LMFDB - Embedded newform 150.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,6,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, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("150.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$24.0575729719$
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)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -32.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -32.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +12.0000 q^{11} -144.000 q^{12} +154.000 q^{13} +128.000 q^{14} +256.000 q^{16} +918.000 q^{17} -324.000 q^{18} -1060.00 q^{19} +288.000 q^{21} -48.0000 q^{22} +4224.00 q^{23} +576.000 q^{24} -616.000 q^{26} -729.000 q^{27} -512.000 q^{28} -7890.00 q^{29} +5192.00 q^{31} -1024.00 q^{32} -108.000 q^{33} -3672.00 q^{34} +1296.00 q^{36} -16382.0 q^{37} +4240.00 q^{38} -1386.00 q^{39} +3642.00 q^{41} -1152.00 q^{42} -15116.0 q^{43} +192.000 q^{44} -16896.0 q^{46} -23592.0 q^{47} -2304.00 q^{48} -15783.0 q^{49} -8262.00 q^{51} +2464.00 q^{52} +16074.0 q^{53} +2916.00 q^{54} +2048.00 q^{56} +9540.00 q^{57} +31560.0 q^{58} -14340.0 q^{59} -47938.0 q^{61} -20768.0 q^{62} -2592.00 q^{63} +4096.00 q^{64} +432.000 q^{66} -33092.0 q^{67} +14688.0 q^{68} -38016.0 q^{69} +51912.0 q^{71} -5184.00 q^{72} -12026.0 q^{73} +65528.0 q^{74} -16960.0 q^{76} -384.000 q^{77} +5544.00 q^{78} +25160.0 q^{79} +6561.00 q^{81} -14568.0 q^{82} -35796.0 q^{83} +4608.00 q^{84} +60464.0 q^{86} +71010.0 q^{87} -768.000 q^{88} -75510.0 q^{89} -4928.00 q^{91} +67584.0 q^{92} -46728.0 q^{93} +94368.0 q^{94} +9216.00 q^{96} +44158.0 q^{97} +63132.0 q^{98} +972.000 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	−32.0000	−0.246834	−0.123417	−	0.992355i	$-0.539385\pi$
$7$	−32.0000	−0.246834	−0.123417	+	0.992355i	$0.539385\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	0	0
$11$	12.0000	0.0299020	0.0149510	−	0.999888i	$-0.495241\pi$
$11$	12.0000	0.0299020	0.0149510	+	0.999888i	$0.495241\pi$
$12$	−144.000	−0.288675
$13$	154.000	0.252733	0.126367	−	0.991984i	$-0.459668\pi$
$13$	154.000	0.252733	0.126367	+	0.991984i	$0.459668\pi$
$14$	128.000	0.174538
$15$	0	0
$16$	256.000	0.250000
$17$	918.000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	918.000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	−324.000	−0.235702
$19$	−1060.00	−0.673631	−0.336815	−	0.941571i	$-0.609350\pi$
$19$	−1060.00	−0.673631	−0.336815	+	0.941571i	$0.609350\pi$
$20$	0	0
$21$	288.000	0.142510
$22$	−48.0000	−0.0211439
$23$	4224.00	1.66496	0.832481	−	0.554054i	$-0.186920\pi$
$23$	4224.00	1.66496	0.832481	+	0.554054i	$0.186920\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−616.000	−0.178709
$27$	−729.000	−0.192450
$28$	−512.000	−0.123417
$29$	−7890.00	−1.74214	−0.871068	−	0.491163i	$-0.836572\pi$
$29$	−7890.00	−1.74214	−0.871068	+	0.491163i	$0.836572\pi$
$30$	0	0
$31$	5192.00	0.970355	0.485177	−	0.874416i	$-0.338755\pi$
$31$	5192.00	0.970355	0.485177	+	0.874416i	$0.338755\pi$
$32$	−1024.00	−0.176777
$33$	−108.000	−0.0172639
$34$	−3672.00	−0.544760
$35$	0	0
$36$	1296.00	0.166667
$37$	−16382.0	−1.96726	−0.983632	−	0.180189i	$-0.942329\pi$
$37$	−16382.0	−1.96726	−0.983632	+	0.180189i	$0.942329\pi$
$38$	4240.00	0.476329
$39$	−1386.00	−0.145916
$40$	0	0
$41$	3642.00	0.338361	0.169180	−	0.985585i	$-0.445888\pi$
$41$	3642.00	0.338361	0.169180	+	0.985585i	$0.445888\pi$
$42$	−1152.00	−0.100770
$43$	−15116.0	−1.24671	−0.623355	−	0.781939i	$-0.714231\pi$
$43$	−15116.0	−1.24671	−0.623355	+	0.781939i	$0.714231\pi$
$44$	192.000	0.0149510
$45$	0	0
$46$	−16896.0	−1.17731
$47$	−23592.0	−1.55783	−0.778915	−	0.627129i	$-0.784230\pi$
$47$	−23592.0	−1.55783	−0.778915	+	0.627129i	$0.784230\pi$
$48$	−2304.00	−0.144338
$49$	−15783.0	−0.939073
$50$	0	0
$51$	−8262.00	−0.444795
$52$	2464.00	0.126367
$53$	16074.0	0.786021	0.393011	−	0.919534i	$-0.371434\pi$
$53$	16074.0	0.786021	0.393011	+	0.919534i	$0.371434\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	2048.00	0.0872690
$57$	9540.00	0.388921
$58$	31560.0	1.23188
$59$	−14340.0	−0.536314	−0.268157	−	0.963375i	$-0.586415\pi$
$59$	−14340.0	−0.536314	−0.268157	+	0.963375i	$0.586415\pi$
$60$	0	0
$61$	−47938.0	−1.64951	−0.824756	−	0.565489i	$-0.808687\pi$
$61$	−47938.0	−1.64951	−0.824756	+	0.565489i	$0.808687\pi$
$62$	−20768.0	−0.686144
$63$	−2592.00	−0.0822780
$64$	4096.00	0.125000
$65$	0	0
$66$	432.000	0.0122074
$67$	−33092.0	−0.900608	−0.450304	−	0.892875i	$-0.648684\pi$
$67$	−33092.0	−0.900608	−0.450304	+	0.892875i	$0.648684\pi$
$68$	14688.0	0.385204
$69$	−38016.0	−0.961266
$70$	0	0
$71$	51912.0	1.22214	0.611071	−	0.791576i	$-0.290739\pi$
$71$	51912.0	1.22214	0.611071	+	0.791576i	$0.290739\pi$
$72$	−5184.00	−0.117851
$73$	−12026.0	−0.264128	−0.132064	−	0.991241i	$-0.542160\pi$
$73$	−12026.0	−0.264128	−0.132064	+	0.991241i	$0.542160\pi$
$74$	65528.0	1.39107
$75$	0	0
$76$	−16960.0	−0.336815
$77$	−384.000	−0.00738082
$78$	5544.00	0.103178
$79$	25160.0	0.453569	0.226784	−	0.973945i	$-0.427179\pi$
$79$	25160.0	0.453569	0.226784	+	0.973945i	$0.427179\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−14568.0	−0.239257
