LMFDB - Embedded newform 1352.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,4,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1352.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:	$79.7705823278$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{3} +2.00000 q^{5} -24.0000 q^{7} -11.0000 q^{9} +44.0000 q^{11} -8.00000 q^{15} +50.0000 q^{17} -44.0000 q^{19} +96.0000 q^{21} -56.0000 q^{23} -121.000 q^{25} +152.000 q^{27} +198.000 q^{29} +160.000 q^{31} -176.000 q^{33} -48.0000 q^{35} +162.000 q^{37} +198.000 q^{41} +52.0000 q^{43} -22.0000 q^{45} -528.000 q^{47} +233.000 q^{49} -200.000 q^{51} -242.000 q^{53} +88.0000 q^{55} +176.000 q^{57} +668.000 q^{59} +550.000 q^{61} +264.000 q^{63} -188.000 q^{67} +224.000 q^{69} -728.000 q^{71} -154.000 q^{73} +484.000 q^{75} -1056.00 q^{77} -656.000 q^{79} -311.000 q^{81} -236.000 q^{83} +100.000 q^{85} -792.000 q^{87} -714.000 q^{89} -640.000 q^{93} -88.0000 q^{95} +478.000 q^{97} -484.000 q^{99} +O(q^{100})$

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
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	0	0
$7$	−24.0000	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$7$	−24.0000	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	44.0000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	44.0000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−8.00000	−0.137706
$16$	0	0
$17$	50.0000	0.713340	0.356670	−	0.934230i	$-0.383912\pi$
$17$	50.0000	0.713340	0.356670	+	0.934230i	$0.383912\pi$
$18$	0	0
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	0	0
$23$	−56.0000	−0.507687	−0.253844	−	0.967245i	$-0.581695\pi$
$23$	−56.0000	−0.507687	−0.253844	+	0.967245i	$0.581695\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	198.000	1.26785	0.633925	−	0.773394i	$-0.281443\pi$
$29$	198.000	1.26785	0.633925	+	0.773394i	$0.281443\pi$
$30$	0	0
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	0	0
$33$	−176.000	−0.928414
$34$	0	0
$35$	−48.0000	−0.231814
$36$	0	0
$37$	162.000	0.719801	0.359900	−	0.932991i	$-0.382811\pi$
$37$	162.000	0.719801	0.359900	+	0.932991i	$0.382811\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	198.000	0.754205	0.377102	−	0.926172i	$-0.376920\pi$
$41$	198.000	0.754205	0.377102	+	0.926172i	$0.376920\pi$
$42$	0	0
$43$	52.0000	0.184417	0.0922084	−	0.995740i	$-0.470607\pi$
$43$	52.0000	0.184417	0.0922084	+	0.995740i	$0.470607\pi$
$44$	0	0
$45$	−22.0000	−0.0728793
$46$	0	0
$47$	−528.000	−1.63865	−0.819327	−	0.573327i	$-0.805653\pi$
$47$	−528.000	−1.63865	−0.819327	+	0.573327i	$0.805653\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	−200.000	−0.549129
$52$	0	0
$53$	−242.000	−0.627194	−0.313597	−	0.949556i	$-0.601534\pi$
$53$	−242.000	−0.627194	−0.313597	+	0.949556i	$0.601534\pi$
$54$	0	0
$55$	88.0000	0.215744
$56$	0	0
$57$	176.000	0.408978
$58$	0	0
$59$	668.000	1.47400	0.737002	−	0.675891i	$-0.236241\pi$
$59$	668.000	1.47400	0.737002	+	0.675891i	$0.236241\pi$
$60$	0	0
$61$	550.000	1.15443	0.577215	−	0.816592i	$-0.304139\pi$
$61$	550.000	1.15443	0.577215	+	0.816592i	$0.304139\pi$
$62$	0	0
$63$	264.000	0.527950
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−188.000	−0.342804	−0.171402	−	0.985201i	$-0.554830\pi$
$67$	−188.000	−0.342804	−0.171402	+	0.985201i	$0.554830\pi$
$68$	0	0
$69$	224.000	0.390818
$70$	0	0
$71$	−728.000	−1.21687	−0.608435	−	0.793604i	$-0.708202\pi$
$71$	−728.000	−1.21687	−0.608435	+	0.793604i	$0.708202\pi$
$72$	0	0
$73$	−154.000	−0.246909	−0.123454	−	0.992350i	$-0.539397\pi$
$73$	−154.000	−0.246909	−0.123454	+	0.992350i	$0.539397\pi$
$74$	0	0
$75$	484.000	0.745167
$76$	0	0
$77$	−1056.00	−1.56289
$78$	0	0
$79$	−656.000	−0.934250	−0.467125	−	0.884191i	$-0.654710\pi$
$79$	−656.000	−0.934250	−0.467125	+	0.884191i	$0.654710\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	−236.000	−0.312101	−0.156050	−	0.987749i	$-0.549876\pi$
$83$	−236.000	−0.312101	−0.156050	+	0.987749i	$0.549876\pi$
$84$	0	0
$85$	100.000	0.127606
$86$	0	0
$87$	−792.000	−0.975992
$88$	0	0
$89$	−714.000	−0.850380	−0.425190	−	0.905104i	$-0.639793\pi$
$89$	−714.000	−0.850380	−0.425190	+	0.905104i	$0.639793\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−640.000	−0.713601
$94$	0	0
$95$	−88.0000	−0.0950380
$96$	0	0
$97$	478.000	0.500346	0.250173	−	0.968201i	$-0.419513\pi$
$97$	478.000	0.500346	0.250173	+	0.968201i	$0.419513\pi$
$98$	0	0
$99$	−484.000	−0.491352
$100$	0	0
$101$	1566.00	1.54280	0.771400	−	0.636350i	$-0.219557\pi$
$101$	1566.00	1.54280	0.771400	+	0.636350i	$0.219557\pi$
$102$	0	0
