LMFDB - Embedded newform 845.2.c.d.506.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [845,2,Mod(506,845)] 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("845.506"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.4
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.d.506.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.30278i q^{2} +1.00000 q^{3} -3.30278 q^{4} +1.00000i q^{5} +2.30278i q^{6} -1.00000i q^{7} -3.00000i q^{8} -2.00000 q^{9} -2.30278 q^{10} +1.60555i q^{11} -3.30278 q^{12} +2.30278 q^{14} +1.00000i q^{15} +0.302776 q^{16} -7.60555 q^{17} -4.60555i q^{18} +5.60555i q^{19} -3.30278i q^{20} -1.00000i q^{21} -3.69722 q^{22} +3.00000 q^{23} -3.00000i q^{24} -1.00000 q^{25} -5.00000 q^{27} +3.30278i q^{28} -6.21110 q^{29} -2.30278 q^{30} +4.00000i q^{31} -5.30278i q^{32} +1.60555i q^{33} -17.5139i q^{34} +1.00000 q^{35} +6.60555 q^{36} +3.60555i q^{37} -12.9083 q^{38} +3.00000 q^{40} -3.00000i q^{41} +2.30278 q^{42} +10.2111 q^{43} -5.30278i q^{44} -2.00000i q^{45} +6.90833i q^{46} +9.21110i q^{47} +0.302776 q^{48} +6.00000 q^{49} -2.30278i q^{50} -7.60555 q^{51} -3.21110 q^{53} -11.5139i q^{54} -1.60555 q^{55} -3.00000 q^{56} +5.60555i q^{57} -14.3028i q^{58} -10.8167i q^{59} -3.30278i q^{60} -1.00000 q^{61} -9.21110 q^{62} +2.00000i q^{63} +12.8167 q^{64} -3.69722 q^{66} +7.00000i q^{67} +25.1194 q^{68} +3.00000 q^{69} +2.30278i q^{70} -4.81665i q^{71} +6.00000i q^{72} -0.788897i q^{73} -8.30278 q^{74} -1.00000 q^{75} -18.5139i q^{76} +1.60555 q^{77} +5.21110 q^{79} +0.302776i q^{80} +1.00000 q^{81} +6.90833 q^{82} +9.21110i q^{83} +3.30278i q^{84} -7.60555i q^{85} +23.5139i q^{86} -6.21110 q^{87} +4.81665 q^{88} -6.21110i q^{89} +4.60555 q^{90} -9.90833 q^{92} +4.00000i q^{93} -21.2111 q^{94} -5.60555 q^{95} -5.30278i q^{96} +8.39445i q^{97} +13.8167i q^{98} -3.21110i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9} - 2 q^{10} - 6 q^{12} + 2 q^{14} - 6 q^{16} - 16 q^{17} - 22 q^{22} + 12 q^{23} - 4 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 12 q^{36} - 30 q^{38} + 12 q^{40}+ \cdots - 8 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 6 * q^4 - 8 * q^9 - 2 * q^10 - 6 * q^12 + 2 * q^14 - 6 * q^16 - 16 * q^17 - 22 * q^22 + 12 * q^23 - 4 * q^25 - 20 * q^27 + 4 * q^29 - 2 * q^30 + 4 * q^35 + 12 * q^36 - 30 * q^38 + 12 * q^40 + 2 * q^42 + 12 * q^43 - 6 * q^48 + 24 * q^49 - 16 * q^51 + 16 * q^53 + 8 * q^55 - 12 * q^56 - 4 * q^61 - 8 * q^62 + 8 * q^64 - 22 * q^66 + 50 * q^68 + 12 * q^69 - 26 * q^74 - 4 * q^75 - 8 * q^77 - 8 * q^79 + 4 * q^81 + 6 * q^82 + 4 * q^87 - 24 * q^88 + 4 * q^90 - 18 * q^92 - 56 * q^94 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.30278i	1.62831i	0.580649	+	0.814154i	$0.302799\pi$
$2$	2.30278i	1.62831i	−0.580649	+	0.814154i	$0.697201\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−3.30278	−1.65139
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	2.30278i	0.940104i
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−2.00000	−0.666667
$10$	−2.30278	−0.728202
$11$	1.60555i	0.484092i	0.970265	+	0.242046i	$0.0778185\pi$
$11$	1.60555i	0.484092i	−0.970265	+	0.242046i	$0.922182\pi$
$12$	−3.30278	−0.953429
$13$	0	0
$13$	0	0
$14$	2.30278	0.615443
$15$	1.00000i	0.258199i
$16$	0.302776	0.0756939
$17$	−7.60555	−1.84462	−0.922309	−	0.386454i	$-0.873700\pi$
$17$	−7.60555	−1.84462	−0.922309	+	0.386454i	$0.873700\pi$
$18$	− 4.60555i	− 1.08554i
$19$	5.60555i	1.28600i	0.765865	+	0.643001i	$0.222311\pi$
$19$	5.60555i	1.28600i	−0.765865	+	0.643001i	$0.777689\pi$
$20$	− 3.30278i	− 0.738523i
$21$	− 1.00000i	− 0.218218i
$22$	−3.69722	−0.788251
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	− 3.00000i	− 0.612372i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	3.30278i	0.624166i
$29$	−6.21110	−1.15337	−0.576686	−	0.816966i	$-0.695655\pi$
$29$	−6.21110	−1.15337	−0.576686	+	0.816966i	$0.695655\pi$
$30$	−2.30278	−0.420427
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 5.30278i	− 0.937407i
$33$	1.60555i	0.279491i
$34$	− 17.5139i	− 3.00361i
$35$	1.00000	0.169031
$36$	6.60555	1.10093
$37$	3.60555i	0.592749i	0.955072	+	0.296374i	$0.0957776\pi$
$37$	3.60555i	0.592749i	−0.955072	+	0.296374i	$0.904222\pi$
$38$	−12.9083	−2.09401
$39$	0	0
$40$	3.00000	0.474342
$41$	− 3.00000i	− 0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$	− 3.00000i	− 0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$	2.30278	0.355326
$43$	10.2111	1.55718	0.778589	−	0.627534i	$-0.215936\pi$
$43$	10.2111	1.55718	0.778589	+	0.627534i	$0.215936\pi$
$44$	− 5.30278i	− 0.799424i
$45$	− 2.00000i	− 0.298142i
$46$	6.90833i	1.01858i
$47$	9.21110i	1.34358i	0.740743	+	0.671789i	$0.234474\pi$
$47$	9.21110i	1.34358i	−0.740743	+	0.671789i	$0.765526\pi$
$48$	0.302776	0.0437019
$49$	6.00000	0.857143
$50$	− 2.30278i	− 0.325662i
$51$	−7.60555	−1.06499
$52$	0	0
$53$	−3.21110	−0.441079	−0.220539	−	0.975378i	$-0.570782\pi$
$53$	−3.21110	−0.441079	−0.220539	+	0.975378i	$0.570782\pi$
$54$	− 11.5139i	− 1.56684i
$55$	−1.60555	−0.216492
$56$	−3.00000	−0.400892
$57$	5.60555i	0.742473i
$58$	− 14.3028i	− 1.87805i
$59$	− 10.8167i	− 1.40821i	−0.710097	−	0.704104i	$-0.751349\pi$
$59$	− 10.8167i	− 1.40821i	0.710097	−	0.704104i	$-0.248651\pi$
$60$	− 3.30278i	− 0.426387i
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−9.21110	−1.16981
$63$	2.00000i	0.251976i
$64$	12.8167	1.60208
$65$	0	0
$66$	−3.69722	−0.455097
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	25.1194	3.04618
$69$	3.00000	0.361158
$70$	2.30278i	0.275234i
$71$	− 4.81665i	− 0.571632i	−0.958285	−	0.285816i	$-0.907735\pi$
$71$	− 4.81665i	− 0.571632i	0.958285	−	0.285816i	$-0.0922646\pi$
$72$	6.00000i	0.707107i
$73$	− 0.788897i	− 0.0923335i	−0.998934	−	0.0461667i	$-0.985299\pi$
