LMFDB - Embedded newform 6223.2.a.k.1.9

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)

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);

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")

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

Newform invariants

comment:select newform

sage:traces = [16,-2,4,12,9] 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.170601$ of defining polynomial
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)

$f(q)$	$=$	\(q+0.170601 q^{2} -1.60825 q^{3} -1.97090 q^{4} +0.206353 q^{5} -0.274369 q^{6} -0.677439 q^{8} -0.413531 q^{9} +0.0352041 q^{10} +2.95191 q^{11} +3.16969 q^{12} +6.02043 q^{13} -0.331868 q^{15} +3.82622 q^{16} +6.47770 q^{17} -0.0705489 q^{18} +5.49983 q^{19} -0.406701 q^{20} +0.503600 q^{22} +7.87219 q^{23} +1.08949 q^{24} -4.95742 q^{25} +1.02709 q^{26} +5.48981 q^{27} +3.67353 q^{29} -0.0566170 q^{30} -0.182599 q^{31} +2.00764 q^{32} -4.74741 q^{33} +1.10510 q^{34} +0.815026 q^{36} -9.06492 q^{37} +0.938277 q^{38} -9.68236 q^{39} -0.139792 q^{40} +0.717792 q^{41} +4.41197 q^{43} -5.81791 q^{44} -0.0853335 q^{45} +1.34301 q^{46} +3.34219 q^{47} -6.15352 q^{48} -0.845741 q^{50} -10.4178 q^{51} -11.8656 q^{52} -6.62197 q^{53} +0.936568 q^{54} +0.609137 q^{55} -8.84510 q^{57} +0.626708 q^{58} -7.35511 q^{59} +0.654077 q^{60} +15.3403 q^{61} -0.0311515 q^{62} -7.30993 q^{64} +1.24234 q^{65} -0.809914 q^{66} +6.35295 q^{67} -12.7669 q^{68} -12.6605 q^{69} -7.06485 q^{71} +0.280142 q^{72} -8.10741 q^{73} -1.54649 q^{74} +7.97277 q^{75} -10.8396 q^{76} -1.65182 q^{78} +5.73229 q^{79} +0.789553 q^{80} -7.58840 q^{81} +0.122456 q^{82} -9.79819 q^{83} +1.33670 q^{85} +0.752687 q^{86} -5.90796 q^{87} -1.99974 q^{88} +14.8905 q^{89} -0.0145580 q^{90} -15.5153 q^{92} +0.293664 q^{93} +0.570182 q^{94} +1.13491 q^{95} -3.22878 q^{96} +14.8333 q^{97} -1.22071 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) $ 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * 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 $

$2$	0.170601	0.120633	0.0603166	−	0.998179i	$-0.480789\pi$
$2$	0.170601	0.120633	0.0603166	+	0.998179i	$0.480789\pi$
$3$	−1.60825	−0.928524	−0.464262	−	0.885698i	$-0.653680\pi$
$3$	−1.60825	−0.928524	−0.464262	+	0.885698i	$0.653680\pi$
$4$	−1.97090	−0.985448
$5$	0.206353	0.0922840	0.0461420	−	0.998935i	$-0.485307\pi$
$5$	0.206353	0.0922840	0.0461420	+	0.998935i	$0.485307\pi$
$6$	−0.274369	−0.112011
$7$	0	0
$7$	0	0
$8$	−0.677439	−0.239511
$9$	−0.413531	−0.137844
$10$	0.0352041	0.0111325
$11$	2.95191	0.890035	0.445018	−	0.895522i	$-0.353197\pi$
$11$	2.95191	0.890035	0.445018	+	0.895522i	$0.353197\pi$
$12$	3.16969	0.915012
$13$	6.02043	1.66977	0.834883	−	0.550427i	$-0.185535\pi$
$13$	6.02043	1.66977	0.834883	+	0.550427i	$0.185535\pi$
$14$	0	0
$15$	−0.331868	−0.0856879
$16$	3.82622	0.956555
$17$	6.47770	1.57107	0.785537	−	0.618815i	$-0.212387\pi$
$17$	6.47770	1.57107	0.785537	+	0.618815i	$0.212387\pi$
$18$	−0.0705489	−0.0166285
$19$	5.49983	1.26175	0.630874	−	0.775886i	$-0.282697\pi$
$19$	5.49983	1.26175	0.630874	+	0.775886i	$0.282697\pi$
$20$	−0.406701	−0.0909410
$21$	0	0
$22$	0.503600	0.107368
$23$	7.87219	1.64147	0.820733	−	0.571312i	$-0.193565\pi$
$23$	7.87219	1.64147	0.820733	+	0.571312i	$0.193565\pi$
$24$	1.08949	0.222392
$25$	−4.95742	−0.991484
$26$	1.02709	0.201429
$27$	5.48981	1.05651
$28$	0	0
$29$	3.67353	0.682157	0.341079	−	0.940035i	$-0.389208\pi$
$29$	3.67353	0.682157	0.341079	+	0.940035i	$0.389208\pi$
$30$	−0.0566170	−0.0103368
$31$	−0.182599	−0.0327957	−0.0163978	−	0.999866i	$-0.505220\pi$
$31$	−0.182599	−0.0327957	−0.0163978	+	0.999866i	$0.505220\pi$
$32$	2.00764	0.354903
$33$	−4.74741	−0.826419
$34$	1.10510	0.189524
$35$	0	0
$36$	0.815026	0.135838
$37$	−9.06492	−1.49026	−0.745132	−	0.666917i	$-0.767614\pi$
$37$	−9.06492	−1.49026	−0.745132	+	0.666917i	$0.767614\pi$
$38$	0.938277	0.152209
$39$	−9.68236	−1.55042
$40$	−0.139792	−0.0221030
$41$	0.717792	0.112100	0.0560501	−	0.998428i	$-0.482149\pi$
$41$	0.717792	0.112100	0.0560501	+	0.998428i	$0.482149\pi$
$42$	0	0
$43$	4.41197	0.672819	0.336410	−	0.941716i	$-0.390787\pi$
$43$	4.41197	0.672819	0.336410	+	0.941716i	$0.390787\pi$
$44$	−5.81791	−0.877083
$45$	−0.0853335	−0.0127208
$46$	1.34301	0.198015
$47$	3.34219	0.487509	0.243754	−	0.969837i	$-0.421621\pi$
$47$	3.34219	0.487509	0.243754	+	0.969837i	$0.421621\pi$
$48$	−6.15352	−0.888184
$49$	0	0
$50$	−0.845741	−0.119606
$51$	−10.4178	−1.45878
$52$	−11.8656	−1.64547
$53$	−6.62197	−0.909597	−0.454799	−	0.890594i	$-0.650289\pi$
$53$	−6.62197	−0.909597	−0.454799	+	0.890594i	$0.650289\pi$
$54$	0.936568	0.127451
$55$	0.609137	0.0821360
$56$	0	0
$57$	−8.84510	−1.17156
$58$	0.626708	0.0822909
$59$	−7.35511	−0.957553	−0.478777	−	0.877937i	$-0.658920\pi$
$59$	−7.35511	−0.957553	−0.478777	+	0.877937i	$0.658920\pi$
$60$	0.654077	0.0844409
$61$	15.3403	1.96413	0.982064	−	0.188549i	$-0.0603785\pi$
$61$	15.3403	1.96413	0.982064	+	0.188549i	$0.0603785\pi$
$62$	−0.0311515	−0.00395625
$63$	0	0
$64$	−7.30993	−0.913742
$65$	1.24234	0.154093
$66$	−0.809914	−0.0996936
$67$	6.35295	0.776137	0.388068	−	0.921631i	$-0.373142\pi$
$67$	6.35295	0.776137	0.388068	+	0.921631i	$0.373142\pi$
$68$	−12.7669	−1.54821
$69$	−12.6605	−1.52414
$70$	0	0
