LMFDB - Embedded newform 112.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(1,112)] 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("112.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-10,0,84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.9629878191$
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	$\chi$	$=$	112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-10.0000 q^{3} +84.0000 q^{5} -49.0000 q^{7} -143.000 q^{9} +336.000 q^{11} +584.000 q^{13} -840.000 q^{15} -1458.00 q^{17} -470.000 q^{19} +490.000 q^{21} +4200.00 q^{23} +3931.00 q^{25} +3860.00 q^{27} +4866.00 q^{29} +7372.00 q^{31} -3360.00 q^{33} -4116.00 q^{35} +14330.0 q^{37} -5840.00 q^{39} +6222.00 q^{41} -3704.00 q^{43} -12012.0 q^{45} +1812.00 q^{47} +2401.00 q^{49} +14580.0 q^{51} -37242.0 q^{53} +28224.0 q^{55} +4700.00 q^{57} -34302.0 q^{59} +24476.0 q^{61} +7007.00 q^{63} +49056.0 q^{65} +17452.0 q^{67} -42000.0 q^{69} -28224.0 q^{71} +3602.00 q^{73} -39310.0 q^{75} -16464.0 q^{77} -42872.0 q^{79} -3851.00 q^{81} +35202.0 q^{83} -122472. q^{85} -48660.0 q^{87} +26730.0 q^{89} -28616.0 q^{91} -73720.0 q^{93} -39480.0 q^{95} -16978.0 q^{97} -48048.0 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−10.0000	−0.641500	−0.320750	−	0.947164i	$-0.603935\pi$
$3$	−10.0000	−0.641500	−0.320750	+	0.947164i	$0.603935\pi$
$4$	0	0
$5$	84.0000	1.50264	0.751319	−	0.659939i	$-0.229418\pi$
$5$	84.0000	1.50264	0.751319	+	0.659939i	$0.229418\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−143.000	−0.588477
$10$	0	0
$11$	336.000	0.837255	0.418627	−	0.908158i	$-0.362511\pi$
$11$	336.000	0.837255	0.418627	+	0.908158i	$0.362511\pi$
$12$	0	0
$13$	584.000	0.958417	0.479208	−	0.877701i	$-0.340924\pi$
$13$	584.000	0.958417	0.479208	+	0.877701i	$0.340924\pi$
$14$	0	0
$15$	−840.000	−0.963943
$16$	0	0
$17$	−1458.00	−1.22359	−0.611794	−	0.791017i	$-0.709552\pi$
$17$	−1458.00	−1.22359	−0.611794	+	0.791017i	$0.709552\pi$
$18$	0	0
$19$	−470.000	−0.298685	−0.149343	−	0.988786i	$-0.547716\pi$
$19$	−470.000	−0.298685	−0.149343	+	0.988786i	$0.547716\pi$
$20$	0	0
$21$	490.000	0.242464
$22$	0	0
$23$	4200.00	1.65550	0.827751	−	0.561096i	$-0.189620\pi$
$23$	4200.00	1.65550	0.827751	+	0.561096i	$0.189620\pi$
$24$	0	0
$25$	3931.00	1.25792
$26$	0	0
$27$	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	$0.319477\pi$
$30$	0	0
$31$	7372.00	1.37778	0.688892	−	0.724864i	$-0.258097\pi$
$31$	7372.00	1.37778	0.688892	+	0.724864i	$0.258097\pi$
$32$	0	0
$33$	−3360.00	−0.537099
$34$	0	0
$35$	−4116.00	−0.567944
$36$	0	0
$37$	14330.0	1.72085	0.860423	−	0.509581i	$-0.170200\pi$
$37$	14330.0	1.72085	0.860423	+	0.509581i	$0.170200\pi$
$38$	0	0
$39$	−5840.00	−0.614825
$40$	0	0
$41$	6222.00	0.578057	0.289028	−	0.957321i	$-0.406668\pi$
$41$	6222.00	0.578057	0.289028	+	0.957321i	$0.406668\pi$
$42$	0	0
$43$	−3704.00	−0.305492	−0.152746	−	0.988265i	$-0.548812\pi$
$43$	−3704.00	−0.305492	−0.152746	+	0.988265i	$0.548812\pi$
$44$	0	0
$45$	−12012.0	−0.884268
$46$	0	0
$47$	1812.00	0.119650	0.0598251	−	0.998209i	$-0.480946\pi$
$47$	1812.00	0.119650	0.0598251	+	0.998209i	$0.480946\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	14580.0	0.784932
$52$	0	0
$53$	−37242.0	−1.82114	−0.910570	−	0.413355i	$-0.864357\pi$
$53$	−37242.0	−1.82114	−0.910570	+	0.413355i	$0.864357\pi$
$54$	0	0
$55$	28224.0	1.25809
$56$	0	0
$57$	4700.00	0.191607
$58$	0	0
$59$	−34302.0	−1.28289	−0.641445	−	0.767169i	$-0.721665\pi$
$59$	−34302.0	−1.28289	−0.641445	+	0.767169i	$0.721665\pi$
$60$	0	0
$61$	24476.0	0.842201	0.421101	−	0.907014i	$-0.361644\pi$
$61$	24476.0	0.842201	0.421101	+	0.907014i	$0.361644\pi$
$62$	0	0
$63$	7007.00	0.222424
$64$	0	0
$65$	49056.0	1.44015
$66$	0	0
$67$	17452.0	0.474961	0.237481	−	0.971392i	$-0.423678\pi$
$67$	17452.0	0.474961	0.237481	+	0.971392i	$0.423678\pi$
$68$	0	0
$69$	−42000.0	−1.06201
$70$	0	0
$71$	−28224.0	−0.664466	−0.332233	−	0.943197i	$-0.607802\pi$
$71$	−28224.0	−0.664466	−0.332233	+	0.943197i	$0.607802\pi$
$72$	0	0
$73$	3602.00	0.0791109	0.0395555	−	0.999217i	$-0.487406\pi$
$73$	3602.00	0.0791109	0.0395555	+	0.999217i	$0.487406\pi$
$74$	0	0
$75$	−39310.0	−0.806956
$76$	0	0
$77$	−16464.0	−0.316453
$78$	0	0
$79$	−42872.0	−0.772869	−0.386435	−	0.922317i	$-0.626294\pi$
