LMFDB - Embedded newform 6223.2.a.k.1.6

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			1.6
Root			$1.24549$ 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)

$f(q)$	$=$	\(q-1.24549 q^{2} -3.30309 q^{3} -0.448744 q^{4} +2.43368 q^{5} +4.11399 q^{6} +3.04990 q^{8} +7.91044 q^{9} -3.03114 q^{10} -4.22568 q^{11} +1.48224 q^{12} +5.67912 q^{13} -8.03869 q^{15} -2.90114 q^{16} +1.72874 q^{17} -9.85240 q^{18} +2.50366 q^{19} -1.09210 q^{20} +5.26306 q^{22} -3.83361 q^{23} -10.0741 q^{24} +0.922820 q^{25} -7.07331 q^{26} -16.2196 q^{27} +5.96884 q^{29} +10.0121 q^{30} +5.47757 q^{31} -2.48644 q^{32} +13.9578 q^{33} -2.15314 q^{34} -3.54976 q^{36} +6.12052 q^{37} -3.11830 q^{38} -18.7587 q^{39} +7.42249 q^{40} +4.31444 q^{41} +4.93422 q^{43} +1.89625 q^{44} +19.2515 q^{45} +4.77474 q^{46} +10.9899 q^{47} +9.58275 q^{48} -1.14937 q^{50} -5.71020 q^{51} -2.54847 q^{52} -4.00048 q^{53} +20.2015 q^{54} -10.2840 q^{55} -8.26984 q^{57} -7.43416 q^{58} +14.3016 q^{59} +3.60731 q^{60} -3.21888 q^{61} -6.82229 q^{62} +8.89913 q^{64} +13.8212 q^{65} -17.3844 q^{66} +7.39502 q^{67} -0.775763 q^{68} +12.6628 q^{69} +7.63071 q^{71} +24.1260 q^{72} -7.53588 q^{73} -7.62307 q^{74} -3.04816 q^{75} -1.12350 q^{76} +23.3638 q^{78} -15.4421 q^{79} -7.06046 q^{80} +29.8437 q^{81} -5.37361 q^{82} +12.7925 q^{83} +4.20722 q^{85} -6.14555 q^{86} -19.7157 q^{87} -12.8879 q^{88} -3.88584 q^{89} -23.9776 q^{90} +1.72031 q^{92} -18.0929 q^{93} -13.6879 q^{94} +6.09313 q^{95} +8.21294 q^{96} -4.24345 q^{97} -33.4270 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.24549	−0.880698	−0.440349	−	0.897827i	$-0.645145\pi$
$2$	−1.24549	−0.880698	−0.440349	+	0.897827i	$0.645145\pi$
$3$	−3.30309	−1.90704	−0.953521	−	0.301325i	$-0.902571\pi$
$3$	−3.30309	−1.90704	−0.953521	+	0.301325i	$0.902571\pi$
$4$	−0.448744	−0.224372
$5$	2.43368	1.08838	0.544188	−	0.838963i	$-0.316838\pi$
$5$	2.43368	1.08838	0.544188	+	0.838963i	$0.316838\pi$
$6$	4.11399	1.67953
$7$	0	0
$7$	0	0
$8$	3.04990	1.07830
$9$	7.91044	2.63681
$10$	−3.03114	−0.958531
$11$	−4.22568	−1.27409	−0.637046	−	0.770826i	$-0.719844\pi$
$11$	−4.22568	−1.27409	−0.637046	+	0.770826i	$0.719844\pi$
$12$	1.48224	0.427887
$13$	5.67912	1.57510	0.787552	−	0.616248i	$-0.211348\pi$
$13$	5.67912	1.57510	0.787552	+	0.616248i	$0.211348\pi$
$14$	0	0
$15$	−8.03869	−2.07558
$16$	−2.90114	−0.725285
$17$	1.72874	0.419282	0.209641	−	0.977778i	$-0.432771\pi$
$17$	1.72874	0.419282	0.209641	+	0.977778i	$0.432771\pi$
$18$	−9.85240	−2.32223
$19$	2.50366	0.574380	0.287190	−	0.957874i	$-0.407279\pi$
$19$	2.50366	0.574380	0.287190	+	0.957874i	$0.407279\pi$
$20$	−1.09210	−0.244201
$21$	0	0
$22$	5.26306	1.12209
$23$	−3.83361	−0.799364	−0.399682	−	0.916654i	$-0.630879\pi$
$23$	−3.83361	−0.799364	−0.399682	+	0.916654i	$0.630879\pi$
$24$	−10.0741	−2.05637
$25$	0.922820	0.184564
$26$	−7.07331	−1.38719
$27$	−16.2196	−3.12147
$28$	0	0
$29$	5.96884	1.10839	0.554193	−	0.832388i	$-0.313027\pi$
$29$	5.96884	1.10839	0.554193	+	0.832388i	$0.313027\pi$
$30$	10.0121	1.82796
$31$	5.47757	0.983801	0.491901	−	0.870651i	$-0.336302\pi$
$31$	5.47757	0.983801	0.491901	+	0.870651i	$0.336302\pi$
$32$	−2.48644	−0.439544
$33$	13.9578	2.42975
$34$	−2.15314	−0.369260
$35$	0	0
$36$	−3.54976	−0.591626
$37$	6.12052	1.00621	0.503103	−	0.864226i	$-0.332191\pi$
$37$	6.12052	1.00621	0.503103	+	0.864226i	$0.332191\pi$
$38$	−3.11830	−0.505855
$39$	−18.7587	−3.00379
$40$	7.42249	1.17360
$41$	4.31444	0.673802	0.336901	−	0.941540i	$-0.390621\pi$
$41$	4.31444	0.673802	0.336901	+	0.941540i	$0.390621\pi$
$42$	0	0
$43$	4.93422	0.752462	0.376231	−	0.926526i	$-0.377220\pi$
$43$	4.93422	0.752462	0.376231	+	0.926526i	$0.377220\pi$
$44$	1.89625	0.285870
$45$	19.2515	2.86984
$46$	4.77474	0.703998
$47$	10.9899	1.60304	0.801521	−	0.597966i	$-0.204024\pi$
$47$	10.9899	1.60304	0.801521	+	0.597966i	$0.204024\pi$
$48$	9.58275	1.38315
$49$	0	0
$50$	−1.14937	−0.162545
$51$	−5.71020	−0.799588
$52$	−2.54847	−0.353409
$53$	−4.00048	−0.549508	−0.274754	−	0.961514i	$-0.588596\pi$
$53$	−4.00048	−0.549508	−0.274754	+	0.961514i	$0.588596\pi$
$54$	20.2015	2.74907
$55$	−10.2840	−1.38669
$56$	0	0
$57$	−8.26984	−1.09537
$58$	−7.43416	−0.976153
$59$	14.3016	1.86191	0.930955	−	0.365133i	$-0.118977\pi$
$59$	14.3016	1.86191	0.930955	+	0.365133i	$0.118977\pi$
$60$	3.60731	0.465702
$61$	−3.21888	−0.412136	−0.206068	−	0.978538i	$-0.566067\pi$
$61$	−3.21888	−0.412136	−0.206068	+	0.978538i	$0.566067\pi$
$62$	−6.82229	−0.866431
$63$	0	0
$64$	8.89913	1.11239
$65$	13.8212	1.71431
$66$	−17.3844	−2.13987
$67$	7.39502	0.903445	0.451723	−	0.892158i	$-0.350810\pi$
$67$	7.39502	0.903445	0.451723	+	0.892158i	$0.350810\pi$
$68$	−0.775763	−0.0940750
$69$	12.6628	1.52442
$70$	0	0
$71$	7.63071	0.905598	0.452799	−	0.891612i	$-0.350425\pi$
$71$	7.63071	0.905598	0.452799	+	0.891612i	$0.350425\pi$
$72$	24.1260	2.84328
$73$	−7.53588	−0.882008	−0.441004	−	0.897505i	$-0.645378\pi$
$73$	−7.53588	−0.882008	−0.441004	+	0.897505i	$0.645378\pi$
$74$	−7.62307	−0.886164
$75$	−3.04816	−0.351972
$76$	−1.12350	−0.128875
$77$	0	0
$78$	23.3638	2.64543
$79$	−15.4421	−1.73737	−0.868685	−	0.495365i	$-0.835034\pi$
