LMFDB - Embedded newform 6223.2.a.t.1.29

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 = [74,18,0,86,0] 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:	$74$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
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)

$f(q)$	$=$	\(q-0.487212 q^{2} -3.12936 q^{3} -1.76262 q^{4} +1.94942 q^{5} +1.52466 q^{6} +1.83320 q^{8} +6.79290 q^{9} -0.949783 q^{10} -2.93120 q^{11} +5.51589 q^{12} -4.46990 q^{13} -6.10045 q^{15} +2.63209 q^{16} -5.90096 q^{17} -3.30958 q^{18} -7.92999 q^{19} -3.43610 q^{20} +1.42812 q^{22} -4.06312 q^{23} -5.73673 q^{24} -1.19975 q^{25} +2.17779 q^{26} -11.8694 q^{27} -0.270357 q^{29} +2.97221 q^{30} +2.04938 q^{31} -4.94878 q^{32} +9.17279 q^{33} +2.87502 q^{34} -11.9733 q^{36} +5.73409 q^{37} +3.86359 q^{38} +13.9879 q^{39} +3.57367 q^{40} -0.689971 q^{41} -10.5411 q^{43} +5.16661 q^{44} +13.2422 q^{45} +1.97960 q^{46} +4.34888 q^{47} -8.23677 q^{48} +0.584533 q^{50} +18.4662 q^{51} +7.87876 q^{52} -6.13952 q^{53} +5.78290 q^{54} -5.71415 q^{55} +24.8158 q^{57} +0.131721 q^{58} -9.18657 q^{59} +10.7528 q^{60} -10.8520 q^{61} -0.998484 q^{62} -2.85308 q^{64} -8.71373 q^{65} -4.46909 q^{66} -3.47659 q^{67} +10.4012 q^{68} +12.7150 q^{69} +8.18612 q^{71} +12.4527 q^{72} +12.3028 q^{73} -2.79372 q^{74} +3.75445 q^{75} +13.9776 q^{76} -6.81509 q^{78} -12.7087 q^{79} +5.13106 q^{80} +16.7648 q^{81} +0.336162 q^{82} -14.7939 q^{83} -11.5035 q^{85} +5.13577 q^{86} +0.846045 q^{87} -5.37347 q^{88} -6.04620 q^{89} -6.45178 q^{90} +7.16176 q^{92} -6.41326 q^{93} -2.11883 q^{94} -15.4589 q^{95} +15.4865 q^{96} +7.38376 q^{97} -19.9114 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * 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 $

$2$	−0.487212	−0.344511	−0.172256	−	0.985052i	$-0.555105\pi$
$2$	−0.487212	−0.344511	−0.172256	+	0.985052i	$0.555105\pi$
$3$	−3.12936	−1.80674	−0.903369	−	0.428865i	$-0.858914\pi$
$3$	−3.12936	−1.80674	−0.903369	+	0.428865i	$0.858914\pi$
$4$	−1.76262	−0.881312
$5$	1.94942	0.871808	0.435904	−	0.899993i	$-0.356429\pi$
$5$	1.94942	0.871808	0.435904	+	0.899993i	$0.356429\pi$
$6$	1.52466	0.622441
$7$	0	0
$7$	0	0
$8$	1.83320	0.648133
$9$	6.79290	2.26430
$10$	−0.949783	−0.300348
$11$	−2.93120	−0.883790	−0.441895	−	0.897067i	$-0.645694\pi$
$11$	−2.93120	−0.883790	−0.441895	+	0.897067i	$0.645694\pi$
$12$	5.51589	1.59230
$13$	−4.46990	−1.23973	−0.619864	−	0.784709i	$-0.712812\pi$
$13$	−4.46990	−1.23973	−0.619864	+	0.784709i	$0.712812\pi$
$14$	0	0
$15$	−6.10045	−1.57513
$16$	2.63209	0.658023
$17$	−5.90096	−1.43119	−0.715596	−	0.698514i	$-0.753845\pi$
$17$	−5.90096	−1.43119	−0.715596	+	0.698514i	$0.753845\pi$
$18$	−3.30958	−0.780077
$19$	−7.92999	−1.81927	−0.909633	−	0.415414i	$-0.863637\pi$
$19$	−7.92999	−1.81927	−0.909633	+	0.415414i	$0.863637\pi$
$20$	−3.43610	−0.768335
$21$	0	0
$22$	1.42812	0.304475
$23$	−4.06312	−0.847220	−0.423610	−	0.905845i	$-0.639237\pi$
$23$	−4.06312	−0.847220	−0.423610	+	0.905845i	$0.639237\pi$
$24$	−5.73673	−1.17101
$25$	−1.19975	−0.239950
$26$	2.17779	0.427100
$27$	−11.8694	−2.28426
$28$	0	0
$29$	−0.270357	−0.0502041	−0.0251020	−	0.999685i	$-0.507991\pi$
$29$	−0.270357	−0.0502041	−0.0251020	+	0.999685i	$0.507991\pi$
$30$	2.97221	0.542649
$31$	2.04938	0.368080	0.184040	−	0.982919i	$-0.441082\pi$
$31$	2.04938	0.368080	0.184040	+	0.982919i	$0.441082\pi$
$32$	−4.94878	−0.874829
$33$	9.17279	1.59678
$34$	2.87502	0.493062
$35$	0	0
$36$	−11.9733	−1.99556
$37$	5.73409	0.942678	0.471339	−	0.881952i	$-0.343771\pi$
$37$	5.73409	0.942678	0.471339	+	0.881952i	$0.343771\pi$
$38$	3.86359	0.626757
$39$	13.9879	2.23986
$40$	3.57367	0.565048
$41$	−0.689971	−0.107755	−0.0538776	−	0.998548i	$-0.517158\pi$
$41$	−0.689971	−0.107755	−0.0538776	+	0.998548i	$0.517158\pi$
$42$	0	0
$43$	−10.5411	−1.60751	−0.803755	−	0.594961i	$-0.797168\pi$
$43$	−10.5411	−1.60751	−0.803755	+	0.594961i	$0.797168\pi$
$44$	5.16661	0.778895
$45$	13.2422	1.97404
$46$	1.97960	0.291876
$47$	4.34888	0.634350	0.317175	−	0.948367i	$-0.397266\pi$
$47$	4.34888	0.634350	0.317175	+	0.948367i	$0.397266\pi$
$48$	−8.23677	−1.18888
$49$	0	0
$50$	0.584533	0.0826655
$51$	18.4662	2.58579
$52$	7.87876	1.09259
$53$	−6.13952	−0.843328	−0.421664	−	0.906752i	$-0.638554\pi$
$53$	−6.13952	−0.843328	−0.421664	+	0.906752i	$0.638554\pi$
$54$	5.78290	0.786953
$55$	−5.71415	−0.770496
$56$	0	0
$57$	24.8158	3.28694
$58$	0.131721	0.0172959
$59$	−9.18657	−1.19599	−0.597995	−	0.801500i	$-0.704036\pi$
$59$	−9.18657	−1.19599	−0.597995	+	0.801500i	$0.704036\pi$
$60$	10.7528	1.38818
$61$	−10.8520	−1.38946	−0.694728	−	0.719273i	$-0.744475\pi$
$61$	−10.8520	−1.38946	−0.694728	+	0.719273i	$0.744475\pi$
$62$	−0.998484	−0.126808
$63$	0	0
$64$	−2.85308	−0.356635
$65$	−8.71373	−1.08080
$66$	−4.46909	−0.550107
$67$	−3.47659	−0.424734	−0.212367	−	0.977190i	$-0.568117\pi$
$67$	−3.47659	−0.424734	−0.212367	+	0.977190i	$0.568117\pi$
$68$	10.4012	1.26133
$69$	12.7150	1.53070
$70$	0	0
$71$	8.18612	0.971514	0.485757	−	0.874094i	$-0.338544\pi$
$71$	8.18612	0.971514	0.485757	+	0.874094i	$0.338544\pi$
$72$	12.4527	1.46757
$73$	12.3028	1.43993	0.719965	−	0.694010i	$-0.244158\pi$
$73$	12.3028	1.43993	0.719965	+	0.694010i	$0.244158\pi$
