LMFDB - Embedded newform 8464.2.a.cf.1.11

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 = [12,0,-8,0,0,0,0,0,8,0,0,0,16] 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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 $ x^12 - 20x^10 + 157x^8 - 616x^6 + 1264x^4 - 1272*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.52595$ 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)

$f(q)$	$=$	$q+2.38043 q^{3} -0.0992850 q^{5} -0.236341 q^{7} +2.66645 q^{9} +2.12038 q^{11} -5.09308 q^{13} -0.236341 q^{15} -6.44776 q^{17} +0.582013 q^{19} -0.562594 q^{21} -4.99014 q^{25} -0.793990 q^{27} +3.65591 q^{29} +3.91182 q^{31} +5.04741 q^{33} +0.0234651 q^{35} +7.09213 q^{37} -12.1237 q^{39} -7.57960 q^{41} +10.7311 q^{43} -0.264739 q^{45} -10.8700 q^{47} -6.94414 q^{49} -15.3485 q^{51} +6.54082 q^{53} -0.210522 q^{55} +1.38544 q^{57} -8.89024 q^{59} -4.79354 q^{61} -0.630192 q^{63} +0.505666 q^{65} +3.29978 q^{67} +12.2161 q^{71} -3.90127 q^{73} -11.8787 q^{75} -0.501132 q^{77} -11.7440 q^{79} -9.88939 q^{81} +7.31907 q^{83} +0.640167 q^{85} +8.70263 q^{87} -3.04701 q^{89} +1.20370 q^{91} +9.31181 q^{93} -0.0577851 q^{95} +4.49485 q^{97} +5.65388 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} - 8 q^{27} - 8 q^{31} - 56 q^{35} - 64 q^{39} - 40 q^{41} - 32 q^{47} + 28 q^{49} - 64 q^{55} - 60 q^{59} + 32 q^{71} + 28 q^{73} - 16 q^{75} + 24 q^{77}+ \cdots - 16 q^{95}+O(q^{100}) $ 12 * q - 8 * q^3 + 8 * q^9 + 16 * q^13 + 4 * q^25 - 8 * q^27 - 8 * q^31 - 56 * q^35 - 64 * q^39 - 40 * q^41 - 32 * q^47 + 28 * q^49 - 64 * q^55 - 60 * q^59 + 32 * q^71 + 28 * q^73 - 16 * q^75 + 24 * q^77 + 4 * q^81 + 64 * q^85 + 16 * q^87 + 24 * q^93 - 16 * q^95

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.38043	1.37434	0.687171	−	0.726496i	$-0.258852\pi$
$3$	2.38043	1.37434	0.687171	+	0.726496i	$0.258852\pi$
$4$	0	0
$5$	−0.0992850	−0.0444016	−0.0222008	−	0.999754i	$-0.507067\pi$
$5$	−0.0992850	−0.0444016	−0.0222008	+	0.999754i	$0.507067\pi$
$6$	0	0
$7$	−0.236341	−0.0893286	−0.0446643	−	0.999002i	$-0.514222\pi$
$7$	−0.236341	−0.0893286	−0.0446643	+	0.999002i	$0.514222\pi$
$8$	0	0
$9$	2.66645	0.888817
$10$	0	0
$11$	2.12038	0.639318	0.319659	−	0.947533i	$-0.396432\pi$
$11$	2.12038	0.639318	0.319659	+	0.947533i	$0.396432\pi$
$12$	0	0
$13$	−5.09308	−1.41257	−0.706283	−	0.707930i	$-0.749629\pi$
$13$	−5.09308	−1.41257	−0.706283	+	0.707930i	$0.749629\pi$
$14$	0	0
$15$	−0.236341	−0.0610230
$16$	0	0
$17$	−6.44776	−1.56381	−0.781906	−	0.623396i	$-0.785752\pi$
$17$	−6.44776	−1.56381	−0.781906	+	0.623396i	$0.785752\pi$
$18$	0	0
$19$	0.582013	0.133523	0.0667614	−	0.997769i	$-0.478733\pi$
$19$	0.582013	0.133523	0.0667614	+	0.997769i	$0.478733\pi$
$20$	0	0
$21$	−0.562594	−0.122768
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.99014	−0.998028
$26$	0	0
$27$	−0.793990	−0.152803
$28$	0	0
$29$	3.65591	0.678885	0.339442	−	0.940627i	$-0.389762\pi$
$29$	3.65591	0.678885	0.339442	+	0.940627i	$0.389762\pi$
$30$	0	0
$31$	3.91182	0.702583	0.351292	−	0.936266i	$-0.385743\pi$
$31$	3.91182	0.702583	0.351292	+	0.936266i	$0.385743\pi$
$32$	0	0
$33$	5.04741	0.878641
$34$	0	0
$35$	0.0234651	0.00396633
$36$	0	0
$37$	7.09213	1.16594	0.582970	−	0.812494i	$-0.301891\pi$
$37$	7.09213	1.16594	0.582970	+	0.812494i	$0.301891\pi$
$38$	0	0
$39$	−12.1237	−1.94135
$40$	0	0
$41$	−7.57960	−1.18374	−0.591868	−	0.806035i	$-0.701609\pi$
$41$	−7.57960	−1.18374	−0.591868	+	0.806035i	$0.701609\pi$
$42$	0	0
$43$	10.7311	1.63648	0.818242	−	0.574873i	$-0.194949\pi$
$43$	10.7311	1.63648	0.818242	+	0.574873i	$0.194949\pi$
$44$	0	0
$45$	−0.264739	−0.0394649
$46$	0	0
$47$	−10.8700	−1.58555	−0.792774	−	0.609515i	$-0.791364\pi$
$47$	−10.8700	−1.58555	−0.792774	+	0.609515i	$0.791364\pi$
$48$	0	0
$49$	−6.94414	−0.992020
$50$	0	0
$51$	−15.3485	−2.14921
$52$	0	0
$53$	6.54082	0.898450	0.449225	−	0.893419i	$-0.351700\pi$
$53$	6.54082	0.898450	0.449225	+	0.893419i	$0.351700\pi$
$54$	0	0
$55$	−0.210522	−0.0283867
$56$	0	0
$57$	1.38544	0.183506
$58$	0	0
$59$	−8.89024	−1.15741	−0.578706	−	0.815537i	$-0.696442\pi$
$59$	−8.89024	−1.15741	−0.578706	+	0.815537i	$0.696442\pi$
$60$	0	0
$61$	−4.79354	−0.613751	−0.306875	−	0.951750i	$-0.599283\pi$
$61$	−4.79354	−0.613751	−0.306875	+	0.951750i	$0.599283\pi$
$62$	0	0
$63$	−0.630192	−0.0793968
$64$	0	0
$65$	0.505666	0.0627202
$66$	0	0
$67$	3.29978	0.403132	0.201566	−	0.979475i	$-0.435397\pi$
$67$	3.29978	0.403132	0.201566	+	0.979475i	$0.435397\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.2161	1.44978	0.724892	−	0.688862i	$-0.241889\pi$
$71$	12.2161	1.44978	0.724892	+	0.688862i	$0.241889\pi$
