LMFDB - Embedded newform 150.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,4,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	150.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:	$8.85028650086$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +1.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +1.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.0000 q^{11} +12.0000 q^{12} +67.0000 q^{13} +2.00000 q^{14} +16.0000 q^{16} -54.0000 q^{17} +18.0000 q^{18} -115.000 q^{19} +3.00000 q^{21} +84.0000 q^{22} +162.000 q^{23} +24.0000 q^{24} +134.000 q^{26} +27.0000 q^{27} +4.00000 q^{28} -210.000 q^{29} -193.000 q^{31} +32.0000 q^{32} +126.000 q^{33} -108.000 q^{34} +36.0000 q^{36} +286.000 q^{37} -230.000 q^{38} +201.000 q^{39} +12.0000 q^{41} +6.00000 q^{42} -263.000 q^{43} +168.000 q^{44} +324.000 q^{46} -414.000 q^{47} +48.0000 q^{48} -342.000 q^{49} -162.000 q^{51} +268.000 q^{52} +192.000 q^{53} +54.0000 q^{54} +8.00000 q^{56} -345.000 q^{57} -420.000 q^{58} +690.000 q^{59} -733.000 q^{61} -386.000 q^{62} +9.00000 q^{63} +64.0000 q^{64} +252.000 q^{66} -299.000 q^{67} -216.000 q^{68} +486.000 q^{69} -228.000 q^{71} +72.0000 q^{72} -938.000 q^{73} +572.000 q^{74} -460.000 q^{76} +42.0000 q^{77} +402.000 q^{78} -160.000 q^{79} +81.0000 q^{81} +24.0000 q^{82} +462.000 q^{83} +12.0000 q^{84} -526.000 q^{86} -630.000 q^{87} +336.000 q^{88} -240.000 q^{89} +67.0000 q^{91} +648.000 q^{92} -579.000 q^{93} -828.000 q^{94} +96.0000 q^{96} +511.000 q^{97} -684.000 q^{98} +378.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	1.00000	0.0539949	0.0269975	−	0.999636i	$-0.491405\pi$
$7$	1.00000	0.0539949	0.0269975	+	0.999636i	$0.491405\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	42.0000	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$11$	42.0000	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$12$	12.0000	0.288675
$13$	67.0000	1.42942	0.714710	−	0.699421i	$-0.246559\pi$
$13$	67.0000	1.42942	0.714710	+	0.699421i	$0.246559\pi$
$14$	2.00000	0.0381802
$15$	0	0
$16$	16.0000	0.250000
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	18.0000	0.235702
$19$	−115.000	−1.38857	−0.694284	−	0.719701i	$-0.744279\pi$
$19$	−115.000	−1.38857	−0.694284	+	0.719701i	$0.744279\pi$
$20$	0	0
$21$	3.00000	0.0311740
$22$	84.0000	0.814039
$23$	162.000	1.46867	0.734333	−	0.678789i	$-0.237495\pi$
$23$	162.000	1.46867	0.734333	+	0.678789i	$0.237495\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	134.000	1.01075
$27$	27.0000	0.192450
$28$	4.00000	0.0269975
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−193.000	−1.11819	−0.559094	−	0.829104i	$-0.688851\pi$
$31$	−193.000	−1.11819	−0.559094	+	0.829104i	$0.688851\pi$
$32$	32.0000	0.176777
$33$	126.000	0.664660
$34$	−108.000	−0.544760
$35$	0	0
$36$	36.0000	0.166667
$37$	286.000	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$37$	286.000	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$38$	−230.000	−0.981866
$39$	201.000	0.825276
$40$	0	0
$41$	12.0000	0.0457094	0.0228547	−	0.999739i	$-0.492724\pi$
$41$	12.0000	0.0457094	0.0228547	+	0.999739i	$0.492724\pi$
$42$	6.00000	0.0220433
$43$	−263.000	−0.932724	−0.466362	−	0.884594i	$-0.654436\pi$
$43$	−263.000	−0.932724	−0.466362	+	0.884594i	$0.654436\pi$
$44$	168.000	0.575613
$45$	0	0
$46$	324.000	1.03850
$47$	−414.000	−1.28485	−0.642427	−	0.766347i	$-0.722072\pi$
$47$	−414.000	−1.28485	−0.642427	+	0.766347i	$0.722072\pi$
$48$	48.0000	0.144338
$49$	−342.000	−0.997085
$50$	0	0
$51$	−162.000	−0.444795
$52$	268.000	0.714710
$53$	192.000	0.497608	0.248804	−	0.968554i	$-0.419962\pi$
$53$	192.000	0.497608	0.248804	+	0.968554i	$0.419962\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	8.00000	0.0190901
$57$	−345.000	−0.801691
$58$	−420.000	−0.950840
$59$	690.000	1.52255	0.761274	−	0.648430i	$-0.224574\pi$
$59$	690.000	1.52255	0.761274	+	0.648430i	$0.224574\pi$
$60$	0	0
$61$	−733.000	−1.53854	−0.769271	−	0.638923i	$-0.779380\pi$
$61$	−733.000	−1.53854	−0.769271	+	0.638923i	$0.779380\pi$
$62$	−386.000	−0.790678
$63$	9.00000	0.0179983
$64$	64.0000	0.125000
$65$	0	0
$66$	252.000	0.469986
$67$	−299.000	−0.545204	−0.272602	−	0.962127i	$-0.587884\pi$
$67$	−299.000	−0.545204	−0.272602	+	0.962127i	$0.587884\pi$
$68$	−216.000	−0.385204
$69$	486.000	0.847935
$70$	0	0
$71$	−228.000	−0.381107	−0.190554	−	0.981677i	$-0.561028\pi$
$71$	−228.000	−0.381107	−0.190554	+	0.981677i	$0.561028\pi$
$72$	72.0000	0.117851
$73$	−938.000	−1.50390	−0.751949	−	0.659221i	$-0.770886\pi$
$73$	−938.000	−1.50390	−0.751949	+	0.659221i	$0.770886\pi$
$74$	572.000	0.898563
$75$	0	0
$76$	−460.000	−0.694284
$77$	42.0000	0.0621603
$78$	402.000	0.583558
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	24.0000	0.0323214
$83$	462.000	0.610977	0.305488	−	0.952196i	$-0.401180\pi$
$83$	462.000	0.610977	0.305488	+	0.952196i	$0.401180\pi$
$84$	12.0000	0.0155870
$85$	0	0
$86$	−526.000	−0.659535
$87$	−630.000	−0.776357
