LMFDB - Embedded newform 6223.2.a.p.1.12

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 = [38,-2,11,38,16] 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:	$38$
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	\(q-1.31336 q^{2} +2.15446 q^{3} -0.275086 q^{4} -2.87032 q^{5} -2.82958 q^{6} +2.98801 q^{8} +1.64168 q^{9} +3.76976 q^{10} +1.59111 q^{11} -0.592661 q^{12} -4.65625 q^{13} -6.18398 q^{15} -3.37416 q^{16} -1.70368 q^{17} -2.15612 q^{18} -4.60752 q^{19} +0.789584 q^{20} -2.08970 q^{22} -0.103769 q^{23} +6.43753 q^{24} +3.23873 q^{25} +6.11533 q^{26} -2.92644 q^{27} -7.33090 q^{29} +8.12178 q^{30} +5.98705 q^{31} -1.54453 q^{32} +3.42798 q^{33} +2.23754 q^{34} -0.451604 q^{36} +2.47284 q^{37} +6.05133 q^{38} -10.0317 q^{39} -8.57653 q^{40} +10.0813 q^{41} +2.98728 q^{43} -0.437692 q^{44} -4.71215 q^{45} +0.136285 q^{46} -5.25039 q^{47} -7.26947 q^{48} -4.25361 q^{50} -3.67050 q^{51} +1.28087 q^{52} -8.35024 q^{53} +3.84346 q^{54} -4.56699 q^{55} -9.92671 q^{57} +9.62811 q^{58} +6.58904 q^{59} +1.70112 q^{60} +5.23865 q^{61} -7.86316 q^{62} +8.77684 q^{64} +13.3649 q^{65} -4.50217 q^{66} -7.44431 q^{67} +0.468658 q^{68} -0.223565 q^{69} -14.2742 q^{71} +4.90536 q^{72} +9.12572 q^{73} -3.24773 q^{74} +6.97769 q^{75} +1.26746 q^{76} +13.1752 q^{78} -10.0197 q^{79} +9.68490 q^{80} -11.2299 q^{81} -13.2403 q^{82} +7.66914 q^{83} +4.89010 q^{85} -3.92338 q^{86} -15.7941 q^{87} +4.75425 q^{88} +8.35255 q^{89} +6.18875 q^{90} +0.0285453 q^{92} +12.8988 q^{93} +6.89565 q^{94} +13.2251 q^{95} -3.32763 q^{96} +7.47347 q^{97} +2.61210 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{2} + 11 q^{3} + 38 q^{4} + 16 q^{5} + 11 q^{6} + 47 q^{9} + 12 q^{10} - 2 q^{11} + 30 q^{12} + 21 q^{13} + 7 q^{15} + 46 q^{16} + 58 q^{17} - 13 q^{18} + 17 q^{19} + 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) $ 38 * q - 2 * q^2 + 11 * q^3 + 38 * q^4 + 16 * q^5 + 11 * q^6 + 47 * q^9 + 12 * q^10 - 2 * q^11 + 30 * q^12 + 21 * q^13 + 7 * q^15 + 46 * q^16 + 58 * q^17 - 13 * q^18 + 17 * q^19 + 44 * q^20 + 21 * q^22 + 7 * q^23 + 21 * q^24 + 52 * q^25 + 49 * q^26 + 35 * q^27 - 2 * q^29 + 6 * q^30 + 2 * q^31 - 22 * q^32 + 29 * q^33 + 37 * q^34 + 25 * q^36 + 11 * q^37 + 40 * q^38 + 2 * q^39 - 8 * q^40 + 68 * q^41 - 38 * q^43 + 43 * q^44 + 60 * q^45 - 28 * q^46 + 35 * q^47 + 66 * q^48 - 25 * q^50 - 3 * q^51 + 32 * q^52 - 18 * q^53 - 68 * q^54 + 19 * q^55 - 31 * q^57 + 55 * q^58 + 21 * q^59 - 11 * q^60 + 5 * q^61 + 38 * q^62 + 18 * q^64 - 9 * q^65 + 20 * q^66 + 6 * q^67 + 74 * q^68 + 53 * q^69 - 26 * q^71 + 24 * q^72 + 22 * q^73 - 2 * q^74 + 38 * q^75 + 39 * q^76 - 114 * q^78 - 14 * q^79 + 140 * q^80 + 58 * q^81 - 8 * q^82 + 46 * q^83 - 11 * q^85 + 51 * q^86 + 59 * q^87 - 4 * q^88 + 74 * q^89 + q^90 - 81 * q^92 + 92 * q^93 - 36 * q^94 + 20 * q^95 - 108 * q^96 + 66 * q^97 + 16 * 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 $

$2$	−1.31336	−0.928686	−0.464343	−	0.885656i	$-0.653709\pi$
$2$	−1.31336	−0.928686	−0.464343	+	0.885656i	$0.653709\pi$
$3$	2.15446	1.24388	0.621938	−	0.783066i	$-0.286346\pi$
$3$	2.15446	1.24388	0.621938	+	0.783066i	$0.286346\pi$
$4$	−0.275086	−0.137543
$5$	−2.87032	−1.28365	−0.641823	−	0.766853i	$-0.721821\pi$
$5$	−2.87032	−1.28365	−0.641823	+	0.766853i	$0.721821\pi$
$6$	−2.82958	−1.15517
$7$	0	0
$7$	0	0
$8$	2.98801	1.05642
$9$	1.64168	0.547228
$10$	3.76976	1.19210
$11$	1.59111	0.479738	0.239869	−	0.970805i	$-0.422896\pi$
$11$	1.59111	0.479738	0.239869	+	0.970805i	$0.422896\pi$
$12$	−0.592661	−0.171086
$13$	−4.65625	−1.29141	−0.645705	−	0.763587i	$-0.723436\pi$
$13$	−4.65625	−1.29141	−0.645705	+	0.763587i	$0.723436\pi$
$14$	0	0
$15$	−6.18398	−1.59670
$16$	−3.37416	−0.843539
$17$	−1.70368	−0.413203	−0.206601	−	0.978425i	$-0.566240\pi$
$17$	−1.70368	−0.413203	−0.206601	+	0.978425i	$0.566240\pi$
$18$	−2.15612	−0.508202
$19$	−4.60752	−1.05704	−0.528519	−	0.848921i	$-0.677252\pi$
$19$	−4.60752	−1.05704	−0.528519	+	0.848921i	$0.677252\pi$
$20$	0.789584	0.176556
$21$	0	0
$22$	−2.08970	−0.445525
$23$	−0.103769	−0.0216372	−0.0108186	−	0.999941i	$-0.503444\pi$
$23$	−0.103769	−0.0216372	−0.0108186	+	0.999941i	$0.503444\pi$
$24$	6.43753	1.31406
$25$	3.23873	0.647745
$26$	6.11533	1.19931
$27$	−2.92644	−0.563193
$28$	0	0
$29$	−7.33090	−1.36131	−0.680657	−	0.732602i	$-0.738306\pi$
$29$	−7.33090	−1.36131	−0.680657	+	0.732602i	$0.738306\pi$
$30$	8.12178	1.48283
$31$	5.98705	1.07531	0.537653	−	0.843166i	$-0.319311\pi$
$31$	5.98705	1.07531	0.537653	+	0.843166i	$0.319311\pi$
$32$	−1.54453	−0.273037
$33$	3.42798	0.596734
$34$	2.23754	0.383736
$35$	0	0
$36$	−0.451604	−0.0752673
$37$	2.47284	0.406532	0.203266	−	0.979124i	$-0.434844\pi$
$37$	2.47284	0.406532	0.203266	+	0.979124i	$0.434844\pi$
$38$	6.05133	0.981656
$39$	−10.0317	−1.60635
$40$	−8.57653	−1.35607
$41$	10.0813	1.57443	0.787214	−	0.616680i	$-0.211523\pi$
$41$	10.0813	1.57443	0.787214	+	0.616680i	$0.211523\pi$
$42$	0	0
$43$	2.98728	0.455556	0.227778	−	0.973713i	$-0.426854\pi$
$43$	2.98728	0.455556	0.227778	+	0.973713i	$0.426854\pi$
$44$	−0.437692	−0.0659846
$45$	−4.71215	−0.702446
$46$	0.136285	0.0200942
$47$	−5.25039	−0.765848	−0.382924	−	0.923780i	$-0.625083\pi$
$47$	−5.25039	−0.765848	−0.382924	+	0.923780i	$0.625083\pi$
$48$	−7.26947	−1.04926
$49$	0	0
$50$	−4.25361	−0.601551
$51$	−3.67050	−0.513973
$52$	1.28087	0.177624
$53$	−8.35024	−1.14699	−0.573497	−	0.819208i	$-0.694413\pi$
$53$	−8.35024	−1.14699	−0.573497	+	0.819208i	$0.694413\pi$
$54$	3.84346	0.523029
$55$	−4.56699	−0.615813
