LMFDB - Embedded newform 150.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,4,Mod(1,150)] 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("150.1"); 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, 0])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,2,-3,4,0,-6,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.85028650086$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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)

$f(q)$

$=$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -32.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -60.0000 q^{11} -12.0000 q^{12} +34.0000 q^{13} -64.0000 q^{14} +16.0000 q^{16} -42.0000 q^{17} +18.0000 q^{18} -76.0000 q^{19} +96.0000 q^{21} -120.000 q^{22} -24.0000 q^{24} +68.0000 q^{26} -27.0000 q^{27} -128.000 q^{28} +6.00000 q^{29} -232.000 q^{31} +32.0000 q^{32} +180.000 q^{33} -84.0000 q^{34} +36.0000 q^{36} -134.000 q^{37} -152.000 q^{38} -102.000 q^{39} +234.000 q^{41} +192.000 q^{42} +412.000 q^{43} -240.000 q^{44} +360.000 q^{47} -48.0000 q^{48} +681.000 q^{49} +126.000 q^{51} +136.000 q^{52} -222.000 q^{53} -54.0000 q^{54} -256.000 q^{56} +228.000 q^{57} +12.0000 q^{58} +660.000 q^{59} -490.000 q^{61} -464.000 q^{62} -288.000 q^{63} +64.0000 q^{64} +360.000 q^{66} -812.000 q^{67} -168.000 q^{68} +120.000 q^{71} +72.0000 q^{72} -746.000 q^{73} -268.000 q^{74} -304.000 q^{76} +1920.00 q^{77} -204.000 q^{78} +152.000 q^{79} +81.0000 q^{81} +468.000 q^{82} +804.000 q^{83} +384.000 q^{84} +824.000 q^{86} -18.0000 q^{87} -480.000 q^{88} -678.000 q^{89} -1088.00 q^{91} +696.000 q^{93} +720.000 q^{94} -96.0000 q^{96} -194.000 q^{97} +1362.00 q^{98} -540.000 q^{99} +O(q^{100})\)

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
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	−12.0000	−0.288675
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	−64.0000	−1.22177
$15$	0	0
$16$	16.0000	0.250000
$17$	−42.0000	−0.599206	−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	18.0000	0.235702
$19$	−76.0000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−76.0000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	−120.000	−1.16291
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−24.0000	−0.204124
$25$	0	0
$26$	68.0000	0.512919
$27$	−27.0000	−0.192450
$28$	−128.000	−0.863919
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	32.0000	0.176777
$33$	180.000	0.949514
$34$	−84.0000	−0.423702
$35$	0	0
$36$	36.0000	0.166667
$37$	−134.000	−0.595391	−0.297695	−	0.954661i	$-0.596218\pi$
$37$	−134.000	−0.595391	−0.297695	+	0.954661i	$0.596218\pi$
$38$	−152.000	−0.648886
$39$	−102.000	−0.418797
$40$	0	0
$41$	234.000	0.891333	0.445667	−	0.895199i	$-0.352967\pi$
$41$	234.000	0.891333	0.445667	+	0.895199i	$0.352967\pi$
$42$	192.000	0.705387
$43$	412.000	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$43$	412.000	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$44$	−240.000	−0.822304
$45$	0	0
$46$	0	0
$47$	360.000	1.11726	0.558632	−	0.829416i	$-0.311326\pi$
$47$	360.000	1.11726	0.558632	+	0.829416i	$0.311326\pi$
$48$	−48.0000	−0.144338
$49$	681.000	1.98542
$50$	0	0
$51$	126.000	0.345952
$52$	136.000	0.362689
$53$	−222.000	−0.575359	−0.287680	−	0.957727i	$-0.592884\pi$
$53$	−222.000	−0.575359	−0.287680	+	0.957727i	$0.592884\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	−256.000	−0.610883
$57$	228.000	0.529813
$58$	12.0000	0.0271668
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	−490.000	−1.02849	−0.514246	−	0.857642i	$-0.671928\pi$
$61$	−490.000	−1.02849	−0.514246	+	0.857642i	$0.671928\pi$
$62$	−464.000	−0.950453
$63$	−288.000	−0.575946
$64$	64.0000	0.125000
$65$	0	0
$66$	360.000	0.671408
$67$	−812.000	−1.48062	−0.740310	−	0.672265i	$-0.765321\pi$
$67$	−812.000	−1.48062	−0.740310	+	0.672265i	$0.765321\pi$
$68$	−168.000	−0.299603
$69$	0	0
$70$	0	0
$71$	120.000	0.200583	0.100291	−	0.994958i	$-0.468022\pi$
$71$	120.000	0.200583	0.100291	+	0.994958i	$0.468022\pi$
$72$	72.0000	0.117851
$73$	−746.000	−1.19606	−0.598032	−	0.801472i	$-0.704051\pi$
$73$	−746.000	−1.19606	−0.598032	+	0.801472i	$0.704051\pi$
$74$	−268.000	−0.421005
$75$	0	0
$76$	−304.000	−0.458831
$77$	1920.00	2.84161
$78$	−204.000	−0.296134
$79$	152.000	0.216473	0.108236	−	0.994125i	$-0.465480\pi$
$79$	152.000	0.216473	0.108236	+	0.994125i	$0.465480\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	468.000	0.630268
$83$	804.000	1.06326	0.531629	−	0.846977i	$-0.321580\pi$
$83$	804.000	1.06326	0.531629	+	0.846977i	$0.321580\pi$
$84$	384.000	0.498784
$85$	0	0
$86$	824.000	1.03319
$87$	−18.0000	−0.0221816
$88$	−480.000	−0.581456
$89$	−678.000	−0.807504	−0.403752	−	0.914868i	$-0.632294\pi$
$89$	−678.000	−0.807504	−0.403752	+	0.914868i	$0.632294\pi$
$90$	0	0
$91$	−1088.00	−1.25333
$92$	0	0
$93$	696.000	0.776041
$94$	720.000	0.790025
