LMFDB - Embedded newform 7623.2.a.cs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
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			1.1
Root			$2.38595$ 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)

$f(q)$	$=$	$q-1.38595 q^{2} -0.0791355 q^{4} +0.133004 q^{5} -1.00000 q^{7} +2.88158 q^{8} +O(q^{10})$	$q-1.38595 q^{2} -0.0791355 q^{4} +0.133004 q^{5} -1.00000 q^{7} +2.88158 q^{8} -0.184338 q^{10} +0.641436 q^{13} +1.38595 q^{14} -3.83547 q^{16} -1.42482 q^{17} -7.18297 q^{19} -0.0105254 q^{20} -1.66655 q^{23} -4.98231 q^{25} -0.889000 q^{26} +0.0791355 q^{28} +4.47657 q^{29} -6.83182 q^{31} -0.447392 q^{32} +1.97473 q^{34} -0.133004 q^{35} +3.76864 q^{37} +9.95526 q^{38} +0.383263 q^{40} +6.17286 q^{41} +1.03970 q^{43} +2.30976 q^{46} -9.48937 q^{47} +1.00000 q^{49} +6.90524 q^{50} -0.0507604 q^{52} +0.666549 q^{53} -2.88158 q^{56} -6.20431 q^{58} -8.18695 q^{59} -9.49807 q^{61} +9.46858 q^{62} +8.29100 q^{64} +0.0853139 q^{65} +12.0398 q^{67} +0.112754 q^{68} +0.184338 q^{70} -4.83418 q^{71} +8.85748 q^{73} -5.22316 q^{74} +0.568428 q^{76} -11.4907 q^{79} -0.510134 q^{80} -8.55529 q^{82} +9.57998 q^{83} -0.189507 q^{85} -1.44098 q^{86} +17.7001 q^{89} -0.641436 q^{91} +0.131883 q^{92} +13.1518 q^{94} -0.955367 q^{95} +6.58139 q^{97} -1.38595 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.38595	−0.980016	−0.490008	−	0.871718i	$-0.663006\pi$
$2$	−1.38595	−0.980016	−0.490008	+	0.871718i	$0.663006\pi$
$3$	0	0
$3$	0	0
$4$	−0.0791355	−0.0395678
$5$	0.133004	0.0594814	0.0297407	−	0.999558i	$-0.490532\pi$
$5$	0.133004	0.0594814	0.0297407	+	0.999558i	$0.490532\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.88158	1.01879
$9$	0	0
$10$	−0.184338	−0.0582928
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.641436	0.177902	0.0889512	−	0.996036i	$-0.471648\pi$
$13$	0.641436	0.177902	0.0889512	+	0.996036i	$0.471648\pi$
$14$	1.38595	0.370411
$15$	0	0
$16$	−3.83547	−0.958867
$17$	−1.42482	−0.345570	−0.172785	−	0.984960i	$-0.555277\pi$
$17$	−1.42482	−0.345570	−0.172785	+	0.984960i	$0.555277\pi$
$18$	0	0
$19$	−7.18297	−1.64789	−0.823943	−	0.566672i	$-0.808231\pi$
$19$	−7.18297	−1.64789	−0.823943	+	0.566672i	$0.808231\pi$
$20$	−0.0105254	−0.00235355
$21$	0	0
$22$	0	0
$23$	−1.66655	−0.347500	−0.173750	−	0.984790i	$-0.555588\pi$
$23$	−1.66655	−0.347500	−0.173750	+	0.984790i	$0.555588\pi$
$24$	0	0
$25$	−4.98231	−0.996462
$26$	−0.889000	−0.174347
$27$	0	0
$28$	0.0791355	0.0149552
$29$	4.47657	0.831278	0.415639	−	0.909530i	$-0.363558\pi$
$29$	4.47657	0.831278	0.415639	+	0.909530i	$0.363558\pi$
$30$	0	0
$31$	−6.83182	−1.22703	−0.613516	−	0.789682i	$-0.710245\pi$
$31$	−6.83182	−1.22703	−0.613516	+	0.789682i	$0.710245\pi$
$32$	−0.447392	−0.0790885
$33$	0	0
$34$	1.97473	0.338664
$35$	−0.133004	−0.0224819
$36$	0	0
$37$	3.76864	0.619561	0.309781	−	0.950808i	$-0.399744\pi$
$37$	3.76864	0.619561	0.309781	+	0.950808i	$0.399744\pi$
$38$	9.95526	1.61496
$39$	0	0
$40$	0.383263	0.0605993
$41$	6.17286	0.964039	0.482019	−	0.876161i	$-0.339904\pi$
$41$	6.17286	0.964039	0.482019	+	0.876161i	$0.339904\pi$
$42$	0	0
$43$	1.03970	0.158553	0.0792764	−	0.996853i	$-0.474739\pi$
$43$	1.03970	0.158553	0.0792764	+	0.996853i	$0.474739\pi$
$44$	0	0
$45$	0	0
$46$	2.30976	0.340555
$47$	−9.48937	−1.38417	−0.692084	−	0.721817i	$-0.743307\pi$
$47$	−9.48937	−1.38417	−0.692084	+	0.721817i	$0.743307\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.90524	0.976549
$51$	0	0
$52$	−0.0507604	−0.00703920
$53$	0.666549	0.0915576	0.0457788	−	0.998952i	$-0.485423\pi$
$53$	0.666549	0.0915576	0.0457788	+	0.998952i	$0.485423\pi$
$54$	0	0
$55$	0	0
$56$	−2.88158	−0.385068
$57$	0	0
$58$	−6.20431	−0.814666
$59$	−8.18695	−1.06585	−0.532925	−	0.846162i	$-0.678907\pi$
$59$	−8.18695	−1.06585	−0.532925	+	0.846162i	$0.678907\pi$
$60$	0	0
$61$	−9.49807	−1.21610	−0.608051	−	0.793898i	$-0.708049\pi$
$61$	−9.49807	−1.21610	−0.608051	+	0.793898i	$0.708049\pi$
$62$	9.46858	1.20251
$63$	0	0
$64$	8.29100	1.03637
$65$	0.0853139	0.0105819
$66$	0	0
$67$	12.0398	1.47089	0.735446	−	0.677583i	$-0.236973\pi$
$67$	12.0398	1.47089	0.735446	+	0.677583i	$0.236973\pi$
$68$	0.112754	0.0136734
$69$	0	0
$70$	0.184338	0.0220326
$71$	−4.83418	−0.573711	−0.286856	−	0.957974i	$-0.592610\pi$
$71$	−4.83418	−0.573711	−0.286856	+	0.957974i	$0.592610\pi$
$72$	0	0
$73$	8.85748	1.03669	0.518345	−	0.855172i	$-0.326548\pi$
$73$	8.85748	1.03669	0.518345	+	0.855172i	$0.326548\pi$
$74$	−5.22316	−0.607180
$75$	0	0
$76$	0.568428	0.0652032
$77$	0	0
$78$	0	0
$79$	−11.4907	−1.29280	−0.646400	−	0.762998i	$-0.723726\pi$
$79$	−11.4907	−1.29280	−0.646400	+	0.762998i	$0.723726\pi$
$80$	−0.510134	−0.0570347
$81$	0	0