$56$	0	0
$57$	−9.92671	−1.31482
$58$	9.62811	1.26423
$59$	6.58904	0.857820	0.428910	−	0.903347i	$-0.358898\pi$
$59$	6.58904	0.857820	0.428910	+	0.903347i	$0.358898\pi$
$60$	1.70112	0.219614
$61$	5.23865	0.670740	0.335370	−	0.942086i	$-0.391139\pi$
$61$	5.23865	0.670740	0.335370	+	0.942086i	$0.391139\pi$
$62$	−7.86316	−0.998622
$63$	0	0
$64$	8.77684	1.09710
$65$	13.3649	1.65771
$66$	−4.50217	−0.554178
$67$	−7.44431	−0.909467	−0.454734	−	0.890627i	$-0.650266\pi$
$67$	−7.44431	−0.909467	−0.454734	+	0.890627i	$0.650266\pi$
$68$	0.468658	0.0568332
$69$	−0.223565	−0.0269141
$70$	0	0
$71$	−14.2742	−1.69404	−0.847019	−	0.531563i	$-0.821605\pi$
$71$	−14.2742	−1.69404	−0.847019	+	0.531563i	$0.821605\pi$
$72$	4.90536	0.578102
$73$	9.12572	1.06808	0.534042	−	0.845458i	$-0.320672\pi$
$73$	9.12572	1.06808	0.534042	+	0.845458i	$0.320672\pi$
$74$	−3.24773	−0.377541
$75$	6.97769	0.805714
$76$	1.26746	0.145388
$77$	0	0
$78$	13.1752	1.49180
$79$	−10.0197	−1.12730	−0.563650	−	0.826014i	$-0.690603\pi$
$79$	−10.0197	−1.12730	−0.563650	+	0.826014i	$0.690603\pi$
$80$	9.68490	1.08280
$81$	−11.2299	−1.24777
$82$	−13.2403	−1.46215
$83$	7.66914	0.841797	0.420898	−	0.907108i	$-0.361715\pi$
$83$	7.66914	0.841797	0.420898	+	0.907108i	$0.361715\pi$
$84$	0	0
$85$	4.89010	0.530406
$86$	−3.92338	−0.423068
$87$	−15.7941	−1.69331
$88$	4.75425	0.506804
$89$	8.35255	0.885368	0.442684	−	0.896678i	$-0.354026\pi$
$89$	8.35255	0.885368	0.442684	+	0.896678i	$0.354026\pi$
$90$	6.18875	0.652352
$91$	0	0
$92$	0.0285453	0.00297605
$93$	12.8988	1.33755
$94$	6.89565	0.711232
$95$	13.2251	1.35686
$96$	−3.32763	−0.339625
$97$	7.47347	0.758816	0.379408	−	0.925229i	$-0.376128\pi$
$97$	7.47347	0.758816	0.379408	+	0.925229i	$0.376128\pi$
$98$	0	0
$99$	2.61210	0.262526
$100$	−0.890928	−0.0890928
$101$	11.4744	1.14174	0.570872	−	0.821039i	$-0.306605\pi$
$101$	11.4744	1.14174	0.570872	+	0.821039i	$0.306605\pi$
$102$	4.82069	0.477319
$103$	−3.03163	−0.298716	−0.149358	−	0.988783i	$-0.547721\pi$
$103$	−3.03163	−0.298716	−0.149358	+	0.988783i	$0.547721\pi$
$104$	−13.9129	−1.36427
$105$	0	0
$106$	10.9669	1.06520
$107$	10.3035	0.996073	0.498037	−	0.867156i	$-0.334055\pi$
$107$	10.3035	0.996073	0.498037	+	0.867156i	$0.334055\pi$
$108$	0.805021	0.0774632
$109$	5.60984	0.537325	0.268663	−	0.963234i	$-0.413418\pi$
$109$	5.60984	0.537325	0.268663	+	0.963234i	$0.413418\pi$
$110$	5.99810	0.571897
$111$	5.32763	0.505676
$112$	0	0
$113$	14.7725	1.38968	0.694841	−	0.719163i	$-0.255475\pi$
$113$	14.7725	1.38968	0.694841	+	0.719163i	$0.255475\pi$
$114$	13.0373	1.22106
$115$	0.297849	0.0277745
$116$	2.01663	0.187239
$117$	−7.64408	−0.706695
$118$	−8.65378	−0.796645
$119$	0	0
$120$	−18.4778	−1.68678
$121$	−8.46837	−0.769852
$122$	−6.88023	−0.622907
$123$	21.7196	1.95839
$124$	−1.64695	−0.147901
$125$	5.05542	0.452170
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−8.43808	−0.745828
$129$	6.43597	0.566655
$130$	−17.5529	−1.53949
$131$	−19.9988	−1.74730	−0.873650	−	0.486555i	$-0.838253\pi$
$131$	−19.9988	−1.74730	−0.873650	+	0.486555i	$0.838253\pi$
$132$	−0.942988	−0.0820766
$133$	0	0
$134$	9.77706	0.844609
$135$	8.39980	0.722940
$136$	−5.09060	−0.436516
$137$	−4.51559	−0.385793	−0.192896	−	0.981219i	$-0.561788\pi$
$137$	−4.51559	−0.385793	−0.192896	+	0.981219i	$0.561788\pi$
$138$	0.293621	0.0249947
$139$	8.75094	0.742245	0.371122	−	0.928584i	$-0.378973\pi$
$139$	8.75094	0.742245	0.371122	+	0.928584i	$0.378973\pi$
$140$	0	0
$141$	−11.3117	−0.952620
$142$	18.7472	1.57323
$143$	−7.40860	−0.619538
$144$	−5.53929	−0.461608
$145$	21.0420	1.74744
$146$	−11.9854	−0.991915
$147$	0	0
$148$	−0.680244	−0.0559157
$149$	−15.1936	−1.24471	−0.622354	−	0.782736i	$-0.713823\pi$
$149$	−15.1936	−1.24471	−0.622354	+	0.782736i	$0.713823\pi$
$150$	−9.16422	−0.748255
$151$	11.7614	0.957126	0.478563	−	0.878053i	$-0.341158\pi$
$151$	11.7614	0.957126	0.478563	+	0.878053i	$0.341158\pi$
$152$	−13.7673	−1.11668
$153$	−2.79690	−0.226116
$154$	0	0
$155$	−17.1847	−1.38031
$156$	2.75957	0.220943
$157$	11.2842	0.900576	0.450288	−	0.892883i	$-0.351321\pi$
$157$	11.2842	0.900576	0.450288	+	0.892883i	$0.351321\pi$
$158$	13.1594	1.04691
$159$	−17.9902	−1.42672
$160$	4.43330	0.350483
$161$	0	0
$162$	14.7489	1.15879
$163$	−18.2298	−1.42786	−0.713932	−	0.700215i	$-0.753088\pi$
$163$	−18.2298	−1.42786	−0.713932	+	0.700215i	$0.753088\pi$
$164$	−2.77321	−0.216551
$165$	−9.83938	−0.765995
$166$	−10.0723	−0.781765
$167$	7.66446	0.593094	0.296547	−	0.955018i	$-0.404165\pi$
$167$	7.66446	0.593094	0.296547	+	0.955018i	$0.404165\pi$
$168$	0	0
$169$	8.68062	0.667740
$170$	−6.42246	−0.492580
$171$	−7.56409	−0.578440
$172$	−0.821759	−0.0626586
$173$	2.59760	0.197492	0.0987460	−	0.995113i	$-0.468517\pi$
$173$	2.59760	0.197492	0.0987460	+	0.995113i	$0.468517\pi$
$174$	20.7434	1.57255
$175$	0	0
$176$	−5.36865	−0.404677
$177$	14.1958	1.06702
$178$	−10.9699	−0.822229
$179$	18.9673	1.41768	0.708842	−	0.705367i	$-0.249218\pi$
$179$	18.9673	1.41768	0.708842	+	0.705367i	$0.249218\pi$
$180$	1.29625	0.0966165
$181$	−3.85473	−0.286520	−0.143260	−	0.989685i	$-0.545758\pi$
$181$	−3.85473	−0.286520	−0.143260	+	0.989685i	$0.545758\pi$
$182$	0	0
$183$	11.2864	0.834318
$184$	−0.310061	−0.0228580
$185$	−7.09784	−0.521843
$186$	−16.9408	−1.24216
$187$	−2.71074	−0.198229
$188$	1.44431	0.105337
$189$	0	0
$190$	−17.3693	−1.26010
$191$	−4.74571	−0.343387	−0.171694	−	0.985150i	$-0.554924\pi$
$191$	−4.74571	−0.343387	−0.171694	+	0.985150i	$0.554924\pi$
$192$	18.9093	1.36466
