LMFDB - Embedded newform 5292.2.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.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:	$42.2568327497$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.28825$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} +O(q^{10})$	$q+2.23607 q^{5} +3.70246 q^{11} -6.86474 q^{13} -7.47214 q^{17} +5.45052 q^{19} +3.16228 q^{23} -3.70246 q^{29} -6.86474 q^{31} -8.70820 q^{37} +2.23607 q^{41} -4.70820 q^{43} -13.4721 q^{47} +5.32300 q^{53} +8.27895 q^{55} +5.94427 q^{59} -0.206331 q^{61} -15.3500 q^{65} -9.70820 q^{67} +5.24419 q^{71} +1.41421 q^{73} +5.00000 q^{79} -8.23607 q^{83} -16.7082 q^{85} -11.2361 q^{89} +12.1877 q^{95} +0.206331 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{17} - 8 q^{37} + 8 q^{43} - 36 q^{47} - 12 q^{59} - 12 q^{67} + 20 q^{79} - 24 q^{83} - 40 q^{85} - 36 q^{89}+O(q^{100}) $ 4 * q - 12 * q^17 - 8 * q^37 + 8 * q^43 - 36 * q^47 - 12 * q^59 - 12 * q^67 + 20 * q^79 - 24 * q^83 - 40 * q^85 - 36 * q^89

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.70246	1.11633	0.558167	−	0.829729i	$-0.311505\pi$
$11$	3.70246	1.11633	0.558167	+	0.829729i	$0.311505\pi$
$12$	0	0
$13$	−6.86474	−1.90394	−0.951968	−	0.306198i	$-0.900943\pi$
$13$	−6.86474	−1.90394	−0.951968	+	0.306198i	$0.900943\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.47214	−1.81226	−0.906130	−	0.423000i	$-0.860977\pi$
$17$	−7.47214	−1.81226	−0.906130	+	0.423000i	$0.860977\pi$
$18$	0	0
$19$	5.45052	1.25044	0.625218	−	0.780450i	$-0.285010\pi$
$19$	5.45052	1.25044	0.625218	+	0.780450i	$0.285010\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.16228	0.659380	0.329690	−	0.944089i	$-0.393056\pi$
$23$	3.16228	0.659380	0.329690	+	0.944089i	$0.393056\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.70246	−0.687529	−0.343765	−	0.939056i	$-0.611702\pi$
$29$	−3.70246	−0.687529	−0.343765	+	0.939056i	$0.611702\pi$
$30$	0	0
$31$	−6.86474	−1.23294	−0.616472	−	0.787377i	$-0.711438\pi$
$31$	−6.86474	−1.23294	−0.616472	+	0.787377i	$0.711438\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.23607	0.349215	0.174608	−	0.984638i	$-0.444134\pi$
$41$	2.23607	0.349215	0.174608	+	0.984638i	$0.444134\pi$
$42$	0	0
$43$	−4.70820	−0.717994	−0.358997	−	0.933339i	$-0.616881\pi$
$43$	−4.70820	−0.717994	−0.358997	+	0.933339i	$0.616881\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.4721	−1.96511	−0.982556	−	0.185964i	$-0.940459\pi$
$47$	−13.4721	−1.96511	−0.982556	+	0.185964i	$0.940459\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.32300	0.731171	0.365585	−	0.930778i	$-0.380869\pi$
$53$	5.32300	0.731171	0.365585	+	0.930778i	$0.380869\pi$
$54$	0	0
$55$	8.27895	1.11633
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.94427	0.773878	0.386939	−	0.922105i	$-0.373532\pi$
$59$	5.94427	0.773878	0.386939	+	0.922105i	$0.373532\pi$
$60$	0	0
$61$	−0.206331	−0.0264180	−0.0132090	−	0.999913i	$-0.504205\pi$
$61$	−0.206331	−0.0264180	−0.0132090	+	0.999913i	$0.504205\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.3500	−1.90394
$66$	0	0
$67$	−9.70820	−1.18605	−0.593023	−	0.805186i	$-0.702066\pi$
$67$	−9.70820	−1.18605	−0.593023	+	0.805186i	$0.702066\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.24419	0.622371	0.311186	−	0.950349i	$-0.399274\pi$
$71$	5.24419	0.622371	0.311186	+	0.950349i	$0.399274\pi$
$72$	0	0
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.23607	−0.904026	−0.452013	−	0.892011i	$-0.649294\pi$
$83$	−8.23607	−0.904026	−0.452013	+	0.892011i	$0.649294\pi$
$84$	0	0
$85$	−16.7082	−1.81226
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.2361	−1.19102	−0.595510	−	0.803348i	$-0.703050\pi$
$89$	−11.2361	−1.19102	−0.595510	+	0.803348i	$0.703050\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.1877	1.25044
$96$	0	0
$97$	0.206331	0.0209497	0.0104749	−	0.999945i	$-0.496666\pi$
