LMFDB - Embedded newform 784.6.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{113})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 59x^{2} + 60x + 674 $ x^4 - 2x^3 - 59x^2 + 60*x + 674
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.40086$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.54802 q^{3} -45.9910 q^{5} -200.123 q^{9} +O(q^{10})$	$q-6.54802 q^{3} -45.9910 q^{5} -200.123 q^{9} +551.781 q^{11} +1094.10 q^{13} +301.150 q^{15} -1180.71 q^{17} +1166.13 q^{19} -44.3851 q^{23} -1009.82 q^{25} +2901.58 q^{27} +3329.02 q^{29} -8784.01 q^{31} -3613.07 q^{33} -2557.12 q^{37} -7164.17 q^{39} -12761.3 q^{41} +96.7714 q^{43} +9203.89 q^{45} -7679.15 q^{47} +7731.33 q^{51} -11953.3 q^{53} -25377.0 q^{55} -7635.81 q^{57} -9857.24 q^{59} -38517.9 q^{61} -50318.7 q^{65} +67548.9 q^{67} +290.634 q^{69} +61374.6 q^{71} +1850.40 q^{73} +6612.34 q^{75} +8.52913 q^{79} +29630.4 q^{81} +95039.3 q^{83} +54302.3 q^{85} -21798.5 q^{87} +53605.6 q^{89} +57517.9 q^{93} -53631.4 q^{95} +3110.79 q^{97} -110424. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 220 q^{9}+O(q^{10}) $ 4 * q + 220 * q^9	$ 4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100}) $ 4 * q + 220 * q^9 + 1952 * q^11 + 4096 * q^15 + 7136 * q^23 + 2764 * q^25 - 3352 * q^29 - 9208 * q^37 - 2464 * q^39 - 20448 * q^43 + 67408 * q^51 - 102920 * q^53 - 15576 * q^57 - 63168 * q^65 + 22896 * q^67 + 153824 * q^71 + 90688 * q^79 - 17204 * q^81 + 272656 * q^85 + 247760 * q^93 - 108224 * q^95 + 42272 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−6.54802	−0.420055	−0.210028	−	0.977695i	$-0.567355\pi$
$3$	−6.54802	−0.420055	−0.210028	+	0.977695i	$0.567355\pi$
$4$	0	0
$5$	−45.9910	−0.822713	−0.411356	−	0.911475i	$-0.634945\pi$
$5$	−45.9910	−0.822713	−0.411356	+	0.911475i	$0.634945\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−200.123	−0.823553
$10$	0	0
$11$	551.781	1.37494	0.687472	−	0.726211i	$-0.258720\pi$
$11$	551.781	1.37494	0.687472	+	0.726211i	$0.258720\pi$
$12$	0	0
$13$	1094.10	1.79555	0.897776	−	0.440453i	$-0.145182\pi$
$13$	1094.10	1.79555	0.897776	+	0.440453i	$0.145182\pi$
$14$	0	0
$15$	301.150	0.345585
$16$	0	0
$17$	−1180.71	−0.990883	−0.495442	−	0.868641i	$-0.664994\pi$
$17$	−1180.71	−0.990883	−0.495442	+	0.868641i	$0.664994\pi$
$18$	0	0
$19$	1166.13	0.741074	0.370537	−	0.928818i	$-0.379174\pi$
$19$	1166.13	0.741074	0.370537	+	0.928818i	$0.379174\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−44.3851	−0.0174951	−0.00874757	−	0.999962i	$-0.502784\pi$
$23$	−44.3851	−0.0174951	−0.00874757	+	0.999962i	$0.502784\pi$
$24$	0	0
$25$	−1009.82	−0.323143
$26$	0	0
$27$	2901.58	0.765993
$28$	0	0
$29$	3329.02	0.735057	0.367529	−	0.930012i	$-0.380204\pi$
$29$	3329.02	0.735057	0.367529	+	0.930012i	$0.380204\pi$
$30$	0	0
$31$	−8784.01	−1.64168	−0.820841	−	0.571157i	$-0.806495\pi$
$31$	−8784.01	−1.64168	−0.820841	+	0.571157i	$0.806495\pi$
$32$	0	0
$33$	−3613.07	−0.577552
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2557.12	−0.307077	−0.153539	−	0.988143i	$-0.549067\pi$
$37$	−2557.12	−0.307077	−0.153539	+	0.988143i	$0.549067\pi$
$38$	0	0
$39$	−7164.17	−0.754231
$40$	0	0
$41$	−12761.3	−1.18559	−0.592794	−	0.805354i	$-0.701975\pi$
$41$	−12761.3	−1.18559	−0.592794	+	0.805354i	$0.701975\pi$
$42$	0	0
$43$	96.7714	0.00798135	0.00399067	−	0.999992i	$-0.498730\pi$
$43$	96.7714	0.00798135	0.00399067	+	0.999992i	$0.498730\pi$
$44$	0	0
$45$	9203.89	0.677548
$46$	0	0
$47$	−7679.15	−0.507071	−0.253535	−	0.967326i	$-0.581593\pi$
$47$	−7679.15	−0.507071	−0.253535	+	0.967326i	$0.581593\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7731.33	0.416226
$52$	0	0
$53$	−11953.3	−0.584520	−0.292260	−	0.956339i	$-0.594407\pi$
$53$	−11953.3	−0.584520	−0.292260	+	0.956339i	$0.594407\pi$
$54$	0	0
$55$	−25377.0	−1.13118
$56$	0	0
$57$	−7635.81	−0.311292
$58$	0	0
$59$	−9857.24	−0.368659	−0.184330	−	0.982864i	$-0.559011\pi$
$59$	−9857.24	−0.368659	−0.184330	+	0.982864i	$0.559011\pi$
$60$	0	0
$61$	−38517.9	−1.32537	−0.662686	−	0.748897i	$-0.730584\pi$
$61$	−38517.9	−1.32537	−0.662686	+	0.748897i	$0.730584\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−50318.7	−1.47722
$66$	0	0
$67$	67548.9	1.83836	0.919182	−	0.393833i	$-0.128851\pi$
$67$	67548.9	1.83836	0.919182	+	0.393833i	$0.128851\pi$
$68$	0	0
$69$	290.634	0.00734893
$70$	0	0
$71$	61374.6	1.44492	0.722458	−	0.691415i	$-0.243012\pi$
$71$	61374.6	1.44492	0.722458	+	0.691415i	$0.243012\pi$
$72$	0	0
$73$	1850.40	0.0406404	0.0203202	−	0.999794i	$-0.493531\pi$
$73$	1850.40	0.0406404	0.0203202	+	0.999794i	$0.493531\pi$
$74$	0	0
$75$	6612.34	0.135738
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.52913	0.000153758 0	7.68788e−5	−	1.00000i	$-0.499976\pi$
