LMFDB - Embedded newform 342.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,6,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	342.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:	$54.8512663760$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} +91.0000 q^{5} -33.0000 q^{7} -64.0000 q^{8} +O(q^{10})$

\(q-4.00000 q^{2} +16.0000 q^{4} +91.0000 q^{5} -33.0000 q^{7} -64.0000 q^{8} -364.000 q^{10} +91.0000 q^{11} -610.000 q^{13} +132.000 q^{14} +256.000 q^{16} +1833.00 q^{17} -361.000 q^{19} +1456.00 q^{20} -364.000 q^{22} +3436.00 q^{23} +5156.00 q^{25} +2440.00 q^{26} -528.000 q^{28} -3562.00 q^{29} +322.000 q^{31} -1024.00 q^{32} -7332.00 q^{34} -3003.00 q^{35} +7216.00 q^{37} +1444.00 q^{38} -5824.00 q^{40} +13664.0 q^{41} -3701.00 q^{43} +1456.00 q^{44} -13744.0 q^{46} -9203.00 q^{47} -15718.0 q^{49} -20624.0 q^{50} -9760.00 q^{52} -29186.0 q^{53} +8281.00 q^{55} +2112.00 q^{56} +14248.0 q^{58} +27804.0 q^{59} +43127.0 q^{61} -1288.00 q^{62} +4096.00 q^{64} -55510.0 q^{65} -19428.0 q^{67} +29328.0 q^{68} +12012.0 q^{70} -7040.00 q^{71} +37341.0 q^{73} -28864.0 q^{74} -5776.00 q^{76} -3003.00 q^{77} -4972.00 q^{79} +23296.0 q^{80} -54656.0 q^{82} +71196.0 q^{83} +166803. q^{85} +14804.0 q^{86} -5824.00 q^{88} +3654.00 q^{89} +20130.0 q^{91} +54976.0 q^{92} +36812.0 q^{94} -32851.0 q^{95} +62362.0 q^{97} +62872.0 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	91.0000	1.62786	0.813929	−	0.580965i	$-0.197325\pi$
$5$	91.0000	1.62786	0.813929	+	0.580965i	$0.197325\pi$
$6$	0	0
$7$	−33.0000	−0.254548	−0.127274	−	0.991868i	$-0.540623\pi$
$7$	−33.0000	−0.254548	−0.127274	+	0.991868i	$0.540623\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−364.000	−1.15107
$11$	91.0000	0.226756	0.113378	−	0.993552i	$-0.463833\pi$
$11$	91.0000	0.226756	0.113378	+	0.993552i	$0.463833\pi$
$12$	0	0
$13$	−610.000	−1.00109	−0.500543	−	0.865712i	$-0.666866\pi$
$13$	−610.000	−1.00109	−0.500543	+	0.865712i	$0.666866\pi$
$14$	132.000	0.179992
$15$	0	0
$16$	256.000	0.250000
$17$	1833.00	1.53830	0.769148	−	0.639070i	$-0.220681\pi$
$17$	1833.00	1.53830	0.769148	+	0.639070i	$0.220681\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	1456.00	0.813929
$21$	0	0
$22$	−364.000	−0.160341
$23$	3436.00	1.35436	0.677179	−	0.735818i	$-0.263202\pi$
$23$	3436.00	1.35436	0.677179	+	0.735818i	$0.263202\pi$
$24$	0	0
$25$	5156.00	1.64992
$26$	2440.00	0.707875
$27$	0	0
$28$	−528.000	−0.127274
$29$	−3562.00	−0.786500	−0.393250	−	0.919432i	$-0.628649\pi$
$29$	−3562.00	−0.786500	−0.393250	+	0.919432i	$0.628649\pi$
$30$	0	0
$31$	322.000	0.0601799	0.0300900	−	0.999547i	$-0.490421\pi$
$31$	322.000	0.0601799	0.0300900	+	0.999547i	$0.490421\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−7332.00	−1.08774
$35$	−3003.00	−0.414367
$36$	0	0
$37$	7216.00	0.866547	0.433274	−	0.901262i	$-0.357358\pi$
$37$	7216.00	0.866547	0.433274	+	0.901262i	$0.357358\pi$
$38$	1444.00	0.162221
$39$	0	0
$40$	−5824.00	−0.575535
$41$	13664.0	1.26946	0.634729	−	0.772735i	$-0.281112\pi$
$41$	13664.0	1.26946	0.634729	+	0.772735i	$0.281112\pi$
$42$	0	0
$43$	−3701.00	−0.305245	−0.152622	−	0.988285i	$-0.548772\pi$
$43$	−3701.00	−0.305245	−0.152622	+	0.988285i	$0.548772\pi$
$44$	1456.00	0.113378
$45$	0	0
$46$	−13744.0	−0.957676
$47$	−9203.00	−0.607694	−0.303847	−	0.952721i	$-0.598271\pi$
$47$	−9203.00	−0.607694	−0.303847	+	0.952721i	$0.598271\pi$
$48$	0	0
$49$	−15718.0	−0.935206
$50$	−20624.0	−1.16667
$51$	0	0
$52$	−9760.00	−0.500543
$53$	−29186.0	−1.42720	−0.713600	−	0.700553i	$-0.752937\pi$
$53$	−29186.0	−1.42720	−0.713600	+	0.700553i	$0.752937\pi$
$54$	0	0
$55$	8281.00	0.369127
$56$	2112.00	0.0899961
$57$	0	0
$58$	14248.0	0.556140
$59$	27804.0	1.03987	0.519933	−	0.854207i	$-0.325957\pi$
$59$	27804.0	1.03987	0.519933	+	0.854207i	$0.325957\pi$
$60$	0	0
$61$	43127.0	1.48397	0.741984	−	0.670417i	$-0.233885\pi$
$61$	43127.0	1.48397	0.741984	+	0.670417i	$0.233885\pi$
$62$	−1288.00	−0.0425536
$63$	0	0
$64$	4096.00	0.125000
$65$	−55510.0	−1.62963
$66$	0	0
$67$	−19428.0	−0.528739	−0.264369	−	0.964422i	$-0.585164\pi$
$67$	−19428.0	−0.528739	−0.264369	+	0.964422i	$0.585164\pi$
$68$	29328.0	0.769148
$69$	0	0
$70$	12012.0	0.293002
$71$	−7040.00	−0.165740	−0.0828699	−	0.996560i	$-0.526409\pi$
$71$	−7040.00	−0.165740	−0.0828699	+	0.996560i	$0.526409\pi$
$72$	0	0
$73$	37341.0	0.820123	0.410061	−	0.912058i	$-0.365507\pi$
$73$	37341.0	0.820123	0.410061	+	0.912058i	$0.365507\pi$
$74$	−28864.0	−0.612741
$75$	0	0
$76$	−5776.00	−0.114708
$77$	−3003.00	−0.0577203
$78$	0	0
$79$	−4972.00	−0.0896321	−0.0448160	−	0.998995i	$-0.514270\pi$
$79$	−4972.00	−0.0896321	−0.0448160	+	0.998995i	$0.514270\pi$
$80$	23296.0	0.406964
$81$	0	0
$82$	−54656.0	−0.897642
$83$	71196.0	1.13438	0.567192	−	0.823585i	$-0.308030\pi$
