LMFDB - Embedded newform 350.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +17.0000 q^{3} +16.0000 q^{4} -68.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +46.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +17.0000 q^{3} +16.0000 q^{4} -68.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +46.0000 q^{9} -715.000 q^{11} +272.000 q^{12} -331.000 q^{13} -196.000 q^{14} +256.000 q^{16} +1699.00 q^{17} -184.000 q^{18} -1718.00 q^{19} +833.000 q^{21} +2860.00 q^{22} +3950.00 q^{23} -1088.00 q^{24} +1324.00 q^{26} -3349.00 q^{27} +784.000 q^{28} +4579.00 q^{29} +6756.00 q^{31} -1024.00 q^{32} -12155.0 q^{33} -6796.00 q^{34} +736.000 q^{36} +16518.0 q^{37} +6872.00 q^{38} -5627.00 q^{39} +18876.0 q^{41} -3332.00 q^{42} -2258.00 q^{43} -11440.0 q^{44} -15800.0 q^{46} +537.000 q^{47} +4352.00 q^{48} +2401.00 q^{49} +28883.0 q^{51} -5296.00 q^{52} +10984.0 q^{53} +13396.0 q^{54} -3136.00 q^{56} -29206.0 q^{57} -18316.0 q^{58} -25956.0 q^{59} +39188.0 q^{61} -27024.0 q^{62} +2254.00 q^{63} +4096.00 q^{64} +48620.0 q^{66} -4416.00 q^{67} +27184.0 q^{68} +67150.0 q^{69} -31880.0 q^{71} -2944.00 q^{72} +5018.00 q^{73} -66072.0 q^{74} -27488.0 q^{76} -35035.0 q^{77} +22508.0 q^{78} -27977.0 q^{79} -68111.0 q^{81} -75504.0 q^{82} -37644.0 q^{83} +13328.0 q^{84} +9032.00 q^{86} +77843.0 q^{87} +45760.0 q^{88} -17216.0 q^{89} -16219.0 q^{91} +63200.0 q^{92} +114852. q^{93} -2148.00 q^{94} -17408.0 q^{96} +63175.0 q^{97} -9604.00 q^{98} -32890.0 q^{99} +O(q^{100})\)

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
$3$	17.0000	1.09055	0.545275	−	0.838257i	$-0.316425\pi$
$3$	17.0000	1.09055	0.545275	+	0.838257i	$0.316425\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−68.0000	−0.771136
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	46.0000	0.189300
$10$	0	0
$11$	−715.000	−1.78166	−0.890829	−	0.454339i	$-0.849876\pi$
$11$	−715.000	−1.78166	−0.890829	+	0.454339i	$0.849876\pi$
$12$	272.000	0.545275
$13$	−331.000	−0.543212	−0.271606	−	0.962408i	$-0.587555\pi$
$13$	−331.000	−0.543212	−0.271606	+	0.962408i	$0.587555\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1699.00	1.42584	0.712920	−	0.701245i	$-0.247372\pi$
$17$	1699.00	1.42584	0.712920	+	0.701245i	$0.247372\pi$
$18$	−184.000	−0.133856
$19$	−1718.00	−1.09179	−0.545895	−	0.837854i	$-0.683810\pi$
$19$	−1718.00	−1.09179	−0.545895	+	0.837854i	$0.683810\pi$
$20$	0	0
$21$	833.000	0.412189
$22$	2860.00	1.25982
$23$	3950.00	1.55696	0.778480	−	0.627669i	$-0.215991\pi$
$23$	3950.00	1.55696	0.778480	+	0.627669i	$0.215991\pi$
$24$	−1088.00	−0.385568
$25$	0	0
$26$	1324.00	0.384109
$27$	−3349.00	−0.884109
$28$	784.000	0.188982
$29$	4579.00	1.01106	0.505529	−	0.862810i	$-0.331298\pi$
$29$	4579.00	1.01106	0.505529	+	0.862810i	$0.331298\pi$
$30$	0	0
$31$	6756.00	1.26266	0.631329	−	0.775516i	$-0.282510\pi$
$31$	6756.00	1.26266	0.631329	+	0.775516i	$0.282510\pi$
$32$	−1024.00	−0.176777
$33$	−12155.0	−1.94299
$34$	−6796.00	−1.00822
$35$	0	0
$36$	736.000	0.0946502
$37$	16518.0	1.98360	0.991798	−	0.127816i	$-0.0407969\pi$
$37$	16518.0	1.98360	0.991798	+	0.127816i	$0.0407969\pi$
$38$	6872.00	0.772012
$39$	−5627.00	−0.592400
$40$	0	0
$41$	18876.0	1.75368	0.876840	−	0.480782i	$-0.159647\pi$
$41$	18876.0	1.75368	0.876840	+	0.480782i	$0.159647\pi$
$42$	−3332.00	−0.291462
$43$	−2258.00	−0.186231	−0.0931157	−	0.995655i	$-0.529683\pi$
$43$	−2258.00	−0.186231	−0.0931157	+	0.995655i	$0.529683\pi$
$44$	−11440.0	−0.890829
$45$	0	0
$46$	−15800.0	−1.10094
$47$	537.000	0.0354593	0.0177296	−	0.999843i	$-0.494356\pi$
$47$	537.000	0.0354593	0.0177296	+	0.999843i	$0.494356\pi$
$48$	4352.00	0.272638
$49$	2401.00	0.142857
$50$	0	0
$51$	28883.0	1.55495
$52$	−5296.00	−0.271606
$53$	10984.0	0.537119	0.268560	−	0.963263i	$-0.413452\pi$
$53$	10984.0	0.537119	0.268560	+	0.963263i	$0.413452\pi$
$54$	13396.0	0.625159
$55$	0	0
$56$	−3136.00	−0.133631
$57$	−29206.0	−1.19065
$58$	−18316.0	−0.714925
$59$	−25956.0	−0.970751	−0.485375	−	0.874306i	$-0.661317\pi$
$59$	−25956.0	−0.970751	−0.485375	+	0.874306i	$0.661317\pi$
$60$	0	0
$61$	39188.0	1.34843	0.674215	−	0.738535i	$-0.264482\pi$
$61$	39188.0	1.34843	0.674215	+	0.738535i	$0.264482\pi$
$62$	−27024.0	−0.892833
$63$	2254.00	0.0715488
$64$	4096.00	0.125000
$65$	0	0
$66$	48620.0	1.37390
$67$	−4416.00	−0.120183	−0.0600914	−	0.998193i	$-0.519139\pi$
$67$	−4416.00	−0.120183	−0.0600914	+	0.998193i	$0.519139\pi$
$68$	27184.0	0.712920
$69$	67150.0	1.69794
$70$	0	0
$71$	−31880.0	−0.750538	−0.375269	−	0.926916i	$-0.622450\pi$
$71$	−31880.0	−0.750538	−0.375269	+	0.926916i	$0.622450\pi$
$72$	−2944.00	−0.0669278
$73$	5018.00	0.110211	0.0551053	−	0.998481i	$-0.482451\pi$
$73$	5018.00	0.110211	0.0551053	+	0.998481i	$0.482451\pi$
$74$	−66072.0	−1.40261
$75$	0	0
$76$	−27488.0	−0.545895
$77$	−35035.0	−0.673403
$78$	22508.0	0.418890
$79$	−27977.0	−0.504352	−0.252176	−	0.967681i	$-0.581146\pi$
$79$	−27977.0	−0.504352	−0.252176	+	0.967681i	$0.581146\pi$
$80$	0	0
$81$	−68111.0	−1.15347
$82$	−75504.0	−1.24004
$83$	−37644.0	−0.599792	−0.299896	−	0.953972i	$-0.596952\pi$