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−194.000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−194.000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1362.00	1.40391
$99$	−540.000	−0.548202
$100$	0	0
$101$	798.000	0.786178	0.393089	−	0.919500i	$-0.371406\pi$
$101$	798.000	0.786178	0.393089	+	0.919500i	$0.371406\pi$
$102$	252.000	0.244625
$103$	−1088.00	−1.04081	−0.520407	−	0.853918i	$-0.674220\pi$
$103$	−1088.00	−1.04081	−0.520407	+	0.853918i	$0.674220\pi$
$104$	272.000	0.256460
$105$	0	0
$106$	−444.000	−0.406840
$107$	−1716.00	−1.55039	−0.775196	−	0.631721i	$-0.782349\pi$
$107$	−1716.00	−1.55039	−0.775196	+	0.631721i	$0.782349\pi$
$108$	−108.000	−0.0962250
$109$	−970.000	−0.852378	−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000	−0.852378	−0.426189	+	0.904634i	$0.640144\pi$
$110$	0	0
$111$	402.000	0.343749
$112$	−512.000	−0.431959
$113$	−426.000	−0.354643	−0.177322	−	0.984153i	$-0.556743\pi$
$113$	−426.000	−0.354643	−0.177322	+	0.984153i	$0.556743\pi$
$114$	456.000	0.374634
$115$	0	0
$116$	24.0000	0.0192099
$117$	306.000	0.241792
$118$	1320.00	1.02980
$119$	1344.00	1.03533
$120$	0	0
$121$	2269.00	1.70473
$122$	−980.000	−0.727254
$123$	−702.000	−0.514611
$124$	−928.000	−0.672071
$125$	0	0
$126$	−576.000	−0.407255
$127$	−200.000	−0.139741	−0.0698706	−	0.997556i	$-0.522259\pi$
$127$	−200.000	−0.139741	−0.0698706	+	0.997556i	$0.522259\pi$
$128$	128.000	0.0883883
$129$	−1236.00	−0.843595
$130$	0	0
$131$	60.0000	0.0400170	0.0200085	−	0.999800i	$-0.493631\pi$
$131$	60.0000	0.0400170	0.0200085	+	0.999800i	$0.493631\pi$
$132$	720.000	0.474757
$133$	2432.00	1.58557
$134$	−1624.00	−1.04696
$135$	0	0
$136$	−336.000	−0.211851
$137$	−642.000	−0.400363	−0.200182	−	0.979759i	$-0.564153\pi$
$137$	−642.000	−0.400363	−0.200182	+	0.979759i	$0.564153\pi$
$138$	0	0
$139$	−2836.00	−1.73055	−0.865275	−	0.501298i	$-0.832856\pi$
$139$	−2836.00	−1.73055	−0.865275	+	0.501298i	$0.832856\pi$
$140$	0	0
$141$	−1080.00	−0.645053
$142$	240.000	0.141833
$143$	−2040.00	−1.19296
$144$	144.000	0.0833333
$145$	0	0
$146$	−1492.00	−0.845745
$147$	−2043.00	−1.14628
$148$	−536.000	−0.297695
$149$	−1554.00	−0.854420	−0.427210	−	0.904152i	$-0.640504\pi$
$149$	−1554.00	−0.854420	−0.427210	+	0.904152i	$0.640504\pi$
$150$	0	0
$151$	−2272.00	−1.22446	−0.612228	−	0.790682i	$-0.709726\pi$
$151$	−2272.00	−1.22446	−0.612228	+	0.790682i	$0.709726\pi$
$152$	−608.000	−0.324443
$153$	−378.000	−0.199735
$154$	3840.00	2.00932
$155$	0	0
$156$	−408.000	−0.209398
$157$	−1694.00	−0.861120	−0.430560	−	0.902562i	$-0.641684\pi$
$157$	−1694.00	−0.861120	−0.430560	+	0.902562i	$0.641684\pi$
$158$	304.000	0.153069
$159$	666.000	0.332184
$160$	0	0
$161$	0	0
$162$	162.000	0.0785674
$163$	52.0000	0.0249874	0.0124937	−	0.999922i	$-0.496023\pi$
$163$	52.0000	0.0249874	0.0124937	+	0.999922i	$0.496023\pi$
$164$	936.000	0.445667
$165$	0	0
$166$	1608.00	0.751837
$167$	1200.00	0.556041	0.278020	−	0.960575i	$-0.410322\pi$
$167$	1200.00	0.556041	0.278020	+	0.960575i	$0.410322\pi$
$168$	768.000	0.352693
$169$	−1041.00	−0.473828
$170$	0	0
$171$	−684.000	−0.305888
$172$	1648.00	0.730575
$173$	−54.0000	−0.0237315	−0.0118657	−	0.999930i	$-0.503777\pi$
$173$	−54.0000	−0.0237315	−0.0118657	+	0.999930i	$0.503777\pi$
$174$	−36.0000	−0.0156848
$175$	0	0
$176$	−960.000	−0.411152
$177$	−1980.00	−0.840824
$178$	−1356.00	−0.570992
$179$	876.000	0.365784	0.182892	−	0.983133i	$-0.441454\pi$
$179$	876.000	0.365784	0.182892	+	0.983133i	$0.441454\pi$
$180$	0	0
$181$	3854.00	1.58268	0.791341	−	0.611375i	$-0.209383\pi$
$181$	3854.00	1.58268	0.791341	+	0.611375i	$0.209383\pi$
$182$	−2176.00	−0.886241
$183$	1470.00	0.593801
$184$	0	0
$185$	0	0
$186$	1392.00	0.548744
$187$	2520.00	0.985458
$188$	1440.00	0.558632
$189$	864.000	0.332522
$190$	0	0
$191$	−2784.00	−1.05468	−0.527338	−	0.849656i	$-0.676810\pi$
$191$	−2784.00	−1.05468	−0.527338	+	0.849656i	$0.676810\pi$
$192$	−192.000	−0.0721688
$193$	−914.000	−0.340887	−0.170443	−	0.985367i	$-0.554520\pi$
$193$	−914.000	−0.340887	−0.170443	+	0.985367i	$0.554520\pi$
$194$	−388.000	−0.143592
$195$	0	0
$196$	2724.00	0.992711
$197$	5202.00	1.88136	0.940678	−	0.339300i	$-0.110190\pi$
$197$	5202.00	1.88136	0.940678	+	0.339300i	$0.110190\pi$
$198$	−1080.00	−0.387638
$199$	3152.00	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$199$	3152.00	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$200$	0	0
$201$	2436.00	0.854837
$202$	1596.00	0.555912
$203$	−192.000	−0.0663830
$204$	504.000	0.172976
$205$	0	0
$206$	−2176.00	−0.735967
$207$	0	0
$208$	544.000	0.181344
$209$	4560.00	1.50920
$210$	0	0
$211$	740.000	0.241439	0.120720	−	0.992687i	$-0.461480\pi$
$211$	740.000	0.241439	0.120720	+	0.992687i	$0.461480\pi$
$212$	−888.000	−0.287680
$213$	−360.000	−0.115807
$214$	−3432.00	−1.09629
$215$	0	0
$216$	−216.000	−0.0680414
$217$	7424.00	2.32246
$218$	−1940.00	−0.602722
$219$	2238.00	0.690548
$220$	0	0
$221$	−1428.00	−0.434650
$222$	804.000	0.243067