$72$	0	0
$73$	−3.90127	−0.456609	−0.228305	−	0.973590i	$-0.573318\pi$
$73$	−3.90127	−0.456609	−0.228305	+	0.973590i	$0.573318\pi$
$74$	0	0
$75$	−11.8787	−1.37163
$76$	0	0
$77$	−0.501132	−0.0571093
$78$	0	0
$79$	−11.7440	−1.32131	−0.660653	−	0.750691i	$-0.729721\pi$
$79$	−11.7440	−1.32131	−0.660653	+	0.750691i	$0.729721\pi$
$80$	0	0
$81$	−9.88939	−1.09882
$82$	0	0
$83$	7.31907	0.803372	0.401686	−	0.915778i	$-0.368424\pi$
$83$	7.31907	0.803372	0.401686	+	0.915778i	$0.368424\pi$
$84$	0	0
$85$	0.640167	0.0694358
$86$	0	0
$87$	8.70263	0.933020
$88$	0	0
$89$	−3.04701	−0.322983	−0.161491	−	0.986874i	$-0.551630\pi$
$89$	−3.04701	−0.322983	−0.161491	+	0.986874i	$0.551630\pi$
$90$	0	0
$91$	1.20370	0.126182
$92$	0	0
$93$	9.31181	0.965590
$94$	0	0
$95$	−0.0577851	−0.00592863
$96$	0	0
$97$	4.49485	0.456383	0.228191	−	0.973616i	$-0.426719\pi$
$97$	4.49485	0.456383	0.228191	+	0.973616i	$0.426719\pi$
$98$	0	0
$99$	5.65388	0.568236
$100$	0	0
$101$	4.04504	0.402496	0.201248	−	0.979540i	$-0.435500\pi$
$101$	4.04504	0.402496	0.201248	+	0.979540i	$0.435500\pi$
$102$	0	0
$103$	−18.6554	−1.83817	−0.919085	−	0.394059i	$-0.871070\pi$
$103$	−18.6554	−1.83817	−0.919085	+	0.394059i	$0.871070\pi$
$104$	0	0
$105$	0.0558572	0.00545110
$106$	0	0
$107$	−6.34320	−0.613220	−0.306610	−	0.951835i	$-0.599195\pi$
$107$	−6.34320	−0.613220	−0.306610	+	0.951835i	$0.599195\pi$
$108$	0	0
$109$	−16.4103	−1.57182	−0.785909	−	0.618342i	$-0.787805\pi$
$109$	−16.4103	−1.57182	−0.785909	+	0.618342i	$0.787805\pi$
$110$	0	0
$111$	16.8823	1.60240
$112$	0	0
$113$	−14.7067	−1.38349	−0.691743	−	0.722143i	$-0.743157\pi$
$113$	−14.7067	−1.38349	−0.691743	+	0.722143i	$0.743157\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13.5804	−1.25551
$118$	0	0
$119$	1.52387	0.139693
$120$	0	0
$121$	−6.50400	−0.591273
$122$	0	0
$123$	−18.0427	−1.62686
$124$	0	0
$125$	0.991872	0.0887157
$126$	0	0
$127$	−6.10279	−0.541535	−0.270767	−	0.962645i	$-0.587277\pi$
$127$	−6.10279	−0.541535	−0.270767	+	0.962645i	$0.587277\pi$
$128$	0	0
$129$	25.5448	2.24909
$130$	0	0
$131$	−13.2167	−1.15475	−0.577376	−	0.816478i	$-0.695923\pi$
$131$	−13.2167	−1.15475	−0.577376	+	0.816478i	$0.695923\pi$
$132$	0	0
$133$	−0.137554	−0.0119274
$134$	0	0
$135$	0.0788313	0.00678472
$136$	0	0
$137$	−22.2634	−1.90209	−0.951047	−	0.309046i	$-0.899990\pi$
$137$	−22.2634	−1.90209	−0.951047	+	0.309046i	$0.899990\pi$
$138$	0	0
$139$	−10.8927	−0.923911	−0.461955	−	0.886903i	$-0.652852\pi$
$139$	−10.8927	−0.923911	−0.461955	+	0.886903i	$0.652852\pi$
$140$	0	0
$141$	−25.8752	−2.17909
$142$	0	0
$143$	−10.7992	−0.903078
$144$	0	0
$145$	−0.362977	−0.0301436
$146$	0	0
$147$	−16.5301	−1.36338
$148$	0	0
$149$	−12.4169	−1.01723	−0.508616	−	0.860994i	$-0.669843\pi$
$149$	−12.4169	−1.01723	−0.508616	+	0.860994i	$0.669843\pi$
$150$	0	0
$151$	5.99101	0.487541	0.243771	−	0.969833i	$-0.421616\pi$
$151$	5.99101	0.487541	0.243771	+	0.969833i	$0.421616\pi$
$152$	0	0
$153$	−17.1926	−1.38994
$154$	0	0
$155$	−0.388385	−0.0311958
$156$	0	0
$157$	24.8510	1.98333	0.991664	−	0.128852i	$-0.0411291\pi$
$157$	24.8510	1.98333	0.991664	+	0.128852i	$0.0411291\pi$
$158$	0	0
$159$	15.5700	1.23478
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.20270	−0.564159	−0.282080	−	0.959391i	$-0.591024\pi$
$163$	−7.20270	−0.564159	−0.282080	+	0.959391i	$0.591024\pi$
$164$	0	0
$165$	−0.501132	−0.0390131
$166$	0	0
$167$	−1.92084	−0.148639	−0.0743196	−	0.997234i	$-0.523678\pi$
$167$	−1.92084	−0.148639	−0.0743196	+	0.997234i	$0.523678\pi$
$168$	0	0
$169$	12.9394	0.995340
$170$	0	0
$171$	1.55191	0.118677
$172$	0	0
$173$	5.01664	0.381408	0.190704	−	0.981648i	$-0.438923\pi$
$173$	5.01664	0.381408	0.190704	+	0.981648i	$0.438923\pi$
$174$	0	0
$175$	1.17938	0.0891525
$176$	0	0
$177$	−21.1626	−1.59068
$178$	0	0
$179$	−13.0341	−0.974217	−0.487109	−	0.873341i	$-0.661948\pi$
$179$	−13.0341	−0.974217	−0.487109	+	0.873341i	$0.661948\pi$
$180$	0	0
$181$	11.3024	0.840103	0.420051	−	0.907500i	$-0.362012\pi$
$181$	11.3024	0.840103	0.420051	+	0.907500i	$0.362012\pi$
$182$	0	0
$183$	−11.4107	−0.843503
$184$	0	0
$185$	−0.704143	−0.0517696
$186$	0	0
$187$	−13.6717	−0.999773
$188$	0	0
$189$	0.187653	0.0136497
$190$	0	0
$191$	20.2168	1.46284	0.731418	−	0.681930i	$-0.238859\pi$
$191$	20.2168	1.46284	0.731418	+	0.681930i	$0.238859\pi$
$192$	0	0
$193$	10.2442	0.737393	0.368696	−	0.929550i	$-0.379804\pi$
$193$	10.2442	0.737393	0.368696	+	0.929550i	$0.379804\pi$
$194$	0	0
$195$	1.20370	0.0861990
$196$	0	0
$197$	−0.242683	−0.0172904	−0.00864522	−	0.999963i	$-0.502752\pi$
$197$	−0.242683	−0.0172904	−0.00864522	+	0.999963i	$0.502752\pi$
