LMFDB - Embedded newform 4600.2.e.s.4049.4

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 = [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,-8] 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:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$1.80194i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.s.4049.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554958i q^{3} -1.10992i q^{7} +2.69202 q^{9} +2.71379 q^{11} -1.69202i q^{13} -1.10992i q^{17} +4.98792 q^{19} +0.615957 q^{21} -1.00000i q^{23} +3.15883i q^{27} -6.34481 q^{29} +5.13706 q^{31} +1.50604i q^{33} -5.70171i q^{37} +0.939001 q^{39} +1.26875 q^{41} -1.90217i q^{43} -4.08815i q^{47} +5.76809 q^{49} +0.615957 q^{51} +3.78017i q^{53} +2.76809i q^{57} -5.59179 q^{59} +8.37196 q^{61} -2.98792i q^{63} -10.7681i q^{67} +0.554958 q^{69} -12.1075 q^{71} +15.3327i q^{73} -3.01208i q^{77} -11.0858 q^{79} +6.32304 q^{81} -7.64742i q^{83} -3.52111i q^{87} +4.17629 q^{89} -1.87800 q^{91} +2.85086i q^{93} -9.50604i q^{97} +7.30559 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9} - 8 q^{19} + 24 q^{21} + 8 q^{29} + 20 q^{31} - 14 q^{39} - 8 q^{41} - 6 q^{49} + 24 q^{51} + 22 q^{59} - 8 q^{61} + 4 q^{69} + 8 q^{71} + 8 q^{79} - 2 q^{81} + 40 q^{89} + 28 q^{91} - 28 q^{99}+O(q^{100}) $ 6 * q + 6 * q^9 - 8 * q^19 + 24 * q^21 + 8 * q^29 + 20 * q^31 - 14 * q^39 - 8 * q^41 - 6 * q^49 + 24 * q^51 + 22 * q^59 - 8 * q^61 + 4 * q^69 + 8 * q^71 + 8 * q^79 - 2 * q^81 + 40 * q^89 + 28 * q^91 - 28 * 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
$3$	0.554958i	0.320405i	0.987084	+	0.160203i	$0.0512148\pi$
$3$	0.554958i	0.320405i	−0.987084	+	0.160203i	$0.948785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.10992i	− 0.419509i	−0.977754	−	0.209754i	$-0.932734\pi$
$7$	− 1.10992i	− 0.419509i	0.977754	−	0.209754i	$-0.0672665\pi$
$8$	0	0
$9$	2.69202	0.897340
$10$	0	0
$11$	2.71379	0.818239	0.409119	−	0.912481i	$-0.365836\pi$
$11$	2.71379	0.818239	0.409119	+	0.912481i	$0.365836\pi$
$12$	0	0
$13$	− 1.69202i	− 0.469282i	−0.972082	−	0.234641i	$-0.924608\pi$
$13$	− 1.69202i	− 0.469282i	0.972082	−	0.234641i	$-0.0753915\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.10992i	− 0.269194i	−0.990900	−	0.134597i	$-0.957026\pi$
$17$	− 1.10992i	− 0.269194i	0.990900	−	0.134597i	$-0.0429740\pi$
$18$	0	0
$19$	4.98792	1.14431	0.572153	−	0.820147i	$-0.306108\pi$
$19$	4.98792	1.14431	0.572153	+	0.820147i	$0.306108\pi$
$20$	0	0
$21$	0.615957	0.134413
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.15883i	0.607918i
$28$	0	0
$29$	−6.34481	−1.17820	−0.589101	−	0.808059i	$-0.700518\pi$
$29$	−6.34481	−1.17820	−0.589101	+	0.808059i	$0.700518\pi$
$30$	0	0
$31$	5.13706	0.922644	0.461322	−	0.887233i	$-0.347375\pi$
$31$	5.13706	0.922644	0.461322	+	0.887233i	$0.347375\pi$
$32$	0	0
$33$	1.50604i	0.262168i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.70171i	− 0.937355i	−0.883369	−	0.468678i	$-0.844731\pi$
$37$	− 5.70171i	− 0.937355i	0.883369	−	0.468678i	$-0.155269\pi$
$38$	0	0
$39$	0.939001	0.150361
$40$	0	0
$41$	1.26875	0.198145	0.0990727	−	0.995080i	$-0.468412\pi$
$41$	1.26875	0.198145	0.0990727	+	0.995080i	$0.468412\pi$
$42$	0	0
$43$	− 1.90217i	− 0.290077i	−0.989426	−	0.145039i	$-0.953669\pi$
$43$	− 1.90217i	− 0.290077i	0.989426	−	0.145039i	$-0.0463307\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.08815i	− 0.596317i	−0.954516	−	0.298159i	$-0.903628\pi$
$47$	− 4.08815i	− 0.596317i	0.954516	−	0.298159i	$-0.0963725\pi$
$48$	0	0
$49$	5.76809	0.824012
$50$	0	0
$51$	0.615957	0.0862512
$52$	0	0
$53$	3.78017i	0.519246i	0.965710	+	0.259623i	$0.0835983\pi$
$53$	3.78017i	0.519246i	−0.965710	+	0.259623i	$0.916402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.76809i	0.366642i
$58$	0	0
$59$	−5.59179	−0.727990	−0.363995	−	0.931401i	$-0.618587\pi$
$59$	−5.59179	−0.727990	−0.363995	+	0.931401i	$0.618587\pi$
$60$	0	0
$61$	8.37196	1.07192	0.535960	−	0.844243i	$-0.319950\pi$
$61$	8.37196	1.07192	0.535960	+	0.844243i	$0.319950\pi$
$62$	0	0
$63$	− 2.98792i	− 0.376442i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.7681i	− 1.31553i	−0.753223	−	0.657766i	$-0.771502\pi$
$67$	− 10.7681i	− 1.31553i	0.753223	−	0.657766i	$-0.228498\pi$
$68$	0	0
$69$	0.554958	0.0668091
$70$	0	0
$71$	−12.1075	−1.43690	−0.718449	−	0.695579i	$-0.755148\pi$
$71$	−12.1075	−1.43690	−0.718449	+	0.695579i	$0.755148\pi$
$72$	0	0
$73$	15.3327i	1.79456i	0.441461	+	0.897280i	$0.354460\pi$
$73$	15.3327i	1.79456i	−0.441461	+	0.897280i	$0.645540\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.01208i	− 0.343259i
$78$	0	0
$79$	−11.0858	−1.24724	−0.623622	−	0.781726i	$-0.714340\pi$
