LMFDB - Embedded newform 1350.4.a.bv.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,4,Mod(1,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,8,0,16,0,0,0,32,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29021904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 50x^{2} + 109 $ x^4 - 50*x^2 + 109
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.51141$ of defining polynomial
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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -22.9416 q^{7} +8.00000 q^{8} -22.9416 q^{11} -92.3072 q^{13} -45.8832 q^{14} +16.0000 q^{16} +112.684 q^{17} -74.6836 q^{19} -45.8832 q^{22} -54.6836 q^{23} -184.614 q^{26} -91.7663 q^{28} +276.921 q^{29} +270.367 q^{31} +32.0000 q^{32} +225.367 q^{34} -44.8016 q^{37} -149.367 q^{38} +228.875 q^{41} +322.805 q^{43} -91.7663 q^{44} -109.367 q^{46} +363.367 q^{47} +183.316 q^{49} -369.229 q^{52} -247.735 q^{53} -183.533 q^{56} +553.843 q^{58} +412.408 q^{59} -35.4181 q^{61} +540.735 q^{62} +64.0000 q^{64} +689.329 q^{67} +450.735 q^{68} -461.536 q^{71} -623.749 q^{73} -89.6031 q^{74} -298.735 q^{76} +526.316 q^{77} +1070.05 q^{79} +457.750 q^{82} +195.102 q^{83} +645.609 q^{86} -183.533 q^{88} -137.109 q^{89} +2117.67 q^{91} -218.735 q^{92} +726.735 q^{94} -25.6456 q^{97} +366.633 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} + 64 q^{16} + 110 q^{17} + 42 q^{19} + 122 q^{23} + 400 q^{31} + 128 q^{32} + 220 q^{34} + 84 q^{38} + 244 q^{46} + 772 q^{47} + 1074 q^{49} + 372 q^{53} + 1562 q^{61}+ \cdots + 2148 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 + 64 * q^16 + 110 * q^17 + 42 * q^19 + 122 * q^23 + 400 * q^31 + 128 * q^32 + 220 * q^34 + 84 * q^38 + 244 * q^46 + 772 * q^47 + 1074 * q^49 + 372 * q^53 + 1562 * q^61 + 800 * q^62 + 256 * q^64 + 440 * q^68 + 168 * q^76 + 2446 * q^77 + 3258 * q^79 - 1264 * q^83 + 1656 * q^91 + 488 * q^92 + 1544 * q^94 + 2148 * q^98

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
$7$	−22.9416	−1.23873	−0.619365	−	0.785103i	$-0.712610\pi$
$7$	−22.9416	−1.23873	−0.619365	+	0.785103i	$0.712610\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−22.9416	−0.628832	−0.314416	−	0.949285i	$-0.601809\pi$
$11$	−22.9416	−0.628832	−0.314416	+	0.949285i	$0.601809\pi$
$12$	0	0
$13$	−92.3072	−1.96934	−0.984669	−	0.174432i	$-0.944191\pi$
$13$	−92.3072	−1.96934	−0.984669	+	0.174432i	$0.944191\pi$
$14$	−45.8832	−0.875914
$15$	0	0
$16$	16.0000	0.250000
$17$	112.684	1.60763	0.803817	−	0.594876i	$-0.202799\pi$
$17$	112.684	1.60763	0.803817	+	0.594876i	$0.202799\pi$
$18$	0	0
$19$	−74.6836	−0.901768	−0.450884	−	0.892582i	$-0.648891\pi$
$19$	−74.6836	−0.901768	−0.450884	+	0.892582i	$0.648891\pi$
$20$	0	0
$21$	0	0
$22$	−45.8832	−0.444651
$23$	−54.6836	−0.495753	−0.247877	−	0.968792i	$-0.579733\pi$
$23$	−54.6836	−0.495753	−0.247877	+	0.968792i	$0.579733\pi$
$24$	0	0
$25$	0	0
$26$	−184.614	−1.39253
$27$	0	0
$28$	−91.7663	−0.619365
$29$	276.921	1.77321	0.886604	−	0.462530i	$-0.153058\pi$
$29$	276.921	1.77321	0.886604	+	0.462530i	$0.153058\pi$
$30$	0	0
$31$	270.367	1.56643	0.783216	−	0.621750i	$-0.213578\pi$
$31$	270.367	1.56643	0.783216	+	0.621750i	$0.213578\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	225.367	1.13677
$35$	0	0
$36$	0	0
$37$	−44.8016	−0.199063	−0.0995315	−	0.995034i	$-0.531734\pi$
$37$	−44.8016	−0.199063	−0.0995315	+	0.995034i	$0.531734\pi$
$38$	−149.367	−0.637647
$39$	0	0
$40$	0	0
$41$	228.875	0.871812	0.435906	−	0.899992i	$-0.356428\pi$
$41$	228.875	0.871812	0.435906	+	0.899992i	$0.356428\pi$
$42$	0	0
$43$	322.805	1.14482	0.572410	−	0.819968i	$-0.306009\pi$
$43$	322.805	1.14482	0.572410	+	0.819968i	$0.306009\pi$
$44$	−91.7663	−0.314416
$45$	0	0
$46$	−109.367	−0.350550
$47$	363.367	1.12771	0.563857	−	0.825872i	$-0.309317\pi$
$47$	363.367	1.12771	0.563857	+	0.825872i	$0.309317\pi$
$48$	0	0
$49$	183.316	0.534450
$50$	0	0
$51$	0	0
$52$	−369.229	−0.984669
$53$	−247.735	−0.642056	−0.321028	−	0.947070i	$-0.604028\pi$
$53$	−247.735	−0.642056	−0.321028	+	0.947070i	$0.604028\pi$
$54$	0	0
$55$	0	0
$56$	−183.533	−0.437957
$57$	0	0
$58$	553.843	1.25385
$59$	412.408	0.910016	0.455008	−	0.890487i	$-0.349636\pi$
$59$	412.408	0.910016	0.455008	+	0.890487i	$0.349636\pi$
$60$	0	0
$61$	−35.4181	−0.0743414	−0.0371707	−	0.999309i	$-0.511835\pi$
$61$	−35.4181	−0.0743414	−0.0371707	+	0.999309i	$0.511835\pi$
$62$	540.735	1.10763
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	689.329	1.25694	0.628470	−	0.777834i	$-0.283682\pi$
$67$	689.329	1.25694	0.628470	+	0.777834i	$0.283682\pi$
$68$	450.735	0.803817
$69$	0	0
$70$	0	0
$71$	−461.536	−0.771468	−0.385734	−	0.922610i	$-0.626052\pi$
$71$	−461.536	−0.771468	−0.385734	+	0.922610i	$0.626052\pi$
