LMFDB - Embedded newform 7623.2.a.cr.1.4

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:	$6$
Coefficient field:	6.6.3829849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 15x^{3} - 5x^{2} - 3x + 1 $ x^6 - 2x^5 - 6x^4 + 15x^3 - 5x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.725011$ 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)

$f(q)$	$=$	$q+0.297484 q^{2} -1.91150 q^{4} -3.06404 q^{5} +1.00000 q^{7} -1.16361 q^{8} +O(q^{10})$	$q+0.297484 q^{2} -1.91150 q^{4} -3.06404 q^{5} +1.00000 q^{7} -1.16361 q^{8} -0.911503 q^{10} +5.38835 q^{13} +0.297484 q^{14} +3.47685 q^{16} +6.72305 q^{17} -0.434652 q^{19} +5.85693 q^{20} +2.92219 q^{23} +4.38835 q^{25} +1.60295 q^{26} -1.91150 q^{28} -6.86491 q^{29} +4.95370 q^{31} +3.36153 q^{32} +2.00000 q^{34} -3.06404 q^{35} -3.38835 q^{37} -0.129302 q^{38} +3.56535 q^{40} -9.92895 q^{41} -1.82301 q^{43} +0.869304 q^{46} +7.45988 q^{47} +1.00000 q^{49} +1.30547 q^{50} -10.2999 q^{52} -9.33398 q^{53} -1.16361 q^{56} -2.04220 q^{58} -7.17616 q^{59} -6.77671 q^{61} +1.47365 q^{62} -5.95370 q^{64} -16.5101 q^{65} +3.56535 q^{67} -12.8511 q^{68} -0.911503 q^{70} +8.50796 q^{71} +2.61165 q^{73} -1.00798 q^{74} +0.830838 q^{76} +11.6460 q^{79} -10.6532 q^{80} -2.95370 q^{82} +1.73225 q^{83} -20.5997 q^{85} -0.542315 q^{86} +7.31802 q^{89} +5.38835 q^{91} -5.58577 q^{92} +2.21919 q^{94} +1.33179 q^{95} -6.95370 q^{97} +0.297484 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 6 q^{7}+O(q^{10}) $ 6 * q + 4 * q^4 + 6 * q^7	$ 6 q + 4 q^{4} + 6 q^{7} + 10 q^{10} + 4 q^{13} + 8 q^{16} - 2 q^{25} + 4 q^{28} + 4 q^{31} + 12 q^{34} + 8 q^{37} + 24 q^{40} + 20 q^{43} + 6 q^{49} - 18 q^{52} - 2 q^{58} + 16 q^{61} - 10 q^{64} + 24 q^{67} + 10 q^{70} + 44 q^{73} + 54 q^{76} + 8 q^{79} + 8 q^{82} - 36 q^{85} + 4 q^{91} + 34 q^{94} - 16 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 + 6 * q^7 + 10 * q^10 + 4 * q^13 + 8 * q^16 - 2 * q^25 + 4 * q^28 + 4 * q^31 + 12 * q^34 + 8 * q^37 + 24 * q^40 + 20 * q^43 + 6 * q^49 - 18 * q^52 - 2 * q^58 + 16 * q^61 - 10 * q^64 + 24 * q^67 + 10 * q^70 + 44 * q^73 + 54 * q^76 + 8 * q^79 + 8 * q^82 - 36 * q^85 + 4 * q^91 + 34 * q^94 - 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.297484	0.210353	0.105176	−	0.994454i	$-0.466459\pi$
$2$	0.297484	0.210353	0.105176	+	0.994454i	$0.466459\pi$
$3$	0	0
$3$	0	0
$4$	−1.91150	−0.955752
$5$	−3.06404	−1.37028	−0.685141	−	0.728411i	$-0.740259\pi$
$5$	−3.06404	−1.37028	−0.685141	+	0.728411i	$0.740259\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.16361	−0.411398
$9$	0	0
$10$	−0.911503	−0.288243
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.38835	1.49446	0.747230	−	0.664565i	$-0.231383\pi$
$13$	5.38835	1.49446	0.747230	+	0.664565i	$0.231383\pi$
$14$	0.297484	0.0795059
$15$	0	0
$16$	3.47685	0.869213
$17$	6.72305	1.63058	0.815290	−	0.579053i	$-0.196578\pi$
$17$	6.72305	1.63058	0.815290	+	0.579053i	$0.196578\pi$
$18$	0	0
$19$	−0.434652	−0.0997160	−0.0498580	−	0.998756i	$-0.515877\pi$
$19$	−0.434652	−0.0997160	−0.0498580	+	0.998756i	$0.515877\pi$
$20$	5.85693	1.30965
$21$	0	0
$22$	0	0
$23$	2.92219	0.609318	0.304659	−	0.952461i	$-0.401457\pi$
$23$	2.92219	0.609318	0.304659	+	0.952461i	$0.401457\pi$
$24$	0	0
$25$	4.38835	0.877671
$26$	1.60295	0.314364
$27$	0	0
$28$	−1.91150	−0.361240
$29$	−6.86491	−1.27478	−0.637391	−	0.770541i	$-0.719986\pi$
$29$	−6.86491	−1.27478	−0.637391	+	0.770541i	$0.719986\pi$
$30$	0	0
$31$	4.95370	0.889711	0.444856	−	0.895602i	$-0.353255\pi$
$31$	4.95370	0.889711	0.444856	+	0.895602i	$0.353255\pi$
$32$	3.36153	0.594239
$33$	0	0
$34$	2.00000	0.342997
$35$	−3.06404	−0.517918
$36$	0	0
$37$	−3.38835	−0.557042	−0.278521	−	0.960430i	$-0.589844\pi$
$37$	−3.38835	−0.557042	−0.278521	+	0.960430i	$0.589844\pi$
$38$	−0.129302	−0.0209755
$39$	0	0
$40$	3.56535	0.563731
$41$	−9.92895	−1.55064	−0.775321	−	0.631568i	$-0.782412\pi$
$41$	−9.92895	−1.55064	−0.775321	+	0.631568i	$0.782412\pi$
$42$	0	0
$43$	−1.82301	−0.278006	−0.139003	−	0.990292i	$-0.544390\pi$
$43$	−1.82301	−0.278006	−0.139003	+	0.990292i	$0.544390\pi$
$44$	0	0
$45$	0	0
$46$	0.869304	0.128172
$47$	7.45988	1.08813	0.544067	−	0.839042i	$-0.316884\pi$
$47$	7.45988	1.08813	0.544067	+	0.839042i	$0.316884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.30547	0.184621
$51$	0	0
$52$	−10.2999	−1.42833
$53$	−9.33398	−1.28212	−0.641061	−	0.767490i	$-0.721505\pi$
$53$	−9.33398	−1.28212	−0.641061	+	0.767490i	$0.721505\pi$
$54$	0	0
$55$	0	0
$56$	−1.16361	−0.155494
$57$	0	0
$58$	−2.04220	−0.268154
$59$	−7.17616	−0.934257	−0.467129	−	0.884189i	$-0.654711\pi$
$59$	−7.17616	−0.934257	−0.467129	+	0.884189i	$0.654711\pi$
$60$	0	0
$61$	−6.77671	−0.867669	−0.433834	−	0.900993i	$-0.642840\pi$
$61$	−6.77671	−0.867669	−0.433834	+	0.900993i	$0.642840\pi$
$62$	1.47365	0.187153
$63$	0	0
$64$	−5.95370	−0.744213
$65$	−16.5101	−2.04783
$66$	0	0