$74$	−2.79372	−0.324763
$75$	3.75445	0.433527
$76$	13.9776	1.60334
$77$	0	0
$78$	−6.81509	−0.771657
$79$	−12.7087	−1.42984	−0.714919	−	0.699207i	$-0.753537\pi$
$79$	−12.7087	−1.42984	−0.714919	+	0.699207i	$0.753537\pi$
$80$	5.13106	0.573670
$81$	16.7648	1.86276
$82$	0.336162	0.0371229
$83$	−14.7939	−1.62384	−0.811918	−	0.583771i	$-0.801577\pi$
$83$	−14.7939	−1.62384	−0.811918	+	0.583771i	$0.801577\pi$
$84$	0	0
$85$	−11.5035	−1.24773
$86$	5.13577	0.553805
$87$	0.846045	0.0907056
$88$	−5.37347	−0.572813
$89$	−6.04620	−0.640896	−0.320448	−	0.947266i	$-0.603833\pi$
$89$	−6.04620	−0.640896	−0.320448	+	0.947266i	$0.603833\pi$
$90$	−6.45178	−0.680077
$91$	0	0
$92$	7.16176	0.746665
$93$	−6.41326	−0.665024
$94$	−2.11883	−0.218540
$95$	−15.4589	−1.58605
$96$	15.4865	1.58059
$97$	7.38376	0.749708	0.374854	−	0.927084i	$-0.377693\pi$
$97$	7.38376	0.749708	0.374854	+	0.927084i	$0.377693\pi$
$98$	0	0
$99$	−19.9114	−2.00117
$100$	2.11471	0.211471
$101$	−2.70009	−0.268669	−0.134334	−	0.990936i	$-0.542890\pi$
$101$	−2.70009	−0.268669	−0.134334	+	0.990936i	$0.542890\pi$
$102$	−8.99697	−0.890833
$103$	−8.54458	−0.841923	−0.420961	−	0.907079i	$-0.638307\pi$
$103$	−8.54458	−0.841923	−0.420961	+	0.907079i	$0.638307\pi$
$104$	−8.19421	−0.803508
$105$	0	0
$106$	2.99125	0.290536
$107$	16.7345	1.61779	0.808893	−	0.587956i	$-0.200067\pi$
$107$	16.7345	1.61779	0.808893	+	0.587956i	$0.200067\pi$
$108$	20.9212	2.01315
$109$	−8.30584	−0.795555	−0.397778	−	0.917482i	$-0.630218\pi$
$109$	−8.30584	−0.795555	−0.397778	+	0.917482i	$0.630218\pi$
$110$	2.78400	0.265444
$111$	−17.9440	−1.70317
$112$	0	0
$113$	14.7041	1.38324	0.691622	−	0.722260i	$-0.256896\pi$
$113$	14.7041	1.38324	0.691622	+	0.722260i	$0.256896\pi$
$114$	−12.0906	−1.13239
$115$	−7.92074	−0.738613
$116$	0.476538	0.0442454
$117$	−30.3636	−2.80712
$118$	4.47581	0.412032
$119$	0	0
$120$	−11.1833	−1.02089
$121$	−2.40806	−0.218915
$122$	5.28722	0.478683
$123$	2.15917	0.194686
$124$	−3.61229	−0.324393
$125$	−12.0859	−1.08100
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	11.2876	0.997694
$129$	32.9870	2.90435
$130$	4.24543	0.372349
$131$	−15.0892	−1.31835	−0.659173	−	0.751991i	$-0.729093\pi$
$131$	−15.0892	−1.31835	−0.659173	+	0.751991i	$0.729093\pi$
$132$	−16.1682	−1.40726
$133$	0	0
$134$	1.69384	0.146325
$135$	−23.1384	−1.99144
$136$	−10.8176	−0.927603
$137$	2.52659	0.215862	0.107931	−	0.994158i	$-0.465578\pi$
$137$	2.52659	0.215862	0.107931	+	0.994158i	$0.465578\pi$
$138$	−6.19489	−0.527344
$139$	−15.2877	−1.29669	−0.648344	−	0.761348i	$-0.724538\pi$
$139$	−15.2877	−1.29669	−0.648344	+	0.761348i	$0.724538\pi$
$140$	0	0
$141$	−13.6092	−1.14610
$142$	−3.98838	−0.334697
$143$	13.1022	1.09566
$144$	17.8796	1.48996
$145$	−0.527040	−0.0437683
$146$	−5.99406	−0.496072
$147$	0	0
$148$	−10.1070	−0.830794
$149$	0.741156	0.0607179	0.0303590	−	0.999539i	$-0.490335\pi$
$149$	0.741156	0.0607179	0.0303590	+	0.999539i	$0.490335\pi$
$150$	−1.82922	−0.149355
$151$	−9.85028	−0.801604	−0.400802	−	0.916165i	$-0.631269\pi$
$151$	−9.85028	−0.801604	−0.400802	+	0.916165i	$0.631269\pi$
$152$	−14.5372	−1.17913
$153$	−40.0846	−3.24065
$154$	0	0
$155$	3.99511	0.320895
$156$	−24.6555	−1.97402
$157$	3.45496	0.275736	0.137868	−	0.990451i	$-0.455975\pi$
$157$	3.45496	0.275736	0.137868	+	0.990451i	$0.455975\pi$
$158$	6.19183	0.492595
$159$	19.2128	1.52367
$160$	−9.64727	−0.762683
$161$	0	0
$162$	−8.16802	−0.641740
$163$	9.81989	0.769153	0.384577	−	0.923093i	$-0.374347\pi$
$163$	9.81989	0.769153	0.384577	+	0.923093i	$0.374347\pi$
$164$	1.21616	0.0949660
$165$	17.8816	1.39208
$166$	7.20775	0.559430
$167$	−16.0396	−1.24118	−0.620590	−	0.784135i	$-0.713107\pi$
$167$	−16.0396	−1.24118	−0.620590	+	0.784135i	$0.713107\pi$
$168$	0	0
$169$	6.98002	0.536924
$170$	5.60463	0.429855
$171$	−53.8677	−4.11936
$172$	18.5801	1.41672
$173$	−20.5728	−1.56412	−0.782059	−	0.623204i	$-0.785831\pi$
$173$	−20.5728	−1.56412	−0.782059	+	0.623204i	$0.785831\pi$
$174$	−0.412204	−0.0312491
$175$	0	0
$176$	−7.71519	−0.581555
$177$	28.7481	2.16084
$178$	2.94578	0.220796
$179$	−5.96103	−0.445548	−0.222774	−	0.974870i	$-0.571511\pi$
$179$	−5.96103	−0.445548	−0.222774	+	0.974870i	$0.571511\pi$
$180$	−23.3411	−1.73974
$181$	−8.44643	−0.627818	−0.313909	−	0.949453i	$-0.601639\pi$
$181$	−8.44643	−0.627818	−0.313909	+	0.949453i	$0.601639\pi$
$182$	0	0
$183$	33.9598	2.51038
$184$	−7.44850	−0.549111
$185$	11.1782	0.821835
$186$	3.12462	0.229108
$187$	17.2969	1.26487
$188$	−7.66544	−0.559060
$189$	0	0
$190$	7.53177	0.546412
$191$	−25.2932	−1.83015	−0.915075	−	0.403285i	$-0.867868\pi$
$191$	−25.2932	−1.83015	−0.915075	+	0.403285i	$0.867868\pi$
$192$	8.92832	0.644346
$193$	−18.3105	−1.31802	−0.659010	−	0.752134i	$-0.729025\pi$
$193$	−18.3105	−1.31802	−0.659010	+	0.752134i	$0.729025\pi$
$194$	−3.59746	−0.258283
$195$	27.2684	1.95273
$196$	0	0
$197$	−1.42059	−0.101213	−0.0506063	−	0.998719i	$-0.516115\pi$
$197$	−1.42059	−0.101213	−0.0506063	+	0.998719i	$0.516115\pi$
$198$	9.70106	0.689424
$199$	−6.83535	−0.484545	−0.242273	−	0.970208i	$-0.577893\pi$
$199$	−6.83535	−0.484545	−0.242273	+	0.970208i	$0.577893\pi$
$200$	−2.19938	−0.155520
$201$	10.8795	0.767382
