LMFDB - Embedded newform 7623.2.a.cx.1.9

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.39396$ 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)

$f(q)$	$=$	$q+2.39396 q^{2} +3.73106 q^{4} +3.93829 q^{5} -1.00000 q^{7} +4.14409 q^{8} +O(q^{10})$	$q+2.39396 q^{2} +3.73106 q^{4} +3.93829 q^{5} -1.00000 q^{7} +4.14409 q^{8} +9.42812 q^{10} -2.99890 q^{13} -2.39396 q^{14} +2.45868 q^{16} +6.60457 q^{17} -5.90310 q^{19} +14.6940 q^{20} +6.02551 q^{23} +10.5101 q^{25} -7.17925 q^{26} -3.73106 q^{28} +1.52075 q^{29} +8.46902 q^{31} -2.40219 q^{32} +15.8111 q^{34} -3.93829 q^{35} -0.607840 q^{37} -14.1318 q^{38} +16.3206 q^{40} +1.70333 q^{41} -3.23364 q^{43} +14.4249 q^{46} +4.80237 q^{47} +1.00000 q^{49} +25.1609 q^{50} -11.1891 q^{52} +6.12834 q^{53} -4.14409 q^{56} +3.64062 q^{58} -6.23883 q^{59} -2.08899 q^{61} +20.2745 q^{62} -10.6681 q^{64} -11.8105 q^{65} -0.599719 q^{67} +24.6420 q^{68} -9.42812 q^{70} -1.40968 q^{71} +7.08851 q^{73} -1.45515 q^{74} -22.0248 q^{76} -1.02327 q^{79} +9.68299 q^{80} +4.07770 q^{82} -3.08729 q^{83} +26.0107 q^{85} -7.74120 q^{86} +2.48531 q^{89} +2.99890 q^{91} +22.4815 q^{92} +11.4967 q^{94} -23.2481 q^{95} +2.55296 q^{97} +2.39396 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8	$ 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37} - 8 q^{38} + 5 q^{40} + 19 q^{41} + 6 q^{43} + q^{46} - 15 q^{47} + 10 q^{49} - q^{50} + 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} + 11 q^{62} + 53 q^{64} - 29 q^{65} + 38 q^{67} + 87 q^{68} - 6 q^{70} - 26 q^{71} + q^{73} - 39 q^{74} + 2 q^{76} - 5 q^{79} - 6 q^{80} + 5 q^{82} + 6 q^{83} + q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} - 42 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^13 + 38 * q^16 + 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 - 18 * q^28 - 14 * q^29 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 24 * q^37 - 8 * q^38 + 5 * q^40 + 19 * q^41 + 6 * q^43 + q^46 - 15 * q^47 + 10 * q^49 - q^50 + 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 + 11 * q^62 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 - 26 * q^71 + q^73 - 39 * q^74 + 2 * q^76 - 5 * q^79 - 6 * q^80 + 5 * q^82 + 6 * q^83 + q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 - 42 * q^95 + 24 * 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 $

$2$	2.39396	1.69279	0.846394	−	0.532558i	$-0.178769\pi$
$2$	2.39396	1.69279	0.846394	+	0.532558i	$0.178769\pi$
$3$	0	0
$3$	0	0
$4$	3.73106	1.86553
$5$	3.93829	1.76126	0.880629	−	0.473807i	$-0.157121\pi$
$5$	3.93829	1.76126	0.880629	+	0.473807i	$0.157121\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	4.14409	1.46516
$9$	0	0
$10$	9.42812	2.98143
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.99890	−0.831744	−0.415872	−	0.909423i	$-0.636524\pi$
$13$	−2.99890	−0.831744	−0.415872	+	0.909423i	$0.636524\pi$
$14$	−2.39396	−0.639813
$15$	0	0
$16$	2.45868	0.614669
$17$	6.60457	1.60184	0.800922	−	0.598769i	$-0.204343\pi$
$17$	6.60457	1.60184	0.800922	+	0.598769i	$0.204343\pi$
$18$	0	0
$19$	−5.90310	−1.35426	−0.677132	−	0.735861i	$-0.736778\pi$
$19$	−5.90310	−1.35426	−0.677132	+	0.735861i	$0.736778\pi$
$20$	14.6940	3.28568
$21$	0	0
$22$	0	0
$23$	6.02551	1.25641	0.628203	−	0.778049i	$-0.283791\pi$
$23$	6.02551	1.25641	0.628203	+	0.778049i	$0.283791\pi$
$24$	0	0
$25$	10.5101	2.10203
$26$	−7.17925	−1.40797
$27$	0	0
$28$	−3.73106	−0.705104
$29$	1.52075	0.282397	0.141198	−	0.989981i	$-0.454904\pi$
$29$	1.52075	0.282397	0.141198	+	0.989981i	$0.454904\pi$
$30$	0	0
$31$	8.46902	1.52108	0.760541	−	0.649290i	$-0.224934\pi$
$31$	8.46902	1.52108	0.760541	+	0.649290i	$0.224934\pi$
$32$	−2.40219	−0.424652
$33$	0	0
$34$	15.8111	2.71158
$35$	−3.93829	−0.665693
$36$	0	0
$37$	−0.607840	−0.0999283	−0.0499641	−	0.998751i	$-0.515911\pi$
$37$	−0.607840	−0.0999283	−0.0499641	+	0.998751i	$0.515911\pi$
$38$	−14.1318	−2.29248
$39$	0	0
$40$	16.3206	2.58052
$41$	1.70333	0.266015	0.133007	−	0.991115i	$-0.457537\pi$
$41$	1.70333	0.266015	0.133007	+	0.991115i	$0.457537\pi$
$42$	0	0
$43$	−3.23364	−0.493125	−0.246562	−	0.969127i	$-0.579301\pi$
$43$	−3.23364	−0.493125	−0.246562	+	0.969127i	$0.579301\pi$
$44$	0	0
$45$	0	0
$46$	14.4249	2.12683
$47$	4.80237	0.700498	0.350249	−	0.936657i	$-0.386097\pi$
$47$	4.80237	0.700498	0.350249	+	0.936657i	$0.386097\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	25.1609	3.55829
$51$	0	0
$52$	−11.1891	−1.55164
$53$	6.12834	0.841793	0.420896	−	0.907109i	$-0.361716\pi$
$53$	6.12834	0.841793	0.420896	+	0.907109i	$0.361716\pi$
$54$	0	0
$55$	0	0
$56$	−4.14409	−0.553777
$57$	0	0
$58$	3.64062	0.478037
$59$	−6.23883	−0.812227	−0.406113	−	0.913823i	$-0.633116\pi$
$59$	−6.23883	−0.812227	−0.406113	+	0.913823i	$0.633116\pi$
$60$	0	0
$61$	−2.08899	−0.267467	−0.133734	−	0.991017i	$-0.542697\pi$
$61$	−2.08899	−0.267467	−0.133734	+	0.991017i	$0.542697\pi$
$62$	20.2745	2.57487
$63$	0	0
$64$	−10.6681	−1.33351
$65$	−11.8105	−1.46492
$66$	0	0