$79$	−11.0858	−1.24724	−0.623622	+	0.781726i	$0.714340\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	0	0
$83$	− 7.64742i	− 0.839413i	−0.907660	−	0.419706i	$-0.862133\pi$
$83$	− 7.64742i	− 0.839413i	0.907660	−	0.419706i	$-0.137867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.52111i	− 0.377502i
$88$	0	0
$89$	4.17629	0.442686	0.221343	−	0.975196i	$-0.428956\pi$
$89$	4.17629	0.442686	0.221343	+	0.975196i	$0.428956\pi$
$90$	0	0
$91$	−1.87800	−0.196868
$92$	0	0
$93$	2.85086i	0.295620i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 9.50604i	− 0.965192i	−0.875843	−	0.482596i	$-0.839694\pi$
$97$	− 9.50604i	− 0.965192i	0.875843	−	0.482596i	$-0.160306\pi$
$98$	0	0
$99$	7.30559	0.734239
$100$	0	0
$101$	−6.97584	−0.694122	−0.347061	−	0.937843i	$-0.612820\pi$
$101$	−6.97584	−0.694122	−0.347061	+	0.937843i	$0.612820\pi$
$102$	0	0
$103$	2.05429i	0.202416i	0.994865	+	0.101208i	$0.0322707\pi$
$103$	2.05429i	0.202416i	−0.994865	+	0.101208i	$0.967729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.90217i	− 0.377237i	−0.982050	−	0.188618i	$-0.939599\pi$
$107$	− 3.90217i	− 0.377237i	0.982050	−	0.188618i	$-0.0604009\pi$
$108$	0	0
$109$	−2.81163	−0.269305	−0.134652	−	0.990893i	$-0.542992\pi$
$109$	−2.81163	−0.269305	−0.134652	+	0.990893i	$0.542992\pi$
$110$	0	0
$111$	3.16421	0.300334
$112$	0	0
$113$	0.792249i	0.0745285i	0.999305	+	0.0372643i	$0.0118643\pi$
$113$	0.792249i	0.0745285i	−0.999305	+	0.0372643i	$0.988136\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.55496i	− 0.421106i
$118$	0	0
$119$	−1.23191	−0.112929
$120$	0	0
$121$	−3.63533	−0.330485
$122$	0	0
$123$	0.704103i	0.0634868i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.85086i	− 0.696651i	−0.937374	−	0.348325i	$-0.886750\pi$
$127$	− 7.85086i	− 0.696651i	0.937374	−	0.348325i	$-0.113250\pi$
$128$	0	0
$129$	1.05562	0.0929423
$130$	0	0
$131$	13.1903	1.15244	0.576221	−	0.817294i	$-0.304527\pi$
$131$	13.1903	1.15244	0.576221	+	0.817294i	$0.304527\pi$
$132$	0	0
$133$	− 5.53617i	− 0.480047i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.49396i	0.725688i	0.931850	+	0.362844i	$0.118194\pi$
$137$	8.49396i	0.725688i	−0.931850	+	0.362844i	$0.881806\pi$
$138$	0	0
$139$	12.0881	1.02530	0.512652	−	0.858597i	$-0.328663\pi$
$139$	12.0881	1.02530	0.512652	+	0.858597i	$0.328663\pi$
$140$	0	0
$141$	2.26875	0.191063
$142$	0	0
$143$	− 4.59179i	− 0.383985i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.20105i	0.264018i
$148$	0	0
$149$	20.5133	1.68052	0.840259	−	0.542185i	$-0.182403\pi$
$149$	20.5133	1.68052	0.840259	+	0.542185i	$0.182403\pi$
$150$	0	0
$151$	8.28382	0.674127	0.337064	−	0.941482i	$-0.390566\pi$
$151$	8.28382	0.674127	0.337064	+	0.941482i	$0.390566\pi$
$152$	0	0
$153$	− 2.98792i	− 0.241559i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.16421i	0.0929141i	0.998920	+	0.0464571i	$0.0147931\pi$
$157$	1.16421i	0.0929141i	−0.998920	+	0.0464571i	$0.985207\pi$
$158$	0	0
$159$	−2.09783	−0.166369
$160$	0	0
$161$	−1.10992	−0.0874737
$162$	0	0
$163$	8.83877i	0.692306i	0.938178	+	0.346153i	$0.112512\pi$
$163$	8.83877i	0.692306i	−0.938178	+	0.346153i	$0.887488\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 19.2392i	− 1.48877i	−0.667748	−	0.744387i	$-0.732742\pi$
$167$	− 19.2392i	− 1.48877i	0.667748	−	0.744387i	$-0.267258\pi$
$168$	0	0
$169$	10.1371	0.779774
$170$	0	0
$171$	13.4276	1.02683
$172$	0	0
$173$	1.00000i	0.0760286i	0.999277	+	0.0380143i	$0.0121032\pi$
$173$	1.00000i	0.0760286i	−0.999277	+	0.0380143i	$0.987897\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.10321i	− 0.233252i
$178$	0	0
$179$	−8.64310	−0.646016	−0.323008	−	0.946396i	$-0.604694\pi$
$179$	−8.64310	−0.646016	−0.323008	+	0.946396i	$0.604694\pi$
$180$	0	0
$181$	−7.90217	−0.587363	−0.293682	−	0.955903i	$-0.594881\pi$
$181$	−7.90217	−0.587363	−0.293682	+	0.955903i	$0.594881\pi$
$182$	0	0
$183$	4.64609i	0.343449i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.01208i	− 0.220265i
$188$	0	0
$189$	3.50604	0.255027
$190$	0	0
$191$	7.34050	0.531140	0.265570	−	0.964092i	$-0.414440\pi$
$191$	7.34050	0.531140	0.265570	+	0.964092i	$0.414440\pi$
$192$	0	0
$193$	24.3545i	1.75308i	0.481333	+	0.876538i	$0.340153\pi$
$193$	24.3545i	1.75308i	−0.481333	+	0.876538i	$0.659847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.51142i	− 0.606413i	−0.952925	−	0.303207i	$-0.901943\pi$
$197$	− 8.51142i	− 0.606413i	0.952925	−	0.303207i	$-0.0980573\pi$
$198$	0	0
$199$	−13.9758	−0.990721	−0.495360	−	0.868688i	$-0.664964\pi$
$199$	−13.9758	−0.990721	−0.495360	+	0.868688i	$0.664964\pi$
$200$	0	0
$201$	5.97584	0.421503
$202$	0	0
$203$	7.04221i	0.494266i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.69202i	− 0.187108i