$223$	520.000	0.156151	0.0780757	−	0.996947i	$-0.475122\pi$
$223$	520.000	0.156151	0.0780757	+	0.996947i	$0.475122\pi$
$224$	−1024.00	−0.305441
$225$	0	0
$226$	−852.000	−0.250771
$227$	−396.000	−0.115786	−0.0578930	−	0.998323i	$-0.518438\pi$
$227$	−396.000	−0.115786	−0.0578930	+	0.998323i	$0.518438\pi$
$228$	912.000	0.264906
$229$	−1330.00	−0.383794	−0.191897	−	0.981415i	$-0.561464\pi$
$229$	−1330.00	−0.383794	−0.191897	+	0.981415i	$0.561464\pi$
$230$	0	0
$231$	−5760.00	−1.64061
$232$	48.0000	0.0135834
$233$	−4866.00	−1.36816	−0.684082	−	0.729405i	$-0.739797\pi$
$233$	−4866.00	−1.36816	−0.684082	+	0.729405i	$0.739797\pi$
$234$	612.000	0.170973
$235$	0	0
$236$	2640.00	0.728175
$237$	−456.000	−0.124981
$238$	2688.00	0.732089
$239$	−1824.00	−0.493660	−0.246830	−	0.969059i	$-0.579389\pi$
$239$	−1824.00	−0.493660	−0.246830	+	0.969059i	$0.579389\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	4538.00	1.20543
$243$	−243.000	−0.0641500
$244$	−1960.00	−0.514246
$245$	0	0
$246$	−1404.00	−0.363885
$247$	−2584.00	−0.665652
$248$	−1856.00	−0.475226
$249$	−2412.00	−0.613873
$250$	0	0
$251$	1476.00	0.371172	0.185586	−	0.982628i	$-0.440582\pi$
$251$	1476.00	0.371172	0.185586	+	0.982628i	$0.440582\pi$
$252$	−1152.00	−0.287973
$253$	0	0
$254$	−400.000	−0.0988119
$255$	0	0
$256$	256.000	0.0625000
$257$	−4314.00	−1.04708	−0.523541	−	0.852001i	$-0.675389\pi$
$257$	−4314.00	−1.04708	−0.523541	+	0.852001i	$0.675389\pi$
$258$	−2472.00	−0.596512
$259$	4288.00	1.02874
$260$	0	0
$261$	54.0000	0.0128066
$262$	120.000	0.0282963
$263$	5280.00	1.23794	0.618971	−	0.785414i	$-0.287550\pi$
$263$	5280.00	1.23794	0.618971	+	0.785414i	$0.287550\pi$
$264$	1440.00	0.335704
$265$	0	0
$266$	4864.00	1.12117
$267$	2034.00	0.466213
$268$	−3248.00	−0.740310
$269$	5526.00	1.25251	0.626257	−	0.779617i	$-0.284586\pi$
$269$	5526.00	1.25251	0.626257	+	0.779617i	$0.284586\pi$
$270$	0	0
$271$	2024.00	0.453687	0.226844	−	0.973931i	$-0.427159\pi$
$271$	2024.00	0.453687	0.226844	+	0.973931i	$0.427159\pi$
$272$	−672.000	−0.149801
$273$	3264.00	0.723613
$274$	−1284.00	−0.283100
$275$	0	0
$276$	0	0
$277$	−2054.00	−0.445534	−0.222767	−	0.974872i	$-0.571509\pi$
$277$	−2054.00	−0.445534	−0.222767	+	0.974872i	$0.571509\pi$
$278$	−5672.00	−1.22368
$279$	−2088.00	−0.448048
$280$	0	0
$281$	−7302.00	−1.55018	−0.775090	−	0.631850i	$-0.782296\pi$
$281$	−7302.00	−1.55018	−0.775090	+	0.631850i	$0.782296\pi$
$282$	−2160.00	−0.456121
$283$	3724.00	0.782222	0.391111	−	0.920344i	$-0.372091\pi$
$283$	3724.00	0.782222	0.391111	+	0.920344i	$0.372091\pi$
$284$	480.000	0.100291
$285$	0	0
$286$	−4080.00	−0.843551
$287$	−7488.00	−1.54008
$288$	288.000	0.0589256
$289$	−3149.00	−0.640953
$290$	0	0
$291$	582.000	0.117242
$292$	−2984.00	−0.598032
$293$	7218.00	1.43918	0.719591	−	0.694399i	$-0.244330\pi$
$293$	7218.00	1.43918	0.719591	+	0.694399i	$0.244330\pi$
$294$	−4086.00	−0.810545
$295$	0	0
$296$	−1072.00	−0.210502
$297$	1620.00	0.316505
$298$	−3108.00	−0.604166
$299$	0	0
$300$	0	0
$301$	−13184.0	−2.52463
$302$	−4544.00	−0.865821
$303$	−2394.00	−0.453900
$304$	−1216.00	−0.229416
$305$	0	0
$306$	−756.000	−0.141234
$307$	−2540.00	−0.472200	−0.236100	−	0.971729i	$-0.575869\pi$
$307$	−2540.00	−0.472200	−0.236100	+	0.971729i	$0.575869\pi$
$308$	7680.00	1.42081
$309$	3264.00	0.600914
$310$	0	0
$311$	1560.00	0.284436	0.142218	−	0.989835i	$-0.454577\pi$
$311$	1560.00	0.284436	0.142218	+	0.989835i	$0.454577\pi$
$312$	−816.000	−0.148067
$313$	934.000	0.168667	0.0843335	−	0.996438i	$-0.473124\pi$
$313$	934.000	0.168667	0.0843335	+	0.996438i	$0.473124\pi$
$314$	−3388.00	−0.608904
$315$	0	0
$316$	608.000	0.108236
$317$	1674.00	0.296597	0.148298	−	0.988943i	$-0.452620\pi$
$317$	1674.00	0.296597	0.148298	+	0.988943i	$0.452620\pi$
$318$	1332.00	0.234889
$319$	−360.000	−0.0631854
$320$	0	0
$321$	5148.00	0.895119
$322$	0	0
$323$	3192.00	0.549869
$324$	324.000	0.0555556
$325$	0	0
$326$	104.000	0.0176688
$327$	2910.00	0.492120
$328$	1872.00	0.315134
$329$	−11520.0	−1.93045
$330$	0	0
$331$	−3988.00	−0.662237	−0.331118	−	0.943589i	$-0.607426\pi$
$331$	−3988.00	−0.662237	−0.331118	+	0.943589i	$0.607426\pi$
$332$	3216.00	0.531629
$333$	−1206.00	−0.198464
$334$	2400.00	0.393180
$335$	0	0
$336$	1536.00	0.249392
$337$	−2.00000	−0.000323285 0	−0.000161642	−	1.00000i	$-0.500051\pi$
$337$	−2.00000	−0.000323285 0	−0.000161642		1.00000i	$0.500051\pi$
$338$	−2082.00	−0.335047
$339$	1278.00	0.204753
$340$	0	0
$341$	13920.0	2.21059
$342$	−1368.00	−0.216295
$343$	−10816.0	−1.70265
$344$	3296.00	0.516594
$345$	0	0
$346$	−108.000	−0.0167807
$347$	−1764.00	−0.272901	−0.136450	−	0.990647i	$-0.543569\pi$
$347$	−1764.00	−0.272901	−0.136450	+	0.990647i	$0.543569\pi$
$348$	−72.0000	−0.0110908
$349$	4310.00	0.661057	0.330529	−	0.943796i	$-0.392773\pi$
$349$	4310.00	0.661057	0.330529	+	0.943796i	$0.392773\pi$
$350$	0	0
$351$	−918.000	−0.139599
$352$	−1920.00	−0.290728
