LMFDB - Embedded newform 1350.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{7} -8.00000 q^{8} +22.0000 q^{11} -30.0000 q^{13} -28.0000 q^{14} +16.0000 q^{16} +7.00000 q^{17} -81.0000 q^{19} -44.0000 q^{22} +151.000 q^{23} +60.0000 q^{26} +56.0000 q^{28} -270.000 q^{29} -113.000 q^{31} -32.0000 q^{32} -14.0000 q^{34} -88.0000 q^{37} +162.000 q^{38} +406.000 q^{41} -442.000 q^{43} +88.0000 q^{44} -302.000 q^{46} +56.0000 q^{47} -147.000 q^{49} -120.000 q^{52} -141.000 q^{53} -112.000 q^{56} +540.000 q^{58} -274.000 q^{59} +41.0000 q^{61} +226.000 q^{62} +64.0000 q^{64} +328.000 q^{67} +28.0000 q^{68} -390.000 q^{71} +626.000 q^{73} +176.000 q^{74} -324.000 q^{76} +308.000 q^{77} -1215.00 q^{79} -812.000 q^{82} +505.000 q^{83} +884.000 q^{86} -176.000 q^{88} +514.000 q^{89} -420.000 q^{91} +604.000 q^{92} -112.000 q^{94} +1816.00 q^{97} +294.000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	14.0000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	14.0000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	22.0000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	22.0000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−30.0000	−0.640039	−0.320019	−	0.947411i	$-0.603689\pi$
$13$	−30.0000	−0.640039	−0.320019	+	0.947411i	$0.603689\pi$
$14$	−28.0000	−0.534522
$15$	0	0
$16$	16.0000	0.250000
$17$	7.00000	0.0998676	0.0499338	−	0.998753i	$-0.484099\pi$
$17$	7.00000	0.0998676	0.0499338	+	0.998753i	$0.484099\pi$
$18$	0	0
$19$	−81.0000	−0.978035	−0.489018	−	0.872274i	$-0.662645\pi$
$19$	−81.0000	−0.978035	−0.489018	+	0.872274i	$0.662645\pi$
$20$	0	0
$21$	0	0
$22$	−44.0000	−0.426401
$23$	151.000	1.36894	0.684471	−	0.729040i	$-0.260033\pi$
$23$	151.000	1.36894	0.684471	+	0.729040i	$0.260033\pi$
$24$	0	0
$25$	0	0
$26$	60.0000	0.452576
$27$	0	0
$28$	56.0000	0.377964
$29$	−270.000	−1.72889	−0.864444	−	0.502729i	$-0.832329\pi$
$29$	−270.000	−1.72889	−0.864444	+	0.502729i	$0.832329\pi$
$30$	0	0
$31$	−113.000	−0.654690	−0.327345	−	0.944905i	$-0.606154\pi$
$31$	−113.000	−0.654690	−0.327345	+	0.944905i	$0.606154\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−14.0000	−0.0706171
$35$	0	0
$36$	0	0
$37$	−88.0000	−0.391003	−0.195501	−	0.980703i	$-0.562633\pi$
$37$	−88.0000	−0.391003	−0.195501	+	0.980703i	$0.562633\pi$
$38$	162.000	0.691576
$39$	0	0
$40$	0	0
$41$	406.000	1.54650	0.773251	−	0.634101i	$-0.218629\pi$
$41$	406.000	1.54650	0.773251	+	0.634101i	$0.218629\pi$
$42$	0	0
$43$	−442.000	−1.56754	−0.783772	−	0.621049i	$-0.786707\pi$
$43$	−442.000	−1.56754	−0.783772	+	0.621049i	$0.786707\pi$
$44$	88.0000	0.301511
$45$	0	0
$46$	−302.000	−0.967988
$47$	56.0000	0.173797	0.0868983	−	0.996217i	$-0.472304\pi$
$47$	56.0000	0.173797	0.0868983	+	0.996217i	$0.472304\pi$
$48$	0	0
$49$	−147.000	−0.428571
$50$	0	0
$51$	0	0
$52$	−120.000	−0.320019
$53$	−141.000	−0.365431	−0.182715	−	0.983166i	$-0.558489\pi$
$53$	−141.000	−0.365431	−0.182715	+	0.983166i	$0.558489\pi$
$54$	0	0
$55$	0	0
$56$	−112.000	−0.267261
$57$	0	0
$58$	540.000	1.22251
$59$	−274.000	−0.604606	−0.302303	−	0.953212i	$-0.597755\pi$
$59$	−274.000	−0.604606	−0.302303	+	0.953212i	$0.597755\pi$
$60$	0	0
$61$	41.0000	0.0860576	0.0430288	−	0.999074i	$-0.486299\pi$
$61$	41.0000	0.0860576	0.0430288	+	0.999074i	$0.486299\pi$
$62$	226.000	0.462936
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	328.000	0.598083	0.299042	−	0.954240i	$-0.403333\pi$
$67$	328.000	0.598083	0.299042	+	0.954240i	$0.403333\pi$
$68$	28.0000	0.0499338
$69$	0	0
$70$	0	0
$71$	−390.000	−0.651894	−0.325947	−	0.945388i	$-0.605683\pi$
$71$	−390.000	−0.651894	−0.325947	+	0.945388i	$0.605683\pi$
$72$	0	0
$73$	626.000	1.00367	0.501834	−	0.864964i	$-0.332659\pi$
$73$	626.000	1.00367	0.501834	+	0.864964i	$0.332659\pi$
$74$	176.000	0.276481
$75$	0	0
$76$	−324.000	−0.489018
$77$	308.000	0.455842
$78$	0	0
$79$	−1215.00	−1.73036	−0.865178	−	0.501464i	$-0.832795\pi$
$79$	−1215.00	−1.73036	−0.865178	+	0.501464i	$0.832795\pi$
$80$	0	0
$81$	0	0
$82$	−812.000	−1.09354
$83$	505.000	0.667843	0.333921	−	0.942601i	$-0.391628\pi$
$83$	505.000	0.667843	0.333921	+	0.942601i	$0.391628\pi$
$84$	0	0
$85$	0	0
$86$	884.000	1.10842
$87$	0	0
$88$	−176.000	−0.213201
$89$	514.000	0.612179	0.306089	−	0.952003i	$-0.400979\pi$
$89$	514.000	0.612179	0.306089	+	0.952003i	$0.400979\pi$
$90$	0	0
$91$	−420.000	−0.483824
$92$	604.000	0.684471
$93$	0	0
