LMFDB - Embedded newform 7623.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -1.46410 q^{4} +0.267949 q^{5} -1.00000 q^{7} -2.53590 q^{8} +O(q^{10})$	$q+0.732051 q^{2} -1.46410 q^{4} +0.267949 q^{5} -1.00000 q^{7} -2.53590 q^{8} +0.196152 q^{10} +2.73205 q^{13} -0.732051 q^{14} +1.07180 q^{16} +3.73205 q^{17} -2.19615 q^{19} -0.392305 q^{20} -3.26795 q^{23} -4.92820 q^{25} +2.00000 q^{26} +1.46410 q^{28} -1.26795 q^{29} +7.46410 q^{31} +5.85641 q^{32} +2.73205 q^{34} -0.267949 q^{35} -9.46410 q^{37} -1.60770 q^{38} -0.679492 q^{40} -4.00000 q^{41} +0.464102 q^{43} -2.39230 q^{46} +9.19615 q^{47} +1.00000 q^{49} -3.60770 q^{50} -4.00000 q^{52} -2.53590 q^{53} +2.53590 q^{56} -0.928203 q^{58} +10.1244 q^{59} +8.19615 q^{61} +5.46410 q^{62} +2.14359 q^{64} +0.732051 q^{65} -7.00000 q^{67} -5.46410 q^{68} -0.196152 q^{70} -9.12436 q^{71} -6.73205 q^{73} -6.92820 q^{74} +3.21539 q^{76} -0.535898 q^{79} +0.287187 q^{80} -2.92820 q^{82} -4.26795 q^{83} +1.00000 q^{85} +0.339746 q^{86} +14.6603 q^{89} -2.73205 q^{91} +4.78461 q^{92} +6.73205 q^{94} -0.588457 q^{95} +3.26795 q^{97} +0.732051 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^4 + 4 * q^5 - 2 * q^7 - 12 * q^8	$ 2 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 2 q^{7} - 12 q^{8} - 10 q^{10} + 2 q^{13} + 2 q^{14} + 16 q^{16} + 4 q^{17} + 6 q^{19} + 20 q^{20} - 10 q^{23} + 4 q^{25} + 4 q^{26} - 4 q^{28} - 6 q^{29} + 8 q^{31} - 16 q^{32} + 2 q^{34} - 4 q^{35} - 12 q^{37} - 24 q^{38} - 36 q^{40} - 8 q^{41} - 6 q^{43} + 16 q^{46} + 8 q^{47} + 2 q^{49} - 28 q^{50} - 8 q^{52} - 12 q^{53} + 12 q^{56} + 12 q^{58} - 4 q^{59} + 6 q^{61} + 4 q^{62} + 32 q^{64} - 2 q^{65} - 14 q^{67} - 4 q^{68} + 10 q^{70} + 6 q^{71} - 10 q^{73} + 48 q^{76} - 8 q^{79} + 56 q^{80} + 8 q^{82} - 12 q^{83} + 2 q^{85} + 18 q^{86} + 12 q^{89} - 2 q^{91} - 32 q^{92} + 10 q^{94} + 30 q^{95} + 10 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 + 4 * q^5 - 2 * q^7 - 12 * q^8 - 10 * q^10 + 2 * q^13 + 2 * q^14 + 16 * q^16 + 4 * q^17 + 6 * q^19 + 20 * q^20 - 10 * q^23 + 4 * q^25 + 4 * q^26 - 4 * q^28 - 6 * q^29 + 8 * q^31 - 16 * q^32 + 2 * q^34 - 4 * q^35 - 12 * q^37 - 24 * q^38 - 36 * q^40 - 8 * q^41 - 6 * q^43 + 16 * q^46 + 8 * q^47 + 2 * q^49 - 28 * q^50 - 8 * q^52 - 12 * q^53 + 12 * q^56 + 12 * q^58 - 4 * q^59 + 6 * q^61 + 4 * q^62 + 32 * q^64 - 2 * q^65 - 14 * q^67 - 4 * q^68 + 10 * q^70 + 6 * q^71 - 10 * q^73 + 48 * q^76 - 8 * q^79 + 56 * q^80 + 8 * q^82 - 12 * q^83 + 2 * q^85 + 18 * q^86 + 12 * q^89 - 2 * q^91 - 32 * q^92 + 10 * q^94 + 30 * q^95 + 10 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.53590	−0.896575
$9$	0	0
$10$	0.196152	0.0620288
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.73205	0.757735	0.378867	−	0.925451i	$-0.376314\pi$
$13$	2.73205	0.757735	0.378867	+	0.925451i	$0.376314\pi$
$14$	−0.732051	−0.195649
$15$	0	0
$16$	1.07180	0.267949
$17$	3.73205	0.905155	0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205	0.905155	0.452578	+	0.891725i	$0.350505\pi$
$18$	0	0
$19$	−2.19615	−0.503832	−0.251916	−	0.967749i	$-0.581061\pi$
$19$	−2.19615	−0.503832	−0.251916	+	0.967749i	$0.581061\pi$
$20$	−0.392305	−0.0877220
$21$	0	0
$22$	0	0
$23$	−3.26795	−0.681415	−0.340707	−	0.940169i	$-0.610666\pi$
$23$	−3.26795	−0.681415	−0.340707	+	0.940169i	$0.610666\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	2.00000	0.392232
$27$	0	0
$28$	1.46410	0.276689
$29$	−1.26795	−0.235452	−0.117726	−	0.993046i	$-0.537560\pi$
$29$	−1.26795	−0.235452	−0.117726	+	0.993046i	$0.537560\pi$
$30$	0	0
$31$	7.46410	1.34059	0.670296	−	0.742094i	$-0.266167\pi$
$31$	7.46410	1.34059	0.670296	+	0.742094i	$0.266167\pi$
$32$	5.85641	1.03528
$33$	0	0
$34$	2.73205	0.468543
$35$	−0.267949	−0.0452917
$36$	0	0
$37$	−9.46410	−1.55589	−0.777944	−	0.628333i	$-0.783737\pi$
$37$	−9.46410	−1.55589	−0.777944	+	0.628333i	$0.783737\pi$
$38$	−1.60770	−0.260803
$39$	0	0
$40$	−0.679492	−0.107437
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	0.464102	0.0707748	0.0353874	−	0.999374i	$-0.488733\pi$
$43$	0.464102	0.0707748	0.0353874	+	0.999374i	$0.488733\pi$
$44$	0	0
$45$	0	0
$46$	−2.39230	−0.352726
$47$	9.19615	1.34140	0.670698	−	0.741730i	$-0.265995\pi$
$47$	9.19615	1.34140	0.670698	+	0.741730i	$0.265995\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.60770	−0.510205
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−2.53590	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$53$	−2.53590	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$54$	0	0
$55$	0	0
$56$	2.53590	0.338874
$57$	0	0
$58$	−0.928203	−0.121879
$59$	10.1244	1.31808	0.659039	−	0.752108i	$-0.270963\pi$
$59$	10.1244	1.31808	0.659039	+	0.752108i	$0.270963\pi$
$60$	0	0
$61$	8.19615	1.04941	0.524705	−	0.851284i	$-0.324176\pi$
$61$	8.19615	1.04941	0.524705	+	0.851284i	$0.324176\pi$
$62$	5.46410	0.693942
$63$	0	0
$64$	2.14359	0.267949