$353$	−138.000	−0.0208074	−0.0104037	−	0.999946i	$-0.503312\pi$
$353$	−138.000	−0.0208074	−0.0104037	+	0.999946i	$0.503312\pi$
$354$	−3960.00	−0.594553
$355$	0	0
$356$	−2712.00	−0.403752
$357$	−4032.00	−0.597748
$358$	1752.00	0.258648
$359$	−11976.0	−1.76064	−0.880319	−	0.474382i	$-0.842672\pi$
$359$	−11976.0	−1.76064	−0.880319	+	0.474382i	$0.842672\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	7708.00	1.11913
$363$	−6807.00	−0.984228
$364$	−4352.00	−0.626667
$365$	0	0
$366$	2940.00	0.419880
$367$	−9704.00	−1.38023	−0.690115	−	0.723699i	$-0.742440\pi$
$367$	−9704.00	−1.38023	−0.690115	+	0.723699i	$0.742440\pi$
$368$	0	0
$369$	2106.00	0.297111
$370$	0	0
$371$	7104.00	0.994128
$372$	2784.00	0.388021
$373$	8122.00	1.12746	0.563728	−	0.825960i	$-0.309367\pi$
$373$	8122.00	1.12746	0.563728	+	0.825960i	$0.309367\pi$
$374$	5040.00	0.696824
$375$	0	0
$376$	2880.00	0.395012
$377$	204.000	0.0278688
$378$	1728.00	0.235129
$379$	3404.00	0.461350	0.230675	−	0.973031i	$-0.425907\pi$
$379$	3404.00	0.461350	0.230675	+	0.973031i	$0.425907\pi$
$380$	0	0
$381$	600.000	0.0806796
$382$	−5568.00	−0.745769
$383$	2520.00	0.336204	0.168102	−	0.985770i	$-0.446236\pi$
$383$	2520.00	0.336204	0.168102	+	0.985770i	$0.446236\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	−1828.00	−0.241043
$387$	3708.00	0.487050
$388$	−776.000	−0.101535
$389$	1566.00	0.204111	0.102056	−	0.994779i	$-0.467458\pi$
$389$	1566.00	0.204111	0.102056	+	0.994779i	$0.467458\pi$
$390$	0	0
$391$	0	0
$392$	5448.00	0.701953
$393$	−180.000	−0.0231038
$394$	10404.0	1.33032
$395$	0	0
$396$	−2160.00	−0.274101
$397$	4354.00	0.550431	0.275215	−	0.961383i	$-0.411251\pi$
$397$	4354.00	0.550431	0.275215	+	0.961383i	$0.411251\pi$
$398$	6304.00	0.793947
$399$	−7296.00	−0.915431
$400$	0	0
$401$	−8046.00	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$401$	−8046.00	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$402$	4872.00	0.604461
$403$	−7888.00	−0.975011
$404$	3192.00	0.393089
$405$	0	0
$406$	−384.000	−0.0469399
$407$	8040.00	0.979184
$408$	1008.00	0.122312
$409$	−2806.00	−0.339237	−0.169618	−	0.985510i	$-0.554253\pi$
$409$	−2806.00	−0.339237	−0.169618	+	0.985510i	$0.554253\pi$
$410$	0	0
$411$	1926.00	0.231150
$412$	−4352.00	−0.520407
$413$	−21120.0	−2.51634
$414$	0	0
$415$	0	0
$416$	1088.00	0.128230
$417$	8508.00	0.999133
$418$	9120.00	1.06716
$419$	11580.0	1.35017	0.675084	−	0.737741i	$-0.264108\pi$
$419$	11580.0	1.35017	0.675084	+	0.737741i	$0.264108\pi$
$420$	0	0
$421$	−370.000	−0.0428330	−0.0214165	−	0.999771i	$-0.506818\pi$
$421$	−370.000	−0.0428330	−0.0214165	+	0.999771i	$0.506818\pi$
$422$	1480.00	0.170723
$423$	3240.00	0.372421
$424$	−1776.00	−0.203420
$425$	0	0
$426$	−720.000	−0.0818876
$427$	15680.0	1.77707
$428$	−6864.00	−0.775196
$429$	6120.00	0.688756
$430$	0	0
$431$	5040.00	0.563267	0.281634	−	0.959522i	$-0.409124\pi$
$431$	5040.00	0.563267	0.281634	+	0.959522i	$0.409124\pi$
$432$	−432.000	−0.0481125
$433$	3742.00	0.415310	0.207655	−	0.978202i	$-0.433417\pi$
$433$	3742.00	0.415310	0.207655	+	0.978202i	$0.433417\pi$
$434$	14848.0	1.64223
$435$	0	0
$436$	−3880.00	−0.426189
$437$	0	0
$438$	4476.00	0.488291
$439$	−6208.00	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$439$	−6208.00	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	−2856.00	−0.307344
$443$	15564.0	1.66923	0.834614	−	0.550835i	$-0.185691\pi$
$443$	15564.0	1.66923	0.834614	+	0.550835i	$0.185691\pi$
$444$	1608.00	0.171875
$445$	0	0
$446$	1040.00	0.110416
$447$	4662.00	0.493300
$448$	−2048.00	−0.215980
$449$	−15774.0	−1.65795	−0.828977	−	0.559283i	$-0.811076\pi$
$449$	−15774.0	−1.65795	−0.828977	+	0.559283i	$0.811076\pi$
$450$	0	0
$451$	−14040.0	−1.46589
$452$	−1704.00	−0.177322
$453$	6816.00	0.706940
$454$	−792.000	−0.0818731
$455$	0	0
$456$	1824.00	0.187317
$457$	−9722.00	−0.995133	−0.497567	−	0.867426i	$-0.665773\pi$
$457$	−9722.00	−0.995133	−0.497567	+	0.867426i	$0.665773\pi$
$458$	−2660.00	−0.271383
$459$	1134.00	0.115317
$460$	0	0
$461$	−10890.0	−1.10021	−0.550106	−	0.835095i	$-0.685413\pi$
$461$	−10890.0	−1.10021	−0.550106	+	0.835095i	$0.685413\pi$
$462$	−11520.0	−1.16008
$463$	−15128.0	−1.51848	−0.759242	−	0.650809i	$-0.774430\pi$
$463$	−15128.0	−1.51848	−0.759242	+	0.650809i	$0.774430\pi$
$464$	96.0000	0.00960493
$465$	0	0
$466$	−9732.00	−0.967438
$467$	−10668.0	−1.05708	−0.528540	−	0.848909i	$-0.677260\pi$
$467$	−10668.0	−1.05708	−0.528540	+	0.848909i	$0.677260\pi$
$468$	1224.00	0.120896
$469$	25984.0	2.55827
$470$	0	0
$471$	5082.00	0.497168
$472$	5280.00	0.514898
$473$	−24720.0	−2.40302
$474$	−912.000	−0.0883746
$475$	0	0
$476$	5376.00	0.517665
$477$	−1998.00	−0.191786
$478$	−3648.00	−0.349070
$479$	15264.0	1.45601	0.728006	−	0.685571i	$-0.240447\pi$
$479$	15264.0	1.45601	0.728006	+	0.685571i	$0.240447\pi$
$480$	0	0
$481$	−4556.00	−0.431883
$482$	12964.0	1.22509
$483$	0	0
$484$	9076.00	0.852367
$485$	0	0
