LMFDB - Embedded newform 147.6.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$147 = 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	147.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:	$23.5764215125$
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			1.4
Root			$-9.22385$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.2239 q^{2} +9.00000 q^{3} +72.5272 q^{4} +23.7528 q^{5} +92.0147 q^{6} +414.344 q^{8} +81.0000 q^{9} +242.845 q^{10} +465.526 q^{11} +652.744 q^{12} -1019.30 q^{13} +213.775 q^{15} +1915.32 q^{16} +561.757 q^{17} +828.132 q^{18} -1387.58 q^{19} +1722.72 q^{20} +4759.47 q^{22} +4113.62 q^{23} +3729.09 q^{24} -2560.80 q^{25} -10421.2 q^{26} +729.000 q^{27} -2381.37 q^{29} +2185.61 q^{30} +2950.66 q^{31} +6322.95 q^{32} +4189.73 q^{33} +5743.32 q^{34} +5874.70 q^{36} -9908.95 q^{37} -14186.4 q^{38} -9173.70 q^{39} +9841.83 q^{40} -4477.13 q^{41} +5181.48 q^{43} +33763.3 q^{44} +1923.98 q^{45} +42057.0 q^{46} -3121.59 q^{47} +17237.9 q^{48} -26181.3 q^{50} +5055.81 q^{51} -73926.9 q^{52} +1141.00 q^{53} +7453.19 q^{54} +11057.6 q^{55} -12488.2 q^{57} -24346.8 q^{58} +27497.1 q^{59} +15504.5 q^{60} -21103.5 q^{61} +30167.1 q^{62} +3354.67 q^{64} -24211.2 q^{65} +42835.2 q^{66} -55588.4 q^{67} +40742.6 q^{68} +37022.6 q^{69} -6076.90 q^{71} +33561.8 q^{72} -16779.6 q^{73} -101308. q^{74} -23047.2 q^{75} -100637. q^{76} -93790.5 q^{78} -4845.26 q^{79} +45494.2 q^{80} +6561.00 q^{81} -45773.5 q^{82} +60145.4 q^{83} +13343.3 q^{85} +52974.7 q^{86} -21432.4 q^{87} +192888. q^{88} -62497.4 q^{89} +19670.5 q^{90} +298349. q^{92} +26555.9 q^{93} -31914.7 q^{94} -32958.9 q^{95} +56906.5 q^{96} -63653.8 q^{97} +37707.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22}+ \cdots + 32562 q^{99}+O(q^{100})$ 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9 + 283 * q^10 + 402 * q^11 + 621 * q^12 + 462 * q^13 + 3273 * q^16 + 276 * q^17 + 243 * q^18 + 510 * q^19 + 4719 * q^20 + 1375 * q^22 + 6900 * q^23 + 1107 * q^24 + 2814 * q^25 - 15138 * q^26 + 2916 * q^27 + 540 * q^29 + 2547 * q^30 - 6410 * q^31 + 15519 * q^32 + 3618 * q^33 + 21144 * q^34 + 5589 * q^36 + 15250 * q^37 - 41250 * q^38 + 4158 * q^39 - 8547 * q^40 + 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 29457 * q^48 - 7302 * q^50 + 2484 * q^51 - 47476 * q^52 + 13692 * q^53 + 2187 * q^54 + 73124 * q^55 + 4590 * q^57 + 52309 * q^58 + 34830 * q^59 + 42471 * q^60 - 5364 * q^61 + 16029 * q^62 - 73487 * q^64 + 66864 * q^65 + 12375 * q^66 - 5994 * q^67 - 58272 * q^68 + 62100 * q^69 + 89268 * q^71 + 9963 * q^72 + 59638 * q^73 - 185442 * q^74 + 25326 * q^75 + 21308 * q^76 - 136242 * q^78 - 44062 * q^79 - 33381 * q^80 + 26244 * q^81 + 57596 * q^82 - 208446 * q^83 + 36324 * q^85 - 136968 * q^86 + 4860 * q^87 + 87597 * q^88 - 77520 * q^89 + 22923 * q^90 + 158256 * q^92 - 57690 * q^93 - 73722 * q^94 - 221376 * q^95 + 139671 * q^96 - 188630 * q^97 + 32562 * q^99

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	$-0.140863\pi$
$2$	10.2239	1.80734	0.903669	+	0.428231i	$0.140863\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	72.5272	2.26647
$5$	23.7528	0.424903	0.212452	−	0.977172i	$-0.431855\pi$
$5$	23.7528	0.424903	0.212452	+	0.977172i	$0.431855\pi$
$6$	92.0147	1.04347
$7$	0	0
$7$	0	0
$8$	414.344	2.28895
$9$	81.0000	0.333333
$10$	242.845	0.767944
$11$	465.526	1.16001	0.580006	−	0.814612i	$-0.303050\pi$
$11$	465.526	1.16001	0.580006	+	0.814612i	$0.303050\pi$
$12$	652.744	1.30855
$13$	−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
$15$	213.775	0.245318
$16$	1915.32	1.87043
$17$	561.757	0.471440	0.235720	−	0.971821i	$-0.424255\pi$
$17$	561.757	0.471440	0.235720	+	0.971821i	$0.424255\pi$
$18$	828.132	0.602446
$19$	−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	$-0.199072\pi$
$23$	4113.62	1.62145	0.810727	+	0.585425i	$0.199072\pi$
$24$	3729.09	1.32152
$25$	−2560.80	−0.819457
$26$	−10421.2	−3.02331
$27$	729.000	0.192450
$28$	0	0
$29$	−2381.37	−0.525814	−0.262907	−	0.964821i	$-0.584681\pi$
$29$	−2381.37	−0.525814	−0.262907	+	0.964821i	$0.584681\pi$
$30$	2185.61	0.443373
$31$	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
$33$	4189.73	0.669733
$34$	5743.32	0.852051
$35$	0	0
$36$	5874.70	0.755491
$37$	−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
$39$	−9173.70	−0.965791
$40$	9841.83	0.972581
