LMFDB - Embedded newform 4761.2.a.bt.1.7

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:	$0$
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			1.7
Root			$0.871916$ 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)

$f(q)$	$=$	$q+0.871916 q^{2} -1.23976 q^{4} -1.25508 q^{5} -2.98232 q^{7} -2.82480 q^{8} +O(q^{10})$	$q+0.871916 q^{2} -1.23976 q^{4} -1.25508 q^{5} -2.98232 q^{7} -2.82480 q^{8} -1.09432 q^{10} -2.17282 q^{11} -3.87407 q^{13} -2.60033 q^{14} +0.0165341 q^{16} -0.672983 q^{17} +4.12239 q^{19} +1.55600 q^{20} -1.89452 q^{22} -3.42478 q^{25} -3.37786 q^{26} +3.69737 q^{28} -8.75589 q^{29} -5.03969 q^{31} +5.66402 q^{32} -0.586785 q^{34} +3.74305 q^{35} +8.92313 q^{37} +3.59438 q^{38} +3.54535 q^{40} -5.60073 q^{41} -8.50086 q^{43} +2.69378 q^{44} -8.72071 q^{47} +1.89424 q^{49} -2.98612 q^{50} +4.80292 q^{52} -11.0723 q^{53} +2.72706 q^{55} +8.42447 q^{56} -7.63440 q^{58} +5.81811 q^{59} +4.46592 q^{61} -4.39419 q^{62} +4.90548 q^{64} +4.86226 q^{65} +5.63671 q^{67} +0.834339 q^{68} +3.26362 q^{70} +7.92300 q^{71} +6.38691 q^{73} +7.78022 q^{74} -5.11078 q^{76} +6.48005 q^{77} +7.19156 q^{79} -0.0207516 q^{80} -4.88337 q^{82} -11.9087 q^{83} +0.844646 q^{85} -7.41203 q^{86} +6.13778 q^{88} -0.361929 q^{89} +11.5537 q^{91} -7.60373 q^{94} -5.17392 q^{95} +12.2591 q^{97} +1.65162 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.871916	0.616538	0.308269	−	0.951299i	$-0.400250\pi$
$2$	0.871916	0.616538	0.308269	+	0.951299i	$0.400250\pi$
$3$	0	0
$3$	0	0
$4$	−1.23976	−0.619881
$5$	−1.25508	−0.561288	−0.280644	−	0.959812i	$-0.590548\pi$
$5$	−1.25508	−0.561288	−0.280644	+	0.959812i	$0.590548\pi$
$6$	0	0
$7$	−2.98232	−1.12721	−0.563606	−	0.826044i	$-0.690586\pi$
$7$	−2.98232	−1.12721	−0.563606	+	0.826044i	$0.690586\pi$
$8$	−2.82480	−0.998718
$9$	0	0
$10$	−1.09432	−0.346055
$11$	−2.17282	−0.655130	−0.327565	−	0.944829i	$-0.606228\pi$
$11$	−2.17282	−0.655130	−0.327565	+	0.944829i	$0.606228\pi$
$12$	0	0
$13$	−3.87407	−1.07447	−0.537236	−	0.843432i	$-0.680532\pi$
$13$	−3.87407	−1.07447	−0.537236	+	0.843432i	$0.680532\pi$
$14$	−2.60033	−0.694969
$15$	0	0
$16$	0.0165341	0.00413354
$17$	−0.672983	−0.163222	−0.0816112	−	0.996664i	$-0.526007\pi$
$17$	−0.672983	−0.163222	−0.0816112	+	0.996664i	$0.526007\pi$
$18$	0	0
$19$	4.12239	0.945741	0.472871	−	0.881132i	$-0.343218\pi$
$19$	4.12239	0.945741	0.472871	+	0.881132i	$0.343218\pi$
$20$	1.55600	0.347932
$21$	0	0
$22$	−1.89452	−0.403912
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.42478	−0.684956
$26$	−3.37786	−0.662453
$27$	0	0
$28$	3.69737	0.698737
$29$	−8.75589	−1.62593	−0.812964	−	0.582314i	$-0.802147\pi$
$29$	−8.75589	−1.62593	−0.812964	+	0.582314i	$0.802147\pi$
$30$	0	0
$31$	−5.03969	−0.905155	−0.452578	−	0.891725i	$-0.649495\pi$
$31$	−5.03969	−0.905155	−0.452578	+	0.891725i	$0.649495\pi$
$32$	5.66402	1.00127
$33$	0	0
$34$	−0.586785	−0.100633
$35$	3.74305	0.632691
$36$	0	0
$37$	8.92313	1.46695	0.733477	−	0.679714i	$-0.237896\pi$
$37$	8.92313	1.46695	0.733477	+	0.679714i	$0.237896\pi$
$38$	3.59438	0.583085
$39$	0	0
$40$	3.54535	0.560569
$41$	−5.60073	−0.874687	−0.437344	−	0.899294i	$-0.644081\pi$
$41$	−5.60073	−0.874687	−0.437344	+	0.899294i	$0.644081\pi$
$42$	0	0
$43$	−8.50086	−1.29637	−0.648184	−	0.761484i	$-0.724471\pi$
$43$	−8.50086	−1.29637	−0.648184	+	0.761484i	$0.724471\pi$
$44$	2.69378	0.406103
$45$	0	0
$46$	0	0
$47$	−8.72071	−1.27205	−0.636023	−	0.771670i	$-0.719422\pi$
$47$	−8.72071	−1.27205	−0.636023	+	0.771670i	$0.719422\pi$
$48$	0	0
$49$	1.89424	0.270606
$50$	−2.98612	−0.422301
$51$	0	0
$52$	4.80292	0.666045
$53$	−11.0723	−1.52089	−0.760446	−	0.649401i	$-0.775020\pi$
$53$	−11.0723	−1.52089	−0.760446	+	0.649401i	$0.775020\pi$
$54$	0	0
$55$	2.72706	0.367717
$56$	8.42447	1.12577
$57$	0	0
$58$	−7.63440	−1.00245
$59$	5.81811	0.757454	0.378727	−	0.925509i	$-0.376362\pi$
$59$	5.81811	0.757454	0.378727	+	0.925509i	$0.376362\pi$
$60$	0	0
$61$	4.46592	0.571803	0.285901	−	0.958259i	$-0.407707\pi$
$61$	4.46592	0.571803	0.285901	+	0.958259i	$0.407707\pi$
$62$	−4.39419	−0.558062
$63$	0	0
$64$	4.90548	0.613185
$65$	4.86226	0.603089
$66$	0	0
$67$	5.63671	0.688634	0.344317	−	0.938854i	$-0.388111\pi$
$67$	5.63671	0.688634	0.344317	+	0.938854i	$0.388111\pi$
$68$	0.834339	0.101178
$69$	0	0
$70$	3.26362	0.390078
$71$	7.92300	0.940287	0.470144	−	0.882590i	$-0.344202\pi$
$71$	7.92300	0.940287	0.470144	+	0.882590i	$0.344202\pi$
$72$	0	0
$73$	6.38691	0.747531	0.373765	−	0.927523i	$-0.378067\pi$
$73$	6.38691	0.747531	0.373765	+	0.927523i	$0.378067\pi$
$74$	7.78022	0.904433
$75$	0	0
$76$	−5.11078	−0.586247
$77$	6.48005	0.738470
$78$	0	0
$79$	7.19156	0.809114	0.404557	−	0.914513i	$-0.367426\pi$
$79$	7.19156	0.809114	0.404557	+	0.914513i	$0.367426\pi$
$80$	−0.0207516	−0.00232010
$81$	0	0
$82$	−4.88337	−0.539278