$79$	−42872.0	−0.772869	−0.386435	+	0.922317i	$0.626294\pi$
$80$	0	0
$81$	−3851.00	−0.0652170
$82$	0	0
$83$	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
$86$	0	0
$87$	−48660.0	−0.689246
$88$	0	0
$89$	26730.0	0.357704	0.178852	−	0.983876i	$-0.442762\pi$
$89$	26730.0	0.357704	0.178852	+	0.983876i	$0.442762\pi$
$90$	0	0
$91$	−28616.0	−0.362248
$92$	0	0
$93$	−73720.0	−0.883849
$94$	0	0
$95$	−39480.0	−0.448816
$96$	0	0
$97$	−16978.0	−0.183213	−0.0916067	−	0.995795i	$-0.529200\pi$
$97$	−16978.0	−0.183213	−0.0916067	+	0.995795i	$0.529200\pi$
$98$	0	0
$99$	−48048.0	−0.492705
$100$	0	0
$101$	99204.0	0.967667	0.483833	−	0.875160i	$-0.339244\pi$
$101$	99204.0	0.967667	0.483833	+	0.875160i	$0.339244\pi$
$102$	0	0
$103$	131644.	1.22267	0.611333	−	0.791373i	$-0.290634\pi$
$103$	131644.	1.22267	0.611333	+	0.791373i	$0.290634\pi$
$104$	0	0
$105$	41160.0	0.364336
$106$	0	0
$107$	−48852.0	−0.412499	−0.206250	−	0.978499i	$-0.566126\pi$
$107$	−48852.0	−0.412499	−0.206250	+	0.978499i	$0.566126\pi$
$108$	0	0
$109$	−56374.0	−0.454478	−0.227239	−	0.973839i	$-0.572970\pi$
$109$	−56374.0	−0.454478	−0.227239	+	0.973839i	$0.572970\pi$
$110$	0	0
$111$	−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	$0.489748\pi$
$114$	0	0
$115$	352800.	2.48762
$116$	0	0
$117$	−83512.0	−0.564007
$118$	0	0
$119$	71442.0	0.462473
$120$	0	0
$121$	−48155.0	−0.299005
$122$	0	0
$123$	−62220.0	−0.370823
$124$	0	0
$125$	67704.0	0.387560
$126$	0	0
$127$	−315992.	−1.73847	−0.869234	−	0.494401i	$-0.835388\pi$
$127$	−315992.	−1.73847	−0.869234	+	0.494401i	$0.835388\pi$
$128$	0	0
$129$	37040.0	0.195973
$130$	0	0
$131$	24666.0	0.125580	0.0627900	−	0.998027i	$-0.480000\pi$
$131$	24666.0	0.125580	0.0627900	+	0.998027i	$0.480000\pi$
$132$	0	0
$133$	23030.0	0.112892
$134$	0	0
$135$	324240.	1.53120
$136$	0	0
$137$	303234.	1.38031	0.690155	−	0.723662i	$-0.257542\pi$
$137$	303234.	1.38031	0.690155	+	0.723662i	$0.257542\pi$
$138$	0	0
$139$	−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
$142$	0	0
$143$	196224.	0.802439
$144$	0	0
$145$	408744.	1.61448
$146$	0	0
$147$	−24010.0	−0.0916429
$148$	0	0
$149$	−60594.0	−0.223596	−0.111798	−	0.993731i	$-0.535661\pi$
$149$	−60594.0	−0.223596	−0.111798	+	0.993731i	$0.535661\pi$
$150$	0	0
$151$	−124448.	−0.444166	−0.222083	−	0.975028i	$-0.571286\pi$
$151$	−124448.	−0.444166	−0.222083	+	0.975028i	$0.571286\pi$
$152$	0	0
$153$	208494.	0.720054
$154$	0	0
$155$	619248.	2.07031
$156$	0	0
$157$	76040.0	0.246203	0.123101	−	0.992394i	$-0.460716\pi$
$157$	76040.0	0.246203	0.123101	+	0.992394i	$0.460716\pi$
$158$	0	0
$159$	372420.	1.16826
$160$	0	0
$161$	−205800.	−0.625721
$162$	0	0
$163$	−124256.	−0.366310	−0.183155	−	0.983084i	$-0.558631\pi$
$163$	−124256.	−0.366310	−0.183155	+	0.983084i	$0.558631\pi$
$164$	0	0
$165$	−282240.	−0.807065
$166$	0	0
$167$	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
$170$	0	0
$171$	67210.0	0.175770
$172$	0	0
$173$	−441552.	−1.12167	−0.560837	−	0.827926i	$-0.689521\pi$
$173$	−441552.	−1.12167	−0.560837	+	0.827926i	$0.689521\pi$
$174$	0	0
$175$	−192619.	−0.475449
$176$	0	0
$177$	343020.	0.822974
$178$	0	0
$179$	10692.0	0.0249417	0.0124709	−	0.999922i	$-0.496030\pi$
$179$	10692.0	0.0249417	0.0124709	+	0.999922i	$0.496030\pi$
$180$	0	0
$181$	−546064.	−1.23893	−0.619465	−	0.785024i	$-0.712651\pi$
$181$	−546064.	−1.23893	−0.619465	+	0.785024i	$0.712651\pi$
$182$	0	0
$183$	−244760.	−0.540272
$184$	0	0
$185$	1.20372e6	2.58581
$186$	0	0
$187$	−489888.	−1.02445
$188$	0	0
$189$	−189140.	−0.385149
$190$	0	0
$191$	575976.	1.14241	0.571204	−	0.820808i	$-0.306477\pi$
$191$	575976.	1.14241	0.571204	+	0.820808i	$0.306477\pi$
$192$	0	0
$193$	−413938.	−0.799912	−0.399956	−	0.916534i	$-0.630975\pi$
$193$	−413938.	−0.799912	−0.399956	+	0.916534i	$0.630975\pi$
$194$	0	0
$195$	−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	$0.650118\pi$
$198$	0	0
$199$	−520364.	−0.931482	−0.465741	−	0.884921i	$-0.654212\pi$
$199$	−520364.	−0.931482	−0.465741	+	0.884921i	$0.654212\pi$
$200$	0	0
$201$	−174520.	−0.304688
$202$	0	0
$203$	−238434.	−0.406095
$204$	0	0
$205$	522648.	0.868610
$206$	0	0
$207$	−600600.	−0.974225
$208$	0	0
$209$	−157920.	−0.250076
$210$	0	0
$211$	−183284.	−0.283412	−0.141706	−	0.989909i	$-0.545259\pi$
