LMFDB - Embedded newform 8464.2.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,4,0,-2,0,0,0,6,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{3} -1.00000 q^{5} +4.24264 q^{7} -2.65685 q^{9} +0.585786 q^{11} +3.82843 q^{13} -0.585786 q^{15} +6.82843 q^{17} -5.41421 q^{19} +2.48528 q^{21} -4.00000 q^{25} -3.31371 q^{27} +0.171573 q^{29} +8.82843 q^{31} +0.343146 q^{33} -4.24264 q^{35} +5.65685 q^{37} +2.24264 q^{39} -3.82843 q^{41} -6.48528 q^{43} +2.65685 q^{45} -2.24264 q^{47} +11.0000 q^{49} +4.00000 q^{51} +5.00000 q^{53} -0.585786 q^{55} -3.17157 q^{57} +11.0711 q^{59} +9.82843 q^{61} -11.2721 q^{63} -3.82843 q^{65} +9.65685 q^{67} +3.07107 q^{71} -9.48528 q^{73} -2.34315 q^{75} +2.48528 q^{77} -7.31371 q^{79} +6.02944 q^{81} -3.17157 q^{83} -6.82843 q^{85} +0.100505 q^{87} -7.00000 q^{89} +16.2426 q^{91} +5.17157 q^{93} +5.41421 q^{95} -15.4853 q^{97} -1.55635 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9} + 4 q^{11} + 2 q^{13} - 4 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{21} - 8 q^{25} + 16 q^{27} + 6 q^{29} + 12 q^{31} + 12 q^{33} - 4 q^{39} - 2 q^{41} + 4 q^{43} - 6 q^{45}+ \cdots + 28 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^5 + 6 * q^9 + 4 * q^11 + 2 * q^13 - 4 * q^15 + 8 * q^17 - 8 * q^19 - 12 * q^21 - 8 * q^25 + 16 * q^27 + 6 * q^29 + 12 * q^31 + 12 * q^33 - 4 * q^39 - 2 * q^41 + 4 * q^43 - 6 * q^45 + 4 * q^47 + 22 * q^49 + 8 * q^51 + 10 * q^53 - 4 * q^55 - 12 * q^57 + 8 * q^59 + 14 * q^61 - 48 * q^63 - 2 * q^65 + 8 * q^67 - 8 * q^71 - 2 * q^73 - 16 * q^75 - 12 * q^77 + 8 * q^79 + 46 * q^81 - 12 * q^83 - 8 * q^85 + 20 * q^87 - 14 * q^89 + 24 * q^91 + 16 * q^93 + 8 * q^95 - 14 * q^97 + 28 * q^99

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
$3$	0.585786	0.338204	0.169102	−	0.985599i	$-0.445913\pi$
$3$	0.585786	0.338204	0.169102	+	0.985599i	$0.445913\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	4.24264	1.60357	0.801784	−	0.597614i	$-0.203885\pi$
$7$	4.24264	1.60357	0.801784	+	0.597614i	$0.203885\pi$
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	−0.585786	−0.151249
$16$	0	0
$17$	6.82843	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$17$	6.82843	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$18$	0	0
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	0	0
$21$	2.48528	0.542333
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−3.31371	−0.637723
$28$	0	0
$29$	0.171573	0.0318603	0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573	0.0318603	0.0159301	+	0.999873i	$0.494929\pi$
$30$	0	0
$31$	8.82843	1.58563	0.792816	−	0.609461i	$-0.208614\pi$
$31$	8.82843	1.58563	0.792816	+	0.609461i	$0.208614\pi$
$32$	0	0
$33$	0.343146	0.0597340
$34$	0	0
$35$	−4.24264	−0.717137
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	2.24264	0.359110
$40$	0	0
$41$	−3.82843	−0.597900	−0.298950	−	0.954269i	$-0.596636\pi$
$41$	−3.82843	−0.597900	−0.298950	+	0.954269i	$0.596636\pi$
$42$	0	0
$43$	−6.48528	−0.988996	−0.494498	−	0.869179i	$-0.664648\pi$
$43$	−6.48528	−0.988996	−0.494498	+	0.869179i	$0.664648\pi$
$44$	0	0
$45$	2.65685	0.396060
$46$	0	0
$47$	−2.24264	−0.327123	−0.163561	−	0.986533i	$-0.552298\pi$
$47$	−2.24264	−0.327123	−0.163561	+	0.986533i	$0.552298\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−0.585786	−0.0789874
$56$	0	0
$57$	−3.17157	−0.420085
$58$	0	0
$59$	11.0711	1.44133	0.720665	−	0.693283i	$-0.243837\pi$
$59$	11.0711	1.44133	0.720665	+	0.693283i	$0.243837\pi$
$60$	0	0
$61$	9.82843	1.25840	0.629201	−	0.777243i	$-0.283382\pi$
$61$	9.82843	1.25840	0.629201	+	0.777243i	$0.283382\pi$
$62$	0	0
$63$	−11.2721	−1.42015
$64$	0	0
$65$	−3.82843	−0.474858
$66$	0	0
$67$	9.65685	1.17977	0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685	1.17977	0.589886	+	0.807486i	$0.299173\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.07107	0.364469	0.182234	−	0.983255i	$-0.441667\pi$
$71$	3.07107	0.364469	0.182234	+	0.983255i	$0.441667\pi$
$72$	0	0
$73$	−9.48528	−1.11017	−0.555084	−	0.831794i	$-0.687314\pi$
$73$	−9.48528	−1.11017	−0.555084	+	0.831794i	$0.687314\pi$
$74$	0	0
$75$	−2.34315	−0.270563
$76$	0	0
$77$	2.48528	0.283224
$78$	0	0
$79$	−7.31371	−0.822856	−0.411428	−	0.911442i	$-0.634970\pi$
$79$	−7.31371	−0.822856	−0.411428	+	0.911442i	$0.634970\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	−3.17157	−0.348125	−0.174063	−	0.984735i	$-0.555690\pi$
$83$	−3.17157	−0.348125	−0.174063	+	0.984735i	$0.555690\pi$
