LMFDB - Embedded newform 1872.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1872 = 2^{4} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 234)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1872.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{5} +26.0000 q^{7} +52.0000 q^{11} -13.0000 q^{13} -48.0000 q^{17} -18.0000 q^{19} -52.0000 q^{23} -121.000 q^{25} -224.000 q^{29} -310.000 q^{31} -52.0000 q^{35} -18.0000 q^{37} -330.000 q^{41} -328.000 q^{43} +616.000 q^{47} +333.000 q^{49} +324.000 q^{53} -104.000 q^{55} +188.000 q^{59} -110.000 q^{61} +26.0000 q^{65} -118.000 q^{67} -656.000 q^{71} -178.000 q^{73} +1352.00 q^{77} -836.000 q^{79} +60.0000 q^{83} +96.0000 q^{85} -870.000 q^{89} -338.000 q^{91} +36.0000 q^{95} +1238.00 q^{97} +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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.00000	−0.178885	−0.0894427	−	0.995992i	$-0.528509\pi$
$5$	−2.00000	−0.178885	−0.0894427	+	0.995992i	$0.528509\pi$
$6$	0	0
$7$	26.0000	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$7$	26.0000	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	52.0000	1.42533	0.712663	−	0.701506i	$-0.247489\pi$
$11$	52.0000	1.42533	0.712663	+	0.701506i	$0.247489\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−48.0000	−0.684806	−0.342403	−	0.939553i	$-0.611241\pi$
$17$	−48.0000	−0.684806	−0.342403	+	0.939553i	$0.611241\pi$
$18$	0	0
$19$	−18.0000	−0.217341	−0.108671	−	0.994078i	$-0.534659\pi$
$19$	−18.0000	−0.217341	−0.108671	+	0.994078i	$0.534659\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−52.0000	−0.471424	−0.235712	−	0.971823i	$-0.575742\pi$
$23$	−52.0000	−0.471424	−0.235712	+	0.971823i	$0.575742\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−224.000	−1.43434	−0.717168	−	0.696900i	$-0.754562\pi$
$29$	−224.000	−1.43434	−0.717168	+	0.696900i	$0.754562\pi$
$30$	0	0
$31$	−310.000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−310.000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−52.0000	−0.251132
$36$	0	0
$37$	−18.0000	−0.0799779	−0.0399889	−	0.999200i	$-0.512732\pi$
$37$	−18.0000	−0.0799779	−0.0399889	+	0.999200i	$0.512732\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−330.000	−1.25701	−0.628504	−	0.777806i	$-0.716332\pi$
$41$	−330.000	−1.25701	−0.628504	+	0.777806i	$0.716332\pi$
$42$	0	0
$43$	−328.000	−1.16324	−0.581622	−	0.813459i	$-0.697582\pi$
$43$	−328.000	−1.16324	−0.581622	+	0.813459i	$0.697582\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	616.000	1.91176	0.955881	−	0.293753i	$-0.0949045\pi$
$47$	616.000	1.91176	0.955881	+	0.293753i	$0.0949045\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	0	0
$51$	0	0
$52$	0	0
$53$	324.000	0.839714	0.419857	−	0.907590i	$-0.362080\pi$
$53$	324.000	0.839714	0.419857	+	0.907590i	$0.362080\pi$
$54$	0	0
$55$	−104.000	−0.254970
$56$	0	0
$57$	0	0
$58$	0	0
$59$	188.000	0.414839	0.207420	−	0.978252i	$-0.433493\pi$
$59$	188.000	0.414839	0.207420	+	0.978252i	$0.433493\pi$
$60$	0	0
$61$	−110.000	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$61$	−110.000	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	26.0000	0.0496139
$66$	0	0
$67$	−118.000	−0.215164	−0.107582	−	0.994196i	$-0.534311\pi$
$67$	−118.000	−0.215164	−0.107582	+	0.994196i	$0.534311\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−656.000	−1.09652	−0.548260	−	0.836308i	$-0.684709\pi$
$71$	−656.000	−1.09652	−0.548260	+	0.836308i	$0.684709\pi$
$72$	0	0
$73$	−178.000	−0.285388	−0.142694	−	0.989767i	$-0.545576\pi$
$73$	−178.000	−0.285388	−0.142694	+	0.989767i	$0.545576\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1352.00	2.00097
$78$	0	0
$79$	−836.000	−1.19060	−0.595300	−	0.803504i	$-0.702967\pi$
$79$	−836.000	−1.19060	−0.595300	+	0.803504i	$0.702967\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	60.0000	0.0793477	0.0396738	−	0.999213i	$-0.487368\pi$
$83$	60.0000	0.0793477	0.0396738	+	0.999213i	$0.487368\pi$
$84$	0	0
$85$	96.0000	0.122502
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−870.000	−1.03618	−0.518089	−	0.855327i	$-0.673356\pi$
$89$	−870.000	−1.03618	−0.518089	+	0.855327i	$0.673356\pi$
$90$	0	0
$91$	−338.000	−0.389363
$92$	0	0
$93$	0	0
$94$	0	0
$95$	36.0000	0.0388792
$96$	0	0
$97$	1238.00	1.29587	0.647937	−	0.761694i	$-0.275632\pi$