$97$	0.206331	0.0209497	0.0104749	+	0.999945i	$0.496666\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.70820	0.368980	0.184490	−	0.982834i	$-0.440937\pi$
$101$	3.70820	0.368980	0.184490	+	0.982834i	$0.440937\pi$
$102$	0	0
$103$	9.89949	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$103$	9.89949	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.5672	−1.02157	−0.510785	−	0.859709i	$-0.670645\pi$
$107$	−10.5672	−1.02157	−0.510785	+	0.859709i	$0.670645\pi$
$108$	0	0
$109$	10.7082	1.02566	0.512830	−	0.858490i	$-0.328597\pi$
$109$	10.7082	1.02566	0.512830	+	0.858490i	$0.328597\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.5942	−1.93734	−0.968670	−	0.248351i	$-0.920111\pi$
$113$	−20.5942	−1.93734	−0.968670	+	0.248351i	$0.920111\pi$
$114$	0	0
$115$	7.07107	0.659380
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.70820	0.246200
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.1058	0.863399	0.431699	−	0.902017i	$-0.357914\pi$
$137$	10.1058	0.863399	0.431699	+	0.902017i	$0.357914\pi$
$138$	0	0
$139$	−3.03476	−0.257405	−0.128702	−	0.991683i	$-0.541081\pi$
$139$	−3.03476	−0.257405	−0.128702	+	0.991683i	$0.541081\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.4164	−2.12543
$144$	0	0
$145$	−8.27895	−0.687529
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.2681	−1.08697	−0.543483	−	0.839420i	$-0.682895\pi$
$149$	−13.2681	−1.08697	−0.543483	+	0.839420i	$0.682895\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.3500	−1.23294
$156$	0	0
$157$	10.6947	0.853531	0.426766	−	0.904362i	$-0.359653\pi$
$157$	10.6947	0.853531	0.426766	+	0.904362i	$0.359653\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.7082	0.838731	0.419366	−	0.907817i	$-0.362253\pi$
$163$	10.7082	0.838731	0.419366	+	0.907817i	$0.362253\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.18034	0.400867	0.200433	−	0.979707i	$-0.435765\pi$
$167$	5.18034	0.400867	0.200433	+	0.979707i	$0.435765\pi$
$168$	0	0
$169$	34.1246	2.62497
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.41641	0.563859	0.281930	−	0.959435i	$-0.409026\pi$
$173$	7.41641	0.563859	0.281930	+	0.959435i	$0.409026\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.1058	0.755345	0.377672	−	0.925939i	$-0.376725\pi$
$179$	10.1058	0.755345	0.377672	+	0.925939i	$0.376725\pi$
$180$	0	0
$181$	−13.7295	−1.02050	−0.510252	−	0.860025i	$-0.670448\pi$
$181$	−13.7295	−1.02050	−0.510252	+	0.860025i	$0.670448\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−19.4721	−1.43162
$186$	0	0
$187$	−27.6653	−2.02309
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.8918	1.22224	0.611122	−	0.791536i	$-0.290718\pi$
$191$	16.8918	1.22224	0.611122	+	0.791536i	$0.290718\pi$
$192$	0	0
$193$	22.1246	1.59256	0.796282	−	0.604925i	$-0.206797\pi$
$193$	22.1246	1.59256	0.796282	+	0.604925i	$0.206797\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.2712	1.08803	0.544014	−	0.839076i	$-0.316904\pi$
$197$	15.2712	1.08803	0.544014	+	0.839076i	$0.316904\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.00000	0.349215
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.1803	1.39590
$210$	0	0
$211$	1.41641	0.0975095	0.0487548	−	0.998811i	$-0.484475\pi$
$211$	1.41641	0.0975095	0.0487548	+	0.998811i	$0.484475\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.5279	−0.717994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	51.2942	3.45042
$222$	0	0
$223$	20.5942	1.37909	0.689545	−	0.724243i	$-0.257810\pi$
$223$	20.5942	1.37909	0.689545	+	0.724243i	$0.257810\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.7082	−1.44082	−0.720412	−	0.693546i	$-0.756047\pi$
$227$	−21.7082	−1.44082	−0.720412	+	0.693546i	$0.756047\pi$
$228$	0	0
$229$	25.4558	1.68217	0.841085	−	0.540903i	$-0.181918\pi$
$229$	25.4558	1.68217	0.841085	+	0.540903i	$0.181918\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.1058	−0.662055	−0.331027	−	0.943621i	$-0.607395\pi$