$202$	1.31552	0.0925593
$203$	0	0
$204$	−32.5490	−2.27889
$205$	−1.34504	−0.0939420
$206$	4.16303	0.290052
$207$	−27.6004	−1.91836
$208$	−11.7652	−0.815770
$209$	23.2444	1.60785
$210$	0	0
$211$	25.4738	1.75369	0.876843	−	0.480776i	$-0.159645\pi$
$211$	25.4738	1.75369	0.876843	+	0.480776i	$0.159645\pi$
$212$	10.8217	0.743235
$213$	−25.6173	−1.75527
$214$	−8.15326	−0.557345
$215$	−20.5491	−1.40144
$216$	−21.7589	−1.48050
$217$	0	0
$218$	4.04671	0.274078
$219$	−38.4998	−2.60158
$220$	10.0719	0.679047
$221$	26.3767	1.77429
$222$	8.74255	0.586762
$223$	26.6886	1.78720	0.893600	−	0.448865i	$-0.148172\pi$
$223$	26.6886	1.78720	0.893600	+	0.448865i	$0.148172\pi$
$224$	0	0
$225$	−8.14979	−0.543319
$226$	−7.16401	−0.476543
$227$	−26.8645	−1.78306	−0.891530	−	0.452961i	$-0.850368\pi$
$227$	−26.8645	−1.78306	−0.891530	+	0.452961i	$0.850368\pi$
$228$	−43.7410	−2.89682
$229$	3.87878	0.256317	0.128159	−	0.991754i	$-0.459093\pi$
$229$	3.87878	0.256317	0.128159	+	0.991754i	$0.459093\pi$
$230$	3.85908	0.254460
$231$	0	0
$232$	−0.495618	−0.0325389
$233$	13.3694	0.875860	0.437930	−	0.899009i	$-0.355712\pi$
$233$	13.3694	0.875860	0.437930	+	0.899009i	$0.355712\pi$
$234$	14.7935	0.967082
$235$	8.47781	0.553031
$236$	16.1925	1.05404
$237$	39.7701	2.58334
$238$	0	0
$239$	4.59873	0.297467	0.148733	−	0.988877i	$-0.452480\pi$
$239$	4.59873	0.297467	0.148733	+	0.988877i	$0.452480\pi$
$240$	−16.0569	−1.03647
$241$	13.4732	0.867886	0.433943	−	0.900940i	$-0.357122\pi$
$241$	13.4732	0.867886	0.433943	+	0.900940i	$0.357122\pi$
$242$	1.17324	0.0754186
$243$	−16.8551	−1.08125
$244$	19.1280	1.22454
$245$	0	0
$246$	−1.05197	−0.0670713
$247$	35.4463	2.25539
$248$	3.75692	0.238565
$249$	46.2953	2.93385
$250$	5.88841	0.372416
$251$	27.0338	1.70636	0.853179	−	0.521617i	$-0.174671\pi$
$251$	27.0338	1.70636	0.853179	+	0.521617i	$0.174671\pi$
$252$	0	0
$253$	11.9098	0.748764
$254$	−0.487212	−0.0305704
$255$	35.9985	2.25431
$256$	0.206696	0.0129185
$257$	−3.34236	−0.208491	−0.104245	−	0.994552i	$-0.533243\pi$
$257$	−3.34236	−0.208491	−0.104245	+	0.994552i	$0.533243\pi$
$258$	−16.0717	−1.00058
$259$	0	0
$260$	15.3590	0.952526
$261$	−1.83651	−0.113677
$262$	7.35162	0.454185
$263$	6.83074	0.421201	0.210601	−	0.977572i	$-0.432458\pi$
$263$	6.83074	0.421201	0.210601	+	0.977572i	$0.432458\pi$
$264$	16.8155	1.03492
$265$	−11.9685	−0.735220
$266$	0	0
$267$	18.9207	1.15793
$268$	6.12793	0.374323
$269$	−13.6114	−0.829904	−0.414952	−	0.909843i	$-0.636202\pi$
$269$	−13.6114	−0.829904	−0.414952	+	0.909843i	$0.636202\pi$
$270$	11.2733	0.686072
$271$	−16.6620	−1.01215	−0.506073	−	0.862491i	$-0.668903\pi$
$271$	−16.6620	−1.01215	−0.506073	+	0.862491i	$0.668903\pi$
$272$	−15.5319	−0.941758
$273$	0	0
$274$	−1.23099	−0.0743667
$275$	3.51671	0.212066
$276$	−22.4117	−1.34903
$277$	6.51350	0.391359	0.195679	−	0.980668i	$-0.437309\pi$
$277$	6.51350	0.391359	0.195679	+	0.980668i	$0.437309\pi$
$278$	7.44836	0.446723
$279$	13.9212	0.833443
$280$	0	0
$281$	11.2016	0.668233	0.334116	−	0.942532i	$-0.391562\pi$
$281$	11.2016	0.668233	0.334116	+	0.942532i	$0.391562\pi$
$282$	6.63058	0.394845
$283$	4.15129	0.246768	0.123384	−	0.992359i	$-0.460625\pi$
$283$	4.15129	0.246768	0.123384	+	0.992359i	$0.460625\pi$
$284$	−14.4291	−0.856207
$285$	48.3765	2.86558
$286$	−6.38354	−0.377467
$287$	0	0
$288$	−33.6166	−1.98088
$289$	17.8213	1.04831
$290$	0.256780	0.0150787
$291$	−23.1065	−1.35453
$292$	−21.6852	−1.26903
$293$	−21.9452	−1.28205	−0.641025	−	0.767520i	$-0.721490\pi$
$293$	−21.9452	−1.28205	−0.641025	+	0.767520i	$0.721490\pi$
$294$	0	0
$295$	−17.9085	−1.04267
$296$	10.5117	0.610981
$297$	34.7915	2.01881
$298$	−0.361100	−0.0209180
$299$	18.1618	1.05032
$300$	−6.61769	−0.382072
$301$	0	0
$302$	4.79918	0.276162
$303$	8.44955	0.485414
$304$	−20.8725	−1.19712
$305$	−21.1551	−1.21134
$306$	19.5297	1.11644
$307$	3.32170	0.189580	0.0947898	−	0.995497i	$-0.469782\pi$
$307$	3.32170	0.189580	0.0947898	+	0.995497i	$0.469782\pi$
$308$	0	0
$309$	26.7391	1.52113
$310$	−1.94647	−0.110552
$311$	−5.52835	−0.313484	−0.156742	−	0.987640i	$-0.550099\pi$
$311$	−5.52835	−0.313484	−0.156742	+	0.987640i	$0.550099\pi$
$312$	25.6426	1.45173
$313$	17.6066	0.995182	0.497591	−	0.867412i	$-0.334218\pi$
$313$	17.6066	0.995182	0.497591	+	0.867412i	$0.334218\pi$
$314$	−1.68330	−0.0949939
$315$	0	0
$316$	22.4006	1.26013
$317$	−24.1894	−1.35861	−0.679305	−	0.733856i	$-0.737719\pi$
$317$	−24.1894	−1.35861	−0.679305	+	0.733856i	$0.737719\pi$
$318$	−9.36069	−0.524922
$319$	0.792471	0.0443699
$320$	−5.56186	−0.310917
$321$	−52.3683	−2.92292
$322$	0	0
$323$	46.7946	2.60372
$324$	−29.5501	−1.64167
$325$	5.36277	0.297473
$326$	−4.78437	−0.264982
$327$	25.9920	1.43736
$328$	−1.26485	−0.0698397
$329$	0	0
$330$	−8.71215	−0.479588
$331$	0.874401	0.0480614	0.0240307	−	0.999711i	$-0.492350\pi$
$331$	0.874401	0.0480614	0.0240307	+	0.999711i	$0.492350\pi$
$332$	26.0760	1.43111
$333$	38.9511	2.13451
$334$	7.81468	0.427600
$335$	−6.77735	−0.370286
$336$	0	0
$337$	9.93486	0.541186	0.270593	−	0.962694i	$-0.412780\pi$
$337$	9.93486	0.541186	0.270593	+	0.962694i	$0.412780\pi$
$338$	−3.40075	−0.184976
$339$	−46.0144	−2.49916
$340$	20.2763	1.09964
$341$	−6.00715	−0.325305
$342$	26.2450	1.41917
