LMFDB - Embedded newform 6223.2.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [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			1.3
Root			$2.18919$ 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)

$f(q)$	$=$	\(q-2.18919 q^{2} +0.0358012 q^{3} +2.79257 q^{4} -0.868150 q^{5} -0.0783758 q^{6} -1.73509 q^{8} -2.99872 q^{9} +1.90055 q^{10} +0.280542 q^{11} +0.0999773 q^{12} -3.69064 q^{13} -0.0310808 q^{15} -1.78669 q^{16} +1.92048 q^{17} +6.56478 q^{18} -1.42421 q^{19} -2.42437 q^{20} -0.614161 q^{22} -6.52129 q^{23} -0.0621183 q^{24} -4.24632 q^{25} +8.07954 q^{26} -0.214761 q^{27} -3.07387 q^{29} +0.0680419 q^{30} +0.0163952 q^{31} +7.38160 q^{32} +0.0100437 q^{33} -4.20429 q^{34} -8.37413 q^{36} +0.359357 q^{37} +3.11787 q^{38} -0.132129 q^{39} +1.50632 q^{40} -8.75364 q^{41} +4.09079 q^{43} +0.783433 q^{44} +2.60334 q^{45} +14.2764 q^{46} -4.32210 q^{47} -0.0639657 q^{48} +9.29601 q^{50} +0.0687553 q^{51} -10.3064 q^{52} -14.1066 q^{53} +0.470154 q^{54} -0.243552 q^{55} -0.0509884 q^{57} +6.72929 q^{58} +13.7905 q^{59} -0.0867953 q^{60} -0.486241 q^{61} -0.0358923 q^{62} -12.5864 q^{64} +3.20403 q^{65} -0.0219877 q^{66} +0.701192 q^{67} +5.36306 q^{68} -0.233470 q^{69} -9.14768 q^{71} +5.20305 q^{72} +0.554740 q^{73} -0.786702 q^{74} -0.152023 q^{75} -3.97721 q^{76} +0.289257 q^{78} -14.4508 q^{79} +1.55112 q^{80} +8.98847 q^{81} +19.1634 q^{82} +11.1517 q^{83} -1.66726 q^{85} -8.95554 q^{86} -0.110048 q^{87} -0.486765 q^{88} +16.6821 q^{89} -5.69921 q^{90} -18.2111 q^{92} +0.000586969 q^{93} +9.46192 q^{94} +1.23643 q^{95} +0.264270 q^{96} -7.95594 q^{97} -0.841266 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.18919	−1.54799	−0.773997	−	0.633189i	$-0.781745\pi$
$2$	−2.18919	−1.54799	−0.773997	+	0.633189i	$0.781745\pi$
$3$	0.0358012	0.0206698	0.0103349	−	0.999947i	$-0.496710\pi$
$3$	0.0358012	0.0206698	0.0103349	+	0.999947i	$0.496710\pi$
$4$	2.79257	1.39629
$5$	−0.868150	−0.388249	−0.194124	−	0.980977i	$-0.562186\pi$
$5$	−0.868150	−0.388249	−0.194124	+	0.980977i	$0.562186\pi$
$6$	−0.0783758	−0.0319968
$7$	0	0
$7$	0	0
$8$	−1.73509	−0.613447
$9$	−2.99872	−0.999573
$10$	1.90055	0.601006
$11$	0.280542	0.0845866	0.0422933	−	0.999105i	$-0.486534\pi$
$11$	0.280542	0.0845866	0.0422933	+	0.999105i	$0.486534\pi$
$12$	0.0999773	0.0288610
$13$	−3.69064	−1.02360	−0.511800	−	0.859104i	$-0.671021\pi$
$13$	−3.69064	−1.02360	−0.511800	+	0.859104i	$0.671021\pi$
$14$	0	0
$15$	−0.0310808	−0.00802503
$16$	−1.78669	−0.446673
$17$	1.92048	0.465784	0.232892	−	0.972503i	$-0.425181\pi$
$17$	1.92048	0.465784	0.232892	+	0.972503i	$0.425181\pi$
$18$	6.56478	1.54733
$19$	−1.42421	−0.326736	−0.163368	−	0.986565i	$-0.552236\pi$
$19$	−1.42421	−0.326736	−0.163368	+	0.986565i	$0.552236\pi$
$20$	−2.42437	−0.542106
$21$	0	0
$22$	−0.614161	−0.130939
$23$	−6.52129	−1.35978	−0.679891	−	0.733313i	$-0.737973\pi$
$23$	−6.52129	−1.35978	−0.679891	+	0.733313i	$0.737973\pi$
$24$	−0.0621183	−0.0126798
$25$	−4.24632	−0.849263
$26$	8.07954	1.58453
$27$	−0.214761	−0.0413308
$28$	0	0
$29$	−3.07387	−0.570803	−0.285402	−	0.958408i	$-0.592127\pi$
$29$	−3.07387	−0.570803	−0.285402	+	0.958408i	$0.592127\pi$
$30$	0.0680419	0.0124227
$31$	0.0163952	0.00294467	0.00147234	−	0.999999i	$-0.499531\pi$
$31$	0.0163952	0.00294467	0.00147234	+	0.999999i	$0.499531\pi$
$32$	7.38160	1.30489
$33$	0.0100437	0.00174839
$34$	−4.20429	−0.721031
$35$	0	0
$36$	−8.37413	−1.39569
$37$	0.359357	0.0590779	0.0295389	−	0.999564i	$-0.490596\pi$
$37$	0.359357	0.0590779	0.0295389	+	0.999564i	$0.490596\pi$
$38$	3.11787	0.505786
$39$	−0.132129	−0.0211576
$40$	1.50632	0.238170
$41$	−8.75364	−1.36709	−0.683544	−	0.729909i	$-0.739562\pi$
$41$	−8.75364	−1.36709	−0.683544	+	0.729909i	$0.739562\pi$
$42$	0	0
$43$	4.09079	0.623840	0.311920	−	0.950108i	$-0.399028\pi$
$43$	4.09079	0.623840	0.311920	+	0.950108i	$0.399028\pi$
$44$	0.783433	0.118107
$45$	2.60334	0.388083
$46$	14.2764	2.10493
$47$	−4.32210	−0.630443	−0.315222	−	0.949018i	$-0.602079\pi$
$47$	−4.32210	−0.630443	−0.315222	+	0.949018i	$0.602079\pi$
$48$	−0.0639657	−0.00923266
$49$	0	0
$50$	9.29601	1.31465
$51$	0.0687553	0.00962767
$52$	−10.3064	−1.42924
$53$	−14.1066	−1.93769	−0.968846	−	0.247662i	$-0.920338\pi$
$53$	−14.1066	−1.93769	−0.968846	+	0.247662i	$0.920338\pi$
$54$	0.470154	0.0639799
$55$	−0.243552	−0.0328406
$56$	0	0
$57$	−0.0509884	−0.00675359
$58$	6.72929	0.883600
$59$	13.7905	1.79537	0.897685	−	0.440638i	$-0.145248\pi$
$59$	13.7905	1.79537	0.897685	+	0.440638i	$0.145248\pi$
$60$	−0.0867953	−0.0112052
$61$	−0.486241	−0.0622568	−0.0311284	−	0.999515i	$-0.509910\pi$
$61$	−0.486241	−0.0622568	−0.0311284	+	0.999515i	$0.509910\pi$
$62$	−0.0358923	−0.00455833
$63$	0	0
$64$	−12.5864	−1.57329
$65$	3.20403	0.397411
$66$	−0.0219877	−0.00270650
$67$	0.701192	0.0856643	0.0428321	−	0.999082i	$-0.486362\pi$
$67$	0.701192	0.0856643	0.0428321	+	0.999082i	$0.486362\pi$
$68$	5.36306	0.650367
$69$	−0.233470	−0.0281065
$70$	0	0
$71$	−9.14768	−1.08563	−0.542815	−	0.839852i	$-0.682642\pi$
$71$	−9.14768	−1.08563	−0.542815	+	0.839852i	$0.682642\pi$
$72$	5.20305	0.613185
$73$	0.554740	0.0649273	0.0324637	−	0.999473i	$-0.489665\pi$
$73$	0.554740	0.0649273	0.0324637	+	0.999473i	$0.489665\pi$
$74$	−0.786702	−0.0914522
$75$	−0.152023	−0.0175541
$76$	−3.97721	−0.456217