$84$	0	0
$85$	−6.82843	−0.740647
$86$	0	0
$87$	0.100505	0.0107753
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	16.2426	1.70269
$92$	0	0
$93$	5.17157	0.536267
$94$	0	0
$95$	5.41421	0.555487
$96$	0	0
$97$	−15.4853	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$97$	−15.4853	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$98$	0	0
$99$	−1.55635	−0.156419
$100$	0	0
$101$	18.6569	1.85643	0.928213	−	0.372049i	$-0.121345\pi$
$101$	18.6569	1.85643	0.928213	+	0.372049i	$0.121345\pi$
$102$	0	0
$103$	3.17157	0.312504	0.156252	−	0.987717i	$-0.450059\pi$
$103$	3.17157	0.312504	0.156252	+	0.987717i	$0.450059\pi$
$104$	0	0
$105$	−2.48528	−0.242539
$106$	0	0
$107$	17.8995	1.73041	0.865205	−	0.501419i	$-0.167188\pi$
$107$	17.8995	1.73041	0.865205	+	0.501419i	$0.167188\pi$
$108$	0	0
$109$	−2.65685	−0.254480	−0.127240	−	0.991872i	$-0.540612\pi$
$109$	−2.65685	−0.254480	−0.127240	+	0.991872i	$0.540612\pi$
$110$	0	0
$111$	3.31371	0.314523
$112$	0	0
$113$	−0.656854	−0.0617916	−0.0308958	−	0.999523i	$-0.509836\pi$
$113$	−0.656854	−0.0617916	−0.0308958	+	0.999523i	$0.509836\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−10.1716	−0.940362
$118$	0	0
$119$	28.9706	2.65573
$120$	0	0
$121$	−10.6569	−0.968805
$122$	0	0
$123$	−2.24264	−0.202212
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	6.48528	0.575476	0.287738	−	0.957709i	$-0.407097\pi$
$127$	6.48528	0.575476	0.287738	+	0.957709i	$0.407097\pi$
$128$	0	0
$129$	−3.79899	−0.334482
$130$	0	0
$131$	11.7574	1.02725	0.513623	−	0.858016i	$-0.328303\pi$
$131$	11.7574	1.02725	0.513623	+	0.858016i	$0.328303\pi$
$132$	0	0
$133$	−22.9706	−1.99180
$134$	0	0
$135$	3.31371	0.285199
$136$	0	0
$137$	−15.8284	−1.35231	−0.676157	−	0.736758i	$-0.736356\pi$
$137$	−15.8284	−1.35231	−0.676157	+	0.736758i	$0.736356\pi$
$138$	0	0
$139$	−6.48528	−0.550074	−0.275037	−	0.961434i	$-0.588690\pi$
$139$	−6.48528	−0.550074	−0.275037	+	0.961434i	$0.588690\pi$
$140$	0	0
$141$	−1.31371	−0.110634
$142$	0	0
$143$	2.24264	0.187539
$144$	0	0
$145$	−0.171573	−0.0142484
$146$	0	0
$147$	6.44365	0.531463
$148$	0	0
$149$	−23.1421	−1.89588	−0.947939	−	0.318453i	$-0.896837\pi$
$149$	−23.1421	−1.89588	−0.947939	+	0.318453i	$0.896837\pi$
$150$	0	0
$151$	12.5858	1.02422	0.512108	−	0.858921i	$-0.328865\pi$
$151$	12.5858	1.02422	0.512108	+	0.858921i	$0.328865\pi$
$152$	0	0
$153$	−18.1421	−1.46670
$154$	0	0
$155$	−8.82843	−0.709116
$156$	0	0
$157$	10.3137	0.823124	0.411562	−	0.911382i	$-0.364983\pi$
$157$	10.3137	0.823124	0.411562	+	0.911382i	$0.364983\pi$
$158$	0	0
$159$	2.92893	0.232279
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.89949	−0.775388	−0.387694	−	0.921788i	$-0.626728\pi$
$163$	−9.89949	−0.775388	−0.387694	+	0.921788i	$0.626728\pi$
$164$	0	0
$165$	−0.343146	−0.0267139
$166$	0	0
$167$	−10.4853	−0.811375	−0.405688	−	0.914012i	$-0.632968\pi$
$167$	−10.4853	−0.811375	−0.405688	+	0.914012i	$0.632968\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	14.3848	1.10003
$172$	0	0
$173$	15.9706	1.21422	0.607110	−	0.794618i	$-0.292329\pi$
$173$	15.9706	1.21422	0.607110	+	0.794618i	$0.292329\pi$
$174$	0	0
$175$	−16.9706	−1.28285
$176$	0	0
$177$	6.48528	0.487464
$178$	0	0
$179$	22.4853	1.68063	0.840314	−	0.542099i	$-0.182370\pi$
$179$	22.4853	1.68063	0.840314	+	0.542099i	$0.182370\pi$
$180$	0	0
$181$	2.82843	0.210235	0.105118	−	0.994460i	$-0.466478\pi$
$181$	2.82843	0.210235	0.105118	+	0.994460i	$0.466478\pi$
$182$	0	0
$183$	5.75736	0.425596
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	−14.0589	−1.02263
$190$	0	0
$191$	−1.51472	−0.109601	−0.0548006	−	0.998497i	$-0.517452\pi$
$191$	−1.51472	−0.109601	−0.0548006	+	0.998497i	$0.517452\pi$
$192$	0	0
$193$	−3.82843	−0.275576	−0.137788	−	0.990462i	$-0.543999\pi$
$193$	−3.82843	−0.275576	−0.137788	+	0.990462i	$0.543999\pi$
$194$	0	0
$195$	−2.24264	−0.160599
$196$	0	0
$197$	−15.1421	−1.07883	−0.539416	−	0.842039i	$-0.681355\pi$
$197$	−15.1421	−1.07883	−0.539416	+	0.842039i	$0.681355\pi$
$198$	0	0
$199$	−22.0416	−1.56249	−0.781245	−	0.624225i	$-0.785415\pi$
$199$	−22.0416	−1.56249	−0.781245	+	0.624225i	$0.785415\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	0	0
$203$	0.727922	0.0510901
$204$	0	0
$205$	3.82843	0.267389
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.17157	−0.219382
$210$	0	0
$211$	2.24264	0.154390	0.0771949	−	0.997016i	$-0.475404\pi$
$211$	2.24264	0.154390	0.0771949	+	0.997016i	$0.475404\pi$