$486$	−486.000	−0.0453609
$487$	5776.00	0.537445	0.268722	−	0.963218i	$-0.413399\pi$
$487$	5776.00	0.537445	0.268722	+	0.963218i	$0.413399\pi$
$488$	−3920.00	−0.363627
$489$	−156.000	−0.0144265
$490$	0	0
$491$	14244.0	1.30921	0.654606	−	0.755971i	$-0.272835\pi$
$491$	14244.0	1.30921	0.654606	+	0.755971i	$0.272835\pi$
$492$	−2808.00	−0.257306
$493$	−252.000	−0.0230213
$494$	−5168.00	−0.470687
$495$	0	0
$496$	−3712.00	−0.336036
$497$	−3840.00	−0.346575
$498$	−4824.00	−0.434074
$499$	−17116.0	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$499$	−17116.0	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$500$	0	0
$501$	−3600.00	−0.321030
$502$	2952.00	0.262459
$503$	16848.0	1.49347	0.746735	−	0.665122i	$-0.231620\pi$
$503$	16848.0	1.49347	0.746735	+	0.665122i	$0.231620\pi$
$504$	−2304.00	−0.203628
$505$	0	0
$506$	0	0
$507$	3123.00	0.273565
$508$	−800.000	−0.0698706
$509$	−3834.00	−0.333868	−0.166934	−	0.985968i	$-0.553387\pi$
$509$	−3834.00	−0.333868	−0.166934	+	0.985968i	$0.553387\pi$
$510$	0	0
$511$	23872.0	2.06660
$512$	512.000	0.0441942
$513$	2052.00	0.176604
$514$	−8628.00	−0.740398
$515$	0	0
$516$	−4944.00	−0.421797
$517$	−21600.0	−1.83746
$518$	8576.00	0.727428
$519$	162.000	0.0137014
$520$	0	0
$521$	−18822.0	−1.58274	−0.791369	−	0.611338i	$-0.790631\pi$
$521$	−18822.0	−1.58274	−0.791369	+	0.611338i	$0.790631\pi$
$522$	108.000	0.00905562
$523$	15340.0	1.28255	0.641273	−	0.767313i	$-0.278407\pi$
$523$	15340.0	1.28255	0.641273	+	0.767313i	$0.278407\pi$
$524$	240.000	0.0200085
$525$	0	0
$526$	10560.0	0.875357
$527$	9744.00	0.805418
$528$	2880.00	0.237379
$529$	−12167.0	−1.00000
$530$	0	0
$531$	5940.00	0.485450
$532$	9728.00	0.792786
$533$	7956.00	0.646553
$534$	4068.00	0.329662
$535$	0	0
$536$	−6496.00	−0.523478
$537$	−2628.00	−0.211185
$538$	11052.0	0.885661
$539$	−40860.0	−3.26524
$540$	0	0
$541$	18950.0	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$541$	18950.0	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$542$	4048.00	0.320805
$543$	−11562.0	−0.913762
$544$	−1344.00	−0.105926
$545$	0	0
$546$	6528.00	0.511671
$547$	10036.0	0.784476	0.392238	−	0.919864i	$-0.371701\pi$
$547$	10036.0	0.784476	0.392238	+	0.919864i	$0.371701\pi$
$548$	−2568.00	−0.200182
$549$	−4410.00	−0.342831
$550$	0	0
$551$	−456.000	−0.0352564
$552$	0	0
$553$	−4864.00	−0.374030
$554$	−4108.00	−0.315040
$555$	0	0
$556$	−11344.0	−0.865275
$557$	−10326.0	−0.785506	−0.392753	−	0.919644i	$-0.628477\pi$
$557$	−10326.0	−0.785506	−0.392753	+	0.919644i	$0.628477\pi$
$558$	−4176.00	−0.316818
$559$	14008.0	1.05988
$560$	0	0
$561$	−7560.00	−0.568954
$562$	−14604.0	−1.09614
$563$	−4524.00	−0.338657	−0.169328	−	0.985560i	$-0.554160\pi$
$563$	−4524.00	−0.338657	−0.169328	+	0.985560i	$0.554160\pi$
$564$	−4320.00	−0.322526
$565$	0	0
$566$	7448.00	0.553114
$567$	−2592.00	−0.191982
$568$	960.000	0.0709167
$569$	16362.0	1.20550	0.602751	−	0.797929i	$-0.294071\pi$
$569$	16362.0	1.20550	0.602751	+	0.797929i	$0.294071\pi$
$570$	0	0
$571$	6620.00	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$571$	6620.00	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$572$	−8160.00	−0.596480
$573$	8352.00	0.608918
$574$	−14976.0	−1.08900
$575$	0	0
$576$	576.000	0.0416667
$577$	−8834.00	−0.637373	−0.318687	−	0.947860i	$-0.603242\pi$
$577$	−8834.00	−0.637373	−0.318687	+	0.947860i	$0.603242\pi$
$578$	−6298.00	−0.453222
$579$	2742.00	0.196811
$580$	0	0
$581$	−25728.0	−1.83714
$582$	1164.00	0.0829027
$583$	13320.0	0.946240
$584$	−5968.00	−0.422873
$585$	0	0
$586$	14436.0	1.01765
$587$	−3636.00	−0.255662	−0.127831	−	0.991796i	$-0.540802\pi$
$587$	−3636.00	−0.255662	−0.127831	+	0.991796i	$0.540802\pi$
$588$	−8172.00	−0.573142
$589$	17632.0	1.23347
$590$	0	0
$591$	−15606.0	−1.08620
$592$	−2144.00	−0.148848
$593$	−6570.00	−0.454971	−0.227485	−	0.973782i	$-0.573050\pi$
$593$	−6570.00	−0.454971	−0.227485	+	0.973782i	$0.573050\pi$
$594$	3240.00	0.223803
$595$	0	0
$596$	−6216.00	−0.427210
$597$	−9456.00	−0.648255
$598$	0	0
$599$	16584.0	1.13123	0.565613	−	0.824671i	$-0.308640\pi$
$599$	16584.0	1.13123	0.565613	+	0.824671i	$0.308640\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	−26368.0	−1.78518
$603$	−7308.00	−0.493540
$604$	−9088.00	−0.612228
$605$	0	0
$606$	−4788.00	−0.320956
$607$	18568.0	1.24160	0.620801	−	0.783969i	$-0.286808\pi$
$607$	18568.0	1.24160	0.620801	+	0.783969i	$0.286808\pi$
$608$	−2432.00	−0.162221
$609$	576.000	0.0383263
$610$	0	0
$611$	12240.0	0.810438
$612$	−1512.00	−0.0998676
$613$	13114.0	0.864061	0.432031	−	0.901859i	$-0.357797\pi$
$613$	13114.0	0.864061	0.432031	+	0.901859i	$0.357797\pi$
$614$	−5080.00	−0.333896
$615$	0	0
$616$	15360.0	1.00466
$617$	−5250.00	−0.342556	−0.171278	−	0.985223i	$-0.554790\pi$
$617$	−5250.00	−0.342556	−0.171278	+	0.985223i	$0.554790\pi$
$618$	6528.00	0.424910
$619$	−10804.0	−0.701534	−0.350767	−	0.936463i	$-0.614079\pi$
