LMFDB - Embedded newform 150.4.c.e.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,4,Mod(49,150)] 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("150.49"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,12,0,0,-18,0,84] 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:	no
Analytic conductor:	$8.85028650086$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	150.49
Dual form			150.4.c.e.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} +1.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +42.0000 q^{11} +12.0000i q^{12} -67.0000i q^{13} -2.00000 q^{14} +16.0000 q^{16} -54.0000i q^{17} -18.0000i q^{18} +115.000 q^{19} +3.00000 q^{21} +84.0000i q^{22} -162.000i q^{23} -24.0000 q^{24} +134.000 q^{26} +27.0000i q^{27} -4.00000i q^{28} +210.000 q^{29} -193.000 q^{31} +32.0000i q^{32} -126.000i q^{33} +108.000 q^{34} +36.0000 q^{36} +286.000i q^{37} +230.000i q^{38} -201.000 q^{39} +12.0000 q^{41} +6.00000i q^{42} +263.000i q^{43} -168.000 q^{44} +324.000 q^{46} -414.000i q^{47} -48.0000i q^{48} +342.000 q^{49} -162.000 q^{51} +268.000i q^{52} -192.000i q^{53} -54.0000 q^{54} +8.00000 q^{56} -345.000i q^{57} +420.000i q^{58} -690.000 q^{59} -733.000 q^{61} -386.000i q^{62} -9.00000i q^{63} -64.0000 q^{64} +252.000 q^{66} -299.000i q^{67} +216.000i q^{68} -486.000 q^{69} -228.000 q^{71} +72.0000i q^{72} +938.000i q^{73} -572.000 q^{74} -460.000 q^{76} +42.0000i q^{77} -402.000i q^{78} +160.000 q^{79} +81.0000 q^{81} +24.0000i q^{82} -462.000i q^{83} -12.0000 q^{84} -526.000 q^{86} -630.000i q^{87} -336.000i q^{88} +240.000 q^{89} +67.0000 q^{91} +648.000i q^{92} +579.000i q^{93} +828.000 q^{94} +96.0000 q^{96} +511.000i q^{97} +684.000i q^{98} -378.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} + 84 q^{11} - 4 q^{14} + 32 q^{16} + 230 q^{19} + 6 q^{21} - 48 q^{24} + 268 q^{26} + 420 q^{29} - 386 q^{31} + 216 q^{34} + 72 q^{36} - 402 q^{39} + 24 q^{41} - 336 q^{44}+ \cdots - 756 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 + 12 * q^6 - 18 * q^9 + 84 * q^11 - 4 * q^14 + 32 * q^16 + 230 * q^19 + 6 * q^21 - 48 * q^24 + 268 * q^26 + 420 * q^29 - 386 * q^31 + 216 * q^34 + 72 * q^36 - 402 * q^39 + 24 * q^41 - 336 * q^44 + 648 * q^46 + 684 * q^49 - 324 * q^51 - 108 * q^54 + 16 * q^56 - 1380 * q^59 - 1466 * q^61 - 128 * q^64 + 504 * q^66 - 972 * q^69 - 456 * q^71 - 1144 * q^74 - 920 * q^76 + 320 * q^79 + 162 * q^81 - 24 * q^84 - 1052 * q^86 + 480 * q^89 + 134 * q^91 + 1656 * q^94 + 192 * q^96 - 756 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	1.00000i	0.0539949i	0.999636	+	0.0269975i	$0.00859460\pi$
$7$	1.00000i	0.0539949i	−0.999636	+	0.0269975i	$0.991405\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−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	$0.304764\pi$
$12$	12.0000i	0.288675i
$13$	− 67.0000i	− 1.42942i	−0.699421	−	0.714710i	$-0.746559\pi$
$13$	− 67.0000i	− 1.42942i	0.699421	−	0.714710i	$-0.253441\pi$
$14$	−2.00000	−0.0381802
$15$	0	0
$16$	16.0000	0.250000
$17$	− 54.0000i	− 0.770407i	−0.922832	−	0.385204i	$-0.874131\pi$
$17$	− 54.0000i	− 0.770407i	0.922832	−	0.385204i	$-0.125869\pi$
$18$	− 18.0000i	− 0.235702i
$19$	115.000	1.38857	0.694284	−	0.719701i	$-0.255721\pi$
$19$	115.000	1.38857	0.694284	+	0.719701i	$0.255721\pi$
$20$	0	0
$21$	3.00000	0.0311740
$22$	84.0000i	0.814039i
$23$	− 162.000i	− 1.46867i	−0.678789	−	0.734333i	$-0.737495\pi$
$23$	− 162.000i	− 1.46867i	0.678789	−	0.734333i	$-0.262505\pi$
$24$	−24.0000	−0.204124
$25$	0	0
$26$	134.000	1.01075
$27$	27.0000i	0.192450i
$28$	− 4.00000i	− 0.0269975i
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	0	0
$31$	−193.000	−1.11819	−0.559094	−	0.829104i	$-0.688851\pi$
$31$	−193.000	−1.11819	−0.559094	+	0.829104i	$0.688851\pi$
$32$	32.0000i	0.176777i
$33$	− 126.000i	− 0.664660i
$34$	108.000	0.544760
$35$	0	0
$36$	36.0000	0.166667
$37$	286.000i	1.27076i	0.772200	+	0.635380i	$0.219156\pi$
$37$	286.000i	1.27076i	−0.772200	+	0.635380i	$0.780844\pi$
$38$	230.000i	0.981866i
$39$	−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	$0.492724\pi$
$42$	6.00000i	0.0220433i
$43$	263.000i	0.932724i	0.884594	+	0.466362i	$0.154436\pi$
$43$	263.000i	0.932724i	−0.884594	+	0.466362i	$0.845564\pi$
$44$	−168.000	−0.575613
$45$	0	0
$46$	324.000	1.03850
$47$	− 414.000i	− 1.28485i	−0.766347	−	0.642427i	$-0.777928\pi$
$47$	− 414.000i	− 1.28485i	0.766347	−	0.642427i	$-0.222072\pi$
$48$	− 48.0000i	− 0.144338i
$49$	342.000	0.997085
$50$	0	0
$51$	−162.000	−0.444795
$52$	268.000i	0.714710i
$53$	− 192.000i	− 0.497608i	−0.968554	−	0.248804i	$-0.919962\pi$
$53$	− 192.000i	− 0.497608i	0.968554	−	0.248804i	$-0.0800375\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	8.00000	0.0190901
$57$	− 345.000i	− 0.801691i
$58$	420.000i	0.950840i
$59$	−690.000	−1.52255	−0.761274	−	0.648430i	$-0.775426\pi$
$59$	−690.000	−1.52255	−0.761274	+	0.648430i	$0.775426\pi$
$60$	0	0
$61$	−733.000	−1.53854	−0.769271	−	0.638923i	$-0.779380\pi$
$61$	−733.000	−1.53854	−0.769271	+	0.638923i	$0.779380\pi$
$62$	− 386.000i	− 0.790678i
$63$	− 9.00000i	− 0.0179983i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	252.000	0.469986
