LMFDB - Embedded newform 1200.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1200.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+3.00000 q^{3} -7.00000 q^{7} +9.00000 q^{9} +54.0000 q^{11} +55.0000 q^{13} -18.0000 q^{17} +25.0000 q^{19} -21.0000 q^{21} -18.0000 q^{23} +27.0000 q^{27} -54.0000 q^{29} +271.000 q^{31} +162.000 q^{33} -314.000 q^{37} +165.000 q^{39} -360.000 q^{41} -163.000 q^{43} +522.000 q^{47} -294.000 q^{49} -54.0000 q^{51} +36.0000 q^{53} +75.0000 q^{57} -126.000 q^{59} +47.0000 q^{61} -63.0000 q^{63} -343.000 q^{67} -54.0000 q^{69} +1080.00 q^{71} +1054.00 q^{73} -378.000 q^{77} +568.000 q^{79} +81.0000 q^{81} +1422.00 q^{83} -162.000 q^{87} +1440.00 q^{89} -385.000 q^{91} +813.000 q^{93} +439.000 q^{97} +486.000 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−7.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	54.0000	1.48015	0.740073	−	0.672526i	$-0.234791\pi$
$11$	54.0000	1.48015	0.740073	+	0.672526i	$0.234791\pi$
$12$	0	0
$13$	55.0000	1.17340	0.586702	−	0.809803i	$-0.300426\pi$
$13$	55.0000	1.17340	0.586702	+	0.809803i	$0.300426\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−18.0000	−0.256802	−0.128401	−	0.991722i	$-0.540985\pi$
$17$	−18.0000	−0.256802	−0.128401	+	0.991722i	$0.540985\pi$
$18$	0	0
$19$	25.0000	0.301863	0.150931	−	0.988544i	$-0.451773\pi$
$19$	25.0000	0.301863	0.150931	+	0.988544i	$0.451773\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−18.0000	−0.163185	−0.0815926	−	0.996666i	$-0.526001\pi$
$23$	−18.0000	−0.163185	−0.0815926	+	0.996666i	$0.526001\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	271.000	1.57010	0.785049	−	0.619434i	$-0.212638\pi$
$31$	271.000	1.57010	0.785049	+	0.619434i	$0.212638\pi$
$32$	0	0
$33$	162.000	0.854563
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−314.000	−1.39517	−0.697585	−	0.716502i	$-0.745742\pi$
$37$	−314.000	−1.39517	−0.697585	+	0.716502i	$0.745742\pi$
$38$	0	0
$39$	165.000	0.677465
$40$	0	0
$41$	−360.000	−1.37128	−0.685641	−	0.727940i	$-0.740478\pi$
$41$	−360.000	−1.37128	−0.685641	+	0.727940i	$0.740478\pi$
$42$	0	0
$43$	−163.000	−0.578076	−0.289038	−	0.957318i	$-0.593335\pi$
$43$	−163.000	−0.578076	−0.289038	+	0.957318i	$0.593335\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	522.000	1.62003	0.810016	−	0.586407i	$-0.199458\pi$
$47$	522.000	1.62003	0.810016	+	0.586407i	$0.199458\pi$
$48$	0	0
$49$	−294.000	−0.857143
$50$	0	0
$51$	−54.0000	−0.148265
$52$	0	0
$53$	36.0000	0.0933015	0.0466508	−	0.998911i	$-0.485145\pi$
$53$	36.0000	0.0933015	0.0466508	+	0.998911i	$0.485145\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	75.0000	0.174281
$58$	0	0
$59$	−126.000	−0.278031	−0.139015	−	0.990290i	$-0.544394\pi$
$59$	−126.000	−0.278031	−0.139015	+	0.990290i	$0.544394\pi$
$60$	0	0
$61$	47.0000	0.0986514	0.0493257	−	0.998783i	$-0.484293\pi$
$61$	47.0000	0.0986514	0.0493257	+	0.998783i	$0.484293\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−343.000	−0.625435	−0.312717	−	0.949846i	$-0.601239\pi$
$67$	−343.000	−0.625435	−0.312717	+	0.949846i	$0.601239\pi$
$68$	0	0
$69$	−54.0000	−0.0942150
$70$	0	0
$71$	1080.00	1.80525	0.902623	−	0.430433i	$-0.141639\pi$
$71$	1080.00	1.80525	0.902623	+	0.430433i	$0.141639\pi$
$72$	0	0
$73$	1054.00	1.68988	0.844941	−	0.534860i	$-0.179636\pi$
$73$	1054.00	1.68988	0.844941	+	0.534860i	$0.179636\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−378.000	−0.559443
$78$	0	0
$79$	568.000	0.808924	0.404462	−	0.914555i	$-0.367459\pi$
$79$	568.000	0.808924	0.404462	+	0.914555i	$0.367459\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1422.00	1.88054	0.940270	−	0.340430i	$-0.110573\pi$
$83$	1422.00	1.88054	0.940270	+	0.340430i	$0.110573\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−162.000	−0.199635
$88$	0	0
$89$	1440.00	1.71505	0.857526	−	0.514440i	$-0.172000\pi$
$89$	1440.00	1.71505	0.857526	+	0.514440i	$0.172000\pi$
$90$	0	0
$91$	−385.000	−0.443505
$92$	0	0
$93$	813.000	0.906496
$94$	0	0
$95$	0	0
$96$	0	0
$97$	439.000	0.459523	0.229761	−	0.973247i	$-0.426205\pi$
$97$	439.000	0.459523	0.229761	+	0.973247i	$0.426205\pi$
$98$	0	0
$99$	486.000	0.493382
$100$	0	0
$101$	828.000	0.815733	0.407867	−	0.913041i	$-0.366273\pi$
$101$	828.000	0.815733	0.407867	+	0.913041i	$0.366273\pi$
$102$	0	0
$103$	548.000	0.524233	0.262117	−	0.965036i	$-0.415579\pi$