$83$	−11.9087	−1.30715	−0.653575	−	0.756862i	$-0.726732\pi$
$83$	−11.9087	−1.30715	−0.653575	+	0.756862i	$0.726732\pi$
$84$	0	0
$85$	0.844646	0.0916148
$86$	−7.41203	−0.799260
$87$	0	0
$88$	6.13778	0.654290
$89$	−0.361929	−0.0383644	−0.0191822	−	0.999816i	$-0.506106\pi$
$89$	−0.361929	−0.0383644	−0.0191822	+	0.999816i	$0.506106\pi$
$90$	0	0
$91$	11.5537	1.21116
$92$	0	0
$93$	0	0
$94$	−7.60373	−0.784264
$95$	−5.17392	−0.530833
$96$	0	0
$97$	12.2591	1.24472	0.622361	−	0.782731i	$-0.286174\pi$
$97$	12.2591	1.24472	0.622361	+	0.782731i	$0.286174\pi$
$98$	1.65162	0.166839
$99$	0	0
$100$	4.24591	0.424591
$101$	15.9040	1.58251	0.791253	−	0.611489i	$-0.209429\pi$
$101$	15.9040	1.58251	0.791253	+	0.611489i	$0.209429\pi$
$102$	0	0
$103$	18.6867	1.84125	0.920625	−	0.390447i	$-0.127679\pi$
$103$	18.6867	1.84125	0.920625	+	0.390447i	$0.127679\pi$
$104$	10.9435	1.07310
$105$	0	0
$106$	−9.65408	−0.937688
$107$	−6.09854	−0.589569	−0.294784	−	0.955564i	$-0.595248\pi$
$107$	−6.09854	−0.589569	−0.294784	+	0.955564i	$0.595248\pi$
$108$	0	0
$109$	9.11013	0.872592	0.436296	−	0.899803i	$-0.356290\pi$
$109$	9.11013	0.872592	0.436296	+	0.899803i	$0.356290\pi$
$110$	2.37777	0.226711
$111$	0	0
$112$	−0.0493101	−0.00465937
$113$	9.42612	0.886735	0.443367	−	0.896340i	$-0.353784\pi$
$113$	9.42612	0.886735	0.443367	+	0.896340i	$0.353784\pi$
$114$	0	0
$115$	0	0
$116$	10.8552	1.00788
$117$	0	0
$118$	5.07291	0.466999
$119$	2.00705	0.183986
$120$	0	0
$121$	−6.27885	−0.570805
$122$	3.89391	0.352538
$123$	0	0
$124$	6.24802	0.561089
$125$	10.5738	0.945746
$126$	0	0
$127$	5.86595	0.520519	0.260259	−	0.965539i	$-0.416192\pi$
$127$	5.86595	0.520519	0.260259	+	0.965539i	$0.416192\pi$
$128$	−7.05087	−0.623215
$129$	0	0
$130$	4.23948	0.371827
$131$	−8.62635	−0.753687	−0.376844	−	0.926277i	$-0.622991\pi$
$131$	−8.62635	−0.753687	−0.376844	+	0.926277i	$0.622991\pi$
$132$	0	0
$133$	−12.2943	−1.06605
$134$	4.91474	0.424569
$135$	0	0
$136$	1.90104	0.163013
$137$	−1.98038	−0.169195	−0.0845977	−	0.996415i	$-0.526960\pi$
$137$	−1.98038	−0.169195	−0.0845977	+	0.996415i	$0.526960\pi$
$138$	0	0
$139$	−5.82311	−0.493910	−0.246955	−	0.969027i	$-0.579430\pi$
$139$	−5.82311	−0.493910	−0.246955	+	0.969027i	$0.579430\pi$
$140$	−4.64049	−0.392193
$141$	0	0
$142$	6.90819	0.579723
$143$	8.41765	0.703919
$144$	0	0
$145$	10.9893	0.912614
$146$	5.56885	0.460881
$147$	0	0
$148$	−11.0626	−0.909337
$149$	17.4261	1.42760	0.713800	−	0.700349i	$-0.246972\pi$
$149$	17.4261	1.42760	0.713800	+	0.700349i	$0.246972\pi$
$150$	0	0
$151$	13.0457	1.06165	0.530823	−	0.847483i	$-0.321883\pi$
$151$	13.0457	1.06165	0.530823	+	0.847483i	$0.321883\pi$
$152$	−11.6449	−0.944529
$153$	0	0
$154$	5.65006	0.455295
$155$	6.32521	0.508053
$156$	0	0
$157$	8.78611	0.701208	0.350604	−	0.936524i	$-0.385976\pi$
$157$	8.78611	0.701208	0.350604	+	0.936524i	$0.385976\pi$
$158$	6.27044	0.498849
$159$	0	0
$160$	−7.10879	−0.561999
$161$	0	0
$162$	0	0
$163$	−8.34009	−0.653246	−0.326623	−	0.945155i	$-0.605911\pi$
$163$	−8.34009	−0.653246	−0.326623	+	0.945155i	$0.605911\pi$
$164$	6.94357	0.542202
$165$	0	0
$166$	−10.3834	−0.805908
$167$	−11.8008	−0.913176	−0.456588	−	0.889678i	$-0.650929\pi$
$167$	−11.8008	−0.913176	−0.456588	+	0.889678i	$0.650929\pi$
$168$	0	0
$169$	2.00840	0.154492
$170$	0.736461	0.0564840
$171$	0	0
$172$	10.5390	0.803594
$173$	−15.4544	−1.17498	−0.587489	−	0.809232i	$-0.699884\pi$
$173$	−15.4544	−1.17498	−0.587489	+	0.809232i	$0.699884\pi$
$174$	0	0
$175$	10.2138	0.772090
$176$	−0.0359257	−0.00270800
$177$	0	0
$178$	−0.315572	−0.0236531
$179$	18.4577	1.37960	0.689798	−	0.724002i	$-0.257699\pi$
$179$	18.4577	1.37960	0.689798	+	0.724002i	$0.257699\pi$
$180$	0	0
$181$	6.96026	0.517352	0.258676	−	0.965964i	$-0.416714\pi$
$181$	6.96026	0.517352	0.258676	+	0.965964i	$0.416714\pi$
$182$	10.0739	0.746725
$183$	0	0
$184$	0	0
$185$	−11.1992	−0.823384
$186$	0	0
$187$	1.46227	0.106932
$188$	10.8116	0.788517
$189$	0	0
$190$	−4.51123	−0.327279
$191$	−14.9768	−1.08368	−0.541842	−	0.840481i	$-0.682273\pi$
$191$	−14.9768	−1.08368	−0.541842	+	0.840481i	$0.682273\pi$
$192$	0	0
$193$	17.6171	1.26810	0.634052	−	0.773291i	$-0.281391\pi$
$193$	17.6171	1.26810	0.634052	+	0.773291i	$0.281391\pi$
$194$	10.6889	0.767418
$195$	0	0
$196$	−2.34841	−0.167744
$197$	−18.3679	−1.30866	−0.654330	−	0.756209i	$-0.727049\pi$
$197$	−18.3679	−1.30866	−0.654330	+	0.756209i	$0.727049\pi$
$198$	0	0
$199$	−18.1695	−1.28800	−0.644002	−	0.765024i	$-0.722727\pi$
$199$	−18.1695	−1.28800	−0.644002	+	0.765024i	$0.722727\pi$
$200$	9.67432	0.684078
$201$	0	0
$202$	13.8669	0.975675
$203$	26.1129	1.83276
$204$	0	0
$205$	7.02936	0.490952
$206$	16.2932	1.13520
$207$	0	0
$208$	−0.0640544	−0.00444137
$209$	−8.95721	−0.619583
$210$	0	0
$211$	−10.5646	−0.727295	−0.363647	−	0.931537i	$-0.618469\pi$
