LMFDB - Embedded newform 294.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,6,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, 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("294.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	294.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:	$47.1528430250$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	294.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -26.0000 q^{5} +36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -26.0000 q^{5} +36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +104.000 q^{10} +664.000 q^{11} -144.000 q^{12} -318.000 q^{13} +234.000 q^{15} +256.000 q^{16} -1582.00 q^{17} -324.000 q^{18} -236.000 q^{19} -416.000 q^{20} -2656.00 q^{22} +2212.00 q^{23} +576.000 q^{24} -2449.00 q^{25} +1272.00 q^{26} -729.000 q^{27} -4954.00 q^{29} -936.000 q^{30} +7128.00 q^{31} -1024.00 q^{32} -5976.00 q^{33} +6328.00 q^{34} +1296.00 q^{36} +4358.00 q^{37} +944.000 q^{38} +2862.00 q^{39} +1664.00 q^{40} -10542.0 q^{41} -8452.00 q^{43} +10624.0 q^{44} -2106.00 q^{45} -8848.00 q^{46} -5352.00 q^{47} -2304.00 q^{48} +9796.00 q^{50} +14238.0 q^{51} -5088.00 q^{52} -33354.0 q^{53} +2916.00 q^{54} -17264.0 q^{55} +2124.00 q^{57} +19816.0 q^{58} +15436.0 q^{59} +3744.00 q^{60} +36762.0 q^{61} -28512.0 q^{62} +4096.00 q^{64} +8268.00 q^{65} +23904.0 q^{66} +40972.0 q^{67} -25312.0 q^{68} -19908.0 q^{69} -9092.00 q^{71} -5184.00 q^{72} +73454.0 q^{73} -17432.0 q^{74} +22041.0 q^{75} -3776.00 q^{76} -11448.0 q^{78} +89400.0 q^{79} -6656.00 q^{80} +6561.00 q^{81} +42168.0 q^{82} +6428.00 q^{83} +41132.0 q^{85} +33808.0 q^{86} +44586.0 q^{87} -42496.0 q^{88} +122658. q^{89} +8424.00 q^{90} +35392.0 q^{92} -64152.0 q^{93} +21408.0 q^{94} +6136.00 q^{95} +9216.00 q^{96} -21370.0 q^{97} +53784.0 q^{99} +O(q^{100})\)

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
$5$	−26.0000	−0.465102	−0.232551	−	0.972584i	$-0.574707\pi$
$5$	−26.0000	−0.465102	−0.232551	+	0.972584i	$0.574707\pi$
$6$	36.0000	0.408248
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	104.000	0.328877
$11$	664.000	1.65457	0.827287	−	0.561779i	$-0.189883\pi$
$11$	664.000	1.65457	0.827287	+	0.561779i	$0.189883\pi$
$12$	−144.000	−0.288675
$13$	−318.000	−0.521878	−0.260939	−	0.965355i	$-0.584032\pi$
$13$	−318.000	−0.521878	−0.260939	+	0.965355i	$0.584032\pi$
$14$	0	0
$15$	234.000	0.268527
$16$	256.000	0.250000
$17$	−1582.00	−1.32765	−0.663826	−	0.747887i	$-0.731068\pi$
$17$	−1582.00	−1.32765	−0.663826	+	0.747887i	$0.731068\pi$
$18$	−324.000	−0.235702
$19$	−236.000	−0.149978	−0.0749891	−	0.997184i	$-0.523892\pi$
$19$	−236.000	−0.149978	−0.0749891	+	0.997184i	$0.523892\pi$
$20$	−416.000	−0.232551
$21$	0	0
$22$	−2656.00	−1.16996
$23$	2212.00	0.871898	0.435949	−	0.899971i	$-0.356413\pi$
$23$	2212.00	0.871898	0.435949	+	0.899971i	$0.356413\pi$
$24$	576.000	0.204124
$25$	−2449.00	−0.783680
$26$	1272.00	0.369023
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4954.00	−1.09386	−0.546929	−	0.837179i	$-0.684203\pi$
$29$	−4954.00	−1.09386	−0.546929	+	0.837179i	$0.684203\pi$
$30$	−936.000	−0.189877
$31$	7128.00	1.33218	0.666091	−	0.745871i	$-0.267966\pi$
$31$	7128.00	1.33218	0.666091	+	0.745871i	$0.267966\pi$
$32$	−1024.00	−0.176777
$33$	−5976.00	−0.955269
$34$	6328.00	0.938792
$35$	0	0
$36$	1296.00	0.166667
$37$	4358.00	0.523339	0.261669	−	0.965158i	$-0.415727\pi$
$37$	4358.00	0.523339	0.261669	+	0.965158i	$0.415727\pi$
$38$	944.000	0.106051
$39$	2862.00	0.301306
$40$	1664.00	0.164438
$41$	−10542.0	−0.979407	−0.489704	−	0.871889i	$-0.662895\pi$
$41$	−10542.0	−0.979407	−0.489704	+	0.871889i	$0.662895\pi$
$42$	0	0
$43$	−8452.00	−0.697089	−0.348545	−	0.937292i	$-0.613324\pi$
$43$	−8452.00	−0.697089	−0.348545	+	0.937292i	$0.613324\pi$
$44$	10624.0	0.827287
$45$	−2106.00	−0.155034
$46$	−8848.00	−0.616525
$47$	−5352.00	−0.353404	−0.176702	−	0.984264i	$-0.556543\pi$
$47$	−5352.00	−0.353404	−0.176702	+	0.984264i	$0.556543\pi$
$48$	−2304.00	−0.144338
$49$	0	0
$50$	9796.00	0.554145
$51$	14238.0	0.766520
$52$	−5088.00	−0.260939
$53$	−33354.0	−1.63102	−0.815508	−	0.578746i	$-0.803542\pi$
$53$	−33354.0	−1.63102	−0.815508	+	0.578746i	$0.803542\pi$
$54$	2916.00	0.136083
$55$	−17264.0	−0.769546
$56$	0	0
$57$	2124.00	0.0865899
$58$	19816.0	0.773475
$59$	15436.0	0.577304	0.288652	−	0.957434i	$-0.406793\pi$
$59$	15436.0	0.577304	0.288652	+	0.957434i	$0.406793\pi$
$60$	3744.00	0.134263
$61$	36762.0	1.26495	0.632477	−	0.774579i	$-0.282038\pi$
$61$	36762.0	1.26495	0.632477	+	0.774579i	$0.282038\pi$
$62$	−28512.0	−0.941995
$63$	0	0
$64$	4096.00	0.125000
$65$	8268.00	0.242726
$66$	23904.0	0.675477
$67$	40972.0	1.11506	0.557532	−	0.830155i	$-0.311748\pi$
$67$	40972.0	1.11506	0.557532	+	0.830155i	$0.311748\pi$
$68$	−25312.0	−0.663826
$69$	−19908.0	−0.503390
$70$	0	0
$71$	−9092.00	−0.214049	−0.107025	−	0.994256i	$-0.534132\pi$
$71$	−9092.00	−0.214049	−0.107025	+	0.994256i	$0.534132\pi$
$72$	−5184.00	−0.117851
$73$	73454.0	1.61327	0.806637	−	0.591047i	$-0.201285\pi$
$73$	73454.0	1.61327	0.806637	+	0.591047i	$0.201285\pi$
$74$	−17432.0	−0.370056
$75$	22041.0	0.452458
$76$	−3776.00	−0.0749891
$77$	0	0
$78$	−11448.0	−0.213056
$79$	89400.0	1.61165	0.805823	−	0.592156i	$-0.201723\pi$