$88$	336.000	0.407020
$89$	−240.000	−0.285842	−0.142921	−	0.989734i	$-0.545650\pi$
$89$	−240.000	−0.285842	−0.142921	+	0.989734i	$0.545650\pi$
$90$	0	0
$91$	67.0000	0.0771814
$92$	648.000	0.734333
$93$	−579.000	−0.645586
$94$	−828.000	−0.908529
$95$	0	0
$96$	96.0000	0.102062
$97$	511.000	0.534889	0.267444	−	0.963573i	$-0.413821\pi$
$97$	511.000	0.534889	0.267444	+	0.963573i	$0.413821\pi$
$98$	−684.000	−0.705045
$99$	378.000	0.383742
$100$	0	0
$101$	912.000	0.898489	0.449245	−	0.893409i	$-0.351693\pi$
$101$	912.000	0.898489	0.449245	+	0.893409i	$0.351693\pi$
$102$	−324.000	−0.314517
$103$	−668.000	−0.639029	−0.319515	−	0.947581i	$-0.603520\pi$
$103$	−668.000	−0.639029	−0.319515	+	0.947581i	$0.603520\pi$
$104$	536.000	0.505376
$105$	0	0
$106$	384.000	0.351862
$107$	1296.00	1.17093	0.585463	−	0.810699i	$-0.300913\pi$
$107$	1296.00	1.17093	0.585463	+	0.810699i	$0.300913\pi$
$108$	108.000	0.0962250
$109$	−1735.00	−1.52461	−0.762307	−	0.647216i	$-0.775933\pi$
$109$	−1735.00	−1.52461	−0.762307	+	0.647216i	$0.775933\pi$
$110$	0	0
$111$	858.000	0.733673
$112$	16.0000	0.0134987
$113$	1092.00	0.909086	0.454543	−	0.890725i	$-0.349803\pi$
$113$	1092.00	0.909086	0.454543	+	0.890725i	$0.349803\pi$
$114$	−690.000	−0.566881
$115$	0	0
$116$	−840.000	−0.672345
$117$	603.000	0.476473
$118$	1380.00	1.07660
$119$	−54.0000	−0.0415981
$120$	0	0
$121$	433.000	0.325319
$122$	−1466.00	−1.08791
$123$	36.0000	0.0263903
$124$	−772.000	−0.559094
$125$	0	0
$126$	18.0000	0.0127267
$127$	16.0000	0.0111793	0.00558965	−	0.999984i	$-0.498221\pi$
$127$	16.0000	0.0111793	0.00558965	+	0.999984i	$0.498221\pi$
$128$	128.000	0.0883883
$129$	−789.000	−0.538508
$130$	0	0
$131$	1992.00	1.32856	0.664282	−	0.747482i	$-0.268737\pi$
$131$	1992.00	1.32856	0.664282	+	0.747482i	$0.268737\pi$
$132$	504.000	0.332330
$133$	−115.000	−0.0749757
$134$	−598.000	−0.385517
$135$	0	0
$136$	−432.000	−0.272380
$137$	2346.00	1.46301	0.731505	−	0.681836i	$-0.238818\pi$
$137$	2346.00	1.46301	0.731505	+	0.681836i	$0.238818\pi$
$138$	972.000	0.599581
$139$	2900.00	1.76960	0.884801	−	0.465968i	$-0.154294\pi$
$139$	2900.00	1.76960	0.884801	+	0.465968i	$0.154294\pi$
$140$	0	0
$141$	−1242.00	−0.741810
$142$	−456.000	−0.269484
$143$	2814.00	1.64558
$144$	144.000	0.0833333
$145$	0	0
$146$	−1876.00	−1.06342
$147$	−1026.00	−0.575667
$148$	1144.00	0.635380
$149$	−2070.00	−1.13813	−0.569064	−	0.822293i	$-0.692694\pi$
$149$	−2070.00	−1.13813	−0.569064	+	0.822293i	$0.692694\pi$
$150$	0	0
$151$	2237.00	1.20559	0.602796	−	0.797895i	$-0.294053\pi$
$151$	2237.00	1.20559	0.602796	+	0.797895i	$0.294053\pi$
$152$	−920.000	−0.490933
$153$	−486.000	−0.256802
$154$	84.0000	0.0439540
$155$	0	0
$156$	804.000	0.412638
$157$	241.000	0.122509	0.0612544	−	0.998122i	$-0.480490\pi$
$157$	241.000	0.122509	0.0612544	+	0.998122i	$0.480490\pi$
$158$	−320.000	−0.161126
$159$	576.000	0.287294
$160$	0	0
$161$	162.000	0.0793006
$162$	162.000	0.0785674
$163$	3547.00	1.70443	0.852216	−	0.523190i	$-0.175258\pi$
$163$	3547.00	1.70443	0.852216	+	0.523190i	$0.175258\pi$
$164$	48.0000	0.0228547
$165$	0	0
$166$	924.000	0.432026
$167$	−984.000	−0.455953	−0.227977	−	0.973667i	$-0.573211\pi$
$167$	−984.000	−0.455953	−0.227977	+	0.973667i	$0.573211\pi$
$168$	24.0000	0.0110217
$169$	2292.00	1.04324
$170$	0	0
$171$	−1035.00	−0.462856
$172$	−1052.00	−0.466362
$173$	−3618.00	−1.59001	−0.795004	−	0.606604i	$-0.792531\pi$
$173$	−3618.00	−1.59001	−0.795004	+	0.606604i	$0.792531\pi$
$174$	−1260.00	−0.548968
$175$	0	0
$176$	672.000	0.287806
$177$	2070.00	0.879044
$178$	−480.000	−0.202121
$179$	−150.000	−0.0626342	−0.0313171	−	0.999509i	$-0.509970\pi$
$179$	−150.000	−0.0626342	−0.0313171	+	0.999509i	$0.509970\pi$
$180$	0	0
$181$	197.000	0.0809000	0.0404500	−	0.999182i	$-0.487121\pi$
$181$	197.000	0.0809000	0.0404500	+	0.999182i	$0.487121\pi$
$182$	134.000	0.0545755
$183$	−2199.00	−0.888277
$184$	1296.00	0.519252
$185$	0	0
$186$	−1158.00	−0.456498
$187$	−2268.00	−0.886912
$188$	−1656.00	−0.642427
$189$	27.0000	0.0103913
$190$	0	0
$191$	1302.00	0.493243	0.246622	−	0.969112i	$-0.420680\pi$
$191$	1302.00	0.493243	0.246622	+	0.969112i	$0.420680\pi$
$192$	192.000	0.0721688
$193$	−4163.00	−1.55264	−0.776319	−	0.630340i	$-0.782916\pi$
$193$	−4163.00	−1.55264	−0.776319	+	0.630340i	$0.782916\pi$
$194$	1022.00	0.378223
$195$	0	0
$196$	−1368.00	−0.498542
$197$	−3054.00	−1.10451	−0.552255	−	0.833675i	$-0.686233\pi$
$197$	−3054.00	−1.10451	−0.552255	+	0.833675i	$0.686233\pi$
$198$	756.000	0.271346
$199$	3425.00	1.22006	0.610030	−	0.792379i	$-0.291158\pi$
$199$	3425.00	1.22006	0.610030	+	0.792379i	$0.291158\pi$
$200$	0	0
$201$	−897.000	−0.314774
$202$	1824.00	0.635328
$203$	−210.000	−0.0726065
$204$	−648.000	−0.222397
$205$	0	0
$206$	−1336.00	−0.451862
$207$	1458.00	0.489556
$208$	1072.00	0.357355
$209$	−4830.00	−1.59856
$210$	0	0
$211$	−2443.00	−0.797076	−0.398538	−	0.917152i	$-0.630482\pi$
$211$	−2443.00	−0.797076	−0.398538	+	0.917152i	$0.630482\pi$
