LMFDB - Embedded newform 7623.2.a.cp.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$6$
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			1.5
Root			$1.82356$ 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)

$f(q)$	$=$	$q+0.823556 q^{2} -1.32176 q^{4} -2.98565 q^{5} +1.00000 q^{7} -2.73565 q^{8} +O(q^{10})$	$q+0.823556 q^{2} -1.32176 q^{4} -2.98565 q^{5} +1.00000 q^{7} -2.73565 q^{8} -2.45885 q^{10} +2.20144 q^{13} +0.823556 q^{14} +0.390549 q^{16} -4.40669 q^{17} -1.72563 q^{19} +3.94630 q^{20} +8.39774 q^{23} +3.91410 q^{25} +1.81301 q^{26} -1.32176 q^{28} -3.29295 q^{29} +7.47573 q^{31} +5.79294 q^{32} -3.62916 q^{34} -2.98565 q^{35} -8.78071 q^{37} -1.42115 q^{38} +8.16769 q^{40} -5.39351 q^{41} +9.44629 q^{43} +6.91601 q^{46} +5.39667 q^{47} +1.00000 q^{49} +3.22348 q^{50} -2.90977 q^{52} -9.39774 q^{53} -2.73565 q^{56} -2.71193 q^{58} -3.47462 q^{59} +12.9942 q^{61} +6.15668 q^{62} +3.98971 q^{64} -6.57274 q^{65} +4.32138 q^{67} +5.82457 q^{68} -2.45885 q^{70} -4.40046 q^{71} +14.7509 q^{73} -7.23140 q^{74} +2.28086 q^{76} -7.18768 q^{79} -1.16604 q^{80} -4.44186 q^{82} -7.63445 q^{83} +13.1568 q^{85} +7.77955 q^{86} -10.8428 q^{89} +2.20144 q^{91} -11.0998 q^{92} +4.44446 q^{94} +5.15211 q^{95} +2.85498 q^{97} +0.823556 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.823556	0.582342	0.291171	−	0.956671i	$-0.405955\pi$
$2$	0.823556	0.582342	0.291171	+	0.956671i	$0.405955\pi$
$3$	0	0
$3$	0	0
$4$	−1.32176	−0.660878
$5$	−2.98565	−1.33522	−0.667611	−	0.744510i	$-0.732683\pi$
$5$	−2.98565	−1.33522	−0.667611	+	0.744510i	$0.732683\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.73565	−0.967199
$9$	0	0
$10$	−2.45885	−0.777556
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.20144	0.610571	0.305285	−	0.952261i	$-0.401248\pi$
$13$	2.20144	0.610571	0.305285	+	0.952261i	$0.401248\pi$
$14$	0.823556	0.220105
$15$	0	0
$16$	0.390549	0.0976372
$17$	−4.40669	−1.06878	−0.534390	−	0.845238i	$-0.679459\pi$
$17$	−4.40669	−1.06878	−0.534390	+	0.845238i	$0.679459\pi$
$18$	0	0
$19$	−1.72563	−0.395886	−0.197943	−	0.980214i	$-0.563426\pi$
$19$	−1.72563	−0.395886	−0.197943	+	0.980214i	$0.563426\pi$
$20$	3.94630	0.882419
$21$	0	0
$22$	0	0
$23$	8.39774	1.75105	0.875525	−	0.483173i	$-0.160516\pi$
$23$	8.39774	1.75105	0.875525	+	0.483173i	$0.160516\pi$
$24$	0	0
$25$	3.91410	0.782819
$26$	1.81301	0.355561
$27$	0	0
$28$	−1.32176	−0.249788
$29$	−3.29295	−0.611485	−0.305743	−	0.952114i	$-0.598905\pi$
$29$	−3.29295	−0.611485	−0.305743	+	0.952114i	$0.598905\pi$
$30$	0	0
$31$	7.47573	1.34268	0.671341	−	0.741149i	$-0.265719\pi$
$31$	7.47573	1.34268	0.671341	+	0.741149i	$0.265719\pi$
$32$	5.79294	1.02406
$33$	0	0
$34$	−3.62916	−0.622396
$35$	−2.98565	−0.504667
$36$	0	0
$37$	−8.78071	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$37$	−8.78071	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$38$	−1.42115	−0.230541
$39$	0	0
$40$	8.16769	1.29143
$41$	−5.39351	−0.842325	−0.421163	−	0.906985i	$-0.638378\pi$
$41$	−5.39351	−0.842325	−0.421163	+	0.906985i	$0.638378\pi$
$42$	0	0
$43$	9.44629	1.44055	0.720273	−	0.693691i	$-0.244017\pi$
$43$	9.44629	1.44055	0.720273	+	0.693691i	$0.244017\pi$
$44$	0	0
$45$	0	0
$46$	6.91601	1.01971
$47$	5.39667	0.787185	0.393593	−	0.919285i	$-0.371232\pi$
$47$	5.39667	0.787185	0.393593	+	0.919285i	$0.371232\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.22348	0.455869
$51$	0	0
$52$	−2.90977	−0.403513
$53$	−9.39774	−1.29088	−0.645439	−	0.763812i	$-0.723326\pi$
$53$	−9.39774	−1.29088	−0.645439	+	0.763812i	$0.723326\pi$
$54$	0	0
$55$	0	0
$56$	−2.73565	−0.365567
$57$	0	0
$58$	−2.71193	−0.356094
$59$	−3.47462	−0.452357	−0.226178	−	0.974086i	$-0.572623\pi$
$59$	−3.47462	−0.452357	−0.226178	+	0.974086i	$0.572623\pi$
$60$	0	0
$61$	12.9942	1.66374	0.831870	−	0.554970i	$-0.187270\pi$
$61$	12.9942	1.66374	0.831870	+	0.554970i	$0.187270\pi$
$62$	6.15668	0.781900
$63$	0	0
$64$	3.98971	0.498714
$65$	−6.57274	−0.815248
$66$	0	0
$67$	4.32138	0.527940	0.263970	−	0.964531i	$-0.414968\pi$
$67$	4.32138	0.527940	0.263970	+	0.964531i	$0.414968\pi$
$68$	5.82457	0.706333
$69$	0	0
$70$	−2.45885	−0.293889
$71$	−4.40046	−0.522238	−0.261119	−	0.965307i	$-0.584092\pi$
$71$	−4.40046	−0.522238	−0.261119	+	0.965307i	$0.584092\pi$
$72$	0	0
$73$	14.7509	1.72647	0.863233	−	0.504805i	$-0.168436\pi$
$73$	14.7509	1.72647	0.863233	+	0.504805i	$0.168436\pi$
$74$	−7.23140	−0.840634
$75$	0	0
$76$	2.28086	0.261632
$77$	0	0
$78$	0	0
$79$	−7.18768	−0.808677	−0.404338	−	0.914609i	$-0.632498\pi$
$79$	−7.18768	−0.808677	−0.404338	+	0.914609i	$0.632498\pi$
$80$	−1.16604	−0.130367
$81$	0	0
$82$	−4.44186	−0.490521
$83$	−7.63445	−0.837990	−0.418995	−	0.907989i	$-0.637617\pi$
$83$	−7.63445	−0.837990	−0.418995	+	0.907989i	$0.637617\pi$
$84$	0	0