$79$	89400.0	1.61165	0.805823	+	0.592156i	$0.201723\pi$
$80$	−6656.00	−0.116276
$81$	6561.00	0.111111
$82$	42168.0	0.692546
$83$	6428.00	0.102419	0.0512095	−	0.998688i	$-0.483692\pi$
$83$	6428.00	0.102419	0.0512095	+	0.998688i	$0.483692\pi$
$84$	0	0
$85$	41132.0	0.617494
$86$	33808.0	0.492916
$87$	44586.0	0.631539
$88$	−42496.0	−0.584980
$89$	122658.	1.64142	0.820712	−	0.571342i	$-0.193577\pi$
$89$	122658.	1.64142	0.820712	+	0.571342i	$0.193577\pi$
$90$	8424.00	0.109626
$91$	0	0
$92$	35392.0	0.435949
$93$	−64152.0	−0.769135
$94$	21408.0	0.249894
$95$	6136.00	0.0697552
$96$	9216.00	0.102062
$97$	−21370.0	−0.230608	−0.115304	−	0.993330i	$-0.536784\pi$
$97$	−21370.0	−0.230608	−0.115304	+	0.993330i	$0.536784\pi$
$98$	0	0
$99$	53784.0	0.551525
$100$	−39184.0	−0.391840
$101$	36814.0	0.359095	0.179548	−	0.983749i	$-0.442537\pi$
$101$	36814.0	0.359095	0.179548	+	0.983749i	$0.442537\pi$
$102$	−56952.0	−0.542012
$103$	−104528.	−0.970822	−0.485411	−	0.874286i	$-0.661330\pi$
$103$	−104528.	−0.970822	−0.485411	+	0.874286i	$0.661330\pi$
$104$	20352.0	0.184512
$105$	0	0
$106$	133416.	1.15330
$107$	214440.	1.81070	0.905350	−	0.424667i	$-0.139609\pi$
$107$	214440.	1.81070	0.905350	+	0.424667i	$0.139609\pi$
$108$	−11664.0	−0.0962250
$109$	28798.0	0.232165	0.116082	−	0.993240i	$-0.462966\pi$
$109$	28798.0	0.232165	0.116082	+	0.993240i	$0.462966\pi$
$110$	69056.0	0.544151
$111$	−39222.0	−0.302150
$112$	0	0
$113$	−56014.0	−0.412668	−0.206334	−	0.978482i	$-0.566153\pi$
$113$	−56014.0	−0.412668	−0.206334	+	0.978482i	$0.566153\pi$
$114$	−8496.00	−0.0612283
$115$	−57512.0	−0.405521
$116$	−79264.0	−0.546929
$117$	−25758.0	−0.173959
$118$	−61744.0	−0.408216
$119$	0	0
$120$	−14976.0	−0.0949386
$121$	279845.	1.73762
$122$	−147048.	−0.894457
$123$	94878.0	0.565461
$124$	114048.	0.666091
$125$	144924.	0.829593
$126$	0	0
$127$	185400.	1.02000	0.510000	−	0.860174i	$-0.329645\pi$
$127$	185400.	1.02000	0.510000	+	0.860174i	$0.329645\pi$
$128$	−16384.0	−0.0883883
$129$	76068.0	0.402465
$130$	−33072.0	−0.171634
$131$	−64532.0	−0.328547	−0.164273	−	0.986415i	$-0.552528\pi$
$131$	−64532.0	−0.328547	−0.164273	+	0.986415i	$0.552528\pi$
$132$	−95616.0	−0.477635
$133$	0	0
$134$	−163888.	−0.788470
$135$	18954.0	0.0895089
$136$	101248.	0.469396
$137$	152930.	0.696131	0.348066	−	0.937470i	$-0.386839\pi$
$137$	152930.	0.696131	0.348066	+	0.937470i	$0.386839\pi$
$138$	79632.0	0.355951
$139$	343460.	1.50778	0.753892	−	0.656998i	$-0.228174\pi$
$139$	343460.	1.50778	0.753892	+	0.656998i	$0.228174\pi$
$140$	0	0
$141$	48168.0	0.204038
$142$	36368.0	0.151356
$143$	−211152.	−0.863486
$144$	20736.0	0.0833333
$145$	128804.	0.508756
$146$	−293816.	−1.14076
$147$	0	0
$148$	69728.0	0.261669
$149$	−174858.	−0.645238	−0.322619	−	0.946529i	$-0.604563\pi$
$149$	−174858.	−0.645238	−0.322619	+	0.946529i	$0.604563\pi$
$150$	−88164.0	−0.319936
$151$	−452552.	−1.61520	−0.807600	−	0.589731i	$-0.799234\pi$
$151$	−452552.	−1.61520	−0.807600	+	0.589731i	$0.799234\pi$
$152$	15104.0	0.0530253
$153$	−128142.	−0.442551
$154$	0	0
$155$	−185328.	−0.619601
$156$	45792.0	0.150653
$157$	499066.	1.61588	0.807940	−	0.589265i	$-0.200583\pi$
$157$	499066.	1.61588	0.807940	+	0.589265i	$0.200583\pi$
$158$	−357600.	−1.13961
$159$	300186.	0.941668
$160$	26624.0	0.0822192
$161$	0	0
$162$	−26244.0	−0.0785674
$163$	−475588.	−1.40204	−0.701022	−	0.713139i	$-0.747273\pi$
$163$	−475588.	−1.40204	−0.701022	+	0.713139i	$0.747273\pi$
$164$	−168672.	−0.489704
$165$	155376.	0.444298
$166$	−25712.0	−0.0724212
$167$	−120224.	−0.333580	−0.166790	−	0.985992i	$-0.553340\pi$
$167$	−120224.	−0.333580	−0.166790	+	0.985992i	$0.553340\pi$
$168$	0	0
$169$	−270169.	−0.727644
$170$	−164528.	−0.436634
$171$	−19116.0	−0.0499927
$172$	−135232.	−0.348545
$173$	−508874.	−1.29269	−0.646346	−	0.763045i	$-0.723704\pi$
$173$	−508874.	−1.29269	−0.646346	+	0.763045i	$0.723704\pi$
$174$	−178344.	−0.446566
$175$	0	0
$176$	169984.	0.413644
$177$	−138924.	−0.333307
$178$	−490632.	−1.16066
$179$	487560.	1.13735	0.568677	−	0.822561i	$-0.307456\pi$
$179$	487560.	1.13735	0.568677	+	0.822561i	$0.307456\pi$
$180$	−33696.0	−0.0775170
$181$	544410.	1.23518	0.617589	−	0.786501i	$-0.288109\pi$
$181$	544410.	1.23518	0.617589	+	0.786501i	$0.288109\pi$
$182$	0	0
$183$	−330858.	−0.730321
$184$	−141568.	−0.308262
$185$	−113308.	−0.243406
$186$	256608.	0.543861
$187$	−1.05045e6	−2.19670
$188$	−85632.0	−0.176702
$189$	0	0
$190$	−24544.0	−0.0493243
$191$	376404.	0.746570	0.373285	−	0.927717i	$-0.378231\pi$
$191$	376404.	0.746570	0.373285	+	0.927717i	$0.378231\pi$
$192$	−36864.0	−0.0721688
$193$	844946.	1.63281	0.816405	−	0.577480i	$-0.195964\pi$
$193$	844946.	1.63281	0.816405	+	0.577480i	$0.195964\pi$
$194$	85480.0	0.163065
$195$	−74412.0	−0.140138
$196$	0	0
$197$	−492794.	−0.904690	−0.452345	−	0.891843i	$-0.649412\pi$
$197$	−492794.	−0.904690	−0.452345	+	0.891843i	$0.649412\pi$
$198$	−215136.	−0.389987
$199$	914776.	1.63750	0.818751	−	0.574148i	$-0.194667\pi$
$199$	914776.	1.63750	0.818751	+	0.574148i	$0.194667\pi$
$200$	156736.	0.277073
$201$	−368748.	−0.643783
$202$	−147256.	−0.253919
$203$	0	0
$204$	227808.	0.383260
$205$	274092.	0.455524
$206$	418112.	0.686475
$207$	179172.	0.290633
$208$	−81408.0	−0.130469
$209$	−156704.	−0.248150
$210$	0	0