$82$	−8.55529	−0.944774
$83$	9.57998	1.05154	0.525770	−	0.850627i	$-0.323777\pi$
$83$	9.57998	1.05154	0.525770	+	0.850627i	$0.323777\pi$
$84$	0	0
$85$	−0.189507	−0.0205550
$86$	−1.44098	−0.155384
$87$	0	0
$88$	0	0
$89$	17.7001	1.87621	0.938104	−	0.346353i	$-0.112580\pi$
$89$	17.7001	1.87621	0.938104	+	0.346353i	$0.112580\pi$
$90$	0	0
$91$	−0.641436	−0.0672408
$92$	0.131883	0.0137498
$93$	0	0
$94$	13.1518	1.35651
$95$	−0.955367	−0.0980186
$96$	0	0
$97$	6.58139	0.668239	0.334119	−	0.942531i	$-0.391561\pi$
$97$	6.58139	0.668239	0.334119	+	0.942531i	$0.391561\pi$
$98$	−1.38595	−0.140002
$99$	0	0
$100$	0.394278	0.0394278
$101$	18.6204	1.85280	0.926400	−	0.376541i	$-0.122887\pi$
$101$	18.6204	1.85280	0.926400	+	0.376541i	$0.122887\pi$
$102$	0	0
$103$	−6.92356	−0.682199	−0.341099	−	0.940027i	$-0.610799\pi$
$103$	−6.92356	−0.682199	−0.341099	+	0.940027i	$0.610799\pi$
$104$	1.84835	0.181246
$105$	0	0
$106$	−0.923806	−0.0897279
$107$	−5.44446	−0.526336	−0.263168	−	0.964750i	$-0.584767\pi$
$107$	−5.44446	−0.526336	−0.263168	+	0.964750i	$0.584767\pi$
$108$	0	0
$109$	9.22316	0.883419	0.441709	−	0.897158i	$-0.354372\pi$
$109$	9.22316	0.883419	0.441709	+	0.897158i	$0.354372\pi$
$110$	0	0
$111$	0	0
$112$	3.83547	0.362418
$113$	−9.38223	−0.882606	−0.441303	−	0.897358i	$-0.645484\pi$
$113$	−9.38223	−0.882606	−0.441303	+	0.897358i	$0.645484\pi$
$114$	0	0
$115$	−0.221659	−0.0206698
$116$	−0.354256	−0.0328918
$117$	0	0
$118$	11.3467	1.04455
$119$	1.42482	0.130613
$120$	0	0
$121$	0	0
$122$	13.1639	1.19180
$123$	0	0
$124$	0.540640	0.0485509
$125$	−1.32769	−0.118752
$126$	0	0
$127$	−15.4560	−1.37150	−0.685749	−	0.727838i	$-0.740525\pi$
$127$	−15.4560	−1.37150	−0.685749	+	0.727838i	$0.740525\pi$
$128$	−10.5961	−0.936576
$129$	0	0
$130$	−0.118241	−0.0103704
$131$	17.6675	1.54362	0.771810	−	0.635854i	$-0.219352\pi$
$131$	17.6675	1.54362	0.771810	+	0.635854i	$0.219352\pi$
$132$	0	0
$133$	7.18297	0.622843
$134$	−16.6866	−1.44150
$135$	0	0
$136$	−4.10574	−0.352064
$137$	−19.0323	−1.62604	−0.813021	−	0.582235i	$-0.802178\pi$
$137$	−19.0323	−1.62604	−0.813021	+	0.582235i	$0.802178\pi$
$138$	0	0
$139$	14.0665	1.19310	0.596552	−	0.802574i	$-0.296537\pi$
$139$	14.0665	1.19310	0.596552	+	0.802574i	$0.296537\pi$
$140$	0.0105254	0.000889557 0
$141$	0	0
$142$	6.69994	0.562247
$143$	0	0
$144$	0	0
$145$	0.595404	0.0494456
$146$	−12.2760	−1.01597
$147$	0	0
$148$	−0.298234	−0.0245147
$149$	−11.0367	−0.904160	−0.452080	−	0.891978i	$-0.649318\pi$
$149$	−11.0367	−0.904160	−0.452080	+	0.891978i	$0.649318\pi$
$150$	0	0
$151$	16.1350	1.31305	0.656523	−	0.754306i	$-0.272026\pi$
$151$	16.1350	1.31305	0.656523	+	0.754306i	$0.272026\pi$
$152$	−20.6983	−1.67886
$153$	0	0
$154$	0	0
$155$	−0.908663	−0.0729856
$156$	0	0
$157$	−14.8699	−1.18674	−0.593372	−	0.804928i	$-0.702204\pi$
$157$	−14.8699	−1.18674	−0.593372	+	0.804928i	$0.702204\pi$
$158$	15.9255	1.26697
$159$	0	0
$160$	−0.0595051	−0.00470429
$161$	1.66655	0.131342
$162$	0	0
$163$	−16.4539	−1.28877	−0.644385	−	0.764701i	$-0.722887\pi$
$163$	−16.4539	−1.28877	−0.644385	+	0.764701i	$0.722887\pi$
$164$	−0.488493	−0.0381449
$165$	0	0
$166$	−13.2774	−1.03053
$167$	−19.1519	−1.48201	−0.741007	−	0.671497i	$-0.765652\pi$
$167$	−19.1519	−1.48201	−0.741007	+	0.671497i	$0.765652\pi$
$168$	0	0
$169$	−12.5886	−0.968351
$170$	0.262648	0.0201442
$171$	0	0
$172$	−0.0822773	−0.00627358
$173$	15.2246	1.15750	0.578752	−	0.815504i	$-0.303540\pi$
$173$	15.2246	1.15750	0.578752	+	0.815504i	$0.303540\pi$
$174$	0	0
$175$	4.98231	0.376627
$176$	0	0
$177$	0	0
$178$	−24.5315	−1.83872
$179$	−1.12862	−0.0843574	−0.0421787	−	0.999110i	$-0.513430\pi$
$179$	−1.12862	−0.0843574	−0.0421787	+	0.999110i	$0.513430\pi$
$180$	0	0
$181$	18.4775	1.37342	0.686711	−	0.726931i	$-0.259054\pi$
$181$	18.4775	1.37342	0.686711	+	0.726931i	$0.259054\pi$
$182$	0.889000	0.0658971
$183$	0	0
$184$	−4.80230	−0.354030
$185$	0.501246	0.0368524
$186$	0	0
$187$	0	0
$188$	0.750947	0.0547684
$189$	0	0
$190$	1.32409	0.0960599
$191$	2.42981	0.175815	0.0879076	−	0.996129i	$-0.471982\pi$
$191$	2.42981	0.175815	0.0879076	+	0.996129i	$0.471982\pi$
$192$	0	0
$193$	0.263324	0.0189545	0.00947725	−	0.999955i	$-0.496983\pi$
$193$	0.263324	0.0189545	0.00947725	+	0.999955i	$0.496983\pi$
$194$	−9.12149	−0.654885
$195$	0	0
$196$	−0.0791355	−0.00565254
$197$	17.4681	1.24455	0.622276	−	0.782798i	$-0.286208\pi$
$197$	17.4681	1.24455	0.622276	+	0.782798i	$0.286208\pi$
$198$	0	0
$199$	−20.1415	−1.42780	−0.713898	−	0.700250i	$-0.753072\pi$
$199$	−20.1415	−1.42780	−0.713898	+	0.700250i	$0.753072\pi$
$200$	−14.3569	−1.01519
$201$	0	0
$202$	−25.8070	−1.81577
$203$	−4.47657	−0.314193
$204$	0	0
$205$	0.821018	0.0573424
$206$	9.59572	0.668566