$97$	1238.00	1.29587	0.647937	+	0.761694i	$0.275632\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	920.000	0.906371	0.453185	−	0.891416i	$-0.350288\pi$
$101$	920.000	0.906371	0.453185	+	0.891416i	$0.350288\pi$
$102$	0	0
$103$	836.000	0.799743	0.399871	−	0.916571i	$-0.369055\pi$
$103$	836.000	0.799743	0.399871	+	0.916571i	$0.369055\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	260.000	0.234908	0.117454	−	0.993078i	$-0.462527\pi$
$107$	260.000	0.234908	0.117454	+	0.993078i	$0.462527\pi$
$108$	0	0
$109$	−2078.00	−1.82602	−0.913011	−	0.407936i	$-0.866249\pi$
$109$	−2078.00	−1.82602	−0.913011	+	0.407936i	$0.866249\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1156.00	0.962366	0.481183	−	0.876620i	$-0.340207\pi$
$113$	1156.00	0.962366	0.481183	+	0.876620i	$0.340207\pi$
$114$	0	0
$115$	104.000	0.0843309
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1248.00	−0.961378
$120$	0	0
$121$	1373.00	1.03156
$122$	0	0
$123$	0	0
$124$	0	0
$125$	492.000	0.352047
$126$	0	0
$127$	752.000	0.525427	0.262713	−	0.964874i	$-0.415383\pi$
$127$	752.000	0.525427	0.262713	+	0.964874i	$0.415383\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1972.00	−1.31522	−0.657612	−	0.753356i	$-0.728434\pi$
$131$	−1972.00	−1.31522	−0.657612	+	0.753356i	$0.728434\pi$
$132$	0	0
$133$	−468.000	−0.305118
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−258.000	−0.160894	−0.0804468	−	0.996759i	$-0.525635\pi$
$137$	−258.000	−0.160894	−0.0804468	+	0.996759i	$0.525635\pi$
$138$	0	0
$139$	−1468.00	−0.895785	−0.447893	−	0.894087i	$-0.647825\pi$
$139$	−1468.00	−0.895785	−0.447893	+	0.894087i	$0.647825\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−676.000	−0.395314
$144$	0	0
$145$	448.000	0.256582
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3414.00	−1.87709	−0.938543	−	0.345163i	$-0.887824\pi$
$149$	−3414.00	−1.87709	−0.938543	+	0.345163i	$0.887824\pi$
$150$	0	0
$151$	2214.00	1.19320	0.596599	−	0.802540i	$-0.296518\pi$
$151$	2214.00	1.19320	0.596599	+	0.802540i	$0.296518\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	620.000	0.321288
$156$	0	0
$157$	−2230.00	−1.13359	−0.566794	−	0.823859i	$-0.691817\pi$
$157$	−2230.00	−1.13359	−0.566794	+	0.823859i	$0.691817\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1352.00	−0.661817
$162$	0	0
$163$	−1118.00	−0.537230	−0.268615	−	0.963248i	$-0.586566\pi$
$163$	−1118.00	−0.537230	−0.268615	+	0.963248i	$0.586566\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−288.000	−0.133450	−0.0667249	−	0.997771i	$-0.521255\pi$
$167$	−288.000	−0.133450	−0.0667249	+	0.997771i	$0.521255\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	328.000	0.144147	0.0720733	−	0.997399i	$-0.477038\pi$
$173$	328.000	0.144147	0.0720733	+	0.997399i	$0.477038\pi$
$174$	0	0
$175$	−3146.00	−1.35894
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2820.00	1.17752	0.588762	−	0.808307i	$-0.299616\pi$
$179$	2820.00	1.17752	0.588762	+	0.808307i	$0.299616\pi$
$180$	0	0
$181$	−2250.00	−0.923984	−0.461992	−	0.886884i	$-0.652865\pi$
$181$	−2250.00	−0.923984	−0.461992	+	0.886884i	$0.652865\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	36.0000	0.0143069
$186$	0	0
$187$	−2496.00	−0.976073
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3604.00	1.36532	0.682660	−	0.730736i	$-0.260823\pi$
$191$	3604.00	1.36532	0.682660	+	0.730736i	$0.260823\pi$
$192$	0	0
$193$	−1198.00	−0.446808	−0.223404	−	0.974726i	$-0.571717\pi$
$193$	−1198.00	−0.446808	−0.223404	+	0.974726i	$0.571717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1026.00	−0.371063	−0.185532	−	0.982638i	$-0.559401\pi$
$197$	−1026.00	−0.371063	−0.185532	+	0.982638i	$0.559401\pi$
$198$	0	0
$199$	−2528.00	−0.900528	−0.450264	−	0.892895i	$-0.648670\pi$
$199$	−2528.00	−0.900528	−0.450264	+	0.892895i	$0.648670\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5824.00	−2.01362
$204$	0	0
$205$	660.000	0.224860
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−936.000	−0.309782
$210$	0	0
$211$	3160.00	1.03101	0.515506	−	0.856886i	$-0.327604\pi$
$211$	3160.00	1.03101	0.515506	+	0.856886i	$0.327604\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	656.000	0.208088
$216$	0	0
$217$	−8060.00	−2.52142
$218$	0	0
$219$	0	0
$220$	0	0
$221$	624.000	0.189931
