LMFDB - Embedded newform 6223.2.a.n.1.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [34,6,0,46,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	\(q+1.70641 q^{2} -3.21112 q^{3} +0.911827 q^{4} +1.53579 q^{5} -5.47948 q^{6} -1.85687 q^{8} +7.31128 q^{9} +2.62068 q^{10} +5.61106 q^{11} -2.92798 q^{12} -5.24831 q^{13} -4.93160 q^{15} -4.99223 q^{16} +4.61362 q^{17} +12.4760 q^{18} +3.11627 q^{19} +1.40037 q^{20} +9.57475 q^{22} +6.41879 q^{23} +5.96262 q^{24} -2.64135 q^{25} -8.95575 q^{26} -13.8440 q^{27} +2.89372 q^{29} -8.41533 q^{30} -4.61069 q^{31} -4.80504 q^{32} -18.0178 q^{33} +7.87272 q^{34} +6.66662 q^{36} +0.0252223 q^{37} +5.31762 q^{38} +16.8529 q^{39} -2.85176 q^{40} +1.87857 q^{41} +2.40675 q^{43} +5.11631 q^{44} +11.2286 q^{45} +10.9531 q^{46} +6.54578 q^{47} +16.0306 q^{48} -4.50722 q^{50} -14.8149 q^{51} -4.78555 q^{52} +3.89106 q^{53} -23.6236 q^{54} +8.61740 q^{55} -10.0067 q^{57} +4.93786 q^{58} -7.83767 q^{59} -4.49677 q^{60} -10.9905 q^{61} -7.86772 q^{62} +1.78510 q^{64} -8.06030 q^{65} -30.7457 q^{66} -13.5525 q^{67} +4.20682 q^{68} -20.6115 q^{69} -13.6677 q^{71} -13.5761 q^{72} +10.7232 q^{73} +0.0430395 q^{74} +8.48168 q^{75} +2.84150 q^{76} +28.7580 q^{78} +14.0466 q^{79} -7.66701 q^{80} +22.5210 q^{81} +3.20561 q^{82} +7.16815 q^{83} +7.08556 q^{85} +4.10690 q^{86} -9.29206 q^{87} -10.4190 q^{88} -11.8620 q^{89} +19.1606 q^{90} +5.85282 q^{92} +14.8055 q^{93} +11.1698 q^{94} +4.78593 q^{95} +15.4295 q^{96} +10.5446 q^{97} +41.0240 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 6 q^{2} + 46 q^{4} + 30 q^{8} + 66 q^{9} + 16 q^{11} + 12 q^{15} + 54 q^{16} + 48 q^{18} + 30 q^{22} + 46 q^{23} + 80 q^{25} + 28 q^{29} - 2 q^{30} + 36 q^{32} + 98 q^{36} + 52 q^{37} - 2 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) $ 34 * q + 6 * q^2 + 46 * q^4 + 30 * q^8 + 66 * q^9 + 16 * q^11 + 12 * q^15 + 54 * q^16 + 48 * q^18 + 30 * q^22 + 46 * q^23 + 80 * q^25 + 28 * q^29 - 2 * q^30 + 36 * q^32 + 98 * q^36 + 52 * q^37 - 2 * q^39 + 10 * q^43 + 16 * q^44 - 36 * q^46 + 74 * q^50 + 56 * q^51 + 64 * q^53 + 44 * q^57 + 32 * q^58 - 132 * q^60 + 74 * q^64 + 108 * q^65 + 14 * q^67 + 6 * q^71 + 172 * q^72 - 6 * q^74 + 52 * q^78 + 46 * q^81 - 2 * q^85 + 96 * q^86 - 170 * q^88 + 130 * q^92 + 6 * q^93 + 14 * q^95 + 2 * 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 $

$2$	1.70641	1.20661	0.603306	−	0.797510i	$-0.293850\pi$
$2$	1.70641	1.20661	0.603306	+	0.797510i	$0.293850\pi$
$3$	−3.21112	−1.85394	−0.926970	−	0.375135i	$-0.877596\pi$
$3$	−3.21112	−1.85394	−0.926970	+	0.375135i	$0.877596\pi$
$4$	0.911827	0.455913
$5$	1.53579	0.686826	0.343413	−	0.939184i	$-0.388417\pi$
$5$	1.53579	0.686826	0.343413	+	0.939184i	$0.388417\pi$
$6$	−5.47948	−2.23699
$7$	0	0
$7$	0	0
$8$	−1.85687	−0.656502
$9$	7.31128	2.43709
$10$	2.62068	0.828733
$11$	5.61106	1.69180	0.845899	−	0.533344i	$-0.179065\pi$
$11$	5.61106	1.69180	0.845899	+	0.533344i	$0.179065\pi$
$12$	−2.92798	−0.845236
$13$	−5.24831	−1.45562	−0.727809	−	0.685780i	$-0.759461\pi$
$13$	−5.24831	−1.45562	−0.727809	+	0.685780i	$0.759461\pi$
$14$	0	0
$15$	−4.93160	−1.27333
$16$	−4.99223	−1.24806
$17$	4.61362	1.11897	0.559484	−	0.828841i	$-0.310999\pi$
$17$	4.61362	1.11897	0.559484	+	0.828841i	$0.310999\pi$
$18$	12.4760	2.94063
$19$	3.11627	0.714921	0.357460	−	0.933928i	$-0.383643\pi$
$19$	3.11627	0.714921	0.357460	+	0.933928i	$0.383643\pi$
$20$	1.40037	0.313133
$21$	0	0
$22$	9.57475	2.04134
$23$	6.41879	1.33841	0.669205	−	0.743078i	$-0.266635\pi$
$23$	6.41879	1.33841	0.669205	+	0.743078i	$0.266635\pi$
$24$	5.96262	1.21711
$25$	−2.64135	−0.528270
$26$	−8.95575	−1.75637
$27$	−13.8440	−2.66429
$28$	0	0
$29$	2.89372	0.537349	0.268675	−	0.963231i	$-0.413414\pi$
$29$	2.89372	0.537349	0.268675	+	0.963231i	$0.413414\pi$
$30$	−8.41533	−1.53642
$31$	−4.61069	−0.828104	−0.414052	−	0.910253i	$-0.635887\pi$
$31$	−4.61069	−0.828104	−0.414052	+	0.910253i	$0.635887\pi$
$32$	−4.80504	−0.849418
$33$	−18.0178	−3.13649
$34$	7.87272	1.35016
$35$	0	0
$36$	6.66662	1.11110
$37$	0.0252223	0.00414652	0.00207326	−	0.999998i	$-0.499340\pi$
$37$	0.0252223	0.00414652	0.00207326	+	0.999998i	$0.499340\pi$
$38$	5.31762	0.862632
$39$	16.8529	2.69863
$40$	−2.85176	−0.450903
$41$	1.87857	0.293383	0.146692	−	0.989182i	$-0.453137\pi$
$41$	1.87857	0.293383	0.146692	+	0.989182i	$0.453137\pi$
$42$	0	0
$43$	2.40675	0.367026	0.183513	−	0.983017i	$-0.441253\pi$
$43$	2.40675	0.367026	0.183513	+	0.983017i	$0.441253\pi$
$44$	5.11631	0.771313
$45$	11.2286	1.67386
$46$	10.9531	1.61494
$47$	6.54578	0.954800	0.477400	−	0.878686i	$-0.341579\pi$
$47$	6.54578	0.954800	0.477400	+	0.878686i	$0.341579\pi$
$48$	16.0306	2.31382
$49$	0	0
$50$	−4.50722	−0.637417
$51$	−14.8149	−2.07450
$52$	−4.78555	−0.663636
$53$	3.89106	0.534478	0.267239	−	0.963630i	$-0.413889\pi$
$53$	3.89106	0.534478	0.267239	+	0.963630i	$0.413889\pi$
$54$	−23.6236	−3.21476
$55$	8.61740	1.16197
$56$	0	0
$57$	−10.0067	−1.32542
$58$	4.93786	0.648372
$59$	−7.83767	−1.02038	−0.510189	−	0.860063i	$-0.670424\pi$
$59$	−7.83767	−1.02038	−0.510189	+	0.860063i	$0.670424\pi$
$60$	−4.49677	−0.580530
$61$	−10.9905	−1.40718	−0.703592	−	0.710604i	$-0.748422\pi$
$61$	−10.9905	−1.40718	−0.703592	+	0.710604i	$0.748422\pi$
$62$	−7.86772	−0.999201
$63$	0	0
$64$	1.78510	0.223138
$65$	−8.06030	−0.999757
$66$	−30.7457	−3.78453