$67$	−0.599719	−0.0732673	−0.0366337	−	0.999329i	$-0.511663\pi$
$67$	−0.599719	−0.0732673	−0.0366337	+	0.999329i	$0.511663\pi$
$68$	24.6420	2.98829
$69$	0	0
$70$	−9.42812	−1.12688
$71$	−1.40968	−0.167298	−0.0836489	−	0.996495i	$-0.526657\pi$
$71$	−1.40968	−0.167298	−0.0836489	+	0.996495i	$0.526657\pi$
$72$	0	0
$73$	7.08851	0.829648	0.414824	−	0.909902i	$-0.363843\pi$
$73$	7.08851	0.829648	0.414824	+	0.909902i	$0.363843\pi$
$74$	−1.45515	−0.169157
$75$	0	0
$76$	−22.0248	−2.52642
$77$	0	0
$78$	0	0
$79$	−1.02327	−0.115126	−0.0575632	−	0.998342i	$-0.518333\pi$
$79$	−1.02327	−0.115126	−0.0575632	+	0.998342i	$0.518333\pi$
$80$	9.68299	1.08259
$81$	0	0
$82$	4.07770	0.450307
$83$	−3.08729	−0.338874	−0.169437	−	0.985541i	$-0.554195\pi$
$83$	−3.08729	−0.338874	−0.169437	+	0.985541i	$0.554195\pi$
$84$	0	0
$85$	26.0107	2.82126
$86$	−7.74120	−0.834755
$87$	0	0
$88$	0	0
$89$	2.48531	0.263442	0.131721	−	0.991287i	$-0.457950\pi$
$89$	2.48531	0.263442	0.131721	+	0.991287i	$0.457950\pi$
$90$	0	0
$91$	2.99890	0.314370
$92$	22.4815	2.34386
$93$	0	0
$94$	11.4967	1.18579
$95$	−23.2481	−2.38521
$96$	0	0
$97$	2.55296	0.259214	0.129607	−	0.991565i	$-0.458628\pi$
$97$	2.55296	0.259214	0.129607	+	0.991565i	$0.458628\pi$
$98$	2.39396	0.241827
$99$	0	0
$100$	39.2139	3.92139
$101$	15.1573	1.50821	0.754106	−	0.656753i	$-0.228071\pi$
$101$	15.1573	1.50821	0.754106	+	0.656753i	$0.228071\pi$
$102$	0	0
$103$	−13.8364	−1.36334	−0.681670	−	0.731660i	$-0.738746\pi$
$103$	−13.8364	−1.36334	−0.681670	+	0.731660i	$0.738746\pi$
$104$	−12.4277	−1.21864
$105$	0	0
$106$	14.6710	1.42498
$107$	−0.445651	−0.0430827	−0.0215413	−	0.999768i	$-0.506857\pi$
$107$	−0.445651	−0.0430827	−0.0215413	+	0.999768i	$0.506857\pi$
$108$	0	0
$109$	11.0988	1.06307	0.531536	−	0.847036i	$-0.321615\pi$
$109$	11.0988	1.06307	0.531536	+	0.847036i	$0.321615\pi$
$110$	0	0
$111$	0	0
$112$	−2.45868	−0.232323
$113$	8.08938	0.760985	0.380492	−	0.924784i	$-0.375755\pi$
$113$	8.08938	0.760985	0.380492	+	0.924784i	$0.375755\pi$
$114$	0	0
$115$	23.7302	2.21285
$116$	5.67401	0.526819
$117$	0	0
$118$	−14.9355	−1.37493
$119$	−6.60457	−0.605440
$120$	0	0
$121$	0	0
$122$	−5.00096	−0.452765
$123$	0	0
$124$	31.5984	2.83762
$125$	21.7005	1.94095
$126$	0	0
$127$	0.179659	0.0159422	0.00797109	−	0.999968i	$-0.497463\pi$
$127$	0.179659	0.0159422	0.00797109	+	0.999968i	$0.497463\pi$
$128$	−20.7347	−1.83271
$129$	0	0
$130$	−28.2740	−2.47979
$131$	−9.38232	−0.819737	−0.409868	−	0.912145i	$-0.634425\pi$
$131$	−9.38232	−0.819737	−0.409868	+	0.912145i	$0.634425\pi$
$132$	0	0
$133$	5.90310	0.511864
$134$	−1.43570	−0.124026
$135$	0	0
$136$	27.3699	2.34695
$137$	−0.893617	−0.0763468	−0.0381734	−	0.999271i	$-0.512154\pi$
$137$	−0.893617	−0.0763468	−0.0381734	+	0.999271i	$0.512154\pi$
$138$	0	0
$139$	2.12281	0.180054	0.0900271	−	0.995939i	$-0.471305\pi$
$139$	2.12281	0.180054	0.0900271	+	0.995939i	$0.471305\pi$
$140$	−14.6940	−1.24187
$141$	0	0
$142$	−3.37471	−0.283199
$143$	0	0
$144$	0	0
$145$	5.98916	0.497373
$146$	16.9696	1.40442
$147$	0	0
$148$	−2.26789	−0.186419
$149$	−14.3220	−1.17331	−0.586653	−	0.809838i	$-0.699555\pi$
$149$	−14.3220	−1.17331	−0.586653	+	0.809838i	$0.699555\pi$
$150$	0	0
$151$	−13.7837	−1.12170	−0.560850	−	0.827918i	$-0.689525\pi$
$151$	−13.7837	−1.12170	−0.560850	+	0.827918i	$0.689525\pi$
$152$	−24.4630	−1.98421
$153$	0	0
$154$	0	0
$155$	33.3535	2.67902
$156$	0	0
$157$	−11.4608	−0.914668	−0.457334	−	0.889295i	$-0.651196\pi$
$157$	−11.4608	−0.914668	−0.457334	+	0.889295i	$0.651196\pi$
$158$	−2.44966	−0.194884
$159$	0	0
$160$	−9.46054	−0.747922
$161$	−6.02551	−0.474877
$162$	0	0
$163$	4.94262	0.387136	0.193568	−	0.981087i	$-0.437994\pi$
$163$	4.94262	0.387136	0.193568	+	0.981087i	$0.437994\pi$
$164$	6.35521	0.496258
$165$	0	0
$166$	−7.39086	−0.573642
$167$	−8.53421	−0.660397	−0.330198	−	0.943912i	$-0.607116\pi$
$167$	−8.53421	−0.660397	−0.330198	+	0.943912i	$0.607116\pi$
$168$	0	0
$169$	−4.00662	−0.308201
$170$	62.2687	4.77579
$171$	0	0
$172$	−12.0649	−0.919939
$173$	−13.0876	−0.995031	−0.497515	−	0.867455i	$-0.665754\pi$
$173$	−13.0876	−0.995031	−0.497515	+	0.867455i	$0.665754\pi$
$174$	0	0
$175$	−10.5101	−0.794492
$176$	0	0
$177$	0	0
$178$	5.94973	0.445951
$179$	−25.8622	−1.93303	−0.966514	−	0.256612i	$-0.917394\pi$
$179$	−25.8622	−1.93303	−0.966514	+	0.256612i	$0.917394\pi$
$180$	0	0
$181$	21.4202	1.59215	0.796077	−	0.605195i	$-0.206905\pi$
$181$	21.4202	1.59215	0.796077	+	0.605195i	$0.206905\pi$
$182$	7.17925	0.532161
$183$	0	0
$184$	24.9703	1.84083
$185$	−2.39385	−0.175999
$186$	0	0
$187$	0	0
$188$	17.9179	1.30680
$189$	0	0
$190$	−55.6552	−4.03765
$191$	3.73988	0.270608	0.135304	−	0.990804i	$-0.456799\pi$
$191$	3.73988	0.270608	0.135304	+	0.990804i	$0.456799\pi$
$192$	0	0
$193$	6.07932	0.437599	0.218799	−	0.975770i	$-0.429786\pi$
$193$	6.07932	0.437599	0.218799	+	0.975770i	$0.429786\pi$
$194$	6.11169	0.438794
$195$	0	0
$196$	3.73106	0.266504
$197$	−21.5458	−1.53508	−0.767538	−	0.641003i	$-0.778518\pi$
