LMFDB - Embedded newform 1200.4.a.bd.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 600)
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)

$f(q)$

$=$

$q+3.00000 q^{3} +5.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +5.00000 q^{7} +9.00000 q^{9} -14.0000 q^{11} -1.00000 q^{13} -46.0000 q^{17} -19.0000 q^{19} +15.0000 q^{21} -46.0000 q^{23} +27.0000 q^{27} +14.0000 q^{29} -133.000 q^{31} -42.0000 q^{33} -258.000 q^{37} -3.00000 q^{39} +84.0000 q^{41} -167.000 q^{43} +410.000 q^{47} -318.000 q^{49} -138.000 q^{51} -456.000 q^{53} -57.0000 q^{57} +194.000 q^{59} -17.0000 q^{61} +45.0000 q^{63} +653.000 q^{67} -138.000 q^{69} -828.000 q^{71} -570.000 q^{73} -70.0000 q^{77} +552.000 q^{79} +81.0000 q^{81} +142.000 q^{83} +42.0000 q^{87} -1104.00 q^{89} -5.00000 q^{91} -399.000 q^{93} -841.000 q^{97} -126.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000	0.269975	0.134987	−	0.990847i	$-0.456901\pi$
$7$	5.00000	0.269975	0.134987	+	0.990847i	$0.456901\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−14.0000	−0.383742	−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000	−0.383742	−0.191871	+	0.981420i	$0.561455\pi$
$12$	0	0
$13$	−1.00000	−0.0213346	−0.0106673	−	0.999943i	$-0.503396\pi$
$13$	−1.00000	−0.0213346	−0.0106673	+	0.999943i	$0.503396\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−46.0000	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$17$	−46.0000	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$18$	0	0
$19$	−19.0000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−19.0000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	15.0000	0.155870
$22$	0	0
$23$	−46.0000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−46.0000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	14.0000	0.0896460	0.0448230	−	0.998995i	$-0.485728\pi$
$29$	14.0000	0.0896460	0.0448230	+	0.998995i	$0.485728\pi$
$30$	0	0
$31$	−133.000	−0.770565	−0.385282	−	0.922799i	$-0.625896\pi$
$31$	−133.000	−0.770565	−0.385282	+	0.922799i	$0.625896\pi$
$32$	0	0
$33$	−42.0000	−0.221553
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−258.000	−1.14635	−0.573175	−	0.819433i	$-0.694288\pi$
$37$	−258.000	−1.14635	−0.573175	+	0.819433i	$0.694288\pi$
$38$	0	0
$39$	−3.00000	−0.0123176
$40$	0	0
$41$	84.0000	0.319966	0.159983	−	0.987120i	$-0.448856\pi$
$41$	84.0000	0.319966	0.159983	+	0.987120i	$0.448856\pi$
$42$	0	0
$43$	−167.000	−0.592262	−0.296131	−	0.955147i	$-0.595696\pi$
$43$	−167.000	−0.592262	−0.296131	+	0.955147i	$0.595696\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	410.000	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$47$	410.000	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$48$	0	0
$49$	−318.000	−0.927114
$50$	0	0
$51$	−138.000	−0.378899
$52$	0	0
$53$	−456.000	−1.18182	−0.590910	−	0.806738i	$-0.701231\pi$
$53$	−456.000	−1.18182	−0.590910	+	0.806738i	$0.701231\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−57.0000	−0.132453
$58$	0	0
$59$	194.000	0.428079	0.214039	−	0.976825i	$-0.431338\pi$
$59$	194.000	0.428079	0.214039	+	0.976825i	$0.431338\pi$
$60$	0	0
$61$	−17.0000	−0.0356824	−0.0178412	−	0.999841i	$-0.505679\pi$
$61$	−17.0000	−0.0356824	−0.0178412	+	0.999841i	$0.505679\pi$
$62$	0	0
$63$	45.0000	0.0899915
$64$	0	0
$65$	0	0
$66$	0	0
$67$	653.000	1.19070	0.595348	−	0.803468i	$-0.297014\pi$
$67$	653.000	1.19070	0.595348	+	0.803468i	$0.297014\pi$
$68$	0	0
$69$	−138.000	−0.240772
$70$	0	0
$71$	−828.000	−1.38402	−0.692011	−	0.721887i	$-0.743275\pi$
$71$	−828.000	−1.38402	−0.692011	+	0.721887i	$0.743275\pi$
$72$	0	0
$73$	−570.000	−0.913883	−0.456941	−	0.889497i	$-0.651055\pi$
$73$	−570.000	−0.913883	−0.456941	+	0.889497i	$0.651055\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−70.0000	−0.103601
$78$	0	0
$79$	552.000	0.786137	0.393069	−	0.919509i	$-0.371413\pi$
$79$	552.000	0.786137	0.393069	+	0.919509i	$0.371413\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	142.000	0.187789	0.0938947	−	0.995582i	$-0.470068\pi$
$83$	142.000	0.187789	0.0938947	+	0.995582i	$0.470068\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	42.0000	0.0517572
$88$	0	0
$89$	−1104.00	−1.31487	−0.657437	−	0.753510i	$-0.728359\pi$
$89$	−1104.00	−1.31487	−0.657437	+	0.753510i	$0.728359\pi$
$90$	0	0
$91$	−5.00000	−0.00575981
$92$	0	0
$93$	−399.000	−0.444886