$79$	−15.4421	−1.73737	−0.868685	+	0.495365i	$0.835034\pi$
$80$	−7.06046	−0.789384
$81$	29.8437	3.31597
$82$	−5.37361	−0.593416
$83$	12.7925	1.40415	0.702077	−	0.712101i	$-0.252256\pi$
$83$	12.7925	1.40415	0.702077	+	0.712101i	$0.252256\pi$
$84$	0	0
$85$	4.20722	0.456337
$86$	−6.14555	−0.662691
$87$	−19.7157	−2.11374
$88$	−12.8879	−1.37385
$89$	−3.88584	−0.411898	−0.205949	−	0.978563i	$-0.566028\pi$
$89$	−3.88584	−0.411898	−0.205949	+	0.978563i	$0.566028\pi$
$90$	−23.9776	−2.52747
$91$	0	0
$92$	1.72031	0.179355
$93$	−18.0929	−1.87615
$94$	−13.6879	−1.41180
$95$	6.09313	0.625142
$96$	8.21294	0.838229
$97$	−4.24345	−0.430857	−0.215429	−	0.976520i	$-0.569115\pi$
$97$	−4.24345	−0.430857	−0.215429	+	0.976520i	$0.569115\pi$
$98$	0	0
$99$	−33.4270	−3.35954
$100$	−0.414110	−0.0414110
$101$	17.4200	1.73335	0.866677	−	0.498870i	$-0.166251\pi$
$101$	17.4200	1.73335	0.866677	+	0.498870i	$0.166251\pi$
$102$	7.11203	0.704196
$103$	4.42750	0.436254	0.218127	−	0.975920i	$-0.430005\pi$
$103$	4.42750	0.436254	0.218127	+	0.975920i	$0.430005\pi$
$104$	17.3207	1.69844
$105$	0	0
$106$	4.98258	0.483951
$107$	−5.71893	−0.552869	−0.276435	−	0.961033i	$-0.589153\pi$
$107$	−5.71893	−0.552869	−0.276435	+	0.961033i	$0.589153\pi$
$108$	7.27846	0.700370
$109$	14.5703	1.39558	0.697790	−	0.716302i	$-0.254167\pi$
$109$	14.5703	1.39558	0.697790	+	0.716302i	$0.254167\pi$
$110$	12.8086	1.22126
$111$	−20.2167	−1.91888
$112$	0	0
$113$	5.91261	0.556212	0.278106	−	0.960550i	$-0.410293\pi$
$113$	5.91261	0.556212	0.278106	+	0.960550i	$0.410293\pi$
$114$	10.3000	0.964687
$115$	−9.32981	−0.870009
$116$	−2.67848	−0.248691
$117$	44.9243	4.15325
$118$	−17.8126	−1.63978
$119$	0	0
$120$	−24.5172	−2.23810
$121$	6.85640	0.623309
$122$	4.00910	0.362967
$123$	−14.2510	−1.28497
$124$	−2.45803	−0.220737
$125$	−9.92257	−0.887502
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−6.11094	−0.540136
$129$	−16.2982	−1.43498
$130$	−17.2142	−1.50979
$131$	−19.7982	−1.72978	−0.864890	−	0.501962i	$-0.832612\pi$
$131$	−19.7982	−1.72978	−0.864890	+	0.501962i	$0.832612\pi$
$132$	−6.26349	−0.545167
$133$	0	0
$134$	−9.21045	−0.795662
$135$	−39.4735	−3.39734
$136$	5.27249	0.452112
$137$	8.97609	0.766879	0.383439	−	0.923566i	$-0.374739\pi$
$137$	8.97609	0.766879	0.383439	+	0.923566i	$0.374739\pi$
$138$	−15.7714	−1.34255
$139$	−5.86647	−0.497587	−0.248794	−	0.968557i	$-0.580034\pi$
$139$	−5.86647	−0.497587	−0.248794	+	0.968557i	$0.580034\pi$
$140$	0	0
$141$	−36.3007	−3.05707
$142$	−9.50400	−0.797558
$143$	−23.9982	−2.00683
$144$	−22.9493	−1.91244
$145$	14.5263	1.20634
$146$	9.38590	0.776782
$147$	0	0
$148$	−2.74654	−0.225764
$149$	−21.5978	−1.76936	−0.884679	−	0.466201i	$-0.845622\pi$
$149$	−21.5978	−1.76936	−0.884679	+	0.466201i	$0.845622\pi$
$150$	3.79647	0.309981
$151$	−10.8490	−0.882877	−0.441439	−	0.897291i	$-0.645532\pi$
$151$	−10.8490	−0.882877	−0.441439	+	0.897291i	$0.645532\pi$
$152$	7.63591	0.619354
$153$	13.6751	1.10557
$154$	0	0
$155$	13.3307	1.07075
$156$	8.41783	0.673966
$157$	3.59825	0.287171	0.143586	−	0.989638i	$-0.454137\pi$
$157$	3.59825	0.287171	0.143586	+	0.989638i	$0.454137\pi$
$158$	19.2330	1.53010
$159$	13.2140	1.04794
$160$	−6.05120	−0.478390
$161$	0	0
$162$	−37.1701	−2.92036
$163$	−5.95370	−0.466330	−0.233165	−	0.972437i	$-0.574908\pi$
$163$	−5.95370	−0.466330	−0.233165	+	0.972437i	$0.574908\pi$
$164$	−1.93608	−0.151182
$165$	33.9690	2.64448
$166$	−15.9329	−1.23663
$167$	−15.7608	−1.21961	−0.609805	−	0.792552i	$-0.708752\pi$
$167$	−15.7608	−1.21961	−0.609805	+	0.792552i	$0.708752\pi$
$168$	0	0
$169$	19.2524	1.48095
$170$	−5.24006	−0.401895
$171$	19.8051	1.51453
$172$	−2.21420	−0.168831
$173$	17.0465	1.29602	0.648010	−	0.761632i	$-0.275602\pi$
$173$	17.0465	1.29602	0.648010	+	0.761632i	$0.275602\pi$
$174$	24.5557	1.86157
$175$	0	0
$176$	12.2593	0.924080
$177$	−47.2396	−3.55074
$178$	4.83979	0.362758
$179$	6.68379	0.499570	0.249785	−	0.968301i	$-0.419640\pi$
$179$	6.68379	0.499570	0.249785	+	0.968301i	$0.419640\pi$
$180$	−8.63899	−0.643912
$181$	−13.1820	−0.979813	−0.489906	−	0.871775i	$-0.662969\pi$
$181$	−13.1820	−0.979813	−0.489906	+	0.871775i	$0.662969\pi$
$182$	0	0
$183$	10.6323	0.785961
$184$	−11.6921	−0.861955
$185$	14.8954	1.09513
$186$	22.5347	1.65232
$187$	−7.30512	−0.534203
$188$	−4.93165	−0.359678
$189$	0	0
$190$	−7.58896	−0.550561
$191$	17.4273	1.26100	0.630498	−	0.776191i	$-0.282851\pi$
$191$	17.4273	1.26100	0.630498	+	0.776191i	$0.282851\pi$
$192$	−29.3947	−2.12138
$193$	8.81879	0.634790	0.317395	−	0.948293i	$-0.397192\pi$
$193$	8.81879	0.634790	0.317395	+	0.948293i	$0.397192\pi$
$194$	5.28520	0.379455
$195$	−45.6527	−3.26926
$196$	0	0
$197$	−22.4395	−1.59874	−0.799372	−	0.600836i	$-0.794834\pi$
$197$	−22.4395	−1.59874	−0.799372	+	0.600836i	$0.794834\pi$
$198$	41.6331	2.95874
$199$	−6.40127	−0.453774	−0.226887	−	0.973921i	$-0.572855\pi$
$199$	−6.40127	−0.453774	−0.226887	+	0.973921i	$0.572855\pi$
$200$	2.81451	0.199016
$201$	−24.4264	−1.72291
$202$	−21.6965	−1.52656
$203$	0	0
$204$	2.56242	0.179405
$205$	10.5000	0.733351
$206$	−5.51443	−0.384208
$207$	−30.3256	−2.10777
$208$	−16.4759	−1.14240
$209$	−10.5797	−0.731812