$193$	1.81361	0.130547	0.0652734	−	0.997867i	$-0.479208\pi$
$193$	1.81361	0.130547	0.0652734	+	0.997867i	$0.479208\pi$
$194$	−9.81536	−0.704702
$195$	28.7941	2.06199
$196$	0	0
$197$	−18.0861	−1.28858	−0.644292	−	0.764780i	$-0.722848\pi$
$197$	−18.0861	−1.28858	−0.644292	+	0.764780i	$0.722848\pi$
$198$	−3.43062	−0.243804
$199$	−7.28332	−0.516301	−0.258150	−	0.966105i	$-0.583113\pi$
$199$	−7.28332	−0.516301	−0.258150	+	0.966105i	$0.583113\pi$
$200$	9.67733	0.684291
$201$	−16.0384	−1.13126
$202$	−15.0700	−1.06032
$203$	0	0
$204$	1.00970	0.0706934
$205$	−28.9364	−2.02101
$206$	3.98163	0.277413
$207$	−0.170355	−0.0118405
$208$	15.7109	1.08935
$209$	−7.33107	−0.507101
$210$	0	0
$211$	4.93418	0.339683	0.169841	−	0.985471i	$-0.445674\pi$
$211$	4.93418	0.339683	0.169841	+	0.985471i	$0.445674\pi$
$212$	2.29703	0.157761
$213$	−30.7532	−2.10717
$214$	−13.5322	−0.925039
$215$	−8.57445	−0.584772
$216$	−8.74421	−0.594968
$217$	0	0
$218$	−7.36774	−0.499006
$219$	19.6610	1.32857
$220$	1.25632	0.0847008
$221$	7.93275	0.533614
$222$	−6.99709	−0.469614
$223$	−7.52334	−0.503800	−0.251900	−	0.967753i	$-0.581055\pi$
$223$	−7.52334	−0.503800	−0.251900	+	0.967753i	$0.581055\pi$
$224$	0	0
$225$	5.31696	0.354464
$226$	−19.4016	−1.29058
$227$	−0.262476	−0.0174212	−0.00871058	−	0.999962i	$-0.502773\pi$
$227$	−0.262476	−0.0174212	−0.00871058	+	0.999962i	$0.502773\pi$
$228$	2.73070	0.180845
$229$	4.30565	0.284525	0.142263	−	0.989829i	$-0.454562\pi$
$229$	4.30565	0.284525	0.142263	+	0.989829i	$0.454562\pi$
$230$	−0.391183	−0.0257938
$231$	0	0
$232$	−21.9048	−1.43812
$233$	13.0087	0.852229	0.426115	−	0.904669i	$-0.359882\pi$
$233$	13.0087	0.852229	0.426115	+	0.904669i	$0.359882\pi$
$234$	10.0394	0.656298
$235$	15.0703	0.983077
$236$	−1.81255	−0.117987
$237$	−21.5869	−1.40222
$238$	0	0
$239$	29.3685	1.89969	0.949844	−	0.312723i	$-0.101241\pi$
$239$	29.3685	1.89969	0.949844	+	0.312723i	$0.101241\pi$
$240$	20.8657	1.34687
$241$	−26.6496	−1.71665	−0.858325	−	0.513107i	$-0.828494\pi$
$241$	−26.6496	−1.71665	−0.858325	+	0.513107i	$0.828494\pi$
$242$	11.1220	0.714950
$243$	−15.4151	−0.988878
$244$	−1.44108	−0.0922556
$245$	0	0
$246$	−28.5257	−1.81873
$247$	21.4538	1.36507
$248$	17.8894	1.13598
$249$	16.5228	1.04709
$250$	−6.63958	−0.419924
$251$	−0.00157963	−9.97052e−5 0	−4.98526e−5	−	1.00000i	$-0.500016\pi$
$251$	−0.00157963	−9.97052e−5 0	−4.98526e−5		1.00000i	$0.500016\pi$
$252$	0	0
$253$	−0.165107	−0.0103802
$254$	−1.31336	−0.0824075
$255$	10.5355	0.659759
$256$	−6.47144	−0.404465
$257$	11.1076	0.692872	0.346436	−	0.938074i	$-0.387392\pi$
$257$	11.1076	0.692872	0.346436	+	0.938074i	$0.387392\pi$
$258$	−8.45274	−0.526245
$259$	0	0
$260$	−3.67650	−0.228007
$261$	−12.0350	−0.744949
$262$	26.2656	1.62269
$263$	−0.562872	−0.0347082	−0.0173541	−	0.999849i	$-0.505524\pi$
$263$	−0.562872	−0.0347082	−0.0173541	+	0.999849i	$0.505524\pi$
$264$	10.2428	0.630402
$265$	23.9678	1.47233
$266$	0	0
$267$	17.9952	1.10129
$268$	2.04783	0.125091
$269$	27.3716	1.66888	0.834439	−	0.551100i	$-0.185792\pi$
$269$	27.3716	1.66888	0.834439	+	0.551100i	$0.185792\pi$
$270$	−11.0320	−0.671384
$271$	19.9530	1.21206	0.606030	−	0.795442i	$-0.292761\pi$
$271$	19.9530	1.21206	0.606030	+	0.795442i	$0.292761\pi$
$272$	5.74848	0.348553
$273$	0	0
$274$	5.93059	0.358280
$275$	5.15317	0.310748
$276$	0.0614996	0.00370184
$277$	−12.2014	−0.733109	−0.366555	−	0.930397i	$-0.619463\pi$
$277$	−12.2014	−0.733109	−0.366555	+	0.930397i	$0.619463\pi$
$278$	−11.4931	−0.689312
$279$	9.82884	0.588438
$280$	0	0
$281$	22.9501	1.36909	0.684543	−	0.728972i	$-0.260002\pi$
$281$	22.9501	1.36909	0.684543	+	0.728972i	$0.260002\pi$
$282$	14.8564	0.884685
$283$	16.2550	0.966261	0.483131	−	0.875548i	$-0.339500\pi$
$283$	16.2550	0.966261	0.483131	+	0.875548i	$0.339500\pi$
$284$	3.92664	0.233003
$285$	28.4928	1.68777
$286$	9.73016	0.575356
$287$	0	0
$288$	−2.53563	−0.149414
$289$	−14.0975	−0.829263
$290$	−27.6357	−1.62283
$291$	16.1013	0.943873
$292$	−2.51036	−0.146908
$293$	23.1704	1.35363	0.676813	−	0.736155i	$-0.263360\pi$
$293$	23.1704	1.35363	0.676813	+	0.736155i	$0.263360\pi$
$294$	0	0
$295$	−18.9126	−1.10114
$296$	7.38886	0.429469
$297$	−4.65628	−0.270185
$298$	19.9546	1.15594
$299$	0.483172	0.0279426
$300$	−1.91947	−0.110820
$301$	0	0
$302$	−15.4469	−0.888869
$303$	24.7210	1.42019
$304$	15.5465	0.891653
$305$	−15.0366	−0.860993
$306$	3.67334	0.209991
$307$	14.4329	0.823731	0.411866	−	0.911245i	$-0.364877\pi$
$307$	14.4329	0.823731	0.411866	+	0.911245i	$0.364877\pi$
$308$	0	0
$309$	−6.53153	−0.371566
$310$	22.5698	1.28188
$311$	−16.6407	−0.943608	−0.471804	−	0.881703i	$-0.656397\pi$
$311$	−16.6407	−0.943608	−0.471804	+	0.881703i	$0.656397\pi$
$312$	−29.9747	−1.69698
$313$	26.0908	1.47474	0.737370	−	0.675489i	$-0.236068\pi$
$313$	26.0908	1.47474	0.737370	+	0.675489i	$0.236068\pi$
$314$	−14.8202	−0.836352
$315$	0	0
$316$	2.75627	0.155052
$317$	25.1016	1.40985	0.704924	−	0.709283i	$-0.250981\pi$
$317$	25.1016	1.40985	0.704924	+	0.709283i	$0.250981\pi$
$318$	23.6276	1.32497
$319$	−11.6643	−0.653074
$320$	−25.1923	−1.40829
$321$	22.1984	1.23899
$322$	0	0
$323$	7.84974	0.436771
$324$	3.08920	0.171622
$325$	−15.0803	−0.836505
$326$	23.9422	1.32604
$327$	12.0862	0.668366
$328$	30.1229	1.66326
$329$	0	0
$330$	12.9227	0.711369
$331$	30.5855	1.68113	0.840566	−	0.541710i	$-0.182223\pi$
$331$	30.5855	1.68113	0.840566	+	0.541710i	$0.182223\pi$
$332$	−2.10967	−0.115783
$333$	4.05962	0.222466
$334$	−10.0662	−0.550797
$335$	21.3675	1.16743
$336$	0	0