$73$	− 0.788897i	− 0.0923335i	0.998934	−	0.0461667i	$-0.0147006\pi$
$74$	−8.30278	−0.965178
$75$	−1.00000	−0.115470
$76$	− 18.5139i	− 2.12369i
$77$	1.60555	0.182970
$78$	0	0
$79$	5.21110	0.586295	0.293147	−	0.956067i	$-0.405297\pi$
$79$	5.21110	0.586295	0.293147	+	0.956067i	$0.405297\pi$
$80$	0.302776i	0.0338513i
$81$	1.00000	0.111111
$82$	6.90833	0.762897
$83$	9.21110i	1.01105i	0.862812	+	0.505525i	$0.168701\pi$
$83$	9.21110i	1.01105i	−0.862812	+	0.505525i	$0.831299\pi$
$84$	3.30278i	0.360362i
$85$	− 7.60555i	− 0.824938i
$86$	23.5139i	2.53557i
$87$	−6.21110	−0.665900
$88$	4.81665	0.513457
$89$	− 6.21110i	− 0.658376i	−0.944264	−	0.329188i	$-0.893225\pi$
$89$	− 6.21110i	− 0.658376i	0.944264	−	0.329188i	$-0.106775\pi$
$90$	4.60555	0.485468
$91$	0	0
$92$	−9.90833	−1.03301
$93$	4.00000i	0.414781i
$94$	−21.2111	−2.18776
$95$	−5.60555	−0.575117
$96$	− 5.30278i	− 0.541212i
$97$	8.39445i	0.852327i	0.904646	+	0.426164i	$0.140135\pi$
$97$	8.39445i	0.852327i	−0.904646	+	0.426164i	$0.859865\pi$
$98$	13.8167i	1.39569i
$99$	− 3.21110i	− 0.322728i
$100$	3.30278	0.330278
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	− 17.5139i	− 1.73413i
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	− 7.39445i	− 0.718212i
$107$	6.21110	0.600450	0.300225	−	0.953868i	$-0.402938\pi$
$107$	6.21110	0.600450	0.300225	+	0.953868i	$0.402938\pi$
$108$	16.5139	1.58905
$109$	19.2111i	1.84009i	0.391813	+	0.920045i	$0.371848\pi$
$109$	19.2111i	1.84009i	−0.391813	+	0.920045i	$0.628152\pi$
$110$	− 3.69722i	− 0.352517i
$111$	3.60555i	0.342224i
$112$	− 0.302776i	− 0.0286096i
$113$	−1.60555	−0.151038	−0.0755188	−	0.997144i	$-0.524061\pi$
$113$	−1.60555	−0.151038	−0.0755188	+	0.997144i	$0.524061\pi$
$114$	−12.9083	−1.20898
$115$	3.00000i	0.279751i
$116$	20.5139	1.90467
$117$	0	0
$118$	24.9083	2.29300
$119$	7.60555i	0.697200i
$120$	3.00000	0.273861
$121$	8.42221	0.765655
$122$	− 2.30278i	− 0.208484i
$123$	− 3.00000i	− 0.270501i
$124$	− 13.2111i	− 1.18639i
$125$	− 1.00000i	− 0.0894427i
$126$	−4.60555	−0.410295
$127$	4.21110	0.373675	0.186837	−	0.982391i	$-0.440176\pi$
$127$	4.21110	0.373675	0.186837	+	0.982391i	$0.440176\pi$
$128$	18.9083i	1.67128i
$129$	10.2111	0.899037
$130$	0	0
$131$	−21.2111	−1.85322	−0.926611	−	0.376021i	$-0.877292\pi$
$131$	−21.2111	−1.85322	−0.926611	+	0.376021i	$0.877292\pi$
$132$	− 5.30278i	− 0.461547i
$133$	5.60555	0.486063
$134$	−16.1194	−1.39251
$135$	− 5.00000i	− 0.430331i
$136$	22.8167i	1.95651i
$137$	− 1.60555i	− 0.137172i	−0.997645	−	0.0685858i	$-0.978151\pi$
$137$	− 1.60555i	− 0.137172i	0.997645	−	0.0685858i	$-0.0218487\pi$
$138$	6.90833i	0.588076i
$139$	6.39445	0.542370	0.271185	−	0.962527i	$-0.412584\pi$
$139$	6.39445	0.542370	0.271185	+	0.962527i	$0.412584\pi$
$140$	−3.30278	−0.279135
$141$	9.21110i	0.775715i
$142$	11.0917	0.930793
$143$	0	0
$144$	−0.605551	−0.0504626
$145$	− 6.21110i	− 0.515804i
$146$	1.81665	0.150347
$147$	6.00000	0.494872
$148$	− 11.9083i	− 0.978858i
$149$	− 3.00000i	− 0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$	− 3.00000i	− 0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$	− 2.30278i	− 0.188021i
$151$	− 1.21110i	− 0.0985581i	−0.998785	−	0.0492791i	$-0.984308\pi$
$151$	− 1.21110i	− 0.0985581i	0.998785	−	0.0492791i	$-0.0156924\pi$
$152$	16.8167	1.36401
$153$	15.2111	1.22974
$154$	3.69722i	0.297931i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	11.2111	0.894743	0.447372	−	0.894348i	$-0.352360\pi$
$157$	11.2111	0.894743	0.447372	+	0.894348i	$0.352360\pi$
$158$	12.0000i	0.954669i
$159$	−3.21110	−0.254657
$160$	5.30278	0.419221
$161$	− 3.00000i	− 0.236433i
$162$	2.30278i	0.180923i
$163$	− 3.78890i	− 0.296769i	−0.988930	−	0.148385i	$-0.952593\pi$
$163$	− 3.78890i	− 0.296769i	0.988930	−	0.148385i	$-0.0474074\pi$
$164$	9.90833i	0.773710i
$165$	−1.60555	−0.124992
$166$	−21.2111	−1.64630
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	17.5139	1.34325
$171$	− 11.2111i	− 0.857334i
$172$	−33.7250	−2.57151
$173$	−4.81665	−0.366203	−0.183102	−	0.983094i	$-0.558614\pi$
$173$	−4.81665	−0.366203	−0.183102	+	0.983094i	$0.558614\pi$
$174$	− 14.3028i	− 1.08429i
$175$	1.00000i	0.0755929i
$176$	0.486122i	0.0366428i
$177$	− 10.8167i	− 0.813029i
$178$	14.3028	1.07204
$179$	−22.8167	−1.70540	−0.852698	−	0.522404i	$-0.825035\pi$
$179$	−22.8167	−1.70540	−0.852698	+	0.522404i	$0.825035\pi$
$180$	6.60555i	0.492349i
$181$	−17.6333	−1.31067	−0.655337	−	0.755337i	$-0.727473\pi$
$181$	−17.6333	−1.31067	−0.655337	+	0.755337i	$0.727473\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	− 9.00000i	− 0.663489i
$185$	−3.60555	−0.265085
$186$	−9.21110	−0.675391
$187$	− 12.2111i	− 0.892964i
$188$	− 30.4222i	− 2.21877i
$189$	5.00000i	0.363696i
$190$	− 12.9083i	− 0.936468i
$191$	16.8167	1.21681	0.608405	−	0.793627i	$-0.291810\pi$
$191$	16.8167	1.21681	0.608405	+	0.793627i	$0.291810\pi$
$192$	12.8167	0.924962
$193$	15.6056i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$193$	15.6056i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$194$	−19.3305	−1.38785
$195$	0	0
$196$	−19.8167	−1.41548
$197$	− 1.18335i	− 0.0843099i	−0.999111	−	0.0421550i	$-0.986578\pi$
$197$	− 1.18335i	− 0.0843099i	0.999111	−	0.0421550i	$-0.0134223\pi$
$198$	7.39445	0.525501
$199$	−12.8167	−0.908549	−0.454274	−	0.890862i	$-0.650101\pi$
$199$	−12.8167	−0.908549	−0.454274	+	0.890862i	$0.650101\pi$
$200$	3.00000i	0.212132i
$201$	7.00000i	0.493742i
$202$	20.7250i	1.45820i
$203$	6.21110i	0.435934i
$204$	25.1194	1.75871
$205$	3.00000	0.209529
$206$	9.21110i	0.641768i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−9.00000	−0.622543
$210$	2.30278i	0.158907i
$211$	−23.6056	−1.62507	−0.812537	−	0.582910i	$-0.801914\pi$
$211$	−23.6056	−1.62507	−0.812537	+	0.582910i	$0.801914\pi$
