LMFDB - Embedded newform 4761.2.a.bu.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	10.10.5791333887977.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 $ x^10 - 2x^9 - 12x^8 + 22x^7 + 49x^6 - 84x^5 - 73x^4 + 132x^3 + 17x^2 - 74*x + 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.568983$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.568983 q^{2} -1.67626 q^{4} +2.70241 q^{5} -3.38601 q^{7} -2.09173 q^{8} +O(q^{10})$	$q+0.568983 q^{2} -1.67626 q^{4} +2.70241 q^{5} -3.38601 q^{7} -2.09173 q^{8} +1.53762 q^{10} -4.48823 q^{11} +6.26113 q^{13} -1.92658 q^{14} +2.16236 q^{16} +3.06906 q^{17} -1.04840 q^{19} -4.52993 q^{20} -2.55372 q^{22} +2.30301 q^{25} +3.56248 q^{26} +5.67582 q^{28} +2.21800 q^{29} -6.48711 q^{31} +5.41380 q^{32} +1.74624 q^{34} -9.15037 q^{35} -0.279560 q^{37} -0.596524 q^{38} -5.65270 q^{40} -1.41972 q^{41} -1.83959 q^{43} +7.52343 q^{44} -4.74565 q^{47} +4.46503 q^{49} +1.31037 q^{50} -10.4953 q^{52} +5.39864 q^{53} -12.1290 q^{55} +7.08260 q^{56} +1.26200 q^{58} +10.8910 q^{59} -4.68450 q^{61} -3.69105 q^{62} -1.24436 q^{64} +16.9201 q^{65} +0.201133 q^{67} -5.14454 q^{68} -5.20640 q^{70} +3.01784 q^{71} -12.0906 q^{73} -0.159065 q^{74} +1.75740 q^{76} +15.1972 q^{77} -7.76749 q^{79} +5.84358 q^{80} -0.807796 q^{82} -14.7611 q^{83} +8.29386 q^{85} -1.04669 q^{86} +9.38814 q^{88} -0.873672 q^{89} -21.2002 q^{91} -2.70019 q^{94} -2.83322 q^{95} -11.5724 q^{97} +2.54053 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31} + 34 q^{32} - 4 q^{34} - 9 q^{35} - 35 q^{37} - 9 q^{38} - 21 q^{40} - 28 q^{43} - 7 q^{44} - 9 q^{47} + 17 q^{49} - 52 q^{52} + 34 q^{53} - 14 q^{55} - 30 q^{56} - 24 q^{58} + 2 q^{59} - 49 q^{61} + 28 q^{62} + 10 q^{64} + 2 q^{65} - 26 q^{67} - 6 q^{68} + 16 q^{70} - 15 q^{71} + 14 q^{73} - 25 q^{74} - 19 q^{76} + 33 q^{77} - 43 q^{79} - 49 q^{80} + 24 q^{82} - 15 q^{83} - 21 q^{85} - 49 q^{86} - 15 q^{88} - 15 q^{89} - 4 q^{91} - 28 q^{94} - 28 q^{95} - 22 q^{97} - 15 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8 - 13 * q^10 + 3 * q^11 - 4 * q^13 - 4 * q^16 + 11 * q^17 - 22 * q^19 + q^20 - 13 * q^22 - 2 * q^25 - 4 * q^26 - 26 * q^28 + 5 * q^29 - 7 * q^31 + 34 * q^32 - 4 * q^34 - 9 * q^35 - 35 * q^37 - 9 * q^38 - 21 * q^40 - 28 * q^43 - 7 * q^44 - 9 * q^47 + 17 * q^49 - 52 * q^52 + 34 * q^53 - 14 * q^55 - 30 * q^56 - 24 * q^58 + 2 * q^59 - 49 * q^61 + 28 * q^62 + 10 * q^64 + 2 * q^65 - 26 * q^67 - 6 * q^68 + 16 * q^70 - 15 * q^71 + 14 * q^73 - 25 * q^74 - 19 * q^76 + 33 * q^77 - 43 * q^79 - 49 * q^80 + 24 * q^82 - 15 * q^83 - 21 * q^85 - 49 * q^86 - 15 * q^88 - 15 * q^89 - 4 * q^91 - 28 * q^94 - 28 * q^95 - 22 * q^97 - 15 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.568983	0.402331	0.201166	−	0.979557i	$-0.435527\pi$
$2$	0.568983	0.402331	0.201166	+	0.979557i	$0.435527\pi$
$3$	0	0
$3$	0	0
$4$	−1.67626	−0.838129
$5$	2.70241	1.20855	0.604277	−	0.796775i	$-0.293462\pi$
$5$	2.70241	1.20855	0.604277	+	0.796775i	$0.293462\pi$
$6$	0	0
$7$	−3.38601	−1.27979	−0.639895	−	0.768462i	$-0.721022\pi$
$7$	−3.38601	−1.27979	−0.639895	+	0.768462i	$0.721022\pi$
$8$	−2.09173	−0.739537
$9$	0	0
$10$	1.53762	0.486239
$11$	−4.48823	−1.35325	−0.676625	−	0.736327i	$-0.736558\pi$
$11$	−4.48823	−1.35325	−0.676625	+	0.736327i	$0.736558\pi$
$12$	0	0
$13$	6.26113	1.73653	0.868263	−	0.496104i	$-0.165237\pi$
$13$	6.26113	1.73653	0.868263	+	0.496104i	$0.165237\pi$
$14$	−1.92658	−0.514900
$15$	0	0
$16$	2.16236	0.540590
$17$	3.06906	0.744357	0.372179	−	0.928161i	$-0.378611\pi$
$17$	3.06906	0.744357	0.372179	+	0.928161i	$0.378611\pi$
$18$	0	0
$19$	−1.04840	−0.240521	−0.120260	−	0.992742i	$-0.538373\pi$
$19$	−1.04840	−0.240521	−0.120260	+	0.992742i	$0.538373\pi$
$20$	−4.52993	−1.01292
$21$	0	0
$22$	−2.55372	−0.544455
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.30301	0.460601
$26$	3.56248	0.698659
$27$	0	0
$28$	5.67582	1.07263
$29$	2.21800	0.411872	0.205936	−	0.978565i	$-0.433976\pi$
$29$	2.21800	0.411872	0.205936	+	0.978565i	$0.433976\pi$
$30$	0	0
$31$	−6.48711	−1.16512	−0.582559	−	0.812788i	$-0.697949\pi$
$31$	−6.48711	−1.16512	−0.582559	+	0.812788i	$0.697949\pi$
$32$	5.41380	0.957034
$33$	0	0
$34$	1.74624	0.299478
$35$	−9.15037	−1.54669
$36$	0	0
$37$	−0.279560	−0.0459594	−0.0229797	−	0.999736i	$-0.507315\pi$
$37$	−0.279560	−0.0459594	−0.0229797	+	0.999736i	$0.507315\pi$
$38$	−0.596524	−0.0967690
$39$	0	0
$40$	−5.65270	−0.893770
$41$	−1.41972	−0.221723	−0.110861	−	0.993836i	$-0.535361\pi$
$41$	−1.41972	−0.221723	−0.110861	+	0.993836i	$0.535361\pi$
$42$	0	0
$43$	−1.83959	−0.280534	−0.140267	−	0.990114i	$-0.544796\pi$
$43$	−1.83959	−0.280534	−0.140267	+	0.990114i	$0.544796\pi$
$44$	7.52343	1.13420
$45$	0	0
$46$	0	0
$47$	−4.74565	−0.692225	−0.346112	−	0.938193i	$-0.612498\pi$
$47$	−4.74565	−0.692225	−0.346112	+	0.938193i	$0.612498\pi$
$48$	0	0
$49$	4.46503	0.637862
$50$	1.31037	0.185314
$51$	0	0
$52$	−10.4953	−1.45543
$53$	5.39864	0.741560	0.370780	−	0.928721i	$-0.379090\pi$
$53$	5.39864	0.741560	0.370780	+	0.928721i	$0.379090\pi$
$54$	0	0
$55$	−12.1290	−1.63548
$56$	7.08260	0.946452
$57$	0	0
$58$	1.26200	0.165709
