LMFDB - Embedded newform 336.8.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,8,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	336.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:	$104.961368563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} +100.000 q^{5} +343.000 q^{7} +729.000 q^{9} -2774.00 q^{11} -3294.00 q^{13} +2700.00 q^{15} +5900.00 q^{17} -6644.00 q^{19} +9261.00 q^{21} -1982.00 q^{23} -68125.0 q^{25} +19683.0 q^{27} -208106. q^{29} +117792. q^{31} -74898.0 q^{33} +34300.0 q^{35} -335686. q^{37} -88938.0 q^{39} -265488. q^{41} +93292.0 q^{43} +72900.0 q^{45} +657516. q^{47} +117649. q^{49} +159300. q^{51} -608718. q^{53} -277400. q^{55} -179388. q^{57} +536120. q^{59} -1.79709e6 q^{61} +250047. q^{63} -329400. q^{65} -2.12318e6 q^{67} -53514.0 q^{69} +1.19121e6 q^{71} +1.05643e6 q^{73} -1.83938e6 q^{75} -951482. q^{77} -998484. q^{79} +531441. q^{81} -3.89800e6 q^{83} +590000. q^{85} -5.61886e6 q^{87} -4.62235e6 q^{89} -1.12984e6 q^{91} +3.18038e6 q^{93} -664400. q^{95} +1.52877e7 q^{97} -2.02225e6 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	100.000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	100.000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−2774.00	−0.628394	−0.314197	−	0.949358i	$-0.601735\pi$
$11$	−2774.00	−0.628394	−0.314197	+	0.949358i	$0.601735\pi$
$12$	0	0
$13$	−3294.00	−0.415836	−0.207918	−	0.978146i	$-0.566669\pi$
$13$	−3294.00	−0.415836	−0.207918	+	0.978146i	$0.566669\pi$
$14$	0	0
$15$	2700.00	0.206559
$16$	0	0
$17$	5900.00	0.291260	0.145630	−	0.989339i	$-0.453479\pi$
$17$	5900.00	0.291260	0.145630	+	0.989339i	$0.453479\pi$
$18$	0	0
$19$	−6644.00	−0.222225	−0.111112	−	0.993808i	$-0.535441\pi$
$19$	−6644.00	−0.222225	−0.111112	+	0.993808i	$0.535441\pi$
$20$	0	0
$21$	9261.00	0.218218
$22$	0	0
$23$	−1982.00	−0.0339669	−0.0169835	−	0.999856i	$-0.505406\pi$
$23$	−1982.00	−0.0339669	−0.0169835	+	0.999856i	$0.505406\pi$
$24$	0	0
$25$	−68125.0	−0.872000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−208106.	−1.58450	−0.792249	−	0.610198i	$-0.791090\pi$
$29$	−208106.	−1.58450	−0.792249	+	0.610198i	$0.791090\pi$
$30$	0	0
$31$	117792.	0.710150	0.355075	−	0.934838i	$-0.384455\pi$
$31$	117792.	0.710150	0.355075	+	0.934838i	$0.384455\pi$
$32$	0	0
$33$	−74898.0	−0.362803
$34$	0	0
$35$	34300.0	0.135225
$36$	0	0
$37$	−335686.	−1.08950	−0.544750	−	0.838599i	$-0.683375\pi$
$37$	−335686.	−1.08950	−0.544750	+	0.838599i	$0.683375\pi$
$38$	0	0
$39$	−88938.0	−0.240083
$40$	0	0
$41$	−265488.	−0.601591	−0.300796	−	0.953689i	$-0.597252\pi$
$41$	−265488.	−0.601591	−0.300796	+	0.953689i	$0.597252\pi$
$42$	0	0
$43$	93292.0	0.178939	0.0894695	−	0.995990i	$-0.471483\pi$
$43$	93292.0	0.178939	0.0894695	+	0.995990i	$0.471483\pi$
$44$	0	0
$45$	72900.0	0.119257
$46$	0	0
$47$	657516.	0.923770	0.461885	−	0.886940i	$-0.347173\pi$
$47$	657516.	0.923770	0.461885	+	0.886940i	$0.347173\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	159300.	0.168159
$52$	0	0
$53$	−608718.	−0.561630	−0.280815	−	0.959762i	$-0.590605\pi$
$53$	−608718.	−0.561630	−0.280815	+	0.959762i	$0.590605\pi$
$54$	0	0
$55$	−277400.	−0.224821
$56$	0	0
$57$	−179388.	−0.128301
$58$	0	0
$59$	536120.	0.339844	0.169922	−	0.985457i	$-0.445648\pi$
$59$	536120.	0.339844	0.169922	+	0.985457i	$0.445648\pi$
$60$	0	0
$61$	−1.79709e6	−1.01371	−0.506857	−	0.862030i	$-0.669193\pi$
$61$	−1.79709e6	−1.01371	−0.506857	+	0.862030i	$0.669193\pi$
$62$	0	0
$63$	250047.	0.125988
$64$	0	0
$65$	−329400.	−0.148774
$66$	0	0
$67$	−2.12318e6	−0.862431	−0.431215	−	0.902249i	$-0.641915\pi$
$67$	−2.12318e6	−0.862431	−0.431215	+	0.902249i	$0.641915\pi$
$68$	0	0
$69$	−53514.0	−0.0196108
$70$	0	0
$71$	1.19121e6	0.394990	0.197495	−	0.980304i	$-0.436719\pi$
$71$	1.19121e6	0.394990	0.197495	+	0.980304i	$0.436719\pi$
$72$	0	0
$73$	1.05643e6	0.317842	0.158921	−	0.987291i	$-0.449199\pi$
$73$	1.05643e6	0.317842	0.158921	+	0.987291i	$0.449199\pi$
$74$	0	0
$75$	−1.83938e6	−0.503449
$76$	0	0
$77$	−951482.	−0.237511
$78$	0	0
$79$	−998484.	−0.227849	−0.113924	−	0.993489i	$-0.536342\pi$
$79$	−998484.	−0.227849	−0.113924	+	0.993489i	$0.536342\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−3.89800e6	−0.748288	−0.374144	−	0.927371i	$-0.622063\pi$
$83$	−3.89800e6	−0.748288	−0.374144	+	0.927371i	$0.622063\pi$
$84$	0	0
$85$	590000.	0.104204
$86$	0	0
$87$	−5.61886e6	−0.914810
$88$	0	0
$89$	−4.62235e6	−0.695021	−0.347511	−	0.937676i	$-0.612973\pi$
$89$	−4.62235e6	−0.695021	−0.347511	+	0.937676i	$0.612973\pi$
$90$	0	0