$211$	311780.	0.482106	0.241053	−	0.970512i	$-0.422507\pi$
$211$	311780.	0.482106	0.241053	+	0.970512i	$0.422507\pi$
$212$	−533664.	−0.815508
$213$	81828.0	0.123581
$214$	−857760.	−1.28036
$215$	219752.	0.324218
$216$	46656.0	0.0680414
$217$	0	0
$218$	−115192.	−0.164165
$219$	−661086.	−0.931425
$220$	−276224.	−0.384773
$221$	503076.	0.692872
$222$	156888.	0.213652
$223$	1.28776e6	1.73409	0.867047	−	0.498226i	$-0.166015\pi$
$223$	1.28776e6	1.73409	0.867047	+	0.498226i	$0.166015\pi$
$224$	0	0
$225$	−198369.	−0.261227
$226$	224056.	0.291800
$227$	−1.28905e6	−1.66037	−0.830187	−	0.557485i	$-0.811766\pi$
$227$	−1.28905e6	−1.66037	−0.830187	+	0.557485i	$0.811766\pi$
$228$	33984.0	0.0432950
$229$	−678214.	−0.854630	−0.427315	−	0.904103i	$-0.640540\pi$
$229$	−678214.	−0.854630	−0.427315	+	0.904103i	$0.640540\pi$
$230$	230048.	0.286747
$231$	0	0
$232$	317056.	0.386737
$233$	−1.11731e6	−1.34829	−0.674146	−	0.738598i	$-0.735488\pi$
$233$	−1.11731e6	−1.34829	−0.674146	+	0.738598i	$0.735488\pi$
$234$	103032.	0.123008
$235$	139152.	0.164369
$236$	246976.	0.288652
$237$	−804600.	−0.930485
$238$	0	0
$239$	−1.26196e6	−1.42906	−0.714528	−	0.699606i	$-0.753359\pi$
$239$	−1.26196e6	−1.42906	−0.714528	+	0.699606i	$0.753359\pi$
$240$	59904.0	0.0671317
$241$	−948218.	−1.05164	−0.525818	−	0.850597i	$-0.676241\pi$
$241$	−948218.	−1.05164	−0.525818	+	0.850597i	$0.676241\pi$
$242$	−1.11938e6	−1.22868
$243$	−59049.0	−0.0641500
$244$	588192.	0.632477
$245$	0	0
$246$	−379512.	−0.399841
$247$	75048.0	0.0782703
$248$	−456192.	−0.470997
$249$	−57852.0	−0.0591317
$250$	−579696.	−0.586611
$251$	486396.	0.487310	0.243655	−	0.969862i	$-0.421653\pi$
$251$	486396.	0.487310	0.243655	+	0.969862i	$0.421653\pi$
$252$	0	0
$253$	1.46877e6	1.44262
$254$	−741600.	−0.721249
$255$	−370188.	−0.356510
$256$	65536.0	0.0625000
$257$	1.03910e6	0.981349	0.490675	−	0.871343i	$-0.336750\pi$
$257$	1.03910e6	0.981349	0.490675	+	0.871343i	$0.336750\pi$
$258$	−304272.	−0.284585
$259$	0	0
$260$	132288.	0.121363
$261$	−401274.	−0.364619
$262$	258128.	0.232317
$263$	1.35104e6	1.20443	0.602213	−	0.798335i	$-0.294286\pi$
$263$	1.35104e6	1.20443	0.602213	+	0.798335i	$0.294286\pi$
$264$	382464.	0.337739
$265$	867204.	0.758589
$266$	0	0
$267$	−1.10392e6	−0.947677
$268$	655552.	0.557532
$269$	1.11811e6	0.942115	0.471057	−	0.882103i	$-0.343872\pi$
$269$	1.11811e6	0.942115	0.471057	+	0.882103i	$0.343872\pi$
$270$	−75816.0	−0.0632924
$271$	190104.	0.157242	0.0786209	−	0.996905i	$-0.474948\pi$
$271$	190104.	0.157242	0.0786209	+	0.996905i	$0.474948\pi$
$272$	−404992.	−0.331913
$273$	0	0
$274$	−611720.	−0.492239
$275$	−1.62614e6	−1.29666
$276$	−318528.	−0.251695
$277$	−200506.	−0.157010	−0.0785051	−	0.996914i	$-0.525015\pi$
$277$	−200506.	−0.157010	−0.0785051	+	0.996914i	$0.525015\pi$
$278$	−1.37384e6	−1.06616
$279$	577368.	0.444061
$280$	0	0
$281$	1.09237e6	0.825285	0.412643	−	0.910893i	$-0.364606\pi$
$281$	1.09237e6	0.825285	0.412643	+	0.910893i	$0.364606\pi$
$282$	−192672.	−0.144277
$283$	−1.81258e6	−1.34534	−0.672669	−	0.739944i	$-0.734852\pi$
$283$	−1.81258e6	−1.34534	−0.672669	+	0.739944i	$0.734852\pi$
$284$	−145472.	−0.107025
$285$	−55224.0	−0.0402732
$286$	844608.	0.610577
$287$	0	0
$288$	−82944.0	−0.0589256
$289$	1.08287e6	0.762659
$290$	−515216.	−0.359745
$291$	192330.	0.133142
$292$	1.17526e6	0.806637
$293$	−2.10031e6	−1.42927	−0.714634	−	0.699499i	$-0.753407\pi$
$293$	−2.10031e6	−1.42927	−0.714634	+	0.699499i	$0.753407\pi$
$294$	0	0
$295$	−401336.	−0.268505
$296$	−278912.	−0.185028
$297$	−484056.	−0.318423
$298$	699432.	0.456252
$299$	−703416.	−0.455024
$300$	352656.	0.226229
$301$	0	0
$302$	1.81021e6	1.14212
$303$	−331326.	−0.207324
$304$	−60416.0	−0.0374945
$305$	−955812.	−0.588333
$306$	512568.	0.312931
$307$	1.64104e6	0.993743	0.496872	−	0.867824i	$-0.334482\pi$
$307$	1.64104e6	0.993743	0.496872	+	0.867824i	$0.334482\pi$
$308$	0	0
$309$	940752.	0.560504
$310$	741312.	0.438124
$311$	945232.	0.554163	0.277081	−	0.960846i	$-0.410633\pi$
$311$	945232.	0.554163	0.277081	+	0.960846i	$0.410633\pi$
$312$	−183168.	−0.106528
$313$	−415354.	−0.239639	−0.119820	−	0.992796i	$-0.538232\pi$
$313$	−415354.	−0.239639	−0.119820	+	0.992796i	$0.538232\pi$
$314$	−1.99626e6	−1.14260
$315$	0	0
$316$	1.43040e6	0.805823
$317$	1.18481e6	0.662220	0.331110	−	0.943592i	$-0.392577\pi$
$317$	1.18481e6	0.662220	0.331110	+	0.943592i	$0.392577\pi$
$318$	−1.20074e6	−0.665860
$319$	−3.28946e6	−1.80987
$320$	−106496.	−0.0581378
$321$	−1.92996e6	−1.04541
$322$	0	0
$323$	373352.	0.199119
$324$	104976.	0.0555556
$325$	778782.	0.408985
$326$	1.90235e6	0.991395
$327$	−259182.	−0.134040
$328$	674688.	0.346273
$329$	0	0
$330$	−621504.	−0.314166
$331$	1.37155e6	0.688083	0.344042	−	0.938954i	$-0.388204\pi$
$331$	1.37155e6	0.688083	0.344042	+	0.938954i	$0.388204\pi$
$332$	102848.	0.0512095
$333$	352998.	0.174446
$334$	480896.	0.235877
$335$	−1.06527e6	−0.518619
$336$	0	0
$337$	963522.	0.462154	0.231077	−	0.972935i	$-0.425775\pi$
$337$	963522.	0.462154	0.231077	+	0.972935i	$0.425775\pi$
$338$	1.08068e6	0.514522
$339$	504126.	0.238254
$340$	658112.	0.308747
$341$	4.73299e6	2.20419
$342$	76464.0	0.0353502
$343$	0	0
$344$	540928.	0.246458
$345$	517608.	0.234128
$346$	2.03550e6	0.914071
$347$	2.57731e6	1.14906	0.574531	−	0.818483i	$-0.305185\pi$