$59$	10.8910	1.41789	0.708944	−	0.705265i	$-0.249172\pi$
$59$	10.8910	1.41789	0.708944	+	0.705265i	$0.249172\pi$
$60$	0	0
$61$	−4.68450	−0.599789	−0.299894	−	0.953972i	$-0.596951\pi$
$61$	−4.68450	−0.599789	−0.299894	+	0.953972i	$0.596951\pi$
$62$	−3.69105	−0.468764
$63$	0	0
$64$	−1.24436	−0.155546
$65$	16.9201	2.09868
$66$	0	0
$67$	0.201133	0.0245724	0.0122862	−	0.999925i	$-0.496089\pi$
$67$	0.201133	0.0245724	0.0122862	+	0.999925i	$0.496089\pi$
$68$	−5.14454	−0.623868
$69$	0	0
$70$	−5.20640	−0.622284
$71$	3.01784	0.358151	0.179076	−	0.983835i	$-0.442689\pi$
$71$	3.01784	0.358151	0.179076	+	0.983835i	$0.442689\pi$
$72$	0	0
$73$	−12.0906	−1.41510	−0.707548	−	0.706666i	$-0.750198\pi$
$73$	−12.0906	−1.41510	−0.707548	+	0.706666i	$0.750198\pi$
$74$	−0.159065	−0.0184909
$75$	0	0
$76$	1.75740	0.201587
$77$	15.1972	1.73188
$78$	0	0
$79$	−7.76749	−0.873911	−0.436956	−	0.899483i	$-0.643943\pi$
$79$	−7.76749	−0.873911	−0.436956	+	0.899483i	$0.643943\pi$
$80$	5.84358	0.653332
$81$	0	0
$82$	−0.807796	−0.0892061
$83$	−14.7611	−1.62024	−0.810122	−	0.586261i	$-0.800599\pi$
$83$	−14.7611	−1.62024	−0.810122	+	0.586261i	$0.800599\pi$
$84$	0	0
$85$	8.29386	0.899595
$86$	−1.04669	−0.112868
$87$	0	0
$88$	9.38814	1.00078
$89$	−0.873672	−0.0926090	−0.0463045	−	0.998927i	$-0.514744\pi$
$89$	−0.873672	−0.0926090	−0.0463045	+	0.998927i	$0.514744\pi$
$90$	0	0
$91$	−21.2002	−2.22239
$92$	0	0
$93$	0	0
$94$	−2.70019	−0.278504
$95$	−2.83322	−0.290682
$96$	0	0
$97$	−11.5724	−1.17500	−0.587502	−	0.809223i	$-0.699888\pi$
$97$	−11.5724	−1.17500	−0.587502	+	0.809223i	$0.699888\pi$
$98$	2.54053	0.256632
$99$	0	0
$100$	−3.86044	−0.386044
$101$	−9.43695	−0.939011	−0.469506	−	0.882929i	$-0.655568\pi$
$101$	−9.43695	−0.939011	−0.469506	+	0.882929i	$0.655568\pi$
$102$	0	0
$103$	−4.01274	−0.395387	−0.197693	−	0.980264i	$-0.563345\pi$
$103$	−4.01274	−0.395387	−0.197693	+	0.980264i	$0.563345\pi$
$104$	−13.0966	−1.28423
$105$	0	0
$106$	3.07173	0.298353
$107$	3.67412	0.355190	0.177595	−	0.984104i	$-0.443168\pi$
$107$	3.67412	0.355190	0.177595	+	0.984104i	$0.443168\pi$
$108$	0	0
$109$	−13.4116	−1.28460	−0.642299	−	0.766454i	$-0.722019\pi$
$109$	−13.4116	−1.28460	−0.642299	+	0.766454i	$0.722019\pi$
$110$	−6.90120	−0.658003
$111$	0	0
$112$	−7.32177	−0.691842
$113$	0.694886	0.0653694	0.0326847	−	0.999466i	$-0.489594\pi$
$113$	0.694886	0.0653694	0.0326847	+	0.999466i	$0.489594\pi$
$114$	0	0
$115$	0	0
$116$	−3.71794	−0.345202
$117$	0	0
$118$	6.19679	0.570461
$119$	−10.3919	−0.952621
$120$	0	0
$121$	9.14417	0.831288
$122$	−2.66540	−0.241314
$123$	0	0
$124$	10.8741	0.976520
$125$	−7.28837	−0.651892
$126$	0	0
$127$	−10.9440	−0.971127	−0.485563	−	0.874202i	$-0.661385\pi$
$127$	−10.9440	−0.971127	−0.485563	+	0.874202i	$0.661385\pi$
$128$	−11.5356	−1.01961
$129$	0	0
$130$	9.62726	0.844367
$131$	3.53749	0.309072	0.154536	−	0.987987i	$-0.450612\pi$
$131$	3.53749	0.309072	0.154536	+	0.987987i	$0.450612\pi$
$132$	0	0
$133$	3.54990	0.307816
$134$	0.114441	0.00988623
$135$	0	0
$136$	−6.41964	−0.550480
$137$	12.7925	1.09294	0.546469	−	0.837479i	$-0.315972\pi$
$137$	12.7925	1.09294	0.546469	+	0.837479i	$0.315972\pi$
$138$	0	0
$139$	8.21741	0.696992	0.348496	−	0.937310i	$-0.386693\pi$
$139$	8.21741	0.696992	0.348496	+	0.937310i	$0.386693\pi$
$140$	15.3384	1.29633
$141$	0	0
$142$	1.71710	0.144095
$143$	−28.1014	−2.34996
$144$	0	0
$145$	5.99393	0.497769
$146$	−6.87933	−0.569337
$147$	0	0
$148$	0.468615	0.0385199
$149$	−18.6176	−1.52521	−0.762607	−	0.646862i	$-0.776081\pi$
$149$	−18.6176	−1.52521	−0.762607	+	0.646862i	$0.776081\pi$
$150$	0	0
$151$	−5.46938	−0.445092	−0.222546	−	0.974922i	$-0.571437\pi$
$151$	−5.46938	−0.445092	−0.222546	+	0.974922i	$0.571437\pi$
$152$	2.19298	0.177874
$153$	0	0
$154$	8.64692	0.696788
$155$	−17.5308	−1.40811
$156$	0	0
$157$	−16.4366	−1.31178	−0.655890	−	0.754856i	$-0.727706\pi$
$157$	−16.4366	−1.31178	−0.655890	+	0.754856i	$0.727706\pi$
$158$	−4.41957	−0.351602
$159$	0	0
$160$	14.6303	1.15663
$161$	0	0
$162$	0	0
$163$	10.0160	0.784511	0.392256	−	0.919856i	$-0.371695\pi$
$163$	10.0160	0.784511	0.392256	+	0.919856i	$0.371695\pi$
$164$	2.37982	0.185833
$165$	0	0
$166$	−8.39883	−0.651875
$167$	7.39906	0.572557	0.286278	−	0.958147i	$-0.407582\pi$
$167$	7.39906	0.572557	0.286278	+	0.958147i	$0.407582\pi$
$168$	0	0
$169$	26.2018	2.01552
$170$	4.71906	0.361936
$171$	0	0
$172$	3.08362	0.235124
$173$	−5.61509	−0.426908	−0.213454	−	0.976953i	$-0.568471\pi$
$173$	−5.61509	−0.426908	−0.213454	+	0.976953i	$0.568471\pi$
$174$	0	0
$175$	−7.79799	−0.589473
$176$	−9.70517	−0.731554
$177$	0	0
$178$	−0.497104	−0.0372595
$179$	−21.5449	−1.61034	−0.805169	−	0.593046i	$-0.797925\pi$
$179$	−21.5449	−1.61034	−0.805169	+	0.593046i	$0.797925\pi$
$180$	0	0
$181$	−10.2613	−0.762719	−0.381360	−	0.924427i	$-0.624544\pi$
$181$	−10.2613	−0.762719	−0.381360	+	0.924427i	$0.624544\pi$
$182$	−12.0626	−0.894137
$183$	0	0
$184$	0	0
$185$	−0.755486	−0.0555444
$186$	0	0
$187$	−13.7746	−1.00730
$188$	7.95494	0.580174
$189$	0	0
$190$	−1.61205	−0.116950
$191$	8.45001	0.611421	0.305710	−	0.952125i	$-0.401106\pi$
$191$	8.45001	0.611421	0.305710	+	0.952125i	$0.401106\pi$
