LMFDB - Embedded newform 144.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, 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("144.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	144.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:	$8.49627504083$
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	$\chi$	$=$	144.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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
$5$	2.00000	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
$9$	0	0
$10$	0	0
$11$	−44.0000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−44.0000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	22.0000	0.469362	0.234681	−	0.972072i	$-0.424595\pi$
$13$	22.0000	0.469362	0.234681	+	0.972072i	$0.424595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−50.0000	−0.713340	−0.356670	−	0.934230i	$-0.616088\pi$
$17$	−50.0000	−0.713340	−0.356670	+	0.934230i	$0.616088\pi$
$18$	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
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	−198.000	−1.26785	−0.633925	−	0.773394i	$-0.718557\pi$
$29$	−198.000	−1.26785	−0.633925	+	0.773394i	$0.718557\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	−48.0000	−0.231814
$36$	0	0
$37$	−162.000	−0.719801	−0.359900	−	0.932991i	$-0.617189\pi$
$37$	−162.000	−0.719801	−0.359900	+	0.932991i	$0.617189\pi$
$38$	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	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	528.000	1.63865	0.819327	−	0.573327i	$-0.194347\pi$
$47$	528.000	1.63865	0.819327	+	0.573327i	$0.194347\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	0	0
$52$	0	0
$53$	242.000	0.627194	0.313597	−	0.949556i	$-0.398466\pi$
$53$	242.000	0.627194	0.313597	+	0.949556i	$0.398466\pi$
$54$	0	0
$55$	−88.0000	−0.215744
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−668.000	−1.47400	−0.737002	−	0.675891i	$-0.763759\pi$
$59$	−668.000	−1.47400	−0.737002	+	0.675891i	$0.763759\pi$
$60$	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
$63$	0	0
$64$	0	0
$65$	44.0000	0.0839620
$66$	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
$69$	0	0
$70$	0	0
$71$	728.000	1.21687	0.608435	−	0.793604i	$-0.291798\pi$
$71$	728.000	1.21687	0.608435	+	0.793604i	$0.291798\pi$
$72$	0	0
$73$	154.000	0.246909	0.123454	−	0.992350i	$-0.460603\pi$
$73$	154.000	0.246909	0.123454	+	0.992350i	$0.460603\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1056.00	1.56289
$78$	0	0
$79$	656.000	0.934250	0.467125	−	0.884191i	$-0.345290\pi$
$79$	656.000	0.934250	0.467125	+	0.884191i	$0.345290\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	236.000	0.312101	0.156050	−	0.987749i	$-0.450124\pi$
$83$	236.000	0.312101	0.156050	+	0.987749i	$0.450124\pi$
$84$	0	0
$85$	−100.000	−0.127606
$86$	0	0
$87$	0	0
$88$	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
$91$	−528.000	−0.608236
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−88.0000	−0.0950380
$96$	0	0
$97$	−478.000	−0.500346	−0.250173	−	0.968201i	$-0.580487\pi$
$97$	−478.000	−0.500346	−0.250173	+	0.968201i	$0.580487\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1566.00	−1.54280	−0.771400	−	0.636350i	$-0.780443\pi$
$101$	−1566.00	−1.54280	−0.771400	+	0.636350i	$0.780443\pi$
$102$	0	0
$103$	968.000	0.926018	0.463009	−	0.886354i	$-0.346770\pi$
$103$	968.000	0.926018	0.463009	+	0.886354i	$0.346770\pi$
