LMFDB - Embedded newform 441.6.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
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:	$ 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			$7.22929$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.81507 q^{2} -24.0754 q^{4} +45.9910 q^{5} +157.856 q^{8} -129.468 q^{10} +551.781 q^{11} +1094.10 q^{13} +326.035 q^{16} +1180.71 q^{17} -1166.13 q^{19} -1107.25 q^{20} -1553.30 q^{22} -44.3851 q^{23} -1009.82 q^{25} -3079.97 q^{26} -3329.02 q^{29} +8784.01 q^{31} -5969.21 q^{32} -3323.80 q^{34} -2557.12 q^{37} +3282.73 q^{38} +7259.97 q^{40} +12761.3 q^{41} -96.7714 q^{43} -13284.3 q^{44} +124.947 q^{46} -7679.15 q^{47} +2842.73 q^{50} -26340.8 q^{52} +11953.3 q^{53} +25377.0 q^{55} +9371.43 q^{58} -9857.24 q^{59} -38517.9 q^{61} -24727.6 q^{62} +6370.65 q^{64} +50318.7 q^{65} -67548.9 q^{67} -28426.1 q^{68} +61374.6 q^{71} +1850.40 q^{73} +7198.49 q^{74} +28074.9 q^{76} -8.52913 q^{79} +14994.7 q^{80} -35923.9 q^{82} +95039.3 q^{83} +54302.3 q^{85} +272.419 q^{86} +87102.1 q^{88} -53605.6 q^{89} +1068.59 q^{92} +21617.4 q^{94} -53631.4 q^{95} +3110.79 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50}+ \cdots - 108224 q^{95}+O(q^{100}) $ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 + 1952 * q^11 - 1566 * q^16 + 3524 * q^22 + 7136 * q^23 + 2764 * q^25 + 3352 * q^29 - 27810 * q^32 - 9208 * q^37 + 20448 * q^43 - 1900 * q^44 + 56712 * q^46 + 43070 * q^50 + 102920 * q^53 + 96972 * q^58 - 40318 * q^64 + 63168 * q^65 - 22896 * q^67 + 153824 * q^71 - 17596 * q^74 - 90688 * q^79 + 272656 * q^85 + 161860 * q^86 + 154812 * q^88 + 212200 * q^92 - 108224 * q^95

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 $

$2$	−2.81507	−0.497639	−0.248820	−	0.968550i	$-0.580043\pi$
$2$	−2.81507	−0.497639	−0.248820	+	0.968550i	$0.580043\pi$
$3$	0	0
$3$	0	0
$4$	−24.0754	−0.752355
$5$	45.9910	0.822713	0.411356	−	0.911475i	$-0.365055\pi$
$5$	45.9910	0.822713	0.411356	+	0.911475i	$0.365055\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	157.856	0.872041
$9$	0	0
$10$	−129.468	−0.409414
$11$	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
$15$	0	0
$16$	326.035	0.318393
$17$	1180.71	0.990883	0.495442	−	0.868641i	$-0.335006\pi$
$17$	1180.71	0.990883	0.495442	+	0.868641i	$0.335006\pi$
$18$	0	0
$19$	−1166.13	−0.741074	−0.370537	−	0.928818i	$-0.620826\pi$
$19$	−1166.13	−0.741074	−0.370537	+	0.928818i	$0.620826\pi$
$20$	−1107.25	−0.618972
$21$	0	0
$22$	−1553.30	−0.684226
$23$	−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
$26$	−3079.97	−0.893537
$27$	0	0
$28$	0	0
$29$	−3329.02	−0.735057	−0.367529	−	0.930012i	$-0.619796\pi$
$29$	−3329.02	−0.735057	−0.367529	+	0.930012i	$0.619796\pi$
$30$	0	0
$31$	8784.01	1.64168	0.820841	−	0.571157i	$-0.193505\pi$
$31$	8784.01	1.64168	0.820841	+	0.571157i	$0.193505\pi$
$32$	−5969.21	−1.03049
$33$	0	0
$34$	−3323.80	−0.493102
$35$	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	$0.549067\pi$
$38$	3282.73	0.368787
$39$	0	0
$40$	7259.97	0.717439
$41$	12761.3	1.18559	0.592794	−	0.805354i	$-0.298025\pi$
$41$	12761.3	1.18559	0.592794	+	0.805354i	$0.298025\pi$
$42$	0	0
$43$	−96.7714	−0.00798135	−0.00399067	−	0.999992i	$-0.501270\pi$
$43$	−96.7714	−0.00798135	−0.00399067	+	0.999992i	$0.501270\pi$
$44$	−13284.3	−1.03445
$45$	0	0
$46$	124.947	0.00870627
$47$	−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
$50$	2842.73	0.160809
$51$	0	0
$52$	−26340.8	−1.35089
$53$	11953.3	0.584520	0.292260	−	0.956339i	$-0.405593\pi$
$53$	11953.3	0.584520	0.292260	+	0.956339i	$0.405593\pi$
$54$	0	0
$55$	25377.0	1.13118
$56$	0	0
$57$	0	0
$58$	9371.43	0.365793
$59$	−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	$0.730584\pi$
$62$	−24727.6	−0.816965
$63$	0	0
$64$	6370.65	0.194417
$65$	50318.7	1.47722
$66$	0	0
$67$	−67548.9	−1.83836	−0.919182	−	0.393833i	$-0.871149\pi$
$67$	−67548.9	−1.83836	−0.919182	+	0.393833i	$0.871149\pi$
$68$	−28426.1	−0.745496
$69$	0	0