$343$	0	0
$344$	−19.3240	−1.04188
$345$	24.7869	1.33448
$346$	10.0233	0.538856
$347$	11.6505	0.625432	0.312716	−	0.949847i	$-0.398761\pi$
$347$	11.6505	0.625432	0.312716	+	0.949847i	$0.398761\pi$
$348$	−1.49126	−0.0799399
$349$	−24.4120	−1.30675	−0.653373	−	0.757036i	$-0.726647\pi$
$349$	−24.4120	−1.30675	−0.653373	+	0.757036i	$0.726647\pi$
$350$	0	0
$351$	53.0549	2.83186
$352$	14.5059	0.773165
$353$	0.401347	0.0213615	0.0106808	−	0.999943i	$-0.496600\pi$
$353$	0.401347	0.0213615	0.0106808	+	0.999943i	$0.496600\pi$
$354$	−14.0064	−0.744433
$355$	15.9582	0.846974
$356$	10.6572	0.564829
$357$	0	0
$358$	2.90428	0.153496
$359$	−27.9197	−1.47354	−0.736772	−	0.676141i	$-0.763651\pi$
$359$	−27.9197	−1.47354	−0.736772	+	0.676141i	$0.763651\pi$
$360$	24.2756	1.27944
$361$	43.8848	2.30973
$362$	4.11520	0.216290
$363$	7.53570	0.395522
$364$	0	0
$365$	23.9833	1.25534
$366$	−16.5456	−0.864854
$367$	10.0533	0.524778	0.262389	−	0.964962i	$-0.415490\pi$
$367$	10.0533	0.524778	0.262389	+	0.964962i	$0.415490\pi$
$368$	−10.6945	−0.557490
$369$	−4.68690	−0.243990
$370$	−5.44614	−0.283131
$371$	0	0
$372$	11.3042	0.586093
$373$	26.1519	1.35409	0.677047	−	0.735940i	$-0.263259\pi$
$373$	26.1519	1.35409	0.677047	+	0.735940i	$0.263259\pi$
$374$	−8.42726	−0.435763
$375$	37.8213	1.95308
$376$	7.97235	0.411143
$377$	1.20847	0.0622394
$378$	0	0
$379$	−22.6948	−1.16575	−0.582876	−	0.812561i	$-0.698073\pi$
$379$	−22.6948	−1.16575	−0.582876	+	0.812561i	$0.698073\pi$
$380$	27.2483	1.39781
$381$	−3.12936	−0.160322
$382$	12.3231	0.630507
$383$	−1.01626	−0.0519283	−0.0259641	−	0.999663i	$-0.508266\pi$
$383$	−1.01626	−0.0519283	−0.0259641	+	0.999663i	$0.508266\pi$
$384$	−35.3230	−1.80257
$385$	0	0
$386$	8.92111	0.454073
$387$	−71.6050	−3.63988
$388$	−13.0148	−0.660726
$389$	−15.3857	−0.780087	−0.390044	−	0.920796i	$-0.627540\pi$
$389$	−15.3857	−0.780087	−0.390044	+	0.920796i	$0.627540\pi$
$390$	−13.2855	−0.672737
$391$	23.9763	1.21253
$392$	0	0
$393$	47.2194	2.38191
$394$	0.692127	0.0348688
$395$	−24.7746	−1.24655
$396$	35.0962	1.76365
$397$	18.6680	0.936920	0.468460	−	0.883485i	$-0.344809\pi$
$397$	18.6680	0.936920	0.468460	+	0.883485i	$0.344809\pi$
$398$	3.33027	0.166931
$399$	0	0
$400$	−3.15785	−0.157893
$401$	22.1798	1.10761	0.553804	−	0.832647i	$-0.313176\pi$
$401$	22.1798	1.10761	0.553804	+	0.832647i	$0.313176\pi$
$402$	−5.30064	−0.264372
$403$	−9.16053	−0.456319
$404$	4.75924	0.236781
$405$	32.6817	1.62397
$406$	0	0
$407$	−16.8078	−0.833130
$408$	33.8522	1.67594
$409$	−17.9815	−0.889128	−0.444564	−	0.895747i	$-0.646641\pi$
$409$	−17.9815	−0.889128	−0.444564	+	0.895747i	$0.646641\pi$
$410$	0.655322	0.0323640
$411$	−7.90663	−0.390005
$412$	15.0609	0.741997
$413$	0	0
$414$	13.4472	0.660896
$415$	−28.8395	−1.41567
$416$	22.1206	1.08455
$417$	47.8408	2.34277
$418$	−11.3250	−0.553922
$419$	2.51341	0.122788	0.0613940	−	0.998114i	$-0.480445\pi$
$419$	2.51341	0.122788	0.0613940	+	0.998114i	$0.480445\pi$
$420$	0	0
$421$	6.65174	0.324186	0.162093	−	0.986775i	$-0.448176\pi$
$421$	6.65174	0.324186	0.162093	+	0.986775i	$0.448176\pi$
$422$	−12.4111	−0.604164
$423$	29.5415	1.43636
$424$	−11.2549	−0.546588
$425$	7.07968	0.343415
$426$	12.4811	0.604710
$427$	0	0
$428$	−29.4967	−1.42577
$429$	−41.0014	−1.97957
$430$	10.0118	0.482812
$431$	9.62036	0.463397	0.231698	−	0.972788i	$-0.425572\pi$
$431$	9.62036	0.463397	0.231698	+	0.972788i	$0.425572\pi$
$432$	−31.2413	−1.50310
$433$	−5.50278	−0.264447	−0.132223	−	0.991220i	$-0.542212\pi$
$433$	−5.50278	−0.264447	−0.132223	+	0.991220i	$0.542212\pi$
$434$	0	0
$435$	1.64930	0.0790779
$436$	14.6401	0.701133
$437$	32.2205	1.54132
$438$	18.7576	0.896271
$439$	11.3503	0.541721	0.270860	−	0.962619i	$-0.412692\pi$
$439$	11.3503	0.541721	0.270860	+	0.962619i	$0.412692\pi$
$440$	−10.4752	−0.499384
$441$	0	0
$442$	−12.8511	−0.611262
$443$	28.6618	1.36177	0.680883	−	0.732392i	$-0.261596\pi$
$443$	28.6618	1.36177	0.680883	+	0.732392i	$0.261596\pi$
$444$	31.6286	1.50103
$445$	−11.7866	−0.558738
$446$	−13.0030	−0.615710
$447$	−2.31935	−0.109701
$448$	0	0
$449$	0.290000	0.0136859	0.00684297	−	0.999977i	$-0.497822\pi$
$449$	0.290000	0.0136859	0.00684297	+	0.999977i	$0.497822\pi$
$450$	3.97068	0.187179
$451$	2.02244	0.0952331
$452$	−25.9178	−1.21907
$453$	30.8251	1.44829
$454$	13.0887	0.614284
$455$	0	0
$456$	45.4923	2.13037
$457$	−29.8701	−1.39726	−0.698631	−	0.715482i	$-0.746207\pi$
$457$	−29.8701	−1.39726	−0.698631	+	0.715482i	$0.746207\pi$
$458$	−1.88979	−0.0883041
$459$	70.0406	3.26922
$460$	13.9613	0.650949
$461$	33.4965	1.56009	0.780044	−	0.625725i	$-0.215197\pi$
$461$	33.4965	1.56009	0.780044	+	0.625725i	$0.215197\pi$
$462$	0	0
$463$	−33.7046	−1.56639	−0.783193	−	0.621778i	$-0.786411\pi$
$463$	−33.7046	−1.56639	−0.783193	+	0.621778i	$0.786411\pi$
$464$	−0.711605	−0.0330354
$465$	−12.5021	−0.579773
$466$	−6.51375	−0.301743
$467$	34.8687	1.61353	0.806765	−	0.590872i	$-0.201216\pi$
$467$	34.8687	1.61353	0.806765	+	0.590872i	$0.201216\pi$
$468$	53.5196	2.47395
$469$	0	0
$470$	−4.13049	−0.190525
$471$	−10.8118	−0.498182
$472$	−16.8408	−0.775160
$473$	30.8982	1.42070
$474$	−19.3765	−0.889990
$475$	9.51401	0.436533
$476$	0	0
$477$	−41.7051	−1.90955