$94$	−112.000	−0.122893
$95$	0	0
$96$	0	0
$97$	1816.00	1.90090	0.950448	−	0.310884i	$-0.100625\pi$
$97$	1816.00	1.90090	0.950448	+	0.310884i	$0.100625\pi$
$98$	294.000	0.303046
$99$	0	0
$100$	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
$103$	−848.000	−0.811223	−0.405611	−	0.914046i	$-0.632941\pi$
$103$	−848.000	−0.811223	−0.405611	+	0.914046i	$0.632941\pi$
$104$	240.000	0.226288
$105$	0	0
$106$	282.000	0.258399
$107$	1500.00	1.35524	0.677619	−	0.735413i	$-0.263012\pi$
$107$	1500.00	1.35524	0.677619	+	0.735413i	$0.263012\pi$
$108$	0	0
$109$	−935.000	−0.821622	−0.410811	−	0.911721i	$-0.634754\pi$
$109$	−935.000	−0.821622	−0.410811	+	0.911721i	$0.634754\pi$
$110$	0	0
$111$	0	0
$112$	224.000	0.188982
$113$	−1266.00	−1.05394	−0.526970	−	0.849884i	$-0.676672\pi$
$113$	−1266.00	−1.05394	−0.526970	+	0.849884i	$0.676672\pi$
$114$	0	0
$115$	0	0
$116$	−1080.00	−0.864444
$117$	0	0
$118$	548.000	0.427521
$119$	98.0000	0.0754928
$120$	0	0
$121$	−847.000	−0.636364
$122$	−82.0000	−0.0608519
$123$	0	0
$124$	−452.000	−0.327345
$125$	0	0
$126$	0	0
$127$	−16.0000	−0.0111793	−0.00558965	−	0.999984i	$-0.501779\pi$
$127$	−16.0000	−0.0111793	−0.00558965	+	0.999984i	$0.501779\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−2304.00	−1.53665	−0.768326	−	0.640059i	$-0.778910\pi$
$131$	−2304.00	−1.53665	−0.768326	+	0.640059i	$0.778910\pi$
$132$	0	0
$133$	−1134.00	−0.739325
$134$	−656.000	−0.422909
$135$	0	0
$136$	−56.0000	−0.0353085
$137$	1341.00	0.836273	0.418136	−	0.908384i	$-0.362683\pi$
$137$	1341.00	0.836273	0.418136	+	0.908384i	$0.362683\pi$
$138$	0	0
$139$	−2564.00	−1.56457	−0.782286	−	0.622919i	$-0.785947\pi$
$139$	−2564.00	−1.56457	−0.782286	+	0.622919i	$0.785947\pi$
$140$	0	0
$141$	0	0
$142$	780.000	0.460959
$143$	−660.000	−0.385958
$144$	0	0
$145$	0	0
$146$	−1252.00	−0.709700
$147$	0	0
$148$	−352.000	−0.195501
$149$	3122.00	1.71654	0.858269	−	0.513200i	$-0.171540\pi$
$149$	3122.00	1.71654	0.858269	+	0.513200i	$0.171540\pi$
$150$	0	0
$151$	3004.00	1.61895	0.809477	−	0.587152i	$-0.199751\pi$
$151$	3004.00	1.61895	0.809477	+	0.587152i	$0.199751\pi$
$152$	648.000	0.345788
$153$	0	0
$154$	−616.000	−0.322329
$155$	0	0
$156$	0	0
$157$	−3566.00	−1.81272	−0.906362	−	0.422501i	$-0.861152\pi$
$157$	−3566.00	−1.81272	−0.906362	+	0.422501i	$0.861152\pi$
$158$	2430.00	1.22355
$159$	0	0
$160$	0	0
$161$	2114.00	1.03482
$162$	0	0
$163$	556.000	0.267174	0.133587	−	0.991037i	$-0.457351\pi$
$163$	556.000	0.267174	0.133587	+	0.991037i	$0.457351\pi$
$164$	1624.00	0.773251
$165$	0	0
$166$	−1010.00	−0.472236
$167$	−129.000	−0.0597744	−0.0298872	−	0.999553i	$-0.509515\pi$
$167$	−129.000	−0.0597744	−0.0298872	+	0.999553i	$0.509515\pi$
$168$	0	0
$169$	−1297.00	−0.590350
$170$	0	0
$171$	0	0
$172$	−1768.00	−0.783772
$173$	1277.00	0.561205	0.280603	−	0.959824i	$-0.409466\pi$
$173$	1277.00	0.561205	0.280603	+	0.959824i	$0.409466\pi$
$174$	0	0
$175$	0	0
$176$	352.000	0.150756
$177$	0	0
$178$	−1028.00	−0.432876
$179$	−1180.00	−0.492723	−0.246361	−	0.969178i	$-0.579235\pi$
$179$	−1180.00	−0.492723	−0.246361	+	0.969178i	$0.579235\pi$
$180$	0	0
$181$	−2527.00	−1.03774	−0.518869	−	0.854854i	$-0.673647\pi$
$181$	−2527.00	−1.03774	−0.518869	+	0.854854i	$0.673647\pi$
$182$	840.000	0.342115
$183$	0	0
$184$	−1208.00	−0.483994
$185$	0	0
$186$	0	0
$187$	154.000	0.0602224
$188$	224.000	0.0868983
$189$	0	0
$190$	0	0
$191$	3480.00	1.31835	0.659173	−	0.751992i	$-0.270907\pi$
$191$	3480.00	1.31835	0.659173	+	0.751992i	$0.270907\pi$
$192$	0	0
$193$	3170.00	1.18229	0.591144	−	0.806566i	$-0.298676\pi$
$193$	3170.00	1.18229	0.591144	+	0.806566i	$0.298676\pi$
$194$	−3632.00	−1.34414
$195$	0	0
$196$	−588.000	−0.214286
$197$	−2335.00	−0.844476	−0.422238	−	0.906485i	$-0.638755\pi$
$197$	−2335.00	−0.844476	−0.422238	+	0.906485i	$0.638755\pi$
$198$	0	0
$199$	−1512.00	−0.538607	−0.269304	−	0.963055i	$-0.586794\pi$
$199$	−1512.00	−0.538607	−0.269304	+	0.963055i	$0.586794\pi$
$200$	0	0
$201$	0	0
$202$	1184.00	0.412406
$203$	−3780.00	−1.30692
$204$	0	0
$205$	0	0
$206$	1696.00	0.573621
$207$	0	0
$208$	−480.000	−0.160010
$209$	−1782.00	−0.589778
$210$	0	0
$211$	−5919.00	−1.93119	−0.965594	−	0.260053i	$-0.916260\pi$
$211$	−5919.00	−1.93119	−0.965594	+	0.260053i	$0.916260\pi$
$212$	−564.000	−0.182715
$213$	0	0
$214$	−3000.00	−0.958298
$215$	0	0
$216$	0	0
$217$	−1582.00	−0.494899
$218$	1870.00	0.580974
$219$	0	0
$220$	0	0