$347$	2.57731e6	1.14906	0.574531	+	0.818483i	$0.305185\pi$
$348$	713376.	0.315770
$349$	3.06751e6	1.34810	0.674051	−	0.738684i	$-0.264553\pi$
$349$	3.06751e6	1.34810	0.674051	+	0.738684i	$0.264553\pi$
$350$	0	0
$351$	231822.	0.100435
$352$	−679936.	−0.292490
$353$	3.10144e6	1.32473	0.662364	−	0.749182i	$-0.269553\pi$
$353$	3.10144e6	1.32473	0.662364	+	0.749182i	$0.269553\pi$
$354$	555696.	0.235683
$355$	236392.	0.0995547
$356$	1.96253e6	0.820712
$357$	0	0
$358$	−1.95024e6	−0.804230
$359$	−327508.	−0.134118	−0.0670588	−	0.997749i	$-0.521362\pi$
$359$	−327508.	−0.134118	−0.0670588	+	0.997749i	$0.521362\pi$
$360$	134784.	0.0548128
$361$	−2.42040e6	−0.977507
$362$	−2.17764e6	−0.873403
$363$	−2.51860e6	−1.00321
$364$	0	0
$365$	−1.90980e6	−0.750337
$366$	1.32343e6	0.516415
$367$	2.86739e6	1.11128	0.555638	−	0.831424i	$-0.312474\pi$
$367$	2.86739e6	1.11128	0.555638	+	0.831424i	$0.312474\pi$
$368$	566272.	0.217974
$369$	−853902.	−0.326469
$370$	453232.	0.172114
$371$	0	0
$372$	−1.02643e6	−0.384568
$373$	3.58029e6	1.33244	0.666218	−	0.745757i	$-0.267912\pi$
$373$	3.58029e6	1.33244	0.666218	+	0.745757i	$0.267912\pi$
$374$	4.20179e6	1.55330
$375$	−1.30432e6	−0.478966
$376$	342528.	0.124947
$377$	1.57537e6	0.570860
$378$	0	0
$379$	1.64235e6	0.587310	0.293655	−	0.955912i	$-0.405128\pi$
$379$	1.64235e6	0.587310	0.293655	+	0.955912i	$0.405128\pi$
$380$	98176.0	0.0348776
$381$	−1.66860e6	−0.588898
$382$	−1.50562e6	−0.527905
$383$	2.05698e6	0.716527	0.358263	−	0.933621i	$-0.383369\pi$
$383$	2.05698e6	0.716527	0.358263	+	0.933621i	$0.383369\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−3.37978e6	−1.15457
$387$	−684612.	−0.232363
$388$	−341920.	−0.115304
$389$	616142.	0.206446	0.103223	−	0.994658i	$-0.467084\pi$
$389$	616142.	0.206446	0.103223	+	0.994658i	$0.467084\pi$
$390$	297648.	0.0990927
$391$	−3.49938e6	−1.15758
$392$	0	0
$393$	580788.	0.189686
$394$	1.97118e6	0.639713
$395$	−2.32440e6	−0.749580
$396$	860544.	0.275762
$397$	−2.19212e6	−0.698052	−0.349026	−	0.937113i	$-0.613487\pi$
$397$	−2.19212e6	−0.698052	−0.349026	+	0.937113i	$0.613487\pi$
$398$	−3.65910e6	−1.15789
$399$	0	0
$400$	−626944.	−0.195920
$401$	3.28454e6	1.02003	0.510015	−	0.860165i	$-0.329640\pi$
$401$	3.28454e6	1.02003	0.510015	+	0.860165i	$0.329640\pi$
$402$	1.47499e6	0.455223
$403$	−2.26670e6	−0.695236
$404$	589024.	0.179548
$405$	−170586.	−0.0516780
$406$	0	0
$407$	2.89371e6	0.865903
$408$	−911232.	−0.271006
$409$	3.61219e6	1.06773	0.533866	−	0.845569i	$-0.320739\pi$
$409$	3.61219e6	1.06773	0.533866	+	0.845569i	$0.320739\pi$
$410$	−1.09637e6	−0.322104
$411$	−1.37637e6	−0.401912
$412$	−1.67245e6	−0.485411
$413$	0	0
$414$	−716688.	−0.205508
$415$	−167128.	−0.0476353
$416$	325632.	0.0922558
$417$	−3.09114e6	−0.870520
$418$	626816.	0.175469
$419$	−5.41489e6	−1.50680	−0.753398	−	0.657564i	$-0.771587\pi$
$419$	−5.41489e6	−1.50680	−0.753398	+	0.657564i	$0.771587\pi$
$420$	0	0
$421$	3.60629e6	0.991644	0.495822	−	0.868424i	$-0.334867\pi$
$421$	3.60629e6	0.991644	0.495822	+	0.868424i	$0.334867\pi$
$422$	−1.24712e6	−0.340900
$423$	−433512.	−0.117801
$424$	2.13466e6	0.576651
$425$	3.87432e6	1.04045
$426$	−327312.	−0.0873852
$427$	0	0
$428$	3.43104e6	0.905350
$429$	1.90037e6	0.498534
$430$	−879008.	−0.229257
$431$	−2.78214e6	−0.721416	−0.360708	−	0.932679i	$-0.617465\pi$
$431$	−2.78214e6	−0.721416	−0.360708	+	0.932679i	$0.617465\pi$
$432$	−186624.	−0.0481125
$433$	−6.27619e6	−1.60871	−0.804353	−	0.594152i	$-0.797488\pi$
$433$	−6.27619e6	−1.60871	−0.804353	+	0.594152i	$0.797488\pi$
$434$	0	0
$435$	−1.15924e6	−0.293730
$436$	460768.	0.116082
$437$	−522032.	−0.130766
$438$	2.64434e6	0.658617
$439$	−641592.	−0.158890	−0.0794452	−	0.996839i	$-0.525315\pi$
$439$	−641592.	−0.158890	−0.0794452	+	0.996839i	$0.525315\pi$
$440$	1.10490e6	0.272076
$441$	0	0
$442$	−2.01230e6	−0.489934
$443$	6.05546e6	1.46601	0.733006	−	0.680222i	$-0.238117\pi$
$443$	6.05546e6	1.46601	0.733006	+	0.680222i	$0.238117\pi$
$444$	−627552.	−0.151075
$445$	−3.18911e6	−0.763430
$446$	−5.15104e6	−1.22619
$447$	1.57372e6	0.372528
$448$	0	0
$449$	−5.16681e6	−1.20950	−0.604752	−	0.796414i	$-0.706728\pi$
$449$	−5.16681e6	−1.20950	−0.604752	+	0.796414i	$0.706728\pi$
$450$	793476.	0.184715
$451$	−6.99989e6	−1.62050
$452$	−896224.	−0.206334
$453$	4.07297e6	0.932536
$454$	5.15621e6	1.17406
$455$	0	0
$456$	−135936.	−0.0306142
$457$	−227798.	−0.0510222	−0.0255111	−	0.999675i	$-0.508121\pi$
$457$	−227798.	−0.0510222	−0.0255111	+	0.999675i	$0.508121\pi$
$458$	2.71286e6	0.604315
$459$	1.15328e6	0.255507
$460$	−920192.	−0.202761
$461$	−585146.	−0.128237	−0.0641183	−	0.997942i	$-0.520423\pi$
$461$	−585146.	−0.128237	−0.0641183	+	0.997942i	$0.520423\pi$
$462$	0	0
$463$	−3.41454e6	−0.740251	−0.370126	−	0.928982i	$-0.620685\pi$
$463$	−3.41454e6	−0.740251	−0.370126	+	0.928982i	$0.620685\pi$
$464$	−1.26822e6	−0.273465
$465$	1.66795e6	0.357727
$466$	4.46924e6	0.953386
$467$	−716300.	−0.151986	−0.0759929	−	0.997108i	$-0.524213\pi$
$467$	−716300.	−0.151986	−0.0759929	+	0.997108i	$0.524213\pi$
$468$	−412128.	−0.0869796
$469$	0	0
$470$	−556608.	−0.116226
$471$	−4.49159e6	−0.932928
$472$	−987904.	−0.204108
$473$	−5.61213e6	−1.15339
$474$	3.21840e6	0.657952
$475$	577964.	0.117535
$476$	0	0
$477$	−2.70167e6	−0.543672