$211$	−183284.	−0.283412	−0.141706	+	0.989909i	$0.545259\pi$
$212$	0	0
$213$	282240.	0.426255
$214$	0	0
$215$	−311136.	−0.459044
$216$	0	0
$217$	−361228.	−0.520753
$218$	0	0
$219$	−36020.0	−0.0507497
$220$	0	0
$221$	−851472.	−1.17271
$222$	0	0
$223$	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
$226$	0	0
$227$	1.28764e6	1.65856	0.829279	−	0.558835i	$-0.188752\pi$
$227$	1.28764e6	1.65856	0.829279	+	0.558835i	$0.188752\pi$
$228$	0	0
$229$	350936.	0.442221	0.221110	−	0.975249i	$-0.429032\pi$
$229$	350936.	0.442221	0.221110	+	0.975249i	$0.429032\pi$
$230$	0	0
$231$	164640.	0.203004
$232$	0	0
$233$	836154.	1.00901	0.504506	−	0.863408i	$-0.331675\pi$
$233$	836154.	1.00901	0.504506	+	0.863408i	$0.331675\pi$
$234$	0	0
$235$	152208.	0.179791
$236$	0	0
$237$	428720.	0.495796
$238$	0	0
$239$	−774336.	−0.876869	−0.438434	−	0.898763i	$-0.644467\pi$
$239$	−774336.	−0.876869	−0.438434	+	0.898763i	$0.644467\pi$
$240$	0	0
$241$	−1.15285e6	−1.27859	−0.639293	−	0.768963i	$-0.720773\pi$
$241$	−1.15285e6	−1.27859	−0.639293	+	0.768963i	$0.720773\pi$
$242$	0	0
$243$	−899470.	−0.977172
$244$	0	0
$245$	201684.	0.214663
$246$	0	0
$247$	−274480.	−0.286265
$248$	0	0
$249$	−352020.	−0.359806
$250$	0	0
$251$	−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
$254$	0	0
$255$	1.22472e6	1.17947
$256$	0	0
$257$	−317742.	−0.300083	−0.150042	−	0.988680i	$-0.547941\pi$
$257$	−317742.	−0.300083	−0.150042	+	0.988680i	$0.547941\pi$
$258$	0	0
$259$	−702170.	−0.650418
$260$	0	0
$261$	−695838.	−0.632276
$262$	0	0
$263$	−1.05101e6	−0.936951	−0.468475	−	0.883477i	$-0.655196\pi$
$263$	−1.05101e6	−0.936951	−0.468475	+	0.883477i	$0.655196\pi$
$264$	0	0
$265$	−3.12833e6	−2.73651
$266$	0	0
$267$	−267300.	−0.229467
$268$	0	0
$269$	1.18958e6	1.00234	0.501169	−	0.865349i	$-0.332903\pi$
$269$	1.18958e6	1.00234	0.501169	+	0.865349i	$0.332903\pi$
$270$	0	0
$271$	1.43008e6	1.18287	0.591435	−	0.806353i	$-0.298562\pi$
$271$	1.43008e6	1.18287	0.591435	+	0.806353i	$0.298562\pi$
$272$	0	0
$273$	286160.	0.232382
$274$	0	0
$275$	1.32082e6	1.05320
$276$	0	0
$277$	63302.0	0.0495699	0.0247849	−	0.999693i	$-0.492110\pi$
$277$	63302.0	0.0495699	0.0247849	+	0.999693i	$0.492110\pi$
$278$	0	0
$279$	−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	$0.560070\pi$
$282$	0	0
$283$	1.15842e6	0.859803	0.429902	−	0.902876i	$-0.358548\pi$
$283$	1.15842e6	0.859803	0.429902	+	0.902876i	$0.358548\pi$
$284$	0	0
$285$	394800.	0.287915
$286$	0	0
$287$	−304878.	−0.218485
$288$	0	0
$289$	705907.	0.497168
$290$	0	0
$291$	169780.	0.117531
$292$	0	0
$293$	1.43886e6	0.979151	0.489575	−	0.871961i	$-0.337152\pi$
$293$	1.43886e6	0.979151	0.489575	+	0.871961i	$0.337152\pi$
$294$	0	0
$295$	−2.88137e6	−1.92772
$296$	0	0
$297$	1.29696e6	0.853170
$298$	0	0
$299$	2.45280e6	1.58666
$300$	0	0
$301$	181496.	0.115465
$302$	0	0
$303$	−992040.	−0.620758
$304$	0	0
$305$	2.05598e6	1.26552
$306$	0	0
$307$	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
$310$	0	0
$311$	2.22050e6	1.30182	0.650909	−	0.759155i	$-0.274388\pi$
$311$	2.22050e6	1.30182	0.650909	+	0.759155i	$0.274388\pi$
$312$	0	0
$313$	2.33008e6	1.34434	0.672171	−	0.740396i	$-0.265362\pi$
$313$	2.33008e6	1.34434	0.672171	+	0.740396i	$0.265362\pi$
$314$	0	0
$315$	588588.	0.334222
$316$	0	0
$317$	427542.	0.238963	0.119481	−	0.992836i	$-0.461877\pi$
$317$	427542.	0.238963	0.119481	+	0.992836i	$0.461877\pi$
$318$	0	0
$319$	1.63498e6	0.899569
$320$	0	0
$321$	488520.	0.264618
$322$	0	0
$323$	685260.	0.365468
$324$	0	0
$325$	2.29570e6	1.20561
$326$	0	0
$327$	563740.	0.291548
$328$	0	0
$329$	−88788.0	−0.0452235
$330$	0	0
$331$	396616.	0.198976	0.0994879	−	0.995039i	$-0.468280\pi$
$331$	396616.	0.198976	0.0994879	+	0.995039i	$0.468280\pi$
$332$	0	0
$333$	−2.04919e6	−1.01268
$334$	0	0
$335$	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	$0.780646\pi$
$338$	0	0
$339$	−87420.0	−0.0413154
$340$	0	0
$341$	2.47699e6	1.15356
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−3.52800e6	−1.59581
$346$	0	0
$347$	−2.78018e6	−1.23951	−0.619755	−	0.784796i	$-0.712768\pi$
$347$	−2.78018e6	−1.23951	−0.619755	+	0.784796i	$0.712768\pi$
$348$	0	0
$349$	−338800.	−0.148895	−0.0744475	−	0.997225i	$-0.523719\pi$
$349$	−338800.	−0.148895	−0.0744475	+	0.997225i	$0.523719\pi$
