LMFDB - Embedded newform 98.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,6,Mod(1,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	98.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,-10,16,-84,40,0,-64,-143] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.7176143417$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	98.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-4.00000 q^{2} -10.0000 q^{3} +16.0000 q^{4} -84.0000 q^{5} +40.0000 q^{6} -64.0000 q^{8} -143.000 q^{9} +336.000 q^{10} -336.000 q^{11} -160.000 q^{12} -584.000 q^{13} +840.000 q^{15} +256.000 q^{16} +1458.00 q^{17} +572.000 q^{18} -470.000 q^{19} -1344.00 q^{20} +1344.00 q^{22} -4200.00 q^{23} +640.000 q^{24} +3931.00 q^{25} +2336.00 q^{26} +3860.00 q^{27} +4866.00 q^{29} -3360.00 q^{30} +7372.00 q^{31} -1024.00 q^{32} +3360.00 q^{33} -5832.00 q^{34} -2288.00 q^{36} +14330.0 q^{37} +1880.00 q^{38} +5840.00 q^{39} +5376.00 q^{40} -6222.00 q^{41} +3704.00 q^{43} -5376.00 q^{44} +12012.0 q^{45} +16800.0 q^{46} +1812.00 q^{47} -2560.00 q^{48} -15724.0 q^{50} -14580.0 q^{51} -9344.00 q^{52} -37242.0 q^{53} -15440.0 q^{54} +28224.0 q^{55} +4700.00 q^{57} -19464.0 q^{58} -34302.0 q^{59} +13440.0 q^{60} -24476.0 q^{61} -29488.0 q^{62} +4096.00 q^{64} +49056.0 q^{65} -13440.0 q^{66} -17452.0 q^{67} +23328.0 q^{68} +42000.0 q^{69} +28224.0 q^{71} +9152.00 q^{72} -3602.00 q^{73} -57320.0 q^{74} -39310.0 q^{75} -7520.00 q^{76} -23360.0 q^{78} +42872.0 q^{79} -21504.0 q^{80} -3851.00 q^{81} +24888.0 q^{82} +35202.0 q^{83} -122472. q^{85} -14816.0 q^{86} -48660.0 q^{87} +21504.0 q^{88} -26730.0 q^{89} -48048.0 q^{90} -67200.0 q^{92} -73720.0 q^{93} -7248.00 q^{94} +39480.0 q^{95} +10240.0 q^{96} +16978.0 q^{97} +48048.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−10.0000	−0.641500	−0.320750	−	0.947164i	$-0.603935\pi$
$3$	−10.0000	−0.641500	−0.320750	+	0.947164i	$0.603935\pi$
$4$	16.0000	0.500000
$5$	−84.0000	−1.50264	−0.751319	−	0.659939i	$-0.770582\pi$
$5$	−84.0000	−1.50264	−0.751319	+	0.659939i	$0.770582\pi$
$6$	40.0000	0.453609
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	−143.000	−0.588477
$10$	336.000	1.06253
$11$	−336.000	−0.837255	−0.418627	−	0.908158i	$-0.637489\pi$
$11$	−336.000	−0.837255	−0.418627	+	0.908158i	$0.637489\pi$
$12$	−160.000	−0.320750
$13$	−584.000	−0.958417	−0.479208	−	0.877701i	$-0.659076\pi$
$13$	−584.000	−0.958417	−0.479208	+	0.877701i	$0.659076\pi$
$14$	0	0
$15$	840.000	0.963943
$16$	256.000	0.250000
$17$	1458.00	1.22359	0.611794	−	0.791017i	$-0.290448\pi$
$17$	1458.00	1.22359	0.611794	+	0.791017i	$0.290448\pi$
$18$	572.000	0.416116
$19$	−470.000	−0.298685	−0.149343	−	0.988786i	$-0.547716\pi$
$19$	−470.000	−0.298685	−0.149343	+	0.988786i	$0.547716\pi$
$20$	−1344.00	−0.751319
$21$	0	0
$22$	1344.00	0.592028
$23$	−4200.00	−1.65550	−0.827751	−	0.561096i	$-0.810380\pi$
$23$	−4200.00	−1.65550	−0.827751	+	0.561096i	$0.810380\pi$
$24$	640.000	0.226805
$25$	3931.00	1.25792
$26$	2336.00	0.677703
$27$	3860.00	1.01901
$28$	0	0
$29$	4866.00	1.07443	0.537214	−	0.843446i	$-0.319477\pi$
$29$	4866.00	1.07443	0.537214	+	0.843446i	$0.319477\pi$
$30$	−3360.00	−0.681610
$31$	7372.00	1.37778	0.688892	−	0.724864i	$-0.258097\pi$
$31$	7372.00	1.37778	0.688892	+	0.724864i	$0.258097\pi$
$32$	−1024.00	−0.176777
$33$	3360.00	0.537099
$34$	−5832.00	−0.865207
$35$	0	0
$36$	−2288.00	−0.294239
$37$	14330.0	1.72085	0.860423	−	0.509581i	$-0.170200\pi$
$37$	14330.0	1.72085	0.860423	+	0.509581i	$0.170200\pi$
$38$	1880.00	0.211202
$39$	5840.00	0.614825
$40$	5376.00	0.531263
$41$	−6222.00	−0.578057	−0.289028	−	0.957321i	$-0.593332\pi$
$41$	−6222.00	−0.578057	−0.289028	+	0.957321i	$0.593332\pi$
$42$	0	0
$43$	3704.00	0.305492	0.152746	−	0.988265i	$-0.451188\pi$
$43$	3704.00	0.305492	0.152746	+	0.988265i	$0.451188\pi$
$44$	−5376.00	−0.418627
$45$	12012.0	0.884268
$46$	16800.0	1.17062
$47$	1812.00	0.119650	0.0598251	−	0.998209i	$-0.480946\pi$
$47$	1812.00	0.119650	0.0598251	+	0.998209i	$0.480946\pi$
$48$	−2560.00	−0.160375
$49$	0	0
$50$	−15724.0	−0.889484
$51$	−14580.0	−0.784932
$52$	−9344.00	−0.479208
$53$	−37242.0	−1.82114	−0.910570	−	0.413355i	$-0.864357\pi$
$53$	−37242.0	−1.82114	−0.910570	+	0.413355i	$0.864357\pi$
$54$	−15440.0	−0.720548
$55$	28224.0	1.25809
$56$	0	0
$57$	4700.00	0.191607
$58$	−19464.0	−0.759735
$59$	−34302.0	−1.28289	−0.641445	−	0.767169i	$-0.721665\pi$
$59$	−34302.0	−1.28289	−0.641445	+	0.767169i	$0.721665\pi$
$60$	13440.0	0.481971
$61$	−24476.0	−0.842201	−0.421101	−	0.907014i	$-0.638356\pi$
$61$	−24476.0	−0.842201	−0.421101	+	0.907014i	$0.638356\pi$
$62$	−29488.0	−0.974240
$63$	0	0
$64$	4096.00	0.125000
$65$	49056.0	1.44015
$66$	−13440.0	−0.379786
$67$	−17452.0	−0.474961	−0.237481	−	0.971392i	$-0.576322\pi$
$67$	−17452.0	−0.474961	−0.237481	+	0.971392i	$0.576322\pi$
$68$	23328.0	0.611794
$69$	42000.0	1.06201
$70$	0	0
$71$	28224.0	0.664466	0.332233	−	0.943197i	$-0.392198\pi$
$71$	28224.0	0.664466	0.332233	+	0.943197i	$0.392198\pi$
$72$	9152.00	0.208058
$73$	−3602.00	−0.0791109	−0.0395555	−	0.999217i	$-0.512594\pi$
$73$	−3602.00	−0.0791109	−0.0395555	+	0.999217i	$0.512594\pi$
$74$	−57320.0	−1.21682
$75$	−39310.0	−0.806956
$76$	−7520.00	−0.149343
$77$	0	0
$78$	−23360.0	−0.434747
$79$	42872.0	0.772869	0.386435	−	0.922317i	$-0.373706\pi$