$198$	0	0
$199$	9.47032	0.671334	0.335667	−	0.941981i	$-0.391038\pi$
$199$	9.47032	0.671334	0.335667	+	0.941981i	$0.391038\pi$
$200$	0	0
$201$	7.85490	0.554042
$202$	0	0
$203$	−0.864041	−0.0606438
$204$	0	0
$205$	0.752541	0.0525598
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.23409	0.0853635
$210$	0	0
$211$	0.784217	0.0539877	0.0269939	−	0.999636i	$-0.491407\pi$
$211$	0.784217	0.0539877	0.0269939	+	0.999636i	$0.491407\pi$
$212$	0	0
$213$	29.0796	1.99250
$214$	0	0
$215$	−1.06544	−0.0726626
$216$	0	0
$217$	−0.924524	−0.0627607
$218$	0	0
$219$	−9.28671	−0.627538
$220$	0	0
$221$	32.8390	2.20899
$222$	0	0
$223$	−26.4758	−1.77295	−0.886476	−	0.462774i	$-0.846854\pi$
$223$	−26.4758	−1.77295	−0.886476	+	0.462774i	$0.846854\pi$
$224$	0	0
$225$	−13.3060	−0.887065
$226$	0	0
$227$	−21.2261	−1.40883	−0.704414	−	0.709789i	$-0.748790\pi$
$227$	−21.2261	−1.40883	−0.704414	+	0.709789i	$0.748790\pi$
$228$	0	0
$229$	10.8477	0.716839	0.358419	−	0.933561i	$-0.383316\pi$
$229$	10.8477	0.716839	0.358419	+	0.933561i	$0.383316\pi$
$230$	0	0
$231$	−1.19291	−0.0784878
$232$	0	0
$233$	7.94236	0.520321	0.260161	−	0.965565i	$-0.416224\pi$
$233$	7.94236	0.520321	0.260161	+	0.965565i	$0.416224\pi$
$234$	0	0
$235$	1.07923	0.0704009
$236$	0	0
$237$	−27.9559	−1.81593
$238$	0	0
$239$	5.23432	0.338580	0.169290	−	0.985566i	$-0.445853\pi$
$239$	5.23432	0.338580	0.169290	+	0.985566i	$0.445853\pi$
$240$	0	0
$241$	15.7419	1.01403	0.507014	−	0.861938i	$-0.330749\pi$
$241$	15.7419	1.01403	0.507014	+	0.861938i	$0.330749\pi$
$242$	0	0
$243$	−21.1590	−1.35735
$244$	0	0
$245$	0.689450	0.0440473
$246$	0	0
$247$	−2.96423	−0.188610
$248$	0	0
$249$	17.4225	1.10411
$250$	0	0
$251$	22.2514	1.40449	0.702247	−	0.711933i	$-0.252180\pi$
$251$	22.2514	1.40449	0.702247	+	0.711933i	$0.252180\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.52387	0.0954286
$256$	0	0
$257$	11.5977	0.723443	0.361721	−	0.932286i	$-0.382189\pi$
$257$	11.5977	0.723443	0.361721	+	0.932286i	$0.382189\pi$
$258$	0	0
$259$	−1.67616	−0.104152
$260$	0	0
$261$	9.74829	0.603404
$262$	0	0
$263$	0.242289	0.0149402	0.00747008	−	0.999972i	$-0.497622\pi$
$263$	0.242289	0.0149402	0.00747008	+	0.999972i	$0.497622\pi$
$264$	0	0
$265$	−0.649405	−0.0398927
$266$	0	0
$267$	−7.25320	−0.443889
$268$	0	0
$269$	−3.21805	−0.196208	−0.0981039	−	0.995176i	$-0.531278\pi$
$269$	−3.21805	−0.196208	−0.0981039	+	0.995176i	$0.531278\pi$
$270$	0	0
$271$	−21.4031	−1.30015	−0.650074	−	0.759871i	$-0.725262\pi$
$271$	−21.4031	−1.30015	−0.650074	+	0.759871i	$0.725262\pi$
$272$	0	0
$273$	2.86533	0.173418
$274$	0	0
$275$	−10.5810	−0.638057
$276$	0	0
$277$	15.9502	0.958353	0.479176	−	0.877719i	$-0.340935\pi$
$277$	15.9502	0.958353	0.479176	+	0.877719i	$0.340935\pi$
$278$	0	0
$279$	10.4307	0.624468
$280$	0	0
$281$	−10.6086	−0.632853	−0.316427	−	0.948617i	$-0.602483\pi$
$281$	−10.6086	−0.632853	−0.316427	+	0.948617i	$0.602483\pi$
$282$	0	0
$283$	26.3547	1.56663	0.783313	−	0.621628i	$-0.213528\pi$
$283$	26.3547	1.56663	0.783313	+	0.621628i	$0.213528\pi$
$284$	0	0
$285$	−0.137554	−0.00814797
$286$	0	0
$287$	1.79137	0.105741
$288$	0	0
$289$	24.5737	1.44551
$290$	0	0
$291$	10.6997	0.627226
$292$	0	0
$293$	1.33270	0.0778573	0.0389287	−	0.999242i	$-0.487605\pi$
$293$	1.33270	0.0778573	0.0389287	+	0.999242i	$0.487605\pi$
$294$	0	0
$295$	0.882668	0.0513909
$296$	0	0
$297$	−1.68356	−0.0976899
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2.53621	−0.146185
$302$	0	0
$303$	9.62894	0.553168
$304$	0	0
$305$	0.475927	0.0272515
$306$	0	0
$307$	−24.5255	−1.39975	−0.699873	−	0.714268i	$-0.746760\pi$
$307$	−24.5255	−1.39975	−0.699873	+	0.714268i	$0.746760\pi$
$308$	0	0
$309$	−44.4079	−2.52628
$310$	0	0
$311$	−2.30695	−0.130815	−0.0654077	−	0.997859i	$-0.520835\pi$
$311$	−2.30695	−0.130815	−0.0654077	+	0.997859i	$0.520835\pi$
$312$	0	0
$313$	−12.3019	−0.695347	−0.347673	−	0.937616i	$-0.613028\pi$
$313$	−12.3019	−0.695347	−0.347673	+	0.937616i	$0.613028\pi$
$314$	0	0
$315$	0.0625687	0.00352534
$316$	0	0
$317$	25.6309	1.43958	0.719788	−	0.694194i	$-0.244239\pi$
$317$	25.6309	1.43958	0.719788	+	0.694194i	$0.244239\pi$
$318$	0	0
$319$	7.75190	0.434023
$320$	0	0
$321$	−15.0996	−0.842775
$322$	0	0
$323$	−3.75268	−0.208805
$324$	0	0
$325$	25.4152	1.40978
$326$	0	0
$327$	−39.0635	−2.16022
$328$	0	0
$329$	2.56902	0.141635
$330$	0	0
$331$	−1.76844	−0.0972023	−0.0486011	−	0.998818i	$-0.515476\pi$
$331$	−1.76844	−0.0972023	−0.0486011	+	0.998818i	$0.515476\pi$
$332$	0	0
$333$	18.9108	1.03631
$334$	0	0
$335$	−0.327619	−0.0178997
$336$	0	0
$337$	5.45219	0.296999	0.148500	−	0.988912i	$-0.452556\pi$
