LMFDB - Embedded newform 4140.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.14345904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 $ x^5 - 2x^4 - 13x^3 + 34x^2 - 11x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.83694$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -0.113901 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -0.113901 q^{7} -3.39991 q^{11} +6.27397 q^{13} -3.61662 q^{17} -1.39991 q^{19} -1.00000 q^{23} +1.00000 q^{25} +6.38787 q^{29} +9.45717 q^{31} +0.113901 q^{35} -7.78778 q^{37} -11.0563 q^{41} +1.55936 q^{43} -1.70196 q^{47} -6.98703 q^{49} +8.71849 q^{53} +3.39991 q^{55} +8.95988 q^{59} -1.83333 q^{61} -6.27397 q^{65} +9.21577 q^{67} +6.82852 q^{71} -2.27397 q^{73} +0.387254 q^{77} +11.7703 q^{79} -1.16101 q^{83} +3.61662 q^{85} +1.77220 q^{89} -0.714615 q^{91} +1.39991 q^{95} -0.131365 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 + 4 * q^7	$ 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * q^97

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.113901	−0.0430507	−0.0215254	−	0.999768i	$-0.506852\pi$
$7$	−0.113901	−0.0430507	−0.0215254	+	0.999768i	$0.506852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.39991	−1.02511	−0.512555	−	0.858654i	$-0.671301\pi$
$11$	−3.39991	−1.02511	−0.512555	+	0.858654i	$0.671301\pi$
$12$	0	0
$13$	6.27397	1.74009	0.870043	−	0.492975i	$-0.164091\pi$
$13$	6.27397	1.74009	0.870043	+	0.492975i	$0.164091\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.61662	−0.877158	−0.438579	−	0.898693i	$-0.644518\pi$
$17$	−3.61662	−0.877158	−0.438579	+	0.898693i	$0.644518\pi$
$18$	0	0
$19$	−1.39991	−0.321160	−0.160580	−	0.987023i	$-0.551337\pi$
$19$	−1.39991	−0.321160	−0.160580	+	0.987023i	$0.551337\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.38787	1.18620	0.593099	−	0.805129i	$-0.297904\pi$
$29$	6.38787	1.18620	0.593099	+	0.805129i	$0.297904\pi$
$30$	0	0
$31$	9.45717	1.69856	0.849279	−	0.527945i	$-0.177037\pi$
$31$	9.45717	1.69856	0.849279	+	0.527945i	$0.177037\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.113901	0.0192529
$36$	0	0
$37$	−7.78778	−1.28030	−0.640151	−	0.768249i	$-0.721128\pi$
$37$	−7.78778	−1.28030	−0.640151	+	0.768249i	$0.721128\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.0563	−1.72671	−0.863353	−	0.504600i	$-0.831640\pi$
$41$	−11.0563	−1.72671	−0.863353	+	0.504600i	$0.831640\pi$
$42$	0	0
$43$	1.55936	0.237800	0.118900	−	0.992906i	$-0.462063\pi$
$43$	1.55936	0.237800	0.118900	+	0.992906i	$0.462063\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.70196	−0.248257	−0.124128	−	0.992266i	$-0.539613\pi$
$47$	−1.70196	−0.248257	−0.124128	+	0.992266i	$0.539613\pi$
$48$	0	0
$49$	−6.98703	−0.998147
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.71849	1.19758	0.598788	−	0.800908i	$-0.295649\pi$
$53$	8.71849	1.19758	0.598788	+	0.800908i	$0.295649\pi$
$54$	0	0
$55$	3.39991	0.458443
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.95988	1.16648	0.583239	−	0.812301i	$-0.301785\pi$
$59$	8.95988	1.16648	0.583239	+	0.812301i	$0.301785\pi$
$60$	0	0
$61$	−1.83333	−0.234734	−0.117367	−	0.993089i	$-0.537445\pi$
$61$	−1.83333	−0.234734	−0.117367	+	0.993089i	$0.537445\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.27397	−0.778190
$66$	0	0
$67$	9.21577	1.12589	0.562943	−	0.826496i	$-0.309669\pi$
$67$	9.21577	1.12589	0.562943	+	0.826496i	$0.309669\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.82852	0.810396	0.405198	−	0.914229i	$-0.367203\pi$
$71$	6.82852	0.810396	0.405198	+	0.914229i	$0.367203\pi$
$72$	0	0
$73$	−2.27397	−0.266148	−0.133074	−	0.991106i	$-0.542485\pi$
$73$	−2.27397	−0.266148	−0.133074	+	0.991106i	$0.542485\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.387254	0.0441317
$78$	0	0
$79$	11.7703	1.32426	0.662132	−	0.749387i	$-0.269652\pi$
$79$	11.7703	1.32426	0.662132	+	0.749387i	$0.269652\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.16101	−0.127437	−0.0637187	−	0.997968i	$-0.520296\pi$
$83$	−1.16101	−0.127437	−0.0637187	+	0.997968i	$0.520296\pi$
$84$	0	0
$85$	3.61662	0.392277
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.77220	0.187853	0.0939263	−	0.995579i	$-0.470058\pi$
$89$	1.77220	0.187853	0.0939263	+	0.995579i	$0.470058\pi$
$90$	0	0
$91$	−0.714615	−0.0749120