$70$	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	$0.493531\pi$
$74$	7198.49	0.152814
$75$	0	0
$76$	28074.9	0.557551
$77$	0	0
$78$	0	0
$79$	−8.52913	−0.000153758 0	−7.68788e−5	−	1.00000i	$-0.500024\pi$
$79$	−8.52913	−0.000153758 0	−7.68788e−5		1.00000i	$0.500024\pi$
$80$	14994.7	0.261946
$81$	0	0
$82$	−35923.9	−0.589996
$83$	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
$86$	272.419	0.00397183
$87$	0	0
$88$	87102.1	1.19901
$89$	−53605.6	−0.717357	−0.358678	−	0.933461i	$-0.616773\pi$
$89$	−53605.6	−0.717357	−0.358678	+	0.933461i	$0.616773\pi$
$90$	0	0
$91$	0	0
$92$	1068.59	0.0131626
$93$	0	0
$94$	21617.4	0.252338
$95$	−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
$99$	0	0
$100$	24311.9	0.243119
$101$	−21835.9	−0.212994	−0.106497	−	0.994313i	$-0.533963\pi$
$101$	−21835.9	−0.212994	−0.106497	+	0.994313i	$0.533963\pi$
$102$	0	0
$103$	65341.4	0.606870	0.303435	−	0.952852i	$-0.401866\pi$
$103$	65341.4	0.606870	0.303435	+	0.952852i	$0.401866\pi$
$104$	172710.	1.56579
$105$	0	0
$106$	−33649.5	−0.290880
$107$	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	$0.386319\pi$
$110$	−71438.1	−0.562922
$111$	0	0
$112$	0	0
$113$	101496.	0.747746	0.373873	−	0.927480i	$-0.378030\pi$
$113$	101496.	0.747746	0.373873	+	0.927480i	$0.378030\pi$
$114$	0	0
$115$	−2041.32	−0.0143935
$116$	80147.3	0.553024
$117$	0	0
$118$	27748.9	0.183459
$119$	0	0
$120$	0	0
$121$	143411.	0.890470
$122$	108431.	0.659557
$123$	0	0
$124$	−211478.	−1.23513
$125$	−190165.	−1.08857
$126$	0	0
$127$	−3094.61	−0.0170253	−0.00851267	−	0.999964i	$-0.502710\pi$
$127$	−3094.61	−0.0170253	−0.00851267	+	0.999964i	$0.502710\pi$
$128$	173081.	0.933736
$129$	0	0
$130$	−141651.	−0.735124
$131$	−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
$134$	190155.	0.914842
$135$	0	0
$136$	186383.	0.864091
$137$	97152.9	0.442236	0.221118	−	0.975247i	$-0.429029\pi$
$137$	97152.9	0.442236	0.221118	+	0.975247i	$0.429029\pi$
$138$	0	0
$139$	210308.	0.923249	0.461624	−	0.887076i	$-0.347267\pi$
$139$	210308.	0.923249	0.461624	+	0.887076i	$0.347267\pi$
$140$	0	0
$141$	0	0
$142$	−172774.	−0.719047
$143$	603702.	2.46878
$144$	0	0
$145$	−153105.	−0.604741
$146$	−5209.01	−0.0202243
$147$	0	0
$148$	61563.7	0.231031
$149$	−140406.	−0.518109	−0.259055	−	0.965863i	$-0.583411\pi$
$149$	−140406.	−0.518109	−0.259055	+	0.965863i	$0.583411\pi$
$150$	0	0
$151$	119696.	0.427205	0.213603	−	0.976921i	$-0.431480\pi$
$151$	119696.	0.427205	0.213603	+	0.976921i	$0.431480\pi$
$152$	−184080.	−0.646247
$153$	0	0
$154$	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	$0.449485\pi$
$158$	24.0101	7.65159e−5 0
$159$	0	0
$160$	−274530.	−0.847794
$161$	0	0
$162$	0	0
$163$	182678.	0.538539	0.269270	−	0.963065i	$-0.413218\pi$
$163$	182678.	0.538539	0.269270	+	0.963065i	$0.413218\pi$
$164$	−307232.	−0.891984
$165$	0	0
$166$	−267542.	−0.753568
$167$	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
$170$	−152865.	−0.405682
$171$	0	0
$172$	2329.81	0.00600481
$173$	−371647.	−0.944095	−0.472047	−	0.881573i	$-0.656485\pi$
$173$	−371647.	−0.944095	−0.472047	+	0.881573i	$0.656485\pi$
$174$	0	0
$175$	0	0
$176$	179900.	0.437773
$177$	0	0
$178$	150904.	0.356985
$179$	−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
$183$	0	0
$184$	−7006.46	−0.0152565
$185$	−117605.	−0.252636
$186$	0	0
$187$	651496.	1.36241
$188$	184878.	0.381497
$189$	0	0
$190$	150976.	0.303406
$191$	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	$0.337710\pi$
$194$	−8757.11	−0.0167054
$195$	0	0
$196$	0	0
$197$	−251505.	−0.461723	−0.230861	−	0.972987i	$-0.574154\pi$
$197$	−251505.	−0.461723	−0.230861	+	0.972987i	$0.574154\pi$
$198$	0	0
$199$	−208033.	−0.372392	−0.186196	−	0.982513i	$-0.559616\pi$
$199$	−208033.	−0.372392	−0.186196	+	0.982513i	$0.559616\pi$
$200$	−159407.	−0.281794
$201$	0	0
$202$	61469.7	0.105994
$203$	0	0
$204$	0	0
$205$	586904.	0.975399
$206$	−183941.	−0.302002
$207$	0	0
$208$	356714.	0.571692
$209$	−643446.	−1.01893
$210$	0	0
$211$	−640577.	−0.990525	−0.495262	−	0.868744i	$-0.664928\pi$