$79$	8.52913	0.000153758 0	7.68788e−5		1.00000i	$0.499976\pi$
$80$	0	0
$81$	29630.4	0.501794
$82$	0	0
$83$	95039.3	1.51429	0.757143	−	0.653249i	$-0.226595\pi$
$83$	95039.3	1.51429	0.757143	+	0.653249i	$0.226595\pi$
$84$	0	0
$85$	54302.3	0.815212
$86$	0	0
$87$	−21798.5	−0.308765
$88$	0	0
$89$	53605.6	0.717357	0.358678	−	0.933461i	$-0.383227\pi$
$89$	53605.6	0.717357	0.358678	+	0.933461i	$0.383227\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	57517.9	0.689597
$94$	0	0
$95$	−53631.4	−0.609691
$96$	0	0
$97$	3110.79	0.0335693	0.0167846	−	0.999859i	$-0.494657\pi$
$97$	3110.79	0.0335693	0.0167846	+	0.999859i	$0.494657\pi$
$98$	0	0
$99$	−110424.	−1.13234
$100$	0	0
$101$	21835.9	0.212994	0.106497	−	0.994313i	$-0.466037\pi$
$101$	21835.9	0.212994	0.106497	+	0.994313i	$0.466037\pi$
$102$	0	0
$103$	−65341.4	−0.606870	−0.303435	−	0.952852i	$-0.598134\pi$
$103$	−65341.4	−0.606870	−0.303435	+	0.952852i	$0.598134\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	108957.	0.920020	0.460010	−	0.887914i	$-0.347846\pi$
$107$	108957.	0.920020	0.460010	+	0.887914i	$0.347846\pi$
$108$	0	0
$109$	86728.7	0.699192	0.349596	−	0.936901i	$-0.386319\pi$
$109$	86728.7	0.699192	0.349596	+	0.936901i	$0.386319\pi$
$110$	0	0
$111$	16744.1	0.128989
$112$	0	0
$113$	−101496.	−0.747746	−0.373873	−	0.927480i	$-0.621970\pi$
$113$	−101496.	−0.747746	−0.373873	+	0.927480i	$0.621970\pi$
$114$	0	0
$115$	2041.32	0.0143935
$116$	0	0
$117$	−218955.	−1.47873
$118$	0	0
$119$	0	0
$120$	0	0
$121$	143411.	0.890470
$122$	0	0
$123$	83560.9	0.498013
$124$	0	0
$125$	190165.	1.08857
$126$	0	0
$127$	3094.61	0.0170253	0.00851267	−	0.999964i	$-0.497290\pi$
$127$	3094.61	0.0170253	0.00851267	+	0.999964i	$0.497290\pi$
$128$	0	0
$129$	−633.661	−0.00335261
$130$	0	0
$131$	−253431.	−1.29027	−0.645136	−	0.764067i	$-0.723199\pi$
$131$	−253431.	−1.29027	−0.645136	+	0.764067i	$0.723199\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−133447.	−0.630193
$136$	0	0
$137$	−97152.9	−0.442236	−0.221118	−	0.975247i	$-0.570971\pi$
$137$	−97152.9	−0.442236	−0.221118	+	0.975247i	$0.570971\pi$
$138$	0	0
$139$	−210308.	−0.923249	−0.461624	−	0.887076i	$-0.652733\pi$
$139$	−210308.	−0.923249	−0.461624	+	0.887076i	$0.652733\pi$
$140$	0	0
$141$	50283.2	0.212998
$142$	0	0
$143$	603702.	2.46878
$144$	0	0
$145$	−153105.	−0.604741
$146$	0	0
$147$	0	0
$148$	0	0
$149$	140406.	0.518109	0.259055	−	0.965863i	$-0.416589\pi$
$149$	140406.	0.518109	0.259055	+	0.965863i	$0.416589\pi$
$150$	0	0
$151$	−119696.	−0.427205	−0.213603	−	0.976921i	$-0.568520\pi$
$151$	−119696.	−0.427205	−0.213603	+	0.976921i	$0.568520\pi$
$152$	0	0
$153$	236289.	0.816045
$154$	0	0
$155$	403986.	1.35063
$156$	0	0
$157$	97616.9	0.316065	0.158032	−	0.987434i	$-0.449485\pi$
$157$	97616.9	0.316065	0.158032	+	0.987434i	$0.449485\pi$
$158$	0	0
$159$	78270.6	0.245531
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−182678.	−0.538539	−0.269270	−	0.963065i	$-0.586782\pi$
$163$	−182678.	−0.538539	−0.269270	+	0.963065i	$0.586782\pi$
$164$	0	0
$165$	166169.	0.475160
$166$	0	0
$167$	451674.	1.25324	0.626619	−	0.779326i	$-0.284438\pi$
$167$	451674.	1.25324	0.626619	+	0.779326i	$0.284438\pi$
$168$	0	0
$169$	825757.	2.22400
$170$	0	0
$171$	−233369.	−0.610314
$172$	0	0
$173$	371647.	0.944095	0.472047	−	0.881573i	$-0.343515\pi$
$173$	371647.	0.944095	0.472047	+	0.881573i	$0.343515\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	64545.4	0.154857
$178$	0	0
$179$	−85003.4	−0.198291	−0.0991457	−	0.995073i	$-0.531611\pi$
$179$	−85003.4	−0.198291	−0.0991457	+	0.995073i	$0.531611\pi$
$180$	0	0
$181$	−379442.	−0.860892	−0.430446	−	0.902616i	$-0.641644\pi$
$181$	−379442.	−0.860892	−0.430446	+	0.902616i	$0.641644\pi$
$182$	0	0
$183$	252216.	0.556730
$184$	0	0
$185$	117605.	0.252636
$186$	0	0
$187$	−651496.	−1.36241
$188$	0	0
$189$	0	0
$190$	0	0
$191$	922196.	1.82911	0.914555	−	0.404462i	$-0.132541\pi$
$191$	922196.	1.82911	0.914555	+	0.404462i	$0.132541\pi$
$192$	0	0
$193$	505107.	0.976090	0.488045	−	0.872818i	$-0.337710\pi$
$193$	505107.	0.976090	0.488045	+	0.872818i	$0.337710\pi$
$194$	0	0
$195$	329488.	0.620516
$196$	0	0
$197$	251505.	0.461723	0.230861	−	0.972987i	$-0.425846\pi$
$197$	251505.	0.461723	0.230861	+	0.972987i	$0.425846\pi$
$198$	0	0
$199$	208033.	0.372392	0.186196	−	0.982513i	$-0.440384\pi$
$199$	208033.	0.372392	0.186196	+	0.982513i	$0.440384\pi$
$200$	0	0
$201$	−442311.	−0.772215
$202$	0	0
$203$	0	0
$204$	0	0
$205$	586904.	0.975399
$206$	0	0
$207$	8882.50	0.0144082
$208$	0	0
$209$	643446.	1.01893
$210$	0	0
$211$	640577.	0.990525	0.495262	−	0.868744i	$-0.335072\pi$
$211$	640577.	0.990525	0.495262	+	0.868744i	$0.335072\pi$
