LMFDB - Embedded newform 1200.4.a.h.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 30)
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)

$f(q)$

$=$

$q-3.00000 q^{3} -2.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -2.00000 q^{7} +9.00000 q^{9} -70.0000 q^{11} -54.0000 q^{13} +22.0000 q^{17} -24.0000 q^{19} +6.00000 q^{21} -100.000 q^{23} -27.0000 q^{27} +216.000 q^{29} -208.000 q^{31} +210.000 q^{33} +254.000 q^{37} +162.000 q^{39} -206.000 q^{41} +292.000 q^{43} -320.000 q^{47} -339.000 q^{49} -66.0000 q^{51} +402.000 q^{53} +72.0000 q^{57} +370.000 q^{59} -550.000 q^{61} -18.0000 q^{63} +728.000 q^{67} +300.000 q^{69} +540.000 q^{71} -604.000 q^{73} +140.000 q^{77} -792.000 q^{79} +81.0000 q^{81} +404.000 q^{83} -648.000 q^{87} -938.000 q^{89} +108.000 q^{91} +624.000 q^{93} -56.0000 q^{97} -630.000 q^{99} +O(q^{100})$

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
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−70.0000	−1.91871	−0.959354	−	0.282204i	$-0.908934\pi$
$11$	−70.0000	−1.91871	−0.959354	+	0.282204i	$0.908934\pi$
$12$	0	0
$13$	−54.0000	−1.15207	−0.576035	−	0.817425i	$-0.695401\pi$
$13$	−54.0000	−1.15207	−0.576035	+	0.817425i	$0.695401\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	22.0000	0.313870	0.156935	−	0.987609i	$-0.449839\pi$
$17$	22.0000	0.313870	0.156935	+	0.987609i	$0.449839\pi$
$18$	0	0
$19$	−24.0000	−0.289788	−0.144894	−	0.989447i	$-0.546284\pi$
$19$	−24.0000	−0.289788	−0.144894	+	0.989447i	$0.546284\pi$
$20$	0	0
$21$	6.00000	0.0623480
$22$	0	0
$23$	−100.000	−0.906584	−0.453292	−	0.891362i	$-0.649751\pi$
$23$	−100.000	−0.906584	−0.453292	+	0.891362i	$0.649751\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	216.000	1.38311	0.691555	−	0.722324i	$-0.256926\pi$
$29$	216.000	1.38311	0.691555	+	0.722324i	$0.256926\pi$
$30$	0	0
$31$	−208.000	−1.20509	−0.602547	−	0.798084i	$-0.705847\pi$
$31$	−208.000	−1.20509	−0.602547	+	0.798084i	$0.705847\pi$
$32$	0	0
$33$	210.000	1.10777
$34$	0	0
$35$	0	0
$36$	0	0
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	0	0
$39$	162.000	0.665148
$40$	0	0
$41$	−206.000	−0.784678	−0.392339	−	0.919821i	$-0.628334\pi$
$41$	−206.000	−0.784678	−0.392339	+	0.919821i	$0.628334\pi$
$42$	0	0
$43$	292.000	1.03557	0.517786	−	0.855510i	$-0.326756\pi$
$43$	292.000	1.03557	0.517786	+	0.855510i	$0.326756\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−320.000	−0.993123	−0.496562	−	0.868001i	$-0.665404\pi$
$47$	−320.000	−0.993123	−0.496562	+	0.868001i	$0.665404\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	−66.0000	−0.181213
$52$	0	0
$53$	402.000	1.04187	0.520933	−	0.853597i	$-0.325584\pi$
$53$	402.000	1.04187	0.520933	+	0.853597i	$0.325584\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	72.0000	0.167309
$58$	0	0
$59$	370.000	0.816439	0.408219	−	0.912884i	$-0.366150\pi$
$59$	370.000	0.816439	0.408219	+	0.912884i	$0.366150\pi$
$60$	0	0
$61$	−550.000	−1.15443	−0.577215	−	0.816592i	$-0.695861\pi$
$61$	−550.000	−1.15443	−0.577215	+	0.816592i	$0.695861\pi$
$62$	0	0
$63$	−18.0000	−0.0359966
$64$	0	0
$65$	0	0
$66$	0	0
$67$	728.000	1.32745	0.663727	−	0.747975i	$-0.268974\pi$
$67$	728.000	1.32745	0.663727	+	0.747975i	$0.268974\pi$
$68$	0	0
$69$	300.000	0.523417
$70$	0	0
$71$	540.000	0.902623	0.451311	−	0.892367i	$-0.350956\pi$
$71$	540.000	0.902623	0.451311	+	0.892367i	$0.350956\pi$
$72$	0	0
$73$	−604.000	−0.968395	−0.484198	−	0.874959i	$-0.660888\pi$
$73$	−604.000	−0.968395	−0.484198	+	0.874959i	$0.660888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	140.000	0.207201
$78$	0	0
$79$	−792.000	−1.12794	−0.563968	−	0.825797i	$-0.690726\pi$
$79$	−792.000	−1.12794	−0.563968	+	0.825797i	$0.690726\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	404.000	0.534274	0.267137	−	0.963659i	$-0.413922\pi$
$83$	404.000	0.534274	0.267137	+	0.963659i	$0.413922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−648.000	−0.798539
$88$	0	0
$89$	−938.000	−1.11717	−0.558583	−	0.829449i	$-0.688655\pi$
$89$	−938.000	−1.11717	−0.558583	+	0.829449i	$0.688655\pi$
$90$	0	0
$91$	108.000	0.124412
$92$	0	0
$93$	624.000	0.695761
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−56.0000	−0.0586179	−0.0293090	−	0.999570i	$-0.509331\pi$
$97$	−56.0000	−0.0586179	−0.0293090	+	0.999570i	$0.509331\pi$
$98$	0	0
$99$	−630.000	−0.639570
$100$	0	0