$208$	0	0
$209$	13.5362	0.936317
$210$	0	0
$211$	6.62804	0.456293	0.228146	−	0.973627i	$-0.426733\pi$
$211$	6.62804	0.456293	0.228146	+	0.973627i	$0.426733\pi$
$212$	0	0
$213$	− 6.71917i	− 0.460390i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.70171i	− 0.387057i
$218$	0	0
$219$	−8.50902	−0.574987
$220$	0	0
$221$	−1.87800	−0.126328
$222$	0	0
$223$	− 12.1642i	− 0.814576i	−0.913300	−	0.407288i	$-0.866475\pi$
$223$	− 12.1642i	− 0.814576i	0.913300	−	0.407288i	$-0.133525\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.4276i	− 1.42220i	−0.703092	−	0.711099i	$-0.748198\pi$
$227$	− 21.4276i	− 1.42220i	0.703092	−	0.711099i	$-0.251802\pi$
$228$	0	0
$229$	28.7875	1.90233	0.951165	−	0.308684i	$-0.0998886\pi$
$229$	28.7875	1.90233	0.951165	+	0.308684i	$0.0998886\pi$
$230$	0	0
$231$	1.67158	0.109982
$232$	0	0
$233$	14.5036i	0.950166i	0.879941	+	0.475083i	$0.157582\pi$
$233$	14.5036i	0.950166i	−0.879941	+	0.475083i	$0.842418\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.15213i	− 0.399624i
$238$	0	0
$239$	19.8364	1.28311	0.641554	−	0.767078i	$-0.278290\pi$
$239$	19.8364	1.28311	0.641554	+	0.767078i	$0.278290\pi$
$240$	0	0
$241$	−12.7332	−0.820216	−0.410108	−	0.912037i	$-0.634509\pi$
$241$	−12.7332	−0.820216	−0.410108	+	0.912037i	$0.634509\pi$
$242$	0	0
$243$	12.9855i	0.833022i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.43967i	− 0.537003i
$248$	0	0
$249$	4.24400	0.268952
$250$	0	0
$251$	13.4383	0.848220	0.424110	−	0.905611i	$-0.360587\pi$
$251$	13.4383	0.848220	0.424110	+	0.905611i	$0.360587\pi$
$252$	0	0
$253$	− 2.71379i	− 0.170615i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.2325i	− 1.01256i	−0.862370	−	0.506278i	$-0.831021\pi$
$257$	− 16.2325i	− 1.01256i	0.862370	−	0.506278i	$-0.168979\pi$
$258$	0	0
$259$	−6.32842	−0.393229
$260$	0	0
$261$	−17.0804	−1.05725
$262$	0	0
$263$	− 5.87800i	− 0.362453i	−0.983441	−	0.181227i	$-0.941993\pi$
$263$	− 5.87800i	− 0.362453i	0.983441	−	0.181227i	$-0.0580067\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.31767i	0.141839i
$268$	0	0
$269$	−7.66786	−0.467518	−0.233759	−	0.972295i	$-0.575103\pi$
$269$	−7.66786	−0.467518	−0.233759	+	0.972295i	$0.575103\pi$
$270$	0	0
$271$	4.60388	0.279666	0.139833	−	0.990175i	$-0.455344\pi$
$271$	4.60388	0.279666	0.139833	+	0.990175i	$0.455344\pi$
$272$	0	0
$273$	− 1.04221i	− 0.0630776i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.03684i	0.242550i	0.992619	+	0.121275i	$0.0386983\pi$
$277$	4.03684i	0.242550i	−0.992619	+	0.121275i	$0.961302\pi$
$278$	0	0
$279$	13.8291	0.827926
$280$	0	0
$281$	20.1414	1.20153	0.600767	−	0.799424i	$-0.294862\pi$
$281$	20.1414	1.20153	0.600767	+	0.799424i	$0.294862\pi$
$282$	0	0
$283$	− 13.8538i	− 0.823525i	−0.911291	−	0.411763i	$-0.864913\pi$
$283$	− 13.8538i	− 0.823525i	0.911291	−	0.411763i	$-0.135087\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.40821i	− 0.0831238i
$288$	0	0
$289$	15.7681	0.927534
$290$	0	0
$291$	5.27545	0.309253
$292$	0	0
$293$	20.7573i	1.21266i	0.795215	+	0.606328i	$0.207358\pi$
$293$	20.7573i	1.21266i	−0.795215	+	0.606328i	$0.792642\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.57242i	0.497422i
$298$	0	0
$299$	−1.69202	−0.0978521
$300$	0	0
$301$	−2.11124	−0.121690
$302$	0	0
$303$	− 3.87130i	− 0.222400i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 0.987918i	− 0.0563835i	−0.999603	−	0.0281917i	$-0.991025\pi$
$307$	− 0.987918i	− 0.0563835i	0.999603	−	0.0281917i	$-0.00897490\pi$
$308$	0	0
$309$	−1.14005	−0.0648550
$310$	0	0
$311$	−7.93794	−0.450119	−0.225060	−	0.974345i	$-0.572258\pi$
$311$	−7.93794	−0.450119	−0.225060	+	0.974345i	$0.572258\pi$
$312$	0	0
$313$	− 8.23921i	− 0.465708i	−0.972512	−	0.232854i	$-0.925194\pi$
$313$	− 8.23921i	− 0.465708i	0.972512	−	0.232854i	$-0.0748064\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.0629i	0.902183i	0.892478	+	0.451092i	$0.148965\pi$
$317$	16.0629i	0.902183i	−0.892478	+	0.451092i	$0.851035\pi$
$318$	0	0
$319$	−17.2185	−0.964051
$320$	0	0
$321$	2.16554	0.120869
$322$	0	0
$323$	− 5.53617i	− 0.308041i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.56033i	− 0.0862867i
$328$	0	0
$329$	−4.53750	−0.250160
$330$	0	0
$331$	25.0683	1.37788	0.688939	−	0.724819i	$-0.258077\pi$
$331$	25.0683	1.37788	0.688939	+	0.724819i	$0.258077\pi$
$332$	0	0
$333$	− 15.3491i	− 0.841127i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.5690i	1.50178i	0.660429	+	0.750888i	$0.270374\pi$
$337$	27.5690i	1.50178i	−0.660429	+	0.750888i	$0.729626\pi$
$338$	0	0
$339$	−0.439665	−0.0238793
$340$	0	0
$341$	13.9409	0.754943
$342$	0	0