$77$	0	0
$78$	0.289257	0.0327519
$79$	−14.4508	−1.62584	−0.812921	−	0.582374i	$-0.802124\pi$
$79$	−14.4508	−1.62584	−0.812921	+	0.582374i	$0.802124\pi$
$80$	1.55112	0.173420
$81$	8.98847	0.998718
$82$	19.1634	2.11624
$83$	11.1517	1.22405	0.612027	−	0.790837i	$-0.290354\pi$
$83$	11.1517	1.22405	0.612027	+	0.790837i	$0.290354\pi$
$84$	0	0
$85$	−1.66726	−0.180840
$86$	−8.95554	−0.965700
$87$	−0.110048	−0.0117984
$88$	−0.486765	−0.0518894
$89$	16.6821	1.76830	0.884149	−	0.467205i	$-0.154739\pi$
$89$	16.6821	1.76830	0.884149	+	0.467205i	$0.154739\pi$
$90$	−5.69921	−0.600750
$91$	0	0
$92$	−18.2111	−1.89864
$93$	0.000586969 0	6.08658e−5 0
$94$	9.46192	0.975922
$95$	1.23643	0.126855
$96$	0.264270	0.0269719
$97$	−7.95594	−0.807803	−0.403902	−	0.914802i	$-0.632346\pi$
$97$	−7.95594	−0.807803	−0.403902	+	0.914802i	$0.632346\pi$
$98$	0	0
$99$	−0.841266	−0.0845504
$100$	−11.8581	−1.18581
$101$	−5.03438	−0.500939	−0.250470	−	0.968124i	$-0.580585\pi$
$101$	−5.03438	−0.500939	−0.250470	+	0.968124i	$0.580585\pi$
$102$	−0.150519	−0.0149036
$103$	−2.57322	−0.253547	−0.126774	−	0.991932i	$-0.540462\pi$
$103$	−2.57322	−0.253547	−0.126774	+	0.991932i	$0.540462\pi$
$104$	6.40360	0.627925
$105$	0	0
$106$	30.8821	2.99954
$107$	−6.71132	−0.648808	−0.324404	−	0.945919i	$-0.605164\pi$
$107$	−6.71132	−0.648808	−0.324404	+	0.945919i	$0.605164\pi$
$108$	−0.599736	−0.0577096
$109$	1.76219	0.168787	0.0843935	−	0.996433i	$-0.473105\pi$
$109$	1.76219	0.168787	0.0843935	+	0.996433i	$0.473105\pi$
$110$	0.533184	0.0508371
$111$	0.0128654	0.00122113
$112$	0	0
$113$	−8.28904	−0.779768	−0.389884	−	0.920864i	$-0.627485\pi$
$113$	−8.28904	−0.779768	−0.389884	+	0.920864i	$0.627485\pi$
$114$	0.111624	0.0104545
$115$	5.66145	0.527933
$116$	−8.58399	−0.797004
$117$	11.0672	1.02316
$118$	−30.1901	−2.77922
$119$	0	0
$120$	0.0539280	0.00492293
$121$	−10.9213	−0.992845
$122$	1.06448	0.0963732
$123$	−0.313391	−0.0282575
$124$	0.0457848	0.00411160
$125$	8.02719	0.717974
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	12.7908	1.13056
$129$	0.146455	0.0128947
$130$	−7.01425	−0.615190
$131$	15.0169	1.31203	0.656015	−	0.754748i	$-0.272241\pi$
$131$	15.0169	1.31203	0.656015	+	0.754748i	$0.272241\pi$
$132$	0.0280478	0.00244125
$133$	0	0
$134$	−1.53505	−0.132608
$135$	0.186445	0.0160466
$136$	−3.33220	−0.285734
$137$	16.5669	1.41540	0.707702	−	0.706511i	$-0.249732\pi$
$137$	16.5669	1.41540	0.707702	+	0.706511i	$0.249732\pi$
$138$	0.511111	0.0435086
$139$	14.5682	1.23566	0.617830	−	0.786312i	$-0.288012\pi$
$139$	14.5682	1.23566	0.617830	+	0.786312i	$0.288012\pi$
$140$	0	0
$141$	−0.154736	−0.0130312
$142$	20.0261	1.68055
$143$	−1.03538	−0.0865828
$144$	5.35779	0.446482
$145$	2.66858	0.221613
$146$	−1.21443	−0.100507
$147$	0	0
$148$	1.00353	0.0824896
$149$	−11.3687	−0.931364	−0.465682	−	0.884952i	$-0.654191\pi$
$149$	−11.3687	−0.931364	−0.465682	+	0.884952i	$0.654191\pi$
$150$	0.332808	0.0271737
$151$	−9.59570	−0.780887	−0.390443	−	0.920627i	$-0.627678\pi$
$151$	−9.59570	−0.780887	−0.390443	+	0.920627i	$0.627678\pi$
$152$	2.47113	0.200435
$153$	−5.75897	−0.465585
$154$	0	0
$155$	−0.0142335	−0.00114326
$156$	−0.368981	−0.0295421
$157$	−10.8227	−0.863743	−0.431871	−	0.901935i	$-0.642147\pi$
$157$	−10.8227	−0.863743	−0.431871	+	0.901935i	$0.642147\pi$
$158$	31.6356	2.51679
$159$	−0.505034	−0.0400518
$160$	−6.40833	−0.506623
$161$	0	0
$162$	−19.6775	−1.54601
$163$	24.3493	1.90718	0.953590	−	0.301107i	$-0.0973561\pi$
$163$	24.3493	1.90718	0.953590	+	0.301107i	$0.0973561\pi$
$164$	−24.4451	−1.90885
$165$	−0.00871947	−0.000678810 0
$166$	−24.4131	−1.89483
$167$	−14.3860	−1.11322	−0.556611	−	0.830773i	$-0.687899\pi$
$167$	−14.3860	−1.11322	−0.556611	+	0.830773i	$0.687899\pi$
$168$	0	0
$169$	0.620853	0.0477579
$170$	3.64996	0.279939
$171$	4.27081	0.326597
$172$	11.4238	0.871058
$173$	20.4335	1.55353	0.776765	−	0.629790i	$-0.216859\pi$
$173$	20.4335	1.55353	0.776765	+	0.629790i	$0.216859\pi$
$174$	0.240917	0.0182639
$175$	0	0
$176$	−0.501242	−0.0377825
$177$	0.493716	0.0371100
$178$	−36.5203	−2.73731
$179$	−1.77187	−0.132436	−0.0662178	−	0.997805i	$-0.521093\pi$
$179$	−1.77187	−0.132436	−0.0662178	+	0.997805i	$0.521093\pi$
$180$	7.27000	0.541874
$181$	−13.7325	−1.02073	−0.510365	−	0.859958i	$-0.670490\pi$
$181$	−13.7325	−1.02073	−0.510365	+	0.859958i	$0.670490\pi$
$182$	0	0
$183$	−0.0174080	−0.00128684
$184$	11.3150	0.834154
$185$	−0.311976	−0.0229369
$186$	−0.00128499	−9.42199e−5 0
$187$	0.538774	0.0393991
$188$	−12.0698	−0.880279
$189$	0	0
$190$	−2.70678	−0.196371
$191$	7.22638	0.522883	0.261441	−	0.965219i	$-0.415802\pi$
$191$	7.22638	0.522883	0.261441	+	0.965219i	$0.415802\pi$
$192$	−0.450607	−0.0325197
$193$	13.0486	0.939255	0.469628	−	0.882865i	$-0.344388\pi$
$193$	13.0486	0.939255	0.469628	+	0.882865i	$0.344388\pi$
$194$	17.4171	1.25047
$195$	0.114708	0.00821442
$196$	0	0
$197$	−3.12406	−0.222580	−0.111290	−	0.993788i	$-0.535498\pi$
$197$	−3.12406	−0.222580	−0.111290	+	0.993788i	$0.535498\pi$
$198$	1.84169	0.130884
$199$	10.9671	0.777438	0.388719	−	0.921356i	$-0.372918\pi$
$199$	10.9671	0.777438	0.388719	+	0.921356i	$0.372918\pi$
$200$	7.36774	0.520978
$201$	0.0251035	0.00177067
$202$	11.0212	0.775451
$203$	0	0
$204$	0.192004	0.0134430
$205$	7.59947	0.530770
$206$	5.63328	0.392489
$207$	19.5555	1.35920