$197$	−21.5458	−1.53508	−0.767538	+	0.641003i	$0.778518\pi$
$198$	0	0
$199$	−14.8965	−1.05598	−0.527992	−	0.849249i	$-0.677055\pi$
$199$	−14.8965	−1.05598	−0.527992	+	0.849249i	$0.677055\pi$
$200$	43.5549	3.07980
$201$	0	0
$202$	36.2861	2.55308
$203$	−1.52075	−0.106736
$204$	0	0
$205$	6.70819	0.468521
$206$	−33.1238	−2.30784
$207$	0	0
$208$	−7.37332	−0.511248
$209$	0	0
$210$	0	0
$211$	16.9594	1.16753	0.583766	−	0.811922i	$-0.301579\pi$
$211$	16.9594	1.16753	0.583766	+	0.811922i	$0.301579\pi$
$212$	22.8652	1.57039
$213$	0	0
$214$	−1.06687	−0.0729298
$215$	−12.7350	−0.868520
$216$	0	0
$217$	−8.46902	−0.574915
$218$	26.5701	1.79955
$219$	0	0
$220$	0	0
$221$	−19.8064	−1.33232
$222$	0	0
$223$	−13.8125	−0.924954	−0.462477	−	0.886631i	$-0.653039\pi$
$223$	−13.8125	−0.924954	−0.462477	+	0.886631i	$0.653039\pi$
$224$	2.40219	0.160503
$225$	0	0
$226$	19.3657	1.28819
$227$	7.25345	0.481428	0.240714	−	0.970596i	$-0.422618\pi$
$227$	7.25345	0.481428	0.240714	+	0.970596i	$0.422618\pi$
$228$	0	0
$229$	−0.327136	−0.0216178	−0.0108089	−	0.999942i	$-0.503441\pi$
$229$	−0.327136	−0.0216178	−0.0108089	+	0.999942i	$0.503441\pi$
$230$	56.8093	3.74589
$231$	0	0
$232$	6.30213	0.413755
$233$	−16.2452	−1.06426	−0.532128	−	0.846664i	$-0.678607\pi$
$233$	−16.2452	−1.06426	−0.532128	+	0.846664i	$0.678607\pi$
$234$	0	0
$235$	18.9131	1.23376
$236$	−23.2774	−1.51523
$237$	0	0
$238$	−15.8111	−1.02488
$239$	29.0009	1.87591	0.937955	−	0.346756i	$-0.112717\pi$
$239$	29.0009	1.87591	0.937955	+	0.346756i	$0.112717\pi$
$240$	0	0
$241$	−29.3358	−1.88968	−0.944841	−	0.327530i	$-0.893784\pi$
$241$	−29.3358	−1.88968	−0.944841	+	0.327530i	$0.893784\pi$
$242$	0	0
$243$	0	0
$244$	−7.79413	−0.498968
$245$	3.93829	0.251608
$246$	0	0
$247$	17.7028	1.12640
$248$	35.0964	2.22862
$249$	0	0
$250$	51.9502	3.28562
$251$	−12.8316	−0.809925	−0.404963	−	0.914333i	$-0.632715\pi$
$251$	−12.8316	−0.809925	−0.404963	+	0.914333i	$0.632715\pi$
$252$	0	0
$253$	0	0
$254$	0.430097	0.0269867
$255$	0	0
$256$	−28.3018	−1.76887
$257$	−19.5210	−1.21769	−0.608844	−	0.793290i	$-0.708367\pi$
$257$	−19.5210	−1.21769	−0.608844	+	0.793290i	$0.708367\pi$
$258$	0	0
$259$	0.607840	0.0377693
$260$	−44.0658	−2.73284
$261$	0	0
$262$	−22.4609	−1.38764
$263$	−23.9606	−1.47747	−0.738736	−	0.673995i	$-0.764577\pi$
$263$	−23.9606	−1.47747	−0.738736	+	0.673995i	$0.764577\pi$
$264$	0	0
$265$	24.1352	1.48261
$266$	14.1318	0.866477
$267$	0	0
$268$	−2.23759	−0.136682
$269$	−11.3260	−0.690558	−0.345279	−	0.938500i	$-0.612216\pi$
$269$	−11.3260	−0.690558	−0.345279	+	0.938500i	$0.612216\pi$
$270$	0	0
$271$	6.12278	0.371932	0.185966	−	0.982556i	$-0.440459\pi$
$271$	6.12278	0.371932	0.185966	+	0.982556i	$0.440459\pi$
$272$	16.2385	0.984604
$273$	0	0
$274$	−2.13929	−0.129239
$275$	0	0
$276$	0	0
$277$	0.600165	0.0360604	0.0180302	−	0.999837i	$-0.494260\pi$
$277$	0.600165	0.0360604	0.0180302	+	0.999837i	$0.494260\pi$
$278$	5.08192	0.304793
$279$	0	0
$280$	−16.3206	−0.975344
$281$	−23.8022	−1.41992	−0.709960	−	0.704242i	$-0.751287\pi$
$281$	−23.8022	−1.41992	−0.709960	+	0.704242i	$0.751287\pi$
$282$	0	0
$283$	−17.3400	−1.03076	−0.515379	−	0.856962i	$-0.672349\pi$
$283$	−17.3400	−1.03076	−0.515379	+	0.856962i	$0.672349\pi$
$284$	−5.25958	−0.312099
$285$	0	0
$286$	0	0
$287$	−1.70333	−0.100544
$288$	0	0
$289$	26.6204	1.56590
$290$	14.3378	0.841947
$291$	0	0
$292$	26.4477	1.54773
$293$	14.9122	0.871179	0.435589	−	0.900145i	$-0.356540\pi$
$293$	14.9122	0.871179	0.435589	+	0.900145i	$0.356540\pi$
$294$	0	0
$295$	−24.5703	−1.43054
$296$	−2.51894	−0.146411
$297$	0	0
$298$	−34.2864	−1.98616
$299$	−18.0699	−1.04501
$300$	0	0
$301$	3.23364	0.186384
$302$	−32.9976	−1.89880
$303$	0	0
$304$	−14.5138	−0.832425
$305$	−8.22704	−0.471079
$306$	0	0
$307$	29.0453	1.65771	0.828853	−	0.559467i	$-0.188994\pi$
$307$	29.0453	1.65771	0.828853	+	0.559467i	$0.188994\pi$
$308$	0	0
$309$	0	0
$310$	79.8470	4.53500
$311$	20.3651	1.15480	0.577399	−	0.816462i	$-0.304068\pi$
$311$	20.3651	1.15480	0.577399	+	0.816462i	$0.304068\pi$
$312$	0	0
$313$	−29.4807	−1.66635	−0.833174	−	0.553010i	$-0.813479\pi$
$313$	−29.4807	−1.66635	−0.833174	+	0.553010i	$0.813479\pi$
$314$	−27.4366	−1.54834
$315$	0	0
$316$	−3.81786	−0.214772
$317$	−7.62530	−0.428280	−0.214140	−	0.976803i	$-0.568695\pi$
$317$	−7.62530	−0.428280	−0.214140	+	0.976803i	$0.568695\pi$
$318$	0	0
$319$	0	0
$320$	−42.0142	−2.34866
$321$	0	0
$322$	−14.4249	−0.803865
$323$	−38.9875	−2.16932
$324$	0	0
$325$	−31.5188	−1.74835
$326$	11.8324	0.655338
$327$	0	0
$328$	7.05873	0.389753
$329$	−4.80237	−0.264763
$330$	0	0
$331$	20.6731	1.13630	0.568148	−	0.822926i	$-0.307660\pi$
$331$	20.6731	1.13630	0.568148	+	0.822926i	$0.307660\pi$
$332$	−11.5189	−0.632180
$333$	0	0
$334$	−20.4306	−1.11791
$335$	−2.36187	−0.129043
$336$	0	0
$337$	−5.46255	−0.297564	−0.148782	−	0.988870i	$-0.547535\pi$
$337$	−5.46255	−0.297564	−0.148782	+	0.988870i	$0.547535\pi$
$338$	−9.59169	−0.521719
$339$	0	0