$212$	768.000	0.248804
$213$	−684.000	−0.220032
$214$	2592.00	0.827969
$215$	0	0
$216$	216.000	0.0680414
$217$	−193.000	−0.0603765
$218$	−3470.00	−1.07806
$219$	−2814.00	−0.868276
$220$	0	0
$221$	−3618.00	−1.10124
$222$	1716.00	0.518785
$223$	−23.0000	−0.00690670	−0.00345335	−	0.999994i	$-0.501099\pi$
$223$	−23.0000	−0.00690670	−0.00345335	+	0.999994i	$0.501099\pi$
$224$	32.0000	0.00954504
$225$	0	0
$226$	2184.00	0.642821
$227$	1956.00	0.571913	0.285957	−	0.958243i	$-0.407689\pi$
$227$	1956.00	0.571913	0.285957	+	0.958243i	$0.407689\pi$
$228$	−1380.00	−0.400845
$229$	1805.00	0.520864	0.260432	−	0.965492i	$-0.416135\pi$
$229$	1805.00	0.520864	0.260432	+	0.965492i	$0.416135\pi$
$230$	0	0
$231$	126.000	0.0358883
$232$	−1680.00	−0.475420
$233$	−3468.00	−0.975091	−0.487546	−	0.873098i	$-0.662108\pi$
$233$	−3468.00	−0.975091	−0.487546	+	0.873098i	$0.662108\pi$
$234$	1206.00	0.336917
$235$	0	0
$236$	2760.00	0.761274
$237$	−480.000	−0.131558
$238$	−108.000	−0.0294143
$239$	2640.00	0.714508	0.357254	−	0.934007i	$-0.383713\pi$
$239$	2640.00	0.714508	0.357254	+	0.934007i	$0.383713\pi$
$240$	0	0
$241$	−5383.00	−1.43879	−0.719397	−	0.694599i	$-0.755582\pi$
$241$	−5383.00	−1.43879	−0.719397	+	0.694599i	$0.755582\pi$
$242$	866.000	0.230035
$243$	243.000	0.0641500
$244$	−2932.00	−0.769271
$245$	0	0
$246$	72.0000	0.0186608
$247$	−7705.00	−1.98485
$248$	−1544.00	−0.395339
$249$	1386.00	0.352748
$250$	0	0
$251$	−5028.00	−1.26440	−0.632200	−	0.774805i	$-0.717848\pi$
$251$	−5028.00	−1.26440	−0.632200	+	0.774805i	$0.717848\pi$
$252$	36.0000	0.00899915
$253$	6804.00	1.69077
$254$	32.0000	0.00790496
$255$	0	0
$256$	256.000	0.0625000
$257$	−564.000	−0.136892	−0.0684462	−	0.997655i	$-0.521804\pi$
$257$	−564.000	−0.136892	−0.0684462	+	0.997655i	$0.521804\pi$
$258$	−1578.00	−0.380783
$259$	286.000	0.0686146
$260$	0	0
$261$	−1890.00	−0.448230
$262$	3984.00	0.939436
$263$	1812.00	0.424839	0.212420	−	0.977179i	$-0.431866\pi$
$263$	1812.00	0.424839	0.212420	+	0.977179i	$0.431866\pi$
$264$	1008.00	0.234993
$265$	0	0
$266$	−230.000	−0.0530158
$267$	−720.000	−0.165031
$268$	−1196.00	−0.272602
$269$	−5190.00	−1.17636	−0.588178	−	0.808731i	$-0.700155\pi$
$269$	−5190.00	−1.17636	−0.588178	+	0.808731i	$0.700155\pi$
$270$	0	0
$271$	4592.00	1.02931	0.514657	−	0.857396i	$-0.327919\pi$
$271$	4592.00	1.02931	0.514657	+	0.857396i	$0.327919\pi$
$272$	−864.000	−0.192602
$273$	201.000	0.0445607
$274$	4692.00	1.03450
$275$	0	0
$276$	1944.00	0.423968
$277$	2191.00	0.475251	0.237625	−	0.971357i	$-0.423631\pi$
$277$	2191.00	0.475251	0.237625	+	0.971357i	$0.423631\pi$
$278$	5800.00	1.25130
$279$	−1737.00	−0.372729
$280$	0	0
$281$	7842.00	1.66482	0.832410	−	0.554160i	$-0.186960\pi$
$281$	7842.00	1.66482	0.832410	+	0.554160i	$0.186960\pi$
$282$	−2484.00	−0.524539
$283$	247.000	0.0518821	0.0259410	−	0.999663i	$-0.491742\pi$
$283$	247.000	0.0518821	0.0259410	+	0.999663i	$0.491742\pi$
$284$	−912.000	−0.190554
$285$	0	0
$286$	5628.00	1.16360
$287$	12.0000	0.00246808
$288$	288.000	0.0589256
$289$	−1997.00	−0.406473
$290$	0	0
$291$	1533.00	0.308818
$292$	−3752.00	−0.751949
$293$	5442.00	1.08507	0.542534	−	0.840034i	$-0.317465\pi$
$293$	5442.00	1.08507	0.542534	+	0.840034i	$0.317465\pi$
$294$	−2052.00	−0.407058
$295$	0	0
$296$	2288.00	0.449281
$297$	1134.00	0.221553
$298$	−4140.00	−0.804778
$299$	10854.0	2.09934
$300$	0	0
$301$	−263.000	−0.0503624
$302$	4474.00	0.852483
$303$	2736.00	0.518743
$304$	−1840.00	−0.347142
$305$	0	0
$306$	−972.000	−0.181587
$307$	3871.00	0.719641	0.359820	−	0.933022i	$-0.382838\pi$
$307$	3871.00	0.719641	0.359820	+	0.933022i	$0.382838\pi$
$308$	168.000	0.0310802
$309$	−2004.00	−0.368944
$310$	0	0
$311$	−5718.00	−1.04257	−0.521283	−	0.853384i	$-0.674546\pi$
$311$	−5718.00	−1.04257	−0.521283	+	0.853384i	$0.674546\pi$
$312$	1608.00	0.291779
$313$	3637.00	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$313$	3637.00	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$314$	482.000	0.0866269
$315$	0	0
$316$	−640.000	−0.113933
$317$	1296.00	0.229623	0.114812	−	0.993387i	$-0.463374\pi$
$317$	1296.00	0.229623	0.114812	+	0.993387i	$0.463374\pi$
$318$	1152.00	0.203148
$319$	−8820.00	−1.54804
$320$	0	0
$321$	3888.00	0.676034
$322$	324.000	0.0560740
$323$	6210.00	1.06976
$324$	324.000	0.0555556
$325$	0	0
$326$	7094.00	1.20522
$327$	−5205.00	−0.880236
$328$	96.0000	0.0161607
$329$	−414.000	−0.0693756
$330$	0	0
$331$	5132.00	0.852206	0.426103	−	0.904675i	$-0.359886\pi$
$331$	5132.00	0.852206	0.426103	+	0.904675i	$0.359886\pi$
$332$	1848.00	0.305488
$333$	2574.00	0.423587
$334$	−1968.00	−0.322408
$335$	0	0
$336$	48.0000	0.00779350
$337$	6751.00	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$337$	6751.00	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$338$	4584.00	0.737683
$339$	3276.00	0.524861
$340$	0	0
$341$	−8106.00	−1.28729
$342$	−2070.00	−0.327289
$343$	−685.000	−0.107832
$344$	−2104.00	−0.329768
$345$	0	0