$192$	0	0
$193$	0.392108	0.0282246	0.0141123	−	0.999900i	$-0.495508\pi$
$193$	0.392108	0.0282246	0.0141123	+	0.999900i	$0.495508\pi$
$194$	−6.58452	−0.472741
$195$	0	0
$196$	−7.48455	−0.534611
$197$	22.5670	1.60783	0.803915	−	0.594744i	$-0.202747\pi$
$197$	22.5670	1.60783	0.803915	+	0.594744i	$0.202747\pi$
$198$	0	0
$199$	−13.3644	−0.947375	−0.473687	−	0.880693i	$-0.657077\pi$
$199$	−13.3644	−0.947375	−0.473687	+	0.880693i	$0.657077\pi$
$200$	−4.81726	−0.340632
$201$	0	0
$202$	−5.36946	−0.377794
$203$	−7.51015	−0.527109
$204$	0	0
$205$	−3.83666	−0.267964
$206$	−2.28318	−0.159076
$207$	0	0
$208$	13.5388	0.938749
$209$	4.70548	0.325485
$210$	0	0
$211$	−0.559001	−0.0384832	−0.0192416	−	0.999815i	$-0.506125\pi$
$211$	−0.559001	−0.0384832	−0.0192416	+	0.999815i	$0.506125\pi$
$212$	−9.04951	−0.621523
$213$	0	0
$214$	2.09051	0.142904
$215$	−4.97131	−0.339041
$216$	0	0
$217$	21.9654	1.49111
$218$	−7.63097	−0.516834
$219$	0	0
$220$	20.3314	1.37074
$221$	19.2158	1.29260
$222$	0	0
$223$	−29.6961	−1.98860	−0.994299	−	0.106631i	$-0.965994\pi$
$223$	−29.6961	−1.98860	−0.994299	+	0.106631i	$0.965994\pi$
$224$	−18.3312	−1.22480
$225$	0	0
$226$	0.395378	0.0263002
$227$	−24.6703	−1.63743	−0.818713	−	0.574203i	$-0.805312\pi$
$227$	−24.6703	−1.63743	−0.818713	+	0.574203i	$0.805312\pi$
$228$	0	0
$229$	28.4810	1.88207	0.941037	−	0.338302i	$-0.109853\pi$
$229$	28.4810	1.88207	0.941037	+	0.338302i	$0.109853\pi$
$230$	0	0
$231$	0	0
$232$	−4.63945	−0.304595
$233$	15.0630	0.986812	0.493406	−	0.869799i	$-0.335752\pi$
$233$	15.0630	0.986812	0.493406	+	0.869799i	$0.335752\pi$
$234$	0	0
$235$	−12.8247	−0.836591
$236$	−18.2561	−1.18837
$237$	0	0
$238$	−5.91279	−0.383269
$239$	26.9624	1.74405	0.872027	−	0.489457i	$-0.162805\pi$
$239$	26.9624	1.74405	0.872027	+	0.489457i	$0.162805\pi$
$240$	0	0
$241$	−21.1091	−1.35976	−0.679879	−	0.733324i	$-0.737968\pi$
$241$	−21.1091	−1.35976	−0.679879	+	0.733324i	$0.737968\pi$
$242$	5.20287	0.334453
$243$	0	0
$244$	7.85243	0.502701
$245$	12.0663	0.770890
$246$	0	0
$247$	−6.56420	−0.417670
$248$	13.5693	0.861649
$249$	0	0
$250$	−4.14696	−0.262277
$251$	−2.32705	−0.146882	−0.0734411	−	0.997300i	$-0.523398\pi$
$251$	−2.32705	−0.146882	−0.0734411	+	0.997300i	$0.523398\pi$
$252$	0	0
$253$	0	0
$254$	−6.22697	−0.390715
$255$	0	0
$256$	−4.07484	−0.254677
$257$	−26.6953	−1.66520	−0.832602	−	0.553871i	$-0.813150\pi$
$257$	−26.6953	−1.66520	−0.832602	+	0.553871i	$0.813150\pi$
$258$	0	0
$259$	0.946593	0.0588184
$260$	−28.3625	−1.75897
$261$	0	0
$262$	2.01277	0.124349
$263$	10.0028	0.616801	0.308401	−	0.951257i	$-0.400206\pi$
$263$	10.0028	0.616801	0.308401	+	0.951257i	$0.400206\pi$
$264$	0	0
$265$	14.5893	0.896215
$266$	2.01983	0.123844
$267$	0	0
$268$	−0.337152	−0.0205948
$269$	17.5445	1.06971	0.534853	−	0.844945i	$-0.320367\pi$
$269$	17.5445	1.06971	0.534853	+	0.844945i	$0.320367\pi$
$270$	0	0
$271$	1.13946	0.0692174	0.0346087	−	0.999401i	$-0.488982\pi$
$271$	1.13946	0.0692174	0.0346087	+	0.999401i	$0.488982\pi$
$272$	6.63642	0.402392
$273$	0	0
$274$	7.27872	0.439724
$275$	−10.3364	−0.623309
$276$	0	0
$277$	−21.3674	−1.28384	−0.641920	−	0.766772i	$-0.721862\pi$
$277$	−21.3674	−1.28384	−0.641920	+	0.766772i	$0.721862\pi$
$278$	4.67556	0.280422
$279$	0	0
$280$	19.1401	1.14384
$281$	−12.3835	−0.738737	−0.369369	−	0.929283i	$-0.620426\pi$
$281$	−12.3835	−0.738737	−0.369369	+	0.929283i	$0.620426\pi$
$282$	0	0
$283$	7.11666	0.423041	0.211521	−	0.977374i	$-0.432158\pi$
$283$	7.11666	0.423041	0.211521	+	0.977374i	$0.432158\pi$
$284$	−5.05867	−0.300177
$285$	0	0
$286$	−15.9892	−0.945461
$287$	4.80718	0.283759
$288$	0	0
$289$	−7.58085	−0.445932
$290$	3.41044	0.200268
$291$	0	0
$292$	20.2669	1.18603
$293$	−17.9845	−1.05066	−0.525332	−	0.850898i	$-0.676059\pi$
$293$	−17.9845	−1.05066	−0.525332	+	0.850898i	$0.676059\pi$
$294$	0	0
$295$	29.4319	1.71359
$296$	0.584764	0.0339887
$297$	0	0
$298$	−10.5931	−0.613641
$299$	0	0
$300$	0	0
$301$	6.22885	0.359025
$302$	−3.11198	−0.179074
$303$	0	0
$304$	−2.26703	−0.130023
$305$	−12.6594	−0.724877
$306$	0	0
$307$	1.83085	0.104492	0.0522460	−	0.998634i	$-0.483362\pi$
$307$	1.83085	0.104492	0.0522460	+	0.998634i	$0.483362\pi$
$308$	−25.4744	−1.45154
$309$	0	0
$310$	−9.97472	−0.566526
$311$	9.26634	0.525446	0.262723	−	0.964871i	$-0.415379\pi$
$311$	9.26634	0.525446	0.262723	+	0.964871i	$0.415379\pi$
$312$	0	0
$313$	−13.9170	−0.786634	−0.393317	−	0.919403i	$-0.628673\pi$
$313$	−13.9170	−0.786634	−0.393317	+	0.919403i	$0.628673\pi$
$314$	−9.35211	−0.527770
$315$	0	0
$316$	13.0203	0.732451
$317$	27.0529	1.51944	0.759722	−	0.650248i	$-0.225335\pi$
$317$	27.0529	1.51944	0.759722	+	0.650248i	$0.225335\pi$
$318$	0	0
$319$	−9.95488	−0.557366
$320$	−3.36278	−0.187985
$321$	0	0
$322$	0	0
$323$	−3.21762	−0.179033
$324$	0	0
$325$	14.4194	0.799846
$326$	5.69891	0.315633
$327$	0	0
$328$	2.96967	0.163972
$329$	16.0688	0.885902
$330$	0	0
$331$	16.4225	0.902661	0.451331	−	0.892357i	$-0.350949\pi$
$331$	16.4225	0.902661	0.451331	+	0.892357i	$0.350949\pi$
$332$	24.7435	1.35797
$333$	0	0
$334$	4.20994	0.230358
$335$	0.543544	0.0296970
$336$	0	0
