LMFDB - Embedded newform 7623.2.a.cs.1.6

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:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.70320$ 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)

$f(q)$	$=$	$q+2.70320 q^{2} +5.30727 q^{4} +0.445072 q^{5} -1.00000 q^{7} +8.94020 q^{8} +O(q^{10})$	$q+2.70320 q^{2} +5.30727 q^{4} +0.445072 q^{5} -1.00000 q^{7} +8.94020 q^{8} +1.20312 q^{10} -0.450933 q^{13} -2.70320 q^{14} +13.5526 q^{16} +4.83117 q^{17} +1.08644 q^{19} +2.36212 q^{20} -4.57222 q^{23} -4.80191 q^{25} -1.21896 q^{26} -5.30727 q^{28} -1.98431 q^{29} +8.25861 q^{31} +18.7549 q^{32} +13.0596 q^{34} -0.445072 q^{35} +7.31725 q^{37} +2.93685 q^{38} +3.97903 q^{40} +1.77073 q^{41} +11.4084 q^{43} -12.3596 q^{46} -1.02259 q^{47} +1.00000 q^{49} -12.9805 q^{50} -2.39322 q^{52} +3.57222 q^{53} -8.94020 q^{56} -5.36399 q^{58} +14.3996 q^{59} -4.92965 q^{61} +22.3246 q^{62} +23.5929 q^{64} -0.200698 q^{65} -6.18858 q^{67} +25.6403 q^{68} -1.20312 q^{70} +5.92165 q^{71} +1.65776 q^{73} +19.7800 q^{74} +5.76602 q^{76} +3.60833 q^{79} +6.03187 q^{80} +4.78663 q^{82} -10.8048 q^{83} +2.15022 q^{85} +30.8392 q^{86} +5.21170 q^{89} +0.450933 q^{91} -24.2660 q^{92} -2.76425 q^{94} +0.483543 q^{95} -5.30985 q^{97} +2.70320 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 6 * q^7 + 12 * q^8	$ 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8} + 8 q^{10} - 4 q^{13} - 4 q^{14} + 8 q^{16} + 22 q^{17} - 6 q^{19} - 2 q^{20} - 2 q^{23} + 4 q^{25} - 6 q^{26} - 4 q^{28} + 12 q^{29} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 14 q^{37} + 22 q^{38} - 18 q^{40} + 26 q^{41} + 4 q^{43} - 12 q^{46} + 16 q^{47} + 6 q^{49} - 4 q^{50} - 12 q^{52} - 4 q^{53} - 12 q^{56} - 2 q^{58} + 4 q^{59} + 8 q^{61} + 20 q^{62} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 8 q^{70} - 22 q^{71} - 14 q^{73} + 44 q^{74} + 30 q^{76} + 28 q^{79} + 4 q^{80} - 4 q^{82} + 22 q^{83} + 24 q^{85} + 30 q^{86} + 4 q^{91} - 10 q^{92} + 38 q^{94} - 24 q^{95} - 4 q^{97} + 4 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 6 * q^7 + 12 * q^8 + 8 * q^10 - 4 * q^13 - 4 * q^14 + 8 * q^16 + 22 * q^17 - 6 * q^19 - 2 * q^20 - 2 * q^23 + 4 * q^25 - 6 * q^26 - 4 * q^28 + 12 * q^29 - 2 * q^31 + 8 * q^32 + 24 * q^34 - 4 * q^35 + 14 * q^37 + 22 * q^38 - 18 * q^40 + 26 * q^41 + 4 * q^43 - 12 * q^46 + 16 * q^47 + 6 * q^49 - 4 * q^50 - 12 * q^52 - 4 * q^53 - 12 * q^56 - 2 * q^58 + 4 * q^59 + 8 * q^61 + 20 * q^62 + 26 * q^64 + 24 * q^65 + 6 * q^67 + 12 * q^68 - 8 * q^70 - 22 * q^71 - 14 * q^73 + 44 * q^74 + 30 * q^76 + 28 * q^79 + 4 * q^80 - 4 * q^82 + 22 * q^83 + 24 * q^85 + 30 * q^86 + 4 * q^91 - 10 * q^92 + 38 * q^94 - 24 * q^95 - 4 * q^97 + 4 * 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 $

$2$	2.70320	1.91145	0.955724	−	0.294264i	$-0.0950745\pi$
$2$	2.70320	1.91145	0.955724	+	0.294264i	$0.0950745\pi$
$3$	0	0
$3$	0	0
$4$	5.30727	2.65363
$5$	0.445072	0.199042	0.0995212	−	0.995035i	$-0.468269\pi$
$5$	0.445072	0.199042	0.0995212	+	0.995035i	$0.468269\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	8.94020	3.16084
$9$	0	0
$10$	1.20312	0.380459
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.450933	−0.125066	−0.0625332	−	0.998043i	$-0.519918\pi$
$13$	−0.450933	−0.125066	−0.0625332	+	0.998043i	$0.519918\pi$
$14$	−2.70320	−0.722460
$15$	0	0
$16$	13.5526	3.38814
$17$	4.83117	1.17173	0.585866	−	0.810408i	$-0.300755\pi$
$17$	4.83117	1.17173	0.585866	+	0.810408i	$0.300755\pi$
$18$	0	0
$19$	1.08644	0.249246	0.124623	−	0.992204i	$-0.460228\pi$
$19$	1.08644	0.249246	0.124623	+	0.992204i	$0.460228\pi$
$20$	2.36212	0.528186
$21$	0	0
$22$	0	0
$23$	−4.57222	−0.953373	−0.476687	−	0.879073i	$-0.658162\pi$
$23$	−4.57222	−0.953373	−0.476687	+	0.879073i	$0.658162\pi$
$24$	0	0
$25$	−4.80191	−0.960382
$26$	−1.21896	−0.239058
$27$	0	0
$28$	−5.30727	−1.00298
$29$	−1.98431	−0.368478	−0.184239	−	0.982881i	$-0.558982\pi$
$29$	−1.98431	−0.368478	−0.184239	+	0.982881i	$0.558982\pi$
$30$	0	0
$31$	8.25861	1.48329	0.741645	−	0.670793i	$-0.234046\pi$
$31$	8.25861	1.48329	0.741645	+	0.670793i	$0.234046\pi$
$32$	18.7549	3.31542
$33$	0	0
$34$	13.0596	2.23970
$35$	−0.445072	−0.0752309
$36$	0	0
$37$	7.31725	1.20295	0.601474	−	0.798892i	$-0.294580\pi$
$37$	7.31725	1.20295	0.601474	+	0.798892i	$0.294580\pi$
$38$	2.93685	0.476421
$39$	0	0
$40$	3.97903	0.629140
$41$	1.77073	0.276542	0.138271	−	0.990394i	$-0.455846\pi$
$41$	1.77073	0.276542	0.138271	+	0.990394i	$0.455846\pi$
$42$	0	0
$43$	11.4084	1.73977	0.869884	−	0.493257i	$-0.164194\pi$
$43$	11.4084	1.73977	0.869884	+	0.493257i	$0.164194\pi$
$44$	0	0
$45$	0	0
$46$	−12.3596	−1.82232
$47$	−1.02259	−0.149159	−0.0745797	−	0.997215i	$-0.523762\pi$
$47$	−1.02259	−0.149159	−0.0745797	+	0.997215i	$0.523762\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−12.9805	−1.83572
$51$	0	0
$52$	−2.39322	−0.331880
$53$	3.57222	0.490682	0.245341	−	0.969437i	$-0.421100\pi$
$53$	3.57222	0.490682	0.245341	+	0.969437i	$0.421100\pi$
$54$	0	0
$55$	0	0
$56$	−8.94020	−1.19468
$57$	0	0
$58$	−5.36399	−0.704326
$59$	14.3996	1.87467	0.937333	−	0.348435i	$-0.113287\pi$