$211$	−640577.	−0.990525	−0.495262	+	0.868744i	$0.664928\pi$
$212$	−287781.	−0.439767
$213$	0	0
$214$	−306723.	−0.457838
$215$	−4450.62	−0.00656636
$216$	0	0
$217$	0	0
$218$	−244148.	−0.347945
$219$	0	0
$220$	−610960.	−0.851052
$221$	1.29182e6	1.77918
$222$	0	0
$223$	390135.	0.525354	0.262677	−	0.964884i	$-0.415395\pi$
$223$	390135.	0.525354	0.262677	+	0.964884i	$0.415395\pi$
$224$	0	0
$225$	0	0
$226$	−285720.	−0.372108
$227$	−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	$0.782365\pi$
$230$	5746.45	0.00716276
$231$	0	0
$232$	−525506.	−0.641000
$233$	−114279.	−0.137903	−0.0689517	−	0.997620i	$-0.521965\pi$
$233$	−114279.	−0.137903	−0.0689517	+	0.997620i	$0.521965\pi$
$234$	0	0
$235$	−353172.	−0.417174
$236$	237317.	0.277363
$237$	0	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	$0.351184\pi$
$242$	−403713.	−0.443133
$243$	0	0
$244$	927332.	0.997150
$245$	0	0
$246$	0	0
$247$	−1.27586e6	−1.33064
$248$	1.38661e6	1.43161
$249$	0	0
$250$	535328.	0.541714
$251$	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
$254$	8711.54	0.00847248
$255$	0	0
$256$	−691096.	−0.659081
$257$	1.69712e6	1.60281	0.801403	−	0.598125i	$-0.204087\pi$
$257$	1.69712e6	1.60281	0.801403	+	0.598125i	$0.204087\pi$
$258$	0	0
$259$	0	0
$260$	−1.21144e6	−1.11140
$261$	0	0
$262$	713427.	0.642090
$263$	−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
$267$	0	0
$268$	1.62627e6	1.38310
$269$	−1.73425e6	−1.46127	−0.730635	−	0.682769i	$-0.760776\pi$
$269$	−1.73425e6	−1.46127	−0.730635	+	0.682769i	$0.760776\pi$
$270$	0	0
$271$	369042.	0.305248	0.152624	−	0.988284i	$-0.451228\pi$
$271$	369042.	0.305248	0.152624	+	0.988284i	$0.451228\pi$
$272$	384954.	0.315491
$273$	0	0
$274$	−273492.	−0.220074
$275$	−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	$0.340391\pi$
$278$	−592032.	−0.459445
$279$	0	0
$280$	0	0
$281$	−2.00671e6	−1.51607	−0.758035	−	0.652214i	$-0.773841\pi$
$281$	−2.00671e6	−1.51607	−0.758035	+	0.652214i	$0.773841\pi$
$282$	0	0
$283$	−1.78581e6	−1.32547	−0.662734	−	0.748855i	$-0.730604\pi$
$283$	−1.78581e6	−1.32547	−0.662734	+	0.748855i	$0.730604\pi$
$284$	−1.47762e6	−1.08709
$285$	0	0
$286$	−1.69947e6	−1.22856
$287$	0	0
$288$	0	0
$289$	−25770.8	−0.0181503
$290$	431002.	0.300943
$291$	0	0
$292$	−44549.0	−0.0305760
$293$	−853248.	−0.580639	−0.290319	−	0.956930i	$-0.593762\pi$
$293$	−853248.	−0.580639	−0.290319	+	0.956930i	$0.593762\pi$
$294$	0	0
$295$	−453345.	−0.303301
$296$	−403658.	−0.267784
$297$	0	0
$298$	395254.	0.257831
$299$	−48561.6	−0.0314134
$300$	0	0
$301$	0	0
$302$	−336952.	−0.212594
$303$	0	0
$304$	−380198.	−0.235953
$305$	−1.77148e6	−1.09040
$306$	0	0
$307$	−1.96068e6	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$307$	−1.96068e6	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$308$	0	0
$309$	0	0
$310$	−1.13725e6	−0.672128
$311$	−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	$0.397170\pi$
$314$	−274799.	−0.157286
$315$	0	0
$316$	205.342	0.000115680 0
$317$	−1.49591e6	−0.836097	−0.418048	−	0.908425i	$-0.637286\pi$
$317$	−1.49591e6	−0.836097	−0.418048	+	0.908425i	$0.637286\pi$
$318$	0	0
$319$	−1.83689e6	−1.01066
$320$	292993.	0.159949
$321$	0	0
$322$	0	0
$323$	−1.37686e6	−0.734318
$324$	0	0
$325$	−1.10485e6	−0.580221
$326$	−514252.	−0.267998
$327$	0	0
$328$	2.01445e6	1.03388
$329$	0	0
$330$	0	0
$331$	−2.74015e6	−1.37469	−0.687345	−	0.726331i	$-0.741224\pi$
$331$	−2.74015e6	−1.37469	−0.687345	+	0.726331i	$0.741224\pi$
$332$	−2.28810e6	−1.13928
$333$	0	0
$334$	−1.27149e6	−0.623661
$335$	−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	$0.687221\pi$
$338$	−2.32457e6	−1.10675
$339$	0	0
$340$	−1.30735e6	−0.613329
$341$	4.84685e6	2.25722
$342$	0	0
$343$	0	0
$344$	−15276.0	−0.00696006
$345$	0	0
$346$	1.04621e6	0.469819
$347$	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
$351$	0	0
$352$	−3.29370e6	−1.41686
$353$	3.76790e6	1.60939	0.804697	−	0.593686i	$-0.202328\pi$
$353$	3.76790e6	1.60939	0.804697	+	0.593686i	$0.202328\pi$