$103$	548.000	0.524233	0.262117	+	0.965036i	$0.415579\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1476.00	−1.33355	−0.666777	−	0.745257i	$-0.732327\pi$
$107$	−1476.00	−1.33355	−0.666777	+	0.745257i	$0.732327\pi$
$108$	0	0
$109$	1277.00	1.12215	0.561075	−	0.827765i	$-0.310388\pi$
$109$	1277.00	1.12215	0.561075	+	0.827765i	$0.310388\pi$
$110$	0	0
$111$	−942.000	−0.805502
$112$	0	0
$113$	−1836.00	−1.52846	−0.764232	−	0.644942i	$-0.776882\pi$
$113$	−1836.00	−1.52846	−0.764232	+	0.644942i	$0.776882\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	495.000	0.391135
$118$	0	0
$119$	126.000	0.0970622
$120$	0	0
$121$	1585.00	1.19083
$122$	0	0
$123$	−1080.00	−0.791710
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−592.000	−0.413634	−0.206817	−	0.978380i	$-0.566310\pi$
$127$	−592.000	−0.413634	−0.206817	+	0.978380i	$0.566310\pi$
$128$	0	0
$129$	−489.000	−0.333752
$130$	0	0
$131$	468.000	0.312132	0.156066	−	0.987747i	$-0.450119\pi$
$131$	468.000	0.312132	0.156066	+	0.987747i	$0.450119\pi$
$132$	0	0
$133$	−175.000	−0.114093
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2574.00	−1.60519	−0.802597	−	0.596521i	$-0.796549\pi$
$137$	−2574.00	−1.60519	−0.802597	+	0.596521i	$0.796549\pi$
$138$	0	0
$139$	1756.00	1.07153	0.535763	−	0.844369i	$-0.320024\pi$
$139$	1756.00	1.07153	0.535763	+	0.844369i	$0.320024\pi$
$140$	0	0
$141$	1566.00	0.935326
$142$	0	0
$143$	2970.00	1.73681
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−882.000	−0.494872
$148$	0	0
$149$	2682.00	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	2682.00	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−3395.00	−1.82968	−0.914838	−	0.403820i	$-0.867682\pi$
$151$	−3395.00	−1.82968	−0.914838	+	0.403820i	$0.867682\pi$
$152$	0	0
$153$	−162.000	−0.0856008
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1549.00	0.787412	0.393706	−	0.919236i	$-0.371193\pi$
$157$	1549.00	0.787412	0.393706	+	0.919236i	$0.371193\pi$
$158$	0	0
$159$	108.000	0.0538677
$160$	0	0
$161$	126.000	0.0616782
$162$	0	0
$163$	−505.000	−0.242667	−0.121333	−	0.992612i	$-0.538717\pi$
$163$	−505.000	−0.242667	−0.121333	+	0.992612i	$0.538717\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1476.00	−0.683930	−0.341965	−	0.939713i	$-0.611092\pi$
$167$	−1476.00	−0.683930	−0.341965	+	0.939713i	$0.611092\pi$
$168$	0	0
$169$	828.000	0.376878
$170$	0	0
$171$	225.000	0.100621
$172$	0	0
$173$	−2358.00	−1.03627	−0.518137	−	0.855298i	$-0.673374\pi$
$173$	−2358.00	−1.03627	−0.518137	+	0.855298i	$0.673374\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−378.000	−0.160521
$178$	0	0
$179$	−1566.00	−0.653901	−0.326951	−	0.945041i	$-0.606021\pi$
$179$	−1566.00	−0.653901	−0.326951	+	0.945041i	$0.606021\pi$
$180$	0	0
$181$	305.000	0.125251	0.0626256	−	0.998037i	$-0.480053\pi$
$181$	305.000	0.125251	0.0626256	+	0.998037i	$0.480053\pi$
$182$	0	0
$183$	141.000	0.0569564
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−972.000	−0.380105
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	1746.00	0.661446	0.330723	−	0.943728i	$-0.392707\pi$
$191$	1746.00	0.661446	0.330723	+	0.943728i	$0.392707\pi$
$192$	0	0
$193$	3877.00	1.44597	0.722986	−	0.690863i	$-0.242769\pi$
$193$	3877.00	1.44597	0.722986	+	0.690863i	$0.242769\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2142.00	−0.774676	−0.387338	−	0.921938i	$-0.626605\pi$
$197$	−2142.00	−0.774676	−0.387338	+	0.921938i	$0.626605\pi$
$198$	0	0
$199$	4033.00	1.43664	0.718321	−	0.695712i	$-0.244911\pi$
$199$	4033.00	1.43664	0.718321	+	0.695712i	$0.244911\pi$
$200$	0	0
$201$	−1029.00	−0.361095
$202$	0	0
$203$	378.000	0.130692
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−162.000	−0.0543951
$208$	0	0
$209$	1350.00	0.446801
$210$	0	0
$211$	4105.00	1.33934	0.669668	−	0.742661i	$-0.266436\pi$
$211$	4105.00	1.33934	0.669668	+	0.742661i	$0.266436\pi$
$212$	0	0
$213$	3240.00	1.04226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1897.00	−0.593441
$218$	0	0
$219$	3162.00	0.975654
$220$	0	0
$221$	−990.000	−0.301333
$222$	0	0
$223$	1385.00	0.415903	0.207952	−	0.978139i	$-0.433320\pi$
$223$	1385.00	0.415903	0.207952	+	0.978139i	$0.433320\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2520.00	−0.736821	−0.368410	−	0.929663i	$-0.620098\pi$
$227$	−2520.00	−0.736821	−0.368410	+	0.929663i	$0.620098\pi$
