LMFDB - Embedded newform 8280.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 16x^{4} + 26x^{3} + 52x^{2} - 48x + 9 $ x^6 - 2x^5 - 16x^4 + 26x^3 + 52x^2 - 48*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.287750$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.73629 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.73629 q^{7} -5.97718 q^{11} -5.40168 q^{13} -6.94546 q^{17} -3.97718 q^{19} +1.00000 q^{23} +1.00000 q^{25} +9.29741 q^{29} -3.54378 q^{31} -2.73629 q^{35} +7.17782 q^{37} -10.1945 q^{41} +1.42450 q^{43} -10.4187 q^{47} +0.487260 q^{49} +4.04839 q^{53} -5.97718 q^{55} +8.13797 q^{59} -3.34516 q^{61} -5.40168 q^{65} -1.70525 q^{67} -7.56247 q^{71} +7.85722 q^{73} +16.3553 q^{77} -1.36798 q^{79} +13.9709 q^{83} -6.94546 q^{85} -6.49805 q^{89} +14.7805 q^{91} -3.97718 q^{95} -13.8769 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 2 q^{7}+O(q^{10}) $ 6 * q + 6 * q^5 - 2 * q^7	$ 6 q + 6 q^{5} - 2 q^{7} - 4 q^{11} + 4 q^{17} + 8 q^{19} + 6 q^{23} + 6 q^{25} + 18 q^{29} - 8 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 2 q^{59} - 8 q^{61} - 2 q^{67} + 2 q^{71} + 8 q^{73} + 16 q^{77} - 28 q^{79} + 24 q^{83} + 4 q^{85} + 4 q^{89} - 8 q^{91} + 8 q^{95} + 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^5 - 2 * q^7 - 4 * q^11 + 4 * q^17 + 8 * q^19 + 6 * q^23 + 6 * q^25 + 18 * q^29 - 8 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^41 + 8 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 2 * q^59 - 8 * q^61 - 2 * q^67 + 2 * q^71 + 8 * q^73 + 16 * q^77 - 28 * q^79 + 24 * q^83 + 4 * q^85 + 4 * q^89 - 8 * q^91 + 8 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.73629	−1.03422	−0.517109	−	0.855919i	$-0.672992\pi$
$7$	−2.73629	−1.03422	−0.517109	+	0.855919i	$0.672992\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.97718	−1.80219	−0.901094	−	0.433624i	$-0.857235\pi$
$11$	−5.97718	−1.80219	−0.901094	+	0.433624i	$0.857235\pi$
$12$	0	0
$13$	−5.40168	−1.49816	−0.749078	−	0.662482i	$-0.769503\pi$
$13$	−5.40168	−1.49816	−0.749078	+	0.662482i	$0.769503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.94546	−1.68452	−0.842261	−	0.539070i	$-0.818776\pi$
$17$	−6.94546	−1.68452	−0.842261	+	0.539070i	$0.818776\pi$
$18$	0	0
$19$	−3.97718	−0.912428	−0.456214	−	0.889870i	$-0.650795\pi$
$19$	−3.97718	−0.912428	−0.456214	+	0.889870i	$0.650795\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.29741	1.72649	0.863243	−	0.504788i	$-0.168429\pi$
$29$	9.29741	1.72649	0.863243	+	0.504788i	$0.168429\pi$
$30$	0	0
$31$	−3.54378	−0.636482	−0.318241	−	0.948010i	$-0.603092\pi$
$31$	−3.54378	−0.636482	−0.318241	+	0.948010i	$0.603092\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.73629	−0.462517
$36$	0	0
$37$	7.17782	1.18003	0.590013	−	0.807394i	$-0.299123\pi$
$37$	7.17782	1.18003	0.590013	+	0.807394i	$0.299123\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.1945	−1.59211	−0.796056	−	0.605223i	$-0.793084\pi$
$41$	−10.1945	−1.59211	−0.796056	+	0.605223i	$0.793084\pi$
$42$	0	0
$43$	1.42450	0.217234	0.108617	−	0.994084i	$-0.465358\pi$
$43$	1.42450	0.217234	0.108617	+	0.994084i	$0.465358\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.4187	−1.51973	−0.759863	−	0.650084i	$-0.774734\pi$
$47$	−10.4187	−1.51973	−0.759863	+	0.650084i	$0.774734\pi$
$48$	0	0
$49$	0.487260	0.0696085