$212$	10.6056	0.728392
$213$	− 4.81665i	− 0.330032i
$214$	14.3028i	0.977718i
$215$	10.2111i	0.696391i
$216$	15.0000i	1.02062i
$217$	4.00000	0.271538
$218$	−44.2389	−2.99623
$219$	− 0.788897i	− 0.0533087i
$220$	5.30278	0.357513
$221$	0	0
$222$	−8.30278	−0.557246
$223$	4.21110i	0.281996i	0.990010	+	0.140998i	$0.0450312\pi$
$223$	4.21110i	0.281996i	−0.990010	+	0.140998i	$0.954969\pi$
$224$	−5.30278	−0.354307
$225$	2.00000	0.133333
$226$	− 3.69722i	− 0.245936i
$227$	27.4222i	1.82008i	0.414525	+	0.910038i	$0.363948\pi$
$227$	27.4222i	1.82008i	−0.414525	+	0.910038i	$0.636052\pi$
$228$	− 18.5139i	− 1.22611i
$229$	14.0000i	0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$	14.0000i	0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	−6.90833	−0.455522
$231$	1.60555	0.105638
$232$	18.6333i	1.22334i
$233$	−15.2111	−0.996512	−0.498256	−	0.867030i	$-0.666026\pi$
$233$	−15.2111	−0.996512	−0.498256	+	0.867030i	$0.666026\pi$
$234$	0	0
$235$	−9.21110	−0.600866
$236$	35.7250i	2.32550i
$237$	5.21110	0.338497
$238$	−17.5139	−1.13526
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0.302776i	0.0195441i
$241$	1.78890i	0.115233i	0.998339	+	0.0576165i	$0.0183501\pi$
$241$	1.78890i	0.115233i	−0.998339	+	0.0576165i	$0.981650\pi$
$242$	19.3944i	1.24672i
$243$	16.0000	1.02640
$244$	3.30278	0.211439
$245$	6.00000i	0.383326i
$246$	6.90833	0.440459
$247$	0	0
$248$	12.0000	0.762001
$249$	9.21110i	0.583730i
$250$	2.30278	0.145640
$251$	7.18335	0.453409	0.226704	−	0.973964i	$-0.427205\pi$
$251$	7.18335	0.453409	0.226704	+	0.973964i	$0.427205\pi$
$252$	− 6.60555i	− 0.416111i
$253$	4.81665i	0.302820i
$254$	9.69722i	0.608458i
$255$	− 7.60555i	− 0.476278i
$256$	−17.9083	−1.11927
$257$	16.3944	1.02266	0.511329	−	0.859385i	$-0.329153\pi$
$257$	16.3944	1.02266	0.511329	+	0.859385i	$0.329153\pi$
$258$	23.5139i	1.46391i
$259$	3.60555	0.224038
$260$	0	0
$261$	12.4222	0.768915
$262$	− 48.8444i	− 3.01762i
$263$	11.7889	0.726935	0.363467	−	0.931607i	$-0.381593\pi$
$263$	11.7889	0.726935	0.363467	+	0.931607i	$0.381593\pi$
$264$	4.81665	0.296445
$265$	− 3.21110i	− 0.197256i
$266$	12.9083i	0.791460i
$267$	− 6.21110i	− 0.380113i
$268$	− 23.1194i	− 1.41224i
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	11.5139	0.700712
$271$	− 20.8167i	− 1.26452i	−0.774756	−	0.632261i	$-0.782127\pi$
$271$	− 20.8167i	− 1.26452i	0.774756	−	0.632261i	$-0.217873\pi$
$272$	−2.30278	−0.139626
$273$	0	0
$274$	3.69722	0.223357
$275$	− 1.60555i	− 0.0968184i
$276$	−9.90833	−0.596411
$277$	−27.6056	−1.65866	−0.829328	−	0.558761i	$-0.811277\pi$
$277$	−27.6056	−1.65866	−0.829328	+	0.558761i	$0.811277\pi$
$278$	14.7250i	0.883146i
$279$	− 8.00000i	− 0.478947i
$280$	− 3.00000i	− 0.179284i
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	−21.2111	−1.26310
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	15.9083i	0.943986i
$285$	−5.60555	−0.332044
$286$	0	0
$287$	−3.00000	−0.177084
$288$	10.6056i	0.624938i
$289$	40.8444	2.40261
$290$	14.3028	0.839888
$291$	8.39445i	0.492091i
$292$	2.60555i	0.152478i
$293$	10.3944i	0.607250i	0.952792	+	0.303625i	$0.0981970\pi$
$293$	10.3944i	0.607250i	−0.952792	+	0.303625i	$0.901803\pi$
$294$	13.8167i	0.805804i
$295$	10.8167	0.629770
$296$	10.8167	0.628705
$297$	− 8.02776i	− 0.465818i
$298$	6.90833	0.400189
$299$	0	0
$300$	3.30278	0.190686
$301$	− 10.2111i	− 0.588558i
$302$	2.78890	0.160483
$303$	9.00000	0.517036
$304$	1.69722i	0.0973425i
$305$	− 1.00000i	− 0.0572598i
$306$	35.0278i	2.00240i
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	−5.30278	−0.302154
$309$	4.00000	0.227552
$310$	− 9.21110i	− 0.523155i
$311$	9.21110	0.522314	0.261157	−	0.965296i	$-0.415896\pi$
$311$	9.21110	0.522314	0.261157	+	0.965296i	$0.415896\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	25.8167i	1.45692i
$315$	−2.00000	−0.112687
$316$	−17.2111	−0.968200
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	− 7.39445i	− 0.414660i
$319$	− 9.97224i	− 0.558338i
$320$	12.8167i	0.716473i
$321$	6.21110	0.346670
$322$	6.90833	0.384986
$323$	− 42.6333i	− 2.37218i
$324$	−3.30278	−0.183488
$325$	0	0
$326$	8.72498	0.483232
$327$	19.2111i	1.06238i
$328$	−9.00000	−0.496942
$329$	9.21110	0.507825
$330$	− 3.69722i	− 0.203526i
$331$	− 10.0278i	− 0.551175i	−0.961276	−	0.275588i	$-0.911128\pi$
$331$	− 10.0278i	− 0.551175i	0.961276	−	0.275588i	$-0.0888724\pi$
$332$	− 30.4222i	− 1.66964i
$333$	− 7.21110i	− 0.395166i
$334$	−20.7250	−1.13402
$335$	−7.00000	−0.382451
$336$	− 0.302776i	− 0.0165178i
$337$	25.6333	1.39634	0.698168	−	0.715934i	$-0.253999\pi$
$337$	25.6333	1.39634	0.698168	+	0.715934i	$0.253999\pi$
$338$	0	0
$339$	−1.60555	−0.0872016
$340$	25.1194i	1.36229i
$341$	−6.42221	−0.347782
$342$	25.8167	1.39600
$343$	− 13.0000i	− 0.701934i
$344$	− 30.6333i	− 1.65164i
$345$	3.00000i	0.161515i
$346$	− 11.0917i	− 0.596292i
$347$	5.78890	0.310764	0.155382	−	0.987854i	$-0.450339\pi$
$347$	5.78890	0.310764	0.155382	+	0.987854i	$0.450339\pi$
$348$	20.5139	1.09966
$349$	− 3.78890i	− 0.202815i	−0.994845	−	0.101408i	$-0.967665\pi$
$349$	− 3.78890i	− 0.202815i	0.994845	−	0.101408i	$-0.0323346\pi$
$350$	−2.30278	−0.123089
$351$	0	0
$352$	8.51388	0.453791
$353$	− 16.8167i	− 0.895060i	−0.894269	−	0.447530i	$-0.852304\pi$
$353$	− 16.8167i	− 0.895060i	0.894269	−	0.447530i	$-0.147696\pi$
$354$	24.9083	1.32386
$355$	4.81665	0.255641
$356$	20.5139i	1.08723i
$357$	7.60555i	0.402528i
$358$	− 52.5416i	− 2.77691i
$359$	18.4222i	0.972287i	0.873879	+	0.486143i	$0.161597\pi$
$359$	18.4222i	0.972287i	−0.873879	+	0.486143i	$0.838403\pi$
$360$	−6.00000	−0.316228
$361$	−12.4222	−0.653800
$362$	− 40.6056i	− 2.13418i
$363$	8.42221	0.442051
$364$	0	0
$365$	0.788897	0.0412928
$366$	− 2.30278i	− 0.120368i
$367$	11.4222	0.596234	0.298117	−	0.954529i	$-0.403641\pi$
