LMFDB - Embedded newform 5243.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5243,2,Mod(1,5243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5243.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5243 = 7^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5243.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.8655657798$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 29x^{3} - 12x^{2} - 20x + 8 $ x^7 - x^6 - 10x^5 + 7x^4 + 29x^3 - 12x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 107)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.400681$ of defining polynomial
Character	$\chi$	$=$	5243.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.400681 q^{2} -2.05242 q^{3} -1.83945 q^{4} -0.858034 q^{5} +0.822367 q^{6} +1.53840 q^{8} +1.21244 q^{9} +0.343798 q^{10} -1.56312 q^{11} +3.77534 q^{12} +4.04020 q^{13} +1.76105 q^{15} +3.06250 q^{16} +0.343798 q^{17} -0.485802 q^{18} -1.63409 q^{19} +1.57831 q^{20} +0.626314 q^{22} -7.84845 q^{23} -3.15744 q^{24} -4.26378 q^{25} -1.61883 q^{26} +3.66883 q^{27} -3.98461 q^{29} -0.705619 q^{30} -6.10060 q^{31} -4.30388 q^{32} +3.20819 q^{33} -0.137753 q^{34} -2.23023 q^{36} +7.64770 q^{37} +0.654748 q^{38} -8.29220 q^{39} -1.32000 q^{40} -0.592274 q^{41} +3.65195 q^{43} +2.87530 q^{44} -1.04031 q^{45} +3.14473 q^{46} +3.80419 q^{47} -6.28555 q^{48} +1.70842 q^{50} -0.705619 q^{51} -7.43177 q^{52} +12.6409 q^{53} -1.47003 q^{54} +1.34121 q^{55} +3.35384 q^{57} +1.59656 q^{58} +8.55816 q^{59} -3.23937 q^{60} -8.95223 q^{61} +2.44439 q^{62} -4.40052 q^{64} -3.46663 q^{65} -1.28546 q^{66} +13.1646 q^{67} -0.632401 q^{68} +16.1083 q^{69} +7.08789 q^{71} +1.86521 q^{72} -3.52777 q^{73} -3.06429 q^{74} +8.75108 q^{75} +3.00583 q^{76} +3.32253 q^{78} +3.98353 q^{79} -2.62773 q^{80} -11.1673 q^{81} +0.237313 q^{82} +17.2874 q^{83} -0.294990 q^{85} -1.46327 q^{86} +8.17811 q^{87} -2.40470 q^{88} -2.34142 q^{89} +0.416834 q^{90} +14.4369 q^{92} +12.5210 q^{93} -1.52427 q^{94} +1.40210 q^{95} +8.83338 q^{96} -2.44185 q^{97} -1.89519 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} - 3 q^{3} + 7 q^{4} - 5 q^{5} + 5 q^{6} - 6 q^{8} + 6 q^{9} + q^{10} - 2 q^{11} - 6 q^{12} - 18 q^{13} - 9 q^{15} - q^{16} + q^{17} - 17 q^{18} + 4 q^{19} + 10 q^{20} - 5 q^{22} + 31 q^{24}+ \cdots - 52 q^{99}+O(q^{100}) $ 7 * q - q^2 - 3 * q^3 + 7 * q^4 - 5 * q^5 + 5 * q^6 - 6 * q^8 + 6 * q^9 + q^10 - 2 * q^11 - 6 * q^12 - 18 * q^13 - 9 * q^15 - q^16 + q^17 - 17 * q^18 + 4 * q^19 + 10 * q^20 - 5 * q^22 + 31 * q^24 + 8 * q^25 + 22 * q^26 - 3 * q^27 - 3 * q^29 - 8 * q^30 - 4 * q^31 - 13 * q^32 + 6 * q^33 + 14 * q^36 + 25 * q^37 - 18 * q^38 + 5 * q^39 - 13 * q^40 + 11 * q^43 - 17 * q^44 - 25 * q^45 + q^46 + 9 * q^47 - 19 * q^48 + 7 * q^50 - 8 * q^51 - 44 * q^52 + 8 * q^53 + 35 * q^54 + 13 * q^55 - 3 * q^57 + 35 * q^58 + 19 * q^59 + 9 * q^60 - 25 * q^61 - 23 * q^62 - 18 * q^64 + 8 * q^65 - 63 * q^66 - 24 * q^67 - 11 * q^68 - 19 * q^71 - 15 * q^72 - 30 * q^73 - 17 * q^74 + 41 * q^75 - 4 * q^76 - 18 * q^78 - 21 * q^79 - 19 * q^80 - 17 * q^81 - 6 * q^82 - 12 * q^83 + 2 * q^85 + 31 * q^86 + 15 * q^87 - 8 * q^88 + 22 * q^89 + 36 * q^90 - 7 * q^92 + 9 * q^93 - 16 * q^94 - 44 * q^95 - 2 * q^96 + 4 * q^97 - 52 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.400681	−0.283324	−0.141662	−	0.989915i	$-0.545245\pi$
$2$	−0.400681	−0.283324	−0.141662	+	0.989915i	$0.545245\pi$
$3$	−2.05242	−1.18497	−0.592483	−	0.805583i	$-0.701852\pi$
$3$	−2.05242	−1.18497	−0.592483	+	0.805583i	$0.701852\pi$
$4$	−1.83945	−0.919727
$5$	−0.858034	−0.383724	−0.191862	−	0.981422i	$-0.561453\pi$
$5$	−0.858034	−0.383724	−0.191862	+	0.981422i	$0.561453\pi$
$6$	0.822367	0.335730
$7$	0	0
$7$	0	0
$8$	1.53840	0.543905
$9$	1.21244	0.404147
$10$	0.343798	0.108718
$11$	−1.56312	−0.471300	−0.235650	−	0.971838i	$-0.575722\pi$
$11$	−1.56312	−0.471300	−0.235650	+	0.971838i	$0.575722\pi$
$12$	3.77534	1.08985
$13$	4.04020	1.12055	0.560275	−	0.828307i	$-0.310695\pi$
$13$	4.04020	1.12055	0.560275	+	0.828307i	$0.310695\pi$
$14$	0	0
$15$	1.76105	0.454701
$16$	3.06250	0.765626
$17$	0.343798	0.0833832	0.0416916	−	0.999131i	$-0.486725\pi$
$17$	0.343798	0.0833832	0.0416916	+	0.999131i	$0.486725\pi$
$18$	−0.485802	−0.114505
$19$	−1.63409	−0.374886	−0.187443	−	0.982276i	$-0.560020\pi$
$19$	−1.63409	−0.374886	−0.187443	+	0.982276i	$0.560020\pi$
$20$	1.57831	0.352922
$21$	0	0
$22$	0.626314	0.133531
$23$	−7.84845	−1.63651	−0.818257	−	0.574852i	$-0.805060\pi$
$23$	−7.84845	−1.63651	−0.818257	+	0.574852i	$0.805060\pi$
$24$	−3.15744	−0.644510
$25$	−4.26378	−0.852756
$26$	−1.61883	−0.317479
$27$	3.66883	0.706067
$28$	0	0
$29$	−3.98461	−0.739924	−0.369962	−	0.929047i	$-0.620629\pi$
$29$	−3.98461	−0.739924	−0.369962	+	0.929047i	$0.620629\pi$
$30$	−0.705619	−0.128828
$31$	−6.10060	−1.09570	−0.547850	−	0.836577i	$-0.684553\pi$
$31$	−6.10060	−1.09570	−0.547850	+	0.836577i	$0.684553\pi$
$32$	−4.30388	−0.760826
$33$	3.20819	0.558474
$34$	−0.137753	−0.0236245
$35$	0	0
$36$	−2.23023	−0.371705
$37$	7.64770	1.25727	0.628637	−	0.777699i	$-0.283613\pi$
$37$	7.64770	1.25727	0.628637	+	0.777699i	$0.283613\pi$
$38$	0.654748	0.106214
$39$	−8.29220	−1.32782
$40$	−1.32000	−0.208710
$41$	−0.592274	−0.0924976	−0.0462488	−	0.998930i	$-0.514727\pi$
$41$	−0.592274	−0.0924976	−0.0462488	+	0.998930i	$0.514727\pi$
$42$	0	0
$43$	3.65195	0.556917	0.278459	−	0.960448i	$-0.410176\pi$
$43$	3.65195	0.556917	0.278459	+	0.960448i	$0.410176\pi$
$44$	2.87530	0.433467
$45$	−1.04031	−0.155081
$46$	3.14473	0.463664
$47$	3.80419	0.554897	0.277449	−	0.960740i	$-0.410511\pi$
$47$	3.80419	0.554897	0.277449	+	0.960740i	$0.410511\pi$
