LMFDB - Embedded newform 4761.2.a.bh.1.1

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:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.517638$ 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)

$f(q)$	$=$	$q-2.44949 q^{2} +4.00000 q^{4} -4.24264 q^{5} +1.73205 q^{7} -4.89898 q^{8} +O(q^{10})$	$q-2.44949 q^{2} +4.00000 q^{4} -4.24264 q^{5} +1.73205 q^{7} -4.89898 q^{8} +10.3923 q^{10} -4.24264 q^{11} +1.00000 q^{13} -4.24264 q^{14} +4.00000 q^{16} +4.24264 q^{17} -6.92820 q^{19} -16.9706 q^{20} +10.3923 q^{22} +13.0000 q^{25} -2.44949 q^{26} +6.92820 q^{28} -2.44949 q^{29} -2.00000 q^{31} -10.3923 q^{34} -7.34847 q^{35} -5.19615 q^{37} +16.9706 q^{38} +20.7846 q^{40} +4.89898 q^{41} -8.66025 q^{43} -16.9706 q^{44} -2.44949 q^{47} -4.00000 q^{49} -31.8434 q^{50} +4.00000 q^{52} +8.48528 q^{53} +18.0000 q^{55} -8.48528 q^{56} +6.00000 q^{58} -7.34847 q^{59} -5.19615 q^{61} +4.89898 q^{62} -8.00000 q^{64} -4.24264 q^{65} +8.66025 q^{67} +16.9706 q^{68} +18.0000 q^{70} +2.44949 q^{71} +4.00000 q^{73} +12.7279 q^{74} -27.7128 q^{76} -7.34847 q^{77} -6.92820 q^{79} -16.9706 q^{80} -12.0000 q^{82} +8.48528 q^{83} -18.0000 q^{85} +21.2132 q^{86} +20.7846 q^{88} -12.7279 q^{89} +1.73205 q^{91} +6.00000 q^{94} +29.3939 q^{95} -13.8564 q^{97} +9.79796 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4}+O(q^{10}) $ 4 * q + 16 * q^4	$ 4 q + 16 q^{4} + 4 q^{13} + 16 q^{16} + 52 q^{25} - 8 q^{31} - 16 q^{49} + 16 q^{52} + 72 q^{55} + 24 q^{58} - 32 q^{64} + 72 q^{70} + 16 q^{73} - 48 q^{82} - 72 q^{85} + 24 q^{94}+O(q^{100}) $ 4 * q + 16 * q^4 + 4 * q^13 + 16 * q^16 + 52 * q^25 - 8 * q^31 - 16 * q^49 + 16 * q^52 + 72 * q^55 + 24 * q^58 - 32 * q^64 + 72 * q^70 + 16 * q^73 - 48 * q^82 - 72 * q^85 + 24 * q^94

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 $

$2$	−2.44949	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−2.44949	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	2.00000
$5$	−4.24264	−1.89737	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−4.24264	−1.89737	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	−4.89898	−1.73205
$9$	0	0
$10$	10.3923	3.28634
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−4.24264	−1.13389
$15$	0	0
$16$	4.00000	1.00000
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	−16.9706	−3.79473
$21$	0	0
$22$	10.3923	2.21565
$23$	0	0
$23$	0	0
$24$	0	0
$25$	13.0000	2.60000
$26$	−2.44949	−0.480384
$27$	0	0
$28$	6.92820	1.30931
$29$	−2.44949	−0.454859	−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949	−0.454859	−0.227429	+	0.973795i	$0.573032\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	−10.3923	−1.78227
$35$	−7.34847	−1.24212
$36$	0	0
$37$	−5.19615	−0.854242	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	−5.19615	−0.854242	−0.427121	+	0.904194i	$0.640472\pi$
$38$	16.9706	2.75299
$39$	0	0
$40$	20.7846	3.28634
$41$	4.89898	0.765092	0.382546	−	0.923936i	$-0.375047\pi$
$41$	4.89898	0.765092	0.382546	+	0.923936i	$0.375047\pi$
$42$	0	0
$43$	−8.66025	−1.32068	−0.660338	−	0.750968i	$-0.729587\pi$
$43$	−8.66025	−1.32068	−0.660338	+	0.750968i	$0.729587\pi$
$44$	−16.9706	−2.55841
$45$	0	0
$46$	0	0
$47$	−2.44949	−0.357295	−0.178647	−	0.983913i	$-0.557172\pi$
$47$	−2.44949	−0.357295	−0.178647	+	0.983913i	$0.557172\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	−31.8434	−4.50333
$51$	0	0
$52$	4.00000	0.554700
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	18.0000	2.42712
$56$	−8.48528	−1.13389
$57$	0	0
$58$	6.00000	0.787839
$59$	−7.34847	−0.956689	−0.478345	−	0.878172i	$-0.658763\pi$
$59$	−7.34847	−0.956689	−0.478345	+	0.878172i	$0.658763\pi$
$60$	0	0
$61$	−5.19615	−0.665299	−0.332650	−	0.943051i	$-0.607943\pi$
$61$	−5.19615	−0.665299	−0.332650	+	0.943051i	$0.607943\pi$
$62$	4.89898	0.622171
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−4.24264	−0.526235
$66$	0	0
$67$	8.66025	1.05802	0.529009	−	0.848616i	$-0.322564\pi$
$67$	8.66025	1.05802	0.529009	+	0.848616i	$0.322564\pi$
$68$	16.9706	2.05798
$69$	0	0
$70$	18.0000	2.15141
$71$	2.44949	0.290701	0.145350	−	0.989380i	$-0.453569\pi$
$71$	2.44949	0.290701	0.145350	+	0.989380i	$0.453569\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	12.7279	1.47959
$75$	0	0
$76$	−27.7128	−3.17888
$77$	−7.34847	−0.837436
$78$	0	0
$79$	−6.92820	−0.779484	−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820	−0.779484	−0.389742	+	0.920924i	$0.627436\pi$
