LMFDB - Embedded newform 1350.4.a.bb.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:	$0$
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)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +34.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} +34.0000 q^{7} +8.00000 q^{8} +48.0000 q^{11} +70.0000 q^{13} +68.0000 q^{14} +16.0000 q^{16} -27.0000 q^{17} +119.000 q^{19} +96.0000 q^{22} -51.0000 q^{23} +140.000 q^{26} +136.000 q^{28} +30.0000 q^{29} -133.000 q^{31} +32.0000 q^{32} -54.0000 q^{34} -218.000 q^{37} +238.000 q^{38} -156.000 q^{41} +88.0000 q^{43} +192.000 q^{44} -102.000 q^{46} -516.000 q^{47} +813.000 q^{49} +280.000 q^{52} -639.000 q^{53} +272.000 q^{56} +60.0000 q^{58} +654.000 q^{59} +461.000 q^{61} -266.000 q^{62} +64.0000 q^{64} -182.000 q^{67} -108.000 q^{68} -900.000 q^{71} -704.000 q^{73} -436.000 q^{74} +476.000 q^{76} +1632.00 q^{77} -1375.00 q^{79} -312.000 q^{82} +915.000 q^{83} +176.000 q^{86} +384.000 q^{88} +1116.00 q^{89} +2380.00 q^{91} -204.000 q^{92} -1032.00 q^{94} +16.0000 q^{97} +1626.00 q^{98} +O(q^{100})$

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
$7$	34.0000	1.83583	0.917914	−	0.396780i	$-0.129872\pi$
$7$	34.0000	1.83583	0.917914	+	0.396780i	$0.129872\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	0	0
$13$	70.0000	1.49342	0.746712	−	0.665148i	$-0.231631\pi$
$13$	70.0000	1.49342	0.746712	+	0.665148i	$0.231631\pi$
$14$	68.0000	1.29813
$15$	0	0
$16$	16.0000	0.250000
$17$	−27.0000	−0.385204	−0.192602	−	0.981277i	$-0.561693\pi$
$17$	−27.0000	−0.385204	−0.192602	+	0.981277i	$0.561693\pi$
$18$	0	0
$19$	119.000	1.43687	0.718433	−	0.695596i	$-0.244859\pi$
$19$	119.000	1.43687	0.718433	+	0.695596i	$0.244859\pi$
$20$	0	0
$21$	0	0
$22$	96.0000	0.930330
$23$	−51.0000	−0.462358	−0.231179	−	0.972911i	$-0.574258\pi$
$23$	−51.0000	−0.462358	−0.231179	+	0.972911i	$0.574258\pi$
$24$	0	0
$25$	0	0
$26$	140.000	1.05601
$27$	0	0
$28$	136.000	0.917914
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	−133.000	−0.770565	−0.385282	−	0.922799i	$-0.625896\pi$
$31$	−133.000	−0.770565	−0.385282	+	0.922799i	$0.625896\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−54.0000	−0.272380
$35$	0	0
$36$	0	0
$37$	−218.000	−0.968621	−0.484311	−	0.874896i	$-0.660930\pi$
$37$	−218.000	−0.968621	−0.484311	+	0.874896i	$0.660930\pi$
$38$	238.000	1.01602
$39$	0	0
$40$	0	0
$41$	−156.000	−0.594222	−0.297111	−	0.954843i	$-0.596023\pi$
$41$	−156.000	−0.594222	−0.297111	+	0.954843i	$0.596023\pi$
$42$	0	0
$43$	88.0000	0.312090	0.156045	−	0.987750i	$-0.450125\pi$
$43$	88.0000	0.312090	0.156045	+	0.987750i	$0.450125\pi$
$44$	192.000	0.657843
$45$	0	0
$46$	−102.000	−0.326937
$47$	−516.000	−1.60141	−0.800706	−	0.599058i	$-0.795542\pi$
$47$	−516.000	−1.60141	−0.800706	+	0.599058i	$0.795542\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	0	0
$51$	0	0
$52$	280.000	0.746712
$53$	−639.000	−1.65610	−0.828051	−	0.560653i	$-0.810550\pi$
$53$	−639.000	−1.65610	−0.828051	+	0.560653i	$0.810550\pi$
$54$	0	0
$55$	0	0
$56$	272.000	0.649063
$57$	0	0
$58$	60.0000	0.135834
$59$	654.000	1.44311	0.721555	−	0.692357i	$-0.243427\pi$
$59$	654.000	1.44311	0.721555	+	0.692357i	$0.243427\pi$
$60$	0	0
$61$	461.000	0.967623	0.483811	−	0.875172i	$-0.339252\pi$
$61$	461.000	0.967623	0.483811	+	0.875172i	$0.339252\pi$
$62$	−266.000	−0.544872
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−182.000	−0.331863	−0.165932	−	0.986137i	$-0.553063\pi$
$67$	−182.000	−0.331863	−0.165932	+	0.986137i	$0.553063\pi$
$68$	−108.000	−0.192602
$69$	0	0
$70$	0	0
$71$	−900.000	−1.50437	−0.752186	−	0.658951i	$-0.771000\pi$
$71$	−900.000	−1.50437	−0.752186	+	0.658951i	$0.771000\pi$
$72$	0	0
$73$	−704.000	−1.12873	−0.564363	−	0.825527i	$-0.690878\pi$
$73$	−704.000	−1.12873	−0.564363	+	0.825527i	$0.690878\pi$
$74$	−436.000	−0.684919
$75$	0	0
$76$	476.000	0.718433
$77$	1632.00	2.41537
$78$	0	0
$79$	−1375.00	−1.95822	−0.979111	−	0.203325i	$-0.934825\pi$
$79$	−1375.00	−1.95822	−0.979111	+	0.203325i	$0.934825\pi$
$80$	0	0
$81$	0	0
$82$	−312.000	−0.420178
$83$	915.000	1.21005	0.605026	−	0.796206i	$-0.293163\pi$
$83$	915.000	1.21005	0.605026	+	0.796206i	$0.293163\pi$
$84$	0	0
$85$	0	0
$86$	176.000	0.220681
$87$	0	0
$88$	384.000	0.465165
$89$	1116.00	1.32917	0.664583	−	0.747215i	$-0.268609\pi$
$89$	1116.00	1.32917	0.664583	+	0.747215i	$0.268609\pi$
$90$	0	0
$91$	2380.00	2.74167
$92$	−204.000	−0.231179
$93$	0	0
$94$	−1032.00	−1.13237