$67$	3.56535	0.435577	0.217788	−	0.975996i	$-0.430116\pi$
$67$	3.56535	0.435577	0.217788	+	0.975996i	$0.430116\pi$
$68$	−12.8511	−1.55843
$69$	0	0
$70$	−0.911503	−0.108945
$71$	8.50796	1.00971	0.504854	−	0.863205i	$-0.331546\pi$
$71$	8.50796	1.00971	0.504854	+	0.863205i	$0.331546\pi$
$72$	0	0
$73$	2.61165	0.305670	0.152835	−	0.988252i	$-0.451160\pi$
$73$	2.61165	0.305670	0.152835	+	0.988252i	$0.451160\pi$
$74$	−1.00798	−0.117175
$75$	0	0
$76$	0.830838	0.0953037
$77$	0	0
$78$	0	0
$79$	11.6460	1.31028	0.655139	−	0.755508i	$-0.272610\pi$
$79$	11.6460	1.31028	0.655139	+	0.755508i	$0.272610\pi$
$80$	−10.6532	−1.19107
$81$	0	0
$82$	−2.95370	−0.326182
$83$	1.73225	0.190139	0.0950696	−	0.995471i	$-0.469693\pi$
$83$	1.73225	0.190139	0.0950696	+	0.995471i	$0.469693\pi$
$84$	0	0
$85$	−20.5997	−2.23435
$86$	−0.542315	−0.0584793
$87$	0	0
$88$	0	0
$89$	7.31802	0.775709	0.387854	−	0.921721i	$-0.373216\pi$
$89$	7.31802	0.775709	0.387854	+	0.921721i	$0.373216\pi$
$90$	0	0
$91$	5.38835	0.564853
$92$	−5.58577	−0.582357
$93$	0	0
$94$	2.21919	0.228892
$95$	1.33179	0.136639
$96$	0	0
$97$	−6.95370	−0.706042	−0.353021	−	0.935615i	$-0.614846\pi$
$97$	−6.95370	−0.706042	−0.353021	+	0.935615i	$0.614846\pi$
$98$	0.297484	0.0300504
$99$	0	0
$100$	−8.38835	−0.838835
$101$	−7.91299	−0.787372	−0.393686	−	0.919245i	$-0.628800\pi$
$101$	−7.91299	−0.787372	−0.393686	+	0.919245i	$0.628800\pi$
$102$	0	0
$103$	18.4227	1.81524	0.907622	−	0.419788i	$-0.137895\pi$
$103$	18.4227	1.81524	0.907622	+	0.419788i	$0.137895\pi$
$104$	−6.26994	−0.614818
$105$	0	0
$106$	−2.77671	−0.269698
$107$	−13.3042	−1.28617	−0.643085	−	0.765795i	$-0.722346\pi$
$107$	−13.3042	−1.28617	−0.643085	+	0.765795i	$0.722346\pi$
$108$	0	0
$109$	−0.953703	−0.0913482	−0.0456741	−	0.998956i	$-0.514544\pi$
$109$	−0.953703	−0.0913482	−0.0456741	+	0.998956i	$0.514544\pi$
$110$	0	0
$111$	0	0
$112$	3.47685	0.328532
$113$	−14.9198	−1.40353	−0.701766	−	0.712407i	$-0.747605\pi$
$113$	−14.9198	−1.40353	−0.701766	+	0.712407i	$0.747605\pi$
$114$	0	0
$115$	−8.95370	−0.834937
$116$	13.1223	1.21837
$117$	0	0
$118$	−2.13479	−0.196524
$119$	6.72305	0.616301
$120$	0	0
$121$	0	0
$122$	−2.01596	−0.182517
$123$	0	0
$124$	−9.46902	−0.850343
$125$	1.87411	0.167625
$126$	0	0
$127$	19.7304	1.75079	0.875396	−	0.483407i	$-0.160601\pi$
$127$	19.7304	1.75079	0.875396	+	0.483407i	$0.160601\pi$
$128$	−8.49418	−0.750787
$129$	0	0
$130$	−4.91150	−0.430767
$131$	7.31802	0.639378	0.319689	−	0.947522i	$-0.396422\pi$
$131$	7.31802	0.639378	0.319689	+	0.947522i	$0.396422\pi$
$132$	0	0
$133$	−0.434652	−0.0376891
$134$	1.06063	0.0916248
$135$	0	0
$136$	−7.82301	−0.670817
$137$	17.5582	1.50010	0.750050	−	0.661381i	$-0.230029\pi$
$137$	17.5582	1.50010	0.750050	+	0.661381i	$0.230029\pi$
$138$	0	0
$139$	19.6460	1.66635	0.833177	−	0.553007i	$-0.186520\pi$
$139$	19.6460	1.66635	0.833177	+	0.553007i	$0.186520\pi$
$140$	5.85693	0.495001
$141$	0	0
$142$	2.53098	0.212395
$143$	0	0
$144$	0	0
$145$	21.0344	1.74681
$146$	0.776922	0.0642986
$147$	0	0
$148$	6.47685	0.532394
$149$	−11.5193	−0.943702	−0.471851	−	0.881678i	$-0.656414\pi$
$149$	−11.5193	−0.943702	−0.471851	+	0.881678i	$0.656414\pi$
$150$	0	0
$151$	−20.5997	−1.67638	−0.838191	−	0.545377i	$-0.816386\pi$
$151$	−20.5997	−1.67638	−0.838191	+	0.545377i	$0.816386\pi$
$152$	0.505765	0.0410229
$153$	0	0
$154$	0	0
$155$	−15.1784	−1.21915
$156$	0	0
$157$	−5.13070	−0.409474	−0.204737	−	0.978817i	$-0.565634\pi$
$157$	−5.13070	−0.409474	−0.204737	+	0.978817i	$0.565634\pi$
$158$	3.46450	0.275621
$159$	0	0
$160$	−10.2999	−0.814275
$161$	2.92219	0.230301
$162$	0	0
$163$	3.56535	0.279260	0.139630	−	0.990204i	$-0.455409\pi$
$163$	3.56535	0.279260	0.139630	+	0.990204i	$0.455409\pi$
$164$	18.9792	1.48203
$165$	0	0
$166$	0.515317	0.0399963
$167$	21.0227	1.62679	0.813394	−	0.581713i	$-0.197617\pi$
$167$	21.0227	1.62679	0.813394	+	0.581713i	$0.197617\pi$
$168$	0	0
$169$	16.0344	1.23341
$170$	−6.12808	−0.470003
$171$	0	0
$172$	3.48468	0.265705
$173$	3.80087	0.288974	0.144487	−	0.989507i	$-0.453847\pi$
$173$	3.80087	0.288974	0.144487	+	0.989507i	$0.453847\pi$
$174$	0	0
$175$	4.38835	0.331728
$176$	0	0
$177$	0	0
$178$	2.17699	0.163173
$179$	24.5123	1.83214	0.916069	−	0.401021i	$-0.131344\pi$
$179$	24.5123	1.83214	0.916069	+	0.401021i	$0.131344\pi$
$180$	0	0
$181$	−5.04630	−0.375088	−0.187544	−	0.982256i	$-0.560053\pi$
$181$	−5.04630	−0.375088	−0.187544	+	0.982256i	$0.560053\pi$
$182$	1.60295	0.118818
$183$	0	0
$184$	−3.40028	−0.250672
$185$	10.3821	0.763304
$186$	0	0
$187$	0	0
$188$	−14.2596	−1.03999
$189$	0	0
$190$	0.396187	0.0287424
$191$	0.826027	0.0597692	0.0298846	−	0.999553i	$-0.490486\pi$
$191$	0.826027	0.0597692	0.0298846	+	0.999553i	$0.490486\pi$
$192$	0	0
$193$	7.04630	0.507204	0.253602	−	0.967309i	$-0.418385\pi$
$193$	7.04630	0.507204	0.253602	+	0.967309i	$0.418385\pi$
$194$	−2.06861	−0.148518
$195$	0	0
$196$	−1.91150	−0.136536