$350$	0	0
$351$	2.25424e6	0.976635
$352$	0	0
$353$	−362046.	−0.154642	−0.0773209	−	0.997006i	$-0.524637\pi$
$353$	−362046.	−0.154642	−0.0773209	+	0.997006i	$0.524637\pi$
$354$	0	0
$355$	−2.37082e6	−0.998451
$356$	0	0
$357$	−714420.	−0.296676
$358$	0	0
$359$	−876528.	−0.358946	−0.179473	−	0.983763i	$-0.557439\pi$
$359$	−876528.	−0.358946	−0.179473	+	0.983763i	$0.557439\pi$
$360$	0	0
$361$	−2.25520e6	−0.910787
$362$	0	0
$363$	481550.	0.191812
$364$	0	0
$365$	302568.	0.118875
$366$	0	0
$367$	−2.98062e6	−1.15516	−0.577578	−	0.816335i	$-0.696002\pi$
$367$	−2.98062e6	−1.15516	−0.577578	+	0.816335i	$0.696002\pi$
$368$	0	0
$369$	−889746.	−0.340173
$370$	0	0
$371$	1.82486e6	0.688326
$372$	0	0
$373$	3.91441e6	1.45678	0.728391	−	0.685162i	$-0.240268\pi$
$373$	3.91441e6	1.45678	0.728391	+	0.685162i	$0.240268\pi$
$374$	0	0
$375$	−677040.	−0.248620
$376$	0	0
$377$	2.84174e6	1.02975
$378$	0	0
$379$	−3.60661e6	−1.28974	−0.644868	−	0.764294i	$-0.723088\pi$
$379$	−3.60661e6	−1.28974	−0.644868	+	0.764294i	$0.723088\pi$
$380$	0	0
$381$	3.15992e6	1.11523
$382$	0	0
$383$	2.66644e6	0.928826	0.464413	−	0.885619i	$-0.346265\pi$
$383$	2.66644e6	0.928826	0.464413	+	0.885619i	$0.346265\pi$
$384$	0	0
$385$	−1.38298e6	−0.475513
$386$	0	0
$387$	529672.	0.179775
$388$	0	0
$389$	−213366.	−0.0714910	−0.0357455	−	0.999361i	$-0.511381\pi$
$389$	−213366.	−0.0714910	−0.0357455	+	0.999361i	$0.511381\pi$
$390$	0	0
$391$	−6.12360e6	−2.02565
$392$	0	0
$393$	−246660.	−0.0805596
$394$	0	0
$395$	−3.60125e6	−1.16134
$396$	0	0
$397$	−4.09408e6	−1.30371	−0.651854	−	0.758345i	$-0.726008\pi$
$397$	−4.09408e6	−1.30371	−0.651854	+	0.758345i	$0.726008\pi$
$398$	0	0
$399$	−230300.	−0.0724205
$400$	0	0
$401$	942366.	0.292657	0.146328	−	0.989236i	$-0.453254\pi$
$401$	942366.	0.292657	0.146328	+	0.989236i	$0.453254\pi$
$402$	0	0
$403$	4.30525e6	1.32049
$404$	0	0
$405$	−323484.	−0.0979976
$406$	0	0
$407$	4.81488e6	1.44079
$408$	0	0
$409$	−4.84561e6	−1.43232	−0.716160	−	0.697936i	$-0.754102\pi$
$409$	−4.84561e6	−1.43232	−0.716160	+	0.697936i	$0.754102\pi$
$410$	0	0
$411$	−3.03234e6	−0.885469
$412$	0	0
$413$	1.68080e6	0.484887
$414$	0	0
$415$	2.95697e6	0.842804
$416$	0	0
$417$	2.50586e6	0.705694
$418$	0	0
$419$	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	$0.572910\pi$
$422$	0	0
$423$	−259116.	−0.0704115
$424$	0	0
$425$	−5.73140e6	−1.53918
$426$	0	0
$427$	−1.19932e6	−0.318322
$428$	0	0
$429$	−1.96224e6	−0.514765
$430$	0	0
$431$	−4.14360e6	−1.07445	−0.537223	−	0.843440i	$-0.680527\pi$
$431$	−4.14360e6	−1.07445	−0.537223	+	0.843440i	$0.680527\pi$
$432$	0	0
$433$	−3.03966e6	−0.779121	−0.389561	−	0.921001i	$-0.627373\pi$
$433$	−3.03966e6	−0.779121	−0.389561	+	0.921001i	$0.627373\pi$
$434$	0	0
$435$	−4.08744e6	−1.03569
$436$	0	0
$437$	−1.97400e6	−0.494474
$438$	0	0
$439$	−2.54271e6	−0.629703	−0.314852	−	0.949141i	$-0.601955\pi$
$439$	−2.54271e6	−0.629703	−0.314852	+	0.949141i	$0.601955\pi$
$440$	0	0
$441$	−343343.	−0.0840682
$442$	0	0
$443$	2.43210e6	0.588806	0.294403	−	0.955681i	$-0.404879\pi$
$443$	2.43210e6	0.588806	0.294403	+	0.955681i	$0.404879\pi$
$444$	0	0
$445$	2.24532e6	0.537500
$446$	0	0
$447$	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	$0.431344\pi$
$450$	0	0
$451$	2.09059e6	0.483981
$452$	0	0
$453$	1.24448e6	0.284933
$454$	0	0
$455$	−2.40374e6	−0.544327
$456$	0	0
$457$	1.58063e6	0.354030	0.177015	−	0.984208i	$-0.443356\pi$
$457$	1.58063e6	0.354030	0.177015	+	0.984208i	$0.443356\pi$
$458$	0	0
$459$	−5.62788e6	−1.24685
$460$	0	0
$461$	5.09604e6	1.11681	0.558407	−	0.829567i	$-0.311413\pi$
$461$	5.09604e6	1.11681	0.558407	+	0.829567i	$0.311413\pi$
$462$	0	0
$463$	7.02338e6	1.52263	0.761313	−	0.648384i	$-0.224555\pi$
$463$	7.02338e6	1.52263	0.761313	+	0.648384i	$0.224555\pi$
$464$	0	0
$465$	−6.19248e6	−1.32810
$466$	0	0
$467$	4.24845e6	0.901443	0.450722	−	0.892665i	$-0.351167\pi$
$467$	4.24845e6	0.901443	0.450722	+	0.892665i	$0.351167\pi$
$468$	0	0
$469$	−855148.	−0.179518
$470$	0	0
$471$	−760400.	−0.157939
$472$	0	0
$473$	−1.24454e6	−0.255775
$474$	0	0
$475$	−1.84757e6	−0.375722
$476$	0	0
$477$	5.32561e6	1.07170
$478$	0	0
$479$	−559284.	−0.111377	−0.0556883	−	0.998448i	$-0.517735\pi$
$479$	−559284.	−0.111377	−0.0556883	+	0.998448i	$0.517735\pi$
$480$	0	0
