LMFDB - Embedded newform 441.6.a.w.1.1

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:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 97x^{2} + 7x + 294 $ x^4 - x^3 - 97x^2 + 7x + 294
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-9.22385$ 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)

$f(q)$	$=$	$q-10.2239 q^{2} +72.5272 q^{4} -23.7528 q^{5} -414.344 q^{8} +O(q^{10})$	$q-10.2239 q^{2} +72.5272 q^{4} -23.7528 q^{5} -414.344 q^{8} +242.845 q^{10} -465.526 q^{11} -1019.30 q^{13} +1915.32 q^{16} -561.757 q^{17} -1387.58 q^{19} -1722.72 q^{20} +4759.47 q^{22} -4113.62 q^{23} -2560.80 q^{25} +10421.2 q^{26} +2381.37 q^{29} +2950.66 q^{31} -6322.95 q^{32} +5743.32 q^{34} -9908.95 q^{37} +14186.4 q^{38} +9841.83 q^{40} +4477.13 q^{41} +5181.48 q^{43} -33763.3 q^{44} +42057.0 q^{46} +3121.59 q^{47} +26181.3 q^{50} -73926.9 q^{52} -1141.00 q^{53} +11057.6 q^{55} -24346.8 q^{58} -27497.1 q^{59} -21103.5 q^{61} -30167.1 q^{62} +3354.67 q^{64} +24211.2 q^{65} -55588.4 q^{67} -40742.6 q^{68} +6076.90 q^{71} -16779.6 q^{73} +101308. q^{74} -100637. q^{76} -4845.26 q^{79} -45494.2 q^{80} -45773.5 q^{82} -60145.4 q^{83} +13343.3 q^{85} -52974.7 q^{86} +192888. q^{88} +62497.4 q^{89} -298349. q^{92} -31914.7 q^{94} +32958.9 q^{95} -63653.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8	$ 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97}+O(q^{100}) $ 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8 + 283 * q^10 - 402 * q^11 + 462 * q^13 + 3273 * q^16 - 276 * q^17 + 510 * q^19 - 4719 * q^20 + 1375 * q^22 - 6900 * q^23 + 2814 * q^25 + 15138 * q^26 - 540 * q^29 - 6410 * q^31 - 15519 * q^32 + 21144 * q^34 + 15250 * q^37 + 41250 * q^38 - 8547 * q^40 - 4308 * q^41 + 29198 * q^43 - 70743 * q^44 + 61800 * q^46 + 15060 * q^47 + 7302 * q^50 - 47476 * q^52 - 13692 * q^53 + 73124 * q^55 + 52309 * q^58 - 34830 * q^59 - 5364 * q^61 - 16029 * q^62 - 73487 * q^64 - 66864 * q^65 - 5994 * q^67 + 58272 * q^68 - 89268 * q^71 + 59638 * q^73 + 185442 * q^74 + 21308 * q^76 - 44062 * q^79 + 33381 * q^80 + 57596 * q^82 + 208446 * q^83 + 36324 * q^85 + 136968 * q^86 + 87597 * q^88 + 77520 * q^89 - 158256 * q^92 - 73722 * q^94 + 221376 * q^95 - 188630 * q^97

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 $

$2$	−10.2239	−1.80734	−0.903669	−	0.428231i	$-0.859137\pi$
$2$	−10.2239	−1.80734	−0.903669	+	0.428231i	$0.859137\pi$
$3$	0	0
$3$	0	0
$4$	72.5272	2.26647
$5$	−23.7528	−0.424903	−0.212452	−	0.977172i	$-0.568145\pi$
$5$	−23.7528	−0.424903	−0.212452	+	0.977172i	$0.568145\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−414.344	−2.28895
$9$	0	0
$10$	242.845	0.767944
$11$	−465.526	−1.16001	−0.580006	−	0.814612i	$-0.696950\pi$
$11$	−465.526	−1.16001	−0.580006	+	0.814612i	$0.696950\pi$
$12$	0	0
$13$	−1019.30	−1.67280	−0.836399	−	0.548121i	$-0.815343\pi$
$13$	−1019.30	−1.67280	−0.836399	+	0.548121i	$0.815343\pi$
$14$	0	0
$15$	0	0
$16$	1915.32	1.87043
$17$	−561.757	−0.471440	−0.235720	−	0.971821i	$-0.575745\pi$
$17$	−561.757	−0.471440	−0.235720	+	0.971821i	$0.575745\pi$
$18$	0	0
$19$	−1387.58	−0.881807	−0.440903	−	0.897555i	$-0.645342\pi$
$19$	−1387.58	−0.881807	−0.440903	+	0.897555i	$0.645342\pi$
$20$	−1722.72	−0.963032
$21$	0	0
$22$	4759.47	2.09653
$23$	−4113.62	−1.62145	−0.810727	−	0.585425i	$-0.800928\pi$
$23$	−4113.62	−1.62145	−0.810727	+	0.585425i	$0.800928\pi$
$24$	0	0
$25$	−2560.80	−0.819457
$26$	10421.2	3.02331
$27$	0	0
$28$	0	0
$29$	2381.37	0.525814	0.262907	−	0.964821i	$-0.415319\pi$
$29$	2381.37	0.525814	0.262907	+	0.964821i	$0.415319\pi$
$30$	0	0
$31$	2950.66	0.551460	0.275730	−	0.961235i	$-0.411080\pi$
$31$	2950.66	0.551460	0.275730	+	0.961235i	$0.411080\pi$
$32$	−6322.95	−1.09155
$33$	0	0
$34$	5743.32	0.852051
$35$	0	0
$36$	0	0
$37$	−9908.95	−1.18994	−0.594968	−	0.803750i	$-0.702835\pi$
$37$	−9908.95	−1.18994	−0.594968	+	0.803750i	$0.702835\pi$
$38$	14186.4	1.59372
$39$	0	0
$40$	9841.83	0.972581
$41$	4477.13	0.415949	0.207974	−	0.978134i	$-0.433313\pi$
$41$	4477.13	0.415949	0.207974	+	0.978134i	$0.433313\pi$
$42$	0	0
$43$	5181.48	0.427349	0.213675	−	0.976905i	$-0.431457\pi$
$43$	5181.48	0.427349	0.213675	+	0.976905i	$0.431457\pi$
$44$	−33763.3	−2.62914
$45$	0	0
$46$	42057.0	2.93052
$47$	3121.59	0.206125	0.103063	−	0.994675i	$-0.467136\pi$
$47$	3121.59	0.206125	0.103063	+	0.994675i	$0.467136\pi$
$48$	0	0
$49$	0	0
$50$	26181.3	1.48104
$51$	0	0
$52$	−73926.9	−3.79135
$53$	−1141.00	−0.0557950	−0.0278975	−	0.999611i	$-0.508881\pi$
$53$	−1141.00	−0.0557950	−0.0278975	+	0.999611i	$0.508881\pi$
$54$	0	0
$55$	11057.6	0.492893
$56$	0	0
$57$	0	0
$58$	−24346.8	−0.950325
$59$	−27497.1	−1.02839	−0.514194	−	0.857674i	$-0.671909\pi$
$59$	−27497.1	−1.02839	−0.514194	+	0.857674i	$0.671909\pi$
$60$	0	0
$61$	−21103.5	−0.726157	−0.363078	−	0.931759i	$-0.618274\pi$
$61$	−21103.5	−0.726157	−0.363078	+	0.931759i	$0.618274\pi$
$62$	−30167.1	−0.996676
$63$	0	0
$64$	3354.67	0.102376
$65$	24211.2	0.710778
$66$	0	0
$67$	−55588.4	−1.51286	−0.756428	−	0.654077i	$-0.773057\pi$