$79$	42872.0	0.772869	0.386435	+	0.922317i	$0.373706\pi$
$80$	−21504.0	−0.375659
$81$	−3851.00	−0.0652170
$82$	24888.0	0.408748
$83$	35202.0	0.560883	0.280441	−	0.959871i	$-0.409519\pi$
$83$	35202.0	0.560883	0.280441	+	0.959871i	$0.409519\pi$
$84$	0	0
$85$	−122472.	−1.83861
$86$	−14816.0	−0.216015
$87$	−48660.0	−0.689246
$88$	21504.0	0.296014
$89$	−26730.0	−0.357704	−0.178852	−	0.983876i	$-0.557238\pi$
$89$	−26730.0	−0.357704	−0.178852	+	0.983876i	$0.557238\pi$
$90$	−48048.0	−0.625272
$91$	0	0
$92$	−67200.0	−0.827751
$93$	−73720.0	−0.883849
$94$	−7248.00	−0.0846055
$95$	39480.0	0.448816
$96$	10240.0	0.113402
$97$	16978.0	0.183213	0.0916067	−	0.995795i	$-0.470800\pi$
$97$	16978.0	0.183213	0.0916067	+	0.995795i	$0.470800\pi$
$98$	0	0
$99$	48048.0	0.492705
$100$	62896.0	0.628960
$101$	−99204.0	−0.967667	−0.483833	−	0.875160i	$-0.660756\pi$
$101$	−99204.0	−0.967667	−0.483833	+	0.875160i	$0.660756\pi$
$102$	58320.0	0.555031
$103$	131644.	1.22267	0.611333	−	0.791373i	$-0.290634\pi$
$103$	131644.	1.22267	0.611333	+	0.791373i	$0.290634\pi$
$104$	37376.0	0.338852
$105$	0	0
$106$	148968.	1.28774
$107$	48852.0	0.412499	0.206250	−	0.978499i	$-0.433874\pi$
$107$	48852.0	0.412499	0.206250	+	0.978499i	$0.433874\pi$
$108$	61760.0	0.509504
$109$	−56374.0	−0.454478	−0.227239	−	0.973839i	$-0.572970\pi$
$109$	−56374.0	−0.454478	−0.227239	+	0.973839i	$0.572970\pi$
$110$	−112896.	−0.889604
$111$	−143300.	−1.10392
$112$	0	0
$113$	8742.00	0.0644043	0.0322021	−	0.999481i	$-0.489748\pi$
$113$	8742.00	0.0644043	0.0322021	+	0.999481i	$0.489748\pi$
$114$	−18800.0	−0.135486
$115$	352800.	2.48762
$116$	77856.0	0.537214
$117$	83512.0	0.564007
$118$	137208.	0.907140
$119$	0	0
$120$	−53760.0	−0.340805
$121$	−48155.0	−0.299005
$122$	97904.0	0.595526
$123$	62220.0	0.370823
$124$	117952.	0.688892
$125$	−67704.0	−0.387560
$126$	0	0
$127$	315992.	1.73847	0.869234	−	0.494401i	$-0.164612\pi$
$127$	315992.	1.73847	0.869234	+	0.494401i	$0.164612\pi$
$128$	−16384.0	−0.0883883
$129$	−37040.0	−0.195973
$130$	−196224.	−1.01834
$131$	24666.0	0.125580	0.0627900	−	0.998027i	$-0.480000\pi$
$131$	24666.0	0.125580	0.0627900	+	0.998027i	$0.480000\pi$
$132$	53760.0	0.268550
$133$	0	0
$134$	69808.0	0.335848
$135$	−324240.	−1.53120
$136$	−93312.0	−0.432604
$137$	303234.	1.38031	0.690155	−	0.723662i	$-0.257542\pi$
$137$	303234.	1.38031	0.690155	+	0.723662i	$0.257542\pi$
$138$	−168000.	−0.750951
$139$	−250586.	−1.10007	−0.550034	−	0.835142i	$-0.685385\pi$
$139$	−250586.	−1.10007	−0.550034	+	0.835142i	$0.685385\pi$
$140$	0	0
$141$	−18120.0	−0.0767557
$142$	−112896.	−0.469848
$143$	196224.	0.802439
$144$	−36608.0	−0.147119
$145$	−408744.	−1.61448
$146$	14408.0	0.0559399
$147$	0	0
$148$	229280.	0.860423
$149$	−60594.0	−0.223596	−0.111798	−	0.993731i	$-0.535661\pi$
$149$	−60594.0	−0.223596	−0.111798	+	0.993731i	$0.535661\pi$
$150$	157240.	0.570604
$151$	124448.	0.444166	0.222083	−	0.975028i	$-0.428714\pi$
$151$	124448.	0.444166	0.222083	+	0.975028i	$0.428714\pi$
$152$	30080.0	0.105601
$153$	−208494.	−0.720054
$154$	0	0
$155$	−619248.	−2.07031
$156$	93440.0	0.307412
$157$	−76040.0	−0.246203	−0.123101	−	0.992394i	$-0.539284\pi$
$157$	−76040.0	−0.246203	−0.123101	+	0.992394i	$0.539284\pi$
$158$	−171488.	−0.546501
$159$	372420.	1.16826
$160$	86016.0	0.265631
$161$	0	0
$162$	15404.0	0.0461154
$163$	124256.	0.366310	0.183155	−	0.983084i	$-0.441369\pi$
$163$	124256.	0.366310	0.183155	+	0.983084i	$0.441369\pi$
$164$	−99552.0	−0.289028
$165$	−282240.	−0.807065
$166$	−140808.	−0.396604
$167$	72420.0	0.200940	0.100470	−	0.994940i	$-0.467965\pi$
$167$	72420.0	0.200940	0.100470	+	0.994940i	$0.467965\pi$
$168$	0	0
$169$	−30237.0	−0.0814370
$170$	489888.	1.30009
$171$	67210.0	0.175770
$172$	59264.0	0.152746
$173$	441552.	1.12167	0.560837	−	0.827926i	$-0.310479\pi$
$173$	441552.	1.12167	0.560837	+	0.827926i	$0.310479\pi$
$174$	194640.	0.487370
$175$	0	0
$176$	−86016.0	−0.209314
$177$	343020.	0.822974
$178$	106920.	0.252935
$179$	−10692.0	−0.0249417	−0.0124709	−	0.999922i	$-0.503970\pi$
$179$	−10692.0	−0.0249417	−0.0124709	+	0.999922i	$0.503970\pi$
$180$	192192.	0.442134
$181$	546064.	1.23893	0.619465	−	0.785024i	$-0.287349\pi$
$181$	546064.	1.23893	0.619465	+	0.785024i	$0.287349\pi$
$182$	0	0
$183$	244760.	0.540272
$184$	268800.	0.585308
$185$	−1.20372e6	−2.58581
$186$	294880.	0.624975
$187$	−489888.	−1.02445
$188$	28992.0	0.0598251
$189$	0	0
$190$	−157920.	−0.317361
$191$	−575976.	−1.14241	−0.571204	−	0.820808i	$-0.693523\pi$
$191$	−575976.	−1.14241	−0.571204	+	0.820808i	$0.693523\pi$
$192$	−40960.0	−0.0801875
$193$	−413938.	−0.799912	−0.399956	−	0.916534i	$-0.630975\pi$
$193$	−413938.	−0.799912	−0.399956	+	0.916534i	$0.630975\pi$
$194$	−67912.0	−0.129551
$195$	−490560.	−0.923859
$196$	0	0
$197$	−494946.	−0.908641	−0.454320	−	0.890838i	$-0.650118\pi$
$197$	−494946.	−0.908641	−0.454320	+	0.890838i	$0.650118\pi$
$198$	−192192.	−0.348395
$199$	−520364.	−0.931482	−0.465741	−	0.884921i	$-0.654212\pi$
$199$	−520364.	−0.931482	−0.465741	+	0.884921i	$0.654212\pi$
$200$	−251584.	−0.444742
$201$	174520.	0.304688
$202$	396816.	0.684244
$203$	0	0
$204$	−233280.	−0.392466
$205$	522648.	0.868610
$206$	−526576.	−0.864556
$207$	600600.	0.974225
$208$	−149504.	−0.239604
$209$	157920.	0.250076
$210$	0	0