$221$	−210.000	−0.0639191
$222$	0	0
$223$	−412.000	−0.123720	−0.0618600	−	0.998085i	$-0.519703\pi$
$223$	−412.000	−0.123720	−0.0618600	+	0.998085i	$0.519703\pi$
$224$	−448.000	−0.133631
$225$	0	0
$226$	2532.00	0.745248
$227$	−5843.00	−1.70843	−0.854215	−	0.519920i	$-0.825962\pi$
$227$	−5843.00	−1.70843	−0.854215	+	0.519920i	$0.825962\pi$
$228$	0	0
$229$	−4153.00	−1.19842	−0.599210	−	0.800592i	$-0.704518\pi$
$229$	−4153.00	−1.19842	−0.599210	+	0.800592i	$0.704518\pi$
$230$	0	0
$231$	0	0
$232$	2160.00	0.611254
$233$	−562.000	−0.158016	−0.0790082	−	0.996874i	$-0.525175\pi$
$233$	−562.000	−0.158016	−0.0790082	+	0.996874i	$0.525175\pi$
$234$	0	0
$235$	0	0
$236$	−1096.00	−0.302303
$237$	0	0
$238$	−196.000	−0.0533815
$239$	−836.000	−0.226261	−0.113130	−	0.993580i	$-0.536088\pi$
$239$	−836.000	−0.226261	−0.113130	+	0.993580i	$0.536088\pi$
$240$	0	0
$241$	−1529.00	−0.408679	−0.204339	−	0.978900i	$-0.565505\pi$
$241$	−1529.00	−0.408679	−0.204339	+	0.978900i	$0.565505\pi$
$242$	1694.00	0.449977
$243$	0	0
$244$	164.000	0.0430288
$245$	0	0
$246$	0	0
$247$	2430.00	0.625981
$248$	904.000	0.231468
$249$	0	0
$250$	0	0
$251$	−252.000	−0.0633709	−0.0316855	−	0.999498i	$-0.510087\pi$
$251$	−252.000	−0.0633709	−0.0316855	+	0.999498i	$0.510087\pi$
$252$	0	0
$253$	3322.00	0.825503
$254$	32.0000	0.00790496
$255$	0	0
$256$	256.000	0.0625000
$257$	−6111.00	−1.48324	−0.741622	−	0.670818i	$-0.765943\pi$
$257$	−6111.00	−1.48324	−0.741622	+	0.670818i	$0.765943\pi$
$258$	0	0
$259$	−1232.00	−0.295570
$260$	0	0
$261$	0	0
$262$	4608.00	1.08658
$263$	−3104.00	−0.727760	−0.363880	−	0.931446i	$-0.618548\pi$
$263$	−3104.00	−0.727760	−0.363880	+	0.931446i	$0.618548\pi$
$264$	0	0
$265$	0	0
$266$	2268.00	0.522782
$267$	0	0
$268$	1312.00	0.299042
$269$	−3240.00	−0.734373	−0.367186	−	0.930147i	$-0.619679\pi$
$269$	−3240.00	−0.734373	−0.367186	+	0.930147i	$0.619679\pi$
$270$	0	0
$271$	2395.00	0.536848	0.268424	−	0.963301i	$-0.413497\pi$
$271$	2395.00	0.536848	0.268424	+	0.963301i	$0.413497\pi$
$272$	112.000	0.0249669
$273$	0	0
$274$	−2682.00	−0.591334
$275$	0	0
$276$	0	0
$277$	−6574.00	−1.42597	−0.712984	−	0.701180i	$-0.752657\pi$
$277$	−6574.00	−1.42597	−0.712984	+	0.701180i	$0.752657\pi$
$278$	5128.00	1.10632
$279$	0	0
$280$	0	0
$281$	4230.00	0.898009	0.449005	−	0.893529i	$-0.351779\pi$
$281$	4230.00	0.898009	0.449005	+	0.893529i	$0.351779\pi$
$282$	0	0
$283$	−5678.00	−1.19266	−0.596329	−	0.802740i	$-0.703375\pi$
$283$	−5678.00	−1.19266	−0.596329	+	0.802740i	$0.703375\pi$
$284$	−1560.00	−0.325947
$285$	0	0
$286$	1320.00	0.272913
$287$	5684.00	1.16904
$288$	0	0
$289$	−4864.00	−0.990026
$290$	0	0
$291$	0	0
$292$	2504.00	0.501834
$293$	−4587.00	−0.914592	−0.457296	−	0.889315i	$-0.651182\pi$
$293$	−4587.00	−0.914592	−0.457296	+	0.889315i	$0.651182\pi$
$294$	0	0
$295$	0	0
$296$	704.000	0.138240
$297$	0	0
$298$	−6244.00	−1.21378
$299$	−4530.00	−0.876176
$300$	0	0
$301$	−6188.00	−1.18495
$302$	−6008.00	−1.14477
$303$	0	0
$304$	−1296.00	−0.244509
$305$	0	0
$306$	0	0
$307$	7140.00	1.32737	0.663683	−	0.748014i	$-0.268992\pi$
$307$	7140.00	1.32737	0.663683	+	0.748014i	$0.268992\pi$
$308$	1232.00	0.227921
$309$	0	0
$310$	0	0
$311$	−7122.00	−1.29856	−0.649279	−	0.760550i	$-0.724929\pi$
$311$	−7122.00	−1.29856	−0.649279	+	0.760550i	$0.724929\pi$
$312$	0	0
$313$	6740.00	1.21715	0.608574	−	0.793497i	$-0.291742\pi$
$313$	6740.00	1.21715	0.608574	+	0.793497i	$0.291742\pi$
$314$	7132.00	1.28179
$315$	0	0
$316$	−4860.00	−0.865178
$317$	−5441.00	−0.964028	−0.482014	−	0.876163i	$-0.660095\pi$
$317$	−5441.00	−0.964028	−0.482014	+	0.876163i	$0.660095\pi$
$318$	0	0
$319$	−5940.00	−1.04256
$320$	0	0
$321$	0	0
$322$	−4228.00	−0.731731
$323$	−567.000	−0.0976741
$324$	0	0
$325$	0	0
$326$	−1112.00	−0.188920
$327$	0	0
$328$	−3248.00	−0.546771
$329$	784.000	0.131378
$330$	0	0
$331$	−380.000	−0.0631018	−0.0315509	−	0.999502i	$-0.510045\pi$
$331$	−380.000	−0.0631018	−0.0315509	+	0.999502i	$0.510045\pi$
$332$	2020.00	0.333921
$333$	0	0
$334$	258.000	0.0422669
$335$	0	0
$336$	0	0
$337$	3146.00	0.508527	0.254263	−	0.967135i	$-0.418167\pi$
$337$	3146.00	0.508527	0.254263	+	0.967135i	$0.418167\pi$
$338$	2594.00	0.417441
$339$	0	0
$340$	0	0
$341$	−2486.00	−0.394793
$342$	0	0
$343$	−6860.00	−1.07990
$344$	3536.00	0.554210
$345$	0	0
$346$	−2554.00	−0.396832
$347$	−8904.00	−1.37750	−0.688749	−	0.725000i	$-0.741840\pi$
