$226$	− 10.8648i	− 0.722718i
$227$	3.55438i	0.235912i	0.993019	+	0.117956i	$0.0376342\pi$
$227$	3.55438i	0.235912i	−0.993019	+	0.117956i	$0.962366\pi$
$228$	1.12061i	0.0742145i
$229$	2.04189	0.134932	0.0674659	−	0.997722i	$-0.478509\pi$
$229$	2.04189	0.134932	0.0674659	+	0.997722i	$0.478509\pi$
$230$	−0.487511	−0.0321456
$231$	− 12.3354i	− 0.811612i
$232$	8.78106i	0.576505i
$233$	3.91622i	0.256560i	0.991738	+	0.128280i	$0.0409457\pi$
$233$	3.91622i	0.256560i	−0.991738	+	0.128280i	$0.959054\pi$
$234$	−2.57398	−0.168266
$235$	9.31820i	0.607852i
$236$	6.41147	0.417351
$237$	42.8384	2.78266
$238$	0	0
$239$	−21.5544	−1.39424	−0.697118	−	0.716956i	$-0.745535\pi$
$239$	−21.5544	−1.39424	−0.697118	+	0.716956i	$0.745535\pi$
$240$	−3.82295	−0.246770
$241$	25.1925i	1.62279i	0.584496	+	0.811397i	$0.301292\pi$
$241$	25.1925i	1.62279i	−0.584496	+	0.811397i	$0.698708\pi$
$242$	−0.638156	−0.0410222
$243$	− 33.7547i	− 2.16536i
$244$	− 5.70233i	− 0.365054i
$245$	− 6.80571i	− 0.434801i
$246$	8.51249	0.542736
$247$	−0.120615	−0.00767453
$248$	− 8.71688i	− 0.553523i
$249$	− 24.9718i	− 1.58253i
$250$	− 10.1848i	− 0.644143i
$251$	3.07604	0.194158	0.0970789	−	0.995277i	$-0.469050\pi$
$251$	3.07604	0.194158	0.0970789	+	0.995277i	$0.469050\pi$
$252$	− 8.30541i	− 0.523191i
$253$	−1.40373	−0.0882520
$254$	6.55169	0.411090
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.9394	1.55568	0.777840	−	0.628462i	$-0.216315\pi$
$257$	24.9394	1.55568	0.777840	+	0.628462i	$0.216315\pi$
$258$	− 30.1138i	− 1.87480i
$259$	−0.533023	−0.0331204
$260$	− 0.411474i	− 0.0255185i
$261$	− 65.0806i	− 4.02839i
$262$	− 0.0496299i	− 0.00306614i
$263$	15.2422	0.939872	0.469936	−	0.882700i	$-0.344277\pi$
$263$	15.2422	0.939872	0.469936	+	0.882700i	$0.344277\pi$
$264$	−11.0077	−0.677480
$265$	9.97502i	0.612761i
$266$	− 0.389185i	− 0.0238625i
$267$	− 23.1830i	− 1.41878i
$268$	7.31315	0.446722
$269$	3.77332i	0.230063i	0.993362	+	0.115032i	$0.0366969\pi$
$269$	3.77332i	0.230063i	−0.993362	+	0.115032i	$0.963303\pi$
$270$	16.8648	1.02636
$271$	4.72462	0.287000	0.143500	−	0.989650i	$-0.454164\pi$
$271$	4.72462	0.287000	0.143500	+	0.989650i	$0.454164\pi$
$272$	0	0
$273$	1.25578	0.0760031
$274$	−8.13341	−0.491357
$275$	− 12.2686i	− 0.739823i
$276$	−1.32770	−0.0799179
$277$	− 18.2276i	− 1.09519i	−0.836743	−	0.547596i	$-0.815543\pi$
$277$	− 18.2276i	− 1.09519i	0.836743	−	0.547596i	$-0.184457\pi$
$278$	15.3824i	0.922574i
$279$	64.6049i	3.86780i
$280$	1.32770	0.0793450
$281$	−25.3773	−1.51388	−0.756942	−	0.653482i	$-0.773308\pi$
$281$	−25.3773	−1.51388	−0.756942	+	0.653482i	$0.773308\pi$
$282$	25.3773i	1.51120i
$283$	13.0446i	0.775420i	0.921781	+	0.387710i	$0.126734\pi$
$283$	13.0446i	0.775420i	−0.921781	+	0.387710i	$0.873266\pi$
$284$	− 7.59627i	− 0.450755i
$285$	1.32770	0.0786459
$286$	− 1.18479i	− 0.0700583i
$287$	−2.95636	−0.174508
$288$	−7.41147	−0.436725
$289$	0	0
$290$	10.4037	0.610928
$291$	−0.715127	−0.0419215
$292$	− 9.04189i	− 0.529137i
$293$	3.80571	0.222332	0.111166	−	0.993802i	$-0.464541\pi$
$293$	3.80571	0.222332	0.111166	+	0.993802i	$0.464541\pi$
$294$	− 18.5348i	− 1.08097i
$295$	− 7.59627i	− 0.442272i
$296$	0.475652i	0.0276467i
$297$	48.5604	2.81776
$298$	−16.1088	−0.933155
$299$	− 0.142903i	− 0.00826432i
$300$	− 11.6040i	− 0.669958i
$301$	10.4584i	0.602813i
$302$	2.70233	0.155502
$303$	15.6973i	0.901785i
$304$	−0.347296	−0.0199188
$305$	−6.75608	−0.386852
$306$	0	0
$307$	13.1070	0.748056	0.374028	−	0.927417i	$-0.377976\pi$
$307$	13.1070	0.748056	0.374028	+	0.927417i	$0.377976\pi$
$308$	3.82295	0.217833
$309$	− 40.0009i	− 2.27557i
$310$	−10.3277	−0.586574
$311$	− 10.3277i	− 0.585630i	−0.956169	−	0.292815i	$-0.905408\pi$
$311$	− 10.3277i	− 0.585630i	0.956169	−	0.292815i	$-0.0945920\pi$
$312$	− 1.12061i	− 0.0634423i
$313$	− 21.7178i	− 1.22756i	−0.789476	−	0.613782i	$-0.789647\pi$
$313$	− 21.7178i	− 1.22756i	0.789476	−	0.613782i	$-0.210353\pi$
$314$	−23.5594	−1.32954
$315$	−9.84018	−0.554432
$316$	13.2763i	0.746851i
$317$	11.1830i	0.628102i	0.949406	+	0.314051i	$0.101686\pi$
$317$	11.1830i	0.628102i	−0.949406	+	0.314051i	$0.898314\pi$
$318$	27.1661i	1.52340i
$319$	29.9564	1.67723
$320$	− 1.18479i	− 0.0662319i
$321$	28.5790	1.59513
$322$	0.461104	0.0256963
$323$	0	0
$324$	23.6955	1.31642
$325$	1.24897	0.0692804
$326$	1.87433i	0.103810i
$327$	38.3090	2.11849
$328$	2.63816i	0.145668i