$65$	0.732051	0.0907997
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−5.46410	−0.662620
$69$	0	0
$70$	−0.196152	−0.0234447
$71$	−9.12436	−1.08286	−0.541431	−	0.840745i	$-0.682117\pi$
$71$	−9.12436	−1.08286	−0.541431	+	0.840745i	$0.682117\pi$
$72$	0	0
$73$	−6.73205	−0.787927	−0.393963	−	0.919126i	$-0.628896\pi$
$73$	−6.73205	−0.787927	−0.393963	+	0.919126i	$0.628896\pi$
$74$	−6.92820	−0.805387
$75$	0	0
$76$	3.21539	0.368831
$77$	0	0
$78$	0	0
$79$	−0.535898	−0.0602933	−0.0301466	−	0.999545i	$-0.509597\pi$
$79$	−0.535898	−0.0602933	−0.0301466	+	0.999545i	$0.509597\pi$
$80$	0.287187	0.0321085
$81$	0	0
$82$	−2.92820	−0.323366
$83$	−4.26795	−0.468468	−0.234234	−	0.972180i	$-0.575258\pi$
$83$	−4.26795	−0.468468	−0.234234	+	0.972180i	$0.575258\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0.339746	0.0366357
$87$	0	0
$88$	0	0
$89$	14.6603	1.55398	0.776992	−	0.629511i	$-0.216745\pi$
$89$	14.6603	1.55398	0.776992	+	0.629511i	$0.216745\pi$
$90$	0	0
$91$	−2.73205	−0.286397
$92$	4.78461	0.498830
$93$	0	0
$94$	6.73205	0.694358
$95$	−0.588457	−0.0603744
$96$	0	0
$97$	3.26795	0.331810	0.165905	−	0.986142i	$-0.446945\pi$
$97$	3.26795	0.331810	0.165905	+	0.986142i	$0.446945\pi$
$98$	0.732051	0.0739483
$99$	0	0
$100$	7.21539	0.721539
$101$	−9.73205	−0.968375	−0.484188	−	0.874964i	$-0.660885\pi$
$101$	−9.73205	−0.968375	−0.484188	+	0.874964i	$0.660885\pi$
$102$	0	0
$103$	4.19615	0.413459	0.206730	−	0.978398i	$-0.433718\pi$
$103$	4.19615	0.413459	0.206730	+	0.978398i	$0.433718\pi$
$104$	−6.92820	−0.679366
$105$	0	0
$106$	−1.85641	−0.180310
$107$	−18.5885	−1.79701	−0.898507	−	0.438959i	$-0.855347\pi$
$107$	−18.5885	−1.79701	−0.898507	+	0.438959i	$0.855347\pi$
$108$	0	0
$109$	17.3923	1.66588	0.832940	−	0.553363i	$-0.186656\pi$
$109$	17.3923	1.66588	0.832940	+	0.553363i	$0.186656\pi$
$110$	0	0
$111$	0	0
$112$	−1.07180	−0.101275
$113$	−8.53590	−0.802990	−0.401495	−	0.915861i	$-0.631509\pi$
$113$	−8.53590	−0.802990	−0.401495	+	0.915861i	$0.631509\pi$
$114$	0	0
$115$	−0.875644	−0.0816543
$116$	1.85641	0.172363
$117$	0	0
$118$	7.41154	0.682288
$119$	−3.73205	−0.342117
$120$	0	0
$121$	0	0
$122$	6.00000	0.543214
$123$	0	0
$124$	−10.9282	−0.981382
$125$	−2.66025	−0.237940
$126$	0	0
$127$	−3.53590	−0.313760	−0.156880	−	0.987618i	$-0.550144\pi$
$127$	−3.53590	−0.313760	−0.156880	+	0.987618i	$0.550144\pi$
$128$	−10.1436	−0.896575
$129$	0	0
$130$	0.535898	0.0470014
$131$	−5.19615	−0.453990	−0.226995	−	0.973896i	$-0.572890\pi$
$131$	−5.19615	−0.453990	−0.226995	+	0.973896i	$0.572890\pi$
$132$	0	0
$133$	2.19615	0.190431
$134$	−5.12436	−0.442677
$135$	0	0
$136$	−9.46410	−0.811540
$137$	−13.6603	−1.16707	−0.583537	−	0.812086i	$-0.698332\pi$
$137$	−13.6603	−1.16707	−0.583537	+	0.812086i	$0.698332\pi$
$138$	0	0
$139$	7.80385	0.661914	0.330957	−	0.943646i	$-0.392629\pi$
$139$	7.80385	0.661914	0.330957	+	0.943646i	$0.392629\pi$
$140$	0.392305	0.0331558
$141$	0	0
$142$	−6.67949	−0.560531
$143$	0	0
$144$	0	0
$145$	−0.339746	−0.0282144
$146$	−4.92820	−0.407861
$147$	0	0
$148$	13.8564	1.13899
$149$	2.73205	0.223818	0.111909	−	0.993718i	$-0.464303\pi$
$149$	2.73205	0.223818	0.111909	+	0.993718i	$0.464303\pi$
$150$	0	0
$151$	9.39230	0.764335	0.382167	−	0.924093i	$-0.375178\pi$
$151$	9.39230	0.764335	0.382167	+	0.924093i	$0.375178\pi$
$152$	5.56922	0.451723
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−6.53590	−0.521621	−0.260811	−	0.965390i	$-0.583990\pi$
$157$	−6.53590	−0.521621	−0.260811	+	0.965390i	$0.583990\pi$
$158$	−0.392305	−0.0312101
$159$	0	0
$160$	1.56922	0.124058
$161$	3.26795	0.257550
$162$	0	0
$163$	7.46410	0.584634	0.292317	−	0.956322i	$-0.405574\pi$
$163$	7.46410	0.584634	0.292317	+	0.956322i	$0.405574\pi$
$164$	5.85641	0.457309
$165$	0	0
$166$	−3.12436	−0.242497
$167$	−25.0526	−1.93863	−0.969313	−	0.245831i	$-0.920939\pi$
$167$	−25.0526	−1.93863	−0.969313	+	0.245831i	$0.920939\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0.732051	0.0561457
$171$	0	0
$172$	−0.679492	−0.0518108
$173$	−5.19615	−0.395056	−0.197528	−	0.980297i	$-0.563291\pi$
$173$	−5.19615	−0.395056	−0.197528	+	0.980297i	$0.563291\pi$
$174$	0	0
$175$	4.92820	0.372537
$176$	0	0
$177$	0	0
$178$	10.7321	0.804401
$179$	4.39230	0.328296	0.164148	−	0.986436i	$-0.447512\pi$
$179$	4.39230	0.328296	0.164148	+	0.986436i	$0.447512\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	8.28719	0.610940
$185$	−2.53590	−0.186443
$186$	0	0
$187$	0	0
$188$	−13.4641	−0.981971
$189$	0	0
$190$	−0.430781	−0.0312521
$191$	−16.1962	−1.17191	−0.585956	−	0.810343i	$-0.699281\pi$
$191$	−16.1962	−1.17191	−0.585956	+	0.810343i	$0.699281\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	2.39230	0.171757
$195$	0	0
$196$	−1.46410	−0.104579
$197$	−12.5359	−0.893146	−0.446573	−	0.894747i	$-0.647356\pi$
$197$	−12.5359	−0.893146	−0.446573	+	0.894747i	$0.647356\pi$
$198$	0	0
$199$	−20.1962	−1.43167	−0.715834	−	0.698271i	$-0.753953\pi$