$103$	−968.000	−0.926018	−0.463009	−	0.886354i	$-0.653230\pi$
$103$	−968.000	−0.926018	−0.463009	+	0.886354i	$0.653230\pi$
$104$	0	0
$105$	192.000	0.178450
$106$	0	0
$107$	−780.000	−0.704724	−0.352362	−	0.935864i	$-0.614621\pi$
$107$	−780.000	−0.704724	−0.352362	+	0.935864i	$0.614621\pi$
$108$	0	0
$109$	1994.00	1.75221	0.876103	−	0.482123i	$-0.160134\pi$
$109$	1994.00	1.75221	0.876103	+	0.482123i	$0.160134\pi$
$110$	0	0
$111$	−648.000	−0.554103
$112$	0	0
$113$	−942.000	−0.784212	−0.392106	−	0.919920i	$-0.628253\pi$
$113$	−942.000	−0.784212	−0.392106	+	0.919920i	$0.628253\pi$
$114$	0	0
$115$	−112.000	−0.0908179
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1200.00	−0.924402
$120$	0	0
$121$	605.000	0.454545
$122$	0	0
$123$	−792.000	−0.580587
$124$	0	0
$125$	−492.000	−0.352047
$126$	0	0
$127$	1408.00	0.983778	0.491889	−	0.870658i	$-0.336307\pi$
$127$	1408.00	0.983778	0.491889	+	0.870658i	$0.336307\pi$
$128$	0	0
$129$	−208.000	−0.141964
$130$	0	0
$131$	−2692.00	−1.79543	−0.897714	−	0.440578i	$-0.854773\pi$
$131$	−2692.00	−1.79543	−0.897714	+	0.440578i	$0.854773\pi$
$132$	0	0
$133$	1056.00	0.688472
$134$	0	0
$135$	304.000	0.193809
$136$	0	0
$137$	−1626.00	−1.01400	−0.507002	−	0.861945i	$-0.669246\pi$
$137$	−1626.00	−1.01400	−0.507002	+	0.861945i	$0.669246\pi$
$138$	0	0
$139$	−684.000	−0.417382	−0.208691	−	0.977982i	$-0.566920\pi$
$139$	−684.000	−0.417382	−0.208691	+	0.977982i	$0.566920\pi$
$140$	0	0
$141$	2112.00	1.26144
$142$	0	0
$143$	0	0
$144$	0	0
$145$	396.000	0.226800
$146$	0	0
$147$	−932.000	−0.522926
$148$	0	0
$149$	−302.000	−0.166046	−0.0830228	−	0.996548i	$-0.526457\pi$
$149$	−302.000	−0.166046	−0.0830228	+	0.996548i	$0.526457\pi$
$150$	0	0
$151$	−1352.00	−0.728637	−0.364319	−	0.931274i	$-0.618698\pi$
$151$	−1352.00	−0.728637	−0.364319	+	0.931274i	$0.618698\pi$
$152$	0	0
$153$	−550.000	−0.290620
$154$	0	0
$155$	320.000	0.165826
$156$	0	0
$157$	3142.00	1.59719	0.798595	−	0.601868i	$-0.205577\pi$
$157$	3142.00	1.59719	0.798595	+	0.601868i	$0.205577\pi$
$158$	0	0
$159$	968.000	0.482814
$160$	0	0
$161$	1344.00	0.657901
$162$	0	0
$163$	−3036.00	−1.45888	−0.729441	−	0.684043i	$-0.760220\pi$
$163$	−3036.00	−1.45888	−0.729441	+	0.684043i	$0.760220\pi$
$164$	0	0
$165$	−352.000	−0.166080
$166$	0	0
$167$	264.000	0.122329	0.0611645	−	0.998128i	$-0.480519\pi$
$167$	264.000	0.122329	0.0611645	+	0.998128i	$0.480519\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	484.000	0.216447
$172$	0	0
$173$	−2826.00	−1.24195	−0.620973	−	0.783832i	$-0.713263\pi$
$173$	−2826.00	−1.24195	−0.620973	+	0.783832i	$0.713263\pi$
$174$	0	0
$175$	2904.00	1.25441
$176$	0	0
$177$	−2672.00	−1.13469
$178$	0	0
$179$	3084.00	1.28776	0.643880	−	0.765127i	$-0.277324\pi$
$179$	3084.00	1.28776	0.643880	+	0.765127i	$0.277324\pi$
$180$	0	0
$181$	−2418.00	−0.992975	−0.496488	−	0.868044i	$-0.665377\pi$
$181$	−2418.00	−0.992975	−0.496488	+	0.868044i	$0.665377\pi$
$182$	0	0
$183$	−2200.00	−0.888681
$184$	0	0
$185$	324.000	0.128762
$186$	0	0
$187$	2200.00	0.860320
$188$	0	0
$189$	−3648.00	−1.40398
$190$	0	0
$191$	−960.000	−0.363681	−0.181841	−	0.983328i	$-0.558206\pi$
$191$	−960.000	−0.363681	−0.181841	+	0.983328i	$0.558206\pi$
$192$	0	0
$193$	−2882.00	−1.07488	−0.537438	−	0.843304i	$-0.680608\pi$
$193$	−2882.00	−1.07488	−0.537438	+	0.843304i	$0.680608\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1086.00	−0.392763	−0.196381	−	0.980528i	$-0.562919\pi$
$197$	−1086.00	−0.392763	−0.196381	+	0.980528i	$0.562919\pi$
$198$	0	0
$199$	88.0000	0.0313475	0.0156738	−	0.999877i	$-0.495011\pi$
$199$	88.0000	0.0313475	0.0156738	+	0.999877i	$0.495011\pi$
$200$	0	0
$201$	752.000	0.263890
$202$	0	0
$203$	−4752.00	−1.64298
$204$	0	0
$205$	396.000	0.134916
$206$	0	0
$207$	616.000	0.206836
$208$	0	0
$209$	−1936.00	−0.640746
$210$	0	0
$211$	−3476.00	−1.13411	−0.567056	−	0.823679i	$-0.691918\pi$
$211$	−3476.00	−1.13411	−0.567056	+	0.823679i	$0.691918\pi$
$212$	0	0
$213$	2912.00	0.936746
$214$	0	0
$215$	104.000	0.0329895
$216$	0	0
$217$	−3840.00	−1.20127
$218$	0	0
$219$	616.000	0.190070
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−928.000	−0.278670	−0.139335	−	0.990245i	$-0.544497\pi$
$223$	−928.000	−0.278670	−0.139335	+	0.990245i	$0.544497\pi$
$224$	0	0
$225$	1331.00	0.394370
$226$	0	0
$227$	−156.000	−0.0456127	−0.0228064	−	0.999740i	$-0.507260\pi$
$227$	−156.000	−0.0456127	−0.0228064	+	0.999740i	$0.507260\pi$
$228$	0	0
$229$	1634.00	0.471519	0.235759	−	0.971811i	$-0.424242\pi$
$229$	1634.00	0.471519	0.235759	+	0.971811i	$0.424242\pi$
$230$	0	0
$231$	4224.00	1.20311