$329$	− 8.81345i	− 0.485901i
$330$	13.0419i	0.717932i
$331$	−18.0847	−0.994026	−0.497013	−	0.867743i	$-0.665570\pi$
$331$	−18.0847	−0.994026	−0.497013	+	0.867743i	$0.665570\pi$
$332$	7.73917	0.424742
$333$	− 3.52528i	− 0.193184i
$334$	− 8.85710i	− 0.484639i
$335$	− 8.66456i	− 0.473396i
$336$	3.61587	0.197262
$337$	1.68004i	0.0915179i	0.998953	+	0.0457589i	$0.0145706\pi$
$337$	1.68004i	0.0915179i	−0.998953	+	0.0457589i	$0.985429\pi$
$338$	−12.8794	−0.700546
$339$	−35.0574	−1.90406
$340$	0	0
$341$	−29.7374	−1.61037
$342$	2.57398	0.139185
$343$	14.2814i	0.771121i
$344$	9.33275	0.503188
$345$	1.57304i	0.0846899i
$346$	6.41147i	0.344683i
$347$	− 22.4766i	− 1.20661i	−0.797512	−	0.603303i	$-0.793851\pi$
$347$	− 22.4766i	− 1.20661i	0.797512	−	0.603303i	$-0.206149\pi$
$348$	28.3337	1.51884
$349$	21.1070	1.12983	0.564916	−	0.825148i	$-0.308909\pi$
$349$	21.1070	1.12983	0.564916	+	0.825148i	$0.308909\pi$
$350$	4.03003i	0.215414i
$351$	4.94356i	0.263868i
$352$	− 3.41147i	− 0.181832i
$353$	3.86659	0.205798	0.102899	−	0.994692i	$-0.467188\pi$
$353$	3.86659	0.205798	0.102899	+	0.994692i	$0.467188\pi$
$354$	− 20.6878i	− 1.09954i
$355$	−9.00000	−0.477670
$356$	7.18479	0.380793
$357$	0	0
$358$	1.26083	0.0666369
$359$	−36.4347	−1.92295	−0.961475	−	0.274893i	$-0.911358\pi$
$359$	−36.4347	−1.92295	−0.961475	+	0.274893i	$0.911358\pi$
$360$	8.78106i	0.462802i
$361$	−18.8794	−0.993652
$362$	15.3209i	0.805248i
$363$	2.05913i	0.108076i
$364$	0.389185i	0.0203989i
$365$	−10.7128	−0.560732
$366$	−18.3996	−0.961763
$367$	13.7570i	0.718110i	0.933316	+	0.359055i	$0.116901\pi$
$367$	13.7570i	0.718110i	−0.933316	+	0.359055i	$0.883099\pi$
$368$	− 0.411474i	− 0.0214496i
$369$	− 19.5526i	− 1.01787i
$370$	0.563549	0.0292975
$371$	− 9.43470i	− 0.489825i
$372$	−28.1266	−1.45830
$373$	22.7638	1.17867	0.589333	−	0.807890i	$-0.299391\pi$
$373$	22.7638	1.17867	0.589333	+	0.807890i	$0.299391\pi$
$374$	0	0
$375$	−32.8631	−1.69704
$376$	−7.86484	−0.405598
$377$	3.04963i	0.157064i
$378$	−15.9513	−0.820447
$379$	− 8.91859i	− 0.458117i	−0.973413	−	0.229058i	$-0.926435\pi$
$379$	− 8.91859i	− 0.458117i	0.973413	−	0.229058i	$-0.0735647\pi$
$380$	0.411474i	0.0211082i
$381$	− 21.1402i	− 1.08305i
$382$	10.3601	0.530068
$383$	−19.4097	−0.991790	−0.495895	−	0.868382i	$-0.665160\pi$
$383$	−19.4097	−0.991790	−0.495895	+	0.868382i	$0.665160\pi$
$384$	− 3.22668i	− 0.164661i
$385$	− 4.52940i	− 0.230840i
$386$	15.2540i	0.776409i
$387$	−69.1694	−3.51608
$388$	− 0.221629i	− 0.0112515i
$389$	−35.6614	−1.80810	−0.904052	−	0.427423i	$-0.859422\pi$
$389$	−35.6614	−1.80810	−0.904052	+	0.427423i	$0.859422\pi$
$390$	−1.32770	−0.0672305
$391$	0	0
$392$	5.74422	0.290127
$393$	−0.160140	−0.00807798
$394$	21.2422i	1.07016i
$395$	15.7297	0.791446
$396$	25.2841i	1.27057i
$397$	7.44562i	0.373685i	0.982390	+	0.186843i	$0.0598254\pi$
$397$	7.44562i	0.373685i	−0.982390	+	0.186843i	$0.940175\pi$
$398$	− 13.0838i	− 0.655831i
$399$	−1.25578	−0.0628675
$400$	3.59627	0.179813
$401$	2.36959i	0.118331i	0.998248	+	0.0591657i	$0.0188440\pi$
$401$	2.36959i	0.118331i	−0.998248	+	0.0591657i	$0.981156\pi$
$402$	− 23.5972i	− 1.17692i
$403$	− 3.02734i	− 0.150803i
$404$	−4.86484	−0.242035
$405$	− 28.0743i	− 1.39502i
$406$	−9.84018	−0.488360
$407$	1.62267	0.0804330
$408$	0	0
$409$	−28.8357	−1.42584	−0.712918	−	0.701248i	$-0.752627\pi$
$409$	−28.8357	−1.42584	−0.712918	+	0.701248i	$0.752627\pi$
$410$	3.12567	0.154366
$411$	26.2439i	1.29452i
$412$	12.3969	0.610753
$413$	7.18479i	0.353541i
$414$	3.04963i	0.149881i
$415$	− 9.16931i	− 0.450104i
$416$	0.347296	0.0170276
$417$	49.6340	2.43059
$418$	1.18479i	0.0579501i
$419$	30.6800i	1.49882i	0.662107	+	0.749409i	$0.269662\pi$
$419$	30.6800i	1.49882i	−0.662107	+	0.749409i	$0.730338\pi$
$420$	− 4.28405i	− 0.209040i
$421$	−6.20708	−0.302515	−0.151257	−	0.988494i	$-0.548332\pi$
$421$	−6.20708	−0.302515	−0.151257	+	0.988494i	$0.548332\pi$
$422$	9.95130i	0.484422i
$423$	58.2900	2.83416
$424$	−8.41921	−0.408873
$425$	0	0
$426$	−24.5107	−1.18755
$427$	6.39012	0.309240
$428$	8.85710i	0.428124i
$429$	−3.82295	−0.184574
$430$	− 11.0574i	− 0.533234i
$431$	8.73143i	0.420578i	0.977639	+	0.210289i	$0.0674405\pi$
$431$	8.73143i	0.420578i	−0.977639	+	0.210289i	$0.932559\pi$
$432$	14.2344i	0.684854i
$433$	−20.9855	−1.00850	−0.504248	−	0.863559i	$-0.668230\pi$
$433$	−20.9855	−1.00850	−0.504248	+	0.863559i	$0.668230\pi$