$346$	−7236.00	−1.12431
$347$	5226.00	0.808491	0.404246	−	0.914651i	$-0.367534\pi$
$347$	5226.00	0.808491	0.404246	+	0.914651i	$0.367534\pi$
$348$	−2520.00	−0.388179
$349$	−6190.00	−0.949407	−0.474704	−	0.880146i	$-0.657445\pi$
$349$	−6190.00	−0.949407	−0.474704	+	0.880146i	$0.657445\pi$
$350$	0	0
$351$	1809.00	0.275092
$352$	1344.00	0.203510
$353$	−6618.00	−0.997849	−0.498924	−	0.866646i	$-0.666271\pi$
$353$	−6618.00	−0.997849	−0.498924	+	0.866646i	$0.666271\pi$
$354$	4140.00	0.621578
$355$	0	0
$356$	−960.000	−0.142921
$357$	−162.000	−0.0240167
$358$	−300.000	−0.0442891
$359$	−3420.00	−0.502787	−0.251394	−	0.967885i	$-0.580889\pi$
$359$	−3420.00	−0.502787	−0.251394	+	0.967885i	$0.580889\pi$
$360$	0	0
$361$	6366.00	0.928124
$362$	394.000	0.0572049
$363$	1299.00	0.187823
$364$	268.000	0.0385907
$365$	0	0
$366$	−4398.00	−0.628107
$367$	871.000	0.123885	0.0619425	−	0.998080i	$-0.480270\pi$
$367$	871.000	0.123885	0.0619425	+	0.998080i	$0.480270\pi$
$368$	2592.00	0.367167
$369$	108.000	0.0152365
$370$	0	0
$371$	192.000	0.0268683
$372$	−2316.00	−0.322793
$373$	−6383.00	−0.886057	−0.443028	−	0.896508i	$-0.646096\pi$
$373$	−6383.00	−0.886057	−0.443028	+	0.896508i	$0.646096\pi$
$374$	−4536.00	−0.627142
$375$	0	0
$376$	−3312.00	−0.454264
$377$	−14070.0	−1.92213
$378$	54.0000	0.00734778
$379$	−9865.00	−1.33702	−0.668511	−	0.743703i	$-0.733068\pi$
$379$	−9865.00	−1.33702	−0.668511	+	0.743703i	$0.733068\pi$
$380$	0	0
$381$	48.0000	0.00645437
$382$	2604.00	0.348775
$383$	−9828.00	−1.31119	−0.655597	−	0.755111i	$-0.727583\pi$
$383$	−9828.00	−1.31119	−0.655597	+	0.755111i	$0.727583\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−8326.00	−1.09788
$387$	−2367.00	−0.310908
$388$	2044.00	0.267444
$389$	12540.0	1.63446	0.817228	−	0.576315i	$-0.195510\pi$
$389$	12540.0	1.63446	0.817228	+	0.576315i	$0.195510\pi$
$390$	0	0
$391$	−8748.00	−1.13147
$392$	−2736.00	−0.352523
$393$	5976.00	0.767047
$394$	−6108.00	−0.781007
$395$	0	0
$396$	1512.00	0.191871
$397$	1381.00	0.174585	0.0872927	−	0.996183i	$-0.472178\pi$
$397$	1381.00	0.174585	0.0872927	+	0.996183i	$0.472178\pi$
$398$	6850.00	0.862712
$399$	−345.000	−0.0432872
$400$	0	0
$401$	14232.0	1.77235	0.886175	−	0.463351i	$-0.153353\pi$
$401$	14232.0	1.77235	0.886175	+	0.463351i	$0.153353\pi$
$402$	−1794.00	−0.222579
$403$	−12931.0	−1.59836
$404$	3648.00	0.449245
$405$	0	0
$406$	−420.000	−0.0513405
$407$	12012.0	1.46293
$408$	−1296.00	−0.157259
$409$	2645.00	0.319772	0.159886	−	0.987135i	$-0.448887\pi$
$409$	2645.00	0.319772	0.159886	+	0.987135i	$0.448887\pi$
$410$	0	0
$411$	7038.00	0.844669
$412$	−2672.00	−0.319515
$413$	690.000	0.0822099
$414$	2916.00	0.346168
$415$	0	0
$416$	2144.00	0.252688
$417$	8700.00	1.02168
$418$	−9660.00	−1.13035
$419$	3000.00	0.349784	0.174892	−	0.984588i	$-0.444042\pi$
$419$	3000.00	0.349784	0.174892	+	0.984588i	$0.444042\pi$
$420$	0	0
$421$	−11338.0	−1.31254	−0.656271	−	0.754525i	$-0.727867\pi$
$421$	−11338.0	−1.31254	−0.656271	+	0.754525i	$0.727867\pi$
$422$	−4886.00	−0.563618
$423$	−3726.00	−0.428284
$424$	1536.00	0.175931
$425$	0	0
$426$	−1368.00	−0.155586
$427$	−733.000	−0.0830734
$428$	5184.00	0.585463
$429$	8442.00	0.950078
$430$	0	0
$431$	−3258.00	−0.364112	−0.182056	−	0.983288i	$-0.558275\pi$
$431$	−3258.00	−0.364112	−0.182056	+	0.983288i	$0.558275\pi$
$432$	432.000	0.0481125
$433$	−1163.00	−0.129077	−0.0645384	−	0.997915i	$-0.520557\pi$
$433$	−1163.00	−0.129077	−0.0645384	+	0.997915i	$0.520557\pi$
$434$	−386.000	−0.0426926
$435$	0	0
$436$	−6940.00	−0.762307
$437$	−18630.0	−2.03934
$438$	−5628.00	−0.613964
$439$	6695.00	0.727870	0.363935	−	0.931424i	$-0.381433\pi$
$439$	6695.00	0.727870	0.363935	+	0.931424i	$0.381433\pi$
$440$	0	0
$441$	−3078.00	−0.332362
$442$	−7236.00	−0.778691
$443$	−16368.0	−1.75546	−0.877728	−	0.479159i	$-0.840942\pi$
$443$	−16368.0	−1.75546	−0.877728	+	0.479159i	$0.840942\pi$
$444$	3432.00	0.366837
$445$	0	0
$446$	−46.0000	−0.00488377
$447$	−6210.00	−0.657098
$448$	64.0000	0.00674937
$449$	16380.0	1.72165	0.860824	−	0.508903i	$-0.169949\pi$
$449$	16380.0	1.72165	0.860824	+	0.508903i	$0.169949\pi$
$450$	0	0
$451$	504.000	0.0526218
$452$	4368.00	0.454543
$453$	6711.00	0.696049
$454$	3912.00	0.404404
$455$	0	0
$456$	−2760.00	−0.283440
$457$	13786.0	1.41112	0.705560	−	0.708650i	$-0.250696\pi$
$457$	13786.0	1.41112	0.705560	+	0.708650i	$0.250696\pi$
$458$	3610.00	0.368306
$459$	−1458.00	−0.148265
$460$	0	0
$461$	11832.0	1.19538	0.597691	−	0.801726i	$-0.296085\pi$
$461$	11832.0	1.19538	0.597691	+	0.801726i	$0.296085\pi$
$462$	252.000	0.0253768
$463$	−3008.00	−0.301930	−0.150965	−	0.988539i	$-0.548238\pi$
$463$	−3008.00	−0.301930	−0.150965	+	0.988539i	$0.548238\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	−6936.00	−0.689494
$467$	−4434.00	−0.439360	−0.219680	−	0.975572i	$-0.570501\pi$
$467$	−4434.00	−0.439360	−0.219680	+	0.975572i	$0.570501\pi$
$468$	2412.00	0.238237
$469$	−299.000	−0.0294382
$470$	0	0
