LMFDB - Embedded newform 7623.2.a.x.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.73205 q^{2} +5.46410 q^{4} +3.73205 q^{5} -1.00000 q^{7} -9.46410 q^{8} +O(q^{10})$	$q-2.73205 q^{2} +5.46410 q^{4} +3.73205 q^{5} -1.00000 q^{7} -9.46410 q^{8} -10.1962 q^{10} -0.732051 q^{13} +2.73205 q^{14} +14.9282 q^{16} +0.267949 q^{17} +8.19615 q^{19} +20.3923 q^{20} -6.73205 q^{23} +8.92820 q^{25} +2.00000 q^{26} -5.46410 q^{28} -4.73205 q^{29} +0.535898 q^{31} -21.8564 q^{32} -0.732051 q^{34} -3.73205 q^{35} -2.53590 q^{37} -22.3923 q^{38} -35.3205 q^{40} -4.00000 q^{41} -6.46410 q^{43} +18.3923 q^{46} -1.19615 q^{47} +1.00000 q^{49} -24.3923 q^{50} -4.00000 q^{52} -9.46410 q^{53} +9.46410 q^{56} +12.9282 q^{58} -14.1244 q^{59} -2.19615 q^{61} -1.46410 q^{62} +29.8564 q^{64} -2.73205 q^{65} -7.00000 q^{67} +1.46410 q^{68} +10.1962 q^{70} +15.1244 q^{71} -3.26795 q^{73} +6.92820 q^{74} +44.7846 q^{76} -7.46410 q^{79} +55.7128 q^{80} +10.9282 q^{82} -7.73205 q^{83} +1.00000 q^{85} +17.6603 q^{86} -2.66025 q^{89} +0.732051 q^{91} -36.7846 q^{92} +3.26795 q^{94} +30.5885 q^{95} +6.73205 q^{97} -2.73205 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.73205	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	−2.73205	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$3$	0	0
$3$	0	0
$4$	5.46410	2.73205
$5$	3.73205	1.66902	0.834512	−	0.550990i	$-0.185750\pi$
$5$	3.73205	1.66902	0.834512	+	0.550990i	$0.185750\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−9.46410	−3.34607
$9$	0	0
$10$	−10.1962	−3.22431
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.732051	−0.203034	−0.101517	−	0.994834i	$-0.532370\pi$
$13$	−0.732051	−0.203034	−0.101517	+	0.994834i	$0.532370\pi$
$14$	2.73205	0.730171
$15$	0	0
$16$	14.9282	3.73205
$17$	0.267949	0.0649872	0.0324936	−	0.999472i	$-0.489655\pi$
$17$	0.267949	0.0649872	0.0324936	+	0.999472i	$0.489655\pi$
$18$	0	0
$19$	8.19615	1.88033	0.940163	−	0.340725i	$-0.110672\pi$
$19$	8.19615	1.88033	0.940163	+	0.340725i	$0.110672\pi$
$20$	20.3923	4.55986
$21$	0	0
$22$	0	0
$23$	−6.73205	−1.40373	−0.701865	−	0.712310i	$-0.747649\pi$
$23$	−6.73205	−1.40373	−0.701865	+	0.712310i	$0.747649\pi$
$24$	0	0
$25$	8.92820	1.78564
$26$	2.00000	0.392232
$27$	0	0
$28$	−5.46410	−1.03262
$29$	−4.73205	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$29$	−4.73205	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$30$	0	0
$31$	0.535898	0.0962502	0.0481251	−	0.998841i	$-0.484675\pi$
$31$	0.535898	0.0962502	0.0481251	+	0.998841i	$0.484675\pi$
$32$	−21.8564	−3.86370
$33$	0	0
$34$	−0.732051	−0.125546
$35$	−3.73205	−0.630832
$36$	0	0
$37$	−2.53590	−0.416899	−0.208450	−	0.978033i	$-0.566842\pi$
$37$	−2.53590	−0.416899	−0.208450	+	0.978033i	$0.566842\pi$
$38$	−22.3923	−3.63251
$39$	0	0
$40$	−35.3205	−5.58466
$41$	−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
$43$	−6.46410	−0.985766	−0.492883	−	0.870096i	$-0.664057\pi$
$43$	−6.46410	−0.985766	−0.492883	+	0.870096i	$0.664057\pi$
$44$	0	0
$45$	0	0
$46$	18.3923	2.71180
$47$	−1.19615	−0.174477	−0.0872384	−	0.996187i	$-0.527804\pi$
$47$	−1.19615	−0.174477	−0.0872384	+	0.996187i	$0.527804\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−24.3923	−3.44959
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−9.46410	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$53$	−9.46410	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$54$	0	0
$55$	0	0
$56$	9.46410	1.26469
$57$	0	0
$58$	12.9282	1.69756
$59$	−14.1244	−1.83883	−0.919417	−	0.393284i	$-0.871339\pi$
$59$	−14.1244	−1.83883	−0.919417	+	0.393284i	$0.871339\pi$
$60$	0	0
$61$	−2.19615	−0.281189	−0.140594	−	0.990067i	$-0.544901\pi$
$61$	−2.19615	−0.281189	−0.140594	+	0.990067i	$0.544901\pi$
$62$	−1.46410	−0.185941
$63$	0	0
$64$	29.8564	3.73205
$65$	−2.73205	−0.338869
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	1.46410	0.177548
$69$	0	0
$70$	10.1962	1.21867
$71$	15.1244	1.79493	0.897465	−	0.441085i	$-0.145406\pi$
$71$	15.1244	1.79493	0.897465	+	0.441085i	$0.145406\pi$
$72$	0	0
$73$	−3.26795	−0.382485	−0.191242	−	0.981543i	$-0.561252\pi$
$73$	−3.26795	−0.382485	−0.191242	+	0.981543i	$0.561252\pi$
$74$	6.92820	0.805387
$75$	0	0
$76$	44.7846	5.13715
$77$	0	0
$78$	0	0
$79$	−7.46410	−0.839777	−0.419889	−	0.907576i	$-0.637931\pi$
$79$	−7.46410	−0.839777	−0.419889	+	0.907576i	$0.637931\pi$
$80$	55.7128	6.22888
$81$	0	0
$82$	10.9282	1.20682
$83$	−7.73205	−0.848703	−0.424351	−	0.905498i	$-0.639498\pi$