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.39991	0.143627
$96$	0	0
$97$	−0.131365	−0.0133380	−0.00666902	−	0.999978i	$-0.502123\pi$
$97$	−0.131365	−0.0133380	−0.00666902	+	0.999978i	$0.502123\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.40052	0.935387	0.467694	−	0.883891i	$-0.345085\pi$
$101$	9.40052	0.935387	0.467694	+	0.883891i	$0.345085\pi$
$102$	0	0
$103$	9.24045	0.910489	0.455244	−	0.890366i	$-0.349552\pi$
$103$	9.24045	0.910489	0.455244	+	0.890366i	$0.349552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.28506	−0.220905	−0.110453	−	0.993881i	$-0.535230\pi$
$107$	−2.28506	−0.220905	−0.110453	+	0.993881i	$0.535230\pi$
$108$	0	0
$109$	−1.39991	−0.134087	−0.0670433	−	0.997750i	$-0.521357\pi$
$109$	−1.39991	−0.134087	−0.0670433	+	0.997750i	$0.521357\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.59977	0.903071	0.451535	−	0.892253i	$-0.350876\pi$
$113$	9.59977	0.903071	0.451535	+	0.892253i	$0.350876\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.411938	0.0377623
$120$	0	0
$121$	0.559357	0.0508506
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.73146	0.597320	0.298660	−	0.954360i	$-0.403460\pi$
$127$	6.73146	0.597320	0.298660	+	0.954360i	$0.403460\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.22780	−0.369385	−0.184692	−	0.982796i	$-0.559129\pi$
$131$	−4.22780	−0.369385	−0.184692	+	0.982796i	$0.559129\pi$
$132$	0	0
$133$	0.159451	0.0138262
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.4496	−1.23451	−0.617257	−	0.786761i	$-0.711756\pi$
$137$	−14.4496	−1.23451	−0.617257	+	0.786761i	$0.711756\pi$
$138$	0	0
$139$	11.9870	1.01673	0.508363	−	0.861143i	$-0.330251\pi$
$139$	11.9870	1.01673	0.508363	+	0.861143i	$0.330251\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−21.3309	−1.78378
$144$	0	0
$145$	−6.38787	−0.530484
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.19972	−0.344054	−0.172027	−	0.985092i	$-0.555032\pi$
$149$	−4.19972	−0.344054	−0.172027	+	0.985092i	$0.555032\pi$
$150$	0	0
$151$	7.55936	0.615172	0.307586	−	0.951520i	$-0.400479\pi$
$151$	7.55936	0.615172	0.307586	+	0.951520i	$0.400479\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.45717	−0.759618
$156$	0	0
$157$	5.86203	0.467841	0.233921	−	0.972256i	$-0.424844\pi$
$157$	5.86203	0.467841	0.233921	+	0.972256i	$0.424844\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.113901	0.00897669
$162$	0	0
$163$	0.766767	0.0600578	0.0300289	−	0.999549i	$-0.490440\pi$
$163$	0.766767	0.0600578	0.0300289	+	0.999549i	$0.490440\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.0443	1.08678	0.543390	−	0.839481i	$-0.317141\pi$
$167$	14.0443	1.08678	0.543390	+	0.839481i	$0.317141\pi$
$168$	0	0
$169$	26.3627	2.02790
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.13859	0.162594	0.0812968	−	0.996690i	$-0.474094\pi$
$173$	2.13859	0.162594	0.0812968	+	0.996690i	$0.474094\pi$
$174$	0	0
$175$	−0.113901	−0.00861014
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.93118	−0.667547	−0.333774	−	0.942653i	$-0.608322\pi$
$179$	−8.93118	−0.667547	−0.333774	+	0.942653i	$0.608322\pi$
$180$	0	0
$181$	25.4623	1.89260	0.946298	−	0.323296i	$-0.104791\pi$
$181$	25.4623	1.89260	0.946298	+	0.323296i	$0.104791\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.78778	0.572569
$186$	0	0
$187$	12.2962	0.899184
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.3813	0.895877	0.447939	−	0.894064i	$-0.352158\pi$
$191$	12.3813	0.895877	0.447939	+	0.894064i	$0.352158\pi$
$192$	0	0
$193$	3.31471	0.238598	0.119299	−	0.992858i	$-0.461935\pi$
$193$	3.31471	0.238598	0.119299	+	0.992858i	$0.461935\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.3201	1.16276	0.581381	−	0.813632i	$-0.302513\pi$
$197$	16.3201	1.16276	0.581381	+	0.813632i	$0.302513\pi$
$198$	0	0
$199$	−9.80336	−0.694942	−0.347471	−	0.937691i	$-0.612959\pi$
$199$	−9.80336	−0.694942	−0.347471	+	0.937691i	$0.612959\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.727588	−0.0510667
$204$	0	0
$205$	11.0563	0.772207
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.75955	0.329225
$210$	0	0
$211$	1.34987	0.0929286	0.0464643	−	0.998920i	$-0.485205\pi$
$211$	1.34987	0.0929286	0.0464643	+	0.998920i	$0.485205\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.55936	−0.106347
$216$	0	0
$217$	−1.07719	−0.0731241