$207$	0	0
$208$	−2.46021	−0.170585
$209$	0	0
$210$	0	0
$211$	14.0802	0.969320	0.484660	−	0.874702i	$-0.338943\pi$
$211$	14.0802	0.969320	0.484660	+	0.874702i	$0.338943\pi$
$212$	−0.0527477	−0.00362273
$213$	0	0
$214$	7.54576	0.515817
$215$	0.138285	0.00943095
$216$	0	0
$217$	6.83182	0.463774
$218$	−12.7829	−0.865765
$219$	0	0
$220$	0	0
$221$	−0.913931	−0.0614777
$222$	0	0
$223$	−9.63364	−0.645116	−0.322558	−	0.946550i	$-0.604543\pi$
$223$	−9.63364	−0.645116	−0.322558	+	0.946550i	$0.604543\pi$
$224$	0.447392	0.0298926
$225$	0	0
$226$	13.0033	0.864969
$227$	−10.4092	−0.690880	−0.345440	−	0.938441i	$-0.612270\pi$
$227$	−10.4092	−0.690880	−0.345440	+	0.938441i	$0.612270\pi$
$228$	0	0
$229$	−9.32529	−0.616232	−0.308116	−	0.951349i	$-0.599699\pi$
$229$	−9.32529	−0.616232	−0.308116	+	0.951349i	$0.599699\pi$
$230$	0.307208	0.0202567
$231$	0	0
$232$	12.8996	0.846900
$233$	16.9560	1.11082	0.555412	−	0.831575i	$-0.312560\pi$
$233$	16.9560	1.11082	0.555412	+	0.831575i	$0.312560\pi$
$234$	0	0
$235$	−1.26213	−0.0823322
$236$	0.647879	0.0421733
$237$	0	0
$238$	−1.97473	−0.128003
$239$	8.27964	0.535565	0.267783	−	0.963479i	$-0.413709\pi$
$239$	8.27964	0.535565	0.267783	+	0.963479i	$0.413709\pi$
$240$	0	0
$241$	−9.31212	−0.599846	−0.299923	−	0.953963i	$-0.596961\pi$
$241$	−9.31212	−0.599846	−0.299923	+	0.953963i	$0.596961\pi$
$242$	0	0
$243$	0	0
$244$	0.751635	0.0481185
$245$	0.133004	0.00849734
$246$	0	0
$247$	−4.60742	−0.293163
$248$	−19.6865	−1.25009
$249$	0	0
$250$	1.84012	0.116379
$251$	4.93532	0.311514	0.155757	−	0.987795i	$-0.450218\pi$
$251$	4.93532	0.311514	0.155757	+	0.987795i	$0.450218\pi$
$252$	0	0
$253$	0	0
$254$	21.4213	1.34409
$255$	0	0
$256$	−1.89624	−0.118515
$257$	16.8159	1.04895	0.524474	−	0.851426i	$-0.324262\pi$
$257$	16.8159	1.04895	0.524474	+	0.851426i	$0.324262\pi$
$258$	0	0
$259$	−3.76864	−0.234172
$260$	−0.00675136	−0.000418702 0
$261$	0	0
$262$	−24.4864	−1.51277
$263$	4.11162	0.253533	0.126767	−	0.991933i	$-0.459540\pi$
$263$	4.11162	0.253533	0.126767	+	0.991933i	$0.459540\pi$
$264$	0	0
$265$	0.0886540	0.00544597
$266$	−9.95526	−0.610396
$267$	0	0
$268$	−0.952774	−0.0581999
$269$	23.9684	1.46138	0.730690	−	0.682710i	$-0.239199\pi$
$269$	23.9684	1.46138	0.730690	+	0.682710i	$0.239199\pi$
$270$	0	0
$271$	6.00791	0.364954	0.182477	−	0.983210i	$-0.441588\pi$
$271$	6.00791	0.364954	0.182477	+	0.983210i	$0.441588\pi$
$272$	5.46485	0.331355
$273$	0	0
$274$	26.3779	1.59355
$275$	0	0
$276$	0	0
$277$	9.16206	0.550495	0.275247	−	0.961373i	$-0.411240\pi$
$277$	9.16206	0.550495	0.275247	+	0.961373i	$0.411240\pi$
$278$	−19.4955	−1.16926
$279$	0	0
$280$	−0.383263	−0.0229044
$281$	12.9579	0.773006	0.386503	−	0.922288i	$-0.373683\pi$
$281$	12.9579	0.773006	0.386503	+	0.922288i	$0.373683\pi$
$282$	0	0
$283$	3.78781	0.225162	0.112581	−	0.993643i	$-0.464088\pi$
$283$	3.78781	0.225162	0.112581	+	0.993643i	$0.464088\pi$
$284$	0.382555	0.0227005
$285$	0	0
$286$	0	0
$287$	−6.17286	−0.364372
$288$	0	0
$289$	−14.9699	−0.880582
$290$	−0.825201	−0.0484575
$291$	0	0
$292$	−0.700941	−0.0410195
$293$	21.3690	1.24839	0.624195	−	0.781269i	$-0.285427\pi$
$293$	21.3690	1.24839	0.624195	+	0.781269i	$0.285427\pi$
$294$	0	0
$295$	−1.08890	−0.0633983
$296$	10.8597	0.631205
$297$	0	0
$298$	15.2963	0.886091
$299$	−1.06898	−0.0618210
$300$	0	0
$301$	−1.03970	−0.0599274
$302$	−22.3623	−1.28681
$303$	0	0
$304$	27.5500	1.58010
$305$	−1.26329	−0.0723355
$306$	0	0
$307$	8.44677	0.482082	0.241041	−	0.970515i	$-0.422511\pi$
$307$	8.44677	0.482082	0.241041	+	0.970515i	$0.422511\pi$
$308$	0	0
$309$	0	0
$310$	1.25936	0.0715271
$311$	−18.0190	−1.02176	−0.510882	−	0.859651i	$-0.670681\pi$
$311$	−18.0190	−1.02176	−0.510882	+	0.859651i	$0.670681\pi$
$312$	0	0
$313$	0.236340	0.0133587	0.00667937	−	0.999978i	$-0.497874\pi$
$313$	0.236340	0.0133587	0.00667937	+	0.999978i	$0.497874\pi$
$314$	20.6089	1.16303
$315$	0	0
$316$	0.909320	0.0511532
$317$	−17.0931	−0.960044	−0.480022	−	0.877256i	$-0.659371\pi$
$317$	−17.0931	−0.960044	−0.480022	+	0.877256i	$0.659371\pi$
$318$	0	0
$319$	0	0
$320$	1.10274	0.0616450
$321$	0	0
$322$	−2.30976	−0.128718
$323$	10.2344	0.569460
$324$	0	0
$325$	−3.19583	−0.177273
$326$	22.8043	1.26302
$327$	0	0
$328$	17.7876	0.982156
$329$	9.48937	0.523166
$330$	0	0
$331$	4.41186	0.242498	0.121249	−	0.992622i	$-0.461310\pi$
$331$	4.41186	0.242498	0.121249	+	0.992622i	$0.461310\pi$
$332$	−0.758117	−0.0416071
$333$	0	0
$334$	26.5436	1.45240
$335$	1.60134	0.0874908
$336$	0	0
$337$	−27.7879	−1.51370	−0.756852	−	0.653586i	$-0.773264\pi$
$337$	−27.7879	−1.51370	−0.756852	+	0.653586i	$0.773264\pi$
$338$	17.4471	0.949000
$339$	0	0