$212$	0	0
$213$	1.79899	0.123265
$214$	0	0
$215$	6.48528	0.442293
$216$	0	0
$217$	37.4558	2.54267
$218$	0	0
$219$	−5.55635	−0.375463
$220$	0	0
$221$	26.1421	1.75851
$222$	0	0
$223$	13.0711	0.875303	0.437652	−	0.899145i	$-0.355810\pi$
$223$	13.0711	0.875303	0.437652	+	0.899145i	$0.355810\pi$
$224$	0	0
$225$	10.6274	0.708494
$226$	0	0
$227$	−14.3431	−0.951988	−0.475994	−	0.879449i	$-0.657912\pi$
$227$	−14.3431	−0.951988	−0.475994	+	0.879449i	$0.657912\pi$
$228$	0	0
$229$	−7.31371	−0.483303	−0.241652	−	0.970363i	$-0.577689\pi$
$229$	−7.31371	−0.483303	−0.241652	+	0.970363i	$0.577689\pi$
$230$	0	0
$231$	1.45584	0.0957875
$232$	0	0
$233$	29.9706	1.96344	0.981718	−	0.190339i	$-0.0609587\pi$
$233$	29.9706	1.96344	0.981718	+	0.190339i	$0.0609587\pi$
$234$	0	0
$235$	2.24264	0.146294
$236$	0	0
$237$	−4.28427	−0.278293
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	−0.656854	−0.0423117	−0.0211559	−	0.999776i	$-0.506735\pi$
$241$	−0.656854	−0.0423117	−0.0211559	+	0.999776i	$0.506735\pi$
$242$	0	0
$243$	13.4731	0.864299
$244$	0	0
$245$	−11.0000	−0.702764
$246$	0	0
$247$	−20.7279	−1.31889
$248$	0	0
$249$	−1.85786	−0.117737
$250$	0	0
$251$	−11.5563	−0.729430	−0.364715	−	0.931119i	$-0.618834\pi$
$251$	−11.5563	−0.729430	−0.364715	+	0.931119i	$0.618834\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	30.4558	1.89978	0.949892	−	0.312579i	$-0.101193\pi$
$257$	30.4558	1.89978	0.949892	+	0.312579i	$0.101193\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	0	0
$261$	−0.455844	−0.0282160
$262$	0	0
$263$	31.0711	1.91592	0.957962	−	0.286895i	$-0.0926231\pi$
$263$	31.0711	1.91592	0.957962	+	0.286895i	$0.0926231\pi$
$264$	0	0
$265$	−5.00000	−0.307148
$266$	0	0
$267$	−4.10051	−0.250947
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	22.2426	1.35114	0.675572	−	0.737294i	$-0.263897\pi$
$271$	22.2426	1.35114	0.675572	+	0.737294i	$0.263897\pi$
$272$	0	0
$273$	9.51472	0.575857
$274$	0	0
$275$	−2.34315	−0.141297
$276$	0	0
$277$	−0.686292	−0.0412353	−0.0206176	−	0.999787i	$-0.506563\pi$
$277$	−0.686292	−0.0412353	−0.0206176	+	0.999787i	$0.506563\pi$
$278$	0	0
$279$	−23.4558	−1.40426
$280$	0	0
$281$	−18.6274	−1.11122	−0.555609	−	0.831444i	$-0.687515\pi$
$281$	−18.6274	−1.11122	−0.555609	+	0.831444i	$0.687515\pi$
$282$	0	0
$283$	7.17157	0.426306	0.213153	−	0.977019i	$-0.431627\pi$
$283$	7.17157	0.426306	0.213153	+	0.977019i	$0.431627\pi$
$284$	0	0
$285$	3.17157	0.187868
$286$	0	0
$287$	−16.2426	−0.958773
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	−9.07107	−0.531755
$292$	0	0
$293$	7.97056	0.465645	0.232823	−	0.972519i	$-0.425204\pi$
$293$	7.97056	0.465645	0.232823	+	0.972519i	$0.425204\pi$
$294$	0	0
$295$	−11.0711	−0.644582
$296$	0	0
$297$	−1.94113	−0.112636
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−27.5147	−1.58592
$302$	0	0
$303$	10.9289	0.627851
$304$	0	0
$305$	−9.82843	−0.562774
$306$	0	0
$307$	19.8995	1.13572	0.567862	−	0.823124i	$-0.307771\pi$
$307$	19.8995	1.13572	0.567862	+	0.823124i	$0.307771\pi$
$308$	0	0
$309$	1.85786	0.105690
$310$	0	0
$311$	−17.8995	−1.01499	−0.507494	−	0.861656i	$-0.669428\pi$
$311$	−17.8995	−1.01499	−0.507494	+	0.861656i	$0.669428\pi$
$312$	0	0
$313$	3.00000	0.169570	0.0847850	−	0.996399i	$-0.472980\pi$
$313$	3.00000	0.169570	0.0847850	+	0.996399i	$0.472980\pi$
$314$	0	0
$315$	11.2721	0.635110
$316$	0	0
$317$	23.1421	1.29979	0.649896	−	0.760023i	$-0.274812\pi$
$317$	23.1421	1.29979	0.649896	+	0.760023i	$0.274812\pi$
$318$	0	0
$319$	0.100505	0.00562720
$320$	0	0
$321$	10.4853	0.585231
$322$	0	0
$323$	−36.9706	−2.05710
$324$	0	0
$325$	−15.3137	−0.849452
$326$	0	0
$327$	−1.55635	−0.0860663
$328$	0	0
$329$	−9.51472	−0.524563
$330$	0	0
$331$	18.4853	1.01604	0.508021	−	0.861344i	$-0.330377\pi$
$331$	18.4853	1.01604	0.508021	+	0.861344i	$0.330377\pi$
$332$	0	0
$333$	−15.0294	−0.823608
$334$	0	0
$335$	−9.65685	−0.527610
$336$	0	0
$337$	−10.1716	−0.554081	−0.277040	−	0.960858i	$-0.589354\pi$
$337$	−10.1716	−0.554081	−0.277040	+	0.960858i	$0.589354\pi$
$338$	0	0
$339$	−0.384776	−0.0208982
$340$	0	0
$341$	5.17157	0.280056
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.5563	1.80140	0.900700	−	0.434442i	$-0.143055\pi$
$347$	33.5563	1.80140	0.900700	+	0.434442i	$0.143055\pi$
$348$	0	0
$349$	8.48528	0.454207	0.227103	−	0.973871i	$-0.427074\pi$
$349$	8.48528	0.454207	0.227103	+	0.973871i	$0.427074\pi$
$350$	0	0
$351$	−12.6863	−0.677144
$352$	0	0
