LMFDB - Embedded newform 208.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{3} +19.0000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-5.00000 q^{3} +19.0000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} -13.0000 q^{13} -95.0000 q^{15} +77.0000 q^{17} +58.0000 q^{19} -15.0000 q^{21} -76.0000 q^{23} +236.000 q^{25} +145.000 q^{27} -6.00000 q^{29} +292.000 q^{31} -10.0000 q^{33} +57.0000 q^{35} +207.000 q^{37} +65.0000 q^{39} +240.000 q^{41} +317.000 q^{43} -38.0000 q^{45} +375.000 q^{47} -334.000 q^{49} -385.000 q^{51} -692.000 q^{53} +38.0000 q^{55} -290.000 q^{57} -214.000 q^{59} -488.000 q^{61} -6.00000 q^{63} -247.000 q^{65} -782.000 q^{67} +380.000 q^{69} +1057.00 q^{71} +1174.00 q^{73} -1180.00 q^{75} +6.00000 q^{77} -892.000 q^{79} -671.000 q^{81} -704.000 q^{83} +1463.00 q^{85} +30.0000 q^{87} +6.00000 q^{89} -39.0000 q^{91} -1460.00 q^{93} +1102.00 q^{95} +830.000 q^{97} -4.00000 q^{99} +O(q^{100})$

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
$3$	−5.00000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−5.00000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	0	0
$5$	19.0000	1.69941	0.849706	−	0.527257i	$-0.176780\pi$
$5$	19.0000	1.69941	0.849706	+	0.527257i	$0.176780\pi$
$6$	0	0
$7$	3.00000	0.161985	0.0809924	−	0.996715i	$-0.474191\pi$
$7$	3.00000	0.161985	0.0809924	+	0.996715i	$0.474191\pi$
$8$	0	0
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	2.00000	0.0548202	0.0274101	−	0.999624i	$-0.491274\pi$
$11$	2.00000	0.0548202	0.0274101	+	0.999624i	$0.491274\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−95.0000	−1.63526
$16$	0	0
$17$	77.0000	1.09854	0.549272	−	0.835644i	$-0.314905\pi$
$17$	77.0000	1.09854	0.549272	+	0.835644i	$0.314905\pi$
$18$	0	0
$19$	58.0000	0.700322	0.350161	−	0.936690i	$-0.386127\pi$
$19$	58.0000	0.700322	0.350161	+	0.936690i	$0.386127\pi$
$20$	0	0
$21$	−15.0000	−0.155870
$22$	0	0
$23$	−76.0000	−0.689004	−0.344502	−	0.938786i	$-0.611952\pi$
$23$	−76.0000	−0.689004	−0.344502	+	0.938786i	$0.611952\pi$
$24$	0	0
$25$	236.000	1.88800
$26$	0	0
$27$	145.000	1.03353
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	292.000	1.69177	0.845883	−	0.533368i	$-0.179074\pi$
$31$	292.000	1.69177	0.845883	+	0.533368i	$0.179074\pi$
$32$	0	0
$33$	−10.0000	−0.0527508
$34$	0	0
$35$	57.0000	0.275279
$36$	0	0
$37$	207.000	0.919746	0.459873	−	0.887985i	$-0.347895\pi$
$37$	207.000	0.919746	0.459873	+	0.887985i	$0.347895\pi$
$38$	0	0
$39$	65.0000	0.266880
$40$	0	0
$41$	240.000	0.914188	0.457094	−	0.889418i	$-0.348890\pi$
$41$	240.000	0.914188	0.457094	+	0.889418i	$0.348890\pi$
$42$	0	0
$43$	317.000	1.12423	0.562117	−	0.827058i	$-0.309987\pi$
$43$	317.000	1.12423	0.562117	+	0.827058i	$0.309987\pi$
$44$	0	0
$45$	−38.0000	−0.125882
$46$	0	0
$47$	375.000	1.16382	0.581908	−	0.813254i	$-0.302306\pi$
$47$	375.000	1.16382	0.581908	+	0.813254i	$0.302306\pi$
$48$	0	0
$49$	−334.000	−0.973761
$50$	0	0
$51$	−385.000	−1.05707
$52$	0	0
$53$	−692.000	−1.79346	−0.896731	−	0.442576i	$-0.854065\pi$
$53$	−692.000	−1.79346	−0.896731	+	0.442576i	$0.854065\pi$
$54$	0	0
$55$	38.0000	0.0931622
$56$	0	0
$57$	−290.000	−0.673885
$58$	0	0
$59$	−214.000	−0.472211	−0.236105	−	0.971727i	$-0.575871\pi$
$59$	−214.000	−0.472211	−0.236105	+	0.971727i	$0.575871\pi$
$60$	0	0
$61$	−488.000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−488.000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−6.00000	−0.0119989
$64$	0	0
$65$	−247.000	−0.471332
$66$	0	0
$67$	−782.000	−1.42592	−0.712959	−	0.701206i	$-0.752645\pi$
$67$	−782.000	−1.42592	−0.712959	+	0.701206i	$0.752645\pi$
$68$	0	0
$69$	380.000	0.662995
$70$	0	0
$71$	1057.00	1.76680	0.883400	−	0.468619i	$-0.155248\pi$
$71$	1057.00	1.76680	0.883400	+	0.468619i	$0.155248\pi$
$72$	0	0
$73$	1174.00	1.88228	0.941139	−	0.338020i	$-0.109757\pi$
$73$	1174.00	1.88228	0.941139	+	0.338020i	$0.109757\pi$
$74$	0	0
$75$	−1180.00	−1.81673
$76$	0	0
$77$	6.00000	0.00888004
$78$	0	0
$79$	−892.000	−1.27035	−0.635176	−	0.772367i	$-0.719073\pi$
$79$	−892.000	−1.27035	−0.635176	+	0.772367i	$0.719073\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	−704.000	−0.931013	−0.465506	−	0.885045i	$-0.654128\pi$
$83$	−704.000	−0.931013	−0.465506	+	0.885045i	$0.654128\pi$
$84$	0	0
$85$	1463.00	1.86688
$86$	0	0
$87$	30.0000	0.0369694
$88$	0	0