$67$	− 299.000i	− 0.545204i	−0.962127	−	0.272602i	$-0.912116\pi$
$67$	− 299.000i	− 0.545204i	0.962127	−	0.272602i	$-0.0878842\pi$
$68$	216.000i	0.385204i
$69$	−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	$0.561028\pi$
$72$	72.0000i	0.117851i
$73$	938.000i	1.50390i	0.659221	+	0.751949i	$0.270886\pi$
$73$	938.000i	1.50390i	−0.659221	+	0.751949i	$0.729114\pi$
$74$	−572.000	−0.898563
$75$	0	0
$76$	−460.000	−0.694284
$77$	42.0000i	0.0621603i
$78$	− 402.000i	− 0.583558i
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	24.0000i	0.0323214i
$83$	− 462.000i	− 0.610977i	−0.952196	−	0.305488i	$-0.901180\pi$
$83$	− 462.000i	− 0.610977i	0.952196	−	0.305488i	$-0.0988197\pi$
$84$	−12.0000	−0.0155870
$85$	0	0
$86$	−526.000	−0.659535
$87$	− 630.000i	− 0.776357i
$88$	− 336.000i	− 0.407020i
$89$	240.000	0.285842	0.142921	−	0.989734i	$-0.454350\pi$
$89$	240.000	0.285842	0.142921	+	0.989734i	$0.454350\pi$
$90$	0	0
$91$	67.0000	0.0771814
$92$	648.000i	0.734333i
$93$	579.000i	0.645586i
$94$	828.000	0.908529
$95$	0	0
$96$	96.0000	0.102062
$97$	511.000i	0.534889i	0.963573	+	0.267444i	$0.0861791\pi$
$97$	511.000i	0.534889i	−0.963573	+	0.267444i	$0.913821\pi$
$98$	684.000i	0.705045i
$99$	−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	$0.351693\pi$
$102$	− 324.000i	− 0.314517i
$103$	668.000i	0.639029i	0.947581	+	0.319515i	$0.103520\pi$
$103$	668.000i	0.639029i	−0.947581	+	0.319515i	$0.896480\pi$
$104$	−536.000	−0.505376
$105$	0	0
$106$	384.000	0.351862
$107$	1296.00i	1.17093i	0.810699	+	0.585463i	$0.199087\pi$
$107$	1296.00i	1.17093i	−0.810699	+	0.585463i	$0.800913\pi$
$108$	− 108.000i	− 0.0962250i
$109$	1735.00	1.52461	0.762307	−	0.647216i	$-0.224067\pi$
$109$	1735.00	1.52461	0.762307	+	0.647216i	$0.224067\pi$
$110$	0	0
$111$	858.000	0.733673
$112$	16.0000i	0.0134987i
$113$	− 1092.00i	− 0.909086i	−0.890725	−	0.454543i	$-0.849803\pi$
$113$	− 1092.00i	− 0.909086i	0.890725	−	0.454543i	$-0.150197\pi$
$114$	690.000	0.566881
$115$	0	0
$116$	−840.000	−0.672345
$117$	603.000i	0.476473i
$118$	− 1380.00i	− 1.07660i
$119$	54.0000	0.0415981
$120$	0	0
$121$	433.000	0.325319
$122$	− 1466.00i	− 1.08791i
$123$	− 36.0000i	− 0.0263903i
$124$	772.000	0.559094
$125$	0	0
$126$	18.0000	0.0127267
$127$	16.0000i	0.0111793i	0.999984	+	0.00558965i	$0.00177925\pi$
$127$	16.0000i	0.0111793i	−0.999984	+	0.00558965i	$0.998221\pi$
$128$	− 128.000i	− 0.0883883i
$129$	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	$0.268737\pi$
$132$	504.000i	0.332330i
$133$	115.000i	0.0749757i
$134$	598.000	0.385517
$135$	0	0
$136$	−432.000	−0.272380
$137$	2346.00i	1.46301i	0.681836	+	0.731505i	$0.261182\pi$
$137$	2346.00i	1.46301i	−0.681836	+	0.731505i	$0.738818\pi$
$138$	− 972.000i	− 0.599581i
$139$	−2900.00	−1.76960	−0.884801	−	0.465968i	$-0.845706\pi$
$139$	−2900.00	−1.76960	−0.884801	+	0.465968i	$0.845706\pi$
$140$	0	0
$141$	−1242.00	−0.741810
$142$	− 456.000i	− 0.269484i
$143$	− 2814.00i	− 1.64558i
$144$	−144.000	−0.0833333
$145$	0	0
$146$	−1876.00	−1.06342
$147$	− 1026.00i	− 0.575667i
$148$	− 1144.00i	− 0.635380i
$149$	2070.00	1.13813	0.569064	−	0.822293i	$-0.307306\pi$
$149$	2070.00	1.13813	0.569064	+	0.822293i	$0.307306\pi$
$150$	0	0
$151$	2237.00	1.20559	0.602796	−	0.797895i	$-0.294053\pi$
$151$	2237.00	1.20559	0.602796	+	0.797895i	$0.294053\pi$
$152$	− 920.000i	− 0.490933i
$153$	486.000i	0.256802i
$154$	−84.0000	−0.0439540
$155$	0	0
$156$	804.000	0.412638
$157$	241.000i	0.122509i	0.998122	+	0.0612544i	$0.0195101\pi$
$157$	241.000i	0.122509i	−0.998122	+	0.0612544i	$0.980490\pi$
$158$	320.000i	0.161126i
$159$	−576.000	−0.287294
$160$	0	0
$161$	162.000	0.0793006
$162$	162.000i	0.0785674i
$163$	− 3547.00i	− 1.70443i	−0.523190	−	0.852216i	$-0.675258\pi$
$163$	− 3547.00i	− 1.70443i	0.523190	−	0.852216i	$-0.324742\pi$
$164$	−48.0000	−0.0228547
$165$	0	0
$166$	924.000	0.432026
$167$	− 984.000i	− 0.455953i	−0.973667	−	0.227977i	$-0.926789\pi$
$167$	− 984.000i	− 0.455953i	0.973667	−	0.227977i	$-0.0732110\pi$
$168$	− 24.0000i	− 0.0110217i
$169$	−2292.00	−1.04324
$170$	0	0
$171$	−1035.00	−0.462856
$172$	− 1052.00i	− 0.466362i
$173$	3618.00i	1.59001i	0.606604	+	0.795004i	$0.292531\pi$
$173$	3618.00i	1.59001i	−0.606604	+	0.795004i	$0.707469\pi$
$174$	1260.00	0.548968
$175$	0	0
$176$	672.000	0.287806
$177$	2070.00i	0.879044i
$178$	480.000i	0.202121i
$179$	150.000	0.0626342	0.0313171	−	0.999509i	$-0.490030\pi$
$179$	150.000	0.0626342	0.0313171	+	0.999509i	$0.490030\pi$
$180$	0	0
$181$	197.000	0.0809000	0.0404500	−	0.999182i	$-0.487121\pi$
$181$	197.000	0.0809000	0.0404500	+	0.999182i	$0.487121\pi$
$182$	134.000i	0.0545755i
$183$	2199.00i	0.888277i
$184$	−1296.00	−0.519252
$185$	0	0
$186$	−1158.00	−0.456498
$187$	− 2268.00i	− 0.886912i
$188$	1656.00i	0.642427i
$189$	−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	$0.420680\pi$
$192$	192.000i	0.0721688i
$193$	4163.00i	1.55264i	0.630340	+	0.776319i	$0.282916\pi$
$193$	4163.00i	1.55264i	−0.630340	+	0.776319i	$0.717084\pi$
$194$	−1022.00	−0.378223
$195$	0	0
$196$	−1368.00	−0.498542
$197$	− 3054.00i	− 1.10451i	−0.833675	−	0.552255i	$-0.813767\pi$
$197$	− 3054.00i	− 1.10451i	0.833675	−	0.552255i	$-0.186233\pi$