$80$	−16.9706	−1.89737
$81$	0	0
$82$	−12.0000	−1.32518
$83$	8.48528	0.931381	0.465690	−	0.884948i	$-0.345806\pi$
$83$	8.48528	0.931381	0.465690	+	0.884948i	$0.345806\pi$
$84$	0	0
$85$	−18.0000	−1.95237
$86$	21.2132	2.28748
$87$	0	0
$88$	20.7846	2.21565
$89$	−12.7279	−1.34916	−0.674579	−	0.738203i	$-0.735675\pi$
$89$	−12.7279	−1.34916	−0.674579	+	0.738203i	$0.735675\pi$
$90$	0	0
$91$	1.73205	0.181568
$92$	0	0
$93$	0	0
$94$	6.00000	0.618853
$95$	29.3939	3.01575
$96$	0	0
$97$	−13.8564	−1.40690	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564	−1.40690	−0.703452	+	0.710742i	$0.748359\pi$
$98$	9.79796	0.989743
$99$	0	0
$100$	52.0000	5.20000
$101$	−7.34847	−0.731200	−0.365600	−	0.930772i	$-0.619136\pi$
$101$	−7.34847	−0.731200	−0.365600	+	0.930772i	$0.619136\pi$
$102$	0	0
$103$	−5.19615	−0.511992	−0.255996	−	0.966678i	$-0.582403\pi$
$103$	−5.19615	−0.511992	−0.255996	+	0.966678i	$0.582403\pi$
$104$	−4.89898	−0.480384
$105$	0	0
$106$	−20.7846	−2.01878
$107$	−8.48528	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$108$	0	0
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	−44.0908	−4.20389
$111$	0	0
$112$	6.92820	0.654654
$113$	8.48528	0.798228	0.399114	−	0.916901i	$-0.369318\pi$
$113$	8.48528	0.798228	0.399114	+	0.916901i	$0.369318\pi$
$114$	0	0
$115$	0	0
$116$	−9.79796	−0.909718
$117$	0	0
$118$	18.0000	1.65703
$119$	7.34847	0.673633
$120$	0	0
$121$	7.00000	0.636364
$122$	12.7279	1.15233
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−33.9411	−3.03579
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	19.5959	1.73205
$129$	0	0
$130$	10.3923	0.911465
$131$	9.79796	0.856052	0.428026	−	0.903767i	$-0.359209\pi$
$131$	9.79796	0.856052	0.428026	+	0.903767i	$0.359209\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	−21.2132	−1.83254
$135$	0	0
$136$	−20.7846	−1.78227
$137$	−4.24264	−0.362473	−0.181237	−	0.983440i	$-0.558010\pi$
$137$	−4.24264	−0.362473	−0.181237	+	0.983440i	$0.558010\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	−29.3939	−2.48424
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−4.24264	−0.354787
$144$	0	0
$145$	10.3923	0.863034
$146$	−9.79796	−0.810885
$147$	0	0
$148$	−20.7846	−1.70848
$149$	−8.48528	−0.695141	−0.347571	−	0.937654i	$-0.612993\pi$
$149$	−8.48528	−0.695141	−0.347571	+	0.937654i	$0.612993\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	33.9411	2.75299
$153$	0	0
$154$	18.0000	1.45048
$155$	8.48528	0.681554
$156$	0	0
$157$	10.3923	0.829396	0.414698	−	0.909959i	$-0.363887\pi$
$157$	10.3923	0.829396	0.414698	+	0.909959i	$0.363887\pi$
$158$	16.9706	1.35011
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.0000	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	−17.0000	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$164$	19.5959	1.53018
$165$	0	0
$166$	−20.7846	−1.61320
$167$	−14.6969	−1.13728	−0.568642	−	0.822585i	$-0.692531\pi$
$167$	−14.6969	−1.13728	−0.568642	+	0.822585i	$0.692531\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	44.0908	3.38161
$171$	0	0
$172$	−34.6410	−2.64135
$173$	−22.0454	−1.67608	−0.838041	−	0.545608i	$-0.816299\pi$
$173$	−22.0454	−1.67608	−0.838041	+	0.545608i	$0.816299\pi$
$174$	0	0
$175$	22.5167	1.70210
$176$	−16.9706	−1.27920
$177$	0	0
$178$	31.1769	2.33681
$179$	2.44949	0.183083	0.0915417	−	0.995801i	$-0.470821\pi$
$179$	2.44949	0.183083	0.0915417	+	0.995801i	$0.470821\pi$
$180$	0	0
$181$	3.46410	0.257485	0.128742	−	0.991678i	$-0.458906\pi$
$181$	3.46410	0.257485	0.128742	+	0.991678i	$0.458906\pi$
$182$	−4.24264	−0.314485
$183$	0	0
$184$	0	0
$185$	22.0454	1.62081
$186$	0	0
$187$	−18.0000	−1.31629
$188$	−9.79796	−0.714590
$189$	0	0
$190$	−72.0000	−5.22343
$191$	8.48528	0.613973	0.306987	−	0.951714i	$-0.400679\pi$
$191$	8.48528	0.613973	0.306987	+	0.951714i	$0.400679\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	33.9411	2.43683
$195$	0	0
$196$	−16.0000	−1.14286
$197$	−4.89898	−0.349038	−0.174519	−	0.984654i	$-0.555837\pi$
$197$	−4.89898	−0.349038	−0.174519	+	0.984654i	$0.555837\pi$
$198$	0	0
$199$	8.66025	0.613909	0.306955	−	0.951724i	$-0.400690\pi$
$199$	8.66025	0.613909	0.306955	+	0.951724i	$0.400690\pi$
$200$	−63.6867	−4.50333
$201$	0	0
$202$	18.0000	1.26648
$203$	−4.24264	−0.297775
$204$	0	0
$205$	−20.7846	−1.45166
$206$	12.7279	0.886796
$207$	0	0
$208$	4.00000	0.277350
$209$	29.3939	2.03322
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	33.9411	2.33109