$233$	−10.1058	−0.662055	−0.331027	+	0.943621i	$0.607395\pi$
$234$	0	0
$235$	−30.1246	−1.96511
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.3500	−0.992910	−0.496455	−	0.868062i	$-0.665365\pi$
$239$	−15.3500	−0.992910	−0.496455	+	0.868062i	$0.665365\pi$
$240$	0	0
$241$	−6.86474	−0.442197	−0.221098	−	0.975252i	$-0.570964\pi$
$241$	−6.86474	−0.442197	−0.221098	+	0.975252i	$0.570964\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−37.4164	−2.38075
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.2361	−0.898573	−0.449286	−	0.893388i	$-0.648322\pi$
$251$	−14.2361	−0.898573	−0.449286	+	0.893388i	$0.648322\pi$
$252$	0	0
$253$	11.7082	0.736088
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.4164	−1.58543	−0.792716	−	0.609591i	$-0.791334\pi$
$257$	−25.4164	−1.58543	−0.792716	+	0.609591i	$0.791334\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.3500	0.946523	0.473261	−	0.880922i	$-0.343077\pi$
$263$	15.3500	0.946523	0.473261	+	0.880922i	$0.343077\pi$
$264$	0	0
$265$	11.9026	0.731171
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.47214	0.455584	0.227792	−	0.973710i	$-0.426849\pi$
$269$	7.47214	0.455584	0.227792	+	0.973710i	$0.426849\pi$
$270$	0	0
$271$	−28.8732	−1.75392	−0.876960	−	0.480564i	$-0.840432\pi$
$271$	−28.8732	−1.75392	−0.876960	+	0.480564i	$0.840432\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.7082	−0.883730	−0.441865	−	0.897081i	$-0.645683\pi$
$277$	−14.7082	−0.883730	−0.441865	+	0.897081i	$0.645683\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.2967	1.44942	0.724709	−	0.689055i	$-0.241974\pi$
$281$	24.2967	1.44942	0.724709	+	0.689055i	$0.241974\pi$
$282$	0	0
$283$	−22.4211	−1.33280	−0.666398	−	0.745597i	$-0.732165\pi$
$283$	−22.4211	−1.33280	−0.666398	+	0.745597i	$0.732165\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	38.8328	2.28428
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.94427	−0.347268	−0.173634	−	0.984810i	$-0.555551\pi$
$293$	−5.94427	−0.347268	−0.173634	+	0.984810i	$0.555551\pi$
$294$	0	0
$295$	13.2918	0.773878
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.7082	−1.25542
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.461370	−0.0264180
$306$	0	0
$307$	1.41421	0.0807134	0.0403567	−	0.999185i	$-0.487151\pi$
$307$	1.41421	0.0807134	0.0403567	+	0.999185i	$0.487151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.18034	0.293750	0.146875	−	0.989155i	$-0.453078\pi$
$311$	5.18034	0.293750	0.146875	+	0.989155i	$0.453078\pi$
$312$	0	0
$313$	−25.4558	−1.43885	−0.719425	−	0.694570i	$-0.755594\pi$
$313$	−25.4558	−1.43885	−0.719425	+	0.694570i	$0.755594\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.40337	−0.359649	−0.179824	−	0.983699i	$-0.557553\pi$
$317$	−6.40337	−0.359649	−0.179824	+	0.983699i	$0.557553\pi$
$318$	0	0
$319$	−13.7082	−0.767512
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−40.7271	−2.26611
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−21.7082	−1.18605
$336$	0	0
$337$	−15.2918	−0.832997	−0.416499	−	0.909136i	$-0.636743\pi$
$337$	−15.2918	−0.832997	−0.416499	+	0.909136i	$0.636743\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−25.4164	−1.37638
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.8384	−1.38708	−0.693539	−	0.720419i	$-0.743950\pi$
$347$	−25.8384	−1.38708	−0.693539	+	0.720419i	$0.743950\pi$
$348$	0	0
$349$	11.7264	0.627698	0.313849	−	0.949473i	$-0.398381\pi$
$349$	11.7264	0.627698	0.313849	+	0.949473i	$0.398381\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.76393	−0.519682	−0.259841	−	0.965651i	$-0.583670\pi$
$353$	−9.76393	−0.519682	−0.259841	+	0.965651i	$0.583670\pi$
$354$	0	0
$355$	11.7264	0.622371
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.6212	−1.61613	−0.808063	−	0.589096i	$-0.799484\pi$
$359$	−30.6212	−1.61613	−0.808063	+	0.589096i	$0.799484\pi$
$360$	0	0
$361$	10.7082	0.563590
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.16228	0.165521