$340$	97.0475	5.26314
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−13.4005	−0.722505
$345$	0	0
$346$	−31.3312	−1.68438
$347$	−9.58110	−0.514341	−0.257170	−	0.966366i	$-0.582790\pi$
$347$	−9.58110	−0.514341	−0.257170	+	0.966366i	$0.582790\pi$
$348$	0	0
$349$	9.36045	0.501054	0.250527	−	0.968110i	$-0.419396\pi$
$349$	9.36045	0.501054	0.250527	+	0.968110i	$0.419396\pi$
$350$	−25.1609	−1.34491
$351$	0	0
$352$	0	0
$353$	27.6629	1.47235	0.736173	−	0.676794i	$-0.236631\pi$
$353$	27.6629	1.47235	0.736173	+	0.676794i	$0.236631\pi$
$354$	0	0
$355$	−5.55171	−0.294654
$356$	9.27282	0.491459
$357$	0	0
$358$	−61.9131	−3.27221
$359$	2.60258	0.137359	0.0686796	−	0.997639i	$-0.478121\pi$
$359$	2.60258	0.137359	0.0686796	+	0.997639i	$0.478121\pi$
$360$	0	0
$361$	15.8466	0.834033
$362$	51.2793	2.69518
$363$	0	0
$364$	11.1891	0.586466
$365$	27.9166	1.46122
$366$	0	0
$367$	−10.3394	−0.539710	−0.269855	−	0.962901i	$-0.586976\pi$
$367$	−10.3394	−0.539710	−0.269855	+	0.962901i	$0.586976\pi$
$368$	14.8148	0.772274
$369$	0	0
$370$	−5.73079	−0.297930
$371$	−6.12834	−0.318168
$372$	0	0
$373$	−5.15587	−0.266961	−0.133480	−	0.991051i	$-0.542615\pi$
$373$	−5.15587	−0.266961	−0.133480	+	0.991051i	$0.542615\pi$
$374$	0	0
$375$	0	0
$376$	19.9015	1.02634
$377$	−4.56058	−0.234882
$378$	0	0
$379$	−35.4223	−1.81952	−0.909761	−	0.415133i	$-0.863735\pi$
$379$	−35.4223	−1.81952	−0.909761	+	0.415133i	$0.863735\pi$
$380$	−86.7401	−4.44968
$381$	0	0
$382$	8.95313	0.458082
$383$	31.4076	1.60486	0.802428	−	0.596750i	$-0.203541\pi$
$383$	31.4076	1.60486	0.802428	+	0.596750i	$0.203541\pi$
$384$	0	0
$385$	0	0
$386$	14.5537	0.740762
$387$	0	0
$388$	9.52524	0.483571
$389$	−20.0789	−1.01804	−0.509020	−	0.860755i	$-0.669992\pi$
$389$	−20.0789	−1.01804	−0.509020	+	0.860755i	$0.669992\pi$
$390$	0	0
$391$	39.7959	2.01257
$392$	4.14409	0.209308
$393$	0	0
$394$	−51.5799	−2.59856
$395$	−4.02992	−0.202767
$396$	0	0
$397$	3.63351	0.182361	0.0911804	−	0.995834i	$-0.470936\pi$
$397$	3.63351	0.182361	0.0911804	+	0.995834i	$0.470936\pi$
$398$	−35.6616	−1.78756
$399$	0	0
$400$	25.8410	1.29205
$401$	−37.6309	−1.87920	−0.939598	−	0.342279i	$-0.888801\pi$
$401$	−37.6309	−1.87920	−0.939598	+	0.342279i	$0.888801\pi$
$402$	0	0
$403$	−25.3977	−1.26515
$404$	56.5529	2.81361
$405$	0	0
$406$	−3.64062	−0.180681
$407$	0	0
$408$	0	0
$409$	16.1059	0.796386	0.398193	−	0.917302i	$-0.369637\pi$
$409$	16.1059	0.796386	0.398193	+	0.917302i	$0.369637\pi$
$410$	16.0592	0.793106
$411$	0	0
$412$	−51.6243	−2.54335
$413$	6.23883	0.306993
$414$	0	0
$415$	−12.1587	−0.596845
$416$	7.20393	0.353202
$417$	0	0
$418$	0	0
$419$	17.7128	0.865325	0.432662	−	0.901556i	$-0.357574\pi$
$419$	17.7128	0.865325	0.432662	+	0.901556i	$0.357574\pi$
$420$	0	0
$421$	30.2721	1.47537	0.737686	−	0.675144i	$-0.235919\pi$
$421$	30.2721	1.47537	0.737686	+	0.675144i	$0.235919\pi$
$422$	40.6001	1.97638
$423$	0	0
$424$	25.3964	1.23336
$425$	69.4149	3.36712
$426$	0	0
$427$	2.08899	0.101093
$428$	−1.66275	−0.0803720
$429$	0	0
$430$	−30.4871	−1.47022
$431$	−19.1918	−0.924438	−0.462219	−	0.886766i	$-0.652947\pi$
$431$	−19.1918	−0.924438	−0.462219	+	0.886766i	$0.652947\pi$
$432$	0	0
$433$	21.7538	1.04542	0.522711	−	0.852510i	$-0.324921\pi$
$433$	21.7538	1.04542	0.522711	+	0.852510i	$0.324921\pi$
$434$	−20.2745	−0.973208
$435$	0	0
$436$	41.4102	1.98319
$437$	−35.5692	−1.70151
$438$	0	0
$439$	2.70519	0.129112	0.0645558	−	0.997914i	$-0.479437\pi$
$439$	2.70519	0.129112	0.0645558	+	0.997914i	$0.479437\pi$
$440$	0	0
$441$	0	0
$442$	−47.4159	−2.25534
$443$	−38.2274	−1.81624	−0.908118	−	0.418713i	$-0.862481\pi$
$443$	−38.2274	−1.81624	−0.908118	+	0.418713i	$0.862481\pi$
$444$	0	0
$445$	9.78786	0.463989
$446$	−33.0666	−1.56575
$447$	0	0
$448$	10.6681	0.504021
$449$	−22.7112	−1.07181	−0.535904	−	0.844279i	$-0.680029\pi$
$449$	−22.7112	−1.07181	−0.535904	+	0.844279i	$0.680029\pi$
$450$	0	0
$451$	0	0
$452$	30.1819	1.41964
$453$	0	0
$454$	17.3645	0.814956
$455$	11.8105	0.553686
$456$	0	0
$457$	−1.57183	−0.0735271	−0.0367636	−	0.999324i	$-0.511705\pi$
$457$	−1.57183	−0.0735271	−0.0367636	+	0.999324i	$0.511705\pi$
$458$	−0.783152	−0.0365943
$459$	0	0
$460$	88.5388	4.12814
$461$	28.1276	1.31003	0.655017	−	0.755614i	$-0.272661\pi$
$461$	28.1276	1.31003	0.655017	+	0.755614i	$0.272661\pi$
$462$	0	0
$463$	−4.69202	−0.218057	−0.109028	−	0.994039i	$-0.534774\pi$
$463$	−4.69202	−0.218057	−0.109028	+	0.994039i	$0.534774\pi$
$464$	3.73904	0.173580
$465$	0	0
$466$	−38.8903	−1.80156
$467$	17.6135	0.815054	0.407527	−	0.913193i	$-0.366391\pi$
$467$	17.6135	0.815054	0.407527	+	0.913193i	$0.366391\pi$
$468$	0	0
$469$	0.599719	0.0276924
$470$	45.2773	2.08849
$471$	0	0
$472$	−25.8543	−1.19004
$473$	0	0
$474$	0	0
$475$	−62.0424	−2.84670
$476$	−24.6420	−1.12947
$477$	0	0
$478$	69.4270	3.17552
$479$	10.7070	0.489215	0.244608	−	0.969622i	$-0.421341\pi$
$479$	10.7070	0.489215	0.244608	+	0.969622i	$0.421341\pi$
$480$	0	0
$481$	1.82285	0.0831148