$211$	183284.	0.283412	0.141706	−	0.989909i	$-0.454741\pi$
$211$	183284.	0.283412	0.141706	+	0.989909i	$0.454741\pi$
$212$	−595872.	−0.910570
$213$	−282240.	−0.426255
$214$	−195408.	−0.291681
$215$	−311136.	−0.459044
$216$	−247040.	−0.360274
$217$	0	0
$218$	225496.	0.321364
$219$	36020.0	0.0507497
$220$	451584.	0.629045
$221$	−851472.	−1.17271
$222$	573200.	0.780591
$223$	1.27746e6	1.72023	0.860115	−	0.510100i	$-0.170392\pi$
$223$	1.27746e6	1.72023	0.860115	+	0.510100i	$0.170392\pi$
$224$	0	0
$225$	−562133.	−0.740257
$226$	−34968.0	−0.0455407
$227$	1.28764e6	1.65856	0.829279	−	0.558835i	$-0.188752\pi$
$227$	1.28764e6	1.65856	0.829279	+	0.558835i	$0.188752\pi$
$228$	75200.0	0.0958034
$229$	−350936.	−0.442221	−0.221110	−	0.975249i	$-0.570968\pi$
$229$	−350936.	−0.442221	−0.221110	+	0.975249i	$0.570968\pi$
$230$	−1.41120e6	−1.75901
$231$	0	0
$232$	−311424.	−0.379867
$233$	836154.	1.00901	0.504506	−	0.863408i	$-0.331675\pi$
$233$	836154.	1.00901	0.504506	+	0.863408i	$0.331675\pi$
$234$	−334048.	−0.398813
$235$	−152208.	−0.179791
$236$	−548832.	−0.641445
$237$	−428720.	−0.495796
$238$	0	0
$239$	774336.	0.876869	0.438434	−	0.898763i	$-0.355533\pi$
$239$	774336.	0.876869	0.438434	+	0.898763i	$0.355533\pi$
$240$	215040.	0.240986
$241$	1.15285e6	1.27859	0.639293	−	0.768963i	$-0.279227\pi$
$241$	1.15285e6	1.27859	0.639293	+	0.768963i	$0.279227\pi$
$242$	192620.	0.211428
$243$	−899470.	−0.977172
$244$	−391616.	−0.421101
$245$	0	0
$246$	−248880.	−0.262212
$247$	274480.	0.286265
$248$	−471808.	−0.487120
$249$	−352020.	−0.359806
$250$	270816.	0.274047
$251$	−1.35801e6	−1.36056	−0.680282	−	0.732951i	$-0.738142\pi$
$251$	−1.35801e6	−1.36056	−0.680282	+	0.732951i	$0.738142\pi$
$252$	0	0
$253$	1.41120e6	1.38608
$254$	−1.26397e6	−1.22928
$255$	1.22472e6	1.17947
$256$	65536.0	0.0625000
$257$	317742.	0.300083	0.150042	−	0.988680i	$-0.452059\pi$
$257$	317742.	0.300083	0.150042	+	0.988680i	$0.452059\pi$
$258$	148160.	0.138574
$259$	0	0
$260$	784896.	0.720077
$261$	−695838.	−0.632276
$262$	−98664.0	−0.0887985
$263$	1.05101e6	0.936951	0.468475	−	0.883477i	$-0.344804\pi$
$263$	1.05101e6	0.936951	0.468475	+	0.883477i	$0.344804\pi$
$264$	−215040.	−0.189893
$265$	3.12833e6	2.73651
$266$	0	0
$267$	267300.	0.229467
$268$	−279232.	−0.237481
$269$	−1.18958e6	−1.00234	−0.501169	−	0.865349i	$-0.667097\pi$
$269$	−1.18958e6	−1.00234	−0.501169	+	0.865349i	$0.667097\pi$
$270$	1.29696e6	1.08272
$271$	1.43008e6	1.18287	0.591435	−	0.806353i	$-0.298562\pi$
$271$	1.43008e6	1.18287	0.591435	+	0.806353i	$0.298562\pi$
$272$	373248.	0.305897
$273$	0	0
$274$	−1.21294e6	−0.976026
$275$	−1.32082e6	−1.05320
$276$	672000.	0.531003
$277$	63302.0	0.0495699	0.0247849	−	0.999693i	$-0.492110\pi$
$277$	63302.0	0.0495699	0.0247849	+	0.999693i	$0.492110\pi$
$278$	1.00234e6	0.777866
$279$	−1.05420e6	−0.810795
$280$	0	0
$281$	−496614.	−0.375192	−0.187596	−	0.982246i	$-0.560070\pi$
$281$	−496614.	−0.375192	−0.187596	+	0.982246i	$0.560070\pi$
$282$	72480.0	0.0542744
$283$	1.15842e6	0.859803	0.429902	−	0.902876i	$-0.358548\pi$
$283$	1.15842e6	0.859803	0.429902	+	0.902876i	$0.358548\pi$
$284$	451584.	0.332233
$285$	−394800.	−0.287915
$286$	−784896.	−0.567410
$287$	0	0
$288$	146432.	0.104029
$289$	705907.	0.497168
$290$	1.63498e6	1.14161
$291$	−169780.	−0.117531
$292$	−57632.0	−0.0395555
$293$	−1.43886e6	−0.979151	−0.489575	−	0.871961i	$-0.662848\pi$
$293$	−1.43886e6	−0.979151	−0.489575	+	0.871961i	$0.662848\pi$
$294$	0	0
$295$	2.88137e6	1.92772
$296$	−917120.	−0.608411
$297$	−1.29696e6	−0.853170
$298$	242376.	0.158106
$299$	2.45280e6	1.58666
$300$	−628960.	−0.403478
$301$	0	0
$302$	−497792.	−0.314073
$303$	992040.	0.620758
$304$	−120320.	−0.0746713
$305$	2.05598e6	1.26552
$306$	833976.	0.509155
$307$	989098.	0.598954	0.299477	−	0.954104i	$-0.403188\pi$
$307$	989098.	0.598954	0.299477	+	0.954104i	$0.403188\pi$
$308$	0	0
$309$	−1.31644e6	−0.784341
$310$	2.47699e6	1.46393
$311$	2.22050e6	1.30182	0.650909	−	0.759155i	$-0.274388\pi$
$311$	2.22050e6	1.30182	0.650909	+	0.759155i	$0.274388\pi$
$312$	−373760.	−0.217373
$313$	−2.33008e6	−1.34434	−0.672171	−	0.740396i	$-0.734638\pi$
$313$	−2.33008e6	−1.34434	−0.672171	+	0.740396i	$0.734638\pi$
$314$	304160.	0.174092
$315$	0	0
$316$	685952.	0.386435
$317$	427542.	0.238963	0.119481	−	0.992836i	$-0.461877\pi$
$317$	427542.	0.238963	0.119481	+	0.992836i	$0.461877\pi$
$318$	−1.48968e6	−0.826086
$319$	−1.63498e6	−0.899569
$320$	−344064.	−0.187830
$321$	−488520.	−0.264618
$322$	0	0
$323$	−685260.	−0.365468
$324$	−61616.0	−0.0326085
$325$	−2.29570e6	−1.20561
$326$	−497024.	−0.259020
$327$	563740.	0.291548
$328$	398208.	0.204374
$329$	0	0
$330$	1.12896e6	0.570681
$331$	−396616.	−0.198976	−0.0994879	−	0.995039i	$-0.531720\pi$
$331$	−396616.	−0.198976	−0.0994879	+	0.995039i	$0.531720\pi$
$332$	563232.	0.280441
$333$	−2.04919e6	−1.01268
$334$	−289680.	−0.142086
$335$	1.46597e6	0.713695
$336$	0	0
$337$	−3.21819e6	−1.54361	−0.771805	−	0.635860i	$-0.780646\pi$
$337$	−3.21819e6	−1.54361	−0.771805	+	0.635860i	$0.780646\pi$
$338$	120948.	0.0575847
$339$	−87420.0	−0.0413154
$340$	−1.95955e6	−0.919305
$341$	−2.47699e6	−1.15356
$342$	−268840.	−0.124288
$343$	0	0
$344$	−237056.	−0.108008
$345$	−3.52800e6	−1.59581
$346$	−1.76621e6	−0.793143