$337$	19.1142	1.04122	0.520608	−	0.853796i	$-0.325705\pi$
$337$	19.1142	1.04122	0.520608	+	0.853796i	$0.325705\pi$
$338$	−11.4008	−0.620121
$339$	31.8268	1.72859
$340$	−1.34520	−0.0729536
$341$	9.52606	0.515865
$342$	9.93437	0.537189
$343$	0	0
$344$	8.92602	0.481259
$345$	0.641702	0.0345481
$346$	−3.41158	−0.183408
$347$	−6.14957	−0.330126	−0.165063	−	0.986283i	$-0.552783\pi$
$347$	−6.14957	−0.330126	−0.165063	+	0.986283i	$0.552783\pi$
$348$	4.34474	0.232902
$349$	−15.5329	−0.831454	−0.415727	−	0.909489i	$-0.636473\pi$
$349$	−15.5329	−0.831454	−0.415727	+	0.909489i	$0.636473\pi$
$350$	0	0
$351$	13.6262	0.727313
$352$	−2.45752	−0.130986
$353$	−14.9628	−0.796389	−0.398194	−	0.917301i	$-0.630363\pi$
$353$	−14.9628	−0.796389	−0.398194	+	0.917301i	$0.630363\pi$
$354$	−18.6442	−0.990928
$355$	40.9715	2.17454
$356$	−2.29767	−0.121776
$357$	0	0
$358$	−24.9109	−1.31658
$359$	−19.5952	−1.03419	−0.517097	−	0.855927i	$-0.672987\pi$
$359$	−19.5952	−1.03419	−0.517097	+	0.855927i	$0.672987\pi$
$360$	−14.0799	−0.742078
$361$	2.22926	0.117330
$362$	5.06265	0.266087
$363$	−18.2447	−0.957600
$364$	0	0
$365$	−26.1937	−1.37104
$366$	−14.8232	−0.774819
$367$	−1.41231	−0.0737220	−0.0368610	−	0.999320i	$-0.511736\pi$
$367$	−1.41231	−0.0737220	−0.0368610	+	0.999320i	$0.511736\pi$
$368$	0.350131	0.0182519
$369$	16.5502	0.861570
$370$	9.32201	0.484628
$371$	0	0
$372$	−3.54829	−0.183970
$373$	33.8601	1.75321	0.876606	−	0.481209i	$-0.159802\pi$
$373$	33.8601	1.75321	0.876606	+	0.481209i	$0.159802\pi$
$374$	3.56018	0.184092
$375$	10.8917	0.562444
$376$	−15.6882	−0.809057
$377$	34.1345	1.75802
$378$	0	0
$379$	−20.8193	−1.06942	−0.534708	−	0.845037i	$-0.679578\pi$
$379$	−20.8193	−1.06942	−0.534708	+	0.845037i	$0.679578\pi$
$380$	−3.63803	−0.186627
$381$	2.15446	0.110376
$382$	6.23282	0.318899
$383$	18.2299	0.931503	0.465751	−	0.884916i	$-0.345784\pi$
$383$	18.2299	0.931503	0.465751	+	0.884916i	$0.345784\pi$
$384$	−18.1795	−0.927718
$385$	0	0
$386$	−2.38193	−0.121237
$387$	4.90417	0.249293
$388$	−2.05585	−0.104370
$389$	−13.1892	−0.668717	−0.334358	−	0.942446i	$-0.608520\pi$
$389$	−13.1892	−0.668717	−0.334358	+	0.942446i	$0.608520\pi$
$390$	−37.8170	−1.91494
$391$	0.176788	0.00894057
$392$	0	0
$393$	−43.0865	−2.17342
$394$	23.7536	1.19669
$395$	28.7596	1.44705
$396$	−0.718552	−0.0361086
$397$	−13.6246	−0.683798	−0.341899	−	0.939737i	$-0.611070\pi$
$397$	−13.6246	−0.683798	−0.341899	+	0.939737i	$0.611070\pi$
$398$	9.56562	0.479481
$399$	0	0
$400$	−10.9280	−0.546398
$401$	10.0237	0.500560	0.250280	−	0.968174i	$-0.419477\pi$
$401$	10.0237	0.500560	0.250280	+	0.968174i	$0.419477\pi$
$402$	21.0642	1.05059
$403$	−27.8772	−1.38866
$404$	−3.15644	−0.157039
$405$	32.2335	1.60169
$406$	0	0
$407$	3.93456	0.195029
$408$	−10.9675	−0.542971
$409$	13.2007	0.652731	0.326366	−	0.945244i	$-0.394176\pi$
$409$	13.2007	0.652731	0.326366	+	0.945244i	$0.394176\pi$
$410$	38.0039	1.87688
$411$	−9.72864	−0.479878
$412$	0.833960	0.0410863
$413$	0	0
$414$	0.223738	0.0109961
$415$	−22.0129	−1.08057
$416$	7.19172	0.352603
$417$	18.8535	0.923260
$418$	9.62834	0.470937
$419$	−37.1218	−1.81352	−0.906760	−	0.421647i	$-0.861452\pi$
$419$	−37.1218	−1.81352	−0.906760	+	0.421647i	$0.861452\pi$
$420$	0	0
$421$	34.8887	1.70037	0.850186	−	0.526482i	$-0.176489\pi$
$421$	34.8887	1.70037	0.850186	+	0.526482i	$0.176489\pi$
$422$	−6.48035	−0.315459
$423$	−8.61948	−0.419093
$424$	−24.9506	−1.21171
$425$	−5.51775	−0.267650
$426$	40.3900	1.95690
$427$	0	0
$428$	−2.83434	−0.137003
$429$	−15.9615	−0.770629
$430$	11.2613	0.543070
$431$	−12.7253	−0.612954	−0.306477	−	0.951878i	$-0.599150\pi$
$431$	−12.7253	−0.612954	−0.306477	+	0.951878i	$0.599150\pi$
$432$	9.87425	0.475075
$433$	16.7833	0.806553	0.403276	−	0.915078i	$-0.367871\pi$
$433$	16.7833	0.806553	0.403276	+	0.915078i	$0.367871\pi$
$434$	0	0
$435$	45.3341	2.17360
$436$	−1.54319	−0.0739054
$437$	0.478116	0.0228714
$438$	−25.8219	−1.23382
$439$	−7.34357	−0.350490	−0.175245	−	0.984525i	$-0.556072\pi$
$439$	−7.34357	−0.350490	−0.175245	+	0.984525i	$0.556072\pi$
$440$	−13.6462	−0.650557
$441$	0	0
$442$	−10.4186	−0.495560
$443$	4.94929	0.235148	0.117574	−	0.993064i	$-0.462488\pi$
$443$	4.94929	0.235148	0.117574	+	0.993064i	$0.462488\pi$
$444$	−1.46556	−0.0695522
$445$	−23.9745	−1.13650
$446$	9.88085	0.467872
$447$	−32.7339	−1.54826
$448$	0	0
$449$	14.6272	0.690298	0.345149	−	0.938548i	$-0.387828\pi$
$449$	14.6272	0.690298	0.345149	+	0.938548i	$0.387828\pi$
$450$	−6.98308	−0.329186
$451$	16.0404	0.755312
$452$	−4.06371	−0.191141
$453$	25.3393	1.19055
$454$	0.344726	0.0161788
$455$	0	0
$456$	−29.6611	−1.38901
$457$	−21.2857	−0.995705	−0.497852	−	0.867262i	$-0.665878\pi$
$457$	−21.2857	−0.995705	−0.497852	+	0.867262i	$0.665878\pi$
$458$	−5.65486	−0.264234
$459$	4.98571	0.232713
$460$	−0.0819340	−0.00382019
$461$	35.4400	1.65060	0.825302	−	0.564692i	$-0.191005\pi$
$461$	35.4400	1.65060	0.825302	+	0.564692i	$0.191005\pi$
$462$	0	0
$463$	13.5341	0.628984	0.314492	−	0.949260i	$-0.398166\pi$
$463$	13.5341	0.628984	0.314492	+	0.949260i	$0.398166\pi$
$464$	24.7356	1.14832
$465$	−37.0238	−1.71694
$466$	−17.0851	−0.791453
$467$	37.1297	1.71816	0.859078	−	0.511845i	$-0.171038\pi$
$467$	37.1297	1.71816	0.859078	+	0.511845i	$0.171038\pi$
$468$	2.10278	0.0972010
$469$	0	0
$470$	−19.7927	−0.912970
$471$	24.3113	1.12020
$472$	19.6881	0.906218
$473$	4.75309	0.218547
$474$	28.3514	1.30222
$475$	−14.9225	−0.684691
$476$	0	0
$477$	−13.7084	−0.627666
$478$	−38.5714	−1.76421
$479$	10.6647	0.487284	0.243642	−	0.969865i	$-0.421658\pi$