$228$	0	0
$229$	5129.00	1.48006	0.740030	−	0.672574i	$-0.234811\pi$
$229$	5129.00	1.48006	0.740030	+	0.672574i	$0.234811\pi$
$230$	0	0
$231$	−1134.00	−0.322994
$232$	0	0
$233$	3240.00	0.910985	0.455492	−	0.890240i	$-0.349463\pi$
$233$	3240.00	0.910985	0.455492	+	0.890240i	$0.349463\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1704.00	0.467032
$238$	0	0
$239$	2988.00	0.808693	0.404347	−	0.914606i	$-0.367499\pi$
$239$	2988.00	0.808693	0.404347	+	0.914606i	$0.367499\pi$
$240$	0	0
$241$	−2647.00	−0.707503	−0.353752	−	0.935339i	$-0.615094\pi$
$241$	−2647.00	−0.707503	−0.353752	+	0.935339i	$0.615094\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1375.00	0.354207
$248$	0	0
$249$	4266.00	1.08573
$250$	0	0
$251$	4212.00	1.05920	0.529600	−	0.848248i	$-0.322342\pi$
$251$	4212.00	1.05920	0.529600	+	0.848248i	$0.322342\pi$
$252$	0	0
$253$	−972.000	−0.241538
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5724.00	−1.38931	−0.694656	−	0.719342i	$-0.744444\pi$
$257$	−5724.00	−1.38931	−0.694656	+	0.719342i	$0.744444\pi$
$258$	0	0
$259$	2198.00	0.527325
$260$	0	0
$261$	−486.000	−0.115259
$262$	0	0
$263$	−4608.00	−1.08039	−0.540193	−	0.841541i	$-0.681649\pi$
$263$	−4608.00	−1.08039	−0.540193	+	0.841541i	$0.681649\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4320.00	0.990186
$268$	0	0
$269$	−6426.00	−1.45651	−0.728253	−	0.685308i	$-0.759667\pi$
$269$	−6426.00	−1.45651	−0.728253	+	0.685308i	$0.759667\pi$
$270$	0	0
$271$	3376.00	0.756743	0.378372	−	0.925654i	$-0.376484\pi$
$271$	3376.00	0.756743	0.378372	+	0.925654i	$0.376484\pi$
$272$	0	0
$273$	−1155.00	−0.256058
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5381.00	−1.16719	−0.583597	−	0.812043i	$-0.698355\pi$
$277$	−5381.00	−1.16719	−0.583597	+	0.812043i	$0.698355\pi$
$278$	0	0
$279$	2439.00	0.523366
$280$	0	0
$281$	−3474.00	−0.737514	−0.368757	−	0.929526i	$-0.620217\pi$
$281$	−3474.00	−0.737514	−0.368757	+	0.929526i	$0.620217\pi$
$282$	0	0
$283$	−2269.00	−0.476601	−0.238300	−	0.971191i	$-0.576590\pi$
$283$	−2269.00	−0.476601	−0.238300	+	0.971191i	$0.576590\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2520.00	0.518296
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	1317.00	0.265306
$292$	0	0
$293$	1674.00	0.333775	0.166888	−	0.985976i	$-0.446628\pi$
$293$	1674.00	0.333775	0.166888	+	0.985976i	$0.446628\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1458.00	0.284854
$298$	0	0
$299$	−990.000	−0.191482
$300$	0	0
$301$	1141.00	0.218492
$302$	0	0
$303$	2484.00	0.470964
$304$	0	0
$305$	0	0
$306$	0	0
$307$	539.000	0.100203	0.0501016	−	0.998744i	$-0.484046\pi$
$307$	539.000	0.100203	0.0501016	+	0.998744i	$0.484046\pi$
$308$	0	0
$309$	1644.00	0.302666
$310$	0	0
$311$	−1494.00	−0.272402	−0.136201	−	0.990681i	$-0.543489\pi$
$311$	−1494.00	−0.272402	−0.136201	+	0.990681i	$0.543489\pi$
$312$	0	0
$313$	3997.00	0.721801	0.360901	−	0.932604i	$-0.382469\pi$
$313$	3997.00	0.721801	0.360901	+	0.932604i	$0.382469\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3672.00	0.650600	0.325300	−	0.945611i	$-0.394535\pi$
$317$	3672.00	0.650600	0.325300	+	0.945611i	$0.394535\pi$
$318$	0	0
$319$	−2916.00	−0.511801
$320$	0	0
$321$	−4428.00	−0.769928
$322$	0	0
$323$	−450.000	−0.0775191
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3831.00	0.647874
$328$	0	0
$329$	−3654.00	−0.612315
$330$	0	0
$331$	−1052.00	−0.174692	−0.0873461	−	0.996178i	$-0.527839\pi$
$331$	−1052.00	−0.174692	−0.0873461	+	0.996178i	$0.527839\pi$
$332$	0	0
$333$	−2826.00	−0.465057
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3871.00	0.625718	0.312859	−	0.949800i	$-0.398713\pi$
$337$	3871.00	0.625718	0.312859	+	0.949800i	$0.398713\pi$
$338$	0	0
$339$	−5508.00	−0.882459
$340$	0	0
$341$	14634.0	2.32398
$342$	0	0
$343$	4459.00	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7686.00	−1.18907	−0.594533	−	0.804071i	$-0.702663\pi$
$347$	−7686.00	−1.18907	−0.594533	+	0.804071i	$0.702663\pi$
$348$	0	0
$349$	−46.0000	−0.00705537	−0.00352768	−	0.999994i	$-0.501123\pi$
$349$	−46.0000	−0.00705537	−0.00352768	+	0.999994i	$0.501123\pi$
$350$	0	0
$351$	1485.00	0.225822
$352$	0	0
$353$	6714.00	1.01232	0.506162	−	0.862439i	$-0.331064\pi$
$353$	6714.00	1.01232	0.506162	+	0.862439i	$0.331064\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	378.000	0.0560389
