LMFDB - Embedded newform 6223.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6223,2,Mod(1,6223)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6223, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6223.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.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:	$49.6909051778$
Analytic rank:	$1$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6223.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55719 q^{2} -3.02541 q^{3} +4.53922 q^{4} -0.525434 q^{5} +7.73654 q^{6} -6.49328 q^{8} +6.15308 q^{9} +O(q^{10})$	\(q-2.55719 q^{2} -3.02541 q^{3} +4.53922 q^{4} -0.525434 q^{5} +7.73654 q^{6} -6.49328 q^{8} +6.15308 q^{9} +1.34364 q^{10} -4.42977 q^{11} -13.7330 q^{12} -3.39489 q^{13} +1.58965 q^{15} +7.52610 q^{16} +2.70492 q^{17} -15.7346 q^{18} +4.77157 q^{19} -2.38506 q^{20} +11.3278 q^{22} -6.89370 q^{23} +19.6448 q^{24} -4.72392 q^{25} +8.68138 q^{26} -9.53934 q^{27} +1.09077 q^{29} -4.06504 q^{30} -2.36460 q^{31} -6.25912 q^{32} +13.4019 q^{33} -6.91698 q^{34} +27.9302 q^{36} +6.00818 q^{37} -12.2018 q^{38} +10.2709 q^{39} +3.41179 q^{40} -0.204776 q^{41} +4.35925 q^{43} -20.1077 q^{44} -3.23304 q^{45} +17.6285 q^{46} +2.94217 q^{47} -22.7695 q^{48} +12.0800 q^{50} -8.18347 q^{51} -15.4102 q^{52} -14.1433 q^{53} +24.3939 q^{54} +2.32756 q^{55} -14.4359 q^{57} -2.78930 q^{58} +9.54194 q^{59} +7.21579 q^{60} +10.5961 q^{61} +6.04674 q^{62} +0.953566 q^{64} +1.78379 q^{65} -34.2711 q^{66} -5.03179 q^{67} +12.2782 q^{68} +20.8562 q^{69} +8.93842 q^{71} -39.9536 q^{72} -6.94434 q^{73} -15.3641 q^{74} +14.2918 q^{75} +21.6592 q^{76} -26.2647 q^{78} +10.1317 q^{79} -3.95447 q^{80} +10.4011 q^{81} +0.523650 q^{82} -1.24723 q^{83} -1.42126 q^{85} -11.1474 q^{86} -3.30001 q^{87} +28.7638 q^{88} -5.80806 q^{89} +8.26750 q^{90} -31.2920 q^{92} +7.15388 q^{93} -7.52369 q^{94} -2.50715 q^{95} +18.9364 q^{96} -9.20742 q^{97} -27.2567 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 14 q^{2} + 46 q^{4} - 42 q^{8} + 14 q^{9}+O(q^{10}) $ 54 * q - 14 * q^2 + 46 * q^4 - 42 * q^8 + 14 * q^9	$ 54 q - 14 q^{2} + 46 q^{4} - 42 q^{8} + 14 q^{9} - 20 q^{11} - 16 q^{15} + 38 q^{16} - 42 q^{18} - 28 q^{22} - 64 q^{23} - 10 q^{25} - 44 q^{29} - 12 q^{30} - 78 q^{32} - 38 q^{36} - 32 q^{37} - 40 q^{39} - 52 q^{43} - 48 q^{44} - 8 q^{46} - 66 q^{50} - 24 q^{51} - 136 q^{53} - 80 q^{57} - 4 q^{58} - 28 q^{60} - 10 q^{64} - 88 q^{65} - 48 q^{67} - 52 q^{71} - 22 q^{72} - 28 q^{74} - 36 q^{78} + 24 q^{79} - 66 q^{81} - 48 q^{85} + 64 q^{86} - 68 q^{88} - 192 q^{92} - 88 q^{93} - 112 q^{95} - 72 q^{99}+O(q^{100}) $ 54 * q - 14 * q^2 + 46 * q^4 - 42 * q^8 + 14 * q^9 - 20 * q^11 - 16 * q^15 + 38 * q^16 - 42 * q^18 - 28 * q^22 - 64 * q^23 - 10 * q^25 - 44 * q^29 - 12 * q^30 - 78 * q^32 - 38 * q^36 - 32 * q^37 - 40 * q^39 - 52 * q^43 - 48 * q^44 - 8 * q^46 - 66 * q^50 - 24 * q^51 - 136 * q^53 - 80 * q^57 - 4 * q^58 - 28 * q^60 - 10 * q^64 - 88 * q^65 - 48 * q^67 - 52 * q^71 - 22 * q^72 - 28 * q^74 - 36 * q^78 + 24 * q^79 - 66 * q^81 - 48 * q^85 + 64 * q^86 - 68 * q^88 - 192 * q^92 - 88 * q^93 - 112 * q^95 - 72 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.55719	−1.80821	−0.904103	−	0.427314i	$-0.859460\pi$
$2$	−2.55719	−1.80821	−0.904103	+	0.427314i	$0.859460\pi$
$3$	−3.02541	−1.74672	−0.873359	−	0.487076i	$-0.838063\pi$
$3$	−3.02541	−1.74672	−0.873359	+	0.487076i	$0.838063\pi$
$4$	4.53922	2.26961
$5$	−0.525434	−0.234981	−0.117491	−	0.993074i	$-0.537485\pi$
$5$	−0.525434	−0.234981	−0.117491	+	0.993074i	$0.537485\pi$
$6$	7.73654	3.15843
$7$	0	0
$7$	0	0
$8$	−6.49328	−2.29572
$9$	6.15308	2.05103
$10$	1.34364	0.424895
$11$	−4.42977	−1.33563	−0.667814	−	0.744329i	$-0.732770\pi$
$11$	−4.42977	−1.33563	−0.667814	+	0.744329i	$0.732770\pi$
$12$	−13.7330	−3.96437
$13$	−3.39489	−0.941573	−0.470787	−	0.882247i	$-0.656030\pi$
$13$	−3.39489	−0.941573	−0.470787	+	0.882247i	$0.656030\pi$
$14$	0	0
$15$	1.58965	0.410446
$16$	7.52610	1.88153
$17$	2.70492	0.656038	0.328019	−	0.944671i	$-0.393619\pi$
$17$	2.70492	0.656038	0.328019	+	0.944671i	$0.393619\pi$
$18$	−15.7346	−3.70868
$19$	4.77157	1.09467	0.547336	−	0.836913i	$-0.315642\pi$
$19$	4.77157	1.09467	0.547336	+	0.836913i	$0.315642\pi$
$20$	−2.38506	−0.533317
$21$	0	0
$22$	11.3278	2.41509
$23$	−6.89370	−1.43744	−0.718718	−	0.695302i	$-0.755271\pi$
$23$	−6.89370	−1.43744	−0.718718	+	0.695302i	$0.755271\pi$
$24$	19.6448	4.00998
$25$	−4.72392	−0.944784
$26$	8.68138	1.70256
$27$	−9.53934	−1.83585
$28$	0	0
$29$	1.09077	0.202550	0.101275	−	0.994858i	$-0.467708\pi$
$29$	1.09077	0.202550	0.101275	+	0.994858i	$0.467708\pi$
$30$	−4.06504	−0.742172
$31$	−2.36460	−0.424695	−0.212347	−	0.977194i	$-0.568111\pi$
$31$	−2.36460	−0.424695	−0.212347	+	0.977194i	$0.568111\pi$
$32$	−6.25912	−1.10647
$33$	13.4019	2.33297
$34$	−6.91698	−1.18625
$35$	0	0
$36$	27.9302	4.65503
$37$	6.00818	0.987739	0.493870	−	0.869536i	$-0.335582\pi$
$37$	6.00818	0.987739	0.493870	+	0.869536i	$0.335582\pi$
$38$	−12.2018	−1.97939
$39$	10.2709	1.64466
$40$	3.41179	0.539452
$41$	−0.204776	−0.0319806	−0.0159903	−	0.999872i	$-0.505090\pi$
$41$	−0.204776	−0.0319806	−0.0159903	+	0.999872i	$0.505090\pi$
$42$	0	0
$43$	4.35925	0.664779	0.332390	−	0.943142i	$-0.392145\pi$
$43$	4.35925	0.664779	0.332390	+	0.943142i	$0.392145\pi$
$44$	−20.1077	−3.03136
$45$	−3.23304	−0.481953
$46$	17.6285	2.59918
$47$	2.94217	0.429160	0.214580	−	0.976706i	$-0.431162\pi$
$47$	2.94217	0.429160	0.214580	+	0.976706i	$0.431162\pi$
$48$	−22.7695	−3.28650