$89$	6.00000	0.00714605	0.00357303	−	0.999994i	$-0.498863\pi$
$89$	6.00000	0.00714605	0.00357303	+	0.999994i	$0.498863\pi$
$90$	0	0
$91$	−39.0000	−0.0449265
$92$	0	0
$93$	−1460.00	−1.62790
$94$	0	0
$95$	1102.00	1.19013
$96$	0	0
$97$	830.000	0.868801	0.434401	−	0.900720i	$-0.356960\pi$
$97$	830.000	0.868801	0.434401	+	0.900720i	$0.356960\pi$
$98$	0	0
$99$	−4.00000	−0.00406076
$100$	0	0
$101$	276.000	0.271911	0.135956	−	0.990715i	$-0.456590\pi$
$101$	276.000	0.271911	0.135956	+	0.990715i	$0.456590\pi$
$102$	0	0
$103$	1160.00	1.10969	0.554846	−	0.831953i	$-0.312777\pi$
$103$	1160.00	1.10969	0.554846	+	0.831953i	$0.312777\pi$
$104$	0	0
$105$	−285.000	−0.264887
$106$	0	0
$107$	−324.000	−0.292731	−0.146366	−	0.989231i	$-0.546758\pi$
$107$	−324.000	−0.292731	−0.146366	+	0.989231i	$0.546758\pi$
$108$	0	0
$109$	199.000	0.174869	0.0874346	−	0.996170i	$-0.472133\pi$
$109$	199.000	0.174869	0.0874346	+	0.996170i	$0.472133\pi$
$110$	0	0
$111$	−1035.00	−0.885026
$112$	0	0
$113$	−1662.00	−1.38361	−0.691804	−	0.722085i	$-0.743184\pi$
$113$	−1662.00	−1.38361	−0.691804	+	0.722085i	$0.743184\pi$
$114$	0	0
$115$	−1444.00	−1.17090
$116$	0	0
$117$	26.0000	0.0205445
$118$	0	0
$119$	231.000	0.177947
$120$	0	0
$121$	−1327.00	−0.996995
$122$	0	0
$123$	−1200.00	−0.879678
$124$	0	0
$125$	2109.00	1.50908
$126$	0	0
$127$	−1840.00	−1.28562	−0.642809	−	0.766026i	$-0.722231\pi$
$127$	−1840.00	−1.28562	−0.642809	+	0.766026i	$0.722231\pi$
$128$	0	0
$129$	−1585.00	−1.08179
$130$	0	0
$131$	−95.0000	−0.0633602	−0.0316801	−	0.999498i	$-0.510086\pi$
$131$	−95.0000	−0.0633602	−0.0316801	+	0.999498i	$0.510086\pi$
$132$	0	0
$133$	174.000	0.113441
$134$	0	0
$135$	2755.00	1.75639
$136$	0	0
$137$	1308.00	0.815693	0.407847	−	0.913050i	$-0.366280\pi$
$137$	1308.00	0.815693	0.407847	+	0.913050i	$0.366280\pi$
$138$	0	0
$139$	−279.000	−0.170248	−0.0851240	−	0.996370i	$-0.527129\pi$
$139$	−279.000	−0.170248	−0.0851240	+	0.996370i	$0.527129\pi$
$140$	0	0
$141$	−1875.00	−1.11988
$142$	0	0
$143$	−26.0000	−0.0152044
$144$	0	0
$145$	−114.000	−0.0652909
$146$	0	0
$147$	1670.00	0.937002
$148$	0	0
$149$	−730.000	−0.401369	−0.200684	−	0.979656i	$-0.564317\pi$
$149$	−730.000	−0.401369	−0.200684	+	0.979656i	$0.564317\pi$
$150$	0	0
$151$	−2387.00	−1.28643	−0.643216	−	0.765685i	$-0.722400\pi$
$151$	−2387.00	−1.28643	−0.643216	+	0.765685i	$0.722400\pi$
$152$	0	0
$153$	−154.000	−0.0813736
$154$	0	0
$155$	5548.00	2.87501
$156$	0	0
$157$	−674.000	−0.342618	−0.171309	−	0.985217i	$-0.554800\pi$
$157$	−674.000	−0.342618	−0.171309	+	0.985217i	$0.554800\pi$
$158$	0	0
$159$	3460.00	1.72576
$160$	0	0
$161$	−228.000	−0.111608
$162$	0	0
$163$	180.000	0.0864950	0.0432475	−	0.999064i	$-0.486230\pi$
$163$	180.000	0.0864950	0.0432475	+	0.999064i	$0.486230\pi$
$164$	0	0
$165$	−190.000	−0.0896453
$166$	0	0
$167$	888.000	0.411470	0.205735	−	0.978608i	$-0.434041\pi$
$167$	888.000	0.411470	0.205735	+	0.978608i	$0.434041\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−116.000	−0.0518757
$172$	0	0
$173$	2868.00	1.26040	0.630202	−	0.776431i	$-0.282972\pi$
$173$	2868.00	1.26040	0.630202	+	0.776431i	$0.282972\pi$
$174$	0	0
$175$	708.000	0.305827
$176$	0	0
$177$	1070.00	0.454385
$178$	0	0
$179$	−4431.00	−1.85022	−0.925108	−	0.379705i	$-0.876025\pi$
$179$	−4431.00	−1.85022	−0.925108	+	0.379705i	$0.876025\pi$
$180$	0	0
$181$	−672.000	−0.275963	−0.137982	−	0.990435i	$-0.544062\pi$
$181$	−672.000	−0.275963	−0.137982	+	0.990435i	$0.544062\pi$
$182$	0	0
$183$	2440.00	0.985628
$184$	0	0
$185$	3933.00	1.56303
$186$	0	0
$187$	154.000	0.0602224
$188$	0	0
$189$	435.000	0.167416
$190$	0	0
$191$	2094.00	0.793280	0.396640	−	0.917974i	$-0.370176\pi$
$191$	2094.00	0.793280	0.396640	+	0.917974i	$0.370176\pi$
$192$	0	0
$193$	−3712.00	−1.38443	−0.692217	−	0.721690i	$-0.743366\pi$
$193$	−3712.00	−1.38443	−0.692217	+	0.721690i	$0.743366\pi$
$194$	0	0
$195$	1235.00	0.453539
$196$	0	0
$197$	3117.00	1.12729	0.563647	−	0.826016i	$-0.309398\pi$
$197$	3117.00	1.12729	0.563647	+	0.826016i	$0.309398\pi$
$198$	0	0
$199$	−4702.00	−1.67495	−0.837477	−	0.546472i	$-0.815970\pi$
$199$	−4702.00	−1.67495	−0.837477	+	0.546472i	$0.815970\pi$
$200$	0	0
$201$	3910.00	1.37209
$202$	0	0
$203$	−18.0000	−0.00622341
$204$	0	0
$205$	4560.00	1.55358
$206$	0	0
$207$	152.000	0.0510373
$208$	0	0
$209$	116.000	0.0383918
$210$	0	0