$83$	71196.0	1.13438	0.567192	+	0.823585i	$0.308030\pi$
$84$	0	0
$85$	166803.	2.50413
$86$	14804.0	0.215841
$87$	0	0
$88$	−5824.00	−0.0801705
$89$	3654.00	0.0488983	0.0244491	−	0.999701i	$-0.492217\pi$
$89$	3654.00	0.0488983	0.0244491	+	0.999701i	$0.492217\pi$
$90$	0	0
$91$	20130.0	0.254824
$92$	54976.0	0.677179
$93$	0	0
$94$	36812.0	0.429704
$95$	−32851.0	−0.373456
$96$	0	0
$97$	62362.0	0.672962	0.336481	−	0.941690i	$-0.390763\pi$
$97$	62362.0	0.672962	0.336481	+	0.941690i	$0.390763\pi$
$98$	62872.0	0.661290
$99$	0	0
$100$	82496.0	0.824960
$101$	171190.	1.66984	0.834920	−	0.550371i	$-0.185514\pi$
$101$	171190.	1.66984	0.834920	+	0.550371i	$0.185514\pi$
$102$	0	0
$103$	88590.0	0.822795	0.411397	−	0.911456i	$-0.365041\pi$
$103$	88590.0	0.822795	0.411397	+	0.911456i	$0.365041\pi$
$104$	39040.0	0.353937
$105$	0	0
$106$	116744.	1.00918
$107$	−117758.	−0.994331	−0.497165	−	0.867656i	$-0.665626\pi$
$107$	−117758.	−0.994331	−0.497165	+	0.867656i	$0.665626\pi$
$108$	0	0
$109$	82416.0	0.664424	0.332212	−	0.943205i	$-0.392205\pi$
$109$	82416.0	0.664424	0.332212	+	0.943205i	$0.392205\pi$
$110$	−33124.0	−0.261012
$111$	0	0
$112$	−8448.00	−0.0636369
$113$	−80414.0	−0.592428	−0.296214	−	0.955122i	$-0.595724\pi$
$113$	−80414.0	−0.592428	−0.296214	+	0.955122i	$0.595724\pi$
$114$	0	0
$115$	312676.	2.20470
$116$	−56992.0	−0.393250
$117$	0	0
$118$	−111216.	−0.735296
$119$	−60489.0	−0.391570
$120$	0	0
$121$	−152770.	−0.948582
$122$	−172508.	−1.04932
$123$	0	0
$124$	5152.00	0.0300900
$125$	184821.	1.05798
$126$	0	0
$127$	−138942.	−0.764406	−0.382203	−	0.924078i	$-0.624835\pi$
$127$	−138942.	−0.764406	−0.382203	+	0.924078i	$0.624835\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	222040.	1.15232
$131$	318813.	1.62315	0.811573	−	0.584251i	$-0.198611\pi$
$131$	318813.	1.62315	0.811573	+	0.584251i	$0.198611\pi$
$132$	0	0
$133$	11913.0	0.0583972
$134$	77712.0	0.373875
$135$	0	0
$136$	−117312.	−0.543870
$137$	363929.	1.65659	0.828295	−	0.560292i	$-0.189311\pi$
$137$	363929.	1.65659	0.828295	+	0.560292i	$0.189311\pi$
$138$	0	0
$139$	−309105.	−1.35697	−0.678483	−	0.734616i	$-0.737362\pi$
$139$	−309105.	−1.35697	−0.678483	+	0.734616i	$0.737362\pi$
$140$	−48048.0	−0.207184
$141$	0	0
$142$	28160.0	0.117196
$143$	−55510.0	−0.227003
$144$	0	0
$145$	−324142.	−1.28031
$146$	−149364.	−0.579914
$147$	0	0
$148$	115456.	0.433274
$149$	436653.	1.61128	0.805640	−	0.592406i	$-0.201822\pi$
$149$	436653.	1.61128	0.805640	+	0.592406i	$0.201822\pi$
$150$	0	0
$151$	466100.	1.66355	0.831777	−	0.555110i	$-0.187324\pi$
$151$	466100.	1.66355	0.831777	+	0.555110i	$0.187324\pi$
$152$	23104.0	0.0811107
$153$	0	0
$154$	12012.0	0.0408144
$155$	29302.0	0.0979643
$156$	0	0
$157$	218686.	0.708063	0.354031	−	0.935234i	$-0.384811\pi$
$157$	218686.	0.708063	0.354031	+	0.935234i	$0.384811\pi$
$158$	19888.0	0.0633794
$159$	0	0
$160$	−93184.0	−0.287767
$161$	−113388.	−0.344749
$162$	0	0
$163$	−279304.	−0.823395	−0.411697	−	0.911321i	$-0.635064\pi$
$163$	−279304.	−0.823395	−0.411697	+	0.911321i	$0.635064\pi$
$164$	218624.	0.634729
$165$	0	0
$166$	−284784.	−0.802131
$167$	−457854.	−1.27039	−0.635193	−	0.772353i	$-0.719080\pi$
$167$	−457854.	−1.27039	−0.635193	+	0.772353i	$0.719080\pi$
$168$	0	0
$169$	807.000	0.00217349
$170$	−667212.	−1.77069
$171$	0	0
$172$	−59216.0	−0.152622
$173$	733002.	1.86204	0.931022	−	0.364963i	$-0.118918\pi$
$173$	733002.	1.86204	0.931022	+	0.364963i	$0.118918\pi$
$174$	0	0
$175$	−170148.	−0.419983
$176$	23296.0	0.0566891
$177$	0	0
$178$	−14616.0	−0.0345763
$179$	247518.	0.577397	0.288698	−	0.957420i	$-0.406778\pi$
$179$	247518.	0.577397	0.288698	+	0.957420i	$0.406778\pi$
$180$	0	0
$181$	−189158.	−0.429169	−0.214584	−	0.976705i	$-0.568840\pi$
$181$	−189158.	−0.429169	−0.214584	+	0.976705i	$0.568840\pi$
$182$	−80520.0	−0.180188
$183$	0	0
$184$	−219904.	−0.478838
$185$	656656.	1.41062
$186$	0	0
$187$	166803.	0.348819
$188$	−147248.	−0.303847
$189$	0	0
$190$	131404.	0.264073
$191$	−330733.	−0.655985	−0.327993	−	0.944680i	$-0.606372\pi$
$191$	−330733.	−0.655985	−0.327993	+	0.944680i	$0.606372\pi$
$192$	0	0
$193$	674472.	1.30338	0.651689	−	0.758486i	$-0.274061\pi$
$193$	674472.	1.30338	0.651689	+	0.758486i	$0.274061\pi$
$194$	−249448.	−0.475856
$195$	0	0
$196$	−251488.	−0.467603
$197$	942346.	1.72999	0.864997	−	0.501776i	$-0.167320\pi$
$197$	942346.	1.72999	0.864997	+	0.501776i	$0.167320\pi$
$198$	0	0
$199$	−429505.	−0.768839	−0.384420	−	0.923158i	$-0.625598\pi$
$199$	−429505.	−0.768839	−0.384420	+	0.923158i	$0.625598\pi$
$200$	−329984.	−0.583335
$201$	0	0
$202$	−684760.	−1.18076
$203$	117546.	0.200202
$204$	0	0
$205$	1.24342e6	2.06650
$206$	−354360.	−0.581804
$207$	0	0
$208$	−156160.	−0.250272
$209$	−32851.0	−0.0520215
$210$	0	0