$340$	0.0149968	0.000813315 0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.99598	0.161533
$345$	0	0
$346$	−21.1006	−1.13437
$347$	31.2515	1.67767	0.838834	−	0.544388i	$-0.183238\pi$
$347$	31.2515	1.67767	0.838834	+	0.544388i	$0.183238\pi$
$348$	0	0
$349$	0.543438	0.0290896	0.0145448	−	0.999894i	$-0.495370\pi$
$349$	0.543438	0.0290896	0.0145448	+	0.999894i	$0.495370\pi$
$350$	−6.90524	−0.369101
$351$	0	0
$352$	0	0
$353$	17.8517	0.950148	0.475074	−	0.879946i	$-0.342421\pi$
$353$	17.8517	0.950148	0.475074	+	0.879946i	$0.342421\pi$
$354$	0	0
$355$	−0.642967	−0.0341252
$356$	−1.40071	−0.0742374
$357$	0	0
$358$	1.56422	0.0826716
$359$	−11.9909	−0.632854	−0.316427	−	0.948617i	$-0.602483\pi$
$359$	−11.9909	−0.632854	−0.316427	+	0.948617i	$0.602483\pi$
$360$	0	0
$361$	32.5951	1.71553
$362$	−25.6089	−1.34598
$363$	0	0
$364$	0.0507604	0.00266057
$365$	1.17808	0.0616637
$366$	0	0
$367$	29.5958	1.54489	0.772444	−	0.635083i	$-0.219034\pi$
$367$	29.5958	1.54489	0.772444	+	0.635083i	$0.219034\pi$
$368$	6.39199	0.333206
$369$	0	0
$370$	−0.694704	−0.0361159
$371$	−0.666549	−0.0346055
$372$	0	0
$373$	−10.5209	−0.544753	−0.272377	−	0.962191i	$-0.587810\pi$
$373$	−10.5209	−0.544753	−0.272377	+	0.962191i	$0.587810\pi$
$374$	0	0
$375$	0	0
$376$	−27.3444	−1.41018
$377$	2.87143	0.147886
$378$	0	0
$379$	−8.34913	−0.428866	−0.214433	−	0.976739i	$-0.568790\pi$
$379$	−8.34913	−0.428866	−0.214433	+	0.976739i	$0.568790\pi$
$380$	0.0756035	0.00387838
$381$	0	0
$382$	−3.36761	−0.172302
$383$	18.4737	0.943965	0.471982	−	0.881608i	$-0.343539\pi$
$383$	18.4737	0.943965	0.471982	+	0.881608i	$0.343539\pi$
$384$	0	0
$385$	0	0
$386$	−0.364955	−0.0185757
$387$	0	0
$388$	−0.520822	−0.0264407
$389$	−23.2560	−1.17913	−0.589564	−	0.807722i	$-0.700700\pi$
$389$	−23.2560	−1.17913	−0.589564	+	0.807722i	$0.700700\pi$
$390$	0	0
$391$	2.37453	0.120085
$392$	2.88158	0.145542
$393$	0	0
$394$	−24.2100	−1.21968
$395$	−1.52831	−0.0768976
$396$	0	0
$397$	21.6794	1.08806	0.544029	−	0.839066i	$-0.316898\pi$
$397$	21.6794	1.08806	0.544029	+	0.839066i	$0.316898\pi$
$398$	27.9152	1.39926
$399$	0	0
$400$	19.1095	0.955474
$401$	34.8075	1.73821	0.869103	−	0.494631i	$-0.164697\pi$
$401$	34.8075	1.73821	0.869103	+	0.494631i	$0.164697\pi$
$402$	0	0
$403$	−4.38218	−0.218292
$404$	−1.47354	−0.0733112
$405$	0	0
$406$	6.20431	0.307915
$407$	0	0
$408$	0	0
$409$	−16.2927	−0.805620	−0.402810	−	0.915284i	$-0.631967\pi$
$409$	−16.2927	−0.805620	−0.402810	+	0.915284i	$0.631967\pi$
$410$	−1.13789	−0.0561965
$411$	0	0
$412$	0.547900	0.0269931
$413$	8.18695	0.402853
$414$	0	0
$415$	1.27418	0.0625471
$416$	−0.286973	−0.0140700
$417$	0	0
$418$	0	0
$419$	22.6536	1.10670	0.553350	−	0.832949i	$-0.313349\pi$
$419$	22.6536	1.10670	0.553350	+	0.832949i	$0.313349\pi$
$420$	0	0
$421$	−15.2061	−0.741102	−0.370551	−	0.928812i	$-0.620831\pi$
$421$	−15.2061	−0.741102	−0.370551	+	0.928812i	$0.620831\pi$
$422$	−19.5145	−0.949950
$423$	0	0
$424$	1.92072	0.0932783
$425$	7.09890	0.344347
$426$	0	0
$427$	9.49807	0.459644
$428$	0.430850	0.0208259
$429$	0	0
$430$	−0.191656	−0.00924249
$431$	−3.20691	−0.154471	−0.0772356	−	0.997013i	$-0.524609\pi$
$431$	−3.20691	−0.154471	−0.0772356	+	0.997013i	$0.524609\pi$
$432$	0	0
$433$	5.99246	0.287979	0.143990	−	0.989579i	$-0.454007\pi$
$433$	5.99246	0.287979	0.143990	+	0.989579i	$0.454007\pi$
$434$	−9.46858	−0.454507
$435$	0	0
$436$	−0.729880	−0.0349549
$437$	11.9708	0.572640
$438$	0	0
$439$	7.68712	0.366886	0.183443	−	0.983030i	$-0.441276\pi$
$439$	7.68712	0.366886	0.183443	+	0.983030i	$0.441276\pi$
$440$	0	0
$441$	0	0
$442$	1.26667	0.0602491
$443$	−37.2499	−1.76980	−0.884898	−	0.465784i	$-0.845772\pi$
$443$	−37.2499	−1.76980	−0.884898	+	0.465784i	$0.845772\pi$
$444$	0	0
$445$	2.35419	0.111600
$446$	13.3518	0.632224
$447$	0	0
$448$	−8.29100	−0.391713
$449$	−11.0724	−0.522539	−0.261269	−	0.965266i	$-0.584141\pi$
$449$	−11.0724	−0.522539	−0.261269	+	0.965266i	$0.584141\pi$
$450$	0	0
$451$	0	0
$452$	0.742468	0.0349228
$453$	0	0
$454$	14.4266	0.677074
$455$	−0.0853139	−0.00399958
$456$	0	0
$457$	−28.3961	−1.32831	−0.664157	−	0.747593i	$-0.731209\pi$
$457$	−28.3961	−1.32831	−0.664157	+	0.747593i	$0.731209\pi$
$458$	12.9244	0.603918
$459$	0	0
$460$	0.0175411	0.000817856 0
$461$	−25.7420	−1.19892	−0.599462	−	0.800403i	$-0.704619\pi$
$461$	−25.7420	−1.19892	−0.599462	+	0.800403i	$0.704619\pi$
$462$	0	0
$463$	37.6543	1.74995	0.874973	−	0.484172i	$-0.160879\pi$
$463$	37.6543	1.74995	0.874973	+	0.484172i	$0.160879\pi$
$464$	−17.1697	−0.797084
$465$	0	0
$466$	−23.5002	−1.08863
$467$	15.3875	0.712046	0.356023	−	0.934477i	$-0.384132\pi$
$467$	15.3875	0.712046	0.356023	+	0.934477i	$0.384132\pi$
$468$	0	0
$469$	−12.0398	−0.555945
$470$	1.74925	0.0806869