$83$	−7.73205	−0.848703	−0.424351	+	0.905498i	$0.639498\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	17.6603	1.90435
$87$	0	0
$88$	0	0
$89$	−2.66025	−0.281986	−0.140993	−	0.990011i	$-0.545030\pi$
$89$	−2.66025	−0.281986	−0.140993	+	0.990011i	$0.545030\pi$
$90$	0	0
$91$	0.732051	0.0767398
$92$	−36.7846	−3.83506
$93$	0	0
$94$	3.26795	0.337063
$95$	30.5885	3.13831
$96$	0	0
$97$	6.73205	0.683536	0.341768	−	0.939784i	$-0.388974\pi$
$97$	6.73205	0.683536	0.341768	+	0.939784i	$0.388974\pi$
$98$	−2.73205	−0.275979
$99$	0	0
$100$	48.7846	4.87846
$101$	−6.26795	−0.623684	−0.311842	−	0.950134i	$-0.600946\pi$
$101$	−6.26795	−0.623684	−0.311842	+	0.950134i	$0.600946\pi$
$102$	0	0
$103$	−6.19615	−0.610525	−0.305263	−	0.952268i	$-0.598744\pi$
$103$	−6.19615	−0.610525	−0.305263	+	0.952268i	$0.598744\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	25.8564	2.51140
$107$	12.5885	1.21697	0.608486	−	0.793565i	$-0.291777\pi$
$107$	12.5885	1.21697	0.608486	+	0.793565i	$0.291777\pi$
$108$	0	0
$109$	−3.39230	−0.324924	−0.162462	−	0.986715i	$-0.551943\pi$
$109$	−3.39230	−0.324924	−0.162462	+	0.986715i	$0.551943\pi$
$110$	0	0
$111$	0	0
$112$	−14.9282	−1.41058
$113$	−15.4641	−1.45474	−0.727370	−	0.686245i	$-0.759258\pi$
$113$	−15.4641	−1.45474	−0.727370	+	0.686245i	$0.759258\pi$
$114$	0	0
$115$	−25.1244	−2.34286
$116$	−25.8564	−2.40071
$117$	0	0
$118$	38.5885	3.55236
$119$	−0.267949	−0.0245629
$120$	0	0
$121$	0	0
$122$	6.00000	0.543214
$123$	0	0
$124$	2.92820	0.262960
$125$	14.6603	1.31125
$126$	0	0
$127$	−10.4641	−0.928539	−0.464269	−	0.885694i	$-0.653683\pi$
$127$	−10.4641	−0.928539	−0.464269	+	0.885694i	$0.653683\pi$
$128$	−37.8564	−3.34607
$129$	0	0
$130$	7.46410	0.654645
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	0	0
$133$	−8.19615	−0.710697
$134$	19.1244	1.65209
$135$	0	0
$136$	−2.53590	−0.217451
$137$	3.66025	0.312717	0.156358	−	0.987700i	$-0.450025\pi$
$137$	3.66025	0.312717	0.156358	+	0.987700i	$0.450025\pi$
$138$	0	0
$139$	18.1962	1.54338	0.771689	−	0.636000i	$-0.219412\pi$
$139$	18.1962	1.54338	0.771689	+	0.636000i	$0.219412\pi$
$140$	−20.3923	−1.72346
$141$	0	0
$142$	−41.3205	−3.46754
$143$	0	0
$144$	0	0
$145$	−17.6603	−1.46660
$146$	8.92820	0.738903
$147$	0	0
$148$	−13.8564	−1.13899
$149$	−0.732051	−0.0599719	−0.0299860	−	0.999550i	$-0.509546\pi$
$149$	−0.732051	−0.0599719	−0.0299860	+	0.999550i	$0.509546\pi$
$150$	0	0
$151$	−11.3923	−0.927093	−0.463546	−	0.886073i	$-0.653423\pi$
$151$	−11.3923	−0.927093	−0.463546	+	0.886073i	$0.653423\pi$
$152$	−77.5692	−6.29169
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−13.4641	−1.07455	−0.537276	−	0.843406i	$-0.680547\pi$
$157$	−13.4641	−1.07455	−0.537276	+	0.843406i	$0.680547\pi$
$158$	20.3923	1.62232
$159$	0	0
$160$	−81.5692	−6.44861
$161$	6.73205	0.530560
$162$	0	0
$163$	0.535898	0.0419748	0.0209874	−	0.999780i	$-0.493319\pi$
$163$	0.535898	0.0419748	0.0209874	+	0.999780i	$0.493319\pi$
$164$	−21.8564	−1.70670
$165$	0	0
$166$	21.1244	1.63957
$167$	13.0526	1.01004	0.505019	−	0.863108i	$-0.331486\pi$
$167$	13.0526	1.01004	0.505019	+	0.863108i	$0.331486\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	−2.73205	−0.209539
$171$	0	0
$172$	−35.3205	−2.69316
$173$	5.19615	0.395056	0.197528	−	0.980297i	$-0.436709\pi$
$173$	5.19615	0.395056	0.197528	+	0.980297i	$0.436709\pi$
$174$	0	0
$175$	−8.92820	−0.674909
$176$	0	0
$177$	0	0
$178$	7.26795	0.544756
$179$	−16.3923	−1.22522	−0.612609	−	0.790386i	$-0.709880\pi$
$179$	−16.3923	−1.22522	−0.612609	+	0.790386i	$0.709880\pi$
$180$	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
$184$	63.7128	4.69697
$185$	−9.46410	−0.695815
$186$	0	0
$187$	0	0
$188$	−6.53590	−0.476679
$189$	0	0
$190$	−83.5692	−6.06275
$191$	−5.80385	−0.419952	−0.209976	−	0.977707i	$-0.567339\pi$
$191$	−5.80385	−0.419952	−0.209976	+	0.977707i	$0.567339\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−18.3923	−1.32049
$195$	0	0
$196$	5.46410	0.390293
$197$	−19.4641	−1.38676	−0.693380	−	0.720572i	$-0.743879\pi$
$197$	−19.4641	−1.38676	−0.693380	+	0.720572i	$0.743879\pi$
$198$	0	0
$199$	−9.80385	−0.694976	−0.347488	−	0.937684i	$-0.612965\pi$
$199$	−9.80385	−0.694976	−0.347488	+	0.937684i	$0.612965\pi$
$200$	−84.4974	−5.97487
$201$	0	0
$202$	17.1244	1.20487
$203$	4.73205	0.332125
$204$	0	0
$205$	−14.9282	−1.04263
$206$	16.9282	1.17944
$207$	0	0
$208$	−10.9282	−0.757735
$209$	0	0
$210$	0	0
$211$	23.7846	1.63740	0.818700	−	0.574221i	$-0.194695\pi$
$211$	23.7846	1.63740	0.818700	+	0.574221i	$0.194695\pi$