$337$	5.45219	0.296999	0.148500	+	0.988912i	$0.452556\pi$
$338$	0	0
$339$	−35.0082	−1.90138
$340$	0	0
$341$	8.29453	0.449174
$342$	0	0
$343$	3.29558	0.177944
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.5002	−1.31524	−0.657619	−	0.753351i	$-0.728436\pi$
$347$	−24.5002	−1.31524	−0.657619	+	0.753351i	$0.728436\pi$
$348$	0	0
$349$	15.7494	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$349$	15.7494	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$350$	0	0
$351$	4.04385	0.215845
$352$	0	0
$353$	34.2192	1.82130	0.910652	−	0.413173i	$-0.135580\pi$
$353$	34.2192	1.82130	0.910652	+	0.413173i	$0.135580\pi$
$354$	0	0
$355$	−1.21288	−0.0643728
$356$	0	0
$357$	3.62747	0.191986
$358$	0	0
$359$	1.52796	0.0806429	0.0403214	−	0.999187i	$-0.487162\pi$
$359$	1.52796	0.0806429	0.0403214	+	0.999187i	$0.487162\pi$
$360$	0	0
$361$	−18.6613	−0.982172
$362$	0	0
$363$	−15.4823	−0.812612
$364$	0	0
$365$	0.387338	0.0202742
$366$	0	0
$367$	12.8439	0.670447	0.335223	−	0.942139i	$-0.391188\pi$
$367$	12.8439	0.670447	0.335223	+	0.942139i	$0.391188\pi$
$368$	0	0
$369$	−20.2106	−1.05212
$370$	0	0
$371$	−1.54586	−0.0802573
$372$	0	0
$373$	−35.1483	−1.81991	−0.909954	−	0.414709i	$-0.863883\pi$
$373$	−35.1483	−1.81991	−0.909954	+	0.414709i	$0.863883\pi$
$374$	0	0
$375$	2.36108	0.121926
$376$	0	0
$377$	−18.6198	−0.958969
$378$	0	0
$379$	−13.1007	−0.672938	−0.336469	−	0.941694i	$-0.609233\pi$
$379$	−13.1007	−0.672938	−0.336469	+	0.941694i	$0.609233\pi$
$380$	0	0
$381$	−14.5273	−0.744254
$382$	0	0
$383$	−10.3137	−0.527006	−0.263503	−	0.964659i	$-0.584878\pi$
$383$	−10.3137	−0.527006	−0.263503	+	0.964659i	$0.584878\pi$
$384$	0	0
$385$	0.0497549	0.00253575
$386$	0	0
$387$	28.6141	1.45454
$388$	0	0
$389$	2.12314	0.107648	0.0538238	−	0.998550i	$-0.482859\pi$
$389$	2.12314	0.107648	0.0538238	+	0.998550i	$0.482859\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−31.4616	−1.58703
$394$	0	0
$395$	1.16601	0.0586682
$396$	0	0
$397$	19.6886	0.988140	0.494070	−	0.869422i	$-0.335509\pi$
$397$	19.6886	0.988140	0.494070	+	0.869422i	$0.335509\pi$
$398$	0	0
$399$	−0.327437	−0.0163923
$400$	0	0
$401$	17.1764	0.857749	0.428874	−	0.903364i	$-0.358910\pi$
$401$	17.1764	0.857749	0.428874	+	0.903364i	$0.358910\pi$
$402$	0	0
$403$	−19.9232	−0.992444
$404$	0	0
$405$	0.981869	0.0487894
$406$	0	0
$407$	15.0380	0.745405
$408$	0	0
$409$	−21.9967	−1.08766	−0.543832	−	0.839194i	$-0.683027\pi$
$409$	−21.9967	−1.08766	−0.543832	+	0.839194i	$0.683027\pi$
$410$	0	0
$411$	−52.9966	−2.61413
$412$	0	0
$413$	2.10113	0.103390
$414$	0	0
$415$	−0.726674	−0.0356710
$416$	0	0
$417$	−25.9294	−1.26977
$418$	0	0
$419$	0.397751	0.0194314	0.00971571	−	0.999953i	$-0.496907\pi$
$419$	0.397751	0.0194314	0.00971571	+	0.999953i	$0.496907\pi$
$420$	0	0
$421$	18.2499	0.889447	0.444723	−	0.895668i	$-0.353302\pi$
$421$	18.2499	0.889447	0.444723	+	0.895668i	$0.353302\pi$
$422$	0	0
$423$	−28.9842	−1.40926
$424$	0	0
$425$	32.1753	1.56073
$426$	0	0
$427$	1.13291	0.0548255
$428$	0	0
$429$	−25.7068	−1.24114
$430$	0	0
$431$	33.0213	1.59058	0.795290	−	0.606229i	$-0.207319\pi$
$431$	33.0213	1.59058	0.795290	+	0.606229i	$0.207319\pi$
$432$	0	0
$433$	−7.05797	−0.339184	−0.169592	−	0.985514i	$-0.554245\pi$
$433$	−7.05797	−0.339184	−0.169592	+	0.985514i	$0.554245\pi$
$434$	0	0
$435$	−0.864041	−0.0414276
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−19.5722	−0.934132	−0.467066	−	0.884222i	$-0.654689\pi$
$439$	−19.5722	−0.934132	−0.467066	+	0.884222i	$0.654689\pi$
$440$	0	0
$441$	−18.5162	−0.881725
$442$	0	0
$443$	33.7424	1.60315	0.801574	−	0.597895i	$-0.203996\pi$
$443$	33.7424	1.60315	0.801574	+	0.597895i	$0.203996\pi$
$444$	0	0
$445$	0.302523	0.0143410
$446$	0	0
$447$	−29.5576	−1.39802
$448$	0	0
$449$	−38.3502	−1.80986	−0.904929	−	0.425562i	$-0.860077\pi$
$449$	−38.3502	−1.80986	−0.904929	+	0.425562i	$0.860077\pi$
$450$	0	0
$451$	−16.0716	−0.756783
$452$	0	0
$453$	14.2612	0.670049
$454$	0	0
$455$	−0.119510	−0.00560270
$456$	0	0
$457$	−11.3553	−0.531178	−0.265589	−	0.964086i	$-0.585566\pi$
$457$	−11.3553	−0.531178	−0.265589	+	0.964086i	$0.585566\pi$
$458$	0	0
$459$	5.11946	0.238956
$460$	0	0
$461$	0.344483	0.0160442	0.00802209	−	0.999968i	$-0.497446\pi$
$461$	0.344483	0.0160442	0.00802209	+	0.999968i	$0.497446\pi$
$462$	0	0
$463$	−18.4711	−0.858427	−0.429214	−	0.903203i	$-0.641209\pi$
$463$	−18.4711	−0.858427	−0.429214	+	0.903203i	$0.641209\pi$
$464$	0	0
$465$	−0.924524	−0.0428738
$466$	0	0
$467$	20.0220	0.926507	0.463253	−	0.886226i	$-0.346682\pi$
$467$	20.0220	0.926507	0.463253	+	0.886226i	$0.346682\pi$