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−841.000	−0.880316	−0.440158	−	0.897920i	$-0.645077\pi$
$97$	−841.000	−0.880316	−0.440158	+	0.897920i	$0.645077\pi$
$98$	0	0
$99$	−126.000	−0.127914
$100$	0	0
$101$	552.000	0.543822	0.271911	−	0.962322i	$-0.412344\pi$
$101$	552.000	0.543822	0.271911	+	0.962322i	$0.412344\pi$
$102$	0	0
$103$	−308.000	−0.294642	−0.147321	−	0.989089i	$-0.547065\pi$
$103$	−308.000	−0.294642	−0.147321	+	0.989089i	$0.547065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	984.000	0.889036	0.444518	−	0.895770i	$-0.353375\pi$
$107$	984.000	0.889036	0.444518	+	0.895770i	$0.353375\pi$
$108$	0	0
$109$	−1843.00	−1.61952	−0.809759	−	0.586763i	$-0.800402\pi$
$109$	−1843.00	−1.61952	−0.809759	+	0.586763i	$0.800402\pi$
$110$	0	0
$111$	−774.000	−0.661845
$112$	0	0
$113$	−876.000	−0.729267	−0.364633	−	0.931151i	$-0.618806\pi$
$113$	−876.000	−0.729267	−0.364633	+	0.931151i	$0.618806\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.00000	−0.00711154
$118$	0	0
$119$	−230.000	−0.177177
$120$	0	0
$121$	−1135.00	−0.852742
$122$	0	0
$123$	252.000	0.184732
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2376.00	1.66013	0.830063	−	0.557670i	$-0.188305\pi$
$127$	2376.00	1.66013	0.830063	+	0.557670i	$0.188305\pi$
$128$	0	0
$129$	−501.000	−0.341943
$130$	0	0
$131$	−1056.00	−0.704299	−0.352149	−	0.935944i	$-0.614549\pi$
$131$	−1056.00	−0.704299	−0.352149	+	0.935944i	$0.614549\pi$
$132$	0	0
$133$	−95.0000	−0.0619364
$134$	0	0
$135$	0	0
$136$	0	0
$137$	778.000	0.485175	0.242588	−	0.970129i	$-0.422004\pi$
$137$	778.000	0.485175	0.242588	+	0.970129i	$0.422004\pi$
$138$	0	0
$139$	−1692.00	−1.03247	−0.516236	−	0.856446i	$-0.672667\pi$
$139$	−1692.00	−1.03247	−0.516236	+	0.856446i	$0.672667\pi$
$140$	0	0
$141$	1230.00	0.734643
$142$	0	0
$143$	14.0000	0.00818698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−954.000	−0.535269
$148$	0	0
$149$	−494.000	−0.271611	−0.135806	−	0.990736i	$-0.543362\pi$
$149$	−494.000	−0.271611	−0.135806	+	0.990736i	$0.543362\pi$
$150$	0	0
$151$	841.000	0.453242	0.226621	−	0.973983i	$-0.427232\pi$
$151$	841.000	0.453242	0.226621	+	0.973983i	$0.427232\pi$
$152$	0	0
$153$	−414.000	−0.218758
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.0000	−0.00965838	−0.00482919	−	0.999988i	$-0.501537\pi$
$157$	−19.0000	−0.00965838	−0.00482919	+	0.999988i	$0.501537\pi$
$158$	0	0
$159$	−1368.00	−0.682324
$160$	0	0
$161$	−230.000	−0.112587
$162$	0	0
$163$	−2261.00	−1.08647	−0.543237	−	0.839580i	$-0.682801\pi$
$163$	−2261.00	−1.08647	−0.543237	+	0.839580i	$0.682801\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2112.00	0.978632	0.489316	−	0.872107i	$-0.337247\pi$
$167$	2112.00	0.978632	0.489316	+	0.872107i	$0.337247\pi$
$168$	0	0
$169$	−2196.00	−0.999545
$170$	0	0
$171$	−171.000	−0.0764719
$172$	0	0
$173$	−562.000	−0.246983	−0.123492	−	0.992346i	$-0.539409\pi$
$173$	−562.000	−0.246983	−0.123492	+	0.992346i	$0.539409\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	582.000	0.247151
$178$	0	0
$179$	−3718.00	−1.55249	−0.776247	−	0.630429i	$-0.782879\pi$
$179$	−3718.00	−1.55249	−0.776247	+	0.630429i	$0.782879\pi$
$180$	0	0
$181$	−1639.00	−0.673071	−0.336536	−	0.941671i	$-0.609255\pi$
$181$	−1639.00	−0.673071	−0.336536	+	0.941671i	$0.609255\pi$
$182$	0	0
$183$	−51.0000	−0.0206012
$184$	0	0
$185$	0	0
$186$	0	0
$187$	644.000	0.251839
$188$	0	0
$189$	135.000	0.0519566
$190$	0	0
$191$	−2410.00	−0.912992	−0.456496	−	0.889725i	$-0.650896\pi$
$191$	−2410.00	−0.912992	−0.456496	+	0.889725i	$0.650896\pi$
$192$	0	0
$193$	2621.00	0.977532	0.488766	−	0.872415i	$-0.337447\pi$
$193$	2621.00	0.977532	0.488766	+	0.872415i	$0.337447\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4954.00	1.79166	0.895832	−	0.444392i	$-0.146580\pi$
$197$	4954.00	1.79166	0.895832	+	0.444392i	$0.146580\pi$
$198$	0	0
$199$	−1739.00	−0.619470	−0.309735	−	0.950823i	$-0.600240\pi$
$199$	−1739.00	−0.619470	−0.309735	+	0.950823i	$0.600240\pi$
$200$	0	0
$201$	1959.00	0.687449
$202$	0	0
$203$	70.0000	0.0242022
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−414.000	−0.139010
$208$	0	0
$209$	266.000	0.0880364
$210$	0	0
$211$	4525.00	1.47637	0.738184	−	0.674599i	$-0.235683\pi$
$211$	4525.00	1.47637	0.738184	+	0.674599i	$0.235683\pi$
$212$	0	0
$213$	−2484.00	−0.799065
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−665.000	−0.208033