$367$	11.4222	0.596234	0.298117	+	0.954529i	$0.403641\pi$
$368$	0.908327	0.0473498
$369$	6.00000i	0.312348i
$370$	− 8.30278i	− 0.431641i
$371$	3.21110i	0.166712i
$372$	− 13.2111i	− 0.684964i
$373$	−20.3944	−1.05598	−0.527992	−	0.849249i	$-0.677055\pi$
$373$	−20.3944	−1.05598	−0.527992	+	0.849249i	$0.677055\pi$
$374$	28.1194	1.45402
$375$	− 1.00000i	− 0.0516398i
$376$	27.6333	1.42508
$377$	0	0
$378$	−11.5139	−0.592210
$379$	− 9.60555i	− 0.493404i	−0.969091	−	0.246702i	$-0.920653\pi$
$379$	− 9.60555i	− 0.493404i	0.969091	−	0.246702i	$-0.0793469\pi$
$380$	18.5139	0.949742
$381$	4.21110	0.215741
$382$	38.7250i	1.98134i
$383$	24.6333i	1.25870i	0.777121	+	0.629352i	$0.216679\pi$
$383$	24.6333i	1.25870i	−0.777121	+	0.629352i	$0.783321\pi$
$384$	18.9083i	0.964912i
$385$	1.60555i	0.0818265i
$386$	−35.9361	−1.82910
$387$	−20.4222	−1.03812
$388$	− 27.7250i	− 1.40752i
$389$	15.2111	0.771234	0.385617	−	0.922659i	$-0.373989\pi$
$389$	15.2111	0.771234	0.385617	+	0.922659i	$0.373989\pi$
$390$	0	0
$391$	−22.8167	−1.15389
$392$	− 18.0000i	− 0.909137i
$393$	−21.2111	−1.06996
$394$	2.72498	0.137283
$395$	5.21110i	0.262199i
$396$	10.6056i	0.532949i
$397$	22.0278i	1.10554i	0.833333	+	0.552771i	$0.186429\pi$
$397$	22.0278i	1.10554i	−0.833333	+	0.552771i	$0.813571\pi$
$398$	− 29.5139i	− 1.47940i
$399$	5.60555	0.280629
$400$	−0.302776	−0.0151388
$401$	12.2111i	0.609793i	0.952385	+	0.304897i	$0.0986219\pi$
$401$	12.2111i	0.609793i	−0.952385	+	0.304897i	$0.901378\pi$
$402$	−16.1194	−0.803964
$403$	0	0
$404$	−29.7250	−1.47887
$405$	1.00000i	0.0496904i
$406$	−14.3028	−0.709835
$407$	−5.78890	−0.286945
$408$	22.8167i	1.12959i
$409$	− 8.21110i	− 0.406013i	−0.979177	−	0.203006i	$-0.934929\pi$
$409$	− 8.21110i	− 0.406013i	0.979177	−	0.203006i	$-0.0650712\pi$
$410$	6.90833i	0.341178i
$411$	− 1.60555i	− 0.0791960i
$412$	−13.2111	−0.650864
$413$	−10.8167	−0.532253
$414$	− 13.8167i	− 0.679051i
$415$	−9.21110	−0.452155
$416$	0	0
$417$	6.39445	0.313138
$418$	− 20.7250i	− 1.01369i
$419$	17.2389	0.842173	0.421087	−	0.907020i	$-0.361649\pi$
$419$	17.2389	0.842173	0.421087	+	0.907020i	$0.361649\pi$
$420$	−3.30278	−0.161159
$421$	− 32.4222i	− 1.58016i	−0.613003	−	0.790081i	$-0.710039\pi$
$421$	− 32.4222i	− 1.58016i	0.613003	−	0.790081i	$-0.289961\pi$
$422$	− 54.3583i	− 2.64612i
$423$	− 18.4222i	− 0.895718i
$424$	9.63331i	0.467835i
$425$	7.60555	0.368923
$426$	11.0917	0.537393
$427$	1.00000i	0.0483934i
$428$	−20.5139	−0.991576
$429$	0	0
$430$	−23.5139	−1.13394
$431$	− 29.2389i	− 1.40839i	−0.710008	−	0.704193i	$-0.751309\pi$
$431$	− 29.2389i	− 1.40839i	0.710008	−	0.704193i	$-0.248691\pi$
$432$	−1.51388	−0.0728365
$433$	−3.60555	−0.173272	−0.0866359	−	0.996240i	$-0.527612\pi$
$433$	−3.60555	−0.173272	−0.0866359	+	0.996240i	$0.527612\pi$
$434$	9.21110i	0.442147i
$435$	− 6.21110i	− 0.297800i
$436$	− 63.4500i	− 3.03870i
$437$	16.8167i	0.804450i
$438$	1.81665	0.0868031
$439$	27.2389	1.30004	0.650020	−	0.759917i	$-0.274761\pi$
$439$	27.2389	1.30004	0.650020	+	0.759917i	$0.274761\pi$
$440$	4.81665i	0.229625i
$441$	−12.0000	−0.571429
$442$	0	0
$443$	6.42221	0.305128	0.152564	−	0.988294i	$-0.451247\pi$
$443$	6.42221	0.305128	0.152564	+	0.988294i	$0.451247\pi$
$444$	− 11.9083i	− 0.565144i
$445$	6.21110	0.294434
$446$	−9.69722	−0.459177
$447$	− 3.00000i	− 0.141895i
$448$	− 12.8167i	− 0.605530i
$449$	30.6333i	1.44568i	0.691018	+	0.722838i	$0.257163\pi$
$449$	30.6333i	1.44568i	−0.691018	+	0.722838i	$0.742837\pi$
$450$	4.60555i	0.217108i
$451$	4.81665	0.226807
$452$	5.30278	0.249422
$453$	− 1.21110i	− 0.0569026i
$454$	−63.1472	−2.96364
$455$	0	0
$456$	16.8167	0.787512
$457$	26.8167i	1.25443i	0.778846	+	0.627215i	$0.215805\pi$
$457$	26.8167i	1.25443i	−0.778846	+	0.627215i	$0.784195\pi$
$458$	−32.2389	−1.50642
$459$	38.0278	1.77498
$460$	− 9.90833i	− 0.461978i
$461$	− 36.2111i	− 1.68652i	−0.537507	−	0.843260i	$-0.680634\pi$
$461$	− 36.2111i	− 1.68652i	0.537507	−	0.843260i	$-0.319366\pi$
$462$	3.69722i	0.172010i
$463$	− 34.4222i	− 1.59974i	−0.600176	−	0.799868i	$-0.704903\pi$
$463$	− 34.4222i	− 1.59974i	0.600176	−	0.799868i	$-0.295097\pi$
$464$	−1.88057	−0.0873033
$465$	−4.00000	−0.185496
$466$	− 35.0278i	− 1.62263i
$467$	−2.78890	−0.129055	−0.0645274	−	0.997916i	$-0.520554\pi$
$467$	−2.78890	−0.129055	−0.0645274	+	0.997916i	$0.520554\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	− 21.2111i	− 0.978395i
$471$	11.2111	0.516580
$472$	−32.4500	−1.49363
$473$	16.3944i	0.753818i
$474$	12.0000i	0.551178i
$475$	− 5.60555i	− 0.257200i
$476$	− 25.1194i	− 1.15135i
$477$	6.42221	0.294053
$478$	0	0
$479$	− 28.8167i	− 1.31667i	−0.752727	−	0.658333i	$-0.771262\pi$
$479$	− 28.8167i	− 1.31667i	0.752727	−	0.658333i	$-0.228738\pi$
$480$	5.30278	0.242037
$481$	0	0
$482$	−4.11943	−0.187635
$483$	− 3.00000i	− 0.136505i
$484$	−27.8167	−1.26439
$485$	−8.39445	−0.381172
$486$	36.8444i	1.67130i
$487$	1.00000i	0.0453143i	0.999743	+	0.0226572i	$0.00721262\pi$
$487$	1.00000i	0.0453143i	−0.999743	+	0.0226572i	$0.992787\pi$
$488$	3.00000i	0.135804i
$489$	− 3.78890i	− 0.171340i
$490$	−13.8167	−0.624173
$491$	16.8167	0.758925	0.379462	−	0.925207i	$-0.376109\pi$
$491$	16.8167	0.758925	0.379462	+	0.925207i	$0.376109\pi$
$492$	9.90833i	0.446702i
$493$	47.2389	2.12753
$494$	0	0
$495$	3.21110	0.144328
$496$	1.21110i	0.0543801i
$497$	−4.81665	−0.216056
$498$	−21.2111	−0.950492
$499$	− 2.42221i	− 0.108433i	−0.998529	−	0.0542164i	$-0.982734\pi$
$499$	− 2.42221i	− 0.108433i	0.998529	−	0.0542164i	$-0.0172661\pi$
$500$	3.30278i	0.147705i
$501$	9.00000i	0.402090i
$502$	16.5416i	0.738289i
$503$	3.00000	0.133763	0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000	0.133763	0.0668817	+	0.997761i	$0.478695\pi$
$504$	6.00000	0.267261
$505$	9.00000i	0.400495i