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.6905	−1.52633
$222$	0	0
$223$	15.2332	1.02009	0.510046	−	0.860147i	$-0.329628\pi$
$223$	15.2332	1.02009	0.510046	+	0.860147i	$0.329628\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.8921	1.25392	0.626958	−	0.779053i	$-0.284300\pi$
$227$	18.8921	1.25392	0.626958	+	0.779053i	$0.284300\pi$
$228$	0	0
$229$	7.57744	0.500731	0.250366	−	0.968151i	$-0.419449\pi$
$229$	7.57744	0.500731	0.250366	+	0.968151i	$0.419449\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.0035	1.11394	0.556970	−	0.830533i	$-0.311964\pi$
$233$	17.0035	1.11394	0.556970	+	0.830533i	$0.311964\pi$
$234$	0	0
$235$	1.70196	0.111024
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.9508	1.48456	0.742281	−	0.670089i	$-0.233744\pi$
$239$	22.9508	1.48456	0.742281	+	0.670089i	$0.233744\pi$
$240$	0	0
$241$	4.51862	0.291070	0.145535	−	0.989353i	$-0.453510\pi$
$241$	4.51862	0.291070	0.145535	+	0.989353i	$0.453510\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.98703	0.446385
$246$	0	0
$247$	−8.78297	−0.558847
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.0257	−1.57961	−0.789805	−	0.613358i	$-0.789818\pi$
$251$	−25.0257	−1.57961	−0.789805	+	0.613358i	$0.789818\pi$
$252$	0	0
$253$	3.39991	0.213750
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.05160	−0.190354	−0.0951768	−	0.995460i	$-0.530342\pi$
$257$	−3.05160	−0.190354	−0.0951768	+	0.995460i	$0.530342\pi$
$258$	0	0
$259$	0.887039	0.0551180
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.11437	0.377028	0.188514	−	0.982070i	$-0.439633\pi$
$263$	6.11437	0.377028	0.188514	+	0.982070i	$0.439633\pi$
$264$	0	0
$265$	−8.71849	−0.535572
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.7080	−0.652879	−0.326440	−	0.945218i	$-0.605849\pi$
$269$	−10.7080	−0.652879	−0.326440	+	0.945218i	$0.605849\pi$
$270$	0	0
$271$	15.0980	0.917138	0.458569	−	0.888659i	$-0.348362\pi$
$271$	15.0980	0.917138	0.458569	+	0.888659i	$0.348362\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.39991	−0.205022
$276$	0	0
$277$	−26.0886	−1.56751	−0.783755	−	0.621070i	$-0.786698\pi$
$277$	−26.0886	−1.56751	−0.783755	+	0.621070i	$0.786698\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.29082	−0.255969	−0.127984	−	0.991776i	$-0.540851\pi$
$281$	−4.29082	−0.255969	−0.127984	+	0.991776i	$0.540851\pi$
$282$	0	0
$283$	−20.6057	−1.22488	−0.612440	−	0.790517i	$-0.709812\pi$
$283$	−20.6057	−1.22488	−0.612440	+	0.790517i	$0.709812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.25933	0.0743360
$288$	0	0
$289$	−3.92008	−0.230593
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.7238	−1.32754	−0.663770	−	0.747937i	$-0.731045\pi$
$293$	−22.7238	−1.32754	−0.663770	+	0.747937i	$0.731045\pi$
$294$	0	0
$295$	−8.95988	−0.521664
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.27397	−0.362833
$300$	0	0
$301$	−0.177613	−0.0102374
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.83333	0.104976
$306$	0	0
$307$	20.7645	1.18509	0.592546	−	0.805536i	$-0.298123\pi$
$307$	20.7645	1.18509	0.592546	+	0.805536i	$0.298123\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−33.3237	−1.88961	−0.944806	−	0.327629i	$-0.893750\pi$
$311$	−33.3237	−1.88961	−0.944806	+	0.327629i	$0.893750\pi$
$312$	0	0
$313$	−5.25070	−0.296787	−0.148393	−	0.988928i	$-0.547410\pi$
$313$	−5.25070	−0.296787	−0.148393	+	0.988928i	$0.547410\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.8495	1.33952	0.669761	−	0.742576i	$-0.266396\pi$
$317$	23.8495	1.33952	0.669761	+	0.742576i	$0.266396\pi$
$318$	0	0
$319$	−21.7182	−1.21598
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.06292	0.281708
$324$	0	0
$325$	6.27397	0.348017
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.193856	0.0106876
$330$	0	0
$331$	−20.8049	−1.14354	−0.571771	−	0.820413i	$-0.693743\pi$
$331$	−20.8049	−1.14354	−0.571771	+	0.820413i	$0.693743\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.21577	−0.503511
$336$	0	0
$337$	4.97050	0.270761	0.135380	−	0.990794i	$-0.456774\pi$
$337$	4.97050	0.270761	0.135380	+	0.990794i	$0.456774\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−32.1535	−1.74121
$342$	0	0
$343$	1.59314	0.0860216
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.3478	1.46810	0.734052	−	0.679093i	$-0.237627\pi$
$347$	27.3478	1.46810	0.734052	+	0.679093i	$0.237627\pi$
