LMFDB - Embedded newform 6223.2.a.k.1.16

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			1.16
Root			$-2.57541$ 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)

$f(q)$	$=$	\(q+2.57541 q^{2} +1.31654 q^{3} +4.63271 q^{4} +3.61540 q^{5} +3.39062 q^{6} +6.78030 q^{8} -1.26673 q^{9} +9.31112 q^{10} -4.80505 q^{11} +6.09914 q^{12} +3.02226 q^{13} +4.75981 q^{15} +8.19660 q^{16} +5.99473 q^{17} -3.26234 q^{18} -5.27314 q^{19} +16.7491 q^{20} -12.3750 q^{22} +5.63109 q^{23} +8.92653 q^{24} +8.07110 q^{25} +7.78354 q^{26} -5.61731 q^{27} -5.62713 q^{29} +12.2584 q^{30} -4.07939 q^{31} +7.54897 q^{32} -6.32603 q^{33} +15.4389 q^{34} -5.86839 q^{36} +3.58935 q^{37} -13.5805 q^{38} +3.97892 q^{39} +24.5135 q^{40} +9.98794 q^{41} +4.00758 q^{43} -22.2604 q^{44} -4.57973 q^{45} +14.5023 q^{46} +9.98363 q^{47} +10.7911 q^{48} +20.7864 q^{50} +7.89230 q^{51} +14.0013 q^{52} +1.00415 q^{53} -14.4668 q^{54} -17.3722 q^{55} -6.94229 q^{57} -14.4922 q^{58} -2.21412 q^{59} +22.0508 q^{60} -6.57832 q^{61} -10.5061 q^{62} +3.04845 q^{64} +10.9267 q^{65} -16.2921 q^{66} -5.13122 q^{67} +27.7719 q^{68} +7.41354 q^{69} -13.8883 q^{71} -8.58880 q^{72} +10.0400 q^{73} +9.24403 q^{74} +10.6259 q^{75} -24.4290 q^{76} +10.2473 q^{78} -17.1340 q^{79} +29.6340 q^{80} -3.59522 q^{81} +25.7230 q^{82} +5.98037 q^{83} +21.6733 q^{85} +10.3211 q^{86} -7.40834 q^{87} -32.5797 q^{88} -15.9290 q^{89} -11.7947 q^{90} +26.0872 q^{92} -5.37067 q^{93} +25.7119 q^{94} -19.0645 q^{95} +9.93851 q^{96} +16.1200 q^{97} +6.08669 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.57541	1.82109	0.910543	−	0.413414i	$-0.135664\pi$
$2$	2.57541	1.82109	0.910543	+	0.413414i	$0.135664\pi$
$3$	1.31654	0.760104	0.380052	−	0.924965i	$-0.375906\pi$
$3$	1.31654	0.760104	0.380052	+	0.924965i	$0.375906\pi$
$4$	4.63271	2.31636
$5$	3.61540	1.61686	0.808428	−	0.588596i	$-0.200319\pi$
$5$	3.61540	1.61686	0.808428	+	0.588596i	$0.200319\pi$
$6$	3.39062	1.38421
$7$	0	0
$7$	0	0
$8$	6.78030	2.39720
$9$	−1.26673	−0.422243
$10$	9.31112	2.94443
$11$	−4.80505	−1.44878	−0.724389	−	0.689392i	$-0.757878\pi$
$11$	−4.80505	−1.44878	−0.724389	+	0.689392i	$0.757878\pi$
$12$	6.09914	1.76067
$13$	3.02226	0.838224	0.419112	−	0.907935i	$-0.362341\pi$
$13$	3.02226	0.838224	0.419112	+	0.907935i	$0.362341\pi$
$14$	0	0
$15$	4.75981	1.22898
$16$	8.19660	2.04915
$17$	5.99473	1.45394	0.726968	−	0.686671i	$-0.240929\pi$
$17$	5.99473	1.45394	0.726968	+	0.686671i	$0.240929\pi$
$18$	−3.26234	−0.768940
$19$	−5.27314	−1.20974	−0.604871	−	0.796324i	$-0.706775\pi$
$19$	−5.27314	−1.20974	−0.604871	+	0.796324i	$0.706775\pi$
$20$	16.7491	3.74521
$21$	0	0
$22$	−12.3750	−2.63835
$23$	5.63109	1.17416	0.587082	−	0.809528i	$-0.300277\pi$
$23$	5.63109	1.17416	0.587082	+	0.809528i	$0.300277\pi$
$24$	8.92653	1.82212
$25$	8.07110	1.61422
$26$	7.78354	1.52648
$27$	−5.61731	−1.08105
$28$	0	0
$29$	−5.62713	−1.04493	−0.522466	−	0.852660i	$-0.674988\pi$
$29$	−5.62713	−1.04493	−0.522466	+	0.852660i	$0.674988\pi$
$30$	12.2584	2.23807
$31$	−4.07939	−0.732680	−0.366340	−	0.930481i	$-0.619389\pi$
$31$	−4.07939	−0.732680	−0.366340	+	0.930481i	$0.619389\pi$
$32$	7.54897	1.33448
$33$	−6.32603	−1.10122
$34$	15.4389	2.64774
$35$	0	0
$36$	−5.86839	−0.978065
$37$	3.58935	0.590085	0.295043	−	0.955484i	$-0.404666\pi$
$37$	3.58935	0.590085	0.295043	+	0.955484i	$0.404666\pi$
$38$	−13.5805	−2.20304
$39$	3.97892	0.637137
$40$	24.5135	3.87592
$41$	9.98794	1.55985	0.779927	−	0.625870i	$-0.215256\pi$
$41$	9.98794	1.55985	0.779927	+	0.625870i	$0.215256\pi$
$42$	0	0
$43$	4.00758	0.611150	0.305575	−	0.952168i	$-0.401151\pi$
$43$	4.00758	0.611150	0.305575	+	0.952168i	$0.401151\pi$
$44$	−22.2604	−3.35588
$45$	−4.57973	−0.682705
$46$	14.5023	2.13825
$47$	9.98363	1.45626	0.728131	−	0.685438i	$-0.240389\pi$
$47$	9.98363	1.45626	0.728131	+	0.685438i	$0.240389\pi$
$48$	10.7911	1.55757
$49$	0	0
$50$	20.7864	2.93964
$51$	7.89230	1.10514
$52$	14.0013	1.94163
$53$	1.00415	0.137931	0.0689655	−	0.997619i	$-0.478030\pi$
$53$	1.00415	0.137931	0.0689655	+	0.997619i	$0.478030\pi$
$54$	−14.4668	−1.96869
$55$	−17.3722	−2.34246
$56$	0	0
$57$	−6.94229	−0.919529
$58$	−14.4922	−1.90291
$59$	−2.21412	−0.288254	−0.144127	−	0.989559i	$-0.546037\pi$
$59$	−2.21412	−0.288254	−0.144127	+	0.989559i	$0.546037\pi$
$60$	22.0508	2.84675
$61$	−6.57832	−0.842267	−0.421133	−	0.906999i	$-0.638368\pi$
$61$	−6.57832	−0.842267	−0.421133	+	0.906999i	$0.638368\pi$
$62$	−10.5061	−1.33427
$63$	0	0
$64$	3.04845	0.381056
$65$	10.9267	1.35529
$66$	−16.2921	−2.00542
$67$	−5.13122	−0.626879	−0.313439	−	0.949608i	$-0.601481\pi$
$67$	−5.13122	−0.626879	−0.313439	+	0.949608i	$0.601481\pi$
$68$	27.7719	3.36784
$69$	7.41354	0.892485
$70$	0	0
$71$	−13.8883	−1.64824	−0.824121	−	0.566414i	$-0.808330\pi$
$71$	−13.8883	−1.64824	−0.824121	+	0.566414i	$0.808330\pi$
$72$	−8.58880	−1.01220
$73$	10.0400	1.17509	0.587544	−	0.809192i	$-0.300095\pi$
$73$	10.0400	1.17509	0.587544	+	0.809192i	$0.300095\pi$
$74$	9.24403	1.07460
$75$	10.6259	1.22697
$76$	−24.4290	−2.80219
$77$	0	0
$78$	10.2473	1.16028
$79$	−17.1340	−1.92772	−0.963861	−	0.266405i	$-0.914164\pi$