$210$	0	0
$211$	17.0999	1.17721	0.588604	−	0.808422i	$-0.299678\pi$
$211$	17.0999	1.17721	0.588604	+	0.808422i	$0.299678\pi$
$212$	1.79519	0.123294
$213$	−25.2050	−1.72702
$214$	7.12289	0.486911
$215$	12.0083	0.818962
$216$	−49.4682	−3.36589
$217$	0	0
$218$	−18.1472	−1.22908
$219$	24.8917	1.68203
$220$	4.61487	0.311135
$221$	9.81774	0.660413
$222$	25.1797	1.68995
$223$	3.55249	0.237892	0.118946	−	0.992901i	$-0.462048\pi$
$223$	3.55249	0.237892	0.118946	+	0.992901i	$0.462048\pi$
$224$	0	0
$225$	7.29991	0.486661
$226$	−7.36413	−0.489854
$227$	0.461961	0.0306614	0.0153307	−	0.999882i	$-0.495120\pi$
$227$	0.461961	0.0306614	0.0153307	+	0.999882i	$0.495120\pi$
$228$	3.71104	0.245769
$229$	−22.8631	−1.51084	−0.755418	−	0.655244i	$-0.772566\pi$
$229$	−22.8631	−1.51084	−0.755418	+	0.655244i	$0.772566\pi$
$230$	11.6202	0.766215
$231$	0	0
$232$	18.2044	1.19517
$233$	2.25856	0.147963	0.0739816	−	0.997260i	$-0.476429\pi$
$233$	2.25856	0.147963	0.0739816	+	0.997260i	$0.476429\pi$
$234$	−55.9530	−3.65776
$235$	26.7460	1.74471
$236$	−6.41776	−0.417760
$237$	51.0066	3.31324
$238$	0	0
$239$	−28.5593	−1.84735	−0.923673	−	0.383182i	$-0.874828\pi$
$239$	−28.5593	−1.84735	−0.923673	+	0.383182i	$0.874828\pi$
$240$	23.3214	1.50539
$241$	13.6906	0.881888	0.440944	−	0.897535i	$-0.354644\pi$
$241$	13.6906	0.881888	0.440944	+	0.897535i	$0.354644\pi$
$242$	−8.53960	−0.548947
$243$	−49.9176	−3.20222
$244$	1.44445	0.0924717
$245$	0	0
$246$	17.7495	1.13167
$247$	14.2186	0.904708
$248$	16.7060	1.06083
$249$	−42.2547	−2.67778
$250$	12.3585	0.781620
$251$	−7.41487	−0.468022	−0.234011	−	0.972234i	$-0.575185\pi$
$251$	−7.41487	−0.468022	−0.234011	+	0.972234i	$0.575185\pi$
$252$	0	0
$253$	16.1996	1.01846
$254$	1.24549	0.0781493
$255$	−13.8968	−0.870253
$256$	−10.1871	−0.636695
$257$	−16.3544	−1.02016	−0.510080	−	0.860127i	$-0.670384\pi$
$257$	−16.3544	−1.02016	−0.510080	+	0.860127i	$0.670384\pi$
$258$	20.2993	1.26378
$259$	0	0
$260$	−6.20217	−0.384642
$261$	47.2161	2.92261
$262$	24.6586	1.52341
$263$	4.68637	0.288974	0.144487	−	0.989507i	$-0.453847\pi$
$263$	4.68637	0.288974	0.144487	+	0.989507i	$0.453847\pi$
$264$	42.5699	2.62000
$265$	−9.73591	−0.598072
$266$	0	0
$267$	12.8353	0.785508
$268$	−3.31847	−0.202708
$269$	14.6818	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$269$	14.6818	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$270$	49.1640	2.99203
$271$	10.7458	0.652764	0.326382	−	0.945238i	$-0.394170\pi$
$271$	10.7458	0.652764	0.326382	+	0.945238i	$0.394170\pi$
$272$	−5.01533	−0.304099
$273$	0	0
$274$	−11.1797	−0.675388
$275$	−3.89955	−0.235152
$276$	−5.68235	−0.342037
$277$	0.856063	0.0514358	0.0257179	−	0.999669i	$-0.491813\pi$
$277$	0.856063	0.0514358	0.0257179	+	0.999669i	$0.491813\pi$
$278$	7.30665	0.438224
$279$	43.3300	2.59410
$280$	0	0
$281$	14.3744	0.857504	0.428752	−	0.903422i	$-0.358953\pi$
$281$	14.3744	0.857504	0.428752	+	0.903422i	$0.358953\pi$
$282$	45.2123	2.69235
$283$	−18.7855	−1.11668	−0.558341	−	0.829612i	$-0.688562\pi$
$283$	−18.7855	−1.11668	−0.558341	+	0.829612i	$0.688562\pi$
$284$	−3.42423	−0.203191
$285$	−20.1262	−1.19217
$286$	29.8896	1.76741
$287$	0	0
$288$	−19.6688	−1.15900
$289$	−14.0114	−0.824203
$290$	−18.0924	−1.06242
$291$	14.0165	0.821663
$292$	3.38168	0.197898
$293$	14.6548	0.856145	0.428072	−	0.903744i	$-0.359193\pi$
$293$	14.6548	0.856145	0.428072	+	0.903744i	$0.359193\pi$
$294$	0	0
$295$	34.8056	2.02646
$296$	18.6669	1.08499
$297$	68.5390	3.97704
$298$	26.8999	1.55827
$299$	−21.7715	−1.25908
$300$	1.36784	0.0789725
$301$	0	0
$302$	13.5123	0.777548
$303$	−57.5399	−3.30558
$304$	−7.26348	−0.416589
$305$	−7.83375	−0.448559
$306$	−17.0323	−0.973670
$307$	11.1328	0.635385	0.317692	−	0.948194i	$-0.397092\pi$
$307$	11.1328	0.635385	0.317692	+	0.948194i	$0.397092\pi$
$308$	0	0
$309$	−14.6245	−0.831956
$310$	−16.6033	−0.943004
$311$	16.8169	0.953602	0.476801	−	0.879011i	$-0.341796\pi$
$311$	16.8169	0.953602	0.476801	+	0.879011i	$0.341796\pi$
$312$	−57.2120	−3.23899
$313$	13.4057	0.757732	0.378866	−	0.925451i	$-0.376314\pi$
$313$	13.4057	0.757732	0.378866	+	0.925451i	$0.376314\pi$
$314$	−4.48160	−0.252911
$315$	0	0
$316$	6.92953	0.389817
$317$	−18.6801	−1.04918	−0.524588	−	0.851356i	$-0.675781\pi$
$317$	−18.6801	−1.04918	−0.524588	+	0.851356i	$0.675781\pi$
$318$	−16.4579	−0.922915
$319$	−25.2224	−1.41219
$320$	21.6577	1.21070
$321$	18.8902	1.05435
$322$	0	0
$323$	4.32819	0.240827
$324$	−13.3922	−0.744009
$325$	5.24081	0.290708
$326$	7.41530	0.410696
$327$	−48.1270	−2.66143
$328$	13.1586	0.726562
$329$	0	0
$330$	−42.3082	−2.32899
$331$	2.41940	0.132982	0.0664910	−	0.997787i	$-0.478820\pi$
$331$	2.41940	0.132982	0.0664910	+	0.997787i	$0.478820\pi$
$332$	−5.74053	−0.315053
$333$	48.4160	2.65318
$334$	19.6300	1.07411
$335$	17.9971	0.983289
$336$	0	0
$337$	−17.2871	−0.941687	−0.470844	−	0.882217i	$-0.656050\pi$
$337$	−17.2871	−0.941687	−0.470844	+	0.882217i	$0.656050\pi$
$338$	−23.9788	−1.30427
$339$	−19.5299	−1.06072
$340$	−1.88796	−0.102389
$341$	−23.1465	−1.25345
$342$	−24.6671	−1.33384
$343$	0	0
$344$	15.0489	0.811380
$345$	30.8172	1.65914
$346$	−21.2313	−1.14140