$104$	0	0
$105$	0	0
$106$	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	$-0.839866\pi$
$109$	−1994.00	−1.75221	−0.876103	+	0.482123i	$0.839866\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	942.000	0.784212	0.392106	−	0.919920i	$-0.371747\pi$
$113$	942.000	0.784212	0.392106	+	0.919920i	$0.371747\pi$
$114$	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
$123$	0	0
$124$	0	0
$125$	−492.000	−0.352047
$126$	0	0
$127$	−1408.00	−0.983778	−0.491889	−	0.870658i	$-0.663693\pi$
$127$	−1408.00	−0.983778	−0.491889	+	0.870658i	$0.663693\pi$
$128$	0	0
$129$	0	0
$130$	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
$135$	0	0
$136$	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	$-0.433080\pi$
$139$	684.000	0.417382	0.208691	+	0.977982i	$0.433080\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−968.000	−0.566072
$144$	0	0
$145$	−396.000	−0.226800
$146$	0	0
$147$	0	0
$148$	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
$153$	0	0
$154$	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
$159$	0	0
$160$	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
$165$	0	0
$166$	0	0
$167$	−264.000	−0.122329	−0.0611645	−	0.998128i	$-0.519481\pi$
$167$	−264.000	−0.122329	−0.0611645	+	0.998128i	$0.519481\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2826.00	1.24195	0.620973	−	0.783832i	$-0.286737\pi$
$173$	2826.00	1.24195	0.620973	+	0.783832i	$0.286737\pi$
$174$	0	0
$175$	2904.00	1.25441
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	−324.000	−0.128762
$186$	0	0
$187$	2200.00	0.860320
$188$	0	0
$189$	0	0
$190$	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	$-0.319392\pi$
$193$	2882.00	1.07488	0.537438	+	0.843304i	$0.319392\pi$
$194$	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	$-0.504989\pi$
$199$	−88.0000	−0.0313475	−0.0156738	+	0.999877i	$0.504989\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4752.00	1.64298
$204$	0	0
$205$	396.000	0.134916
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1936.00	0.640746
$210$	0	0
$211$	3476.00	1.13411	0.567056	−	0.823679i	$-0.308082\pi$
$211$	3476.00	1.13411	0.567056	+	0.823679i	$0.308082\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−104.000	−0.0329895
$216$	0	0
$217$	−3840.00	−1.20127
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1100.00	−0.334815
$222$	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
$225$	0	0
$226$	0	0
$227$	156.000	0.0456127	0.0228064	−	0.999740i	$-0.492740\pi$
$227$	156.000	0.0456127	0.0228064	+	0.999740i	$0.492740\pi$
$228$	0	0
$229$	−1634.00	−0.471519	−0.235759	−	0.971811i	$-0.575758\pi$
$229$	−1634.00	−0.471519	−0.235759	+	0.971811i	$0.575758\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	902.000	0.253614	0.126807	−	0.991927i	$-0.459527\pi$
$233$	902.000	0.253614	0.126807	+	0.991927i	$0.459527\pi$
$234$	0	0
$235$	1056.00	0.293131
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1616.00	0.437365	0.218683	−	0.975796i	$-0.429824\pi$
$239$	1616.00	0.437365	0.218683	+	0.975796i	$0.429824\pi$
$240$	0	0
$241$	4818.00	1.28778	0.643889	−	0.765119i	$-0.277320\pi$
$241$	4818.00	1.28778	0.643889	+	0.765119i	$0.277320\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	466.000	0.121517
$246$	0	0
$247$	−968.000	−0.249362
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	−770.000	−0.186892	−0.0934461	−	0.995624i	$-0.529788\pi$