$91$	−1.12984e6	−0.157171
$92$	0	0
$93$	3.18038e6	0.410005
$94$	0	0
$95$	−664400.	−0.0795055
$96$	0	0
$97$	1.52877e7	1.70075	0.850377	−	0.526174i	$-0.176374\pi$
$97$	1.52877e7	1.70075	0.850377	+	0.526174i	$0.176374\pi$
$98$	0	0
$99$	−2.02225e6	−0.209465
$100$	0	0
$101$	2.23869e6	0.216207	0.108103	−	0.994140i	$-0.465522\pi$
$101$	2.23869e6	0.216207	0.108103	+	0.994140i	$0.465522\pi$
$102$	0	0
$103$	−1.24502e7	−1.12266	−0.561328	−	0.827593i	$-0.689709\pi$
$103$	−1.24502e7	−1.12266	−0.561328	+	0.827593i	$0.689709\pi$
$104$	0	0
$105$	926100.	0.0780720
$106$	0	0
$107$	7.58860e6	0.598851	0.299425	−	0.954120i	$-0.403205\pi$
$107$	7.58860e6	0.598851	0.299425	+	0.954120i	$0.403205\pi$
$108$	0	0
$109$	5.17097e6	0.382454	0.191227	−	0.981546i	$-0.438753\pi$
$109$	5.17097e6	0.382454	0.191227	+	0.981546i	$0.438753\pi$
$110$	0	0
$111$	−9.06352e6	−0.629023
$112$	0	0
$113$	−9.63868e6	−0.628410	−0.314205	−	0.949355i	$-0.601738\pi$
$113$	−9.63868e6	−0.628410	−0.314205	+	0.949355i	$0.601738\pi$
$114$	0	0
$115$	−198200.	−0.0121524
$116$	0	0
$117$	−2.40133e6	−0.138612
$118$	0	0
$119$	2.02370e6	0.110086
$120$	0	0
$121$	−1.17921e7	−0.605121
$122$	0	0
$123$	−7.16818e6	−0.347329
$124$	0	0
$125$	−1.46250e7	−0.669747
$126$	0	0
$127$	−8.08309e6	−0.350158	−0.175079	−	0.984554i	$-0.556018\pi$
$127$	−8.08309e6	−0.350158	−0.175079	+	0.984554i	$0.556018\pi$
$128$	0	0
$129$	2.51888e6	0.103310
$130$	0	0
$131$	1.97585e7	0.767901	0.383951	−	0.923354i	$-0.374563\pi$
$131$	1.97585e7	0.767901	0.383951	+	0.923354i	$0.374563\pi$
$132$	0	0
$133$	−2.27889e6	−0.0839930
$134$	0	0
$135$	1.96830e6	0.0688530
$136$	0	0
$137$	−4.80896e7	−1.59783	−0.798913	−	0.601447i	$-0.794591\pi$
$137$	−4.80896e7	−1.59783	−0.798913	+	0.601447i	$0.794591\pi$
$138$	0	0
$139$	−1.37173e7	−0.433229	−0.216615	−	0.976257i	$-0.569501\pi$
$139$	−1.37173e7	−0.433229	−0.216615	+	0.976257i	$0.569501\pi$
$140$	0	0
$141$	1.77529e7	0.533339
$142$	0	0
$143$	9.13756e6	0.261309
$144$	0	0
$145$	−2.08106e7	−0.566887
$146$	0	0
$147$	3.17652e6	0.0824786
$148$	0	0
$149$	−7.07104e7	−1.75118	−0.875591	−	0.483053i	$-0.839528\pi$
$149$	−7.07104e7	−1.75118	−0.875591	+	0.483053i	$0.839528\pi$
$150$	0	0
$151$	2.56970e7	0.607383	0.303691	−	0.952770i	$-0.401781\pi$
$151$	2.56970e7	0.607383	0.303691	+	0.952770i	$0.401781\pi$
$152$	0	0
$153$	4.30110e6	0.0970867
$154$	0	0
$155$	1.17792e7	0.254071
$156$	0	0
$157$	2.09180e7	0.431390	0.215695	−	0.976461i	$-0.430798\pi$
$157$	2.09180e7	0.431390	0.215695	+	0.976461i	$0.430798\pi$
$158$	0	0
$159$	−1.64354e7	−0.324257
$160$	0	0
$161$	−679826.	−0.0128383
$162$	0	0
$163$	−5.44160e7	−0.984169	−0.492084	−	0.870547i	$-0.663765\pi$
$163$	−5.44160e7	−0.984169	−0.492084	+	0.870547i	$0.663765\pi$
$164$	0	0
$165$	−7.48980e6	−0.129801
$166$	0	0
$167$	−6.41888e6	−0.106648	−0.0533238	−	0.998577i	$-0.516982\pi$
$167$	−6.41888e6	−0.106648	−0.0533238	+	0.998577i	$0.516982\pi$
$168$	0	0
$169$	−5.18981e7	−0.827081
$170$	0	0
$171$	−4.84348e6	−0.0740748
$172$	0	0
$173$	−5.46532e7	−0.802517	−0.401259	−	0.915965i	$-0.631427\pi$
$173$	−5.46532e7	−0.802517	−0.401259	+	0.915965i	$0.631427\pi$
$174$	0	0
$175$	−2.33669e7	−0.329585
$176$	0	0
$177$	1.44752e7	0.196209
$178$	0	0
$179$	3.76658e7	0.490865	0.245433	−	0.969414i	$-0.421070\pi$
$179$	3.76658e7	0.490865	0.245433	+	0.969414i	$0.421070\pi$
$180$	0	0
$181$	1.76788e7	0.221604	0.110802	−	0.993842i	$-0.464658\pi$
$181$	1.76788e7	0.221604	0.110802	+	0.993842i	$0.464658\pi$
$182$	0	0
$183$	−4.85214e7	−0.585268
$184$	0	0
$185$	−3.35686e7	−0.389791
$186$	0	0
$187$	−1.63666e7	−0.183026
$188$	0	0
$189$	6.75127e6	0.0727393
$190$	0	0
$191$	−4.09177e7	−0.424907	−0.212454	−	0.977171i	$-0.568145\pi$
$191$	−4.09177e7	−0.424907	−0.212454	+	0.977171i	$0.568145\pi$
$192$	0	0
$193$	−1.63827e8	−1.64034	−0.820171	−	0.572118i	$-0.806122\pi$
$193$	−1.63827e8	−1.64034	−0.820171	+	0.572118i	$0.806122\pi$
$194$	0	0
$195$	−8.89380e6	−0.0858947
$196$	0	0
$197$	8.02076e7	0.747453	0.373726	−	0.927539i	$-0.378080\pi$
$197$	8.02076e7	0.747453	0.373726	+	0.927539i	$0.378080\pi$
$198$	0	0
$199$	−9.83219e7	−0.884432	−0.442216	−	0.896908i	$-0.645808\pi$
$199$	−9.83219e7	−0.884432	−0.442216	+	0.896908i	$0.645808\pi$
$200$	0	0
$201$	−5.73258e7	−0.497925
$202$	0	0
$203$	−7.13804e7	−0.598884
$204$	0	0
$205$	−2.65488e7	−0.215232
$206$	0	0
$207$	−1.44488e6	−0.0113223
$208$	0	0
$209$	1.84305e7	0.139645
$210$	0	0
$211$	−1.36321e8	−0.999021	−0.499510	−	0.866308i	$-0.666487\pi$
$211$	−1.36321e8	−0.999021	−0.499510	+	0.866308i	$0.666487\pi$
$212$	0	0
$213$	3.21628e7	0.228047