$85$	13.1568	1.42706
$86$	7.77955	0.838890
$87$	0	0
$88$	0	0
$89$	−10.8428	−1.14934	−0.574668	−	0.818386i	$-0.694869\pi$
$89$	−10.8428	−1.14934	−0.574668	+	0.818386i	$0.694869\pi$
$90$	0	0
$91$	2.20144	0.230774
$92$	−11.0998	−1.15723
$93$	0	0
$94$	4.44446	0.458411
$95$	5.15211	0.528596
$96$	0	0
$97$	2.85498	0.289879	0.144940	−	0.989440i	$-0.453701\pi$
$97$	2.85498	0.289879	0.144940	+	0.989440i	$0.453701\pi$
$98$	0.823556	0.0831917
$99$	0	0
$100$	−5.17348	−0.517348
$101$	14.6011	1.45287	0.726434	−	0.687236i	$-0.241176\pi$
$101$	14.6011	1.45287	0.726434	+	0.687236i	$0.241176\pi$
$102$	0	0
$103$	−0.107767	−0.0106186	−0.00530932	−	0.999986i	$-0.501690\pi$
$103$	−0.107767	−0.0106186	−0.00530932	+	0.999986i	$0.501690\pi$
$104$	−6.02238	−0.590543
$105$	0	0
$106$	−7.73956	−0.751733
$107$	4.64902	0.449438	0.224719	−	0.974424i	$-0.427854\pi$
$107$	4.64902	0.449438	0.224719	+	0.974424i	$0.427854\pi$
$108$	0	0
$109$	3.23140	0.309512	0.154756	−	0.987953i	$-0.450541\pi$
$109$	3.23140	0.309512	0.154756	+	0.987953i	$0.450541\pi$
$110$	0	0
$111$	0	0
$112$	0.390549	0.0369034
$113$	11.3194	1.06484	0.532420	−	0.846480i	$-0.321283\pi$
$113$	11.3194	1.06484	0.532420	+	0.846480i	$0.321283\pi$
$114$	0	0
$115$	−25.0727	−2.33804
$116$	4.35247	0.404117
$117$	0	0
$118$	−2.86154	−0.263426
$119$	−4.40669	−0.403961
$120$	0	0
$121$	0	0
$122$	10.7015	0.968866
$123$	0	0
$124$	−9.88109	−0.887348
$125$	3.24213	0.289985
$126$	0	0
$127$	−19.9802	−1.77295	−0.886476	−	0.462774i	$-0.846854\pi$
$127$	−19.9802	−1.77295	−0.886476	+	0.462774i	$0.846854\pi$
$128$	−8.30013	−0.733635
$129$	0	0
$130$	−5.41302	−0.474753
$131$	−8.45523	−0.738737	−0.369368	−	0.929283i	$-0.620426\pi$
$131$	−8.45523	−0.738737	−0.369368	+	0.929283i	$0.620426\pi$
$132$	0	0
$133$	−1.72563	−0.149631
$134$	3.55890	0.307442
$135$	0	0
$136$	12.0552	1.03372
$137$	−8.35456	−0.713779	−0.356889	−	0.934147i	$-0.616163\pi$
$137$	−8.35456	−0.713779	−0.356889	+	0.934147i	$0.616163\pi$
$138$	0	0
$139$	−14.7134	−1.24797	−0.623986	−	0.781436i	$-0.714488\pi$
$139$	−14.7134	−1.24797	−0.623986	+	0.781436i	$0.714488\pi$
$140$	3.94630	0.333523
$141$	0	0
$142$	−3.62402	−0.304121
$143$	0	0
$144$	0	0
$145$	9.83159	0.816469
$146$	12.1482	1.00539
$147$	0	0
$148$	11.6059	0.954003
$149$	−8.84271	−0.724423	−0.362212	−	0.932096i	$-0.617978\pi$
$149$	−8.84271	−0.724423	−0.362212	+	0.932096i	$0.617978\pi$
$150$	0	0
$151$	−19.9699	−1.62513	−0.812563	−	0.582873i	$-0.801929\pi$
$151$	−19.9699	−1.62513	−0.812563	+	0.582873i	$0.801929\pi$
$152$	4.72071	0.382900
$153$	0	0
$154$	0	0
$155$	−22.3199	−1.79278
$156$	0	0
$157$	13.0517	1.04164	0.520818	−	0.853668i	$-0.325627\pi$
$157$	13.0517	1.04164	0.520818	+	0.853668i	$0.325627\pi$
$158$	−5.91945	−0.470926
$159$	0	0
$160$	−17.2957	−1.36734
$161$	8.39774	0.661834
$162$	0	0
$163$	5.96447	0.467174	0.233587	−	0.972336i	$-0.424954\pi$
$163$	5.96447	0.467174	0.233587	+	0.972336i	$0.424954\pi$
$164$	7.12891	0.556674
$165$	0	0
$166$	−6.28740	−0.487997
$167$	−13.2472	−1.02510	−0.512548	−	0.858659i	$-0.671298\pi$
$167$	−13.2472	−1.02510	−0.512548	+	0.858659i	$0.671298\pi$
$168$	0	0
$169$	−8.15365	−0.627204
$170$	10.8354	0.831037
$171$	0	0
$172$	−12.4857	−0.952025
$173$	0.517716	0.0393612	0.0196806	−	0.999806i	$-0.493735\pi$
$173$	0.517716	0.0393612	0.0196806	+	0.999806i	$0.493735\pi$
$174$	0	0
$175$	3.91410	0.295878
$176$	0	0
$177$	0	0
$178$	−8.92967	−0.669307
$179$	−1.69615	−0.126776	−0.0633881	−	0.997989i	$-0.520191\pi$
$179$	−1.69615	−0.126776	−0.0633881	+	0.997989i	$0.520191\pi$
$180$	0	0
$181$	−9.88599	−0.734820	−0.367410	−	0.930059i	$-0.619755\pi$
$181$	−9.88599	−0.734820	−0.367410	+	0.930059i	$0.619755\pi$
$182$	1.81301	0.134389
$183$	0	0
$184$	−22.9733	−1.69361
$185$	26.2161	1.92745
$186$	0	0
$187$	0	0
$188$	−7.13308	−0.520233
$189$	0	0
$190$	4.24305	0.307823
$191$	−23.1329	−1.67384	−0.836918	−	0.547328i	$-0.815645\pi$
$191$	−23.1329	−1.67384	−0.836918	+	0.547328i	$0.815645\pi$
$192$	0	0
$193$	−23.0888	−1.66197	−0.830983	−	0.556297i	$-0.812222\pi$
$193$	−23.0888	−1.66197	−0.830983	+	0.556297i	$0.812222\pi$
$194$	2.35124	0.168809
$195$	0	0
$196$	−1.32176	−0.0944111
$197$	−6.68989	−0.476635	−0.238318	−	0.971187i	$-0.576596\pi$
$197$	−6.68989	−0.476635	−0.238318	+	0.971187i	$0.576596\pi$
$198$	0	0
$199$	−15.8233	−1.12169	−0.560844	−	0.827922i	$-0.689523\pi$
$199$	−15.8233	−1.12169	−0.560844	+	0.827922i	$0.689523\pi$
$200$	−10.7076	−0.757142
$201$	0	0
$202$	12.0249	0.846066
$203$	−3.29295	−0.231120
$204$	0	0
$205$	16.1031	1.12469
$206$	−0.0887525	−0.00618368
$207$	0	0
$208$	0.859771	0.0596144
$209$	0	0
$210$	0	0
$211$	11.3525	0.781538	0.390769	−	0.920489i	$-0.372209\pi$
$211$	11.3525	0.781538	0.390769	+	0.920489i	$0.372209\pi$
$212$	12.4215	0.853113
$213$	0	0
$214$	3.82873	0.261727
$215$	−28.2033	−1.92345
$216$	0	0