$83$	−35796.0	−0.570347	−0.285174	−	0.958476i	$-0.592051\pi$
$83$	−35796.0	−0.570347	−0.285174	+	0.958476i	$0.592051\pi$
$84$	4608.00	0.0712548
$85$	0	0
$86$	60464.0	0.881558
$87$	71010.0	1.00582
$88$	−768.000	−0.0105719
$89$	−75510.0	−1.01048	−0.505242	−	0.862978i	$-0.668597\pi$
$89$	−75510.0	−1.01048	−0.505242	+	0.862978i	$0.668597\pi$
$90$	0	0
$91$	−4928.00	−0.0623831
$92$	67584.0	0.832481
$93$	−46728.0	−0.560234
$94$	94368.0	1.10155
$95$	0	0
$96$	9216.00	0.102062
$97$	44158.0	0.476519	0.238259	−	0.971202i	$-0.423423\pi$
$97$	44158.0	0.476519	0.238259	+	0.971202i	$0.423423\pi$
$98$	63132.0	0.664025
$99$	972.000	0.00996732
$100$	0	0
$101$	88662.0	0.864837	0.432418	−	0.901673i	$-0.357660\pi$
$101$	88662.0	0.864837	0.432418	+	0.901673i	$0.357660\pi$
$102$	33048.0	0.314517
$103$	−149696.	−1.39033	−0.695164	−	0.718851i	$-0.744668\pi$
$103$	−149696.	−1.39033	−0.695164	+	0.718851i	$0.744668\pi$
$104$	−9856.00	−0.0893547
$105$	0	0
$106$	−64296.0	−0.555801
$107$	189828.	1.60288	0.801440	−	0.598076i	$-0.204068\pi$
$107$	189828.	1.60288	0.801440	+	0.598076i	$0.204068\pi$
$108$	−11664.0	−0.0962250
$109$	18110.0	0.146000	0.0729999	−	0.997332i	$-0.476743\pi$
$109$	18110.0	0.146000	0.0729999	+	0.997332i	$0.476743\pi$
$110$	0	0
$111$	147438.	1.13580
$112$	−8192.00	−0.0617085
$113$	−2346.00	−0.0172835	−0.00864175	−	0.999963i	$-0.502751\pi$
$113$	−2346.00	−0.0172835	−0.00864175	+	0.999963i	$0.502751\pi$
$114$	−38160.0	−0.275009
$115$	0	0
$116$	−126240.	−0.871068
$117$	12474.0	0.0842444
$118$	57360.0	0.379231
$119$	−29376.0	−0.190163
$120$	0	0
$121$	−160907.	−0.999106
$122$	191752.	1.16638
$123$	−32778.0	−0.195353
$124$	83072.0	0.485177
$125$	0	0
$126$	10368.0	0.0581793
$127$	−43832.0	−0.241147	−0.120574	−	0.992704i	$-0.538473\pi$
$127$	−43832.0	−0.241147	−0.120574	+	0.992704i	$0.538473\pi$
$128$	−16384.0	−0.0883883
$129$	136044.	0.719789
$130$	0	0
$131$	87252.0	0.444219	0.222110	−	0.975022i	$-0.428706\pi$
$131$	87252.0	0.444219	0.222110	+	0.975022i	$0.428706\pi$
$132$	−1728.00	−0.00863195
$133$	33920.0	0.166275
$134$	132368.	0.636826
$135$	0	0
$136$	−58752.0	−0.272380
$137$	−118002.	−0.537141	−0.268570	−	0.963260i	$-0.586551\pi$
$137$	−118002.	−0.537141	−0.268570	+	0.963260i	$0.586551\pi$
$138$	152064.	0.679718
$139$	260900.	1.14535	0.572673	−	0.819784i	$-0.305906\pi$
$139$	260900.	1.14535	0.572673	+	0.819784i	$0.305906\pi$
$140$	0	0
$141$	212328.	0.899414
$142$	−207648.	−0.864185
$143$	1848.00	0.00755722
$144$	20736.0	0.0833333
$145$	0	0
$146$	48104.0	0.186767
$147$	142047.	0.542174
$148$	−262112.	−0.983632
$149$	94470.0	0.348601	0.174300	−	0.984693i	$-0.444234\pi$
$149$	94470.0	0.348601	0.174300	+	0.984693i	$0.444234\pi$
$150$	0	0
$151$	−306688.	−1.09460	−0.547299	−	0.836937i	$-0.684344\pi$
$151$	−306688.	−1.09460	−0.547299	+	0.836937i	$0.684344\pi$
$152$	67840.0	0.238164
$153$	74358.0	0.256802
$154$	1536.00	0.00521903
$155$	0	0
$156$	−22176.0	−0.0729578
$157$	−433862.	−1.40476	−0.702381	−	0.711802i	$-0.747880\pi$
$157$	−433862.	−1.40476	−0.702381	+	0.711802i	$0.747880\pi$
$158$	−100640.	−0.320721
$159$	−144666.	−0.453810
$160$	0	0
$161$	−135168.	−0.410969
$162$	−26244.0	−0.0785674
$163$	203164.	0.598932	0.299466	−	0.954107i	$-0.403191\pi$
$163$	203164.	0.598932	0.299466	+	0.954107i	$0.403191\pi$
$164$	58272.0	0.169180
$165$	0	0
$166$	143184.	0.403296
$167$	96528.0	0.267832	0.133916	−	0.990993i	$-0.457245\pi$
$167$	96528.0	0.267832	0.133916	+	0.990993i	$0.457245\pi$
$168$	−18432.0	−0.0503848
$169$	−347577.	−0.936126
$170$	0	0
$171$	−85860.0	−0.224544
$172$	−241856.	−0.623355
$173$	−255486.	−0.649011	−0.324505	−	0.945884i	$-0.605198\pi$
$173$	−255486.	−0.649011	−0.324505	+	0.945884i	$0.605198\pi$
$174$	−284040.	−0.711224
$175$	0	0
$176$	3072.00	0.00747549
$177$	129060.	0.309641
$178$	302040.	0.714520
$179$	−639900.	−1.49272	−0.746362	−	0.665540i	$-0.768201\pi$
$179$	−639900.	−1.49272	−0.746362	+	0.665540i	$0.768201\pi$
$180$	0	0
$181$	−550618.	−1.24926	−0.624632	−	0.780920i	$-0.714751\pi$
$181$	−550618.	−1.24926	−0.624632	+	0.780920i	$0.714751\pi$
$182$	19712.0	0.0441115
$183$	431442.	0.952346
$184$	−270336.	−0.588653
$185$	0	0
$186$	186912.	0.396146
$187$	11016.0	0.0230367
$188$	−377472.	−0.778915
$189$	23328.0	0.0475032
$190$	0	0
$191$	16032.0	0.0317983	0.0158992	−	0.999874i	$-0.494939\pi$
$191$	16032.0	0.0317983	0.0158992	+	0.999874i	$0.494939\pi$
$192$	−36864.0	−0.0721688
$193$	137134.	0.265004	0.132502	−	0.991183i	$-0.457699\pi$
$193$	137134.	0.265004	0.132502	+	0.991183i	$0.457699\pi$
$194$	−176632.	−0.336950
$195$	0	0
$196$	−252528.	−0.469537
$197$	−893862.	−1.64099	−0.820493	−	0.571656i	$-0.806301\pi$
$197$	−893862.	−1.64099	−0.820493	+	0.571656i	$0.806301\pi$
$198$	−3888.00	−0.00704796
$199$	189200.	0.338679	0.169340	−	0.985558i	$-0.445837\pi$
$199$	189200.	0.338679	0.169340	+	0.985558i	$0.445837\pi$
$200$	0	0
$201$	297828.	0.519967
$202$	−354648.	−0.611532
$203$	252480.	0.430018
$204$	−132192.	−0.222397
$205$	0	0
$206$	598784.	0.983110
$207$	342144.	0.554987
$208$	39424.0	0.0631833
$209$	−12720.0	−0.0201429
$210$	0	0
$211$	940652.	1.45453	0.727265	−	0.686356i	$-0.240791\pi$