$198$	− 756.000i	− 0.271346i
$199$	−3425.00	−1.22006	−0.610030	−	0.792379i	$-0.708842\pi$
$199$	−3425.00	−1.22006	−0.610030	+	0.792379i	$0.708842\pi$
$200$	0	0
$201$	−897.000	−0.314774
$202$	1824.00i	0.635328i
$203$	210.000i	0.0726065i
$204$	648.000	0.222397
$205$	0	0
$206$	−1336.00	−0.451862
$207$	1458.00i	0.489556i
$208$	− 1072.00i	− 0.357355i
$209$	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	$0.630482\pi$
$212$	768.000i	0.248804i
$213$	684.000i	0.220032i
$214$	−2592.00	−0.827969
$215$	0	0
$216$	216.000	0.0680414
$217$	− 193.000i	− 0.0603765i
$218$	3470.00i	1.07806i
$219$	2814.00	0.868276
$220$	0	0
$221$	−3618.00	−1.10124
$222$	1716.00i	0.518785i
$223$	23.0000i	0.00690670i	0.999994	+	0.00345335i	$0.00109924\pi$
$223$	23.0000i	0.00690670i	−0.999994	+	0.00345335i	$0.998901\pi$
$224$	−32.0000	−0.00954504
$225$	0	0
$226$	2184.00	0.642821
$227$	1956.00i	0.571913i	0.958243	+	0.285957i	$0.0923113\pi$
$227$	1956.00i	0.571913i	−0.958243	+	0.285957i	$0.907689\pi$
$228$	1380.00i	0.400845i
$229$	−1805.00	−0.520864	−0.260432	−	0.965492i	$-0.583865\pi$
$229$	−1805.00	−0.520864	−0.260432	+	0.965492i	$0.583865\pi$
$230$	0	0
$231$	126.000	0.0358883
$232$	− 1680.00i	− 0.475420i
$233$	3468.00i	0.975091i	0.873098	+	0.487546i	$0.162108\pi$
$233$	3468.00i	0.975091i	−0.873098	+	0.487546i	$0.837892\pi$
$234$	−1206.00	−0.336917
$235$	0	0
$236$	2760.00	0.761274
$237$	− 480.000i	− 0.131558i
$238$	108.000i	0.0294143i
$239$	−2640.00	−0.714508	−0.357254	−	0.934007i	$-0.616287\pi$
$239$	−2640.00	−0.714508	−0.357254	+	0.934007i	$0.616287\pi$
$240$	0	0
$241$	−5383.00	−1.43879	−0.719397	−	0.694599i	$-0.755582\pi$
$241$	−5383.00	−1.43879	−0.719397	+	0.694599i	$0.755582\pi$
$242$	866.000i	0.230035i
$243$	− 243.000i	− 0.0641500i
$244$	2932.00	0.769271
$245$	0	0
$246$	72.0000	0.0186608
$247$	− 7705.00i	− 1.98485i
$248$	1544.00i	0.395339i
$249$	−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	$0.717848\pi$
$252$	36.0000i	0.00899915i
$253$	− 6804.00i	− 1.69077i
$254$	−32.0000	−0.00790496
$255$	0	0
$256$	256.000	0.0625000
$257$	− 564.000i	− 0.136892i	−0.997655	−	0.0684462i	$-0.978196\pi$
$257$	− 564.000i	− 0.136892i	0.997655	−	0.0684462i	$-0.0218042\pi$
$258$	1578.00i	0.380783i
$259$	−286.000	−0.0686146
$260$	0	0
$261$	−1890.00	−0.448230
$262$	3984.00i	0.939436i
$263$	− 1812.00i	− 0.424839i	−0.977179	−	0.212420i	$-0.931866\pi$
$263$	− 1812.00i	− 0.424839i	0.977179	−	0.212420i	$-0.0681344\pi$
$264$	−1008.00	−0.234993
$265$	0	0
$266$	−230.000	−0.0530158
$267$	− 720.000i	− 0.165031i
$268$	1196.00i	0.272602i
$269$	5190.00	1.17636	0.588178	−	0.808731i	$-0.299845\pi$
$269$	5190.00	1.17636	0.588178	+	0.808731i	$0.299845\pi$
$270$	0	0
$271$	4592.00	1.02931	0.514657	−	0.857396i	$-0.327919\pi$
$271$	4592.00	1.02931	0.514657	+	0.857396i	$0.327919\pi$
$272$	− 864.000i	− 0.192602i
$273$	− 201.000i	− 0.0445607i
$274$	−4692.00	−1.03450
$275$	0	0
$276$	1944.00	0.423968
$277$	2191.00i	0.475251i	0.971357	+	0.237625i	$0.0763690\pi$
$277$	2191.00i	0.475251i	−0.971357	+	0.237625i	$0.923631\pi$
$278$	− 5800.00i	− 1.25130i
$279$	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	$0.186960\pi$
$282$	− 2484.00i	− 0.524539i
$283$	− 247.000i	− 0.0518821i	−0.999663	−	0.0259410i	$-0.991742\pi$
$283$	− 247.000i	− 0.0518821i	0.999663	−	0.0259410i	$-0.00825821\pi$
$284$	912.000	0.190554
$285$	0	0
$286$	5628.00	1.16360
$287$	12.0000i	0.00246808i
$288$	− 288.000i	− 0.0589256i
$289$	1997.00	0.406473
$290$	0	0
$291$	1533.00	0.308818
$292$	− 3752.00i	− 0.751949i
$293$	− 5442.00i	− 1.08507i	−0.840034	−	0.542534i	$-0.817465\pi$
$293$	− 5442.00i	− 1.08507i	0.840034	−	0.542534i	$-0.182535\pi$
$294$	2052.00	0.407058
$295$	0	0
$296$	2288.00	0.449281
$297$	1134.00i	0.221553i
$298$	4140.00i	0.804778i
$299$	−10854.0	−2.09934
$300$	0	0
$301$	−263.000	−0.0503624
$302$	4474.00i	0.852483i
$303$	− 2736.00i	− 0.518743i
$304$	1840.00	0.347142
$305$	0	0
$306$	−972.000	−0.181587
$307$	3871.00i	0.719641i	0.933022	+	0.359820i	$0.117162\pi$
$307$	3871.00i	0.719641i	−0.933022	+	0.359820i	$0.882838\pi$
$308$	− 168.000i	− 0.0310802i
$309$	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	$0.674546\pi$
$312$	1608.00i	0.291779i
$313$	− 3637.00i	− 0.656790i	−0.944540	−	0.328395i	$-0.893492\pi$
$313$	− 3637.00i	− 0.656790i	0.944540	−	0.328395i	$-0.106508\pi$
$314$	−482.000	−0.0866269
$315$	0	0
$316$	−640.000	−0.113933
$317$	1296.00i	0.229623i	0.993387	+	0.114812i	$0.0366265\pi$
$317$	1296.00i	0.229623i	−0.993387	+	0.114812i	$0.963374\pi$
$318$	− 1152.00i	− 0.203148i
$319$	8820.00	1.54804
$320$	0	0
$321$	3888.00	0.676034
$322$	324.000i	0.0560740i
$323$	− 6210.00i	− 1.06976i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	7094.00	1.20522
$327$	− 5205.00i	− 0.880236i
$328$	− 96.0000i	− 0.0161607i
$329$	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	$0.359886\pi$
$332$	1848.00i	0.305488i
$333$	− 2574.00i	− 0.423587i
$334$	1968.00	0.322408
$335$	0	0
$336$	48.0000	0.00779350
$337$	6751.00i	1.09125i	0.838030	+	0.545624i	$0.183707\pi$
$337$	6751.00i	1.09125i	−0.838030	+	0.545624i	$0.816293\pi$
$338$	− 4584.00i	− 0.737683i
$339$	−3276.00	−0.524861