$71$	−7.06485	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$71$	−7.06485	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$72$	0.280142	0.0330151
$73$	−8.10741	−0.948900	−0.474450	−	0.880282i	$-0.657353\pi$
$73$	−8.10741	−0.948900	−0.474450	+	0.880282i	$0.657353\pi$
$74$	−1.54649	−0.179775
$75$	7.97277	0.920616
$76$	−10.8396	−1.24339
$77$	0	0
$78$	−1.65182	−0.187032
$79$	5.73229	0.644933	0.322467	−	0.946581i	$-0.395488\pi$
$79$	5.73229	0.644933	0.322467	+	0.946581i	$0.395488\pi$
$80$	0.789553	0.0882747
$81$	−7.58840	−0.843155
$82$	0.122456	0.0135230
$83$	−9.79819	−1.07549	−0.537746	−	0.843107i	$-0.680724\pi$
$83$	−9.79819	−1.07549	−0.537746	+	0.843107i	$0.680724\pi$
$84$	0	0
$85$	1.33670	0.144985
$86$	0.752687	0.0811644
$87$	−5.90796	−0.633399
$88$	−1.99974	−0.213173
$89$	14.8905	1.57839	0.789193	−	0.614145i	$-0.210499\pi$
$89$	14.8905	1.57839	0.789193	+	0.614145i	$0.210499\pi$
$90$	−0.0145580	−0.00153455
$91$	0	0
$92$	−15.5153	−1.61758
$93$	0.293664	0.0304516
$94$	0.570182	0.0588098
$95$	1.13491	0.116439
$96$	−3.22878	−0.329536
$97$	14.8333	1.50609	0.753046	−	0.657967i	$-0.228584\pi$
$97$	14.8333	1.50609	0.753046	+	0.657967i	$0.228584\pi$
$98$	0	0
$99$	−1.22071	−0.122686
$100$	9.77055	0.977055
$101$	−15.4154	−1.53389	−0.766943	−	0.641716i	$-0.778223\pi$
$101$	−15.4154	−1.53389	−0.766943	+	0.641716i	$0.778223\pi$
$102$	−1.77728	−0.175977
$103$	−12.6158	−1.24307	−0.621534	−	0.783387i	$-0.713490\pi$
$103$	−12.6158	−1.24307	−0.621534	+	0.783387i	$0.713490\pi$
$104$	−4.07848	−0.399927
$105$	0	0
$106$	−1.12972	−0.109728
$107$	18.0545	1.74539	0.872697	−	0.488262i	$-0.162369\pi$
$107$	18.0545	1.74539	0.872697	+	0.488262i	$0.162369\pi$
$108$	−10.8198	−1.04114
$109$	5.67574	0.543638	0.271819	−	0.962348i	$-0.412375\pi$
$109$	5.67574	0.543638	0.271819	+	0.962348i	$0.412375\pi$
$110$	0.103919	0.00990833
$111$	14.5787	1.38375
$112$	0	0
$113$	−0.331384	−0.0311740	−0.0155870	−	0.999879i	$-0.504962\pi$
$113$	−0.331384	−0.0311740	−0.0155870	+	0.999879i	$0.504962\pi$
$114$	−1.50898	−0.141329
$115$	1.62445	0.151481
$116$	−7.24014	−0.672230
$117$	−2.48963	−0.230167
$118$	−1.25479	−0.115513
$119$	0	0
$120$	0.224820	0.0205232
$121$	−2.28621	−0.207838
$122$	2.61708	0.236939
$123$	−1.15439	−0.104088
$124$	0.359883	0.0323184
$125$	−2.05475	−0.183782
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−5.26235	−0.465131
$129$	−7.09555	−0.624729
$130$	0.211944	0.0185887
$131$	9.55770	0.835061	0.417530	−	0.908663i	$-0.362896\pi$
$131$	9.55770	0.835061	0.417530	+	0.908663i	$0.362896\pi$
$132$	9.35665	0.814392
$133$	0	0
$134$	1.08382	0.0936279
$135$	1.13284	0.0974994
$136$	−4.38825	−0.376289
$137$	19.1217	1.63367	0.816837	−	0.576868i	$-0.195725\pi$
$137$	19.1217	1.63367	0.816837	+	0.576868i	$0.195725\pi$
$138$	−2.15989	−0.183862
$139$	19.5559	1.65871	0.829353	−	0.558725i	$-0.188709\pi$
$139$	19.5559	1.65871	0.829353	+	0.558725i	$0.188709\pi$
$140$	0	0
$141$	−5.37508	−0.452664
$142$	−1.20527	−0.101144
$143$	17.7718	1.48615
$144$	−1.58226	−0.131855
$145$	0.758045	0.0629522
$146$	−1.38313	−0.114469
$147$	0	0
$148$	17.8660	1.46858
$149$	−8.65860	−0.709340	−0.354670	−	0.934992i	$-0.615407\pi$
$149$	−8.65860	−0.709340	−0.354670	+	0.934992i	$0.615407\pi$
$150$	1.36016	0.111057
$151$	−14.9600	−1.21743	−0.608715	−	0.793389i	$-0.708315\pi$
$151$	−14.9600	−1.21743	−0.608715	+	0.793389i	$0.708315\pi$
$152$	−3.72580	−0.302202
$153$	−2.67873	−0.216563
$154$	0	0
$155$	−0.0376798	−0.00302652
$156$	19.0829	1.52786
$157$	−13.5556	−1.08185	−0.540926	−	0.841070i	$-0.681926\pi$
$157$	−13.5556	−1.08185	−0.540926	+	0.841070i	$0.681926\pi$
$158$	0.977936	0.0778004
$159$	10.6498	0.844583
$160$	0.414282	0.0327519
$161$	0	0
$162$	−1.29459	−0.101713
$163$	13.3149	1.04291	0.521453	−	0.853280i	$-0.325390\pi$
$163$	13.3149	1.04291	0.521453	+	0.853280i	$0.325390\pi$
$164$	−1.41469	−0.110469
$165$	−0.979644	−0.0762652
$166$	−1.67158	−0.129740
$167$	−5.40055	−0.417907	−0.208953	−	0.977926i	$-0.567006\pi$
$167$	−5.40055	−0.417907	−0.208953	+	0.977926i	$0.567006\pi$
$168$	0	0
$169$	23.2456	1.78812
$170$	0.228042	0.0174900
$171$	−2.27435	−0.173924
$172$	−8.69553	−0.663028
$173$	−0.466383	−0.0354584	−0.0177292	−	0.999843i	$-0.505644\pi$
$173$	−0.466383	−0.0354584	−0.0177292	+	0.999843i	$0.505644\pi$
$174$	−1.00790	−0.0764090
$175$	0	0
$176$	11.2947	0.851367
$177$	11.8289	0.889111
$178$	2.54033	0.190406
$179$	10.0960	0.754611	0.377306	−	0.926089i	$-0.376851\pi$
$179$	10.0960	0.754611	0.377306	+	0.926089i	$0.376851\pi$
$180$	0.168183	0.0125356
$181$	22.2609	1.65464	0.827319	−	0.561732i	$-0.189865\pi$
$181$	22.2609	1.65464	0.827319	+	0.561732i	$0.189865\pi$
$182$	0	0
$183$	−24.6711	−1.82374
$184$	−5.33293	−0.393149
$185$	−1.87058	−0.137528
$186$	0.0500995	0.00367347
$187$	19.1216	1.39831
$188$	−6.58711	−0.480414
$189$	0	0
$190$	0.193617	0.0140464
$191$	−8.20390	−0.593613	−0.296807	−	0.954938i	$-0.595922\pi$
$191$	−8.20390	−0.593613	−0.296807	+	0.954938i	$0.595922\pi$
$192$	11.7562	0.848431
$193$	7.11018	0.511802	0.255901	−	0.966703i	$-0.417628\pi$
$193$	7.11018	0.511802	0.255901	+	0.966703i	$0.417628\pi$
$194$	2.53058	0.181685
$195$	−1.99799	−0.143079
$196$	0	0
$197$	−12.9689	−0.923994	−0.461997	−	0.886882i	$-0.652867\pi$
$197$	−12.9689	−0.923994	−0.461997	+	0.886882i	$0.652867\pi$
$198$	−0.208254	−0.0148000