$48$	−6.28555	−0.907241
$49$	0	0
$50$	1.70842	0.241606
$51$	−0.705619	−0.0988064
$52$	−7.43177	−1.03060
$53$	12.6409	1.73636	0.868180	−	0.496249i	$-0.165290\pi$
$53$	12.6409	1.73636	0.868180	+	0.496249i	$0.165290\pi$
$54$	−1.47003	−0.200046
$55$	1.34121	0.180849
$56$	0	0
$57$	3.35384	0.444227
$58$	1.59656	0.209639
$59$	8.55816	1.11418	0.557089	−	0.830453i	$-0.311918\pi$
$59$	8.55816	1.11418	0.557089	+	0.830453i	$0.311918\pi$
$60$	−3.23937	−0.418201
$61$	−8.95223	−1.14622	−0.573108	−	0.819480i	$-0.694262\pi$
$61$	−8.95223	−1.14622	−0.573108	+	0.819480i	$0.694262\pi$
$62$	2.44439	0.310438
$63$	0	0
$64$	−4.40052	−0.550065
$65$	−3.46663	−0.429982
$66$	−1.28546	−0.158229
$67$	13.1646	1.60832	0.804158	−	0.594415i	$-0.202616\pi$
$67$	13.1646	1.60832	0.804158	+	0.594415i	$0.202616\pi$
$68$	−0.632401	−0.0766898
$69$	16.1083	1.93922
$70$	0	0
$71$	7.08789	0.841177	0.420589	−	0.907251i	$-0.361824\pi$
$71$	7.08789	0.841177	0.420589	+	0.907251i	$0.361824\pi$
$72$	1.86521	0.219818
$73$	−3.52777	−0.412895	−0.206447	−	0.978458i	$-0.566190\pi$
$73$	−3.52777	−0.412895	−0.206447	+	0.978458i	$0.566190\pi$
$74$	−3.06429	−0.356216
$75$	8.75108	1.01049
$76$	3.00583	0.344792
$77$	0	0
$78$	3.32253	0.376202
$79$	3.98353	0.448182	0.224091	−	0.974568i	$-0.428059\pi$
$79$	3.98353	0.448182	0.224091	+	0.974568i	$0.428059\pi$
$80$	−2.62773	−0.293789
$81$	−11.1673	−1.24081
$82$	0.237313	0.0262068
$83$	17.2874	1.89754	0.948770	−	0.315966i	$-0.102329\pi$
$83$	17.2874	1.89754	0.948770	+	0.315966i	$0.102329\pi$
$84$	0	0
$85$	−0.294990	−0.0319962
$86$	−1.46327	−0.157788
$87$	8.17811	0.876786
$88$	−2.40470	−0.256342
$89$	−2.34142	−0.248190	−0.124095	−	0.992270i	$-0.539603\pi$
$89$	−2.34142	−0.248190	−0.124095	+	0.992270i	$0.539603\pi$
$90$	0.416834	0.0439382
$91$	0	0
$92$	14.4369	1.50515
$93$	12.5210	1.29837
$94$	−1.52427	−0.157216
$95$	1.40210	0.143853
$96$	8.83338	0.901553
$97$	−2.44185	−0.247932	−0.123966	−	0.992286i	$-0.539561\pi$
$97$	−2.44185	−0.247932	−0.123966	+	0.992286i	$0.539561\pi$
$98$	0	0
$99$	−1.89519	−0.190474
$100$	7.84303	0.784303
$101$	−14.8606	−1.47868	−0.739340	−	0.673332i	$-0.764862\pi$
$101$	−14.8606	−1.47868	−0.739340	+	0.673332i	$0.764862\pi$
$102$	0.282728	0.0279942
$103$	14.0057	1.38002	0.690012	−	0.723798i	$-0.257605\pi$
$103$	14.0057	1.38002	0.690012	+	0.723798i	$0.257605\pi$
$104$	6.21543	0.609473
$105$	0	0
$106$	−5.06497	−0.491953
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	−6.74865	−0.649389
$109$	−7.27844	−0.697148	−0.348574	−	0.937281i	$-0.613334\pi$
$109$	−7.27844	−0.697148	−0.348574	+	0.937281i	$0.613334\pi$
$110$	−0.537399	−0.0512390
$111$	−15.6963	−1.48983
$112$	0	0
$113$	9.98665	0.939465	0.469732	−	0.882809i	$-0.344350\pi$
$113$	9.98665	0.939465	0.469732	+	0.882809i	$0.344350\pi$
$114$	−1.34382	−0.125860
$115$	6.73423	0.627971
$116$	7.32952	0.680529
$117$	4.89850	0.452867
$118$	−3.42909	−0.315674
$119$	0	0
$120$	2.70919	0.247314
$121$	−8.55664	−0.777877
$122$	3.58699	0.324751
$123$	1.21560	0.109607
$124$	11.2218	1.00774
$125$	7.94863	0.710947
$126$	0	0
$127$	−20.1182	−1.78520	−0.892601	−	0.450848i	$-0.851122\pi$
$127$	−20.1182	−1.78520	−0.892601	+	0.450848i	$0.851122\pi$
$128$	10.3710	0.916673
$129$	−7.49535	−0.659929
$130$	1.38901	0.121824
$131$	19.0407	1.66360	0.831798	−	0.555078i	$-0.187312\pi$
$131$	19.0407	1.66360	0.831798	+	0.555078i	$0.187312\pi$
$132$	−5.90132	−0.513644
$133$	0	0
$134$	−5.27482	−0.455675
$135$	−3.14798	−0.270935
$136$	0.528898	0.0453526
$137$	10.4337	0.891409	0.445705	−	0.895180i	$-0.352953\pi$
$137$	10.4337	0.891409	0.445705	+	0.895180i	$0.352953\pi$
$138$	−6.45431	−0.549427
$139$	8.04313	0.682209	0.341105	−	0.940025i	$-0.389199\pi$
$139$	8.04313	0.682209	0.341105	+	0.940025i	$0.389199\pi$
$140$	0	0
$141$	−7.80780	−0.657535
$142$	−2.83998	−0.238326
$143$	−6.31534	−0.528115
$144$	3.71310	0.309425
$145$	3.41893	0.283927
$146$	1.41351	0.116983
$147$	0	0
$148$	−14.0676	−1.15635
$149$	5.43794	0.445493	0.222747	−	0.974876i	$-0.428498\pi$
$149$	5.43794	0.445493	0.222747	+	0.974876i	$0.428498\pi$
$150$	−3.50639	−0.286296
$151$	2.26716	0.184499	0.0922494	−	0.995736i	$-0.470594\pi$
$151$	2.26716	0.184499	0.0922494	+	0.995736i	$0.470594\pi$
$152$	−2.51388	−0.203902
$153$	0.416834	0.0336990
$154$	0	0
$155$	5.23452	0.420446
$156$	15.2531	1.22123
$157$	−23.6776	−1.88968	−0.944840	−	0.327531i	$-0.893783\pi$
$157$	−23.6776	−1.88968	−0.944840	+	0.327531i	$0.893783\pi$
$158$	−1.59613	−0.126981
$159$	−25.9445	−2.05753
$160$	3.69287	0.291947
$161$	0	0
$162$	4.47453	0.351552
$163$	7.44006	0.582751	0.291375	−	0.956609i	$-0.405887\pi$
$163$	7.44006	0.582751	0.291375	+	0.956609i	$0.405887\pi$
$164$	1.08946	0.0850726
$165$	−2.75274	−0.214300
$166$	−6.92674	−0.537619
$167$	−16.5490	−1.28060	−0.640299	−	0.768125i	$-0.721190\pi$
$167$	−16.5490	−1.28060	−0.640299	+	0.768125i	$0.721190\pi$
$168$	0	0
$169$	3.32323	0.255633
$170$	0.118197	0.00906529
$171$	−1.98123	−0.151509
$172$	−6.71760	−0.512212
$173$	−19.9068	−1.51348	−0.756742	−	0.653714i	$-0.773210\pi$
$173$	−19.9068	−1.51348	−0.756742	+	0.653714i	$0.773210\pi$
$174$	−3.27682	−0.248415
$175$	0	0
$176$	−4.78707	−0.360839
$177$	−17.5650	−1.32026
$178$	0.938162	0.0703182
$179$	2.86289	0.213982	0.106991	−	0.994260i	$-0.465878\pi$
$179$	2.86289	0.213982	0.106991	+	0.994260i	$0.465878\pi$
$180$	1.91361	0.142632
$181$	3.68801	0.274128	0.137064	−	0.990562i	$-0.456233\pi$
$181$	3.68801	0.274128	0.137064	+	0.990562i	$0.456233\pi$
$182$	0	0
$183$	18.3738	1.35823
$184$	−12.0740	−0.890109
$185$	−6.56199	−0.482447
$186$	−5.01693	−0.367859
$187$	−0.537399	−0.0392985
$188$	−6.99763	−0.510354