$353$	−5.82843	−0.310216	−0.155108	−	0.987898i	$-0.549573\pi$
$353$	−5.82843	−0.310216	−0.155108	+	0.987898i	$0.549573\pi$
$354$	0	0
$355$	−3.07107	−0.162995
$356$	0	0
$357$	16.9706	0.898177
$358$	0	0
$359$	18.3848	0.970311	0.485156	−	0.874428i	$-0.338763\pi$
$359$	18.3848	0.970311	0.485156	+	0.874428i	$0.338763\pi$
$360$	0	0
$361$	10.3137	0.542827
$362$	0	0
$363$	−6.24264	−0.327654
$364$	0	0
$365$	9.48528	0.496482
$366$	0	0
$367$	31.4142	1.63981	0.819904	−	0.572501i	$-0.194027\pi$
$367$	31.4142	1.63981	0.819904	+	0.572501i	$0.194027\pi$
$368$	0	0
$369$	10.1716	0.529511
$370$	0	0
$371$	21.2132	1.10133
$372$	0	0
$373$	−19.7990	−1.02515	−0.512576	−	0.858642i	$-0.671309\pi$
$373$	−19.7990	−1.02515	−0.512576	+	0.858642i	$0.671309\pi$
$374$	0	0
$375$	5.27208	0.272249
$376$	0	0
$377$	0.656854	0.0338297
$378$	0	0
$379$	−29.4142	−1.51091	−0.755453	−	0.655202i	$-0.772583\pi$
$379$	−29.4142	−1.51091	−0.755453	+	0.655202i	$0.772583\pi$
$380$	0	0
$381$	3.79899	0.194628
$382$	0	0
$383$	−13.7990	−0.705095	−0.352548	−	0.935794i	$-0.614685\pi$
$383$	−13.7990	−0.705095	−0.352548	+	0.935794i	$0.614685\pi$
$384$	0	0
$385$	−2.48528	−0.126662
$386$	0	0
$387$	17.2304	0.875873
$388$	0	0
$389$	24.4853	1.24145	0.620727	−	0.784027i	$-0.286838\pi$
$389$	24.4853	1.24145	0.620727	+	0.784027i	$0.286838\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6.88730	0.347418
$394$	0	0
$395$	7.31371	0.367993
$396$	0	0
$397$	−7.68629	−0.385764	−0.192882	−	0.981222i	$-0.561783\pi$
$397$	−7.68629	−0.385764	−0.192882	+	0.981222i	$0.561783\pi$
$398$	0	0
$399$	−13.4558	−0.673635
$400$	0	0
$401$	−1.14214	−0.0570355	−0.0285178	−	0.999593i	$-0.509079\pi$
$401$	−1.14214	−0.0570355	−0.0285178	+	0.999593i	$0.509079\pi$
$402$	0	0
$403$	33.7990	1.68365
$404$	0	0
$405$	−6.02944	−0.299605
$406$	0	0
$407$	3.31371	0.164254
$408$	0	0
$409$	−22.8284	−1.12879	−0.564397	−	0.825504i	$-0.690891\pi$
$409$	−22.8284	−1.12879	−0.564397	+	0.825504i	$0.690891\pi$
$410$	0	0
$411$	−9.27208	−0.457358
$412$	0	0
$413$	46.9706	2.31127
$414$	0	0
$415$	3.17157	0.155686
$416$	0	0
$417$	−3.79899	−0.186037
$418$	0	0
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	9.31371	0.453922	0.226961	−	0.973904i	$-0.427121\pi$
$421$	9.31371	0.453922	0.226961	+	0.973904i	$0.427121\pi$
$422$	0	0
$423$	5.95837	0.289706
$424$	0	0
$425$	−27.3137	−1.32491
$426$	0	0
$427$	41.6985	2.01793
$428$	0	0
$429$	1.31371	0.0634264
$430$	0	0
$431$	−4.58579	−0.220890	−0.110445	−	0.993882i	$-0.535228\pi$
$431$	−4.58579	−0.220890	−0.110445	+	0.993882i	$0.535228\pi$
$432$	0	0
$433$	12.5147	0.601419	0.300709	−	0.953716i	$-0.402777\pi$
$433$	12.5147	0.601419	0.300709	+	0.953716i	$0.402777\pi$
$434$	0	0
$435$	−0.100505	−0.00481885
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.48528	−0.118616	−0.0593080	−	0.998240i	$-0.518889\pi$
$439$	−2.48528	−0.118616	−0.0593080	+	0.998240i	$0.518889\pi$
$440$	0	0
$441$	−29.2254	−1.39169
$442$	0	0
$443$	−9.79899	−0.465564	−0.232782	−	0.972529i	$-0.574783\pi$
$443$	−9.79899	−0.465564	−0.232782	+	0.972529i	$0.574783\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	−13.5563	−0.641193
$448$	0	0
$449$	−10.7990	−0.509636	−0.254818	−	0.966989i	$-0.582016\pi$
$449$	−10.7990	−0.509636	−0.254818	+	0.966989i	$0.582016\pi$
$450$	0	0
$451$	−2.24264	−0.105602
$452$	0	0
$453$	7.37258	0.346394
$454$	0	0
$455$	−16.2426	−0.761467
$456$	0	0
$457$	−8.51472	−0.398302	−0.199151	−	0.979969i	$-0.563818\pi$
$457$	−8.51472	−0.398302	−0.199151	+	0.979969i	$0.563818\pi$
$458$	0	0
$459$	−22.6274	−1.05616
$460$	0	0
$461$	−21.6274	−1.00729	−0.503645	−	0.863911i	$-0.668008\pi$
$461$	−21.6274	−1.00729	−0.503645	+	0.863911i	$0.668008\pi$
$462$	0	0
$463$	15.3137	0.711688	0.355844	−	0.934545i	$-0.384193\pi$
$463$	15.3137	0.711688	0.355844	+	0.934545i	$0.384193\pi$
$464$	0	0
$465$	−5.17157	−0.239826
$466$	0	0
$467$	4.97056	0.230010	0.115005	−	0.993365i	$-0.463312\pi$
$467$	4.97056	0.230010	0.115005	+	0.993365i	$0.463312\pi$
$468$	0	0
$469$	40.9706	1.89184
$470$	0	0
$471$	6.04163	0.278384
$472$	0	0
$473$	−3.79899	−0.174678
$474$	0	0
$475$	21.6569	0.993685
$476$	0	0
$477$	−13.2843	−0.608245
$478$	0	0
$479$	−21.7990	−0.996021	−0.498011	−	0.867171i	$-0.665936\pi$
$479$	−21.7990	−0.996021	−0.498011	+	0.867171i	$0.665936\pi$
$480$	0	0
$481$	21.6569	0.987468
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.4853	0.703150
$486$	0	0
$487$	−17.2132	−0.780005	−0.390002	−	0.920814i	$-0.627526\pi$