$59$	14.3996	1.87467	0.937333	+	0.348435i	$0.113287\pi$
$60$	0	0
$61$	−4.92965	−0.631177	−0.315588	−	0.948896i	$-0.602202\pi$
$61$	−4.92965	−0.631177	−0.315588	+	0.948896i	$0.602202\pi$
$62$	22.3246	2.83523
$63$	0	0
$64$	23.5929	2.94911
$65$	−0.200698	−0.0248935
$66$	0	0
$67$	−6.18858	−0.756055	−0.378028	−	0.925794i	$-0.623398\pi$
$67$	−6.18858	−0.756055	−0.378028	+	0.925794i	$0.623398\pi$
$68$	25.6403	3.10935
$69$	0	0
$70$	−1.20312	−0.143800
$71$	5.92165	0.702771	0.351385	−	0.936231i	$-0.385711\pi$
$71$	5.92165	0.702771	0.351385	+	0.936231i	$0.385711\pi$
$72$	0	0
$73$	1.65776	0.194027	0.0970133	−	0.995283i	$-0.469071\pi$
$73$	1.65776	0.194027	0.0970133	+	0.995283i	$0.469071\pi$
$74$	19.7800	2.29937
$75$	0	0
$76$	5.76602	0.661407
$77$	0	0
$78$	0	0
$79$	3.60833	0.405969	0.202984	−	0.979182i	$-0.434936\pi$
$79$	3.60833	0.405969	0.202984	+	0.979182i	$0.434936\pi$
$80$	6.03187	0.674384
$81$	0	0
$82$	4.78663	0.528595
$83$	−10.8048	−1.18598	−0.592992	−	0.805208i	$-0.702053\pi$
$83$	−10.8048	−1.18598	−0.592992	+	0.805208i	$0.702053\pi$
$84$	0	0
$85$	2.15022	0.233224
$86$	30.8392	3.32548
$87$	0	0
$88$	0	0
$89$	5.21170	0.552439	0.276220	−	0.961095i	$-0.410918\pi$
$89$	5.21170	0.552439	0.276220	+	0.961095i	$0.410918\pi$
$90$	0	0
$91$	0.450933	0.0472706
$92$	−24.2660	−2.52990
$93$	0	0
$94$	−2.76425	−0.285110
$95$	0.483543	0.0496105
$96$	0	0
$97$	−5.30985	−0.539133	−0.269567	−	0.962982i	$-0.586880\pi$
$97$	−5.30985	−0.539133	−0.269567	+	0.962982i	$0.586880\pi$
$98$	2.70320	0.273064
$99$	0	0
$100$	−25.4850	−2.54850
$101$	15.5604	1.54832	0.774158	−	0.632992i	$-0.218174\pi$
$101$	15.5604	1.54832	0.774158	+	0.632992i	$0.218174\pi$
$102$	0	0
$103$	−14.1713	−1.39634	−0.698172	−	0.715930i	$-0.746003\pi$
$103$	−14.1713	−1.39634	−0.698172	+	0.715930i	$0.746003\pi$
$104$	−4.03143	−0.395314
$105$	0	0
$106$	9.65640	0.937913
$107$	−11.7551	−1.13641	−0.568205	−	0.822887i	$-0.692362\pi$
$107$	−11.7551	−1.13641	−0.568205	+	0.822887i	$0.692362\pi$
$108$	0	0
$109$	−15.7800	−1.51145	−0.755723	−	0.654891i	$-0.772714\pi$
$109$	−15.7800	−1.51145	−0.755723	+	0.654891i	$0.772714\pi$
$110$	0	0
$111$	0	0
$112$	−13.5526	−1.28060
$113$	5.92347	0.557233	0.278617	−	0.960402i	$-0.410124\pi$
$113$	5.92347	0.557233	0.278617	+	0.960402i	$0.410124\pi$
$114$	0	0
$115$	−2.03497	−0.189762
$116$	−10.5313	−0.977805
$117$	0	0
$118$	38.9249	3.58333
$119$	−4.83117	−0.442873
$120$	0	0
$121$	0	0
$122$	−13.3258	−1.20646
$123$	0	0
$124$	43.8306	3.93611
$125$	−4.36256	−0.390199
$126$	0	0
$127$	−10.1337	−0.899222	−0.449611	−	0.893224i	$-0.648437\pi$
$127$	−10.1337	−0.899222	−0.449611	+	0.893224i	$0.648437\pi$
$128$	26.2666	2.32166
$129$	0	0
$130$	−0.542526	−0.0475826
$131$	−6.83340	−0.597037	−0.298519	−	0.954404i	$-0.596492\pi$
$131$	−6.83340	−0.597037	−0.298519	+	0.954404i	$0.596492\pi$
$132$	0	0
$133$	−1.08644	−0.0942061
$134$	−16.7289	−1.44516
$135$	0	0
$136$	43.1916	3.70365
$137$	−14.4673	−1.23602	−0.618012	−	0.786169i	$-0.712062\pi$
$137$	−14.4673	−1.23602	−0.618012	+	0.786169i	$0.712062\pi$
$138$	0	0
$139$	14.6957	1.24647	0.623235	−	0.782035i	$-0.285818\pi$
$139$	14.6957	1.24647	0.623235	+	0.782035i	$0.285818\pi$
$140$	−2.36212	−0.199635
$141$	0	0
$142$	16.0074	1.34331
$143$	0	0
$144$	0	0
$145$	−0.883163	−0.0733427
$146$	4.48126	0.370872
$147$	0	0
$148$	38.8346	3.19219
$149$	1.00140	0.0820377	0.0410189	−	0.999158i	$-0.486940\pi$
$149$	1.00140	0.0820377	0.0410189	+	0.999158i	$0.486940\pi$
$150$	0	0
$151$	−1.37539	−0.111927	−0.0559636	−	0.998433i	$-0.517823\pi$
$151$	−1.37539	−0.111927	−0.0559636	+	0.998433i	$0.517823\pi$
$152$	9.71297	0.787826
$153$	0	0
$154$	0	0
$155$	3.67568	0.295237
$156$	0	0
$157$	5.37668	0.429106	0.214553	−	0.976712i	$-0.431171\pi$
$157$	5.37668	0.429106	0.214553	+	0.976712i	$0.431171\pi$
$158$	9.75403	0.775989
$159$	0	0
$160$	8.34726	0.659909
$161$	4.57222	0.360341
$162$	0	0
$163$	−9.42513	−0.738233	−0.369116	−	0.929383i	$-0.620340\pi$
$163$	−9.42513	−0.738233	−0.369116	+	0.929383i	$0.620340\pi$
$164$	9.39775	0.733841
$165$	0	0
$166$	−29.2076	−2.26695
$167$	20.0118	1.54856	0.774281	−	0.632842i	$-0.218112\pi$
$167$	20.0118	1.54856	0.774281	+	0.632842i	$0.218112\pi$
$168$	0	0
$169$	−12.7967	−0.984358
$170$	5.81247	0.445796
$171$	0	0
$172$	60.5476	4.61671
$173$	10.8465	0.824647	0.412324	−	0.911037i	$-0.364717\pi$
$173$	10.8465	0.824647	0.412324	+	0.911037i	$0.364717\pi$
$174$	0	0
$175$	4.80191	0.362990
$176$	0	0
$177$	0	0
$178$	14.0883	1.05596
$179$	−16.8484	−1.25931	−0.629654	−	0.776876i	$-0.716803\pi$
$179$	−16.8484	−1.25931	−0.629654	+	0.776876i	$0.716803\pi$
$180$	0	0
$181$	−20.3041	−1.50919	−0.754597	−	0.656188i	$-0.772168\pi$
$181$	−20.3041	−1.50919	−0.754597	+	0.656188i	$0.772168\pi$
$182$	1.21896	0.0903554
$183$	0	0
$184$	−40.8765	−3.01346
$185$	3.25671	0.239438
$186$	0	0
$187$	0	0
$188$	−5.42714	−0.395815
$189$	0	0
$190$	1.30711	0.0948279
$191$	−2.54435	−0.184103	−0.0920513	−	0.995754i	$-0.529342\pi$
$191$	−2.54435	−0.184103	−0.0920513	+	0.995754i	$0.529342\pi$
$192$	0	0
$193$	−17.5086	−1.26030	−0.630148	−	0.776475i	$-0.717006\pi$
$193$	−17.5086	−1.26030	−0.630148	+	0.776475i	$0.717006\pi$