$95$	0	0
$96$	0	0
$97$	16.0000	0.0167480	0.00837399	−	0.999965i	$-0.497334\pi$
$97$	16.0000	0.0167480	0.00837399	+	0.999965i	$0.497334\pi$
$98$	1626.00	1.67603
$99$	0	0
$100$	0	0
$101$	−348.000	−0.342844	−0.171422	−	0.985198i	$-0.554836\pi$
$101$	−348.000	−0.342844	−0.171422	+	0.985198i	$0.554836\pi$
$102$	0	0
$103$	412.000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	412.000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	560.000	0.528005
$105$	0	0
$106$	−1278.00	−1.17104
$107$	−900.000	−0.813143	−0.406571	−	0.913619i	$-0.633276\pi$
$107$	−900.000	−0.813143	−0.406571	+	0.913619i	$0.633276\pi$
$108$	0	0
$109$	−115.000	−0.101055	−0.0505275	−	0.998723i	$-0.516090\pi$
$109$	−115.000	−0.101055	−0.0505275	+	0.998723i	$0.516090\pi$
$110$	0	0
$111$	0	0
$112$	544.000	0.458957
$113$	966.000	0.804191	0.402096	−	0.915598i	$-0.368282\pi$
$113$	966.000	0.804191	0.402096	+	0.915598i	$0.368282\pi$
$114$	0	0
$115$	0	0
$116$	120.000	0.0960493
$117$	0	0
$118$	1308.00	1.02043
$119$	−918.000	−0.707167
$120$	0	0
$121$	973.000	0.731029
$122$	922.000	0.684213
$123$	0	0
$124$	−532.000	−0.385282
$125$	0	0
$126$	0	0
$127$	−1406.00	−0.982381	−0.491190	−	0.871052i	$-0.663438\pi$
$127$	−1406.00	−0.982381	−0.491190	+	0.871052i	$0.663438\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	−246.000	−0.164070	−0.0820348	−	0.996629i	$-0.526142\pi$
$131$	−246.000	−0.164070	−0.0820348	+	0.996629i	$0.526142\pi$
$132$	0	0
$133$	4046.00	2.63784
$134$	−364.000	−0.234663
$135$	0	0
$136$	−216.000	−0.136190
$137$	519.000	0.323658	0.161829	−	0.986819i	$-0.448261\pi$
$137$	519.000	0.323658	0.161829	+	0.986819i	$0.448261\pi$
$138$	0	0
$139$	1316.00	0.803034	0.401517	−	0.915852i	$-0.368483\pi$
$139$	1316.00	0.803034	0.401517	+	0.915852i	$0.368483\pi$
$140$	0	0
$141$	0	0
$142$	−1800.00	−1.06375
$143$	3360.00	1.96488
$144$	0	0
$145$	0	0
$146$	−1408.00	−0.798130
$147$	0	0
$148$	−872.000	−0.484311
$149$	−372.000	−0.204533	−0.102267	−	0.994757i	$-0.532609\pi$
$149$	−372.000	−0.204533	−0.102267	+	0.994757i	$0.532609\pi$
$150$	0	0
$151$	−1456.00	−0.784686	−0.392343	−	0.919819i	$-0.628335\pi$
$151$	−1456.00	−0.784686	−0.392343	+	0.919819i	$0.628335\pi$
$152$	952.000	0.508009
$153$	0	0
$154$	3264.00	1.70793
$155$	0	0
$156$	0	0
$157$	−956.000	−0.485969	−0.242984	−	0.970030i	$-0.578126\pi$
$157$	−956.000	−0.485969	−0.242984	+	0.970030i	$0.578126\pi$
$158$	−2750.00	−1.38467
$159$	0	0
$160$	0	0
$161$	−1734.00	−0.848810
$162$	0	0
$163$	2446.00	1.17537	0.587686	−	0.809089i	$-0.300039\pi$
$163$	2446.00	1.17537	0.587686	+	0.809089i	$0.300039\pi$
$164$	−624.000	−0.297111
$165$	0	0
$166$	1830.00	0.855636
$167$	−3111.00	−1.44154	−0.720768	−	0.693177i	$-0.756211\pi$
$167$	−3111.00	−1.44154	−0.720768	+	0.693177i	$0.756211\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	0	0
$171$	0	0
$172$	352.000	0.156045
$173$	−2397.00	−1.05341	−0.526707	−	0.850047i	$-0.676573\pi$
$173$	−2397.00	−1.05341	−0.526707	+	0.850047i	$0.676573\pi$
$174$	0	0
$175$	0	0
$176$	768.000	0.328921
$177$	0	0
$178$	2232.00	0.939862
$179$	540.000	0.225483	0.112742	−	0.993624i	$-0.464037\pi$
$179$	540.000	0.225483	0.112742	+	0.993624i	$0.464037\pi$
$180$	0	0
$181$	2333.00	0.958069	0.479035	−	0.877796i	$-0.340987\pi$
$181$	2333.00	0.958069	0.479035	+	0.877796i	$0.340987\pi$
$182$	4760.00	1.93865
$183$	0	0
$184$	−408.000	−0.163468
$185$	0	0
$186$	0	0
$187$	−1296.00	−0.506807
$188$	−2064.00	−0.800706
$189$	0	0
$190$	0	0
$191$	2730.00	1.03422	0.517110	−	0.855919i	$-0.327008\pi$
$191$	2730.00	1.03422	0.517110	+	0.855919i	$0.327008\pi$
$192$	0	0
$193$	4570.00	1.70443	0.852217	−	0.523188i	$-0.175258\pi$
$193$	4570.00	1.70443	0.852217	+	0.523188i	$0.175258\pi$
$194$	32.0000	0.0118426
$195$	0	0
$196$	3252.00	1.18513
$197$	675.000	0.244121	0.122060	−	0.992523i	$-0.461050\pi$
$197$	675.000	0.244121	0.122060	+	0.992523i	$0.461050\pi$
$198$	0	0
$199$	−3112.00	−1.10856	−0.554281	−	0.832330i	$-0.687007\pi$
$199$	−3112.00	−1.10856	−0.554281	+	0.832330i	$0.687007\pi$
$200$	0	0
$201$	0	0
$202$	−696.000	−0.242428
$203$	1020.00	0.352660
$204$	0	0
$205$	0	0
$206$	824.000	0.278693
$207$	0	0
$208$	1120.00	0.373356
$209$	5712.00	1.89047
$210$	0	0
$211$	2441.00	0.796424	0.398212	−	0.917294i	$-0.369631\pi$
$211$	2441.00	0.796424	0.398212	+	0.917294i	$0.369631\pi$
$212$	−2556.00	−0.828051
$213$	0	0
$214$	−1800.00	−0.574979
$215$	0	0
$216$	0	0
$217$	−4522.00	−1.41462
$218$	−230.000	−0.0714567
$219$	0	0