$487$	−17.2132	−0.780005	−0.390002	+	0.920814i	$0.627526\pi$
$488$	0	0
$489$	−5.79899	−0.262239
$490$	0	0
$491$	11.8579	0.535138	0.267569	−	0.963539i	$-0.413780\pi$
$491$	11.8579	0.535138	0.267569	+	0.963539i	$0.413780\pi$
$492$	0	0
$493$	1.17157	0.0527650
$494$	0	0
$495$	1.55635	0.0699527
$496$	0	0
$497$	13.0294	0.584450
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−6.14214	−0.274410
$502$	0	0
$503$	18.2426	0.813399	0.406700	−	0.913562i	$-0.366680\pi$
$503$	18.2426	0.813399	0.406700	+	0.913562i	$0.366680\pi$
$504$	0	0
$505$	−18.6569	−0.830219
$506$	0	0
$507$	0.970563	0.0431042
$508$	0	0
$509$	29.6274	1.31321	0.656606	−	0.754234i	$-0.271991\pi$
$509$	29.6274	1.31321	0.656606	+	0.754234i	$0.271991\pi$
$510$	0	0
$511$	−40.2426	−1.78023
$512$	0	0
$513$	17.9411	0.792120
$514$	0	0
$515$	−3.17157	−0.139756
$516$	0	0
$517$	−1.31371	−0.0577768
$518$	0	0
$519$	9.35534	0.410654
$520$	0	0
$521$	15.7990	0.692166	0.346083	−	0.938204i	$-0.387512\pi$
$521$	15.7990	0.692166	0.346083	+	0.938204i	$0.387512\pi$
$522$	0	0
$523$	15.3137	0.669622	0.334811	−	0.942285i	$-0.391328\pi$
$523$	15.3137	0.669622	0.334811	+	0.942285i	$0.391328\pi$
$524$	0	0
$525$	−9.94113	−0.433866
$526$	0	0
$527$	60.2843	2.62602
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−29.4142	−1.27647
$532$	0	0
$533$	−14.6569	−0.634859
$534$	0	0
$535$	−17.8995	−0.773863
$536$	0	0
$537$	13.1716	0.568395
$538$	0	0
$539$	6.44365	0.277548
$540$	0	0
$541$	−30.6569	−1.31804	−0.659021	−	0.752125i	$-0.729029\pi$
$541$	−30.6569	−1.31804	−0.659021	+	0.752125i	$0.729029\pi$
$542$	0	0
$543$	1.65685	0.0711024
$544$	0	0
$545$	2.65685	0.113807
$546$	0	0
$547$	−7.17157	−0.306634	−0.153317	−	0.988177i	$-0.548996\pi$
$547$	−7.17157	−0.306634	−0.153317	+	0.988177i	$0.548996\pi$
$548$	0	0
$549$	−26.1127	−1.11446
$550$	0	0
$551$	−0.928932	−0.0395738
$552$	0	0
$553$	−31.0294	−1.31951
$554$	0	0
$555$	−3.31371	−0.140659
$556$	0	0
$557$	33.1421	1.40428	0.702139	−	0.712040i	$-0.252229\pi$
$557$	33.1421	1.40428	0.702139	+	0.712040i	$0.252229\pi$
$558$	0	0
$559$	−24.8284	−1.05013
$560$	0	0
$561$	2.34315	0.0989277
$562$	0	0
$563$	0.828427	0.0349140	0.0174570	−	0.999848i	$-0.494443\pi$
$563$	0.828427	0.0349140	0.0174570	+	0.999848i	$0.494443\pi$
$564$	0	0
$565$	0.656854	0.0276341
$566$	0	0
$567$	25.5807	1.07429
$568$	0	0
$569$	5.14214	0.215570	0.107785	−	0.994174i	$-0.465624\pi$
$569$	5.14214	0.215570	0.107785	+	0.994174i	$0.465624\pi$
$570$	0	0
$571$	42.5269	1.77970	0.889848	−	0.456257i	$-0.150810\pi$
$571$	42.5269	1.77970	0.889848	+	0.456257i	$0.150810\pi$
$572$	0	0
$573$	−0.887302	−0.0370676
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.31371	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$577$	5.31371	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$578$	0	0
$579$	−2.24264	−0.0932010
$580$	0	0
$581$	−13.4558	−0.558242
$582$	0	0
$583$	2.92893	0.121304
$584$	0	0
$585$	10.1716	0.420543
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−47.7990	−1.96952
$590$	0	0
$591$	−8.87006	−0.364865
$592$	0	0
$593$	−11.5147	−0.472853	−0.236426	−	0.971649i	$-0.575976\pi$
$593$	−11.5147	−0.472853	−0.236426	+	0.971649i	$0.575976\pi$
$594$	0	0
$595$	−28.9706	−1.18768
$596$	0	0
$597$	−12.9117	−0.528440
$598$	0	0
$599$	−8.72792	−0.356613	−0.178307	−	0.983975i	$-0.557062\pi$
$599$	−8.72792	−0.356613	−0.178307	+	0.983975i	$0.557062\pi$
$600$	0	0
$601$	−2.65685	−0.108375	−0.0541877	−	0.998531i	$-0.517257\pi$
$601$	−2.65685	−0.108375	−0.0541877	+	0.998531i	$0.517257\pi$
$602$	0	0
$603$	−25.6569	−1.04483
$604$	0	0
$605$	10.6569	0.433263
$606$	0	0
$607$	−44.2426	−1.79575	−0.897877	−	0.440247i	$-0.854891\pi$
$607$	−44.2426	−1.79575	−0.897877	+	0.440247i	$0.854891\pi$
$608$	0	0
$609$	0.426407	0.0172789
$610$	0	0
$611$	−8.58579	−0.347344
$612$	0	0
$613$	18.5147	0.747802	0.373901	−	0.927469i	$-0.378020\pi$
$613$	18.5147	0.747802	0.373901	+	0.927469i	$0.378020\pi$
$614$	0	0
$615$	2.24264	0.0904320
$616$	0	0
$617$	−20.6274	−0.830429	−0.415214	−	0.909724i	$-0.636293\pi$
$617$	−20.6274	−0.830429	−0.415214	+	0.909724i	$0.636293\pi$
$618$	0	0
$619$	−29.6569	−1.19201	−0.596005	−	0.802981i	$-0.703246\pi$
$619$	−29.6569	−1.19201	−0.596005	+	0.802981i	$0.703246\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−29.6985	−1.18984
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−1.85786	−0.0741960
$628$	0	0
$629$	38.6274	1.54018
$630$	0	0