$212$	−51.7128	−3.55165
$213$	0	0
$214$	−34.3923	−2.35101
$215$	−24.1244	−1.64527
$216$	0	0
$217$	−0.535898	−0.0363792
$218$	9.26795	0.627705
$219$	0	0
$220$	0	0
$221$	−0.196152	−0.0131946
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	21.8564	1.46034
$225$	0	0
$226$	42.2487	2.81034
$227$	2.66025	0.176567	0.0882836	−	0.996095i	$-0.471862\pi$
$227$	2.66025	0.176567	0.0882836	+	0.996095i	$0.471862\pi$
$228$	0	0
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	68.6410	4.52605
$231$	0	0
$232$	44.7846	2.94025
$233$	−9.80385	−0.642271	−0.321136	−	0.947033i	$-0.604065\pi$
$233$	−9.80385	−0.642271	−0.321136	+	0.947033i	$0.604065\pi$
$234$	0	0
$235$	−4.46410	−0.291206
$236$	−77.1769	−5.02379
$237$	0	0
$238$	0.732051	0.0474518
$239$	−11.0718	−0.716175	−0.358087	−	0.933688i	$-0.616571\pi$
$239$	−11.0718	−0.716175	−0.358087	+	0.933688i	$0.616571\pi$
$240$	0	0
$241$	−1.66025	−0.106946	−0.0534732	−	0.998569i	$-0.517029\pi$
$241$	−1.66025	−0.106946	−0.0534732	+	0.998569i	$0.517029\pi$
$242$	0	0
$243$	0	0
$244$	−12.0000	−0.768221
$245$	3.73205	0.238432
$246$	0	0
$247$	−6.00000	−0.381771
$248$	−5.07180	−0.322059
$249$	0	0
$250$	−40.0526	−2.53315
$251$	14.9282	0.942260	0.471130	−	0.882064i	$-0.343846\pi$
$251$	14.9282	0.942260	0.471130	+	0.882064i	$0.343846\pi$
$252$	0	0
$253$	0	0
$254$	28.5885	1.79380
$255$	0	0
$256$	43.7128	2.73205
$257$	20.1244	1.25532	0.627661	−	0.778486i	$-0.284012\pi$
$257$	20.1244	1.25532	0.627661	+	0.778486i	$0.284012\pi$
$258$	0	0
$259$	2.53590	0.157573
$260$	−14.9282	−0.925808
$261$	0	0
$262$	−14.1962	−0.877041
$263$	−16.1962	−0.998698	−0.499349	−	0.866401i	$-0.666427\pi$
$263$	−16.1962	−0.998698	−0.499349	+	0.866401i	$0.666427\pi$
$264$	0	0
$265$	−35.3205	−2.16972
$266$	22.3923	1.37296
$267$	0	0
$268$	−38.2487	−2.33641
$269$	3.85641	0.235129	0.117565	−	0.993065i	$-0.462491\pi$
$269$	3.85641	0.235129	0.117565	+	0.993065i	$0.462491\pi$
$270$	0	0
$271$	24.3923	1.48173	0.740863	−	0.671656i	$-0.234417\pi$
$271$	24.3923	1.48173	0.740863	+	0.671656i	$0.234417\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	15.7846	0.948405	0.474203	−	0.880416i	$-0.342736\pi$
$277$	15.7846	0.948405	0.474203	+	0.880416i	$0.342736\pi$
$278$	−49.7128	−2.98158
$279$	0	0
$280$	35.3205	2.11080
$281$	−27.7128	−1.65321	−0.826604	−	0.562784i	$-0.809730\pi$
$281$	−27.7128	−1.65321	−0.826604	+	0.562784i	$0.809730\pi$
$282$	0	0
$283$	22.2487	1.32255	0.661274	−	0.750144i	$-0.270016\pi$
$283$	22.2487	1.32255	0.661274	+	0.750144i	$0.270016\pi$
$284$	82.6410	4.90384
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−16.9282	−0.995777
$290$	48.2487	2.83326
$291$	0	0
$292$	−17.8564	−1.04497
$293$	19.7321	1.15276	0.576379	−	0.817182i	$-0.304465\pi$
$293$	19.7321	1.15276	0.576379	+	0.817182i	$0.304465\pi$
$294$	0	0
$295$	−52.7128	−3.06906
$296$	24.0000	1.39497
$297$	0	0
$298$	2.00000	0.115857
$299$	4.92820	0.285005
$300$	0	0
$301$	6.46410	0.372585
$302$	31.1244	1.79101
$303$	0	0
$304$	122.354	7.01747
$305$	−8.19615	−0.469310
$306$	0	0
$307$	−4.58846	−0.261877	−0.130939	−	0.991390i	$-0.541799\pi$
$307$	−4.58846	−0.261877	−0.130939	+	0.991390i	$0.541799\pi$
$308$	0	0
$309$	0	0
$310$	−5.46410	−0.310340
$311$	19.0526	1.08037	0.540186	−	0.841546i	$-0.318354\pi$
$311$	19.0526	1.08037	0.540186	+	0.841546i	$0.318354\pi$
$312$	0	0
$313$	−20.3923	−1.15264	−0.576321	−	0.817224i	$-0.695512\pi$
$313$	−20.3923	−1.15264	−0.576321	+	0.817224i	$0.695512\pi$
$314$	36.7846	2.07588
$315$	0	0
$316$	−40.7846	−2.29431
$317$	15.5167	0.871502	0.435751	−	0.900067i	$-0.356483\pi$
$317$	15.5167	0.871502	0.435751	+	0.900067i	$0.356483\pi$
$318$	0	0
$319$	0	0
$320$	111.426	6.22888
$321$	0	0
$322$	−18.3923	−1.02496
$323$	2.19615	0.122197
$324$	0	0
$325$	−6.53590	−0.362546
$326$	−1.46410	−0.0810891
$327$	0	0
$328$	37.8564	2.09027
$329$	1.19615	0.0659460
$330$	0	0
$331$	−36.1769	−1.98846	−0.994232	−	0.107255i	$-0.965794\pi$
$331$	−36.1769	−1.98846	−0.994232	+	0.107255i	$0.965794\pi$
$332$	−42.2487	−2.31870
$333$	0	0
$334$	−35.6603	−1.95124
$335$	−26.1244	−1.42733
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	34.0526	1.85222
$339$	0	0
$340$	5.46410	0.296333
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	61.1769	3.29844
$345$	0	0
$346$	−14.1962	−0.763190
$347$	12.0526	0.647015	0.323508	−	0.946226i	$-0.395138\pi$
$347$	12.0526	0.647015	0.323508	+	0.946226i	$0.395138\pi$
$348$	0	0
$349$	22.3923	1.19863	0.599316	−	0.800512i	$-0.295439\pi$