$49$	0	0
$50$	12.0800	1.70836
$51$	−8.18347	−1.14591
$52$	−15.4102	−2.13701
$53$	−14.1433	−1.94273	−0.971366	−	0.237589i	$-0.923643\pi$
$53$	−14.1433	−1.94273	−0.971366	+	0.237589i	$0.923643\pi$
$54$	24.3939	3.31959
$55$	2.32756	0.313848
$56$	0	0
$57$	−14.4359	−1.91208
$58$	−2.78930	−0.366253
$59$	9.54194	1.24226	0.621128	−	0.783709i	$-0.286675\pi$
$59$	9.54194	1.24226	0.621128	+	0.783709i	$0.286675\pi$
$60$	7.21579	0.931554
$61$	10.5961	1.35669	0.678344	−	0.734745i	$-0.262698\pi$
$61$	10.5961	1.35669	0.678344	+	0.734745i	$0.262698\pi$
$62$	6.04674	0.767936
$63$	0	0
$64$	0.953566	0.119196
$65$	1.78379	0.221252
$66$	−34.2711	−4.21848
$67$	−5.03179	−0.614731	−0.307366	−	0.951592i	$-0.599447\pi$
$67$	−5.03179	−0.614731	−0.307366	+	0.951592i	$0.599447\pi$
$68$	12.2782	1.48895
$69$	20.8562	2.51079
$70$	0	0
$71$	8.93842	1.06079	0.530397	−	0.847749i	$-0.322043\pi$
$71$	8.93842	1.06079	0.530397	+	0.847749i	$0.322043\pi$
$72$	−39.9536	−4.70858
$73$	−6.94434	−0.812773	−0.406387	−	0.913701i	$-0.633211\pi$
$73$	−6.94434	−0.812773	−0.406387	+	0.913701i	$0.633211\pi$
$74$	−15.3641	−1.78604
$75$	14.2918	1.65027
$76$	21.6592	2.48448
$77$	0	0
$78$	−26.2647	−2.97389
$79$	10.1317	1.13990	0.569952	−	0.821678i	$-0.306962\pi$
$79$	10.1317	1.13990	0.569952	+	0.821678i	$0.306962\pi$
$80$	−3.95447	−0.442124
$81$	10.4011	1.15568
$82$	0.523650	0.0578275
$83$	−1.24723	−0.136901	−0.0684506	−	0.997655i	$-0.521806\pi$
$83$	−1.24723	−0.136901	−0.0684506	+	0.997655i	$0.521806\pi$
$84$	0	0
$85$	−1.42126	−0.154157
$86$	−11.1474	−1.20206
$87$	−3.30001	−0.353798
$88$	28.7638	3.06623
$89$	−5.80806	−0.615653	−0.307827	−	0.951442i	$-0.599602\pi$
$89$	−5.80806	−0.615653	−0.307827	+	0.951442i	$0.599602\pi$
$90$	8.26750	0.871471
$91$	0	0
$92$	−31.2920	−3.26242
$93$	7.15388	0.741823
$94$	−7.52369	−0.776009
$95$	−2.50715	−0.257228
$96$	18.9364	1.93269
$97$	−9.20742	−0.934872	−0.467436	−	0.884027i	$-0.654822\pi$
$97$	−9.20742	−0.934872	−0.467436	+	0.884027i	$0.654822\pi$
$98$	0	0
$99$	−27.2567	−2.73941
$100$	−21.4429	−2.14429
$101$	−13.7266	−1.36584	−0.682921	−	0.730492i	$-0.739291\pi$
$101$	−13.7266	−1.36584	−0.682921	+	0.730492i	$0.739291\pi$
$102$	20.9267	2.07205
$103$	13.7804	1.35782	0.678912	−	0.734219i	$-0.262452\pi$
$103$	13.7804	1.35782	0.678912	+	0.734219i	$0.262452\pi$
$104$	22.0440	2.16159
$105$	0	0
$106$	36.1671	3.51286
$107$	−2.66363	−0.257502	−0.128751	−	0.991677i	$-0.541097\pi$
$107$	−2.66363	−0.257502	−0.128751	+	0.991677i	$0.541097\pi$
$108$	−43.3012	−4.16666
$109$	−15.9811	−1.53071	−0.765356	−	0.643607i	$-0.777437\pi$
$109$	−15.9811	−1.53071	−0.765356	+	0.643607i	$0.777437\pi$
$110$	−5.95200	−0.567501
$111$	−18.1772	−1.72530
$112$	0	0
$113$	−10.6420	−1.00112	−0.500558	−	0.865703i	$-0.666872\pi$
$113$	−10.6420	−1.00112	−0.500558	+	0.865703i	$0.666872\pi$
$114$	36.9154	3.45744
$115$	3.62219	0.337771
$116$	4.95123	0.459710
$117$	−20.8890	−1.93119
$118$	−24.4006	−2.24625
$119$	0	0
$120$	−10.3221	−0.942270
$121$	8.62290	0.783900
$122$	−27.0962	−2.45317
$123$	0.619529	0.0558611
$124$	−10.7335	−0.963893
$125$	5.10928	0.456988
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	10.0798	0.890937
$129$	−13.1885	−1.16118
$130$	−4.56150	−0.400070
$131$	14.4994	1.26682	0.633410	−	0.773816i	$-0.281654\pi$
$131$	14.4994	1.26682	0.633410	+	0.773816i	$0.281654\pi$
$132$	60.8341	5.29492
$133$	0	0
$134$	12.8672	1.11156
$135$	5.01230	0.431390
$136$	−17.5638	−1.50608
$137$	−5.96712	−0.509805	−0.254903	−	0.966967i	$-0.582043\pi$
$137$	−5.96712	−0.509805	−0.254903	+	0.966967i	$0.582043\pi$
$138$	−53.3333	−4.54004
$139$	1.53694	0.130362	0.0651808	−	0.997873i	$-0.479238\pi$
$139$	1.53694	0.130362	0.0651808	+	0.997873i	$0.479238\pi$
$140$	0	0
$141$	−8.90126	−0.749621
$142$	−22.8572	−1.91814
$143$	15.0386	1.25759
$144$	46.3087	3.85906
$145$	−0.573126	−0.0475955
$146$	17.7580	1.46966
$147$	0	0
$148$	27.2725	2.24178
$149$	−12.6353	−1.03513	−0.517563	−	0.855645i	$-0.673161\pi$
$149$	−12.6353	−1.03513	−0.517563	+	0.855645i	$0.673161\pi$
$150$	−36.5468	−2.98403
$151$	0.659652	0.0536817	0.0268409	−	0.999640i	$-0.491455\pi$
$151$	0.659652	0.0536817	0.0268409	+	0.999640i	$0.491455\pi$
$152$	−30.9831	−2.51306
$153$	16.6436	1.34555
$154$	0	0
$155$	1.24244	0.0997954
$156$	46.6220	3.73275
$157$	24.4894	1.95446	0.977232	−	0.212174i	$-0.0680543\pi$
$157$	24.4894	1.95446	0.977232	+	0.212174i	$0.0680543\pi$
$158$	−25.9086	−2.06118
$159$	42.7892	3.39341
$160$	3.28876	0.259999
$161$	0	0
$162$	−26.5977	−2.08971
$163$	18.2408	1.42873	0.714364	−	0.699774i	$-0.246716\pi$
$163$	18.2408	1.42873	0.714364	+	0.699774i	$0.246716\pi$
$164$	−0.929522	−0.0725835
$165$	−7.04180	−0.548203
$166$	3.18940	0.247546
$167$	−0.0726873	−0.00562471	−0.00281236	−	0.999996i	$-0.500895\pi$
$167$	−0.0726873	−0.00562471	−0.00281236	+	0.999996i	$0.500895\pi$
$168$	0	0
$169$	−1.47472	−0.113440
$170$	3.63442	0.278747
$171$	29.3598	2.24520
$172$	19.7876	1.50879
$173$	−11.4255	−0.868666	−0.434333	−	0.900752i	$-0.643016\pi$
$173$	−11.4255	−0.868666	−0.434333	+	0.900752i	$0.643016\pi$
$174$	8.43875	0.639740
$175$	0	0
$176$	−33.3389	−2.51302
$177$	−28.8682	−2.16987
$178$	14.8523	1.11323
$179$	15.2650	1.14096	0.570479	−	0.821312i	$-0.306758\pi$
$179$	15.2650	1.14096	0.570479	+	0.821312i	$0.306758\pi$
$180$	−14.6755	−1.09385
$181$	18.7528	1.39389	0.696943	−	0.717127i	$-0.254543\pi$
$181$	18.7528	1.39389	0.696943	+	0.717127i	$0.254543\pi$
$182$	0	0
$183$	−32.0574	−2.36975
$184$	44.7627	3.29995
$185$	−3.15691	−0.232100
$186$	−18.2938	−1.34137