$101$	−592.000	−0.583230	−0.291615	−	0.956536i	$-0.594193\pi$
$101$	−592.000	−0.583230	−0.291615	+	0.956536i	$0.594193\pi$
$102$	0	0
$103$	62.0000	0.0593111	0.0296555	−	0.999560i	$-0.490559\pi$
$103$	62.0000	0.0593111	0.0296555	+	0.999560i	$0.490559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	84.0000	0.0758933	0.0379467	−	0.999280i	$-0.487918\pi$
$107$	84.0000	0.0758933	0.0379467	+	0.999280i	$0.487918\pi$
$108$	0	0
$109$	370.000	0.325134	0.162567	−	0.986698i	$-0.448023\pi$
$109$	370.000	0.325134	0.162567	+	0.986698i	$0.448023\pi$
$110$	0	0
$111$	−762.000	−0.651584
$112$	0	0
$113$	1746.00	1.45354	0.726769	−	0.686882i	$-0.241021\pi$
$113$	1746.00	1.45354	0.726769	+	0.686882i	$0.241021\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−486.000	−0.384023
$118$	0	0
$119$	−44.0000	−0.0338947
$120$	0	0
$121$	3569.00	2.68144
$122$	0	0
$123$	618.000	0.453034
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1630.00	1.13889	0.569445	−	0.822029i	$-0.307158\pi$
$127$	1630.00	1.13889	0.569445	+	0.822029i	$0.307158\pi$
$128$	0	0
$129$	−876.000	−0.597888
$130$	0	0
$131$	870.000	0.580246	0.290123	−	0.956989i	$-0.406304\pi$
$131$	870.000	0.580246	0.290123	+	0.956989i	$0.406304\pi$
$132$	0	0
$133$	48.0000	0.0312942
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−918.000	−0.572482	−0.286241	−	0.958158i	$-0.592406\pi$
$137$	−918.000	−0.572482	−0.286241	+	0.958158i	$0.592406\pi$
$138$	0	0
$139$	596.000	0.363684	0.181842	−	0.983328i	$-0.441794\pi$
$139$	596.000	0.363684	0.181842	+	0.983328i	$0.441794\pi$
$140$	0	0
$141$	960.000	0.573380
$142$	0	0
$143$	3780.00	2.21049
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1017.00	0.570617
$148$	0	0
$149$	1076.00	0.591606	0.295803	−	0.955249i	$-0.404413\pi$
$149$	1076.00	0.591606	0.295803	+	0.955249i	$0.404413\pi$
$150$	0	0
$151$	32.0000	0.0172458	0.00862292	−	0.999963i	$-0.497255\pi$
$151$	32.0000	0.0172458	0.00862292	+	0.999963i	$0.497255\pi$
$152$	0	0
$153$	198.000	0.104623
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2554.00	1.29829	0.649145	−	0.760665i	$-0.275127\pi$
$157$	2554.00	1.29829	0.649145	+	0.760665i	$0.275127\pi$
$158$	0	0
$159$	−1206.00	−0.601522
$160$	0	0
$161$	200.000	0.0979019
$162$	0	0
$163$	752.000	0.361357	0.180678	−	0.983542i	$-0.442171\pi$
$163$	752.000	0.361357	0.180678	+	0.983542i	$0.442171\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2700.00	1.25109	0.625546	−	0.780188i	$-0.284876\pi$
$167$	2700.00	1.25109	0.625546	+	0.780188i	$0.284876\pi$
$168$	0	0
$169$	719.000	0.327264
$170$	0	0
$171$	−216.000	−0.0965961
$172$	0	0
$173$	1334.00	0.586255	0.293128	−	0.956073i	$-0.405304\pi$
$173$	1334.00	0.586255	0.293128	+	0.956073i	$0.405304\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1110.00	−0.471371
$178$	0	0
$179$	1714.00	0.715700	0.357850	−	0.933779i	$-0.383510\pi$
$179$	1714.00	0.715700	0.357850	+	0.933779i	$0.383510\pi$
$180$	0	0
$181$	−4006.00	−1.64510	−0.822551	−	0.568691i	$-0.807450\pi$
$181$	−4006.00	−1.64510	−0.822551	+	0.568691i	$0.807450\pi$
$182$	0	0
$183$	1650.00	0.666511
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1540.00	−0.602224
$188$	0	0
$189$	54.0000	0.0207827
$190$	0	0
$191$	684.000	0.259123	0.129562	−	0.991571i	$-0.458643\pi$
$191$	684.000	0.259123	0.129562	+	0.991571i	$0.458643\pi$
$192$	0	0
$193$	4484.00	1.67236	0.836180	−	0.548455i	$-0.184784\pi$
$193$	4484.00	1.67236	0.836180	+	0.548455i	$0.184784\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1058.00	0.382636	0.191318	−	0.981528i	$-0.438724\pi$
$197$	1058.00	0.382636	0.191318	+	0.981528i	$0.438724\pi$
$198$	0	0
$199$	1128.00	0.401818	0.200909	−	0.979610i	$-0.435610\pi$
$199$	1128.00	0.401818	0.200909	+	0.979610i	$0.435610\pi$
$200$	0	0
$201$	−2184.00	−0.766405
$202$	0	0
$203$	−432.000	−0.149362
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−900.000	−0.302195
$208$	0	0
$209$	1680.00	0.556019
$210$	0	0
$211$	−780.000	−0.254490	−0.127245	−	0.991871i	$-0.540613\pi$
$211$	−780.000	−0.254490	−0.127245	+	0.991871i	$0.540613\pi$
$212$	0	0
$213$	−1620.00	−0.521129
$214$	0	0
$215$	0	0
$216$	0	0
$217$	416.000	0.130138
$218$	0	0
$219$	1812.00	0.559103
$220$	0	0
$221$	−1188.00	−0.361600
$222$	0	0
$223$	−2570.00	−0.771749	−0.385874	−	0.922551i	$-0.626100\pi$
$223$	−2570.00	−0.771749	−0.385874	+	0.922551i	$0.626100\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2836.00	−0.829216	−0.414608	−	0.910000i	$-0.636081\pi$