$72$	0	0
$73$	−623.749	−1.00006	−0.500030	−	0.866008i	$-0.666678\pi$
$73$	−623.749	−1.00006	−0.500030	+	0.866008i	$0.666678\pi$
$74$	−89.6031	−0.140759
$75$	0	0
$76$	−298.735	−0.450884
$77$	526.316	0.778952
$78$	0	0
$79$	1070.05	1.52393	0.761963	−	0.647621i	$-0.224236\pi$
$79$	1070.05	1.52393	0.761963	+	0.647621i	$0.224236\pi$
$80$	0	0
$81$	0	0
$82$	457.750	0.616464
$83$	195.102	0.258014	0.129007	−	0.991644i	$-0.458821\pi$
$83$	195.102	0.258014	0.129007	+	0.991644i	$0.458821\pi$
$84$	0	0
$85$	0	0
$86$	645.609	0.809510
$87$	0	0
$88$	−183.533	−0.222326
$89$	−137.109	−0.163298	−0.0816488	−	0.996661i	$-0.526019\pi$
$89$	−137.109	−0.163298	−0.0816488	+	0.996661i	$0.526019\pi$
$90$	0	0
$91$	2117.67	2.43948
$92$	−218.735	−0.247877
$93$	0	0
$94$	726.735	0.797414
$95$	0	0
$96$	0	0
$97$	−25.6456	−0.0268445	−0.0134223	−	0.999910i	$-0.504273\pi$
$97$	−25.6456	−0.0268445	−0.0134223	+	0.999910i	$0.504273\pi$
$98$	366.633	0.377913
$99$	0	0
$100$	0	0
$101$	−569.754	−0.561313	−0.280657	−	0.959808i	$-0.590552\pi$
$101$	−569.754	−0.561313	−0.280657	+	0.959808i	$0.590552\pi$
$102$	0	0
$103$	−1156.81	−1.10664	−0.553322	−	0.832968i	$-0.686640\pi$
$103$	−1156.81	−1.10664	−0.553322	+	0.832968i	$0.686640\pi$
$104$	−738.457	−0.696266
$105$	0	0
$106$	−495.469	−0.454002
$107$	−327.153	−0.295580	−0.147790	−	0.989019i	$-0.547216\pi$
$107$	−327.153	−0.295580	−0.147790	+	0.989019i	$0.547216\pi$
$108$	0	0
$109$	−372.254	−0.327115	−0.163557	−	0.986534i	$-0.552297\pi$
$109$	−372.254	−0.327115	−0.163557	+	0.986534i	$0.552297\pi$
$110$	0	0
$111$	0	0
$112$	−367.065	−0.309682
$113$	628.735	0.523419	0.261710	−	0.965147i	$-0.415714\pi$
$113$	628.735	0.523419	0.261710	+	0.965147i	$0.415714\pi$
$114$	0	0
$115$	0	0
$116$	1107.69	0.886604
$117$	0	0
$118$	824.815	0.643478
$119$	−2585.14	−1.99142
$120$	0	0
$121$	−804.684	−0.604571
$122$	−70.8363	−0.0525673
$123$	0	0
$124$	1081.47	0.783216
$125$	0	0
$126$	0	0
$127$	1368.16	0.955938	0.477969	−	0.878377i	$-0.341373\pi$
$127$	1368.16	0.955938	0.477969	+	0.878377i	$0.341373\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	1675.28	1.11733	0.558663	−	0.829395i	$-0.311315\pi$
$131$	1675.28	1.11733	0.558663	+	0.829395i	$0.311315\pi$
$132$	0	0
$133$	1713.36	1.11705
$134$	1378.66	0.888791
$135$	0	0
$136$	901.469	0.568385
$137$	−1012.25	−0.631261	−0.315630	−	0.948882i	$-0.602216\pi$
$137$	−1012.25	−0.631261	−0.315630	+	0.948882i	$0.602216\pi$
$138$	0	0
$139$	−158.938	−0.0969852	−0.0484926	−	0.998824i	$-0.515442\pi$
$139$	−158.938	−0.0969852	−0.0484926	+	0.998824i	$0.515442\pi$
$140$	0	0
$141$	0	0
$142$	−923.072	−0.545510
$143$	2117.67	1.23838
$144$	0	0
$145$	0	0
$146$	−1247.50	−0.707149
$147$	0	0
$148$	−179.206	−0.0995315
$149$	996.223	0.547743	0.273872	−	0.961766i	$-0.411696\pi$
$149$	996.223	0.547743	0.273872	+	0.961766i	$0.411696\pi$
$150$	0	0
$151$	242.113	0.130483	0.0652413	−	0.997870i	$-0.479218\pi$
$151$	242.113	0.130483	0.0652413	+	0.997870i	$0.479218\pi$
$152$	−597.469	−0.318823
$153$	0	0
$154$	1052.63	0.550802
$155$	0	0
$156$	0	0
$157$	−1470.97	−0.747744	−0.373872	−	0.927480i	$-0.621970\pi$
$157$	−1470.97	−0.747744	−0.373872	+	0.927480i	$0.621970\pi$
$158$	2140.10	1.07758
$159$	0	0
$160$	0	0
$161$	1254.53	0.614104
$162$	0	0
$163$	−1381.90	−0.664043	−0.332022	−	0.943272i	$-0.607731\pi$
$163$	−1381.90	−0.664043	−0.332022	+	0.943272i	$0.607731\pi$
$164$	915.500	0.435906
$165$	0	0
$166$	390.204	0.182444
$167$	3313.72	1.53547	0.767735	−	0.640767i	$-0.221384\pi$
$167$	3313.72	1.53547	0.767735	+	0.640767i	$0.221384\pi$
$168$	0	0
$169$	6323.61	2.87829
$170$	0	0
$171$	0	0
$172$	1291.22	0.572410
$173$	−1739.63	−0.764519	−0.382260	−	0.924055i	$-0.624854\pi$
$173$	−1739.63	−0.764519	−0.382260	+	0.924055i	$0.624854\pi$
$174$	0	0
$175$	0	0
$176$	−367.065	−0.157208
$177$	0	0
$178$	−274.217	−0.115469
$179$	−942.227	−0.393438	−0.196719	−	0.980460i	$-0.563029\pi$
$179$	−942.227	−0.393438	−0.196719	+	0.980460i	$0.563029\pi$
$180$	0	0
$181$	−1253.09	−0.514594	−0.257297	−	0.966332i	$-0.582832\pi$
$181$	−1253.09	−0.514594	−0.257297	+	0.966332i	$0.582832\pi$
$182$	4235.35	1.72497
$183$	0	0
$184$	−437.469	−0.175275
$185$	0	0
$186$	0	0
$187$	−2585.14	−1.01093
$188$	1453.47	0.563857
$189$	0	0
$190$	0	0
$191$	2021.02	0.765634	0.382817	−	0.923824i	$-0.374954\pi$
$191$	2021.02	0.765634	0.382817	+	0.923824i	$0.374954\pi$
$192$	0	0
$193$	2377.04	0.886546	0.443273	−	0.896387i	$-0.353817\pi$
$193$	2377.04	0.886546	0.443273	+	0.896387i	$0.353817\pi$
$194$	−51.2913	−0.0189819
$195$	0	0
$196$	733.265	0.267225
$197$	−2616.90	−0.946428	−0.473214	−	0.880948i	$-0.656906\pi$
$197$	−2616.90	−0.946428	−0.473214	+	0.880948i	$0.656906\pi$
$198$	0	0
$199$	3354.32	1.19488	0.597440	−	0.801913i	$-0.296185\pi$