$211$	−2743.00	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−2743.00	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	−5285.00	−1.70010
$214$	0	0
$215$	6023.00	1.91054
$216$	0	0
$217$	876.000	0.274040
$218$	0	0
$219$	−5870.00	−1.81122
$220$	0	0
$221$	−1001.00	−0.304681
$222$	0	0
$223$	−4303.00	−1.29215	−0.646077	−	0.763273i	$-0.723591\pi$
$223$	−4303.00	−1.29215	−0.646077	+	0.763273i	$0.723591\pi$
$224$	0	0
$225$	−472.000	−0.139852
$226$	0	0
$227$	−2136.00	−0.624543	−0.312272	−	0.949993i	$-0.601090\pi$
$227$	−2136.00	−0.624543	−0.312272	+	0.949993i	$0.601090\pi$
$228$	0	0
$229$	205.000	0.0591563	0.0295781	−	0.999562i	$-0.490584\pi$
$229$	205.000	0.0591563	0.0295781	+	0.999562i	$0.490584\pi$
$230$	0	0
$231$	−30.0000	−0.00854483
$232$	0	0
$233$	37.0000	0.0104032	0.00520161	−	0.999986i	$-0.498344\pi$
$233$	37.0000	0.0104032	0.00520161	+	0.999986i	$0.498344\pi$
$234$	0	0
$235$	7125.00	1.97780
$236$	0	0
$237$	4460.00	1.22240
$238$	0	0
$239$	4099.00	1.10938	0.554691	−	0.832056i	$-0.312836\pi$
$239$	4099.00	1.10938	0.554691	+	0.832056i	$0.312836\pi$
$240$	0	0
$241$	6258.00	1.67267	0.836334	−	0.548220i	$-0.184694\pi$
$241$	6258.00	1.67267	0.836334	+	0.548220i	$0.184694\pi$
$242$	0	0
$243$	−560.000	−0.147835
$244$	0	0
$245$	−6346.00	−1.65482
$246$	0	0
$247$	−754.000	−0.194234
$248$	0	0
$249$	3520.00	0.895867
$250$	0	0
$251$	−3992.00	−1.00388	−0.501938	−	0.864904i	$-0.667379\pi$
$251$	−3992.00	−1.00388	−0.501938	+	0.864904i	$0.667379\pi$
$252$	0	0
$253$	−152.000	−0.0377714
$254$	0	0
$255$	−7315.00	−1.79640
$256$	0	0
$257$	4097.00	0.994412	0.497206	−	0.867633i	$-0.334359\pi$
$257$	4097.00	0.994412	0.497206	+	0.867633i	$0.334359\pi$
$258$	0	0
$259$	621.000	0.148985
$260$	0	0
$261$	12.0000	0.00284591
$262$	0	0
$263$	6308.00	1.47897	0.739483	−	0.673175i	$-0.235070\pi$
$263$	6308.00	1.47897	0.739483	+	0.673175i	$0.235070\pi$
$264$	0	0
$265$	−13148.0	−3.04783
$266$	0	0
$267$	−30.0000	−0.00687629
$268$	0	0
$269$	−4792.00	−1.08615	−0.543073	−	0.839685i	$-0.682739\pi$
$269$	−4792.00	−1.08615	−0.543073	+	0.839685i	$0.682739\pi$
$270$	0	0
$271$	−2585.00	−0.579438	−0.289719	−	0.957112i	$-0.593562\pi$
$271$	−2585.00	−0.579438	−0.289719	+	0.957112i	$0.593562\pi$
$272$	0	0
$273$	195.000	0.0432305
$274$	0	0
$275$	472.000	0.103501
$276$	0	0
$277$	−5156.00	−1.11839	−0.559195	−	0.829036i	$-0.688890\pi$
$277$	−5156.00	−1.11839	−0.559195	+	0.829036i	$0.688890\pi$
$278$	0	0
$279$	−584.000	−0.125316
$280$	0	0
$281$	−2650.00	−0.562583	−0.281291	−	0.959622i	$-0.590763\pi$
$281$	−2650.00	−0.562583	−0.281291	+	0.959622i	$0.590763\pi$
$282$	0	0
$283$	2596.00	0.545287	0.272643	−	0.962115i	$-0.412102\pi$
$283$	2596.00	0.545287	0.272643	+	0.962115i	$0.412102\pi$
$284$	0	0
$285$	−5510.00	−1.14521
$286$	0	0
$287$	720.000	0.148085
$288$	0	0
$289$	1016.00	0.206798
$290$	0	0
$291$	−4150.00	−0.836004
$292$	0	0
$293$	435.000	0.0867337	0.0433668	−	0.999059i	$-0.486192\pi$
$293$	435.000	0.0867337	0.0433668	+	0.999059i	$0.486192\pi$
$294$	0	0
$295$	−4066.00	−0.802480
$296$	0	0
$297$	290.000	0.0566583
$298$	0	0
$299$	988.000	0.191095
$300$	0	0
$301$	951.000	0.182109
$302$	0	0
$303$	−1380.00	−0.261647
$304$	0	0
$305$	−9272.00	−1.74070
$306$	0	0
$307$	3090.00	0.574448	0.287224	−	0.957863i	$-0.407268\pi$
$307$	3090.00	0.574448	0.287224	+	0.957863i	$0.407268\pi$
$308$	0	0
$309$	−5800.00	−1.06780
$310$	0	0
$311$	−2682.00	−0.489011	−0.244505	−	0.969648i	$-0.578626\pi$
$311$	−2682.00	−0.489011	−0.244505	+	0.969648i	$0.578626\pi$
$312$	0	0
$313$	−801.000	−0.144649	−0.0723246	−	0.997381i	$-0.523042\pi$
$313$	−801.000	−0.144649	−0.0723246	+	0.997381i	$0.523042\pi$
$314$	0	0
$315$	−114.000	−0.0203910
$316$	0	0
$317$	9750.00	1.72749	0.863745	−	0.503929i	$-0.168113\pi$
$317$	9750.00	1.72749	0.863745	+	0.503929i	$0.168113\pi$
$318$	0	0
$319$	−12.0000	−0.00210618
$320$	0	0
$321$	1620.00	0.281681
$322$	0	0
$323$	4466.00	0.769334
$324$	0	0
$325$	−3068.00	−0.523637
$326$	0	0
$327$	−995.000	−0.168268
$328$	0	0
$329$	1125.00	0.188521
$330$	0	0
$331$	−6620.00	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−6620.00	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	−414.000	−0.0681293
$334$	0	0
$335$	−14858.0	−2.42322
$336$	0	0
$337$	399.000	0.0644953	0.0322476	−	0.999480i	$-0.489733\pi$
$337$	399.000	0.0644953	0.0322476	+	0.999480i	$0.489733\pi$
$338$	0	0
$339$	8310.00	1.33138
