LMFDB - Embedded newform 845.2.c.a.506.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(506,845)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.a.506.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000i q^{11} -2.00000 q^{12} +4.00000 q^{14} -2.00000i q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} +8.00000i q^{21} -2.00000 q^{22} +6.00000 q^{23} -6.00000i q^{24} -1.00000 q^{25} +4.00000 q^{27} -4.00000i q^{28} +2.00000 q^{29} +2.00000 q^{30} +10.0000i q^{31} +5.00000i q^{32} -4.00000i q^{33} -2.00000i q^{34} +4.00000 q^{35} +1.00000 q^{36} -2.00000i q^{37} -6.00000 q^{38} -3.00000 q^{40} +6.00000i q^{41} -8.00000 q^{42} -10.0000 q^{43} +2.00000i q^{44} +1.00000i q^{45} +6.00000i q^{46} +4.00000i q^{47} +2.00000 q^{48} -9.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} +2.00000 q^{53} +4.00000i q^{54} -2.00000 q^{55} +12.0000 q^{56} -12.0000i q^{57} +2.00000i q^{58} +6.00000i q^{59} -2.00000i q^{60} +2.00000 q^{61} -10.0000 q^{62} -4.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} -2.00000 q^{68} -12.0000 q^{69} +4.00000i q^{70} -6.00000i q^{71} +3.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} +2.00000 q^{75} +6.00000i q^{76} +8.00000 q^{77} -12.0000 q^{79} -1.00000i q^{80} -11.0000 q^{81} -6.00000 q^{82} +16.0000i q^{83} +8.00000i q^{84} -2.00000i q^{85} -10.0000i q^{86} -4.00000 q^{87} -6.00000 q^{88} +2.00000i q^{89} -1.00000 q^{90} +6.00000 q^{92} -20.0000i q^{93} -4.00000 q^{94} -6.00000 q^{95} -10.0000i q^{96} +2.00000i q^{97} -9.00000i q^{98} +2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^4 + 2 * q^9	$ 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{10} - 4 q^{12} + 8 q^{14} - 2 q^{16} - 4 q^{17} - 4 q^{22} + 12 q^{23} - 2 q^{25} + 8 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{35} + 2 q^{36} - 12 q^{38} - 6 q^{40} - 16 q^{42} - 20 q^{43} + 4 q^{48} - 18 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 24 q^{56} + 4 q^{61} - 20 q^{62} - 14 q^{64} + 8 q^{66} - 4 q^{68} - 24 q^{69} + 4 q^{74} + 4 q^{75} + 16 q^{77} - 24 q^{79} - 22 q^{81} - 12 q^{82} - 8 q^{87} - 12 q^{88} - 2 q^{90} + 12 q^{92} - 8 q^{94} - 12 q^{95}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^10 - 4 * q^12 + 8 * q^14 - 2 * q^16 - 4 * q^17 - 4 * q^22 + 12 * q^23 - 2 * q^25 + 8 * q^27 + 4 * q^29 + 4 * q^30 + 8 * q^35 + 2 * q^36 - 12 * q^38 - 6 * q^40 - 16 * q^42 - 20 * q^43 + 4 * q^48 - 18 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 24 * q^56 + 4 * q^61 - 20 * q^62 - 14 * q^64 + 8 * q^66 - 4 * q^68 - 24 * q^69 + 4 * q^74 + 4 * q^75 + 16 * q^77 - 24 * q^79 - 22 * q^81 - 12 * q^82 - 8 * q^87 - 12 * q^88 - 2 * q^90 + 12 * q^92 - 8 * q^94 - 12 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 2.00000i	− 0.816497i
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	3.00000i	1.06066i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	4.00000	1.06904
$15$	− 2.00000i	− 0.516398i
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000i	0.235702i
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	1.00000i	0.223607i
$21$	8.00000i	1.74574i
$22$	−2.00000	−0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	− 6.00000i	− 1.22474i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	− 4.00000i	− 0.755929i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	10.0000i	1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$	10.0000i	1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$	5.00000i	0.883883i
$33$	− 4.00000i	− 0.696311i
$34$	− 2.00000i	− 0.342997i
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−3.00000	−0.474342
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	−8.00000	−1.23443
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.00000i	0.301511i
$45$	1.00000i	0.149071i
$46$	6.00000i	0.884652i
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	2.00000	0.288675
$49$	−9.00000	−1.28571
$50$	− 1.00000i	− 0.141421i
$51$	4.00000	0.560112
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	4.00000i	0.544331i
$55$	−2.00000	−0.269680
$56$	12.0000	1.60357
$57$	− 12.0000i	− 1.58944i
$58$	2.00000i	0.262613i
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	− 2.00000i	− 0.258199i
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−10.0000	−1.27000
$63$	− 4.00000i	− 0.503953i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	−2.00000	−0.242536
$69$	−12.0000	−1.44463
$70$	4.00000i	0.478091i
$71$	− 6.00000i	− 0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$	− 6.00000i	− 0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$	3.00000i	0.353553i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	2.00000	0.232495