$347$	−8904.00	−1.37750	−0.688749	+	0.725000i	$0.741840\pi$
$348$	0	0
$349$	10429.0	1.59957	0.799787	−	0.600283i	$-0.204946\pi$
$349$	10429.0	1.59957	0.799787	+	0.600283i	$0.204946\pi$
$350$	0	0
$351$	0	0
$352$	−704.000	−0.106600
$353$	−10018.0	−1.51049	−0.755247	−	0.655440i	$-0.772483\pi$
$353$	−10018.0	−1.51049	−0.755247	+	0.655440i	$0.772483\pi$
$354$	0	0
$355$	0	0
$356$	2056.00	0.306089
$357$	0	0
$358$	2360.00	0.348407
$359$	8532.00	1.25432	0.627161	−	0.778889i	$-0.284217\pi$
$359$	8532.00	1.25432	0.627161	+	0.778889i	$0.284217\pi$
$360$	0	0
$361$	−298.000	−0.0434466
$362$	5054.00	0.733791
$363$	0	0
$364$	−1680.00	−0.241912
$365$	0	0
$366$	0	0
$367$	−3586.00	−0.510048	−0.255024	−	0.966935i	$-0.582083\pi$
$367$	−3586.00	−0.510048	−0.255024	+	0.966935i	$0.582083\pi$
$368$	2416.00	0.342236
$369$	0	0
$370$	0	0
$371$	−1974.00	−0.276240
$372$	0	0
$373$	10500.0	1.45756	0.728779	−	0.684749i	$-0.240088\pi$
$373$	10500.0	1.45756	0.728779	+	0.684749i	$0.240088\pi$
$374$	−308.000	−0.0425837
$375$	0	0
$376$	−448.000	−0.0614464
$377$	8100.00	1.10655
$378$	0	0
$379$	−6811.00	−0.923107	−0.461554	−	0.887112i	$-0.652708\pi$
$379$	−6811.00	−0.923107	−0.461554	+	0.887112i	$0.652708\pi$
$380$	0	0
$381$	0	0
$382$	−6960.00	−0.932211
$383$	−6913.00	−0.922292	−0.461146	−	0.887324i	$-0.652562\pi$
$383$	−6913.00	−0.922292	−0.461146	+	0.887324i	$0.652562\pi$
$384$	0	0
$385$	0	0
$386$	−6340.00	−0.836004
$387$	0	0
$388$	7264.00	0.950448
$389$	5998.00	0.781776	0.390888	−	0.920438i	$-0.372168\pi$
$389$	5998.00	0.781776	0.390888	+	0.920438i	$0.372168\pi$
$390$	0	0
$391$	1057.00	0.136713
$392$	1176.00	0.151523
$393$	0	0
$394$	4670.00	0.597135
$395$	0	0
$396$	0	0
$397$	−5902.00	−0.746128	−0.373064	−	0.927806i	$-0.621693\pi$
$397$	−5902.00	−0.746128	−0.373064	+	0.927806i	$0.621693\pi$
$398$	3024.00	0.380853
$399$	0	0
$400$	0	0
$401$	2396.00	0.298380	0.149190	−	0.988809i	$-0.452333\pi$
$401$	2396.00	0.298380	0.149190	+	0.988809i	$0.452333\pi$
$402$	0	0
$403$	3390.00	0.419027
$404$	−2368.00	−0.291615
$405$	0	0
$406$	7560.00	0.924129
$407$	−1936.00	−0.235784
$408$	0	0
$409$	8141.00	0.984221	0.492111	−	0.870533i	$-0.336226\pi$
$409$	8141.00	0.984221	0.492111	+	0.870533i	$0.336226\pi$
$410$	0	0
$411$	0	0
$412$	−3392.00	−0.405611
$413$	−3836.00	−0.457039
$414$	0	0
$415$	0	0
$416$	960.000	0.113144
$417$	0	0
$418$	3564.00	0.417036
$419$	9094.00	1.06031	0.530156	−	0.847900i	$-0.322133\pi$
$419$	9094.00	1.06031	0.530156	+	0.847900i	$0.322133\pi$
$420$	0	0
$421$	6751.00	0.781529	0.390764	−	0.920491i	$-0.372211\pi$
$421$	6751.00	0.781529	0.390764	+	0.920491i	$0.372211\pi$
$422$	11838.0	1.36556
$423$	0	0
$424$	1128.00	0.129199
$425$	0	0
$426$	0	0
$427$	574.000	0.0650534
$428$	6000.00	0.677619
$429$	0	0
$430$	0	0
$431$	−5594.00	−0.625182	−0.312591	−	0.949888i	$-0.601197\pi$
$431$	−5594.00	−0.625182	−0.312591	+	0.949888i	$0.601197\pi$
$432$	0	0
$433$	−15610.0	−1.73249	−0.866246	−	0.499618i	$-0.833474\pi$
$433$	−15610.0	−1.73249	−0.866246	+	0.499618i	$0.833474\pi$
$434$	3164.00	0.349947
$435$	0	0
$436$	−3740.00	−0.410811
$437$	−12231.0	−1.33887
$438$	0	0
$439$	5751.00	0.625240	0.312620	−	0.949878i	$-0.398793\pi$
$439$	5751.00	0.625240	0.312620	+	0.949878i	$0.398793\pi$
$440$	0	0
$441$	0	0
$442$	420.000	0.0451977
$443$	−2331.00	−0.249998	−0.124999	−	0.992157i	$-0.539893\pi$
$443$	−2331.00	−0.249998	−0.124999	+	0.992157i	$0.539893\pi$
$444$	0	0
$445$	0	0
$446$	824.000	0.0874833
$447$	0	0
$448$	896.000	0.0944911
$449$	−7158.00	−0.752354	−0.376177	−	0.926548i	$-0.622762\pi$
$449$	−7158.00	−0.752354	−0.376177	+	0.926548i	$0.622762\pi$
$450$	0	0
$451$	8932.00	0.932575
$452$	−5064.00	−0.526970
$453$	0	0
$454$	11686.0	1.20804
$455$	0	0
$456$	0	0
$457$	−3196.00	−0.327139	−0.163570	−	0.986532i	$-0.552301\pi$
$457$	−3196.00	−0.327139	−0.163570	+	0.986532i	$0.552301\pi$
$458$	8306.00	0.847410
$459$	0	0
$460$	0	0
$461$	−13368.0	−1.35056	−0.675282	−	0.737560i	$-0.735978\pi$
$461$	−13368.0	−1.35056	−0.675282	+	0.737560i	$0.735978\pi$
$462$	0	0
$463$	−9912.00	−0.994924	−0.497462	−	0.867486i	$-0.665735\pi$
$463$	−9912.00	−0.994924	−0.497462	+	0.867486i	$0.665735\pi$
$464$	−4320.00	−0.432222
$465$	0	0
$466$	1124.00	0.111735
$467$	13137.0	1.30173	0.650865	−	0.759194i	$-0.274406\pi$
$467$	13137.0	1.30173	0.650865	+	0.759194i	$0.274406\pi$
$468$	0	0
$469$	4592.00	0.452108
$470$	0	0
$471$	0	0
$472$	2192.00	0.213761
