LMFDB - Embedded newform 4600.2.e.j.4049.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,6,0,-12,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7311849298$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.j.4049.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{7} +3.00000 q^{9} -6.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +6.00000 q^{19} -1.00000i q^{23} -3.00000 q^{29} -3.00000 q^{31} +1.00000i q^{37} +9.00000 q^{41} +8.00000i q^{43} +4.00000i q^{47} +6.00000 q^{49} -1.00000i q^{53} -1.00000 q^{59} +8.00000 q^{61} +3.00000i q^{63} -7.00000i q^{67} -5.00000 q^{71} +6.00000i q^{73} -6.00000i q^{77} +9.00000 q^{81} +11.0000i q^{83} -4.00000 q^{89} -2.00000 q^{91} +6.00000i q^{97} -18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{9} - 12 q^{11} + 12 q^{19} - 6 q^{29} - 6 q^{31} + 18 q^{41} + 12 q^{49} - 2 q^{59} + 16 q^{61} - 10 q^{71} + 18 q^{81} - 8 q^{89} - 4 q^{91} - 36 q^{99}+O(q^{100}) $ 2 * q + 6 * q^9 - 12 * q^11 + 12 * q^19 - 6 * q^29 - 6 * q^31 + 18 * q^41 + 12 * q^49 - 2 * q^59 + 16 * q^61 - 10 * q^71 + 18 * q^81 - 8 * q^89 - 4 * q^91 - 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.00000i	− 0.137361i	−0.997639	−	0.0686803i	$-0.978121\pi$
$53$	− 1.00000i	− 0.137361i	0.997639	−	0.0686803i	$-0.0218788\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.00000	−0.130189	−0.0650945	−	0.997879i	$-0.520735\pi$
$59$	−1.00000	−0.130189	−0.0650945	+	0.997879i	$0.520735\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	3.00000i	0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	11.0000i	1.20741i	0.797209	+	0.603703i	$0.206309\pi$
$83$	11.0000i	1.20741i	−0.797209	+	0.603703i	$0.793691\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	0	0
$99$	−18.0000	−1.80907
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.00000i	0.483368i	0.970355	+	0.241684i	$0.0776998\pi$
$107$	5.00000i	0.483368i	−0.970355	+	0.241684i	$0.922300\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.00000i	0.658505i	0.944242	+	0.329252i	$0.106797\pi$
$113$	7.00000i	0.658505i	−0.944242	+	0.329252i	$0.893203\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	6.00000i	0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 12.0000i	− 1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	− 9.00000i	− 0.727607i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.00000i	0.239426i	0.992809	+	0.119713i	$0.0381975\pi$
$157$	3.00000i	0.239426i	−0.992809	+	0.119713i	$0.961803\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	18.0000	1.37649
$172$	0	0
$173$	16.0000i	1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$	16.0000i	1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.0000i	1.31629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.00000i	− 0.208514i
$208$	0	0
$209$	−36.0000	−2.49017
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.00000i	− 0.203653i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0000i	1.24757i	0.781598	+	0.623783i	$0.214405\pi$
$257$	20.0000i	1.24757i	−0.781598	+	0.623783i	$0.785595\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	− 3.00000i	− 0.184988i	−0.995713	−	0.0924940i	$-0.970516\pi$
$263$	− 3.00000i	− 0.184988i	0.995713	−	0.0924940i	$-0.0294839\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.0000	1.89010	0.945052	−	0.326921i	$-0.106011\pi$
$269$	31.0000	1.89010	0.945052	+	0.326921i	$0.106011\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	−9.00000	−0.538816
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	17.0000i	1.01055i	0.862960	+	0.505273i	$0.168608\pi$
$283$	17.0000i	1.01055i	−0.862960	+	0.505273i	$0.831392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000i	0.531253i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.0000i	1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$	21.0000i	1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	29.0000i	1.63918i	0.572953	+	0.819588i	$0.305798\pi$
$313$	29.0000i	1.63918i	−0.572953	+	0.819588i	$0.694202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.00000i	− 0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$	− 8.00000i	− 0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 18.0000i	− 1.00155i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	3.00000i	0.164399i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.0000	0.974755
$342$	0	0
$343$	13.0000i	0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	35.0000i	1.82699i	0.406855	+	0.913493i	$0.366625\pi$
$367$	35.0000i	1.82699i	−0.406855	+	0.913493i	$0.633375\pi$
$368$	0	0
$369$	27.0000	1.40556
$370$	0	0
$371$	1.00000	0.0519174
$372$	0	0
$373$	38.0000i	1.96757i	0.179364	+	0.983783i	$0.442596\pi$
$373$	38.0000i	1.96757i	−0.179364	+	0.983783i	$0.557404\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.0000i	1.21999i
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 32.0000i	− 1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$	− 32.0000i	− 1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.00000i	− 0.0492068i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 17.0000i	− 0.816968i	−0.912766	−	0.408484i	$-0.866058\pi$