$67$	−55588.4	−1.51286	−0.756428	+	0.654077i	$0.773057\pi$
$68$	−40742.6	−1.06851
$69$	0	0
$70$	0	0
$71$	6076.90	0.143066	0.0715330	−	0.997438i	$-0.477211\pi$
$71$	6076.90	0.143066	0.0715330	+	0.997438i	$0.477211\pi$
$72$	0	0
$73$	−16779.6	−0.368532	−0.184266	−	0.982876i	$-0.558991\pi$
$73$	−16779.6	−0.368532	−0.184266	+	0.982876i	$0.558991\pi$
$74$	101308.	2.15062
$75$	0	0
$76$	−100637.	−1.99859
$77$	0	0
$78$	0	0
$79$	−4845.26	−0.0873473	−0.0436737	−	0.999046i	$-0.513906\pi$
$79$	−4845.26	−0.0873473	−0.0436737	+	0.999046i	$0.513906\pi$
$80$	−45494.2	−0.794752
$81$	0	0
$82$	−45773.5	−0.751760
$83$	−60145.4	−0.958313	−0.479156	−	0.877730i	$-0.659057\pi$
$83$	−60145.4	−0.958313	−0.479156	+	0.877730i	$0.659057\pi$
$84$	0	0
$85$	13343.3	0.200316
$86$	−52974.7	−0.772365
$87$	0	0
$88$	192888.	2.65520
$89$	62497.4	0.836348	0.418174	−	0.908367i	$-0.362670\pi$
$89$	62497.4	0.836348	0.418174	+	0.908367i	$0.362670\pi$
$90$	0	0
$91$	0	0
$92$	−298349.	−3.67498
$93$	0	0
$94$	−31914.7	−0.372539
$95$	32958.9	0.374683
$96$	0	0
$97$	−63653.8	−0.686903	−0.343451	−	0.939170i	$-0.611596\pi$
$97$	−63653.8	−0.686903	−0.343451	+	0.939170i	$0.611596\pi$
$98$	0	0
$99$	0	0
$100$	−185728.	−1.85728
$101$	−184623.	−1.80087	−0.900434	−	0.434993i	$-0.856751\pi$
$101$	−184623.	−1.80087	−0.900434	+	0.434993i	$0.856751\pi$
$102$	0	0
$103$	52043.7	0.483365	0.241683	−	0.970355i	$-0.422301\pi$
$103$	52043.7	0.483365	0.241683	+	0.970355i	$0.422301\pi$
$104$	422341.	3.82895
$105$	0	0
$106$	11665.4	0.100840
$107$	−48177.8	−0.406806	−0.203403	−	0.979095i	$-0.565200\pi$
$107$	−48177.8	−0.406806	−0.203403	+	0.979095i	$0.565200\pi$
$108$	0	0
$109$	−36435.6	−0.293737	−0.146869	−	0.989156i	$-0.546919\pi$
$109$	−36435.6	−0.293737	−0.146869	+	0.989156i	$0.546919\pi$
$110$	−113051.	−0.890824
$111$	0	0
$112$	0	0
$113$	96711.1	0.712492	0.356246	−	0.934392i	$-0.384056\pi$
$113$	96711.1	0.712492	0.356246	+	0.934392i	$0.384056\pi$
$114$	0	0
$115$	97710.0	0.688961
$116$	172714.	1.19174
$117$	0	0
$118$	281126.	1.85864
$119$	0	0
$120$	0	0
$121$	55663.5	0.345626
$122$	215759.	1.31241
$123$	0	0
$124$	214003.	1.24987
$125$	135054.	0.773093
$126$	0	0
$127$	23322.9	0.128314	0.0641568	−	0.997940i	$-0.479564\pi$
$127$	23322.9	0.128314	0.0641568	+	0.997940i	$0.479564\pi$
$128$	168037.	0.906524
$129$	0	0
$130$	−247532.	−1.28462
$131$	338758.	1.72469	0.862345	−	0.506321i	$-0.168995\pi$
$131$	338758.	1.72469	0.862345	+	0.506321i	$0.168995\pi$
$132$	0	0
$133$	0	0
$134$	568328.	2.73424
$135$	0	0
$136$	232760.	1.07910
$137$	62876.7	0.286213	0.143106	−	0.989707i	$-0.454291\pi$
$137$	62876.7	0.286213	0.143106	+	0.989707i	$0.454291\pi$
$138$	0	0
$139$	211927.	0.930356	0.465178	−	0.885217i	$-0.345990\pi$
$139$	211927.	0.930356	0.465178	+	0.885217i	$0.345990\pi$
$140$	0	0
$141$	0	0
$142$	−62129.4	−0.258569
$143$	474511.	1.94047
$144$	0	0
$145$	−56564.3	−0.223420
$146$	171553.	0.666062
$147$	0	0
$148$	−718668.	−2.69696
$149$	−140273.	−0.517616	−0.258808	−	0.965929i	$-0.583330\pi$
$149$	−140273.	−0.517616	−0.258808	+	0.965929i	$0.583330\pi$
$150$	0	0
$151$	−163991.	−0.585300	−0.292650	−	0.956220i	$-0.594537\pi$
$151$	−163991.	−0.585300	−0.292650	+	0.956220i	$0.594537\pi$
$152$	574934.	2.01841
$153$	0	0
$154$	0	0
$155$	−70086.4	−0.234317
$156$	0	0
$157$	556543.	1.80198	0.900990	−	0.433840i	$-0.142842\pi$
$157$	556543.	1.80198	0.900990	+	0.433840i	$0.142842\pi$
$158$	49537.2	0.157866
$159$	0	0
$160$	150188.	0.463804
$161$	0	0
$162$	0	0
$163$	−19726.6	−0.0581546	−0.0290773	−	0.999577i	$-0.509257\pi$
$163$	−19726.6	−0.0581546	−0.0290773	+	0.999577i	$0.509257\pi$
$164$	324713.	0.942736
$165$	0	0
$166$	614918.	1.73200
$167$	−94776.2	−0.262971	−0.131486	−	0.991318i	$-0.541975\pi$
$167$	−94776.2	−0.262971	−0.131486	+	0.991318i	$0.541975\pi$
$168$	0	0
$169$	667679.	1.79825
$170$	−136420.	−0.362039
$171$	0	0
$172$	375798.	0.968576
$173$	−338841.	−0.860757	−0.430379	−	0.902649i	$-0.641620\pi$
$173$	−338841.	−0.860757	−0.430379	+	0.902649i	$0.641620\pi$
$174$	0	0
$175$	0	0
$176$	−891631.	−2.16972
$177$	0	0
$178$	−638965.	−1.51156
$179$	−776193.	−1.81066	−0.905330	−	0.424708i	$-0.860377\pi$
$179$	−776193.	−1.81066	−0.905330	+	0.424708i	$0.860377\pi$
$180$	0	0
$181$	−132697.	−0.301067	−0.150534	−	0.988605i	$-0.548099\pi$
$181$	−132697.	−0.301067	−0.150534	+	0.988605i	$0.548099\pi$
$182$	0	0
$183$	0	0
$184$	1.70445e6	3.71142
$185$	235366.	0.505607
$186$	0	0
$187$	261512.	0.546875
$188$	226400.	0.467178
$189$	0	0
$190$	−336967.	−0.677178
$191$	−6637.41	−0.0131648	−0.00658242	−	0.999978i	$-0.502095\pi$
$191$	−6637.41	−0.0131648	−0.00658242	+	0.999978i	$0.502095\pi$
$192$	0	0
$193$	452590.	0.874604	0.437302	−	0.899315i	$-0.355934\pi$
$193$	452590.	0.874604	0.437302	+	0.899315i	$0.355934\pi$
$194$	650787.	1.24147
$195$	0	0
$196$	0	0
$197$	−816952.	−1.49979	−0.749896	−	0.661556i	$-0.769896\pi$
$197$	−816952.	−1.49979	−0.749896	+	0.661556i	$0.769896\pi$
$198$	0	0
$199$	806417.	1.44353	0.721767	−	0.692136i	$-0.243330\pi$
$199$	806417.	1.44353	0.721767	+	0.692136i	$0.243330\pi$