$79$	−17.1340	−1.92772	−0.963861	+	0.266405i	$0.914164\pi$
$80$	29.6340	3.31318
$81$	−3.59522	−0.399468
$82$	25.7230	2.84063
$83$	5.98037	0.656431	0.328216	−	0.944603i	$-0.393553\pi$
$83$	5.98037	0.656431	0.328216	+	0.944603i	$0.393553\pi$
$84$	0	0
$85$	21.6733	2.35080
$86$	10.3211	1.11296
$87$	−7.40834	−0.794257
$88$	−32.5797	−3.47301
$89$	−15.9290	−1.68847	−0.844237	−	0.535970i	$-0.819946\pi$
$89$	−15.9290	−1.68847	−0.844237	+	0.535970i	$0.819946\pi$
$90$	−11.7947	−1.24327
$91$	0	0
$92$	26.0872	2.71978
$93$	−5.37067	−0.556913
$94$	25.7119	2.65198
$95$	−19.0645	−1.95598
$96$	9.93851	1.01434
$97$	16.1200	1.63673	0.818367	−	0.574695i	$-0.194879\pi$
$97$	16.1200	1.63673	0.818367	+	0.574695i	$0.194879\pi$
$98$	0	0
$99$	6.08669	0.611736
$100$	37.3911	3.73911
$101$	−15.0825	−1.50077	−0.750385	−	0.661001i	$-0.770132\pi$
$101$	−15.0825	−1.50077	−0.750385	+	0.661001i	$0.770132\pi$
$102$	20.3259	2.01256
$103$	−13.8383	−1.36353	−0.681764	−	0.731572i	$-0.738787\pi$
$103$	−13.8383	−1.36353	−0.681764	+	0.731572i	$0.738787\pi$
$104$	20.4918	2.00939
$105$	0	0
$106$	2.58610	0.251184
$107$	14.4768	1.39952	0.699761	−	0.714377i	$-0.253290\pi$
$107$	14.4768	1.39952	0.699761	+	0.714377i	$0.253290\pi$
$108$	−26.0234	−2.50410
$109$	5.17787	0.495950	0.247975	−	0.968766i	$-0.420235\pi$
$109$	5.17787	0.495950	0.247975	+	0.968766i	$0.420235\pi$
$110$	−44.7404	−4.26583
$111$	4.72551	0.448526
$112$	0	0
$113$	−2.17656	−0.204753	−0.102377	−	0.994746i	$-0.532645\pi$
$113$	−2.17656	−0.204753	−0.102377	+	0.994746i	$0.532645\pi$
$114$	−17.8792	−1.67454
$115$	20.3586	1.89845
$116$	−26.0689	−2.42044
$117$	−3.82838	−0.353934
$118$	−5.70227	−0.524936
$119$	0	0
$120$	32.2729	2.94610
$121$	12.0885	1.09896
$122$	−16.9418	−1.53384
$123$	13.1495	1.18565
$124$	−18.8986	−1.69715
$125$	11.1033	0.993105
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−7.24694	−0.640545
$129$	5.27613	0.464537
$130$	28.1406	2.46809
$131$	−1.35652	−0.118520	−0.0592598	−	0.998243i	$-0.518874\pi$
$131$	−1.35652	−0.118520	−0.0592598	+	0.998243i	$0.518874\pi$
$132$	−29.3067	−2.55082
$133$	0	0
$134$	−13.2150	−1.14160
$135$	−20.3088	−1.74790
$136$	40.6461	3.48538
$137$	7.96241	0.680274	0.340137	−	0.940376i	$-0.389526\pi$
$137$	7.96241	0.680274	0.340137	+	0.940376i	$0.389526\pi$
$138$	19.0929	1.62529
$139$	−3.78746	−0.321248	−0.160624	−	0.987016i	$-0.551351\pi$
$139$	−3.78746	−0.321248	−0.160624	+	0.987016i	$0.551351\pi$
$140$	0	0
$141$	13.1438	1.10691
$142$	−35.7681	−3.00159
$143$	−14.5221	−1.21440
$144$	−10.3829	−0.865239
$145$	−20.3443	−1.68950
$146$	25.8570	2.13994
$147$	0	0
$148$	16.6284	1.36685
$149$	−10.5322	−0.862835	−0.431417	−	0.902152i	$-0.641986\pi$
$149$	−10.5322	−0.862835	−0.431417	+	0.902152i	$0.641986\pi$
$150$	27.3660	2.23443
$151$	−8.21168	−0.668257	−0.334128	−	0.942528i	$-0.608442\pi$
$151$	−8.21168	−0.668257	−0.334128	+	0.942528i	$0.608442\pi$
$152$	−35.7535	−2.89999
$153$	−7.59370	−0.613914
$154$	0	0
$155$	−14.7486	−1.18464
$156$	18.4332	1.47584
$157$	−4.22460	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$157$	−4.22460	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$158$	−44.1269	−3.51055
$159$	1.32200	0.104842
$160$	27.2925	2.15766
$161$	0	0
$162$	−9.25914	−0.727467
$163$	0.350024	0.0274160	0.0137080	−	0.999906i	$-0.495636\pi$
$163$	0.350024	0.0274160	0.0137080	+	0.999906i	$0.495636\pi$
$164$	46.2713	3.61318
$165$	−22.8711	−1.78051
$166$	15.4019	1.19542
$167$	−0.955076	−0.0739060	−0.0369530	−	0.999317i	$-0.511765\pi$
$167$	−0.955076	−0.0739060	−0.0369530	+	0.999317i	$0.511765\pi$
$168$	0	0
$169$	−3.86595	−0.297381
$170$	55.8177	4.28102
$171$	6.67964	0.510805
$172$	18.5660	1.41564
$173$	6.62247	0.503497	0.251749	−	0.967793i	$-0.418994\pi$
$173$	6.62247	0.503497	0.251749	+	0.967793i	$0.418994\pi$
$174$	−19.0795	−1.44641
$175$	0	0
$176$	−39.3851	−2.96876
$177$	−2.91498	−0.219103
$178$	−41.0237	−3.07486
$179$	−12.6159	−0.942958	−0.471479	−	0.881877i	$-0.656280\pi$
$179$	−12.6159	−0.942958	−0.471479	+	0.881877i	$0.656280\pi$
$180$	−21.2166	−1.58139
$181$	11.9530	0.888460	0.444230	−	0.895913i	$-0.353477\pi$
$181$	11.9530	0.888460	0.444230	+	0.895913i	$0.353477\pi$
$182$	0	0
$183$	−8.66060	−0.640210
$184$	38.1805	2.81470
$185$	12.9769	0.954082
$186$	−13.8317	−1.01419
$187$	−28.8050	−2.10643
$188$	46.2513	3.37322
$189$	0	0
$190$	−49.0988	−3.56200
$191$	−21.1585	−1.53098	−0.765489	−	0.643449i	$-0.777503\pi$
$191$	−21.1585	−1.53098	−0.765489	+	0.643449i	$0.777503\pi$
$192$	4.01340	0.289642
$193$	−20.2465	−1.45737	−0.728686	−	0.684848i	$-0.759869\pi$
$193$	−20.2465	−1.45737	−0.728686	+	0.684848i	$0.759869\pi$
$194$	41.5155	2.98064
$195$	14.3854	1.03016
$196$	0	0
$197$	7.09872	0.505763	0.252881	−	0.967497i	$-0.418622\pi$
$197$	7.09872	0.505763	0.252881	+	0.967497i	$0.418622\pi$
$198$	15.6757	1.11402
$199$	16.6871	1.18292	0.591458	−	0.806336i	$-0.298553\pi$
$199$	16.6871	1.18292	0.591458	+	0.806336i	$0.298553\pi$
$200$	54.7245	3.86961
$201$	−6.75545	−0.476493
$202$	−38.8437	−2.73303
$203$	0	0
$204$	36.5627	2.55990
$205$	36.1104	2.52206
$206$	−35.6392	−2.48310
$207$	−7.13306	−0.495782
$208$	24.7723	1.71765
$209$	25.3377	1.75265
$210$	0	0