$218$	0	0
$219$	−1710.00	−0.527631
$220$	0	0
$221$	46.0000	0.0140013
$222$	0	0
$223$	−3211.00	−0.964235	−0.482118	−	0.876106i	$-0.660132\pi$
$223$	−3211.00	−0.964235	−0.482118	+	0.876106i	$0.660132\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2484.00	−0.726295	−0.363147	−	0.931732i	$-0.618298\pi$
$227$	−2484.00	−0.726295	−0.363147	+	0.931732i	$0.618298\pi$
$228$	0	0
$229$	−1847.00	−0.532983	−0.266492	−	0.963837i	$-0.585864\pi$
$229$	−1847.00	−0.532983	−0.266492	+	0.963837i	$0.585864\pi$
$230$	0	0
$231$	−210.000	−0.0598138
$232$	0	0
$233$	1020.00	0.286792	0.143396	−	0.989665i	$-0.454198\pi$
$233$	1020.00	0.286792	0.143396	+	0.989665i	$0.454198\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1656.00	0.453877
$238$	0	0
$239$	−1176.00	−0.318281	−0.159140	−	0.987256i	$-0.550872\pi$
$239$	−1176.00	−0.318281	−0.159140	+	0.987256i	$0.550872\pi$
$240$	0	0
$241$	−6967.00	−1.86217	−0.931087	−	0.364797i	$-0.881138\pi$
$241$	−6967.00	−1.86217	−0.931087	+	0.364797i	$0.881138\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.0000	0.00489450
$248$	0	0
$249$	426.000	0.108420
$250$	0	0
$251$	−1380.00	−0.347031	−0.173516	−	0.984831i	$-0.555513\pi$
$251$	−1380.00	−0.347031	−0.173516	+	0.984831i	$0.555513\pi$
$252$	0	0
$253$	644.000	0.160031
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6924.00	1.68057	0.840286	−	0.542143i	$-0.182387\pi$
$257$	6924.00	1.68057	0.840286	+	0.542143i	$0.182387\pi$
$258$	0	0
$259$	−1290.00	−0.309485
$260$	0	0
$261$	126.000	0.0298820
$262$	0	0
$263$	1884.00	0.441720	0.220860	−	0.975305i	$-0.429114\pi$
$263$	1884.00	0.441720	0.220860	+	0.975305i	$0.429114\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3312.00	−0.759143
$268$	0	0
$269$	3610.00	0.818236	0.409118	−	0.912481i	$-0.365836\pi$
$269$	3610.00	0.818236	0.409118	+	0.912481i	$0.365836\pi$
$270$	0	0
$271$	6072.00	1.36106	0.680531	−	0.732719i	$-0.261749\pi$
$271$	6072.00	1.36106	0.680531	+	0.732719i	$0.261749\pi$
$272$	0	0
$273$	−15.0000	−0.00332543
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2803.00	0.608000	0.304000	−	0.952672i	$-0.401678\pi$
$277$	2803.00	0.608000	0.304000	+	0.952672i	$0.401678\pi$
$278$	0	0
$279$	−1197.00	−0.256855
$280$	0	0
$281$	−6694.00	−1.42111	−0.710553	−	0.703644i	$-0.751555\pi$
$281$	−6694.00	−1.42111	−0.710553	+	0.703644i	$0.751555\pi$
$282$	0	0
$283$	−6481.00	−1.36133	−0.680663	−	0.732596i	$-0.738308\pi$
$283$	−6481.00	−1.36133	−0.680663	+	0.732596i	$0.738308\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	420.000	0.0863826
$288$	0	0
$289$	−2797.00	−0.569306
$290$	0	0
$291$	−2523.00	−0.508250
$292$	0	0
$293$	−3014.00	−0.600955	−0.300477	−	0.953789i	$-0.597146\pi$
$293$	−3014.00	−0.600955	−0.300477	+	0.953789i	$0.597146\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−378.000	−0.0738511
$298$	0	0
$299$	46.0000	0.00889715
$300$	0	0
$301$	−835.000	−0.159896
$302$	0	0
$303$	1656.00	0.313976
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5369.00	−0.998127	−0.499064	−	0.866565i	$-0.666323\pi$
$307$	−5369.00	−0.998127	−0.499064	+	0.866565i	$0.666323\pi$
$308$	0	0
$309$	−924.000	−0.170112
$310$	0	0
$311$	−4846.00	−0.883574	−0.441787	−	0.897120i	$-0.645655\pi$
$311$	−4846.00	−0.883574	−0.441787	+	0.897120i	$0.645655\pi$
$312$	0	0
$313$	757.000	0.136703	0.0683517	−	0.997661i	$-0.478226\pi$
$313$	757.000	0.136703	0.0683517	+	0.997661i	$0.478226\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7632.00	1.35223	0.676113	−	0.736798i	$-0.263663\pi$
$317$	7632.00	1.35223	0.676113	+	0.736798i	$0.263663\pi$
$318$	0	0
$319$	−196.000	−0.0344009
$320$	0	0
$321$	2952.00	0.513285
$322$	0	0
$323$	874.000	0.150559
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5529.00	−0.935029
$328$	0	0
$329$	2050.00	0.343526
$330$	0	0
$331$	6780.00	1.12587	0.562934	−	0.826502i	$-0.309672\pi$
$331$	6780.00	1.12587	0.562934	+	0.826502i	$0.309672\pi$
$332$	0	0
$333$	−2322.00	−0.382117
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7849.00	−1.26873	−0.634365	−	0.773033i	$-0.718739\pi$
$337$	−7849.00	−1.26873	−0.634365	+	0.773033i	$0.718739\pi$
$338$	0	0
$339$	−2628.00	−0.421042
$340$	0	0
$341$	1862.00	0.295698
$342$	0	0
$343$	−3305.00	−0.520272
$344$	0	0
$345$	0	0
$346$	0	0
$347$	634.000	0.0980833	0.0490416	−	0.998797i	$-0.484383\pi$
$347$	634.000	0.0980833	0.0490416	+	0.998797i	$0.484383\pi$