$631$	19.4558	0.774525	0.387262	−	0.921970i	$-0.373421\pi$
$631$	19.4558	0.774525	0.387262	+	0.921970i	$0.373421\pi$
$632$	0	0
$633$	1.31371	0.0522152
$634$	0	0
$635$	−6.48528	−0.257361
$636$	0	0
$637$	42.1127	1.66857
$638$	0	0
$639$	−8.15938	−0.322780
$640$	0	0
$641$	−16.5147	−0.652292	−0.326146	−	0.945319i	$-0.605750\pi$
$641$	−16.5147	−0.652292	−0.326146	+	0.945319i	$0.605750\pi$
$642$	0	0
$643$	32.7279	1.29066	0.645332	−	0.763903i	$-0.276719\pi$
$643$	32.7279	1.29066	0.645332	+	0.763903i	$0.276719\pi$
$644$	0	0
$645$	3.79899	0.149585
$646$	0	0
$647$	34.0416	1.33831	0.669157	−	0.743121i	$-0.266655\pi$
$647$	34.0416	1.33831	0.669157	+	0.743121i	$0.266655\pi$
$648$	0	0
$649$	6.48528	0.254570
$650$	0	0
$651$	21.9411	0.859941
$652$	0	0
$653$	−33.4853	−1.31038	−0.655190	−	0.755464i	$-0.727412\pi$
$653$	−33.4853	−1.31038	−0.655190	+	0.755464i	$0.727412\pi$
$654$	0	0
$655$	−11.7574	−0.459398
$656$	0	0
$657$	25.2010	0.983185
$658$	0	0
$659$	5.65685	0.220360	0.110180	−	0.993912i	$-0.464857\pi$
$659$	5.65685	0.220360	0.110180	+	0.993912i	$0.464857\pi$
$660$	0	0
$661$	−10.8579	−0.422322	−0.211161	−	0.977451i	$-0.567724\pi$
$661$	−10.8579	−0.422322	−0.211161	+	0.977451i	$0.567724\pi$
$662$	0	0
$663$	15.3137	0.594735
$664$	0	0
$665$	22.9706	0.890760
$666$	0	0
$667$	0	0
$668$	0	0
$669$	7.65685	0.296031
$670$	0	0
$671$	5.75736	0.222260
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	0	0
$675$	13.2548	0.510179
$676$	0	0
$677$	16.1127	0.619261	0.309631	−	0.950857i	$-0.399795\pi$
$677$	16.1127	0.619261	0.309631	+	0.950857i	$0.399795\pi$
$678$	0	0
$679$	−65.6985	−2.52128
$680$	0	0
$681$	−8.40202	−0.321966
$682$	0	0
$683$	28.9706	1.10853	0.554264	−	0.832341i	$-0.313000\pi$
$683$	28.9706	1.10853	0.554264	+	0.832341i	$0.313000\pi$
$684$	0	0
$685$	15.8284	0.604773
$686$	0	0
$687$	−4.28427	−0.163455
$688$	0	0
$689$	19.1421	0.729257
$690$	0	0
$691$	13.1127	0.498831	0.249415	−	0.968397i	$-0.419762\pi$
$691$	13.1127	0.498831	0.249415	+	0.968397i	$0.419762\pi$
$692$	0	0
$693$	−6.60303	−0.250828
$694$	0	0
$695$	6.48528	0.246001
$696$	0	0
$697$	−26.1421	−0.990204
$698$	0	0
$699$	17.5563	0.664042
$700$	0	0
$701$	26.6274	1.00570	0.502852	−	0.864373i	$-0.332284\pi$
$701$	26.6274	1.00570	0.502852	+	0.864373i	$0.332284\pi$
$702$	0	0
$703$	−30.6274	−1.15513
$704$	0	0
$705$	1.31371	0.0494771
$706$	0	0
$707$	79.1543	2.97690
$708$	0	0
$709$	0.0294373	0.00110554	0.000552770	−	1.00000i	$-0.499824\pi$
$709$	0.0294373	0.00110554	0.000552770		1.00000i	$0.499824\pi$
$710$	0	0
$711$	19.4315	0.728737
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−2.24264	−0.0838700
$716$	0	0
$717$	9.94113	0.371258
$718$	0	0
$719$	−32.1421	−1.19870	−0.599350	−	0.800487i	$-0.704574\pi$
$719$	−32.1421	−1.19870	−0.599350	+	0.800487i	$0.704574\pi$
$720$	0	0
$721$	13.4558	0.501122
$722$	0	0
$723$	−0.384776	−0.0143100
$724$	0	0
$725$	−0.686292	−0.0254882
$726$	0	0
$727$	40.2426	1.49252	0.746258	−	0.665656i	$-0.231848\pi$
$727$	40.2426	1.49252	0.746258	+	0.665656i	$0.231848\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	−44.2843	−1.63791
$732$	0	0
$733$	−33.9706	−1.25473	−0.627366	−	0.778725i	$-0.715867\pi$
$733$	−33.9706	−1.25473	−0.627366	+	0.778725i	$0.715867\pi$
$734$	0	0
$735$	−6.44365	−0.237678
$736$	0	0
$737$	5.65685	0.208373
$738$	0	0
$739$	−8.72792	−0.321062	−0.160531	−	0.987031i	$-0.551321\pi$
$739$	−8.72792	−0.321062	−0.160531	+	0.987031i	$0.551321\pi$
$740$	0	0
$741$	−12.1421	−0.446052
$742$	0	0
$743$	4.38478	0.160862	0.0804309	−	0.996760i	$-0.474370\pi$
$743$	4.38478	0.160862	0.0804309	+	0.996760i	$0.474370\pi$
$744$	0	0
$745$	23.1421	0.847862
$746$	0	0
$747$	8.42641	0.308306
$748$	0	0
$749$	75.9411	2.77483
$750$	0	0
$751$	−41.5563	−1.51641	−0.758206	−	0.652015i	$-0.773924\pi$
$751$	−41.5563	−1.51641	−0.758206	+	0.652015i	$0.773924\pi$
$752$	0	0
$753$	−6.76955	−0.246696
$754$	0	0
$755$	−12.5858	−0.458044
$756$	0	0
$757$	−12.8579	−0.467327	−0.233664	−	0.972318i	$-0.575071\pi$
$757$	−12.8579	−0.467327	−0.233664	+	0.972318i	$0.575071\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.00000	0.0362500	0.0181250	−	0.999836i	$-0.494230\pi$
$761$	1.00000	0.0362500	0.0181250	+	0.999836i	$0.494230\pi$
$762$	0	0
$763$	−11.2721	−0.408077
$764$	0	0
$765$	18.1421	0.655930
$766$	0	0
$767$	42.3848	1.53043
$768$	0	0
$769$	−37.4558	−1.35069	−0.675346	−	0.737501i	$-0.736006\pi$
$769$	−37.4558	−1.35069	−0.675346	+	0.737501i	$0.736006\pi$
$770$	0	0
$771$	17.8406	0.642514