$349$	22.3923	1.19863	0.599316	+	0.800512i	$0.295439\pi$
$350$	24.3923	1.30382
$351$	0	0
$352$	0	0
$353$	24.9282	1.32679	0.663397	−	0.748267i	$-0.269114\pi$
$353$	24.9282	1.32679	0.663397	+	0.748267i	$0.269114\pi$
$354$	0	0
$355$	56.4449	2.99578
$356$	−14.5359	−0.770401
$357$	0	0
$358$	44.7846	2.36694
$359$	−0.875644	−0.0462147	−0.0231074	−	0.999733i	$-0.507356\pi$
$359$	−0.875644	−0.0462147	−0.0231074	+	0.999733i	$0.507356\pi$
$360$	0	0
$361$	48.1769	2.53563
$362$	21.8564	1.14875
$363$	0	0
$364$	4.00000	0.209657
$365$	−12.1962	−0.638376
$366$	0	0
$367$	−9.26795	−0.483783	−0.241892	−	0.970303i	$-0.577768\pi$
$367$	−9.26795	−0.483783	−0.241892	+	0.970303i	$0.577768\pi$
$368$	−100.497	−5.23879
$369$	0	0
$370$	25.8564	1.34421
$371$	9.46410	0.491352
$372$	0	0
$373$	−10.0718	−0.521498	−0.260749	−	0.965407i	$-0.583969\pi$
$373$	−10.0718	−0.521498	−0.260749	+	0.965407i	$0.583969\pi$
$374$	0	0
$375$	0	0
$376$	11.3205	0.583811
$377$	3.46410	0.178410
$378$	0	0
$379$	10.4641	0.537505	0.268752	−	0.963209i	$-0.413389\pi$
$379$	10.4641	0.537505	0.268752	+	0.963209i	$0.413389\pi$
$380$	167.138	8.57402
$381$	0	0
$382$	15.8564	0.811284
$383$	9.87564	0.504622	0.252311	−	0.967646i	$-0.418809\pi$
$383$	9.87564	0.504622	0.252311	+	0.967646i	$0.418809\pi$
$384$	0	0
$385$	0	0
$386$	13.6603	0.695289
$387$	0	0
$388$	36.7846	1.86746
$389$	4.39230	0.222699	0.111349	−	0.993781i	$-0.464483\pi$
$389$	4.39230	0.222699	0.111349	+	0.993781i	$0.464483\pi$
$390$	0	0
$391$	−1.80385	−0.0912245
$392$	−9.46410	−0.478009
$393$	0	0
$394$	53.1769	2.67901
$395$	−27.8564	−1.40161
$396$	0	0
$397$	21.3205	1.07005	0.535023	−	0.844838i	$-0.320303\pi$
$397$	21.3205	1.07005	0.535023	+	0.844838i	$0.320303\pi$
$398$	26.7846	1.34259
$399$	0	0
$400$	133.282	6.66410
$401$	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
$403$	−0.392305	−0.0195421
$404$	−34.2487	−1.70394
$405$	0	0
$406$	−12.9282	−0.641616
$407$	0	0
$408$	0	0
$409$	−6.33975	−0.313480	−0.156740	−	0.987640i	$-0.550099\pi$
$409$	−6.33975	−0.313480	−0.156740	+	0.987640i	$0.550099\pi$
$410$	40.7846	2.01421
$411$	0	0
$412$	−33.8564	−1.66799
$413$	14.1244	0.695014
$414$	0	0
$415$	−28.8564	−1.41651
$416$	16.0000	0.784465
$417$	0	0
$418$	0	0
$419$	−6.26795	−0.306209	−0.153105	−	0.988210i	$-0.548927\pi$
$419$	−6.26795	−0.306209	−0.153105	+	0.988210i	$0.548927\pi$
$420$	0	0
$421$	11.1436	0.543106	0.271553	−	0.962424i	$-0.412463\pi$
$421$	11.1436	0.543106	0.271553	+	0.962424i	$0.412463\pi$
$422$	−64.9808	−3.16321
$423$	0	0
$424$	89.5692	4.34987
$425$	2.39230	0.116044
$426$	0	0
$427$	2.19615	0.106279
$428$	68.7846	3.32483
$429$	0	0
$430$	65.9090	3.17841
$431$	11.3205	0.545290	0.272645	−	0.962115i	$-0.412102\pi$
$431$	11.3205	0.545290	0.272645	+	0.962115i	$0.412102\pi$
$432$	0	0
$433$	−31.9090	−1.53345	−0.766724	−	0.641977i	$-0.778114\pi$
$433$	−31.9090	−1.53345	−0.766724	+	0.641977i	$0.778114\pi$
$434$	1.46410	0.0702791
$435$	0	0
$436$	−18.5359	−0.887709
$437$	−55.1769	−2.63947
$438$	0	0
$439$	−27.8564	−1.32951	−0.664757	−	0.747060i	$-0.731465\pi$
$439$	−27.8564	−1.32951	−0.664757	+	0.747060i	$0.731465\pi$
$440$	0	0
$441$	0	0
$442$	0.535898	0.0254901
$443$	16.1962	0.769502	0.384751	−	0.923020i	$-0.374287\pi$
$443$	16.1962	0.769502	0.384751	+	0.923020i	$0.374287\pi$
$444$	0	0
$445$	−9.92820	−0.470642
$446$	−43.7128	−2.06986
$447$	0	0
$448$	−29.8564	−1.41058
$449$	−32.2487	−1.52191	−0.760955	−	0.648804i	$-0.775269\pi$
$449$	−32.2487	−1.52191	−0.760955	+	0.648804i	$0.775269\pi$
$450$	0	0
$451$	0	0
$452$	−84.4974	−3.97442
$453$	0	0
$454$	−7.26795	−0.341102
$455$	2.73205	0.128081
$456$	0	0
$457$	−20.0718	−0.938919	−0.469460	−	0.882954i	$-0.655551\pi$
$457$	−20.0718	−0.938919	−0.469460	+	0.882954i	$0.655551\pi$
$458$	18.9282	0.884457
$459$	0	0
$460$	−137.282	−6.40081
$461$	−3.58846	−0.167131	−0.0835656	−	0.996502i	$-0.526631\pi$
$461$	−3.58846	−0.167131	−0.0835656	+	0.996502i	$0.526631\pi$
$462$	0	0
$463$	24.5359	1.14028	0.570140	−	0.821548i	$-0.306889\pi$
$463$	24.5359	1.14028	0.570140	+	0.821548i	$0.306889\pi$
$464$	−70.6410	−3.27943
$465$	0	0
$466$	26.7846	1.24077
$467$	13.6077	0.629689	0.314845	−	0.949143i	$-0.398048\pi$
$467$	13.6077	0.629689	0.314845	+	0.949143i	$0.398048\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	12.1962	0.562567
$471$	0	0
$472$	133.674	6.15286
$473$	0	0
$474$	0	0
$475$	73.1769	3.35759
$476$	−1.46410	−0.0671070
$477$	0	0
$478$	30.2487	1.38354
$479$	40.3731	1.84469	0.922346	−	0.386364i	$-0.126269\pi$
$479$	40.3731	1.84469	0.922346	+	0.386364i	$0.126269\pi$
