LMFDB - Embedded newform 4761.2.a.bt.1.2

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			1.2
Root			$-1.98594$ 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)

$f(q)$	$=$	$q-1.98594 q^{2} +1.94394 q^{4} -2.04789 q^{5} +3.23167 q^{7} +0.111323 q^{8} +O(q^{10})$	$q-1.98594 q^{2} +1.94394 q^{4} -2.04789 q^{5} +3.23167 q^{7} +0.111323 q^{8} +4.06699 q^{10} -1.62535 q^{11} -2.59558 q^{13} -6.41790 q^{14} -4.10897 q^{16} -4.64517 q^{17} -6.39588 q^{19} -3.98099 q^{20} +3.22785 q^{22} -0.806128 q^{25} +5.15465 q^{26} +6.28220 q^{28} -7.72445 q^{29} -8.76531 q^{31} +7.93751 q^{32} +9.22502 q^{34} -6.61813 q^{35} +7.97053 q^{37} +12.7018 q^{38} -0.227978 q^{40} +2.86353 q^{41} -3.72797 q^{43} -3.15960 q^{44} +8.70030 q^{47} +3.44372 q^{49} +1.60092 q^{50} -5.04566 q^{52} +1.50346 q^{53} +3.32855 q^{55} +0.359761 q^{56} +15.3403 q^{58} -5.02560 q^{59} -8.45927 q^{61} +17.4073 q^{62} -7.54545 q^{64} +5.31547 q^{65} +3.72964 q^{67} -9.02995 q^{68} +13.1432 q^{70} +1.23327 q^{71} +11.8109 q^{73} -15.8290 q^{74} -12.4332 q^{76} -5.25261 q^{77} +8.98797 q^{79} +8.41474 q^{80} -5.68678 q^{82} +15.3360 q^{83} +9.51282 q^{85} +7.40350 q^{86} -0.180940 q^{88} +2.61216 q^{89} -8.38806 q^{91} -17.2783 q^{94} +13.0981 q^{95} +1.21486 q^{97} -6.83901 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.98594	−1.40427	−0.702135	−	0.712044i	$-0.747770\pi$
$2$	−1.98594	−1.40427	−0.702135	+	0.712044i	$0.747770\pi$
$3$	0	0
$3$	0	0
$4$	1.94394	0.971972
$5$	−2.04789	−0.915846	−0.457923	−	0.888992i	$-0.651407\pi$
$5$	−2.04789	−0.915846	−0.457923	+	0.888992i	$0.651407\pi$
$6$	0	0
$7$	3.23167	1.22146	0.610729	−	0.791840i	$-0.290876\pi$
$7$	3.23167	1.22146	0.610729	+	0.791840i	$0.290876\pi$
$8$	0.111323	0.0393587
$9$	0	0
$10$	4.06699	1.28609
$11$	−1.62535	−0.490062	−0.245031	−	0.969515i	$-0.578798\pi$
$11$	−1.62535	−0.490062	−0.245031	+	0.969515i	$0.578798\pi$
$12$	0	0
$13$	−2.59558	−0.719884	−0.359942	−	0.932975i	$-0.617203\pi$
$13$	−2.59558	−0.719884	−0.359942	+	0.932975i	$0.617203\pi$
$14$	−6.41790	−1.71526
$15$	0	0
$16$	−4.10897	−1.02724
$17$	−4.64517	−1.12662	−0.563310	−	0.826246i	$-0.690472\pi$
$17$	−4.64517	−1.12662	−0.563310	+	0.826246i	$0.690472\pi$
$18$	0	0
$19$	−6.39588	−1.46732	−0.733658	−	0.679519i	$-0.762188\pi$
$19$	−6.39588	−1.46732	−0.733658	+	0.679519i	$0.762188\pi$
$20$	−3.98099	−0.890177
$21$	0	0
$22$	3.22785	0.688180
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−0.806128	−0.161226
$26$	5.15465	1.01091
$27$	0	0
$28$	6.28220	1.18722
$29$	−7.72445	−1.43439	−0.717197	−	0.696870i	$-0.754575\pi$
$29$	−7.72445	−1.43439	−0.717197	+	0.696870i	$0.754575\pi$
$30$	0	0
$31$	−8.76531	−1.57430	−0.787148	−	0.616764i	$-0.788443\pi$
$31$	−8.76531	−1.57430	−0.787148	+	0.616764i	$0.788443\pi$
$32$	7.93751	1.40317
$33$	0	0
$34$	9.22502	1.58208
$35$	−6.61813	−1.11867
$36$	0	0
$37$	7.97053	1.31035	0.655174	−	0.755478i	$-0.272596\pi$
$37$	7.97053	1.31035	0.655174	+	0.755478i	$0.272596\pi$
$38$	12.7018	2.06051
$39$	0	0
$40$	−0.227978	−0.0360465
$41$	2.86353	0.447208	0.223604	−	0.974680i	$-0.428218\pi$
$41$	2.86353	0.447208	0.223604	+	0.974680i	$0.428218\pi$
$42$	0	0
$43$	−3.72797	−0.568510	−0.284255	−	0.958749i	$-0.591746\pi$
$43$	−3.72797	−0.568510	−0.284255	+	0.958749i	$0.591746\pi$
$44$	−3.15960	−0.476327
$45$	0	0
$46$	0	0
$47$	8.70030	1.26907	0.634535	−	0.772894i	$-0.281192\pi$
$47$	8.70030	1.26907	0.634535	+	0.772894i	$0.281192\pi$
$48$	0	0
$49$	3.44372	0.491960
$50$	1.60092	0.226404
$51$	0	0
$52$	−5.04566	−0.699707
$53$	1.50346	0.206516	0.103258	−	0.994655i	$-0.467073\pi$
$53$	1.50346	0.206516	0.103258	+	0.994655i	$0.467073\pi$
$54$	0	0
$55$	3.32855	0.448822
$56$	0.359761	0.0480750
$57$	0	0
$58$	15.3403	2.01428
$59$	−5.02560	−0.654277	−0.327139	−	0.944976i	$-0.606084\pi$
$59$	−5.02560	−0.654277	−0.327139	+	0.944976i	$0.606084\pi$
$60$	0	0
$61$	−8.45927	−1.08310	−0.541549	−	0.840669i	$-0.682162\pi$
$61$	−8.45927	−1.08310	−0.541549	+	0.840669i	$0.682162\pi$
$62$	17.4073	2.21074
$63$	0	0
$64$	−7.54545	−0.943181
$65$	5.31547	0.659303
$66$	0	0
$67$	3.72964	0.455648	0.227824	−	0.973702i	$-0.426839\pi$
$67$	3.72964	0.455648	0.227824	+	0.973702i	$0.426839\pi$
$68$	−9.02995	−1.09504
$69$	0	0
$70$	13.1432	1.57091
$71$	1.23327	0.146362	0.0731809	−	0.997319i	$-0.476685\pi$
$71$	1.23327	0.146362	0.0731809	+	0.997319i	$0.476685\pi$
$72$	0	0
$73$	11.8109	1.38236	0.691182	−	0.722681i	$-0.257090\pi$
$73$	11.8109	1.38236	0.691182	+	0.722681i	$0.257090\pi$
$74$	−15.8290	−1.84008
$75$	0	0
$76$	−12.4332	−1.42619
$77$	−5.25261	−0.598591
$78$	0	0
$79$	8.98797	1.01123	0.505613	−	0.862761i	$-0.331266\pi$
$79$	8.98797	1.01123	0.505613	+	0.862761i	$0.331266\pi$
$80$	8.41474	0.940796
$81$	0	0
$82$	−5.68678	−0.628000
$83$	15.3360	1.68334	0.841672	−	0.539989i	$-0.181572\pi$