$208$	6.59404	0.457215
$209$	−0.399551	−0.0276375
$210$	0	0
$211$	−13.8674	−0.954668	−0.477334	−	0.878722i	$-0.658397\pi$
$211$	−13.8674	−0.954668	−0.477334	+	0.878722i	$0.658397\pi$
$212$	−39.3937	−2.70557
$213$	−0.327498	−0.0224398
$214$	14.6924	1.00435
$215$	−3.55142	−0.242205
$216$	0.372630	0.0253543
$217$	0	0
$218$	−3.85777	−0.261281
$219$	0.0198603	0.00134204
$220$	−0.680137	−0.0458548
$221$	−7.08779	−0.476777
$222$	−0.0281649	−0.00189030
$223$	−9.77971	−0.654898	−0.327449	−	0.944869i	$-0.606189\pi$
$223$	−9.77971	−0.654898	−0.327449	+	0.944869i	$0.606189\pi$
$224$	0	0
$225$	12.7335	0.848900
$226$	18.1463	1.20708
$227$	12.3932	0.822563	0.411282	−	0.911508i	$-0.365081\pi$
$227$	12.3932	0.822563	0.411282	+	0.911508i	$0.365081\pi$
$228$	−0.142389	−0.00942993
$229$	−24.5147	−1.61998	−0.809988	−	0.586447i	$-0.800526\pi$
$229$	−24.5147	−1.61998	−0.809988	+	0.586447i	$0.800526\pi$
$230$	−12.3940	−0.817238
$231$	0	0
$232$	5.33344	0.350157
$233$	−7.75695	−0.508175	−0.254087	−	0.967181i	$-0.581775\pi$
$233$	−7.75695	−0.508175	−0.254087	+	0.967181i	$0.581775\pi$
$234$	−24.2283	−1.58385
$235$	3.75223	0.244769
$236$	38.5109	2.50685
$237$	−0.517356	−0.0336059
$238$	0	0
$239$	12.1161	0.783723	0.391862	−	0.920024i	$-0.371831\pi$
$239$	12.1161	0.783723	0.391862	+	0.920024i	$0.371831\pi$
$240$	0.0555318	0.00358456
$241$	−3.36131	−0.216521	−0.108261	−	0.994123i	$-0.534528\pi$
$241$	−3.36131	−0.216521	−0.108261	+	0.994123i	$0.534528\pi$
$242$	23.9088	1.53692
$243$	0.966082	0.0619742
$244$	−1.35786	−0.0869283
$245$	0	0
$246$	0.686073	0.0437424
$247$	5.25626	0.334448
$248$	−0.0284472	−0.00180640
$249$	0.399243	0.0253010
$250$	−17.5731	−1.11142
$251$	23.2358	1.46663	0.733314	−	0.679890i	$-0.237973\pi$
$251$	23.2358	1.46663	0.733314	+	0.679890i	$0.237973\pi$
$252$	0	0
$253$	−1.82949	−0.115019
$254$	2.18919	0.137362
$255$	−0.0596900	−0.00373793
$256$	−2.82880	−0.176800
$257$	−19.4430	−1.21282	−0.606411	−	0.795151i	$-0.707392\pi$
$257$	−19.4430	−1.21282	−0.606411	+	0.795151i	$0.707392\pi$
$258$	−0.320619	−0.0199609
$259$	0	0
$260$	8.94749	0.554900
$261$	9.21767	0.570559
$262$	−32.8749	−2.03102
$263$	−4.25647	−0.262465	−0.131233	−	0.991352i	$-0.541894\pi$
$263$	−4.25647	−0.262465	−0.131233	+	0.991352i	$0.541894\pi$
$264$	−0.0174268	−0.00107254
$265$	12.2467	0.752306
$266$	0	0
$267$	0.597239	0.0365504
$268$	1.95813	0.119612
$269$	−4.00443	−0.244154	−0.122077	−	0.992521i	$-0.538956\pi$
$269$	−4.00443	−0.244154	−0.122077	+	0.992521i	$0.538956\pi$
$270$	−0.408164	−0.0248401
$271$	−0.684262	−0.0415659	−0.0207830	−	0.999784i	$-0.506616\pi$
$271$	−0.684262	−0.0415659	−0.0207830	+	0.999784i	$0.506616\pi$
$272$	−3.43130	−0.208053
$273$	0	0
$274$	−36.2681	−2.19104
$275$	−1.19127	−0.0718362
$276$	−0.651981	−0.0392446
$277$	5.19540	0.312162	0.156081	−	0.987744i	$-0.450114\pi$
$277$	5.19540	0.312162	0.156081	+	0.987744i	$0.450114\pi$
$278$	−31.8926	−1.91279
$279$	−0.0491647	−0.00294341
$280$	0	0
$281$	23.1569	1.38143	0.690713	−	0.723129i	$-0.257297\pi$
$281$	23.1569	1.38143	0.690713	+	0.723129i	$0.257297\pi$
$282$	0.338748	0.0201721
$283$	0.107190	0.00637180	0.00318590	−	0.999995i	$-0.498986\pi$
$283$	0.107190	0.00637180	0.00318590	+	0.999995i	$0.498986\pi$
$284$	−25.5455	−1.51585
$285$	0.0442656	0.00262207
$286$	2.26665	0.134030
$287$	0	0
$288$	−22.1353	−1.30434
$289$	−13.3118	−0.783045
$290$	−5.84204	−0.343056
$291$	−0.284832	−0.0166972
$292$	1.54915	0.0906571
$293$	−10.5671	−0.617337	−0.308668	−	0.951170i	$-0.599883\pi$
$293$	−10.5671	−0.617337	−0.308668	+	0.951170i	$0.599883\pi$
$294$	0	0
$295$	−11.9722	−0.697050
$296$	−0.623516	−0.0362411
$297$	−0.0602495	−0.00349603
$298$	24.8884	1.44175
$299$	24.0677	1.39187
$300$	−0.424535	−0.0245106
$301$	0	0
$302$	21.0068	1.20881
$303$	−0.180237	−0.0103543
$304$	2.54463	0.145944
$305$	0.422130	0.0241711
$306$	12.6075	0.720723
$307$	3.41074	0.194661	0.0973307	−	0.995252i	$-0.468970\pi$
$307$	3.41074	0.194661	0.0973307	+	0.995252i	$0.468970\pi$
$308$	0	0
$309$	−0.0921244	−0.00524078
$310$	0.0311599	0.00176977
$311$	−29.7332	−1.68602	−0.843008	−	0.537901i	$-0.819218\pi$
$311$	−29.7332	−1.68602	−0.843008	+	0.537901i	$0.819218\pi$
$312$	0.229256	0.0129791
$313$	−21.5328	−1.21710	−0.608552	−	0.793514i	$-0.708249\pi$
$313$	−21.5328	−1.21710	−0.608552	+	0.793514i	$0.708249\pi$
$314$	23.6929	1.33707
$315$	0	0
$316$	−40.3549	−2.27014
$317$	28.3682	1.59332	0.796660	−	0.604428i	$-0.206598\pi$
$317$	28.3682	1.59332	0.796660	+	0.604428i	$0.206598\pi$
$318$	1.10562	0.0619999
$319$	−0.862349	−0.0482823
$320$	10.9268	0.610829
$321$	−0.240273	−0.0134107
$322$	0	0
$323$	−2.73516	−0.152189
$324$	25.1009	1.39450
$325$	15.6716	0.869306
$326$	−53.3052	−2.95230
$327$	0.0630884	0.00348880
$328$	15.1883	0.838636
$329$	0	0
$330$	0.0190886	0.00105079
$331$	−7.53693	−0.414267	−0.207134	−	0.978313i	$-0.566414\pi$
$331$	−7.53693	−0.414267	−0.207134	+	0.978313i	$0.566414\pi$
$332$	31.1418	1.70913
$333$	−1.07761	−0.0590527
$334$	31.4938	1.72326
$335$	−0.608740	−0.0332590
$336$	0	0
$337$	11.9671	0.651888	0.325944	−	0.945389i	$-0.394318\pi$
$337$	11.9671	0.651888	0.325944	+	0.945389i	$0.394318\pi$
$338$	−1.35917	−0.0739289
$339$	−0.296758	−0.0161177
$340$	−4.65594	−0.252504
$341$	0.00459955	0.000249080 0
$342$	−9.34963	−0.505570
$343$	0	0
$344$	−7.09789	−0.382693
$345$	0.202687	0.0109123