$211$	940652.	1.45453	0.727265	+	0.686356i	$0.240791\pi$
$212$	257184.	0.393011
$213$	−467208.	−0.705604
$214$	−759312.	−1.13341
$215$	0	0
$216$	46656.0	0.0680414
$217$	−166144.	−0.239516
$218$	−72440.0	−0.103237
$219$	108234.	0.152494
$220$	0	0
$221$	141372.	0.194708
$222$	−589752.	−0.803132
$223$	170104.	0.229062	0.114531	−	0.993420i	$-0.463464\pi$
$223$	170104.	0.229062	0.114531	+	0.993420i	$0.463464\pi$
$224$	32768.0	0.0436345
$225$	0	0
$226$	9384.00	0.0122213
$227$	−19812.0	−0.0255190	−0.0127595	−	0.999919i	$-0.504062\pi$
$227$	−19812.0	−0.0255190	−0.0127595	+	0.999919i	$0.504062\pi$
$228$	152640.	0.194460
$229$	1.53287e6	1.93160	0.965799	−	0.259293i	$-0.0834895\pi$
$229$	1.53287e6	1.93160	0.965799	+	0.259293i	$0.0834895\pi$
$230$	0	0
$231$	3456.00	0.00426132
$232$	504960.	0.615938
$233$	892494.	1.07700	0.538500	−	0.842626i	$-0.318991\pi$
$233$	892494.	1.07700	0.538500	+	0.842626i	$0.318991\pi$
$234$	−49896.0	−0.0595698
$235$	0	0
$236$	−229440.	−0.268157
$237$	−226440.	−0.261868
$238$	117504.	0.134465
$239$	−654240.	−0.740871	−0.370435	−	0.928858i	$-0.620791\pi$
$239$	−654240.	−0.740871	−0.370435	+	0.928858i	$0.620791\pi$
$240$	0	0
$241$	−937678.	−1.03995	−0.519973	−	0.854182i	$-0.674058\pi$
$241$	−937678.	−1.03995	−0.519973	+	0.854182i	$0.674058\pi$
$242$	643628.	0.706475
$243$	−59049.0	−0.0641500
$244$	−767008.	−0.824756
$245$	0	0
$246$	131112.	0.138135
$247$	−163240.	−0.170249
$248$	−332288.	−0.343072
$249$	322164.	0.329290
$250$	0	0
$251$	−733428.	−0.734807	−0.367403	−	0.930062i	$-0.619753\pi$
$251$	−733428.	−0.734807	−0.367403	+	0.930062i	$0.619753\pi$
$252$	−41472.0	−0.0411390
$253$	50688.0	0.0497856
$254$	175328.	0.170517
$255$	0	0
$256$	65536.0	0.0625000
$257$	−139002.	−0.131277	−0.0656384	−	0.997843i	$-0.520908\pi$
$257$	−139002.	−0.131277	−0.0656384	+	0.997843i	$0.520908\pi$
$258$	−544176.	−0.508968
$259$	524224.	0.485587
$260$	0	0
$261$	−639090.	−0.580712
$262$	−349008.	−0.314110
$263$	1.65494e6	1.47535	0.737673	−	0.675158i	$-0.235925\pi$
$263$	1.65494e6	1.47535	0.737673	+	0.675158i	$0.235925\pi$
$264$	6912.00	0.00610371
$265$	0	0
$266$	−135680.	−0.117574
$267$	679590.	0.583403
$268$	−529472.	−0.450304
$269$	562590.	0.474036	0.237018	−	0.971505i	$-0.423830\pi$
$269$	562590.	0.474036	0.237018	+	0.971505i	$0.423830\pi$
$270$	0	0
$271$	1.49823e6	1.23924	0.619621	−	0.784901i	$-0.287286\pi$
$271$	1.49823e6	1.23924	0.619621	+	0.784901i	$0.287286\pi$
$272$	235008.	0.192602
$273$	44352.0	0.0360169
$274$	472008.	0.379816
$275$	0	0
$276$	−608256.	−0.480633
$277$	2.10282e6	1.64665	0.823327	−	0.567568i	$-0.192115\pi$
$277$	2.10282e6	1.64665	0.823327	+	0.567568i	$0.192115\pi$
$278$	−1.04360e6	−0.809883
$279$	420552.	0.323452
$280$	0	0
$281$	−1.88540e6	−1.42442	−0.712209	−	0.701968i	$-0.752305\pi$
$281$	−1.88540e6	−1.42442	−0.712209	+	0.701968i	$0.752305\pi$
$282$	−849312.	−0.635982
$283$	−2.11008e6	−1.56615	−0.783073	−	0.621930i	$-0.786349\pi$
$283$	−2.11008e6	−1.56615	−0.783073	+	0.621930i	$0.786349\pi$
$284$	830592.	0.611071
$285$	0	0
$286$	−7392.00	−0.00534376
$287$	−116544.	−0.0835190
$288$	−82944.0	−0.0589256
$289$	−577133.	−0.406473
$290$	0	0
$291$	−397422.	−0.275118
$292$	−192416.	−0.132064
$293$	−345606.	−0.235186	−0.117593	−	0.993062i	$-0.537518\pi$
$293$	−345606.	−0.235186	−0.117593	+	0.993062i	$0.537518\pi$
$294$	−568188.	−0.383375
$295$	0	0
$296$	1.04845e6	0.695533
$297$	−8748.00	−0.00575463
$298$	−377880.	−0.246498
$299$	650496.	0.420791
$300$	0	0
$301$	483712.	0.307731
$302$	1.22675e6	0.773997
$303$	−797958.	−0.499314
$304$	−271360.	−0.168408
$305$	0	0
$306$	−297432.	−0.181587
$307$	2.22003e6	1.34435	0.672175	−	0.740392i	$-0.265360\pi$
$307$	2.22003e6	1.34435	0.672175	+	0.740392i	$0.265360\pi$
$308$	−6144.00	−0.00369041
$309$	1.34726e6	0.802706
$310$	0	0
$311$	3.02935e6	1.77602	0.888012	−	0.459821i	$-0.152086\pi$
$311$	3.02935e6	1.77602	0.888012	+	0.459821i	$0.152086\pi$
$312$	88704.0	0.0515890
$313$	999094.	0.576429	0.288214	−	0.957566i	$-0.406938\pi$
$313$	999094.	0.576429	0.288214	+	0.957566i	$0.406938\pi$
$314$	1.73545e6	0.993316
$315$	0	0
$316$	402560.	0.226784
$317$	−2.80918e6	−1.57012	−0.785058	−	0.619422i	$-0.787367\pi$
$317$	−2.80918e6	−1.57012	−0.785058	+	0.619422i	$0.787367\pi$
$318$	578664.	0.320892
$319$	−94680.0	−0.0520933
$320$	0	0
$321$	−1.70845e6	−0.925423
$322$	540672.	0.290599
$323$	−973080.	−0.518970
$324$	104976.	0.0555556
$325$	0	0
$326$	−812656.	−0.423509
$327$	−162990.	−0.0842930
$328$	−233088.	−0.119629
$329$	754944.	0.384525
$330$	0	0
$331$	2.41149e6	1.20981	0.604903	−	0.796299i	$-0.293212\pi$
$331$	2.41149e6	1.20981	0.604903	+	0.796299i	$0.293212\pi$
$332$	−572736.	−0.285174
$333$	−1.32694e6	−0.655755
$334$	−386112.	−0.189386
$335$	0	0
$336$	73728.0	0.0356274
$337$	3.41904e6	1.63994	0.819972	−	0.572403i	$-0.193989\pi$
$337$	3.41904e6	1.63994	0.819972	+	0.572403i	$0.193989\pi$
$338$	1.39031e6	0.661941
$339$	21114.0	0.00997864
$340$	0	0
$341$	62304.0	0.0290155
$342$	343440.	0.158776
$343$	1.04288e6	0.478629
$344$	967424.	0.440779
$345$	0	0
$346$	1.02194e6	0.458920
$347$	−1.42177e6	−0.633879	−0.316939	−	0.948446i	$-0.602655\pi$