$257$	−770.000	−0.186892	−0.0934461	+	0.995624i	$0.529788\pi$
$258$	0	0
$259$	3888.00	0.932774
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	2794.00	0.633283	0.316642	−	0.948545i	$-0.397445\pi$
$269$	2794.00	0.633283	0.316642	+	0.948545i	$0.397445\pi$
$270$	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
$279$	0	0
$280$	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	$-0.771508\pi$
$283$	−7172.00	−1.50647	−0.753235	+	0.657751i	$0.771508\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4752.00	−0.977358
$288$	0	0
$289$	−2413.00	−0.491146
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	0	0
$299$	−1232.00	−0.238289
$300$	0	0
$301$	1248.00	0.238982
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	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
$315$	0	0
$316$	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
$321$	0	0
$322$	0	0
$323$	2200.00	0.378982
$324$	0	0
$325$	−2662.00	−0.454342
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	−7040.00	−1.11800
$342$	0	0
$343$	2640.00	0.415588
$344$	0	0
$345$	0	0
$346$	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	$-0.666727\pi$
$349$	−6522.00	−1.00033	−0.500164	+	0.865931i	$0.666727\pi$
$350$	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
$357$	0	0
$358$	0	0
$359$	1848.00	0.271682	0.135841	−	0.990731i	$-0.456626\pi$
$359$	1848.00	0.271682	0.135841	+	0.990731i	$0.456626\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	0	0
$364$	0	0
$365$	308.000	0.0441684
$366$	0	0
$367$	−7120.00	−1.01270	−0.506350	−	0.862328i	$-0.669006\pi$
$367$	−7120.00	−1.01270	−0.506350	+	0.862328i	$0.669006\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	−4356.00	−0.595081
$378$	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
$381$	0	0
$382$	0	0
$383$	10368.0	1.38324	0.691619	−	0.722263i	$-0.256898\pi$
$383$	10368.0	1.38324	0.691619	+	0.722263i	$0.256898\pi$
$384$	0	0
$385$	2112.00	0.279578
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8830.00	−1.15090	−0.575448	−	0.817838i	$-0.695172\pi$
$389$	−8830.00	−1.15090	−0.575448	+	0.817838i	$0.695172\pi$
$390$	0	0
$391$	2800.00	0.362154
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1312.00	0.167124
$396$	0	0
$397$	9878.00	1.24877	0.624386	−	0.781116i	$-0.285349\pi$
$397$	9878.00	1.24877	0.624386	+	0.781116i	$0.285349\pi$
$398$	0	0
$399$	0	0
$400$	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
$403$	3520.00	0.435096
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7128.00	0.868113
$408$	0	0
$409$	906.000	0.109533	0.0547663	−	0.998499i	$-0.482559\pi$
$409$	906.000	0.109533	0.0547663	+	0.998499i	$0.482559\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16032.0	1.91013
$414$	0	0
$415$	472.000	0.0558303
$416$	0	0
$417$	0	0
$418$	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	$-0.586591\pi$
$421$	−4642.00	−0.537381	−0.268690	+	0.963227i	$0.586591\pi$
$422$	0	0
$423$	0	0
$424$	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	$-0.488329\pi$
$431$	656.000	0.0733142	0.0366571	+	0.999328i	$0.488329\pi$
$432$	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
$435$	0	0
$436$	0	0
$437$	2464.00	0.269723
$438$	0	0
$439$	−5544.00	−0.602735	−0.301368	−	0.953508i	$-0.597443\pi$
$439$	−5544.00	−0.602735	−0.301368	+	0.953508i	$0.597443\pi$
$440$	0	0
$441$	0	0
$442$	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
$447$	0	0