$433$	− 17.0000i	− 0.816968i	0.912766	−	0.408484i	$-0.133942\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 25.0000i	− 1.16945i	−0.811231	−	0.584725i	$-0.801202\pi$
$457$	− 25.0000i	− 1.16945i	0.811231	−	0.584725i	$-0.198798\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.0000i	− 1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$	− 27.0000i	− 1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 48.0000i	− 2.20704i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.00000i	− 0.137361i
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11.0000	0.496423	0.248212	−	0.968706i	$-0.420157\pi$
$491$	11.0000	0.496423	0.248212	+	0.968706i	$0.420157\pi$
$492$	0	0
$493$	9.00000i	0.405340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.00000i	− 0.224281i
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.0000i	0.757993i	0.925398	+	0.378996i	$0.123731\pi$
$503$	17.0000i	0.757993i	−0.925398	+	0.378996i	$0.876269\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 24.0000i	− 1.05552i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.00000i	0.392046i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	18.0000i	0.779667i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−36.0000	−1.55063
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	24.0000	1.02430
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.0000i	− 1.31351i	−0.754103	−	0.656756i	$-0.771928\pi$
$557$	− 31.0000i	− 1.31351i	0.754103	−	0.656756i	$-0.228072\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 11.0000i	− 0.463595i	−0.972764	−	0.231797i	$-0.925539\pi$
$563$	− 11.0000i	− 0.463595i	0.972764	−	0.231797i	$-0.0744606\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.00000i	0.377964i
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.0000i	0.499567i	0.968302	+	0.249783i	$0.0803594\pi$
$577$	12.0000i	0.499567i	−0.968302	+	0.249783i	$0.919641\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11.0000	−0.456357
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 28.0000i	− 1.14982i	−0.818216	−	0.574911i	$-0.805037\pi$
$593$	− 28.0000i	− 1.14982i	0.818216	−	0.574911i	$-0.194963\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	− 21.0000i	− 0.855186i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21.0000i	− 0.845428i	−0.906263	−	0.422714i	$-0.861077\pi$
$617$	− 21.0000i	− 0.845428i	0.906263	−	0.422714i	$-0.138923\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 4.00000i	− 0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.00000	0.119618
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	− 37.0000i	− 1.45914i	−0.683907	−	0.729569i	$-0.739721\pi$
$643$	− 37.0000i	− 1.45914i	0.683907	−	0.729569i	$-0.260279\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.0000i	0.702247i
$658$	0	0
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.00000i	0.116160i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−48.0000	−1.85302
$672$	0	0
$673$	− 28.0000i	− 1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$	− 28.0000i	− 1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.0000i	1.80636i	0.429265	+	0.903178i	$0.358772\pi$
$677$	47.0000i	1.80636i	−0.429265	+	0.903178i	$0.641228\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	− 18.0000i	− 0.683763i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 27.0000i	− 1.02270i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	6.00000i	0.226294i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.00000i	− 0.338480i
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.00000i	0.112351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13.0000	−0.484818	−0.242409	−	0.970174i	$-0.577938\pi$
$719$	−13.0000	−0.484818	−0.242409	+	0.970174i	$0.577938\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 21.0000i	− 0.778847i	−0.921059	−	0.389423i	$-0.872674\pi$
$727$	− 21.0000i	− 0.778847i	0.921059	−	0.389423i	$-0.127326\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 25.0000i	− 0.923396i	−0.887037	−	0.461698i	$-0.847240\pi$
$733$	− 25.0000i	− 0.923396i	0.887037	−	0.461698i	$-0.152760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.0000i	1.54709i
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	33.0000i	1.20741i
$748$	0	0
$749$	−5.00000	−0.182696
$750$	0	0
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.0000i	0.545184i	0.962130	+	0.272592i	$0.0878810\pi$
$757$	15.0000i	0.545184i	−0.962130	+	0.272592i	$0.912119\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.0000	1.05125	0.525625	−	0.850717i	$-0.323832\pi$
$761$	29.0000	1.05125	0.525625	+	0.850717i	$0.323832\pi$
$762$	0	0
$763$	6.00000i	0.217215i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.00000i	− 0.0722158i
$768$	0	0
$769$	−44.0000	−1.58668	−0.793340	−	0.608778i	$-0.791660\pi$
$769$	−44.0000	−1.58668	−0.793340	+	0.608778i	$0.791660\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0000i	0.791285i	0.918405	+	0.395643i	$0.129478\pi$
$773$	22.0000i	0.791285i	−0.918405	+	0.395643i	$0.870522\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	54.0000	1.93475
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.0000i	0.463400i	0.972787	+	0.231700i	$0.0744288\pi$