$340$	0	0
$341$	584.000	0.0927430
$342$	0	0
$343$	−2031.00	−0.319719
$344$	0	0
$345$	7220.00	1.12670
$346$	0	0
$347$	12201.0	1.88756	0.943781	−	0.330571i	$-0.107241\pi$
$347$	12201.0	1.88756	0.943781	+	0.330571i	$0.107241\pi$
$348$	0	0
$349$	−189.000	−0.0289884	−0.0144942	−	0.999895i	$-0.504614\pi$
$349$	−189.000	−0.0289884	−0.0144942	+	0.999895i	$0.504614\pi$
$350$	0	0
$351$	−1885.00	−0.286649
$352$	0	0
$353$	−10444.0	−1.57473	−0.787363	−	0.616490i	$-0.788554\pi$
$353$	−10444.0	−1.57473	−0.787363	+	0.616490i	$0.788554\pi$
$354$	0	0
$355$	20083.0	3.00252
$356$	0	0
$357$	−1155.00	−0.171230
$358$	0	0
$359$	−9792.00	−1.43956	−0.719780	−	0.694202i	$-0.755757\pi$
$359$	−9792.00	−1.43956	−0.719780	+	0.694202i	$0.755757\pi$
$360$	0	0
$361$	−3495.00	−0.509549
$362$	0	0
$363$	6635.00	0.959359
$364$	0	0
$365$	22306.0	3.19877
$366$	0	0
$367$	11906.0	1.69343	0.846714	−	0.532048i	$-0.178577\pi$
$367$	11906.0	1.69343	0.846714	+	0.532048i	$0.178577\pi$
$368$	0	0
$369$	−480.000	−0.0677176
$370$	0	0
$371$	−2076.00	−0.290514
$372$	0	0
$373$	−6628.00	−0.920067	−0.460033	−	0.887902i	$-0.652163\pi$
$373$	−6628.00	−0.920067	−0.460033	+	0.887902i	$0.652163\pi$
$374$	0	0
$375$	−10545.0	−1.45211
$376$	0	0
$377$	78.0000	0.0106557
$378$	0	0
$379$	3760.00	0.509600	0.254800	−	0.966994i	$-0.417990\pi$
$379$	3760.00	0.509600	0.254800	+	0.966994i	$0.417990\pi$
$380$	0	0
$381$	9200.00	1.23709
$382$	0	0
$383$	10065.0	1.34281	0.671407	−	0.741089i	$-0.265690\pi$
$383$	10065.0	1.34281	0.671407	+	0.741089i	$0.265690\pi$
$384$	0	0
$385$	114.000	0.0150909
$386$	0	0
$387$	−634.000	−0.0832766
$388$	0	0
$389$	3994.00	0.520575	0.260288	−	0.965531i	$-0.416183\pi$
$389$	3994.00	0.520575	0.260288	+	0.965531i	$0.416183\pi$
$390$	0	0
$391$	−5852.00	−0.756901
$392$	0	0
$393$	475.000	0.0609684
$394$	0	0
$395$	−16948.0	−2.15885
$396$	0	0
$397$	8930.00	1.12893	0.564463	−	0.825458i	$-0.309083\pi$
$397$	8930.00	1.12893	0.564463	+	0.825458i	$0.309083\pi$
$398$	0	0
$399$	−870.000	−0.109159
$400$	0	0
$401$	5592.00	0.696387	0.348193	−	0.937423i	$-0.386795\pi$
$401$	5592.00	0.696387	0.348193	+	0.937423i	$0.386795\pi$
$402$	0	0
$403$	−3796.00	−0.469211
$404$	0	0
$405$	−12749.0	−1.56420
$406$	0	0
$407$	414.000	0.0504207
$408$	0	0
$409$	6276.00	0.758749	0.379374	−	0.925243i	$-0.376139\pi$
$409$	6276.00	0.758749	0.379374	+	0.925243i	$0.376139\pi$
$410$	0	0
$411$	−6540.00	−0.784901
$412$	0	0
$413$	−642.000	−0.0764909
$414$	0	0
$415$	−13376.0	−1.58217
$416$	0	0
$417$	1395.00	0.163821
$418$	0	0
$419$	−429.000	−0.0500191	−0.0250096	−	0.999687i	$-0.507962\pi$
$419$	−429.000	−0.0500191	−0.0250096	+	0.999687i	$0.507962\pi$
$420$	0	0
$421$	−97.0000	−0.0112292	−0.00561460	−	0.999984i	$-0.501787\pi$
$421$	−97.0000	−0.0112292	−0.00561460	+	0.999984i	$0.501787\pi$
$422$	0	0
$423$	−750.000	−0.0862086
$424$	0	0
$425$	18172.0	2.07405
$426$	0	0
$427$	−1464.00	−0.165920
$428$	0	0
$429$	130.000	0.0146304
$430$	0	0
$431$	−12293.0	−1.37386	−0.686929	−	0.726724i	$-0.741042\pi$
$431$	−12293.0	−1.37386	−0.686929	+	0.726724i	$0.741042\pi$
$432$	0	0
$433$	−13553.0	−1.50419	−0.752097	−	0.659053i	$-0.770957\pi$
$433$	−13553.0	−1.50419	−0.752097	+	0.659053i	$0.770957\pi$
$434$	0	0
$435$	570.000	0.0628262
$436$	0	0
$437$	−4408.00	−0.482525
$438$	0	0
$439$	7182.00	0.780816	0.390408	−	0.920642i	$-0.372334\pi$
$439$	7182.00	0.780816	0.390408	+	0.920642i	$0.372334\pi$
$440$	0	0
$441$	668.000	0.0721304
$442$	0	0
$443$	8815.00	0.945402	0.472701	−	0.881223i	$-0.343279\pi$
$443$	8815.00	0.945402	0.472701	+	0.881223i	$0.343279\pi$
$444$	0	0
$445$	114.000	0.0121441
$446$	0	0
$447$	3650.00	0.386217
$448$	0	0
$449$	6550.00	0.688449	0.344225	−	0.938887i	$-0.388142\pi$
$449$	6550.00	0.688449	0.344225	+	0.938887i	$0.388142\pi$
$450$	0	0
$451$	480.000	0.0501160
$452$	0	0
$453$	11935.0	1.23787
$454$	0	0
$455$	−741.000	−0.0763486
$456$	0	0
$457$	−934.000	−0.0956032	−0.0478016	−	0.998857i	$-0.515222\pi$
$457$	−934.000	−0.0956032	−0.0478016	+	0.998857i	$0.515222\pi$
$458$	0	0
$459$	11165.0	1.13538
$460$	0	0
$461$	−3809.00	−0.384822	−0.192411	−	0.981314i	$-0.561631\pi$
$461$	−3809.00	−0.384822	−0.192411	+	0.981314i	$0.561631\pi$
$462$	0	0
$463$	17368.0	1.74332	0.871662	−	0.490107i	$-0.163042\pi$
$463$	17368.0	1.74332	0.871662	+	0.490107i	$0.163042\pi$
$464$	0	0
$465$	−27740.0	−2.76648
$466$	0	0