$83$	−37644.0	−0.599792	−0.299896	+	0.953972i	$0.596952\pi$
$84$	13328.0	0.206095
$85$	0	0
$86$	9032.00	0.131685
$87$	77843.0	1.10261
$88$	45760.0	0.629911
$89$	−17216.0	−0.230387	−0.115193	−	0.993343i	$-0.536749\pi$
$89$	−17216.0	−0.230387	−0.115193	+	0.993343i	$0.536749\pi$
$90$	0	0
$91$	−16219.0	−0.205315
$92$	63200.0	0.778480
$93$	114852.	1.37699
$94$	−2148.00	−0.0250735
$95$	0	0
$96$	−17408.0	−0.192784
$97$	63175.0	0.681736	0.340868	−	0.940111i	$-0.389279\pi$
$97$	63175.0	0.681736	0.340868	+	0.940111i	$0.389279\pi$
$98$	−9604.00	−0.101015
$99$	−32890.0	−0.337269
$100$	0	0
$101$	−29250.0	−0.285314	−0.142657	−	0.989772i	$-0.545565\pi$
$101$	−29250.0	−0.285314	−0.142657	+	0.989772i	$0.545565\pi$
$102$	−115532.	−1.09952
$103$	149189.	1.38562	0.692809	−	0.721121i	$-0.256373\pi$
$103$	149189.	1.38562	0.692809	+	0.721121i	$0.256373\pi$
$104$	21184.0	0.192055
$105$	0	0
$106$	−43936.0	−0.379801
$107$	−83742.0	−0.707105	−0.353552	−	0.935415i	$-0.615026\pi$
$107$	−83742.0	−0.707105	−0.353552	+	0.935415i	$0.615026\pi$
$108$	−53584.0	−0.442054
$109$	105377.	0.849532	0.424766	−	0.905303i	$-0.360356\pi$
$109$	105377.	0.849532	0.424766	+	0.905303i	$0.360356\pi$
$110$	0	0
$111$	280806.	2.16321
$112$	12544.0	0.0944911
$113$	122754.	0.904356	0.452178	−	0.891928i	$-0.350647\pi$
$113$	122754.	0.904356	0.452178	+	0.891928i	$0.350647\pi$
$114$	116824.	0.841918
$115$	0	0
$116$	73264.0	0.505529
$117$	−15226.0	−0.102830
$118$	103824.	0.686424
$119$	83251.0	0.538917
$120$	0	0
$121$	350174.	2.17431
$122$	−156752.	−0.953484
$123$	320892.	1.91248
$124$	108096.	0.631329
$125$	0	0
$126$	−9016.00	−0.0505927
$127$	219196.	1.20593	0.602967	−	0.797766i	$-0.293985\pi$
$127$	219196.	1.20593	0.602967	+	0.797766i	$0.293985\pi$
$128$	−16384.0	−0.0883883
$129$	−38386.0	−0.203095
$130$	0	0
$131$	−96682.0	−0.492229	−0.246115	−	0.969241i	$-0.579154\pi$
$131$	−96682.0	−0.492229	−0.246115	+	0.969241i	$0.579154\pi$
$132$	−194480.	−0.971494
$133$	−84182.0	−0.412658
$134$	17664.0	0.0849820
$135$	0	0
$136$	−108736.	−0.504111
$137$	−187288.	−0.852528	−0.426264	−	0.904599i	$-0.640170\pi$
$137$	−187288.	−0.852528	−0.426264	+	0.904599i	$0.640170\pi$
$138$	−268600.	−1.20063
$139$	176894.	0.776562	0.388281	−	0.921541i	$-0.373069\pi$
$139$	176894.	0.776562	0.388281	+	0.921541i	$0.373069\pi$
$140$	0	0
$141$	9129.00	0.0386701
$142$	127520.	0.530710
$143$	236665.	0.967819
$144$	11776.0	0.0473251
$145$	0	0
$146$	−20072.0	−0.0779307
$147$	40817.0	0.155793
$148$	264288.	0.991798
$149$	−199078.	−0.734611	−0.367306	−	0.930100i	$-0.619720\pi$
$149$	−199078.	−0.734611	−0.367306	+	0.930100i	$0.619720\pi$
$150$	0	0
$151$	471583.	1.68312	0.841561	−	0.540162i	$-0.181637\pi$
$151$	471583.	1.68312	0.841561	+	0.540162i	$0.181637\pi$
$152$	109952.	0.386006
$153$	78154.0	0.269912
$154$	140140.	0.476168
$155$	0	0
$156$	−90032.0	−0.296200
$157$	72054.0	0.233297	0.116648	−	0.993173i	$-0.462785\pi$
$157$	72054.0	0.233297	0.116648	+	0.993173i	$0.462785\pi$
$158$	111908.	0.356630
$159$	186728.	0.585756
$160$	0	0
$161$	193550.	0.588476
$162$	272444.	0.815623
$163$	−385334.	−1.13597	−0.567987	−	0.823038i	$-0.692278\pi$
$163$	−385334.	−1.13597	−0.567987	+	0.823038i	$0.692278\pi$
$164$	302016.	0.876840
$165$	0	0
$166$	150576.	0.424117
$167$	542957.	1.50652	0.753259	−	0.657724i	$-0.228481\pi$
$167$	542957.	1.50652	0.753259	+	0.657724i	$0.228481\pi$
$168$	−53312.0	−0.145731
$169$	−261732.	−0.704920
$170$	0	0
$171$	−79028.0	−0.206676
$172$	−36128.0	−0.0931157
$173$	−370953.	−0.942331	−0.471166	−	0.882045i	$-0.656167\pi$
$173$	−370953.	−0.942331	−0.471166	+	0.882045i	$0.656167\pi$
$174$	−311372.	−0.779662
$175$	0	0
$176$	−183040.	−0.445414
$177$	−441252.	−1.05865
$178$	68864.0	0.162908
$179$	−754172.	−1.75929	−0.879646	−	0.475629i	$-0.842220\pi$
$179$	−754172.	−1.75929	−0.879646	+	0.475629i	$0.842220\pi$
$180$	0	0
$181$	303840.	0.689364	0.344682	−	0.938720i	$-0.387987\pi$
$181$	303840.	0.689364	0.344682	+	0.938720i	$0.387987\pi$
$182$	64876.0	0.145180
$183$	666196.	1.47053
$184$	−252800.	−0.550469
$185$	0	0
$186$	−459408.	−0.973680
$187$	−1.21478e6	−2.54036
$188$	8592.00	0.0177296
$189$	−164101.	−0.334162
$190$	0	0
$191$	−186271.	−0.369455	−0.184728	−	0.982790i	$-0.559140\pi$
$191$	−186271.	−0.369455	−0.184728	+	0.982790i	$0.559140\pi$
$192$	69632.0	0.136319
$193$	−92504.0	−0.178759	−0.0893794	−	0.995998i	$-0.528488\pi$
$193$	−92504.0	−0.178759	−0.0893794	+	0.995998i	$0.528488\pi$
$194$	−252700.	−0.482060
$195$	0	0
$196$	38416.0	0.0714286
$197$	736368.	1.35185	0.675926	−	0.736969i	$-0.263744\pi$
$197$	736368.	1.35185	0.675926	+	0.736969i	$0.263744\pi$
$198$	131560.	0.238485
$199$	−481620.	−0.862128	−0.431064	−	0.902321i	$-0.641862\pi$
$199$	−481620.	−0.862128	−0.431064	+	0.902321i	$0.641862\pi$
$200$	0	0
$201$	−75072.0	−0.131065
$202$	117000.	0.201747
$203$	224371.	0.382144
$204$	462128.	0.777476
$205$	0	0
$206$	−596756.	−0.979780
$207$	181700.	0.294733
$208$	−84736.0	−0.135803
$209$	1.22837e6	1.94520
$210$	0	0
$211$	189531.	0.293072	0.146536	−	0.989205i	$-0.453188\pi$
$211$	189531.	0.293072	0.146536	+	0.989205i	$0.453188\pi$