$506$	−11.0917	−0.493085
$507$	0	0
$508$	−13.9083	−0.617082
$509$	− 3.00000i	− 0.132973i	−0.997787	−	0.0664863i	$-0.978821\pi$
$509$	− 3.00000i	− 0.132973i	0.997787	−	0.0664863i	$-0.0211789\pi$
$510$	17.5139	0.775528
$511$	−0.788897	−0.0348988
$512$	− 3.42221i	− 0.151242i
$513$	− 28.0278i	− 1.23746i
$514$	37.7527i	1.66520i
$515$	4.00000i	0.176261i
$516$	−33.7250	−1.48466
$517$	−14.7889	−0.650415
$518$	8.30278i	0.364803i
$519$	−4.81665	−0.211428
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	28.6056i	1.25203i
$523$	−1.42221	−0.0621887	−0.0310943	−	0.999516i	$-0.509899\pi$
$523$	−1.42221	−0.0621887	−0.0310943	+	0.999516i	$0.509899\pi$
$524$	70.0555	3.06039
$525$	1.00000i	0.0436436i
$526$	27.1472i	1.18367i
$527$	− 30.4222i	− 1.32521i
$528$	0.486122i	0.0211557i
$529$	−14.0000	−0.608696
$530$	7.39445	0.321194
$531$	21.6333i	0.938806i
$532$	−18.5139	−0.802678
$533$	0	0
$534$	14.3028	0.618942
$535$	6.21110i	0.268529i
$536$	21.0000	0.907062
$537$	−22.8167	−0.984611
$538$	− 20.7250i	− 0.893517i
$539$	9.63331i	0.414936i
$540$	16.5139i	0.710644i
$541$	− 25.6333i	− 1.10206i	−0.834485	−	0.551031i	$-0.814235\pi$
$541$	− 25.6333i	− 1.10206i	0.834485	−	0.551031i	$-0.185765\pi$
$542$	47.9361	2.05903
$543$	−17.6333	−0.756718
$544$	40.3305i	1.72916i
$545$	−19.2111	−0.822913
$546$	0	0
$547$	32.8444	1.40433	0.702163	−	0.712016i	$-0.252218\pi$
$547$	32.8444	1.40433	0.702163	+	0.712016i	$0.252218\pi$
$548$	5.30278i	0.226523i
$549$	2.00000	0.0853579
$550$	3.69722	0.157650
$551$	− 34.8167i	− 1.48324i
$552$	− 9.00000i	− 0.383065i
$553$	− 5.21110i	− 0.221599i
$554$	− 63.5694i	− 2.70080i
$555$	−3.60555	−0.153047
$556$	−21.1194	−0.895663
$557$	− 1.60555i	− 0.0680294i	−0.999421	−	0.0340147i	$-0.989171\pi$
$557$	− 1.60555i	− 0.0680294i	0.999421	−	0.0340147i	$-0.0108293\pi$
$558$	18.4222	0.779874
$559$	0	0
$560$	0.302776	0.0127946
$561$	− 12.2111i	− 0.515553i
$562$	13.8167	0.582820
$563$	−9.42221	−0.397099	−0.198549	−	0.980091i	$-0.563623\pi$
$563$	−9.42221	−0.397099	−0.198549	+	0.980091i	$0.563623\pi$
$564$	− 30.4222i	− 1.28101i
$565$	− 1.60555i	− 0.0675460i
$566$	− 11.5139i	− 0.483964i
$567$	− 1.00000i	− 0.0419961i
$568$	−14.4500	−0.606307
$569$	27.4222	1.14960	0.574799	−	0.818294i	$-0.305080\pi$
$569$	27.4222	1.14960	0.574799	+	0.818294i	$0.305080\pi$
$570$	− 12.9083i	− 0.540670i
$571$	−20.8444	−0.872311	−0.436156	−	0.899871i	$-0.643660\pi$
$571$	−20.8444	−0.872311	−0.436156	+	0.899871i	$0.643660\pi$
$572$	0	0
$573$	16.8167	0.702526
$574$	− 6.90833i	− 0.288348i
$575$	−3.00000	−0.125109
$576$	−25.6333	−1.06805
$577$	13.6333i	0.567562i	0.958889	+	0.283781i	$0.0915889\pi$
$577$	13.6333i	0.567562i	−0.958889	+	0.283781i	$0.908411\pi$
$578$	94.0555i	3.91219i
$579$	15.6056i	0.648545i
$580$	20.5139i	0.851792i
$581$	9.21110	0.382141
$582$	−19.3305	−0.801276
$583$	− 5.15559i	− 0.213523i
$584$	−2.36669	−0.0979344
$585$	0	0
$586$	−23.9361	−0.988790
$587$	33.4222i	1.37948i	0.724056	+	0.689741i	$0.242276\pi$
$587$	33.4222i	1.37948i	−0.724056	+	0.689741i	$0.757724\pi$
$588$	−19.8167	−0.817225
$589$	−22.4222	−0.923891
$590$	24.9083i	1.02546i
$591$	− 1.18335i	− 0.0486764i
$592$	1.09167i	0.0448675i
$593$	20.7889i	0.853698i	0.904323	+	0.426849i	$0.140376\pi$
$593$	20.7889i	0.853698i	−0.904323	+	0.426849i	$0.859624\pi$
$594$	18.4861	0.758495
$595$	−7.60555	−0.311797
$596$	9.90833i	0.405861i
$597$	−12.8167	−0.524551
$598$	0	0
$599$	−21.2111	−0.866662	−0.433331	−	0.901235i	$-0.642662\pi$
$599$	−21.2111	−0.866662	−0.433331	+	0.901235i	$0.642662\pi$
$600$	3.00000i	0.122474i
$601$	13.7889	0.562461	0.281230	−	0.959640i	$-0.409257\pi$
$601$	13.7889	0.562461	0.281230	+	0.959640i	$0.409257\pi$
$602$	23.5139	0.958354
$603$	− 14.0000i	− 0.570124i
$604$	4.00000i	0.162758i
$605$	8.42221i	0.342411i
$606$	20.7250i	0.841895i
$607$	−34.2111	−1.38859	−0.694293	−	0.719693i	$-0.744283\pi$
$607$	−34.2111	−1.38859	−0.694293	+	0.719693i	$0.744283\pi$
$608$	29.7250	1.20551
$609$	6.21110i	0.251687i
$610$	2.30278	0.0932367
$611$	0	0
$612$	−50.2389	−2.03079
$613$	5.60555i	0.226406i	0.993572	+	0.113203i	$0.0361111\pi$
$613$	5.60555i	0.226406i	−0.993572	+	0.113203i	$0.963889\pi$
$614$	36.8444	1.48692
$615$	3.00000	0.120972
$616$	− 4.81665i	− 0.194069i
$617$	38.4500i	1.54794i	0.633224	+	0.773969i	$0.281731\pi$
$617$	38.4500i	1.54794i	−0.633224	+	0.773969i	$0.718269\pi$
$618$	9.21110i	0.370525i
$619$	14.4222i	0.579677i	0.957076	+	0.289839i	$0.0936017\pi$
$619$	14.4222i	0.579677i	−0.957076	+	0.289839i	$0.906398\pi$
$620$	13.2111	0.530571
$621$	−15.0000	−0.601929
$622$	21.2111i	0.850488i
$623$	−6.21110	−0.248843
$624$	0	0
$625$	1.00000	0.0400000
$626$	32.2389i	1.28852i
$627$	−9.00000	−0.359425
$628$	−37.0278	−1.47757
$629$	− 27.4222i	− 1.09339i
$630$	− 4.60555i	− 0.183490i
$631$	− 36.0278i	− 1.43424i	−0.696949	−	0.717121i	$-0.745459\pi$
$631$	− 36.0278i	− 1.43424i	0.696949	−	0.717121i	$-0.254541\pi$
$632$	− 15.6333i	− 0.621860i
$633$	−23.6056	−0.938236
$634$	13.8167	0.548729
$635$	4.21110i	0.167113i
$636$	10.6056	0.420537
$637$	0	0
$638$	22.9638	0.909147
$639$	9.63331i	0.381088i
$640$	−18.9083	−0.747417
$641$	−9.42221	−0.372155	−0.186077	−	0.982535i	$-0.559578\pi$
$641$	−9.42221	−0.372155	−0.186077	+	0.982535i	$0.559578\pi$
$642$	14.3028i	0.564486i
$643$	− 2.63331i	− 0.103848i	−0.998651	−	0.0519238i	$-0.983465\pi$
$643$	− 2.63331i	− 0.103848i	0.998651	−	0.0519238i	$-0.0165353\pi$
$644$	9.90833i	0.390443i
$645$	10.2111i	0.402062i
$646$	98.1749	3.86264
$647$	−39.4222	−1.54985	−0.774923	−	0.632055i	$-0.782212\pi$
$647$	−39.4222	−1.54985	−0.774923	+	0.632055i	$0.782212\pi$
$648$	− 3.00000i	− 0.117851i
$649$	17.3667	0.681702
$650$	0	0
$651$	4.00000	0.156772
$652$	12.5139i	0.490081i
$653$	−7.18335	−0.281106	−0.140553	−	0.990073i	$-0.544888\pi$