$468$	0	0
$469$	−0.779874	−0.0360112
$470$	0	0
$471$	59.1561	2.72577
$472$	0	0
$473$	22.7541	1.04623
$474$	0	0
$475$	−2.90433	−0.133260
$476$	0	0
$477$	17.4408	0.798558
$478$	0	0
$479$	−31.8289	−1.45430	−0.727149	−	0.686479i	$-0.759155\pi$
$479$	−31.8289	−1.45430	−0.727149	+	0.686479i	$0.759155\pi$
$480$	0	0
$481$	−36.1208	−1.64696
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.446271	−0.0202641
$486$	0	0
$487$	29.2812	1.32686	0.663429	−	0.748239i	$-0.269100\pi$
$487$	29.2812	1.32686	0.663429	+	0.748239i	$0.269100\pi$
$488$	0	0
$489$	−17.1455	−0.775348
$490$	0	0
$491$	23.5596	1.06323	0.531616	−	0.846986i	$-0.321585\pi$
$491$	23.5596	1.06323	0.531616	+	0.846986i	$0.321585\pi$
$492$	0	0
$493$	−23.5724	−1.06165
$494$	0	0
$495$	−0.561346	−0.0252306
$496$	0	0
$497$	−2.88717	−0.129507
$498$	0	0
$499$	26.8538	1.20214	0.601071	−	0.799196i	$-0.294741\pi$
$499$	26.8538	1.20214	0.601071	+	0.799196i	$0.294741\pi$
$500$	0	0
$501$	−4.57243	−0.204281
$502$	0	0
$503$	31.2902	1.39516	0.697581	−	0.716506i	$-0.254260\pi$
$503$	31.2902	1.39516	0.697581	+	0.716506i	$0.254260\pi$
$504$	0	0
$505$	−0.401612	−0.0178715
$506$	0	0
$507$	30.8014	1.36794
$508$	0	0
$509$	−9.24251	−0.409667	−0.204834	−	0.978797i	$-0.565665\pi$
$509$	−9.24251	−0.409667	−0.204834	+	0.978797i	$0.565665\pi$
$510$	0	0
$511$	0.922031	0.0407883
$512$	0	0
$513$	−0.462112	−0.0204028
$514$	0	0
$515$	1.85220	0.0816177
$516$	0	0
$517$	−23.0484	−1.01367
$518$	0	0
$519$	11.9418	0.524185
$520$	0	0
$521$	7.46630	0.327105	0.163552	−	0.986535i	$-0.447705\pi$
$521$	7.46630	0.327105	0.163552	+	0.986535i	$0.447705\pi$
$522$	0	0
$523$	24.3192	1.06340	0.531701	−	0.846932i	$-0.321553\pi$
$523$	24.3192	1.06340	0.531701	+	0.846932i	$0.321553\pi$
$524$	0	0
$525$	2.80742	0.122526
$526$	0	0
$527$	−25.2225	−1.09871
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−23.7054	−1.02873
$532$	0	0
$533$	38.6035	1.67210
$534$	0	0
$535$	0.629785	0.0272280
$536$	0	0
$537$	−31.0269	−1.33891
$538$	0	0
$539$	−14.7242	−0.634216
$540$	0	0
$541$	−4.78128	−0.205563	−0.102782	−	0.994704i	$-0.532774\pi$
$541$	−4.78128	−0.205563	−0.102782	+	0.994704i	$0.532774\pi$
$542$	0	0
$543$	26.9046	1.15459
$544$	0	0
$545$	1.62929	0.0697913
$546$	0	0
$547$	15.8829	0.679106	0.339553	−	0.940587i	$-0.389724\pi$
$547$	15.8829	0.679106	0.339553	+	0.940587i	$0.389724\pi$
$548$	0	0
$549$	−12.7818	−0.545512
$550$	0	0
$551$	2.12778	0.0906466
$552$	0	0
$553$	2.77560	0.118030
$554$	0	0
$555$	−1.67616	−0.0711491
$556$	0	0
$557$	14.1802	0.600835	0.300418	−	0.953808i	$-0.402874\pi$
$557$	14.1802	0.600835	0.300418	+	0.953808i	$0.402874\pi$
$558$	0	0
$559$	−54.6546	−2.31164
$560$	0	0
$561$	−32.5445	−1.37403
$562$	0	0
$563$	−5.30011	−0.223373	−0.111686	−	0.993743i	$-0.535625\pi$
$563$	−5.30011	−0.223373	−0.111686	+	0.993743i	$0.535625\pi$
$564$	0	0
$565$	1.46015	0.0614291
$566$	0	0
$567$	2.33727	0.0981561
$568$	0	0
$569$	26.5092	1.11132	0.555661	−	0.831409i	$-0.312465\pi$
$569$	26.5092	1.11132	0.555661	+	0.831409i	$0.312465\pi$
$570$	0	0
$571$	−13.9334	−0.583096	−0.291548	−	0.956556i	$-0.594170\pi$
$571$	−13.9334	−0.583096	−0.291548	+	0.956556i	$0.594170\pi$
$572$	0	0
$573$	48.1247	2.01044
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.40068	0.224833	0.112417	−	0.993661i	$-0.464141\pi$
$577$	5.40068	0.224833	0.112417	+	0.993661i	$0.464141\pi$
$578$	0	0
$579$	24.3856	1.01343
$580$	0	0
$581$	−1.72980	−0.0717641
$582$	0	0
$583$	13.8690	0.574395
$584$	0	0
$585$	1.34833	0.0557468
$586$	0	0
$587$	−18.8138	−0.776527	−0.388263	−	0.921548i	$-0.626925\pi$
$587$	−18.8138	−0.776527	−0.388263	+	0.921548i	$0.626925\pi$
$588$	0	0
$589$	2.27673	0.0938109
$590$	0	0
$591$	−0.577690	−0.0237630
$592$	0	0
$593$	−21.0990	−0.866434	−0.433217	−	0.901290i	$-0.642622\pi$
$593$	−21.0990	−0.866434	−0.433217	+	0.901290i	$0.642622\pi$
$594$	0	0
$595$	−0.151298	−0.00620260
$596$	0	0
$597$	22.5435	0.922642
$598$	0	0
$599$	−19.2225	−0.785410	−0.392705	−	0.919664i	$-0.628461\pi$
$599$	−19.2225	−0.785410	−0.392705	+	0.919664i	$0.628461\pi$
$600$	0	0
$601$	31.8298	1.29836	0.649182	−	0.760633i	$-0.275111\pi$
$601$	31.8298	1.29836	0.649182	+	0.760633i	$0.275111\pi$
$602$	0	0
$603$	8.79870	0.358311
$604$	0	0
$605$	0.645750	0.0262535
$606$	0	0
$607$	−35.2173	−1.42942	−0.714712	−	0.699418i	$-0.753442\pi$
$607$	−35.2173	−1.42942	−0.714712	+	0.699418i	$0.753442\pi$
$608$	0	0
$609$	−2.05679	−0.0833453
$610$	0	0
$611$	55.3616	2.23969
$612$	0	0
$613$	41.4856	1.67559	0.837795	−	0.545985i	$-0.183845\pi$
$613$	41.4856	1.67559	0.837795	+	0.545985i	$0.183845\pi$
$614$	0	0
$615$	1.79137	0.0722351
$616$	0	0