$478$	5.04782e6	1.01050
$479$	−5.24092e6	−1.04368	−0.521842	−	0.853042i	$-0.674755\pi$
$479$	−5.24092e6	−1.04368	−0.521842	+	0.853042i	$0.674755\pi$
$480$	−239616.	−0.0474693
$481$	−1.38584e6	−0.273119
$482$	3.79287e6	0.743619
$483$	0	0
$484$	4.47752e6	0.868809
$485$	555620.	0.107256
$486$	236196.	0.0453609
$487$	1.11702e6	0.213421	0.106710	−	0.994290i	$-0.465968\pi$
$487$	1.11702e6	0.213421	0.106710	+	0.994290i	$0.465968\pi$
$488$	−2.35277e6	−0.447229
$489$	4.28029e6	0.809471
$490$	0	0
$491$	1.34458e6	0.251699	0.125850	−	0.992049i	$-0.459834\pi$
$491$	1.34458e6	0.251699	0.125850	+	0.992049i	$0.459834\pi$
$492$	1.51805e6	0.282731
$493$	7.83723e6	1.45226
$494$	−300192.	−0.0553454
$495$	−1.39838e6	−0.256515
$496$	1.82477e6	0.333045
$497$	0	0
$498$	231408.	0.0418124
$499$	−6.54648e6	−1.17695	−0.588473	−	0.808517i	$-0.700271\pi$
$499$	−6.54648e6	−1.17695	−0.588473	+	0.808517i	$0.700271\pi$
$500$	2.31878e6	0.414797
$501$	1.08202e6	0.192592
$502$	−1.94558e6	−0.344580
$503$	8.22050e6	1.44870	0.724350	−	0.689432i	$-0.242140\pi$
$503$	8.22050e6	1.44870	0.724350	+	0.689432i	$0.242140\pi$
$504$	0	0
$505$	−957164.	−0.167016
$506$	−5.87507e6	−1.02009
$507$	2.43152e6	0.420105
$508$	2.96640e6	0.510000
$509$	5.11045e6	0.874308	0.437154	−	0.899387i	$-0.355987\pi$
$509$	5.11045e6	0.874308	0.437154	+	0.899387i	$0.355987\pi$
$510$	1.48075e6	0.252091
$511$	0	0
$512$	−262144.	−0.0441942
$513$	172044.	0.0288633
$514$	−4.15639e6	−0.693919
$515$	2.71773e6	0.451531
$516$	1.21709e6	0.201232
$517$	−3.55373e6	−0.584733
$518$	0	0
$519$	4.57987e6	0.746336
$520$	−529152.	−0.0858168
$521$	−9.69999e6	−1.56559	−0.782793	−	0.622282i	$-0.786206\pi$
$521$	−9.69999e6	−1.56559	−0.782793	+	0.622282i	$0.786206\pi$
$522$	1.60510e6	0.257825
$523$	3.17295e6	0.507234	0.253617	−	0.967305i	$-0.418380\pi$
$523$	3.17295e6	0.507234	0.253617	+	0.967305i	$0.418380\pi$
$524$	−1.03251e6	−0.164273
$525$	0	0
$526$	−5.40418e6	−0.851658
$527$	−1.12765e7	−1.76867
$528$	−1.52986e6	−0.238817
$529$	−1.54340e6	−0.239794
$530$	−3.46882e6	−0.536403
$531$	1.25032e6	0.192435
$532$	0	0
$533$	3.35236e6	0.511131
$534$	4.41569e6	0.670109
$535$	−5.57544e6	−0.842160
$536$	−2.62221e6	−0.394235
$537$	−4.38804e6	−0.656651
$538$	−4.47244e6	−0.666176
$539$	0	0
$540$	303264.	0.0447545
$541$	−6.62575e6	−0.973289	−0.486644	−	0.873600i	$-0.661779\pi$
$541$	−6.62575e6	−0.973289	−0.486644	+	0.873600i	$0.661779\pi$
$542$	−760416.	−0.111187
$543$	−4.89969e6	−0.713131
$544$	1.61997e6	0.234698
$545$	−748748.	−0.107980
$546$	0	0
$547$	3.84707e6	0.549745	0.274873	−	0.961481i	$-0.411364\pi$
$547$	3.84707e6	0.549745	0.274873	+	0.961481i	$0.411364\pi$
$548$	2.44688e6	0.348066
$549$	2.97772e6	0.421651
$550$	6.50454e6	0.916875
$551$	1.16914e6	0.164055
$552$	1.27411e6	0.177975
$553$	0	0
$554$	802024.	0.111023
$555$	1.01977e6	0.140531
$556$	5.49536e6	0.753892
$557$	5.00176e6	0.683101	0.341550	−	0.939863i	$-0.389048\pi$
$557$	5.00176e6	0.683101	0.341550	+	0.939863i	$0.389048\pi$
$558$	−2.30947e6	−0.313998
$559$	2.68774e6	0.363795
$560$	0	0
$561$	9.45403e6	1.26826
$562$	−4.36948e6	−0.583565
$563$	−2.27772e6	−0.302852	−0.151426	−	0.988469i	$-0.548386\pi$
$563$	−2.27772e6	−0.302852	−0.151426	+	0.988469i	$0.548386\pi$
$564$	770688.	0.102019
$565$	1.45636e6	0.191933
$566$	7.25032e6	0.951297
$567$	0	0
$568$	581888.	0.0756778
$569$	8.86979e6	1.14850	0.574252	−	0.818678i	$-0.305293\pi$
$569$	8.86979e6	1.14850	0.574252	+	0.818678i	$0.305293\pi$
$570$	220896.	0.0284774
$571$	1.40102e7	1.79826	0.899132	−	0.437678i	$-0.144199\pi$
$571$	1.40102e7	1.79826	0.899132	+	0.437678i	$0.144199\pi$
$572$	−3.37843e6	−0.431743
$573$	−3.38764e6	−0.431033
$574$	0	0
$575$	−5.41719e6	−0.683289
$576$	331776.	0.0416667
$577$	−8.75327e6	−1.09454	−0.547269	−	0.836957i	$-0.684332\pi$
$577$	−8.75327e6	−1.09454	−0.547269	+	0.836957i	$0.684332\pi$
$578$	−4.33147e6	−0.539281
$579$	−7.60451e6	−0.942703
$580$	2.06086e6	0.254378
$581$	0	0
$582$	−769320.	−0.0941455
$583$	−2.21471e7	−2.69864
$584$	−4.70106e6	−0.570379
$585$	669708.	0.0809088
$586$	8.40122e6	1.01064
$587$	1.06117e7	1.27113	0.635564	−	0.772048i	$-0.280768\pi$
$587$	1.06117e7	1.27113	0.635564	+	0.772048i	$0.280768\pi$
$588$	0	0
$589$	−1.68221e6	−0.199798
$590$	1.60534e6	0.189862
$591$	4.43515e6	0.522323
$592$	1.11565e6	0.130835
$593$	−1.88552e6	−0.220188	−0.110094	−	0.993921i	$-0.535115\pi$
$593$	−1.88552e6	−0.220188	−0.110094	+	0.993921i	$0.535115\pi$
$594$	1.93622e6	0.225159
$595$	0	0
$596$	−2.79773e6	−0.322619
$597$	−8.23298e6	−0.945413
$598$	2.81366e6	0.321751
$599$	1.27256e7	1.44915	0.724573	−	0.689198i	$-0.242037\pi$
$599$	1.27256e7	1.44915	0.724573	+	0.689198i	$0.242037\pi$
$600$	−1.41062e6	−0.159968
$601$	−7.18846e6	−0.811801	−0.405900	−	0.913917i	$-0.633042\pi$
$601$	−7.18846e6	−0.811801	−0.405900	+	0.913917i	$0.633042\pi$
$602$	0	0
$603$	3.31873e6	0.371688
$604$	−7.24083e6	−0.807600
$605$	−7.27597e6	−0.808170
$606$	1.32530e6	0.146600
$607$	−1.08494e7	−1.19519	−0.597593	−	0.801800i	$-0.703876\pi$
$607$	−1.08494e7	−1.19519	−0.597593	+	0.801800i	$0.703876\pi$
$608$	241664.	0.0265126
$609$	0	0
$610$	3.82325e6	0.416014
$611$	1.70194e6	0.184434
$612$	−2.05027e6	−0.221275
$613$	−4.90511e6	−0.527227	−0.263614	−	0.964628i	$-0.584914\pi$
$613$	−4.90511e6	−0.527227	−0.263614	+	0.964628i	$0.584914\pi$