$197$	−14.5834	−1.03902	−0.519512	−	0.854463i	$-0.673886\pi$
$197$	−14.5834	−1.03902	−0.519512	+	0.854463i	$0.673886\pi$
$198$	0	0
$199$	−15.7304	−1.11510	−0.557550	−	0.830144i	$-0.688258\pi$
$199$	−15.7304	−1.11510	−0.557550	+	0.830144i	$0.688258\pi$
$200$	−5.10633	−0.361072
$201$	0	0
$202$	−2.35399	−0.165626
$203$	−6.86491	−0.481822
$204$	0	0
$205$	30.4227	2.12482
$206$	5.48046	0.381842
$207$	0	0
$208$	18.7345	1.29900
$209$	0	0
$210$	0	0
$211$	−13.5534	−0.933056	−0.466528	−	0.884506i	$-0.654495\pi$
$211$	−13.5534	−0.933056	−0.466528	+	0.884506i	$0.654495\pi$
$212$	17.8419	1.22539
$213$	0	0
$214$	−3.95780	−0.270550
$215$	5.58577	0.380946
$216$	0	0
$217$	4.95370	0.336279
$218$	−0.283711	−0.0192154
$219$	0	0
$220$	0	0
$221$	36.2262	2.43684
$222$	0	0
$223$	−15.6460	−1.04773	−0.523867	−	0.851800i	$-0.675511\pi$
$223$	−15.6460	−1.04773	−0.523867	+	0.851800i	$0.675511\pi$
$224$	3.36153	0.224601
$225$	0	0
$226$	−4.43839	−0.295237
$227$	−0.906224	−0.0601482	−0.0300741	−	0.999548i	$-0.509574\pi$
$227$	−0.906224	−0.0601482	−0.0300741	+	0.999548i	$0.509574\pi$
$228$	0	0
$229$	15.4690	1.02222	0.511111	−	0.859515i	$-0.329234\pi$
$229$	15.4690	1.02222	0.511111	+	0.859515i	$0.329234\pi$
$230$	−2.66358	−0.175631
$231$	0	0
$232$	7.98807	0.524443
$233$	17.5307	1.14847	0.574237	−	0.818689i	$-0.305299\pi$
$233$	17.5307	1.14847	0.574237	+	0.818689i	$0.305299\pi$
$234$	0	0
$235$	−22.8574	−1.49105
$236$	13.7173	0.892918
$237$	0	0
$238$	2.00000	0.129641
$239$	2.23802	0.144765	0.0723826	−	0.997377i	$-0.476940\pi$
$239$	2.23802	0.144765	0.0723826	+	0.997377i	$0.476940\pi$
$240$	0	0
$241$	17.0344	1.09728	0.548640	−	0.836059i	$-0.315146\pi$
$241$	17.0344	1.09728	0.548640	+	0.836059i	$0.315146\pi$
$242$	0	0
$243$	0	0
$244$	12.9537	0.829276
$245$	−3.06404	−0.195754
$246$	0	0
$247$	−2.34206	−0.149022
$248$	−5.76418	−0.366025
$249$	0	0
$250$	0.557517	0.0352604
$251$	19.4323	1.22656	0.613279	−	0.789866i	$-0.289850\pi$
$251$	19.4323	1.22656	0.613279	+	0.789866i	$0.289850\pi$
$252$	0	0
$253$	0	0
$254$	5.86948	0.368284
$255$	0	0
$256$	9.38052	0.586283
$257$	0.400459	0.0249800	0.0124900	−	0.999922i	$-0.496024\pi$
$257$	0.400459	0.0249800	0.0124900	+	0.999922i	$0.496024\pi$
$258$	0	0
$259$	−3.38835	−0.210542
$260$	31.5592	1.95722
$261$	0	0
$262$	2.17699	0.134495
$263$	5.98623	0.369127	0.184563	−	0.982821i	$-0.440913\pi$
$263$	5.98623	0.369127	0.184563	+	0.982821i	$0.440913\pi$
$264$	0	0
$265$	28.5997	1.75687
$266$	−0.129302	−0.00792801
$267$	0	0
$268$	−6.81517	−0.416303
$269$	−11.3499	−0.692018	−0.346009	−	0.938231i	$-0.612463\pi$
$269$	−11.3499	−0.692018	−0.346009	+	0.938231i	$0.612463\pi$
$270$	0	0
$271$	23.5653	1.43149	0.715746	−	0.698360i	$-0.246087\pi$
$271$	23.5653	1.43149	0.715746	+	0.698360i	$0.246087\pi$
$272$	23.3751	1.41732
$273$	0	0
$274$	5.22329	0.315551
$275$	0	0
$276$	0	0
$277$	15.1307	0.909115	0.454558	−	0.890717i	$-0.349797\pi$
$277$	15.1307	0.909115	0.454558	+	0.890717i	$0.349797\pi$
$278$	5.84437	0.350522
$279$	0	0
$280$	3.56535	0.213070
$281$	−26.1554	−1.56030	−0.780150	−	0.625593i	$-0.784857\pi$
$281$	−26.1554	−1.56030	−0.780150	+	0.625593i	$0.784857\pi$
$282$	0	0
$283$	−20.0807	−1.19367	−0.596836	−	0.802363i	$-0.703576\pi$
$283$	−20.0807	−1.19367	−0.596836	+	0.802363i	$0.703576\pi$
$284$	−16.2630	−0.965031
$285$	0	0
$286$	0	0
$287$	−9.92895	−0.586087
$288$	0	0
$289$	28.1994	1.65879
$290$	6.25739	0.367446
$291$	0	0
$292$	−4.99217	−0.292145
$293$	−19.5215	−1.14046	−0.570230	−	0.821485i	$-0.693146\pi$
$293$	−19.5215	−1.14046	−0.570230	+	0.821485i	$0.693146\pi$
$294$	0	0
$295$	21.9881	1.28020
$296$	3.94272	0.229166
$297$	0	0
$298$	−3.42682	−0.198510
$299$	15.7458	0.910602
$300$	0	0
$301$	−1.82301	−0.105076
$302$	−6.12808	−0.352632
$303$	0	0
$304$	−1.51122	−0.0866744
$305$	20.7641	1.18895
$306$	0	0
$307$	23.2920	1.32935	0.664673	−	0.747134i	$-0.268571\pi$
$307$	23.2920	1.32935	0.664673	+	0.747134i	$0.268571\pi$
$308$	0	0
$309$	0	0
$310$	−4.51532	−0.256453
$311$	−8.79167	−0.498530	−0.249265	−	0.968435i	$-0.580189\pi$
$311$	−8.79167	−0.498530	−0.249265	+	0.968435i	$0.580189\pi$
$312$	0	0
$313$	25.3764	1.43436	0.717180	−	0.696888i	$-0.245432\pi$
$313$	25.3764	1.43436	0.717180	+	0.696888i	$0.245432\pi$
$314$	−1.52630	−0.0861341
$315$	0	0
$316$	−22.2614	−1.25230
$317$	0.258604	0.0145246	0.00726232	−	0.999974i	$-0.497688\pi$
$317$	0.258604	0.0145246	0.00726232	+	0.999974i	$0.497688\pi$
$318$	0	0
$319$	0	0
$320$	18.2424	1.01978
$321$	0	0
$322$	0.869304	0.0484444
$323$	−2.92219	−0.162595
$324$	0	0
$325$	23.6460	1.31164
$326$	1.06063	0.0587431
$327$	0	0
$328$	11.5534	0.637931
$329$	7.45988	0.411276
$330$	0	0
$331$	−27.2920	−1.50011	−0.750053	−	0.661378i	$-0.769972\pi$
$331$	−27.2920	−1.50011	−0.750053	+	0.661378i	$0.769972\pi$
$332$	−3.31120	−0.181726
$333$	0	0
$334$	6.25392	0.342199
$335$	−10.9244	−0.596863
$336$	0	0
$337$	22.3383	1.21685	0.608423	−	0.793613i	$-0.291802\pi$