$211$	17.6748	1.21678	0.608391	−	0.793637i	$-0.291815\pi$
$211$	17.6748	1.21678	0.608391	+	0.793637i	$0.291815\pi$
$212$	4.65195	0.319497
$213$	−18.2845	−1.25283
$214$	37.2835	2.54865
$215$	14.4890	0.988141
$216$	−38.0871	−2.59150
$217$	0	0
$218$	13.3351	0.903167
$219$	13.2180	0.893189
$220$	−80.4803	−5.42598
$221$	18.1176	1.21872
$222$	12.1701	0.816804
$223$	−2.34873	−0.157283	−0.0786414	−	0.996903i	$-0.525058\pi$
$223$	−2.34873	−0.157283	−0.0786414	+	0.996903i	$0.525058\pi$
$224$	0	0
$225$	−10.2239	−0.681593
$226$	−5.60551	−0.372873
$227$	−8.80210	−0.584216	−0.292108	−	0.956385i	$-0.594357\pi$
$227$	−8.80210	−0.584216	−0.292108	+	0.956385i	$0.594357\pi$
$228$	−32.1616	−2.12996
$229$	0.995190	0.0657640	0.0328820	−	0.999459i	$-0.489531\pi$
$229$	0.995190	0.0657640	0.0328820	+	0.999459i	$0.489531\pi$
$230$	52.4317	3.45724
$231$	0	0
$232$	−38.1537	−2.50491
$233$	−16.8582	−1.10442	−0.552209	−	0.833706i	$-0.686215\pi$
$233$	−16.8582	−1.10442	−0.552209	+	0.833706i	$0.686215\pi$
$234$	−9.85963	−0.644544
$235$	36.0948	2.35456
$236$	−10.2574	−0.667700
$237$	−22.5575	−1.46527
$238$	0	0
$239$	−5.12666	−0.331616	−0.165808	−	0.986158i	$-0.553023\pi$
$239$	−5.12666	−0.331616	−0.165808	+	0.986158i	$0.553023\pi$
$240$	39.0143	2.51836
$241$	−8.86855	−0.571273	−0.285637	−	0.958338i	$-0.592205\pi$
$241$	−8.86855	−0.571273	−0.285637	+	0.958338i	$0.592205\pi$
$242$	31.1328	2.00129
$243$	12.1187	0.777414
$244$	−30.4754	−1.95099
$245$	0	0
$246$	33.8653	2.15917
$247$	−15.9368	−1.01403
$248$	−27.6595	−1.75638
$249$	7.87339	0.498956
$250$	28.5954	1.80853
$251$	8.12236	0.512679	0.256339	−	0.966587i	$-0.417484\pi$
$251$	8.12236	0.512679	0.256339	+	0.966587i	$0.417484\pi$
$252$	0	0
$253$	−27.0577	−1.70110
$254$	−2.57541	−0.161595
$255$	28.5338	1.78686
$256$	−24.7607	−1.54754
$257$	−23.4193	−1.46086	−0.730429	−	0.682989i	$-0.760680\pi$
$257$	−23.4193	−1.46086	−0.730429	+	0.682989i	$0.760680\pi$
$258$	13.5882	0.845962
$259$	0	0
$260$	50.6201	3.13933
$261$	7.12805	0.441215
$262$	−3.49358	−0.215834
$263$	−8.96856	−0.553025	−0.276513	−	0.961010i	$-0.589179\pi$
$263$	−8.96856	−0.553025	−0.276513	+	0.961010i	$0.589179\pi$
$264$	−42.8924	−2.63985
$265$	3.63041	0.223014
$266$	0	0
$267$	−20.9712	−1.28341
$268$	−23.7715	−1.45207
$269$	−12.9224	−0.787892	−0.393946	−	0.919134i	$-0.628890\pi$
$269$	−12.9224	−0.787892	−0.393946	+	0.919134i	$0.628890\pi$
$270$	−52.3034	−3.18308
$271$	7.70820	0.468240	0.234120	−	0.972208i	$-0.424779\pi$
$271$	7.70820	0.468240	0.234120	+	0.972208i	$0.424779\pi$
$272$	49.1365	2.97934
$273$	0	0
$274$	20.5064	1.23884
$275$	−38.7821	−2.33865
$276$	34.3448	2.06731
$277$	−8.26154	−0.496388	−0.248194	−	0.968710i	$-0.579837\pi$
$277$	−8.26154	−0.496388	−0.248194	+	0.968710i	$0.579837\pi$
$278$	−9.75425	−0.585021
$279$	5.16748	0.309369
$280$	0	0
$281$	−11.9817	−0.714766	−0.357383	−	0.933958i	$-0.616331\pi$
$281$	−11.9817	−0.714766	−0.357383	+	0.933958i	$0.616331\pi$
$282$	33.8507	2.01578
$283$	5.98073	0.355517	0.177759	−	0.984074i	$-0.443115\pi$
$283$	5.98073	0.355517	0.177759	+	0.984074i	$0.443115\pi$
$284$	−64.3407	−3.81792
$285$	−25.0991	−1.48675
$286$	−37.4003	−2.21153
$287$	0	0
$288$	−9.56249	−0.563475
$289$	18.9368	1.11393
$290$	−52.3949	−3.07673
$291$	21.2226	1.24409
$292$	46.5123	2.72192
$293$	−9.49311	−0.554593	−0.277297	−	0.960784i	$-0.589439\pi$
$293$	−9.49311	−0.554593	−0.277297	+	0.960784i	$0.589439\pi$
$294$	0	0
$295$	−8.00494	−0.466066
$296$	24.3369	1.41455
$297$	26.9915	1.56620
$298$	−27.1248	−1.57130
$299$	17.0186	0.984212
$300$	49.2268	2.84211
$301$	0	0
$302$	−21.1484	−1.21695
$303$	−19.8567	−1.14074
$304$	−43.2219	−2.47894
$305$	−23.7832	−1.36182
$306$	−19.5569	−1.11799
$307$	20.3886	1.16364	0.581820	−	0.813318i	$-0.302341\pi$
$307$	20.3886	1.16364	0.581820	+	0.813318i	$0.302341\pi$
$308$	0	0
$309$	−18.2186	−1.03642
$310$	−37.9837	−2.15733
$311$	−13.4256	−0.761298	−0.380649	−	0.924720i	$-0.624299\pi$
$311$	−13.4256	−0.761298	−0.380649	+	0.924720i	$0.624299\pi$
$312$	26.9783	1.52734
$313$	8.23860	0.465673	0.232837	−	0.972516i	$-0.425199\pi$
$313$	8.23860	0.465673	0.232837	+	0.972516i	$0.425199\pi$
$314$	−10.8801	−0.613997
$315$	0	0
$316$	−79.3768	−4.46529
$317$	18.9347	1.06348	0.531739	−	0.846908i	$-0.321539\pi$
$317$	18.9347	1.06348	0.531739	+	0.846908i	$0.321539\pi$
$318$	3.40470	0.190926
$319$	27.0387	1.51387
$320$	11.0214	0.616113
$321$	19.0592	1.06378
$322$	0	0
$323$	−31.6111	−1.75889
$324$	−16.6556	−0.925311
$325$	24.3930	1.35308
$326$	0.901455	0.0499270
$327$	6.81686	0.376973
$328$	67.7213	3.73928
$329$	0	0
$330$	−58.9024	−3.24247
$331$	10.9260	0.600547	0.300274	−	0.953853i	$-0.402922\pi$
$331$	10.9260	0.600547	0.300274	+	0.953853i	$0.402922\pi$
$332$	27.7054	1.52053
$333$	−4.54673	−0.249159
$334$	−2.45971	−0.134589
$335$	−18.5514	−1.01357
$336$	0	0
$337$	19.2079	1.04632	0.523162	−	0.852233i	$-0.324752\pi$
$337$	19.2079	1.04632	0.523162	+	0.852233i	$0.324752\pi$
$338$	−9.95638	−0.541556
$339$	−2.86552	−0.155634
$340$	100.406	5.44530
$341$	19.6017	1.06149
$342$	17.2028	0.930219
$343$	0	0
$344$	27.1726	1.46505
$345$	26.8029	1.44302
$346$	17.0555	0.916912
$347$	2.30435	0.123704	0.0618519	−	0.998085i	$-0.480299\pi$