$212$	0	0
$213$	−401882.	−0.606945
$214$	0	0
$215$	−4450.62	−0.00656636
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12116.4	−0.0170712
$220$	0	0
$221$	−1.29182e6	−1.77918
$222$	0	0
$223$	−390135.	−0.525354	−0.262677	−	0.964884i	$-0.584605\pi$
$223$	−390135.	−0.525354	−0.262677	+	0.964884i	$0.584605\pi$
$224$	0	0
$225$	202089.	0.266126
$226$	0	0
$227$	−291353.	−0.375279	−0.187639	−	0.982238i	$-0.560084\pi$
$227$	−291353.	−0.375279	−0.187639	+	0.982238i	$0.560084\pi$
$228$	0	0
$229$	−1.23040e6	−1.55045	−0.775227	−	0.631682i	$-0.782365\pi$
$229$	−1.23040e6	−1.55045	−0.775227	+	0.631682i	$0.782365\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	114279.	0.137903	0.0689517	−	0.997620i	$-0.478035\pi$
$233$	114279.	0.137903	0.0689517	+	0.997620i	$0.478035\pi$
$234$	0	0
$235$	353172.	0.417174
$236$	0	0
$237$	−55.8489	−6.45867e−5 0
$238$	0	0
$239$	1.14782e6	1.29981	0.649906	−	0.760014i	$-0.274808\pi$
$239$	1.14782e6	1.29981	0.649906	+	0.760014i	$0.274808\pi$
$240$	0	0
$241$	812708.	0.901346	0.450673	−	0.892689i	$-0.351184\pi$
$241$	812708.	0.901346	0.450673	+	0.892689i	$0.351184\pi$
$242$	0	0
$243$	−899104.	−0.976775
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.27586e6	1.33064
$248$	0	0
$249$	−622318.	−0.636084
$250$	0	0
$251$	406772.	0.407537	0.203768	−	0.979019i	$-0.434681\pi$
$251$	406772.	0.407537	0.203768	+	0.979019i	$0.434681\pi$
$252$	0	0
$253$	−24490.8	−0.0240548
$254$	0	0
$255$	−355572.	−0.342434
$256$	0	0
$257$	−1.69712e6	−1.60281	−0.801403	−	0.598125i	$-0.795913\pi$
$257$	−1.69712e6	−1.60281	−0.801403	+	0.598125i	$0.795913\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−666215.	−0.605359
$262$	0	0
$263$	−205694.	−0.183372	−0.0916859	−	0.995788i	$-0.529226\pi$
$263$	−205694.	−0.183372	−0.0916859	+	0.995788i	$0.529226\pi$
$264$	0	0
$265$	549746.	0.480892
$266$	0	0
$267$	−351010.	−0.301330
$268$	0	0
$269$	1.73425e6	1.46127	0.730635	−	0.682769i	$-0.239224\pi$
$269$	1.73425e6	1.46127	0.730635	+	0.682769i	$0.239224\pi$
$270$	0	0
$271$	−369042.	−0.305248	−0.152624	−	0.988284i	$-0.548772\pi$
$271$	−369042.	−0.305248	−0.152624	+	0.988284i	$0.548772\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−557201.	−0.444304
$276$	0	0
$277$	1.22767e6	0.961351	0.480676	−	0.876899i	$-0.340391\pi$
$277$	1.22767e6	0.961351	0.480676	+	0.876899i	$0.340391\pi$
$278$	0	0
$279$	1.75789e6	1.35201
$280$	0	0
$281$	2.00671e6	1.51607	0.758035	−	0.652214i	$-0.226159\pi$
$281$	2.00671e6	1.51607	0.758035	+	0.652214i	$0.226159\pi$
$282$	0	0
$283$	1.78581e6	1.32547	0.662734	−	0.748855i	$-0.269396\pi$
$283$	1.78581e6	1.32547	0.662734	+	0.748855i	$0.269396\pi$
$284$	0	0
$285$	351179.	0.256104
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−25770.8	−0.0181503
$290$	0	0
$291$	−20369.5	−0.0141009
$292$	0	0
$293$	853248.	0.580639	0.290319	−	0.956930i	$-0.406238\pi$
$293$	853248.	0.580639	0.290319	+	0.956930i	$0.406238\pi$
$294$	0	0
$295$	453345.	0.303301
$296$	0	0
$297$	1.60104e6	1.05320
$298$	0	0
$299$	−48561.6	−0.0314134
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−142982.	−0.0894693
$304$	0	0
$305$	1.77148e6	1.09040
$306$	0	0
$307$	1.96068e6	1.18730	0.593652	−	0.804722i	$-0.297686\pi$
$307$	1.96068e6	1.18730	0.593652	+	0.804722i	$0.297686\pi$
$308$	0	0
$309$	427857.	0.254919
$310$	0	0
$311$	−863604.	−0.506307	−0.253153	−	0.967426i	$-0.581468\pi$
$311$	−863604.	−0.506307	−0.253153	+	0.967426i	$0.581468\pi$
$312$	0	0
$313$	1.10047e6	0.634918	0.317459	−	0.948272i	$-0.397170\pi$
$313$	1.10047e6	0.634918	0.317459	+	0.948272i	$0.397170\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.49591e6	0.836097	0.418048	−	0.908425i	$-0.362714\pi$
$317$	1.49591e6	0.836097	0.418048	+	0.908425i	$0.362714\pi$
$318$	0	0
$319$	1.83689e6	1.01066
$320$	0	0
$321$	−713454.	−0.386459
$322$	0	0
$323$	−1.37686e6	−0.734318
$324$	0	0
$325$	−1.10485e6	−0.580221
$326$	0	0
$327$	−567901.	−0.293699
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.74015e6	1.37469	0.687345	−	0.726331i	$-0.258776\pi$
$331$	2.74015e6	1.37469	0.687345	+	0.726331i	$0.258776\pi$
$332$	0	0
$333$	511741.	0.252894
$334$	0	0
$335$	−3.10665e6	−1.51245
$336$	0	0
$337$	−2.31353e6	−1.10968	−0.554842	−	0.831956i	$-0.687221\pi$
$337$	−2.31353e6	−1.10968	−0.554842	+	0.831956i	$0.687221\pi$
$338$	0	0
$339$	664600.	0.314095
$340$	0	0
$341$	−4.84685e6	−2.25722
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−13366.6	−0.00604606
$346$	0	0
$347$	3.05926e6	1.36393	0.681966	−	0.731384i	$-0.261125\pi$
$347$	3.05926e6	1.36393	0.681966	+	0.731384i	$0.261125\pi$
$348$	0	0
$349$	−210232.	−0.0923921	−0.0461961	−	0.998932i	$-0.514710\pi$
