LMFDB - Embedded newform 114.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,6,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	114.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:	$18.2837554587$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -91.0000 q^{5} +36.0000 q^{6} -33.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -91.0000 q^{5} +36.0000 q^{6} -33.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -364.000 q^{10} -91.0000 q^{11} +144.000 q^{12} -610.000 q^{13} -132.000 q^{14} -819.000 q^{15} +256.000 q^{16} -1833.00 q^{17} +324.000 q^{18} -361.000 q^{19} -1456.00 q^{20} -297.000 q^{21} -364.000 q^{22} -3436.00 q^{23} +576.000 q^{24} +5156.00 q^{25} -2440.00 q^{26} +729.000 q^{27} -528.000 q^{28} +3562.00 q^{29} -3276.00 q^{30} +322.000 q^{31} +1024.00 q^{32} -819.000 q^{33} -7332.00 q^{34} +3003.00 q^{35} +1296.00 q^{36} +7216.00 q^{37} -1444.00 q^{38} -5490.00 q^{39} -5824.00 q^{40} -13664.0 q^{41} -1188.00 q^{42} -3701.00 q^{43} -1456.00 q^{44} -7371.00 q^{45} -13744.0 q^{46} +9203.00 q^{47} +2304.00 q^{48} -15718.0 q^{49} +20624.0 q^{50} -16497.0 q^{51} -9760.00 q^{52} +29186.0 q^{53} +2916.00 q^{54} +8281.00 q^{55} -2112.00 q^{56} -3249.00 q^{57} +14248.0 q^{58} -27804.0 q^{59} -13104.0 q^{60} +43127.0 q^{61} +1288.00 q^{62} -2673.00 q^{63} +4096.00 q^{64} +55510.0 q^{65} -3276.00 q^{66} -19428.0 q^{67} -29328.0 q^{68} -30924.0 q^{69} +12012.0 q^{70} +7040.00 q^{71} +5184.00 q^{72} +37341.0 q^{73} +28864.0 q^{74} +46404.0 q^{75} -5776.00 q^{76} +3003.00 q^{77} -21960.0 q^{78} -4972.00 q^{79} -23296.0 q^{80} +6561.00 q^{81} -54656.0 q^{82} -71196.0 q^{83} -4752.00 q^{84} +166803. q^{85} -14804.0 q^{86} +32058.0 q^{87} -5824.00 q^{88} -3654.00 q^{89} -29484.0 q^{90} +20130.0 q^{91} -54976.0 q^{92} +2898.00 q^{93} +36812.0 q^{94} +32851.0 q^{95} +9216.00 q^{96} +62362.0 q^{97} -62872.0 q^{98} -7371.00 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−91.0000	−1.62786	−0.813929	−	0.580965i	$-0.802675\pi$
$5$	−91.0000	−1.62786	−0.813929	+	0.580965i	$0.802675\pi$
$6$	36.0000	0.408248
$7$	−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
$9$	81.0000	0.333333
$10$	−364.000	−1.15107
$11$	−91.0000	−0.226756	−0.113378	−	0.993552i	$-0.536167\pi$
$11$	−91.0000	−0.226756	−0.113378	+	0.993552i	$0.536167\pi$
$12$	144.000	0.288675
$13$	−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
$15$	−819.000	−0.939844
$16$	256.000	0.250000
$17$	−1833.00	−1.53830	−0.769148	−	0.639070i	$-0.779319\pi$
$17$	−1833.00	−1.53830	−0.769148	+	0.639070i	$0.779319\pi$
$18$	324.000	0.235702
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	−1456.00	−0.813929
$21$	−297.000	−0.146963
$22$	−364.000	−0.160341
$23$	−3436.00	−1.35436	−0.677179	−	0.735818i	$-0.736798\pi$
$23$	−3436.00	−1.35436	−0.677179	+	0.735818i	$0.736798\pi$
$24$	576.000	0.204124
$25$	5156.00	1.64992
$26$	−2440.00	−0.707875
$27$	729.000	0.192450
$28$	−528.000	−0.127274
$29$	3562.00	0.786500	0.393250	−	0.919432i	$-0.371351\pi$
$29$	3562.00	0.786500	0.393250	+	0.919432i	$0.371351\pi$
$30$	−3276.00	−0.664570
$31$	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
$33$	−819.000	−0.130918
$34$	−7332.00	−1.08774
$35$	3003.00	0.414367
$36$	1296.00	0.166667
$37$	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
$39$	−5490.00	−0.577977
$40$	−5824.00	−0.575535
$41$	−13664.0	−1.26946	−0.634729	−	0.772735i	$-0.718888\pi$
$41$	−13664.0	−1.26946	−0.634729	+	0.772735i	$0.718888\pi$
$42$	−1188.00	−0.103919
$43$	−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
$45$	−7371.00	−0.542619
$46$	−13744.0	−0.957676
$47$	9203.00	0.607694	0.303847	−	0.952721i	$-0.401729\pi$
$47$	9203.00	0.607694	0.303847	+	0.952721i	$0.401729\pi$
$48$	2304.00	0.144338
$49$	−15718.0	−0.935206
$50$	20624.0	1.16667
$51$	−16497.0	−0.888136
$52$	−9760.00	−0.500543
$53$	29186.0	1.42720	0.713600	−	0.700553i	$-0.247063\pi$
$53$	29186.0	1.42720	0.713600	+	0.700553i	$0.247063\pi$
$54$	2916.00	0.136083
$55$	8281.00	0.369127
$56$	−2112.00	−0.0899961
$57$	−3249.00	−0.132453
$58$	14248.0	0.556140
$59$	−27804.0	−1.03987	−0.519933	−	0.854207i	$-0.674043\pi$
$59$	−27804.0	−1.03987	−0.519933	+	0.854207i	$0.674043\pi$
$60$	−13104.0	−0.469922
$61$	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
$63$	−2673.00	−0.0848492
$64$	4096.00	0.125000
$65$	55510.0	1.62963
$66$	−3276.00	−0.0925729
$67$	−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
$69$	−30924.0	−0.781939
$70$	12012.0	0.293002
$71$	7040.00	0.165740	0.0828699	−	0.996560i	$-0.473591\pi$
$71$	7040.00	0.165740	0.0828699	+	0.996560i	$0.473591\pi$
$72$	5184.00	0.117851