$227$	−2836.00	−0.829216	−0.414608	+	0.910000i	$0.636081\pi$
$228$	0	0
$229$	−610.000	−0.176026	−0.0880130	−	0.996119i	$-0.528052\pi$
$229$	−610.000	−0.176026	−0.0880130	+	0.996119i	$0.528052\pi$
$230$	0	0
$231$	−420.000	−0.119628
$232$	0	0
$233$	−3514.00	−0.988025	−0.494012	−	0.869455i	$-0.664470\pi$
$233$	−3514.00	−0.988025	−0.494012	+	0.869455i	$0.664470\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2376.00	0.651214
$238$	0	0
$239$	1844.00	0.499073	0.249536	−	0.968365i	$-0.419722\pi$
$239$	1844.00	0.499073	0.249536	+	0.968365i	$0.419722\pi$
$240$	0	0
$241$	982.000	0.262474	0.131237	−	0.991351i	$-0.458105\pi$
$241$	982.000	0.262474	0.131237	+	0.991351i	$0.458105\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1296.00	0.333856
$248$	0	0
$249$	−1212.00	−0.308463
$250$	0	0
$251$	3174.00	0.798172	0.399086	−	0.916914i	$-0.369328\pi$
$251$	3174.00	0.798172	0.399086	+	0.916914i	$0.369328\pi$
$252$	0	0
$253$	7000.00	1.73947
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1194.00	0.289804	0.144902	−	0.989446i	$-0.453713\pi$
$257$	1194.00	0.289804	0.144902	+	0.989446i	$0.453713\pi$
$258$	0	0
$259$	−508.000	−0.121875
$260$	0	0
$261$	1944.00	0.461037
$262$	0	0
$263$	140.000	0.0328242	0.0164121	−	0.999865i	$-0.494776\pi$
$263$	140.000	0.0328242	0.0164121	+	0.999865i	$0.494776\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2814.00	0.644996
$268$	0	0
$269$	5256.00	1.19132	0.595658	−	0.803238i	$-0.296891\pi$
$269$	5256.00	1.19132	0.595658	+	0.803238i	$0.296891\pi$
$270$	0	0
$271$	−544.000	−0.121940	−0.0609698	−	0.998140i	$-0.519419\pi$
$271$	−544.000	−0.121940	−0.0609698	+	0.998140i	$0.519419\pi$
$272$	0	0
$273$	−324.000	−0.0718292
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−946.000	−0.205197	−0.102599	−	0.994723i	$-0.532716\pi$
$277$	−946.000	−0.205197	−0.102599	+	0.994723i	$0.532716\pi$
$278$	0	0
$279$	−1872.00	−0.401698
$280$	0	0
$281$	1278.00	0.271313	0.135657	−	0.990756i	$-0.456686\pi$
$281$	1278.00	0.271313	0.135657	+	0.990756i	$0.456686\pi$
$282$	0	0
$283$	7424.00	1.55940	0.779701	−	0.626152i	$-0.215371\pi$
$283$	7424.00	1.55940	0.779701	+	0.626152i	$0.215371\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	412.000	0.0847373
$288$	0	0
$289$	−4429.00	−0.901486
$290$	0	0
$291$	168.000	0.0338431
$292$	0	0
$293$	1362.00	0.271566	0.135783	−	0.990739i	$-0.456645\pi$
$293$	1362.00	0.271566	0.135783	+	0.990739i	$0.456645\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1890.00	0.369256
$298$	0	0
$299$	5400.00	1.04445
$300$	0	0
$301$	−584.000	−0.111831
$302$	0	0
$303$	1776.00	0.336728
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7740.00	−1.43891	−0.719455	−	0.694539i	$-0.755608\pi$
$307$	−7740.00	−1.43891	−0.719455	+	0.694539i	$0.755608\pi$
$308$	0	0
$309$	−186.000	−0.0342433
$310$	0	0
$311$	−4980.00	−0.908006	−0.454003	−	0.891000i	$-0.650004\pi$
$311$	−4980.00	−0.908006	−0.454003	+	0.891000i	$0.650004\pi$
$312$	0	0
$313$	−604.000	−0.109074	−0.0545369	−	0.998512i	$-0.517368\pi$
$313$	−604.000	−0.109074	−0.0545369	+	0.998512i	$0.517368\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8566.00	1.51771	0.758856	−	0.651259i	$-0.225759\pi$
$317$	8566.00	1.51771	0.758856	+	0.651259i	$0.225759\pi$
$318$	0	0
$319$	−15120.0	−2.65379
$320$	0	0
$321$	−252.000	−0.0438170
$322$	0	0
$323$	−528.000	−0.0909557
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1110.00	−0.187716
$328$	0	0
$329$	640.000	0.107247
$330$	0	0
$331$	−3472.00	−0.576551	−0.288275	−	0.957548i	$-0.593082\pi$
$331$	−3472.00	−0.576551	−0.288275	+	0.957548i	$0.593082\pi$
$332$	0	0
$333$	2286.00	0.376192
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5668.00	−0.916189	−0.458094	−	0.888904i	$-0.651468\pi$
$337$	−5668.00	−0.916189	−0.458094	+	0.888904i	$0.651468\pi$
$338$	0	0
$339$	−5238.00	−0.839201
$340$	0	0
$341$	14560.0	2.31222
$342$	0	0
$343$	1364.00	0.214720
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10836.0	1.67639	0.838194	−	0.545371i	$-0.183611\pi$
$347$	10836.0	1.67639	0.838194	+	0.545371i	$0.183611\pi$
$348$	0	0
$349$	−8990.00	−1.37886	−0.689432	−	0.724350i	$-0.742140\pi$
$349$	−8990.00	−1.37886	−0.689432	+	0.724350i	$0.742140\pi$
$350$	0	0
$351$	1458.00	0.221716
$352$	0	0
$353$	5078.00	0.765651	0.382825	−	0.923821i	$-0.374951\pi$