$220$	0	0
$221$	−1890.00	−0.575272
$222$	0	0
$223$	3418.00	1.02640	0.513198	−	0.858270i	$-0.328461\pi$
$223$	3418.00	1.02640	0.513198	+	0.858270i	$0.328461\pi$
$224$	1088.00	0.324532
$225$	0	0
$226$	1932.00	0.568649
$227$	−4377.00	−1.27979	−0.639894	−	0.768464i	$-0.721022\pi$
$227$	−4377.00	−1.27979	−0.639894	+	0.768464i	$0.721022\pi$
$228$	0	0
$229$	4187.00	1.20823	0.604115	−	0.796897i	$-0.293527\pi$
$229$	4187.00	1.20823	0.604115	+	0.796897i	$0.293527\pi$
$230$	0	0
$231$	0	0
$232$	240.000	0.0679171
$233$	−1098.00	−0.308723	−0.154361	−	0.988014i	$-0.549332\pi$
$233$	−1098.00	−0.308723	−0.154361	+	0.988014i	$0.549332\pi$
$234$	0	0
$235$	0	0
$236$	2616.00	0.721555
$237$	0	0
$238$	−1836.00	−0.500043
$239$	−6474.00	−1.75217	−0.876084	−	0.482158i	$-0.839853\pi$
$239$	−6474.00	−1.75217	−0.876084	+	0.482158i	$0.839853\pi$
$240$	0	0
$241$	3251.00	0.868943	0.434472	−	0.900686i	$-0.356935\pi$
$241$	3251.00	0.868943	0.434472	+	0.900686i	$0.356935\pi$
$242$	1946.00	0.516916
$243$	0	0
$244$	1844.00	0.483811
$245$	0	0
$246$	0	0
$247$	8330.00	2.14585
$248$	−1064.00	−0.272436
$249$	0	0
$250$	0	0
$251$	−1728.00	−0.434543	−0.217272	−	0.976111i	$-0.569716\pi$
$251$	−1728.00	−0.434543	−0.217272	+	0.976111i	$0.569716\pi$
$252$	0	0
$253$	−2448.00	−0.608318
$254$	−2812.00	−0.694648
$255$	0	0
$256$	256.000	0.0625000
$257$	−5469.00	−1.32742	−0.663710	−	0.747990i	$-0.731019\pi$
$257$	−5469.00	−1.32742	−0.663710	+	0.747990i	$0.731019\pi$
$258$	0	0
$259$	−7412.00	−1.77822
$260$	0	0
$261$	0	0
$262$	−492.000	−0.116015
$263$	−3216.00	−0.754019	−0.377010	−	0.926209i	$-0.623048\pi$
$263$	−3216.00	−0.754019	−0.377010	+	0.926209i	$0.623048\pi$
$264$	0	0
$265$	0	0
$266$	8092.00	1.86523
$267$	0	0
$268$	−728.000	−0.165932
$269$	8010.00	1.81553	0.907766	−	0.419476i	$-0.137786\pi$
$269$	8010.00	1.81553	0.907766	+	0.419476i	$0.137786\pi$
$270$	0	0
$271$	−3805.00	−0.852905	−0.426453	−	0.904510i	$-0.640237\pi$
$271$	−3805.00	−0.852905	−0.426453	+	0.904510i	$0.640237\pi$
$272$	−432.000	−0.0963009
$273$	0	0
$274$	1038.00	0.228861
$275$	0	0
$276$	0	0
$277$	−3224.00	−0.699319	−0.349660	−	0.936877i	$-0.613703\pi$
$277$	−3224.00	−0.699319	−0.349660	+	0.936877i	$0.613703\pi$
$278$	2632.00	0.567830
$279$	0	0
$280$	0	0
$281$	4530.00	0.961698	0.480849	−	0.876803i	$-0.340329\pi$
$281$	4530.00	0.961698	0.480849	+	0.876803i	$0.340329\pi$
$282$	0	0
$283$	3292.00	0.691481	0.345740	−	0.938330i	$-0.387628\pi$
$283$	3292.00	0.691481	0.345740	+	0.938330i	$0.387628\pi$
$284$	−3600.00	−0.752186
$285$	0	0
$286$	6720.00	1.38938
$287$	−5304.00	−1.09089
$288$	0	0
$289$	−4184.00	−0.851618
$290$	0	0
$291$	0	0
$292$	−2816.00	−0.564363
$293$	−7953.00	−1.58573	−0.792866	−	0.609397i	$-0.791412\pi$
$293$	−7953.00	−1.58573	−0.792866	+	0.609397i	$0.791412\pi$
$294$	0	0
$295$	0	0
$296$	−1744.00	−0.342459
$297$	0	0
$298$	−744.000	−0.144627
$299$	−3570.00	−0.690496
$300$	0	0
$301$	2992.00	0.572944
$302$	−2912.00	−0.554857
$303$	0	0
$304$	1904.00	0.359217
$305$	0	0
$306$	0	0
$307$	5290.00	0.983441	0.491720	−	0.870753i	$-0.336368\pi$
$307$	5290.00	0.983441	0.491720	+	0.870753i	$0.336368\pi$
$308$	6528.00	1.20769
$309$	0	0
$310$	0	0
$311$	−5358.00	−0.976927	−0.488464	−	0.872584i	$-0.662442\pi$
$311$	−5358.00	−0.976927	−0.488464	+	0.872584i	$0.662442\pi$
$312$	0	0
$313$	−5600.00	−1.01128	−0.505640	−	0.862744i	$-0.668744\pi$
$313$	−5600.00	−1.01128	−0.505640	+	0.862744i	$0.668744\pi$
$314$	−1912.00	−0.343632
$315$	0	0
$316$	−5500.00	−0.979111
$317$	7341.00	1.30067	0.650334	−	0.759649i	$-0.274629\pi$
$317$	7341.00	1.30067	0.650334	+	0.759649i	$0.274629\pi$
$318$	0	0
$319$	1440.00	0.252741
$320$	0	0
$321$	0	0
$322$	−3468.00	−0.600199
$323$	−3213.00	−0.553486
$324$	0	0
$325$	0	0
$326$	4892.00	0.831113
$327$	0	0
$328$	−1248.00	−0.210089
$329$	−17544.0	−2.93991
$330$	0	0
$331$	380.000	0.0631018	0.0315509	−	0.999502i	$-0.489955\pi$
$331$	380.000	0.0631018	0.0315509	+	0.999502i	$0.489955\pi$
$332$	3660.00	0.605026
$333$	0	0
$334$	−6222.00	−1.01932
$335$	0	0
$336$	0	0
$337$	−434.000	−0.0701528	−0.0350764	−	0.999385i	$-0.511167\pi$
$337$	−434.000	−0.0701528	−0.0350764	+	0.999385i	$0.511167\pi$
$338$	5406.00	0.869963
$339$	0	0
$340$	0	0
$341$	−6384.00	−1.01382
$342$	0	0
$343$	15980.0	2.51557
$344$	704.000	0.110341
$345$	0	0
$346$	−4794.00	−0.744876
$347$	8004.00	1.23826	0.619131	−	0.785287i	$-0.287485\pi$
$347$	8004.00	1.23826	0.619131	+	0.785287i	$0.287485\pi$
$348$	0	0