$214$	0	0
$215$	9.32920e6	0.0640191
$216$	0	0
$217$	4.04027e7	0.268411
$218$	0	0
$219$	2.85236e7	0.183506
$220$	0	0
$221$	−1.94346e7	−0.121116
$222$	0	0
$223$	−1.26358e8	−0.763019	−0.381510	−	0.924365i	$-0.624596\pi$
$223$	−1.26358e8	−0.763019	−0.381510	+	0.924365i	$0.624596\pi$
$224$	0	0
$225$	−4.96631e7	−0.290667
$226$	0	0
$227$	−2.34007e8	−1.32782	−0.663909	−	0.747814i	$-0.731104\pi$
$227$	−2.34007e8	−1.32782	−0.663909	+	0.747814i	$0.731104\pi$
$228$	0	0
$229$	6.83606e7	0.376168	0.188084	−	0.982153i	$-0.439772\pi$
$229$	6.83606e7	0.376168	0.188084	+	0.982153i	$0.439772\pi$
$230$	0	0
$231$	−2.56900e7	−0.137127
$232$	0	0
$233$	1.80385e8	0.934230	0.467115	−	0.884197i	$-0.345293\pi$
$233$	1.80385e8	0.934230	0.467115	+	0.884197i	$0.345293\pi$
$234$	0	0
$235$	6.57516e7	0.330498
$236$	0	0
$237$	−2.69591e7	−0.131548
$238$	0	0
$239$	−2.61131e8	−1.23727	−0.618637	−	0.785677i	$-0.712315\pi$
$239$	−2.61131e8	−1.23727	−0.618637	+	0.785677i	$0.712315\pi$
$240$	0	0
$241$	1.97756e7	0.0910061	0.0455031	−	0.998964i	$-0.485511\pi$
$241$	1.97756e7	0.0910061	0.0455031	+	0.998964i	$0.485511\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	1.17649e7	0.0511101
$246$	0	0
$247$	2.18853e7	0.0924089
$248$	0	0
$249$	−1.05246e8	−0.432024
$250$	0	0
$251$	−1.53770e8	−0.613779	−0.306890	−	0.951745i	$-0.599288\pi$
$251$	−1.53770e8	−0.613779	−0.306890	+	0.951745i	$0.599288\pi$
$252$	0	0
$253$	5.49807e6	0.0213446
$254$	0	0
$255$	1.59300e7	0.0601624
$256$	0	0
$257$	−1.43360e8	−0.526821	−0.263411	−	0.964684i	$-0.584847\pi$
$257$	−1.43360e8	−0.526821	−0.263411	+	0.964684i	$0.584847\pi$
$258$	0	0
$259$	−1.15140e8	−0.411792
$260$	0	0
$261$	−1.51709e8	−0.528166
$262$	0	0
$263$	5.07301e8	1.71957	0.859786	−	0.510654i	$-0.170597\pi$
$263$	5.07301e8	1.71957	0.859786	+	0.510654i	$0.170597\pi$
$264$	0	0
$265$	−6.08718e7	−0.200935
$266$	0	0
$267$	−1.24804e8	−0.401271
$268$	0	0
$269$	5.27886e8	1.65351	0.826755	−	0.562562i	$-0.190184\pi$
$269$	5.27886e8	1.65351	0.826755	+	0.562562i	$0.190184\pi$
$270$	0	0
$271$	5.05836e8	1.54389	0.771947	−	0.635687i	$-0.219283\pi$
$271$	5.05836e8	1.54389	0.771947	+	0.635687i	$0.219283\pi$
$272$	0	0
$273$	−3.05057e7	−0.0907428
$274$	0	0
$275$	1.88979e8	0.547960
$276$	0	0
$277$	−5.88586e8	−1.66391	−0.831956	−	0.554841i	$-0.812779\pi$
$277$	−5.88586e8	−1.66391	−0.831956	+	0.554841i	$0.812779\pi$
$278$	0	0
$279$	8.58704e7	0.236717
$280$	0	0
$281$	8.65142e7	0.232603	0.116301	−	0.993214i	$-0.462896\pi$
$281$	8.65142e7	0.232603	0.116301	+	0.993214i	$0.462896\pi$
$282$	0	0
$283$	1.57301e8	0.412552	0.206276	−	0.978494i	$-0.433866\pi$
$283$	1.57301e8	0.412552	0.206276	+	0.978494i	$0.433866\pi$
$284$	0	0
$285$	−1.79388e7	−0.0459025
$286$	0	0
$287$	−9.10624e7	−0.227380
$288$	0	0
$289$	−3.75529e8	−0.915168
$290$	0	0
$291$	4.12768e8	0.981930
$292$	0	0
$293$	6.80964e8	1.58157	0.790783	−	0.612097i	$-0.209674\pi$
$293$	6.80964e8	1.58157	0.790783	+	0.612097i	$0.209674\pi$
$294$	0	0
$295$	5.36120e7	0.121586
$296$	0	0
$297$	−5.46006e7	−0.120934
$298$	0	0
$299$	6.52871e6	0.0141247
$300$	0	0
$301$	3.19992e7	0.0676326
$302$	0	0
$303$	6.04446e7	0.124827
$304$	0	0
$305$	−1.79709e8	−0.362677
$306$	0	0
$307$	−2.81734e8	−0.555718	−0.277859	−	0.960622i	$-0.589625\pi$
$307$	−2.81734e8	−0.555718	−0.277859	+	0.960622i	$0.589625\pi$
$308$	0	0
$309$	−3.36156e8	−0.648166
$310$	0	0
$311$	4.19719e8	0.791220	0.395610	−	0.918419i	$-0.370533\pi$
$311$	4.19719e8	0.791220	0.395610	+	0.918419i	$0.370533\pi$
$312$	0	0
$313$	−2.28684e8	−0.421532	−0.210766	−	0.977537i	$-0.567596\pi$
$313$	−2.28684e8	−0.421532	−0.210766	+	0.977537i	$0.567596\pi$
$314$	0	0
$315$	2.50047e7	0.0450749
$316$	0	0
$317$	2.66003e8	0.469006	0.234503	−	0.972115i	$-0.424654\pi$
$317$	2.66003e8	0.469006	0.234503	+	0.972115i	$0.424654\pi$
$318$	0	0
$319$	5.77286e8	0.995689
$320$	0	0
$321$	2.04892e8	0.345747
$322$	0	0
$323$	−3.91996e7	−0.0647251
$324$	0	0
$325$	2.24404e8	0.362609
$326$	0	0
$327$	1.39616e8	0.220810
$328$	0	0
$329$	2.25528e8	0.349152
$330$	0	0
$331$	7.16107e8	1.08538	0.542688	−	0.839935i	$-0.317407\pi$
$331$	7.16107e8	1.08538	0.542688	+	0.839935i	$0.317407\pi$
$332$	0	0
$333$	−2.44715e8	−0.363167
$334$	0	0
$335$	−2.12318e8	−0.308553
$336$	0	0
$337$	5.70266e8	0.811657	0.405829	−	0.913949i	$-0.366983\pi$
$337$	5.70266e8	0.811657	0.405829	+	0.913949i	$0.366983\pi$
$338$	0	0
$339$	−2.60244e8	−0.362813
$340$	0	0
$341$	−3.26755e8	−0.446254
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−5.35140e6	−0.00701618
$346$	0	0
$347$	1.64641e8	0.211536	0.105768	−	0.994391i	$-0.466270\pi$