$348$	0	0
$349$	24.7307	1.32380	0.661901	−	0.749591i	$-0.269750\pi$
$349$	24.7307	1.32380	0.661901	+	0.749591i	$0.269750\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.5053	1.35751	0.678756	−	0.734364i	$-0.262520\pi$
$353$	25.5053	1.35751	0.678756	+	0.734364i	$0.262520\pi$
$354$	0	0
$355$	−6.82852	−0.362420
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.2644	−0.594514	−0.297257	−	0.954797i	$-0.596072\pi$
$359$	−11.2644	−0.594514	−0.297257	+	0.954797i	$0.596072\pi$
$360$	0	0
$361$	−17.0403	−0.896856
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.27397	0.119025
$366$	0	0
$367$	36.6209	1.91160	0.955799	−	0.294022i	$-0.0949939\pi$
$367$	36.6209	1.91160	0.955799	+	0.294022i	$0.0949939\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.993048	−0.0515565
$372$	0	0
$373$	−38.4718	−1.99199	−0.995997	−	0.0893834i	$-0.971510\pi$
$373$	−38.4718	−1.99199	−0.995997	+	0.0893834i	$0.971510\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.0773	2.06409
$378$	0	0
$379$	−8.63716	−0.443661	−0.221831	−	0.975085i	$-0.571203\pi$
$379$	−8.63716	−0.443661	−0.221831	+	0.975085i	$0.571203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.4684	0.688203	0.344102	−	0.938932i	$-0.388183\pi$
$383$	13.4684	0.688203	0.344102	+	0.938932i	$0.388183\pi$
$384$	0	0
$385$	−0.387254	−0.0197363
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.46292	0.175577	0.0877885	−	0.996139i	$-0.472020\pi$
$389$	3.46292	0.175577	0.0877885	+	0.996139i	$0.472020\pi$
$390$	0	0
$391$	3.61662	0.182900
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.7703	−0.592229
$396$	0	0
$397$	26.5791	1.33397	0.666984	−	0.745072i	$-0.267585\pi$
$397$	26.5791	1.33397	0.666984	+	0.745072i	$0.267585\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.4575	−0.622097	−0.311049	−	0.950394i	$-0.600680\pi$
$401$	−12.4575	−0.622097	−0.311049	+	0.950394i	$0.600680\pi$
$402$	0	0
$403$	59.3340	2.95564
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.4777	1.31245
$408$	0	0
$409$	2.58310	0.127726	0.0638630	−	0.997959i	$-0.479658\pi$
$409$	2.58310	0.127726	0.0638630	+	0.997959i	$0.479658\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.02054	−0.0502177
$414$	0	0
$415$	1.16101	0.0569918
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.6181	−1.78891	−0.894454	−	0.447159i	$-0.852436\pi$
$419$	−36.6181	−1.78891	−0.894454	+	0.447159i	$0.852436\pi$
$420$	0	0
$421$	−9.60553	−0.468145	−0.234072	−	0.972219i	$-0.575205\pi$
$421$	−9.60553	−0.468145	−0.234072	+	0.972219i	$0.575205\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.61662	−0.175432
$426$	0	0
$427$	0.208819	0.0101054
$428$	0	0
$429$	0	0
$430$	0	0
$431$	39.4934	1.90233	0.951166	−	0.308680i	$-0.0998872\pi$
$431$	39.4934	1.90233	0.951166	+	0.308680i	$0.0998872\pi$
$432$	0	0
$433$	3.32495	0.159787	0.0798935	−	0.996803i	$-0.474542\pi$
$433$	3.32495	0.159787	0.0798935	+	0.996803i	$0.474542\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.39991	0.0669666
$438$	0	0
$439$	−14.5882	−0.696257	−0.348129	−	0.937447i	$-0.613183\pi$
$439$	−14.5882	−0.696257	−0.348129	+	0.937447i	$0.613183\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.0497	−1.38019	−0.690097	−	0.723717i	$-0.742432\pi$
$443$	−29.0497	−1.38019	−0.690097	+	0.723717i	$0.742432\pi$
$444$	0	0
$445$	−1.77220	−0.0840102
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.07505	0.145120	0.0725602	−	0.997364i	$-0.476883\pi$
$449$	3.07505	0.145120	0.0725602	+	0.997364i	$0.476883\pi$
$450$	0	0
$451$	37.5904	1.77006
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.714615	0.0335017
$456$	0	0
$457$	6.49525	0.303835	0.151918	−	0.988393i	$-0.451455\pi$
$457$	6.49525	0.303835	0.151918	+	0.988393i	$0.451455\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−28.1367	−1.31046	−0.655228	−	0.755431i	$-0.727428\pi$
$461$	−28.1367	−1.31046	−0.655228	+	0.755431i	$0.727428\pi$
$462$	0	0
$463$	22.7626	1.05787	0.528934	−	0.848663i	$-0.322592\pi$
$463$	22.7626	1.05787	0.528934	+	0.848663i	$0.322592\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.0296	0.510391	0.255196	−	0.966889i	$-0.417860\pi$
$467$	11.0296	0.510391	0.255196	+	0.966889i	$0.417860\pi$
$468$	0	0
$469$	−1.04969	−0.0484702
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.30167	−0.243771
$474$	0	0
$475$	−1.39991	−0.0642321