$448$	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
$453$	0	0
$454$	0	0
$455$	−1056.00	−0.108804
$456$	0	0
$457$	1562.00	0.159885	0.0799423	−	0.996799i	$-0.474526\pi$
$457$	1562.00	0.159885	0.0799423	+	0.996799i	$0.474526\pi$
$458$	0	0
$459$	0	0
$460$	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
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	2288.00	0.222415
$474$	0	0
$475$	5324.00	0.514278
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−352.000	−0.0335768	−0.0167884	−	0.999859i	$-0.505344\pi$
$479$	−352.000	−0.0335768	−0.0167884	+	0.999859i	$0.505344\pi$
$480$	0	0
$481$	−3564.00	−0.337847
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	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
$495$	0	0
$496$	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
$501$	0	0
$502$	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
$513$	0	0
$514$	0	0
$515$	1936.00	0.165651
$516$	0	0
$517$	−23232.0	−1.97629
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10970.0	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$521$	−10970.0	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$522$	0	0
$523$	16940.0	1.41632	0.708159	−	0.706053i	$-0.249526\pi$
$523$	16940.0	1.41632	0.708159	+	0.706053i	$0.249526\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8000.00	−0.661263
$528$	0	0
$529$	−9031.00	−0.742254
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4356.00	0.353995
$534$	0	0
$535$	−1560.00	−0.126065
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10252.0	−0.819267
$540$	0	0
$541$	198.000	0.0157351	0.00786755	−	0.999969i	$-0.497496\pi$
$541$	198.000	0.0157351	0.00786755	+	0.999969i	$0.497496\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3988.00	−0.313444
$546$	0	0
$547$	15268.0	1.19344	0.596721	−	0.802449i	$-0.296470\pi$
$547$	15268.0	1.19344	0.596721	+	0.802449i	$0.296470\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8712.00	0.673582
$552$	0	0
$553$	−15744.0	−1.21067
$554$	0	0
$555$	0	0
$556$	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
$559$	−1144.00	−0.0865582
$560$	0	0
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	−7018.00	−0.517065	−0.258532	−	0.966003i	$-0.583239\pi$
$569$	−7018.00	−0.517065	−0.258532	+	0.966003i	$0.583239\pi$
$570$	0	0
$571$	−24420.0	−1.78975	−0.894873	−	0.446320i	$-0.852734\pi$
$571$	−24420.0	−1.78975	−0.894873	+	0.446320i	$0.852734\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6776.00	0.491441
$576$	0	0
$577$	23234.0	1.67633	0.838166	−	0.545415i	$-0.183628\pi$
$577$	23234.0	1.67633	0.838166	+	0.545415i	$0.183628\pi$
$578$	0	0
$579$	0	0
$580$	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	$-0.621604\pi$
$587$	−10604.0	−0.745611	−0.372806	+	0.927909i	$0.621604\pi$
$588$	0	0
$589$	−7040.00	−0.492493
$590$	0	0
$591$	0	0
$592$	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
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	1210.00	0.0813116
$606$	0	0
$607$	3040.00	0.203278	0.101639	−	0.994821i	$-0.467591\pi$
$607$	3040.00	0.203278	0.101639	+	0.994821i	$0.467591\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11616.0	0.769121
$612$	0	0
$613$	−2530.00	−0.166698	−0.0833489	−	0.996520i	$-0.526562\pi$
$613$	−2530.00	−0.166698	−0.0833489	+	0.996520i	$0.526562\pi$
$614$	0	0
$615$	0	0
$616$	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
$621$	0	0
$622$	0	0
$623$	17136.0	1.10199