$199$	−20.1962	−1.43167	−0.715834	+	0.698271i	$0.753953\pi$
$200$	12.4974	0.883701
$201$	0	0
$202$	−7.12436	−0.501268
$203$	1.26795	0.0889926
$204$	0	0
$205$	−1.07180	−0.0748575
$206$	3.07180	0.214022
$207$	0	0
$208$	2.92820	0.203034
$209$	0	0
$210$	0	0
$211$	−17.7846	−1.22434	−0.612172	−	0.790725i	$-0.709704\pi$
$211$	−17.7846	−1.22434	−0.612172	+	0.790725i	$0.709704\pi$
$212$	3.71281	0.254997
$213$	0	0
$214$	−13.6077	−0.930203
$215$	0.124356	0.00848099
$216$	0	0
$217$	−7.46410	−0.506696
$218$	12.7321	0.862323
$219$	0	0
$220$	0	0
$221$	10.1962	0.685867
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−5.85641	−0.391298
$225$	0	0
$226$	−6.24871	−0.415658
$227$	−14.6603	−0.973035	−0.486518	−	0.873671i	$-0.661733\pi$
$227$	−14.6603	−0.973035	−0.486518	+	0.873671i	$0.661733\pi$
$228$	0	0
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	−0.641016	−0.0422674
$231$	0	0
$232$	3.21539	0.211101
$233$	−20.1962	−1.32309	−0.661547	−	0.749904i	$-0.730100\pi$
$233$	−20.1962	−1.32309	−0.661547	+	0.749904i	$0.730100\pi$
$234$	0	0
$235$	2.46410	0.160740
$236$	−14.8231	−0.964901
$237$	0	0
$238$	−2.73205	−0.177093
$239$	−24.9282	−1.61247	−0.806236	−	0.591594i	$-0.798499\pi$
$239$	−24.9282	−1.61247	−0.806236	+	0.591594i	$0.798499\pi$
$240$	0	0
$241$	15.6603	1.00877	0.504383	−	0.863480i	$-0.331720\pi$
$241$	15.6603	1.00877	0.504383	+	0.863480i	$0.331720\pi$
$242$	0	0
$243$	0	0
$244$	−12.0000	−0.768221
$245$	0.267949	0.0171186
$246$	0	0
$247$	−6.00000	−0.381771
$248$	−18.9282	−1.20194
$249$	0	0
$250$	−1.94744	−0.123167
$251$	1.07180	0.0676512	0.0338256	−	0.999428i	$-0.489231\pi$
$251$	1.07180	0.0676512	0.0338256	+	0.999428i	$0.489231\pi$
$252$	0	0
$253$	0	0
$254$	−2.58846	−0.162414
$255$	0	0
$256$	−11.7128	−0.732051
$257$	−4.12436	−0.257270	−0.128635	−	0.991692i	$-0.541060\pi$
$257$	−4.12436	−0.257270	−0.128635	+	0.991692i	$0.541060\pi$
$258$	0	0
$259$	9.46410	0.588071
$260$	−1.07180	−0.0664700
$261$	0	0
$262$	−3.80385	−0.235002
$263$	−5.80385	−0.357881	−0.178940	−	0.983860i	$-0.557267\pi$
$263$	−5.80385	−0.357881	−0.178940	+	0.983860i	$0.557267\pi$
$264$	0	0
$265$	−0.679492	−0.0417409
$266$	1.60770	0.0985741
$267$	0	0
$268$	10.2487	0.626040
$269$	−23.8564	−1.45455	−0.727275	−	0.686346i	$-0.759214\pi$
$269$	−23.8564	−1.45455	−0.727275	+	0.686346i	$0.759214\pi$
$270$	0	0
$271$	3.60770	0.219152	0.109576	−	0.993978i	$-0.465051\pi$
$271$	3.60770	0.219152	0.109576	+	0.993978i	$0.465051\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	−25.7846	−1.54925	−0.774624	−	0.632423i	$-0.782061\pi$
$277$	−25.7846	−1.54925	−0.774624	+	0.632423i	$0.782061\pi$
$278$	5.71281	0.342632
$279$	0	0
$280$	0.679492	0.0406074
$281$	27.7128	1.65321	0.826604	−	0.562784i	$-0.190270\pi$
$281$	27.7128	1.65321	0.826604	+	0.562784i	$0.190270\pi$
$282$	0	0
$283$	−26.2487	−1.56032	−0.780162	−	0.625578i	$-0.784863\pi$
$283$	−26.2487	−1.56032	−0.780162	+	0.625578i	$0.784863\pi$
$284$	13.3590	0.792710
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−3.07180	−0.180694
$290$	−0.248711	−0.0146048
$291$	0	0
$292$	9.85641	0.576803
$293$	16.2679	0.950384	0.475192	−	0.879882i	$-0.342379\pi$
$293$	16.2679	0.950384	0.475192	+	0.879882i	$0.342379\pi$
$294$	0	0
$295$	2.71281	0.157946
$296$	24.0000	1.39497
$297$	0	0
$298$	2.00000	0.115857
$299$	−8.92820	−0.516331
$300$	0	0
$301$	−0.464102	−0.0267504
$302$	6.87564	0.395649
$303$	0	0
$304$	−2.35383	−0.135001
$305$	2.19615	0.125751
$306$	0	0
$307$	26.5885	1.51748	0.758742	−	0.651392i	$-0.225814\pi$
$307$	26.5885	1.51748	0.758742	+	0.651392i	$0.225814\pi$
$308$	0	0
$309$	0	0
$310$	1.46410	0.0831554
$311$	−19.0526	−1.08037	−0.540186	−	0.841546i	$-0.681646\pi$
$311$	−19.0526	−1.08037	−0.540186	+	0.841546i	$0.681646\pi$
$312$	0	0
$313$	0.392305	0.0221744	0.0110872	−	0.999939i	$-0.496471\pi$
$313$	0.392305	0.0221744	0.0110872	+	0.999939i	$0.496471\pi$
$314$	−4.78461	−0.270011
$315$	0	0
$316$	0.784610	0.0441377
$317$	−29.5167	−1.65782	−0.828910	−	0.559381i	$-0.811039\pi$
$317$	−29.5167	−1.65782	−0.828910	+	0.559381i	$0.811039\pi$
$318$	0	0
$319$	0	0
$320$	0.574374	0.0321085
$321$	0	0
$322$	2.39230	0.133318
$323$	−8.19615	−0.456046
$324$	0	0
$325$	−13.4641	−0.746854
$326$	5.46410	0.302629
$327$	0	0
$328$	10.1436	0.560086
$329$	−9.19615	−0.507000
$330$	0	0
$331$	26.1769	1.43881	0.719407	−	0.694589i	$-0.244414\pi$
$331$	26.1769	1.43881	0.719407	+	0.694589i	$0.244414\pi$
$332$	6.24871	0.342943
$333$	0	0
$334$	−18.3397	−1.00351
$335$	−1.87564	−0.102477
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−4.05256	−0.220430
$339$	0	0
$340$	−1.46410	−0.0794021
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.17691	−0.0634550
$345$	0	0
$346$	−3.80385	−0.204496
$347$	−26.0526	−1.39857	−0.699287	−	0.714841i	$-0.746499\pi$
$347$	−26.0526	−1.39857	−0.699287	+	0.714841i	$0.746499\pi$
$348$	0	0