$232$	0	0
$233$	−902.000	−0.253614	−0.126807	−	0.991927i	$-0.540473\pi$
$233$	−902.000	−0.253614	−0.126807	+	0.991927i	$0.540473\pi$
$234$	0	0
$235$	−1056.00	−0.293131
$236$	0	0
$237$	2624.00	0.719186
$238$	0	0
$239$	−1616.00	−0.437365	−0.218683	−	0.975796i	$-0.570176\pi$
$239$	−1616.00	−0.437365	−0.218683	+	0.975796i	$0.570176\pi$
$240$	0	0
$241$	−4818.00	−1.28778	−0.643889	−	0.765119i	$-0.722680\pi$
$241$	−4818.00	−1.28778	−0.643889	+	0.765119i	$0.722680\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	466.000	0.121517
$246$	0	0
$247$	0	0
$248$	0	0
$249$	944.000	0.240255
$250$	0	0
$251$	−2140.00	−0.538150	−0.269075	−	0.963119i	$-0.586718\pi$
$251$	−2140.00	−0.538150	−0.269075	+	0.963119i	$0.586718\pi$
$252$	0	0
$253$	−2464.00	−0.612294
$254$	0	0
$255$	−400.000	−0.0982313
$256$	0	0
$257$	770.000	0.186892	0.0934461	−	0.995624i	$-0.470212\pi$
$257$	770.000	0.186892	0.0934461	+	0.995624i	$0.470212\pi$
$258$	0	0
$259$	−3888.00	−0.932774
$260$	0	0
$261$	−2178.00	−0.516532
$262$	0	0
$263$	−7400.00	−1.73499	−0.867497	−	0.497442i	$-0.834273\pi$
$263$	−7400.00	−1.73499	−0.867497	+	0.497442i	$0.834273\pi$
$264$	0	0
$265$	−484.000	−0.112196
$266$	0	0
$267$	2856.00	0.654623
$268$	0	0
$269$	−2794.00	−0.633283	−0.316642	−	0.948545i	$-0.602555\pi$
$269$	−2794.00	−0.633283	−0.316642	+	0.948545i	$0.602555\pi$
$270$	0	0
$271$	−8624.00	−1.93310	−0.966551	−	0.256474i	$-0.917439\pi$
$271$	−8624.00	−1.93310	−0.966551	+	0.256474i	$0.917439\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5324.00	−1.16745
$276$	0	0
$277$	−1874.00	−0.406490	−0.203245	−	0.979128i	$-0.565149\pi$
$277$	−1874.00	−0.406490	−0.203245	+	0.979128i	$0.565149\pi$
$278$	0	0
$279$	−1760.00	−0.377665
$280$	0	0
$281$	−3338.00	−0.708642	−0.354321	−	0.935124i	$-0.615288\pi$
$281$	−3338.00	−0.708642	−0.354321	+	0.935124i	$0.615288\pi$
$282$	0	0
$283$	7172.00	1.50647	0.753235	−	0.657751i	$-0.228492\pi$
$283$	7172.00	1.50647	0.753235	+	0.657751i	$0.228492\pi$
$284$	0	0
$285$	352.000	0.0731603
$286$	0	0
$287$	−4752.00	−0.977358
$288$	0	0
$289$	−2413.00	−0.491146
$290$	0	0
$291$	−1912.00	−0.385166
$292$	0	0
$293$	−5214.00	−1.03961	−0.519804	−	0.854286i	$-0.673995\pi$
$293$	−5214.00	−1.03961	−0.519804	+	0.854286i	$0.673995\pi$
$294$	0	0
$295$	1336.00	0.263678
$296$	0	0
$297$	6688.00	1.30666
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1248.00	−0.238982
$302$	0	0
$303$	−6264.00	−1.18765
$304$	0	0
$305$	1100.00	0.206511
$306$	0	0
$307$	−396.000	−0.0736186	−0.0368093	−	0.999322i	$-0.511719\pi$
$307$	−396.000	−0.0736186	−0.0368093	+	0.999322i	$0.511719\pi$
$308$	0	0
$309$	3872.00	0.712849
$310$	0	0
$311$	−4056.00	−0.739533	−0.369766	−	0.929125i	$-0.620562\pi$
$311$	−4056.00	−0.739533	−0.369766	+	0.929125i	$0.620562\pi$
$312$	0	0
$313$	2154.00	0.388982	0.194491	−	0.980904i	$-0.437695\pi$
$313$	2154.00	0.388982	0.194491	+	0.980904i	$0.437695\pi$
$314$	0	0
$315$	528.000	0.0944426
$316$	0	0
$317$	7386.00	1.30864	0.654320	−	0.756217i	$-0.272955\pi$
$317$	7386.00	1.30864	0.654320	+	0.756217i	$0.272955\pi$
$318$	0	0
$319$	8712.00	1.52909
$320$	0	0
$321$	3120.00	0.542497
$322$	0	0
$323$	−2200.00	−0.378982
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7976.00	−1.34885
$328$	0	0
$329$	12672.0	2.12350
$330$	0	0
$331$	1132.00	0.187977	0.0939884	−	0.995573i	$-0.470038\pi$
$331$	1132.00	0.187977	0.0939884	+	0.995573i	$0.470038\pi$
$332$	0	0
$333$	−1782.00	−0.293252
$334$	0	0
$335$	−376.000	−0.0613226
$336$	0	0
$337$	−3342.00	−0.540209	−0.270104	−	0.962831i	$-0.587058\pi$
$337$	−3342.00	−0.540209	−0.270104	+	0.962831i	$0.587058\pi$
$338$	0	0
$339$	3768.00	0.603686
$340$	0	0
$341$	7040.00	1.11800
$342$	0	0
$343$	2640.00	0.415588
$344$	0	0
$345$	448.000	0.0699116
$346$	0	0
$347$	2244.00	0.347159	0.173580	−	0.984820i	$-0.444467\pi$
$347$	2244.00	0.347159	0.173580	+	0.984820i	$0.444467\pi$
$348$	0	0
$349$	6522.00	1.00033	0.500164	−	0.865931i	$-0.333273\pi$
$349$	6522.00	1.00033	0.500164	+	0.865931i	$0.333273\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11230.0	1.69324	0.846618	−	0.532200i	$-0.178635\pi$
$353$	11230.0	1.69324	0.846618	+	0.532200i	$0.178635\pi$
$354$	0	0
$355$	−1456.00	−0.217680
$356$	0	0
$357$	4800.00	0.711605
$358$	0	0
$359$	−1848.00	−0.271682	−0.135841	−	0.990731i	$-0.543374\pi$
$359$	−1848.00	−0.271682	−0.135841	+	0.990731i	$0.543374\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	−2420.00	−0.349909
$364$	0	0
$365$	−308.000	−0.0441684
$366$	0	0
$367$	7120.00	1.01270	0.506350	−	0.862328i	$-0.330994\pi$