$617$	0.478933	0.0192811	0.00964056	−	0.999954i	$-0.496931\pi$
$617$	0.478933	0.0192811	0.00964056	+	0.999954i	$0.496931\pi$
$618$	0	0
$619$	2.35398	0.0946144	0.0473072	−	0.998880i	$-0.484936\pi$
$619$	2.35398	0.0946144	0.0473072	+	0.998880i	$0.484936\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.720135	0.0288516
$624$	0	0
$625$	24.8522	0.994089
$626$	0	0
$627$	2.93766	0.117319
$628$	0	0
$629$	−45.7284	−1.82331
$630$	0	0
$631$	−29.2574	−1.16472	−0.582358	−	0.812932i	$-0.697870\pi$
$631$	−29.2574	−1.16472	−0.582358	+	0.812932i	$0.697870\pi$
$632$	0	0
$633$	1.86677	0.0741976
$634$	0	0
$635$	0.605915	0.0240450
$636$	0	0
$637$	35.3670	1.40129
$638$	0	0
$639$	32.5736	1.28859
$640$	0	0
$641$	−13.1285	−0.518544	−0.259272	−	0.965804i	$-0.583483\pi$
$641$	−13.1285	−0.518544	−0.259272	+	0.965804i	$0.583483\pi$
$642$	0	0
$643$	−29.5408	−1.16497	−0.582487	−	0.812840i	$-0.697920\pi$
$643$	−29.5408	−1.16497	−0.582487	+	0.812840i	$0.697920\pi$
$644$	0	0
$645$	−2.53621	−0.0998633
$646$	0	0
$647$	37.7876	1.48558	0.742792	−	0.669523i	$-0.233501\pi$
$647$	37.7876	1.48558	0.742792	+	0.669523i	$0.233501\pi$
$648$	0	0
$649$	−18.8507	−0.739953
$650$	0	0
$651$	−2.20076	−0.0862548
$652$	0	0
$653$	−36.6336	−1.43358	−0.716791	−	0.697288i	$-0.754390\pi$
$653$	−36.6336	−1.43358	−0.716791	+	0.697288i	$0.754390\pi$
$654$	0	0
$655$	1.31223	0.0512729
$656$	0	0
$657$	−10.4026	−0.405842
$658$	0	0
$659$	13.4095	0.522361	0.261181	−	0.965290i	$-0.415888\pi$
$659$	13.4095	0.522361	0.261181	+	0.965290i	$0.415888\pi$
$660$	0	0
$661$	−40.8965	−1.59069	−0.795344	−	0.606158i	$-0.792710\pi$
$661$	−40.8965	−1.59069	−0.795344	+	0.606158i	$0.792710\pi$
$662$	0	0
$663$	78.1709	3.03590
$664$	0	0
$665$	0.0136570	0.000529596 0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−63.0239	−2.43664
$670$	0	0
$671$	−10.1641	−0.392382
$672$	0	0
$673$	33.4358	1.28886	0.644429	−	0.764665i	$-0.277095\pi$
$673$	33.4358	1.28886	0.644429	+	0.764665i	$0.277095\pi$
$674$	0	0
$675$	3.96212	0.152502
$676$	0	0
$677$	−21.8451	−0.839576	−0.419788	−	0.907622i	$-0.637896\pi$
$677$	−21.8451	−0.839576	−0.419788	+	0.907622i	$0.637896\pi$
$678$	0	0
$679$	−1.06232	−0.0407680
$680$	0	0
$681$	−50.5274	−1.93621
$682$	0	0
$683$	−39.9627	−1.52913	−0.764566	−	0.644546i	$-0.777047\pi$
$683$	−39.9627	−1.52913	−0.764566	+	0.644546i	$0.777047\pi$
$684$	0	0
$685$	2.21043	0.0844561
$686$	0	0
$687$	25.8223	0.985182
$688$	0	0
$689$	−33.3129	−1.26912
$690$	0	0
$691$	19.9533	0.759059	0.379529	−	0.925180i	$-0.376086\pi$
$691$	19.9533	0.759059	0.379529	+	0.925180i	$0.376086\pi$
$692$	0	0
$693$	−1.33624	−0.0507597
$694$	0	0
$695$	1.08149	0.0410231
$696$	0	0
$697$	48.8715	1.85114
$698$	0	0
$699$	18.9062	0.715100
$700$	0	0
$701$	−18.5699	−0.701377	−0.350688	−	0.936492i	$-0.614052\pi$
$701$	−18.5699	−0.701377	−0.350688	+	0.936492i	$0.614052\pi$
$702$	0	0
$703$	4.12771	0.155679
$704$	0	0
$705$	2.56902	0.0967550
$706$	0	0
$707$	−0.956009	−0.0359544
$708$	0	0
$709$	−8.53099	−0.320388	−0.160194	−	0.987086i	$-0.551212\pi$
$709$	−8.53099	−0.320388	−0.160194	+	0.987086i	$0.551212\pi$
$710$	0	0
$711$	−31.3149	−1.17440
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.07220	0.0400981
$716$	0	0
$717$	12.4599	0.465325
$718$	0	0
$719$	9.86849	0.368032	0.184016	−	0.982923i	$-0.441090\pi$
$719$	9.86849	0.368032	0.184016	+	0.982923i	$0.441090\pi$
$720$	0	0
$721$	4.40904	0.164201
$722$	0	0
$723$	37.4726	1.39362
$724$	0	0
$725$	−18.2435	−0.677546
$726$	0	0
$727$	26.4891	0.982427	0.491213	−	0.871039i	$-0.336554\pi$
$727$	26.4891	0.982427	0.491213	+	0.871039i	$0.336554\pi$
$728$	0	0
$729$	−20.6995	−0.766647
$730$	0	0
$731$	−69.1919	−2.55916
$732$	0	0
$733$	−32.0429	−1.18353	−0.591766	−	0.806110i	$-0.701569\pi$
$733$	−32.0429	−1.18353	−0.591766	+	0.806110i	$0.701569\pi$
$734$	0	0
$735$	1.64119	0.0605361
$736$	0	0
$737$	6.99678	0.257730
$738$	0	0
$739$	−12.3822	−0.455487	−0.227744	−	0.973721i	$-0.573135\pi$
$739$	−12.3822	−0.455487	−0.227744	+	0.973721i	$0.573135\pi$
$740$	0	0
$741$	−7.05615	−0.259214
$742$	0	0
$743$	−40.5823	−1.48882	−0.744410	−	0.667723i	$-0.767269\pi$
$743$	−40.5823	−1.48882	−0.744410	+	0.667723i	$0.767269\pi$
$744$	0	0
$745$	1.23281	0.0451667
$746$	0	0
$747$	19.5159	0.714051
$748$	0	0
$749$	1.49916	0.0547781
$750$	0	0
$751$	−27.7103	−1.01116	−0.505582	−	0.862779i	$-0.668722\pi$
$751$	−27.7103	−1.01116	−0.505582	+	0.862779i	$0.668722\pi$
$752$	0	0
$753$	52.9679	1.93026
$754$	0	0
$755$	−0.594817	−0.0216476
$756$	0	0
$757$	51.4698	1.87070	0.935350	−	0.353722i	$-0.115084\pi$
$757$	51.4698	1.87070	0.935350	+	0.353722i	$0.115084\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.2127	0.442710	0.221355	−	0.975193i	$-0.428952\pi$