$349$	1109.00	0.170096	0.0850479	−	0.996377i	$-0.472896\pi$
$349$	1109.00	0.170096	0.0850479	+	0.996377i	$0.472896\pi$
$350$	0	0
$351$	0	0
$352$	1536.00	0.232583
$353$	−7662.00	−1.15526	−0.577630	−	0.816298i	$-0.696023\pi$
$353$	−7662.00	−1.15526	−0.577630	+	0.816298i	$0.696023\pi$
$354$	0	0
$355$	0	0
$356$	4464.00	0.664583
$357$	0	0
$358$	1080.00	0.159441
$359$	8478.00	1.24638	0.623192	−	0.782069i	$-0.285836\pi$
$359$	8478.00	1.24638	0.623192	+	0.782069i	$0.285836\pi$
$360$	0	0
$361$	7302.00	1.06459
$362$	4666.00	0.677457
$363$	0	0
$364$	9520.00	1.37083
$365$	0	0
$366$	0	0
$367$	−13286.0	−1.88971	−0.944855	−	0.327489i	$-0.893798\pi$
$367$	−13286.0	−1.88971	−0.944855	+	0.327489i	$0.893798\pi$
$368$	−816.000	−0.115590
$369$	0	0
$370$	0	0
$371$	−21726.0	−3.04032
$372$	0	0
$373$	−3080.00	−0.427551	−0.213775	−	0.976883i	$-0.568576\pi$
$373$	−3080.00	−0.427551	−0.213775	+	0.976883i	$0.568576\pi$
$374$	−2592.00	−0.358367
$375$	0	0
$376$	−4128.00	−0.566184
$377$	2100.00	0.286885
$378$	0	0
$379$	10109.0	1.37009	0.685045	−	0.728500i	$-0.259782\pi$
$379$	10109.0	1.37009	0.685045	+	0.728500i	$0.259782\pi$
$380$	0	0
$381$	0	0
$382$	5460.00	0.731303
$383$	−8727.00	−1.16431	−0.582153	−	0.813080i	$-0.697789\pi$
$383$	−8727.00	−1.16431	−0.582153	+	0.813080i	$0.697789\pi$
$384$	0	0
$385$	0	0
$386$	9140.00	1.20522
$387$	0	0
$388$	64.0000	0.00837399
$389$	2712.00	0.353480	0.176740	−	0.984258i	$-0.443445\pi$
$389$	2712.00	0.353480	0.176740	+	0.984258i	$0.443445\pi$
$390$	0	0
$391$	1377.00	0.178102
$392$	6504.00	0.838014
$393$	0	0
$394$	1350.00	0.172619
$395$	0	0
$396$	0	0
$397$	8818.00	1.11477	0.557384	−	0.830255i	$-0.311805\pi$
$397$	8818.00	1.11477	0.557384	+	0.830255i	$0.311805\pi$
$398$	−6224.00	−0.783872
$399$	0	0
$400$	0	0
$401$	−3306.00	−0.411705	−0.205853	−	0.978583i	$-0.565997\pi$
$401$	−3306.00	−0.411705	−0.205853	+	0.978583i	$0.565997\pi$
$402$	0	0
$403$	−9310.00	−1.15078
$404$	−1392.00	−0.171422
$405$	0	0
$406$	2040.00	0.249368
$407$	−10464.0	−1.27440
$408$	0	0
$409$	6401.00	0.773861	0.386930	−	0.922109i	$-0.373535\pi$
$409$	6401.00	0.773861	0.386930	+	0.922109i	$0.373535\pi$
$410$	0	0
$411$	0	0
$412$	1648.00	0.197066
$413$	22236.0	2.64930
$414$	0	0
$415$	0	0
$416$	2240.00	0.264002
$417$	0	0
$418$	11424.0	1.33676
$419$	2256.00	0.263038	0.131519	−	0.991314i	$-0.458015\pi$
$419$	2256.00	0.263038	0.131519	+	0.991314i	$0.458015\pi$
$420$	0	0
$421$	1811.00	0.209650	0.104825	−	0.994491i	$-0.466572\pi$
$421$	1811.00	0.209650	0.104825	+	0.994491i	$0.466572\pi$
$422$	4882.00	0.563156
$423$	0	0
$424$	−5112.00	−0.585520
$425$	0	0
$426$	0	0
$427$	15674.0	1.77639
$428$	−3600.00	−0.406571
$429$	0	0
$430$	0	0
$431$	5454.00	0.609536	0.304768	−	0.952427i	$-0.401421\pi$
$431$	5454.00	0.609536	0.304768	+	0.952427i	$0.401421\pi$
$432$	0	0
$433$	−2990.00	−0.331848	−0.165924	−	0.986139i	$-0.553061\pi$
$433$	−2990.00	−0.331848	−0.165924	+	0.986139i	$0.553061\pi$
$434$	−9044.00	−1.00029
$435$	0	0
$436$	−460.000	−0.0505275
$437$	−6069.00	−0.664347
$438$	0	0
$439$	9371.00	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$439$	9371.00	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$440$	0	0
$441$	0	0
$442$	−3780.00	−0.406779
$443$	6171.00	0.661835	0.330918	−	0.943660i	$-0.392642\pi$
$443$	6171.00	0.661835	0.330918	+	0.943660i	$0.392642\pi$
$444$	0	0
$445$	0	0
$446$	6836.00	0.725771
$447$	0	0
$448$	2176.00	0.229478
$449$	−4122.00	−0.433250	−0.216625	−	0.976255i	$-0.569505\pi$
$449$	−4122.00	−0.433250	−0.216625	+	0.976255i	$0.569505\pi$
$450$	0	0
$451$	−7488.00	−0.781810
$452$	3864.00	0.402096
$453$	0	0
$454$	−8754.00	−0.904946
$455$	0	0
$456$	0	0
$457$	−7076.00	−0.724292	−0.362146	−	0.932121i	$-0.617956\pi$
$457$	−7076.00	−0.724292	−0.362146	+	0.932121i	$0.617956\pi$
$458$	8374.00	0.854348
$459$	0	0
$460$	0	0
$461$	−762.000	−0.0769846	−0.0384923	−	0.999259i	$-0.512255\pi$
$461$	−762.000	−0.0769846	−0.0384923	+	0.999259i	$0.512255\pi$
$462$	0	0
$463$	−8822.00	−0.885514	−0.442757	−	0.896642i	$-0.646000\pi$
$463$	−8822.00	−0.885514	−0.442757	+	0.896642i	$0.646000\pi$
$464$	480.000	0.0480247
$465$	0	0
$466$	−2196.00	−0.218300
$467$	−4977.00	−0.493165	−0.246583	−	0.969122i	$-0.579308\pi$
$467$	−4977.00	−0.493165	−0.246583	+	0.969122i	$0.579308\pi$
$468$	0	0
$469$	−6188.00	−0.609244
$470$	0	0
$471$	0	0
$472$	5232.00	0.510217
$473$	4224.00	0.410613
$474$	0	0
$475$	0	0
$476$	−3672.00	−0.353584
$477$	0	0
$478$	−12948.0	−1.23897