$83$	15.3360	1.68334	0.841672	+	0.539989i	$0.181572\pi$
$84$	0	0
$85$	9.51282	1.03181
$86$	7.40350	0.798340
$87$	0	0
$88$	−0.180940	−0.0192882
$89$	2.61216	0.276888	0.138444	−	0.990370i	$-0.455790\pi$
$89$	2.61216	0.276888	0.138444	+	0.990370i	$0.455790\pi$
$90$	0	0
$91$	−8.38806	−0.879308
$92$	0	0
$93$	0	0
$94$	−17.2783	−1.78212
$95$	13.0981	1.34384
$96$	0	0
$97$	1.21486	0.123351	0.0616753	−	0.998096i	$-0.480356\pi$
$97$	1.21486	0.123351	0.0616753	+	0.998096i	$0.480356\pi$
$98$	−6.83901	−0.690845
$99$	0	0
$100$	−1.56707	−0.156707
$101$	−7.61351	−0.757572	−0.378786	−	0.925484i	$-0.623658\pi$
$101$	−7.61351	−0.757572	−0.378786	+	0.925484i	$0.623658\pi$
$102$	0	0
$103$	10.8432	1.06842	0.534208	−	0.845353i	$-0.320610\pi$
$103$	10.8432	1.06842	0.534208	+	0.845353i	$0.320610\pi$
$104$	−0.288948	−0.0283337
$105$	0	0
$106$	−2.98577	−0.290004
$107$	−4.59294	−0.444016	−0.222008	−	0.975045i	$-0.571261\pi$
$107$	−4.59294	−0.444016	−0.222008	+	0.975045i	$0.571261\pi$
$108$	0	0
$109$	0.844304	0.0808697	0.0404348	−	0.999182i	$-0.487126\pi$
$109$	0.844304	0.0808697	0.0404348	+	0.999182i	$0.487126\pi$
$110$	−6.61029	−0.630267
$111$	0	0
$112$	−13.2789	−1.25473
$113$	−11.1311	−1.04712	−0.523561	−	0.851988i	$-0.675397\pi$
$113$	−11.1311	−1.04712	−0.523561	+	0.851988i	$0.675397\pi$
$114$	0	0
$115$	0	0
$116$	−15.0159	−1.39419
$117$	0	0
$118$	9.98052	0.918782
$119$	−15.0117	−1.37612
$120$	0	0
$121$	−8.35823	−0.759839
$122$	16.7996	1.52096
$123$	0	0
$124$	−17.0393	−1.53017
$125$	11.8903	1.06350
$126$	0	0
$127$	−18.2170	−1.61649	−0.808247	−	0.588843i	$-0.799584\pi$
$127$	−18.2170	−1.61649	−0.808247	+	0.588843i	$0.799584\pi$
$128$	−0.890237	−0.0786865
$129$	0	0
$130$	−10.5562	−0.925839
$131$	−1.45244	−0.126901	−0.0634503	−	0.997985i	$-0.520210\pi$
$131$	−1.45244	−0.126901	−0.0634503	+	0.997985i	$0.520210\pi$
$132$	0	0
$133$	−20.6694	−1.79226
$134$	−7.40683	−0.639853
$135$	0	0
$136$	−0.517116	−0.0443423
$137$	4.47536	0.382356	0.191178	−	0.981555i	$-0.438769\pi$
$137$	4.47536	0.382356	0.191178	+	0.981555i	$0.438769\pi$
$138$	0	0
$139$	5.81596	0.493303	0.246652	−	0.969104i	$-0.420670\pi$
$139$	5.81596	0.493303	0.246652	+	0.969104i	$0.420670\pi$
$140$	−12.8653	−1.08731
$141$	0	0
$142$	−2.44919	−0.205531
$143$	4.21873	0.352788
$144$	0	0
$145$	15.8189	1.31368
$146$	−23.4557	−1.94121
$147$	0	0
$148$	15.4943	1.27362
$149$	0.0331955	0.00271948	0.00135974	−	0.999999i	$-0.499567\pi$
$149$	0.0331955	0.00271948	0.00135974	+	0.999999i	$0.499567\pi$
$150$	0	0
$151$	15.0623	1.22575	0.612877	−	0.790178i	$-0.290012\pi$
$151$	15.0623	1.22575	0.612877	+	0.790178i	$0.290012\pi$
$152$	−0.712010	−0.0577517
$153$	0	0
$154$	10.4314	0.840583
$155$	17.9504	1.44181
$156$	0	0
$157$	−15.6067	−1.24555	−0.622774	−	0.782402i	$-0.713994\pi$
$157$	−15.6067	−1.24555	−0.622774	+	0.782402i	$0.713994\pi$
$158$	−17.8495	−1.42003
$159$	0	0
$160$	−16.2552	−1.28508
$161$	0	0
$162$	0	0
$163$	13.8252	1.08287	0.541436	−	0.840742i	$-0.317881\pi$
$163$	13.8252	1.08287	0.541436	+	0.840742i	$0.317881\pi$
$164$	5.56654	0.434674
$165$	0	0
$166$	−30.4563	−2.36387
$167$	−2.60289	−0.201418	−0.100709	−	0.994916i	$-0.532111\pi$
$167$	−2.60289	−0.201418	−0.100709	+	0.994916i	$0.532111\pi$
$168$	0	0
$169$	−6.26298	−0.481768
$170$	−18.8919	−1.44894
$171$	0	0
$172$	−7.24696	−0.552575
$173$	16.2980	1.23911	0.619557	−	0.784951i	$-0.287312\pi$
$173$	16.2980	1.23911	0.619557	+	0.784951i	$0.287312\pi$
$174$	0	0
$175$	−2.60514	−0.196930
$176$	6.67853	0.503413
$177$	0	0
$178$	−5.18758	−0.388826
$179$	5.74006	0.429032	0.214516	−	0.976720i	$-0.431183\pi$
$179$	5.74006	0.429032	0.214516	+	0.976720i	$0.431183\pi$
$180$	0	0
$181$	11.6412	0.865285	0.432642	−	0.901566i	$-0.357581\pi$
$181$	11.6412	0.865285	0.432642	+	0.901566i	$0.357581\pi$
$182$	16.6582	1.23478
$183$	0	0
$184$	0	0
$185$	−16.3228	−1.20008
$186$	0	0
$187$	7.55004	0.552114
$188$	16.9129	1.23350
$189$	0	0
$190$	−26.0120	−1.88711
$191$	−3.22928	−0.233663	−0.116831	−	0.993152i	$-0.537274\pi$
$191$	−3.22928	−0.233663	−0.116831	+	0.993152i	$0.537274\pi$
$192$	0	0
$193$	4.36918	0.314501	0.157250	−	0.987559i	$-0.449737\pi$
$193$	4.36918	0.314501	0.157250	+	0.987559i	$0.449737\pi$
$194$	−2.41264	−0.173218
$195$	0	0
$196$	6.69440	0.478172
$197$	21.4758	1.53009	0.765044	−	0.643978i	$-0.222717\pi$
$197$	21.4758	1.53009	0.765044	+	0.643978i	$0.222717\pi$
$198$	0	0
$199$	6.34454	0.449752	0.224876	−	0.974387i	$-0.427802\pi$
$199$	6.34454	0.449752	0.224876	+	0.974387i	$0.427802\pi$
$200$	−0.0897408	−0.00634563
$201$	0	0
$202$	15.1199	1.06384
$203$	−24.9629	−1.75205
$204$	0	0
$205$	−5.86420	−0.409574
$206$	−21.5340	−1.50034
$207$	0	0
$208$	10.6651	0.739495
$209$	10.3956	0.719076
$210$	0	0
$211$	−14.7659	−1.01653	−0.508263	−	0.861202i	$-0.669712\pi$
$211$	−14.7659	−1.01653	−0.508263	+	0.861202i	$0.669712\pi$