$348$	0	0
$349$	930.000	0.142641	0.0713206	−	0.997453i	$-0.477279\pi$
$349$	930.000	0.142641	0.0713206	+	0.997453i	$0.477279\pi$
$350$	0	0
$351$	−27.0000	−0.00410585
$352$	0	0
$353$	4286.00	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$353$	4286.00	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−690.000	−0.102293
$358$	0	0
$359$	4236.00	0.622751	0.311375	−	0.950287i	$-0.399210\pi$
$359$	4236.00	0.622751	0.311375	+	0.950287i	$0.399210\pi$
$360$	0	0
$361$	−6498.00	−0.947368
$362$	0	0
$363$	−3405.00	−0.492331
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1451.00	0.206380	0.103190	−	0.994662i	$-0.467095\pi$
$367$	1451.00	0.206380	0.103190	+	0.994662i	$0.467095\pi$
$368$	0	0
$369$	756.000	0.106655
$370$	0	0
$371$	−2280.00	−0.319061
$372$	0	0
$373$	−3115.00	−0.432409	−0.216205	−	0.976348i	$-0.569368\pi$
$373$	−3115.00	−0.432409	−0.216205	+	0.976348i	$0.569368\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.0000	−0.00191256
$378$	0	0
$379$	1415.00	0.191777	0.0958887	−	0.995392i	$-0.469431\pi$
$379$	1415.00	0.191777	0.0958887	+	0.995392i	$0.469431\pi$
$380$	0	0
$381$	7128.00	0.958474
$382$	0	0
$383$	−180.000	−0.0240145	−0.0120073	−	0.999928i	$-0.503822\pi$
$383$	−180.000	−0.0240145	−0.0120073	+	0.999928i	$0.503822\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1503.00	−0.197421
$388$	0	0
$389$	12372.0	1.61256	0.806279	−	0.591535i	$-0.201478\pi$
$389$	12372.0	1.61256	0.806279	+	0.591535i	$0.201478\pi$
$390$	0	0
$391$	2116.00	0.273685
$392$	0	0
$393$	−3168.00	−0.406627
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5767.00	−0.729062	−0.364531	−	0.931191i	$-0.618771\pi$
$397$	−5767.00	−0.729062	−0.364531	+	0.931191i	$0.618771\pi$
$398$	0	0
$399$	−285.000	−0.0357590
$400$	0	0
$401$	−3120.00	−0.388542	−0.194271	−	0.980948i	$-0.562234\pi$
$401$	−3120.00	−0.388542	−0.194271	+	0.980948i	$0.562234\pi$
$402$	0	0
$403$	133.000	0.0164397
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3612.00	0.439902
$408$	0	0
$409$	1501.00	0.181466	0.0907331	−	0.995875i	$-0.471079\pi$
$409$	1501.00	0.181466	0.0907331	+	0.995875i	$0.471079\pi$
$410$	0	0
$411$	2334.00	0.280116
$412$	0	0
$413$	970.000	0.115570
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5076.00	−0.596098
$418$	0	0
$419$	9072.00	1.05775	0.528874	−	0.848701i	$-0.322614\pi$
$419$	9072.00	1.05775	0.528874	+	0.848701i	$0.322614\pi$
$420$	0	0
$421$	7350.00	0.850872	0.425436	−	0.904989i	$-0.360121\pi$
$421$	7350.00	0.850872	0.425436	+	0.904989i	$0.360121\pi$
$422$	0	0
$423$	3690.00	0.424146
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−85.0000	−0.00963334
$428$	0	0
$429$	42.0000	0.00472676
$430$	0	0
$431$	−5962.00	−0.666310	−0.333155	−	0.942872i	$-0.608113\pi$
$431$	−5962.00	−0.666310	−0.333155	+	0.942872i	$0.608113\pi$
$432$	0	0
$433$	10093.0	1.12018	0.560091	−	0.828431i	$-0.310766\pi$
$433$	10093.0	1.12018	0.560091	+	0.828431i	$0.310766\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	874.000	0.0956730
$438$	0	0
$439$	2555.00	0.277776	0.138888	−	0.990308i	$-0.455647\pi$
$439$	2555.00	0.277776	0.138888	+	0.990308i	$0.455647\pi$
$440$	0	0
$441$	−2862.00	−0.309038
$442$	0	0
$443$	6240.00	0.669236	0.334618	−	0.942354i	$-0.391393\pi$
$443$	6240.00	0.669236	0.334618	+	0.942354i	$0.391393\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1482.00	−0.156815
$448$	0	0
$449$	3324.00	0.349375	0.174687	−	0.984624i	$-0.444109\pi$
$449$	3324.00	0.349375	0.174687	+	0.984624i	$0.444109\pi$
$450$	0	0
$451$	−1176.00	−0.122784
$452$	0	0
$453$	2523.00	0.261680
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16774.0	−1.71697	−0.858484	−	0.512840i	$-0.828593\pi$
$457$	−16774.0	−1.71697	−0.858484	+	0.512840i	$0.828593\pi$
$458$	0	0
$459$	−1242.00	−0.126300
$460$	0	0
$461$	14304.0	1.44513	0.722564	−	0.691304i	$-0.242964\pi$
$461$	14304.0	1.44513	0.722564	+	0.691304i	$0.242964\pi$
$462$	0	0
$463$	6936.00	0.696206	0.348103	−	0.937456i	$-0.386826\pi$
$463$	6936.00	0.696206	0.348103	+	0.937456i	$0.386826\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15622.0	1.54797	0.773983	−	0.633207i	$-0.218262\pi$
$467$	15622.0	1.54797	0.773983	+	0.633207i	$0.218262\pi$
$468$	0	0
$469$	3265.00	0.321458
$470$	0	0
$471$	−57.0000	−0.00557627
$472$	0	0
$473$	2338.00	0.227276
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4104.00	−0.393940