$353$	5078.00	0.765651	0.382825	+	0.923821i	$0.374951\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	132.000	0.0195691
$358$	0	0
$359$	3696.00	0.543363	0.271682	−	0.962387i	$-0.412420\pi$
$359$	3696.00	0.543363	0.271682	+	0.962387i	$0.412420\pi$
$360$	0	0
$361$	−6283.00	−0.916023
$362$	0	0
$363$	−10707.0	−1.54813
$364$	0	0
$365$	0	0
$366$	0	0
$367$	286.000	0.0406787	0.0203393	−	0.999793i	$-0.493525\pi$
$367$	286.000	0.0406787	0.0203393	+	0.999793i	$0.493525\pi$
$368$	0	0
$369$	−1854.00	−0.261559
$370$	0	0
$371$	−804.000	−0.112511
$372$	0	0
$373$	−8262.00	−1.14689	−0.573445	−	0.819244i	$-0.694393\pi$
$373$	−8262.00	−1.14689	−0.573445	+	0.819244i	$0.694393\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11664.0	−1.59344
$378$	0	0
$379$	2956.00	0.400632	0.200316	−	0.979731i	$-0.435803\pi$
$379$	2956.00	0.400632	0.200316	+	0.979731i	$0.435803\pi$
$380$	0	0
$381$	−4890.00	−0.657539
$382$	0	0
$383$	−5240.00	−0.699090	−0.349545	−	0.936920i	$-0.613664\pi$
$383$	−5240.00	−0.699090	−0.349545	+	0.936920i	$0.613664\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2628.00	0.345191
$388$	0	0
$389$	−884.000	−0.115220	−0.0576100	−	0.998339i	$-0.518348\pi$
$389$	−884.000	−0.115220	−0.0576100	+	0.998339i	$0.518348\pi$
$390$	0	0
$391$	−2200.00	−0.284549
$392$	0	0
$393$	−2610.00	−0.335005
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3394.00	−0.429068	−0.214534	−	0.976717i	$-0.568823\pi$
$397$	−3394.00	−0.429068	−0.214534	+	0.976717i	$0.568823\pi$
$398$	0	0
$399$	−144.000	−0.0180677
$400$	0	0
$401$	−6826.00	−0.850060	−0.425030	−	0.905179i	$-0.639737\pi$
$401$	−6826.00	−0.850060	−0.425030	+	0.905179i	$0.639737\pi$
$402$	0	0
$403$	11232.0	1.38835
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17780.0	−2.16541
$408$	0	0
$409$	7814.00	0.944688	0.472344	−	0.881414i	$-0.343408\pi$
$409$	7814.00	0.944688	0.472344	+	0.881414i	$0.343408\pi$
$410$	0	0
$411$	2754.00	0.330523
$412$	0	0
$413$	−740.000	−0.0881671
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1788.00	−0.209973
$418$	0	0
$419$	−8290.00	−0.966570	−0.483285	−	0.875463i	$-0.660557\pi$
$419$	−8290.00	−0.966570	−0.483285	+	0.875463i	$0.660557\pi$
$420$	0	0
$421$	2110.00	0.244264	0.122132	−	0.992514i	$-0.461027\pi$
$421$	2110.00	0.244264	0.122132	+	0.992514i	$0.461027\pi$
$422$	0	0
$423$	−2880.00	−0.331041
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1100.00	0.124667
$428$	0	0
$429$	−11340.0	−1.27622
$430$	0	0
$431$	12080.0	1.35005	0.675027	−	0.737793i	$-0.264132\pi$
$431$	12080.0	1.35005	0.675027	+	0.737793i	$0.264132\pi$
$432$	0	0
$433$	−16492.0	−1.83038	−0.915190	−	0.403022i	$-0.867960\pi$
$433$	−16492.0	−1.83038	−0.915190	+	0.403022i	$0.867960\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2400.00	0.262718
$438$	0	0
$439$	15048.0	1.63600	0.817998	−	0.575222i	$-0.195084\pi$
$439$	15048.0	1.63600	0.817998	+	0.575222i	$0.195084\pi$
$440$	0	0
$441$	−3051.00	−0.329446
$442$	0	0
$443$	−9876.00	−1.05919	−0.529597	−	0.848249i	$-0.677657\pi$
$443$	−9876.00	−1.05919	−0.529597	+	0.848249i	$0.677657\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3228.00	−0.341564
$448$	0	0
$449$	17166.0	1.80426	0.902131	−	0.431462i	$-0.142002\pi$
$449$	17166.0	1.80426	0.902131	+	0.431462i	$0.142002\pi$
$450$	0	0
$451$	14420.0	1.50557
$452$	0	0
$453$	−96.0000	−0.00995690
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14848.0	−1.51983	−0.759913	−	0.650025i	$-0.774758\pi$
$457$	−14848.0	−1.51983	−0.759913	+	0.650025i	$0.774758\pi$
$458$	0	0
$459$	−594.000	−0.0604042
$460$	0	0
$461$	−1260.00	−0.127297	−0.0636486	−	0.997972i	$-0.520274\pi$
$461$	−1260.00	−0.127297	−0.0636486	+	0.997972i	$0.520274\pi$
$462$	0	0
$463$	−11238.0	−1.12802	−0.564011	−	0.825767i	$-0.690742\pi$
$463$	−11238.0	−1.12802	−0.564011	+	0.825767i	$0.690742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14772.0	1.46374	0.731870	−	0.681444i	$-0.238648\pi$
$467$	14772.0	1.46374	0.731870	+	0.681444i	$0.238648\pi$
$468$	0	0
$469$	−1456.00	−0.143351
$470$	0	0
$471$	−7662.00	−0.749568
$472$	0	0
$473$	−20440.0	−1.98696
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3618.00	0.347289
$478$	0	0
$479$	6116.00	0.583397	0.291699	−	0.956510i	$-0.405780\pi$
$479$	6116.00	0.583397	0.291699	+	0.956510i	$0.405780\pi$
$480$	0	0
$481$	−13716.0	−1.30020
$482$	0	0
$483$	−600.000	−0.0565237
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15906.0	1.48002	0.740010	−	0.672596i	$-0.234821\pi$