$212$	2.92264	0.200728
$213$	0	0
$214$	9.12129	0.623519
$215$	7.63448	0.520667
$216$	0	0
$217$	−28.3266	−1.92294
$218$	−1.67674	−0.113563
$219$	0	0
$220$	6.47052	0.436242
$221$	12.0569	0.811035
$222$	0	0
$223$	−19.3458	−1.29549	−0.647747	−	0.761856i	$-0.724288\pi$
$223$	−19.3458	−1.29549	−0.647747	+	0.761856i	$0.724288\pi$
$224$	25.6514	1.71391
$225$	0	0
$226$	22.1056	1.47044
$227$	9.71543	0.644836	0.322418	−	0.946597i	$-0.395504\pi$
$227$	9.71543	0.644836	0.322418	+	0.946597i	$0.395504\pi$
$228$	0	0
$229$	5.31591	0.351285	0.175642	−	0.984454i	$-0.443800\pi$
$229$	5.31591	0.351285	0.175642	+	0.984454i	$0.443800\pi$
$230$	0	0
$231$	0	0
$232$	−0.859911	−0.0564559
$233$	16.8475	1.10372	0.551858	−	0.833938i	$-0.313919\pi$
$233$	16.8475	1.10372	0.551858	+	0.833938i	$0.313919\pi$
$234$	0	0
$235$	−17.8173	−1.16227
$236$	−9.76949	−0.635939
$237$	0	0
$238$	29.8123	1.93244
$239$	−14.2509	−0.921816	−0.460908	−	0.887448i	$-0.652476\pi$
$239$	−14.2509	−0.921816	−0.460908	+	0.887448i	$0.652476\pi$
$240$	0	0
$241$	26.8989	1.73271	0.866356	−	0.499426i	$-0.166456\pi$
$241$	26.8989	1.73271	0.866356	+	0.499426i	$0.166456\pi$
$242$	16.5989	1.06702
$243$	0	0
$244$	−16.4444	−1.05274
$245$	−7.05238	−0.450560
$246$	0	0
$247$	16.6010	1.05630
$248$	−0.975783	−0.0619623
$249$	0	0
$250$	−23.6135	−1.49345
$251$	−6.17502	−0.389764	−0.194882	−	0.980827i	$-0.562432\pi$
$251$	−6.17502	−0.389764	−0.194882	+	0.980827i	$0.562432\pi$
$252$	0	0
$253$	0	0
$254$	36.1777	2.26999
$255$	0	0
$256$	16.8588	1.05368
$257$	9.54985	0.595703	0.297851	−	0.954612i	$-0.403730\pi$
$257$	9.54985	0.595703	0.297851	+	0.954612i	$0.403730\pi$
$258$	0	0
$259$	25.7582	1.60053
$260$	10.3330	0.640824
$261$	0	0
$262$	2.88446	0.178203
$263$	−2.87937	−0.177550	−0.0887748	−	0.996052i	$-0.528295\pi$
$263$	−2.87937	−0.177550	−0.0887748	+	0.996052i	$0.528295\pi$
$264$	0	0
$265$	−3.07892	−0.189137
$266$	41.0481	2.51682
$267$	0	0
$268$	7.25022	0.442877
$269$	−8.62891	−0.526114	−0.263057	−	0.964780i	$-0.584731\pi$
$269$	−8.62891	−0.526114	−0.263057	+	0.964780i	$0.584731\pi$
$270$	0	0
$271$	−7.50580	−0.455945	−0.227972	−	0.973668i	$-0.573210\pi$
$271$	−7.50580	−0.455945	−0.227972	+	0.973668i	$0.573210\pi$
$272$	19.0869	1.15731
$273$	0	0
$274$	−8.88777	−0.536930
$275$	1.31024	0.0790106
$276$	0	0
$277$	−7.67215	−0.460975	−0.230487	−	0.973075i	$-0.574032\pi$
$277$	−7.67215	−0.460975	−0.230487	+	0.973075i	$0.574032\pi$
$278$	−11.5501	−0.692731
$279$	0	0
$280$	−0.736752	−0.0440293
$281$	23.0513	1.37513	0.687564	−	0.726124i	$-0.258680\pi$
$281$	23.0513	1.37513	0.687564	+	0.726124i	$0.258680\pi$
$282$	0	0
$283$	7.75889	0.461218	0.230609	−	0.973046i	$-0.425928\pi$
$283$	7.75889	0.461218	0.230609	+	0.973046i	$0.425928\pi$
$284$	2.39740	0.142260
$285$	0	0
$286$	−8.37813	−0.495409
$287$	9.25399	0.546246
$288$	0	0
$289$	4.57762	0.269272
$290$	−31.4153	−1.84477
$291$	0	0
$292$	22.9598	1.34362
$293$	−10.4739	−0.611890	−0.305945	−	0.952049i	$-0.598972\pi$
$293$	−10.4739	−0.611890	−0.305945	+	0.952049i	$0.598972\pi$
$294$	0	0
$295$	10.2919	0.599217
$296$	0.887306	0.0515736
$297$	0	0
$298$	−0.0659242	−0.00381888
$299$	0	0
$300$	0	0
$301$	−12.0476	−0.694411
$302$	−29.9128	−1.72129
$303$	0	0
$304$	26.2805	1.50729
$305$	17.3237	0.991952
$306$	0	0
$307$	−20.1047	−1.14743	−0.573717	−	0.819054i	$-0.694499\pi$
$307$	−20.1047	−1.14743	−0.573717	+	0.819054i	$0.694499\pi$
$308$	−10.2108	−0.581814
$309$	0	0
$310$	−35.6484	−2.02469
$311$	17.0820	0.968629	0.484315	−	0.874894i	$-0.339069\pi$
$311$	17.0820	0.968629	0.484315	+	0.874894i	$0.339069\pi$
$312$	0	0
$313$	10.6397	0.601389	0.300694	−	0.953721i	$-0.402782\pi$
$313$	10.6397	0.601389	0.300694	+	0.953721i	$0.402782\pi$
$314$	30.9938	1.74908
$315$	0	0
$316$	17.4721	0.982883
$317$	−15.7745	−0.885987	−0.442993	−	0.896525i	$-0.646083\pi$
$317$	−15.7745	−0.885987	−0.442993	+	0.896525i	$0.646083\pi$
$318$	0	0
$319$	12.5550	0.702943
$320$	15.4523	0.863808
$321$	0	0
$322$	0	0
$323$	29.7100	1.65311
$324$	0	0
$325$	2.09237	0.116064
$326$	−27.4559	−1.52064
$327$	0	0
$328$	0.318777	0.0176015
$329$	28.1166	1.55012
$330$	0	0
$331$	−12.3289	−0.677656	−0.338828	−	0.940848i	$-0.610030\pi$
$331$	−12.3289	−0.677656	−0.338828	+	0.940848i	$0.610030\pi$
$332$	29.8123	1.63616
$333$	0	0
$334$	5.16918	0.282845
$335$	−7.63791	−0.417304
$336$	0	0
$337$	29.3948	1.60124	0.800618	−	0.599175i	$-0.204505\pi$
$337$	29.3948	1.60124	0.800618	+	0.599175i	$0.204505\pi$
$338$	12.4379	0.676531
$339$	0	0
$340$	18.4924	1.00289
$341$	14.2467	0.771503
$342$	0	0
$343$	−11.4927	−0.620549
$344$	−0.415009	−0.0223758
$345$	0	0
$346$	−32.3668	−1.74005
$347$	−29.9843	−1.60964	−0.804822	−	0.593517i	$-0.797739\pi$
$347$	−29.9843	−1.60964	−0.804822	+	0.593517i	$0.797739\pi$
$348$	0	0
$349$	8.95829	0.479526	0.239763	−	0.970831i	$-0.422930\pi$
