LMFDB - Embedded newform 4600.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554958 q^{3} +1.10992 q^{7} -2.69202 q^{9} +O(q^{10})$	$q+0.554958 q^{3} +1.10992 q^{7} -2.69202 q^{9} +2.71379 q^{11} -1.69202 q^{13} +1.10992 q^{17} -4.98792 q^{19} +0.615957 q^{21} -1.00000 q^{23} -3.15883 q^{27} +6.34481 q^{29} +5.13706 q^{31} +1.50604 q^{33} +5.70171 q^{37} -0.939001 q^{39} +1.26875 q^{41} -1.90217 q^{43} +4.08815 q^{47} -5.76809 q^{49} +0.615957 q^{51} +3.78017 q^{53} -2.76809 q^{57} +5.59179 q^{59} +8.37196 q^{61} -2.98792 q^{63} +10.7681 q^{67} -0.554958 q^{69} -12.1075 q^{71} +15.3327 q^{73} +3.01208 q^{77} +11.0858 q^{79} +6.32304 q^{81} -7.64742 q^{83} +3.52111 q^{87} -4.17629 q^{89} -1.87800 q^{91} +2.85086 q^{93} +9.50604 q^{97} -7.30559 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 4 * q^7 - 3 * q^9	$ 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9} + 4 q^{17} + 4 q^{19} + 12 q^{21} - 3 q^{23} - q^{27} - 4 q^{29} + 10 q^{31} + 14 q^{33} - 10 q^{37} + 7 q^{39} - 4 q^{41} - 24 q^{43} + 16 q^{47} + 3 q^{49} + 12 q^{51} + 10 q^{53} + 12 q^{57} - 11 q^{59} - 4 q^{61} + 10 q^{63} + 12 q^{67} - 2 q^{69} + 4 q^{71} + 4 q^{73} + 28 q^{77} - 4 q^{79} - q^{81} - 8 q^{83} - 5 q^{87} - 20 q^{89} + 14 q^{91} - 5 q^{93} + 38 q^{97} + 14 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 4 * q^7 - 3 * q^9 + 4 * q^17 + 4 * q^19 + 12 * q^21 - 3 * q^23 - q^27 - 4 * q^29 + 10 * q^31 + 14 * q^33 - 10 * q^37 + 7 * q^39 - 4 * q^41 - 24 * q^43 + 16 * q^47 + 3 * q^49 + 12 * q^51 + 10 * q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 10 * q^63 + 12 * q^67 - 2 * q^69 + 4 * q^71 + 4 * q^73 + 28 * q^77 - 4 * q^79 - q^81 - 8 * q^83 - 5 * q^87 - 20 * q^89 + 14 * q^91 - 5 * q^93 + 38 * q^97 + 14 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.554958	0.320405	0.160203	−	0.987084i	$-0.448785\pi$
$3$	0.554958	0.320405	0.160203	+	0.987084i	$0.448785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.10992	0.419509	0.209754	−	0.977754i	$-0.432734\pi$
$7$	1.10992	0.419509	0.209754	+	0.977754i	$0.432734\pi$
$8$	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
$13$	−1.69202	−0.469282	−0.234641	−	0.972082i	$-0.575392\pi$
$13$	−1.69202	−0.469282	−0.234641	+	0.972082i	$0.575392\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.10992	0.269194	0.134597	−	0.990900i	$-0.457026\pi$
$17$	1.10992	0.269194	0.134597	+	0.990900i	$0.457026\pi$
$18$	0	0
$19$	−4.98792	−1.14431	−0.572153	−	0.820147i	$-0.693892\pi$
$19$	−4.98792	−1.14431	−0.572153	+	0.820147i	$0.693892\pi$
$20$	0	0
$21$	0.615957	0.134413
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.15883	−0.607918
$28$	0	0
$29$	6.34481	1.17820	0.589101	−	0.808059i	$-0.299482\pi$
$29$	6.34481	1.17820	0.589101	+	0.808059i	$0.299482\pi$
$30$	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
$33$	1.50604	0.262168
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.70171	0.937355	0.468678	−	0.883369i	$-0.344731\pi$
$37$	5.70171	0.937355	0.468678	+	0.883369i	$0.344731\pi$
$38$	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
$43$	−1.90217	−0.290077	−0.145039	−	0.989426i	$-0.546331\pi$
$43$	−1.90217	−0.290077	−0.145039	+	0.989426i	$0.546331\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.08815	0.596317	0.298159	−	0.954516i	$-0.403628\pi$
$47$	4.08815	0.596317	0.298159	+	0.954516i	$0.403628\pi$
$48$	0	0
$49$	−5.76809	−0.824012
$50$	0	0
$51$	0.615957	0.0862512
$52$	0	0
$53$	3.78017	0.519246	0.259623	−	0.965710i	$-0.416402\pi$
$53$	3.78017	0.519246	0.259623	+	0.965710i	$0.416402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.76809	−0.366642
$58$	0	0
$59$	5.59179	0.727990	0.363995	−	0.931401i	$-0.381413\pi$
$59$	5.59179	0.727990	0.363995	+	0.931401i	$0.381413\pi$
$60$	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
$63$	−2.98792	−0.376442
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.7681	1.31553	0.657766	−	0.753223i	$-0.271502\pi$
$67$	10.7681	1.31553	0.657766	+	0.753223i	$0.271502\pi$
$68$	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
$73$	15.3327	1.79456	0.897280	−	0.441461i	$-0.145540\pi$