$471$	0	0
$472$	−23.5914	−1.08588
$473$	0	0
$474$	0	0
$475$	35.7878	1.64206
$476$	−0.112754	−0.00516807
$477$	0	0
$478$	−11.4752	−0.524863
$479$	−0.600178	−0.0274228	−0.0137114	−	0.999906i	$-0.504365\pi$
$479$	−0.600178	−0.0274228	−0.0137114	+	0.999906i	$0.504365\pi$
$480$	0	0
$481$	2.41734	0.110221
$482$	12.9062	0.587859
$483$	0	0
$484$	0	0
$485$	0.875354	0.0397478
$486$	0	0
$487$	10.2991	0.466698	0.233349	−	0.972393i	$-0.425032\pi$
$487$	10.2991	0.466698	0.233349	+	0.972393i	$0.425032\pi$
$488$	−27.3695	−1.23896
$489$	0	0
$490$	−0.184338	−0.00832754
$491$	−10.0925	−0.455466	−0.227733	−	0.973724i	$-0.573131\pi$
$491$	−10.0925	−0.455466	−0.227733	+	0.973724i	$0.573131\pi$
$492$	0	0
$493$	−6.37830	−0.287264
$494$	6.38566	0.287304
$495$	0	0
$496$	26.2032	1.17656
$497$	4.83418	0.216842
$498$	0	0
$499$	19.7955	0.886170	0.443085	−	0.896480i	$-0.353884\pi$
$499$	19.7955	0.886170	0.443085	+	0.896480i	$0.353884\pi$
$500$	0.105068	0.00469877
$501$	0	0
$502$	−6.84012	−0.305289
$503$	−14.3034	−0.637757	−0.318879	−	0.947796i	$-0.603306\pi$
$503$	−14.3034	−0.637757	−0.318879	+	0.947796i	$0.603306\pi$
$504$	0	0
$505$	2.47660	0.110207
$506$	0	0
$507$	0	0
$508$	1.22312	0.0542671
$509$	42.0975	1.86594	0.932970	−	0.359953i	$-0.117207\pi$
$509$	42.0975	1.86594	0.932970	+	0.359953i	$0.117207\pi$
$510$	0	0
$511$	−8.85748	−0.391832
$512$	23.8204	1.05272
$513$	0	0
$514$	−23.3061	−1.02799
$515$	−0.920864	−0.0405781
$516$	0	0
$517$	0	0
$518$	5.22316	0.229492
$519$	0	0
$520$	0.245839	0.0107808
$521$	21.4472	0.939619	0.469810	−	0.882768i	$-0.344323\pi$
$521$	21.4472	0.939619	0.469810	+	0.882768i	$0.344323\pi$
$522$	0	0
$523$	35.5946	1.55644	0.778222	−	0.627989i	$-0.216122\pi$
$523$	35.5946	1.55644	0.778222	+	0.627989i	$0.216122\pi$
$524$	−1.39813	−0.0610776
$525$	0	0
$526$	−5.69851	−0.248467
$527$	9.73412	0.424025
$528$	0	0
$529$	−20.2226	−0.879244
$530$	−0.122870	−0.00533714
$531$	0	0
$532$	−0.568428	−0.0246445
$533$	3.95949	0.171505
$534$	0	0
$535$	−0.724137	−0.0313072
$536$	34.6936	1.49854
$537$	0	0
$538$	−33.2191	−1.43218
$539$	0	0
$540$	0	0
$541$	−16.9350	−0.728092	−0.364046	−	0.931381i	$-0.618605\pi$
$541$	−16.9350	−0.728092	−0.364046	+	0.931381i	$0.618605\pi$
$542$	−8.32667	−0.357661
$543$	0	0
$544$	0.637453	0.0273306
$545$	1.22672	0.0525470
$546$	0	0
$547$	8.79082	0.375868	0.187934	−	0.982182i	$-0.439821\pi$
$547$	8.79082	0.375868	0.187934	+	0.982182i	$0.439821\pi$
$548$	1.50613	0.0643388
$549$	0	0
$550$	0	0
$551$	−32.1551	−1.36985
$552$	0	0
$553$	11.4907	0.488633
$554$	−12.6982	−0.539494
$555$	0	0
$556$	−1.11316	−0.0472085
$557$	21.7276	0.920629	0.460315	−	0.887756i	$-0.347737\pi$
$557$	21.7276	0.920629	0.460315	+	0.887756i	$0.347737\pi$
$558$	0	0
$559$	0.666902	0.0282069
$560$	0.510134	0.0215571
$561$	0	0
$562$	−17.9591	−0.757559
$563$	−1.82134	−0.0767605	−0.0383803	−	0.999263i	$-0.512220\pi$
$563$	−1.82134	−0.0767605	−0.0383803	+	0.999263i	$0.512220\pi$
$564$	0	0
$565$	−1.24788	−0.0524987
$566$	−5.24972	−0.220662
$567$	0	0
$568$	−13.9301	−0.584493
$569$	32.9418	1.38099	0.690496	−	0.723336i	$-0.257392\pi$
$569$	32.9418	1.38099	0.690496	+	0.723336i	$0.257392\pi$
$570$	0	0
$571$	23.4394	0.980908	0.490454	−	0.871467i	$-0.336831\pi$
$571$	23.4394	0.980908	0.490454	+	0.871467i	$0.336831\pi$
$572$	0	0
$573$	0	0
$574$	8.55529	0.357091
$575$	8.30326	0.346270
$576$	0	0
$577$	34.4539	1.43433	0.717166	−	0.696902i	$-0.245439\pi$
$577$	34.4539	1.43433	0.717166	+	0.696902i	$0.245439\pi$
$578$	20.7476	0.862984
$579$	0	0
$580$	−0.0471176	−0.00195645
$581$	−9.57998	−0.397445
$582$	0	0
$583$	0	0
$584$	25.5236	1.05617
$585$	0	0
$586$	−29.6164	−1.22344
$587$	20.8670	0.861274	0.430637	−	0.902525i	$-0.358289\pi$
$587$	20.8670	0.861274	0.430637	+	0.902525i	$0.358289\pi$
$588$	0	0
$589$	49.0728	2.02201
$590$	1.50916	0.0621313
$591$	0	0
$592$	−14.4545	−0.594076
$593$	26.6381	1.09389	0.546947	−	0.837167i	$-0.315790\pi$
$593$	26.6381	1.09389	0.546947	+	0.837167i	$0.315790\pi$
$594$	0	0
$595$	0.189507	0.00776905
$596$	0.873393	0.0357756
$597$	0	0
$598$	1.48156	0.0605856
$599$	20.6712	0.844602	0.422301	−	0.906456i	$-0.361223\pi$
$599$	20.6712	0.844602	0.422301	+	0.906456i	$0.361223\pi$
$600$	0	0
$601$	4.94812	0.201838	0.100919	−	0.994895i	$-0.467822\pi$
$601$	4.94812	0.201838	0.100919	+	0.994895i	$0.467822\pi$
$602$	1.44098	0.0587298
$603$	0	0
$604$	−1.27685	−0.0519543
$605$	0	0
$606$	0	0
$607$	27.6086	1.12060	0.560300	−	0.828290i	$-0.310686\pi$
$607$	27.6086	1.12060	0.560300	+	0.828290i	$0.310686\pi$
$608$	3.21360	0.130329
$609$	0	0
$610$	1.75085	0.0708900
$611$	−6.08683	−0.246247
$612$	0	0
$613$	−7.18919	−0.290368	−0.145184	−	0.989405i	$-0.546377\pi$
$613$	−7.18919	−0.290368	−0.145184	+	0.989405i	$0.546377\pi$