$340$	0	0
$341$	−8106.00	−1.28729
$342$	− 2070.00i	− 0.327289i
$343$	685.000i	0.107832i
$344$	2104.00	0.329768
$345$	0	0
$346$	−7236.00	−1.12431
$347$	5226.00i	0.808491i	0.914651	+	0.404246i	$0.132466\pi$
$347$	5226.00i	0.808491i	−0.914651	+	0.404246i	$0.867534\pi$
$348$	2520.00i	0.388179i
$349$	6190.00	0.949407	0.474704	−	0.880146i	$-0.342555\pi$
$349$	6190.00	0.949407	0.474704	+	0.880146i	$0.342555\pi$
$350$	0	0
$351$	1809.00	0.275092
$352$	1344.00i	0.203510i
$353$	6618.00i	0.997849i	0.866646	+	0.498924i	$0.166271\pi$
$353$	6618.00i	0.997849i	−0.866646	+	0.498924i	$0.833729\pi$
$354$	−4140.00	−0.621578
$355$	0	0
$356$	−960.000	−0.142921
$357$	− 162.000i	− 0.0240167i
$358$	300.000i	0.0442891i
$359$	3420.00	0.502787	0.251394	−	0.967885i	$-0.419111\pi$
$359$	3420.00	0.502787	0.251394	+	0.967885i	$0.419111\pi$
$360$	0	0
$361$	6366.00	0.928124
$362$	394.000i	0.0572049i
$363$	− 1299.00i	− 0.187823i
$364$	−268.000	−0.0385907
$365$	0	0
$366$	−4398.00	−0.628107
$367$	871.000i	0.123885i	0.998080	+	0.0619425i	$0.0197296\pi$
$367$	871.000i	0.123885i	−0.998080	+	0.0619425i	$0.980270\pi$
$368$	− 2592.00i	− 0.367167i
$369$	−108.000	−0.0152365
$370$	0	0
$371$	192.000	0.0268683
$372$	− 2316.00i	− 0.322793i
$373$	6383.00i	0.886057i	0.896508	+	0.443028i	$0.146096\pi$
$373$	6383.00i	0.886057i	−0.896508	+	0.443028i	$0.853904\pi$
$374$	4536.00	0.627142
$375$	0	0
$376$	−3312.00	−0.454264
$377$	− 14070.0i	− 1.92213i
$378$	− 54.0000i	− 0.00734778i
$379$	9865.00	1.33702	0.668511	−	0.743703i	$-0.266932\pi$
$379$	9865.00	1.33702	0.668511	+	0.743703i	$0.266932\pi$
$380$	0	0
$381$	48.0000	0.00645437
$382$	2604.00i	0.348775i
$383$	9828.00i	1.31119i	0.755111	+	0.655597i	$0.227583\pi$
$383$	9828.00i	1.31119i	−0.755111	+	0.655597i	$0.772417\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	−8326.00	−1.09788
$387$	− 2367.00i	− 0.310908i
$388$	− 2044.00i	− 0.267444i
$389$	−12540.0	−1.63446	−0.817228	−	0.576315i	$-0.804490\pi$
$389$	−12540.0	−1.63446	−0.817228	+	0.576315i	$0.804490\pi$
$390$	0	0
$391$	−8748.00	−1.13147
$392$	− 2736.00i	− 0.352523i
$393$	− 5976.00i	− 0.767047i
$394$	6108.00	0.781007
$395$	0	0
$396$	1512.00	0.191871
$397$	1381.00i	0.174585i	0.996183	+	0.0872927i	$0.0278215\pi$
$397$	1381.00i	0.174585i	−0.996183	+	0.0872927i	$0.972178\pi$
$398$	− 6850.00i	− 0.862712i
$399$	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	$0.153353\pi$
$402$	− 1794.00i	− 0.222579i
$403$	12931.0i	1.59836i
$404$	−3648.00	−0.449245
$405$	0	0
$406$	−420.000	−0.0513405
$407$	12012.0i	1.46293i
$408$	1296.00i	0.157259i
$409$	−2645.00	−0.319772	−0.159886	−	0.987135i	$-0.551113\pi$
$409$	−2645.00	−0.319772	−0.159886	+	0.987135i	$0.551113\pi$
$410$	0	0
$411$	7038.00	0.844669
$412$	− 2672.00i	− 0.319515i
$413$	− 690.000i	− 0.0822099i
$414$	−2916.00	−0.346168
$415$	0	0
$416$	2144.00	0.252688
$417$	8700.00i	1.02168i
$418$	9660.00i	1.13035i
$419$	−3000.00	−0.349784	−0.174892	−	0.984588i	$-0.555958\pi$
$419$	−3000.00	−0.349784	−0.174892	+	0.984588i	$0.555958\pi$
$420$	0	0
$421$	−11338.0	−1.31254	−0.656271	−	0.754525i	$-0.727867\pi$
$421$	−11338.0	−1.31254	−0.656271	+	0.754525i	$0.727867\pi$
$422$	− 4886.00i	− 0.563618i
$423$	3726.00i	0.428284i
$424$	−1536.00	−0.175931
$425$	0	0
$426$	−1368.00	−0.155586
$427$	− 733.000i	− 0.0830734i
$428$	− 5184.00i	− 0.585463i
$429$	−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	$0.558275\pi$
$432$	432.000i	0.0481125i
$433$	1163.00i	0.129077i	0.997915	+	0.0645384i	$0.0205575\pi$
$433$	1163.00i	0.129077i	−0.997915	+	0.0645384i	$0.979443\pi$
$434$	386.000	0.0426926
$435$	0	0
$436$	−6940.00	−0.762307
$437$	− 18630.0i	− 2.03934i
$438$	5628.00i	0.613964i
$439$	−6695.00	−0.727870	−0.363935	−	0.931424i	$-0.618567\pi$
$439$	−6695.00	−0.727870	−0.363935	+	0.931424i	$0.618567\pi$
$440$	0	0
$441$	−3078.00	−0.332362
$442$	− 7236.00i	− 0.778691i
$443$	16368.0i	1.75546i	0.479159	+	0.877728i	$0.340942\pi$
$443$	16368.0i	1.75546i	−0.479159	+	0.877728i	$0.659058\pi$
$444$	−3432.00	−0.366837
$445$	0	0
$446$	−46.0000	−0.00488377
$447$	− 6210.00i	− 0.657098i
$448$	− 64.0000i	− 0.00674937i
$449$	−16380.0	−1.72165	−0.860824	−	0.508903i	$-0.830051\pi$
$449$	−16380.0	−1.72165	−0.860824	+	0.508903i	$0.830051\pi$
$450$	0	0
$451$	504.000	0.0526218
$452$	4368.00i	0.454543i
$453$	− 6711.00i	− 0.696049i
$454$	−3912.00	−0.404404
$455$	0	0
$456$	−2760.00	−0.283440
$457$	13786.0i	1.41112i	0.708650	+	0.705560i	$0.249304\pi$
$457$	13786.0i	1.41112i	−0.708650	+	0.705560i	$0.750696\pi$
$458$	− 3610.00i	− 0.368306i
$459$	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	$0.296085\pi$
$462$	252.000i	0.0253768i
$463$	3008.00i	0.301930i	0.988539	+	0.150965i	$0.0482381\pi$
$463$	3008.00i	0.301930i	−0.988539	+	0.150965i	$0.951762\pi$
$464$	3360.00	0.336173
$465$	0	0
$466$	−6936.00	−0.689494
$467$	− 4434.00i	− 0.439360i	−0.975572	−	0.219680i	$-0.929499\pi$
$467$	− 4434.00i	− 0.439360i	0.975572	−	0.219680i	$-0.0705013\pi$
$468$	− 2412.00i	− 0.238237i
$469$	299.000	0.0294382
$470$	0	0
$471$	723.000	0.0707305
$472$	5520.00i	0.538302i
$473$	11046.0i	1.07378i
$474$	960.000	0.0930259
$475$	0	0
$476$	−216.000	−0.0207990
$477$	1728.00i	0.165869i