$200$	1.06105e6	1.87569
$201$	0	0
$202$	1.88756e6	3.25478
$203$	0	0
$204$	0	0
$205$	−106344.	−0.176738
$206$	−532087.	−0.873605
$207$	0	0
$208$	−1.95229e6	−3.12885
$209$	645954.	1.02291
$210$	0	0
$211$	68773.9	0.106345	0.0531726	−	0.998585i	$-0.483067\pi$
$211$	68773.9	0.106345	0.0531726	+	0.998585i	$0.483067\pi$
$212$	−82753.3	−0.126458
$213$	0	0
$214$	492563.	0.735237
$215$	−123075.	−0.181582
$216$	0	0
$217$	0	0
$218$	372512.	0.530883
$219$	0	0
$220$	801973.	1.11713
$221$	572599.	0.788623
$222$	0	0
$223$	−620227.	−0.835196	−0.417598	−	0.908632i	$-0.637128\pi$
$223$	−620227.	−0.835196	−0.417598	+	0.908632i	$0.637128\pi$
$224$	0	0
$225$	0	0
$226$	−988760.	−1.28771
$227$	−1.00223e6	−1.29093	−0.645467	−	0.763788i	$-0.723337\pi$
$227$	−1.00223e6	−1.29093	−0.645467	+	0.763788i	$0.723337\pi$
$228$	0	0
$229$	885861.	1.11629	0.558145	−	0.829743i	$-0.311513\pi$
$229$	885861.	1.11629	0.558145	+	0.829743i	$0.311513\pi$
$230$	−998973.	−1.24519
$231$	0	0
$232$	−986707.	−1.20356
$233$	−596163.	−0.719408	−0.359704	−	0.933066i	$-0.617122\pi$
$233$	−596163.	−0.719408	−0.359704	+	0.933066i	$0.617122\pi$
$234$	0	0
$235$	−74146.6	−0.0875834
$236$	−1.99429e6	−2.33081
$237$	0	0
$238$	0	0
$239$	−743111.	−0.841509	−0.420754	−	0.907175i	$-0.638235\pi$
$239$	−743111.	−0.841509	−0.420754	+	0.907175i	$0.638235\pi$
$240$	0	0
$241$	−1.17484e6	−1.30297	−0.651487	−	0.758660i	$-0.725854\pi$
$241$	−1.17484e6	−1.30297	−0.651487	+	0.758660i	$0.725854\pi$
$242$	−569095.	−0.624664
$243$	0	0
$244$	−1.53058e6	−1.64582
$245$	0	0
$246$	0	0
$247$	1.41436e6	1.47508
$248$	−1.22259e6	−1.26226
$249$	0	0
$250$	−1.38077e6	−1.39724
$251$	−352992.	−0.353655	−0.176828	−	0.984242i	$-0.556584\pi$
$251$	−352992.	−0.353655	−0.176828	+	0.984242i	$0.556584\pi$
$252$	0	0
$253$	1.91500e6	1.88090
$254$	−238450.	−0.231906
$255$	0	0
$256$	−1.82533e6	−1.74077
$257$	−10012.7	−0.00945621	−0.00472811	−	0.999989i	$-0.501505\pi$
$257$	−10012.7	−0.00945621	−0.00472811	+	0.999989i	$0.501505\pi$
$258$	0	0
$259$	0	0
$260$	1.75597e6	1.61096
$261$	0	0
$262$	−3.46341e6	−3.11710
$263$	−1.68180e6	−1.49929	−0.749644	−	0.661841i	$-0.769775\pi$
$263$	−1.68180e6	−1.49929	−0.749644	+	0.661841i	$0.769775\pi$
$264$	0	0
$265$	27101.9	0.0237075
$266$	0	0
$267$	0	0
$268$	−4.03167e6	−3.42885
$269$	1.91717e6	1.61540	0.807702	−	0.589591i	$-0.200711\pi$
$269$	1.91717e6	1.61540	0.807702	+	0.589591i	$0.200711\pi$
$270$	0	0
$271$	2.29000e6	1.89414	0.947069	−	0.321029i	$-0.104029\pi$
$271$	2.29000e6	1.89414	0.947069	+	0.321029i	$0.104029\pi$
$272$	−1.07594e6	−0.881795
$273$	0	0
$274$	−642843.	−0.517283
$275$	1.19212e6	0.950580
$276$	0	0
$277$	−394802.	−0.309157	−0.154579	−	0.987980i	$-0.549402\pi$
$277$	−394802.	−0.309157	−0.154579	+	0.987980i	$0.549402\pi$
$278$	−2.16671e6	−1.68147
$279$	0	0
$280$	0	0
$281$	1.77699e6	1.34252	0.671259	−	0.741223i	$-0.265754\pi$
$281$	1.77699e6	1.34252	0.671259	+	0.741223i	$0.265754\pi$
$282$	0	0
$283$	1.21524e6	0.901981	0.450991	−	0.892529i	$-0.351071\pi$
$283$	1.21524e6	0.901981	0.450991	+	0.892529i	$0.351071\pi$
$284$	440741.	0.324255
$285$	0	0
$286$	−4.85133e6	−3.50708
$287$	0	0
$288$	0	0
$289$	−1.10429e6	−0.777745
$290$	578305.	0.403796
$291$	0	0
$292$	−1.21698e6	−0.835268
$293$	1.48897e6	1.01325	0.506627	−	0.862165i	$-0.330892\pi$
$293$	1.48897e6	1.01325	0.506627	+	0.862165i	$0.330892\pi$
$294$	0	0
$295$	653133.	0.436965
$296$	4.10571e6	2.72370
$297$	0	0
$298$	1.43413e6	0.935508
$299$	4.19301e6	2.71236
$300$	0	0
$301$	0	0
$302$	1.67662e6	1.05784
$303$	0	0
$304$	−2.65766e6	−1.64936
$305$	501268.	0.308546
$306$	0	0
$307$	2.03109e6	1.22994	0.614968	−	0.788552i	$-0.289169\pi$
$307$	2.03109e6	1.22994	0.614968	+	0.788552i	$0.289169\pi$
$308$	0	0
$309$	0	0
$310$	716553.	0.423491
$311$	−289785.	−0.169893	−0.0849463	−	0.996386i	$-0.527072\pi$
$311$	−289785.	−0.169893	−0.0849463	+	0.996386i	$0.527072\pi$
$312$	0	0
$313$	218211.	0.125897	0.0629484	−	0.998017i	$-0.479950\pi$
$313$	218211.	0.125897	0.0629484	+	0.998017i	$0.479950\pi$
$314$	−5.69002e6	−3.25679
$315$	0	0
$316$	−351413.	−0.197970
$317$	−1.29063e6	−0.721363	−0.360681	−	0.932689i	$-0.617456\pi$
$317$	−1.29063e6	−0.721363	−0.360681	+	0.932689i	$0.617456\pi$
$318$	0	0
$319$	−1.10859e6	−0.609951
$320$	−79682.8	−0.0435000
$321$	0	0
$322$	0	0
$323$	779481.	0.415719
$324$	0	0
$325$	2.61023e6	1.37079
$326$	201682.	0.105105
$327$	0	0
$328$	−1.85507e6	−0.952084
$329$	0	0
$330$	0	0
$331$	−3.48561e6	−1.74867	−0.874336	−	0.485321i	$-0.838703\pi$
$331$	−3.48561e6	−1.74867	−0.874336	+	0.485321i	$0.838703\pi$
$332$	−4.36218e6	−2.17199
$333$	0	0
$334$	968978.	0.475278
$335$	1.32038e6	0.642817
$336$	0	0
$337$	249198.	0.119528	0.0597641	−	0.998213i	$-0.480965\pi$
$337$	249198.	0.119528	0.0597641	+	0.998213i	$0.480965\pi$
$338$	−6.82625e6	−3.25005
$339$	0	0
$340$	967752.	0.454012
$341$	−1.37361e6	−0.639700
$342$	0	0
$343$	0	0
$344$	−2.14692e6	−0.978180
$345$	0	0
$346$	3.46426e6	1.55568
$347$	1.50601e6	0.671437	0.335719	−	0.941962i	$-0.391021\pi$