$213$	0	0
$214$	20.7846	1.42081
$215$	36.7423	2.50581
$216$	0	0
$217$	−3.46410	−0.235159
$218$	25.4558	1.72409
$219$	0	0
$220$	72.0000	4.85424
$221$	4.24264	0.285391
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	0	0
$225$	0	0
$226$	−20.7846	−1.38257
$227$	−4.24264	−0.281594	−0.140797	−	0.990038i	$-0.544966\pi$
$227$	−4.24264	−0.281594	−0.140797	+	0.990038i	$0.544966\pi$
$228$	0	0
$229$	19.0526	1.25903	0.629514	−	0.776989i	$-0.283254\pi$
$229$	19.0526	1.25903	0.629514	+	0.776989i	$0.283254\pi$
$230$	0	0
$231$	0	0
$232$	12.0000	0.787839
$233$	−9.79796	−0.641886	−0.320943	−	0.947099i	$-0.604000\pi$
$233$	−9.79796	−0.641886	−0.320943	+	0.947099i	$0.604000\pi$
$234$	0	0
$235$	10.3923	0.677919
$236$	−29.3939	−1.91338
$237$	0	0
$238$	−18.0000	−1.16677
$239$	2.44949	0.158444	0.0792222	−	0.996857i	$-0.474756\pi$
$239$	2.44949	0.158444	0.0792222	+	0.996857i	$0.474756\pi$
$240$	0	0
$241$	15.5885	1.00414	0.502070	−	0.864827i	$-0.332572\pi$
$241$	15.5885	1.00414	0.502070	+	0.864827i	$0.332572\pi$
$242$	−17.1464	−1.10221
$243$	0	0
$244$	−20.7846	−1.33060
$245$	16.9706	1.08421
$246$	0	0
$247$	−6.92820	−0.440831
$248$	9.79796	0.622171
$249$	0	0
$250$	83.1384	5.25814
$251$	−8.48528	−0.535586	−0.267793	−	0.963476i	$-0.586294\pi$
$251$	−8.48528	−0.535586	−0.267793	+	0.963476i	$0.586294\pi$
$252$	0	0
$253$	0	0
$254$	−31.8434	−1.99803
$255$	0	0
$256$	−32.0000	−2.00000
$257$	−4.89898	−0.305590	−0.152795	−	0.988258i	$-0.548827\pi$
$257$	−4.89898	−0.305590	−0.152795	+	0.988258i	$0.548827\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	−16.9706	−1.05247
$261$	0	0
$262$	−24.0000	−1.48272
$263$	4.24264	0.261612	0.130806	−	0.991408i	$-0.458243\pi$
$263$	4.24264	0.261612	0.130806	+	0.991408i	$0.458243\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	29.3939	1.80225
$267$	0	0
$268$	34.6410	2.11604
$269$	4.89898	0.298696	0.149348	−	0.988785i	$-0.452283\pi$
$269$	4.89898	0.298696	0.149348	+	0.988785i	$0.452283\pi$
$270$	0	0
$271$	25.0000	1.51864	0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000	1.51864	0.759321	+	0.650716i	$0.225531\pi$
$272$	16.9706	1.02899
$273$	0	0
$274$	10.3923	0.627822
$275$	−55.1543	−3.32593
$276$	0	0
$277$	−29.0000	−1.74244	−0.871221	−	0.490892i	$-0.836671\pi$
$277$	−29.0000	−1.74244	−0.871221	+	0.490892i	$0.836671\pi$
$278$	−26.9444	−1.61602
$279$	0	0
$280$	36.0000	2.15141
$281$	−25.4558	−1.51857	−0.759284	−	0.650759i	$-0.774451\pi$
$281$	−25.4558	−1.51857	−0.759284	+	0.650759i	$0.774451\pi$
$282$	0	0
$283$	8.66025	0.514799	0.257399	−	0.966305i	$-0.417134\pi$
$283$	8.66025	0.514799	0.257399	+	0.966305i	$0.417134\pi$
$284$	9.79796	0.581402
$285$	0	0
$286$	10.3923	0.614510
$287$	8.48528	0.500870
$288$	0	0
$289$	1.00000	0.0588235
$290$	−25.4558	−1.49482
$291$	0	0
$292$	16.0000	0.936329
$293$	4.24264	0.247858	0.123929	−	0.992291i	$-0.460451\pi$
$293$	4.24264	0.247858	0.123929	+	0.992291i	$0.460451\pi$
$294$	0	0
$295$	31.1769	1.81519
$296$	25.4558	1.47959
$297$	0	0
$298$	20.7846	1.20402
$299$	0	0
$300$	0	0
$301$	−15.0000	−0.864586
$302$	−26.9444	−1.55048
$303$	0	0
$304$	−27.7128	−1.58944
$305$	22.0454	1.26232
$306$	0	0
$307$	19.0000	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$308$	−29.3939	−1.67487
$309$	0	0
$310$	−20.7846	−1.18049
$311$	−24.4949	−1.38898	−0.694489	−	0.719503i	$-0.744370\pi$
$311$	−24.4949	−1.38898	−0.694489	+	0.719503i	$0.744370\pi$
$312$	0	0
$313$	5.19615	0.293704	0.146852	−	0.989158i	$-0.453086\pi$
$313$	5.19615	0.293704	0.146852	+	0.989158i	$0.453086\pi$
$314$	−25.4558	−1.43656
$315$	0	0
$316$	−27.7128	−1.55897
$317$	29.3939	1.65092	0.825462	−	0.564457i	$-0.190915\pi$
$317$	29.3939	1.65092	0.825462	+	0.564457i	$0.190915\pi$
$318$	0	0
$319$	10.3923	0.581857
$320$	33.9411	1.89737
$321$	0	0
$322$	0	0
$323$	−29.3939	−1.63552
$324$	0	0
$325$	13.0000	0.721110
$326$	41.6413	2.30630
$327$	0	0
$328$	−24.0000	−1.32518
$329$	−4.24264	−0.233904
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	33.9411	1.86276
$333$	0	0
$334$	36.0000	1.96983
$335$	−36.7423	−2.00745
$336$	0	0
$337$	22.5167	1.22656	0.613280	−	0.789865i	$-0.289850\pi$
$337$	22.5167	1.22656	0.613280	+	0.789865i	$0.289850\pi$
$338$	29.3939	1.59882
$339$	0	0
$340$	−72.0000	−3.90475
$341$	8.48528	0.459504
$342$	0	0
$343$	−19.0526	−1.02874
$344$	42.4264	2.28748
$345$	0	0
$346$	54.0000	2.90306
$347$	7.34847	0.394486	0.197243	−	0.980355i	$-0.436801\pi$
$347$	7.34847	0.394486	0.197243	+	0.980355i	$0.436801\pi$