$73$	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
$75$	46404.0	0.952582
$76$	−5776.00	−0.114708
$77$	3003.00	0.0577203
$78$	−21960.0	−0.408692
$79$	−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
$81$	6561.00	0.111111
$82$	−54656.0	−0.897642
$83$	−71196.0	−1.13438	−0.567192	−	0.823585i	$-0.691970\pi$
$83$	−71196.0	−1.13438	−0.567192	+	0.823585i	$0.691970\pi$
$84$	−4752.00	−0.0734815
$85$	166803.	2.50413
$86$	−14804.0	−0.215841
$87$	32058.0	0.454086
$88$	−5824.00	−0.0801705
$89$	−3654.00	−0.0488983	−0.0244491	−	0.999701i	$-0.507783\pi$
$89$	−3654.00	−0.0488983	−0.0244491	+	0.999701i	$0.507783\pi$
$90$	−29484.0	−0.383690
$91$	20130.0	0.254824
$92$	−54976.0	−0.677179
$93$	2898.00	0.0347449
$94$	36812.0	0.429704
$95$	32851.0	0.373456
$96$	9216.00	0.102062
$97$	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
$99$	−7371.00	−0.0755855
$100$	82496.0	0.824960
$101$	−171190.	−1.66984	−0.834920	−	0.550371i	$-0.814486\pi$
$101$	−171190.	−1.66984	−0.834920	+	0.550371i	$0.814486\pi$
$102$	−65988.0	−0.628007
$103$	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
$105$	27027.0	0.239235
$106$	116744.	1.00918
$107$	117758.	0.994331	0.497165	−	0.867656i	$-0.334374\pi$
$107$	117758.	0.994331	0.497165	+	0.867656i	$0.334374\pi$
$108$	11664.0	0.0962250
$109$	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
$111$	64944.0	0.500301
$112$	−8448.00	−0.0636369
$113$	80414.0	0.592428	0.296214	−	0.955122i	$-0.404276\pi$
$113$	80414.0	0.592428	0.296214	+	0.955122i	$0.404276\pi$
$114$	−12996.0	−0.0936586
$115$	312676.	2.20470
$116$	56992.0	0.393250
$117$	−49410.0	−0.333695
$118$	−111216.	−0.735296
$119$	60489.0	0.391570
$120$	−52416.0	−0.332285
$121$	−152770.	−0.948582
$122$	172508.	1.04932
$123$	−122976.	−0.732922
$124$	5152.00	0.0300900
$125$	−184821.	−1.05798
$126$	−10692.0	−0.0599974
$127$	−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
$129$	−33309.0	−0.176233
$130$	222040.	1.15232
$131$	−318813.	−1.62315	−0.811573	−	0.584251i	$-0.801389\pi$
$131$	−318813.	−1.62315	−0.811573	+	0.584251i	$0.801389\pi$
$132$	−13104.0	−0.0654590
$133$	11913.0	0.0583972
$134$	−77712.0	−0.373875
$135$	−66339.0	−0.313281
$136$	−117312.	−0.543870
$137$	−363929.	−1.65659	−0.828295	−	0.560292i	$-0.810689\pi$
$137$	−363929.	−1.65659	−0.828295	+	0.560292i	$0.810689\pi$
$138$	−123696.	−0.552914
$139$	−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
$141$	82827.0	0.350852
$142$	28160.0	0.117196
$143$	55510.0	0.227003
$144$	20736.0	0.0833333
$145$	−324142.	−1.28031
$146$	149364.	0.579914
$147$	−141462.	−0.539941
$148$	115456.	0.433274
$149$	−436653.	−1.61128	−0.805640	−	0.592406i	$-0.798178\pi$
$149$	−436653.	−1.61128	−0.805640	+	0.592406i	$0.798178\pi$
$150$	185616.	0.673577
$151$	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
$153$	−148473.	−0.512766
$154$	12012.0	0.0408144
$155$	−29302.0	−0.0979643
$156$	−87840.0	−0.288989
$157$	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
$159$	262674.	0.823994
$160$	−93184.0	−0.287767
$161$	113388.	0.344749
$162$	26244.0	0.0785674
$163$	−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
$165$	74529.0	0.213116
$166$	−284784.	−0.802131
$167$	457854.	1.27039	0.635193	−	0.772353i	$-0.280920\pi$
$167$	457854.	1.27039	0.635193	+	0.772353i	$0.280920\pi$
$168$	−19008.0	−0.0519593
$169$	807.000	0.00217349
$170$	667212.	1.77069
$171$	−29241.0	−0.0764719
$172$	−59216.0	−0.152622
$173$	−733002.	−1.86204	−0.931022	−	0.364963i	$-0.881082\pi$
$173$	−733002.	−1.86204	−0.931022	+	0.364963i	$0.881082\pi$
$174$	128232.	0.321087
$175$	−170148.	−0.419983
$176$	−23296.0	−0.0566891
$177$	−250236.	−0.600367
$178$	−14616.0	−0.0345763
$179$	−247518.	−0.577397	−0.288698	−	0.957420i	$-0.593222\pi$
$179$	−247518.	−0.577397	−0.288698	+	0.957420i	$0.593222\pi$
$180$	−117936.	−0.271310
$181$	−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
$183$	388143.	0.856770
$184$	−219904.	−0.478838
$185$	−656656.	−1.41062
$186$	11592.0	0.0245684
$187$	166803.	0.348819
$188$	147248.	0.303847
$189$	−24057.0	−0.0489877
$190$	131404.	0.264073
$191$	330733.	0.655985	0.327993	−	0.944680i	$-0.393628\pi$
$191$	330733.	0.655985	0.327993	+	0.944680i	$0.393628\pi$
$192$	36864.0	0.0721688
$193$	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
$195$	499590.	0.940865
$196$	−251488.	−0.467603
$197$	−942346.	−1.72999	−0.864997	−	0.501776i	$-0.832680\pi$
$197$	−942346.	−1.72999	−0.864997	+	0.501776i	$0.832680\pi$
$198$	−29484.0	−0.0534470
$199$	−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
$201$	−174852.	−0.305267
$202$	−684760.	−1.18076