$347$	2.78018e6	1.23951	0.619755	−	0.784796i	$-0.287232\pi$
$347$	2.78018e6	1.23951	0.619755	+	0.784796i	$0.287232\pi$
$348$	−778560.	−0.344623
$349$	338800.	0.148895	0.0744475	−	0.997225i	$-0.476281\pi$
$349$	338800.	0.148895	0.0744475	+	0.997225i	$0.476281\pi$
$350$	0	0
$351$	−2.25424e6	−0.976635
$352$	344064.	0.148007
$353$	362046.	0.154642	0.0773209	−	0.997006i	$-0.475363\pi$
$353$	362046.	0.154642	0.0773209	+	0.997006i	$0.475363\pi$
$354$	−1.37208e6	−0.581931
$355$	−2.37082e6	−0.998451
$356$	−427680.	−0.178852
$357$	0	0
$358$	42768.0	0.0176365
$359$	876528.	0.358946	0.179473	−	0.983763i	$-0.442561\pi$
$359$	876528.	0.358946	0.179473	+	0.983763i	$0.442561\pi$
$360$	−768768.	−0.312636
$361$	−2.25520e6	−0.910787
$362$	−2.18426e6	−0.876056
$363$	481550.	0.191812
$364$	0	0
$365$	302568.	0.118875
$366$	−979040.	−0.382030
$367$	−2.98062e6	−1.15516	−0.577578	−	0.816335i	$-0.696002\pi$
$367$	−2.98062e6	−1.15516	−0.577578	+	0.816335i	$0.696002\pi$
$368$	−1.07520e6	−0.413875
$369$	889746.	0.340173
$370$	4.81488e6	1.82844
$371$	0	0
$372$	−1.17952e6	−0.441924
$373$	3.91441e6	1.45678	0.728391	−	0.685162i	$-0.240268\pi$
$373$	3.91441e6	1.45678	0.728391	+	0.685162i	$0.240268\pi$
$374$	1.95955e6	0.724399
$375$	677040.	0.248620
$376$	−115968.	−0.0423027
$377$	−2.84174e6	−1.02975
$378$	0	0
$379$	3.60661e6	1.28974	0.644868	−	0.764294i	$-0.276912\pi$
$379$	3.60661e6	1.28974	0.644868	+	0.764294i	$0.276912\pi$
$380$	631680.	0.224408
$381$	−3.15992e6	−1.11523
$382$	2.30390e6	0.807804
$383$	2.66644e6	0.928826	0.464413	−	0.885619i	$-0.346265\pi$
$383$	2.66644e6	0.928826	0.464413	+	0.885619i	$0.346265\pi$
$384$	163840.	0.0567012
$385$	0	0
$386$	1.65575e6	0.565623
$387$	−529672.	−0.179775
$388$	271648.	0.0916067
$389$	−213366.	−0.0714910	−0.0357455	−	0.999361i	$-0.511381\pi$
$389$	−213366.	−0.0714910	−0.0357455	+	0.999361i	$0.511381\pi$
$390$	1.96224e6	0.653267
$391$	−6.12360e6	−2.02565
$392$	0	0
$393$	−246660.	−0.0805596
$394$	1.97978e6	0.642506
$395$	−3.60125e6	−1.16134
$396$	768768.	0.246353
$397$	4.09408e6	1.30371	0.651854	−	0.758345i	$-0.273992\pi$
$397$	4.09408e6	1.30371	0.651854	+	0.758345i	$0.273992\pi$
$398$	2.08146e6	0.658657
$399$	0	0
$400$	1.00634e6	0.314480
$401$	942366.	0.292657	0.146328	−	0.989236i	$-0.453254\pi$
$401$	942366.	0.292657	0.146328	+	0.989236i	$0.453254\pi$
$402$	−698080.	−0.215447
$403$	−4.30525e6	−1.32049
$404$	−1.58726e6	−0.483833
$405$	323484.	0.0979976
$406$	0	0
$407$	−4.81488e6	−1.44079
$408$	933120.	0.277515
$409$	4.84561e6	1.43232	0.716160	−	0.697936i	$-0.245898\pi$
$409$	4.84561e6	1.43232	0.716160	+	0.697936i	$0.245898\pi$
$410$	−2.09059e6	−0.614200
$411$	−3.03234e6	−0.885469
$412$	2.10630e6	0.611333
$413$	0	0
$414$	−2.40240e6	−0.688881
$415$	−2.95697e6	−0.842804
$416$	598016.	0.169426
$417$	2.50586e6	0.705694
$418$	−631680.	−0.176830
$419$	1.73485e6	0.482754	0.241377	−	0.970431i	$-0.422401\pi$
$419$	1.73485e6	0.482754	0.241377	+	0.970431i	$0.422401\pi$
$420$	0	0
$421$	−1.65145e6	−0.454109	−0.227055	−	0.973882i	$-0.572910\pi$
$421$	−1.65145e6	−0.454109	−0.227055	+	0.973882i	$0.572910\pi$
$422$	−733136.	−0.200403
$423$	−259116.	−0.0704115
$424$	2.38349e6	0.643870
$425$	5.73140e6	1.53918
$426$	1.12896e6	0.301408
$427$	0	0
$428$	781632.	0.206250
$429$	−1.96224e6	−0.514765
$430$	1.24454e6	0.324593
$431$	4.14360e6	1.07445	0.537223	−	0.843440i	$-0.319473\pi$
$431$	4.14360e6	1.07445	0.537223	+	0.843440i	$0.319473\pi$
$432$	988160.	0.254752
$433$	3.03966e6	0.779121	0.389561	−	0.921001i	$-0.372627\pi$
$433$	3.03966e6	0.779121	0.389561	+	0.921001i	$0.372627\pi$
$434$	0	0
$435$	4.08744e6	1.03569
$436$	−901984.	−0.227239
$437$	1.97400e6	0.494474
$438$	−144080.	−0.0358855
$439$	−2.54271e6	−0.629703	−0.314852	−	0.949141i	$-0.601955\pi$
$439$	−2.54271e6	−0.629703	−0.314852	+	0.949141i	$0.601955\pi$
$440$	−1.80634e6	−0.444802
$441$	0	0
$442$	3.40589e6	0.829229
$443$	−2.43210e6	−0.588806	−0.294403	−	0.955681i	$-0.595121\pi$
$443$	−2.43210e6	−0.588806	−0.294403	+	0.955681i	$0.595121\pi$
$444$	−2.29280e6	−0.551961
$445$	2.24532e6	0.537500
$446$	−5.10986e6	−1.21639
$447$	605940.	0.143437
$448$	0	0
$449$	1.82853e6	0.428042	0.214021	−	0.976829i	$-0.431344\pi$
$449$	1.82853e6	0.428042	0.214021	+	0.976829i	$0.431344\pi$
$450$	2.24853e6	0.523441
$451$	2.09059e6	0.483981
$452$	139872.	0.0322021
$453$	−1.24448e6	−0.284933
$454$	−5.15057e6	−1.17278
$455$	0	0
$456$	−300800.	−0.0677432
$457$	1.58063e6	0.354030	0.177015	−	0.984208i	$-0.443356\pi$
$457$	1.58063e6	0.354030	0.177015	+	0.984208i	$0.443356\pi$
$458$	1.40374e6	0.312697
$459$	5.62788e6	1.24685
$460$	5.64480e6	1.24381
$461$	−5.09604e6	−1.11681	−0.558407	−	0.829567i	$-0.688587\pi$
$461$	−5.09604e6	−1.11681	−0.558407	+	0.829567i	$0.688587\pi$
$462$	0	0
$463$	−7.02338e6	−1.52263	−0.761313	−	0.648384i	$-0.775445\pi$
$463$	−7.02338e6	−1.52263	−0.761313	+	0.648384i	$0.775445\pi$
$464$	1.24570e6	0.268607
$465$	6.19248e6	1.32810
$466$	−3.34462e6	−0.713479
$467$	4.24845e6	0.901443	0.450722	−	0.892665i	$-0.351167\pi$
$467$	4.24845e6	0.901443	0.450722	+	0.892665i	$0.351167\pi$
$468$	1.33619e6	0.282003
$469$	0	0
$470$	608832.	0.127131
$471$	760400.	0.157939
$472$	2.19533e6	0.453570
$473$	−1.24454e6	−0.255775
$474$	1.71488e6	0.350581
$475$	−1.84757e6	−0.375722
$476$	0	0
$477$	5.32561e6	1.07170
$478$	−3.09734e6	−0.620040