$478$	−2.24056	−0.102481
$479$	10.0174	0.457705	0.228852	−	0.973461i	$-0.426503\pi$
$479$	10.0174	0.457705	0.228852	+	0.973461i	$0.426503\pi$
$480$	30.1898	1.37797
$481$	−25.6308	−1.16866
$482$	−6.56431	−0.298996
$483$	0	0
$484$	4.24451	0.192932
$485$	14.3941	0.653601
$486$	8.21200	0.372504
$487$	−41.0784	−1.86144	−0.930719	−	0.365735i	$-0.880818\pi$
$487$	−41.0784	−1.86144	−0.930719	+	0.365735i	$0.880818\pi$
$488$	−19.8938	−0.900552
$489$	−30.7300	−1.38966
$490$	0	0
$491$	−36.8842	−1.66456	−0.832281	−	0.554353i	$-0.812966\pi$
$491$	−36.8842	−1.66456	−0.832281	+	0.554353i	$0.812966\pi$
$492$	−3.80580	−0.171579
$493$	1.59537	0.0718517
$494$	−17.2699	−0.777008
$495$	−38.8157	−1.74463
$496$	5.39416	0.242205
$497$	0	0
$498$	−22.5556	−1.01074
$499$	7.42065	0.332194	0.166097	−	0.986109i	$-0.446884\pi$
$499$	7.42065	0.332194	0.166097	+	0.986109i	$0.446884\pi$
$500$	21.3030	0.952697
$501$	50.1937	2.24249
$502$	−13.1712	−0.587859
$503$	−14.2514	−0.635440	−0.317720	−	0.948185i	$-0.602917\pi$
$503$	−14.2514	−0.635440	−0.317720	+	0.948185i	$0.602917\pi$
$504$	0	0
$505$	−5.26361	−0.234228
$506$	−5.80261	−0.257958
$507$	−21.8430	−0.970081
$508$	−1.76262	−0.0782038
$509$	5.33791	0.236599	0.118299	−	0.992978i	$-0.462256\pi$
$509$	5.33791	0.236599	0.118299	+	0.992978i	$0.462256\pi$
$510$	−17.5389	−0.776636
$511$	0	0
$512$	−22.6759	−1.00214
$513$	94.1240	4.15567
$514$	1.62844	0.0718273
$515$	−16.6570	−0.733995
$516$	−58.1438	−2.55964
$517$	−12.7474	−0.560632
$518$	0	0
$519$	64.3796	2.82595
$520$	−15.9740	−0.700505
$521$	−3.09969	−0.135800	−0.0679000	−	0.997692i	$-0.521630\pi$
$521$	−3.09969	−0.135800	−0.0679000	+	0.997692i	$0.521630\pi$
$522$	0.894770	0.0391630
$523$	−10.4788	−0.458206	−0.229103	−	0.973402i	$-0.573579\pi$
$523$	−10.4788	−0.458206	−0.229103	+	0.973402i	$0.573579\pi$
$524$	26.5965	1.16187
$525$	0	0
$526$	−3.32802	−0.145109
$527$	−12.0933	−0.526793
$528$	24.1436	1.05072
$529$	−6.49104	−0.282219
$530$	5.83121	0.253291
$531$	−62.4035	−2.70808
$532$	0	0
$533$	3.08410	0.133587
$534$	−9.21841	−0.398920
$535$	32.6226	1.41040
$536$	−6.37328	−0.275284
$537$	18.6542	0.804988
$538$	6.63166	0.285911
$539$	0	0
$540$	40.7843	1.75508
$541$	−25.1403	−1.08086	−0.540432	−	0.841387i	$-0.681739\pi$
$541$	−25.1403	−1.08086	−0.540432	+	0.841387i	$0.681739\pi$
$542$	8.11794	0.348695
$543$	26.4319	1.13430
$544$	29.2026	1.25205
$545$	−16.1916	−0.693572
$546$	0	0
$547$	8.00878	0.342431	0.171215	−	0.985234i	$-0.445231\pi$
$547$	8.00878	0.342431	0.171215	+	0.985234i	$0.445231\pi$
$548$	−4.45344	−0.190241
$549$	−73.7165	−3.14615
$550$	−1.71338	−0.0730589
$551$	2.14393	0.0913345
$552$	23.3091	0.992099
$553$	0	0
$554$	−3.17346	−0.134827
$555$	−34.9805	−1.48484
$556$	26.9465	1.14279
$557$	17.5786	0.744831	0.372415	−	0.928066i	$-0.378530\pi$
$557$	17.5786	0.744831	0.372415	+	0.928066i	$0.378530\pi$
$558$	−6.78260	−0.287130
$559$	47.1179	1.99287
$560$	0	0
$561$	−54.1282	−2.28530
$562$	−5.45757	−0.230214
$563$	−15.9436	−0.671944	−0.335972	−	0.941872i	$-0.609065\pi$
$563$	−15.9436	−0.671944	−0.335972	+	0.941872i	$0.609065\pi$
$564$	23.9879	1.01007
$565$	28.6645	1.20592
$566$	−2.02256	−0.0850145
$567$	0	0
$568$	15.0068	0.629670
$569$	−23.3560	−0.979136	−0.489568	−	0.871965i	$-0.662845\pi$
$569$	−23.3560	−0.979136	−0.489568	+	0.871965i	$0.662845\pi$
$570$	−23.5696	−0.987223
$571$	−1.75673	−0.0735168	−0.0367584	−	0.999324i	$-0.511703\pi$
$571$	−1.75673	−0.0735168	−0.0367584	+	0.999324i	$0.511703\pi$
$572$	−23.0942	−0.965618
$573$	79.1515	3.30660
$574$	0	0
$575$	4.87473	0.203290
$576$	−19.3807	−0.807529
$577$	6.36994	0.265184	0.132592	−	0.991171i	$-0.457670\pi$
$577$	6.36994	0.265184	0.132592	+	0.991171i	$0.457670\pi$
$578$	−8.68276	−0.361155
$579$	57.3002	2.38132
$580$	0.928974	0.0385736
$581$	0	0
$582$	11.2578	0.466649
$583$	17.9962	0.745325
$584$	22.5534	0.933266
$585$	−59.1915	−2.44727
$586$	10.6919	0.441680
$587$	−39.1239	−1.61481	−0.807407	−	0.589994i	$-0.799130\pi$
$587$	−39.1239	−1.61481	−0.807407	+	0.589994i	$0.799130\pi$
$588$	0	0
$589$	−16.2516	−0.669635
$590$	8.72525	0.359213
$591$	4.44553	0.182865
$592$	15.0927	0.620304
$593$	−4.40855	−0.181037	−0.0905187	−	0.995895i	$-0.528852\pi$
$593$	−4.40855	−0.181037	−0.0905187	+	0.995895i	$0.528852\pi$
$594$	−16.9508	−0.695501
$595$	0	0
$596$	−1.30638	−0.0535114
$597$	21.3903	0.875446
$598$	−8.84863	−0.361847
$599$	−13.6284	−0.556842	−0.278421	−	0.960459i	$-0.589811\pi$
$599$	−13.6284	−0.556842	−0.278421	+	0.960459i	$0.589811\pi$
$600$	6.88265	0.280983
$601$	9.40876	0.383791	0.191896	−	0.981415i	$-0.438536\pi$
$601$	9.40876	0.383791	0.191896	+	0.981415i	$0.438536\pi$
$602$	0	0
$603$	−23.6162	−0.961725
$604$	17.3623	0.706464
$605$	−4.69433	−0.190852
$606$	−4.11672	−0.167230
$607$	44.6157	1.81089	0.905447	−	0.424458i	$-0.139535\pi$
$607$	44.6157	1.81089	0.905447	+	0.424458i	$0.139535\pi$
$608$	39.2438	1.59155
$609$	0	0
$610$	10.3070	0.417320
$611$	−19.4391	−0.786421
$612$	70.6542	2.85602
$613$	4.48586	0.181182	0.0905911	−	0.995888i	$-0.471124\pi$
$613$	4.48586	0.181182	0.0905911	+	0.995888i	$0.471124\pi$
$614$	−1.61837	−0.0653123
$615$	4.20913	0.169728
$616$	0	0
$617$	−0.109855	−0.00442260	−0.00221130	−	0.999998i	$-0.500704\pi$