$367$	7120.00	1.01270	0.506350	+	0.862328i	$0.330994\pi$
$368$	0	0
$369$	−2178.00	−0.307269
$370$	0	0
$371$	5808.00	0.812766
$372$	0	0
$373$	6350.00	0.881476	0.440738	−	0.897636i	$-0.354717\pi$
$373$	6350.00	0.881476	0.440738	+	0.897636i	$0.354717\pi$
$374$	0	0
$375$	1968.00	0.271006
$376$	0	0
$377$	0	0
$378$	0	0
$379$	7900.00	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$379$	7900.00	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$380$	0	0
$381$	−5632.00	−0.757313
$382$	0	0
$383$	−10368.0	−1.38324	−0.691619	−	0.722263i	$-0.743102\pi$
$383$	−10368.0	−1.38324	−0.691619	+	0.722263i	$0.743102\pi$
$384$	0	0
$385$	−2112.00	−0.279578
$386$	0	0
$387$	−572.000	−0.0751328
$388$	0	0
$389$	8830.00	1.15090	0.575448	−	0.817838i	$-0.304828\pi$
$389$	8830.00	1.15090	0.575448	+	0.817838i	$0.304828\pi$
$390$	0	0
$391$	−2800.00	−0.362154
$392$	0	0
$393$	10768.0	1.38212
$394$	0	0
$395$	−1312.00	−0.167124
$396$	0	0
$397$	−9878.00	−1.24877	−0.624386	−	0.781116i	$-0.714651\pi$
$397$	−9878.00	−1.24877	−0.624386	+	0.781116i	$0.714651\pi$
$398$	0	0
$399$	−4224.00	−0.529986
$400$	0	0
$401$	13134.0	1.63561	0.817806	−	0.575494i	$-0.195190\pi$
$401$	13134.0	1.63561	0.817806	+	0.575494i	$0.195190\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−622.000	−0.0763146
$406$	0	0
$407$	7128.00	0.868113
$408$	0	0
$409$	−906.000	−0.109533	−0.0547663	−	0.998499i	$-0.517441\pi$
$409$	−906.000	−0.109533	−0.0547663	+	0.998499i	$0.517441\pi$
$410$	0	0
$411$	6504.00	0.780581
$412$	0	0
$413$	−16032.0	−1.91013
$414$	0	0
$415$	−472.000	−0.0558303
$416$	0	0
$417$	2736.00	0.321301
$418$	0	0
$419$	−5412.00	−0.631011	−0.315505	−	0.948924i	$-0.602174\pi$
$419$	−5412.00	−0.631011	−0.315505	+	0.948924i	$0.602174\pi$
$420$	0	0
$421$	4642.00	0.537381	0.268690	−	0.963227i	$-0.413409\pi$
$421$	4642.00	0.537381	0.268690	+	0.963227i	$0.413409\pi$
$422$	0	0
$423$	5808.00	0.667600
$424$	0	0
$425$	−6050.00	−0.690513
$426$	0	0
$427$	−13200.0	−1.49600
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−656.000	−0.0733142	−0.0366571	−	0.999328i	$-0.511671\pi$
$431$	−656.000	−0.0733142	−0.0366571	+	0.999328i	$0.511671\pi$
$432$	0	0
$433$	9490.00	1.05326	0.526629	−	0.850096i	$-0.323456\pi$
$433$	9490.00	1.05326	0.526629	+	0.850096i	$0.323456\pi$
$434$	0	0
$435$	−1584.00	−0.174591
$436$	0	0
$437$	2464.00	0.269723
$438$	0	0
$439$	5544.00	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$439$	5544.00	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$440$	0	0
$441$	−2563.00	−0.276752
$442$	0	0
$443$	7652.00	0.820672	0.410336	−	0.911935i	$-0.365412\pi$
$443$	7652.00	0.820672	0.410336	+	0.911935i	$0.365412\pi$
$444$	0	0
$445$	−1428.00	−0.152121
$446$	0	0
$447$	1208.00	0.127822
$448$	0	0
$449$	446.000	0.0468776	0.0234388	−	0.999725i	$-0.492539\pi$
$449$	446.000	0.0468776	0.0234388	+	0.999725i	$0.492539\pi$
$450$	0	0
$451$	8712.00	0.909605
$452$	0	0
$453$	5408.00	0.560905
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1562.00	−0.159885	−0.0799423	−	0.996799i	$-0.525474\pi$
$457$	−1562.00	−0.159885	−0.0799423	+	0.996799i	$0.525474\pi$
$458$	0	0
$459$	7600.00	0.772849
$460$	0	0
$461$	−10582.0	−1.06910	−0.534548	−	0.845138i	$-0.679518\pi$
$461$	−10582.0	−1.06910	−0.534548	+	0.845138i	$0.679518\pi$
$462$	0	0
$463$	10768.0	1.08085	0.540423	−	0.841394i	$-0.318264\pi$
$463$	10768.0	1.08085	0.540423	+	0.841394i	$0.318264\pi$
$464$	0	0
$465$	−1280.00	−0.127653
$466$	0	0
$467$	−9876.00	−0.978601	−0.489301	−	0.872115i	$-0.662748\pi$
$467$	−9876.00	−0.978601	−0.489301	+	0.872115i	$0.662748\pi$
$468$	0	0
$469$	4512.00	0.444232
$470$	0	0
$471$	−12568.0	−1.22952
$472$	0	0
$473$	2288.00	0.222415
$474$	0	0
$475$	5324.00	0.514278
$476$	0	0
$477$	2662.00	0.255523
$478$	0	0
$479$	352.000	0.0335768	0.0167884	−	0.999859i	$-0.494656\pi$
$479$	352.000	0.0335768	0.0167884	+	0.999859i	$0.494656\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−5376.00	−0.506452
$484$	0	0
$485$	956.000	0.0895046
$486$	0	0
$487$	15176.0	1.41209	0.706047	−	0.708165i	$-0.250477\pi$
$487$	15176.0	1.41209	0.706047	+	0.708165i	$0.250477\pi$
$488$	0	0
$489$	12144.0	1.12305
$490$	0	0
$491$	−8844.00	−0.812880	−0.406440	−	0.913677i	$-0.633230\pi$
$491$	−8844.00	−0.812880	−0.406440	+	0.913677i	$0.633230\pi$
$492$	0	0
$493$	9900.00	0.904409
$494$	0	0
$495$	−968.000	−0.0878957
$496$	0	0
$497$	17472.0	1.57691
$498$	0	0
$499$	−19404.0	−1.74077	−0.870383	−	0.492375i	$-0.836129\pi$
$499$	−19404.0	−1.74077	−0.870383	+	0.492375i	$0.836129\pi$
$500$	0	0
$501$	−1056.00	−0.0941689
$502$	0	0
$503$	16488.0	1.46156	0.730779	−	0.682614i	$-0.239157\pi$