$354$	0	0
$355$	2.82268e6	1.18875
$356$	1.29057e6	0.539707
$357$	0	0
$358$	239291.	0.0986776
$359$	−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
$362$	1.06816e6	0.428414
$363$	0	0
$364$	0	0
$365$	85101.8	0.0334354
$366$	0	0
$367$	−1.52650e6	−0.591603	−0.295802	−	0.955249i	$-0.595587\pi$
$367$	−1.52650e6	−0.591603	−0.295802	+	0.955249i	$0.595587\pi$
$368$	−14471.1	−0.00557034
$369$	0	0
$370$	331066.	0.125722
$371$	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	$0.139947\pi$
$374$	−1.83401e6	−0.677988
$375$	0	0
$376$	−1.21220e6	−0.442186
$377$	−3.64227e6	−1.31983
$378$	0	0
$379$	630878.	0.225604	0.112802	−	0.993617i	$-0.464017\pi$
$379$	630878.	0.225604	0.112802	+	0.993617i	$0.464017\pi$
$380$	1.29119e6	0.458704
$381$	0	0
$382$	−2.59605e6	−0.910237
$383$	−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
$386$	−1.42191e6	−0.485741
$387$	0	0
$388$	−74893.5	−0.0252560
$389$	−592212.	−0.198428	−0.0992140	−	0.995066i	$-0.531633\pi$
$389$	−592212.	−0.198428	−0.0992140	+	0.995066i	$0.531633\pi$
$390$	0	0
$391$	−52406.1	−0.0173356
$392$	0	0
$393$	0	0
$394$	708005.	0.229771
$395$	−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	$0.431400\pi$
$398$	585629.	0.185317
$399$	0	0
$400$	−329238.	−0.102887
$401$	3.68716e6	1.14507	0.572534	−	0.819881i	$-0.305960\pi$
$401$	3.68716e6	1.14507	0.572534	+	0.819881i	$0.305960\pi$
$402$	0	0
$403$	9.61057e6	2.94772
$404$	525707.	0.160247
$405$	0	0
$406$	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	$0.430948\pi$
$410$	−1.65218e6	−0.485397
$411$	0	0
$412$	−1.57312e6	−0.456582
$413$	0	0
$414$	0	0
$415$	4.37096e6	1.24582
$416$	−6.53090e6	−1.85029
$417$	0	0
$418$	1.81135e6	0.507062
$419$	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	$0.410420\pi$
$422$	1.80327e6	0.492924
$423$	0	0
$424$	1.88691e6	0.509725
$425$	−1.19231e6	−0.320197
$426$	0	0
$427$	0	0
$428$	−2.62319e6	−0.692182
$429$	0	0
$430$	12528.8	0.00326768
$431$	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
$435$	0	0
$436$	−2.08802e6	−0.526041
$437$	51758.6	0.0129652
$438$	0	0
$439$	2.35150e6	0.582350	0.291175	−	0.956670i	$-0.405954\pi$
$439$	2.35150e6	0.582350	0.291175	+	0.956670i	$0.405954\pi$
$440$	4.00591e6	0.986439
$441$	0	0
$442$	−3.63656e6	−0.885391
$443$	−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
$446$	−1.09826e6	−0.261437
$447$	0	0
$448$	0	0
$449$	2.76805e6	0.647975	0.323987	−	0.946061i	$-0.394976\pi$
$449$	2.76805e6	0.647975	0.323987	+	0.946061i	$0.394976\pi$
$450$	0	0
$451$	7.04142e6	1.63012
$452$	−2.44356e6	−0.562571
$453$	0	0
$454$	820179.	0.186754
$455$	0	0
$456$	0	0
$457$	241566.	0.0541061	0.0270530	−	0.999634i	$-0.491388\pi$
$457$	241566.	0.0541061	0.0270530	+	0.999634i	$0.491388\pi$
$458$	3.46368e6	0.771567
$459$	0	0
$460$	49145.4	0.0108290
$461$	−990579.	−0.217088	−0.108544	−	0.994092i	$-0.534619\pi$
$461$	−990579.	−0.217088	−0.108544	+	0.994092i	$0.534619\pi$
$462$	0	0
$463$	6.20488e6	1.34518	0.672591	−	0.740014i	$-0.265181\pi$
$463$	6.20488e6	1.34518	0.672591	+	0.740014i	$0.265181\pi$
$464$	−1.08538e6	−0.234037
$465$	0	0
$466$	321702.	0.0686261
$467$	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
$470$	994206.	0.207602
$471$	0	0
$472$	−1.55603e6	−0.321486
$473$	−53396.6	−0.0109739
$474$	0	0
$475$	1.17758e6	0.239473
$476$	0	0
$477$	0	0
$478$	−3.23121e6	−0.646838
$479$	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
$482$	−2.28783e6	−0.448545
$483$	0	0
$484$	−3.45268e6	−0.669950
$485$	143069.	0.0276179
$486$	0	0
$487$	−3.01043e6	−0.575182	−0.287591	−	0.957753i	$-0.592854\pi$
$487$	−3.01043e6	−0.575182	−0.287591	+	0.957753i	$0.592854\pi$
$488$	−6.08029e6	−1.15578
$489$	0	0
$490$	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
$494$	3.59163e6	0.662177
$495$	0	0
$496$	2.86389e6	0.522700
$497$	0	0
$498$	0	0
$499$	−5.12788e6	−0.921906	−0.460953	−	0.887425i	$-0.652492\pi$
$499$	−5.12788e6	−0.921906	−0.460953	+	0.887425i	$0.652492\pi$
$500$	4.57829e6	0.818989
$501$	0	0