$349$	1.60770	0.0860579	0.0430290	−	0.999074i	$-0.486299\pi$
$349$	1.60770	0.0860579	0.0430290	+	0.999074i	$0.486299\pi$
$350$	3.60770	0.192839
$351$	0	0
$352$	0	0
$353$	11.0718	0.589292	0.294646	−	0.955606i	$-0.404798\pi$
$353$	11.0718	0.589292	0.294646	+	0.955606i	$0.404798\pi$
$354$	0	0
$355$	−2.44486	−0.129760
$356$	−21.4641	−1.13760
$357$	0	0
$358$	3.21539	0.169939
$359$	−25.1244	−1.32601	−0.663006	−	0.748614i	$-0.730720\pi$
$359$	−25.1244	−1.32601	−0.663006	+	0.748614i	$0.730720\pi$
$360$	0	0
$361$	−14.1769	−0.746153
$362$	−5.85641	−0.307806
$363$	0	0
$364$	4.00000	0.209657
$365$	−1.80385	−0.0944177
$366$	0	0
$367$	−12.7321	−0.664608	−0.332304	−	0.943172i	$-0.607826\pi$
$367$	−12.7321	−0.664608	−0.332304	+	0.943172i	$0.607826\pi$
$368$	−3.50258	−0.182584
$369$	0	0
$370$	−1.85641	−0.0965100
$371$	2.53590	0.131657
$372$	0	0
$373$	−23.9282	−1.23896	−0.619478	−	0.785014i	$-0.712656\pi$
$373$	−23.9282	−1.23896	−0.619478	+	0.785014i	$0.712656\pi$
$374$	0	0
$375$	0	0
$376$	−23.3205	−1.20266
$377$	−3.46410	−0.178410
$378$	0	0
$379$	3.53590	0.181627	0.0908135	−	0.995868i	$-0.471053\pi$
$379$	3.53590	0.181627	0.0908135	+	0.995868i	$0.471053\pi$
$380$	0.861561	0.0441972
$381$	0	0
$382$	−11.8564	−0.606627
$383$	34.1244	1.74367	0.871837	−	0.489797i	$-0.162929\pi$
$383$	34.1244	1.74367	0.871837	+	0.489797i	$0.162929\pi$
$384$	0	0
$385$	0	0
$386$	−3.66025	−0.186302
$387$	0	0
$388$	−4.78461	−0.242902
$389$	−16.3923	−0.831123	−0.415561	−	0.909565i	$-0.636415\pi$
$389$	−16.3923	−0.831123	−0.415561	+	0.909565i	$0.636415\pi$
$390$	0	0
$391$	−12.1962	−0.616786
$392$	−2.53590	−0.128082
$393$	0	0
$394$	−9.17691	−0.462326
$395$	−0.143594	−0.00722498
$396$	0	0
$397$	−13.3205	−0.668537	−0.334269	−	0.942478i	$-0.608489\pi$
$397$	−13.3205	−0.668537	−0.334269	+	0.942478i	$0.608489\pi$
$398$	−14.7846	−0.741086
$399$	0	0
$400$	−5.28203	−0.264102
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	20.3923	1.01581
$404$	14.2487	0.708900
$405$	0	0
$406$	0.928203	0.0460660
$407$	0	0
$408$	0	0
$409$	−23.6603	−1.16992	−0.584962	−	0.811061i	$-0.698891\pi$
$409$	−23.6603	−1.16992	−0.584962	+	0.811061i	$0.698891\pi$
$410$	−0.784610	−0.0387491
$411$	0	0
$412$	−6.14359	−0.302673
$413$	−10.1244	−0.498187
$414$	0	0
$415$	−1.14359	−0.0561368
$416$	16.0000	0.784465
$417$	0	0
$418$	0	0
$419$	−9.73205	−0.475442	−0.237721	−	0.971334i	$-0.576400\pi$
$419$	−9.73205	−0.475442	−0.237721	+	0.971334i	$0.576400\pi$
$420$	0	0
$421$	38.8564	1.89375	0.946873	−	0.321609i	$-0.104224\pi$
$421$	38.8564	1.89375	0.946873	+	0.321609i	$0.104224\pi$
$422$	−13.0192	−0.633767
$423$	0	0
$424$	6.43078	0.312306
$425$	−18.3923	−0.892158
$426$	0	0
$427$	−8.19615	−0.396640
$428$	27.2154	1.31551
$429$	0	0
$430$	0.0910347	0.00439008
$431$	−23.3205	−1.12331	−0.561655	−	0.827372i	$-0.689835\pi$
$431$	−23.3205	−1.12331	−0.561655	+	0.827372i	$0.689835\pi$
$432$	0	0
$433$	33.9090	1.62956	0.814780	−	0.579770i	$-0.196857\pi$
$433$	33.9090	1.62956	0.814780	+	0.579770i	$0.196857\pi$
$434$	−5.46410	−0.262285
$435$	0	0
$436$	−25.4641	−1.21951
$437$	7.17691	0.343318
$438$	0	0
$439$	−0.143594	−0.00685335	−0.00342667	−	0.999994i	$-0.501091\pi$
$439$	−0.143594	−0.00685335	−0.00342667	+	0.999994i	$0.501091\pi$
$440$	0	0
$441$	0	0
$442$	7.46410	0.355031
$443$	5.80385	0.275749	0.137875	−	0.990450i	$-0.455973\pi$
$443$	5.80385	0.275749	0.137875	+	0.990450i	$0.455973\pi$
$444$	0	0
$445$	3.92820	0.186215
$446$	11.7128	0.554618
$447$	0	0
$448$	−2.14359	−0.101275
$449$	16.2487	0.766824	0.383412	−	0.923577i	$-0.374749\pi$
$449$	16.2487	0.766824	0.383412	+	0.923577i	$0.374749\pi$
$450$	0	0
$451$	0	0
$452$	12.4974	0.587829
$453$	0	0
$454$	−10.7321	−0.503680
$455$	−0.732051	−0.0343191
$456$	0	0
$457$	−33.9282	−1.58709	−0.793547	−	0.608509i	$-0.791768\pi$
$457$	−33.9282	−1.58709	−0.793547	+	0.608509i	$0.791768\pi$
$458$	5.07180	0.236989
$459$	0	0
$460$	1.28203	0.0597751
$461$	27.5885	1.28492	0.642461	−	0.766318i	$-0.277913\pi$
$461$	27.5885	1.28492	0.642461	+	0.766318i	$0.277913\pi$
$462$	0	0
$463$	31.4641	1.46226	0.731130	−	0.682238i	$-0.238993\pi$
$463$	31.4641	1.46226	0.731130	+	0.682238i	$0.238993\pi$
$464$	−1.35898	−0.0630892
$465$	0	0
$466$	−14.7846	−0.684884
$467$	34.3923	1.59149	0.795743	−	0.605634i	$-0.207081\pi$
$467$	34.3923	1.59149	0.795743	+	0.605634i	$0.207081\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	1.80385	0.0832053
$471$	0	0
$472$	−25.6743	−1.18176
$473$	0	0
$474$	0	0
$475$	10.8231	0.496597
$476$	5.46410	0.250447
$477$	0	0
$478$	−18.2487	−0.834677
$479$	−32.3731	−1.47916	−0.739582	−	0.673067i	$-0.764977\pi$
$479$	−32.3731	−1.47916	−0.739582	+	0.673067i	$0.764977\pi$
$480$	0	0
$481$	−25.8564	−1.17895
$482$	11.4641	0.522176
$483$	0	0
$484$	0	0
$485$	0.875644	0.0397610
$486$	0	0
$487$	42.1769	1.91122	0.955609	−	0.294637i	$-0.0951988\pi$
$487$	42.1769	1.91122	0.955609	+	0.294637i	$0.0951988\pi$
$488$	−20.7846	−0.940875
$489$	0	0