$199$	3354.32	1.19488	0.597440	+	0.801913i	$0.296185\pi$
$200$	0	0
$201$	0	0
$202$	−1139.51	−0.396908
$203$	−6353.02	−2.19652
$204$	0	0
$205$	0	0
$206$	−2313.63	−0.782515
$207$	0	0
$208$	−1476.91	−0.492335
$209$	1713.36	0.567061
$210$	0	0
$211$	−457.113	−0.149142	−0.0745710	−	0.997216i	$-0.523759\pi$
$211$	−457.113	−0.149142	−0.0745710	+	0.997216i	$0.523759\pi$
$212$	−990.938	−0.321028
$213$	0	0
$214$	−654.305	−0.209006
$215$	0	0
$216$	0	0
$217$	−6202.65	−1.94039
$218$	−744.509	−0.231305
$219$	0	0
$220$	0	0
$221$	−10401.5	−3.16598
$222$	0	0
$223$	−1256.15	−0.377211	−0.188606	−	0.982053i	$-0.560397\pi$
$223$	−1256.15	−0.377211	−0.188606	+	0.982053i	$0.560397\pi$
$224$	−734.131	−0.218978
$225$	0	0
$226$	1257.47	0.370113
$227$	−4313.09	−1.26110	−0.630550	−	0.776148i	$-0.717171\pi$
$227$	−4313.09	−1.26110	−0.630550	+	0.776148i	$0.717171\pi$
$228$	0	0
$229$	6814.33	1.96639	0.983196	−	0.182552i	$-0.0584358\pi$
$229$	6814.33	1.96639	0.983196	+	0.182552i	$0.0584358\pi$
$230$	0	0
$231$	0	0
$232$	2215.37	0.626924
$233$	−5086.20	−1.43008	−0.715039	−	0.699084i	$-0.753591\pi$
$233$	−5086.20	−1.43008	−0.715039	+	0.699084i	$0.753591\pi$
$234$	0	0
$235$	0	0
$236$	1649.63	0.455008
$237$	0	0
$238$	−5170.28	−1.40815
$239$	1905.01	0.515584	0.257792	−	0.966200i	$-0.417005\pi$
$239$	1905.01	0.515584	0.257792	+	0.966200i	$0.417005\pi$
$240$	0	0
$241$	5688.05	1.52033	0.760165	−	0.649730i	$-0.225118\pi$
$241$	5688.05	1.52033	0.760165	+	0.649730i	$0.225118\pi$
$242$	−1609.37	−0.427496
$243$	0	0
$244$	−141.673	−0.0371707
$245$	0	0
$246$	0	0
$247$	6893.83	1.77589
$248$	2162.94	0.553817
$249$	0	0
$250$	0	0
$251$	1595.09	0.401121	0.200561	−	0.979681i	$-0.435724\pi$
$251$	1595.09	0.401121	0.200561	+	0.979681i	$0.435724\pi$
$252$	0	0
$253$	1254.53	0.311745
$254$	2736.31	0.675950
$255$	0	0
$256$	256.000	0.0625000
$257$	4413.60	1.07126	0.535628	−	0.844454i	$-0.320075\pi$
$257$	4413.60	1.07126	0.535628	+	0.844454i	$0.320075\pi$
$258$	0	0
$259$	1027.82	0.246585
$260$	0	0
$261$	0	0
$262$	3350.55	0.790068
$263$	−2312.43	−0.542169	−0.271085	−	0.962556i	$-0.587382\pi$
$263$	−2312.43	−0.542169	−0.271085	+	0.962556i	$0.587382\pi$
$264$	0	0
$265$	0	0
$266$	3426.72	0.789871
$267$	0	0
$268$	2757.32	0.628470
$269$	−1671.26	−0.378806	−0.189403	−	0.981899i	$-0.560655\pi$
$269$	−1671.26	−0.378806	−0.189403	+	0.981899i	$0.560655\pi$
$270$	0	0
$271$	1727.43	0.387210	0.193605	−	0.981080i	$-0.437982\pi$
$271$	1727.43	0.387210	0.193605	+	0.981080i	$0.437982\pi$
$272$	1802.94	0.401909
$273$	0	0
$274$	−2024.51	−0.446369
$275$	0	0
$276$	0	0
$277$	7181.03	1.55764	0.778820	−	0.627248i	$-0.215819\pi$
$277$	7181.03	1.55764	0.778820	+	0.627248i	$0.215819\pi$
$278$	−317.876	−0.0685789
$279$	0	0
$280$	0	0
$281$	4134.35	0.877704	0.438852	−	0.898559i	$-0.355385\pi$
$281$	4134.35	0.877704	0.438852	+	0.898559i	$0.355385\pi$
$282$	0	0
$283$	5027.77	1.05608	0.528038	−	0.849221i	$-0.322928\pi$
$283$	5027.77	1.05608	0.528038	+	0.849221i	$0.322928\pi$
$284$	−1846.14	−0.385734
$285$	0	0
$286$	4235.35	0.875669
$287$	−5250.76	−1.07994
$288$	0	0
$289$	7784.60	1.58449
$290$	0	0
$291$	0	0
$292$	−2495.00	−0.500030
$293$	−135.644	−0.0270457	−0.0135229	−	0.999909i	$-0.504305\pi$
$293$	−135.644	−0.0270457	−0.0135229	+	0.999909i	$0.504305\pi$
$294$	0	0
$295$	0	0
$296$	−358.412	−0.0703794
$297$	0	0
$298$	1992.45	0.387313
$299$	5047.69	0.976306
$300$	0	0
$301$	−7405.65	−1.41812
$302$	484.226	0.0922651
$303$	0	0
$304$	−1194.94	−0.225442
$305$	0	0
$306$	0	0
$307$	2594.33	0.482301	0.241151	−	0.970488i	$-0.422475\pi$
$307$	2594.33	0.482301	0.241151	+	0.970488i	$0.422475\pi$
$308$	2105.27	0.389476
$309$	0	0
$310$	0	0
$311$	−518.235	−0.0944901	−0.0472450	−	0.998883i	$-0.515044\pi$
$311$	−518.235	−0.0944901	−0.0472450	+	0.998883i	$0.515044\pi$
$312$	0	0
$313$	−682.071	−0.123172	−0.0615862	−	0.998102i	$-0.519616\pi$
$313$	−682.071	−0.123172	−0.0615862	+	0.998102i	$0.519616\pi$
$314$	−2941.93	−0.528735
$315$	0	0
$316$	4280.20	0.761963
$317$	−1901.43	−0.336892	−0.168446	−	0.985711i	$-0.553875\pi$
$317$	−1901.43	−0.336892	−0.168446	+	0.985711i	$0.553875\pi$
$318$	0	0
$319$	−6353.02	−1.11505
$320$	0	0
$321$	0	0
$322$	2509.06	0.434237
$323$	−8415.62	−1.44971
$324$	0	0
$325$	0	0
$326$	−2763.81	−0.469549
$327$	0	0
$328$	1831.00	0.308232
$329$	−8336.22	−1.39693
$330$	0	0
$331$	11227.1	1.86435	0.932173	−	0.362014i	$-0.117911\pi$
$331$	11227.1	1.86435	0.932173	+	0.362014i	$0.117911\pi$
$332$	780.407	0.129007
$333$	0	0
$334$	6627.45	1.08574
$335$	0	0
$336$	0	0
$337$	−6467.45	−1.04541	−0.522707	−	0.852512i	$-0.675078\pi$
$337$	−6467.45	−1.04541	−0.522707	+	0.852512i	$0.675078\pi$
$338$	12647.2	2.03526
$339$	0	0
$340$	0	0
$341$	−6202.65	−0.985022