$217$	7.47573	0.507486
$218$	2.66124	0.180242
$219$	0	0
$220$	0	0
$221$	−9.70109	−0.652566
$222$	0	0
$223$	−10.6899	−0.715846	−0.357923	−	0.933751i	$-0.616515\pi$
$223$	−10.6899	−0.715846	−0.357923	+	0.933751i	$0.616515\pi$
$224$	5.79294	0.387057
$225$	0	0
$226$	9.32217	0.620102
$227$	3.73783	0.248089	0.124044	−	0.992277i	$-0.460414\pi$
$227$	3.73783	0.248089	0.124044	+	0.992277i	$0.460414\pi$
$228$	0	0
$229$	−23.6585	−1.56340	−0.781699	−	0.623656i	$-0.785647\pi$
$229$	−23.6585	−1.56340	−0.781699	+	0.623656i	$0.785647\pi$
$230$	−20.6488	−1.36154
$231$	0	0
$232$	9.00836	0.591428
$233$	−3.86675	−0.253319	−0.126660	−	0.991946i	$-0.540426\pi$
$233$	−3.86675	−0.253319	−0.126660	+	0.991946i	$0.540426\pi$
$234$	0	0
$235$	−16.1126	−1.05107
$236$	4.59259	0.298952
$237$	0	0
$238$	−3.62916	−0.235243
$239$	10.0717	0.651482	0.325741	−	0.945459i	$-0.394386\pi$
$239$	10.0717	0.651482	0.325741	+	0.945459i	$0.394386\pi$
$240$	0	0
$241$	13.4265	0.864878	0.432439	−	0.901663i	$-0.357653\pi$
$241$	13.4265	0.864878	0.432439	+	0.901663i	$0.357653\pi$
$242$	0	0
$243$	0	0
$244$	−17.1752	−1.09953
$245$	−2.98565	−0.190746
$246$	0	0
$247$	−3.79887	−0.241716
$248$	−20.4510	−1.29864
$249$	0	0
$250$	2.67007	0.168870
$251$	−0.0764471	−0.00482530	−0.00241265	−	0.999997i	$-0.500768\pi$
$251$	−0.0764471	−0.00482530	−0.00241265	+	0.999997i	$0.500768\pi$
$252$	0	0
$253$	0	0
$254$	−16.4548	−1.03246
$255$	0	0
$256$	−14.8151	−0.925941
$257$	−9.51194	−0.593338	−0.296669	−	0.954980i	$-0.595876\pi$
$257$	−9.51194	−0.593338	−0.296669	+	0.954980i	$0.595876\pi$
$258$	0	0
$259$	−8.78071	−0.545607
$260$	8.68755	0.538779
$261$	0	0
$262$	−6.96335	−0.430197
$263$	−14.7919	−0.912107	−0.456053	−	0.889952i	$-0.650737\pi$
$263$	−14.7919	−0.912107	−0.456053	+	0.889952i	$0.650737\pi$
$264$	0	0
$265$	28.0583	1.72361
$266$	−1.42115	−0.0871363
$267$	0	0
$268$	−5.71181	−0.348904
$269$	30.0556	1.83252	0.916262	−	0.400580i	$-0.131191\pi$
$269$	30.0556	1.83252	0.916262	+	0.400580i	$0.131191\pi$
$270$	0	0
$271$	−10.3918	−0.631260	−0.315630	−	0.948882i	$-0.602216\pi$
$271$	−10.3918	−0.631260	−0.315630	+	0.948882i	$0.602216\pi$
$272$	−1.72103	−0.104353
$273$	0	0
$274$	−6.88045	−0.415663
$275$	0	0
$276$	0	0
$277$	28.1138	1.68919	0.844597	−	0.535403i	$-0.179840\pi$
$277$	28.1138	1.68919	0.844597	+	0.535403i	$0.179840\pi$
$278$	−12.1173	−0.726747
$279$	0	0
$280$	8.16769	0.488113
$281$	−9.57010	−0.570904	−0.285452	−	0.958393i	$-0.592144\pi$
$281$	−9.57010	−0.570904	−0.285452	+	0.958393i	$0.592144\pi$
$282$	0	0
$283$	−1.65526	−0.0983947	−0.0491974	−	0.998789i	$-0.515666\pi$
$283$	−1.65526	−0.0983947	−0.0491974	+	0.998789i	$0.515666\pi$
$284$	5.81633	0.345136
$285$	0	0
$286$	0	0
$287$	−5.39351	−0.318369
$288$	0	0
$289$	2.41896	0.142292
$290$	8.09686	0.475464
$291$	0	0
$292$	−19.4971	−1.14098
$293$	−4.68188	−0.273519	−0.136759	−	0.990604i	$-0.543669\pi$
$293$	−4.68188	−0.273519	−0.136759	+	0.990604i	$0.543669\pi$
$294$	0	0
$295$	10.3740	0.603997
$296$	24.0210	1.39619
$297$	0	0
$298$	−7.28247	−0.421862
$299$	18.4871	1.06914
$300$	0	0
$301$	9.44629	0.544475
$302$	−16.4463	−0.946379
$303$	0	0
$304$	−0.673941	−0.0386532
$305$	−38.7962	−2.22146
$306$	0	0
$307$	−16.9829	−0.969266	−0.484633	−	0.874718i	$-0.661047\pi$
$307$	−16.9829	−0.969266	−0.484633	+	0.874718i	$0.661047\pi$
$308$	0	0
$309$	0	0
$310$	−18.3817	−1.04401
$311$	21.9257	1.24329	0.621645	−	0.783299i	$-0.286465\pi$
$311$	21.9257	1.24329	0.621645	+	0.783299i	$0.286465\pi$
$312$	0	0
$313$	10.0555	0.568368	0.284184	−	0.958770i	$-0.408277\pi$
$313$	10.0555	0.568368	0.284184	+	0.958770i	$0.408277\pi$
$314$	10.7488	0.606589
$315$	0	0
$316$	9.50035	0.534436
$317$	−23.5602	−1.32327	−0.661635	−	0.749826i	$-0.730137\pi$
$317$	−23.5602	−1.32327	−0.661635	+	0.749826i	$0.730137\pi$
$318$	0	0
$319$	0	0
$320$	−11.9119	−0.665895
$321$	0	0
$322$	6.91601	0.385414
$323$	7.60431	0.423115
$324$	0	0
$325$	8.61666	0.477966
$326$	4.91208	0.272055
$327$	0	0
$328$	14.7548	0.814696
$329$	5.39667	0.297528
$330$	0	0
$331$	−20.8607	−1.14661	−0.573304	−	0.819343i	$-0.694339\pi$
$331$	−20.8607	−1.14661	−0.573304	+	0.819343i	$0.694339\pi$
$332$	10.0909	0.553809
$333$	0	0
$334$	−10.9098	−0.596956
$335$	−12.9021	−0.704918
$336$	0	0
$337$	13.3544	0.727458	0.363729	−	0.931505i	$-0.381503\pi$
$337$	13.3544	0.727458	0.363729	+	0.931505i	$0.381503\pi$
$338$	−6.71498	−0.365247
$339$	0	0
$340$	−17.3901	−0.943112
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−25.8418	−1.39329
$345$	0	0
$346$	0.426368	0.0229217
$347$	1.59256	0.0854933	0.0427466	−	0.999086i	$-0.486389\pi$
$347$	1.59256	0.0854933	0.0427466	+	0.999086i	$0.486389\pi$
$348$	0	0
$349$	6.63475	0.355150	0.177575	−	0.984107i	$-0.443175\pi$
$349$	6.63475	0.355150	0.177575	+	0.984107i	$0.443175\pi$
$350$	3.22348	0.172302
$351$	0	0
$352$	0	0