$212$	175744.	0.268560
$213$	−541960.	−0.818499
$214$	334968.	0.499999
$215$	0	0
$216$	214336.	0.312580
$217$	331044.	0.477240
$218$	−421508.	−0.600710
$219$	85306.0	0.120190
$220$	0	0
$221$	−562369.	−0.774534
$222$	−1.12322e6	−1.52962
$223$	22597.0	0.0304291	0.0152145	−	0.999884i	$-0.495157\pi$
$223$	22597.0	0.0304291	0.0152145	+	0.999884i	$0.495157\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	−491016.	−0.639476
$227$	998117.	1.28563	0.642816	−	0.766020i	$-0.277766\pi$
$227$	998117.	1.28563	0.642816	+	0.766020i	$0.277766\pi$
$228$	−467296.	−0.595326
$229$	−854644.	−1.07695	−0.538476	−	0.842641i	$-0.681000\pi$
$229$	−854644.	−1.07695	−0.538476	+	0.842641i	$0.681000\pi$
$230$	0	0
$231$	−595595.	−0.734380
$232$	−293056.	−0.357463
$233$	−1.25818e6	−1.51829	−0.759144	−	0.650922i	$-0.774382\pi$
$233$	−1.25818e6	−1.51829	−0.759144	+	0.650922i	$0.774382\pi$
$234$	60904.0	0.0727120
$235$	0	0
$236$	−415296.	−0.485375
$237$	−475609.	−0.550021
$238$	−333004.	−0.381072
$239$	−706581.	−0.800142	−0.400071	−	0.916484i	$-0.631015\pi$
$239$	−706581.	−0.800142	−0.400071	+	0.916484i	$0.631015\pi$
$240$	0	0
$241$	616330.	0.683551	0.341775	−	0.939782i	$-0.388972\pi$
$241$	616330.	0.683551	0.341775	+	0.939782i	$0.388972\pi$
$242$	−1.40070e6	−1.53747
$243$	−344080.	−0.373804
$244$	627008.	0.674215
$245$	0	0
$246$	−1.28357e6	−1.35233
$247$	568658.	0.593074
$248$	−432384.	−0.446417
$249$	−639948.	−0.654103
$250$	0	0
$251$	190842.	0.191201	0.0956004	−	0.995420i	$-0.469523\pi$
$251$	190842.	0.191201	0.0956004	+	0.995420i	$0.469523\pi$
$252$	36064.0	0.0357744
$253$	−2.82425e6	−2.77397
$254$	−876784.	−0.852724
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.13094e6	1.06809	0.534045	−	0.845456i	$-0.320671\pi$
$257$	1.13094e6	1.06809	0.534045	+	0.845456i	$0.320671\pi$
$258$	153544.	0.143610
$259$	809382.	0.749729
$260$	0	0
$261$	210634.	0.191394
$262$	386728.	0.348059
$263$	1.67377e6	1.49213	0.746065	−	0.665874i	$-0.231941\pi$
$263$	1.67377e6	1.49213	0.746065	+	0.665874i	$0.231941\pi$
$264$	777920.	0.686950
$265$	0	0
$266$	336728.	0.291793
$267$	−292672.	−0.251248
$268$	−70656.0	−0.0600914
$269$	−630942.	−0.531629	−0.265815	−	0.964024i	$-0.585641\pi$
$269$	−630942.	−0.531629	−0.265815	+	0.964024i	$0.585641\pi$
$270$	0	0
$271$	−372476.	−0.308088	−0.154044	−	0.988064i	$-0.549230\pi$
$271$	−372476.	−0.308088	−0.154044	+	0.988064i	$0.549230\pi$
$272$	434944.	0.356460
$273$	−275723.	−0.223906
$274$	749152.	0.602828
$275$	0	0
$276$	1.07440e6	0.848972
$277$	−867010.	−0.678930	−0.339465	−	0.940619i	$-0.610246\pi$
$277$	−867010.	−0.678930	−0.339465	+	0.940619i	$0.610246\pi$
$278$	−707576.	−0.549112
$279$	310776.	0.239021
$280$	0	0
$281$	−1.94498e6	−1.46943	−0.734716	−	0.678375i	$-0.762685\pi$
$281$	−1.94498e6	−1.46943	−0.734716	+	0.678375i	$0.762685\pi$
$282$	−36516.0	−0.0273439
$283$	−1.18501e6	−0.879543	−0.439771	−	0.898110i	$-0.644941\pi$
$283$	−1.18501e6	−0.879543	−0.439771	+	0.898110i	$0.644941\pi$
$284$	−510080.	−0.375269
$285$	0	0
$286$	−946660.	−0.684351
$287$	924924.	0.662829
$288$	−47104.0	−0.0334639
$289$	1.46674e6	1.03302
$290$	0	0
$291$	1.07398e6	0.743467
$292$	80288.0	0.0551053
$293$	−33669.0	−0.0229119	−0.0114560	−	0.999934i	$-0.503647\pi$
$293$	−33669.0	−0.0229119	−0.0114560	+	0.999934i	$0.503647\pi$
$294$	−163268.	−0.110162
$295$	0	0
$296$	−1.05715e6	−0.701307
$297$	2.39454e6	1.57518
$298$	796312.	0.519449
$299$	−1.30745e6	−0.845760
$300$	0	0
$301$	−110642.	−0.0703888
$302$	−1.88633e6	−1.19015
$303$	−497250.	−0.311149
$304$	−439808.	−0.272948
$305$	0	0
$306$	−312616.	−0.190857
$307$	27043.0	0.0163760	0.00818802	−	0.999966i	$-0.497394\pi$
$307$	27043.0	0.0163760	0.00818802	+	0.999966i	$0.497394\pi$
$308$	−560560.	−0.336702
$309$	2.53621e6	1.51109
$310$	0	0
$311$	2.14919e6	1.26001	0.630004	−	0.776592i	$-0.283053\pi$
$311$	2.14919e6	1.26001	0.630004	+	0.776592i	$0.283053\pi$
$312$	360128.	0.209445
$313$	2.67052e6	1.54076	0.770381	−	0.637583i	$-0.220066\pi$
$313$	2.67052e6	1.54076	0.770381	+	0.637583i	$0.220066\pi$
$314$	−288216.	−0.164966
$315$	0	0
$316$	−447632.	−0.252176
$317$	250514.	0.140018	0.0700090	−	0.997546i	$-0.477697\pi$
$317$	250514.	0.140018	0.0700090	+	0.997546i	$0.477697\pi$
$318$	−746912.	−0.414192
$319$	−3.27398e6	−1.80136
$320$	0	0
$321$	−1.42361e6	−0.771134
$322$	−774200.	−0.416115
$323$	−2.91888e6	−1.55672
$324$	−1.08978e6	−0.576733
$325$	0	0
$326$	1.54134e6	0.803255
$327$	1.79141e6	0.926457
$328$	−1.20806e6	−0.620019
$329$	26313.0	0.0134023
$330$	0	0
$331$	1.05899e6	0.531277	0.265639	−	0.964073i	$-0.414417\pi$
$331$	1.05899e6	0.531277	0.265639	+	0.964073i	$0.414417\pi$
$332$	−602304.	−0.299896
$333$	759828.	0.375495
$334$	−2.17183e6	−1.06527
$335$	0	0
$336$	213248.	0.103047
$337$	2.85025e6	1.36712	0.683562	−	0.729893i	$-0.260430\pi$
$337$	2.85025e6	1.36712	0.683562	+	0.729893i	$0.260430\pi$
$338$	1.04693e6	0.498454
$339$	2.08682e6	0.986246
$340$	0	0
$341$	−4.83054e6	−2.24962
$342$	316112.	0.146142
$343$	117649.	0.0539949
$344$	144512.	0.0658427
$345$	0	0
$346$	1.48381e6	0.666329
$347$	−1.89141e6	−0.843259	−0.421630	−	0.906768i	$-0.638542\pi$