$487$	15906.0	1.48002	0.740010	+	0.672596i	$0.234821\pi$
$488$	0	0
$489$	−2256.00	−0.208630
$490$	0	0
$491$	−18714.0	−1.72006	−0.860032	−	0.510241i	$-0.829556\pi$
$491$	−18714.0	−1.72006	−0.860032	+	0.510241i	$0.829556\pi$
$492$	0	0
$493$	4752.00	0.434116
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1080.00	−0.0974741
$498$	0	0
$499$	4056.00	0.363871	0.181935	−	0.983310i	$-0.441764\pi$
$499$	4056.00	0.363871	0.181935	+	0.983310i	$0.441764\pi$
$500$	0	0
$501$	−8100.00	−0.722318
$502$	0	0
$503$	6288.00	0.557392	0.278696	−	0.960379i	$-0.410098\pi$
$503$	6288.00	0.557392	0.278696	+	0.960379i	$0.410098\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2157.00	−0.188946
$508$	0	0
$509$	2856.00	0.248703	0.124352	−	0.992238i	$-0.460315\pi$
$509$	2856.00	0.248703	0.124352	+	0.992238i	$0.460315\pi$
$510$	0	0
$511$	1208.00	0.104577
$512$	0	0
$513$	648.000	0.0557698
$514$	0	0
$515$	0	0
$516$	0	0
$517$	22400.0	1.90551
$518$	0	0
$519$	−4002.00	−0.338475
$520$	0	0
$521$	17078.0	1.43609	0.718043	−	0.695999i	$-0.245038\pi$
$521$	17078.0	1.43609	0.718043	+	0.695999i	$0.245038\pi$
$522$	0	0
$523$	−8560.00	−0.715684	−0.357842	−	0.933782i	$-0.616487\pi$
$523$	−8560.00	−0.715684	−0.357842	+	0.933782i	$0.616487\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4576.00	−0.378242
$528$	0	0
$529$	−2167.00	−0.178105
$530$	0	0
$531$	3330.00	0.272146
$532$	0	0
$533$	11124.0	0.904004
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5142.00	−0.413210
$538$	0	0
$539$	23730.0	1.89633
$540$	0	0
$541$	15970.0	1.26914	0.634569	−	0.772866i	$-0.281178\pi$
$541$	15970.0	1.26914	0.634569	+	0.772866i	$0.281178\pi$
$542$	0	0
$543$	12018.0	0.949801
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15524.0	−1.21345	−0.606726	−	0.794911i	$-0.707518\pi$
$547$	−15524.0	−1.21345	−0.606726	+	0.794911i	$0.707518\pi$
$548$	0	0
$549$	−4950.00	−0.384810
$550$	0	0
$551$	−5184.00	−0.400809
$552$	0	0
$553$	1584.00	0.121806
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6774.00	−0.515303	−0.257651	−	0.966238i	$-0.582949\pi$
$557$	−6774.00	−0.515303	−0.257651	+	0.966238i	$0.582949\pi$
$558$	0	0
$559$	−15768.0	−1.19305
$560$	0	0
$561$	4620.00	0.347694
$562$	0	0
$563$	−10484.0	−0.784810	−0.392405	−	0.919793i	$-0.628357\pi$
$563$	−10484.0	−0.784810	−0.392405	+	0.919793i	$0.628357\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−162.000	−0.0119989
$568$	0	0
$569$	23302.0	1.71682	0.858410	−	0.512964i	$-0.171453\pi$
$569$	23302.0	1.71682	0.858410	+	0.512964i	$0.171453\pi$
$570$	0	0
$571$	−21520.0	−1.57720	−0.788602	−	0.614903i	$-0.789195\pi$
$571$	−21520.0	−1.57720	−0.788602	+	0.614903i	$0.789195\pi$
$572$	0	0
$573$	−2052.00	−0.149605
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3856.00	−0.278210	−0.139105	−	0.990278i	$-0.544423\pi$
$577$	−3856.00	−0.278210	−0.139105	+	0.990278i	$0.544423\pi$
$578$	0	0
$579$	−13452.0	−0.965537
$580$	0	0
$581$	−808.000	−0.0576962
$582$	0	0
$583$	−28140.0	−1.99904
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26796.0	−1.88414	−0.942069	−	0.335418i	$-0.891122\pi$
$587$	−26796.0	−1.88414	−0.942069	+	0.335418i	$0.891122\pi$
$588$	0	0
$589$	4992.00	0.349222
$590$	0	0
$591$	−3174.00	−0.220915
$592$	0	0
$593$	−9870.00	−0.683495	−0.341747	−	0.939792i	$-0.611019\pi$
$593$	−9870.00	−0.683495	−0.341747	+	0.939792i	$0.611019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3384.00	−0.231990
$598$	0	0
$599$	13296.0	0.906945	0.453472	−	0.891270i	$-0.350185\pi$
$599$	13296.0	0.906945	0.453472	+	0.891270i	$0.350185\pi$
$600$	0	0
$601$	−9262.00	−0.628627	−0.314314	−	0.949319i	$-0.601774\pi$
$601$	−9262.00	−0.628627	−0.314314	+	0.949319i	$0.601774\pi$
$602$	0	0
$603$	6552.00	0.442484
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5498.00	0.367639	0.183820	−	0.982960i	$-0.441154\pi$
$607$	5498.00	0.367639	0.183820	+	0.982960i	$0.441154\pi$
$608$	0	0
$609$	1296.00	0.0862341
$610$	0	0
$611$	17280.0	1.14415
$612$	0	0
$613$	−394.000	−0.0259600	−0.0129800	−	0.999916i	$-0.504132\pi$
$613$	−394.000	−0.0259600	−0.0129800	+	0.999916i	$0.504132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7370.00	−0.480883	−0.240442	−	0.970664i	$-0.577292\pi$
$617$	−7370.00	−0.480883	−0.240442	+	0.970664i	$0.577292\pi$
$618$	0	0
$619$	−25316.0	−1.64384	−0.821919	−	0.569604i	$-0.807097\pi$
$619$	−25316.0	−1.64384	−0.821919	+	0.569604i	$0.807097\pi$