$348$	0	0
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	−55.1543	−2.94812
$351$	0	0
$352$	0	0
$353$	−9.79796	−0.521493	−0.260746	−	0.965407i	$-0.583969\pi$
$353$	−9.79796	−0.521493	−0.260746	+	0.965407i	$0.583969\pi$
$354$	0	0
$355$	−10.3923	−0.551566
$356$	−50.9117	−2.69831
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−16.9706	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	−8.48528	−0.445976
$363$	0	0
$364$	6.92820	0.363137
$365$	−16.9706	−0.888280
$366$	0	0
$367$	−13.8564	−0.723299	−0.361649	−	0.932314i	$-0.617786\pi$
$367$	−13.8564	−0.723299	−0.361649	+	0.932314i	$0.617786\pi$
$368$	0	0
$369$	0	0
$370$	−54.0000	−2.80733
$371$	14.6969	0.763027
$372$	0	0
$373$	−5.19615	−0.269047	−0.134523	−	0.990910i	$-0.542950\pi$
$373$	−5.19615	−0.269047	−0.134523	+	0.990910i	$0.542950\pi$
$374$	44.0908	2.27988
$375$	0	0
$376$	12.0000	0.618853
$377$	−2.44949	−0.126155
$378$	0	0
$379$	25.9808	1.33454	0.667271	−	0.744815i	$-0.267462\pi$
$379$	25.9808	1.33454	0.667271	+	0.744815i	$0.267462\pi$
$380$	117.576	6.03150
$381$	0	0
$382$	−20.7846	−1.06343
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	31.1769	1.58892
$386$	17.1464	0.872730
$387$	0	0
$388$	−55.4256	−2.81381
$389$	38.1838	1.93599	0.967997	−	0.250962i	$-0.0807470\pi$
$389$	38.1838	1.93599	0.967997	+	0.250962i	$0.0807470\pi$
$390$	0	0
$391$	0	0
$392$	19.5959	0.989743
$393$	0	0
$394$	12.0000	0.604551
$395$	29.3939	1.47897
$396$	0	0
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	−21.2132	−1.06332
$399$	0	0
$400$	52.0000	2.60000
$401$	−8.48528	−0.423735	−0.211867	−	0.977298i	$-0.567954\pi$
$401$	−8.48528	−0.423735	−0.211867	+	0.977298i	$0.567954\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	−29.3939	−1.46240
$405$	0	0
$406$	10.3923	0.515761
$407$	22.0454	1.09275
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	50.9117	2.51435
$411$	0	0
$412$	−20.7846	−1.02398
$413$	−12.7279	−0.626300
$414$	0	0
$415$	−36.0000	−1.76717
$416$	0	0
$417$	0	0
$418$	−72.0000	−3.52164
$419$	21.2132	1.03633	0.518166	−	0.855280i	$-0.326615\pi$
$419$	21.2132	1.03633	0.518166	+	0.855280i	$0.326615\pi$
$420$	0	0
$421$	−32.9090	−1.60388	−0.801942	−	0.597401i	$-0.796200\pi$
$421$	−32.9090	−1.60388	−0.801942	+	0.597401i	$0.796200\pi$
$422$	−31.8434	−1.55011
$423$	0	0
$424$	−41.5692	−2.01878
$425$	55.1543	2.67538
$426$	0	0
$427$	−9.00000	−0.435541
$428$	−33.9411	−1.64061
$429$	0	0
$430$	−90.0000	−4.34019
$431$	−25.4558	−1.22616	−0.613082	−	0.790019i	$-0.710071\pi$
$431$	−25.4558	−1.22616	−0.613082	+	0.790019i	$0.710071\pi$
$432$	0	0
$433$	−12.1244	−0.582659	−0.291330	−	0.956623i	$-0.594098\pi$
$433$	−12.1244	−0.582659	−0.291330	+	0.956623i	$0.594098\pi$
$434$	8.48528	0.407307
$435$	0	0
$436$	−41.5692	−1.99080
$437$	0	0
$438$	0	0
$439$	29.0000	1.38409	0.692047	−	0.721852i	$-0.256709\pi$
$439$	29.0000	1.38409	0.692047	+	0.721852i	$0.256709\pi$
$440$	−88.1816	−4.20389
$441$	0	0
$442$	−10.3923	−0.494312
$443$	−41.6413	−1.97844	−0.989220	−	0.146440i	$-0.953218\pi$
$443$	−41.6413	−1.97844	−0.989220	+	0.146440i	$0.953218\pi$
$444$	0	0
$445$	54.0000	2.55985
$446$	−56.3383	−2.66769
$447$	0	0
$448$	−13.8564	−0.654654
$449$	7.34847	0.346796	0.173398	−	0.984852i	$-0.444525\pi$
$449$	7.34847	0.346796	0.173398	+	0.984852i	$0.444525\pi$
$450$	0	0
$451$	−20.7846	−0.978709
$452$	33.9411	1.59646
$453$	0	0
$454$	10.3923	0.487735
$455$	−7.34847	−0.344502
$456$	0	0
$457$	−1.73205	−0.0810219	−0.0405110	−	0.999179i	$-0.512899\pi$
$457$	−1.73205	−0.0810219	−0.0405110	+	0.999179i	$0.512899\pi$
$458$	−46.6690	−2.18070
$459$	0	0
$460$	0	0
$461$	31.8434	1.48309	0.741547	−	0.670901i	$-0.234093\pi$
$461$	31.8434	1.48309	0.741547	+	0.670901i	$0.234093\pi$
$462$	0	0
$463$	41.0000	1.90543	0.952716	−	0.303863i	$-0.0982765\pi$
$463$	41.0000	1.90543	0.952716	+	0.303863i	$0.0982765\pi$
$464$	−9.79796	−0.454859
$465$	0	0
$466$	24.0000	1.11178
$467$	21.2132	0.981630	0.490815	−	0.871264i	$-0.336699\pi$
$467$	21.2132	0.981630	0.490815	+	0.871264i	$0.336699\pi$
$468$	0	0
$469$	15.0000	0.692636
$470$	−25.4558	−1.17419
$471$	0	0
$472$	36.0000	1.65703
$473$	36.7423	1.68941
$474$	0	0
$475$	−90.0666	−4.13254
$476$	29.3939	1.34727
$477$	0	0
$478$	−6.00000	−0.274434
$479$	−16.9706	−0.775405	−0.387702	−	0.921785i	$-0.626731\pi$
$479$	−16.9706	−0.775405	−0.387702	+	0.921785i	$0.626731\pi$
$480$	0	0
$481$	−5.19615	−0.236924
$482$	−38.1838	−1.73922