$211$	−569088.	−0.879981	−0.439990	−	0.898002i	$-0.645018\pi$
$211$	−569088.	−0.879981	−0.439990	+	0.898002i	$0.645018\pi$
$212$	−466976.	−0.713600
$213$	0	0
$214$	471032.	0.703098
$215$	−336791.	−0.496895
$216$	0	0
$217$	−10626.0	−0.0153186
$218$	−329664.	−0.469819
$219$	0	0
$220$	132496.	0.184564
$221$	−1.11813e6	−1.53997
$222$	0	0
$223$	1.00132e6	1.34838	0.674190	−	0.738558i	$-0.264493\pi$
$223$	1.00132e6	1.34838	0.674190	+	0.738558i	$0.264493\pi$
$224$	33792.0	0.0449981
$225$	0	0
$226$	321656.	0.418910
$227$	−169582.	−0.218431	−0.109216	−	0.994018i	$-0.534834\pi$
$227$	−169582.	−0.218431	−0.109216	+	0.994018i	$0.534834\pi$
$228$	0	0
$229$	−405367.	−0.510810	−0.255405	−	0.966834i	$-0.582209\pi$
$229$	−405367.	−0.510810	−0.255405	+	0.966834i	$0.582209\pi$
$230$	−1.25070e6	−1.55896
$231$	0	0
$232$	227968.	0.278070
$233$	−506649.	−0.611389	−0.305694	−	0.952130i	$-0.598889\pi$
$233$	−506649.	−0.611389	−0.305694	+	0.952130i	$0.598889\pi$
$234$	0	0
$235$	−837473.	−0.989239
$236$	444864.	0.519933
$237$	0	0
$238$	241956.	0.276882
$239$	−1.34766e6	−1.52611	−0.763053	−	0.646336i	$-0.776300\pi$
$239$	−1.34766e6	−1.52611	−0.763053	+	0.646336i	$0.776300\pi$
$240$	0	0
$241$	−840812.	−0.932516	−0.466258	−	0.884649i	$-0.654398\pi$
$241$	−840812.	−0.932516	−0.466258	+	0.884649i	$0.654398\pi$
$242$	611080.	0.670748
$243$	0	0
$244$	690032.	0.741984
$245$	−1.43034e6	−1.52238
$246$	0	0
$247$	220210.	0.229665
$248$	−20608.0	−0.0212768
$249$	0	0
$250$	−739284.	−0.748103
$251$	−1.08289e6	−1.08493	−0.542463	−	0.840079i	$-0.682508\pi$
$251$	−1.08289e6	−1.08493	−0.542463	+	0.840079i	$0.682508\pi$
$252$	0	0
$253$	312676.	0.307109
$254$	555768.	0.540517
$255$	0	0
$256$	65536.0	0.0625000
$257$	−522416.	−0.493382	−0.246691	−	0.969094i	$-0.579343\pi$
$257$	−522416.	−0.493382	−0.246691	+	0.969094i	$0.579343\pi$
$258$	0	0
$259$	−238128.	−0.220577
$260$	−888160.	−0.814813
$261$	0	0
$262$	−1.27525e6	−1.14774
$263$	1.08895e6	0.970774	0.485387	−	0.874299i	$-0.338679\pi$
$263$	1.08895e6	0.970774	0.485387	+	0.874299i	$0.338679\pi$
$264$	0	0
$265$	−2.65593e6	−2.32328
$266$	−47652.0	−0.0412931
$267$	0	0
$268$	−310848.	−0.264369
$269$	924702.	0.779150	0.389575	−	0.920995i	$-0.372622\pi$
$269$	924702.	0.779150	0.389575	+	0.920995i	$0.372622\pi$
$270$	0	0
$271$	−1.19270e6	−0.986525	−0.493262	−	0.869881i	$-0.664196\pi$
$271$	−1.19270e6	−0.986525	−0.493262	+	0.869881i	$0.664196\pi$
$272$	469248.	0.384574
$273$	0	0
$274$	−1.45572e6	−1.17139
$275$	469196.	0.374130
$276$	0	0
$277$	−1.90691e6	−1.49324	−0.746621	−	0.665250i	$-0.768325\pi$
$277$	−1.90691e6	−1.49324	−0.746621	+	0.665250i	$0.768325\pi$
$278$	1.23642e6	0.959520
$279$	0	0
$280$	192192.	0.146501
$281$	−19066.0	−0.0144044	−0.00720218	−	0.999974i	$-0.502293\pi$
$281$	−19066.0	−0.0144044	−0.00720218	+	0.999974i	$0.502293\pi$
$282$	0	0
$283$	−667833.	−0.495680	−0.247840	−	0.968801i	$-0.579721\pi$
$283$	−667833.	−0.495680	−0.247840	+	0.968801i	$0.579721\pi$
$284$	−112640.	−0.0828699
$285$	0	0
$286$	222040.	0.160515
$287$	−450912.	−0.323137
$288$	0	0
$289$	1.94003e6	1.36636
$290$	1.29657e6	0.905316
$291$	0	0
$292$	597456.	0.410061
$293$	1.43226e6	0.974659	0.487330	−	0.873218i	$-0.337971\pi$
$293$	1.43226e6	0.974659	0.487330	+	0.873218i	$0.337971\pi$
$294$	0	0
$295$	2.53016e6	1.69275
$296$	−461824.	−0.306371
$297$	0	0
$298$	−1.74661e6	−1.13935
$299$	−2.09596e6	−1.35583
$300$	0	0
$301$	122133.	0.0776992
$302$	−1.86440e6	−1.17631
$303$	0	0
$304$	−92416.0	−0.0573539
$305$	3.92456e6	2.41569
$306$	0	0
$307$	3.08911e6	1.87063	0.935315	−	0.353817i	$-0.115116\pi$
$307$	3.08911e6	1.87063	0.935315	+	0.353817i	$0.115116\pi$
$308$	−48048.0	−0.0288601
$309$	0	0
$310$	−117208.	−0.0692712
$311$	2.78248e6	1.63129	0.815645	−	0.578553i	$-0.196382\pi$
$311$	2.78248e6	1.63129	0.815645	+	0.578553i	$0.196382\pi$
$312$	0	0
$313$	−1.09383e6	−0.631089	−0.315544	−	0.948911i	$-0.602187\pi$
$313$	−1.09383e6	−0.631089	−0.315544	+	0.948911i	$0.602187\pi$
$314$	−874744.	−0.500676
$315$	0	0
$316$	−79552.0	−0.0448160
$317$	−1.55578e6	−0.869559	−0.434779	−	0.900537i	$-0.643174\pi$
$317$	−1.55578e6	−0.869559	−0.434779	+	0.900537i	$0.643174\pi$
$318$	0	0
$319$	−324142.	−0.178344
$320$	372736.	0.203482
$321$	0	0
$322$	453552.	0.243774
$323$	−661713.	−0.352910
$324$	0	0
$325$	−3.14516e6	−1.65171
$326$	1.11722e6	0.582228
$327$	0	0
$328$	−874496.	−0.448821
$329$	303699.	0.154687
$330$	0	0
$331$	35240.0	0.0176793	0.00883967	−	0.999961i	$-0.497186\pi$
$331$	35240.0	0.0176793	0.00883967	+	0.999961i	$0.497186\pi$
$332$	1.13914e6	0.567192
$333$	0	0
$334$	1.83142e6	0.898299
$335$	−1.76795e6	−0.860711
$336$	0	0
$337$	−1.64825e6	−0.790585	−0.395292	−	0.918555i	$-0.629357\pi$
$337$	−1.64825e6	−0.790585	−0.395292	+	0.918555i	$0.629357\pi$
$338$	−3228.00	−0.00153689
$339$	0	0