$480$	0	0
$481$	1.85641	0.0846448
$482$	4.53590	0.206605
$483$	0	0
$484$	0	0
$485$	25.1244	1.14084
$486$	0	0
$487$	−20.1769	−0.914303	−0.457152	−	0.889389i	$-0.651130\pi$
$487$	−20.1769	−0.914303	−0.457152	+	0.889389i	$0.651130\pi$
$488$	20.7846	0.940875
$489$	0	0
$490$	−10.1962	−0.460615
$491$	−38.0526	−1.71729	−0.858644	−	0.512572i	$-0.828693\pi$
$491$	−38.0526	−1.71729	−0.858644	+	0.512572i	$0.828693\pi$
$492$	0	0
$493$	−1.26795	−0.0571056
$494$	16.3923	0.737525
$495$	0	0
$496$	8.00000	0.359211
$497$	−15.1244	−0.678420
$498$	0	0
$499$	5.67949	0.254249	0.127124	−	0.991887i	$-0.459425\pi$
$499$	5.67949	0.254249	0.127124	+	0.991887i	$0.459425\pi$
$500$	80.1051	3.58241
$501$	0	0
$502$	−40.7846	−1.82031
$503$	−26.5167	−1.18232	−0.591160	−	0.806555i	$-0.701330\pi$
$503$	−26.5167	−1.18232	−0.591160	+	0.806555i	$0.701330\pi$
$504$	0	0
$505$	−23.3923	−1.04094
$506$	0	0
$507$	0	0
$508$	−57.1769	−2.53682
$509$	6.80385	0.301575	0.150788	−	0.988566i	$-0.451819\pi$
$509$	6.80385	0.301575	0.150788	+	0.988566i	$0.451819\pi$
$510$	0	0
$511$	3.26795	0.144566
$512$	−43.7128	−1.93185
$513$	0	0
$514$	−54.9808	−2.42510
$515$	−23.1244	−1.01898
$516$	0	0
$517$	0	0
$518$	−6.92820	−0.304408
$519$	0	0
$520$	25.8564	1.13388
$521$	−31.9808	−1.40110	−0.700551	−	0.713602i	$-0.747063\pi$
$521$	−31.9808	−1.40110	−0.700551	+	0.713602i	$0.747063\pi$
$522$	0	0
$523$	−4.58846	−0.200639	−0.100320	−	0.994955i	$-0.531987\pi$
$523$	−4.58846	−0.200639	−0.100320	+	0.994955i	$0.531987\pi$
$524$	28.3923	1.24032
$525$	0	0
$526$	44.2487	1.92934
$527$	0.143594	0.00625503
$528$	0	0
$529$	22.3205	0.970457
$530$	96.4974	4.19158
$531$	0	0
$532$	−44.7846	−1.94166
$533$	2.92820	0.126835
$534$	0	0
$535$	46.9808	2.03116
$536$	66.2487	2.86151
$537$	0	0
$538$	−10.5359	−0.454235
$539$	0	0
$540$	0	0
$541$	20.6077	0.885994	0.442997	−	0.896523i	$-0.353915\pi$
$541$	20.6077	0.885994	0.442997	+	0.896523i	$0.353915\pi$
$542$	−66.6410	−2.86248
$543$	0	0
$544$	−5.85641	−0.251091
$545$	−12.6603	−0.542306
$546$	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
$551$	−38.7846	−1.65228
$552$	0	0
$553$	7.46410	0.317406
$554$	−43.1244	−1.83218
$555$	0	0
$556$	99.4256	4.21659
$557$	−3.07180	−0.130156	−0.0650781	−	0.997880i	$-0.520730\pi$
$557$	−3.07180	−0.130156	−0.0650781	+	0.997880i	$0.520730\pi$
$558$	0	0
$559$	4.73205	0.200144
$560$	−55.7128	−2.35430
$561$	0	0
$562$	75.7128	3.19375
$563$	6.26795	0.264163	0.132081	−	0.991239i	$-0.457834\pi$
$563$	6.26795	0.264163	0.132081	+	0.991239i	$0.457834\pi$
$564$	0	0
$565$	−57.7128	−2.42800
$566$	−60.7846	−2.55497
$567$	0	0
$568$	−143.138	−6.00596
$569$	−28.7321	−1.20451	−0.602255	−	0.798304i	$-0.705731\pi$
$569$	−28.7321	−1.20451	−0.602255	+	0.798304i	$0.705731\pi$
$570$	0	0
$571$	14.9282	0.624726	0.312363	−	0.949963i	$-0.398880\pi$
$571$	14.9282	0.624726	0.312363	+	0.949963i	$0.398880\pi$
$572$	0	0
$573$	0	0
$574$	−10.9282	−0.456134
$575$	−60.1051	−2.50656
$576$	0	0
$577$	44.4449	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$577$	44.4449	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$578$	46.2487	1.92369
$579$	0	0
$580$	−96.4974	−4.00684
$581$	7.73205	0.320780
$582$	0	0
$583$	0	0
$584$	30.9282	1.27982
$585$	0	0
$586$	−53.9090	−2.22696
$587$	−16.1244	−0.665523	−0.332762	−	0.943011i	$-0.607980\pi$
$587$	−16.1244	−0.665523	−0.332762	+	0.943011i	$0.607980\pi$
$588$	0	0
$589$	4.39230	0.180982
$590$	144.014	5.92897
$591$	0	0
$592$	−37.8564	−1.55589
$593$	−4.41154	−0.181160	−0.0905802	−	0.995889i	$-0.528872\pi$
$593$	−4.41154	−0.181160	−0.0905802	+	0.995889i	$0.528872\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−4.00000	−0.163846
$597$	0	0
$598$	−13.4641	−0.550588
$599$	39.1769	1.60073	0.800363	−	0.599516i	$-0.204640\pi$
$599$	39.1769	1.60073	0.800363	+	0.599516i	$0.204640\pi$
$600$	0	0
$601$	−33.9090	−1.38318	−0.691588	−	0.722292i	$-0.743088\pi$
$601$	−33.9090	−1.38318	−0.691588	+	0.722292i	$0.743088\pi$
$602$	−17.6603	−0.719778
$603$	0	0
$604$	−62.2487	−2.53286
$605$	0	0
$606$	0	0
$607$	26.5885	1.07919	0.539596	−	0.841924i	$-0.318577\pi$
$607$	26.5885	1.07919	0.539596	+	0.841924i	$0.318577\pi$
$608$	−179.138	−7.26502
$609$	0	0
$610$	22.3923	0.906638
$611$	0.875644	0.0354248
$612$	0	0
$613$	−35.3923	−1.42948	−0.714741	−	0.699389i	$-0.753455\pi$
$613$	−35.3923	−1.42948	−0.714741	+	0.699389i	$0.753455\pi$
$614$	12.5359	0.505908
$615$	0	0
$616$	0	0
$617$	32.5359	1.30985	0.654923	−	0.755696i	$-0.272701\pi$