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−35.3137	−1.26851
$776$	0	0
$777$	14.0589	0.504359
$778$	0	0
$779$	20.7279	0.742655
$780$	0	0
$781$	1.79899	0.0643729
$782$	0	0
$783$	−0.568542	−0.0203181
$784$	0	0
$785$	−10.3137	−0.368112
$786$	0	0
$787$	−8.87006	−0.316183	−0.158092	−	0.987424i	$-0.550534\pi$
$787$	−8.87006	−0.316183	−0.158092	+	0.987424i	$0.550534\pi$
$788$	0	0
$789$	18.2010	0.647973
$790$	0	0
$791$	−2.78680	−0.0990871
$792$	0	0
$793$	37.6274	1.33619
$794$	0	0
$795$	−2.92893	−0.103879
$796$	0	0
$797$	−25.6569	−0.908812	−0.454406	−	0.890795i	$-0.650149\pi$
$797$	−25.6569	−0.908812	−0.454406	+	0.890795i	$0.650149\pi$
$798$	0	0
$799$	−15.3137	−0.541760
$800$	0	0
$801$	18.5980	0.657127
$802$	0	0
$803$	−5.55635	−0.196079
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.0589	−0.494896
$808$	0	0
$809$	47.9411	1.68552	0.842760	−	0.538289i	$-0.180929\pi$
$809$	47.9411	1.68552	0.842760	+	0.538289i	$0.180929\pi$
$810$	0	0
$811$	48.3848	1.69902	0.849510	−	0.527573i	$-0.176898\pi$
$811$	48.3848	1.69902	0.849510	+	0.527573i	$0.176898\pi$
$812$	0	0
$813$	13.0294	0.456962
$814$	0	0
$815$	9.89949	0.346764
$816$	0	0
$817$	35.1127	1.22844
$818$	0	0
$819$	−43.1543	−1.50793
$820$	0	0
$821$	29.4853	1.02904	0.514522	−	0.857477i	$-0.327969\pi$
$821$	29.4853	1.02904	0.514522	+	0.857477i	$0.327969\pi$
$822$	0	0
$823$	−28.8701	−1.00635	−0.503173	−	0.864185i	$-0.667834\pi$
$823$	−28.8701	−1.00635	−0.503173	+	0.864185i	$0.667834\pi$
$824$	0	0
$825$	−1.37258	−0.0477872
$826$	0	0
$827$	−36.3848	−1.26522	−0.632611	−	0.774469i	$-0.718017\pi$
$827$	−36.3848	−1.26522	−0.632611	+	0.774469i	$0.718017\pi$
$828$	0	0
$829$	−21.3431	−0.741278	−0.370639	−	0.928777i	$-0.620861\pi$
$829$	−21.3431	−0.741278	−0.370639	+	0.928777i	$0.620861\pi$
$830$	0	0
$831$	−0.402020	−0.0139459
$832$	0	0
$833$	75.1127	2.60250
$834$	0	0
$835$	10.4853	0.362858
$836$	0	0
$837$	−29.2548	−1.01119
$838$	0	0
$839$	−8.52691	−0.294382	−0.147191	−	0.989108i	$-0.547023\pi$
$839$	−8.52691	−0.294382	−0.147191	+	0.989108i	$0.547023\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	0	0
$843$	−10.9117	−0.375819
$844$	0	0
$845$	−1.65685	−0.0569975
$846$	0	0
$847$	−45.2132	−1.55354
$848$	0	0
$849$	4.20101	0.144178
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−27.7990	−0.951819	−0.475910	−	0.879494i	$-0.657881\pi$
$853$	−27.7990	−0.951819	−0.475910	+	0.879494i	$0.657881\pi$
$854$	0	0
$855$	−14.3848	−0.491949
$856$	0	0
$857$	35.2843	1.20529	0.602644	−	0.798010i	$-0.294114\pi$
$857$	35.2843	1.20529	0.602644	+	0.798010i	$0.294114\pi$
$858$	0	0
$859$	−36.1838	−1.23457	−0.617287	−	0.786738i	$-0.711768\pi$
$859$	−36.1838	−1.23457	−0.617287	+	0.786738i	$0.711768\pi$
$860$	0	0
$861$	−9.51472	−0.324261
$862$	0	0
$863$	39.4142	1.34168	0.670838	−	0.741604i	$-0.265935\pi$
$863$	39.4142	1.34168	0.670838	+	0.741604i	$0.265935\pi$
$864$	0	0
$865$	−15.9706	−0.543015
$866$	0	0
$867$	17.3553	0.589418
$868$	0	0
$869$	−4.28427	−0.145334
$870$	0	0
$871$	36.9706	1.25270
$872$	0	0
$873$	41.1421	1.39245
$874$	0	0
$875$	38.1838	1.29085
$876$	0	0
$877$	−5.17157	−0.174632	−0.0873158	−	0.996181i	$-0.527829\pi$
$877$	−5.17157	−0.174632	−0.0873158	+	0.996181i	$0.527829\pi$
$878$	0	0
$879$	4.66905	0.157483
$880$	0	0
$881$	10.7990	0.363827	0.181914	−	0.983315i	$-0.441771\pi$
$881$	10.7990	0.363827	0.181914	+	0.983315i	$0.441771\pi$
$882$	0	0
$883$	55.3553	1.86286	0.931428	−	0.363926i	$-0.118564\pi$
$883$	55.3553	1.86286	0.931428	+	0.363926i	$0.118564\pi$
$884$	0	0
$885$	−6.48528	−0.218000
$886$	0	0
$887$	−16.9706	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$888$	0	0
$889$	27.5147	0.922814
$890$	0	0
$891$	3.53196	0.118325
$892$	0	0
$893$	12.1421	0.406321
$894$	0	0
$895$	−22.4853	−0.751600
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.51472	0.0505187
$900$	0	0
$901$	34.1421	1.13744
$902$	0	0
$903$	−16.1177	−0.536365
$904$	0	0
$905$	−2.82843	−0.0940201
$906$	0	0
$907$	40.3848	1.34095	0.670477	−	0.741930i	$-0.266089\pi$
$907$	40.3848	1.34095	0.670477	+	0.741930i	$0.266089\pi$
$908$	0	0
$909$	−49.5685	−1.64408
$910$	0	0
$911$	15.1716	0.502657	0.251328	−	0.967902i	$-0.419133\pi$
$911$	15.1716	0.502657	0.251328	+	0.967902i	$0.419133\pi$
$912$	0	0
$913$	−1.85786	−0.0614863
$914$	0	0
$915$	−5.75736	−0.190332
$916$	0	0
$917$	49.8823	1.64726
$918$	0	0
$919$	−35.8406	−1.18227	−0.591136	−	0.806572i	$-0.701321\pi$
$919$	−35.8406	−1.18227	−0.591136	+	0.806572i	$0.701321\pi$