$620$	0	0
$621$	2700.00	0.174472
$622$	0	0
$623$	1876.00	0.120643
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5040.00	−0.321018
$628$	0	0
$629$	5588.00	0.354226
$630$	0	0
$631$	−2552.00	−0.161004	−0.0805020	−	0.996754i	$-0.525652\pi$
$631$	−2552.00	−0.161004	−0.0805020	+	0.996754i	$0.525652\pi$
$632$	0	0
$633$	2340.00	0.146930
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18306.0	1.13863
$638$	0	0
$639$	4860.00	0.300874
$640$	0	0
$641$	8050.00	0.496031	0.248016	−	0.968756i	$-0.420222\pi$
$641$	8050.00	0.496031	0.248016	+	0.968756i	$0.420222\pi$
$642$	0	0
$643$	−19368.0	−1.18787	−0.593934	−	0.804514i	$-0.702426\pi$
$643$	−19368.0	−1.18787	−0.593934	+	0.804514i	$0.702426\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9912.00	0.602289	0.301144	−	0.953579i	$-0.402631\pi$
$647$	9912.00	0.602289	0.301144	+	0.953579i	$0.402631\pi$
$648$	0	0
$649$	−25900.0	−1.56651
$650$	0	0
$651$	−1248.00	−0.0751351
$652$	0	0
$653$	−27986.0	−1.67715	−0.838573	−	0.544789i	$-0.816610\pi$
$653$	−27986.0	−1.67715	−0.838573	+	0.544789i	$0.816610\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5436.00	−0.322798
$658$	0	0
$659$	−7562.00	−0.447001	−0.223501	−	0.974704i	$-0.571748\pi$
$659$	−7562.00	−0.447001	−0.223501	+	0.974704i	$0.571748\pi$
$660$	0	0
$661$	20234.0	1.19064	0.595319	−	0.803490i	$-0.297026\pi$
$661$	20234.0	1.19064	0.595319	+	0.803490i	$0.297026\pi$
$662$	0	0
$663$	3564.00	0.208770
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−21600.0	−1.25391
$668$	0	0
$669$	7710.00	0.445569
$670$	0	0
$671$	38500.0	2.21502
$672$	0	0
$673$	−25332.0	−1.45093	−0.725466	−	0.688258i	$-0.758376\pi$
$673$	−25332.0	−1.45093	−0.725466	+	0.688258i	$0.758376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18358.0	1.04218	0.521090	−	0.853502i	$-0.325526\pi$
$677$	18358.0	1.04218	0.521090	+	0.853502i	$0.325526\pi$
$678$	0	0
$679$	112.000	0.00633014
$680$	0	0
$681$	8508.00	0.478748
$682$	0	0
$683$	−124.000	−0.00694689	−0.00347345	−	0.999994i	$-0.501106\pi$
$683$	−124.000	−0.00694689	−0.00347345	+	0.999994i	$0.501106\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1830.00	0.101629
$688$	0	0
$689$	−21708.0	−1.20030
$690$	0	0
$691$	17456.0	0.961009	0.480505	−	0.876992i	$-0.340453\pi$
$691$	17456.0	0.961009	0.480505	+	0.876992i	$0.340453\pi$
$692$	0	0
$693$	1260.00	0.0690670
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4532.00	−0.246287
$698$	0	0
$699$	10542.0	0.570436
$700$	0	0
$701$	−17816.0	−0.959916	−0.479958	−	0.877291i	$-0.659348\pi$
$701$	−17816.0	−0.959916	−0.479958	+	0.877291i	$0.659348\pi$
$702$	0	0
$703$	−6096.00	−0.327048
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1184.00	0.0629829
$708$	0	0
$709$	−14298.0	−0.757366	−0.378683	−	0.925526i	$-0.623623\pi$
$709$	−14298.0	−0.757366	−0.378683	+	0.925526i	$0.623623\pi$
$710$	0	0
$711$	−7128.00	−0.375979
$712$	0	0
$713$	20800.0	1.09252
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5532.00	−0.288140
$718$	0	0
$719$	18440.0	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$719$	18440.0	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$720$	0	0
$721$	−124.000	−0.00640499
$722$	0	0
$723$	−2946.00	−0.151539
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9666.00	0.493112	0.246556	−	0.969129i	$-0.420701\pi$
$727$	9666.00	0.493112	0.246556	+	0.969129i	$0.420701\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	6424.00	0.325035
$732$	0	0
$733$	6094.00	0.307076	0.153538	−	0.988143i	$-0.450933\pi$
$733$	6094.00	0.307076	0.153538	+	0.988143i	$0.450933\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−50960.0	−2.54700
$738$	0	0
$739$	−9952.00	−0.495386	−0.247693	−	0.968839i	$-0.579672\pi$
$739$	−9952.00	−0.495386	−0.247693	+	0.968839i	$0.579672\pi$
$740$	0	0
$741$	−3888.00	−0.192752
$742$	0	0
$743$	−2208.00	−0.109022	−0.0545112	−	0.998513i	$-0.517360\pi$
$743$	−2208.00	−0.109022	−0.0545112	+	0.998513i	$0.517360\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3636.00	0.178091
$748$	0	0
$749$	−168.000	−0.00819571
$750$	0	0
$751$	9400.00	0.456739	0.228369	−	0.973575i	$-0.426661\pi$
$751$	9400.00	0.456739	0.228369	+	0.973575i	$0.426661\pi$
$752$	0	0
$753$	−9522.00	−0.460825
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22574.0	1.08384	0.541919	−	0.840430i	$-0.317698\pi$
$757$	22574.0	1.08384	0.541919	+	0.840430i	$0.317698\pi$