$67$	−13.5525	−1.65570	−0.827849	−	0.560951i	$-0.810436\pi$
$67$	−13.5525	−1.65570	−0.827849	+	0.560951i	$0.810436\pi$
$68$	4.20682	0.510152
$69$	−20.6115	−2.48133
$70$	0	0
$71$	−13.6677	−1.62206	−0.811028	−	0.585007i	$-0.801092\pi$
$71$	−13.6677	−1.62206	−0.811028	+	0.585007i	$0.801092\pi$
$72$	−13.5761	−1.59996
$73$	10.7232	1.25505	0.627526	−	0.778595i	$-0.284068\pi$
$73$	10.7232	1.25505	0.627526	+	0.778595i	$0.284068\pi$
$74$	0.0430395	0.00500324
$75$	8.48168	0.979381
$76$	2.84150	0.325942
$77$	0	0
$78$	28.7580	3.25620
$79$	14.0466	1.58036	0.790182	−	0.612872i	$-0.209986\pi$
$79$	14.0466	1.58036	0.790182	+	0.612872i	$0.209986\pi$
$80$	−7.66701	−0.857198
$81$	22.5210	2.50233
$82$	3.20561	0.354000
$83$	7.16815	0.786807	0.393404	−	0.919366i	$-0.371298\pi$
$83$	7.16815	0.786807	0.393404	+	0.919366i	$0.371298\pi$
$84$	0	0
$85$	7.08556	0.768536
$86$	4.10690	0.442858
$87$	−9.29206	−0.996214
$88$	−10.4190	−1.11067
$89$	−11.8620	−1.25737	−0.628687	−	0.777659i	$-0.716407\pi$
$89$	−11.8620	−1.25737	−0.628687	+	0.777659i	$0.716407\pi$
$90$	19.1606	2.01970
$91$	0	0
$92$	5.85282	0.610199
$93$	14.8055	1.53526
$94$	11.1698	1.15207
$95$	4.78593	0.491026
$96$	15.4295	1.57477
$97$	10.5446	1.07065	0.535323	−	0.844647i	$-0.320190\pi$
$97$	10.5446	1.07065	0.535323	+	0.844647i	$0.320190\pi$
$98$	0	0
$99$	41.0240	4.12307
$100$	−2.40845	−0.240845
$101$	11.3347	1.12784	0.563922	−	0.825828i	$-0.309292\pi$
$101$	11.3347	1.12784	0.563922	+	0.825828i	$0.309292\pi$
$102$	−25.2802	−2.50312
$103$	−8.34729	−0.822483	−0.411241	−	0.911526i	$-0.634905\pi$
$103$	−8.34729	−0.822483	−0.411241	+	0.911526i	$0.634905\pi$
$104$	9.74541	0.955616
$105$	0	0
$106$	6.63973	0.644908
$107$	8.69302	0.840386	0.420193	−	0.907435i	$-0.361962\pi$
$107$	8.69302	0.840386	0.420193	+	0.907435i	$0.361962\pi$
$108$	−12.6234	−1.21468
$109$	−6.58318	−0.630555	−0.315277	−	0.949000i	$-0.602098\pi$
$109$	−6.58318	−0.630555	−0.315277	+	0.949000i	$0.602098\pi$
$110$	14.7048	1.40205
$111$	−0.0809917	−0.00768740
$112$	0	0
$113$	18.2357	1.71547	0.857737	−	0.514089i	$-0.171870\pi$
$113$	18.2357	1.71547	0.857737	+	0.514089i	$0.171870\pi$
$114$	−17.0755	−1.59927
$115$	9.85791	0.919255
$116$	2.63857	0.244985
$117$	−38.3719	−3.54748
$118$	−13.3743	−1.23120
$119$	0	0
$120$	9.15733	0.835946
$121$	20.4840	1.86218
$122$	−18.7542	−1.69793
$123$	−6.03231	−0.543915
$124$	−4.20415	−0.377544
$125$	−11.7355	−1.04966
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	12.6562	1.11866
$129$	−7.72836	−0.680444
$130$	−13.7542	−1.20632
$131$	1.03939	0.0908122	0.0454061	−	0.998969i	$-0.485542\pi$
$131$	1.03939	0.0908122	0.0454061	+	0.998969i	$0.485542\pi$
$132$	−16.4291	−1.42997
$133$	0	0
$134$	−23.1260	−1.99779
$135$	−21.2615	−1.82990
$136$	−8.56689	−0.734604
$137$	11.6097	0.991881	0.495940	−	0.868357i	$-0.334823\pi$
$137$	11.6097	0.991881	0.495940	+	0.868357i	$0.334823\pi$
$138$	−35.1716	−2.99401
$139$	−7.71479	−0.654360	−0.327180	−	0.944962i	$-0.606098\pi$
$139$	−7.71479	−0.654360	−0.327180	+	0.944962i	$0.606098\pi$
$140$	0	0
$141$	−21.0193	−1.77014
$142$	−23.3227	−1.95719
$143$	−29.4486	−2.46261
$144$	−36.4996	−3.04163
$145$	4.44414	0.369066
$146$	18.2981	1.51436
$147$	0	0
$148$	0.0229983	0.00189045
$149$	−10.4504	−0.856128	−0.428064	−	0.903748i	$-0.640804\pi$
$149$	−10.4504	−0.856128	−0.428064	+	0.903748i	$0.640804\pi$
$150$	14.4732	1.18173
$151$	8.92385	0.726212	0.363106	−	0.931748i	$-0.381716\pi$
$151$	8.92385	0.726212	0.363106	+	0.931748i	$0.381716\pi$
$152$	−5.78650	−0.469347
$153$	33.7315	2.72703
$154$	0	0
$155$	−7.08105	−0.568764
$156$	15.3670	1.23034
$157$	10.2878	0.821056	0.410528	−	0.911848i	$-0.365344\pi$
$157$	10.2878	0.821056	0.410528	+	0.911848i	$0.365344\pi$
$158$	23.9692	1.90689
$159$	−12.4947	−0.990890
$160$	−7.37953	−0.583403
$161$	0	0
$162$	38.4300	3.01935
$163$	16.0332	1.25581	0.627907	−	0.778288i	$-0.283912\pi$
$163$	16.0332	1.25581	0.627907	+	0.778288i	$0.283912\pi$
$164$	1.71293	0.133757
$165$	−27.6715	−2.15422
$166$	12.2318	0.949371
$167$	3.60583	0.279028	0.139514	−	0.990220i	$-0.455446\pi$
$167$	3.60583	0.279028	0.139514	+	0.990220i	$0.455446\pi$
$168$	0	0
$169$	14.5447	1.11883
$170$	12.0908	0.927326
$171$	22.7839	1.74233
$172$	2.19454	0.167332
$173$	8.55738	0.650605	0.325303	−	0.945610i	$-0.394534\pi$
$173$	8.55738	0.650605	0.325303	+	0.945610i	$0.394534\pi$
$174$	−15.8560	−1.20204
$175$	0	0
$176$	−28.0117	−2.11146
$177$	25.1677	1.89172
$178$	−20.2415	−1.51716
$179$	−11.2089	−0.837796	−0.418898	−	0.908033i	$-0.637584\pi$
$179$	−11.2089	−0.837796	−0.418898	+	0.908033i	$0.637584\pi$
$180$	10.2385	0.763135
$181$	16.2092	1.20482	0.602410	−	0.798187i	$-0.294207\pi$
$181$	16.2092	1.20482	0.602410	+	0.798187i	$0.294207\pi$
$182$	0	0
$183$	35.2917	2.60883
$184$	−11.9188	−0.878669
$185$	0.0387361	0.00284794
$186$	25.2642	1.85246
$187$	25.8873	1.89307
$188$	5.96862	0.435306
$189$	0	0
$190$	8.16675	0.592479
$191$	−19.9706	−1.44502	−0.722510	−	0.691360i	$-0.757012\pi$
$191$	−19.9706	−1.44502	−0.722510	+	0.691360i	$0.757012\pi$
$192$	−5.73217	−0.413684
$193$	−3.06104	−0.220339	−0.110169	−	0.993913i	$-0.535139\pi$
$193$	−3.06104	−0.220339	−0.110169	+	0.993913i	$0.535139\pi$
$194$	17.9934	1.29185
$195$	25.8826	1.85349
$196$	0	0
$197$	−15.1641	−1.08040	−0.540198	−	0.841538i	$-0.681651\pi$
$197$	−15.1641	−1.08040	−0.540198	+	0.841538i	$0.681651\pi$
