LMFDB - Embedded newform 4600.2.e.c.4049.1

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,-2,0,-10,0,0,0,0,0,0,0,-14] 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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} +1.00000i q^{13} +4.00000i q^{17} -7.00000 q^{19} -2.00000 q^{21} -1.00000i q^{23} -4.00000i q^{27} -5.00000 q^{29} +2.00000 q^{31} +10.0000i q^{33} +2.00000i q^{37} +2.00000 q^{39} +11.0000 q^{41} +1.00000i q^{43} +8.00000i q^{47} +6.00000 q^{49} +8.00000 q^{51} +14.0000i q^{57} +14.0000 q^{59} +10.0000 q^{61} +1.00000i q^{63} +8.00000i q^{67} -2.00000 q^{69} -10.0000 q^{71} +7.00000i q^{73} +5.00000i q^{77} -7.00000 q^{79} -11.0000 q^{81} -15.0000i q^{83} +10.0000i q^{87} -10.0000 q^{89} +1.00000 q^{91} -4.00000i q^{93} +4.00000i q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 10 q^{11} - 14 q^{19} - 4 q^{21} - 10 q^{29} + 4 q^{31} + 4 q^{39} + 22 q^{41} + 12 q^{49} + 16 q^{51} + 28 q^{59} + 20 q^{61} - 4 q^{69} - 20 q^{71} - 14 q^{79} - 22 q^{81} - 20 q^{89}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 10 * q^11 - 14 * q^19 - 4 * q^21 - 10 * q^29 + 4 * q^31 + 4 * q^39 + 22 * q^41 + 12 * q^49 + 16 * q^51 + 28 * q^59 + 20 * q^61 - 4 * q^69 - 20 * q^71 - 14 * q^79 - 22 * q^81 - 20 * q^89 + 2 * q^91 + 10 * 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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	10.0000i	1.74078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.0000i	1.85435i
$58$	0	0
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.00000i	0.569803i
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 15.0000i	− 1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	− 15.0000i	− 1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.0000i	1.07211i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	11.0000i	1.08386i	0.840423	+	0.541931i	$0.182307\pi$
$103$	11.0000i	1.08386i	−0.840423	+	0.541931i	$0.817693\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	− 22.0000i	− 1.98367i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	22.0000i	1.95218i	0.217357	+	0.976092i	$0.430256\pi$
$127$	22.0000i	1.95218i	−0.217357	+	0.976092i	$0.569744\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	7.00000i	0.606977i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	− 5.00000i	− 0.418121i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 12.0000i	− 0.989743i
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.0000i	1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$	22.0000i	1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	− 9.00000i	− 0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$	− 9.00000i	− 0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 28.0000i	− 2.10461i
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	− 20.0000i	− 1.47844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.0000i	− 1.46254i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	0	0
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.0000i	1.49619i	0.663593	+	0.748094i	$0.269031\pi$
$197$	21.0000i	1.49619i	−0.663593	+	0.748094i	$0.730969\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	5.00000i	0.350931i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	35.0000	2.42100
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	20.0000i	1.37038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.00000i	− 0.135769i
$218$	0	0
$219$	14.0000	0.946032
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	10.0000	0.657952
$232$	0	0
$233$	− 15.0000i	− 0.982683i	−0.870967	−	0.491341i	$-0.836507\pi$
$233$	− 15.0000i	− 0.982683i	0.870967	−	0.491341i	$-0.163493\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.0000i	0.909398i
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7.00000i	− 0.445399i
$248$	0	0
$249$	−30.0000	−1.90117
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	5.00000i	0.314347i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	5.00000	0.309492
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.0000i	1.22398i
$268$	0	0
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	− 2.00000i	− 0.121046i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.00000i	0.0600842i	0.999549	+	0.0300421i	$0.00956413\pi$
$277$	1.00000i	0.0600842i	−0.999549	+	0.0300421i	$0.990436\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11.0000i	− 0.649309i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	20.0000i	1.16052i
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	− 36.0000i	− 2.06815i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	22.0000	1.25154
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.00000i	0.280828i	0.990093	+	0.140414i	$0.0448433\pi$
$317$	5.00000i	0.280828i	−0.990093	+	0.140414i	$0.955157\pi$
$318$	0	0
$319$	25.0000	1.39973