$73$	15.3327	1.79456	0.897280	+	0.441461i	$0.145540\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.01208	0.343259
$78$	0	0
$79$	11.0858	1.24724	0.623622	−	0.781726i	$-0.285660\pi$
$79$	11.0858	1.24724	0.623622	+	0.781726i	$0.285660\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	0	0
$83$	−7.64742	−0.839413	−0.419706	−	0.907660i	$-0.637867\pi$
$83$	−7.64742	−0.839413	−0.419706	+	0.907660i	$0.637867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.52111	0.377502
$88$	0	0
$89$	−4.17629	−0.442686	−0.221343	−	0.975196i	$-0.571044\pi$
$89$	−4.17629	−0.442686	−0.221343	+	0.975196i	$0.571044\pi$
$90$	0	0
$91$	−1.87800	−0.196868
$92$	0	0
$93$	2.85086	0.295620
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.50604	0.965192	0.482596	−	0.875843i	$-0.339694\pi$
$97$	9.50604	0.965192	0.482596	+	0.875843i	$0.339694\pi$
$98$	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
$103$	2.05429	0.202416	0.101208	−	0.994865i	$-0.467729\pi$
$103$	2.05429	0.202416	0.101208	+	0.994865i	$0.467729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.90217	0.377237	0.188618	−	0.982050i	$-0.439599\pi$
$107$	3.90217	0.377237	0.188618	+	0.982050i	$0.439599\pi$
$108$	0	0
$109$	2.81163	0.269305	0.134652	−	0.990893i	$-0.457008\pi$
$109$	2.81163	0.269305	0.134652	+	0.990893i	$0.457008\pi$
$110$	0	0
$111$	3.16421	0.300334
$112$	0	0
$113$	0.792249	0.0745285	0.0372643	−	0.999305i	$-0.488136\pi$
$113$	0.792249	0.0745285	0.0372643	+	0.999305i	$0.488136\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.55496	0.421106
$118$	0	0
$119$	1.23191	0.112929
$120$	0	0
$121$	−3.63533	−0.330485
$122$	0	0
$123$	0.704103	0.0634868
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.85086	0.696651	0.348325	−	0.937374i	$-0.386750\pi$
$127$	7.85086	0.696651	0.348325	+	0.937374i	$0.386750\pi$
$128$	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
$133$	−5.53617	−0.480047
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.49396	−0.725688	−0.362844	−	0.931850i	$-0.618194\pi$
$137$	−8.49396	−0.725688	−0.362844	+	0.931850i	$0.618194\pi$
$138$	0	0
$139$	−12.0881	−1.02530	−0.512652	−	0.858597i	$-0.671337\pi$
$139$	−12.0881	−1.02530	−0.512652	+	0.858597i	$0.671337\pi$
$140$	0	0
$141$	2.26875	0.191063
$142$	0	0
$143$	−4.59179	−0.383985
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.20105	−0.264018
$148$	0	0
$149$	−20.5133	−1.68052	−0.840259	−	0.542185i	$-0.817597\pi$
$149$	−20.5133	−1.68052	−0.840259	+	0.542185i	$0.817597\pi$
$150$	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
$153$	−2.98792	−0.241559
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.16421	−0.0929141	−0.0464571	−	0.998920i	$-0.514793\pi$
$157$	−1.16421	−0.0929141	−0.0464571	+	0.998920i	$0.514793\pi$
$158$	0	0
$159$	2.09783	0.166369
$160$	0	0
$161$	−1.10992	−0.0874737
$162$	0	0
$163$	8.83877	0.692306	0.346153	−	0.938178i	$-0.387488\pi$
$163$	8.83877	0.692306	0.346153	+	0.938178i	$0.387488\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.2392	1.48877	0.744387	−	0.667748i	$-0.232742\pi$
$167$	19.2392	1.48877	0.744387	+	0.667748i	$0.232742\pi$
$168$	0	0
$169$	−10.1371	−0.779774
$170$	0	0
$171$	13.4276	1.02683
$172$	0	0
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.10321	0.233252
$178$	0	0
$179$	8.64310	0.646016	0.323008	−	0.946396i	$-0.395306\pi$
$179$	8.64310	0.646016	0.323008	+	0.946396i	$0.395306\pi$
$180$	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
$183$	4.64609	0.343449
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.01208	0.220265
$188$	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
$193$	24.3545	1.75308	0.876538	−	0.481333i	$-0.159847\pi$
$193$	24.3545	1.75308	0.876538	+	0.481333i	$0.159847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.51142	0.606413	0.303207	−	0.952925i	$-0.401943\pi$
$197$	8.51142	0.606413	0.303207	+	0.952925i	$0.401943\pi$
$198$	0	0
$199$	13.9758	0.990721	0.495360	−	0.868688i	$-0.335036\pi$
$199$	13.9758	0.990721	0.495360	+	0.868688i	$0.335036\pi$
$200$	0	0
$201$	5.97584	0.421503
$202$	0	0
$203$	7.04221	0.494266