$614$	−6.56418e6	−0.702683
$615$	−2.46683e6	−0.262997
$616$	0	0
$617$	2.58445e6	0.273310	0.136655	−	0.990619i	$-0.456365\pi$
$617$	2.58445e6	0.273310	0.136655	+	0.990619i	$0.456365\pi$
$618$	−3.76301e6	−0.396336
$619$	4.99336e6	0.523801	0.261901	−	0.965095i	$-0.415651\pi$
$619$	4.99336e6	0.523801	0.261901	+	0.965095i	$0.415651\pi$
$620$	−2.96525e6	−0.309800
$621$	−1.61255e6	−0.167797
$622$	−3.78093e6	−0.391852
$623$	0	0
$624$	732672.	0.0753266
$625$	3.88510e6	0.397834
$626$	1.66142e6	0.169450
$627$	1.41034e6	0.143269
$628$	7.98506e6	0.807940
$629$	−6.89436e6	−0.694812
$630$	0	0
$631$	−1.18219e7	−1.18199	−0.590997	−	0.806674i	$-0.701265\pi$
$631$	−1.18219e7	−1.18199	−0.590997	+	0.806674i	$0.701265\pi$
$632$	−5.72160e6	−0.569803
$633$	−2.80602e6	−0.278344
$634$	−4.73926e6	−0.468260
$635$	−4.82040e6	−0.474404
$636$	4.80298e6	0.470834
$637$	0	0
$638$	1.31578e7	1.27977
$639$	−736452.	−0.0713497
$640$	425984.	0.0411096
$641$	−5.47007e6	−0.525833	−0.262916	−	0.964819i	$-0.584684\pi$
$641$	−5.47007e6	−0.525833	−0.262916	+	0.964819i	$0.584684\pi$
$642$	7.71984e6	0.739215
$643$	−9.64934e6	−0.920386	−0.460193	−	0.887819i	$-0.652220\pi$
$643$	−9.64934e6	−0.920386	−0.460193	+	0.887819i	$0.652220\pi$
$644$	0	0
$645$	−1.97777e6	−0.187187
$646$	−1.49341e6	−0.140798
$647$	−292368.	−0.0274580	−0.0137290	−	0.999906i	$-0.504370\pi$
$647$	−292368.	−0.0274580	−0.0137290	+	0.999906i	$0.504370\pi$
$648$	−419904.	−0.0392837
$649$	1.02495e7	0.955193
$650$	−3.11513e6	−0.289196
$651$	0	0
$652$	−7.60941e6	−0.701022
$653$	6.94081e6	0.636982	0.318491	−	0.947926i	$-0.396824\pi$
$653$	6.94081e6	0.636982	0.318491	+	0.947926i	$0.396824\pi$
$654$	1.03673e6	0.0947808
$655$	1.67783e6	0.152808
$656$	−2.69875e6	−0.244852
$657$	5.94977e6	0.537758
$658$	0	0
$659$	−1.32912e7	−1.19221	−0.596104	−	0.802908i	$-0.703285\pi$
$659$	−1.32912e7	−1.19221	−0.596104	+	0.802908i	$0.703285\pi$
$660$	2.48602e6	0.222149
$661$	−2.05219e6	−0.182690	−0.0913448	−	0.995819i	$-0.529117\pi$
$661$	−2.05219e6	−0.182690	−0.0913448	+	0.995819i	$0.529117\pi$
$662$	−5.48619e6	−0.486548
$663$	−4.52768e6	−0.400030
$664$	−411392.	−0.0362106
$665$	0	0
$666$	−1.41199e6	−0.123352
$667$	−1.09582e7	−0.953732
$668$	−1.92358e6	−0.166790
$669$	−1.15898e7	−1.00118
$670$	4.26109e6	0.366719
$671$	2.44100e7	2.09296
$672$	0	0
$673$	−1.57039e7	−1.33650	−0.668252	−	0.743935i	$-0.732957\pi$
$673$	−1.57039e7	−1.33650	−0.668252	+	0.743935i	$0.732957\pi$
$674$	−3.85409e6	−0.326792
$675$	1.78532e6	0.150819
$676$	−4.32270e6	−0.363822
$677$	969534.	0.0813002	0.0406501	−	0.999173i	$-0.487057\pi$
$677$	969534.	0.0813002	0.0406501	+	0.999173i	$0.487057\pi$
$678$	−2.01650e6	−0.168471
$679$	0	0
$680$	−2.63245e6	−0.218317
$681$	1.16015e7	0.958617
$682$	−1.89320e7	−1.55860
$683$	−1.49908e7	−1.22962	−0.614812	−	0.788673i	$-0.710768\pi$
$683$	−1.49908e7	−1.22962	−0.614812	+	0.788673i	$0.710768\pi$
$684$	−305856.	−0.0249964
$685$	−3.97618e6	−0.323772
$686$	0	0
$687$	6.10393e6	0.493421
$688$	−2.16371e6	−0.174272
$689$	1.06066e7	0.851191
$690$	−2.07043e6	−0.165553
$691$	7.16038e6	0.570481	0.285240	−	0.958456i	$-0.407927\pi$
$691$	7.16038e6	0.570481	0.285240	+	0.958456i	$0.407927\pi$
$692$	−8.14198e6	−0.646346
$693$	0	0
$694$	−1.03092e7	−0.812509
$695$	−8.92996e6	−0.701274
$696$	−2.85350e6	−0.223283
$697$	1.66774e7	1.30031
$698$	−1.22701e7	−0.953253
$699$	1.00558e7	0.778437
$700$	0	0
$701$	−91834.0	−0.00705844	−0.00352922	−	0.999994i	$-0.501123\pi$
$701$	−91834.0	−0.00705844	−0.00352922	+	0.999994i	$0.501123\pi$
$702$	−927288.	−0.0710186
$703$	−1.02849e6	−0.0784894
$704$	2.71974e6	0.206822
$705$	−1.25237e6	−0.0948985
$706$	−1.24058e7	−0.936725
$707$	0	0
$708$	−2.22278e6	−0.166653
$709$	2.20981e7	1.65097	0.825487	−	0.564422i	$-0.190901\pi$
$709$	2.20981e7	1.65097	0.825487	+	0.564422i	$0.190901\pi$
$710$	−945568.	−0.0703958
$711$	7.24140e6	0.537216
$712$	−7.85011e6	−0.580331
$713$	1.57671e7	1.16153
$714$	0	0
$715$	5.48995e6	0.401609
$716$	7.80096e6	0.568677
$717$	1.13576e7	0.825066
$718$	1.31003e6	0.0948355
$719$	−1.58388e7	−1.14262	−0.571308	−	0.820736i	$-0.693564\pi$
$719$	−1.58388e7	−1.14262	−0.571308	+	0.820736i	$0.693564\pi$
$720$	−539136.	−0.0387585
$721$	0	0
$722$	9.68161e6	0.691202
$723$	8.53396e6	0.607163
$724$	8.71056e6	0.617589
$725$	1.21323e7	0.857235
$726$	1.00744e7	0.709379
$727$	−6.31418e6	−0.443078	−0.221539	−	0.975151i	$-0.571108\pi$
$727$	−6.31418e6	−0.443078	−0.221539	+	0.975151i	$0.571108\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	7.63922e6	0.530569
$731$	1.33711e7	0.925492
$732$	−5.29373e6	−0.365161
$733$	−6.93003e6	−0.476404	−0.238202	−	0.971216i	$-0.576558\pi$
$733$	−6.93003e6	−0.476404	−0.238202	+	0.971216i	$0.576558\pi$
$734$	−1.14696e7	−0.785791
$735$	0	0
$736$	−2.26509e6	−0.154131
$737$	2.72054e7	1.84496
$738$	3.41561e6	0.230849
$739$	1.42331e7	0.958714	0.479357	−	0.877620i	$-0.340870\pi$
$739$	1.42331e7	0.958714	0.479357	+	0.877620i	$0.340870\pi$
$740$	−1.81293e6	−0.121703
$741$	−675432.	−0.0451894
$742$	0	0
$743$	−5.94460e6	−0.395048	−0.197524	−	0.980298i	$-0.563290\pi$
$743$	−5.94460e6	−0.395048	−0.197524	+	0.980298i	$0.563290\pi$
$744$	4.10573e6	0.271930
$745$	4.54631e6	0.300102
$746$	−1.43212e7	−0.942175
$747$	520668.	0.0341397
$748$	−1.68072e7	−1.09835