$41$	−4477.13	−0.415949	−0.207974	−	0.978134i	$-0.566687\pi$
$41$	−4477.13	−0.415949	−0.207974	+	0.978134i	$0.566687\pi$
$42$	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
$45$	1923.98	0.141634
$46$	42057.0	2.93052
$47$	−3121.59	−0.206125	−0.103063	−	0.994675i	$-0.532864\pi$
$47$	−3121.59	−0.206125	−0.103063	+	0.994675i	$0.532864\pi$
$48$	17237.9	1.07989
$49$	0	0
$50$	−26181.3	−1.48104
$51$	5055.81	0.272186
$52$	−73926.9	−3.79135
$53$	1141.00	0.0557950	0.0278975	−	0.999611i	$-0.491119\pi$
$53$	1141.00	0.0557950	0.0278975	+	0.999611i	$0.491119\pi$
$54$	7453.19	0.347823
$55$	11057.6	0.492893
$56$	0	0
$57$	−12488.2	−0.509111
$58$	−24346.8	−0.950325
$59$	27497.1	1.02839	0.514194	−	0.857674i	$-0.328091\pi$
$59$	27497.1	1.02839	0.514194	+	0.857674i	$0.328091\pi$
$60$	15504.5	0.556007
$61$	−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
$66$	42835.2	1.21043
$67$	−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
$69$	37022.6	0.936146
$70$	0	0
$71$	−6076.90	−0.143066	−0.0715330	−	0.997438i	$-0.522789\pi$
$71$	−6076.90	−0.143066	−0.0715330	+	0.997438i	$0.522789\pi$
$72$	33561.8	0.762982
$73$	−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
$75$	−23047.2	−0.473114
$76$	−100637.	−1.99859
$77$	0	0
$78$	−93790.5	−1.74551
$79$	−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
$81$	6561.00	0.111111
$82$	−45773.5	−0.751760
$83$	60145.4	0.958313	0.479156	−	0.877730i	$-0.340943\pi$
$83$	60145.4	0.958313	0.479156	+	0.877730i	$0.340943\pi$
$84$	0	0
$85$	13343.3	0.200316
$86$	52974.7	0.772365
$87$	−21432.4	−0.303579
$88$	192888.	2.65520
$89$	−62497.4	−0.836348	−0.418174	−	0.908367i	$-0.637330\pi$
$89$	−62497.4	−0.836348	−0.418174	+	0.908367i	$0.637330\pi$
$90$	19670.5	0.255981
$91$	0	0
$92$	298349.	3.67498
$93$	26555.9	0.318386
$94$	−31914.7	−0.372539
$95$	−32958.9	−0.374683
$96$	56906.5	0.630208
$97$	−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
$99$	37707.6	0.386670
$100$	−185728.	−1.85728
$101$	184623.	1.80087	0.900434	−	0.434993i	$-0.143249\pi$
$101$	184623.	1.80087	0.900434	+	0.434993i	$0.143249\pi$
$102$	51689.9	0.491932
$103$	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	$-0.434800\pi$
$107$	48177.8	0.406806	0.203403	+	0.979095i	$0.434800\pi$
$108$	52872.3	0.436183
$109$	−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
$111$	−89180.6	−0.687010
$112$	0	0
$113$	−96711.1	−0.712492	−0.356246	−	0.934392i	$-0.615944\pi$
$113$	−96711.1	−0.712492	−0.356246	+	0.934392i	$0.615944\pi$
$114$	−127678.	−0.920137
$115$	97710.0	0.688961
$116$	−172714.	−1.19174
$117$	−82563.3	−0.557599
$118$	281126.	1.85864
$119$	0	0
$120$	88576.5	0.561520
$121$	55663.5	0.345626
$122$	−215759.	−1.31241
$123$	−40294.1	−0.240148
$124$	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
$129$	46633.4	0.246730
$130$	−247532.	−1.28462
$131$	−338758.	−1.72469	−0.862345	−	0.506321i	$-0.831005\pi$
$131$	−338758.	−1.72469	−0.862345	+	0.506321i	$0.831005\pi$
$132$	303870.	1.51793
$133$	0	0
$134$	−568328.	−2.73424
$135$	17315.8	0.0817727
$136$	232760.	1.07910
$137$	−62876.7	−0.286213	−0.143106	−	0.989707i	$-0.545709\pi$
$137$	−62876.7	−0.286213	−0.143106	+	0.989707i	$0.545709\pi$
$138$	378513.	1.69193
$139$	211927.	0.930356	0.465178	−	0.885217i	$-0.345990\pi$
$139$	211927.	0.930356	0.465178	+	0.885217i	$0.345990\pi$
$140$	0	0
$141$	−28094.3	−0.119007
$142$	−62129.4	−0.258569
$143$	−474511.	−1.94047
$144$	155141.	0.623477
$145$	−56564.3	−0.223420
$146$	−171553.	−0.666062
$147$	0	0
$148$	−718668.	−2.69696
$149$	140273.	0.517616	0.258808	−	0.965929i	$-0.416670\pi$
$149$	140273.	0.517616	0.258808	+	0.965929i	$0.416670\pi$
$150$	−235632.	−0.855077
$151$	−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
$153$	45502.3	0.157147
$154$	0	0
$155$	70086.4	0.234317
$156$	−665342.	−2.18894
$157$	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
$159$	10269.0	0.0322132
$160$	150188.	0.463804
$161$	0	0
$162$	67078.7	0.200815
$163$	−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
$165$	99518.0	0.284572
$166$	614918.	1.73200
$167$	94776.2	0.262971	0.131486	−	0.991318i	$-0.458025\pi$
$167$	94776.2	0.262971	0.131486	+	0.991318i	$0.458025\pi$
$168$	0	0
$169$	667679.	1.79825
$170$	136420.	0.362039
$171$	−112394.	−0.293936
$172$	375798.	0.968576
$173$	338841.	0.860757	0.430379	−	0.902649i	$-0.358380\pi$
$173$	338841.	0.860757	0.430379	+	0.902649i	$0.358380\pi$