$503$	16488.0	1.46156	0.730779	+	0.682614i	$0.239157\pi$
$504$	0	0
$505$	3132.00	0.275984
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12954.0	1.12805	0.564024	−	0.825759i	$-0.309253\pi$
$509$	12954.0	1.12805	0.564024	+	0.825759i	$0.309253\pi$
$510$	0	0
$511$	3696.00	0.319964
$512$	0	0
$513$	−6688.00	−0.575599
$514$	0	0
$515$	−1936.00	−0.165651
$516$	0	0
$517$	−23232.0	−1.97629
$518$	0	0
$519$	11304.0	0.956051
$520$	0	0
$521$	10970.0	0.922465	0.461233	−	0.887279i	$-0.347407\pi$
$521$	10970.0	0.922465	0.461233	+	0.887279i	$0.347407\pi$
$522$	0	0
$523$	−16940.0	−1.41632	−0.708159	−	0.706053i	$-0.750474\pi$
$523$	−16940.0	−1.41632	−0.708159	+	0.706053i	$0.750474\pi$
$524$	0	0
$525$	−11616.0	−0.965645
$526$	0	0
$527$	8000.00	0.661263
$528$	0	0
$529$	−9031.00	−0.742254
$530$	0	0
$531$	−7348.00	−0.600520
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1560.00	−0.126065
$536$	0	0
$537$	−12336.0	−0.991318
$538$	0	0
$539$	10252.0	0.819267
$540$	0	0
$541$	−198.000	−0.0157351	−0.00786755	−	0.999969i	$-0.502504\pi$
$541$	−198.000	−0.0157351	−0.00786755	+	0.999969i	$0.502504\pi$
$542$	0	0
$543$	9672.00	0.764393
$544$	0	0
$545$	3988.00	0.313444
$546$	0	0
$547$	−15268.0	−1.19344	−0.596721	−	0.802449i	$-0.703530\pi$
$547$	−15268.0	−1.19344	−0.596721	+	0.802449i	$0.703530\pi$
$548$	0	0
$549$	−6050.00	−0.470324
$550$	0	0
$551$	−8712.00	−0.673582
$552$	0	0
$553$	15744.0	1.21067
$554$	0	0
$555$	−1296.00	−0.0991210
$556$	0	0
$557$	−20854.0	−1.58638	−0.793189	−	0.608976i	$-0.791581\pi$
$557$	−20854.0	−1.58638	−0.793189	+	0.608976i	$0.791581\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8800.00	−0.662275
$562$	0	0
$563$	−19316.0	−1.44595	−0.722977	−	0.690872i	$-0.757227\pi$
$563$	−19316.0	−1.44595	−0.722977	+	0.690872i	$0.757227\pi$
$564$	0	0
$565$	−1884.00	−0.140284
$566$	0	0
$567$	7464.00	0.552837
$568$	0	0
$569$	7018.00	0.517065	0.258532	−	0.966003i	$-0.416761\pi$
$569$	7018.00	0.517065	0.258532	+	0.966003i	$0.416761\pi$
$570$	0	0
$571$	24420.0	1.78975	0.894873	−	0.446320i	$-0.147266\pi$
$571$	24420.0	1.78975	0.894873	+	0.446320i	$0.147266\pi$
$572$	0	0
$573$	3840.00	0.279962
$574$	0	0
$575$	6776.00	0.491441
$576$	0	0
$577$	−23234.0	−1.67633	−0.838166	−	0.545415i	$-0.816372\pi$
$577$	−23234.0	−1.67633	−0.838166	+	0.545415i	$0.816372\pi$
$578$	0	0
$579$	11528.0	0.827439
$580$	0	0
$581$	5664.00	0.404445
$582$	0	0
$583$	−10648.0	−0.756424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10604.0	0.745611	0.372806	−	0.927909i	$-0.378396\pi$
$587$	10604.0	0.745611	0.372806	+	0.927909i	$0.378396\pi$
$588$	0	0
$589$	−7040.00	−0.492493
$590$	0	0
$591$	4344.00	0.302349
$592$	0	0
$593$	13838.0	0.958277	0.479139	−	0.877739i	$-0.340949\pi$
$593$	13838.0	0.958277	0.479139	+	0.877739i	$0.340949\pi$
$594$	0	0
$595$	−2400.00	−0.165362
$596$	0	0
$597$	−352.000	−0.0241313
$598$	0	0
$599$	−3960.00	−0.270119	−0.135059	−	0.990837i	$-0.543123\pi$
$599$	−3960.00	−0.270119	−0.135059	+	0.990837i	$0.543123\pi$
$600$	0	0
$601$	−5942.00	−0.403293	−0.201647	−	0.979458i	$-0.564629\pi$
$601$	−5942.00	−0.403293	−0.201647	+	0.979458i	$0.564629\pi$
$602$	0	0
$603$	2068.00	0.139661
$604$	0	0
$605$	1210.00	0.0813116
$606$	0	0
$607$	−3040.00	−0.203278	−0.101639	−	0.994821i	$-0.532409\pi$
$607$	−3040.00	−0.203278	−0.101639	+	0.994821i	$0.532409\pi$
$608$	0	0
$609$	19008.0	1.26477
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2530.00	0.166698	0.0833489	−	0.996520i	$-0.473438\pi$
$613$	2530.00	0.166698	0.0833489	+	0.996520i	$0.473438\pi$
$614$	0	0
$615$	−1584.00	−0.103859
$616$	0	0
$617$	19206.0	1.25317	0.626584	−	0.779354i	$-0.284453\pi$
$617$	19206.0	1.25317	0.626584	+	0.779354i	$0.284453\pi$
$618$	0	0
$619$	−10996.0	−0.714001	−0.357000	−	0.934104i	$-0.616201\pi$
$619$	−10996.0	−0.714001	−0.357000	+	0.934104i	$0.616201\pi$
$620$	0	0
$621$	−8512.00	−0.550040
$622$	0	0
$623$	17136.0	1.10199
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	7744.00	0.493247
$628$	0	0
$629$	8100.00	0.513463
$630$	0	0
$631$	6680.00	0.421437	0.210718	−	0.977547i	$-0.432420\pi$
$631$	6680.00	0.421437	0.210718	+	0.977547i	$0.432420\pi$
$632$	0	0
$633$	13904.0	0.873040
$634$	0	0
$635$	2816.00	0.175984
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8008.00	0.495761
$640$	0	0
$641$	6274.00	0.386596	0.193298	−	0.981140i	$-0.438082\pi$
$641$	6274.00	0.386596	0.193298	+	0.981140i	$0.438082\pi$
$642$	0	0
$643$	−9084.00	−0.557135	−0.278568	−	0.960417i	$-0.589860\pi$
$643$	−9084.00	−0.557135	−0.278568	+	0.960417i	$0.589860\pi$
$644$	0	0