$619$	−10804.0	−0.701534	−0.350767	+	0.936463i	$0.614079\pi$
$620$	0	0
$621$	0	0
$622$	3120.00	0.201126
$623$	21696.0	1.39524
$624$	−1632.00	−0.104699
$625$	0	0
$626$	1868.00	0.119266
$627$	−13680.0	−0.871334
$628$	−6776.00	−0.430560
$629$	5628.00	0.356762
$630$	0	0
$631$	−27088.0	−1.70896	−0.854482	−	0.519481i	$-0.826125\pi$
$631$	−27088.0	−1.70896	−0.854482	+	0.519481i	$0.826125\pi$
$632$	1216.00	0.0765346
$633$	−2220.00	−0.139395
$634$	3348.00	0.209726
$635$	0	0
$636$	2664.00	0.166092
$637$	23154.0	1.44018
$638$	−720.000	−0.0446788
$639$	1080.00	0.0668609
$640$	0	0
$641$	18930.0	1.16644	0.583222	−	0.812313i	$-0.301792\pi$
$641$	18930.0	1.16644	0.583222	+	0.812313i	$0.301792\pi$
$642$	10296.0	0.632945
$643$	−20108.0	−1.23325	−0.616627	−	0.787256i	$-0.711501\pi$
$643$	−20108.0	−1.23325	−0.616627	+	0.787256i	$0.711501\pi$
$644$	0	0
$645$	0	0
$646$	6384.00	0.388816
$647$	7152.00	0.434581	0.217291	−	0.976107i	$-0.430278\pi$
$647$	7152.00	0.434581	0.217291	+	0.976107i	$0.430278\pi$
$648$	648.000	0.0392837
$649$	−39600.0	−2.39512
$650$	0	0
$651$	−22272.0	−1.34087
$652$	208.000	0.0124937
$653$	31626.0	1.89528	0.947642	−	0.319333i	$-0.103459\pi$
$653$	31626.0	1.89528	0.947642	+	0.319333i	$0.103459\pi$
$654$	5820.00	0.347982
$655$	0	0
$656$	3744.00	0.222833
$657$	−6714.00	−0.398688
$658$	−23040.0	−1.36503
$659$	28092.0	1.66056	0.830280	−	0.557347i	$-0.188181\pi$
$659$	28092.0	1.66056	0.830280	+	0.557347i	$0.188181\pi$
$660$	0	0
$661$	−13186.0	−0.775909	−0.387955	−	0.921678i	$-0.626818\pi$
$661$	−13186.0	−0.775909	−0.387955	+	0.921678i	$0.626818\pi$
$662$	−7976.00	−0.468272
$663$	4284.00	0.250945
$664$	6432.00	0.375919
$665$	0	0
$666$	−2412.00	−0.140335
$667$	0	0
$668$	4800.00	0.278020
$669$	−1560.00	−0.0901541
$670$	0	0
$671$	29400.0	1.69147
$672$	3072.00	0.176347
$673$	−5138.00	−0.294287	−0.147144	−	0.989115i	$-0.547008\pi$
$673$	−5138.00	−0.294287	−0.147144	+	0.989115i	$0.547008\pi$
$674$	−4.00000	−0.000228597 0
$675$	0	0
$676$	−4164.00	−0.236914
$677$	−6078.00	−0.345047	−0.172523	−	0.985005i	$-0.555192\pi$
$677$	−6078.00	−0.345047	−0.172523	+	0.985005i	$0.555192\pi$
$678$	2556.00	0.144783
$679$	6208.00	0.350871
$680$	0	0
$681$	1188.00	0.0668491
$682$	27840.0	1.56312
$683$	−32244.0	−1.80642	−0.903208	−	0.429203i	$-0.858795\pi$
$683$	−32244.0	−1.80642	−0.903208	+	0.429203i	$0.858795\pi$
$684$	−2736.00	−0.152944
$685$	0	0
$686$	−21632.0	−1.20396
$687$	3990.00	0.221584
$688$	6592.00	0.365287
$689$	−7548.00	−0.417353
$690$	0	0
$691$	4484.00	0.246859	0.123429	−	0.992353i	$-0.460611\pi$
$691$	4484.00	0.246859	0.123429	+	0.992353i	$0.460611\pi$
$692$	−216.000	−0.0118657
$693$	17280.0	0.947205
$694$	−3528.00	−0.192970
$695$	0	0
$696$	−144.000	−0.00784239
$697$	−9828.00	−0.534092
$698$	8620.00	0.467438
$699$	14598.0	0.789910
$700$	0	0
$701$	−30426.0	−1.63934	−0.819668	−	0.572839i	$-0.805842\pi$
$701$	−30426.0	−1.63934	−0.819668	+	0.572839i	$0.805842\pi$
$702$	−1836.00	−0.0987113
$703$	10184.0	0.546368
$704$	−3840.00	−0.205576
$705$	0	0
$706$	−276.000	−0.0147130
$707$	−25536.0	−1.35839
$708$	−7920.00	−0.420412
$709$	13262.0	0.702489	0.351245	−	0.936284i	$-0.385759\pi$
$709$	13262.0	0.702489	0.351245	+	0.936284i	$0.385759\pi$
$710$	0	0
$711$	1368.00	0.0721575
$712$	−5424.00	−0.285496
$713$	0	0
$714$	−8064.00	−0.422672
$715$	0	0
$716$	3504.00	0.182892
$717$	5472.00	0.285015
$718$	−23952.0	−1.24496
$719$	13920.0	0.722014	0.361007	−	0.932563i	$-0.382433\pi$
$719$	13920.0	0.722014	0.361007	+	0.932563i	$0.382433\pi$
$720$	0	0
$721$	34816.0	1.79836
$722$	−2166.00	−0.111648
$723$	−19446.0	−1.00028
$724$	15416.0	0.791341
$725$	0	0
$726$	−13614.0	−0.695954
$727$	9376.00	0.478317	0.239159	−	0.970981i	$-0.423128\pi$
$727$	9376.00	0.478317	0.239159	+	0.970981i	$0.423128\pi$
$728$	−8704.00	−0.443120
$729$	729.000	0.0370370
$730$	0	0
$731$	−17304.0	−0.875529
$732$	5880.00	0.296900
$733$	−6014.00	−0.303045	−0.151523	−	0.988454i	$-0.548418\pi$
$733$	−6014.00	−0.303045	−0.151523	+	0.988454i	$0.548418\pi$
$734$	−19408.0	−0.975971
$735$	0	0
$736$	0	0
$737$	48720.0	2.43504
$738$	4212.00	0.210089
$739$	−7468.00	−0.371739	−0.185869	−	0.982574i	$-0.559510\pi$
$739$	−7468.00	−0.371739	−0.185869	+	0.982574i	$0.559510\pi$
$740$	0	0
$741$	7752.00	0.384314
$742$	14208.0	0.702954
$743$	−31248.0	−1.54290	−0.771452	−	0.636287i	$-0.780469\pi$
$743$	−31248.0	−1.54290	−0.771452	+	0.636287i	$0.780469\pi$
$744$	5568.00	0.274372
$745$	0	0
$746$	16244.0	0.797232
$747$	7236.00	0.354420
$748$	10080.0	0.492729
$749$	54912.0	2.67883
$750$	0	0
$751$	32840.0	1.59567	0.797835	−	0.602875i	$-0.205978\pi$
$751$	32840.0	1.59567	0.797835	+	0.602875i	$0.205978\pi$
$752$	5760.00	0.279316
$753$	−4428.00	−0.214297
$754$	408.000	0.0197062
$755$	0	0
$756$	3456.00	0.166261
$757$	19066.0	0.915410	0.457705	−	0.889104i	$-0.348672\pi$
$757$	19066.0	0.915410	0.457705	+	0.889104i	$0.348672\pi$
$758$	6808.00	0.326224
$759$	0	0
$760$	0	0