$434$	9.76827	0.468892
$435$	− 33.5695i	− 1.60954i
$436$	11.8726i	0.568593i
$437$	0.142903i	0.00683600i
$438$	−29.1753	−1.39405
$439$	8.46616i	0.404068i	0.979379	+	0.202034i	$0.0647551\pi$
$439$	8.46616i	0.404068i	−0.979379	+	0.202034i	$0.935245\pi$
$440$	−4.04189	−0.192690
$441$	−42.5732	−2.02729
$442$	0	0
$443$	23.9564	1.13820	0.569100	−	0.822268i	$-0.307292\pi$
$443$	23.9564	1.13820	0.569100	+	0.822268i	$0.307292\pi$
$444$	1.53478	0.0728373
$445$	− 8.51249i	− 0.403531i
$446$	−19.7638	−0.935844
$447$	51.9778i	2.45847i
$448$	1.12061i	0.0529441i
$449$	6.12567i	0.289088i	0.989498	+	0.144544i	$0.0461715\pi$
$449$	6.12567i	0.289088i	−0.989498	+	0.144544i	$0.953828\pi$
$450$	−26.6536	−1.25646
$451$	9.00000	0.423793
$452$	− 10.8648i	− 0.511039i
$453$	− 8.71957i	− 0.409681i
$454$	3.55438i	0.166815i
$455$	0.461104	0.0216169
$456$	1.12061i	0.0524776i
$457$	−1.01691	−0.0475691	−0.0237846	−	0.999717i	$-0.507572\pi$
$457$	−1.01691	−0.0475691	−0.0237846	+	0.999717i	$0.507572\pi$
$458$	2.04189	0.0954112
$459$	0	0
$460$	−0.487511	−0.0227303
$461$	23.0838	1.07512	0.537559	−	0.843226i	$-0.319346\pi$
$461$	23.0838	1.07512	0.537559	+	0.843226i	$0.319346\pi$
$462$	− 12.3354i	− 0.573896i
$463$	19.3628	0.899865	0.449932	−	0.893063i	$-0.351448\pi$
$463$	19.3628	0.899865	0.449932	+	0.893063i	$0.351448\pi$
$464$	8.78106i	0.407650i
$465$	33.3242i	1.54537i
$466$	3.91622i	0.181415i
$467$	10.6895	0.494653	0.247326	−	0.968932i	$-0.420448\pi$
$467$	10.6895	0.494653	0.247326	+	0.968932i	$0.420448\pi$
$468$	−2.57398	−0.118982
$469$	8.19522i	0.378420i
$470$	9.31820i	0.429817i
$471$	76.0188i	3.50276i
$472$	6.41147	0.295112
$473$	− 31.8384i	− 1.46393i
$474$	42.8384	1.96763
$475$	−1.24897	−0.0573067
$476$	0	0
$477$	62.3988	2.85704
$478$	−21.5544	−0.985874
$479$	− 19.0155i	− 0.868840i	−0.900711	−	0.434420i	$-0.856953\pi$
$479$	− 19.0155i	− 0.868840i	0.900711	−	0.434420i	$-0.143047\pi$
$480$	−3.82295	−0.174493
$481$	0.165192i	0.00753211i
$482$	25.1925i	1.14749i
$483$	− 1.48784i	− 0.0676989i
$484$	−0.638156	−0.0290071
$485$	−0.262585	−0.0119234
$486$	− 33.7547i	− 1.53114i
$487$	− 3.13104i	− 0.141881i	−0.997481	−	0.0709406i	$-0.977400\pi$
$487$	− 3.13104i	− 0.141881i	0.997481	−	0.0709406i	$-0.0226001\pi$
$488$	− 5.70233i	− 0.258133i
$489$	6.04788	0.273494
$490$	− 6.80571i	− 0.307451i
$491$	−33.3851	−1.50665	−0.753323	−	0.657650i	$-0.771551\pi$
$491$	−33.3851	−1.50665	−0.753323	+	0.657650i	$0.771551\pi$
$492$	8.51249	0.383773
$493$	0	0
$494$	−0.120615	−0.00542671
$495$	29.9564	1.34644
$496$	− 8.71688i	− 0.391400i
$497$	8.51249	0.381837
$498$	− 24.9718i	− 1.11901i
$499$	− 3.93077i	− 0.175965i	−0.996122	−	0.0879827i	$-0.971958\pi$
$499$	− 3.93077i	− 0.175965i	0.996122	−	0.0879827i	$-0.0280420\pi$
$500$	− 10.1848i	− 0.455478i
$501$	−28.5790	−1.27682
$502$	3.07604	0.137290
$503$	4.47977i	0.199743i	0.995000	+	0.0998716i	$0.0318432\pi$
$503$	4.47977i	0.199743i	−0.995000	+	0.0998716i	$0.968157\pi$
$504$	− 8.30541i	− 0.369952i
$505$	5.76382i	0.256487i
$506$	−1.40373	−0.0624036
$507$	41.5577i	1.84564i
$508$	6.55169	0.290684
$509$	−2.58853	−0.114734	−0.0573672	−	0.998353i	$-0.518271\pi$
$509$	−2.58853	−0.114734	−0.0573672	+	0.998353i	$0.518271\pi$
$510$	0	0
$511$	10.1325	0.448234
$512$	1.00000	0.0441942
$513$	− 4.94356i	− 0.218264i
$514$	24.9394	1.10003
$515$	− 14.6878i	− 0.647221i
$516$	− 30.1138i	− 1.32569i
$517$	26.8307i	1.18001i
$518$	−0.533023	−0.0234197
$519$	20.6878	0.908093
$520$	− 0.411474i	− 0.0180443i
$521$	− 38.4020i	− 1.68242i	−0.540708	−	0.841211i	$-0.681843\pi$
$521$	− 38.4020i	− 1.68242i	0.540708	−	0.841211i	$-0.318157\pi$
$522$	− 65.0806i	− 2.84850i
$523$	31.6614	1.38446	0.692228	−	0.721679i	$-0.256629\pi$
$523$	31.6614	1.38446	0.692228	+	0.721679i	$0.256629\pi$
$524$	− 0.0496299i	− 0.00216809i
$525$	13.0036	0.567525
$526$	15.2422	0.664590
$527$	0	0
$528$	−11.0077	−0.479050
$529$	22.8307	0.992639
$530$	9.97502i	0.433287i
$531$	−47.5185	−2.06213
$532$	− 0.389185i	− 0.0168733i
$533$	0.916222i	0.0396860i
$534$	− 23.1830i	− 1.00323i
$535$	10.4938	0.453687
$536$	7.31315	0.315880
$537$	− 4.06830i	− 0.175560i
$538$	3.77332i	0.162679i
$539$	− 19.5963i	− 0.844071i
$540$	16.8648	0.725747
$541$	37.9394i	1.63114i	0.578656	+	0.815572i	$0.303578\pi$
$541$	37.9394i	1.63114i	−0.578656	+	0.815572i	$0.696422\pi$
$542$	4.72462	0.202940
$543$	49.4356	2.12149
$544$	0	0
$545$	14.0665	0.602544
$546$	1.25578	0.0537423