$349$	−210232.	−0.0923921	−0.0461961	+	0.998932i	$0.514710\pi$
$350$	0	0
$351$	3.17461e6	1.37538
$352$	0	0
$353$	−3.76790e6	−1.60939	−0.804697	−	0.593686i	$-0.797672\pi$
$353$	−3.76790e6	−1.60939	−0.804697	+	0.593686i	$0.797672\pi$
$354$	0	0
$355$	−2.82268e6	−1.18875
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.00722e6	−0.412465	−0.206232	−	0.978503i	$-0.566120\pi$
$359$	−1.00722e6	−0.412465	−0.206232	+	0.978503i	$0.566120\pi$
$360$	0	0
$361$	−1.11625e6	−0.450809
$362$	0	0
$363$	−939058.	−0.374047
$364$	0	0
$365$	−85101.8	−0.0334354
$366$	0	0
$367$	1.52650e6	0.591603	0.295802	−	0.955249i	$-0.404413\pi$
$367$	1.52650e6	0.591603	0.295802	+	0.955249i	$0.404413\pi$
$368$	0	0
$369$	2.55383e6	0.976396
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.86297e6	1.80980	0.904898	−	0.425629i	$-0.139947\pi$
$373$	4.86297e6	1.80980	0.904898	+	0.425629i	$0.139947\pi$
$374$	0	0
$375$	−1.24520e6	−0.457258
$376$	0	0
$377$	3.64227e6	1.31983
$378$	0	0
$379$	−630878.	−0.225604	−0.112802	−	0.993617i	$-0.535983\pi$
$379$	−630878.	−0.225604	−0.112802	+	0.993617i	$0.535983\pi$
$380$	0	0
$381$	−20263.5	−0.00715159
$382$	0	0
$383$	−565644.	−0.197036	−0.0985182	−	0.995135i	$-0.531410\pi$
$383$	−565644.	−0.197036	−0.0985182	+	0.995135i	$0.531410\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19366.2	−0.00657306
$388$	0	0
$389$	592212.	0.198428	0.0992140	−	0.995066i	$-0.468367\pi$
$389$	592212.	0.198428	0.0992140	+	0.995066i	$0.468367\pi$
$390$	0	0
$391$	52406.1	0.0173356
$392$	0	0
$393$	1.65947e6	0.541986
$394$	0	0
$395$	−392.264	−0.000126498 0
$396$	0	0
$397$	1.34312e6	0.427698	0.213849	−	0.976867i	$-0.431400\pi$
$397$	1.34312e6	0.427698	0.213849	+	0.976867i	$0.431400\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.68716e6	−1.14507	−0.572534	−	0.819881i	$-0.694040\pi$
$401$	−3.68716e6	−1.14507	−0.572534	+	0.819881i	$0.694040\pi$
$402$	0	0
$403$	−9.61057e6	−2.94772
$404$	0	0
$405$	−1.36273e6	−0.412832
$406$	0	0
$407$	−1.41097e6	−0.422214
$408$	0	0
$409$	1.45630e6	0.430470	0.215235	−	0.976562i	$-0.430948\pi$
$409$	1.45630e6	0.430470	0.215235	+	0.976562i	$0.430948\pi$
$410$	0	0
$411$	636159.	0.185764
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.37096e6	−1.24582
$416$	0	0
$417$	1.37710e6	0.387816
$418$	0	0
$419$	2.92192e6	0.813080	0.406540	−	0.913633i	$-0.366735\pi$
$419$	2.92192e6	0.813080	0.406540	+	0.913633i	$0.366735\pi$
$420$	0	0
$421$	2.01999e6	0.555450	0.277725	−	0.960661i	$-0.410420\pi$
$421$	2.01999e6	0.555450	0.277725	+	0.960661i	$0.410420\pi$
$422$	0	0
$423$	1.53678e6	0.417600
$424$	0	0
$425$	1.19231e6	0.320197
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.95305e6	−1.03703
$430$	0	0
$431$	5.43800e6	1.41009	0.705043	−	0.709164i	$-0.250928\pi$
$431$	5.43800e6	1.41009	0.705043	+	0.709164i	$0.250928\pi$
$432$	0	0
$433$	−3.77335e6	−0.967179	−0.483590	−	0.875295i	$-0.660667\pi$
$433$	−3.77335e6	−0.967179	−0.483590	+	0.875295i	$0.660667\pi$
$434$	0	0
$435$	1.00253e6	0.254025
$436$	0	0
$437$	−51758.6	−0.0129652
$438$	0	0
$439$	−2.35150e6	−0.582350	−0.291175	−	0.956670i	$-0.594046\pi$
$439$	−2.35150e6	−0.582350	−0.291175	+	0.956670i	$0.594046\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.80377e6	−1.16298	−0.581491	−	0.813553i	$-0.697530\pi$
$443$	−4.80377e6	−1.16298	−0.581491	+	0.813553i	$0.697530\pi$
$444$	0	0
$445$	−2.46538e6	−0.590179
$446$	0	0
$447$	−919383.	−0.217634
$448$	0	0
$449$	−2.76805e6	−0.647975	−0.323987	−	0.946061i	$-0.605024\pi$
$449$	−2.76805e6	−0.647975	−0.323987	+	0.946061i	$0.605024\pi$
$450$	0	0
$451$	−7.04142e6	−1.63012
$452$	0	0
$453$	783770.	0.179450
$454$	0	0
$455$	0	0
$456$	0	0
$457$	241566.	0.0541061	0.0270530	−	0.999634i	$-0.491388\pi$
$457$	241566.	0.0541061	0.0270530	+	0.999634i	$0.491388\pi$
$458$	0	0
$459$	−3.42594e6	−0.759010
$460$	0	0
$461$	990579.	0.217088	0.108544	−	0.994092i	$-0.465381\pi$
$461$	990579.	0.217088	0.108544	+	0.994092i	$0.465381\pi$
$462$	0	0
$463$	−6.20488e6	−1.34518	−0.672591	−	0.740014i	$-0.734819\pi$
$463$	−6.20488e6	−1.34518	−0.672591	+	0.740014i	$0.734819\pi$
$464$	0	0
$465$	−2.64531e6	−0.567340
$466$	0	0
$467$	6.88497e6	1.46086	0.730432	−	0.682985i	$-0.239319\pi$
$467$	6.88497e6	1.46086	0.730432	+	0.682985i	$0.239319\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−639197.	−0.132765
$472$	0	0
$473$	53396.6	0.0109739
$474$	0	0
$475$	−1.17758e6	−0.239473
$476$	0	0
$477$	2.39214e6	0.481383
$478$	0	0
$479$	5.41288e6	1.07793	0.538963	−	0.842329i	$-0.318816\pi$
$479$	5.41288e6	1.07793	0.538963	+	0.842329i	$0.318816\pi$
$480$	0	0
$481$	−2.79774e6	−0.551373
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−143069.	−0.0276179
$486$	0	0
$487$	3.01043e6	0.575182	0.287591	−	0.957753i	$-0.407146\pi$