$347$	1.64641e8	0.211536	0.105768	+	0.994391i	$0.466270\pi$
$348$	0	0
$349$	1.31564e9	1.65671	0.828357	−	0.560200i	$-0.189276\pi$
$349$	1.31564e9	1.65671	0.828357	+	0.560200i	$0.189276\pi$
$350$	0	0
$351$	−6.48358e7	−0.0800276
$352$	0	0
$353$	1.56118e9	1.88904	0.944518	−	0.328459i	$-0.106529\pi$
$353$	1.56118e9	1.88904	0.944518	+	0.328459i	$0.106529\pi$
$354$	0	0
$355$	1.19121e8	0.141316
$356$	0	0
$357$	5.46399e7	0.0635581
$358$	0	0
$359$	8.60716e8	0.981814	0.490907	−	0.871212i	$-0.336665\pi$
$359$	8.60716e8	0.981814	0.490907	+	0.871212i	$0.336665\pi$
$360$	0	0
$361$	−8.49729e8	−0.950616
$362$	0	0
$363$	−3.18387e8	−0.349367
$364$	0	0
$365$	1.05643e8	0.113714
$366$	0	0
$367$	−9.12133e8	−0.963223	−0.481612	−	0.876385i	$-0.659948\pi$
$367$	−9.12133e8	−0.963223	−0.481612	+	0.876385i	$0.659948\pi$
$368$	0	0
$369$	−1.93541e8	−0.200530
$370$	0	0
$371$	−2.08790e8	−0.212276
$372$	0	0
$373$	2.36004e8	0.235472	0.117736	−	0.993045i	$-0.462436\pi$
$373$	2.36004e8	0.235472	0.117736	+	0.993045i	$0.462436\pi$
$374$	0	0
$375$	−3.94875e8	−0.386679
$376$	0	0
$377$	6.85501e8	0.658891
$378$	0	0
$379$	−8.29313e8	−0.782495	−0.391247	−	0.920286i	$-0.627956\pi$
$379$	−8.29313e8	−0.782495	−0.391247	+	0.920286i	$0.627956\pi$
$380$	0	0
$381$	−2.18243e8	−0.202164
$382$	0	0
$383$	1.21663e9	1.10653	0.553265	−	0.833005i	$-0.313382\pi$
$383$	1.21663e9	1.10653	0.553265	+	0.833005i	$0.313382\pi$
$384$	0	0
$385$	−9.51482e7	−0.0849744
$386$	0	0
$387$	6.80099e7	0.0596463
$388$	0	0
$389$	−1.18213e8	−0.101822	−0.0509109	−	0.998703i	$-0.516212\pi$
$389$	−1.18213e8	−0.101822	−0.0509109	+	0.998703i	$0.516212\pi$
$390$	0	0
$391$	−1.16938e7	−0.00989321
$392$	0	0
$393$	5.33481e8	0.443348
$394$	0	0
$395$	−9.98484e7	−0.0815176
$396$	0	0
$397$	3.62565e7	0.0290817	0.0145408	−	0.999894i	$-0.495371\pi$
$397$	3.62565e7	0.0290817	0.0145408	+	0.999894i	$0.495371\pi$
$398$	0	0
$399$	−6.15301e7	−0.0484934
$400$	0	0
$401$	5.06850e8	0.392531	0.196265	−	0.980551i	$-0.437119\pi$
$401$	5.06850e8	0.392531	0.196265	+	0.980551i	$0.437119\pi$
$402$	0	0
$403$	−3.88007e8	−0.295306
$404$	0	0
$405$	5.31441e7	0.0397523
$406$	0	0
$407$	9.31193e8	0.684635
$408$	0	0
$409$	1.68121e9	1.21504	0.607519	−	0.794305i	$-0.292165\pi$
$409$	1.68121e9	1.21504	0.607519	+	0.794305i	$0.292165\pi$
$410$	0	0
$411$	−1.29842e9	−0.922505
$412$	0	0
$413$	1.83889e8	0.128449
$414$	0	0
$415$	−3.89800e8	−0.267716
$416$	0	0
$417$	−3.70368e8	−0.250125
$418$	0	0
$419$	−4.94962e8	−0.328718	−0.164359	−	0.986401i	$-0.552556\pi$
$419$	−4.94962e8	−0.328718	−0.164359	+	0.986401i	$0.552556\pi$
$420$	0	0
$421$	−6.57487e7	−0.0429437	−0.0214719	−	0.999769i	$-0.506835\pi$
$421$	−6.57487e7	−0.0429437	−0.0214719	+	0.999769i	$0.506835\pi$
$422$	0	0
$423$	4.79329e8	0.307923
$424$	0	0
$425$	−4.01938e8	−0.253979
$426$	0	0
$427$	−6.16402e8	−0.383148
$428$	0	0
$429$	2.46714e8	0.150867
$430$	0	0
$431$	1.33418e9	0.802682	0.401341	−	0.915929i	$-0.368544\pi$
$431$	1.33418e9	0.802682	0.401341	+	0.915929i	$0.368544\pi$
$432$	0	0
$433$	−2.47903e7	−0.0146749	−0.00733745	−	0.999973i	$-0.502336\pi$
$433$	−2.47903e7	−0.0146749	−0.00733745	+	0.999973i	$0.502336\pi$
$434$	0	0
$435$	−5.61886e8	−0.327292
$436$	0	0
$437$	1.31684e7	0.00754828
$438$	0	0
$439$	6.89327e8	0.388866	0.194433	−	0.980916i	$-0.437713\pi$
$439$	6.89327e8	0.388866	0.194433	+	0.980916i	$0.437713\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	0	0
$443$	5.74617e8	0.314026	0.157013	−	0.987597i	$-0.449814\pi$
$443$	5.74617e8	0.314026	0.157013	+	0.987597i	$0.449814\pi$
$444$	0	0
$445$	−4.62235e8	−0.248658
$446$	0	0
$447$	−1.90918e9	−1.01105
$448$	0	0
$449$	9.56884e7	0.0498881	0.0249440	−	0.999689i	$-0.492059\pi$
$449$	9.56884e7	0.0498881	0.0249440	+	0.999689i	$0.492059\pi$
$450$	0	0
$451$	7.36464e8	0.378036
$452$	0	0
$453$	6.93818e8	0.350672
$454$	0	0
$455$	−1.12984e8	−0.0562313
$456$	0	0
$457$	−4.73089e7	−0.0231866	−0.0115933	−	0.999933i	$-0.503690\pi$
$457$	−4.73089e7	−0.0231866	−0.0115933	+	0.999933i	$0.503690\pi$
$458$	0	0
$459$	1.16130e8	0.0560530
$460$	0	0
$461$	−2.13567e9	−1.01527	−0.507634	−	0.861573i	$-0.669480\pi$
$461$	−2.13567e9	−1.01527	−0.507634	+	0.861573i	$0.669480\pi$
$462$	0	0
$463$	−2.92675e8	−0.137042	−0.0685208	−	0.997650i	$-0.521828\pi$
$463$	−2.92675e8	−0.137042	−0.0685208	+	0.997650i	$0.521828\pi$
$464$	0	0
$465$	3.18038e8	0.146688
$466$	0	0
$467$	3.53834e9	1.60765	0.803824	−	0.594868i	$-0.202796\pi$
$467$	3.53834e9	1.60765	0.803824	+	0.594868i	$0.202796\pi$
$468$	0	0
$469$	−7.28249e8	−0.325968