$547$	− 1.48845i	− 0.0636413i	−0.999494	−	0.0318207i	$-0.989869\pi$
$547$	− 1.48845i	− 0.0636413i	0.999494	−	0.0318207i	$-0.0101305\pi$
$548$	−8.13341	−0.347442
$549$	42.2627i	1.80373i
$550$	− 12.2686i	− 0.523134i
$551$	− 3.04963i	− 0.129919i
$552$	−1.32770	−0.0565105
$553$	−14.8776	−0.632661
$554$	− 18.2276i	− 0.774417i
$555$	− 1.81839i	− 0.0771865i
$556$	15.3824i	0.652358i
$557$	23.3259	0.988352	0.494176	−	0.869362i	$-0.335470\pi$
$557$	23.3259	0.988352	0.494176	+	0.869362i	$0.335470\pi$
$558$	64.6049i	2.73495i
$559$	3.24123	0.137089
$560$	1.32770	0.0561054
$561$	0	0
$562$	−25.3773	−1.07048
$563$	4.37733	0.184482	0.0922411	−	0.995737i	$-0.470597\pi$
$563$	4.37733	0.184482	0.0922411	+	0.995737i	$0.470597\pi$
$564$	25.3773i	1.06858i
$565$	−12.8726	−0.541553
$566$	13.0446i	0.548304i
$567$	26.5536i	1.11514i
$568$	− 7.59627i	− 0.318732i
$569$	−4.69553	−0.196847	−0.0984234	−	0.995145i	$-0.531380\pi$
$569$	−4.69553	−0.196847	−0.0984234	+	0.995145i	$0.531380\pi$
$570$	1.32770	0.0556111
$571$	− 13.4679i	− 0.563615i	−0.959471	−	0.281807i	$-0.909066\pi$
$571$	− 13.4679i	− 0.563615i	0.959471	−	0.281807i	$-0.0909339\pi$
$572$	− 1.18479i	− 0.0495387i
$573$	− 33.4287i	− 1.39650i
$574$	−2.95636	−0.123396
$575$	− 1.47977i	− 0.0617107i
$576$	−7.41147	−0.308811
$577$	22.8066	0.949453	0.474727	−	0.880133i	$-0.342547\pi$
$577$	22.8066	0.949453	0.474727	+	0.880133i	$0.342547\pi$
$578$	0	0
$579$	49.2199	2.04551
$580$	10.4037	0.431992
$581$	8.67263i	0.359801i
$582$	−0.715127	−0.0296430
$583$	28.7219i	1.18954i
$584$	− 9.04189i	− 0.374156i
$585$	3.04963i	0.126087i
$586$	3.80571	0.157213
$587$	−38.5945	−1.59297	−0.796483	−	0.604661i	$-0.793309\pi$
$587$	−38.5945	−1.59297	−0.796483	+	0.604661i	$0.793309\pi$
$588$	− 18.5348i	− 0.764361i
$589$	3.02734i	0.124739i
$590$	− 7.59627i	− 0.312733i
$591$	68.5417	2.81943
$592$	0.475652i	0.0195492i
$593$	−2.30272	−0.0945613	−0.0472807	−	0.998882i	$-0.515056\pi$
$593$	−2.30272	−0.0945613	−0.0472807	+	0.998882i	$0.515056\pi$
$594$	48.5604	1.99246
$595$	0	0
$596$	−16.1088	−0.659840
$597$	−42.2172	−1.72783
$598$	− 0.142903i	− 0.00584376i
$599$	−29.1830	−1.19239	−0.596193	−	0.802841i	$-0.703321\pi$
$599$	−29.1830	−1.19239	−0.596193	+	0.802841i	$0.703321\pi$
$600$	− 11.6040i	− 0.473732i
$601$	14.0428i	0.572819i	0.958107	+	0.286409i	$0.0924617\pi$
$601$	14.0428i	0.572819i	−0.958107	+	0.286409i	$0.907538\pi$
$602$	10.4584i	0.426253i
$603$	−54.2012	−2.20724
$604$	2.70233	0.109956
$605$	0.756082i	0.0307391i
$606$	15.6973i	0.637658i
$607$	− 10.2635i	− 0.416583i	−0.978067	−	0.208292i	$-0.933210\pi$
$607$	− 10.2635i	− 0.416583i	0.978067	−	0.208292i	$-0.0667903\pi$
$608$	−0.347296	−0.0140847
$609$	31.7511i	1.28662i
$610$	−6.75608	−0.273546
$611$	−2.73143	−0.110502
$612$	0	0
$613$	2.66725	0.107729	0.0538646	−	0.998548i	$-0.482846\pi$
$613$	2.66725	0.107729	0.0538646	+	0.998548i	$0.482846\pi$
$614$	13.1070	0.528955
$615$	− 10.0855i	− 0.406688i
$616$	3.82295	0.154031
$617$	− 39.5544i	− 1.59240i	−0.605034	−	0.796200i	$-0.706840\pi$
$617$	− 39.5544i	− 1.59240i	0.605034	−	0.796200i	$-0.293160\pi$
$618$	− 40.0009i	− 1.60907i
$619$	− 22.3851i	− 0.899732i	−0.893096	−	0.449866i	$-0.851472\pi$
$619$	− 22.3851i	− 0.899732i	0.893096	−	0.449866i	$-0.148528\pi$
$620$	−10.3277	−0.414770
$621$	5.85710	0.235037
$622$	− 10.3277i	− 0.414103i
$623$	8.05138i	0.322572i
$624$	− 1.12061i	− 0.0448605i
$625$	5.91447	0.236579
$626$	− 21.7178i	− 0.868018i
$627$	3.82295	0.152674
$628$	−23.5594	−0.940124
$629$	0	0
$630$	−9.84018	−0.392042
$631$	13.9727	0.556243	0.278121	−	0.960546i	$-0.410288\pi$
$631$	13.9727	0.556243	0.278121	+	0.960546i	$0.410288\pi$
$632$	13.2763i	0.528103i
$633$	32.1097	1.27625
$634$	11.1830i	0.444135i
$635$	− 7.76239i	− 0.308041i
$636$	27.1661i	1.07721i
$637$	1.99495	0.0790427
$638$	29.9564	1.18598
$639$	56.2995i	2.22718i
$640$	− 1.18479i	− 0.0468330i
$641$	24.5868i	0.971119i	0.874204	+	0.485560i	$0.161384\pi$
$641$	24.5868i	0.971119i	−0.874204	+	0.485560i	$0.838616\pi$
$642$	28.5790	1.12792
$643$	− 29.6996i	− 1.17124i	−0.810586	−	0.585620i	$-0.800851\pi$
$643$	− 29.6996i	− 1.17124i	0.810586	−	0.585620i	$-0.199149\pi$
$644$	0.461104	0.0181700
$645$	−35.6786	−1.40484
$646$	0	0
$647$	16.3946	0.644537	0.322268	−	0.946648i	$-0.395555\pi$
$647$	16.3946	0.644537	0.322268	+	0.946648i	$0.395555\pi$
$648$	23.6955	0.930848
$649$	− 21.8726i	− 0.858574i