$482$	−70.2287	−3.19883
$483$	0	0
$484$	0	0
$485$	10.0543	0.456542
$486$	0	0
$487$	38.7602	1.75639	0.878197	−	0.478299i	$-0.158746\pi$
$487$	38.7602	1.75639	0.878197	+	0.478299i	$0.158746\pi$
$488$	−8.65695	−0.391882
$489$	0	0
$490$	9.42812	0.425919
$491$	6.00061	0.270804	0.135402	−	0.990791i	$-0.456767\pi$
$491$	6.00061	0.270804	0.135402	+	0.990791i	$0.456767\pi$
$492$	0	0
$493$	10.0439	0.452355
$494$	42.3798	1.90676
$495$	0	0
$496$	20.8226	0.934962
$497$	1.40968	0.0632326
$498$	0	0
$499$	−28.9776	−1.29722	−0.648608	−	0.761123i	$-0.724648\pi$
$499$	−28.9776	−1.29722	−0.648608	+	0.761123i	$0.724648\pi$
$500$	80.9659	3.62091
$501$	0	0
$502$	−30.7184	−1.37103
$503$	12.8657	0.573653	0.286827	−	0.957982i	$-0.407400\pi$
$503$	12.8657	0.573653	0.286827	+	0.957982i	$0.407400\pi$
$504$	0	0
$505$	59.6940	2.65635
$506$	0	0
$507$	0	0
$508$	0.670319	0.0297406
$509$	−19.7361	−0.874788	−0.437394	−	0.899270i	$-0.644098\pi$
$509$	−19.7361	−0.874788	−0.437394	+	0.899270i	$0.644098\pi$
$510$	0	0
$511$	−7.08851	−0.313577
$512$	−26.2842	−1.16161
$513$	0	0
$514$	−46.7326	−2.06129
$515$	−54.4917	−2.40119
$516$	0	0
$517$	0	0
$518$	1.45515	0.0639355
$519$	0	0
$520$	−48.9439	−2.14633
$521$	23.3926	1.02485	0.512423	−	0.858733i	$-0.328748\pi$
$521$	23.3926	1.02485	0.512423	+	0.858733i	$0.328748\pi$
$522$	0	0
$523$	−10.7796	−0.471361	−0.235680	−	0.971831i	$-0.575732\pi$
$523$	−10.7796	−0.471361	−0.235680	+	0.971831i	$0.575732\pi$
$524$	−35.0060	−1.52924
$525$	0	0
$526$	−57.3607	−2.50105
$527$	55.9343	2.43653
$528$	0	0
$529$	13.3068	0.578556
$530$	57.7788	2.50975
$531$	0	0
$532$	22.0248	0.954897
$533$	−5.10810	−0.221256
$534$	0	0
$535$	−1.75510	−0.0758797
$536$	−2.48529	−0.107348
$537$	0	0
$538$	−27.1140	−1.16897
$539$	0	0
$540$	0	0
$541$	−6.42155	−0.276084	−0.138042	−	0.990426i	$-0.544081\pi$
$541$	−6.42155	−0.276084	−0.138042	+	0.990426i	$0.544081\pi$
$542$	14.6577	0.629602
$543$	0	0
$544$	−15.8655	−0.680226
$545$	43.7103	1.87234
$546$	0	0
$547$	−28.0529	−1.19946	−0.599728	−	0.800204i	$-0.704724\pi$
$547$	−28.0529	−1.19946	−0.599728	+	0.800204i	$0.704724\pi$
$548$	−3.33414	−0.142427
$549$	0	0
$550$	0	0
$551$	−8.97715	−0.382440
$552$	0	0
$553$	1.02327	0.0435137
$554$	1.43677	0.0610427
$555$	0	0
$556$	7.92031	0.335896
$557$	−23.1102	−0.979211	−0.489606	−	0.871944i	$-0.662859\pi$
$557$	−23.1102	−0.979211	−0.489606	+	0.871944i	$0.662859\pi$
$558$	0	0
$559$	9.69734	0.410154
$560$	−9.68299	−0.409181
$561$	0	0
$562$	−56.9815	−2.40362
$563$	28.8436	1.21561	0.607806	−	0.794086i	$-0.292050\pi$
$563$	28.8436	1.21561	0.607806	+	0.794086i	$0.292050\pi$
$564$	0	0
$565$	31.8583	1.34029
$566$	−41.5114	−1.74485
$567$	0	0
$568$	−5.84182	−0.245117
$569$	17.7760	0.745208	0.372604	−	0.927991i	$-0.378465\pi$
$569$	17.7760	0.745208	0.372604	+	0.927991i	$0.378465\pi$
$570$	0	0
$571$	17.8981	0.749012	0.374506	−	0.927225i	$-0.377812\pi$
$571$	17.8981	0.749012	0.374506	+	0.927225i	$0.377812\pi$
$572$	0	0
$573$	0	0
$574$	−4.07770	−0.170200
$575$	63.3289	2.64100
$576$	0	0
$577$	21.0769	0.877441	0.438721	−	0.898624i	$-0.355432\pi$
$577$	21.0769	0.877441	0.438721	+	0.898624i	$0.355432\pi$
$578$	63.7281	2.65074
$579$	0	0
$580$	22.3459	0.927864
$581$	3.08729	0.128083
$582$	0	0
$583$	0	0
$584$	29.3754	1.21556
$585$	0	0
$586$	35.6992	1.47472
$587$	−17.3136	−0.714609	−0.357305	−	0.933988i	$-0.616304\pi$
$587$	−17.3136	−0.714609	−0.357305	+	0.933988i	$0.616304\pi$
$588$	0	0
$589$	−49.9935	−2.05995
$590$	−58.8205	−2.42160
$591$	0	0
$592$	−1.49448	−0.0614228
$593$	26.3881	1.08363	0.541814	−	0.840498i	$-0.317738\pi$
$593$	26.3881	1.08363	0.541814	+	0.840498i	$0.317738\pi$
$594$	0	0
$595$	−26.0107	−1.06634
$596$	−53.4363	−2.18884
$597$	0	0
$598$	−43.2586	−1.76898
$599$	28.2593	1.15465	0.577323	−	0.816516i	$-0.304097\pi$
$599$	28.2593	1.15465	0.577323	+	0.816516i	$0.304097\pi$
$600$	0	0
$601$	−0.440908	−0.0179850	−0.00899251	−	0.999960i	$-0.502862\pi$
$601$	−0.440908	−0.0179850	−0.00899251	+	0.999960i	$0.502862\pi$
$602$	7.74120	0.315508
$603$	0	0
$604$	−51.4277	−2.09256
$605$	0	0
$606$	0	0
$607$	0.559085	0.0226925	0.0113463	−	0.999936i	$-0.496388\pi$
$607$	0.559085	0.0226925	0.0113463	+	0.999936i	$0.496388\pi$
$608$	14.1804	0.575091
$609$	0	0
$610$	−19.6952	−0.797436
$611$	−14.4018	−0.582635
$612$	0	0
$613$	30.3820	1.22712	0.613559	−	0.789649i	$-0.289737\pi$
$613$	30.3820	1.22712	0.613559	+	0.789649i	$0.289737\pi$
$614$	69.5335	2.80614
$615$	0	0
$616$	0	0
$617$	−2.35080	−0.0946395	−0.0473198	−	0.998880i	$-0.515068\pi$
$617$	−2.35080	−0.0946395	−0.0473198	+	0.998880i	$0.515068\pi$
$618$	0	0
$619$	6.11038	0.245597	0.122798	−	0.992432i	$-0.460813\pi$
$619$	6.11038	0.245597	0.122798	+	0.992432i	$0.460813\pi$
$620$	124.444	4.99778
$621$	0	0
$622$	48.7532	1.95483
$623$	−2.48531	−0.0995717
$624$	0	0
$625$	32.9123	1.31649
$626$	−70.5757	−2.82077
$627$	0	0
$628$	−42.7608	−1.70634
$629$	−4.01452	−0.160069
$630$	0	0
$631$	−28.8822	−1.14978	−0.574891	−	0.818230i	$-0.694955\pi$