$479$	−559284.	−0.111377	−0.0556883	−	0.998448i	$-0.517735\pi$
$479$	−559284.	−0.111377	−0.0556883	+	0.998448i	$0.517735\pi$
$480$	−860160.	−0.170403
$481$	−8.36872e6	−1.64929
$482$	−4.61140e6	−0.904097
$483$	0	0
$484$	−770480.	−0.149502
$485$	−1.42615e6	−0.275303
$486$	3.59788e6	0.690965
$487$	−1.32057e6	−0.252312	−0.126156	−	0.992010i	$-0.540264\pi$
$487$	−1.32057e6	−0.252312	−0.126156	+	0.992010i	$0.540264\pi$
$488$	1.56646e6	0.297763
$489$	−1.24256e6	−0.234988
$490$	0	0
$491$	6.27193e6	1.17408	0.587040	−	0.809558i	$-0.300293\pi$
$491$	6.27193e6	1.17408	0.587040	+	0.809558i	$0.300293\pi$
$492$	995520.	0.185412
$493$	7.09463e6	1.31466
$494$	−1.09792e6	−0.202420
$495$	−4.03603e6	−0.740358
$496$	1.88723e6	0.344446
$497$	0	0
$498$	1.40808e6	0.254422
$499$	−3.93785e6	−0.707959	−0.353979	−	0.935253i	$-0.615172\pi$
$499$	−3.93785e6	−0.707959	−0.353979	+	0.935253i	$0.615172\pi$
$500$	−1.08326e6	−0.193780
$501$	−724200.	−0.128903
$502$	5.43204e6	0.962063
$503$	7.59830e6	1.33905	0.669525	−	0.742790i	$-0.266498\pi$
$503$	7.59830e6	1.33905	0.669525	+	0.742790i	$0.266498\pi$
$504$	0	0
$505$	8.33314e6	1.45405
$506$	−5.64480e6	−0.980104
$507$	302370.	0.0522419
$508$	5.05587e6	0.869234
$509$	7.82664e6	1.33900	0.669501	−	0.742812i	$-0.266508\pi$
$509$	7.82664e6	1.33900	0.669501	+	0.742812i	$0.266508\pi$
$510$	−4.89888e6	−0.834010
$511$	0	0
$512$	−262144.	−0.0441942
$513$	−1.81420e6	−0.304363
$514$	−1.27097e6	−0.212191
$515$	−1.10581e7	−1.83722
$516$	−592640.	−0.0979866
$517$	−608832.	−0.100178
$518$	0	0
$519$	−4.41552e6	−0.719554
$520$	−3.13958e6	−0.509171
$521$	−8.94454e6	−1.44366	−0.721828	−	0.692072i	$-0.756698\pi$
$521$	−8.94454e6	−1.44366	−0.721828	+	0.692072i	$0.756698\pi$
$522$	2.78335e6	0.447087
$523$	−4.07481e6	−0.651407	−0.325704	−	0.945472i	$-0.605601\pi$
$523$	−4.07481e6	−0.651407	−0.325704	+	0.945472i	$0.605601\pi$
$524$	394656.	0.0627900
$525$	0	0
$526$	−4.20403e6	−0.662524
$527$	1.07484e7	1.68584
$528$	860160.	0.134275
$529$	1.12037e7	1.74069
$530$	−1.25133e7	−1.93501
$531$	4.90519e6	0.754952
$532$	0	0
$533$	3.63365e6	0.554019
$534$	−1.06920e6	−0.162258
$535$	−4.10357e6	−0.619837
$536$	1.11693e6	0.167924
$537$	106920.	0.0160001
$538$	4.75834e6	0.708760
$539$	0	0
$540$	−5.18784e6	−0.765600
$541$	−1.18676e7	−1.74329	−0.871644	−	0.490140i	$-0.836946\pi$
$541$	−1.18676e7	−1.74329	−0.871644	+	0.490140i	$0.836946\pi$
$542$	−5.72032e6	−0.836416
$543$	−5.46064e6	−0.794775
$544$	−1.49299e6	−0.216302
$545$	4.73542e6	0.682915
$546$	0	0
$547$	−5.37801e6	−0.768516	−0.384258	−	0.923226i	$-0.625543\pi$
$547$	−5.37801e6	−0.768516	−0.384258	+	0.923226i	$0.625543\pi$
$548$	4.85174e6	0.690155
$549$	3.50007e6	0.495616
$550$	5.28326e6	0.744724
$551$	−2.28702e6	−0.320916
$552$	−2.68800e6	−0.375475
$553$	0	0
$554$	−253208.	−0.0350512
$555$	1.20372e7	1.65880
$556$	−4.00938e6	−0.550034
$557$	−5.64878e6	−0.771466	−0.385733	−	0.922611i	$-0.626051\pi$
$557$	−5.64878e6	−0.771466	−0.385733	+	0.922611i	$0.626051\pi$
$558$	4.21678e6	0.573318
$559$	−2.16314e6	−0.292789
$560$	0	0
$561$	4.89888e6	0.657188
$562$	1.98646e6	0.265301
$563$	−4.56407e6	−0.606850	−0.303425	−	0.952855i	$-0.598130\pi$
$563$	−4.56407e6	−0.606850	−0.303425	+	0.952855i	$0.598130\pi$
$564$	−289920.	−0.0383778
$565$	−734328.	−0.0967763
$566$	−4.63367e6	−0.607973
$567$	0	0
$568$	−1.80634e6	−0.234924
$569$	8.00165e6	1.03609	0.518047	−	0.855352i	$-0.326659\pi$
$569$	8.00165e6	1.03609	0.518047	+	0.855352i	$0.326659\pi$
$570$	1.57920e6	0.203587
$571$	−1.37164e7	−1.76055	−0.880275	−	0.474464i	$-0.842642\pi$
$571$	−1.37164e7	−1.76055	−0.880275	+	0.474464i	$0.842642\pi$
$572$	3.13958e6	0.401220
$573$	5.75976e6	0.732855
$574$	0	0
$575$	−1.65102e7	−2.08249
$576$	−585728.	−0.0735597
$577$	−6.09797e6	−0.762510	−0.381255	−	0.924470i	$-0.624508\pi$
$577$	−6.09797e6	−0.762510	−0.381255	+	0.924470i	$0.624508\pi$
$578$	−2.82363e6	−0.351551
$579$	4.13938e6	0.513144
$580$	−6.53990e6	−0.807238
$581$	0	0
$582$	679120.	0.0831073
$583$	1.25133e7	1.52476
$584$	230528.	0.0279699
$585$	−7.01501e6	−0.847498
$586$	5.75544e6	0.692364
$587$	8.08462e6	0.968422	0.484211	−	0.874951i	$-0.339107\pi$
$587$	8.08462e6	0.968422	0.484211	+	0.874951i	$0.339107\pi$
$588$	0	0
$589$	−3.46484e6	−0.411524
$590$	−1.15255e7	−1.36310
$591$	4.94946e6	0.582893
$592$	3.66848e6	0.430211
$593$	−1.41575e6	−0.165330	−0.0826649	−	0.996577i	$-0.526343\pi$
$593$	−1.41575e6	−0.165330	−0.0826649	+	0.996577i	$0.526343\pi$
$594$	5.18784e6	0.603282
$595$	0	0
$596$	−969504.	−0.111798
$597$	5.20364e6	0.597546
$598$	−9.81120e6	−1.12194
$599$	8.75460e6	0.996941	0.498470	−	0.866907i	$-0.333895\pi$
$599$	8.75460e6	0.996941	0.498470	+	0.866907i	$0.333895\pi$
$600$	2.51584e6	0.285302
$601$	−8.70276e6	−0.982813	−0.491407	−	0.870930i	$-0.663517\pi$
$601$	−8.70276e6	−0.982813	−0.491407	+	0.870930i	$0.663517\pi$
$602$	0	0
$603$	2.49564e6	0.279504
$604$	1.99117e6	0.222083
$605$	4.04502e6	0.449296
$606$	−3.96816e6	−0.438942
$607$	1.69578e7	1.86809	0.934045	−	0.357157i	$-0.116254\pi$
$607$	1.69578e7	1.86809	0.934045	+	0.357157i	$0.116254\pi$
$608$	481280.	0.0528006
$609$	0	0
$610$	−8.22394e6	−0.894860
$611$	−1.05821e6	−0.114675
$612$	−3.33590e6	−0.360027
$613$	1.76743e7	1.89973	0.949866	−	0.312658i	$-0.101220\pi$
$613$	1.76743e7	1.89973	0.949866	+	0.312658i	$0.101220\pi$