$617$	32.5359	1.30985	0.654923	+	0.755696i	$0.272701\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	10.9282	0.438887
$621$	0	0
$622$	−52.0526	−2.08712
$623$	2.66025	0.106581
$624$	0	0
$625$	10.0718	0.402872
$626$	55.7128	2.22673
$627$	0	0
$628$	−73.5692	−2.93573
$629$	−0.679492	−0.0270931
$630$	0	0
$631$	26.8564	1.06914	0.534568	−	0.845125i	$-0.320474\pi$
$631$	26.8564	1.06914	0.534568	+	0.845125i	$0.320474\pi$
$632$	70.6410	2.80995
$633$	0	0
$634$	−42.3923	−1.68361
$635$	−39.0526	−1.54975
$636$	0	0
$637$	−0.732051	−0.0290049
$638$	0	0
$639$	0	0
$640$	−141.282	−5.58466
$641$	29.9090	1.18133	0.590667	−	0.806916i	$-0.298865\pi$
$641$	29.9090	1.18133	0.590667	+	0.806916i	$0.298865\pi$
$642$	0	0
$643$	−14.1436	−0.557769	−0.278884	−	0.960325i	$-0.589965\pi$
$643$	−14.1436	−0.557769	−0.278884	+	0.960325i	$0.589965\pi$
$644$	36.7846	1.44952
$645$	0	0
$646$	−6.00000	−0.236067
$647$	16.8038	0.660627	0.330314	−	0.943871i	$-0.392845\pi$
$647$	16.8038	0.660627	0.330314	+	0.943871i	$0.392845\pi$
$648$	0	0
$649$	0	0
$650$	17.8564	0.700386
$651$	0	0
$652$	2.92820	0.114677
$653$	30.9808	1.21237	0.606185	−	0.795323i	$-0.292699\pi$
$653$	30.9808	1.21237	0.606185	+	0.795323i	$0.292699\pi$
$654$	0	0
$655$	19.3923	0.757720
$656$	−59.7128	−2.33139
$657$	0	0
$658$	−3.26795	−0.127398
$659$	−41.5167	−1.61726	−0.808630	−	0.588318i	$-0.799790\pi$
$659$	−41.5167	−1.61726	−0.808630	+	0.588318i	$0.799790\pi$
$660$	0	0
$661$	−18.1436	−0.705704	−0.352852	−	0.935679i	$-0.614788\pi$
$661$	−18.1436	−0.705704	−0.352852	+	0.935679i	$0.614788\pi$
$662$	98.8372	3.84142
$663$	0	0
$664$	73.1769	2.83982
$665$	−30.5885	−1.18617
$666$	0	0
$667$	31.8564	1.23348
$668$	71.3205	2.75947
$669$	0	0
$670$	71.3731	2.75738
$671$	0	0
$672$	0	0
$673$	−29.3923	−1.13299	−0.566495	−	0.824065i	$-0.691701\pi$
$673$	−29.3923	−1.13299	−0.566495	+	0.824065i	$0.691701\pi$
$674$	76.4974	2.94657
$675$	0	0
$676$	−68.1051	−2.61943
$677$	3.58846	0.137916	0.0689578	−	0.997620i	$-0.478033\pi$
$677$	3.58846	0.137916	0.0689578	+	0.997620i	$0.478033\pi$
$678$	0	0
$679$	−6.73205	−0.258352
$680$	−9.46410	−0.362932
$681$	0	0
$682$	0	0
$683$	−20.2487	−0.774795	−0.387398	−	0.921913i	$-0.626626\pi$
$683$	−20.2487	−0.774795	−0.387398	+	0.921913i	$0.626626\pi$
$684$	0	0
$685$	13.6603	0.521931
$686$	2.73205	0.104310
$687$	0	0
$688$	−96.4974	−3.67893
$689$	6.92820	0.263944
$690$	0	0
$691$	12.3923	0.471425	0.235713	−	0.971823i	$-0.424258\pi$
$691$	12.3923	0.471425	0.235713	+	0.971823i	$0.424258\pi$
$692$	28.3923	1.07931
$693$	0	0
$694$	−32.9282	−1.24994
$695$	67.9090	2.57593
$696$	0	0
$697$	−1.07180	−0.0405972
$698$	−61.1769	−2.31558
$699$	0	0
$700$	−48.7846	−1.84388
$701$	3.66025	0.138246	0.0691229	−	0.997608i	$-0.477980\pi$
$701$	3.66025	0.138246	0.0691229	+	0.997608i	$0.477980\pi$
$702$	0	0
$703$	−20.7846	−0.783906
$704$	0	0
$705$	0	0
$706$	−68.1051	−2.56317
$707$	6.26795	0.235730
$708$	0	0
$709$	−29.3923	−1.10385	−0.551926	−	0.833893i	$-0.686107\pi$
$709$	−29.3923	−1.10385	−0.551926	+	0.833893i	$0.686107\pi$
$710$	−154.210	−5.78741
$711$	0	0
$712$	25.1769	0.943545
$713$	−3.60770	−0.135109
$714$	0	0
$715$	0	0
$716$	−89.5692	−3.34736
$717$	0	0
$718$	2.39230	0.0892800
$719$	−32.1051	−1.19732	−0.598659	−	0.801004i	$-0.704300\pi$
$719$	−32.1051	−1.19732	−0.598659	+	0.801004i	$0.704300\pi$
$720$	0	0
$721$	6.19615	0.230757
$722$	−131.622	−4.89846
$723$	0	0
$724$	−43.7128	−1.62457
$725$	−42.2487	−1.56908
$726$	0	0
$727$	−7.51666	−0.278778	−0.139389	−	0.990238i	$-0.544514\pi$
$727$	−7.51666	−0.278778	−0.139389	+	0.990238i	$0.544514\pi$
$728$	−6.92820	−0.256776
$729$	0	0
$730$	33.3205	1.23325
$731$	−1.73205	−0.0640622
$732$	0	0
$733$	44.4449	1.64161	0.820804	−	0.571210i	$-0.193526\pi$
$733$	44.4449	1.64161	0.820804	+	0.571210i	$0.193526\pi$
$734$	25.3205	0.934597
$735$	0	0
$736$	147.138	5.42359
$737$	0	0
$738$	0	0
$739$	−33.3205	−1.22571	−0.612857	−	0.790194i	$-0.709980\pi$
$739$	−33.3205	−1.22571	−0.612857	+	0.790194i	$0.709980\pi$
$740$	−51.7128	−1.90100
$741$	0	0
$742$	−25.8564	−0.949219
$743$	−7.12436	−0.261367	−0.130684	−	0.991424i	$-0.541717\pi$
$743$	−7.12436	−0.261367	−0.130684	+	0.991424i	$0.541717\pi$
$744$	0	0
$745$	−2.73205	−0.100095
$746$	27.5167	1.00746
$747$	0	0
$748$	0	0
$749$	−12.5885	−0.459972
$750$	0	0
$751$	−9.39230	−0.342730	−0.171365	−	0.985208i	$-0.554818\pi$
$751$	−9.39230	−0.342730	−0.171365	+	0.985208i	$0.554818\pi$
$752$	−17.8564	−0.651156
$753$	0	0
$754$	−9.46410	−0.344662
$755$	−42.5167	−1.54734
$756$	0	0