$617$	−0.109855	−0.00442260	−0.00221130	+	0.999998i	$0.500704\pi$
$618$	−13.0276	−0.524047
$619$	−25.6036	−1.02909	−0.514547	−	0.857462i	$-0.672040\pi$
$619$	−25.6036	−1.02909	−0.514547	+	0.857462i	$0.672040\pi$
$620$	−7.04188	−0.282809
$621$	48.2267	1.93527
$622$	2.69348	0.107999
$623$	0	0
$624$	36.8175	1.47388
$625$	−17.5618	−0.702474
$626$	−8.57814	−0.342851
$627$	−72.7401	−2.90496
$628$	−6.08979	−0.243009
$629$	−33.8366	−1.34915
$630$	0	0
$631$	−18.3712	−0.731344	−0.365672	−	0.930744i	$-0.619161\pi$
$631$	−18.3712	−0.731344	−0.365672	+	0.930744i	$0.619161\pi$
$632$	−23.2975	−0.926725
$633$	−79.7166	−3.16845
$634$	11.7853	0.468056
$635$	1.94942	0.0773605
$636$	−33.8649	−1.34283
$637$	0	0
$638$	−0.386102	−0.0152859
$639$	55.6075	2.19980
$640$	22.0043	0.869798
$641$	12.0560	0.476184	0.238092	−	0.971243i	$-0.423478\pi$
$641$	12.0560	0.476184	0.238092	+	0.971243i	$0.423478\pi$
$642$	25.5145	1.00698
$643$	−9.37553	−0.369735	−0.184867	−	0.982763i	$-0.559186\pi$
$643$	−9.37553	−0.369735	−0.184867	+	0.982763i	$0.559186\pi$
$644$	0	0
$645$	64.3057	2.53203
$646$	−22.7989	−0.897010
$647$	−37.1619	−1.46099	−0.730493	−	0.682920i	$-0.760710\pi$
$647$	−37.1619	−1.46099	−0.730493	+	0.682920i	$0.760710\pi$
$648$	30.7332	1.20731
$649$	26.9277	1.05700
$650$	−2.61280	−0.102483
$651$	0	0
$652$	−17.3088	−0.677864
$653$	−9.38565	−0.367289	−0.183644	−	0.982993i	$-0.558790\pi$
$653$	−9.38565	−0.367289	−0.183644	+	0.982993i	$0.558790\pi$
$654$	−12.6636	−0.495186
$655$	−29.4151	−1.14935
$656$	−1.81607	−0.0709055
$657$	83.5715	3.26043
$658$	0	0
$659$	−27.1795	−1.05876	−0.529381	−	0.848384i	$-0.677576\pi$
$659$	−27.1795	−1.05876	−0.529381	+	0.848384i	$0.677576\pi$
$660$	−31.5186	−1.22686
$661$	32.7142	1.27243	0.636217	−	0.771510i	$-0.280498\pi$
$661$	32.7142	1.27243	0.636217	+	0.771510i	$0.280498\pi$
$662$	−0.426019	−0.0165577
$663$	−82.5422	−3.20567
$664$	−27.1200	−1.05246
$665$	0	0
$666$	−18.9774	−0.735361
$667$	1.09849	0.0425339
$668$	28.2718	1.09387
$669$	−83.5182	−3.22900
$670$	3.30201	0.127568
$671$	31.8094	1.22799
$672$	0	0
$673$	20.8849	0.805056	0.402528	−	0.915408i	$-0.368132\pi$
$673$	20.8849	0.805056	0.402528	+	0.915408i	$0.368132\pi$
$674$	−4.84038	−0.186445
$675$	14.2403	0.548108
$676$	−12.3031	−0.473198
$677$	14.1437	0.543588	0.271794	−	0.962355i	$-0.412383\pi$
$677$	14.1437	0.543588	0.271794	+	0.962355i	$0.412383\pi$
$678$	22.4188	0.860988
$679$	0	0
$680$	−21.0881	−0.808692
$681$	84.0688	3.22152
$682$	2.92676	0.112071
$683$	−22.6385	−0.866239	−0.433119	−	0.901337i	$-0.642587\pi$
$683$	−22.6385	−0.866239	−0.433119	+	0.901337i	$0.642587\pi$
$684$	94.9485	3.63045
$685$	4.92540	0.188190
$686$	0	0
$687$	−12.1381	−0.463098
$688$	−27.7453	−1.05778
$689$	27.4430	1.04550
$690$	−12.0765	−0.459743
$691$	0.730142	0.0277759	0.0138880	−	0.999904i	$-0.495579\pi$
$691$	0.730142	0.0277759	0.0138880	+	0.999904i	$0.495579\pi$
$692$	36.2621	1.37848
$693$	0	0
$694$	−5.67627	−0.215468
$695$	−29.8022	−1.13046
$696$	1.55097	0.0587893
$697$	4.07149	0.154219
$698$	11.8938	0.450189
$699$	−41.8378	−1.58245
$700$	0	0
$701$	28.5503	1.07833	0.539165	−	0.842200i	$-0.318740\pi$
$701$	28.5503	1.07833	0.539165	+	0.842200i	$0.318740\pi$
$702$	−25.8490	−0.975607
$703$	−45.4713	−1.71498
$704$	8.36295	0.315191
$705$	−26.5301	−0.999182
$706$	−0.195541	−0.00735928
$707$	0	0
$708$	−50.6721	−1.90437
$709$	32.5286	1.22164	0.610818	−	0.791771i	$-0.290841\pi$
$709$	32.5286	1.22164	0.610818	+	0.791771i	$0.290841\pi$
$710$	−7.77504	−0.291792
$711$	−86.3288	−3.23758
$712$	−11.0839	−0.415385
$713$	−8.32689	−0.311844
$714$	0	0
$715$	25.5417	0.955205
$716$	10.5070	0.392667
$717$	−14.3911	−0.537444
$718$	13.6028	0.507652
$719$	25.5471	0.952748	0.476374	−	0.879243i	$-0.341951\pi$
$719$	25.5471	0.952748	0.476374	+	0.879243i	$0.341951\pi$
$720$	34.8548	1.29896
$721$	0	0
$722$	−21.3812	−0.795726
$723$	−42.1625	−1.56804
$724$	14.8879	0.553304
$725$	0.324361	0.0120465
$726$	−3.67148	−0.136262
$727$	21.8717	0.811176	0.405588	−	0.914056i	$-0.367067\pi$
$727$	21.8717	0.811176	0.405588	+	0.914056i	$0.367067\pi$
$728$	0	0
$729$	2.45119	0.0907847
$730$	−11.6850	−0.432480
$731$	62.2029	2.30066
$732$	−59.8584	−2.21243
$733$	−25.1299	−0.928195	−0.464097	−	0.885784i	$-0.653621\pi$
$733$	−25.1299	−0.928195	−0.464097	+	0.885784i	$0.653621\pi$
$734$	−4.89809	−0.180792
$735$	0	0
$736$	20.1075	0.741172
$737$	10.1906	0.375375
$738$	2.28352	0.0840574
$739$	41.0024	1.50830	0.754149	−	0.656703i	$-0.228050\pi$
$739$	41.0024	1.50830	0.754149	+	0.656703i	$0.228050\pi$
$740$	−19.7029	−0.724293
$741$	−110.924	−4.07490
$742$	0	0
$743$	15.1924	0.557356	0.278678	−	0.960385i	$-0.410104\pi$
$743$	15.1924	0.557356	0.278678	+	0.960385i	$0.410104\pi$
$744$	−11.7568	−0.431024
$745$	1.44483	0.0529344
$746$	−12.7415	−0.466500
$747$	−100.493	−3.67685
$748$	−30.4879	−1.11475
$749$	0	0
$750$	−18.4270	−0.672858
$751$	−6.47338	−0.236217	−0.118108	−	0.993001i	$-0.537683\pi$
$751$	−6.47338	−0.236217	−0.118108	+	0.993001i	$0.537683\pi$
$752$	11.4467	0.417417
$753$	−84.5986	−3.08294
$754$	−0.588781	−0.0214421
$755$	−19.2024	−0.698845
$756$	0	0
$757$	47.6567	1.73211	0.866057	−	0.499946i	$-0.166647\pi$
$757$	47.6567	1.73211	0.866057	+	0.499946i	$0.166647\pi$
$758$	11.0572	0.401615