$349$	8.95829	0.479526	0.239763	+	0.970831i	$0.422930\pi$
$350$	5.17365	0.276543
$351$	0	0
$352$	−12.9013	−0.687639
$353$	9.57827	0.509800	0.254900	−	0.966967i	$-0.417957\pi$
$353$	9.57827	0.509800	0.254900	+	0.966967i	$0.417957\pi$
$354$	0	0
$355$	−2.52560	−0.134045
$356$	5.07789	0.269128
$357$	0	0
$358$	−11.3994	−0.602477
$359$	14.0907	0.743680	0.371840	−	0.928297i	$-0.378727\pi$
$359$	14.0907	0.743680	0.371840	+	0.928297i	$0.378727\pi$
$360$	0	0
$361$	21.9073	1.15301
$362$	−23.1187	−1.21509
$363$	0	0
$364$	−16.3059	−0.854663
$365$	−24.1875	−1.26603
$366$	0	0
$367$	8.38697	0.437796	0.218898	−	0.975748i	$-0.429754\pi$
$367$	8.38697	0.437796	0.218898	+	0.975748i	$0.429754\pi$
$368$	0	0
$369$	0	0
$370$	32.4161	1.68523
$371$	4.85869	0.252251
$372$	0	0
$373$	17.8476	0.924112	0.462056	−	0.886851i	$-0.347112\pi$
$373$	17.8476	0.924112	0.462056	+	0.886851i	$0.347112\pi$
$374$	−14.9939	−0.775317
$375$	0	0
$376$	0.968546	0.0499490
$377$	20.0494	1.03260
$378$	0	0
$379$	−22.0466	−1.13246	−0.566228	−	0.824249i	$-0.691598\pi$
$379$	−22.0466	−1.13246	−0.566228	+	0.824249i	$0.691598\pi$
$380$	25.4619	1.30617
$381$	0	0
$382$	6.41315	0.328125
$383$	9.26608	0.473475	0.236737	−	0.971574i	$-0.423922\pi$
$383$	9.26608	0.473475	0.236737	+	0.971574i	$0.423922\pi$
$384$	0	0
$385$	10.7568	0.548217
$386$	−8.67692	−0.441643
$387$	0	0
$388$	2.36163	0.119893
$389$	−24.3669	−1.23545	−0.617725	−	0.786394i	$-0.711946\pi$
$389$	−24.3669	−1.23545	−0.617725	+	0.786394i	$0.711946\pi$
$390$	0	0
$391$	0	0
$392$	0.383367	0.0193629
$393$	0	0
$394$	−42.6496	−2.14866
$395$	−18.4064	−0.926127
$396$	0	0
$397$	28.0529	1.40794	0.703968	−	0.710232i	$-0.251410\pi$
$397$	28.0529	1.40794	0.703968	+	0.710232i	$0.251410\pi$
$398$	−12.5998	−0.631573
$399$	0	0
$400$	3.31236	0.165618
$401$	−13.2018	−0.659268	−0.329634	−	0.944109i	$-0.606925\pi$
$401$	−13.2018	−0.659268	−0.329634	+	0.944109i	$0.606925\pi$
$402$	0	0
$403$	22.7510	1.13331
$404$	−14.8002	−0.736339
$405$	0	0
$406$	49.5748	2.46035
$407$	−12.9549	−0.642152
$408$	0	0
$409$	19.7955	0.978824	0.489412	−	0.872053i	$-0.337211\pi$
$409$	19.7955	0.978824	0.489412	+	0.872053i	$0.337211\pi$
$410$	11.6459	0.575152
$411$	0	0
$412$	21.0786	1.03847
$413$	−16.2411	−0.799172
$414$	0	0
$415$	−31.4065	−1.54168
$416$	−20.6024	−1.01012
$417$	0	0
$418$	−20.6449	−1.00978
$419$	−4.57599	−0.223552	−0.111776	−	0.993733i	$-0.535654\pi$
$419$	−4.57599	−0.223552	−0.111776	+	0.993733i	$0.535654\pi$
$420$	0	0
$421$	6.40985	0.312397	0.156199	−	0.987726i	$-0.450076\pi$
$421$	6.40985	0.312397	0.156199	+	0.987726i	$0.450076\pi$
$422$	29.3241	1.42748
$423$	0	0
$424$	0.167370	0.00812820
$425$	3.74460	0.181640
$426$	0	0
$427$	−27.3376	−1.32296
$428$	−8.92842	−0.431572
$429$	0	0
$430$	−15.1616	−0.731157
$431$	11.2893	0.543784	0.271892	−	0.962328i	$-0.412351\pi$
$431$	11.2893	0.543784	0.271892	+	0.962328i	$0.412351\pi$
$432$	0	0
$433$	−4.57383	−0.219804	−0.109902	−	0.993942i	$-0.535054\pi$
$433$	−4.57383	−0.219804	−0.109902	+	0.993942i	$0.535054\pi$
$434$	56.2549	2.70032
$435$	0	0
$436$	1.64128	0.0786031
$437$	0	0
$438$	0	0
$439$	−1.72786	−0.0824663	−0.0412331	−	0.999150i	$-0.513129\pi$
$439$	−1.72786	−0.0824663	−0.0412331	+	0.999150i	$0.513129\pi$
$440$	0.370545	0.0176651
$441$	0	0
$442$	−23.9442	−1.13891
$443$	−11.0181	−0.523484	−0.261742	−	0.965138i	$-0.584297\pi$
$443$	−11.0181	−0.523484	−0.261742	+	0.965138i	$0.584297\pi$
$444$	0	0
$445$	−5.34943	−0.253587
$446$	38.4196	1.81922
$447$	0	0
$448$	−24.3844	−1.15206
$449$	−13.6045	−0.642037	−0.321018	−	0.947073i	$-0.604025\pi$
$449$	−13.6045	−0.642037	−0.321018	+	0.947073i	$0.604025\pi$
$450$	0	0
$451$	−4.65424	−0.219160
$452$	−21.6382	−1.01777
$453$	0	0
$454$	−19.2942	−0.905523
$455$	17.1779	0.805311
$456$	0	0
$457$	25.7894	1.20638	0.603189	−	0.797599i	$-0.293897\pi$
$457$	25.7894	1.20638	0.603189	+	0.797599i	$0.293897\pi$
$458$	−10.5571	−0.493299
$459$	0	0
$460$	0	0
$461$	−38.6842	−1.80170	−0.900851	−	0.434129i	$-0.857056\pi$
$461$	−38.6842	−1.80170	−0.900851	+	0.434129i	$0.857056\pi$
$462$	0	0
$463$	10.7842	0.501183	0.250592	−	0.968093i	$-0.419375\pi$
$463$	10.7842	0.501183	0.250592	+	0.968093i	$0.419375\pi$
$464$	31.7395	1.47347
$465$	0	0
$466$	−33.4581	−1.54992
$467$	−14.9570	−0.692127	−0.346064	−	0.938211i	$-0.612482\pi$
$467$	−14.9570	−0.692127	−0.346064	+	0.938211i	$0.612482\pi$
$468$	0	0
$469$	12.0530	0.556555
$470$	35.3840	1.63214
$471$	0	0
$472$	−0.559466	−0.0257515
$473$	6.05926	0.278605
$474$	0	0
$475$	5.15590	0.236569
$476$	−29.1819	−1.33755
$477$	0	0
$478$	28.3014	1.29448
$479$	42.6289	1.94776	0.973882	−	0.227053i	$-0.0729091\pi$
$479$	42.6289	1.94776	0.973882	+	0.227053i	$0.0729091\pi$
$480$	0	0
$481$	−20.6881	−0.943297
$482$	−53.4196	−2.43320
$483$	0	0
$484$	−16.2479	−0.738542