$478$	− 5280.00i	− 0.505233i
$479$	−7410.00	−0.706830	−0.353415	−	0.935467i	$-0.614980\pi$
$479$	−7410.00	−0.706830	−0.353415	+	0.935467i	$0.614980\pi$
$480$	0	0
$481$	19162.0	1.81645
$482$	− 10766.0i	− 1.01738i
$483$	− 486.000i	− 0.0457842i
$484$	−1732.00	−0.162660
$485$	0	0
$486$	486.000	0.0453609
$487$	8671.00i	0.806818i	0.915020	+	0.403409i	$0.132175\pi$
$487$	8671.00i	0.806818i	−0.915020	+	0.403409i	$0.867825\pi$
$488$	5864.00i	0.543957i
$489$	−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	$0.849357\pi$
$492$	144.000i	0.0131952i
$493$	− 11340.0i	− 1.03596i
$494$	15410.0	1.40350
$495$	0	0
$496$	−3088.00	−0.279547
$497$	− 228.000i	− 0.0205779i
$498$	− 2772.00i	− 0.249430i
$499$	8875.00	0.796192	0.398096	−	0.917344i	$-0.369671\pi$
$499$	8875.00	0.796192	0.398096	+	0.917344i	$0.369671\pi$
$500$	0	0
$501$	−2952.00	−0.263245
$502$	− 10056.0i	− 0.894066i
$503$	− 10452.0i	− 0.926504i	−0.886227	−	0.463252i	$-0.846682\pi$
$503$	− 10452.0i	− 0.926504i	0.886227	−	0.463252i	$-0.153318\pi$
$504$	−72.0000	−0.00636336
$505$	0	0
$506$	13608.0	1.19555
$507$	6876.00i	0.602315i
$508$	− 64.0000i	− 0.00558965i
$509$	19770.0	1.72159	0.860796	−	0.508951i	$-0.169967\pi$
$509$	19770.0	1.72159	0.860796	+	0.508951i	$0.169967\pi$
$510$	0	0
$511$	−938.000	−0.0812029
$512$	512.000i	0.0441942i
$513$	3105.00i	0.267230i
$514$	1128.00	0.0967976
$515$	0	0
$516$	−3156.00	−0.269254
$517$	− 17388.0i	− 1.47916i
$518$	− 572.000i	− 0.0485178i
$519$	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	$0.656649\pi$
$522$	− 3780.00i	− 0.316947i
$523$	− 7447.00i	− 0.622628i	−0.950307	−	0.311314i	$-0.899231\pi$
$523$	− 7447.00i	− 0.622628i	0.950307	−	0.311314i	$-0.100769\pi$
$524$	−7968.00	−0.664282
$525$	0	0
$526$	3624.00	0.300407
$527$	10422.0i	0.861460i
$528$	− 2016.00i	− 0.166165i
$529$	−14077.0	−1.15698
$530$	0	0
$531$	6210.00	0.507516
$532$	− 460.000i	− 0.0374878i
$533$	− 804.000i	− 0.0653379i
$534$	1440.00	0.116695
$535$	0	0
$536$	−2392.00	−0.192759
$537$	− 450.000i	− 0.0361619i
$538$	10380.0i	0.831810i
$539$	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	$0.746929\pi$
$542$	9184.00i	0.727835i
$543$	− 591.000i	− 0.0467076i
$544$	1728.00	0.136190
$545$	0	0
$546$	402.000	0.0315092
$547$	10096.0i	0.789166i	0.918860	+	0.394583i	$0.129111\pi$
$547$	10096.0i	0.789166i	−0.918860	+	0.394583i	$0.870889\pi$
$548$	− 9384.00i	− 0.731505i
$549$	6597.00	0.512847
$550$	0	0
$551$	24150.0	1.86720
$552$	3888.00i	0.299790i
$553$	160.000i	0.0123036i
$554$	−4382.00	−0.336053
$555$	0	0
$556$	11600.0	0.884801
$557$	− 14514.0i	− 1.10409i	−0.833814	−	0.552045i	$-0.813848\pi$
$557$	− 14514.0i	− 1.10409i	0.833814	−	0.552045i	$-0.186152\pi$
$558$	3474.00i	0.263559i
$559$	17621.0	1.33325
$560$	0	0
$561$	−6804.00	−0.512059
$562$	15684.0i	1.17721i
$563$	− 10242.0i	− 0.766694i	−0.923604	−	0.383347i	$-0.874771\pi$
$563$	− 10242.0i	− 0.766694i	0.923604	−	0.383347i	$-0.125229\pi$
$564$	4968.00	0.370905
$565$	0	0
$566$	494.000	0.0366862
$567$	81.0000i	0.00599944i
$568$	1824.00i	0.134742i
$569$	6750.00	0.497319	0.248660	−	0.968591i	$-0.420010\pi$
$569$	6750.00	0.497319	0.248660	+	0.968591i	$0.420010\pi$
$570$	0	0
$571$	17117.0	1.25451	0.627254	−	0.778815i	$-0.284179\pi$
$571$	17117.0	1.25451	0.627254	+	0.778815i	$0.284179\pi$
$572$	11256.0i	0.822792i
$573$	− 3906.00i	− 0.284774i
$574$	−24.0000	−0.00174519
$575$	0	0
$576$	576.000	0.0416667
$577$	301.000i	0.0217171i	0.999941	+	0.0108586i	$0.00345646\pi$
$577$	301.000i	0.0217171i	−0.999941	+	0.0108586i	$0.996544\pi$
$578$	3994.00i	0.287420i
$579$	12489.0	0.896416
$580$	0	0
$581$	462.000	0.0329897
$582$	3066.00i	0.218367i
$583$	− 8064.00i	− 0.572859i
$584$	7504.00	0.531708
$585$	0	0
$586$	10884.0	0.767259
$587$	15456.0i	1.08678i	0.839482	+	0.543388i	$0.182859\pi$
$587$	15456.0i	1.08678i	−0.839482	+	0.543388i	$0.817141\pi$
$588$	4104.00i	0.287834i
$589$	−22195.0	−1.55268
$590$	0	0
$591$	−9162.00	−0.637689
$592$	4576.00i	0.317690i
$593$	− 9492.00i	− 0.657318i	−0.944449	−	0.328659i	$-0.893403\pi$
$593$	− 9492.00i	− 0.657318i	0.944449	−	0.328659i	$-0.106597\pi$
$594$	−2268.00	−0.156662
$595$	0	0
$596$	−8280.00	−0.569064
$597$	10275.0i	0.704402i
$598$	− 21708.0i	− 1.48446i
$599$	−1500.00	−0.102318	−0.0511589	−	0.998691i	$-0.516291\pi$
$599$	−1500.00	−0.102318	−0.0511589	+	0.998691i	$0.516291\pi$
$600$	0	0
$601$	14627.0	0.992758	0.496379	−	0.868106i	$-0.334663\pi$
$601$	14627.0	0.992758	0.496379	+	0.868106i	$0.334663\pi$
$602$	− 526.000i	− 0.0356116i
$603$	2691.00i	0.181735i
$604$	−8948.00	−0.602796
$605$	0	0
$606$	5472.00	0.366807
$607$	− 16184.0i	− 1.08219i	−0.840962	−	0.541094i	$-0.818010\pi$
$607$	− 16184.0i	− 1.08219i	0.840962	−	0.541094i	$-0.181990\pi$
$608$	3680.00i	0.245467i
$609$	630.000	0.0419194
$610$	0	0
$611$	−27738.0	−1.83659
$612$	− 1944.00i	− 0.128401i
$613$	− 18502.0i	− 1.21907i	−0.792760	−	0.609534i	$-0.791357\pi$
$613$	− 18502.0i	− 1.21907i	0.792760	−	0.609534i	$-0.208643\pi$
$614$	−7742.00	−0.508863
$615$	0	0
$616$	336.000	0.0219770
$617$	13896.0i	0.906697i	0.891333	+	0.453348i	$0.149771\pi$
$617$	13896.0i	0.906697i	−0.891333	+	0.453348i	$0.850229\pi$
$618$	4008.00i	0.260883i