$358$	0	0
$359$	−1296.00	−0.190530	−0.0952650	−	0.995452i	$-0.530370\pi$
$359$	−1296.00	−0.190530	−0.0952650	+	0.995452i	$0.530370\pi$
$360$	0	0
$361$	−6234.00	−0.908879
$362$	0	0
$363$	4755.00	0.687528
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5903.00	0.839602	0.419801	−	0.907616i	$-0.362100\pi$
$367$	5903.00	0.839602	0.419801	+	0.907616i	$0.362100\pi$
$368$	0	0
$369$	−3240.00	−0.457094
$370$	0	0
$371$	−252.000	−0.0352647
$372$	0	0
$373$	−8867.00	−1.23087	−0.615437	−	0.788186i	$-0.711020\pi$
$373$	−8867.00	−1.23087	−0.615437	+	0.788186i	$0.711020\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2970.00	−0.405737
$378$	0	0
$379$	−8837.00	−1.19769	−0.598847	−	0.800863i	$-0.704374\pi$
$379$	−8837.00	−1.19769	−0.598847	+	0.800863i	$0.704374\pi$
$380$	0	0
$381$	−1776.00	−0.238812
$382$	0	0
$383$	−10044.0	−1.34001	−0.670006	−	0.742356i	$-0.733708\pi$
$383$	−10044.0	−1.34001	−0.670006	+	0.742356i	$0.733708\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1467.00	−0.192692
$388$	0	0
$389$	−2736.00	−0.356609	−0.178304	−	0.983975i	$-0.557061\pi$
$389$	−2736.00	−0.356609	−0.178304	+	0.983975i	$0.557061\pi$
$390$	0	0
$391$	324.000	0.0419064
$392$	0	0
$393$	1404.00	0.180210
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12241.0	1.54750	0.773751	−	0.633490i	$-0.218378\pi$
$397$	12241.0	1.54750	0.773751	+	0.633490i	$0.218378\pi$
$398$	0	0
$399$	−525.000	−0.0658719
$400$	0	0
$401$	9036.00	1.12528	0.562639	−	0.826703i	$-0.309786\pi$
$401$	9036.00	1.12528	0.562639	+	0.826703i	$0.309786\pi$
$402$	0	0
$403$	14905.0	1.84236
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16956.0	−2.06506
$408$	0	0
$409$	8549.00	1.03355	0.516774	−	0.856122i	$-0.327133\pi$
$409$	8549.00	1.03355	0.516774	+	0.856122i	$0.327133\pi$
$410$	0	0
$411$	−7722.00	−0.926760
$412$	0	0
$413$	882.000	0.105086
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5268.00	0.618645
$418$	0	0
$419$	−1548.00	−0.180489	−0.0902443	−	0.995920i	$-0.528765\pi$
$419$	−1548.00	−0.180489	−0.0902443	+	0.995920i	$0.528765\pi$
$420$	0	0
$421$	6110.00	0.707323	0.353662	−	0.935373i	$-0.384936\pi$
$421$	6110.00	0.707323	0.353662	+	0.935373i	$0.384936\pi$
$422$	0	0
$423$	4698.00	0.540011
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−329.000	−0.0372867
$428$	0	0
$429$	8910.00	1.00275
$430$	0	0
$431$	−5958.00	−0.665863	−0.332931	−	0.942951i	$-0.608038\pi$
$431$	−5958.00	−0.665863	−0.332931	+	0.942951i	$0.608038\pi$
$432$	0	0
$433$	−7163.00	−0.794993	−0.397496	−	0.917604i	$-0.630121\pi$
$433$	−7163.00	−0.794993	−0.397496	+	0.917604i	$0.630121\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−450.000	−0.0492595
$438$	0	0
$439$	−17.0000	−0.00184821	−0.000924107	−	1.00000i	$-0.500294\pi$
$439$	−17.0000	−0.00184821	−0.000924107		1.00000i	$0.500294\pi$
$440$	0	0
$441$	−2646.00	−0.285714
$442$	0	0
$443$	−9432.00	−1.01158	−0.505788	−	0.862658i	$-0.668798\pi$
$443$	−9432.00	−1.01158	−0.505788	+	0.862658i	$0.668798\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8046.00	0.851371
$448$	0	0
$449$	−15228.0	−1.60057	−0.800283	−	0.599623i	$-0.795317\pi$
$449$	−15228.0	−1.60057	−0.800283	+	0.599623i	$0.795317\pi$
$450$	0	0
$451$	−19440.0	−2.02970
$452$	0	0
$453$	−10185.0	−1.05636
$454$	0	0
$455$	0	0
$456$	0	0
$457$	250.000	0.0255897	0.0127949	−	0.999918i	$-0.495927\pi$
$457$	250.000	0.0255897	0.0127949	+	0.999918i	$0.495927\pi$
$458$	0	0
$459$	−486.000	−0.0494217
$460$	0	0
$461$	16956.0	1.71306	0.856529	−	0.516099i	$-0.172616\pi$
$461$	16956.0	1.71306	0.856529	+	0.516099i	$0.172616\pi$
$462$	0	0
$463$	−4384.00	−0.440047	−0.220023	−	0.975495i	$-0.570613\pi$
$463$	−4384.00	−0.440047	−0.220023	+	0.975495i	$0.570613\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5166.00	−0.511893	−0.255946	−	0.966691i	$-0.582387\pi$
$467$	−5166.00	−0.511893	−0.255946	+	0.966691i	$0.582387\pi$
$468$	0	0
$469$	2401.00	0.236392
$470$	0	0
$471$	4647.00	0.454612
$472$	0	0
$473$	−8802.00	−0.855637
$474$	0	0
$475$	0	0
$476$	0	0
$477$	324.000	0.0311005
$478$	0	0
$479$	−6966.00	−0.664477	−0.332239	−	0.943195i	$-0.607804\pi$
$479$	−6966.00	−0.664477	−0.332239	+	0.943195i	$0.607804\pi$
$480$	0	0
$481$	−17270.0	−1.63710
$482$	0	0
$483$	378.000	0.0356099
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18431.0	1.71497	0.857483	−	0.514512i	$-0.172027\pi$