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.69202	0.187108
$208$	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
$213$	−6.71917	−0.460390
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.70171	0.387057
$218$	0	0
$219$	8.50902	0.574987
$220$	0	0
$221$	−1.87800	−0.126328
$222$	0	0
$223$	−12.1642	−0.814576	−0.407288	−	0.913300i	$-0.633525\pi$
$223$	−12.1642	−0.814576	−0.407288	+	0.913300i	$0.633525\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.4276	1.42220	0.711099	−	0.703092i	$-0.248198\pi$
$227$	21.4276	1.42220	0.711099	+	0.703092i	$0.248198\pi$
$228$	0	0
$229$	−28.7875	−1.90233	−0.951165	−	0.308684i	$-0.900111\pi$
$229$	−28.7875	−1.90233	−0.951165	+	0.308684i	$0.900111\pi$
$230$	0	0
$231$	1.67158	0.109982
$232$	0	0
$233$	14.5036	0.950166	0.475083	−	0.879941i	$-0.342418\pi$
$233$	14.5036	0.950166	0.475083	+	0.879941i	$0.342418\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.15213	0.399624
$238$	0	0
$239$	−19.8364	−1.28311	−0.641554	−	0.767078i	$-0.721710\pi$
$239$	−19.8364	−1.28311	−0.641554	+	0.767078i	$0.721710\pi$
$240$	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
$243$	12.9855	0.833022
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.43967	0.537003
$248$	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
$253$	−2.71379	−0.170615
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.2325	1.01256	0.506278	−	0.862370i	$-0.331021\pi$
$257$	16.2325	1.01256	0.506278	+	0.862370i	$0.331021\pi$
$258$	0	0
$259$	6.32842	0.393229
$260$	0	0
$261$	−17.0804	−1.05725
$262$	0	0
$263$	−5.87800	−0.362453	−0.181227	−	0.983441i	$-0.558007\pi$
$263$	−5.87800	−0.362453	−0.181227	+	0.983441i	$0.558007\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.31767	−0.141839
$268$	0	0
$269$	7.66786	0.467518	0.233759	−	0.972295i	$-0.424897\pi$
$269$	7.66786	0.467518	0.233759	+	0.972295i	$0.424897\pi$
$270$	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
$273$	−1.04221	−0.0630776
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.03684	−0.242550	−0.121275	−	0.992619i	$-0.538698\pi$
$277$	−4.03684	−0.242550	−0.121275	+	0.992619i	$0.538698\pi$
$278$	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
$283$	−13.8538	−0.823525	−0.411763	−	0.911291i	$-0.635087\pi$
$283$	−13.8538	−0.823525	−0.411763	+	0.911291i	$0.635087\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.40821	0.0831238
$288$	0	0
$289$	−15.7681	−0.927534
$290$	0	0
$291$	5.27545	0.309253
$292$	0	0
$293$	20.7573	1.21266	0.606328	−	0.795215i	$-0.292642\pi$
$293$	20.7573	1.21266	0.606328	+	0.795215i	$0.292642\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.57242	−0.497422
$298$	0	0
$299$	1.69202	0.0978521
$300$	0	0
$301$	−2.11124	−0.121690
$302$	0	0
$303$	−3.87130	−0.222400
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.987918	0.0563835	0.0281917	−	0.999603i	$-0.491025\pi$
$307$	0.987918	0.0563835	0.0281917	+	0.999603i	$0.491025\pi$
$308$	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
$313$	−8.23921	−0.465708	−0.232854	−	0.972512i	$-0.574806\pi$
$313$	−8.23921	−0.465708	−0.232854	+	0.972512i	$0.574806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.0629	−0.902183	−0.451092	−	0.892478i	$-0.648965\pi$
$317$	−16.0629	−0.902183	−0.451092	+	0.892478i	$0.648965\pi$
$318$	0	0
$319$	17.2185	0.964051
$320$	0	0
$321$	2.16554	0.120869
$322$	0	0
$323$	−5.53617	−0.308041
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.56033	0.0862867
$328$	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
$333$	−15.3491	−0.841127
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−27.5690	−1.50178	−0.750888	−	0.660429i	$-0.770374\pi$
$337$	−27.5690	−1.50178	−0.750888	+	0.660429i	$0.770374\pi$
$338$	0	0
$339$	0.439665	0.0238793
$340$	0	0
$341$	13.9409	0.754943
$342$	0	0
$343$	−14.1715	−0.765189
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.47112	0.454754	0.227377	−	0.973807i	$-0.426985\pi$
$347$	8.47112	0.454754	0.227377	+	0.973807i	$0.426985\pi$
$348$	0	0
$349$	−19.2959	−1.03289	−0.516443	−	0.856322i	$-0.672744\pi$
$349$	−19.2959	−1.03289	−0.516443	+	0.856322i	$0.672744\pi$