$479$	−10104.0	−0.963807	−0.481903	−	0.876224i	$-0.660054\pi$
$479$	−10104.0	−0.963807	−0.481903	+	0.876224i	$0.660054\pi$
$480$	0	0
$481$	−15260.0	−1.44656
$482$	6502.00	0.614436
$483$	0	0
$484$	3892.00	0.365515
$485$	0	0
$486$	0	0
$487$	−14924.0	−1.38865	−0.694323	−	0.719663i	$-0.744296\pi$
$487$	−14924.0	−1.38865	−0.694323	+	0.719663i	$0.744296\pi$
$488$	3688.00	0.342106
$489$	0	0
$490$	0	0
$491$	−1146.00	−0.105332	−0.0526662	−	0.998612i	$-0.516772\pi$
$491$	−1146.00	−0.105332	−0.0526662	+	0.998612i	$0.516772\pi$
$492$	0	0
$493$	−810.000	−0.0739971
$494$	16660.0	1.51735
$495$	0	0
$496$	−2128.00	−0.192641
$497$	−30600.0	−2.76177
$498$	0	0
$499$	−14965.0	−1.34254	−0.671268	−	0.741215i	$-0.734250\pi$
$499$	−14965.0	−1.34254	−0.671268	+	0.741215i	$0.734250\pi$
$500$	0	0
$501$	0	0
$502$	−3456.00	−0.307269
$503$	15525.0	1.37619	0.688097	−	0.725619i	$-0.258446\pi$
$503$	15525.0	1.37619	0.688097	+	0.725619i	$0.258446\pi$
$504$	0	0
$505$	0	0
$506$	−4896.00	−0.430146
$507$	0	0
$508$	−5624.00	−0.491190
$509$	8196.00	0.713716	0.356858	−	0.934159i	$-0.383848\pi$
$509$	8196.00	0.713716	0.356858	+	0.934159i	$0.383848\pi$
$510$	0	0
$511$	−23936.0	−2.07215
$512$	512.000	0.0441942
$513$	0	0
$514$	−10938.0	−0.938627
$515$	0	0
$516$	0	0
$517$	−24768.0	−2.10695
$518$	−14824.0	−1.25739
$519$	0	0
$520$	0	0
$521$	−4932.00	−0.414731	−0.207365	−	0.978264i	$-0.566489\pi$
$521$	−4932.00	−0.414731	−0.207365	+	0.978264i	$0.566489\pi$
$522$	0	0
$523$	5938.00	0.496464	0.248232	−	0.968701i	$-0.420150\pi$
$523$	5938.00	0.496464	0.248232	+	0.968701i	$0.420150\pi$
$524$	−984.000	−0.0820348
$525$	0	0
$526$	−6432.00	−0.533172
$527$	3591.00	0.296824
$528$	0	0
$529$	−9566.00	−0.786225
$530$	0	0
$531$	0	0
$532$	16184.0	1.31892
$533$	−10920.0	−0.887425
$534$	0	0
$535$	0	0
$536$	−1456.00	−0.117331
$537$	0	0
$538$	16020.0	1.28378
$539$	39024.0	3.11852
$540$	0	0
$541$	−6730.00	−0.534834	−0.267417	−	0.963581i	$-0.586170\pi$
$541$	−6730.00	−0.534834	−0.267417	+	0.963581i	$0.586170\pi$
$542$	−7610.00	−0.603095
$543$	0	0
$544$	−864.000	−0.0680950
$545$	0	0
$546$	0	0
$547$	17656.0	1.38010	0.690051	−	0.723761i	$-0.257588\pi$
$547$	17656.0	1.38010	0.690051	+	0.723761i	$0.257588\pi$
$548$	2076.00	0.161829
$549$	0	0
$550$	0	0
$551$	3570.00	0.276020
$552$	0	0
$553$	−46750.0	−3.59496
$554$	−6448.00	−0.494493
$555$	0	0
$556$	5264.00	0.401517
$557$	7974.00	0.606587	0.303294	−	0.952897i	$-0.401914\pi$
$557$	7974.00	0.606587	0.303294	+	0.952897i	$0.401914\pi$
$558$	0	0
$559$	6160.00	0.466083
$560$	0	0
$561$	0	0
$562$	9060.00	0.680023
$563$	−25332.0	−1.89630	−0.948150	−	0.317824i	$-0.897048\pi$
$563$	−25332.0	−1.89630	−0.948150	+	0.317824i	$0.897048\pi$
$564$	0	0
$565$	0	0
$566$	6584.00	0.488951
$567$	0	0
$568$	−7200.00	−0.531876
$569$	1038.00	0.0764767	0.0382383	−	0.999269i	$-0.487825\pi$
$569$	1038.00	0.0764767	0.0382383	+	0.999269i	$0.487825\pi$
$570$	0	0
$571$	15671.0	1.14853	0.574265	−	0.818669i	$-0.305288\pi$
$571$	15671.0	1.14853	0.574265	+	0.818669i	$0.305288\pi$
$572$	13440.0	0.982438
$573$	0	0
$574$	−10608.0	−0.771375
$575$	0	0
$576$	0	0
$577$	916.000	0.0660894	0.0330447	−	0.999454i	$-0.489480\pi$
$577$	916.000	0.0660894	0.0330447	+	0.999454i	$0.489480\pi$
$578$	−8368.00	−0.602185
$579$	0	0
$580$	0	0
$581$	31110.0	2.22145
$582$	0	0
$583$	−30672.0	−2.17891
$584$	−5632.00	−0.399065
$585$	0	0
$586$	−15906.0	−1.12128
$587$	9141.00	0.642742	0.321371	−	0.946953i	$-0.395856\pi$
$587$	9141.00	0.642742	0.321371	+	0.946953i	$0.395856\pi$
$588$	0	0
$589$	−15827.0	−1.10720
$590$	0	0
$591$	0	0
$592$	−3488.00	−0.242155
$593$	5247.00	0.363353	0.181677	−	0.983358i	$-0.441848\pi$
$593$	5247.00	0.363353	0.181677	+	0.983358i	$0.441848\pi$
$594$	0	0
$595$	0	0
$596$	−1488.00	−0.102267
$597$	0	0
$598$	−7140.00	−0.488255
$599$	24162.0	1.64813	0.824067	−	0.566492i	$-0.191700\pi$
$599$	24162.0	1.64813	0.824067	+	0.566492i	$0.191700\pi$
$600$	0	0
$601$	14357.0	0.974433	0.487217	−	0.873281i	$-0.338012\pi$
$601$	14357.0	0.974433	0.487217	+	0.873281i	$0.338012\pi$
$602$	5984.00	0.405132
$603$	0	0
$604$	−5824.00	−0.392343
$605$	0	0
$606$	0	0
$607$	−3152.00	−0.210767	−0.105384	−	0.994432i	$-0.533607\pi$
$607$	−3152.00	−0.210767	−0.105384	+	0.994432i	$0.533607\pi$
$608$	3808.00	0.254005
$609$	0	0
$610$	0	0
$611$	−36120.0	−2.39159
$612$	0	0
$613$	−4592.00	−0.302560	−0.151280	−	0.988491i	$-0.548339\pi$
$613$	−4592.00	−0.302560	−0.151280	+	0.988491i	$0.548339\pi$
$614$	10580.0	0.695397
$615$	0	0