$337$	22.3383	1.21685	0.608423	+	0.793613i	$0.291802\pi$
$338$	4.76997	0.259452
$339$	0	0
$340$	39.3764	2.13549
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.12127	0.114371
$345$	0	0
$346$	1.13070	0.0607866
$347$	−35.6839	−1.91561	−0.957805	−	0.287417i	$-0.907203\pi$
$347$	−35.6839	−1.91561	−0.957805	+	0.287417i	$0.907203\pi$
$348$	0	0
$349$	9.03437	0.483599	0.241799	−	0.970326i	$-0.422262\pi$
$349$	9.03437	0.483599	0.241799	+	0.970326i	$0.422262\pi$
$350$	1.30547	0.0697800
$351$	0	0
$352$	0	0
$353$	−36.0843	−1.92058	−0.960288	−	0.279012i	$-0.909993\pi$
$353$	−36.0843	−1.92058	−0.960288	+	0.279012i	$0.909993\pi$
$354$	0	0
$355$	−26.0687	−1.38358
$356$	−13.9884	−0.741385
$357$	0	0
$358$	7.29203	0.385396
$359$	23.3224	1.23091	0.615455	−	0.788172i	$-0.288972\pi$
$359$	23.3224	1.23091	0.615455	+	0.788172i	$0.288972\pi$
$360$	0	0
$361$	−18.8111	−0.990057
$362$	−1.50119	−0.0789009
$363$	0	0
$364$	−10.2999	−0.539859
$365$	−8.00219	−0.418854
$366$	0	0
$367$	−21.1994	−1.10660	−0.553301	−	0.832982i	$-0.686632\pi$
$367$	−21.1994	−1.10660	−0.553301	+	0.832982i	$0.686632\pi$
$368$	10.1600	0.529627
$369$	0	0
$370$	3.08850	0.160563
$371$	−9.33398	−0.484596
$372$	0	0
$373$	9.82301	0.508616	0.254308	−	0.967123i	$-0.418152\pi$
$373$	9.82301	0.508616	0.254308	+	0.967123i	$0.418152\pi$
$374$	0	0
$375$	0	0
$376$	−8.68038	−0.447656
$377$	−36.9906	−1.90511
$378$	0	0
$379$	24.0807	1.23694	0.618470	−	0.785808i	$-0.287753\pi$
$379$	24.0807	1.23694	0.618470	+	0.785808i	$0.287753\pi$
$380$	−2.54572	−0.130593
$381$	0	0
$382$	0.245730	0.0125726
$383$	−32.7366	−1.67276	−0.836381	−	0.548149i	$-0.815333\pi$
$383$	−32.7366	−1.67276	−0.836381	+	0.548149i	$0.815333\pi$
$384$	0	0
$385$	0	0
$386$	2.09616	0.106692
$387$	0	0
$388$	13.2920	0.674800
$389$	6.69551	0.339476	0.169738	−	0.985489i	$-0.445708\pi$
$389$	6.69551	0.339476	0.169738	+	0.985489i	$0.445708\pi$
$390$	0	0
$391$	19.6460	0.993542
$392$	−1.16361	−0.0587711
$393$	0	0
$394$	−4.33832	−0.218562
$395$	−35.6839	−1.79545
$396$	0	0
$397$	29.0224	1.45659	0.728297	−	0.685261i	$-0.240312\pi$
$397$	29.0224	1.45659	0.728297	+	0.685261i	$0.240312\pi$
$398$	−4.67954	−0.234564
$399$	0	0
$400$	15.2577	0.762883
$401$	32.4780	1.62187	0.810936	−	0.585134i	$-0.198958\pi$
$401$	32.4780	1.62187	0.810936	+	0.585134i	$0.198958\pi$
$402$	0	0
$403$	26.6923	1.32964
$404$	15.1257	0.752532
$405$	0	0
$406$	−2.04220	−0.101353
$407$	0	0
$408$	0	0
$409$	−10.4227	−0.515370	−0.257685	−	0.966229i	$-0.582960\pi$
$409$	−10.4227	−0.515370	−0.257685	+	0.966229i	$0.582960\pi$
$410$	9.05027	0.446961
$411$	0	0
$412$	−35.2151	−1.73492
$413$	−7.17616	−0.353116
$414$	0	0
$415$	−5.30769	−0.260544
$416$	18.1131	0.888068
$417$	0	0
$418$	0	0
$419$	−28.5077	−1.39269	−0.696346	−	0.717706i	$-0.745192\pi$
$419$	−28.5077	−1.39269	−0.696346	+	0.717706i	$0.745192\pi$
$420$	0	0
$421$	−6.16506	−0.300467	−0.150233	−	0.988651i	$-0.548003\pi$
$421$	−6.16506	−0.300467	−0.150233	+	0.988651i	$0.548003\pi$
$422$	−4.03192	−0.196271
$423$	0	0
$424$	10.8611	0.527462
$425$	29.5031	1.43111
$426$	0	0
$427$	−6.77671	−0.327948
$428$	25.4311	1.22926
$429$	0	0
$430$	1.66168	0.0801332
$431$	26.4666	1.27485	0.637427	−	0.770511i	$-0.279999\pi$
$431$	26.4666	1.27485	0.637427	+	0.770511i	$0.279999\pi$
$432$	0	0
$433$	−18.8693	−0.906801	−0.453400	−	0.891307i	$-0.649789\pi$
$433$	−18.8693	−0.906801	−0.453400	+	0.891307i	$0.649789\pi$
$434$	1.47365	0.0707373
$435$	0	0
$436$	1.82301	0.0873062
$437$	−1.27013	−0.0607587
$438$	0	0
$439$	14.8574	0.709104	0.354552	−	0.935036i	$-0.384633\pi$
$439$	14.8574	0.709104	0.354552	+	0.935036i	$0.384633\pi$
$440$	0	0
$441$	0	0
$442$	10.7767	0.512596
$443$	−20.7641	−0.986533	−0.493267	−	0.869878i	$-0.664197\pi$
$443$	−20.7641	−0.986533	−0.493267	+	0.869878i	$0.664197\pi$
$444$	0	0
$445$	−22.4227	−1.06294
$446$	−4.65444	−0.220394
$447$	0	0
$448$	−5.95370	−0.281286
$449$	33.5877	1.58510	0.792551	−	0.609805i	$-0.208752\pi$
$449$	33.5877	1.58510	0.792551	+	0.609805i	$0.208752\pi$
$450$	0	0
$451$	0	0
$452$	28.5192	1.34143
$453$	0	0
$454$	−0.269587	−0.0126524
$455$	−16.5101	−0.774008
$456$	0	0
$457$	30.5071	1.42706	0.713532	−	0.700623i	$-0.247095\pi$
$457$	30.5071	1.42706	0.713532	+	0.700623i	$0.247095\pi$
$458$	4.60178	0.215027
$459$	0	0
$460$	17.1150	0.797993
$461$	10.5515	0.491431	0.245715	−	0.969342i	$-0.420977\pi$
$461$	10.5515	0.491431	0.245715	+	0.969342i	$0.420977\pi$
$462$	0	0
$463$	−14.6960	−0.682983	−0.341492	−	0.939885i	$-0.610932\pi$
$463$	−14.6960	−0.682983	−0.341492	+	0.939885i	$0.610932\pi$
$464$	−23.8683	−1.10806
$465$	0	0
$466$	5.21510	0.241585
$467$	9.83975	0.455329	0.227665	−	0.973740i	$-0.426891\pi$
$467$	9.83975	0.455329	0.227665	+	0.973740i	$0.426891\pi$
$468$	0	0
$469$	3.56535	0.164632
$470$	−6.79970	−0.313647
$471$	0	0
$472$	8.35025	0.384352
$473$	0	0
$474$	0	0
$475$	−1.90741	−0.0875178
$476$	−12.8511	−0.589031
$477$	0	0
$478$	0.665774	0.0304518