$198$	70.0037	4.97495
$199$	−3.42644	−0.242894	−0.121447	−	0.992598i	$-0.538753\pi$
$199$	−3.42644	−0.242894	−0.121447	+	0.992598i	$0.538753\pi$
$200$	4.90463	0.346810
$201$	43.5186	3.06957
$202$	19.3416	1.36087
$203$	0	0
$204$	−13.5086	−0.945792
$205$	2.88509	0.201503
$206$	−14.2439	−0.992418
$207$	46.9296	3.26183
$208$	26.2007	1.81669
$209$	17.4856	1.20950
$210$	0	0
$211$	−0.101067	−0.00695772	−0.00347886	−	0.999994i	$-0.501107\pi$
$211$	−0.101067	−0.00695772	−0.00347886	+	0.999994i	$0.501107\pi$
$212$	3.54797	0.243676
$213$	43.8886	3.00720
$214$	14.8338	1.01402
$215$	3.69626	0.252083
$216$	25.7065	1.74911
$217$	0	0
$218$	−11.2336	−0.760835
$219$	−34.4334	−2.32679
$220$	7.85758	0.529758
$221$	−24.2137	−1.62879
$222$	−0.138205	−0.00927571
$223$	3.32773	0.222841	0.111421	−	0.993773i	$-0.464460\pi$
$223$	3.32773	0.222841	0.111421	+	0.993773i	$0.464460\pi$
$224$	0	0
$225$	−19.3116	−1.28744
$226$	31.1176	2.06991
$227$	20.5038	1.36089	0.680444	−	0.732800i	$-0.261787\pi$
$227$	20.5038	1.36089	0.680444	+	0.732800i	$0.261787\pi$
$228$	−9.12438	−0.604277
$229$	2.28861	0.151236	0.0756179	−	0.997137i	$-0.475907\pi$
$229$	2.28861	0.151236	0.0756179	+	0.997137i	$0.475907\pi$
$230$	16.8216	1.10918
$231$	0	0
$232$	−5.37325	−0.352771
$233$	14.7266	0.964770	0.482385	−	0.875959i	$-0.339771\pi$
$233$	14.7266	0.964770	0.482385	+	0.875959i	$0.339771\pi$
$234$	−65.4780	−4.28043
$235$	10.0529	0.655782
$236$	−7.14659	−0.465204
$237$	−45.1053	−2.92990
$238$	0	0
$239$	−2.76845	−0.179076	−0.0895382	−	0.995983i	$-0.528539\pi$
$239$	−2.76845	−0.179076	−0.0895382	+	0.995983i	$0.528539\pi$
$240$	24.6197	1.58919
$241$	28.3577	1.82668	0.913341	−	0.407196i	$-0.133493\pi$
$241$	28.3577	1.82668	0.913341	+	0.407196i	$0.133493\pi$
$242$	34.9540	2.24693
$243$	−30.7855	−1.97489
$244$	−10.0214	−0.641554
$245$	0	0
$246$	−10.2936	−0.656295
$247$	−16.3551	−1.04065
$248$	8.56144	0.543652
$249$	−23.0178	−1.45869
$250$	−20.0256	−1.26653
$251$	−4.44874	−0.280802	−0.140401	−	0.990095i	$-0.544839\pi$
$251$	−4.44874	−0.280802	−0.140401	+	0.990095i	$0.544839\pi$
$252$	0	0
$253$	36.0162	2.26432
$254$	−1.70641	−0.107070
$255$	−22.7526	−1.42482
$256$	18.0264	1.12665
$257$	−11.6922	−0.729336	−0.364668	−	0.931138i	$-0.618818\pi$
$257$	−11.6922	−0.729336	−0.364668	+	0.931138i	$0.618818\pi$
$258$	−13.1877	−0.821032
$259$	0	0
$260$	−7.34959	−0.455803
$261$	21.1568	1.30957
$262$	1.77363	0.109575
$263$	1.35931	0.0838184	0.0419092	−	0.999121i	$-0.486656\pi$
$263$	1.35931	0.0838184	0.0419092	+	0.999121i	$0.486656\pi$
$264$	33.4566	2.05911
$265$	5.97585	0.367094
$266$	0	0
$267$	38.0904	2.33110
$268$	−12.3575	−0.754855
$269$	2.80946	0.171296	0.0856480	−	0.996325i	$-0.472704\pi$
$269$	2.80946	0.171296	0.0856480	+	0.996325i	$0.472704\pi$
$270$	−36.2809	−2.20798
$271$	15.0390	0.913555	0.456778	−	0.889581i	$-0.349004\pi$
$271$	15.0390	0.913555	0.456778	+	0.889581i	$0.349004\pi$
$272$	−23.0322	−1.39653
$273$	0	0
$274$	19.8108	1.19682
$275$	−14.8208	−0.893725
$276$	−18.7941	−1.13127
$277$	25.8698	1.55437	0.777184	−	0.629274i	$-0.216648\pi$
$277$	25.8698	1.55437	0.777184	+	0.629274i	$0.216648\pi$
$278$	−13.1646	−0.789558
$279$	−33.7101	−2.01817
$280$	0	0
$281$	17.4161	1.03896	0.519478	−	0.854484i	$-0.326126\pi$
$281$	17.4161	1.03896	0.519478	+	0.854484i	$0.326126\pi$
$282$	−35.8675	−2.13588
$283$	25.3533	1.50710	0.753548	−	0.657393i	$-0.228341\pi$
$283$	25.3533	1.50710	0.753548	+	0.657393i	$0.228341\pi$
$284$	−12.4626	−0.739517
$285$	−15.3682	−0.910334
$286$	−50.2512	−2.97142
$287$	0	0
$288$	−35.1310	−2.07011
$289$	4.28551	0.252089
$290$	7.58351	0.445319
$291$	−33.8601	−1.98491
$292$	9.77767	0.572195
$293$	32.1952	1.88086	0.940432	−	0.339982i	$-0.110421\pi$
$293$	32.1952	1.88086	0.940432	+	0.339982i	$0.110421\pi$
$294$	0	0
$295$	−12.0370	−0.700822
$296$	−0.0468344	−0.00272220
$297$	−77.6797	−4.50743
$298$	−17.8326	−1.03301
$299$	−33.6878	−1.94822
$300$	7.73382	0.446513
$301$	0	0
$302$	15.2277	0.876257
$303$	−36.3970	−2.09095
$304$	−15.5571	−0.892262
$305$	−16.8790	−0.966491
$306$	57.5597	3.29047
$307$	−10.5275	−0.600838	−0.300419	−	0.953807i	$-0.597126\pi$
$307$	−10.5275	−0.600838	−0.300419	+	0.953807i	$0.597126\pi$
$308$	0	0
$309$	26.8041	1.52483
$310$	−12.0832	−0.686277
$311$	22.9148	1.29938	0.649689	−	0.760200i	$-0.274899\pi$
$311$	22.9148	1.29938	0.649689	+	0.760200i	$0.274899\pi$
$312$	−31.2937	−1.77166
$313$	28.7454	1.62479	0.812393	−	0.583111i	$-0.198165\pi$
$313$	28.7454	1.62479	0.812393	+	0.583111i	$0.198165\pi$
$314$	17.5552	0.990696
$315$	0	0
$316$	12.8080	0.720509
$317$	−24.5948	−1.38138	−0.690690	−	0.723151i	$-0.742693\pi$
$317$	−24.5948	−1.38138	−0.690690	+	0.723151i	$0.742693\pi$
$318$	−21.3210	−1.19562
$319$	16.2368	0.909086
$320$	2.74154	0.153257
$321$	−27.9143	−1.55803
$322$	0	0
$323$	14.3773	0.799974
$324$	20.5353	1.14085
$325$	13.8626	0.768959
$326$	27.3591	1.51528
$327$	21.1394	1.16901
$328$	−3.48826	−0.192607
$329$	0	0
$330$	−47.2189	−2.59931
$331$	0.720308	0.0395917	0.0197959	−	0.999804i	$-0.493698\pi$
$331$	0.720308	0.0395917	0.0197959	+	0.999804i	$0.493698\pi$
$332$	6.53611	0.358716
$333$	0.184407	0.0101055
$334$	6.15302	0.336678
$335$	−20.8138	−1.13718
$336$	0	0
$337$	20.6866	1.12687	0.563437	−	0.826159i	$-0.309479\pi$
$337$	20.6866	1.12687	0.563437	+	0.826159i	$0.309479\pi$
$338$	24.8192	1.34999
$339$	−58.5571	−3.18039
$340$	6.46080	0.350386