$187$	−11.9822	−0.876223
$188$	13.3552	0.974026
$189$	0	0
$190$	6.41125	0.465121
$191$	−22.9506	−1.66065	−0.830324	−	0.557281i	$-0.811844\pi$
$191$	−22.9506	−1.66065	−0.830324	+	0.557281i	$0.811844\pi$
$192$	−2.88492	−0.208201
$193$	17.3794	1.25100	0.625498	−	0.780226i	$-0.284896\pi$
$193$	17.3794	1.25100	0.625498	+	0.780226i	$0.284896\pi$
$194$	23.5451	1.69044
$195$	−5.39670	−0.386465
$196$	0	0
$197$	12.3790	0.881968	0.440984	−	0.897515i	$-0.354629\pi$
$197$	12.3790	0.881968	0.440984	+	0.897515i	$0.354629\pi$
$198$	69.7007	4.95341
$199$	27.2074	1.92868	0.964341	−	0.264662i	$-0.0852602\pi$
$199$	27.2074	1.92868	0.964341	+	0.264662i	$0.0852602\pi$
$200$	30.6737	2.16896
$201$	15.2232	1.07376
$202$	35.1014	2.46973
$203$	0	0
$204$	−37.1466	−2.60078
$205$	0.107596	0.00751484
$206$	−35.2391	−2.45523
$207$	−42.4175	−2.94822
$208$	−25.5503	−1.77159
$209$	−21.1370	−1.46207
$210$	0	0
$211$	21.3462	1.46954	0.734768	−	0.678319i	$-0.237291\pi$
$211$	21.3462	1.46954	0.734768	+	0.678319i	$0.237291\pi$
$212$	−64.1996	−4.40925
$213$	−27.0423	−1.85291
$214$	6.81140	0.465618
$215$	−2.29050	−0.156211
$216$	61.9416	4.21459
$217$	0	0
$218$	40.8667	2.76784
$219$	21.0094	1.41969
$220$	10.5653	0.712312
$221$	−9.18289	−0.617708
$222$	46.4825	3.11970
$223$	10.1320	0.678491	0.339245	−	0.940698i	$-0.389828\pi$
$223$	10.1320	0.678491	0.339245	+	0.940698i	$0.389828\pi$
$224$	0	0
$225$	−29.0666	−1.93778
$226$	27.2137	1.81023
$227$	17.4400	1.15754	0.578768	−	0.815492i	$-0.303534\pi$
$227$	17.4400	1.15754	0.578768	+	0.815492i	$0.303534\pi$
$228$	−65.5279	−4.33969
$229$	16.0340	1.05955	0.529777	−	0.848137i	$-0.322275\pi$
$229$	16.0340	1.05955	0.529777	+	0.848137i	$0.322275\pi$
$230$	−9.26262	−0.610759
$231$	0	0
$232$	−7.08265	−0.464999
$233$	−18.1204	−1.18710	−0.593552	−	0.804796i	$-0.702275\pi$
$233$	−18.1204	−1.18710	−0.593552	+	0.804796i	$0.702275\pi$
$234$	53.4172	3.49199
$235$	−1.54592	−0.100845
$236$	43.3130	2.81944
$237$	−30.6525	−1.99109
$238$	0	0
$239$	−6.51237	−0.421250	−0.210625	−	0.977567i	$-0.567550\pi$
$239$	−6.51237	−0.421250	−0.210625	+	0.977567i	$0.567550\pi$
$240$	11.9639	0.772266
$241$	−2.43601	−0.156917	−0.0784587	−	0.996917i	$-0.525000\pi$
$241$	−2.43601	−0.156917	−0.0784587	+	0.996917i	$0.525000\pi$
$242$	−22.0504	−1.41745
$243$	−2.84963	−0.182804
$244$	48.0979	3.07915
$245$	0	0
$246$	−1.58425	−0.101008
$247$	−16.1989	−1.03071
$248$	15.3540	0.974981
$249$	3.77337	0.239128
$250$	−13.0654	−0.826329
$251$	10.7672	0.679620	0.339810	−	0.940494i	$-0.389637\pi$
$251$	10.7672	0.679620	0.339810	+	0.940494i	$0.389637\pi$
$252$	0	0
$253$	30.5375	1.91988
$254$	2.55719	0.160452
$255$	4.29987	0.269269
$256$	−27.6831	−1.73019
$257$	15.1862	0.947290	0.473645	−	0.880716i	$-0.342938\pi$
$257$	15.1862	0.947290	0.473645	+	0.880716i	$0.342938\pi$
$258$	33.7255	2.09966
$259$	0	0
$260$	8.09703	0.502157
$261$	6.71157	0.415436
$262$	−37.0778	−2.29067
$263$	22.2192	1.37009	0.685046	−	0.728500i	$-0.259782\pi$
$263$	22.2192	1.37009	0.685046	+	0.728500i	$0.259782\pi$
$264$	−87.0220	−5.35584
$265$	7.43138	0.456506
$266$	0	0
$267$	17.5717	1.07537
$268$	−22.8404	−1.39520
$269$	5.88542	0.358840	0.179420	−	0.983773i	$-0.442578\pi$
$269$	5.88542	0.358840	0.179420	+	0.983773i	$0.442578\pi$
$270$	−12.8174	−0.780042
$271$	−9.14243	−0.555363	−0.277681	−	0.960673i	$-0.589566\pi$
$271$	−9.14243	−0.555363	−0.277681	+	0.960673i	$0.589566\pi$
$272$	20.3575	1.23435
$273$	0	0
$274$	15.2591	0.921833
$275$	20.9259	1.26188
$276$	94.6711	5.69853
$277$	−16.2334	−0.975373	−0.487686	−	0.873019i	$-0.662159\pi$
$277$	−16.2334	−0.975373	−0.487686	+	0.873019i	$0.662159\pi$
$278$	−3.93025	−0.235721
$279$	−14.5496	−0.871060
$280$	0	0
$281$	11.3987	0.679988	0.339994	−	0.940428i	$-0.389575\pi$
$281$	11.3987	0.679988	0.339994	+	0.940428i	$0.389575\pi$
$282$	22.7622	1.35547
$283$	−18.8394	−1.11989	−0.559944	−	0.828530i	$-0.689178\pi$
$283$	−18.8394	−1.11989	−0.559944	+	0.828530i	$0.689178\pi$
$284$	40.5735	2.40759
$285$	7.58513	0.449304
$286$	−38.4566	−2.27398
$287$	0	0
$288$	−38.5129	−2.26939
$289$	−9.68343	−0.569614
$290$	1.46559	0.0860626
$291$	27.8562	1.63296
$292$	−31.5219	−1.84468
$293$	25.5564	1.49302	0.746509	−	0.665375i	$-0.231728\pi$
$293$	25.5564	1.49302	0.746509	+	0.665375i	$0.231728\pi$
$294$	0	0
$295$	−5.01367	−0.291907
$296$	−39.0128	−2.26757
$297$	42.2571	2.45201
$298$	32.3109	1.87172
$299$	23.4033	1.35345
$300$	64.8735	3.74548
$301$	0	0
$302$	−1.68686	−0.0970677
$303$	41.5284	2.38574
$304$	35.9113	2.05965
$305$	−5.56754	−0.318796
$306$	−42.5607	−2.43304
$307$	−25.4178	−1.45067	−0.725335	−	0.688396i	$-0.758315\pi$
$307$	−25.4178	−1.45067	−0.725335	+	0.688396i	$0.758315\pi$
$308$	0	0
$309$	−41.6913	−2.37174
$310$	−3.17716	−0.180451
$311$	−14.9145	−0.845722	−0.422861	−	0.906195i	$-0.638974\pi$
$311$	−14.9145	−0.845722	−0.422861	+	0.906195i	$0.638974\pi$
$312$	−66.6920	−3.77569
$313$	−4.78914	−0.270698	−0.135349	−	0.990798i	$-0.543216\pi$
$313$	−4.78914	−0.270698	−0.135349	+	0.990798i	$0.543216\pi$
$314$	−62.6240	−3.53407
$315$	0	0
$316$	45.9900	2.58714
$317$	−11.2364	−0.631096	−0.315548	−	0.948910i	$-0.602188\pi$
$317$	−11.2364	−0.631096	−0.315548	+	0.948910i	$0.602188\pi$
$318$	−109.420	−6.13598
$319$	−4.83185	−0.270532
$320$	−0.501036	−0.0280088
$321$	8.05855	0.449784
$322$	0	0
$323$	12.9067	0.718147
$324$	47.2131	2.62295
$325$	16.0372	0.889583
$326$	−46.6452	−2.58344
$327$	48.3493	2.67372
$328$	1.32967	0.0734185
$329$	0	0
$330$	18.0072	0.991265
$331$	8.07064	0.443603	0.221801	−	0.975092i	$-0.428806\pi$