$340$	2.66885e6	1.25206
$341$	29302.0	0.0136462
$342$	0	0
$343$	1.07332e6	0.492602
$344$	236864.	0.107920
$345$	0	0
$346$	−2.93201e6	−1.31666
$347$	3.74631e6	1.67024	0.835122	−	0.550065i	$-0.185397\pi$
$347$	3.74631e6	1.67024	0.835122	+	0.550065i	$0.185397\pi$
$348$	0	0
$349$	1.15723e6	0.508578	0.254289	−	0.967128i	$-0.418159\pi$
$349$	1.15723e6	0.508578	0.254289	+	0.967128i	$0.418159\pi$
$350$	680592.	0.296973
$351$	0	0
$352$	−93184.0	−0.0400853
$353$	648018.	0.276790	0.138395	−	0.990377i	$-0.455806\pi$
$353$	648018.	0.276790	0.138395	+	0.990377i	$0.455806\pi$
$354$	0	0
$355$	−640640.	−0.269801
$356$	58464.0	0.0244491
$357$	0	0
$358$	−990072.	−0.408281
$359$	−3.31969e6	−1.35944	−0.679722	−	0.733470i	$-0.737900\pi$
$359$	−3.31969e6	−1.35944	−0.679722	+	0.733470i	$0.737900\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	756632.	0.303468
$363$	0	0
$364$	322080.	0.127412
$365$	3.39803e6	1.33504
$366$	0	0
$367$	3.30592e6	1.28123	0.640615	−	0.767862i	$-0.278679\pi$
$367$	3.30592e6	1.28123	0.640615	+	0.767862i	$0.278679\pi$
$368$	879616.	0.338590
$369$	0	0
$370$	−2.62662e6	−0.997456
$371$	963138.	0.363290
$372$	0	0
$373$	1.95786e6	0.728633	0.364316	−	0.931275i	$-0.381303\pi$
$373$	1.95786e6	0.728633	0.364316	+	0.931275i	$0.381303\pi$
$374$	−667212.	−0.246652
$375$	0	0
$376$	588992.	0.214852
$377$	2.17282e6	0.787355
$378$	0	0
$379$	−1.12179e6	−0.401156	−0.200578	−	0.979678i	$-0.564282\pi$
$379$	−1.12179e6	−0.401156	−0.200578	+	0.979678i	$0.564282\pi$
$380$	−525616.	−0.186728
$381$	0	0
$382$	1.32293e6	0.463852
$383$	−1.03305e6	−0.359854	−0.179927	−	0.983680i	$-0.557586\pi$
$383$	−1.03305e6	−0.359854	−0.179927	+	0.983680i	$0.557586\pi$
$384$	0	0
$385$	−273273.	−0.0939604
$386$	−2.69789e6	−0.921628
$387$	0	0
$388$	997792.	0.336481
$389$	−4.66876e6	−1.56433	−0.782164	−	0.623072i	$-0.785884\pi$
$389$	−4.66876e6	−1.56433	−0.782164	+	0.623072i	$0.785884\pi$
$390$	0	0
$391$	6.29819e6	2.08341
$392$	1.00595e6	0.330645
$393$	0	0
$394$	−3.76938e6	−1.22329
$395$	−452452.	−0.145908
$396$	0	0
$397$	−3.00310e6	−0.956300	−0.478150	−	0.878278i	$-0.658692\pi$
$397$	−3.00310e6	−0.956300	−0.478150	+	0.878278i	$0.658692\pi$
$398$	1.71802e6	0.543651
$399$	0	0
$400$	1.31994e6	0.412480
$401$	−3.94648e6	−1.22560	−0.612800	−	0.790238i	$-0.709957\pi$
$401$	−3.94648e6	−1.22560	−0.612800	+	0.790238i	$0.709957\pi$
$402$	0	0
$403$	−196420.	−0.0602453
$404$	2.73904e6	0.834920
$405$	0	0
$406$	−470184.	−0.141564
$407$	656656.	0.196495
$408$	0	0
$409$	4.38003e6	1.29470	0.647350	−	0.762193i	$-0.275877\pi$
$409$	4.38003e6	1.29470	0.647350	+	0.762193i	$0.275877\pi$
$410$	−4.97370e6	−1.46123
$411$	0	0
$412$	1.41744e6	0.411397
$413$	−917532.	−0.264695
$414$	0	0
$415$	6.47884e6	1.84662
$416$	624640.	0.176969
$417$	0	0
$418$	131404.	0.0367848
$419$	−872676.	−0.242839	−0.121419	−	0.992601i	$-0.538745\pi$
$419$	−872676.	−0.242839	−0.121419	+	0.992601i	$0.538745\pi$
$420$	0	0
$421$	−1.95854e6	−0.538552	−0.269276	−	0.963063i	$-0.586784\pi$
$421$	−1.95854e6	−0.538552	−0.269276	+	0.963063i	$0.586784\pi$
$422$	2.27635e6	0.622241
$423$	0	0
$424$	1.86790e6	0.504591
$425$	9.45095e6	2.53807
$426$	0	0
$427$	−1.42319e6	−0.377740
$428$	−1.88413e6	−0.497165
$429$	0	0
$430$	1.34716e6	0.351358
$431$	−90666.0	−0.0235099	−0.0117550	−	0.999931i	$-0.503742\pi$
$431$	−90666.0	−0.0235099	−0.0117550	+	0.999931i	$0.503742\pi$
$432$	0	0
$433$	3.50825e6	0.899230	0.449615	−	0.893222i	$-0.351561\pi$
$433$	3.50825e6	0.899230	0.449615	+	0.893222i	$0.351561\pi$
$434$	42504.0	0.0108319
$435$	0	0
$436$	1.31866e6	0.332212
$437$	−1.24040e6	−0.310711
$438$	0	0
$439$	−4.91970e6	−1.21836	−0.609182	−	0.793030i	$-0.708502\pi$
$439$	−4.91970e6	−1.21836	−0.609182	+	0.793030i	$0.708502\pi$
$440$	−529984.	−0.130506
$441$	0	0
$442$	4.47252e6	1.08892
$443$	−7.92687e6	−1.91908	−0.959539	−	0.281577i	$-0.909143\pi$
$443$	−7.92687e6	−1.91908	−0.959539	+	0.281577i	$0.909143\pi$
$444$	0	0
$445$	332514.	0.0795994
$446$	−4.00530e6	−0.953449
$447$	0	0
$448$	−135168.	−0.0318184
$449$	−1.48280e6	−0.347109	−0.173554	−	0.984824i	$-0.555525\pi$
$449$	−1.48280e6	−0.347109	−0.173554	+	0.984824i	$0.555525\pi$
$450$	0	0
$451$	1.24342e6	0.287858
$452$	−1.28662e6	−0.296214
$453$	0	0
$454$	678328.	0.154454
$455$	1.83183e6	0.414817
$456$	0	0
$457$	1.72825e6	0.387094	0.193547	−	0.981091i	$-0.438001\pi$
$457$	1.72825e6	0.387094	0.193547	+	0.981091i	$0.438001\pi$
$458$	1.62147e6	0.361197
$459$	0	0
$460$	5.00282e6	1.10235
$461$	552109.	0.120996	0.0604982	−	0.998168i	$-0.480731\pi$
$461$	552109.	0.120996	0.0604982	+	0.998168i	$0.480731\pi$
$462$	0	0
$463$	−5.54929e6	−1.20305	−0.601527	−	0.798853i	$-0.705441\pi$
$463$	−5.54929e6	−1.20305	−0.601527	+	0.798853i	$0.705441\pi$
$464$	−911872.	−0.196625
$465$	0	0
$466$	2.02660e6	0.432317