$478$	0	0
$479$	13354.0	1.27382	0.636910	−	0.770938i	$-0.280212\pi$
$479$	13354.0	1.27382	0.636910	+	0.770938i	$0.280212\pi$
$480$	0	0
$481$	258.000	0.0244569
$482$	0	0
$483$	−690.000	−0.0650023
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−461.000	−0.0428951	−0.0214475	−	0.999770i	$-0.506827\pi$
$487$	−461.000	−0.0428951	−0.0214475	+	0.999770i	$0.506827\pi$
$488$	0	0
$489$	−6783.00	−0.627276
$490$	0	0
$491$	−3768.00	−0.346329	−0.173164	−	0.984893i	$-0.555399\pi$
$491$	−3768.00	−0.346329	−0.173164	+	0.984893i	$0.555399\pi$
$492$	0	0
$493$	−644.000	−0.0588323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4140.00	−0.373651
$498$	0	0
$499$	14317.0	1.28440	0.642201	−	0.766536i	$-0.278021\pi$
$499$	14317.0	1.28440	0.642201	+	0.766536i	$0.278021\pi$
$500$	0	0
$501$	6336.00	0.565013
$502$	0	0
$503$	9228.00	0.818004	0.409002	−	0.912533i	$-0.365877\pi$
$503$	9228.00	0.818004	0.409002	+	0.912533i	$0.365877\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6588.00	−0.577087
$508$	0	0
$509$	4574.00	0.398308	0.199154	−	0.979968i	$-0.436181\pi$
$509$	4574.00	0.398308	0.199154	+	0.979968i	$0.436181\pi$
$510$	0	0
$511$	−2850.00	−0.246725
$512$	0	0
$513$	−513.000	−0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5740.00	−0.488288
$518$	0	0
$519$	−1686.00	−0.142596
$520$	0	0
$521$	−8494.00	−0.714259	−0.357129	−	0.934055i	$-0.616245\pi$
$521$	−8494.00	−0.714259	−0.357129	+	0.934055i	$0.616245\pi$
$522$	0	0
$523$	8263.00	0.690852	0.345426	−	0.938446i	$-0.387734\pi$
$523$	8263.00	0.690852	0.345426	+	0.938446i	$0.387734\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6118.00	0.505701
$528$	0	0
$529$	−10051.0	−0.826087
$530$	0	0
$531$	1746.00	0.142693
$532$	0	0
$533$	−84.0000	−0.00682635
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11154.0	−0.896333
$538$	0	0
$539$	4452.00	0.355772
$540$	0	0
$541$	21157.0	1.68135	0.840675	−	0.541540i	$-0.182158\pi$
$541$	21157.0	1.68135	0.840675	+	0.541540i	$0.182158\pi$
$542$	0	0
$543$	−4917.00	−0.388598
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4048.00	−0.316417	−0.158208	−	0.987406i	$-0.550572\pi$
$547$	−4048.00	−0.316417	−0.158208	+	0.987406i	$0.550572\pi$
$548$	0	0
$549$	−153.000	−0.0118941
$550$	0	0
$551$	−266.000	−0.0205662
$552$	0	0
$553$	2760.00	0.212237
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6758.00	0.514086	0.257043	−	0.966400i	$-0.417252\pi$
$557$	6758.00	0.514086	0.257043	+	0.966400i	$0.417252\pi$
$558$	0	0
$559$	167.000	0.0126357
$560$	0	0
$561$	1932.00	0.145399
$562$	0	0
$563$	24506.0	1.83447	0.917233	−	0.398350i	$-0.130417\pi$
$563$	24506.0	1.83447	0.917233	+	0.398350i	$0.130417\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	405.000	0.0299972
$568$	0	0
$569$	−1430.00	−0.105358	−0.0526790	−	0.998611i	$-0.516776\pi$
$569$	−1430.00	−0.105358	−0.0526790	+	0.998611i	$0.516776\pi$
$570$	0	0
$571$	−3691.00	−0.270514	−0.135257	−	0.990811i	$-0.543186\pi$
$571$	−3691.00	−0.270514	−0.135257	+	0.990811i	$0.543186\pi$
$572$	0	0
$573$	−7230.00	−0.527116
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4571.00	−0.329798	−0.164899	−	0.986310i	$-0.552730\pi$
$577$	−4571.00	−0.329798	−0.164899	+	0.986310i	$0.552730\pi$
$578$	0	0
$579$	7863.00	0.564378
$580$	0	0
$581$	710.000	0.0506984
$582$	0	0
$583$	6384.00	0.453513
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14808.0	1.04121	0.520606	−	0.853797i	$-0.325706\pi$
$587$	14808.0	1.04121	0.520606	+	0.853797i	$0.325706\pi$
$588$	0	0
$589$	2527.00	0.176780
$590$	0	0
$591$	14862.0	1.03442
$592$	0	0
$593$	24588.0	1.70271	0.851356	−	0.524588i	$-0.175781\pi$
$593$	24588.0	1.70271	0.851356	+	0.524588i	$0.175781\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5217.00	−0.357651
$598$	0	0
$599$	27564.0	1.88019	0.940096	−	0.340911i	$-0.110735\pi$
$599$	27564.0	1.88019	0.940096	+	0.340911i	$0.110735\pi$
$600$	0	0
$601$	10987.0	0.745706	0.372853	−	0.927891i	$-0.378380\pi$
$601$	10987.0	0.745706	0.372853	+	0.927891i	$0.378380\pi$
$602$	0	0
$603$	5877.00	0.396899
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13200.0	−0.882655	−0.441327	−	0.897346i	$-0.645492\pi$
$607$	−13200.0	−0.882655	−0.441327	+	0.897346i	$0.645492\pi$
$608$	0	0
$609$	210.000	0.0139731
$610$	0	0
$611$	−410.000	−0.0271470
$612$	0	0
$613$	−21066.0	−1.38801	−0.694003	−	0.719972i	$-0.744155\pi$