$761$	6858.00	0.326678	0.163339	−	0.986570i	$-0.447773\pi$
$761$	6858.00	0.326678	0.163339	+	0.986570i	$0.447773\pi$
$762$	1200.00	0.0570491
$763$	31040.0	1.47277
$764$	−11136.0	−0.527338
$765$	0	0
$766$	5040.00	0.237732
$767$	22440.0	1.05640
$768$	−768.000	−0.0360844
$769$	22178.0	1.04000	0.519999	−	0.854167i	$-0.325932\pi$
$769$	22178.0	1.04000	0.519999	+	0.854167i	$0.325932\pi$
$770$	0	0
$771$	12942.0	0.604533
$772$	−3656.00	−0.170443
$773$	−14286.0	−0.664724	−0.332362	−	0.943152i	$-0.607846\pi$
$773$	−14286.0	−0.664724	−0.332362	+	0.943152i	$0.607846\pi$
$774$	7416.00	0.344396
$775$	0	0
$776$	−1552.00	−0.0717958
$777$	−12864.0	−0.593943
$778$	3132.00	0.144329
$779$	−17784.0	−0.817943
$780$	0	0
$781$	−7200.00	−0.329880
$782$	0	0
$783$	−162.000	−0.00739388
$784$	10896.0	0.496356
$785$	0	0
$786$	−360.000	−0.0163369
$787$	18868.0	0.854602	0.427301	−	0.904109i	$-0.359465\pi$
$787$	18868.0	0.854602	0.427301	+	0.904109i	$0.359465\pi$
$788$	20808.0	0.940678
$789$	−15840.0	−0.714726
$790$	0	0
$791$	13632.0	0.612766
$792$	−4320.00	−0.193819
$793$	−16660.0	−0.746045
$794$	8708.00	0.389213
$795$	0	0
$796$	12608.0	0.561405
$797$	21690.0	0.963989	0.481994	−	0.876174i	$-0.339913\pi$
$797$	21690.0	0.963989	0.481994	+	0.876174i	$0.339913\pi$
$798$	−14592.0	−0.647307
$799$	−15120.0	−0.669471
$800$	0	0
$801$	−6102.00	−0.269168
$802$	−16092.0	−0.708514
$803$	44760.0	1.96706
$804$	9744.00	0.427418
$805$	0	0
$806$	−15776.0	−0.689437
$807$	−16578.0	−0.723139
$808$	6384.00	0.277956
$809$	−24726.0	−1.07456	−0.537281	−	0.843404i	$-0.680548\pi$
$809$	−24726.0	−1.07456	−0.537281	+	0.843404i	$0.680548\pi$
$810$	0	0
$811$	−2644.00	−0.114480	−0.0572401	−	0.998360i	$-0.518230\pi$
$811$	−2644.00	−0.114480	−0.0572401	+	0.998360i	$0.518230\pi$
$812$	−768.000	−0.0331915
$813$	−6072.00	−0.261936
$814$	16080.0	0.692388
$815$	0	0
$816$	2016.00	0.0864879
$817$	−31312.0	−1.34084
$818$	−5612.00	−0.239877
$819$	−9792.00	−0.417778
$820$	0	0
$821$	−37842.0	−1.60864	−0.804321	−	0.594195i	$-0.797471\pi$
$821$	−37842.0	−1.60864	−0.804321	+	0.594195i	$0.797471\pi$
$822$	3852.00	0.163448
$823$	880.000	0.0372720	0.0186360	−	0.999826i	$-0.494068\pi$
$823$	880.000	0.0372720	0.0186360	+	0.999826i	$0.494068\pi$
$824$	−8704.00	−0.367983
$825$	0	0
$826$	−42240.0	−1.77932
$827$	12876.0	0.541406	0.270703	−	0.962663i	$-0.412744\pi$
$827$	12876.0	0.541406	0.270703	+	0.962663i	$0.412744\pi$
$828$	0	0
$829$	−25498.0	−1.06825	−0.534127	−	0.845404i	$-0.679359\pi$
$829$	−25498.0	−1.06825	−0.534127	+	0.845404i	$0.679359\pi$
$830$	0	0
$831$	6162.00	0.257229
$832$	2176.00	0.0906721
$833$	−28602.0	−1.18968
$834$	17016.0	0.706494
$835$	0	0
$836$	18240.0	0.754598
$837$	6264.00	0.258680
$838$	23160.0	0.954712
$839$	−40584.0	−1.66998	−0.834991	−	0.550263i	$-0.814527\pi$
$839$	−40584.0	−1.66998	−0.834991	+	0.550263i	$0.814527\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	−740.000	−0.0302875
$843$	21906.0	0.894997
$844$	2960.00	0.120720
$845$	0	0
$846$	6480.00	0.263342
$847$	−72608.0	−2.94550
$848$	−3552.00	−0.143840
$849$	−11172.0	−0.451616
$850$	0	0
$851$	0	0
$852$	−1440.00	−0.0579033
$853$	25738.0	1.03312	0.516561	−	0.856251i	$-0.327212\pi$
$853$	25738.0	1.03312	0.516561	+	0.856251i	$0.327212\pi$
$854$	31360.0	1.25658
$855$	0	0
$856$	−13728.0	−0.548146
$857$	−13314.0	−0.530686	−0.265343	−	0.964154i	$-0.585485\pi$
$857$	−13314.0	−0.530686	−0.265343	+	0.964154i	$0.585485\pi$
$858$	12240.0	0.487024
$859$	24524.0	0.974096	0.487048	−	0.873375i	$-0.338074\pi$
$859$	24524.0	0.974096	0.487048	+	0.873375i	$0.338074\pi$
$860$	0	0
$861$	22464.0	0.889165
$862$	10080.0	0.398290
$863$	−5592.00	−0.220572	−0.110286	−	0.993900i	$-0.535177\pi$
$863$	−5592.00	−0.220572	−0.110286	+	0.993900i	$0.535177\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	7484.00	0.293668
$867$	9447.00	0.370054
$868$	29696.0	1.16123
$869$	−9120.00	−0.356012
$870$	0	0
$871$	−27608.0	−1.07401
$872$	−7760.00	−0.301361
$873$	−1746.00	−0.0676897
$874$	0	0
$875$	0	0
$876$	8952.00	0.345274
$877$	14386.0	0.553912	0.276956	−	0.960883i	$-0.410674\pi$
$877$	14386.0	0.553912	0.276956	+	0.960883i	$0.410674\pi$
$878$	−12416.0	−0.477243
$879$	−21654.0	−0.830912
$880$	0	0
$881$	47106.0	1.80141	0.900705	−	0.434432i	$-0.143051\pi$
$881$	47106.0	1.80141	0.900705	+	0.434432i	$0.143051\pi$
$882$	12258.0	0.467969
$883$	−51548.0	−1.96458	−0.982292	−	0.187354i	$-0.940009\pi$
$883$	−51548.0	−1.96458	−0.982292	+	0.187354i	$0.940009\pi$
$884$	−5712.00	−0.217325
$885$	0	0
$886$	31128.0	1.18032
$887$	−34080.0	−1.29007	−0.645036	−	0.764152i	$-0.723158\pi$
$887$	−34080.0	−1.29007	−0.645036	+	0.764152i	$0.723158\pi$
$888$	3216.00	0.121534
$889$	6400.00	0.241450
$890$	0	0
$891$	−4860.00	−0.182734
$892$	2080.00	0.0780757
$893$	−27360.0	−1.02527
$894$	9324.00	0.348816
$895$	0	0
$896$	−4096.00	−0.152721
$897$	0	0
$898$	−31548.0	−1.17235
$899$	−1392.00	−0.0516416
$900$	0	0
$901$	9324.00	0.344759