$174$	−219121.	−0.548670
$175$	0	0
$176$	891631.	2.16972
$177$	247474.	0.593740
$178$	−638965.	−1.51156
$179$	776193.	1.81066	0.905330	−	0.424708i	$-0.139623\pi$
$179$	776193.	1.81066	0.905330	+	0.424708i	$0.139623\pi$
$180$	139541.	0.321011
$181$	−132697.	−0.301067	−0.150534	−	0.988605i	$-0.548099\pi$
$181$	−132697.	−0.301067	−0.150534	+	0.988605i	$0.548099\pi$
$182$	0	0
$183$	−189932.	−0.419247
$184$	1.70445e6	3.71142
$185$	−235366.	−0.505607
$186$	271504.	0.575431
$187$	261512.	0.546875
$188$	−226400.	−0.467178
$189$	0	0
$190$	−336967.	−0.677178
$191$	6637.41	0.0131648	0.00658242	−	0.999978i	$-0.497905\pi$
$191$	6637.41	0.0131648	0.00658242	+	0.999978i	$0.497905\pi$
$192$	30192.0	0.0591070
$193$	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
$195$	−217901.	−0.410368
$196$	0	0
$197$	816952.	1.49979	0.749896	−	0.661556i	$-0.230104\pi$
$197$	816952.	1.49979	0.749896	+	0.661556i	$0.230104\pi$
$198$	385517.	0.698845
$199$	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
$201$	−500296.	−0.873447
$202$	1.88756e6	3.25478
$203$	0	0
$204$	366684.	0.616902
$205$	−106344.	−0.176738
$206$	532087.	0.873605
$207$	333203.	0.540484
$208$	−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
$213$	−54692.1	−0.0825992
$214$	492563.	0.735237
$215$	123075.	0.181582
$216$	302057.	0.440508
$217$	0	0
$218$	−372512.	−0.530883
$219$	−151017.	−0.212772
$220$	801973.	1.11713
$221$	−572599.	−0.788623
$222$	−911769.	−1.24166
$223$	−620227.	−0.835196	−0.417598	−	0.908632i	$-0.637128\pi$
$223$	−620227.	−0.835196	−0.417598	+	0.908632i	$0.637128\pi$
$224$	0	0
$225$	−207425.	−0.273152
$226$	−988760.	−1.28771
$227$	1.00223e6	1.29093	0.645467	−	0.763788i	$-0.276663\pi$
$227$	1.00223e6	1.29093	0.645467	+	0.763788i	$0.276663\pi$
$228$	−905734.	−1.15389
$229$	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	$-0.382878\pi$
$233$	596163.	0.719408	0.359704	+	0.933066i	$0.382878\pi$
$234$	−844115.	−1.00777
$235$	−74146.6	−0.0875834
$236$	1.99429e6	2.33081
$237$	−43607.4	−0.0504300
$238$	0	0
$239$	743111.	0.841509	0.420754	−	0.907175i	$-0.361765\pi$
$239$	743111.	0.841509	0.420754	+	0.907175i	$0.361765\pi$
$240$	409448.	0.458850
$241$	−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
$243$	59049.0	0.0641500
$244$	−1.53058e6	−1.64582
$245$	0	0
$246$	−411961.	−0.434029
$247$	1.41436e6	1.47508
$248$	1.22259e6	1.26226
$249$	541309.	0.553282
$250$	−1.38077e6	−1.39724
$251$	352992.	0.353655	0.176828	−	0.984242i	$-0.443416\pi$
$251$	352992.	0.353655	0.176828	+	0.984242i	$0.443416\pi$
$252$	0	0
$253$	1.91500e6	1.88090
$254$	238450.	0.231906
$255$	120090.	0.115653
$256$	−1.82533e6	−1.74077
$257$	10012.7	0.00945621	0.00472811	−	0.999989i	$-0.498495\pi$
$257$	10012.7	0.00945621	0.00472811	+	0.999989i	$0.498495\pi$
$258$	476773.	0.445925
$259$	0	0
$260$	−1.75597e6	−1.61096
$261$	−192891.	−0.175271
$262$	−3.46341e6	−3.11710
$263$	1.68180e6	1.49929	0.749644	−	0.661841i	$-0.230225\pi$
$263$	1.68180e6	1.49929	0.749644	+	0.661841i	$0.230225\pi$
$264$	1.73599e6	1.53298
$265$	27101.9	0.0237075
$266$	0	0
$267$	−562477.	−0.482866
$268$	−4.03167e6	−3.42885
$269$	−1.91717e6	−1.61540	−0.807702	−	0.589591i	$-0.799289\pi$
$269$	−1.91717e6	−1.61540	−0.807702	+	0.589591i	$0.799289\pi$
$270$	177034.	0.147791
$271$	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
$276$	2.68514e6	2.12175
$277$	−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
$279$	239003.	0.183820
$280$	0	0
$281$	−1.77699e6	−1.34252	−0.671259	−	0.741223i	$-0.734246\pi$
$281$	−1.77699e6	−1.34252	−0.671259	+	0.741223i	$0.734246\pi$
$282$	−287232.	−0.215085
$283$	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
$285$	−296630.	−0.216323
$286$	−4.85133e6	−3.50708
$287$	0	0
$288$	512159.	0.363851
$289$	−1.10429e6	−0.777745
$290$	−578305.	−0.403796
$291$	−572884.	−0.396583
$292$	−1.21698e6	−0.835268
$293$	−1.48897e6	−1.01325	−0.506627	−	0.862165i	$-0.669108\pi$
$293$	−1.48897e6	−1.01325	−0.506627	+	0.862165i	$0.669108\pi$
$294$	0	0
$295$	653133.	0.436965
$296$	−4.10571e6	−2.72370
$297$	339368.	0.223244
$298$	1.43413e6	0.935508
$299$	−4.19301e6	−2.71236
$300$	−1.67155e6	−1.07230
$301$	0	0
$302$	−1.67662e6	−1.05784
$303$	1.66161e6	1.03973
$304$	−2.65766e6	−1.64936
$305$	−501268.	−0.308546
$306$	465209.	0.284017
$307$	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
$309$	468394.	0.279071