$470$	0	0
$471$	5.64785e8	0.249063
$472$	0	0
$473$	−2.58792e8	−0.112444
$474$	0	0
$475$	4.52622e8	0.193780
$476$	0	0
$477$	−4.43755e8	−0.187210
$478$	0	0
$479$	4.15698e9	1.72824	0.864120	−	0.503286i	$-0.167876\pi$
$479$	4.15698e9	1.72824	0.864120	+	0.503286i	$0.167876\pi$
$480$	0	0
$481$	1.10575e9	0.453053
$482$	0	0
$483$	−1.83553e7	−0.00741219
$484$	0	0
$485$	1.52877e9	0.608480
$486$	0	0
$487$	−1.24092e8	−0.0486847	−0.0243423	−	0.999704i	$-0.507749\pi$
$487$	−1.24092e8	−0.0486847	−0.0243423	+	0.999704i	$0.507749\pi$
$488$	0	0
$489$	−1.46923e9	−0.568210
$490$	0	0
$491$	3.29218e9	1.25516	0.627578	−	0.778554i	$-0.284046\pi$
$491$	3.29218e9	1.25516	0.627578	+	0.778554i	$0.284046\pi$
$492$	0	0
$493$	−1.22783e9	−0.461501
$494$	0	0
$495$	−2.02225e8	−0.0749404
$496$	0	0
$497$	4.08586e8	0.149292
$498$	0	0
$499$	3.60439e8	0.129861	0.0649306	−	0.997890i	$-0.479317\pi$
$499$	3.60439e8	0.129861	0.0649306	+	0.997890i	$0.479317\pi$
$500$	0	0
$501$	−1.73310e8	−0.0615731
$502$	0	0
$503$	4.29760e9	1.50570	0.752849	−	0.658193i	$-0.228679\pi$
$503$	4.29760e9	1.50570	0.752849	+	0.658193i	$0.228679\pi$
$504$	0	0
$505$	2.23869e8	0.0773524
$506$	0	0
$507$	−1.40125e9	−0.477515
$508$	0	0
$509$	−5.42588e8	−0.182372	−0.0911860	−	0.995834i	$-0.529066\pi$
$509$	−5.42588e8	−0.182372	−0.0911860	+	0.995834i	$0.529066\pi$
$510$	0	0
$511$	3.62355e8	0.120133
$512$	0	0
$513$	−1.30774e8	−0.0427671
$514$	0	0
$515$	−1.24502e9	−0.401654
$516$	0	0
$517$	−1.82395e9	−0.580492
$518$	0	0
$519$	−1.47564e9	−0.463334
$520$	0	0
$521$	−2.36755e8	−0.0733445	−0.0366722	−	0.999327i	$-0.511676\pi$
$521$	−2.36755e8	−0.0733445	−0.0366722	+	0.999327i	$0.511676\pi$
$522$	0	0
$523$	−6.48324e9	−1.98169	−0.990846	−	0.135000i	$-0.956896\pi$
$523$	−6.48324e9	−1.98169	−0.990846	+	0.135000i	$0.956896\pi$
$524$	0	0
$525$	−6.30906e8	−0.190286
$526$	0	0
$527$	6.94973e8	0.206838
$528$	0	0
$529$	−3.40090e9	−0.998846
$530$	0	0
$531$	3.90831e8	0.113281
$532$	0	0
$533$	8.74517e8	0.250163
$534$	0	0
$535$	7.58860e8	0.214251
$536$	0	0
$537$	1.01698e9	0.283401
$538$	0	0
$539$	−3.26358e8	−0.0897706
$540$	0	0
$541$	−1.50099e8	−0.0407556	−0.0203778	−	0.999792i	$-0.506487\pi$
$541$	−1.50099e8	−0.0407556	−0.0203778	+	0.999792i	$0.506487\pi$
$542$	0	0
$543$	4.77328e8	0.127943
$544$	0	0
$545$	5.17097e8	0.136831
$546$	0	0
$547$	−3.32631e9	−0.868974	−0.434487	−	0.900678i	$-0.643070\pi$
$547$	−3.32631e9	−0.868974	−0.434487	+	0.900678i	$0.643070\pi$
$548$	0	0
$549$	−1.31008e9	−0.337905
$550$	0	0
$551$	1.38266e9	0.352114
$552$	0	0
$553$	−3.42480e8	−0.0861187
$554$	0	0
$555$	−9.06352e8	−0.225046
$556$	0	0
$557$	4.72512e9	1.15856	0.579282	−	0.815127i	$-0.303333\pi$
$557$	4.72512e9	1.15856	0.579282	+	0.815127i	$0.303333\pi$
$558$	0	0
$559$	−3.07304e8	−0.0744092
$560$	0	0
$561$	−4.41898e8	−0.105670
$562$	0	0
$563$	−7.11044e9	−1.67926	−0.839628	−	0.543162i	$-0.817227\pi$
$563$	−7.11044e9	−1.67926	−0.839628	+	0.543162i	$0.817227\pi$
$564$	0	0
$565$	−9.63868e8	−0.224827
$566$	0	0
$567$	1.82284e8	0.0419961
$568$	0	0
$569$	6.55315e9	1.49127	0.745636	−	0.666353i	$-0.232146\pi$
$569$	6.55315e9	1.49127	0.745636	+	0.666353i	$0.232146\pi$
$570$	0	0
$571$	−8.29277e9	−1.86412	−0.932059	−	0.362306i	$-0.881990\pi$
$571$	−8.29277e9	−1.86412	−0.932059	+	0.362306i	$0.881990\pi$
$572$	0	0
$573$	−1.10478e9	−0.245320
$574$	0	0
$575$	1.35024e8	0.0296192
$576$	0	0
$577$	−5.88000e9	−1.27427	−0.637135	−	0.770752i	$-0.719881\pi$
$577$	−5.88000e9	−1.27427	−0.637135	+	0.770752i	$0.719881\pi$
$578$	0	0
$579$	−4.42333e9	−0.947052
$580$	0	0
$581$	−1.33702e9	−0.282826
$582$	0	0
$583$	1.68858e9	0.352925
$584$	0	0
$585$	−2.40133e8	−0.0495913
$586$	0	0
$587$	1.43194e9	0.292207	0.146103	−	0.989269i	$-0.453327\pi$
$587$	1.43194e9	0.292207	0.146103	+	0.989269i	$0.453327\pi$
$588$	0	0
$589$	−7.82610e8	−0.157813
$590$	0	0
$591$	2.16561e9	0.431542
$592$	0	0
$593$	3.48138e9	0.685584	0.342792	−	0.939411i	$-0.388627\pi$
$593$	3.48138e9	0.685584	0.342792	+	0.939411i	$0.388627\pi$
$594$	0	0
$595$	2.02370e8	0.0393855
$596$	0	0
$597$	−2.65469e9	−0.510627
$598$	0	0
$599$	−7.93597e9	−1.50871	−0.754356	−	0.656466i	$-0.772051\pi$
$599$	−7.93597e9	−1.50871	−0.754356	+	0.656466i	$0.772051\pi$
$600$	0	0
$601$	−4.05169e9	−0.761335	−0.380668	−	0.924712i	$-0.624306\pi$
$601$	−4.05169e9	−0.761335	−0.380668	+	0.924712i	$0.624306\pi$
$602$	0	0
$603$	−1.54780e9	−0.287477
$604$	0	0
$605$	−1.17921e9	−0.216495
$606$	0	0
$607$	−5.19159e9	−0.942194	−0.471097	−	0.882081i	$-0.656142\pi$
$607$	−5.19159e9	−0.942194	−0.471097	+	0.882081i	$0.656142\pi$