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.9755	0.501482	0.250741	−	0.968054i	$-0.419326\pi$
$479$	10.9755	0.501482	0.250741	+	0.968054i	$0.419326\pi$
$480$	0	0
$481$	−48.8603	−2.22784
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.131365	0.00596496
$486$	0	0
$487$	5.47416	0.248058	0.124029	−	0.992279i	$-0.460418\pi$
$487$	5.47416	0.248058	0.124029	+	0.992279i	$0.460418\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.5766	−0.928611	−0.464306	−	0.885675i	$-0.653696\pi$
$491$	−20.5766	−0.928611	−0.464306	+	0.885675i	$0.653696\pi$
$492$	0	0
$493$	−23.1025	−1.04048
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.777778	−0.0348881
$498$	0	0
$499$	21.2605	0.951752	0.475876	−	0.879512i	$-0.342131\pi$
$499$	21.2605	0.951752	0.475876	+	0.879512i	$0.342131\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.7049	−0.967773	−0.483887	−	0.875131i	$-0.660775\pi$
$503$	−21.7049	−0.967773	−0.483887	+	0.875131i	$0.660775\pi$
$504$	0	0
$505$	−9.40052	−0.418318
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.6961	0.695716	0.347858	−	0.937547i	$-0.386909\pi$
$509$	15.6961	0.695716	0.347858	+	0.937547i	$0.386909\pi$
$510$	0	0
$511$	0.259009	0.0114579
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.24045	−0.407183
$516$	0	0
$517$	5.78651	0.254491
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.207458	0.00908890	0.00454445	−	0.999990i	$-0.498553\pi$
$521$	0.207458	0.00908890	0.00454445	+	0.999990i	$0.498553\pi$
$522$	0	0
$523$	26.1457	1.14327	0.571635	−	0.820508i	$-0.306309\pi$
$523$	26.1457	1.14327	0.571635	+	0.820508i	$0.306309\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.2029	−1.48990
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−69.3670	−3.00462
$534$	0	0
$535$	2.28506	0.0987919
$536$	0	0
$537$	0	0
$538$	0	0
$539$	23.7552	1.02321
$540$	0	0
$541$	−36.9994	−1.59073	−0.795365	−	0.606130i	$-0.792721\pi$
$541$	−36.9994	−1.59073	−0.795365	+	0.606130i	$0.792721\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.39991	0.0599654
$546$	0	0
$547$	3.24644	0.138808	0.0694038	−	0.997589i	$-0.477890\pi$
$547$	3.24644	0.138808	0.0694038	+	0.997589i	$0.477890\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.94242	−0.380960
$552$	0	0
$553$	−1.34066	−0.0570105
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.8366	1.94216	0.971079	−	0.238760i	$-0.0767410\pi$
$557$	45.8366	1.94216	0.971079	+	0.238760i	$0.0767410\pi$
$558$	0	0
$559$	9.78336	0.413792
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.4399	1.57791	0.788953	−	0.614454i	$-0.210624\pi$
$563$	37.4399	1.57791	0.788953	+	0.614454i	$0.210624\pi$
$564$	0	0
$565$	−9.59977	−0.403865
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.8277	0.663533	0.331766	−	0.943362i	$-0.392355\pi$
$569$	15.8277	0.663533	0.331766	+	0.943362i	$0.392355\pi$
$570$	0	0
$571$	−33.8423	−1.41626	−0.708128	−	0.706084i	$-0.750460\pi$
$571$	−33.8423	−1.41626	−0.708128	+	0.706084i	$0.750460\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−37.3490	−1.55486	−0.777429	−	0.628970i	$-0.783477\pi$
$577$	−37.3490	−1.55486	−0.777429	+	0.628970i	$0.783477\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.132241	0.00548627
$582$	0	0
$583$	−29.6420	−1.22765
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.28941	0.300866	0.150433	−	0.988620i	$-0.451933\pi$
$587$	7.28941	0.300866	0.150433	+	0.988620i	$0.451933\pi$
$588$	0	0
$589$	−13.2391	−0.545509
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.4510	0.963018	0.481509	−	0.876441i	$-0.340089\pi$
$593$	23.4510	0.963018	0.481509	+	0.876441i	$0.340089\pi$
$594$	0	0
$595$	−0.411938	−0.0168878
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.6570	−0.558011	−0.279006	−	0.960289i	$-0.590005\pi$
$599$	−13.6570	−0.558011	−0.279006	+	0.960289i	$0.590005\pi$
$600$	0	0
$601$	28.5752	1.16561	0.582804	−	0.812613i	$-0.301956\pi$
$601$	28.5752	1.16561	0.582804	+	0.812613i	$0.301956\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.559357	−0.0227411
$606$	0	0
$607$	−32.1335	−1.30426	−0.652129	−	0.758108i	$-0.726124\pi$
$607$	−32.1335	−1.30426	−0.652129	+	0.758108i	$0.726124\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.6781	−0.431988
$612$	0	0
$613$	21.0366	0.849660	0.424830	−	0.905273i	$-0.360334\pi$