$75$	2.00000	0.230940
$76$	6.00000i	0.688247i
$77$	8.00000	0.911685
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	− 1.00000i	− 0.111803i
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	8.00000i	0.872872i
$85$	− 2.00000i	− 0.216930i
$86$	− 10.0000i	− 1.07833i
$87$	−4.00000	−0.428845
$88$	−6.00000	−0.639602
$89$	2.00000i	0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$	2.00000i	0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	− 20.0000i	− 2.07390i
$94$	−4.00000	−0.412568
$95$	−6.00000	−0.615587
$96$	− 10.0000i	− 1.02062i
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 9.00000i	− 0.909137i
$99$	2.00000i	0.201008i
$100$	−1.00000	−0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	4.00000i	0.396059i
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	2.00000i	0.194257i
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	4.00000	0.384900
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	− 2.00000i	− 0.190693i
$111$	4.00000i	0.379663i
$112$	4.00000i	0.377964i
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	12.0000	1.12390
$115$	6.00000i	0.559503i
$116$	2.00000	0.185695
$117$	0	0
$118$	−6.00000	−0.552345
$119$	8.00000i	0.733359i
$120$	6.00000	0.547723
$121$	7.00000	0.636364
$122$	2.00000i	0.181071i
$123$	− 12.0000i	− 1.08200i
$124$	10.0000i	0.898027i
$125$	− 1.00000i	− 0.0894427i
$126$	4.00000	0.356348
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	3.00000i	0.265165i
$129$	20.0000	1.76090
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	− 4.00000i	− 0.348155i
$133$	24.0000	2.08106
$134$	−4.00000	−0.345547
$135$	4.00000i	0.344265i
$136$	− 6.00000i	− 0.514496i
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	− 12.0000i	− 1.02151i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	4.00000	0.338062
$141$	− 8.00000i	− 0.673722i
$142$	6.00000	0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	2.00000i	0.166091i
$146$	6.00000	0.496564
$147$	18.0000	1.48461
$148$	− 2.00000i	− 0.164399i
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	2.00000i	0.163299i
$151$	10.0000i	0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$	10.0000i	0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	−18.0000	−1.45999
$153$	−2.00000	−0.161690
$154$	8.00000i	0.644658i
$155$	−10.0000	−0.803219
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	− 12.0000i	− 0.954669i
$159$	−4.00000	−0.317221
$160$	−5.00000	−0.395285
$161$	− 24.0000i	− 1.89146i
$162$	− 11.0000i	− 0.864242i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	6.00000i	0.468521i
$165$	4.00000	0.311400
$166$	−16.0000	−1.24184
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	−24.0000	−1.85164
$169$	0	0
$170$	2.00000	0.153393
$171$	6.00000i	0.458831i
$172$	−10.0000	−0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	− 4.00000i	− 0.303239i
$175$	4.00000i	0.302372i
$176$	− 2.00000i	− 0.150756i
$177$	− 12.0000i	− 0.901975i
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000i	0.0745356i
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	18.0000i	1.32698i
$185$	2.00000	0.147043
$186$	20.0000	1.46647
$187$	− 4.00000i	− 0.292509i
$188$	4.00000i	0.291730i
$189$	− 16.0000i	− 1.16383i
$190$	− 6.00000i	− 0.435286i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	14.0000	1.01036
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	−2.00000	−0.142134
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	− 3.00000i	− 0.212132i
$201$	− 8.00000i	− 0.564276i
$202$	18.0000i	1.26648i
$203$	− 8.00000i	− 0.561490i
$204$	4.00000	0.280056
$205$	−6.00000	−0.419058
$206$	− 2.00000i	− 0.139347i
$207$	6.00000	0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	− 8.00000i	− 0.552052i
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	2.00000	0.137361
$213$	12.0000i	0.822226i
$214$	10.0000i	0.683586i
$215$	− 10.0000i	− 0.681994i
$216$	12.0000i	0.816497i
$217$	40.0000	2.71538
$218$	10.0000	0.677285
$219$	12.0000i	0.810885i
$220$	−2.00000	−0.134840
$221$	0	0
$222$	−4.00000	−0.268462
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	20.0000	1.33631
$225$	−1.00000	−0.0666667
$226$	− 14.0000i	− 0.931266i
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	− 12.0000i	− 0.794719i
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	−6.00000	−0.395628
$231$	−16.0000	−1.05272
$232$	6.00000i	0.393919i
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	6.00000i	0.390567i
$237$	24.0000	1.55897
$238$	−8.00000	−0.518563