$337$	16.5746	0.902878	0.451439	−	0.892302i	$-0.350911\pi$
$337$	16.5746	0.902878	0.451439	+	0.892302i	$0.350911\pi$
$338$	14.9084	0.810908
$339$	0	0
$340$	−13.9027	−0.753977
$341$	29.1156	1.57670
$342$	0	0
$343$	8.58341	0.463461
$344$	3.84791	0.207466
$345$	0	0
$346$	−3.19489	−0.171758
$347$	0.186730	0.0100242	0.00501208	−	0.999987i	$-0.498405\pi$
$347$	0.186730	0.0100242	0.00501208	+	0.999987i	$0.498405\pi$
$348$	0	0
$349$	10.5157	0.562892	0.281446	−	0.959577i	$-0.409186\pi$
$349$	10.5157	0.562892	0.281446	+	0.959577i	$0.409186\pi$
$350$	−4.43692	−0.237163
$351$	0	0
$352$	−24.2984	−1.29511
$353$	20.6655	1.09991	0.549956	−	0.835194i	$-0.314645\pi$
$353$	20.6655	1.09991	0.549956	+	0.835194i	$0.314645\pi$
$354$	0	0
$355$	8.15542	0.432845
$356$	1.46450	0.0776183
$357$	0	0
$358$	−12.2586	−0.647889
$359$	15.6775	0.827425	0.413712	−	0.910408i	$-0.364232\pi$
$359$	15.6775	0.827425	0.413712	+	0.910408i	$0.364232\pi$
$360$	0	0
$361$	−17.9008	−0.942150
$362$	−5.83852	−0.306866
$363$	0	0
$364$	35.5371	1.86265
$365$	−32.6737	−1.71022
$366$	0	0
$367$	−5.44563	−0.284259	−0.142130	−	0.989848i	$-0.545395\pi$
$367$	−5.44563	−0.284259	−0.142130	+	0.989848i	$0.545395\pi$
$368$	0	0
$369$	0	0
$370$	−0.429858	−0.0223473
$371$	−18.2798	−0.949041
$372$	0	0
$373$	4.02738	0.208530	0.104265	−	0.994550i	$-0.466751\pi$
$373$	4.02738	0.208530	0.104265	+	0.994550i	$0.466751\pi$
$374$	−7.83753	−0.405269
$375$	0	0
$376$	9.92661	0.511926
$377$	13.8872	0.715226
$378$	0	0
$379$	11.1490	0.572684	0.286342	−	0.958127i	$-0.407561\pi$
$379$	11.1490	0.572684	0.286342	+	0.958127i	$0.407561\pi$
$380$	4.74921	0.243629
$381$	0	0
$382$	4.80791	0.245994
$383$	−15.0122	−0.767090	−0.383545	−	0.923522i	$-0.625297\pi$
$383$	−15.0122	−0.767090	−0.383545	+	0.923522i	$0.625297\pi$
$384$	0	0
$385$	41.0689	2.09307
$386$	0.223103	0.0113556
$387$	0	0
$388$	19.3984	0.984805
$389$	−33.1611	−1.68134	−0.840668	−	0.541551i	$-0.817837\pi$
$389$	−33.1611	−1.68134	−0.840668	+	0.541551i	$0.817837\pi$
$390$	0	0
$391$	0	0
$392$	−9.33963	−0.471723
$393$	0	0
$394$	12.8402	0.646881
$395$	−20.9909	−1.05617
$396$	0	0
$397$	−6.67063	−0.334789	−0.167395	−	0.985890i	$-0.553535\pi$
$397$	−6.67063	−0.334789	−0.167395	+	0.985890i	$0.553535\pi$
$398$	−7.60409	−0.381159
$399$	0	0
$400$	4.97993	0.248997
$401$	−6.22639	−0.310931	−0.155466	−	0.987841i	$-0.549688\pi$
$401$	−6.22639	−0.310931	−0.155466	+	0.987841i	$0.549688\pi$
$402$	0	0
$403$	−40.6166	−2.02326
$404$	15.8188	0.787013
$405$	0	0
$406$	−4.27315	−0.212073
$407$	1.25473	0.0621946
$408$	0	0
$409$	−1.95994	−0.0969128	−0.0484564	−	0.998825i	$-0.515430\pi$
$409$	−1.95994	−0.0969128	−0.0484564	+	0.998825i	$0.515430\pi$
$410$	−2.18299	−0.107810
$411$	0	0
$412$	6.72638	0.331385
$413$	−36.8770	−1.81460
$414$	0	0
$415$	−39.8906	−1.95815
$416$	33.8965	1.66191
$417$	0	0
$418$	2.67733	0.130953
$419$	8.26445	0.403745	0.201872	−	0.979412i	$-0.435297\pi$
$419$	8.26445	0.403745	0.201872	+	0.979412i	$0.435297\pi$
$420$	0	0
$421$	1.77108	0.0863170	0.0431585	−	0.999068i	$-0.486258\pi$
$421$	1.77108	0.0863170	0.0431585	+	0.999068i	$0.486258\pi$
$422$	−0.318062	−0.0154830
$423$	0	0
$424$	−11.2925	−0.548411
$425$	7.06807	0.342852
$426$	0	0
$427$	15.8617	0.767604
$428$	−6.15877	−0.297695
$429$	0	0
$430$	−2.82859	−0.136407
$431$	26.7569	1.28884	0.644418	−	0.764674i	$-0.277100\pi$
$431$	26.7569	1.28884	0.644418	+	0.764674i	$0.277100\pi$
$432$	0	0
$433$	−24.5762	−1.18106	−0.590529	−	0.807017i	$-0.701081\pi$
$433$	−24.5762	−1.18106	−0.590529	+	0.807017i	$0.701081\pi$
$434$	12.4979	0.599919
$435$	0	0
$436$	22.4813	1.07666
$437$	0	0
$438$	0	0
$439$	5.59553	0.267060	0.133530	−	0.991045i	$-0.457369\pi$
$439$	5.59553	0.267060	0.133530	+	0.991045i	$0.457369\pi$
$440$	25.3706	1.20950
$441$	0	0
$442$	10.9335	0.520052
$443$	−12.5170	−0.594701	−0.297351	−	0.954768i	$-0.596103\pi$
$443$	−12.5170	−0.594701	−0.297351	+	0.954768i	$0.596103\pi$
$444$	0	0
$445$	−2.36102	−0.111923
$446$	−16.8966	−0.800075
$447$	0	0
$448$	4.21343	0.199066
$449$	8.14963	0.384605	0.192302	−	0.981336i	$-0.438405\pi$
$449$	8.14963	0.384605	0.192302	+	0.981336i	$0.438405\pi$
$450$	0	0
$451$	6.37202	0.300047
$452$	−1.16481	−0.0547880
$453$	0	0
$454$	−14.0370	−0.658788
$455$	−57.2917	−2.68587
$456$	0	0
$457$	18.1009	0.846723	0.423361	−	0.905961i	$-0.360850\pi$
$457$	18.1009	0.846723	0.423361	+	0.905961i	$0.360850\pi$
$458$	16.2052	0.757218
$459$	0	0
$460$	0	0
$461$	−13.1011	−0.610180	−0.305090	−	0.952324i	$-0.598687\pi$
$461$	−13.1011	−0.610180	−0.305090	+	0.952324i	$0.598687\pi$
$462$	0	0
$463$	−19.4661	−0.904668	−0.452334	−	0.891849i	$-0.649409\pi$
$463$	−19.4661	−0.904668	−0.452334	+	0.891849i	$0.649409\pi$
$464$	4.79611	0.222654
$465$	0	0
$466$	8.57060	0.397025
$467$	−1.19583	−0.0553365	−0.0276682	−	0.999617i	$-0.508808\pi$
$467$	−1.19583	−0.0553365	−0.0276682	+	0.999617i	$0.508808\pi$
$468$	0	0
$469$	−0.681039	−0.0314474
$470$	−7.29702	−0.336587
$471$	0	0
$472$	−22.7810	−1.04858
$473$	8.25648	0.379633
$474$	0	0
$475$	−2.41448	−0.110784
$476$	17.4195	0.798419
$477$	0	0
$478$	15.3412	0.701688
$479$	−17.4856	−0.798937	−0.399468	−	0.916747i	$-0.630805\pi$