$487$	18431.0	1.71497	0.857483	+	0.514512i	$0.172027\pi$
$488$	0	0
$489$	−1515.00	−0.140104
$490$	0	0
$491$	−1224.00	−0.112502	−0.0562509	−	0.998417i	$-0.517915\pi$
$491$	−1224.00	−0.112502	−0.0562509	+	0.998417i	$0.517915\pi$
$492$	0	0
$493$	972.000	0.0887965
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7560.00	−0.682319
$498$	0	0
$499$	2449.00	0.219704	0.109852	−	0.993948i	$-0.464962\pi$
$499$	2449.00	0.219704	0.109852	+	0.993948i	$0.464962\pi$
$500$	0	0
$501$	−4428.00	−0.394867
$502$	0	0
$503$	−3312.00	−0.293588	−0.146794	−	0.989167i	$-0.546895\pi$
$503$	−3312.00	−0.293588	−0.146794	+	0.989167i	$0.546895\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2484.00	0.217590
$508$	0	0
$509$	−9162.00	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−9162.00	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−7378.00	−0.638715
$512$	0	0
$513$	675.000	0.0580935
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28188.0	2.39789
$518$	0	0
$519$	−7074.00	−0.598293
$520$	0	0
$521$	−5418.00	−0.455599	−0.227799	−	0.973708i	$-0.573153\pi$
$521$	−5418.00	−0.455599	−0.227799	+	0.973708i	$0.573153\pi$
$522$	0	0
$523$	−6829.00	−0.570959	−0.285479	−	0.958385i	$-0.592153\pi$
$523$	−6829.00	−0.570959	−0.285479	+	0.958385i	$0.592153\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4878.00	−0.403205
$528$	0	0
$529$	−11843.0	−0.973371
$530$	0	0
$531$	−1134.00	−0.0926769
$532$	0	0
$533$	−19800.0	−1.60907
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4698.00	−0.377530
$538$	0	0
$539$	−15876.0	−1.26870
$540$	0	0
$541$	3053.00	0.242622	0.121311	−	0.992615i	$-0.461290\pi$
$541$	3053.00	0.242622	0.121311	+	0.992615i	$0.461290\pi$
$542$	0	0
$543$	915.000	0.0723138
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20456.0	1.59897	0.799484	−	0.600687i	$-0.205106\pi$
$547$	20456.0	1.59897	0.799484	+	0.600687i	$0.205106\pi$
$548$	0	0
$549$	423.000	0.0328838
$550$	0	0
$551$	−1350.00	−0.104377
$552$	0	0
$553$	−3976.00	−0.305745
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10854.0	0.825671	0.412835	−	0.910806i	$-0.364538\pi$
$557$	10854.0	0.825671	0.412835	+	0.910806i	$0.364538\pi$
$558$	0	0
$559$	−8965.00	−0.678317
$560$	0	0
$561$	−2916.00	−0.219454
$562$	0	0
$563$	−24930.0	−1.86621	−0.933103	−	0.359609i	$-0.882910\pi$
$563$	−24930.0	−1.86621	−0.933103	+	0.359609i	$0.882910\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−24786.0	−1.82616	−0.913078	−	0.407784i	$-0.866302\pi$
$569$	−24786.0	−1.82616	−0.913078	+	0.407784i	$0.866302\pi$
$570$	0	0
$571$	14785.0	1.08360	0.541798	−	0.840509i	$-0.317744\pi$
$571$	14785.0	1.08360	0.541798	+	0.840509i	$0.317744\pi$
$572$	0	0
$573$	5238.00	0.381886
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15851.0	−1.14365	−0.571825	−	0.820376i	$-0.693764\pi$
$577$	−15851.0	−1.14365	−0.571825	+	0.820376i	$0.693764\pi$
$578$	0	0
$579$	11631.0	0.834832
$580$	0	0
$581$	−9954.00	−0.710777
$582$	0	0
$583$	1944.00	0.138100
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23148.0	1.62763	0.813816	−	0.581122i	$-0.197386\pi$
$587$	23148.0	1.62763	0.813816	+	0.581122i	$0.197386\pi$
$588$	0	0
$589$	6775.00	0.473954
$590$	0	0
$591$	−6426.00	−0.447259
$592$	0	0
$593$	21888.0	1.51574	0.757869	−	0.652407i	$-0.226241\pi$
$593$	21888.0	1.51574	0.757869	+	0.652407i	$0.226241\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12099.0	0.829446
$598$	0	0
$599$	−10764.0	−0.734232	−0.367116	−	0.930175i	$-0.619655\pi$
$599$	−10764.0	−0.734232	−0.367116	+	0.930175i	$0.619655\pi$
$600$	0	0
$601$	−25597.0	−1.73731	−0.868655	−	0.495417i	$-0.835015\pi$
$601$	−25597.0	−1.73731	−0.868655	+	0.495417i	$0.835015\pi$
$602$	0	0
$603$	−3087.00	−0.208478
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24976.0	−1.67009	−0.835045	−	0.550182i	$-0.814558\pi$
$607$	−24976.0	−1.67009	−0.835045	+	0.550182i	$0.814558\pi$
$608$	0	0
$609$	1134.00	0.0754548
$610$	0	0
$611$	28710.0	1.90095
$612$	0	0
$613$	2134.00	0.140606	0.0703030	−	0.997526i	$-0.477603\pi$
$613$	2134.00	0.140606	0.0703030	+	0.997526i	$0.477603\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13932.0	−0.909046	−0.454523	−	0.890735i	$-0.650190\pi$
$617$	−13932.0	−0.909046	−0.454523	+	0.890735i	$0.650190\pi$
$618$	0	0
$619$	10429.0	0.677184	0.338592	−	0.940933i	$-0.390049\pi$