$347$	−20.7673	−1.11485	−0.557424	−	0.830228i	$-0.688210\pi$
$347$	−20.7673	−1.11485	−0.557424	+	0.830228i	$0.688210\pi$
$348$	8.84727	0.474264
$349$	−23.6254	−1.26464	−0.632318	−	0.774709i	$-0.717897\pi$
$349$	−23.6254	−1.26464	−0.632318	+	0.774709i	$0.717897\pi$
$350$	0	0
$351$	−92.1132	−4.91664
$352$	10.5069	0.560019
$353$	15.5597	0.828161	0.414080	−	0.910240i	$-0.364103\pi$
$353$	15.5597	0.828161	0.414080	+	0.910240i	$0.364103\pi$
$354$	58.8366	3.12713
$355$	18.5707	0.985632
$356$	1.74375	0.0924184
$357$	0	0
$358$	−8.32462	−0.439970
$359$	−7.03515	−0.371301	−0.185651	−	0.982616i	$-0.559439\pi$
$359$	−7.03515	−0.371301	−0.185651	+	0.982616i	$0.559439\pi$
$360$	58.7151	3.09456
$361$	−12.7317	−0.670088
$362$	16.4181	0.862919
$363$	−22.6473	−1.18868
$364$	0	0
$365$	−18.3400	−0.959957
$366$	−13.2424	−0.692194
$367$	−28.6265	−1.49429	−0.747145	−	0.664661i	$-0.768576\pi$
$367$	−28.6265	−1.49429	−0.747145	+	0.664661i	$0.768576\pi$
$368$	11.1219	0.579767
$369$	34.1291	1.77669
$370$	−18.5521	−0.964480
$371$	0	0
$372$	8.11909	0.420955
$373$	3.75654	0.194506	0.0972531	−	0.995260i	$-0.468994\pi$
$373$	3.75654	0.194506	0.0972531	+	0.995260i	$0.468994\pi$
$374$	9.09849	0.470472
$375$	32.7752	1.69250
$376$	33.5181	1.72856
$377$	33.8978	1.74582
$378$	0	0
$379$	20.7800	1.06740	0.533699	−	0.845675i	$-0.320802\pi$
$379$	20.7800	1.06740	0.533699	+	0.845675i	$0.320802\pi$
$380$	−2.73425	−0.140264
$381$	3.30309	0.169223
$382$	−21.7056	−1.11056
$383$	−17.6927	−0.904054	−0.452027	−	0.892004i	$-0.649299\pi$
$383$	−17.6927	−0.904054	−0.452027	+	0.892004i	$0.649299\pi$
$384$	20.1850	1.03006
$385$	0	0
$386$	−10.9838	−0.559058
$387$	39.0318	1.98410
$388$	1.90422	0.0966722
$389$	27.3076	1.38455	0.692275	−	0.721634i	$-0.256609\pi$
$389$	27.3076	1.38455	0.692275	+	0.721634i	$0.256609\pi$
$390$	56.8602	2.87923
$391$	−6.62733	−0.335159
$392$	0	0
$393$	65.3954	3.29876
$394$	27.9482	1.40801
$395$	−37.5811	−1.89091
$396$	15.0002	0.753786
$397$	12.7728	0.641046	0.320523	−	0.947241i	$-0.396141\pi$
$397$	12.7728	0.641046	0.320523	+	0.947241i	$0.396141\pi$
$398$	7.97275	0.399638
$399$	0	0
$400$	−2.67723	−0.133862
$401$	10.4514	0.521919	0.260960	−	0.965350i	$-0.415961\pi$
$401$	10.4514	0.521919	0.260960	+	0.965350i	$0.415961\pi$
$402$	30.4230	1.51736
$403$	31.1078	1.54959
$404$	−7.81711	−0.388916
$405$	72.6301	3.60902
$406$	0	0
$407$	−25.8634	−1.28200
$408$	−17.4155	−0.862197
$409$	10.0572	0.497297	0.248648	−	0.968594i	$-0.420014\pi$
$409$	10.0572	0.497297	0.248648	+	0.968594i	$0.420014\pi$
$410$	−13.0777	−0.645860
$411$	−29.6489	−1.46247
$412$	−1.98681	−0.0978832
$413$	0	0
$414$	37.7703	1.85631
$415$	31.1328	1.52825
$416$	−14.1208	−0.692328
$417$	19.3775	0.948920
$418$	13.1769	0.644505
$419$	4.78230	0.233631	0.116815	−	0.993154i	$-0.462731\pi$
$419$	4.78230	0.233631	0.116815	+	0.993154i	$0.462731\pi$
$420$	0	0
$421$	−13.2705	−0.646766	−0.323383	−	0.946268i	$-0.604820\pi$
$421$	−13.2705	−0.646766	−0.323383	+	0.946268i	$0.604820\pi$
$422$	−21.2979	−1.03676
$423$	86.9350	4.22692
$424$	−12.2011	−0.592536
$425$	1.59532	0.0773844
$426$	31.3926	1.52098
$427$	0	0
$428$	2.56633	0.124048
$429$	79.2682	3.82710
$430$	−14.9563	−0.721258
$431$	−21.8535	−1.05264	−0.526322	−	0.850285i	$-0.676429\pi$
$431$	−21.8535	−1.05264	−0.526322	+	0.850285i	$0.676429\pi$
$432$	47.0555	2.26396
$433$	28.0481	1.34791	0.673953	−	0.738774i	$-0.264595\pi$
$433$	28.0481	1.34791	0.673953	+	0.738774i	$0.264595\pi$
$434$	0	0
$435$	−47.9817	−2.30055
$436$	−6.53832	−0.313129
$437$	−9.59808	−0.459138
$438$	−31.0025	−1.48136
$439$	0.864598	0.0412650	0.0206325	−	0.999787i	$-0.493432\pi$
$439$	0.864598	0.0412650	0.0206325	+	0.999787i	$0.493432\pi$
$440$	−31.3651	−1.49527
$441$	0	0
$442$	−12.2279	−0.581624
$443$	17.3136	0.822592	0.411296	−	0.911502i	$-0.365076\pi$
$443$	17.3136	0.822592	0.411296	+	0.911502i	$0.365076\pi$
$444$	9.07209	0.430543
$445$	−9.45691	−0.448301
$446$	−4.42460	−0.209511
$447$	71.3395	3.37424
$448$	0	0
$449$	8.34805	0.393969	0.196985	−	0.980407i	$-0.436885\pi$
$449$	8.34805	0.393969	0.196985	+	0.980407i	$0.436885\pi$
$450$	−9.09200	−0.428601
$451$	−18.2315	−0.858486
$452$	−2.65325	−0.124798
$453$	35.8352	1.68368
$454$	−0.575369	−0.0270034
$455$	0	0
$456$	−25.2222	−1.18114
$457$	31.6135	1.47882	0.739408	−	0.673258i	$-0.235106\pi$
$457$	31.6135	1.47882	0.739408	+	0.673258i	$0.235106\pi$
$458$	28.4759	1.33059
$459$	−28.0396	−1.30878
$460$	4.18669	0.195205
$461$	−14.9040	−0.694148	−0.347074	−	0.937838i	$-0.612825\pi$
$461$	−14.9040	−0.694148	−0.347074	+	0.937838i	$0.612825\pi$
$462$	0	0
$463$	−6.32576	−0.293983	−0.146992	−	0.989138i	$-0.546959\pi$
$463$	−6.32576	−0.293983	−0.146992	+	0.989138i	$0.546959\pi$
$464$	−17.3165	−0.803896
$465$	−44.0325	−2.04196
$466$	−2.81303	−0.130311
$467$	25.5517	1.18239	0.591196	−	0.806528i	$-0.298656\pi$
$467$	25.5517	1.18239	0.591196	+	0.806528i	$0.298656\pi$
$468$	−20.1595	−0.931873
$469$	0	0
$470$	−33.3120	−1.53657
$471$	−11.8854	−0.547648
$472$	43.6184	2.00770
$473$	−20.8505	−0.958705
$474$	−63.5285	−2.91796
$475$	2.31043	0.106010
$476$	0	0
$477$	−31.6456	−1.44895
$478$	35.5704	1.62695
$479$	5.61743	0.256667	0.128333	−	0.991731i	$-0.459037\pi$