$211$	−10.5646	−0.727295	−0.363647	+	0.931537i	$0.618469\pi$
$212$	13.7270	0.942772
$213$	0	0
$214$	−5.31742	−0.363491
$215$	10.6692	0.727636
$216$	0	0
$217$	15.0300	1.02030
$218$	7.94327	0.537986
$219$	0	0
$220$	−3.38090	−0.227941
$221$	2.60718	0.175378
$222$	0	0
$223$	−6.29166	−0.421321	−0.210660	−	0.977559i	$-0.567561\pi$
$223$	−6.29166	−0.421321	−0.210660	+	0.977559i	$0.567561\pi$
$224$	−16.8919	−1.12864
$225$	0	0
$226$	8.21879	0.546706
$227$	7.02645	0.466362	0.233181	−	0.972433i	$-0.425087\pi$
$227$	7.02645	0.466362	0.233181	+	0.972433i	$0.425087\pi$
$228$	0	0
$229$	0.341930	0.0225954	0.0112977	−	0.999936i	$-0.496404\pi$
$229$	0.341930	0.0225954	0.0112977	+	0.999936i	$0.496404\pi$
$230$	0	0
$231$	0	0
$232$	24.7336	1.62384
$233$	11.0494	0.723872	0.361936	−	0.932203i	$-0.382116\pi$
$233$	11.0494	0.723872	0.361936	+	0.932203i	$0.382116\pi$
$234$	0	0
$235$	10.9452	0.713984
$236$	−7.21307	−0.469531
$237$	0	0
$238$	1.74998	0.113434
$239$	5.11750	0.331024	0.165512	−	0.986208i	$-0.447072\pi$
$239$	5.11750	0.331024	0.165512	+	0.986208i	$0.447072\pi$
$240$	0	0
$241$	−16.6023	−1.06945	−0.534725	−	0.845026i	$-0.679585\pi$
$241$	−16.6023	−1.06945	−0.534725	+	0.845026i	$0.679585\pi$
$242$	−5.47463	−0.351923
$243$	0	0
$244$	−5.53668	−0.354450
$245$	−2.37742	−0.151888
$246$	0	0
$247$	−15.9704	−1.01617
$248$	14.2361	0.903995
$249$	0	0
$250$	9.21943	0.583088
$251$	−16.2243	−1.02407	−0.512035	−	0.858965i	$-0.671108\pi$
$251$	−16.2243	−1.02407	−0.512035	+	0.858965i	$0.671108\pi$
$252$	0	0
$253$	0	0
$254$	5.11462	0.320920
$255$	0	0
$256$	−15.9587	−0.997421
$257$	−17.6391	−1.10030	−0.550150	−	0.835066i	$-0.685429\pi$
$257$	−17.6391	−1.10030	−0.550150	+	0.835066i	$0.685429\pi$
$258$	0	0
$259$	−26.6116	−1.65357
$260$	−6.02804	−0.373843
$261$	0	0
$262$	−7.52145	−0.464677
$263$	−16.7409	−1.03229	−0.516143	−	0.856502i	$-0.672633\pi$
$263$	−16.7409	−1.03229	−0.516143	+	0.856502i	$0.672633\pi$
$264$	0	0
$265$	13.8966	0.853659
$266$	−10.7196	−0.657261
$267$	0	0
$268$	−6.98818	−0.426871
$269$	−7.91874	−0.482814	−0.241407	−	0.970424i	$-0.577609\pi$
$269$	−7.91874	−0.482814	−0.241407	+	0.970424i	$0.577609\pi$
$270$	0	0
$271$	−2.85867	−0.173652	−0.0868260	−	0.996223i	$-0.527672\pi$
$271$	−2.85867	−0.173652	−0.0868260	+	0.996223i	$0.527672\pi$
$272$	−0.0111272	−0.000674685 0
$273$	0	0
$274$	−1.72673	−0.104315
$275$	7.44143	0.448735
$276$	0	0
$277$	−3.94930	−0.237290	−0.118645	−	0.992937i	$-0.537855\pi$
$277$	−3.94930	−0.237290	−0.118645	+	0.992937i	$0.537855\pi$
$278$	−5.07727	−0.304514
$279$	0	0
$280$	−10.5734	−0.631879
$281$	−12.5613	−0.749344	−0.374672	−	0.927158i	$-0.622245\pi$
$281$	−12.5613	−0.749344	−0.374672	+	0.927158i	$0.622245\pi$
$282$	0	0
$283$	−15.4248	−0.916908	−0.458454	−	0.888718i	$-0.651597\pi$
$283$	−15.4248	−0.916908	−0.458454	+	0.888718i	$0.651597\pi$
$284$	−9.82264	−0.582866
$285$	0	0
$286$	7.33949	0.433993
$287$	16.7032	0.985958
$288$	0	0
$289$	−16.5471	−0.973358
$290$	9.58177	0.562661
$291$	0	0
$292$	−7.91824	−0.463380
$293$	20.3585	1.18935	0.594677	−	0.803964i	$-0.297280\pi$
$293$	20.3585	1.18935	0.594677	+	0.803964i	$0.297280\pi$
$294$	0	0
$295$	−7.30219	−0.425150
$296$	−25.2061	−1.46507
$297$	0	0
$298$	15.1941	0.880170
$299$	0	0
$300$	0	0
$301$	25.3523	1.46128
$302$	11.3748	0.654545
$303$	0	0
$304$	0.0681602	0.00390926
$305$	−5.60508	−0.320946
$306$	0	0
$307$	−4.64072	−0.264860	−0.132430	−	0.991192i	$-0.542278\pi$
$307$	−4.64072	−0.264860	−0.132430	+	0.991192i	$0.542278\pi$
$308$	−8.03372	−0.457764
$309$	0	0
$310$	5.51505	0.313234
$311$	10.4856	0.594585	0.297293	−	0.954786i	$-0.403916\pi$
$311$	10.4856	0.594585	0.297293	+	0.954786i	$0.403916\pi$
$312$	0	0
$313$	−30.9855	−1.75140	−0.875702	−	0.482852i	$-0.839601\pi$
$313$	−30.9855	−1.75140	−0.875702	+	0.482852i	$0.839601\pi$
$314$	7.66075	0.432321
$315$	0	0
$316$	−8.91583	−0.501554
$317$	−13.4374	−0.754719	−0.377360	−	0.926067i	$-0.623168\pi$
$317$	−13.4374	−0.754719	−0.377360	+	0.926067i	$0.623168\pi$
$318$	0	0
$319$	19.0250	1.06519
$320$	−6.15676	−0.344174
$321$	0	0
$322$	0	0
$323$	−2.77430	−0.154366
$324$	0	0
$325$	13.2678	0.735966
$326$	−7.27186	−0.402751
$327$	0	0
$328$	15.8210	0.873566
$329$	26.0080	1.43386
$330$	0	0
$331$	2.68810	0.147751	0.0738756	−	0.997267i	$-0.476463\pi$
$331$	2.68810	0.147751	0.0738756	+	0.997267i	$0.476463\pi$
$332$	14.7640	0.810278
$333$	0	0
$334$	−10.2893	−0.563008
$335$	−7.07451	−0.386522
$336$	0	0
$337$	10.2419	0.557912	0.278956	−	0.960304i	$-0.410012\pi$
$337$	10.2419	0.557912	0.278956	+	0.960304i	$0.410012\pi$
$338$	1.75115	0.0952502
$339$	0	0
$340$	−1.04716	−0.0567902
$341$	10.9503	0.592994
$342$	0	0
$343$	15.2270	0.822181
$344$	24.0132	1.29471
$345$	0	0
$346$	−13.4750	−0.724419
$347$	26.4171	1.41815	0.709073	−	0.705135i	$-0.249114\pi$
$347$	26.4171	1.41815	0.709073	+	0.705135i	$0.249114\pi$
$348$	0	0