$203$	−117546.	−0.200202
$204$	−263952.	−0.444068
$205$	1.24342e6	2.06650
$206$	354360.	0.581804
$207$	−278316.	−0.451453
$208$	−156160.	−0.250272
$209$	32851.0	0.0520215
$210$	108108.	0.169165
$211$	−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
$213$	63360.0	0.0956899
$214$	471032.	0.703098
$215$	336791.	0.496895
$216$	46656.0	0.0680414
$217$	−10626.0	−0.0153186
$218$	329664.	0.469819
$219$	336069.	0.473498
$220$	132496.	0.184564
$221$	1.11813e6	1.53997
$222$	259776.	0.353766
$223$	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
$225$	417636.	0.549973
$226$	321656.	0.418910
$227$	169582.	0.218431	0.109216	−	0.994018i	$-0.465166\pi$
$227$	169582.	0.218431	0.109216	+	0.994018i	$0.465166\pi$
$228$	−51984.0	−0.0662266
$229$	−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
$231$	27027.0	0.0333248
$232$	227968.	0.278070
$233$	506649.	0.611389	0.305694	−	0.952130i	$-0.401111\pi$
$233$	506649.	0.611389	0.305694	+	0.952130i	$0.401111\pi$
$234$	−197640.	−0.235958
$235$	−837473.	−0.989239
$236$	−444864.	−0.519933
$237$	−44748.0	−0.0517491
$238$	241956.	0.276882
$239$	1.34766e6	1.52611	0.763053	−	0.646336i	$-0.223700\pi$
$239$	1.34766e6	1.52611	0.763053	+	0.646336i	$0.223700\pi$
$240$	−209664.	−0.234961
$241$	−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
$243$	59049.0	0.0641500
$244$	690032.	0.741984
$245$	1.43034e6	1.52238
$246$	−491904.	−0.518254
$247$	220210.	0.229665
$248$	20608.0	0.0212768
$249$	−640764.	−0.654937
$250$	−739284.	−0.748103
$251$	1.08289e6	1.08493	0.542463	−	0.840079i	$-0.317492\pi$
$251$	1.08289e6	1.08493	0.542463	+	0.840079i	$0.317492\pi$
$252$	−42768.0	−0.0424246
$253$	312676.	0.307109
$254$	−555768.	−0.540517
$255$	1.50123e6	1.44576
$256$	65536.0	0.0625000
$257$	522416.	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$257$	522416.	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$258$	−133236.	−0.124616
$259$	−238128.	−0.220577
$260$	888160.	0.814813
$261$	288522.	0.262167
$262$	−1.27525e6	−1.14774
$263$	−1.08895e6	−0.970774	−0.485387	−	0.874299i	$-0.661321\pi$
$263$	−1.08895e6	−0.970774	−0.485387	+	0.874299i	$0.661321\pi$
$264$	−52416.0	−0.0462865
$265$	−2.65593e6	−2.32328
$266$	47652.0	0.0412931
$267$	−32886.0	−0.0282314
$268$	−310848.	−0.264369
$269$	−924702.	−0.779150	−0.389575	−	0.920995i	$-0.627378\pi$
$269$	−924702.	−0.779150	−0.389575	+	0.920995i	$0.627378\pi$
$270$	−265356.	−0.221523
$271$	−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
$273$	181170.	0.147123
$274$	−1.45572e6	−1.17139
$275$	−469196.	−0.374130
$276$	−494784.	−0.390970
$277$	−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
$279$	26082.0	0.0200600
$280$	192192.	0.146501
$281$	19066.0	0.0144044	0.00720218	−	0.999974i	$-0.497707\pi$
$281$	19066.0	0.0144044	0.00720218	+	0.999974i	$0.497707\pi$
$282$	331308.	0.248090
$283$	−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
$285$	295659.	0.215615
$286$	222040.	0.160515
$287$	450912.	0.323137
$288$	82944.0	0.0589256
$289$	1.94003e6	1.36636
$290$	−1.29657e6	−0.905316
$291$	561258.	0.388535
$292$	597456.	0.410061
$293$	−1.43226e6	−0.974659	−0.487330	−	0.873218i	$-0.662029\pi$
$293$	−1.43226e6	−0.974659	−0.487330	+	0.873218i	$0.662029\pi$
$294$	−565848.	−0.381796
$295$	2.53016e6	1.69275
$296$	461824.	0.306371
$297$	−66339.0	−0.0436393
$298$	−1.74661e6	−1.13935
$299$	2.09596e6	1.35583
$300$	742464.	0.476291
$301$	122133.	0.0776992
$302$	1.86440e6	1.17631
$303$	−1.54071e6	−0.964083
$304$	−92416.0	−0.0573539
$305$	−3.92456e6	−2.41569
$306$	−593892.	−0.362580
$307$	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
$309$	797310.	0.475041
$310$	−117208.	−0.0692712
$311$	−2.78248e6	−1.63129	−0.815645	−	0.578553i	$-0.803618\pi$
$311$	−2.78248e6	−1.63129	−0.815645	+	0.578553i	$0.803618\pi$
$312$	−351360.	−0.204346
$313$	−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
$315$	243243.	0.138122
$316$	−79552.0	−0.0448160
$317$	1.55578e6	0.869559	0.434779	−	0.900537i	$-0.356826\pi$
$317$	1.55578e6	0.869559	0.434779	+	0.900537i	$0.356826\pi$
$318$	1.05070e6	0.582652
$319$	−324142.	−0.178344
$320$	−372736.	−0.203482
$321$	1.05982e6	0.574077
$322$	453552.	0.243774
$323$	661713.	0.352910
$324$	104976.	0.0555556
$325$	−3.14516e6	−1.65171
$326$	−1.11722e6	−0.582228
$327$	741744.	0.383605
$328$	−874496.	−0.448821
$329$	−303699.	−0.154687
$330$	298116.	0.150696
$331$	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
$333$	584496.	0.288849
$334$	1.83142e6	0.898299
$335$	1.76795e6	0.860711
$336$	−76032.0	−0.0367408
$337$	−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
$339$	723726.	0.342038