$485$	−2.48791	−0.112970
$486$	0	0
$487$	−32.9996	−1.49536	−0.747678	−	0.664061i	$-0.768831\pi$
$487$	−32.9996	−1.49536	−0.747678	+	0.664061i	$0.768831\pi$
$488$	−0.941714	−0.0426294
$489$	0	0
$490$	14.0056	0.632708
$491$	−30.4743	−1.37529	−0.687644	−	0.726048i	$-0.741355\pi$
$491$	−30.4743	−1.37529	−0.687644	+	0.726048i	$0.741355\pi$
$492$	0	0
$493$	35.8814	1.61602
$494$	−32.9685	−1.48332
$495$	0	0
$496$	36.0164	1.61718
$497$	3.98552	0.178775
$498$	0	0
$499$	−26.7393	−1.19702	−0.598508	−	0.801117i	$-0.704239\pi$
$499$	−26.7393	−1.19702	−0.598508	+	0.801117i	$0.704239\pi$
$500$	23.1142	1.03370
$501$	0	0
$502$	12.2632	0.547333
$503$	−30.3753	−1.35437	−0.677184	−	0.735814i	$-0.736800\pi$
$503$	−30.3753	−1.35437	−0.677184	+	0.735814i	$0.736800\pi$
$504$	0	0
$505$	15.5917	0.693820
$506$	0	0
$507$	0	0
$508$	−35.4128	−1.57119
$509$	−16.7509	−0.742470	−0.371235	−	0.928539i	$-0.621066\pi$
$509$	−16.7509	−0.742470	−0.371235	+	0.928539i	$0.621066\pi$
$510$	0	0
$511$	38.1691	1.68850
$512$	−31.7001	−1.40096
$513$	0	0
$514$	−18.9654	−0.836527
$515$	−22.2058	−0.978504
$516$	0	0
$517$	−14.1411	−0.621923
$518$	−51.1541	−2.24758
$519$	0	0
$520$	0.591735	0.0259493
$521$	11.8427	0.518838	0.259419	−	0.965765i	$-0.416469\pi$
$521$	11.8427	0.518838	0.259419	+	0.965765i	$0.416469\pi$
$522$	0	0
$523$	8.68182	0.379630	0.189815	−	0.981820i	$-0.439211\pi$
$523$	8.68182	0.379630	0.189815	+	0.981820i	$0.439211\pi$
$524$	−2.82347	−0.123344
$525$	0	0
$526$	5.71825	0.249327
$527$	40.7164	1.77363
$528$	0	0
$529$	0	0
$530$	6.11455	0.265599
$531$	0	0
$532$	−40.1802	−1.74203
$533$	−7.43251	−0.321938
$534$	0	0
$535$	9.40586	0.406651
$536$	0.415196	0.0179337
$537$	0	0
$538$	17.1365	0.738806
$539$	−5.59727	−0.241091
$540$	0	0
$541$	−31.9956	−1.37560	−0.687798	−	0.725902i	$-0.741423\pi$
$541$	−31.9956	−1.37560	−0.687798	+	0.725902i	$0.741423\pi$
$542$	14.9060	0.640269
$543$	0	0
$544$	−36.8711	−1.58083
$545$	−1.72905	−0.0740642
$546$	0	0
$547$	20.7026	0.885179	0.442590	−	0.896724i	$-0.354060\pi$
$547$	20.7026	0.885179	0.442590	+	0.896724i	$0.354060\pi$
$548$	8.69984	0.371639
$549$	0	0
$550$	−2.60206	−0.110952
$551$	49.4047	2.10471
$552$	0	0
$553$	29.0462	1.23517
$554$	15.2364	0.647333
$555$	0	0
$556$	11.3059	0.479477
$557$	23.4293	0.992730	0.496365	−	0.868114i	$-0.334668\pi$
$557$	23.4293	0.992730	0.496365	+	0.868114i	$0.334668\pi$
$558$	0	0
$559$	9.67622	0.409261
$560$	27.1937	1.14914
$561$	0	0
$562$	−45.7785	−1.93105
$563$	19.0895	0.804526	0.402263	−	0.915524i	$-0.368224\pi$
$563$	19.0895	0.804526	0.402263	+	0.915524i	$0.368224\pi$
$564$	0	0
$565$	22.7953	0.959003
$566$	−15.4087	−0.647675
$567$	0	0
$568$	0.137291	0.00576062
$569$	−15.2668	−0.640017	−0.320009	−	0.947415i	$-0.603686\pi$
$569$	−15.2668	−0.640017	−0.320009	+	0.947415i	$0.603686\pi$
$570$	0	0
$571$	32.5593	1.36256	0.681282	−	0.732021i	$-0.261423\pi$
$571$	32.5593	1.36256	0.681282	+	0.732021i	$0.261423\pi$
$572$	8.20098	0.342900
$573$	0	0
$574$	−18.3778	−0.767076
$575$	0	0
$576$	0	0
$577$	38.7216	1.61200	0.806000	−	0.591916i	$-0.201628\pi$
$577$	38.7216	1.61200	0.806000	+	0.591916i	$0.201628\pi$
$578$	−9.09086	−0.378130
$579$	0	0
$580$	30.7510	1.27687
$581$	49.5609	2.05613
$582$	0	0
$583$	−2.44365	−0.101206
$584$	1.31483	0.0544081
$585$	0	0
$586$	20.8004	0.859258
$587$	4.80794	0.198445	0.0992224	−	0.995065i	$-0.468364\pi$
$587$	4.80794	0.198445	0.0992224	+	0.995065i	$0.468364\pi$
$588$	0	0
$589$	56.0618	2.30999
$590$	−20.4391	−0.841463
$591$	0	0
$592$	−32.7507	−1.34604
$593$	31.4147	1.29005	0.645023	−	0.764163i	$-0.276848\pi$
$593$	31.4147	1.29005	0.645023	+	0.764163i	$0.276848\pi$
$594$	0	0
$595$	30.7423	1.26031
$596$	0.0645302	0.00264326
$597$	0	0
$598$	0	0
$599$	−38.0521	−1.55477	−0.777384	−	0.629026i	$-0.783454\pi$
$599$	−38.0521	−1.55477	−0.777384	+	0.629026i	$0.783454\pi$
$600$	0	0
$601$	−6.66607	−0.271914	−0.135957	−	0.990715i	$-0.543411\pi$
$601$	−6.66607	−0.271914	−0.135957	+	0.990715i	$0.543411\pi$
$602$	23.9257	0.975140
$603$	0	0
$604$	29.2803	1.19140
$605$	17.1168	0.695896
$606$	0	0
$607$	6.62208	0.268782	0.134391	−	0.990928i	$-0.457092\pi$
$607$	6.62208	0.268782	0.134391	+	0.990928i	$0.457092\pi$
$608$	−50.7673	−2.05889
$609$	0	0
$610$	−34.4038	−1.39297
$611$	−22.5823	−0.913582
$612$	0	0
$613$	29.7324	1.20088	0.600440	−	0.799670i	$-0.294992\pi$
$613$	29.7324	1.20088	0.600440	+	0.799670i	$0.294992\pi$
$614$	39.9266	1.61131
$615$	0	0
$616$	−0.584738	−0.0235598
$617$	−48.6614	−1.95903	−0.979516	−	0.201365i	$-0.935462\pi$
$617$	−48.6614	−1.95903	−0.979516	+	0.201365i	$0.935462\pi$
$618$	0	0
$619$	26.5122	1.06562	0.532808	−	0.846236i	$-0.321137\pi$
$619$	26.5122	1.06562	0.532808	+	0.846236i	$0.321137\pi$
$620$	34.8946	1.40140
$621$	0	0
$622$	−33.9237	−1.36022
$623$	8.44165	0.338208
$624$	0	0
$625$	−20.3195	−0.812781