$341$	−25.8708	−1.40098
$342$	38.8786	2.10232
$343$	0	0
$344$	−4.46902	−0.240953
$345$	−31.6549	−1.70424
$346$	14.6024	0.785028
$347$	6.39249	0.343167	0.171583	−	0.985170i	$-0.445112\pi$
$347$	6.39249	0.343167	0.171583	+	0.985170i	$0.445112\pi$
$348$	−8.47275	−0.454187
$349$	15.4965	0.829510	0.414755	−	0.909933i	$-0.363867\pi$
$349$	15.4965	0.829510	0.414755	+	0.909933i	$0.363867\pi$
$350$	0	0
$351$	72.6578	3.87819
$352$	−26.9613	−1.43704
$353$	33.1686	1.76539	0.882694	−	0.469948i	$-0.155727\pi$
$353$	33.1686	1.76539	0.882694	+	0.469948i	$0.155727\pi$
$354$	42.9463	2.28257
$355$	−20.9907	−1.11407
$356$	−10.8161	−0.573253
$357$	0	0
$358$	−19.1270	−1.01089
$359$	−25.8554	−1.36460	−0.682298	−	0.731074i	$-0.739020\pi$
$359$	−25.8554	−1.36460	−0.682298	+	0.731074i	$0.739020\pi$
$360$	−20.8500	−1.09889
$361$	−9.28887	−0.488888
$362$	27.6595	1.45375
$363$	−65.7764	−3.45237
$364$	0	0
$365$	16.4685	0.862003
$366$	60.2219	3.14785
$367$	−16.6625	−0.869776	−0.434888	−	0.900485i	$-0.643212\pi$
$367$	−16.6625	−0.869776	−0.434888	+	0.900485i	$0.643212\pi$
$368$	−32.0440	−1.67041
$369$	13.7348	0.715003
$370$	0.0660996	0.00343636
$371$	0	0
$372$	13.5000	0.699944
$373$	−24.6440	−1.27602	−0.638010	−	0.770028i	$-0.720242\pi$
$373$	−24.6440	−1.27602	−0.638010	+	0.770028i	$0.720242\pi$
$374$	44.1743	2.28420
$375$	37.6841	1.94600
$376$	−12.1546	−0.626828
$377$	−15.1871	−0.782176
$378$	0	0
$379$	28.8190	1.48033	0.740165	−	0.672425i	$-0.234747\pi$
$379$	28.8190	1.48033	0.740165	+	0.672425i	$0.234747\pi$
$380$	4.36394	0.223865
$381$	3.21112	0.164511
$382$	−34.0779	−1.74358
$383$	11.0284	0.563527	0.281764	−	0.959484i	$-0.409081\pi$
$383$	11.0284	0.563527	0.281764	+	0.959484i	$0.409081\pi$
$384$	−40.6405	−2.07393
$385$	0	0
$386$	−5.22339	−0.265863
$387$	17.5964	0.894477
$388$	9.61488	0.488122
$389$	5.51478	0.279610	0.139805	−	0.990179i	$-0.455352\pi$
$389$	5.51478	0.279610	0.139805	+	0.990179i	$0.455352\pi$
$390$	44.1662	2.23644
$391$	29.6139	1.49764
$392$	0	0
$393$	−3.33761	−0.168360
$394$	−25.8761	−1.30362
$395$	21.5726	1.08544
$396$	37.4068	1.87976
$397$	17.7881	0.892757	0.446378	−	0.894844i	$-0.352714\pi$
$397$	17.7881	0.892757	0.446378	+	0.894844i	$0.352714\pi$
$398$	−5.84690	−0.293079
$399$	0	0
$400$	13.1862	0.659310
$401$	22.6550	1.13134	0.565668	−	0.824633i	$-0.308618\pi$
$401$	22.6550	1.13134	0.565668	+	0.824633i	$0.308618\pi$
$402$	74.2605	3.70378
$403$	24.1983	1.20540
$404$	10.3353	0.514199
$405$	34.5875	1.71867
$406$	0	0
$407$	0.141524	0.00701507
$408$	27.5093	1.36191
$409$	14.6979	0.726764	0.363382	−	0.931640i	$-0.381622\pi$
$409$	14.6979	0.726764	0.363382	+	0.931640i	$0.381622\pi$
$410$	4.92314	0.243136
$411$	−37.2800	−1.83889
$412$	−7.61128	−0.374981
$413$	0	0
$414$	80.0810	3.93577
$415$	11.0088	0.540400
$416$	25.2183	1.23643
$417$	24.7731	1.21314
$418$	29.8375	1.45940
$419$	−2.39319	−0.116915	−0.0584575	−	0.998290i	$-0.518618\pi$
$419$	−2.39319	−0.116915	−0.0584575	+	0.998290i	$0.518618\pi$
$420$	0	0
$421$	5.58715	0.272301	0.136151	−	0.990688i	$-0.456527\pi$
$421$	5.58715	0.272301	0.136151	+	0.990688i	$0.456527\pi$
$422$	−0.172461	−0.00839527
$423$	47.8581	2.32694
$424$	−7.22518	−0.350886
$425$	−12.1862	−0.591117
$426$	74.8918	3.62852
$427$	0	0
$428$	7.92653	0.383143
$429$	94.5628	4.56553
$430$	6.30733	0.304167
$431$	15.0018	0.722611	0.361306	−	0.932447i	$-0.382331\pi$
$431$	15.0018	0.722611	0.361306	+	0.932447i	$0.382331\pi$
$432$	69.1126	3.32518
$433$	−21.5115	−1.03378	−0.516888	−	0.856053i	$-0.672910\pi$
$433$	−21.5115	−1.03378	−0.516888	+	0.856053i	$0.672910\pi$
$434$	0	0
$435$	−14.2707	−0.684226
$436$	−6.00272	−0.287478
$437$	20.0027	0.956858
$438$	−58.7574	−2.80754
$439$	−21.7928	−1.04011	−0.520057	−	0.854131i	$-0.674089\pi$
$439$	−21.7928	−1.04011	−0.520057	+	0.854131i	$0.674089\pi$
$440$	−16.0014	−0.762836
$441$	0	0
$442$	−41.3185	−1.96532
$443$	−1.14984	−0.0546307	−0.0273153	−	0.999627i	$-0.508696\pi$
$443$	−1.14984	−0.0546307	−0.0273153	+	0.999627i	$0.508696\pi$
$444$	−0.0738504	−0.00350479
$445$	−18.2176	−0.863597
$446$	5.67846	0.268883
$447$	33.5574	1.58721
$448$	0	0
$449$	−18.3374	−0.865395	−0.432697	−	0.901539i	$-0.642438\pi$
$449$	−18.3374	−0.865395	−0.432697	+	0.901539i	$0.642438\pi$
$450$	−32.9535	−1.55344
$451$	10.5408	0.496345
$452$	16.6278	0.782107
$453$	−28.6555	−1.34635
$454$	34.9879	1.64206
$455$	0	0
$456$	18.5811	0.870141
$457$	−18.3394	−0.857883	−0.428941	−	0.903332i	$-0.641113\pi$
$457$	−18.3394	−0.857883	−0.428941	+	0.903332i	$0.641113\pi$
$458$	3.90531	0.182483
$459$	−63.8712	−2.98125
$460$	8.98871	0.419101
$461$	−36.0527	−1.67914	−0.839570	−	0.543251i	$-0.817193\pi$
$461$	−36.0527	−1.67914	−0.839570	+	0.543251i	$0.817193\pi$
$462$	0	0
$463$	−37.6647	−1.75043	−0.875215	−	0.483735i	$-0.839280\pi$
$463$	−37.6647	−1.75043	−0.875215	+	0.483735i	$0.839280\pi$
$464$	−14.4461	−0.670642
$465$	22.7381	1.05445
$466$	25.1295	1.16410
$467$	−34.0050	−1.57356	−0.786782	−	0.617231i	$-0.788255\pi$
$467$	−34.0050	−1.57356	−0.786782	+	0.617231i	$0.788255\pi$
$468$	−34.9885	−1.61734
$469$	0	0
$470$	17.1544	0.791274
$471$	−33.0354	−1.52219
$472$	14.5535	0.669879
$473$	13.5044	0.620934
$474$	−76.9679	−3.53525
$475$	−8.23115	−0.377671
$476$	0	0
$477$	28.4486	1.30257
$478$	−4.72411	−0.216076
$479$	26.7696	1.22314	0.611568	−	0.791192i	$-0.290539\pi$
$479$	26.7696	1.22314	0.611568	+	0.791192i	$0.290539\pi$