$481$	8.36872e6	1.64929
$482$	0	0
$483$	2.05800e6	0.401400
$484$	0	0
$485$	−1.42615e6	−0.275303
$486$	0	0
$487$	1.32057e6	0.252312	0.126156	−	0.992010i	$-0.459736\pi$
$487$	1.32057e6	0.252312	0.126156	+	0.992010i	$0.459736\pi$
$488$	0	0
$489$	1.24256e6	0.234988
$490$	0	0
$491$	−6.27193e6	−1.17408	−0.587040	−	0.809558i	$-0.699707\pi$
$491$	−6.27193e6	−1.17408	−0.587040	+	0.809558i	$0.699707\pi$
$492$	0	0
$493$	−7.09463e6	−1.31466
$494$	0	0
$495$	−4.03603e6	−0.740358
$496$	0	0
$497$	1.38298e6	0.251144
$498$	0	0
$499$	3.93785e6	0.707959	0.353979	−	0.935253i	$-0.384828\pi$
$499$	3.93785e6	0.707959	0.353979	+	0.935253i	$0.384828\pi$
$500$	0	0
$501$	−724200.	−0.128903
$502$	0	0
$503$	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
$506$	0	0
$507$	302370.	0.0522419
$508$	0	0
$509$	−7.82664e6	−1.33900	−0.669501	−	0.742812i	$-0.733492\pi$
$509$	−7.82664e6	−1.33900	−0.669501	+	0.742812i	$0.733492\pi$
$510$	0	0
$511$	−176498.	−0.0299011
$512$	0	0
$513$	−1.81420e6	−0.304363
$514$	0	0
$515$	1.10581e7	1.83722
$516$	0	0
$517$	608832.	0.100178
$518$	0	0
$519$	4.41552e6	0.719554
$520$	0	0
$521$	8.94454e6	1.44366	0.721828	−	0.692072i	$-0.243302\pi$
$521$	8.94454e6	1.44366	0.721828	+	0.692072i	$0.243302\pi$
$522$	0	0
$523$	−4.07481e6	−0.651407	−0.325704	−	0.945472i	$-0.605601\pi$
$523$	−4.07481e6	−0.651407	−0.325704	+	0.945472i	$0.605601\pi$
$524$	0	0
$525$	1.92619e6	0.305001
$526$	0	0
$527$	−1.07484e7	−1.68584
$528$	0	0
$529$	1.12037e7	1.74069
$530$	0	0
$531$	4.90519e6	0.754952
$532$	0	0
$533$	3.63365e6	0.554019
$534$	0	0
$535$	−4.10357e6	−0.619837
$536$	0	0
$537$	−106920.	−0.0160001
$538$	0	0
$539$	806736.	0.119608
$540$	0	0
$541$	−1.18676e7	−1.74329	−0.871644	−	0.490140i	$-0.836946\pi$
$541$	−1.18676e7	−1.74329	−0.871644	+	0.490140i	$0.836946\pi$
$542$	0	0
$543$	5.46064e6	0.794775
$544$	0	0
$545$	−4.73542e6	−0.682915
$546$	0	0
$547$	5.37801e6	0.768516	0.384258	−	0.923226i	$-0.374457\pi$
$547$	5.37801e6	0.768516	0.384258	+	0.923226i	$0.374457\pi$
$548$	0	0
$549$	−3.50007e6	−0.495616
$550$	0	0
$551$	−2.28702e6	−0.320916
$552$	0	0
$553$	2.10073e6	0.292117
$554$	0	0
$555$	−1.20372e7	−1.65880
$556$	0	0
$557$	−5.64878e6	−0.771466	−0.385733	−	0.922611i	$-0.626051\pi$
$557$	−5.64878e6	−0.771466	−0.385733	+	0.922611i	$0.626051\pi$
$558$	0	0
$559$	−2.16314e6	−0.292789
$560$	0	0
$561$	4.89888e6	0.657188
$562$	0	0
$563$	−4.56407e6	−0.606850	−0.303425	−	0.952855i	$-0.598130\pi$
$563$	−4.56407e6	−0.606850	−0.303425	+	0.952855i	$0.598130\pi$
$564$	0	0
$565$	734328.	0.0967763
$566$	0	0
$567$	188699.	0.0246497
$568$	0	0
$569$	8.00165e6	1.03609	0.518047	−	0.855352i	$-0.326659\pi$
$569$	8.00165e6	1.03609	0.518047	+	0.855352i	$0.326659\pi$
$570$	0	0
$571$	1.37164e7	1.76055	0.880275	−	0.474464i	$-0.157358\pi$
$571$	1.37164e7	1.76055	0.880275	+	0.474464i	$0.157358\pi$
$572$	0	0
$573$	−5.75976e6	−0.732855
$574$	0	0
$575$	1.65102e7	2.08249
$576$	0	0
$577$	6.09797e6	0.762510	0.381255	−	0.924470i	$-0.375492\pi$
$577$	6.09797e6	0.762510	0.381255	+	0.924470i	$0.375492\pi$
$578$	0	0
$579$	4.13938e6	0.513144
$580$	0	0
$581$	−1.72490e6	−0.211994
$582$	0	0
$583$	−1.25133e7	−1.52476
$584$	0	0
$585$	−7.01501e6	−0.847498
$586$	0	0
$587$	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
$590$	0	0
$591$	4.94946e6	0.582893
$592$	0	0
$593$	1.41575e6	0.165330	0.0826649	−	0.996577i	$-0.473657\pi$
$593$	1.41575e6	0.165330	0.0826649	+	0.996577i	$0.473657\pi$
$594$	0	0
$595$	6.00113e6	0.694929
$596$	0	0
$597$	5.20364e6	0.597546
$598$	0	0
$599$	−8.75460e6	−0.996941	−0.498470	−	0.866907i	$-0.666105\pi$
$599$	−8.75460e6	−0.996941	−0.498470	+	0.866907i	$0.666105\pi$
$600$	0	0
$601$	8.70276e6	0.982813	0.491407	−	0.870930i	$-0.336483\pi$
$601$	8.70276e6	0.982813	0.491407	+	0.870930i	$0.336483\pi$
$602$	0	0
$603$	−2.49564e6	−0.279504
$604$	0	0
$605$	−4.04502e6	−0.449296
$606$	0	0
$607$	1.69578e7	1.86809	0.934045	−	0.357157i	$-0.116254\pi$
$607$	1.69578e7	1.86809	0.934045	+	0.357157i	$0.116254\pi$
$608$	0	0
$609$	2.38434e6	0.260510
$610$	0	0
$611$	1.05821e6	0.114675
$612$	0	0
$613$	1.76743e7	1.89973	0.949866	−	0.312658i	$-0.101220\pi$
$613$	1.76743e7	1.89973	0.949866	+	0.312658i	$0.101220\pi$
$614$	0	0
$615$	−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	$0.671552\pi$
$618$	0	0