$346$	−44.7329	−2.40486
$347$	4.07527	0.218772	0.109386	−	0.993999i	$-0.465112\pi$
$347$	4.07527	0.218772	0.109386	+	0.993999i	$0.465112\pi$
$348$	−0.307317	−0.0164739
$349$	2.38630	0.127736	0.0638679	−	0.997958i	$-0.479656\pi$
$349$	2.38630	0.127736	0.0638679	+	0.997958i	$0.479656\pi$
$350$	0	0
$351$	0.792607	0.0423063
$352$	2.07085	0.110376
$353$	−19.0735	−1.01518	−0.507591	−	0.861598i	$-0.669464\pi$
$353$	−19.0735	−1.01518	−0.507591	+	0.861598i	$0.669464\pi$
$354$	−1.08084	−0.0574460
$355$	7.94156	0.421494
$356$	46.5859	2.46905
$357$	0	0
$358$	3.87896	0.205010
$359$	1.49830	0.0790775	0.0395387	−	0.999218i	$-0.487411\pi$
$359$	1.49830	0.0790775	0.0395387	+	0.999218i	$0.487411\pi$
$360$	−4.51702	−0.238068
$361$	−16.9716	−0.893243
$362$	30.0631	1.58008
$363$	−0.390995	−0.0205219
$364$	0	0
$365$	−0.481597	−0.0252079
$366$	0.0381095	0.00199202
$367$	−10.1206	−0.528291	−0.264146	−	0.964483i	$-0.585090\pi$
$367$	−10.1206	−0.528291	−0.264146	+	0.964483i	$0.585090\pi$
$368$	11.6515	0.607378
$369$	26.2497	1.36650
$370$	0.682975	0.0355062
$371$	0	0
$372$	0.00163915	8.49861e−5 0
$373$	−25.6071	−1.32588	−0.662942	−	0.748671i	$-0.730693\pi$
$373$	−25.6071	−1.32588	−0.662942	+	0.748671i	$0.730693\pi$
$374$	−1.17948	−0.0609895
$375$	0.287383	0.0148404
$376$	7.49923	0.386743
$377$	11.3446	0.584274
$378$	0	0
$379$	−13.9735	−0.717771	−0.358886	−	0.933382i	$-0.616843\pi$
$379$	−13.9735	−0.717771	−0.358886	+	0.933382i	$0.616843\pi$
$380$	3.45281	0.177126
$381$	−0.0358012	−0.00183415
$382$	−15.8199	−0.809419
$383$	25.7039	1.31341	0.656705	−	0.754147i	$-0.271950\pi$
$383$	25.7039	1.31341	0.656705	+	0.754147i	$0.271950\pi$
$384$	0.457926	0.0233684
$385$	0	0
$386$	−28.5658	−1.45396
$387$	−12.2671	−0.623573
$388$	−22.2175	−1.12792
$389$	4.06415	0.206061	0.103030	−	0.994678i	$-0.467146\pi$
$389$	4.06415	0.206061	0.103030	+	0.994678i	$0.467146\pi$
$390$	−0.251118	−0.0127159
$391$	−12.5240	−0.633365
$392$	0	0
$393$	0.537622	0.0271194
$394$	6.83918	0.344553
$395$	12.5455	0.631231
$396$	−2.34929	−0.118056
$397$	−1.03102	−0.0517453	−0.0258726	−	0.999665i	$-0.508236\pi$
$397$	−1.03102	−0.0517453	−0.0258726	+	0.999665i	$0.508236\pi$
$398$	−24.0091	−1.20347
$399$	0	0
$400$	7.58686	0.379343
$401$	−3.59965	−0.179758	−0.0898789	−	0.995953i	$-0.528648\pi$
$401$	−3.59965	−0.179758	−0.0898789	+	0.995953i	$0.528648\pi$
$402$	−0.0549565	−0.00274098
$403$	−0.0605090	−0.00301417
$404$	−14.0588	−0.699454
$405$	−7.80334	−0.387751
$406$	0	0
$407$	0.100815	0.00499720
$408$	−0.119297	−0.00590607
$409$	21.8694	1.08137	0.540685	−	0.841225i	$-0.318165\pi$
$409$	21.8694	1.08137	0.540685	+	0.841225i	$0.318165\pi$
$410$	−16.6367	−0.821629
$411$	0.593114	0.0292562
$412$	−7.18590	−0.354024
$413$	0	0
$414$	−42.8108	−2.10403
$415$	−9.68131	−0.475237
$416$	−27.2428	−1.33569
$417$	0.521559	0.0255409
$418$	0.874694	0.0427827
$419$	14.5188	0.709288	0.354644	−	0.935001i	$-0.384602\pi$
$419$	14.5188	0.709288	0.354644	+	0.935001i	$0.384602\pi$
$420$	0	0
$421$	1.74372	0.0849836	0.0424918	−	0.999097i	$-0.486470\pi$
$421$	1.74372	0.0849836	0.0424918	+	0.999097i	$0.486470\pi$
$422$	30.3583	1.47782
$423$	12.9608	0.630174
$424$	24.4762	1.18867
$425$	−8.15495	−0.395573
$426$	0.716957	0.0347367
$427$	0	0
$428$	−18.7418	−0.905920
$429$	−0.0370678	−0.00178965
$430$	7.77475	0.374932
$431$	24.6850	1.18904	0.594518	−	0.804082i	$-0.297343\pi$
$431$	24.6850	1.18904	0.594518	+	0.804082i	$0.297343\pi$
$432$	0.383712	0.0184614
$433$	16.9930	0.816633	0.408316	−	0.912840i	$-0.366116\pi$
$433$	16.9930	0.816633	0.408316	+	0.912840i	$0.366116\pi$
$434$	0	0
$435$	0.0955383	0.00458071
$436$	4.92103	0.235675
$437$	9.28769	0.444290
$438$	−0.0434781	−0.00207747
$439$	−9.88431	−0.471752	−0.235876	−	0.971783i	$-0.575796\pi$
$439$	−9.88431	−0.471752	−0.235876	+	0.971783i	$0.575796\pi$
$440$	0.422585	0.0201460
$441$	0	0
$442$	15.5166	0.738047
$443$	39.8440	1.89305	0.946523	−	0.322636i	$-0.104569\pi$
$443$	39.8440	1.89305	0.946523	+	0.322636i	$0.104569\pi$
$444$	0.0359275	0.00170505
$445$	−14.4826	−0.686539
$446$	21.4097	1.01378
$447$	−0.407015	−0.0192511
$448$	0	0
$449$	24.0250	1.13381	0.566904	−	0.823784i	$-0.308141\pi$
$449$	24.0250	1.13381	0.566904	+	0.823784i	$0.308141\pi$
$450$	−27.8761	−1.31409
$451$	−2.45576	−0.115637
$452$	−23.1477	−1.08878
$453$	−0.343537	−0.0161408
$454$	−27.1310	−1.27332
$455$	0	0
$456$	0.0884695	0.00414297
$457$	3.81846	0.178620	0.0893101	−	0.996004i	$-0.471534\pi$
$457$	3.81846	0.178620	0.0893101	+	0.996004i	$0.471534\pi$
$458$	53.6674	2.50771
$459$	−0.412444	−0.0192512
$460$	15.8100	0.737145
$461$	27.3775	1.27510	0.637549	−	0.770409i	$-0.279948\pi$
$461$	27.3775	1.27510	0.637549	+	0.770409i	$0.279948\pi$
$462$	0	0
$463$	−6.11153	−0.284027	−0.142013	−	0.989865i	$-0.545358\pi$
$463$	−6.11153	−0.284027	−0.142013	+	0.989865i	$0.545358\pi$
$464$	5.49206	0.254962
$465$	−0.000509577 0	−2.36311e−5 0
$466$	16.9815	0.786652
$467$	−23.4738	−1.08624	−0.543118	−	0.839656i	$-0.682756\pi$
$467$	−23.4738	−1.08624	−0.543118	+	0.839656i	$0.682756\pi$
$468$	30.9059	1.42863
$469$	0	0
$470$	−8.21437	−0.378900
$471$	−0.387464	−0.0178534
$472$	−23.9278	−1.10136
$473$	1.14764	0.0527685
$474$	1.13259	0.0520217
$475$	6.04765	0.277485
$476$	0	0
$477$	42.3018	1.93687
$478$	−26.5244	−1.21320
$479$	30.4428	1.39097	0.695484	−	0.718541i	$-0.255190\pi$