$342$	0	0
$343$	3663.40	0.576690
$344$	2582.44	0.404755
$345$	0	0
$346$	−3479.27	−0.540597
$347$	−2046.90	−0.316667	−0.158334	−	0.987386i	$-0.550612\pi$
$347$	−2046.90	−0.316667	−0.158334	+	0.987386i	$0.550612\pi$
$348$	0	0
$349$	966.073	0.148174	0.0740870	−	0.997252i	$-0.476396\pi$
$349$	966.073	0.148174	0.0740870	+	0.997252i	$0.476396\pi$
$350$	0	0
$351$	0	0
$352$	−734.131	−0.111163
$353$	1787.88	0.269572	0.134786	−	0.990875i	$-0.456965\pi$
$353$	1787.88	0.269572	0.134786	+	0.990875i	$0.456965\pi$
$354$	0	0
$355$	0	0
$356$	−548.435	−0.0816488
$357$	0	0
$358$	−1884.45	−0.278203
$359$	9893.00	1.45441	0.727204	−	0.686421i	$-0.240819\pi$
$359$	9893.00	1.45441	0.727204	+	0.686421i	$0.240819\pi$
$360$	0	0
$361$	−1281.36	−0.186814
$362$	−2506.18	−0.363873
$363$	0	0
$364$	8470.69	1.21974
$365$	0	0
$366$	0	0
$367$	−9933.71	−1.41290	−0.706451	−	0.707762i	$-0.749705\pi$
$367$	−9933.71	−1.41290	−0.706451	+	0.707762i	$0.749705\pi$
$368$	−874.938	−0.123938
$369$	0	0
$370$	0	0
$371$	5683.42	0.795333
$372$	0	0
$373$	−3663.08	−0.508491	−0.254246	−	0.967140i	$-0.581827\pi$
$373$	−3663.08	−0.508491	−0.254246	+	0.967140i	$0.581827\pi$
$374$	−5170.28	−0.714837
$375$	0	0
$376$	2906.94	0.398707
$377$	−25561.8	−3.49205
$378$	0	0
$379$	−1754.60	−0.237805	−0.118902	−	0.992906i	$-0.537938\pi$
$379$	−1754.60	−0.237805	−0.118902	+	0.992906i	$0.537938\pi$
$380$	0	0
$381$	0	0
$382$	4042.05	0.541385
$383$	1779.83	0.237454	0.118727	−	0.992927i	$-0.462119\pi$
$383$	1779.83	0.237454	0.118727	+	0.992927i	$0.462119\pi$
$384$	0	0
$385$	0	0
$386$	4754.09	0.626883
$387$	0	0
$388$	−102.583	−0.0134223
$389$	−9870.69	−1.28654	−0.643270	−	0.765640i	$-0.722423\pi$
$389$	−9870.69	−1.28654	−0.643270	+	0.765640i	$0.722423\pi$
$390$	0	0
$391$	−6161.95	−0.796990
$392$	1466.53	0.188957
$393$	0	0
$394$	−5233.80	−0.669226
$395$	0	0
$396$	0	0
$397$	−8433.48	−1.06616	−0.533078	−	0.846066i	$-0.678965\pi$
$397$	−8433.48	−1.06616	−0.533078	+	0.846066i	$0.678965\pi$
$398$	6708.63	0.844908
$399$	0	0
$400$	0	0
$401$	14407.5	1.79420	0.897102	−	0.441824i	$-0.145668\pi$
$401$	14407.5	1.79420	0.897102	+	0.441824i	$0.145668\pi$
$402$	0	0
$403$	−24956.8	−3.08483
$404$	−2279.02	−0.280657
$405$	0	0
$406$	−12706.0	−1.55318
$407$	1027.82	0.125177
$408$	0	0
$409$	−2569.33	−0.310624	−0.155312	−	0.987865i	$-0.549638\pi$
$409$	−2569.33	−0.310624	−0.155312	+	0.987865i	$0.549638\pi$
$410$	0	0
$411$	0	0
$412$	−4627.26	−0.553322
$413$	−9461.29	−1.12726
$414$	0	0
$415$	0	0
$416$	−2953.83	−0.348133
$417$	0	0
$418$	3426.72	0.400972
$419$	12267.6	1.43033	0.715167	−	0.698953i	$-0.246351\pi$
$419$	12267.6	1.43033	0.715167	+	0.698953i	$0.246351\pi$
$420$	0	0
$421$	−1203.72	−0.139349	−0.0696745	−	0.997570i	$-0.522196\pi$
$421$	−1203.72	−0.139349	−0.0696745	+	0.997570i	$0.522196\pi$
$422$	−914.226	−0.105459
$423$	0	0
$424$	−1981.88	−0.227001
$425$	0	0
$426$	0	0
$427$	812.548	0.0920889
$428$	−1308.61	−0.147790
$429$	0	0
$430$	0	0
$431$	6975.41	0.779568	0.389784	−	0.920906i	$-0.372550\pi$
$431$	6975.41	0.779568	0.389784	+	0.920906i	$0.372550\pi$
$432$	0	0
$433$	4595.66	0.510054	0.255027	−	0.966934i	$-0.417916\pi$
$433$	4595.66	0.510054	0.255027	+	0.966934i	$0.417916\pi$
$434$	−12405.3	−1.37206
$435$	0	0
$436$	−1489.02	−0.163557
$437$	4083.97	0.447055
$438$	0	0
$439$	−4709.81	−0.512044	−0.256022	−	0.966671i	$-0.582412\pi$
$439$	−4709.81	−0.512044	−0.256022	+	0.966671i	$0.582412\pi$
$440$	0	0
$441$	0	0
$442$	−20803.0	−2.23868
$443$	10640.9	1.14123	0.570617	−	0.821216i	$-0.306704\pi$
$443$	10640.9	1.14123	0.570617	+	0.821216i	$0.306704\pi$
$444$	0	0
$445$	0	0
$446$	−2512.30	−0.266729
$447$	0	0
$448$	−1468.26	−0.154841
$449$	−4074.41	−0.428248	−0.214124	−	0.976807i	$-0.568690\pi$
$449$	−4074.41	−0.428248	−0.214124	+	0.976807i	$0.568690\pi$
$450$	0	0
$451$	−5250.76	−0.548223
$452$	2514.94	0.261710
$453$	0	0
$454$	−8626.18	−0.891733
$455$	0	0
$456$	0	0
$457$	14066.1	1.43979	0.719894	−	0.694084i	$-0.244191\pi$
$457$	14066.1	1.43979	0.719894	+	0.694084i	$0.244191\pi$
$458$	13628.7	1.39045
$459$	0	0
$460$	0	0
$461$	−13134.3	−1.32696	−0.663479	−	0.748195i	$-0.730921\pi$
$461$	−13134.3	−1.32696	−0.663479	+	0.748195i	$0.730921\pi$
$462$	0	0
$463$	139.955	0.0140481	0.00702403	−	0.999975i	$-0.497764\pi$
$463$	139.955	0.0140481	0.00702403	+	0.999975i	$0.497764\pi$
$464$	4430.74	0.443302
$465$	0	0
$466$	−10172.4	−1.01122
$467$	−587.270	−0.0581919	−0.0290960	−	0.999577i	$-0.509263\pi$
$467$	−587.270	−0.0581919	−0.0290960	+	0.999577i	$0.509263\pi$
$468$	0	0
$469$	−15814.3	−1.55701
$470$	0	0
$471$	0	0
$472$	3299.26	0.321739
$473$	−7405.65	−0.719899
$474$	0	0
$475$	0	0
$476$	−10340.6	−0.995712
$477$	0	0
$478$	3810.01	0.364573