$749$	0	0
$750$	5.21726e6	0.338680
$751$	−682752.	−0.0441736	−0.0220868	−	0.999756i	$-0.507031\pi$
$751$	−682752.	−0.0441736	−0.0220868	+	0.999756i	$0.507031\pi$
$752$	−1.37011e6	−0.0883510
$753$	−4.37756e6	−0.281349
$754$	−6.30149e6	−0.403659
$755$	1.17664e7	0.751233
$756$	0	0
$757$	1.46333e7	0.928116	0.464058	−	0.885805i	$-0.346393\pi$
$757$	1.46333e7	0.928116	0.464058	+	0.885805i	$0.346393\pi$
$758$	−6.56939e6	−0.415291
$759$	−1.32189e7	−0.832897
$760$	−392704.	−0.0246622
$761$	1.16367e7	0.728399	0.364200	−	0.931321i	$-0.381343\pi$
$761$	1.16367e7	0.728399	0.364200	+	0.931321i	$0.381343\pi$
$762$	6.67440e6	0.416414
$763$	0	0
$764$	6.02246e6	0.373285
$765$	3.33169e6	0.205831
$766$	−8.22790e6	−0.506661
$767$	−4.90865e6	−0.301282
$768$	−589824.	−0.0360844
$769$	−1.91472e7	−1.16759	−0.583793	−	0.811902i	$-0.698432\pi$
$769$	−1.91472e7	−1.16759	−0.583793	+	0.811902i	$0.698432\pi$
$770$	0	0
$771$	−9.35188e6	−0.566582
$772$	1.35191e7	0.816405
$773$	5.39261e6	0.324601	0.162301	−	0.986741i	$-0.448109\pi$
$773$	5.39261e6	0.324601	0.162301	+	0.986741i	$0.448109\pi$
$774$	2.73845e6	0.164305
$775$	−1.74565e7	−1.04400
$776$	1.36768e6	0.0815324
$777$	0	0
$778$	−2.46457e6	−0.145979
$779$	2.48791e6	0.146890
$780$	−1.19059e6	−0.0700691
$781$	−6.03709e6	−0.354160
$782$	1.39975e7	0.818530
$783$	3.61147e6	0.210513
$784$	0	0
$785$	−1.29757e7	−0.751549
$786$	−2.32315e6	−0.134129
$787$	−3.04348e6	−0.175159	−0.0875796	−	0.996158i	$-0.527913\pi$
$787$	−3.04348e6	−0.175159	−0.0875796	+	0.996158i	$0.527913\pi$
$788$	−7.88470e6	−0.452345
$789$	−1.21594e7	−0.695376
$790$	9.29760e6	0.530033
$791$	0	0
$792$	−3.44218e6	−0.194993
$793$	−1.16903e7	−0.660151
$794$	8.76847e6	0.493597
$795$	−7.80484e6	−0.437972
$796$	1.46364e7	0.818751
$797$	−2.29652e7	−1.28063	−0.640316	−	0.768111i	$-0.721197\pi$
$797$	−2.29652e7	−1.28063	−0.640316	+	0.768111i	$0.721197\pi$
$798$	0	0
$799$	8.46686e6	0.469197
$800$	2.50778e6	0.138536
$801$	9.93530e6	0.547141
$802$	−1.31382e7	−0.721271
$803$	4.87735e7	2.66928
$804$	−5.89997e6	−0.321892
$805$	0	0
$806$	9.06682e6	0.491606
$807$	−1.00630e7	−0.543930
$808$	−2.35610e6	−0.126959
$809$	1.90787e7	1.02489	0.512445	−	0.858720i	$-0.328740\pi$
$809$	1.90787e7	1.02489	0.512445	+	0.858720i	$0.328740\pi$
$810$	682344.	0.0365419
$811$	−1.09414e7	−0.584147	−0.292074	−	0.956396i	$-0.594345\pi$
$811$	−1.09414e7	−0.584147	−0.292074	+	0.956396i	$0.594345\pi$
$812$	0	0
$813$	−1.71094e6	−0.0907836
$814$	−1.15748e7	−0.612286
$815$	1.23653e7	0.652094
$816$	3.64493e6	0.191630
$817$	1.99467e6	0.104548
$818$	−1.44488e7	−0.755001
$819$	0	0
$820$	4.38547e6	0.227762
$821$	2.12594e7	1.10076	0.550380	−	0.834914i	$-0.314483\pi$
$821$	2.12594e7	1.10076	0.550380	+	0.834914i	$0.314483\pi$
$822$	5.50548e6	0.284194
$823$	−1.42256e7	−0.732103	−0.366052	−	0.930595i	$-0.619291\pi$
$823$	−1.42256e7	−0.732103	−0.366052	+	0.930595i	$0.619291\pi$
$824$	6.68979e6	0.343237
$825$	1.46352e7	0.748625
$826$	0	0
$827$	2.76103e6	0.140381	0.0701904	−	0.997534i	$-0.477639\pi$
$827$	2.76103e6	0.140381	0.0701904	+	0.997534i	$0.477639\pi$
$828$	2.86675e6	0.145316
$829$	3.82147e7	1.93127	0.965637	−	0.259895i	$-0.0836880\pi$
$829$	3.82147e7	1.93127	0.965637	+	0.259895i	$0.0836880\pi$
$830$	668512.	0.0336832
$831$	1.80455e6	0.0906499
$832$	−1.30253e6	−0.0652347
$833$	0	0
$834$	1.23646e7	0.615550
$835$	3.12582e6	0.155149
$836$	−2.50726e6	−0.124075
$837$	−5.19631e6	−0.256378
$838$	2.16596e7	1.06547
$839$	−1.06044e7	−0.520094	−0.260047	−	0.965596i	$-0.583738\pi$
$839$	−1.06044e7	−0.520094	−0.260047	+	0.965596i	$0.583738\pi$
$840$	0	0
$841$	4.03097e6	0.196526
$842$	−1.44252e7	−0.701198
$843$	−9.83133e6	−0.476479
$844$	4.98848e6	0.241053
$845$	7.02439e6	0.338429
$846$	1.73405e6	0.0832981
$847$	0	0
$848$	−8.53862e6	−0.407754
$849$	1.63132e7	0.776731
$850$	−1.54973e7	−0.735712
$851$	9.63990e6	0.456298
$852$	1.30925e6	0.0617907
$853$	4.07009e7	1.91527	0.957637	−	0.287977i	$-0.0929826\pi$
$853$	4.07009e7	1.91527	0.957637	+	0.287977i	$0.0929826\pi$
$854$	0	0
$855$	497016.	0.0232517
$856$	−1.37242e7	−0.640179
$857$	3.10120e7	1.44237	0.721187	−	0.692741i	$-0.243597\pi$
$857$	3.10120e7	1.44237	0.721187	+	0.692741i	$0.243597\pi$
$858$	−7.60147e6	−0.352517
$859$	−1.09104e7	−0.504495	−0.252247	−	0.967663i	$-0.581170\pi$
$859$	−1.09104e7	−0.504495	−0.252247	+	0.967663i	$0.581170\pi$
$860$	3.51603e6	0.162109
$861$	0	0
$862$	1.11286e7	0.510118
$863$	1.04089e7	0.475751	0.237875	−	0.971296i	$-0.423549\pi$
$863$	1.04089e7	0.475751	0.237875	+	0.971296i	$0.423549\pi$
$864$	746496.	0.0340207
$865$	1.32307e7	0.601234
$866$	2.51048e7	1.13753
$867$	−9.74580e6	−0.440321
$868$	0	0
$869$	5.93616e7	2.66659
$870$	4.63694e6	0.207699
$871$	−1.30291e7	−0.581928
$872$	−1.84307e6	−0.0820826
$873$	−1.73097e6	−0.0768695
$874$	2.08813e6	0.0924652
$875$	0	0
$876$	−1.05774e7	−0.465712
$877$	1.64064e7	0.720299	0.360150	−	0.932895i	$-0.382726\pi$
$877$	1.64064e7	0.720299	0.360150	+	0.932895i	$0.382726\pi$
$878$	2.56637e6	0.112352
$879$	1.89028e7	0.825188
$880$	−4.41958e6	−0.192387
$881$	−1.48577e7	−0.644927	−0.322464	−	0.946582i	$-0.604511\pi$
$881$	−1.48577e7	−0.644927	−0.322464	+	0.946582i	$0.604511\pi$
$882$	0	0
$883$	−2.72018e7	−1.17407	−0.587037	−	0.809560i	$-0.699706\pi$
$883$	−2.72018e7	−1.17407	−0.587037	+	0.809560i	$0.699706\pi$