$619$	−1.48739e7	−1.56027	−0.780133	−	0.625613i	$-0.784849\pi$
$619$	−1.48739e7	−1.56027	−0.780133	+	0.625613i	$0.784849\pi$
$620$	0	0
$621$	1.62120e7	1.68697
$622$	0	0
$623$	−1.30977e6	−0.135199
$624$	0	0
$625$	−6.59724e6	−0.675557
$626$	0	0
$627$	1.57920e6	0.160424
$628$	0	0
$629$	−2.08931e7	−2.10561
$630$	0	0
$631$	−1.26353e7	−1.26331	−0.631656	−	0.775248i	$-0.717625\pi$
$631$	−1.26353e7	−1.26331	−0.631656	+	0.775248i	$0.717625\pi$
$632$	0	0
$633$	1.83284e6	0.181809
$634$	0	0
$635$	−2.65433e7	−2.61229
$636$	0	0
$637$	1.40218e6	0.136917
$638$	0	0
$639$	4.03603e6	0.391023
$640$	0	0
$641$	6.23398e6	0.599267	0.299634	−	0.954054i	$-0.403136\pi$
$641$	6.23398e6	0.599267	0.299634	+	0.954054i	$0.403136\pi$
$642$	0	0
$643$	−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
$646$	0	0
$647$	−1.83258e7	−1.72109	−0.860544	−	0.509376i	$-0.829876\pi$
$647$	−1.83258e7	−1.72109	−0.860544	+	0.509376i	$0.829876\pi$
$648$	0	0
$649$	−1.15255e7	−1.07411
$650$	0	0
$651$	3.61228e6	0.334063
$652$	0	0
$653$	−7.28857e6	−0.668897	−0.334448	−	0.942414i	$-0.608550\pi$
$653$	−7.28857e6	−0.668897	−0.334448	+	0.942414i	$0.608550\pi$
$654$	0	0
$655$	2.07194e6	0.188701
$656$	0	0
$657$	−515086.	−0.0465550
$658$	0	0
$659$	−4.54337e6	−0.407534	−0.203767	−	0.979019i	$-0.565319\pi$
$659$	−4.54337e6	−0.407534	−0.203767	+	0.979019i	$0.565319\pi$
$660$	0	0
$661$	−2.10021e7	−1.86964	−0.934821	−	0.355120i	$-0.884440\pi$
$661$	−2.10021e7	−1.86964	−0.934821	+	0.355120i	$0.884440\pi$
$662$	0	0
$663$	8.51472e6	0.752292
$664$	0	0
$665$	1.93452e6	0.169636
$666$	0	0
$667$	2.04372e7	1.77872
$668$	0	0
$669$	−1.27746e7	−1.10353
$670$	0	0
$671$	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	$0.452837\pi$
$674$	0	0
$675$	1.51737e7	1.28183
$676$	0	0
$677$	−1.80916e7	−1.51707	−0.758536	−	0.651631i	$-0.774085\pi$
$677$	−1.80916e7	−1.51707	−0.758536	+	0.651631i	$0.774085\pi$
$678$	0	0
$679$	831922.	0.0692481
$680$	0	0
$681$	−1.28764e7	−1.06397
$682$	0	0
$683$	−4.67752e6	−0.383675	−0.191838	−	0.981427i	$-0.561445\pi$
$683$	−4.67752e6	−0.383675	−0.191838	+	0.981427i	$0.561445\pi$
$684$	0	0
$685$	2.54717e7	2.07411
$686$	0	0
$687$	−3.50936e6	−0.283685
$688$	0	0
$689$	−2.17493e7	−1.74541
$690$	0	0
$691$	−1.68960e7	−1.34614	−0.673069	−	0.739579i	$-0.735024\pi$
$691$	−1.68960e7	−1.34614	−0.673069	+	0.739579i	$0.735024\pi$
$692$	0	0
$693$	2.35435e6	0.186225
$694$	0	0
$695$	−2.10492e7	−1.65300
$696$	0	0
$697$	−9.07168e6	−0.707303
$698$	0	0
$699$	−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	$0.470481\pi$
$702$	0	0
$703$	−6.73510e6	−0.513991
$704$	0	0
$705$	−1.52208e6	−0.115336
$706$	0	0
$707$	−4.86100e6	−0.365744
$708$	0	0
$709$	−5.77010e6	−0.431090	−0.215545	−	0.976494i	$-0.569153\pi$
$709$	−5.77010e6	−0.431090	−0.215545	+	0.976494i	$0.569153\pi$
$710$	0	0
$711$	6.13070e6	0.454816
$712$	0	0
$713$	3.09624e7	2.28092
$714$	0	0
$715$	1.64828e7	1.20578
$716$	0	0
$717$	7.74336e6	0.562512
$718$	0	0
$719$	1.43716e7	1.03677	0.518385	−	0.855147i	$-0.326533\pi$
$719$	1.43716e7	1.03677	0.518385	+	0.855147i	$0.326533\pi$
$720$	0	0
$721$	−6.45056e6	−0.462124
$722$	0	0
$723$	1.15285e7	0.820214
$724$	0	0
$725$	1.91282e7	1.35154
$726$	0	0
$727$	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
$730$	0	0
$731$	5.40043e6	0.373796
$732$	0	0
$733$	−3.75000e6	−0.257793	−0.128897	−	0.991658i	$-0.541144\pi$
$733$	−3.75000e6	−0.257793	−0.128897	+	0.991658i	$0.541144\pi$
$734$	0	0
$735$	−2.01684e6	−0.137706
$736$	0	0
$737$	5.86387e6	0.397664
$738$	0	0
$739$	−2.61318e7	−1.76019	−0.880093	−	0.474802i	$-0.842520\pi$
$739$	−2.61318e7	−1.76019	−0.880093	+	0.474802i	$0.842520\pi$
$740$	0	0
$741$	2.74480e6	0.183639
$742$	0	0
$743$	159072.	0.0105711	0.00528557	−	0.999986i	$-0.498318\pi$
$743$	159072.	0.0105711	0.00528557	+	0.999986i	$0.498318\pi$
$744$	0	0
$745$	−5.08990e6	−0.335984
$746$	0	0
$747$	−5.03389e6	−0.330067
$748$	0	0
$749$	2.39375e6	0.155910
$750$	0	0
$751$	2.65311e7	1.71654	0.858272	−	0.513196i	$-0.171539\pi$
$751$	2.65311e7	1.71654	0.858272	+	0.513196i	$0.171539\pi$
$752$	0	0
$753$	1.35801e7	0.872802
$754$	0	0
$755$	−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	$0.660137\pi$
$758$	0	0
$759$	−1.41120e7	−0.889169
$760$	0	0
$761$	4.71380e6	0.295059	0.147530	−	0.989058i	$-0.452868\pi$