$479$	−15456.7	−1.47439	−0.737196	−	0.675679i	$-0.763851\pi$
$479$	−15456.7	−1.47439	−0.737196	+	0.675679i	$0.763851\pi$
$480$	0	0
$481$	4135.50	0.392022
$482$	11376.1	1.07504
$483$	0	0
$484$	−3218.73	−0.302285
$485$	0	0
$486$	0	0
$487$	18448.4	1.71658	0.858291	−	0.513164i	$-0.171527\pi$
$487$	18448.4	1.71658	0.858291	+	0.513164i	$0.171527\pi$
$488$	−283.345	−0.0262837
$489$	0	0
$490$	0	0
$491$	518.918	0.0476954	0.0238477	−	0.999716i	$-0.492408\pi$
$491$	518.918	0.0476954	0.0238477	+	0.999716i	$0.492408\pi$
$492$	0	0
$493$	31204.5	2.85067
$494$	13787.7	1.25574
$495$	0	0
$496$	4325.88	0.391608
$497$	10588.4	0.955640
$498$	0	0
$499$	−5853.01	−0.525084	−0.262542	−	0.964921i	$-0.584561\pi$
$499$	−5853.01	−0.525084	−0.262542	+	0.964921i	$0.584561\pi$
$500$	0	0
$501$	0	0
$502$	3190.19	0.283636
$503$	−15178.2	−1.34546	−0.672728	−	0.739890i	$-0.734877\pi$
$503$	−15178.2	−1.34546	−0.672728	+	0.739890i	$0.734877\pi$
$504$	0	0
$505$	0	0
$506$	2509.06	0.220437
$507$	0	0
$508$	5472.62	0.477969
$509$	5421.02	0.472068	0.236034	−	0.971745i	$-0.424152\pi$
$509$	5421.02	0.472068	0.236034	+	0.971745i	$0.424152\pi$
$510$	0	0
$511$	14309.8	1.23880
$512$	512.000	0.0441942
$513$	0	0
$514$	8827.20	0.757492
$515$	0	0
$516$	0	0
$517$	−8336.22	−0.709142
$518$	2055.64	0.174362
$519$	0	0
$520$	0	0
$521$	8494.96	0.714340	0.357170	−	0.934039i	$-0.383742\pi$
$521$	8494.96	0.714340	0.357170	+	0.934039i	$0.383742\pi$
$522$	0	0
$523$	−4363.31	−0.364808	−0.182404	−	0.983224i	$-0.558388\pi$
$523$	−4363.31	−0.364808	−0.182404	+	0.983224i	$0.558388\pi$
$524$	6701.11	0.558663
$525$	0	0
$526$	−4624.86	−0.383372
$527$	30466.0	2.51825
$528$	0	0
$529$	−9176.70	−0.754229
$530$	0	0
$531$	0	0
$532$	6853.44	0.558523
$533$	−21126.8	−1.71689
$534$	0	0
$535$	0	0
$536$	5514.63	0.444395
$537$	0	0
$538$	−3342.53	−0.267856
$539$	−4205.57	−0.336079
$540$	0	0
$541$	−16112.3	−1.28044	−0.640222	−	0.768190i	$-0.721158\pi$
$541$	−16112.3	−1.28044	−0.640222	+	0.768190i	$0.721158\pi$
$542$	3454.86	0.273799
$543$	0	0
$544$	3605.88	0.284192
$545$	0	0
$546$	0	0
$547$	9522.61	0.744346	0.372173	−	0.928163i	$-0.378613\pi$
$547$	9522.61	0.744346	0.372173	+	0.928163i	$0.378613\pi$
$548$	−4049.02	−0.315630
$549$	0	0
$550$	0	0
$551$	−20681.5	−1.59902
$552$	0	0
$553$	−24548.7	−1.88773
$554$	14362.1	1.10142
$555$	0	0
$556$	−635.752	−0.0484926
$557$	−21740.5	−1.65381	−0.826907	−	0.562339i	$-0.809902\pi$
$557$	−21740.5	−1.65381	−0.826907	+	0.562339i	$0.809902\pi$
$558$	0	0
$559$	−29797.2	−2.25454
$560$	0	0
$561$	0	0
$562$	8268.71	0.620630
$563$	6514.93	0.487693	0.243847	−	0.969814i	$-0.421591\pi$
$563$	6514.93	0.487693	0.243847	+	0.969814i	$0.421591\pi$
$564$	0	0
$565$	0	0
$566$	10055.5	0.746759
$567$	0	0
$568$	−3692.29	−0.272755
$569$	25108.1	1.84989	0.924944	−	0.380105i	$-0.124112\pi$
$569$	25108.1	1.84989	0.924944	+	0.380105i	$0.124112\pi$
$570$	0	0
$571$	1863.83	0.136600	0.0683000	−	0.997665i	$-0.478242\pi$
$571$	1863.83	0.136600	0.0683000	+	0.997665i	$0.478242\pi$
$572$	8470.69	0.619191
$573$	0	0
$574$	−10501.5	−0.763632
$575$	0	0
$576$	0	0
$577$	20829.5	1.50285	0.751424	−	0.659819i	$-0.229367\pi$
$577$	20829.5	1.50285	0.751424	+	0.659819i	$0.229367\pi$
$578$	15569.2	1.12040
$579$	0	0
$580$	0	0
$581$	−4475.94	−0.319610
$582$	0	0
$583$	5683.42	0.403745
$584$	−4989.99	−0.353574
$585$	0	0
$586$	−271.288	−0.0191242
$587$	15537.3	1.09249	0.546247	−	0.837624i	$-0.316056\pi$
$587$	15537.3	1.09249	0.546247	+	0.837624i	$0.316056\pi$
$588$	0	0
$589$	−20192.0	−1.41256
$590$	0	0
$591$	0	0
$592$	−716.825	−0.0497657
$593$	−4495.44	−0.311308	−0.155654	−	0.987812i	$-0.549748\pi$
$593$	−4495.44	−0.311308	−0.155654	+	0.987812i	$0.549748\pi$
$594$	0	0
$595$	0	0
$596$	3984.89	0.273872
$597$	0	0
$598$	10095.4	0.690352
$599$	−27270.9	−1.86020	−0.930100	−	0.367307i	$-0.880280\pi$
$599$	−27270.9	−1.86020	−0.930100	+	0.367307i	$0.880280\pi$
$600$	0	0
$601$	−22616.8	−1.53504	−0.767520	−	0.641025i	$-0.778509\pi$
$601$	−22616.8	−1.53504	−0.767520	+	0.641025i	$0.778509\pi$
$602$	−14811.3	−1.00276
$603$	0	0
$604$	968.451	0.0652413
$605$	0	0
$606$	0	0
$607$	−19720.3	−1.31865	−0.659326	−	0.751857i	$-0.729158\pi$
$607$	−19720.3	−1.31865	−0.659326	+	0.751857i	$0.729158\pi$
$608$	−2389.88	−0.159412
$609$	0	0
$610$	0	0
$611$	−33541.4	−2.22085
$612$	0	0
$613$	−5447.29	−0.358913	−0.179457	−	0.983766i	$-0.557434\pi$
$613$	−5447.29	−0.358913	−0.179457	+	0.983766i	$0.557434\pi$
$614$	5188.67	0.341039
$615$	0	0
$616$	4210.53	0.275401
$617$	12015.4	0.783993	0.391996	−	0.919967i	$-0.371785\pi$
$617$	12015.4	0.783993	0.391996	+	0.919967i	$0.371785\pi$
$618$	0	0
$619$	2316.36	0.150408	0.0752039	−	0.997168i	$-0.476039\pi$
$619$	2316.36	0.150408	0.0752039	+	0.997168i	$0.476039\pi$
$620$	0	0
$621$	0	0