$340$	2.66885e6	1.25206
$341$	−29302.0	−0.0136462
$342$	−116964.	−0.0540738
$343$	1.07332e6	0.492602
$344$	−236864.	−0.107920
$345$	2.81408e6	1.27289
$346$	−2.93201e6	−1.31666
$347$	−3.74631e6	−1.67024	−0.835122	−	0.550065i	$-0.814603\pi$
$347$	−3.74631e6	−1.67024	−0.835122	+	0.550065i	$0.814603\pi$
$348$	512928.	0.227043
$349$	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
$351$	−444690.	−0.192659
$352$	−93184.0	−0.0400853
$353$	−648018.	−0.276790	−0.138395	−	0.990377i	$-0.544194\pi$
$353$	−648018.	−0.276790	−0.138395	+	0.990377i	$0.544194\pi$
$354$	−1.00094e6	−0.424523
$355$	−640640.	−0.269801
$356$	−58464.0	−0.0244491
$357$	544401.	0.226073
$358$	−990072.	−0.408281
$359$	3.31969e6	1.35944	0.679722	−	0.733470i	$-0.262100\pi$
$359$	3.31969e6	1.35944	0.679722	+	0.733470i	$0.262100\pi$
$360$	−471744.	−0.191845
$361$	130321.	0.0526316
$362$	−756632.	−0.303468
$363$	−1.37493e6	−0.547664
$364$	322080.	0.127412
$365$	−3.39803e6	−1.33504
$366$	1.55257e6	0.605828
$367$	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
$369$	−1.10678e6	−0.423153
$370$	−2.62662e6	−0.997456
$371$	−963138.	−0.363290
$372$	46368.0	0.0173724
$373$	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
$375$	−1.66339e6	−0.610823
$376$	588992.	0.214852
$377$	−2.17282e6	−0.787355
$378$	−96228.0	−0.0346395
$379$	−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
$381$	−1.25048e6	−0.441330
$382$	1.32293e6	0.463852
$383$	1.03305e6	0.359854	0.179927	−	0.983680i	$-0.442414\pi$
$383$	1.03305e6	0.359854	0.179927	+	0.983680i	$0.442414\pi$
$384$	147456.	0.0510310
$385$	−273273.	−0.0939604
$386$	2.69789e6	0.921628
$387$	−299781.	−0.101748
$388$	997792.	0.336481
$389$	4.66876e6	1.56433	0.782164	−	0.623072i	$-0.214116\pi$
$389$	4.66876e6	1.56433	0.782164	+	0.623072i	$0.214116\pi$
$390$	1.99836e6	0.665292
$391$	6.29819e6	2.08341
$392$	−1.00595e6	−0.330645
$393$	−2.86932e6	−0.937124
$394$	−3.76938e6	−1.22329
$395$	452452.	0.145908
$396$	−117936.	−0.0377927
$397$	−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
$399$	107217.	0.0337156
$400$	1.31994e6	0.412480
$401$	3.94648e6	1.22560	0.612800	−	0.790238i	$-0.290043\pi$
$401$	3.94648e6	1.22560	0.612800	+	0.790238i	$0.290043\pi$
$402$	−699408.	−0.215857
$403$	−196420.	−0.0602453
$404$	−2.73904e6	−0.834920
$405$	−597051.	−0.180873
$406$	−470184.	−0.141564
$407$	−656656.	−0.196495
$408$	−1.05581e6	−0.314004
$409$	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
$411$	−3.27536e6	−0.956433
$412$	1.41744e6	0.411397
$413$	917532.	0.264695
$414$	−1.11326e6	−0.319225
$415$	6.47884e6	1.84662
$416$	−624640.	−0.176969
$417$	−2.78194e6	−0.783445
$418$	131404.	0.0367848
$419$	872676.	0.242839	0.121419	−	0.992601i	$-0.461255\pi$
$419$	872676.	0.242839	0.121419	+	0.992601i	$0.461255\pi$
$420$	432432.	0.119617
$421$	−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
$423$	745443.	0.202565
$424$	1.86790e6	0.504591
$425$	−9.45095e6	−2.53807
$426$	253440.	0.0676630
$427$	−1.42319e6	−0.377740
$428$	1.88413e6	0.497165
$429$	499590.	0.131060
$430$	1.34716e6	0.351358
$431$	90666.0	0.0235099	0.0117550	−	0.999931i	$-0.496258\pi$
$431$	90666.0	0.0235099	0.0117550	+	0.999931i	$0.496258\pi$
$432$	186624.	0.0481125
$433$	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
$435$	−2.91728e6	−0.739188
$436$	1.31866e6	0.332212
$437$	1.24040e6	0.310711
$438$	1.34428e6	0.334814
$439$	−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
$441$	−1.27316e6	−0.311735
$442$	4.47252e6	1.08892
$443$	7.92687e6	1.91908	0.959539	−	0.281577i	$-0.0908575\pi$
$443$	7.92687e6	1.91908	0.959539	+	0.281577i	$0.0908575\pi$
$444$	1.03910e6	0.250151
$445$	332514.	0.0795994
$446$	4.00530e6	0.953449
$447$	−3.92988e6	−0.930272
$448$	−135168.	−0.0318184
$449$	1.48280e6	0.347109	0.173554	−	0.984824i	$-0.444475\pi$
$449$	1.48280e6	0.347109	0.173554	+	0.984824i	$0.444475\pi$
$450$	1.67054e6	0.388890
$451$	1.24342e6	0.287858
$452$	1.28662e6	0.296214
$453$	4.19490e6	0.960453
$454$	678328.	0.154454
$455$	−1.83183e6	−0.414817
$456$	−207936.	−0.0468293
$457$	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
$459$	−1.33626e6	−0.296045
$460$	5.00282e6	1.10235
$461$	−552109.	−0.120996	−0.0604982	−	0.998168i	$-0.519269\pi$
$461$	−552109.	−0.120996	−0.0604982	+	0.998168i	$0.519269\pi$
$462$	108108.	0.0235642
$463$	−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
$465$	−263718.	−0.0565597
$466$	2.02660e6	0.432317
$467$	2.05633e6	0.436315	0.218157	−	0.975914i	$-0.429995\pi$
$467$	2.05633e6	0.436315	0.218157	+	0.975914i	$0.429995\pi$
$468$	−790560.	−0.166848
$469$	641124.	0.134589