$467$	−2.05633e6	−0.436315	−0.218157	−	0.975914i	$-0.570005\pi$
$467$	−2.05633e6	−0.436315	−0.218157	+	0.975914i	$0.570005\pi$
$468$	0	0
$469$	641124.	0.134589
$470$	3.34989e6	0.699497
$471$	0	0
$472$	−1.77946e6	−0.367648
$473$	−336791.	−0.0692162
$474$	0	0
$475$	−1.86132e6	−0.378518
$476$	−967824.	−0.195785
$477$	0	0
$478$	5.39063e6	1.07912
$479$	2.04279e6	0.406804	0.203402	−	0.979095i	$-0.434800\pi$
$479$	2.04279e6	0.406804	0.203402	+	0.979095i	$0.434800\pi$
$480$	0	0
$481$	−4.40176e6	−0.867488
$482$	3.36325e6	0.659388
$483$	0	0
$484$	−2.44432e6	−0.474291
$485$	5.67494e6	1.09549
$486$	0	0
$487$	−6.58564e6	−1.25828	−0.629138	−	0.777294i	$-0.716592\pi$
$487$	−6.58564e6	−1.25828	−0.629138	+	0.777294i	$0.716592\pi$
$488$	−2.76013e6	−0.524662
$489$	0	0
$490$	5.72135e6	1.07649
$491$	3.96714e6	0.742633	0.371316	−	0.928506i	$-0.378906\pi$
$491$	3.96714e6	0.742633	0.371316	+	0.928506i	$0.378906\pi$
$492$	0	0
$493$	−6.52915e6	−1.20987
$494$	−880840.	−0.162398
$495$	0	0
$496$	82432.0	0.0150450
$497$	232320.	0.0421886
$498$	0	0
$499$	−2.69611e6	−0.484715	−0.242357	−	0.970187i	$-0.577921\pi$
$499$	−2.69611e6	−0.484715	−0.242357	+	0.970187i	$0.577921\pi$
$500$	2.95714e6	0.528989
$501$	0	0
$502$	4.33156e6	0.767159
$503$	−8.31756e6	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$503$	−8.31756e6	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$504$	0	0
$505$	1.55783e7	2.71826
$506$	−1.25070e6	−0.217159
$507$	0	0
$508$	−2.22307e6	−0.382203
$509$	−1.00197e7	−1.71420	−0.857098	−	0.515153i	$-0.827735\pi$
$509$	−1.00197e7	−1.71420	−0.857098	+	0.515153i	$0.827735\pi$
$510$	0	0
$511$	−1.23225e6	−0.208760
$512$	−262144.	−0.0441942
$513$	0	0
$514$	2.08966e6	0.348874
$515$	8.06169e6	1.33939
$516$	0	0
$517$	−837473.	−0.137798
$518$	952512.	0.155972
$519$	0	0
$520$	3.55264e6	0.576160
$521$	235936.	0.0380803	0.0190401	−	0.999819i	$-0.493939\pi$
$521$	235936.	0.0380803	0.0190401	+	0.999819i	$0.493939\pi$
$522$	0	0
$523$	914870.	0.146253	0.0731266	−	0.997323i	$-0.476702\pi$
$523$	914870.	0.146253	0.0731266	+	0.997323i	$0.476702\pi$
$524$	5.10101e6	0.811573
$525$	0	0
$526$	−4.35580e6	−0.686441
$527$	590226.	0.0925746
$528$	0	0
$529$	5.36975e6	0.834286
$530$	1.06237e7	1.64281
$531$	0	0
$532$	190608.	0.0291986
$533$	−8.33504e6	−1.27084
$534$	0	0
$535$	−1.07160e7	−1.61863
$536$	1.24339e6	0.186937
$537$	0	0
$538$	−3.69881e6	−0.550942
$539$	−1.43034e6	−0.212064
$540$	0	0
$541$	−4.04192e6	−0.593738	−0.296869	−	0.954918i	$-0.595943\pi$
$541$	−4.04192e6	−0.593738	−0.296869	+	0.954918i	$0.595943\pi$
$542$	4.77080e6	0.697578
$543$	0	0
$544$	−1.87699e6	−0.271935
$545$	7.49986e6	1.08159
$546$	0	0
$547$	8.18293e6	1.16934	0.584670	−	0.811271i	$-0.301224\pi$
$547$	8.18293e6	1.16934	0.584670	+	0.811271i	$0.301224\pi$
$548$	5.82286e6	0.828295
$549$	0	0
$550$	−1.87678e6	−0.264550
$551$	1.28588e6	0.180436
$552$	0	0
$553$	164076.	0.0228156
$554$	7.62763e6	1.05588
$555$	0	0
$556$	−4.94568e6	−0.678483
$557$	1.22357e7	1.67105	0.835527	−	0.549449i	$-0.185162\pi$
$557$	1.22357e7	1.67105	0.835527	+	0.549449i	$0.185162\pi$
$558$	0	0
$559$	2.25761e6	0.305576
$560$	−768768.	−0.103592
$561$	0	0
$562$	76264.0	0.0101854
$563$	−1.07830e7	−1.43374	−0.716870	−	0.697207i	$-0.754426\pi$
$563$	−1.07830e7	−1.43374	−0.716870	+	0.697207i	$0.754426\pi$
$564$	0	0
$565$	−7.31767e6	−0.964388
$566$	2.67133e6	0.350499
$567$	0	0
$568$	450560.	0.0585979
$569$	−1.10760e7	−1.43418	−0.717088	−	0.696982i	$-0.754526\pi$
$569$	−1.10760e7	−1.43418	−0.717088	+	0.696982i	$0.754526\pi$
$570$	0	0
$571$	−5.85570e6	−0.751604	−0.375802	−	0.926700i	$-0.622633\pi$
$571$	−5.85570e6	−0.751604	−0.375802	+	0.926700i	$0.622633\pi$
$572$	−888160.	−0.113501
$573$	0	0
$574$	1.80365e6	0.228493
$575$	1.77160e7	2.23458
$576$	0	0
$577$	−7.26541e6	−0.908491	−0.454246	−	0.890876i	$-0.650091\pi$
$577$	−7.26541e6	−0.908491	−0.454246	+	0.890876i	$0.650091\pi$
$578$	−7.76013e6	−0.966161
$579$	0	0
$580$	−5.18627e6	−0.640155
$581$	−2.34947e6	−0.288755
$582$	0	0
$583$	−2.65593e6	−0.323627
$584$	−2.38982e6	−0.289957
$585$	0	0
$586$	−5.72904e6	−0.689188
$587$	−1.19721e7	−1.43409	−0.717045	−	0.697027i	$-0.754506\pi$
$587$	−1.19721e7	−1.43409	−0.717045	+	0.697027i	$0.754506\pi$
$588$	0	0
$589$	−116242.	−0.0138062
$590$	−1.01207e7	−1.19696
$591$	0	0
$592$	1.84730e6	0.216637
$593$	2.07789e6	0.242653	0.121326	−	0.992613i	$-0.461285\pi$
$593$	2.07789e6	0.242653	0.121326	+	0.992613i	$0.461285\pi$
$594$	0	0
$595$	−5.50450e6	−0.637420
$596$	6.98645e6	0.805640
$597$	0	0
$598$	8.38384e6	0.958716
$599$	−1.05217e7	−1.19817	−0.599085	−	0.800686i	$-0.704469\pi$
$599$	−1.05217e7	−1.19817	−0.599085	+	0.800686i	$0.704469\pi$
$600$	0	0
$601$	3.58294e6	0.404626	0.202313	−	0.979321i	$-0.435154\pi$
$601$	3.58294e6	0.404626	0.202313	+	0.979321i	$0.435154\pi$