$619$	9895.00	0.642510	0.321255	−	0.946993i	$-0.395895\pi$
$619$	9895.00	0.642510	0.321255	+	0.946993i	$0.395895\pi$
$620$	0	0
$621$	4374.00	0.282645
$622$	− 11436.0i	− 0.737206i
$623$	240.000i	0.0154340i
$624$	−3216.00	−0.206319
$625$	0	0
$626$	7274.00	0.464421
$627$	− 14490.0i	− 0.922926i
$628$	− 964.000i	− 0.0612544i
$629$	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	$0.495311\pi$
$632$	− 1280.00i	− 0.0805628i
$633$	7329.00i	0.460192i
$634$	−2592.00	−0.162368
$635$	0	0
$636$	2304.00	0.143647
$637$	− 22914.0i	− 1.42525i
$638$	17640.0i	1.09463i
$639$	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	$0.107860\pi$
$642$	7776.00i	0.478028i
$643$	− 1852.00i	− 0.113586i	−0.998386	−	0.0567930i	$-0.981913\pi$
$643$	− 1852.00i	− 0.113586i	0.998386	−	0.0567930i	$-0.0180875\pi$
$644$	−648.000	−0.0396503
$645$	0	0
$646$	12420.0	0.756437
$647$	21156.0i	1.28551i	0.766070	+	0.642757i	$0.222210\pi$
$647$	21156.0i	1.28551i	−0.766070	+	0.642757i	$0.777790\pi$
$648$	− 648.000i	− 0.0392837i
$649$	−28980.0	−1.75280
$650$	0	0
$651$	−579.000	−0.0348584
$652$	14188.0i	0.852216i
$653$	− 9702.00i	− 0.581422i	−0.956811	−	0.290711i	$-0.906108\pi$
$653$	− 9702.00i	− 0.581422i	0.956811	−	0.290711i	$-0.0938918\pi$
$654$	10410.0	0.622421
$655$	0	0
$656$	192.000	0.0114273
$657$	− 8442.00i	− 0.501300i
$658$	828.000i	0.0490559i
$659$	−1980.00	−0.117041	−0.0585204	−	0.998286i	$-0.518638\pi$
$659$	−1980.00	−0.117041	−0.0585204	+	0.998286i	$0.518638\pi$
$660$	0	0
$661$	−20158.0	−1.18617	−0.593083	−	0.805142i	$-0.702089\pi$
$661$	−20158.0	−1.18617	−0.593083	+	0.805142i	$0.702089\pi$
$662$	10264.0i	0.602601i
$663$	10854.0i	0.635799i
$664$	−3696.00	−0.216013
$665$	0	0
$666$	5148.00	0.299521
$667$	− 34020.0i	− 1.97490i
$668$	3936.00i	0.227977i
$669$	69.0000	0.00398758
$670$	0	0
$671$	−30786.0	−1.77121
$672$	96.0000i	0.00551083i
$673$	− 16882.0i	− 0.966944i	−0.875360	−	0.483472i	$-0.839376\pi$
$673$	− 16882.0i	− 0.966944i	0.875360	−	0.483472i	$-0.160624\pi$
$674$	−13502.0	−0.771628
$675$	0	0
$676$	9168.00	0.521620
$677$	− 20934.0i	− 1.18842i	−0.804311	−	0.594209i	$-0.797465\pi$
$677$	− 20934.0i	− 1.18842i	0.804311	−	0.594209i	$-0.202535\pi$
$678$	− 6552.00i	− 0.371133i
$679$	−511.000	−0.0288813
$680$	0	0
$681$	5868.00	0.330194
$682$	− 16212.0i	− 0.910249i
$683$	− 8712.00i	− 0.488075i	−0.969766	−	0.244038i	$-0.921528\pi$
$683$	− 8712.00i	− 0.488075i	0.969766	−	0.244038i	$-0.0784720\pi$
$684$	4140.00	0.231428
$685$	0	0
$686$	−1370.00	−0.0762490
$687$	5415.00i	0.300721i
$688$	4208.00i	0.233181i
$689$	−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	$0.627144\pi$
$692$	− 14472.0i	− 0.795004i
$693$	− 378.000i	− 0.0207201i
$694$	−10452.0	−0.571689
$695$	0	0
$696$	−5040.00	−0.274484
$697$	− 648.000i	− 0.0352148i
$698$	12380.0i	0.671332i
$699$	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	$0.775685\pi$
$702$	3618.00i	0.194519i
$703$	32890.0i	1.76454i
$704$	−2688.00	−0.143903
$705$	0	0
$706$	−13236.0	−0.705586
$707$	912.000i	0.0485138i
$708$	− 8280.00i	− 0.439522i
$709$	−8885.00	−0.470639	−0.235320	−	0.971918i	$-0.575614\pi$
$709$	−8885.00	−0.470639	−0.235320	+	0.971918i	$0.575614\pi$
$710$	0	0
$711$	−1440.00	−0.0759553
$712$	− 1920.00i	− 0.101060i
$713$	31266.0i	1.64225i
$714$	324.000	0.0169823
$715$	0	0
$716$	−600.000	−0.0313171
$717$	7920.00i	0.412521i
$718$	6840.00i	0.355524i
$719$	7530.00	0.390572	0.195286	−	0.980746i	$-0.437436\pi$
$719$	7530.00	0.390572	0.195286	+	0.980746i	$0.437436\pi$
$720$	0	0
$721$	−668.000	−0.0345043
$722$	12732.0i	0.656283i
$723$	16149.0i	0.830688i
$724$	−788.000	−0.0404500
$725$	0	0
$726$	2598.00	0.132811
$727$	1801.00i	0.0918781i	0.998944	+	0.0459391i	$0.0146280\pi$
$727$	1801.00i	0.0918781i	−0.998944	+	0.0459391i	$0.985372\pi$
$728$	− 536.000i	− 0.0272877i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	14202.0	0.718577
$732$	− 8796.00i	− 0.444139i
$733$	− 7882.00i	− 0.397174i	−0.980083	−	0.198587i	$-0.936365\pi$
$733$	− 7882.00i	− 0.397174i	0.980083	−	0.198587i	$-0.0636352\pi$
$734$	−1742.00	−0.0876000
$735$	0	0
$736$	5184.00	0.259626
$737$	− 12558.0i	− 0.627652i
$738$	− 216.000i	− 0.0107738i
$739$	−33860.0	−1.68547	−0.842734	−	0.538331i	$-0.819055\pi$
$739$	−33860.0	−1.68547	−0.842734	+	0.538331i	$0.819055\pi$
$740$	0	0
$741$	−23115.0	−1.14595
$742$	384.000i	0.0189988i
$743$	− 20652.0i	− 1.01972i	−0.860259	−	0.509858i	$-0.829698\pi$
$743$	− 20652.0i	− 1.01972i	0.860259	−	0.509858i	$-0.170302\pi$
$744$	4632.00	0.228249
$745$	0	0
$746$	−12766.0	−0.626537
$747$	4158.00i	0.203659i
$748$	9072.00i	0.443456i
$749$	−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	$0.441895\pi$
$752$	− 6624.00i	− 0.321213i
$753$	15084.0i	0.730002i
$754$	28140.0	1.35915
$755$	0	0
$756$	108.000	0.00519566
$757$	32251.0i	1.54846i	0.632906	+	0.774229i	$0.281862\pi$
$757$	32251.0i	1.54846i	−0.632906	+	0.774229i	$0.718138\pi$
$758$	19730.0i	0.945417i
$759$	−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	$0.368865\pi$
$762$	96.0000i	0.00456393i
$763$	1735.00i	0.0823214i
$764$	−5208.00	−0.246622
$765$	0	0
$766$	−19656.0	−0.927154
$767$	46230.0i	2.17636i
$768$	− 768.000i	− 0.0360844i