$490$	0.196152	0.00886126
$491$	0.0525589	0.00237195	0.00118597	−	0.999999i	$-0.499622\pi$
$491$	0.0525589	0.00237195	0.00118597	+	0.999999i	$0.499622\pi$
$492$	0	0
$493$	−4.73205	−0.213121
$494$	−4.39230	−0.197619
$495$	0	0
$496$	8.00000	0.359211
$497$	9.12436	0.409283
$498$	0	0
$499$	40.3205	1.80499	0.902497	−	0.430696i	$-0.141732\pi$
$499$	40.3205	1.80499	0.902497	+	0.430696i	$0.141732\pi$
$500$	3.89488	0.174184
$501$	0	0
$502$	0.784610	0.0350188
$503$	18.5167	0.825617	0.412809	−	0.910818i	$-0.364548\pi$
$503$	18.5167	0.825617	0.412809	+	0.910818i	$0.364548\pi$
$504$	0	0
$505$	−2.60770	−0.116041
$506$	0	0
$507$	0	0
$508$	5.17691	0.229688
$509$	17.1962	0.762206	0.381103	−	0.924533i	$-0.375544\pi$
$509$	17.1962	0.762206	0.381103	+	0.924533i	$0.375544\pi$
$510$	0	0
$511$	6.73205	0.297808
$512$	11.7128	0.517638
$513$	0	0
$514$	−3.01924	−0.133173
$515$	1.12436	0.0495450
$516$	0	0
$517$	0	0
$518$	6.92820	0.304408
$519$	0	0
$520$	−1.85641	−0.0814088
$521$	19.9808	0.875373	0.437687	−	0.899128i	$-0.355798\pi$
$521$	19.9808	0.875373	0.437687	+	0.899128i	$0.355798\pi$
$522$	0	0
$523$	26.5885	1.16263	0.581316	−	0.813678i	$-0.302538\pi$
$523$	26.5885	1.16263	0.581316	+	0.813678i	$0.302538\pi$
$524$	7.60770	0.332344
$525$	0	0
$526$	−4.24871	−0.185253
$527$	27.8564	1.21344
$528$	0	0
$529$	−12.3205	−0.535674
$530$	−0.497423	−0.0216067
$531$	0	0
$532$	−3.21539	−0.139405
$533$	−10.9282	−0.473353
$534$	0	0
$535$	−4.98076	−0.215337
$536$	17.7513	0.766739
$537$	0	0
$538$	−17.4641	−0.752931
$539$	0	0
$540$	0	0
$541$	41.3923	1.77959	0.889797	−	0.456356i	$-0.150846\pi$
$541$	41.3923	1.77959	0.889797	+	0.456356i	$0.150846\pi$
$542$	2.64102	0.113441
$543$	0	0
$544$	21.8564	0.937086
$545$	4.66025	0.199623
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	20.0000	0.854358
$549$	0	0
$550$	0	0
$551$	2.78461	0.118628
$552$	0	0
$553$	0.535898	0.0227887
$554$	−18.8756	−0.801949
$555$	0	0
$556$	−11.4256	−0.484554
$557$	−16.9282	−0.717271	−0.358635	−	0.933478i	$-0.616758\pi$
$557$	−16.9282	−0.717271	−0.358635	+	0.933478i	$0.616758\pi$
$558$	0	0
$559$	1.26795	0.0536285
$560$	−0.287187	−0.0121359
$561$	0	0
$562$	20.2872	0.855763
$563$	9.73205	0.410157	0.205079	−	0.978746i	$-0.434255\pi$
$563$	9.73205	0.410157	0.205079	+	0.978746i	$0.434255\pi$
$564$	0	0
$565$	−2.28719	−0.0962227
$566$	−19.2154	−0.807683
$567$	0	0
$568$	23.1384	0.970867
$569$	−25.2679	−1.05929	−0.529644	−	0.848220i	$-0.677674\pi$
$569$	−25.2679	−1.05929	−0.529644	+	0.848220i	$0.677674\pi$
$570$	0	0
$571$	1.07180	0.0448533	0.0224266	−	0.999748i	$-0.492861\pi$
$571$	1.07180	0.0448533	0.0224266	+	0.999748i	$0.492861\pi$
$572$	0	0
$573$	0	0
$574$	2.92820	0.122221
$575$	16.1051	0.671630
$576$	0	0
$577$	−14.4449	−0.601348	−0.300674	−	0.953727i	$-0.597212\pi$
$577$	−14.4449	−0.601348	−0.300674	+	0.953727i	$0.597212\pi$
$578$	−2.24871	−0.0935341
$579$	0	0
$580$	0.497423	0.0206543
$581$	4.26795	0.177064
$582$	0	0
$583$	0	0
$584$	17.0718	0.706436
$585$	0	0
$586$	11.9090	0.491955
$587$	8.12436	0.335328	0.167664	−	0.985844i	$-0.446378\pi$
$587$	8.12436	0.335328	0.167664	+	0.985844i	$0.446378\pi$
$588$	0	0
$589$	−16.3923	−0.675433
$590$	1.98592	0.0817589
$591$	0	0
$592$	−10.1436	−0.416899
$593$	−35.5885	−1.46144	−0.730721	−	0.682676i	$-0.760816\pi$
$593$	−35.5885	−1.46144	−0.730721	+	0.682676i	$0.760816\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−4.00000	−0.163846
$597$	0	0
$598$	−6.53590	−0.267273
$599$	−23.1769	−0.946983	−0.473492	−	0.880798i	$-0.657007\pi$
$599$	−23.1769	−0.946983	−0.473492	+	0.880798i	$0.657007\pi$
$600$	0	0
$601$	31.9090	1.30159	0.650797	−	0.759252i	$-0.274435\pi$
$601$	31.9090	1.30159	0.650797	+	0.759252i	$0.274435\pi$
$602$	−0.339746	−0.0138470
$603$	0	0
$604$	−13.7513	−0.559532
$605$	0	0
$606$	0	0
$607$	−4.58846	−0.186240	−0.0931199	−	0.995655i	$-0.529684\pi$
$607$	−4.58846	−0.186240	−0.0931199	+	0.995655i	$0.529684\pi$
$608$	−12.8616	−0.521605
$609$	0	0
$610$	1.60770	0.0650937
$611$	25.1244	1.01642
$612$	0	0
$613$	−14.6077	−0.589999	−0.295000	−	0.955497i	$-0.595320\pi$
$613$	−14.6077	−0.589999	−0.295000	+	0.955497i	$0.595320\pi$
$614$	19.4641	0.785507
$615$	0	0
$616$	0	0
$617$	39.4641	1.58876	0.794382	−	0.607418i	$-0.207795\pi$
$617$	39.4641	1.58876	0.794382	+	0.607418i	$0.207795\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−2.92820	−0.117599
$621$	0	0
$622$	−13.9474	−0.559241
$623$	−14.6603	−0.587351
$624$	0	0
$625$	23.9282	0.957128
$626$	0.287187	0.0114783
$627$	0	0
$628$	9.56922	0.381853
$629$	−35.3205	−1.40832
$630$	0	0
$631$	−0.856406	−0.0340930	−0.0170465	−	0.999855i	$-0.505426\pi$
$631$	−0.856406	−0.0340930	−0.0170465	+	0.999855i	$0.505426\pi$
$632$	1.35898	0.0540575
$633$	0	0
$634$	−21.6077	−0.858151
$635$	−0.947441	−0.0375981
$636$	0	0
$637$	2.73205	0.108248
$638$	0	0
$639$	0	0
$640$	−2.71797	−0.107437
$641$	−35.9090	−1.41832	−0.709159	−	0.705048i	$-0.750925\pi$