$480$	23.6965	1.08159
$481$	−0.132374	−0.00603575
$482$	48.3898	2.20410
$483$	0	0
$484$	18.6778	0.848992
$485$	16.1944	0.735348
$486$	−52.5326	−2.38293
$487$	−21.9852	−0.996247	−0.498123	−	0.867106i	$-0.665977\pi$
$487$	−21.9852	−0.996247	−0.498123	+	0.867106i	$0.665977\pi$
$488$	20.4078	0.923819
$489$	−51.4844	−2.32821
$490$	0	0
$491$	−28.8601	−1.30244	−0.651218	−	0.758890i	$-0.725742\pi$
$491$	−28.8601	−1.30244	−0.651218	+	0.758890i	$0.725742\pi$
$492$	−5.50042	−0.247978
$493$	13.3505	0.601277
$494$	−27.9085	−1.25566
$495$	63.0043	2.83183
$496$	23.0176	1.03352
$497$	0	0
$498$	−39.2777	−1.76008
$499$	19.4190	0.869315	0.434657	−	0.900596i	$-0.356870\pi$
$499$	19.4190	0.869315	0.434657	+	0.900596i	$0.356870\pi$
$500$	−10.7007	−0.478552
$501$	−11.5788	−0.517300
$502$	−7.59136	−0.338819
$503$	−10.4500	−0.465943	−0.232972	−	0.972484i	$-0.574845\pi$
$503$	−10.4500	−0.465943	−0.232972	+	0.972484i	$0.574845\pi$
$504$	0	0
$505$	17.4077	0.774632
$506$	61.4583	2.73215
$507$	−46.7049	−2.07424
$508$	−0.911827	−0.0404558
$509$	3.45926	0.153329	0.0766644	−	0.997057i	$-0.475573\pi$
$509$	3.45926	0.153329	0.0766644	+	0.997057i	$0.475573\pi$
$510$	−38.8251	−1.71921
$511$	0	0
$512$	5.44802	0.240771
$513$	−43.1417	−1.90475
$514$	−19.9516	−0.880026
$515$	−12.8197	−0.564903
$516$	−7.04693	−0.310224
$517$	36.7287	1.61533
$518$	0	0
$519$	−27.4787	−1.20618
$520$	14.9669	0.656342
$521$	7.98506	0.349832	0.174916	−	0.984583i	$-0.444035\pi$
$521$	7.98506	0.349832	0.174916	+	0.984583i	$0.444035\pi$
$522$	36.1021	1.58014
$523$	8.23346	0.360024	0.180012	−	0.983664i	$-0.442386\pi$
$523$	8.23346	0.360024	0.180012	+	0.983664i	$0.442386\pi$
$524$	0.947746	0.0414025
$525$	0	0
$526$	2.31953	0.101136
$527$	−21.2720	−0.926622
$528$	89.9488	3.91452
$529$	18.2009	0.791342
$530$	10.1972	0.442940
$531$	−57.3034	−2.48676
$532$	0	0
$533$	−9.85932	−0.427054
$534$	64.9978	2.81273
$535$	13.3507	0.577199
$536$	25.1652	1.08697
$537$	35.9933	1.55322
$538$	4.79409	0.206688
$539$	0	0
$540$	−19.3868	−0.834277
$541$	−34.7984	−1.49610	−0.748050	−	0.663642i	$-0.769010\pi$
$541$	−34.7984	−1.49610	−0.748050	+	0.663642i	$0.769010\pi$
$542$	25.6627	1.10231
$543$	−52.0497	−2.23366
$544$	−22.1686	−0.950472
$545$	−10.1104	−0.433081
$546$	0	0
$547$	−1.59444	−0.0681732	−0.0340866	−	0.999419i	$-0.510852\pi$
$547$	−1.59444	−0.0681732	−0.0340866	+	0.999419i	$0.510852\pi$
$548$	10.5860	0.452212
$549$	−80.3543	−3.42944
$550$	−25.2903	−1.07838
$551$	9.01759	0.384162
$552$	38.2728	1.62900
$553$	0	0
$554$	44.1445	1.87552
$555$	−0.124386	−0.00527991
$556$	−7.03455	−0.298331
$557$	30.3687	1.28676	0.643382	−	0.765545i	$-0.277531\pi$
$557$	30.3687	1.28676	0.643382	+	0.765545i	$0.277531\pi$
$558$	−57.5231	−2.43515
$559$	−12.6314	−0.534250
$560$	0	0
$561$	−83.1272	−3.50963
$562$	29.7189	1.25362
$563$	−34.4555	−1.45212	−0.726062	−	0.687629i	$-0.758652\pi$
$563$	−34.4555	−1.45212	−0.726062	+	0.687629i	$0.758652\pi$
$564$	−19.1659	−0.807031
$565$	28.0063	1.17823
$566$	43.2630	1.81848
$567$	0	0
$568$	25.3791	1.06488
$569$	14.0779	0.590177	0.295089	−	0.955470i	$-0.404651\pi$
$569$	14.0779	0.590177	0.295089	+	0.955470i	$0.404651\pi$
$570$	−26.2244	−1.09842
$571$	−34.6936	−1.45188	−0.725942	−	0.687756i	$-0.758596\pi$
$571$	−34.6936	−1.45188	−0.725942	+	0.687756i	$0.758596\pi$
$572$	−26.8520	−1.12274
$573$	64.1279	2.67898
$574$	0	0
$575$	−16.9543	−0.707042
$576$	13.0514	0.543807
$577$	−45.2875	−1.88534	−0.942671	−	0.333722i	$-0.891695\pi$
$577$	−45.2875	−1.88534	−0.942671	+	0.333722i	$0.891695\pi$
$578$	7.31283	0.304174
$579$	9.82937	0.408495
$580$	4.05228	0.168262
$581$	0	0
$582$	−57.7791	−2.39502
$583$	21.8330	0.904228
$584$	−19.9115	−0.823944
$585$	−58.9311	−2.43650
$586$	54.9381	2.26947
$587$	−15.8746	−0.655215	−0.327608	−	0.944814i	$-0.606242\pi$
$587$	−15.8746	−0.655215	−0.327608	+	0.944814i	$0.606242\pi$
$588$	0	0
$589$	−14.3681	−0.592029
$590$	−20.5400	−0.845620
$591$	48.6937	2.00299
$592$	−0.125915	−0.00517509
$593$	0.506556	0.0208018	0.0104009	−	0.999946i	$-0.496689\pi$
$593$	0.506556	0.0208018	0.0104009	+	0.999946i	$0.496689\pi$
$594$	−132.553	−5.43872
$595$	0	0
$596$	−9.52893	−0.390320
$597$	11.0027	0.450311
$598$	−57.4851	−2.35074
$599$	−39.6058	−1.61825	−0.809125	−	0.587636i	$-0.800059\pi$
$599$	−39.6058	−1.61825	−0.809125	+	0.587636i	$0.800059\pi$
$600$	−15.7494	−0.642965
$601$	2.20248	0.0898409	0.0449205	−	0.998991i	$-0.485697\pi$
$601$	2.20248	0.0898409	0.0449205	+	0.998991i	$0.485697\pi$
$602$	0	0
$603$	−99.0860	−4.03509
$604$	8.13700	0.331090
$605$	31.4591	1.27899
$606$	−62.1081	−2.52297
$607$	34.0903	1.38368	0.691840	−	0.722051i	$-0.256800\pi$
$607$	34.0903	1.38368	0.691840	+	0.722051i	$0.256800\pi$
$608$	−14.9738	−0.607267
$609$	0	0
$610$	−28.8025	−1.16618
$611$	−34.3543	−1.38983
$612$	30.7573	1.24329
$613$	−9.33234	−0.376930	−0.188465	−	0.982080i	$-0.560351\pi$
$613$	−9.33234	−0.376930	−0.188465	+	0.982080i	$0.560351\pi$
$614$	−17.9642	−0.724978
$615$	−9.26437	−0.373575
$616$	0	0
$617$	16.2640	0.654764	0.327382	−	0.944892i	$-0.393834\pi$
$617$	16.2640	0.654764	0.327382	+	0.944892i	$0.393834\pi$
$618$	45.7388	1.83988
$619$	42.4015	1.70426	0.852130	−	0.523331i	$-0.175311\pi$
$619$	42.4015	1.70426	0.852130	+	0.523331i	$0.175311\pi$
$620$	−6.45669	−0.259307
$621$	−88.8620	−3.56591
$622$	39.1020	1.56785
$623$	0	0