$614$	−11.7068	−0.472449
$615$	0	0
$616$	0	0
$617$	−21.1215	−0.850320	−0.425160	−	0.905118i	$-0.639782\pi$
$617$	−21.1215	−0.850320	−0.425160	+	0.905118i	$0.639782\pi$
$618$	0	0
$619$	2.77412	0.111501	0.0557506	−	0.998445i	$-0.482245\pi$
$619$	2.77412	0.111501	0.0557506	+	0.998445i	$0.482245\pi$
$620$	0.0719076	0.00288788
$621$	0	0
$622$	24.9735	1.00134
$623$	−17.7001	−0.709140
$624$	0	0
$625$	24.7350	0.989398
$626$	−0.327556	−0.0130918
$627$	0	0
$628$	1.17673	0.0469568
$629$	−5.36964	−0.214102
$630$	0	0
$631$	−24.2090	−0.963743	−0.481872	−	0.876242i	$-0.660043\pi$
$631$	−24.2090	−0.963743	−0.481872	+	0.876242i	$0.660043\pi$
$632$	−33.1113	−1.31710
$633$	0	0
$634$	23.6902	0.940859
$635$	−2.05572	−0.0815786
$636$	0	0
$637$	0.641436	0.0254146
$638$	0	0
$639$	0	0
$640$	−1.40933	−0.0557088
$641$	−43.4897	−1.71774	−0.858871	−	0.512192i	$-0.828834\pi$
$641$	−43.4897	−1.71774	−0.858871	+	0.512192i	$0.828834\pi$
$642$	0	0
$643$	−17.1517	−0.676397	−0.338199	−	0.941075i	$-0.609818\pi$
$643$	−17.1517	−0.676397	−0.338199	+	0.941075i	$0.609818\pi$
$644$	−0.131883	−0.00519693
$645$	0	0
$646$	−14.1845	−0.558080
$647$	−14.2909	−0.561833	−0.280916	−	0.959732i	$-0.590638\pi$
$647$	−14.2909	−0.561833	−0.280916	+	0.959732i	$0.590638\pi$
$648$	0	0
$649$	0	0
$650$	4.42927	0.173730
$651$	0	0
$652$	1.30209	0.0509938
$653$	37.8518	1.48126	0.740628	−	0.671915i	$-0.234528\pi$
$653$	37.8518	1.48126	0.740628	+	0.671915i	$0.234528\pi$
$654$	0	0
$655$	2.34986	0.0918166
$656$	−23.6758	−0.924384
$657$	0	0
$658$	−13.1518	−0.512711
$659$	8.13829	0.317023	0.158511	−	0.987357i	$-0.449331\pi$
$659$	8.13829	0.317023	0.158511	+	0.987357i	$0.449331\pi$
$660$	0	0
$661$	−5.33161	−0.207375	−0.103688	−	0.994610i	$-0.533064\pi$
$661$	−5.33161	−0.207375	−0.103688	+	0.994610i	$0.533064\pi$
$662$	−6.11463	−0.237652
$663$	0	0
$664$	27.6055	1.07130
$665$	0.955367	0.0370476
$666$	0	0
$667$	−7.46042	−0.288869
$668$	1.51559	0.0586400
$669$	0	0
$670$	−2.21939	−0.0857424
$671$	0	0
$672$	0	0
$673$	19.4495	0.749724	0.374862	−	0.927081i	$-0.377690\pi$
$673$	19.4495	0.749724	0.374862	+	0.927081i	$0.377690\pi$
$674$	38.5127	1.48345
$675$	0	0
$676$	0.996203	0.0383155
$677$	−36.3042	−1.39528	−0.697642	−	0.716446i	$-0.745768\pi$
$677$	−36.3042	−1.39528	−0.697642	+	0.716446i	$0.745768\pi$
$678$	0	0
$679$	−6.58139	−0.252571
$680$	−0.546082	−0.0209413
$681$	0	0
$682$	0	0
$683$	20.7805	0.795142	0.397571	−	0.917571i	$-0.369853\pi$
$683$	20.7805	0.795142	0.397571	+	0.917571i	$0.369853\pi$
$684$	0	0
$685$	−2.53138	−0.0967192
$686$	1.38595	0.0529159
$687$	0	0
$688$	−3.98774	−0.152031
$689$	0.427549	0.0162883
$690$	0	0
$691$	−50.1050	−1.90608	−0.953042	−	0.302838i	$-0.902066\pi$
$691$	−50.1050	−1.90608	−0.953042	+	0.302838i	$0.902066\pi$
$692$	−1.20481	−0.0457998
$693$	0	0
$694$	−43.3131	−1.64414
$695$	1.87091	0.0709675
$696$	0	0
$697$	−8.79521	−0.333143
$698$	−0.753179	−0.0285083
$699$	0	0
$700$	−0.394278	−0.0149023
$701$	−16.3229	−0.616508	−0.308254	−	0.951304i	$-0.599745\pi$
$701$	−16.3229	−0.616508	−0.308254	+	0.951304i	$0.599745\pi$
$702$	0	0
$703$	−27.0701	−1.02097
$704$	0	0
$705$	0	0
$706$	−24.7416	−0.931161
$707$	−18.6204	−0.700292
$708$	0	0
$709$	33.2299	1.24798	0.623988	−	0.781434i	$-0.285511\pi$
$709$	33.2299	1.24798	0.623988	+	0.781434i	$0.285511\pi$
$710$	0.891122	0.0334432
$711$	0	0
$712$	51.0044	1.91147
$713$	11.3856	0.426393
$714$	0	0
$715$	0	0
$716$	0.0893143	0.00333783
$717$	0	0
$718$	16.6188	0.620207
$719$	25.2455	0.941498	0.470749	−	0.882267i	$-0.343984\pi$
$719$	25.2455	0.941498	0.470749	+	0.882267i	$0.343984\pi$
$720$	0	0
$721$	6.92356	0.257847
$722$	−45.1752	−1.68125
$723$	0	0
$724$	−1.46223	−0.0543432
$725$	−22.3036	−0.828337
$726$	0	0
$727$	1.86242	0.0690733	0.0345366	−	0.999403i	$-0.489004\pi$
$727$	1.86242	0.0690733	0.0345366	+	0.999403i	$0.489004\pi$
$728$	−1.84835	−0.0685045
$729$	0	0
$730$	−1.63277	−0.0604315
$731$	−1.48139	−0.0547911
$732$	0	0
$733$	−20.0651	−0.741123	−0.370561	−	0.928808i	$-0.620835\pi$
$733$	−20.0651	−0.741123	−0.370561	+	0.928808i	$0.620835\pi$
$734$	−41.0184	−1.51402
$735$	0	0
$736$	0.745601	0.0274832
$737$	0	0
$738$	0	0
$739$	22.3970	0.823886	0.411943	−	0.911210i	$-0.364850\pi$
$739$	22.3970	0.823886	0.411943	+	0.911210i	$0.364850\pi$
$740$	−0.0396664	−0.00145817
$741$	0	0
$742$	0.923806	0.0339140
$743$	−36.2705	−1.33063	−0.665317	−	0.746561i	$-0.731704\pi$
$743$	−36.2705	−1.33063	−0.665317	+	0.746561i	$0.731704\pi$
$744$	0	0
$745$	−1.46793	−0.0537807
$746$	14.5815	0.533867
$747$	0	0
$748$	0	0
$749$	5.44446	0.198936
$750$	0	0
$751$	44.6237	1.62834	0.814170	−	0.580626i	$-0.197192\pi$
$751$	44.6237	1.62834	0.814170	+	0.580626i	$0.197192\pi$