$366$	0	0
$367$	32.1142	1.67635	0.838175	−	0.545401i	$-0.183623\pi$
$367$	32.1142	1.67635	0.838175	+	0.545401i	$0.183623\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.7082	−1.07223	−0.536115	−	0.844145i	$-0.680109\pi$
$373$	−20.7082	−1.07223	−0.536115	+	0.844145i	$0.680109\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.4164	1.30901
$378$	0	0
$379$	−18.4164	−0.945987	−0.472994	−	0.881066i	$-0.656827\pi$
$379$	−18.4164	−0.945987	−0.472994	+	0.881066i	$0.656827\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.5967	0.643664	0.321832	−	0.946797i	$-0.395701\pi$
$383$	12.5967	0.643664	0.321832	+	0.946797i	$0.395701\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.7534	1.10294	0.551470	−	0.834195i	$-0.314067\pi$
$389$	21.7534	1.10294	0.551470	+	0.834195i	$0.314067\pi$
$390$	0	0
$391$	−23.6290	−1.19497
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.1803	0.562544
$396$	0	0
$397$	12.3153	0.618085	0.309043	−	0.951048i	$-0.399991\pi$
$397$	12.3153	0.618085	0.309043	+	0.951048i	$0.399991\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.7000	−1.53309	−0.766543	−	0.642193i	$-0.778025\pi$
$401$	−30.7000	−1.53309	−0.766543	+	0.642193i	$0.778025\pi$
$402$	0	0
$403$	47.1246	2.34744
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.2418	−1.59817
$408$	0	0
$409$	−13.7295	−0.678879	−0.339439	−	0.940628i	$-0.610237\pi$
$409$	−13.7295	−0.678879	−0.339439	+	0.940628i	$0.610237\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−18.4164	−0.904026
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.1803	−1.13243	−0.566217	−	0.824256i	$-0.691594\pi$
$419$	−23.1803	−1.13243	−0.566217	+	0.824256i	$0.691594\pi$
$420$	0	0
$421$	−1.41641	−0.0690315	−0.0345157	−	0.999404i	$-0.510989\pi$
$421$	−1.41641	−0.0690315	−0.0345157	+	0.999404i	$0.510989\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.24419	0.252604	0.126302	−	0.991992i	$-0.459689\pi$
$431$	5.24419	0.252604	0.126302	+	0.991992i	$0.459689\pi$
$432$	0	0
$433$	−6.86474	−0.329898	−0.164949	−	0.986302i	$-0.552746\pi$
$433$	−6.86474	−0.329898	−0.164949	+	0.986302i	$0.552746\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.2361	0.824513
$438$	0	0
$439$	15.7326	0.750875	0.375437	−	0.926848i	$-0.377492\pi$
$439$	15.7326	0.750875	0.375437	+	0.926848i	$0.377492\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.4350	−0.923386	−0.461693	−	0.887040i	$-0.652758\pi$
$443$	−19.4350	−0.923386	−0.461693	+	0.887040i	$0.652758\pi$
$444$	0	0
$445$	−25.1246	−1.19102
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.86163	0.229435	0.114717	−	0.993398i	$-0.463404\pi$
$449$	4.86163	0.229435	0.114717	+	0.993398i	$0.463404\pi$
$450$	0	0
$451$	8.27895	0.389841
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.05573	0.282043	0.141022	−	0.990007i	$-0.454961\pi$
$461$	6.05573	0.282043	0.141022	+	0.990007i	$0.454961\pi$
$462$	0	0
$463$	−38.7082	−1.79892	−0.899461	−	0.437000i	$-0.856041\pi$
$463$	−38.7082	−1.79892	−0.899461	+	0.437000i	$0.856041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.76393	−0.312997	−0.156499	−	0.987678i	$-0.550021\pi$
$467$	−6.76393	−0.312997	−0.156499	+	0.987678i	$0.550021\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17.4319	−0.801521
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.1803	−0.784990	−0.392495	−	0.919754i	$-0.628388\pi$
$479$	−17.1803	−0.784990	−0.392495	+	0.919754i	$0.628388\pi$
$480$	0	0
$481$	59.7795	2.72571
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.461370	0.0209497
$486$	0	0
$487$	38.8328	1.75968	0.879841	−	0.475267i	$-0.157649\pi$
$487$	38.8328	1.75968	0.879841	+	0.475267i	$0.157649\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.0826	1.40274	0.701369	−	0.712798i	$-0.252573\pi$
$491$	31.0826	1.40274	0.701369	+	0.712798i	$0.252573\pi$
$492$	0	0
$493$	27.6653	1.24598
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.4164	0.555835	0.277917	−	0.960605i	$-0.410356\pi$