$619$	10429.0	0.677184	0.338592	+	0.940933i	$0.390049\pi$
$620$	0	0
$621$	−486.000	−0.0314050
$622$	0	0
$623$	−10080.0	−0.648229
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4050.00	0.257961
$628$	0	0
$629$	5652.00	0.358283
$630$	0	0
$631$	6283.00	0.396390	0.198195	−	0.980163i	$-0.436492\pi$
$631$	6283.00	0.396390	0.198195	+	0.980163i	$0.436492\pi$
$632$	0	0
$633$	12315.0	0.773266
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−16170.0	−1.00578
$638$	0	0
$639$	9720.00	0.601748
$640$	0	0
$641$	−20916.0	−1.28882	−0.644409	−	0.764681i	$-0.722897\pi$
$641$	−20916.0	−1.28882	−0.644409	+	0.764681i	$0.722897\pi$
$642$	0	0
$643$	−23452.0	−1.43835	−0.719173	−	0.694831i	$-0.755479\pi$
$643$	−23452.0	−1.43835	−0.719173	+	0.694831i	$0.755479\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11916.0	−0.724059	−0.362030	−	0.932167i	$-0.617916\pi$
$647$	−11916.0	−0.724059	−0.362030	+	0.932167i	$0.617916\pi$
$648$	0	0
$649$	−6804.00	−0.411526
$650$	0	0
$651$	−5691.00	−0.342623
$652$	0	0
$653$	−31842.0	−1.90823	−0.954115	−	0.299441i	$-0.903200\pi$
$653$	−31842.0	−1.90823	−0.954115	+	0.299441i	$0.903200\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9486.00	0.563294
$658$	0	0
$659$	−3672.00	−0.217057	−0.108529	−	0.994093i	$-0.534614\pi$
$659$	−3672.00	−0.217057	−0.108529	+	0.994093i	$0.534614\pi$
$660$	0	0
$661$	5138.00	0.302337	0.151169	−	0.988508i	$-0.451696\pi$
$661$	5138.00	0.302337	0.151169	+	0.988508i	$0.451696\pi$
$662$	0	0
$663$	−2970.00	−0.173975
$664$	0	0
$665$	0	0
$666$	0	0
$667$	972.000	0.0564258
$668$	0	0
$669$	4155.00	0.240122
$670$	0	0
$671$	2538.00	0.146018
$672$	0	0
$673$	5050.00	0.289247	0.144623	−	0.989487i	$-0.453803\pi$
$673$	5050.00	0.289247	0.144623	+	0.989487i	$0.453803\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28458.0	−1.61555	−0.807777	−	0.589488i	$-0.799329\pi$
$677$	−28458.0	−1.61555	−0.807777	+	0.589488i	$0.799329\pi$
$678$	0	0
$679$	−3073.00	−0.173683
$680$	0	0
$681$	−7560.00	−0.425404
$682$	0	0
$683$	24408.0	1.36742	0.683709	−	0.729755i	$-0.260366\pi$
$683$	24408.0	1.36742	0.683709	+	0.729755i	$0.260366\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15387.0	0.854513
$688$	0	0
$689$	1980.00	0.109480
$690$	0	0
$691$	−16328.0	−0.898909	−0.449455	−	0.893303i	$-0.648382\pi$
$691$	−16328.0	−0.898909	−0.449455	+	0.893303i	$0.648382\pi$
$692$	0	0
$693$	−3402.00	−0.186481
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6480.00	0.352148
$698$	0	0
$699$	9720.00	0.525957
$700$	0	0
$701$	6246.00	0.336531	0.168265	−	0.985742i	$-0.446183\pi$
$701$	6246.00	0.336531	0.168265	+	0.985742i	$0.446183\pi$
$702$	0	0
$703$	−7850.00	−0.421150
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5796.00	−0.308318
$708$	0	0
$709$	−30679.0	−1.62507	−0.812535	−	0.582913i	$-0.801913\pi$
$709$	−30679.0	−1.62507	−0.812535	+	0.582913i	$0.801913\pi$
$710$	0	0
$711$	5112.00	0.269641
$712$	0	0
$713$	−4878.00	−0.256217
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8964.00	0.466899
$718$	0	0
$719$	−15462.0	−0.801996	−0.400998	−	0.916079i	$-0.631337\pi$
$719$	−15462.0	−0.801996	−0.400998	+	0.916079i	$0.631337\pi$
$720$	0	0
$721$	−3836.00	−0.198142
$722$	0	0
$723$	−7941.00	−0.408477
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26801.0	1.36725	0.683627	−	0.729831i	$-0.260401\pi$
$727$	26801.0	1.36725	0.683627	+	0.729831i	$0.260401\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2934.00	0.148451
$732$	0	0
$733$	22858.0	1.15181	0.575907	−	0.817515i	$-0.304649\pi$
$733$	22858.0	1.15181	0.575907	+	0.817515i	$0.304649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18522.0	−0.925735
$738$	0	0
$739$	16900.0	0.841240	0.420620	−	0.907237i	$-0.361813\pi$
$739$	16900.0	0.841240	0.420620	+	0.907237i	$0.361813\pi$
$740$	0	0
$741$	4125.00	0.204502
$742$	0	0
$743$	8028.00	0.396391	0.198196	−	0.980162i	$-0.436492\pi$
$743$	8028.00	0.396391	0.198196	+	0.980162i	$0.436492\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12798.0	0.626846
$748$	0	0
$749$	10332.0	0.504036
$750$	0	0
$751$	12448.0	0.604839	0.302419	−	0.953175i	$-0.402206\pi$
$751$	12448.0	0.604839	0.302419	+	0.953175i	$0.402206\pi$
$752$	0	0
$753$	12636.0	0.611529
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30103.0	1.44533	0.722663	−	0.691200i	$-0.242918\pi$
$757$	30103.0	1.44533	0.722663	+	0.691200i	$0.242918\pi$