$758$	0	0
$759$	−21000.0	−1.00428
$760$	0	0
$761$	7278.00	0.346685	0.173343	−	0.984862i	$-0.444543\pi$
$761$	7278.00	0.346685	0.173343	+	0.984862i	$0.444543\pi$
$762$	0	0
$763$	−740.000	−0.0351111
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19980.0	−0.940595
$768$	0	0
$769$	−16542.0	−0.775708	−0.387854	−	0.921721i	$-0.626784\pi$
$769$	−16542.0	−0.775708	−0.387854	+	0.921721i	$0.626784\pi$
$770$	0	0
$771$	−3582.00	−0.167319
$772$	0	0
$773$	28926.0	1.34592	0.672960	−	0.739679i	$-0.265023\pi$
$773$	28926.0	1.34592	0.672960	+	0.739679i	$0.265023\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1524.00	0.0703645
$778$	0	0
$779$	4944.00	0.227390
$780$	0	0
$781$	−37800.0	−1.73187
$782$	0	0
$783$	−5832.00	−0.266180
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20608.0	0.933413	0.466706	−	0.884412i	$-0.345440\pi$
$787$	20608.0	0.933413	0.466706	+	0.884412i	$0.345440\pi$
$788$	0	0
$789$	−420.000	−0.0189511
$790$	0	0
$791$	−3492.00	−0.156967
$792$	0	0
$793$	29700.0	1.32998
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41350.0	1.83776	0.918878	−	0.394541i	$-0.129096\pi$
$797$	41350.0	1.83776	0.918878	+	0.394541i	$0.129096\pi$
$798$	0	0
$799$	−7040.00	−0.311711
$800$	0	0
$801$	−8442.00	−0.372389
$802$	0	0
$803$	42280.0	1.85807
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15768.0	−0.687807
$808$	0	0
$809$	1794.00	0.0779650	0.0389825	−	0.999240i	$-0.487588\pi$
$809$	1794.00	0.0779650	0.0389825	+	0.999240i	$0.487588\pi$
$810$	0	0
$811$	−22756.0	−0.985291	−0.492646	−	0.870230i	$-0.663970\pi$
$811$	−22756.0	−0.985291	−0.492646	+	0.870230i	$0.663970\pi$
$812$	0	0
$813$	1632.00	0.0704019
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7008.00	−0.300097
$818$	0	0
$819$	972.000	0.0414706
$820$	0	0
$821$	−23632.0	−1.00458	−0.502291	−	0.864698i	$-0.667510\pi$
$821$	−23632.0	−1.00458	−0.502291	+	0.864698i	$0.667510\pi$
$822$	0	0
$823$	33210.0	1.40660	0.703298	−	0.710896i	$-0.251710\pi$
$823$	33210.0	1.40660	0.703298	+	0.710896i	$0.251710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30476.0	1.28144	0.640722	−	0.767773i	$-0.278635\pi$
$827$	30476.0	1.28144	0.640722	+	0.767773i	$0.278635\pi$
$828$	0	0
$829$	29802.0	1.24857	0.624286	−	0.781196i	$-0.285390\pi$
$829$	29802.0	1.24857	0.624286	+	0.781196i	$0.285390\pi$
$830$	0	0
$831$	2838.00	0.118471
$832$	0	0
$833$	−7458.00	−0.310209
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5616.00	0.231920
$838$	0	0
$839$	28024.0	1.15315	0.576577	−	0.817043i	$-0.304388\pi$
$839$	28024.0	1.15315	0.576577	+	0.817043i	$0.304388\pi$
$840$	0	0
$841$	22267.0	0.912994
$842$	0	0
$843$	−3834.00	−0.156643
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7138.00	−0.289569
$848$	0	0
$849$	−22272.0	−0.900322
$850$	0	0
$851$	−25400.0	−1.02315
$852$	0	0
$853$	−3938.00	−0.158071	−0.0790355	−	0.996872i	$-0.525184\pi$
$853$	−3938.00	−0.158071	−0.0790355	+	0.996872i	$0.525184\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8094.00	0.322621	0.161310	−	0.986904i	$-0.448428\pi$
$857$	8094.00	0.322621	0.161310	+	0.986904i	$0.448428\pi$
$858$	0	0
$859$	−9044.00	−0.359229	−0.179614	−	0.983737i	$-0.557485\pi$
$859$	−9044.00	−0.359229	−0.179614	+	0.983737i	$0.557485\pi$
$860$	0	0
$861$	−1236.00	−0.0489231
$862$	0	0
$863$	−6252.00	−0.246606	−0.123303	−	0.992369i	$-0.539349\pi$
$863$	−6252.00	−0.246606	−0.123303	+	0.992369i	$0.539349\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13287.0	0.520473
$868$	0	0
$869$	55440.0	2.16418
$870$	0	0
$871$	−39312.0	−1.52932
$872$	0	0
$873$	−504.000	−0.0195393
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−40166.0	−1.54653	−0.773267	−	0.634081i	$-0.781379\pi$
$877$	−40166.0	−1.54653	−0.773267	+	0.634081i	$0.781379\pi$
$878$	0	0
$879$	−4086.00	−0.156789
$880$	0	0
$881$	−12834.0	−0.490793	−0.245396	−	0.969423i	$-0.578918\pi$
$881$	−12834.0	−0.490793	−0.245396	+	0.969423i	$0.578918\pi$
$882$	0	0
$883$	27192.0	1.03633	0.518167	−	0.855279i	$-0.326614\pi$
$883$	27192.0	1.03633	0.518167	+	0.855279i	$0.326614\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42060.0	1.59215	0.796075	−	0.605198i	$-0.206906\pi$
$887$	42060.0	1.59215	0.796075	+	0.605198i	$0.206906\pi$
$888$	0	0
$889$	−3260.00	−0.122989
$890$	0	0
$891$	−5670.00	−0.213190
$892$	0	0
$893$	7680.00	0.287796
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16200.0	−0.603013
$898$	0	0