$479$	5.61743	0.256667	0.128333	+	0.991731i	$0.459037\pi$
$480$	19.9877	0.912309
$481$	34.7592	1.58488
$482$	−17.0516	−0.776677
$483$	0	0
$484$	−3.07676	−0.139853
$485$	−10.3272	−0.468935
$486$	62.1721	2.82018
$487$	25.1524	1.13976	0.569881	−	0.821727i	$-0.306989\pi$
$487$	25.1524	1.13976	0.569881	+	0.821727i	$0.306989\pi$
$488$	−9.81726	−0.444407
$489$	19.6656	0.889311
$490$	0	0
$491$	−30.3800	−1.37103	−0.685515	−	0.728058i	$-0.740423\pi$
$491$	−30.3800	−1.37103	−0.685515	+	0.728058i	$0.740423\pi$
$492$	6.39505	0.288311
$493$	10.3186	0.464726
$494$	−17.7092	−0.796774
$495$	−81.3508	−3.65644
$496$	−15.8912	−0.713537
$497$	0	0
$498$	52.6280	2.35832
$499$	−28.6191	−1.28116	−0.640582	−	0.767889i	$-0.721307\pi$
$499$	−28.6191	−1.28116	−0.640582	+	0.767889i	$0.721307\pi$
$500$	4.45269	0.199130
$501$	52.0595	2.32585
$502$	9.23518	0.412186
$503$	−13.1761	−0.587493	−0.293746	−	0.955883i	$-0.594902\pi$
$503$	−13.1761	−0.587493	−0.293746	+	0.955883i	$0.594902\pi$
$504$	0	0
$505$	42.3948	1.88654
$506$	−20.1766	−0.896957
$507$	−63.5925	−2.82424
$508$	0.448744	0.0199098
$509$	31.0674	1.37704	0.688518	−	0.725219i	$-0.258262\pi$
$509$	31.0674	1.37704	0.688518	+	0.725219i	$0.258262\pi$
$510$	17.3084	0.766430
$511$	0	0
$512$	24.9099	1.10087
$513$	−40.6085	−1.79291
$514$	20.3693	0.898452
$515$	10.7751	0.474809
$516$	7.31371	0.321968
$517$	−46.4399	−2.04242
$518$	0	0
$519$	−56.3061	−2.47157
$520$	42.1532	1.84854
$521$	−21.6092	−0.946714	−0.473357	−	0.880871i	$-0.656958\pi$
$521$	−21.6092	−0.946714	−0.473357	+	0.880871i	$0.656958\pi$
$522$	−58.8074	−2.57393
$523$	11.8477	0.518063	0.259032	−	0.965869i	$-0.416597\pi$
$523$	11.8477	0.518063	0.259032	+	0.965869i	$0.416597\pi$
$524$	8.88433	0.388114
$525$	0	0
$526$	−5.83685	−0.254499
$527$	9.46932	0.412490
$528$	−40.4937	−1.76226
$529$	−8.30341	−0.361018
$530$	12.1260	0.526721
$531$	113.132	4.90951
$532$	0	0
$533$	24.5022	1.06131
$534$	−15.9863	−0.691795
$535$	−13.9181	−0.601730
$536$	22.5540	0.974186
$537$	−22.0772	−0.952701
$538$	−18.2861	−0.788372
$539$	0	0
$540$	17.7135	0.762266
$541$	−13.3103	−0.572253	−0.286126	−	0.958192i	$-0.592368\pi$
$541$	−13.3103	−0.572253	−0.286126	+	0.958192i	$0.592368\pi$
$542$	−13.3839	−0.574887
$543$	43.5415	1.86854
$544$	−4.29841	−0.184293
$545$	35.4595	1.51892
$546$	0	0
$547$	0.103601	0.00442966	0.00221483	−	0.999998i	$-0.499295\pi$
$547$	0.103601	0.00442966	0.00221483	+	0.999998i	$0.499295\pi$
$548$	−4.02796	−0.172066
$549$	−25.4628	−1.08672
$550$	4.85686	0.207097
$551$	14.9440	0.636635
$552$	38.6202	1.64378
$553$	0	0
$554$	−1.06622	−0.0452994
$555$	−49.2010	−2.08846
$556$	2.63254	0.111645
$557$	3.49486	0.148082	0.0740410	−	0.997255i	$-0.476410\pi$
$557$	3.49486	0.148082	0.0740410	+	0.997255i	$0.476410\pi$
$558$	−53.9673	−2.28462
$559$	28.0220	1.18521
$560$	0	0
$561$	24.1295	1.01875
$562$	−17.9032	−0.755202
$563$	35.0406	1.47679	0.738393	−	0.674370i	$-0.235585\pi$
$563$	35.0406	1.47679	0.738393	+	0.674370i	$0.235585\pi$
$564$	16.2897	0.685921
$565$	14.3894	0.605368
$566$	23.3972	0.983459
$567$	0	0
$568$	23.2729	0.976508
$569$	9.51641	0.398949	0.199474	−	0.979903i	$-0.436077\pi$
$569$	9.51641	0.398949	0.199474	+	0.979903i	$0.436077\pi$
$570$	25.0670	1.04994
$571$	−41.7756	−1.74825	−0.874127	−	0.485697i	$-0.838566\pi$
$571$	−41.7756	−1.74825	−0.874127	+	0.485697i	$0.838566\pi$
$572$	10.7690	0.450275
$573$	−57.5641	−2.40477
$574$	0	0
$575$	−3.53774	−0.147534
$576$	70.3960	2.93317
$577$	18.2875	0.761320	0.380660	−	0.924715i	$-0.375697\pi$
$577$	18.2875	0.761320	0.380660	+	0.924715i	$0.375697\pi$
$578$	17.4512	0.725873
$579$	−29.1293	−1.21057
$580$	−6.51858	−0.270669
$581$	0	0
$582$	−17.4575	−0.723637
$583$	16.9048	0.700124
$584$	−22.9837	−0.951070
$585$	109.332	4.52031
$586$	−18.2525	−0.754005
$587$	−2.18408	−0.0901465	−0.0450732	−	0.998984i	$-0.514352\pi$
$587$	−2.18408	−0.0901465	−0.0450732	+	0.998984i	$0.514352\pi$
$588$	0	0
$589$	13.7140	0.565076
$590$	−43.3502	−1.78470
$591$	74.1196	3.04887
$592$	−17.7565	−0.729787
$593$	31.5402	1.29520	0.647600	−	0.761981i	$-0.275773\pi$
$593$	31.5402	1.29520	0.647600	+	0.761981i	$0.275773\pi$
$594$	−85.3650	−3.50257
$595$	0	0
$596$	9.69186	0.396994
$597$	21.1440	0.865367
$598$	27.1163	1.10887
$599$	−42.0406	−1.71773	−0.858866	−	0.512200i	$-0.828831\pi$
$599$	−42.0406	−1.71773	−0.858866	+	0.512200i	$0.828831\pi$
$600$	−9.29658	−0.379531
$601$	13.4807	0.549891	0.274945	−	0.961460i	$-0.411340\pi$
$601$	13.4807	0.549891	0.274945	+	0.961460i	$0.411340\pi$
$602$	0	0
$603$	58.4978	2.38222
$604$	4.86841	0.198093
$605$	16.6863	0.678395
$606$	71.6656	2.91122
$607$	6.69701	0.271823	0.135912	−	0.990721i	$-0.456604\pi$
$607$	6.69701	0.271823	0.135912	+	0.990721i	$0.456604\pi$
$608$	−6.22520	−0.252465
$609$	0	0
$610$	9.75689	0.395045
$611$	62.4130	2.52496
$612$	−6.13662	−0.248058
$613$	−36.2623	−1.46462	−0.732311	−	0.680970i	$-0.761558\pi$
$613$	−36.2623	−1.46462	−0.732311	+	0.680970i	$0.761558\pi$
$614$	−13.8659	−0.559582
$615$	−34.6824	−1.39853
$616$	0	0
$617$	38.6679	1.55671	0.778356	−	0.627823i	$-0.216054\pi$
$617$	38.6679	1.55671	0.778356	+	0.627823i	$0.216054\pi$
$618$	18.2147	0.732702