$479$	−17.4856	−0.798937	−0.399468	+	0.916747i	$0.630805\pi$
$480$	0	0
$481$	−1.75036	−0.0798097
$482$	−12.0107	−0.547073
$483$	0	0
$484$	−15.3280	−0.696727
$485$	−31.2735	−1.42005
$486$	0	0
$487$	−22.5133	−1.02018	−0.510088	−	0.860122i	$-0.670387\pi$
$487$	−22.5133	−1.02018	−0.510088	+	0.860122i	$0.670387\pi$
$488$	9.79870	0.443566
$489$	0	0
$490$	6.86554	0.310153
$491$	−29.4342	−1.32835	−0.664174	−	0.747578i	$-0.731217\pi$
$491$	−29.4342	−1.32835	−0.664174	+	0.747578i	$0.731217\pi$
$492$	0	0
$493$	6.80718	0.306580
$494$	−3.73492	−0.168042
$495$	0	0
$496$	−14.0275	−0.629852
$497$	−10.2184	−0.458358
$498$	0	0
$499$	−12.9838	−0.581233	−0.290616	−	0.956840i	$-0.593860\pi$
$499$	−12.9838	−0.581233	−0.290616	+	0.956840i	$0.593860\pi$
$500$	12.2172	0.546370
$501$	0	0
$502$	−1.32405	−0.0590953
$503$	−0.735172	−0.0327797	−0.0163898	−	0.999866i	$-0.505217\pi$
$503$	−0.735172	−0.0327797	−0.0163898	+	0.999866i	$0.505217\pi$
$504$	0	0
$505$	−25.5025	−1.13485
$506$	0	0
$507$	0	0
$508$	18.3450	0.813930
$509$	−2.76431	−0.122526	−0.0612629	−	0.998122i	$-0.519513\pi$
$509$	−2.76431	−0.122526	−0.0612629	+	0.998122i	$0.519513\pi$
$510$	0	0
$511$	40.9388	1.81102
$512$	20.7527	0.917150
$513$	0	0
$514$	−15.1891	−0.669964
$515$	−10.8440	−0.477846
$516$	0	0
$517$	21.2996	0.936754
$518$	0.538595	0.0236645
$519$	0	0
$520$	−35.3923	−1.55206
$521$	−16.3874	−0.717944	−0.358972	−	0.933348i	$-0.616873\pi$
$521$	−16.3874	−0.717944	−0.358972	+	0.933348i	$0.616873\pi$
$522$	0	0
$523$	−24.8590	−1.08701	−0.543504	−	0.839406i	$-0.682903\pi$
$523$	−24.8590	−1.08701	−0.543504	+	0.839406i	$0.682903\pi$
$524$	−5.92975	−0.259042
$525$	0	0
$526$	5.69144	0.248159
$527$	−19.9093	−0.867265
$528$	0	0
$529$	0	0
$530$	8.30107	0.360575
$531$	0	0
$532$	−5.95056	−0.257989
$533$	−8.88905	−0.385028
$534$	0	0
$535$	9.92896	0.429267
$536$	−0.420716	−0.0181722
$537$	0	0
$538$	9.98250	0.430376
$539$	−20.0401	−0.863187
$540$	0	0
$541$	38.3120	1.64716	0.823581	−	0.567199i	$-0.191973\pi$
$541$	38.3120	1.64716	0.823581	+	0.567199i	$0.191973\pi$
$542$	0.648334	0.0278483
$543$	0	0
$544$	16.6153	0.712375
$545$	−36.2436	−1.55251
$546$	0	0
$547$	−18.8583	−0.806324	−0.403162	−	0.915129i	$-0.632089\pi$
$547$	−18.8583	−0.806324	−0.403162	+	0.915129i	$0.632089\pi$
$548$	−21.4436	−0.916024
$549$	0	0
$550$	−5.88124	−0.250777
$551$	−2.32536	−0.0990637
$552$	0	0
$553$	26.3008	1.11842
$554$	−12.1577	−0.516529
$555$	0	0
$556$	−13.7745	−0.584169
$557$	22.7686	0.964736	0.482368	−	0.875969i	$-0.339777\pi$
$557$	22.7686	0.964736	0.482368	+	0.875969i	$0.339777\pi$
$558$	0	0
$559$	−11.5179	−0.487155
$560$	−19.7864	−0.836128
$561$	0	0
$562$	−7.04599	−0.297217
$563$	−18.2054	−0.767264	−0.383632	−	0.923486i	$-0.625327\pi$
$563$	−18.2054	−0.767264	−0.383632	+	0.923486i	$0.625327\pi$
$564$	0	0
$565$	1.87787	0.0790024
$566$	4.04925	0.170203
$567$	0	0
$568$	−6.31249	−0.264866
$569$	−31.5087	−1.32091	−0.660456	−	0.750865i	$-0.729637\pi$
$569$	−31.5087	−1.32091	−0.660456	+	0.750865i	$0.729637\pi$
$570$	0	0
$571$	−4.84071	−0.202577	−0.101289	−	0.994857i	$-0.532297\pi$
$571$	−4.84071	−0.202577	−0.101289	+	0.994857i	$0.532297\pi$
$572$	47.1052	1.96957
$573$	0	0
$574$	2.73520	0.114165
$575$	0	0
$576$	0	0
$577$	25.1384	1.04653	0.523263	−	0.852171i	$-0.324715\pi$
$577$	25.1384	1.04653	0.523263	+	0.852171i	$0.324715\pi$
$578$	−4.31337	−0.179413
$579$	0	0
$580$	−10.0474	−0.417195
$581$	49.9813	2.07357
$582$	0	0
$583$	−24.2303	−1.00352
$584$	25.2902	1.04652
$585$	0	0
$586$	−10.2328	−0.422715
$587$	−31.1328	−1.28499	−0.642495	−	0.766290i	$-0.722101\pi$
$587$	−31.1328	−1.28499	−0.642495	+	0.766290i	$0.722101\pi$
$588$	0	0
$589$	6.80111	0.280235
$590$	16.7462	0.689432
$591$	0	0
$592$	−0.604510	−0.0248452
$593$	25.8767	1.06263	0.531315	−	0.847174i	$-0.321698\pi$
$593$	25.8767	1.06263	0.531315	+	0.847174i	$0.321698\pi$
$594$	0	0
$595$	−28.0831	−1.15129
$596$	31.2079	1.27833
$597$	0	0
$598$	0	0
$599$	43.7707	1.78842	0.894210	−	0.447647i	$-0.147738\pi$
$599$	43.7707	1.78842	0.894210	+	0.447647i	$0.147738\pi$
$600$	0	0
$601$	13.0905	0.533972	0.266986	−	0.963700i	$-0.413972\pi$
$601$	13.0905	0.533972	0.266986	+	0.963700i	$0.413972\pi$
$602$	3.54411	0.144447
$603$	0	0
$604$	9.16810	0.373045
$605$	24.7113	1.00466
$606$	0	0
$607$	12.1643	0.493732	0.246866	−	0.969050i	$-0.420599\pi$
$607$	12.1643	0.493732	0.246866	+	0.969050i	$0.420599\pi$
$608$	−5.67585	−0.230186
$609$	0	0
$610$	−7.20299	−0.291641
$611$	−29.7132	−1.20207
$612$	0	0
$613$	−42.6873	−1.72412	−0.862062	−	0.506802i	$-0.830827\pi$
$613$	−42.6873	−1.72412	−0.862062	+	0.506802i	$0.830827\pi$
$614$	1.04172	0.0420404
$615$	0	0
$616$	−31.7883	−1.28079
$617$	0.397854	0.0160170	0.00800849	−	0.999968i	$-0.497451\pi$
$617$	0.397854	0.0160170	0.00800849	+	0.999968i	$0.497451\pi$
$618$	0	0
$619$	30.0498	1.20780	0.603902	−	0.797058i	$-0.293612\pi$
$619$	30.0498	1.20780	0.603902	+	0.797058i	$0.293612\pi$
$620$	29.3862	1.18018
$621$	0	0
$622$	5.27239	0.211403
$623$	2.95826	0.118520
$624$	0	0
$625$	−31.2112	−1.24845
$626$	−7.91852	−0.316488
$627$	0	0
$628$	27.5519	1.09944
$629$	−0.857988	−0.0342102
$630$	0	0
$631$	24.1375	0.960900	0.480450	−	0.877022i	$-0.340473\pi$