$347$	−1.42177e6	−0.633879	−0.316939	+	0.948446i	$0.602655\pi$
$348$	1.13616e6	0.502911
$349$	−2.01901e6	−0.887309	−0.443655	−	0.896198i	$-0.646318\pi$
$349$	−2.01901e6	−0.887309	−0.443655	+	0.896198i	$0.646318\pi$
$350$	0	0
$351$	−112266.	−0.0486385
$352$	−12288.0	−0.00528597
$353$	74454.0	0.0318018	0.0159009	−	0.999874i	$-0.494938\pi$
$353$	74454.0	0.0318018	0.0159009	+	0.999874i	$0.494938\pi$
$354$	−516240.	−0.218949
$355$	0	0
$356$	−1.20816e6	−0.505242
$357$	264384.	0.109790
$358$	2.55960e6	1.05552
$359$	−1.72908e6	−0.708075	−0.354037	−	0.935231i	$-0.615191\pi$
$359$	−1.72908e6	−0.708075	−0.354037	+	0.935231i	$0.615191\pi$
$360$	0	0
$361$	−1.35250e6	−0.546222
$362$	2.20247e6	0.883363
$363$	1.44816e6	0.576834
$364$	−78848.0	−0.0311916
$365$	0	0
$366$	−1.72577e6	−0.673410
$367$	2.52177e6	0.977327	0.488664	−	0.872472i	$-0.337485\pi$
$367$	2.52177e6	0.977327	0.488664	+	0.872472i	$0.337485\pi$
$368$	1.08134e6	0.416240
$369$	295002.	0.112787
$370$	0	0
$371$	−514368.	−0.194017
$372$	−747648.	−0.280117
$373$	−2.65917e6	−0.989631	−0.494816	−	0.868998i	$-0.664764\pi$
$373$	−2.65917e6	−0.989631	−0.494816	+	0.868998i	$0.664764\pi$
$374$	−44064.0	−0.0162894
$375$	0	0
$376$	1.50989e6	0.550776
$377$	−1.21506e6	−0.440296
$378$	−93312.0	−0.0335898
$379$	−3.43366e6	−1.22789	−0.613945	−	0.789349i	$-0.710418\pi$
$379$	−3.43366e6	−1.22789	−0.613945	+	0.789349i	$0.710418\pi$
$380$	0	0
$381$	394488.	0.139226
$382$	−64128.0	−0.0224848
$383$	−2.26198e6	−0.787936	−0.393968	−	0.919124i	$-0.628898\pi$
$383$	−2.26198e6	−0.787936	−0.393968	+	0.919124i	$0.628898\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−548536.	−0.187386
$387$	−1.22440e6	−0.415570
$388$	706528.	0.238259
$389$	−1.77009e6	−0.593091	−0.296546	−	0.955019i	$-0.595835\pi$
$389$	−1.77009e6	−0.593091	−0.296546	+	0.955019i	$0.595835\pi$
$390$	0	0
$391$	3.87763e6	1.28270
$392$	1.01011e6	0.332012
$393$	−785268.	−0.256470
$394$	3.57545e6	1.16035
$395$	0	0
$396$	15552.0	0.00498366
$397$	−5.95098e6	−1.89501	−0.947507	−	0.319735i	$-0.896406\pi$
$397$	−5.95098e6	−1.89501	−0.947507	+	0.319735i	$0.896406\pi$
$398$	−756800.	−0.239482
$399$	−305280.	−0.0959989
$400$	0	0
$401$	−2.01992e6	−0.627296	−0.313648	−	0.949539i	$-0.601551\pi$
$401$	−2.01992e6	−0.627296	−0.313648	+	0.949539i	$0.601551\pi$
$402$	−1.19131e6	−0.367672
$403$	799568.	0.245241
$404$	1.41859e6	0.432418
$405$	0	0
$406$	−1.00992e6	−0.304069
$407$	−196584.	−0.0588250
$408$	528768.	0.157259
$409$	1.66553e6	0.492316	0.246158	−	0.969230i	$-0.420832\pi$
$409$	1.66553e6	0.492316	0.246158	+	0.969230i	$0.420832\pi$
$410$	0	0
$411$	1.06202e6	0.310118
$412$	−2.39514e6	−0.695164
$413$	458880.	0.132380
$414$	−1.36858e6	−0.392435
$415$	0	0
$416$	−157696.	−0.0446773
$417$	−2.34810e6	−0.661266
$418$	50880.0	0.0142432
$419$	−4.99710e6	−1.39054	−0.695269	−	0.718749i	$-0.744715\pi$
$419$	−4.99710e6	−1.39054	−0.695269	+	0.718749i	$0.744715\pi$
$420$	0	0
$421$	653702.	0.179752	0.0898762	−	0.995953i	$-0.471353\pi$
$421$	653702.	0.179752	0.0898762	+	0.995953i	$0.471353\pi$
$422$	−3.76261e6	−1.02851
$423$	−1.91095e6	−0.519277
$424$	−1.02874e6	−0.277900
$425$	0	0
$426$	1.86883e6	0.498938
$427$	1.53402e6	0.407155
$428$	3.03725e6	0.801440
$429$	−16632.0	−0.00436316
$430$	0	0
$431$	−4.05605e6	−1.05174	−0.525872	−	0.850564i	$-0.676261\pi$
$431$	−4.05605e6	−1.05174	−0.525872	+	0.850564i	$0.676261\pi$
$432$	−186624.	−0.0481125
$433$	493054.	0.126379	0.0631895	−	0.998002i	$-0.479873\pi$
$433$	493054.	0.126379	0.0631895	+	0.998002i	$0.479873\pi$
$434$	664576.	0.169364
$435$	0	0
$436$	289760.	0.0729999
$437$	−4.47744e6	−1.12157
$438$	−432936.	−0.107830
$439$	−3.71632e6	−0.920347	−0.460174	−	0.887829i	$-0.652213\pi$
$439$	−3.71632e6	−0.920347	−0.460174	+	0.887829i	$0.652213\pi$
$440$	0	0
$441$	−1.27842e6	−0.313024
$442$	−565488.	−0.137679
$443$	−710076.	−0.171908	−0.0859539	−	0.996299i	$-0.527394\pi$
$443$	−710076.	−0.171908	−0.0859539	+	0.996299i	$0.527394\pi$
$444$	2.35901e6	0.567900
$445$	0	0
$446$	−680416.	−0.161971
$447$	−850230.	−0.201265
$448$	−131072.	−0.0308542
$449$	−6.14403e6	−1.43826	−0.719130	−	0.694875i	$-0.755459\pi$
$449$	−6.14403e6	−1.43826	−0.719130	+	0.694875i	$0.755459\pi$
$450$	0	0
$451$	43704.0	0.0101177
$452$	−37536.0	−0.00864175
$453$	2.76019e6	0.631966
$454$	79248.0	0.0180447
$455$	0	0
$456$	−610560.	−0.137504
$457$	−6.11220e6	−1.36901	−0.684506	−	0.729007i	$-0.739982\pi$
$457$	−6.11220e6	−1.36901	−0.684506	+	0.729007i	$0.739982\pi$
$458$	−6.13148e6	−1.36585
$459$	−669222.	−0.148265
$460$	0	0
$461$	−3.42170e6	−0.749876	−0.374938	−	0.927050i	$-0.622336\pi$
$461$	−3.42170e6	−0.749876	−0.374938	+	0.927050i	$0.622336\pi$
$462$	−13824.0	−0.00301321
$463$	7.51650e6	1.62953	0.814767	−	0.579789i	$-0.196865\pi$
$463$	7.51650e6	1.62953	0.814767	+	0.579789i	$0.196865\pi$
$464$	−2.01984e6	−0.435534
$465$	0	0
$466$	−3.56998e6	−0.761554
$467$	6.13523e6	1.30178	0.650891	−	0.759171i	$-0.274395\pi$
$467$	6.13523e6	1.30178	0.650891	+	0.759171i	$0.274395\pi$
$468$	199584.	0.0421222
$469$	1.05894e6	0.222301
$470$	0	0
$471$	3.90476e6	0.811039
$472$	917760.	0.189616
$473$	−181392.	−0.0372791