$619$	8.11816	0.326296	0.163148	−	0.986602i	$-0.447835\pi$
$619$	8.11816	0.326296	0.163148	+	0.986602i	$0.447835\pi$
$620$	−5.98206	−0.240245
$621$	62.1798	2.49519
$622$	−20.9454	−0.839835
$623$	0	0
$624$	54.4216	2.17861
$625$	−28.7625	−1.15050
$626$	−16.6967	−0.667333
$627$	34.9457	1.39560
$628$	−1.61469	−0.0644332
$629$	10.5808	0.421884
$630$	0	0
$631$	−19.4072	−0.772587	−0.386293	−	0.922376i	$-0.626245\pi$
$631$	−19.4072	−0.772587	−0.386293	+	0.922376i	$0.626245\pi$
$632$	−47.0967	−1.87341
$633$	−56.4827	−2.24499
$634$	23.2659	0.924007
$635$	−2.43368	−0.0965778
$636$	−5.92969	−0.235127
$637$	0	0
$638$	31.4144	1.24371
$639$	60.3622	2.38789
$640$	−14.8721	−0.587871
$641$	11.5636	0.456736	0.228368	−	0.973575i	$-0.426661\pi$
$641$	11.5636	0.456736	0.228368	+	0.973575i	$0.426661\pi$
$642$	−23.5276	−0.928560
$643$	5.96147	0.235098	0.117549	−	0.993067i	$-0.462496\pi$
$643$	5.96147	0.235098	0.117549	+	0.993067i	$0.462496\pi$
$644$	0	0
$645$	−39.6647	−1.56180
$646$	−5.39074	−0.212096
$647$	50.1845	1.97296	0.986478	−	0.163892i	$-0.0524050\pi$
$647$	50.1845	1.97296	0.986478	+	0.163892i	$0.0524050\pi$
$648$	91.0202	3.57561
$649$	−60.4341	−2.37224
$650$	−6.52740	−0.256026
$651$	0	0
$652$	2.67169	0.104631
$653$	7.04866	0.275835	0.137918	−	0.990444i	$-0.455959\pi$
$653$	7.04866	0.275835	0.137918	+	0.990444i	$0.455959\pi$
$654$	59.9420	2.34392
$655$	−48.1826	−1.88265
$656$	−12.5168	−0.488699
$657$	−59.6121	−2.32569
$658$	0	0
$659$	−48.8016	−1.90104	−0.950521	−	0.310660i	$-0.899450\pi$
$659$	−48.8016	−1.90104	−0.950521	+	0.310660i	$0.899450\pi$
$660$	−15.2434	−0.593347
$661$	7.03546	0.273648	0.136824	−	0.990595i	$-0.456311\pi$
$661$	7.03546	0.273648	0.136824	+	0.990595i	$0.456311\pi$
$662$	−3.01334	−0.117117
$663$	−32.4289	−1.25944
$664$	39.0156	1.51410
$665$	0	0
$666$	−60.3018	−2.33665
$667$	−22.8822	−0.886004
$668$	7.07257	0.273646
$669$	−11.7342	−0.453670
$670$	−22.4153	−0.865980
$671$	13.6020	0.525099
$672$	0	0
$673$	−48.0452	−1.85201	−0.926004	−	0.377514i	$-0.876779\pi$
$673$	−48.0452	−1.85201	−0.926004	+	0.377514i	$0.876779\pi$
$674$	21.5310	0.829342
$675$	−14.9678	−0.576111
$676$	−8.63939	−0.332284
$677$	−8.31499	−0.319571	−0.159786	−	0.987152i	$-0.551080\pi$
$677$	−8.31499	−0.319571	−0.159786	+	0.987152i	$0.551080\pi$
$678$	24.3244	0.934173
$679$	0	0
$680$	12.8316	0.492068
$681$	−1.52590	−0.0584726
$682$	28.8288	1.10391
$683$	5.68344	0.217471	0.108735	−	0.994071i	$-0.465320\pi$
$683$	5.68344	0.217471	0.108735	+	0.994071i	$0.465320\pi$
$684$	−8.88740	−0.339818
$685$	21.8450	0.834653
$686$	0	0
$687$	75.5190	2.88123
$688$	−14.3149	−0.545750
$689$	−22.7192	−0.865533
$690$	−38.3827	−1.46120
$691$	−6.26192	−0.238215	−0.119107	−	0.992881i	$-0.538003\pi$
$691$	−6.26192	−0.238215	−0.119107	+	0.992881i	$0.538003\pi$
$692$	−7.64950	−0.290790
$693$	0	0
$694$	25.8656	0.981844
$695$	−14.2771	−0.541562
$696$	−60.1307	−2.27925
$697$	7.45856	0.282513
$698$	29.4253	1.11376
$699$	−7.46024	−0.282172
$700$	0	0
$701$	9.37500	0.354089	0.177044	−	0.984203i	$-0.443346\pi$
$701$	9.37500	0.354089	0.177044	+	0.984203i	$0.443346\pi$
$702$	114.727	4.33007
$703$	15.3237	0.577945
$704$	−37.6049	−1.41729
$705$	−88.3445	−3.32725
$706$	−19.3796	−0.729359
$707$	0	0
$708$	21.1985	0.796687
$709$	−34.9615	−1.31301	−0.656504	−	0.754323i	$-0.727965\pi$
$709$	−34.9615	−1.31301	−0.656504	+	0.754323i	$0.727965\pi$
$710$	−23.1297	−0.868044
$711$	−122.154	−4.58112
$712$	−11.8514	−0.444151
$713$	−20.9989	−0.786415
$714$	0	0
$715$	−58.4039	−2.18418
$716$	−2.99931	−0.112089
$717$	94.3340	3.52297
$718$	8.76224	0.327004
$719$	36.7725	1.37138	0.685691	−	0.727893i	$-0.259500\pi$
$719$	36.7725	1.37138	0.685691	+	0.727893i	$0.259500\pi$
$720$	−55.8513	−2.08146
$721$	0	0
$722$	15.8572	0.590145
$723$	−45.2213	−1.68180
$724$	5.91535	0.219842
$725$	5.50817	0.204568
$726$	28.2071	1.04686
$727$	9.14372	0.339122	0.169561	−	0.985520i	$-0.445765\pi$
$727$	9.14372	0.339122	0.169561	+	0.985520i	$0.445765\pi$
$728$	0	0
$729$	75.3516	2.79080
$730$	22.8423	0.845432
$731$	8.53000	0.315494
$732$	−4.77117	−0.176347
$733$	−29.1847	−1.07796	−0.538981	−	0.842318i	$-0.681191\pi$
$733$	−29.1847	−1.07796	−0.538981	+	0.842318i	$0.681191\pi$
$734$	35.6541	1.31602
$735$	0	0
$736$	9.53204	0.351356
$737$	−31.2490	−1.15107
$738$	−42.5076	−1.56473
$739$	−0.392476	−0.0144375	−0.00721873	−	0.999974i	$-0.502298\pi$
$739$	−0.392476	−0.0144375	−0.00721873	+	0.999974i	$0.502298\pi$
$740$	−6.68422	−0.245717
$741$	−46.9654	−1.72532
$742$	0	0
$743$	−1.15399	−0.0423359	−0.0211680	−	0.999776i	$-0.506738\pi$
$743$	−1.15399	−0.0423359	−0.0211680	+	0.999776i	$0.506738\pi$
$744$	−55.1816	−2.02306
$745$	−52.5622	−1.92573
$746$	−4.67875	−0.171301
$747$	101.194	3.70249
$748$	3.27813	0.119860
$749$	0	0
$750$	−40.8213	−1.49058
$751$	48.0457	1.75321	0.876607	−	0.481208i	$-0.159802\pi$
$751$	48.0457	1.75321	0.876607	+	0.481208i	$0.159802\pi$
$752$	−31.8833	−1.16266
$753$	24.4920	0.892538
$754$	−42.2195	−1.53754
$755$	−26.4030	−0.960903
$756$	0	0
$757$	40.4145	1.46889	0.734445	−	0.678668i	$-0.237443\pi$
$757$	40.4145	1.46889	0.734445	+	0.678668i	$0.237443\pi$
$758$	−25.8814	−0.940054
$759$	−53.5089	−1.94225
$760$	18.5834	0.674091