$347$	1.50601e6	0.671437	0.335719	+	0.941962i	$0.391021\pi$
$348$	0	0
$349$	1.54370e6	0.678423	0.339212	−	0.940710i	$-0.389840\pi$
$349$	1.54370e6	0.678423	0.339212	+	0.940710i	$0.389840\pi$
$350$	0	0
$351$	0	0
$352$	2.94350e6	1.26621
$353$	1.67796e6	0.716710	0.358355	−	0.933585i	$-0.383338\pi$
$353$	1.67796e6	0.716710	0.358355	+	0.933585i	$0.383338\pi$
$354$	0	0
$355$	−144344.	−0.0607892
$356$	4.53276e6	1.89556
$357$	0	0
$358$	7.93568e6	3.27248
$359$	2.49973e6	1.02366	0.511832	−	0.859086i	$-0.328967\pi$
$359$	2.49973e6	1.02366	0.511832	+	0.859086i	$0.328967\pi$
$360$	0	0
$361$	−550727.	−0.222417
$362$	1.35667e6	0.544130
$363$	0	0
$364$	0	0
$365$	398564.	0.156591
$366$	0	0
$367$	−2.10741e6	−0.816740	−0.408370	−	0.912816i	$-0.633903\pi$
$367$	−2.10741e6	−0.816740	−0.408370	+	0.912816i	$0.633903\pi$
$368$	−7.87890e6	−3.03281
$369$	0	0
$370$	−2.40634e6	−0.913804
$371$	0	0
$372$	0	0
$373$	2.73225e6	1.01683	0.508414	−	0.861113i	$-0.330232\pi$
$373$	2.73225e6	1.01683	0.508414	+	0.861113i	$0.330232\pi$
$374$	−2.67366e6	−0.988389
$375$	0	0
$376$	−1.29341e6	−0.471810
$377$	−2.42733e6	−0.879581
$378$	0	0
$379$	−2.04295e6	−0.730567	−0.365283	−	0.930896i	$-0.619028\pi$
$379$	−2.04295e6	−0.730567	−0.365283	+	0.930896i	$0.619028\pi$
$380$	2.39041e6	0.849208
$381$	0	0
$382$	67859.9	0.0237933
$383$	−1.50899e6	−0.525643	−0.262821	−	0.964845i	$-0.584653\pi$
$383$	−1.50899e6	−0.525643	−0.262821	+	0.964845i	$0.584653\pi$
$384$	0	0
$385$	0	0
$386$	−4.62721e6	−1.58071
$387$	0	0
$388$	−4.61663e6	−1.55685
$389$	−4.50065e6	−1.50800	−0.754000	−	0.656874i	$-0.771878\pi$
$389$	−4.50065e6	−1.50800	−0.754000	+	0.656874i	$0.771878\pi$
$390$	0	0
$391$	2.31085e6	0.764417
$392$	0	0
$393$	0	0
$394$	8.35240e6	2.71063
$395$	115089.	0.0371142
$396$	0	0
$397$	−3.52136e6	−1.12133	−0.560665	−	0.828042i	$-0.689455\pi$
$397$	−3.52136e6	−1.12133	−0.560665	+	0.828042i	$0.689455\pi$
$398$	−8.24469e6	−2.60895
$399$	0	0
$400$	−4.90476e6	−1.53274
$401$	907107.	0.281707	0.140853	−	0.990030i	$-0.455015\pi$
$401$	907107.	0.281707	0.140853	+	0.990030i	$0.455015\pi$
$402$	0	0
$403$	−3.00760e6	−0.922482
$404$	−1.33902e7	−4.08162
$405$	0	0
$406$	0	0
$407$	4.61287e6	1.38034
$408$	0	0
$409$	4.31853e6	1.27652	0.638260	−	0.769821i	$-0.279654\pi$
$409$	4.31853e6	1.27652	0.638260	+	0.769821i	$0.279654\pi$
$410$	1.08725e6	0.319425
$411$	0	0
$412$	3.77458e6	1.09553
$413$	0	0
$414$	0	0
$415$	1.42862e6	0.407190
$416$	6.44498e6	1.82595
$417$	0	0
$418$	−6.60413e6	−1.84874
$419$	1.52041e6	0.423083	0.211541	−	0.977369i	$-0.432152\pi$
$419$	1.52041e6	0.423083	0.211541	+	0.977369i	$0.432152\pi$
$420$	0	0
$421$	−4.42050e6	−1.21553	−0.607766	−	0.794116i	$-0.707934\pi$
$421$	−4.42050e6	−1.21553	−0.607766	+	0.794116i	$0.707934\pi$
$422$	−703134.	−0.192202
$423$	0	0
$424$	472765.	0.127712
$425$	1.43855e6	0.386325
$426$	0	0
$427$	0	0
$428$	−3.49420e6	−0.922016
$429$	0	0
$430$	1.25830e6	0.328180
$431$	6.84785e6	1.77567	0.887833	−	0.460166i	$-0.152210\pi$
$431$	6.84785e6	1.77567	0.887833	+	0.460166i	$0.152210\pi$
$432$	0	0
$433$	−3.99328e6	−1.02355	−0.511777	−	0.859119i	$-0.671012\pi$
$433$	−3.99328e6	−1.02355	−0.511777	+	0.859119i	$0.671012\pi$
$434$	0	0
$435$	0	0
$436$	−2.64257e6	−0.665748
$437$	5.70797e6	1.42981
$438$	0	0
$439$	1.48557e6	0.367901	0.183950	−	0.982936i	$-0.441111\pi$
$439$	1.48557e6	0.367901	0.183950	+	0.982936i	$0.441111\pi$
$440$	−4.58163e6	−1.12821
$441$	0	0
$442$	−5.85416e6	−1.42531
$443$	5.62193e6	1.36106	0.680528	−	0.732722i	$-0.261750\pi$
$443$	5.62193e6	1.36106	0.680528	+	0.732722i	$0.261750\pi$
$444$	0	0
$445$	−1.48449e6	−0.355367
$446$	6.34111e6	1.50948
$447$	0	0
$448$	0	0
$449$	−1.94883e6	−0.456202	−0.228101	−	0.973637i	$-0.573252\pi$
$449$	−1.94883e6	−0.456202	−0.228101	+	0.973637i	$0.573252\pi$
$450$	0	0
$451$	−2.08422e6	−0.482505
$452$	7.01418e6	1.61484
$453$	0	0
$454$	1.02467e7	2.33315
$455$	0	0
$456$	0	0
$457$	−2.67312e6	−0.598726	−0.299363	−	0.954139i	$-0.596774\pi$
$457$	−2.67312e6	−0.598726	−0.299363	+	0.954139i	$0.596774\pi$
$458$	−9.05691e6	−2.01751
$459$	0	0
$460$	7.08663e6	1.56151
$461$	4.65262e6	1.01964	0.509819	−	0.860282i	$-0.329713\pi$
$461$	4.65262e6	1.01964	0.509819	+	0.860282i	$0.329713\pi$
$462$	0	0
$463$	−5.18586e6	−1.12426	−0.562132	−	0.827047i	$-0.690019\pi$
$463$	−5.18586e6	−1.12426	−0.562132	+	0.827047i	$0.690019\pi$
$464$	4.56109e6	0.983499
$465$	0	0
$466$	6.09509e6	1.30021
$467$	5.05980e6	1.07360	0.536799	−	0.843710i	$-0.319633\pi$
$467$	5.05980e6	1.07360	0.536799	+	0.843710i	$0.319633\pi$
$468$	0	0
$469$	0	0
$470$	758064.	0.158293
$471$	0	0
$472$	1.13932e7	2.35392
$473$	−2.41212e6	−0.495730
$474$	0	0
$475$	3.55331e6	0.722603
$476$	0	0
$477$	0	0
$478$	7.59745e6	1.52089
$479$	5.55085e6	1.10540	0.552702	−	0.833379i	$-0.313597\pi$
$479$	5.55085e6	1.10540	0.552702	+	0.833379i	$0.313597\pi$
$480$	0	0
$481$	1.01002e7	1.99052
$482$	1.20114e7	2.35491
$483$	0	0
$484$	4.03711e6	0.783353
$485$	1.51196e6	0.291867
$486$	0	0
$487$	−6.87810e6	−1.31415	−0.657077	−	0.753824i	$-0.728207\pi$
$487$	−6.87810e6	−1.31415	−0.657077	+	0.753824i	$0.728207\pi$