$199$	−20.6661	−1.46498	−0.732491	−	0.680777i	$-0.761642\pi$
$199$	−20.6661	−1.46498	−0.732491	+	0.680777i	$0.761642\pi$
$200$	3.35835	0.237471
$201$	−10.2171	−0.720662
$202$	−2.62988	−0.185038
$203$	0	0
$204$	20.5323	1.43755
$205$	0.148119	0.0103451
$206$	−2.15226	−0.149955
$207$	−3.25540	−0.226266
$208$	23.0355	1.59722
$209$	16.2350	1.12300
$210$	0	0
$211$	2.16808	0.149257	0.0746283	−	0.997211i	$-0.476223\pi$
$211$	2.16808	0.149257	0.0746283	+	0.997211i	$0.476223\pi$
$212$	13.0512	0.896360
$213$	11.3621	0.778515
$214$	3.08012	0.210552
$215$	0.910425	0.0620905
$216$	−3.71901	−0.253047
$217$	0	0
$218$	0.968289	0.0655808
$219$	13.0387	0.881076
$220$	−1.20054	−0.0809407
$221$	38.9986	2.62333
$222$	2.48714	0.166926
$223$	1.10332	0.0738836	0.0369418	−	0.999317i	$-0.488238\pi$
$223$	1.10332	0.0738836	0.0369418	+	0.999317i	$0.488238\pi$
$224$	0	0
$225$	2.05005	0.136670
$226$	−0.0565346	−0.00376062
$227$	8.00286	0.531169	0.265584	−	0.964088i	$-0.414435\pi$
$227$	8.00286	0.531169	0.265584	+	0.964088i	$0.414435\pi$
$228$	17.4328	1.15451
$229$	−0.363896	−0.0240469	−0.0120235	−	0.999928i	$-0.503827\pi$
$229$	−0.363896	−0.0240469	−0.0120235	+	0.999928i	$0.503827\pi$
$230$	0.277134	0.0182736
$231$	0	0
$232$	−2.48859	−0.163384
$233$	−28.6092	−1.87425	−0.937125	−	0.348994i	$-0.886523\pi$
$233$	−28.6092	−1.87425	−0.937125	+	0.348994i	$0.886523\pi$
$234$	−0.424734	−0.0277658
$235$	0.689672	0.0449893
$236$	14.4961	0.943619
$237$	−9.21896	−0.598836
$238$	0	0
$239$	−28.4902	−1.84288	−0.921438	−	0.388525i	$-0.872985\pi$
$239$	−28.4902	−1.84288	−0.921438	+	0.388525i	$0.872985\pi$
$240$	−1.26980	−0.0819651
$241$	2.12643	0.136975	0.0684877	−	0.997652i	$-0.478183\pi$
$241$	2.12643	0.136975	0.0684877	+	0.997652i	$0.478183\pi$
$242$	−0.390031	−0.0250721
$243$	−4.26539	−0.273625
$244$	−30.2342	−1.93554
$245$	0	0
$246$	−0.196940	−0.0125564
$247$	33.1113	2.10682
$248$	0.123700	0.00785493
$249$	15.7579	0.998619
$250$	−0.350542	−0.0221702
$251$	−12.3182	−0.777519	−0.388759	−	0.921339i	$-0.627096\pi$
$251$	−12.3182	−0.777519	−0.388759	+	0.921339i	$0.627096\pi$
$252$	0	0
$253$	23.2380	1.46096
$254$	−0.170601	−0.0107045
$255$	−2.14974	−0.134622
$256$	13.7221	0.857631
$257$	4.35584	0.271710	0.135855	−	0.990729i	$-0.456622\pi$
$257$	4.35584	0.271710	0.135855	+	0.990729i	$0.456622\pi$
$258$	−1.21051	−0.0753631
$259$	0	0
$260$	−2.44851	−0.151850
$261$	−1.51912	−0.0940311
$262$	1.63056	0.100736
$263$	−7.24987	−0.447046	−0.223523	−	0.974699i	$-0.571756\pi$
$263$	−7.24987	−0.447046	−0.223523	+	0.974699i	$0.571756\pi$
$264$	3.21608	0.197936
$265$	−1.36646	−0.0839413
$266$	0	0
$267$	−23.9476	−1.46557
$268$	−12.5210	−0.764842
$269$	1.81003	0.110360	0.0551798	−	0.998476i	$-0.482427\pi$
$269$	1.81003	0.110360	0.0551798	+	0.998476i	$0.482427\pi$
$270$	0.193264	0.0117617
$271$	−15.9554	−0.969219	−0.484609	−	0.874731i	$-0.661038\pi$
$271$	−15.9554	−0.969219	−0.484609	+	0.874731i	$0.661038\pi$
$272$	24.7851	1.50282
$273$	0	0
$274$	3.26218	0.197075
$275$	−14.6339	−0.882455
$276$	24.9524	1.50196
$277$	−10.5622	−0.634621	−0.317310	−	0.948322i	$-0.602780\pi$
$277$	−10.5622	−0.634621	−0.317310	+	0.948322i	$0.602780\pi$
$278$	3.33625	0.200095
$279$	0.0755102	0.00452068
$280$	0	0
$281$	8.87376	0.529364	0.264682	−	0.964336i	$-0.414733\pi$
$281$	8.87376	0.529364	0.264682	+	0.964336i	$0.414733\pi$
$282$	−0.916995	−0.0546063
$283$	13.8510	0.823359	0.411680	−	0.911329i	$-0.364942\pi$
$283$	13.8510	0.823359	0.411680	+	0.911329i	$0.364942\pi$
$284$	13.9241	0.826243
$285$	−1.82522	−0.108116
$286$	3.03189	0.179279
$287$	0	0
$288$	−0.830220	−0.0489212
$289$	24.9606	1.46827
$290$	0.129323	0.00759413
$291$	−23.8557	−1.39844
$292$	15.9788	0.935091
$293$	9.28962	0.542705	0.271353	−	0.962480i	$-0.412529\pi$
$293$	9.28962	0.542705	0.271353	+	0.962480i	$0.412529\pi$
$294$	0	0
$295$	−1.51775	−0.0883669
$296$	6.14094	0.356935
$297$	16.2054	0.940335
$298$	−1.47717	−0.0855700
$299$	47.3940	2.74087
$300$	−15.7135	−0.907219
$301$	0	0
$302$	−2.55220	−0.146863
$303$	24.7918	1.42425
$304$	21.0435	1.20693
$305$	3.16553	0.181258
$306$	−0.456995	−0.0261246
$307$	−28.5119	−1.62726	−0.813631	−	0.581382i	$-0.802512\pi$
$307$	−28.5119	−1.62726	−0.813631	+	0.581382i	$0.802512\pi$
$308$	0	0
$309$	20.2893	1.15422
$310$	−0.00642822	−0.000365099 0
$311$	−7.51913	−0.426371	−0.213185	−	0.977012i	$-0.568384\pi$
$311$	−7.51913	−0.426371	−0.213185	+	0.977012i	$0.568384\pi$
$312$	6.55921	0.371342
$313$	−15.1262	−0.854983	−0.427492	−	0.904019i	$-0.640603\pi$
$313$	−15.1262	−0.854983	−0.427492	+	0.904019i	$0.640603\pi$
$314$	−2.31259	−0.130507
$315$	0	0
$316$	−11.2977	−0.635548
$317$	−10.3384	−0.580661	−0.290331	−	0.956926i	$-0.593765\pi$
$317$	−10.3384	−0.580661	−0.290331	+	0.956926i	$0.593765\pi$
$318$	1.81687	0.101885
$319$	10.8439	0.607144
$320$	−1.50843	−0.0843237
$321$	−29.0361	−1.62064
$322$	0	0
$323$	35.6263	1.98230
$324$	14.9559	0.830886
$325$	−29.8458	−1.65555
$326$	2.27154	0.125809
$327$	−9.12802	−0.504781
$328$	−0.486260	−0.0268492
$329$	0	0
$330$	−0.167128	−0.00920012
$331$	1.33558	0.0734100	0.0367050	−	0.999326i	$-0.488314\pi$
$331$	1.33558	0.0734100	0.0367050	+	0.999326i	$0.488314\pi$
$332$	19.3112	1.05984
$333$	3.74863	0.205424
$334$	−0.921339	−0.0504135
$335$	1.31095	0.0716250
$336$	0	0
$337$	−12.6188	−0.687388	−0.343694	−	0.939082i	$-0.611678\pi$