$613$	−21066.0	−1.38801	−0.694003	+	0.719972i	$0.744155\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12336.0	−0.804909	−0.402454	−	0.915440i	$-0.631843\pi$
$617$	−12336.0	−0.804909	−0.402454	+	0.915440i	$0.631843\pi$
$618$	0	0
$619$	1441.00	0.0935681	0.0467841	−	0.998905i	$-0.485103\pi$
$619$	1441.00	0.0935681	0.0467841	+	0.998905i	$0.485103\pi$
$620$	0	0
$621$	−1242.00	−0.0802572
$622$	0	0
$623$	−5520.00	−0.354983
$624$	0	0
$625$	0	0
$626$	0	0
$627$	798.000	0.0508278
$628$	0	0
$629$	11868.0	0.752318
$630$	0	0
$631$	9839.00	0.620736	0.310368	−	0.950616i	$-0.399548\pi$
$631$	9839.00	0.620736	0.310368	+	0.950616i	$0.399548\pi$
$632$	0	0
$633$	13575.0	0.852382
$634$	0	0
$635$	0	0
$636$	0	0
$637$	318.000	0.0197796
$638$	0	0
$639$	−7452.00	−0.461340
$640$	0	0
$641$	−21564.0	−1.32875	−0.664373	−	0.747401i	$-0.731302\pi$
$641$	−21564.0	−1.32875	−0.664373	+	0.747401i	$0.731302\pi$
$642$	0	0
$643$	8604.00	0.527696	0.263848	−	0.964564i	$-0.415008\pi$
$643$	8604.00	0.527696	0.263848	+	0.964564i	$0.415008\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3444.00	0.209270	0.104635	−	0.994511i	$-0.466633\pi$
$647$	3444.00	0.209270	0.104635	+	0.994511i	$0.466633\pi$
$648$	0	0
$649$	−2716.00	−0.164272
$650$	0	0
$651$	−1995.00	−0.120108
$652$	0	0
$653$	3518.00	0.210827	0.105413	−	0.994428i	$-0.466383\pi$
$653$	3518.00	0.210827	0.105413	+	0.994428i	$0.466383\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5130.00	−0.304628
$658$	0	0
$659$	12612.0	0.745514	0.372757	−	0.927929i	$-0.378412\pi$
$659$	12612.0	0.745514	0.372757	+	0.927929i	$0.378412\pi$
$660$	0	0
$661$	27090.0	1.59407	0.797034	−	0.603935i	$-0.206401\pi$
$661$	27090.0	1.59407	0.797034	+	0.603935i	$0.206401\pi$
$662$	0	0
$663$	138.000	0.00808367
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−644.000	−0.0373850
$668$	0	0
$669$	−9633.00	−0.556701
$670$	0	0
$671$	238.000	0.0136928
$672$	0	0
$673$	5442.00	0.311699	0.155850	−	0.987781i	$-0.450188\pi$
$673$	5442.00	0.311699	0.155850	+	0.987781i	$0.450188\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15226.0	0.864376	0.432188	−	0.901783i	$-0.357742\pi$
$677$	15226.0	0.864376	0.432188	+	0.901783i	$0.357742\pi$
$678$	0	0
$679$	−4205.00	−0.237663
$680$	0	0
$681$	−7452.00	−0.419326
$682$	0	0
$683$	−552.000	−0.0309249	−0.0154624	−	0.999880i	$-0.504922\pi$
$683$	−552.000	−0.0309249	−0.0154624	+	0.999880i	$0.504922\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5541.00	−0.307718
$688$	0	0
$689$	456.000	0.0252137
$690$	0	0
$691$	−9776.00	−0.538201	−0.269100	−	0.963112i	$-0.586726\pi$
$691$	−9776.00	−0.538201	−0.269100	+	0.963112i	$0.586726\pi$
$692$	0	0
$693$	−630.000	−0.0345335
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3864.00	−0.209985
$698$	0	0
$699$	3060.00	0.165579
$700$	0	0
$701$	13066.0	0.703989	0.351994	−	0.936002i	$-0.385504\pi$
$701$	13066.0	0.703989	0.351994	+	0.936002i	$0.385504\pi$
$702$	0	0
$703$	4902.00	0.262991
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2760.00	0.146818
$708$	0	0
$709$	28985.0	1.53534	0.767669	−	0.640847i	$-0.221417\pi$
$709$	28985.0	1.53534	0.767669	+	0.640847i	$0.221417\pi$
$710$	0	0
$711$	4968.00	0.262046
$712$	0	0
$713$	6118.00	0.321348
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3528.00	−0.183760
$718$	0	0
$719$	−15722.0	−0.815482	−0.407741	−	0.913098i	$-0.633683\pi$
$719$	−15722.0	−0.815482	−0.407741	+	0.913098i	$0.633683\pi$
$720$	0	0
$721$	−1540.00	−0.0795459
$722$	0	0
$723$	−20901.0	−1.07513
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−32611.0	−1.66365	−0.831826	−	0.555036i	$-0.812704\pi$
$727$	−32611.0	−1.66365	−0.831826	+	0.555036i	$0.812704\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7682.00	0.388685
$732$	0	0
$733$	−8358.00	−0.421159	−0.210580	−	0.977577i	$-0.567535\pi$
$733$	−8358.00	−0.421159	−0.210580	+	0.977577i	$0.567535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9142.00	−0.456920
$738$	0	0
$739$	−20604.0	−1.02562	−0.512808	−	0.858503i	$-0.671395\pi$
$739$	−20604.0	−1.02562	−0.512808	+	0.858503i	$0.671395\pi$
$740$	0	0
$741$	57.0000	0.00282584
$742$	0	0
$743$	−19476.0	−0.961649	−0.480824	−	0.876817i	$-0.659663\pi$
$743$	−19476.0	−0.961649	−0.480824	+	0.876817i	$0.659663\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1278.00	0.0625965
$748$	0	0
$749$	4920.00	0.240017
$750$	0	0
$751$	−3864.00	−0.187749	−0.0938744	−	0.995584i	$-0.529925\pi$