$222$	0	0
$223$	−2670.00	−0.801778	−0.400889	−	0.916127i	$-0.631299\pi$
$223$	−2670.00	−0.801778	−0.400889	+	0.916127i	$0.631299\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5220.00	1.52627	0.763136	−	0.646238i	$-0.223659\pi$
$227$	5220.00	1.52627	0.763136	+	0.646238i	$0.223659\pi$
$228$	0	0
$229$	−6246.00	−1.80239	−0.901195	−	0.433414i	$-0.857309\pi$
$229$	−6246.00	−1.80239	−0.901195	+	0.433414i	$0.857309\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2860.00	−0.804141	−0.402070	−	0.915609i	$-0.631709\pi$
$233$	−2860.00	−0.804141	−0.402070	+	0.915609i	$0.631709\pi$
$234$	0	0
$235$	−1232.00	−0.341986
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7176.00	1.94216	0.971081	−	0.238749i	$-0.0767374\pi$
$239$	7176.00	1.94216	0.971081	+	0.238749i	$0.0767374\pi$
$240$	0	0
$241$	−2922.00	−0.781006	−0.390503	−	0.920602i	$-0.627699\pi$
$241$	−2922.00	−0.781006	−0.390503	+	0.920602i	$0.627699\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−666.000	−0.173670
$246$	0	0
$247$	234.000	0.0602796
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1440.00	−0.362119	−0.181060	−	0.983472i	$-0.557953\pi$
$251$	−1440.00	−0.362119	−0.181060	+	0.983472i	$0.557953\pi$
$252$	0	0
$253$	−2704.00	−0.671933
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3852.00	0.934946	0.467473	−	0.884007i	$-0.345164\pi$
$257$	3852.00	0.934946	0.467473	+	0.884007i	$0.345164\pi$
$258$	0	0
$259$	−468.000	−0.112278
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7008.00	1.64309	0.821543	−	0.570146i	$-0.193113\pi$
$263$	7008.00	1.64309	0.821543	+	0.570146i	$0.193113\pi$
$264$	0	0
$265$	−648.000	−0.150213
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1992.00	0.451503	0.225752	−	0.974185i	$-0.427516\pi$
$269$	1992.00	0.451503	0.225752	+	0.974185i	$0.427516\pi$
$270$	0	0
$271$	3362.00	0.753605	0.376803	−	0.926294i	$-0.377024\pi$
$271$	3362.00	0.753605	0.376803	+	0.926294i	$0.377024\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6292.00	−1.37972
$276$	0	0
$277$	1506.00	0.326667	0.163334	−	0.986571i	$-0.447775\pi$
$277$	1506.00	0.326667	0.163334	+	0.986571i	$0.447775\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3638.00	0.772331	0.386165	−	0.922430i	$-0.373799\pi$
$281$	3638.00	0.772331	0.386165	+	0.922430i	$0.373799\pi$
$282$	0	0
$283$	−5404.00	−1.13510	−0.567552	−	0.823338i	$-0.692109\pi$
$283$	−5404.00	−1.13510	−0.567552	+	0.823338i	$0.692109\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8580.00	−1.76467
$288$	0	0
$289$	−2609.00	−0.531040
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4994.00	0.995743	0.497871	−	0.867251i	$-0.334115\pi$
$293$	4994.00	0.995743	0.497871	+	0.867251i	$0.334115\pi$
$294$	0	0
$295$	−376.000	−0.0742087
$296$	0	0
$297$	0	0
$298$	0	0
$299$	676.000	0.130749
$300$	0	0
$301$	−8528.00	−1.63304
$302$	0	0
$303$	0	0
$304$	0	0
$305$	220.000	0.0413022
$306$	0	0
$307$	−2474.00	−0.459930	−0.229965	−	0.973199i	$-0.573861\pi$
$307$	−2474.00	−0.459930	−0.229965	+	0.973199i	$0.573861\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9156.00	−1.66942	−0.834709	−	0.550691i	$-0.814364\pi$
$311$	−9156.00	−1.66942	−0.834709	+	0.550691i	$0.814364\pi$
$312$	0	0
$313$	3818.00	0.689476	0.344738	−	0.938699i	$-0.387968\pi$
$313$	3818.00	0.689476	0.344738	+	0.938699i	$0.387968\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6234.00	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$317$	−6234.00	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$318$	0	0
$319$	−11648.0	−2.04440
$320$	0	0
$321$	0	0
$322$	0	0
$323$	864.000	0.148837
$324$	0	0
$325$	1573.00	0.268475
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16016.0	2.68386
$330$	0	0
$331$	−5054.00	−0.839254	−0.419627	−	0.907697i	$-0.637839\pi$
$331$	−5054.00	−0.839254	−0.419627	+	0.907697i	$0.637839\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	236.000	0.0384897
$336$	0	0
$337$	10238.0	1.65489	0.827447	−	0.561544i	$-0.189792\pi$
$337$	10238.0	1.65489	0.827447	+	0.561544i	$0.189792\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16120.0	−2.55996
$342$	0	0
$343$	−260.000	−0.0409291
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2184.00	−0.337877	−0.168938	−	0.985627i	$-0.554034\pi$
$347$	−2184.00	−0.337877	−0.168938	+	0.985627i	$0.554034\pi$