$626$	−21.1297	−0.844512
$627$	0	0
$628$	−30.3385	−1.21064
$629$	−37.0245	−1.47626
$630$	0	0
$631$	30.1729	1.20117	0.600583	−	0.799562i	$-0.294935\pi$
$631$	30.1729	1.20117	0.600583	+	0.799562i	$0.294935\pi$
$632$	1.00057	0.0398005
$633$	0	0
$634$	31.3272	1.24416
$635$	37.3064	1.48046
$636$	0	0
$637$	−8.93845	−0.354154
$638$	−24.9334	−0.987121
$639$	0	0
$640$	1.82311	0.0720648
$641$	43.1659	1.70495	0.852475	−	0.522768i	$-0.175100\pi$
$641$	43.1659	1.70495	0.852475	+	0.522768i	$0.175100\pi$
$642$	0	0
$643$	−7.57458	−0.298712	−0.149356	−	0.988783i	$-0.547720\pi$
$643$	−7.57458	−0.298712	−0.149356	+	0.988783i	$0.547720\pi$
$644$	0	0
$645$	0	0
$646$	−59.0021	−2.32141
$647$	−25.8899	−1.01784	−0.508918	−	0.860815i	$-0.669955\pi$
$647$	−25.8899	−1.01784	−0.508918	+	0.860815i	$0.669955\pi$
$648$	0	0
$649$	8.16837	0.320637
$650$	−4.15531	−0.162985
$651$	0	0
$652$	26.8754	1.05252
$653$	45.0028	1.76110	0.880548	−	0.473958i	$-0.157175\pi$
$653$	45.0028	1.76110	0.880548	+	0.473958i	$0.157175\pi$
$654$	0	0
$655$	2.97445	0.116221
$656$	−11.7661	−0.459391
$657$	0	0
$658$	−55.8377	−2.17678
$659$	30.9168	1.20435	0.602173	−	0.798366i	$-0.294302\pi$
$659$	30.9168	1.20435	0.602173	+	0.798366i	$0.294302\pi$
$660$	0	0
$661$	−11.2118	−0.436087	−0.218043	−	0.975939i	$-0.569967\pi$
$661$	−11.2118	−0.436087	−0.218043	+	0.975939i	$0.569967\pi$
$662$	24.4844	0.951612
$663$	0	0
$664$	1.70725	0.0662543
$665$	42.3287	1.64144
$666$	0	0
$667$	0	0
$668$	−5.05988	−0.195773
$669$	0	0
$670$	15.1684	0.586007
$671$	13.7493	0.530786
$672$	0	0
$673$	6.60149	0.254469	0.127234	−	0.991873i	$-0.459390\pi$
$673$	6.60149	0.254469	0.127234	+	0.991873i	$0.459390\pi$
$674$	−58.3762	−2.24857
$675$	0	0
$676$	−12.1749	−0.468265
$677$	29.0671	1.11714	0.558570	−	0.829458i	$-0.311350\pi$
$677$	29.0671	1.11714	0.558570	+	0.829458i	$0.311350\pi$
$678$	0	0
$679$	3.92604	0.150668
$680$	1.05900	0.0406107
$681$	0	0
$682$	−28.2931	−1.08340
$683$	12.3662	0.473178	0.236589	−	0.971610i	$-0.423970\pi$
$683$	12.3662	0.473178	0.236589	+	0.971610i	$0.423970\pi$
$684$	0	0
$685$	−9.16506	−0.350179
$686$	22.8238	0.871418
$687$	0	0
$688$	15.3181	0.583997
$689$	−3.90234	−0.148667
$690$	0	0
$691$	28.7504	1.09372	0.546858	−	0.837225i	$-0.315824\pi$
$691$	28.7504	1.09372	0.546858	+	0.837225i	$0.315824\pi$
$692$	31.6824	1.20438
$693$	0	0
$694$	59.5470	2.26037
$695$	−11.9105	−0.451790
$696$	0	0
$697$	−13.3016	−0.503833
$698$	−17.7906	−0.673384
$699$	0	0
$700$	−5.06425	−0.191411
$701$	−28.5719	−1.07915	−0.539573	−	0.841939i	$-0.681414\pi$
$701$	−28.5719	−1.07915	−0.539573	+	0.841939i	$0.681414\pi$
$702$	0	0
$703$	−50.9785	−1.92269
$704$	12.2640	0.462217
$705$	0	0
$706$	−19.0218	−0.715897
$707$	−24.6044	−0.925343
$708$	0	0
$709$	27.3419	1.02685	0.513424	−	0.858135i	$-0.328377\pi$
$709$	27.3419	1.02685	0.513424	+	0.858135i	$0.328377\pi$
$710$	5.01568	0.188235
$711$	0	0
$712$	0.290794	0.0108980
$713$	0	0
$714$	0	0
$715$	−8.63951	−0.323099
$716$	11.1584	0.417007
$717$	0	0
$718$	−27.9833	−1.04433
$719$	33.3145	1.24242	0.621211	−	0.783643i	$-0.286641\pi$
$719$	33.3145	1.24242	0.621211	+	0.783643i	$0.286641\pi$
$720$	0	0
$721$	35.0418	1.30503
$722$	−43.5064	−1.61914
$723$	0	0
$724$	22.6299	0.841032
$725$	6.22690	0.231261
$726$	0	0
$727$	−23.0443	−0.854665	−0.427333	−	0.904095i	$-0.640547\pi$
$727$	−23.0443	−0.854665	−0.427333	+	0.904095i	$0.640547\pi$
$728$	−0.933787	−0.0346084
$729$	0	0
$730$	48.0349	1.77785
$731$	17.3170	0.640494
$732$	0	0
$733$	−10.1108	−0.373450	−0.186725	−	0.982412i	$-0.559787\pi$
$733$	−10.1108	−0.373450	−0.186725	+	0.982412i	$0.559787\pi$
$734$	−16.6560	−0.614784
$735$	0	0
$736$	0	0
$737$	−6.06199	−0.223296
$738$	0	0
$739$	4.81027	0.176949	0.0884744	−	0.996078i	$-0.471801\pi$
$739$	4.81027	0.176949	0.0884744	+	0.996078i	$0.471801\pi$
$740$	−31.7306	−1.16644
$741$	0	0
$742$	−9.64905	−0.354228
$743$	26.6850	0.978978	0.489489	−	0.872010i	$-0.337183\pi$
$743$	26.6850	0.978978	0.489489	+	0.872010i	$0.337183\pi$
$744$	0	0
$745$	−0.0679809	−0.00249063
$746$	−35.4441	−1.29770
$747$	0	0
$748$	14.6769	0.536639
$749$	−14.8429	−0.542348
$750$	0	0
$751$	20.8396	0.760447	0.380223	−	0.924895i	$-0.375847\pi$
$751$	20.8396	0.760447	0.380223	+	0.924895i	$0.375847\pi$
$752$	−35.7493	−1.30364
$753$	0	0
$754$	−39.8169	−1.45004
$755$	−30.8460	−1.12260
$756$	0	0
$757$	14.6445	0.532262	0.266131	−	0.963937i	$-0.414255\pi$
$757$	14.6445	0.532262	0.266131	+	0.963937i	$0.414255\pi$
$758$	43.7831	1.59027
$759$	0	0
$760$	1.45812	0.0528916
$761$	−24.8121	−0.899436	−0.449718	−	0.893171i	$-0.648476\pi$
$761$	−24.8121	−0.899436	−0.449718	+	0.893171i	$0.648476\pi$
$762$	0	0
$763$	2.72852	0.0987790
$764$	−6.27755	−0.227114
$765$	0	0
$766$	−18.4019	−0.664886
$767$	13.0443	0.471004
$768$	0	0
$769$	−27.3778	−0.987269	−0.493635	−	0.869669i	$-0.664332\pi$