$751$	−3864.00	−0.187749	−0.0938744	+	0.995584i	$0.529925\pi$
$752$	0	0
$753$	−4140.00	−0.200359
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18871.0	0.906048	0.453024	−	0.891498i	$-0.350345\pi$
$757$	18871.0	0.906048	0.453024	+	0.891498i	$0.350345\pi$
$758$	0	0
$759$	1932.00	0.0923941
$760$	0	0
$761$	−36372.0	−1.73257	−0.866284	−	0.499552i	$-0.833498\pi$
$761$	−36372.0	−1.73257	−0.866284	+	0.499552i	$0.833498\pi$
$762$	0	0
$763$	−9215.00	−0.437229
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−194.000	−0.00913290
$768$	0	0
$769$	4603.00	0.215850	0.107925	−	0.994159i	$-0.465579\pi$
$769$	4603.00	0.215850	0.107925	+	0.994159i	$0.465579\pi$
$770$	0	0
$771$	20772.0	0.970279
$772$	0	0
$773$	36.0000	0.00167507	0.000837536	−	1.00000i	$-0.499733\pi$
$773$	36.0000	0.00167507	0.000837536		1.00000i	$0.499733\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3870.00	−0.178681
$778$	0	0
$779$	−1596.00	−0.0734052
$780$	0	0
$781$	11592.0	0.531107
$782$	0	0
$783$	378.000	0.0172524
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20281.0	−0.918602	−0.459301	−	0.888281i	$-0.651900\pi$
$787$	−20281.0	−0.918602	−0.459301	+	0.888281i	$0.651900\pi$
$788$	0	0
$789$	5652.00	0.255027
$790$	0	0
$791$	−4380.00	−0.196884
$792$	0	0
$793$	17.0000	0.000761271 0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37524.0	1.66771	0.833857	−	0.551980i	$-0.186128\pi$
$797$	37524.0	1.66771	0.833857	+	0.551980i	$0.186128\pi$
$798$	0	0
$799$	−18860.0	−0.835067
$800$	0	0
$801$	−9936.00	−0.438291
$802$	0	0
$803$	7980.00	0.350695
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10830.0	0.472409
$808$	0	0
$809$	31224.0	1.35696	0.678478	−	0.734621i	$-0.262640\pi$
$809$	31224.0	1.35696	0.678478	+	0.734621i	$0.262640\pi$
$810$	0	0
$811$	−32579.0	−1.41061	−0.705304	−	0.708905i	$-0.749190\pi$
$811$	−32579.0	−1.41061	−0.705304	+	0.708905i	$0.749190\pi$
$812$	0	0
$813$	18216.0	0.785809
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3173.00	0.135874
$818$	0	0
$819$	−45.0000	−0.00191994
$820$	0	0
$821$	−19810.0	−0.842112	−0.421056	−	0.907035i	$-0.638340\pi$
$821$	−19810.0	−0.842112	−0.421056	+	0.907035i	$0.638340\pi$
$822$	0	0
$823$	10273.0	0.435108	0.217554	−	0.976048i	$-0.430192\pi$
$823$	10273.0	0.435108	0.217554	+	0.976048i	$0.430192\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16656.0	0.700346	0.350173	−	0.936685i	$-0.386123\pi$
$827$	16656.0	0.700346	0.350173	+	0.936685i	$0.386123\pi$
$828$	0	0
$829$	−4790.00	−0.200680	−0.100340	−	0.994953i	$-0.531993\pi$
$829$	−4790.00	−0.200680	−0.100340	+	0.994953i	$0.531993\pi$
$830$	0	0
$831$	8409.00	0.351029
$832$	0	0
$833$	14628.0	0.608440
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3591.00	−0.148295
$838$	0	0
$839$	−7414.00	−0.305077	−0.152539	−	0.988298i	$-0.548745\pi$
$839$	−7414.00	−0.305077	−0.152539	+	0.988298i	$0.548745\pi$
$840$	0	0
$841$	−24193.0	−0.991964
$842$	0	0
$843$	−20082.0	−0.820475
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5675.00	−0.230219
$848$	0	0
$849$	−19443.0	−0.785962
$850$	0	0
$851$	11868.0	0.478061
$852$	0	0
$853$	−30155.0	−1.21042	−0.605210	−	0.796066i	$-0.706911\pi$
$853$	−30155.0	−1.21042	−0.605210	+	0.796066i	$0.706911\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8244.00	0.328599	0.164300	−	0.986410i	$-0.447464\pi$
$857$	8244.00	0.328599	0.164300	+	0.986410i	$0.447464\pi$
$858$	0	0
$859$	−17552.0	−0.697167	−0.348584	−	0.937278i	$-0.613337\pi$
$859$	−17552.0	−0.697167	−0.348584	+	0.937278i	$0.613337\pi$
$860$	0	0
$861$	1260.00	0.0498730
$862$	0	0
$863$	−34104.0	−1.34521	−0.672604	−	0.740003i	$-0.734824\pi$
$863$	−34104.0	−1.34521	−0.672604	+	0.740003i	$0.734824\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8391.00	−0.328689
$868$	0	0
$869$	−7728.00	−0.301674
$870$	0	0
$871$	−653.000	−0.0254031
$872$	0	0
$873$	−7569.00	−0.293439
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46229.0	1.77998	0.889990	−	0.455980i	$-0.150711\pi$
$877$	46229.0	1.77998	0.889990	+	0.455980i	$0.150711\pi$
$878$	0	0
$879$	−9042.00	−0.346961
$880$	0	0
$881$	−22440.0	−0.858142	−0.429071	−	0.903271i	$-0.641159\pi$
$881$	−22440.0	−0.858142	−0.429071	+	0.903271i	$0.641159\pi$
$882$	0	0
$883$	−17143.0	−0.653350	−0.326675	−	0.945137i	$-0.605928\pi$