$189$	0	0
$190$	−0.561796	−0.0407570
$191$	2.54895	0.184436	0.0922179	−	0.995739i	$-0.470604\pi$
$191$	2.54895	0.184436	0.0922179	+	0.995739i	$0.470604\pi$
$192$	9.03173	0.651809
$193$	−2.29500	−0.165198	−0.0825988	−	0.996583i	$-0.526322\pi$
$193$	−2.29500	−0.165198	−0.0825988	+	0.996583i	$0.526322\pi$
$194$	0.978401	0.0702451
$195$	7.11499	0.509515
$196$	0	0
$197$	−23.8635	−1.70021	−0.850103	−	0.526617i	$-0.823460\pi$
$197$	−23.8635	−1.70021	−0.850103	+	0.526617i	$0.823460\pi$
$198$	0.759368	0.0539660
$199$	−19.9864	−1.41680	−0.708400	−	0.705811i	$-0.750583\pi$
$199$	−19.9864	−1.41680	−0.708400	+	0.705811i	$0.750583\pi$
$200$	−6.55938	−0.463818
$201$	−27.0194	−1.90580
$202$	5.95434	0.418946
$203$	0	0
$204$	1.29795	0.0908749
$205$	0.508191	0.0354936
$206$	−5.61182	−0.390994
$207$	−9.51577	−0.661392
$208$	12.3731	0.857922
$209$	2.55428	0.176683
$210$	0	0
$211$	−13.9182	−0.958171	−0.479086	−	0.877768i	$-0.659032\pi$
$211$	−13.9182	−0.958171	−0.479086	+	0.877768i	$0.659032\pi$
$212$	−23.2523	−1.59698
$213$	−14.5473	−0.996767
$214$	−0.400681	−0.0273900
$215$	−3.13350	−0.213703
$216$	5.64412	0.384033
$217$	0	0
$218$	2.91633	0.197519
$219$	7.24048	0.489267
$220$	−2.46710	−0.166332
$221$	1.38901	0.0934351
$222$	6.28922	0.422105
$223$	0.937849	0.0628030	0.0314015	−	0.999507i	$-0.490003\pi$
$223$	0.937849	0.0628030	0.0314015	+	0.999507i	$0.490003\pi$
$224$	0	0
$225$	−5.16957	−0.344638
$226$	−4.00146	−0.266173
$227$	13.5014	0.896122	0.448061	−	0.894003i	$-0.352115\pi$
$227$	13.5014	0.896122	0.448061	+	0.894003i	$0.352115\pi$
$228$	−6.16924	−0.408568
$229$	−22.1489	−1.46364	−0.731820	−	0.681498i	$-0.761329\pi$
$229$	−22.1489	−1.46364	−0.731820	+	0.681498i	$0.761329\pi$
$230$	−2.69828	−0.177919
$231$	0	0
$232$	−6.12992	−0.402449
$233$	4.30702	0.282162	0.141081	−	0.989998i	$-0.454942\pi$
$233$	4.30702	0.282162	0.141081	+	0.989998i	$0.454942\pi$
$234$	−1.96274	−0.128308
$235$	−3.26412	−0.212928
$236$	−15.7423	−1.02474
$237$	−8.17589	−0.531081
$238$	0	0
$239$	−12.2987	−0.795538	−0.397769	−	0.917486i	$-0.630215\pi$
$239$	−12.2987	−0.795538	−0.397769	+	0.917486i	$0.630215\pi$
$240$	5.39321	0.348130
$241$	21.9315	1.41274	0.706368	−	0.707845i	$-0.250333\pi$
$241$	21.9315	1.41274	0.706368	+	0.707845i	$0.250333\pi$
$242$	3.42849	0.220391
$243$	11.9136	0.764255
$244$	16.4672	1.05421
$245$	0	0
$246$	−0.487066	−0.0310542
$247$	−6.60205	−0.420078
$248$	−9.38514	−0.595957
$249$	−35.4811	−2.24852
$250$	−3.18487	−0.201429
$251$	−15.5593	−0.982092	−0.491046	−	0.871134i	$-0.663385\pi$
$251$	−15.5593	−0.982092	−0.491046	+	0.871134i	$0.663385\pi$
$252$	0	0
$253$	12.2681	0.771289
$254$	8.06098	0.505791
$255$	0.605445	0.0379144
$256$	4.64559	0.290350
$257$	21.0087	1.31049	0.655243	−	0.755418i	$-0.272566\pi$
$257$	21.0087	1.31049	0.655243	+	0.755418i	$0.272566\pi$
$258$	3.00324	0.186974
$259$	0	0
$260$	6.37671	0.395467
$261$	−4.83111	−0.299038
$262$	−7.62926	−0.471337
$263$	30.3403	1.87086	0.935432	−	0.353508i	$-0.115011\pi$
$263$	30.3403	1.87086	0.935432	+	0.353508i	$0.115011\pi$
$264$	4.93547	0.303757
$265$	−10.8463	−0.666284
$266$	0	0
$267$	4.80558	0.294097
$268$	−24.2158	−1.47921
$269$	−4.44775	−0.271184	−0.135592	−	0.990765i	$-0.543294\pi$
$269$	−4.44775	−0.271184	−0.135592	+	0.990765i	$0.543294\pi$
$270$	1.26134	0.0767624
$271$	3.02256	0.183608	0.0918038	−	0.995777i	$-0.470737\pi$
$271$	3.02256	0.183608	0.0918038	+	0.995777i	$0.470737\pi$
$272$	1.05288	0.0638403
$273$	0	0
$274$	−4.18058	−0.252558
$275$	6.66481	0.401903
$276$	−29.6306	−1.78355
$277$	15.5342	0.933360	0.466680	−	0.884426i	$-0.345450\pi$
$277$	15.5342	0.933360	0.466680	+	0.884426i	$0.345450\pi$
$278$	−3.22273	−0.193286
$279$	−7.39660	−0.442823
$280$	0	0
$281$	−23.9934	−1.43132	−0.715662	−	0.698447i	$-0.753875\pi$
$281$	−23.9934	−1.43132	−0.715662	+	0.698447i	$0.753875\pi$
$282$	3.12844	0.186296
$283$	0.375834	0.0223410	0.0111705	−	0.999938i	$-0.496444\pi$
$283$	0.375834	0.0223410	0.0111705	+	0.999938i	$0.496444\pi$
$284$	−13.0378	−0.773654
$285$	−2.87771	−0.170461
$286$	2.53044	0.149628
$287$	0	0
$288$	−5.21820	−0.307485
$289$	−16.8818	−0.993047
$290$	−1.36990	−0.0804434
$291$	5.01170	0.293791
$292$	6.48918	0.379751
$293$	1.19887	0.0700389	0.0350194	−	0.999387i	$-0.488851\pi$
$293$	1.19887	0.0700389	0.0350194	+	0.999387i	$0.488851\pi$
$294$	0	0
$295$	−7.34319	−0.427537
$296$	11.7652	0.683838
$297$	−5.73483	−0.332769
$298$	−2.17888	−0.126219
$299$	−31.7093	−1.83380
$300$	−16.0972	−0.929373
$301$	0	0
$302$	−0.908408	−0.0522730
$303$	30.5001	1.75219
$304$	−5.00440	−0.287022
$305$	7.68132	0.439831
$306$	−0.167018	−0.00954776
$307$	3.53770	0.201907	0.100954	−	0.994891i	$-0.467811\pi$
$307$	3.53770	0.201907	0.100954	+	0.994891i	$0.467811\pi$
$308$	0	0
$309$	−28.7456	−1.63528
$310$	−2.09737	−0.119123
$311$	27.8999	1.58206	0.791030	−	0.611777i	$-0.209545\pi$
$311$	27.8999	1.58206	0.791030	+	0.611777i	$0.209545\pi$
$312$	−12.7567	−0.722206
$313$	4.32943	0.244714	0.122357	−	0.992486i	$-0.460955\pi$
$313$	4.32943	0.244714	0.122357	+	0.992486i	$0.460955\pi$
$314$	9.48718	0.535392
$315$	0	0
$316$	−7.32752	−0.412205
$317$	1.93923	0.108918	0.0544589	−	0.998516i	$-0.482657\pi$
$317$	1.93923	0.108918	0.0544589	+	0.998516i	$0.482657\pi$
$318$	10.3955	0.582948
$319$	6.22845	0.348726
$320$	3.77580	0.211073
$321$	−2.05242	−0.114555
$322$	0	0
$323$	−0.561796	−0.0312592
$324$	20.5418	1.14121
$325$	−17.2265	−0.955556
$326$	−2.98109	−0.165107
$327$	14.9384	0.826097
$328$	−0.911152	−0.0503099
$329$	0	0
$330$	1.10297	0.0607165
$331$	24.7851	1.36231	0.681157	−	0.732138i	$-0.261477\pi$
$331$	24.7851	1.36231	0.681157	+	0.732138i	$0.261477\pi$