$631$	−28.8822	−1.14978	−0.574891	+	0.818230i	$0.694955\pi$
$632$	−4.24050	−0.168678
$633$	0	0
$634$	−18.2547	−0.724987
$635$	0.707550	0.0280783
$636$	0	0
$637$	−2.99890	−0.118821
$638$	0	0
$639$	0	0
$640$	−81.6592	−3.22787
$641$	39.8256	1.57302	0.786508	−	0.617579i	$-0.211887\pi$
$641$	39.8256	1.57302	0.786508	+	0.617579i	$0.211887\pi$
$642$	0	0
$643$	5.85723	0.230987	0.115493	−	0.993308i	$-0.463155\pi$
$643$	5.85723	0.230987	0.115493	+	0.993308i	$0.463155\pi$
$644$	−22.4815	−0.885896
$645$	0	0
$646$	−93.3345	−3.67220
$647$	−22.5855	−0.887926	−0.443963	−	0.896045i	$-0.646428\pi$
$647$	−22.5855	−0.887926	−0.443963	+	0.896045i	$0.646428\pi$
$648$	0	0
$649$	0	0
$650$	−75.4549	−2.95958
$651$	0	0
$652$	18.4412	0.722213
$653$	40.8369	1.59807	0.799035	−	0.601284i	$-0.205344\pi$
$653$	40.8369	1.59807	0.799035	+	0.601284i	$0.205344\pi$
$654$	0	0
$655$	−36.9503	−1.44377
$656$	4.18793	0.163511
$657$	0	0
$658$	−11.4967	−0.448188
$659$	−32.5869	−1.26941	−0.634703	−	0.772756i	$-0.718878\pi$
$659$	−32.5869	−1.26941	−0.634703	+	0.772756i	$0.718878\pi$
$660$	0	0
$661$	43.6393	1.69737	0.848686	−	0.528898i	$-0.177395\pi$
$661$	43.6393	1.69737	0.848686	+	0.528898i	$0.177395\pi$
$662$	49.4906	1.92351
$663$	0	0
$664$	−12.7940	−0.496504
$665$	23.2481	0.901524
$666$	0	0
$667$	9.16331	0.354805
$668$	−31.8416	−1.23199
$669$	0	0
$670$	−5.65422	−0.218442
$671$	0	0
$672$	0	0
$673$	−23.8718	−0.920190	−0.460095	−	0.887870i	$-0.652185\pi$
$673$	−23.8718	−0.920190	−0.460095	+	0.887870i	$0.652185\pi$
$674$	−13.0771	−0.503712
$675$	0	0
$676$	−14.9489	−0.574958
$677$	4.26651	0.163975	0.0819877	−	0.996633i	$-0.473873\pi$
$677$	4.26651	0.163975	0.0819877	+	0.996633i	$0.473873\pi$
$678$	0	0
$679$	−2.55296	−0.0979736
$680$	107.791	4.13359
$681$	0	0
$682$	0	0
$683$	−5.88217	−0.225075	−0.112537	−	0.993647i	$-0.535898\pi$
$683$	−5.88217	−0.225075	−0.112537	+	0.993647i	$0.535898\pi$
$684$	0	0
$685$	−3.51932	−0.134466
$686$	−2.39396	−0.0914019
$687$	0	0
$688$	−7.95047	−0.303109
$689$	−18.3783	−0.700156
$690$	0	0
$691$	−0.237559	−0.00903716	−0.00451858	−	0.999990i	$-0.501438\pi$
$691$	−0.237559	−0.00903716	−0.00451858	+	0.999990i	$0.501438\pi$
$692$	−48.8305	−1.85626
$693$	0	0
$694$	−22.9368	−0.870669
$695$	8.36023	0.317122
$696$	0	0
$697$	11.2497	0.426114
$698$	22.4086	0.848177
$699$	0	0
$700$	−39.2139	−1.48215
$701$	10.6036	0.400494	0.200247	−	0.979745i	$-0.435826\pi$
$701$	10.6036	0.400494	0.200247	+	0.979745i	$0.435826\pi$
$702$	0	0
$703$	3.58814	0.135329
$704$	0	0
$705$	0	0
$706$	66.2239	2.49237
$707$	−15.1573	−0.570050
$708$	0	0
$709$	−48.3236	−1.81483	−0.907415	−	0.420235i	$-0.861947\pi$
$709$	−48.3236	−1.81483	−0.907415	+	0.420235i	$0.861947\pi$
$710$	−13.2906	−0.498787
$711$	0	0
$712$	10.2993	0.385984
$713$	51.0302	1.91110
$714$	0	0
$715$	0	0
$716$	−96.4932	−3.60612
$717$	0	0
$718$	6.23049	0.232520
$719$	36.2637	1.35241	0.676203	−	0.736715i	$-0.263624\pi$
$719$	36.2637	1.35241	0.676203	+	0.736715i	$0.263624\pi$
$720$	0	0
$721$	13.8364	0.515294
$722$	37.9362	1.41184
$723$	0	0
$724$	79.9202	2.97021
$725$	15.9833	0.593605
$726$	0	0
$727$	1.15184	0.0427195	0.0213597	−	0.999772i	$-0.493200\pi$
$727$	1.15184	0.0427195	0.0213597	+	0.999772i	$0.493200\pi$
$728$	12.4277	0.460601
$729$	0	0
$730$	66.8314	2.47354
$731$	−21.3568	−0.789909
$732$	0	0
$733$	−27.1042	−1.00111	−0.500557	−	0.865703i	$-0.666872\pi$
$733$	−27.1042	−1.00111	−0.500557	+	0.865703i	$0.666872\pi$
$734$	−24.7520	−0.913614
$735$	0	0
$736$	−14.4745	−0.533535
$737$	0	0
$738$	0	0
$739$	−23.8018	−0.875563	−0.437782	−	0.899081i	$-0.644236\pi$
$739$	−23.8018	−0.875563	−0.437782	+	0.899081i	$0.644236\pi$
$740$	−8.93160	−0.328332
$741$	0	0
$742$	−14.6710	−0.538590
$743$	−15.0656	−0.552704	−0.276352	−	0.961056i	$-0.589126\pi$
$743$	−15.0656	−0.552704	−0.276352	+	0.961056i	$0.589126\pi$
$744$	0	0
$745$	−56.4043	−2.06650
$746$	−12.3430	−0.451908
$747$	0	0
$748$	0	0
$749$	0.445651	0.0162837
$750$	0	0
$751$	10.6107	0.387191	0.193596	−	0.981081i	$-0.437985\pi$
$751$	10.6107	0.387191	0.193596	+	0.981081i	$0.437985\pi$
$752$	11.8075	0.430575
$753$	0	0
$754$	−10.9179	−0.397605
$755$	−54.2841	−1.97560
$756$	0	0
$757$	−35.0862	−1.27523	−0.637615	−	0.770355i	$-0.720079\pi$
$757$	−35.0862	−1.27523	−0.637615	+	0.770355i	$0.720079\pi$
$758$	−84.7997	−3.08006
$759$	0	0
$760$	−96.3423	−3.49470
$761$	−37.2895	−1.35174	−0.675872	−	0.737019i	$-0.736233\pi$
$761$	−37.2895	−1.35174	−0.675872	+	0.737019i	$0.736233\pi$
$762$	0	0
$763$	−11.0988	−0.401803
$764$	13.9537	0.504827
$765$	0	0
$766$	75.1887	2.71668
$767$	18.7096	0.675565
$768$	0	0
$769$	21.8827	0.789110	0.394555	−	0.918872i	$-0.370899\pi$
$769$	21.8827	0.789110	0.394555	+	0.918872i	$0.370899\pi$
$770$	0	0
$771$	0	0
$772$	22.6823	0.816353
$773$	34.4742	1.23995	0.619975	−	0.784622i	$-0.287143\pi$
$773$	34.4742	1.23995	0.619975	+	0.784622i	$0.287143\pi$
$774$	0	0
$775$	89.0106	3.19735
$776$	10.5797	0.379789
$777$	0	0
$778$	−48.0681	−1.72332
$779$	−10.0549	−0.360254
$780$	0	0