$471$	723.000	0.0707305
$472$	5520.00	0.538302
$473$	−11046.0	−1.07378
$474$	−960.000	−0.0930259
$475$	0	0
$476$	−216.000	−0.0207990
$477$	1728.00	0.165869
$478$	5280.00	0.505233
$479$	7410.00	0.706830	0.353415	−	0.935467i	$-0.385020\pi$
$479$	7410.00	0.706830	0.353415	+	0.935467i	$0.385020\pi$
$480$	0	0
$481$	19162.0	1.81645
$482$	−10766.0	−1.01738
$483$	486.000	0.0457842
$484$	1732.00	0.162660
$485$	0	0
$486$	486.000	0.0453609
$487$	8671.00	0.806818	0.403409	−	0.915020i	$-0.367825\pi$
$487$	8671.00	0.806818	0.403409	+	0.915020i	$0.367825\pi$
$488$	−5864.00	−0.543957
$489$	10641.0	0.984055
$490$	0	0
$491$	−19368.0	−1.78017	−0.890087	−	0.455790i	$-0.849357\pi$
$491$	−19368.0	−1.78017	−0.890087	+	0.455790i	$0.849357\pi$
$492$	144.000	0.0131952
$493$	11340.0	1.03596
$494$	−15410.0	−1.40350
$495$	0	0
$496$	−3088.00	−0.279547
$497$	−228.000	−0.0205779
$498$	2772.00	0.249430
$499$	−8875.00	−0.796192	−0.398096	−	0.917344i	$-0.630329\pi$
$499$	−8875.00	−0.796192	−0.398096	+	0.917344i	$0.630329\pi$
$500$	0	0
$501$	−2952.00	−0.263245
$502$	−10056.0	−0.894066
$503$	10452.0	0.926504	0.463252	−	0.886227i	$-0.346682\pi$
$503$	10452.0	0.926504	0.463252	+	0.886227i	$0.346682\pi$
$504$	72.0000	0.00636336
$505$	0	0
$506$	13608.0	1.19555
$507$	6876.00	0.602315
$508$	64.0000	0.00558965
$509$	−19770.0	−1.72159	−0.860796	−	0.508951i	$-0.830033\pi$
$509$	−19770.0	−1.72159	−0.860796	+	0.508951i	$0.830033\pi$
$510$	0	0
$511$	−938.000	−0.0812029
$512$	512.000	0.0441942
$513$	−3105.00	−0.267230
$514$	−1128.00	−0.0967976
$515$	0	0
$516$	−3156.00	−0.269254
$517$	−17388.0	−1.47916
$518$	572.000	0.0485178
$519$	−10854.0	−0.917992
$520$	0	0
$521$	−11238.0	−0.945001	−0.472501	−	0.881330i	$-0.656649\pi$
$521$	−11238.0	−0.945001	−0.472501	+	0.881330i	$0.656649\pi$
$522$	−3780.00	−0.316947
$523$	7447.00	0.622628	0.311314	−	0.950307i	$-0.399231\pi$
$523$	7447.00	0.622628	0.311314	+	0.950307i	$0.399231\pi$
$524$	7968.00	0.664282
$525$	0	0
$526$	3624.00	0.300407
$527$	10422.0	0.861460
$528$	2016.00	0.166165
$529$	14077.0	1.15698
$530$	0	0
$531$	6210.00	0.507516
$532$	−460.000	−0.0374878
$533$	804.000	0.0653379
$534$	−1440.00	−0.116695
$535$	0	0
$536$	−2392.00	−0.192759
$537$	−450.000	−0.0361619
$538$	−10380.0	−0.831810
$539$	−14364.0	−1.14787
$540$	0	0
$541$	−17623.0	−1.40050	−0.700251	−	0.713896i	$-0.746929\pi$
$541$	−17623.0	−1.40050	−0.700251	+	0.713896i	$0.746929\pi$
$542$	9184.00	0.727835
$543$	591.000	0.0467076
$544$	−1728.00	−0.136190
$545$	0	0
$546$	402.000	0.0315092
$547$	10096.0	0.789166	0.394583	−	0.918860i	$-0.370889\pi$
$547$	10096.0	0.789166	0.394583	+	0.918860i	$0.370889\pi$
$548$	9384.00	0.731505
$549$	−6597.00	−0.512847
$550$	0	0
$551$	24150.0	1.86720
$552$	3888.00	0.299790
$553$	−160.000	−0.0123036
$554$	4382.00	0.336053
$555$	0	0
$556$	11600.0	0.884801
$557$	−14514.0	−1.10409	−0.552045	−	0.833814i	$-0.686152\pi$
$557$	−14514.0	−1.10409	−0.552045	+	0.833814i	$0.686152\pi$
$558$	−3474.00	−0.263559
$559$	−17621.0	−1.33325
$560$	0	0
$561$	−6804.00	−0.512059
$562$	15684.0	1.17721
$563$	10242.0	0.766694	0.383347	−	0.923604i	$-0.374771\pi$
$563$	10242.0	0.766694	0.383347	+	0.923604i	$0.374771\pi$
$564$	−4968.00	−0.370905
$565$	0	0
$566$	494.000	0.0366862
$567$	81.0000	0.00599944
$568$	−1824.00	−0.134742
$569$	−6750.00	−0.497319	−0.248660	−	0.968591i	$-0.579990\pi$
$569$	−6750.00	−0.497319	−0.248660	+	0.968591i	$0.579990\pi$
$570$	0	0
$571$	17117.0	1.25451	0.627254	−	0.778815i	$-0.284179\pi$
$571$	17117.0	1.25451	0.627254	+	0.778815i	$0.284179\pi$
$572$	11256.0	0.822792
$573$	3906.00	0.284774
$574$	24.0000	0.00174519
$575$	0	0
$576$	576.000	0.0416667
$577$	301.000	0.0217171	0.0108586	−	0.999941i	$-0.496544\pi$
$577$	301.000	0.0217171	0.0108586	+	0.999941i	$0.496544\pi$
$578$	−3994.00	−0.287420
$579$	−12489.0	−0.896416
$580$	0	0
$581$	462.000	0.0329897
$582$	3066.00	0.218367
$583$	8064.00	0.572859
$584$	−7504.00	−0.531708
$585$	0	0
$586$	10884.0	0.767259
$587$	15456.0	1.08678	0.543388	−	0.839482i	$-0.317141\pi$
$587$	15456.0	1.08678	0.543388	+	0.839482i	$0.317141\pi$
$588$	−4104.00	−0.287834
$589$	22195.0	1.55268
$590$	0	0
$591$	−9162.00	−0.637689
$592$	4576.00	0.317690
$593$	9492.00	0.657318	0.328659	−	0.944449i	$-0.393403\pi$
$593$	9492.00	0.657318	0.328659	+	0.944449i	$0.393403\pi$
$594$	2268.00	0.156662
$595$	0	0
$596$	−8280.00	−0.569064
$597$	10275.0	0.704402
$598$	21708.0	1.48446
$599$	1500.00	0.102318	0.0511589	−	0.998691i	$-0.483709\pi$
$599$	1500.00	0.102318	0.0511589	+	0.998691i	$0.483709\pi$
$600$	0	0
$601$	14627.0	0.992758	0.496379	−	0.868106i	$-0.334663\pi$
$601$	14627.0	0.992758	0.496379	+	0.868106i	$0.334663\pi$
$602$	−526.000	−0.0356116
$603$	−2691.00	−0.181735
$604$	8948.00	0.602796
$605$	0	0
$606$	5472.00	0.366807
$607$	−16184.0	−1.08219	−0.541094	−	0.840962i	$-0.681990\pi$
$607$	−16184.0	−1.08219	−0.541094	+	0.840962i	$0.681990\pi$
$608$	−3680.00	−0.245467
$609$	−630.000	−0.0419194
$610$	0	0
$611$	−27738.0	−1.83659