$332$	−31.7994	−1.74522
$333$	9.27238	0.508123
$334$	6.63086	0.362825
$335$	−11.2957	−0.617150
$336$	0	0
$337$	−26.2295	−1.42881	−0.714406	−	0.699731i	$-0.753303\pi$
$337$	−26.2295	−1.42881	−0.714406	+	0.699731i	$0.753303\pi$
$338$	−1.33156	−0.0724271
$339$	−20.4968	−1.11323
$340$	0.542621	0.0294278
$341$	9.53599	0.516403
$342$	0.793843	0.0429261
$343$	0	0
$344$	5.61815	0.302910
$345$	−13.8215	−0.744124
$346$	7.97627	0.428807
$347$	16.2012	0.869726	0.434863	−	0.900497i	$-0.356797\pi$
$347$	16.2012	0.869726	0.434863	+	0.900497i	$0.356797\pi$
$348$	−15.0433	−0.806404
$349$	9.68150	0.518239	0.259119	−	0.965845i	$-0.416568\pi$
$349$	9.68150	0.518239	0.259119	+	0.965845i	$0.416568\pi$
$350$	0	0
$351$	14.8228	0.791183
$352$	6.72750	0.358577
$353$	−0.951831	−0.0506609	−0.0253304	−	0.999679i	$-0.508064\pi$
$353$	−0.951831	−0.0506609	−0.0253304	+	0.999679i	$0.508064\pi$
$354$	7.03795	0.374063
$355$	−6.08164	−0.322780
$356$	4.30693	0.228267
$357$	0	0
$358$	−1.14710	−0.0606263
$359$	−15.4999	−0.818052	−0.409026	−	0.912523i	$-0.634131\pi$
$359$	−15.4999	−0.818052	−0.409026	+	0.912523i	$0.634131\pi$
$360$	−1.60042	−0.0843493
$361$	−16.3298	−0.859461
$362$	−1.47772	−0.0776671
$363$	17.5619	0.921758
$364$	0	0
$365$	3.02695	0.158438
$366$	−7.36202	−0.384819
$367$	2.63718	0.137660	0.0688299	−	0.997628i	$-0.478073\pi$
$367$	2.63718	0.137660	0.0688299	+	0.997628i	$0.478073\pi$
$368$	−24.0359	−1.25296
$369$	−0.718096	−0.0373826
$370$	2.62926	0.136689
$371$	0	0
$372$	−23.0318	−1.19414
$373$	8.98443	0.465196	0.232598	−	0.972573i	$-0.425277\pi$
$373$	8.98443	0.465196	0.232598	+	0.972573i	$0.425277\pi$
$374$	0.215325	0.0111342
$375$	−16.3140	−0.842449
$376$	5.85235	0.301812
$377$	−16.0986	−0.829123
$378$	0	0
$379$	1.01229	0.0519977	0.0259988	−	0.999662i	$-0.491723\pi$
$379$	1.01229	0.0519977	0.0259988	+	0.999662i	$0.491723\pi$
$380$	−2.57910	−0.132305
$381$	41.2910	2.11540
$382$	−1.02132	−0.0522551
$383$	−4.16498	−0.212821	−0.106410	−	0.994322i	$-0.533936\pi$
$383$	−4.16498	−0.212821	−0.106410	+	0.994322i	$0.533936\pi$
$384$	−21.2856	−1.08623
$385$	0	0
$386$	0.919562	0.0468045
$387$	4.42777	0.225076
$388$	4.49166	0.228030
$389$	2.14928	0.108973	0.0544865	−	0.998515i	$-0.482648\pi$
$389$	2.14928	0.108973	0.0544865	+	0.998515i	$0.482648\pi$
$390$	−2.85084	−0.144358
$391$	−2.69828	−0.136458
$392$	0	0
$393$	−39.0796	−1.97131
$394$	9.56166	0.481710
$395$	−3.41800	−0.171978
$396$	3.48612	0.175184
$397$	−10.2162	−0.512738	−0.256369	−	0.966579i	$-0.582526\pi$
$397$	−10.2162	−0.512738	−0.256369	+	0.966579i	$0.582526\pi$
$398$	8.00818	0.401414
$399$	0	0
$400$	−13.0578	−0.652892
$401$	8.64190	0.431556	0.215778	−	0.976442i	$-0.430771\pi$
$401$	8.64190	0.431556	0.215778	+	0.976442i	$0.430771\pi$
$402$	10.8262	0.539960
$403$	−24.6476	−1.22779
$404$	27.3353	1.35998
$405$	9.58193	0.476130
$406$	0	0
$407$	−11.9543	−0.592553
$408$	−1.08552	−0.0537413
$409$	21.2166	1.04909	0.524546	−	0.851382i	$-0.324235\pi$
$409$	21.2166	1.04909	0.524546	+	0.851382i	$0.324235\pi$
$410$	−0.203622	−0.0100562
$411$	−21.4143	−1.05629
$412$	−25.7629	−1.26925
$413$	0	0
$414$	3.81279	0.187388
$415$	−14.8332	−0.728133
$416$	−17.3885	−0.852544
$417$	−16.5079	−0.808396
$418$	−1.02345	−0.0500587
$419$	16.5527	0.808650	0.404325	−	0.914615i	$-0.367506\pi$
$419$	16.5527	0.808650	0.404325	+	0.914615i	$0.367506\pi$
$420$	0	0
$421$	8.84459	0.431059	0.215529	−	0.976497i	$-0.430852\pi$
$421$	8.84459	0.431059	0.215529	+	0.976497i	$0.430852\pi$
$422$	5.57678	0.271473
$423$	4.61235	0.224260
$424$	19.4467	0.944416
$425$	−1.46588	−0.0711055
$426$	5.82884	0.282408
$427$	0	0
$428$	−1.83945	−0.0889134
$429$	12.9617	0.625799
$430$	1.25553	0.0605472
$431$	−28.3430	−1.36523	−0.682616	−	0.730777i	$-0.739158\pi$
$431$	−28.3430	−1.36523	−0.682616	+	0.730777i	$0.739158\pi$
$432$	11.2358	0.540583
$433$	−29.9692	−1.44023	−0.720114	−	0.693855i	$-0.755911\pi$
$433$	−29.9692	−1.44023	−0.720114	+	0.693855i	$0.755911\pi$
$434$	0	0
$435$	−7.01710	−0.336444
$436$	13.3884	0.641186
$437$	12.8251	0.613506
$438$	−2.90113	−0.138621
$439$	−13.6019	−0.649183	−0.324592	−	0.945854i	$-0.605227\pi$
$439$	−13.6019	−0.649183	−0.324592	+	0.945854i	$0.605227\pi$
$440$	2.06332	0.0983648
$441$	0	0
$442$	−0.556551	−0.0264724
$443$	−30.1900	−1.43437	−0.717185	−	0.696883i	$-0.754570\pi$
$443$	−30.1900	−1.43437	−0.717185	+	0.696883i	$0.754570\pi$
$444$	28.8727	1.37024
$445$	2.00902	0.0952365
$446$	−0.375779	−0.0177936
$447$	−11.1609	−0.527895
$448$	0	0
$449$	−15.9602	−0.753207	−0.376603	−	0.926375i	$-0.622908\pi$
$449$	−15.9602	−0.753207	−0.376603	+	0.926375i	$0.622908\pi$
$450$	2.07135	0.0976444
$451$	0.925797	0.0435941
$452$	−18.3700	−0.864052
$453$	−4.65317	−0.218625
$454$	−5.40977	−0.253893
$455$	0	0
$456$	5.15954	0.241617
$457$	10.6928	0.500187	0.250093	−	0.968222i	$-0.419539\pi$
$457$	10.6928	0.500187	0.250093	+	0.968222i	$0.419539\pi$
$458$	8.87464	0.414685
$459$	1.26134	0.0588741
$460$	−12.3873	−0.577562
$461$	30.1188	1.40277	0.701387	−	0.712781i	$-0.252565\pi$
$461$	30.1188	1.40277	0.701387	+	0.712781i	$0.252565\pi$
$462$	0	0
$463$	−16.5292	−0.768176	−0.384088	−	0.923296i	$-0.625484\pi$
$463$	−16.5292	−0.768176	−0.384088	+	0.923296i	$0.625484\pi$
$464$	−12.2029	−0.566505
$465$	−10.7434	−0.498215
$466$	−1.72574	−0.0799434
$467$	−39.1468	−1.81150	−0.905750	−	0.423813i	$-0.860692\pi$
$467$	−39.1468	−1.81150	−0.905750	+	0.423813i	$0.860692\pi$
$468$	−9.01057	−0.416514
$469$	0	0
$470$	1.30787	0.0603276
$471$	48.5965	2.23921
$472$	13.1658	0.606007
$473$	−5.70845	−0.262475
$474$	3.27592	0.150468
$475$	6.96739	0.319686
$476$	0	0
$477$	15.3263	0.701744