$337$	−12.6188	−0.687388	−0.343694	+	0.939082i	$0.611678\pi$
$338$	3.96572	0.215707
$339$	0.532949	0.0289458
$340$	−2.63449	−0.142875
$341$	−0.539015	−0.0291893
$342$	−0.388007	−0.0209810
$343$	0	0
$344$	−2.98884	−0.161148
$345$	−2.61253	−0.140654
$346$	−0.0795655	−0.00427747
$347$	−2.34794	−0.126044	−0.0630221	−	0.998012i	$-0.520074\pi$
$347$	−2.34794	−0.126044	−0.0630221	+	0.998012i	$0.520074\pi$
$348$	11.6440	0.624182
$349$	14.3453	0.767885	0.383943	−	0.923357i	$-0.374566\pi$
$349$	14.3453	0.767885	0.383943	+	0.923357i	$0.374566\pi$
$350$	0	0
$351$	33.0510	1.76413
$352$	5.92637	0.315876
$353$	−18.7887	−1.00002	−0.500011	−	0.866019i	$-0.666671\pi$
$353$	−18.7887	−1.00002	−0.500011	+	0.866019i	$0.666671\pi$
$354$	2.01802	0.107256
$355$	−1.45786	−0.0773750
$356$	−29.3476	−1.55542
$357$	0	0
$358$	1.72239	0.0910312
$359$	−11.5788	−0.611107	−0.305553	−	0.952175i	$-0.598841\pi$
$359$	−11.5788	−0.611107	−0.305553	+	0.952175i	$0.598841\pi$
$360$	0.0578082	0.00304676
$361$	11.2481	0.592006
$362$	3.79773	0.199604
$363$	3.67681	0.192982
$364$	0	0
$365$	−1.67299	−0.0875683
$366$	−4.20892	−0.220004
$367$	−5.31371	−0.277373	−0.138687	−	0.990336i	$-0.544288\pi$
$367$	−5.31371	−0.277373	−0.138687	+	0.990336i	$0.544288\pi$
$368$	30.1207	1.57015
$369$	−0.296829	−0.0154523
$370$	−0.319123	−0.0165904
$371$	0	0
$372$	−0.578782	−0.0300084
$373$	−36.9922	−1.91539	−0.957693	−	0.287792i	$-0.907079\pi$
$373$	−36.9922	−1.91539	−0.957693	+	0.287792i	$0.907079\pi$
$374$	3.26217	0.168683
$375$	3.30455	0.170646
$376$	−2.26413	−0.116764
$377$	22.1162	1.13904
$378$	0	0
$379$	−14.5568	−0.747730	−0.373865	−	0.927483i	$-0.621968\pi$
$379$	−14.5568	−0.747730	−0.373865	+	0.927483i	$0.621968\pi$
$380$	−2.23678	−0.114745
$381$	1.60825	0.0823932
$382$	−1.39959	−0.0716095
$383$	16.5337	0.844833	0.422416	−	0.906402i	$-0.361182\pi$
$383$	16.5337	0.844833	0.422416	+	0.906402i	$0.361182\pi$
$384$	8.46318	0.431885
$385$	0	0
$386$	1.21301	0.0617403
$387$	−1.82449	−0.0927439
$388$	−29.2349	−1.48418
$389$	0.903542	0.0458114	0.0229057	−	0.999738i	$-0.492708\pi$
$389$	0.903542	0.0458114	0.0229057	+	0.999738i	$0.492708\pi$
$390$	−0.340859	−0.0172601
$391$	50.9937	2.57886
$392$	0	0
$393$	−15.3712	−0.775374
$394$	−2.21250	−0.111464
$395$	1.18288	0.0595170
$396$	2.40589	0.120900
$397$	19.6722	0.987318	0.493659	−	0.869656i	$-0.335659\pi$
$397$	19.6722	0.987318	0.493659	+	0.869656i	$0.335659\pi$
$398$	−3.52566	−0.176725
$399$	0	0
$400$	−18.9682	−0.948408
$401$	19.5957	0.978562	0.489281	−	0.872126i	$-0.337259\pi$
$401$	19.5957	0.978562	0.489281	+	0.872126i	$0.337259\pi$
$402$	−1.74306	−0.0869357
$403$	−1.09932	−0.0547612
$404$	30.3821	1.51156
$405$	−1.56589	−0.0778098
$406$	0	0
$407$	−26.7589	−1.32639
$408$	7.05741	0.349394
$409$	16.0598	0.794106	0.397053	−	0.917796i	$-0.370033\pi$
$409$	16.0598	0.794106	0.397053	+	0.917796i	$0.370033\pi$
$410$	0.0252692	0.00124796
$411$	−30.7524	−1.51691
$412$	24.8643	1.22498
$413$	0	0
$414$	−0.555374	−0.0272952
$415$	−2.02189	−0.0992506
$416$	12.0868	0.592606
$417$	−31.4507	−1.54015
$418$	2.76971	0.135471
$419$	12.9330	0.631818	0.315909	−	0.948790i	$-0.397691\pi$
$419$	12.9330	0.631818	0.315909	+	0.948790i	$0.397691\pi$
$420$	0	0
$421$	−24.9490	−1.21594	−0.607970	−	0.793960i	$-0.708016\pi$
$421$	−24.9490	−1.21594	−0.607970	+	0.793960i	$0.708016\pi$
$422$	0.369876	0.0180053
$423$	−1.38210	−0.0672000
$424$	4.48598	0.217859
$425$	−32.1127	−1.55769
$426$	1.93838	0.0939148
$427$	0	0
$428$	−35.5835	−1.71999
$429$	−28.5815	−1.37993
$430$	0.155320	0.00749017
$431$	−7.50132	−0.361326	−0.180663	−	0.983545i	$-0.557824\pi$
$431$	−7.50132	−0.361326	−0.180663	+	0.983545i	$0.557824\pi$
$432$	21.0052	1.01061
$433$	14.1428	0.679661	0.339831	−	0.940487i	$-0.389630\pi$
$433$	14.1428	0.679661	0.339831	+	0.940487i	$0.389630\pi$
$434$	0	0
$435$	−1.21913	−0.0584526
$436$	−11.1863	−0.535726
$437$	43.2957	2.07112
$438$	2.22442	0.106287
$439$	−9.05446	−0.432146	−0.216073	−	0.976377i	$-0.569325\pi$
$439$	−9.05446	−0.432146	−0.216073	+	0.976377i	$0.569325\pi$
$440$	−0.412653	−0.0196725
$441$	0	0
$442$	6.65320	0.316460
$443$	−29.2026	−1.38746	−0.693729	−	0.720236i	$-0.744033\pi$
$443$	−29.2026	−1.38746	−0.693729	+	0.720236i	$0.744033\pi$
$444$	−28.7330	−1.36361
$445$	3.07270	0.145660
$446$	0.188227	0.00891282
$447$	13.9252	0.658639
$448$	0	0
$449$	1.14072	0.0538341	0.0269170	−	0.999638i	$-0.491431\pi$
$449$	1.14072	0.0538341	0.0269170	+	0.999638i	$0.491431\pi$
$450$	0.349740	0.0164869
$451$	2.11886	0.0997732
$452$	0.653124	0.0307204
$453$	24.0595	1.13041
$454$	1.36530	0.0640766
$455$	0	0
$456$	5.99202	0.280602
$457$	−21.9756	−1.02798	−0.513988	−	0.857797i	$-0.671832\pi$
$457$	−21.9756	−1.02798	−0.513988	+	0.857797i	$0.671832\pi$
$458$	−0.0620810	−0.00290086
$459$	35.5614	1.65986
$460$	−3.20163	−0.149277
$461$	22.1987	1.03390	0.516948	−	0.856017i	$-0.327068\pi$
$461$	22.1987	1.03390	0.516948	+	0.856017i	$0.327068\pi$
$462$	0	0
$463$	3.57602	0.166192	0.0830958	−	0.996542i	$-0.473519\pi$
$463$	3.57602	0.166192	0.0830958	+	0.996542i	$0.473519\pi$
$464$	14.0557	0.652521
$465$	0.0605986	0.00281019
$466$	−4.88076	−0.226097
$467$	5.02468	0.232514	0.116257	−	0.993219i	$-0.462910\pi$
$467$	5.02468	0.232514	0.116257	+	0.993219i	$0.462910\pi$
$468$	4.90681	0.226817
$469$	0	0
$470$	0.117659	0.00542720