$487$	3.01043e6	0.575182	0.287591	+	0.957753i	$0.407146\pi$
$488$	0	0
$489$	1.19618e6	0.226216
$490$	0	0
$491$	7.24498e6	1.35623	0.678115	−	0.734956i	$-0.262797\pi$
$491$	7.24498e6	1.35623	0.678115	+	0.734956i	$0.262797\pi$
$492$	0	0
$493$	−3.93062e6	−0.728356
$494$	0	0
$495$	5.07853e6	0.931591
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.12788e6	0.921906	0.460953	−	0.887425i	$-0.347508\pi$
$499$	5.12788e6	0.921906	0.460953	+	0.887425i	$0.347508\pi$
$500$	0	0
$501$	−2.95757e6	−0.526429
$502$	0	0
$503$	−1.05978e7	−1.86766	−0.933830	−	0.357718i	$-0.883555\pi$
$503$	−1.05978e7	−1.86766	−0.933830	+	0.357718i	$0.883555\pi$
$504$	0	0
$505$	−1.00426e6	−0.175233
$506$	0	0
$507$	−5.40707e6	−0.934205
$508$	0	0
$509$	8.78840e6	1.50354	0.751770	−	0.659425i	$-0.229200\pi$
$509$	8.78840e6	1.50354	0.751770	+	0.659425i	$0.229200\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.38361e6	0.567658
$514$	0	0
$515$	3.00512e6	0.499280
$516$	0	0
$517$	−4.23721e6	−0.697194
$518$	0	0
$519$	−2.43355e6	−0.396572
$520$	0	0
$521$	162133.	0.0261684	0.0130842	−	0.999914i	$-0.495835\pi$
$521$	162133.	0.0261684	0.0130842	+	0.999914i	$0.495835\pi$
$522$	0	0
$523$	7.14844e6	1.14277	0.571383	−	0.820684i	$-0.306407\pi$
$523$	7.14844e6	1.14277	0.571383	+	0.820684i	$0.306407\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.03714e7	1.62671
$528$	0	0
$529$	−6.43437e6	−0.999694
$530$	0	0
$531$	1.97267e6	0.303611
$532$	0	0
$533$	−1.39621e7	−2.12879
$534$	0	0
$535$	−5.01106e6	−0.756912
$536$	0	0
$537$	556604.	0.0832934
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.83604e6	−0.710390	−0.355195	−	0.934792i	$-0.615586\pi$
$541$	−4.83604e6	−0.710390	−0.355195	+	0.934792i	$0.615586\pi$
$542$	0	0
$543$	2.48459e6	0.361622
$544$	0	0
$545$	−3.98874e6	−0.575234
$546$	0	0
$547$	−9.98777e6	−1.42725	−0.713626	−	0.700527i	$-0.752948\pi$
$547$	−9.98777e6	−1.42725	−0.713626	+	0.700527i	$0.752948\pi$
$548$	0	0
$549$	7.70833e6	1.09151
$550$	0	0
$551$	3.88205e6	0.544732
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−770078.	−0.106121
$556$	0	0
$557$	1.74619e6	0.238481	0.119241	−	0.992865i	$-0.461954\pi$
$557$	1.74619e6	0.238481	0.119241	+	0.992865i	$0.461954\pi$
$558$	0	0
$559$	105877.	0.0143309
$560$	0	0
$561$	4.26600e6	0.572287
$562$	0	0
$563$	755218.	0.100416	0.0502078	−	0.998739i	$-0.484012\pi$
$563$	755218.	0.100416	0.0502078	+	0.998739i	$0.484012\pi$
$564$	0	0
$565$	4.66792e6	0.615181
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.39534e6	−0.569131	−0.284565	−	0.958657i	$-0.591849\pi$
$569$	−4.39534e6	−0.569131	−0.284565	+	0.958657i	$0.591849\pi$
$570$	0	0
$571$	−1.16104e7	−1.49024	−0.745121	−	0.666930i	$-0.767608\pi$
$571$	−1.16104e7	−1.49024	−0.745121	+	0.666930i	$0.767608\pi$
$572$	0	0
$573$	−6.03855e6	−0.768327
$574$	0	0
$575$	44821.1	0.00565344
$576$	0	0
$577$	1.06643e7	1.33350	0.666748	−	0.745283i	$-0.267686\pi$
$577$	1.06643e7	1.33350	0.666748	+	0.745283i	$0.267686\pi$
$578$	0	0
$579$	−3.30745e6	−0.410012
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.59562e6	−0.803682
$584$	0	0
$585$	1.00700e7	1.21657
$586$	0	0
$587$	1.39482e7	1.67079	0.835396	−	0.549648i	$-0.185238\pi$
$587$	1.39482e7	1.67079	0.835396	+	0.549648i	$0.185238\pi$
$588$	0	0
$589$	−1.02433e7	−1.21661
$590$	0	0
$591$	−1.64686e6	−0.193949
$592$	0	0
$593$	1.17933e7	1.37720	0.688600	−	0.725142i	$-0.258226\pi$
$593$	1.17933e7	1.37720	0.688600	+	0.725142i	$0.258226\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.36220e6	−0.156425
$598$	0	0
$599$	4.38057e6	0.498843	0.249421	−	0.968395i	$-0.419760\pi$
$599$	4.38057e6	0.498843	0.249421	+	0.968395i	$0.419760\pi$
$600$	0	0
$601$	688570.	0.0777610	0.0388805	−	0.999244i	$-0.487621\pi$
$601$	688570.	0.0777610	0.0388805	+	0.999244i	$0.487621\pi$
$602$	0	0
$603$	−1.35181e7	−1.51399
$604$	0	0
$605$	−6.59563e6	−0.732601
$606$	0	0
$607$	9.37319e6	1.03256	0.516281	−	0.856420i	$-0.327316\pi$
$607$	9.37319e6	1.03256	0.516281	+	0.856420i	$0.327316\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.40174e6	−0.910472
$612$	0	0
$613$	−2.16685e6	−0.232904	−0.116452	−	0.993196i	$-0.537152\pi$
$613$	−2.16685e6	−0.232904	−0.116452	+	0.993196i	$0.537152\pi$
$614$	0	0
$615$	−3.84306e6	−0.409722
$616$	0	0
$617$	−5.07951e6	−0.537166	−0.268583	−	0.963256i	$-0.586555\pi$
$617$	−5.07951e6	−0.537166	−0.268583	+	0.963256i	$0.586555\pi$
$618$	0	0
$619$	2.19034e6	0.229766	0.114883	−	0.993379i	$-0.463351\pi$
$619$	2.19034e6	0.229766	0.114883	+	0.993379i	$0.463351\pi$
$620$	0	0
$621$	−128787.	−0.0134012
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−5.59018e6	−0.572435
$626$	0	0
$627$	−4.21329e6	−0.428009
$628$	0	0
$629$	3.01923e6	0.304278