$602$	−488532.	−0.0549417
$603$	0	0
$604$	7.45760e6	0.831777
$605$	−1.39021e7	−1.54416
$606$	0	0
$607$	−4.38625e6	−0.483194	−0.241597	−	0.970377i	$-0.577671\pi$
$607$	−4.38625e6	−0.483194	−0.241597	+	0.970377i	$0.577671\pi$
$608$	369664.	0.0405554
$609$	0	0
$610$	−1.56982e7	−1.70815
$611$	5.61383e6	0.608354
$612$	0	0
$613$	3.85958e6	0.414848	0.207424	−	0.978251i	$-0.433492\pi$
$613$	3.85958e6	0.414848	0.207424	+	0.978251i	$0.433492\pi$
$614$	−1.23564e7	−1.32273
$615$	0	0
$616$	192192.	0.0204072
$617$	−1.17256e7	−1.24000	−0.620000	−	0.784602i	$-0.712867\pi$
$617$	−1.17256e7	−1.24000	−0.620000	+	0.784602i	$0.712867\pi$
$618$	0	0
$619$	6.81869e6	0.715277	0.357639	−	0.933860i	$-0.383582\pi$
$619$	6.81869e6	0.715277	0.357639	+	0.933860i	$0.383582\pi$
$620$	468832.	0.0489822
$621$	0	0
$622$	−1.11299e7	−1.15350
$623$	−120582.	−0.0124469
$624$	0	0
$625$	706211.	0.0723160
$626$	4.37534e6	0.446247
$627$	0	0
$628$	3.49898e6	0.354031
$629$	1.32269e7	1.33301
$630$	0	0
$631$	−9.81980e6	−0.981815	−0.490907	−	0.871212i	$-0.663335\pi$
$631$	−9.81980e6	−0.981815	−0.490907	+	0.871212i	$0.663335\pi$
$632$	318208.	0.0316897
$633$	0	0
$634$	6.22310e6	0.614871
$635$	−1.26437e7	−1.24434
$636$	0	0
$637$	9.58798e6	0.936221
$638$	1.29657e6	0.126108
$639$	0	0
$640$	−1.49094e6	−0.143884
$641$	1.96019e7	1.88432	0.942158	−	0.335170i	$-0.108794\pi$
$641$	1.96019e7	1.88432	0.942158	+	0.335170i	$0.108794\pi$
$642$	0	0
$643$	1.23252e7	1.17562	0.587810	−	0.808999i	$-0.299990\pi$
$643$	1.23252e7	1.17562	0.587810	+	0.808999i	$0.299990\pi$
$644$	−1.81421e6	−0.172374
$645$	0	0
$646$	2.64685e6	0.249545
$647$	−968621.	−0.0909689	−0.0454845	−	0.998965i	$-0.514483\pi$
$647$	−968621.	−0.0909689	−0.0454845	+	0.998965i	$0.514483\pi$
$648$	0	0
$649$	2.53016e6	0.235796
$650$	1.25806e7	1.16794
$651$	0	0
$652$	−4.46886e6	−0.411697
$653$	517653.	0.0475068	0.0237534	−	0.999718i	$-0.492438\pi$
$653$	517653.	0.0475068	0.0237534	+	0.999718i	$0.492438\pi$
$654$	0	0
$655$	2.90120e7	2.64225
$656$	3.49798e6	0.317364
$657$	0	0
$658$	−1.21480e6	−0.109380
$659$	−7.30548e6	−0.655293	−0.327646	−	0.944800i	$-0.606255\pi$
$659$	−7.30548e6	−0.655293	−0.327646	+	0.944800i	$0.606255\pi$
$660$	0	0
$661$	−2.12076e7	−1.88794	−0.943971	−	0.330028i	$-0.892942\pi$
$661$	−2.12076e7	−1.88794	−0.943971	+	0.330028i	$0.892942\pi$
$662$	−140960.	−0.0125012
$663$	0	0
$664$	−4.55654e6	−0.401066
$665$	1.08408e6	0.0950623
$666$	0	0
$667$	−1.22390e7	−1.06520
$668$	−7.32566e6	−0.635193
$669$	0	0
$670$	7.07179e6	0.608615
$671$	3.92456e6	0.336499
$672$	0	0
$673$	−5.20143e6	−0.442675	−0.221337	−	0.975197i	$-0.571042\pi$
$673$	−5.20143e6	−0.442675	−0.221337	+	0.975197i	$0.571042\pi$
$674$	6.59300e6	0.559028
$675$	0	0
$676$	12912.0	0.00108674
$677$	8.90338e6	0.746592	0.373296	−	0.927712i	$-0.378228\pi$
$677$	8.90338e6	0.746592	0.373296	+	0.927712i	$0.378228\pi$
$678$	0	0
$679$	−2.05795e6	−0.171301
$680$	−1.06754e7	−0.885343
$681$	0	0
$682$	−117208.	−0.00964931
$683$	1.40518e7	1.15260	0.576302	−	0.817237i	$-0.304495\pi$
$683$	1.40518e7	1.15260	0.576302	+	0.817237i	$0.304495\pi$
$684$	0	0
$685$	3.31175e7	2.69669
$686$	−4.29330e6	−0.348322
$687$	0	0
$688$	−947456.	−0.0763111
$689$	1.78035e7	1.42875
$690$	0	0
$691$	−5.38920e6	−0.429368	−0.214684	−	0.976684i	$-0.568872\pi$
$691$	−5.38920e6	−0.429368	−0.214684	+	0.976684i	$0.568872\pi$
$692$	1.17280e7	0.931022
$693$	0	0
$694$	−1.49852e7	−1.18104
$695$	−2.81286e7	−2.20895
$696$	0	0
$697$	2.50461e7	1.95280
$698$	−4.62893e6	−0.359619
$699$	0	0
$700$	−2.72237e6	−0.209992
$701$	−2.30897e7	−1.77469	−0.887347	−	0.461103i	$-0.847454\pi$
$701$	−2.30897e7	−1.77469	−0.887347	+	0.461103i	$0.847454\pi$
$702$	0	0
$703$	−2.60498e6	−0.198800
$704$	372736.	0.0283446
$705$	0	0
$706$	−2.59207e6	−0.195720
$707$	−5.64927e6	−0.425054
$708$	0	0
$709$	4.25355e6	0.317787	0.158893	−	0.987296i	$-0.449207\pi$
$709$	4.25355e6	0.317787	0.158893	+	0.987296i	$0.449207\pi$
$710$	2.56256e6	0.190778
$711$	0	0
$712$	−233856.	−0.0172882
$713$	1.10639e6	0.0815052
$714$	0	0
$715$	−5.05141e6	−0.369528
$716$	3.96029e6	0.288698
$717$	0	0
$718$	1.32788e7	0.961272
$719$	818051.	0.0590144	0.0295072	−	0.999565i	$-0.490606\pi$
$719$	818051.	0.0590144	0.0295072	+	0.999565i	$0.490606\pi$
$720$	0	0
$721$	−2.92347e6	−0.209440
$722$	−521284.	−0.0372161
$723$	0	0
$724$	−3.02653e6	−0.214584
$725$	−1.83657e7	−1.29766
$726$	0	0
$727$	−2.45644e7	−1.72373	−0.861866	−	0.507136i	$-0.830704\pi$
$727$	−2.45644e7	−1.72373	−0.861866	+	0.507136i	$0.830704\pi$
$728$	−1.28832e6	−0.0900939
$729$	0	0
$730$	−1.35921e7	−0.944018
$731$	−6.78393e6	−0.469557
$732$	0	0
$733$	368658.	0.0253433	0.0126717	−	0.999920i	$-0.495966\pi$
$733$	368658.	0.0253433	0.0126717	+	0.999920i	$0.495966\pi$
$734$	−1.32237e7	−0.905967
$735$	0	0
$736$	−3.51846e6	−0.239419