$310$	716553.	0.423491
$311$	289785.	0.169893	0.0849463	−	0.996386i	$-0.472928\pi$
$311$	289785.	0.169893	0.0849463	+	0.996386i	$0.472928\pi$
$312$	−3.80106e6	−2.21064
$313$	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	$-0.382544\pi$
$317$	1.29063e6	0.721363	0.360681	+	0.932689i	$0.382544\pi$
$318$	104988.	0.0582202
$319$	−1.10859e6	−0.609951
$320$	79682.8	0.0435000
$321$	433600.	0.234870
$322$	0	0
$323$	−779481.	−0.415719
$324$	475851.	0.251830
$325$	2.61023e6	1.37079
$326$	−201682.	−0.105105
$327$	−327920.	−0.169589
$328$	−1.85507e6	−0.952084
$329$	0	0
$330$	1.01746e6	0.514318
$331$	−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
$333$	−802625.	−0.396645
$334$	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
$339$	−870400.	−0.411358
$340$	967752.	0.454012
$341$	1.37361e6	0.639700
$342$	−1.14910e6	−0.531241
$343$	0	0
$344$	2.14692e6	0.978180
$345$	879390.	0.397772
$346$	3.46426e6	1.55568
$347$	−1.50601e6	−0.671437	−0.335719	−	0.941962i	$-0.608979\pi$
$347$	−1.50601e6	−0.671437	−0.335719	+	0.941962i	$0.608979\pi$
$348$	−1.55443e6	−0.688054
$349$	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
$351$	−743070.	−0.321930
$352$	2.94350e6	1.26621
$353$	−1.67796e6	−0.716710	−0.358355	−	0.933585i	$-0.616662\pi$
$353$	−1.67796e6	−0.716710	−0.358355	+	0.933585i	$0.616662\pi$
$354$	2.53014e6	1.07309
$355$	−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	$-0.671033\pi$
$359$	−2.49973e6	−1.02366	−0.511832	+	0.859086i	$0.671033\pi$
$360$	797188.	0.324194
$361$	−550727.	−0.222417
$362$	−1.35667e6	−0.544130
$363$	500971.	0.199547
$364$	0	0
$365$	−398564.	−0.156591
$366$	−1.94183e6	−0.757721
$367$	−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
$369$	−362647.	−0.138650
$370$	−2.40634e6	−0.913804
$371$	0	0
$372$	1.92602e6	0.721613
$373$	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
$375$	−1.21548e6	−0.446346
$376$	−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
$381$	209906.	0.0740819
$382$	67859.9	0.0237933
$383$	1.50899e6	0.525643	0.262821	−	0.964845i	$-0.415347\pi$
$383$	1.50899e6	0.525643	0.262821	+	0.964845i	$0.415347\pi$
$384$	−1.51233e6	−0.523382
$385$	0	0
$386$	4.62721e6	1.58071
$387$	419700.	0.142450
$388$	−4.61663e6	−1.55685
$389$	4.50065e6	1.50800	0.754000	−	0.656874i	$-0.228122\pi$
$389$	4.50065e6	1.50800	0.754000	+	0.656874i	$0.228122\pi$
$390$	−2.22779e6	−0.741673
$391$	2.31085e6	0.764417
$392$	0	0
$393$	−3.04882e6	−0.995751
$394$	8.35240e6	2.71063
$395$	−115089.	−0.0371142
$396$	2.73483e6	0.876378
$397$	−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	$-0.544985\pi$
$401$	−907107.	−0.281707	−0.140853	+	0.990030i	$0.544985\pi$
$402$	−5.11495e6	−1.57862
$403$	−3.00760e6	−0.922482
$404$	1.33902e7	4.08162
$405$	155842.	0.0472115
$406$	0	0
$407$	−4.61287e6	−1.38034
$408$	2.09484e6	0.623019
$409$	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
$411$	−565891.	−0.165245
$412$	3.77458e6	1.09553
$413$	0	0
$414$	3.40662e6	0.976838
$415$	1.42862e6	0.407190
$416$	−6.44498e6	−1.82595
$417$	1.90734e6	0.537141
$418$	−6.60413e6	−1.84874
$419$	−1.52041e6	−0.423083	−0.211541	−	0.977369i	$-0.567848\pi$
$419$	−1.52041e6	−0.423083	−0.211541	+	0.977369i	$0.567848\pi$
$420$	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
$423$	−252849.	−0.0687085
$424$	472765.	0.127712
$425$	−1.43855e6	−0.386325
$426$	−559164.	−0.149285
$427$	0	0
$428$	3.49420e6	0.922016
$429$	−4.27060e6	−1.12033
$430$	1.25830e6	0.328180
$431$	−6.84785e6	−1.77567	−0.887833	−	0.460166i	$-0.847790\pi$
$431$	−6.84785e6	−1.77567	−0.887833	+	0.460166i	$0.847790\pi$
$432$	1.39627e6	0.359964
$433$	−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
$435$	−509079.	−0.128992
$436$	−2.64257e6	−0.665748
$437$	−5.70797e6	−1.42981
$438$	−1.54397e6	−0.384551
$439$	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	$-0.738250\pi$
$443$	−5.62193e6	−1.36106	−0.680528	+	0.732722i	$0.738250\pi$
$444$	−6.46801e6	−1.55709
$445$	−1.48449e6	−0.355367
$446$	−6.34111e6	−1.50948
$447$	1.26246e6	0.298846
$448$	0	0
$449$	1.94883e6	0.456202	0.228101	−	0.973637i	$-0.426748\pi$
$449$	1.94883e6	0.456202	0.228101	+	0.973637i	$0.426748\pi$
$450$	−2.12068e6	−0.493679
$451$	−2.08422e6	−0.482505
$452$	−7.01418e6	−1.61484
$453$	−1.47592e6	−0.337923
$454$	1.02467e7	2.33315
$455$	0	0
$456$	−5.17441e6	−1.16533
$457$	−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