$473$	−9724.00	−0.945264
$474$	0	0
$475$	0	0
$476$	392.000	0.0377464
$477$	0	0
$478$	1672.00	0.159991
$479$	−4946.00	−0.471792	−0.235896	−	0.971778i	$-0.575803\pi$
$479$	−4946.00	−0.471792	−0.235896	+	0.971778i	$0.575803\pi$
$480$	0	0
$481$	2640.00	0.250257
$482$	3058.00	0.288979
$483$	0	0
$484$	−3388.00	−0.318182
$485$	0	0
$486$	0	0
$487$	15966.0	1.48560	0.742801	−	0.669512i	$-0.233497\pi$
$487$	15966.0	1.48560	0.742801	+	0.669512i	$0.233497\pi$
$488$	−328.000	−0.0304259
$489$	0	0
$490$	0	0
$491$	11196.0	1.02906	0.514530	−	0.857472i	$-0.327966\pi$
$491$	11196.0	1.02906	0.514530	+	0.857472i	$0.327966\pi$
$492$	0	0
$493$	−1890.00	−0.172660
$494$	−4860.00	−0.442635
$495$	0	0
$496$	−1808.00	−0.163673
$497$	−5460.00	−0.492786
$498$	0	0
$499$	−1485.00	−0.133222	−0.0666110	−	0.997779i	$-0.521219\pi$
$499$	−1485.00	−0.133222	−0.0666110	+	0.997779i	$0.521219\pi$
$500$	0	0
$501$	0	0
$502$	504.000	0.0448100
$503$	10635.0	0.942726	0.471363	−	0.881939i	$-0.343762\pi$
$503$	10635.0	0.942726	0.471363	+	0.881939i	$0.343762\pi$
$504$	0	0
$505$	0	0
$506$	−6644.00	−0.583719
$507$	0	0
$508$	−64.0000	−0.00558965
$509$	−13176.0	−1.14738	−0.573690	−	0.819073i	$-0.694488\pi$
$509$	−13176.0	−1.14738	−0.573690	+	0.819073i	$0.694488\pi$
$510$	0	0
$511$	8764.00	0.758702
$512$	−512.000	−0.0441942
$513$	0	0
$514$	12222.0	1.04881
$515$	0	0
$516$	0	0
$517$	1232.00	0.104803
$518$	2464.00	0.209000
$519$	0	0
$520$	0	0
$521$	12542.0	1.05465	0.527327	−	0.849662i	$-0.323194\pi$
$521$	12542.0	1.05465	0.527327	+	0.849662i	$0.323194\pi$
$522$	0	0
$523$	3778.00	0.315871	0.157935	−	0.987449i	$-0.449516\pi$
$523$	3778.00	0.315871	0.157935	+	0.987449i	$0.449516\pi$
$524$	−9216.00	−0.768326
$525$	0	0
$526$	6208.00	0.514604
$527$	−791.000	−0.0653824
$528$	0	0
$529$	10634.0	0.874003
$530$	0	0
$531$	0	0
$532$	−4536.00	−0.369663
$533$	−12180.0	−0.989821
$534$	0	0
$535$	0	0
$536$	−2624.00	−0.211454
$537$	0	0
$538$	6480.00	0.519280
$539$	−3234.00	−0.258438
$540$	0	0
$541$	1150.00	0.0913907	0.0456953	−	0.998955i	$-0.485450\pi$
$541$	1150.00	0.0913907	0.0456953	+	0.998955i	$0.485450\pi$
$542$	−4790.00	−0.379609
$543$	0	0
$544$	−224.000	−0.0176543
$545$	0	0
$546$	0	0
$547$	23726.0	1.85457	0.927286	−	0.374355i	$-0.122136\pi$
$547$	23726.0	1.85457	0.927286	+	0.374355i	$0.122136\pi$
$548$	5364.00	0.418136
$549$	0	0
$550$	0	0
$551$	21870.0	1.69091
$552$	0	0
$553$	−17010.0	−1.30803
$554$	13148.0	1.00831
$555$	0	0
$556$	−10256.0	−0.782286
$557$	15786.0	1.20085	0.600426	−	0.799681i	$-0.294998\pi$
$557$	15786.0	1.20085	0.600426	+	0.799681i	$0.294998\pi$
$558$	0	0
$559$	13260.0	1.00329
$560$	0	0
$561$	0	0
$562$	−8460.00	−0.634989
$563$	−4468.00	−0.334465	−0.167232	−	0.985917i	$-0.553483\pi$
$563$	−4468.00	−0.334465	−0.167232	+	0.985917i	$0.553483\pi$
$564$	0	0
$565$	0	0
$566$	11356.0	0.843336
$567$	0	0
$568$	3120.00	0.230479
$569$	13102.0	0.965315	0.482658	−	0.875809i	$-0.339672\pi$
$569$	13102.0	0.965315	0.482658	+	0.875809i	$0.339672\pi$
$570$	0	0
$571$	13711.0	1.00488	0.502441	−	0.864612i	$-0.332435\pi$
$571$	13711.0	1.00488	0.502441	+	0.864612i	$0.332435\pi$
$572$	−2640.00	−0.192979
$573$	0	0
$574$	−11368.0	−0.826640
$575$	0	0
$576$	0	0
$577$	−8374.00	−0.604184	−0.302092	−	0.953279i	$-0.597685\pi$
$577$	−8374.00	−0.604184	−0.302092	+	0.953279i	$0.597685\pi$
$578$	9728.00	0.700054
$579$	0	0
$580$	0	0
$581$	7070.00	0.504842
$582$	0	0
$583$	−3102.00	−0.220363
$584$	−5008.00	−0.354850
$585$	0	0
$586$	9174.00	0.646714
$587$	11499.0	0.808543	0.404271	−	0.914639i	$-0.367525\pi$
$587$	11499.0	0.808543	0.404271	+	0.914639i	$0.367525\pi$
$588$	0	0
$589$	9153.00	0.640310
$590$	0	0
$591$	0	0
$592$	−1408.00	−0.0977507
$593$	24753.0	1.71414	0.857069	−	0.515202i	$-0.172283\pi$
$593$	24753.0	1.71414	0.857069	+	0.515202i	$0.172283\pi$
$594$	0	0
$595$	0	0
$596$	12488.0	0.858269
$597$	0	0
$598$	9060.00	0.619550
$599$	378.000	0.0257841	0.0128920	−	0.999917i	$-0.495896\pi$
$599$	378.000	0.0257841	0.0128920	+	0.999917i	$0.495896\pi$
$600$	0	0
$601$	15917.0	1.08031	0.540156	−	0.841565i	$-0.318365\pi$
$601$	15917.0	1.08031	0.540156	+	0.841565i	$0.318365\pi$
$602$	12376.0	0.837887
$603$	0	0
$604$	12016.0	0.809477
$605$	0	0
$606$	0	0
$607$	718.000	0.0480111	0.0240055	−	0.999712i	$-0.492358\pi$
$607$	718.000	0.0480111	0.0240055	+	0.999712i	$0.492358\pi$
$608$	2592.00	0.172894
$609$	0	0
$610$	0	0
$611$	−1680.00	−0.111237
$612$	0	0