$331$	8.07064	0.443603	0.221801	+	0.975092i	$0.428806\pi$
$332$	−5.66145	−0.310713
$333$	36.9688	2.02588
$334$	0.185875	0.0101706
$335$	2.64388	0.144450
$336$	0	0
$337$	−28.3377	−1.54365	−0.771825	−	0.635835i	$-0.780656\pi$
$337$	−28.3377	−1.54365	−0.771825	+	0.635835i	$0.780656\pi$
$338$	3.77113	0.205123
$339$	32.1964	1.74867
$340$	−6.45140	−0.349876
$341$	10.4747	0.567234
$342$	−75.0787	−4.05979
$343$	0	0
$344$	−28.3058	−1.52615
$345$	−10.9586	−0.589990
$346$	29.2172	1.57073
$347$	27.0870	1.45410	0.727052	−	0.686582i	$-0.240890\pi$
$347$	27.0870	1.45410	0.727052	+	0.686582i	$0.240890\pi$
$348$	−14.9795	−0.802985
$349$	−4.49985	−0.240871	−0.120436	−	0.992721i	$-0.538429\pi$
$349$	−4.49985	−0.240871	−0.120436	+	0.992721i	$0.538429\pi$
$350$	0	0
$351$	32.3850	1.72858
$352$	27.7265	1.47783
$353$	35.3814	1.88316	0.941580	−	0.336790i	$-0.109341\pi$
$353$	35.3814	1.88316	0.941580	+	0.336790i	$0.109341\pi$
$354$	73.8216	3.92357
$355$	−4.69655	−0.249267
$356$	−26.3641	−1.39729
$357$	0	0
$358$	−39.0355	−2.06309
$359$	−0.855349	−0.0451436	−0.0225718	−	0.999745i	$-0.507185\pi$
$359$	−0.855349	−0.0451436	−0.0225718	+	0.999745i	$0.507185\pi$
$360$	20.9930	1.10643
$361$	3.76784	0.198308
$362$	−47.9545	−2.52043
$363$	−26.0878	−1.36925
$364$	0	0
$365$	3.64879	0.190987
$366$	81.9769	4.28500
$367$	−22.2643	−1.16218	−0.581092	−	0.813838i	$-0.697375\pi$
$367$	−22.2643	−1.16218	−0.581092	+	0.813838i	$0.697375\pi$
$368$	−51.8827	−2.70457
$369$	−1.26000	−0.0655930
$370$	8.07281	0.419685
$371$	0	0
$372$	32.4731	1.68365
$373$	23.8476	1.23478	0.617391	−	0.786656i	$-0.288189\pi$
$373$	23.8476	1.23478	0.617391	+	0.786656i	$0.288189\pi$
$374$	30.6407	1.58439
$375$	−15.4576	−0.798230
$376$	−19.1043	−0.985231
$377$	−3.70303	−0.190716
$378$	0	0
$379$	34.3437	1.76412	0.882058	−	0.471141i	$-0.156158\pi$
$379$	34.3437	1.76412	0.882058	+	0.471141i	$0.156158\pi$
$380$	−11.3805	−0.583807
$381$	3.02541	0.154996
$382$	58.6891	3.00279
$383$	−19.4171	−0.992169	−0.496084	−	0.868274i	$-0.665229\pi$
$383$	−19.4171	−0.992169	−0.496084	+	0.868274i	$0.665229\pi$
$384$	−30.4955	−1.55622
$385$	0	0
$386$	−44.4424	−2.26206
$387$	26.8228	1.36348
$388$	−41.7946	−2.12180
$389$	29.7459	1.50818	0.754089	−	0.656772i	$-0.228079\pi$
$389$	29.7459	1.50818	0.754089	+	0.656772i	$0.228079\pi$
$390$	13.8004	0.698809
$391$	−18.6469	−0.943013
$392$	0	0
$393$	−43.8667	−2.21278
$394$	−31.6555	−1.59478
$395$	−5.32354	−0.267856
$396$	−123.724	−6.21739
$397$	−10.2565	−0.514757	−0.257378	−	0.966311i	$-0.582859\pi$
$397$	−10.2565	−0.514757	−0.257378	+	0.966311i	$0.582859\pi$
$398$	−69.5746	−3.48746
$399$	0	0
$400$	−35.5527	−1.77764
$401$	31.8605	1.59104	0.795519	−	0.605928i	$-0.207198\pi$
$401$	31.8605	1.59104	0.795519	+	0.605928i	$0.207198\pi$
$402$	−38.9286	−1.94158
$403$	8.02756	0.399881
$404$	−62.3079	−3.09993
$405$	−5.46511	−0.271564
$406$	0	0
$407$	−26.6149	−1.31925
$408$	53.1375	2.63070
$409$	−10.0752	−0.498189	−0.249094	−	0.968479i	$-0.580133\pi$
$409$	−10.0752	−0.498189	−0.249094	+	0.968479i	$0.580133\pi$
$410$	−0.275144	−0.0135884
$411$	18.0529	0.890486
$412$	62.5524	3.08173
$413$	0	0
$414$	108.470	5.33099
$415$	0.655337	0.0321692
$416$	21.2490	1.04182
$417$	−4.64987	−0.227705
$418$	54.0512	2.64373
$419$	19.8772	0.971066	0.485533	−	0.874218i	$-0.338626\pi$
$419$	19.8772	0.971066	0.485533	+	0.874218i	$0.338626\pi$
$420$	0	0
$421$	−1.40256	−0.0683566	−0.0341783	−	0.999416i	$-0.510881\pi$
$421$	−1.40256	−0.0683566	−0.0341783	+	0.999416i	$0.510881\pi$
$422$	−54.5864	−2.65722
$423$	18.1034	0.880218
$424$	91.8364	4.45997
$425$	−12.7778	−0.619814
$426$	69.1524	3.35044
$427$	0	0
$428$	−12.0908	−0.584430
$429$	−45.4979	−2.19666
$430$	5.85725	0.282461
$431$	−36.3763	−1.75218	−0.876092	−	0.482145i	$-0.839858\pi$
$431$	−36.3763	−1.75218	−0.876092	+	0.482145i	$0.839858\pi$
$432$	−71.7941	−3.45419
$433$	31.6643	1.52169	0.760845	−	0.648934i	$-0.224785\pi$
$433$	31.6643	1.52169	0.760845	+	0.648934i	$0.224785\pi$
$434$	0	0
$435$	1.73394	0.0831360
$436$	−72.5418	−3.47412
$437$	−32.8937	−1.57352
$438$	−53.7251	−2.56709
$439$	−3.07089	−0.146566	−0.0732828	−	0.997311i	$-0.523348\pi$
$439$	−3.07089	−0.146566	−0.0732828	+	0.997311i	$0.523348\pi$
$440$	−15.1135	−0.720506
$441$	0	0
$442$	23.4824	1.11694
$443$	−18.4972	−0.878830	−0.439415	−	0.898284i	$-0.644814\pi$
$443$	−18.4972	−0.878830	−0.439415	+	0.898284i	$0.644814\pi$
$444$	−82.5103	−3.91577
$445$	3.05176	0.144667
$446$	−25.9095	−1.22685
$447$	38.2270	1.80807
$448$	0	0
$449$	−28.7644	−1.35748	−0.678738	−	0.734380i	$-0.737473\pi$
$449$	−28.7644	−1.35748	−0.678738	+	0.734380i	$0.737473\pi$
$450$	74.3289	3.50390
$451$	0.907110	0.0427141
$452$	−48.3065	−2.27215
$453$	−1.99571	−0.0937669
$454$	−44.5975	−2.09307
$455$	0	0
$456$	93.7365	4.38961
$457$	−8.52606	−0.398832	−0.199416	−	0.979915i	$-0.563905\pi$
$457$	−8.52606	−0.398832	−0.199416	+	0.979915i	$0.563905\pi$
$458$	−41.0019	−1.91589
$459$	−25.8031	−1.20439
$460$	16.4419	0.766608
$461$	−10.7872	−0.502408	−0.251204	−	0.967934i	$-0.580827\pi$
$461$	−10.7872	−0.502408	−0.251204	+	0.967934i	$0.580827\pi$
$462$	0	0
$463$	12.4621	0.579161	0.289581	−	0.957154i	$-0.406484\pi$
$463$	12.4621	0.579161	0.289581	+	0.957154i	$0.406484\pi$
$464$	8.20922	0.381103
$465$	−3.75889	−0.174315
$466$	46.3372	2.14653
$467$	−36.6307	−1.69507	−0.847533	−	0.530742i	$-0.821913\pi$
$467$	−36.6307	−1.69507	−0.847533	+	0.530742i	$0.821913\pi$
$468$	−94.8200	−4.38305
$469$	0	0
$470$	3.95321	0.182348
$471$	−74.0902	−3.41390
$472$	−61.9585	−2.85187
$473$	−19.3105	−0.887897