$194$	−14.3536	−1.03053
$195$	0	0
$196$	5.30727	0.379091
$197$	−2.16558	−0.154291	−0.0771457	−	0.997020i	$-0.524581\pi$
$197$	−2.16558	−0.154291	−0.0771457	+	0.997020i	$0.524581\pi$
$198$	0	0
$199$	−14.5756	−1.03324	−0.516620	−	0.856215i	$-0.672810\pi$
$199$	−14.5756	−1.03324	−0.516620	+	0.856215i	$0.672810\pi$
$200$	−42.9300	−3.03561
$201$	0	0
$202$	42.0628	2.95953
$203$	1.98431	0.139272
$204$	0	0
$205$	0.788103	0.0550435
$206$	−38.3079	−2.66904
$207$	0	0
$208$	−6.11130	−0.423743
$209$	0	0
$210$	0	0
$211$	3.63034	0.249923	0.124961	−	0.992162i	$-0.460119\pi$
$211$	3.63034	0.249923	0.124961	+	0.992162i	$0.460119\pi$
$212$	18.9587	1.30209
$213$	0	0
$214$	−31.7764	−2.17219
$215$	5.07757	0.346287
$216$	0	0
$217$	−8.25861	−0.560631
$218$	−42.6563	−2.88905
$219$	0	0
$220$	0	0
$221$	−2.17854	−0.146544
$222$	0	0
$223$	−4.84062	−0.324152	−0.162076	−	0.986778i	$-0.551819\pi$
$223$	−4.84062	−0.324152	−0.162076	+	0.986778i	$0.551819\pi$
$224$	−18.7549	−1.25311
$225$	0	0
$226$	16.0123	1.06512
$227$	22.7306	1.50868	0.754342	−	0.656482i	$-0.227956\pi$
$227$	22.7306	1.50868	0.754342	+	0.656482i	$0.227956\pi$
$228$	0	0
$229$	−22.6732	−1.49828	−0.749142	−	0.662409i	$-0.769534\pi$
$229$	−22.6732	−1.49828	−0.749142	+	0.662409i	$0.769534\pi$
$230$	−5.50092	−0.362720
$231$	0	0
$232$	−17.7402	−1.16470
$233$	−0.587282	−0.0384741	−0.0192371	−	0.999815i	$-0.506124\pi$
$233$	−0.587282	−0.0384741	−0.0192371	+	0.999815i	$0.506124\pi$
$234$	0	0
$235$	−0.455124	−0.0296890
$236$	76.4225	4.97468
$237$	0	0
$238$	−13.0596	−0.846529
$239$	−14.9721	−0.968465	−0.484233	−	0.874939i	$-0.660901\pi$
$239$	−14.9721	−0.968465	−0.484233	+	0.874939i	$0.660901\pi$
$240$	0	0
$241$	18.2746	1.17717	0.588586	−	0.808435i	$-0.299685\pi$
$241$	18.2746	1.17717	0.588586	+	0.808435i	$0.299685\pi$
$242$	0	0
$243$	0	0
$244$	−26.1630	−1.67491
$245$	0.445072	0.0284346
$246$	0	0
$247$	−0.489911	−0.0311723
$248$	73.8336	4.68844
$249$	0	0
$250$	−11.7929	−0.745845
$251$	−10.5649	−0.666850	−0.333425	−	0.942777i	$-0.608204\pi$
$251$	−10.5649	−0.666850	−0.333425	+	0.942777i	$0.608204\pi$
$252$	0	0
$253$	0	0
$254$	−27.3934	−1.71882
$255$	0	0
$256$	23.8178	1.48861
$257$	19.4680	1.21438	0.607189	−	0.794557i	$-0.292297\pi$
$257$	19.4680	1.21438	0.607189	+	0.794557i	$0.292297\pi$
$258$	0	0
$259$	−7.31725	−0.454672
$260$	−1.06516	−0.0660583
$261$	0	0
$262$	−18.4720	−1.14121
$263$	21.1885	1.30654	0.653269	−	0.757126i	$-0.273397\pi$
$263$	21.1885	1.30654	0.653269	+	0.757126i	$0.273397\pi$
$264$	0	0
$265$	1.58989	0.0976665
$266$	−2.93685	−0.180070
$267$	0	0
$268$	−32.8444	−2.00629
$269$	−23.1564	−1.41187	−0.705935	−	0.708276i	$-0.749473\pi$
$269$	−23.1564	−1.41187	−0.705935	+	0.708276i	$0.749473\pi$
$270$	0	0
$271$	−26.8232	−1.62939	−0.814696	−	0.579888i	$-0.803096\pi$
$271$	−26.8232	−1.62939	−0.814696	+	0.579888i	$0.803096\pi$
$272$	65.4748	3.96999
$273$	0	0
$274$	−39.1079	−2.36260
$275$	0	0
$276$	0	0
$277$	14.0508	0.844230	0.422115	−	0.906542i	$-0.361288\pi$
$277$	14.0508	0.844230	0.422115	+	0.906542i	$0.361288\pi$
$278$	39.7253	2.38256
$279$	0	0
$280$	−3.97903	−0.237793
$281$	−1.74538	−0.104120	−0.0520602	−	0.998644i	$-0.516579\pi$
$281$	−1.74538	−0.104120	−0.0520602	+	0.998644i	$0.516579\pi$
$282$	0	0
$283$	4.66531	0.277324	0.138662	−	0.990340i	$-0.455720\pi$
$283$	4.66531	0.277324	0.138662	+	0.990340i	$0.455720\pi$
$284$	31.4278	1.86490
$285$	0	0
$286$	0	0
$287$	−1.77073	−0.104523
$288$	0	0
$289$	6.34024	0.372955
$290$	−2.38736	−0.140191
$291$	0	0
$292$	8.79820	0.514876
$293$	14.5092	0.847637	0.423819	−	0.905747i	$-0.360689\pi$
$293$	14.5092	0.847637	0.423819	+	0.905747i	$0.360689\pi$
$294$	0	0
$295$	6.40886	0.373138
$296$	65.4177	3.80232
$297$	0	0
$298$	2.70698	0.156811
$299$	2.06176	0.119235
$300$	0	0
$301$	−11.4084	−0.657570
$302$	−3.71794	−0.213943
$303$	0	0
$304$	14.7240	0.844480
$305$	−2.19405	−0.125631
$306$	0	0
$307$	−2.35679	−0.134509	−0.0672547	−	0.997736i	$-0.521424\pi$
$307$	−2.35679	−0.134509	−0.0672547	+	0.997736i	$0.521424\pi$
$308$	0	0
$309$	0	0
$310$	9.93608	0.564331
$311$	20.0050	1.13438	0.567190	−	0.823587i	$-0.308031\pi$
$311$	20.0050	1.13438	0.567190	+	0.823587i	$0.308031\pi$
$312$	0	0
$313$	18.7530	1.05998	0.529992	−	0.848003i	$-0.322195\pi$
$313$	18.7530	1.05998	0.529992	+	0.848003i	$0.322195\pi$
$314$	14.5342	0.820214
$315$	0	0
$316$	19.1504	1.07729
$317$	−4.20203	−0.236009	−0.118005	−	0.993013i	$-0.537650\pi$
$317$	−4.20203	−0.236009	−0.118005	+	0.993013i	$0.537650\pi$
$318$	0	0
$319$	0	0
$320$	10.5005	0.586999
$321$	0	0
$322$	12.3596	0.688774
$323$	5.24877	0.292049
$324$	0	0
$325$	2.16534	0.120112
$326$	−25.4780	−1.41109
$327$	0	0
$328$	15.8307	0.874103
$329$	1.02259	0.0563770
$330$	0	0
$331$	−0.0682694	−0.00375242	−0.00187621	−	0.999998i	$-0.500597\pi$
$331$	−0.0682694	−0.00375242	−0.00187621	+	0.999998i	$0.500597\pi$
$332$	−57.3441	−3.14717
$333$	0	0
$334$	54.0959	2.96000
$335$	−2.75436	−0.150487
$336$	0	0
$337$	−21.7461	−1.18459	−0.592294	−	0.805722i	$-0.701777\pi$
$337$	−21.7461	−1.18459	−0.592294	+	0.805722i	$0.701777\pi$
$338$	−34.5919	−1.88155
$339$	0	0
$340$	11.4118	0.618892