$478$	4.92786	0.225395
$479$	−2.08979	−0.0954849	−0.0477425	−	0.998860i	$-0.515203\pi$
$479$	−2.08979	−0.0954849	−0.0477425	+	0.998860i	$0.515203\pi$
$480$	−7.57934	−0.345948
$481$	30.8983	1.40884
$482$	−8.78756	−0.400262
$483$	0	0
$484$	15.7396	0.715434
$485$	2.09519	0.0951375
$486$	−4.77354	−0.216532
$487$	−33.4056	−1.51375	−0.756877	−	0.653557i	$-0.773276\pi$
$487$	−33.4056	−1.51375	−0.756877	+	0.653557i	$0.773276\pi$
$488$	−13.7721	−0.623433
$489$	−15.2702	−0.690540
$490$	0	0
$491$	16.1338	0.728110	0.364055	−	0.931377i	$-0.381392\pi$
$491$	16.1338	0.728110	0.364055	+	0.931377i	$0.381392\pi$
$492$	−2.23603	−0.100808
$493$	−1.36990	−0.0616973
$494$	2.64532	0.119018
$495$	1.62614	0.0730896
$496$	−18.6831	−0.838896
$497$	0	0
$498$	14.2166	0.637061
$499$	−21.5343	−0.964007	−0.482003	−	0.876169i	$-0.660091\pi$
$499$	−21.5343	−0.964007	−0.482003	+	0.876169i	$0.660091\pi$
$500$	−14.6212	−0.653878
$501$	33.9655	1.51747
$502$	6.23430	0.278251
$503$	19.6399	0.875700	0.437850	−	0.899048i	$-0.355740\pi$
$503$	19.6399	0.875700	0.437850	+	0.899048i	$0.355740\pi$
$504$	0	0
$505$	12.7509	0.567406
$506$	−4.91560	−0.218525
$507$	−6.82067	−0.302917
$508$	37.0065	1.64190
$509$	−33.7799	−1.49727	−0.748633	−	0.662984i	$-0.769290\pi$
$509$	−33.7799	−1.49727	−0.748633	+	0.662984i	$0.769290\pi$
$510$	−0.242590	−0.0107421
$511$	0	0
$512$	−22.6033	−0.998936
$513$	−5.99519	−0.264694
$514$	−8.41778	−0.371292
$515$	−12.0174	−0.529549
$516$	13.7874	0.606954
$517$	−5.94641	−0.261523
$518$	0	0
$519$	40.8571	1.79343
$520$	−5.33305	−0.233870
$521$	−27.3215	−1.19698	−0.598489	−	0.801131i	$-0.704232\pi$
$521$	−27.3215	−1.19698	−0.598489	+	0.801131i	$0.704232\pi$
$522$	1.93573	0.0847247
$523$	4.31540	0.188699	0.0943496	−	0.995539i	$-0.469923\pi$
$523$	4.31540	0.188699	0.0943496	+	0.995539i	$0.469923\pi$
$524$	−35.0246	−1.53006
$525$	0	0
$526$	−12.1568	−0.530061
$527$	−2.09737	−0.0913629
$528$	9.82509	0.427582
$529$	38.5982	1.67818
$530$	4.34591	0.188774
$531$	10.3763	0.450291
$532$	0	0
$533$	−2.39291	−0.103648
$534$	−1.92551	−0.0833248
$535$	−0.858034	−0.0370960
$536$	20.2524	0.874772
$537$	−5.87585	−0.253562
$538$	1.78213	0.0768331
$539$	0	0
$540$	5.79057	0.249186
$541$	−38.0043	−1.63393	−0.816967	−	0.576685i	$-0.804346\pi$
$541$	−38.0043	−1.63393	−0.816967	+	0.576685i	$0.804346\pi$
$542$	−1.21108	−0.0520205
$543$	−7.56936	−0.324832
$544$	−1.47966	−0.0634401
$545$	6.24515	0.267513
$546$	0	0
$547$	2.79784	0.119627	0.0598135	−	0.998210i	$-0.480949\pi$
$547$	2.79784	0.119627	0.0598135	+	0.998210i	$0.480949\pi$
$548$	−19.1923	−0.819853
$549$	−10.8540	−0.463239
$550$	−2.67046	−0.113869
$551$	6.51121	0.277387
$552$	24.7810	1.05475
$553$	0	0
$554$	−6.22426	−0.264444
$555$	13.4680	0.571684
$556$	−14.7950	−0.627447
$557$	−41.9379	−1.77697	−0.888483	−	0.458910i	$-0.848240\pi$
$557$	−41.9379	−1.77697	−0.888483	+	0.458910i	$0.848240\pi$
$558$	2.96368	0.125463
$559$	14.7546	0.624054
$560$	0	0
$561$	1.10297	0.0465674
$562$	9.61368	0.405529
$563$	−12.3159	−0.519054	−0.259527	−	0.965736i	$-0.583567\pi$
$563$	−12.3159	−0.519054	−0.259527	+	0.965736i	$0.583567\pi$
$564$	14.3621	0.604753
$565$	−8.56888	−0.360496
$566$	−0.150590	−0.00632975
$567$	0	0
$568$	10.9040	0.457521
$569$	5.52854	0.231769	0.115884	−	0.993263i	$-0.463030\pi$
$569$	5.52854	0.231769	0.115884	+	0.993263i	$0.463030\pi$
$570$	1.15304	0.0482957
$571$	−28.7910	−1.20487	−0.602433	−	0.798169i	$-0.705802\pi$
$571$	−28.7910	−1.20487	−0.602433	+	0.798169i	$0.705802\pi$
$572$	11.6168	0.485722
$573$	−5.23153	−0.218550
$574$	0	0
$575$	33.4641	1.39555
$576$	−5.33537	−0.222307
$577$	−2.16381	−0.0900807	−0.0450404	−	0.998985i	$-0.514342\pi$
$577$	−2.16381	−0.0900807	−0.0450404	+	0.998985i	$0.514342\pi$
$578$	6.76422	0.281354
$579$	4.71031	0.195754
$580$	−6.28897	−0.261135
$581$	0	0
$582$	−2.00809	−0.0832382
$583$	−19.7593	−0.818346
$584$	−5.42712	−0.224576
$585$	−4.20308	−0.173776
$586$	−0.480366	−0.0198437
$587$	−34.4416	−1.42156	−0.710778	−	0.703416i	$-0.751657\pi$
$587$	−34.4416	−1.42156	−0.710778	+	0.703416i	$0.751657\pi$
$588$	0	0
$589$	9.96891	0.410762
$590$	2.94228	0.121132
$591$	48.9780	2.01469
$592$	23.4211	0.962602
$593$	−23.5676	−0.967806	−0.483903	−	0.875122i	$-0.660781\pi$
$593$	−23.5676	−0.967806	−0.483903	+	0.875122i	$0.660781\pi$
$594$	2.29784	0.0942815
$595$	0	0
$596$	−10.0028	−0.409732
$597$	41.0206	1.67886
$598$	12.7053	0.519559
$599$	−34.3462	−1.40335	−0.701675	−	0.712497i	$-0.747564\pi$
$599$	−34.3462	−1.40335	−0.701675	+	0.712497i	$0.747564\pi$
$600$	13.4626	0.549609
$601$	−5.58348	−0.227755	−0.113878	−	0.993495i	$-0.536327\pi$
$601$	−5.58348	−0.227755	−0.113878	+	0.993495i	$0.536327\pi$
$602$	0	0
$603$	15.9613	0.649996
$604$	−4.17034	−0.169689
$605$	7.34189	0.298490
$606$	−12.2208	−0.496437
$607$	−6.51339	−0.264370	−0.132185	−	0.991225i	$-0.542199\pi$
$607$	−6.51339	−0.264370	−0.132185	+	0.991225i	$0.542199\pi$
$608$	7.03292	0.285223
$609$	0	0
$610$	−3.07776	−0.124615
$611$	15.3697	0.621790
$612$	−0.766748	−0.0309939
$613$	31.5232	1.27321	0.636606	−	0.771189i	$-0.280338\pi$
$613$	31.5232	1.27321	0.636606	+	0.771189i	$0.280338\pi$
$614$	−1.41749	−0.0572052
$615$	−1.04302	−0.0420587
$616$	0	0
$617$	23.6858	0.953553	0.476776	−	0.879025i	$-0.341805\pi$
$617$	23.6858	0.953553	0.476776	+	0.879025i	$0.341805\pi$
$618$	11.5178	0.463315
$619$	−25.6089	−1.02931	−0.514653	−	0.857398i	$-0.672079\pi$
$619$	−25.6089	−1.02931	−0.514653	+	0.857398i	$0.672079\pi$
$620$	−9.62866	−0.386696
$621$	−28.7946	−1.15549
$622$	−11.1790	−0.448236
$623$	0	0
$624$	−25.3949	−1.01661
$625$	14.4987	0.579948
$626$	−1.73472	−0.0693334
$627$	−5.24247	−0.209364