$761$	4.71380e6	0.295059	0.147530	+	0.989058i	$0.452868\pi$
$762$	0	0
$763$	2.76233e6	0.171776
$764$	0	0
$765$	1.75135e7	1.08198
$766$	0	0
$767$	−2.00324e7	−1.22954
$768$	0	0
$769$	−1.58977e6	−0.0969434	−0.0484717	−	0.998825i	$-0.515435\pi$
$769$	−1.58977e6	−0.0969434	−0.0484717	+	0.998825i	$0.515435\pi$
$770$	0	0
$771$	3.17742e6	0.192504
$772$	0	0
$773$	−9.69095e6	−0.583334	−0.291667	−	0.956520i	$-0.594210\pi$
$773$	−9.69095e6	−0.583334	−0.291667	+	0.956520i	$0.594210\pi$
$774$	0	0
$775$	2.89793e7	1.73314
$776$	0	0
$777$	7.02170e6	0.417244
$778$	0	0
$779$	−2.92434e6	−0.172657
$780$	0	0
$781$	−9.48326e6	−0.556327
$782$	0	0
$783$	1.87828e7	1.09485
$784$	0	0
$785$	6.38736e6	0.369954
$786$	0	0
$787$	1.57170e6	0.0904549	0.0452275	−	0.998977i	$-0.485599\pi$
$787$	1.57170e6	0.0904549	0.0452275	+	0.998977i	$0.485599\pi$
$788$	0	0
$789$	1.05101e7	0.601054
$790$	0	0
$791$	−428358.	−0.0243425
$792$	0	0
$793$	1.42940e7	0.807180
$794$	0	0
$795$	3.12833e7	1.75547
$796$	0	0
$797$	−2.25298e6	−0.125635	−0.0628175	−	0.998025i	$-0.520009\pi$
$797$	−2.25298e6	−0.125635	−0.0628175	+	0.998025i	$0.520009\pi$
$798$	0	0
$799$	−2.64190e6	−0.146403
$800$	0	0
$801$	−3.82239e6	−0.210501
$802$	0	0
$803$	1.21027e6	0.0662360
$804$	0	0
$805$	−1.72872e7	−0.940232
$806$	0	0
$807$	−1.18958e7	−0.643000
$808$	0	0
$809$	−2.37938e7	−1.27818	−0.639090	−	0.769132i	$-0.720689\pi$
$809$	−2.37938e7	−1.27818	−0.639090	+	0.769132i	$0.720689\pi$
$810$	0	0
$811$	−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
$814$	0	0
$815$	−1.04375e7	−0.550431
$816$	0	0
$817$	1.74088e6	0.0912460
$818$	0	0
$819$	4.09209e6	0.213174
$820$	0	0
$821$	1.48802e7	0.770464	0.385232	−	0.922820i	$-0.374121\pi$
$821$	1.48802e7	0.770464	0.385232	+	0.922820i	$0.374121\pi$
$822$	0	0
$823$	−2.00601e7	−1.03236	−0.516182	−	0.856479i	$-0.672647\pi$
$823$	−2.00601e7	−1.03236	−0.516182	+	0.856479i	$0.672647\pi$
$824$	0	0
$825$	−1.32082e7	−0.675628
$826$	0	0
$827$	−1.21539e7	−0.617949	−0.308975	−	0.951070i	$-0.599986\pi$
$827$	−1.21539e7	−0.617949	−0.308975	+	0.951070i	$0.599986\pi$
$828$	0	0
$829$	3.21197e7	1.62325	0.811625	−	0.584179i	$-0.198583\pi$
$829$	3.21197e7	1.62325	0.811625	+	0.584179i	$0.198583\pi$
$830$	0	0
$831$	−633020.	−0.0317991
$832$	0	0
$833$	−3.50066e6	−0.174798
$834$	0	0
$835$	6.08328e6	0.301941
$836$	0	0
$837$	2.84559e7	1.40397
$838$	0	0
$839$	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
$842$	0	0
$843$	4.96614e6	0.240686
$844$	0	0
$845$	−2.53991e6	−0.122370
$846$	0	0
$847$	2.35960e6	0.113013
$848$	0	0
$849$	−1.15842e7	−0.551564
$850$	0	0
$851$	6.01860e7	2.84886
$852$	0	0
$853$	234824.	0.0110502	0.00552510	−	0.999985i	$-0.498241\pi$
$853$	234824.	0.0110502	0.00552510	+	0.999985i	$0.498241\pi$
$854$	0	0
$855$	5.64564e6	0.264118
$856$	0	0
$857$	2.83802e7	1.31997	0.659985	−	0.751279i	$-0.270563\pi$
$857$	2.83802e7	1.31997	0.659985	+	0.751279i	$0.270563\pi$
$858$	0	0
$859$	−4.00081e7	−1.84997	−0.924986	−	0.380001i	$-0.875924\pi$
$859$	−4.00081e7	−1.84997	−0.924986	+	0.380001i	$0.875924\pi$
$860$	0	0
$861$	3.04878e6	0.140158
$862$	0	0
$863$	2.08030e7	0.950823	0.475411	−	0.879764i	$-0.342299\pi$
$863$	2.08030e7	0.950823	0.475411	+	0.879764i	$0.342299\pi$
$864$	0	0
$865$	−3.70904e7	−1.68547
$866$	0	0
$867$	−7.05907e6	−0.318933
$868$	0	0
$869$	−1.44050e7	−0.647088
$870$	0	0
$871$	1.01920e7	0.455211
$872$	0	0
$873$	2.42785e6	0.107817
$874$	0	0
$875$	−3.31750e6	−0.146484
$876$	0	0
$877$	3.03559e7	1.33273	0.666367	−	0.745624i	$-0.267848\pi$
$877$	3.03559e7	1.33273	0.666367	+	0.745624i	$0.267848\pi$
$878$	0	0
$879$	−1.43886e7	−0.628125
$880$	0	0
$881$	−2.58936e7	−1.12396	−0.561981	−	0.827150i	$-0.689961\pi$
$881$	−2.58936e7	−1.12396	−0.561981	+	0.827150i	$0.689961\pi$
$882$	0	0
$883$	1.88813e7	0.814950	0.407475	−	0.913216i	$-0.366409\pi$
$883$	1.88813e7	0.814950	0.407475	+	0.913216i	$0.366409\pi$
$884$	0	0
$885$	2.88137e7	1.23663
$886$	0	0
$887$	2.34431e7	1.00048	0.500238	−	0.865888i	$-0.333246\pi$
$887$	2.34431e7	1.00048	0.500238	+	0.865888i	$0.333246\pi$
$888$	0	0
$889$	1.54836e7	0.657079
$890$	0	0
$891$	−1.29394e6	−0.0546033
$892$	0	0
$893$	−851640.	−0.0357378
$894$	0	0
$895$	898128.	0.0374784
$896$	0	0
$897$	−2.45280e7	−1.01784
$898$	0	0
$899$	3.58722e7	1.48033