$759$	−37.2702	−1.35282
$760$	−28.3392	−1.02797
$761$	−28.4280	−1.03051	−0.515257	−	0.857036i	$-0.672304\pi$
$761$	−28.4280	−1.03051	−0.515257	+	0.857036i	$0.672304\pi$
$762$	1.52466	0.0552327
$763$	0	0
$764$	44.5823	1.61293
$765$	−78.1419	−2.82523
$766$	0.495133	0.0178899
$767$	41.0631	1.48270
$768$	−0.646825	−0.0233403
$769$	−50.3189	−1.81455	−0.907274	−	0.420541i	$-0.861840\pi$
$769$	−50.3189	−1.81455	−0.907274	+	0.420541i	$0.861840\pi$
$770$	0	0
$771$	10.4594	0.376688
$772$	32.2746	1.16159
$773$	49.6267	1.78495	0.892474	−	0.451100i	$-0.148968\pi$
$773$	49.6267	1.78495	0.892474	+	0.451100i	$0.148968\pi$
$774$	34.8868	1.25398
$775$	−2.45875	−0.0883208
$776$	13.5359	0.485910
$777$	0	0
$778$	7.49611	0.268749
$779$	5.47146	0.196035
$780$	−48.0639	−1.72097
$781$	−23.9952	−0.858615
$782$	−11.6816	−0.417732
$783$	3.20897	0.114679
$784$	0	0
$785$	6.73517	0.240389
$786$	−23.0059	−0.820593
$787$	51.2377	1.82643	0.913214	−	0.407481i	$-0.133593\pi$
$787$	51.2377	1.82643	0.913214	+	0.407481i	$0.133593\pi$
$788$	2.50396	0.0891999
$789$	−21.3758	−0.761000
$790$	12.0705	0.429449
$791$	0	0
$792$	−36.5014	−1.29702
$793$	48.5073	1.72255
$794$	−9.09528	−0.322779
$795$	37.4538	1.32835
$796$	12.0482	0.427036
$797$	−0.672658	−0.0238268	−0.0119134	−	0.999929i	$-0.503792\pi$
$797$	−0.672658	−0.0238268	−0.0119134	+	0.999929i	$0.503792\pi$
$798$	0	0
$799$	−25.6626	−0.907877
$800$	5.93730	0.209915
$801$	−41.0712	−1.45118
$802$	−10.8063	−0.381583
$803$	−36.0619	−1.27260
$804$	−19.1765	−0.676303
$805$	0	0
$806$	4.46312	0.157207
$807$	42.5951	1.49942
$808$	−4.94979	−0.174133
$809$	5.71038	0.200766	0.100383	−	0.994949i	$-0.467993\pi$
$809$	5.71038	0.200766	0.100383	+	0.994949i	$0.467993\pi$
$810$	−15.9229	−0.559475
$811$	14.5296	0.510204	0.255102	−	0.966914i	$-0.417891\pi$
$811$	14.5296	0.510204	0.255102	+	0.966914i	$0.417891\pi$
$812$	0	0
$813$	52.1415	1.82868
$814$	8.18895	0.287022
$815$	19.1431	0.670554
$816$	48.6048	1.70151
$817$	83.5912	2.92449
$818$	8.76081	0.306315
$819$	0	0
$820$	2.37081	0.0827922
$821$	48.5350	1.69388	0.846941	−	0.531686i	$-0.178441\pi$
$821$	48.5350	1.69388	0.846941	+	0.531686i	$0.178441\pi$
$822$	3.85220	0.134361
$823$	−30.1898	−1.05235	−0.526175	−	0.850376i	$-0.676374\pi$
$823$	−30.1898	−1.05235	−0.526175	+	0.850376i	$0.676374\pi$
$824$	−15.6639	−0.545678
$825$	−11.0051	−0.383147
$826$	0	0
$827$	4.67682	0.162629	0.0813144	−	0.996688i	$-0.474088\pi$
$827$	4.67682	0.162629	0.0813144	+	0.996688i	$0.474088\pi$
$828$	48.6491	1.69067
$829$	−11.2623	−0.391155	−0.195577	−	0.980688i	$-0.562658\pi$
$829$	−11.2623	−0.391155	−0.195577	+	0.980688i	$0.562658\pi$
$830$	14.0509	0.487715
$831$	−20.3831	−0.707082
$832$	12.7530	0.442130
$833$	0	0
$834$	−23.3086	−0.807111
$835$	−31.2679	−1.08207
$836$	−40.9711	−1.41702
$837$	−24.3249	−0.840790
$838$	−1.22456	−0.0423018
$839$	45.7068	1.57798	0.788988	−	0.614409i	$-0.210605\pi$
$839$	45.7068	1.57798	0.788988	+	0.614409i	$0.210605\pi$
$840$	0	0
$841$	−28.9269	−0.997480
$842$	−3.24081	−0.111686
$843$	−35.0539	−1.20732
$844$	−44.9007	−1.54555
$845$	13.6070	0.468095
$846$	−14.3930	−0.494841
$847$	0	0
$848$	−16.1598	−0.554929
$849$	−12.9909	−0.445846
$850$	−3.44931	−0.118310
$851$	−23.2983	−0.798655
$852$	45.1537	1.54694
$853$	43.0308	1.47335	0.736674	−	0.676248i	$-0.236395\pi$
$853$	43.0308	1.47335	0.736674	+	0.676248i	$0.236395\pi$
$854$	0	0
$855$	−105.011	−3.59130
$856$	30.6776	1.04854
$857$	−0.992817	−0.0339140	−0.0169570	−	0.999856i	$-0.505398\pi$
$857$	−0.992817	−0.0339140	−0.0169570	+	0.999856i	$0.505398\pi$
$858$	19.9764	0.681983
$859$	41.9473	1.43122	0.715612	−	0.698498i	$-0.246148\pi$
$859$	41.9473	1.43122	0.715612	+	0.698498i	$0.246148\pi$
$860$	36.2204	1.23511
$861$	0	0
$862$	−4.68716	−0.159645
$863$	−4.49436	−0.152990	−0.0764949	−	0.997070i	$-0.524373\pi$
$863$	−4.49436	−0.152990	−0.0764949	+	0.997070i	$0.524373\pi$
$864$	58.7389	1.99834
$865$	−40.1050	−1.36361
$866$	2.68102	0.0911048
$867$	−55.7693	−1.89403
$868$	0	0
$869$	37.2517	1.26368
$870$	−0.803559	−0.0272432
$871$	15.5400	0.526554
$872$	−15.2262	−0.515625
$873$	50.1572	1.69756
$874$	−15.6982	−0.531001
$875$	0	0
$876$	67.8607	2.29280
$877$	18.4253	0.622178	0.311089	−	0.950381i	$-0.399306\pi$
$877$	18.4253	0.622178	0.311089	+	0.950381i	$0.399306\pi$
$878$	−5.53001	−0.186629
$879$	68.6743	2.31633
$880$	−15.0402	−0.507004
$881$	32.5195	1.09561	0.547805	−	0.836606i	$-0.315464\pi$
$881$	32.5195	1.09561	0.547805	+	0.836606i	$0.315464\pi$
$882$	0	0
$883$	−41.8716	−1.40909	−0.704546	−	0.709658i	$-0.748849\pi$
$883$	−41.8716	−1.40909	−0.704546	+	0.709658i	$0.748849\pi$
$884$	−46.4922	−1.56370
$885$	56.0422	1.88384
$886$	−13.9644	−0.469143
$887$	25.8805	0.868981	0.434491	−	0.900676i	$-0.356928\pi$
$887$	25.8805	0.868981	0.434491	+	0.900676i	$0.356928\pi$
$888$	−32.8949	−1.10388
$889$	0	0
$890$	5.74257	0.192491
$891$	−49.1410	−1.64629
$892$	−47.0419	−1.57508
$893$	−34.4866	−1.15405
$894$	1.13001	0.0377933
$895$	−11.6206	−0.388433
$896$	0	0
$897$	−56.8347	−1.89766
$898$	−0.141291	−0.00471496
$899$	−0.554065	−0.0184791
$900$	14.3650	0.478834
$901$	36.2290	1.20696
$902$	−0.985358	−0.0328088
$903$	0	0
$904$	26.9555	0.896526
$905$	−16.4657	−0.547337