$628$	43.5539	1.73799
$629$	2.62926	0.104836
$630$	0	0
$631$	−18.7159	−0.745067	−0.372534	−	0.928019i	$-0.621511\pi$
$631$	−18.7159	−0.745067	−0.372534	+	0.928019i	$0.621511\pi$
$632$	6.12825	0.243769
$633$	28.5661	1.13540
$634$	−0.777011	−0.0308591
$635$	17.2621	0.685025
$636$	47.7236	1.89237
$637$	0	0
$638$	−2.49562	−0.0988026
$639$	8.59363	0.339959
$640$	−8.89864	−0.351750
$641$	11.5357	0.455632	0.227816	−	0.973704i	$-0.426841\pi$
$641$	11.5357	0.455632	0.227816	+	0.973704i	$0.426841\pi$
$642$	0.822367	0.0324562
$643$	−12.5190	−0.493701	−0.246851	−	0.969054i	$-0.579396\pi$
$643$	−12.5190	−0.493701	−0.246851	+	0.969054i	$0.579396\pi$
$644$	0	0
$645$	6.43126	0.253231
$646$	0.225101	0.00885648
$647$	22.1925	0.872478	0.436239	−	0.899831i	$-0.356310\pi$
$647$	22.1925	0.872478	0.436239	+	0.899831i	$0.356310\pi$
$648$	−17.1798	−0.674884
$649$	−13.3775	−0.525111
$650$	6.90234	0.270732
$651$	0	0
$652$	−13.6857	−0.535972
$653$	−0.545398	−0.0213431	−0.0106715	−	0.999943i	$-0.503397\pi$
$653$	−0.545398	−0.0213431	−0.0106715	+	0.999943i	$0.503397\pi$
$654$	−5.98555	−0.234053
$655$	−16.3376	−0.638363
$656$	−1.81384	−0.0708185
$657$	−4.27721	−0.166870
$658$	0	0
$659$	−0.416896	−0.0162400	−0.00811999	−	0.999967i	$-0.502585\pi$
$659$	−0.416896	−0.0162400	−0.00811999	+	0.999967i	$0.502585\pi$
$660$	5.06353	0.197098
$661$	−28.6717	−1.11520	−0.557600	−	0.830110i	$-0.688278\pi$
$661$	−28.6717	−1.11520	−0.557600	+	0.830110i	$0.688278\pi$
$662$	−9.93093	−0.385976
$663$	−2.85084	−0.110718
$664$	26.5949	1.03208
$665$	0	0
$666$	−3.71527	−0.143964
$667$	31.2730	1.21090
$668$	30.4411	1.17780
$669$	−1.92486	−0.0744195
$670$	4.52598	0.174854
$671$	13.9935	0.540211
$672$	0	0
$673$	6.25365	0.241060	0.120530	−	0.992710i	$-0.461541\pi$
$673$	6.25365	0.241060	0.120530	+	0.992710i	$0.461541\pi$
$674$	10.5097	0.404817
$675$	−15.6431	−0.602102
$676$	−6.11293	−0.235113
$677$	9.78422	0.376038	0.188019	−	0.982165i	$-0.439793\pi$
$677$	9.78422	0.376038	0.188019	+	0.982165i	$0.439793\pi$
$678$	8.21269	0.315406
$679$	0	0
$680$	−0.453812	−0.0174029
$681$	−27.7107	−1.06188
$682$	−3.82089	−0.146309
$683$	24.7545	0.947205	0.473602	−	0.880739i	$-0.342953\pi$
$683$	24.7545	0.947205	0.473602	+	0.880739i	$0.342953\pi$
$684$	3.64439	0.139347
$685$	−8.95245	−0.342055
$686$	0	0
$687$	45.4589	1.73436
$688$	11.1841	0.426390
$689$	51.0718	1.94568
$690$	5.53801	0.210829
$691$	41.1103	1.56391	0.781955	−	0.623334i	$-0.214222\pi$
$691$	41.1103	1.56391	0.781955	+	0.623334i	$0.214222\pi$
$692$	36.6176	1.39199
$693$	0	0
$694$	−6.49152	−0.246415
$695$	−6.90128	−0.261780
$696$	12.5812	0.476889
$697$	−0.203622	−0.00771275
$698$	−3.87919	−0.146830
$699$	−8.83983	−0.334353
$700$	0	0
$701$	20.9689	0.791986	0.395993	−	0.918254i	$-0.370400\pi$
$701$	20.9689	0.791986	0.395993	+	0.918254i	$0.370400\pi$
$702$	−5.93922	−0.224161
$703$	−12.4970	−0.471334
$704$	6.87856	0.259246
$705$	6.69935	0.252312
$706$	0.381381	0.0143535
$707$	0	0
$708$	32.3100	1.21428
$709$	−33.2811	−1.24990	−0.624948	−	0.780666i	$-0.714880\pi$
$709$	−33.2811	−1.24990	−0.624948	+	0.780666i	$0.714880\pi$
$710$	2.43680	0.0914515
$711$	4.82979	0.181131
$712$	−3.60203	−0.134992
$713$	47.8802	1.79313
$714$	0	0
$715$	5.41877	0.202651
$716$	−5.26615	−0.196805
$717$	25.2422	0.942686
$718$	6.21050	0.231774
$719$	−49.7380	−1.85492	−0.927458	−	0.373928i	$-0.878011\pi$
$719$	−49.7380	−1.85492	−0.927458	+	0.373928i	$0.878011\pi$
$720$	−3.18597	−0.118734
$721$	0	0
$722$	6.54302	0.243506
$723$	−45.0128	−1.67404
$724$	−6.78393	−0.252123
$725$	16.9895	0.630975
$726$	−7.03670	−0.261157
$727$	20.6762	0.766838	0.383419	−	0.923575i	$-0.374747\pi$
$727$	20.6762	0.766838	0.383419	+	0.923575i	$0.374747\pi$
$728$	0	0
$729$	9.05028	0.335195
$730$	−1.21284	−0.0448893
$731$	1.25553	0.0464376
$732$	−33.7977	−1.24920
$733$	−2.70726	−0.0999949	−0.0499975	−	0.998749i	$-0.515921\pi$
$733$	−2.70726	−0.0999949	−0.0499975	+	0.998749i	$0.515921\pi$
$734$	−1.05667	−0.0390024
$735$	0	0
$736$	33.7788	1.24510
$737$	−20.5780	−0.757999
$738$	0.287728	0.0105914
$739$	−4.12593	−0.151775	−0.0758873	−	0.997116i	$-0.524179\pi$
$739$	−4.12593	−0.151775	−0.0758873	+	0.997116i	$0.524179\pi$
$740$	12.0705	0.443720
$741$	13.5502	0.497779
$742$	0	0
$743$	−31.7860	−1.16611	−0.583057	−	0.812431i	$-0.698143\pi$
$743$	−31.7860	−1.16611	−0.583057	+	0.812431i	$0.698143\pi$
$744$	19.2623	0.706189
$745$	−4.66593	−0.170947
$746$	−3.59989	−0.131801
$747$	20.9600	0.766885
$748$	0.988520	0.0361439
$749$	0	0
$750$	6.53669	0.238686
$751$	−46.7601	−1.70630	−0.853150	−	0.521666i	$-0.825311\pi$
$751$	−46.7601	−1.70630	−0.853150	+	0.521666i	$0.825311\pi$
$752$	11.6503	0.424844
$753$	31.9342	1.16375
$754$	6.45042	0.234911
$755$	−1.94530	−0.0707967
$756$	0	0
$757$	30.2678	1.10010	0.550051	−	0.835131i	$-0.314608\pi$
$757$	30.2678	1.10010	0.550051	+	0.835131i	$0.314608\pi$
$758$	−0.405604	−0.0147322
$759$	−25.1793	−0.913952
$760$	2.15699	0.0782423
$761$	−9.68827	−0.351200	−0.175600	−	0.984462i	$-0.556186\pi$
$761$	−9.68827	−0.351200	−0.175600	+	0.984462i	$0.556186\pi$
$762$	−16.5445	−0.599345
$763$	0	0
$764$	−4.68868	−0.169631
$765$	−0.357658	−0.0129311
$766$	1.66883	0.0602973
$767$	34.5767	1.24849
$768$	−9.53472	−0.344055
$769$	−23.5689	−0.849915	−0.424958	−	0.905213i	$-0.639711\pi$
$769$	−23.5689	−0.849915	−0.424958	+	0.905213i	$0.639711\pi$
$770$	0	0
$771$	−43.1187	−1.55288
$772$	4.22154	0.151937
$773$	−5.10704	−0.183687	−0.0918436	−	0.995773i	$-0.529276\pi$
$773$	−5.10704	−0.183687	−0.0918436	+	0.995773i	$0.529276\pi$
$774$	−1.77412	−0.0637696
$775$	26.0116	0.934364
$776$	−3.75653	−0.134851
$777$	0	0
$778$	−0.861178	−0.0308747