$474$	78.3842	3.60030
$475$	−22.5405	−1.03423
$476$	0	0
$477$	−87.0248	−3.98459
$478$	16.6534	0.761708
$479$	−5.58604	−0.255233	−0.127616	−	0.991824i	$-0.540733\pi$
$479$	−5.58604	−0.255233	−0.127616	+	0.991824i	$0.540733\pi$
$480$	−9.94983	−0.454146
$481$	−20.3971	−0.930029
$482$	6.22935	0.283739
$483$	0	0
$484$	39.1413	1.77915
$485$	4.83790	0.219678
$486$	7.28705	0.330547
$487$	−22.0622	−0.999734	−0.499867	−	0.866102i	$-0.666618\pi$
$487$	−22.0622	−0.999734	−0.499867	+	0.866102i	$0.666618\pi$
$488$	−68.8032	−3.11458
$489$	−55.1858	−2.49559
$490$	0	0
$491$	−1.50823	−0.0680655	−0.0340328	−	0.999421i	$-0.510835\pi$
$491$	−1.50823	−0.0680655	−0.0340328	+	0.999421i	$0.510835\pi$
$492$	2.81218	0.126783
$493$	2.95043	0.132881
$494$	41.4238	1.86374
$495$	14.3216	0.643710
$496$	−17.7962	−0.799075
$497$	0	0
$498$	−9.64924	−0.432393
$499$	14.9720	0.670237	0.335119	−	0.942176i	$-0.391224\pi$
$499$	14.9720	0.670237	0.335119	+	0.942176i	$0.391224\pi$
$500$	23.1922	1.03719
$501$	0.219909	0.00982479
$502$	−27.5338	−1.22889
$503$	−29.9513	−1.33546	−0.667731	−	0.744402i	$-0.732735\pi$
$503$	−29.9513	−1.33546	−0.667731	+	0.744402i	$0.732735\pi$
$504$	0	0
$505$	7.21240	0.320948
$506$	−78.0903	−3.47154
$507$	4.46162	0.198147
$508$	−4.53922	−0.201395
$509$	−27.7075	−1.22811	−0.614057	−	0.789262i	$-0.710464\pi$
$509$	−27.7075	−1.22811	−0.614057	+	0.789262i	$0.710464\pi$
$510$	−10.9956	−0.486893
$511$	0	0
$512$	50.6314	2.23761
$513$	−45.5176	−2.00965
$514$	−38.8341	−1.71290
$515$	−7.24071	−0.319064
$516$	−59.8655	−2.63543
$517$	−13.0332	−0.573197
$518$	0	0
$519$	34.5668	1.51731
$520$	−11.5827	−0.507933
$521$	15.5960	0.683271	0.341636	−	0.939832i	$-0.389019\pi$
$521$	15.5960	0.683271	0.341636	+	0.939832i	$0.389019\pi$
$522$	−17.1628	−0.751194
$523$	−12.5494	−0.548749	−0.274374	−	0.961623i	$-0.588471\pi$
$523$	−12.5494	−0.548749	−0.274374	+	0.961623i	$0.588471\pi$
$524$	65.8161	2.87519
$525$	0	0
$526$	−56.8186	−2.47741
$527$	−6.39605	−0.278616
$528$	100.864	4.38953
$529$	24.5231	1.06622
$530$	−19.0034	−0.825457
$531$	58.7123	2.54790
$532$	0	0
$533$	0.695191	0.0301121
$534$	−44.9343	−1.94450
$535$	1.39956	0.0605083
$536$	32.6728	1.41125
$537$	−46.1827	−1.99293
$538$	−15.0501	−0.648857
$539$	0	0
$540$	22.7519	0.979088
$541$	−19.4861	−0.837774	−0.418887	−	0.908038i	$-0.637580\pi$
$541$	−19.4861	−0.837774	−0.418887	+	0.908038i	$0.637580\pi$
$542$	23.3789	1.00421
$543$	−56.7349	−2.43473
$544$	−16.9304	−0.725885
$545$	8.39702	0.359689
$546$	0	0
$547$	−15.0948	−0.645409	−0.322705	−	0.946500i	$-0.604592\pi$
$547$	−15.0948	−0.645409	−0.322705	+	0.946500i	$0.604592\pi$
$548$	−27.0861	−1.15706
$549$	65.1984	2.78260
$550$	−53.5115	−2.28174
$551$	5.20466	0.221726
$552$	−135.425	−5.76408
$553$	0	0
$554$	41.5120	1.76368
$555$	9.55092	0.405414
$556$	6.97652	0.295870
$557$	−3.24015	−0.137290	−0.0686449	−	0.997641i	$-0.521868\pi$
$557$	−3.24015	−0.137290	−0.0686449	+	0.997641i	$0.521868\pi$
$558$	37.2060	1.57506
$559$	−14.7992	−0.625938
$560$	0	0
$561$	36.2509	1.53051
$562$	−29.1486	−1.22956
$563$	−26.3904	−1.11222	−0.556112	−	0.831107i	$-0.687707\pi$
$563$	−26.3904	−1.11222	−0.556112	+	0.831107i	$0.687707\pi$
$564$	−40.4048	−1.70135
$565$	5.59168	0.235244
$566$	48.1761	2.02499
$567$	0	0
$568$	−58.0396	−2.43529
$569$	−32.5629	−1.36511	−0.682555	−	0.730835i	$-0.739131\pi$
$569$	−32.5629	−1.36511	−0.682555	+	0.730835i	$0.739131\pi$
$570$	−19.3966	−0.812435
$571$	9.74642	0.407875	0.203937	−	0.978984i	$-0.434626\pi$
$571$	9.74642	0.407875	0.203937	+	0.978984i	$0.434626\pi$
$572$	68.2636	2.85424
$573$	69.4349	2.90068
$574$	0	0
$575$	32.5653	1.35807
$576$	5.86736	0.244474
$577$	−1.66180	−0.0691815	−0.0345908	−	0.999402i	$-0.511013\pi$
$577$	−1.66180	−0.0691815	−0.0345908	+	0.999402i	$0.511013\pi$
$578$	24.7624	1.02998
$579$	−52.5797	−2.18514
$580$	−2.60155	−0.108023
$581$	0	0
$582$	−71.2336	−2.95273
$583$	62.6516	2.59477
$584$	45.0915	1.86590
$585$	10.9758	0.453794
$586$	−65.3525	−2.69969
$587$	−22.4857	−0.928084	−0.464042	−	0.885813i	$-0.653601\pi$
$587$	−22.4857	−0.928084	−0.464042	+	0.885813i	$0.653601\pi$
$588$	0	0
$589$	−11.2829	−0.464902
$590$	12.8209	0.527828
$591$	−37.4515	−1.54055
$592$	45.2182	1.85846
$593$	−33.9552	−1.39437	−0.697186	−	0.716891i	$-0.745565\pi$
$593$	−33.9552	−1.39437	−0.697186	+	0.716891i	$0.745565\pi$
$594$	−108.060	−4.43374
$595$	0	0
$596$	−57.3546	−2.34933
$597$	−82.3135	−3.36887
$598$	−59.8468	−2.44732
$599$	40.9169	1.67182	0.835909	−	0.548868i	$-0.184941\pi$
$599$	40.9169	1.67182	0.835909	+	0.548868i	$0.184941\pi$
$600$	−92.8004	−3.78856
$601$	25.6712	1.04715	0.523575	−	0.851979i	$-0.324598\pi$
$601$	25.6712	1.04715	0.523575	+	0.851979i	$0.324598\pi$
$602$	0	0
$603$	−30.9610	−1.26083
$604$	2.99431	0.121837
$605$	−4.53077	−0.184202
$606$	−106.196	−4.31392
$607$	32.5637	1.32172	0.660860	−	0.750509i	$-0.270192\pi$
$607$	32.5637	1.32172	0.660860	+	0.750509i	$0.270192\pi$
$608$	−29.8658	−1.21122
$609$	0	0
$610$	14.2373	0.576450
$611$	−9.98835	−0.404085
$612$	75.5488	3.05388
$613$	17.9447	0.724781	0.362391	−	0.932026i	$-0.381961\pi$
$613$	17.9447	0.724781	0.362391	+	0.932026i	$0.381961\pi$
$614$	64.9981	2.62311
$615$	−0.325522	−0.0131263
$616$	0	0
$617$	12.3265	0.496247	0.248124	−	0.968728i	$-0.420186\pi$
$617$	12.3265	0.496247	0.248124	+	0.968728i	$0.420186\pi$
$618$	106.613	4.28859
$619$	5.51580	0.221699	0.110849	−	0.993837i	$-0.464643\pi$
$619$	5.51580	0.221699	0.110849	+	0.993837i	$0.464643\pi$
$620$	5.63973	0.226497
$621$	65.7613	2.63891
$622$	38.1391	1.52924