$502$	−1.14509e6	−0.202806
$503$	−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
$506$	68943.5	0.0119706
$507$	0	0
$508$	74503.8	0.0128091
$509$	−8.78840e6	−1.50354	−0.751770	−	0.659425i	$-0.770800\pi$
$509$	−8.78840e6	−1.50354	−0.751770	+	0.659425i	$0.770800\pi$
$510$	0	0
$511$	0	0
$512$	−3.59310e6	−0.605752
$513$	0	0
$514$	−4.77753e6	−0.797619
$515$	3.00512e6	0.499280
$516$	0	0
$517$	−4.23721e6	−0.697194
$518$	0	0
$519$	0	0
$520$	7.94312e6	1.28820
$521$	−162133.	−0.0261684	−0.0130842	−	0.999914i	$-0.504165\pi$
$521$	−162133.	−0.0261684	−0.0130842	+	0.999914i	$0.504165\pi$
$522$	0	0
$523$	−7.14844e6	−1.14277	−0.571383	−	0.820684i	$-0.693593\pi$
$523$	−7.14844e6	−1.14277	−0.571383	+	0.820684i	$0.693593\pi$
$524$	6.10144e6	0.970743
$525$	0	0
$526$	579044.	0.0912530
$527$	1.03714e7	1.62671
$528$	0	0
$529$	−6.43437e6	−0.999694
$530$	−1.54758e6	−0.239311
$531$	0	0
$532$	0	0
$533$	1.39621e7	2.12879
$534$	0	0
$535$	5.01106e6	0.756912
$536$	−1.06630e7	−1.60313
$537$	0	0
$538$	4.88203e6	0.727185
$539$	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	$0.615586\pi$
$542$	−1.03888e6	−0.151903
$543$	0	0
$544$	−7.04793e6	−1.02109
$545$	3.98874e6	0.575234
$546$	0	0
$547$	9.98777e6	1.42725	0.713626	−	0.700527i	$-0.247052\pi$
$547$	9.98777e6	1.42725	0.713626	+	0.700527i	$0.247052\pi$
$548$	−2.33899e6	−0.332719
$549$	0	0
$550$	1.56856e6	0.221103
$551$	3.88205e6	0.544732
$552$	0	0
$553$	0	0
$554$	−3.45598e6	−0.478406
$555$	0	0
$556$	−5.06324e6	−0.694611
$557$	−1.74619e6	−0.238481	−0.119241	−	0.992865i	$-0.538046\pi$
$557$	−1.74619e6	−0.238481	−0.119241	+	0.992865i	$0.538046\pi$
$558$	0	0
$559$	−105877.	−0.0143309
$560$	0	0
$561$	0	0
$562$	5.64904e6	0.754456
$563$	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
$566$	5.02719e6	0.659605
$567$	0	0
$568$	9.68836e6	1.26003
$569$	4.39534e6	0.569131	0.284565	−	0.958657i	$-0.408151\pi$
$569$	4.39534e6	0.569131	0.284565	+	0.958657i	$0.408151\pi$
$570$	0	0
$571$	1.16104e7	1.49024	0.745121	−	0.666930i	$-0.232392\pi$
$571$	1.16104e7	1.49024	0.745121	+	0.666930i	$0.232392\pi$
$572$	−1.45344e7	−1.85740
$573$	0	0
$574$	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	$0.267686\pi$
$578$	72546.6	0.00903228
$579$	0	0
$580$	3.68606e6	0.454980
$581$	0	0
$582$	0	0
$583$	6.59562e6	0.803682
$584$	292097.	0.0354401
$585$	0	0
$586$	2.40195e6	0.288949
$587$	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
$590$	1.27620e6	0.150934
$591$	0	0
$592$	−833711.	−0.0977713
$593$	−1.17933e7	−1.37720	−0.688600	−	0.725142i	$-0.741774\pi$
$593$	−1.17933e7	−1.37720	−0.688600	+	0.725142i	$0.741774\pi$
$594$	0	0
$595$	0	0
$596$	3.38033e6	0.389802
$597$	0	0
$598$	136705.	0.0156326
$599$	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
$603$	0	0
$604$	−2.88172e6	−0.321410
$605$	6.59563e6	0.732601
$606$	0	0
$607$	−9.37319e6	−1.03256	−0.516281	−	0.856420i	$-0.672684\pi$
$607$	−9.37319e6	−1.03256	−0.516281	+	0.856420i	$0.672684\pi$
$608$	6.96085e6	0.763666
$609$	0	0
$610$	4.98684e6	0.542626
$611$	−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	$0.537152\pi$
$614$	5.51947e6	0.590849
$615$	0	0
$616$	0	0
$617$	5.07951e6	0.537166	0.268583	−	0.963256i	$-0.413445\pi$
$617$	5.07951e6	0.537166	0.268583	+	0.963256i	$0.413445\pi$
$618$	0	0
$619$	−2.19034e6	−0.229766	−0.114883	−	0.993379i	$-0.536649\pi$
$619$	−2.19034e6	−0.229766	−0.114883	+	0.993379i	$0.536649\pi$
$620$	−9.72611e6	−1.01616
$621$	0	0
$622$	2.43111e6	0.251958
$623$	0	0
$624$	0	0
$625$	−5.59018e6	−0.572435
$626$	−3.09790e6	−0.315960
$627$	0	0
$628$	−2.35016e6	−0.237793
$629$	−3.01923e6	−0.304278
$630$	0	0
$631$	−7.18693e6	−0.718572	−0.359286	−	0.933228i	$-0.616980\pi$
$631$	−7.18693e6	−0.718572	−0.359286	+	0.933228i	$0.616980\pi$
$632$	−1346.38	−0.000134083 0
$633$	0	0
$634$	4.21109e6	0.416075
$635$	−142324.	−0.0140070
$636$	0	0
$637$	0	0
$638$	5.17097e6	0.502945
$639$	0	0
$640$	7.96017e6	0.768197
$641$	1.76500e7	1.69668	0.848340	−	0.529452i	$-0.177602\pi$
$641$	1.76500e7	1.69668	0.848340	+	0.529452i	$0.177602\pi$