$653$	−7.18335	−0.281106	−0.140553	+	0.990073i	$0.544888\pi$
$654$	−44.2389	−1.72988
$655$	− 21.2111i	− 0.828786i
$656$	− 0.908327i	− 0.0354642i
$657$	1.57779i	0.0615556i
$658$	21.2111i	0.826895i
$659$	−34.8167	−1.35626	−0.678132	−	0.734940i	$-0.737210\pi$
$659$	−34.8167	−1.35626	−0.678132	+	0.734940i	$0.737210\pi$
$660$	5.30278	0.206410
$661$	− 4.63331i	− 0.180215i	−0.995932	−	0.0901074i	$-0.971279\pi$
$661$	− 4.63331i	− 0.180215i	0.995932	−	0.0901074i	$-0.0287210\pi$
$662$	23.0917	0.897483
$663$	0	0
$664$	27.6333	1.07238
$665$	5.60555i	0.217374i
$666$	16.6056	0.643452
$667$	−18.6333	−0.721485
$668$	− 29.7250i	− 1.15009i
$669$	4.21110i	0.162811i
$670$	− 16.1194i	− 0.622748i
$671$	− 1.60555i	− 0.0619816i
$672$	−5.30278	−0.204559
$673$	17.6056	0.678644	0.339322	−	0.940670i	$-0.389802\pi$
$673$	17.6056	0.678644	0.339322	+	0.940670i	$0.389802\pi$
$674$	59.0278i	2.27366i
$675$	5.00000	0.192450
$676$	0	0
$677$	−9.63331	−0.370238	−0.185119	−	0.982716i	$-0.559267\pi$
$677$	−9.63331	−0.370238	−0.185119	+	0.982716i	$0.559267\pi$
$678$	− 3.69722i	− 0.141991i
$679$	8.39445	0.322149
$680$	−22.8167	−0.874979
$681$	27.4222i	1.05082i
$682$	− 14.7889i	− 0.566296i
$683$	36.2111i	1.38558i	0.721140	+	0.692790i	$0.243619\pi$
$683$	36.2111i	1.38558i	−0.721140	+	0.692790i	$0.756381\pi$
$684$	37.0278i	1.41579i
$685$	1.60555	0.0613450
$686$	29.9361	1.14296
$687$	14.0000i	0.534133i
$688$	3.09167	0.117869
$689$	0	0
$690$	−6.90833	−0.262996
$691$	30.0278i	1.14231i	0.820842	+	0.571155i	$0.193504\pi$
$691$	30.0278i	1.14231i	−0.820842	+	0.571155i	$0.806496\pi$
$692$	15.9083	0.604744
$693$	−3.21110	−0.121980
$694$	13.3305i	0.506020i
$695$	6.39445i	0.242555i
$696$	18.6333i	0.706294i
$697$	22.8167i	0.864242i
$698$	8.72498	0.330245
$699$	−15.2111	−0.575337
$700$	− 3.30278i	− 0.124833i
$701$	36.4222	1.37565	0.687824	−	0.725878i	$-0.258566\pi$
$701$	36.4222	1.37565	0.687824	+	0.725878i	$0.258566\pi$
$702$	0	0
$703$	−20.2111	−0.762276
$704$	20.5778i	0.775555i
$705$	−9.21110	−0.346910
$706$	38.7250	1.45743
$707$	− 9.00000i	− 0.338480i
$708$	35.7250i	1.34263i
$709$	− 13.8444i	− 0.519938i	−0.965617	−	0.259969i	$-0.916288\pi$
$709$	− 13.8444i	− 0.519938i	0.965617	−	0.259969i	$-0.0837123\pi$
$710$	11.0917i	0.416263i
$711$	−10.4222	−0.390863
$712$	−18.6333	−0.698313
$713$	12.0000i	0.449404i
$714$	−17.5139	−0.655440
$715$	0	0
$716$	75.3583	2.81627
$717$	0	0
$718$	−42.4222	−1.58318
$719$	25.6056	0.954926	0.477463	−	0.878652i	$-0.341556\pi$
$719$	25.6056	0.954926	0.477463	+	0.878652i	$0.341556\pi$
$720$	− 0.605551i	− 0.0225676i
$721$	− 4.00000i	− 0.148968i
$722$	− 28.6056i	− 1.06459i
$723$	1.78890i	0.0665298i
$724$	58.2389	2.16443
$725$	6.21110	0.230675
$726$	19.3944i	0.719796i
$727$	−13.5778	−0.503573	−0.251786	−	0.967783i	$-0.581018\pi$
$727$	−13.5778	−0.503573	−0.251786	+	0.967783i	$0.581018\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	1.81665i	0.0672374i
$731$	−77.6611	−2.87240
$732$	3.30278	0.122074
$733$	46.8444i	1.73024i	0.501567	+	0.865119i	$0.332757\pi$
$733$	46.8444i	1.73024i	−0.501567	+	0.865119i	$0.667243\pi$
$734$	26.3028i	0.970853i
$735$	6.00000i	0.221313i
$736$	− 15.9083i	− 0.586389i
$737$	−11.2389	−0.413989
$738$	−13.8167	−0.508598
$739$	− 35.6056i	− 1.30977i	−0.755728	−	0.654886i	$-0.772717\pi$
$739$	− 35.6056i	− 1.30977i	0.755728	−	0.654886i	$-0.227283\pi$
$740$	11.9083	0.437759
$741$	0	0
$742$	−7.39445	−0.271459
$743$	− 36.6333i	− 1.34395i	−0.740576	−	0.671973i	$-0.765447\pi$
$743$	− 36.6333i	− 1.34395i	0.740576	−	0.671973i	$-0.234553\pi$
$744$	12.0000	0.439941
$745$	3.00000	0.109911
$746$	− 46.9638i	− 1.71947i
$747$	− 18.4222i	− 0.674033i
$748$	40.3305i	1.47463i
$749$	− 6.21110i	− 0.226949i
$750$	2.30278	0.0840855
$751$	−46.4500	−1.69498	−0.847492	−	0.530809i	$-0.821888\pi$
$751$	−46.4500	−1.69498	−0.847492	+	0.530809i	$0.821888\pi$
$752$	2.78890i	0.101701i
$753$	7.18335	0.261776
$754$	0	0
$755$	1.21110	0.0440765
$756$	− 16.5139i	− 0.600604i
$757$	0.816654	0.0296818	0.0148409	−	0.999890i	$-0.495276\pi$
$757$	0.816654	0.0296818	0.0148409	+	0.999890i	$0.495276\pi$
$758$	22.1194	0.803414
$759$	4.81665i	0.174833i
$760$	16.8167i	0.610004i
$761$	18.6333i	0.675457i	0.941244	+	0.337728i	$0.109659\pi$
$761$	18.6333i	0.675457i	−0.941244	+	0.337728i	$0.890341\pi$
$762$	9.69722i	0.351293i
$763$	19.2111	0.695489
$764$	−55.5416	−2.00943
$765$	15.2111i	0.549959i
$766$	−56.7250	−2.04956
$767$	0	0
$768$	−17.9083	−0.646211
$769$	− 11.0000i	− 0.396670i	−0.980134	−	0.198335i	$-0.936447\pi$
$769$	− 11.0000i	− 0.396670i	0.980134	−	0.198335i	$-0.0635534\pi$
$770$	−3.69722	−0.133239
$771$	16.3944	0.590432
$772$	− 51.5416i	− 1.85502i
$773$	− 22.3944i	− 0.805472i	−0.915316	−	0.402736i	$-0.868059\pi$
$773$	− 22.3944i	− 0.805472i	0.915316	−	0.402736i	$-0.131941\pi$
$774$	− 47.0278i	− 1.69038i
$775$	− 4.00000i	− 0.143684i
$776$	25.1833	0.904029
$777$	3.60555	0.129348
$778$	35.0278i	1.25581i
$779$	16.8167	0.602519
$780$	0	0
$781$	7.73338	0.276722
$782$	− 52.5416i	− 1.87889i
$783$	31.0555	1.10983
$784$	1.81665	0.0648805
$785$	11.2111i	0.400141i
$786$	− 48.8444i	− 1.74222i
$787$	14.6333i	0.521621i	0.965390	+	0.260811i	$0.0839898\pi$
$787$	14.6333i	0.521621i	−0.965390	+	0.260811i	$0.916010\pi$
$788$	3.90833i	0.139228i
$789$	11.7889	0.419696
$790$	−12.0000	−0.426941
$791$	1.60555i	0.0570868i
$792$	−9.63331	−0.342305
$793$	0	0
$794$	−50.7250	−1.80016
$795$	− 3.21110i	− 0.113886i
$796$	42.3305	1.50037
$797$	14.4500	0.511844	0.255922	−	0.966697i	$-0.417621\pi$
$797$	14.4500	0.511844	0.255922	+	0.966697i	$0.417621\pi$
$798$	12.9083i	0.456950i
$799$	− 70.0555i	− 2.47839i
$800$	5.30278i	0.187481i
$801$	12.4222i	0.438917i
$802$	−28.1194	−0.992932
$803$	1.26662	0.0446979
$804$	− 23.1194i	− 0.815359i
$805$	3.00000	0.105736
$806$	0	0