$459$	409521.	0.0907286
$460$	7.08663e6	1.56151
$461$	−4.65262e6	−1.01964	−0.509819	−	0.860282i	$-0.670287\pi$
$461$	−4.65262e6	−1.01964	−0.509819	+	0.860282i	$0.670287\pi$
$462$	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
$465$	630777.	0.135283
$466$	6.09509e6	1.30021
$467$	−5.05980e6	−1.07360	−0.536799	−	0.843710i	$-0.680367\pi$
$467$	−5.05980e6	−1.07360	−0.536799	+	0.843710i	$0.680367\pi$
$468$	−5.98808e6	−1.26378
$469$	0	0
$470$	−758064.	−0.158293
$471$	5.00889e6	1.04037
$472$	1.13932e7	2.35392
$473$	2.41212e6	0.495730
$474$	−445835.	−0.0911441
$475$	3.55331e6	0.722603
$476$	0	0
$477$	92420.8	0.0185983
$478$	7.59745e6	1.52089
$479$	−5.55085e6	−1.10540	−0.552702	−	0.833379i	$-0.686403\pi$
$479$	−5.55085e6	−1.10540	−0.552702	+	0.833379i	$0.686403\pi$
$480$	1.35169e6	0.267778
$481$	1.01002e7	1.99052
$482$	−1.20114e7	−2.35491
$483$	0	0
$484$	4.03711e6	0.783353
$485$	−1.51196e6	−0.291867
$486$	603708.	0.115941
$487$	−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
$489$	−177540.	−0.0335756
$490$	0	0
$491$	7.44459e6	1.39360	0.696798	−	0.717267i	$-0.254607\pi$
$491$	7.44459e6	1.39360	0.696798	+	0.717267i	$0.254607\pi$
$492$	−2.92242e6	−0.544289
$493$	−1.33775e6	−0.247890
$494$	1.44602e7	2.66598
$495$	895662.	0.164298
$496$	5.65145e6	1.03147
$497$	0	0
$498$	5.53426e6	0.999968
$499$	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
$501$	852986.	0.151826
$502$	3.60894e6	0.639175
$503$	8.61012e6	1.51736	0.758681	−	0.651463i	$-0.225844\pi$
$503$	8.61012e6	1.51736	0.758681	+	0.651463i	$0.225844\pi$
$504$	0	0
$505$	4.38531e6	0.765195
$506$	1.95786e7	3.39943
$507$	6.00911e6	1.03822
$508$	1.69154e6	0.290820
$509$	3.34201e6	0.571759	0.285879	−	0.958266i	$-0.407714\pi$
$509$	3.34201e6	0.571759	0.285879	+	0.958266i	$0.407714\pi$
$510$	1.22778e6	0.209024
$511$	0	0
$512$	−1.32848e7	−2.23964
$513$	−1.01154e6	−0.169704
$514$	102368.	0.0170906
$515$	1.23619e6	0.205383
$516$	3.38219e6	0.559208
$517$	−1.45318e6	−0.239108
$518$	0	0
$519$	3.04957e6	0.496958
$520$	−1.00318e7	−1.62693
$521$	−2.68305e6	−0.433046	−0.216523	−	0.976278i	$-0.569472\pi$
$521$	−2.68305e6	−0.433046	−0.216523	+	0.976278i	$0.569472\pi$
$522$	−1.97209e6	−0.316775
$523$	−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
$528$	8.02468e6	1.25269
$529$	1.04855e7	1.62911
$530$	277086.	0.0428474
$531$	2.22726e6	0.342796
$532$	0	0
$533$	4.56353e6	0.695798
$534$	−5.75068e6	−0.872702
$535$	1.14436e6	0.172853
$536$	−2.30327e7	−3.46285
$537$	6.98574e6	1.04539
$538$	−1.96009e7	−2.91958
$539$	0	0
$540$	1.25587e6	0.185336
$541$	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
$543$	−1.19427e6	−0.173821
$544$	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
$549$	−1.70939e6	−0.242052
$550$	−1.21881e7	−1.71802
$551$	3.30434e6	0.463667
$552$	1.53401e7	2.14279
$553$	0	0
$554$	−4.03639e6	−0.558752
$555$	−2.11829e6	−0.291913
$556$	1.53705e7	2.10863
$557$	5.47656e6	0.747946	0.373973	−	0.927440i	$-0.377995\pi$
$557$	5.47656e6	0.747946	0.373973	+	0.927440i	$0.377995\pi$
$558$	2.44353e6	0.332225
$559$	−5.28149e6	−0.714869
$560$	0	0
$561$	2.35361e6	0.315739
$562$	−1.81677e7	−2.42638
$563$	398976.	0.0530488	0.0265244	−	0.999648i	$-0.491556\pi$
$563$	398976.	0.0530488	0.0265244	+	0.999648i	$0.491556\pi$
$564$	−2.03760e6	−0.269725
$565$	−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	$-0.588325\pi$
$569$	−4.23113e6	−0.547868	−0.273934	+	0.961749i	$0.588325\pi$
$570$	−3.03270e6	−0.390969
$571$	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
$573$	59736.7	0.00760072
$574$	0	0
$575$	−1.05342e7	−1.32871
$576$	271728.	0.0341254
$577$	−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
$579$	4.07331e6	0.504953
$580$	−4.10245e6	−0.506376
$581$	0	0
$582$	−5.85709e6	−0.716761
$583$	531164.	0.0647228
$584$	−6.95254e6	−0.843551
$585$	−1.96111e6	−0.236926
$586$	−1.52231e7	−1.83129
$587$	−8.65009e6	−1.03616	−0.518078	−	0.855333i	$-0.673352\pi$
$587$	−8.65009e6	−1.03616	−0.518078	+	0.855333i	$0.673352\pi$
$588$	0	0
$589$	−4.09426e6	−0.486281
$590$	6.67754e6	0.789744
$591$	7.35257e6	0.865905
$592$	−1.89788e7	−2.22569
$593$	1.54514e7	1.80439	0.902194	−	0.431329i	$-0.141955\pi$
$593$	1.54514e7	1.80439	0.902194	+	0.431329i	$0.141955\pi$
$594$	3.46965e6	0.403478
$595$	0	0
$596$	1.01736e7	1.17316
$597$	7.25775e6	0.833425
$598$	−4.28687e7	−4.90216