$350$	0	0
$351$	5.34481	0.285285
$352$	0	0
$353$	−0.829085	−0.0441277	−0.0220639	−	0.999757i	$-0.507024\pi$
$353$	−0.829085	−0.0441277	−0.0220639	+	0.999757i	$0.507024\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.683661	0.0361832
$358$	0	0
$359$	−9.87800	−0.521341	−0.260671	−	0.965428i	$-0.583944\pi$
$359$	−9.87800	−0.521341	−0.260671	+	0.965428i	$0.583944\pi$
$360$	0	0
$361$	5.87933	0.309438
$362$	0	0
$363$	−2.01746	−0.105889
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.0930	−0.944449	−0.472225	−	0.881478i	$-0.656549\pi$
$367$	−18.0930	−0.944449	−0.472225	+	0.881478i	$0.656549\pi$
$368$	0	0
$369$	−3.41550	−0.177804
$370$	0	0
$371$	4.19567	0.217828
$372$	0	0
$373$	18.3370	0.949456	0.474728	−	0.880133i	$-0.342546\pi$
$373$	18.3370	0.949456	0.474728	+	0.880133i	$0.342546\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.7356	−0.552910
$378$	0	0
$379$	−3.05562	−0.156957	−0.0784784	−	0.996916i	$-0.525006\pi$
$379$	−3.05562	−0.156957	−0.0784784	+	0.996916i	$0.525006\pi$
$380$	0	0
$381$	4.35690	0.223211
$382$	0	0
$383$	34.8611	1.78132	0.890660	−	0.454669i	$-0.150243\pi$
$383$	34.8611	1.78132	0.890660	+	0.454669i	$0.150243\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.12067	0.260298
$388$	0	0
$389$	1.42758	0.0723814	0.0361907	−	0.999345i	$-0.488478\pi$
$389$	1.42758	0.0723814	0.0361907	+	0.999345i	$0.488478\pi$
$390$	0	0
$391$	−1.10992	−0.0561309
$392$	0	0
$393$	7.32006	0.369248
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.9474	1.35245	0.676225	−	0.736695i	$-0.263615\pi$
$397$	26.9474	1.35245	0.676225	+	0.736695i	$0.263615\pi$
$398$	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
$403$	−8.69202	−0.432980
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.4733	0.766981
$408$	0	0
$409$	8.79417	0.434844	0.217422	−	0.976078i	$-0.430235\pi$
$409$	8.79417	0.434844	0.217422	+	0.976078i	$0.430235\pi$
$410$	0	0
$411$	−4.71379	−0.232514
$412$	0	0
$413$	6.20642	0.305398
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.70841	−0.328512
$418$	0	0
$419$	24.0844	1.17660	0.588301	−	0.808642i	$-0.299797\pi$
$419$	24.0844	1.17660	0.588301	+	0.808642i	$0.299797\pi$
$420$	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
$423$	−11.0054	−0.535100
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.29218	0.449680
$428$	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
$433$	5.76941	0.277260	0.138630	−	0.990344i	$-0.455730\pi$
$433$	5.76941	0.277260	0.138630	+	0.990344i	$0.455730\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.98792	0.238604
$438$	0	0
$439$	−3.51871	−0.167939	−0.0839695	−	0.996468i	$-0.526760\pi$
$439$	−3.51871	−0.167939	−0.0839695	+	0.996468i	$0.526760\pi$
$440$	0	0
$441$	15.5278	0.739420
$442$	0	0
$443$	−17.6437	−0.838277	−0.419139	−	0.907922i	$-0.637668\pi$
$443$	−17.6437	−0.838277	−0.419139	+	0.907922i	$0.637668\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−11.3840	−0.538447
$448$	0	0
$449$	5.13275	0.242230	0.121115	−	0.992639i	$-0.461353\pi$
$449$	5.13275	0.242230	0.121115	+	0.992639i	$0.461353\pi$
$450$	0	0
$451$	3.44312	0.162130
$452$	0	0
$453$	4.59717	0.215994
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.15213	−0.194228	−0.0971142	−	0.995273i	$-0.530961\pi$
$457$	−4.15213	−0.194228	−0.0971142	+	0.995273i	$0.530961\pi$
$458$	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
$463$	27.5676	1.28118	0.640588	−	0.767885i	$-0.278691\pi$
$463$	27.5676	1.28118	0.640588	+	0.767885i	$0.278691\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.6125	1.27775	0.638877	−	0.769309i	$-0.279399\pi$
$467$	27.6125	1.27775	0.638877	+	0.769309i	$0.279399\pi$
$468$	0	0
$469$	11.9517	0.551877
$470$	0	0
$471$	−0.646088	−0.0297702
$472$	0	0
$473$	−5.16208	−0.237353
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.1763	−0.465940
$478$	0	0
$479$	19.6233	0.896609	0.448305	−	0.893881i	$-0.352028\pi$
$479$	19.6233	0.896609	0.448305	+	0.893881i	$0.352028\pi$
$480$	0	0
$481$	−9.64742	−0.439884
$482$	0	0
$483$	−0.615957	−0.0280270
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.3709	−0.923093	−0.461547	−	0.887116i	$-0.652705\pi$