$900$	0	0
$901$	5.42988e7	2.22833
$902$	0	0
$903$	−1.81496e6	−0.0740709
$904$	0	0
$905$	−4.58694e7	−1.86166
$906$	0	0
$907$	5.60873e6	0.226384	0.113192	−	0.993573i	$-0.463892\pi$
$907$	5.60873e6	0.226384	0.113192	+	0.993573i	$0.463892\pi$
$908$	0	0
$909$	−1.41862e7	−0.569450
$910$	0	0
$911$	−2.16215e7	−0.863156	−0.431578	−	0.902076i	$-0.642043\pi$
$911$	−2.16215e7	−0.863156	−0.431578	+	0.902076i	$0.642043\pi$
$912$	0	0
$913$	1.18279e7	0.469602
$914$	0	0
$915$	−2.05598e7	−0.811834
$916$	0	0
$917$	−1.20863e6	−0.0474648
$918$	0	0
$919$	−4.51695e7	−1.76424	−0.882119	−	0.471028i	$-0.843883\pi$
$919$	−4.51695e7	−1.76424	−0.882119	+	0.471028i	$0.843883\pi$
$920$	0	0
$921$	−9.89098e6	−0.384229
$922$	0	0
$923$	−1.64828e7	−0.636835
$924$	0	0
$925$	5.63312e7	2.16469
$926$	0	0
$927$	−1.88251e7	−0.719512
$928$	0	0
$929$	−2.28729e7	−0.869524	−0.434762	−	0.900545i	$-0.643168\pi$
$929$	−2.28729e7	−0.869524	−0.434762	+	0.900545i	$0.643168\pi$
$930$	0	0
$931$	−1.12847e6	−0.0426693
$932$	0	0
$933$	−2.22050e7	−0.835117
$934$	0	0
$935$	−4.11506e7	−1.53938
$936$	0	0
$937$	−1.79616e7	−0.668336	−0.334168	−	0.942514i	$-0.608455\pi$
$937$	−1.79616e7	−0.668336	−0.334168	+	0.942514i	$0.608455\pi$
$938$	0	0
$939$	−2.33008e7	−0.862395
$940$	0	0
$941$	−1.79697e7	−0.661558	−0.330779	−	0.943708i	$-0.607311\pi$
$941$	−1.79697e7	−0.661558	−0.330779	+	0.943708i	$0.607311\pi$
$942$	0	0
$943$	2.61324e7	0.956974
$944$	0	0
$945$	−1.58878e7	−0.578740
$946$	0	0
$947$	−4.32115e7	−1.56576	−0.782879	−	0.622174i	$-0.786250\pi$
$947$	−4.32115e7	−1.56576	−0.782879	+	0.622174i	$0.786250\pi$
$948$	0	0
$949$	2.10357e6	0.0758213
$950$	0	0
$951$	−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	$0.542758\pi$
$954$	0	0
$955$	4.83820e7	1.71662
$956$	0	0
$957$	−1.63498e7	−0.577074
$958$	0	0
$959$	−1.48585e7	−0.521708
$960$	0	0
$961$	2.57172e7	0.898288
$962$	0	0
$963$	6.98584e6	0.242746
$964$	0	0
$965$	−3.47708e7	−1.20198
$966$	0	0
$967$	1.69305e7	0.582242	0.291121	−	0.956686i	$-0.405972\pi$
$967$	1.69305e7	0.582242	0.291121	+	0.956686i	$0.405972\pi$
$968$	0	0
$969$	−6.85260e6	−0.234448
$970$	0	0
$971$	−2.86144e7	−0.973949	−0.486974	−	0.873416i	$-0.661899\pi$
$971$	−2.86144e7	−0.973949	−0.486974	+	0.873416i	$0.661899\pi$
$972$	0	0
$973$	1.22787e7	0.415787
$974$	0	0
$975$	−2.29570e7	−0.773400
$976$	0	0
$977$	3.69445e7	1.23826	0.619132	−	0.785287i	$-0.287485\pi$
$977$	3.69445e7	1.23826	0.619132	+	0.785287i	$0.287485\pi$
$978$	0	0
$979$	8.98128e6	0.299489
$980$	0	0
$981$	8.06148e6	0.267450
$982$	0	0
$983$	3.88787e7	1.28330	0.641650	−	0.766998i	$-0.278250\pi$
$983$	3.88787e7	1.28330	0.641650	+	0.766998i	$0.278250\pi$
$984$	0	0
$985$	−4.15755e7	−1.36536
$986$	0	0
$987$	887880.	0.0290109
$988$	0	0
$989$	−1.55568e7	−0.505743
$990$	0	0
$991$	−2.49212e7	−0.806092	−0.403046	−	0.915180i	$-0.632049\pi$
$991$	−2.49212e7	−0.806092	−0.403046	+	0.915180i	$0.632049\pi$
$992$	0	0
$993$	−3.96616e6	−0.127643
$994$	0	0
$995$	−4.37106e7	−1.39968
$996$	0	0
$997$	1.01956e7	0.324845	0.162422	−	0.986721i	$-0.448069\pi$
$997$	1.01956e7	0.324845	0.162422	+	0.986721i	$0.448069\pi$
$998$	0	0
$999$	5.53138e7	1.75356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.a.c.1.1		1
3.2	odd	2		1008.6.a.b.1.1		1
4.3	odd	2		14.6.a.a.1.1	✓	1
7.6	odd	2		784.6.a.i.1.1		1
8.3	odd	2		448.6.a.e.1.1		1
8.5	even	2		448.6.a.l.1.1		1
12.11	even	2		126.6.a.f.1.1		1
20.3	even	4		350.6.c.d.99.2		2
20.7	even	4		350.6.c.d.99.1		2
20.19	odd	2		350.6.a.i.1.1		1
28.3	even	6		98.6.c.d.79.1		2
28.11	odd	6		98.6.c.c.79.1		2
28.19	even	6		98.6.c.d.67.1		2
28.23	odd	6		98.6.c.c.67.1		2
28.27	even	2		98.6.a.a.1.1		1
84.83	odd	2		882.6.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.a.1.1	✓	1	4.3	odd	2
98.6.a.a.1.1		1	28.27	even	2
98.6.c.c.67.1		2	28.23	odd	6
98.6.c.c.79.1		2	28.11	odd	6
98.6.c.d.67.1		2	28.19	even	6
98.6.c.d.79.1		2	28.3	even	6
112.6.a.c.1.1		1	1.1	even	1	trivial
126.6.a.f.1.1		1	12.11	even	2
350.6.a.i.1.1		1	20.19	odd	2
350.6.c.d.99.1		2	20.7	even	4
350.6.c.d.99.2		2	20.3	even	4
448.6.a.e.1.1		1	8.3	odd	2
448.6.a.l.1.1		1	8.5	even	2
784.6.a.i.1.1		1	7.6	odd	2
882.6.a.x.1.1		1	84.83	odd	2
1008.6.a.b.1.1		1	3.2	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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