$239$	6.00000i	0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$	6.00000i	0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$	2.00000i	0.129099i
$241$	10.0000i	0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$	10.0000i	0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$	7.00000i	0.449977i
$243$	10.0000	0.641500
$244$	2.00000	0.128037
$245$	− 9.00000i	− 0.574989i
$246$	12.0000	0.765092
$247$	0	0
$248$	−30.0000	−1.90500
$249$	− 32.0000i	− 2.02792i
$250$	1.00000	0.0632456
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	− 4.00000i	− 0.251976i
$253$	12.0000i	0.754434i
$254$	2.00000i	0.125491i
$255$	4.00000i	0.250490i
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	20.0000i	1.24515i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	20.0000i	1.23560i
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	12.0000	0.738549
$265$	2.00000i	0.122859i
$266$	24.0000i	1.47153i
$267$	− 4.00000i	− 0.244796i
$268$	4.00000i	0.244339i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−4.00000	−0.243432
$271$	2.00000i	0.121491i	0.998153	+	0.0607457i	$0.0193479\pi$
$271$	2.00000i	0.121491i	−0.998153	+	0.0607457i	$0.980652\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	− 2.00000i	− 0.120605i
$276$	−12.0000	−0.722315
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	10.0000i	0.598684i
$280$	12.0000i	0.717137i
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	8.00000	0.476393
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	− 6.00000i	− 0.356034i
$285$	12.0000	0.710819
$286$	0	0
$287$	24.0000	1.41668
$288$	5.00000i	0.294628i
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	− 4.00000i	− 0.234484i
$292$	− 6.00000i	− 0.351123i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	18.0000i	1.04978i
$295$	−6.00000	−0.349334
$296$	6.00000	0.348743
$297$	8.00000i	0.464207i
$298$	18.0000	1.04271
$299$	0	0
$300$	2.00000	0.115470
$301$	40.0000i	2.30556i
$302$	−10.0000	−0.575435
$303$	−36.0000	−2.06815
$304$	− 6.00000i	− 0.344124i
$305$	2.00000i	0.114520i
$306$	− 2.00000i	− 0.114332i
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	8.00000	0.455842
$309$	4.00000	0.227552
$310$	− 10.0000i	− 0.567962i
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	− 6.00000i	− 0.338600i
$315$	4.00000	0.225374
$316$	−12.0000	−0.675053
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 4.00000i	− 0.224309i
$319$	4.00000i	0.223957i
$320$	− 7.00000i	− 0.391312i
$321$	−20.0000	−1.11629
$322$	24.0000	1.33747
$323$	− 12.0000i	− 0.667698i
$324$	−11.0000	−0.611111
$325$	0	0
$326$	12.0000	0.664619
$327$	20.0000i	1.10600i
$328$	−18.0000	−0.993884
$329$	16.0000	0.882109
$330$	4.00000i	0.220193i
$331$	18.0000i	0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$	18.0000i	0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$	16.0000i	0.878114i
$333$	− 2.00000i	− 0.109599i
$334$	12.0000	0.656611
$335$	−4.00000	−0.218543
$336$	− 8.00000i	− 0.436436i
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	28.0000	1.52075
$340$	− 2.00000i	− 0.108465i
$341$	−20.0000	−1.08306
$342$	−6.00000	−0.324443
$343$	8.00000i	0.431959i
$344$	− 30.0000i	− 1.61749i
$345$	− 12.0000i	− 0.646058i
$346$	6.00000i	0.322562i
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	−4.00000	−0.214423
$349$	− 30.0000i	− 1.60586i	−0.596071	−	0.802932i	$-0.703272\pi$
$349$	− 30.0000i	− 1.60586i	0.596071	−	0.802932i	$-0.296728\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−10.0000	−0.533002
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	12.0000	0.637793
$355$	6.00000	0.318447
$356$	2.00000i	0.106000i
$357$	− 16.0000i	− 0.846810i
$358$	− 12.0000i	− 0.634220i
$359$	− 10.0000i	− 0.527780i	−0.964553	−	0.263890i	$-0.914994\pi$
$359$	− 10.0000i	− 0.527780i	0.964553	−	0.263890i	$-0.0850056\pi$
$360$	−3.00000	−0.158114
$361$	−17.0000	−0.894737
$362$	22.0000i	1.15629i
$363$	−14.0000	−0.734809
$364$	0	0
$365$	6.00000	0.314054
$366$	− 4.00000i	− 0.209083i
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−6.00000	−0.312772
$369$	6.00000i	0.312348i
$370$	2.00000i	0.103975i
$371$	− 8.00000i	− 0.415339i
$372$	− 20.0000i	− 1.03695i
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	4.00000	0.206835
$375$	2.00000i	0.103280i
$376$	−12.0000	−0.618853
$377$	0	0
$378$	16.0000	0.822951
$379$	− 10.0000i	− 0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$	− 10.0000i	− 0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$	−6.00000	−0.307794
$381$	−4.00000	−0.204926
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	− 6.00000i	− 0.306186i
$385$	8.00000i	0.407718i
$386$	2.00000	0.101797