$467$	−348.000	−0.0344829	−0.0172415	−	0.999851i	$-0.505488\pi$
$467$	−348.000	−0.0344829	−0.0172415	+	0.999851i	$0.505488\pi$
$468$	0	0
$469$	−2346.00	−0.230977
$470$	0	0
$471$	3370.00	0.329685
$472$	0	0
$473$	634.000	0.0616308
$474$	0	0
$475$	13688.0	1.32221
$476$	0	0
$477$	1384.00	0.132849
$478$	0	0
$479$	19111.0	1.82297	0.911486	−	0.411330i	$-0.134936\pi$
$479$	19111.0	1.82297	0.911486	+	0.411330i	$0.134936\pi$
$480$	0	0
$481$	−2691.00	−0.255092
$482$	0	0
$483$	1140.00	0.107395
$484$	0	0
$485$	15770.0	1.47645
$486$	0	0
$487$	3608.00	0.335717	0.167858	−	0.985811i	$-0.446315\pi$
$487$	3608.00	0.335717	0.167858	+	0.985811i	$0.446315\pi$
$488$	0	0
$489$	−900.000	−0.0832299
$490$	0	0
$491$	−6291.00	−0.578226	−0.289113	−	0.957295i	$-0.593360\pi$
$491$	−6291.00	−0.578226	−0.289113	+	0.957295i	$0.593360\pi$
$492$	0	0
$493$	−462.000	−0.0422057
$494$	0	0
$495$	−76.0000	−0.00690090
$496$	0	0
$497$	3171.00	0.286195
$498$	0	0
$499$	−14736.0	−1.32199	−0.660996	−	0.750389i	$-0.729866\pi$
$499$	−14736.0	−1.32199	−0.660996	+	0.750389i	$0.729866\pi$
$500$	0	0
$501$	−4440.00	−0.395937
$502$	0	0
$503$	2982.00	0.264336	0.132168	−	0.991227i	$-0.457806\pi$
$503$	2982.00	0.264336	0.132168	+	0.991227i	$0.457806\pi$
$504$	0	0
$505$	5244.00	0.462089
$506$	0	0
$507$	−845.000	−0.0740193
$508$	0	0
$509$	−3078.00	−0.268035	−0.134018	−	0.990979i	$-0.542788\pi$
$509$	−3078.00	−0.268035	−0.134018	+	0.990979i	$0.542788\pi$
$510$	0	0
$511$	3522.00	0.304900
$512$	0	0
$513$	8410.00	0.723802
$514$	0	0
$515$	22040.0	1.88582
$516$	0	0
$517$	750.000	0.0638007
$518$	0	0
$519$	−14340.0	−1.21282
$520$	0	0
$521$	3815.00	0.320803	0.160401	−	0.987052i	$-0.448721\pi$
$521$	3815.00	0.320803	0.160401	+	0.987052i	$0.448721\pi$
$522$	0	0
$523$	5180.00	0.433089	0.216545	−	0.976273i	$-0.430521\pi$
$523$	5180.00	0.433089	0.216545	+	0.976273i	$0.430521\pi$
$524$	0	0
$525$	−3540.00	−0.294282
$526$	0	0
$527$	22484.0	1.85848
$528$	0	0
$529$	−6391.00	−0.525273
$530$	0	0
$531$	428.000	0.0349786
$532$	0	0
$533$	−3120.00	−0.253550
$534$	0	0
$535$	−6156.00	−0.497471
$536$	0	0
$537$	22155.0	1.78037
$538$	0	0
$539$	−668.000	−0.0533818
$540$	0	0
$541$	717.000	0.0569801	0.0284901	−	0.999594i	$-0.490930\pi$
$541$	717.000	0.0569801	0.0284901	+	0.999594i	$0.490930\pi$
$542$	0	0
$543$	3360.00	0.265546
$544$	0	0
$545$	3781.00	0.297175
$546$	0	0
$547$	−13949.0	−1.09034	−0.545170	−	0.838325i	$-0.683535\pi$
$547$	−13949.0	−1.09034	−0.545170	+	0.838325i	$0.683535\pi$
$548$	0	0
$549$	976.000	0.0758737
$550$	0	0
$551$	−348.000	−0.0269062
$552$	0	0
$553$	−2676.00	−0.205778
$554$	0	0
$555$	−19665.0	−1.50402
$556$	0	0
$557$	−22499.0	−1.71151	−0.855757	−	0.517378i	$-0.826908\pi$
$557$	−22499.0	−1.71151	−0.855757	+	0.517378i	$0.826908\pi$
$558$	0	0
$559$	−4121.00	−0.311806
$560$	0	0
$561$	−770.000	−0.0579491
$562$	0	0
$563$	−9715.00	−0.727244	−0.363622	−	0.931547i	$-0.618460\pi$
$563$	−9715.00	−0.727244	−0.363622	+	0.931547i	$0.618460\pi$
$564$	0	0
$565$	−31578.0	−2.35132
$566$	0	0
$567$	−2013.00	−0.149097
$568$	0	0
$569$	−13481.0	−0.993239	−0.496619	−	0.867968i	$-0.665426\pi$
$569$	−13481.0	−0.993239	−0.496619	+	0.867968i	$0.665426\pi$
$570$	0	0
$571$	−17361.0	−1.27239	−0.636195	−	0.771528i	$-0.719493\pi$
$571$	−17361.0	−1.27239	−0.636195	+	0.771528i	$0.719493\pi$
$572$	0	0
$573$	−10470.0	−0.763334
$574$	0	0
$575$	−17936.0	−1.30084
$576$	0	0
$577$	−15790.0	−1.13925	−0.569624	−	0.821905i	$-0.692911\pi$
$577$	−15790.0	−1.13925	−0.569624	+	0.821905i	$0.692911\pi$
$578$	0	0
$579$	18560.0	1.33217
$580$	0	0
$581$	−2112.00	−0.150810
$582$	0	0
$583$	−1384.00	−0.0983181
$584$	0	0
$585$	494.000	0.0349135
$586$	0	0
$587$	−10084.0	−0.709048	−0.354524	−	0.935047i	$-0.615357\pi$
$587$	−10084.0	−0.709048	−0.354524	+	0.935047i	$0.615357\pi$
$588$	0	0
$589$	16936.0	1.18478
$590$	0	0
$591$	−15585.0	−1.08474
$592$	0	0
$593$	−19622.0	−1.35882	−0.679409	−	0.733760i	$-0.737764\pi$
$593$	−19622.0	−1.35882	−0.679409	+	0.733760i	$0.737764\pi$
$594$	0	0
$595$	4389.00	0.302406
$596$	0	0
$597$	23510.0	1.61173
$598$	0	0
$599$	10746.0	0.733004	0.366502	−	0.930417i	$-0.380555\pi$
$599$	10746.0	0.733004	0.366502	+	0.930417i	$0.380555\pi$
$600$	0	0
$601$	16453.0	1.11669	0.558346	−	0.829608i	$-0.311436\pi$
$601$	16453.0	1.11669	0.558346	+	0.829608i	$0.311436\pi$
$602$	0	0
$603$	1564.00	0.105624
$604$	0	0
$605$	−25213.0	−1.69430
$606$	0	0