$347$	2.30435	0.123704	0.0618519	+	0.998085i	$0.480299\pi$
$348$	−34.3207	−1.83978
$349$	10.0945	0.540347	0.270173	−	0.962812i	$-0.412919\pi$
$349$	10.0945	0.540347	0.270173	+	0.962812i	$0.412919\pi$
$350$	0	0
$351$	−16.9770	−0.906163
$352$	−36.2732	−1.93337
$353$	20.0098	1.06501	0.532506	−	0.846426i	$-0.321250\pi$
$353$	20.0098	1.06501	0.532506	+	0.846426i	$0.321250\pi$
$354$	−7.50725	−0.399006
$355$	−50.2118	−2.66497
$356$	−73.7946	−3.91111
$357$	0	0
$358$	−32.4911	−1.71721
$359$	24.9914	1.31899	0.659497	−	0.751707i	$-0.270769\pi$
$359$	24.9914	1.31899	0.659497	+	0.751707i	$0.270769\pi$
$360$	−31.0519	−1.63658
$361$	8.80603	0.463475
$362$	30.7838	1.61796
$363$	15.9150	0.835320
$364$	0	0
$365$	36.2985	1.89995
$366$	−22.3046	−1.16588
$367$	−37.5916	−1.96226	−0.981132	−	0.193340i	$-0.938068\pi$
$367$	−37.5916	−1.96226	−0.981132	+	0.193340i	$0.938068\pi$
$368$	46.1558	2.40604
$369$	−12.6520	−0.658637
$370$	33.4208	1.73747
$371$	0	0
$372$	−24.8808	−1.29001
$373$	21.3793	1.10698	0.553489	−	0.832856i	$-0.313296\pi$
$373$	21.3793	1.10698	0.553489	+	0.832856i	$0.313296\pi$
$374$	−74.1846	−3.83599
$375$	14.6179	0.754863
$376$	67.6920	3.49095
$377$	−17.0067	−0.875887
$378$	0	0
$379$	−24.9688	−1.28256	−0.641279	−	0.767308i	$-0.721596\pi$
$379$	−24.9688	−1.28256	−0.641279	+	0.767308i	$0.721596\pi$
$380$	−88.3204	−4.53074
$381$	−1.31654	−0.0674483
$382$	−54.4918	−2.78804
$383$	−21.9425	−1.12121	−0.560605	−	0.828084i	$-0.689431\pi$
$383$	−21.9425	−1.12121	−0.560605	+	0.828084i	$0.689431\pi$
$384$	−9.54087	−0.486881
$385$	0	0
$386$	−52.1428	−2.65400
$387$	−5.07651	−0.258054
$388$	74.6792	3.79126
$389$	−8.14442	−0.412938	−0.206469	−	0.978453i	$-0.566197\pi$
$389$	−8.14442	−0.412938	−0.206469	+	0.978453i	$0.566197\pi$
$390$	37.0482	1.87601
$391$	33.7569	1.70716
$392$	0	0
$393$	−1.78591	−0.0900871
$394$	18.2821	0.921038
$395$	−61.9461	−3.11685
$396$	28.1979	1.41700
$397$	24.5214	1.23069	0.615347	−	0.788257i	$-0.289016\pi$
$397$	24.5214	1.23069	0.615347	+	0.788257i	$0.289016\pi$
$398$	42.9760	2.15419
$399$	0	0
$400$	66.1556	3.30778
$401$	17.6974	0.883763	0.441882	−	0.897073i	$-0.354311\pi$
$401$	17.6974	0.883763	0.441882	+	0.897073i	$0.354311\pi$
$402$	−17.3980	−0.867734
$403$	−12.3290	−0.614150
$404$	−69.8731	−3.47632
$405$	−12.9981	−0.645883
$406$	0	0
$407$	−17.2470	−0.854902
$408$	53.5122	2.64925
$409$	3.61653	0.178826	0.0894130	−	0.995995i	$-0.471501\pi$
$409$	3.61653	0.178826	0.0894130	+	0.995995i	$0.471501\pi$
$410$	92.9989	4.59289
$411$	10.4828	0.517079
$412$	−64.1088	−3.15842
$413$	0	0
$414$	−18.3705	−0.902861
$415$	21.6214	1.06135
$416$	22.8149	1.11859
$417$	−4.98634	−0.244182
$418$	65.2549	3.19172
$419$	7.39479	0.361259	0.180630	−	0.983551i	$-0.442186\pi$
$419$	7.39479	0.361259	0.180630	+	0.983551i	$0.442186\pi$
$420$	0	0
$421$	−14.6420	−0.713608	−0.356804	−	0.934179i	$-0.616134\pi$
$421$	−14.6420	−0.713608	−0.356804	+	0.934179i	$0.616134\pi$
$422$	45.5197	2.21587
$423$	−12.6465	−0.614896
$424$	6.80846	0.330648
$425$	48.3841	2.34697
$426$	−47.0900	−2.28152
$427$	0	0
$428$	67.0667	3.24179
$429$	−19.1189	−0.923070
$430$	37.3150	1.79949
$431$	−20.5154	−0.988194	−0.494097	−	0.869407i	$-0.664501\pi$
$431$	−20.5154	−0.988194	−0.494097	+	0.869407i	$0.664501\pi$
$432$	−46.0429	−2.21524
$433$	−15.2805	−0.734333	−0.367167	−	0.930155i	$-0.619672\pi$
$433$	−15.2805	−0.734333	−0.367167	+	0.930155i	$0.619672\pi$
$434$	0	0
$435$	−26.7841	−1.28420
$436$	23.9876	1.14880
$437$	−29.6935	−1.42043
$438$	34.0417	1.62657
$439$	12.3526	0.589556	0.294778	−	0.955566i	$-0.404754\pi$
$439$	12.3526	0.589556	0.294778	+	0.955566i	$0.404754\pi$
$440$	−117.789	−5.61535
$441$	0	0
$442$	46.6603	2.21940
$443$	23.3013	1.10708	0.553540	−	0.832823i	$-0.313277\pi$
$443$	23.3013	1.10708	0.553540	+	0.832823i	$0.313277\pi$
$444$	21.8919	1.03895
$445$	−57.5898	−2.73002
$446$	−6.04894	−0.286426
$447$	−13.8661	−0.655844
$448$	0	0
$449$	41.9961	1.98192	0.990959	−	0.134165i	$-0.0428351\pi$
$449$	41.9961	1.98192	0.990959	+	0.134165i	$0.0428351\pi$
$450$	−26.3307	−1.24124
$451$	−47.9926	−2.25988
$452$	−10.0834	−0.474281
$453$	−10.8110	−0.507944
$454$	−22.6690	−1.06391
$455$	0	0
$456$	−47.0708	−2.20429
$457$	16.9573	0.793231	0.396615	−	0.917985i	$-0.370185\pi$
$457$	16.9573	0.793231	0.396615	+	0.917985i	$0.370185\pi$
$458$	2.56302	0.119762
$459$	−33.6743	−1.57178
$460$	94.3157	4.39749
$461$	22.4259	1.04448	0.522239	−	0.852799i	$-0.325097\pi$
$461$	22.4259	1.04448	0.522239	+	0.852799i	$0.325097\pi$
$462$	0	0
$463$	4.90815	0.228101	0.114051	−	0.993475i	$-0.463617\pi$
$463$	4.90815	0.228101	0.114051	+	0.993475i	$0.463617\pi$
$464$	−46.1234	−2.14122
$465$	−19.4171	−0.900447
$466$	−43.4167	−2.01124
$467$	−10.0412	−0.464654	−0.232327	−	0.972638i	$-0.574634\pi$
$467$	−10.0412	−0.464654	−0.232327	+	0.972638i	$0.574634\pi$
$468$	−17.7358	−0.819837
$469$	0	0
$470$	92.9587	4.28787
$471$	−5.56185	−0.256276
$472$	−15.0124	−0.691003
$473$	−19.2566	−0.885420
$474$	−58.0948	−2.66838
$475$	−42.5601	−1.95279
$476$	0	0
$477$	−1.27199	−0.0582403
$478$	−13.2032	−0.603901
$479$	25.4444	1.16258	0.581291	−	0.813695i	$-0.302548\pi$