$348$	0	0
$349$	6766.00	1.03775	0.518876	−	0.854849i	$-0.326351\pi$
$349$	6766.00	1.03775	0.518876	+	0.854849i	$0.326351\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8290.00	−1.24995	−0.624975	−	0.780645i	$-0.714891\pi$
$353$	−8290.00	−1.24995	−0.624975	+	0.780645i	$0.714891\pi$
$354$	0	0
$355$	1312.00	0.196151
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9456.00	−1.39016	−0.695082	−	0.718931i	$-0.744632\pi$
$359$	−9456.00	−1.39016	−0.695082	+	0.718931i	$0.744632\pi$
$360$	0	0
$361$	−6535.00	−0.952763
$362$	0	0
$363$	0	0
$364$	0	0
$365$	356.000	0.0510518
$366$	0	0
$367$	10096.0	1.43599	0.717993	−	0.696050i	$-0.245061\pi$
$367$	10096.0	1.43599	0.717993	+	0.696050i	$0.245061\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8424.00	1.17885
$372$	0	0
$373$	−1234.00	−0.171298	−0.0856489	−	0.996325i	$-0.527296\pi$
$373$	−1234.00	−0.171298	−0.0856489	+	0.996325i	$0.527296\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2912.00	0.397813
$378$	0	0
$379$	−11218.0	−1.52040	−0.760198	−	0.649692i	$-0.774898\pi$
$379$	−11218.0	−1.52040	−0.760198	+	0.649692i	$0.774898\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7488.00	−0.999005	−0.499503	−	0.866312i	$-0.666484\pi$
$383$	−7488.00	−0.999005	−0.499503	+	0.866312i	$0.666484\pi$
$384$	0	0
$385$	−2704.00	−0.357944
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14364.0	1.87219	0.936097	−	0.351741i	$-0.114410\pi$
$389$	14364.0	1.87219	0.936097	+	0.351741i	$0.114410\pi$
$390$	0	0
$391$	2496.00	0.322834
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1672.00	0.212981
$396$	0	0
$397$	7454.00	0.942331	0.471166	−	0.882045i	$-0.343833\pi$
$397$	7454.00	0.942331	0.471166	+	0.882045i	$0.343833\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5766.00	−0.718056	−0.359028	−	0.933327i	$-0.616892\pi$
$401$	−5766.00	−0.718056	−0.359028	+	0.933327i	$0.616892\pi$
$402$	0	0
$403$	4030.00	0.498135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−936.000	−0.113995
$408$	0	0
$409$	11926.0	1.44182	0.720908	−	0.693031i	$-0.243725\pi$
$409$	11926.0	1.44182	0.720908	+	0.693031i	$0.243725\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4888.00	0.582380
$414$	0	0
$415$	−120.000	−0.0141941
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9996.00	1.16548	0.582741	−	0.812658i	$-0.301980\pi$
$419$	9996.00	1.16548	0.582741	+	0.812658i	$0.301980\pi$
$420$	0	0
$421$	−2030.00	−0.235003	−0.117501	−	0.993073i	$-0.537488\pi$
$421$	−2030.00	−0.235003	−0.117501	+	0.993073i	$0.537488\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5808.00	0.662893
$426$	0	0
$427$	−2860.00	−0.324134
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15888.0	1.77563	0.887817	−	0.460197i	$-0.152221\pi$
$431$	15888.0	1.77563	0.887817	+	0.460197i	$0.152221\pi$
$432$	0	0
$433$	−14402.0	−1.59842	−0.799210	−	0.601052i	$-0.794749\pi$
$433$	−14402.0	−1.59842	−0.799210	+	0.601052i	$0.794749\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	936.000	0.102460
$438$	0	0
$439$	−404.000	−0.0439223	−0.0219611	−	0.999759i	$-0.506991\pi$
$439$	−404.000	−0.0439223	−0.0219611	+	0.999759i	$0.506991\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16692.0	−1.79021	−0.895103	−	0.445860i	$-0.852898\pi$
$443$	−16692.0	−1.79021	−0.895103	+	0.445860i	$0.852898\pi$
$444$	0	0
$445$	1740.00	0.185357
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4058.00	0.426523	0.213262	−	0.976995i	$-0.431591\pi$
$449$	4058.00	0.426523	0.213262	+	0.976995i	$0.431591\pi$
$450$	0	0
$451$	−17160.0	−1.79165
$452$	0	0
$453$	0	0
$454$	0	0
$455$	676.000	0.0696514
$456$	0	0
$457$	−14642.0	−1.49874	−0.749370	−	0.662152i	$-0.769643\pi$
$457$	−14642.0	−1.49874	−0.749370	+	0.662152i	$0.769643\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17970.0	−1.81550	−0.907751	−	0.419510i	$-0.862202\pi$
$461$	−17970.0	−1.81550	−0.907751	+	0.419510i	$0.862202\pi$
$462$	0	0
$463$	6242.00	0.626545	0.313273	−	0.949663i	$-0.398575\pi$
$463$	6242.00	0.626545	0.313273	+	0.949663i	$0.398575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5000.00	0.495444	0.247722	−	0.968831i	$-0.420318\pi$
$467$	5000.00	0.495444	0.247722	+	0.968831i	$0.420318\pi$
$468$	0	0
$469$	−3068.00	−0.302062
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17056.0	−1.65800