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 28.0000i	− 1.55796i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.00000i	0.221201i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.0000i	1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$	20.0000i	1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0000i	0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$	16.0000i	0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$	0	0
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	11.0000i	0.585471i	0.956193	+	0.292735i	$0.0945655\pi$
$353$	11.0000i	0.585471i	−0.956193	+	0.292735i	$0.905434\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.00000i	− 0.423405i
$358$	0	0
$359$	33.0000	1.74167	0.870837	−	0.491572i	$-0.163578\pi$
$359$	33.0000	1.74167	0.870837	+	0.491572i	$0.163578\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	− 28.0000i	− 1.46962i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.00000i	0.156599i	0.996930	+	0.0782994i	$0.0249490\pi$
$367$	3.00000i	0.156599i	−0.996930	+	0.0782994i	$0.975051\pi$
$368$	0	0
$369$	−11.0000	−0.572637
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.00000i	− 0.257513i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	44.0000	2.25419
$382$	0	0
$383$	− 19.0000i	− 0.970855i	−0.874277	−	0.485427i	$-0.838664\pi$
$383$	− 19.0000i	− 0.970855i	0.874277	−	0.485427i	$-0.161336\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.00000i	− 0.0508329i
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	− 4.00000i	− 0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	14.0000	0.700877
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	2.00000i	0.0996271i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.0000i	− 0.495682i
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	− 14.0000i	− 0.688895i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.00000i	0.391762i
$418$	0	0
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	− 8.00000i	− 0.388973i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	− 10.0000i	− 0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$	− 10.0000i	− 0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.00000i	0.334855i
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	30.0000i	1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$	30.0000i	1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.00000i	− 0.189194i
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−55.0000	−2.58985
$452$	0	0
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	16.0000	0.746816
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	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
$469$	8.00000	0.369406
$470$	0	0
$471$	8.00000	0.368621
$472$	0	0
$473$	− 5.00000i	− 0.229900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.0000	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$479$	35.0000	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	− 20.0000i	− 0.900755i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.0000i	0.448561i
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	44.0000	1.96578
$502$	0	0
$503$	15.0000i	0.668817i	0.942428	+	0.334408i	$0.108537\pi$
$503$	15.0000i	0.668817i	−0.942428	+	0.334408i	$0.891463\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 24.0000i	− 1.06588i
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	28.0000i	1.23623i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 40.0000i	− 1.75920i
$518$	0	0
$519$	−18.0000	−0.790112
$520$	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
$523$	− 1.00000i	− 0.0437269i	−0.999761	−	0.0218635i	$-0.993040\pi$
$523$	− 1.00000i	− 0.0437269i	0.999761	−	0.0218635i	$-0.00695991\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00000i	0.348485i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−14.0000	−0.607548
$532$	0	0
$533$	11.0000i	0.476463i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.0000i	1.38090i
$538$	0	0
$539$	−30.0000	−1.29219
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	0	0
$543$	24.0000i	1.02994i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	35.0000	1.49105
$552$	0	0
$553$	7.00000i	0.297670i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	−40.0000	−1.68880
$562$	0	0
$563$	33.0000i	1.39078i	0.718631	+	0.695392i	$0.244769\pi$
$563$	33.0000i	1.39078i	−0.718631	+	0.695392i	$0.755231\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.0000i	0.461957i
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	54.0000i	2.25588i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.0000i	1.70685i	0.521214	+	0.853426i	$0.325479\pi$
$577$	41.0000i	1.70685i	−0.521214	+	0.853426i	$0.674521\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	−15.0000	−0.622305
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.00000i	− 0.0825488i	−0.999148	−	0.0412744i	$-0.986858\pi$
$587$	− 2.00000i	− 0.0825488i	0.999148	−	0.0412744i	$-0.0131418\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	0	0
$591$	42.0000	1.72765