$752$	36.3962	1.32723
$753$	0	0
$754$	−3.97967	−0.144931
$755$	2.14603	0.0781019
$756$	0	0
$757$	8.99964	0.327097	0.163549	−	0.986535i	$-0.447706\pi$
$757$	8.99964	0.327097	0.163549	+	0.986535i	$0.447706\pi$
$758$	11.5715	0.420296
$759$	0	0
$760$	−2.75297	−0.0998607
$761$	5.15155	0.186743	0.0933717	−	0.995631i	$-0.470235\pi$
$761$	5.15155	0.186743	0.0933717	+	0.995631i	$0.470235\pi$
$762$	0	0
$763$	−9.22316	−0.333901
$764$	−0.192285	−0.00695661
$765$	0	0
$766$	−25.6037	−0.925101
$767$	−5.25140	−0.189617
$768$	0	0
$769$	38.5242	1.38922	0.694608	−	0.719388i	$-0.255578\pi$
$769$	38.5242	1.38922	0.694608	+	0.719388i	$0.255578\pi$
$770$	0	0
$771$	0	0
$772$	−0.0208383	−0.000749987 0
$773$	6.99024	0.251421	0.125711	−	0.992067i	$-0.459879\pi$
$773$	6.99024	0.251421	0.125711	+	0.992067i	$0.459879\pi$
$774$	0	0
$775$	34.0383	1.22269
$776$	18.9648	0.680797
$777$	0	0
$778$	32.2317	1.15556
$779$	−44.3395	−1.58863
$780$	0	0
$781$	0	0
$782$	−3.29099	−0.117686
$783$	0	0
$784$	−3.83547	−0.136981
$785$	−1.97776	−0.0705892
$786$	0	0
$787$	−31.3859	−1.11879	−0.559393	−	0.828903i	$-0.688966\pi$
$787$	−31.3859	−1.11879	−0.559393	+	0.828903i	$0.688966\pi$
$788$	−1.38235	−0.0492441
$789$	0	0
$790$	2.11816	0.0753609
$791$	9.38223	0.333594
$792$	0	0
$793$	−6.09240	−0.216348
$794$	−30.0466	−1.06631
$795$	0	0
$796$	1.59391	0.0564947
$797$	−26.2972	−0.931496	−0.465748	−	0.884917i	$-0.654215\pi$
$797$	−26.2972	−0.931496	−0.465748	+	0.884917i	$0.654215\pi$
$798$	0	0
$799$	13.5207	0.478326
$800$	2.22904	0.0788086
$801$	0	0
$802$	−48.2416	−1.70347
$803$	0	0
$804$	0	0
$805$	0.221659	0.00781244
$806$	6.07349	0.213930
$807$	0	0
$808$	53.6562	1.88762
$809$	11.7634	0.413578	0.206789	−	0.978386i	$-0.433699\pi$
$809$	11.7634	0.413578	0.206789	+	0.978386i	$0.433699\pi$
$810$	0	0
$811$	8.27304	0.290506	0.145253	−	0.989395i	$-0.453600\pi$
$811$	8.27304	0.290506	0.145253	+	0.989395i	$0.453600\pi$
$812$	0.354256	0.0124319
$813$	0	0
$814$	0	0
$815$	−2.18844	−0.0766579
$816$	0	0
$817$	−7.46814	−0.261277
$818$	22.5809	0.789521
$819$	0	0
$820$	−0.0649717	−0.00226891
$821$	−7.38513	−0.257743	−0.128871	−	0.991661i	$-0.541135\pi$
$821$	−7.38513	−0.257743	−0.128871	+	0.991661i	$0.541135\pi$
$822$	0	0
$823$	49.4433	1.72349	0.861743	−	0.507345i	$-0.169373\pi$
$823$	49.4433	1.72349	0.861743	+	0.507345i	$0.169373\pi$
$824$	−19.9508	−0.695019
$825$	0	0
$826$	−11.3467	−0.394803
$827$	23.2339	0.807922	0.403961	−	0.914776i	$-0.367633\pi$
$827$	23.2339	0.807922	0.403961	+	0.914776i	$0.367633\pi$
$828$	0	0
$829$	30.1938	1.04867	0.524336	−	0.851511i	$-0.324314\pi$
$829$	30.1938	1.04867	0.524336	+	0.851511i	$0.324314\pi$
$830$	−1.76595	−0.0612972
$831$	0	0
$832$	5.31814	0.184374
$833$	−1.42482	−0.0493671
$834$	0	0
$835$	−2.54728	−0.0881523
$836$	0	0
$837$	0	0
$838$	−31.3968	−1.08458
$839$	−16.6125	−0.573529	−0.286764	−	0.958001i	$-0.592580\pi$
$839$	−16.6125	−0.573529	−0.286764	+	0.958001i	$0.592580\pi$
$840$	0	0
$841$	−8.96035	−0.308977
$842$	21.0750	0.726292
$843$	0	0
$844$	−1.11424	−0.0383539
$845$	−1.67433	−0.0575989
$846$	0	0
$847$	0	0
$848$	−2.55653	−0.0877915
$849$	0	0
$850$	−9.83873	−0.337466
$851$	−6.28063	−0.215297
$852$	0	0
$853$	40.0507	1.37131	0.685654	−	0.727927i	$-0.259516\pi$
$853$	40.0507	1.37131	0.685654	+	0.727927i	$0.259516\pi$
$854$	−13.1639	−0.450458
$855$	0	0
$856$	−15.6887	−0.536227
$857$	−31.7070	−1.08309	−0.541546	−	0.840671i	$-0.682161\pi$
$857$	−31.7070	−1.08309	−0.541546	+	0.840671i	$0.682161\pi$
$858$	0	0
$859$	1.63654	0.0558381	0.0279190	−	0.999610i	$-0.491112\pi$
$859$	1.63654	0.0558381	0.0279190	+	0.999610i	$0.491112\pi$
$860$	−0.0109432	−0.000373162 0
$861$	0	0
$862$	4.44462	0.151384
$863$	17.4055	0.592489	0.296244	−	0.955112i	$-0.404266\pi$
$863$	17.4055	0.592489	0.296244	+	0.955112i	$0.404266\pi$
$864$	0	0
$865$	2.02494	0.0688500
$866$	−8.30526	−0.282224
$867$	0	0
$868$	−0.540640	−0.0183505
$869$	0	0
$870$	0	0
$871$	7.72275	0.261675
$872$	26.5773	0.900021
$873$	0	0
$874$	−16.5909	−0.561196
$875$	1.32769	0.0448842
$876$	0	0
$877$	48.1667	1.62647	0.813237	−	0.581933i	$-0.197703\pi$
$877$	48.1667	1.62647	0.813237	+	0.581933i	$0.197703\pi$
$878$	−10.6540	−0.359555
$879$	0	0
$880$	0	0
$881$	30.0141	1.01120	0.505601	−	0.862767i	$-0.331271\pi$
$881$	30.0141	1.01120	0.505601	+	0.862767i	$0.331271\pi$
$882$	0	0
$883$	56.3355	1.89584	0.947921	−	0.318506i	$-0.103181\pi$
$883$	56.3355	1.89584	0.947921	+	0.318506i	$0.103181\pi$
$884$	0.0723244	0.00243253
$885$	0	0
$886$	51.6266	1.73443
$887$	−4.53095	−0.152135	−0.0760673	−	0.997103i	$-0.524236\pi$
$887$	−4.53095	−0.152135	−0.0760673	+	0.997103i	$0.524236\pi$
$888$	0	0
$889$	15.4560	0.518377
$890$	−3.26280	−0.109369
$891$	0	0