$631$	24.1375	0.960900	0.480450	+	0.877022i	$0.340473\pi$
$632$	16.2475	0.646290
$633$	0	0
$634$	15.3926	0.611320
$635$	−29.5753	−1.17366
$636$	0	0
$637$	27.9562	1.10766
$638$	−5.66415	−0.224246
$639$	0	0
$640$	−31.1740	−1.23226
$641$	32.9635	1.30198	0.650990	−	0.759086i	$-0.274354\pi$
$641$	32.9635	1.30198	0.650990	+	0.759086i	$0.274354\pi$
$642$	0	0
$643$	−7.50786	−0.296081	−0.148040	−	0.988981i	$-0.547297\pi$
$643$	−7.50786	−0.296081	−0.148040	+	0.988981i	$0.547297\pi$
$644$	0	0
$645$	0	0
$646$	−1.83077	−0.0720307
$647$	−13.8074	−0.542824	−0.271412	−	0.962463i	$-0.587491\pi$
$647$	−13.8074	−0.542824	−0.271412	+	0.962463i	$0.587491\pi$
$648$	0	0
$649$	−48.8813	−1.91876
$650$	8.20441	0.321803
$651$	0	0
$652$	−16.7894	−0.657522
$653$	−23.8790	−0.934458	−0.467229	−	0.884136i	$-0.654748\pi$
$653$	−23.8790	−0.934458	−0.467229	+	0.884136i	$0.654748\pi$
$654$	0	0
$655$	9.55974	0.373530
$656$	−3.06995	−0.119861
$657$	0	0
$658$	9.14287	0.356426
$659$	20.6448	0.804207	0.402104	−	0.915594i	$-0.368279\pi$
$659$	20.6448	0.804207	0.402104	+	0.915594i	$0.368279\pi$
$660$	0	0
$661$	45.8471	1.78325	0.891624	−	0.452777i	$-0.149567\pi$
$661$	45.8471	1.78325	0.891624	+	0.452777i	$0.149567\pi$
$662$	9.34411	0.363169
$663$	0	0
$664$	30.8763	1.19823
$665$	9.59329	0.372012
$666$	0	0
$667$	0	0
$668$	−12.4027	−0.479877
$669$	0	0
$670$	0.309267	0.0119480
$671$	21.0251	0.811665
$672$	0	0
$673$	34.7483	1.33945	0.669725	−	0.742609i	$-0.266412\pi$
$673$	34.7483	1.33945	0.669725	+	0.742609i	$0.266412\pi$
$674$	9.43068	0.363256
$675$	0	0
$676$	−43.9210	−1.68927
$677$	31.2288	1.20022	0.600109	−	0.799918i	$-0.295124\pi$
$677$	31.2288	1.20022	0.600109	+	0.799918i	$0.295124\pi$
$678$	0	0
$679$	39.1844	1.50376
$680$	−17.3485	−0.665284
$681$	0	0
$682$	16.5663	0.634355
$683$	−25.0559	−0.958738	−0.479369	−	0.877613i	$-0.659134\pi$
$683$	−25.0559	−0.958738	−0.479369	+	0.877613i	$0.659134\pi$
$684$	0	0
$685$	34.5706	1.32087
$686$	4.88381	0.186465
$687$	0	0
$688$	−3.97785	−0.151654
$689$	33.8016	1.28774
$690$	0	0
$691$	47.0943	1.79155	0.895775	−	0.444507i	$-0.146621\pi$
$691$	47.0943	1.79155	0.895775	+	0.444507i	$0.146621\pi$
$692$	9.41235	0.357804
$693$	0	0
$694$	0.106246	0.00403304
$695$	22.2068	0.842351
$696$	0	0
$697$	−4.35721	−0.165041
$698$	5.98325	0.226469
$699$	0	0
$700$	13.0715	0.494055
$701$	25.5709	0.965799	0.482900	−	0.875676i	$-0.339584\pi$
$701$	25.5709	0.965799	0.482900	+	0.875676i	$0.339584\pi$
$702$	0	0
$703$	0.293092	0.0110542
$704$	5.58499	0.210492
$705$	0	0
$706$	11.7583	0.442529
$707$	31.9536	1.20174
$708$	0	0
$709$	21.3040	0.800089	0.400045	−	0.916496i	$-0.368995\pi$
$709$	21.3040	0.800089	0.400045	+	0.916496i	$0.368995\pi$
$710$	4.64029	0.174147
$711$	0	0
$712$	1.82748	0.0684878
$713$	0	0
$714$	0	0
$715$	−75.9414	−2.84005
$716$	36.1147	1.34967
$717$	0	0
$718$	8.92020	0.332899
$719$	16.8868	0.629771	0.314885	−	0.949130i	$-0.398034\pi$
$719$	16.8868	0.629771	0.314885	+	0.949130i	$0.398034\pi$
$720$	0	0
$721$	13.5871	0.506012
$722$	−10.1853	−0.379057
$723$	0	0
$724$	17.2007	0.639257
$725$	5.10807	0.189709
$726$	0	0
$727$	46.8022	1.73580	0.867898	−	0.496742i	$-0.165471\pi$
$727$	46.8022	1.73580	0.867898	+	0.496742i	$0.165471\pi$
$728$	44.3451	1.64354
$729$	0	0
$730$	−18.5907	−0.688074
$731$	−5.64581	−0.208818
$732$	0	0
$733$	1.21960	0.0450470	0.0225235	−	0.999746i	$-0.492830\pi$
$733$	1.21960	0.0450470	0.0225235	+	0.999746i	$0.492830\pi$
$734$	−3.09847	−0.114367
$735$	0	0
$736$	0	0
$737$	−0.902732	−0.0332526
$738$	0	0
$739$	−29.0460	−1.06848	−0.534238	−	0.845334i	$-0.679401\pi$
$739$	−29.0460	−1.06848	−0.534238	+	0.845334i	$0.679401\pi$
$740$	1.26639	0.0465534
$741$	0	0
$742$	−10.4009	−0.381829
$743$	−26.6891	−0.979128	−0.489564	−	0.871967i	$-0.662844\pi$
$743$	−26.6891	−0.979128	−0.489564	+	0.871967i	$0.662844\pi$
$744$	0	0
$745$	−50.3124	−1.84330
$746$	2.29151	0.0838982
$747$	0	0
$748$	23.0899	0.844249
$749$	−12.4406	−0.454569
$750$	0	0
$751$	−1.36537	−0.0498232	−0.0249116	−	0.999690i	$-0.507930\pi$
$751$	−1.36537	−0.0498232	−0.0249116	+	0.999690i	$0.507930\pi$
$752$	−10.2618	−0.374210
$753$	0	0
$754$	7.90156	0.287758
$755$	−14.7805	−0.537917
$756$	0	0
$757$	−52.1755	−1.89635	−0.948176	−	0.317746i	$-0.897074\pi$
$757$	−52.1755	−1.89635	−0.948176	+	0.317746i	$0.897074\pi$
$758$	6.34357	0.230409
$759$	0	0
$760$	5.92632	0.214970
$761$	−21.7889	−0.789847	−0.394923	−	0.918714i	$-0.629229\pi$
$761$	−21.7889	−0.789847	−0.394923	+	0.918714i	$0.629229\pi$
$762$	0	0
$763$	45.4118	1.64402
$764$	−14.1644	−0.512450
$765$	0	0
$766$	−8.54170	−0.308624
$767$	68.1900	2.46220
$768$	0	0
$769$	−37.6505	−1.35771	−0.678856	−	0.734271i	$-0.737524\pi$
$769$	−37.6505	−1.35771	−0.678856	+	0.734271i	$0.737524\pi$
$770$	23.3675	0.842106
$771$	0	0
$772$	−0.657275	−0.0236559
$773$	1.82449	0.0656225	0.0328112	−	0.999462i	$-0.489554\pi$
$773$	1.82449	0.0656225	0.0328112	+	0.999462i	$0.489554\pi$
$774$	0	0
$775$	−14.9399	−0.536655
$776$	24.2064	0.868959
$777$	0	0
$778$	−18.8681	−0.676454
$779$	1.48844	0.0533289
$780$	0	0
$781$	−13.5447	−0.484668
$782$	0	0
$783$	0	0
$784$	9.65501	0.344822
$785$	−44.4183	−1.58536
$786$	0	0