$343$	− 14.1715i	− 0.765189i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.47112i	− 0.454754i	−0.973807	−	0.227377i	$-0.926985\pi$
$347$	− 8.47112i	− 0.454754i	0.973807	−	0.227377i	$-0.0730149\pi$
$348$	0	0
$349$	19.2959	1.03289	0.516443	−	0.856322i	$-0.327256\pi$
$349$	19.2959	1.03289	0.516443	+	0.856322i	$0.327256\pi$
$350$	0	0
$351$	5.34481	0.285285
$352$	0	0
$353$	− 0.829085i	− 0.0441277i	−0.999757	−	0.0220639i	$-0.992976\pi$
$353$	− 0.829085i	− 0.0441277i	0.999757	−	0.0220639i	$-0.00702372\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.683661i	− 0.0361832i
$358$	0	0
$359$	9.87800	0.521341	0.260671	−	0.965428i	$-0.416056\pi$
$359$	9.87800	0.521341	0.260671	+	0.965428i	$0.416056\pi$
$360$	0	0
$361$	5.87933	0.309438
$362$	0	0
$363$	− 2.01746i	− 0.105889i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.0930i	0.944449i	0.881478	+	0.472225i	$0.156549\pi$
$367$	18.0930i	0.944449i	−0.881478	+	0.472225i	$0.843451\pi$
$368$	0	0
$369$	3.41550	0.177804
$370$	0	0
$371$	4.19567	0.217828
$372$	0	0
$373$	18.3370i	0.949456i	0.880133	+	0.474728i	$0.157454\pi$
$373$	18.3370i	0.949456i	−0.880133	+	0.474728i	$0.842546\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.7356i	0.552910i
$378$	0	0
$379$	3.05562	0.156957	0.0784784	−	0.996916i	$-0.474994\pi$
$379$	3.05562	0.156957	0.0784784	+	0.996916i	$0.474994\pi$
$380$	0	0
$381$	4.35690	0.223211
$382$	0	0
$383$	34.8611i	1.78132i	0.454669	+	0.890660i	$0.349757\pi$
$383$	34.8611i	1.78132i	−0.454669	+	0.890660i	$0.650243\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.12067i	− 0.260298i
$388$	0	0
$389$	−1.42758	−0.0723814	−0.0361907	−	0.999345i	$-0.511522\pi$
$389$	−1.42758	−0.0723814	−0.0361907	+	0.999345i	$0.511522\pi$
$390$	0	0
$391$	−1.10992	−0.0561309
$392$	0	0
$393$	7.32006i	0.369248i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 26.9474i	− 1.35245i	−0.736695	−	0.676225i	$-0.763615\pi$
$397$	− 26.9474i	− 1.35245i	0.736695	−	0.676225i	$-0.236385\pi$
$398$	0	0
$399$	3.07234	0.153810
$400$	0	0
$401$	−6.19567	−0.309397	−0.154698	−	0.987962i	$-0.549441\pi$
$401$	−6.19567	−0.309397	−0.154698	+	0.987962i	$0.549441\pi$
$402$	0	0
$403$	− 8.69202i	− 0.432980i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 15.4733i	− 0.766981i
$408$	0	0
$409$	−8.79417	−0.434844	−0.217422	−	0.976078i	$-0.569765\pi$
$409$	−8.79417	−0.434844	−0.217422	+	0.976078i	$0.569765\pi$
$410$	0	0
$411$	−4.71379	−0.232514
$412$	0	0
$413$	6.20642i	0.305398i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.70841i	0.328512i
$418$	0	0
$419$	−24.0844	−1.17660	−0.588301	−	0.808642i	$-0.700203\pi$
$419$	−24.0844	−1.17660	−0.588301	+	0.808642i	$0.700203\pi$
$420$	0	0
$421$	−11.5690	−0.563837	−0.281918	−	0.959438i	$-0.590971\pi$
$421$	−11.5690	−0.563837	−0.281918	+	0.959438i	$0.590971\pi$
$422$	0	0
$423$	− 11.0054i	− 0.535100i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 9.29218i	− 0.449680i
$428$	0	0
$429$	2.54825	0.123031
$430$	0	0
$431$	−17.4577	−0.840909	−0.420454	−	0.907314i	$-0.638129\pi$
$431$	−17.4577	−0.840909	−0.420454	+	0.907314i	$0.638129\pi$
$432$	0	0
$433$	5.76941i	0.277260i	0.990344	+	0.138630i	$0.0442699\pi$
$433$	5.76941i	0.277260i	−0.990344	+	0.138630i	$0.955730\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.98792i	− 0.238604i
$438$	0	0
$439$	3.51871	0.167939	0.0839695	−	0.996468i	$-0.473240\pi$
$439$	3.51871	0.167939	0.0839695	+	0.996468i	$0.473240\pi$
$440$	0	0
$441$	15.5278	0.739420
$442$	0	0
$443$	− 17.6437i	− 0.838277i	−0.907922	−	0.419139i	$-0.862332\pi$
$443$	− 17.6437i	− 0.838277i	0.907922	−	0.419139i	$-0.137668\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.3840i	0.538447i
$448$	0	0
$449$	−5.13275	−0.242230	−0.121115	−	0.992639i	$-0.538647\pi$
$449$	−5.13275	−0.242230	−0.121115	+	0.992639i	$0.538647\pi$
$450$	0	0
$451$	3.44312	0.162130
$452$	0	0
$453$	4.59717i	0.215994i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.15213i	0.194228i	0.995273	+	0.0971142i	$0.0309612\pi$
$457$	4.15213i	0.194228i	−0.995273	+	0.0971142i	$0.969039\pi$
$458$	0	0
$459$	3.50604	0.163648
$460$	0	0
$461$	−27.1002	−1.26218	−0.631092	−	0.775708i	$-0.717393\pi$
$461$	−27.1002	−1.26218	−0.631092	+	0.775708i	$0.717393\pi$
$462$	0	0
$463$	27.5676i	1.28118i	0.767885	+	0.640588i	$0.221309\pi$
$463$	27.5676i	1.28118i	−0.767885	+	0.640588i	$0.778691\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.6125i	− 1.27775i	−0.769309	−	0.638877i	$-0.779399\pi$
$467$	− 27.6125i	− 1.27775i	0.769309	−	0.638877i	$-0.220601\pi$
$468$	0	0
$469$	−11.9517	−0.551877
$470$	0	0
$471$	−0.646088	−0.0297702
$472$	0	0
$473$	− 5.16208i	− 0.237353i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.1763i	0.465940i
$478$	0	0