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	101.994	5.49912
$345$	0	0
$346$	29.3203	1.57627
$347$	23.9351	1.28490	0.642452	−	0.766326i	$-0.277917\pi$
$347$	23.9351	1.28490	0.642452	+	0.766326i	$0.277917\pi$
$348$	0	0
$349$	26.4151	1.41397	0.706983	−	0.707231i	$-0.250056\pi$
$349$	26.4151	1.41397	0.706983	+	0.707231i	$0.250056\pi$
$350$	12.9805	0.693837
$351$	0	0
$352$	0	0
$353$	4.44182	0.236414	0.118207	−	0.992989i	$-0.462285\pi$
$353$	4.44182	0.236414	0.118207	+	0.992989i	$0.462285\pi$
$354$	0	0
$355$	2.63556	0.139881
$356$	27.6599	1.46597
$357$	0	0
$358$	−45.5445	−2.40710
$359$	−14.5325	−0.766997	−0.383499	−	0.923541i	$-0.625281\pi$
$359$	−14.5325	−0.766997	−0.383499	+	0.923541i	$0.625281\pi$
$360$	0	0
$361$	−17.8197	−0.937876
$362$	−54.8860	−2.88475
$363$	0	0
$364$	2.39322	0.125439
$365$	0.737825	0.0386195
$366$	0	0
$367$	4.98158	0.260037	0.130018	−	0.991512i	$-0.458496\pi$
$367$	4.98158	0.260037	0.130018	+	0.991512i	$0.458496\pi$
$368$	−61.9653	−3.23016
$369$	0	0
$370$	8.80351	0.457673
$371$	−3.57222	−0.185460
$372$	0	0
$373$	13.6638	0.707485	0.353743	−	0.935343i	$-0.384909\pi$
$373$	13.6638	0.707485	0.353743	+	0.935343i	$0.384909\pi$
$374$	0	0
$375$	0	0
$376$	−9.14211	−0.471469
$377$	0.894793	0.0460842
$378$	0	0
$379$	−8.36232	−0.429544	−0.214772	−	0.976664i	$-0.568901\pi$
$379$	−8.36232	−0.429544	−0.214772	+	0.976664i	$0.568901\pi$
$380$	2.56629	0.131648
$381$	0	0
$382$	−6.87787	−0.351902
$383$	1.98679	0.101520	0.0507601	−	0.998711i	$-0.483836\pi$
$383$	1.98679	0.101520	0.0507601	+	0.998711i	$0.483836\pi$
$384$	0	0
$385$	0	0
$386$	−47.3292	−2.40899
$387$	0	0
$388$	−28.1808	−1.43066
$389$	−14.8325	−0.752039	−0.376019	−	0.926612i	$-0.622707\pi$
$389$	−14.8325	−0.752039	−0.376019	+	0.926612i	$0.622707\pi$
$390$	0	0
$391$	−22.0892	−1.11710
$392$	8.94020	0.451548
$393$	0	0
$394$	−5.85400	−0.294920
$395$	1.60597	0.0808050
$396$	0	0
$397$	11.4630	0.575314	0.287657	−	0.957734i	$-0.407124\pi$
$397$	11.4630	0.575314	0.287657	+	0.957734i	$0.407124\pi$
$398$	−39.4008	−1.97498
$399$	0	0
$400$	−65.0782	−3.25391
$401$	−4.31030	−0.215246	−0.107623	−	0.994192i	$-0.534324\pi$
$401$	−4.31030	−0.215246	−0.107623	+	0.994192i	$0.534324\pi$
$402$	0	0
$403$	−3.72408	−0.185510
$404$	82.5831	4.10867
$405$	0	0
$406$	5.36399	0.266210
$407$	0	0
$408$	0	0
$409$	9.31202	0.460450	0.230225	−	0.973137i	$-0.426054\pi$
$409$	9.31202	0.460450	0.230225	+	0.973137i	$0.426054\pi$
$410$	2.13040	0.105213
$411$	0	0
$412$	−75.2111	−3.70538
$413$	−14.3996	−0.708557
$414$	0	0
$415$	−4.80893	−0.236061
$416$	−8.45719	−0.414648
$417$	0	0
$418$	0	0
$419$	26.3424	1.28691	0.643454	−	0.765485i	$-0.277501\pi$
$419$	26.3424	1.28691	0.643454	+	0.765485i	$0.277501\pi$
$420$	0	0
$421$	14.1792	0.691053	0.345526	−	0.938409i	$-0.387700\pi$
$421$	14.1792	0.691053	0.345526	+	0.938409i	$0.387700\pi$
$422$	9.81352	0.477715
$423$	0	0
$424$	31.9363	1.55097
$425$	−23.1989	−1.12531
$426$	0	0
$427$	4.92965	0.238562
$428$	−62.3876	−3.01562
$429$	0	0
$430$	13.7257	0.661911
$431$	7.59531	0.365853	0.182927	−	0.983127i	$-0.441443\pi$
$431$	7.59531	0.365853	0.182927	+	0.983127i	$0.441443\pi$
$432$	0	0
$433$	22.4008	1.07652	0.538258	−	0.842780i	$-0.319083\pi$
$433$	22.4008	1.07652	0.538258	+	0.842780i	$0.319083\pi$
$434$	−22.3246	−1.07162
$435$	0	0
$436$	−83.7485	−4.01083
$437$	−4.96743	−0.237624
$438$	0	0
$439$	−15.4051	−0.735244	−0.367622	−	0.929975i	$-0.619828\pi$
$439$	−15.4051	−0.735244	−0.367622	+	0.929975i	$0.619828\pi$
$440$	0	0
$441$	0	0
$442$	−5.88901	−0.280112
$443$	−21.2099	−1.00771	−0.503857	−	0.863787i	$-0.668086\pi$
$443$	−21.2099	−1.00771	−0.503857	+	0.863787i	$0.668086\pi$
$444$	0	0
$445$	2.31958	0.109959
$446$	−13.0851	−0.619599
$447$	0	0
$448$	−23.5929	−1.11466
$449$	11.6316	0.548931	0.274466	−	0.961597i	$-0.411499\pi$
$449$	11.6316	0.548931	0.274466	+	0.961597i	$0.411499\pi$
$450$	0	0
$451$	0	0
$452$	31.4375	1.47869
$453$	0	0
$454$	61.4453	2.88377
$455$	0.200698	0.00940886
$456$	0	0
$457$	−31.5976	−1.47807	−0.739037	−	0.673665i	$-0.764719\pi$
$457$	−31.5976	−1.47807	−0.739037	+	0.673665i	$0.764719\pi$
$458$	−61.2900	−2.86389
$459$	0	0
$460$	−10.8001	−0.503558
$461$	−8.86324	−0.412802	−0.206401	−	0.978467i	$-0.566175\pi$
$461$	−8.86324	−0.412802	−0.206401	+	0.978467i	$0.566175\pi$
$462$	0	0
$463$	−25.6132	−1.19035	−0.595173	−	0.803597i	$-0.702917\pi$
$463$	−25.6132	−1.19035	−0.595173	+	0.803597i	$0.702917\pi$
$464$	−26.8925	−1.24846
$465$	0	0
$466$	−1.58754	−0.0735413
$467$	−20.5402	−0.950486	−0.475243	−	0.879855i	$-0.657640\pi$
$467$	−20.5402	−0.950486	−0.475243	+	0.879855i	$0.657640\pi$
$468$	0	0
$469$	6.18858	0.285762
$470$	−1.23029	−0.0567491
$471$	0	0
$472$	128.735	5.92551
$473$	0	0
$474$	0	0
$475$	−5.21698	−0.239371
$476$	−25.6403	−1.17522
$477$	0	0
$478$	−40.4726	−1.85117
$479$	−27.5487	−1.25873	−0.629367	−	0.777108i	$-0.716686\pi$
$479$	−27.5487	−1.25873	−0.629367	+	0.777108i	$0.716686\pi$
$480$	0	0
$481$	−3.29959	−0.150448
$482$	49.3999	2.25010
$483$	0	0
$484$	0	0
$485$	−2.36327	−0.107310
$486$	0	0
$487$	−12.1013	−0.548364	−0.274182	−	0.961678i	$-0.588407\pi$