$387$	−10.0000	−0.508329
$388$	2.00000i	0.101535i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	− 27.0000i	− 1.36371i
$393$	−40.0000	−2.01773
$394$	−6.00000	−0.302276
$395$	− 12.0000i	− 0.603786i
$396$	2.00000i	0.100504i
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	16.0000i	0.802008i
$399$	−48.0000	−2.40301
$400$	1.00000	0.0500000
$401$	10.0000i	0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$	10.0000i	0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	18.0000	0.895533
$405$	− 11.0000i	− 0.546594i
$406$	8.00000	0.397033
$407$	4.00000	0.198273
$408$	12.0000i	0.594089i
$409$	− 18.0000i	− 0.890043i	−0.895520	−	0.445021i	$-0.853196\pi$
$409$	− 18.0000i	− 0.890043i	0.895520	−	0.445021i	$-0.146804\pi$
$410$	− 6.00000i	− 0.296319i
$411$	4.00000i	0.197305i
$412$	−2.00000	−0.0985329
$413$	24.0000	1.18096
$414$	6.00000i	0.294884i
$415$	−16.0000	−0.785409
$416$	0	0
$417$	0	0
$418$	− 12.0000i	− 0.586939i
$419$	−40.0000	−1.95413	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000	−1.95413	−0.977064	+	0.212946i	$0.931694\pi$
$420$	−8.00000	−0.390360
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	12.0000i	0.584151i
$423$	4.00000i	0.194487i
$424$	6.00000i	0.291386i
$425$	2.00000	0.0970143
$426$	−12.0000	−0.581402
$427$	− 8.00000i	− 0.387147i
$428$	10.0000	0.483368
$429$	0	0
$430$	10.0000	0.482243
$431$	− 14.0000i	− 0.674356i	−0.941441	−	0.337178i	$-0.890528\pi$
$431$	− 14.0000i	− 0.674356i	0.941441	−	0.337178i	$-0.109472\pi$
$432$	−4.00000	−0.192450
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	40.0000i	1.92006i
$435$	− 4.00000i	− 0.191785i
$436$	− 10.0000i	− 0.478913i
$437$	36.0000i	1.72211i
$438$	−12.0000	−0.573382
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	− 6.00000i	− 0.286039i
$441$	−9.00000	−0.428571
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	4.00000i	0.189832i
$445$	−2.00000	−0.0948091
$446$	4.00000	0.189405
$447$	36.0000i	1.70274i
$448$	28.0000i	1.32288i
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−12.0000	−0.565058
$452$	−14.0000	−0.658505
$453$	− 20.0000i	− 0.939682i
$454$	4.00000	0.187729
$455$	0	0
$456$	36.0000	1.68585
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	22.0000	1.02799
$459$	−8.00000	−0.373408
$460$	6.00000i	0.279751i
$461$	− 10.0000i	− 0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$	− 10.0000i	− 0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$	− 16.0000i	− 0.744387i
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−2.00000	−0.0928477
$465$	20.0000	0.927478
$466$	− 10.0000i	− 0.463241i
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	− 4.00000i	− 0.184506i
$471$	12.0000	0.552931
$472$	−18.0000	−0.828517
$473$	− 20.0000i	− 0.919601i
$474$	24.0000i	1.10236i
$475$	− 6.00000i	− 0.275299i
$476$	8.00000i	0.366679i
$477$	2.00000	0.0915737
$478$	−6.00000	−0.274434
$479$	30.0000i	1.37073i	0.728197	+	0.685367i	$0.240358\pi$
$479$	30.0000i	1.37073i	−0.728197	+	0.685367i	$0.759642\pi$
$480$	10.0000	0.456435
$481$	0	0
$482$	−10.0000	−0.455488
$483$	48.0000i	2.18408i
$484$	7.00000	0.318182
$485$	−2.00000	−0.0908153
$486$	10.0000i	0.453609i
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	6.00000i	0.271607i
$489$	24.0000i	1.08532i
$490$	9.00000	0.406579
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	− 12.0000i	− 0.541002i
$493$	−4.00000	−0.180151
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	− 10.0000i	− 0.449013i
$497$	−24.0000	−1.07655
$498$	32.0000	1.43395
$499$	− 10.0000i	− 0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$	− 10.0000i	− 0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	24.0000i	1.07224i
$502$	− 24.0000i	− 1.07117i
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	12.0000	0.534522
$505$	18.0000i	0.800989i
$506$	−12.0000	−0.533465
$507$	0	0
$508$	2.00000	0.0887357
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	−4.00000	−0.177123
$511$	−24.0000	−1.06170
$512$	− 11.0000i	− 0.486136i
$513$	24.0000i	1.05963i
$514$	− 2.00000i	− 0.0882162i
$515$	− 2.00000i	− 0.0881305i
$516$	20.0000	0.880451
$517$	−8.00000	−0.351840
$518$	− 8.00000i	− 0.351500i
$519$	−12.0000	−0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	2.00000i	0.0875376i
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	20.0000	0.873704
$525$	− 8.00000i	− 0.349149i
$526$	14.0000i	0.610429i
$527$	− 20.0000i	− 0.871214i
$528$	4.00000i	0.174078i
$529$	13.0000	0.565217
$530$	−2.00000	−0.0868744
$531$	6.00000i	0.260378i
$532$	24.0000	1.04053
$533$	0	0
$534$	4.00000	0.173097
$535$	10.0000i	0.432338i