$650$	1.24897	0.0489886
$651$	− 31.5191i	− 1.23533i
$652$	1.87433i	0.0734045i
$653$	35.8307i	1.40216i	0.713081	+	0.701082i	$0.247299\pi$
$653$	35.8307i	1.40216i	−0.713081	+	0.701082i	$0.752701\pi$
$654$	38.3090	1.49800
$655$	−0.0588011	−0.00229755
$656$	2.63816i	0.103003i
$657$	67.0137i	2.61445i
$658$	− 8.81345i	− 0.343584i
$659$	−28.5354	−1.11158	−0.555790	−	0.831322i	$-0.687584\pi$
$659$	−28.5354	−1.11158	−0.555790	+	0.831322i	$0.687584\pi$
$660$	13.0419i	0.507655i
$661$	45.1462	1.75598	0.877992	−	0.478676i	$-0.158883\pi$
$661$	45.1462	1.75598	0.877992	+	0.478676i	$0.158883\pi$
$662$	−18.0847	−0.702882
$663$	0	0
$664$	7.73917	0.300338
$665$	−0.461104	−0.0178808
$666$	− 3.52528i	− 0.136602i
$667$	3.61318	0.139903
$668$	− 8.85710i	− 0.342691i
$669$	63.7716i	2.46555i
$670$	− 8.66456i	− 0.334741i
$671$	−19.4534	−0.750989
$672$	3.61587	0.139485
$673$	− 4.91178i	− 0.189335i	−0.995509	−	0.0946676i	$-0.969821\pi$
$673$	− 4.91178i	− 0.189335i	0.995509	−	0.0946676i	$-0.0301788\pi$
$674$	1.68004i	0.0647129i
$675$	51.1908i	1.97033i
$676$	−12.8794	−0.495361
$677$	18.7885i	0.722100i	0.932547	+	0.361050i	$0.117582\pi$
$677$	18.7885i	0.722100i	−0.932547	+	0.361050i	$0.882418\pi$
$678$	−35.0574	−1.34637
$679$	0.248361	0.00953122
$680$	0	0
$681$	11.4688	0.439487
$682$	−29.7374	−1.13870
$683$	− 25.5699i	− 0.978403i	−0.872171	−	0.489202i	$-0.837288\pi$
$683$	− 25.5699i	− 0.978403i	0.872171	−	0.489202i	$-0.162712\pi$
$684$	2.57398	0.0984185
$685$	9.63640i	0.368188i
$686$	14.2814i	0.545265i
$687$	− 6.58853i	− 0.251368i
$688$	9.33275	0.355808
$689$	−2.92396	−0.111394
$690$	1.57304i	0.0598848i
$691$	− 41.6091i	− 1.58288i	−0.611245	−	0.791442i	$-0.709331\pi$
$691$	− 41.6091i	− 1.58288i	0.611245	−	0.791442i	$-0.290669\pi$
$692$	6.41147i	0.243728i
$693$	−28.3337	−1.07631
$694$	− 22.4766i	− 0.853200i
$695$	18.2249	0.691311
$696$	28.3337	1.07399
$697$	0	0
$698$	21.1070	0.798912
$699$	12.6364	0.477953
$700$	4.03003i	0.152321i
$701$	−0.746911	−0.0282104	−0.0141052	−	0.999901i	$-0.504490\pi$
$701$	−0.746911	−0.0282104	−0.0141052	+	0.999901i	$0.504490\pi$
$702$	4.94356i	0.186583i
$703$	− 0.165192i	− 0.00623034i
$704$	− 3.41147i	− 0.128575i
$705$	30.0669	1.13238
$706$	3.86659	0.145521
$707$	− 5.45161i	− 0.205029i
$708$	− 20.6878i	− 0.777495i
$709$	21.1310i	0.793593i	0.917907	+	0.396797i	$0.129878\pi$
$709$	21.1310i	0.793593i	−0.917907	+	0.396797i	$0.870122\pi$
$710$	−9.00000	−0.337764
$711$	− 98.3970i	− 3.69018i
$712$	7.18479	0.269261
$713$	−3.58677	−0.134326
$714$	0	0
$715$	−1.40373	−0.0524967
$716$	1.26083	0.0471194
$717$	69.5491i	2.59736i
$718$	−36.4347	−1.35973
$719$	− 3.46110i	− 0.129077i	−0.997915	−	0.0645387i	$-0.979442\pi$
$719$	− 3.46110i	− 0.129077i	0.997915	−	0.0645387i	$-0.0205576\pi$
$720$	8.78106i	0.327251i
$721$	13.8922i	0.517372i
$722$	−18.8794	−0.702618
$723$	81.2883	3.02314
$724$	15.3209i	0.569396i
$725$	31.5790i	1.17282i
$726$	2.05913i	0.0764213i
$727$	−36.3327	−1.34751	−0.673754	−	0.738956i	$-0.735319\pi$
$727$	−36.3327	−1.34751	−0.673754	+	0.738956i	$0.735319\pi$
$728$	0.389185i	0.0144242i
$729$	−37.8289	−1.40107
$730$	−10.7128	−0.396497
$731$	0	0
$732$	−18.3996	−0.680069
$733$	−6.90167	−0.254919	−0.127460	−	0.991844i	$-0.540682\pi$
$733$	−6.90167	−0.254919	−0.127460	+	0.991844i	$0.540682\pi$
$734$	13.7570i	0.507781i
$735$	−21.9599	−0.810002
$736$	− 0.411474i	− 0.0151671i
$737$	− 24.9486i	− 0.918994i
$738$	− 19.5526i	− 0.719742i
$739$	36.3651	1.33771	0.668857	−	0.743391i	$-0.266784\pi$
$739$	36.3651	1.33771	0.668857	+	0.743391i	$0.266784\pi$
$740$	0.563549	0.0207165
$741$	0.389185i	0.0142971i
$742$	− 9.43470i	− 0.346359i
$743$	− 48.3150i	− 1.77251i	−0.463201	−	0.886253i	$-0.653299\pi$
$743$	− 48.3150i	− 1.77251i	0.463201	−	0.886253i	$-0.346701\pi$
$744$	−28.1266	−1.03117
$745$	19.0855i	0.699240i
$746$	22.7638	0.833443
$747$	−57.3587	−2.09864
$748$	0	0
$749$	−9.92539	−0.362666
$750$	−32.8631	−1.19999
$751$	− 10.9436i	− 0.399336i	−0.979864	−	0.199668i	$-0.936014\pi$
$751$	− 10.9436i	− 0.399336i	0.979864	−	0.199668i	$-0.0639864\pi$
$752$	−7.86484	−0.286801
$753$	− 9.92539i	− 0.361701i
$754$	3.04963i	0.111061i
$755$	− 3.20170i	− 0.116522i
$756$	−15.9513	−0.580143
$757$	−20.0601	−0.729095	−0.364548	−	0.931185i	$-0.618776\pi$
$757$	−20.0601	−0.729095	−0.364548	+	0.931185i	$0.618776\pi$
$758$	− 8.91859i	− 0.323938i
$759$	4.52940i	0.164407i
$760$	0.411474i	0.0149257i