$479$	13.8100	0.630996	0.315498	−	0.948926i	$-0.397829\pi$
$479$	13.8100	0.630996	0.315498	+	0.948926i	$0.397829\pi$
$480$	0	0
$481$	−18.2577	−0.832478
$482$	5.06745	0.230816
$483$	0	0
$484$	0	0
$485$	21.3064	0.967476
$486$	0	0
$487$	−6.26139	−0.283731	−0.141865	−	0.989886i	$-0.545310\pi$
$487$	−6.26139	−0.283731	−0.141865	+	0.989886i	$0.545310\pi$
$488$	7.88544	0.356957
$489$	0	0
$490$	−0.911503	−0.0411775
$491$	−13.8717	−0.626020	−0.313010	−	0.949750i	$-0.601337\pi$
$491$	−13.8717	−0.626020	−0.313010	+	0.949750i	$0.601337\pi$
$492$	0	0
$493$	−46.1531	−2.07863
$494$	−0.696725	−0.0313471
$495$	0	0
$496$	17.2233	0.773349
$497$	8.50796	0.381634
$498$	0	0
$499$	18.6960	0.836950	0.418475	−	0.908228i	$-0.362565\pi$
$499$	18.6960	0.836950	0.418475	+	0.908228i	$0.362565\pi$
$500$	−3.58236	−0.160208
$501$	0	0
$502$	5.78081	0.258010
$503$	11.4301	0.509645	0.254822	−	0.966988i	$-0.417983\pi$
$503$	11.4301	0.509645	0.254822	+	0.966988i	$0.417983\pi$
$504$	0	0
$505$	24.2457	1.07892
$506$	0	0
$507$	0	0
$508$	−37.7147	−1.67332
$509$	14.8144	0.656639	0.328319	−	0.944567i	$-0.393518\pi$
$509$	14.8144	0.656639	0.328319	+	0.944567i	$0.393518\pi$
$510$	0	0
$511$	2.61165	0.115532
$512$	19.7789	0.874113
$513$	0	0
$514$	0.119130	0.00525461
$515$	−56.4480	−2.48740
$516$	0	0
$517$	0	0
$518$	−1.00798	−0.0442881
$519$	0	0
$520$	19.2114	0.842474
$521$	−19.3521	−0.847832	−0.423916	−	0.905701i	$-0.639345\pi$
$521$	−19.3521	−0.847832	−0.423916	+	0.905701i	$0.639345\pi$
$522$	0	0
$523$	23.7267	1.03750	0.518748	−	0.854927i	$-0.326398\pi$
$523$	23.7267	1.03750	0.518748	+	0.854927i	$0.326398\pi$
$524$	−13.9884	−0.611087
$525$	0	0
$526$	1.78081	0.0776469
$527$	33.3040	1.45075
$528$	0	0
$529$	−14.4608	−0.628732
$530$	8.50796	0.369562
$531$	0	0
$532$	0.830838	0.0360214
$533$	−53.5007	−2.31737
$534$	0	0
$535$	40.7648	1.76242
$536$	−4.14867	−0.179195
$537$	0	0
$538$	−3.37643	−0.145568
$539$	0	0
$540$	0	0
$541$	20.5153	0.882022	0.441011	−	0.897502i	$-0.354620\pi$
$541$	20.5153	0.882022	0.441011	+	0.897502i	$0.354620\pi$
$542$	7.01031	0.301119
$543$	0	0
$544$	22.5997	0.968955
$545$	2.92219	0.125173
$546$	0	0
$547$	28.6841	1.22644	0.613222	−	0.789911i	$-0.289873\pi$
$547$	28.6841	1.22644	0.613222	+	0.789911i	$0.289873\pi$
$548$	−33.5626	−1.43372
$549$	0	0
$550$	0	0
$551$	2.98384	0.127116
$552$	0	0
$553$	11.6460	0.495239
$554$	4.50114	0.191235
$555$	0	0
$556$	−37.5534	−1.59262
$557$	−26.7228	−1.13228	−0.566141	−	0.824309i	$-0.691564\pi$
$557$	−26.7228	−1.13228	−0.566141	+	0.824309i	$0.691564\pi$
$558$	0	0
$559$	−9.82301	−0.415469
$560$	−10.6532	−0.450181
$561$	0	0
$562$	−7.78081	−0.328214
$563$	20.6839	0.871724	0.435862	−	0.900014i	$-0.356444\pi$
$563$	20.6839	0.871724	0.435862	+	0.900014i	$0.356444\pi$
$564$	0	0
$565$	45.7147	1.92323
$566$	−5.97368	−0.251092
$567$	0	0
$568$	−9.89994	−0.415392
$569$	7.54908	0.316474	0.158237	−	0.987401i	$-0.449419\pi$
$569$	7.54908	0.316474	0.158237	+	0.987401i	$0.449419\pi$
$570$	0	0
$571$	−26.7767	−1.12057	−0.560285	−	0.828300i	$-0.689308\pi$
$571$	−26.7767	−1.12057	−0.560285	+	0.828300i	$0.689308\pi$
$572$	0	0
$573$	0	0
$574$	−2.95370	−0.123285
$575$	12.8236	0.534781
$576$	0	0
$577$	−13.5616	−0.564577	−0.282289	−	0.959330i	$-0.591094\pi$
$577$	−13.5616	−0.564577	−0.282289	+	0.959330i	$0.591094\pi$
$578$	8.38888	0.348931
$579$	0	0
$580$	−40.2073	−1.66952
$581$	1.73225	0.0718659
$582$	0	0
$583$	0	0
$584$	−3.03893	−0.125752
$585$	0	0
$586$	−5.80734	−0.239899
$587$	−16.2515	−0.670773	−0.335386	−	0.942081i	$-0.608867\pi$
$587$	−16.2515	−0.670773	−0.335386	+	0.942081i	$0.608867\pi$
$588$	0	0
$589$	−2.15314	−0.0887184
$590$	6.54110	0.269293
$591$	0	0
$592$	−11.7808	−0.484188
$593$	−25.6496	−1.05330	−0.526652	−	0.850081i	$-0.676553\pi$
$593$	−25.6496	−1.05330	−0.526652	+	0.850081i	$0.676553\pi$
$594$	0	0
$595$	−20.5997	−0.844506
$596$	22.0193	0.901944
$597$	0	0
$598$	4.68412	0.191548
$599$	6.41180	0.261979	0.130989	−	0.991384i	$-0.458185\pi$
$599$	6.41180	0.261979	0.130989	+	0.991384i	$0.458185\pi$
$600$	0	0
$601$	20.5191	0.836990	0.418495	−	0.908219i	$-0.362558\pi$
$601$	20.5191	0.836990	0.418495	+	0.908219i	$0.362558\pi$
$602$	−0.542315	−0.0221031
$603$	0	0
$604$	39.3764	1.60220
$605$	0	0
$606$	0	0
$607$	−10.1733	−0.412920	−0.206460	−	0.978455i	$-0.566194\pi$
$607$	−10.1733	−0.412920	−0.206460	+	0.978455i	$0.566194\pi$
$608$	−1.46109	−0.0592552
$609$	0	0
$610$	6.17699	0.250099
$611$	40.1965	1.62617
$612$	0	0
$613$	−5.46902	−0.220892	−0.110446	−	0.993882i	$-0.535228\pi$
$613$	−5.46902	−0.220892	−0.110446	+	0.993882i	$0.535228\pi$
$614$	6.92900	0.279632
$615$	0	0
$616$	0	0
$617$	3.46450	0.139476	0.0697378	−	0.997565i	$-0.477784\pi$
$617$	3.46450	0.139476	0.0697378	+	0.997565i	$0.477784\pi$
$618$	0	0
$619$	8.08440	0.324939	0.162470	−	0.986714i	$-0.448054\pi$
$619$	8.08440	0.324939	0.162470	+	0.986714i	$0.448054\pi$
$620$	29.0135	1.16521
$621$	0	0