$607$	−14078.0	−0.941365	−0.470682	−	0.882303i	$-0.655992\pi$
$607$	−14078.0	−0.941365	−0.470682	+	0.882303i	$0.655992\pi$
$608$	0	0
$609$	90.0000	0.00598848
$610$	0	0
$611$	−4875.00	−0.322785
$612$	0	0
$613$	194.000	0.0127824	0.00639118	−	0.999980i	$-0.497966\pi$
$613$	194.000	0.0127824	0.00639118	+	0.999980i	$0.497966\pi$
$614$	0	0
$615$	−22800.0	−1.49493
$616$	0	0
$617$	21792.0	1.42190	0.710950	−	0.703242i	$-0.248265\pi$
$617$	21792.0	1.42190	0.710950	+	0.703242i	$0.248265\pi$
$618$	0	0
$619$	5276.00	0.342585	0.171293	−	0.985220i	$-0.445206\pi$
$619$	5276.00	0.342585	0.171293	+	0.985220i	$0.445206\pi$
$620$	0	0
$621$	−11020.0	−0.712105
$622$	0	0
$623$	18.0000	0.00115755
$624$	0	0
$625$	10571.0	0.676544
$626$	0	0
$627$	−580.000	−0.0369425
$628$	0	0
$629$	15939.0	1.01038
$630$	0	0
$631$	3305.00	0.208510	0.104255	−	0.994551i	$-0.466754\pi$
$631$	3305.00	0.208510	0.104255	+	0.994551i	$0.466754\pi$
$632$	0	0
$633$	13715.0	0.861173
$634$	0	0
$635$	−34960.0	−2.18480
$636$	0	0
$637$	4342.00	0.270073
$638$	0	0
$639$	−2114.00	−0.130874
$640$	0	0
$641$	−21122.0	−1.30151	−0.650756	−	0.759287i	$-0.725548\pi$
$641$	−21122.0	−1.30151	−0.650756	+	0.759287i	$0.725548\pi$
$642$	0	0
$643$	4098.00	0.251336	0.125668	−	0.992072i	$-0.459893\pi$
$643$	4098.00	0.251336	0.125668	+	0.992072i	$0.459893\pi$
$644$	0	0
$645$	−30115.0	−1.83841
$646$	0	0
$647$	15454.0	0.939041	0.469520	−	0.882922i	$-0.344427\pi$
$647$	15454.0	0.939041	0.469520	+	0.882922i	$0.344427\pi$
$648$	0	0
$649$	−428.000	−0.0258867
$650$	0	0
$651$	−4380.00	−0.263695
$652$	0	0
$653$	26136.0	1.56628	0.783140	−	0.621846i	$-0.213617\pi$
$653$	26136.0	1.56628	0.783140	+	0.621846i	$0.213617\pi$
$654$	0	0
$655$	−1805.00	−0.107675
$656$	0	0
$657$	−2348.00	−0.139428
$658$	0	0
$659$	−7884.00	−0.466035	−0.233017	−	0.972473i	$-0.574860\pi$
$659$	−7884.00	−0.466035	−0.233017	+	0.972473i	$0.574860\pi$
$660$	0	0
$661$	−20818.0	−1.22500	−0.612501	−	0.790470i	$-0.709836\pi$
$661$	−20818.0	−1.22500	−0.612501	+	0.790470i	$0.709836\pi$
$662$	0	0
$663$	5005.00	0.293180
$664$	0	0
$665$	3306.00	0.192784
$666$	0	0
$667$	456.000	0.0264714
$668$	0	0
$669$	21515.0	1.24338
$670$	0	0
$671$	−976.000	−0.0561521
$672$	0	0
$673$	−7595.00	−0.435016	−0.217508	−	0.976059i	$-0.569793\pi$
$673$	−7595.00	−0.435016	−0.217508	+	0.976059i	$0.569793\pi$
$674$	0	0
$675$	34220.0	1.95130
$676$	0	0
$677$	8428.00	0.478455	0.239228	−	0.970963i	$-0.423106\pi$
$677$	8428.00	0.478455	0.239228	+	0.970963i	$0.423106\pi$
$678$	0	0
$679$	2490.00	0.140733
$680$	0	0
$681$	10680.0	0.600967
$682$	0	0
$683$	−12676.0	−0.710152	−0.355076	−	0.934837i	$-0.615545\pi$
$683$	−12676.0	−0.710152	−0.355076	+	0.934837i	$0.615545\pi$
$684$	0	0
$685$	24852.0	1.38620
$686$	0	0
$687$	−1025.00	−0.0569231
$688$	0	0
$689$	8996.00	0.497417
$690$	0	0
$691$	9736.00	0.535998	0.267999	−	0.963419i	$-0.413638\pi$
$691$	9736.00	0.535998	0.267999	+	0.963419i	$0.413638\pi$
$692$	0	0
$693$	−12.0000	−0.000657781 0
$694$	0	0
$695$	−5301.00	−0.289321
$696$	0	0
$697$	18480.0	1.00428
$698$	0	0
$699$	−185.000	−0.0100105
$700$	0	0
$701$	−11052.0	−0.595475	−0.297738	−	0.954648i	$-0.596232\pi$
$701$	−11052.0	−0.595475	−0.297738	+	0.954648i	$0.596232\pi$
$702$	0	0
$703$	12006.0	0.644118
$704$	0	0
$705$	−35625.0	−1.90314
$706$	0	0
$707$	828.000	0.0440455
$708$	0	0
$709$	13214.0	0.699947	0.349973	−	0.936760i	$-0.386191\pi$
$709$	13214.0	0.699947	0.349973	+	0.936760i	$0.386191\pi$
$710$	0	0
$711$	1784.00	0.0941002
$712$	0	0
$713$	−22192.0	−1.16563
$714$	0	0
$715$	−494.000	−0.0258385
$716$	0	0
$717$	−20495.0	−1.06750
$718$	0	0
$719$	5150.00	0.267125	0.133562	−	0.991040i	$-0.457358\pi$
$719$	5150.00	0.267125	0.133562	+	0.991040i	$0.457358\pi$
$720$	0	0
$721$	3480.00	0.179753
$722$	0	0
$723$	−31290.0	−1.60953
$724$	0	0
$725$	−1416.00	−0.0725364
$726$	0	0
$727$	−5178.00	−0.264156	−0.132078	−	0.991239i	$-0.542165\pi$
$727$	−5178.00	−0.264156	−0.132078	+	0.991239i	$0.542165\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	24409.0	1.23502
$732$	0	0
$733$	26777.0	1.34929	0.674646	−	0.738141i	$-0.264296\pi$
$733$	26777.0	1.34929	0.674646	+	0.738141i	$0.264296\pi$
$734$	0	0
$735$	31730.0	1.59235
$736$	0	0
$737$	−1564.00	−0.0781692
$738$	0	0
$739$	2500.00	0.124444	0.0622219	−	0.998062i	$-0.480181\pi$
$739$	2500.00	0.124444	0.0622219	+	0.998062i	$0.480181\pi$
$740$	0	0