$349$	−10.1713	−0.544459	−0.272229	−	0.962232i	$-0.587761\pi$
$349$	−10.1713	−0.544459	−0.272229	+	0.962232i	$0.587761\pi$
$350$	8.90557	0.476023
$351$	0	0
$352$	−12.3069	−0.655960
$353$	5.80219	0.308819	0.154410	−	0.988007i	$-0.450652\pi$
$353$	5.80219	0.308819	0.154410	+	0.988007i	$0.450652\pi$
$354$	0	0
$355$	−9.94399	−0.527772
$356$	0.448706	0.0237814
$357$	0	0
$358$	16.0936	0.850573
$359$	25.1348	1.32656	0.663281	−	0.748370i	$-0.269163\pi$
$359$	25.1348	1.32656	0.663281	+	0.748370i	$0.269163\pi$
$360$	0	0
$361$	−2.00589	−0.105573
$362$	6.06876	0.318967
$363$	0	0
$364$	−14.3239	−0.750774
$365$	−8.01607	−0.419580
$366$	0	0
$367$	3.87312	0.202175	0.101087	−	0.994878i	$-0.467768\pi$
$367$	3.87312	0.202175	0.101087	+	0.994878i	$0.467768\pi$
$368$	0	0
$369$	0	0
$370$	−9.76479	−0.507647
$371$	33.0210	1.71437
$372$	0	0
$373$	−3.93817	−0.203911	−0.101955	−	0.994789i	$-0.532510\pi$
$373$	−3.93817	−0.203911	−0.101955	+	0.994789i	$0.532510\pi$
$374$	1.27498	0.0659275
$375$	0	0
$376$	24.6343	1.27042
$377$	33.9209	1.74701
$378$	0	0
$379$	−33.2476	−1.70782	−0.853908	−	0.520425i	$-0.825774\pi$
$379$	−33.2476	−1.70782	−0.853908	+	0.520425i	$0.825774\pi$
$380$	6.41444	0.329054
$381$	0	0
$382$	−13.0585	−0.668132
$383$	33.6095	1.71736	0.858682	−	0.512508i	$-0.171284\pi$
$383$	33.6095	1.71736	0.858682	+	0.512508i	$0.171284\pi$
$384$	0	0
$385$	−8.13297	−0.414494
$386$	15.3606	0.781834
$387$	0	0
$388$	−15.1983	−0.771579
$389$	8.16521	0.413993	0.206996	−	0.978342i	$-0.433631\pi$
$389$	8.16521	0.413993	0.206996	+	0.978342i	$0.433631\pi$
$390$	0	0
$391$	0	0
$392$	−5.35086	−0.270259
$393$	0	0
$394$	−16.0153	−0.806838
$395$	−9.02597	−0.454146
$396$	0	0
$397$	−17.3532	−0.870934	−0.435467	−	0.900205i	$-0.643417\pi$
$397$	−17.3532	−0.870934	−0.435467	+	0.900205i	$0.643417\pi$
$398$	−15.8423	−0.794104
$399$	0	0
$400$	−0.0566258	−0.00283129
$401$	−20.9122	−1.04430	−0.522152	−	0.852853i	$-0.674870\pi$
$401$	−20.9122	−1.04430	−0.522152	+	0.852853i	$0.674870\pi$
$402$	0	0
$403$	19.5241	0.972565
$404$	−19.7172	−0.980966
$405$	0	0
$406$	22.7682	1.12997
$407$	−19.3884	−0.961045
$408$	0	0
$409$	11.8065	0.583792	0.291896	−	0.956450i	$-0.405714\pi$
$409$	11.8065	0.583792	0.291896	+	0.956450i	$0.405714\pi$
$410$	6.12901	0.302690
$411$	0	0
$412$	−23.1670	−1.14136
$413$	−17.3515	−0.853810
$414$	0	0
$415$	14.9464	0.733688
$416$	−21.9428	−1.07583
$417$	0	0
$418$	−7.80994	−0.381997
$419$	28.2557	1.38038	0.690191	−	0.723627i	$-0.257526\pi$
$419$	28.2557	1.38038	0.690191	+	0.723627i	$0.257526\pi$
$420$	0	0
$421$	10.3907	0.506412	0.253206	−	0.967412i	$-0.418515\pi$
$421$	10.3907	0.506412	0.253206	+	0.967412i	$0.418515\pi$
$422$	−9.21142	−0.448405
$423$	0	0
$424$	31.2769	1.51894
$425$	2.30482	0.111800
$426$	0	0
$427$	−13.3188	−0.644543
$428$	7.56074	0.365462
$429$	0	0
$430$	9.30268	0.448615
$431$	9.29261	0.447609	0.223805	−	0.974634i	$-0.428152\pi$
$431$	9.29261	0.447609	0.223805	+	0.974634i	$0.428152\pi$
$432$	0	0
$433$	−4.36317	−0.209681	−0.104840	−	0.994489i	$-0.533433\pi$
$433$	−4.36317	−0.209681	−0.104840	+	0.994489i	$0.533433\pi$
$434$	13.1049	0.629054
$435$	0	0
$436$	−11.2944	−0.540903
$437$	0	0
$438$	0	0
$439$	7.29588	0.348213	0.174107	−	0.984727i	$-0.444296\pi$
$439$	7.29588	0.348213	0.174107	+	0.984727i	$0.444296\pi$
$440$	−7.70340	−0.367245
$441$	0	0
$442$	2.27324	0.108127
$443$	−21.0222	−0.998794	−0.499397	−	0.866373i	$-0.666445\pi$
$443$	−21.0222	−0.998794	−0.499397	+	0.866373i	$0.666445\pi$
$444$	0	0
$445$	0.454249	0.0215335
$446$	−5.48580	−0.259760
$447$	0	0
$448$	−14.6297	−0.691189
$449$	−10.7789	−0.508688	−0.254344	−	0.967114i	$-0.581859\pi$
$449$	−10.7789	−0.508688	−0.254344	+	0.967114i	$0.581859\pi$
$450$	0	0
$451$	12.1694	0.573034
$452$	−11.6861	−0.549670
$453$	0	0
$454$	6.12648	0.287530
$455$	−14.5008	−0.679809
$456$	0	0
$457$	−3.32365	−0.155474	−0.0777369	−	0.996974i	$-0.524769\pi$
$457$	−3.32365	−0.155474	−0.0777369	+	0.996974i	$0.524769\pi$
$458$	0.298134	0.0139309
$459$	0	0
$460$	0	0
$461$	16.7849	0.781749	0.390874	−	0.920444i	$-0.372173\pi$
$461$	16.7849	0.781749	0.390874	+	0.920444i	$0.372173\pi$
$462$	0	0
$463$	−14.0457	−0.652761	−0.326380	−	0.945239i	$-0.605829\pi$
$463$	−14.0457	−0.652761	−0.326380	+	0.945239i	$0.605829\pi$
$464$	−0.144771	−0.00672083
$465$	0	0
$466$	9.63418	0.446295
$467$	20.6413	0.955165	0.477582	−	0.878587i	$-0.341513\pi$
$467$	20.6413	0.955165	0.477582	+	0.878587i	$0.341513\pi$
$468$	0	0
$469$	−16.8105	−0.776236
$470$	9.54327	0.440198
$471$	0	0
$472$	−16.4350	−0.756482
$473$	18.4708	0.849290
$474$	0	0
$475$	−14.1183	−0.647791
$476$	−2.48827	−0.114049
$477$	0	0
$478$	4.46203	0.204089
$479$	2.86558	0.130932	0.0654658	−	0.997855i	$-0.479147\pi$
$479$	2.86558	0.130932	0.0654658	+	0.997855i	$0.479147\pi$
$480$	0	0
$481$	−34.5688	−1.57620
$482$	−14.4758	−0.659357
$483$	0	0