$622$	−2.61538	−0.104867
$623$	7.31802	0.293190
$624$	0	0
$625$	−27.6841	−1.10736
$626$	7.54908	0.301722
$627$	0	0
$628$	9.80734	0.391356
$629$	−22.7801	−0.908302
$630$	0	0
$631$	23.2920	0.927241	0.463620	−	0.886034i	$-0.346550\pi$
$631$	23.2920	0.927241	0.463620	+	0.886034i	$0.346550\pi$
$632$	−13.5514	−0.539046
$633$	0	0
$634$	0.0769305	0.00305530
$635$	−60.4548	−2.39908
$636$	0	0
$637$	5.38835	0.213494
$638$	0	0
$639$	0	0
$640$	26.0265	1.02879
$641$	36.2262	1.43085	0.715424	−	0.698690i	$-0.246233\pi$
$641$	36.2262	1.43085	0.715424	+	0.698690i	$0.246233\pi$
$642$	0	0
$643$	−27.5616	−1.08692	−0.543462	−	0.839434i	$-0.682887\pi$
$643$	−27.5616	−1.08692	−0.543462	+	0.839434i	$0.682887\pi$
$644$	−5.58577	−0.220110
$645$	0	0
$646$	−0.869304	−0.0342023
$647$	29.3086	1.15224	0.576121	−	0.817365i	$-0.304566\pi$
$647$	29.3086	1.15224	0.576121	+	0.817365i	$0.304566\pi$
$648$	0	0
$649$	0	0
$650$	7.03431	0.275908
$651$	0	0
$652$	−6.81517	−0.266903
$653$	39.1484	1.53199	0.765997	−	0.642844i	$-0.222246\pi$
$653$	39.1484	1.53199	0.765997	+	0.642844i	$0.222246\pi$
$654$	0	0
$655$	−22.4227	−0.876128
$656$	−34.5215	−1.34784
$657$	0	0
$658$	2.21919	0.0865132
$659$	1.33179	0.0518792	0.0259396	−	0.999664i	$-0.491742\pi$
$659$	1.33179	0.0518792	0.0259396	+	0.999664i	$0.491742\pi$
$660$	0	0
$661$	31.6378	1.23057	0.615284	−	0.788305i	$-0.289041\pi$
$661$	31.6378	1.23057	0.615284	+	0.788305i	$0.289041\pi$
$662$	−8.11894	−0.315552
$663$	0	0
$664$	−2.01566	−0.0782229
$665$	1.33179	0.0516447
$666$	0	0
$667$	−20.0605	−0.776747
$668$	−40.1850	−1.55480
$669$	0	0
$670$	−3.24983	−0.125552
$671$	0	0
$672$	0	0
$673$	32.0844	1.23676	0.618381	−	0.785878i	$-0.287789\pi$
$673$	32.0844	1.23676	0.618381	+	0.785878i	$0.287789\pi$
$674$	6.64529	0.255967
$675$	0	0
$676$	−30.6497	−1.17884
$677$	−15.1508	−0.582293	−0.291146	−	0.956678i	$-0.594037\pi$
$677$	−15.1508	−0.582293	−0.291146	+	0.956678i	$0.594037\pi$
$678$	0	0
$679$	−6.95370	−0.266859
$680$	23.9700	0.919208
$681$	0	0
$682$	0	0
$683$	30.0981	1.15167	0.575836	−	0.817565i	$-0.304677\pi$
$683$	30.0981	1.15167	0.575836	+	0.817565i	$0.304677\pi$
$684$	0	0
$685$	−53.7991	−2.05556
$686$	0.297484	0.0113580
$687$	0	0
$688$	−6.33832	−0.241646
$689$	−50.2948	−1.91608
$690$	0	0
$691$	25.1994	0.958632	0.479316	−	0.877643i	$-0.340885\pi$
$691$	25.1994	0.958632	0.479316	+	0.877643i	$0.340885\pi$
$692$	−7.26537	−0.276188
$693$	0	0
$694$	−10.6154	−0.402954
$695$	−60.1962	−2.28337
$696$	0	0
$697$	−66.7529	−2.52844
$698$	2.68758	0.101726
$699$	0	0
$700$	−8.38835	−0.317050
$701$	39.8235	1.50411	0.752057	−	0.659098i	$-0.229062\pi$
$701$	39.8235	1.50411	0.752057	+	0.659098i	$0.229062\pi$
$702$	0	0
$703$	1.47275	0.0555460
$704$	0	0
$705$	0	0
$706$	−10.7345	−0.403999
$707$	−7.91299	−0.297599
$708$	0	0
$709$	40.9418	1.53760	0.768800	−	0.639489i	$-0.220854\pi$
$709$	40.9418	1.53760	0.768800	+	0.639489i	$0.220854\pi$
$710$	−7.75503	−0.291041
$711$	0	0
$712$	−8.51532	−0.319125
$713$	14.4756	0.542117
$714$	0	0
$715$	0	0
$716$	−46.8554	−1.75107
$717$	0	0
$718$	6.93804	0.258925
$719$	3.14424	0.117260	0.0586302	−	0.998280i	$-0.481327\pi$
$719$	3.14424	0.117260	0.0586302	+	0.998280i	$0.481327\pi$
$720$	0	0
$721$	18.4227	0.686098
$722$	−5.59599	−0.208261
$723$	0	0
$724$	9.64601	0.358491
$725$	−30.1257	−1.11884
$726$	0	0
$727$	45.9843	1.70546	0.852732	−	0.522348i	$-0.174944\pi$
$727$	45.9843	1.70546	0.852732	+	0.522348i	$0.174944\pi$
$728$	−6.26994	−0.232379
$729$	0	0
$730$	−2.38052	−0.0881071
$731$	−12.2562	−0.453311
$732$	0	0
$733$	−14.7767	−0.545790	−0.272895	−	0.962044i	$-0.587981\pi$
$733$	−14.7767	−0.545790	−0.272895	+	0.962044i	$0.587981\pi$
$734$	−6.30649	−0.232777
$735$	0	0
$736$	9.82301	0.362081
$737$	0	0
$738$	0	0
$739$	51.3526	1.88903	0.944517	−	0.328461i	$-0.106530\pi$
$739$	51.3526	1.88903	0.944517	+	0.328461i	$0.106530\pi$
$740$	−19.8453	−0.729529
$741$	0	0
$742$	−2.77671	−0.101936
$743$	16.4299	0.602756	0.301378	−	0.953505i	$-0.402553\pi$
$743$	16.4299	0.602756	0.301378	+	0.953505i	$0.402553\pi$
$744$	0	0
$745$	35.2958	1.29314
$746$	2.92219	0.106989
$747$	0	0
$748$	0	0
$749$	−13.3042	−0.486127
$750$	0	0
$751$	28.9500	1.05640	0.528200	−	0.849120i	$-0.322867\pi$
$751$	28.9500	1.05640	0.528200	+	0.849120i	$0.322867\pi$
$752$	25.9369	0.945821
$753$	0	0
$754$	−11.0041	−0.400746
$755$	63.1184	2.29711
$756$	0	0
$757$	36.0725	1.31108	0.655538	−	0.755162i	$-0.272442\pi$
$757$	36.0725	1.31108	0.655538	+	0.755162i	$0.272442\pi$
$758$	7.16361	0.260194
$759$	0	0
$760$	−1.54968	−0.0562130
$761$	−34.8303	−1.26260	−0.631299	−	0.775540i	$-0.717478\pi$
$761$	−34.8303	−1.26260	−0.631299	+	0.775540i	$0.717478\pi$
$762$	0	0
$763$	−0.953703	−0.0345264
$764$	−1.57895	−0.0571245
$765$	0	0
$766$	−9.73861	−0.351870
$767$	−38.6677	−1.39621
$768$	0	0
$769$	−4.35025	−0.156874	−0.0784371	−	0.996919i	$-0.524993\pi$
$769$	−4.35025	−0.156874	−0.0784371	+	0.996919i	$0.524993\pi$