$892$	0.762363	0.0255258
$893$	68.1619	2.28095
$894$	0	0
$895$	−0.150112	−0.00501770
$896$	10.5961	0.353992
$897$	0	0
$898$	15.3458	0.512097
$899$	−30.5831	−1.02000
$900$	0	0
$901$	−0.949713	−0.0316395
$902$	0	0
$903$	0	0
$904$	−27.0357	−0.899193
$905$	2.45759	0.0816930
$906$	0	0
$907$	−20.7609	−0.689355	−0.344678	−	0.938721i	$-0.612012\pi$
$907$	−20.7609	−0.689355	−0.344678	+	0.938721i	$0.612012\pi$
$908$	0.823735	0.0273366
$909$	0	0
$910$	0.118241	0.00391965
$911$	15.3952	0.510066	0.255033	−	0.966932i	$-0.417914\pi$
$911$	15.3952	0.510066	0.255033	+	0.966932i	$0.417914\pi$
$912$	0	0
$913$	0	0
$914$	39.3557	1.30177
$915$	0	0
$916$	0.737962	0.0243829
$917$	−17.6675	−0.583433
$918$	0	0
$919$	−21.3581	−0.704540	−0.352270	−	0.935898i	$-0.614590\pi$
$919$	−21.3581	−0.704540	−0.352270	+	0.935898i	$0.614590\pi$
$920$	−0.638727	−0.0210582
$921$	0	0
$922$	35.6772	1.17496
$923$	−3.10082	−0.102065
$924$	0	0
$925$	−18.7765	−0.617369
$926$	−52.1871	−1.71498
$927$	0	0
$928$	−2.00278	−0.0657445
$929$	−16.1336	−0.529325	−0.264662	−	0.964341i	$-0.585261\pi$
$929$	−16.1336	−0.529325	−0.264662	+	0.964341i	$0.585261\pi$
$930$	0	0
$931$	−7.18297	−0.235412
$932$	−1.34182	−0.0439529
$933$	0	0
$934$	−21.3263	−0.697817
$935$	0	0
$936$	0	0
$937$	4.56306	0.149069	0.0745343	−	0.997218i	$-0.476253\pi$
$937$	4.56306	0.149069	0.0745343	+	0.997218i	$0.476253\pi$
$938$	16.6866	0.544835
$939$	0	0
$940$	0.0998793	0.00325770
$941$	12.5504	0.409130	0.204565	−	0.978853i	$-0.434422\pi$
$941$	12.5504	0.409130	0.204565	+	0.978853i	$0.434422\pi$
$942$	0	0
$943$	−10.2874	−0.335003
$944$	31.4008	1.02201
$945$	0	0
$946$	0	0
$947$	−51.6790	−1.67934	−0.839672	−	0.543094i	$-0.817253\pi$
$947$	−51.6790	−1.67934	−0.839672	+	0.543094i	$0.817253\pi$
$948$	0	0
$949$	5.68151	0.184429
$950$	−49.6002	−1.60924
$951$	0	0
$952$	4.10574	0.133068
$953$	1.32251	0.0428403	0.0214202	−	0.999771i	$-0.493181\pi$
$953$	1.32251	0.0428403	0.0214202	+	0.999771i	$0.493181\pi$
$954$	0	0
$955$	0.323176	0.0104577
$956$	−0.655213	−0.0211911
$957$	0	0
$958$	0.831818	0.0268748
$959$	19.0323	0.614586
$960$	0	0
$961$	15.6738	0.505607
$962$	−3.35032	−0.108019
$963$	0	0
$964$	0.736919	0.0237346
$965$	0.0350233	0.00112744
$966$	0	0
$967$	44.1214	1.41885	0.709425	−	0.704781i	$-0.248955\pi$
$967$	44.1214	1.41885	0.709425	+	0.704781i	$0.248955\pi$
$968$	0	0
$969$	0	0
$970$	−1.21320	−0.0389535
$971$	1.31170	0.0420944	0.0210472	−	0.999778i	$-0.493300\pi$
$971$	1.31170	0.0420944	0.0210472	+	0.999778i	$0.493300\pi$
$972$	0	0
$973$	−14.0665	−0.450951
$974$	−14.2741	−0.457371
$975$	0	0
$976$	36.4295	1.16608
$977$	12.4102	0.397036	0.198518	−	0.980097i	$-0.436387\pi$
$977$	12.4102	0.397036	0.198518	+	0.980097i	$0.436387\pi$
$978$	0	0
$979$	0	0
$980$	−0.0105254	−0.000336221 0
$981$	0	0
$982$	13.9877	0.446364
$983$	31.9565	1.01925	0.509627	−	0.860395i	$-0.329783\pi$
$983$	31.9565	1.01925	0.509627	+	0.860395i	$0.329783\pi$
$984$	0	0
$985$	2.32334	0.0740277
$986$	8.84003	0.281524
$987$	0	0
$988$	0.364610	0.0115998
$989$	−1.73271	−0.0550970
$990$	0	0
$991$	−9.17926	−0.291589	−0.145794	−	0.989315i	$-0.546574\pi$
$991$	−9.17926	−0.291589	−0.145794	+	0.989315i	$0.546574\pi$
$992$	3.05650	0.0970440
$993$	0	0
$994$	−6.69994	−0.212509
$995$	−2.67891	−0.0849273
$996$	0	0
$997$	−14.3428	−0.454242	−0.227121	−	0.973867i	$-0.572931\pi$
$997$	−14.3428	−0.454242	−0.227121	+	0.973867i	$0.572931\pi$
$998$	−27.4357	−0.868461
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.38595 q^{2} -0.0791355 q^{4} +0.133004 q^{5} -1.00000 q^{7} +2.88158 q^{8} +O(q^{10})\)	\(q-1.38595 q^{2} -0.0791355 q^{4} +0.133004 q^{5} -1.00000 q^{7} +2.88158 q^{8} -0.184338 q^{10} +0.641436 q^{13} +1.38595 q^{14} -3.83547 q^{16} -1.42482 q^{17} -7.18297 q^{19} -0.0105254 q^{20} -1.66655 q^{23} -4.98231 q^{25} -0.889000 q^{26} +0.0791355 q^{28} +4.47657 q^{29} -6.83182 q^{31} -0.447392 q^{32} +1.97473 q^{34} -0.133004 q^{35} +3.76864 q^{37} +9.95526 q^{38} +0.383263 q^{40} +6.17286 q^{41} +1.03970 q^{43} +2.30976 q^{46} -9.48937 q^{47} +1.00000 q^{49} +6.90524 q^{50} -0.0507604 q^{52} +0.666549 q^{53} -2.88158 q^{56} -6.20431 q^{58} -8.18695 q^{59} -9.49807 q^{61} +9.46858 q^{62} +8.29100 q^{64} +0.0853139 q^{65} +12.0398 q^{67} +0.112754 q^{68} +0.184338 q^{70} -4.83418 q^{71} +8.85748 q^{73} -5.22316 q^{74} +0.568428 q^{76} -11.4907 q^{79} -0.510134 q^{80} -8.55529 q^{82} +9.57998 q^{83} -0.189507 q^{85} -1.44098 q^{86} +17.7001 q^{89} -0.641436 q^{91} +0.131883 q^{92} +13.1518 q^{94} -0.955367 q^{95} +6.58139 q^{97} -1.38595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 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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