$779$	0.967827	0.0346760
$780$	−13.0877	−0.468615
$781$	−11.0792	−0.396446
$782$	1.08115	0.0386618
$783$	−14.6189	−0.522436
$784$	0	0
$785$	20.3162	0.725116
$786$	15.6585	0.558519
$787$	−40.3189	−1.43721	−0.718606	−	0.695417i	$-0.755219\pi$
$787$	−40.3189	−1.43721	−0.718606	+	0.695417i	$0.755219\pi$
$788$	43.8959	1.56373
$789$	−62.2711	−2.21691
$790$	1.36953	0.0487257
$791$	0	0
$792$	−2.91556	−0.103600
$793$	−36.1688	−1.28439
$794$	4.09345	0.145271
$795$	22.2612	0.789524
$796$	36.7641	1.30307
$797$	−32.5063	−1.15143	−0.575715	−	0.817650i	$-0.695276\pi$
$797$	−32.5063	−1.15143	−0.575715	+	0.817650i	$0.695276\pi$
$798$	0	0
$799$	1.30787	0.0462691
$800$	18.3508	0.648799
$801$	−2.83883	−0.100305
$802$	−3.46265	−0.122270
$803$	5.51435	0.194597
$804$	49.7010	1.75282
$805$	0	0
$806$	9.87584	0.347862
$807$	9.12867	0.321345
$808$	−22.8614	−0.804262
$809$	40.5004	1.42392	0.711959	−	0.702221i	$-0.247808\pi$
$809$	40.5004	1.42392	0.711959	+	0.702221i	$0.247808\pi$
$810$	−3.83930	−0.134899
$811$	−5.17273	−0.181639	−0.0908196	−	0.995867i	$-0.528949\pi$
$811$	−5.17273	−0.181639	−0.0908196	+	0.995867i	$0.528949\pi$
$812$	0	0
$813$	−6.20358	−0.217569
$814$	4.78986	0.167885
$815$	−6.38382	−0.223616
$816$	−2.16096	−0.0756487
$817$	−5.96761	−0.208780
$818$	−8.50107	−0.297233
$819$	0	0
$820$	−0.934794	−0.0326444
$821$	6.50706	0.227098	0.113549	−	0.993532i	$-0.463778\pi$
$821$	6.50706	0.227098	0.113549	+	0.993532i	$0.463778\pi$
$822$	8.58031	0.299273
$823$	42.8219	1.49268	0.746339	−	0.665566i	$-0.231810\pi$
$823$	42.8219	1.49268	0.746339	+	0.665566i	$0.231810\pi$
$824$	21.5463	0.750602
$825$	−13.6790	−0.476242
$826$	0	0
$827$	27.2737	0.948401	0.474200	−	0.880417i	$-0.342737\pi$
$827$	27.2737	0.948401	0.474200	+	0.880417i	$0.342737\pi$
$828$	17.5038	0.608300
$829$	45.7752	1.58984	0.794919	−	0.606715i	$-0.207513\pi$
$829$	45.7752	1.58984	0.794919	+	0.606715i	$0.207513\pi$
$830$	5.94338	0.206298
$831$	−31.8828	−1.10600
$832$	−17.7790	−0.616376
$833$	0	0
$834$	6.61440	0.229038
$835$	14.1996	0.491397
$836$	−4.69849	−0.162501
$837$	−22.3820	−0.773637
$838$	−6.63234	−0.229110
$839$	50.1044	1.72980	0.864898	−	0.501948i	$-0.167383\pi$
$839$	50.1044	1.72980	0.864898	+	0.501948i	$0.167383\pi$
$840$	0	0
$841$	−13.1228	−0.452512
$842$	−3.54386	−0.122129
$843$	49.2445	1.69607
$844$	25.6020	0.881256
$845$	−2.85144	−0.0980926
$846$	−1.84808	−0.0635383
$847$	0	0
$848$	38.7128	1.32940
$849$	−0.771370	−0.0264734
$850$	0.587350	0.0201459
$851$	−60.0226	−2.05755
$852$	26.7592	0.916754
$853$	−31.6634	−1.08413	−0.542067	−	0.840335i	$-0.682358\pi$
$853$	−31.6634	−1.08413	−0.542067	+	0.840335i	$0.682358\pi$
$854$	0	0
$855$	1.69997	0.0581376
$856$	1.53840	0.0525813
$857$	33.3618	1.13962	0.569809	−	0.821777i	$-0.307017\pi$
$857$	33.3618	1.13962	0.569809	+	0.821777i	$0.307017\pi$
$858$	−5.19352	−0.177304
$859$	25.5566	0.871981	0.435990	−	0.899951i	$-0.356398\pi$
$859$	25.5566	0.871981	0.435990	+	0.899951i	$0.356398\pi$
$860$	5.76393	0.196548
$861$	0	0
$862$	11.3565	0.386803
$863$	12.6757	0.431486	0.215743	−	0.976450i	$-0.430783\pi$
$863$	12.6757	0.431486	0.215743	+	0.976450i	$0.430783\pi$
$864$	−15.7902	−0.537194
$865$	17.0807	0.580761
$866$	12.0081	0.408052
$867$	34.6486	1.17673
$868$	0	0
$869$	−6.22675	−0.211228
$870$	2.81162	0.0953228
$871$	53.1878	1.80220
$872$	−11.1971	−0.379183
$873$	−2.96059	−0.100201
$874$	−5.13876	−0.173821
$875$	0	0
$876$	−13.3185	−0.449992
$877$	−51.9996	−1.75590	−0.877950	−	0.478752i	$-0.841089\pi$
$877$	−51.9996	−1.75590	−0.877950	+	0.478752i	$0.841089\pi$
$878$	5.45002	0.183929
$879$	−2.46059	−0.0829938
$880$	4.10747	0.138463
$881$	−24.5788	−0.828081	−0.414040	−	0.910258i	$-0.635883\pi$
$881$	−24.5788	−0.828081	−0.414040	+	0.910258i	$0.635883\pi$
$882$	0	0
$883$	6.62659	0.223002	0.111501	−	0.993764i	$-0.464434\pi$
$883$	6.62659	0.223002	0.111501	+	0.993764i	$0.464434\pi$
$884$	−2.55503	−0.0859348
$885$	15.0713	0.506617
$886$	12.0966	0.406392
$887$	−14.7796	−0.496250	−0.248125	−	0.968728i	$-0.579814\pi$
$887$	−14.7796	−0.496250	−0.248125	+	0.968728i	$0.579814\pi$
$888$	−24.1472	−0.810326
$889$	0	0
$890$	−0.804975	−0.0269828
$891$	17.4559	0.584794
$892$	−1.72513	−0.0577617
$893$	−6.21637	−0.208023
$894$	4.47198	0.149565
$895$	−2.45645	−0.0821101
$896$	0	0
$897$	65.0809	2.17299
$898$	6.39494	0.213402
$899$	24.3085	0.810735
$900$	9.50920	0.316973
$901$	4.34591	0.144783
$902$	−0.370949	−0.0123513
$903$	0	0
$904$	15.3634	0.510980
$905$	−3.16444	−0.105189
$906$	1.86444	0.0619418
$907$	−21.5048	−0.714054	−0.357027	−	0.934094i	$-0.616210\pi$
$907$	−21.5048	−0.714054	−0.357027	+	0.934094i	$0.616210\pi$
$908$	−24.8353	−0.824188
$909$	−18.0175	−0.597604
$910$	0	0
$911$	25.4159	0.842065	0.421033	−	0.907045i	$-0.361668\pi$
$911$	25.4159	0.842065	0.421033	+	0.907045i	$0.361668\pi$
$912$	10.2711	0.340112
$913$	−27.0224	−0.894310
$914$	−4.28439	−0.141715
$915$	−15.7653	−0.521185
$916$	40.7419	1.34615
$917$	0	0
$918$	−0.505393	−0.0166805
$919$	−20.7751	−0.685306	−0.342653	−	0.939462i	$-0.611325\pi$
$919$	−20.7751	−0.685306	−0.342653	+	0.939462i	$0.611325\pi$
$920$	10.3599	0.341557
$921$	−7.26086	−0.239253
$922$	−12.0680	−0.397440
$923$	28.6365	0.942581
$924$	0	0
$925$	−32.6081	−1.07215
$926$	6.62293	0.217643
$927$	16.9811	0.557732
$928$	17.1493	0.562954
$929$	−0.734942	−0.0241127	−0.0120563	−	0.999927i	$-0.503838\pi$
$929$	−0.734942	−0.0241127	−0.0120563	+	0.999927i	$0.503838\pi$
$930$	4.30469	0.141156
$931$	0	0
$932$	−7.92257	−0.259512
$933$	−57.2625	−1.87469
$934$	15.6854	0.513242
$935$	0.461106	0.0150798
$936$	7.53584	0.246317
$937$	18.0314	0.589061	0.294531	−	0.955642i	$-0.404837\pi$