$630$	0	0
$631$	7.18693e6	0.718572	0.359286	−	0.933228i	$-0.383020\pi$
$631$	7.18693e6	0.718572	0.359286	+	0.933228i	$0.383020\pi$
$632$	0	0
$633$	−4.19451e6	−0.416075
$634$	0	0
$635$	−142324.	−0.0140070
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.22825e7	−1.18997
$640$	0	0
$641$	−1.76500e7	−1.69668	−0.848340	−	0.529452i	$-0.822398\pi$
$641$	−1.76500e7	−1.69668	−0.848340	+	0.529452i	$0.822398\pi$
$642$	0	0
$643$	−898309.	−0.0856837	−0.0428419	−	0.999082i	$-0.513641\pi$
$643$	−898309.	−0.0856837	−0.0428419	+	0.999082i	$0.513641\pi$
$644$	0	0
$645$	29142.7	0.00275823
$646$	0	0
$647$	1.38642e6	0.130207	0.0651035	−	0.997879i	$-0.479262\pi$
$647$	1.38642e6	0.130207	0.0651035	+	0.997879i	$0.479262\pi$
$648$	0	0
$649$	−5.43904e6	−0.506886
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.75425e7	−1.60994	−0.804968	−	0.593318i	$-0.797818\pi$
$653$	−1.75425e7	−1.60994	−0.804968	+	0.593318i	$0.797818\pi$
$654$	0	0
$655$	1.16556e7	1.06152
$656$	0	0
$657$	−370308.	−0.0334696
$658$	0	0
$659$	−9.87522e6	−0.885795	−0.442898	−	0.896572i	$-0.646050\pi$
$659$	−9.87522e6	−0.885795	−0.442898	+	0.896572i	$0.646050\pi$
$660$	0	0
$661$	−8.06792e6	−0.718221	−0.359110	−	0.933295i	$-0.616920\pi$
$661$	−8.06792e6	−0.718221	−0.359110	+	0.933295i	$0.616920\pi$
$662$	0	0
$663$	8.45884e6	0.747355
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−147759.	−0.0128599
$668$	0	0
$669$	2.55461e6	0.220678
$670$	0	0
$671$	−2.12534e7	−1.82231
$672$	0	0
$673$	−1.12772e7	−0.959762	−0.479881	−	0.877334i	$-0.659320\pi$
$673$	−1.12772e7	−0.959762	−0.479881	+	0.877334i	$0.659320\pi$
$674$	0	0
$675$	−2.93008e6	−0.247526
$676$	0	0
$677$	5.20372e6	0.436357	0.218179	−	0.975909i	$-0.429988\pi$
$677$	5.20372e6	0.436357	0.218179	+	0.975909i	$0.429988\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.90778e6	0.157638
$682$	0	0
$683$	−6.05915e6	−0.497004	−0.248502	−	0.968631i	$-0.579938\pi$
$683$	−6.05915e6	−0.497004	−0.248502	+	0.968631i	$0.579938\pi$
$684$	0	0
$685$	4.46816e6	0.363833
$686$	0	0
$687$	8.05671e6	0.651277
$688$	0	0
$689$	−1.30781e7	−1.04954
$690$	0	0
$691$	−7.36498e6	−0.586781	−0.293391	−	0.955993i	$-0.594784\pi$
$691$	−7.36498e6	−0.586781	−0.293391	+	0.955993i	$0.594784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.67228e6	0.759569
$696$	0	0
$697$	1.50674e7	1.17478
$698$	0	0
$699$	−748298.	−0.0579271
$700$	0	0
$701$	7.80919e6	0.600221	0.300110	−	0.953904i	$-0.402977\pi$
$701$	7.80919e6	0.600221	0.300110	+	0.953904i	$0.402977\pi$
$702$	0	0
$703$	−2.98193e6	−0.227567
$704$	0	0
$705$	−2.31258e6	−0.175236
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.75650e7	1.31230	0.656150	−	0.754631i	$-0.272184\pi$
$709$	1.75650e7	1.31230	0.656150	+	0.754631i	$0.272184\pi$
$710$	0	0
$711$	−1706.88	−0.000126628 0
$712$	0	0
$713$	389879.	0.0287214
$714$	0	0
$715$	−2.77649e7	−2.03110
$716$	0	0
$717$	−7.51597e6	−0.545993
$718$	0	0
$719$	8.09220e6	0.583773	0.291887	−	0.956453i	$-0.405717\pi$
$719$	8.09220e6	0.583773	0.291887	+	0.956453i	$0.405717\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5.32162e6	−0.378615
$724$	0	0
$725$	−3.36172e6	−0.237529
$726$	0	0
$727$	−1.51986e7	−1.06652	−0.533258	−	0.845952i	$-0.679033\pi$
$727$	−1.51986e7	−1.06652	−0.533258	+	0.845952i	$0.679033\pi$
$728$	0	0
$729$	−1.31285e6	−0.0914944
$730$	0	0
$731$	−114259.	−0.00790858
$732$	0	0
$733$	5.83402e6	0.401059	0.200530	−	0.979688i	$-0.435734\pi$
$733$	5.83402e6	0.401059	0.200530	+	0.979688i	$0.435734\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.72722e7	2.52765
$738$	0	0
$739$	−6.47719e6	−0.436290	−0.218145	−	0.975916i	$-0.570001\pi$
$739$	−6.47719e6	−0.436290	−0.218145	+	0.975916i	$0.570001\pi$
$740$	0	0
$741$	−8.35433e6	−0.558941
$742$	0	0
$743$	1.50899e7	1.00280	0.501401	−	0.865215i	$-0.332818\pi$
$743$	1.50899e7	1.00280	0.501401	+	0.865215i	$0.332818\pi$
$744$	0	0
$745$	−6.45744e6	−0.426255
$746$	0	0
$747$	−1.90196e7	−1.24710
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.13997e6	0.138455	0.0692273	−	0.997601i	$-0.477947\pi$
$751$	2.13997e6	0.138455	0.0692273	+	0.997601i	$0.477947\pi$
$752$	0	0
$753$	−2.66355e6	−0.171188
$754$	0	0
$755$	5.50494e6	0.351467
$756$	0	0
$757$	2.10943e7	1.33791	0.668954	−	0.743304i	$-0.266742\pi$
$757$	2.10943e7	1.33791	0.668954	+	0.743304i	$0.266742\pi$
$758$	0	0
$759$	160366.	0.0101044
$760$	0	0
$761$	9.79958e6	0.613403	0.306701	−	0.951806i	$-0.400775\pi$
$761$	9.79958e6	0.613403	0.306701	+	0.951806i	$0.400775\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.08672e7	−0.671371
$766$	0	0
$767$	−1.07848e7	−0.661947
$768$	0	0
$769$	−3.23493e7	−1.97265	−0.986323	−	0.164825i	$-0.947294\pi$