$347$	−1.89141e6	−0.843259	−0.421630	+	0.906768i	$0.638542\pi$
$348$	1.24549e6	0.551304
$349$	−1.04232e6	−0.458075	−0.229038	−	0.973418i	$-0.573558\pi$
$349$	−1.04232e6	−0.458075	−0.229038	+	0.973418i	$0.573558\pi$
$350$	0	0
$351$	1.10852e6	0.480259
$352$	732160.	0.314956
$353$	2.30309e6	0.983725	0.491862	−	0.870673i	$-0.336316\pi$
$353$	2.30309e6	0.983725	0.491862	+	0.870673i	$0.336316\pi$
$354$	1.76501e6	0.748581
$355$	0	0
$356$	−275456.	−0.115193
$357$	1.41527e6	0.587716
$358$	3.01669e6	1.24401
$359$	−1.67594e6	−0.686315	−0.343157	−	0.939278i	$-0.611496\pi$
$359$	−1.67594e6	−0.686315	−0.343157	+	0.939278i	$0.611496\pi$
$360$	0	0
$361$	475425.	0.192006
$362$	−1.21536e6	−0.487454
$363$	5.95296e6	2.37119
$364$	−259504.	−0.102657
$365$	0	0
$366$	−2.66478e6	−1.03982
$367$	−94663.0	−0.0366872	−0.0183436	−	0.999832i	$-0.505839\pi$
$367$	−94663.0	−0.0366872	−0.0183436	+	0.999832i	$0.505839\pi$
$368$	1.01120e6	0.389240
$369$	868296.	0.331972
$370$	0	0
$371$	538216.	0.203012
$372$	1.83763e6	0.688496
$373$	953536.	0.354867	0.177433	−	0.984133i	$-0.443221\pi$
$373$	953536.	0.354867	0.177433	+	0.984133i	$0.443221\pi$
$374$	4.85914e6	1.79631
$375$	0	0
$376$	−34368.0	−0.0125367
$377$	−1.51565e6	−0.549219
$378$	656404.	0.236288
$379$	3.88824e6	1.39045	0.695225	−	0.718792i	$-0.255305\pi$
$379$	3.88824e6	1.39045	0.695225	+	0.718792i	$0.255305\pi$
$380$	0	0
$381$	3.72633e6	1.31513
$382$	745084.	0.261244
$383$	−2.93636e6	−1.02285	−0.511425	−	0.859328i	$-0.670882\pi$
$383$	−2.93636e6	−1.02285	−0.511425	+	0.859328i	$0.670882\pi$
$384$	−278528.	−0.0963920
$385$	0	0
$386$	370016.	0.126402
$387$	−103868.	−0.0352537
$388$	1.01080e6	0.340868
$389$	1.70377e6	0.570871	0.285435	−	0.958398i	$-0.407862\pi$
$389$	1.70377e6	0.570871	0.285435	+	0.958398i	$0.407862\pi$
$390$	0	0
$391$	6.71105e6	2.21998
$392$	−153664.	−0.0505076
$393$	−1.64359e6	−0.536801
$394$	−2.94547e6	−0.955904
$395$	0	0
$396$	−526240.	−0.168634
$397$	−1.19110e6	−0.379292	−0.189646	−	0.981853i	$-0.560734\pi$
$397$	−1.19110e6	−0.379292	−0.189646	+	0.981853i	$0.560734\pi$
$398$	1.92648e6	0.609617
$399$	−1.43109e6	−0.450024
$400$	0	0
$401$	−3.38330e6	−1.05070	−0.525351	−	0.850885i	$-0.676066\pi$
$401$	−3.38330e6	−1.05070	−0.525351	+	0.850885i	$0.676066\pi$
$402$	300288.	0.0926772
$403$	−2.23624e6	−0.685891
$404$	−468000.	−0.142657
$405$	0	0
$406$	−897484.	−0.270216
$407$	−1.18104e7	−3.53409
$408$	−1.84851e6	−0.549758
$409$	−1.33185e6	−0.393682	−0.196841	−	0.980435i	$-0.563068\pi$
$409$	−1.33185e6	−0.393682	−0.196841	+	0.980435i	$0.563068\pi$
$410$	0	0
$411$	−3.18390e6	−0.929725
$412$	2.38702e6	0.692809
$413$	−1.27184e6	−0.366909
$414$	−726800.	−0.208408
$415$	0	0
$416$	338944.	0.0960273
$417$	3.00720e6	0.846880
$418$	−4.91348e6	−1.37546
$419$	5.82786e6	1.62171	0.810856	−	0.585246i	$-0.199002\pi$
$419$	5.82786e6	1.62171	0.810856	+	0.585246i	$0.199002\pi$
$420$	0	0
$421$	2.47430e6	0.680374	0.340187	−	0.940358i	$-0.389510\pi$
$421$	2.47430e6	0.680374	0.340187	+	0.940358i	$0.389510\pi$
$422$	−758124.	−0.207233
$423$	24702.0	0.00671245
$424$	−702976.	−0.189900
$425$	0	0
$426$	2.16784e6	0.578766
$427$	1.92021e6	0.509659
$428$	−1.33987e6	−0.353552
$429$	4.02331e6	1.05546
$430$	0	0
$431$	4.61851e6	1.19759	0.598796	−	0.800902i	$-0.295646\pi$
$431$	4.61851e6	1.19759	0.598796	+	0.800902i	$0.295646\pi$
$432$	−857344.	−0.221027
$433$	−58606.0	−0.0150218	−0.00751091	−	0.999972i	$-0.502391\pi$
$433$	−58606.0	−0.0150218	−0.00751091	+	0.999972i	$0.502391\pi$
$434$	−1.32418e6	−0.337459
$435$	0	0
$436$	1.68603e6	0.424766
$437$	−6.78610e6	−1.69987
$438$	−341224.	−0.0849874
$439$	7.04298e6	1.74419	0.872097	−	0.489332i	$-0.162759\pi$
$439$	7.04298e6	1.74419	0.872097	+	0.489332i	$0.162759\pi$
$440$	0	0
$441$	110446.	0.0270429
$442$	2.24948e6	0.547678
$443$	−1.46894e6	−0.355627	−0.177813	−	0.984064i	$-0.556902\pi$
$443$	−1.46894e6	−0.355627	−0.177813	+	0.984064i	$0.556902\pi$
$444$	4.49290e6	1.08161
$445$	0	0
$446$	−90388.0	−0.0215166
$447$	−3.38433e6	−0.801131
$448$	200704.	0.0472456
$449$	−7.48414e6	−1.75197	−0.875983	−	0.482341i	$-0.839787\pi$
$449$	−7.48414e6	−1.75197	−0.875983	+	0.482341i	$0.839787\pi$
$450$	0	0
$451$	−1.34963e7	−3.12446
$452$	1.96406e6	0.452178
$453$	8.01691e6	1.83553
$454$	−3.99247e6	−0.909079
$455$	0	0
$456$	1.86918e6	0.420959
$457$	−170320.	−0.0381483	−0.0190741	−	0.999818i	$-0.506072\pi$
$457$	−170320.	−0.0381483	−0.0190741	+	0.999818i	$0.506072\pi$
$458$	3.41858e6	0.761520
$459$	−5.68995e6	−1.26060
$460$	0	0
$461$	−4.28685e6	−0.939476	−0.469738	−	0.882806i	$-0.655652\pi$
$461$	−4.28685e6	−0.939476	−0.469738	+	0.882806i	$0.655652\pi$
$462$	2.38238e6	0.519285
$463$	−3.38317e6	−0.733452	−0.366726	−	0.930329i	$-0.619521\pi$
$463$	−3.38317e6	−0.733452	−0.366726	+	0.930329i	$0.619521\pi$
$464$	1.17222e6	0.252764
$465$	0	0
$466$	5.03274e6	1.07359
$467$	−5.18029e6	−1.09916	−0.549581	−	0.835440i	$-0.685213\pi$
$467$	−5.18029e6	−1.09916	−0.549581	+	0.835440i	$0.685213\pi$
$468$	−243616.	−0.0514152
$469$	−216384.	−0.0454248
$470$	0	0
$471$	1.22492e6	0.254422