$642$	0	0
$643$	898309.	0.0856837	0.0428419	−	0.999082i	$-0.486359\pi$
$643$	898309.	0.0856837	0.0428419	+	0.999082i	$0.486359\pi$
$644$	0	0
$645$	0	0
$646$	3.87597e6	0.365425
$647$	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
$650$	3.11022e6	0.288741
$651$	0	0
$652$	−4.39804e6	−0.405173
$653$	1.75425e7	1.60994	0.804968	−	0.593318i	$-0.202182\pi$
$653$	1.75425e7	1.60994	0.804968	+	0.593318i	$0.202182\pi$
$654$	0	0
$655$	−1.16556e7	−1.06152
$656$	4.16062e6	0.377484
$657$	0	0
$658$	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	$0.616920\pi$
$662$	7.71373e6	0.684100
$663$	0	0
$664$	1.50025e7	1.32052
$665$	0	0
$666$	0	0
$667$	147759.	0.0128599
$668$	−1.08742e7	−0.942880
$669$	0	0
$670$	8.74544e6	0.752652
$671$	−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	$0.659320\pi$
$674$	6.51274e6	0.552223
$675$	0	0
$676$	−1.98804e7	−1.67324
$677$	−5.20372e6	−0.436357	−0.218179	−	0.975909i	$-0.570012\pi$
$677$	−5.20372e6	−0.436357	−0.218179	+	0.975909i	$0.570012\pi$
$678$	0	0
$679$	0	0
$680$	8.57196e6	0.710899
$681$	0	0
$682$	−1.36442e7	−1.12328
$683$	−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
$687$	0	0
$688$	−31550.9	−0.00254121
$689$	1.30781e7	1.04954
$690$	0	0
$691$	7.36498e6	0.586781	0.293391	−	0.955993i	$-0.405216\pi$
$691$	7.36498e6	0.586781	0.293391	+	0.955993i	$0.405216\pi$
$692$	8.94754e6	0.710295
$693$	0	0
$694$	−8.61204e6	−0.678746
$695$	9.67228e6	0.759569
$696$	0	0
$697$	1.50674e7	1.17478
$698$	591818.	0.0459779
$699$	0	0
$700$	0	0
$701$	−7.80919e6	−0.600221	−0.300110	−	0.953904i	$-0.597023\pi$
$701$	−7.80919e6	−0.600221	−0.300110	+	0.953904i	$0.597023\pi$
$702$	0	0
$703$	2.98193e6	0.227567
$704$	3.51520e6	0.267312
$705$	0	0
$706$	−1.06069e7	−0.800898
$707$	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	$0.272184\pi$
$710$	−7.94606e6	−0.591569
$711$	0	0
$712$	−8.46198e6	−0.625564
$713$	−389879.	−0.0287214
$714$	0	0
$715$	2.77649e7	2.03110
$716$	2.04649e6	0.149186
$717$	0	0
$718$	2.83539e6	0.205259
$719$	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
$722$	3.14232e6	0.224341
$723$	0	0
$724$	9.13520e6	0.647697
$725$	3.36172e6	0.237529
$726$	0	0
$727$	1.51986e7	1.06652	0.533258	−	0.845952i	$-0.320967\pi$
$727$	1.51986e7	1.06652	0.533258	+	0.845952i	$0.320967\pi$
$728$	0	0
$729$	0	0
$730$	−239568.	−0.0166388
$731$	−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	$0.435734\pi$
$734$	4.29720e6	0.294405
$735$	0	0
$736$	264944.	0.0180285
$737$	−3.72722e7	−2.52765
$738$	0	0
$739$	6.47719e6	0.436290	0.218145	−	0.975916i	$-0.429999\pi$
$739$	6.47719e6	0.436290	0.218145	+	0.975916i	$0.429999\pi$
$740$	2.83138e6	0.190072
$741$	0	0
$742$	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
$746$	−1.36896e7	−0.900626
$747$	0	0
$748$	−1.56850e7	−1.02502
$749$	0	0
$750$	0	0
$751$	−2.13997e6	−0.138455	−0.0692273	−	0.997601i	$-0.522053\pi$
$751$	−2.13997e6	−0.138455	−0.0692273	+	0.997601i	$0.522053\pi$
$752$	−2.50367e6	−0.161448
$753$	0	0
$754$	1.02533e7	0.656801
$755$	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	$0.266742\pi$
$758$	−1.77597e6	−0.112270
$759$	0	0
$760$	−8.46605e6	−0.531675
$761$	−9.79958e6	−0.613403	−0.306701	−	0.951806i	$-0.599225\pi$
$761$	−9.79958e6	−0.613403	−0.306701	+	0.951806i	$0.599225\pi$
$762$	0	0
$763$	0	0
$764$	−2.22022e7	−1.37614
$765$	0	0
$766$	1.59233e6	0.0980530
$767$	−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
$771$	0	0
$772$	−1.21606e7	−0.734366
$773$	−9.19713e6	−0.553609	−0.276805	−	0.960926i	$-0.589276\pi$
$773$	−9.19713e6	−0.553609	−0.276805	+	0.960926i	$0.589276\pi$
$774$	0	0
$775$	−8.87030e6	−0.530499
$776$	491058.	0.0292738
$777$	0	0
$778$	1.66712e6	0.0987456
$779$	−1.48812e7	−0.878609
$780$	0	0
$781$	3.38653e7	1.98668
$782$	147527.	0.00862690
$783$	0	0
$784$	0	0
$785$	4.48950e6	0.260030
$786$	0	0
$787$	1.46950e7	0.845731	0.422866	−	0.906192i	$-0.361024\pi$
$787$	1.46950e7	0.845731	0.422866	+	0.906192i	$0.361024\pi$
$788$	6.05508e6	0.347379