$769$	−27.3778	−0.987269	−0.493635	+	0.869669i	$0.664332\pi$
$770$	−21.3623	−0.769844
$771$	0	0
$772$	8.49344	0.305686
$773$	9.00208	0.323782	0.161891	−	0.986809i	$-0.448241\pi$
$773$	9.00208	0.323782	0.161891	+	0.986809i	$0.448241\pi$
$774$	0	0
$775$	7.06596	0.253817
$776$	0.135243	0.00485492
$777$	0	0
$778$	48.3911	1.73491
$779$	−18.3148	−0.656195
$780$	0	0
$781$	−2.00449	−0.0717264
$782$	0	0
$783$	0	0
$784$	−14.1501	−0.505362
$785$	31.9608	1.14073
$786$	0	0
$787$	−21.7249	−0.774410	−0.387205	−	0.921994i	$-0.626559\pi$
$787$	−21.7249	−0.774410	−0.387205	+	0.921994i	$0.626559\pi$
$788$	41.7478	1.48720
$789$	0	0
$790$	36.5540	1.30053
$791$	−35.9720	−1.27902
$792$	0	0
$793$	21.9567	0.779705
$794$	−55.7113	−1.97712
$795$	0	0
$796$	12.3334	0.437147
$797$	−28.8566	−1.02215	−0.511077	−	0.859535i	$-0.670753\pi$
$797$	−28.8566	−1.02215	−0.511077	+	0.859535i	$0.670753\pi$
$798$	0	0
$799$	−40.4144	−1.42976
$800$	−6.39865	−0.226226
$801$	0	0
$802$	26.2180	0.925789
$803$	−19.1969	−0.677445
$804$	0	0
$805$	0	0
$806$	−45.1821	−1.59147
$807$	0	0
$808$	−0.847561	−0.0298171
$809$	−34.7863	−1.22302	−0.611510	−	0.791236i	$-0.709438\pi$
$809$	−34.7863	−1.22302	−0.611510	+	0.791236i	$0.709438\pi$
$810$	0	0
$811$	11.0278	0.387238	0.193619	−	0.981077i	$-0.437977\pi$
$811$	11.0278	0.387238	0.193619	+	0.981077i	$0.437977\pi$
$812$	−48.5265	−1.70295
$813$	0	0
$814$	25.7277	0.901754
$815$	−28.3125	−0.991744
$816$	0	0
$817$	23.8436	0.834183
$818$	−39.3126	−1.37453
$819$	0	0
$820$	−11.3997	−0.398094
$821$	−13.7936	−0.481402	−0.240701	−	0.970599i	$-0.577377\pi$
$821$	−13.7936	−0.481402	−0.240701	+	0.970599i	$0.577377\pi$
$822$	0	0
$823$	−24.9701	−0.870404	−0.435202	−	0.900333i	$-0.643323\pi$
$823$	−24.9701	−0.870404	−0.435202	+	0.900333i	$0.643323\pi$
$824$	1.20710	0.0420515
$825$	0	0
$826$	32.2538	1.12225
$827$	29.2203	1.01609	0.508045	−	0.861331i	$-0.330369\pi$
$827$	29.2203	1.01609	0.508045	+	0.861331i	$0.330369\pi$
$828$	0	0
$829$	16.9029	0.587063	0.293531	−	0.955949i	$-0.405169\pi$
$829$	16.9029	0.587063	0.293531	+	0.955949i	$0.405169\pi$
$830$	62.3713	2.16494
$831$	0	0
$832$	19.5848	0.678980
$833$	−15.9967	−0.554252
$834$	0	0
$835$	5.33045	0.184468
$836$	20.2084	0.698922
$837$	0	0
$838$	9.08763	0.313927
$839$	1.58954	0.0548769	0.0274385	−	0.999623i	$-0.491265\pi$
$839$	1.58954	0.0548769	0.0274385	+	0.999623i	$0.491265\pi$
$840$	0	0
$841$	30.6671	1.05749
$842$	−12.7296	−0.438690
$843$	0	0
$844$	−28.7041	−0.988035
$845$	12.8259	0.441225
$846$	0	0
$847$	−27.0111	−0.928111
$848$	−6.17766	−0.212142
$849$	0	0
$850$	−7.43654	−0.255071
$851$	0	0
$852$	0	0
$853$	18.2259	0.624042	0.312021	−	0.950075i	$-0.398994\pi$
$853$	18.2259	0.624042	0.312021	+	0.950075i	$0.398994\pi$
$854$	54.2908	1.85779
$855$	0	0
$856$	−0.511301	−0.0174759
$857$	30.7156	1.04923	0.524613	−	0.851341i	$-0.324210\pi$
$857$	30.7156	1.04923	0.524613	+	0.851341i	$0.324210\pi$
$858$	0	0
$859$	11.0405	0.376695	0.188348	−	0.982102i	$-0.439687\pi$
$859$	11.0405	0.376695	0.188348	+	0.982102i	$0.439687\pi$
$860$	14.8410	0.506074
$861$	0	0
$862$	−22.4197	−0.763619
$863$	−15.9877	−0.544229	−0.272115	−	0.962265i	$-0.587723\pi$
$863$	−15.9877	−0.544229	−0.272115	+	0.962265i	$0.587723\pi$
$864$	0	0
$865$	−33.3766	−1.13484
$866$	9.08334	0.308665
$867$	0	0
$868$	−55.0654	−1.86904
$869$	−14.6086	−0.495564
$870$	0	0
$871$	−9.68057	−0.328014
$872$	0.0939907	0.00318293
$873$	0	0
$874$	0	0
$875$	38.4257	1.29903
$876$	0	0
$877$	−33.1090	−1.11801	−0.559006	−	0.829163i	$-0.688817\pi$
$877$	−33.1090	−1.11801	−0.559006	+	0.829163i	$0.688817\pi$
$878$	3.43142	0.115805
$879$	0	0
$880$	−13.6769	−0.461049
$881$	33.2335	1.11967	0.559833	−	0.828605i	$-0.310865\pi$
$881$	33.2335	1.11967	0.559833	+	0.828605i	$0.310865\pi$
$882$	0	0
$883$	−29.6421	−0.997538	−0.498769	−	0.866735i	$-0.666214\pi$
$883$	−29.6421	−0.997538	−0.498769	+	0.866735i	$0.666214\pi$
$884$	23.4379	0.788303
$885$	0	0
$886$	21.8812	0.735113
$887$	−9.86211	−0.331137	−0.165569	−	0.986198i	$-0.552946\pi$
$887$	−9.86211	−0.331137	−0.165569	+	0.986198i	$0.552946\pi$
$888$	0	0
$889$	−58.8713	−1.97448
$890$	10.6236	0.356105
$891$	0	0
$892$	−37.6072	−1.25918
$893$	−55.6461	−1.86212
$894$	0	0
$895$	−11.7550	−0.392927
$896$	−2.87695	−0.0961123
$897$	0	0
$898$	27.0177	0.901593
$899$	67.7072	2.25816
$900$	0	0
$901$	−6.98382	−0.232665
$902$	9.24303	0.307759
$903$	0	0
$904$	−1.23915	−0.0412134
$905$	−23.8400	−0.792468
$906$	0	0
$907$	39.0547	1.29679	0.648394	−	0.761305i	$-0.275441\pi$
$907$	39.0547	1.29679	0.648394	+	0.761305i	$0.275441\pi$
$908$	18.8862	0.626762
$909$	0	0
$910$	−34.1142	−1.13087
$911$	−8.59017	−0.284605	−0.142302	−	0.989823i	$-0.545451\pi$
$911$	−8.59017	−0.284605	−0.142302	+	0.989823i	$0.545451\pi$
$912$	0	0
$913$	−24.9264	−0.824944