$472$	1.66118e6	0.343212
$473$	1.61447e6	0.331801
$474$	1.90244e6	0.388924
$475$	0	0
$476$	1.33202e6	0.269459
$477$	505264.	0.101677
$478$	2.82632e6	0.565786
$479$	−8.76779e6	−1.74603	−0.873014	−	0.487695i	$-0.837838\pi$
$479$	−8.76779e6	−1.74603	−0.873014	+	0.487695i	$0.837838\pi$
$480$	0	0
$481$	−5.46746e6	−1.07751
$482$	−2.46532e6	−0.483343
$483$	3.29035e6	0.641762
$484$	5.60278e6	1.08715
$485$	0	0
$486$	1.37632e6	0.264319
$487$	−270154.	−0.0516166	−0.0258083	−	0.999667i	$-0.508216\pi$
$487$	−270154.	−0.0516166	−0.0258083	+	0.999667i	$0.508216\pi$
$488$	−2.50803e6	−0.476742
$489$	−6.55068e6	−1.23884
$490$	0	0
$491$	4.85550e6	0.908930	0.454465	−	0.890765i	$-0.349830\pi$
$491$	4.85550e6	0.908930	0.454465	+	0.890765i	$0.349830\pi$
$492$	5.13427e6	0.956238
$493$	7.77972e6	1.44161
$494$	−2.27463e6	−0.419367
$495$	0	0
$496$	1.72954e6	0.315664
$497$	−1.56212e6	−0.283677
$498$	2.55979e6	0.462521
$499$	−2.98576e6	−0.536789	−0.268394	−	0.963309i	$-0.586493\pi$
$499$	−2.98576e6	−0.536789	−0.268394	+	0.963309i	$0.586493\pi$
$500$	0	0
$501$	9.23027e6	1.64293
$502$	−763368.	−0.135199
$503$	8.28783e6	1.46057	0.730283	−	0.683145i	$-0.239388\pi$
$503$	8.28783e6	1.46057	0.730283	+	0.683145i	$0.239388\pi$
$504$	−144256.	−0.0252963
$505$	0	0
$506$	1.12970e7	1.96149
$507$	−4.44944e6	−0.768751
$508$	3.50714e6	0.602967
$509$	6.24307e6	1.06808	0.534040	−	0.845459i	$-0.320673\pi$
$509$	6.24307e6	1.06808	0.534040	+	0.845459i	$0.320673\pi$
$510$	0	0
$511$	245882.	0.0416557
$512$	−262144.	−0.0441942
$513$	5.75358e6	0.965261
$514$	−4.52377e6	−0.755253
$515$	0	0
$516$	−614176.	−0.101547
$517$	−383955.	−0.0631763
$518$	−3.23753e6	−0.530138
$519$	−6.30620e6	−1.02766
$520$	0	0
$521$	7.49509e6	1.20971	0.604856	−	0.796335i	$-0.293230\pi$
$521$	7.49509e6	1.20971	0.604856	+	0.796335i	$0.293230\pi$
$522$	−842536.	−0.135336
$523$	−3.80957e6	−0.609007	−0.304503	−	0.952511i	$-0.598490\pi$
$523$	−3.80957e6	−0.609007	−0.304503	+	0.952511i	$0.598490\pi$
$524$	−1.54691e6	−0.246115
$525$	0	0
$526$	−6.69508e6	−1.05509
$527$	1.14784e7	1.80035
$528$	−3.11168e6	−0.485747
$529$	9.16616e6	1.42413
$530$	0	0
$531$	−1.19398e6	−0.183764
$532$	−1.34691e6	−0.206329
$533$	−6.24796e6	−0.952621
$534$	1.17069e6	0.177659
$535$	0	0
$536$	282624.	0.0424910
$537$	−1.28209e7	−1.91860
$538$	2.52377e6	0.375919
$539$	−1.71672e6	−0.254523
$540$	0	0
$541$	7.67156e6	1.12691	0.563457	−	0.826145i	$-0.309471\pi$
$541$	7.67156e6	1.12691	0.563457	+	0.826145i	$0.309471\pi$
$542$	1.48990e6	0.217851
$543$	5.16528e6	0.751786
$544$	−1.73978e6	−0.252055
$545$	0	0
$546$	1.10289e6	0.158326
$547$	9.53845e6	1.36304	0.681522	−	0.731798i	$-0.261319\pi$
$547$	9.53845e6	1.36304	0.681522	+	0.731798i	$0.261319\pi$
$548$	−2.99661e6	−0.426264
$549$	1.80265e6	0.255258
$550$	0	0
$551$	−7.86672e6	−1.10386
$552$	−4.29760e6	−0.600314
$553$	−1.37087e6	−0.190627
$554$	3.46804e6	0.480076
$555$	0	0
$556$	2.83030e6	0.388281
$557$	7.45022e6	1.01749	0.508746	−	0.860916i	$-0.330109\pi$
$557$	7.45022e6	1.01749	0.508746	+	0.860916i	$0.330109\pi$
$558$	−1.24310e6	−0.169014
$559$	747398.	0.101163
$560$	0	0
$561$	−2.06513e7	−2.77039
$562$	7.77992e6	1.03905
$563$	−3.36698e6	−0.447682	−0.223841	−	0.974626i	$-0.571860\pi$
$563$	−3.36698e6	−0.447682	−0.223841	+	0.974626i	$0.571860\pi$
$564$	146064.	0.0193351
$565$	0	0
$566$	4.74005e6	0.621931
$567$	−3.33744e6	−0.435969
$568$	2.04032e6	0.265355
$569$	−4.05501e6	−0.525063	−0.262532	−	0.964923i	$-0.584557\pi$
$569$	−4.05501e6	−0.525063	−0.262532	+	0.964923i	$0.584557\pi$
$570$	0	0
$571$	7.31585e6	0.939020	0.469510	−	0.882927i	$-0.344431\pi$
$571$	7.31585e6	0.939020	0.469510	+	0.882927i	$0.344431\pi$
$572$	3.78664e6	0.483909
$573$	−3.16661e6	−0.402910
$574$	−3.69970e6	−0.468691
$575$	0	0
$576$	188416.	0.0236626
$577$	9.76895e6	1.22154	0.610771	−	0.791807i	$-0.290860\pi$
$577$	9.76895e6	1.22154	0.610771	+	0.791807i	$0.290860\pi$
$578$	−5.86698e6	−0.730457
$579$	−1.57257e6	−0.194945
$580$	0	0
$581$	−1.84456e6	−0.226700
$582$	−4.29590e6	−0.525711
$583$	−7.85356e6	−0.956963
$584$	−321152.	−0.0389653
$585$	0	0
$586$	134676.	0.0162012
$587$	−3.75689e6	−0.450021	−0.225011	−	0.974356i	$-0.572242\pi$
$587$	−3.75689e6	−0.450021	−0.225011	+	0.974356i	$0.572242\pi$
$588$	653072.	0.0778965
$589$	−1.16068e7	−1.37856
$590$	0	0
$591$	1.25183e7	1.47426
$592$	4.22861e6	0.495899
$593$	−2.89048e6	−0.337546	−0.168773	−	0.985655i	$-0.553980\pi$
$593$	−2.89048e6	−0.337546	−0.168773	+	0.985655i	$0.553980\pi$
$594$	−9.57814e6	−1.11382
$595$	0	0
$596$	−3.18525e6	−0.367306
$597$	−8.18754e6	−0.940194
$598$	5.22980e6	0.598043
$599$	1.32233e7	1.50582	0.752910	−	0.658124i	$-0.228650\pi$
$599$	1.32233e7	1.50582	0.752910	+	0.658124i	$0.228650\pi$
$600$	0	0
$601$	3.47399e6	0.392321	0.196161	−	0.980572i	$-0.437153\pi$
$601$	3.47399e6	0.392321	0.196161	+	0.980572i	$0.437153\pi$
$602$	442568.	0.0497724
$603$	−203136.	−0.0227506
$604$	7.54533e6	0.841561
$605$	0	0
$606$	1.98900e6	0.220015
$607$	−6.45088e6	−0.710636	−0.355318	−	0.934746i	$-0.615627\pi$
$607$	−6.45088e6	−0.710636	−0.355318	+	0.934746i	$0.615627\pi$
$608$	1.75923e6	0.193003