$353$	25.4141	1.35265	0.676327	−	0.736601i	$-0.263570\pi$
$353$	25.4141	1.35265	0.676327	+	0.736601i	$0.263570\pi$
$354$	0	0
$355$	13.1382	0.697304
$356$	14.3316	0.759571
$357$	0	0
$358$	−1.39688	−0.0738271
$359$	−16.0792	−0.848629	−0.424314	−	0.905515i	$-0.639485\pi$
$359$	−16.0792	−0.848629	−0.424314	+	0.905515i	$0.639485\pi$
$360$	0	0
$361$	−16.0222	−0.843274
$362$	−8.14167	−0.427916
$363$	0	0
$364$	−2.90977	−0.152513
$365$	−44.0411	−2.30522
$366$	0	0
$367$	−3.02606	−0.157959	−0.0789796	−	0.996876i	$-0.525166\pi$
$367$	−3.02606	−0.157959	−0.0789796	+	0.996876i	$0.525166\pi$
$368$	3.27973	0.170968
$369$	0	0
$370$	21.5904	1.12243
$371$	−9.39774	−0.487906
$372$	0	0
$373$	1.73856	0.0900192	0.0450096	−	0.998987i	$-0.485668\pi$
$373$	1.73856	0.0900192	0.0450096	+	0.998987i	$0.485668\pi$
$374$	0	0
$375$	0	0
$376$	−14.7634	−0.761365
$377$	−7.24924	−0.373355
$378$	0	0
$379$	19.5728	1.00538	0.502692	−	0.864465i	$-0.332343\pi$
$379$	19.5728	1.00538	0.502692	+	0.864465i	$0.332343\pi$
$380$	−6.80983	−0.349337
$381$	0	0
$382$	−19.0512	−0.974746
$383$	−4.76472	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$383$	−4.76472	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$384$	0	0
$385$	0	0
$386$	−19.0149	−0.967833
$387$	0	0
$388$	−3.77359	−0.191575
$389$	1.42208	0.0721024	0.0360512	−	0.999350i	$-0.488522\pi$
$389$	1.42208	0.0721024	0.0360512	+	0.999350i	$0.488522\pi$
$390$	0	0
$391$	−37.0063	−1.87149
$392$	−2.73565	−0.138171
$393$	0	0
$394$	−5.50950	−0.277565
$395$	21.4599	1.07976
$396$	0	0
$397$	−18.3969	−0.923314	−0.461657	−	0.887059i	$-0.652745\pi$
$397$	−18.3969	−0.923314	−0.461657	+	0.887059i	$0.652745\pi$
$398$	−13.0314	−0.653206
$399$	0	0
$400$	1.52865	0.0764323
$401$	1.65268	0.0825308	0.0412654	−	0.999148i	$-0.486861\pi$
$401$	1.65268	0.0825308	0.0412654	+	0.999148i	$0.486861\pi$
$402$	0	0
$403$	16.4574	0.819802
$404$	−19.2991	−0.960168
$405$	0	0
$406$	−2.71193	−0.134591
$407$	0	0
$408$	0	0
$409$	−36.6858	−1.81400	−0.906999	−	0.421133i	$-0.861632\pi$
$409$	−36.6858	−1.81400	−0.906999	+	0.421133i	$0.861632\pi$
$410$	13.2618	0.654955
$411$	0	0
$412$	0.142442	0.00701762
$413$	−3.47462	−0.170975
$414$	0	0
$415$	22.7938	1.11890
$416$	12.7528	0.625259
$417$	0	0
$418$	0	0
$419$	17.3452	0.847366	0.423683	−	0.905810i	$-0.360737\pi$
$419$	17.3452	0.847366	0.423683	+	0.905810i	$0.360737\pi$
$420$	0	0
$421$	5.32683	0.259614	0.129807	−	0.991539i	$-0.458564\pi$
$421$	5.32683	0.259614	0.129807	+	0.991539i	$0.458564\pi$
$422$	9.34941	0.455122
$423$	0	0
$424$	25.7089	1.24854
$425$	−17.2482	−0.836662
$426$	0	0
$427$	12.9942	0.628835
$428$	−6.14487	−0.297024
$429$	0	0
$430$	−23.2270	−1.12011
$431$	28.2081	1.35873	0.679367	−	0.733798i	$-0.262254\pi$
$431$	28.2081	1.35873	0.679367	+	0.733798i	$0.262254\pi$
$432$	0	0
$433$	17.7060	0.850895	0.425447	−	0.904983i	$-0.360117\pi$
$433$	17.7060	0.850895	0.425447	+	0.904983i	$0.360117\pi$
$434$	6.15668	0.295530
$435$	0	0
$436$	−4.27113	−0.204550
$437$	−14.4914	−0.693216
$438$	0	0
$439$	−25.1189	−1.19886	−0.599430	−	0.800427i	$-0.704606\pi$
$439$	−25.1189	−1.19886	−0.599430	+	0.800427i	$0.704606\pi$
$440$	0	0
$441$	0	0
$442$	−7.98939	−0.380017
$443$	26.9567	1.28075	0.640376	−	0.768061i	$-0.278778\pi$
$443$	26.9567	1.28075	0.640376	+	0.768061i	$0.278778\pi$
$444$	0	0
$445$	32.3728	1.53462
$446$	−8.80369	−0.416867
$447$	0	0
$448$	3.98971	0.188496
$449$	21.0964	0.995599	0.497799	−	0.867292i	$-0.334142\pi$
$449$	21.0964	0.995599	0.497799	+	0.867292i	$0.334142\pi$
$450$	0	0
$451$	0	0
$452$	−14.9615	−0.703730
$453$	0	0
$454$	3.07831	0.144472
$455$	−6.57274	−0.308135
$456$	0	0
$457$	−1.90356	−0.0890448	−0.0445224	−	0.999008i	$-0.514177\pi$
$457$	−1.90356	−0.0890448	−0.0445224	+	0.999008i	$0.514177\pi$
$458$	−19.4841	−0.910432
$459$	0	0
$460$	33.1400	1.54516
$461$	−25.0440	−1.16641	−0.583207	−	0.812324i	$-0.698202\pi$
$461$	−25.0440	−1.16641	−0.583207	+	0.812324i	$0.698202\pi$
$462$	0	0
$463$	−21.1721	−0.983952	−0.491976	−	0.870609i	$-0.663725\pi$
$463$	−21.1721	−0.983952	−0.491976	+	0.870609i	$0.663725\pi$
$464$	−1.28606	−0.0597037
$465$	0	0
$466$	−3.18449	−0.147519
$467$	−16.6484	−0.770398	−0.385199	−	0.922834i	$-0.625867\pi$
$467$	−16.6484	−0.770398	−0.385199	+	0.922834i	$0.625867\pi$
$468$	0	0
$469$	4.32138	0.199543
$470$	−13.2696	−0.612081
$471$	0	0
$472$	9.50534	0.437519
$473$	0	0
$474$	0	0
$475$	−6.75427	−0.309907
$476$	5.82457	0.266969
$477$	0	0
$478$	8.29459	0.379386
$479$	−21.6613	−0.989730	−0.494865	−	0.868970i	$-0.664783\pi$
$479$	−21.6613	−0.989730	−0.494865	+	0.868970i	$0.664783\pi$
$480$	0	0
$481$	−19.3302	−0.881383
$482$	11.0575	0.503655
$483$	0	0
$484$	0	0
$485$	−8.52397	−0.387053
$486$	0	0
$487$	−2.81335	−0.127485	−0.0637426	−	0.997966i	$-0.520304\pi$
$487$	−2.81335	−0.127485	−0.0637426	+	0.997966i	$0.520304\pi$