$914$	−51.2161	−1.69408
$915$	0	0
$916$	10.3338	0.341439
$917$	−4.69383	−0.155004
$918$	0	0
$919$	6.75276	0.222753	0.111377	−	0.993778i	$-0.464474\pi$
$919$	6.75276	0.222753	0.111377	+	0.993778i	$0.464474\pi$
$920$	0	0
$921$	0	0
$922$	76.8243	2.53007
$923$	−3.20104	−0.105363
$924$	0	0
$925$	−6.42527	−0.211262
$926$	−21.4167	−0.703796
$927$	0	0
$928$	−61.3129	−2.01269
$929$	42.7013	1.40099	0.700493	−	0.713660i	$-0.252964\pi$
$929$	42.7013	1.40099	0.700493	+	0.713660i	$0.252964\pi$
$930$	0	0
$931$	−22.0256	−0.721861
$932$	32.7506	1.07278
$933$	0	0
$934$	29.7036	0.971933
$935$	−15.4617	−0.505651
$936$	0	0
$937$	7.43666	0.242945	0.121473	−	0.992595i	$-0.461238\pi$
$937$	7.43666	0.242945	0.121473	+	0.992595i	$0.461238\pi$
$938$	−23.9365	−0.781554
$939$	0	0
$940$	−34.6358	−1.12970
$941$	−46.4633	−1.51466	−0.757329	−	0.653033i	$-0.773496\pi$
$941$	−46.4633	−1.51466	−0.757329	+	0.653033i	$0.773496\pi$
$942$	0	0
$943$	0	0
$944$	20.6500	0.672101
$945$	0	0
$946$	−12.0333	−0.391237
$947$	11.1030	0.360798	0.180399	−	0.983594i	$-0.442261\pi$
$947$	11.1030	0.360798	0.180399	+	0.983594i	$0.442261\pi$
$948$	0	0
$949$	−30.6562	−0.995141
$950$	−10.2393	−0.332206
$951$	0	0
$952$	−1.67115	−0.0541623
$953$	−22.6088	−0.732369	−0.366185	−	0.930542i	$-0.619336\pi$
$953$	−22.6088	−0.732369	−0.366185	+	0.930542i	$0.619336\pi$
$954$	0	0
$955$	6.61323	0.213999
$956$	−27.7030	−0.895979
$957$	0	0
$958$	−84.6583	−2.73519
$959$	14.4629	0.467031
$960$	0	0
$961$	45.8306	1.47841
$962$	41.0853	1.32464
$963$	0	0
$964$	52.2900	1.68415
$965$	−8.94762	−0.288034
$966$	0	0
$967$	0.0544688	0.00175160	0.000875800	−	1.00000i	$-0.499721\pi$
$967$	0.0544688	0.00175160	0.000875800		1.00000i	$0.499721\pi$
$968$	−0.930465	−0.0299063
$969$	0	0
$970$	4.94083	0.158641
$971$	48.6670	1.56180	0.780899	−	0.624658i	$-0.214761\pi$
$971$	48.6670	1.56180	0.780899	+	0.624658i	$0.214761\pi$
$972$	0	0
$973$	18.7953	0.602549
$974$	65.5352	2.09988
$975$	0	0
$976$	34.7589	1.11261
$977$	−55.1331	−1.76386	−0.881931	−	0.471378i	$-0.843757\pi$
$977$	−55.1331	−1.76386	−0.881931	+	0.471378i	$0.843757\pi$
$978$	0	0
$979$	−4.24568	−0.135693
$980$	−13.7094	−0.437932
$981$	0	0
$982$	60.5201	1.93127
$983$	47.9686	1.52996	0.764981	−	0.644053i	$-0.222749\pi$
$983$	47.9686	1.52996	0.764981	+	0.644053i	$0.222749\pi$
$984$	0	0
$985$	−43.9802	−1.40133
$986$	−71.2582	−2.26932
$987$	0	0
$988$	32.2714	1.02669
$989$	0	0
$990$	0	0
$991$	−26.2692	−0.834469	−0.417234	−	0.908799i	$-0.637001\pi$
$991$	−26.2692	−0.834469	−0.417234	+	0.908799i	$0.637001\pi$
$992$	−69.5747	−2.20900
$993$	0	0
$994$	−7.91498	−0.251048
$995$	−12.9929	−0.411904
$996$	0	0
$997$	23.9595	0.758805	0.379403	−	0.925232i	$-0.376130\pi$
$997$	23.9595	0.758805	0.379403	+	0.925232i	$0.376130\pi$
$998$	53.1026	1.68093
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bt.1.2		10
3.2	odd	2		1587.2.a.u.1.9		10
23.3	even	11		207.2.i.d.55.2		20
23.8	even	11		207.2.i.d.64.2		20
23.22	odd	2		4761.2.a.bu.1.2		10
69.8	odd	22		69.2.e.c.64.1	yes	20
69.26	odd	22		69.2.e.c.55.1	✓	20
69.68	even	2		1587.2.a.t.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.c.55.1	✓	20	69.26	odd	22
69.2.e.c.64.1	yes	20	69.8	odd	22
207.2.i.d.55.2		20	23.3	even	11
207.2.i.d.64.2		20	23.8	even	11
1587.2.a.t.1.9		10	69.68	even	2
1587.2.a.u.1.9		10	3.2	odd	2
4761.2.a.bt.1.2		10	1.1	even	1	trivial
4761.2.a.bu.1.2		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)

\(f(q)\)	\(=\)	\(q-1.98594 q^{2} +1.94394 q^{4} -2.04789 q^{5} +3.23167 q^{7} +0.111323 q^{8} +O(q^{10})\)	\(q-1.98594 q^{2} +1.94394 q^{4} -2.04789 q^{5} +3.23167 q^{7} +0.111323 q^{8} +4.06699 q^{10} -1.62535 q^{11} -2.59558 q^{13} -6.41790 q^{14} -4.10897 q^{16} -4.64517 q^{17} -6.39588 q^{19} -3.98099 q^{20} +3.22785 q^{22} -0.806128 q^{25} +5.15465 q^{26} +6.28220 q^{28} -7.72445 q^{29} -8.76531 q^{31} +7.93751 q^{32} +9.22502 q^{34} -6.61813 q^{35} +7.97053 q^{37} +12.7018 q^{38} -0.227978 q^{40} +2.86353 q^{41} -3.72797 q^{43} -3.15960 q^{44} +8.70030 q^{47} +3.44372 q^{49} +1.60092 q^{50} -5.04566 q^{52} +1.50346 q^{53} +3.32855 q^{55} +0.359761 q^{56} +15.3403 q^{58} -5.02560 q^{59} -8.45927 q^{61} +17.4073 q^{62} -7.54545 q^{64} +5.31547 q^{65} +3.72964 q^{67} -9.02995 q^{68} +13.1432 q^{70} +1.23327 q^{71} +11.8109 q^{73} -15.8290 q^{74} -12.4332 q^{76} -5.25261 q^{77} +8.98797 q^{79} +8.41474 q^{80} -5.68678 q^{82} +15.3360 q^{83} +9.51282 q^{85} +7.40350 q^{86} -0.180940 q^{88} +2.61216 q^{89} -8.38806 q^{91} -17.2783 q^{94} +13.0981 q^{95} +1.21486 q^{97} -6.83901 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} - 8 q^{5} + 19 q^{7} + 6 q^{8}+O(q^{10}) \) 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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Database

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