$769$	34645.0	1.62462	0.812309	−	0.583228i	$-0.198211\pi$
$769$	34645.0	1.62462	0.812309	+	0.583228i	$0.198211\pi$
$770$	0	0
$771$	−1692.00	−0.0790349
$772$	− 16652.0i	− 0.776319i
$773$	− 8412.00i	− 0.391408i	−0.980663	−	0.195704i	$-0.937301\pi$
$773$	− 8412.00i	− 0.391408i	0.980663	−	0.195704i	$-0.0626992\pi$
$774$	4734.00	0.219845
$775$	0	0
$776$	4088.00	0.189112
$777$	858.000i	0.0396146i
$778$	− 25080.0i	− 1.15573i
$779$	1380.00	0.0634706
$780$	0	0
$781$	−9576.00	−0.438740
$782$	− 17496.0i	− 0.800071i
$783$	5670.00i	0.258786i
$784$	5472.00	0.249271
$785$	0	0
$786$	11952.0	0.542384
$787$	− 18329.0i	− 0.830188i	−0.909778	−	0.415094i	$-0.863749\pi$
$787$	− 18329.0i	− 0.830188i	0.909778	−	0.415094i	$-0.136251\pi$
$788$	12216.0i	0.552255i
$789$	−5436.00	−0.245281
$790$	0	0
$791$	1092.00	0.0490860
$792$	3024.00i	0.135673i
$793$	49111.0i	2.19922i
$794$	−2762.00	−0.123451
$795$	0	0
$796$	13700.0	0.610030
$797$	− 16044.0i	− 0.713059i	−0.934284	−	0.356529i	$-0.883960\pi$
$797$	− 16044.0i	− 0.713059i	0.934284	−	0.356529i	$-0.116040\pi$
$798$	690.000i	0.0306087i
$799$	−22356.0	−0.989860
$800$	0	0
$801$	−2160.00	−0.0952807
$802$	28464.0i	1.25324i
$803$	39396.0i	1.73133i
$804$	3588.00	0.157387
$805$	0	0
$806$	−25862.0	−1.13021
$807$	− 15570.0i	− 0.679170i
$808$	− 7296.00i	− 0.317664i
$809$	24000.0	1.04301	0.521505	−	0.853248i	$-0.325371\pi$
$809$	24000.0	1.04301	0.521505	+	0.853248i	$0.325371\pi$
$810$	0	0
$811$	5117.00	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$811$	5117.00	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$812$	− 840.000i	− 0.0363032i
$813$	− 13776.0i	− 0.594275i
$814$	−24024.0	−1.03445
$815$	0	0
$816$	−2592.00	−0.111199
$817$	30245.0i	1.29515i
$818$	− 5290.00i	− 0.226113i
$819$	−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	$0.407066\pi$
$822$	14076.0i	0.597271i
$823$	1283.00i	0.0543409i	0.999631	+	0.0271705i	$0.00864969\pi$
$823$	1283.00i	0.0543409i	−0.999631	+	0.0271705i	$0.991350\pi$
$824$	5344.00	0.225931
$825$	0	0
$826$	1380.00	0.0581312
$827$	− 16344.0i	− 0.687227i	−0.939111	−	0.343613i	$-0.888349\pi$
$827$	− 16344.0i	− 0.687227i	0.939111	−	0.343613i	$-0.111651\pi$
$828$	− 5832.00i	− 0.244778i
$829$	790.000	0.0330975	0.0165488	−	0.999863i	$-0.494732\pi$
$829$	790.000	0.0330975	0.0165488	+	0.999863i	$0.494732\pi$
$830$	0	0
$831$	6573.00	0.274386
$832$	4288.00i	0.178677i
$833$	− 18468.0i	− 0.768161i
$834$	−17400.0	−0.722437
$835$	0	0
$836$	−19320.0	−0.799278
$837$	− 5211.00i	− 0.215195i
$838$	− 6000.00i	− 0.247335i
$839$	9990.00	0.411076	0.205538	−	0.978649i	$-0.434106\pi$
$839$	9990.00	0.411076	0.205538	+	0.978649i	$0.434106\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	− 22676.0i	− 0.928108i
$843$	− 23526.0i	− 0.961184i
$844$	9772.00	0.398538
$845$	0	0
$846$	−7452.00	−0.302843
$847$	433.000i	0.0175656i
$848$	− 3072.00i	− 0.124402i
$849$	−741.000	−0.0299541
$850$	0	0
$851$	46332.0	1.86632
$852$	− 2736.00i	− 0.110016i
$853$	24743.0i	0.993182i	0.867985	+	0.496591i	$0.165415\pi$
$853$	24743.0i	0.993182i	−0.867985	+	0.496591i	$0.834585\pi$
$854$	1466.00	0.0587418
$855$	0	0
$856$	10368.0	0.413985
$857$	23556.0i	0.938924i	0.882953	+	0.469462i	$0.155552\pi$
$857$	23556.0i	0.938924i	−0.882953	+	0.469462i	$0.844448\pi$
$858$	− 16884.0i	− 0.671807i
$859$	34000.0	1.35048	0.675242	−	0.737597i	$-0.264039\pi$
$859$	34000.0	1.35048	0.675242	+	0.737597i	$0.264039\pi$
$860$	0	0
$861$	36.0000	0.00142494
$862$	− 6516.00i	− 0.257466i
$863$	− 37032.0i	− 1.46070i	−0.683073	−	0.730350i	$-0.739357\pi$
$863$	− 37032.0i	− 1.46070i	0.683073	−	0.730350i	$-0.260643\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	−2326.00	−0.0912710
$867$	− 5991.00i	− 0.234677i
$868$	772.000i	0.0301882i
$869$	6720.00	0.262325
$870$	0	0
$871$	−20033.0	−0.779325
$872$	− 13880.0i	− 0.539032i
$873$	− 4599.00i	− 0.178296i
$874$	37260.0	1.44203
$875$	0	0
$876$	−11256.0	−0.434138
$877$	− 2519.00i	− 0.0969904i	−0.998823	−	0.0484952i	$-0.984557\pi$
$877$	− 2519.00i	− 0.0969904i	0.998823	−	0.0484952i	$-0.0154426\pi$
$878$	− 13390.0i	− 0.514682i
$879$	−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	$0.181872\pi$
$882$	− 6156.00i	− 0.235015i
$883$	− 19177.0i	− 0.730869i	−0.930837	−	0.365435i	$-0.880920\pi$
$883$	− 19177.0i	− 0.730869i	0.930837	−	0.365435i	$-0.119080\pi$
$884$	14472.0	0.550618
$885$	0	0
$886$	−32736.0	−1.24130
$887$	− 44994.0i	− 1.70321i	−0.524181	−	0.851607i	$-0.675628\pi$
$887$	− 44994.0i	− 1.70321i	0.524181	−	0.851607i	$-0.324372\pi$
$888$	− 6864.00i	− 0.259393i
$889$	−16.0000	−0.000603625 0
$890$	0	0
$891$	3402.00	0.127914
$892$	− 92.0000i	− 0.00345335i
$893$	− 47610.0i	− 1.78411i
$894$	12420.0	0.464639
$895$	0	0
$896$	128.000	0.00477252
$897$	32562.0i	1.21206i
$898$	− 32760.0i	− 1.21739i
$899$	−40530.0	−1.50362
$900$	0	0
$901$	−10368.0	−0.383361
$902$	1008.00i	0.0372092i
$903$	789.000i	0.0290767i
$904$	−8736.00	−0.321410
$905$	0	0
$906$	13422.0	0.492181
$907$	52396.0i	1.91817i	0.283117	+	0.959085i	$0.408631\pi$
$907$	52396.0i	1.91817i	−0.283117	+	0.959085i	$0.591369\pi$
$908$	− 7824.00i	− 0.285957i
$909$	−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	$0.457960\pi$
$912$	− 5520.00i	− 0.200423i