$741$	3770.00	0.186902
$742$	0	0
$743$	20267.0	1.00071	0.500353	−	0.865822i	$-0.333204\pi$
$743$	20267.0	1.00071	0.500353	+	0.865822i	$0.333204\pi$
$744$	0	0
$745$	−13870.0	−0.682091
$746$	0	0
$747$	1408.00	0.0689639
$748$	0	0
$749$	−972.000	−0.0474180
$750$	0	0
$751$	14280.0	0.693854	0.346927	−	0.937892i	$-0.387225\pi$
$751$	14280.0	0.693854	0.346927	+	0.937892i	$0.387225\pi$
$752$	0	0
$753$	19960.0	0.965980
$754$	0	0
$755$	−45353.0	−2.18618
$756$	0	0
$757$	3332.00	0.159978	0.0799892	−	0.996796i	$-0.474511\pi$
$757$	3332.00	0.159978	0.0799892	+	0.996796i	$0.474511\pi$
$758$	0	0
$759$	760.000	0.0363455
$760$	0	0
$761$	−33178.0	−1.58042	−0.790211	−	0.612835i	$-0.790029\pi$
$761$	−33178.0	−1.58042	−0.790211	+	0.612835i	$0.790029\pi$
$762$	0	0
$763$	597.000	0.0283261
$764$	0	0
$765$	−2926.00	−0.138287
$766$	0	0
$767$	2782.00	0.130968
$768$	0	0
$769$	10784.0	0.505697	0.252848	−	0.967506i	$-0.418633\pi$
$769$	10784.0	0.505697	0.252848	+	0.967506i	$0.418633\pi$
$770$	0	0
$771$	−20485.0	−0.956873
$772$	0	0
$773$	−23169.0	−1.07805	−0.539024	−	0.842290i	$-0.681207\pi$
$773$	−23169.0	−1.07805	−0.539024	+	0.842290i	$0.681207\pi$
$774$	0	0
$775$	68912.0	3.19405
$776$	0	0
$777$	−3105.00	−0.143361
$778$	0	0
$779$	13920.0	0.640226
$780$	0	0
$781$	2114.00	0.0968564
$782$	0	0
$783$	−870.000	−0.0397079
$784$	0	0
$785$	−12806.0	−0.582249
$786$	0	0
$787$	−17776.0	−0.805141	−0.402571	−	0.915389i	$-0.631883\pi$
$787$	−17776.0	−0.805141	−0.402571	+	0.915389i	$0.631883\pi$
$788$	0	0
$789$	−31540.0	−1.42314
$790$	0	0
$791$	−4986.00	−0.224124
$792$	0	0
$793$	6344.00	0.284088
$794$	0	0
$795$	65740.0	2.93278
$796$	0	0
$797$	−17858.0	−0.793680	−0.396840	−	0.917888i	$-0.629893\pi$
$797$	−17858.0	−0.793680	−0.396840	+	0.917888i	$0.629893\pi$
$798$	0	0
$799$	28875.0	1.27850
$800$	0	0
$801$	−12.0000	−0.000529337 0
$802$	0	0
$803$	2348.00	0.103187
$804$	0	0
$805$	−4332.00	−0.189668
$806$	0	0
$807$	23960.0	1.04514
$808$	0	0
$809$	21623.0	0.939709	0.469854	−	0.882744i	$-0.344306\pi$
$809$	21623.0	0.939709	0.469854	+	0.882744i	$0.344306\pi$
$810$	0	0
$811$	−39652.0	−1.71686	−0.858428	−	0.512934i	$-0.828558\pi$
$811$	−39652.0	−1.71686	−0.858428	+	0.512934i	$0.828558\pi$
$812$	0	0
$813$	12925.0	0.557564
$814$	0	0
$815$	3420.00	0.146991
$816$	0	0
$817$	18386.0	0.787325
$818$	0	0
$819$	78.0000	0.00332789
$820$	0	0
$821$	7707.00	0.327620	0.163810	−	0.986492i	$-0.447622\pi$
$821$	7707.00	0.327620	0.163810	+	0.986492i	$0.447622\pi$
$822$	0	0
$823$	−1358.00	−0.0575175	−0.0287588	−	0.999586i	$-0.509155\pi$
$823$	−1358.00	−0.0575175	−0.0287588	+	0.999586i	$0.509155\pi$
$824$	0	0
$825$	−2360.00	−0.0995935
$826$	0	0
$827$	−29774.0	−1.25193	−0.625963	−	0.779852i	$-0.715294\pi$
$827$	−29774.0	−1.25193	−0.625963	+	0.779852i	$0.715294\pi$
$828$	0	0
$829$	35170.0	1.47347	0.736734	−	0.676183i	$-0.236367\pi$
$829$	35170.0	1.47347	0.736734	+	0.676183i	$0.236367\pi$
$830$	0	0
$831$	25780.0	1.07617
$832$	0	0
$833$	−25718.0	−1.06972
$834$	0	0
$835$	16872.0	0.699257
$836$	0	0
$837$	42340.0	1.74849
$838$	0	0
$839$	−12480.0	−0.513537	−0.256768	−	0.966473i	$-0.582658\pi$
$839$	−12480.0	−0.513537	−0.256768	+	0.966473i	$0.582658\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	13250.0	0.541345
$844$	0	0
$845$	3211.00	0.130724
$846$	0	0
$847$	−3981.00	−0.161498
$848$	0	0
$849$	−12980.0	−0.524703
$850$	0	0
$851$	−15732.0	−0.633709
$852$	0	0
$853$	37133.0	1.49052	0.745258	−	0.666777i	$-0.232326\pi$
$853$	37133.0	1.49052	0.745258	+	0.666777i	$0.232326\pi$
$854$	0	0
$855$	−2204.00	−0.0881581
$856$	0	0
$857$	−3618.00	−0.144211	−0.0721053	−	0.997397i	$-0.522972\pi$
$857$	−3618.00	−0.144211	−0.0721053	+	0.997397i	$0.522972\pi$
$858$	0	0
$859$	−46140.0	−1.83269	−0.916343	−	0.400395i	$-0.868873\pi$
$859$	−46140.0	−1.83269	−0.916343	+	0.400395i	$0.868873\pi$
$860$	0	0
$861$	−3600.00	−0.142494
$862$	0	0
$863$	−1753.00	−0.0691458	−0.0345729	−	0.999402i	$-0.511007\pi$
$863$	−1753.00	−0.0691458	−0.0345729	+	0.999402i	$0.511007\pi$
$864$	0	0
$865$	54492.0	2.14195
$866$	0	0
$867$	−5080.00	−0.198992
$868$	0	0
$869$	−1784.00	−0.0696410
$870$	0	0
$871$	10166.0	0.395478
$872$	0	0
$873$	−1660.00	−0.0643557
$874$	0	0
$875$	6327.00	0.244448
$876$	0	0
$877$	19485.0	0.750241	0.375121	−	0.926976i	$-0.377601\pi$
$877$	19485.0	0.750241	0.375121	+	0.926976i	$0.377601\pi$
$878$	0	0