$487$	−12.1013	−0.548364	−0.274182	+	0.961678i	$0.588407\pi$
$488$	−44.0720	−1.99505
$489$	0	0
$490$	1.20312	0.0543513
$491$	−39.3867	−1.77750	−0.888749	−	0.458394i	$-0.848425\pi$
$491$	−39.3867	−1.77750	−0.888749	+	0.458394i	$0.848425\pi$
$492$	0	0
$493$	−9.58656	−0.431757
$494$	−1.32432	−0.0595842
$495$	0	0
$496$	111.925	5.02560
$497$	−5.92165	−0.265622
$498$	0	0
$499$	−34.7832	−1.55711	−0.778555	−	0.627576i	$-0.784047\pi$
$499$	−34.7832	−1.55711	−0.778555	+	0.627576i	$0.784047\pi$
$500$	−23.1533	−1.03545
$501$	0	0
$502$	−28.5590	−1.27465
$503$	3.23224	0.144119	0.0720593	−	0.997400i	$-0.477043\pi$
$503$	3.23224	0.144119	0.0720593	+	0.997400i	$0.477043\pi$
$504$	0	0
$505$	6.92550	0.308181
$506$	0	0
$507$	0	0
$508$	−53.7824	−2.38621
$509$	13.2217	0.586042	0.293021	−	0.956106i	$-0.405339\pi$
$509$	13.2217	0.586042	0.293021	+	0.956106i	$0.405339\pi$
$510$	0	0
$511$	−1.65776	−0.0733351
$512$	11.8512	0.523752
$513$	0	0
$514$	52.6257	2.32122
$515$	−6.30727	−0.277931
$516$	0	0
$517$	0	0
$518$	−19.7800	−0.869082
$519$	0	0
$520$	−1.79428	−0.0786843
$521$	−7.30239	−0.319924	−0.159962	−	0.987123i	$-0.551137\pi$
$521$	−7.30239	−0.319924	−0.159962	+	0.987123i	$0.551137\pi$
$522$	0	0
$523$	−8.38007	−0.366435	−0.183217	−	0.983072i	$-0.558651\pi$
$523$	−8.38007	−0.366435	−0.183217	+	0.983072i	$0.558651\pi$
$524$	−36.2667	−1.58432
$525$	0	0
$526$	57.2767	2.49738
$527$	39.8988	1.73802
$528$	0	0
$529$	−2.09483	−0.0910794
$530$	4.29780	0.186684
$531$	0	0
$532$	−5.76602	−0.249989
$533$	−0.798481	−0.0345861
$534$	0	0
$535$	−5.23188	−0.226194
$536$	−55.3271	−2.38977
$537$	0	0
$538$	−62.5963	−2.69872
$539$	0	0
$540$	0	0
$541$	3.79768	0.163275	0.0816375	−	0.996662i	$-0.473985\pi$
$541$	3.79768	0.163275	0.0816375	+	0.996662i	$0.473985\pi$
$542$	−72.5083	−3.11450
$543$	0	0
$544$	90.6080	3.88478
$545$	−7.02323	−0.300842
$546$	0	0
$547$	−9.08985	−0.388654	−0.194327	−	0.980937i	$-0.562252\pi$
$547$	−9.08985	−0.388654	−0.194327	+	0.980937i	$0.562252\pi$
$548$	−76.7818	−3.27996
$549$	0	0
$550$	0	0
$551$	−2.15583	−0.0918416
$552$	0	0
$553$	−3.60833	−0.153442
$554$	37.9820	1.61370
$555$	0	0
$556$	77.9939	3.30768
$557$	18.2372	0.772736	0.386368	−	0.922345i	$-0.373730\pi$
$557$	18.2372	0.772736	0.386368	+	0.922345i	$0.373730\pi$
$558$	0	0
$559$	−5.14444	−0.217586
$560$	−6.03187	−0.254893
$561$	0	0
$562$	−4.71809	−0.199021
$563$	−15.5744	−0.656384	−0.328192	−	0.944611i	$-0.606439\pi$
$563$	−15.5744	−0.656384	−0.328192	+	0.944611i	$0.606439\pi$
$564$	0	0
$565$	2.63637	0.110913
$566$	12.6112	0.530090
$567$	0	0
$568$	52.9407	2.22134
$569$	24.4254	1.02397	0.511984	−	0.858995i	$-0.328911\pi$
$569$	24.4254	1.02397	0.511984	+	0.858995i	$0.328911\pi$
$570$	0	0
$571$	14.9541	0.625808	0.312904	−	0.949785i	$-0.398698\pi$
$571$	14.9541	0.625808	0.312904	+	0.949785i	$0.398698\pi$
$572$	0	0
$573$	0	0
$574$	−4.78663	−0.199790
$575$	21.9554	0.915603
$576$	0	0
$577$	−21.4682	−0.893734	−0.446867	−	0.894600i	$-0.647460\pi$
$577$	−21.4682	−0.893734	−0.446867	+	0.894600i	$0.647460\pi$
$578$	17.1389	0.712885
$579$	0	0
$580$	−4.68718	−0.194625
$581$	10.8048	0.448260
$582$	0	0
$583$	0	0
$584$	14.8207	0.613286
$585$	0	0
$586$	39.2212	1.62021
$587$	20.6760	0.853391	0.426695	−	0.904395i	$-0.359678\pi$
$587$	20.6760	0.853391	0.426695	+	0.904395i	$0.359678\pi$
$588$	0	0
$589$	8.97246	0.369704
$590$	17.3244	0.713234
$591$	0	0
$592$	99.1675	4.07576
$593$	3.30237	0.135612	0.0678060	−	0.997699i	$-0.478400\pi$
$593$	3.30237	0.135612	0.0678060	+	0.997699i	$0.478400\pi$
$594$	0	0
$595$	−2.15022	−0.0881505
$596$	5.31469	0.217698
$597$	0	0
$598$	5.57335	0.227911
$599$	−38.9212	−1.59028	−0.795140	−	0.606426i	$-0.792602\pi$
$599$	−38.9212	−1.59028	−0.795140	+	0.606426i	$0.792602\pi$
$600$	0	0
$601$	−44.3783	−1.81023	−0.905115	−	0.425168i	$-0.860215\pi$
$601$	−44.3783	−1.81023	−0.905115	+	0.425168i	$0.860215\pi$
$602$	−30.8392	−1.25691
$603$	0	0
$604$	−7.29954	−0.297014
$605$	0	0
$606$	0	0
$607$	18.9059	0.767365	0.383683	−	0.923465i	$-0.374656\pi$
$607$	18.9059	0.767365	0.383683	+	0.923465i	$0.374656\pi$
$608$	20.3760	0.826355
$609$	0	0
$610$	−5.93095	−0.240137
$611$	0.461118	0.0186548
$612$	0	0
$613$	9.68116	0.391018	0.195509	−	0.980702i	$-0.437364\pi$
$613$	9.68116	0.391018	0.195509	+	0.980702i	$0.437364\pi$
$614$	−6.37088	−0.257108
$615$	0	0
$616$	0	0
$617$	26.2775	1.05789	0.528947	−	0.848655i	$-0.322587\pi$
$617$	26.2775	1.05789	0.528947	+	0.848655i	$0.322587\pi$
$618$	0	0
$619$	−24.7954	−0.996610	−0.498305	−	0.867002i	$-0.666044\pi$
$619$	−24.7954	−0.996610	−0.498305	+	0.867002i	$0.666044\pi$
$620$	19.5078	0.783452
$621$	0	0
$622$	54.0774	2.16831
$623$	−5.21170	−0.208802
$624$	0	0
$625$	22.0679	0.882716
$626$	50.6931	2.02611
$627$	0	0
$628$	28.5355	1.13869
$629$	35.3509	1.40953
$630$	0	0
$631$	−11.8107	−0.470178	−0.235089	−	0.971974i	$-0.575538\pi$
$631$	−11.8107	−0.470178	−0.235089	+	0.971974i	$0.575538\pi$
$632$	32.2592	1.28320
$633$	0	0
$634$	−11.3589	−0.451120
$635$	−4.51024	−0.178983
$636$	0	0
$637$	−0.450933	−0.0178666
$638$	0	0
$639$	0	0
$640$	11.6905	0.462108
$641$	22.5182	0.889416	0.444708	−	0.895676i	$-0.353307\pi$