$781$	0	0
$782$	95.2699	3.40685
$783$	0	0
$784$	2.45868	0.0878099
$785$	−45.1358	−1.61097
$786$	0	0
$787$	11.6668	0.415878	0.207939	−	0.978142i	$-0.433324\pi$
$787$	11.6668	0.415878	0.207939	+	0.978142i	$0.433324\pi$
$788$	−80.3887	−2.86373
$789$	0	0
$790$	−9.64747	−0.343242
$791$	−8.08938	−0.287625
$792$	0	0
$793$	6.26466	0.222464
$794$	8.69849	0.308698
$795$	0	0
$796$	−55.5797	−1.96997
$797$	35.1322	1.24445	0.622223	−	0.782840i	$-0.286230\pi$
$797$	35.1322	1.24445	0.622223	+	0.782840i	$0.286230\pi$
$798$	0	0
$799$	31.7176	1.12209
$800$	−25.2474	−0.892630
$801$	0	0
$802$	−90.0869	−3.18108
$803$	0	0
$804$	0	0
$805$	−23.7302	−0.836380
$806$	−60.8012	−2.14163
$807$	0	0
$808$	62.8133	2.20977
$809$	14.5235	0.510620	0.255310	−	0.966859i	$-0.417822\pi$
$809$	14.5235	0.510620	0.255310	+	0.966859i	$0.417822\pi$
$810$	0	0
$811$	1.76417	0.0619483	0.0309741	−	0.999520i	$-0.490139\pi$
$811$	1.76417	0.0619483	0.0309741	+	0.999520i	$0.490139\pi$
$812$	−5.67401	−0.199119
$813$	0	0
$814$	0	0
$815$	19.4655	0.681846
$816$	0	0
$817$	19.0885	0.667821
$818$	38.5570	1.34811
$819$	0	0
$820$	25.0287	0.874039
$821$	9.76074	0.340652	0.170326	−	0.985388i	$-0.445518\pi$
$821$	9.76074	0.340652	0.170326	+	0.985388i	$0.445518\pi$
$822$	0	0
$823$	−45.6322	−1.59064	−0.795319	−	0.606191i	$-0.792697\pi$
$823$	−45.6322	−1.59064	−0.795319	+	0.606191i	$0.792697\pi$
$824$	−57.3392	−1.99750
$825$	0	0
$826$	14.9355	0.519674
$827$	−20.7517	−0.721608	−0.360804	−	0.932642i	$-0.617498\pi$
$827$	−20.7517	−0.721608	−0.360804	+	0.932642i	$0.617498\pi$
$828$	0	0
$829$	16.3868	0.569137	0.284568	−	0.958656i	$-0.408150\pi$
$829$	16.3868	0.569137	0.284568	+	0.958656i	$0.408150\pi$
$830$	−29.1074	−1.01033
$831$	0	0
$832$	31.9926	1.10914
$833$	6.60457	0.228835
$834$	0	0
$835$	−33.6102	−1.16313
$836$	0	0
$837$	0	0
$838$	42.4037	1.46481
$839$	−35.0935	−1.21156	−0.605782	−	0.795631i	$-0.707140\pi$
$839$	−35.0935	−1.21156	−0.605782	+	0.795631i	$0.707140\pi$
$840$	0	0
$841$	−26.6873	−0.920252
$842$	72.4703	2.49749
$843$	0	0
$844$	63.2764	2.17806
$845$	−15.7792	−0.542822
$846$	0	0
$847$	0	0
$848$	15.0676	0.517424
$849$	0	0
$850$	166.177	5.69982
$851$	−3.66255	−0.125550
$852$	0	0
$853$	−11.3030	−0.387007	−0.193504	−	0.981100i	$-0.561985\pi$
$853$	−11.3030	−0.387007	−0.193504	+	0.981100i	$0.561985\pi$
$854$	5.00096	0.171129
$855$	0	0
$856$	−1.84682	−0.0631229
$857$	−13.8531	−0.473214	−0.236607	−	0.971605i	$-0.576035\pi$
$857$	−13.8531	−0.473214	−0.236607	+	0.971605i	$0.576035\pi$
$858$	0	0
$859$	−5.23831	−0.178729	−0.0893645	−	0.995999i	$-0.528484\pi$
$859$	−5.23831	−0.178729	−0.0893645	+	0.995999i	$0.528484\pi$
$860$	−47.5150	−1.62025
$861$	0	0
$862$	−45.9445	−1.56488
$863$	41.1289	1.40004	0.700022	−	0.714121i	$-0.253174\pi$
$863$	41.1289	1.40004	0.700022	+	0.714121i	$0.253174\pi$
$864$	0	0
$865$	−51.5427	−1.75250
$866$	52.0779	1.76968
$867$	0	0
$868$	−31.5984	−1.07252
$869$	0	0
$870$	0	0
$871$	1.79849	0.0609397
$872$	45.9944	1.55757
$873$	0	0
$874$	−85.1514	−2.88029
$875$	−21.7005	−0.733612
$876$	0	0
$877$	−5.41488	−0.182848	−0.0914238	−	0.995812i	$-0.529142\pi$
$877$	−5.41488	−0.182848	−0.0914238	+	0.995812i	$0.529142\pi$
$878$	6.47612	0.218558
$879$	0	0
$880$	0	0
$881$	−11.0961	−0.373837	−0.186918	−	0.982375i	$-0.559850\pi$
$881$	−11.0961	−0.373837	−0.186918	+	0.982375i	$0.559850\pi$
$882$	0	0
$883$	−51.4596	−1.73175	−0.865877	−	0.500256i	$-0.833239\pi$
$883$	−51.4596	−1.73175	−0.865877	+	0.500256i	$0.833239\pi$
$884$	−73.8989	−2.48549
$885$	0	0
$886$	−91.5149	−3.07450
$887$	19.9938	0.671325	0.335662	−	0.941982i	$-0.391040\pi$
$887$	19.9938	0.671325	0.335662	+	0.941982i	$0.391040\pi$
$888$	0	0
$889$	−0.179659	−0.00602558
$890$	23.4318	0.785435
$891$	0	0
$892$	−51.5353	−1.72553
$893$	−28.3489	−0.948659
$894$	0	0
$895$	−101.853	−3.40456
$896$	20.7347	0.692697
$897$	0	0
$898$	−54.3697	−1.81434
$899$	12.8793	0.429548
$900$	0	0
$901$	40.4751	1.34842
$902$	0	0
$903$	0	0
$904$	33.5231	1.11496
$905$	84.3591	2.80419
$906$	0	0
$907$	−5.23855	−0.173943	−0.0869716	−	0.996211i	$-0.527719\pi$
$907$	−5.23855	−0.173943	−0.0869716	+	0.996211i	$0.527719\pi$
$908$	27.0630	0.898119
$909$	0	0
$910$	28.2740	0.937273
$911$	11.0131	0.364882	0.182441	−	0.983217i	$-0.441600\pi$
$911$	11.0131	0.364882	0.182441	+	0.983217i	$0.441600\pi$
$912$	0	0
$913$	0	0
$914$	−3.76290	−0.124466
$915$	0	0
$916$	−1.22056	−0.0403286
$917$	9.38232	0.309831
$918$	0	0
$919$	−28.8457	−0.951533	−0.475767	−	0.879572i	$-0.657829\pi$
$919$	−28.8457	−0.951533	−0.475767	+	0.879572i	$0.657829\pi$
$920$	98.3401	3.24218
$921$	0	0
$922$	67.3365	2.21761
$923$	4.22747	0.139149
$924$	0	0
$925$	−6.38848	−0.210052
$926$	−11.2325	−0.369123
$927$	0	0
$928$	−3.65314	−0.119920
$929$	−16.2208	−0.532186	−0.266093	−	0.963947i	$-0.585733\pi$
$929$	−16.2208	−0.532186	−0.266093	+	0.963947i	$0.585733\pi$
$930$	0	0
$931$	−5.90310	−0.193466
$932$	−60.6116	−1.98540
$933$	0	0
$934$	42.1660	1.37971
$935$	0	0
$936$	0	0