$758$	0	0
$759$	−2916.00	−0.139452
$760$	0	0
$761$	−17748.0	−0.845420	−0.422710	−	0.906265i	$-0.638921\pi$
$761$	−17748.0	−0.845420	−0.422710	+	0.906265i	$0.638921\pi$
$762$	0	0
$763$	−8939.00	−0.424133
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6930.00	−0.326242
$768$	0	0
$769$	13283.0	0.622883	0.311442	−	0.950265i	$-0.399188\pi$
$769$	13283.0	0.622883	0.311442	+	0.950265i	$0.399188\pi$
$770$	0	0
$771$	−17172.0	−0.802120
$772$	0	0
$773$	−26424.0	−1.22950	−0.614751	−	0.788721i	$-0.710744\pi$
$773$	−26424.0	−1.22950	−0.614751	+	0.788721i	$0.710744\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6594.00	0.304451
$778$	0	0
$779$	−9000.00	−0.413939
$780$	0	0
$781$	58320.0	2.67203
$782$	0	0
$783$	−1458.00	−0.0665449
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9709.00	−0.439757	−0.219878	−	0.975527i	$-0.570566\pi$
$787$	−9709.00	−0.439757	−0.219878	+	0.975527i	$0.570566\pi$
$788$	0	0
$789$	−13824.0	−0.623761
$790$	0	0
$791$	12852.0	0.577705
$792$	0	0
$793$	2585.00	0.115758
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7884.00	0.350396	0.175198	−	0.984533i	$-0.443943\pi$
$797$	7884.00	0.350396	0.175198	+	0.984533i	$0.443943\pi$
$798$	0	0
$799$	−9396.00	−0.416028
$800$	0	0
$801$	12960.0	0.571684
$802$	0	0
$803$	56916.0	2.50127
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19278.0	−0.840914
$808$	0	0
$809$	−1476.00	−0.0641451	−0.0320726	−	0.999486i	$-0.510211\pi$
$809$	−1476.00	−0.0641451	−0.0320726	+	0.999486i	$0.510211\pi$
$810$	0	0
$811$	−21455.0	−0.928960	−0.464480	−	0.885583i	$-0.653759\pi$
$811$	−21455.0	−0.928960	−0.464480	+	0.885583i	$0.653759\pi$
$812$	0	0
$813$	10128.0	0.436906
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4075.00	−0.174500
$818$	0	0
$819$	−3465.00	−0.147835
$820$	0	0
$821$	31086.0	1.32145	0.660724	−	0.750629i	$-0.270249\pi$
$821$	31086.0	1.32145	0.660724	+	0.750629i	$0.270249\pi$
$822$	0	0
$823$	23381.0	0.990292	0.495146	−	0.868810i	$-0.335115\pi$
$823$	23381.0	0.990292	0.495146	+	0.868810i	$0.335115\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3636.00	0.152885	0.0764426	−	0.997074i	$-0.475644\pi$
$827$	3636.00	0.152885	0.0764426	+	0.997074i	$0.475644\pi$
$828$	0	0
$829$	4058.00	0.170012	0.0850061	−	0.996380i	$-0.472909\pi$
$829$	4058.00	0.170012	0.0850061	+	0.996380i	$0.472909\pi$
$830$	0	0
$831$	−16143.0	−0.673880
$832$	0	0
$833$	5292.00	0.220116
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7317.00	0.302165
$838$	0	0
$839$	−306.000	−0.0125915	−0.00629576	−	0.999980i	$-0.502004\pi$
$839$	−306.000	−0.0125915	−0.00629576	+	0.999980i	$0.502004\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	−10422.0	−0.425804
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11095.0	−0.450093
$848$	0	0
$849$	−6807.00	−0.275166
$850$	0	0
$851$	5652.00	0.227671
$852$	0	0
$853$	−42299.0	−1.69788	−0.848939	−	0.528491i	$-0.822758\pi$
$853$	−42299.0	−1.69788	−0.848939	+	0.528491i	$0.822758\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11484.0	0.457743	0.228872	−	0.973457i	$-0.426496\pi$
$857$	11484.0	0.457743	0.228872	+	0.973457i	$0.426496\pi$
$858$	0	0
$859$	−33560.0	−1.33301	−0.666503	−	0.745502i	$-0.732210\pi$
$859$	−33560.0	−1.33301	−0.666503	+	0.745502i	$0.732210\pi$
$860$	0	0
$861$	7560.00	0.299238
$862$	0	0
$863$	−14976.0	−0.590717	−0.295359	−	0.955386i	$-0.595439\pi$
$863$	−14976.0	−0.590717	−0.295359	+	0.955386i	$0.595439\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13767.0	−0.539275
$868$	0	0
$869$	30672.0	1.19733
$870$	0	0
$871$	−18865.0	−0.733888
$872$	0	0
$873$	3951.00	0.153174
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41893.0	1.61303	0.806514	−	0.591215i	$-0.201351\pi$
$877$	41893.0	1.61303	0.806514	+	0.591215i	$0.201351\pi$
$878$	0	0
$879$	5022.00	0.192705
$880$	0	0
$881$	−720.000	−0.0275340	−0.0137670	−	0.999905i	$-0.504382\pi$
$881$	−720.000	−0.0275340	−0.0137670	+	0.999905i	$0.504382\pi$
$882$	0	0
$883$	17309.0	0.659676	0.329838	−	0.944037i	$-0.393006\pi$
$883$	17309.0	0.659676	0.329838	+	0.944037i	$0.393006\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7002.00	−0.265055	−0.132528	−	0.991179i	$-0.542309\pi$
$887$	−7002.00	−0.265055	−0.132528	+	0.991179i	$0.542309\pi$
$888$	0	0
$889$	4144.00	0.156339
$890$	0	0
$891$	4374.00	0.164461
$892$	0	0
$893$	13050.0	0.489028