$761$	−36.4671	−1.32193	−0.660965	−	0.750416i	$-0.729853\pi$
$761$	−36.4671	−1.32193	−0.660965	+	0.750416i	$0.729853\pi$
$762$	− 21.1402i	− 0.765829i
$763$	−13.3046	−0.481658
$764$	10.3601	0.374815
$765$	0	0
$766$	−19.4097	−0.701302
$767$	2.22668	0.0804008
$768$	− 3.22668i	− 0.116433i
$769$	45.8120	1.65202	0.826012	−	0.563653i	$-0.190604\pi$
$769$	45.8120	1.65202	0.826012	+	0.563653i	$0.190604\pi$
$770$	− 4.52940i	− 0.163228i
$771$	− 80.4716i	− 2.89812i
$772$	15.2540i	0.549004i
$773$	42.1549	1.51621	0.758103	−	0.652135i	$-0.226127\pi$
$773$	42.1549	1.51621	0.758103	+	0.652135i	$0.226127\pi$
$774$	−69.1694	−2.48624
$775$	− 31.3482i	− 1.12606i
$776$	− 0.221629i	− 0.00795602i
$777$	1.71989i	0.0617009i
$778$	−35.6614	−1.27852
$779$	− 0.916222i	− 0.0328271i
$780$	−1.32770	−0.0475391
$781$	−25.9145	−0.927293
$782$	0	0
$783$	−124.993	−4.46690
$784$	5.74422	0.205151
$785$	27.9130i	0.996259i
$786$	−0.160140	−0.00571200
$787$	22.2145i	0.791861i	0.918280	+	0.395931i	$0.129578\pi$
$787$	22.2145i	0.791861i	−0.918280	+	0.395931i	$0.870422\pi$
$788$	21.2422i	0.756721i
$789$	− 49.1816i	− 1.75091i
$790$	15.7297	0.559637
$791$	12.1753	0.432904
$792$	25.2841i	0.898430i
$793$	− 1.98040i	− 0.0703261i
$794$	7.44562i	0.264235i
$795$	32.1862	1.14153
$796$	− 13.0838i	− 0.463742i
$797$	−5.68180	−0.201260	−0.100630	−	0.994924i	$-0.532086\pi$
$797$	−5.68180	−0.201260	−0.100630	+	0.994924i	$0.532086\pi$
$798$	−1.25578	−0.0444540
$799$	0	0
$800$	3.59627	0.127147
$801$	−53.2499	−1.88149
$802$	2.36959i	0.0836730i
$803$	−30.8462	−1.08854
$804$	− 23.5972i	− 0.832209i
$805$	− 0.546313i	− 0.0192550i
$806$	− 3.02734i	− 0.106634i
$807$	12.1753	0.428591
$808$	−4.86484	−0.171144
$809$	17.8631i	0.628033i	0.949418	+	0.314016i	$0.101675\pi$
$809$	17.8631i	0.628033i	−0.949418	+	0.314016i	$0.898325\pi$
$810$	− 28.0743i	− 0.986430i
$811$	− 2.49794i	− 0.0877146i	−0.999038	−	0.0438573i	$-0.986035\pi$
$811$	− 2.49794i	− 0.0877146i	0.999038	−	0.0438573i	$-0.0139647\pi$
$812$	−9.84018	−0.345323
$813$	− 15.2449i	− 0.534660i
$814$	1.62267	0.0568747
$815$	2.22070	0.0777876
$816$	0	0
$817$	−3.24123	−0.113396
$818$	−28.8357	−1.00822
$819$	− 2.88444i	− 0.100790i
$820$	3.12567	0.109153
$821$	− 25.7128i	− 0.897382i	−0.893687	−	0.448691i	$-0.851890\pi$
$821$	− 25.7128i	− 0.897382i	0.893687	−	0.448691i	$-0.148110\pi$
$822$	26.2439i	0.915362i
$823$	13.3304i	0.464668i	0.972636	+	0.232334i	$0.0746362\pi$
$823$	13.3304i	0.464668i	−0.972636	+	0.232334i	$0.925364\pi$
$824$	12.3969	0.431867
$825$	−39.5868	−1.37823
$826$	7.18479i	0.249991i
$827$	− 11.2855i	− 0.392435i	−0.980560	−	0.196217i	$-0.937134\pi$
$827$	− 11.2855i	− 0.392435i	0.980560	−	0.196217i	$-0.0628658\pi$
$828$	3.04963i	0.105982i
$829$	−25.9786	−0.902276	−0.451138	−	0.892454i	$-0.648982\pi$
$829$	−25.9786	−0.902276	−0.451138	+	0.892454i	$0.648982\pi$
$830$	− 9.16931i	− 0.318271i
$831$	−58.8147	−2.04026
$832$	0.347296	0.0120403
$833$	0	0
$834$	49.6340	1.71869
$835$	−10.4938	−0.363154
$836$	1.18479i	0.0409769i
$837$	124.080	4.28882
$838$	30.6800i	1.05982i
$839$	− 20.9736i	− 0.724089i	−0.932161	−	0.362044i	$-0.882079\pi$
$839$	− 20.9736i	− 0.724089i	0.932161	−	0.362044i	$-0.117921\pi$
$840$	− 4.28405i	− 0.147814i
$841$	−48.1070	−1.65886
$842$	−6.20708	−0.213910
$843$	81.8846i	2.82025i
$844$	9.95130i	0.342538i
$845$	15.2594i	0.524939i
$846$	58.2900	2.00405
$847$	− 0.715127i	− 0.0245720i
$848$	−8.41921	−0.289117
$849$	42.0907	1.44455
$850$	0	0
$851$	0.195718	0.00670914
$852$	−24.5107	−0.839724
$853$	− 50.9650i	− 1.74501i	−0.488606	−	0.872505i	$-0.662494\pi$
$853$	− 50.9650i	− 1.74501i	0.488606	−	0.872505i	$-0.337506\pi$
$854$	6.39012	0.218665
$855$	− 3.04963i	− 0.104295i
$856$	8.85710i	0.302729i
$857$	39.0060i	1.33242i	0.745765	+	0.666210i	$0.232084\pi$
$857$	39.0060i	1.33242i	−0.745765	+	0.666210i	$0.767916\pi$
$858$	−3.82295	−0.130513
$859$	12.6742	0.432437	0.216219	−	0.976345i	$-0.430628\pi$
$859$	12.6742	0.432437	0.216219	+	0.976345i	$0.430628\pi$
$860$	− 11.0574i	− 0.377053i
$861$	9.53922i	0.325096i
$862$	8.73143i	0.297394i
$863$	31.9905	1.08897	0.544485	−	0.838771i	$-0.316725\pi$
$863$	31.9905	1.08897	0.544485	+	0.838771i	$0.316725\pi$
$864$	14.2344i	0.484265i
$865$	7.59627	0.258281
$866$	−20.9855	−0.713115
$867$	0	0
$868$	9.76827	0.331557
$869$	45.2918	1.53642
$870$	− 33.5695i	− 1.13811i
$871$	2.53983	0.0860588
$872$	11.8726i	0.402056i