$470$	−3.34989e6	−0.699497
$471$	1.96817e6	0.408800
$472$	−1.77946e6	−0.367648
$473$	336791.	0.0692162
$474$	−178992.	−0.0365921
$475$	−1.86132e6	−0.378518
$476$	967824.	0.195785
$477$	2.36407e6	0.475733
$478$	5.39063e6	1.07912
$479$	−2.04279e6	−0.406804	−0.203402	−	0.979095i	$-0.565200\pi$
$479$	−2.04279e6	−0.406804	−0.203402	+	0.979095i	$0.565200\pi$
$480$	−838656.	−0.166143
$481$	−4.40176e6	−0.867488
$482$	−3.36325e6	−0.659388
$483$	1.02049e6	0.199041
$484$	−2.44432e6	−0.474291
$485$	−5.67494e6	−1.09549
$486$	236196.	0.0453609
$487$	−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
$489$	−2.51374e6	−0.475387
$490$	5.72135e6	1.07649
$491$	−3.96714e6	−0.742633	−0.371316	−	0.928506i	$-0.621094\pi$
$491$	−3.96714e6	−0.742633	−0.371316	+	0.928506i	$0.621094\pi$
$492$	−1.96762e6	−0.366461
$493$	−6.52915e6	−1.20987
$494$	880840.	0.162398
$495$	670761.	0.123042
$496$	82432.0	0.0150450
$497$	−232320.	−0.0421886
$498$	−2.56306e6	−0.463111
$499$	−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
$501$	4.12069e6	0.733458
$502$	4.33156e6	0.767159
$503$	8.31756e6	1.46580	0.732902	−	0.680334i	$-0.238165\pi$
$503$	8.31756e6	1.46580	0.732902	+	0.680334i	$0.238165\pi$
$504$	−171072.	−0.0299987
$505$	1.55783e7	2.71826
$506$	1.25070e6	0.217159
$507$	7263.00	0.00125486
$508$	−2.22307e6	−0.382203
$509$	1.00197e7	1.71420	0.857098	−	0.515153i	$-0.172265\pi$
$509$	1.00197e7	1.71420	0.857098	+	0.515153i	$0.172265\pi$
$510$	6.00491e6	1.02231
$511$	−1.23225e6	−0.208760
$512$	262144.	0.0441942
$513$	−263169.	−0.0441511
$514$	2.08966e6	0.348874
$515$	−8.06169e6	−1.33939
$516$	−532944.	−0.0881165
$517$	−837473.	−0.137798
$518$	−952512.	−0.155972
$519$	−6.59702e6	−1.07505
$520$	3.55264e6	0.576160
$521$	−235936.	−0.0380803	−0.0190401	−	0.999819i	$-0.506061\pi$
$521$	−235936.	−0.0380803	−0.0190401	+	0.999819i	$0.506061\pi$
$522$	1.15409e6	0.185380
$523$	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
$525$	−1.53133e6	−0.242477
$526$	−4.35580e6	−0.686441
$527$	−590226.	−0.0925746
$528$	−209664.	−0.0327295
$529$	5.36975e6	0.834286
$530$	−1.06237e7	−1.64281
$531$	−2.25212e6	−0.346622
$532$	190608.	0.0291986
$533$	8.33504e6	1.27084
$534$	−131544.	−0.0199626
$535$	−1.07160e7	−1.61863
$536$	−1.24339e6	−0.186937
$537$	−2.22766e6	−0.333360
$538$	−3.69881e6	−0.550942
$539$	1.43034e6	0.212064
$540$	−1.06142e6	−0.156641
$541$	−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
$543$	−1.70242e6	−0.247781
$544$	−1.87699e6	−0.271935
$545$	−7.49986e6	−1.08159
$546$	724680.	0.104031
$547$	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
$549$	3.49329e6	0.494656
$550$	−1.87678e6	−0.264550
$551$	−1.28588e6	−0.180436
$552$	−1.97914e6	−0.276457
$553$	164076.	0.0228156
$554$	−7.62763e6	−1.05588
$555$	−5.90990e6	−0.814419
$556$	−4.94568e6	−0.678483
$557$	−1.22357e7	−1.67105	−0.835527	−	0.549449i	$-0.814838\pi$
$557$	−1.22357e7	−1.67105	−0.835527	+	0.549449i	$0.814838\pi$
$558$	104328.	0.0141845
$559$	2.25761e6	0.305576
$560$	768768.	0.103592
$561$	1.50123e6	0.201391
$562$	76264.0	0.0101854
$563$	1.07830e7	1.43374	0.716870	−	0.697207i	$-0.245574\pi$
$563$	1.07830e7	1.43374	0.716870	+	0.697207i	$0.245574\pi$
$564$	1.32523e6	0.175426
$565$	−7.31767e6	−0.964388
$566$	−2.67133e6	−0.350499
$567$	−216513.	−0.0282831
$568$	450560.	0.0585979
$569$	1.10760e7	1.43418	0.717088	−	0.696982i	$-0.245474\pi$
$569$	1.10760e7	1.43418	0.717088	+	0.696982i	$0.245474\pi$
$570$	1.18264e6	0.152463
$571$	−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
$573$	2.97660e6	0.378733
$574$	1.80365e6	0.228493
$575$	−1.77160e7	−2.23458
$576$	331776.	0.0416667
$577$	−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
$579$	6.07025e6	0.752506
$580$	−5.18627e6	−0.640155
$581$	2.34947e6	0.288755
$582$	2.24503e6	0.274736
$583$	−2.65593e6	−0.323627
$584$	2.38982e6	0.289957
$585$	4.49631e6	0.543209
$586$	−5.72904e6	−0.689188
$587$	1.19721e7	1.43409	0.717045	−	0.697027i	$-0.245494\pi$
$587$	1.19721e7	1.43409	0.717045	+	0.697027i	$0.245494\pi$
$588$	−2.26339e6	−0.269971
$589$	−116242.	−0.0138062
$590$	1.01207e7	1.19696
$591$	−8.48111e6	−0.998813
$592$	1.84730e6	0.216637
$593$	−2.07789e6	−0.242653	−0.121326	−	0.992613i	$-0.538715\pi$
$593$	−2.07789e6	−0.242653	−0.121326	+	0.992613i	$0.538715\pi$
$594$	−265356.	−0.0308576
$595$	−5.50450e6	−0.637420
$596$	−6.98645e6	−0.805640
$597$	−3.86554e6	−0.443890
$598$	8.38384e6	0.958716
$599$	1.05217e7	1.19817	0.599085	−	0.800686i	$-0.295531\pi$
$599$	1.05217e7	1.19817	0.599085	+	0.800686i	$0.295531\pi$
$600$	2.96986e6	0.336789
$601$	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