$787$	13.0000i	0.463400i	−0.972787	+	0.231700i	$0.925571\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.00000	−0.248891
$792$	0	0
$793$	16.0000i	0.568177i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 33.0000i	− 1.16892i	−0.811423	−	0.584460i	$-0.801306\pi$
$797$	− 33.0000i	− 1.16892i	0.811423	−	0.584460i	$-0.198694\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	− 36.0000i	− 1.27041i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−47.0000	−1.65243	−0.826216	−	0.563353i	$-0.809511\pi$
$809$	−47.0000	−1.65243	−0.826216	+	0.563353i	$0.809511\pi$
$810$	0	0
$811$	37.0000	1.29925	0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000	1.29925	0.649623	+	0.760257i	$0.274927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	48.0000i	1.67931i
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	− 24.0000i	− 0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$	− 24.0000i	− 0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 57.0000i	− 1.98208i	−0.133550	−	0.991042i	$-0.542638\pi$
$827$	− 57.0000i	− 1.98208i	0.133550	−	0.991042i	$-0.457362\pi$
$828$	0	0
$829$	−33.0000	−1.14614	−0.573069	−	0.819507i	$-0.694247\pi$
$829$	−33.0000	−1.14614	−0.573069	+	0.819507i	$0.694247\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.0000i	0.859010i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.00000	0.0342796
$852$	0	0
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 22.0000i	− 0.748889i	−0.927249	−	0.374444i	$-0.877833\pi$
$863$	− 22.0000i	− 0.748889i	0.927249	−	0.374444i	$-0.122167\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	14.0000	0.474372
$872$	0	0
$873$	18.0000i	0.609208i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 30.0000i	− 1.00730i	−0.863907	−	0.503651i	$-0.831990\pi$
$887$	− 30.0000i	− 1.00730i	0.863907	−	0.503651i	$-0.168010\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−54.0000	−1.80907
$892$	0	0
$893$	24.0000i	0.803129i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7.00000i	− 0.232431i	−0.993224	−	0.116216i	$-0.962924\pi$
$907$	− 7.00000i	− 0.232431i	0.993224	−	0.116216i	$-0.0370764\pi$
$908$	0	0
$909$	−27.0000	−0.895533
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	− 66.0000i	− 2.18428i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	48.0000i	1.57653i
$928$	0	0
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	− 9.00000i	− 0.293080i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 54.0000i	− 1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$	− 54.0000i	− 1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	15.0000i	0.483368i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 56.0000i	− 1.80084i	−0.435023	−	0.900419i	$-0.643260\pi$
$967$	− 56.0000i	− 1.80084i	0.435023	−	0.900419i	$-0.356740\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	0	0
$973$	− 7.00000i	− 0.224410i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.0000i	− 1.05576i	−0.849318	−	0.527882i	$-0.822986\pi$
$977$	− 33.0000i	− 1.05576i	0.849318	−	0.527882i	$-0.177014\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	− 1.00000i	− 0.0318950i	−0.999873	−	0.0159475i	$-0.994924\pi$
$983$	− 1.00000i	− 0.0318950i	0.999873	−	0.0159475i	$-0.00507647\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	−15.0000	−0.476491	−0.238245	−	0.971205i	$-0.576572\pi$
$991$	−15.0000	−0.476491	−0.238245	+	0.971205i	$0.576572\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 32.0000i	− 1.01345i	−0.862108	−	0.506725i	$-0.830856\pi$
$997$	− 32.0000i	− 1.01345i	0.862108	−	0.506725i	$-0.169144\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.j.4049.2		2
5.2	odd	4		4600.2.a.h.1.1		1
5.3	odd	4		920.2.a.c.1.1	✓	1
5.4	even	2	inner	4600.2.e.j.4049.1		2
15.8	even	4		8280.2.a.j.1.1		1
20.3	even	4		1840.2.a.f.1.1		1
20.7	even	4		9200.2.a.w.1.1		1
40.3	even	4		7360.2.a.k.1.1		1
40.13	odd	4		7360.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.c.1.1	✓	1	5.3	odd	4
1840.2.a.f.1.1		1	20.3	even	4
4600.2.a.h.1.1		1	5.2	odd	4
4600.2.e.j.4049.1		2	5.4	even	2	inner
4600.2.e.j.4049.2		2	1.1	even	1	trivial
7360.2.a.k.1.1		1	40.3	even	4
7360.2.a.l.1.1		1	40.13	odd	4
8280.2.a.j.1.1		1	15.8	even	4
9200.2.a.w.1.1		1	20.7	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{7} +3.00000 q^{9} -6.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +6.00000 q^{19} -1.00000i q^{23} -3.00000 q^{29} -3.00000 q^{31} +1.00000i q^{37} +9.00000 q^{41} +8.00000i q^{43} +4.00000i q^{47} +6.00000 q^{49} -1.00000i q^{53} -1.00000 q^{59} +8.00000 q^{61} +3.00000i q^{63} -7.00000i q^{67} -5.00000 q^{71} +6.00000i q^{73} -6.00000i q^{77} +9.00000 q^{81} +11.0000i q^{83} -4.00000 q^{89} -2.00000 q^{91} +6.00000i q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{9} - 12 q^{11} + 12 q^{19} - 6 q^{29} - 6 q^{31} + 18 q^{41} + 12 q^{49} - 2 q^{59} + 16 q^{61} - 10 q^{71} + 18 q^{81} - 8 q^{89} - 4 q^{91} - 36 q^{99}+O(q^{100}) \) 2 * q + 6 * q^9 - 12 * q^11 + 12 * q^19 - 6 * q^29 - 6 * q^31 + 18 * q^41 + 12 * q^49 - 2 * q^59 + 16 * q^61 - 10 * q^71 + 18 * q^81 - 8 * q^89 - 4 * q^91 - 36 * q^99

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