$474$	0	0
$475$	2178.00	0.210386
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9408.00	0.897416	0.448708	−	0.893678i	$-0.351884\pi$
$479$	9408.00	0.897416	0.448708	+	0.893678i	$0.351884\pi$
$480$	0	0
$481$	234.000	0.0221819
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2476.00	−0.231813
$486$	0	0
$487$	7090.00	0.659710	0.329855	−	0.944032i	$-0.393000\pi$
$487$	7090.00	0.659710	0.329855	+	0.944032i	$0.393000\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8176.00	0.751482	0.375741	−	0.926725i	$-0.377388\pi$
$491$	8176.00	0.751482	0.375741	+	0.926725i	$0.377388\pi$
$492$	0	0
$493$	10752.0	0.982243
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17056.0	−1.53937
$498$	0	0
$499$	−9646.00	−0.865359	−0.432680	−	0.901548i	$-0.642432\pi$
$499$	−9646.00	−0.865359	−0.432680	+	0.901548i	$0.642432\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4352.00	−0.385778	−0.192889	−	0.981221i	$-0.561786\pi$
$503$	−4352.00	−0.385778	−0.192889	+	0.981221i	$0.561786\pi$
$504$	0	0
$505$	−1840.00	−0.162136
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14582.0	1.26981	0.634907	−	0.772588i	$-0.281038\pi$
$509$	14582.0	1.26981	0.634907	+	0.772588i	$0.281038\pi$
$510$	0	0
$511$	−4628.00	−0.400647
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1672.00	−0.143062
$516$	0	0
$517$	32032.0	2.72489
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11820.0	0.993942	0.496971	−	0.867767i	$-0.334446\pi$
$521$	11820.0	0.993942	0.496971	+	0.867767i	$0.334446\pi$
$522$	0	0
$523$	−2676.00	−0.223735	−0.111867	−	0.993723i	$-0.535683\pi$
$523$	−2676.00	−0.223735	−0.111867	+	0.993723i	$0.535683\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14880.0	1.22995
$528$	0	0
$529$	−9463.00	−0.777760
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4290.00	0.348631
$534$	0	0
$535$	−520.000	−0.0420216
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17316.0	1.38377
$540$	0	0
$541$	842.000	0.0669139	0.0334569	−	0.999440i	$-0.489348\pi$
$541$	842.000	0.0669139	0.0334569	+	0.999440i	$0.489348\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4156.00	0.326649
$546$	0	0
$547$	−1036.00	−0.0809802	−0.0404901	−	0.999180i	$-0.512892\pi$
$547$	−1036.00	−0.0809802	−0.0404901	+	0.999180i	$0.512892\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4032.00	0.311740
$552$	0	0
$553$	−21736.0	−1.67144
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3342.00	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−3342.00	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	4264.00	0.322626
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12412.0	−0.929136	−0.464568	−	0.885538i	$-0.653790\pi$
$563$	−12412.0	−0.929136	−0.464568	+	0.885538i	$0.653790\pi$
$564$	0	0
$565$	−2312.00	−0.172153
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21556.0	−1.58818	−0.794090	−	0.607800i	$-0.792052\pi$
$569$	−21556.0	−1.58818	−0.794090	+	0.607800i	$0.792052\pi$
$570$	0	0
$571$	−9012.00	−0.660491	−0.330246	−	0.943895i	$-0.607132\pi$
$571$	−9012.00	−0.660491	−0.330246	+	0.943895i	$0.607132\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6292.00	0.456338
$576$	0	0
$577$	−10446.0	−0.753679	−0.376839	−	0.926279i	$-0.622989\pi$
$577$	−10446.0	−0.753679	−0.376839	+	0.926279i	$0.622989\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1560.00	0.111394
$582$	0	0
$583$	16848.0	1.19687
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23452.0	−1.64901	−0.824504	−	0.565856i	$-0.808546\pi$
$587$	−23452.0	−1.64901	−0.824504	+	0.565856i	$0.808546\pi$
$588$	0	0
$589$	5580.00	0.390356
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23914.0	1.65604	0.828019	−	0.560700i	$-0.189468\pi$
$593$	23914.0	1.65604	0.828019	+	0.560700i	$0.189468\pi$
$594$	0	0
$595$	2496.00	0.171977
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2552.00	−0.174077	−0.0870383	−	0.996205i	$-0.527740\pi$
$599$	−2552.00	−0.174077	−0.0870383	+	0.996205i	$0.527740\pi$
$600$	0	0
$601$	3418.00	0.231985	0.115993	−	0.993250i	$-0.462995\pi$
$601$	3418.00	0.231985	0.115993	+	0.993250i	$0.462995\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2746.00	−0.184530
$606$	0	0
$607$	−12336.0	−0.824881	−0.412441	−	0.910985i	$-0.635324\pi$
$607$	−12336.0	−0.824881	−0.412441	+	0.910985i	$0.635324\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8008.00	−0.530228