$487$	−20.3709	−0.923093	−0.461547	+	0.887116i	$0.652705\pi$
$488$	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
$493$	7.04221	0.317165
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.4383	−0.602792
$498$	0	0
$499$	1.10321	0.0493865	0.0246933	−	0.999695i	$-0.492139\pi$
$499$	1.10321	0.0493865	0.0246933	+	0.999695i	$0.492139\pi$
$500$	0	0
$501$	10.6770	0.477011
$502$	0	0
$503$	1.06638	0.0475473	0.0237737	−	0.999717i	$-0.492432\pi$
$503$	1.06638	0.0475473	0.0237737	+	0.999717i	$0.492432\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.62565	−0.249844
$508$	0	0
$509$	6.02608	0.267101	0.133551	−	0.991042i	$-0.457362\pi$
$509$	6.02608	0.267101	0.133551	+	0.991042i	$0.457362\pi$
$510$	0	0
$511$	17.0180	0.752834
$512$	0	0
$513$	15.7560	0.695645
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.0944	0.487930
$518$	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
$523$	−0.846543	−0.0370167	−0.0185084	−	0.999829i	$-0.505892\pi$
$523$	−0.846543	−0.0370167	−0.0185084	+	0.999829i	$0.505892\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.70171	0.248370
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−15.0532	−0.653255
$532$	0	0
$533$	−2.14675	−0.0929862
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.79656	0.206987
$538$	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
$543$	−4.38537	−0.188194
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.57135	0.109943	0.0549715	−	0.998488i	$-0.482493\pi$
$547$	2.57135	0.109943	0.0549715	+	0.998488i	$0.482493\pi$
$548$	0	0
$549$	−22.5375	−0.961877
$550$	0	0
$551$	−31.6474	−1.34823
$552$	0	0
$553$	12.3043	0.523230
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.94092	0.421210	0.210605	−	0.977571i	$-0.432457\pi$
$557$	9.94092	0.421210	0.210605	+	0.977571i	$0.432457\pi$
$558$	0	0
$559$	3.21850	0.136128
$560$	0	0
$561$	1.67158	0.0705741
$562$	0	0
$563$	13.7017	0.577458	0.288729	−	0.957411i	$-0.406767\pi$
$563$	13.7017	0.577458	0.288729	+	0.957411i	$0.406767\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.01805	0.294730
$568$	0	0
$569$	−27.8974	−1.16952	−0.584759	−	0.811207i	$-0.698811\pi$
$569$	−27.8974	−1.16952	−0.584759	+	0.811207i	$0.698811\pi$
$570$	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
$573$	4.07367	0.170180
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−43.0006	−1.79014	−0.895069	−	0.445928i	$-0.852874\pi$
$577$	−43.0006	−1.79014	−0.895069	+	0.445928i	$0.852874\pi$
$578$	0	0
$579$	13.5157	0.561695
$580$	0	0
$581$	−8.48799	−0.352141
$582$	0	0
$583$	10.2586	0.424867
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.3080	0.590553	0.295277	−	0.955412i	$-0.404588\pi$
$587$	14.3080	0.590553	0.295277	+	0.955412i	$0.404588\pi$
$588$	0	0
$589$	−25.6233	−1.05579
$590$	0	0
$591$	4.72348	0.194298
$592$	0	0
$593$	37.5991	1.54401	0.772005	−	0.635617i	$-0.219254\pi$
$593$	37.5991	1.54401	0.772005	+	0.635617i	$0.219254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.75600	0.317432
$598$	0	0
$599$	23.2634	0.950516	0.475258	−	0.879847i	$-0.342355\pi$
$599$	23.2634	0.950516	0.475258	+	0.879847i	$0.342355\pi$
$600$	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
$603$	−28.9879	−1.18048
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.0508	0.935605	0.467802	−	0.883833i	$-0.345046\pi$
$607$	23.0508	0.935605	0.467802	+	0.883833i	$0.345046\pi$
$608$	0	0
$609$	3.90813	0.158366
$610$	0	0
$611$	−6.91723	−0.279841
$612$	0	0
$613$	−42.8961	−1.73256	−0.866278	−	0.499563i	$-0.833494\pi$
$613$	−42.8961	−1.73256	−0.866278	+	0.499563i	$0.833494\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.87933	0.357468	0.178734	−	0.983897i	$-0.442800\pi$
$617$	8.87933	0.357468	0.178734	+	0.983897i	$0.442800\pi$
$618$	0	0
$619$	−14.9831	−0.602223	−0.301111	−	0.953589i	$-0.597358\pi$
$619$	−14.9831	−0.602223	−0.301111	+	0.953589i	$0.597358\pi$
$620$	0	0
$621$	3.15883	0.126760
$622$	0	0
$623$	−4.63533	−0.185711
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−7.51201	−0.300001
$628$	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