$484$	7.78428	0.353831
$485$	−15.3861	−0.698647
$486$	0	0
$487$	19.2582	0.872670	0.436335	−	0.899784i	$-0.356276\pi$
$487$	19.2582	0.872670	0.436335	+	0.899784i	$0.356276\pi$
$488$	−12.6153	−0.571070
$489$	0	0
$490$	−2.07291	−0.0936447
$491$	16.7684	0.756746	0.378373	−	0.925653i	$-0.376484\pi$
$491$	16.7684	0.756746	0.378373	+	0.925653i	$0.376484\pi$
$492$	0	0
$493$	5.89256	0.265388
$494$	−13.9249	−0.626510
$495$	0	0
$496$	−0.0833270	−0.00374149
$497$	−23.6289	−1.05990
$498$	0	0
$499$	−20.5251	−0.918830	−0.459415	−	0.888222i	$-0.651941\pi$
$499$	−20.5251	−0.918830	−0.459415	+	0.888222i	$0.651941\pi$
$500$	−13.1089	−0.586250
$501$	0	0
$502$	−14.1462	−0.631377
$503$	11.3484	0.505998	0.252999	−	0.967467i	$-0.418583\pi$
$503$	11.3484	0.505998	0.252999	+	0.967467i	$0.418583\pi$
$504$	0	0
$505$	−19.9608	−0.888242
$506$	0	0
$507$	0	0
$508$	−7.27238	−0.322660
$509$	−1.84023	−0.0815665	−0.0407833	−	0.999168i	$-0.512985\pi$
$509$	−1.84023	−0.0815665	−0.0407833	+	0.999168i	$0.512985\pi$
$510$	0	0
$511$	−19.0478	−0.842625
$512$	0.187061	0.00826701
$513$	0	0
$514$	−15.3799	−0.678376
$515$	−23.4532	−1.03347
$516$	0	0
$517$	18.9485	0.833355
$518$	−23.2031	−1.01949
$519$	0	0
$520$	−13.7349	−0.602316
$521$	−0.998799	−0.0437582	−0.0218791	−	0.999761i	$-0.506965\pi$
$521$	−0.998799	−0.0437582	−0.0218791	+	0.999761i	$0.506965\pi$
$522$	0	0
$523$	−26.9172	−1.17700	−0.588502	−	0.808495i	$-0.700282\pi$
$523$	−26.9172	−1.17700	−0.588502	+	0.808495i	$0.700282\pi$
$524$	10.6946	0.467196
$525$	0	0
$526$	−14.5966	−0.636444
$527$	3.39163	0.147742
$528$	0	0
$529$	0	0
$530$	12.1166	0.526313
$531$	0	0
$532$	15.2420	0.660825
$533$	21.6976	0.939828
$534$	0	0
$535$	7.65415	0.330918
$536$	−15.9226	−0.687751
$537$	0	0
$538$	−6.90448	−0.297673
$539$	−4.11585	−0.177282
$540$	0	0
$541$	43.6033	1.87465	0.937325	−	0.348456i	$-0.113294\pi$
$541$	43.6033	1.87465	0.937325	+	0.348456i	$0.113294\pi$
$542$	−2.49252	−0.107063
$543$	0	0
$544$	−3.81179	−0.163429
$545$	−11.4339	−0.489776
$546$	0	0
$547$	−9.46390	−0.404647	−0.202324	−	0.979319i	$-0.564849\pi$
$547$	−9.46390	−0.404647	−0.202324	+	0.979319i	$0.564849\pi$
$548$	2.45520	0.104881
$549$	0	0
$550$	6.48830	0.276662
$551$	−36.0952	−1.53771
$552$	0	0
$553$	−21.4476	−0.912043
$554$	−3.44346	−0.146299
$555$	0	0
$556$	7.21928	0.306166
$557$	−36.2655	−1.53662	−0.768309	−	0.640079i	$-0.778902\pi$
$557$	−36.2655	−1.53662	−0.768309	+	0.640079i	$0.778902\pi$
$558$	0	0
$559$	32.9329	1.39291
$560$	0.0618881	0.00261525
$561$	0	0
$562$	−10.9524	−0.461999
$563$	11.1898	0.471594	0.235797	−	0.971802i	$-0.424230\pi$
$563$	11.1898	0.471594	0.235797	+	0.971802i	$0.424230\pi$
$564$	0	0
$565$	−11.8305	−0.497714
$566$	−13.4491	−0.565309
$567$	0	0
$568$	−22.3809	−0.939082
$569$	2.31335	0.0969807	0.0484904	−	0.998824i	$-0.484559\pi$
$569$	2.31335	0.0969807	0.0484904	+	0.998824i	$0.484559\pi$
$570$	0	0
$571$	4.02140	0.168290	0.0841452	−	0.996454i	$-0.473184\pi$
$571$	4.02140	0.168290	0.0841452	+	0.996454i	$0.473184\pi$
$572$	−10.4359	−0.436346
$573$	0	0
$574$	14.5638	0.607880
$575$	0	0
$576$	0	0
$577$	−31.9515	−1.33016	−0.665079	−	0.746773i	$-0.731602\pi$
$577$	−31.9515	−1.33016	−0.665079	+	0.746773i	$0.731602\pi$
$578$	−14.4277	−0.600112
$579$	0	0
$580$	−13.6241	−0.565712
$581$	35.5156	1.47344
$582$	0	0
$583$	24.0580	0.996382
$584$	−18.0417	−0.746572
$585$	0	0
$586$	17.7509	0.733282
$587$	20.0703	0.828390	0.414195	−	0.910188i	$-0.364063\pi$
$587$	20.0703	0.828390	0.414195	+	0.910188i	$0.364063\pi$
$588$	0	0
$589$	−20.7756	−0.856043
$590$	−6.36689	−0.262121
$591$	0	0
$592$	0.147536	0.00606371
$593$	−11.2694	−0.462778	−0.231389	−	0.972861i	$-0.574327\pi$
$593$	−11.2694	−0.462778	−0.231389	+	0.972861i	$0.574327\pi$
$594$	0	0
$595$	−2.51901	−0.103269
$596$	−21.6042	−0.884942
$597$	0	0
$598$	0	0
$599$	−31.2141	−1.27537	−0.637687	−	0.770295i	$-0.720109\pi$
$599$	−31.2141	−1.27537	−0.637687	+	0.770295i	$0.720109\pi$
$600$	0	0
$601$	31.9065	1.30149	0.650746	−	0.759296i	$-0.274456\pi$
$601$	31.9065	1.30149	0.650746	+	0.759296i	$0.274456\pi$
$602$	22.1051	0.900935
$603$	0	0
$604$	−16.1736	−0.658094
$605$	7.88045	0.320386
$606$	0	0
$607$	20.9489	0.850290	0.425145	−	0.905125i	$-0.360223\pi$
$607$	20.9489	0.850290	0.425145	+	0.905125i	$0.360223\pi$
$608$	23.3493	0.946939
$609$	0	0
$610$	−4.88716	−0.197875
$611$	33.7846	1.36678
$612$	0	0
$613$	19.7729	0.798619	0.399309	−	0.916816i	$-0.369250\pi$
$613$	19.7729	0.798619	0.399309	+	0.916816i	$0.369250\pi$
$614$	−4.04632	−0.163296
$615$	0	0
$616$	−18.3048	−0.737523
$617$	−35.6781	−1.43635	−0.718173	−	0.695864i	$-0.755022\pi$
$617$	−35.6781	−1.43635	−0.718173	+	0.695864i	$0.755022\pi$
$618$	0	0
$619$	14.6223	0.587718	0.293859	−	0.955849i	$-0.405060\pi$
$619$	14.6223	0.587718	0.293859	+	0.955849i	$0.405060\pi$
$620$	−7.84175	−0.314932
$621$	0	0