$623$	0	0
$624$	77.3000	3.09448
$625$	20.9350	0.837400
$626$	12.2467	0.489478
$627$	63.9479	2.55383
$628$	111.163	4.43587
$629$	16.2516	0.647995
$630$	0	0
$631$	44.4450	1.76933	0.884664	−	0.466230i	$-0.154388\pi$
$631$	44.4450	1.76933	0.884664	+	0.466230i	$0.154388\pi$
$632$	−65.7878	−2.61690
$633$	−64.5810	−2.56687
$634$	28.7335	1.14115
$635$	0.525434	0.0208512
$636$	194.230	7.70171
$637$	0	0
$638$	12.3560	0.489177
$639$	54.9988	2.17572
$640$	−5.29627	−0.209354
$641$	−27.8117	−1.09850	−0.549248	−	0.835659i	$-0.685086\pi$
$641$	−27.8117	−1.09850	−0.549248	+	0.835659i	$0.685086\pi$
$642$	−20.6072	−0.813303
$643$	31.4933	1.24197	0.620987	−	0.783821i	$-0.286732\pi$
$643$	31.4933	1.24197	0.620987	+	0.783821i	$0.286732\pi$
$644$	0	0
$645$	6.92969	0.272856
$646$	−33.0048	−1.29856
$647$	−32.5581	−1.27999	−0.639995	−	0.768379i	$-0.721064\pi$
$647$	−32.5581	−1.27999	−0.639995	+	0.768379i	$0.721064\pi$
$648$	−67.5375	−2.65312
$649$	−42.2687	−1.65919
$650$	−41.0101	−1.60855
$651$	0	0
$652$	82.7990	3.24266
$653$	−5.59029	−0.218765	−0.109382	−	0.994000i	$-0.534887\pi$
$653$	−5.59029	−0.218765	−0.109382	+	0.994000i	$0.534887\pi$
$654$	−123.638	−4.83465
$655$	−7.61850	−0.297679
$656$	−1.54116	−0.0601723
$657$	−42.7290	−1.66702
$658$	0	0
$659$	20.5083	0.798892	0.399446	−	0.916757i	$-0.369203\pi$
$659$	20.5083	0.798892	0.399446	+	0.916757i	$0.369203\pi$
$660$	−31.9643	−1.24421
$661$	−12.8001	−0.497867	−0.248933	−	0.968521i	$-0.580080\pi$
$661$	−12.8001	−0.497867	−0.248933	+	0.968521i	$0.580080\pi$
$662$	−20.6382	−0.802125
$663$	27.7820	1.07896
$664$	8.09861	0.314287
$665$	0	0
$666$	−94.5363	−3.66321
$667$	−7.51941	−0.291153
$668$	−0.329944	−0.0127659
$669$	−30.6535	−1.18513
$670$	−6.76090	−0.261196
$671$	−46.9382	−1.81203
$672$	0	0
$673$	15.1243	0.582998	0.291499	−	0.956571i	$-0.405846\pi$
$673$	15.1243	0.582998	0.291499	+	0.956571i	$0.405846\pi$
$674$	72.4648	2.79124
$675$	45.0631	1.73448
$676$	−6.69407	−0.257464
$677$	38.6595	1.48580	0.742902	−	0.669400i	$-0.233449\pi$
$677$	38.6595	1.48580	0.742902	+	0.669400i	$0.233449\pi$
$678$	−82.3324	−3.16196
$679$	0	0
$680$	9.22861	0.353901
$681$	−52.7632	−2.02189
$682$	−26.7857	−1.02568
$683$	26.4301	1.01132	0.505661	−	0.862732i	$-0.331249\pi$
$683$	26.4301	1.01132	0.505661	+	0.862732i	$0.331249\pi$
$684$	133.271	5.09574
$685$	3.13533	0.119795
$686$	0	0
$687$	−48.5093	−1.85074
$688$	32.8082	1.25080
$689$	48.0150	1.82922
$690$	28.0232	1.06682
$691$	−29.4452	−1.12015	−0.560075	−	0.828442i	$-0.689228\pi$
$691$	−29.4452	−1.12015	−0.560075	+	0.828442i	$0.689228\pi$
$692$	−51.8630	−1.97153
$693$	0	0
$694$	−69.2665	−2.62932
$695$	−0.807562	−0.0306326
$696$	21.4279	0.812222
$697$	−0.553901	−0.0209805
$698$	11.5070	0.435545
$699$	54.8214	2.07354
$700$	0	0
$701$	18.0426	0.681461	0.340730	−	0.940161i	$-0.389326\pi$
$701$	18.0426	0.681461	0.340730	+	0.940161i	$0.389326\pi$
$702$	−82.8147	−3.12564
$703$	28.6684	1.08125
$704$	−4.22408	−0.159201
$705$	4.67703	0.176147
$706$	−90.4769	−3.40514
$707$	0	0
$708$	−131.039	−4.92476
$709$	33.2955	1.25044	0.625219	−	0.780450i	$-0.285010\pi$
$709$	33.2955	1.25044	0.625219	+	0.780450i	$0.285010\pi$
$710$	12.0100	0.450726
$711$	62.3410	2.33797
$712$	37.7134	1.41337
$713$	16.3008	0.610471
$714$	0	0
$715$	−7.90180	−0.295511
$716$	69.2911	2.58953
$717$	19.7026	0.735806
$718$	2.18729	0.0816289
$719$	−13.1731	−0.491274	−0.245637	−	0.969362i	$-0.578997\pi$
$719$	−13.1731	−0.491274	−0.245637	+	0.969362i	$0.578997\pi$
$720$	−24.3322	−0.906807
$721$	0	0
$722$	−9.63510	−0.358581
$723$	7.36992	0.274090
$724$	85.1232	3.16358
$725$	−5.15269	−0.191366
$726$	66.7114	2.47589
$727$	−16.9162	−0.627387	−0.313693	−	0.949524i	$-0.601566\pi$
$727$	−16.9162	−0.627387	−0.313693	+	0.949524i	$0.601566\pi$
$728$	0	0
$729$	−22.5821	−0.836375
$730$	−9.33066	−0.345343
$731$	11.7914	0.436121
$732$	−145.516	−5.37842
$733$	27.2599	1.00687	0.503433	−	0.864034i	$-0.332070\pi$
$733$	27.2599	1.00687	0.503433	+	0.864034i	$0.332070\pi$
$734$	56.9339	2.10147
$735$	0	0
$736$	43.1485	1.59047
$737$	22.2897	0.821052
$738$	3.22206	0.118606
$739$	−32.2540	−1.18648	−0.593242	−	0.805024i	$-0.702152\pi$
$739$	−32.2540	−1.18648	−0.593242	+	0.805024i	$0.702152\pi$
$740$	−14.3299	−0.526778
$741$	49.0084	1.80037
$742$	0	0
$743$	−35.9911	−1.32039	−0.660193	−	0.751096i	$-0.729526\pi$
$743$	−35.9911	−1.32039	−0.660193	+	0.751096i	$0.729526\pi$
$744$	−46.4521	−1.70302
$745$	6.63903	0.243235
$746$	−60.9829	−2.23274
$747$	−7.67430	−0.280788
$748$	−54.3897	−1.98869
$749$	0	0
$750$	39.5282	1.44336
$751$	3.75523	0.137030	0.0685152	−	0.997650i	$-0.478174\pi$
$751$	3.75523	0.137030	0.0685152	+	0.997650i	$0.478174\pi$
$752$	22.1431	0.807475
$753$	−32.5752	−1.18710
$754$	9.46936	0.344854
$755$	−0.346604	−0.0126142
$756$	0	0
$757$	−23.6497	−0.859563	−0.429781	−	0.902933i	$-0.641409\pi$
$757$	−23.6497	−0.859563	−0.429781	+	0.902933i	$0.641409\pi$
$758$	−87.8233	−3.18989
$759$	−92.3884	−3.35349
$760$	16.2796	0.590523
$761$	19.7413	0.715621	0.357810	−	0.933794i	$-0.383523\pi$
$761$	19.7413	0.715621	0.357810	+	0.933794i	$0.383523\pi$
$762$	−7.73654	−0.280265
$763$	0	0
$764$	−104.178	−3.76903
$765$	−8.74510	−0.316180
$766$	49.6533	1.79405
$767$	−32.3939	−1.16967
$768$	83.7526	3.02216
$769$	−51.0927	−1.84245	−0.921225	−	0.389030i	$-0.872810\pi$
$769$	−51.0927	−1.84245	−0.921225	+	0.389030i	$0.872810\pi$
$770$	0	0
$771$	−45.9445	−1.65465
$772$	78.8889	2.83927
$773$	0.939084	0.0337765	0.0168883	−	0.999857i	$-0.494624\pi$
$773$	0.939084	0.0337765	0.0168883	+	0.999857i	$0.494624\pi$