$613$	24848.0	1.63720	0.818598	−	0.574367i	$-0.194752\pi$
$613$	24848.0	1.63720	0.818598	+	0.574367i	$0.194752\pi$
$614$	−14280.0	−0.938589
$615$	0	0
$616$	−2464.00	−0.161165
$617$	−29681.0	−1.93665	−0.968324	−	0.249696i	$-0.919669\pi$
$617$	−29681.0	−1.93665	−0.968324	+	0.249696i	$0.919669\pi$
$618$	0	0
$619$	30308.0	1.96798	0.983991	−	0.178216i	$-0.0570325\pi$
$619$	30308.0	1.96798	0.983991	+	0.178216i	$0.0570325\pi$
$620$	0	0
$621$	0	0
$622$	14244.0	0.918219
$623$	7196.00	0.462763
$624$	0	0
$625$	0	0
$626$	−13480.0	−0.860654
$627$	0	0
$628$	−14264.0	−0.906362
$629$	−616.000	−0.0390485
$630$	0	0
$631$	−15475.0	−0.976307	−0.488154	−	0.872758i	$-0.662329\pi$
$631$	−15475.0	−0.976307	−0.488154	+	0.872758i	$0.662329\pi$
$632$	9720.00	0.611773
$633$	0	0
$634$	10882.0	0.681671
$635$	0	0
$636$	0	0
$637$	4410.00	0.274302
$638$	11880.0	0.737200
$639$	0	0
$640$	0	0
$641$	−29984.0	−1.84758	−0.923788	−	0.382903i	$-0.874924\pi$
$641$	−29984.0	−1.84758	−0.923788	+	0.382903i	$0.874924\pi$
$642$	0	0
$643$	22530.0	1.38180	0.690899	−	0.722951i	$-0.257215\pi$
$643$	22530.0	1.38180	0.690899	+	0.722951i	$0.257215\pi$
$644$	8456.00	0.517412
$645$	0	0
$646$	1134.00	0.0690660
$647$	31179.0	1.89455	0.947274	−	0.320424i	$-0.103825\pi$
$647$	31179.0	1.89455	0.947274	+	0.320424i	$0.103825\pi$
$648$	0	0
$649$	−6028.00	−0.364591
$650$	0	0
$651$	0	0
$652$	2224.00	0.133587
$653$	−5.00000	−0.000299640 0	−0.000149820	−	1.00000i	$-0.500048\pi$
$653$	−5.00000	−0.000299640 0	−0.000149820		1.00000i	$0.500048\pi$
$654$	0	0
$655$	0	0
$656$	6496.00	0.386625
$657$	0	0
$658$	−1568.00	−0.0928982
$659$	−17938.0	−1.06034	−0.530171	−	0.847891i	$-0.677872\pi$
$659$	−17938.0	−1.06034	−0.530171	+	0.847891i	$0.677872\pi$
$660$	0	0
$661$	−8722.00	−0.513232	−0.256616	−	0.966513i	$-0.582608\pi$
$661$	−8722.00	−0.513232	−0.256616	+	0.966513i	$0.582608\pi$
$662$	760.000	0.0446197
$663$	0	0
$664$	−4040.00	−0.236118
$665$	0	0
$666$	0	0
$667$	−40770.0	−2.36675
$668$	−516.000	−0.0298872
$669$	0	0
$670$	0	0
$671$	902.000	0.0518947
$672$	0	0
$673$	−14598.0	−0.836124	−0.418062	−	0.908418i	$-0.637291\pi$
$673$	−14598.0	−0.836124	−0.418062	+	0.908418i	$0.637291\pi$
$674$	−6292.00	−0.359583
$675$	0	0
$676$	−5188.00	−0.295175
$677$	−8258.00	−0.468805	−0.234402	−	0.972140i	$-0.575313\pi$
$677$	−8258.00	−0.468805	−0.234402	+	0.972140i	$0.575313\pi$
$678$	0	0
$679$	25424.0	1.43694
$680$	0	0
$681$	0	0
$682$	4972.00	0.279161
$683$	−11075.0	−0.620458	−0.310229	−	0.950662i	$-0.600406\pi$
$683$	−11075.0	−0.620458	−0.310229	+	0.950662i	$0.600406\pi$
$684$	0	0
$685$	0	0
$686$	13720.0	0.763604
$687$	0	0
$688$	−7072.00	−0.391886
$689$	4230.00	0.233890
$690$	0	0
$691$	−2315.00	−0.127448	−0.0637241	−	0.997968i	$-0.520298\pi$
$691$	−2315.00	−0.127448	−0.0637241	+	0.997968i	$0.520298\pi$
$692$	5108.00	0.280603
$693$	0	0
$694$	17808.0	0.974038
$695$	0	0
$696$	0	0
$697$	2842.00	0.154445
$698$	−20858.0	−1.13107
$699$	0	0
$700$	0	0
$701$	4180.00	0.225216	0.112608	−	0.993639i	$-0.464080\pi$
$701$	4180.00	0.225216	0.112608	+	0.993639i	$0.464080\pi$
$702$	0	0
$703$	7128.00	0.382415
$704$	1408.00	0.0753778
$705$	0	0
$706$	20036.0	1.06808
$707$	−8288.00	−0.440880
$708$	0	0
$709$	−222.000	−0.0117594	−0.00587968	−	0.999983i	$-0.501872\pi$
$709$	−222.000	−0.0117594	−0.00587968	+	0.999983i	$0.501872\pi$
$710$	0	0
$711$	0	0
$712$	−4112.00	−0.216438
$713$	−17063.0	−0.896233
$714$	0	0
$715$	0	0
$716$	−4720.00	−0.246361
$717$	0	0
$718$	−17064.0	−0.886940
$719$	16816.0	0.872227	0.436113	−	0.899892i	$-0.356355\pi$
$719$	16816.0	0.872227	0.436113	+	0.899892i	$0.356355\pi$
$720$	0	0
$721$	−11872.0	−0.613227
$722$	596.000	0.0307214
$723$	0	0
$724$	−10108.0	−0.518869
$725$	0	0
$726$	0	0
$727$	−21996.0	−1.12213	−0.561064	−	0.827773i	$-0.689608\pi$
$727$	−21996.0	−1.12213	−0.561064	+	0.827773i	$0.689608\pi$
$728$	3360.00	0.171058
$729$	0	0
$730$	0	0
$731$	−3094.00	−0.156547
$732$	0	0
$733$	21112.0	1.06383	0.531916	−	0.846797i	$-0.321472\pi$
$733$	21112.0	1.06383	0.531916	+	0.846797i	$0.321472\pi$
$734$	7172.00	0.360659
$735$	0	0
$736$	−4832.00	−0.241997
$737$	7216.00	0.360658
$738$	0	0
$739$	49.0000	0.00243910	0.00121955	−	0.999999i	$-0.499612\pi$
$739$	49.0000	0.00243910	0.00121955	+	0.999999i	$0.499612\pi$
$740$	0	0
$741$	0	0
$742$	3948.00	0.195331
$743$	3876.00	0.191382	0.0956909	−	0.995411i	$-0.469494\pi$
$743$	3876.00	0.191382	0.0956909	+	0.995411i	$0.469494\pi$