$483$	0	0
$484$	28.0000	1.27273
$485$	58.7878	2.66941
$486$	0	0
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	25.4558	1.15233
$489$	0	0
$490$	−41.5692	−1.87791
$491$	31.8434	1.43707	0.718536	−	0.695490i	$-0.244813\pi$
$491$	31.8434	1.43707	0.718536	+	0.695490i	$0.244813\pi$
$492$	0	0
$493$	−10.3923	−0.468046
$494$	16.9706	0.763542
$495$	0	0
$496$	−8.00000	−0.359211
$497$	4.24264	0.190308
$498$	0	0
$499$	19.0000	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$499$	19.0000	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$500$	−135.765	−6.07157
$501$	0	0
$502$	20.7846	0.927663
$503$	38.1838	1.70253	0.851265	−	0.524736i	$-0.175836\pi$
$503$	38.1838	1.70253	0.851265	+	0.524736i	$0.175836\pi$
$504$	0	0
$505$	31.1769	1.38735
$506$	0	0
$507$	0	0
$508$	52.0000	2.30713
$509$	−19.5959	−0.868574	−0.434287	−	0.900775i	$-0.643000\pi$
$509$	−19.5959	−0.868574	−0.434287	+	0.900775i	$0.643000\pi$
$510$	0	0
$511$	6.92820	0.306486
$512$	39.1918	1.73205
$513$	0	0
$514$	12.0000	0.529297
$515$	22.0454	0.971437
$516$	0	0
$517$	10.3923	0.457053
$518$	22.0454	0.968620
$519$	0	0
$520$	20.7846	0.911465
$521$	42.4264	1.85873	0.929367	−	0.369156i	$-0.120353\pi$
$521$	42.4264	1.85873	0.929367	+	0.369156i	$0.120353\pi$
$522$	0	0
$523$	27.7128	1.21180	0.605898	−	0.795542i	$-0.292814\pi$
$523$	27.7128	1.21180	0.605898	+	0.795542i	$0.292814\pi$
$524$	39.1918	1.71210
$525$	0	0
$526$	−10.3923	−0.453126
$527$	−8.48528	−0.369625
$528$	0	0
$529$	0	0
$530$	88.1816	3.83037
$531$	0	0
$532$	−48.0000	−2.08106
$533$	4.89898	0.212198
$534$	0	0
$535$	36.0000	1.55642
$536$	−42.4264	−1.83254
$537$	0	0
$538$	−12.0000	−0.517357
$539$	16.9706	0.730974
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	−61.2372	−2.63036
$543$	0	0
$544$	0	0
$545$	44.0908	1.88864
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−16.9706	−0.724947
$549$	0	0
$550$	135.100	5.76068
$551$	16.9706	0.722970
$552$	0	0
$553$	−12.0000	−0.510292
$554$	71.0352	3.01800
$555$	0	0
$556$	44.0000	1.86602
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−8.66025	−0.366290
$560$	−29.3939	−1.24212
$561$	0	0
$562$	62.3538	2.63024
$563$	33.9411	1.43045	0.715224	−	0.698895i	$-0.246325\pi$
$563$	33.9411	1.43045	0.715224	+	0.698895i	$0.246325\pi$
$564$	0	0
$565$	−36.0000	−1.51453
$566$	−21.2132	−0.891657
$567$	0	0
$568$	−12.0000	−0.503509
$569$	38.1838	1.60075	0.800373	−	0.599502i	$-0.204635\pi$
$569$	38.1838	1.60075	0.800373	+	0.599502i	$0.204635\pi$
$570$	0	0
$571$	25.9808	1.08726	0.543631	−	0.839325i	$-0.317049\pi$
$571$	25.9808	1.08726	0.543631	+	0.839325i	$0.317049\pi$
$572$	−16.9706	−0.709575
$573$	0	0
$574$	−20.7846	−0.867533
$575$	0	0
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−2.44949	−0.101885
$579$	0	0
$580$	41.5692	1.72607
$581$	14.6969	0.609732
$582$	0	0
$583$	−36.0000	−1.49097
$584$	−19.5959	−0.810885
$585$	0	0
$586$	−10.3923	−0.429302
$587$	31.8434	1.31432	0.657158	−	0.753753i	$-0.271758\pi$
$587$	31.8434	1.31432	0.657158	+	0.753753i	$0.271758\pi$
$588$	0	0
$589$	13.8564	0.570943
$590$	−76.3675	−3.14400
$591$	0	0
$592$	−20.7846	−0.854242
$593$	7.34847	0.301765	0.150883	−	0.988552i	$-0.451788\pi$
$593$	7.34847	0.301765	0.150883	+	0.988552i	$0.451788\pi$
$594$	0	0
$595$	−31.1769	−1.27813
$596$	−33.9411	−1.39028
$597$	0	0
$598$	0	0
$599$	36.7423	1.50125	0.750626	−	0.660728i	$-0.229752\pi$
$599$	36.7423	1.50125	0.750626	+	0.660728i	$0.229752\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	36.7423	1.49751
$603$	0	0
$604$	44.0000	1.79033
$605$	−29.6985	−1.20742
$606$	0	0
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	0	0
$609$	0	0
$610$	−54.0000	−2.18640
$611$	−2.44949	−0.0990957
$612$	0	0
$613$	15.5885	0.629612	0.314806	−	0.949156i	$-0.398061\pi$
$613$	15.5885	0.629612	0.314806	+	0.949156i	$0.398061\pi$
$614$	−46.5403	−1.87821
$615$	0	0
$616$	36.0000	1.45048
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−43.3013	−1.74042	−0.870212	−	0.492677i	$-0.836019\pi$
$619$	−43.3013	−1.74042	−0.870212	+	0.492677i	$0.836019\pi$
$620$	33.9411	1.36311
$621$	0	0
$622$	60.0000	2.40578
$623$	−22.0454	−0.883231
$624$	0	0
$625$	79.0000	3.16000
$626$	−12.7279	−0.508710
$627$	0	0
$628$	41.5692	1.65879
$629$	−22.0454	−0.879008
$630$	0	0
$631$	22.5167	0.896374	0.448187	−	0.893940i	$-0.352070\pi$
$631$	22.5167	0.896374	0.448187	+	0.893940i	$0.352070\pi$
$632$	33.9411	1.35011
$633$	0	0
$634$	−72.0000	−2.85949
$635$	−55.1543	−2.18873