$479$	−19.6233	−0.896609	−0.448305	−	0.893881i	$-0.647972\pi$
$479$	−19.6233	−0.896609	−0.448305	+	0.893881i	$0.647972\pi$
$480$	0	0
$481$	−9.64742	−0.439884
$482$	0	0
$483$	− 0.615957i	− 0.0280270i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.3709i	0.923093i	0.887116	+	0.461547i	$0.152705\pi$
$487$	20.3709i	0.923093i	−0.887116	+	0.461547i	$0.847295\pi$
$488$	0	0
$489$	−4.90515	−0.221819
$490$	0	0
$491$	−34.0640	−1.53729	−0.768643	−	0.639678i	$-0.779068\pi$
$491$	−34.0640	−1.53729	−0.768643	+	0.639678i	$0.779068\pi$
$492$	0	0
$493$	7.04221i	0.317165i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.4383i	0.602792i
$498$	0	0
$499$	−1.10321	−0.0493865	−0.0246933	−	0.999695i	$-0.507861\pi$
$499$	−1.10321	−0.0493865	−0.0246933	+	0.999695i	$0.507861\pi$
$500$	0	0
$501$	10.6770	0.477011
$502$	0	0
$503$	1.06638i	0.0475473i	0.999717	+	0.0237737i	$0.00756811\pi$
$503$	1.06638i	0.0475473i	−0.999717	+	0.0237737i	$0.992432\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.62565i	0.249844i
$508$	0	0
$509$	−6.02608	−0.267101	−0.133551	−	0.991042i	$-0.542638\pi$
$509$	−6.02608	−0.267101	−0.133551	+	0.991042i	$0.542638\pi$
$510$	0	0
$511$	17.0180	0.752834
$512$	0	0
$513$	15.7560i	0.695645i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 11.0944i	− 0.487930i
$518$	0	0
$519$	−0.554958	−0.0243600
$520$	0	0
$521$	19.5555	0.856744	0.428372	−	0.903602i	$-0.359087\pi$
$521$	19.5555	0.856744	0.428372	+	0.903602i	$0.359087\pi$
$522$	0	0
$523$	− 0.846543i	− 0.0370167i	−0.999829	−	0.0185084i	$-0.994108\pi$
$523$	− 0.846543i	− 0.0370167i	0.999829	−	0.0185084i	$-0.00589174\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.70171i	− 0.248370i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−15.0532	−0.653255
$532$	0	0
$533$	− 2.14675i	− 0.0929862i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.79656i	− 0.206987i
$538$	0	0
$539$	15.6534	0.674239
$540$	0	0
$541$	−16.8291	−0.723539	−0.361769	−	0.932268i	$-0.617827\pi$
$541$	−16.8291	−0.723539	−0.361769	+	0.932268i	$0.617827\pi$
$542$	0	0
$543$	− 4.38537i	− 0.188194i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.57135i	− 0.109943i	−0.998488	−	0.0549715i	$-0.982493\pi$
$547$	− 2.57135i	− 0.109943i	0.998488	−	0.0549715i	$-0.0175068\pi$
$548$	0	0
$549$	22.5375	0.961877
$550$	0	0
$551$	−31.6474	−1.34823
$552$	0	0
$553$	12.3043i	0.523230i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.94092i	− 0.421210i	−0.977571	−	0.210605i	$-0.932457\pi$
$557$	− 9.94092i	− 0.421210i	0.977571	−	0.210605i	$-0.0675434\pi$
$558$	0	0
$559$	−3.21850	−0.136128
$560$	0	0
$561$	1.67158	0.0705741
$562$	0	0
$563$	13.7017i	0.577458i	0.957411	+	0.288729i	$0.0932327\pi$
$563$	13.7017i	0.577458i	−0.957411	+	0.288729i	$0.906767\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.01805i	− 0.294730i
$568$	0	0
$569$	27.8974	1.16952	0.584759	−	0.811207i	$-0.301189\pi$
$569$	27.8974	1.16952	0.584759	+	0.811207i	$0.301189\pi$
$570$	0	0
$571$	−31.5120	−1.31874	−0.659368	−	0.751820i	$-0.729176\pi$
$571$	−31.5120	−1.31874	−0.659368	+	0.751820i	$0.729176\pi$
$572$	0	0
$573$	4.07367i	0.170180i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	43.0006i	1.79014i	0.445928	+	0.895069i	$0.352874\pi$
$577$	43.0006i	1.79014i	−0.445928	+	0.895069i	$0.647126\pi$
$578$	0	0
$579$	−13.5157	−0.561695
$580$	0	0
$581$	−8.48799	−0.352141
$582$	0	0
$583$	10.2586i	0.424867i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.3080i	− 0.590553i	−0.955412	−	0.295277i	$-0.904588\pi$
$587$	− 14.3080i	− 0.590553i	0.955412	−	0.295277i	$-0.0954118\pi$
$588$	0	0
$589$	25.6233	1.05579
$590$	0	0
$591$	4.72348	0.194298
$592$	0	0
$593$	37.5991i	1.54401i	0.635617	+	0.772005i	$0.280746\pi$
$593$	37.5991i	1.54401i	−0.635617	+	0.772005i	$0.719254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7.75600i	− 0.317432i
$598$	0	0
$599$	−23.2634	−0.950516	−0.475258	−	0.879847i	$-0.657645\pi$
$599$	−23.2634	−0.950516	−0.475258	+	0.879847i	$0.657645\pi$
$600$	0	0
$601$	17.9922	0.733918	0.366959	−	0.930237i	$-0.380399\pi$
$601$	17.9922	0.733918	0.366959	+	0.930237i	$0.380399\pi$
$602$	0	0
$603$	− 28.9879i	− 1.18048i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 23.0508i	− 0.935605i	−0.883833	−	0.467802i	$-0.845046\pi$
$607$	− 23.0508i	− 0.935605i	0.883833	−	0.467802i	$-0.154954\pi$
$608$	0	0
$609$	−3.90813	−0.158366
$610$	0	0
$611$	−6.91723	−0.279841
$612$	0	0
$613$	− 42.8961i	− 1.73256i	−0.499563	−	0.866278i	$-0.666506\pi$
$613$	− 42.8961i	− 1.73256i	0.499563	−	0.866278i	$-0.333494\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 8.87933i	− 0.357468i	−0.983897	−	0.178734i	$-0.942800\pi$
$617$	− 8.87933i	− 0.357468i	0.983897	−	0.178734i	$-0.0572002\pi$