$937$	18.0314	0.589061	0.294531	+	0.955642i	$0.404837\pi$
$938$	0	0
$939$	−8.88583	−0.289978
$940$	6.00420	0.195835
$941$	−31.5482	−1.02844	−0.514221	−	0.857658i	$-0.671919\pi$
$941$	−31.5482	−1.02844	−0.514221	+	0.857658i	$0.671919\pi$
$942$	−19.4717	−0.634422
$943$	4.64843	0.151374
$944$	26.2094	0.853043
$945$	0	0
$946$	2.28727	0.0743655
$947$	−31.4358	−1.02153	−0.510764	−	0.859721i	$-0.670637\pi$
$947$	−31.4358	−1.02153	−0.510764	+	0.859721i	$0.670637\pi$
$948$	15.0392	0.488450
$949$	−14.2529	−0.462669
$950$	−2.79170	−0.0905748
$951$	−3.98011	−0.129064
$952$	0	0
$953$	42.2834	1.36969	0.684847	−	0.728687i	$-0.259869\pi$
$953$	42.2834	1.36969	0.684847	+	0.728687i	$0.259869\pi$
$954$	−6.14097	−0.198821
$955$	−2.18709	−0.0707725
$956$	22.6229	0.731678
$957$	−12.7834	−0.413229
$958$	0.837339	0.0270532
$959$	0	0
$960$	−7.74953	−0.250115
$961$	6.21726	0.200557
$962$	−12.3803	−0.399158
$963$	1.21244	0.0390703
$964$	−40.3421	−1.29933
$965$	1.96919	0.0633903
$966$	0	0
$967$	30.8891	0.993328	0.496664	−	0.867943i	$-0.334558\pi$
$967$	30.8891	0.993328	0.496664	+	0.867943i	$0.334558\pi$
$968$	−13.1635	−0.423091
$969$	1.15304	0.0370411
$970$	−0.839501	−0.0269548
$971$	24.5489	0.787811	0.393906	−	0.919151i	$-0.371124\pi$
$971$	24.5489	0.787811	0.393906	+	0.919151i	$0.371124\pi$
$972$	−21.9144	−0.702906
$973$	0	0
$974$	13.3850	0.428883
$975$	35.3561	1.13230
$976$	−27.4162	−0.877572
$977$	−23.9232	−0.765373	−0.382686	−	0.923878i	$-0.625001\pi$
$977$	−23.9232	−0.765373	−0.382686	+	0.923878i	$0.625001\pi$
$978$	6.11846	0.195647
$979$	3.65993	0.116972
$980$	0	0
$981$	−8.82467	−0.281750
$982$	−6.46453	−0.206291
$983$	3.26031	0.103988	0.0519939	−	0.998647i	$-0.483442\pi$
$983$	3.26031	0.103988	0.0519939	+	0.998647i	$0.483442\pi$
$984$	1.87007	0.0596156
$985$	20.4757	0.652410
$986$	0.548894	0.0174803
$987$	0	0
$988$	12.1442	0.386357
$989$	−28.6622	−0.911404
$990$	−0.651564	−0.0207080
$991$	−7.31531	−0.232379	−0.116189	−	0.993227i	$-0.537068\pi$
$991$	−7.31531	−0.232379	−0.116189	+	0.993227i	$0.537068\pi$
$992$	26.2562	0.833636
$993$	−50.8695	−1.61430
$994$	0	0
$995$	17.1490	0.543661
$996$	65.2659	2.06803
$997$	−28.0435	−0.888146	−0.444073	−	0.895991i	$-0.646467\pi$
$997$	−28.0435	−0.888146	−0.444073	+	0.895991i	$0.646467\pi$
$998$	8.62838	0.273127
$999$	28.0581	0.887720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5243.2.a.g.1.4		7
7.6	odd	2		107.2.a.b.1.4	✓	7
21.20	even	2		963.2.a.f.1.4		7
28.27	even	2		1712.2.a.t.1.2		7
35.34	odd	2		2675.2.a.g.1.4		7
56.13	odd	2		6848.2.a.bu.1.2		7
56.27	even	2		6848.2.a.bv.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
107.2.a.b.1.4	✓	7	7.6	odd	2
963.2.a.f.1.4		7	21.20	even	2
1712.2.a.t.1.2		7	28.27	even	2
2675.2.a.g.1.4		7	35.34	odd	2
5243.2.a.g.1.4		7	1.1	even	1	trivial
6848.2.a.bu.1.2		7	56.13	odd	2
6848.2.a.bv.1.6		7	56.27	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:	\( N \)	\(=\)	\( 5243 = 7^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5243.a (trivial)

\(f(q)\)	\(=\)	\(q-0.400681 q^{2} -2.05242 q^{3} -1.83945 q^{4} -0.858034 q^{5} +0.822367 q^{6} +1.53840 q^{8} +1.21244 q^{9} +0.343798 q^{10} -1.56312 q^{11} +3.77534 q^{12} +4.04020 q^{13} +1.76105 q^{15} +3.06250 q^{16} +0.343798 q^{17} -0.485802 q^{18} -1.63409 q^{19} +1.57831 q^{20} +0.626314 q^{22} -7.84845 q^{23} -3.15744 q^{24} -4.26378 q^{25} -1.61883 q^{26} +3.66883 q^{27} -3.98461 q^{29} -0.705619 q^{30} -6.10060 q^{31} -4.30388 q^{32} +3.20819 q^{33} -0.137753 q^{34} -2.23023 q^{36} +7.64770 q^{37} +0.654748 q^{38} -8.29220 q^{39} -1.32000 q^{40} -0.592274 q^{41} +3.65195 q^{43} +2.87530 q^{44} -1.04031 q^{45} +3.14473 q^{46} +3.80419 q^{47} -6.28555 q^{48} +1.70842 q^{50} -0.705619 q^{51} -7.43177 q^{52} +12.6409 q^{53} -1.47003 q^{54} +1.34121 q^{55} +3.35384 q^{57} +1.59656 q^{58} +8.55816 q^{59} -3.23937 q^{60} -8.95223 q^{61} +2.44439 q^{62} -4.40052 q^{64} -3.46663 q^{65} -1.28546 q^{66} +13.1646 q^{67} -0.632401 q^{68} +16.1083 q^{69} +7.08789 q^{71} +1.86521 q^{72} -3.52777 q^{73} -3.06429 q^{74} +8.75108 q^{75} +3.00583 q^{76} +3.32253 q^{78} +3.98353 q^{79} -2.62773 q^{80} -11.1673 q^{81} +0.237313 q^{82} +17.2874 q^{83} -0.294990 q^{85} -1.46327 q^{86} +8.17811 q^{87} -2.40470 q^{88} -2.34142 q^{89} +0.416834 q^{90} +14.4369 q^{92} +12.5210 q^{93} -1.52427 q^{94} +1.40210 q^{95} +8.83338 q^{96} -2.44185 q^{97} -1.89519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} - 3 q^{3} + 7 q^{4} - 5 q^{5} + 5 q^{6} - 6 q^{8} + 6 q^{9} + q^{10} - 2 q^{11} - 6 q^{12} - 18 q^{13} - 9 q^{15} - q^{16} + q^{17} - 17 q^{18} + 4 q^{19} + 10 q^{20} - 5 q^{22} + 31 q^{24}+ \cdots - 52 q^{99}+O(q^{100}) \) 7 * q - q^2 - 3 * q^3 + 7 * q^4 - 5 * q^5 + 5 * q^6 - 6 * q^8 + 6 * q^9 + q^10 - 2 * q^11 - 6 * q^12 - 18 * q^13 - 9 * q^15 - q^16 + q^17 - 17 * q^18 + 4 * q^19 + 10 * q^20 - 5 * q^22 + 31 * q^24 + 8 * q^25 + 22 * q^26 - 3 * q^27 - 3 * q^29 - 8 * q^30 - 4 * q^31 - 13 * q^32 + 6 * q^33 + 14 * q^36 + 25 * q^37 - 18 * q^38 + 5 * q^39 - 13 * q^40 + 11 * q^43 - 17 * q^44 - 25 * q^45 + q^46 + 9 * q^47 - 19 * q^48 + 7 * q^50 - 8 * q^51 - 44 * q^52 + 8 * q^53 + 35 * q^54 + 13 * q^55 - 3 * q^57 + 35 * q^58 + 19 * q^59 + 9 * q^60 - 25 * q^61 - 23 * q^62 - 18 * q^64 + 8 * q^65 - 63 * q^66 - 24 * q^67 - 11 * q^68 - 19 * q^71 - 15 * q^72 - 30 * q^73 - 17 * q^74 + 41 * q^75 - 4 * q^76 - 18 * q^78 - 21 * q^79 - 19 * q^80 - 17 * q^81 - 6 * q^82 - 12 * q^83 + 2 * q^85 + 31 * q^86 + 15 * q^87 - 8 * q^88 + 22 * q^89 + 36 * q^90 - 7 * q^92 + 9 * q^93 - 16 * q^94 - 44 * q^95 - 2 * q^96 + 4 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more