$636$	0	0
$637$	−4.00000	−0.158486
$638$	−25.4558	−1.00781
$639$	0	0
$640$	−83.1384	−3.28634
$641$	33.9411	1.34059	0.670297	−	0.742093i	$-0.266167\pi$
$641$	33.9411	1.34059	0.670297	+	0.742093i	$0.266167\pi$
$642$	0	0
$643$	43.3013	1.70764	0.853818	−	0.520572i	$-0.174281\pi$
$643$	43.3013	1.70764	0.853818	+	0.520572i	$0.174281\pi$
$644$	0	0
$645$	0	0
$646$	72.0000	2.83280
$647$	41.6413	1.63709	0.818545	−	0.574443i	$-0.194781\pi$
$647$	41.6413	1.63709	0.818545	+	0.574443i	$0.194781\pi$
$648$	0	0
$649$	31.1769	1.22380
$650$	−31.8434	−1.24900
$651$	0	0
$652$	−68.0000	−2.66309
$653$	9.79796	0.383424	0.191712	−	0.981451i	$-0.438596\pi$
$653$	9.79796	0.383424	0.191712	+	0.981451i	$0.438596\pi$
$654$	0	0
$655$	−41.5692	−1.62424
$656$	19.5959	0.765092
$657$	0	0
$658$	10.3923	0.405134
$659$	−21.2132	−0.826349	−0.413175	−	0.910652i	$-0.635580\pi$
$659$	−21.2132	−0.826349	−0.413175	+	0.910652i	$0.635580\pi$
$660$	0	0
$661$	−34.6410	−1.34738	−0.673690	−	0.739014i	$-0.735292\pi$
$661$	−34.6410	−1.34738	−0.673690	+	0.739014i	$0.735292\pi$
$662$	26.9444	1.04722
$663$	0	0
$664$	−41.5692	−1.61320
$665$	50.9117	1.97427
$666$	0	0
$667$	0	0
$668$	−58.7878	−2.27457
$669$	0	0
$670$	90.0000	3.47700
$671$	22.0454	0.851054
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	−55.1543	−2.12447
$675$	0	0
$676$	−48.0000	−1.84615
$677$	−12.7279	−0.489174	−0.244587	−	0.969627i	$-0.578652\pi$
$677$	−12.7279	−0.489174	−0.244587	+	0.969627i	$0.578652\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	88.1816	3.38161
$681$	0	0
$682$	−20.7846	−0.795884
$683$	41.6413	1.59336	0.796681	−	0.604401i	$-0.206587\pi$
$683$	41.6413	1.59336	0.796681	+	0.604401i	$0.206587\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	46.6690	1.78183
$687$	0	0
$688$	−34.6410	−1.32068
$689$	8.48528	0.323263
$690$	0	0
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	−88.1816	−3.35216
$693$	0	0
$694$	−18.0000	−0.683271
$695$	−46.6690	−1.77026
$696$	0	0
$697$	20.7846	0.787273
$698$	17.1464	0.649002
$699$	0	0
$700$	90.0666	3.40420
$701$	8.48528	0.320485	0.160242	−	0.987078i	$-0.448772\pi$
$701$	8.48528	0.320485	0.160242	+	0.987078i	$0.448772\pi$
$702$	0	0
$703$	36.0000	1.35777
$704$	33.9411	1.27920
$705$	0	0
$706$	24.0000	0.903252
$707$	−12.7279	−0.478683
$708$	0	0
$709$	−12.1244	−0.455340	−0.227670	−	0.973738i	$-0.573111\pi$
$709$	−12.1244	−0.455340	−0.227670	+	0.973738i	$0.573111\pi$
$710$	25.4558	0.955341
$711$	0	0
$712$	62.3538	2.33681
$713$	0	0
$714$	0	0
$715$	18.0000	0.673162
$716$	9.79796	0.366167
$717$	0	0
$718$	41.5692	1.55135
$719$	36.7423	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$719$	36.7423	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$720$	0	0
$721$	−9.00000	−0.335178
$722$	−71.0352	−2.64366
$723$	0	0
$724$	13.8564	0.514969
$725$	−31.8434	−1.18263
$726$	0	0
$727$	−20.7846	−0.770859	−0.385429	−	0.922737i	$-0.625947\pi$
$727$	−20.7846	−0.770859	−0.385429	+	0.922737i	$0.625947\pi$
$728$	−8.48528	−0.314485
$729$	0	0
$730$	41.5692	1.53855
$731$	−36.7423	−1.35896
$732$	0	0
$733$	−39.8372	−1.47142	−0.735710	−	0.677297i	$-0.763151\pi$
$733$	−39.8372	−1.47142	−0.735710	+	0.677297i	$0.763151\pi$
$734$	33.9411	1.25279
$735$	0	0
$736$	0	0
$737$	−36.7423	−1.35342
$738$	0	0
$739$	−49.0000	−1.80249	−0.901247	−	0.433306i	$-0.857347\pi$
$739$	−49.0000	−1.80249	−0.901247	+	0.433306i	$0.857347\pi$
$740$	88.1816	3.24162
$741$	0	0
$742$	−36.0000	−1.32160
$743$	42.4264	1.55647	0.778237	−	0.627971i	$-0.216114\pi$
$743$	42.4264	1.55647	0.778237	+	0.627971i	$0.216114\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	12.7279	0.466002
$747$	0	0
$748$	−72.0000	−2.63258
$749$	−14.6969	−0.537014
$750$	0	0
$751$	6.92820	0.252814	0.126407	−	0.991978i	$-0.459656\pi$
$751$	6.92820	0.252814	0.126407	+	0.991978i	$0.459656\pi$
$752$	−9.79796	−0.357295
$753$	0	0
$754$	6.00000	0.218507
$755$	−46.6690	−1.69846
$756$	0	0
$757$	3.46410	0.125905	0.0629525	−	0.998017i	$-0.479948\pi$
$757$	3.46410	0.125905	0.0629525	+	0.998017i	$0.479948\pi$
$758$	−63.6396	−2.31149
$759$	0	0
$760$	−144.000	−5.22343
$761$	−46.5403	−1.68708	−0.843542	−	0.537063i	$-0.819534\pi$
$761$	−46.5403	−1.68708	−0.843542	+	0.537063i	$0.819534\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	33.9411	1.22795
$765$	0	0
$766$	0	0
$767$	−7.34847	−0.265338
$768$	0	0
$769$	32.9090	1.18673	0.593364	−	0.804934i	$-0.297800\pi$
$769$	32.9090	1.18673	0.593364	+	0.804934i	$0.297800\pi$