$645$	−416.000	−0.0253953
$646$	0	0
$647$	−23656.0	−1.43742	−0.718712	−	0.695308i	$-0.755268\pi$
$647$	−23656.0	−1.43742	−0.718712	+	0.695308i	$0.755268\pi$
$648$	0	0
$649$	29392.0	1.77771
$650$	0	0
$651$	15360.0	0.924740
$652$	0	0
$653$	−6762.00	−0.405234	−0.202617	−	0.979258i	$-0.564945\pi$
$653$	−6762.00	−0.405234	−0.202617	+	0.979258i	$0.564945\pi$
$654$	0	0
$655$	−5384.00	−0.321176
$656$	0	0
$657$	1694.00	0.100592
$658$	0	0
$659$	15276.0	0.902987	0.451494	−	0.892274i	$-0.350891\pi$
$659$	15276.0	0.902987	0.451494	+	0.892274i	$0.350891\pi$
$660$	0	0
$661$	−11054.0	−0.650455	−0.325228	−	0.945636i	$-0.605441\pi$
$661$	−11054.0	−0.650455	−0.325228	+	0.945636i	$0.605441\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2112.00	0.123158
$666$	0	0
$667$	−11088.0	−0.643672
$668$	0	0
$669$	3712.00	0.214520
$670$	0	0
$671$	24200.0	1.39230
$672$	0	0
$673$	−21278.0	−1.21873	−0.609366	−	0.792889i	$-0.708576\pi$
$673$	−21278.0	−1.21873	−0.609366	+	0.792889i	$0.708576\pi$
$674$	0	0
$675$	−18392.0	−1.04875
$676$	0	0
$677$	8926.00	0.506727	0.253363	−	0.967371i	$-0.418463\pi$
$677$	8926.00	0.506727	0.253363	+	0.967371i	$0.418463\pi$
$678$	0	0
$679$	−11472.0	−0.648387
$680$	0	0
$681$	624.000	0.0351127
$682$	0	0
$683$	−8116.00	−0.454685	−0.227343	−	0.973815i	$-0.573004\pi$
$683$	−8116.00	−0.454685	−0.227343	+	0.973815i	$0.573004\pi$
$684$	0	0
$685$	−3252.00	−0.181391
$686$	0	0
$687$	−6536.00	−0.362975
$688$	0	0
$689$	0	0
$690$	0	0
$691$	11764.0	0.647646	0.323823	−	0.946118i	$-0.395032\pi$
$691$	11764.0	0.647646	0.323823	+	0.946118i	$0.395032\pi$
$692$	0	0
$693$	11616.0	0.636732
$694$	0	0
$695$	−1368.00	−0.0746636
$696$	0	0
$697$	9900.00	0.538005
$698$	0	0
$699$	3608.00	0.195232
$700$	0	0
$701$	−4698.00	−0.253126	−0.126563	−	0.991959i	$-0.540395\pi$
$701$	−4698.00	−0.253126	−0.126563	+	0.991959i	$0.540395\pi$
$702$	0	0
$703$	−7128.00	−0.382415
$704$	0	0
$705$	4224.00	0.225653
$706$	0	0
$707$	−37584.0	−1.99928
$708$	0	0
$709$	−24638.0	−1.30508	−0.652538	−	0.757756i	$-0.726296\pi$
$709$	−24638.0	−1.30508	−0.652538	+	0.757756i	$0.726296\pi$
$710$	0	0
$711$	7216.00	0.380620
$712$	0	0
$713$	−8960.00	−0.470624
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6464.00	0.336684
$718$	0	0
$719$	16624.0	0.862268	0.431134	−	0.902288i	$-0.358114\pi$
$719$	16624.0	0.862268	0.431134	+	0.902288i	$0.358114\pi$
$720$	0	0
$721$	23232.0	1.20001
$722$	0	0
$723$	19272.0	0.991332
$724$	0	0
$725$	−23958.0	−1.22728
$726$	0	0
$727$	30216.0	1.54147	0.770735	−	0.637155i	$-0.219889\pi$
$727$	30216.0	1.54147	0.770735	+	0.637155i	$0.219889\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	2600.00	0.131552
$732$	0	0
$733$	3322.00	0.167395	0.0836977	−	0.996491i	$-0.473327\pi$
$733$	3322.00	0.167395	0.0836977	+	0.996491i	$0.473327\pi$
$734$	0	0
$735$	−1864.00	−0.0935438
$736$	0	0
$737$	−8272.00	−0.413437
$738$	0	0
$739$	14692.0	0.731331	0.365666	−	0.930746i	$-0.380841\pi$
$739$	14692.0	0.731331	0.365666	+	0.930746i	$0.380841\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28600.0	−1.41216	−0.706078	−	0.708134i	$-0.749537\pi$
$743$	−28600.0	−1.41216	−0.706078	+	0.708134i	$0.749537\pi$
$744$	0	0
$745$	−604.000	−0.0297032
$746$	0	0
$747$	2596.00	0.127152
$748$	0	0
$749$	18720.0	0.913236
$750$	0	0
$751$	−29616.0	−1.43902	−0.719509	−	0.694483i	$-0.755633\pi$
$751$	−29616.0	−1.43902	−0.719509	+	0.694483i	$0.755633\pi$
$752$	0	0
$753$	8560.00	0.414268
$754$	0	0
$755$	−2704.00	−0.130343
$756$	0	0
$757$	2894.00	0.138949	0.0694744	−	0.997584i	$-0.477868\pi$
$757$	2894.00	0.138949	0.0694744	+	0.997584i	$0.477868\pi$
$758$	0	0
$759$	9856.00	0.471344
$760$	0	0
$761$	−14762.0	−0.703183	−0.351591	−	0.936154i	$-0.614359\pi$
$761$	−14762.0	−0.703183	−0.351591	+	0.936154i	$0.614359\pi$
$762$	0	0
$763$	−47856.0	−2.27065
$764$	0	0
$765$	−1100.00	−0.0519877
$766$	0	0
$767$	0	0
$768$	0	0
$769$	7678.00	0.360047	0.180023	−	0.983662i	$-0.442383\pi$
$769$	7678.00	0.360047	0.180023	+	0.983662i	$0.442383\pi$
$770$	0	0
$771$	−3080.00	−0.143870
$772$	0	0
$773$	−27390.0	−1.27445	−0.637225	−	0.770678i	$-0.719918\pi$
$773$	−27390.0	−1.27445	−0.637225	+	0.770678i	$0.719918\pi$
$774$	0	0
$775$	−19360.0	−0.897331
$776$	0	0
$777$	15552.0	0.718050
$778$	0	0
$779$	−8712.00	−0.400693
$780$	0	0
$781$	−32032.0	−1.46760
$782$	0	0
$783$	30096.0	1.37362
$784$	0	0
$785$	6284.00	0.285714
$786$	0	0
$787$	−19756.0	−0.894823	−0.447411	−	0.894328i	$-0.647654\pi$
$787$	−19756.0	−0.894823	−0.447411	+	0.894328i	$0.647654\pi$
$788$	0	0
$789$	29600.0	1.33560
$790$	0	0
$791$	22608.0	1.01624