$873$	1.64260i	0.0555935i
$874$	0.142903i	0.00483378i
$875$	11.4132	0.385838
$876$	−29.1753	−0.985742
$877$	− 28.6212i	− 0.966471i	−0.875490	−	0.483235i	$-0.839462\pi$
$877$	− 28.6212i	− 0.966471i	0.875490	−	0.483235i	$-0.160538\pi$
$878$	8.46616i	0.285719i
$879$	− 12.2798i	− 0.414188i
$880$	−4.04189	−0.136252
$881$	18.2594i	0.615175i	0.951520	+	0.307587i	$0.0995216\pi$
$881$	18.2594i	0.615175i	−0.951520	+	0.307587i	$0.900478\pi$
$882$	−42.5732	−1.43351
$883$	−18.4688	−0.621526	−0.310763	−	0.950487i	$-0.600585\pi$
$883$	−18.4688	−0.621526	−0.310763	+	0.950487i	$0.600585\pi$
$884$	0	0
$885$	−24.5107	−0.823919
$886$	23.9564	0.804830
$887$	− 43.4858i	− 1.46011i	−0.683389	−	0.730054i	$-0.739495\pi$
$887$	− 43.4858i	− 1.46011i	0.683389	−	0.730054i	$-0.260505\pi$
$888$	1.53478	0.0515038
$889$	7.34192i	0.246240i
$890$	− 8.51249i	− 0.285339i
$891$	− 80.8367i	− 2.70813i
$892$	−19.7638	−0.661742
$893$	2.73143	0.0914038
$894$	51.9778i	1.73840i
$895$	− 1.49382i	− 0.0499330i
$896$	1.12061i	0.0374371i
$897$	−0.461104	−0.0153958
$898$	6.12567i	0.204416i
$899$	76.5435	2.55287
$900$	−26.6536	−0.888455
$901$	0	0
$902$	9.00000	0.299667
$903$	33.7460	1.12300
$904$	− 10.8648i	− 0.361359i
$905$	18.1521	0.603395
$906$	− 8.71957i	− 0.289688i
$907$	− 26.3993i	− 0.876574i	−0.898835	−	0.438287i	$-0.855585\pi$
$907$	− 26.3993i	− 0.876574i	0.898835	−	0.438287i	$-0.144415\pi$
$908$	3.55438i	0.117956i
$909$	36.0556	1.19589
$910$	0.461104	0.0152854
$911$	− 2.66138i	− 0.0881754i	−0.999028	−	0.0440877i	$-0.985962\pi$
$911$	− 2.66138i	− 0.0881754i	0.999028	−	0.0440877i	$-0.0140381\pi$
$912$	1.12061i	0.0371073i
$913$	− 26.4020i	− 0.873778i
$914$	−1.01691	−0.0336365
$915$	21.7997i	0.720677i
$916$	2.04189	0.0674659
$917$	0.0556159	0.00183660
$918$	0	0
$919$	15.1088	0.498392	0.249196	−	0.968453i	$-0.419834\pi$
$919$	15.1088	0.498392	0.249196	+	0.968453i	$0.419834\pi$
$920$	−0.487511	−0.0160728
$921$	− 42.2921i	− 1.39357i
$922$	23.0838	0.760224
$923$	− 2.63816i	− 0.0868360i
$924$	− 12.3354i	− 0.405806i
$925$	1.71057i	0.0562432i
$926$	19.3628	0.636300
$927$	−91.8795	−3.01772
$928$	8.78106i	0.288252i
$929$	4.94087i	0.162105i	0.996710	+	0.0810524i	$0.0258281\pi$
$929$	4.94087i	0.162105i	−0.996710	+	0.0810524i	$0.974172\pi$
$930$	33.3242i	1.09274i
$931$	−1.99495	−0.0653818
$932$	3.91622i	0.128280i
$933$	−33.3242	−1.09098
$934$	10.6895	0.349772
$935$	0	0
$936$	−2.57398	−0.0841331
$937$	−3.41416	−0.111536	−0.0557679	−	0.998444i	$-0.517761\pi$
$937$	−3.41416	−0.111536	−0.0557679	+	0.998444i	$0.517761\pi$
$938$	8.19522i	0.267583i
$939$	−70.0765	−2.28686
$940$	9.31820i	0.303926i
$941$	− 28.2608i	− 0.921277i	−0.887588	−	0.460638i	$-0.847621\pi$
$941$	− 28.2608i	− 0.921277i	0.887588	−	0.460638i	$-0.152379\pi$
$942$	76.0188i	2.47683i
$943$	1.08553	0.0353498
$944$	6.41147	0.208676
$945$	18.8990i	0.614784i
$946$	− 31.8384i	− 1.03516i
$947$	6.08410i	0.197707i	0.995102	+	0.0988534i	$0.0315175\pi$
$947$	6.08410i	0.197707i	−0.995102	+	0.0988534i	$0.968483\pi$
$948$	42.8384	1.39133
$949$	− 3.14022i	− 0.101936i
$950$	−1.24897	−0.0405219
$951$	36.0841	1.17011
$952$	0	0
$953$	26.7110	0.865254	0.432627	−	0.901573i	$-0.357587\pi$
$953$	26.7110	0.865254	0.432627	+	0.901573i	$0.357587\pi$
$954$	62.3988	2.02024
$955$	− 12.2746i	− 0.397195i
$956$	−21.5544	−0.697118
$957$	− 96.6596i	− 3.12456i
$958$	− 19.0155i	− 0.614362i
$959$	− 9.11442i	− 0.294320i
$960$	−3.82295	−0.123385
$961$	−44.9840	−1.45110
$962$	0.165192i	0.00532601i
$963$	− 65.6441i	− 2.11535i
$964$	25.1925i	0.811397i
$965$	18.0729	0.581786
$966$	− 1.48784i	− 0.0478703i
$967$	27.1070	0.871702	0.435851	−	0.900019i	$-0.356447\pi$
$967$	27.1070	0.871702	0.435851	+	0.900019i	$0.356447\pi$
$968$	−0.638156	−0.0205111
$969$	0	0
$970$	−0.262585	−0.00843108
$971$	−50.0725	−1.60690	−0.803452	−	0.595370i	$-0.797006\pi$
$971$	−50.0725	−1.60690	−0.803452	+	0.595370i	$0.797006\pi$
$972$	− 33.7547i	− 1.08268i
$973$	−17.2377	−0.552616
$974$	− 3.13104i	− 0.100325i
$975$	− 4.03003i	− 0.129064i
$976$	− 5.70233i	− 0.182527i
$977$	−2.25309	−0.0720827	−0.0360414	−	0.999350i	$-0.511475\pi$
$977$	−2.25309	−0.0720827	−0.0360414	+	0.999350i	$0.511475\pi$
$978$	6.04788	0.193390
$979$	− 24.5107i	− 0.783366i
$980$	− 6.80571i	− 0.217400i
$981$	− 87.9933i	− 2.80941i
$982$	−33.3851	−1.06536
$983$	− 34.7047i	− 1.10691i	−0.832880	−	0.553454i	$-0.813309\pi$