$761$	12.2127	0.442710	0.221355	+	0.975193i	$0.428952\pi$
$762$	0	0
$763$	3.87842	0.140408
$764$	0	0
$765$	1.70697	0.0617157
$766$	0	0
$767$	45.2787	1.63492
$768$	0	0
$769$	8.39614	0.302773	0.151386	−	0.988475i	$-0.451626\pi$
$769$	8.39614	0.302773	0.151386	+	0.988475i	$0.451626\pi$
$770$	0	0
$771$	27.6074	0.994258
$772$	0	0
$773$	−47.5714	−1.71102	−0.855512	−	0.517783i	$-0.826758\pi$
$773$	−47.5714	−1.71102	−0.855512	+	0.517783i	$0.826758\pi$
$774$	0	0
$775$	−19.5205	−0.701198
$776$	0	0
$777$	−3.98999	−0.143140
$778$	0	0
$779$	−4.41142	−0.158056
$780$	0	0
$781$	25.9027	0.926873
$782$	0	0
$783$	−2.90275	−0.103736
$784$	0	0
$785$	−2.46734	−0.0880630
$786$	0	0
$787$	38.0529	1.35644	0.678220	−	0.734859i	$-0.262752\pi$
$787$	38.0529	1.35644	0.678220	+	0.734859i	$0.262752\pi$
$788$	0	0
$789$	0.576751	0.0205329
$790$	0	0
$791$	3.47579	0.123585
$792$	0	0
$793$	24.4139	0.866963
$794$	0	0
$795$	−1.54586	−0.0548262
$796$	0	0
$797$	34.9130	1.23668	0.618341	−	0.785910i	$-0.287805\pi$
$797$	34.9130	1.23668	0.618341	+	0.785910i	$0.287805\pi$
$798$	0	0
$799$	70.0870	2.47950
$800$	0	0
$801$	−8.12471	−0.287073
$802$	0	0
$803$	−8.27217	−0.291918
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.66034	−0.269657
$808$	0	0
$809$	0.218265	0.00767378	0.00383689	−	0.999993i	$-0.498779\pi$
$809$	0.218265	0.00767378	0.00383689	+	0.999993i	$0.498779\pi$
$810$	0	0
$811$	47.9443	1.68355	0.841776	−	0.539827i	$-0.181510\pi$
$811$	47.9443	1.68355	0.841776	+	0.539827i	$0.181510\pi$
$812$	0	0
$813$	−50.9487	−1.78685
$814$	0	0
$815$	0.715120	0.0250496
$816$	0	0
$817$	6.24566	0.218508
$818$	0	0
$819$	3.20962	0.112153
$820$	0	0
$821$	−21.2732	−0.742440	−0.371220	−	0.928545i	$-0.621060\pi$
$821$	−21.2732	−0.742440	−0.371220	+	0.928545i	$0.621060\pi$
$822$	0	0
$823$	39.8753	1.38996	0.694982	−	0.719027i	$-0.255412\pi$
$823$	39.8753	1.38996	0.694982	+	0.719027i	$0.255412\pi$
$824$	0	0
$825$	−25.1873	−0.876909
$826$	0	0
$827$	9.34574	0.324983	0.162492	−	0.986710i	$-0.448047\pi$
$827$	9.34574	0.324983	0.162492	+	0.986710i	$0.448047\pi$
$828$	0	0
$829$	13.0782	0.454225	0.227113	−	0.973869i	$-0.427071\pi$
$829$	13.0782	0.454225	0.227113	+	0.973869i	$0.427071\pi$
$830$	0	0
$831$	37.9683	1.31710
$832$	0	0
$833$	44.7742	1.55133
$834$	0	0
$835$	0.190711	0.00659982
$836$	0	0
$837$	−3.10594	−0.107357
$838$	0	0
$839$	23.6560	0.816696	0.408348	−	0.912826i	$-0.366105\pi$
$839$	23.6560	0.816696	0.408348	+	0.912826i	$0.366105\pi$
$840$	0	0
$841$	−15.6344	−0.539116
$842$	0	0
$843$	−25.2529	−0.869757
$844$	0	0
$845$	−1.28469	−0.0441947
$846$	0	0
$847$	1.53716	0.0528176
$848$	0	0
$849$	62.7356	2.15308
$850$	0	0
$851$	0	0
$852$	0	0
$853$	48.7271	1.66838	0.834192	−	0.551474i	$-0.185935\pi$
$853$	48.7271	1.66838	0.834192	+	0.551474i	$0.185935\pi$
$854$	0	0
$855$	−0.154081	−0.00526947
$856$	0	0
$857$	−41.7822	−1.42725	−0.713627	−	0.700526i	$-0.752949\pi$
$857$	−41.7822	−1.42725	−0.713627	+	0.700526i	$0.752949\pi$
$858$	0	0
$859$	−11.4386	−0.390280	−0.195140	−	0.980775i	$-0.562516\pi$
$859$	−11.4386	−0.390280	−0.195140	+	0.980775i	$0.562516\pi$
$860$	0	0
$861$	4.26424	0.145325
$862$	0	0
$863$	11.0784	0.377112	0.188556	−	0.982062i	$-0.439619\pi$
$863$	11.0784	0.377112	0.188556	+	0.982062i	$0.439619\pi$
$864$	0	0
$865$	−0.498077	−0.0169351
$866$	0	0
$867$	58.4959	1.98663
$868$	0	0
$869$	−24.9018	−0.844735
$870$	0	0
$871$	−16.8060	−0.569451
$872$	0	0
$873$	11.9853	0.405641
$874$	0	0
$875$	−0.234420	−0.00792485
$876$	0	0
$877$	16.1877	0.546620	0.273310	−	0.961926i	$-0.411881\pi$
$877$	16.1877	0.546620	0.273310	+	0.961926i	$0.411881\pi$
$878$	0	0
$879$	3.17241	0.107003
$880$	0	0
$881$	35.8636	1.20827	0.604137	−	0.796880i	$-0.293518\pi$
$881$	35.8636	1.20827	0.604137	+	0.796880i	$0.293518\pi$
$882$	0	0
$883$	16.5361	0.556485	0.278243	−	0.960511i	$-0.410248\pi$
$883$	16.5361	0.556485	0.278243	+	0.960511i	$0.410248\pi$
$884$	0	0
$885$	2.10113	0.0706287
$886$	0	0
$887$	32.2034	1.08129	0.540643	−	0.841252i	$-0.318181\pi$
$887$	32.2034	1.08129	0.540643	+	0.841252i	$0.318181\pi$
$888$	0	0
$889$	1.44234	0.0483745
$890$	0	0
$891$	−20.9692	−0.702496
$892$	0	0
$893$	−6.32646	−0.211707
$894$	0	0
$895$	1.29409	0.0432568
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.3012	0.476973
$900$	0	0
$901$	−42.1737	−1.40501
$902$	0	0
$903$	−6.03728	−0.200908
$904$	0	0
$905$	−1.12216	−0.0373019
$906$	0	0
$907$	−31.6218	−1.04999	−0.524993	−	0.851107i	$-0.675932\pi$
$907$	−31.6218	−1.04999	−0.524993	+	0.851107i	$0.675932\pi$
$908$	0	0
$909$	10.7859	0.357746
$910$	0	0