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2970.00	−0.110552
$898$	0	0
$899$	−14634.0	−0.542905
$900$	0	0
$901$	−648.000	−0.0239601
$902$	0	0
$903$	3423.00	0.126147
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1484.00	0.0543279	0.0271640	−	0.999631i	$-0.491352\pi$
$907$	1484.00	0.0543279	0.0271640	+	0.999631i	$0.491352\pi$
$908$	0	0
$909$	7452.00	0.271911
$910$	0	0
$911$	18882.0	0.686705	0.343353	−	0.939207i	$-0.388437\pi$
$911$	18882.0	0.686705	0.343353	+	0.939207i	$0.388437\pi$
$912$	0	0
$913$	76788.0	2.78347
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3276.00	−0.117975
$918$	0	0
$919$	50653.0	1.81816	0.909080	−	0.416622i	$-0.136786\pi$
$919$	50653.0	1.81816	0.909080	+	0.416622i	$0.136786\pi$
$920$	0	0
$921$	1617.00	0.0578523
$922$	0	0
$923$	59400.0	2.11828
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4932.00	0.174744
$928$	0	0
$929$	35262.0	1.24533	0.622663	−	0.782490i	$-0.286051\pi$
$929$	35262.0	1.24533	0.622663	+	0.782490i	$0.286051\pi$
$930$	0	0
$931$	−7350.00	−0.258740
$932$	0	0
$933$	−4482.00	−0.157271
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20279.0	−0.707029	−0.353514	−	0.935429i	$-0.615013\pi$
$937$	−20279.0	−0.707029	−0.353514	+	0.935429i	$0.615013\pi$
$938$	0	0
$939$	11991.0	0.416732
$940$	0	0
$941$	42390.0	1.46852	0.734259	−	0.678870i	$-0.237530\pi$
$941$	42390.0	1.46852	0.734259	+	0.678870i	$0.237530\pi$
$942$	0	0
$943$	6480.00	0.223773
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42426.0	−1.45582	−0.727909	−	0.685674i	$-0.759508\pi$
$947$	−42426.0	−1.45582	−0.727909	+	0.685674i	$0.759508\pi$
$948$	0	0
$949$	57970.0	1.98291
$950$	0	0
$951$	11016.0	0.375624
$952$	0	0
$953$	−48168.0	−1.63727	−0.818633	−	0.574317i	$-0.805268\pi$
$953$	−48168.0	−1.63727	−0.818633	+	0.574317i	$0.805268\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8748.00	−0.295489
$958$	0	0
$959$	18018.0	0.606707
$960$	0	0
$961$	43650.0	1.46521
$962$	0	0
$963$	−13284.0	−0.444518
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−51400.0	−1.70932	−0.854660	−	0.519188i	$-0.826234\pi$
$967$	−51400.0	−1.70932	−0.854660	+	0.519188i	$0.826234\pi$
$968$	0	0
$969$	−1350.00	−0.0447557
$970$	0	0
$971$	−33858.0	−1.11901	−0.559503	−	0.828828i	$-0.689008\pi$
$971$	−33858.0	−1.11901	−0.559503	+	0.828828i	$0.689008\pi$
$972$	0	0
$973$	−12292.0	−0.404998
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47106.0	−1.54253	−0.771266	−	0.636513i	$-0.780376\pi$
$977$	−47106.0	−1.54253	−0.771266	+	0.636513i	$0.780376\pi$
$978$	0	0
$979$	77760.0	2.53853
$980$	0	0
$981$	11493.0	0.374050
$982$	0	0
$983$	−20844.0	−0.676318	−0.338159	−	0.941089i	$-0.609804\pi$
$983$	−20844.0	−0.676318	−0.338159	+	0.941089i	$0.609804\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−10962.0	−0.353520
$988$	0	0
$989$	2934.00	0.0943334
$990$	0	0
$991$	−4133.00	−0.132481	−0.0662407	−	0.997804i	$-0.521101\pi$
$991$	−4133.00	−0.132481	−0.0662407	+	0.997804i	$0.521101\pi$
$992$	0	0
$993$	−3156.00	−0.100859
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33122.0	−1.05214	−0.526070	−	0.850441i	$-0.676335\pi$
$997$	−33122.0	−1.05214	−0.526070	+	0.850441i	$0.676335\pi$
$998$	0	0
$999$	−8478.00	−0.268501

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.y.1.1		1
4.3	odd	2		300.4.a.c.1.1	✓	1
5.2	odd	4		1200.4.f.s.49.1		2
5.3	odd	4		1200.4.f.s.49.2		2
5.4	even	2		1200.4.a.m.1.1		1
12.11	even	2		900.4.a.k.1.1		1
20.3	even	4		300.4.d.a.49.1		2
20.7	even	4		300.4.d.a.49.2		2
20.19	odd	2		300.4.a.g.1.1	yes	1
60.23	odd	4		900.4.d.j.649.1		2
60.47	odd	4		900.4.d.j.649.2		2
60.59	even	2		900.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.4.a.c.1.1	✓	1	4.3	odd	2
300.4.a.g.1.1	yes	1	20.19	odd	2
300.4.d.a.49.1		2	20.3	even	4
300.4.d.a.49.2		2	20.7	even	4
900.4.a.h.1.1		1	60.59	even	2
900.4.a.k.1.1		1	12.11	even	2
900.4.d.j.649.1		2	60.23	odd	4
900.4.d.j.649.2		2	60.47	odd	4
1200.4.a.m.1.1		1	5.4	even	2
1200.4.a.y.1.1		1	1.1	even	1	trivial
1200.4.f.s.49.1		2	5.2	odd	4
1200.4.f.s.49.2		2	5.3	odd	4

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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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	1200.4.a.y.1.1

Level	$1200$
Weight	$4$
Character	1200.1
Self dual	yes
Analytic conductor	$70.802$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$