$488$	8.74411e6	1.66213
$489$	0	0
$490$	0	0
$491$	−7.44459e6	−1.39360	−0.696798	−	0.717267i	$-0.745393\pi$
$491$	−7.44459e6	−1.39360	−0.696798	+	0.717267i	$0.745393\pi$
$492$	0	0
$493$	−1.33775e6	−0.247890
$494$	−1.44602e7	−2.66598
$495$	0	0
$496$	5.65145e6	1.03147
$497$	0	0
$498$	0	0
$499$	2.15329e6	0.387126	0.193563	−	0.981088i	$-0.437996\pi$
$499$	2.15329e6	0.387126	0.193563	+	0.981088i	$0.437996\pi$
$500$	9.79507e6	1.75220
$501$	0	0
$502$	3.60894e6	0.639175
$503$	−8.61012e6	−1.51736	−0.758681	−	0.651463i	$-0.774156\pi$
$503$	−8.61012e6	−1.51736	−0.758681	+	0.651463i	$0.774156\pi$
$504$	0	0
$505$	4.38531e6	0.765195
$506$	−1.95786e7	−3.39943
$507$	0	0
$508$	1.69154e6	0.290820
$509$	−3.34201e6	−0.571759	−0.285879	−	0.958266i	$-0.592286\pi$
$509$	−3.34201e6	−0.571759	−0.285879	+	0.958266i	$0.592286\pi$
$510$	0	0
$511$	0	0
$512$	1.32848e7	2.23964
$513$	0	0
$514$	102368.	0.0170906
$515$	−1.23619e6	−0.205383
$516$	0	0
$517$	−1.45318e6	−0.239108
$518$	0	0
$519$	0	0
$520$	−1.00318e7	−1.62693
$521$	2.68305e6	0.433046	0.216523	−	0.976278i	$-0.430528\pi$
$521$	2.68305e6	0.433046	0.216523	+	0.976278i	$0.430528\pi$
$522$	0	0
$523$	−5.42355e6	−0.867021	−0.433511	−	0.901148i	$-0.642725\pi$
$523$	−5.42355e6	−0.867021	−0.433511	+	0.901148i	$0.642725\pi$
$524$	2.45692e7	3.90897
$525$	0	0
$526$	1.71945e7	2.70972
$527$	−1.65755e6	−0.259980
$528$	0	0
$529$	1.04855e7	1.62911
$530$	−277086.	−0.0428474
$531$	0	0
$532$	0	0
$533$	−4.56353e6	−0.695798
$534$	0	0
$535$	1.14436e6	0.172853
$536$	2.30327e7	3.46285
$537$	0	0
$538$	−1.96009e7	−2.91958
$539$	0	0
$540$	0	0
$541$	2.18456e6	0.320900	0.160450	−	0.987044i	$-0.448705\pi$
$541$	2.18456e6	0.320900	0.160450	+	0.987044i	$0.448705\pi$
$542$	−2.34126e7	−3.42335
$543$	0	0
$544$	3.55196e6	0.514601
$545$	865447.	0.124810
$546$	0	0
$547$	−691437.	−0.0988062	−0.0494031	−	0.998779i	$-0.515732\pi$
$547$	−691437.	−0.0988062	−0.0494031	+	0.998779i	$0.515732\pi$
$548$	4.56027e6	0.648693
$549$	0	0
$550$	−1.21881e7	−1.71802
$551$	−3.30434e6	−0.463667
$552$	0	0
$553$	0	0
$554$	4.03639e6	0.558752
$555$	0	0
$556$	1.53705e7	2.10863
$557$	−5.47656e6	−0.747946	−0.373973	−	0.927440i	$-0.622005\pi$
$557$	−5.47656e6	−0.747946	−0.373973	+	0.927440i	$0.622005\pi$
$558$	0	0
$559$	−5.28149e6	−0.714869
$560$	0	0
$561$	0	0
$562$	−1.81677e7	−2.42638
$563$	−398976.	−0.0530488	−0.0265244	−	0.999648i	$-0.508444\pi$
$563$	−398976.	−0.0530488	−0.0265244	+	0.999648i	$0.508444\pi$
$564$	0	0
$565$	−2.29716e6	−0.302740
$566$	−1.24245e7	−1.63019
$567$	0	0
$568$	−2.51793e6	−0.327471
$569$	4.23113e6	0.547868	0.273934	−	0.961749i	$-0.411675\pi$
$569$	4.23113e6	0.547868	0.273934	+	0.961749i	$0.411675\pi$
$570$	0	0
$571$	8.75102e6	1.12323	0.561615	−	0.827399i	$-0.310180\pi$
$571$	8.75102e6	1.12323	0.561615	+	0.827399i	$0.310180\pi$
$572$	3.44149e7	4.39801
$573$	0	0
$574$	0	0
$575$	1.05342e7	1.32871
$576$	0	0
$577$	−1.76712e6	−0.220966	−0.110483	−	0.993878i	$-0.535240\pi$
$577$	−1.76712e6	−0.220966	−0.110483	+	0.993878i	$0.535240\pi$
$578$	1.12901e7	1.40565
$579$	0	0
$580$	−4.10245e6	−0.506376
$581$	0	0
$582$	0	0
$583$	531164.	0.0647228
$584$	6.95254e6	0.843551
$585$	0	0
$586$	−1.52231e7	−1.83129
$587$	8.65009e6	1.03616	0.518078	−	0.855333i	$-0.326648\pi$
$587$	8.65009e6	1.03616	0.518078	+	0.855333i	$0.326648\pi$
$588$	0	0
$589$	−4.09426e6	−0.486281
$590$	−6.67754e6	−0.789744
$591$	0	0
$592$	−1.89788e7	−2.22569
$593$	−1.54514e7	−1.80439	−0.902194	−	0.431329i	$-0.858045\pi$
$593$	−1.54514e7	−1.80439	−0.902194	+	0.431329i	$0.858045\pi$
$594$	0	0
$595$	0	0
$596$	−1.01736e7	−1.17316
$597$	0	0
$598$	−4.28687e7	−4.90216
$599$	3.84801e6	0.438196	0.219098	−	0.975703i	$-0.429688\pi$
$599$	3.84801e6	0.438196	0.219098	+	0.975703i	$0.429688\pi$
$600$	0	0
$601$	1.27578e7	1.44075	0.720376	−	0.693583i	$-0.243969\pi$
$601$	1.27578e7	1.44075	0.720376	+	0.693583i	$0.243969\pi$
$602$	0	0
$603$	0	0
$604$	−1.18938e7	−1.32657
$605$	−1.32216e6	−0.146858
$606$	0	0
$607$	−1.36207e7	−1.50047	−0.750233	−	0.661173i	$-0.770059\pi$
$607$	−1.36207e7	−1.50047	−0.750233	+	0.661173i	$0.770059\pi$
$608$	8.77359e6	0.962539
$609$	0	0
$610$	−5.12489e6	−0.557648
$611$	−3.18184e6	−0.344806
$612$	0	0
$613$	1.53474e7	1.64962	0.824812	−	0.565407i	$-0.191281\pi$
$613$	1.53474e7	1.64962	0.824812	+	0.565407i	$0.191281\pi$
$614$	−2.07655e7	−2.22291
$615$	0	0
$616$	0	0
$617$	−1.18485e7	−1.25300	−0.626500	−	0.779421i	$-0.715513\pi$
$617$	−1.18485e7	−1.25300	−0.626500	+	0.779421i	$0.715513\pi$
$618$	0	0
$619$	−874460.	−0.0917304	−0.0458652	−	0.998948i	$-0.514604\pi$
$619$	−874460.	−0.0917304	−0.0458652	+	0.998948i	$0.514604\pi$
$620$	−5.08317e6	−0.531074
$621$	0	0
$622$	2.96272e6	0.307054
$623$	0	0
$624$	0	0
$625$	4.79460e6	0.490967
$626$	−2.23095e6	−0.227538
$627$	0	0
$628$	4.03645e7	4.08414
$629$	5.56642e6	0.560983
$630$	0	0
$631$	−4.43859e6	−0.443784	−0.221892	−	0.975071i	$-0.571223\pi$
$631$	−4.43859e6	−0.443784	−0.221892	+	0.975071i	$0.571223\pi$
$632$	2.00760e6	0.199933
$633$	0	0
$634$	1.31952e7	1.30375