$612$	0	0
$613$	−21946.0	−1.44599	−0.722994	−	0.690854i	$-0.757235\pi$
$613$	−21946.0	−1.44599	−0.722994	+	0.690854i	$0.757235\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23890.0	−1.55879	−0.779396	−	0.626531i	$-0.784474\pi$
$617$	−23890.0	−1.55879	−0.779396	+	0.626531i	$0.784474\pi$
$618$	0	0
$619$	−10042.0	−0.652055	−0.326028	−	0.945360i	$-0.605710\pi$
$619$	−10042.0	−0.652055	−0.326028	+	0.945360i	$0.605710\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−22620.0	−1.45466
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	0	0
$628$	0	0
$629$	864.000	0.0547694
$630$	0	0
$631$	21830.0	1.37724	0.688620	−	0.725122i	$-0.258217\pi$
$631$	21830.0	1.37724	0.688620	+	0.725122i	$0.258217\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1504.00	−0.0939912
$636$	0	0
$637$	−4329.00	−0.269264
$638$	0	0
$639$	0	0
$640$	0	0
$641$	776.000	0.0478162	0.0239081	−	0.999714i	$-0.492389\pi$
$641$	776.000	0.0478162	0.0239081	+	0.999714i	$0.492389\pi$
$642$	0	0
$643$	−4094.00	−0.251091	−0.125546	−	0.992088i	$-0.540068\pi$
$643$	−4094.00	−0.251091	−0.125546	+	0.992088i	$0.540068\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13064.0	−0.793816	−0.396908	−	0.917858i	$-0.629917\pi$
$647$	−13064.0	−0.793816	−0.396908	+	0.917858i	$0.629917\pi$
$648$	0	0
$649$	9776.00	0.591281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24504.0	1.46848	0.734239	−	0.678891i	$-0.237539\pi$
$653$	24504.0	1.46848	0.734239	+	0.678891i	$0.237539\pi$
$654$	0	0
$655$	3944.00	0.235275
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4752.00	0.280898	0.140449	−	0.990088i	$-0.455145\pi$
$659$	4752.00	0.280898	0.140449	+	0.990088i	$0.455145\pi$
$660$	0	0
$661$	−9514.00	−0.559836	−0.279918	−	0.960024i	$-0.590307\pi$
$661$	−9514.00	−0.559836	−0.279918	+	0.960024i	$0.590307\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	936.000	0.0545812
$666$	0	0
$667$	11648.0	0.676180
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5720.00	−0.329088
$672$	0	0
$673$	8374.00	0.479634	0.239817	−	0.970818i	$-0.422912\pi$
$673$	8374.00	0.479634	0.239817	+	0.970818i	$0.422912\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14556.0	0.826341	0.413170	−	0.910654i	$-0.364421\pi$
$677$	14556.0	0.826341	0.413170	+	0.910654i	$0.364421\pi$
$678$	0	0
$679$	32188.0	1.81924
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23276.0	−1.30400	−0.652000	−	0.758219i	$-0.726070\pi$
$683$	−23276.0	−1.30400	−0.652000	+	0.758219i	$0.726070\pi$
$684$	0	0
$685$	516.000	0.0287815
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4212.00	−0.232895
$690$	0	0
$691$	29522.0	1.62528	0.812641	−	0.582765i	$-0.198029\pi$
$691$	29522.0	1.62528	0.812641	+	0.582765i	$0.198029\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2936.00	0.160243
$696$	0	0
$697$	15840.0	0.860807
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27252.0	−1.46832	−0.734161	−	0.678975i	$-0.762424\pi$
$701$	−27252.0	−1.46832	−0.734161	+	0.678975i	$0.762424\pi$
$702$	0	0
$703$	324.000	0.0173825
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23920.0	1.27242
$708$	0	0
$709$	2162.00	0.114521	0.0572607	−	0.998359i	$-0.481763\pi$
$709$	2162.00	0.114521	0.0572607	+	0.998359i	$0.481763\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16120.0	0.846702
$714$	0	0
$715$	1352.00	0.0707160
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8044.00	−0.417233	−0.208617	−	0.977998i	$-0.566896\pi$
$719$	−8044.00	−0.417233	−0.208617	+	0.977998i	$0.566896\pi$
$720$	0	0
$721$	21736.0	1.12273
$722$	0	0
$723$	0	0
$724$	0	0
$725$	27104.0	1.38844
$726$	0	0
$727$	22496.0	1.14763	0.573817	−	0.818983i	$-0.305462\pi$
$727$	22496.0	1.14763	0.573817	+	0.818983i	$0.305462\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15744.0	0.796598
$732$	0	0
$733$	−15934.0	−0.802914	−0.401457	−	0.915878i	$-0.631496\pi$
$733$	−15934.0	−0.802914	−0.401457	+	0.915878i	$0.631496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6136.00	−0.306679
$738$	0	0
$739$	29838.0	1.48526	0.742631	−	0.669701i	$-0.233578\pi$
$739$	29838.0	1.48526	0.742631	+	0.669701i	$0.233578\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10760.0	−0.531287	−0.265643	−	0.964071i	$-0.585584\pi$