$769$	−3.23493e7	−1.97265	−0.986323	+	0.164825i	$0.947294\pi$
$770$	0	0
$771$	1.11128e7	0.673267
$772$	0	0
$773$	9.19713e6	0.553609	0.276805	−	0.960926i	$-0.410724\pi$
$773$	9.19713e6	0.553609	0.276805	+	0.960926i	$0.410724\pi$
$774$	0	0
$775$	8.87030e6	0.530499
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.48812e7	−0.878609
$780$	0	0
$781$	3.38653e7	1.98668
$782$	0	0
$783$	9.65941e6	0.563049
$784$	0	0
$785$	−4.48950e6	−0.260030
$786$	0	0
$787$	−1.46950e7	−0.845731	−0.422866	−	0.906192i	$-0.638976\pi$
$787$	−1.46950e7	−0.845731	−0.422866	+	0.906192i	$0.638976\pi$
$788$	0	0
$789$	1.34689e6	0.0770263
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.21423e7	−2.37977
$794$	0	0
$795$	−3.59975e6	−0.202001
$796$	0	0
$797$	2.33344e7	1.30122	0.650610	−	0.759412i	$-0.274513\pi$
$797$	2.33344e7	1.30122	0.650610	+	0.759412i	$0.274513\pi$
$798$	0	0
$799$	9.06688e6	0.502448
$800$	0	0
$801$	−1.07277e7	−0.590782
$802$	0	0
$803$	1.02101e6	0.0558783
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.13559e7	−0.613814
$808$	0	0
$809$	1.69301e6	0.0909468	0.0454734	−	0.998966i	$-0.485520\pi$
$809$	1.69301e6	0.0909468	0.0454734	+	0.998966i	$0.485520\pi$
$810$	0	0
$811$	2.12400e7	1.13397	0.566987	−	0.823727i	$-0.308109\pi$
$811$	2.12400e7	1.13397	0.566987	+	0.823727i	$0.308109\pi$
$812$	0	0
$813$	2.41649e6	0.128221
$814$	0	0
$815$	8.40155e6	0.443063
$816$	0	0
$817$	112848.	0.00591477
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.73550e6	0.452304	0.226152	−	0.974092i	$-0.427385\pi$
$821$	8.73550e6	0.452304	0.226152	+	0.974092i	$0.427385\pi$
$822$	0	0
$823$	3.27964e7	1.68782	0.843910	−	0.536485i	$-0.180248\pi$
$823$	3.27964e7	1.68782	0.843910	+	0.536485i	$0.180248\pi$
$824$	0	0
$825$	3.64856e6	0.186632
$826$	0	0
$827$	−1.31248e7	−0.667311	−0.333656	−	0.942695i	$-0.608282\pi$
$827$	−1.31248e7	−0.667311	−0.333656	+	0.942695i	$0.608282\pi$
$828$	0	0
$829$	2.05402e7	1.03805	0.519026	−	0.854759i	$-0.326295\pi$
$829$	2.05402e7	1.03805	0.519026	+	0.854759i	$0.326295\pi$
$830$	0	0
$831$	−8.03880e6	−0.403821
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.07729e7	−1.03106
$836$	0	0
$837$	−2.54875e7	−1.25752
$838$	0	0
$839$	−2.83736e7	−1.39159	−0.695793	−	0.718243i	$-0.744947\pi$
$839$	−2.83736e7	−1.39159	−0.695793	+	0.718243i	$0.744947\pi$
$840$	0	0
$841$	−9.42879e6	−0.459691
$842$	0	0
$843$	−1.31400e7	−0.636834
$844$	0	0
$845$	−3.79774e7	−1.82972
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.16935e7	−0.556770
$850$	0	0
$851$	113498.	0.00537236
$852$	0	0
$853$	1.02093e7	0.480424	0.240212	−	0.970720i	$-0.422783\pi$
$853$	1.02093e7	0.480424	0.240212	+	0.970720i	$0.422783\pi$
$854$	0	0
$855$	1.07329e7	0.502113
$856$	0	0
$857$	−8.33206e6	−0.387525	−0.193763	−	0.981048i	$-0.562069\pi$
$857$	−8.33206e6	−0.387525	−0.193763	+	0.981048i	$0.562069\pi$
$858$	0	0
$859$	−3.12766e7	−1.44623	−0.723113	−	0.690729i	$-0.757290\pi$
$859$	−3.12766e7	−1.44623	−0.723113	+	0.690729i	$0.757290\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.73573e7	1.70745	0.853726	−	0.520722i	$-0.174337\pi$
$863$	3.73573e7	1.70745	0.853726	+	0.520722i	$0.174337\pi$
$864$	0	0
$865$	−1.70924e7	−0.776719
$866$	0	0
$867$	168747.	0.00762411
$868$	0	0
$869$	4706.21	0.000211408 0
$870$	0	0
$871$	7.39051e7	3.30088
$872$	0	0
$873$	−622543.	−0.0276461
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38996.7	0.00171210	0.000856049	−	1.00000i	$-0.499728\pi$
$877$	38996.7	0.00171210	0.000856049		1.00000i	$0.499728\pi$
$878$	0	0
$879$	−5.58708e6	−0.243900
$880$	0	0
$881$	3.15554e7	1.36973	0.684864	−	0.728671i	$-0.259862\pi$
$881$	3.15554e7	1.36973	0.684864	+	0.728671i	$0.259862\pi$
$882$	0	0
$883$	3.42253e7	1.47722	0.738611	−	0.674132i	$-0.235482\pi$
$883$	3.42253e7	1.47722	0.738611	+	0.674132i	$0.235482\pi$
$884$	0	0
$885$	−2.96851e6	−0.127403
$886$	0	0
$887$	2.69886e7	1.15178	0.575892	−	0.817526i	$-0.304655\pi$
$887$	2.69886e7	1.15178	0.575892	+	0.817526i	$0.304655\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.63495e7	0.689938
$892$	0	0
$893$	−8.95486e6	−0.375777
$894$	0	0
$895$	3.90940e6	0.163137
$896$	0	0
$897$	317982.	0.0131954
$898$	0	0
$899$	−2.92421e7	−1.20673
$900$	0	0
$901$	1.41135e7	0.579191
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.74509e7	0.708267
$906$	0	0
$907$	−1.92103e7	−0.775381	−0.387690	−	0.921790i	$-0.626727\pi$
$907$	−1.92103e7	−0.775381	−0.387690	+	0.921790i	$0.626727\pi$
$908$	0	0
$909$	−4.36988e6	−0.175412
$910$	0	0
$911$	−2.86013e7	−1.14180	−0.570899	−	0.821020i	$-0.693405\pi$
$911$	−2.86013e7	−1.14180	−0.570899	+	0.821020i	$0.693405\pi$
$912$	0	0
$913$	5.24408e7	2.08206
$914$	0	0
$915$	−1.15997e7	−0.458029