$603$	−1.57367e6	−0.176246
$604$	7.45760e6	0.831777
$605$	1.39021e7	1.54416
$606$	−6.16284e6	−0.681709
$607$	−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
$609$	−1.05791e6	−0.115587
$610$	−1.56982e7	−1.70815
$611$	−5.61383e6	−0.608354
$612$	−2.37557e6	−0.256383
$613$	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
$615$	1.11908e7	1.19309
$616$	192192.	0.0204072
$617$	1.17256e7	1.24000	0.620000	−	0.784602i	$-0.287133\pi$
$617$	1.17256e7	1.24000	0.620000	+	0.784602i	$0.287133\pi$
$618$	3.18924e6	0.335905
$619$	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
$621$	−2.50484e6	−0.260646
$622$	−1.11299e7	−1.15350
$623$	120582.	0.0124469
$624$	−1.40544e6	−0.144494
$625$	706211.	0.0723160
$626$	−4.37534e6	−0.446247
$627$	295659.	0.0300346
$628$	3.49898e6	0.354031
$629$	−1.32269e7	−1.33301
$630$	972972.	0.0976673
$631$	−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
$633$	−5.12179e6	−0.508057
$634$	6.22310e6	0.614871
$635$	1.26437e7	1.24434
$636$	4.20278e6	0.411997
$637$	9.58798e6	0.936221
$638$	−1.29657e6	−0.126108
$639$	570240.	0.0552466
$640$	−1.49094e6	−0.143884
$641$	−1.96019e7	−1.88432	−0.942158	−	0.335170i	$-0.891206\pi$
$641$	−1.96019e7	−1.88432	−0.942158	+	0.335170i	$0.891206\pi$
$642$	4.23929e6	0.405934
$643$	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
$645$	3.03112e6	0.286882
$646$	2.64685e6	0.249545
$647$	968621.	0.0909689	0.0454845	−	0.998965i	$-0.485517\pi$
$647$	968621.	0.0909689	0.0454845	+	0.998965i	$0.485517\pi$
$648$	419904.	0.0392837
$649$	2.53016e6	0.235796
$650$	−1.25806e7	−1.16794
$651$	−95634.0	−0.00884423
$652$	−4.46886e6	−0.411697
$653$	−517653.	−0.0475068	−0.0237534	−	0.999718i	$-0.507562\pi$
$653$	−517653.	−0.0475068	−0.0237534	+	0.999718i	$0.507562\pi$
$654$	2.96698e6	0.271250
$655$	2.90120e7	2.64225
$656$	−3.49798e6	−0.317364
$657$	3.02462e6	0.273374
$658$	−1.21480e6	−0.109380
$659$	7.30548e6	0.655293	0.327646	−	0.944800i	$-0.393745\pi$
$659$	7.30548e6	0.655293	0.327646	+	0.944800i	$0.393745\pi$
$660$	1.19246e6	0.106558
$661$	−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
$663$	1.00632e7	0.889101
$664$	−4.55654e6	−0.401066
$665$	−1.08408e6	−0.0950623
$666$	2.33798e6	0.204247
$667$	−1.22390e7	−1.06520
$668$	7.32566e6	0.635193
$669$	9.01192e6	0.778488
$670$	7.07179e6	0.608615
$671$	−3.92456e6	−0.336499
$672$	−304128.	−0.0259796
$673$	−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
$675$	3.75872e6	0.317527
$676$	12912.0	0.00108674
$677$	−8.90338e6	−0.746592	−0.373296	−	0.927712i	$-0.621772\pi$
$677$	−8.90338e6	−0.746592	−0.373296	+	0.927712i	$0.621772\pi$
$678$	2.89490e6	0.241858
$679$	−2.05795e6	−0.171301
$680$	1.06754e7	0.885343
$681$	1.52624e6	0.126111
$682$	−117208.	−0.00964931
$683$	−1.40518e7	−1.15260	−0.576302	−	0.817237i	$-0.695505\pi$
$683$	−1.40518e7	−1.15260	−0.576302	+	0.817237i	$0.695505\pi$
$684$	−467856.	−0.0382360
$685$	3.31175e7	2.69669
$686$	4.29330e6	0.348322
$687$	−3.64830e6	−0.294916
$688$	−947456.	−0.0763111
$689$	−1.78035e7	−1.42875
$690$	1.12563e7	0.900066
$691$	−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
$693$	243243.	0.0192401
$694$	−1.49852e7	−1.18104
$695$	2.81286e7	2.20895
$696$	2.05171e6	0.160544
$697$	2.50461e7	1.95280
$698$	4.62893e6	0.359619
$699$	4.55984e6	0.352985
$700$	−2.72237e6	−0.209992
$701$	2.30897e7	1.77469	0.887347	−	0.461103i	$-0.152546\pi$
$701$	2.30897e7	1.77469	0.887347	+	0.461103i	$0.152546\pi$
$702$	−1.77876e6	−0.136231
$703$	−2.60498e6	−0.198800
$704$	−372736.	−0.0283446
$705$	−7.53726e6	−0.571137
$706$	−2.59207e6	−0.195720
$707$	5.64927e6	0.425054
$708$	−4.00378e6	−0.300183
$709$	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
$711$	−402732.	−0.0298774
$712$	−233856.	−0.0172882
$713$	−1.10639e6	−0.0815052
$714$	2.17760e6	0.159858
$715$	−5.05141e6	−0.369528
$716$	−3.96029e6	−0.288698
$717$	1.21289e7	0.881098
$718$	1.32788e7	0.961272
$719$	−818051.	−0.0590144	−0.0295072	−	0.999565i	$-0.509394\pi$
$719$	−818051.	−0.0590144	−0.0295072	+	0.999565i	$0.509394\pi$
$720$	−1.88698e6	−0.135655
$721$	−2.92347e6	−0.209440
$722$	521284.	0.0372161
$723$	−7.56731e6	−0.538388
$724$	−3.02653e6	−0.214584
$725$	1.83657e7	1.29766
$726$	−5.49972e6	−0.387257
$727$	−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
$729$	531441.	0.0370370
$730$	−1.35921e7	−0.944018
$731$	6.78393e6	0.469557
$732$	6.21029e6	0.428385
$733$	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
$735$	1.28730e7	0.878947
$736$	−3.51846e6	−0.239419
$737$	1.76795e6	0.119895
$738$	−4.42714e6	−0.299214
$739$	−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