$599$	−3.84801e6	−0.438196	−0.219098	−	0.975703i	$-0.570312\pi$
$599$	−3.84801e6	−0.438196	−0.219098	+	0.975703i	$0.570312\pi$
$600$	−9.54948e6	−1.08293
$601$	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
$603$	−4.50266e6	−0.504285
$604$	−1.18938e7	−1.32657
$605$	1.32216e6	0.146858
$606$	1.69880e7	1.87915
$607$	−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
$612$	3.30015e6	0.356168
$613$	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
$615$	−957099.	−0.102040
$616$	0	0
$617$	1.18485e7	1.25300	0.626500	−	0.779421i	$-0.284487\pi$
$617$	1.18485e7	1.25300	0.626500	+	0.779421i	$0.284487\pi$
$618$	4.78879e6	0.504376
$619$	−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
$621$	2.99883e6	0.312049
$622$	2.96272e6	0.307054
$623$	0	0
$624$	−1.75706e7	−1.80644
$625$	4.79460e6	0.490967
$626$	2.23095e6	0.227538
$627$	−5.81358e6	−0.590575
$628$	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
$633$	618965.	0.0613984
$634$	1.31952e7	1.30375
$635$	553984.	0.0545209
$636$	744780.	0.0730104
$637$	0	0
$638$	−1.13341e7	−1.10239
$639$	−492229.	−0.0476887
$640$	−3.99135e6	−0.385185
$641$	812551.	0.0781098	0.0390549	−	0.999237i	$-0.487565\pi$
$641$	812551.	0.0781098	0.0390549	+	0.999237i	$0.487565\pi$
$642$	4.43306e6	0.424489
$643$	−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
$645$	1.10767e6	0.104837
$646$	−7.96930e6	−0.751344
$647$	−2.43380e6	−0.228573	−0.114286	−	0.993448i	$-0.536458\pi$
$647$	−2.43380e6	−0.228573	−0.114286	+	0.993448i	$0.536458\pi$
$648$	2.71851e6	0.254327
$649$	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	$-0.643037\pi$
$653$	−9.46664e6	−0.868786	−0.434393	+	0.900723i	$0.643037\pi$
$654$	−3.35261e6	−0.306506
$655$	−8.04646e6	−0.732827
$656$	−8.57513e6	−0.778003
$657$	−1.35915e6	−0.122844
$658$	0	0
$659$	1.40662e7	1.26172	0.630860	−	0.775896i	$-0.282702\pi$
$659$	1.40662e7	1.26172	0.630860	+	0.775896i	$0.282702\pi$
$660$	7.21776e6	0.644974
$661$	−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
$663$	−5.15339e6	−0.455312
$664$	2.49209e7	2.19353
$665$	0	0
$666$	−8.20592e6	−0.716872
$667$	−9.79606e6	−0.852583
$668$	6.87385e6	0.596017
$669$	−5.58204e6	−0.482201
$670$	−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
$675$	−1.86683e6	−0.157705
$676$	4.84249e7	4.07570
$677$	1.39274e7	1.16788	0.583938	−	0.811798i	$-0.301511\pi$
$677$	1.39274e7	1.16788	0.583938	+	0.811798i	$0.301511\pi$
$678$	−8.89884e6	−0.743462
$679$	0	0
$680$	5.52872e6	0.458513
$681$	9.02009e6	0.745321
$682$	1.40436e7	1.15615
$683$	1.83101e7	1.50189	0.750945	−	0.660365i	$-0.229598\pi$
$683$	1.83101e7	1.50189	0.750945	+	0.660365i	$0.229598\pi$
$684$	−8.15160e6	−0.666197
$685$	−1.49350e6	−0.121613
$686$	0	0
$687$	7.97275e6	0.644490
$688$	9.92420e6	0.799327
$689$	−1.16302e6	−0.0933337
$690$	8.99076e6	0.718908
$691$	−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
$696$	−8.88036e6	−0.694877
$697$	−2.51506e6	−0.196095
$698$	1.57826e7	1.22614
$699$	5.36547e6	0.415351
$700$	0	0
$701$	8.22993e6	0.632559	0.316279	−	0.948666i	$-0.397566\pi$
$701$	8.22993e6	0.632559	0.316279	+	0.948666i	$0.397566\pi$
$702$	−7.59703e6	−0.581837
$703$	1.37494e7	1.04929
$704$	1.56169e6	0.118758
$705$	−667320.	−0.0505663
$706$	−1.71552e7	−1.29534
$707$	0	0
$708$	1.79486e7	1.34570
$709$	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
$711$	−392466.	−0.0291158
$712$	−2.58954e7	−1.91436
$713$	1.21379e7	0.894167
$714$	0	0
$715$	−1.12710e7	−0.824510
$716$	5.62951e7	4.10382
$717$	6.68800e6	0.485845
$718$	−2.55569e7	−1.85011
$719$	−1.88254e7	−1.35807	−0.679034	−	0.734107i	$-0.737601\pi$
$719$	−1.88254e7	−1.35807	−0.679034	+	0.734107i	$0.737601\pi$
$720$	3.68503e6	0.264917
$721$	0	0
$722$	−5.63055e6	−0.401983
$723$	−1.05735e7	−0.752272
$724$	−9.62411e6	−0.682361
$725$	6.09823e6	0.430882
$726$	5.12186e6	0.360650
$727$	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
$729$	531441.	0.0370370
$730$	−4.07486e6	−0.283012
$731$	2.91073e6	0.201469
$732$	−1.37752e7	−0.950212
$733$	−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
$738$	−3.70765e6	−0.250587
$739$	−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
$741$	1.27292e7	0.851641
$742$	0	0
$743$	−1.36977e7	−0.910281	−0.455141	−	0.890420i	$-0.650411\pi$
$743$	−1.36977e7	−0.910281	−0.455141	+	0.890420i	$0.650411\pi$
$744$	1.10033e7	0.728768
$745$	3.33188e6	0.219937
$746$	2.79341e7	1.83775