$641$	22.5182	0.889416	0.444708	+	0.895676i	$0.353307\pi$
$642$	0	0
$643$	13.8901	0.547774	0.273887	−	0.961762i	$-0.411691\pi$
$643$	13.8901	0.547774	0.273887	+	0.961762i	$0.411691\pi$
$644$	24.2660	0.956214
$645$	0	0
$646$	14.1885	0.558237
$647$	−50.8116	−1.99761	−0.998805	−	0.0488792i	$-0.984435\pi$
$647$	−50.8116	−1.99761	−0.998805	+	0.0488792i	$0.984435\pi$
$648$	0	0
$649$	0	0
$650$	5.85334	0.229587
$651$	0	0
$652$	−50.0217	−1.95900
$653$	4.81411	0.188391	0.0941953	−	0.995554i	$-0.469972\pi$
$653$	4.81411	0.188391	0.0941953	+	0.995554i	$0.469972\pi$
$654$	0	0
$655$	−3.04136	−0.118836
$656$	23.9979	0.936962
$657$	0	0
$658$	2.76425	0.107762
$659$	−28.0409	−1.09232	−0.546159	−	0.837681i	$-0.683911\pi$
$659$	−28.0409	−1.09232	−0.546159	+	0.837681i	$0.683911\pi$
$660$	0	0
$661$	−41.7390	−1.62346	−0.811730	−	0.584033i	$-0.801474\pi$
$661$	−41.7390	−1.62346	−0.811730	+	0.584033i	$0.801474\pi$
$662$	−0.184545	−0.00717256
$663$	0	0
$664$	−96.5973	−3.74870
$665$	−0.483543	−0.0187510
$666$	0	0
$667$	9.07271	0.351297
$668$	106.208	4.10932
$669$	0	0
$670$	−7.44559	−0.287648
$671$	0	0
$672$	0	0
$673$	28.7747	1.10918	0.554592	−	0.832123i	$-0.312874\pi$
$673$	28.7747	1.10918	0.554592	+	0.832123i	$0.312874\pi$
$674$	−58.7841	−2.26428
$675$	0	0
$676$	−67.9153	−2.61213
$677$	−12.2862	−0.472195	−0.236098	−	0.971729i	$-0.575869\pi$
$677$	−12.2862	−0.472195	−0.236098	+	0.971729i	$0.575869\pi$
$678$	0	0
$679$	5.30985	0.203773
$680$	19.2234	0.737184
$681$	0	0
$682$	0	0
$683$	−30.5940	−1.17065	−0.585323	−	0.810800i	$-0.699032\pi$
$683$	−30.5940	−1.17065	−0.585323	+	0.810800i	$0.699032\pi$
$684$	0	0
$685$	−6.43899	−0.246021
$686$	−2.70320	−0.103209
$687$	0	0
$688$	154.613	5.89458
$689$	−1.61083	−0.0613678
$690$	0	0
$691$	−17.1121	−0.650976	−0.325488	−	0.945546i	$-0.605529\pi$
$691$	−17.1121	−0.650976	−0.325488	+	0.945546i	$0.605529\pi$
$692$	57.5655	2.18831
$693$	0	0
$694$	64.7013	2.45603
$695$	6.54063	0.248100
$696$	0	0
$697$	8.55471	0.324033
$698$	71.4051	2.70272
$699$	0	0
$700$	25.4850	0.963244
$701$	13.5936	0.513423	0.256712	−	0.966488i	$-0.417361\pi$
$701$	13.5936	0.513423	0.256712	+	0.966488i	$0.417361\pi$
$702$	0	0
$703$	7.94974	0.299830
$704$	0	0
$705$	0	0
$706$	12.0071	0.451893
$707$	−15.5604	−0.585208
$708$	0	0
$709$	29.5995	1.11163	0.555816	−	0.831305i	$-0.312406\pi$
$709$	29.5995	1.11163	0.555816	+	0.831305i	$0.312406\pi$
$710$	7.12444	0.267376
$711$	0	0
$712$	46.5936	1.74617
$713$	−37.7601	−1.41413
$714$	0	0
$715$	0	0
$716$	−89.4190	−3.34174
$717$	0	0
$718$	−39.2842	−1.46608
$719$	−45.5407	−1.69838	−0.849190	−	0.528087i	$-0.822909\pi$
$719$	−45.5407	−1.69838	−0.849190	+	0.528087i	$0.822909\pi$
$720$	0	0
$721$	14.1713	0.527768
$722$	−48.1700	−1.79270
$723$	0	0
$724$	−107.759	−4.00485
$725$	9.52850	0.353880
$726$	0	0
$727$	−43.8796	−1.62740	−0.813702	−	0.581283i	$-0.802551\pi$
$727$	−43.8796	−1.62740	−0.813702	+	0.581283i	$0.802551\pi$
$728$	4.03143	0.149415
$729$	0	0
$730$	1.99448	0.0738192
$731$	55.1161	2.03854
$732$	0	0
$733$	−14.0878	−0.520345	−0.260172	−	0.965562i	$-0.583779\pi$
$733$	−14.0878	−0.520345	−0.260172	+	0.965562i	$0.583779\pi$
$734$	13.4662	0.497046
$735$	0	0
$736$	−85.7513	−3.16083
$737$	0	0
$738$	0	0
$739$	39.4975	1.45294	0.726470	−	0.687199i	$-0.241160\pi$
$739$	39.4975	1.45294	0.726470	+	0.687199i	$0.241160\pi$
$740$	17.2842	0.635380
$741$	0	0
$742$	−9.65640	−0.354498
$743$	18.6162	0.682962	0.341481	−	0.939889i	$-0.389072\pi$
$743$	18.6162	0.682962	0.341481	+	0.939889i	$0.389072\pi$
$744$	0	0
$745$	0.445694	0.0163290
$746$	36.9360	1.35232
$747$	0	0
$748$	0	0
$749$	11.7551	0.429523
$750$	0	0
$751$	−15.8060	−0.576768	−0.288384	−	0.957515i	$-0.593118\pi$
$751$	−15.8060	−0.576768	−0.288384	+	0.957515i	$0.593118\pi$
$752$	−13.8587	−0.505373
$753$	0	0
$754$	2.41880	0.0880875
$755$	−0.612146	−0.0222783
$756$	0	0
$757$	−3.61112	−0.131248	−0.0656241	−	0.997844i	$-0.520904\pi$
$757$	−3.61112	−0.131248	−0.0656241	+	0.997844i	$0.520904\pi$
$758$	−22.6050	−0.821051
$759$	0	0
$760$	4.32297	0.156811
$761$	39.5819	1.43484	0.717422	−	0.696639i	$-0.245322\pi$
$761$	39.5819	1.43484	0.717422	+	0.696639i	$0.245322\pi$
$762$	0	0
$763$	15.7800	0.571273
$764$	−13.5035	−0.488541
$765$	0	0
$766$	5.37068	0.194051
$767$	−6.49325	−0.234458
$768$	0	0
$769$	10.5472	0.380341	0.190171	−	0.981751i	$-0.439096\pi$
$769$	10.5472	0.380341	0.190171	+	0.981751i	$0.439096\pi$
$770$	0	0
$771$	0	0
$772$	−92.9228	−3.34437
$773$	−41.5632	−1.49492	−0.747462	−	0.664305i	$-0.768728\pi$
$773$	−41.5632	−1.49492	−0.747462	+	0.664305i	$0.768728\pi$
$774$	0	0
$775$	−39.6571	−1.42452
$776$	−47.4711	−1.70411
$777$	0	0
$778$	−40.0952	−1.43748
$779$	1.92379	0.0689269
$780$	0	0
$781$	0	0
$782$	−59.7114	−2.13527
$783$	0	0
$784$	13.5526	0.484020
$785$	2.39301	0.0854103
$786$	0	0
$787$	−11.5474	−0.411620	−0.205810	−	0.978592i	$-0.565983\pi$
$787$	−11.5474	−0.411620	−0.205810	+	0.978592i	$0.565983\pi$
$788$	−11.4933	−0.409433
$789$	0	0
$790$	4.34125	0.154455
$791$	−5.92347	−0.210614
$792$	0	0
$793$	2.22294	0.0789390
$794$	30.9869	1.09968
$795$	0	0
$796$	−77.3569	−2.74184