$884$	8.04922e6	0.346436
$885$	3.61202e6	0.155022
$886$	−2.42218e7	−1.03663
$887$	−2.71242e7	−1.15757	−0.578785	−	0.815480i	$-0.696473\pi$
$887$	−2.71242e7	−1.15757	−0.578785	+	0.815480i	$0.696473\pi$
$888$	2.51021e6	0.106826
$889$	0	0
$890$	1.27564e7	0.539827
$891$	4.35650e6	0.183842
$892$	2.06042e7	0.867047
$893$	1.26307e6	0.0530029
$894$	−6.29489e6	−0.263417
$895$	−1.26766e7	−0.528986
$896$	0	0
$897$	6.33074e6	0.262708
$898$	2.06673e7	0.855248
$899$	−3.53121e7	−1.45722
$900$	−3.17390e6	−0.130613
$901$	5.27660e7	2.16542
$902$	2.79996e7	1.14587
$903$	0	0
$904$	3.58490e6	0.145900
$905$	−1.41547e7	−0.574484
$906$	−1.62919e7	−0.659402
$907$	−8.42269e6	−0.339964	−0.169982	−	0.985447i	$-0.554371\pi$
$907$	−8.42269e6	−0.339964	−0.169982	+	0.985447i	$0.554371\pi$
$908$	−2.06248e7	−0.830187
$909$	2.98193e6	0.119698
$910$	0	0
$911$	3.08637e7	1.23212	0.616060	−	0.787700i	$-0.288728\pi$
$911$	3.08637e7	1.23212	0.616060	+	0.787700i	$0.288728\pi$
$912$	543744.	0.0216475
$913$	4.26819e6	0.169460
$914$	911192.	0.0360782
$915$	8.60231e6	0.339674
$916$	−1.08514e7	−0.427315
$917$	0	0
$918$	−4.61311e6	−0.180671
$919$	4.93895e6	0.192906	0.0964531	−	0.995338i	$-0.469250\pi$
$919$	4.93895e6	0.192906	0.0964531	+	0.995338i	$0.469250\pi$
$920$	3.68077e6	0.143373
$921$	−1.47694e7	−0.573738
$922$	2.34058e6	0.0906770
$923$	2.89126e6	0.111707
$924$	0	0
$925$	−1.06727e7	−0.410130
$926$	1.36581e7	0.523437
$927$	−8.46677e6	−0.323607
$928$	5.07290e6	0.193369
$929$	−5.62575e6	−0.213866	−0.106933	−	0.994266i	$-0.534103\pi$
$929$	−5.62575e6	−0.213866	−0.106933	+	0.994266i	$0.534103\pi$
$930$	−6.67181e6	−0.252951
$931$	0	0
$932$	−1.78770e7	−0.674146
$933$	−8.50709e6	−0.319946
$934$	2.86520e6	0.107470
$935$	2.73116e7	1.02169
$936$	1.64851e6	0.0615039
$937$	−2.60073e7	−0.967714	−0.483857	−	0.875147i	$-0.660764\pi$
$937$	−2.60073e7	−0.967714	−0.483857	+	0.875147i	$0.660764\pi$
$938$	0	0
$939$	3.73819e6	0.138356
$940$	2.22643e6	0.0821845
$941$	−3.02160e6	−0.111241	−0.0556203	−	0.998452i	$-0.517714\pi$
$941$	−3.02160e6	−0.111241	−0.0556203	+	0.998452i	$0.517714\pi$
$942$	1.79664e7	0.659680
$943$	−2.33189e7	−0.853943
$944$	3.95162e6	0.144326
$945$	0	0
$946$	2.24485e7	0.815567
$947$	−3.48282e7	−1.26199	−0.630995	−	0.775787i	$-0.717353\pi$
$947$	−3.48282e7	−1.26199	−0.630995	+	0.775787i	$0.717353\pi$
$948$	−1.28736e7	−0.465242
$949$	−2.33584e7	−0.841932
$950$	−2.31186e6	−0.0831097
$951$	−1.06633e7	−0.382333
$952$	0	0
$953$	−9.39009e6	−0.334917	−0.167459	−	0.985879i	$-0.553556\pi$
$953$	−9.39009e6	−0.334917	−0.167459	+	0.985879i	$0.553556\pi$
$954$	1.08067e7	0.384434
$955$	−9.78650e6	−0.347232
$956$	−2.01913e7	−0.714528
$957$	2.96051e7	1.04493
$958$	2.09637e7	0.737996
$959$	0	0
$960$	958464.	0.0335659
$961$	2.21792e7	0.774708
$962$	5.54338e6	0.193124
$963$	1.73696e7	0.603566
$964$	−1.51715e7	−0.525818
$965$	−2.19686e7	−0.759423
$966$	0	0
$967$	1.44768e7	0.497860	0.248930	−	0.968521i	$-0.419921\pi$
$967$	1.44768e7	0.497860	0.248930	+	0.968521i	$0.419921\pi$
$968$	−1.79101e7	−0.614340
$969$	−3.36017e6	−0.114961
$970$	−2.22248e6	−0.0758418
$971$	−9.24976e6	−0.314834	−0.157417	−	0.987532i	$-0.550317\pi$
$971$	−9.24976e6	−0.314834	−0.157417	+	0.987532i	$0.550317\pi$
$972$	−944784.	−0.0320750
$973$	0	0
$974$	−4.46806e6	−0.150911
$975$	−7.00904e6	−0.236128
$976$	9.41107e6	0.316238
$977$	−4.97780e7	−1.66840	−0.834202	−	0.551459i	$-0.814071\pi$
$977$	−4.97780e7	−1.66840	−0.834202	+	0.551459i	$0.814071\pi$
$978$	−1.71212e7	−0.572382
$979$	8.14449e7	2.71586
$980$	0	0
$981$	2.33264e6	0.0773882
$982$	−5.37830e6	−0.177978
$983$	8.95601e6	0.295618	0.147809	−	0.989016i	$-0.452778\pi$
$983$	8.95601e6	0.295618	0.147809	+	0.989016i	$0.452778\pi$
$984$	−6.07219e6	−0.199921
$985$	1.28126e7	0.420773
$986$	−3.13489e7	−1.02690
$987$	0	0
$988$	1.20077e6	0.0391351
$989$	−1.86958e7	−0.607790
$990$	5.59354e6	0.181384
$991$	2.62400e7	0.848751	0.424376	−	0.905486i	$-0.360494\pi$
$991$	2.62400e7	0.848751	0.424376	+	0.905486i	$0.360494\pi$
$992$	−7.29907e6	−0.235499
$993$	−1.23439e7	−0.397265
$994$	0	0
$995$	−2.37842e7	−0.761606
$996$	−925632.	−0.0295658
$997$	−2.80506e7	−0.893727	−0.446863	−	0.894602i	$-0.647459\pi$
$997$	−2.80506e7	−0.893727	−0.446863	+	0.894602i	$0.647459\pi$
$998$	2.61859e7	0.832226
$999$	−3.17698e6	−0.100717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.6.a.b.1.1		1
3.2	odd	2		882.6.a.s.1.1		1
7.2	even	3		294.6.e.p.67.1		2
7.3	odd	6		294.6.e.i.79.1		2
7.4	even	3		294.6.e.p.79.1		2
7.5	odd	6		294.6.e.i.67.1		2
7.6	odd	2		42.6.a.d.1.1	✓	1
21.20	even	2		126.6.a.i.1.1		1
28.27	even	2		336.6.a.h.1.1		1
35.13	even	4		1050.6.g.i.799.2		2
35.27	even	4		1050.6.g.i.799.1		2
35.34	odd	2		1050.6.a.k.1.1		1
84.83	odd	2		1008.6.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.d.1.1	✓	1	7.6	odd	2
126.6.a.i.1.1		1	21.20	even	2
294.6.a.b.1.1		1	1.1	even	1	trivial
294.6.e.i.67.1		2	7.5	odd	6
294.6.e.i.79.1		2	7.3	odd	6
294.6.e.p.67.1		2	7.2	even	3
294.6.e.p.79.1		2	7.4	even	3
336.6.a.h.1.1		1	28.27	even	2
882.6.a.s.1.1		1	3.2	odd	2
1008.6.a.j.1.1		1	84.83	odd	2
1050.6.a.k.1.1		1	35.34	odd	2
1050.6.g.i.799.1		2	35.27	even	4
1050.6.g.i.799.2		2	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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