$747$	4.87178e6	0.319438
$748$	1.89668e7	1.23948
$749$	0	0
$750$	−1.24269e7	−0.806698
$751$	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
$753$	3.17693e6	0.204183
$754$	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
$759$	1.72350e7	1.08594
$760$	−1.36563e7	−0.857629
$761$	−3.02125e7	−1.89115	−0.945574	−	0.325406i	$-0.894499\pi$
$761$	−3.02125e7	−1.89115	−0.945574	+	0.325406i	$0.894499\pi$
$762$	2.14605e6	0.133891
$763$	0	0
$764$	481393.	0.0298378
$765$	1.08081e6	0.0667721
$766$	1.54277e7	0.950014
$767$	−2.80278e7	−1.72028
$768$	−1.64280e7	−1.00504
$769$	−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
$771$	90114.1	0.00545955
$772$	3.28251e7	1.98227
$773$	2.55409e7	1.53740	0.768701	−	0.639608i	$-0.220903\pi$
$773$	2.55409e7	1.53740	0.768701	+	0.639608i	$0.220903\pi$
$774$	4.29095e6	0.257455
$775$	−7.55605e6	−0.451898
$776$	−2.63746e7	−1.57228
$777$	0	0
$778$	4.60140e7	2.72547
$779$	6.21236e6	0.366786
$780$	−1.58038e7	−0.930087
$781$	−2.82896e6	−0.165958
$782$	2.36258e7	1.38156
$783$	−1.73602e6	−0.101193
$784$	0	0
$785$	1.32195e7	0.765667
$786$	−3.11707e7	−1.79966
$787$	−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
$789$	1.51362e7	0.865614
$790$	−1.17665e6	−0.0670779
$791$	0	0
$792$	1.56239e7	0.885068
$793$	2.15108e7	1.21471
$794$	−3.60018e7	−2.02662
$795$	243917.	0.0136875
$796$	5.84871e7	3.27173
$797$	−2.07624e7	−1.15780	−0.578899	−	0.815399i	$-0.696517\pi$
$797$	−2.07624e7	−1.15780	−0.578899	+	0.815399i	$0.696517\pi$
$798$	0	0
$799$	−1.75358e6	−0.0971757
$800$	−1.61918e7	−0.894481
$801$	−5.06229e6	−0.278783
$802$	−9.27413e6	−0.509140
$803$	−7.81136e6	−0.427501
$804$	−3.62850e7	−1.97965
$805$	0	0
$806$	−3.07493e7	−1.66724
$807$	−1.72546e7	−0.932653
$808$	7.64973e7	4.12209
$809$	1.37021e7	0.736067	0.368034	−	0.929813i	$-0.380031\pi$
$809$	1.37021e7	0.736067	0.368034	+	0.929813i	$0.380031\pi$
$810$	1.59331e6	0.0853271
$811$	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
$813$	2.06100e7	1.09358
$814$	−4.71613e7	−2.49474
$815$	−468563.	−0.0247101
$816$	9.68350e6	0.509104
$817$	−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	$-0.374549\pi$
$821$	1.48323e7	0.767982	0.383991	+	0.923337i	$0.374549\pi$
$822$	−5.78558e6	−0.298653
$823$	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
$825$	−1.07291e7	−0.548817
$826$	0	0
$827$	−1.34267e7	−0.682662	−0.341331	−	0.939943i	$-0.610878\pi$
$827$	−1.34267e7	−0.682662	−0.341331	+	0.939943i	$0.610878\pi$
$828$	2.41663e7	1.22499
$829$	−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
$831$	−3.55322e6	−0.178492
$832$	−3.41941e6	−0.171255
$833$	0	0
$834$	1.95004e7	0.970796
$835$	2.25120e6	0.111737
$836$	−4.68492e7	−2.31839
$837$	2.15103e6	0.106129
$838$	−1.55444e7	−0.764654
$839$	2.76635e7	1.35676	0.678379	−	0.734712i	$-0.262683\pi$
$839$	2.76635e7	1.35676	0.678379	+	0.734712i	$0.262683\pi$
$840$	0	0
$841$	−1.48402e7	−0.723519
$842$	−4.51946e7	−2.19688
$843$	−1.59929e7	−0.775103
$844$	4.98798e6	0.241028
$845$	1.58593e7	0.764084
$846$	−2.58509e6	−0.124180
$847$	0	0
$848$	2.18537e6	0.104361
$849$	1.09372e7	0.520759
$850$	−1.47075e7	−0.698219
$851$	−4.07616e7	−1.92942
$852$	−3.96667e6	−0.187209
$853$	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
$855$	−2.66967e6	−0.124894
$856$	1.99622e7	0.931158
$857$	−6.40626e6	−0.297956	−0.148978	−	0.988840i	$-0.547598\pi$
$857$	−6.40626e6	−0.297956	−0.148978	+	0.988840i	$0.547598\pi$
$858$	−4.36619e7	−2.02481
$859$	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	$-0.494044\pi$
$863$	818774.	0.0374229	0.0187114	+	0.999825i	$0.494044\pi$
$864$	4.60943e6	0.210069
$865$	8.04843e6	0.365739
$866$	−4.08267e7	−1.84991
$867$	−9.93858e6	−0.449031
$868$	0	0
$869$	−2.25560e6	−0.101324
$870$	−5.20475e6	−0.233132
$871$	5.66613e7	2.53070
$872$	−1.50968e7	−0.672350
$873$	−5.15596e6	−0.228968
$874$	−5.83574e7	−2.58415
$875$	0	0
$876$	−1.09528e7	−0.482242
$877$	−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
$879$	−1.34008e7	−0.585002
$880$	2.11788e7	0.921921
$881$	−1.24325e7	−0.539658	−0.269829	−	0.962908i	$-0.586967\pi$
$881$	−1.24325e7	−0.539658	−0.269829	+	0.962908i	$0.586967\pi$
$882$	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
$885$	5.87820e6	0.252282
$886$	−5.74778e7	−2.45989
$887$	−4.75528e6	−0.202940	−0.101470	−	0.994839i	$-0.532355\pi$
$887$	−4.75528e6	−0.202940	−0.101470	+	0.994839i	$0.532355\pi$