$622$	9.14258	0.366584
$623$	1.07939	0.0432448
$624$	0	0
$625$	3.85300	0.154120
$626$	−27.0168	−1.07981
$627$	0	0
$628$	−10.8927	−0.434666
$629$	−6.00511	−0.239440
$630$	0	0
$631$	−17.2411	−0.686356	−0.343178	−	0.939270i	$-0.611503\pi$
$631$	−17.2411	−0.686356	−0.343178	+	0.939270i	$0.611503\pi$
$632$	−20.3147	−0.808077
$633$	0	0
$634$	−11.7163	−0.465313
$635$	−7.36223	−0.292161
$636$	0	0
$637$	−7.33842	−0.290759
$638$	16.5882	0.656732
$639$	0	0
$640$	8.84939	0.349803
$641$	16.8452	0.665347	0.332673	−	0.943042i	$-0.392049\pi$
$641$	16.8452	0.665347	0.332673	+	0.943042i	$0.392049\pi$
$642$	0	0
$643$	−22.2079	−0.875796	−0.437898	−	0.899025i	$-0.644277\pi$
$643$	−22.2079	−0.875796	−0.437898	+	0.899025i	$0.644277\pi$
$644$	0	0
$645$	0	0
$646$	−2.41896	−0.0951726
$647$	−5.40423	−0.212462	−0.106231	−	0.994341i	$-0.533878\pi$
$647$	−5.40423	−0.212462	−0.106231	+	0.994341i	$0.533878\pi$
$648$	0	0
$649$	−12.6417	−0.496230
$650$	11.5684	0.453751
$651$	0	0
$652$	10.3397	0.404935
$653$	20.5805	0.805379	0.402689	−	0.915337i	$-0.368075\pi$
$653$	20.5805	0.805379	0.402689	+	0.915337i	$0.368075\pi$
$654$	0	0
$655$	10.8267	0.423036
$656$	−0.0926033	−0.00361555
$657$	0	0
$658$	22.6768	0.884032
$659$	25.6384	0.998730	0.499365	−	0.866392i	$-0.333567\pi$
$659$	25.6384	0.998730	0.499365	+	0.866392i	$0.333567\pi$
$660$	0	0
$661$	47.5943	1.85120	0.925601	−	0.378501i	$-0.123560\pi$
$661$	47.5943	1.85120	0.925601	+	0.378501i	$0.123560\pi$
$662$	2.34380	0.0910943
$663$	0	0
$664$	33.6397	1.30547
$665$	15.4303	0.598362
$666$	0	0
$667$	0	0
$668$	14.6302	0.566061
$669$	0	0
$670$	−6.16838	−0.238305
$671$	−9.70365	−0.374605
$672$	0	0
$673$	15.1619	0.584447	0.292223	−	0.956350i	$-0.405605\pi$
$673$	15.1619	0.584447	0.292223	+	0.956350i	$0.405605\pi$
$674$	8.93009	0.343974
$675$	0	0
$676$	−2.48993	−0.0957667
$677$	−27.1287	−1.04264	−0.521320	−	0.853361i	$-0.674560\pi$
$677$	−27.1287	−1.04264	−0.521320	+	0.853361i	$0.674560\pi$
$678$	0	0
$679$	−36.5605	−1.40306
$680$	−2.38596	−0.0914973
$681$	0	0
$682$	9.54778	0.365603
$683$	20.6439	0.789918	0.394959	−	0.918699i	$-0.370759\pi$
$683$	20.6439	0.789918	0.394959	+	0.918699i	$0.370759\pi$
$684$	0	0
$685$	2.48553	0.0949673
$686$	13.2767	0.506906
$687$	0	0
$688$	−0.140554	−0.00535858
$689$	42.8947	1.63416
$690$	0	0
$691$	−17.3303	−0.659276	−0.329638	−	0.944107i	$-0.606927\pi$
$691$	−17.3303	−0.659276	−0.329638	+	0.944107i	$0.606927\pi$
$692$	19.1598	0.728347
$693$	0	0
$694$	23.0335	0.874341
$695$	7.30847	0.277226
$696$	0	0
$697$	3.76920	0.142769
$698$	−8.86855	−0.335679
$699$	0	0
$700$	−12.6627	−0.478604
$701$	−30.9574	−1.16925	−0.584623	−	0.811305i	$-0.698758\pi$
$701$	−30.9574	−1.16925	−0.584623	+	0.811305i	$0.698758\pi$
$702$	0	0
$703$	36.7846	1.38736
$704$	−10.6587	−0.401716
$705$	0	0
$706$	5.05902	0.190399
$707$	−47.4308	−1.78382
$708$	0	0
$709$	−2.45353	−0.0921444	−0.0460722	−	0.998938i	$-0.514670\pi$
$709$	−2.45353	−0.0921444	−0.0460722	+	0.998938i	$0.514670\pi$
$710$	−8.67033	−0.325392
$711$	0	0
$712$	1.02238	0.0383152
$713$	0	0
$714$	0	0
$715$	−10.5648	−0.395102
$716$	−22.8832	−0.855186
$717$	0	0
$718$	21.9154	0.817876
$719$	36.2875	1.35330	0.676648	−	0.736306i	$-0.263432\pi$
$719$	36.2875	1.35330	0.676648	+	0.736306i	$0.263432\pi$
$720$	0	0
$721$	−55.7296	−2.07548
$722$	−1.74897	−0.0650898
$723$	0	0
$724$	−8.62906	−0.320697
$725$	29.9870	1.11369
$726$	0	0
$727$	33.8569	1.25568	0.627841	−	0.778341i	$-0.283939\pi$
$727$	33.8569	1.25568	0.627841	+	0.778341i	$0.283939\pi$
$728$	−32.6369	−1.20961
$729$	0	0
$730$	−6.98934	−0.258687
$731$	5.72093	0.211596
$732$	0	0
$733$	−6.53573	−0.241403	−0.120701	−	0.992689i	$-0.538514\pi$
$733$	−6.53573	−0.241403	−0.120701	+	0.992689i	$0.538514\pi$
$734$	3.37703	0.124649
$735$	0	0
$736$	0	0
$737$	−12.2476	−0.451144
$738$	0	0
$739$	40.3922	1.48585	0.742926	−	0.669373i	$-0.233437\pi$
$739$	40.3922	1.48585	0.742926	+	0.669373i	$0.233437\pi$
$740$	13.8844	0.510400
$741$	0	0
$742$	28.7916	1.05697
$743$	5.76900	0.211644	0.105822	−	0.994385i	$-0.466253\pi$
$743$	5.76900	0.211644	0.105822	+	0.994385i	$0.466253\pi$
$744$	0	0
$745$	−21.8711	−0.801295
$746$	−3.43375	−0.125719
$747$	0	0
$748$	−1.81287	−0.0662850
$749$	18.1878	0.664569
$750$	0	0
$751$	7.43055	0.271145	0.135572	−	0.990767i	$-0.456713\pi$
$751$	7.43055	0.271145	0.135572	+	0.990767i	$0.456713\pi$
$752$	−0.144189	−0.00525805
$753$	0	0
$754$	29.5762	1.07710
$755$	−16.3734	−0.595889
$756$	0	0
$757$	9.11587	0.331322	0.165661	−	0.986183i	$-0.447024\pi$
$757$	9.11587	0.331322	0.165661	+	0.986183i	$0.447024\pi$
$758$	−28.9891	−1.05293
$759$	0	0
$760$	14.6153	0.530153
$761$	−10.8763	−0.394264	−0.197132	−	0.980377i	$-0.563163\pi$
$761$	−10.8763	−0.394264	−0.197132	+	0.980377i	$0.563163\pi$
$762$	0	0
$763$	−27.1693	−0.983596
$764$	18.5677	0.671755
$765$	0	0
$766$	29.3047	1.05882