$614$	−3.95639e6	−0.423524
$615$	−5.22648e6	−0.557213
$616$	0	0
$617$	−9.70636e6	−1.02646	−0.513232	−	0.858250i	$-0.671552\pi$
$617$	−9.70636e6	−1.02646	−0.513232	+	0.858250i	$0.671552\pi$
$618$	5.26576e6	0.554613
$619$	−1.48739e7	−1.56027	−0.780133	−	0.625613i	$-0.784849\pi$
$619$	−1.48739e7	−1.56027	−0.780133	+	0.625613i	$0.784849\pi$
$620$	−9.90797e6	−1.03515
$621$	−1.62120e7	−1.68697
$622$	−8.88202e6	−0.920525
$623$	0	0
$624$	1.49504e6	0.153706
$625$	−6.59724e6	−0.675557
$626$	9.32031e6	0.950593
$627$	−1.57920e6	−0.160424
$628$	−1.21664e6	−0.123101
$629$	2.08931e7	2.10561
$630$	0	0
$631$	1.26353e7	1.26331	0.631656	−	0.775248i	$-0.282375\pi$
$631$	1.26353e7	1.26331	0.631656	+	0.775248i	$0.282375\pi$
$632$	−2.74381e6	−0.273251
$633$	−1.83284e6	−0.181809
$634$	−1.71017e6	−0.168972
$635$	−2.65433e7	−2.61229
$636$	5.95872e6	0.584131
$637$	0	0
$638$	6.53990e6	0.636092
$639$	−4.03603e6	−0.391023
$640$	1.37626e6	0.132816
$641$	6.23398e6	0.599267	0.299634	−	0.954054i	$-0.403136\pi$
$641$	6.23398e6	0.599267	0.299634	+	0.954054i	$0.403136\pi$
$642$	1.95408e6	0.187113
$643$	−1.06874e7	−1.01940	−0.509701	−	0.860352i	$-0.670244\pi$
$643$	−1.06874e7	−1.01940	−0.509701	+	0.860352i	$0.670244\pi$
$644$	0	0
$645$	3.11136e6	0.294477
$646$	2.74104e6	0.258425
$647$	−1.83258e7	−1.72109	−0.860544	−	0.509376i	$-0.829876\pi$
$647$	−1.83258e7	−1.72109	−0.860544	+	0.509376i	$0.829876\pi$
$648$	246464.	0.0230577
$649$	1.15255e7	1.07411
$650$	9.18282e6	0.852496
$651$	0	0
$652$	1.98810e6	0.183155
$653$	−7.28857e6	−0.668897	−0.334448	−	0.942414i	$-0.608550\pi$
$653$	−7.28857e6	−0.668897	−0.334448	+	0.942414i	$0.608550\pi$
$654$	−2.25496e6	−0.206155
$655$	−2.07194e6	−0.188701
$656$	−1.59283e6	−0.144514
$657$	515086.	0.0465550
$658$	0	0
$659$	4.54337e6	0.407534	0.203767	−	0.979019i	$-0.434681\pi$
$659$	4.54337e6	0.407534	0.203767	+	0.979019i	$0.434681\pi$
$660$	−4.51584e6	−0.403533
$661$	2.10021e7	1.86964	0.934821	−	0.355120i	$-0.115560\pi$
$661$	2.10021e7	1.86964	0.934821	+	0.355120i	$0.115560\pi$
$662$	1.58646e6	0.140697
$663$	8.51472e6	0.752292
$664$	−2.25293e6	−0.198302
$665$	0	0
$666$	8.19676e6	0.716072
$667$	−2.04372e7	−1.77872
$668$	1.15872e6	0.100470
$669$	−1.27746e7	−1.10353
$670$	−5.86387e6	−0.504658
$671$	8.22394e6	0.705137
$672$	0	0
$673$	3.46923e6	0.295253	0.147627	−	0.989043i	$-0.452837\pi$
$673$	3.46923e6	0.295253	0.147627	+	0.989043i	$0.452837\pi$
$674$	1.28728e7	1.09150
$675$	1.51737e7	1.28183
$676$	−483792.	−0.0407185
$677$	1.80916e7	1.51707	0.758536	−	0.651631i	$-0.225915\pi$
$677$	1.80916e7	1.51707	0.758536	+	0.651631i	$0.225915\pi$
$678$	349680.	0.0292144
$679$	0	0
$680$	7.83821e6	0.650047
$681$	−1.28764e7	−1.06397
$682$	9.90797e6	0.815687
$683$	4.67752e6	0.383675	0.191838	−	0.981427i	$-0.438555\pi$
$683$	4.67752e6	0.383675	0.191838	+	0.981427i	$0.438555\pi$
$684$	1.07536e6	0.0878848
$685$	−2.54717e7	−2.07411
$686$	0	0
$687$	3.50936e6	0.283685
$688$	948224.	0.0763730
$689$	2.17493e7	1.74541
$690$	1.41120e7	1.12841
$691$	−1.68960e7	−1.34614	−0.673069	−	0.739579i	$-0.735024\pi$
$691$	−1.68960e7	−1.34614	−0.673069	+	0.739579i	$0.735024\pi$
$692$	7.06483e6	0.560837
$693$	0	0
$694$	−1.11207e7	−0.876466
$695$	2.10492e7	1.65300
$696$	3.11424e6	0.243685
$697$	−9.07168e6	−0.707303
$698$	−1.35520e6	−0.105285
$699$	−8.36154e6	−0.647282
$700$	0	0
$701$	2.40964e6	0.185207	0.0926035	−	0.995703i	$-0.470481\pi$
$701$	2.40964e6	0.185207	0.0926035	+	0.995703i	$0.470481\pi$
$702$	9.01696e6	0.690585
$703$	−6.73510e6	−0.513991
$704$	−1.37626e6	−0.104657
$705$	1.52208e6	0.115336
$706$	−1.44818e6	−0.109348
$707$	0	0
$708$	5.48832e6	0.411487
$709$	−5.77010e6	−0.431090	−0.215545	−	0.976494i	$-0.569153\pi$
$709$	−5.77010e6	−0.431090	−0.215545	+	0.976494i	$0.569153\pi$
$710$	9.48326e6	0.706012
$711$	−6.13070e6	−0.454816
$712$	1.71072e6	0.126468
$713$	−3.09624e7	−2.28092
$714$	0	0
$715$	−1.64828e7	−1.20578
$716$	−171072.	−0.0124709
$717$	−7.74336e6	−0.562512
$718$	−3.50611e6	−0.253813
$719$	1.43716e7	1.03677	0.518385	−	0.855147i	$-0.326533\pi$
$719$	1.43716e7	1.03677	0.518385	+	0.855147i	$0.326533\pi$
$720$	3.07507e6	0.221067
$721$	0	0
$722$	9.02080e6	0.644024
$723$	−1.15285e7	−0.820214
$724$	8.73702e6	0.619465
$725$	1.91282e7	1.35154
$726$	−1.92620e6	−0.135631
$727$	1.40705e7	0.987353	0.493676	−	0.869646i	$-0.335653\pi$
$727$	1.40705e7	0.987353	0.493676	+	0.869646i	$0.335653\pi$
$728$	0	0
$729$	9.93049e6	0.692073
$730$	−1.21027e6	−0.0840574
$731$	5.40043e6	0.373796
$732$	3.91616e6	0.270136
$733$	3.75000e6	0.257793	0.128897	−	0.991658i	$-0.458856\pi$
$733$	3.75000e6	0.257793	0.128897	+	0.991658i	$0.458856\pi$
$734$	1.19225e7	0.816819
$735$	0	0
$736$	4.30080e6	0.292654
$737$	5.86387e6	0.397664
$738$	−3.55898e6	−0.240539
$739$	2.61318e7	1.76019	0.880093	−	0.474802i	$-0.157480\pi$
$739$	2.61318e7	1.76019	0.880093	+	0.474802i	$0.157480\pi$
$740$	−1.92595e7	−1.29290
$741$	−2.74480e6	−0.183639
$742$	0	0
$743$	−159072.	−0.0105711	−0.00528557	−	0.999986i	$-0.501682\pi$
$743$	−159072.	−0.0105711	−0.00528557	+	0.999986i	$0.501682\pi$
$744$	4.71808e6	0.312488
$745$	5.08990e6	0.335984
$746$	−1.56577e7	−1.03010
$747$	−5.03389e6	−0.330067
$748$	−7.83821e6	−0.512227
$749$	0	0
$750$	−2.70816e6	−0.175801
$751$	−2.65311e7	−1.71654	−0.858272	−	0.513196i	$-0.828461\pi$