$499$	12.4164	0.555835	0.277917	+	0.960605i	$0.410356\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.9443	−1.60268	−0.801338	−	0.598212i	$-0.795878\pi$
$503$	−35.9443	−1.60268	−0.801338	+	0.598212i	$0.795878\pi$
$504$	0	0
$505$	8.29180	0.368980
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.0557	0.534361	0.267180	−	0.963647i	$-0.413908\pi$
$509$	12.0557	0.534361	0.267180	+	0.963647i	$0.413908\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	22.1359	0.975426
$516$	0	0
$517$	−49.8800	−2.19372
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−0.0557281	−0.00244149	−0.00122075	−	0.999999i	$-0.500389\pi$
$521$	−0.0557281	−0.00244149	−0.00122075	+	0.999999i	$0.500389\pi$
$522$	0	0
$523$	8.86784	0.387764	0.193882	−	0.981025i	$-0.437892\pi$
$523$	8.86784	0.387764	0.193882	+	0.981025i	$0.437892\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	51.2942	2.23441
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.3500	−0.664883
$534$	0	0
$535$	−23.6290	−1.02157
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−39.2492	−1.68746	−0.843728	−	0.536771i	$-0.819644\pi$
$541$	−39.2492	−1.68746	−0.843728	+	0.536771i	$0.819644\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	23.9443	1.02566
$546$	0	0
$547$	21.2918	0.910371	0.455186	−	0.890397i	$-0.349573\pi$
$547$	21.2918	0.910371	0.455186	+	0.890397i	$0.349573\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.1803	−0.859711
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.9442	1.52301	0.761503	−	0.648161i	$-0.224462\pi$
$557$	35.9442	1.52301	0.761503	+	0.648161i	$0.224462\pi$
$558$	0	0
$559$	32.3206	1.36701
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.5967	−1.41593	−0.707967	−	0.706245i	$-0.750387\pi$
$563$	−33.5967	−1.41593	−0.707967	+	0.706245i	$0.750387\pi$
$564$	0	0
$565$	−46.0501	−1.93734
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.08502	0.171253	0.0856264	−	0.996327i	$-0.472711\pi$
$569$	4.08502	0.171253	0.0856264	+	0.996327i	$0.472711\pi$
$570$	0	0
$571$	−1.29180	−0.0540600	−0.0270300	−	0.999635i	$-0.508605\pi$
$571$	−1.29180	−0.0540600	−0.0270300	+	0.999635i	$0.508605\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.7264	0.488175	0.244088	−	0.969753i	$-0.421512\pi$
$577$	11.7264	0.488175	0.244088	+	0.969753i	$0.421512\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19.7082	0.816230
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.41641	0.306108	0.153054	−	0.988218i	$-0.451089\pi$
$587$	7.41641	0.306108	0.153054	+	0.988218i	$0.451089\pi$
$588$	0	0
$589$	−37.4164	−1.54172
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.8885	1.35057	0.675285	−	0.737557i	$-0.264020\pi$
$593$	32.8885	1.35057	0.675285	+	0.737557i	$0.264020\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.4319	−0.712249	−0.356125	−	0.934438i	$-0.615902\pi$
$599$	−17.4319	−0.712249	−0.356125	+	0.934438i	$0.615902\pi$
$600$	0	0
$601$	32.3206	1.31838	0.659192	−	0.751975i	$-0.270898\pi$
$601$	32.3206	1.31838	0.659192	+	0.751975i	$0.270898\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.05573	0.246200
$606$	0	0
$607$	32.3206	1.31185	0.655926	−	0.754825i	$-0.272278\pi$
$607$	32.3206	1.31185	0.655926	+	0.754825i	$0.272278\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	92.4827	3.74145
$612$	0	0
$613$	−20.8328	−0.841430	−0.420715	−	0.907193i	$-0.638221\pi$
$613$	−20.8328	−0.841430	−0.420715	+	0.907193i	$0.638221\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.5378	1.10863	0.554314	−	0.832307i	$-0.312981\pi$
$617$	27.5378	1.10863	0.554314	+	0.832307i	$0.312981\pi$
$618$	0	0
$619$	−27.2526	−1.09538	−0.547688	−	0.836683i	$-0.684492\pi$
$619$	−27.2526	−1.09538	−0.547688	+	0.836683i	$0.684492\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	65.0689	2.59447
$630$	0	0
$631$	8.70820	0.346668	0.173334	−	0.984863i	$-0.444546\pi$