$789$	0	0
$790$	1104.25	6.29506e−5 0
$791$	0	0
$792$	0	0
$793$	−4.21423e7	−2.37977
$794$	−3.78097e6	−0.212839
$795$	0	0
$796$	5.00847e6	0.280171
$797$	−2.33344e7	−1.30122	−0.650610	−	0.759412i	$-0.725487\pi$
$797$	−2.33344e7	−1.30122	−0.650610	+	0.759412i	$0.725487\pi$
$798$	0	0
$799$	−9.06688e6	−0.502448
$800$	6.02785e6	0.332995
$801$	0	0
$802$	−1.03796e7	−0.569831
$803$	1.02101e6	0.0558783
$804$	0	0
$805$	0	0
$806$	−2.70545e7	−1.46690
$807$	0	0
$808$	−3.44693e6	−0.185740
$809$	−1.69301e6	−0.0909468	−0.0454734	−	0.998966i	$-0.514480\pi$
$809$	−1.69301e6	−0.0909468	−0.0454734	+	0.998966i	$0.514480\pi$
$810$	0	0
$811$	−2.12400e7	−1.13397	−0.566987	−	0.823727i	$-0.691891\pi$
$811$	−2.12400e7	−1.13397	−0.566987	+	0.823727i	$0.691891\pi$
$812$	0	0
$813$	0	0
$814$	3.97199e6	0.210110
$815$	8.40155e6	0.443063
$816$	0	0
$817$	112848.	0.00591477
$818$	−4.09960e6	−0.214219
$819$	0	0
$820$	−1.41299e7	−0.733847
$821$	−8.73550e6	−0.452304	−0.226152	−	0.974092i	$-0.572615\pi$
$821$	−8.73550e6	−0.452304	−0.226152	+	0.974092i	$0.572615\pi$
$822$	0	0
$823$	−3.27964e7	−1.68782	−0.843910	−	0.536485i	$-0.819752\pi$
$823$	−3.27964e7	−1.68782	−0.843910	+	0.536485i	$0.819752\pi$
$824$	1.03146e7	0.529215
$825$	0	0
$826$	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	$0.326295\pi$
$830$	−1.23046e7	−0.619970
$831$	0	0
$832$	6.97012e6	0.349085
$833$	0	0
$834$	0	0
$835$	2.07729e7	1.03106
$836$	1.54912e7	0.766601
$837$	0	0
$838$	−8.22542e6	−0.404621
$839$	−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
$842$	−5.68643e6	−0.276414
$843$	0	0
$844$	1.54221e7	0.745226
$845$	3.79774e7	1.82972
$846$	0	0
$847$	0	0
$848$	3.89720e6	0.186107
$849$	0	0
$850$	3.35645e6	0.159343
$851$	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
$855$	0	0
$856$	1.71996e7	0.802295
$857$	8.33206e6	0.387525	0.193763	−	0.981048i	$-0.437931\pi$
$857$	8.33206e6	0.387525	0.193763	+	0.981048i	$0.437931\pi$
$858$	0	0
$859$	3.12766e7	1.44623	0.723113	−	0.690729i	$-0.242710\pi$
$859$	3.12766e7	1.44623	0.723113	+	0.690729i	$0.242710\pi$
$860$	107150.	0.00494023
$861$	0	0
$862$	−1.53084e7	−0.701714
$863$	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
$866$	1.06222e7	0.481306
$867$	0	0
$868$	0	0
$869$	−4706.21	−0.000211408 0
$870$	0	0
$871$	−7.39051e7	−3.30088
$872$	1.36907e7	0.609724
$873$	0	0
$874$	−145704.	−0.00645199
$875$	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	$0.499728\pi$
$878$	−6.61965e6	−0.289800
$879$	0	0
$880$	8.27378e6	0.360162
$881$	−3.15554e7	−1.36973	−0.684864	−	0.728671i	$-0.740138\pi$
$881$	−3.15554e7	−1.36973	−0.684864	+	0.728671i	$0.740138\pi$
$882$	0	0
$883$	−3.42253e7	−1.47722	−0.738611	−	0.674132i	$-0.764518\pi$
$883$	−3.42253e7	−1.47722	−0.738611	+	0.674132i	$0.764518\pi$
$884$	−3.11010e7	−1.33858
$885$	0	0
$886$	1.35230e7	0.578745
$887$	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
$890$	6.94022e6	0.293696
$891$	0	0
$892$	−9.39263e6	−0.395253
$893$	8.95486e6	0.375777
$894$	0	0
$895$	−3.90940e6	−0.163137
$896$	0	0
$897$	0	0
$898$	−7.79226e6	−0.322458
$899$	−2.92421e7	−1.20673
$900$	0	0
$901$	1.41135e7	0.579191
$902$	−1.98221e7	−0.811211
$903$	0	0
$904$	1.60218e7	0.652065
$905$	−1.74509e7	−0.708267
$906$	0	0
$907$	1.92103e7	0.775381	0.387690	−	0.921790i	$-0.373273\pi$
$907$	1.92103e7	0.775381	0.387690	+	0.921790i	$0.373273\pi$
$908$	7.01442e6	0.282343
$909$	0	0
$910$	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
$914$	−680027.	−0.0269253
$915$	0	0
$916$	2.96224e7	1.16649
$917$	0	0
$918$	0	0
$919$	−4.21754e7	−1.64729	−0.823645	−	0.567106i	$-0.808063\pi$
$919$	−4.21754e7	−1.64729	−0.823645	+	0.567106i	$0.808063\pi$
$920$	−322235.	−0.0125517
$921$	0	0
$922$	2.78855e6	0.108032
$923$	6.71498e7	2.59442
$924$	0	0
$925$	2.58224e6	0.0992299
$926$	−1.74672e7	−0.669416
$927$	0	0
$928$	1.98716e7	0.757466
$929$	3.01886e7	1.14763	0.573817	−	0.818983i	$-0.305462\pi$
$929$	3.01886e7	1.14763	0.573817	+	0.818983i	$0.305462\pi$
$930$	0	0
$931$	0	0
$932$	2.75130e6	0.103752