$761$	26.8626	0.973769	0.486884	−	0.873466i	$-0.338133\pi$
$761$	26.8626	0.973769	0.486884	+	0.873466i	$0.338133\pi$
$762$	−4.11399	−0.149034
$763$	0	0
$764$	−7.82040	−0.282932
$765$	33.2809	1.20327
$766$	22.0362	0.796199
$767$	81.2205	2.93270
$768$	33.6490	1.21420
$769$	36.9143	1.33117	0.665583	−	0.746324i	$-0.268183\pi$
$769$	36.9143	1.33117	0.665583	+	0.746324i	$0.268183\pi$
$770$	0	0
$771$	54.0201	1.94549
$772$	−3.95737	−0.142429
$773$	−35.0570	−1.26091	−0.630457	−	0.776224i	$-0.717132\pi$
$773$	−35.0570	−1.26091	−0.630457	+	0.776224i	$0.717132\pi$
$774$	−48.6139	−1.74739
$775$	5.05482	0.181574
$776$	−12.9421	−0.464594
$777$	0	0
$778$	−34.0114	−1.21937
$779$	10.8019	0.387018
$780$	20.4864	0.733529
$781$	−32.2450	−1.15382
$782$	8.25431	0.295173
$783$	−96.8124	−3.45979
$784$	0	0
$785$	8.75700	0.312551
$786$	−81.4496	−2.90521
$787$	−4.03988	−0.144006	−0.0720031	−	0.997404i	$-0.522939\pi$
$787$	−4.03988	−0.144006	−0.0720031	+	0.997404i	$0.522939\pi$
$788$	10.0696	0.358713
$789$	−15.4795	−0.551086
$790$	46.8071	1.66532
$791$	0	0
$792$	−101.949	−3.62260
$793$	−18.2804	−0.649157
$794$	−15.9084	−0.564568
$795$	32.1586	1.14055
$796$	2.87253	0.101814
$797$	−21.3487	−0.756208	−0.378104	−	0.925763i	$-0.623424\pi$
$797$	−21.3487	−0.756208	−0.378104	+	0.925763i	$0.623424\pi$
$798$	0	0
$799$	18.9987	0.672127
$800$	−2.29454	−0.0811241
$801$	−30.7387	−1.08610
$802$	−13.0172	−0.459653
$803$	31.8442	1.12376
$804$	10.9612	0.386572
$805$	0	0
$806$	−38.7446	−1.36472
$807$	−48.4955	−1.70712
$808$	53.1292	1.86908
$809$	−26.7700	−0.941182	−0.470591	−	0.882352i	$-0.655959\pi$
$809$	−26.7700	−0.941182	−0.470591	+	0.882352i	$0.655959\pi$
$810$	−90.4604	−3.17845
$811$	−22.1435	−0.777563	−0.388781	−	0.921330i	$-0.627104\pi$
$811$	−22.1435	−0.777563	−0.388781	+	0.921330i	$0.627104\pi$
$812$	0	0
$813$	−35.4946	−1.24485
$814$	32.2127	1.12905
$815$	−14.4894	−0.507543
$816$	16.5661	0.579930
$817$	12.3536	0.432199
$818$	−12.5262	−0.437968
$819$	0	0
$820$	−4.71180	−0.164543
$821$	9.39267	0.327806	0.163903	−	0.986476i	$-0.447592\pi$
$821$	9.39267	0.327806	0.163903	+	0.986476i	$0.447592\pi$
$822$	36.9275	1.28799
$823$	−0.914372	−0.0318730	−0.0159365	−	0.999873i	$-0.505073\pi$
$823$	−0.914372	−0.0318730	−0.0159365	+	0.999873i	$0.505073\pi$
$824$	13.5034	0.470414
$825$	12.8806	0.448444
$826$	0	0
$827$	39.0657	1.35845	0.679224	−	0.733931i	$-0.262317\pi$
$827$	39.0657	1.35845	0.679224	+	0.733931i	$0.262317\pi$
$828$	13.6084	0.472925
$829$	−35.1117	−1.21948	−0.609741	−	0.792601i	$-0.708726\pi$
$829$	−35.1117	−1.21948	−0.609741	+	0.792601i	$0.708726\pi$
$830$	−38.7757	−1.34592
$831$	−2.82766	−0.0980903
$832$	50.5392	1.75213
$833$	0	0
$834$	−24.1346	−0.835712
$835$	−38.3569	−1.32740
$836$	4.74757	0.164198
$837$	−88.8442	−3.07091
$838$	−5.95632	−0.205758
$839$	38.3771	1.32492	0.662462	−	0.749096i	$-0.269512\pi$
$839$	38.3771	1.32492	0.662462	+	0.749096i	$0.269512\pi$
$840$	0	0
$841$	6.62708	0.228520
$842$	16.5284	0.569605
$843$	−47.4800	−1.63530
$844$	−7.67348	−0.264132
$845$	46.8543	1.61184
$846$	−108.277	−3.72264
$847$	0	0
$848$	11.6060	0.398550
$849$	62.0503	2.12956
$850$	−1.98696	−0.0681522
$851$	−23.4637	−0.804325
$852$	11.3106	0.387494
$853$	−46.6670	−1.59785	−0.798924	−	0.601432i	$-0.794597\pi$
$853$	−46.6670	−1.59785	−0.798924	+	0.601432i	$0.794597\pi$
$854$	0	0
$855$	48.1993	1.64838
$856$	−17.4421	−0.596160
$857$	−13.5444	−0.462670	−0.231335	−	0.972874i	$-0.574309\pi$
$857$	−13.5444	−0.462670	−0.231335	+	0.972874i	$0.574309\pi$
$858$	−98.7281	−3.37052
$859$	12.2410	0.417658	0.208829	−	0.977952i	$-0.433035\pi$
$859$	12.2410	0.417658	0.208829	+	0.977952i	$0.433035\pi$
$860$	−5.38867	−0.183752
$861$	0	0
$862$	27.2184	0.927061
$863$	31.6870	1.07864	0.539318	−	0.842102i	$-0.318682\pi$
$863$	31.6870	1.07864	0.539318	+	0.842102i	$0.318682\pi$
$864$	40.3291	1.37202
$865$	41.4858	1.41056
$866$	−34.9338	−1.18710
$867$	46.2811	1.57179
$868$	0	0
$869$	65.2533	2.21357
$870$	59.7609	2.02608
$871$	41.9972	1.42302
$872$	44.4379	1.50486
$873$	−33.5676	−1.13609
$874$	11.9544	0.404362
$875$	0	0
$876$	−11.1700	−0.377399
$877$	34.1129	1.15191	0.575955	−	0.817481i	$-0.304630\pi$
$877$	34.1129	1.15191	0.575955	+	0.817481i	$0.304630\pi$
$878$	−1.07685	−0.0363420
$879$	−48.4063	−1.63270
$880$	29.8353	1.00575
$881$	23.3863	0.787905	0.393952	−	0.919131i	$-0.371107\pi$
$881$	23.3863	0.787905	0.393952	+	0.919131i	$0.371107\pi$
$882$	0	0
$883$	6.39758	0.215296	0.107648	−	0.994189i	$-0.465668\pi$
$883$	6.39758	0.215296	0.107648	+	0.994189i	$0.465668\pi$
$884$	−4.40565	−0.148178
$885$	−114.966	−3.86455
$886$	−21.5640	−0.724455
$887$	−16.3462	−0.548851	−0.274425	−	0.961608i	$-0.588488\pi$
$887$	−16.3462	−0.548851	−0.274425	+	0.961608i	$0.588488\pi$
$888$	−61.6587	−2.06913
$889$	0	0
$890$	11.7785	0.394817
$891$	−126.110	−4.22484
$892$	−1.59416	−0.0533763
$893$	27.5150	0.920755
$894$	−88.8529	−2.97169
$895$	16.2662	0.543720
$896$	0	0
$897$	71.9135	2.40112
$898$	−10.3975	−0.346968
$899$	32.6948	1.09043
$900$	−3.27579	−0.109193
$901$	−6.91581	−0.230399
$902$	22.7072	0.756066
$903$	0	0
$904$	18.0329	0.599764
$905$	−32.0809	−1.06641
$906$	−44.6325	−1.48282