$608$	0	0
$609$	−1.92727e9	−0.345766
$610$	0	0
$611$	−2.16586e9	−0.384137
$612$	0	0
$613$	3.69807e9	0.648430	0.324215	−	0.945983i	$-0.394900\pi$
$613$	3.69807e9	0.648430	0.324215	+	0.945983i	$0.394900\pi$
$614$	0	0
$615$	−7.16818e8	−0.124264
$616$	0	0
$617$	−6.19879e9	−1.06245	−0.531226	−	0.847230i	$-0.678268\pi$
$617$	−6.19879e9	−1.06245	−0.531226	+	0.847230i	$0.678268\pi$
$618$	0	0
$619$	2.85137e9	0.483211	0.241605	−	0.970375i	$-0.422326\pi$
$619$	2.85137e9	0.483211	0.241605	+	0.970375i	$0.422326\pi$
$620$	0	0
$621$	−3.90117e7	−0.00653694
$622$	0	0
$623$	−1.58547e9	−0.262693
$624$	0	0
$625$	3.85977e9	0.632384
$626$	0	0
$627$	4.97622e8	0.0806238
$628$	0	0
$629$	−1.98055e9	−0.317328
$630$	0	0
$631$	7.56414e9	1.19855	0.599276	−	0.800543i	$-0.295455\pi$
$631$	7.56414e9	1.19855	0.599276	+	0.800543i	$0.295455\pi$
$632$	0	0
$633$	−3.68067e9	−0.576785
$634$	0	0
$635$	−8.08309e8	−0.125276
$636$	0	0
$637$	−3.87536e8	−0.0594051
$638$	0	0
$639$	8.68395e8	0.131663
$640$	0	0
$641$	1.12019e10	1.67991	0.839957	−	0.542653i	$-0.182580\pi$
$641$	1.12019e10	1.67991	0.839957	+	0.542653i	$0.182580\pi$
$642$	0	0
$643$	−7.87742e9	−1.16855	−0.584273	−	0.811557i	$-0.698620\pi$
$643$	−7.87742e9	−1.16855	−0.584273	+	0.811557i	$0.698620\pi$
$644$	0	0
$645$	2.51888e8	0.0369615
$646$	0	0
$647$	−1.24945e9	−0.181365	−0.0906825	−	0.995880i	$-0.528905\pi$
$647$	−1.24945e9	−0.181365	−0.0906825	+	0.995880i	$0.528905\pi$
$648$	0	0
$649$	−1.48720e9	−0.213556
$650$	0	0
$651$	1.09087e9	0.154967
$652$	0	0
$653$	7.52696e9	1.05785	0.528924	−	0.848669i	$-0.322596\pi$
$653$	7.52696e9	1.05785	0.528924	+	0.848669i	$0.322596\pi$
$654$	0	0
$655$	1.97585e9	0.274733
$656$	0	0
$657$	7.70137e8	0.105947
$658$	0	0
$659$	−2.29417e9	−0.312268	−0.156134	−	0.987736i	$-0.549903\pi$
$659$	−2.29417e9	−0.312268	−0.156134	+	0.987736i	$0.549903\pi$
$660$	0	0
$661$	−5.08384e9	−0.684679	−0.342339	−	0.939576i	$-0.611219\pi$
$661$	−5.08384e9	−0.684679	−0.342339	+	0.939576i	$0.611219\pi$
$662$	0	0
$663$	−5.24734e8	−0.0699265
$664$	0	0
$665$	−2.27889e8	−0.0300502
$666$	0	0
$667$	4.12466e8	0.0538205
$668$	0	0
$669$	−3.41166e9	−0.440529
$670$	0	0
$671$	4.98513e9	0.637012
$672$	0	0
$673$	−5.62649e9	−0.711516	−0.355758	−	0.934578i	$-0.615777\pi$
$673$	−5.62649e9	−0.711516	−0.355758	+	0.934578i	$0.615777\pi$
$674$	0	0
$675$	−1.34090e9	−0.167816
$676$	0	0
$677$	−1.02745e10	−1.27263	−0.636314	−	0.771430i	$-0.719542\pi$
$677$	−1.02745e10	−1.27263	−0.636314	+	0.771430i	$0.719542\pi$
$678$	0	0
$679$	5.24368e9	0.642824
$680$	0	0
$681$	−6.31819e9	−0.766616
$682$	0	0
$683$	−1.55246e10	−1.86443	−0.932217	−	0.361901i	$-0.882128\pi$
$683$	−1.55246e10	−1.86443	−0.932217	+	0.361901i	$0.882128\pi$
$684$	0	0
$685$	−4.80896e9	−0.571655
$686$	0	0
$687$	1.84574e9	0.217181
$688$	0	0
$689$	2.00512e9	0.233546
$690$	0	0
$691$	−8.23449e9	−0.949432	−0.474716	−	0.880139i	$-0.657449\pi$
$691$	−8.23449e9	−0.949432	−0.474716	+	0.880139i	$0.657449\pi$
$692$	0	0
$693$	−6.93630e8	−0.0791702
$694$	0	0
$695$	−1.37173e9	−0.154997
$696$	0	0
$697$	−1.56638e9	−0.175219
$698$	0	0
$699$	4.87039e9	0.539378
$700$	0	0
$701$	−4.88820e9	−0.535964	−0.267982	−	0.963424i	$-0.586357\pi$
$701$	−4.88820e9	−0.535964	−0.267982	+	0.963424i	$0.586357\pi$
$702$	0	0
$703$	2.23030e9	0.242114
$704$	0	0
$705$	1.77529e9	0.190813
$706$	0	0
$707$	7.67870e8	0.0817184
$708$	0	0
$709$	−7.95106e9	−0.837844	−0.418922	−	0.908022i	$-0.637592\pi$
$709$	−7.95106e9	−0.837844	−0.418922	+	0.908022i	$0.637592\pi$
$710$	0	0
$711$	−7.27895e8	−0.0759495
$712$	0	0
$713$	−2.33464e8	−0.0241216
$714$	0	0
$715$	9.13756e8	0.0934887
$716$	0	0
$717$	−7.05053e9	−0.714340
$718$	0	0
$719$	1.30035e10	1.30469	0.652346	−	0.757921i	$-0.273785\pi$
$719$	1.30035e10	1.30469	0.652346	+	0.757921i	$0.273785\pi$
$720$	0	0
$721$	−4.27043e9	−0.424324
$722$	0	0
$723$	5.33942e8	0.0525424
$724$	0	0
$725$	1.41772e10	1.38168
$726$	0	0
$727$	1.73805e10	1.67761	0.838805	−	0.544431i	$-0.183255\pi$
$727$	1.73805e10	1.67761	0.838805	+	0.544431i	$0.183255\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	5.50423e8	0.0521177
$732$	0	0
$733$	9.47114e9	0.888256	0.444128	−	0.895963i	$-0.353513\pi$
$733$	9.47114e9	0.888256	0.444128	+	0.895963i	$0.353513\pi$
$734$	0	0
$735$	3.17652e8	0.0295084
$736$	0	0
$737$	5.88969e9	0.541946
$738$	0	0
$739$	−5.52914e9	−0.503967	−0.251983	−	0.967732i	$-0.581083\pi$
$739$	−5.52914e9	−0.503967	−0.251983	+	0.967732i	$0.581083\pi$
$740$	0	0
$741$	5.90904e8	0.0533523
$742$	0	0
$743$	1.53701e10	1.37472	0.687362	−	0.726315i	$-0.258769\pi$