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.04839	0.556089	0.278044	−	0.960568i	$-0.410314\pi$
$53$	4.04839	0.556089	0.278044	+	0.960568i	$0.410314\pi$
$54$	0	0
$55$	−5.97718	−0.805963
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.13797	1.05947	0.529736	−	0.848162i	$-0.322291\pi$
$59$	8.13797	1.05947	0.529736	+	0.848162i	$0.322291\pi$
$60$	0	0
$61$	−3.34516	−0.428304	−0.214152	−	0.976800i	$-0.568699\pi$
$61$	−3.34516	−0.428304	−0.214152	+	0.976800i	$0.568699\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.40168	−0.669996
$66$	0	0
$67$	−1.70525	−0.208329	−0.104164	−	0.994560i	$-0.533217\pi$
$67$	−1.70525	−0.208329	−0.104164	+	0.994560i	$0.533217\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.56247	−0.897500	−0.448750	−	0.893657i	$-0.648131\pi$
$71$	−7.56247	−0.897500	−0.448750	+	0.893657i	$0.648131\pi$
$72$	0	0
$73$	7.85722	0.919618	0.459809	−	0.888018i	$-0.347918\pi$
$73$	7.85722	0.919618	0.459809	+	0.888018i	$0.347918\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.3553	1.86386
$78$	0	0
$79$	−1.36798	−0.153910	−0.0769548	−	0.997035i	$-0.524520\pi$
$79$	−1.36798	−0.153910	−0.0769548	+	0.997035i	$0.524520\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.9709	1.53351	0.766755	−	0.641940i	$-0.221870\pi$
$83$	13.9709	1.53351	0.766755	+	0.641940i	$0.221870\pi$
$84$	0	0
$85$	−6.94546	−0.753341
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.49805	−0.688792	−0.344396	−	0.938824i	$-0.611916\pi$
$89$	−6.49805	−0.688792	−0.344396	+	0.938824i	$0.611916\pi$
$90$	0	0
$91$	14.7805	1.54942
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.97718	−0.408050
$96$	0	0
$97$	−13.8769	−1.40899	−0.704493	−	0.709710i	$-0.748826\pi$
$97$	−13.8769	−1.40899	−0.704493	+	0.709710i	$0.748826\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.4293	1.83378	0.916890	−	0.399140i	$-0.130691\pi$
$101$	18.4293	1.83378	0.916890	+	0.399140i	$0.130691\pi$
$102$	0	0
$103$	8.29130	0.816966	0.408483	−	0.912766i	$-0.366058\pi$
$103$	8.29130	0.816966	0.408483	+	0.912766i	$0.366058\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.583632	−0.0564218	−0.0282109	−	0.999602i	$-0.508981\pi$
$107$	−0.583632	−0.0564218	−0.0282109	+	0.999602i	$0.508981\pi$
$108$	0	0
$109$	−4.37064	−0.418631	−0.209316	−	0.977848i	$-0.567124\pi$
$109$	−4.37064	−0.418631	−0.209316	+	0.977848i	$0.567124\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.4504	1.07717	0.538583	−	0.842572i	$-0.318960\pi$
$113$	11.4504	1.07717	0.538583	+	0.842572i	$0.318960\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.0048	1.74216
$120$	0	0
$121$	24.7267	2.24788
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.9710	1.50594	0.752968	−	0.658058i	$-0.228622\pi$
$127$	16.9710	1.50594	0.752968	+	0.658058i	$0.228622\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.3727	−1.60523	−0.802616	−	0.596497i	$-0.796559\pi$
$131$	−18.3727	−1.60523	−0.802616	+	0.596497i	$0.796559\pi$
$132$	0	0
$133$	10.8827	0.943650
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.74607	0.661792	0.330896	−	0.943667i	$-0.392649\pi$
$137$	7.74607	0.661792	0.330896	+	0.943667i	$0.392649\pi$
$138$	0	0
$139$	−10.2906	−0.872839	−0.436419	−	0.899743i	$-0.643754\pi$
$139$	−10.2906	−0.872839	−0.436419	+	0.899743i	$0.643754\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	32.2868	2.69996