$624$	−84.1337	−3.36804
$625$	−4.81653	−0.192661
$626$	49.0513	1.96049
$627$	−56.1482	−2.24234
$628$	9.38069	0.374330
$629$	0.116366	0.00463982
$630$	0	0
$631$	−25.4016	−1.01122	−0.505610	−	0.862762i	$-0.668733\pi$
$631$	−25.4016	−1.01122	−0.505610	+	0.862762i	$0.668733\pi$
$632$	−26.0826	−1.03751
$633$	0.324537	0.0128992
$634$	−41.9687	−1.66679
$635$	−1.53579	−0.0609460
$636$	−11.3930	−0.451760
$637$	0	0
$638$	27.7066	1.09691
$639$	−99.9284	−3.95311
$640$	19.4372	0.768324
$641$	−18.3028	−0.722916	−0.361458	−	0.932388i	$-0.617721\pi$
$641$	−18.3028	−0.722916	−0.361458	+	0.932388i	$0.617721\pi$
$642$	−47.6332	−1.87993
$643$	−36.4025	−1.43557	−0.717786	−	0.696263i	$-0.754845\pi$
$643$	−36.4025	−1.43557	−0.717786	+	0.696263i	$0.754845\pi$
$644$	0	0
$645$	−11.8691	−0.467347
$646$	24.5335	0.965258
$647$	24.7611	0.973461	0.486730	−	0.873552i	$-0.338189\pi$
$647$	24.7611	0.973461	0.486730	+	0.873552i	$0.338189\pi$
$648$	−41.8185	−1.64279
$649$	−43.9776	−1.72627
$650$	23.6553	0.927836
$651$	0	0
$652$	14.6195	0.572543
$653$	29.2537	1.14479	0.572393	−	0.819979i	$-0.306015\pi$
$653$	29.2537	1.14479	0.572393	+	0.819979i	$0.306015\pi$
$654$	36.0724	1.41054
$655$	1.59629	0.0623722
$656$	−9.37825	−0.366159
$657$	78.4001	3.05868
$658$	0	0
$659$	−5.04629	−0.196576	−0.0982879	−	0.995158i	$-0.531337\pi$
$659$	−5.04629	−0.196576	−0.0982879	+	0.995158i	$0.531337\pi$
$660$	−25.2316	−0.982139
$661$	27.6781	1.07655	0.538276	−	0.842768i	$-0.319076\pi$
$661$	27.6781	1.07655	0.538276	+	0.842768i	$0.319076\pi$
$662$	1.22914	0.0477718
$663$	77.7531	3.01968
$664$	−13.3103	−0.516540
$665$	0	0
$666$	0.314674	0.0121934
$667$	18.5742	0.719194
$668$	3.28789	0.127212
$669$	−10.6857	−0.413134
$670$	−35.5168	−1.37213
$671$	−61.6681	−2.38067
$672$	0	0
$673$	42.8493	1.65172	0.825859	−	0.563877i	$-0.190691\pi$
$673$	42.8493	1.65172	0.825859	+	0.563877i	$0.190691\pi$
$674$	35.2998	1.35970
$675$	36.5669	1.40746
$676$	13.2623	0.510087
$677$	−23.1176	−0.888481	−0.444240	−	0.895908i	$-0.646526\pi$
$677$	−23.1176	−0.888481	−0.444240	+	0.895908i	$0.646526\pi$
$678$	−99.9223	−3.83749
$679$	0	0
$680$	−13.1569	−0.504546
$681$	−65.8403	−2.52300
$682$	−44.1462	−1.69045
$683$	44.6201	1.70734	0.853670	−	0.520815i	$-0.174372\pi$
$683$	44.6201	1.70734	0.853670	+	0.520815i	$0.174372\pi$
$684$	20.7750	0.794351
$685$	17.8300	0.681250
$686$	0	0
$687$	−7.34901	−0.280382
$688$	−12.0150	−0.458069
$689$	−20.4215	−0.777996
$690$	−54.0162	−2.05636
$691$	−47.9475	−1.82401	−0.912005	−	0.410178i	$-0.865466\pi$
$691$	−47.9475	−1.82401	−0.912005	+	0.410178i	$0.865466\pi$
$692$	7.80284	0.296620
$693$	0	0
$694$	10.9082	0.414069
$695$	−11.8483	−0.449431
$696$	17.2541	0.654016
$697$	8.66701	0.328287
$698$	26.4434	1.00090
$699$	−47.2888	−1.78863
$700$	0	0
$701$	−23.3536	−0.882055	−0.441027	−	0.897494i	$-0.645386\pi$
$701$	−23.3536	−0.882055	−0.441027	+	0.897494i	$0.645386\pi$
$702$	123.984	4.67947
$703$	0.0785994	0.00296443
$704$	10.0163	0.377504
$705$	−32.2812	−1.21578
$706$	56.5992	2.13014
$707$	0	0
$708$	22.9486	0.862460
$709$	21.2672	0.798706	0.399353	−	0.916797i	$-0.369235\pi$
$709$	21.2672	0.798706	0.399353	+	0.916797i	$0.369235\pi$
$710$	−35.8187	−1.34425
$711$	102.699	3.85150
$712$	22.0262	0.825468
$713$	−29.5951	−1.10834
$714$	0	0
$715$	−45.2268	−1.69139
$716$	−10.2206	−0.381962
$717$	8.88983	0.331997
$718$	−44.1199	−1.64654
$719$	−21.5128	−0.802293	−0.401146	−	0.916014i	$-0.631388\pi$
$719$	−21.5128	−0.802293	−0.401146	+	0.916014i	$0.631388\pi$
$720$	−56.0557	−2.08907
$721$	0	0
$722$	−15.8506	−0.589898
$723$	−91.0600	−3.38656
$724$	14.7800	0.549294
$725$	−7.64331	−0.283865
$726$	−112.241	−4.16567
$727$	1.53402	0.0568938	0.0284469	−	0.999595i	$-0.490944\pi$
$727$	1.53402	0.0568938	0.0284469	+	0.999595i	$0.490944\pi$
$728$	0	0
$729$	31.2929	1.15900
$730$	28.1020	1.04010
$731$	11.1038	0.410690
$732$	32.1799	1.18940
$733$	43.5113	1.60713	0.803563	−	0.595220i	$-0.202935\pi$
$733$	43.5113	1.60713	0.803563	+	0.595220i	$0.202935\pi$
$734$	−28.4330	−1.04948
$735$	0	0
$736$	−30.8425	−1.13687
$737$	−76.0437	−2.80111
$738$	23.4371	0.862731
$739$	−0.633734	−0.0233123	−0.0116561	−	0.999932i	$-0.503710\pi$
$739$	−0.633734	−0.0233123	−0.0116561	+	0.999932i	$0.503710\pi$
$740$	0.0353206	0.00129841
$741$	52.5183	1.92931
$742$	0	0
$743$	18.3207	0.672120	0.336060	−	0.941840i	$-0.390905\pi$
$743$	18.3207	0.672120	0.336060	+	0.941840i	$0.390905\pi$
$744$	−27.4918	−1.00790
$745$	−16.0496	−0.588011
$746$	−42.0528	−1.53966
$747$	52.4084	1.91752
$748$	23.6047	0.863074
$749$	0	0
$750$	64.3044	2.34807
$751$	28.9956	1.05806	0.529032	−	0.848602i	$-0.322555\pi$
$751$	28.9956	1.05806	0.529032	+	0.848602i	$0.322555\pi$
$752$	−32.6780	−1.19164
$753$	14.2854	0.520590
$754$	−25.9154	−0.943783
$755$	13.7052	0.498782
$756$	0	0
$757$	0.371463	0.0135010	0.00675052	−	0.999977i	$-0.497851\pi$
$757$	0.371463	0.0135010	0.00675052	+	0.999977i	$0.497851\pi$
$758$	49.1769	1.78619
$759$	−115.652	−4.19791
$760$	−8.88684	−0.322360
$761$	−14.8411	−0.537989	−0.268994	−	0.963142i	$-0.586691\pi$
$761$	−14.8411	−0.537989	−0.268994	+	0.963142i	$0.586691\pi$
$762$	5.47948	0.198501
$763$	0	0
$764$	−18.2097	−0.658804
$765$	51.8045	1.87300
$766$	18.8190	0.679959
$767$	41.1345	1.48528
$768$	−57.8849	−2.08874
$769$	5.17283	0.186537	0.0932685	−	0.995641i	$-0.470269\pi$
$769$	5.17283	0.186537	0.0932685	+	0.995641i	$0.470269\pi$