$751$	−2.65311e7	−1.71654	−0.858272	+	0.513196i	$0.828461\pi$
$752$	463872.	0.0299126
$753$	1.35801e7	0.872802
$754$	1.13670e7	0.728143
$755$	−1.04536e7	−0.667421
$756$	0	0
$757$	−1.52032e7	−0.964260	−0.482130	−	0.876100i	$-0.660137\pi$
$757$	−1.52032e7	−0.964260	−0.482130	+	0.876100i	$0.660137\pi$
$758$	−1.44264e7	−0.911981
$759$	−1.41120e7	−0.889169
$760$	−2.52672e6	−0.158680
$761$	−4.71380e6	−0.295059	−0.147530	−	0.989058i	$-0.547132\pi$
$761$	−4.71380e6	−0.295059	−0.147530	+	0.989058i	$0.547132\pi$
$762$	1.26397e7	0.788585
$763$	0	0
$764$	−9.21562e6	−0.571204
$765$	1.75135e7	1.08198
$766$	−1.06657e7	−0.656779
$767$	2.00324e7	1.22954
$768$	−655360.	−0.0400938
$769$	1.58977e6	0.0969434	0.0484717	−	0.998825i	$-0.484565\pi$
$769$	1.58977e6	0.0969434	0.0484717	+	0.998825i	$0.484565\pi$
$770$	0	0
$771$	−3.17742e6	−0.192504
$772$	−6.62301e6	−0.399956
$773$	9.69095e6	0.583334	0.291667	−	0.956520i	$-0.405790\pi$
$773$	9.69095e6	0.583334	0.291667	+	0.956520i	$0.405790\pi$
$774$	2.11869e6	0.127120
$775$	2.89793e7	1.73314
$776$	−1.08659e6	−0.0647757
$777$	0	0
$778$	853464.	0.0505518
$779$	2.92434e6	0.172657
$780$	−7.84896e6	−0.461929
$781$	−9.48326e6	−0.556327
$782$	2.44944e7	1.43235
$783$	1.87828e7	1.09485
$784$	0	0
$785$	6.38736e6	0.369954
$786$	986640.	0.0569642
$787$	1.57170e6	0.0904549	0.0452275	−	0.998977i	$-0.485599\pi$
$787$	1.57170e6	0.0904549	0.0452275	+	0.998977i	$0.485599\pi$
$788$	−7.91914e6	−0.454320
$789$	−1.05101e7	−0.601054
$790$	1.44050e7	0.821193
$791$	0	0
$792$	−3.07507e6	−0.174198
$793$	1.42940e7	0.807180
$794$	−1.63763e7	−0.921860
$795$	−3.12833e7	−1.75547
$796$	−8.32582e6	−0.465741
$797$	2.25298e6	0.125635	0.0628175	−	0.998025i	$-0.479991\pi$
$797$	2.25298e6	0.125635	0.0628175	+	0.998025i	$0.479991\pi$
$798$	0	0
$799$	2.64190e6	0.146403
$800$	−4.02534e6	−0.222371
$801$	3.82239e6	0.210501
$802$	−3.76946e6	−0.206940
$803$	1.21027e6	0.0662360
$804$	2.79232e6	0.152344
$805$	0	0
$806$	1.72210e7	0.933728
$807$	1.18958e7	0.643000
$808$	6.34906e6	0.342122
$809$	−2.37938e7	−1.27818	−0.639090	−	0.769132i	$-0.720689\pi$
$809$	−2.37938e7	−1.27818	−0.639090	+	0.769132i	$0.720689\pi$
$810$	−1.29394e6	−0.0692947
$811$	−5.32300e6	−0.284187	−0.142093	−	0.989853i	$-0.545383\pi$
$811$	−5.32300e6	−0.284187	−0.142093	+	0.989853i	$0.545383\pi$
$812$	0	0
$813$	−1.43008e7	−0.758812
$814$	1.92595e7	1.01879
$815$	−1.04375e7	−0.550431
$816$	−3.73248e6	−0.196233
$817$	−1.74088e6	−0.0912460
$818$	−1.93824e7	−1.01280
$819$	0	0
$820$	8.36237e6	0.434305
$821$	1.48802e7	0.770464	0.385232	−	0.922820i	$-0.374121\pi$
$821$	1.48802e7	0.770464	0.385232	+	0.922820i	$0.374121\pi$
$822$	1.21294e7	0.626121
$823$	2.00601e7	1.03236	0.516182	−	0.856479i	$-0.327353\pi$
$823$	2.00601e7	1.03236	0.516182	+	0.856479i	$0.327353\pi$
$824$	−8.42522e6	−0.432278
$825$	1.32082e7	0.675628
$826$	0	0
$827$	1.21539e7	0.617949	0.308975	−	0.951070i	$-0.400014\pi$
$827$	1.21539e7	0.617949	0.308975	+	0.951070i	$0.400014\pi$
$828$	9.60960e6	0.487113
$829$	−3.21197e7	−1.62325	−0.811625	−	0.584179i	$-0.801417\pi$
$829$	−3.21197e7	−1.62325	−0.811625	+	0.584179i	$0.801417\pi$
$830$	1.18279e7	0.595952
$831$	−633020.	−0.0317991
$832$	−2.39206e6	−0.119802
$833$	0	0
$834$	−1.00234e7	−0.499001
$835$	−6.08328e6	−0.301941
$836$	2.52672e6	0.125038
$837$	2.84559e7	1.40397
$838$	−6.93938e6	−0.341359
$839$	1.01320e6	0.0496922	0.0248461	−	0.999691i	$-0.492090\pi$
$839$	1.01320e6	0.0496922	0.0248461	+	0.999691i	$0.492090\pi$
$840$	0	0
$841$	3.16681e6	0.154394
$842$	6.60580e6	0.321104
$843$	4.96614e6	0.240686
$844$	2.93254e6	0.141706
$845$	2.53991e6	0.122370
$846$	1.03646e6	0.0497884
$847$	0	0
$848$	−9.53395e6	−0.455285
$849$	−1.15842e7	−0.551564
$850$	−2.29256e7	−1.08836
$851$	−6.01860e7	−2.84886
$852$	−4.51584e6	−0.213128
$853$	−234824.	−0.0110502	−0.00552510	−	0.999985i	$-0.501759\pi$
$853$	−234824.	−0.0110502	−0.00552510	+	0.999985i	$0.501759\pi$
$854$	0	0
$855$	−5.64564e6	−0.264118
$856$	−3.12653e6	−0.145840
$857$	−2.83802e7	−1.31997	−0.659985	−	0.751279i	$-0.729437\pi$
$857$	−2.83802e7	−1.31997	−0.659985	+	0.751279i	$0.729437\pi$
$858$	7.84896e6	0.363994
$859$	−4.00081e7	−1.84997	−0.924986	−	0.380001i	$-0.875924\pi$
$859$	−4.00081e7	−1.84997	−0.924986	+	0.380001i	$0.875924\pi$
$860$	−4.97818e6	−0.229522
$861$	0	0
$862$	−1.65744e7	−0.759748
$863$	−2.08030e7	−0.950823	−0.475411	−	0.879764i	$-0.657701\pi$
$863$	−2.08030e7	−0.950823	−0.475411	+	0.879764i	$0.657701\pi$
$864$	−3.95264e6	−0.180137
$865$	−3.70904e7	−1.68547
$866$	−1.21586e7	−0.550922
$867$	−7.05907e6	−0.318933
$868$	0	0
$869$	−1.44050e7	−0.647088
$870$	−1.63498e7	−0.732341
$871$	1.01920e7	0.455211
$872$	3.60794e6	0.160682
$873$	−2.42785e6	−0.107817
$874$	−7.89600e6	−0.349646
$875$	0	0
$876$	576320.	0.0253748
$877$	3.03559e7	1.33273	0.666367	−	0.745624i	$-0.267848\pi$
$877$	3.03559e7	1.33273	0.666367	+	0.745624i	$0.267848\pi$
$878$	1.01708e7	0.445267
$879$	1.43886e7	0.628125
$880$	7.22534e6	0.314523
$881$	2.58936e7	1.12396	0.561981	−	0.827150i	$-0.310039\pi$
$881$	2.58936e7	1.12396	0.561981	+	0.827150i	$0.310039\pi$
$882$	0	0
$883$	−1.88813e7	−0.814950	−0.407475	−	0.913216i	$-0.633591\pi$
$883$	−1.88813e7	−0.814950	−0.407475	+	0.913216i	$0.633591\pi$
$884$	−1.36236e7	−0.586354
$885$	−2.88137e7	−1.23663
$886$	9.72840e6	0.416349