$641$	−35.9090	−1.41832	−0.709159	+	0.705048i	$0.750925\pi$
$642$	0	0
$643$	−41.8564	−1.65066	−0.825328	−	0.564654i	$-0.809010\pi$
$643$	−41.8564	−1.65066	−0.825328	+	0.564654i	$0.809010\pi$
$644$	−4.78461	−0.188540
$645$	0	0
$646$	−6.00000	−0.236067
$647$	27.1962	1.06919	0.534596	−	0.845108i	$-0.320464\pi$
$647$	27.1962	1.06919	0.534596	+	0.845108i	$0.320464\pi$
$648$	0	0
$649$	0	0
$650$	−9.85641	−0.386600
$651$	0	0
$652$	−10.9282	−0.427981
$653$	−20.9808	−0.821041	−0.410520	−	0.911851i	$-0.634653\pi$
$653$	−20.9808	−0.821041	−0.410520	+	0.911851i	$0.634653\pi$
$654$	0	0
$655$	−1.39230	−0.0544019
$656$	−4.28719	−0.167387
$657$	0	0
$658$	−6.73205	−0.262443
$659$	3.51666	0.136990	0.0684948	−	0.997651i	$-0.478180\pi$
$659$	3.51666	0.136990	0.0684948	+	0.997651i	$0.478180\pi$
$660$	0	0
$661$	−45.8564	−1.78361	−0.891804	−	0.452422i	$-0.850560\pi$
$661$	−45.8564	−1.78361	−0.891804	+	0.452422i	$0.850560\pi$
$662$	19.1628	0.744785
$663$	0	0
$664$	10.8231	0.420017
$665$	0.588457	0.0228194
$666$	0	0
$667$	4.14359	0.160441
$668$	36.6795	1.41917
$669$	0	0
$670$	−1.37307	−0.0530462
$671$	0	0
$672$	0	0
$673$	−8.60770	−0.331802	−0.165901	−	0.986142i	$-0.553053\pi$
$673$	−8.60770	−0.331802	−0.165901	+	0.986142i	$0.553053\pi$
$674$	−20.4974	−0.789531
$675$	0	0
$676$	8.10512	0.311735
$677$	−27.5885	−1.06031	−0.530155	−	0.847901i	$-0.677866\pi$
$677$	−27.5885	−1.06031	−0.530155	+	0.847901i	$0.677866\pi$
$678$	0	0
$679$	−3.26795	−0.125412
$680$	−2.53590	−0.0972473
$681$	0	0
$682$	0	0
$683$	28.2487	1.08091	0.540453	−	0.841374i	$-0.318253\pi$
$683$	28.2487	1.08091	0.540453	+	0.841374i	$0.318253\pi$
$684$	0	0
$685$	−3.66025	−0.139851
$686$	−0.732051	−0.0279498
$687$	0	0
$688$	0.497423	0.0189641
$689$	−6.92820	−0.263944
$690$	0	0
$691$	−8.39230	−0.319258	−0.159629	−	0.987177i	$-0.551030\pi$
$691$	−8.39230	−0.319258	−0.159629	+	0.987177i	$0.551030\pi$
$692$	7.60770	0.289201
$693$	0	0
$694$	−19.0718	−0.723956
$695$	2.09103	0.0793175
$696$	0	0
$697$	−14.9282	−0.565446
$698$	1.17691	0.0445469
$699$	0	0
$700$	−7.21539	−0.272716
$701$	−13.6603	−0.515941	−0.257970	−	0.966153i	$-0.583054\pi$
$701$	−13.6603	−0.515941	−0.257970	+	0.966153i	$0.583054\pi$
$702$	0	0
$703$	20.7846	0.783906
$704$	0	0
$705$	0	0
$706$	8.10512	0.305040
$707$	9.73205	0.366011
$708$	0	0
$709$	−8.60770	−0.323269	−0.161634	−	0.986851i	$-0.551677\pi$
$709$	−8.60770	−0.323269	−0.161634	+	0.986851i	$0.551677\pi$
$710$	−1.78976	−0.0671687
$711$	0	0
$712$	−37.1769	−1.39326
$713$	−24.3923	−0.913499
$714$	0	0
$715$	0	0
$716$	−6.43078	−0.240330
$717$	0	0
$718$	−18.3923	−0.686395
$719$	44.1051	1.64484	0.822422	−	0.568878i	$-0.192622\pi$
$719$	44.1051	1.64484	0.822422	+	0.568878i	$0.192622\pi$
$720$	0	0
$721$	−4.19615	−0.156273
$722$	−10.3782	−0.386237
$723$	0	0
$724$	11.7128	0.435303
$725$	6.24871	0.232071
$726$	0	0
$727$	37.5167	1.39142	0.695708	−	0.718325i	$-0.255091\pi$
$727$	37.5167	1.39142	0.695708	+	0.718325i	$0.255091\pi$
$728$	6.92820	0.256776
$729$	0	0
$730$	−1.32051	−0.0488742
$731$	1.73205	0.0640622
$732$	0	0
$733$	−14.4449	−0.533533	−0.266767	−	0.963761i	$-0.585955\pi$
$733$	−14.4449	−0.533533	−0.266767	+	0.963761i	$0.585955\pi$
$734$	−9.32051	−0.344026
$735$	0	0
$736$	−19.1384	−0.705452
$737$	0	0
$738$	0	0
$739$	1.32051	0.0485757	0.0242878	−	0.999705i	$-0.492268\pi$
$739$	1.32051	0.0485757	0.0242878	+	0.999705i	$0.492268\pi$
$740$	3.71281	0.136486
$741$	0	0
$742$	1.85641	0.0681508
$743$	17.1244	0.628232	0.314116	−	0.949385i	$-0.398292\pi$
$743$	17.1244	0.628232	0.314116	+	0.949385i	$0.398292\pi$
$744$	0	0
$745$	0.732051	0.0268203
$746$	−17.5167	−0.641331
$747$	0	0
$748$	0	0
$749$	18.5885	0.679207
$750$	0	0
$751$	11.3923	0.415711	0.207856	−	0.978160i	$-0.433352\pi$
$751$	11.3923	0.415711	0.207856	+	0.978160i	$0.433352\pi$
$752$	9.85641	0.359426
$753$	0	0
$754$	−2.53590	−0.0923520
$755$	2.51666	0.0915907
$756$	0	0
$757$	−7.92820	−0.288155	−0.144078	−	0.989566i	$-0.546022\pi$
$757$	−7.92820	−0.288155	−0.144078	+	0.989566i	$0.546022\pi$
$758$	2.58846	0.0940170
$759$	0	0
$760$	1.49227	0.0541302
$761$	−20.9090	−0.757949	−0.378975	−	0.925407i	$-0.623723\pi$
$761$	−20.9090	−0.757949	−0.378975	+	0.925407i	$0.623723\pi$
$762$	0	0
$763$	−17.3923	−0.629644
$764$	23.7128	0.857899
$765$	0	0
$766$	24.9808	0.902592
$767$	27.6603	0.998754
$768$	0	0
$769$	−38.6410	−1.39343	−0.696715	−	0.717348i	$-0.745356\pi$
$769$	−38.6410	−1.39343	−0.696715	+	0.717348i	$0.745356\pi$
$770$	0	0
$771$	0	0
$772$	7.32051	0.263471
$773$	1.73205	0.0622975	0.0311488	−	0.999515i	$-0.490083\pi$
$773$	1.73205	0.0622975	0.0311488	+	0.999515i	$0.490083\pi$
$774$	0	0
$775$	−36.7846	−1.32134
$776$	−8.28719	−0.297493
$777$	0	0
$778$	−12.0000	−0.430221
$779$	8.78461	0.314741
$780$	0	0
$781$	0	0
$782$	−8.92820	−0.319272
$783$	0	0
$784$	1.07180	0.0382785
$785$	−1.75129	−0.0625062
$786$	0	0
$787$	−15.1769	−0.540999	−0.270499	−	0.962720i	$-0.587189\pi$