$770$	0	0
$771$	37.5449	1.35215
$772$	−2.79114	−0.100455
$773$	45.8739	1.64997	0.824984	−	0.565156i	$-0.191184\pi$
$773$	45.8739	1.64997	0.824984	+	0.565156i	$0.191184\pi$
$774$	30.0267	1.07929
$775$	12.1784	0.437463
$776$	−19.5800	−0.702881
$777$	0	0
$778$	9.41046	0.337381
$779$	5.85413	0.209746
$780$	23.6004	0.845031
$781$	−76.6902	−2.74419
$782$	50.5333	1.80707
$783$	−40.0607	−1.43165
$784$	0	0
$785$	15.7999	0.563923
$786$	−5.69533	−0.203146
$787$	−3.57941	−0.127592	−0.0637962	−	0.997963i	$-0.520321\pi$
$787$	−3.57941	−0.127592	−0.0637962	+	0.997963i	$0.520321\pi$
$788$	−13.8270	−0.492567
$789$	−4.36489	−0.155394
$790$	36.8117	1.30970
$791$	0	0
$792$	−76.1762	−2.70680
$793$	57.6813	2.04832
$794$	30.3537	1.07721
$795$	−19.1892	−0.680569
$796$	−3.12432	−0.110738
$797$	−40.4937	−1.43436	−0.717181	−	0.696887i	$-0.754568\pi$
$797$	−40.4937	−1.43436	−0.717181	+	0.696887i	$0.754568\pi$
$798$	0	0
$799$	30.1998	1.06839
$800$	12.6918	0.448722
$801$	−86.7267	−3.06434
$802$	38.6586	1.36508
$803$	60.1683	2.12329
$804$	39.6814	1.39946
$805$	0	0
$806$	41.2922	1.45446
$807$	−9.02152	−0.317572
$808$	−21.0470	−0.740431
$809$	−6.06378	−0.213191	−0.106596	−	0.994302i	$-0.533995\pi$
$809$	−6.06378	−0.213191	−0.106596	+	0.994302i	$0.533995\pi$
$810$	59.0204	2.07377
$811$	−21.2030	−0.744537	−0.372269	−	0.928125i	$-0.621420\pi$
$811$	−21.2030	−0.744537	−0.372269	+	0.928125i	$0.621420\pi$
$812$	0	0
$813$	−48.2921	−1.69368
$814$	0.241497	0.00846447
$815$	24.6236	0.862527
$816$	73.9593	2.58909
$817$	7.50008	0.262395
$818$	25.0806	0.876923
$819$	0	0
$820$	2.63070	0.0918681
$821$	−17.9853	−0.627692	−0.313846	−	0.949474i	$-0.601618\pi$
$821$	−17.9853	−0.627692	−0.313846	+	0.949474i	$0.601618\pi$
$822$	−63.6149	−2.21882
$823$	−54.6425	−1.90472	−0.952359	−	0.304978i	$-0.901351\pi$
$823$	−54.6425	−1.90472	−0.952359	+	0.304978i	$0.901351\pi$
$824$	15.4998	0.539961
$825$	47.5912	1.65691
$826$	0	0
$827$	11.1715	0.388471	0.194236	−	0.980955i	$-0.437777\pi$
$827$	11.1715	0.388471	0.194236	+	0.980955i	$0.437777\pi$
$828$	42.7916	1.48711
$829$	0.257725	0.00895117	0.00447559	−	0.999990i	$-0.498575\pi$
$829$	0.257725	0.00895117	0.00447559	+	0.999990i	$0.498575\pi$
$830$	18.7855	0.652053
$831$	−83.0711	−2.88170
$832$	−9.36876	−0.324803
$833$	0	0
$834$	42.2730	1.46379
$835$	5.53780	0.191643
$836$	15.9438	0.551428
$837$	63.8306	2.20631
$838$	−4.08376	−0.141071
$839$	−23.8829	−0.824529	−0.412264	−	0.911064i	$-0.635262\pi$
$839$	−23.8829	−0.824529	−0.412264	+	0.911064i	$0.635262\pi$
$840$	0	0
$841$	−20.6264	−0.711256
$842$	9.53396	0.328562
$843$	−55.9251	−1.92616
$844$	−0.0921553	−0.00317212
$845$	22.3377	0.768439
$846$	81.6653	2.80771
$847$	0	0
$848$	−19.4250	−0.667059
$849$	−81.4124	−2.79407
$850$	−20.7946	−0.713249
$851$	0.161897	0.00554974
$852$	40.0188	1.37102
$853$	−26.4238	−0.904734	−0.452367	−	0.891832i	$-0.649420\pi$
$853$	−26.4238	−0.904734	−0.452367	+	0.891832i	$0.649420\pi$
$854$	0	0
$855$	34.9913	1.19668
$856$	−16.1418	−0.551715
$857$	−40.2311	−1.37427	−0.687134	−	0.726531i	$-0.741131\pi$
$857$	−40.2311	−1.37427	−0.687134	+	0.726531i	$0.741131\pi$
$858$	161.363	5.50883
$859$	−5.87585	−0.200481	−0.100241	−	0.994963i	$-0.531961\pi$
$859$	−5.87585	−0.200481	−0.100241	+	0.994963i	$0.531961\pi$
$860$	3.37035	0.114928
$861$	0	0
$862$	25.5992	0.871912
$863$	36.5866	1.24542	0.622710	−	0.782452i	$-0.286031\pi$
$863$	36.5866	1.24542	0.622710	+	0.782452i	$0.286031\pi$
$864$	66.5211	2.26309
$865$	13.1423	0.446853
$866$	−36.7074	−1.24737
$867$	−13.7613	−0.467358
$868$	0	0
$869$	78.8162	2.67366
$870$	−24.3516	−0.825595
$871$	71.1276	2.41007
$872$	12.2241	0.413960
$873$	77.0948	2.60926
$874$	34.1327	1.15456
$875$	0	0
$876$	−31.3973	−1.06082
$877$	54.4962	1.84021	0.920103	−	0.391678i	$-0.128105\pi$
$877$	54.4962	1.84021	0.920103	+	0.391678i	$0.128105\pi$
$878$	−37.1874	−1.25501
$879$	−103.383	−3.48701
$880$	−43.0200	−1.45020
$881$	28.8859	0.973192	0.486596	−	0.873627i	$-0.338238\pi$
$881$	28.8859	0.973192	0.486596	+	0.873627i	$0.338238\pi$
$882$	0	0
$883$	40.5720	1.36536	0.682678	−	0.730720i	$-0.260815\pi$
$883$	40.5720	1.36536	0.682678	+	0.730720i	$0.260815\pi$
$884$	−22.0787	−0.742587
$885$	38.6523	1.29928
$886$	−1.96210	−0.0659181
$887$	−1.57560	−0.0529035	−0.0264517	−	0.999650i	$-0.508421\pi$
$887$	−1.57560	−0.0529035	−0.0264517	+	0.999650i	$0.508421\pi$
$888$	0.150391	0.00504679
$889$	0	0
$890$	−31.0867	−1.04203
$891$	126.367	4.23344
$892$	3.03431	0.101596
$893$	20.3984	0.682607
$894$	57.2626	1.91515
$895$	−17.2146	−0.575420
$896$	0	0
$897$	108.175	3.61187
$898$	−31.2910	−1.04420
$899$	−13.3420	−0.444981
$900$	−17.6089	−0.586962
$901$	17.9519	0.598064
$902$	17.9868	0.598896
$903$	0	0
$904$	−33.8613	−1.12621
$905$	24.8939	0.827502
$906$	−48.8980	−1.62453
$907$	−39.5125	−1.31199	−0.655995	−	0.754765i	$-0.727751\pi$
$907$	−39.5125	−1.31199	−0.655995	+	0.754765i	$0.727751\pi$
$908$	18.6959	0.620447
$909$	82.8711	2.74866
$910$	0	0
$911$	−3.02234	−0.100135	−0.0500673	−	0.998746i	$-0.515944\pi$
$911$	−3.02234	−0.100135	−0.0500673	+	0.998746i	$0.515944\pi$
$912$	49.9557	1.65420
$913$	40.2209	1.33112
$914$	−31.2946	−1.03513
$915$	54.2006	1.79182
$916$	2.08682	0.0689504
$917$	0	0
$918$	−108.990	−3.59721
$919$	15.9560	0.526339	0.263170	−	0.964750i	$-0.415232\pi$
$919$	15.9560	0.526339	0.263170	+	0.964750i	$0.415232\pi$