$883$	−17143.0	−0.653350	−0.326675	+	0.945137i	$0.605928\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23626.0	−0.894344	−0.447172	−	0.894448i	$-0.647569\pi$
$887$	−23626.0	−0.894344	−0.447172	+	0.894448i	$0.647569\pi$
$888$	0	0
$889$	11880.0	0.448192
$890$	0	0
$891$	−1134.00	−0.0426380
$892$	0	0
$893$	−7790.00	−0.291918
$894$	0	0
$895$	0	0
$896$	0	0
$897$	138.000	0.00513677
$898$	0	0
$899$	−1862.00	−0.0690781
$900$	0	0
$901$	20976.0	0.775596
$902$	0	0
$903$	−2505.00	−0.0923158
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−11268.0	−0.412511	−0.206256	−	0.978498i	$-0.566128\pi$
$907$	−11268.0	−0.412511	−0.206256	+	0.978498i	$0.566128\pi$
$908$	0	0
$909$	4968.00	0.181274
$910$	0	0
$911$	−10046.0	−0.365355	−0.182678	−	0.983173i	$-0.558476\pi$
$911$	−10046.0	−0.365355	−0.182678	+	0.983173i	$0.558476\pi$
$912$	0	0
$913$	−1988.00	−0.0720626
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5280.00	−0.190143
$918$	0	0
$919$	−15359.0	−0.551302	−0.275651	−	0.961258i	$-0.588893\pi$
$919$	−15359.0	−0.551302	−0.275651	+	0.961258i	$0.588893\pi$
$920$	0	0
$921$	−16107.0	−0.576269
$922$	0	0
$923$	828.000	0.0295276
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2772.00	−0.0982141
$928$	0	0
$929$	−39790.0	−1.40524	−0.702620	−	0.711565i	$-0.747986\pi$
$929$	−39790.0	−1.40524	−0.702620	+	0.711565i	$0.747986\pi$
$930$	0	0
$931$	6042.00	0.212694
$932$	0	0
$933$	−14538.0	−0.510132
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17009.0	0.593020	0.296510	−	0.955030i	$-0.404177\pi$
$937$	17009.0	0.593020	0.296510	+	0.955030i	$0.404177\pi$
$938$	0	0
$939$	2271.00	0.0789258
$940$	0	0
$941$	−11674.0	−0.404422	−0.202211	−	0.979342i	$-0.564813\pi$
$941$	−11674.0	−0.404422	−0.202211	+	0.979342i	$0.564813\pi$
$942$	0	0
$943$	−3864.00	−0.133435
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6026.00	0.206778	0.103389	−	0.994641i	$-0.467031\pi$
$947$	6026.00	0.206778	0.103389	+	0.994641i	$0.467031\pi$
$948$	0	0
$949$	570.000	0.0194973
$950$	0	0
$951$	22896.0	0.780708
$952$	0	0
$953$	14088.0	0.478862	0.239431	−	0.970913i	$-0.423039\pi$
$953$	14088.0	0.478862	0.239431	+	0.970913i	$0.423039\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−588.000	−0.0198614
$958$	0	0
$959$	3890.00	0.130985
$960$	0	0
$961$	−12102.0	−0.406230
$962$	0	0
$963$	8856.00	0.296345
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11208.0	−0.372725	−0.186362	−	0.982481i	$-0.559670\pi$
$967$	−11208.0	−0.372725	−0.186362	+	0.982481i	$0.559670\pi$
$968$	0	0
$969$	2622.00	0.0869255
$970$	0	0
$971$	26054.0	0.861084	0.430542	−	0.902571i	$-0.358322\pi$
$971$	26054.0	0.861084	0.430542	+	0.902571i	$0.358322\pi$
$972$	0	0
$973$	−8460.00	−0.278741
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26870.0	0.879885	0.439942	−	0.898026i	$-0.354999\pi$
$977$	26870.0	0.879885	0.439942	+	0.898026i	$0.354999\pi$
$978$	0	0
$979$	15456.0	0.504572
$980$	0	0
$981$	−16587.0	−0.539839
$982$	0	0
$983$	−23388.0	−0.758862	−0.379431	−	0.925220i	$-0.623880\pi$
$983$	−23388.0	−0.758862	−0.379431	+	0.925220i	$0.623880\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6150.00	0.198335
$988$	0	0
$989$	7682.00	0.246990
$990$	0	0
$991$	−17345.0	−0.555986	−0.277993	−	0.960583i	$-0.589669\pi$
$991$	−17345.0	−0.555986	−0.277993	+	0.960583i	$0.589669\pi$
$992$	0	0
$993$	20340.0	0.650021
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25998.0	0.825842	0.412921	−	0.910767i	$-0.364508\pi$
$997$	25998.0	0.825842	0.412921	+	0.910767i	$0.364508\pi$
$998$	0	0
$999$	−6966.00	−0.220615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bd.1.1		1
4.3	odd	2		600.4.a.e.1.1	✓	1
5.2	odd	4		1200.4.f.i.49.1		2
5.3	odd	4		1200.4.f.i.49.2		2
5.4	even	2		1200.4.a.g.1.1		1
12.11	even	2		1800.4.a.m.1.1		1
20.3	even	4		600.4.f.e.49.1		2
20.7	even	4		600.4.f.e.49.2		2
20.19	odd	2		600.4.a.n.1.1	yes	1
60.23	odd	4		1800.4.f.l.649.2		2
60.47	odd	4		1800.4.f.l.649.1		2
60.59	even	2		1800.4.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.e.1.1	✓	1	4.3	odd	2
600.4.a.n.1.1	yes	1	20.19	odd	2
600.4.f.e.49.1		2	20.3	even	4
600.4.f.e.49.2		2	20.7	even	4
1200.4.a.g.1.1		1	5.4	even	2
1200.4.a.bd.1.1		1	1.1	even	1	trivial
1200.4.f.i.49.1		2	5.2	odd	4
1200.4.f.i.49.2		2	5.3	odd	4
1800.4.a.m.1.1		1	12.11	even	2
1800.4.a.v.1.1		1	60.59	even	2
1800.4.f.l.649.1		2	60.47	odd	4
1800.4.f.l.649.2		2	60.23	odd	4

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more