$906$	−15.0184	−0.498951
$907$	51.2392	1.70137	0.850685	−	0.525676i	$-0.176187\pi$
$907$	51.2392	1.70137	0.850685	+	0.525676i	$0.176187\pi$
$908$	47.3520	1.57143
$909$	−18.3414	−0.608347
$910$	0	0
$911$	−34.3190	−1.13704	−0.568520	−	0.822669i	$-0.692484\pi$
$911$	−34.3190	−1.13704	−0.568520	+	0.822669i	$0.692484\pi$
$912$	65.3175	2.16288
$913$	43.3638	1.43513
$914$	14.5531	0.481372
$915$	66.2020	2.18857
$916$	−6.83683	−0.225895
$917$	0	0
$918$	−34.1246	−1.12628
$919$	51.0333	1.68343	0.841717	−	0.539919i	$-0.181545\pi$
$919$	51.0333	1.68343	0.841717	+	0.539919i	$0.181545\pi$
$920$	−14.5203	−0.478719
$921$	−10.3948	−0.342521
$922$	−16.3199	−0.537467
$923$	−36.5912	−1.20441
$924$	0	0
$925$	−6.87947	−0.226196
$926$	16.4213	0.539638
$927$	−58.0425	−1.90637
$928$	1.33794	0.0439200
$929$	21.9450	0.719991	0.359995	−	0.932954i	$-0.382778\pi$
$929$	21.9450	0.719991	0.359995	+	0.932954i	$0.382778\pi$
$930$	6.09120	0.199738
$931$	0	0
$932$	−23.5653	−0.771906
$933$	17.3002	0.566383
$934$	−16.9885	−0.555879
$935$	33.7190	1.10273
$936$	−55.6624	−1.81938
$937$	−51.8886	−1.69513	−0.847564	−	0.530693i	$-0.821932\pi$
$937$	−51.8886	−1.69513	−0.847564	+	0.530693i	$0.821932\pi$
$938$	0	0
$939$	−55.0973	−1.79803
$940$	−14.9432	−0.487393
$941$	48.9451	1.59557	0.797783	−	0.602945i	$-0.206006\pi$
$941$	48.9451	1.59557	0.797783	+	0.602945i	$0.206006\pi$
$942$	5.26764	0.171629
$943$	2.80343	0.0912924
$944$	−24.1799	−0.786989
$945$	0	0
$946$	−15.0540	−0.489447
$947$	−19.1634	−0.622726	−0.311363	−	0.950291i	$-0.600786\pi$
$947$	−19.1634	−0.622726	−0.311363	+	0.950291i	$0.600786\pi$
$948$	−70.0997	−2.27673
$949$	−54.9922	−1.78512
$950$	−4.63534	−0.150390
$951$	75.6972	2.45465
$952$	0	0
$953$	−24.3027	−0.787240	−0.393620	−	0.919273i	$-0.628777\pi$
$953$	−24.3027	−0.787240	−0.393620	+	0.919273i	$0.628777\pi$
$954$	20.3193	0.657860
$955$	−49.3071	−1.59554
$956$	−8.10583	−0.262161
$957$	−2.47993	−0.0801647
$958$	−4.88058	−0.157684
$959$	0	0
$960$	17.4051	0.561746
$961$	−26.8000	−0.864517
$962$	12.4876	0.402618
$963$	113.676	3.66315
$964$	−23.7482	−0.764878
$965$	−35.6950	−1.14906
$966$	0	0
$967$	53.9167	1.73384	0.866922	−	0.498444i	$-0.166095\pi$
$967$	53.9167	1.73384	0.866922	+	0.498444i	$0.166095\pi$
$968$	−4.41445	−0.141886
$969$	−146.437	−4.70424
$970$	−7.01297	−0.225173
$971$	−24.5164	−0.786769	−0.393385	−	0.919374i	$-0.628696\pi$
$971$	−24.5164	−0.786769	−0.393385	+	0.919374i	$0.628696\pi$
$972$	29.7092	0.952922
$973$	0	0
$974$	20.0139	0.641286
$975$	−16.7820	−0.537455
$976$	−28.5635	−0.914294
$977$	6.22997	0.199314	0.0996572	−	0.995022i	$-0.468225\pi$
$977$	6.22997	0.199314	0.0996572	+	0.995022i	$0.468225\pi$
$978$	14.9720	0.478753
$979$	17.7226	0.566417
$980$	0	0
$981$	−56.4208	−1.80138
$982$	17.9705	0.573460
$983$	32.3189	1.03081	0.515406	−	0.856946i	$-0.327641\pi$
$983$	32.3189	1.03081	0.515406	+	0.856946i	$0.327641\pi$
$984$	3.95818	0.126182
$985$	−2.76932	−0.0882380
$986$	−0.777282	−0.0247537
$987$	0	0
$988$	−62.4785	−1.98771
$989$	42.8300	1.36191
$990$	18.9115	0.601046
$991$	−22.2893	−0.708042	−0.354021	−	0.935238i	$-0.615186\pi$
$991$	−22.2893	−0.708042	−0.354021	+	0.935238i	$0.615186\pi$
$992$	−10.1419	−0.322007
$993$	−2.73632	−0.0868344
$994$	0	0
$995$	−13.3250	−0.422431
$996$	−81.6012	−2.58563
$997$	−2.84934	−0.0902394	−0.0451197	−	0.998982i	$-0.514367\pi$
$997$	−2.84934	−0.0902394	−0.0451197	+	0.998982i	$0.514367\pi$
$998$	−3.61543	−0.114444
$999$	−68.0600	−2.15332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.t.1.29	✓	74
7.6	odd	2	inner	6223.2.a.t.1.30	yes	74

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.t.1.29	✓	74	1.1	even	1	trivial
6223.2.a.t.1.30	yes	74	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)

\(f(q)\)	\(=\)	\(q-0.487212 q^{2} -3.12936 q^{3} -1.76262 q^{4} +1.94942 q^{5} +1.52466 q^{6} +1.83320 q^{8} +6.79290 q^{9} -0.949783 q^{10} -2.93120 q^{11} +5.51589 q^{12} -4.46990 q^{13} -6.10045 q^{15} +2.63209 q^{16} -5.90096 q^{17} -3.30958 q^{18} -7.92999 q^{19} -3.43610 q^{20} +1.42812 q^{22} -4.06312 q^{23} -5.73673 q^{24} -1.19975 q^{25} +2.17779 q^{26} -11.8694 q^{27} -0.270357 q^{29} +2.97221 q^{30} +2.04938 q^{31} -4.94878 q^{32} +9.17279 q^{33} +2.87502 q^{34} -11.9733 q^{36} +5.73409 q^{37} +3.86359 q^{38} +13.9879 q^{39} +3.57367 q^{40} -0.689971 q^{41} -10.5411 q^{43} +5.16661 q^{44} +13.2422 q^{45} +1.97960 q^{46} +4.34888 q^{47} -8.23677 q^{48} +0.584533 q^{50} +18.4662 q^{51} +7.87876 q^{52} -6.13952 q^{53} +5.78290 q^{54} -5.71415 q^{55} +24.8158 q^{57} +0.131721 q^{58} -9.18657 q^{59} +10.7528 q^{60} -10.8520 q^{61} -0.998484 q^{62} -2.85308 q^{64} -8.71373 q^{65} -4.46909 q^{66} -3.47659 q^{67} +10.4012 q^{68} +12.7150 q^{69} +8.18612 q^{71} +12.4527 q^{72} +12.3028 q^{73} -2.79372 q^{74} +3.75445 q^{75} +13.9776 q^{76} -6.81509 q^{78} -12.7087 q^{79} +5.13106 q^{80} +16.7648 q^{81} +0.336162 q^{82} -14.7939 q^{83} -11.5035 q^{85} +5.13577 q^{86} +0.846045 q^{87} -5.37347 q^{88} -6.04620 q^{89} -6.45178 q^{90} +7.16176 q^{92} -6.41326 q^{93} -2.11883 q^{94} -15.4589 q^{95} +15.4865 q^{96} +7.38376 q^{97} -19.9114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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