$479$	30.4428	1.39097	0.695484	+	0.718541i	$0.255190\pi$
$480$	−0.229426	−0.0104718
$481$	−1.32626	−0.0604722
$482$	7.35857	0.335173
$483$	0	0
$484$	−30.4985	−1.38629
$485$	6.90695	0.313628
$486$	−2.11494	−0.0959356
$487$	−43.9485	−1.99150	−0.995749	−	0.0921094i	$-0.970639\pi$
$487$	−43.9485	−1.99150	−0.995749	+	0.0921094i	$0.970639\pi$
$488$	0.843672	0.0381912
$489$	0.871732	0.0394211
$490$	0	0
$491$	−4.76508	−0.215045	−0.107522	−	0.994203i	$-0.534292\pi$
$491$	−4.76508	−0.215045	−0.107522	+	0.994203i	$0.534292\pi$
$492$	−0.875165	−0.0394555
$493$	−5.90329	−0.265871
$494$	−11.5070	−0.517723
$495$	0.730345	0.0328266
$496$	−0.0292932	−0.00131531
$497$	0	0
$498$	−0.874020	−0.0391658
$499$	−9.87038	−0.441859	−0.220929	−	0.975290i	$-0.570909\pi$
$499$	−9.87038	−0.441859	−0.220929	+	0.975290i	$0.570909\pi$
$500$	22.4165	1.00250
$501$	−0.515036	−0.0230101
$502$	−50.8676	−2.27033
$503$	29.9723	1.33640	0.668200	−	0.743982i	$-0.267065\pi$
$503$	29.9723	1.33640	0.668200	+	0.743982i	$0.267065\pi$
$504$	0	0
$505$	4.37059	0.194489
$506$	4.00512	0.178049
$507$	0.0222273	0.000987148 0
$508$	−2.79257	−0.123900
$509$	−33.7727	−1.49695	−0.748474	−	0.663165i	$-0.769213\pi$
$509$	−33.7727	−1.49695	−0.748474	+	0.663165i	$0.769213\pi$
$510$	0.130673	0.00578629
$511$	0	0
$512$	−19.3888	−0.856871
$513$	0.305865	0.0135043
$514$	42.5646	1.87744
$515$	2.23394	0.0984393
$516$	0.408986	0.0180046
$517$	−1.21253	−0.0533270
$518$	0	0
$519$	0.731544	0.0321112
$520$	−5.55928	−0.243791
$521$	−23.4277	−1.02639	−0.513194	−	0.858273i	$-0.671538\pi$
$521$	−23.4277	−1.02639	−0.513194	+	0.858273i	$0.671538\pi$
$522$	−20.1793	−0.883222
$523$	−0.337038	−0.0147376	−0.00736882	−	0.999973i	$-0.502346\pi$
$523$	−0.337038	−0.0147376	−0.00736882	+	0.999973i	$0.502346\pi$
$524$	41.9357	1.83197
$525$	0	0
$526$	9.31825	0.406295
$527$	0.0314867	0.00137158
$528$	−0.0179451	−0.000780959 0
$529$	19.5272	0.849007
$530$	−26.8103	−1.16457
$531$	−41.3538	−1.79460
$532$	0	0
$533$	32.3066	1.39935
$534$	−1.30747	−0.0565798
$535$	5.82643	0.251899
$536$	−1.21663	−0.0525505
$537$	−0.0634350	−0.00273742
$538$	8.76646	0.377949
$539$	0	0
$540$	0.520661	0.0224057
$541$	5.85948	0.251919	0.125959	−	0.992035i	$-0.459799\pi$
$541$	5.85948	0.251919	0.125959	+	0.992035i	$0.459799\pi$
$542$	1.49798	0.0643438
$543$	−0.491640	−0.0210983
$544$	14.1762	0.607799
$545$	−1.52984	−0.0655313
$546$	0	0
$547$	31.8131	1.36023	0.680114	−	0.733106i	$-0.261930\pi$
$547$	31.8131	1.36023	0.680114	+	0.733106i	$0.261930\pi$
$548$	46.2642	1.97631
$549$	1.45810	0.0622302
$550$	2.60792	0.111202
$551$	4.37784	0.186502
$552$	0.405091	0.0172418
$553$	0	0
$554$	−11.3737	−0.483224
$555$	−0.0111691	−0.000474102 0
$556$	40.6827	1.72533
$557$	−7.19289	−0.304773	−0.152386	−	0.988321i	$-0.548696\pi$
$557$	−7.19289	−0.304773	−0.152386	+	0.988321i	$0.548696\pi$
$558$	0.107631	0.00455638
$559$	−15.0977	−0.638563
$560$	0	0
$561$	0.0192887	0.000814372 0
$562$	−50.6950	−2.13844
$563$	−28.3208	−1.19358	−0.596789	−	0.802398i	$-0.703557\pi$
$563$	−28.3208	−1.19358	−0.596789	+	0.802398i	$0.703557\pi$
$564$	−0.432112	−0.0181952
$565$	7.19613	0.302744
$566$	−0.234660	−0.00986351
$567$	0	0
$568$	15.8721	0.665977
$569$	26.6346	1.11658	0.558290	−	0.829646i	$-0.311458\pi$
$569$	26.6346	1.11658	0.558290	+	0.829646i	$0.311458\pi$
$570$	−0.0969060	−0.00405895
$571$	15.4515	0.646625	0.323313	−	0.946292i	$-0.395203\pi$
$571$	15.4515	0.646625	0.323313	+	0.946292i	$0.395203\pi$
$572$	−2.89137	−0.120894
$573$	0.258713	0.0108079
$574$	0	0
$575$	27.6914	1.15481
$576$	37.7429	1.57262
$577$	24.0575	1.00153	0.500763	−	0.865585i	$-0.333053\pi$
$577$	24.0575	1.00153	0.500763	+	0.865585i	$0.333053\pi$
$578$	29.1420	1.21215
$579$	0.467154	0.0194142
$580$	7.45219	0.309436
$581$	0	0
$582$	0.623553	0.0258471
$583$	−3.95750	−0.163903
$584$	−0.962523	−0.0398295
$585$	−9.60799	−0.397242
$586$	23.1334	0.955634
$587$	38.3645	1.58347	0.791736	−	0.610863i	$-0.209177\pi$
$587$	38.3645	1.58347	0.791736	+	0.610863i	$0.209177\pi$
$588$	0	0
$589$	−0.0233503	−0.000962131 0
$590$	26.2095	1.07903
$591$	−0.111845	−0.00460069
$592$	−0.642060	−0.0263885
$593$	39.1039	1.60580	0.802902	−	0.596111i	$-0.203288\pi$
$593$	39.1039	1.60580	0.802902	+	0.596111i	$0.203288\pi$
$594$	0.131898	0.00541184
$595$	0	0
$596$	−31.7480	−1.30045
$597$	0.392636	0.0160695
$598$	−52.6890	−2.15461
$599$	−34.6581	−1.41609	−0.708045	−	0.706167i	$-0.750423\pi$
$599$	−34.6581	−1.41609	−0.708045	+	0.706167i	$0.750423\pi$
$600$	0.263774	0.0107685
$601$	−31.9773	−1.30438	−0.652190	−	0.758056i	$-0.726150\pi$
$601$	−31.9773	−1.30438	−0.652190	+	0.758056i	$0.726150\pi$
$602$	0	0
$603$	−2.10268	−0.0856277
$604$	−26.7967	−1.09034
$605$	9.48132	0.385471
$606$	0.394573	0.0160284
$607$	24.7650	1.00518	0.502591	−	0.864525i	$-0.332380\pi$
$607$	24.7650	1.00518	0.502591	+	0.864525i	$0.332380\pi$
$608$	−10.5129	−0.426356
$609$	0	0
$610$	−0.924125	−0.0374167
$611$	15.9513	0.645322
$612$	−16.0823	−0.650089
$613$	28.3938	1.14681	0.573407	−	0.819271i	$-0.305621\pi$
$613$	28.3938	1.14681	0.573407	+	0.819271i	$0.305621\pi$
$614$	−7.46678	−0.301335
$615$	0.272070	0.0109709
$616$	0	0
$617$	5.98355	0.240889	0.120444	−	0.992720i	$-0.461568\pi$
$617$	5.98355	0.240889	0.120444	+	0.992720i	$0.461568\pi$
$618$	0.201678	0.00811269