$770$	0	0
$771$	0	0
$772$	−13.4690	−0.484761
$773$	−1.41199	−0.0507857	−0.0253929	−	0.999678i	$-0.508084\pi$
$773$	−1.41199	−0.0507857	−0.0253929	+	0.999678i	$0.508084\pi$
$774$	0	0
$775$	21.7386	0.780874
$776$	8.09139	0.290464
$777$	0	0
$778$	1.99181	0.0714097
$779$	4.31564	0.154624
$780$	0	0
$781$	0	0
$782$	5.84437	0.208994
$783$	0	0
$784$	3.47685	0.124173
$785$	15.7207	0.561095
$786$	0	0
$787$	12.7886	0.455866	0.227933	−	0.973677i	$-0.426803\pi$
$787$	12.7886	0.455866	0.227933	+	0.973677i	$0.426803\pi$
$788$	27.8762	0.993048
$789$	0	0
$790$	−10.6154	−0.377678
$791$	−14.9198	−0.530485
$792$	0	0
$793$	−36.5153	−1.29670
$794$	8.63371	0.306399
$795$	0	0
$796$	30.0687	1.06576
$797$	8.90842	0.315552	0.157776	−	0.987475i	$-0.449568\pi$
$797$	8.90842	0.315552	0.157776	+	0.987475i	$0.449568\pi$
$798$	0	0
$799$	50.1531	1.77429
$800$	14.7516	0.521547
$801$	0	0
$802$	9.66168	0.341166
$803$	0	0
$804$	0	0
$805$	−8.95370	−0.315577
$806$	7.94053	0.279693
$807$	0	0
$808$	9.20763	0.323923
$809$	54.9834	1.93311	0.966556	−	0.256456i	$-0.0825548\pi$
$809$	54.9834	1.93311	0.966556	+	0.256456i	$0.0825548\pi$
$810$	0	0
$811$	−3.21136	−0.112766	−0.0563831	−	0.998409i	$-0.517957\pi$
$811$	−3.21136	−0.112766	−0.0563831	+	0.998409i	$0.517957\pi$
$812$	13.1223	0.460502
$813$	0	0
$814$	0	0
$815$	−10.9244	−0.382664
$816$	0	0
$817$	0.792373	0.0277216
$818$	−3.10059	−0.108410
$819$	0	0
$820$	−58.1531	−2.03080
$821$	−25.2492	−0.881202	−0.440601	−	0.897703i	$-0.645235\pi$
$821$	−25.2492	−0.881202	−0.440601	+	0.897703i	$0.645235\pi$
$822$	0	0
$823$	5.82674	0.203107	0.101554	−	0.994830i	$-0.467619\pi$
$823$	5.82674	0.203107	0.101554	+	0.994830i	$0.467619\pi$
$824$	−21.4369	−0.746788
$825$	0	0
$826$	−2.13479	−0.0742790
$827$	−19.7160	−0.685594	−0.342797	−	0.939409i	$-0.611374\pi$
$827$	−19.7160	−0.685594	−0.342797	+	0.939409i	$0.611374\pi$
$828$	0	0
$829$	7.03810	0.244443	0.122222	−	0.992503i	$-0.460998\pi$
$829$	7.03810	0.244443	0.122222	+	0.992503i	$0.460998\pi$
$830$	−1.57895	−0.0548062
$831$	0	0
$832$	−32.0807	−1.11220
$833$	6.72305	0.232940
$834$	0	0
$835$	−64.4145	−2.22916
$836$	0	0
$837$	0	0
$838$	−8.48059	−0.292957
$839$	−22.9471	−0.792220	−0.396110	−	0.918203i	$-0.629640\pi$
$839$	−22.9471	−0.792220	−0.396110	+	0.918203i	$0.629640\pi$
$840$	0	0
$841$	18.1270	0.625068
$842$	−1.83401	−0.0632041
$843$	0	0
$844$	25.9074	0.891770
$845$	−49.1300	−1.69012
$846$	0	0
$847$	0	0
$848$	−32.4529	−1.11444
$849$	0	0
$850$	8.77671	0.301039
$851$	−9.90141	−0.339416
$852$	0	0
$853$	42.5915	1.45831	0.729153	−	0.684351i	$-0.239914\pi$
$853$	42.5915	1.45831	0.729153	+	0.684351i	$0.239914\pi$
$854$	−2.01596	−0.0689848
$855$	0	0
$856$	15.4809	0.529128
$857$	−4.42338	−0.151100	−0.0755499	−	0.997142i	$-0.524071\pi$
$857$	−4.42338	−0.151100	−0.0755499	+	0.997142i	$0.524071\pi$
$858$	0	0
$859$	−10.4466	−0.356433	−0.178216	−	0.983991i	$-0.557033\pi$
$859$	−10.4466	−0.356433	−0.178216	+	0.983991i	$0.557033\pi$
$860$	−10.6772	−0.364090
$861$	0	0
$862$	7.87340	0.268169
$863$	0.258604	0.00880298	0.00440149	−	0.999990i	$-0.498599\pi$
$863$	0.258604	0.00880298	0.00440149	+	0.999990i	$0.498599\pi$
$864$	0	0
$865$	−11.6460	−0.395976
$866$	−5.61331	−0.190748
$867$	0	0
$868$	−9.46902	−0.321399
$869$	0	0
$870$	0	0
$871$	19.2114	0.650952
$872$	1.10974	0.0375805
$873$	0	0
$874$	−0.377844	−0.0127808
$875$	1.87411	0.0633564
$876$	0	0
$877$	−38.7529	−1.30859	−0.654295	−	0.756239i	$-0.727035\pi$
$877$	−38.7529	−1.30859	−0.654295	+	0.756239i	$0.727035\pi$
$878$	4.41983	0.149162
$879$	0	0
$880$	0	0
$881$	−29.3888	−0.990135	−0.495067	−	0.868855i	$-0.664857\pi$
$881$	−29.3888	−0.990135	−0.495067	+	0.868855i	$0.664857\pi$
$882$	0	0
$883$	0.788639	0.0265398	0.0132699	−	0.999912i	$-0.495776\pi$
$883$	0.788639	0.0265398	0.0132699	+	0.999912i	$0.495776\pi$
$884$	−69.2465	−2.32901
$885$	0	0
$886$	−6.17699	−0.207520
$887$	−18.4094	−0.618126	−0.309063	−	0.951042i	$-0.600015\pi$
$887$	−18.4094	−0.618126	−0.309063	+	0.951042i	$0.600015\pi$
$888$	0	0
$889$	19.7304	0.661737
$890$	−6.67040	−0.223592
$891$	0	0
$892$	29.9074	1.00137
$893$	−3.24245	−0.108504
$894$	0	0
$895$	−75.1068	−2.51054
$896$	−8.49418	−0.283771
$897$	0	0
$898$	9.99181	0.333431
$899$	−34.0067	−1.13419
$900$	0	0
$901$	−62.7529	−2.09060
$902$	0	0
$903$	0	0
$904$	17.3608	0.577410
$905$	15.4621	0.513976
$906$	0	0
$907$	−25.3846	−0.842882	−0.421441	−	0.906856i	$-0.638476\pi$
$907$	−25.3846	−0.842882	−0.421441	+	0.906856i	$0.638476\pi$
$908$	1.73225	0.0574868
$909$	0	0
$910$	−4.91150	−0.162815
$911$	−31.9357	−1.05808	−0.529038	−	0.848598i	$-0.677447\pi$
$911$	−31.9357	−1.05808	−0.529038	+	0.848598i	$0.677447\pi$
$912$	0	0
$913$	0	0
$914$	9.07538	0.300187
$915$	0	0
$916$	−29.5691	−0.976990
$917$	7.31802	0.241662
$918$	0	0
$919$	−29.7991	−0.982983	−0.491492	−	0.870882i	$-0.663548\pi$
$919$	−29.7991	−0.982983	−0.491492	+	0.870882i	$0.663548\pi$
$920$	10.4186	0.343491
$921$	0	0