$757$	5.92820	0.215464	0.107732	−	0.994180i	$-0.465641\pi$
$757$	5.92820	0.215464	0.107732	+	0.994180i	$0.465641\pi$
$758$	−28.5885	−1.03838
$759$	0	0
$760$	−289.492	−10.5010
$761$	44.9090	1.62795	0.813974	−	0.580901i	$-0.197300\pi$
$761$	44.9090	1.62795	0.813974	+	0.580901i	$0.197300\pi$
$762$	0	0
$763$	3.39230	0.122810
$764$	−31.7128	−1.14733
$765$	0	0
$766$	−26.9808	−0.974855
$767$	10.3397	0.373347
$768$	0	0
$769$	30.6410	1.10494	0.552472	−	0.833532i	$-0.313685\pi$
$769$	30.6410	1.10494	0.552472	+	0.833532i	$0.313685\pi$
$770$	0	0
$771$	0	0
$772$	−27.3205	−0.983287
$773$	−1.73205	−0.0622975	−0.0311488	−	0.999515i	$-0.509917\pi$
$773$	−1.73205	−0.0622975	−0.0311488	+	0.999515i	$0.509917\pi$
$774$	0	0
$775$	4.78461	0.171868
$776$	−63.7128	−2.28716
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−32.7846	−1.17463
$780$	0	0
$781$	0	0
$782$	4.92820	0.176232
$783$	0	0
$784$	14.9282	0.533150
$785$	−50.2487	−1.79345
$786$	0	0
$787$	47.1769	1.68168	0.840838	−	0.541287i	$-0.182063\pi$
$787$	47.1769	1.68168	0.840838	+	0.541287i	$0.182063\pi$
$788$	−106.354	−3.78870
$789$	0	0
$790$	76.1051	2.70770
$791$	15.4641	0.549840
$792$	0	0
$793$	1.60770	0.0570909
$794$	−58.2487	−2.06717
$795$	0	0
$796$	−53.5692	−1.89871
$797$	−46.9090	−1.66160	−0.830800	−	0.556570i	$-0.812117\pi$
$797$	−46.9090	−1.66160	−0.830800	+	0.556570i	$0.812117\pi$
$798$	0	0
$799$	−0.320508	−0.0113388
$800$	−195.138	−6.89919
$801$	0	0
$802$	−43.7128	−1.54355
$803$	0	0
$804$	0	0
$805$	25.1244	0.885517
$806$	1.07180	0.0377524
$807$	0	0
$808$	59.3205	2.08689
$809$	38.4449	1.35165	0.675825	−	0.737062i	$-0.263788\pi$
$809$	38.4449	1.35165	0.675825	+	0.737062i	$0.263788\pi$
$810$	0	0
$811$	−19.9090	−0.699098	−0.349549	−	0.936918i	$-0.613665\pi$
$811$	−19.9090	−0.699098	−0.349549	+	0.936918i	$0.613665\pi$
$812$	25.8564	0.907382
$813$	0	0
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	−52.9808	−1.85356
$818$	17.3205	0.605597
$819$	0	0
$820$	−81.5692	−2.84852
$821$	3.85641	0.134590	0.0672948	−	0.997733i	$-0.478563\pi$
$821$	3.85641	0.134590	0.0672948	+	0.997733i	$0.478563\pi$
$822$	0	0
$823$	5.32051	0.185461	0.0927306	−	0.995691i	$-0.470440\pi$
$823$	5.32051	0.185461	0.0927306	+	0.995691i	$0.470440\pi$
$824$	58.6410	2.04286
$825$	0	0
$826$	−38.5885	−1.34266
$827$	−15.5167	−0.539567	−0.269784	−	0.962921i	$-0.586952\pi$
$827$	−15.5167	−0.539567	−0.269784	+	0.962921i	$0.586952\pi$
$828$	0	0
$829$	42.1962	1.46553	0.732766	−	0.680480i	$-0.238229\pi$
$829$	42.1962	1.46553	0.732766	+	0.680480i	$0.238229\pi$
$830$	78.8372	2.73648
$831$	0	0
$832$	−21.8564	−0.757735
$833$	0.267949	0.00928389
$834$	0	0
$835$	48.7128	1.68578
$836$	0	0
$837$	0	0
$838$	17.1244	0.591551
$839$	52.2295	1.80316	0.901581	−	0.432611i	$-0.142408\pi$
$839$	52.2295	1.80316	0.901581	+	0.432611i	$0.142408\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	−30.4449	−1.04920
$843$	0	0
$844$	129.962	4.47346
$845$	−46.5167	−1.60022
$846$	0	0
$847$	0	0
$848$	−141.282	−4.85164
$849$	0	0
$850$	−6.53590	−0.224179
$851$	17.0718	0.585214
$852$	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
$856$	−119.138	−4.07207
$857$	−1.87564	−0.0640708	−0.0320354	−	0.999487i	$-0.510199\pi$
$857$	−1.87564	−0.0640708	−0.0320354	+	0.999487i	$0.510199\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	−131.818	−4.49495
$861$	0	0
$862$	−30.9282	−1.05342
$863$	−12.9282	−0.440081	−0.220041	−	0.975491i	$-0.570619\pi$
$863$	−12.9282	−0.440081	−0.220041	+	0.975491i	$0.570619\pi$
$864$	0	0
$865$	19.3923	0.659358
$866$	87.1769	2.96239
$867$	0	0
$868$	−2.92820	−0.0993897
$869$	0	0
$870$	0	0
$871$	5.12436	0.173632
$872$	32.1051	1.08722
$873$	0	0
$874$	150.746	5.09906
$875$	−14.6603	−0.495607
$876$	0	0
$877$	−37.1769	−1.25538	−0.627688	−	0.778465i	$-0.715998\pi$
$877$	−37.1769	−1.25538	−0.627688	+	0.778465i	$0.715998\pi$
$878$	76.1051	2.56842
$879$	0	0
$880$	0	0
$881$	−12.1436	−0.409128	−0.204564	−	0.978853i	$-0.565578\pi$
$881$	−12.1436	−0.409128	−0.204564	+	0.978853i	$0.565578\pi$
$882$	0	0
$883$	−13.7846	−0.463889	−0.231945	−	0.972729i	$-0.574509\pi$
$883$	−13.7846	−0.463889	−0.231945	+	0.972729i	$0.574509\pi$
$884$	−1.07180	−0.0360484
$885$	0	0
$886$	−44.2487	−1.48656
$887$	−14.1244	−0.474249	−0.237125	−	0.971479i	$-0.576205\pi$
$887$	−14.1244	−0.474249	−0.237125	+	0.971479i	$0.576205\pi$
$888$	0	0
$889$	10.4641	0.350955
$890$	27.1244	0.909210
$891$	0	0
$892$	87.4256	2.92723
$893$	−9.80385	−0.328073
$894$	0	0
$895$	−61.1769	−2.04492
$896$	37.8564	1.26469
$897$	0	0