$613$	21.0366	0.849660	0.424830	+	0.905273i	$0.360334\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.7378	−1.55953	−0.779763	−	0.626075i	$-0.784660\pi$
$617$	−38.7378	−1.55953	−0.779763	+	0.626075i	$0.784660\pi$
$618$	0	0
$619$	27.4941	1.10508	0.552540	−	0.833486i	$-0.313659\pi$
$619$	27.4941	1.10508	0.552540	+	0.833486i	$0.313659\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.201856	−0.00808718
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.1654	1.12303
$630$	0	0
$631$	6.22378	0.247765	0.123882	−	0.992297i	$-0.460465\pi$
$631$	6.22378	0.247765	0.123882	+	0.992297i	$0.460465\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.73146	−0.267130
$636$	0	0
$637$	−43.8364	−1.73686
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.6929	−0.856816	−0.428408	−	0.903585i	$-0.640925\pi$
$641$	−21.6929	−0.856816	−0.428408	+	0.903585i	$0.640925\pi$
$642$	0	0
$643$	−23.2411	−0.916538	−0.458269	−	0.888813i	$-0.651530\pi$
$643$	−23.2411	−0.916538	−0.458269	+	0.888813i	$0.651530\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.2830	−0.482893	−0.241446	−	0.970414i	$-0.577622\pi$
$647$	−12.2830	−0.482893	−0.241446	+	0.970414i	$0.577622\pi$
$648$	0	0
$649$	−30.4627	−1.19577
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.12546	−0.161442	−0.0807209	−	0.996737i	$-0.525722\pi$
$653$	−4.12546	−0.161442	−0.0807209	+	0.996737i	$0.525722\pi$
$654$	0	0
$655$	4.22780	0.165194
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.05758	0.119107	0.0595533	−	0.998225i	$-0.481032\pi$
$659$	3.05758	0.119107	0.0595533	+	0.998225i	$0.481032\pi$
$660$	0	0
$661$	−41.6105	−1.61846	−0.809230	−	0.587492i	$-0.800115\pi$
$661$	−41.6105	−1.61846	−0.809230	+	0.587492i	$0.800115\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.159451	−0.00618326
$666$	0	0
$667$	−6.38787	−0.247339
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.23314	0.240628
$672$	0	0
$673$	−21.8495	−0.842237	−0.421119	−	0.907006i	$-0.638362\pi$
$673$	−21.8495	−0.842237	−0.421119	+	0.907006i	$0.638362\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.8625	−0.532779	−0.266390	−	0.963865i	$-0.585831\pi$
$677$	−13.8625	−0.532779	−0.266390	+	0.963865i	$0.585831\pi$
$678$	0	0
$679$	0.0149626	0.000574212 0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.3016	1.04467	0.522333	−	0.852742i	$-0.325062\pi$
$683$	27.3016	1.04467	0.522333	+	0.852742i	$0.325062\pi$
$684$	0	0
$685$	14.4496	0.552092
$686$	0	0
$687$	0	0
$688$	0	0
$689$	54.6995	2.08389
$690$	0	0
$691$	35.7403	1.35962	0.679812	−	0.733387i	$-0.262061\pi$
$691$	35.7403	1.35962	0.679812	+	0.733387i	$0.262061\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.9870	−0.454694
$696$	0	0
$697$	39.9865	1.51460
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.25046	−0.0472292	−0.0236146	−	0.999721i	$-0.507517\pi$
$701$	−1.25046	−0.0472292	−0.0236146	+	0.999721i	$0.507517\pi$
$702$	0	0
$703$	10.9022	0.411182
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.07073	−0.0402691
$708$	0	0
$709$	−38.2680	−1.43718	−0.718592	−	0.695432i	$-0.755213\pi$
$709$	−38.2680	−1.43718	−0.718592	+	0.695432i	$0.755213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.45717	−0.354174
$714$	0	0
$715$	21.3309	0.797731
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.1228	−0.787749	−0.393874	−	0.919164i	$-0.628865\pi$
$719$	−21.1228	−0.787749	−0.393874	+	0.919164i	$0.628865\pi$
$720$	0	0
$721$	−1.05250	−0.0391972
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.38787	0.237240
$726$	0	0
$727$	−29.9281	−1.10997	−0.554986	−	0.831859i	$-0.687277\pi$
$727$	−29.9281	−1.10997	−0.554986	+	0.831859i	$0.687277\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.63960	−0.208588
$732$	0	0
$733$	−39.5600	−1.46118	−0.730591	−	0.682816i	$-0.760755\pi$
$733$	−39.5600	−1.46118	−0.730591	+	0.682816i	$0.760755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.3327	−1.15416
$738$	0	0
$739$	−21.1460	−0.777868	−0.388934	−	0.921266i	$-0.627157\pi$
$739$	−21.1460	−0.777868	−0.388934	+	0.921266i	$0.627157\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.0959	1.50766	0.753831	−	0.657068i	$-0.228204\pi$
$743$	41.0959	1.50766	0.753831	+	0.657068i	$0.228204\pi$
$744$	0	0
$745$	4.19972	0.153866
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.260272	0.00951013
$750$	0	0
$751$	−14.3735	−0.524498	−0.262249	−	0.965000i	$-0.584464\pi$