$479$	25.4444	1.16258	0.581291	+	0.813695i	$0.302548\pi$
$480$	35.9317	1.64005
$481$	10.8479	0.494624
$482$	−22.8401	−1.04034
$483$	0	0
$484$	56.0026	2.54557
$485$	58.2801	2.64636
$486$	31.2105	1.41574
$487$	16.0841	0.728839	0.364419	−	0.931235i	$-0.381267\pi$
$487$	16.0841	0.728839	0.364419	+	0.931235i	$0.381267\pi$
$488$	−44.6030	−2.01908
$489$	0.460820	0.0208390
$490$	0	0
$491$	−28.4648	−1.28460	−0.642300	−	0.766453i	$-0.722020\pi$
$491$	−28.4648	−1.28460	−0.642300	+	0.766453i	$0.722020\pi$
$492$	60.9179	2.74639
$493$	−33.7332	−1.51927
$494$	−41.0437	−1.84664
$495$	22.0058	0.989088
$496$	−33.4372	−1.50137
$497$	0	0
$498$	20.2772	0.908642
$499$	13.5099	0.604786	0.302393	−	0.953183i	$-0.402214\pi$
$499$	13.5099	0.604786	0.302393	+	0.953183i	$0.402214\pi$
$500$	51.4382	2.30039
$501$	−1.25739	−0.0561762
$502$	20.9184	0.933633
$503$	20.6323	0.919949	0.459975	−	0.887932i	$-0.347859\pi$
$503$	20.6323	0.919949	0.459975	+	0.887932i	$0.347859\pi$
$504$	0	0
$505$	−54.5294	−2.42653
$506$	−69.6845	−3.09785
$507$	−5.08967	−0.226040
$508$	−4.63271	−0.205543
$509$	1.46524	0.0649456	0.0324728	−	0.999473i	$-0.489662\pi$
$509$	1.46524	0.0649456	0.0324728	+	0.999473i	$0.489662\pi$
$510$	73.4861	3.25402
$511$	0	0
$512$	−49.2750	−2.17767
$513$	29.6209	1.30779
$514$	−60.3143	−2.66035
$515$	−50.0309	−2.20463
$516$	24.4428	1.07603
$517$	−47.9718	−2.10980
$518$	0	0
$519$	8.71874	0.382710
$520$	74.0861	3.24889
$521$	15.1230	0.662549	0.331275	−	0.943534i	$-0.392521\pi$
$521$	15.1230	0.662549	0.331275	+	0.943534i	$0.392521\pi$
$522$	18.3576	0.803491
$523$	29.7942	1.30281	0.651404	−	0.758731i	$-0.274180\pi$
$523$	29.7942	1.30281	0.651404	+	0.758731i	$0.274180\pi$
$524$	−6.28436	−0.274533
$525$	0	0
$526$	−23.0977	−1.00711
$527$	−24.4549	−1.06527
$528$	−51.8520	−2.25657
$529$	8.70915	0.378659
$530$	9.34978	0.406128
$531$	2.80469	0.121713
$532$	0	0
$533$	30.1862	1.30751
$534$	−54.0093	−2.33721
$535$	52.3392	2.26282
$536$	−34.7912	−1.50275
$537$	−16.6093	−0.716745
$538$	−33.2804	−1.43482
$539$	0	0
$540$	−94.0849	−4.04877
$541$	−9.71401	−0.417638	−0.208819	−	0.977954i	$-0.566962\pi$
$541$	−9.71401	−0.417638	−0.208819	+	0.977954i	$0.566962\pi$
$542$	19.8517	0.852705
$543$	15.7366	0.675321
$544$	45.2541	1.94025
$545$	18.7200	0.801879
$546$	0	0
$547$	10.6353	0.454732	0.227366	−	0.973809i	$-0.426989\pi$
$547$	10.6353	0.454732	0.227366	+	0.973809i	$0.426989\pi$
$548$	36.8876	1.57576
$549$	8.33294	0.355641
$550$	−99.8795	−4.25888
$551$	29.6727	1.26410
$552$	50.2661	2.13947
$553$	0	0
$554$	−21.2768	−0.903966
$555$	17.0846	0.725201
$556$	−17.5462	−0.744125
$557$	8.26750	0.350305	0.175153	−	0.984541i	$-0.443958\pi$
$557$	8.26750	0.350305	0.175153	+	0.984541i	$0.443958\pi$
$558$	13.3084	0.563388
$559$	12.1119	0.512280
$560$	0	0
$561$	−37.9229	−1.60111
$562$	−30.8576	−1.30165
$563$	19.6229	0.827008	0.413504	−	0.910502i	$-0.364305\pi$
$563$	19.6229	0.827008	0.413504	+	0.910502i	$0.364305\pi$
$564$	60.8916	2.56400
$565$	−7.86912	−0.331056
$566$	15.4028	0.647428
$567$	0	0
$568$	−94.1671	−3.95116
$569$	36.0303	1.51047	0.755235	−	0.655454i	$-0.227523\pi$
$569$	36.0303	1.51047	0.755235	+	0.655454i	$0.227523\pi$
$570$	−64.6405	−2.70749
$571$	21.8466	0.914250	0.457125	−	0.889402i	$-0.348879\pi$
$571$	21.8466	0.914250	0.457125	+	0.889402i	$0.348879\pi$
$572$	−67.2768	−2.81298
$573$	−27.8560	−1.16370
$574$	0	0
$575$	45.4491	1.89536
$576$	−3.86156	−0.160898
$577$	19.8478	0.826274	0.413137	−	0.910669i	$-0.364433\pi$
$577$	19.8478	0.826274	0.413137	+	0.910669i	$0.364433\pi$
$578$	48.7700	2.02857
$579$	−26.6552	−1.10775
$580$	−94.2494	−3.91349
$581$	0	0
$582$	54.6567	2.26559
$583$	−4.82500	−0.199831
$584$	68.0740	2.81692
$585$	−13.8411	−0.572260
$586$	−24.4486	−1.00996
$587$	−2.95082	−0.121793	−0.0608966	−	0.998144i	$-0.519396\pi$
$587$	−2.95082	−0.121793	−0.0608966	+	0.998144i	$0.519396\pi$
$588$	0	0
$589$	21.5112	0.886354
$590$	−20.6160	−0.848746
$591$	9.34573	0.384432
$592$	29.4205	1.20917
$593$	16.6183	0.682432	0.341216	−	0.939985i	$-0.389161\pi$
$593$	16.6183	0.682432	0.341216	+	0.939985i	$0.389161\pi$
$594$	69.5139	2.85219
$595$	0	0
$596$	−48.7929	−1.99863
$597$	21.9692	0.899139
$598$	43.8298	1.79233
$599$	−38.1190	−1.55750	−0.778749	−	0.627335i	$-0.784146\pi$
$599$	−38.1190	−1.55750	−0.778749	+	0.627335i	$0.784146\pi$
$600$	72.0469	2.94130
$601$	−23.5289	−0.959763	−0.479881	−	0.877333i	$-0.659320\pi$
$601$	−23.5289	−0.959763	−0.479881	+	0.877333i	$0.659320\pi$
$602$	0	0
$603$	6.49986	0.264695
$604$	−38.0423	−1.54792
$605$	43.7048	1.77685
$606$	−51.1392	−2.07739
$607$	24.9940	1.01448	0.507239	−	0.861806i	$-0.330666\pi$
$607$	24.9940	1.01448	0.507239	+	0.861806i	$0.330666\pi$
$608$	−39.8068	−1.61438
$609$	0	0
$610$	−61.2514	−2.48000
$611$	30.1731	1.22067
$612$	−35.1794	−1.42204
$613$	−24.4873	−0.989033	−0.494516	−	0.869168i	$-0.664655\pi$
$613$	−24.4873	−0.989033	−0.494516	+	0.869168i	$0.664655\pi$
$614$	52.5089	2.11909
$615$	47.5407	1.91703
$616$	0	0
$617$	25.5183	1.02733	0.513664	−	0.857992i	$-0.328288\pi$
$617$	25.5183	1.02733	0.513664	+	0.857992i	$0.328288\pi$
$618$	−46.9204	−1.88741
$619$	8.02886	0.322707	0.161354	−	0.986897i	$-0.448414\pi$