$898$	88.1051	2.94011
$899$	−2.53590	−0.0845769
$900$	0	0
$901$	−2.53590	−0.0844830
$902$	0	0
$903$	0	0
$904$	146.354	4.86766
$905$	−29.8564	−0.992461
$906$	0	0
$907$	−38.7846	−1.28782	−0.643911	−	0.765100i	$-0.722689\pi$
$907$	−38.7846	−1.28782	−0.643911	+	0.765100i	$0.722689\pi$
$908$	14.5359	0.482391
$909$	0	0
$910$	−7.46410	−0.247433
$911$	14.7846	0.489836	0.244918	−	0.969544i	$-0.421239\pi$
$911$	14.7846	0.489836	0.244918	+	0.969544i	$0.421239\pi$
$912$	0	0
$913$	0	0
$914$	54.8372	1.81385
$915$	0	0
$916$	−37.8564	−1.25081
$917$	−5.19615	−0.171592
$918$	0	0
$919$	14.3923	0.474758	0.237379	−	0.971417i	$-0.423712\pi$
$919$	14.3923	0.474758	0.237379	+	0.971417i	$0.423712\pi$
$920$	237.779	7.83936
$921$	0	0
$922$	9.80385	0.322873
$923$	−11.0718	−0.364433
$924$	0	0
$925$	−22.6410	−0.744432
$926$	−67.0333	−2.20285
$927$	0	0
$928$	103.426	3.39511
$929$	2.66025	0.0872801	0.0436401	−	0.999047i	$-0.486105\pi$
$929$	2.66025	0.0872801	0.0436401	+	0.999047i	$0.486105\pi$
$930$	0	0
$931$	8.19615	0.268618
$932$	−53.5692	−1.75472
$933$	0	0
$934$	−37.1769	−1.21647
$935$	0	0
$936$	0	0
$937$	10.4449	0.341219	0.170609	−	0.985339i	$-0.445426\pi$
$937$	10.4449	0.341219	0.170609	+	0.985339i	$0.445426\pi$
$938$	−19.1244	−0.624432
$939$	0	0
$940$	−24.3923	−0.795589
$941$	34.3923	1.12116	0.560579	−	0.828101i	$-0.310579\pi$
$941$	34.3923	1.12116	0.560579	+	0.828101i	$0.310579\pi$
$942$	0	0
$943$	26.9282	0.876903
$944$	−210.851	−6.86262
$945$	0	0
$946$	0	0
$947$	−35.0718	−1.13968	−0.569840	−	0.821756i	$-0.692995\pi$
$947$	−35.0718	−1.13968	−0.569840	+	0.821756i	$0.692995\pi$
$948$	0	0
$949$	2.39230	0.0776575
$950$	−199.923	−6.48636
$951$	0	0
$952$	2.53590	0.0821889
$953$	14.6795	0.475515	0.237758	−	0.971324i	$-0.423588\pi$
$953$	14.6795	0.475515	0.237758	+	0.971324i	$0.423588\pi$
$954$	0	0
$955$	−21.6603	−0.700909
$956$	−60.4974	−1.95663
$957$	0	0
$958$	−110.301	−3.56367
$959$	−3.66025	−0.118196
$960$	0	0
$961$	−30.7128	−0.990736
$962$	−5.07180	−0.163521
$963$	0	0
$964$	−9.07180	−0.292183
$965$	−18.6603	−0.600695
$966$	0	0
$967$	8.60770	0.276805	0.138402	−	0.990376i	$-0.455803\pi$
$967$	8.60770	0.276805	0.138402	+	0.990376i	$0.455803\pi$
$968$	0	0
$969$	0	0
$970$	−68.6410	−2.20393
$971$	46.1244	1.48020	0.740101	−	0.672496i	$-0.234778\pi$
$971$	46.1244	1.48020	0.740101	+	0.672496i	$0.234778\pi$
$972$	0	0
$973$	−18.1962	−0.583342
$974$	55.1244	1.76630
$975$	0	0
$976$	−32.7846	−1.04941
$977$	−39.8038	−1.27344	−0.636719	−	0.771096i	$-0.719709\pi$
$977$	−39.8038	−1.27344	−0.636719	+	0.771096i	$0.719709\pi$
$978$	0	0
$979$	0	0
$980$	20.3923	0.651408
$981$	0	0
$982$	103.962	3.31755
$983$	32.7846	1.04567	0.522833	−	0.852435i	$-0.324875\pi$
$983$	32.7846	1.04567	0.522833	+	0.852435i	$0.324875\pi$
$984$	0	0
$985$	−72.6410	−2.31454
$986$	3.46410	0.110319
$987$	0	0
$988$	−32.7846	−1.04302
$989$	43.5167	1.38375
$990$	0	0
$991$	38.3923	1.21957	0.609786	−	0.792566i	$-0.291255\pi$
$991$	38.3923	1.21957	0.609786	+	0.792566i	$0.291255\pi$
$992$	−11.7128	−0.371882
$993$	0	0
$994$	41.3205	1.31061
$995$	−36.5885	−1.15993
$996$	0	0
$997$	29.0333	0.919495	0.459747	−	0.888050i	$-0.347940\pi$
$997$	29.0333	0.919495	0.459747	+	0.888050i	$0.347940\pi$
$998$	−15.5167	−0.491171
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.73205 q^{2} +5.46410 q^{4} +3.73205 q^{5} -1.00000 q^{7} -9.46410 q^{8} +O(q^{10})\)	\(q-2.73205 q^{2} +5.46410 q^{4} +3.73205 q^{5} -1.00000 q^{7} -9.46410 q^{8} -10.1962 q^{10} -0.732051 q^{13} +2.73205 q^{14} +14.9282 q^{16} +0.267949 q^{17} +8.19615 q^{19} +20.3923 q^{20} -6.73205 q^{23} +8.92820 q^{25} +2.00000 q^{26} -5.46410 q^{28} -4.73205 q^{29} +0.535898 q^{31} -21.8564 q^{32} -0.732051 q^{34} -3.73205 q^{35} -2.53590 q^{37} -22.3923 q^{38} -35.3205 q^{40} -4.00000 q^{41} -6.46410 q^{43} +18.3923 q^{46} -1.19615 q^{47} +1.00000 q^{49} -24.3923 q^{50} -4.00000 q^{52} -9.46410 q^{53} +9.46410 q^{56} +12.9282 q^{58} -14.1244 q^{59} -2.19615 q^{61} -1.46410 q^{62} +29.8564 q^{64} -2.73205 q^{65} -7.00000 q^{67} +1.46410 q^{68} +10.1962 q^{70} +15.1244 q^{71} -3.26795 q^{73} +6.92820 q^{74} +44.7846 q^{76} -7.46410 q^{79} +55.7128 q^{80} +10.9282 q^{82} -7.73205 q^{83} +1.00000 q^{85} +17.6603 q^{86} -2.66025 q^{89} +0.732051 q^{91} -36.7846 q^{92} +3.26795 q^{94} +30.5885 q^{95} +6.73205 q^{97} -2.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) 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

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