$616$	13056.0	0.853963
$617$	−7359.00	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$617$	−7359.00	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$618$	0	0
$619$	−15712.0	−1.02022	−0.510112	−	0.860108i	$-0.670396\pi$
$619$	−15712.0	−1.02022	−0.510112	+	0.860108i	$0.670396\pi$
$620$	0	0
$621$	0	0
$622$	−10716.0	−0.690792
$623$	37944.0	2.44012
$624$	0	0
$625$	0	0
$626$	−11200.0	−0.715083
$627$	0	0
$628$	−3824.00	−0.242984
$629$	5886.00	0.373116
$630$	0	0
$631$	−3175.00	−0.200309	−0.100154	−	0.994972i	$-0.531934\pi$
$631$	−3175.00	−0.200309	−0.100154	+	0.994972i	$0.531934\pi$
$632$	−11000.0	−0.692336
$633$	0	0
$634$	14682.0	0.919711
$635$	0	0
$636$	0	0
$637$	56910.0	3.53981
$638$	2880.00	0.178715
$639$	0	0
$640$	0	0
$641$	−96.0000	−0.00591540	−0.00295770	−	0.999996i	$-0.500941\pi$
$641$	−96.0000	−0.00591540	−0.00295770	+	0.999996i	$0.500941\pi$
$642$	0	0
$643$	18070.0	1.10826	0.554130	−	0.832430i	$-0.313051\pi$
$643$	18070.0	1.10826	0.554130	+	0.832430i	$0.313051\pi$
$644$	−6936.00	−0.424405
$645$	0	0
$646$	−6426.00	−0.391374
$647$	1341.00	0.0814840	0.0407420	−	0.999170i	$-0.487028\pi$
$647$	1341.00	0.0814840	0.0407420	+	0.999170i	$0.487028\pi$
$648$	0	0
$649$	31392.0	1.89868
$650$	0	0
$651$	0	0
$652$	9784.00	0.587686
$653$	−24495.0	−1.46794	−0.733969	−	0.679183i	$-0.762334\pi$
$653$	−24495.0	−1.46794	−0.733969	+	0.679183i	$0.762334\pi$
$654$	0	0
$655$	0	0
$656$	−2496.00	−0.148556
$657$	0	0
$658$	−35088.0	−2.07883
$659$	12378.0	0.731682	0.365841	−	0.930677i	$-0.380781\pi$
$659$	12378.0	0.731682	0.365841	+	0.930677i	$0.380781\pi$
$660$	0	0
$661$	−24442.0	−1.43825	−0.719125	−	0.694880i	$-0.755457\pi$
$661$	−24442.0	−1.43825	−0.719125	+	0.694880i	$0.755457\pi$
$662$	760.000	0.0446197
$663$	0	0
$664$	7320.00	0.427818
$665$	0	0
$666$	0	0
$667$	−1530.00	−0.0888183
$668$	−12444.0	−0.720768
$669$	0	0
$670$	0	0
$671$	22128.0	1.27309
$672$	0	0
$673$	−2378.00	−0.136204	−0.0681019	−	0.997678i	$-0.521694\pi$
$673$	−2378.00	−0.136204	−0.0681019	+	0.997678i	$0.521694\pi$
$674$	−868.000	−0.0496055
$675$	0	0
$676$	10812.0	0.615157
$677$	5478.00	0.310985	0.155492	−	0.987837i	$-0.450304\pi$
$677$	5478.00	0.310985	0.155492	+	0.987837i	$0.450304\pi$
$678$	0	0
$679$	544.000	0.0307464
$680$	0	0
$681$	0	0
$682$	−12768.0	−0.716880
$683$	8595.00	0.481521	0.240760	−	0.970585i	$-0.422603\pi$
$683$	8595.00	0.481521	0.240760	+	0.970585i	$0.422603\pi$
$684$	0	0
$685$	0	0
$686$	31960.0	1.77877
$687$	0	0
$688$	1408.00	0.0780225
$689$	−44730.0	−2.47326
$690$	0	0
$691$	−31615.0	−1.74051	−0.870254	−	0.492603i	$-0.836046\pi$
$691$	−31615.0	−1.74051	−0.870254	+	0.492603i	$0.836046\pi$
$692$	−9588.00	−0.526707
$693$	0	0
$694$	16008.0	0.875584
$695$	0	0
$696$	0	0
$697$	4212.00	0.228897
$698$	2218.00	0.120276
$699$	0	0
$700$	0	0
$701$	−29790.0	−1.60507	−0.802534	−	0.596606i	$-0.796515\pi$
$701$	−29790.0	−1.60507	−0.802534	+	0.596606i	$0.796515\pi$
$702$	0	0
$703$	−25942.0	−1.39178
$704$	3072.00	0.164461
$705$	0	0
$706$	−15324.0	−0.816893
$707$	−11832.0	−0.629403
$708$	0	0
$709$	3818.00	0.202240	0.101120	−	0.994874i	$-0.467757\pi$
$709$	3818.00	0.202240	0.101120	+	0.994874i	$0.467757\pi$
$710$	0	0
$711$	0	0
$712$	8928.00	0.469931
$713$	6783.00	0.356277
$714$	0	0
$715$	0	0
$716$	2160.00	0.112742
$717$	0	0
$718$	16956.0	0.881326
$719$	28314.0	1.46861	0.734307	−	0.678817i	$-0.237507\pi$
$719$	28314.0	1.46861	0.734307	+	0.678817i	$0.237507\pi$
$720$	0	0
$721$	14008.0	0.723558
$722$	14604.0	0.752776
$723$	0	0
$724$	9332.00	0.479035
$725$	0	0
$726$	0	0
$727$	−56.0000	−0.00285684	−0.00142842	−	0.999999i	$-0.500455\pi$
$727$	−56.0000	−0.00285684	−0.00142842	+	0.999999i	$0.500455\pi$
$728$	19040.0	0.969326
$729$	0	0
$730$	0	0
$731$	−2376.00	−0.120218
$732$	0	0
$733$	34432.0	1.73503	0.867514	−	0.497413i	$-0.165717\pi$
$733$	34432.0	1.73503	0.867514	+	0.497413i	$0.165717\pi$
$734$	−26572.0	−1.33623
$735$	0	0
$736$	−1632.00	−0.0817341
$737$	−8736.00	−0.436628
$738$	0	0
$739$	−1051.00	−0.0523162	−0.0261581	−	0.999658i	$-0.508327\pi$
$739$	−1051.00	−0.0523162	−0.0261581	+	0.999658i	$0.508327\pi$
$740$	0	0
$741$	0	0
$742$	−43452.0	−2.14983
$743$	39144.0	1.93278	0.966389	−	0.257084i	$-0.0827618\pi$
$743$	39144.0	1.93278	0.966389	+	0.257084i	$0.0827618\pi$
$744$	0	0
$745$	0	0
$746$	−6160.00	−0.302324
$747$	0	0
$748$	−5184.00	−0.253403
$749$	−30600.0	−1.49279
$750$	0	0
$751$	−1735.00	−0.0843023	−0.0421512	−	0.999111i	$-0.513421\pi$