$618$	0	0
$619$	14.9831	0.602223	0.301111	−	0.953589i	$-0.402642\pi$
$619$	14.9831	0.602223	0.301111	+	0.953589i	$0.402642\pi$
$620$	0	0
$621$	3.15883	0.126760
$622$	0	0
$623$	− 4.63533i	− 0.185711i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.51201i	0.300001i
$628$	0	0
$629$	−6.32842	−0.252331
$630$	0	0
$631$	27.8866	1.11015	0.555075	−	0.831801i	$-0.312690\pi$
$631$	27.8866	1.11015	0.555075	+	0.831801i	$0.312690\pi$
$632$	0	0
$633$	3.67828i	0.146199i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.75973i	− 0.386694i
$638$	0	0
$639$	−32.5937	−1.28939
$640$	0	0
$641$	17.2862	0.682764	0.341382	−	0.939925i	$-0.389105\pi$
$641$	17.2862	0.682764	0.341382	+	0.939925i	$0.389105\pi$
$642$	0	0
$643$	− 39.1379i	− 1.54345i	−0.635957	−	0.771724i	$-0.719394\pi$
$643$	− 39.1379i	− 1.54345i	0.635957	−	0.771724i	$-0.280606\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.97152i	0.274079i	0.990566	+	0.137039i	$0.0437587\pi$
$647$	6.97152i	0.274079i	−0.990566	+	0.137039i	$0.956241\pi$
$648$	0	0
$649$	−15.1750	−0.595669
$650$	0	0
$651$	3.16421	0.124015
$652$	0	0
$653$	− 16.7603i	− 0.655882i	−0.944698	−	0.327941i	$-0.893645\pi$
$653$	− 16.7603i	− 0.655882i	0.944698	−	0.327941i	$-0.106355\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	41.2760i	1.61033i
$658$	0	0
$659$	−34.3612	−1.33852	−0.669261	−	0.743027i	$-0.733389\pi$
$659$	−34.3612	−1.33852	−0.669261	+	0.743027i	$0.733389\pi$
$660$	0	0
$661$	−42.4456	−1.65094	−0.825472	−	0.564443i	$-0.809091\pi$
$661$	−42.4456	−1.65094	−0.825472	+	0.564443i	$0.809091\pi$
$662$	0	0
$663$	− 1.04221i	− 0.0404762i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.34481i	0.245672i
$668$	0	0
$669$	6.75063	0.260994
$670$	0	0
$671$	22.7198	0.877087
$672$	0	0
$673$	− 42.9657i	− 1.65621i	−0.560577	−	0.828103i	$-0.689420\pi$
$673$	− 42.9657i	− 1.65621i	0.560577	−	0.828103i	$-0.310580\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.2150i	0.853794i	0.904300	+	0.426897i	$0.140393\pi$
$677$	22.2150i	0.853794i	−0.904300	+	0.426897i	$0.859607\pi$
$678$	0	0
$679$	−10.5509	−0.404907
$680$	0	0
$681$	11.8914	0.455680
$682$	0	0
$683$	7.85086i	0.300405i	0.988655	+	0.150202i	$0.0479925\pi$
$683$	7.85086i	0.300405i	−0.988655	+	0.150202i	$0.952007\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.9758i	0.609516i
$688$	0	0
$689$	6.39612	0.243673
$690$	0	0
$691$	35.8961	1.36555	0.682775	−	0.730629i	$-0.260773\pi$
$691$	35.8961	1.36555	0.682775	+	0.730629i	$0.260773\pi$
$692$	0	0
$693$	− 8.10859i	− 0.308020i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.40821i	− 0.0533396i
$698$	0	0
$699$	−8.04892	−0.304438
$700$	0	0
$701$	16.4263	0.620411	0.310206	−	0.950670i	$-0.399602\pi$
$701$	16.4263	0.620411	0.310206	+	0.950670i	$0.399602\pi$
$702$	0	0
$703$	− 28.4397i	− 1.07262i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.74259i	0.291190i
$708$	0	0
$709$	42.5978	1.59979	0.799896	−	0.600138i	$-0.204888\pi$
$709$	42.5978	1.59979	0.799896	+	0.600138i	$0.204888\pi$
$710$	0	0
$711$	−29.8431	−1.11920
$712$	0	0
$713$	− 5.13706i	− 0.192385i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.0084i	0.411115i
$718$	0	0
$719$	−13.7802	−0.513914	−0.256957	−	0.966423i	$-0.582720\pi$
$719$	−13.7802	−0.513914	−0.256957	+	0.966423i	$0.582720\pi$
$720$	0	0
$721$	2.28009	0.0849152
$722$	0	0
$723$	− 7.06638i	− 0.262801i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.7138i	0.694056i	0.937855	+	0.347028i	$0.112809\pi$
$727$	18.7138i	0.694056i	−0.937855	+	0.347028i	$0.887191\pi$
$728$	0	0
$729$	11.7627	0.435656
$730$	0	0
$731$	−2.11124	−0.0780872
$732$	0	0
$733$	− 6.09783i	− 0.225229i	−0.993639	−	0.112614i	$-0.964078\pi$
$733$	− 6.09783i	− 0.225229i	0.993639	−	0.112614i	$-0.0359225\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 29.2223i	− 1.07642i
$738$	0	0
$739$	11.3569	0.417770	0.208885	−	0.977940i	$-0.433017\pi$
$739$	11.3569	0.417770	0.208885	+	0.977940i	$0.433017\pi$
$740$	0	0
$741$	4.68366	0.172059
$742$	0	0
$743$	− 0.748709i	− 0.0274675i	−0.999906	−	0.0137337i	$-0.995628\pi$
$743$	− 0.748709i	− 0.0274675i	0.999906	−	0.0137337i	$-0.00437172\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 20.5870i	− 0.753239i
$748$	0	0
$749$	−4.33108	−0.158254
$750$	0	0
$751$	22.1220	0.807243	0.403622	−	0.914926i	$-0.367751\pi$
$751$	22.1220	0.807243	0.403622	+	0.914926i	$0.367751\pi$
$752$	0	0
$753$	7.45771i	0.271774i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.24400i	0.0815594i	0.999168	+	0.0407797i	$0.0129842\pi$
$757$	2.24400i	0.0815594i	−0.999168	+	0.0407797i	$0.987016\pi$
$758$	0	0
$759$	1.50604	0.0546658
$760$	0	0
$761$	−17.8896	−0.648498	−0.324249	−	0.945972i	$-0.605112\pi$