$622$	−1036.47	−0.0668146
$623$	3145.49	0.202282
$624$	0	0
$625$	0	0
$626$	−1364.14	−0.0870960
$627$	0	0
$628$	−5883.86	−0.373872
$629$	−5048.40	−0.320021
$630$	0	0
$631$	7324.32	0.462087	0.231043	−	0.972943i	$-0.425786\pi$
$631$	7324.32	0.462087	0.231043	+	0.972943i	$0.425786\pi$
$632$	8560.41	0.538789
$633$	0	0
$634$	−3802.86	−0.238219
$635$	0	0
$636$	0	0
$637$	−16921.4	−1.05251
$638$	−12706.0	−0.788459
$639$	0	0
$640$	0	0
$641$	−90.2295	−0.00555983	−0.00277991	−	0.999996i	$-0.500885\pi$
$641$	−90.2295	−0.00555983	−0.00277991	+	0.999996i	$0.500885\pi$
$642$	0	0
$643$	11101.0	0.680842	0.340421	−	0.940273i	$-0.389430\pi$
$643$	11101.0	0.680842	0.340421	+	0.940273i	$0.389430\pi$
$644$	5018.12	0.307052
$645$	0	0
$646$	−16831.2	−1.02510
$647$	−5204.38	−0.316237	−0.158118	−	0.987420i	$-0.550543\pi$
$647$	−5204.38	−0.316237	−0.158118	+	0.987420i	$0.550543\pi$
$648$	0	0
$649$	−9461.29	−0.572247
$650$	0	0
$651$	0	0
$652$	−5527.61	−0.332022
$653$	18852.8	1.12981	0.564907	−	0.825155i	$-0.308912\pi$
$653$	18852.8	1.12981	0.564907	+	0.825155i	$0.308912\pi$
$654$	0	0
$655$	0	0
$656$	3662.00	0.217953
$657$	0	0
$658$	−16672.4	−0.987780
$659$	13402.4	0.792238	0.396119	−	0.918199i	$-0.370357\pi$
$659$	13402.4	0.792238	0.396119	+	0.918199i	$0.370357\pi$
$660$	0	0
$661$	−15928.1	−0.937267	−0.468633	−	0.883393i	$-0.655253\pi$
$661$	−15928.1	−0.937267	−0.468633	+	0.883393i	$0.655253\pi$
$662$	22454.2	1.31829
$663$	0	0
$664$	1560.81	0.0912219
$665$	0	0
$666$	0	0
$667$	−15143.1	−0.879073
$668$	13254.9	0.767735
$669$	0	0
$670$	0	0
$671$	812.548	0.0467482
$672$	0	0
$673$	355.395	0.0203558	0.0101779	−	0.999948i	$-0.496760\pi$
$673$	355.395	0.0203558	0.0101779	+	0.999948i	$0.496760\pi$
$674$	−12934.9	−0.739219
$675$	0	0
$676$	25294.4	1.43915
$677$	18243.7	1.03569	0.517846	−	0.855474i	$-0.326734\pi$
$677$	18243.7	1.03569	0.517846	+	0.855474i	$0.326734\pi$
$678$	0	0
$679$	588.351	0.0332531
$680$	0	0
$681$	0	0
$682$	−12405.3	−0.696516
$683$	−5682.31	−0.318342	−0.159171	−	0.987251i	$-0.550882\pi$
$683$	−5682.31	−0.318342	−0.159171	+	0.987251i	$0.550882\pi$
$684$	0	0
$685$	0	0
$686$	7326.79	0.407782
$687$	0	0
$688$	5164.87	0.286205
$689$	22867.7	1.26442
$690$	0	0
$691$	−24469.4	−1.34712	−0.673559	−	0.739133i	$-0.735235\pi$
$691$	−24469.4	−1.34712	−0.673559	+	0.739133i	$0.735235\pi$
$692$	−6958.53	−0.382260
$693$	0	0
$694$	−4093.81	−0.223918
$695$	0	0
$696$	0	0
$697$	25790.5	1.40155
$698$	1932.15	0.104775
$699$	0	0
$700$	0	0
$701$	25286.9	1.36244	0.681222	−	0.732077i	$-0.261449\pi$
$701$	25286.9	1.36244	0.681222	+	0.732077i	$0.261449\pi$
$702$	0	0
$703$	3345.94	0.179509
$704$	−1468.26	−0.0786040
$705$	0	0
$706$	3575.75	0.190616
$707$	13071.1	0.695315
$708$	0	0
$709$	32374.8	1.71490	0.857448	−	0.514570i	$-0.172048\pi$
$709$	32374.8	1.71490	0.857448	+	0.514570i	$0.172048\pi$
$710$	0	0
$711$	0	0
$712$	−1096.87	−0.0577345
$713$	−14784.7	−0.776564
$714$	0	0
$715$	0	0
$716$	−3768.91	−0.196719
$717$	0	0
$718$	19786.0	1.02842
$719$	−22565.9	−1.17047	−0.585233	−	0.810865i	$-0.698997\pi$
$719$	−22565.9	−1.17047	−0.585233	+	0.810865i	$0.698997\pi$
$720$	0	0
$721$	26539.1	1.37083
$722$	−2562.71	−0.132097
$723$	0	0
$724$	−5012.36	−0.257297
$725$	0	0
$726$	0	0
$727$	−24371.4	−1.24331	−0.621654	−	0.783292i	$-0.713539\pi$
$727$	−24371.4	−1.24331	−0.621654	+	0.783292i	$0.713539\pi$
$728$	16941.4	0.862485
$729$	0	0
$730$	0	0
$731$	36374.8	1.84045
$732$	0	0
$733$	−13467.4	−0.678621	−0.339311	−	0.940674i	$-0.610194\pi$
$733$	−13467.4	−0.678621	−0.339311	+	0.940674i	$0.610194\pi$
$734$	−19867.4	−0.999073
$735$	0	0
$736$	−1749.88	−0.0876376
$737$	−15814.3	−0.790404
$738$	0	0
$739$	−16009.5	−0.796911	−0.398456	−	0.917188i	$-0.630454\pi$
$739$	−16009.5	−0.796911	−0.398456	+	0.917188i	$0.630454\pi$
$740$	0	0
$741$	0	0
$742$	11366.8	0.562385
$743$	10714.7	0.529052	0.264526	−	0.964379i	$-0.414785\pi$
$743$	10714.7	0.529052	0.264526	+	0.964379i	$0.414785\pi$
$744$	0	0
$745$	0	0
$746$	−7326.16	−0.359558
$747$	0	0
$748$	−10340.6	−0.505466
$749$	7505.40	0.366143
$750$	0	0
$751$	4772.75	0.231904	0.115952	−	0.993255i	$-0.463008\pi$
$751$	4772.75	0.231904	0.115952	+	0.993255i	$0.463008\pi$
$752$	5813.88	0.281929
$753$	0	0
$754$	−51123.7	−2.46925
$755$	0	0
$756$	0	0
$757$	−24326.3	−1.16797	−0.583985	−	0.811764i	$-0.698507\pi$
$757$	−24326.3	−1.16797	−0.583985	+	0.811764i	$0.698507\pi$
$758$	−3509.21	−0.168153
$759$	0	0
$760$	0	0
$761$	13038.2	0.621070	0.310535	−	0.950562i	$-0.399492\pi$
$761$	13038.2	0.621070	0.310535	+	0.950562i	$0.399492\pi$
$762$	0	0
$763$	8540.11	0.405207
$764$	8084.09	0.382817
$765$	0	0
$766$	3559.65	0.167905
$767$	−38068.2	−1.79213
$768$	0	0
$769$	−21815.7	−1.02301	−0.511505	−	0.859280i	$-0.670912\pi$