$751$	−1735.00	−0.0843023	−0.0421512	+	0.999111i	$0.513421\pi$
$752$	−8256.00	−0.400353
$753$	0	0
$754$	4200.00	0.202858
$755$	0	0
$756$	0	0
$757$	−6698.00	−0.321589	−0.160795	−	0.986988i	$-0.551406\pi$
$757$	−6698.00	−0.321589	−0.160795	+	0.986988i	$0.551406\pi$
$758$	20218.0	0.968801
$759$	0	0
$760$	0	0
$761$	−38766.0	−1.84660	−0.923302	−	0.384074i	$-0.874521\pi$
$761$	−38766.0	−1.84660	−0.923302	+	0.384074i	$0.874521\pi$
$762$	0	0
$763$	−3910.00	−0.185520
$764$	10920.0	0.517110
$765$	0	0
$766$	−17454.0	−0.823288
$767$	45780.0	2.15518
$768$	0	0
$769$	23501.0	1.10204	0.551019	−	0.834492i	$-0.314239\pi$
$769$	23501.0	1.10204	0.551019	+	0.834492i	$0.314239\pi$
$770$	0	0
$771$	0	0
$772$	18280.0	0.852217
$773$	−3591.00	−0.167088	−0.0835442	−	0.996504i	$-0.526624\pi$
$773$	−3591.00	−0.167088	−0.0835442	+	0.996504i	$0.526624\pi$
$774$	0	0
$775$	0	0
$776$	128.000	0.00592130
$777$	0	0
$778$	5424.00	0.249948
$779$	−18564.0	−0.853818
$780$	0	0
$781$	−43200.0	−1.97928
$782$	2754.00	0.125937
$783$	0	0
$784$	13008.0	0.592566
$785$	0	0
$786$	0	0
$787$	20716.0	0.938305	0.469152	−	0.883117i	$-0.344560\pi$
$787$	20716.0	0.938305	0.469152	+	0.883117i	$0.344560\pi$
$788$	2700.00	0.122060
$789$	0	0
$790$	0	0
$791$	32844.0	1.47636
$792$	0	0
$793$	32270.0	1.44507
$794$	17636.0	0.788260
$795$	0	0
$796$	−12448.0	−0.554281
$797$	−42981.0	−1.91024	−0.955122	−	0.296211i	$-0.904277\pi$
$797$	−42981.0	−1.91024	−0.955122	+	0.296211i	$0.904277\pi$
$798$	0	0
$799$	13932.0	0.616869
$800$	0	0
$801$	0	0
$802$	−6612.00	−0.291119
$803$	−33792.0	−1.48505
$804$	0	0
$805$	0	0
$806$	−18620.0	−0.813724
$807$	0	0
$808$	−2784.00	−0.121214
$809$	2268.00	0.0985644	0.0492822	−	0.998785i	$-0.484307\pi$
$809$	2268.00	0.0985644	0.0492822	+	0.998785i	$0.484307\pi$
$810$	0	0
$811$	11756.0	0.509012	0.254506	−	0.967071i	$-0.418087\pi$
$811$	11756.0	0.509012	0.254506	+	0.967071i	$0.418087\pi$
$812$	4080.00	0.176330
$813$	0	0
$814$	−20928.0	−0.901138
$815$	0	0
$816$	0	0
$817$	10472.0	0.448432
$818$	12802.0	0.547202
$819$	0	0
$820$	0	0
$821$	−8646.00	−0.367537	−0.183768	−	0.982970i	$-0.558830\pi$
$821$	−8646.00	−0.367537	−0.183768	+	0.982970i	$0.558830\pi$
$822$	0	0
$823$	−10784.0	−0.456752	−0.228376	−	0.973573i	$-0.573341\pi$
$823$	−10784.0	−0.456752	−0.228376	+	0.973573i	$0.573341\pi$
$824$	3296.00	0.139347
$825$	0	0
$826$	44472.0	1.87334
$827$	42597.0	1.79110	0.895552	−	0.444957i	$-0.146781\pi$
$827$	42597.0	1.79110	0.895552	+	0.444957i	$0.146781\pi$
$828$	0	0
$829$	−26458.0	−1.10847	−0.554237	−	0.832359i	$-0.686990\pi$
$829$	−26458.0	−1.10847	−0.554237	+	0.832359i	$0.686990\pi$
$830$	0	0
$831$	0	0
$832$	4480.00	0.186678
$833$	−21951.0	−0.913034
$834$	0	0
$835$	0	0
$836$	22848.0	0.945233
$837$	0	0
$838$	4512.00	0.185996
$839$	11496.0	0.473046	0.236523	−	0.971626i	$-0.423992\pi$
$839$	11496.0	0.473046	0.236523	+	0.971626i	$0.423992\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	3622.00	0.148245
$843$	0	0
$844$	9764.00	0.398212
$845$	0	0
$846$	0	0
$847$	33082.0	1.34204
$848$	−10224.0	−0.414025
$849$	0	0
$850$	0	0
$851$	11118.0	0.447850
$852$	0	0
$853$	−21548.0	−0.864935	−0.432467	−	0.901650i	$-0.642357\pi$
$853$	−21548.0	−0.864935	−0.432467	+	0.901650i	$0.642357\pi$
$854$	31348.0	1.25610
$855$	0	0
$856$	−7200.00	−0.287489
$857$	6261.00	0.249559	0.124779	−	0.992185i	$-0.460178\pi$
$857$	6261.00	0.249559	0.124779	+	0.992185i	$0.460178\pi$
$858$	0	0
$859$	−3355.00	−0.133261	−0.0666305	−	0.997778i	$-0.521225\pi$
$859$	−3355.00	−0.133261	−0.0666305	+	0.997778i	$0.521225\pi$
$860$	0	0
$861$	0	0
$862$	10908.0	0.431007
$863$	19701.0	0.777091	0.388546	−	0.921429i	$-0.372978\pi$
$863$	19701.0	0.777091	0.388546	+	0.921429i	$0.372978\pi$
$864$	0	0
$865$	0	0
$866$	−5980.00	−0.234652
$867$	0	0
$868$	−18088.0	−0.707312
$869$	−66000.0	−2.57641
$870$	0	0
$871$	−12740.0	−0.495612
$872$	−920.000	−0.0357284
$873$	0	0
$874$	−12138.0	−0.469764
$875$	0	0
$876$	0	0
$877$	−16292.0	−0.627300	−0.313650	−	0.949539i	$-0.601552\pi$
$877$	−16292.0	−0.627300	−0.313650	+	0.949539i	$0.601552\pi$
$878$	18742.0	0.720401
$879$	0	0
$880$	0	0
$881$	−9270.00	−0.354500	−0.177250	−	0.984166i	$-0.556720\pi$
$881$	−9270.00	−0.354500	−0.177250	+	0.984166i	$0.556720\pi$
$882$	0	0
$883$	−38486.0	−1.46677	−0.733384	−	0.679814i	$-0.762060\pi$
$883$	−38486.0	−1.46677	−0.733384	+	0.679814i	$0.762060\pi$
$884$	−7560.00	−0.287636
$885$	0	0
$886$	12342.0	0.467988
$887$	1893.00	0.0716581	0.0358290	−	0.999358i	$-0.488593\pi$