$474$	905760.	0.185169
$475$	0	0
$476$	−470016.	−0.0950813
$477$	1.30199e6	0.262007
$478$	2.61696e6	0.523875
$479$	5.24784e6	1.04506	0.522531	−	0.852620i	$-0.324988\pi$
$479$	5.24784e6	1.04506	0.522531	+	0.852620i	$0.324988\pi$
$480$	0	0
$481$	−2.52283e6	−0.497193
$482$	3.75071e6	0.735353
$483$	1.21651e6	0.237273
$484$	−2.57451e6	−0.499553
$485$	0	0
$486$	236196.	0.0453609
$487$	558448.	0.106699	0.0533495	−	0.998576i	$-0.483010\pi$
$487$	558448.	0.106699	0.0533495	+	0.998576i	$0.483010\pi$
$488$	3.06803e6	0.583190
$489$	−1.82848e6	−0.345794
$490$	0	0
$491$	3.56881e6	0.668067	0.334033	−	0.942561i	$-0.391590\pi$
$491$	3.56881e6	0.668067	0.334033	+	0.942561i	$0.391590\pi$
$492$	−524448.	−0.0976764
$493$	−7.24302e6	−1.34215
$494$	652960.	0.120384
$495$	0	0
$496$	1.32915e6	0.242589
$497$	−1.66118e6	−0.301666
$498$	−1.28866e6	−0.232843
$499$	3.50270e6	0.629726	0.314863	−	0.949137i	$-0.398041\pi$
$499$	3.50270e6	0.629726	0.314863	+	0.949137i	$0.398041\pi$
$500$	0	0
$501$	−868752.	−0.154633
$502$	2.93371e6	0.519587
$503$	−813456.	−0.143355	−0.0716777	−	0.997428i	$-0.522835\pi$
$503$	−813456.	−0.143355	−0.0716777	+	0.997428i	$0.522835\pi$
$504$	165888.	0.0290897
$505$	0	0
$506$	−202752.	−0.0352037
$507$	3.12819e6	0.540473
$508$	−701312.	−0.120574
$509$	8.44107e6	1.44412	0.722060	−	0.691831i	$-0.243196\pi$
$509$	8.44107e6	1.44412	0.722060	+	0.691831i	$0.243196\pi$
$510$	0	0
$511$	384832.	0.0651957
$512$	−262144.	−0.0441942
$513$	772740.	0.129640
$514$	556008.	0.0928268
$515$	0	0
$516$	2.17670e6	0.359894
$517$	−283104.	−0.0465822
$518$	−2.09690e6	−0.343362
$519$	2.29937e6	0.374706
$520$	0	0
$521$	5.24500e6	0.846548	0.423274	−	0.906002i	$-0.360881\pi$
$521$	5.24500e6	0.846548	0.423274	+	0.906002i	$0.360881\pi$
$522$	2.55636e6	0.410625
$523$	−3.32016e6	−0.530767	−0.265384	−	0.964143i	$-0.585499\pi$
$523$	−3.32016e6	−0.530767	−0.265384	+	0.964143i	$0.585499\pi$
$524$	1.39603e6	0.222110
$525$	0	0
$526$	−6.61978e6	−1.04323
$527$	4.76626e6	0.747568
$528$	−27648.0	−0.00431597
$529$	1.14058e7	1.77210
$530$	0	0
$531$	−1.16154e6	−0.178771
$532$	542720.	0.0831375
$533$	560868.	0.0855151
$534$	−2.71836e6	−0.412528
$535$	0	0
$536$	2.11789e6	0.318413
$537$	5.75910e6	0.861825
$538$	−2.25036e6	−0.335194
$539$	−189396.	−0.0280801
$540$	0	0
$541$	934382.	0.137256	0.0686280	−	0.997642i	$-0.478138\pi$
$541$	934382.	0.137256	0.0686280	+	0.997642i	$0.478138\pi$
$542$	−5.99293e6	−0.876276
$543$	4.95556e6	0.721262
$544$	−940032.	−0.136190
$545$	0	0
$546$	−177408.	−0.0254678
$547$	−363812.	−0.0519887	−0.0259943	−	0.999662i	$-0.508275\pi$
$547$	−363812.	−0.0519887	−0.0259943	+	0.999662i	$0.508275\pi$
$548$	−1.88803e6	−0.268570
$549$	−3.88298e6	−0.549837
$550$	0	0
$551$	8.36340e6	1.17356
$552$	2.43302e6	0.339859
$553$	−805120.	−0.111956
$554$	−8.41127e6	−1.16436
$555$	0	0
$556$	4.17440e6	0.572673
$557$	1.17857e7	1.60960	0.804799	−	0.593548i	$-0.202273\pi$
$557$	1.17857e7	1.60960	0.804799	+	0.593548i	$0.202273\pi$
$558$	−1.68221e6	−0.228715
$559$	−2.32786e6	−0.315085
$560$	0	0
$561$	−99144.0	−0.0133002
$562$	7.54159e6	1.00722
$563$	−5.16340e6	−0.686538	−0.343269	−	0.939237i	$-0.611534\pi$
$563$	−5.16340e6	−0.686538	−0.343269	+	0.939237i	$0.611534\pi$
$564$	3.39725e6	0.449707
$565$	0	0
$566$	8.44030e6	1.10743
$567$	−209952.	−0.0274260
$568$	−3.32237e6	−0.432093
$569$	1.32586e7	1.71680	0.858398	−	0.512984i	$-0.171460\pi$
$569$	1.32586e7	1.71680	0.858398	+	0.512984i	$0.171460\pi$
$570$	0	0
$571$	−1.06179e7	−1.36285	−0.681423	−	0.731889i	$-0.738639\pi$
$571$	−1.06179e7	−1.36285	−0.681423	+	0.731889i	$0.738639\pi$
$572$	29568.0	0.00377861
$573$	−144288.	−0.0183588
$574$	466176.	0.0590568
$575$	0	0
$576$	331776.	0.0416667
$577$	−8.23680e6	−1.02996	−0.514978	−	0.857203i	$-0.672200\pi$
$577$	−8.23680e6	−1.02996	−0.514978	+	0.857203i	$0.672200\pi$
$578$	2.30853e6	0.287420
$579$	−1.23421e6	−0.153000
$580$	0	0
$581$	1.14547e6	0.140781
$582$	1.58969e6	0.194538
$583$	192888.	0.0235036
$584$	769664.	0.0933833
$585$	0	0
$586$	1.38242e6	0.166302
$587$	1.23242e7	1.47627	0.738133	−	0.674655i	$-0.235708\pi$
$587$	1.23242e7	1.47627	0.738133	+	0.674655i	$0.235708\pi$
$588$	2.27275e6	0.271087
$589$	−5.50352e6	−0.653661
$590$	0	0
$591$	8.04476e6	0.947424
$592$	−4.19379e6	−0.491816
$593$	9.54869e6	1.11508	0.557542	−	0.830149i	$-0.311745\pi$
$593$	9.54869e6	1.11508	0.557542	+	0.830149i	$0.311745\pi$
$594$	34992.0	0.00406914
$595$	0	0
$596$	1.51152e6	0.174300
$597$	−1.70280e6	−0.195536
$598$	−2.60198e6	−0.297544
$599$	5.69580e6	0.648616	0.324308	−	0.945952i	$-0.394869\pi$
$599$	5.69580e6	0.648616	0.324308	+	0.945952i	$0.394869\pi$
$600$	0	0
$601$	−1.01490e7	−1.14614	−0.573069	−	0.819507i	$-0.694247\pi$
$601$	−1.01490e7	−1.14614	−0.573069	+	0.819507i	$0.694247\pi$
$602$	−1.93485e6	−0.217598
$603$	−2.68045e6	−0.300203
$604$	−4.90701e6	−0.547299
$605$	0	0
$606$	3.19183e6	0.353068
$607$	1.11865e7	1.23232	0.616158	−	0.787623i	$-0.288688\pi$
$607$	1.11865e7	1.23232	0.616158	+	0.787623i	$0.288688\pi$
$608$	1.08544e6	0.119082
$609$	−2.27232e6	−0.248271
$610$	0	0
$611$	−3.63317e6	−0.393715
$612$	1.18973e6	0.128401
$613$	6.22227e6	0.668803	0.334401	−	0.942431i	$-0.391466\pi$