$479$	10.6647	0.487284	0.243642	+	0.969865i	$0.421658\pi$
$480$	9.55135	0.435958
$481$	−11.5142	−0.525000
$482$	35.0005	1.59423
$483$	0	0
$484$	2.32953	0.105888
$485$	−21.4512	−0.974051
$486$	20.2455	0.918357
$487$	0.867698	0.0393192	0.0196596	−	0.999807i	$-0.493742\pi$
$487$	0.867698	0.0393192	0.0196596	+	0.999807i	$0.493742\pi$
$488$	15.6531	0.708583
$489$	−39.2752	−1.77609
$490$	0	0
$491$	−11.2654	−0.508399	−0.254200	−	0.967152i	$-0.581812\pi$
$491$	−11.2654	−0.508399	−0.254200	+	0.967152i	$0.581812\pi$
$492$	−5.97476	−0.269363
$493$	12.4895	0.562499
$494$	−28.1765	−1.26772
$495$	−7.49755	−0.336990
$496$	−20.2013	−0.907063
$497$	0	0
$498$	−21.7004	−0.972418
$499$	5.82592	0.260804	0.130402	−	0.991461i	$-0.458373\pi$
$499$	5.82592	0.260804	0.130402	+	0.991461i	$0.458373\pi$
$500$	−1.39068	−0.0621929
$501$	16.5127	0.737735
$502$	0.00207462	9.25948e−5 0
$503$	−22.4988	−1.00317	−0.501585	−	0.865108i	$-0.667250\pi$
$503$	−22.4988	−1.00317	−0.501585	+	0.865108i	$0.667250\pi$
$504$	0	0
$505$	−32.9351	−1.46559
$506$	0.216845	0.00963994
$507$	18.7020	0.830586
$508$	−0.275086	−0.0122050
$509$	33.9763	1.50597	0.752987	−	0.658035i	$-0.228612\pi$
$509$	33.9763	1.50597	0.752987	+	0.658035i	$0.228612\pi$
$510$	−13.8369	−0.612709
$511$	0	0
$512$	25.3755	1.12145
$513$	13.4836	0.595316
$514$	−14.5882	−0.643460
$515$	8.70176	0.383445
$516$	−1.77044	−0.0779395
$517$	−8.35395	−0.367406
$518$	0	0
$519$	5.59642	0.245655
$520$	39.9344	1.75124
$521$	7.88458	0.345430	0.172715	−	0.984972i	$-0.444746\pi$
$521$	7.88458	0.345430	0.172715	+	0.984972i	$0.444746\pi$
$522$	15.8063	0.691823
$523$	−8.08077	−0.353347	−0.176674	−	0.984269i	$-0.556534\pi$
$523$	−8.08077	−0.353347	−0.176674	+	0.984269i	$0.556534\pi$
$524$	5.50138	0.240329
$525$	0	0
$526$	0.739254	0.0322330
$527$	−10.2000	−0.444320
$528$	−11.5665	−0.503369
$529$	−22.9892	−0.999532
$530$	−31.4784	−1.36733
$531$	10.8171	0.469423
$532$	0	0
$533$	−46.9408	−2.03323
$534$	−23.6342	−1.02275
$535$	−29.5742	−1.27860
$536$	−22.2437	−0.960779
$537$	40.8643	1.76342
$538$	−35.9488	−1.54986
$539$	0	0
$540$	−2.31067	−0.0994353
$541$	−29.0363	−1.24837	−0.624183	−	0.781278i	$-0.714568\pi$
$541$	−29.0363	−1.24837	−0.624183	+	0.781278i	$0.714568\pi$
$542$	−26.2055	−1.12562
$543$	−8.30485	−0.356395
$544$	2.63139	0.112820
$545$	−16.1020	−0.689735
$546$	0	0
$547$	−22.8596	−0.977406	−0.488703	−	0.872450i	$-0.662530\pi$
$547$	−22.8596	−0.977406	−0.488703	+	0.872450i	$0.662530\pi$
$548$	1.24218	0.0530631
$549$	8.60020	0.367048
$550$	−6.76796	−0.288587
$551$	33.7773	1.43896
$552$	−0.668013	−0.0284325
$553$	0	0
$554$	16.0248	0.680828
$555$	−15.2920	−0.649108
$556$	−2.40726	−0.102091
$557$	3.10093	0.131391	0.0656953	−	0.997840i	$-0.479073\pi$
$557$	3.10093	0.131391	0.0656953	+	0.997840i	$0.479073\pi$
$558$	−12.9088	−0.546473
$559$	−13.9095	−0.588310
$560$	0	0
$561$	−5.84017	−0.246572
$562$	−30.1417	−1.27145
$563$	33.4493	1.40972	0.704860	−	0.709347i	$-0.251010\pi$
$563$	33.4493	1.40972	0.704860	+	0.709347i	$0.251010\pi$
$564$	3.11170	0.131026
$565$	−42.4018	−1.78386
$566$	−21.3487	−0.897353
$567$	0	0
$568$	−42.6515	−1.78962
$569$	21.1981	0.888671	0.444335	−	0.895861i	$-0.353440\pi$
$569$	21.1981	0.888671	0.444335	+	0.895861i	$0.353440\pi$
$570$	−37.4213	−1.56741
$571$	−37.8686	−1.58475	−0.792377	−	0.610032i	$-0.791157\pi$
$571$	−37.8686	−1.58475	−0.792377	+	0.610032i	$0.791157\pi$
$572$	2.03800	0.0852131
$573$	−10.2244	−0.427131
$574$	0	0
$575$	−0.336078	−0.0140154
$576$	14.4088	0.600366
$577$	−34.0007	−1.41547	−0.707734	−	0.706479i	$-0.750282\pi$
$577$	−34.0007	−1.41547	−0.707734	+	0.706479i	$0.750282\pi$
$578$	18.5151	0.770125
$579$	3.90735	0.162384
$580$	−5.78837	−0.240349
$581$	0	0
$582$	−21.1468	−0.876562
$583$	−13.2861	−0.550256
$584$	27.2677	1.12835
$585$	21.9409	0.907146
$586$	−30.4310	−1.25709
$587$	13.4607	0.555583	0.277792	−	0.960641i	$-0.410398\pi$
$587$	13.4607	0.555583	0.277792	+	0.960641i	$0.410398\pi$
$588$	0	0
$589$	−27.5855	−1.13664
$590$	24.8391	1.02261
$591$	−38.9658	−1.60284
$592$	−8.34375	−0.342926
$593$	35.6827	1.46531	0.732657	−	0.680598i	$-0.238280\pi$
$593$	35.6827	1.46531	0.732657	+	0.680598i	$0.238280\pi$
$594$	6.11537	0.250917
$595$	0	0
$596$	4.17954	0.171201
$597$	−15.6916	−0.642214
$598$	−0.634579	−0.0259499
$599$	13.0709	0.534061	0.267031	−	0.963688i	$-0.413958\pi$
$599$	13.0709	0.534061	0.267031	+	0.963688i	$0.413958\pi$
$600$	20.8494	0.851173
$601$	24.2381	0.988694	0.494347	−	0.869265i	$-0.335407\pi$
$601$	24.2381	0.988694	0.494347	+	0.869265i	$0.335407\pi$
$602$	0	0
$603$	−12.2212	−0.497686
$604$	−3.23538	−0.131646
$605$	24.3069	0.988217
$606$	−32.4676	−1.31891
$607$	3.03367	0.123133	0.0615665	−	0.998103i	$-0.480390\pi$
$607$	3.03367	0.123133	0.0615665	+	0.998103i	$0.480390\pi$
$608$	7.11647	0.288611
$609$	0	0
$610$	19.7485	0.799591
$611$	24.4471	0.989024
$612$	0.769388	0.0311007
$613$	−13.8288	−0.558539	−0.279269	−	0.960213i	$-0.590092\pi$
$613$	−13.8288	−0.558539	−0.279269	+	0.960213i	$0.590092\pi$
$614$	−18.9556	−0.764987
$615$	−62.3422	−2.51388
$616$	0	0
$617$	−10.0221	−0.403474	−0.201737	−	0.979440i	$-0.564659\pi$
$617$	−10.0221	−0.403474	−0.201737	+	0.979440i	$0.564659\pi$
$618$	8.57824	0.345068
$619$	22.1084	0.888613	0.444306	−	0.895875i	$-0.353450\pi$
$619$	22.1084	0.888613	0.444306	+	0.895875i	$0.353450\pi$
$620$	4.72728	0.189852
$621$	0.303672	0.0121859
$622$	21.8552	0.876315
$623$	0	0
$624$	33.8484	1.35502
$625$	−30.7043	−1.22817
$626$	−34.2666	−1.36957
$627$	−15.7945	−0.630771
$628$	−3.10412	−0.123868