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	−2816.00	−0.175984
$636$	0	0
$637$	5126.00	0.318838
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6274.00	−0.386596	−0.193298	−	0.981140i	$-0.561918\pi$
$641$	−6274.00	−0.386596	−0.193298	+	0.981140i	$0.561918\pi$
$642$	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
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	6762.00	0.405234	0.202617	−	0.979258i	$-0.435055\pi$
$653$	6762.00	0.405234	0.202617	+	0.979258i	$0.435055\pi$
$654$	0	0
$655$	−5384.00	−0.321176
$656$	0	0
$657$	0	0
$658$	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	$-0.394559\pi$
$661$	11054.0	0.650455	0.325228	+	0.945636i	$0.394559\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2112.00	0.123158
$666$	0	0
$667$	11088.0	0.643672
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−8926.00	−0.506727	−0.253363	−	0.967371i	$-0.581537\pi$
$677$	−8926.00	−0.506727	−0.253363	+	0.967371i	$0.581537\pi$
$678$	0	0
$679$	11472.0	0.648387
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8116.00	0.454685	0.227343	−	0.973815i	$-0.426996\pi$
$683$	8116.00	0.454685	0.227343	+	0.973815i	$0.426996\pi$
$684$	0	0
$685$	−3252.00	−0.181391
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5324.00	0.294381
$690$	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
$693$	0	0
$694$	0	0
$695$	1368.00	0.0746636
$696$	0	0
$697$	−9900.00	−0.538005
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4698.00	0.253126	0.126563	−	0.991959i	$-0.459605\pi$
$701$	4698.00	0.253126	0.126563	+	0.991959i	$0.459605\pi$
$702$	0	0
$703$	7128.00	0.382415
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37584.0	1.99928
$708$	0	0
$709$	24638.0	1.30508	0.652538	−	0.757756i	$-0.273704\pi$
$709$	24638.0	1.30508	0.652538	+	0.757756i	$0.273704\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8960.00	−0.470624
$714$	0	0
$715$	−1936.00	−0.101262
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	23958.0	1.22728
$726$	0	0
$727$	−30216.0	−1.54147	−0.770735	−	0.637155i	$-0.780111\pi$
$727$	−30216.0	−1.54147	−0.770735	+	0.637155i	$0.780111\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2600.00	0.131552
$732$	0	0
$733$	−3322.00	−0.167395	−0.0836977	−	0.996491i	$-0.526673\pi$
$733$	−3322.00	−0.167395	−0.0836977	+	0.996491i	$0.526673\pi$
$734$	0	0
$735$	0	0
$736$	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	$-0.250463\pi$
$743$	28600.0	1.41216	0.706078	+	0.708134i	$0.250463\pi$
$744$	0	0
$745$	−604.000	−0.0297032
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18720.0	0.913236
$750$	0	0
$751$	29616.0	1.43902	0.719509	−	0.694483i	$-0.244367\pi$
$751$	29616.0	1.43902	0.719509	+	0.694483i	$0.244367\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	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
$765$	0	0
$766$	0	0
$767$	−14696.0	−0.691841
$768$	0	0
$769$	−7678.00	−0.360047	−0.180023	−	0.983662i	$-0.557617\pi$
$769$	−7678.00	−0.360047	−0.180023	+	0.983662i	$0.557617\pi$
$770$	0	0
$771$	0	0
$772$	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
$777$	0	0
$778$	0	0
$779$	−8712.00	−0.400693
$780$	0	0
$781$	−32032.0	−1.46760
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	−22608.0	−1.01624
$792$	0	0
$793$	12100.0	0.541846
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38854.0	−1.72682	−0.863412	−	0.504499i	$-0.831677\pi$