$609$	3.81431e6	0.416747
$610$	0	0
$611$	−177747.	−0.0192619
$612$	1.25046e6	0.134956
$613$	−8.43820e6	−0.906982	−0.453491	−	0.891261i	$-0.649822\pi$
$613$	−8.43820e6	−0.906982	−0.453491	+	0.891261i	$0.649822\pi$
$614$	−108172.	−0.0115796
$615$	0	0
$616$	2.24224e6	0.238084
$617$	−9.45501e6	−0.999882	−0.499941	−	0.866059i	$-0.666645\pi$
$617$	−9.45501e6	−0.999882	−0.499941	+	0.866059i	$0.666645\pi$
$618$	−1.01449e7	−1.06850
$619$	1.43145e6	0.150158	0.0750790	−	0.997178i	$-0.476079\pi$
$619$	1.43145e6	0.150158	0.0750790	+	0.997178i	$0.476079\pi$
$620$	0	0
$621$	−1.32286e7	−1.37652
$622$	−8.59674e6	−0.890960
$623$	−843584.	−0.0870780
$624$	−1.44051e6	−0.148100
$625$	0	0
$626$	−1.06821e7	−1.08948
$627$	2.08823e7	2.12134
$628$	1.15286e6	0.116648
$629$	2.80641e7	2.82829
$630$	0	0
$631$	−1.01813e7	−1.01795	−0.508977	−	0.860780i	$-0.669976\pi$
$631$	−1.01813e7	−1.01795	−0.508977	+	0.860780i	$0.669976\pi$
$632$	1.79053e6	0.178315
$633$	3.22203e6	0.319610
$634$	−1.00206e6	−0.0990077
$635$	0	0
$636$	2.98765e6	0.292878
$637$	−794731.	−0.0776018
$638$	1.30959e7	1.27375
$639$	−1.46648e6	−0.142077
$640$	0	0
$641$	−1.76908e7	−1.70060	−0.850300	−	0.526298i	$-0.823580\pi$
$641$	−1.76908e7	−1.70060	−0.850300	+	0.526298i	$0.823580\pi$
$642$	5.69446e6	0.545274
$643$	−1.82748e7	−1.74311	−0.871556	−	0.490296i	$-0.836889\pi$
$643$	−1.82748e7	−1.74311	−0.871556	+	0.490296i	$0.836889\pi$
$644$	3.09680e6	0.294238
$645$	0	0
$646$	1.16755e7	1.10077
$647$	−1.52897e6	−0.143594	−0.0717972	−	0.997419i	$-0.522873\pi$
$647$	−1.52897e6	−0.143594	−0.0717972	+	0.997419i	$0.522873\pi$
$648$	4.35910e6	0.407812
$649$	1.85585e7	1.72955
$650$	0	0
$651$	5.62775e6	0.520454
$652$	−6.16534e6	−0.567987
$653$	9.10088e6	0.835219	0.417610	−	0.908627i	$-0.362868\pi$
$653$	9.10088e6	0.835219	0.417610	+	0.908627i	$0.362868\pi$
$654$	−7.16564e6	−0.655104
$655$	0	0
$656$	4.83226e6	0.438420
$657$	230828.	0.0208629
$658$	−105252.	−0.00947689
$659$	−430119.	−0.0385811	−0.0192906	−	0.999814i	$-0.506141\pi$
$659$	−430119.	−0.0385811	−0.0192906	+	0.999814i	$0.506141\pi$
$660$	0	0
$661$	7.65248e6	0.681238	0.340619	−	0.940202i	$-0.389363\pi$
$661$	7.65248e6	0.681238	0.340619	+	0.940202i	$0.389363\pi$
$662$	−4.23595e6	−0.375670
$663$	−9.56027e6	−0.844669
$664$	2.40922e6	0.212058
$665$	0	0
$666$	−3.03931e6	−0.265515
$667$	1.80870e7	1.57418
$668$	8.68731e6	0.753259
$669$	384149.	0.0331844
$670$	0	0
$671$	−2.80194e7	−2.40244
$672$	−852992.	−0.0728655
$673$	2.18404e7	1.85876	0.929378	−	0.369128i	$-0.120344\pi$
$673$	2.18404e7	1.85876	0.929378	+	0.369128i	$0.120344\pi$
$674$	−1.14010e7	−0.966702
$675$	0	0
$676$	−4.18771e6	−0.352460
$677$	−1.39504e7	−1.16981	−0.584905	−	0.811102i	$-0.698868\pi$
$677$	−1.39504e7	−1.16981	−0.584905	+	0.811102i	$0.698868\pi$
$678$	−8.34727e6	−0.697381
$679$	3.09558e6	0.257672
$680$	0	0
$681$	1.69680e7	1.40205
$682$	1.93222e7	1.59072
$683$	−2.29121e7	−1.87937	−0.939686	−	0.342040i	$-0.888882\pi$
$683$	−2.29121e7	−1.87937	−0.939686	+	0.342040i	$0.888882\pi$
$684$	−1.26445e6	−0.103338
$685$	0	0
$686$	−470596.	−0.0381802
$687$	−1.45289e7	−1.17447
$688$	−578048.	−0.0465578
$689$	−3.63570e6	−0.291770
$690$	0	0
$691$	−1.69127e7	−1.34747	−0.673734	−	0.738974i	$-0.735310\pi$
$691$	−1.69127e7	−1.34747	−0.673734	+	0.738974i	$0.735310\pi$
$692$	−5.93525e6	−0.471166
$693$	−1.61161e6	−0.127476
$694$	7.56562e6	0.596274
$695$	0	0
$696$	−4.98195e6	−0.389831
$697$	3.20703e7	2.50047
$698$	4.16927e6	0.323908
$699$	−2.13891e7	−1.65577
$700$	0	0
$701$	−1.90087e7	−1.46102	−0.730510	−	0.682902i	$-0.760718\pi$
$701$	−1.90087e7	−1.46102	−0.730510	+	0.682902i	$0.760718\pi$
$702$	−4.43408e6	−0.339594
$703$	−2.83779e7	−2.16567
$704$	−2.92864e6	−0.222707
$705$	0	0
$706$	−9.21235e6	−0.695598
$707$	−1.43325e6	−0.107838
$708$	−7.06003e6	−0.529326
$709$	1.66079e7	1.24079	0.620396	−	0.784289i	$-0.286972\pi$
$709$	1.66079e7	1.24079	0.620396	+	0.784289i	$0.286972\pi$
$710$	0	0
$711$	−1.28694e6	−0.0954740
$712$	1.10182e6	0.0814540
$713$	2.66862e7	1.96591
$714$	−5.66107e6	−0.415578
$715$	0	0
$716$	−1.20668e7	−0.879646
$717$	−1.20119e7	−0.872596
$718$	6.70378e6	0.485298
$719$	5.93610e6	0.428232	0.214116	−	0.976808i	$-0.431313\pi$
$719$	5.93610e6	0.428232	0.214116	+	0.976808i	$0.431313\pi$
$720$	0	0
$721$	7.31026e6	0.523715
$722$	−1.90170e6	−0.135768
$723$	1.04776e7	0.745447
$724$	4.86144e6	0.344682
$725$	0	0
$726$	−2.38118e7	−1.67668
$727$	−1.73276e7	−1.21591	−0.607957	−	0.793970i	$-0.708011\pi$
$727$	−1.73276e7	−1.21591	−0.607957	+	0.793970i	$0.708011\pi$
$728$	1.03802e6	0.0725898
$729$	1.07016e7	0.745814
$730$	0	0
$731$	−3.83634e6	−0.265536
$732$	1.06591e7	0.735266
$733$	−1.39829e7	−0.961255	−0.480627	−	0.876925i	$-0.659591\pi$
$733$	−1.39829e7	−0.961255	−0.480627	+	0.876925i	$0.659591\pi$
$734$	378652.	0.0259418
$735$	0	0
$736$	−4.04480e6	−0.275234
$737$	3.15744e6	0.214125
$738$	−3.47318e6	−0.234740
$739$	−1.14263e7	−0.769649	−0.384824	−	0.922990i	$-0.625738\pi$
$739$	−1.14263e7	−0.769649	−0.384824	+	0.922990i	$0.625738\pi$
$740$	0	0
$741$	9.66719e6	0.646777
$742$	−2.15286e6	−0.143551