$787$	28.8204	1.02734	0.513668	−	0.857989i	$-0.328286\pi$
$787$	28.8204	1.02734	0.513668	+	0.857989i	$0.328286\pi$
$788$	−37.8281	−1.34757
$789$	0	0
$790$	−11.9435	−0.424930
$791$	−2.35289	−0.0836591
$792$	0	0
$793$	−29.3303	−1.04155
$794$	−3.79547	−0.134696
$795$	0	0
$796$	22.4021	0.794023
$797$	−0.897615	−0.0317952	−0.0158976	−	0.999874i	$-0.505061\pi$
$797$	−0.897615	−0.0317952	−0.0158976	+	0.999874i	$0.505061\pi$
$798$	0	0
$799$	−14.5647	−0.515262
$800$	12.4680	0.440811
$801$	0	0
$802$	−3.54271	−0.125097
$803$	54.2652	1.91498
$804$	0	0
$805$	0	0
$806$	−23.1102	−0.814021
$807$	0	0
$808$	19.7395	0.694434
$809$	37.6547	1.32387	0.661934	−	0.749562i	$-0.269736\pi$
$809$	37.6547	1.32387	0.661934	+	0.749562i	$0.269736\pi$
$810$	0	0
$811$	−20.4727	−0.718893	−0.359446	−	0.933166i	$-0.617034\pi$
$811$	−20.4727	−0.718893	−0.359446	+	0.933166i	$0.617034\pi$
$812$	12.5890	0.441786
$813$	0	0
$814$	0.713919	0.0250229
$815$	27.0672	0.948124
$816$	0	0
$817$	1.92863	0.0674743
$818$	−1.11517	−0.0389911
$819$	0	0
$820$	6.43124	0.224589
$821$	16.1822	0.564761	0.282381	−	0.959302i	$-0.408876\pi$
$821$	16.1822	0.564761	0.282381	+	0.959302i	$0.408876\pi$
$822$	0	0
$823$	−17.8806	−0.623279	−0.311640	−	0.950200i	$-0.600878\pi$
$823$	−17.8806	−0.623279	−0.311640	+	0.950200i	$0.600878\pi$
$824$	8.39355	0.292403
$825$	0	0
$826$	−20.9824	−0.730070
$827$	−8.08676	−0.281204	−0.140602	−	0.990066i	$-0.544904\pi$
$827$	−8.08676	−0.281204	−0.140602	+	0.990066i	$0.544904\pi$
$828$	0	0
$829$	−37.2365	−1.29328	−0.646639	−	0.762796i	$-0.723826\pi$
$829$	−37.2365	−1.29328	−0.646639	+	0.762796i	$0.723826\pi$
$830$	−22.6971	−0.787826
$831$	0	0
$832$	−7.79113	−0.270109
$833$	13.7035	0.474797
$834$	0	0
$835$	19.9953	0.691965
$836$	−7.88760	−0.272798
$837$	0	0
$838$	4.70233	0.162439
$839$	−34.7553	−1.19989	−0.599943	−	0.800043i	$-0.704810\pi$
$839$	−34.7553	−1.19989	−0.599943	+	0.800043i	$0.704810\pi$
$840$	0	0
$841$	−24.0805	−0.830362
$842$	1.00771	0.0347281
$843$	0	0
$844$	0.937030	0.0322539
$845$	70.8079	2.43587
$846$	0	0
$847$	−30.9622	−1.06387
$848$	11.6738	0.400880
$849$	0	0
$850$	4.02161	0.137940
$851$	0	0
$852$	0	0
$853$	−28.2381	−0.966855	−0.483428	−	0.875384i	$-0.660608\pi$
$853$	−28.2381	−0.966855	−0.483428	+	0.875384i	$0.660608\pi$
$854$	9.02505	0.308831
$855$	0	0
$856$	−7.68525	−0.262677
$857$	7.43551	0.253992	0.126996	−	0.991903i	$-0.459466\pi$
$857$	7.43551	0.253992	0.126996	+	0.991903i	$0.459466\pi$
$858$	0	0
$859$	40.8947	1.39531	0.697654	−	0.716435i	$-0.254227\pi$
$859$	40.8947	1.39531	0.697654	+	0.716435i	$0.254227\pi$
$860$	8.33320	0.284160
$861$	0	0
$862$	15.2242	0.518539
$863$	21.2318	0.722738	0.361369	−	0.932423i	$-0.382309\pi$
$863$	21.2318	0.722738	0.361369	+	0.932423i	$0.382309\pi$
$864$	0	0
$865$	−15.1743	−0.515941
$866$	−13.9834	−0.475176
$867$	0	0
$868$	−36.8197	−1.24974
$869$	34.8623	1.18262
$870$	0	0
$871$	1.25932	0.0426705
$872$	28.0534	0.950008
$873$	0	0
$874$	0	0
$875$	24.6785	0.834285
$876$	0	0
$877$	18.7101	0.631795	0.315897	−	0.948793i	$-0.397694\pi$
$877$	18.7101	0.631795	0.315897	+	0.948793i	$0.397694\pi$
$878$	3.18376	0.107447
$879$	0	0
$880$	−26.2273	−0.884123
$881$	35.8925	1.20925	0.604624	−	0.796511i	$-0.293323\pi$
$881$	35.8925	1.20925	0.604624	+	0.796511i	$0.293323\pi$
$882$	0	0
$883$	6.34991	0.213691	0.106846	−	0.994276i	$-0.465925\pi$
$883$	6.34991	0.213691	0.106846	+	0.994276i	$0.465925\pi$
$884$	−32.2107	−1.08336
$885$	0	0
$886$	−7.12196	−0.239267
$887$	−15.9876	−0.536810	−0.268405	−	0.963306i	$-0.586497\pi$
$887$	−15.9876	−0.536810	−0.268405	+	0.963306i	$0.586497\pi$
$888$	0	0
$889$	37.0566	1.24284
$890$	−1.34338	−0.0450301
$891$	0	0
$892$	49.7783	1.66670
$893$	4.97537	0.166494
$894$	0	0
$895$	−58.2230	−1.94618
$896$	39.0597	1.30489
$897$	0	0
$898$	4.63700	0.154739
$899$	−14.3884	−0.479880
$900$	0	0
$901$	16.5688	0.551985
$902$	3.62557	0.120718
$903$	0	0
$904$	−1.45351	−0.0483431
$905$	−27.7303	−0.921787
$906$	0	0
$907$	23.4646	0.779128	0.389564	−	0.920999i	$-0.372626\pi$
$907$	23.4646	0.779128	0.389564	+	0.920999i	$0.372626\pi$
$908$	41.3538	1.37238
$909$	0	0
$910$	−32.5980	−1.08061
$911$	27.8151	0.921557	0.460778	−	0.887515i	$-0.347570\pi$
$911$	27.8151	0.921557	0.460778	+	0.887515i	$0.347570\pi$
$912$	0	0
$913$	66.2513	2.19260
$914$	10.2991	0.340663
$915$	0	0
$916$	−47.7415	−1.57742
$917$	−11.9780	−0.395547
$918$	0	0
$919$	−38.7863	−1.27944	−0.639720	−	0.768608i	$-0.720950\pi$
$919$	−38.7863	−1.27944	−0.639720	+	0.768608i	$0.720950\pi$
$920$	0	0
$921$	0	0
$922$	−7.45431	−0.245495
$923$	18.8951	0.621939
$924$	0	0
$925$	−0.643829	−0.0211690
$926$	−11.0759	−0.363977
$927$	0	0
$928$	12.0078	0.394175
$929$	32.3914	1.06273	0.531364	−	0.847144i	$-0.321680\pi$
$929$	32.3914	1.06273	0.531364	+	0.847144i	$0.321680\pi$
$930$	0	0
$931$	−4.68116	−0.153419
$932$	−25.2495	−0.827076
$933$	0	0
$934$	−0.680407	−0.0222636
$935$	−37.2247	−1.21738
$936$	0	0
$937$	1.43049	0.0467320	0.0233660	−	0.999727i	$-0.492562\pi$
$937$	1.43049	0.0467320	0.0233660	+	0.999727i	$0.492562\pi$
$938$	−0.387499	−0.0126523
$939$	0	0
$940$	21.4975	0.701171
$941$	−0.634645	−0.0206889	−0.0103444	−	0.999946i	$-0.503293\pi$