$937$	−27.4672	−0.897315	−0.448657	−	0.893704i	$-0.648098\pi$
$937$	−27.4672	−0.897315	−0.448657	+	0.893704i	$0.648098\pi$
$938$	1.43570	0.0468774
$939$	0	0
$940$	70.5660	2.30161
$941$	−48.7222	−1.58830	−0.794150	−	0.607722i	$-0.792084\pi$
$941$	−48.7222	−1.58830	−0.794150	+	0.607722i	$0.792084\pi$
$942$	0	0
$943$	10.2634	0.334223
$944$	−15.3393	−0.499251
$945$	0	0
$946$	0	0
$947$	−21.1998	−0.688902	−0.344451	−	0.938804i	$-0.611935\pi$
$947$	−21.1998	−0.688902	−0.344451	+	0.938804i	$0.611935\pi$
$948$	0	0
$949$	−21.2577	−0.690055
$950$	−148.527	−4.81886
$951$	0	0
$952$	−27.3699	−0.887064
$953$	−34.1228	−1.10535	−0.552673	−	0.833398i	$-0.686392\pi$
$953$	−34.1228	−1.10535	−0.552673	+	0.833398i	$0.686392\pi$
$954$	0	0
$955$	14.7287	0.476610
$956$	108.204	3.49957
$957$	0	0
$958$	25.6322	0.828138
$959$	0.893617	0.0288564
$960$	0	0
$961$	40.7243	1.31369
$962$	4.36383	0.140696
$963$	0	0
$964$	−109.453	−3.52526
$965$	23.9421	0.770724
$966$	0	0
$967$	−18.8881	−0.607400	−0.303700	−	0.952768i	$-0.598222\pi$
$967$	−18.8881	−0.607400	−0.303700	+	0.952768i	$0.598222\pi$
$968$	0	0
$969$	0	0
$970$	24.0696	0.772829
$971$	−12.4847	−0.400652	−0.200326	−	0.979729i	$-0.564200\pi$
$971$	−12.4847	−0.400652	−0.200326	+	0.979729i	$0.564200\pi$
$972$	0	0
$973$	−2.12281	−0.0680541
$974$	92.7906	2.97320
$975$	0	0
$976$	−5.13614	−0.164404
$977$	13.8747	0.443890	0.221945	−	0.975059i	$-0.428759\pi$
$977$	13.8747	0.443890	0.221945	+	0.975059i	$0.428759\pi$
$978$	0	0
$979$	0	0
$980$	14.6940	0.469382
$981$	0	0
$982$	14.3652	0.458413
$983$	−43.8326	−1.39804	−0.699021	−	0.715101i	$-0.746381\pi$
$983$	−43.8326	−1.39804	−0.699021	+	0.715101i	$0.746381\pi$
$984$	0	0
$985$	−84.8537	−2.70366
$986$	24.0448	0.765741
$987$	0	0
$988$	66.0502	2.10134
$989$	−19.4843	−0.619565
$990$	0	0
$991$	42.8926	1.36253	0.681264	−	0.732037i	$-0.261430\pi$
$991$	42.8926	1.36253	0.681264	+	0.732037i	$0.261430\pi$
$992$	−20.3442	−0.645930
$993$	0	0
$994$	3.37471	0.107039
$995$	−58.6667	−1.85986
$996$	0	0
$997$	−28.2985	−0.896224	−0.448112	−	0.893977i	$-0.647903\pi$
$997$	−28.2985	−0.896224	−0.448112	+	0.893977i	$0.647903\pi$
$998$	−69.3713	−2.19591
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cx.1.9		10
3.2	odd	2		2541.2.a.bq.1.2		10
11.3	even	5		693.2.m.j.64.5		20
11.4	even	5		693.2.m.j.379.5		20
11.10	odd	2		7623.2.a.cy.1.2		10
33.14	odd	10		231.2.j.g.64.1	✓	20
33.26	odd	10		231.2.j.g.148.1	yes	20
33.32	even	2		2541.2.a.br.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.64.1	✓	20	33.14	odd	10
231.2.j.g.148.1	yes	20	33.26	odd	10
693.2.m.j.64.5		20	11.3	even	5
693.2.m.j.379.5		20	11.4	even	5
2541.2.a.bq.1.2		10	3.2	odd	2
2541.2.a.br.1.9		10	33.32	even	2
7623.2.a.cx.1.9		10	1.1	even	1	trivial
7623.2.a.cy.1.2		10	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)

\(f(q)\)	\(=\)	\(q+2.39396 q^{2} +3.73106 q^{4} +3.93829 q^{5} -1.00000 q^{7} +4.14409 q^{8} +O(q^{10})\)	\(q+2.39396 q^{2} +3.73106 q^{4} +3.93829 q^{5} -1.00000 q^{7} +4.14409 q^{8} +9.42812 q^{10} -2.99890 q^{13} -2.39396 q^{14} +2.45868 q^{16} +6.60457 q^{17} -5.90310 q^{19} +14.6940 q^{20} +6.02551 q^{23} +10.5101 q^{25} -7.17925 q^{26} -3.73106 q^{28} +1.52075 q^{29} +8.46902 q^{31} -2.40219 q^{32} +15.8111 q^{34} -3.93829 q^{35} -0.607840 q^{37} -14.1318 q^{38} +16.3206 q^{40} +1.70333 q^{41} -3.23364 q^{43} +14.4249 q^{46} +4.80237 q^{47} +1.00000 q^{49} +25.1609 q^{50} -11.1891 q^{52} +6.12834 q^{53} -4.14409 q^{56} +3.64062 q^{58} -6.23883 q^{59} -2.08899 q^{61} +20.2745 q^{62} -10.6681 q^{64} -11.8105 q^{65} -0.599719 q^{67} +24.6420 q^{68} -9.42812 q^{70} -1.40968 q^{71} +7.08851 q^{73} -1.45515 q^{74} -22.0248 q^{76} -1.02327 q^{79} +9.68299 q^{80} +4.07770 q^{82} -3.08729 q^{83} +26.0107 q^{85} -7.74120 q^{86} +2.48531 q^{89} +2.99890 q^{91} +22.4815 q^{92} +11.4967 q^{94} -23.2481 q^{95} +2.55296 q^{97} +2.39396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8	\( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37} - 8 q^{38} + 5 q^{40} + 19 q^{41} + 6 q^{43} + q^{46} - 15 q^{47} + 10 q^{49} - q^{50} + 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} + 11 q^{62} + 53 q^{64} - 29 q^{65} + 38 q^{67} + 87 q^{68} - 6 q^{70} - 26 q^{71} + q^{73} - 39 q^{74} + 2 q^{76} - 5 q^{79} - 6 q^{80} + 5 q^{82} + 6 q^{83} + q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} - 42 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^13 + 38 * q^16 + 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 - 18 * q^28 - 14 * q^29 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 24 * q^37 - 8 * q^38 + 5 * q^40 + 19 * q^41 + 6 * q^43 + q^46 - 15 * q^47 + 10 * q^49 - q^50 + 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 + 11 * q^62 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 - 26 * q^71 + q^73 - 39 * q^74 + 2 * q^76 - 5 * q^79 - 6 * q^80 + 5 * q^82 + 6 * q^83 + q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 - 42 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more