$744$	0	0
$745$	0	0
$746$	−21000.0	−1.03065
$747$	0	0
$748$	616.000	0.0301112
$749$	21000.0	1.02446
$750$	0	0
$751$	−8395.00	−0.407907	−0.203953	−	0.978981i	$-0.565379\pi$
$751$	−8395.00	−0.407907	−0.203953	+	0.978981i	$0.565379\pi$
$752$	896.000	0.0434491
$753$	0	0
$754$	−16200.0	−0.782453
$755$	0	0
$756$	0	0
$757$	1892.00	0.0908400	0.0454200	−	0.998968i	$-0.485537\pi$
$757$	1892.00	0.0908400	0.0454200	+	0.998968i	$0.485537\pi$
$758$	13622.0	0.652735
$759$	0	0
$760$	0	0
$761$	−2034.00	−0.0968889	−0.0484444	−	0.998826i	$-0.515426\pi$
$761$	−2034.00	−0.0968889	−0.0484444	+	0.998826i	$0.515426\pi$
$762$	0	0
$763$	−13090.0	−0.621088
$764$	13920.0	0.659173
$765$	0	0
$766$	13826.0	0.652159
$767$	8220.00	0.386971
$768$	0	0
$769$	−5439.00	−0.255052	−0.127526	−	0.991835i	$-0.540704\pi$
$769$	−5439.00	−0.255052	−0.127526	+	0.991835i	$0.540704\pi$
$770$	0	0
$771$	0	0
$772$	12680.0	0.591144
$773$	−8829.00	−0.410811	−0.205406	−	0.978677i	$-0.565851\pi$
$773$	−8829.00	−0.410811	−0.205406	+	0.978677i	$0.565851\pi$
$774$	0	0
$775$	0	0
$776$	−14528.0	−0.672068
$777$	0	0
$778$	−11996.0	−0.552799
$779$	−32886.0	−1.51253
$780$	0	0
$781$	−8580.00	−0.393107
$782$	−2114.00	−0.0966707
$783$	0	0
$784$	−2352.00	−0.107143
$785$	0	0
$786$	0	0
$787$	−15694.0	−0.710839	−0.355420	−	0.934707i	$-0.615662\pi$
$787$	−15694.0	−0.710839	−0.355420	+	0.934707i	$0.615662\pi$
$788$	−9340.00	−0.422238
$789$	0	0
$790$	0	0
$791$	−17724.0	−0.796704
$792$	0	0
$793$	−1230.00	−0.0550802
$794$	11804.0	0.527592
$795$	0	0
$796$	−6048.00	−0.269304
$797$	−4859.00	−0.215953	−0.107977	−	0.994153i	$-0.534437\pi$
$797$	−4859.00	−0.215953	−0.107977	+	0.994153i	$0.534437\pi$
$798$	0	0
$799$	392.000	0.0173566
$800$	0	0
$801$	0	0
$802$	−4792.00	−0.210987
$803$	13772.0	0.605235
$804$	0	0
$805$	0	0
$806$	−6780.00	−0.296297
$807$	0	0
$808$	4736.00	0.206203
$809$	5862.00	0.254755	0.127378	−	0.991854i	$-0.459344\pi$
$809$	5862.00	0.254755	0.127378	+	0.991854i	$0.459344\pi$
$810$	0	0
$811$	376.000	0.0162801	0.00814004	−	0.999967i	$-0.497409\pi$
$811$	376.000	0.0162801	0.00814004	+	0.999967i	$0.497409\pi$
$812$	−15120.0	−0.653458
$813$	0	0
$814$	3872.00	0.166724
$815$	0	0
$816$	0	0
$817$	35802.0	1.53311
$818$	−16282.0	−0.695950
$819$	0	0
$820$	0	0
$821$	−13804.0	−0.586800	−0.293400	−	0.955990i	$-0.594787\pi$
$821$	−13804.0	−0.586800	−0.293400	+	0.955990i	$0.594787\pi$
$822$	0	0
$823$	−18714.0	−0.792623	−0.396312	−	0.918116i	$-0.629710\pi$
$823$	−18714.0	−0.792623	−0.396312	+	0.918116i	$0.629710\pi$
$824$	6784.00	0.286810
$825$	0	0
$826$	7672.00	0.323176
$827$	36163.0	1.52057	0.760285	−	0.649590i	$-0.225059\pi$
$827$	36163.0	1.52057	0.760285	+	0.649590i	$0.225059\pi$
$828$	0	0
$829$	26862.0	1.12540	0.562700	−	0.826662i	$-0.309763\pi$
$829$	26862.0	1.12540	0.562700	+	0.826662i	$0.309763\pi$
$830$	0	0
$831$	0	0
$832$	−1920.00	−0.0800048
$833$	−1029.00	−0.0428004
$834$	0	0
$835$	0	0
$836$	−7128.00	−0.294889
$837$	0	0
$838$	−18188.0	−0.749754
$839$	29564.0	1.21652	0.608261	−	0.793737i	$-0.291867\pi$
$839$	29564.0	1.21652	0.608261	+	0.793737i	$0.291867\pi$
$840$	0	0
$841$	48511.0	1.98905
$842$	−13502.0	−0.552624
$843$	0	0
$844$	−23676.0	−0.965594
$845$	0	0
$846$	0	0
$847$	−11858.0	−0.481046
$848$	−2256.00	−0.0913577
$849$	0	0
$850$	0	0
$851$	−13288.0	−0.535261
$852$	0	0
$853$	−8108.00	−0.325454	−0.162727	−	0.986671i	$-0.552029\pi$
$853$	−8108.00	−0.325454	−0.162727	+	0.986671i	$0.552029\pi$
$854$	−1148.00	−0.0459997
$855$	0	0
$856$	−12000.0	−0.479149
$857$	−861.000	−0.0343188	−0.0171594	−	0.999853i	$-0.505462\pi$
$857$	−861.000	−0.0343188	−0.0171594	+	0.999853i	$0.505462\pi$
$858$	0	0
$859$	12725.0	0.505438	0.252719	−	0.967540i	$-0.418675\pi$
$859$	12725.0	0.505438	0.252719	+	0.967540i	$0.418675\pi$
$860$	0	0
$861$	0	0
$862$	11188.0	0.442071
$863$	−18801.0	−0.741592	−0.370796	−	0.928714i	$-0.620915\pi$
$863$	−18801.0	−0.741592	−0.370796	+	0.928714i	$0.620915\pi$
$864$	0	0
$865$	0	0
$866$	31220.0	1.22506
$867$	0	0
$868$	−6328.00	−0.247450
$869$	−26730.0	−1.04344
$870$	0	0
$871$	−9840.00	−0.382796
$872$	7480.00	0.290487
$873$	0	0
$874$	24462.0	0.946727
$875$	0	0
$876$	0	0
$877$	10438.0	0.401900	0.200950	−	0.979602i	$-0.435597\pi$
$877$	10438.0	0.401900	0.200950	+	0.979602i	$0.435597\pi$
$878$	−11502.0	−0.442111
$879$	0	0
$880$	0	0
$881$	−1170.00	−0.0447427	−0.0223713	−	0.999750i	$-0.507122\pi$