$741$	1.98189e6	0.132597
$742$	−3.85255e6	−0.256885
$743$	2.77051e7	1.84114	0.920572	−	0.390574i	$-0.127723\pi$
$743$	2.77051e7	1.84114	0.920572	+	0.390574i	$0.127723\pi$
$744$	185472.	0.0122842
$745$	3.97354e7	2.62293
$746$	7.83142e6	0.515221
$747$	−5.76688e6	−0.378128
$748$	2.66885e6	0.174409
$749$	−3.88601e6	−0.253104
$750$	−6.65356e6	−0.431917
$751$	−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
$753$	9.74602e6	0.626383
$754$	−8.69128e6	−0.556744
$755$	−4.24151e7	−2.70803
$756$	−384912.	−0.0244938
$757$	−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
$759$	2.81408e6	0.177310
$760$	2.10246e6	0.132037
$761$	3.58869e6	0.224633	0.112317	−	0.993672i	$-0.464173\pi$
$761$	3.58869e6	0.224633	0.112317	+	0.993672i	$0.464173\pi$
$762$	−5.00191e6	−0.312068
$763$	−2.71973e6	−0.169127
$764$	5.29173e6	0.327993
$765$	1.35110e7	0.834709
$766$	4.13222e6	0.254455
$767$	1.69604e7	1.04100
$768$	589824.	0.0360844
$769$	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
$771$	4.70174e6	0.284854
$772$	1.07916e7	0.651689
$773$	−4.33598e6	−0.260999	−0.130500	−	0.991448i	$-0.541658\pi$
$773$	−4.33598e6	−0.260999	−0.130500	+	0.991448i	$0.541658\pi$
$774$	−1.19912e6	−0.0719468
$775$	1.66023e6	0.0992921
$776$	3.99117e6	0.237928
$777$	−2.14315e6	−0.127350
$778$	1.86751e7	1.10615
$779$	4.93270e6	0.291234
$780$	7.99344e6	0.470432
$781$	−640640.	−0.0375826
$782$	2.51928e7	1.47319
$783$	2.59670e6	0.151362
$784$	−4.02381e6	−0.233801
$785$	−1.99004e7	−1.15263
$786$	−1.14773e7	−0.662647
$787$	−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
$789$	−9.80054e6	−0.560477
$790$	1.80981e6	0.103173
$791$	−2.65366e6	−0.150801
$792$	−471744.	−0.0267235
$793$	−2.63075e7	−1.48558
$794$	−1.20124e7	−0.676206
$795$	−2.39033e7	−1.34135
$796$	−6.87208e6	−0.384420
$797$	−2.49744e7	−1.39268	−0.696338	−	0.717714i	$-0.745189\pi$
$797$	−2.49744e7	−1.39268	−0.696338	+	0.717714i	$0.745189\pi$
$798$	428868.	0.0238406
$799$	−1.68691e7	−0.934813
$800$	5.27974e6	0.291667
$801$	−295974.	−0.0162994
$802$	1.57859e7	0.866631
$803$	−3.39803e6	−0.185968
$804$	−2.79763e6	−0.152634
$805$	−1.03183e7	−0.561201
$806$	−785680.	−0.0425999
$807$	−8.32232e6	−0.449842
$808$	−1.09562e7	−0.590378
$809$	−2.33716e7	−1.25550	−0.627750	−	0.778415i	$-0.716024\pi$
$809$	−2.33716e7	−1.25550	−0.627750	+	0.778415i	$0.716024\pi$
$810$	−2.38820e6	−0.127897
$811$	−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
$813$	−1.07343e7	−0.569570
$814$	−2.62662e6	−0.138943
$815$	2.54167e7	1.34037
$816$	−4.22323e6	−0.222034
$817$	1.33606e6	0.0700279
$818$	1.75201e7	0.915491
$819$	1.63053e6	0.0849413
$820$	1.98948e7	1.03325
$821$	−2.26403e7	−1.17226	−0.586131	−	0.810216i	$-0.699350\pi$
$821$	−2.26403e7	−1.17226	−0.586131	+	0.810216i	$0.699350\pi$
$822$	−1.31014e7	−0.676300
$823$	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
$825$	−4.22276e6	−0.216004
$826$	3.67013e6	0.187168
$827$	−1.54979e7	−0.787969	−0.393984	−	0.919117i	$-0.628904\pi$
$827$	−1.54979e7	−0.787969	−0.393984	+	0.919117i	$0.628904\pi$
$828$	−4.45306e6	−0.225726
$829$	−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
$831$	−1.71622e7	−0.862123
$832$	−2.49856e6	−0.125136
$833$	2.88111e7	1.43862
$834$	−1.11278e7	−0.553979
$835$	−4.16647e7	−2.06801
$836$	525616.	0.0260108
$837$	234738.	0.0115816
$838$	3.49070e6	0.171713
$839$	2.10515e7	1.03247	0.516236	−	0.856447i	$-0.327333\pi$
$839$	2.10515e7	1.03247	0.516236	+	0.856447i	$0.327333\pi$
$840$	1.72973e6	0.0845823
$841$	−7.82330e6	−0.381417
$842$	−7.83417e6	−0.380814
$843$	171594.	0.00831636
$844$	−9.10541e6	−0.439990
$845$	−73437.0	−0.00353812
$846$	2.98177e6	0.143235
$847$	5.04141e6	0.241459
$848$	7.47162e6	0.356800
$849$	−6.01050e6	−0.286181
$850$	−3.78038e7	−1.79468
$851$	−2.47942e7	−1.17362
$852$	1.01376e6	0.0478450
$853$	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
$855$	2.66093e6	0.124485
$856$	7.53651e6	0.351549
$857$	−1.40825e7	−0.654979	−0.327490	−	0.944855i	$-0.606203\pi$
$857$	−1.40825e7	−0.654979	−0.327490	+	0.944855i	$0.606203\pi$
$858$	1.99836e6	0.0926735
$859$	−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
$861$	4.05821e6	0.186563
$862$	362664.	0.0166240
$863$	7.83752e6	0.358222	0.179111	−	0.983829i	$-0.442678\pi$
$863$	7.83752e6	0.358222	0.179111	+	0.983829i	$0.442678\pi$
$864$	746496.	0.0340207
$865$	6.67032e7	3.03114
$866$	1.40330e7	0.635852
$867$	1.74603e7	0.788867
$868$	−170016.	−0.00765932
$869$	452452.	0.0203246
$870$	−1.16691e7	−0.522685
$871$	1.18511e7	0.529313
$872$	5.27462e6	0.234909
$873$	5.05132e6	0.224321
$874$	4.96158e6	0.219706
$875$	6.09909e6	0.269305
$876$	5.37710e6	0.236749