$743$	−10760.0	−0.531287	−0.265643	+	0.964071i	$0.585584\pi$
$744$	0	0
$745$	6828.00	0.335783
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6760.00	0.329780
$750$	0	0
$751$	14916.0	0.724757	0.362379	−	0.932031i	$-0.381965\pi$
$751$	14916.0	0.724757	0.362379	+	0.932031i	$0.381965\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4428.00	−0.213446
$756$	0	0
$757$	29046.0	1.39458	0.697289	−	0.716791i	$-0.254390\pi$
$757$	29046.0	1.39458	0.697289	+	0.716791i	$0.254390\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2766.00	0.131757	0.0658787	−	0.997828i	$-0.479015\pi$
$761$	2766.00	0.131757	0.0658787	+	0.997828i	$0.479015\pi$
$762$	0	0
$763$	−54028.0	−2.56349
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2444.00	−0.115056
$768$	0	0
$769$	5394.00	0.252942	0.126471	−	0.991970i	$-0.459635\pi$
$769$	5394.00	0.252942	0.126471	+	0.991970i	$0.459635\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19378.0	0.901654	0.450827	−	0.892611i	$-0.351129\pi$
$773$	19378.0	0.901654	0.450827	+	0.892611i	$0.351129\pi$
$774$	0	0
$775$	37510.0	1.73858
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5940.00	0.273200
$780$	0	0
$781$	−34112.0	−1.56290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4460.00	0.202782
$786$	0	0
$787$	8350.00	0.378202	0.189101	−	0.981958i	$-0.439443\pi$
$787$	8350.00	0.378202	0.189101	+	0.981958i	$0.439443\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	30056.0	1.35103
$792$	0	0
$793$	1430.00	0.0640363
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22116.0	−0.982922	−0.491461	−	0.870900i	$-0.663537\pi$
$797$	−22116.0	−0.982922	−0.491461	+	0.870900i	$0.663537\pi$
$798$	0	0
$799$	−29568.0	−1.30919
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9256.00	−0.406771
$804$	0	0
$805$	2704.00	0.118389
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6444.00	−0.280048	−0.140024	−	0.990148i	$-0.544718\pi$
$809$	−6444.00	−0.280048	−0.140024	+	0.990148i	$0.544718\pi$
$810$	0	0
$811$	8998.00	0.389596	0.194798	−	0.980843i	$-0.437595\pi$
$811$	8998.00	0.389596	0.194798	+	0.980843i	$0.437595\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2236.00	0.0961027
$816$	0	0
$817$	5904.00	0.252821
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11938.0	0.507478	0.253739	−	0.967273i	$-0.418340\pi$
$821$	11938.0	0.507478	0.253739	+	0.967273i	$0.418340\pi$
$822$	0	0
$823$	41028.0	1.73772	0.868862	−	0.495055i	$-0.164852\pi$
$823$	41028.0	1.73772	0.868862	+	0.495055i	$0.164852\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20460.0	0.860295	0.430147	−	0.902759i	$-0.358462\pi$
$827$	20460.0	0.860295	0.430147	+	0.902759i	$0.358462\pi$
$828$	0	0
$829$	27814.0	1.16528	0.582642	−	0.812729i	$-0.302019\pi$
$829$	27814.0	1.16528	0.582642	+	0.812729i	$0.302019\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15984.0	−0.664841
$834$	0	0
$835$	576.000	0.0238722
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24880.0	−1.02378	−0.511891	−	0.859050i	$-0.671055\pi$
$839$	−24880.0	−1.02378	−0.511891	+	0.859050i	$0.671055\pi$
$840$	0	0
$841$	25787.0	1.05732
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−338.000	−0.0137604
$846$	0	0
$847$	35698.0	1.44817
$848$	0	0
$849$	0	0
$850$	0	0
$851$	936.000	0.0377035
$852$	0	0
$853$	37678.0	1.51239	0.756196	−	0.654345i	$-0.227056\pi$
$853$	37678.0	1.51239	0.756196	+	0.654345i	$0.227056\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14708.0	0.586249	0.293125	−	0.956074i	$-0.405305\pi$
$857$	14708.0	0.586249	0.293125	+	0.956074i	$0.405305\pi$
$858$	0	0
$859$	−16472.0	−0.654269	−0.327135	−	0.944978i	$-0.606083\pi$
$859$	−16472.0	−0.654269	−0.327135	+	0.944978i	$0.606083\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13640.0	−0.538020	−0.269010	−	0.963137i	$-0.586696\pi$
$863$	−13640.0	−0.538020	−0.269010	+	0.963137i	$0.586696\pi$
$864$	0	0
$865$	−656.000	−0.0257857
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−43472.0	−1.69699
$870$	0	0
$871$	1534.00	0.0596758
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12792.0	0.494227
$876$	0	0
$877$	44430.0	1.71071	0.855356	−	0.518041i	$-0.173338\pi$
$877$	44430.0	1.71071	0.855356	+	0.518041i	$0.173338\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6684.00	−0.255607	−0.127803	−	0.991800i	$-0.540793\pi$