$797$	−42.3155	−1.49889	−0.749447	−	0.662065i	$-0.769680\pi$
$797$	−42.3155	−1.49889	−0.749447	+	0.662065i	$0.769680\pi$
$798$	0	0
$799$	−4.94029	−0.174775
$800$	−90.0591	−3.18407
$801$	0	0
$802$	−11.6516	−0.411431
$803$	0	0
$804$	0	0
$805$	2.03497	0.0717232
$806$	−10.0669	−0.354592
$807$	0	0
$808$	139.113	4.89397
$809$	−17.9814	−0.632194	−0.316097	−	0.948727i	$-0.602372\pi$
$809$	−17.9814	−0.632194	−0.316097	+	0.948727i	$0.602372\pi$
$810$	0	0
$811$	−31.1200	−1.09277	−0.546386	−	0.837533i	$-0.683997\pi$
$811$	−31.1200	−1.09277	−0.546386	+	0.837533i	$0.683997\pi$
$812$	10.5313	0.369576
$813$	0	0
$814$	0	0
$815$	−4.19486	−0.146940
$816$	0	0
$817$	12.3945	0.433630
$818$	25.1722	0.880126
$819$	0	0
$820$	4.18268	0.146065
$821$	48.8675	1.70549	0.852744	−	0.522329i	$-0.174937\pi$
$821$	48.8675	1.70549	0.852744	+	0.522329i	$0.174937\pi$
$822$	0	0
$823$	29.5821	1.03117	0.515584	−	0.856839i	$-0.327575\pi$
$823$	29.5821	1.03117	0.515584	+	0.856839i	$0.327575\pi$
$824$	−126.695	−4.41361
$825$	0	0
$826$	−38.9249	−1.35437
$827$	25.3924	0.882981	0.441490	−	0.897266i	$-0.354450\pi$
$827$	25.3924	0.882981	0.441490	+	0.897266i	$0.354450\pi$
$828$	0	0
$829$	7.09580	0.246447	0.123224	−	0.992379i	$-0.460677\pi$
$829$	7.09580	0.246447	0.123224	+	0.992379i	$0.460677\pi$
$830$	−12.9995	−0.451219
$831$	0	0
$832$	−10.6388	−0.368835
$833$	4.83117	0.167390
$834$	0	0
$835$	8.90671	0.308230
$836$	0	0
$837$	0	0
$838$	71.2086	2.45986
$839$	23.7285	0.819197	0.409599	−	0.912266i	$-0.365669\pi$
$839$	23.7285	0.819197	0.409599	+	0.912266i	$0.365669\pi$
$840$	0	0
$841$	−25.0625	−0.864224
$842$	38.3292	1.32091
$843$	0	0
$844$	19.2672	0.663204
$845$	−5.69544	−0.195929
$846$	0	0
$847$	0	0
$848$	48.4127	1.66250
$849$	0	0
$850$	−62.7111	−2.15097
$851$	−33.4561	−1.14686
$852$	0	0
$853$	38.6542	1.32350	0.661748	−	0.749726i	$-0.269815\pi$
$853$	38.6542	1.32350	0.661748	+	0.749726i	$0.269815\pi$
$854$	13.3258	0.456000
$855$	0	0
$856$	−105.093	−3.59201
$857$	34.0838	1.16428	0.582139	−	0.813089i	$-0.302216\pi$
$857$	34.0838	1.16428	0.582139	+	0.813089i	$0.302216\pi$
$858$	0	0
$859$	56.7283	1.93555	0.967773	−	0.251825i	$-0.0810308\pi$
$859$	56.7283	1.93555	0.967773	+	0.251825i	$0.0810308\pi$
$860$	26.9480	0.918920
$861$	0	0
$862$	20.5316	0.699310
$863$	−24.3930	−0.830346	−0.415173	−	0.909742i	$-0.636279\pi$
$863$	−24.3930	−0.830346	−0.415173	+	0.909742i	$0.636279\pi$
$864$	0	0
$865$	4.82750	0.164140
$866$	60.5539	2.05770
$867$	0	0
$868$	−43.8306	−1.48771
$869$	0	0
$870$	0	0
$871$	2.79064	0.0945571
$872$	−141.076	−4.77744
$873$	0	0
$874$	−13.4279	−0.454207
$875$	4.36256	0.147481
$876$	0	0
$877$	−30.4950	−1.02974	−0.514871	−	0.857267i	$-0.672160\pi$
$877$	−30.4950	−1.02974	−0.514871	+	0.857267i	$0.672160\pi$
$878$	−41.6429	−1.40538
$879$	0	0
$880$	0	0
$881$	−2.10056	−0.0707697	−0.0353848	−	0.999374i	$-0.511266\pi$
$881$	−2.10056	−0.0707697	−0.0353848	+	0.999374i	$0.511266\pi$
$882$	0	0
$883$	45.1955	1.52095	0.760475	−	0.649367i	$-0.224966\pi$
$883$	45.1955	1.52095	0.760475	+	0.649367i	$0.224966\pi$
$884$	−11.5621	−0.388875
$885$	0	0
$886$	−57.3346	−1.92619
$887$	5.18758	0.174182	0.0870910	−	0.996200i	$-0.472243\pi$
$887$	5.18758	0.174182	0.0870910	+	0.996200i	$0.472243\pi$
$888$	0	0
$889$	10.1337	0.339874
$890$	6.27029	0.210181
$891$	0	0
$892$	−25.6905	−0.860181
$893$	−1.11098	−0.0371774
$894$	0	0
$895$	−7.49875	−0.250656
$896$	−26.2666	−0.877504
$897$	0	0
$898$	31.4426	1.04925
$899$	−16.3877	−0.546559
$900$	0	0
$901$	17.2580	0.574947
$902$	0	0
$903$	0	0
$904$	52.9570	1.76132
$905$	−9.03681	−0.300394
$906$	0	0
$907$	−38.6473	−1.28326	−0.641631	−	0.767013i	$-0.721742\pi$
$907$	−38.6473	−1.28326	−0.641631	+	0.767013i	$0.721742\pi$
$908$	120.637	4.00350
$909$	0	0
$910$	0.542526	0.0179846
$911$	18.3990	0.609585	0.304793	−	0.952419i	$-0.401413\pi$
$911$	18.3990	0.609585	0.304793	+	0.952419i	$0.401413\pi$
$912$	0	0
$913$	0	0
$914$	−85.4145	−2.82526
$915$	0	0
$916$	−120.333	−3.97590
$917$	6.83340	0.225659
$918$	0	0
$919$	−17.6533	−0.582330	−0.291165	−	0.956673i	$-0.594043\pi$
$919$	−17.6533	−0.582330	−0.291165	+	0.956673i	$0.594043\pi$
$920$	−18.1930	−0.599806
$921$	0	0
$922$	−23.9591	−0.789050
$923$	−2.67027	−0.0878930
$924$	0	0
$925$	−35.1368	−1.15529
$926$	−69.2375	−2.27529
$927$	0	0
$928$	−37.2155	−1.22166
$929$	13.4484	0.441226	0.220613	−	0.975361i	$-0.429194\pi$
$929$	13.4484	0.441226	0.220613	+	0.975361i	$0.429194\pi$
$930$	0	0
$931$	1.08644	0.0356066
$932$	−3.11687	−0.102096
$933$	0	0
$934$	−55.5241	−1.81680
$935$	0	0
$936$	0	0
$937$	8.59580	0.280813	0.140406	−	0.990094i	$-0.455159\pi$
$937$	8.59580	0.280813	0.140406	+	0.990094i	$0.455159\pi$
$938$	16.7289	0.546219
$939$	0	0
$940$	−2.41547	−0.0787839
$941$	−33.6975	−1.09851	−0.549254	−	0.835655i	$-0.685088\pi$
$941$	−33.6975	−1.09851	−0.549254	+	0.835655i	$0.685088\pi$
$942$	0	0
$943$	−8.09617	−0.263647
$944$	195.151	6.35163
$945$	0	0
$946$	0	0
$947$	15.3289	0.498121	0.249061	−	0.968488i	$-0.419878\pi$
$947$	15.3289	0.498121	0.249061	+	0.968488i	$0.419878\pi$
$948$	0	0
$949$	−0.747541	−0.0242662
$950$	−14.1025	−0.457546
$951$	0	0
$952$	−43.1916	−1.39985