$807$	−9.00000	−0.316815
$808$	− 27.0000i	− 0.949857i
$809$	−55.0555	−1.93565	−0.967824	−	0.251627i	$-0.919034\pi$
$809$	−55.0555	−1.93565	−0.967824	+	0.251627i	$0.919034\pi$
$810$	−2.30278	−0.0809113
$811$	46.4222i	1.63010i	0.579388	+	0.815052i	$0.303292\pi$
$811$	46.4222i	1.63010i	−0.579388	+	0.815052i	$0.696708\pi$
$812$	− 20.5139i	− 0.719896i
$813$	− 20.8167i	− 0.730072i
$814$	− 13.3305i	− 0.467235i
$815$	3.78890	0.132719
$816$	−2.30278	−0.0806133
$817$	57.2389i	2.00253i
$818$	18.9083	0.661114
$819$	0	0
$820$	−9.90833	−0.346014
$821$	− 21.4222i	− 0.747640i	−0.927501	−	0.373820i	$-0.878048\pi$
$821$	− 21.4222i	− 0.747640i	0.927501	−	0.373820i	$-0.121952\pi$
$822$	3.69722	0.128956
$823$	16.6333	0.579801	0.289900	−	0.957057i	$-0.406378\pi$
$823$	16.6333	0.579801	0.289900	+	0.957057i	$0.406378\pi$
$824$	− 12.0000i	− 0.418040i
$825$	− 1.60555i	− 0.0558981i
$826$	− 24.9083i	− 0.866672i
$827$	− 42.4222i	− 1.47516i	−0.675257	−	0.737582i	$-0.735967\pi$
$827$	− 42.4222i	− 1.47516i	0.675257	−	0.737582i	$-0.264033\pi$
$828$	19.8167	0.688676
$829$	−29.4222	−1.02188	−0.510938	−	0.859618i	$-0.670702\pi$
$829$	−29.4222	−1.02188	−0.510938	+	0.859618i	$0.670702\pi$
$830$	− 21.2111i	− 0.736248i
$831$	−27.6056	−0.957626
$832$	0	0
$833$	−45.6333	−1.58110
$834$	14.7250i	0.509884i
$835$	−9.00000	−0.311458
$836$	29.7250	1.02806
$837$	− 20.0000i	− 0.691301i
$838$	39.6972i	1.37132i
$839$	− 20.0278i	− 0.691435i	−0.938339	−	0.345717i	$-0.887636\pi$
$839$	− 20.0278i	− 0.691435i	0.938339	−	0.345717i	$-0.112364\pi$
$840$	− 3.00000i	− 0.103510i
$841$	9.57779	0.330269
$842$	74.6611	2.57299
$843$	− 6.00000i	− 0.206651i
$844$	77.9638	2.68363
$845$	0	0
$846$	42.4222	1.45851
$847$	− 8.42221i	− 0.289390i
$848$	−0.972244	−0.0333870
$849$	−5.00000	−0.171600
$850$	17.5139i	0.600721i
$851$	10.8167i	0.370790i
$852$	15.9083i	0.545010i
$853$	47.2111i	1.61648i	0.588855	+	0.808239i	$0.299579\pi$
$853$	47.2111i	1.61648i	−0.588855	+	0.808239i	$0.700421\pi$
$854$	−2.30278	−0.0787994
$855$	11.2111	0.383412
$856$	− 18.6333i	− 0.636873i
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	10.7889	0.368112	0.184056	−	0.982916i	$-0.441077\pi$
$859$	10.7889	0.368112	0.184056	+	0.982916i	$0.441077\pi$
$860$	− 33.7250i	− 1.15001i
$861$	−3.00000	−0.102240
$862$	67.3305	2.29329
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	26.5139i	0.902020i
$865$	− 4.81665i	− 0.163771i
$866$	− 8.30278i	− 0.282140i
$867$	40.8444	1.38715
$868$	−13.2111	−0.448414
$869$	8.36669i	0.283821i
$870$	14.3028	0.484910
$871$	0	0
$872$	57.6333	1.95171
$873$	− 16.7889i	− 0.568218i
$874$	−38.7250	−1.30989
$875$	−1.00000	−0.0338062
$876$	2.60555i	0.0880334i
$877$	1.97224i	0.0665979i	0.999445	+	0.0332990i	$0.0106014\pi$
$877$	1.97224i	0.0665979i	−0.999445	+	0.0332990i	$0.989399\pi$
$878$	62.7250i	2.11687i
$879$	10.3944i	0.350596i
$880$	−0.486122	−0.0163872
$881$	21.8444	0.735957	0.367978	−	0.929834i	$-0.380050\pi$
$881$	21.8444	0.735957	0.367978	+	0.929834i	$0.380050\pi$
$882$	− 27.6333i	− 0.930462i
$883$	−11.6333	−0.391492	−0.195746	−	0.980655i	$-0.562713\pi$
$883$	−11.6333	−0.391492	−0.195746	+	0.980655i	$0.562713\pi$
$884$	0	0
$885$	10.8167	0.363598
$886$	14.7889i	0.496843i
$887$	−37.0555	−1.24420	−0.622101	−	0.782937i	$-0.713721\pi$
$887$	−37.0555	−1.24420	−0.622101	+	0.782937i	$0.713721\pi$
$888$	10.8167	0.362983
$889$	− 4.21110i	− 0.141236i
$890$	14.3028i	0.479430i
$891$	1.60555i	0.0537880i
$892$	− 13.9083i	− 0.465685i
$893$	−51.6333	−1.72784
$894$	6.90833	0.231049
$895$	− 22.8167i	− 0.762677i
$896$	18.9083	0.631683
$897$	0	0
$898$	−70.5416	−2.35400
$899$	− 24.8444i	− 0.828607i
$900$	−6.60555	−0.220185
$901$	24.4222	0.813622
$902$	11.0917i	0.369312i
$903$	− 10.2111i	− 0.339804i
$904$	4.81665i	0.160200i
$905$	− 17.6333i	− 0.586151i
$906$	2.78890	0.0926549
$907$	38.2666	1.27062	0.635311	−	0.772256i	$-0.280872\pi$
$907$	38.2666	1.27062	0.635311	+	0.772256i	$0.280872\pi$
$908$	− 90.5694i	− 3.00565i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	1.69722i	0.0562007i
$913$	−14.7889	−0.489441
$914$	−61.7527	−2.04260
$915$	− 1.00000i	− 0.0330590i
$916$	− 46.2389i	− 1.52777i
$917$	21.2111i	0.700452i
$918$	87.5694i	2.89022i
$919$	−38.8167	−1.28044	−0.640222	−	0.768190i	$-0.721157\pi$
$919$	−38.8167	−1.28044	−0.640222	+	0.768190i	$0.721157\pi$
$920$	9.00000	0.296721
$921$	− 16.0000i	− 0.527218i
$922$	83.3860	2.74617
$923$	0	0
$924$	−5.30278	−0.174449
$925$	− 3.60555i	− 0.118550i
$926$	79.2666	2.60486
$927$	−8.00000	−0.262754
$928$	32.9361i	1.08118i
$929$	15.4222i	0.505986i	0.967468	+	0.252993i	$0.0814150\pi$
$929$	15.4222i	0.505986i	−0.967468	+	0.252993i	$0.918585\pi$
$930$	− 9.21110i	− 0.302044i
$931$	33.6333i	1.10229i
$932$	50.2389	1.64563
$933$	9.21110	0.301558
$934$	− 6.42221i	− 0.210141i
$935$	12.2111	0.399346
$936$	0	0
$937$	54.4777	1.77971	0.889855	−	0.456244i	$-0.150806\pi$
$937$	54.4777	1.77971	0.889855	+	0.456244i	$0.150806\pi$
$938$	16.1194i	0.526318i
$939$	14.0000	0.456873
$940$	30.4222	0.992263
$941$	9.63331i	0.314037i	0.987596	+	0.157018i	$0.0501882\pi$
$941$	9.63331i	0.314037i	−0.987596	+	0.157018i	$0.949812\pi$
$942$	25.8167i	0.841152i
$943$	− 9.00000i	− 0.293080i
$944$	− 3.27502i	− 0.106593i
$945$	−5.00000	−0.162650
$946$	−37.7527	−1.22745
$947$	− 18.6333i	− 0.605501i	−0.953070	−	0.302751i	$-0.902095\pi$
$947$	− 18.6333i	− 0.605501i	0.953070	−	0.302751i	$-0.0979049\pi$
$948$	−17.2111	−0.558991
$949$	0	0
$950$	12.9083	0.418801
$951$	− 6.00000i	− 0.194563i
$952$	22.8167	0.739492
$953$	14.4500	0.468080	0.234040	−	0.972227i	$-0.424805\pi$
$953$	14.4500	0.468080	0.234040	+	0.972227i	$0.424805\pi$
$954$	14.7889i	0.478808i
$955$	16.8167i	0.544174i
$956$	0	0
$957$	− 9.97224i	− 0.322357i
$958$	66.3583	2.14394
$959$	−1.60555	−0.0518460
$960$	12.8167i	0.413656i