$881$	−6684.00	−0.255607	−0.127803	+	0.991800i	$0.540793\pi$
$882$	0	0
$883$	−9196.00	−0.350476	−0.175238	−	0.984526i	$-0.556069\pi$
$883$	−9196.00	−0.350476	−0.175238	+	0.984526i	$0.556069\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34852.0	1.31930	0.659648	−	0.751575i	$-0.270705\pi$
$887$	34852.0	1.31930	0.659648	+	0.751575i	$0.270705\pi$
$888$	0	0
$889$	19552.0	0.737630
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11088.0	−0.415505
$894$	0	0
$895$	−5640.00	−0.210642
$896$	0	0
$897$	0	0
$898$	0	0
$899$	69440.0	2.57614
$900$	0	0
$901$	−15552.0	−0.575041
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4500.00	0.165287
$906$	0	0
$907$	−36780.0	−1.34648	−0.673241	−	0.739423i	$-0.735099\pi$
$907$	−36780.0	−1.34648	−0.673241	+	0.739423i	$0.735099\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22980.0	0.835742	0.417871	−	0.908506i	$-0.362776\pi$
$911$	22980.0	0.835742	0.417871	+	0.908506i	$0.362776\pi$
$912$	0	0
$913$	3120.00	0.113096
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−51272.0	−1.84640
$918$	0	0
$919$	19980.0	0.717170	0.358585	−	0.933497i	$-0.383259\pi$
$919$	19980.0	0.717170	0.358585	+	0.933497i	$0.383259\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8528.00	0.304120
$924$	0	0
$925$	2178.00	0.0774186
$926$	0	0
$927$	0	0
$928$	0	0
$929$	51486.0	1.81830	0.909150	−	0.416469i	$-0.136732\pi$
$929$	51486.0	1.81830	0.909150	+	0.416469i	$0.136732\pi$
$930$	0	0
$931$	−5994.00	−0.211005
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4992.00	0.174605
$936$	0	0
$937$	21774.0	0.759152	0.379576	−	0.925161i	$-0.376070\pi$
$937$	21774.0	0.759152	0.379576	+	0.925161i	$0.376070\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29158.0	−1.01012	−0.505060	−	0.863084i	$-0.668530\pi$
$941$	−29158.0	−1.01012	−0.505060	+	0.863084i	$0.668530\pi$
$942$	0	0
$943$	17160.0	0.592584
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47844.0	−1.64173	−0.820866	−	0.571120i	$-0.806509\pi$
$947$	−47844.0	−1.64173	−0.820866	+	0.571120i	$0.806509\pi$
$948$	0	0
$949$	2314.00	0.0791524
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20716.0	0.704152	0.352076	−	0.935971i	$-0.385476\pi$
$953$	20716.0	0.704152	0.352076	+	0.935971i	$0.385476\pi$
$954$	0	0
$955$	−7208.00	−0.244236
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6708.00	−0.225873
$960$	0	0
$961$	66309.0	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2396.00	0.0799274
$966$	0	0
$967$	−1406.00	−0.0467569	−0.0233784	−	0.999727i	$-0.507442\pi$
$967$	−1406.00	−0.0467569	−0.0233784	+	0.999727i	$0.507442\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40176.0	−1.32782	−0.663908	−	0.747814i	$-0.731103\pi$
$971$	−40176.0	−1.32782	−0.663908	+	0.747814i	$0.731103\pi$
$972$	0	0
$973$	−38168.0	−1.25756
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20910.0	−0.684719	−0.342359	−	0.939569i	$-0.611226\pi$
$977$	−20910.0	−0.684719	−0.342359	+	0.939569i	$0.611226\pi$
$978$	0	0
$979$	−45240.0	−1.47689
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26288.0	0.852957	0.426479	−	0.904498i	$-0.359754\pi$
$983$	26288.0	0.852957	0.426479	+	0.904498i	$0.359754\pi$
$984$	0	0
$985$	2052.00	0.0663778
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17056.0	0.548381
$990$	0	0
$991$	25608.0	0.820853	0.410426	−	0.911894i	$-0.365380\pi$
$991$	25608.0	0.820853	0.410426	+	0.911894i	$0.365380\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5056.00	0.161091
$996$	0	0
$997$	−20098.0	−0.638425	−0.319213	−	0.947683i	$-0.603418\pi$
$997$	−20098.0	−0.638425	−0.319213	+	0.947683i	$0.603418\pi$
$998$	0	0
$999$	0	0

Display

a_p

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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

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a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.g.1.1		1
3.2	odd	2		1872.4.a.h.1.1		1
4.3	odd	2		234.4.a.i.1.1	yes	1
12.11	even	2		234.4.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.4.a.d.1.1	✓	1	12.11	even	2
234.4.a.i.1.1	yes	1	4.3	odd	2
1872.4.a.g.1.1		1	1.1	even	1	trivial
1872.4.a.h.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1872.4.a.g.1.1

Level	$1872$
Weight	$4$
Character	1872.1
Self dual	yes
Analytic conductor	$110.452$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$