$767$	−22.5398	−0.813863
$768$	0	0
$769$	−52.3842	−1.88902	−0.944512	−	0.328477i	$-0.893465\pi$
$769$	−52.3842	−1.88902	−0.944512	+	0.328477i	$0.893465\pi$
$770$	−7.09127	−0.255552
$771$	0	0
$772$	−21.8410	−0.786073
$773$	−25.1490	−0.904545	−0.452273	−	0.891880i	$-0.649387\pi$
$773$	−25.1490	−0.904545	−0.452273	+	0.891880i	$0.649387\pi$
$774$	0	0
$775$	17.2598	0.619991
$776$	−34.6295	−1.24313
$777$	0	0
$778$	7.11938	0.255242
$779$	−23.0884	−0.827228
$780$	0	0
$781$	−17.2153	−0.616010
$782$	0	0
$783$	0	0
$784$	0.0313197	0.00111856
$785$	−11.0273	−0.393580
$786$	0	0
$787$	2.11021	0.0752210	0.0376105	−	0.999292i	$-0.488025\pi$
$787$	2.11021	0.0752210	0.0376105	+	0.999292i	$0.488025\pi$
$788$	22.7718	0.811213
$789$	0	0
$790$	−7.86989	−0.279998
$791$	−28.1117	−0.999538
$792$	0	0
$793$	−17.3013	−0.614387
$794$	−15.1306	−0.536964
$795$	0	0
$796$	22.5259	0.798410
$797$	−29.6909	−1.05171	−0.525853	−	0.850575i	$-0.676254\pi$
$797$	−29.6909	−1.05171	−0.525853	+	0.850575i	$0.676254\pi$
$798$	0	0
$799$	5.86889	0.207626
$800$	−19.3980	−0.685823
$801$	0	0
$802$	−18.2336	−0.643852
$803$	−13.8776	−0.489730
$804$	0	0
$805$	0	0
$806$	17.0234	0.599623
$807$	0	0
$808$	−44.9256	−1.58048
$809$	−34.8116	−1.22391	−0.611956	−	0.790892i	$-0.709617\pi$
$809$	−34.8116	−1.22391	−0.611956	+	0.790892i	$0.709617\pi$
$810$	0	0
$811$	29.6881	1.04249	0.521245	−	0.853407i	$-0.325468\pi$
$811$	29.6881	1.04249	0.521245	+	0.853407i	$0.325468\pi$
$812$	−32.3737	−1.13610
$813$	0	0
$814$	−16.9050	−0.592521
$815$	10.4675	0.366659
$816$	0	0
$817$	−35.0439	−1.22603
$818$	10.2942	0.359930
$819$	0	0
$820$	−8.71473	−0.304332
$821$	45.7225	1.59572	0.797862	−	0.602840i	$-0.205964\pi$
$821$	45.7225	1.59572	0.797862	+	0.602840i	$0.205964\pi$
$822$	0	0
$823$	3.88501	0.135423	0.0677114	−	0.997705i	$-0.478430\pi$
$823$	3.88501	0.135423	0.0677114	+	0.997705i	$0.478430\pi$
$824$	−52.7861	−1.83889
$825$	0	0
$826$	−15.1290	−0.526406
$827$	−11.4728	−0.398948	−0.199474	−	0.979903i	$-0.563923\pi$
$827$	−11.4728	−0.398948	−0.199474	+	0.979903i	$0.563923\pi$
$828$	0	0
$829$	30.5516	1.06110	0.530550	−	0.847654i	$-0.321985\pi$
$829$	30.5516	1.06110	0.530550	+	0.847654i	$0.321985\pi$
$830$	13.0320	0.452347
$831$	0	0
$832$	−19.0042	−0.658851
$833$	−1.27479	−0.0441689
$834$	0	0
$835$	14.8110	0.512555
$836$	11.1048	0.384068
$837$	0	0
$838$	24.6366	0.851058
$839$	0.816806	0.0281993	0.0140996	−	0.999901i	$-0.495512\pi$
$839$	0.816806	0.0281993	0.0140996	+	0.999901i	$0.495512\pi$
$840$	0	0
$841$	47.6656	1.64364
$842$	9.05982	0.312222
$843$	0	0
$844$	13.0975	0.450836
$845$	−2.52069	−0.0867145
$846$	0	0
$847$	18.7256	0.643418
$848$	−0.183070	−0.00628666
$849$	0	0
$850$	2.00961	0.0689290
$851$	0	0
$852$	0	0
$853$	27.4246	0.939000	0.469500	−	0.882933i	$-0.344434\pi$
$853$	27.4246	0.939000	0.469500	+	0.882933i	$0.344434\pi$
$854$	−11.6129	−0.397385
$855$	0	0
$856$	17.2272	0.588813
$857$	−57.7032	−1.97110	−0.985552	−	0.169373i	$-0.945826\pi$
$857$	−57.7032	−1.97110	−0.985552	+	0.169373i	$0.945826\pi$
$858$	0	0
$859$	54.3820	1.85549	0.927745	−	0.373214i	$-0.121744\pi$
$859$	54.3820	1.85549	0.927745	+	0.373214i	$0.121744\pi$
$860$	−13.2273	−0.451048
$861$	0	0
$862$	8.10237	0.275968
$863$	24.9368	0.848860	0.424430	−	0.905461i	$-0.360475\pi$
$863$	24.9368	0.848860	0.424430	+	0.905461i	$0.360475\pi$
$864$	0	0
$865$	19.3965	0.659502
$866$	−3.80432	−0.129276
$867$	0	0
$868$	−18.6336	−0.632466
$869$	−15.6260	−0.530075
$870$	0	0
$871$	−21.8370	−0.739918
$872$	−25.7343	−0.871474
$873$	0	0
$874$	0	0
$875$	−31.5343	−1.06606
$876$	0	0
$877$	36.4605	1.23119	0.615593	−	0.788065i	$-0.288917\pi$
$877$	36.4605	1.23119	0.615593	+	0.788065i	$0.288917\pi$
$878$	6.36140	0.214687
$879$	0	0
$880$	0.0450896	0.00151997
$881$	9.35761	0.315266	0.157633	−	0.987498i	$-0.449614\pi$
$881$	9.35761	0.315266	0.157633	+	0.987498i	$0.449614\pi$
$882$	0	0
$883$	17.7981	0.598954	0.299477	−	0.954104i	$-0.403188\pi$
$883$	17.7981	0.598954	0.299477	+	0.954104i	$0.403188\pi$
$884$	−3.23228	−0.108713
$885$	0	0
$886$	−18.3296	−0.615794
$887$	30.5261	1.02497	0.512483	−	0.858697i	$-0.328726\pi$
$887$	30.5261	1.02497	0.512483	+	0.858697i	$0.328726\pi$
$888$	0	0
$889$	−17.4941	−0.586735
$890$	0.396067	0.0132762
$891$	0	0
$892$	7.80016	0.261169
$893$	−35.9502	−1.20303
$894$	0	0
$895$	−23.1659	−0.774351
$896$	21.0280	0.702495
$897$	0	0
$898$	−9.39830	−0.313625
$899$	44.1270	1.47172
$900$	0	0
$901$	7.45144	0.248244
$902$	10.6107	0.353297
$903$	0	0
$904$	−26.6269	−0.885598
$905$	−8.73567	−0.290383
$906$	0	0
$907$	−15.2620	−0.506765	−0.253383	−	0.967366i	$-0.581543\pi$
$907$	−15.2620	−0.506765	−0.253383	+	0.967366i	$0.581543\pi$
$908$	−8.71113	−0.289089
$909$	0	0
$910$	−12.6435	−0.419128
$911$	29.3858	0.973597	0.486798	−	0.873514i	$-0.338165\pi$
$911$	29.3858	0.973597	0.486798	+	0.873514i	$0.338165\pi$
$912$	0	0