$613$	6.22227e6	0.668803	0.334401	+	0.942431i	$0.391466\pi$
$614$	−8.88011e6	−0.950599
$615$	0	0
$616$	24576.0	0.00260951
$617$	1.93352e6	0.204473	0.102236	−	0.994760i	$-0.467400\pi$
$617$	1.93352e6	0.204473	0.102236	+	0.994760i	$0.467400\pi$
$618$	−5.38906e6	−0.567599
$619$	1.30764e7	1.37171	0.685855	−	0.727738i	$-0.259428\pi$
$619$	1.30764e7	1.37171	0.685855	+	0.727738i	$0.259428\pi$
$620$	0	0
$621$	−3.07930e6	−0.320422
$622$	−1.21174e7	−1.25584
$623$	2.41632e6	0.249422
$624$	−354816.	−0.0364789
$625$	0	0
$626$	−3.99638e6	−0.407597
$627$	114480.	0.0116295
$628$	−6.94179e6	−0.702381
$629$	−1.50387e7	−1.51559
$630$	0	0
$631$	−1.59016e7	−1.58990	−0.794948	−	0.606678i	$-0.792502\pi$
$631$	−1.59016e7	−1.58990	−0.794948	+	0.606678i	$0.792502\pi$
$632$	−1.61024e6	−0.160361
$633$	−8.46587e6	−0.839774
$634$	1.12367e7	1.11024
$635$	0	0
$636$	−2.31466e6	−0.226905
$637$	−2.43058e6	−0.237335
$638$	378720.	0.0368355
$639$	4.20487e6	0.407381
$640$	0	0
$641$	5.94424e6	0.571415	0.285707	−	0.958317i	$-0.407771\pi$
$641$	5.94424e6	0.571415	0.285707	+	0.958317i	$0.407771\pi$
$642$	6.83381e6	0.654373
$643$	2.33916e6	0.223117	0.111559	−	0.993758i	$-0.464416\pi$
$643$	2.33916e6	0.223117	0.111559	+	0.993758i	$0.464416\pi$
$644$	−2.16269e6	−0.205485
$645$	0	0
$646$	3.89232e6	0.366967
$647$	−8.26603e6	−0.776312	−0.388156	−	0.921594i	$-0.626888\pi$
$647$	−8.26603e6	−0.776312	−0.388156	+	0.921594i	$0.626888\pi$
$648$	−419904.	−0.0392837
$649$	−172080.	−0.0160368
$650$	0	0
$651$	1.49530e6	0.138285
$652$	3.25062e6	0.299466
$653$	1.03986e7	0.954315	0.477157	−	0.878818i	$-0.341667\pi$
$653$	1.03986e7	0.954315	0.477157	+	0.878818i	$0.341667\pi$
$654$	651960.	0.0596042
$655$	0	0
$656$	932352.	0.0845902
$657$	−974106.	−0.0880426
$658$	−3.01978e6	−0.271900
$659$	1.12096e7	1.00549	0.502745	−	0.864435i	$-0.332324\pi$
$659$	1.12096e7	1.00549	0.502745	+	0.864435i	$0.332324\pi$
$660$	0	0
$661$	−1.04777e7	−0.932747	−0.466374	−	0.884588i	$-0.654440\pi$
$661$	−1.04777e7	−0.932747	−0.466374	+	0.884588i	$0.654440\pi$
$662$	−9.64597e6	−0.855462
$663$	−1.27235e6	−0.112414
$664$	2.29094e6	0.201648
$665$	0	0
$666$	5.30777e6	0.463689
$667$	−3.33274e7	−2.90059
$668$	1.54445e6	0.133916
$669$	−1.53094e6	−0.132249
$670$	0	0
$671$	−575256.	−0.0493236
$672$	−294912.	−0.0251924
$673$	−7.21359e6	−0.613923	−0.306961	−	0.951722i	$-0.599312\pi$
$673$	−7.21359e6	−0.613923	−0.306961	+	0.951722i	$0.599312\pi$
$674$	−1.36762e7	−1.15962
$675$	0	0
$676$	−5.56123e6	−0.468063
$677$	2.28302e6	0.191442	0.0957211	−	0.995408i	$-0.469484\pi$
$677$	2.28302e6	0.191442	0.0957211	+	0.995408i	$0.469484\pi$
$678$	−84456.0	−0.00705596
$679$	−1.41306e6	−0.117621
$680$	0	0
$681$	178308.	0.0147334
$682$	−249216.	−0.0205171
$683$	−1.18048e7	−0.968293	−0.484146	−	0.874987i	$-0.660870\pi$
$683$	−1.18048e7	−0.968293	−0.484146	+	0.874987i	$0.660870\pi$
$684$	−1.37376e6	−0.112272
$685$	0	0
$686$	−4.17152e6	−0.338442
$687$	−1.37958e7	−1.11521
$688$	−3.86970e6	−0.311678
$689$	2.47540e6	0.198654
$690$	0	0
$691$	1.27511e7	1.01590	0.507950	−	0.861387i	$-0.330403\pi$
$691$	1.27511e7	1.01590	0.507950	+	0.861387i	$0.330403\pi$
$692$	−4.08778e6	−0.324505
$693$	−31104.0	−0.00246027
$694$	5.68709e6	0.448220
$695$	0	0
$696$	−4.54464e6	−0.355612
$697$	3.34336e6	0.260676
$698$	8.07604e6	0.627422
$699$	−8.03245e6	−0.621806
$700$	0	0
$701$	5.61742e6	0.431760	0.215880	−	0.976420i	$-0.430738\pi$
$701$	5.61742e6	0.431760	0.215880	+	0.976420i	$0.430738\pi$
$702$	449064.	0.0343926
$703$	1.73649e7	1.32521
$704$	49152.0	0.00373774
$705$	0	0
$706$	−297816.	−0.0224872
$707$	−2.83718e6	−0.213471
$708$	2.06496e6	0.154821
$709$	−4.42201e6	−0.330373	−0.165186	−	0.986262i	$-0.552823\pi$
$709$	−4.42201e6	−0.330373	−0.165186	+	0.986262i	$0.552823\pi$
$710$	0	0
$711$	2.03796e6	0.151190
$712$	4.83264e6	0.357260
$713$	2.19310e7	1.61560
$714$	−1.05754e6	−0.0776336
$715$	0	0
$716$	−1.02384e7	−0.746362
$717$	5.88816e6	0.427742
$718$	6.91632e6	0.500684
$719$	−4.33872e6	−0.312996	−0.156498	−	0.987678i	$-0.550021\pi$
$719$	−4.33872e6	−0.312996	−0.156498	+	0.987678i	$0.550021\pi$
$720$	0	0
$721$	4.79027e6	0.343180
$722$	5.41000e6	0.386237
$723$	8.43910e6	0.600414
$724$	−8.80989e6	−0.624632
$725$	0	0
$726$	−5.79265e6	−0.407883
$727$	−9.60195e6	−0.673788	−0.336894	−	0.941543i	$-0.609376\pi$
$727$	−9.60195e6	−0.673788	−0.336894	+	0.941543i	$0.609376\pi$
$728$	315392.	0.0220558
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.38765e7	−0.960475
$732$	6.90307e6	0.476173
$733$	3.69519e6	0.254025	0.127013	−	0.991901i	$-0.459461\pi$
$733$	3.69519e6	0.254025	0.127013	+	0.991901i	$0.459461\pi$
$734$	−1.00871e7	−0.691075
$735$	0	0
$736$	−4.32538e6	−0.294326
$737$	−397104.	−0.0269299
$738$	−1.18001e6	−0.0797525
$739$	−2.08019e7	−1.40117	−0.700585	−	0.713569i	$-0.747078\pi$
$739$	−2.08019e7	−1.40117	−0.700585	+	0.713569i	$0.747078\pi$
$740$	0	0
$741$	1.46916e6	0.0982932
$742$	2.05747e6	0.137191
$743$	−2.18283e7	−1.45060	−0.725302	−	0.688431i	$-0.758300\pi$
$743$	−2.18283e7	−1.45060	−0.725302	+	0.688431i	$0.758300\pi$
$744$	2.99059e6	0.198073
$745$	0	0
$746$	1.06367e7	0.699775