$619$	−4.70457	−0.189093	−0.0945464	−	0.995520i	$-0.530140\pi$
$619$	−4.70457	−0.189093	−0.0945464	+	0.995520i	$0.530140\pi$
$620$	−0.0397481	−0.00159632
$621$	1.40052	0.0562009
$622$	65.0918	2.60994
$623$	0	0
$624$	0.236075	0.00945055
$625$	14.2628	0.570511
$626$	47.1394	1.88407
$627$	−0.0143044	−0.000571262 0
$628$	−30.2231	−1.20603
$629$	0.690136	0.0275175
$630$	0	0
$631$	−21.1503	−0.841978	−0.420989	−	0.907066i	$-0.638317\pi$
$631$	−21.1503	−0.841978	−0.420989	+	0.907066i	$0.638317\pi$
$632$	25.0734	0.997367
$633$	−0.496468	−0.0197328
$634$	−62.1036	−2.46645
$635$	0.868150	0.0344515
$636$	−1.41034	−0.0559237
$637$	0	0
$638$	1.88785	0.0747406
$639$	27.4313	1.08517
$640$	−11.1043	−0.438937
$641$	8.70143	0.343686	0.171843	−	0.985124i	$-0.445028\pi$
$641$	8.70143	0.343686	0.171843	+	0.985124i	$0.445028\pi$
$642$	0.526005	0.0207597
$643$	12.9855	0.512100	0.256050	−	0.966664i	$-0.417579\pi$
$643$	12.9855	0.512100	0.256050	+	0.966664i	$0.417579\pi$
$644$	0	0
$645$	−0.127145	−0.00500633
$646$	5.98780	0.235587
$647$	29.0762	1.14310	0.571552	−	0.820566i	$-0.306342\pi$
$647$	29.0762	1.14310	0.571552	+	0.820566i	$0.306342\pi$
$648$	−15.5958	−0.612661
$649$	3.86881	0.151864
$650$	−34.3083	−1.34568
$651$	0	0
$652$	67.9970	2.66297
$653$	0.0994735	0.00389270	0.00194635	−	0.999998i	$-0.499380\pi$
$653$	0.0994735	0.00389270	0.00194635	+	0.999998i	$0.499380\pi$
$654$	−0.138113	−0.00540064
$655$	−13.0369	−0.509394
$656$	15.6401	0.610642
$657$	−1.66351	−0.0648996
$658$	0	0
$659$	31.4715	1.22596	0.612978	−	0.790100i	$-0.289971\pi$
$659$	31.4715	1.22596	0.612978	+	0.790100i	$0.289971\pi$
$660$	−0.0243497	−0.000947812 0
$661$	41.6056	1.61827	0.809135	−	0.587623i	$-0.199936\pi$
$661$	41.6056	1.61827	0.809135	+	0.587623i	$0.199936\pi$
$662$	16.4998	0.641283
$663$	−0.253751	−0.00985489
$664$	−19.3491	−0.750892
$665$	0	0
$666$	2.35910	0.0914131
$667$	20.0456	0.776168
$668$	−40.1739	−1.55438
$669$	−0.350125	−0.0135366
$670$	1.33265	0.0514848
$671$	−0.136411	−0.00526609
$672$	0	0
$673$	2.99578	0.115479	0.0577394	−	0.998332i	$-0.481611\pi$
$673$	2.99578	0.115479	0.0577394	+	0.998332i	$0.481611\pi$
$674$	−26.1982	−1.00912
$675$	0.911944	0.0351007
$676$	1.73377	0.0666837
$677$	−48.2569	−1.85466	−0.927332	−	0.374239i	$-0.877904\pi$
$677$	−48.2569	−1.85466	−0.927332	+	0.374239i	$0.877904\pi$
$678$	0.649660	0.0249500
$679$	0	0
$680$	2.89285	0.110936
$681$	0.443690	0.0170022
$682$	−0.0100693	−0.000385574 0
$683$	12.9192	0.494338	0.247169	−	0.968972i	$-0.420500\pi$
$683$	12.9192	0.494338	0.247169	+	0.968972i	$0.420500\pi$
$684$	11.9265	0.456022
$685$	−14.3825	−0.549529
$686$	0	0
$687$	−0.877655	−0.0334846
$688$	−7.30899	−0.278652
$689$	52.0625	1.98342
$690$	−0.443721	−0.0168922
$691$	−20.8880	−0.794616	−0.397308	−	0.917685i	$-0.630055\pi$
$691$	−20.8880	−0.794616	−0.397308	+	0.917685i	$0.630055\pi$
$692$	57.0620	2.16917
$693$	0	0
$694$	−8.92155	−0.338657
$695$	−12.6474	−0.479743
$696$	0.190943	0.00723769
$697$	−16.8111	−0.636768
$698$	−5.22407	−0.197734
$699$	−0.277708	−0.0105039
$700$	0	0
$701$	46.8697	1.77024	0.885122	−	0.465358i	$-0.154074\pi$
$701$	46.8697	1.77024	0.885122	+	0.465358i	$0.154074\pi$
$702$	−1.73517	−0.0654898
$703$	−0.511800	−0.0193029
$704$	−3.53100	−0.133080
$705$	0.134334	0.00505933
$706$	41.7557	1.57150
$707$	0	0
$708$	1.37874	0.0518161
$709$	−33.3937	−1.25413	−0.627064	−	0.778968i	$-0.715744\pi$
$709$	−33.3937	−1.25413	−0.627064	+	0.778968i	$0.715744\pi$
$710$	−17.3856	−0.652471
$711$	43.3339	1.62515
$712$	−28.9449	−1.08476
$713$	−0.106918	−0.00400411
$714$	0	0
$715$	0.898865	0.0336157
$716$	−4.94807	−0.184918
$717$	0.433770	0.0161994
$718$	−3.28008	−0.122411
$719$	14.6697	0.547088	0.273544	−	0.961860i	$-0.411804\pi$
$719$	14.6697	0.547088	0.273544	+	0.961860i	$0.411804\pi$
$720$	−4.65136	−0.173346
$721$	0	0
$722$	37.1542	1.38274
$723$	−0.120339	−0.00447546
$724$	−38.3490	−1.42523
$725$	13.0526	0.484762
$726$	0.855965	0.0317678
$727$	39.9187	1.48050	0.740250	−	0.672331i	$-0.234707\pi$
$727$	39.9187	1.48050	0.740250	+	0.672331i	$0.234707\pi$
$728$	0	0
$729$	−26.9308	−0.997437
$730$	1.05431	0.0390217
$731$	7.85627	0.290575
$732$	−0.0486131	−0.00179679
$733$	−44.3086	−1.63657	−0.818287	−	0.574810i	$-0.805076\pi$
$733$	−44.3086	−1.63657	−0.818287	+	0.574810i	$0.805076\pi$
$734$	22.1560	0.817792
$735$	0	0
$736$	−48.1375	−1.77437
$737$	0.196714	0.00724605
$738$	−57.4657	−2.11534
$739$	−9.90080	−0.364207	−0.182103	−	0.983279i	$-0.558291\pi$
$739$	−9.90080	−0.364207	−0.182103	+	0.983279i	$0.558291\pi$
$740$	−0.871214	−0.0320265
$741$	0.188180	0.00691297
$742$	0	0
$743$	43.7246	1.60410	0.802049	−	0.597258i	$-0.203743\pi$
$743$	43.7246	1.60410	0.802049	+	0.597258i	$0.203743\pi$
$744$	−0.00101844	−3.73380e−5 0
$745$	9.86977	0.361601
$746$	56.0589	2.05246
$747$	−33.4407	−1.22353
$748$	1.50456	0.0550123
$749$	0	0
$750$	−0.629137	−0.0229728
$751$	−1.71759	−0.0626756	−0.0313378	−	0.999509i	$-0.509977\pi$
$751$	−1.71759	−0.0626756	−0.0313378	+	0.999509i	$0.509977\pi$
$752$	7.72227	0.281602
$753$	0.831868	0.0303149
$754$	−24.8354	−0.904453
$755$	8.33050	0.303178
$756$	0	0
$757$	33.7461	1.22652	0.613261	−	0.789880i	$-0.289857\pi$
$757$	33.7461	1.22652	0.613261	+	0.789880i	$0.289857\pi$
$758$	30.5907	1.11111
$759$	−0.0654980	−0.00237743
$760$	−2.14532	−0.0778188
$761$	5.95307	0.215799	0.107899	−	0.994162i	$-0.465588\pi$