$471$	21.8007	1.00452
$472$	4.98264	0.229345
$473$	13.0238	0.598833
$474$	−1.57277	−0.0722395
$475$	−27.2649	−1.25100
$476$	0	0
$477$	2.73839	0.125382
$478$	−4.86046	−0.222312
$479$	39.0051	1.78219	0.891094	−	0.453818i	$-0.149938\pi$
$479$	39.0051	1.78219	0.891094	+	0.453818i	$0.149938\pi$
$480$	−0.666270	−0.0304109
$481$	−54.5747	−2.48839
$482$	0.362771	0.0165238
$483$	0	0
$484$	4.50589	0.204813
$485$	3.06090	0.138988
$486$	−0.727681	−0.0330083
$487$	−1.09342	−0.0495476	−0.0247738	−	0.999693i	$-0.507887\pi$
$487$	−1.09342	−0.0495476	−0.0247738	+	0.999693i	$0.507887\pi$
$488$	−10.3921	−0.470430
$489$	−21.4138	−0.968364
$490$	0	0
$491$	−26.2655	−1.18535	−0.592673	−	0.805443i	$-0.701927\pi$
$491$	−26.2655	−1.18535	−0.592673	+	0.805443i	$0.701927\pi$
$492$	2.27518	0.102573
$493$	23.7960	1.07172
$494$	5.64883	0.254153
$495$	−0.251897	−0.0113219
$496$	−0.698663	−0.0313709
$497$	0	0
$498$	2.68832	0.120467
$499$	−29.5452	−1.32262	−0.661311	−	0.750111i	$-0.730000\pi$
$499$	−29.5452	−1.32262	−0.661311	+	0.750111i	$0.730000\pi$
$500$	4.04969	0.181108
$501$	8.68543	0.388036
$502$	−2.10150	−0.0937946
$503$	42.8080	1.90871	0.954357	−	0.298670i	$-0.0965429\pi$
$503$	42.8080	1.90871	0.954357	+	0.298670i	$0.0965429\pi$
$504$	0	0
$505$	−3.18101	−0.141553
$506$	3.96443	0.176241
$507$	−37.3847	−1.66031
$508$	1.97090	0.0874443
$509$	36.8649	1.63401	0.817004	−	0.576632i	$-0.195633\pi$
$509$	36.8649	1.63401	0.817004	+	0.576632i	$0.195633\pi$
$510$	−0.366748	−0.0162399
$511$	0	0
$512$	12.8657	0.568590
$513$	30.1930	1.33305
$514$	0.743112	0.0327773
$515$	−2.60330	−0.114715
$516$	13.9846	0.615637
$517$	9.86586	0.433900
$518$	0	0
$519$	0.750060	0.0329240
$520$	−0.841607	−0.0369069
$521$	2.49859	0.109465	0.0547327	−	0.998501i	$-0.482569\pi$
$521$	2.49859	0.109465	0.0547327	+	0.998501i	$0.482569\pi$
$522$	−0.259163	−0.0113433
$523$	−17.7123	−0.774506	−0.387253	−	0.921974i	$-0.626576\pi$
$523$	−17.7123	−0.774506	−0.387253	+	0.921974i	$0.626576\pi$
$524$	−18.8372	−0.822908
$525$	0	0
$526$	−1.23684	−0.0539286
$527$	−1.18282	−0.0515245
$528$	−18.1646	−0.790515
$529$	38.9714	1.69441
$530$	−0.233120	−0.0101261
$531$	3.04156	0.131993
$532$	0	0
$533$	4.32142	0.187181
$534$	−4.08549	−0.176796
$535$	3.72560	0.161072
$536$	−4.30374	−0.185893
$537$	−16.2369	−0.700674
$538$	0.308794	0.0133130
$539$	0	0
$540$	−2.23271	−0.0960806
$541$	36.2962	1.56050	0.780248	−	0.625470i	$-0.215093\pi$
$541$	36.2962	1.56050	0.780248	+	0.625470i	$0.215093\pi$
$542$	−2.72200	−0.116920
$543$	−35.8011	−1.53637
$544$	13.0049	0.557579
$545$	1.17121	0.0501691
$546$	0	0
$547$	1.05534	0.0451231	0.0225615	−	0.999745i	$-0.492818\pi$
$547$	1.05534	0.0451231	0.0225615	+	0.999745i	$0.492818\pi$
$548$	−37.6868	−1.60990
$549$	−6.34370	−0.270743
$550$	−2.49655	−0.106453
$551$	20.2038	0.860710
$552$	8.57669	0.365048
$553$	0	0
$554$	−1.80192	−0.0765564
$555$	3.00836	0.127698
$556$	−38.5426	−1.63457
$557$	4.73085	0.200453	0.100226	−	0.994965i	$-0.468043\pi$
$557$	4.73085	0.200453	0.100226	+	0.994965i	$0.468043\pi$
$558$	0.0128821	0.000545344 0
$559$	26.5620	1.12345
$560$	0	0
$561$	−30.7523	−1.29836
$562$	1.51387	0.0638589
$563$	−29.8718	−1.25894	−0.629472	−	0.777023i	$-0.716729\pi$
$563$	−29.8718	−1.25894	−0.629472	+	0.777023i	$0.716729\pi$
$564$	10.5937	0.446076
$565$	−0.0683823	−0.00287686
$566$	2.36300	0.0993245
$567$	0	0
$568$	4.78601	0.200817
$569$	−40.9556	−1.71695	−0.858473	−	0.512859i	$-0.828587\pi$
$569$	−40.9556	−1.71695	−0.858473	+	0.512859i	$0.828587\pi$
$570$	−0.311384	−0.0130424
$571$	28.0854	1.17534	0.587668	−	0.809102i	$-0.300046\pi$
$571$	28.0854	1.17534	0.587668	+	0.809102i	$0.300046\pi$
$572$	−35.0263	−1.46452
$573$	13.1939	0.551184
$574$	0	0
$575$	−39.0258	−1.62749
$576$	3.02288	0.125953
$577$	24.4957	1.01977	0.509884	−	0.860243i	$-0.329688\pi$
$577$	24.4957	1.01977	0.509884	+	0.860243i	$0.329688\pi$
$578$	4.25832	0.177123
$579$	−11.4350	−0.475220
$580$	−1.49403	−0.0620361
$581$	0	0
$582$	−4.06980	−0.168699
$583$	−19.5475	−0.809573
$584$	5.49228	0.227272
$585$	−0.513744	−0.0212407
$586$	1.58482	0.0654683
$587$	−26.5944	−1.09767	−0.548834	−	0.835931i	$-0.684928\pi$
$587$	−26.5944	−1.09767	−0.548834	+	0.835931i	$0.684928\pi$
$588$	0	0
$589$	−1.00426	−0.0413799
$590$	−0.258930	−0.0106600
$591$	20.8572	0.857950
$592$	−34.6844	−1.42552
$593$	−1.49143	−0.0612459	−0.0306229	−	0.999531i	$-0.509749\pi$
$593$	−1.49143	−0.0612459	−0.0306229	+	0.999531i	$0.509749\pi$
$594$	2.76467	0.113436
$595$	0	0
$596$	17.0652	0.699017
$597$	33.2363	1.36027
$598$	8.08547	0.330639
$599$	−28.5590	−1.16689	−0.583445	−	0.812153i	$-0.698296\pi$
$599$	−28.5590	−1.16689	−0.583445	+	0.812153i	$0.698296\pi$
$600$	−5.40107	−0.220498
$601$	7.45416	0.304062	0.152031	−	0.988376i	$-0.451419\pi$
$601$	7.45416	0.304062	0.152031	+	0.988376i	$0.451419\pi$
$602$	0	0
$603$	−2.62714	−0.106986
$604$	29.4847	1.19971
$605$	−0.471768	−0.0191801
$606$	4.22950	0.171812
$607$	−6.18969	−0.251232	−0.125616	−	0.992079i	$-0.540091\pi$
$607$	−6.18969	−0.251232	−0.125616	+	0.992079i	$0.540091\pi$
$608$	11.0417	0.447798
$609$	0	0
$610$	0.540042	0.0218657
$611$	20.1214	0.814026
$612$	5.27950	0.213411
$613$	37.3292	1.50771	0.753857	−	0.657039i	$-0.228191\pi$
$613$	37.3292	1.50771	0.753857	+	0.657039i	$0.228191\pi$
$614$	−4.86417	−0.196302
$615$	−0.238212	−0.00960563