$941$	−0.634645	−0.0206889	−0.0103444	+	0.999946i	$0.503293\pi$
$942$	0	0
$943$	0	0
$944$	23.5503	0.766496
$945$	0	0
$946$	4.69779	0.152738
$947$	33.2751	1.08130	0.540648	−	0.841249i	$-0.318179\pi$
$947$	33.2751	1.08130	0.540648	+	0.841249i	$0.318179\pi$
$948$	0	0
$949$	−75.7007	−2.45735
$950$	−1.37380	−0.0445719
$951$	0	0
$952$	21.7369	0.704498
$953$	−39.9874	−1.29532	−0.647660	−	0.761930i	$-0.724252\pi$
$953$	−39.9874	−1.29532	−0.647660	+	0.761930i	$0.724252\pi$
$954$	0	0
$955$	22.8354	0.738935
$956$	−45.1960	−1.46174
$957$	0	0
$958$	−9.94899	−0.321437
$959$	−43.3155	−1.39873
$960$	0	0
$961$	11.0826	0.357502
$962$	−0.995927	−0.0321100
$963$	0	0
$964$	35.3843	1.13965
$965$	1.05964	0.0341109
$966$	0	0
$967$	50.8204	1.63427	0.817136	−	0.576444i	$-0.195560\pi$
$967$	50.8204	1.63427	0.817136	+	0.576444i	$0.195560\pi$
$968$	−19.1271	−0.614769
$969$	0	0
$970$	−17.7941	−0.571333
$971$	42.9158	1.37723	0.688617	−	0.725125i	$-0.258218\pi$
$971$	42.9158	1.37723	0.688617	+	0.725125i	$0.258218\pi$
$972$	0	0
$973$	−27.8242	−0.892003
$974$	−12.8097	−0.410449
$975$	0	0
$976$	−10.1296	−0.324240
$977$	10.5644	0.337985	0.168992	−	0.985617i	$-0.445949\pi$
$977$	10.5644	0.337985	0.168992	+	0.985617i	$0.445949\pi$
$978$	0	0
$979$	3.92124	0.125323
$980$	−20.2263	−0.646106
$981$	0	0
$982$	−16.7476	−0.534436
$983$	47.3537	1.51035	0.755175	−	0.655524i	$-0.227552\pi$
$983$	47.3537	1.51035	0.755175	+	0.655524i	$0.227552\pi$
$984$	0	0
$985$	60.9852	1.94315
$986$	3.87316	0.123347
$987$	0	0
$988$	11.0033	0.350062
$989$	0	0
$990$	0	0
$991$	24.1058	0.765745	0.382872	−	0.923801i	$-0.374935\pi$
$991$	24.1058	0.765745	0.382872	+	0.923801i	$0.374935\pi$
$992$	−35.1199	−1.11506
$993$	0	0
$994$	−5.81410	−0.184412
$995$	−36.1160	−1.14495
$996$	0	0
$997$	28.2519	0.894747	0.447374	−	0.894347i	$-0.352359\pi$
$997$	28.2519	0.894747	0.447374	+	0.894347i	$0.352359\pi$
$998$	−7.38753	−0.233848
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bu.1.5		10
3.2	odd	2		1587.2.a.t.1.6		10
23.11	odd	22		207.2.i.d.190.1		20
23.21	odd	22		207.2.i.d.73.1		20
23.22	odd	2		4761.2.a.bt.1.5		10
69.11	even	22		69.2.e.c.52.2	yes	20
69.44	even	22		69.2.e.c.4.2	✓	20
69.68	even	2		1587.2.a.u.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.c.4.2	✓	20	69.44	even	22
69.2.e.c.52.2	yes	20	69.11	even	22
207.2.i.d.73.1		20	23.21	odd	22
207.2.i.d.190.1		20	23.11	odd	22
1587.2.a.t.1.6		10	3.2	odd	2
1587.2.a.u.1.6		10	69.68	even	2
4761.2.a.bt.1.5		10	23.22	odd	2
4761.2.a.bu.1.5		10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q+0.568983 q^{2} -1.67626 q^{4} +2.70241 q^{5} -3.38601 q^{7} -2.09173 q^{8} +O(q^{10})\)	\(q+0.568983 q^{2} -1.67626 q^{4} +2.70241 q^{5} -3.38601 q^{7} -2.09173 q^{8} +1.53762 q^{10} -4.48823 q^{11} +6.26113 q^{13} -1.92658 q^{14} +2.16236 q^{16} +3.06906 q^{17} -1.04840 q^{19} -4.52993 q^{20} -2.55372 q^{22} +2.30301 q^{25} +3.56248 q^{26} +5.67582 q^{28} +2.21800 q^{29} -6.48711 q^{31} +5.41380 q^{32} +1.74624 q^{34} -9.15037 q^{35} -0.279560 q^{37} -0.596524 q^{38} -5.65270 q^{40} -1.41972 q^{41} -1.83959 q^{43} +7.52343 q^{44} -4.74565 q^{47} +4.46503 q^{49} +1.31037 q^{50} -10.4953 q^{52} +5.39864 q^{53} -12.1290 q^{55} +7.08260 q^{56} +1.26200 q^{58} +10.8910 q^{59} -4.68450 q^{61} -3.69105 q^{62} -1.24436 q^{64} +16.9201 q^{65} +0.201133 q^{67} -5.14454 q^{68} -5.20640 q^{70} +3.01784 q^{71} -12.0906 q^{73} -0.159065 q^{74} +1.75740 q^{76} +15.1972 q^{77} -7.76749 q^{79} +5.84358 q^{80} -0.807796 q^{82} -14.7611 q^{83} +8.29386 q^{85} -1.04669 q^{86} +9.38814 q^{88} -0.873672 q^{89} -21.2002 q^{91} -2.70019 q^{94} -2.83322 q^{95} -11.5724 q^{97} +2.54053 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31} + 34 q^{32} - 4 q^{34} - 9 q^{35} - 35 q^{37} - 9 q^{38} - 21 q^{40} - 28 q^{43} - 7 q^{44} - 9 q^{47} + 17 q^{49} - 52 q^{52} + 34 q^{53} - 14 q^{55} - 30 q^{56} - 24 q^{58} + 2 q^{59} - 49 q^{61} + 28 q^{62} + 10 q^{64} + 2 q^{65} - 26 q^{67} - 6 q^{68} + 16 q^{70} - 15 q^{71} + 14 q^{73} - 25 q^{74} - 19 q^{76} + 33 q^{77} - 43 q^{79} - 49 q^{80} + 24 q^{82} - 15 q^{83} - 21 q^{85} - 49 q^{86} - 15 q^{88} - 15 q^{89} - 4 q^{91} - 28 q^{94} - 28 q^{95} - 22 q^{97} - 15 q^{98}+O(q^{100}) \) 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8 - 13 * q^10 + 3 * q^11 - 4 * q^13 - 4 * q^16 + 11 * q^17 - 22 * q^19 + q^20 - 13 * q^22 - 2 * q^25 - 4 * q^26 - 26 * q^28 + 5 * q^29 - 7 * q^31 + 34 * q^32 - 4 * q^34 - 9 * q^35 - 35 * q^37 - 9 * q^38 - 21 * q^40 - 28 * q^43 - 7 * q^44 - 9 * q^47 + 17 * q^49 - 52 * q^52 + 34 * q^53 - 14 * q^55 - 30 * q^56 - 24 * q^58 + 2 * q^59 - 49 * q^61 + 28 * q^62 + 10 * q^64 + 2 * q^65 - 26 * q^67 - 6 * q^68 + 16 * q^70 - 15 * q^71 + 14 * q^73 - 25 * q^74 - 19 * q^76 + 33 * q^77 - 43 * q^79 - 49 * q^80 + 24 * q^82 - 15 * q^83 - 21 * q^85 - 49 * q^86 - 15 * q^88 - 15 * q^89 - 4 * q^91 - 28 * q^94 - 28 * q^95 - 22 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

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Representations

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Database

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