$743$	1.53701e10	1.37472	0.687362	+	0.726315i	$0.258769\pi$
$744$	0	0
$745$	−7.07104e9	−0.626522
$746$	0	0
$747$	−2.84164e9	−0.249429
$748$	0	0
$749$	2.60289e9	0.226344
$750$	0	0
$751$	8.51950e9	0.733964	0.366982	−	0.930228i	$-0.380391\pi$
$751$	8.51950e9	0.733964	0.366982	+	0.930228i	$0.380391\pi$
$752$	0	0
$753$	−4.15178e9	−0.354366
$754$	0	0
$755$	2.56970e9	0.217304
$756$	0	0
$757$	−4.72648e8	−0.0396006	−0.0198003	−	0.999804i	$-0.506303\pi$
$757$	−4.72648e8	−0.0396006	−0.0198003	+	0.999804i	$0.506303\pi$
$758$	0	0
$759$	1.48448e8	0.0123233
$760$	0	0
$761$	−1.43465e10	−1.18005	−0.590025	−	0.807385i	$-0.700882\pi$
$761$	−1.43465e10	−1.18005	−0.590025	+	0.807385i	$0.700882\pi$
$762$	0	0
$763$	1.77364e9	0.144554
$764$	0	0
$765$	4.30110e8	0.0347348
$766$	0	0
$767$	−1.76598e9	−0.141319
$768$	0	0
$769$	1.39480e10	1.10603	0.553017	−	0.833170i	$-0.313476\pi$
$769$	1.39480e10	1.10603	0.553017	+	0.833170i	$0.313476\pi$
$770$	0	0
$771$	−3.87073e9	−0.304160
$772$	0	0
$773$	−4.32337e9	−0.336662	−0.168331	−	0.985731i	$-0.553838\pi$
$773$	−4.32337e9	−0.336662	−0.168331	+	0.985731i	$0.553838\pi$
$774$	0	0
$775$	−8.02458e9	−0.619250
$776$	0	0
$777$	−3.10879e9	−0.237748
$778$	0	0
$779$	1.76390e9	0.133688
$780$	0	0
$781$	−3.30443e9	−0.248209
$782$	0	0
$783$	−4.09615e9	−0.304937
$784$	0	0
$785$	2.09180e9	0.154339
$786$	0	0
$787$	−6.71201e9	−0.490842	−0.245421	−	0.969417i	$-0.578926\pi$
$787$	−6.71201e9	−0.490842	−0.245421	+	0.969417i	$0.578926\pi$
$788$	0	0
$789$	1.36971e10	0.992796
$790$	0	0
$791$	−3.30607e9	−0.237517
$792$	0	0
$793$	5.91961e9	0.421539
$794$	0	0
$795$	−1.64354e9	−0.116010
$796$	0	0
$797$	5.61333e9	0.392750	0.196375	−	0.980529i	$-0.437083\pi$
$797$	5.61333e9	0.392750	0.196375	+	0.980529i	$0.437083\pi$
$798$	0	0
$799$	3.87934e9	0.269057
$800$	0	0
$801$	−3.36969e9	−0.231674
$802$	0	0
$803$	−2.93054e9	−0.199730
$804$	0	0
$805$	−6.79826e7	−0.00459317
$806$	0	0
$807$	1.42529e10	0.954655
$808$	0	0
$809$	4.15844e9	0.276128	0.138064	−	0.990423i	$-0.455912\pi$
$809$	4.15844e9	0.276128	0.138064	+	0.990423i	$0.455912\pi$
$810$	0	0
$811$	1.11994e10	0.737259	0.368629	−	0.929576i	$-0.379827\pi$
$811$	1.11994e10	0.737259	0.368629	+	0.929576i	$0.379827\pi$
$812$	0	0
$813$	1.36576e10	0.891367
$814$	0	0
$815$	−5.44160e9	−0.352107
$816$	0	0
$817$	−6.19832e8	−0.0397646
$818$	0	0
$819$	−8.23655e8	−0.0523904
$820$	0	0
$821$	2.32872e10	1.46864	0.734322	−	0.678801i	$-0.237500\pi$
$821$	2.32872e10	1.46864	0.734322	+	0.678801i	$0.237500\pi$
$822$	0	0
$823$	−9.31943e9	−0.582761	−0.291380	−	0.956607i	$-0.594114\pi$
$823$	−9.31943e9	−0.582761	−0.291380	+	0.956607i	$0.594114\pi$
$824$	0	0
$825$	5.10243e9	0.316365
$826$	0	0
$827$	1.16608e10	0.716902	0.358451	−	0.933548i	$-0.383305\pi$
$827$	1.16608e10	0.716902	0.358451	+	0.933548i	$0.383305\pi$
$828$	0	0
$829$	−2.31007e8	−0.0140826	−0.00704132	−	0.999975i	$-0.502241\pi$
$829$	−2.31007e8	−0.0140826	−0.00704132	+	0.999975i	$0.502241\pi$
$830$	0	0
$831$	−1.58918e10	−0.960660
$832$	0	0
$833$	6.94129e8	0.0416086
$834$	0	0
$835$	−6.41888e8	−0.0381554
$836$	0	0
$837$	2.31850e9	0.136668
$838$	0	0
$839$	2.77032e10	1.61943	0.809716	−	0.586822i	$-0.199621\pi$
$839$	2.77032e10	1.61943	0.809716	+	0.586822i	$0.199621\pi$
$840$	0	0
$841$	2.60582e10	1.51063
$842$	0	0
$843$	2.33588e9	0.134293
$844$	0	0
$845$	−5.18981e9	−0.295905
$846$	0	0
$847$	−4.04469e9	−0.228714
$848$	0	0
$849$	4.24712e9	0.238187
$850$	0	0
$851$	6.65330e8	0.0370070
$852$	0	0
$853$	2.87010e10	1.58335	0.791673	−	0.610944i	$-0.209210\pi$
$853$	2.87010e10	1.58335	0.791673	+	0.610944i	$0.209210\pi$
$854$	0	0
$855$	−4.84348e8	−0.0265018
$856$	0	0
$857$	1.44528e9	0.0784366	0.0392183	−	0.999231i	$-0.487513\pi$
$857$	1.44528e9	0.0784366	0.0392183	+	0.999231i	$0.487513\pi$
$858$	0	0
$859$	−2.94906e10	−1.58748	−0.793738	−	0.608260i	$-0.791868\pi$
$859$	−2.94906e10	−1.58748	−0.793738	+	0.608260i	$0.791868\pi$
$860$	0	0
$861$	−2.45868e9	−0.131278
$862$	0	0
$863$	−8.87258e9	−0.469907	−0.234954	−	0.972007i	$-0.575494\pi$
$863$	−8.87258e9	−0.469907	−0.234954	+	0.972007i	$0.575494\pi$
$864$	0	0
$865$	−5.46532e9	−0.287117
$866$	0	0
$867$	−1.01393e10	−0.528372
$868$	0	0
$869$	2.76979e9	0.143179
$870$	0	0
$871$	6.99374e9	0.358630
$872$	0	0
$873$	1.11447e10	0.566918
$874$	0	0
$875$	−5.01638e9	−0.253141
$876$	0	0
$877$	1.57308e10	0.787504	0.393752	−	0.919217i	$-0.371177\pi$
$877$	1.57308e10	0.787504	0.393752	+	0.919217i	$0.371177\pi$
$878$	0	0
$879$	1.83860e10	0.913117
$880$	0	0
$881$	−9.12522e9	−0.449602	−0.224801	−	0.974405i	$-0.572173\pi$