$592$	0	0
$593$	− 23.0000i	− 0.944497i	−0.881466	−	0.472248i	$-0.843443\pi$
$593$	− 23.0000i	− 0.944497i	0.881466	−	0.472248i	$-0.156557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 34.0000i	− 1.39153i
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 36.0000i	− 1.46119i	−0.682808	−	0.730597i	$-0.739242\pi$
$607$	− 36.0000i	− 1.46119i	0.682808	−	0.730597i	$-0.260758\pi$
$608$	0	0
$609$	10.0000	0.405220
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 8.00000i	− 0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$	− 8.00000i	− 0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	10.0000i	0.400642i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 70.0000i	− 2.79553i
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	− 4.00000i	− 0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	5.00000i	0.197181i	0.995128	+	0.0985904i	$0.0314334\pi$
$643$	5.00000i	0.197181i	−0.995128	+	0.0985904i	$0.968567\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	−70.0000	−2.74774
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	25.0000i	0.978326i	0.872192	+	0.489163i	$0.162698\pi$
$653$	25.0000i	0.978326i	−0.872192	+	0.489163i	$0.837302\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7.00000i	− 0.273096i
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	8.00000i	0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.00000i	0.193601i
$668$	0	0
$669$	−40.0000	−1.54649
$670$	0	0
$671$	−50.0000	−1.93023
$672$	0	0
$673$	− 23.0000i	− 0.886585i	−0.896377	−	0.443292i	$-0.853810\pi$
$673$	− 23.0000i	− 0.886585i	0.896377	−	0.443292i	$-0.146190\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	6.00000i	0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$	6.00000i	0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 44.0000i	− 1.67870i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	− 5.00000i	− 0.189934i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	44.0000i	1.66662i
$698$	0	0
$699$	−30.0000	−1.13470
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	0	0
$703$	− 14.0000i	− 0.528020i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.0000i	− 0.676960i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	7.00000	0.262521
$712$	0	0
$713$	− 2.00000i	− 0.0749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 32.0000i	− 1.19506i
$718$	0	0
$719$	−38.0000	−1.41716	−0.708580	−	0.705630i	$-0.750664\pi$
$719$	−38.0000	−1.41716	−0.708580	+	0.705630i	$0.750664\pi$
$720$	0	0
$721$	11.0000	0.409661
$722$	0	0
$723$	32.0000i	1.19009i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	− 38.0000i	− 1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$	− 38.0000i	− 1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 40.0000i	− 1.47342i
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	0	0
$743$	− 39.0000i	− 1.43077i	−0.698730	−	0.715386i	$-0.746251\pi$
$743$	− 39.0000i	− 1.43077i	0.698730	−	0.715386i	$-0.253749\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.0000i	0.548821i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	0	0
$753$	8.00000i	0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	10.0000	0.362977
$760$	0	0
$761$	−23.0000	−0.833749	−0.416875	−	0.908964i	$-0.636875\pi$
$761$	−23.0000	−0.833749	−0.416875	+	0.908964i	$0.636875\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.0000i	0.505511i
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.00000i	− 0.143499i
$778$	0	0
$779$	−77.0000	−2.75881
$780$	0	0
$781$	50.0000	1.78914
$782$	0	0
$783$	20.0000i	0.714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 47.0000i	− 1.67537i	−0.546154	−	0.837685i	$-0.683909\pi$
$787$	− 47.0000i	− 1.67537i	0.546154	−	0.837685i	$-0.316091\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	10.0000i	0.355110i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	54.0000i	1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$	54.0000i	1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	− 35.0000i	− 1.23512i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.0000i	1.19686i
$808$	0	0
$809$	−17.0000	−0.597688	−0.298844	−	0.954302i	$-0.596601\pi$
$809$	−17.0000	−0.597688	−0.298844	+	0.954302i	$0.596601\pi$
$810$	0	0
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.00000i	− 0.244899i
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	31.0000	1.08191	0.540954	−	0.841052i	$-0.318063\pi$
$821$	31.0000	1.08191	0.540954	+	0.841052i	$0.318063\pi$
$822$	0	0
$823$	36.0000i	1.25488i	0.778664	+	0.627441i	$0.215897\pi$
$823$	36.0000i	1.25488i	−0.778664	+	0.627441i	$0.784103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.0000i	1.42571i	0.701312	+	0.712855i	$0.252598\pi$
$827$	41.0000i	1.42571i	−0.701312	+	0.712855i	$0.747402\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	24.0000i	0.831551i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	56.0000i	1.92874i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	0	0