$633$	3.67828	0.146199
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.75973	0.386694
$638$	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
$643$	−39.1379	−1.54345	−0.771724	−	0.635957i	$-0.780606\pi$
$643$	−39.1379	−1.54345	−0.771724	+	0.635957i	$0.780606\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.97152	−0.274079	−0.137039	−	0.990566i	$-0.543759\pi$
$647$	−6.97152	−0.274079	−0.137039	+	0.990566i	$0.543759\pi$
$648$	0	0
$649$	15.1750	0.595669
$650$	0	0
$651$	3.16421	0.124015
$652$	0	0
$653$	−16.7603	−0.655882	−0.327941	−	0.944698i	$-0.606355\pi$
$653$	−16.7603	−0.655882	−0.327941	+	0.944698i	$0.606355\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−41.2760	−1.61033
$658$	0	0
$659$	34.3612	1.33852	0.669261	−	0.743027i	$-0.266611\pi$
$659$	34.3612	1.33852	0.669261	+	0.743027i	$0.266611\pi$
$660$	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
$663$	−1.04221	−0.0404762
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.34481	−0.245672
$668$	0	0
$669$	−6.75063	−0.260994
$670$	0	0
$671$	22.7198	0.877087
$672$	0	0
$673$	−42.9657	−1.65621	−0.828103	−	0.560577i	$-0.810580\pi$
$673$	−42.9657	−1.65621	−0.828103	+	0.560577i	$0.810580\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.2150	−0.853794	−0.426897	−	0.904300i	$-0.640393\pi$
$677$	−22.2150	−0.853794	−0.426897	+	0.904300i	$0.640393\pi$
$678$	0	0
$679$	10.5509	0.404907
$680$	0	0
$681$	11.8914	0.455680
$682$	0	0
$683$	7.85086	0.300405	0.150202	−	0.988655i	$-0.452007\pi$
$683$	7.85086	0.300405	0.150202	+	0.988655i	$0.452007\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15.9758	−0.609516
$688$	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
$693$	−8.10859	−0.308020
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.40821	0.0533396
$698$	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
$703$	−28.4397	−1.07262
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.74259	−0.291190
$708$	0	0
$709$	−42.5978	−1.59979	−0.799896	−	0.600138i	$-0.795112\pi$
$709$	−42.5978	−1.59979	−0.799896	+	0.600138i	$0.795112\pi$
$710$	0	0
$711$	−29.8431	−1.11920
$712$	0	0
$713$	−5.13706	−0.192385
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.0084	−0.411115
$718$	0	0
$719$	13.7802	0.513914	0.256957	−	0.966423i	$-0.417280\pi$
$719$	13.7802	0.513914	0.256957	+	0.966423i	$0.417280\pi$
$720$	0	0
$721$	2.28009	0.0849152
$722$	0	0
$723$	−7.06638	−0.262801
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.7138	−0.694056	−0.347028	−	0.937855i	$-0.612809\pi$
$727$	−18.7138	−0.694056	−0.347028	+	0.937855i	$0.612809\pi$
$728$	0	0
$729$	−11.7627	−0.435656
$730$	0	0
$731$	−2.11124	−0.0780872
$732$	0	0
$733$	−6.09783	−0.225229	−0.112614	−	0.993639i	$-0.535922\pi$
$733$	−6.09783	−0.225229	−0.112614	+	0.993639i	$0.535922\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.2223	1.07642
$738$	0	0
$739$	−11.3569	−0.417770	−0.208885	−	0.977940i	$-0.566983\pi$
$739$	−11.3569	−0.417770	−0.208885	+	0.977940i	$0.566983\pi$
$740$	0	0
$741$	4.68366	0.172059
$742$	0	0
$743$	−0.748709	−0.0274675	−0.0137337	−	0.999906i	$-0.504372\pi$
$743$	−0.748709	−0.0274675	−0.0137337	+	0.999906i	$0.504372\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.5870	0.753239
$748$	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
$753$	7.45771	0.271774
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.24400	−0.0815594	−0.0407797	−	0.999168i	$-0.512984\pi$
$757$	−2.24400	−0.0815594	−0.0407797	+	0.999168i	$0.512984\pi$
$758$	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
$763$	3.12067	0.112976
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.46144	−0.341633
$768$	0	0
$769$	10.8465	0.391136	0.195568	−	0.980690i	$-0.437345\pi$
$769$	10.8465	0.391136	0.195568	+	0.980690i	$0.437345\pi$
$770$	0	0
$771$	9.00836	0.324428
$772$	0	0
$773$	34.7633	1.25035	0.625174	−	0.780485i	$-0.285028\pi$
$773$	34.7633	1.25035	0.625174	+	0.780485i	$0.285028\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.51201	0.125993
$778$	0	0
$779$	−6.32842	−0.226739
$780$	0	0
$781$	−32.8573	−1.17573
$782$	0	0
$783$	−20.0422	−0.716250