$488$	−35.5477	−1.60917
$489$	0	0
$490$	−2.45885	−0.111079
$491$	−3.20580	−0.144676	−0.0723379	−	0.997380i	$-0.523046\pi$
$491$	−3.20580	−0.144676	−0.0723379	+	0.997380i	$0.523046\pi$
$492$	0	0
$493$	14.5110	0.653543
$494$	−3.12858	−0.140762
$495$	0	0
$496$	2.91964	0.131096
$497$	−4.40046	−0.197388
$498$	0	0
$499$	−20.0512	−0.897616	−0.448808	−	0.893628i	$-0.648151\pi$
$499$	−20.0512	−0.897616	−0.448808	+	0.893628i	$0.648151\pi$
$500$	−4.28530	−0.191644
$501$	0	0
$502$	−0.0629585	−0.00280997
$503$	−35.5089	−1.58326	−0.791631	−	0.610999i	$-0.790768\pi$
$503$	−35.5089	−1.58326	−0.791631	+	0.610999i	$0.790768\pi$
$504$	0	0
$505$	−43.5939	−1.93990
$506$	0	0
$507$	0	0
$508$	26.4089	1.17170
$509$	−9.18980	−0.407331	−0.203665	−	0.979041i	$-0.565285\pi$
$509$	−9.18980	−0.407331	−0.203665	+	0.979041i	$0.565285\pi$
$510$	0	0
$511$	14.7509	0.652543
$512$	4.39924	0.194421
$513$	0	0
$514$	−7.83361	−0.345526
$515$	0.321756	0.0141782
$516$	0	0
$517$	0	0
$518$	−7.23140	−0.317730
$519$	0	0
$520$	17.9807	0.788507
$521$	−12.3998	−0.543246	−0.271623	−	0.962404i	$-0.587560\pi$
$521$	−12.3998	−0.543246	−0.271623	+	0.962404i	$0.587560\pi$
$522$	0	0
$523$	22.7105	0.993062	0.496531	−	0.868019i	$-0.334607\pi$
$523$	22.7105	0.993062	0.496531	+	0.868019i	$0.334607\pi$
$524$	11.1757	0.488215
$525$	0	0
$526$	−12.1819	−0.531158
$527$	−32.9433	−1.43503
$528$	0	0
$529$	47.5220	2.06617
$530$	23.1076	1.00373
$531$	0	0
$532$	2.28086	0.0988877
$533$	−11.8735	−0.514299
$534$	0	0
$535$	−13.8803	−0.600100
$536$	−11.8218	−0.510623
$537$	0	0
$538$	24.7525	1.06716
$539$	0	0
$540$	0	0
$541$	17.2748	0.742703	0.371351	−	0.928492i	$-0.378895\pi$
$541$	17.2748	0.742703	0.371351	+	0.928492i	$0.378895\pi$
$542$	−8.55827	−0.367609
$543$	0	0
$544$	−25.5277	−1.09449
$545$	−9.64784	−0.413268
$546$	0	0
$547$	−13.1740	−0.563278	−0.281639	−	0.959520i	$-0.590878\pi$
$547$	−13.1740	−0.563278	−0.281639	+	0.959520i	$0.590878\pi$
$548$	11.0427	0.471720
$549$	0	0
$550$	0	0
$551$	5.68240	0.242078
$552$	0	0
$553$	−7.18768	−0.305651
$554$	23.1533	0.983689
$555$	0	0
$556$	19.4475	0.824757
$557$	12.6216	0.534793	0.267396	−	0.963587i	$-0.413837\pi$
$557$	12.6216	0.534793	0.267396	+	0.963587i	$0.413837\pi$
$558$	0	0
$559$	20.7955	0.879555
$560$	−1.16604	−0.0492743
$561$	0	0
$562$	−7.88151	−0.332462
$563$	−9.81924	−0.413832	−0.206916	−	0.978359i	$-0.566343\pi$
$563$	−9.81924	−0.413832	−0.206916	+	0.978359i	$0.566343\pi$
$564$	0	0
$565$	−33.7958	−1.42180
$566$	−1.36320	−0.0572994
$567$	0	0
$568$	12.0381	0.505108
$569$	−20.3544	−0.853301	−0.426650	−	0.904417i	$-0.640307\pi$
$569$	−20.3544	−0.853301	−0.426650	+	0.904417i	$0.640307\pi$
$570$	0	0
$571$	−32.3174	−1.35244	−0.676221	−	0.736699i	$-0.736384\pi$
$571$	−32.3174	−1.35244	−0.676221	+	0.736699i	$0.736384\pi$
$572$	0	0
$573$	0	0
$574$	−4.44186	−0.185400
$575$	32.8696	1.37076
$576$	0	0
$577$	−11.3181	−0.471179	−0.235590	−	0.971853i	$-0.575702\pi$
$577$	−11.3181	−0.471179	−0.235590	+	0.971853i	$0.575702\pi$
$578$	1.99215	0.0828623
$579$	0	0
$580$	−12.9950	−0.539586
$581$	−7.63445	−0.316730
$582$	0	0
$583$	0	0
$584$	−40.3534	−1.66984
$585$	0	0
$586$	−3.85579	−0.159281
$587$	38.2388	1.57828	0.789142	−	0.614211i	$-0.210526\pi$
$587$	38.2388	1.57828	0.789142	+	0.614211i	$0.210526\pi$
$588$	0	0
$589$	−12.9003	−0.531548
$590$	8.54356	0.351733
$591$	0	0
$592$	−3.42930	−0.140943
$593$	26.4263	1.08520	0.542598	−	0.839992i	$-0.317441\pi$
$593$	26.4263	1.08520	0.542598	+	0.839992i	$0.317441\pi$
$594$	0	0
$595$	13.1568	0.539378
$596$	11.6879	0.478755
$597$	0	0
$598$	15.2252	0.622605
$599$	20.2994	0.829413	0.414706	−	0.909955i	$-0.363884\pi$
$599$	20.2994	0.829413	0.414706	+	0.909955i	$0.363884\pi$
$600$	0	0
$601$	−5.09410	−0.207793	−0.103896	−	0.994588i	$-0.533131\pi$
$601$	−5.09410	−0.207793	−0.103896	+	0.994588i	$0.533131\pi$
$602$	7.77955	0.317071
$603$	0	0
$604$	26.3953	1.07401
$605$	0	0
$606$	0	0
$607$	30.6407	1.24367	0.621833	−	0.783150i	$-0.286388\pi$
$607$	30.6407	1.24367	0.621833	+	0.783150i	$0.286388\pi$
$608$	−9.99645	−0.405410
$609$	0	0
$610$	−31.9508	−1.29365
$611$	11.8805	0.480632
$612$	0	0
$613$	−24.8391	−1.00324	−0.501620	−	0.865088i	$-0.667263\pi$
$613$	−24.8391	−1.00324	−0.501620	+	0.865088i	$0.667263\pi$
$614$	−13.9864	−0.564444
$615$	0	0
$616$	0	0
$617$	−0.521714	−0.0210034	−0.0105017	−	0.999945i	$-0.503343\pi$
$617$	−0.521714	−0.0210034	−0.0105017	+	0.999945i	$0.503343\pi$
$618$	0	0
$619$	31.4859	1.26552	0.632762	−	0.774346i	$-0.281921\pi$
$619$	31.4859	1.26552	0.632762	+	0.774346i	$0.281921\pi$
$620$	29.5015	1.18481
$621$	0	0
$622$	18.0570	0.724020
$623$	−10.8428	−0.434408
$624$	0	0
$625$	−29.2503	−1.17001
$626$	8.28124	0.330985
$627$	0	0
$628$	−17.2511	−0.688395
$629$	38.6939	1.54283
$630$	0	0
$631$	−0.114230	−0.00454742	−0.00227371	−	0.999997i	$-0.500724\pi$