$635$	−553984.	−0.0545209
$636$	0	0
$637$	0	0
$638$	1.13341e7	1.10239
$639$	0	0
$640$	−3.99135e6	−0.385185
$641$	−812551.	−0.0781098	−0.0390549	−	0.999237i	$-0.512435\pi$
$641$	−812551.	−0.0781098	−0.0390549	+	0.999237i	$0.512435\pi$
$642$	0	0
$643$	−1.17941e7	−1.12496	−0.562481	−	0.826810i	$-0.690153\pi$
$643$	−1.17941e7	−1.12496	−0.562481	+	0.826810i	$0.690153\pi$
$644$	0	0
$645$	0	0
$646$	−7.96930e6	−0.751344
$647$	2.43380e6	0.228573	0.114286	−	0.993448i	$-0.463542\pi$
$647$	2.43380e6	0.228573	0.114286	+	0.993448i	$0.463542\pi$
$648$	0	0
$649$	1.28006e7	1.19294
$650$	−2.66866e7	−2.47748
$651$	0	0
$652$	−1.43072e6	−0.131806
$653$	9.46664e6	0.868786	0.434393	−	0.900723i	$-0.356963\pi$
$653$	9.46664e6	0.868786	0.434393	+	0.900723i	$0.356963\pi$
$654$	0	0
$655$	−8.04646e6	−0.732827
$656$	8.57513e6	0.778003
$657$	0	0
$658$	0	0
$659$	−1.40662e7	−1.26172	−0.630860	−	0.775896i	$-0.717298\pi$
$659$	−1.40662e7	−1.26172	−0.630860	+	0.775896i	$0.717298\pi$
$660$	0	0
$661$	−1.71606e7	−1.52766	−0.763832	−	0.645415i	$-0.776685\pi$
$661$	−1.71606e7	−1.52766	−0.763832	+	0.645415i	$0.776685\pi$
$662$	3.56363e7	3.16044
$663$	0	0
$664$	2.49209e7	2.19353
$665$	0	0
$666$	0	0
$667$	−9.79606e6	−0.852583
$668$	−6.87385e6	−0.596017
$669$	0	0
$670$	−1.34994e7	−1.16179
$671$	9.82424e6	0.842350
$672$	0	0
$673$	5.87113e6	0.499671	0.249836	−	0.968288i	$-0.419623\pi$
$673$	5.87113e6	0.499671	0.249836	+	0.968288i	$0.419623\pi$
$674$	−2.54777e6	−0.216028
$675$	0	0
$676$	4.84249e7	4.07570
$677$	−1.39274e7	−1.16788	−0.583938	−	0.811798i	$-0.698489\pi$
$677$	−1.39274e7	−1.16788	−0.583938	+	0.811798i	$0.698489\pi$
$678$	0	0
$679$	0	0
$680$	−5.52872e6	−0.458513
$681$	0	0
$682$	1.40436e7	1.15615
$683$	−1.83101e7	−1.50189	−0.750945	−	0.660365i	$-0.770402\pi$
$683$	−1.83101e7	−1.50189	−0.750945	+	0.660365i	$0.770402\pi$
$684$	0	0
$685$	−1.49350e6	−0.121613
$686$	0	0
$687$	0	0
$688$	9.92420e6	0.799327
$689$	1.16302e6	0.0933337
$690$	0	0
$691$	−1.76717e7	−1.40793	−0.703967	−	0.710233i	$-0.748590\pi$
$691$	−1.76717e7	−1.40793	−0.703967	+	0.710233i	$0.748590\pi$
$692$	−2.45752e7	−1.95088
$693$	0	0
$694$	−1.53973e7	−1.21351
$695$	−5.03386e6	−0.395311
$696$	0	0
$697$	−2.51506e6	−0.196095
$698$	−1.57826e7	−1.22614
$699$	0	0
$700$	0	0
$701$	−8.22993e6	−0.632559	−0.316279	−	0.948666i	$-0.602434\pi$
$701$	−8.22993e6	−0.632559	−0.316279	+	0.948666i	$0.602434\pi$
$702$	0	0
$703$	1.37494e7	1.04929
$704$	−1.56169e6	−0.118758
$705$	0	0
$706$	−1.71552e7	−1.29534
$707$	0	0
$708$	0	0
$709$	2.46367e7	1.84063	0.920314	−	0.391180i	$-0.127933\pi$
$709$	2.46367e7	1.84063	0.920314	+	0.391180i	$0.127933\pi$
$710$	1.47575e6	0.109867
$711$	0	0
$712$	−2.58954e7	−1.91436
$713$	−1.21379e7	−0.894167
$714$	0	0
$715$	−1.12710e7	−0.824510
$716$	−5.62951e7	−4.10382
$717$	0	0
$718$	−2.55569e7	−1.85011
$719$	1.88254e7	1.35807	0.679034	−	0.734107i	$-0.262399\pi$
$719$	1.88254e7	1.35807	0.679034	+	0.734107i	$0.262399\pi$
$720$	0	0
$721$	0	0
$722$	5.63055e6	0.401983
$723$	0	0
$724$	−9.62411e6	−0.682361
$725$	−6.09823e6	−0.430882
$726$	0	0
$727$	6.77607e6	0.475491	0.237745	−	0.971328i	$-0.423592\pi$
$727$	6.77607e6	0.475491	0.237745	+	0.971328i	$0.423592\pi$
$728$	0	0
$729$	0	0
$730$	−4.07486e6	−0.283012
$731$	−2.91073e6	−0.201469
$732$	0	0
$733$	−1.72227e7	−1.18397	−0.591986	−	0.805948i	$-0.701656\pi$
$733$	−1.72227e7	−1.18397	−0.591986	+	0.805948i	$0.701656\pi$
$734$	2.15459e7	1.47613
$735$	0	0
$736$	2.60102e7	1.76990
$737$	2.58779e7	1.75493
$738$	0	0
$739$	−1.76534e7	−1.18910	−0.594548	−	0.804060i	$-0.702669\pi$
$739$	−1.76534e7	−1.18910	−0.594548	+	0.804060i	$0.702669\pi$
$740$	1.70704e7	1.14595
$741$	0	0
$742$	0	0
$743$	1.36977e7	0.910281	0.455141	−	0.890420i	$-0.349589\pi$
$743$	1.36977e7	0.910281	0.455141	+	0.890420i	$0.349589\pi$
$744$	0	0
$745$	3.33188e6	0.219937
$746$	−2.79341e7	−1.83775
$747$	0	0
$748$	1.89668e7	1.23948
$749$	0	0
$750$	0	0
$751$	1.16089e7	0.751090	0.375545	−	0.926804i	$-0.377456\pi$
$751$	1.16089e7	0.751090	0.375545	+	0.926804i	$0.377456\pi$
$752$	5.97885e6	0.385543
$753$	0	0
$754$	2.48167e7	1.58970
$755$	3.89526e6	0.248696
$756$	0	0
$757$	6.25226e6	0.396550	0.198275	−	0.980146i	$-0.436466\pi$
$757$	6.25226e6	0.396550	0.198275	+	0.980146i	$0.436466\pi$
$758$	2.08868e7	1.32038
$759$	0	0
$760$	−1.36563e7	−0.857629
$761$	3.02125e7	1.89115	0.945574	−	0.325406i	$-0.105501\pi$
$761$	3.02125e7	1.89115	0.945574	+	0.325406i	$0.105501\pi$
$762$	0	0
$763$	0	0
$764$	−481393.	−0.0298378
$765$	0	0
$766$	1.54277e7	0.950014
$767$	2.80278e7	1.72028
$768$	0	0
$769$	−2.58756e7	−1.57788	−0.788940	−	0.614470i	$-0.789370\pi$
$769$	−2.58756e7	−1.57788	−0.788940	+	0.614470i	$0.789370\pi$
$770$	0	0
$771$	0	0
$772$	3.28251e7	1.98227
$773$	−2.55409e7	−1.53740	−0.768701	−	0.639608i	$-0.779097\pi$
$773$	−2.55409e7	−1.53740	−0.768701	+	0.639608i	$0.779097\pi$
$774$	0	0
$775$	−7.55605e6	−0.451898
$776$	2.63746e7	1.57228
$777$	0	0
$778$	4.60140e7	2.72547
$779$	−6.21236e6	−0.366786
$780$	0	0
$781$	−2.82896e6	−0.165958
$782$	−2.36258e7	−1.38156
$783$	0	0