$769$	−21815.7	−1.02301	−0.511505	+	0.859280i	$0.670912\pi$
$770$	0	0
$771$	0	0
$772$	9508.18	0.443273
$773$	23946.0	1.11420	0.557100	−	0.830446i	$-0.311914\pi$
$773$	23946.0	1.11420	0.557100	+	0.830446i	$0.311914\pi$
$774$	0	0
$775$	0	0
$776$	−205.165	−0.00949097
$777$	0	0
$778$	−19741.4	−0.909721
$779$	−17093.2	−0.786172
$780$	0	0
$781$	10588.4	0.485123
$782$	−12323.9	−0.563557
$783$	0	0
$784$	2933.06	0.133613
$785$	0	0
$786$	0	0
$787$	13567.9	0.614540	0.307270	−	0.951622i	$-0.400584\pi$
$787$	13567.9	0.614540	0.307270	+	0.951622i	$0.400584\pi$
$788$	−10467.6	−0.473214
$789$	0	0
$790$	0	0
$791$	−14424.2	−0.648375
$792$	0	0
$793$	3269.35	0.146403
$794$	−16867.0	−0.753887
$795$	0	0
$796$	13417.3	0.597440
$797$	−39913.7	−1.77392	−0.886962	−	0.461843i	$-0.847188\pi$
$797$	−39913.7	−1.77392	−0.886962	+	0.461843i	$0.847188\pi$
$798$	0	0
$799$	40945.5	1.81295
$800$	0	0
$801$	0	0
$802$	28815.0	1.26869
$803$	14309.8	0.628869
$804$	0	0
$805$	0	0
$806$	−49913.7	−2.18131
$807$	0	0
$808$	−4558.03	−0.198454
$809$	13891.2	0.603696	0.301848	−	0.953356i	$-0.402396\pi$
$809$	13891.2	0.603696	0.301848	+	0.953356i	$0.402396\pi$
$810$	0	0
$811$	−24237.7	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$811$	−24237.7	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$812$	−25412.1	−1.09826
$813$	0	0
$814$	2055.64	0.0885136
$815$	0	0
$816$	0	0
$817$	−24108.2	−1.03236
$818$	−5138.65	−0.219644
$819$	0	0
$820$	0	0
$821$	19978.0	0.849254	0.424627	−	0.905368i	$-0.360405\pi$
$821$	19978.0	0.849254	0.424627	+	0.905368i	$0.360405\pi$
$822$	0	0
$823$	42871.5	1.81580	0.907902	−	0.419182i	$-0.137683\pi$
$823$	42871.5	1.81580	0.907902	+	0.419182i	$0.137683\pi$
$824$	−9254.51	−0.391257
$825$	0	0
$826$	−18922.6	−0.797095
$827$	360.143	0.0151432	0.00757159	−	0.999971i	$-0.497590\pi$
$827$	360.143	0.0151432	0.00757159	+	0.999971i	$0.497590\pi$
$828$	0	0
$829$	−15020.5	−0.629294	−0.314647	−	0.949209i	$-0.601886\pi$
$829$	−15020.5	−0.629294	−0.314647	+	0.949209i	$0.601886\pi$
$830$	0	0
$831$	0	0
$832$	−5907.66	−0.246167
$833$	20656.8	0.859201
$834$	0	0
$835$	0	0
$836$	6853.44	0.283530
$837$	0	0
$838$	24535.1	1.01140
$839$	−29440.2	−1.21143	−0.605714	−	0.795682i	$-0.707113\pi$
$839$	−29440.2	−1.21143	−0.605714	+	0.795682i	$0.707113\pi$
$840$	0	0
$841$	52296.5	2.14427
$842$	−2407.45	−0.0985346
$843$	0	0
$844$	−1828.45	−0.0745710
$845$	0	0
$846$	0	0
$847$	18460.7	0.748899
$848$	−3963.75	−0.160514
$849$	0	0
$850$	0	0
$851$	2449.91	0.0986861
$852$	0	0
$853$	−14688.6	−0.589601	−0.294801	−	0.955559i	$-0.595253\pi$
$853$	−14688.6	−0.589601	−0.294801	+	0.955559i	$0.595253\pi$
$854$	1625.10	0.0651167
$855$	0	0
$856$	−2617.22	−0.104503
$857$	−7812.22	−0.311389	−0.155694	−	0.987805i	$-0.549762\pi$
$857$	−7812.22	−0.311389	−0.155694	+	0.987805i	$0.549762\pi$
$858$	0	0
$859$	4035.68	0.160298	0.0801488	−	0.996783i	$-0.474460\pi$
$859$	4035.68	0.160298	0.0801488	+	0.996783i	$0.474460\pi$
$860$	0	0
$861$	0	0
$862$	13950.8	0.551238
$863$	−32201.0	−1.27015	−0.635073	−	0.772452i	$-0.719030\pi$
$863$	−32201.0	−1.27015	−0.635073	+	0.772452i	$0.719030\pi$
$864$	0	0
$865$	0	0
$866$	9191.32	0.360663
$867$	0	0
$868$	−24810.6	−0.970193
$869$	−24548.7	−0.958293
$870$	0	0
$871$	−63630.0	−2.47534
$872$	−2978.04	−0.115653
$873$	0	0
$874$	8167.94	0.316115
$875$	0	0
$876$	0	0
$877$	−18350.3	−0.706551	−0.353275	−	0.935519i	$-0.614932\pi$
$877$	−18350.3	−0.706551	−0.353275	+	0.935519i	$0.614932\pi$
$878$	−9419.63	−0.362070
$879$	0	0
$880$	0	0
$881$	32914.7	1.25871	0.629356	−	0.777117i	$-0.283319\pi$
$881$	32914.7	1.25871	0.629356	+	0.777117i	$0.283319\pi$
$882$	0	0
$883$	−27122.2	−1.03367	−0.516837	−	0.856084i	$-0.672891\pi$
$883$	−27122.2	−1.03367	−0.516837	+	0.856084i	$0.672891\pi$
$884$	−41606.0	−1.58299
$885$	0	0
$886$	21281.9	0.806974
$887$	35660.0	1.34988	0.674941	−	0.737872i	$-0.264169\pi$
$887$	35660.0	1.34988	0.674941	+	0.737872i	$0.264169\pi$
$888$	0	0
$889$	−31387.7	−1.18415
$890$	0	0
$891$	0	0
$892$	−5024.61	−0.188606
$893$	−27137.6	−1.01694
$894$	0	0
$895$	0	0
$896$	−2936.52	−0.109489
$897$	0	0
$898$	−8148.82	−0.302817
$899$	74870.5	2.77761
$900$	0	0
$901$	−27915.6	−1.03219
$902$	−10501.5	−0.387652
$903$	0	0
$904$	5029.88	0.185057
$905$	0	0
$906$	0	0
$907$	13630.5	0.499002	0.249501	−	0.968375i	$-0.419733\pi$
$907$	13630.5	0.499002	0.249501	+	0.968375i	$0.419733\pi$
$908$	−17252.4	−0.630550
$909$	0	0
$910$	0	0
$911$	−36521.1	−1.32821	−0.664104	−	0.747640i	$-0.731187\pi$
$911$	−36521.1	−1.32821	−0.664104	+	0.747640i	$0.731187\pi$
$912$	0	0
$913$	−4475.94	−0.162248
$914$	28132.1	1.01808
$915$	0	0
$916$	27257.3	0.983196
$917$	−38433.5	−1.38406
$918$	0	0
$919$	13527.6	0.485565	0.242782	−	0.970081i	$-0.421940\pi$