$787$	−15.1769	−0.540999	−0.270499	+	0.962720i	$0.587189\pi$
$788$	18.3538	0.653828
$789$	0	0
$790$	−0.105118	−0.00373992
$791$	8.53590	0.303502
$792$	0	0
$793$	22.3923	0.795174
$794$	−9.75129	−0.346060
$795$	0	0
$796$	29.5692	1.04805
$797$	18.9090	0.669790	0.334895	−	0.942255i	$-0.391299\pi$
$797$	18.9090	0.669790	0.334895	+	0.942255i	$0.391299\pi$
$798$	0	0
$799$	34.3205	1.21417
$800$	−28.8616	−1.02041
$801$	0	0
$802$	11.7128	0.413594
$803$	0	0
$804$	0	0
$805$	0.875644	0.0308624
$806$	14.9282	0.525824
$807$	0	0
$808$	24.6795	0.868221
$809$	−20.4449	−0.718803	−0.359402	−	0.933183i	$-0.617019\pi$
$809$	−20.4449	−0.718803	−0.359402	+	0.933183i	$0.617019\pi$
$810$	0	0
$811$	45.9090	1.61208	0.806041	−	0.591860i	$-0.201606\pi$
$811$	45.9090	1.61208	0.806041	+	0.591860i	$0.201606\pi$
$812$	−1.85641	−0.0651471
$813$	0	0
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	−1.01924	−0.0356586
$818$	−17.3205	−0.605597
$819$	0	0
$820$	1.56922	0.0547995
$821$	−23.8564	−0.832594	−0.416297	−	0.909229i	$-0.636672\pi$
$821$	−23.8564	−0.832594	−0.416297	+	0.909229i	$0.636672\pi$
$822$	0	0
$823$	−29.3205	−1.02205	−0.511024	−	0.859566i	$-0.670734\pi$
$823$	−29.3205	−1.02205	−0.511024	+	0.859566i	$0.670734\pi$
$824$	−10.6410	−0.370697
$825$	0	0
$826$	−7.41154	−0.257881
$827$	29.5167	1.02639	0.513197	−	0.858271i	$-0.328461\pi$
$827$	29.5167	1.02639	0.513197	+	0.858271i	$0.328461\pi$
$828$	0	0
$829$	31.8038	1.10459	0.552297	−	0.833648i	$-0.313752\pi$
$829$	31.8038	1.10459	0.552297	+	0.833648i	$0.313752\pi$
$830$	−0.837169	−0.0290585
$831$	0	0
$832$	5.85641	0.203034
$833$	3.73205	0.129308
$834$	0	0
$835$	−6.71281	−0.232306
$836$	0	0
$837$	0	0
$838$	−7.12436	−0.246107
$839$	−48.2295	−1.66507	−0.832533	−	0.553975i	$-0.813110\pi$
$839$	−48.2295	−1.66507	−0.832533	+	0.553975i	$0.813110\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	28.4449	0.980275
$843$	0	0
$844$	26.0385	0.896281
$845$	−1.48334	−0.0510284
$846$	0	0
$847$	0	0
$848$	−2.71797	−0.0933354
$849$	0	0
$850$	−13.4641	−0.461815
$851$	30.9282	1.06021
$852$	0	0
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	47.1384	1.61116
$857$	−26.1244	−0.892391	−0.446195	−	0.894936i	$-0.647221\pi$
$857$	−26.1244	−0.892391	−0.446195	+	0.894936i	$0.647221\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	−0.182069	−0.00620851
$861$	0	0
$862$	−17.0718	−0.581468
$863$	0.928203	0.0315964	0.0157982	−	0.999875i	$-0.494971\pi$
$863$	0.928203	0.0315964	0.0157982	+	0.999875i	$0.494971\pi$
$864$	0	0
$865$	−1.39230	−0.0473398
$866$	24.8231	0.843523
$867$	0	0
$868$	10.9282	0.370927
$869$	0	0
$870$	0	0
$871$	−19.1244	−0.648004
$872$	−44.1051	−1.49359
$873$	0	0
$874$	5.25387	0.177715
$875$	2.66025	0.0899330
$876$	0	0
$877$	25.1769	0.850164	0.425082	−	0.905155i	$-0.360245\pi$
$877$	25.1769	0.850164	0.425082	+	0.905155i	$0.360245\pi$
$878$	−0.105118	−0.00354755
$879$	0	0
$880$	0	0
$881$	−39.8564	−1.34280	−0.671398	−	0.741097i	$-0.734306\pi$
$881$	−39.8564	−1.34280	−0.671398	+	0.741097i	$0.734306\pi$
$882$	0	0
$883$	27.7846	0.935027	0.467513	−	0.883986i	$-0.345150\pi$
$883$	27.7846	0.935027	0.467513	+	0.883986i	$0.345150\pi$
$884$	−14.9282	−0.502090
$885$	0	0
$886$	4.24871	0.142738
$887$	10.1244	0.339943	0.169971	−	0.985449i	$-0.445632\pi$
$887$	10.1244	0.339943	0.169971	+	0.985449i	$0.445632\pi$
$888$	0	0
$889$	3.53590	0.118590
$890$	2.87564	0.0963918
$891$	0	0
$892$	−23.4256	−0.784348
$893$	−20.1962	−0.675838
$894$	0	0
$895$	1.17691	0.0393399
$896$	10.1436	0.338874
$897$	0	0
$898$	11.8949	0.396937
$899$	−9.46410	−0.315645
$900$	0	0
$901$	−9.46410	−0.315295
$902$	0	0
$903$	0	0
$904$	21.6462	0.719941
$905$	−2.14359	−0.0712555
$906$	0	0
$907$	2.78461	0.0924614	0.0462307	−	0.998931i	$-0.485279\pi$
$907$	2.78461	0.0924614	0.0462307	+	0.998931i	$0.485279\pi$
$908$	21.4641	0.712311
$909$	0	0
$910$	−0.535898	−0.0177649
$911$	−26.7846	−0.887414	−0.443707	−	0.896172i	$-0.646337\pi$
$911$	−26.7846	−0.887414	−0.443707	+	0.896172i	$0.646337\pi$
$912$	0	0
$913$	0	0
$914$	−24.8372	−0.821541
$915$	0	0
$916$	−10.1436	−0.335154
$917$	5.19615	0.171592
$918$	0	0
$919$	−6.39230	−0.210863	−0.105431	−	0.994427i	$-0.533622\pi$
$919$	−6.39230	−0.210863	−0.105431	+	0.994427i	$0.533622\pi$
$920$	2.22055	0.0732092
$921$	0	0
$922$	20.1962	0.665125
$923$	−24.9282	−0.820522
$924$	0	0
$925$	46.6410	1.53355
$926$	23.0333	0.756922
$927$	0	0
$928$	−7.42563	−0.243758
$929$	−14.6603	−0.480987	−0.240494	−	0.970651i	$-0.577309\pi$
$929$	−14.6603	−0.480987	−0.240494	+	0.970651i	$0.577309\pi$
$930$	0	0
$931$	−2.19615	−0.0719760
$932$	29.5692	0.968572
$933$	0	0
$934$	25.1769	0.823814
$935$	0	0
$936$	0	0
$937$	−48.4449	−1.58262	−0.791312	−	0.611412i	$-0.790602\pi$
$937$	−48.4449	−1.58262	−0.791312	+	0.611412i	$0.790602\pi$
$938$	5.12436	0.167316
$939$	0	0
$940$	−3.60770	−0.117670
$941$	13.6077	0.443598	0.221799	−	0.975092i	$-0.428807\pi$