$536$	−12.0000	−0.518321
$537$	24.0000	1.03568
$538$	6.00000i	0.258678i
$539$	− 18.0000i	− 0.775315i
$540$	4.00000i	0.172133i
$541$	− 22.0000i	− 0.945854i	−0.881102	−	0.472927i	$-0.843197\pi$
$541$	− 22.0000i	− 0.945854i	0.881102	−	0.472927i	$-0.156803\pi$
$542$	−2.00000	−0.0859074
$543$	−44.0000	−1.88822
$544$	− 10.0000i	− 0.428746i
$545$	10.0000	0.428353
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	2.00000	0.0853579
$550$	2.00000	0.0852803
$551$	12.0000i	0.511217i
$552$	− 36.0000i	− 1.53226i
$553$	48.0000i	2.04117i
$554$	14.0000i	0.594803i
$555$	−4.00000	−0.169791
$556$	0	0
$557$	− 26.0000i	− 1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$	− 26.0000i	− 1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$	−10.0000	−0.423334
$559$	0	0
$560$	−4.00000	−0.169031
$561$	8.00000i	0.337760i
$562$	6.00000	0.253095
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	− 8.00000i	− 0.336861i
$565$	− 14.0000i	− 0.588984i
$566$	− 2.00000i	− 0.0840663i
$567$	44.0000i	1.84783i
$568$	18.0000	0.755263
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	12.0000i	0.502625i
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	0	0
$574$	24.0000i	1.00174i
$575$	−6.00000	−0.250217
$576$	−7.00000	−0.291667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	− 13.0000i	− 0.540729i
$579$	4.00000i	0.166234i
$580$	2.00000i	0.0830455i
$581$	64.0000	2.65517
$582$	4.00000	0.165805
$583$	4.00000i	0.165663i
$584$	18.0000	0.744845
$585$	0	0
$586$	−22.0000	−0.908812
$587$	44.0000i	1.81607i	0.418890	+	0.908037i	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	18.0000	0.742307
$589$	−60.0000	−2.47226
$590$	− 6.00000i	− 0.247016i
$591$	− 12.0000i	− 0.493614i
$592$	2.00000i	0.0821995i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	−8.00000	−0.328244
$595$	−8.00000	−0.327968
$596$	− 18.0000i	− 0.737309i
$597$	−32.0000	−1.30967
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	6.00000i	0.244949i
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−40.0000	−1.63028
$603$	4.00000i	0.162893i
$604$	10.0000i	0.406894i
$605$	7.00000i	0.284590i
$606$	− 36.0000i	− 1.46240i
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−30.0000	−1.21666
$609$	16.0000i	0.648353i
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−8.00000	−0.322854
$615$	12.0000	0.483887
$616$	24.0000i	0.966988i
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	4.00000i	0.160904i
$619$	− 2.00000i	− 0.0803868i	−0.999192	−	0.0401934i	$-0.987203\pi$
$619$	− 2.00000i	− 0.0803868i	0.999192	−	0.0401934i	$-0.0127974\pi$
$620$	−10.0000	−0.401610
$621$	24.0000	0.963087
$622$	4.00000i	0.160385i
$623$	8.00000	0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 22.0000i	− 0.879297i
$627$	24.0000	0.958468
$628$	−6.00000	−0.239426
$629$	4.00000i	0.159490i
$630$	4.00000i	0.159364i
$631$	− 14.0000i	− 0.557331i	−0.960388	−	0.278666i	$-0.910108\pi$
$631$	− 14.0000i	− 0.557331i	0.960388	−	0.278666i	$-0.0898921\pi$
$632$	− 36.0000i	− 1.43200i
$633$	−24.0000	−0.953914
$634$	−18.0000	−0.714871
$635$	2.00000i	0.0793676i
$636$	−4.00000	−0.158610
$637$	0	0
$638$	−4.00000	−0.158362
$639$	− 6.00000i	− 0.237356i
$640$	−3.00000	−0.118585
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	− 20.0000i	− 0.789337i
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	− 24.0000i	− 0.945732i
$645$	20.0000i	0.787499i
$646$	12.0000	0.472134
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	− 33.0000i	− 1.29636i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−80.0000	−3.13545
$652$	− 12.0000i	− 0.469956i
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−20.0000	−0.782062
$655$	20.0000i	0.781465i
$656$	− 6.00000i	− 0.234261i
$657$	− 6.00000i	− 0.234082i
$658$	16.0000i	0.623745i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	4.00000	0.155700
$661$	2.00000i	0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$	2.00000i	0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$	−18.0000	−0.699590
$663$	0	0
$664$	−48.0000	−1.86276
$665$	24.0000i	0.930680i
$666$	2.00000	0.0774984
$667$	12.0000	0.464642
$668$	− 12.0000i	− 0.464294i
$669$	8.00000i	0.309298i
$670$	− 4.00000i	− 0.154533i
$671$	4.00000i	0.154418i
$672$	−40.0000	−1.54303
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	− 26.0000i	− 1.00148i
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	28.0000i	1.07533i
$679$	8.00000	0.307012
$680$	6.00000	0.230089
$681$	8.00000i	0.306561i
$682$	− 20.0000i	− 0.765840i
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	6.00000i	0.229416i
$685$	2.00000	0.0764161
$686$	−8.00000	−0.305441