$629$	−4.21292	−0.167980
$630$	0	0
$631$	44.9892	1.79099	0.895495	−	0.445071i	$-0.146822\pi$
$631$	44.9892	1.79099	0.895495	+	0.445071i	$0.146822\pi$
$632$	−29.9388	−1.19090
$633$	10.6305	0.422523
$634$	−32.9675	−1.30931
$635$	−2.87032	−0.113905
$636$	4.94886	0.196235
$637$	0	0
$638$	15.3194	0.606500
$639$	−23.4337	−0.927024
$640$	24.2200	0.957379
$641$	26.0573	1.02920	0.514602	−	0.857429i	$-0.327940\pi$
$641$	26.0573	1.02920	0.514602	+	0.857429i	$0.327940\pi$
$642$	−29.1544	−1.15063
$643$	6.31691	0.249115	0.124557	−	0.992212i	$-0.460249\pi$
$643$	6.31691	0.249115	0.124557	+	0.992212i	$0.460249\pi$
$644$	0	0
$645$	−18.4733	−0.727384
$646$	−10.3095	−0.405623
$647$	10.3120	0.405407	0.202704	−	0.979240i	$-0.435027\pi$
$647$	10.3120	0.405407	0.202704	+	0.979240i	$0.435027\pi$
$648$	−33.5551	−1.31817
$649$	10.4839	0.411529
$650$	19.8059	0.776850
$651$	0	0
$652$	5.01475	0.196393
$653$	−18.4623	−0.722485	−0.361243	−	0.932472i	$-0.617647\pi$
$653$	−18.4623	−0.722485	−0.361243	+	0.932472i	$0.617647\pi$
$654$	−15.8735	−0.620702
$655$	57.4028	2.24291
$656$	−34.0157	−1.32809
$657$	14.9815	0.584486
$658$	0	0
$659$	−11.6728	−0.454706	−0.227353	−	0.973812i	$-0.573007\pi$
$659$	−11.6728	−0.454706	−0.227353	+	0.973812i	$0.573007\pi$
$660$	2.70668	0.105357
$661$	−47.4022	−1.84373	−0.921866	−	0.387508i	$-0.873336\pi$
$661$	−47.4022	−1.84373	−0.921866	+	0.387508i	$0.873336\pi$
$662$	−40.1698	−1.56124
$663$	17.0908	0.663750
$664$	22.9154	0.889291
$665$	0	0
$666$	−5.33174	−0.206601
$667$	0.760717	0.0294551
$668$	−2.10838	−0.0815759
$669$	−16.2087	−0.626665
$670$	−28.0633	−1.08418
$671$	8.33527	0.321779
$672$	0	0
$673$	−28.5338	−1.09990	−0.549949	−	0.835199i	$-0.685353\pi$
$673$	−28.5338	−1.09990	−0.549949	+	0.835199i	$0.685353\pi$
$674$	−25.1038	−0.966963
$675$	−9.47792	−0.364805
$676$	−2.38792	−0.0918430
$677$	−44.3296	−1.70373	−0.851863	−	0.523765i	$-0.824527\pi$
$677$	−44.3296	−1.70373	−0.851863	+	0.523765i	$0.824527\pi$
$678$	−41.8000	−1.60532
$679$	0	0
$680$	14.6117	0.560331
$681$	−0.565494	−0.0216698
$682$	−12.5111	−0.479077
$683$	35.3285	1.35181	0.675904	−	0.736990i	$-0.263753\pi$
$683$	35.3285	1.35181	0.675904	+	0.736990i	$0.263753\pi$
$684$	2.08078	0.0795604
$685$	12.9612	0.495221
$686$	0	0
$687$	9.27633	0.353914
$688$	−10.0796	−0.384279
$689$	38.8808	1.48124
$690$	−0.842786	−0.0320843
$691$	14.2738	0.543002	0.271501	−	0.962438i	$-0.412480\pi$
$691$	14.2738	0.543002	0.271501	+	0.962438i	$0.412480\pi$
$692$	−0.714564	−0.0271636
$693$	0	0
$694$	8.07660	0.306584
$695$	−25.1180	−0.952779
$696$	−47.1929	−1.78884
$697$	−17.1752	−0.650558
$698$	20.4002	0.772160
$699$	28.0267	1.06007
$700$	0	0
$701$	19.8244	0.748759	0.374379	−	0.927276i	$-0.377856\pi$
$701$	19.8244	0.748759	0.374379	+	0.927276i	$0.377856\pi$
$702$	−17.8961	−0.675445
$703$	−11.3937	−0.429720
$704$	13.9649	0.526322
$705$	32.4683	1.22283
$706$	19.6515	0.739595
$707$	0	0
$708$	−3.90507	−0.146761
$709$	−44.3096	−1.66408	−0.832041	−	0.554714i	$-0.812828\pi$
$709$	−44.3096	−1.66408	−0.832041	+	0.554714i	$0.812828\pi$
$710$	−53.8104	−2.01947
$711$	−16.4491	−0.616890
$712$	24.9575	0.935321
$713$	−0.621268	−0.0232667
$714$	0	0
$715$	21.2650	0.795267
$716$	−5.21764	−0.194992
$717$	63.2731	2.36298
$718$	25.7355	0.960440
$719$	36.3924	1.35721	0.678603	−	0.734505i	$-0.262586\pi$
$719$	36.3924	1.35721	0.678603	+	0.734505i	$0.262586\pi$
$720$	15.8995	0.592541
$721$	0	0
$722$	−2.92782	−0.108962
$723$	−57.4153	−2.13530
$724$	1.06038	0.0394088
$725$	−23.7428	−0.881785
$726$	23.9619	0.889310
$727$	−28.9004	−1.07186	−0.535929	−	0.844263i	$-0.680038\pi$
$727$	−28.9004	−1.07186	−0.535929	+	0.844263i	$0.680038\pi$
$728$	0	0
$729$	0.478655	0.0177280
$730$	34.4018	1.27327
$731$	−5.08937	−0.188237
$732$	−3.10474	−0.114755
$733$	−16.5945	−0.612934	−0.306467	−	0.951881i	$-0.599147\pi$
$733$	−16.5945	−0.612934	−0.306467	+	0.951881i	$0.599147\pi$
$734$	1.85487	0.0684645
$735$	0	0
$736$	0.160274	0.00590778
$737$	−11.8447	−0.436306
$738$	−21.7364	−0.800128
$739$	32.7856	1.20604	0.603019	−	0.797727i	$-0.293964\pi$
$739$	32.7856	1.20604	0.603019	+	0.797727i	$0.293964\pi$
$740$	1.95252	0.0717759
$741$	46.2212	1.69798
$742$	0	0
$743$	31.3429	1.14986	0.574929	−	0.818203i	$-0.305030\pi$
$743$	31.3429	1.14986	0.574929	+	0.818203i	$0.305030\pi$
$744$	38.5418	1.41301
$745$	43.6104	1.59776
$746$	−44.4705	−1.62818
$747$	12.5903	0.460655
$748$	0.745687	0.0272650
$749$	0	0
$750$	−14.3047	−0.522334
$751$	33.2746	1.21421	0.607104	−	0.794622i	$-0.292331\pi$
$751$	33.2746	1.21421	0.607104	+	0.794622i	$0.292331\pi$
$752$	17.7156	0.646023
$753$	−0.00340324	−0.000124021 0
$754$	−44.8309	−1.63264
$755$	−33.7588	−1.22861
$756$	0	0
$757$	10.1688	0.369591	0.184796	−	0.982777i	$-0.440838\pi$
$757$	10.1688	0.369591	0.184796	+	0.982777i	$0.440838\pi$
$758$	27.3432	0.993151
$759$	−0.355716	−0.0129117
$760$	39.5165	1.43342
$761$	1.93614	0.0701851	0.0350925	−	0.999384i	$-0.488827\pi$
$761$	1.93614	0.0701851	0.0350925	+	0.999384i	$0.488827\pi$
$762$	−2.82958	−0.102505
$763$	0	0
$764$	1.30548	0.0472305
$765$	8.02799	0.290253
$766$	−23.9424	−0.865073
$767$	−30.6802	−1.10780
$768$	−13.9424	−0.503104
$769$	−4.48664	−0.161792	−0.0808961	−	0.996723i	$-0.525778\pi$
$769$	−4.48664	−0.161792	−0.0808961	+	0.996723i	$0.525778\pi$
$770$	0	0
$771$	23.9308	0.861847
$772$	−0.498900	−0.0179558
$773$	54.3911	1.95631	0.978155	−	0.207875i	$-0.0666546\pi$
$773$	54.3911	1.95631	0.978155	+	0.207875i	$0.0666546\pi$
$774$	−6.44094	−0.231515
$775$	19.3904	0.696525
$776$	22.3308	0.801629