$849$	56.0000	1.92192
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	− 7.00000i	− 0.239675i	−0.992793	−	0.119838i	$-0.961763\pi$
$853$	− 7.00000i	− 0.239675i	0.992793	−	0.119838i	$-0.0382374\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	58.0000i	1.98124i	0.136637	+	0.990621i	$0.456370\pi$
$857$	58.0000i	1.98124i	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	0	0
$861$	−22.0000	−0.749758
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.00000i	− 0.0679236i
$868$	0	0
$869$	35.0000	1.18729
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	− 4.00000i	− 0.135379i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	32.0000i	1.07689i	0.842662	+	0.538443i	$0.180987\pi$
$883$	32.0000i	1.07689i	−0.842662	+	0.538443i	$0.819013\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	22.0000	0.737856
$890$	0	0
$891$	55.0000	1.84257
$892$	0	0
$893$	− 56.0000i	− 1.87397i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.00000i	− 0.0667781i
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 2.00000i	− 0.0665558i
$904$	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
$909$	−18.0000	−0.597022
$910$	0	0
$911$	47.0000	1.55718	0.778590	−	0.627533i	$-0.215935\pi$
$911$	47.0000	1.55718	0.778590	+	0.627533i	$0.215935\pi$
$912$	0	0
$913$	75.0000i	2.48214i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.00000i	− 0.0660458i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 11.0000i	− 0.361287i
$928$	0	0
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	−42.0000	−1.37649
$932$	0	0
$933$	− 4.00000i	− 0.130954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 48.0000i	− 1.56809i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	− 48.0000i	− 1.56809i	0.620703	−	0.784046i	$-0.286847\pi$
$938$	0	0
$939$	44.0000	1.43589
$940$	0	0
$941$	−20.0000	−0.651981	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000	−0.651981	−0.325991	+	0.945373i	$0.605698\pi$
$942$	0	0
$943$	− 11.0000i	− 0.358209i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.0000i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$	38.0000i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	38.0000i	1.23094i	0.788160	+	0.615470i	$0.211034\pi$
$953$	38.0000i	1.23094i	−0.788160	+	0.615470i	$0.788966\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 50.0000i	− 1.61627i
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	0	0
$969$	−56.0000	−1.79898
$970$	0	0
$971$	−55.0000	−1.76503	−0.882517	−	0.470281i	$-0.844153\pi$
$971$	−55.0000	−1.76503	−0.882517	+	0.470281i	$0.844153\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.00000i	0.255943i	0.991778	+	0.127971i	$0.0408466\pi$
$977$	8.00000i	0.255943i	−0.991778	+	0.127971i	$0.959153\pi$
$978$	0	0
$979$	50.0000	1.59801
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	− 21.0000i	− 0.669796i	−0.942254	−	0.334898i	$-0.891298\pi$
$983$	− 21.0000i	− 0.669796i	0.942254	−	0.334898i	$-0.108702\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	0	0
$993$	36.0000i	1.14243i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 23.0000i	− 0.728417i	−0.931317	−	0.364209i	$-0.881339\pi$
$997$	− 23.0000i	− 0.728417i	0.931317	−	0.364209i	$-0.118661\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.c.4049.1		2
5.2	odd	4		4600.2.a.c.1.1	✓	1
5.3	odd	4		4600.2.a.n.1.1	yes	1
5.4	even	2	inner	4600.2.e.c.4049.2		2
20.3	even	4		9200.2.a.i.1.1		1
20.7	even	4		9200.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.c.1.1	✓	1	5.2	odd	4
4600.2.a.n.1.1	yes	1	5.3	odd	4
4600.2.e.c.4049.1		2	1.1	even	1	trivial
4600.2.e.c.4049.2		2	5.4	even	2	inner
9200.2.a.i.1.1		1	20.3	even	4
9200.2.a.bd.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)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} +1.00000i q^{13} +4.00000i q^{17} -7.00000 q^{19} -2.00000 q^{21} -1.00000i q^{23} -4.00000i q^{27} -5.00000 q^{29} +2.00000 q^{31} +10.0000i q^{33} +2.00000i q^{37} +2.00000 q^{39} +11.0000 q^{41} +1.00000i q^{43} +8.00000i q^{47} +6.00000 q^{49} +8.00000 q^{51} +14.0000i q^{57} +14.0000 q^{59} +10.0000 q^{61} +1.00000i q^{63} +8.00000i q^{67} -2.00000 q^{69} -10.0000 q^{71} +7.00000i q^{73} +5.00000i q^{77} -7.00000 q^{79} -11.0000 q^{81} -15.0000i q^{83} +10.0000i q^{87} -10.0000 q^{89} +1.00000 q^{91} -4.00000i q^{93} +4.00000i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 10 q^{11} - 14 q^{19} - 4 q^{21} - 10 q^{29} + 4 q^{31} + 4 q^{39} + 22 q^{41} + 12 q^{49} + 16 q^{51} + 28 q^{59} + 20 q^{61} - 4 q^{69} - 20 q^{71} - 14 q^{79} - 22 q^{81} - 20 q^{89}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 10 * q^11 - 14 * q^19 - 4 * q^21 - 10 * q^29 + 4 * q^31 + 4 * q^39 + 22 * q^41 + 12 * q^49 + 16 * q^51 + 28 * q^59 + 20 * q^61 - 4 * q^69 - 20 * q^71 - 14 * q^79 - 22 * q^81 - 20 * q^89 + 2 * q^91 + 10 * q^99

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