$631$	8.70820	0.346668	0.173334	+	0.984863i	$0.444546\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.23607	0.0887357
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.6467	−1.56595	−0.782975	−	0.622053i	$-0.786299\pi$
$641$	−39.6467	−1.56595	−0.782975	+	0.622053i	$0.786299\pi$
$642$	0	0
$643$	−37.3584	−1.47327	−0.736637	−	0.676289i	$-0.763587\pi$
$643$	−37.3584	−1.47327	−0.736637	+	0.676289i	$0.763587\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.41641	0.291569	0.145785	−	0.989316i	$-0.453429\pi$
$647$	7.41641	0.291569	0.145785	+	0.989316i	$0.453429\pi$
$648$	0	0
$649$	22.0084	0.863906
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.0826	−1.21636	−0.608178	−	0.793801i	$-0.708099\pi$
$653$	−31.0826	−1.21636	−0.608178	+	0.793801i	$0.708099\pi$
$654$	0	0
$655$	−40.2492	−1.57267
$656$	0	0
$657$	0	0
$658$	0	0
$659$	41.1884	1.60447	0.802237	−	0.597006i	$-0.203643\pi$
$659$	41.1884	1.60447	0.802237	+	0.597006i	$0.203643\pi$
$660$	0	0
$661$	−18.5911	−0.723110	−0.361555	−	0.932351i	$-0.617754\pi$
$661$	−18.5911	−0.723110	−0.361555	+	0.932351i	$0.617754\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.7082	−0.453343
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.763932	−0.0294913
$672$	0	0
$673$	−11.1246	−0.428822	−0.214411	−	0.976744i	$-0.568783\pi$
$673$	−11.1246	−0.428822	−0.214411	+	0.976744i	$0.568783\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.382559	0.0146382	0.00731910	−	0.999973i	$-0.497670\pi$
$683$	0.382559	0.0146382	0.00731910	+	0.999973i	$0.497670\pi$
$684$	0	0
$685$	22.5973	0.863399
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−36.5410	−1.39210
$690$	0	0
$691$	−32.3206	−1.22953	−0.614766	−	0.788709i	$-0.710750\pi$
$691$	−32.3206	−1.22953	−0.614766	+	0.788709i	$0.710750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.78593	−0.257405
$696$	0	0
$697$	−16.7082	−0.632868
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25.8384	−0.975903	−0.487952	−	0.872871i	$-0.662256\pi$
$701$	−25.8384	−0.975903	−0.487952	+	0.872871i	$0.662256\pi$
$702$	0	0
$703$	−47.4643	−1.79015
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.41641	−0.240973	−0.120487	−	0.992715i	$-0.538445\pi$
$709$	−6.41641	−0.240973	−0.120487	+	0.992715i	$0.538445\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.7082	−0.812979
$714$	0	0
$715$	−56.8328	−2.12543
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.5279	0.392623	0.196312	−	0.980542i	$-0.437104\pi$
$719$	10.5279	0.392623	0.196312	+	0.980542i	$0.437104\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−27.2526	−1.01074	−0.505372	−	0.862902i	$-0.668645\pi$
$727$	−27.2526	−1.01074	−0.505372	+	0.862902i	$0.668645\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	35.1803	1.30119
$732$	0	0
$733$	−29.4621	−1.08821	−0.544103	−	0.839019i	$-0.683130\pi$
$733$	−29.4621	−1.08821	−0.544103	+	0.839019i	$0.683130\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.9442	−1.32402
$738$	0	0
$739$	−16.5836	−0.610037	−0.305019	−	0.952346i	$-0.598663\pi$
$739$	−16.5836	−0.610037	−0.305019	+	0.952346i	$0.598663\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.2681	−0.486760	−0.243380	−	0.969931i	$-0.578256\pi$
$743$	−13.2681	−0.486760	−0.243380	+	0.969931i	$0.578256\pi$
$744$	0	0
$745$	−29.6684	−1.08697
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.12461	0.259981	0.129990	−	0.991515i	$-0.458505\pi$
$751$	7.12461	0.259981	0.129990	+	0.991515i	$0.458505\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.1803	0.406894
$756$	0	0
$757$	36.4164	1.32358	0.661788	−	0.749691i	$-0.269798\pi$
$757$	36.4164	1.32358	0.661788	+	0.749691i	$0.269798\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.0557	0.437020	0.218510	−	0.975835i	$-0.429880\pi$
$761$	12.0557	0.437020	0.218510	+	0.975835i	$0.429880\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.8059	−1.47341
$768$	0	0