$616$	0	0
$617$	45.8234	1.84478	0.922391	−	0.386257i	$-0.126232\pi$
$617$	45.8234	1.84478	0.922391	+	0.386257i	$0.126232\pi$
$618$	3.46138	0.139237
$619$	19.5672	0.786473	0.393236	−	0.919437i	$-0.371355\pi$
$619$	19.5672	0.786473	0.393236	+	0.919437i	$0.371355\pi$
$620$	0.0742630	0.00298247
$621$	43.2169	1.73423
$622$	−1.28277	−0.0514345
$623$	0	0
$624$	−37.0468	−1.48306
$625$	24.3631	0.974524
$626$	−2.58055	−0.103139
$627$	−26.1100	−1.04273
$628$	26.7166	1.06611
$629$	−58.7199	−2.34132
$630$	0	0
$631$	−2.98989	−0.119026	−0.0595129	−	0.998228i	$-0.518955\pi$
$631$	−2.98989	−0.119026	−0.0595129	+	0.998228i	$0.518955\pi$
$632$	−3.88328	−0.154469
$633$	−3.48681	−0.138588
$634$	−1.76374	−0.0700470
$635$	−0.206353	−0.00818888
$636$	−20.9896	−0.832292
$637$	0	0
$638$	1.84999	0.0732417
$639$	2.92154	0.115574
$640$	−1.08590	−0.0429241
$641$	45.0880	1.78087	0.890434	−	0.455112i	$-0.150401\pi$
$641$	45.0880	1.78087	0.890434	+	0.455112i	$0.150401\pi$
$642$	−4.95360	−0.195503
$643$	−43.4923	−1.71517	−0.857585	−	0.514342i	$-0.828036\pi$
$643$	−43.4923	−1.71517	−0.857585	+	0.514342i	$0.828036\pi$
$644$	0	0
$645$	−1.46419	−0.0576525
$646$	6.07788	0.239131
$647$	11.9165	0.468485	0.234242	−	0.972178i	$-0.424739\pi$
$647$	11.9165	0.468485	0.234242	+	0.972178i	$0.424739\pi$
$648$	5.14068	0.201945
$649$	−21.7116	−0.852256
$650$	−5.09173	−0.199714
$651$	0	0
$652$	−26.2424	−1.02773
$653$	−0.522585	−0.0204503	−0.0102252	−	0.999948i	$-0.503255\pi$
$653$	−0.522585	−0.0204503	−0.0102252	+	0.999948i	$0.503255\pi$
$654$	−1.55725	−0.0608933
$655$	1.97226	0.0770627
$656$	2.74643	0.107230
$657$	3.35266	0.130800
$658$	0	0
$659$	18.0786	0.704243	0.352121	−	0.935954i	$-0.385460\pi$
$659$	18.0786	0.704243	0.352121	+	0.935954i	$0.385460\pi$
$660$	1.93078	0.0751554
$661$	10.6760	0.415249	0.207625	−	0.978209i	$-0.433427\pi$
$661$	10.6760	0.415249	0.207625	+	0.978209i	$0.433427\pi$
$662$	0.227851	0.00885569
$663$	−62.7194	−2.43582
$664$	6.63768	0.257592
$665$	0	0
$666$	0.639520	0.0247809
$667$	28.9187	1.11974
$668$	10.6439	0.411825
$669$	−1.77441	−0.0686027
$670$	0.223650	0.00864036
$671$	45.2833	1.74814
$672$	0	0
$673$	44.6274	1.72026	0.860130	−	0.510074i	$-0.170382\pi$
$673$	44.6274	1.72026	0.860130	+	0.510074i	$0.170382\pi$
$674$	−2.15278	−0.0829219
$675$	−27.2153	−1.04752
$676$	−45.8146	−1.76210
$677$	45.2122	1.73765	0.868823	−	0.495123i	$-0.164877\pi$
$677$	45.2122	1.73765	0.868823	+	0.495123i	$0.164877\pi$
$678$	0.0909217	0.00349183
$679$	0	0
$680$	−0.905530	−0.0347255
$681$	−12.8706	−0.493203
$682$	−0.0919566	−0.00352120
$683$	6.72334	0.257261	0.128631	−	0.991693i	$-0.458942\pi$
$683$	6.72334	0.257261	0.128631	+	0.991693i	$0.458942\pi$
$684$	4.48250	0.171393
$685$	3.94582	0.150762
$686$	0	0
$687$	0.585235	0.0223281
$688$	16.8812	0.643589
$689$	−39.8671	−1.51882
$690$	−0.445700	−0.0169675
$691$	9.59035	0.364834	0.182417	−	0.983221i	$-0.441608\pi$
$691$	9.59035	0.364834	0.182417	+	0.983221i	$0.441608\pi$
$692$	0.919192	0.0349424
$693$	0	0
$694$	−0.400562	−0.0152051
$695$	4.03542	0.153072
$696$	4.00228	0.151706
$697$	4.64964	0.176118
$698$	2.44732	0.0926325
$699$	46.0107	1.74029
$700$	0	0
$701$	18.0322	0.681066	0.340533	−	0.940233i	$-0.389393\pi$
$701$	18.0322	0.681066	0.340533	+	0.940233i	$0.389393\pi$
$702$	5.63854	0.212813
$703$	−49.8555	−1.88034
$704$	−21.5783	−0.813262
$705$	−1.10917	−0.0417736
$706$	−3.20538	−0.120636
$707$	0	0
$708$	−23.3134	−0.876172
$709$	29.8573	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$709$	29.8573	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$710$	−0.248712	−0.00933399
$711$	−2.37048	−0.0888999
$712$	−10.0874	−0.378041
$713$	−1.43745	−0.0538330
$714$	0	0
$715$	3.66726	0.137148
$716$	−19.8982	−0.743630
$717$	45.8193	1.71115
$718$	−1.97536	−0.0737198
$719$	−25.0473	−0.934108	−0.467054	−	0.884229i	$-0.654685\pi$
$719$	−25.0473	−0.934108	−0.467054	+	0.884229i	$0.654685\pi$
$720$	−0.326505	−0.0121681
$721$	0	0
$722$	1.91894	0.0714156
$723$	−3.41983	−0.127185
$724$	−43.8739	−1.63056
$725$	−18.2112	−0.676348
$726$	0.627267	0.0232801
$727$	−43.1090	−1.59882	−0.799412	−	0.600784i	$-0.794855\pi$
$727$	−43.1090	−1.59882	−0.799412	+	0.600784i	$0.794855\pi$
$728$	0	0
$729$	29.6250	1.09722
$730$	−0.285414	−0.0105636
$731$	28.5794	1.05705
$732$	48.6241	1.79720
$733$	−5.25092	−0.193947	−0.0969736	−	0.995287i	$-0.530916\pi$
$733$	−5.25092	−0.193947	−0.0969736	+	0.995287i	$0.530916\pi$
$734$	−0.906525	−0.0334605
$735$	0	0
$736$	15.8045	0.582562
$737$	18.7534	0.690789
$738$	−0.0506394	−0.00186406
$739$	5.24635	0.192990	0.0964950	−	0.995333i	$-0.469237\pi$
$739$	5.24635	0.192990	0.0964950	+	0.995333i	$0.469237\pi$
$740$	3.68671	0.135526
$741$	−53.2513	−1.95624
$742$	0	0
$743$	11.7455	0.430899	0.215449	−	0.976515i	$-0.430878\pi$
$743$	11.7455	0.430899	0.215449	+	0.976515i	$0.430878\pi$
$744$	−0.198940	−0.00729349
$745$	−1.78673	−0.0654607
$746$	−6.31092	−0.231059
$747$	4.05186	0.148250
$748$	−37.6867	−1.37796
$749$	0	0
$750$	0.563759	0.0205856
$751$	24.2930	0.886463	0.443231	−	0.896407i	$-0.353832\pi$
$751$	24.2930	0.886463	0.443231	+	0.896407i	$0.353832\pi$
$752$	12.7880	0.466329
$753$	19.8108	0.721945
$754$	3.77305	0.137407
$755$	−3.08705	−0.112349
$756$	0	0
$757$	−28.3351	−1.02986	−0.514929	−	0.857233i	$-0.672182\pi$