$792$	0	0
$793$	0	0
$794$	0	0
$795$	1936.00	0.0863684
$796$	0	0
$797$	38854.0	1.72682	0.863412	−	0.504499i	$-0.168323\pi$
$797$	38854.0	1.72682	0.863412	+	0.504499i	$0.168323\pi$
$798$	0	0
$799$	−26400.0	−1.16892
$800$	0	0
$801$	7854.00	0.346451
$802$	0	0
$803$	−6776.00	−0.297783
$804$	0	0
$805$	2688.00	0.117689
$806$	0	0
$807$	11176.0	0.487502
$808$	0	0
$809$	−14278.0	−0.620504	−0.310252	−	0.950654i	$-0.600413\pi$
$809$	−14278.0	−0.620504	−0.310252	+	0.950654i	$0.600413\pi$
$810$	0	0
$811$	716.000	0.0310014	0.0155007	−	0.999880i	$-0.495066\pi$
$811$	716.000	0.0310014	0.0155007	+	0.999880i	$0.495066\pi$
$812$	0	0
$813$	34496.0	1.48810
$814$	0	0
$815$	−6072.00	−0.260973
$816$	0	0
$817$	−2288.00	−0.0979767
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23538.0	1.00059	0.500293	−	0.865856i	$-0.333225\pi$
$821$	23538.0	1.00059	0.500293	+	0.865856i	$0.333225\pi$
$822$	0	0
$823$	−6616.00	−0.280218	−0.140109	−	0.990136i	$-0.544745\pi$
$823$	−6616.00	−0.280218	−0.140109	+	0.990136i	$0.544745\pi$
$824$	0	0
$825$	21296.0	0.898705
$826$	0	0
$827$	−27236.0	−1.14521	−0.572605	−	0.819831i	$-0.694067\pi$
$827$	−27236.0	−1.14521	−0.572605	+	0.819831i	$0.694067\pi$
$828$	0	0
$829$	12070.0	0.505680	0.252840	−	0.967508i	$-0.418635\pi$
$829$	12070.0	0.505680	0.252840	+	0.967508i	$0.418635\pi$
$830$	0	0
$831$	7496.00	0.312916
$832$	0	0
$833$	11650.0	0.484572
$834$	0	0
$835$	528.000	0.0218829
$836$	0	0
$837$	24320.0	1.00433
$838$	0	0
$839$	42024.0	1.72924	0.864618	−	0.502429i	$-0.167560\pi$
$839$	42024.0	1.72924	0.864618	+	0.502429i	$0.167560\pi$
$840$	0	0
$841$	14815.0	0.607446
$842$	0	0
$843$	13352.0	0.545513
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14520.0	−0.589036
$848$	0	0
$849$	−28688.0	−1.15968
$850$	0	0
$851$	−9072.00	−0.365434
$852$	0	0
$853$	−2414.00	−0.0968978	−0.0484489	−	0.998826i	$-0.515428\pi$
$853$	−2414.00	−0.0968978	−0.0484489	+	0.998826i	$0.515428\pi$
$854$	0	0
$855$	968.000	0.0387192
$856$	0	0
$857$	−37686.0	−1.50213	−0.751067	−	0.660226i	$-0.770461\pi$
$857$	−37686.0	−1.50213	−0.751067	+	0.660226i	$0.770461\pi$
$858$	0	0
$859$	40644.0	1.61438	0.807192	−	0.590289i	$-0.200986\pi$
$859$	40644.0	1.61438	0.807192	+	0.590289i	$0.200986\pi$
$860$	0	0
$861$	19008.0	0.752370
$862$	0	0
$863$	18656.0	0.735872	0.367936	−	0.929851i	$-0.380065\pi$
$863$	18656.0	0.735872	0.367936	+	0.929851i	$0.380065\pi$
$864$	0	0
$865$	−5652.00	−0.222166
$866$	0	0
$867$	9652.00	0.378084
$868$	0	0
$869$	−28864.0	−1.12675
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−5258.00	−0.203845
$874$	0	0
$875$	11808.0	0.456209
$876$	0	0
$877$	13002.0	0.500623	0.250311	−	0.968165i	$-0.419467\pi$
$877$	13002.0	0.500623	0.250311	+	0.968165i	$0.419467\pi$
$878$	0	0
$879$	20856.0	0.800291
$880$	0	0
$881$	49490.0	1.89258	0.946289	−	0.323323i	$-0.104800\pi$
$881$	49490.0	1.89258	0.946289	+	0.323323i	$0.104800\pi$
$882$	0	0
$883$	1100.00	0.0419229	0.0209615	−	0.999780i	$-0.493327\pi$
$883$	1100.00	0.0419229	0.0209615	+	0.999780i	$0.493327\pi$
$884$	0	0
$885$	−5344.00	−0.202979
$886$	0	0
$887$	−14104.0	−0.533896	−0.266948	−	0.963711i	$-0.586015\pi$
$887$	−14104.0	−0.533896	−0.266948	+	0.963711i	$0.586015\pi$
$888$	0	0
$889$	−33792.0	−1.27486
$890$	0	0
$891$	−13684.0	−0.514513
$892$	0	0
$893$	23232.0	0.870581
$894$	0	0
$895$	6168.00	0.230361
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31680.0	1.17529
$900$	0	0
$901$	−12100.0	−0.447402
$902$	0	0
$903$	4992.00	0.183968
$904$	0	0
$905$	−4836.00	−0.177629
$906$	0	0
$907$	−12716.0	−0.465521	−0.232761	−	0.972534i	$-0.574776\pi$
$907$	−12716.0	−0.465521	−0.232761	+	0.972534i	$0.574776\pi$
$908$	0	0
$909$	−17226.0	−0.628548
$910$	0	0
$911$	−39632.0	−1.44135	−0.720673	−	0.693275i	$-0.756167\pi$
$911$	−39632.0	−1.44135	−0.720673	+	0.693275i	$0.756167\pi$
$912$	0	0
$913$	−10384.0	−0.376408
$914$	0	0
$915$	−4400.00	−0.158972
$916$	0	0
$917$	64608.0	2.32666
$918$	0	0
$919$	5704.00	0.204742	0.102371	−	0.994746i	$-0.467357\pi$
$919$	5704.00	0.204742	0.102371	+	0.994746i	$0.467357\pi$
$920$	0	0
$921$	1584.00	0.0566716
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−19602.0	−0.696767
$926$	0	0
$927$	10648.0	0.377267
$928$	0	0
$929$	−8162.00	−0.288252	−0.144126	−	0.989559i	$-0.546037\pi$
$929$	−8162.00	−0.288252	−0.144126	+	0.989559i	$0.546037\pi$
$930$	0	0
$931$	−10252.0	−0.360898
$932$	0	0
$933$	16224.0	0.569293
$934$	0	0
$935$	4400.00	0.153899
$936$	0	0
$937$	−55110.0	−1.92141	−0.960707	−	0.277564i	$-0.910473\pi$
$937$	−55110.0	−1.92141	−0.960707	+	0.277564i	$0.910473\pi$
$938$	0	0