$920$	−18.3048	−0.603493
$921$	33.8051	1.11392
$922$	−61.5206	−2.02607
$923$	71.7323	2.36110
$924$	0	0
$925$	−0.0666209	−0.00219048
$926$	−64.2714	−2.11209
$927$	−61.0294	−2.00447
$928$	−13.9044	−0.456434
$929$	39.2648	1.28824	0.644119	−	0.764926i	$-0.277224\pi$
$929$	39.2648	1.28824	0.644119	+	0.764926i	$0.277224\pi$
$930$	38.8005	1.27232
$931$	0	0
$932$	13.4281	0.439852
$933$	−73.5821	−2.40897
$934$	−58.0264	−1.89868
$935$	39.7575	1.30021
$936$	71.2515	2.32893
$937$	9.53287	0.311425	0.155713	−	0.987802i	$-0.450233\pi$
$937$	9.53287	0.311425	0.155713	+	0.987802i	$0.450233\pi$
$938$	0	0
$939$	−92.3049	−3.01226
$940$	9.16654	0.298980
$941$	−52.6695	−1.71698	−0.858488	−	0.512834i	$-0.828596\pi$
$941$	−52.6695	−1.71698	−0.858488	+	0.512834i	$0.828596\pi$
$942$	−56.3718	−1.83669
$943$	12.0581	0.392667
$944$	39.1274	1.27349
$945$	0	0
$946$	23.0440	0.749226
$947$	−11.4971	−0.373606	−0.186803	−	0.982397i	$-0.559813\pi$
$947$	−11.4971	−0.373606	−0.186803	+	0.982397i	$0.559813\pi$
$948$	−41.1282	−1.33578
$949$	−56.2785	−1.82688
$950$	−14.0457	−0.455703
$951$	78.9768	2.56100
$952$	0	0
$953$	−12.0855	−0.391487	−0.195743	−	0.980655i	$-0.562712\pi$
$953$	−12.0855	−0.391487	−0.195743	+	0.980655i	$0.562712\pi$
$954$	48.5450	1.57170
$955$	−30.6706	−0.992478
$956$	−2.52435	−0.0816433
$957$	−52.1383	−1.68539
$958$	45.6799	1.47585
$959$	0	0
$960$	−8.80341	−0.284129
$961$	−9.74153	−0.314243
$962$	−0.225885	−0.00728281
$963$	63.5571	2.04810
$964$	25.8573	0.832809
$965$	−4.70112	−0.151334
$966$	0	0
$967$	−37.6146	−1.20960	−0.604802	−	0.796376i	$-0.706748\pi$
$967$	−37.6146	−1.20960	−0.604802	+	0.796376i	$0.706748\pi$
$968$	−38.0360	−1.22252
$969$	−46.1672	−1.48310
$970$	27.6342	0.887279
$971$	−9.92063	−0.318368	−0.159184	−	0.987249i	$-0.550886\pi$
$971$	−9.92063	−0.318368	−0.159184	+	0.987249i	$0.550886\pi$
$972$	−28.0710	−0.900379
$973$	0	0
$974$	−37.5158	−1.20208
$975$	−44.5145	−1.42560
$976$	54.8668	1.75624
$977$	−46.0243	−1.47245	−0.736223	−	0.676739i	$-0.763393\pi$
$977$	−46.0243	−1.47245	−0.736223	+	0.676739i	$0.763393\pi$
$978$	−87.8534	−2.80924
$979$	−66.5586	−2.12722
$980$	0	0
$981$	−48.1315	−1.53672
$982$	−49.2470	−1.57154
$983$	−40.8250	−1.30212	−0.651058	−	0.759028i	$-0.725675\pi$
$983$	−40.8250	−1.30212	−0.651058	+	0.759028i	$0.725675\pi$
$984$	11.2012	0.357081
$985$	−23.2889	−0.742045
$986$	22.7814	0.725508
$987$	0	0
$988$	−14.9130	−0.474447
$989$	15.4484	0.491231
$990$	107.511	3.41692
$991$	36.0398	1.14484	0.572421	−	0.819960i	$-0.306004\pi$
$991$	36.0398	1.14484	0.572421	+	0.819960i	$0.306004\pi$
$992$	22.1545	0.703407
$993$	−2.31299	−0.0734006
$994$	0	0
$995$	−5.26229	−0.166826
$996$	−20.9882	−0.665038
$997$	2.70237	0.0855851	0.0427925	−	0.999084i	$-0.486375\pi$
$997$	2.70237	0.0855851	0.0427925	+	0.999084i	$0.486375\pi$
$998$	33.1368	1.04893
$999$	−0.349178	−0.0110475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.n.1.25	✓	34
7.6	odd	2	inner	6223.2.a.n.1.26	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.n.1.25	✓	34	1.1	even	1	trivial
6223.2.a.n.1.26	yes	34	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.70641 q^{2} -3.21112 q^{3} +0.911827 q^{4} +1.53579 q^{5} -5.47948 q^{6} -1.85687 q^{8} +7.31128 q^{9} +2.62068 q^{10} +5.61106 q^{11} -2.92798 q^{12} -5.24831 q^{13} -4.93160 q^{15} -4.99223 q^{16} +4.61362 q^{17} +12.4760 q^{18} +3.11627 q^{19} +1.40037 q^{20} +9.57475 q^{22} +6.41879 q^{23} +5.96262 q^{24} -2.64135 q^{25} -8.95575 q^{26} -13.8440 q^{27} +2.89372 q^{29} -8.41533 q^{30} -4.61069 q^{31} -4.80504 q^{32} -18.0178 q^{33} +7.87272 q^{34} +6.66662 q^{36} +0.0252223 q^{37} +5.31762 q^{38} +16.8529 q^{39} -2.85176 q^{40} +1.87857 q^{41} +2.40675 q^{43} +5.11631 q^{44} +11.2286 q^{45} +10.9531 q^{46} +6.54578 q^{47} +16.0306 q^{48} -4.50722 q^{50} -14.8149 q^{51} -4.78555 q^{52} +3.89106 q^{53} -23.6236 q^{54} +8.61740 q^{55} -10.0067 q^{57} +4.93786 q^{58} -7.83767 q^{59} -4.49677 q^{60} -10.9905 q^{61} -7.86772 q^{62} +1.78510 q^{64} -8.06030 q^{65} -30.7457 q^{66} -13.5525 q^{67} +4.20682 q^{68} -20.6115 q^{69} -13.6677 q^{71} -13.5761 q^{72} +10.7232 q^{73} +0.0430395 q^{74} +8.48168 q^{75} +2.84150 q^{76} +28.7580 q^{78} +14.0466 q^{79} -7.66701 q^{80} +22.5210 q^{81} +3.20561 q^{82} +7.16815 q^{83} +7.08556 q^{85} +4.10690 q^{86} -9.29206 q^{87} -10.4190 q^{88} -11.8620 q^{89} +19.1606 q^{90} +5.85282 q^{92} +14.8055 q^{93} +11.1698 q^{94} +4.78593 q^{95} +15.4295 q^{96} +10.5446 q^{97} +41.0240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 6 q^{2} + 46 q^{4} + 30 q^{8} + 66 q^{9} + 16 q^{11} + 12 q^{15} + 54 q^{16} + 48 q^{18} + 30 q^{22} + 46 q^{23} + 80 q^{25} + 28 q^{29} - 2 q^{30} + 36 q^{32} + 98 q^{36} + 52 q^{37} - 2 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 34 * q + 6 * q^2 + 46 * q^4 + 30 * q^8 + 66 * q^9 + 16 * q^11 + 12 * q^15 + 54 * q^16 + 48 * q^18 + 30 * q^22 + 46 * q^23 + 80 * q^25 + 28 * q^29 - 2 * q^30 + 36 * q^32 + 98 * q^36 + 52 * q^37 - 2 * q^39 + 10 * q^43 + 16 * q^44 - 36 * q^46 + 74 * q^50 + 56 * q^51 + 64 * q^53 + 44 * q^57 + 32 * q^58 - 132 * q^60 + 74 * q^64 + 108 * q^65 + 14 * q^67 + 6 * q^71 + 172 * q^72 - 6 * q^74 + 52 * q^78 + 46 * q^81 - 2 * q^85 + 96 * q^86 - 170 * q^88 + 130 * q^92 + 6 * q^93 + 14 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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