$877$	−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
$879$	−1.28903e7	−0.562720
$880$	2.11994e6	0.0922818
$881$	−1.85223e7	−0.803997	−0.401998	−	0.915640i	$-0.631684\pi$
$881$	−1.85223e7	−0.803997	−0.401998	+	0.915640i	$0.631684\pi$
$882$	−5.09263e6	−0.220430
$883$	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
$885$	2.27715e7	0.977311
$886$	3.17075e7	1.35699
$887$	−4.56817e7	−1.94955	−0.974773	−	0.223199i	$-0.928350\pi$
$887$	−4.56817e7	−1.94955	−0.974773	+	0.223199i	$0.928350\pi$
$888$	4.15642e6	0.176883
$889$	4.58509e6	0.194578
$890$	1.33006e6	0.0562853
$891$	−597051.	−0.0251952
$892$	1.60212e7	0.674190
$893$	−3.32228e6	−0.139415
$894$	−1.57195e7	−0.657802
$895$	2.25241e7	0.939919
$896$	−540672.	−0.0224990
$897$	1.88636e7	0.782788
$898$	5.93118e6	0.245443
$899$	1.14696e6	0.0473315
$900$	6.68218e6	0.274987
$901$	−5.34979e7	−2.19546
$902$	4.97370e6	0.203546
$903$	1.09920e6	0.0448597
$904$	5.14650e6	0.209455
$905$	1.72134e7	0.698626
$906$	1.67796e7	0.679143
$907$	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
$909$	−1.38664e7	−0.556613
$910$	−7.32732e6	−0.293320
$911$	−4.18553e7	−1.67092	−0.835458	−	0.549554i	$-0.814798\pi$
$911$	−4.18553e7	−1.67092	−0.835458	+	0.549554i	$0.814798\pi$
$912$	−831744.	−0.0331133
$913$	6.47884e6	0.257229
$914$	6.91300e6	0.273717
$915$	−3.53210e7	−1.39470
$916$	−6.48587e6	−0.255405
$917$	1.05208e7	0.413168
$918$	−5.34503e6	−0.209336
$919$	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
$921$	2.78020e7	1.08001
$922$	−2.20844e6	−0.0855574
$923$	−4.29440e6	−0.165920
$924$	432432.	0.0166624
$925$	3.72057e7	1.42973
$926$	−2.21972e7	−0.850687
$927$	7.17579e6	0.274265
$928$	3.64749e6	0.139035
$929$	−3.42756e7	−1.30300	−0.651502	−	0.758647i	$-0.725861\pi$
$929$	−3.42756e7	−1.30300	−0.651502	+	0.758647i	$0.725861\pi$
$930$	−1.05487e6	−0.0399938
$931$	5.67420e6	0.214551
$932$	8.10638e6	0.305694
$933$	−2.50423e7	−0.941826
$934$	8.22531e6	0.308521
$935$	−1.51791e7	−0.567827
$936$	−3.16224e6	−0.117979
$937$	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
$939$	−9.84451e6	−0.364359
$940$	−1.33996e7	−0.494619
$941$	4.34881e7	1.60102	0.800510	−	0.599320i	$-0.204562\pi$
$941$	4.34881e7	1.60102	0.800510	+	0.599320i	$0.204562\pi$
$942$	7.87270e6	0.289065
$943$	4.69495e7	1.71930
$944$	−7.11782e6	−0.259966
$945$	2.18919e6	0.0797450
$946$	1.34716e6	0.0489432
$947$	−3.23771e7	−1.17318	−0.586588	−	0.809885i	$-0.699529\pi$
$947$	−3.23771e7	−1.17318	−0.586588	+	0.809885i	$0.699529\pi$
$948$	−715968.	−0.0258745
$949$	−2.27780e7	−0.821013
$950$	−7.44526e6	−0.267652
$951$	1.40020e7	0.502040
$952$	3.87130e6	0.138441
$953$	4.12172e7	1.47010	0.735050	−	0.678013i	$-0.237159\pi$
$953$	4.12172e7	1.47010	0.735050	+	0.678013i	$0.237159\pi$
$954$	9.45626e6	0.336394
$955$	−3.00967e7	−1.06785
$956$	2.15625e7	0.763053
$957$	−2.91728e6	−0.102967
$958$	−8.17117e6	−0.287654
$959$	1.20097e7	0.421681
$960$	−3.35462e6	−0.117480
$961$	−2.85255e7	−0.996378
$962$	−1.76070e7	−0.613407
$963$	9.53840e6	0.331444
$964$	−1.34530e7	−0.466258
$965$	−6.13770e7	−2.12171
$966$	4.08197e6	0.140743
$967$	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
$969$	5.95542e6	0.203752
$970$	−2.26998e7	−0.774626
$971$	−4.03426e7	−1.37314	−0.686571	−	0.727063i	$-0.740885\pi$
$971$	−4.03426e7	−1.37314	−0.686571	+	0.727063i	$0.740885\pi$
$972$	944784.	0.0320750
$973$	1.02005e7	0.345412
$974$	−2.63426e7	−0.889735
$975$	−2.83064e7	−0.953616
$976$	1.10405e7	0.370992
$977$	1.70401e7	0.571130	0.285565	−	0.958359i	$-0.407819\pi$
$977$	1.70401e7	0.571130	0.285565	+	0.958359i	$0.407819\pi$
$978$	−1.00549e7	−0.336150
$979$	332514.	0.0110880
$980$	2.28854e7	0.761191
$981$	6.67570e6	0.221475
$982$	−1.58686e7	−0.525121
$983$	2.20796e7	0.728799	0.364399	−	0.931243i	$-0.381274\pi$
$983$	2.20796e7	0.728799	0.364399	+	0.931243i	$0.381274\pi$
$984$	−7.87046e6	−0.259127
$985$	8.57535e7	2.81619
$986$	−2.61166e7	−0.855508
$987$	−2.73329e6	−0.0893085
$988$	3.52336e6	0.114832
$989$	1.27166e7	0.413411
$990$	2.68304e6	0.0870041
$991$	−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
$993$	317160.	0.0102072
$994$	−929280.	−0.0298319
$995$	3.90850e7	1.25156
$996$	−1.02522e7	−0.327469
$997$	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
$999$	5.26046e6	0.166767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.6.a.d.1.1	✓	1
3.2	odd	2		342.6.a.c.1.1		1
4.3	odd	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	1.1	even	1	trivial
342.6.a.c.1.1		1	3.2	odd	2
912.6.a.b.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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