$769$	−11.7264	−0.422864	−0.211432	−	0.977393i	$-0.567813\pi$
$769$	−11.7264	−0.422864	−0.211432	+	0.977393i	$0.567813\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.1803	1.04954	0.524772	−	0.851243i	$-0.324151\pi$
$773$	29.1803	1.04954	0.524772	+	0.851243i	$0.324151\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.1877	0.436671
$780$	0	0
$781$	19.4164	0.694774
$782$	0	0
$783$	0	0
$784$	0	0
$785$	23.9141	0.853531
$786$	0	0
$787$	11.9026	0.424282	0.212141	−	0.977239i	$-0.431956\pi$
$787$	11.9026	0.424282	0.212141	+	0.977239i	$0.431956\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.41641	0.0502981
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.5410	−0.656757	−0.328378	−	0.944546i	$-0.606502\pi$
$797$	−18.5410	−0.656757	−0.328378	+	0.944546i	$0.606502\pi$
$798$	0	0
$799$	100.666	3.56129
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.23607	0.184777
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.0788114	−0.00277086	−0.00138543	−	0.999999i	$-0.500441\pi$
$809$	−0.0788114	−0.00277086	−0.00138543	+	0.999999i	$0.500441\pi$
$810$	0	0
$811$	4.86163	0.170715	0.0853575	−	0.996350i	$-0.472797\pi$
$811$	4.86163	0.170715	0.0853575	+	0.996350i	$0.472797\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.9443	0.838731
$816$	0	0
$817$	−25.6622	−0.897806
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.24419	−0.183024	−0.0915118	−	0.995804i	$-0.529170\pi$
$821$	−5.24419	−0.183024	−0.0915118	+	0.995804i	$0.529170\pi$
$822$	0	0
$823$	27.8328	0.970191	0.485095	−	0.874461i	$-0.338785\pi$
$823$	27.8328	0.970191	0.485095	+	0.874461i	$0.338785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.0826	−1.08085	−0.540424	−	0.841393i	$-0.681736\pi$
$827$	−31.0826	−1.08085	−0.540424	+	0.841393i	$0.681736\pi$
$828$	0	0
$829$	49.4674	1.71807	0.859036	−	0.511914i	$-0.171064\pi$
$829$	49.4674	1.71807	0.859036	+	0.511914i	$0.171064\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.5836	0.400867
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.5967	−0.849174	−0.424587	−	0.905387i	$-0.639581\pi$
$839$	−24.5967	−0.849174	−0.424587	+	0.905387i	$0.639581\pi$
$840$	0	0
$841$	−15.2918	−0.527303
$842$	0	0
$843$	0	0
$844$	0	0
$845$	76.3050	2.62497
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−27.5378	−0.943982
$852$	0	0
$853$	39.1853	1.34168	0.670840	−	0.741602i	$-0.265934\pi$
$853$	39.1853	1.34168	0.670840	+	0.741602i	$0.265934\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.5967	1.45508	0.727539	−	0.686067i	$-0.240664\pi$
$857$	42.5967	1.45508	0.727539	+	0.686067i	$0.240664\pi$
$858$	0	0
$859$	49.0848	1.67475	0.837376	−	0.546627i	$-0.184088\pi$
$859$	49.0848	1.67475	0.837376	+	0.546627i	$0.184088\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.3445	1.37334	0.686671	−	0.726968i	$-0.259071\pi$
$863$	40.3445	1.37334	0.686671	+	0.726968i	$0.259071\pi$
$864$	0	0
$865$	16.5836	0.563859
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.5123	0.627987
$870$	0	0
$871$	66.6443	2.25815
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.70820	0.0914495	0.0457248	−	0.998954i	$-0.485440\pi$
$877$	2.70820	0.0914495	0.0457248	+	0.998954i	$0.485440\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32.8328	−1.10617	−0.553083	−	0.833126i	$-0.686549\pi$
$881$	−32.8328	−1.10617	−0.553083	+	0.833126i	$0.686549\pi$
$882$	0	0
$883$	39.2492	1.32084	0.660421	−	0.750896i	$-0.270378\pi$
$883$	39.2492	1.32084	0.660421	+	0.750896i	$0.270378\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.3607	−1.05299	−0.526494	−	0.850179i	$-0.676494\pi$
$887$	−31.3607	−1.05299	−0.526494	+	0.850179i	$0.676494\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−73.4302	−2.45725
$894$	0	0
$895$	22.5973	0.755345
$896$	0	0
$897$	0	0
$898$	0	0
$899$	25.4164	0.847685
$900$	0	0
$901$	−39.7742	−1.32507
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−30.7000	−1.02050
$906$	0	0