$751$	−14.3735	−0.524498	−0.262249	+	0.965000i	$0.584464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.55936	−0.275113
$756$	0	0
$757$	−25.7135	−0.934574	−0.467287	−	0.884106i	$-0.654768\pi$
$757$	−25.7135	−0.934574	−0.467287	+	0.884106i	$0.654768\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.4516	−1.06762	−0.533811	−	0.845604i	$-0.679241\pi$
$761$	−29.4516	−1.06762	−0.533811	+	0.845604i	$0.679241\pi$
$762$	0	0
$763$	0.159451	0.00577252
$764$	0	0
$765$	0	0
$766$	0	0
$767$	56.2140	2.02977
$768$	0	0
$769$	7.92378	0.285739	0.142869	−	0.989742i	$-0.454367\pi$
$769$	7.92378	0.285739	0.142869	+	0.989742i	$0.454367\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.74082	0.242450	0.121225	−	0.992625i	$-0.461318\pi$
$773$	6.74082	0.242450	0.121225	+	0.992625i	$0.461318\pi$
$774$	0	0
$775$	9.45717	0.339711
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.4778	0.554550
$780$	0	0
$781$	−23.2163	−0.830745
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.86203	−0.209225
$786$	0	0
$787$	−36.8407	−1.31323	−0.656614	−	0.754226i	$-0.728012\pi$
$787$	−36.8407	−1.31323	−0.656614	+	0.754226i	$0.728012\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.09343	−0.0388778
$792$	0	0
$793$	−11.5022	−0.408457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.4548	1.36214	0.681070	−	0.732219i	$-0.261515\pi$
$797$	38.4548	1.36214	0.681070	+	0.732219i	$0.261515\pi$
$798$	0	0
$799$	6.15535	0.217761
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.73129	0.272831
$804$	0	0
$805$	−0.113901	−0.00401450
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.2100	0.780861	0.390430	−	0.920632i	$-0.372326\pi$
$809$	22.2100	0.780861	0.390430	+	0.920632i	$0.372326\pi$
$810$	0	0
$811$	−18.7647	−0.658916	−0.329458	−	0.944170i	$-0.606866\pi$
$811$	−18.7647	−0.658916	−0.329458	+	0.944170i	$0.606866\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.766767	−0.0268587
$816$	0	0
$817$	−2.18295	−0.0763718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.5882	−1.20714	−0.603568	−	0.797311i	$-0.706255\pi$
$821$	−34.5882	−1.20714	−0.603568	+	0.797311i	$0.706255\pi$
$822$	0	0
$823$	−46.0642	−1.60570	−0.802849	−	0.596183i	$-0.796683\pi$
$823$	−46.0642	−1.60570	−0.802849	+	0.596183i	$0.796683\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.24847	0.252054	0.126027	−	0.992027i	$-0.459777\pi$
$827$	7.24847	0.252054	0.126027	+	0.992027i	$0.459777\pi$
$828$	0	0
$829$	−14.7688	−0.512940	−0.256470	−	0.966552i	$-0.582559\pi$
$829$	−14.7688	−0.512940	−0.256470	+	0.966552i	$0.582559\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.2694	0.875533
$834$	0	0
$835$	−14.0443	−0.486023
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.3292	0.460173	0.230087	−	0.973170i	$-0.426099\pi$
$839$	13.3292	0.460173	0.230087	+	0.973170i	$0.426099\pi$
$840$	0	0
$841$	11.8049	0.407066
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.3627	−0.906905
$846$	0	0
$847$	−0.0637116	−0.00218915
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.78778	0.266962
$852$	0	0
$853$	11.0307	0.377685	0.188843	−	0.982007i	$-0.439526\pi$
$853$	11.0307	0.377685	0.188843	+	0.982007i	$0.439526\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.1979	−0.416672	−0.208336	−	0.978057i	$-0.566805\pi$
$857$	−12.1979	−0.416672	−0.208336	+	0.978057i	$0.566805\pi$
$858$	0	0
$859$	−26.5718	−0.906617	−0.453309	−	0.891354i	$-0.649756\pi$
$859$	−26.5718	−0.906617	−0.453309	+	0.891354i	$0.649756\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.0276	0.783869	0.391935	−	0.919993i	$-0.371806\pi$
$863$	23.0276	0.783869	0.391935	+	0.919993i	$0.371806\pi$
$864$	0	0
$865$	−2.13859	−0.0727141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−40.0180	−1.35752
$870$	0	0
$871$	57.8195	1.95914
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.113901	0.00385057
$876$	0	0
$877$	−46.8332	−1.58144	−0.790722	−	0.612176i	$-0.790294\pi$
$877$	−46.8332	−1.58144	−0.790722	+	0.612176i	$0.790294\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17.0810	−0.575474	−0.287737	−	0.957710i	$-0.592903\pi$
$881$	−17.0810	−0.575474	−0.287737	+	0.957710i	$0.592903\pi$
$882$	0	0
$883$	7.21425	0.242779	0.121389	−	0.992605i	$-0.461265\pi$
$883$	7.21425	0.242779	0.121389	+	0.992605i	$0.461265\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.6625	−0.559471	−0.279735	−	0.960077i	$-0.590247\pi$