$761$	−17.8896	−0.648498	−0.324249	+	0.945972i	$0.605112\pi$
$762$	0	0
$763$	3.12067i	0.112976i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.46144i	0.341633i
$768$	0	0
$769$	−10.8465	−0.391136	−0.195568	−	0.980690i	$-0.562655\pi$
$769$	−10.8465	−0.391136	−0.195568	+	0.980690i	$0.562655\pi$
$770$	0	0
$771$	9.00836	0.324428
$772$	0	0
$773$	34.7633i	1.25035i	0.780485	+	0.625174i	$0.214972\pi$
$773$	34.7633i	1.25035i	−0.780485	+	0.625174i	$0.785028\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.51201i	− 0.125993i
$778$	0	0
$779$	6.32842	0.226739
$780$	0	0
$781$	−32.8573	−1.17573
$782$	0	0
$783$	− 20.0422i	− 0.716250i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.06638i	0.251889i	0.992037	+	0.125945i	$0.0401961\pi$
$787$	7.06638i	0.251889i	−0.992037	+	0.125945i	$0.959804\pi$
$788$	0	0
$789$	3.26205	0.116132
$790$	0	0
$791$	0.879330	0.0312654
$792$	0	0
$793$	− 14.1655i	− 0.503033i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.8672i	0.987109i	0.869715	+	0.493554i	$0.164303\pi$
$797$	27.8672i	0.987109i	−0.869715	+	0.493554i	$0.835697\pi$
$798$	0	0
$799$	−4.53750	−0.160525
$800$	0	0
$801$	11.2427	0.397240
$802$	0	0
$803$	41.6098i	1.46838i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.25534i	− 0.149795i
$808$	0	0
$809$	−39.4155	−1.38578	−0.692888	−	0.721046i	$-0.743662\pi$
$809$	−39.4155	−1.38578	−0.692888	+	0.721046i	$0.743662\pi$
$810$	0	0
$811$	−45.2040	−1.58733	−0.793664	−	0.608356i	$-0.791829\pi$
$811$	−45.2040	−1.58733	−0.793664	+	0.608356i	$0.791829\pi$
$812$	0	0
$813$	2.55496i	0.0896063i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.48785i	− 0.331938i
$818$	0	0
$819$	−5.05562	−0.176658
$820$	0	0
$821$	−40.5603	−1.41557	−0.707783	−	0.706430i	$-0.750304\pi$
$821$	−40.5603	−1.41557	−0.707783	+	0.706430i	$0.750304\pi$
$822$	0	0
$823$	− 29.2640i	− 1.02008i	−0.860151	−	0.510039i	$-0.829631\pi$
$823$	− 29.2640i	− 1.02008i	0.860151	−	0.510039i	$-0.170369\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.0060i	− 0.626129i	−0.949732	−	0.313064i	$-0.898644\pi$
$827$	− 18.0060i	− 0.626129i	0.949732	−	0.313064i	$-0.101356\pi$
$828$	0	0
$829$	−1.39134	−0.0483232	−0.0241616	−	0.999708i	$-0.507692\pi$
$829$	−1.39134	−0.0483232	−0.0241616	+	0.999708i	$0.507692\pi$
$830$	0	0
$831$	−2.24027	−0.0777143
$832$	0	0
$833$	− 6.40209i	− 0.221819i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.2271i	0.560892i
$838$	0	0
$839$	−29.3706	−1.01399	−0.506993	−	0.861950i	$-0.669243\pi$
$839$	−29.3706	−1.01399	−0.506993	+	0.861950i	$0.669243\pi$
$840$	0	0
$841$	11.2567	0.388161
$842$	0	0
$843$	11.1776i	0.384978i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.03492i	0.138641i
$848$	0	0
$849$	7.68830	0.263862
$850$	0	0
$851$	−5.70171	−0.195452
$852$	0	0
$853$	− 31.5147i	− 1.07904i	−0.841972	−	0.539521i	$-0.818605\pi$
$853$	− 31.5147i	− 1.07904i	0.841972	−	0.539521i	$-0.181395\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.3207i	0.967415i	0.875230	+	0.483708i	$0.160710\pi$
$857$	28.3207i	0.967415i	−0.875230	+	0.483708i	$0.839290\pi$
$858$	0	0
$859$	−20.6276	−0.703803	−0.351902	−	0.936037i	$-0.614465\pi$
$859$	−20.6276	−0.703803	−0.351902	+	0.936037i	$0.614465\pi$
$860$	0	0
$861$	0.781495	0.0266333
$862$	0	0
$863$	− 14.6394i	− 0.498330i	−0.968461	−	0.249165i	$-0.919844\pi$
$863$	− 14.6394i	− 0.498330i	0.968461	−	0.249165i	$-0.0801562\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.75063i	0.297187i
$868$	0	0
$869$	−30.0844	−1.02054
$870$	0	0
$871$	−18.2198	−0.617355
$872$	0	0
$873$	− 25.5905i	− 0.866106i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.6112i	0.999898i	0.866055	+	0.499949i	$0.166648\pi$
$877$	29.6112i	0.999898i	−0.866055	+	0.499949i	$0.833352\pi$
$878$	0	0
$879$	−11.5195	−0.388541
$880$	0	0
$881$	−19.7754	−0.666250	−0.333125	−	0.942883i	$-0.608103\pi$
$881$	−19.7754	−0.666250	−0.333125	+	0.942883i	$0.608103\pi$
$882$	0	0
$883$	− 9.78746i	− 0.329374i	−0.986346	−	0.164687i	$-0.947339\pi$
$883$	− 9.78746i	− 0.329374i	0.986346	−	0.164687i	$-0.0526615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.9476i	1.27416i	0.770799	+	0.637078i	$0.219857\pi$
$887$	37.9476i	1.27416i	−0.770799	+	0.637078i	$0.780143\pi$
$888$	0	0
$889$	−8.71379	−0.292251
$890$	0	0
$891$	17.1594	0.574862
$892$	0	0
$893$	− 20.3913i	− 0.682370i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.939001i	− 0.0313523i
$898$	0	0
$899$	−32.5937	−1.08706
$900$	0	0
$901$	4.19567	0.139778
$902$	0	0
$903$	− 1.17165i	− 0.0389901i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	48.6521i	1.61546i	0.589549	+	0.807732i	$0.299305\pi$
$907$	48.6521i	1.61546i	−0.589549	+	0.807732i	$0.700695\pi$
$908$	0	0