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.06638	−0.251889	−0.125945	−	0.992037i	$-0.540196\pi$
$787$	−7.06638	−0.251889	−0.125945	+	0.992037i	$0.540196\pi$
$788$	0	0
$789$	−3.26205	−0.116132
$790$	0	0
$791$	0.879330	0.0312654
$792$	0	0
$793$	−14.1655	−0.503033
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.8672	−0.987109	−0.493554	−	0.869715i	$-0.664303\pi$
$797$	−27.8672	−0.987109	−0.493554	+	0.869715i	$0.664303\pi$
$798$	0	0
$799$	4.53750	0.160525
$800$	0	0
$801$	11.2427	0.397240
$802$	0	0
$803$	41.6098	1.46838
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.25534	0.149795
$808$	0	0
$809$	39.4155	1.38578	0.692888	−	0.721046i	$-0.256338\pi$
$809$	39.4155	1.38578	0.692888	+	0.721046i	$0.256338\pi$
$810$	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
$813$	2.55496	0.0896063
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.48785	0.331938
$818$	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
$823$	−29.2640	−1.02008	−0.510039	−	0.860151i	$-0.670369\pi$
$823$	−29.2640	−1.02008	−0.510039	+	0.860151i	$0.670369\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.0060	0.626129	0.313064	−	0.949732i	$-0.398644\pi$
$827$	18.0060	0.626129	0.313064	+	0.949732i	$0.398644\pi$
$828$	0	0
$829$	1.39134	0.0483232	0.0241616	−	0.999708i	$-0.492308\pi$
$829$	1.39134	0.0483232	0.0241616	+	0.999708i	$0.492308\pi$
$830$	0	0
$831$	−2.24027	−0.0777143
$832$	0	0
$833$	−6.40209	−0.221819
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−16.2271	−0.560892
$838$	0	0
$839$	29.3706	1.01399	0.506993	−	0.861950i	$-0.330757\pi$
$839$	29.3706	1.01399	0.506993	+	0.861950i	$0.330757\pi$
$840$	0	0
$841$	11.2567	0.388161
$842$	0	0
$843$	11.1776	0.384978
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.03492	−0.138641
$848$	0	0
$849$	−7.68830	−0.263862
$850$	0	0
$851$	−5.70171	−0.195452
$852$	0	0
$853$	−31.5147	−1.07904	−0.539521	−	0.841972i	$-0.681395\pi$
$853$	−31.5147	−1.07904	−0.539521	+	0.841972i	$0.681395\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.3207	−0.967415	−0.483708	−	0.875230i	$-0.660710\pi$
$857$	−28.3207	−0.967415	−0.483708	+	0.875230i	$0.660710\pi$
$858$	0	0
$859$	20.6276	0.703803	0.351902	−	0.936037i	$-0.385535\pi$
$859$	20.6276	0.703803	0.351902	+	0.936037i	$0.385535\pi$
$860$	0	0
$861$	0.781495	0.0266333
$862$	0	0
$863$	−14.6394	−0.498330	−0.249165	−	0.968461i	$-0.580156\pi$
$863$	−14.6394	−0.498330	−0.249165	+	0.968461i	$0.580156\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.75063	−0.297187
$868$	0	0
$869$	30.0844	1.02054
$870$	0	0
$871$	−18.2198	−0.617355
$872$	0	0
$873$	−25.5905	−0.866106
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.6112	−0.999898	−0.499949	−	0.866055i	$-0.666648\pi$
$877$	−29.6112	−0.999898	−0.499949	+	0.866055i	$0.666648\pi$
$878$	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
$883$	−9.78746	−0.329374	−0.164687	−	0.986346i	$-0.552661\pi$
$883$	−9.78746	−0.329374	−0.164687	+	0.986346i	$0.552661\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.9476	−1.27416	−0.637078	−	0.770799i	$-0.719857\pi$
$887$	−37.9476	−1.27416	−0.637078	+	0.770799i	$0.719857\pi$
$888$	0	0
$889$	8.71379	0.292251
$890$	0	0
$891$	17.1594	0.574862
$892$	0	0
$893$	−20.3913	−0.682370
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.939001	0.0313523
$898$	0	0
$899$	32.5937	1.08706
$900$	0	0
$901$	4.19567	0.139778
$902$	0	0
$903$	−1.17165	−0.0389901
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−48.6521	−1.61546	−0.807732	−	0.589549i	$-0.799305\pi$
$907$	−48.6521	−1.61546	−0.807732	+	0.589549i	$0.799305\pi$
$908$	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
$913$	−20.7535	−0.686840
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.6401	0.483459
$918$	0	0
$919$	12.8552	0.424053	0.212026	−	0.977264i	$-0.431994\pi$
$919$	12.8552	0.424053	0.212026	+	0.977264i	$0.431994\pi$
$920$	0	0
$921$	0.548253	0.0180656
$922$	0	0
$923$	20.4862	0.674311
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.53020	−0.181636
$928$	0	0
$929$	42.1551	1.38306	0.691532	−	0.722346i	$-0.256936\pi$