$631$	−0.114230	−0.00454742	−0.00227371	+	0.999997i	$0.500724\pi$
$632$	19.6630	0.782151
$633$	0	0
$634$	−19.4031	−0.770596
$635$	59.6537	2.36729
$636$	0	0
$637$	2.20144	0.0872244
$638$	0	0
$639$	0	0
$640$	24.7813	0.979566
$641$	−13.2141	−0.521927	−0.260963	−	0.965349i	$-0.584040\pi$
$641$	−13.2141	−0.521927	−0.260963	+	0.965349i	$0.584040\pi$
$642$	0	0
$643$	2.45599	0.0968550	0.0484275	−	0.998827i	$-0.484579\pi$
$643$	2.45599	0.0968550	0.0484275	+	0.998827i	$0.484579\pi$
$644$	−11.0998	−0.437392
$645$	0	0
$646$	6.26257	0.246398
$647$	18.6116	0.731698	0.365849	−	0.930674i	$-0.380779\pi$
$647$	18.6116	0.731698	0.365849	+	0.930674i	$0.380779\pi$
$648$	0	0
$649$	0	0
$650$	7.09630	0.278340
$651$	0	0
$652$	−7.88357	−0.308745
$653$	−19.4481	−0.761065	−0.380532	−	0.924768i	$-0.624259\pi$
$653$	−19.4481	−0.761065	−0.380532	+	0.924768i	$0.624259\pi$
$654$	0	0
$655$	25.2443	0.986378
$656$	−2.10643	−0.0822423
$657$	0	0
$658$	4.44446	0.173263
$659$	−4.71629	−0.183721	−0.0918603	−	0.995772i	$-0.529281\pi$
$659$	−4.71629	−0.183721	−0.0918603	+	0.995772i	$0.529281\pi$
$660$	0	0
$661$	24.9330	0.969782	0.484891	−	0.874575i	$-0.338859\pi$
$661$	24.9330	0.969782	0.484891	+	0.874575i	$0.338859\pi$
$662$	−17.1800	−0.667718
$663$	0	0
$664$	20.8852	0.810503
$665$	5.15211	0.199790
$666$	0	0
$667$	−27.6533	−1.07074
$668$	17.5095	0.677463
$669$	0	0
$670$	−10.6256	−0.410503
$671$	0	0
$672$	0	0
$673$	−5.12972	−0.197736	−0.0988681	−	0.995101i	$-0.531522\pi$
$673$	−5.12972	−0.197736	−0.0988681	+	0.995101i	$0.531522\pi$
$674$	10.9981	0.423629
$675$	0	0
$676$	10.7771	0.414505
$677$	51.2610	1.97012	0.985060	−	0.172213i	$-0.0550917\pi$
$677$	51.2610	1.97012	0.985060	+	0.172213i	$0.0550917\pi$
$678$	0	0
$679$	2.85498	0.109564
$680$	−35.9925	−1.38025
$681$	0	0
$682$	0	0
$683$	24.9448	0.954485	0.477242	−	0.878772i	$-0.341636\pi$
$683$	24.9448	0.954485	0.477242	+	0.878772i	$0.341636\pi$
$684$	0	0
$685$	24.9438	0.953053
$686$	0.823556	0.0314435
$687$	0	0
$688$	3.68924	0.140651
$689$	−20.6886	−0.788172
$690$	0	0
$691$	11.7847	0.448312	0.224156	−	0.974553i	$-0.428038\pi$
$691$	11.7847	0.448312	0.224156	+	0.974553i	$0.428038\pi$
$692$	−0.684294	−0.0260130
$693$	0	0
$694$	1.31156	0.0497863
$695$	43.9290	1.66632
$696$	0	0
$697$	23.7676	0.900261
$698$	5.46409	0.206819
$699$	0	0
$700$	−5.17348	−0.195539
$701$	31.3172	1.18283	0.591417	−	0.806366i	$-0.298569\pi$
$701$	31.3172	1.18283	0.591417	+	0.806366i	$0.298569\pi$
$702$	0	0
$703$	15.1522	0.571477
$704$	0	0
$705$	0	0
$706$	20.9299	0.787708
$707$	14.6011	0.549133
$708$	0	0
$709$	−41.4751	−1.55763	−0.778815	−	0.627254i	$-0.784179\pi$
$709$	−41.4751	−1.55763	−0.778815	+	0.627254i	$0.784179\pi$
$710$	10.8201	0.406070
$711$	0	0
$712$	29.6622	1.11164
$713$	62.7792	2.35110
$714$	0	0
$715$	0	0
$716$	2.24190	0.0837836
$717$	0	0
$718$	−13.2421	−0.494192
$719$	7.89027	0.294258	0.147129	−	0.989117i	$-0.452997\pi$
$719$	7.89027	0.294258	0.147129	+	0.989117i	$0.452997\pi$
$720$	0	0
$721$	−0.107767	−0.00401347
$722$	−13.1952	−0.491074
$723$	0	0
$724$	13.0669	0.485626
$725$	−12.8889	−0.478682
$726$	0	0
$727$	5.12729	0.190161	0.0950803	−	0.995470i	$-0.469689\pi$
$727$	5.12729	0.190161	0.0950803	+	0.995470i	$0.469689\pi$
$728$	−6.02238	−0.223204
$729$	0	0
$730$	−36.2703	−1.34242
$731$	−41.6269	−1.53963
$732$	0	0
$733$	−33.7736	−1.24746	−0.623728	−	0.781642i	$-0.714383\pi$
$733$	−33.7736	−1.24746	−0.623728	+	0.781642i	$0.714383\pi$
$734$	−2.49213	−0.0919863
$735$	0	0
$736$	48.6476	1.79317
$737$	0	0
$738$	0	0
$739$	−42.7431	−1.57233	−0.786164	−	0.618017i	$-0.787936\pi$
$739$	−42.7431	−1.57233	−0.786164	+	0.618017i	$0.787936\pi$
$740$	−34.6513	−1.27381
$741$	0	0
$742$	−7.73956	−0.284128
$743$	−22.9565	−0.842192	−0.421096	−	0.907016i	$-0.638354\pi$
$743$	−22.9565	−0.842192	−0.421096	+	0.907016i	$0.638354\pi$
$744$	0	0
$745$	26.4012	0.967266
$746$	1.43180	0.0524220
$747$	0	0
$748$	0	0
$749$	4.64902	0.169872
$750$	0	0
$751$	−0.930719	−0.0339624	−0.0169812	−	0.999856i	$-0.505406\pi$
$751$	−0.930719	−0.0339624	−0.0169812	+	0.999856i	$0.505406\pi$
$752$	2.10766	0.0768586
$753$	0	0
$754$	−5.97016	−0.217420
$755$	59.6231	2.16991
$756$	0	0
$757$	−11.4824	−0.417334	−0.208667	−	0.977987i	$-0.566913\pi$
$757$	−11.4824	−0.417334	−0.208667	+	0.977987i	$0.566913\pi$
$758$	16.1193	0.585478
$759$	0	0
$760$	−14.0944	−0.511257
$761$	−38.8640	−1.40882	−0.704410	−	0.709794i	$-0.748788\pi$
$761$	−38.8640	−1.40882	−0.704410	+	0.709794i	$0.748788\pi$
$762$	0	0
$763$	3.23140	0.116985
$764$	30.5760	1.10620
$765$	0	0
$766$	−3.92402	−0.141780
$767$	−7.64917	−0.276196
$768$	0	0
$769$	−35.6509	−1.28560	−0.642802	−	0.766032i	$-0.722228\pi$
$769$	−35.6509	−1.28560	−0.642802	+	0.766032i	$0.722228\pi$
$770$	0	0
$771$	0	0
$772$	30.5177	1.09836