$887$	1893.00	0.0716581	0.0358290	+	0.999358i	$0.488593\pi$
$888$	0	0
$889$	−47804.0	−1.80348
$890$	0	0
$891$	0	0
$892$	13672.0	0.513198
$893$	−61404.0	−2.30102
$894$	0	0
$895$	0	0
$896$	4352.00	0.162266
$897$	0	0
$898$	−8244.00	−0.306354
$899$	−3990.00	−0.148024
$900$	0	0
$901$	17253.0	0.637936
$902$	−14976.0	−0.552823
$903$	0	0
$904$	7728.00	0.284325
$905$	0	0
$906$	0	0
$907$	−44876.0	−1.64287	−0.821435	−	0.570302i	$-0.806826\pi$
$907$	−44876.0	−1.64287	−0.821435	+	0.570302i	$0.806826\pi$
$908$	−17508.0	−0.639894
$909$	0	0
$910$	0	0
$911$	−23802.0	−0.865637	−0.432819	−	0.901481i	$-0.642481\pi$
$911$	−23802.0	−0.865637	−0.432819	+	0.901481i	$0.642481\pi$
$912$	0	0
$913$	43920.0	1.59205
$914$	−14152.0	−0.512152
$915$	0	0
$916$	16748.0	0.604115
$917$	−8364.00	−0.301204
$918$	0	0
$919$	−24784.0	−0.889607	−0.444803	−	0.895628i	$-0.646726\pi$
$919$	−24784.0	−0.889607	−0.444803	+	0.895628i	$0.646726\pi$
$920$	0	0
$921$	0	0
$922$	−1524.00	−0.0544363
$923$	−63000.0	−2.24666
$924$	0	0
$925$	0	0
$926$	−17644.0	−0.626153
$927$	0	0
$928$	960.000	0.0339586
$929$	−9060.00	−0.319967	−0.159983	−	0.987120i	$-0.551144\pi$
$929$	−9060.00	−0.319967	−0.159983	+	0.987120i	$0.551144\pi$
$930$	0	0
$931$	96747.0	3.40575
$932$	−4392.00	−0.154361
$933$	0	0
$934$	−9954.00	−0.348720
$935$	0	0
$936$	0	0
$937$	−6176.00	−0.215327	−0.107663	−	0.994187i	$-0.534337\pi$
$937$	−6176.00	−0.215327	−0.107663	+	0.994187i	$0.534337\pi$
$938$	−12376.0	−0.430800
$939$	0	0
$940$	0	0
$941$	4182.00	0.144877	0.0724385	−	0.997373i	$-0.476922\pi$
$941$	4182.00	0.144877	0.0724385	+	0.997373i	$0.476922\pi$
$942$	0	0
$943$	7956.00	0.274743
$944$	10464.0	0.360778
$945$	0	0
$946$	8448.00	0.290347
$947$	44079.0	1.51254	0.756270	−	0.654260i	$-0.227020\pi$
$947$	44079.0	1.51254	0.756270	+	0.654260i	$0.227020\pi$
$948$	0	0
$949$	−49280.0	−1.68567
$950$	0	0
$951$	0	0
$952$	−7344.00	−0.250021
$953$	12726.0	0.432566	0.216283	−	0.976331i	$-0.430607\pi$
$953$	12726.0	0.432566	0.216283	+	0.976331i	$0.430607\pi$
$954$	0	0
$955$	0	0
$956$	−25896.0	−0.876084
$957$	0	0
$958$	−20208.0	−0.681514
$959$	17646.0	0.594180
$960$	0	0
$961$	−12102.0	−0.406230
$962$	−30520.0	−1.02287
$963$	0	0
$964$	13004.0	0.434472
$965$	0	0
$966$	0	0
$967$	−45218.0	−1.50374	−0.751868	−	0.659314i	$-0.770847\pi$
$967$	−45218.0	−1.50374	−0.751868	+	0.659314i	$0.770847\pi$
$968$	7784.00	0.258458
$969$	0	0
$970$	0	0
$971$	3978.00	0.131473	0.0657364	−	0.997837i	$-0.479060\pi$
$971$	3978.00	0.131473	0.0657364	+	0.997837i	$0.479060\pi$
$972$	0	0
$973$	44744.0	1.47423
$974$	−29848.0	−0.981922
$975$	0	0
$976$	7376.00	0.241906
$977$	23466.0	0.768417	0.384209	−	0.923246i	$-0.374474\pi$
$977$	23466.0	0.768417	0.384209	+	0.923246i	$0.374474\pi$
$978$	0	0
$979$	53568.0	1.74876
$980$	0	0
$981$	0	0
$982$	−2292.00	−0.0744813
$983$	47913.0	1.55462	0.777308	−	0.629120i	$-0.216585\pi$
$983$	47913.0	1.55462	0.777308	+	0.629120i	$0.216585\pi$
$984$	0	0
$985$	0	0
$986$	−1620.00	−0.0523238
$987$	0	0
$988$	33320.0	1.07293
$989$	−4488.00	−0.144297
$990$	0	0
$991$	31997.0	1.02565	0.512825	−	0.858493i	$-0.328599\pi$
$991$	31997.0	1.02565	0.512825	+	0.858493i	$0.328599\pi$
$992$	−4256.00	−0.136218
$993$	0	0
$994$	−61200.0	−1.95286
$995$	0	0
$996$	0	0
$997$	45628.0	1.44940	0.724701	−	0.689064i	$-0.241978\pi$
$997$	45628.0	1.44940	0.724701	+	0.689064i	$0.241978\pi$
$998$	−29930.0	−0.949316
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.a.bb.1.1		1
3.2	odd	2		1350.4.a.n.1.1		1
5.2	odd	4		1350.4.c.r.649.2		2
5.3	odd	4		1350.4.c.r.649.1		2
5.4	even	2		270.4.a.a.1.1	✓	1
15.2	even	4		1350.4.c.c.649.1		2
15.8	even	4		1350.4.c.c.649.2		2
15.14	odd	2		270.4.a.k.1.1	yes	1
20.19	odd	2		2160.4.a.j.1.1		1
45.4	even	6		810.4.e.x.541.1		2
45.14	odd	6		810.4.e.d.541.1		2
45.29	odd	6		810.4.e.d.271.1		2
45.34	even	6		810.4.e.x.271.1		2
60.59	even	2		2160.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.a.1.1	✓	1	5.4	even	2
270.4.a.k.1.1	yes	1	15.14	odd	2
810.4.e.d.271.1		2	45.29	odd	6
810.4.e.d.541.1		2	45.14	odd	6
810.4.e.x.271.1		2	45.34	even	6
810.4.e.x.541.1		2	45.4	even	6
1350.4.a.n.1.1		1	3.2	odd	2
1350.4.a.bb.1.1		1	1.1	even	1	trivial
1350.4.c.c.649.1		2	15.2	even	4
1350.4.c.c.649.2		2	15.8	even	4
1350.4.c.r.649.1		2	5.3	odd	4
1350.4.c.r.649.2		2	5.2	odd	4
2160.4.a.j.1.1		1	20.19	odd	2
2160.4.a.t.1.1		1	60.59	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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