$784$	0	0
$785$	−1.32195e7	−0.765667
$786$	0	0
$787$	−8.81179e6	−0.507139	−0.253570	−	0.967317i	$-0.581605\pi$
$787$	−8.81179e6	−0.507139	−0.253570	+	0.967317i	$0.581605\pi$
$788$	−5.92512e7	−3.39924
$789$	0	0
$790$	−1.17665e6	−0.0670779
$791$	0	0
$792$	0	0
$793$	2.15108e7	1.21471
$794$	3.60018e7	2.02662
$795$	0	0
$796$	5.84871e7	3.27173
$797$	2.07624e7	1.15780	0.578899	−	0.815399i	$-0.303483\pi$
$797$	2.07624e7	1.15780	0.578899	+	0.815399i	$0.303483\pi$
$798$	0	0
$799$	−1.75358e6	−0.0971757
$800$	1.61918e7	0.894481
$801$	0	0
$802$	−9.27413e6	−0.509140
$803$	7.81136e6	0.427501
$804$	0	0
$805$	0	0
$806$	3.07493e7	1.66724
$807$	0	0
$808$	7.64973e7	4.12209
$809$	−1.37021e7	−0.736067	−0.368034	−	0.929813i	$-0.619969\pi$
$809$	−1.37021e7	−0.736067	−0.368034	+	0.929813i	$0.619969\pi$
$810$	0	0
$811$	3.30799e7	1.76609	0.883043	−	0.469292i	$-0.155491\pi$
$811$	3.30799e7	1.76609	0.883043	+	0.469292i	$0.155491\pi$
$812$	0	0
$813$	0	0
$814$	−4.71613e7	−2.49474
$815$	468563.	0.0247101
$816$	0	0
$817$	−7.18971e6	−0.376840
$818$	−4.41520e7	−2.30711
$819$	0	0
$820$	−7.71285e6	−0.400572
$821$	−1.48323e7	−0.767982	−0.383991	−	0.923337i	$-0.625451\pi$
$821$	−1.48323e7	−0.767982	−0.383991	+	0.923337i	$0.625451\pi$
$822$	0	0
$823$	1.08722e7	0.559525	0.279762	−	0.960069i	$-0.409744\pi$
$823$	1.08722e7	0.559525	0.279762	+	0.960069i	$0.409744\pi$
$824$	−2.15640e7	−1.10640
$825$	0	0
$826$	0	0
$827$	1.34267e7	0.682662	0.341331	−	0.939943i	$-0.389122\pi$
$827$	1.34267e7	0.682662	0.341331	+	0.939943i	$0.389122\pi$
$828$	0	0
$829$	−1.46754e7	−0.741658	−0.370829	−	0.928701i	$-0.620926\pi$
$829$	−1.46754e7	−0.741658	−0.370829	+	0.928701i	$0.620926\pi$
$830$	−1.46060e7	−0.735931
$831$	0	0
$832$	−3.41941e6	−0.171255
$833$	0	0
$834$	0	0
$835$	2.25120e6	0.111737
$836$	4.68492e7	2.31839
$837$	0	0
$838$	−1.55444e7	−0.764654
$839$	−2.76635e7	−1.35676	−0.678379	−	0.734712i	$-0.737317\pi$
$839$	−2.76635e7	−1.35676	−0.678379	+	0.734712i	$0.737317\pi$
$840$	0	0
$841$	−1.48402e7	−0.723519
$842$	4.51946e7	2.19688
$843$	0	0
$844$	4.98798e6	0.241028
$845$	−1.58593e7	−0.764084
$846$	0	0
$847$	0	0
$848$	−2.18537e6	−0.104361
$849$	0	0
$850$	−1.47075e7	−0.698219
$851$	4.07616e7	1.92942
$852$	0	0
$853$	3.43515e6	0.161649	0.0808244	−	0.996728i	$-0.474245\pi$
$853$	3.43515e6	0.161649	0.0808244	+	0.996728i	$0.474245\pi$
$854$	0	0
$855$	0	0
$856$	1.99622e7	0.931158
$857$	6.40626e6	0.297956	0.148978	−	0.988840i	$-0.452402\pi$
$857$	6.40626e6	0.297956	0.148978	+	0.988840i	$0.452402\pi$
$858$	0	0
$859$	1.81739e7	0.840360	0.420180	−	0.907441i	$-0.361967\pi$
$859$	1.81739e7	0.840360	0.420180	+	0.907441i	$0.361967\pi$
$860$	−8.92627e6	−0.411551
$861$	0	0
$862$	−7.00114e7	−3.20923
$863$	−818774.	−0.0374229	−0.0187114	−	0.999825i	$-0.505956\pi$
$863$	−818774.	−0.0374229	−0.0187114	+	0.999825i	$0.505956\pi$
$864$	0	0
$865$	8.04843e6	0.365739
$866$	4.08267e7	1.84991
$867$	0	0
$868$	0	0
$869$	2.25560e6	0.101324
$870$	0	0
$871$	5.66613e7	2.53070
$872$	1.50968e7	0.672350
$873$	0	0
$874$	−5.83574e7	−2.58415
$875$	0	0
$876$	0	0
$877$	−3.33962e7	−1.46622	−0.733108	−	0.680113i	$-0.761931\pi$
$877$	−3.33962e7	−1.46622	−0.733108	+	0.680113i	$0.761931\pi$
$878$	−1.51882e7	−0.664921
$879$	0	0
$880$	2.11788e7	0.921921
$881$	1.24325e7	0.539658	0.269829	−	0.962908i	$-0.413033\pi$
$881$	1.24325e7	0.539658	0.269829	+	0.962908i	$0.413033\pi$
$882$	0	0
$883$	−2.33298e7	−1.00695	−0.503476	−	0.864009i	$-0.667946\pi$
$883$	−2.33298e7	−1.00695	−0.503476	+	0.864009i	$0.667946\pi$
$884$	4.15290e7	1.78739
$885$	0	0
$886$	−5.74778e7	−2.45989
$887$	4.75528e6	0.202940	0.101470	−	0.994839i	$-0.467645\pi$
$887$	4.75528e6	0.202940	0.101470	+	0.994839i	$0.467645\pi$
$888$	0	0
$889$	0	0
$890$	1.51772e7	0.642269
$891$	0	0
$892$	−4.49833e7	−1.89295
$893$	−4.33145e6	−0.181763
$894$	0	0
$895$	1.84368e7	0.769356
$896$	0	0
$897$	0	0
$898$	1.99245e7	0.824512
$899$	7.02661e6	0.289966
$900$	0	0
$901$	640963.	0.0263040
$902$	2.13087e7	0.872050
$903$	0	0
$904$	−4.00716e7	−1.63086
$905$	3.15192e6	0.127924
$906$	0	0
$907$	−1.90902e7	−0.770537	−0.385268	−	0.922805i	$-0.625891\pi$
$907$	−1.90902e7	−0.770537	−0.385268	+	0.922805i	$0.625891\pi$
$908$	−7.26891e7	−2.92587
$909$	0	0
$910$	0	0
$911$	1.22591e6	0.0489399	0.0244700	−	0.999701i	$-0.492210\pi$
$911$	1.22591e6	0.0489399	0.0244700	+	0.999701i	$0.492210\pi$
$912$	0	0
$913$	2.79993e7	1.11165
$914$	2.73296e7	1.08210
$915$	0	0
$916$	6.42490e7	2.53004
$917$	0	0
$918$	0	0
$919$	−1.69227e7	−0.660970	−0.330485	−	0.943811i	$-0.607212\pi$
$919$	−1.69227e7	−0.660970	−0.330485	+	0.943811i	$0.607212\pi$
$920$	−4.04855e7	−1.57699
$921$	0	0
$922$	−4.75677e7	−1.84283
$923$	−6.19419e6	−0.239321
$924$	0	0
$925$	2.53749e7	0.975101
$926$	5.30195e7	2.03193
$927$	0	0
$928$	−1.50573e7	−0.573954
$929$	−6.77240e6	−0.257456	−0.128728	−	0.991680i	$-0.541089\pi$
$929$	−6.77240e6	−0.257456	−0.128728	+	0.991680i	$0.541089\pi$
$930$	0	0
$931$	0	0
$932$	−4.32380e7	−1.63052
$933$	0	0
$934$	−5.17307e7	−1.94035
$935$	−6.21166e6	−0.232369
$936$	0	0