$743$	−1.23126e7	−0.818236	−0.409118	−	0.912481i	$-0.634164\pi$
$743$	−1.23126e7	−0.818236	−0.409118	+	0.912481i	$0.634164\pi$
$744$	−7.35053e6	−0.486840
$745$	0	0
$746$	−3.81414e6	−0.250929
$747$	−1.73162e6	−0.113541
$748$	−1.94366e7	−1.27018
$749$	−4.10336e6	−0.267261
$750$	0	0
$751$	−1.43093e7	−0.925806	−0.462903	−	0.886409i	$-0.653192\pi$
$751$	−1.43093e7	−0.925806	−0.462903	+	0.886409i	$0.653192\pi$
$752$	137472.	0.00886481
$753$	3.24431e6	0.208514
$754$	6.06260e6	0.388356
$755$	0	0
$756$	−2.62562e6	−0.167081
$757$	5.34505e6	0.339010	0.169505	−	0.985529i	$-0.445783\pi$
$757$	5.34505e6	0.339010	0.169505	+	0.985529i	$0.445783\pi$
$758$	−1.55530e7	−0.983197
$759$	−4.80122e7	−3.02515
$760$	0	0
$761$	6.22568e6	0.389695	0.194848	−	0.980834i	$-0.437579\pi$
$761$	6.22568e6	0.389695	0.194848	+	0.980834i	$0.437579\pi$
$762$	−1.49053e7	−0.929938
$763$	5.16347e6	0.321093
$764$	−2.98034e6	−0.184728
$765$	0	0
$766$	1.17454e7	0.723264
$767$	8.59144e6	0.527324
$768$	1.11411e6	0.0681594
$769$	1.57888e7	0.962793	0.481397	−	0.876503i	$-0.340130\pi$
$769$	1.57888e7	0.962793	0.481397	+	0.876503i	$0.340130\pi$
$770$	0	0
$771$	1.92260e7	1.16481
$772$	−1.48006e6	−0.0893794
$773$	2.50453e7	1.50757	0.753785	−	0.657121i	$-0.228226\pi$
$773$	2.50453e7	1.50757	0.753785	+	0.657121i	$0.228226\pi$
$774$	415472.	0.0249281
$775$	0	0
$776$	−4.04320e6	−0.241030
$777$	1.37595e7	0.817617
$778$	−6.81509e6	−0.403667
$779$	−3.24290e7	−1.91465
$780$	0	0
$781$	2.27942e7	1.33720
$782$	−2.68442e7	−1.56976
$783$	−1.53351e7	−0.893884
$784$	614656.	0.0357143
$785$	0	0
$786$	6.57438e6	0.379576
$787$	1.28020e6	0.0736784	0.0368392	−	0.999321i	$-0.488271\pi$
$787$	1.28020e6	0.0736784	0.0368392	+	0.999321i	$0.488271\pi$
$788$	1.17819e7	0.675926
$789$	2.84541e7	1.62724
$790$	0	0
$791$	6.01495e6	0.341815
$792$	2.10496e6	0.119242
$793$	−1.29712e7	−0.732484
$794$	4.76442e6	0.268200
$795$	0	0
$796$	−7.70592e6	−0.431064
$797$	1.13798e7	0.634584	0.317292	−	0.948328i	$-0.397226\pi$
$797$	1.13798e7	0.634584	0.317292	+	0.948328i	$0.397226\pi$
$798$	5.72438e6	0.318215
$799$	912363.	0.0505593
$800$	0	0
$801$	−791936.	−0.0436123
$802$	1.35332e7	0.742959
$803$	−3.58787e6	−0.196358
$804$	−1.20115e6	−0.0655327
$805$	0	0
$806$	8.94494e6	0.484998
$807$	−1.07260e7	−0.579768
$808$	1.87200e6	0.100874
$809$	−1.70542e7	−0.916138	−0.458069	−	0.888917i	$-0.651459\pi$
$809$	−1.70542e7	−0.916138	−0.458069	+	0.888917i	$0.651459\pi$
$810$	0	0
$811$	−2.21494e7	−1.18252	−0.591262	−	0.806480i	$-0.701370\pi$
$811$	−2.21494e7	−1.18252	−0.591262	+	0.806480i	$0.701370\pi$
$812$	3.58994e6	0.191072
$813$	−6.33209e6	−0.335986
$814$	4.72415e7	2.49898
$815$	0	0
$816$	7.39405e6	0.388738
$817$	3.87924e6	0.203326
$818$	5.32738e6	0.278375
$819$	−746074.	−0.0388662
$820$	0	0
$821$	−1.01068e7	−0.523307	−0.261654	−	0.965162i	$-0.584268\pi$
$821$	−1.01068e7	−0.523307	−0.261654	+	0.965162i	$0.584268\pi$
$822$	1.27356e7	0.657415
$823$	1.83993e7	0.946895	0.473447	−	0.880822i	$-0.343009\pi$
$823$	1.83993e7	0.946895	0.473447	+	0.880822i	$0.343009\pi$
$824$	−9.54810e6	−0.489890
$825$	0	0
$826$	5.08738e6	0.259444
$827$	2.48056e7	1.26121	0.630604	−	0.776105i	$-0.282807\pi$
$827$	2.48056e7	1.26121	0.630604	+	0.776105i	$0.282807\pi$
$828$	2.90720e6	0.147367
$829$	−1.19708e6	−0.0604976	−0.0302488	−	0.999542i	$-0.509630\pi$
$829$	−1.19708e6	−0.0604976	−0.0302488	+	0.999542i	$0.509630\pi$
$830$	0	0
$831$	−1.47392e7	−0.740407
$832$	−1.35578e6	−0.0679015
$833$	4.07930e6	0.203692
$834$	−1.20288e7	−0.598835
$835$	0	0
$836$	1.96539e7	0.972598
$837$	−2.26258e7	−1.11633
$838$	−2.33114e7	−1.14672
$839$	−3.17171e7	−1.55557	−0.777783	−	0.628533i	$-0.783656\pi$
$839$	−3.17171e7	−1.55557	−0.777783	+	0.628533i	$0.783656\pi$
$840$	0	0
$841$	456092.	0.0222363
$842$	−9.89722e6	−0.481097
$843$	−3.30647e7	−1.60249
$844$	3.03250e6	0.146536
$845$	0	0
$846$	−98808.0	−0.00474642
$847$	1.71585e7	0.821810
$848$	2.81190e6	0.134280
$849$	−2.01452e7	−0.959186
$850$	0	0
$851$	6.52461e7	3.08838
$852$	−8.67136e6	−0.409250
$853$	−3.18237e7	−1.49754	−0.748769	−	0.662831i	$-0.769355\pi$
$853$	−3.18237e7	−1.49754	−0.748769	+	0.662831i	$0.769355\pi$
$854$	−7.68085e6	−0.360383
$855$	0	0
$856$	5.35949e6	0.249999
$857$	−2.27853e7	−1.05975	−0.529874	−	0.848076i	$-0.677761\pi$
$857$	−2.27853e7	−1.05975	−0.529874	+	0.848076i	$0.677761\pi$
$858$	−1.60932e7	−0.746319
$859$	−1.85966e7	−0.859907	−0.429953	−	0.902851i	$-0.641470\pi$
$859$	−1.85966e7	−0.859907	−0.429953	+	0.902851i	$0.641470\pi$
$860$	0	0
$861$	1.57237e7	0.722848
$862$	−1.84740e7	−0.846825
$863$	2.77046e7	1.26627	0.633133	−	0.774043i	$-0.281769\pi$
$863$	2.77046e7	1.26627	0.633133	+	0.774043i	$0.281769\pi$
$864$	3.42938e6	0.156290
$865$	0	0
$866$	234424.	0.0106220
$867$	2.49346e7	1.12656
$868$	5.29670e6	0.238620
$869$	2.00036e7	0.898582
$870$	0	0
$871$	1.46170e6	0.0652847
$872$	−6.74413e6	−0.300355
$873$	2.90605e6	0.129053
$874$	2.71444e7	1.20199
$875$	0	0
$876$	1.36490e6	0.0600951
$877$	2.41150e7	1.05874	0.529370	−	0.848391i	$-0.322428\pi$
$877$	2.41150e7	1.05874	0.529370	+	0.848391i	$0.322428\pi$
$878$	−2.81719e7	−1.23333
$879$	−572373.	−0.0249866