$777$	0	0
$778$	17.3221	0.621028
$779$	−46.4496	−1.66423
$780$	−7.92086	−0.283612
$781$	−22.7118	−0.812694
$782$	−0.232187	−0.00830298
$783$	21.4534	0.766682
$784$	0	0
$785$	−32.3892	−1.15602
$786$	56.5880	2.01843
$787$	−55.2407	−1.96912	−0.984560	−	0.175050i	$-0.943991\pi$
$787$	−55.2407	−1.96912	−0.984560	+	0.175050i	$0.943991\pi$
$788$	4.97524	0.177236
$789$	−1.21268	−0.0431727
$790$	−37.7717	−1.34386
$791$	0	0
$792$	7.80497	0.277337
$793$	−24.3924	−0.866201
$794$	17.8940	0.635034
$795$	51.6377	1.83140
$796$	2.00354	0.0710135
$797$	−5.80327	−0.205562	−0.102781	−	0.994704i	$-0.532774\pi$
$797$	−5.80327	−0.205562	−0.102781	+	0.994704i	$0.532774\pi$
$798$	0	0
$799$	8.94498	0.316451
$800$	−5.00232	−0.176859
$801$	13.7122	0.484498
$802$	−13.1647	−0.464863
$803$	14.5200	0.512401
$804$	4.41195	0.155598
$805$	0	0
$806$	36.6128	1.28963
$807$	58.9710	2.07588
$808$	34.2855	1.20616
$809$	6.34449	0.223060	0.111530	−	0.993761i	$-0.464425\pi$
$809$	6.34449	0.223060	0.111530	+	0.993761i	$0.464425\pi$
$810$	−42.3341	−1.48747
$811$	−0.941916	−0.0330752	−0.0165376	−	0.999863i	$-0.505264\pi$
$811$	−0.941916	−0.0330752	−0.0165376	+	0.999863i	$0.505264\pi$
$812$	0	0
$813$	42.9879	1.50765
$814$	−5.16749	−0.181121
$815$	52.3252	1.83287
$816$	12.3848	0.433556
$817$	−13.7640	−0.481540
$818$	−17.3372	−0.606182
$819$	0	0
$820$	7.96000	0.277975
$821$	−33.4339	−1.16685	−0.583426	−	0.812166i	$-0.698288\pi$
$821$	−33.4339	−1.16685	−0.583426	+	0.812166i	$0.698288\pi$
$822$	12.7772	0.445656
$823$	6.06800	0.211517	0.105759	−	0.994392i	$-0.466273\pi$
$823$	6.06800	0.211517	0.105759	+	0.994392i	$0.466273\pi$
$824$	−9.05854	−0.315569
$825$	11.1023	0.386532
$826$	0	0
$827$	−11.5578	−0.401905	−0.200953	−	0.979601i	$-0.564404\pi$
$827$	−11.5578	−0.401905	−0.200953	+	0.979601i	$0.564404\pi$
$828$	0.0468623	0.00162858
$829$	−2.42424	−0.0841975	−0.0420987	−	0.999113i	$-0.513404\pi$
$829$	−2.42424	−0.0841975	−0.0420987	+	0.999113i	$0.513404\pi$
$830$	28.9108	1.00351
$831$	−26.2873	−0.911897
$832$	−40.8671	−1.41681
$833$	0	0
$834$	−24.7614	−0.857419
$835$	−21.9994	−0.761322
$836$	2.01668	0.0697482
$837$	−17.5207	−0.605605
$838$	48.7543	1.68419
$839$	−29.0493	−1.00289	−0.501447	−	0.865189i	$-0.667199\pi$
$839$	−29.0493	−1.00289	−0.501447	+	0.865189i	$0.667199\pi$
$840$	0	0
$841$	24.7421	0.853177
$842$	−45.8215	−1.57911
$843$	49.4449	1.70297
$844$	−1.35732	−0.0467210
$845$	−24.9162	−0.857142
$846$	11.3205	0.389206
$847$	0	0
$848$	28.1750	0.967533
$849$	35.0208	1.20191
$850$	7.24679	0.248563
$851$	−0.256603	−0.00879624
$852$	8.45977	0.289827
$853$	−41.0726	−1.40630	−0.703149	−	0.711043i	$-0.748223\pi$
$853$	−41.0726	−1.40630	−0.703149	+	0.711043i	$0.748223\pi$
$854$	0	0
$855$	21.7113	0.742512
$856$	30.7868	1.05227
$857$	21.4981	0.734362	0.367181	−	0.930150i	$-0.380323\pi$
$857$	21.4981	0.734362	0.367181	+	0.930150i	$0.380323\pi$
$858$	20.9632	0.715672
$859$	−45.3398	−1.54697	−0.773487	−	0.633812i	$-0.781489\pi$
$859$	−45.3398	−1.54697	−0.773487	+	0.633812i	$0.781489\pi$
$860$	2.35871	0.0804314
$861$	0	0
$862$	16.7128	0.569241
$863$	50.6983	1.72579	0.862896	−	0.505382i	$-0.168648\pi$
$863$	50.6983	1.72579	0.862896	+	0.505382i	$0.168648\pi$
$864$	4.51997	0.153773
$865$	−7.45594	−0.253510
$866$	−22.0425	−0.749034
$867$	−30.3724	−1.03150
$868$	0	0
$869$	−15.9424	−0.540808
$870$	−59.5400	−2.01860
$871$	34.6625	1.17450
$872$	16.7622	0.567641
$873$	12.2691	0.415245
$874$	−0.627938	−0.0212403
$875$	0	0
$876$	−5.40846	−0.182735
$877$	31.1264	1.05106	0.525532	−	0.850774i	$-0.323866\pi$
$877$	31.1264	1.05106	0.525532	+	0.850774i	$0.323866\pi$
$878$	9.64475	0.325495
$879$	49.9195	1.68374
$880$	15.4097	0.519462
$881$	8.88546	0.299359	0.149679	−	0.988735i	$-0.452176\pi$
$881$	8.88546	0.299359	0.149679	+	0.988735i	$0.452176\pi$
$882$	0	0
$883$	32.1160	1.08079	0.540394	−	0.841412i	$-0.318275\pi$
$883$	32.1160	1.08079	0.540394	+	0.841412i	$0.318275\pi$
$884$	−2.18219	−0.0733949
$885$	−40.7465	−1.36968
$886$	−6.50020	−0.218379
$887$	−41.3922	−1.38981	−0.694907	−	0.719099i	$-0.744555\pi$
$887$	−41.3922	−1.38981	−0.694907	+	0.719099i	$0.744555\pi$
$888$	15.9190	0.534206
$889$	0	0
$890$	31.4871	1.05545
$891$	−17.8680	−0.598602
$892$	2.06956	0.0692942
$893$	24.1913	0.809531
$894$	42.9914	1.43785
$895$	−54.4422	−1.81980
$896$	0	0
$897$	1.04097	0.0347571
$898$	−19.2107	−0.641070
$899$	−43.8905	−1.46383
$900$	−1.46262	−0.0487540
$901$	14.2261	0.473941
$902$	−21.0668	−0.701447
$903$	0	0
$904$	44.1404	1.46809
$905$	11.0643	0.367790
$906$	−33.2797	−1.10564
$907$	−39.2293	−1.30259	−0.651295	−	0.758825i	$-0.725774\pi$
$907$	−39.2293	−1.30259	−0.651295	+	0.758825i	$0.725774\pi$
$908$	0.0722036	0.00239616
$909$	18.8373	0.624793
$910$	0	0
$911$	40.2594	1.33385	0.666926	−	0.745124i	$-0.267610\pi$
$911$	40.2594	1.33385	0.666926	+	0.745124i	$0.267610\pi$
$912$	33.4943	1.10911
$913$	12.2024	0.403842
$914$	27.9558	0.924697
$915$	−32.3957	−1.07097
$916$	−1.18442	−0.0391344
$917$	0	0
$918$	−6.54803	−0.216117
$919$	−37.0755	−1.22301	−0.611505	−	0.791241i	$-0.709435\pi$
$919$	−37.0755	−1.22301	−0.611505	+	0.791241i	$0.709435\pi$
$920$	0.889974	0.0293416
$921$	31.0951	1.02462
$922$	−46.5454	−1.53289
$923$	66.4643	2.18770
$924$	0	0
$925$	8.00885	0.263329
$926$	−17.7752	−0.584128
$927$	−4.97698	−0.163466
$928$	11.3228	0.371690
$929$	53.5592	1.75722	0.878610	−	0.477539i	$-0.158471\pi$
$929$	53.5592	1.75722	0.878610	+	0.477539i	$0.158471\pi$
$930$	48.6256	1.59450
$931$	0	0
$932$	−3.57851	−0.117218
$933$	−35.8517	−1.17373