$797$	−38854.0	−1.72682	−0.863412	+	0.504499i	$0.831677\pi$
$798$	0	0
$799$	−26400.0	−1.16892
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6776.00	−0.297783
$804$	0	0
$805$	2688.00	0.117689
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14278.0	0.620504	0.310252	−	0.950654i	$-0.399587\pi$
$809$	14278.0	0.620504	0.310252	+	0.950654i	$0.399587\pi$
$810$	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
$813$	0	0
$814$	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	$-0.455255\pi$
$823$	6616.00	0.280218	0.140109	+	0.990136i	$0.455255\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27236.0	1.14521	0.572605	−	0.819831i	$-0.305933\pi$
$827$	27236.0	1.14521	0.572605	+	0.819831i	$0.305933\pi$
$828$	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
$831$	0	0
$832$	0	0
$833$	−11650.0	−0.484572
$834$	0	0
$835$	−528.000	−0.0218829
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42024.0	−1.72924	−0.864618	−	0.502429i	$-0.832440\pi$
$839$	−42024.0	−1.72924	−0.864618	+	0.502429i	$0.832440\pi$
$840$	0	0
$841$	14815.0	0.607446
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3426.00	−0.139477
$846$	0	0
$847$	−14520.0	−0.589036
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9072.00	0.365434
$852$	0	0
$853$	2414.00	0.0968978	0.0484489	−	0.998826i	$-0.484572\pi$
$853$	2414.00	0.0968978	0.0484489	+	0.998826i	$0.484572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37686.0	1.50213	0.751067	−	0.660226i	$-0.229539\pi$
$857$	37686.0	1.50213	0.751067	+	0.660226i	$0.229539\pi$
$858$	0	0
$859$	−40644.0	−1.61438	−0.807192	−	0.590289i	$-0.799014\pi$
$859$	−40644.0	−1.61438	−0.807192	+	0.590289i	$0.799014\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18656.0	−0.735872	−0.367936	−	0.929851i	$-0.619935\pi$
$863$	−18656.0	−0.735872	−0.367936	+	0.929851i	$0.619935\pi$
$864$	0	0
$865$	5652.00	0.222166
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28864.0	−1.12675
$870$	0	0
$871$	−4136.00	−0.160899
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11808.0	0.456209
$876$	0	0
$877$	−13002.0	−0.500623	−0.250311	−	0.968165i	$-0.580533\pi$
$877$	−13002.0	−0.500623	−0.250311	+	0.968165i	$0.580533\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−49490.0	−1.89258	−0.946289	−	0.323323i	$-0.895200\pi$
$881$	−49490.0	−1.89258	−0.946289	+	0.323323i	$0.895200\pi$
$882$	0	0
$883$	−1100.00	−0.0419229	−0.0209615	−	0.999780i	$-0.506673\pi$
$883$	−1100.00	−0.0419229	−0.0209615	+	0.999780i	$0.506673\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	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
$903$	0	0
$904$	0	0
$905$	−4836.00	−0.177629
$906$	0	0
$907$	12716.0	0.465521	0.232761	−	0.972534i	$-0.425224\pi$
$907$	12716.0	0.465521	0.232761	+	0.972534i	$0.425224\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	64608.0	2.32666
$918$	0	0
$919$	−5704.00	−0.204742	−0.102371	−	0.994746i	$-0.532643\pi$
$919$	−5704.00	−0.204742	−0.102371	+	0.994746i	$0.532643\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16016.0	0.571152
$924$	0	0
$925$	19602.0	0.696767
$926$	0	0
$927$	0	0
$928$	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
$933$	0	0
$934$	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
$939$	0	0
$940$	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
$945$	0	0
$946$	0	0
$947$	8460.00	0.290299	0.145149	−	0.989410i	$-0.453634\pi$