$612$	−1944.00	−0.128401
$613$	18502.0	1.21907	0.609534	−	0.792760i	$-0.291357\pi$
$613$	18502.0	1.21907	0.609534	+	0.792760i	$0.291357\pi$
$614$	7742.00	0.508863
$615$	0	0
$616$	336.000	0.0219770
$617$	13896.0	0.906697	0.453348	−	0.891333i	$-0.350229\pi$
$617$	13896.0	0.906697	0.453348	+	0.891333i	$0.350229\pi$
$618$	−4008.00	−0.260883
$619$	−9895.00	−0.642510	−0.321255	−	0.946993i	$-0.604105\pi$
$619$	−9895.00	−0.642510	−0.321255	+	0.946993i	$0.604105\pi$
$620$	0	0
$621$	4374.00	0.282645
$622$	−11436.0	−0.737206
$623$	−240.000	−0.0154340
$624$	3216.00	0.206319
$625$	0	0
$626$	7274.00	0.464421
$627$	−14490.0	−0.922926
$628$	964.000	0.0612544
$629$	−15444.0	−0.979003
$630$	0	0
$631$	467.000	0.0294627	0.0147314	−	0.999891i	$-0.495311\pi$
$631$	467.000	0.0294627	0.0147314	+	0.999891i	$0.495311\pi$
$632$	−1280.00	−0.0805628
$633$	−7329.00	−0.460192
$634$	2592.00	0.162368
$635$	0	0
$636$	2304.00	0.143647
$637$	−22914.0	−1.42525
$638$	−17640.0	−1.09463
$639$	−2052.00	−0.127036
$640$	0	0
$641$	30612.0	1.88627	0.943137	−	0.332405i	$-0.107860\pi$
$641$	30612.0	1.88627	0.943137	+	0.332405i	$0.107860\pi$
$642$	7776.00	0.478028
$643$	1852.00	0.113586	0.0567930	−	0.998386i	$-0.481913\pi$
$643$	1852.00	0.113586	0.0567930	+	0.998386i	$0.481913\pi$
$644$	648.000	0.0396503
$645$	0	0
$646$	12420.0	0.756437
$647$	21156.0	1.28551	0.642757	−	0.766070i	$-0.277790\pi$
$647$	21156.0	1.28551	0.642757	+	0.766070i	$0.277790\pi$
$648$	648.000	0.0392837
$649$	28980.0	1.75280
$650$	0	0
$651$	−579.000	−0.0348584
$652$	14188.0	0.852216
$653$	9702.00	0.581422	0.290711	−	0.956811i	$-0.406108\pi$
$653$	9702.00	0.581422	0.290711	+	0.956811i	$0.406108\pi$
$654$	−10410.0	−0.622421
$655$	0	0
$656$	192.000	0.0114273
$657$	−8442.00	−0.501300
$658$	−828.000	−0.0490559
$659$	1980.00	0.117041	0.0585204	−	0.998286i	$-0.481362\pi$
$659$	1980.00	0.117041	0.0585204	+	0.998286i	$0.481362\pi$
$660$	0	0
$661$	−20158.0	−1.18617	−0.593083	−	0.805142i	$-0.702089\pi$
$661$	−20158.0	−1.18617	−0.593083	+	0.805142i	$0.702089\pi$
$662$	10264.0	0.602601
$663$	−10854.0	−0.635799
$664$	3696.00	0.216013
$665$	0	0
$666$	5148.00	0.299521
$667$	−34020.0	−1.97490
$668$	−3936.00	−0.227977
$669$	−69.0000	−0.00398758
$670$	0	0
$671$	−30786.0	−1.77121
$672$	96.0000	0.00551083
$673$	16882.0	0.966944	0.483472	−	0.875360i	$-0.339376\pi$
$673$	16882.0	0.966944	0.483472	+	0.875360i	$0.339376\pi$
$674$	13502.0	0.771628
$675$	0	0
$676$	9168.00	0.521620
$677$	−20934.0	−1.18842	−0.594209	−	0.804311i	$-0.702535\pi$
$677$	−20934.0	−1.18842	−0.594209	+	0.804311i	$0.702535\pi$
$678$	6552.00	0.371133
$679$	511.000	0.0288813
$680$	0	0
$681$	5868.00	0.330194
$682$	−16212.0	−0.910249
$683$	8712.00	0.488075	0.244038	−	0.969766i	$-0.421528\pi$
$683$	8712.00	0.488075	0.244038	+	0.969766i	$0.421528\pi$
$684$	−4140.00	−0.231428
$685$	0	0
$686$	−1370.00	−0.0762490
$687$	5415.00	0.300721
$688$	−4208.00	−0.233181
$689$	12864.0	0.711291
$690$	0	0
$691$	−14128.0	−0.777792	−0.388896	−	0.921282i	$-0.627144\pi$
$691$	−14128.0	−0.777792	−0.388896	+	0.921282i	$0.627144\pi$
$692$	−14472.0	−0.795004
$693$	378.000	0.0207201
$694$	10452.0	0.571689
$695$	0	0
$696$	−5040.00	−0.274484
$697$	−648.000	−0.0352148
$698$	−12380.0	−0.671332
$699$	−10404.0	−0.562969
$700$	0	0
$701$	−28278.0	−1.52360	−0.761801	−	0.647811i	$-0.775685\pi$
$701$	−28278.0	−1.52360	−0.761801	+	0.647811i	$0.775685\pi$
$702$	3618.00	0.194519
$703$	−32890.0	−1.76454
$704$	2688.00	0.143903
$705$	0	0
$706$	−13236.0	−0.705586
$707$	912.000	0.0485138
$708$	8280.00	0.439522
$709$	8885.00	0.470639	0.235320	−	0.971918i	$-0.424386\pi$
$709$	8885.00	0.470639	0.235320	+	0.971918i	$0.424386\pi$
$710$	0	0
$711$	−1440.00	−0.0759553
$712$	−1920.00	−0.101060
$713$	−31266.0	−1.64225
$714$	−324.000	−0.0169823
$715$	0	0
$716$	−600.000	−0.0313171
$717$	7920.00	0.412521
$718$	−6840.00	−0.355524
$719$	−7530.00	−0.390572	−0.195286	−	0.980746i	$-0.562564\pi$
$719$	−7530.00	−0.390572	−0.195286	+	0.980746i	$0.562564\pi$
$720$	0	0
$721$	−668.000	−0.0345043
$722$	12732.0	0.656283
$723$	−16149.0	−0.830688
$724$	788.000	0.0404500
$725$	0	0
$726$	2598.00	0.132811
$727$	1801.00	0.0918781	0.0459391	−	0.998944i	$-0.485372\pi$
$727$	1801.00	0.0918781	0.0459391	+	0.998944i	$0.485372\pi$
$728$	536.000	0.0272877
$729$	729.000	0.0370370
$730$	0	0
$731$	14202.0	0.718577
$732$	−8796.00	−0.444139
$733$	7882.00	0.397174	0.198587	−	0.980083i	$-0.436365\pi$
$733$	7882.00	0.397174	0.198587	+	0.980083i	$0.436365\pi$
$734$	1742.00	0.0876000
$735$	0	0
$736$	5184.00	0.259626
$737$	−12558.0	−0.627652
$738$	216.000	0.0107738
$739$	33860.0	1.68547	0.842734	−	0.538331i	$-0.180945\pi$
$739$	33860.0	1.68547	0.842734	+	0.538331i	$0.180945\pi$
$740$	0	0
$741$	−23115.0	−1.14595
$742$	384.000	0.0189988
$743$	20652.0	1.01972	0.509858	−	0.860259i	$-0.329698\pi$
$743$	20652.0	1.01972	0.509858	+	0.860259i	$0.329698\pi$
$744$	−4632.00	−0.228249
$745$	0	0
$746$	−12766.0	−0.626537