$913$	25.8755	0.856353
$914$	−2.89794	−0.0958555
$915$	0	0
$916$	−0.423912	−0.0140064
$917$	25.7265	0.849565
$918$	0	0
$919$	38.9857	1.28602	0.643010	−	0.765858i	$-0.277686\pi$
$919$	38.9857	1.28602	0.643010	+	0.765858i	$0.277686\pi$
$920$	0	0
$921$	0	0
$922$	14.6350	0.481978
$923$	−30.6942	−1.01031
$924$	0	0
$925$	−30.5597	−1.00480
$926$	−12.2467	−0.402452
$927$	0	0
$928$	−49.5935	−1.62799
$929$	−21.5630	−0.707458	−0.353729	−	0.935348i	$-0.615086\pi$
$929$	−21.5630	−0.707458	−0.353729	+	0.935348i	$0.615086\pi$
$930$	0	0
$931$	7.80881	0.255923
$932$	−13.6987	−0.448715
$933$	0	0
$934$	17.9975	0.588895
$935$	−1.83526	−0.0600196
$936$	0	0
$937$	33.8184	1.10480	0.552399	−	0.833580i	$-0.313712\pi$
$937$	33.8184	1.10480	0.552399	+	0.833580i	$0.313712\pi$
$938$	−14.6573	−0.478579
$939$	0	0
$940$	−13.5694	−0.442585
$941$	−3.29510	−0.107417	−0.0537086	−	0.998557i	$-0.517104\pi$
$941$	−3.29510	−0.107417	−0.0537086	+	0.998557i	$0.517104\pi$
$942$	0	0
$943$	0	0
$944$	0.0961975	0.00313096
$945$	0	0
$946$	16.1050	0.523619
$947$	21.5434	0.700067	0.350034	−	0.936737i	$-0.386170\pi$
$947$	21.5434	0.700067	0.350034	+	0.936737i	$0.386170\pi$
$948$	0	0
$949$	−24.7433	−0.803202
$950$	−12.3100	−0.399388
$951$	0	0
$952$	−5.66952	−0.183750
$953$	−34.0815	−1.10401	−0.552004	−	0.833842i	$-0.686137\pi$
$953$	−34.0815	−1.10401	−0.552004	+	0.833842i	$0.686137\pi$
$954$	0	0
$955$	18.7971	0.608259
$956$	−6.34448	−0.205195
$957$	0	0
$958$	2.49854	0.0807243
$959$	5.90613	0.190719
$960$	0	0
$961$	−5.60152	−0.180694
$962$	−30.1411	−0.971788
$963$	0	0
$964$	20.5829	0.662932
$965$	−22.1108	−0.711771
$966$	0	0
$967$	5.29658	0.170327	0.0851633	−	0.996367i	$-0.472859\pi$
$967$	5.29658	0.170327	0.0851633	+	0.996367i	$0.472859\pi$
$968$	17.7365	0.570073
$969$	0	0
$970$	−13.4154	−0.430743
$971$	−56.6496	−1.81797	−0.908986	−	0.416827i	$-0.863142\pi$
$971$	−56.6496	−1.81797	−0.908986	+	0.416827i	$0.863142\pi$
$972$	0	0
$973$	17.3664	0.556741
$974$	16.7915	0.538034
$975$	0	0
$976$	0.0738402	0.00236357
$977$	−54.6608	−1.74875	−0.874376	−	0.485248i	$-0.838729\pi$
$977$	−54.6608	−1.74875	−0.874376	+	0.485248i	$0.838729\pi$
$978$	0	0
$979$	0.786406	0.0251337
$980$	2.94744	0.0941525
$981$	0	0
$982$	14.6206	0.466563
$983$	51.0328	1.62769	0.813846	−	0.581080i	$-0.197370\pi$
$983$	51.0328	1.62769	0.813846	+	0.581080i	$0.197370\pi$
$984$	0	0
$985$	23.0532	0.734535
$986$	5.13782	0.163622
$987$	0	0
$988$	19.7995	0.629907
$989$	0	0
$990$	0	0
$991$	−41.2312	−1.30975	−0.654876	−	0.755736i	$-0.727279\pi$
$991$	−41.2312	−1.30975	−0.654876	+	0.755736i	$0.727279\pi$
$992$	−28.5449	−0.906302
$993$	0	0
$994$	−20.6025	−0.653470
$995$	22.8042	0.722942
$996$	0	0
$997$	−20.8301	−0.659697	−0.329848	−	0.944034i	$-0.606998\pi$
$997$	−20.8301	−0.659697	−0.329848	+	0.944034i	$0.606998\pi$
$998$	−17.8962	−0.566493
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bt.1.7		10
3.2	odd	2		1587.2.a.u.1.4		10
23.9	even	11		207.2.i.d.127.2		20
23.18	even	11		207.2.i.d.163.2		20
23.22	odd	2		4761.2.a.bu.1.7		10
69.32	odd	22		69.2.e.c.58.1	yes	20
69.41	odd	22		69.2.e.c.25.1	✓	20
69.68	even	2		1587.2.a.t.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.c.25.1	✓	20	69.41	odd	22
69.2.e.c.58.1	yes	20	69.32	odd	22
207.2.i.d.127.2		20	23.9	even	11
207.2.i.d.163.2		20	23.18	even	11
1587.2.a.t.1.4		10	69.68	even	2
1587.2.a.u.1.4		10	3.2	odd	2
4761.2.a.bt.1.7		10	1.1	even	1	trivial
4761.2.a.bu.1.7		10	23.22	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.871916 q^{2} -1.23976 q^{4} -1.25508 q^{5} -2.98232 q^{7} -2.82480 q^{8} +O(q^{10})\)	\(q+0.871916 q^{2} -1.23976 q^{4} -1.25508 q^{5} -2.98232 q^{7} -2.82480 q^{8} -1.09432 q^{10} -2.17282 q^{11} -3.87407 q^{13} -2.60033 q^{14} +0.0165341 q^{16} -0.672983 q^{17} +4.12239 q^{19} +1.55600 q^{20} -1.89452 q^{22} -3.42478 q^{25} -3.37786 q^{26} +3.69737 q^{28} -8.75589 q^{29} -5.03969 q^{31} +5.66402 q^{32} -0.586785 q^{34} +3.74305 q^{35} +8.92313 q^{37} +3.59438 q^{38} +3.54535 q^{40} -5.60073 q^{41} -8.50086 q^{43} +2.69378 q^{44} -8.72071 q^{47} +1.89424 q^{49} -2.98612 q^{50} +4.80292 q^{52} -11.0723 q^{53} +2.72706 q^{55} +8.42447 q^{56} -7.63440 q^{58} +5.81811 q^{59} +4.46592 q^{61} -4.39419 q^{62} +4.90548 q^{64} +4.86226 q^{65} +5.63671 q^{67} +0.834339 q^{68} +3.26362 q^{70} +7.92300 q^{71} +6.38691 q^{73} +7.78022 q^{74} -5.11078 q^{76} +6.48005 q^{77} +7.19156 q^{79} -0.0207516 q^{80} -4.88337 q^{82} -11.9087 q^{83} +0.844646 q^{85} -7.41203 q^{86} +6.13778 q^{88} -0.361929 q^{89} +11.5537 q^{91} -7.60373 q^{94} -5.17392 q^{95} +12.2591 q^{97} +1.65162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} - 8 q^{5} + 19 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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Database

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