$773$	1.77045	0.0636788	0.0318394	−	0.999493i	$-0.489863\pi$
$773$	1.77045	0.0636788	0.0318394	+	0.999493i	$0.489863\pi$
$774$	0	0
$775$	29.2607	1.05108
$776$	−7.81023	−0.280371
$777$	0	0
$778$	1.17116	0.0419883
$779$	9.30719	0.333465
$780$	0	0
$781$	0	0
$782$	−30.4767	−1.08985
$783$	0	0
$784$	0.390549	0.0139482
$785$	−38.9677	−1.39082
$786$	0	0
$787$	3.35315	0.119527	0.0597635	−	0.998213i	$-0.480965\pi$
$787$	3.35315	0.119527	0.0597635	+	0.998213i	$0.480965\pi$
$788$	8.84240	0.314998
$789$	0	0
$790$	17.6734	0.628792
$791$	11.3194	0.402472
$792$	0	0
$793$	28.6061	1.01583
$794$	−15.1509	−0.537684
$795$	0	0
$796$	20.9146	0.741298
$797$	−4.92542	−0.174467	−0.0872337	−	0.996188i	$-0.527803\pi$
$797$	−4.92542	−0.174467	−0.0872337	+	0.996188i	$0.527803\pi$
$798$	0	0
$799$	−23.7815	−0.841328
$800$	22.6741	0.801652
$801$	0	0
$802$	1.36107	0.0480612
$803$	0	0
$804$	0	0
$805$	−25.0727	−0.883696
$806$	13.5536	0.477405
$807$	0	0
$808$	−39.9436	−1.40521
$809$	6.97322	0.245165	0.122583	−	0.992458i	$-0.460882\pi$
$809$	6.97322	0.245165	0.122583	+	0.992458i	$0.460882\pi$
$810$	0	0
$811$	5.93377	0.208363	0.104181	−	0.994558i	$-0.466778\pi$
$811$	5.93377	0.208363	0.104181	+	0.994558i	$0.466778\pi$
$812$	4.35247	0.152742
$813$	0	0
$814$	0	0
$815$	−17.8078	−0.623781
$816$	0	0
$817$	−16.3008	−0.570292
$818$	−30.2128	−1.05637
$819$	0	0
$820$	−21.2844	−0.743284
$821$	−41.7468	−1.45697	−0.728487	−	0.685060i	$-0.759776\pi$
$821$	−41.7468	−1.45697	−0.728487	+	0.685060i	$0.759776\pi$
$822$	0	0
$823$	−23.2004	−0.808714	−0.404357	−	0.914601i	$-0.632505\pi$
$823$	−23.2004	−0.808714	−0.404357	+	0.914601i	$0.632505\pi$
$824$	0.294814	0.0102703
$825$	0	0
$826$	−2.86154	−0.0995658
$827$	13.1779	0.458240	0.229120	−	0.973398i	$-0.426415\pi$
$827$	13.1779	0.458240	0.229120	+	0.973398i	$0.426415\pi$
$828$	0	0
$829$	4.29090	0.149029	0.0745146	−	0.997220i	$-0.476259\pi$
$829$	4.29090	0.149029	0.0745146	+	0.997220i	$0.476259\pi$
$830$	18.7720	0.651584
$831$	0	0
$832$	8.78313	0.304500
$833$	−4.40669	−0.152683
$834$	0	0
$835$	39.5513	1.36873
$836$	0	0
$837$	0	0
$838$	14.2847	0.493457
$839$	15.7453	0.543588	0.271794	−	0.962355i	$-0.412383\pi$
$839$	15.7453	0.543588	0.271794	+	0.962355i	$0.412383\pi$
$840$	0	0
$841$	−18.1565	−0.626086
$842$	4.38694	0.151184
$843$	0	0
$844$	−15.0052	−0.516501
$845$	24.3439	0.837456
$846$	0	0
$847$	0	0
$848$	−3.67028	−0.126038
$849$	0	0
$850$	−14.2049	−0.487223
$851$	−73.7381	−2.52771
$852$	0	0
$853$	6.14015	0.210235	0.105117	−	0.994460i	$-0.466478\pi$
$853$	6.14015	0.210235	0.105117	+	0.994460i	$0.466478\pi$
$854$	10.7015	0.366197
$855$	0	0
$856$	−12.7181	−0.434696
$857$	−34.4740	−1.17761	−0.588804	−	0.808276i	$-0.700401\pi$
$857$	−34.4740	−1.17761	−0.588804	+	0.808276i	$0.700401\pi$
$858$	0	0
$859$	8.21553	0.280310	0.140155	−	0.990130i	$-0.455240\pi$
$859$	8.21553	0.280310	0.140155	+	0.990130i	$0.455240\pi$
$860$	37.2779	1.27116
$861$	0	0
$862$	23.2309	0.791248
$863$	−13.0390	−0.443852	−0.221926	−	0.975063i	$-0.571234\pi$
$863$	−13.0390	−0.443852	−0.221926	+	0.975063i	$0.571234\pi$
$864$	0	0
$865$	−1.54572	−0.0525560
$866$	14.5819	0.495512
$867$	0	0
$868$	−9.88109	−0.335386
$869$	0	0
$870$	0	0
$871$	9.51327	0.322345
$872$	−8.84000	−0.299360
$873$	0	0
$874$	−11.9344	−0.403689
$875$	3.24213	0.109604
$876$	0	0
$877$	17.9259	0.605316	0.302658	−	0.953099i	$-0.402126\pi$
$877$	17.9259	0.605316	0.302658	+	0.953099i	$0.402126\pi$
$878$	−20.6868	−0.698147
$879$	0	0
$880$	0	0
$881$	17.3276	0.583780	0.291890	−	0.956452i	$-0.405716\pi$
$881$	17.3276	0.583780	0.291890	+	0.956452i	$0.405716\pi$
$882$	0	0
$883$	21.8740	0.736120	0.368060	−	0.929802i	$-0.380022\pi$
$883$	21.8740	0.736120	0.368060	+	0.929802i	$0.380022\pi$
$884$	12.8225	0.431266
$885$	0	0
$886$	22.2004	0.745836
$887$	−25.6098	−0.859893	−0.429946	−	0.902854i	$-0.641468\pi$
$887$	−25.6098	−0.859893	−0.429946	+	0.902854i	$0.641468\pi$
$888$	0	0
$889$	−19.9802	−0.670113
$890$	26.6608	0.893674
$891$	0	0
$892$	14.1294	0.473087
$893$	−9.31263	−0.311635
$894$	0	0
$895$	5.06411	0.169275
$896$	−8.30013	−0.277288
$897$	0	0
$898$	17.3740	0.579779
$899$	−24.6172	−0.821030
$900$	0	0
$901$	41.4130	1.37967
$902$	0	0
$903$	0	0
$904$	−30.9660	−1.02991
$905$	29.5161	0.981148
$906$	0	0
$907$	−14.4388	−0.479431	−0.239716	−	0.970843i	$-0.577054\pi$
$907$	−14.4388	−0.479431	−0.239716	+	0.970843i	$0.577054\pi$
$908$	−4.94050	−0.163956
$909$	0	0
$910$	−5.41302	−0.179440
$911$	−37.1858	−1.23202	−0.616010	−	0.787739i	$-0.711252\pi$
$911$	−37.1858	−1.23202	−0.616010	+	0.787739i	$0.711252\pi$
$912$	0	0
$913$	0	0
$914$	−1.56769	−0.0518545
$915$	0	0
$916$	31.2708	1.03321
$917$	−8.45523	−0.279216
$918$	0	0
$919$	−16.0496	−0.529426	−0.264713	−	0.964327i	$-0.585277\pi$
$919$	−16.0496	−0.529426	−0.264713	+	0.964327i	$0.585277\pi$
$920$	68.5901	2.26135
$921$	0	0