$747$	−2.89948e6	−0.190116
$748$	176256.	0.0115183
$749$	−6.07450e6	−0.395645
$750$	0	0
$751$	1.99887e6	0.129326	0.0646629	−	0.997907i	$-0.479403\pi$
$751$	1.99887e6	0.129326	0.0646629	+	0.997907i	$0.479403\pi$
$752$	−6.03955e6	−0.389458
$753$	6.60085e6	0.424241
$754$	4.86024e6	0.311336
$755$	0	0
$756$	373248.	0.0237516
$757$	1.17844e7	0.747426	0.373713	−	0.927544i	$-0.378084\pi$
$757$	1.17844e7	0.747426	0.373713	+	0.927544i	$0.378084\pi$
$758$	1.37346e7	0.868249
$759$	−456192.	−0.0287437
$760$	0	0
$761$	2.49819e7	1.56374	0.781868	−	0.623444i	$-0.214267\pi$
$761$	2.49819e7	1.56374	0.781868	+	0.623444i	$0.214267\pi$
$762$	−1.57795e6	−0.0984479
$763$	−579520.	−0.0360377
$764$	256512.	0.0158992
$765$	0	0
$766$	9.04790e6	0.557155
$767$	−2.20836e6	−0.135544
$768$	−589824.	−0.0360844
$769$	−4.46563e6	−0.272312	−0.136156	−	0.990687i	$-0.543475\pi$
$769$	−4.46563e6	−0.272312	−0.136156	+	0.990687i	$0.543475\pi$
$770$	0	0
$771$	1.25102e6	0.0757927
$772$	2.19414e6	0.132502
$773$	−2.15875e7	−1.29943	−0.649717	−	0.760176i	$-0.725113\pi$
$773$	−2.15875e7	−1.29943	−0.649717	+	0.760176i	$0.725113\pi$
$774$	4.89758e6	0.293853
$775$	0	0
$776$	−2.82611e6	−0.168475
$777$	−4.71802e6	−0.280354
$778$	7.08036e6	0.419379
$779$	−3.86052e6	−0.227930
$780$	0	0
$781$	622944.	0.0365444
$782$	−1.55105e7	−0.907005
$783$	5.75181e6	0.335274
$784$	−4.04045e6	−0.234768
$785$	0	0
$786$	3.14107e6	0.181352
$787$	2.79075e6	0.160614	0.0803071	−	0.996770i	$-0.474410\pi$
$787$	2.79075e6	0.160614	0.0803071	+	0.996770i	$0.474410\pi$
$788$	−1.43018e7	−0.820493
$789$	−1.48945e7	−0.851792
$790$	0	0
$791$	75072.0	0.00426616
$792$	−62208.0	−0.00352398
$793$	−7.38245e6	−0.416886
$794$	2.38039e7	1.33998
$795$	0	0
$796$	3.02720e6	0.169340
$797$	−38382.0	−0.00214034	−0.00107017	−	0.999999i	$-0.500341\pi$
$797$	−38382.0	−0.00214034	−0.00107017	+	0.999999i	$0.500341\pi$
$798$	1.22112e6	0.0678815
$799$	−2.16575e7	−1.20016
$800$	0	0
$801$	−6.11631e6	−0.336828
$802$	8.07967e6	0.443566
$803$	−144312.	−0.00789794
$804$	4.76525e6	0.259983
$805$	0	0
$806$	−3.19827e6	−0.173411
$807$	−5.06331e6	−0.273685
$808$	−5.67437e6	−0.305766
$809$	2.05618e7	1.10456	0.552282	−	0.833657i	$-0.313757\pi$
$809$	2.05618e7	1.10456	0.552282	+	0.833657i	$0.313757\pi$
$810$	0	0
$811$	2.27222e7	1.21311	0.606553	−	0.795043i	$-0.292552\pi$
$811$	2.27222e7	1.21311	0.606553	+	0.795043i	$0.292552\pi$
$812$	4.03968e6	0.215009
$813$	−1.34841e7	−0.715476
$814$	786336.	0.0415956
$815$	0	0
$816$	−2.11507e6	−0.111199
$817$	1.60230e7	0.839823
$818$	−6.66212e6	−0.348120
$819$	−399168.	−0.0207944
$820$	0	0
$821$	3.82618e6	0.198111	0.0990553	−	0.995082i	$-0.468418\pi$
$821$	3.82618e6	0.198111	0.0990553	+	0.995082i	$0.468418\pi$
$822$	−4.24807e6	−0.219287
$823$	634384.	0.0326477	0.0163239	−	0.999867i	$-0.494804\pi$
$823$	634384.	0.0326477	0.0163239	+	0.999867i	$0.494804\pi$
$824$	9.58054e6	0.491555
$825$	0	0
$826$	−1.83552e6	−0.0936071
$827$	742788.	0.0377660	0.0188830	−	0.999822i	$-0.493989\pi$
$827$	742788.	0.0377660	0.0188830	+	0.999822i	$0.493989\pi$
$828$	5.47430e6	0.277494
$829$	−2.85450e7	−1.44259	−0.721297	−	0.692626i	$-0.756454\pi$
$829$	−2.85450e7	−1.44259	−0.721297	+	0.692626i	$0.756454\pi$
$830$	0	0
$831$	−1.89254e7	−0.950696
$832$	630784.	0.0315917
$833$	−1.44888e7	−0.723469
$834$	9.39240e6	0.467586
$835$	0	0
$836$	−203520.	−0.0100714
$837$	−3.78497e6	−0.186745
$838$	1.99884e7	0.983259
$839$	4.16244e6	0.204147	0.102074	−	0.994777i	$-0.467452\pi$
$839$	4.16244e6	0.204147	0.102074	+	0.994777i	$0.467452\pi$
$840$	0	0
$841$	4.17410e7	2.03504
$842$	−2.61481e6	−0.127104
$843$	1.69686e7	0.822388
$844$	1.50504e7	0.727265
$845$	0	0
$846$	7.64381e6	0.367184
$847$	5.14902e6	0.246613
$848$	4.11494e6	0.196505
$849$	1.89907e7	0.904214
$850$	0	0
$851$	−6.91976e7	−3.27542
$852$	−7.47533e6	−0.352802
$853$	−1.66504e7	−0.783526	−0.391763	−	0.920066i	$-0.628135\pi$
$853$	−1.66504e7	−0.783526	−0.391763	+	0.920066i	$0.628135\pi$
$854$	−6.13606e6	−0.287902
$855$	0	0
$856$	−1.21490e7	−0.566703
$857$	2.77446e7	1.29040	0.645202	−	0.764012i	$-0.276773\pi$
$857$	2.77446e7	1.29040	0.645202	+	0.764012i	$0.276773\pi$
$858$	66528.0	0.00308522
$859$	4.18553e7	1.93539	0.967693	−	0.252132i	$-0.0811317\pi$
$859$	4.18553e7	1.93539	0.967693	+	0.252132i	$0.0811317\pi$
$860$	0	0
$861$	1.04890e6	0.0482197
$862$	1.62242e7	0.743695
$863$	−2.69606e7	−1.23226	−0.616131	−	0.787644i	$-0.711301\pi$
$863$	−2.69606e7	−1.23226	−0.616131	+	0.787644i	$0.711301\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	−1.97222e6	−0.0893634
$867$	5.19420e6	0.234677
$868$	−2.65830e6	−0.119758
$869$	301920.	0.0135626
$870$	0	0
$871$	−5.09617e6	−0.227614
$872$	−1.15904e6	−0.0516187
$873$	3.57680e6	0.158840
$874$	1.79098e7	0.793069
$875$	0	0
$876$	1.73174e6	0.0762471
$877$	1.95641e7	0.858938	0.429469	−	0.903082i	$-0.358701\pi$
$877$	1.95641e7	0.858938	0.429469	+	0.903082i	$0.358701\pi$
$878$	1.48653e7	0.650784
$879$	3.11045e6	0.135785
$880$	0	0
$881$	−696798.	−0.0302459	−0.0151230	−	0.999886i	$-0.504814\pi$
$881$	−696798.	−0.0302459	−0.0151230	+	0.999886i	$0.504814\pi$
$882$	5.11369e6	0.221342
$883$	−1.88473e7	−0.813482	−0.406741	−	0.913544i	$-0.633335\pi$