$909$	−18.7791	−0.622864
$910$	0	0
$911$	−12.2875	−0.407104	−0.203552	−	0.979064i	$-0.565249\pi$
$911$	−12.2875	−0.407104	−0.203552	+	0.979064i	$0.565249\pi$
$912$	0	0
$913$	− 20.7535i	− 0.686840i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.6401i	− 0.483459i
$918$	0	0
$919$	−12.8552	−0.424053	−0.212026	−	0.977264i	$-0.568006\pi$
$919$	−12.8552	−0.424053	−0.212026	+	0.977264i	$0.568006\pi$
$920$	0	0
$921$	0.548253	0.0180656
$922$	0	0
$923$	20.4862i	0.674311i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.53020i	0.181636i
$928$	0	0
$929$	−42.1551	−1.38306	−0.691532	−	0.722346i	$-0.743064\pi$
$929$	−42.1551	−1.38306	−0.691532	+	0.722346i	$0.743064\pi$
$930$	0	0
$931$	28.7707	0.942923
$932$	0	0
$933$	− 4.40522i	− 0.144221i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13.3056i	0.434675i	0.976097	+	0.217337i	$0.0697371\pi$
$937$	13.3056i	0.434675i	−0.976097	+	0.217337i	$0.930263\pi$
$938$	0	0
$939$	4.57242	0.149215
$940$	0	0
$941$	−15.7802	−0.514419	−0.257209	−	0.966356i	$-0.582803\pi$
$941$	−15.7802	−0.514419	−0.257209	+	0.966356i	$0.582803\pi$
$942$	0	0
$943$	− 1.26875i	− 0.0413162i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.2784i	0.691456i	0.938335	+	0.345728i	$0.112368\pi$
$947$	21.2784i	0.691456i	−0.938335	+	0.345728i	$0.887632\pi$
$948$	0	0
$949$	25.9433	0.842156
$950$	0	0
$951$	−8.91425	−0.289064
$952$	0	0
$953$	45.5905i	1.47682i	0.674352	+	0.738410i	$0.264423\pi$
$953$	45.5905i	1.47682i	−0.674352	+	0.738410i	$0.735577\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.55555i	− 0.308887i
$958$	0	0
$959$	9.42758	0.304433
$960$	0	0
$961$	−4.61058	−0.148728
$962$	0	0
$963$	− 10.5047i	− 0.338510i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	36.2355i	1.16525i	0.812739	+	0.582627i	$0.197975\pi$
$967$	36.2355i	1.16525i	−0.812739	+	0.582627i	$0.802025\pi$
$968$	0	0
$969$	3.07234	0.0986979
$970$	0	0
$971$	−29.6534	−0.951622	−0.475811	−	0.879547i	$-0.657845\pi$
$971$	−29.6534	−0.951622	−0.475811	+	0.879547i	$0.657845\pi$
$972$	0	0
$973$	− 13.4168i	− 0.430124i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.2887i	1.80084i	0.435026	+	0.900418i	$0.356739\pi$
$977$	56.2887i	1.80084i	−0.435026	+	0.900418i	$0.643261\pi$
$978$	0	0
$979$	11.3336	0.362223
$980$	0	0
$981$	−7.56896	−0.241658
$982$	0	0
$983$	21.1535i	0.674690i	0.941381	+	0.337345i	$0.109529\pi$
$983$	21.1535i	0.674690i	−0.941381	+	0.337345i	$0.890471\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.51812i	− 0.0801527i
$988$	0	0
$989$	−1.90217	−0.0604853
$990$	0	0
$991$	50.5555	1.60595	0.802975	−	0.596013i	$-0.203249\pi$
$991$	50.5555	1.60595	0.802975	+	0.596013i	$0.203249\pi$
$992$	0	0
$993$	13.9119i	0.441479i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 28.7560i	− 0.910712i	−0.890310	−	0.455356i	$-0.849512\pi$
$997$	− 28.7560i	− 0.910712i	0.890310	−	0.455356i	$-0.150488\pi$
$998$	0	0
$999$	18.0108	0.569835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.s.4049.4		6
5.2	odd	4		4600.2.a.z.1.2	yes	3
5.3	odd	4		4600.2.a.w.1.2	✓	3
5.4	even	2	inner	4600.2.e.s.4049.3		6
20.3	even	4		9200.2.a.ch.1.2		3
20.7	even	4		9200.2.a.cb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.w.1.2	✓	3	5.3	odd	4
4600.2.a.z.1.2	yes	3	5.2	odd	4
4600.2.e.s.4049.3		6	5.4	even	2	inner
4600.2.e.s.4049.4		6	1.1	even	1	trivial
9200.2.a.cb.1.2		3	20.7	even	4
9200.2.a.ch.1.2		3	20.3	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)

\(f(q)\)	\(=\)	\(q+0.554958i q^{3} -1.10992i q^{7} +2.69202 q^{9} +2.71379 q^{11} -1.69202i q^{13} -1.10992i q^{17} +4.98792 q^{19} +0.615957 q^{21} -1.00000i q^{23} +3.15883i q^{27} -6.34481 q^{29} +5.13706 q^{31} +1.50604i q^{33} -5.70171i q^{37} +0.939001 q^{39} +1.26875 q^{41} -1.90217i q^{43} -4.08815i q^{47} +5.76809 q^{49} +0.615957 q^{51} +3.78017i q^{53} +2.76809i q^{57} -5.59179 q^{59} +8.37196 q^{61} -2.98792i q^{63} -10.7681i q^{67} +0.554958 q^{69} -12.1075 q^{71} +15.3327i q^{73} -3.01208i q^{77} -11.0858 q^{79} +6.32304 q^{81} -7.64742i q^{83} -3.52111i q^{87} +4.17629 q^{89} -1.87800 q^{91} +2.85086i q^{93} -9.50604i q^{97} +7.30559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9} - 8 q^{19} + 24 q^{21} + 8 q^{29} + 20 q^{31} - 14 q^{39} - 8 q^{41} - 6 q^{49} + 24 q^{51} + 22 q^{59} - 8 q^{61} + 4 q^{69} + 8 q^{71} + 8 q^{79} - 2 q^{81} + 40 q^{89} + 28 q^{91} - 28 q^{99}+O(q^{100}) \) 6 * q + 6 * q^9 - 8 * q^19 + 24 * q^21 + 8 * q^29 + 20 * q^31 - 14 * q^39 - 8 * q^41 - 6 * q^49 + 24 * q^51 + 22 * q^59 - 8 * q^61 + 4 * q^69 + 8 * q^71 + 8 * q^79 - 2 * q^81 + 40 * q^89 + 28 * q^91 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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