$937$	−1.41035e7	−0.524779	−0.262389	−	0.964962i	$-0.584511\pi$
$937$	−1.41035e7	−0.524779	−0.262389	+	0.964962i	$0.584511\pi$
$938$	0	0
$939$	0	0
$940$	−5.37765e6	−0.198505
$941$	5.62524e6	0.207094	0.103547	−	0.994625i	$-0.466981\pi$
$941$	5.62524e6	0.207094	0.103547	+	0.994625i	$0.466981\pi$
$942$	0	0
$943$	−1.84172e7	−0.674441
$944$	−5.26657e7	−1.92353
$945$	0	0
$946$	2.46611e7	0.895952
$947$	−2.64126e7	−0.957052	−0.478526	−	0.878073i	$-0.658829\pi$
$947$	−2.64126e7	−0.957052	−0.478526	+	0.878073i	$0.658829\pi$
$948$	0	0
$949$	1.71035e7	0.616480
$950$	−3.63286e7	−1.30599
$951$	0	0
$952$	0	0
$953$	−4.21824e7	−1.50452	−0.752262	−	0.658864i	$-0.771037\pi$
$953$	−4.21824e7	−1.50452	−0.752262	+	0.658864i	$0.771037\pi$
$954$	0	0
$955$	157657.	0.00559378
$956$	−5.38957e7	−1.90726
$957$	0	0
$958$	−5.67511e7	−1.99784
$959$	0	0
$960$	0	0
$961$	−1.99228e7	−0.695892
$962$	−1.03263e8	−3.59755
$963$	0	0
$964$	−8.52077e7	−2.95315
$965$	−1.07503e7	−0.371622
$966$	0	0
$967$	4.01501e7	1.38077	0.690384	−	0.723443i	$-0.257441\pi$
$967$	4.01501e7	1.38077	0.690384	+	0.723443i	$0.257441\pi$
$968$	−2.30638e7	−0.791120
$969$	0	0
$970$	−1.54580e7	−0.527503
$971$	−1.64551e7	−0.560084	−0.280042	−	0.959988i	$-0.590348\pi$
$971$	−1.64551e7	−0.560084	−0.280042	+	0.959988i	$0.590348\pi$
$972$	0	0
$973$	0	0
$974$	7.03207e7	2.37512
$975$	0	0
$976$	−4.04200e7	−1.35823
$977$	−1.71176e7	−0.573729	−0.286865	−	0.957971i	$-0.592613\pi$
$977$	−1.71176e7	−0.573729	−0.286865	+	0.957971i	$0.592613\pi$
$978$	0	0
$979$	−2.90942e7	−0.970174
$980$	0	0
$981$	0	0
$982$	7.61124e7	2.51870
$983$	2.33803e7	0.771730	0.385865	−	0.922555i	$-0.373903\pi$
$983$	2.33803e7	0.771730	0.385865	+	0.922555i	$0.373903\pi$
$984$	0	0
$985$	1.94049e7	0.637266
$986$	1.36770e7	0.448021
$987$	0	0
$988$	1.02579e8	3.34324
$989$	−2.13146e7	−0.692927
$990$	0	0
$991$	−4.70955e6	−0.152333	−0.0761667	−	0.997095i	$-0.524268\pi$
$991$	−4.70955e6	−0.152333	−0.0761667	+	0.997095i	$0.524268\pi$
$992$	−1.86568e7	−0.601948
$993$	0	0
$994$	0	0
$995$	−1.91547e7	−0.613362
$996$	0	0
$997$	−4.55428e7	−1.45105	−0.725524	−	0.688197i	$-0.758403\pi$
$997$	−4.55428e7	−1.45105	−0.725524	+	0.688197i	$0.758403\pi$
$998$	−2.20150e7	−0.699667
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.w.1.1		4
3.2	odd	2		147.6.a.m.1.4		4
7.2	even	3		63.6.e.e.46.4		8
7.4	even	3		63.6.e.e.37.4		8
7.6	odd	2		441.6.a.v.1.1		4
21.2	odd	6		21.6.e.c.4.1	✓	8
21.5	even	6		147.6.e.o.67.1		8
21.11	odd	6		21.6.e.c.16.1	yes	8
21.17	even	6		147.6.e.o.79.1		8
21.20	even	2		147.6.a.l.1.4		4
84.11	even	6		336.6.q.j.289.2		8
84.23	even	6		336.6.q.j.193.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.c.4.1	✓	8	21.2	odd	6
21.6.e.c.16.1	yes	8	21.11	odd	6
63.6.e.e.37.4		8	7.4	even	3
63.6.e.e.46.4		8	7.2	even	3
147.6.a.l.1.4		4	21.20	even	2
147.6.a.m.1.4		4	3.2	odd	2
147.6.e.o.67.1		8	21.5	even	6
147.6.e.o.79.1		8	21.17	even	6
336.6.q.j.193.2		8	84.23	even	6
336.6.q.j.289.2		8	84.11	even	6
441.6.a.v.1.1		4	7.6	odd	2
441.6.a.w.1.1		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-10.2239 q^{2} +72.5272 q^{4} -23.7528 q^{5} -414.344 q^{8} +O(q^{10})\)	\(q-10.2239 q^{2} +72.5272 q^{4} -23.7528 q^{5} -414.344 q^{8} +242.845 q^{10} -465.526 q^{11} -1019.30 q^{13} +1915.32 q^{16} -561.757 q^{17} -1387.58 q^{19} -1722.72 q^{20} +4759.47 q^{22} -4113.62 q^{23} -2560.80 q^{25} +10421.2 q^{26} +2381.37 q^{29} +2950.66 q^{31} -6322.95 q^{32} +5743.32 q^{34} -9908.95 q^{37} +14186.4 q^{38} +9841.83 q^{40} +4477.13 q^{41} +5181.48 q^{43} -33763.3 q^{44} +42057.0 q^{46} +3121.59 q^{47} +26181.3 q^{50} -73926.9 q^{52} -1141.00 q^{53} +11057.6 q^{55} -24346.8 q^{58} -27497.1 q^{59} -21103.5 q^{61} -30167.1 q^{62} +3354.67 q^{64} +24211.2 q^{65} -55588.4 q^{67} -40742.6 q^{68} +6076.90 q^{71} -16779.6 q^{73} +101308. q^{74} -100637. q^{76} -4845.26 q^{79} -45494.2 q^{80} -45773.5 q^{82} -60145.4 q^{83} +13343.3 q^{85} -52974.7 q^{86} +192888. q^{88} +62497.4 q^{89} -298349. q^{92} -31914.7 q^{94} +32958.9 q^{95} -63653.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8	\( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97}+O(q^{100}) \) 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8 + 283 * q^10 - 402 * q^11 + 462 * q^13 + 3273 * q^16 - 276 * q^17 + 510 * q^19 - 4719 * q^20 + 1375 * q^22 - 6900 * q^23 + 2814 * q^25 + 15138 * q^26 - 540 * q^29 - 6410 * q^31 - 15519 * q^32 + 21144 * q^34 + 15250 * q^37 + 41250 * q^38 - 8547 * q^40 - 4308 * q^41 + 29198 * q^43 - 70743 * q^44 + 61800 * q^46 + 15060 * q^47 + 7302 * q^50 - 47476 * q^52 - 13692 * q^53 + 73124 * q^55 + 52309 * q^58 - 34830 * q^59 - 5364 * q^61 - 16029 * q^62 - 73487 * q^64 - 66864 * q^65 - 5994 * q^67 + 58272 * q^68 - 89268 * q^71 + 59638 * q^73 + 185442 * q^74 + 21308 * q^76 - 44062 * q^79 + 33381 * q^80 + 57596 * q^82 + 208446 * q^83 + 36324 * q^85 + 136968 * q^86 + 87597 * q^88 + 77520 * q^89 - 158256 * q^92 - 73722 * q^94 + 221376 * q^95 - 188630 * q^97

Introduction

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