$939$	−8616.00	−0.299438
$940$	0	0
$941$	−16374.0	−0.567245	−0.283622	−	0.958936i	$-0.591536\pi$
$941$	−16374.0	−0.567245	−0.283622	+	0.958936i	$0.591536\pi$
$942$	0	0
$943$	−11088.0	−0.382900
$944$	0	0
$945$	−7296.00	−0.251152
$946$	0	0
$947$	−8460.00	−0.290299	−0.145149	−	0.989410i	$-0.546366\pi$
$947$	−8460.00	−0.290299	−0.145149	+	0.989410i	$0.546366\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−29544.0	−1.00739
$952$	0	0
$953$	−20502.0	−0.696878	−0.348439	−	0.937331i	$-0.613288\pi$
$953$	−20502.0	−0.696878	−0.348439	+	0.937331i	$0.613288\pi$
$954$	0	0
$955$	−1920.00	−0.0650573
$956$	0	0
$957$	−34848.0	−1.17709
$958$	0	0
$959$	39024.0	1.31403
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	8580.00	0.287110
$964$	0	0
$965$	−5764.00	−0.192280
$966$	0	0
$967$	36520.0	1.21448	0.607241	−	0.794518i	$-0.292276\pi$
$967$	36520.0	1.21448	0.607241	+	0.794518i	$0.292276\pi$
$968$	0	0
$969$	8800.00	0.291741
$970$	0	0
$971$	20244.0	0.669064	0.334532	−	0.942384i	$-0.391422\pi$
$971$	20244.0	0.669064	0.334532	+	0.942384i	$0.391422\pi$
$972$	0	0
$973$	16416.0	0.540876
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50034.0	−1.63841	−0.819206	−	0.573499i	$-0.805586\pi$
$977$	−50034.0	−1.63841	−0.819206	+	0.573499i	$0.805586\pi$
$978$	0	0
$979$	−31416.0	−1.02560
$980$	0	0
$981$	−21934.0	−0.713862
$982$	0	0
$983$	−37128.0	−1.20468	−0.602339	−	0.798240i	$-0.705765\pi$
$983$	−37128.0	−1.20468	−0.602339	+	0.798240i	$0.705765\pi$
$984$	0	0
$985$	−2172.00	−0.0702596
$986$	0	0
$987$	−50688.0	−1.63467
$988$	0	0
$989$	−2912.00	−0.0936261
$990$	0	0
$991$	27808.0	0.891373	0.445686	−	0.895189i	$-0.352960\pi$
$991$	27808.0	0.891373	0.445686	+	0.895189i	$0.352960\pi$
$992$	0	0
$993$	−4528.00	−0.144705
$994$	0	0
$995$	176.000	0.00560761
$996$	0	0
$997$	−28514.0	−0.905765	−0.452882	−	0.891570i	$-0.649604\pi$
$997$	−28514.0	−0.905765	−0.452882	+	0.891570i	$0.649604\pi$
$998$	0	0
$999$	24624.0	0.779849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.4.a.a.1.1		1
13.12	even	2		8.4.a.a.1.1	✓	1
39.38	odd	2		72.4.a.c.1.1		1
52.51	odd	2		16.4.a.a.1.1		1
65.12	odd	4		200.4.c.e.49.2		2
65.38	odd	4		200.4.c.e.49.1		2
65.64	even	2		200.4.a.g.1.1		1
91.12	odd	6		392.4.i.b.361.1		2
91.25	even	6		392.4.i.g.177.1		2
91.38	odd	6		392.4.i.b.177.1		2
91.51	even	6		392.4.i.g.361.1		2
91.90	odd	2		392.4.a.e.1.1		1
104.51	odd	2		64.4.a.b.1.1		1
104.77	even	2		64.4.a.d.1.1		1
117.25	even	6		648.4.i.h.433.1		2
117.38	odd	6		648.4.i.e.433.1		2
117.77	odd	6		648.4.i.e.217.1		2
117.103	even	6		648.4.i.h.217.1		2
143.142	odd	2		968.4.a.a.1.1		1
156.155	even	2		144.4.a.e.1.1		1
195.38	even	4		1800.4.f.u.649.1		2
195.77	even	4		1800.4.f.u.649.2		2
195.194	odd	2		1800.4.a.d.1.1		1
208.51	odd	4		256.4.b.g.129.2		2
208.77	even	4		256.4.b.a.129.1		2
208.155	odd	4		256.4.b.g.129.1		2
208.181	even	4		256.4.b.a.129.2		2
221.220	even	2		2312.4.a.a.1.1		1
260.103	even	4		400.4.c.i.49.2		2
260.207	even	4		400.4.c.i.49.1		2
260.259	odd	2		400.4.a.g.1.1		1
312.77	odd	2		576.4.a.k.1.1		1
312.155	even	2		576.4.a.j.1.1		1
364.363	even	2		784.4.a.e.1.1		1
520.259	odd	2		1600.4.a.bm.1.1		1
520.389	even	2		1600.4.a.o.1.1		1
572.571	even	2		1936.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.a.a.1.1	✓	1	13.12	even	2
16.4.a.a.1.1		1	52.51	odd	2
64.4.a.b.1.1		1	104.51	odd	2
64.4.a.d.1.1		1	104.77	even	2
72.4.a.c.1.1		1	39.38	odd	2
144.4.a.e.1.1		1	156.155	even	2
200.4.a.g.1.1		1	65.64	even	2
200.4.c.e.49.1		2	65.38	odd	4
200.4.c.e.49.2		2	65.12	odd	4
256.4.b.a.129.1		2	208.77	even	4
256.4.b.a.129.2		2	208.181	even	4
256.4.b.g.129.1		2	208.155	odd	4
256.4.b.g.129.2		2	208.51	odd	4
392.4.a.e.1.1		1	91.90	odd	2
392.4.i.b.177.1		2	91.38	odd	6
392.4.i.b.361.1		2	91.12	odd	6
392.4.i.g.177.1		2	91.25	even	6
392.4.i.g.361.1		2	91.51	even	6
400.4.a.g.1.1		1	260.259	odd	2
400.4.c.i.49.1		2	260.207	even	4
400.4.c.i.49.2		2	260.103	even	4
576.4.a.j.1.1		1	312.155	even	2
576.4.a.k.1.1		1	312.77	odd	2
648.4.i.e.217.1		2	117.77	odd	6
648.4.i.e.433.1		2	117.38	odd	6
648.4.i.h.217.1		2	117.103	even	6
648.4.i.h.433.1		2	117.25	even	6
784.4.a.e.1.1		1	364.363	even	2
968.4.a.a.1.1		1	143.142	odd	2
1352.4.a.a.1.1		1	1.1	even	1	trivial
1600.4.a.o.1.1		1	520.389	even	2
1600.4.a.bm.1.1		1	520.259	odd	2
1800.4.a.d.1.1		1	195.194	odd	2
1800.4.f.u.649.1		2	195.38	even	4
1800.4.f.u.649.2		2	195.77	even	4
1936.4.a.l.1.1		1	572.571	even	2
2312.4.a.a.1.1		1	221.220	even	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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more