$757$	−28.3351	−1.02986	−0.514929	+	0.857233i	$0.672182\pi$
$758$	−2.48340	−0.0902011
$759$	−37.3726	−1.35654
$760$	−0.768831	−0.0278884
$761$	0.521485	0.0189038	0.00945190	−	0.999955i	$-0.496991\pi$
$761$	0.521485	0.0189038	0.00945190	+	0.999955i	$0.496991\pi$
$762$	0.274369	0.00993935
$763$	0	0
$764$	16.1690	0.584975
$765$	−0.552765	−0.0199853
$766$	2.82067	0.101915
$767$	−44.2809	−1.59889
$768$	−22.0686	−0.796331
$769$	42.0950	1.51798	0.758992	−	0.651100i	$-0.225692\pi$
$769$	42.0950	1.51798	0.758992	+	0.651100i	$0.225692\pi$
$770$	0	0
$771$	−7.00528	−0.252289
$772$	−14.0134	−0.504354
$773$	−44.2658	−1.59213	−0.796066	−	0.605210i	$-0.793089\pi$
$773$	−44.2658	−1.59213	−0.796066	+	0.605210i	$0.793089\pi$
$774$	−0.311260	−0.0111880
$775$	0.905218	0.0325164
$776$	−10.0487	−0.360726
$777$	0	0
$778$	0.154145	0.00552638
$779$	3.94773	0.141442
$780$	3.93782	0.140997
$781$	−20.8548	−0.746244
$782$	8.69959	0.311097
$783$	20.1670	0.720709
$784$	0	0
$785$	−2.79723	−0.0998376
$786$	−2.62234	−0.0935358
$787$	30.4455	1.08526	0.542632	−	0.839971i	$-0.317428\pi$
$787$	30.4455	1.08526	0.542632	+	0.839971i	$0.317428\pi$
$788$	25.5603	0.910547
$789$	11.6596	0.415093
$790$	0.201800	0.00717973
$791$	0	0
$792$	0.826955	0.0293846
$793$	92.3554	3.27963
$794$	3.35610	0.119103
$795$	2.19762	0.0779415
$796$	40.7307	1.44366
$797$	31.4005	1.11226	0.556131	−	0.831095i	$-0.312285\pi$
$797$	31.4005	1.11226	0.556131	+	0.831095i	$0.312285\pi$
$798$	0	0
$799$	21.6497	0.765912
$800$	−9.95269	−0.351881
$801$	−6.15767	−0.217571
$802$	3.34305	0.118047
$803$	−23.9324	−0.844554
$804$	20.1369	0.710174
$805$	0	0
$806$	−0.187546	−0.00660602
$807$	−2.91098	−0.102471
$808$	10.4430	0.367382
$809$	−35.4110	−1.24499	−0.622493	−	0.782626i	$-0.713880\pi$
$809$	−35.4110	−1.24499	−0.622493	+	0.782626i	$0.713880\pi$
$810$	−0.267143	−0.00938644
$811$	33.1853	1.16529	0.582647	−	0.812725i	$-0.302017\pi$
$811$	33.1853	1.16529	0.582647	+	0.812725i	$0.302017\pi$
$812$	0	0
$813$	25.6602	0.899943
$814$	−4.56509	−0.160006
$815$	2.74758	0.0962436
$816$	−39.8607	−1.39540
$817$	24.2651	0.848928
$818$	2.73982	0.0957956
$819$	0	0
$820$	−0.291926	−0.0101945
$821$	−40.6379	−1.41827	−0.709136	−	0.705072i	$-0.750915\pi$
$821$	−40.6379	−1.41827	−0.709136	+	0.705072i	$0.750915\pi$
$822$	−5.24640	−0.182989
$823$	−47.6335	−1.66040	−0.830200	−	0.557465i	$-0.811774\pi$
$823$	−47.6335	−1.66040	−0.830200	+	0.557465i	$0.811774\pi$
$824$	8.54641	0.297728
$825$	23.5349	0.819381
$826$	0	0
$827$	33.7637	1.17408	0.587039	−	0.809559i	$-0.300294\pi$
$827$	33.7637	1.17408	0.587039	+	0.809559i	$0.300294\pi$
$828$	6.41605	0.222973
$829$	−27.3714	−0.950649	−0.475325	−	0.879811i	$-0.657669\pi$
$829$	−27.3714	−0.950649	−0.475325	+	0.879811i	$0.657669\pi$
$830$	−0.344937	−0.0119729
$831$	16.9867	0.589260
$832$	−44.0089	−1.52574
$833$	0	0
$834$	−5.36553	−0.185793
$835$	−1.11442	−0.0385661
$836$	−31.9975	−1.10666
$837$	−1.00243	−0.0346491
$838$	2.20638	0.0762182
$839$	−19.6027	−0.676761	−0.338380	−	0.941009i	$-0.609879\pi$
$839$	−19.6027	−0.676761	−0.338380	+	0.941009i	$0.609879\pi$
$840$	0	0
$841$	−15.5052	−0.534661
$842$	−4.25633	−0.146683
$843$	−14.2712	−0.491527
$844$	−4.27305	−0.147085
$845$	4.79680	0.165015
$846$	−0.235788	−0.00810655
$847$	0	0
$848$	−25.3371	−0.870080
$849$	−22.2759	−0.764509
$850$	−5.47846	−0.187910
$851$	−71.3609	−2.44622
$852$	−22.3934	−0.767186
$853$	−16.1746	−0.553809	−0.276905	−	0.960897i	$-0.589309\pi$
$853$	−16.1746	−0.553809	−0.276905	+	0.960897i	$0.589309\pi$
$854$	0	0
$855$	−0.469319	−0.0160504
$856$	−12.2308	−0.418041
$857$	−21.3510	−0.729335	−0.364668	−	0.931138i	$-0.618817\pi$
$857$	−21.3510	−0.729335	−0.364668	+	0.931138i	$0.618817\pi$
$858$	−4.87603	−0.166465
$859$	−0.615958	−0.0210162	−0.0105081	−	0.999945i	$-0.503345\pi$
$859$	−0.615958	−0.0210162	−0.0105081	+	0.999945i	$0.503345\pi$
$860$	−1.79435	−0.0611869
$861$	0	0
$862$	−1.27973	−0.0435879
$863$	−50.4704	−1.71803	−0.859016	−	0.511949i	$-0.828924\pi$
$863$	−50.4704	−1.71803	−0.859016	+	0.511949i	$0.828924\pi$
$864$	11.0215	0.374961
$865$	−0.0962396	−0.00327225
$866$	2.41278	0.0819897
$867$	−40.1430	−1.36333
$868$	0	0
$869$	16.9212	0.574013
$870$	−0.207984	−0.00705133
$871$	38.2475	1.29597
$872$	−3.84497	−0.130207
$873$	−6.13403	−0.207605
$874$	7.38630	0.249845
$875$	0	0
$876$	−25.6980	−0.868255
$877$	40.6889	1.37397	0.686983	−	0.726673i	$-0.258935\pi$
$877$	40.6889	1.37397	0.686983	+	0.726673i	$0.258935\pi$
$878$	−1.54470	−0.0521312
$879$	−14.9400	−0.503915
$880$	2.33069	0.0785676
$881$	−34.1963	−1.15210	−0.576052	−	0.817413i	$-0.695407\pi$
$881$	−34.1963	−1.15210	−0.576052	+	0.817413i	$0.695407\pi$
$882$	0	0
$883$	−20.2286	−0.680745	−0.340373	−	0.940291i	$-0.610553\pi$
$883$	−20.2286	−0.680745	−0.340373	+	0.940291i	$0.610553\pi$
$884$	−76.8621	−2.58515
$885$	2.44092	0.0820507
$886$	−4.98200	−0.167374
$887$	−43.6034	−1.46406	−0.732030	−	0.681273i	$-0.761427\pi$
$887$	−43.6034	−1.46406	−0.732030	+	0.681273i	$0.761427\pi$
$888$	−9.87616	−0.331422
$889$	0	0
$890$	0.524206	0.0175714
$891$	−22.4003	−0.750438
$892$	−2.17452	−0.0728084
$893$	18.3815	0.615113
$894$	2.37565	0.0794537
$895$	2.08334	0.0696385
$896$	0	0
$897$	−76.2214	−2.54496
$898$	0.194609	0.00649418
$899$	−0.670782	−0.0223718
$900$	−4.04043	−0.134681
$901$	−42.8952	−1.42904
$902$	0.361480	0.0120360