$920$	0	0
$921$	11.6569	0.384106
$922$	0	0
$923$	11.7574	0.386998
$924$	0	0
$925$	−22.6274	−0.743985
$926$	0	0
$927$	−8.42641	−0.276760
$928$	0	0
$929$	3.11270	0.102124	0.0510622	−	0.998695i	$-0.483739\pi$
$929$	3.11270	0.102124	0.0510622	+	0.998695i	$0.483739\pi$
$930$	0	0
$931$	−59.5563	−1.95188
$932$	0	0
$933$	−10.4853	−0.343273
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−38.8284	−1.26847	−0.634235	−	0.773141i	$-0.718685\pi$
$937$	−38.8284	−1.26847	−0.634235	+	0.773141i	$0.718685\pi$
$938$	0	0
$939$	1.75736	0.0573493
$940$	0	0
$941$	−55.2548	−1.80126	−0.900628	−	0.434591i	$-0.856893\pi$
$941$	−55.2548	−1.80126	−0.900628	+	0.434591i	$0.856893\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	14.0589	0.457335
$946$	0	0
$947$	−28.6863	−0.932179	−0.466090	−	0.884738i	$-0.654338\pi$
$947$	−28.6863	−0.932179	−0.466090	+	0.884738i	$0.654338\pi$
$948$	0	0
$949$	−36.3137	−1.17879
$950$	0	0
$951$	13.5563	0.439595
$952$	0	0
$953$	−24.1716	−0.782994	−0.391497	−	0.920179i	$-0.628043\pi$
$953$	−24.1716	−0.782994	−0.391497	+	0.920179i	$0.628043\pi$
$954$	0	0
$955$	1.51472	0.0490151
$956$	0	0
$957$	0.0588745	0.00190314
$958$	0	0
$959$	−67.1543	−2.16853
$960$	0	0
$961$	46.9411	1.51423
$962$	0	0
$963$	−47.5563	−1.53248
$964$	0	0
$965$	3.82843	0.123241
$966$	0	0
$967$	57.2132	1.83985	0.919926	−	0.392091i	$-0.128248\pi$
$967$	57.2132	1.83985	0.919926	+	0.392091i	$0.128248\pi$
$968$	0	0
$969$	−21.6569	−0.695718
$970$	0	0
$971$	11.2132	0.359849	0.179924	−	0.983680i	$-0.442415\pi$
$971$	11.2132	0.359849	0.179924	+	0.983680i	$0.442415\pi$
$972$	0	0
$973$	−27.5147	−0.882081
$974$	0	0
$975$	−8.97056	−0.287288
$976$	0	0
$977$	−56.1127	−1.79520	−0.897602	−	0.440807i	$-0.854692\pi$
$977$	−56.1127	−1.79520	−0.897602	+	0.440807i	$0.854692\pi$
$978$	0	0
$979$	−4.10051	−0.131053
$980$	0	0
$981$	7.05887	0.225373
$982$	0	0
$983$	−55.4558	−1.76877	−0.884383	−	0.466761i	$-0.845421\pi$
$983$	−55.4558	−1.76877	−0.884383	+	0.466761i	$0.845421\pi$
$984$	0	0
$985$	15.1421	0.482469
$986$	0	0
$987$	−5.57359	−0.177409
$988$	0	0
$989$	0	0
$990$	0	0
$991$	5.65685	0.179696	0.0898479	−	0.995955i	$-0.471362\pi$
$991$	5.65685	0.179696	0.0898479	+	0.995955i	$0.471362\pi$
$992$	0	0
$993$	10.8284	0.343630
$994$	0	0
$995$	22.0416	0.698767
$996$	0	0
$997$	−41.9706	−1.32922	−0.664611	−	0.747190i	$-0.731403\pi$
$997$	−41.9706	−1.32922	−0.664611	+	0.747190i	$0.731403\pi$
$998$	0	0
$999$	−18.7452	−0.593071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bg.1.1		2
4.3	odd	2		4232.2.a.k.1.2	✓	2
23.22	odd	2		8464.2.a.bj.1.1		2
92.91	even	2		4232.2.a.l.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.k.1.2	✓	2	4.3	odd	2
4232.2.a.l.1.2	yes	2	92.91	even	2
8464.2.a.bg.1.1		2	1.1	even	1	trivial
8464.2.a.bj.1.1		2	23.22	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.585786 q^{3} -1.00000 q^{5} +4.24264 q^{7} -2.65685 q^{9} +0.585786 q^{11} +3.82843 q^{13} -0.585786 q^{15} +6.82843 q^{17} -5.41421 q^{19} +2.48528 q^{21} -4.00000 q^{25} -3.31371 q^{27} +0.171573 q^{29} +8.82843 q^{31} +0.343146 q^{33} -4.24264 q^{35} +5.65685 q^{37} +2.24264 q^{39} -3.82843 q^{41} -6.48528 q^{43} +2.65685 q^{45} -2.24264 q^{47} +11.0000 q^{49} +4.00000 q^{51} +5.00000 q^{53} -0.585786 q^{55} -3.17157 q^{57} +11.0711 q^{59} +9.82843 q^{61} -11.2721 q^{63} -3.82843 q^{65} +9.65685 q^{67} +3.07107 q^{71} -9.48528 q^{73} -2.34315 q^{75} +2.48528 q^{77} -7.31371 q^{79} +6.02944 q^{81} -3.17157 q^{83} -6.82843 q^{85} +0.100505 q^{87} -7.00000 q^{89} +16.2426 q^{91} +5.17157 q^{93} +5.41421 q^{95} -15.4853 q^{97} -1.55635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9} + 4 q^{11} + 2 q^{13} - 4 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{21} - 8 q^{25} + 16 q^{27} + 6 q^{29} + 12 q^{31} + 12 q^{33} - 4 q^{39} - 2 q^{41} + 4 q^{43} - 6 q^{45}+ \cdots + 28 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^5 + 6 * q^9 + 4 * q^11 + 2 * q^13 - 4 * q^15 + 8 * q^17 - 8 * q^19 - 12 * q^21 - 8 * q^25 + 16 * q^27 + 6 * q^29 + 12 * q^31 + 12 * q^33 - 4 * q^39 - 2 * q^41 + 4 * q^43 - 6 * q^45 + 4 * q^47 + 22 * q^49 + 8 * q^51 + 10 * q^53 - 4 * q^55 - 12 * q^57 + 8 * q^59 + 14 * q^61 - 48 * q^63 - 2 * q^65 + 8 * q^67 - 8 * q^71 - 2 * q^73 - 16 * q^75 - 12 * q^77 + 8 * q^79 + 46 * q^81 - 12 * q^83 - 8 * q^85 + 20 * q^87 - 14 * q^89 + 24 * q^91 + 16 * q^93 + 8 * q^95 - 14 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more