$947$	8460.00	0.290299	0.145149	+	0.989410i	$0.453634\pi$
$948$	0	0
$949$	3388.00	0.115889
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20502.0	0.696878	0.348439	−	0.937331i	$-0.386712\pi$
$953$	20502.0	0.696878	0.348439	+	0.937331i	$0.386712\pi$
$954$	0	0
$955$	−1920.00	−0.0650573
$956$	0	0
$957$	0	0
$958$	0	0
$959$	39024.0	1.31403
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	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
$981$	0	0
$982$	0	0
$983$	37128.0	1.20468	0.602339	−	0.798240i	$-0.294235\pi$
$983$	37128.0	1.20468	0.602339	+	0.798240i	$0.294235\pi$
$984$	0	0
$985$	−2172.00	−0.0702596
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2912.00	0.0936261
$990$	0	0
$991$	−27808.0	−0.891373	−0.445686	−	0.895189i	$-0.647040\pi$
$991$	−27808.0	−0.891373	−0.445686	+	0.895189i	$0.647040\pi$
$992$	0	0
$993$	0	0
$994$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.a.e.1.1		1
3.2	odd	2		16.4.a.a.1.1		1
4.3	odd	2		72.4.a.c.1.1		1
8.3	odd	2		576.4.a.k.1.1		1
8.5	even	2		576.4.a.j.1.1		1
12.11	even	2		8.4.a.a.1.1	✓	1
15.2	even	4		400.4.c.i.49.1		2
15.8	even	4		400.4.c.i.49.2		2
15.14	odd	2		400.4.a.g.1.1		1
20.3	even	4		1800.4.f.u.649.1		2
20.7	even	4		1800.4.f.u.649.2		2
20.19	odd	2		1800.4.a.d.1.1		1
21.20	even	2		784.4.a.e.1.1		1
24.5	odd	2		64.4.a.b.1.1		1
24.11	even	2		64.4.a.d.1.1		1
33.32	even	2		1936.4.a.l.1.1		1
36.7	odd	6		648.4.i.e.433.1		2
36.11	even	6		648.4.i.h.433.1		2
36.23	even	6		648.4.i.h.217.1		2
36.31	odd	6		648.4.i.e.217.1		2
48.5	odd	4		256.4.b.g.129.1		2
48.11	even	4		256.4.b.a.129.2		2
48.29	odd	4		256.4.b.g.129.2		2
48.35	even	4		256.4.b.a.129.1		2
60.23	odd	4		200.4.c.e.49.1		2
60.47	odd	4		200.4.c.e.49.2		2
60.59	even	2		200.4.a.g.1.1		1
84.11	even	6		392.4.i.g.177.1		2
84.23	even	6		392.4.i.g.361.1		2
84.47	odd	6		392.4.i.b.361.1		2
84.59	odd	6		392.4.i.b.177.1		2
84.83	odd	2		392.4.a.e.1.1		1
120.29	odd	2		1600.4.a.bm.1.1		1
120.59	even	2		1600.4.a.o.1.1		1
132.131	odd	2		968.4.a.a.1.1		1
156.155	even	2		1352.4.a.a.1.1		1
204.203	even	2		2312.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.a.a.1.1	✓	1	12.11	even	2
16.4.a.a.1.1		1	3.2	odd	2
64.4.a.b.1.1		1	24.5	odd	2
64.4.a.d.1.1		1	24.11	even	2
72.4.a.c.1.1		1	4.3	odd	2
144.4.a.e.1.1		1	1.1	even	1	trivial
200.4.a.g.1.1		1	60.59	even	2
200.4.c.e.49.1		2	60.23	odd	4
200.4.c.e.49.2		2	60.47	odd	4
256.4.b.a.129.1		2	48.35	even	4
256.4.b.a.129.2		2	48.11	even	4
256.4.b.g.129.1		2	48.5	odd	4
256.4.b.g.129.2		2	48.29	odd	4
392.4.a.e.1.1		1	84.83	odd	2
392.4.i.b.177.1		2	84.59	odd	6
392.4.i.b.361.1		2	84.47	odd	6
392.4.i.g.177.1		2	84.11	even	6
392.4.i.g.361.1		2	84.23	even	6
400.4.a.g.1.1		1	15.14	odd	2
400.4.c.i.49.1		2	15.2	even	4
400.4.c.i.49.2		2	15.8	even	4
576.4.a.j.1.1		1	8.5	even	2
576.4.a.k.1.1		1	8.3	odd	2
648.4.i.e.217.1		2	36.31	odd	6
648.4.i.e.433.1		2	36.7	odd	6
648.4.i.h.217.1		2	36.23	even	6
648.4.i.h.433.1		2	36.11	even	6
784.4.a.e.1.1		1	21.20	even	2
968.4.a.a.1.1		1	132.131	odd	2
1352.4.a.a.1.1		1	156.155	even	2
1600.4.a.o.1.1		1	120.59	even	2
1600.4.a.bm.1.1		1	120.29	odd	2
1800.4.a.d.1.1		1	20.19	odd	2
1800.4.f.u.649.1		2	20.3	even	4
1800.4.f.u.649.2		2	20.7	even	4
1936.4.a.l.1.1		1	33.32	even	2
2312.4.a.a.1.1		1	204.203	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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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