$903$	0	0
$904$	0.224493	0.00746652
$905$	4.59361	0.152697
$906$	4.10457	0.136365
$907$	24.4877	0.813102	0.406551	−	0.913628i	$-0.366731\pi$
$907$	24.4877	0.813102	0.406551	+	0.913628i	$0.366731\pi$
$908$	−15.7728	−0.523439
$909$	6.37473	0.211436
$910$	0	0
$911$	30.1286	0.998206	0.499103	−	0.866543i	$-0.333663\pi$
$911$	30.1286	0.998206	0.499103	+	0.866543i	$0.333663\pi$
$912$	−33.8433	−1.12066
$913$	−28.9234	−0.957225
$914$	−3.74906	−0.124008
$915$	−5.09096	−0.168302
$916$	0.717200	0.0236970
$917$	0	0
$918$	6.06681	0.200235
$919$	−37.4612	−1.23573	−0.617865	−	0.786284i	$-0.712002\pi$
$919$	−37.4612	−1.23573	−0.617865	+	0.786284i	$0.712002\pi$
$920$	−1.10047	−0.0362814
$921$	45.8543	1.51095
$922$	3.78713	0.124722
$923$	−42.5335	−1.40001
$924$	0	0
$925$	44.9386	1.47757
$926$	0.610073	0.0200482
$927$	5.21701	0.171349
$928$	7.37511	0.242100
$929$	10.3163	0.338468	0.169234	−	0.985576i	$-0.445871\pi$
$929$	10.3163	0.338468	0.169234	+	0.985576i	$0.445871\pi$
$930$	0.0103382	0.000339003 0
$931$	0	0
$932$	56.3857	1.84698
$933$	12.0926	0.395895
$934$	0.857216	0.0280490
$935$	3.94581	0.129042
$936$	1.68658	0.0551275
$937$	−24.4343	−0.798235	−0.399117	−	0.916900i	$-0.630683\pi$
$937$	−24.4343	−0.798235	−0.399117	+	0.916900i	$0.630683\pi$
$938$	0	0
$939$	24.3267	0.793872
$940$	−1.35927	−0.0443346
$941$	−10.3783	−0.338324	−0.169162	−	0.985588i	$-0.554106\pi$
$941$	−10.3783	−0.338324	−0.169162	+	0.985588i	$0.554106\pi$
$942$	3.71923	0.121179
$943$	5.65060	0.184009
$944$	−28.1422	−0.915952
$945$	0	0
$946$	2.22187	0.0722391
$947$	20.3460	0.661157	0.330579	−	0.943778i	$-0.392756\pi$
$947$	20.3460	0.661157	0.330579	+	0.943778i	$0.392756\pi$
$948$	18.1696	0.590121
$949$	−48.8101	−1.58444
$950$	−4.65143	−0.150912
$951$	16.6267	0.539158
$952$	0	0
$953$	25.1505	0.814704	0.407352	−	0.913271i	$-0.366452\pi$
$953$	25.1505	0.814704	0.407352	+	0.913271i	$0.366452\pi$
$954$	0.467172	0.0151253
$955$	−1.69290	−0.0547810
$956$	56.1511	1.81606
$957$	−17.4398	−0.563748
$958$	6.65432	0.214991
$959$	0	0
$960$	2.42593	0.0782966
$961$	−30.9667	−0.998924
$962$	−9.31051	−0.300183
$963$	−7.46609	−0.240591
$964$	−4.19097	−0.134982
$965$	1.46721	0.0472311
$966$	0	0
$967$	−27.8536	−0.895711	−0.447855	−	0.894106i	$-0.647812\pi$
$967$	−27.8536	−0.895711	−0.447855	+	0.894106i	$0.647812\pi$
$968$	1.54877	0.0497794
$969$	−57.2959	−1.84061
$970$	0.522193	0.0167666
$971$	2.52631	0.0810732	0.0405366	−	0.999178i	$-0.487093\pi$
$971$	2.52631	0.0810732	0.0405366	+	0.999178i	$0.487093\pi$
$972$	8.40664	0.269643
$973$	0	0
$974$	−0.186539	−0.00597708
$975$	47.9995	1.53721
$976$	58.6954	1.87880
$977$	52.7486	1.68758	0.843789	−	0.536674i	$-0.180320\pi$
$977$	52.7486	1.68758	0.843789	+	0.536674i	$0.180320\pi$
$978$	−3.65321	−0.116817
$979$	43.9554	1.40482
$980$	0	0
$981$	−2.34710	−0.0749370
$982$	−4.48092	−0.142992
$983$	7.36372	0.234866	0.117433	−	0.993081i	$-0.462533\pi$
$983$	7.36372	0.234866	0.117433	+	0.993081i	$0.462533\pi$
$984$	0.782029	0.0249302
$985$	−2.67617	−0.0852698
$986$	4.05963	0.129285
$987$	0	0
$988$	−65.2590	−2.07616
$989$	34.7319	1.10441
$990$	−0.0429739	−0.00136580
$991$	31.3967	0.997350	0.498675	−	0.866789i	$-0.333820\pi$
$991$	31.3967	0.997350	0.498675	+	0.866789i	$0.333820\pi$
$992$	−0.366592	−0.0116393
$993$	−2.14794	−0.0681629
$994$	0	0
$995$	−4.26452	−0.135194
$996$	−31.0573	−0.984087
$997$	16.1969	0.512960	0.256480	−	0.966550i	$-0.417437\pi$
$997$	16.1969	0.512960	0.256480	+	0.966550i	$0.417437\pi$
$998$	−5.04044	−0.159552
$999$	−49.7647	−1.57449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.9		16
7.6	odd	2		889.2.a.c.1.9	✓	16
21.20	even	2		8001.2.a.t.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.9	✓	16	7.6	odd	2
6223.2.a.k.1.9		16	1.1	even	1	trivial
8001.2.a.t.1.8		16	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.170601 q^{2} -1.60825 q^{3} -1.97090 q^{4} +0.206353 q^{5} -0.274369 q^{6} -0.677439 q^{8} -0.413531 q^{9} +0.0352041 q^{10} +2.95191 q^{11} +3.16969 q^{12} +6.02043 q^{13} -0.331868 q^{15} +3.82622 q^{16} +6.47770 q^{17} -0.0705489 q^{18} +5.49983 q^{19} -0.406701 q^{20} +0.503600 q^{22} +7.87219 q^{23} +1.08949 q^{24} -4.95742 q^{25} +1.02709 q^{26} +5.48981 q^{27} +3.67353 q^{29} -0.0566170 q^{30} -0.182599 q^{31} +2.00764 q^{32} -4.74741 q^{33} +1.10510 q^{34} +0.815026 q^{36} -9.06492 q^{37} +0.938277 q^{38} -9.68236 q^{39} -0.139792 q^{40} +0.717792 q^{41} +4.41197 q^{43} -5.81791 q^{44} -0.0853335 q^{45} +1.34301 q^{46} +3.34219 q^{47} -6.15352 q^{48} -0.845741 q^{50} -10.4178 q^{51} -11.8656 q^{52} -6.62197 q^{53} +0.936568 q^{54} +0.609137 q^{55} -8.84510 q^{57} +0.626708 q^{58} -7.35511 q^{59} +0.654077 q^{60} +15.3403 q^{61} -0.0311515 q^{62} -7.30993 q^{64} +1.24234 q^{65} -0.809914 q^{66} +6.35295 q^{67} -12.7669 q^{68} -12.6605 q^{69} -7.06485 q^{71} +0.280142 q^{72} -8.10741 q^{73} -1.54649 q^{74} +7.97277 q^{75} -10.8396 q^{76} -1.65182 q^{78} +5.73229 q^{79} +0.789553 q^{80} -7.58840 q^{81} +0.122456 q^{82} -9.79819 q^{83} +1.33670 q^{85} +0.752687 q^{86} -5.90796 q^{87} -1.99974 q^{88} +14.8905 q^{89} -0.0145580 q^{90} -15.5153 q^{92} +0.293664 q^{93} +0.570182 q^{94} +1.13491 q^{95} -3.22878 q^{96} +14.8333 q^{97} -1.22071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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