$929$	42.1551	1.38306	0.691532	+	0.722346i	$0.256936\pi$
$930$	0	0
$931$	28.7707	0.942923
$932$	0	0
$933$	−4.40522	−0.144221
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.3056	−0.434675	−0.217337	−	0.976097i	$-0.569737\pi$
$937$	−13.3056	−0.434675	−0.217337	+	0.976097i	$0.569737\pi$
$938$	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
$943$	−1.26875	−0.0413162
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.2784	−0.691456	−0.345728	−	0.938335i	$-0.612368\pi$
$947$	−21.2784	−0.691456	−0.345728	+	0.938335i	$0.612368\pi$
$948$	0	0
$949$	−25.9433	−0.842156
$950$	0	0
$951$	−8.91425	−0.289064
$952$	0	0
$953$	45.5905	1.47682	0.738410	−	0.674352i	$-0.235577\pi$
$953$	45.5905	1.47682	0.738410	+	0.674352i	$0.235577\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.55555	0.308887
$958$	0	0
$959$	−9.42758	−0.304433
$960$	0	0
$961$	−4.61058	−0.148728
$962$	0	0
$963$	−10.5047	−0.338510
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.2355	−1.16525	−0.582627	−	0.812739i	$-0.697975\pi$
$967$	−36.2355	−1.16525	−0.582627	+	0.812739i	$0.697975\pi$
$968$	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
$973$	−13.4168	−0.430124
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56.2887	−1.80084	−0.900418	−	0.435026i	$-0.856739\pi$
$977$	−56.2887	−1.80084	−0.900418	+	0.435026i	$0.856739\pi$
$978$	0	0
$979$	−11.3336	−0.362223
$980$	0	0
$981$	−7.56896	−0.241658
$982$	0	0
$983$	21.1535	0.674690	0.337345	−	0.941381i	$-0.390471\pi$
$983$	21.1535	0.674690	0.337345	+	0.941381i	$0.390471\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.51812	0.0801527
$988$	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
$993$	13.9119	0.441479
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.7560	0.910712	0.455356	−	0.890310i	$-0.349512\pi$
$997$	28.7560	0.910712	0.455356	+	0.890310i	$0.349512\pi$
$998$	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.a.z.1.2	yes	3
4.3	odd	2		9200.2.a.cb.1.2		3
5.2	odd	4		4600.2.e.s.4049.3		6
5.3	odd	4		4600.2.e.s.4049.4		6
5.4	even	2		4600.2.a.w.1.2	✓	3
20.19	odd	2		9200.2.a.ch.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.w.1.2	✓	3	5.4	even	2
4600.2.a.z.1.2	yes	3	1.1	even	1	trivial
4600.2.e.s.4049.3		6	5.2	odd	4
4600.2.e.s.4049.4		6	5.3	odd	4
9200.2.a.cb.1.2		3	4.3	odd	2
9200.2.a.ch.1.2		3	20.19	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+0.554958 q^{3} +1.10992 q^{7} -2.69202 q^{9} +O(q^{10})\)	\(q+0.554958 q^{3} +1.10992 q^{7} -2.69202 q^{9} +2.71379 q^{11} -1.69202 q^{13} +1.10992 q^{17} -4.98792 q^{19} +0.615957 q^{21} -1.00000 q^{23} -3.15883 q^{27} +6.34481 q^{29} +5.13706 q^{31} +1.50604 q^{33} +5.70171 q^{37} -0.939001 q^{39} +1.26875 q^{41} -1.90217 q^{43} +4.08815 q^{47} -5.76809 q^{49} +0.615957 q^{51} +3.78017 q^{53} -2.76809 q^{57} +5.59179 q^{59} +8.37196 q^{61} -2.98792 q^{63} +10.7681 q^{67} -0.554958 q^{69} -12.1075 q^{71} +15.3327 q^{73} +3.01208 q^{77} +11.0858 q^{79} +6.32304 q^{81} -7.64742 q^{83} +3.52111 q^{87} -4.17629 q^{89} -1.87800 q^{91} +2.85086 q^{93} +9.50604 q^{97} -7.30559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 4 * q^7 - 3 * q^9	\( 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9} + 4 q^{17} + 4 q^{19} + 12 q^{21} - 3 q^{23} - q^{27} - 4 q^{29} + 10 q^{31} + 14 q^{33} - 10 q^{37} + 7 q^{39} - 4 q^{41} - 24 q^{43} + 16 q^{47} + 3 q^{49} + 12 q^{51} + 10 q^{53} + 12 q^{57} - 11 q^{59} - 4 q^{61} + 10 q^{63} + 12 q^{67} - 2 q^{69} + 4 q^{71} + 4 q^{73} + 28 q^{77} - 4 q^{79} - q^{81} - 8 q^{83} - 5 q^{87} - 20 q^{89} + 14 q^{91} - 5 q^{93} + 38 q^{97} + 14 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 4 * q^7 - 3 * q^9 + 4 * q^17 + 4 * q^19 + 12 * q^21 - 3 * q^23 - q^27 - 4 * q^29 + 10 * q^31 + 14 * q^33 - 10 * q^37 + 7 * q^39 - 4 * q^41 - 24 * q^43 + 16 * q^47 + 3 * q^49 + 12 * q^51 + 10 * q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 10 * q^63 + 12 * q^67 - 2 * q^69 + 4 * q^71 + 4 * q^73 + 28 * q^77 - 4 * q^79 - q^81 - 8 * q^83 - 5 * q^87 - 20 * q^89 + 14 * q^91 - 5 * q^93 + 38 * q^97 + 14 * q^99

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