$941$	13.6077	0.443598	0.221799	+	0.975092i	$0.428807\pi$
$942$	0	0
$943$	13.0718	0.425676
$944$	10.8513	0.353178
$945$	0	0
$946$	0	0
$947$	−48.9282	−1.58995	−0.794976	−	0.606640i	$-0.792517\pi$
$947$	−48.9282	−1.58995	−0.794976	+	0.606640i	$0.792517\pi$
$948$	0	0
$949$	−18.3923	−0.597039
$950$	7.92305	0.257058
$951$	0	0
$952$	9.46410	0.306733
$953$	49.3205	1.59765	0.798824	−	0.601565i	$-0.205456\pi$
$953$	49.3205	1.59765	0.798824	+	0.601565i	$0.205456\pi$
$954$	0	0
$955$	−4.33975	−0.140431
$956$	36.4974	1.18041
$957$	0	0
$958$	−23.6987	−0.765671
$959$	13.6603	0.441113
$960$	0	0
$961$	24.7128	0.797188
$962$	−18.9282	−0.610270
$963$	0	0
$964$	−22.9282	−0.738468
$965$	−1.33975	−0.0431279
$966$	0	0
$967$	29.3923	0.945193	0.472596	−	0.881279i	$-0.343317\pi$
$967$	29.3923	0.945193	0.472596	+	0.881279i	$0.343317\pi$
$968$	0	0
$969$	0	0
$970$	0.641016	0.0205818
$971$	21.8756	0.702023	0.351011	−	0.936371i	$-0.385838\pi$
$971$	21.8756	0.702023	0.351011	+	0.936371i	$0.385838\pi$
$972$	0	0
$973$	−7.80385	−0.250180
$974$	30.8756	0.989319
$975$	0	0
$976$	8.78461	0.281189
$977$	−50.1962	−1.60592	−0.802959	−	0.596035i	$-0.796742\pi$
$977$	−50.1962	−1.60592	−0.802959	+	0.596035i	$0.796742\pi$
$978$	0	0
$979$	0	0
$980$	−0.392305	−0.0125317
$981$	0	0
$982$	0.0384758	0.00122781
$983$	−8.78461	−0.280186	−0.140093	−	0.990138i	$-0.544740\pi$
$983$	−8.78461	−0.280186	−0.140093	+	0.990138i	$0.544740\pi$
$984$	0	0
$985$	−3.35898	−0.107026
$986$	−3.46410	−0.110319
$987$	0	0
$988$	8.78461	0.279476
$989$	−1.51666	−0.0482270
$990$	0	0
$991$	17.6077	0.559327	0.279663	−	0.960098i	$-0.409777\pi$
$991$	17.6077	0.559327	0.279663	+	0.960098i	$0.409777\pi$
$992$	43.7128	1.38788
$993$	0	0
$994$	6.67949	0.211861
$995$	−5.41154	−0.171557
$996$	0	0
$997$	−61.0333	−1.93294	−0.966472	−	0.256771i	$-0.917341\pi$
$997$	−61.0333	−1.93294	−0.966472	+	0.256771i	$0.917341\pi$
$998$	29.5167	0.934334
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.x.1.2		2
3.2	odd	2		2541.2.a.be.1.1	yes	2
11.10	odd	2		7623.2.a.bv.1.1		2
33.32	even	2		2541.2.a.o.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.o.1.2	✓	2	33.32	even	2
2541.2.a.be.1.1	yes	2	3.2	odd	2
7623.2.a.x.1.2		2	1.1	even	1	trivial
7623.2.a.bv.1.1		2	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.46410 q^{4} +0.267949 q^{5} -1.00000 q^{7} -2.53590 q^{8} +O(q^{10})\)	\(q+0.732051 q^{2} -1.46410 q^{4} +0.267949 q^{5} -1.00000 q^{7} -2.53590 q^{8} +0.196152 q^{10} +2.73205 q^{13} -0.732051 q^{14} +1.07180 q^{16} +3.73205 q^{17} -2.19615 q^{19} -0.392305 q^{20} -3.26795 q^{23} -4.92820 q^{25} +2.00000 q^{26} +1.46410 q^{28} -1.26795 q^{29} +7.46410 q^{31} +5.85641 q^{32} +2.73205 q^{34} -0.267949 q^{35} -9.46410 q^{37} -1.60770 q^{38} -0.679492 q^{40} -4.00000 q^{41} +0.464102 q^{43} -2.39230 q^{46} +9.19615 q^{47} +1.00000 q^{49} -3.60770 q^{50} -4.00000 q^{52} -2.53590 q^{53} +2.53590 q^{56} -0.928203 q^{58} +10.1244 q^{59} +8.19615 q^{61} +5.46410 q^{62} +2.14359 q^{64} +0.732051 q^{65} -7.00000 q^{67} -5.46410 q^{68} -0.196152 q^{70} -9.12436 q^{71} -6.73205 q^{73} -6.92820 q^{74} +3.21539 q^{76} -0.535898 q^{79} +0.287187 q^{80} -2.92820 q^{82} -4.26795 q^{83} +1.00000 q^{85} +0.339746 q^{86} +14.6603 q^{89} -2.73205 q^{91} +4.78461 q^{92} +6.73205 q^{94} -0.588457 q^{95} +3.26795 q^{97} +0.732051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^4 + 4 * q^5 - 2 * q^7 - 12 * q^8	\( 2 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 2 q^{7} - 12 q^{8} - 10 q^{10} + 2 q^{13} + 2 q^{14} + 16 q^{16} + 4 q^{17} + 6 q^{19} + 20 q^{20} - 10 q^{23} + 4 q^{25} + 4 q^{26} - 4 q^{28} - 6 q^{29} + 8 q^{31} - 16 q^{32} + 2 q^{34} - 4 q^{35} - 12 q^{37} - 24 q^{38} - 36 q^{40} - 8 q^{41} - 6 q^{43} + 16 q^{46} + 8 q^{47} + 2 q^{49} - 28 q^{50} - 8 q^{52} - 12 q^{53} + 12 q^{56} + 12 q^{58} - 4 q^{59} + 6 q^{61} + 4 q^{62} + 32 q^{64} - 2 q^{65} - 14 q^{67} - 4 q^{68} + 10 q^{70} + 6 q^{71} - 10 q^{73} + 48 q^{76} - 8 q^{79} + 56 q^{80} + 8 q^{82} - 12 q^{83} + 2 q^{85} + 18 q^{86} + 12 q^{89} - 2 q^{91} - 32 q^{92} + 10 q^{94} + 30 q^{95} + 10 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 + 4 * q^5 - 2 * q^7 - 12 * q^8 - 10 * q^10 + 2 * q^13 + 2 * q^14 + 16 * q^16 + 4 * q^17 + 6 * q^19 + 20 * q^20 - 10 * q^23 + 4 * q^25 + 4 * q^26 - 4 * q^28 - 6 * q^29 + 8 * q^31 - 16 * q^32 + 2 * q^34 - 4 * q^35 - 12 * q^37 - 24 * q^38 - 36 * q^40 - 8 * q^41 - 6 * q^43 + 16 * q^46 + 8 * q^47 + 2 * q^49 - 28 * q^50 - 8 * q^52 - 12 * q^53 + 12 * q^56 + 12 * q^58 - 4 * q^59 + 6 * q^61 + 4 * q^62 + 32 * q^64 - 2 * q^65 - 14 * q^67 - 4 * q^68 + 10 * q^70 + 6 * q^71 - 10 * q^73 + 48 * q^76 - 8 * q^79 + 56 * q^80 + 8 * q^82 - 12 * q^83 + 2 * q^85 + 18 * q^86 + 12 * q^89 - 2 * q^91 - 32 * q^92 + 10 * q^94 + 30 * q^95 + 10 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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