$687$	44.0000i	1.67870i
$688$	10.0000	0.381246
$689$	0	0
$690$	12.0000	0.456832
$691$	− 22.0000i	− 0.836919i	−0.908235	−	0.418460i	$-0.862570\pi$
$691$	− 22.0000i	− 0.836919i	0.908235	−	0.418460i	$-0.137430\pi$
$692$	6.00000	0.228086
$693$	8.00000	0.303895
$694$	− 22.0000i	− 0.835109i
$695$	0	0
$696$	− 12.0000i	− 0.454859i
$697$	− 12.0000i	− 0.454532i
$698$	30.0000	1.13552
$699$	20.0000	0.756469
$700$	4.00000i	0.151186i
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	− 14.0000i	− 0.527645i
$705$	8.00000	0.301297
$706$	18.0000	0.677439
$707$	− 72.0000i	− 2.70784i
$708$	− 12.0000i	− 0.450988i
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	6.00000i	0.225176i
$711$	−12.0000	−0.450035
$712$	−6.00000	−0.224860
$713$	60.0000i	2.24702i
$714$	16.0000	0.598785
$715$	0	0
$716$	−12.0000	−0.448461
$717$	− 12.0000i	− 0.448148i
$718$	10.0000	0.373197
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	8.00000i	0.297936i
$722$	− 17.0000i	− 0.632674i
$723$	− 20.0000i	− 0.743808i
$724$	22.0000	0.817624
$725$	−2.00000	−0.0742781
$726$	− 14.0000i	− 0.519589i
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	6.00000i	0.222070i
$731$	20.0000	0.739727
$732$	−4.00000	−0.147844
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	− 14.0000i	− 0.516749i
$735$	18.0000i	0.663940i
$736$	30.0000i	1.10581i
$737$	−8.00000	−0.294684
$738$	−6.00000	−0.220863
$739$	6.00000i	0.220714i	0.993892	+	0.110357i	$0.0351994\pi$
$739$	6.00000i	0.220714i	−0.993892	+	0.110357i	$0.964801\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	8.00000	0.293689
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	60.0000	2.19971
$745$	18.0000	0.659469
$746$	34.0000i	1.24483i
$747$	16.0000i	0.585409i
$748$	− 4.00000i	− 0.146254i
$749$	− 40.0000i	− 1.46157i
$750$	−2.00000	−0.0730297
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	− 4.00000i	− 0.145865i
$753$	48.0000	1.74922
$754$	0	0
$755$	−10.0000	−0.363937
$756$	− 16.0000i	− 0.581914i
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	10.0000	0.363216
$759$	− 24.0000i	− 0.871145i
$760$	− 18.0000i	− 0.652929i
$761$	18.0000i	0.652499i	0.945284	+	0.326250i	$0.105785\pi$
$761$	18.0000i	0.652499i	−0.945284	+	0.326250i	$0.894215\pi$
$762$	− 4.00000i	− 0.144905i
$763$	−40.0000	−1.44810
$764$	0	0
$765$	− 2.00000i	− 0.0723102i
$766$	12.0000	0.433578
$767$	0	0
$768$	34.0000	1.22687
$769$	− 26.0000i	− 0.937584i	−0.883309	−	0.468792i	$-0.844689\pi$
$769$	− 26.0000i	− 0.937584i	0.883309	−	0.468792i	$-0.155311\pi$
$770$	−8.00000	−0.288300
$771$	4.00000	0.144056
$772$	− 2.00000i	− 0.0719816i
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	− 10.0000i	− 0.359443i
$775$	− 10.0000i	− 0.359211i
$776$	−6.00000	−0.215387
$777$	16.0000	0.573997
$778$	10.0000i	0.358517i
$779$	−36.0000	−1.28983
$780$	0	0
$781$	12.0000	0.429394
$782$	− 12.0000i	− 0.429119i
$783$	8.00000	0.285897
$784$	9.00000	0.321429
$785$	− 6.00000i	− 0.214149i
$786$	− 40.0000i	− 1.42675i
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	6.00000i	0.213741i
$789$	−28.0000	−0.996826
$790$	12.0000	0.426941
$791$	56.0000i	1.99113i
$792$	−6.00000	−0.213201
$793$	0	0
$794$	−6.00000	−0.212932
$795$	− 4.00000i	− 0.141865i
$796$	16.0000	0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	− 48.0000i	− 1.69918i
$799$	− 8.00000i	− 0.283020i
$800$	− 5.00000i	− 0.176777i
$801$	2.00000i	0.0706665i
$802$	−10.0000	−0.353112
$803$	12.0000	0.423471
$804$	− 8.00000i	− 0.282138i
$805$	24.0000	0.845889
$806$	0	0
$807$	−12.0000	−0.422420
$808$	54.0000i	1.89971i
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	11.0000	0.386501
$811$	42.0000i	1.47482i	0.675446	+	0.737410i	$0.263951\pi$
$811$	42.0000i	1.47482i	−0.675446	+	0.737410i	$0.736049\pi$
$812$	− 8.00000i	− 0.280745i
$813$	− 4.00000i	− 0.140286i
$814$	4.00000i	0.140200i
$815$	12.0000	0.420342
$816$	−4.00000	−0.140028
$817$	− 60.0000i	− 2.09913i
$818$	18.0000	0.629355
$819$	0	0
$820$	−6.00000	−0.209529
$821$	− 50.0000i	− 1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	− 50.0000i	− 1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$	−4.00000	−0.139516
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	− 6.00000i	− 0.209020i
$825$	4.00000i	0.139262i
$826$	24.0000i	0.835067i
$827$	− 32.0000i	− 1.11275i	−0.830932	−	0.556375i	$-0.812192\pi$
$827$	− 32.0000i	− 1.11275i	0.830932	−	0.556375i	$-0.187808\pi$
$828$	6.00000	0.208514
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	− 16.0000i	− 0.555368i
$831$	−28.0000	−0.971309
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	12.0000	0.415277