$907$	−12.0928	−0.401534	−0.200767	−	0.979639i	$-0.564343\pi$
$907$	−12.0928	−0.401534	−0.200767	+	0.979639i	$0.564343\pi$
$908$	−0.207302	−0.00687955
$909$	137.800	4.57053
$910$	0	0
$911$	−6.38071	−0.211402	−0.105701	−	0.994398i	$-0.533709\pi$
$911$	−6.38071	−0.211402	−0.105701	+	0.994398i	$0.533709\pi$
$912$	23.9920	0.794454
$913$	−54.0568	−1.78902
$914$	−39.3744	−1.30239
$915$	25.8756	0.855421
$916$	10.2597	0.338989
$917$	0	0
$918$	34.9231	1.15264
$919$	−10.8878	−0.359156	−0.179578	−	0.983744i	$-0.557473\pi$
$919$	−10.8878	−0.359156	−0.179578	+	0.983744i	$0.557473\pi$
$920$	−28.4549	−0.938132
$921$	−36.7728	−1.21171
$922$	18.5628	0.611334
$923$	43.3357	1.42641
$924$	0	0
$925$	5.64814	0.185710
$926$	7.87870	0.258910
$927$	35.0234	1.15032
$928$	−14.8412	−0.487185
$929$	15.7838	0.517851	0.258925	−	0.965897i	$-0.416632\pi$
$929$	15.7838	0.517851	0.258925	+	0.965897i	$0.416632\pi$
$930$	54.8423	1.79835
$931$	0	0
$932$	−1.01351	−0.0331988
$933$	−55.5480	−1.81856
$934$	−31.8245	−1.04133
$935$	−17.7784	−0.581415
$936$	137.014	4.47846
$937$	28.4718	0.930134	0.465067	−	0.885275i	$-0.346030\pi$
$937$	28.4718	0.930134	0.465067	+	0.885275i	$0.346030\pi$
$938$	0	0
$939$	−44.2801	−1.44503
$940$	−12.0021	−0.391465
$941$	−47.9062	−1.56170	−0.780849	−	0.624720i	$-0.785213\pi$
$941$	−47.9062	−1.56170	−0.780849	+	0.624720i	$0.785213\pi$
$942$	14.8031	0.482312
$943$	−16.5399	−0.538613
$944$	−41.4910	−1.35042
$945$	0	0
$946$	25.9691	0.844329
$947$	13.1474	0.427233	0.213617	−	0.976918i	$-0.431476\pi$
$947$	13.1474	0.427233	0.213617	+	0.976918i	$0.431476\pi$
$948$	−22.8889	−0.743397
$949$	−42.7972	−1.38925
$950$	−2.87763	−0.0933626
$951$	61.7020	2.00082
$952$	0	0
$953$	4.75915	0.154164	0.0770820	−	0.997025i	$-0.475440\pi$
$953$	4.75915	0.154164	0.0770820	+	0.997025i	$0.475440\pi$
$954$	39.4144	1.27609
$955$	42.4126	1.37244
$956$	12.8158	0.414492
$957$	83.3121	2.69310
$958$	−6.99648	−0.226046
$959$	0	0
$960$	−71.5373	−2.30886
$961$	−0.996189	−0.0321351
$962$	−43.2923	−1.39580
$963$	−45.2392	−1.45781
$964$	−6.14356	−0.197871
$965$	21.4621	0.690891
$966$	0	0
$967$	−14.6017	−0.469558	−0.234779	−	0.972049i	$-0.575437\pi$
$967$	−14.6017	−0.469558	−0.234779	+	0.972049i	$0.575437\pi$
$968$	20.9113	0.672115
$969$	−14.2964	−0.459267
$970$	12.8625	0.412990
$971$	−2.56019	−0.0821603	−0.0410801	−	0.999156i	$-0.513080\pi$
$971$	−2.56019	−0.0821603	−0.0410801	+	0.999156i	$0.513080\pi$
$972$	22.4002	0.718487
$973$	0	0
$974$	−31.3271	−1.00379
$975$	−17.3109	−0.554392
$976$	9.33844	0.298916
$977$	20.9064	0.668856	0.334428	−	0.942421i	$-0.391457\pi$
$977$	20.9064	0.668856	0.334428	+	0.942421i	$0.391457\pi$
$978$	−24.4934	−0.783214
$979$	16.4203	0.524796
$980$	0	0
$981$	115.257	3.67988
$982$	37.8381	1.20746
$983$	0.0609831	0.00194506	0.000972530	−	1.00000i	$-0.499690\pi$
$983$	0.0609831	0.00194506	0.000972530		1.00000i	$0.499690\pi$
$984$	−43.4641	−1.38558
$985$	−54.6105	−1.74004
$986$	−12.8518	−0.409283
$987$	0	0
$988$	−6.38051	−0.202991
$989$	−18.9159	−0.601491
$990$	101.322	3.22022
$991$	6.39312	0.203084	0.101542	−	0.994831i	$-0.467622\pi$
$991$	6.39312	0.203084	0.101542	+	0.994831i	$0.467622\pi$
$992$	−13.6196	−0.432424
$993$	−7.99149	−0.253602
$994$	0	0
$995$	−15.5787	−0.493877
$996$	18.9615	0.600819
$997$	−33.1091	−1.04858	−0.524288	−	0.851541i	$-0.675668\pi$
$997$	−33.1091	−1.04858	−0.524288	+	0.851541i	$0.675668\pi$
$998$	35.6449	1.12832
$999$	−99.2726	−3.14084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.6		16
7.6	odd	2		889.2.a.c.1.6	✓	16
21.20	even	2		8001.2.a.t.1.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.6	✓	16	7.6	odd	2
6223.2.a.k.1.6		16	1.1	even	1	trivial
8001.2.a.t.1.11		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)

\(f(q)\)	\(=\)	\(q-1.24549 q^{2} -3.30309 q^{3} -0.448744 q^{4} +2.43368 q^{5} +4.11399 q^{6} +3.04990 q^{8} +7.91044 q^{9} -3.03114 q^{10} -4.22568 q^{11} +1.48224 q^{12} +5.67912 q^{13} -8.03869 q^{15} -2.90114 q^{16} +1.72874 q^{17} -9.85240 q^{18} +2.50366 q^{19} -1.09210 q^{20} +5.26306 q^{22} -3.83361 q^{23} -10.0741 q^{24} +0.922820 q^{25} -7.07331 q^{26} -16.2196 q^{27} +5.96884 q^{29} +10.0121 q^{30} +5.47757 q^{31} -2.48644 q^{32} +13.9578 q^{33} -2.15314 q^{34} -3.54976 q^{36} +6.12052 q^{37} -3.11830 q^{38} -18.7587 q^{39} +7.42249 q^{40} +4.31444 q^{41} +4.93422 q^{43} +1.89625 q^{44} +19.2515 q^{45} +4.77474 q^{46} +10.9899 q^{47} +9.58275 q^{48} -1.14937 q^{50} -5.71020 q^{51} -2.54847 q^{52} -4.00048 q^{53} +20.2015 q^{54} -10.2840 q^{55} -8.26984 q^{57} -7.43416 q^{58} +14.3016 q^{59} +3.60731 q^{60} -3.21888 q^{61} -6.82229 q^{62} +8.89913 q^{64} +13.8212 q^{65} -17.3844 q^{66} +7.39502 q^{67} -0.775763 q^{68} +12.6628 q^{69} +7.63071 q^{71} +24.1260 q^{72} -7.53588 q^{73} -7.62307 q^{74} -3.04816 q^{75} -1.12350 q^{76} +23.3638 q^{78} -15.4421 q^{79} -7.06046 q^{80} +29.8437 q^{81} -5.37361 q^{82} +12.7925 q^{83} +4.20722 q^{85} -6.14555 q^{86} -19.7157 q^{87} -12.8879 q^{88} -3.88584 q^{89} -23.9776 q^{90} +1.72031 q^{92} -18.0929 q^{93} -13.6879 q^{94} +6.09313 q^{95} +8.21294 q^{96} -4.24345 q^{97} -33.4270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more