$770$	−76.3675	−2.75209
$771$	0	0
$772$	−28.0000	−1.00774
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−26.0000	−0.933948
$776$	67.8823	2.43683
$777$	0	0
$778$	−93.5307	−3.35324
$779$	−33.9411	−1.21607
$780$	0	0
$781$	−10.3923	−0.371866
$782$	0	0
$783$	0	0
$784$	−16.0000	−0.571429
$785$	−44.0908	−1.57367
$786$	0	0
$787$	−6.92820	−0.246964	−0.123482	−	0.992347i	$-0.539406\pi$
$787$	−6.92820	−0.246964	−0.123482	+	0.992347i	$0.539406\pi$
$788$	−19.5959	−0.698076
$789$	0	0
$790$	−72.0000	−2.56165
$791$	14.6969	0.522563
$792$	0	0
$793$	−5.19615	−0.184521
$794$	−56.3383	−1.99937
$795$	0	0
$796$	34.6410	1.22782
$797$	−29.6985	−1.05197	−0.525987	−	0.850493i	$-0.676304\pi$
$797$	−29.6985	−1.05197	−0.525987	+	0.850493i	$0.676304\pi$
$798$	0	0
$799$	−10.3923	−0.367653
$800$	0	0
$801$	0	0
$802$	20.7846	0.733930
$803$	−16.9706	−0.598878
$804$	0	0
$805$	0	0
$806$	4.89898	0.172559
$807$	0	0
$808$	36.0000	1.26648
$809$	−36.7423	−1.29179	−0.645896	−	0.763425i	$-0.723516\pi$
$809$	−36.7423	−1.29179	−0.645896	+	0.763425i	$0.723516\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	−16.9706	−0.595550
$813$	0	0
$814$	−54.0000	−1.89270
$815$	72.1249	2.52642
$816$	0	0
$817$	60.0000	2.09913
$818$	−41.6413	−1.45595
$819$	0	0
$820$	−83.1384	−2.90332
$821$	−14.6969	−0.512927	−0.256463	−	0.966554i	$-0.582557\pi$
$821$	−14.6969	−0.512927	−0.256463	+	0.966554i	$0.582557\pi$
$822$	0	0
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	25.4558	0.886796
$825$	0	0
$826$	31.1769	1.08478
$827$	−50.9117	−1.77037	−0.885186	−	0.465236i	$-0.845969\pi$
$827$	−50.9117	−1.77037	−0.885186	+	0.465236i	$0.845969\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	88.1816	3.06083
$831$	0	0
$832$	−8.00000	−0.277350
$833$	−16.9706	−0.587995
$834$	0	0
$835$	62.3538	2.15784
$836$	117.576	4.06643
$837$	0	0
$838$	−51.9615	−1.79498
$839$	25.4558	0.878833	0.439417	−	0.898283i	$-0.355185\pi$
$839$	25.4558	0.878833	0.439417	+	0.898283i	$0.355185\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	80.6102	2.77801
$843$	0	0
$844$	52.0000	1.78991
$845$	50.9117	1.75142
$846$	0	0
$847$	12.1244	0.416598
$848$	33.9411	1.16554
$849$	0	0
$850$	−135.100	−4.63389
$851$	0	0
$852$	0	0
$853$	−13.0000	−0.445112	−0.222556	−	0.974920i	$-0.571440\pi$
$853$	−13.0000	−0.445112	−0.222556	+	0.974920i	$0.571440\pi$
$854$	22.0454	0.754378
$855$	0	0
$856$	41.5692	1.42081
$857$	17.1464	0.585711	0.292855	−	0.956157i	$-0.405395\pi$
$857$	17.1464	0.585711	0.292855	+	0.956157i	$0.405395\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	146.969	5.01161
$861$	0	0
$862$	62.3538	2.12378
$863$	9.79796	0.333526	0.166763	−	0.985997i	$-0.446668\pi$
$863$	9.79796	0.333526	0.166763	+	0.985997i	$0.446668\pi$
$864$	0	0
$865$	93.5307	3.18014
$866$	29.6985	1.00920
$867$	0	0
$868$	−13.8564	−0.470317
$869$	29.3939	0.997119
$870$	0	0
$871$	8.66025	0.293442
$872$	50.9117	1.72409
$873$	0	0
$874$	0	0
$875$	−58.7878	−1.98739
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−71.0352	−2.39732
$879$	0	0
$880$	72.0000	2.42712
$881$	8.48528	0.285876	0.142938	−	0.989732i	$-0.454345\pi$
$881$	8.48528	0.285876	0.142938	+	0.989732i	$0.454345\pi$
$882$	0	0
$883$	46.0000	1.54802	0.774012	−	0.633171i	$-0.218247\pi$
$883$	46.0000	1.54802	0.774012	+	0.633171i	$0.218247\pi$
$884$	16.9706	0.570782
$885$	0	0
$886$	102.000	3.42676
$887$	−9.79796	−0.328983	−0.164492	−	0.986378i	$-0.552598\pi$
$887$	−9.79796	−0.328983	−0.164492	+	0.986378i	$0.552598\pi$
$888$	0	0
$889$	22.5167	0.755185
$890$	−132.272	−4.43378
$891$	0	0
$892$	92.0000	3.08039
$893$	16.9706	0.567898
$894$	0	0
$895$	−10.3923	−0.347376
$896$	33.9411	1.13389
$897$	0	0
$898$	−18.0000	−0.600668
$899$	4.89898	0.163390
$900$	0	0
$901$	36.0000	1.19933
$902$	50.9117	1.69517
$903$	0	0
$904$	−41.5692	−1.38257
$905$	−14.6969	−0.488543
$906$	0	0
$907$	−25.9808	−0.862677	−0.431339	−	0.902190i	$-0.641959\pi$
$907$	−25.9808	−0.862677	−0.431339	+	0.902190i	$0.641959\pi$
$908$	−16.9706	−0.563188
$909$	0	0
$910$	18.0000	0.596694
$911$	−16.9706	−0.562260	−0.281130	−	0.959670i	$-0.590709\pi$
$911$	−16.9706	−0.562260	−0.281130	+	0.959670i	$0.590709\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	4.24264	0.140334
$915$	0	0
$916$	76.2102	2.51806
$917$	16.9706	0.560417
$918$	0	0
$919$	−19.0526	−0.628486	−0.314243	−	0.949343i	$-0.601751\pi$
$919$	−19.0526	−0.628486	−0.314243	+	0.949343i	$0.601751\pi$
$920$	0	0
$921$	0	0