$933$	0	0
$934$	−1.93817e7	−0.726984
$935$	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
$939$	0	0
$940$	8.50275e6	0.313863
$941$	−1.88601e7	−0.694336	−0.347168	−	0.937803i	$-0.612857\pi$
$941$	−1.88601e7	−0.694336	−0.347168	+	0.937803i	$0.612857\pi$
$942$	0	0
$943$	−566410.	−0.0207420
$944$	−3.21380e6	−0.117379
$945$	0	0
$946$	150315.	0.00546104
$947$	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
$950$	−3.31498e6	−0.119171
$951$	0	0
$952$	0	0
$953$	−2.76898e7	−0.987616	−0.493808	−	0.869571i	$-0.664396\pi$
$953$	−2.76898e7	−0.987616	−0.493808	+	0.869571i	$0.664396\pi$
$954$	0	0
$955$	4.24127e7	1.50483
$956$	−2.76343e7	−0.977921
$957$	0	0
$958$	−1.52376e7	−0.536419
$959$	0	0
$960$	0	0
$961$	4.85298e7	1.69512
$962$	7.87585e6	0.274385
$963$	0	0
$964$	−1.95662e7	−0.678133
$965$	2.32304e7	0.803042
$966$	0	0
$967$	−2.44768e7	−0.841761	−0.420881	−	0.907116i	$-0.638279\pi$
$967$	−2.44768e7	−0.841761	−0.420881	+	0.907116i	$0.638279\pi$
$968$	2.26383e7	0.776526
$969$	0	0
$970$	−402749.	−0.0137437
$971$	−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
$974$	8.47457e6	0.286233
$975$	0	0
$976$	−1.25582e7	−0.421990
$977$	−4.69012e7	−1.57198	−0.785991	−	0.618238i	$-0.787847\pi$
$977$	−4.69012e7	−1.57198	−0.785991	+	0.618238i	$0.787847\pi$
$978$	0	0
$979$	−2.95785e7	−0.986325
$980$	0	0
$981$	0	0
$982$	−2.03951e7	−0.674914
$983$	−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
$986$	1.10650e7	0.362458
$987$	0	0
$988$	3.07167e7	1.00111
$989$	4295.21	0.000139635 0
$990$	0	0
$991$	−2.64104e7	−0.854261	−0.427130	−	0.904190i	$-0.640475\pi$
$991$	−2.64104e7	−0.854261	−0.427130	+	0.904190i	$0.640475\pi$
$992$	−5.24336e7	−1.69173
$993$	0	0
$994$	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	$0.600633\pi$
$998$	1.44354e7	0.458777
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.z.1.2		4
3.2	odd	2		49.6.a.g.1.4	yes	4
7.6	odd	2	inner	441.6.a.z.1.1		4
12.11	even	2		784.6.a.bf.1.2		4
21.2	odd	6		49.6.c.h.18.1		8
21.5	even	6		49.6.c.h.18.2		8
21.11	odd	6		49.6.c.h.30.1		8
21.17	even	6		49.6.c.h.30.2		8
21.20	even	2		49.6.a.g.1.3	✓	4
84.83	odd	2		784.6.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.g.1.3	✓	4	21.20	even	2
49.6.a.g.1.4	yes	4	3.2	odd	2
49.6.c.h.18.1		8	21.2	odd	6
49.6.c.h.18.2		8	21.5	even	6
49.6.c.h.30.1		8	21.11	odd	6
49.6.c.h.30.2		8	21.17	even	6
441.6.a.z.1.1		4	7.6	odd	2	inner
441.6.a.z.1.2		4	1.1	even	1	trivial
784.6.a.bf.1.2		4	12.11	even	2
784.6.a.bf.1.3		4	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-2.81507 q^{2} -24.0754 q^{4} +45.9910 q^{5} +157.856 q^{8} -129.468 q^{10} +551.781 q^{11} +1094.10 q^{13} +326.035 q^{16} +1180.71 q^{17} -1166.13 q^{19} -1107.25 q^{20} -1553.30 q^{22} -44.3851 q^{23} -1009.82 q^{25} -3079.97 q^{26} -3329.02 q^{29} +8784.01 q^{31} -5969.21 q^{32} -3323.80 q^{34} -2557.12 q^{37} +3282.73 q^{38} +7259.97 q^{40} +12761.3 q^{41} -96.7714 q^{43} -13284.3 q^{44} +124.947 q^{46} -7679.15 q^{47} +2842.73 q^{50} -26340.8 q^{52} +11953.3 q^{53} +25377.0 q^{55} +9371.43 q^{58} -9857.24 q^{59} -38517.9 q^{61} -24727.6 q^{62} +6370.65 q^{64} +50318.7 q^{65} -67548.9 q^{67} -28426.1 q^{68} +61374.6 q^{71} +1850.40 q^{73} +7198.49 q^{74} +28074.9 q^{76} -8.52913 q^{79} +14994.7 q^{80} -35923.9 q^{82} +95039.3 q^{83} +54302.3 q^{85} +272.419 q^{86} +87102.1 q^{88} -53605.6 q^{89} +1068.59 q^{92} +21617.4 q^{94} -53631.4 q^{95} +3110.79 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50}+ \cdots - 108224 q^{95}+O(q^{100}) \) 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 + 1952 * q^11 - 1566 * q^16 + 3524 * q^22 + 7136 * q^23 + 2764 * q^25 + 3352 * q^29 - 27810 * q^32 - 9208 * q^37 + 20448 * q^43 - 1900 * q^44 + 56712 * q^46 + 43070 * q^50 + 102920 * q^53 + 96972 * q^58 - 40318 * q^64 + 63168 * q^65 - 22896 * q^67 + 153824 * q^71 - 17596 * q^74 - 90688 * q^79 + 272656 * q^85 + 161860 * q^86 + 154812 * q^88 + 212200 * q^92 - 108224 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more