$836$	−12.0000	−0.415029
$837$	40.0000i	1.38260i
$838$	− 40.0000i	− 1.38178i
$839$	38.0000i	1.31191i	0.754802	+	0.655953i	$0.227733\pi$
$839$	38.0000i	1.31191i	−0.754802	+	0.655953i	$0.772267\pi$
$840$	− 24.0000i	− 0.828079i
$841$	−25.0000	−0.862069
$842$	10.0000	0.344623
$843$	12.0000i	0.413302i
$844$	12.0000	0.413057
$845$	0	0
$846$	−4.00000	−0.137523
$847$	− 28.0000i	− 0.962091i
$848$	−2.00000	−0.0686803
$849$	4.00000	0.137280
$850$	2.00000i	0.0685994i
$851$	− 12.0000i	− 0.411355i
$852$	12.0000i	0.411113i
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	8.00000	0.273754
$855$	−6.00000	−0.205196
$856$	30.0000i	1.02538i
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	− 10.0000i	− 0.340997i
$861$	−48.0000	−1.63584
$862$	14.0000	0.476842
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	20.0000i	0.680414i
$865$	6.00000i	0.204006i
$866$	− 10.0000i	− 0.339814i
$867$	26.0000	0.883006
$868$	40.0000	1.35769
$869$	− 24.0000i	− 0.814144i
$870$	4.00000	0.135613
$871$	0	0
$872$	30.0000	1.01593
$873$	2.00000i	0.0676897i
$874$	−36.0000	−1.21772
$875$	−4.00000	−0.135225
$876$	12.0000i	0.405442i
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	− 44.0000i	− 1.48408i
$880$	2.00000	0.0674200
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	− 9.00000i	− 0.303046i
$883$	22.0000	0.740359	0.370179	−	0.928960i	$-0.379296\pi$
$883$	22.0000	0.740359	0.370179	+	0.928960i	$0.379296\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	14.0000i	0.470339i
$887$	58.0000	1.94745	0.973725	−	0.227728i	$-0.0731298\pi$
$887$	58.0000	1.94745	0.973725	+	0.227728i	$0.0731298\pi$
$888$	−12.0000	−0.402694
$889$	− 8.00000i	− 0.268311i
$890$	− 2.00000i	− 0.0670402i
$891$	− 22.0000i	− 0.737028i
$892$	− 4.00000i	− 0.133930i
$893$	−24.0000	−0.803129
$894$	−36.0000	−1.20402
$895$	− 12.0000i	− 0.401116i
$896$	12.0000	0.400892
$897$	0	0
$898$	6.00000	0.200223
$899$	20.0000i	0.667037i
$900$	−1.00000	−0.0333333
$901$	−4.00000	−0.133259
$902$	− 12.0000i	− 0.399556i
$903$	− 80.0000i	− 2.66223i
$904$	− 42.0000i	− 1.39690i
$905$	22.0000i	0.731305i
$906$	20.0000	0.664455
$907$	34.0000	1.12895	0.564476	−	0.825450i	$-0.309078\pi$
$907$	34.0000	1.12895	0.564476	+	0.825450i	$0.309078\pi$
$908$	− 4.00000i	− 0.132745i
$909$	18.0000	0.597022
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	12.0000i	0.397360i
$913$	−32.0000	−1.05905
$914$	38.0000	1.25693
$915$	− 4.00000i	− 0.132236i
$916$	− 22.0000i	− 0.726900i
$917$	− 80.0000i	− 2.64183i
$918$	− 8.00000i	− 0.264039i
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	−18.0000	−0.593442
$921$	− 16.0000i	− 0.527218i
$922$	10.0000	0.329332
$923$	0	0
$924$	−16.0000	−0.526361
$925$	2.00000i	0.0657596i
$926$	24.0000	0.788689
$927$	−2.00000	−0.0656886
$928$	10.0000i	0.328266i
$929$	38.0000i	1.24674i	0.781927	+	0.623370i	$0.214237\pi$
$929$	38.0000i	1.24674i	−0.781927	+	0.623370i	$0.785763\pi$
$930$	20.0000i	0.655826i
$931$	− 54.0000i	− 1.76978i
$932$	−10.0000	−0.327561
$933$	−8.00000	−0.261908
$934$	10.0000i	0.327210i
$935$	4.00000	0.130814
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	16.0000i	0.522419i
$939$	44.0000	1.43589
$940$	−4.00000	−0.130466
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	12.0000i	0.390981i
$943$	36.0000i	1.17232i
$944$	− 6.00000i	− 0.195283i
$945$	16.0000	0.520480
$946$	20.0000	0.650256
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	24.0000	0.779484
$949$	0	0
$950$	6.00000	0.194666
$951$	− 36.0000i	− 1.16738i
$952$	−24.0000	−0.777844
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	2.00000i	0.0647524i
$955$	0	0
$956$	6.00000i	0.194054i
$957$	− 8.00000i	− 0.258603i
$958$	−30.0000	−0.969256
$959$	−8.00000	−0.258333
$960$	14.0000i	0.451848i
$961$	−69.0000	−2.22581
$962$	0	0
$963$	10.0000	0.322245
$964$	10.0000i	0.322078i
$965$	2.00000	0.0643823
$966$	−48.0000	−1.54437
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	21.0000i	0.674966i
$969$	24.0000i	0.770991i
$970$	− 2.00000i	− 0.0642161i
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	−24.0000	−0.767435
$979$	−4.00000	−0.127841
$980$	− 9.00000i	− 0.287494i
$981$	− 10.0000i	− 0.319275i
$982$	24.0000i	0.765871i
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	36.0000	1.14764
$985$	−6.00000	−0.191176
$986$	− 4.00000i	− 0.127386i
$987$	−32.0000	−1.01857
$988$	0	0
$989$	−60.0000	−1.90789
$990$	− 2.00000i	− 0.0635642i
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	−50.0000	−1.58750
$993$	− 36.0000i	− 1.14243i
$994$	− 24.0000i	− 0.761234i
$995$	16.0000i	0.507234i
$996$	− 32.0000i	− 1.01396i