$880$	0	0
$881$	−1.26207e7	−0.547827	−0.273914	−	0.961754i	$-0.588318\pi$
$881$	−1.26207e7	−0.547827	−0.273914	+	0.961754i	$0.588318\pi$
$882$	−441784.	−0.0191222
$883$	6.01876e6	0.259780	0.129890	−	0.991528i	$-0.458538\pi$
$883$	6.01876e6	0.259780	0.129890	+	0.991528i	$0.458538\pi$
$884$	−8.99790e6	−0.387267
$885$	0	0
$886$	5.87575e6	0.251466
$887$	−2.36901e7	−1.01102	−0.505509	−	0.862821i	$-0.668695\pi$
$887$	−2.36901e7	−1.01102	−0.505509	+	0.862821i	$0.668695\pi$
$888$	−1.79716e7	−0.764811
$889$	1.07406e7	0.455800
$890$	0	0
$891$	4.86994e7	2.05508
$892$	361552.	0.0152145
$893$	−922566.	−0.0387141
$894$	1.35373e7	0.566485
$895$	0	0
$896$	−802816.	−0.0334077
$897$	−2.22266e7	−0.922344
$898$	2.99365e7	1.23883
$899$	3.09357e7	1.27662
$900$	0	0
$901$	1.86618e7	0.765847
$902$	5.39854e7	2.20933
$903$	−1.88091e6	−0.0767626
$904$	−7.85626e6	−0.319738
$905$	0	0
$906$	−3.20676e7	−1.29792
$907$	1.10583e7	0.446346	0.223173	−	0.974779i	$-0.428359\pi$
$907$	1.10583e7	0.446346	0.223173	+	0.974779i	$0.428359\pi$
$908$	1.59699e7	0.642816
$909$	−1.34550e6	−0.0540100
$910$	0	0
$911$	3.07573e6	0.122787	0.0613934	−	0.998114i	$-0.480446\pi$
$911$	3.07573e6	0.122787	0.0613934	+	0.998114i	$0.480446\pi$
$912$	−7.47674e6	−0.297663
$913$	2.69155e7	1.06862
$914$	681280.	0.0269749
$915$	0	0
$916$	−1.36743e7	−0.538476
$917$	−4.73742e6	−0.186045
$918$	2.27598e7	0.891378
$919$	−1.89018e7	−0.738270	−0.369135	−	0.929376i	$-0.620346\pi$
$919$	−1.89018e7	−0.738270	−0.369135	+	0.929376i	$0.620346\pi$
$920$	0	0
$921$	459731.	0.0178589
$922$	1.71474e7	0.664310
$923$	1.05523e7	0.407701
$924$	−9.52952e6	−0.367190
$925$	0	0
$926$	1.35327e7	0.518629
$927$	6.86269e6	0.262298
$928$	−4.68890e6	−0.178731
$929$	−1.81458e7	−0.689821	−0.344911	−	0.938636i	$-0.612091\pi$
$929$	−1.81458e7	−0.689821	−0.344911	+	0.938636i	$0.612091\pi$
$930$	0	0
$931$	−4.12492e6	−0.155970
$932$	−2.01309e7	−0.759144
$933$	3.65362e7	1.37410
$934$	2.07212e7	0.777225
$935$	0	0
$936$	974464.	0.0363560
$937$	2.17350e7	0.808744	0.404372	−	0.914595i	$-0.367490\pi$
$937$	2.17350e7	0.808744	0.404372	+	0.914595i	$0.367490\pi$
$938$	865536.	0.0321202
$939$	4.53989e7	1.68028
$940$	0	0
$941$	1.86808e7	0.687735	0.343868	−	0.939018i	$-0.388263\pi$
$941$	1.86808e7	0.687735	0.343868	+	0.939018i	$0.388263\pi$
$942$	−4.89967e6	−0.179904
$943$	7.45602e7	2.73041
$944$	−6.64474e6	−0.242688
$945$	0	0
$946$	−6.45788e6	−0.234618
$947$	2.24778e6	0.0814476	0.0407238	−	0.999170i	$-0.487034\pi$
$947$	2.24778e6	0.0814476	0.0407238	+	0.999170i	$0.487034\pi$
$948$	−7.60974e6	−0.275010
$949$	−1.66096e6	−0.0598678
$950$	0	0
$951$	4.25874e6	0.152697
$952$	−5.32806e6	−0.190536
$953$	3.73293e7	1.33143	0.665714	−	0.746207i	$-0.268127\pi$
$953$	3.73293e7	1.33143	0.665714	+	0.746207i	$0.268127\pi$
$954$	−2.02106e6	−0.0718964
$955$	0	0
$956$	−1.13053e7	−0.400071
$957$	−5.56577e7	−1.96447
$958$	3.50711e7	1.23463
$959$	−9.17711e6	−0.322225
$960$	0	0
$961$	1.70144e7	0.594303
$962$	2.18698e7	0.761917
$963$	−3.85213e6	−0.133855
$964$	9.86128e6	0.341775
$965$	0	0
$966$	−1.31614e7	−0.453795
$967$	−2.61870e7	−0.900573	−0.450287	−	0.892884i	$-0.648678\pi$
$967$	−2.61870e7	−0.900573	−0.450287	+	0.892884i	$0.648678\pi$
$968$	−2.24111e7	−0.768733
$969$	−4.96210e7	−1.69768
$970$	0	0
$971$	−3.91957e7	−1.33410	−0.667052	−	0.745011i	$-0.732444\pi$
$971$	−3.91957e7	−1.33410	−0.667052	+	0.745011i	$0.732444\pi$
$972$	−5.50528e6	−0.186902
$973$	8.66781e6	0.293513
$974$	1.08062e6	0.0364984
$975$	0	0
$976$	1.00321e7	0.337108
$977$	−3.03935e6	−0.101870	−0.0509348	−	0.998702i	$-0.516220\pi$
$977$	−3.03935e6	−0.101870	−0.0509348	+	0.998702i	$0.516220\pi$
$978$	2.62027e7	0.875990
$979$	1.23094e7	0.410470
$980$	0	0
$981$	4.84734e6	0.160817
$982$	−1.94220e7	−0.642711
$983$	1.59937e7	0.527915	0.263957	−	0.964534i	$-0.414972\pi$
$983$	1.59937e7	0.527915	0.263957	+	0.964534i	$0.414972\pi$
$984$	−2.05371e7	−0.676163
$985$	0	0
$986$	−3.11189e7	−1.01937
$987$	447321.	0.0146159
$988$	9.09853e6	0.296537
$989$	−8.91910e6	−0.289955
$990$	0	0
$991$	3.63186e6	0.117475	0.0587375	−	0.998273i	$-0.481293\pi$
$991$	3.63186e6	0.117475	0.0587375	+	0.998273i	$0.481293\pi$
$992$	−6.91814e6	−0.223208
$993$	1.80028e7	0.579384
$994$	6.24848e6	0.200590
$995$	0	0
$996$	−1.02392e7	−0.327052
$997$	4.33287e7	1.38051	0.690253	−	0.723568i	$-0.257499\pi$
$997$	4.33287e7	1.38051	0.690253	+	0.723568i	$0.257499\pi$
$998$	1.19430e7	0.379567
$999$	−5.53188e7	−1.75371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.e.1.1		1
5.2	odd	4		350.6.c.a.99.1		2
5.3	odd	4		350.6.c.a.99.2		2
5.4	even	2		70.6.a.e.1.1	✓	1
15.14	odd	2		630.6.a.b.1.1		1
20.19	odd	2		560.6.a.h.1.1		1
35.34	odd	2		490.6.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.e.1.1	✓	1	5.4	even	2
350.6.a.e.1.1		1	1.1	even	1	trivial
350.6.c.a.99.1		2	5.2	odd	4
350.6.c.a.99.2		2	5.3	odd	4
490.6.a.m.1.1		1	35.34	odd	2
560.6.a.h.1.1		1	20.19	odd	2
630.6.a.b.1.1		1	15.14	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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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