$619$	8.02886	0.322707	0.161354	+	0.986897i	$0.448414\pi$
$620$	−68.3261	−2.74404
$621$	−31.6316	−1.26933
$622$	−34.5764	−1.38639
$623$	0	0
$624$	32.6136	1.30559
$625$	−0.212826	−0.00851305
$626$	21.2177	0.848031
$627$	33.3581	1.33219
$628$	−19.5714	−0.780983
$629$	21.5172	0.857947
$630$	0	0
$631$	14.9596	0.595534	0.297767	−	0.954639i	$-0.403758\pi$
$631$	14.9596	0.595534	0.297767	+	0.954639i	$0.403758\pi$
$632$	−116.174	−4.62113
$633$	23.2695	0.924881
$634$	48.7645	1.93669
$635$	−3.61540	−0.143473
$636$	6.12447	0.242851
$637$	0	0
$638$	69.6355	2.75690
$639$	17.5927	0.695958
$640$	−26.2006	−1.03567
$641$	40.7640	1.61008	0.805040	−	0.593220i	$-0.202144\pi$
$641$	40.7640	1.61008	0.805040	+	0.593220i	$0.202144\pi$
$642$	49.0852	1.93724
$643$	−36.2589	−1.42991	−0.714956	−	0.699170i	$-0.753553\pi$
$643$	−36.2589	−1.42991	−0.714956	+	0.699170i	$0.753553\pi$
$644$	0	0
$645$	19.0753	0.751089
$646$	−81.4114	−3.20309
$647$	−2.06513	−0.0811887	−0.0405943	−	0.999176i	$-0.512925\pi$
$647$	−2.06513	−0.0811887	−0.0405943	+	0.999176i	$0.512925\pi$
$648$	−24.3767	−0.957606
$649$	10.6390	0.417617
$650$	62.8218	2.46407
$651$	0	0
$652$	1.62156	0.0635053
$653$	23.6696	0.926263	0.463131	−	0.886290i	$-0.346726\pi$
$653$	23.6696	0.926263	0.463131	+	0.886290i	$0.346726\pi$
$654$	17.5562	0.686501
$655$	−4.90435	−0.191629
$656$	81.8672	3.19638
$657$	−12.7179	−0.496173
$658$	0	0
$659$	50.5108	1.96762	0.983812	−	0.179206i	$-0.0573528\pi$
$659$	50.5108	1.96762	0.983812	+	0.179206i	$0.0573528\pi$
$660$	−105.955	−4.12431
$661$	9.85995	0.383508	0.191754	−	0.981443i	$-0.438582\pi$
$661$	9.85995	0.383508	0.191754	+	0.981443i	$0.438582\pi$
$662$	28.1389	1.09365
$663$	23.8526	0.926357
$664$	40.5488	1.57360
$665$	0	0
$666$	−11.7097	−0.453740
$667$	−31.6869	−1.22692
$668$	−4.42460	−0.171193
$669$	−3.09220	−0.119551
$670$	−47.7774	−1.84580
$671$	31.6091	1.22026
$672$	0	0
$673$	33.6849	1.29846	0.649229	−	0.760593i	$-0.275092\pi$
$673$	33.6849	1.29846	0.649229	+	0.760593i	$0.275092\pi$
$674$	49.4683	1.90545
$675$	−45.3379	−1.74506
$676$	−17.9098	−0.688840
$677$	13.9184	0.534927	0.267464	−	0.963568i	$-0.413814\pi$
$677$	13.9184	0.534927	0.267464	+	0.963568i	$0.413814\pi$
$678$	−7.37987	−0.283422
$679$	0	0
$680$	146.952	5.63535
$681$	−11.5883	−0.444065
$682$	50.4823	1.93307
$683$	0.478159	0.0182963	0.00914813	−	0.999958i	$-0.497088\pi$
$683$	0.478159	0.0182963	0.00914813	+	0.999958i	$0.497088\pi$
$684$	30.9448	1.18321
$685$	28.7873	1.09991
$686$	0	0
$687$	1.31021	0.0499875
$688$	32.8485	1.25234
$689$	3.03481	0.115617
$690$	69.0283	2.62786
$691$	23.8010	0.905435	0.452717	−	0.891654i	$-0.350455\pi$
$691$	23.8010	0.905435	0.452717	+	0.891654i	$0.350455\pi$
$692$	30.6800	1.16628
$693$	0	0
$694$	5.93463	0.225275
$695$	−13.6932	−0.519412
$696$	−50.2308	−1.90399
$697$	59.8751	2.26793
$698$	25.9974	0.984018
$699$	−22.1945	−0.839471
$700$	0	0
$701$	12.3729	0.467320	0.233660	−	0.972318i	$-0.424930\pi$
$701$	12.3729	0.467320	0.233660	+	0.972318i	$0.424930\pi$
$702$	−43.7226	−1.65020
$703$	−18.9271	−0.713851
$704$	−14.6480	−0.552066
$705$	47.5202	1.78971
$706$	51.5333	1.93948
$707$	0	0
$708$	−13.5043	−0.507521
$709$	−6.03901	−0.226800	−0.113400	−	0.993549i	$-0.536174\pi$
$709$	−6.03901	−0.226800	−0.113400	+	0.993549i	$0.536174\pi$
$710$	−129.316	−4.85314
$711$	21.7041	0.813967
$712$	−108.004	−4.04761
$713$	−22.9714	−0.860286
$714$	0	0
$715$	−52.5032	−1.96351
$716$	−58.4459	−2.18423
$717$	−6.74944	−0.252062
$718$	64.3629	2.40200
$719$	−38.8243	−1.44790	−0.723951	−	0.689852i	$-0.757676\pi$
$719$	−38.8243	−1.44790	−0.723951	+	0.689852i	$0.757676\pi$
$720$	−37.5382	−1.39897
$721$	0	0
$722$	22.6791	0.844029
$723$	−11.6758	−0.434227
$724$	55.3748	2.05799
$725$	−45.4172	−1.68675
$726$	40.9875	1.52119
$727$	45.7266	1.69591	0.847953	−	0.530071i	$-0.177835\pi$
$727$	45.7266	1.69591	0.847953	+	0.530071i	$0.177835\pi$
$728$	0	0
$729$	26.7404	0.990384
$730$	93.4832	3.45997
$731$	24.0244	0.888573
$732$	−40.1221	−1.48295
$733$	−26.3784	−0.974309	−0.487155	−	0.873316i	$-0.661965\pi$
$733$	−26.3784	−0.974309	−0.487155	+	0.873316i	$0.661965\pi$
$734$	−96.8135	−3.57345
$735$	0	0
$736$	42.5089	1.56690
$737$	24.6558	0.908208
$738$	−32.5840	−1.19944
$739$	−44.0186	−1.61925	−0.809625	−	0.586948i	$-0.800329\pi$
$739$	−44.0186	−1.61925	−0.809625	+	0.586948i	$0.800329\pi$
$740$	60.1184	2.20999
$741$	−20.9814	−0.770771
$742$	0	0
$743$	−39.6817	−1.45578	−0.727891	−	0.685693i	$-0.759499\pi$
$743$	−39.6817	−1.45578	−0.727891	+	0.685693i	$0.759499\pi$
$744$	−36.4148	−1.33503
$745$	−38.0782	−1.39508
$746$	55.0604	2.01590
$747$	−7.57551	−0.277173
$748$	−133.445	−4.87924
$749$	0	0
$750$	37.6469	1.37467
$751$	−40.3374	−1.47193	−0.735965	−	0.677019i	$-0.763271\pi$
$751$	−40.3374	−1.47193	−0.735965	+	0.677019i	$0.763271\pi$
$752$	81.8318	2.98410
$753$	10.6934	0.389689
$754$	−43.7990	−1.59507
$755$	−29.6885	−1.08047
$756$	0	0
$757$	31.1435	1.13193	0.565965	−	0.824429i	$-0.308504\pi$
$757$	31.1435	1.13193	0.565965	+	0.824429i	$0.308504\pi$
$758$	−64.3047	−2.33565
$759$	−35.6224	−1.29301
$760$	−129.263	−4.68887
$761$	−1.19795	−0.0434255	−0.0217128	−	0.999764i	$-0.506912\pi$