$961$	15.0000	0.483871
$962$	0	0
$963$	−12.4222	−0.400300
$964$	− 5.90833i	− 0.190294i
$965$	−15.6056	−0.502360
$966$	6.90833	0.222272
$967$	44.4777i	1.43031i	0.698967	+	0.715153i	$0.253643\pi$
$967$	44.4777i	1.43031i	−0.698967	+	0.715153i	$0.746357\pi$
$968$	− 25.2666i	− 0.812100i
$969$	− 42.6333i	− 1.36958i
$970$	− 19.3305i	− 0.620666i
$971$	−44.0278	−1.41292	−0.706459	−	0.707754i	$-0.749709\pi$
$971$	−44.0278	−1.41292	−0.706459	+	0.707754i	$0.749709\pi$
$972$	−52.8444	−1.69499
$973$	− 6.39445i	− 0.204997i
$974$	−2.30278	−0.0737857
$975$	0	0
$976$	−0.302776	−0.00969161
$977$	− 28.8167i	− 0.921926i	−0.887419	−	0.460963i	$-0.847504\pi$
$977$	− 28.8167i	− 0.921926i	0.887419	−	0.460963i	$-0.152496\pi$
$978$	8.72498	0.278994
$979$	9.97224	0.318714
$980$	− 19.8167i	− 0.633020i
$981$	− 38.4222i	− 1.22673i
$982$	38.7250i	1.23576i
$983$	18.4222i	0.587577i	0.955870	+	0.293789i	$0.0949162\pi$
$983$	18.4222i	0.587577i	−0.955870	+	0.293789i	$0.905084\pi$
$984$	−9.00000	−0.286910
$985$	1.18335	0.0377045
$986$	108.780i	3.46428i
$987$	9.21110	0.293193
$988$	0	0
$989$	30.6333	0.974083
$990$	7.39445i	0.235011i
$991$	40.0278	1.27152	0.635762	−	0.771885i	$-0.280686\pi$
$991$	40.0278	1.27152	0.635762	+	0.771885i	$0.280686\pi$
$992$	21.2111	0.673453
$993$	− 10.0278i	− 0.318221i
$994$	− 11.0917i	− 0.351807i
$995$	− 12.8167i	− 0.406315i
$996$	− 30.4222i	− 0.963964i
$997$	−18.4500	−0.584316	−0.292158	−	0.956370i	$-0.594373\pi$
$997$	−18.4500	−0.584316	−0.292158	+	0.956370i	$0.594373\pi$
$998$	5.57779	0.176562
$999$	− 18.0278i	− 0.570373i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.d.506.4		4
13.2	odd	12		845.2.e.d.191.1		4
13.3	even	3		845.2.m.d.316.1		8
13.4	even	6		845.2.m.d.361.1		8
13.5	odd	4		845.2.a.f.1.2		2
13.6	odd	12		845.2.e.d.146.1		4
13.7	odd	12		65.2.e.b.16.2	✓	4
13.8	odd	4		845.2.a.c.1.1		2
13.9	even	3		845.2.m.d.361.4		8
13.10	even	6		845.2.m.d.316.4		8
13.11	odd	12		65.2.e.b.61.2	yes	4
13.12	even	2	inner	845.2.c.d.506.1		4
39.5	even	4		7605.2.a.bb.1.1		2
39.8	even	4		7605.2.a.bg.1.2		2
39.11	even	12		585.2.j.d.451.1		4
39.20	even	12		585.2.j.d.406.1		4
52.7	even	12		1040.2.q.o.81.2		4
52.11	even	12		1040.2.q.o.321.2		4
65.7	even	12		325.2.o.b.224.1		8
65.24	odd	12		325.2.e.a.126.1		4
65.33	even	12		325.2.o.b.224.4		8
65.34	odd	4		4225.2.a.x.1.2		2
65.37	even	12		325.2.o.b.74.4		8
65.44	odd	4		4225.2.a.t.1.1		2
65.59	odd	12		325.2.e.a.276.1		4
65.63	even	12		325.2.o.b.74.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.b.16.2	✓	4	13.7	odd	12
65.2.e.b.61.2	yes	4	13.11	odd	12
325.2.e.a.126.1		4	65.24	odd	12
325.2.e.a.276.1		4	65.59	odd	12
325.2.o.b.74.1		8	65.63	even	12
325.2.o.b.74.4		8	65.37	even	12
325.2.o.b.224.1		8	65.7	even	12
325.2.o.b.224.4		8	65.33	even	12
585.2.j.d.406.1		4	39.20	even	12
585.2.j.d.451.1		4	39.11	even	12
845.2.a.c.1.1		2	13.8	odd	4
845.2.a.f.1.2		2	13.5	odd	4
845.2.c.d.506.1		4	13.12	even	2	inner
845.2.c.d.506.4		4	1.1	even	1	trivial
845.2.e.d.146.1		4	13.6	odd	12
845.2.e.d.191.1		4	13.2	odd	12
845.2.m.d.316.1		8	13.3	even	3
845.2.m.d.316.4		8	13.10	even	6
845.2.m.d.361.1		8	13.4	even	6
845.2.m.d.361.4		8	13.9	even	3
1040.2.q.o.81.2		4	52.7	even	12
1040.2.q.o.321.2		4	52.11	even	12
4225.2.a.t.1.1		2	65.44	odd	4
4225.2.a.x.1.2		2	65.34	odd	4
7605.2.a.bb.1.1		2	39.5	even	4
7605.2.a.bg.1.2		2	39.8	even	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.30278i q^{2} +1.00000 q^{3} -3.30278 q^{4} +1.00000i q^{5} +2.30278i q^{6} -1.00000i q^{7} -3.00000i q^{8} -2.00000 q^{9} -2.30278 q^{10} +1.60555i q^{11} -3.30278 q^{12} +2.30278 q^{14} +1.00000i q^{15} +0.302776 q^{16} -7.60555 q^{17} -4.60555i q^{18} +5.60555i q^{19} -3.30278i q^{20} -1.00000i q^{21} -3.69722 q^{22} +3.00000 q^{23} -3.00000i q^{24} -1.00000 q^{25} -5.00000 q^{27} +3.30278i q^{28} -6.21110 q^{29} -2.30278 q^{30} +4.00000i q^{31} -5.30278i q^{32} +1.60555i q^{33} -17.5139i q^{34} +1.00000 q^{35} +6.60555 q^{36} +3.60555i q^{37} -12.9083 q^{38} +3.00000 q^{40} -3.00000i q^{41} +2.30278 q^{42} +10.2111 q^{43} -5.30278i q^{44} -2.00000i q^{45} +6.90833i q^{46} +9.21110i q^{47} +0.302776 q^{48} +6.00000 q^{49} -2.30278i q^{50} -7.60555 q^{51} -3.21110 q^{53} -11.5139i q^{54} -1.60555 q^{55} -3.00000 q^{56} +5.60555i q^{57} -14.3028i q^{58} -10.8167i q^{59} -3.30278i q^{60} -1.00000 q^{61} -9.21110 q^{62} +2.00000i q^{63} +12.8167 q^{64} -3.69722 q^{66} +7.00000i q^{67} +25.1194 q^{68} +3.00000 q^{69} +2.30278i q^{70} -4.81665i q^{71} +6.00000i q^{72} -0.788897i q^{73} -8.30278 q^{74} -1.00000 q^{75} -18.5139i q^{76} +1.60555 q^{77} +5.21110 q^{79} +0.302776i q^{80} +1.00000 q^{81} +6.90833 q^{82} +9.21110i q^{83} +3.30278i q^{84} -7.60555i q^{85} +23.5139i q^{86} -6.21110 q^{87} +4.81665 q^{88} -6.21110i q^{89} +4.60555 q^{90} -9.90833 q^{92} +4.00000i q^{93} -21.2111 q^{94} -5.60555 q^{95} -5.30278i q^{96} +8.39445i q^{97} +13.8167i q^{98} -3.21110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9} - 2 q^{10} - 6 q^{12} + 2 q^{14} - 6 q^{16} - 16 q^{17} - 22 q^{22} + 12 q^{23} - 4 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 12 q^{36} - 30 q^{38} + 12 q^{40}+ \cdots - 8 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 6 * q^4 - 8 * q^9 - 2 * q^10 - 6 * q^12 + 2 * q^14 - 6 * q^16 - 16 * q^17 - 22 * q^22 + 12 * q^23 - 4 * q^25 - 20 * q^27 + 4 * q^29 - 2 * q^30 + 4 * q^35 + 12 * q^36 - 30 * q^38 + 12 * q^40 + 2 * q^42 + 12 * q^43 - 6 * q^48 + 24 * q^49 - 16 * q^51 + 16 * q^53 + 8 * q^55 - 12 * q^56 - 4 * q^61 - 8 * q^62 + 8 * q^64 - 22 * q^66 + 50 * q^68 + 12 * q^69 - 26 * q^74 - 4 * q^75 - 8 * q^77 - 8 * q^79 + 4 * q^81 + 6 * q^82 + 4 * q^87 - 24 * q^88 + 4 * q^90 - 18 * q^92 - 56 * q^94 - 8 * q^95

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