$881$	−9.12522e9	−0.449602	−0.224801	+	0.974405i	$0.572173\pi$
$882$	0	0
$883$	−2.24538e10	−1.09756	−0.548778	−	0.835968i	$-0.684907\pi$
$883$	−2.24538e10	−1.09756	−0.548778	+	0.835968i	$0.684907\pi$
$884$	0	0
$885$	1.44752e9	0.0701979
$886$	0	0
$887$	3.43569e10	1.65303	0.826516	−	0.562913i	$-0.190320\pi$
$887$	3.43569e10	1.65303	0.826516	+	0.562913i	$0.190320\pi$
$888$	0	0
$889$	−2.77250e9	−0.132347
$890$	0	0
$891$	−1.47422e9	−0.0698216
$892$	0	0
$893$	−4.36854e9	−0.205284
$894$	0	0
$895$	3.76658e9	0.175617
$896$	0	0
$897$	1.76275e8	0.00815488
$898$	0	0
$899$	−2.45132e10	−1.12523
$900$	0	0
$901$	−3.59144e9	−0.163580
$902$	0	0
$903$	8.63977e8	0.0390477
$904$	0	0
$905$	1.76788e9	0.0792835
$906$	0	0
$907$	4.01062e10	1.78478	0.892392	−	0.451261i	$-0.149026\pi$
$907$	4.01062e10	1.78478	0.892392	+	0.451261i	$0.149026\pi$
$908$	0	0
$909$	1.63200e9	0.0720688
$910$	0	0
$911$	3.70230e10	1.62240	0.811200	−	0.584769i	$-0.198815\pi$
$911$	3.70230e10	1.62240	0.811200	+	0.584769i	$0.198815\pi$
$912$	0	0
$913$	1.08131e10	0.470220
$914$	0	0
$915$	−4.85214e9	−0.209392
$916$	0	0
$917$	6.77718e9	0.290239
$918$	0	0
$919$	3.37645e10	1.43501	0.717507	−	0.696551i	$-0.245283\pi$
$919$	3.37645e10	1.43501	0.717507	+	0.696551i	$0.245283\pi$
$920$	0	0
$921$	−7.60681e9	−0.320844
$922$	0	0
$923$	−3.92386e9	−0.164251
$924$	0	0
$925$	2.28686e10	0.950044
$926$	0	0
$927$	−9.07621e9	−0.374219
$928$	0	0
$929$	−4.77100e10	−1.95233	−0.976167	−	0.217019i	$-0.930367\pi$
$929$	−4.77100e10	−1.95233	−0.976167	+	0.217019i	$0.930367\pi$
$930$	0	0
$931$	−7.81660e8	−0.0317464
$932$	0	0
$933$	1.13324e10	0.456811
$934$	0	0
$935$	−1.63666e9	−0.0654814
$936$	0	0
$937$	3.86353e8	0.0153425	0.00767125	−	0.999971i	$-0.497558\pi$
$937$	3.86353e8	0.0153425	0.00767125	+	0.999971i	$0.497558\pi$
$938$	0	0
$939$	−6.17447e9	−0.243372
$940$	0	0
$941$	−2.79446e10	−1.09329	−0.546643	−	0.837366i	$-0.684095\pi$
$941$	−2.79446e10	−1.09329	−0.546643	+	0.837366i	$0.684095\pi$
$942$	0	0
$943$	5.26197e8	0.0204342
$944$	0	0
$945$	6.75127e8	0.0260240
$946$	0	0
$947$	5.50921e9	0.210797	0.105399	−	0.994430i	$-0.466388\pi$
$947$	5.50921e9	0.210797	0.105399	+	0.994430i	$0.466388\pi$
$948$	0	0
$949$	−3.47988e9	−0.132170
$950$	0	0
$951$	7.18207e9	0.270781
$952$	0	0
$953$	−7.40406e9	−0.277105	−0.138553	−	0.990355i	$-0.544245\pi$
$953$	−7.40406e9	−0.277105	−0.138553	+	0.990355i	$0.544245\pi$
$954$	0	0
$955$	−4.09177e9	−0.152019
$956$	0	0
$957$	1.55867e10	0.574861
$958$	0	0
$959$	−1.64947e10	−0.603921
$960$	0	0
$961$	−1.36377e10	−0.495687
$962$	0	0
$963$	5.53209e9	0.199617
$964$	0	0
$965$	−1.63827e10	−0.586867
$966$	0	0
$967$	−1.97731e10	−0.703204	−0.351602	−	0.936149i	$-0.614363\pi$
$967$	−1.97731e10	−0.703204	−0.351602	+	0.936149i	$0.614363\pi$
$968$	0	0
$969$	−1.05839e9	−0.0373691
$970$	0	0
$971$	2.81294e10	0.986037	0.493018	−	0.870019i	$-0.335894\pi$
$971$	2.81294e10	0.986037	0.493018	+	0.870019i	$0.335894\pi$
$972$	0	0
$973$	−4.70504e9	−0.163745
$974$	0	0
$975$	6.05890e9	0.209352
$976$	0	0
$977$	9.63337e8	0.0330482	0.0165241	−	0.999863i	$-0.494740\pi$
$977$	9.63337e8	0.0330482	0.0165241	+	0.999863i	$0.494740\pi$
$978$	0	0
$979$	1.28224e10	0.436747
$980$	0	0
$981$	3.76963e9	0.127485
$982$	0	0
$983$	−4.20434e10	−1.41176	−0.705879	−	0.708332i	$-0.749448\pi$
$983$	−4.20434e10	−1.41176	−0.705879	+	0.708332i	$0.749448\pi$
$984$	0	0
$985$	8.02076e9	0.267417
$986$	0	0
$987$	6.08926e9	0.201583
$988$	0	0
$989$	−1.84905e8	−0.00607800
$990$	0	0
$991$	2.87955e10	0.939869	0.469934	−	0.882701i	$-0.344278\pi$
$991$	2.87955e10	0.939869	0.469934	+	0.882701i	$0.344278\pi$
$992$	0	0
$993$	1.93349e10	0.626642
$994$	0	0
$995$	−9.83219e9	−0.316424
$996$	0	0
$997$	−3.54828e10	−1.13393	−0.566963	−	0.823743i	$-0.691882\pi$
$997$	−3.54828e10	−1.13393	−0.566963	+	0.823743i	$0.691882\pi$
$998$	0	0
$999$	−6.60731e9	−0.209674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.8.a.j.1.1		1
4.3	odd	2		84.8.a.a.1.1	✓	1
12.11	even	2		252.8.a.a.1.1		1
28.3	even	6		588.8.i.c.373.1		2
28.11	odd	6		588.8.i.f.373.1		2
28.19	even	6		588.8.i.c.361.1		2
28.23	odd	6		588.8.i.f.361.1		2
28.27	even	2		588.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.8.a.a.1.1	✓	1	4.3	odd	2
252.8.a.a.1.1		1	12.11	even	2
336.8.a.j.1.1		1	1.1	even	1	trivial
588.8.a.c.1.1		1	28.27	even	2
588.8.i.c.361.1		2	28.19	even	6
588.8.i.c.373.1		2	28.3	even	6
588.8.i.f.361.1		2	28.23	odd	6
588.8.i.f.373.1		2	28.11	odd	6

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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