$879$	−2175.00	−0.0834595
$880$	0	0
$881$	−3067.00	−0.117287	−0.0586435	−	0.998279i	$-0.518678\pi$
$881$	−3067.00	−0.117287	−0.0586435	+	0.998279i	$0.518678\pi$
$882$	0	0
$883$	−38321.0	−1.46048	−0.730240	−	0.683190i	$-0.760592\pi$
$883$	−38321.0	−1.46048	−0.730240	+	0.683190i	$0.760592\pi$
$884$	0	0
$885$	20330.0	0.772187
$886$	0	0
$887$	−20936.0	−0.792516	−0.396258	−	0.918139i	$-0.629691\pi$
$887$	−20936.0	−0.792516	−0.396258	+	0.918139i	$0.629691\pi$
$888$	0	0
$889$	−5520.00	−0.208251
$890$	0	0
$891$	−1342.00	−0.0504587
$892$	0	0
$893$	21750.0	0.815046
$894$	0	0
$895$	−84189.0	−3.14428
$896$	0	0
$897$	−4940.00	−0.183882
$898$	0	0
$899$	−1752.00	−0.0649972
$900$	0	0
$901$	−53284.0	−1.97020
$902$	0	0
$903$	−4755.00	−0.175234
$904$	0	0
$905$	−12768.0	−0.468975
$906$	0	0
$907$	32321.0	1.18324	0.591621	−	0.806216i	$-0.298488\pi$
$907$	32321.0	1.18324	0.591621	+	0.806216i	$0.298488\pi$
$908$	0	0
$909$	−552.000	−0.0201416
$910$	0	0
$911$	22766.0	0.827960	0.413980	−	0.910286i	$-0.364138\pi$
$911$	22766.0	0.827960	0.413980	+	0.910286i	$0.364138\pi$
$912$	0	0
$913$	−1408.00	−0.0510383
$914$	0	0
$915$	46360.0	1.67499
$916$	0	0
$917$	−285.000	−0.0102634
$918$	0	0
$919$	−3112.00	−0.111703	−0.0558517	−	0.998439i	$-0.517787\pi$
$919$	−3112.00	−0.111703	−0.0558517	+	0.998439i	$0.517787\pi$
$920$	0	0
$921$	−15450.0	−0.552763
$922$	0	0
$923$	−13741.0	−0.490022
$924$	0	0
$925$	48852.0	1.73648
$926$	0	0
$927$	−2320.00	−0.0821994
$928$	0	0
$929$	−8980.00	−0.317141	−0.158571	−	0.987348i	$-0.550689\pi$
$929$	−8980.00	−0.317141	−0.158571	+	0.987348i	$0.550689\pi$
$930$	0	0
$931$	−19372.0	−0.681946
$932$	0	0
$933$	13410.0	0.470551
$934$	0	0
$935$	2926.00	0.102343
$936$	0	0
$937$	−33306.0	−1.16122	−0.580608	−	0.814183i	$-0.697185\pi$
$937$	−33306.0	−1.16122	−0.580608	+	0.814183i	$0.697185\pi$
$938$	0	0
$939$	4005.00	0.139189
$940$	0	0
$941$	−55299.0	−1.91572	−0.957862	−	0.287230i	$-0.907266\pi$
$941$	−55299.0	−1.91572	−0.957862	+	0.287230i	$0.907266\pi$
$942$	0	0
$943$	−18240.0	−0.629879
$944$	0	0
$945$	8265.00	0.284508
$946$	0	0
$947$	28578.0	0.980634	0.490317	−	0.871544i	$-0.336881\pi$
$947$	28578.0	0.980634	0.490317	+	0.871544i	$0.336881\pi$
$948$	0	0
$949$	−15262.0	−0.522050
$950$	0	0
$951$	−48750.0	−1.66228
$952$	0	0
$953$	10623.0	0.361084	0.180542	−	0.983567i	$-0.442215\pi$
$953$	10623.0	0.361084	0.180542	+	0.983567i	$0.442215\pi$
$954$	0	0
$955$	39786.0	1.34811
$956$	0	0
$957$	60.0000	0.00202667
$958$	0	0
$959$	3924.00	0.132130
$960$	0	0
$961$	55473.0	1.86207
$962$	0	0
$963$	648.000	0.0216838
$964$	0	0
$965$	−70528.0	−2.35272
$966$	0	0
$967$	11605.0	0.385927	0.192964	−	0.981206i	$-0.438190\pi$
$967$	11605.0	0.385927	0.192964	+	0.981206i	$0.438190\pi$
$968$	0	0
$969$	−22330.0	−0.740292
$970$	0	0
$971$	−6057.00	−0.200184	−0.100092	−	0.994978i	$-0.531914\pi$
$971$	−6057.00	−0.200184	−0.100092	+	0.994978i	$0.531914\pi$
$972$	0	0
$973$	−837.000	−0.0275776
$974$	0	0
$975$	15340.0	0.503870
$976$	0	0
$977$	−5166.00	−0.169166	−0.0845829	−	0.996416i	$-0.526956\pi$
$977$	−5166.00	−0.169166	−0.0845829	+	0.996416i	$0.526956\pi$
$978$	0	0
$979$	12.0000	0.000391748 0
$980$	0	0
$981$	−398.000	−0.0129533
$982$	0	0
$983$	21075.0	0.683813	0.341906	−	0.939734i	$-0.388927\pi$
$983$	21075.0	0.683813	0.341906	+	0.939734i	$0.388927\pi$
$984$	0	0
$985$	59223.0	1.91574
$986$	0	0
$987$	−5625.00	−0.181404
$988$	0	0
$989$	−24092.0	−0.774602
$990$	0	0
$991$	−29050.0	−0.931184	−0.465592	−	0.884999i	$-0.654159\pi$
$991$	−29050.0	−0.931184	−0.465592	+	0.884999i	$0.654159\pi$
$992$	0	0
$993$	33100.0	1.05780
$994$	0	0
$995$	−89338.0	−2.84644
$996$	0	0
$997$	−56450.0	−1.79317	−0.896584	−	0.442873i	$-0.853959\pi$
$997$	−56450.0	−1.79317	−0.896584	+	0.442873i	$0.853959\pi$
$998$	0	0
$999$	30015.0	0.950583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.a.1.1		1
3.2	odd	2		1872.4.a.a.1.1		1
4.3	odd	2		104.4.a.b.1.1	✓	1
8.3	odd	2		832.4.a.b.1.1		1
8.5	even	2		832.4.a.p.1.1		1
12.11	even	2		936.4.a.a.1.1		1
52.51	odd	2		1352.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.b.1.1	✓	1	4.3	odd	2
208.4.a.a.1.1		1	1.1	even	1	trivial
832.4.a.b.1.1		1	8.3	odd	2
832.4.a.p.1.1		1	8.5	even	2
936.4.a.a.1.1		1	12.11	even	2
1352.4.a.c.1.1		1	52.51	odd	2
1872.4.a.a.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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