$881$	−1170.00	−0.0447427	−0.0223713	+	0.999750i	$0.507122\pi$
$882$	0	0
$883$	20334.0	0.774964	0.387482	−	0.921877i	$-0.373345\pi$
$883$	20334.0	0.774964	0.387482	+	0.921877i	$0.373345\pi$
$884$	−840.000	−0.0319596
$885$	0	0
$886$	4662.00	0.176775
$887$	−30693.0	−1.16186	−0.580930	−	0.813953i	$-0.697311\pi$
$887$	−30693.0	−1.16186	−0.580930	+	0.813953i	$0.697311\pi$
$888$	0	0
$889$	−224.000	−0.00845075
$890$	0	0
$891$	0	0
$892$	−1648.00	−0.0618600
$893$	−4536.00	−0.169979
$894$	0	0
$895$	0	0
$896$	−1792.00	−0.0668153
$897$	0	0
$898$	14316.0	0.531995
$899$	30510.0	1.13189
$900$	0	0
$901$	−987.000	−0.0364947
$902$	−17864.0	−0.659430
$903$	0	0
$904$	10128.0	0.372624
$905$	0	0
$906$	0	0
$907$	−2616.00	−0.0957694	−0.0478847	−	0.998853i	$-0.515248\pi$
$907$	−2616.00	−0.0957694	−0.0478847	+	0.998853i	$0.515248\pi$
$908$	−23372.0	−0.854215
$909$	0	0
$910$	0	0
$911$	47802.0	1.73848	0.869238	−	0.494395i	$-0.164610\pi$
$911$	47802.0	1.73848	0.869238	+	0.494395i	$0.164610\pi$
$912$	0	0
$913$	11110.0	0.402724
$914$	6392.00	0.231322
$915$	0	0
$916$	−16612.0	−0.599210
$917$	−32256.0	−1.16160
$918$	0	0
$919$	−44064.0	−1.58165	−0.790825	−	0.612042i	$-0.790348\pi$
$919$	−44064.0	−1.58165	−0.790825	+	0.612042i	$0.790348\pi$
$920$	0	0
$921$	0	0
$922$	26736.0	0.954993
$923$	11700.0	0.417237
$924$	0	0
$925$	0	0
$926$	19824.0	0.703517
$927$	0	0
$928$	8640.00	0.305627
$929$	−1290.00	−0.0455582	−0.0227791	−	0.999741i	$-0.507251\pi$
$929$	−1290.00	−0.0455582	−0.0227791	+	0.999741i	$0.507251\pi$
$930$	0	0
$931$	11907.0	0.419158
$932$	−2248.00	−0.0790082
$933$	0	0
$934$	−26274.0	−0.920462
$935$	0	0
$936$	0	0
$937$	43804.0	1.52723	0.763615	−	0.645672i	$-0.223423\pi$
$937$	43804.0	1.52723	0.763615	+	0.645672i	$0.223423\pi$
$938$	−9184.00	−0.319689
$939$	0	0
$940$	0	0
$941$	18148.0	0.628701	0.314351	−	0.949307i	$-0.398213\pi$
$941$	18148.0	0.628701	0.314351	+	0.949307i	$0.398213\pi$
$942$	0	0
$943$	61306.0	2.11707
$944$	−4384.00	−0.151152
$945$	0	0
$946$	19448.0	0.668403
$947$	−5079.00	−0.174282	−0.0871411	−	0.996196i	$-0.527773\pi$
$947$	−5079.00	−0.174282	−0.0871411	+	0.996196i	$0.527773\pi$
$948$	0	0
$949$	−18780.0	−0.642386
$950$	0	0
$951$	0	0
$952$	−784.000	−0.0266907
$953$	28394.0	0.965133	0.482567	−	0.875859i	$-0.339705\pi$
$953$	28394.0	0.965133	0.482567	+	0.875859i	$0.339705\pi$
$954$	0	0
$955$	0	0
$956$	−3344.00	−0.113130
$957$	0	0
$958$	9892.00	0.333608
$959$	18774.0	0.632163
$960$	0	0
$961$	−17022.0	−0.571381
$962$	−5280.00	−0.176958
$963$	0	0
$964$	−6116.00	−0.204339
$965$	0	0
$966$	0	0
$967$	−41228.0	−1.37105	−0.685524	−	0.728050i	$-0.740427\pi$
$967$	−41228.0	−1.37105	−0.685524	+	0.728050i	$0.740427\pi$
$968$	6776.00	0.224989
$969$	0	0
$970$	0	0
$971$	−21528.0	−0.711500	−0.355750	−	0.934581i	$-0.615775\pi$
$971$	−21528.0	−0.711500	−0.355750	+	0.934581i	$0.615775\pi$
$972$	0	0
$973$	−35896.0	−1.18271
$974$	−31932.0	−1.05048
$975$	0	0
$976$	656.000	0.0215144
$977$	−49146.0	−1.60933	−0.804667	−	0.593726i	$-0.797656\pi$
$977$	−49146.0	−1.60933	−0.804667	+	0.593726i	$0.797656\pi$
$978$	0	0
$979$	11308.0	0.369158
$980$	0	0
$981$	0	0
$982$	−22392.0	−0.727655
$983$	33467.0	1.08589	0.542946	−	0.839768i	$-0.317309\pi$
$983$	33467.0	1.08589	0.542946	+	0.839768i	$0.317309\pi$
$984$	0	0
$985$	0	0
$986$	3780.00	0.122089
$987$	0	0
$988$	9720.00	0.312990
$989$	−66742.0	−2.14588
$990$	0	0
$991$	−23183.0	−0.743120	−0.371560	−	0.928409i	$-0.621177\pi$
$991$	−23183.0	−0.743120	−0.371560	+	0.928409i	$0.621177\pi$
$992$	3616.00	0.115734
$993$	0	0
$994$	10920.0	0.348452
$995$	0	0
$996$	0	0
$997$	−60282.0	−1.91489	−0.957447	−	0.288608i	$-0.906808\pi$
$997$	−60282.0	−1.91489	−0.957447	+	0.288608i	$0.906808\pi$
$998$	2970.00	0.0942021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.a.j.1.1		1
3.2	odd	2		1350.4.a.x.1.1		1
5.2	odd	4		270.4.c.a.109.1	✓	2
5.3	odd	4		270.4.c.a.109.2	yes	2
5.4	even	2		1350.4.a.s.1.1		1
15.2	even	4		270.4.c.b.109.2	yes	2
15.8	even	4		270.4.c.b.109.1	yes	2
15.14	odd	2		1350.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.c.a.109.1	✓	2	5.2	odd	4
270.4.c.a.109.2	yes	2	5.3	odd	4
270.4.c.b.109.1	yes	2	15.8	even	4
270.4.c.b.109.2	yes	2	15.2	even	4
1350.4.a.d.1.1		1	15.14	odd	2
1350.4.a.j.1.1		1	1.1	even	1	trivial
1350.4.a.s.1.1		1	5.4	even	2
1350.4.a.x.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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