$907$	32.4164	1.07637	0.538185	−	0.842827i	$-0.319110\pi$
$907$	32.4164	1.07637	0.538185	+	0.842827i	$0.319110\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.70246	−0.122668	−0.0613340	−	0.998117i	$-0.519535\pi$
$911$	−3.70246	−0.122668	−0.0613340	+	0.998117i	$0.519535\pi$
$912$	0	0
$913$	−30.4937	−1.00919
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.4164	1.26724	0.633620	−	0.773644i	$-0.281568\pi$
$919$	38.4164	1.26724	0.633620	+	0.773644i	$0.281568\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−36.0000	−1.18495
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	55.3607	1.81632	0.908162	−	0.418618i	$-0.137485\pi$
$929$	55.3607	1.81632	0.908162	+	0.418618i	$0.137485\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−61.8614	−2.02309
$936$	0	0
$937$	−20.0053	−0.653545	−0.326773	−	0.945103i	$-0.605961\pi$
$937$	−20.0053	−0.653545	−0.326773	+	0.945103i	$0.605961\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.8197	0.417909	0.208954	−	0.977925i	$-0.432994\pi$
$941$	12.8197	0.417909	0.208954	+	0.977925i	$0.432994\pi$
$942$	0	0
$943$	7.07107	0.230266
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.4558	0.827204	0.413602	−	0.910458i	$-0.364271\pi$
$947$	25.4558	0.827204	0.413602	+	0.910458i	$0.364271\pi$
$948$	0	0
$949$	−9.70820	−0.315142
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.0230	1.16690	0.583450	−	0.812149i	$-0.301702\pi$
$953$	36.0230	1.16690	0.583450	+	0.812149i	$0.301702\pi$
$954$	0	0
$955$	37.7711	1.22224
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	16.1246	0.520149
$962$	0	0
$963$	0	0
$964$	0	0
$965$	49.4721	1.59256
$966$	0	0
$967$	−16.5836	−0.533292	−0.266646	−	0.963794i	$-0.585916\pi$
$967$	−16.5836	−0.533292	−0.266646	+	0.963794i	$0.585916\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.3050	−1.29345	−0.646724	−	0.762724i	$-0.723861\pi$
$971$	−40.3050	−1.29345	−0.646724	+	0.762724i	$0.723861\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.9768	0.671106	0.335553	−	0.942021i	$-0.391077\pi$
$977$	20.9768	0.671106	0.335553	+	0.942021i	$0.391077\pi$
$978$	0	0
$979$	−41.6011	−1.32958
$980$	0	0
$981$	0	0
$982$	0	0
$983$	38.8885	1.24035	0.620176	−	0.784463i	$-0.287061\pi$
$983$	38.8885	1.24035	0.620176	+	0.784463i	$0.287061\pi$
$984$	0	0
$985$	34.1475	1.08803
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.8886	−0.473431
$990$	0	0
$991$	10.1246	0.321619	0.160809	−	0.986985i	$-0.448590\pi$
$991$	10.1246	0.321619	0.160809	+	0.986985i	$0.448590\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.79677	0.0569043	0.0284522	−	0.999595i	$-0.490942\pi$
$997$	1.79677	0.0569043	0.0284522	+	0.999595i	$0.490942\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.y.1.4	yes	4
3.2	odd	2		5292.2.a.bb.1.1	yes	4
7.6	odd	2		5292.2.a.bb.1.2	yes	4
21.20	even	2	inner	5292.2.a.y.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5292.2.a.y.1.3	✓	4	21.20	even	2	inner
5292.2.a.y.1.4	yes	4	1.1	even	1	trivial
5292.2.a.bb.1.1	yes	4	3.2	odd	2
5292.2.a.bb.1.2	yes	4	7.6	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} +O(q^{10})\)	\(q+2.23607 q^{5} +3.70246 q^{11} -6.86474 q^{13} -7.47214 q^{17} +5.45052 q^{19} +3.16228 q^{23} -3.70246 q^{29} -6.86474 q^{31} -8.70820 q^{37} +2.23607 q^{41} -4.70820 q^{43} -13.4721 q^{47} +5.32300 q^{53} +8.27895 q^{55} +5.94427 q^{59} -0.206331 q^{61} -15.3500 q^{65} -9.70820 q^{67} +5.24419 q^{71} +1.41421 q^{73} +5.00000 q^{79} -8.23607 q^{83} -16.7082 q^{85} -11.2361 q^{89} +12.1877 q^{95} +0.206331 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{17} - 8 q^{37} + 8 q^{43} - 36 q^{47} - 12 q^{59} - 12 q^{67} + 20 q^{79} - 24 q^{83} - 40 q^{85} - 36 q^{89}+O(q^{100}) \) 4 * q - 12 * q^17 - 8 * q^37 + 8 * q^43 - 36 * q^47 - 12 * q^59 - 12 * q^67 + 20 * q^79 - 24 * q^83 - 40 * q^85 - 36 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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