$922$	3.13889	0.103374
$923$	45.8439	1.50897
$924$	0	0
$925$	−14.8693	−0.488900
$926$	−4.37184	−0.143667
$927$	0	0
$928$	−23.0766	−0.757525
$929$	30.5237	1.00145	0.500725	−	0.865607i	$-0.333067\pi$
$929$	30.5237	1.00145	0.500725	+	0.865607i	$0.333067\pi$
$930$	0	0
$931$	−0.434652	−0.0142451
$932$	−33.5100	−1.09766
$933$	0	0
$934$	2.92717	0.0957798
$935$	0	0
$936$	0	0
$937$	−11.2995	−0.369138	−0.184569	−	0.982820i	$-0.559089\pi$
$937$	−11.2995	−0.369138	−0.184569	+	0.982820i	$0.559089\pi$
$938$	1.06063	0.0346309
$939$	0	0
$940$	43.6919	1.42507
$941$	7.99319	0.260570	0.130285	−	0.991477i	$-0.458411\pi$
$941$	7.99319	0.260570	0.130285	+	0.991477i	$0.458411\pi$
$942$	0	0
$943$	−29.0142	−0.944834
$944$	−24.9505	−0.812068
$945$	0	0
$946$	0	0
$947$	49.5921	1.61153	0.805763	−	0.592238i	$-0.201755\pi$
$947$	49.5921	1.61153	0.805763	+	0.592238i	$0.201755\pi$
$948$	0	0
$949$	14.0725	0.456812
$950$	−0.567423	−0.0184096
$951$	0	0
$952$	−7.82301	−0.253545
$953$	−13.7939	−0.446829	−0.223414	−	0.974724i	$-0.571720\pi$
$953$	−13.7939	−0.446829	−0.223414	+	0.974724i	$0.571720\pi$
$954$	0	0
$955$	−2.53098	−0.0819006
$956$	−4.27797	−0.138360
$957$	0	0
$958$	4.10826	0.132732
$959$	17.5582	0.566985
$960$	0	0
$961$	−6.46083	−0.208414
$962$	−5.43136	−0.175114
$963$	0	0
$964$	−32.5613	−1.04873
$965$	−21.5902	−0.695012
$966$	0	0
$967$	−32.1613	−1.03424	−0.517119	−	0.855913i	$-0.672996\pi$
$967$	−32.1613	−1.03424	−0.517119	+	0.855913i	$0.672996\pi$
$968$	0	0
$969$	0	0
$970$	6.33832	0.203511
$971$	2.41642	0.0775467	0.0387733	−	0.999248i	$-0.487655\pi$
$971$	2.41642	0.0775467	0.0387733	+	0.999248i	$0.487655\pi$
$972$	0	0
$973$	19.6460	0.629822
$974$	−1.86266	−0.0596836
$975$	0	0
$976$	−23.5616	−0.754189
$977$	−5.30206	−0.169628	−0.0848139	−	0.996397i	$-0.527030\pi$
$977$	−5.30206	−0.169628	−0.0848139	+	0.996397i	$0.527030\pi$
$978$	0	0
$979$	0	0
$980$	5.85693	0.187093
$981$	0	0
$982$	−4.12660	−0.131685
$983$	4.03192	0.128598	0.0642992	−	0.997931i	$-0.479519\pi$
$983$	4.03192	0.128598	0.0642992	+	0.997931i	$0.479519\pi$
$984$	0	0
$985$	44.6841	1.42375
$986$	−13.7298	−0.437246
$987$	0	0
$988$	4.47685	0.142428
$989$	−5.32717	−0.169394
$990$	0	0
$991$	29.1188	0.924988	0.462494	−	0.886622i	$-0.346955\pi$
$991$	29.1188	0.924988	0.462494	+	0.886622i	$0.346955\pi$
$992$	16.6520	0.528702
$993$	0	0
$994$	2.53098	0.0802778
$995$	48.1986	1.52800
$996$	0	0
$997$	−11.1307	−0.352513	−0.176256	−	0.984344i	$-0.556399\pi$
$997$	−11.1307	−0.352513	−0.176256	+	0.984344i	$0.556399\pi$
$998$	5.56177	0.176055
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cr.1.4	yes	6
3.2	odd	2	inner	7623.2.a.cr.1.3	yes	6
11.10	odd	2		7623.2.a.cq.1.3	✓	6
33.32	even	2		7623.2.a.cq.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cq.1.3	✓	6	11.10	odd	2
7623.2.a.cq.1.4	yes	6	33.32	even	2
7623.2.a.cr.1.3	yes	6	3.2	odd	2	inner
7623.2.a.cr.1.4	yes	6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.297484 q^{2} -1.91150 q^{4} -3.06404 q^{5} +1.00000 q^{7} -1.16361 q^{8} +O(q^{10})\)	\(q+0.297484 q^{2} -1.91150 q^{4} -3.06404 q^{5} +1.00000 q^{7} -1.16361 q^{8} -0.911503 q^{10} +5.38835 q^{13} +0.297484 q^{14} +3.47685 q^{16} +6.72305 q^{17} -0.434652 q^{19} +5.85693 q^{20} +2.92219 q^{23} +4.38835 q^{25} +1.60295 q^{26} -1.91150 q^{28} -6.86491 q^{29} +4.95370 q^{31} +3.36153 q^{32} +2.00000 q^{34} -3.06404 q^{35} -3.38835 q^{37} -0.129302 q^{38} +3.56535 q^{40} -9.92895 q^{41} -1.82301 q^{43} +0.869304 q^{46} +7.45988 q^{47} +1.00000 q^{49} +1.30547 q^{50} -10.2999 q^{52} -9.33398 q^{53} -1.16361 q^{56} -2.04220 q^{58} -7.17616 q^{59} -6.77671 q^{61} +1.47365 q^{62} -5.95370 q^{64} -16.5101 q^{65} +3.56535 q^{67} -12.8511 q^{68} -0.911503 q^{70} +8.50796 q^{71} +2.61165 q^{73} -1.00798 q^{74} +0.830838 q^{76} +11.6460 q^{79} -10.6532 q^{80} -2.95370 q^{82} +1.73225 q^{83} -20.5997 q^{85} -0.542315 q^{86} +7.31802 q^{89} +5.38835 q^{91} -5.58577 q^{92} +2.21919 q^{94} +1.33179 q^{95} -6.95370 q^{97} +0.297484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 6 q^{7}+O(q^{10}) \) 6 * q + 4 * q^4 + 6 * q^7	\( 6 q + 4 q^{4} + 6 q^{7} + 10 q^{10} + 4 q^{13} + 8 q^{16} - 2 q^{25} + 4 q^{28} + 4 q^{31} + 12 q^{34} + 8 q^{37} + 24 q^{40} + 20 q^{43} + 6 q^{49} - 18 q^{52} - 2 q^{58} + 16 q^{61} - 10 q^{64} + 24 q^{67} + 10 q^{70} + 44 q^{73} + 54 q^{76} + 8 q^{79} + 8 q^{82} - 36 q^{85} + 4 q^{91} + 34 q^{94} - 16 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 + 6 * q^7 + 10 * q^10 + 4 * q^13 + 8 * q^16 - 2 * q^25 + 4 * q^28 + 4 * q^31 + 12 * q^34 + 8 * q^37 + 24 * q^40 + 20 * q^43 + 6 * q^49 - 18 * q^52 - 2 * q^58 + 16 * q^61 - 10 * q^64 + 24 * q^67 + 10 * q^70 + 44 * q^73 + 54 * q^76 + 8 * q^79 + 8 * q^82 - 36 * q^85 + 4 * q^91 + 34 * q^94 - 16 * q^97

Introduction

L-functions

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Database

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