$913$	− 19404.0i	− 0.703372i
$914$	−27572.0	−0.997813
$915$	0	0
$916$	7220.00	0.260432
$917$	1992.00i	0.0717357i
$918$	2916.00i	0.104839i
$919$	−4085.00	−0.146629	−0.0733143	−	0.997309i	$-0.523358\pi$
$919$	−4085.00	−0.146629	−0.0733143	+	0.997309i	$0.523358\pi$
$920$	0	0
$921$	11613.0	0.415485
$922$	23664.0i	0.845263i
$923$	15276.0i	0.544762i
$924$	−504.000	−0.0179441
$925$	0	0
$926$	−6016.00	−0.213497
$927$	− 6012.00i	− 0.213010i
$928$	6720.00i	0.237710i
$929$	3030.00	0.107009	0.0535043	−	0.998568i	$-0.482961\pi$
$929$	3030.00	0.107009	0.0535043	+	0.998568i	$0.482961\pi$
$930$	0	0
$931$	39330.0	1.38452
$932$	− 13872.0i	− 0.487546i
$933$	17154.0i	0.601926i
$934$	8868.00	0.310674
$935$	0	0
$936$	4824.00	0.168459
$937$	− 5759.00i	− 0.200788i	−0.994948	−	0.100394i	$-0.967990\pi$
$937$	− 5759.00i	− 0.200788i	0.994948	−	0.100394i	$-0.0320103\pi$
$938$	598.000i	0.0208160i
$939$	−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	$0.501423\pi$
$942$	1446.00i	0.0500140i
$943$	− 1944.00i	− 0.0671319i
$944$	−11040.0	−0.380637
$945$	0	0
$946$	−22092.0	−0.759274
$947$	− 1374.00i	− 0.0471478i	−0.999722	−	0.0235739i	$-0.992495\pi$
$947$	− 1374.00i	− 0.0471478i	0.999722	−	0.0235739i	$-0.00750451\pi$
$948$	1920.00i	0.0657792i
$949$	62846.0	2.14970
$950$	0	0
$951$	3888.00	0.132573
$952$	− 432.000i	− 0.0147071i
$953$	9288.00i	0.315706i	0.987463	+	0.157853i	$0.0504572\pi$
$953$	9288.00i	0.315706i	−0.987463	+	0.157853i	$0.949543\pi$
$954$	−3456.00	−0.117287
$955$	0	0
$956$	10560.0	0.357254
$957$	− 26460.0i	− 0.893762i
$958$	− 14820.0i	− 0.499804i
$959$	−2346.00	−0.0789951
$960$	0	0
$961$	7458.00	0.250344
$962$	38324.0i	1.28442i
$963$	− 11664.0i	− 0.390309i
$964$	21532.0	0.719397
$965$	0	0
$966$	972.000	0.0323743
$967$	21616.0i	0.718846i	0.933175	+	0.359423i	$0.117026\pi$
$967$	21616.0i	0.718846i	−0.933175	+	0.359423i	$0.882974\pi$
$968$	− 3464.00i	− 0.115018i
$969$	−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	$0.602204\pi$
$972$	972.000i	0.0320750i
$973$	− 2900.00i	− 0.0955496i
$974$	−17342.0	−0.570507
$975$	0	0
$976$	−11728.0	−0.384635
$977$	18246.0i	0.597483i	0.954334	+	0.298742i	$0.0965669\pi$
$977$	18246.0i	0.597483i	−0.954334	+	0.298742i	$0.903433\pi$
$978$	− 21282.0i	− 0.695832i
$979$	10080.0	0.329069
$980$	0	0
$981$	−15615.0	−0.508204
$982$	− 38736.0i	− 1.25877i
$983$	− 38772.0i	− 1.25802i	−0.777397	−	0.629011i	$-0.783460\pi$
$983$	− 38772.0i	− 1.25802i	0.777397	−	0.629011i	$-0.216540\pi$
$984$	−288.000	−0.00933039
$985$	0	0
$986$	22680.0	0.732534
$987$	− 1242.00i	− 0.0400540i
$988$	30820.0i	0.992424i
$989$	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	$0.620463\pi$
$992$	− 6176.00i	− 0.197670i
$993$	− 15396.0i	− 0.492021i
$994$	456.000	0.0145507
$995$	0	0
$996$	5544.00	0.176374
$997$	10366.0i	0.329282i	0.986354	+	0.164641i	$0.0526466\pi$
$997$	10366.0i	0.329282i	−0.986354	+	0.164641i	$0.947353\pi$
$998$	17750.0i	0.562992i
$999$	−7722.00	−0.244558

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} +1.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +42.0000 q^{11} +12.0000i q^{12} -67.0000i q^{13} -2.00000 q^{14} +16.0000 q^{16} -54.0000i q^{17} -18.0000i q^{18} +115.000 q^{19} +3.00000 q^{21} +84.0000i q^{22} -162.000i q^{23} -24.0000 q^{24} +134.000 q^{26} +27.0000i q^{27} -4.00000i q^{28} +210.000 q^{29} -193.000 q^{31} +32.0000i q^{32} -126.000i q^{33} +108.000 q^{34} +36.0000 q^{36} +286.000i q^{37} +230.000i q^{38} -201.000 q^{39} +12.0000 q^{41} +6.00000i q^{42} +263.000i q^{43} -168.000 q^{44} +324.000 q^{46} -414.000i q^{47} -48.0000i q^{48} +342.000 q^{49} -162.000 q^{51} +268.000i q^{52} -192.000i q^{53} -54.0000 q^{54} +8.00000 q^{56} -345.000i q^{57} +420.000i q^{58} -690.000 q^{59} -733.000 q^{61} -386.000i q^{62} -9.00000i q^{63} -64.0000 q^{64} +252.000 q^{66} -299.000i q^{67} +216.000i q^{68} -486.000 q^{69} -228.000 q^{71} +72.0000i q^{72} +938.000i q^{73} -572.000 q^{74} -460.000 q^{76} +42.0000i q^{77} -402.000i q^{78} +160.000 q^{79} +81.0000 q^{81} +24.0000i q^{82} -462.000i q^{83} -12.0000 q^{84} -526.000 q^{86} -630.000i q^{87} -336.000i q^{88} +240.000 q^{89} +67.0000 q^{91} +648.000i q^{92} +579.000i q^{93} +828.000 q^{94} +96.0000 q^{96} +511.000i q^{97} +684.000i q^{98} -378.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} + 84 q^{11} - 4 q^{14} + 32 q^{16} + 230 q^{19} + 6 q^{21} - 48 q^{24} + 268 q^{26} + 420 q^{29} - 386 q^{31} + 216 q^{34} + 72 q^{36} - 402 q^{39} + 24 q^{41} - 336 q^{44}+ \cdots - 756 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 + 12 * q^6 - 18 * q^9 + 84 * q^11 - 4 * q^14 + 32 * q^16 + 230 * q^19 + 6 * q^21 - 48 * q^24 + 268 * q^26 + 420 * q^29 - 386 * q^31 + 216 * q^34 + 72 * q^36 - 402 * q^39 + 24 * q^41 - 336 * q^44 + 648 * q^46 + 684 * q^49 - 324 * q^51 - 108 * q^54 + 16 * q^56 - 1380 * q^59 - 1466 * q^61 - 128 * q^64 + 504 * q^66 - 972 * q^69 - 456 * q^71 - 1144 * q^74 - 920 * q^76 + 320 * q^79 + 162 * q^81 - 24 * q^84 - 1052 * q^86 + 480 * q^89 + 134 * q^91 + 1656 * q^94 + 192 * q^96 - 756 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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