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	10.0000	0.316544
$999$	− 8.00000i	− 0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.a.506.2		2
13.2	odd	12		845.2.e.a.191.1		2
13.3	even	3		845.2.m.b.316.1		4
13.4	even	6		845.2.m.b.361.1		4
13.5	odd	4		845.2.a.a.1.1		1
13.6	odd	12		845.2.e.a.146.1		2
13.7	odd	12		845.2.e.b.146.1		2
13.8	odd	4		65.2.a.a.1.1	✓	1
13.9	even	3		845.2.m.b.361.2		4
13.10	even	6		845.2.m.b.316.2		4
13.11	odd	12		845.2.e.b.191.1		2
13.12	even	2	inner	845.2.c.a.506.1		2
39.5	even	4		7605.2.a.f.1.1		1
39.8	even	4		585.2.a.h.1.1		1
52.47	even	4		1040.2.a.f.1.1		1
65.8	even	4		325.2.b.b.274.2		2
65.34	odd	4		325.2.a.d.1.1		1
65.44	odd	4		4225.2.a.g.1.1		1
65.47	even	4		325.2.b.b.274.1		2
91.34	even	4		3185.2.a.e.1.1		1
104.21	odd	4		4160.2.a.q.1.1		1
104.99	even	4		4160.2.a.f.1.1		1
143.21	even	4		7865.2.a.c.1.1		1
156.47	odd	4		9360.2.a.ca.1.1		1
195.8	odd	4		2925.2.c.h.2224.1		2
195.47	odd	4		2925.2.c.h.2224.2		2
195.164	even	4		2925.2.a.f.1.1		1
260.99	even	4		5200.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.a.1.1	✓	1	13.8	odd	4
325.2.a.d.1.1		1	65.34	odd	4
325.2.b.b.274.1		2	65.47	even	4
325.2.b.b.274.2		2	65.8	even	4
585.2.a.h.1.1		1	39.8	even	4
845.2.a.a.1.1		1	13.5	odd	4
845.2.c.a.506.1		2	13.12	even	2	inner
845.2.c.a.506.2		2	1.1	even	1	trivial
845.2.e.a.146.1		2	13.6	odd	12
845.2.e.a.191.1		2	13.2	odd	12
845.2.e.b.146.1		2	13.7	odd	12
845.2.e.b.191.1		2	13.11	odd	12
845.2.m.b.316.1		4	13.3	even	3
845.2.m.b.316.2		4	13.10	even	6
845.2.m.b.361.1		4	13.4	even	6
845.2.m.b.361.2		4	13.9	even	3
1040.2.a.f.1.1		1	52.47	even	4
2925.2.a.f.1.1		1	195.164	even	4
2925.2.c.h.2224.1		2	195.8	odd	4
2925.2.c.h.2224.2		2	195.47	odd	4
3185.2.a.e.1.1		1	91.34	even	4
4160.2.a.f.1.1		1	104.99	even	4
4160.2.a.q.1.1		1	104.21	odd	4
4225.2.a.g.1.1		1	65.44	odd	4
5200.2.a.d.1.1		1	260.99	even	4
7605.2.a.f.1.1		1	39.5	even	4
7865.2.a.c.1.1		1	143.21	even	4
9360.2.a.ca.1.1		1	156.47	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000i q^{11} -2.00000 q^{12} +4.00000 q^{14} -2.00000i q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} +8.00000i q^{21} -2.00000 q^{22} +6.00000 q^{23} -6.00000i q^{24} -1.00000 q^{25} +4.00000 q^{27} -4.00000i q^{28} +2.00000 q^{29} +2.00000 q^{30} +10.0000i q^{31} +5.00000i q^{32} -4.00000i q^{33} -2.00000i q^{34} +4.00000 q^{35} +1.00000 q^{36} -2.00000i q^{37} -6.00000 q^{38} -3.00000 q^{40} +6.00000i q^{41} -8.00000 q^{42} -10.0000 q^{43} +2.00000i q^{44} +1.00000i q^{45} +6.00000i q^{46} +4.00000i q^{47} +2.00000 q^{48} -9.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} +2.00000 q^{53} +4.00000i q^{54} -2.00000 q^{55} +12.0000 q^{56} -12.0000i q^{57} +2.00000i q^{58} +6.00000i q^{59} -2.00000i q^{60} +2.00000 q^{61} -10.0000 q^{62} -4.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} -2.00000 q^{68} -12.0000 q^{69} +4.00000i q^{70} -6.00000i q^{71} +3.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} +2.00000 q^{75} +6.00000i q^{76} +8.00000 q^{77} -12.0000 q^{79} -1.00000i q^{80} -11.0000 q^{81} -6.00000 q^{82} +16.0000i q^{83} +8.00000i q^{84} -2.00000i q^{85} -10.0000i q^{86} -4.00000 q^{87} -6.00000 q^{88} +2.00000i q^{89} -1.00000 q^{90} +6.00000 q^{92} -20.0000i q^{93} -4.00000 q^{94} -6.00000 q^{95} -10.0000i q^{96} +2.00000i q^{97} -9.00000i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^4 + 2 * q^9	\( 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{10} - 4 q^{12} + 8 q^{14} - 2 q^{16} - 4 q^{17} - 4 q^{22} + 12 q^{23} - 2 q^{25} + 8 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{35} + 2 q^{36} - 12 q^{38} - 6 q^{40} - 16 q^{42} - 20 q^{43} + 4 q^{48} - 18 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 24 q^{56} + 4 q^{61} - 20 q^{62} - 14 q^{64} + 8 q^{66} - 4 q^{68} - 24 q^{69} + 4 q^{74} + 4 q^{75} + 16 q^{77} - 24 q^{79} - 22 q^{81} - 12 q^{82} - 8 q^{87} - 12 q^{88} - 2 q^{90} + 12 q^{92} - 8 q^{94} - 12 q^{95}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^10 - 4 * q^12 + 8 * q^14 - 2 * q^16 - 4 * q^17 - 4 * q^22 + 12 * q^23 - 2 * q^25 + 8 * q^27 + 4 * q^29 + 4 * q^30 + 8 * q^35 + 2 * q^36 - 12 * q^38 - 6 * q^40 - 16 * q^42 - 20 * q^43 + 4 * q^48 - 18 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 24 * q^56 + 4 * q^61 - 20 * q^62 - 14 * q^64 + 8 * q^66 - 4 * q^68 - 24 * q^69 + 4 * q^74 + 4 * q^75 + 16 * q^77 - 24 * q^79 - 22 * q^81 - 12 * q^82 - 8 * q^87 - 12 * q^88 - 2 * q^90 + 12 * q^92 - 8 * q^94 - 12 * q^95

Introduction

L-functions

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