$761$	5.95307	0.215799	0.107899	+	0.994162i	$0.465588\pi$
$762$	0.0783758	0.00283925
$763$	0	0
$764$	20.1802	0.730093
$765$	4.99965	0.180763
$766$	−56.2709	−2.03315
$767$	−50.8958	−1.83774
$768$	−0.101275	−0.00365443
$769$	−19.4719	−0.702173	−0.351087	−	0.936343i	$-0.614188\pi$
$769$	−19.4719	−0.702173	−0.351087	+	0.936343i	$0.614188\pi$
$770$	0	0
$771$	−0.696084	−0.0250688
$772$	36.4390	1.31147
$773$	34.4291	1.23833	0.619165	−	0.785261i	$-0.287471\pi$
$773$	34.4291	1.23833	0.619165	+	0.785261i	$0.287471\pi$
$774$	26.8551	0.965288
$775$	−0.0696193	−0.00250080
$776$	13.8043	0.495544
$777$	0	0
$778$	−8.89721	−0.318981
$779$	12.4670	0.446678
$780$	0.320331	0.0114697
$781$	−2.56631	−0.0918297
$782$	27.4174	0.980444
$783$	0.660148	0.0235918
$784$	0	0
$785$	9.39570	0.335347
$786$	−1.17696	−0.0419807
$787$	26.7540	0.953676	0.476838	−	0.878991i	$-0.341783\pi$
$787$	26.7540	0.953676	0.476838	+	0.878991i	$0.341783\pi$
$788$	−8.72416	−0.310785
$789$	−0.152387	−0.00542512
$790$	−27.4644	−0.977141
$791$	0	0
$792$	1.45967	0.0518672
$793$	1.79454	0.0637261
$794$	2.25710	0.0801014
$795$	0.438445	0.0155500
$796$	30.6264	1.08552
$797$	−4.82222	−0.170812	−0.0854058	−	0.996346i	$-0.527219\pi$
$797$	−4.82222	−0.170812	−0.0854058	+	0.996346i	$0.527219\pi$
$798$	0	0
$799$	−8.30049	−0.293650
$800$	−31.3446	−1.10820
$801$	−50.0249	−1.76754
$802$	7.88033	0.278264
$803$	0.155628	0.00549198
$804$	0.0701033	0.00247235
$805$	0	0
$806$	0.132466	0.00466591
$807$	−0.143363	−0.00504662
$808$	8.73509	0.307300
$809$	7.80400	0.274374	0.137187	−	0.990545i	$-0.456194\pi$
$809$	7.80400	0.274374	0.137187	+	0.990545i	$0.456194\pi$
$810$	17.0830	0.600236
$811$	−26.2261	−0.920924	−0.460462	−	0.887679i	$-0.652316\pi$
$811$	−26.2261	−0.920924	−0.460462	+	0.887679i	$0.652316\pi$
$812$	0	0
$813$	−0.0244974	−0.000859161 0
$814$	−0.220703	−0.00773563
$815$	−21.1388	−0.740460
$816$	−0.122845	−0.00430042
$817$	−5.82615	−0.203831
$818$	−47.8762	−1.67395
$819$	0	0
$820$	21.2221	0.741106
$821$	2.81989	0.0984148	0.0492074	−	0.998789i	$-0.484330\pi$
$821$	2.81989	0.0984148	0.0492074	+	0.998789i	$0.484330\pi$
$822$	−1.29844	−0.0452884
$823$	36.8534	1.28463	0.642314	−	0.766441i	$-0.277974\pi$
$823$	36.8534	1.28463	0.642314	+	0.766441i	$0.277974\pi$
$824$	4.46477	0.155538
$825$	−0.0426489	−0.00148484
$826$	0	0
$827$	−17.0927	−0.594371	−0.297186	−	0.954820i	$-0.596048\pi$
$827$	−17.0927	−0.594371	−0.297186	+	0.954820i	$0.596048\pi$
$828$	54.6101	1.89783
$829$	34.5735	1.20079	0.600394	−	0.799704i	$-0.295010\pi$
$829$	34.5735	1.20079	0.600394	+	0.799704i	$0.295010\pi$
$830$	21.1943	0.735664
$831$	0.186002	0.00645233
$832$	46.4518	1.61043
$833$	0	0
$834$	−1.14179	−0.0395371
$835$	12.4892	0.432207
$836$	−1.11577	−0.0385898
$837$	−0.00352106	−0.000121706 0
$838$	−31.7844	−1.09797
$839$	17.1104	0.590718	0.295359	−	0.955386i	$-0.404561\pi$
$839$	17.1104	0.590718	0.295359	+	0.955386i	$0.404561\pi$
$840$	0	0
$841$	−19.5513	−0.674184
$842$	−3.81733	−0.131554
$843$	0.829045	0.0285538
$844$	−38.7256	−1.33299
$845$	−0.538993	−0.0185419
$846$	−28.3736	−0.975505
$847$	0	0
$848$	25.2042	0.865515
$849$	0.00383754	0.000131704 0
$850$	17.8528	0.612345
$851$	−2.34347	−0.0803331
$852$	−0.914561	−0.0313323
$853$	45.5681	1.56022	0.780111	−	0.625642i	$-0.215163\pi$
$853$	45.5681	1.56022	0.780111	+	0.625642i	$0.215163\pi$
$854$	0	0
$855$	−3.70770	−0.126801
$856$	11.6447	0.398009
$857$	30.0586	1.02678	0.513391	−	0.858155i	$-0.328389\pi$
$857$	30.0586	1.02678	0.513391	+	0.858155i	$0.328389\pi$
$858$	0.0811487	0.00277037
$859$	−24.8645	−0.848365	−0.424183	−	0.905577i	$-0.639439\pi$
$859$	−24.8645	−0.848365	−0.424183	+	0.905577i	$0.639439\pi$
$860$	−9.91759	−0.338187
$861$	0	0
$862$	−54.0403	−1.84062
$863$	−14.4612	−0.492264	−0.246132	−	0.969236i	$-0.579160\pi$
$863$	−14.4612	−0.492264	−0.246132	+	0.969236i	$0.579160\pi$
$864$	−1.58528	−0.0539323
$865$	−17.7393	−0.603156
$866$	−37.2010	−1.26414
$867$	−0.476577	−0.0161854
$868$	0	0
$869$	−4.05405	−0.137524
$870$	−0.209152	−0.00709091
$871$	−2.58785	−0.0876860
$872$	−3.05755	−0.103542
$873$	23.8576	0.807458
$874$	−20.3325	−0.687759
$875$	0	0
$876$	0.0554614	0.00187387
$877$	−32.6606	−1.10287	−0.551435	−	0.834218i	$-0.685920\pi$
$877$	−32.6606	−1.10287	−0.551435	+	0.834218i	$0.685920\pi$
$878$	21.6387	0.730269
$879$	−0.378315	−0.0127602
$880$	0.435153	0.0146690
$881$	−11.2108	−0.377700	−0.188850	−	0.982006i	$-0.560476\pi$
$881$	−11.2108	−0.377700	−0.188850	+	0.982006i	$0.560476\pi$
$882$	0	0
$883$	27.6874	0.931754	0.465877	−	0.884849i	$-0.345739\pi$
$883$	27.6874	0.931754	0.465877	+	0.884849i	$0.345739\pi$
$884$	−19.7932	−0.665716
$885$	−0.428620	−0.0144079
$886$	−87.2263	−2.93042
$887$	−9.88366	−0.331861	−0.165930	−	0.986137i	$-0.553063\pi$
$887$	−9.88366	−0.331861	−0.165930	+	0.986137i	$0.553063\pi$
$888$	−0.0223226	−0.000749098 0
$889$	0	0
$890$	31.7051	1.06276
$891$	2.52164	0.0844782
$892$	−27.3105	−0.914424
$893$	6.15558	0.205989
$894$	0.891034	0.0298006
$895$	1.53825	0.0514179
$896$	0	0
$897$	0.861654	0.0287698
$898$	−52.5953	−1.75513
$899$	−0.0503968	−0.00168083
$900$	35.5592	1.18531
$901$	−27.0914	−0.902546
$902$	5.37614	0.179006
$903$	0	0
$904$	14.3822	0.478346
$905$	11.9219	0.396297
$906$	0.752070	0.0249858
$907$	−2.03008	−0.0674076	−0.0337038	−	0.999432i	$-0.510730\pi$