$899$	−44928.0	−1.66678
$900$	0	0
$901$	8844.00	0.327010
$902$	0	0
$903$	1752.00	0.0645658
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41172.0	1.50727	0.753635	−	0.657293i	$-0.228299\pi$
$907$	41172.0	1.50727	0.753635	+	0.657293i	$0.228299\pi$
$908$	0	0
$909$	−5328.00	−0.194410
$910$	0	0
$911$	48.0000	0.00174568	0.000872838	−	1.00000i	$-0.499722\pi$
$911$	48.0000	0.00174568	0.000872838		1.00000i	$0.499722\pi$
$912$	0	0
$913$	−28280.0	−1.02512
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1740.00	−0.0626607
$918$	0	0
$919$	34584.0	1.24137	0.620686	−	0.784059i	$-0.286854\pi$
$919$	34584.0	1.24137	0.620686	+	0.784059i	$0.286854\pi$
$920$	0	0
$921$	23220.0	0.830755
$922$	0	0
$923$	−29160.0	−1.03988
$924$	0	0
$925$	0	0
$926$	0	0
$927$	558.000	0.0197704
$928$	0	0
$929$	−3474.00	−0.122689	−0.0613446	−	0.998117i	$-0.519539\pi$
$929$	−3474.00	−0.122689	−0.0613446	+	0.998117i	$0.519539\pi$
$930$	0	0
$931$	8136.00	0.286409
$932$	0	0
$933$	14940.0	0.524238
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44408.0	−1.54829	−0.774144	−	0.633009i	$-0.781819\pi$
$937$	−44408.0	−1.54829	−0.774144	+	0.633009i	$0.781819\pi$
$938$	0	0
$939$	1812.00	0.0629738
$940$	0	0
$941$	20188.0	0.699373	0.349686	−	0.936867i	$-0.386288\pi$
$941$	20188.0	0.699373	0.349686	+	0.936867i	$0.386288\pi$
$942$	0	0
$943$	20600.0	0.711377
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31212.0	−1.07102	−0.535509	−	0.844530i	$-0.679880\pi$
$947$	−31212.0	−1.07102	−0.535509	+	0.844530i	$0.679880\pi$
$948$	0	0
$949$	32616.0	1.11566
$950$	0	0
$951$	−25698.0	−0.876251
$952$	0	0
$953$	−20182.0	−0.686001	−0.343001	−	0.939335i	$-0.611443\pi$
$953$	−20182.0	−0.686001	−0.343001	+	0.939335i	$0.611443\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	45360.0	1.53216
$958$	0	0
$959$	1836.00	0.0618222
$960$	0	0
$961$	13473.0	0.452251
$962$	0	0
$963$	756.000	0.0252978
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−53722.0	−1.78654	−0.893269	−	0.449522i	$-0.851594\pi$
$967$	−53722.0	−1.78654	−0.893269	+	0.449522i	$0.851594\pi$
$968$	0	0
$969$	1584.00	0.0525133
$970$	0	0
$971$	−22554.0	−0.745409	−0.372705	−	0.927950i	$-0.621570\pi$
$971$	−22554.0	−0.745409	−0.372705	+	0.927950i	$0.621570\pi$
$972$	0	0
$973$	−1192.00	−0.0392742
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18126.0	0.593554	0.296777	−	0.954947i	$-0.404088\pi$
$977$	18126.0	0.593554	0.296777	+	0.954947i	$0.404088\pi$
$978$	0	0
$979$	65660.0	2.14352
$980$	0	0
$981$	3330.00	0.108378
$982$	0	0
$983$	6232.00	0.202207	0.101104	−	0.994876i	$-0.467763\pi$
$983$	6232.00	0.202207	0.101104	+	0.994876i	$0.467763\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1920.00	−0.0619192
$988$	0	0
$989$	−29200.0	−0.938833
$990$	0	0
$991$	−15184.0	−0.486716	−0.243358	−	0.969937i	$-0.578249\pi$
$991$	−15184.0	−0.486716	−0.243358	+	0.969937i	$0.578249\pi$
$992$	0	0
$993$	10416.0	0.332872
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29922.0	0.950491	0.475245	−	0.879853i	$-0.342359\pi$
$997$	29922.0	0.950491	0.475245	+	0.879853i	$0.342359\pi$
$998$	0	0
$999$	−6858.00	−0.217195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.h.1.1		1
4.3	odd	2		150.4.a.d.1.1		1
5.2	odd	4		240.4.f.d.49.2		2
5.3	odd	4		240.4.f.d.49.1		2
5.4	even	2		1200.4.a.bc.1.1		1
12.11	even	2		450.4.a.p.1.1		1
15.2	even	4		720.4.f.c.289.2		2
15.8	even	4		720.4.f.c.289.1		2
20.3	even	4		30.4.c.a.19.2	yes	2
20.7	even	4		30.4.c.a.19.1	✓	2
20.19	odd	2		150.4.a.f.1.1		1
40.3	even	4		960.4.f.c.769.1		2
40.13	odd	4		960.4.f.d.769.2		2
40.27	even	4		960.4.f.c.769.2		2
40.37	odd	4		960.4.f.d.769.1		2
60.23	odd	4		90.4.c.a.19.1		2
60.47	odd	4		90.4.c.a.19.2		2
60.59	even	2		450.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.c.a.19.1	✓	2	20.7	even	4
30.4.c.a.19.2	yes	2	20.3	even	4
90.4.c.a.19.1		2	60.23	odd	4
90.4.c.a.19.2		2	60.47	odd	4
150.4.a.d.1.1		1	4.3	odd	2
150.4.a.f.1.1		1	20.19	odd	2
240.4.f.d.49.1		2	5.3	odd	4
240.4.f.d.49.2		2	5.2	odd	4
450.4.a.e.1.1		1	60.59	even	2
450.4.a.p.1.1		1	12.11	even	2
720.4.f.c.289.1		2	15.8	even	4
720.4.f.c.289.2		2	15.2	even	4
960.4.f.c.769.1		2	40.3	even	4
960.4.f.c.769.2		2	40.27	even	4
960.4.f.d.769.1		2	40.37	odd	4
960.4.f.d.769.2		2	40.13	odd	4
1200.4.a.h.1.1		1	1.1	even	1	trivial
1200.4.a.bc.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
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