$902$	−28080.0	−1.03654
$903$	39552.0	1.45759
$904$	−3408.00	−0.125385
$905$	0	0
$906$	13632.0	0.499882
$907$	−25748.0	−0.942611	−0.471306	−	0.881970i	$-0.656217\pi$
$907$	−25748.0	−0.942611	−0.471306	+	0.881970i	$0.656217\pi$
$908$	−1584.00	−0.0578930
$909$	7182.00	0.262059
$910$	0	0
$911$	−24768.0	−0.900769	−0.450384	−	0.892835i	$-0.648713\pi$
$911$	−24768.0	−0.900769	−0.450384	+	0.892835i	$0.648713\pi$
$912$	3648.00	0.132453
$913$	−48240.0	−1.74864
$914$	−19444.0	−0.703666
$915$	0	0
$916$	−5320.00	−0.191897
$917$	−1920.00	−0.0691428
$918$	2268.00	0.0815416
$919$	−31264.0	−1.12220	−0.561101	−	0.827747i	$-0.689622\pi$
$919$	−31264.0	−1.12220	−0.561101	+	0.827747i	$0.689622\pi$
$920$	0	0
$921$	7620.00	0.272625
$922$	−21780.0	−0.777968
$923$	4080.00	0.145498
$924$	−23040.0	−0.820303
$925$	0	0
$926$	−30256.0	−1.07373
$927$	−9792.00	−0.346938
$928$	192.000	0.00679171
$929$	−6174.00	−0.218043	−0.109022	−	0.994039i	$-0.534772\pi$
$929$	−6174.00	−0.218043	−0.109022	+	0.994039i	$0.534772\pi$
$930$	0	0
$931$	−51756.0	−1.82195
$932$	−19464.0	−0.684082
$933$	−4680.00	−0.164219
$934$	−21336.0	−0.747468
$935$	0	0
$936$	2448.00	0.0854865
$937$	−28922.0	−1.00837	−0.504184	−	0.863596i	$-0.668207\pi$
$937$	−28922.0	−1.00837	−0.504184	+	0.863596i	$0.668207\pi$
$938$	51968.0	1.80897
$939$	−2802.00	−0.0973800
$940$	0	0
$941$	29238.0	1.01289	0.506446	−	0.862272i	$-0.330959\pi$
$941$	29238.0	1.01289	0.506446	+	0.862272i	$0.330959\pi$
$942$	10164.0	0.351551
$943$	0	0
$944$	10560.0	0.364088
$945$	0	0
$946$	−49440.0	−1.69919
$947$	2868.00	0.0984134	0.0492067	−	0.998789i	$-0.484331\pi$
$947$	2868.00	0.0984134	0.0492067	+	0.998789i	$0.484331\pi$
$948$	−1824.00	−0.0624903
$949$	−25364.0	−0.867598
$950$	0	0
$951$	−5022.00	−0.171240
$952$	10752.0	0.366044
$953$	−24018.0	−0.816390	−0.408195	−	0.912895i	$-0.633842\pi$
$953$	−24018.0	−0.816390	−0.408195	+	0.912895i	$0.633842\pi$
$954$	−3996.00	−0.135613
$955$	0	0
$956$	−7296.00	−0.246830
$957$	1080.00	0.0364801
$958$	30528.0	1.02956
$959$	20544.0	0.691763
$960$	0	0
$961$	24033.0	0.806720
$962$	−9112.00	−0.305387
$963$	−15444.0	−0.516797
$964$	25928.0	0.866270
$965$	0	0
$966$	0	0
$967$	−25712.0	−0.855059	−0.427530	−	0.904001i	$-0.640616\pi$
$967$	−25712.0	−0.855059	−0.427530	+	0.904001i	$0.640616\pi$
$968$	18152.0	0.602714
$969$	−9576.00	−0.317467
$970$	0	0
$971$	−12396.0	−0.409688	−0.204844	−	0.978795i	$-0.565669\pi$
$971$	−12396.0	−0.409688	−0.204844	+	0.978795i	$0.565669\pi$
$972$	−972.000	−0.0320750
$973$	90752.0	2.99011
$974$	11552.0	0.380031
$975$	0	0
$976$	−7840.00	−0.257123
$977$	46614.0	1.52642	0.763211	−	0.646150i	$-0.223622\pi$
$977$	46614.0	1.52642	0.763211	+	0.646150i	$0.223622\pi$
$978$	−312.000	−0.0102011
$979$	40680.0	1.32803
$980$	0	0
$981$	−8730.00	−0.284126
$982$	28488.0	0.925752
$983$	672.000	0.0218041	0.0109021	−	0.999941i	$-0.496530\pi$
$983$	672.000	0.0218041	0.0109021	+	0.999941i	$0.496530\pi$
$984$	−5616.00	−0.181943
$985$	0	0
$986$	−504.000	−0.0162785
$987$	34560.0	1.11455
$988$	−10336.0	−0.332826
$989$	0	0
$990$	0	0
$991$	−38776.0	−1.24295	−0.621473	−	0.783435i	$-0.713466\pi$
$991$	−38776.0	−1.24295	−0.621473	+	0.783435i	$0.713466\pi$
$992$	−7424.00	−0.237613
$993$	11964.0	0.382342
$994$	−7680.00	−0.245065
$995$	0	0
$996$	−9648.00	−0.306936
$997$	−30422.0	−0.966374	−0.483187	−	0.875517i	$-0.660521\pi$
$997$	−30422.0	−0.966374	−0.483187	+	0.875517i	$0.660521\pi$
$998$	−34232.0	−1.08577
$999$	3618.00	0.114583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.a.e.1.1		1
3.2	odd	2		450.4.a.b.1.1		1
4.3	odd	2		1200.4.a.bk.1.1		1
5.2	odd	4		150.4.c.a.49.2		2
5.3	odd	4		150.4.c.a.49.1		2
5.4	even	2		30.4.a.a.1.1	✓	1
15.2	even	4		450.4.c.k.199.1		2
15.8	even	4		450.4.c.k.199.2		2
15.14	odd	2		90.4.a.d.1.1		1
20.3	even	4		1200.4.f.u.49.2		2
20.7	even	4		1200.4.f.u.49.1		2
20.19	odd	2		240.4.a.c.1.1		1
35.34	odd	2		1470.4.a.a.1.1		1
40.19	odd	2		960.4.a.s.1.1		1
40.29	even	2		960.4.a.j.1.1		1
45.4	even	6		810.4.e.m.541.1		2
45.14	odd	6		810.4.e.e.541.1		2
45.29	odd	6		810.4.e.e.271.1		2
45.34	even	6		810.4.e.m.271.1		2
60.59	even	2		720.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	5.4	even	2
90.4.a.d.1.1		1	15.14	odd	2
150.4.a.e.1.1		1	1.1	even	1	trivial
150.4.c.a.49.1		2	5.3	odd	4
150.4.c.a.49.2		2	5.2	odd	4
240.4.a.c.1.1		1	20.19	odd	2
450.4.a.b.1.1		1	3.2	odd	2
450.4.c.k.199.1		2	15.2	even	4
450.4.c.k.199.2		2	15.8	even	4
720.4.a.b.1.1		1	60.59	even	2
810.4.e.e.271.1		2	45.29	odd	6
810.4.e.e.541.1		2	45.14	odd	6
810.4.e.m.271.1		2	45.34	even	6
810.4.e.m.541.1		2	45.4	even	6
960.4.a.j.1.1		1	40.29	even	2
960.4.a.s.1.1		1	40.19	odd	2
1200.4.a.bk.1.1		1	4.3	odd	2
1200.4.f.u.49.1		2	20.7	even	4
1200.4.f.u.49.2		2	20.3	even	4
1470.4.a.a.1.1		1	35.34	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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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