$983$	− 34.7047i	− 1.10691i	0.832880	−	0.553454i	$-0.186691\pi$
$984$	8.51249	0.271368
$985$	25.1676	0.801905
$986$	0	0
$987$	−28.4382	−0.905198
$988$	−0.120615	−0.00383727
$989$	− 3.84018i	− 0.122111i
$990$	29.9564	0.952075
$991$	15.3200i	0.486654i	0.969944	+	0.243327i	$0.0782389\pi$
$991$	15.3200i	0.486654i	−0.969944	+	0.243327i	$0.921761\pi$
$992$	− 8.71688i	− 0.276761i
$993$	58.3536i	1.85180i
$994$	8.51249	0.270000
$995$	−15.5016	−0.491433
$996$	− 24.9718i	− 0.791263i
$997$	− 52.6076i	− 1.66610i	−0.553197	−	0.833050i	$-0.686592\pi$
$997$	− 52.6076i	− 1.66610i	0.553197	−	0.833050i	$-0.313408\pi$
$998$	− 3.93077i	− 0.124426i
$999$	−6.77063	−0.214213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	578.2.b.f.577.1		6
17.2	even	8		578.2.c.g.251.1		12
17.3	odd	16		578.2.d.h.399.6		24
17.4	even	4		578.2.a.e.1.1	✓	3
17.5	odd	16		578.2.d.h.179.6		24
17.6	odd	16		578.2.d.h.423.1		24
17.7	odd	16		578.2.d.h.155.1		24
17.8	even	8		578.2.c.g.327.1		12
17.9	even	8		578.2.c.g.327.6		12
17.10	odd	16		578.2.d.h.155.6		24
17.11	odd	16		578.2.d.h.423.6		24
17.12	odd	16		578.2.d.h.179.1		24
17.13	even	4		578.2.a.f.1.3	yes	3
17.14	odd	16		578.2.d.h.399.1		24
17.15	even	8		578.2.c.g.251.6		12
17.16	even	2	inner	578.2.b.f.577.6		6
51.38	odd	4		5202.2.a.bn.1.2		3
51.47	odd	4		5202.2.a.bo.1.2		3
68.47	odd	4		4624.2.a.ba.1.1		3
68.55	odd	4		4624.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
578.2.a.e.1.1	✓	3	17.4	even	4
578.2.a.f.1.3	yes	3	17.13	even	4
578.2.b.f.577.1		6	1.1	even	1	trivial
578.2.b.f.577.6		6	17.16	even	2	inner
578.2.c.g.251.1		12	17.2	even	8
578.2.c.g.251.6		12	17.15	even	8
578.2.c.g.327.1		12	17.8	even	8
578.2.c.g.327.6		12	17.9	even	8
578.2.d.h.155.1		24	17.7	odd	16
578.2.d.h.155.6		24	17.10	odd	16
578.2.d.h.179.1		24	17.12	odd	16
578.2.d.h.179.6		24	17.5	odd	16
578.2.d.h.399.1		24	17.14	odd	16
578.2.d.h.399.6		24	17.3	odd	16
578.2.d.h.423.1		24	17.6	odd	16
578.2.d.h.423.6		24	17.11	odd	16
4624.2.a.ba.1.1		3	68.47	odd	4
4624.2.a.bj.1.3		3	68.55	odd	4
5202.2.a.bn.1.2		3	51.38	odd	4
5202.2.a.bo.1.2		3	51.47	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	578.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.22668i q^{3} +1.00000 q^{4} -1.18479i q^{5} -3.22668i q^{6} +1.12061i q^{7} +1.00000 q^{8} -7.41147 q^{9} -1.18479i q^{10} -3.41147i q^{11} -3.22668i q^{12} +0.347296 q^{13} +1.12061i q^{14} -3.82295 q^{15} +1.00000 q^{16} -7.41147 q^{18} -0.347296 q^{19} -1.18479i q^{20} +3.61587 q^{21} -3.41147i q^{22} -0.411474i q^{23} -3.22668i q^{24} +3.59627 q^{25} +0.347296 q^{26} +14.2344i q^{27} +1.12061i q^{28} +8.78106i q^{29} -3.82295 q^{30} -8.71688i q^{31} +1.00000 q^{32} -11.0077 q^{33} +1.32770 q^{35} -7.41147 q^{36} +0.475652i q^{37} -0.347296 q^{38} -1.12061i q^{39} -1.18479i q^{40} +2.63816i q^{41} +3.61587 q^{42} +9.33275 q^{43} -3.41147i q^{44} +8.78106i q^{45} -0.411474i q^{46} -7.86484 q^{47} -3.22668i q^{48} +5.74422 q^{49} +3.59627 q^{50} +0.347296 q^{52} -8.41921 q^{53} +14.2344i q^{54} -4.04189 q^{55} +1.12061i q^{56} +1.12061i q^{57} +8.78106i q^{58} +6.41147 q^{59} -3.82295 q^{60} -5.70233i q^{61} -8.71688i q^{62} -8.30541i q^{63} +1.00000 q^{64} -0.411474i q^{65} -11.0077 q^{66} +7.31315 q^{67} -1.32770 q^{69} +1.32770 q^{70} -7.59627i q^{71} -7.41147 q^{72} -9.04189i q^{73} +0.475652i q^{74} -11.6040i q^{75} -0.347296 q^{76} +3.82295 q^{77} -1.12061i q^{78} +13.2763i q^{79} -1.18479i q^{80} +23.6955 q^{81} +2.63816i q^{82} +7.73917 q^{83} +3.61587 q^{84} +9.33275 q^{86} +28.3337 q^{87} -3.41147i q^{88} +7.18479 q^{89} +8.78106i q^{90} +0.389185i q^{91} -0.411474i q^{92} -28.1266 q^{93} -7.86484 q^{94} +0.411474i q^{95} -3.22668i q^{96} -0.221629i q^{97} +5.74422 q^{98} +25.2841i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} - 24 q^{9} + 18 q^{15} + 6 q^{16} - 24 q^{18} - 6 q^{25} + 18 q^{30} + 6 q^{32} - 18 q^{33} - 24 q^{36} + 18 q^{43} - 24 q^{49} - 6 q^{50} + 18 q^{53} - 18 q^{55} + 18 q^{59}+ \cdots - 24 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 - 24 * q^9 + 18 * q^15 + 6 * q^16 - 24 * q^18 - 6 * q^25 + 18 * q^30 + 6 * q^32 - 18 * q^33 - 24 * q^36 + 18 * q^43 - 24 * q^49 - 6 * q^50 + 18 * q^53 - 18 * q^55 + 18 * q^59 + 18 * q^60 + 6 * q^64 - 18 * q^66 - 24 * q^72 - 18 * q^77 + 6 * q^81 + 18 * q^83 + 18 * q^86 + 36 * q^89 - 18 * q^93 - 24 * q^98

Introduction

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