$887$	2.34431e7	1.00048	0.500238	−	0.865888i	$-0.333246\pi$
$887$	2.34431e7	1.00048	0.500238	+	0.865888i	$0.333246\pi$
$888$	9.17120e6	0.390296
$889$	0	0
$890$	−8.98128e6	−0.380070
$891$	1.29394e6	0.0546033
$892$	2.04394e7	0.860115
$893$	−851640.	−0.0357378
$894$	−2.42376e6	−0.101425
$895$	898128.	0.0374784
$896$	0	0
$897$	−2.45280e7	−1.01784
$898$	−7.31412e6	−0.302671
$899$	3.58722e7	1.48033
$900$	−8.99413e6	−0.370129
$901$	−5.42988e7	−2.22833
$902$	−8.36237e6	−0.342226
$903$	0	0
$904$	−559488.	−0.0227703
$905$	−4.58694e7	−1.86166
$906$	4.97792e6	0.201478
$907$	−5.60873e6	−0.226384	−0.113192	−	0.993573i	$-0.536108\pi$
$907$	−5.60873e6	−0.226384	−0.113192	+	0.993573i	$0.536108\pi$
$908$	2.06023e7	0.829279
$909$	1.41862e7	0.569450
$910$	0	0
$911$	2.16215e7	0.863156	0.431578	−	0.902076i	$-0.357957\pi$
$911$	2.16215e7	0.863156	0.431578	+	0.902076i	$0.357957\pi$
$912$	1.20320e6	0.0479017
$913$	−1.18279e7	−0.469602
$914$	−6.32252e6	−0.250337
$915$	−2.05598e7	−0.811834
$916$	−5.61498e6	−0.221110
$917$	0	0
$918$	−2.25115e7	−0.881654
$919$	4.51695e7	1.76424	0.882119	−	0.471028i	$-0.156117\pi$
$919$	4.51695e7	1.76424	0.882119	+	0.471028i	$0.156117\pi$
$920$	−2.25792e7	−0.879506
$921$	−9.89098e6	−0.384229
$922$	2.03842e7	0.789706
$923$	−1.64828e7	−0.636835
$924$	0	0
$925$	5.63312e7	2.16469
$926$	2.80935e7	1.07666
$927$	−1.88251e7	−0.719512
$928$	−4.98278e6	−0.189934
$929$	2.28729e7	0.869524	0.434762	−	0.900545i	$-0.356832\pi$
$929$	2.28729e7	0.869524	0.434762	+	0.900545i	$0.356832\pi$
$930$	−2.47699e7	−0.939112
$931$	0	0
$932$	1.33785e7	0.504506
$933$	−2.22050e7	−0.835117
$934$	−1.69938e7	−0.637417
$935$	4.11506e7	1.53938
$936$	−5.34477e6	−0.199406
$937$	1.79616e7	0.668336	0.334168	−	0.942514i	$-0.391545\pi$
$937$	1.79616e7	0.668336	0.334168	+	0.942514i	$0.391545\pi$
$938$	0	0
$939$	2.33008e7	0.862395
$940$	−2.43533e6	−0.0898955
$941$	1.79697e7	0.661558	0.330779	−	0.943708i	$-0.392689\pi$
$941$	1.79697e7	0.661558	0.330779	+	0.943708i	$0.392689\pi$
$942$	−3.04160e6	−0.111680
$943$	2.61324e7	0.956974
$944$	−8.78131e6	−0.320722
$945$	0	0
$946$	4.97818e6	0.180860
$947$	4.32115e7	1.56576	0.782879	−	0.622174i	$-0.213750\pi$
$947$	4.32115e7	1.56576	0.782879	+	0.622174i	$0.213750\pi$
$948$	−6.85952e6	−0.247898
$949$	2.10357e6	0.0758213
$950$	7.39028e6	0.265676
$951$	−4.27542e6	−0.153295
$952$	0	0
$953$	−7.50965e6	−0.267848	−0.133924	−	0.990992i	$-0.542758\pi$
$953$	−7.50965e6	−0.267848	−0.133924	+	0.990992i	$0.542758\pi$
$954$	−2.13024e7	−0.757806
$955$	4.83820e7	1.71662
$956$	1.23894e7	0.438434
$957$	1.63498e7	0.577074
$958$	2.23714e6	0.0787551
$959$	0	0
$960$	3.44064e6	0.120493
$961$	2.57172e7	0.898288
$962$	3.34749e7	1.16622
$963$	−6.98584e6	−0.242746
$964$	1.84456e7	0.639293
$965$	3.47708e7	1.20198
$966$	0	0
$967$	−1.69305e7	−0.582242	−0.291121	−	0.956686i	$-0.594028\pi$
$967$	−1.69305e7	−0.582242	−0.291121	+	0.956686i	$0.594028\pi$
$968$	3.08192e6	0.105714
$969$	6.85260e6	0.234448
$970$	5.70461e6	0.194669
$971$	−2.86144e7	−0.973949	−0.486974	−	0.873416i	$-0.661899\pi$
$971$	−2.86144e7	−0.973949	−0.486974	+	0.873416i	$0.661899\pi$
$972$	−1.43915e7	−0.488586
$973$	0	0
$974$	5.28227e6	0.178412
$975$	2.29570e7	0.773400
$976$	−6.26586e6	−0.210550
$977$	3.69445e7	1.23826	0.619132	−	0.785287i	$-0.287485\pi$
$977$	3.69445e7	1.23826	0.619132	+	0.785287i	$0.287485\pi$
$978$	4.97024e6	0.166161
$979$	8.98128e6	0.299489
$980$	0	0
$981$	8.06148e6	0.267450
$982$	−2.50877e7	−0.830200
$983$	3.88787e7	1.28330	0.641650	−	0.766998i	$-0.278250\pi$
$983$	3.88787e7	1.28330	0.641650	+	0.766998i	$0.278250\pi$
$984$	−3.98208e6	−0.131106
$985$	4.15755e7	1.36536
$986$	−2.83785e7	−0.929603
$987$	0	0
$988$	4.39168e6	0.143133
$989$	−1.55568e7	−0.505743
$990$	1.61441e7	0.523512
$991$	2.49212e7	0.806092	0.403046	−	0.915180i	$-0.367951\pi$
$991$	2.49212e7	0.806092	0.403046	+	0.915180i	$0.367951\pi$
$992$	−7.54893e6	−0.243560
$993$	3.96616e6	0.127643
$994$	0	0
$995$	4.37106e7	1.39968
$996$	−5.63232e6	−0.179903
$997$	−1.01956e7	−0.324845	−0.162422	−	0.986721i	$-0.551931\pi$
$997$	−1.01956e7	−0.324845	−0.162422	+	0.986721i	$0.551931\pi$
$998$	1.57514e7	0.500603
$999$	5.53138e7	1.75356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.6.a.a.1.1		1
3.2	odd	2		882.6.a.x.1.1		1
4.3	odd	2		784.6.a.i.1.1		1
7.2	even	3		98.6.c.d.67.1		2
7.3	odd	6		98.6.c.c.79.1		2
7.4	even	3		98.6.c.d.79.1		2
7.5	odd	6		98.6.c.c.67.1		2
7.6	odd	2		14.6.a.a.1.1	✓	1
21.20	even	2		126.6.a.f.1.1		1
28.27	even	2		112.6.a.c.1.1		1
35.13	even	4		350.6.c.d.99.2		2
35.27	even	4		350.6.c.d.99.1		2
35.34	odd	2		350.6.a.i.1.1		1
56.13	odd	2		448.6.a.e.1.1		1
56.27	even	2		448.6.a.l.1.1		1
84.83	odd	2		1008.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.a.1.1	✓	1	7.6	odd	2
98.6.a.a.1.1		1	1.1	even	1	trivial
98.6.c.c.67.1		2	7.5	odd	6
98.6.c.c.79.1		2	7.3	odd	6
98.6.c.d.67.1		2	7.2	even	3
98.6.c.d.79.1		2	7.4	even	3
112.6.a.c.1.1		1	28.27	even	2
126.6.a.f.1.1		1	21.20	even	2
350.6.a.i.1.1		1	35.34	odd	2
350.6.c.d.99.1		2	35.27	even	4
350.6.c.d.99.2		2	35.13	even	4
448.6.a.e.1.1		1	56.13	odd	2
448.6.a.l.1.1		1	56.27	even	2
784.6.a.i.1.1		1	4.3	odd	2
882.6.a.x.1.1		1	3.2	odd	2
1008.6.a.b.1.1		1	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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