$747$	4158.00	0.203659
$748$	−9072.00	−0.443456
$749$	1296.00	0.0632240
$750$	0	0
$751$	7472.00	0.363059	0.181529	−	0.983386i	$-0.441895\pi$
$751$	7472.00	0.363059	0.181529	+	0.983386i	$0.441895\pi$
$752$	−6624.00	−0.321213
$753$	−15084.0	−0.730002
$754$	−28140.0	−1.35915
$755$	0	0
$756$	108.000	0.00519566
$757$	32251.0	1.54846	0.774229	−	0.632906i	$-0.218138\pi$
$757$	32251.0	1.54846	0.774229	+	0.632906i	$0.218138\pi$
$758$	−19730.0	−0.945417
$759$	20412.0	0.976164
$760$	0	0
$761$	16812.0	0.800834	0.400417	−	0.916333i	$-0.368865\pi$
$761$	16812.0	0.800834	0.400417	+	0.916333i	$0.368865\pi$
$762$	96.0000	0.00456393
$763$	−1735.00	−0.0823214
$764$	5208.00	0.246622
$765$	0	0
$766$	−19656.0	−0.927154
$767$	46230.0	2.17636
$768$	768.000	0.0360844
$769$	−34645.0	−1.62462	−0.812309	−	0.583228i	$-0.801789\pi$
$769$	−34645.0	−1.62462	−0.812309	+	0.583228i	$0.801789\pi$
$770$	0	0
$771$	−1692.00	−0.0790349
$772$	−16652.0	−0.776319
$773$	8412.00	0.391408	0.195704	−	0.980663i	$-0.437301\pi$
$773$	8412.00	0.391408	0.195704	+	0.980663i	$0.437301\pi$
$774$	−4734.00	−0.219845
$775$	0	0
$776$	4088.00	0.189112
$777$	858.000	0.0396146
$778$	25080.0	1.15573
$779$	−1380.00	−0.0634706
$780$	0	0
$781$	−9576.00	−0.438740
$782$	−17496.0	−0.800071
$783$	−5670.00	−0.258786
$784$	−5472.00	−0.249271
$785$	0	0
$786$	11952.0	0.542384
$787$	−18329.0	−0.830188	−0.415094	−	0.909778i	$-0.636251\pi$
$787$	−18329.0	−0.830188	−0.415094	+	0.909778i	$0.636251\pi$
$788$	−12216.0	−0.552255
$789$	5436.00	0.245281
$790$	0	0
$791$	1092.00	0.0490860
$792$	3024.00	0.135673
$793$	−49111.0	−2.19922
$794$	2762.00	0.123451
$795$	0	0
$796$	13700.0	0.610030
$797$	−16044.0	−0.713059	−0.356529	−	0.934284i	$-0.616040\pi$
$797$	−16044.0	−0.713059	−0.356529	+	0.934284i	$0.616040\pi$
$798$	−690.000	−0.0306087
$799$	22356.0	0.989860
$800$	0	0
$801$	−2160.00	−0.0952807
$802$	28464.0	1.25324
$803$	−39396.0	−1.73133
$804$	−3588.00	−0.157387
$805$	0	0
$806$	−25862.0	−1.13021
$807$	−15570.0	−0.679170
$808$	7296.00	0.317664
$809$	−24000.0	−1.04301	−0.521505	−	0.853248i	$-0.674629\pi$
$809$	−24000.0	−1.04301	−0.521505	+	0.853248i	$0.674629\pi$
$810$	0	0
$811$	5117.00	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$811$	5117.00	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$812$	−840.000	−0.0363032
$813$	13776.0	0.594275
$814$	24024.0	1.03445
$815$	0	0
$816$	−2592.00	−0.111199
$817$	30245.0	1.29515
$818$	5290.00	0.226113
$819$	603.000	0.0257271
$820$	0	0
$821$	13542.0	0.575663	0.287831	−	0.957681i	$-0.407066\pi$
$821$	13542.0	0.575663	0.287831	+	0.957681i	$0.407066\pi$
$822$	14076.0	0.597271
$823$	−1283.00	−0.0543409	−0.0271705	−	0.999631i	$-0.508650\pi$
$823$	−1283.00	−0.0543409	−0.0271705	+	0.999631i	$0.508650\pi$
$824$	−5344.00	−0.225931
$825$	0	0
$826$	1380.00	0.0581312
$827$	−16344.0	−0.687227	−0.343613	−	0.939111i	$-0.611651\pi$
$827$	−16344.0	−0.687227	−0.343613	+	0.939111i	$0.611651\pi$
$828$	5832.00	0.244778
$829$	−790.000	−0.0330975	−0.0165488	−	0.999863i	$-0.505268\pi$
$829$	−790.000	−0.0330975	−0.0165488	+	0.999863i	$0.505268\pi$
$830$	0	0
$831$	6573.00	0.274386
$832$	4288.00	0.178677
$833$	18468.0	0.768161
$834$	17400.0	0.722437
$835$	0	0
$836$	−19320.0	−0.799278
$837$	−5211.00	−0.215195
$838$	6000.00	0.247335
$839$	−9990.00	−0.411076	−0.205538	−	0.978649i	$-0.565894\pi$
$839$	−9990.00	−0.411076	−0.205538	+	0.978649i	$0.565894\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	−22676.0	−0.928108
$843$	23526.0	0.961184
$844$	−9772.00	−0.398538
$845$	0	0
$846$	−7452.00	−0.302843
$847$	433.000	0.0175656
$848$	3072.00	0.124402
$849$	741.000	0.0299541
$850$	0	0
$851$	46332.0	1.86632
$852$	−2736.00	−0.110016
$853$	−24743.0	−0.993182	−0.496591	−	0.867985i	$-0.665415\pi$
$853$	−24743.0	−0.993182	−0.496591	+	0.867985i	$0.665415\pi$
$854$	−1466.00	−0.0587418
$855$	0	0
$856$	10368.0	0.413985
$857$	23556.0	0.938924	0.469462	−	0.882953i	$-0.344448\pi$
$857$	23556.0	0.938924	0.469462	+	0.882953i	$0.344448\pi$
$858$	16884.0	0.671807
$859$	−34000.0	−1.35048	−0.675242	−	0.737597i	$-0.735961\pi$
$859$	−34000.0	−1.35048	−0.675242	+	0.737597i	$0.735961\pi$
$860$	0	0
$861$	36.0000	0.00142494
$862$	−6516.00	−0.257466
$863$	37032.0	1.46070	0.730350	−	0.683073i	$-0.239357\pi$
$863$	37032.0	1.46070	0.730350	+	0.683073i	$0.239357\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−2326.00	−0.0912710
$867$	−5991.00	−0.234677
$868$	−772.000	−0.0301882
$869$	−6720.00	−0.262325
$870$	0	0
$871$	−20033.0	−0.779325
$872$	−13880.0	−0.539032
$873$	4599.00	0.178296
$874$	−37260.0	−1.44203
$875$	0	0
$876$	−11256.0	−0.434138
$877$	−2519.00	−0.0969904	−0.0484952	−	0.998823i	$-0.515443\pi$
$877$	−2519.00	−0.0969904	−0.0484952	+	0.998823i	$0.515443\pi$
$878$	13390.0	0.514682
$879$	16326.0	0.626465
$880$	0	0
$881$	43992.0	1.68232	0.841162	−	0.540783i	$-0.181872\pi$
$881$	43992.0	1.68232	0.841162	+	0.540783i	$0.181872\pi$
$882$	−6156.00	−0.235015
$883$	19177.0	0.730869	0.365435	−	0.930837i	$-0.380920\pi$