$774$	−68.5910	−2.46545
$775$	11.1702	0.401245
$776$	59.7864	2.14621
$777$	0	0
$778$	−76.0660	−2.72710
$779$	−0.977101	−0.0350083
$780$	−24.4968	−0.877126
$781$	−39.5952	−1.41683
$782$	47.6836	1.70516
$783$	−10.4052	−0.371851
$784$	0	0
$785$	−12.8676	−0.459263
$786$	112.175	4.00116
$787$	−27.6870	−0.986934	−0.493467	−	0.869764i	$-0.664271\pi$
$787$	−27.6870	−0.986934	−0.493467	+	0.869764i	$0.664271\pi$
$788$	56.1911	2.00173
$789$	−67.2220	−2.39317
$790$	13.6133	0.484339
$791$	0	0
$792$	176.986	6.28891
$793$	−35.9725	−1.27742
$794$	26.2277	0.930787
$795$	−22.4829	−0.797387
$796$	123.501	4.37736
$797$	−11.5777	−0.410102	−0.205051	−	0.978751i	$-0.565736\pi$
$797$	−11.5777	−0.410102	−0.205051	+	0.978751i	$0.565736\pi$
$798$	0	0
$799$	7.95832	0.281545
$800$	29.5676	1.04537
$801$	−35.7375	−1.26272
$802$	−81.4734	−2.87693
$803$	30.7618	1.08556
$804$	69.1015	2.43702
$805$	0	0
$806$	−20.5280	−0.723068
$807$	−17.8058	−0.626793
$808$	89.1303	3.13559
$809$	−6.01779	−0.211574	−0.105787	−	0.994389i	$-0.533736\pi$
$809$	−6.01779	−0.211574	−0.105787	+	0.994389i	$0.533736\pi$
$810$	13.9753	0.491043
$811$	−9.69490	−0.340434	−0.170217	−	0.985407i	$-0.554447\pi$
$811$	−9.69490	−0.340434	−0.170217	+	0.985407i	$0.554447\pi$
$812$	0	0
$813$	27.6595	0.970063
$814$	68.0594	2.38548
$815$	−9.58434	−0.335725
$816$	−61.5896	−2.15607
$817$	20.8004	0.727716
$818$	25.7643	0.900828
$819$	0	0
$820$	0.488403	0.0170558
$821$	−48.5847	−1.69562	−0.847809	−	0.530301i	$-0.822079\pi$
$821$	−48.5847	−1.69562	−0.847809	+	0.530301i	$0.822079\pi$
$822$	−46.1648	−1.61018
$823$	17.3579	0.605057	0.302529	−	0.953140i	$-0.402169\pi$
$823$	17.3579	0.605057	0.302529	+	0.953140i	$0.402169\pi$
$824$	−89.4801	−3.11719
$825$	−63.3093	−2.20415
$826$	0	0
$827$	−35.8750	−1.24750	−0.623748	−	0.781625i	$-0.714391\pi$
$827$	−35.8750	−1.24750	−0.623748	+	0.781625i	$0.714391\pi$
$828$	−192.542	−6.69131
$829$	−29.0542	−1.00909	−0.504547	−	0.863384i	$-0.668340\pi$
$829$	−29.0542	−1.00909	−0.504547	+	0.863384i	$0.668340\pi$
$830$	−1.67582	−0.0581686
$831$	49.1127	1.70370
$832$	−3.23725	−0.112232
$833$	0	0
$834$	11.8906	0.411738
$835$	0.0381924	0.00132170
$836$	−95.9454	−3.31834
$837$	22.5567	0.779675
$838$	−50.8298	−1.75589
$839$	23.1420	0.798952	0.399476	−	0.916744i	$-0.369192\pi$
$839$	23.1420	0.798952	0.399476	+	0.916744i	$0.369192\pi$
$840$	0	0
$841$	−27.8102	−0.958973
$842$	3.58662	0.123603
$843$	−34.4856	−1.18775
$844$	96.8954	3.33528
$845$	0.774867	0.0266562
$846$	−46.2939	−1.59162
$847$	0	0
$848$	−106.444	−3.65530
$849$	56.9970	1.95613
$850$	32.6753	1.12075
$851$	−41.4186	−1.41981
$852$	−122.751	−4.20539
$853$	4.68483	0.160406	0.0802028	−	0.996779i	$-0.474443\pi$
$853$	4.68483	0.160406	0.0802028	+	0.996779i	$0.474443\pi$
$854$	0	0
$855$	−15.4267	−0.527581
$856$	17.2957	0.591154
$857$	48.0548	1.64152	0.820761	−	0.571272i	$-0.193550\pi$
$857$	48.0548	1.64152	0.820761	+	0.571272i	$0.193550\pi$
$858$	116.347	3.97201
$859$	−3.90283	−0.133163	−0.0665814	−	0.997781i	$-0.521209\pi$
$859$	−3.90283	−0.133163	−0.0665814	+	0.997781i	$0.521209\pi$
$860$	−10.3971	−0.354538
$861$	0	0
$862$	93.0210	3.16831
$863$	18.7374	0.637830	0.318915	−	0.947783i	$-0.396682\pi$
$863$	18.7374	0.637830	0.318915	+	0.947783i	$0.396682\pi$
$864$	59.7079	2.03130
$865$	6.00336	0.204120
$866$	−80.9717	−2.75153
$867$	29.2963	0.994955
$868$	0	0
$869$	−44.8811	−1.52249
$870$	−4.43401	−0.150327
$871$	17.0824	0.578814
$872$	103.770	3.51409
$873$	−56.6540	−1.91745
$874$	84.1155	2.84525
$875$	0	0
$876$	95.3665	3.22214
$877$	−13.9680	−0.471667	−0.235833	−	0.971793i	$-0.575782\pi$
$877$	−13.9680	−0.471667	−0.235833	+	0.971793i	$0.575782\pi$
$878$	7.85285	0.265021
$879$	−77.3184	−2.60788
$880$	17.5174	0.590512
$881$	19.6038	0.660469	0.330235	−	0.943899i	$-0.392872\pi$
$881$	19.6038	0.660469	0.330235	+	0.943899i	$0.392872\pi$
$882$	0	0
$883$	−17.4839	−0.588381	−0.294191	−	0.955747i	$-0.595050\pi$
$883$	−17.4839	−0.588381	−0.294191	+	0.955747i	$0.595050\pi$
$884$	−41.6832	−1.40196
$885$	15.1684	0.509879
$886$	47.3009	1.58911
$887$	−12.6826	−0.425839	−0.212920	−	0.977070i	$-0.568297\pi$
$887$	−12.6826	−0.425839	−0.212920	+	0.977070i	$0.568297\pi$
$888$	118.030	3.96081
$889$	0	0
$890$	−7.80392	−0.261588
$891$	−46.0747	−1.54356
$892$	45.9915	1.53991
$893$	14.0388	0.469789
$894$	−97.7537	−3.26937
$895$	−8.02074	−0.268104
$896$	0	0
$897$	−70.8046	−2.36410
$898$	73.5561	2.45460
$899$	−2.57923	−0.0860220
$900$	−131.940	−4.39800
$901$	−38.2564	−1.27451
$902$	−2.31965	−0.0772360
$903$	0	0
$904$	69.1016	2.29828
$905$	−9.85337	−0.327537
$906$	5.10342	0.169550
$907$	−39.1960	−1.30148	−0.650741	−	0.759299i	$-0.725542\pi$
$907$	−39.1960	−1.30148	−0.650741	+	0.759299i	$0.725542\pi$
$908$	79.1643	2.62716
$909$	−84.4605	−2.80138
$910$	0	0
$911$	−0.0121211	−0.000401590 0	−0.000200795	−	1.00000i	$-0.500064\pi$
$911$	−0.0121211	−0.000401590 0	−0.000200795		1.00000i	$0.500064\pi$
$912$	−108.646	−3.59764
$913$	5.52495	0.182849
$914$	21.8028	0.721171
$915$	16.8441	0.556848
$916$	72.7818	2.40478
$917$	0	0
$918$	65.9835	2.17778
$919$	−28.7055	−0.946908	−0.473454	−	0.880818i	$-0.656993\pi$
$919$	−28.7055	−0.946908	−0.473454	+	0.880818i	$0.656993\pi$
$920$	−23.5199	−0.775427
$921$	76.8991	2.53391
$922$	27.5848	0.908458
$923$	−30.3449	−0.998816
$924$	0	0
$925$	−28.3822	−0.933200
$926$	−31.8679	−1.04724
$927$	84.7920	2.78493
$928$	−6.82724	−0.224115
$929$	−12.6020	−0.413457	−0.206728	−	0.978398i	$-0.566282\pi$
$929$	−12.6020	−0.413457	−0.206728	+	0.978398i	$0.566282\pi$