$907$	−2.03008	−0.0674076	−0.0337038	+	0.999432i	$0.510730\pi$
$908$	34.6088	1.14853
$909$	15.0967	0.500725
$910$	0	0
$911$	−29.2269	−0.968330	−0.484165	−	0.874977i	$-0.660877\pi$
$911$	−29.2269	−0.968330	−0.484165	+	0.874977i	$0.660877\pi$
$912$	0.0911007	0.00301664
$913$	3.12851	0.103538
$914$	−8.35936	−0.276503
$915$	0.0151128	0.000499613 0
$916$	−68.4590	−2.26195
$917$	0	0
$918$	0.902920	0.0298008
$919$	15.5165	0.511842	0.255921	−	0.966698i	$-0.417621\pi$
$919$	15.5165	0.511842	0.255921	+	0.966698i	$0.417621\pi$
$920$	−9.82313	−0.323859
$921$	0.122109	0.00402362
$922$	−59.9347	−1.97385
$923$	33.7608	1.11125
$924$	0	0
$925$	−1.52594	−0.0501727
$926$	13.3793	0.439672
$927$	7.71637	0.253439
$928$	−22.6901	−0.744838
$929$	−53.4586	−1.75392	−0.876960	−	0.480562i	$-0.840433\pi$
$929$	−53.4586	−1.75392	−0.876960	+	0.480562i	$0.840433\pi$
$930$	0.00111556	3.65808e−5 0
$931$	0	0
$932$	−21.6618	−0.709557
$933$	−1.06448	−0.0348497
$934$	51.3886	1.68149
$935$	−0.467737	−0.0152966
$936$	−19.2026	−0.627656
$937$	25.9970	0.849285	0.424642	−	0.905361i	$-0.360400\pi$
$937$	25.9970	0.849285	0.424642	+	0.905361i	$0.360400\pi$
$938$	0	0
$939$	−0.770899	−0.0251573
$940$	10.4784	0.341767
$941$	45.2741	1.47589	0.737947	−	0.674858i	$-0.235795\pi$
$941$	45.2741	1.47589	0.737947	+	0.674858i	$0.235795\pi$
$942$	0.848235	0.0276370
$943$	57.0850	1.85894
$944$	−24.6394	−0.801943
$945$	0	0
$946$	−2.51240	−0.0816853
$947$	−24.4990	−0.796111	−0.398056	−	0.917361i	$-0.630315\pi$
$947$	−24.4990	−0.796111	−0.398056	+	0.917361i	$0.630315\pi$
$948$	−1.44475	−0.0469234
$949$	−2.04735	−0.0664597
$950$	−13.2395	−0.429545
$951$	1.01562	0.0329336
$952$	0	0
$953$	11.2283	0.363720	0.181860	−	0.983324i	$-0.441788\pi$
$953$	11.2283	0.363720	0.181860	+	0.983324i	$0.441788\pi$
$954$	−92.6068	−2.99826
$955$	−6.27358	−0.203008
$956$	33.8350	1.09430
$957$	−0.0308731	−0.000997986 0
$958$	−66.6453	−2.15321
$959$	0	0
$960$	0.391194	0.0126257
$961$	−30.9997	−0.999991
$962$	2.90344	0.0936105
$963$	20.1253	0.648530
$964$	−9.38670	−0.302325
$965$	−11.3281	−0.364665
$966$	0	0
$967$	4.25392	0.136797	0.0683984	−	0.997658i	$-0.478211\pi$
$967$	4.25392	0.136797	0.0683984	+	0.997658i	$0.478211\pi$
$968$	18.9494	0.609058
$969$	−0.0979221	−0.00314571
$970$	−15.1206	−0.485495
$971$	33.0193	1.05964	0.529821	−	0.848110i	$-0.322259\pi$
$971$	33.0193	1.05964	0.529821	+	0.848110i	$0.322259\pi$
$972$	2.69785	0.0865336
$973$	0	0
$974$	96.2118	3.08283
$975$	0.561063	0.0179684
$976$	0.868763	0.0278084
$977$	28.1942	0.902013	0.451006	−	0.892521i	$-0.351065\pi$
$977$	28.1942	0.902013	0.451006	+	0.892521i	$0.351065\pi$
$978$	−1.90839	−0.0610236
$979$	4.68002	0.149574
$980$	0	0
$981$	−5.28430	−0.168715
$982$	10.4317	0.332888
$983$	40.8708	1.30358	0.651788	−	0.758401i	$-0.274019\pi$
$983$	40.8708	1.30358	0.651788	+	0.758401i	$0.274019\pi$
$984$	0.543761	0.0173345
$985$	2.71215	0.0864164
$986$	12.9234	0.411566
$987$	0	0
$988$	14.6785	0.466984
$989$	−26.6772	−0.848286
$990$	−1.59887	−0.0508153
$991$	−2.56939	−0.0816194	−0.0408097	−	0.999167i	$-0.512994\pi$
$991$	−2.56939	−0.0816194	−0.0408097	+	0.999167i	$0.512994\pi$
$992$	0.121023	0.00384248
$993$	−0.269831	−0.00856284
$994$	0	0
$995$	−9.52110	−0.301839
$996$	1.11491	0.0353274
$997$	36.9676	1.17077	0.585387	−	0.810754i	$-0.300943\pi$
$997$	36.9676	1.17077	0.585387	+	0.810754i	$0.300943\pi$
$998$	21.6082	0.683995
$999$	−0.0771759	−0.00244174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.3		16
7.6	odd	2		889.2.a.c.1.3	✓	16
21.20	even	2		8001.2.a.t.1.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.3	✓	16	7.6	odd	2
6223.2.a.k.1.3		16	1.1	even	1	trivial
8001.2.a.t.1.14		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)

\(f(q)\)	\(=\)	\(q-2.18919 q^{2} +0.0358012 q^{3} +2.79257 q^{4} -0.868150 q^{5} -0.0783758 q^{6} -1.73509 q^{8} -2.99872 q^{9} +1.90055 q^{10} +0.280542 q^{11} +0.0999773 q^{12} -3.69064 q^{13} -0.0310808 q^{15} -1.78669 q^{16} +1.92048 q^{17} +6.56478 q^{18} -1.42421 q^{19} -2.42437 q^{20} -0.614161 q^{22} -6.52129 q^{23} -0.0621183 q^{24} -4.24632 q^{25} +8.07954 q^{26} -0.214761 q^{27} -3.07387 q^{29} +0.0680419 q^{30} +0.0163952 q^{31} +7.38160 q^{32} +0.0100437 q^{33} -4.20429 q^{34} -8.37413 q^{36} +0.359357 q^{37} +3.11787 q^{38} -0.132129 q^{39} +1.50632 q^{40} -8.75364 q^{41} +4.09079 q^{43} +0.783433 q^{44} +2.60334 q^{45} +14.2764 q^{46} -4.32210 q^{47} -0.0639657 q^{48} +9.29601 q^{50} +0.0687553 q^{51} -10.3064 q^{52} -14.1066 q^{53} +0.470154 q^{54} -0.243552 q^{55} -0.0509884 q^{57} +6.72929 q^{58} +13.7905 q^{59} -0.0867953 q^{60} -0.486241 q^{61} -0.0358923 q^{62} -12.5864 q^{64} +3.20403 q^{65} -0.0219877 q^{66} +0.701192 q^{67} +5.36306 q^{68} -0.233470 q^{69} -9.14768 q^{71} +5.20305 q^{72} +0.554740 q^{73} -0.786702 q^{74} -0.152023 q^{75} -3.97721 q^{76} +0.289257 q^{78} -14.4508 q^{79} +1.55112 q^{80} +8.98847 q^{81} +19.1634 q^{82} +11.1517 q^{83} -1.66726 q^{85} -8.95554 q^{86} -0.110048 q^{87} -0.486765 q^{88} +16.6821 q^{89} -5.69921 q^{90} -18.2111 q^{92} +0.000586969 q^{93} +9.46192 q^{94} +1.23643 q^{95} +0.264270 q^{96} -7.95594 q^{97} -0.841266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 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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