$911$	−33.2874	−1.10286	−0.551430	−	0.834221i	$-0.685918\pi$
$911$	−33.2874	−1.10286	−0.551430	+	0.834221i	$0.685918\pi$
$912$	0	0
$913$	15.5192	0.513610
$914$	0	0
$915$	1.13291	0.0374529
$916$	0	0
$917$	3.12366	0.103152
$918$	0	0
$919$	−13.3563	−0.440582	−0.220291	−	0.975434i	$-0.570701\pi$
$919$	−13.3563	−0.440582	−0.220291	+	0.975434i	$0.570701\pi$
$920$	0	0
$921$	−58.3813	−1.92373
$922$	0	0
$923$	−62.2175	−2.04792
$924$	0	0
$925$	−35.3907	−1.16364
$926$	0	0
$927$	−49.7437	−1.63380
$928$	0	0
$929$	−4.97427	−0.163201	−0.0816003	−	0.996665i	$-0.526003\pi$
$929$	−4.97427	−0.163201	−0.0816003	+	0.996665i	$0.526003\pi$
$930$	0	0
$931$	−4.04158	−0.132457
$932$	0	0
$933$	−5.49155	−0.179785
$934$	0	0
$935$	1.35739	0.0443915
$936$	0	0
$937$	−44.4008	−1.45051	−0.725256	−	0.688479i	$-0.758279\pi$
$937$	−44.4008	−1.45051	−0.725256	+	0.688479i	$0.758279\pi$
$938$	0	0
$939$	−29.2839	−0.955644
$940$	0	0
$941$	−31.2974	−1.02027	−0.510133	−	0.860096i	$-0.670404\pi$
$941$	−31.2974	−1.02027	−0.510133	+	0.860096i	$0.670404\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−0.0186311	−0.000606069 0
$946$	0	0
$947$	19.4869	0.633238	0.316619	−	0.948553i	$-0.397452\pi$
$947$	19.4869	0.633238	0.316619	+	0.948553i	$0.397452\pi$
$948$	0	0
$949$	19.8695	0.644990
$950$	0	0
$951$	61.0126	1.97847
$952$	0	0
$953$	29.9248	0.969360	0.484680	−	0.874692i	$-0.338936\pi$
$953$	29.9248	0.969360	0.484680	+	0.874692i	$0.338936\pi$
$954$	0	0
$955$	−2.00722	−0.0649523
$956$	0	0
$957$	18.4529	0.596496
$958$	0	0
$959$	5.26177	0.169911
$960$	0	0
$961$	−15.6977	−0.506377
$962$	0	0
$963$	−16.9138	−0.545041
$964$	0	0
$965$	−1.01709	−0.0327414
$966$	0	0
$967$	36.4381	1.17177	0.585885	−	0.810394i	$-0.300747\pi$
$967$	36.4381	1.17177	0.585885	+	0.810394i	$0.300747\pi$
$968$	0	0
$969$	−8.93299	−0.286969
$970$	0	0
$971$	−5.03709	−0.161648	−0.0808239	−	0.996728i	$-0.525755\pi$
$971$	−5.03709	−0.161648	−0.0808239	+	0.996728i	$0.525755\pi$
$972$	0	0
$973$	2.57441	0.0825316
$974$	0	0
$975$	60.4991	1.93752
$976$	0	0
$977$	−39.4069	−1.26074	−0.630370	−	0.776295i	$-0.717097\pi$
$977$	−39.4069	−1.26074	−0.630370	+	0.776295i	$0.717097\pi$
$978$	0	0
$979$	−6.46082	−0.206489
$980$	0	0
$981$	−43.7572	−1.39706
$982$	0	0
$983$	−46.9646	−1.49794	−0.748969	−	0.662605i	$-0.769451\pi$
$983$	−46.9646	−1.49794	−0.748969	+	0.662605i	$0.769451\pi$
$984$	0	0
$985$	0.0240948	0.000767724 0
$986$	0	0
$987$	6.11538	0.194655
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.43925	−0.268082	−0.134041	−	0.990976i	$-0.542795\pi$
$991$	−8.43925	−0.268082	−0.134041	+	0.990976i	$0.542795\pi$
$992$	0	0
$993$	−4.20965	−0.133589
$994$	0	0
$995$	−0.940262	−0.0298083
$996$	0	0
$997$	30.4275	0.963648	0.481824	−	0.876268i	$-0.339974\pi$
$997$	30.4275	0.963648	0.481824	+	0.876268i	$0.339974\pi$
$998$	0	0
$999$	−5.63108	−0.178160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.cf.1.11		12
4.3	odd	2		4232.2.a.x.1.1	✓	12
23.22	odd	2	inner	8464.2.a.cf.1.12		12
92.91	even	2		4232.2.a.x.1.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.x.1.1	✓	12	4.3	odd	2
4232.2.a.x.1.2	yes	12	92.91	even	2
8464.2.a.cf.1.11		12	1.1	even	1	trivial
8464.2.a.cf.1.12		12	23.22	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+2.38043 q^{3} -0.0992850 q^{5} -0.236341 q^{7} +2.66645 q^{9} +2.12038 q^{11} -5.09308 q^{13} -0.236341 q^{15} -6.44776 q^{17} +0.582013 q^{19} -0.562594 q^{21} -4.99014 q^{25} -0.793990 q^{27} +3.65591 q^{29} +3.91182 q^{31} +5.04741 q^{33} +0.0234651 q^{35} +7.09213 q^{37} -12.1237 q^{39} -7.57960 q^{41} +10.7311 q^{43} -0.264739 q^{45} -10.8700 q^{47} -6.94414 q^{49} -15.3485 q^{51} +6.54082 q^{53} -0.210522 q^{55} +1.38544 q^{57} -8.89024 q^{59} -4.79354 q^{61} -0.630192 q^{63} +0.505666 q^{65} +3.29978 q^{67} +12.2161 q^{71} -3.90127 q^{73} -11.8787 q^{75} -0.501132 q^{77} -11.7440 q^{79} -9.88939 q^{81} +7.31907 q^{83} +0.640167 q^{85} +8.70263 q^{87} -3.04701 q^{89} +1.20370 q^{91} +9.31181 q^{93} -0.0577851 q^{95} +4.49485 q^{97} +5.65388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} - 8 q^{27} - 8 q^{31} - 56 q^{35} - 64 q^{39} - 40 q^{41} - 32 q^{47} + 28 q^{49} - 64 q^{55} - 60 q^{59} + 32 q^{71} + 28 q^{73} - 16 q^{75} + 24 q^{77}+ \cdots - 16 q^{95}+O(q^{100}) \) 12 * q - 8 * q^3 + 8 * q^9 + 16 * q^13 + 4 * q^25 - 8 * q^27 - 8 * q^31 - 56 * q^35 - 64 * q^39 - 40 * q^41 - 32 * q^47 + 28 * q^49 - 64 * q^55 - 60 * q^59 + 32 * q^71 + 28 * q^73 - 16 * q^75 + 24 * q^77 + 4 * q^81 + 64 * q^85 + 16 * q^87 + 24 * q^93 - 16 * q^95

Introduction

L-functions

Modular forms

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