$953$	44.9376	1.45567	0.727835	−	0.685752i	$-0.240527\pi$
$953$	44.9376	1.45567	0.727835	+	0.685752i	$0.240527\pi$
$954$	0	0
$955$	−1.13242	−0.0366442
$956$	−79.4610	−2.56995
$957$	0	0
$958$	−74.4697	−2.40601
$959$	14.4673	0.467173
$960$	0	0
$961$	37.2046	1.20015
$962$	−8.91944	−0.287574
$963$	0	0
$964$	96.9883	3.12378
$965$	−7.79259	−0.250852
$966$	0	0
$967$	−33.9453	−1.09161	−0.545804	−	0.837913i	$-0.683776\pi$
$967$	−33.9453	−1.09161	−0.545804	+	0.837913i	$0.683776\pi$
$968$	0	0
$969$	0	0
$970$	−6.38837	−0.205118
$971$	55.4600	1.77980	0.889899	−	0.456158i	$-0.150775\pi$
$971$	55.4600	1.77980	0.889899	+	0.456158i	$0.150775\pi$
$972$	0	0
$973$	−14.6957	−0.471121
$974$	−32.7123	−1.04817
$975$	0	0
$976$	−66.8094	−2.13852
$977$	37.9269	1.21339	0.606694	−	0.794936i	$-0.292495\pi$
$977$	37.9269	1.21339	0.606694	+	0.794936i	$0.292495\pi$
$978$	0	0
$979$	0	0
$980$	2.36212	0.0754551
$981$	0	0
$982$	−106.470	−3.39760
$983$	−31.5337	−1.00577	−0.502884	−	0.864354i	$-0.667728\pi$
$983$	−31.5337	−1.00577	−0.502884	+	0.864354i	$0.667728\pi$
$984$	0	0
$985$	−0.963841	−0.0307105
$986$	−25.9144	−0.825281
$987$	0	0
$988$	−2.60009	−0.0827198
$989$	−52.1618	−1.65865
$990$	0	0
$991$	39.8173	1.26484	0.632419	−	0.774627i	$-0.282062\pi$
$991$	39.8173	1.26484	0.632419	+	0.774627i	$0.282062\pi$
$992$	154.889	4.91773
$993$	0	0
$994$	−16.0074	−0.507723
$995$	−6.48721	−0.205659
$996$	0	0
$997$	18.7190	0.592836	0.296418	−	0.955058i	$-0.404208\pi$
$997$	18.7190	0.592836	0.296418	+	0.955058i	$0.404208\pi$
$998$	−94.0258	−2.97634
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cs.1.6		6
3.2	odd	2		847.2.a.m.1.1	✓	6
11.10	odd	2		7623.2.a.cp.1.1		6
21.20	even	2		5929.2.a.bj.1.1		6
33.2	even	10		847.2.f.y.323.1		24
33.5	odd	10		847.2.f.z.729.6		24
33.8	even	10		847.2.f.y.372.6		24
33.14	odd	10		847.2.f.z.372.1		24
33.17	even	10		847.2.f.y.729.1		24
33.20	odd	10		847.2.f.z.323.6		24
33.26	odd	10		847.2.f.z.148.1		24
33.29	even	10		847.2.f.y.148.6		24
33.32	even	2		847.2.a.n.1.6	yes	6
231.230	odd	2		5929.2.a.bm.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.1	✓	6	3.2	odd	2
847.2.a.n.1.6	yes	6	33.32	even	2
847.2.f.y.148.6		24	33.29	even	10
847.2.f.y.323.1		24	33.2	even	10
847.2.f.y.372.6		24	33.8	even	10
847.2.f.y.729.1		24	33.17	even	10
847.2.f.z.148.1		24	33.26	odd	10
847.2.f.z.323.6		24	33.20	odd	10
847.2.f.z.372.1		24	33.14	odd	10
847.2.f.z.729.6		24	33.5	odd	10
5929.2.a.bj.1.1		6	21.20	even	2
5929.2.a.bm.1.6		6	231.230	odd	2
7623.2.a.cp.1.1		6	11.10	odd	2
7623.2.a.cs.1.6		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)

\(f(q)\)	\(=\)	\(q+2.70320 q^{2} +5.30727 q^{4} +0.445072 q^{5} -1.00000 q^{7} +8.94020 q^{8} +O(q^{10})\)	\(q+2.70320 q^{2} +5.30727 q^{4} +0.445072 q^{5} -1.00000 q^{7} +8.94020 q^{8} +1.20312 q^{10} -0.450933 q^{13} -2.70320 q^{14} +13.5526 q^{16} +4.83117 q^{17} +1.08644 q^{19} +2.36212 q^{20} -4.57222 q^{23} -4.80191 q^{25} -1.21896 q^{26} -5.30727 q^{28} -1.98431 q^{29} +8.25861 q^{31} +18.7549 q^{32} +13.0596 q^{34} -0.445072 q^{35} +7.31725 q^{37} +2.93685 q^{38} +3.97903 q^{40} +1.77073 q^{41} +11.4084 q^{43} -12.3596 q^{46} -1.02259 q^{47} +1.00000 q^{49} -12.9805 q^{50} -2.39322 q^{52} +3.57222 q^{53} -8.94020 q^{56} -5.36399 q^{58} +14.3996 q^{59} -4.92965 q^{61} +22.3246 q^{62} +23.5929 q^{64} -0.200698 q^{65} -6.18858 q^{67} +25.6403 q^{68} -1.20312 q^{70} +5.92165 q^{71} +1.65776 q^{73} +19.7800 q^{74} +5.76602 q^{76} +3.60833 q^{79} +6.03187 q^{80} +4.78663 q^{82} -10.8048 q^{83} +2.15022 q^{85} +30.8392 q^{86} +5.21170 q^{89} +0.450933 q^{91} -24.2660 q^{92} -2.76425 q^{94} +0.483543 q^{95} -5.30985 q^{97} +2.70320 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 6 * q^7 + 12 * q^8	\( 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8} + 8 q^{10} - 4 q^{13} - 4 q^{14} + 8 q^{16} + 22 q^{17} - 6 q^{19} - 2 q^{20} - 2 q^{23} + 4 q^{25} - 6 q^{26} - 4 q^{28} + 12 q^{29} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 14 q^{37} + 22 q^{38} - 18 q^{40} + 26 q^{41} + 4 q^{43} - 12 q^{46} + 16 q^{47} + 6 q^{49} - 4 q^{50} - 12 q^{52} - 4 q^{53} - 12 q^{56} - 2 q^{58} + 4 q^{59} + 8 q^{61} + 20 q^{62} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 8 q^{70} - 22 q^{71} - 14 q^{73} + 44 q^{74} + 30 q^{76} + 28 q^{79} + 4 q^{80} - 4 q^{82} + 22 q^{83} + 24 q^{85} + 30 q^{86} + 4 q^{91} - 10 q^{92} + 38 q^{94} - 24 q^{95} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 6 * q^7 + 12 * q^8 + 8 * q^10 - 4 * q^13 - 4 * q^14 + 8 * q^16 + 22 * q^17 - 6 * q^19 - 2 * q^20 - 2 * q^23 + 4 * q^25 - 6 * q^26 - 4 * q^28 + 12 * q^29 - 2 * q^31 + 8 * q^32 + 24 * q^34 - 4 * q^35 + 14 * q^37 + 22 * q^38 - 18 * q^40 + 26 * q^41 + 4 * q^43 - 12 * q^46 + 16 * q^47 + 6 * q^49 - 4 * q^50 - 12 * q^52 - 4 * q^53 - 12 * q^56 - 2 * q^58 + 4 * q^59 + 8 * q^61 + 20 * q^62 + 26 * q^64 + 24 * q^65 + 6 * q^67 + 12 * q^68 - 8 * q^70 - 22 * q^71 - 14 * q^73 + 44 * q^74 + 30 * q^76 + 28 * q^79 + 4 * q^80 - 4 * q^82 + 22 * q^83 + 24 * q^85 + 30 * q^86 + 4 * q^91 - 10 * q^92 + 38 * q^94 - 24 * q^95 - 4 * q^97 + 4 * q^98

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