$737$	−1.76795e6	−0.119895
$738$	0	0
$739$	−2.64037e7	−1.77850	−0.889249	−	0.457424i	$-0.848772\pi$
$739$	−2.64037e7	−1.77850	−0.889249	+	0.457424i	$0.848772\pi$
$740$	1.05065e7	0.705308
$741$	0	0
$742$	−3.85255e6	−0.256885
$743$	−2.77051e7	−1.84114	−0.920572	−	0.390574i	$-0.872277\pi$
$743$	−2.77051e7	−1.84114	−0.920572	+	0.390574i	$0.872277\pi$
$744$	0	0
$745$	3.97354e7	2.62293
$746$	−7.83142e6	−0.515221
$747$	0	0
$748$	2.66885e6	0.174409
$749$	3.88601e6	0.253104
$750$	0	0
$751$	−2.72102e7	−1.76048	−0.880240	−	0.474528i	$-0.842619\pi$
$751$	−2.72102e7	−1.76048	−0.880240	+	0.474528i	$0.842619\pi$
$752$	−2.35597e6	−0.151923
$753$	0	0
$754$	−8.69128e6	−0.556744
$755$	4.24151e7	2.70803
$756$	0	0
$757$	−1.40965e7	−0.894071	−0.447035	−	0.894516i	$-0.647520\pi$
$757$	−1.40965e7	−0.894071	−0.447035	+	0.894516i	$0.647520\pi$
$758$	4.48716e6	0.283660
$759$	0	0
$760$	2.10246e6	0.132037
$761$	−3.58869e6	−0.224633	−0.112317	−	0.993672i	$-0.535827\pi$
$761$	−3.58869e6	−0.224633	−0.112317	+	0.993672i	$0.535827\pi$
$762$	0	0
$763$	−2.71973e6	−0.169127
$764$	−5.29173e6	−0.327993
$765$	0	0
$766$	4.13222e6	0.254455
$767$	−1.69604e7	−1.04100
$768$	0	0
$769$	9.72717e6	0.593158	0.296579	−	0.955008i	$-0.404154\pi$
$769$	9.72717e6	0.593158	0.296579	+	0.955008i	$0.404154\pi$
$770$	1.09309e6	0.0664400
$771$	0	0
$772$	1.07916e7	0.651689
$773$	4.33598e6	0.260999	0.130500	−	0.991448i	$-0.458342\pi$
$773$	4.33598e6	0.260999	0.130500	+	0.991448i	$0.458342\pi$
$774$	0	0
$775$	1.66023e6	0.0992921
$776$	−3.99117e6	−0.237928
$777$	0	0
$778$	1.86751e7	1.10615
$779$	−4.93270e6	−0.291234
$780$	0	0
$781$	−640640.	−0.0375826
$782$	−2.51928e7	−1.47319
$783$	0	0
$784$	−4.02381e6	−0.233801
$785$	1.99004e7	1.15263
$786$	0	0
$787$	−3.07451e7	−1.76946	−0.884728	−	0.466108i	$-0.845656\pi$
$787$	−3.07451e7	−1.76946	−0.884728	+	0.466108i	$0.845656\pi$
$788$	1.50775e7	0.864997
$789$	0	0
$790$	1.80981e6	0.103173
$791$	2.65366e6	0.150801
$792$	0	0
$793$	−2.63075e7	−1.48558
$794$	1.20124e7	0.676206
$795$	0	0
$796$	−6.87208e6	−0.384420
$797$	2.49744e7	1.39268	0.696338	−	0.717714i	$-0.254811\pi$
$797$	2.49744e7	1.39268	0.696338	+	0.717714i	$0.254811\pi$
$798$	0	0
$799$	−1.68691e7	−0.934813
$800$	−5.27974e6	−0.291667
$801$	0	0
$802$	1.57859e7	0.866631
$803$	3.39803e6	0.185968
$804$	0	0
$805$	−1.03183e7	−0.561201
$806$	785680.	0.0425999
$807$	0	0
$808$	−1.09562e7	−0.590378
$809$	2.33716e7	1.25550	0.627750	−	0.778415i	$-0.283976\pi$
$809$	2.33716e7	1.25550	0.627750	+	0.778415i	$0.283976\pi$
$810$	0	0
$811$	−2.33591e7	−1.24711	−0.623554	−	0.781780i	$-0.714312\pi$
$811$	−2.33591e7	−1.24711	−0.623554	+	0.781780i	$0.714312\pi$
$812$	1.88074e6	0.100101
$813$	0	0
$814$	−2.62662e6	−0.138943
$815$	−2.54167e7	−1.34037
$816$	0	0
$817$	1.33606e6	0.0700279
$818$	−1.75201e7	−0.915491
$819$	0	0
$820$	1.98948e7	1.03325
$821$	2.26403e7	1.17226	0.586131	−	0.810216i	$-0.300650\pi$
$821$	2.26403e7	1.17226	0.586131	+	0.810216i	$0.300650\pi$
$822$	0	0
$823$	2.62615e7	1.35151	0.675757	−	0.737125i	$-0.263817\pi$
$823$	2.62615e7	1.35151	0.675757	+	0.737125i	$0.263817\pi$
$824$	−5.66976e6	−0.290902
$825$	0	0
$826$	3.67013e6	0.187168
$827$	1.54979e7	0.787969	0.393984	−	0.919117i	$-0.371096\pi$
$827$	1.54979e7	0.787969	0.393984	+	0.919117i	$0.371096\pi$
$828$	0	0
$829$	−6.51750e6	−0.329378	−0.164689	−	0.986346i	$-0.552662\pi$
$829$	−6.51750e6	−0.329378	−0.164689	+	0.986346i	$0.552662\pi$
$830$	−2.59153e7	−1.30576
$831$	0	0
$832$	−2.49856e6	−0.125136
$833$	−2.88111e7	−1.43862
$834$	0	0
$835$	−4.16647e7	−2.06801
$836$	−525616.	−0.0260108
$837$	0	0
$838$	3.49070e6	0.171713
$839$	−2.10515e7	−1.03247	−0.516236	−	0.856447i	$-0.672667\pi$
$839$	−2.10515e7	−1.03247	−0.516236	+	0.856447i	$0.672667\pi$
$840$	0	0
$841$	−7.82330e6	−0.381417
$842$	7.83417e6	0.380814
$843$	0	0
$844$	−9.10541e6	−0.439990
$845$	73437.0	0.00353812
$846$	0	0
$847$	5.04141e6	0.241459
$848$	−7.47162e6	−0.356800
$849$	0	0
$850$	−3.78038e7	−1.79468
$851$	2.47942e7	1.17362
$852$	0	0
$853$	2.92684e7	1.37729	0.688647	−	0.725097i	$-0.258205\pi$
$853$	2.92684e7	1.37729	0.688647	+	0.725097i	$0.258205\pi$
$854$	5.69276e6	0.267103
$855$	0	0
$856$	7.53651e6	0.351549
$857$	1.40825e7	0.654979	0.327490	−	0.944855i	$-0.393797\pi$
$857$	1.40825e7	0.654979	0.327490	+	0.944855i	$0.393797\pi$
$858$	0	0
$859$	−1.08817e7	−0.503167	−0.251584	−	0.967836i	$-0.580951\pi$
$859$	−1.08817e7	−0.503167	−0.251584	+	0.967836i	$0.580951\pi$
$860$	−5.38866e6	−0.248447
$861$	0	0
$862$	362664.	0.0166240
$863$	−7.83752e6	−0.358222	−0.179111	−	0.983829i	$-0.557322\pi$
$863$	−7.83752e6	−0.358222	−0.179111	+	0.983829i	$0.557322\pi$
$864$	0	0
$865$	6.67032e7	3.03114
$866$	−1.40330e7	−0.635852
$867$	0	0
$868$	−170016.	−0.00765932
$869$	−452452.	−0.0203246
$870$	0	0