$922$	−20.6251	−0.679252
$923$	−9.68736	−0.318863
$924$	0	0
$925$	−34.3685	−1.13003
$926$	−17.4364	−0.572997
$927$	0	0
$928$	−19.0759	−0.626196
$929$	−33.8914	−1.11194	−0.555970	−	0.831202i	$-0.687653\pi$
$929$	−33.8914	−1.11194	−0.555970	+	0.831202i	$0.687653\pi$
$930$	0	0
$931$	−1.72563	−0.0565551
$932$	5.11090	0.167413
$933$	0	0
$934$	−13.7109	−0.448635
$935$	0	0
$936$	0	0
$937$	−9.22624	−0.301408	−0.150704	−	0.988579i	$-0.548154\pi$
$937$	−9.22624	−0.301408	−0.150704	+	0.988579i	$0.548154\pi$
$938$	3.55890	0.116202
$939$	0	0
$940$	21.2969	0.694627
$941$	53.3626	1.73957	0.869786	−	0.493430i	$-0.164257\pi$
$941$	53.3626	1.73957	0.869786	+	0.493430i	$0.164257\pi$
$942$	0	0
$943$	−45.2933	−1.47495
$944$	−1.35701	−0.0441668
$945$	0	0
$946$	0	0
$947$	−31.0986	−1.01057	−0.505285	−	0.862953i	$-0.668612\pi$
$947$	−31.0986	−1.01057	−0.505285	+	0.862953i	$0.668612\pi$
$948$	0	0
$949$	32.4734	1.05413
$950$	−5.56252	−0.180472
$951$	0	0
$952$	12.0552	0.390711
$953$	−34.9286	−1.13145	−0.565724	−	0.824595i	$-0.691403\pi$
$953$	−34.9286	−1.13145	−0.565724	+	0.824595i	$0.691403\pi$
$954$	0	0
$955$	69.0667	2.23494
$956$	−13.3123	−0.430550
$957$	0	0
$958$	−17.8393	−0.576361
$959$	−8.35456	−0.269783
$960$	0	0
$961$	24.8866	0.802793
$962$	−15.9195	−0.513266
$963$	0	0
$964$	−17.7466	−0.571579
$965$	68.9350	2.21910
$966$	0	0
$967$	−20.8029	−0.668975	−0.334488	−	0.942400i	$-0.608563\pi$
$967$	−20.8029	−0.668975	−0.334488	+	0.942400i	$0.608563\pi$
$968$	0	0
$969$	0	0
$970$	−7.01996	−0.225397
$971$	40.0308	1.28465	0.642325	−	0.766433i	$-0.277970\pi$
$971$	40.0308	1.28465	0.642325	+	0.766433i	$0.277970\pi$
$972$	0	0
$973$	−14.7134	−0.471689
$974$	−2.31695	−0.0742400
$975$	0	0
$976$	5.07488	0.162443
$977$	−6.48609	−0.207509	−0.103754	−	0.994603i	$-0.533086\pi$
$977$	−6.48609	−0.207509	−0.103754	+	0.994603i	$0.533086\pi$
$978$	0	0
$979$	0	0
$980$	3.94630	0.126060
$981$	0	0
$982$	−2.64016	−0.0842508
$983$	−10.9782	−0.350150	−0.175075	−	0.984555i	$-0.556017\pi$
$983$	−10.9782	−0.350150	−0.175075	+	0.984555i	$0.556017\pi$
$984$	0	0
$985$	19.9737	0.636414
$986$	11.9506	0.380586
$987$	0	0
$988$	5.02118	0.159745
$989$	79.3275	2.52247
$990$	0	0
$991$	−59.2666	−1.88267	−0.941333	−	0.337479i	$-0.890426\pi$
$991$	−59.2666	−1.88267	−0.941333	+	0.337479i	$0.890426\pi$
$992$	43.3065	1.37498
$993$	0	0
$994$	−3.62402	−0.114947
$995$	47.2430	1.49770
$996$	0	0
$997$	−43.3517	−1.37296	−0.686481	−	0.727148i	$-0.740845\pi$
$997$	−43.3517	−1.37296	−0.686481	+	0.727148i	$0.740845\pi$
$998$	−16.5133	−0.522720
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cp.1.5		6
3.2	odd	2		847.2.a.n.1.2	yes	6
11.10	odd	2		7623.2.a.cs.1.2		6
21.20	even	2		5929.2.a.bm.1.2		6
33.2	even	10		847.2.f.z.323.2		24
33.5	odd	10		847.2.f.y.729.5		24
33.8	even	10		847.2.f.z.372.5		24
33.14	odd	10		847.2.f.y.372.2		24
33.17	even	10		847.2.f.z.729.2		24
33.20	odd	10		847.2.f.y.323.5		24
33.26	odd	10		847.2.f.y.148.2		24
33.29	even	10		847.2.f.z.148.5		24
33.32	even	2		847.2.a.m.1.5	✓	6
231.230	odd	2		5929.2.a.bj.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.5	✓	6	33.32	even	2
847.2.a.n.1.2	yes	6	3.2	odd	2
847.2.f.y.148.2		24	33.26	odd	10
847.2.f.y.323.5		24	33.20	odd	10
847.2.f.y.372.2		24	33.14	odd	10
847.2.f.y.729.5		24	33.5	odd	10
847.2.f.z.148.5		24	33.29	even	10
847.2.f.z.323.2		24	33.2	even	10
847.2.f.z.372.5		24	33.8	even	10
847.2.f.z.729.2		24	33.17	even	10
5929.2.a.bj.1.5		6	231.230	odd	2
5929.2.a.bm.1.2		6	21.20	even	2
7623.2.a.cp.1.5		6	1.1	even	1	trivial
7623.2.a.cs.1.2		6	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.823556 q^{2} -1.32176 q^{4} -2.98565 q^{5} +1.00000 q^{7} -2.73565 q^{8} +O(q^{10})\)	\(q+0.823556 q^{2} -1.32176 q^{4} -2.98565 q^{5} +1.00000 q^{7} -2.73565 q^{8} -2.45885 q^{10} +2.20144 q^{13} +0.823556 q^{14} +0.390549 q^{16} -4.40669 q^{17} -1.72563 q^{19} +3.94630 q^{20} +8.39774 q^{23} +3.91410 q^{25} +1.81301 q^{26} -1.32176 q^{28} -3.29295 q^{29} +7.47573 q^{31} +5.79294 q^{32} -3.62916 q^{34} -2.98565 q^{35} -8.78071 q^{37} -1.42115 q^{38} +8.16769 q^{40} -5.39351 q^{41} +9.44629 q^{43} +6.91601 q^{46} +5.39667 q^{47} +1.00000 q^{49} +3.22348 q^{50} -2.90977 q^{52} -9.39774 q^{53} -2.73565 q^{56} -2.71193 q^{58} -3.47462 q^{59} +12.9942 q^{61} +6.15668 q^{62} +3.98971 q^{64} -6.57274 q^{65} +4.32138 q^{67} +5.82457 q^{68} -2.45885 q^{70} -4.40046 q^{71} +14.7509 q^{73} -7.23140 q^{74} +2.28086 q^{76} -7.18768 q^{79} -1.16604 q^{80} -4.44186 q^{82} -7.63445 q^{83} +13.1568 q^{85} +7.77955 q^{86} -10.8428 q^{89} +2.20144 q^{91} -11.0998 q^{92} +4.44446 q^{94} +5.15211 q^{95} +2.85498 q^{97} +0.823556 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 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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