$934$	−48.7646	−1.59563
$935$	7.78069	0.254456
$936$	−22.8406	−0.746567
$937$	−24.4182	−0.797709	−0.398855	−	0.917014i	$-0.630592\pi$
$937$	−24.4182	−0.797709	−0.398855	+	0.917014i	$0.630592\pi$
$938$	0	0
$939$	56.2115	1.83439
$940$	−4.14563	−0.135215
$941$	14.5597	0.474634	0.237317	−	0.971432i	$-0.423732\pi$
$941$	14.5597	0.474634	0.237317	+	0.971432i	$0.423732\pi$
$942$	−31.9295	−1.04032
$943$	−1.04612	−0.0340663
$944$	−22.2324	−0.723604
$945$	0	0
$946$	−6.24252	−0.202962
$947$	−2.58241	−0.0839172	−0.0419586	−	0.999119i	$-0.513360\pi$
$947$	−2.58241	−0.0839172	−0.0419586	+	0.999119i	$0.513360\pi$
$948$	5.93826	0.192866
$949$	−42.4916	−1.37934
$950$	19.5986	0.635863
$951$	54.0804	1.75368
$952$	0	0
$953$	10.9801	0.355682	0.177841	−	0.984059i	$-0.443089\pi$
$953$	10.9801	0.355682	0.177841	+	0.984059i	$0.443089\pi$
$954$	18.0041	0.582905
$955$	13.6217	0.440787
$956$	−8.07886	−0.261289
$957$	−25.1302	−0.812343
$958$	−14.0066	−0.452533
$959$	0	0
$960$	−54.2758	−1.75174
$961$	4.84482	0.156285
$962$	15.1222	0.487560
$963$	16.9150	0.545079
$964$	7.33092	0.236113
$965$	−5.20565	−0.167576
$966$	0	0
$967$	34.1180	1.09716	0.548580	−	0.836098i	$-0.315168\pi$
$967$	34.1180	1.09716	0.548580	+	0.836098i	$0.315168\pi$
$968$	−25.3035	−0.813287
$969$	16.9119	0.543289
$970$	28.1732	0.904587
$971$	35.4951	1.13909	0.569547	−	0.821959i	$-0.307119\pi$
$971$	35.4951	1.13909	0.569547	+	0.821959i	$0.307119\pi$
$972$	4.24047	0.136013
$973$	0	0
$974$	−1.13960	−0.0365151
$975$	−32.4898	−1.04051
$976$	−17.6760	−0.565796
$977$	−34.9889	−1.11939	−0.559696	−	0.828698i	$-0.689082\pi$
$977$	−34.9889	−1.11939	−0.559696	+	0.828698i	$0.689082\pi$
$978$	51.5825	1.64943
$979$	13.2898	0.424744
$980$	0	0
$981$	9.20958	0.294039
$982$	14.7955	0.472143
$983$	−17.4400	−0.556250	−0.278125	−	0.960545i	$-0.589713\pi$
$983$	−17.4400	−0.556250	−0.278125	+	0.960545i	$0.589713\pi$
$984$	64.8984	2.06888
$985$	51.9130	1.65408
$986$	−16.4032	−0.522385
$987$	0	0
$988$	−5.90163	−0.187756
$989$	−0.309986	−0.00985698
$990$	9.84698	0.312958
$991$	−28.4282	−0.903052	−0.451526	−	0.892258i	$-0.649120\pi$
$991$	−28.4282	−0.903052	−0.451526	+	0.892258i	$0.649120\pi$
$992$	−9.24720	−0.293599
$993$	65.8951	2.09112
$994$	0	0
$995$	20.9054	0.662747
$996$	−4.54520	−0.144020
$997$	−56.7159	−1.79621	−0.898105	−	0.439781i	$-0.855056\pi$
$997$	−56.7159	−1.79621	−0.898105	+	0.439781i	$0.855056\pi$
$998$	−7.65153	−0.242205
$999$	−7.23661	−0.228956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.p.1.12		38
7.2	even	3		889.2.f.c.382.27	yes	76
7.4	even	3		889.2.f.c.128.27	✓	76
7.6	odd	2		6223.2.a.o.1.12		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.f.c.128.27	✓	76	7.4	even	3
889.2.f.c.382.27	yes	76	7.2	even	3
6223.2.a.o.1.12		38	7.6	odd	2
6223.2.a.p.1.12		38	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.31336 q^{2} +2.15446 q^{3} -0.275086 q^{4} -2.87032 q^{5} -2.82958 q^{6} +2.98801 q^{8} +1.64168 q^{9} +3.76976 q^{10} +1.59111 q^{11} -0.592661 q^{12} -4.65625 q^{13} -6.18398 q^{15} -3.37416 q^{16} -1.70368 q^{17} -2.15612 q^{18} -4.60752 q^{19} +0.789584 q^{20} -2.08970 q^{22} -0.103769 q^{23} +6.43753 q^{24} +3.23873 q^{25} +6.11533 q^{26} -2.92644 q^{27} -7.33090 q^{29} +8.12178 q^{30} +5.98705 q^{31} -1.54453 q^{32} +3.42798 q^{33} +2.23754 q^{34} -0.451604 q^{36} +2.47284 q^{37} +6.05133 q^{38} -10.0317 q^{39} -8.57653 q^{40} +10.0813 q^{41} +2.98728 q^{43} -0.437692 q^{44} -4.71215 q^{45} +0.136285 q^{46} -5.25039 q^{47} -7.26947 q^{48} -4.25361 q^{50} -3.67050 q^{51} +1.28087 q^{52} -8.35024 q^{53} +3.84346 q^{54} -4.56699 q^{55} -9.92671 q^{57} +9.62811 q^{58} +6.58904 q^{59} +1.70112 q^{60} +5.23865 q^{61} -7.86316 q^{62} +8.77684 q^{64} +13.3649 q^{65} -4.50217 q^{66} -7.44431 q^{67} +0.468658 q^{68} -0.223565 q^{69} -14.2742 q^{71} +4.90536 q^{72} +9.12572 q^{73} -3.24773 q^{74} +6.97769 q^{75} +1.26746 q^{76} +13.1752 q^{78} -10.0197 q^{79} +9.68490 q^{80} -11.2299 q^{81} -13.2403 q^{82} +7.66914 q^{83} +4.89010 q^{85} -3.92338 q^{86} -15.7941 q^{87} +4.75425 q^{88} +8.35255 q^{89} +6.18875 q^{90} +0.0285453 q^{92} +12.8988 q^{93} +6.89565 q^{94} +13.2251 q^{95} -3.32763 q^{96} +7.47347 q^{97} +2.61210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} + 11 q^{3} + 38 q^{4} + 16 q^{5} + 11 q^{6} + 47 q^{9} + 12 q^{10} - 2 q^{11} + 30 q^{12} + 21 q^{13} + 7 q^{15} + 46 q^{16} + 58 q^{17} - 13 q^{18} + 17 q^{19} + 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 + 11 * q^3 + 38 * q^4 + 16 * q^5 + 11 * q^6 + 47 * q^9 + 12 * q^10 - 2 * q^11 + 30 * q^12 + 21 * q^13 + 7 * q^15 + 46 * q^16 + 58 * q^17 - 13 * q^18 + 17 * q^19 + 44 * q^20 + 21 * q^22 + 7 * q^23 + 21 * q^24 + 52 * q^25 + 49 * q^26 + 35 * q^27 - 2 * q^29 + 6 * q^30 + 2 * q^31 - 22 * q^32 + 29 * q^33 + 37 * q^34 + 25 * q^36 + 11 * q^37 + 40 * q^38 + 2 * q^39 - 8 * q^40 + 68 * q^41 - 38 * q^43 + 43 * q^44 + 60 * q^45 - 28 * q^46 + 35 * q^47 + 66 * q^48 - 25 * q^50 - 3 * q^51 + 32 * q^52 - 18 * q^53 - 68 * q^54 + 19 * q^55 - 31 * q^57 + 55 * q^58 + 21 * q^59 - 11 * q^60 + 5 * q^61 + 38 * q^62 + 18 * q^64 - 9 * q^65 + 20 * q^66 + 6 * q^67 + 74 * q^68 + 53 * q^69 - 26 * q^71 + 24 * q^72 + 22 * q^73 - 2 * q^74 + 38 * q^75 + 39 * q^76 - 114 * q^78 - 14 * q^79 + 140 * q^80 + 58 * q^81 - 8 * q^82 + 46 * q^83 - 11 * q^85 + 51 * q^86 + 59 * q^87 - 4 * q^88 + 74 * q^89 + q^90 - 81 * q^92 + 92 * q^93 - 36 * q^94 + 20 * q^95 - 108 * q^96 + 66 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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