$887$	−16.6625	−0.559471	−0.279735	+	0.960077i	$0.590247\pi$
$888$	0	0
$889$	−0.766723	−0.0257151
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.38259	0.0797303
$894$	0	0
$895$	8.93118	0.298536
$896$	0	0
$897$	0	0
$898$	0	0
$899$	60.4112	2.01483
$900$	0	0
$901$	−31.5314	−1.05046
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25.4623	−0.846395
$906$	0	0
$907$	28.0024	0.929804	0.464902	−	0.885362i	$-0.346090\pi$
$907$	28.0024	0.929804	0.464902	+	0.885362i	$0.346090\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.45915	0.0483439	0.0241720	−	0.999708i	$-0.492305\pi$
$911$	1.45915	0.0483439	0.0241720	+	0.999708i	$0.492305\pi$
$912$	0	0
$913$	3.94733	0.130637
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.481553	0.0159023
$918$	0	0
$919$	26.8825	0.886773	0.443386	−	0.896331i	$-0.353777\pi$
$919$	26.8825	0.886773	0.443386	+	0.896331i	$0.353777\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	42.8419	1.41016
$924$	0	0
$925$	−7.78778	−0.256061
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.3843	0.832833	0.416416	−	0.909174i	$-0.363286\pi$
$929$	25.3843	0.832833	0.416416	+	0.909174i	$0.363286\pi$
$930$	0	0
$931$	9.78118	0.320565
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.2962	−0.402127
$936$	0	0
$937$	13.4201	0.438416	0.219208	−	0.975678i	$-0.429653\pi$
$937$	13.4201	0.438416	0.219208	+	0.975678i	$0.429653\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−40.3482	−1.31531	−0.657657	−	0.753317i	$-0.728452\pi$
$941$	−40.3482	−1.31531	−0.657657	+	0.753317i	$0.728452\pi$
$942$	0	0
$943$	11.0563	0.360043
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.6974	1.32249	0.661244	−	0.750171i	$-0.270029\pi$
$947$	40.6974	1.32249	0.661244	+	0.750171i	$0.270029\pi$
$948$	0	0
$949$	−14.2668	−0.463121
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−41.7552	−1.35258	−0.676292	−	0.736633i	$-0.736414\pi$
$953$	−41.7552	−1.35258	−0.676292	+	0.736633i	$0.736414\pi$
$954$	0	0
$955$	−12.3813	−0.400649
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.64583	0.0531467
$960$	0	0
$961$	58.4380	1.88510
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.31471	−0.106704
$966$	0	0
$967$	−34.4343	−1.10733	−0.553666	−	0.832739i	$-0.686772\pi$
$967$	−34.4343	−1.10733	−0.553666	+	0.832739i	$0.686772\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.96276	−0.223446	−0.111723	−	0.993739i	$-0.535637\pi$
$971$	−6.96276	−0.223446	−0.111723	+	0.993739i	$0.535637\pi$
$972$	0	0
$973$	−1.36534	−0.0437708
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36.7569	1.17596	0.587978	−	0.808877i	$-0.299924\pi$
$977$	36.7569	1.17596	0.587978	+	0.808877i	$0.299924\pi$
$978$	0	0
$979$	−6.02530	−0.192569
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.56822	−0.209494	−0.104747	−	0.994499i	$-0.533403\pi$
$983$	−6.56822	−0.209494	−0.104747	+	0.994499i	$0.533403\pi$
$984$	0	0
$985$	−16.3201	−0.520003
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.55936	−0.0495847
$990$	0	0
$991$	50.0562	1.59009	0.795044	−	0.606551i	$-0.207448\pi$
$991$	50.0562	1.59009	0.795044	+	0.606551i	$0.207448\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.80336	0.310787
$996$	0	0
$997$	14.7902	0.468410	0.234205	−	0.972187i	$-0.424751\pi$
$997$	14.7902	0.468410	0.234205	+	0.972187i	$0.424751\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.t.1.3	✓	5
3.2	odd	2		4140.2.a.u.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.t.1.3	✓	5	1.1	even	1	trivial
4140.2.a.u.1.3	yes	5	3.2	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.113901 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -0.113901 q^{7} -3.39991 q^{11} +6.27397 q^{13} -3.61662 q^{17} -1.39991 q^{19} -1.00000 q^{23} +1.00000 q^{25} +6.38787 q^{29} +9.45717 q^{31} +0.113901 q^{35} -7.78778 q^{37} -11.0563 q^{41} +1.55936 q^{43} -1.70196 q^{47} -6.98703 q^{49} +8.71849 q^{53} +3.39991 q^{55} +8.95988 q^{59} -1.83333 q^{61} -6.27397 q^{65} +9.21577 q^{67} +6.82852 q^{71} -2.27397 q^{73} +0.387254 q^{77} +11.7703 q^{79} -1.16101 q^{83} +3.61662 q^{85} +1.77220 q^{89} -0.714615 q^{91} +1.39991 q^{95} -0.131365 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 + 4 * q^7	\( 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more