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.21754e7	1.64729	0.823645	−	0.567106i	$-0.191937\pi$
$919$	4.21754e7	1.64729	0.823645	+	0.567106i	$0.191937\pi$
$920$	0	0
$921$	−1.28386e7	−0.498733
$922$	0	0
$923$	6.71498e7	2.59442
$924$	0	0
$925$	2.58224e6	0.0992299
$926$	0	0
$927$	1.30764e7	0.499790
$928$	0	0
$929$	−3.01886e7	−1.14763	−0.573817	−	0.818983i	$-0.694538\pi$
$929$	−3.01886e7	−1.14763	−0.573817	+	0.818983i	$0.694538\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	5.65489e6	0.212677
$934$	0	0
$935$	2.99630e7	1.12087
$936$	0	0
$937$	−3.64068e6	−0.135467	−0.0677335	−	0.997703i	$-0.521577\pi$
$937$	−3.64068e6	−0.135467	−0.0677335	+	0.997703i	$0.521577\pi$
$938$	0	0
$939$	−7.20589e6	−0.266701
$940$	0	0
$941$	1.88601e7	0.694336	0.347168	−	0.937803i	$-0.387143\pi$
$941$	1.88601e7	0.694336	0.347168	+	0.937803i	$0.387143\pi$
$942$	0	0
$943$	566410.	0.0207420
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.82172e7	0.660094	0.330047	−	0.943965i	$-0.392935\pi$
$947$	1.82172e7	0.660094	0.330047	+	0.943965i	$0.392935\pi$
$948$	0	0
$949$	2.02452e6	0.0729720
$950$	0	0
$951$	−9.79522e6	−0.351207
$952$	0	0
$953$	2.76898e7	0.987616	0.493808	−	0.869571i	$-0.335604\pi$
$953$	2.76898e7	0.987616	0.493808	+	0.869571i	$0.335604\pi$
$954$	0	0
$955$	−4.24127e7	−1.50483
$956$	0	0
$957$	−1.20280e7	−0.424534
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.85298e7	1.69512
$962$	0	0
$963$	−2.18049e7	−0.757685
$964$	0	0
$965$	−2.32304e7	−0.803042
$966$	0	0
$967$	2.44768e7	0.841761	0.420881	−	0.907116i	$-0.361721\pi$
$967$	2.44768e7	0.841761	0.420881	+	0.907116i	$0.361721\pi$
$968$	0	0
$969$	9.01571e6	0.308454
$970$	0	0
$971$	−9.50151e6	−0.323403	−0.161702	−	0.986840i	$-0.551698\pi$
$971$	−9.50151e6	−0.323403	−0.161702	+	0.986840i	$0.551698\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.23455e6	0.243725
$976$	0	0
$977$	4.69012e7	1.57198	0.785991	−	0.618238i	$-0.212153\pi$
$977$	4.69012e7	1.57198	0.785991	+	0.618238i	$0.212153\pi$
$978$	0	0
$979$	2.95785e7	0.986325
$980$	0	0
$981$	−1.73564e7	−0.575822
$982$	0	0
$983$	−2.35382e7	−0.776945	−0.388473	−	0.921460i	$-0.626997\pi$
$983$	−2.35382e7	−0.776945	−0.388473	+	0.921460i	$0.626997\pi$
$984$	0	0
$985$	−1.15670e7	−0.379865
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4295.21	−0.000139635 0
$990$	0	0
$991$	2.64104e7	0.854261	0.427130	−	0.904190i	$-0.359525\pi$
$991$	2.64104e7	0.854261	0.427130	+	0.904190i	$0.359525\pi$
$992$	0	0
$993$	−1.79426e7	−0.577446
$994$	0	0
$995$	−9.56766e6	−0.306371
$996$	0	0
$997$	−1.95164e7	−0.621815	−0.310907	−	0.950440i	$-0.600633\pi$
$997$	−1.95164e7	−0.621815	−0.310907	+	0.950440i	$0.600633\pi$
$998$	0	0
$999$	−7.41970e6	−0.235219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bf.1.2		4
4.3	odd	2		49.6.a.g.1.4	yes	4
7.6	odd	2	inner	784.6.a.bf.1.3		4
12.11	even	2		441.6.a.z.1.2		4
28.3	even	6		49.6.c.h.30.2		8
28.11	odd	6		49.6.c.h.30.1		8
28.19	even	6		49.6.c.h.18.2		8
28.23	odd	6		49.6.c.h.18.1		8
28.27	even	2		49.6.a.g.1.3	✓	4
84.83	odd	2		441.6.a.z.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.g.1.3	✓	4	28.27	even	2
49.6.a.g.1.4	yes	4	4.3	odd	2
49.6.c.h.18.1		8	28.23	odd	6
49.6.c.h.18.2		8	28.19	even	6
49.6.c.h.30.1		8	28.11	odd	6
49.6.c.h.30.2		8	28.3	even	6
441.6.a.z.1.1		4	84.83	odd	2
441.6.a.z.1.2		4	12.11	even	2
784.6.a.bf.1.2		4	1.1	even	1	trivial
784.6.a.bf.1.3		4	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-6.54802 q^{3} -45.9910 q^{5} -200.123 q^{9} +O(q^{10})\)	\(q-6.54802 q^{3} -45.9910 q^{5} -200.123 q^{9} +551.781 q^{11} +1094.10 q^{13} +301.150 q^{15} -1180.71 q^{17} +1166.13 q^{19} -44.3851 q^{23} -1009.82 q^{25} +2901.58 q^{27} +3329.02 q^{29} -8784.01 q^{31} -3613.07 q^{33} -2557.12 q^{37} -7164.17 q^{39} -12761.3 q^{41} +96.7714 q^{43} +9203.89 q^{45} -7679.15 q^{47} +7731.33 q^{51} -11953.3 q^{53} -25377.0 q^{55} -7635.81 q^{57} -9857.24 q^{59} -38517.9 q^{61} -50318.7 q^{65} +67548.9 q^{67} +290.634 q^{69} +61374.6 q^{71} +1850.40 q^{73} +6612.34 q^{75} +8.52913 q^{79} +29630.4 q^{81} +95039.3 q^{83} +54302.3 q^{85} -21798.5 q^{87} +53605.6 q^{89} +57517.9 q^{93} -53631.4 q^{95} +3110.79 q^{97} -110424. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 220 q^{9}+O(q^{10}) \) 4 * q + 220 * q^9	\( 4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100}) \) 4 * q + 220 * q^9 + 1952 * q^11 + 4096 * q^15 + 7136 * q^23 + 2764 * q^25 - 3352 * q^29 - 9208 * q^37 - 2464 * q^39 - 20448 * q^43 + 67408 * q^51 - 102920 * q^53 - 15576 * q^57 - 63168 * q^65 + 22896 * q^67 + 153824 * q^71 + 90688 * q^79 - 17204 * q^81 + 272656 * q^85 + 247760 * q^93 - 108224 * q^95 + 42272 * q^99

Introduction

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