$761$	−1.19795	−0.0434255	−0.0217128	+	0.999764i	$0.506912\pi$
$762$	−3.39062	−0.122829
$763$	0	0
$764$	−98.0214	−3.54629
$765$	−27.4542	−0.992610
$766$	−56.5108	−2.04182
$767$	−6.69166	−0.241622
$768$	−32.5984	−1.17629
$769$	−9.53802	−0.343950	−0.171975	−	0.985101i	$-0.555015\pi$
$769$	−9.53802	−0.343950	−0.171975	+	0.985101i	$0.555015\pi$
$770$	0	0
$771$	−30.8324	−1.11040
$772$	−93.7960	−3.37579
$773$	27.3573	0.983974	0.491987	−	0.870603i	$-0.336271\pi$
$773$	27.3573	0.983974	0.491987	+	0.870603i	$0.336271\pi$
$774$	−13.0741	−0.469938
$775$	−32.9252	−1.18271
$776$	109.298	3.92358
$777$	0	0
$778$	−20.9752	−0.751996
$779$	−52.6678	−1.88702
$780$	66.6433	2.38621
$781$	66.7341	2.38794
$782$	86.9376	3.10888
$783$	31.6094	1.12963
$784$	0	0
$785$	−15.2736	−0.545139
$786$	−4.59944	−0.164056
$787$	−19.7641	−0.704514	−0.352257	−	0.935903i	$-0.614586\pi$
$787$	−19.7641	−0.704514	−0.352257	+	0.935903i	$0.614586\pi$
$788$	32.8863	1.17153
$789$	−11.8074	−0.420356
$790$	−159.536	−5.67605
$791$	0	0
$792$	41.2696	1.46645
$793$	−19.8814	−0.706008
$794$	63.1525	2.24120
$795$	4.77957	0.169514
$796$	77.3065	2.74005
$797$	48.1120	1.70422	0.852108	−	0.523366i	$-0.175324\pi$
$797$	48.1120	1.70422	0.852108	+	0.523366i	$0.175324\pi$
$798$	0	0
$799$	59.8492	2.11731
$800$	60.9285	2.15415
$801$	20.1777	0.712946
$802$	45.5779	1.60941
$803$	−48.2425	−1.70244
$804$	−31.2961	−1.10373
$805$	0	0
$806$	−31.7521	−1.11842
$807$	−17.0128	−0.598880
$808$	−102.264	−3.59764
$809$	−45.1368	−1.58693	−0.793463	−	0.608618i	$-0.791724\pi$
$809$	−45.1368	−1.58693	−0.793463	+	0.608618i	$0.791724\pi$
$810$	−33.4755	−1.17621
$811$	44.9022	1.57673	0.788366	−	0.615207i	$-0.210928\pi$
$811$	44.9022	1.57673	0.788366	+	0.615207i	$0.210928\pi$
$812$	0	0
$813$	10.1481	0.355911
$814$	−44.4180	−1.55685
$815$	1.26548	0.0443277
$816$	64.6900	2.26460
$817$	−21.1325	−0.739333
$818$	9.31403	0.325658
$819$	0	0
$820$	167.289	5.84199
$821$	−47.1163	−1.64437	−0.822185	−	0.569221i	$-0.807245\pi$
$821$	−47.1163	−1.64437	−0.822185	+	0.569221i	$0.807245\pi$
$822$	26.9975	0.941646
$823$	−3.85102	−0.134238	−0.0671191	−	0.997745i	$-0.521381\pi$
$823$	−3.85102	−0.134238	−0.0671191	+	0.997745i	$0.521381\pi$
$824$	−93.8278	−3.26865
$825$	−51.0580	−1.77761
$826$	0	0
$827$	−3.78077	−0.131470	−0.0657351	−	0.997837i	$-0.520939\pi$
$827$	−3.78077	−0.131470	−0.0657351	+	0.997837i	$0.520939\pi$
$828$	−33.0454	−1.14841
$829$	23.5958	0.819518	0.409759	−	0.912194i	$-0.365613\pi$
$829$	23.5958	0.819518	0.409759	+	0.912194i	$0.365613\pi$
$830$	55.6840	1.93282
$831$	−10.8766	−0.377306
$832$	9.21321	0.319411
$833$	0	0
$834$	−12.8418	−0.444676
$835$	−3.45298	−0.119495
$836$	117.382	4.05975
$837$	22.9152	0.792065
$838$	19.0446	0.657884
$839$	25.2441	0.871522	0.435761	−	0.900062i	$-0.356479\pi$
$839$	25.2441	0.871522	0.435761	+	0.900062i	$0.356479\pi$
$840$	0	0
$841$	2.66464	0.0918840
$842$	−37.7091	−1.29954
$843$	−15.7743	−0.543296
$844$	81.8822	2.81850
$845$	−13.9769	−0.480821
$846$	−32.5700	−1.11978
$847$	0	0
$848$	8.23064	0.282641
$849$	7.87386	0.270230
$850$	124.609	4.27404
$851$	20.2119	0.692856
$852$	−84.7069	−2.90201
$853$	1.23491	0.0422826	0.0211413	−	0.999776i	$-0.493270\pi$
$853$	1.23491	0.0422826	0.0211413	+	0.999776i	$0.493270\pi$
$854$	0	0
$855$	24.1495	0.825897
$856$	98.1568	3.35493
$857$	−21.9609	−0.750169	−0.375085	−	0.926991i	$-0.622386\pi$
$857$	−21.9609	−0.750169	−0.375085	+	0.926991i	$0.622386\pi$
$858$	−49.2389	−1.68099
$859$	−17.3623	−0.592395	−0.296197	−	0.955127i	$-0.595719\pi$
$859$	−17.3623	−0.592395	−0.296197	+	0.955127i	$0.595719\pi$
$860$	67.1233	2.28889
$861$	0	0
$862$	−52.8356	−1.79959
$863$	47.6552	1.62220	0.811102	−	0.584905i	$-0.198868\pi$
$863$	47.6552	1.62220	0.811102	+	0.584905i	$0.198868\pi$
$864$	−42.4049	−1.44264
$865$	23.9429	0.814082
$866$	−39.3535	−1.33728
$867$	24.9311	0.846703
$868$	0	0
$869$	82.3296	2.79284
$870$	−68.9799	−2.33864
$871$	−15.5079	−0.525465
$872$	35.1075	1.18889
$873$	−20.4196	−0.691099
$874$	−76.4729	−2.58673
$875$	0	0
$876$	61.2352	2.06894
$877$	38.2044	1.29007	0.645035	−	0.764153i	$-0.276843\pi$
$877$	38.2044	1.29007	0.645035	+	0.764153i	$0.276843\pi$
$878$	31.8129	1.07363
$879$	−12.4980	−0.421548
$880$	−142.393	−4.80006
$881$	−19.8672	−0.669342	−0.334671	−	0.942335i	$-0.608625\pi$
$881$	−19.8672	−0.669342	−0.334671	+	0.942335i	$0.608625\pi$
$882$	0	0
$883$	9.70573	0.326624	0.163312	−	0.986574i	$-0.447782\pi$
$883$	9.70573	0.326624	0.163312	+	0.986574i	$0.447782\pi$
$884$	83.9338	2.82300
$885$	−10.5388	−0.354258
$886$	60.0104	2.01609
$887$	44.4765	1.49338	0.746688	−	0.665175i	$-0.231643\pi$
$887$	44.4765	1.49338	0.746688	+	0.665175i	$0.231643\pi$
$888$	32.0404	1.07521
$889$	0	0
$890$	−148.317	−4.97160
$891$	17.2752	0.578741
$892$	−10.8810	−0.364323
$893$	−52.6451	−1.76170
$894$	−35.7108	−1.19435
$895$	−45.6115	−1.52463
$896$	0	0
$897$	22.4056	0.748103
$898$	108.157	3.60924
$899$	22.9553	0.765602
$900$	−47.3643	−1.57881
$901$	6.01963	0.200543
$902$	−123.600	−4.11544
$903$	0	0
$904$	−14.7577	−0.490834
$905$	43.2149	1.43651
$906$	−27.8427	−0.925011
$907$	−24.0803	−0.799572	−0.399786	−	0.916608i	$-0.630916\pi$