$919$	13527.6	0.485565	0.242782	+	0.970081i	$0.421940\pi$
$920$	0	0
$921$	0	0
$922$	−26268.7	−0.938301
$923$	42603.1	1.51928
$924$	0	0
$925$	0	0
$926$	279.910	0.00993348
$927$	0	0
$928$	8861.49	0.313462
$929$	38200.4	1.34910	0.674550	−	0.738229i	$-0.264338\pi$
$929$	38200.4	1.34910	0.674550	+	0.738229i	$0.264338\pi$
$930$	0	0
$931$	−13690.7	−0.481950
$932$	−20344.8	−0.715039
$933$	0	0
$934$	−1174.54	−0.0411479
$935$	0	0
$936$	0	0
$937$	45991.8	1.60351	0.801754	−	0.597654i	$-0.203900\pi$
$937$	45991.8	1.60351	0.801754	+	0.597654i	$0.203900\pi$
$938$	−31628.6	−1.10097
$939$	0	0
$940$	0	0
$941$	3180.14	0.110170	0.0550848	−	0.998482i	$-0.482457\pi$
$941$	3180.14	0.110170	0.0550848	+	0.998482i	$0.482457\pi$
$942$	0	0
$943$	−12515.7	−0.432203
$944$	6598.52	0.227504
$945$	0	0
$946$	−14811.3	−0.509045
$947$	56742.3	1.94707	0.973535	−	0.228536i	$-0.0733940\pi$
$947$	56742.3	1.94707	0.973535	+	0.228536i	$0.0733940\pi$
$948$	0	0
$949$	57576.5	1.96946
$950$	0	0
$951$	0	0
$952$	−20681.1	−0.704075
$953$	17765.0	0.603846	0.301923	−	0.953332i	$-0.402371\pi$
$953$	17765.0	0.603846	0.301923	+	0.953332i	$0.402371\pi$
$954$	0	0
$955$	0	0
$956$	7620.02	0.257792
$957$	0	0
$958$	−30913.4	−1.04255
$959$	23222.7	0.781961
$960$	0	0
$961$	43307.5	1.45371
$962$	8271.01	0.277202
$963$	0	0
$964$	22752.2	0.760165
$965$	0	0
$966$	0	0
$967$	35530.1	1.18156	0.590781	−	0.806832i	$-0.298820\pi$
$967$	35530.1	1.18156	0.590781	+	0.806832i	$0.298820\pi$
$968$	−6437.47	−0.213748
$969$	0	0
$970$	0	0
$971$	−26227.6	−0.866822	−0.433411	−	0.901196i	$-0.642690\pi$
$971$	−26227.6	−0.866822	−0.433411	+	0.901196i	$0.642690\pi$
$972$	0	0
$973$	3646.29	0.120138
$974$	36896.7	1.21381
$975$	0	0
$976$	−566.690	−0.0185854
$977$	−13869.8	−0.454182	−0.227091	−	0.973874i	$-0.572921\pi$
$977$	−13869.8	−0.454182	−0.227091	+	0.973874i	$0.572921\pi$
$978$	0	0
$979$	3145.49	0.102687
$980$	0	0
$981$	0	0
$982$	1037.84	0.0337257
$983$	19637.7	0.637177	0.318588	−	0.947893i	$-0.396791\pi$
$983$	19637.7	0.637177	0.318588	+	0.947893i	$0.396791\pi$
$984$	0	0
$985$	0	0
$986$	62409.0	2.01573
$987$	0	0
$988$	27575.3	0.887943
$989$	−17652.1	−0.567548
$990$	0	0
$991$	−5935.20	−0.190250	−0.0951251	−	0.995465i	$-0.530325\pi$
$991$	−5935.20	−0.190250	−0.0951251	+	0.995465i	$0.530325\pi$
$992$	8651.75	0.276909
$993$	0	0
$994$	21176.7	0.675739
$995$	0	0
$996$	0	0
$997$	42278.0	1.34299	0.671493	−	0.741011i	$-0.265653\pi$
$997$	42278.0	1.34299	0.671493	+	0.741011i	$0.265653\pi$
$998$	−11706.0	−0.371290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.a.bv.1.2		4
3.2	odd	2		1350.4.a.bu.1.2		4
5.2	odd	4		270.4.c.e.109.6	yes	8
5.3	odd	4		270.4.c.e.109.2	✓	8
5.4	even	2		1350.4.a.bu.1.3		4
15.2	even	4		270.4.c.e.109.3	yes	8
15.8	even	4		270.4.c.e.109.7	yes	8
15.14	odd	2	inner	1350.4.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.c.e.109.2	✓	8	5.3	odd	4
270.4.c.e.109.3	yes	8	15.2	even	4
270.4.c.e.109.6	yes	8	5.2	odd	4
270.4.c.e.109.7	yes	8	15.8	even	4
1350.4.a.bu.1.2		4	3.2	odd	2
1350.4.a.bu.1.3		4	5.4	even	2
1350.4.a.bv.1.2		4	1.1	even	1	trivial
1350.4.a.bv.1.3		4	15.14	odd	2	inner

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -22.9416 q^{7} +8.00000 q^{8} -22.9416 q^{11} -92.3072 q^{13} -45.8832 q^{14} +16.0000 q^{16} +112.684 q^{17} -74.6836 q^{19} -45.8832 q^{22} -54.6836 q^{23} -184.614 q^{26} -91.7663 q^{28} +276.921 q^{29} +270.367 q^{31} +32.0000 q^{32} +225.367 q^{34} -44.8016 q^{37} -149.367 q^{38} +228.875 q^{41} +322.805 q^{43} -91.7663 q^{44} -109.367 q^{46} +363.367 q^{47} +183.316 q^{49} -369.229 q^{52} -247.735 q^{53} -183.533 q^{56} +553.843 q^{58} +412.408 q^{59} -35.4181 q^{61} +540.735 q^{62} +64.0000 q^{64} +689.329 q^{67} +450.735 q^{68} -461.536 q^{71} -623.749 q^{73} -89.6031 q^{74} -298.735 q^{76} +526.316 q^{77} +1070.05 q^{79} +457.750 q^{82} +195.102 q^{83} +645.609 q^{86} -183.533 q^{88} -137.109 q^{89} +2117.67 q^{91} -218.735 q^{92} +726.735 q^{94} -25.6456 q^{97} +366.633 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} + 64 q^{16} + 110 q^{17} + 42 q^{19} + 122 q^{23} + 400 q^{31} + 128 q^{32} + 220 q^{34} + 84 q^{38} + 244 q^{46} + 772 q^{47} + 1074 q^{49} + 372 q^{53} + 1562 q^{61}+ \cdots + 2148 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 + 64 * q^16 + 110 * q^17 + 42 * q^19 + 122 * q^23 + 400 * q^31 + 128 * q^32 + 220 * q^34 + 84 * q^38 + 244 * q^46 + 772 * q^47 + 1074 * q^49 + 372 * q^53 + 1562 * q^61 + 800 * q^62 + 256 * q^64 + 440 * q^68 + 168 * q^76 + 2446 * q^77 + 3258 * q^79 - 1264 * q^83 + 1656 * q^91 + 488 * q^92 + 1544 * q^94 + 2148 * q^98

Introduction

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Database

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