$922$	−78.0000	−2.56879
$923$	2.44949	0.0806259
$924$	0	0
$925$	−67.5500	−2.22103
$926$	−100.429	−3.30030
$927$	0	0
$928$	0	0
$929$	12.2474	0.401826	0.200913	−	0.979609i	$-0.435609\pi$
$929$	12.2474	0.401826	0.200913	+	0.979609i	$0.435609\pi$
$930$	0	0
$931$	27.7128	0.908251
$932$	−39.1918	−1.28377
$933$	0	0
$934$	−51.9615	−1.70023
$935$	76.3675	2.49749
$936$	0	0
$937$	−36.3731	−1.18826	−0.594128	−	0.804370i	$-0.702503\pi$
$937$	−36.3731	−1.18826	−0.594128	+	0.804370i	$0.702503\pi$
$938$	−36.7423	−1.19968
$939$	0	0
$940$	41.5692	1.35584
$941$	16.9706	0.553225	0.276612	−	0.960982i	$-0.410788\pi$
$941$	16.9706	0.553225	0.276612	+	0.960982i	$0.410788\pi$
$942$	0	0
$943$	0	0
$944$	−29.3939	−0.956689
$945$	0	0
$946$	−90.0000	−2.92615
$947$	−31.8434	−1.03477	−0.517385	−	0.855753i	$-0.673095\pi$
$947$	−31.8434	−1.03477	−0.517385	+	0.855753i	$0.673095\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	220.617	7.15777
$951$	0	0
$952$	−36.0000	−1.16677
$953$	−12.7279	−0.412298	−0.206149	−	0.978521i	$-0.566093\pi$
$953$	−12.7279	−0.412298	−0.206149	+	0.978521i	$0.566093\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	9.79796	0.316889
$957$	0	0
$958$	41.5692	1.34304
$959$	−7.34847	−0.237294
$960$	0	0
$961$	−27.0000	−0.870968
$962$	12.7279	0.410365
$963$	0	0
$964$	62.3538	2.00828
$965$	29.6985	0.956028
$966$	0	0
$967$	25.0000	0.803946	0.401973	−	0.915652i	$-0.368325\pi$
$967$	25.0000	0.803946	0.401973	+	0.915652i	$0.368325\pi$
$968$	−34.2929	−1.10221
$969$	0	0
$970$	−144.000	−4.62356
$971$	8.48528	0.272306	0.136153	−	0.990688i	$-0.456526\pi$
$971$	8.48528	0.272306	0.136153	+	0.990688i	$0.456526\pi$
$972$	0	0
$973$	19.0526	0.610797
$974$	31.8434	1.02033
$975$	0	0
$976$	−20.7846	−0.665299
$977$	33.9411	1.08587	0.542936	−	0.839774i	$-0.317312\pi$
$977$	33.9411	1.08587	0.542936	+	0.839774i	$0.317312\pi$
$978$	0	0
$979$	54.0000	1.72585
$980$	67.8823	2.16842
$981$	0	0
$982$	−78.0000	−2.48908
$983$	−21.2132	−0.676596	−0.338298	−	0.941039i	$-0.609851\pi$
$983$	−21.2132	−0.676596	−0.338298	+	0.941039i	$0.609851\pi$
$984$	0	0
$985$	20.7846	0.662253
$986$	25.4558	0.810679
$987$	0	0
$988$	−27.7128	−0.881662
$989$	0	0
$990$	0	0
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	0	0
$993$	0	0
$994$	−10.3923	−0.329624
$995$	−36.7423	−1.16481
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	−46.5403	−1.47321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bh.1.1	✓	4
3.2	odd	2	inner	4761.2.a.bh.1.4	yes	4
23.22	odd	2	inner	4761.2.a.bh.1.2	yes	4
69.68	even	2	inner	4761.2.a.bh.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4761.2.a.bh.1.1	✓	4	1.1	even	1	trivial
4761.2.a.bh.1.2	yes	4	23.22	odd	2	inner
4761.2.a.bh.1.3	yes	4	69.68	even	2	inner
4761.2.a.bh.1.4	yes	4	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.44949 q^{2} +4.00000 q^{4} -4.24264 q^{5} +1.73205 q^{7} -4.89898 q^{8} +O(q^{10})\)	\(q-2.44949 q^{2} +4.00000 q^{4} -4.24264 q^{5} +1.73205 q^{7} -4.89898 q^{8} +10.3923 q^{10} -4.24264 q^{11} +1.00000 q^{13} -4.24264 q^{14} +4.00000 q^{16} +4.24264 q^{17} -6.92820 q^{19} -16.9706 q^{20} +10.3923 q^{22} +13.0000 q^{25} -2.44949 q^{26} +6.92820 q^{28} -2.44949 q^{29} -2.00000 q^{31} -10.3923 q^{34} -7.34847 q^{35} -5.19615 q^{37} +16.9706 q^{38} +20.7846 q^{40} +4.89898 q^{41} -8.66025 q^{43} -16.9706 q^{44} -2.44949 q^{47} -4.00000 q^{49} -31.8434 q^{50} +4.00000 q^{52} +8.48528 q^{53} +18.0000 q^{55} -8.48528 q^{56} +6.00000 q^{58} -7.34847 q^{59} -5.19615 q^{61} +4.89898 q^{62} -8.00000 q^{64} -4.24264 q^{65} +8.66025 q^{67} +16.9706 q^{68} +18.0000 q^{70} +2.44949 q^{71} +4.00000 q^{73} +12.7279 q^{74} -27.7128 q^{76} -7.34847 q^{77} -6.92820 q^{79} -16.9706 q^{80} -12.0000 q^{82} +8.48528 q^{83} -18.0000 q^{85} +21.2132 q^{86} +20.7846 q^{88} -12.7279 q^{89} +1.73205 q^{91} +6.00000 q^{94} +29.3939 q^{95} -13.8564 q^{97} +9.79796 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4}+O(q^{10}) \) 4 * q + 16 * q^4	\( 4 q + 16 q^{4} + 4 q^{13} + 16 q^{16} + 52 q^{25} - 8 q^{31} - 16 q^{49} + 16 q^{52} + 72 q^{55} + 24 q^{58} - 32 q^{64} + 72 q^{70} + 16 q^{73} - 48 q^{82} - 72 q^{85} + 24 q^{94}+O(q^{100}) \) 4 * q + 16 * q^4 + 4 * q^13 + 16 * q^16 + 52 * q^25 - 8 * q^31 - 16 * q^49 + 16 * q^52 + 72 * q^55 + 24 * q^58 - 32 * q^64 + 72 * q^70 + 16 * q^73 - 48 * q^82 - 72 * q^85 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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