$930$	9.61221	0.315197
$931$	0	0
$932$	−82.2523	−2.69426
$933$	45.1223	1.47724
$934$	93.6717	3.06503
$935$	6.29584	0.205896
$936$	135.638	4.43348
$937$	50.7660	1.65845	0.829227	−	0.558912i	$-0.188781\pi$
$937$	50.7660	1.65845	0.829227	+	0.558912i	$0.188781\pi$
$938$	0	0
$939$	14.4891	0.472833
$940$	−7.01727	−0.228878
$941$	45.9281	1.49721	0.748606	−	0.663015i	$-0.230723\pi$
$941$	45.9281	1.49721	0.748606	+	0.663015i	$0.230723\pi$
$942$	189.463	6.17303
$943$	1.41166	0.0459700
$944$	71.8137	2.33734
$945$	0	0
$946$	49.3806	1.60550
$947$	−36.1696	−1.17535	−0.587677	−	0.809096i	$-0.699957\pi$
$947$	−36.1696	−1.17535	−0.587677	+	0.809096i	$0.699957\pi$
$948$	−139.138	−4.51900
$949$	23.5753	0.765286
$950$	57.6403	1.87010
$951$	33.9945	1.10235
$952$	0	0
$953$	15.0088	0.486182	0.243091	−	0.970004i	$-0.421839\pi$
$953$	15.0088	0.486182	0.243091	+	0.970004i	$0.421839\pi$
$954$	222.539	7.20497
$955$	12.0590	0.390221
$956$	−29.5611	−0.956075
$957$	14.6183	0.472543
$958$	14.2846	0.461513
$959$	0	0
$960$	1.51584	0.0489235
$961$	−25.4087	−0.819634
$962$	52.1593	1.68168
$963$	−16.3895	−0.528144
$964$	−11.0576	−0.356141
$965$	−9.13173	−0.293961
$966$	0	0
$967$	−4.52194	−0.145416	−0.0727078	−	0.997353i	$-0.523164\pi$
$967$	−4.52194	−0.145416	−0.0727078	+	0.997353i	$0.523164\pi$
$968$	−55.9909	−1.79962
$969$	−39.0479	−1.25440
$970$	−12.3714	−0.397223
$971$	−15.0453	−0.482827	−0.241413	−	0.970422i	$-0.577611\pi$
$971$	−15.0453	−0.482827	−0.241413	+	0.970422i	$0.577611\pi$
$972$	−12.9351	−0.414894
$973$	0	0
$974$	56.4173	1.80773
$975$	−48.5190	−1.55385
$976$	79.7471	2.55264
$977$	2.54122	0.0813009	0.0406504	−	0.999173i	$-0.487057\pi$
$977$	2.54122	0.0813009	0.0406504	+	0.999173i	$0.487057\pi$
$978$	141.121	4.51254
$979$	25.7284	0.822283
$980$	0	0
$981$	−98.3330	−3.13953
$982$	3.85684	0.123077
$983$	18.1396	0.578563	0.289282	−	0.957244i	$-0.406584\pi$
$983$	18.1396	0.578563	0.289282	+	0.957244i	$0.406584\pi$
$984$	−4.02278	−0.128241
$985$	−6.50436	−0.207246
$986$	−7.54481	−0.240276
$987$	0	0
$988$	−73.5306	−2.33932
$989$	−30.0513	−0.955577
$990$	−36.6231	−1.16396
$991$	26.6955	0.848010	0.424005	−	0.905660i	$-0.360624\pi$
$991$	26.6955	0.848010	0.424005	+	0.905660i	$0.360624\pi$
$992$	14.8003	0.469911
$993$	−24.4170	−0.774849
$994$	0	0
$995$	−14.2957	−0.453205
$996$	17.1282	0.542728
$997$	−31.0054	−0.981950	−0.490975	−	0.871174i	$-0.663359\pi$
$997$	−31.0054	−0.981950	−0.490975	+	0.871174i	$0.663359\pi$
$998$	−38.2862	−1.21193
$999$	−57.3141	−1.81334

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.s.1.3	✓	54
7.6	odd	2	inner	6223.2.a.s.1.4	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.s.1.3	✓	54	1.1	even	1	trivial
6223.2.a.s.1.4	yes	54	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55719 q^{2} -3.02541 q^{3} +4.53922 q^{4} -0.525434 q^{5} +7.73654 q^{6} -6.49328 q^{8} +6.15308 q^{9} +O(q^{10})\)	\(q-2.55719 q^{2} -3.02541 q^{3} +4.53922 q^{4} -0.525434 q^{5} +7.73654 q^{6} -6.49328 q^{8} +6.15308 q^{9} +1.34364 q^{10} -4.42977 q^{11} -13.7330 q^{12} -3.39489 q^{13} +1.58965 q^{15} +7.52610 q^{16} +2.70492 q^{17} -15.7346 q^{18} +4.77157 q^{19} -2.38506 q^{20} +11.3278 q^{22} -6.89370 q^{23} +19.6448 q^{24} -4.72392 q^{25} +8.68138 q^{26} -9.53934 q^{27} +1.09077 q^{29} -4.06504 q^{30} -2.36460 q^{31} -6.25912 q^{32} +13.4019 q^{33} -6.91698 q^{34} +27.9302 q^{36} +6.00818 q^{37} -12.2018 q^{38} +10.2709 q^{39} +3.41179 q^{40} -0.204776 q^{41} +4.35925 q^{43} -20.1077 q^{44} -3.23304 q^{45} +17.6285 q^{46} +2.94217 q^{47} -22.7695 q^{48} +12.0800 q^{50} -8.18347 q^{51} -15.4102 q^{52} -14.1433 q^{53} +24.3939 q^{54} +2.32756 q^{55} -14.4359 q^{57} -2.78930 q^{58} +9.54194 q^{59} +7.21579 q^{60} +10.5961 q^{61} +6.04674 q^{62} +0.953566 q^{64} +1.78379 q^{65} -34.2711 q^{66} -5.03179 q^{67} +12.2782 q^{68} +20.8562 q^{69} +8.93842 q^{71} -39.9536 q^{72} -6.94434 q^{73} -15.3641 q^{74} +14.2918 q^{75} +21.6592 q^{76} -26.2647 q^{78} +10.1317 q^{79} -3.95447 q^{80} +10.4011 q^{81} +0.523650 q^{82} -1.24723 q^{83} -1.42126 q^{85} -11.1474 q^{86} -3.30001 q^{87} +28.7638 q^{88} -5.80806 q^{89} +8.26750 q^{90} -31.2920 q^{92} +7.15388 q^{93} -7.52369 q^{94} -2.50715 q^{95} +18.9364 q^{96} -9.20742 q^{97} -27.2567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 14 q^{2} + 46 q^{4} - 42 q^{8} + 14 q^{9}+O(q^{10}) \) 54 * q - 14 * q^2 + 46 * q^4 - 42 * q^8 + 14 * q^9	\( 54 q - 14 q^{2} + 46 q^{4} - 42 q^{8} + 14 q^{9} - 20 q^{11} - 16 q^{15} + 38 q^{16} - 42 q^{18} - 28 q^{22} - 64 q^{23} - 10 q^{25} - 44 q^{29} - 12 q^{30} - 78 q^{32} - 38 q^{36} - 32 q^{37} - 40 q^{39} - 52 q^{43} - 48 q^{44} - 8 q^{46} - 66 q^{50} - 24 q^{51} - 136 q^{53} - 80 q^{57} - 4 q^{58} - 28 q^{60} - 10 q^{64} - 88 q^{65} - 48 q^{67} - 52 q^{71} - 22 q^{72} - 28 q^{74} - 36 q^{78} + 24 q^{79} - 66 q^{81} - 48 q^{85} + 64 q^{86} - 68 q^{88} - 192 q^{92} - 88 q^{93} - 112 q^{95} - 72 q^{99}+O(q^{100}) \) 54 * q - 14 * q^2 + 46 * q^4 - 42 * q^8 + 14 * q^9 - 20 * q^11 - 16 * q^15 + 38 * q^16 - 42 * q^18 - 28 * q^22 - 64 * q^23 - 10 * q^25 - 44 * q^29 - 12 * q^30 - 78 * q^32 - 38 * q^36 - 32 * q^37 - 40 * q^39 - 52 * q^43 - 48 * q^44 - 8 * q^46 - 66 * q^50 - 24 * q^51 - 136 * q^53 - 80 * q^57 - 4 * q^58 - 28 * q^60 - 10 * q^64 - 88 * q^65 - 48 * q^67 - 52 * q^71 - 22 * q^72 - 28 * q^74 - 36 * q^78 + 24 * q^79 - 66 * q^81 - 48 * q^85 + 64 * q^86 - 68 * q^88 - 192 * q^92 - 88 * q^93 - 112 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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