$888$	−3.69514e7	−1.57253
$889$	0	0
$890$	−1.51772e7	−0.642269
$891$	3.05432e6	0.128890
$892$	−4.49833e7	−1.89295
$893$	4.33145e6	0.181763
$894$	1.29072e7	0.540116
$895$	1.84368e7	0.769356
$896$	0	0
$897$	−3.77371e7	−1.56598
$898$	1.99245e7	0.824512
$899$	−7.02661e6	−0.289966
$900$	−1.50440e7	−0.619093
$901$	640963.	0.0263040
$902$	−2.13087e7	−0.872050
$903$	0	0
$904$	−4.00716e7	−1.63086
$905$	−3.15192e6	−0.127924
$906$	−1.50896e7	−0.610742
$907$	−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
$909$	1.49544e7	0.600289
$910$	0	0
$911$	−1.22591e6	−0.0489399	−0.0244700	−	0.999701i	$-0.507790\pi$
$911$	−1.22591e6	−0.0489399	−0.0244700	+	0.999701i	$0.507790\pi$
$912$	−2.39189e7	−0.952257
$913$	2.79993e7	1.11165
$914$	−2.73296e7	−1.08210
$915$	−4.51141e6	−0.178139
$916$	6.42490e7	2.53004
$917$	0	0
$918$	4.18688e6	0.163977
$919$	−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
$921$	1.82798e7	0.710103
$922$	−4.75677e7	−1.84283
$923$	6.19419e6	0.239321
$924$	0	0
$925$	2.53749e7	0.975101
$926$	−5.30195e7	−2.03193
$927$	4.21554e6	0.161122
$928$	−1.50573e7	−0.573954
$929$	6.77240e6	0.257456	0.128728	−	0.991680i	$-0.458911\pi$
$929$	6.77240e6	0.257456	0.128728	+	0.991680i	$0.458911\pi$
$930$	6.44897e6	0.244502
$931$	0	0
$932$	4.32380e7	1.63052
$933$	2.60806e6	0.0980875
$934$	−5.17307e7	−1.94035
$935$	6.21166e6	0.232369
$936$	−3.42096e7	−1.27632
$937$	−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
$939$	1.96389e6	0.0726866
$940$	−5.37765e6	−0.198505
$941$	−5.62524e6	−0.207094	−0.103547	−	0.994625i	$-0.533019\pi$
$941$	−5.62524e6	−0.207094	−0.103547	+	0.994625i	$0.533019\pi$
$942$	5.12102e7	1.88031
$943$	−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	$-0.341171\pi$
$947$	2.64126e7	0.957052	0.478526	+	0.878073i	$0.341171\pi$
$948$	−3.16272e6	−0.114298
$949$	1.71035e7	0.616480
$950$	3.63286e7	1.30599
$951$	1.16157e7	0.416479
$952$	0	0
$953$	4.21824e7	1.50452	0.752262	−	0.658864i	$-0.228963\pi$
$953$	4.21824e7	1.50452	0.752262	+	0.658864i	$0.228963\pi$
$954$	944896.	0.0336135
$955$	157657.	0.00559378
$956$	5.38957e7	1.90726
$957$	−9.97732e6	−0.352155
$958$	−5.67511e7	−1.99784
$959$	0	0
$960$	717145.	0.0251148
$961$	−1.99228e7	−0.695892
$962$	1.03263e8	3.59755
$963$	3.90240e6	0.135602
$964$	−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
$969$	−7.01533e6	−0.240015
$970$	−1.54580e7	−0.527503
$971$	1.64551e7	0.560084	0.280042	−	0.959988i	$-0.409652\pi$
$971$	1.64551e7	0.560084	0.280042	+	0.959988i	$0.409652\pi$
$972$	4.28266e6	0.145394
$973$	0	0
$974$	−7.03207e7	−2.37512
$975$	2.34920e7	0.791424
$976$	−4.04200e7	−1.35823
$977$	1.71176e7	0.573729	0.286865	−	0.957971i	$-0.407387\pi$
$977$	1.71176e7	0.573729	0.286865	+	0.957971i	$0.407387\pi$
$978$	−1.81514e6	−0.0606825
$979$	−2.90942e7	−0.970174
$980$	0	0
$981$	−2.95128e6	−0.0979125
$982$	7.61124e7	2.51870
$983$	−2.33803e7	−0.771730	−0.385865	−	0.922555i	$-0.626097\pi$
$983$	−2.33803e7	−0.771730	−0.385865	+	0.922555i	$0.626097\pi$
$984$	−1.66956e7	−0.549686
$985$	1.94049e7	0.637266
$986$	−1.36770e7	−0.448021
$987$	0	0
$988$	1.02579e8	3.34324
$989$	2.13146e7	0.692927
$990$	9.15712e6	0.296941
$991$	−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
$993$	−3.13705e7	−1.00960
$994$	0	0
$995$	1.91547e7	0.613362
$996$	3.92596e7	1.25400
$997$	−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
$999$	−7.22363e6	−0.229003

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.6.a.m.1.4		4
3.2	odd	2		441.6.a.w.1.1		4
7.2	even	3		21.6.e.c.4.1	✓	8
7.3	odd	6		147.6.e.o.79.1		8
7.4	even	3		21.6.e.c.16.1	yes	8
7.5	odd	6		147.6.e.o.67.1		8
7.6	odd	2		147.6.a.l.1.4		4
21.2	odd	6		63.6.e.e.46.4		8
21.11	odd	6		63.6.e.e.37.4		8
21.20	even	2		441.6.a.v.1.1		4
28.11	odd	6		336.6.q.j.289.2		8
28.23	odd	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	7.2	even	3
21.6.e.c.16.1	yes	8	7.4	even	3
63.6.e.e.37.4		8	21.11	odd	6
63.6.e.e.46.4		8	21.2	odd	6
147.6.a.l.1.4		4	7.6	odd	2
147.6.a.m.1.4		4	1.1	even	1	trivial
147.6.e.o.67.1		8	7.5	odd	6
147.6.e.o.79.1		8	7.3	odd	6
336.6.q.j.193.2		8	28.23	odd	6
336.6.q.j.289.2		8	28.11	odd	6
441.6.a.v.1.1		4	21.20	even	2
441.6.a.w.1.1		4	3.2	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	147.6.a.m.1.4

Level	$147$
Weight	$6$
Character	147.1
Self dual	yes
Analytic conductor	$23.576$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$