$883$	−1.88473e7	−0.813482	−0.406741	+	0.913544i	$0.633335\pi$
$884$	2.26195e6	0.0973538
$885$	0	0
$886$	2.84030e6	0.121557
$887$	3.08093e6	0.131484	0.0657419	−	0.997837i	$-0.479059\pi$
$887$	3.08093e6	0.131484	0.0657419	+	0.997837i	$0.479059\pi$
$888$	−9.43603e6	−0.401566
$889$	1.40262e6	0.0595233
$890$	0	0
$891$	78732.0	0.00332244
$892$	2.72166e6	0.114531
$893$	2.50075e7	1.04940
$894$	3.40092e6	0.142316
$895$	0	0
$896$	524288.	0.0218172
$897$	−5.85446e6	−0.242944
$898$	2.45761e7	1.01700
$899$	−4.09649e7	−1.69049
$900$	0	0
$901$	1.47559e7	0.605556
$902$	−174816.	−0.00715426
$903$	−4.35341e6	−0.177668
$904$	150144.	0.00611064
$905$	0	0
$906$	−1.10408e7	−0.446868
$907$	−7.45257e6	−0.300807	−0.150404	−	0.988625i	$-0.548057\pi$
$907$	−7.45257e6	−0.300807	−0.150404	+	0.988625i	$0.548057\pi$
$908$	−316992.	−0.0127595
$909$	7.18162e6	0.288279
$910$	0	0
$911$	−4.04523e7	−1.61491	−0.807453	−	0.589932i	$-0.799154\pi$
$911$	−4.04523e7	−1.61491	−0.807453	+	0.589932i	$0.799154\pi$
$912$	2.44224e6	0.0972302
$913$	−429552.	−0.0170545
$914$	2.44488e7	0.968038
$915$	0	0
$916$	2.45259e7	0.965799
$917$	−2.79206e6	−0.109648
$918$	2.67689e6	0.104839
$919$	2.81446e7	1.09928	0.549638	−	0.835403i	$-0.314766\pi$
$919$	2.81446e7	1.09928	0.549638	+	0.835403i	$0.314766\pi$
$920$	0	0
$921$	−1.99803e7	−0.776161
$922$	1.36868e7	0.530242
$923$	7.99445e6	0.308876
$924$	55296.0	0.00213066
$925$	0	0
$926$	−3.00660e7	−1.15225
$927$	−1.21254e7	−0.463443
$928$	8.07936e6	0.307969
$929$	4.62537e7	1.75836	0.879179	−	0.476491i	$-0.158092\pi$
$929$	4.62537e7	1.75836	0.879179	+	0.476491i	$0.158092\pi$
$930$	0	0
$931$	1.67300e7	0.632588
$932$	1.42799e7	0.538500
$933$	−2.72642e7	−1.02539
$934$	−2.45409e7	−0.920499
$935$	0	0
$936$	−798336.	−0.0297849
$937$	1.83784e7	0.683845	0.341923	−	0.939728i	$-0.388922\pi$
$937$	1.83784e7	0.683845	0.341923	+	0.939728i	$0.388922\pi$
$938$	−4.23578e6	−0.157190
$939$	−8.99185e6	−0.332801
$940$	0	0
$941$	4.54826e7	1.67445	0.837224	−	0.546861i	$-0.184177\pi$
$941$	4.54826e7	1.67445	0.837224	+	0.546861i	$0.184177\pi$
$942$	−1.56190e7	−0.573491
$943$	1.53838e7	0.563358
$944$	−3.67104e6	−0.134078
$945$	0	0
$946$	725568.	0.0263603
$947$	1.36688e7	0.495286	0.247643	−	0.968851i	$-0.420344\pi$
$947$	1.36688e7	0.495286	0.247643	+	0.968851i	$0.420344\pi$
$948$	−3.62304e6	−0.130934
$949$	−1.85200e6	−0.0667539
$950$	0	0
$951$	2.52826e7	0.906507
$952$	1.88006e6	0.0672327
$953$	−1.45580e7	−0.519241	−0.259621	−	0.965711i	$-0.583597\pi$
$953$	−1.45580e7	−0.519241	−0.259621	+	0.965711i	$0.583597\pi$
$954$	−5.20798e6	−0.185267
$955$	0	0
$956$	−1.04678e7	−0.370435
$957$	852120.	0.0300761
$958$	−2.09914e7	−0.738970
$959$	3.77606e6	0.132585
$960$	0	0
$961$	−1.67229e6	−0.0584120
$962$	1.00913e7	0.351568
$963$	1.53761e7	0.534293
$964$	−1.50028e7	−0.519973
$965$	0	0
$966$	−4.86605e6	−0.167777
$967$	−1.78371e7	−0.613419	−0.306710	−	0.951803i	$-0.599228\pi$
$967$	−1.78371e7	−0.613419	−0.306710	+	0.951803i	$0.599228\pi$
$968$	1.02980e7	0.353237
$969$	8.75772e6	0.299627
$970$	0	0
$971$	2.52377e7	0.859017	0.429508	−	0.903063i	$-0.358687\pi$
$971$	2.52377e7	0.859017	0.429508	+	0.903063i	$0.358687\pi$
$972$	−944784.	−0.0320750
$973$	−8.34880e6	−0.282711
$974$	−2.23379e6	−0.0754476
$975$	0	0
$976$	−1.22721e7	−0.412378
$977$	−3.49920e7	−1.17282	−0.586412	−	0.810013i	$-0.699460\pi$
$977$	−3.49920e7	−1.17282	−0.586412	+	0.810013i	$0.699460\pi$
$978$	7.31390e6	0.244513
$979$	−906120.	−0.0302154
$980$	0	0
$981$	1.46691e6	0.0486666
$982$	−1.42752e7	−0.472395
$983$	−7.67338e6	−0.253281	−0.126641	−	0.991949i	$-0.540419\pi$
$983$	−7.67338e6	−0.253281	−0.126641	+	0.991949i	$0.540419\pi$
$984$	2.09779e6	0.0690676
$985$	0	0
$986$	2.89721e7	0.949046
$987$	−6.79450e6	−0.222006
$988$	−2.61184e6	−0.0851244
$989$	−6.38500e7	−2.07573
$990$	0	0
$991$	5.49135e6	0.177621	0.0888107	−	0.996049i	$-0.471693\pi$
$991$	5.49135e6	0.177621	0.0888107	+	0.996049i	$0.471693\pi$
$992$	−5.31661e6	−0.171536
$993$	−2.17034e7	−0.698482
$994$	6.64474e6	0.213310
$995$	0	0
$996$	5.15462e6	0.164645
$997$	5.51157e7	1.75605	0.878026	−	0.478613i	$-0.158860\pi$
$997$	5.51157e7	1.75605	0.878026	+	0.478613i	$0.158860\pi$
$998$	−1.40108e7	−0.445283
$999$	1.19425e7	0.378600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.a.b.1.1		1
3.2	odd	2		450.6.a.q.1.1		1
5.2	odd	4		150.6.c.f.49.1		2
5.3	odd	4		150.6.c.f.49.2		2
5.4	even	2		30.6.a.b.1.1	✓	1
15.2	even	4		450.6.c.i.199.2		2
15.8	even	4		450.6.c.i.199.1		2
15.14	odd	2		90.6.a.a.1.1		1
20.19	odd	2		240.6.a.f.1.1		1
40.19	odd	2		960.6.a.q.1.1		1
40.29	even	2		960.6.a.d.1.1		1
60.59	even	2		720.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.a.b.1.1	✓	1	5.4	even	2
90.6.a.a.1.1		1	15.14	odd	2
150.6.a.b.1.1		1	1.1	even	1	trivial
150.6.c.f.49.1		2	5.2	odd	4
150.6.c.f.49.2		2	5.3	odd	4
240.6.a.f.1.1		1	20.19	odd	2
450.6.a.q.1.1		1	3.2	odd	2
450.6.c.i.199.1		2	15.8	even	4
450.6.c.i.199.2		2	15.2	even	4
720.6.a.e.1.1		1	60.59	even	2
960.6.a.d.1.1		1	40.29	even	2
960.6.a.q.1.1		1	40.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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