$871$	1.18511e7	0.529313
$872$	−5.27462e6	−0.234909
$873$	0	0
$874$	4.96158e6	0.219706
$875$	−6.09909e6	−0.269305
$876$	0	0
$877$	−3.13170e6	−0.137493	−0.0687466	−	0.997634i	$-0.521900\pi$
$877$	−3.13170e6	−0.137493	−0.0687466	+	0.997634i	$0.521900\pi$
$878$	1.96788e7	0.861514
$879$	0	0
$880$	2.11994e6	0.0922818
$881$	1.85223e7	0.803997	0.401998	−	0.915640i	$-0.368316\pi$
$881$	1.85223e7	0.803997	0.401998	+	0.915640i	$0.368316\pi$
$882$	0	0
$883$	5.51096e6	0.237862	0.118931	−	0.992903i	$-0.462053\pi$
$883$	5.51096e6	0.237862	0.118931	+	0.992903i	$0.462053\pi$
$884$	−1.78901e7	−0.769984
$885$	0	0
$886$	3.17075e7	1.35699
$887$	4.56817e7	1.94955	0.974773	−	0.223199i	$-0.0716499\pi$
$887$	4.56817e7	1.94955	0.974773	+	0.223199i	$0.0716499\pi$
$888$	0	0
$889$	4.58509e6	0.194578
$890$	−1.33006e6	−0.0562853
$891$	0	0
$892$	1.60212e7	0.674190
$893$	3.32228e6	0.139415
$894$	0	0
$895$	2.25241e7	0.939919
$896$	540672.	0.0224990
$897$	0	0
$898$	5.93118e6	0.245443
$899$	−1.14696e6	−0.0473315
$900$	0	0
$901$	−5.34979e7	−2.19546
$902$	−4.97370e6	−0.203546
$903$	0	0
$904$	5.14650e6	0.209455
$905$	−1.72134e7	−0.698626
$906$	0	0
$907$	3.54419e7	1.43054	0.715269	−	0.698849i	$-0.246304\pi$
$907$	3.54419e7	1.43054	0.715269	+	0.698849i	$0.246304\pi$
$908$	−2.71331e6	−0.109216
$909$	0	0
$910$	−7.32732e6	−0.293320
$911$	4.18553e7	1.67092	0.835458	−	0.549554i	$-0.185202\pi$
$911$	4.18553e7	1.67092	0.835458	+	0.549554i	$0.185202\pi$
$912$	0	0
$913$	6.47884e6	0.257229
$914$	−6.91300e6	−0.273717
$915$	0	0
$916$	−6.48587e6	−0.255405
$917$	−1.05208e7	−0.413168
$918$	0	0
$919$	1.29489e7	0.505758	0.252879	−	0.967498i	$-0.418622\pi$
$919$	1.29489e7	0.505758	0.252879	+	0.967498i	$0.418622\pi$
$920$	−2.00113e7	−0.779480
$921$	0	0
$922$	−2.20844e6	−0.0855574
$923$	4.29440e6	0.165920
$924$	0	0
$925$	3.72057e7	1.42973
$926$	2.21972e7	0.850687
$927$	0	0
$928$	3.64749e6	0.139035
$929$	3.42756e7	1.30300	0.651502	−	0.758647i	$-0.274139\pi$
$929$	3.42756e7	1.30300	0.651502	+	0.758647i	$0.274139\pi$
$930$	0	0
$931$	5.67420e6	0.214551
$932$	−8.10638e6	−0.305694
$933$	0	0
$934$	8.22531e6	0.308521
$935$	1.51791e7	0.567827
$936$	0	0
$937$	3.81392e7	1.41913	0.709566	−	0.704639i	$-0.248891\pi$
$937$	3.81392e7	1.41913	0.709566	+	0.704639i	$0.248891\pi$
$938$	−2.56450e6	−0.0951689
$939$	0	0
$940$	−1.33996e7	−0.494619
$941$	−4.34881e7	−1.60102	−0.800510	−	0.599320i	$-0.795438\pi$
$941$	−4.34881e7	−1.60102	−0.800510	+	0.599320i	$0.795438\pi$
$942$	0	0
$943$	4.69495e7	1.71930
$944$	7.11782e6	0.259966
$945$	0	0
$946$	1.34716e6	0.0489432
$947$	3.23771e7	1.17318	0.586588	−	0.809885i	$-0.300471\pi$
$947$	3.23771e7	1.17318	0.586588	+	0.809885i	$0.300471\pi$
$948$	0	0
$949$	−2.27780e7	−0.821013
$950$	7.44526e6	0.267652
$951$	0	0
$952$	3.87130e6	0.138441
$953$	−4.12172e7	−1.47010	−0.735050	−	0.678013i	$-0.762841\pi$
$953$	−4.12172e7	−1.47010	−0.735050	+	0.678013i	$0.762841\pi$
$954$	0	0
$955$	−3.00967e7	−1.06785
$956$	−2.15625e7	−0.763053
$957$	0	0
$958$	−8.17117e6	−0.287654
$959$	−1.20097e7	−0.421681
$960$	0	0
$961$	−2.85255e7	−0.996378
$962$	1.76070e7	0.613407
$963$	0	0
$964$	−1.34530e7	−0.466258
$965$	6.13770e7	2.12171
$966$	0	0
$967$	3.40238e7	1.17008	0.585041	−	0.811003i	$-0.301078\pi$
$967$	3.40238e7	1.17008	0.585041	+	0.811003i	$0.301078\pi$
$968$	9.77728e6	0.335374
$969$	0	0
$970$	−2.26998e7	−0.774626
$971$	4.03426e7	1.37314	0.686571	−	0.727063i	$-0.259115\pi$
$971$	4.03426e7	1.37314	0.686571	+	0.727063i	$0.259115\pi$
$972$	0	0
$973$	1.02005e7	0.345412
$974$	2.63426e7	0.889735
$975$	0	0
$976$	1.10405e7	0.370992
$977$	−1.70401e7	−0.571130	−0.285565	−	0.958359i	$-0.592181\pi$
$977$	−1.70401e7	−0.571130	−0.285565	+	0.958359i	$0.592181\pi$
$978$	0	0
$979$	332514.	0.0110880
$980$	−2.28854e7	−0.761191
$981$	0	0
$982$	−1.58686e7	−0.525121
$983$	−2.20796e7	−0.728799	−0.364399	−	0.931243i	$-0.618726\pi$
$983$	−2.20796e7	−0.728799	−0.364399	+	0.931243i	$0.618726\pi$
$984$	0	0
$985$	8.57535e7	2.81619
$986$	2.61166e7	0.855508
$987$	0	0
$988$	3.52336e6	0.114832
$989$	−1.27166e7	−0.413411
$990$	0	0
$991$	−1.70400e7	−0.551171	−0.275585	−	0.961277i	$-0.588872\pi$
$991$	−1.70400e7	−0.551171	−0.275585	+	0.961277i	$0.588872\pi$
$992$	−329728.	−0.0106384
$993$	0	0
$994$	−929280.	−0.0298319
$995$	−3.90850e7	−1.25156
$996$	0	0
$997$	2.01866e7	0.643170	0.321585	−	0.946881i	$-0.395784\pi$
$997$	2.01866e7	0.643170	0.321585	+	0.946881i	$0.395784\pi$
$998$	1.07844e7	0.342745
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.6.a.c.1.1		1
3.2	odd	2		114.6.a.d.1.1	✓	1
12.11	even	2		912.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.6.a.d.1.1	✓	1	3.2	odd	2
342.6.a.c.1.1		1	1.1	even	1	trivial
912.6.a.b.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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