$883$	19177.0	0.730869	0.365435	+	0.930837i	$0.380920\pi$
$884$	−14472.0	−0.550618
$885$	0	0
$886$	−32736.0	−1.24130
$887$	−44994.0	−1.70321	−0.851607	−	0.524181i	$-0.824372\pi$
$887$	−44994.0	−1.70321	−0.851607	+	0.524181i	$0.824372\pi$
$888$	6864.00	0.259393
$889$	16.0000	0.000603625 0
$890$	0	0
$891$	3402.00	0.127914
$892$	−92.0000	−0.00345335
$893$	47610.0	1.78411
$894$	−12420.0	−0.464639
$895$	0	0
$896$	128.000	0.00477252
$897$	32562.0	1.21206
$898$	32760.0	1.21739
$899$	40530.0	1.50362
$900$	0	0
$901$	−10368.0	−0.383361
$902$	1008.00	0.0372092
$903$	−789.000	−0.0290767
$904$	8736.00	0.321410
$905$	0	0
$906$	13422.0	0.492181
$907$	52396.0	1.91817	0.959085	−	0.283117i	$-0.0913686\pi$
$907$	52396.0	1.91817	0.959085	+	0.283117i	$0.0913686\pi$
$908$	7824.00	0.285957
$909$	8208.00	0.299496
$910$	0	0
$911$	7242.00	0.263379	0.131689	−	0.991291i	$-0.457960\pi$
$911$	7242.00	0.263379	0.131689	+	0.991291i	$0.457960\pi$
$912$	−5520.00	−0.200423
$913$	19404.0	0.703372
$914$	27572.0	0.997813
$915$	0	0
$916$	7220.00	0.260432
$917$	1992.00	0.0717357
$918$	−2916.00	−0.104839
$919$	4085.00	0.146629	0.0733143	−	0.997309i	$-0.476642\pi$
$919$	4085.00	0.146629	0.0733143	+	0.997309i	$0.476642\pi$
$920$	0	0
$921$	11613.0	0.415485
$922$	23664.0	0.845263
$923$	−15276.0	−0.544762
$924$	504.000	0.0179441
$925$	0	0
$926$	−6016.00	−0.213497
$927$	−6012.00	−0.213010
$928$	−6720.00	−0.237710
$929$	−3030.00	−0.107009	−0.0535043	−	0.998568i	$-0.517039\pi$
$929$	−3030.00	−0.107009	−0.0535043	+	0.998568i	$0.517039\pi$
$930$	0	0
$931$	39330.0	1.38452
$932$	−13872.0	−0.487546
$933$	−17154.0	−0.601926
$934$	−8868.00	−0.310674
$935$	0	0
$936$	4824.00	0.168459
$937$	−5759.00	−0.200788	−0.100394	−	0.994948i	$-0.532010\pi$
$937$	−5759.00	−0.200788	−0.100394	+	0.994948i	$0.532010\pi$
$938$	−598.000	−0.0208160
$939$	10911.0	0.379198
$940$	0	0
$941$	−258.000	−0.00893790	−0.00446895	−	0.999990i	$-0.501423\pi$
$941$	−258.000	−0.00893790	−0.00446895	+	0.999990i	$0.501423\pi$
$942$	1446.00	0.0500140
$943$	1944.00	0.0671319
$944$	11040.0	0.380637
$945$	0	0
$946$	−22092.0	−0.759274
$947$	−1374.00	−0.0471478	−0.0235739	−	0.999722i	$-0.507505\pi$
$947$	−1374.00	−0.0471478	−0.0235739	+	0.999722i	$0.507505\pi$
$948$	−1920.00	−0.0657792
$949$	−62846.0	−2.14970
$950$	0	0
$951$	3888.00	0.132573
$952$	−432.000	−0.0147071
$953$	−9288.00	−0.315706	−0.157853	−	0.987463i	$-0.550457\pi$
$953$	−9288.00	−0.315706	−0.157853	+	0.987463i	$0.550457\pi$
$954$	3456.00	0.117287
$955$	0	0
$956$	10560.0	0.357254
$957$	−26460.0	−0.893762
$958$	14820.0	0.499804
$959$	2346.00	0.0789951
$960$	0	0
$961$	7458.00	0.250344
$962$	38324.0	1.28442
$963$	11664.0	0.390309
$964$	−21532.0	−0.719397
$965$	0	0
$966$	972.000	0.0323743
$967$	21616.0	0.718846	0.359423	−	0.933175i	$-0.382974\pi$
$967$	21616.0	0.718846	0.359423	+	0.933175i	$0.382974\pi$
$968$	3464.00	0.115018
$969$	18630.0	0.617628
$970$	0	0
$971$	−19098.0	−0.631188	−0.315594	−	0.948894i	$-0.602204\pi$
$971$	−19098.0	−0.631188	−0.315594	+	0.948894i	$0.602204\pi$
$972$	972.000	0.0320750
$973$	2900.00	0.0955496
$974$	17342.0	0.570507
$975$	0	0
$976$	−11728.0	−0.384635
$977$	18246.0	0.597483	0.298742	−	0.954334i	$-0.403433\pi$
$977$	18246.0	0.597483	0.298742	+	0.954334i	$0.403433\pi$
$978$	21282.0	0.695832
$979$	−10080.0	−0.329069
$980$	0	0
$981$	−15615.0	−0.508204
$982$	−38736.0	−1.25877
$983$	38772.0	1.25802	0.629011	−	0.777397i	$-0.283460\pi$
$983$	38772.0	1.25802	0.629011	+	0.777397i	$0.283460\pi$
$984$	288.000	0.00933039
$985$	0	0
$986$	22680.0	0.732534
$987$	−1242.00	−0.0400540
$988$	−30820.0	−0.992424
$989$	−42606.0	−1.36986
$990$	0	0
$991$	−23053.0	−0.738953	−0.369477	−	0.929240i	$-0.620463\pi$
$991$	−23053.0	−0.738953	−0.369477	+	0.929240i	$0.620463\pi$
$992$	−6176.00	−0.197670
$993$	15396.0	0.492021
$994$	−456.000	−0.0145507
$995$	0	0
$996$	5544.00	0.176374
$997$	10366.0	0.329282	0.164641	−	0.986354i	$-0.447353\pi$
$997$	10366.0	0.329282	0.164641	+	0.986354i	$0.447353\pi$
$998$	−17750.0	−0.562992
$999$	7722.00	0.244558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.a.h.1.1	yes	1
3.2	odd	2		450.4.a.f.1.1		1
4.3	odd	2		1200.4.a.i.1.1		1
5.2	odd	4		150.4.c.e.49.2		2
5.3	odd	4		150.4.c.e.49.1		2
5.4	even	2		150.4.a.a.1.1	✓	1
15.2	even	4		450.4.c.a.199.1		2
15.8	even	4		450.4.c.a.199.2		2
15.14	odd	2		450.4.a.o.1.1		1
20.3	even	4		1200.4.f.c.49.1		2
20.7	even	4		1200.4.f.c.49.2		2
20.19	odd	2		1200.4.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.a.1.1	✓	1	5.4	even	2
150.4.a.h.1.1	yes	1	1.1	even	1	trivial
150.4.c.e.49.1		2	5.3	odd	4
150.4.c.e.49.2		2	5.2	odd	4
450.4.a.f.1.1		1	3.2	odd	2
450.4.a.o.1.1		1	15.14	odd	2
450.4.c.a.199.1		2	15.2	even	4
450.4.c.a.199.2		2	15.8	even	4
1200.4.a.i.1.1		1	4.3	odd	2
1200.4.a.bb.1.1		1	20.19	odd	2
1200.4.f.c.49.1		2	20.3	even	4
1200.4.f.c.49.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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