$907$	−24.0803	−0.799572	−0.399786	+	0.916608i	$0.630916\pi$
$908$	−40.7776	−1.35325
$909$	19.1055	0.633689
$910$	0	0
$911$	1.68964	0.0559802	0.0279901	−	0.999608i	$-0.491089\pi$
$911$	1.68964	0.0559802	0.0279901	+	0.999608i	$0.491089\pi$
$912$	−56.9032	−1.88425
$913$	−28.7360	−0.951023
$914$	43.6720	1.44454
$915$	−31.3115	−1.03513
$916$	4.61043	0.152333
$917$	0	0
$918$	−86.7249	−2.86235
$919$	40.5462	1.33750	0.668748	−	0.743489i	$-0.266831\pi$
$919$	40.5462	1.33750	0.668748	+	0.743489i	$0.266831\pi$
$920$	138.038	4.55097
$921$	26.8424	0.884486
$922$	57.7558	1.90208
$923$	−41.9741	−1.38160
$924$	0	0
$925$	28.9700	0.952528
$926$	12.6405	0.415392
$927$	17.5294	0.575739
$928$	−42.4791	−1.39444
$929$	33.1179	1.08656	0.543281	−	0.839551i	$-0.317182\pi$
$929$	33.1179	1.08656	0.543281	+	0.839551i	$0.317182\pi$
$930$	−50.0070	−1.63979
$931$	0	0
$932$	−78.0992	−2.55822
$933$	−17.6754	−0.578665
$934$	−25.8603	−0.846174
$935$	−104.142	−3.40579
$936$	−25.9576	−0.848450
$937$	−29.0490	−0.948991	−0.474495	−	0.880258i	$-0.657369\pi$
$937$	−29.0490	−0.948991	−0.474495	+	0.880258i	$0.657369\pi$
$938$	0	0
$939$	10.8464	0.353960
$940$	167.217	5.45401
$941$	−9.45338	−0.308171	−0.154086	−	0.988057i	$-0.549243\pi$
$941$	−9.45338	−0.308171	−0.154086	+	0.988057i	$0.549243\pi$
$942$	−14.3240	−0.466702
$943$	56.2430	1.83152
$944$	−18.1483	−0.590677
$945$	0	0
$946$	−49.5936	−1.61243
$947$	22.3098	0.724970	0.362485	−	0.931990i	$-0.381928\pi$
$947$	22.3098	0.724970	0.362485	+	0.931990i	$0.381928\pi$
$948$	−104.503	−3.39408
$949$	30.3434	0.984987
$950$	−109.609	−3.55620
$951$	24.9282	0.808354
$952$	0	0
$953$	−49.6079	−1.60696	−0.803478	−	0.595334i	$-0.797020\pi$
$953$	−49.6079	−1.60696	−0.803478	+	0.595334i	$0.797020\pi$
$954$	−3.27588	−0.106061
$955$	−76.4965	−2.47537
$956$	−23.7503	−0.768141
$957$	35.5974	1.15070
$958$	65.5296	2.11716
$959$	0	0
$960$	14.5100	0.468310
$961$	−14.3586	−0.463180
$962$	27.9378	0.900752
$963$	−18.3381	−0.590938
$964$	−41.0854	−1.32327
$965$	−73.1990	−2.35636
$966$	0	0
$967$	44.5415	1.43236	0.716179	−	0.697916i	$-0.245889\pi$
$967$	44.5415	1.43236	0.716179	+	0.697916i	$0.245889\pi$
$968$	81.9638	2.63442
$969$	−41.6172	−1.33694
$970$	150.095	4.81926
$971$	−10.9162	−0.350318	−0.175159	−	0.984540i	$-0.556044\pi$
$971$	−10.9162	−0.350318	−0.175159	+	0.984540i	$0.556044\pi$
$972$	56.1424	1.80077
$973$	0	0
$974$	41.4230	1.32728
$975$	32.1143	1.02848
$976$	−53.9198	−1.72593
$977$	47.4189	1.51706	0.758532	−	0.651636i	$-0.225917\pi$
$977$	47.4189	1.51706	0.758532	+	0.651636i	$0.225917\pi$
$978$	1.18680	0.0379497
$979$	76.5398	2.44622
$980$	0	0
$981$	−6.55895	−0.209411
$982$	−73.3085	−2.33937
$983$	27.6546	0.882046	0.441023	−	0.897496i	$-0.354616\pi$
$983$	27.6546	0.882046	0.441023	+	0.897496i	$0.354616\pi$
$984$	89.1576	2.84224
$985$	25.6647	0.817745
$986$	−86.8766	−2.76671
$987$	0	0
$988$	−73.8306	−2.34887
$989$	22.5670	0.717589
$990$	56.6739	1.80121
$991$	−9.83870	−0.312537	−0.156268	−	0.987715i	$-0.549946\pi$
$991$	−9.83870	−0.312537	−0.156268	+	0.987715i	$0.549946\pi$
$992$	−30.7952	−0.977749
$993$	14.3845	0.456478
$994$	0	0
$995$	60.3304	1.91260
$996$	36.4752	1.15576
$997$	−52.0999	−1.65002	−0.825010	−	0.565119i	$-0.808830\pi$
$997$	−52.0999	−1.65002	−0.825010	+	0.565119i	$0.808830\pi$
$998$	34.7935	1.10137
$999$	−20.1625	−0.637913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.16		16
7.6	odd	2		889.2.a.c.1.16	✓	16
21.20	even	2		8001.2.a.t.1.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.16	✓	16	7.6	odd	2
6223.2.a.k.1.16		16	1.1	even	1	trivial
8001.2.a.t.1.1		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)

\(f(q)\)	\(=\)	\(q+2.57541 q^{2} +1.31654 q^{3} +4.63271 q^{4} +3.61540 q^{5} +3.39062 q^{6} +6.78030 q^{8} -1.26673 q^{9} +9.31112 q^{10} -4.80505 q^{11} +6.09914 q^{12} +3.02226 q^{13} +4.75981 q^{15} +8.19660 q^{16} +5.99473 q^{17} -3.26234 q^{18} -5.27314 q^{19} +16.7491 q^{20} -12.3750 q^{22} +5.63109 q^{23} +8.92653 q^{24} +8.07110 q^{25} +7.78354 q^{26} -5.61731 q^{27} -5.62713 q^{29} +12.2584 q^{30} -4.07939 q^{31} +7.54897 q^{32} -6.32603 q^{33} +15.4389 q^{34} -5.86839 q^{36} +3.58935 q^{37} -13.5805 q^{38} +3.97892 q^{39} +24.5135 q^{40} +9.98794 q^{41} +4.00758 q^{43} -22.2604 q^{44} -4.57973 q^{45} +14.5023 q^{46} +9.98363 q^{47} +10.7911 q^{48} +20.7864 q^{50} +7.89230 q^{51} +14.0013 q^{52} +1.00415 q^{53} -14.4668 q^{54} -17.3722 q^{55} -6.94229 q^{57} -14.4922 q^{58} -2.21412 q^{59} +22.0508 q^{60} -6.57832 q^{61} -10.5061 q^{62} +3.04845 q^{64} +10.9267 q^{65} -16.2921 q^{66} -5.13122 q^{67} +27.7719 q^{68} +7.41354 q^{69} -13.8883 q^{71} -8.58880 q^{72} +10.0400 q^{73} +9.24403 q^{74} +10.6259 q^{75} -24.4290 q^{76} +10.2473 q^{78} -17.1340 q^{79} +29.6340 q^{80} -3.59522 q^{81} +25.7230 q^{82} +5.98037 q^{83} +21.6733 q^{85} +10.3211 q^{86} -7.40834 q^{87} -32.5797 q^{88} -15.9290 q^{89} -11.7947 q^{90} +26.0872 q^{92} -5.37067 q^{93} +25.7119 q^{94} -19.0645 q^{95} +9.93851 q^{96} +16.1200 q^{97} +6.08669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 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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