$144$	0	0
$145$	9.29741	0.772108
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.0647	1.39800	0.698999	−	0.715123i	$-0.253629\pi$
$149$	17.0647	1.39800	0.698999	+	0.715123i	$0.253629\pi$
$150$	0	0
$151$	−15.9141	−1.29507	−0.647536	−	0.762035i	$-0.724200\pi$
$151$	−15.9141	−1.29507	−0.647536	+	0.762035i	$0.724200\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.54378	−0.284643
$156$	0	0
$157$	24.5482	1.95916	0.979581	−	0.201050i	$-0.0644355\pi$
$157$	24.5482	1.95916	0.979581	+	0.201050i	$0.0644355\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.73629	−0.215650
$162$	0	0
$163$	−7.71580	−0.604348	−0.302174	−	0.953253i	$-0.597712\pi$
$163$	−7.71580	−0.604348	−0.302174	+	0.953253i	$0.597712\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.0793410	0.00613959	0.00306979	−	0.999995i	$-0.499023\pi$
$167$	0.0793410	0.00613959	0.00306979	+	0.999995i	$0.499023\pi$
$168$	0	0
$169$	16.1781	1.24447
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.1590	−1.45663	−0.728316	−	0.685242i	$-0.759697\pi$
$173$	−19.1590	−1.45663	−0.728316	+	0.685242i	$0.759697\pi$
$174$	0	0
$175$	−2.73629	−0.206844
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.68009	0.349807	0.174903	−	0.984586i	$-0.444039\pi$
$179$	4.68009	0.349807	0.174903	+	0.984586i	$0.444039\pi$
$180$	0	0
$181$	7.46476	0.554851	0.277426	−	0.960747i	$-0.410519\pi$
$181$	7.46476	0.554851	0.277426	+	0.960747i	$0.410519\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.17782	0.527724
$186$	0	0
$187$	41.5143	3.03582
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.7805	1.93777	0.968886	−	0.247508i	$-0.0796116\pi$
$191$	26.7805	1.93777	0.968886	+	0.247508i	$0.0796116\pi$
$192$	0	0
$193$	12.8196	0.922777	0.461388	−	0.887198i	$-0.347351\pi$
$193$	12.8196	0.922777	0.461388	+	0.887198i	$0.347351\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.60577	0.185654	0.0928268	−	0.995682i	$-0.470410\pi$
$197$	2.60577	0.185654	0.0928268	+	0.995682i	$0.470410\pi$
$198$	0	0
$199$	−12.5686	−0.890963	−0.445482	−	0.895291i	$-0.646968\pi$
$199$	−12.5686	−0.890963	−0.445482	+	0.895291i	$0.646968\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−25.4404	−1.78556
$204$	0	0
$205$	−10.1945	−0.712014
$206$	0	0
$207$	0	0
$208$	0	0
$209$	23.7723	1.64437
$210$	0	0
$211$	−25.1102	−1.72866	−0.864330	−	0.502925i	$-0.832257\pi$
$211$	−25.1102	−1.72866	−0.864330	+	0.502925i	$0.832257\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.42450	0.0971501
$216$	0	0
$217$	9.69680	0.658261
$218$	0	0
$219$	0	0
$220$	0	0
$221$	37.5172	2.52368
$222$	0	0
$223$	−10.4718	−0.701244	−0.350622	−	0.936517i	$-0.614030\pi$
$223$	−10.4718	−0.701244	−0.350622	+	0.936517i	$0.614030\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.2767	−1.14669	−0.573347	−	0.819312i	$-0.694355\pi$
$227$	−17.2767	−1.14669	−0.573347	+	0.819312i	$0.694355\pi$
$228$	0	0
$229$	−12.5065	−0.826453	−0.413226	−	0.910628i	$-0.635598\pi$
$229$	−12.5065	−0.826453	−0.413226	+	0.910628i	$0.635598\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.09678	−0.399413	−0.199707	−	0.979856i	$-0.563999\pi$
$233$	−6.09678	−0.399413	−0.199707	+	0.979856i	$0.563999\pi$
$234$	0	0
$235$	−10.4187	−0.679642
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.57168	−0.295718	−0.147859	−	0.989008i	$-0.547238\pi$
$239$	−4.57168	−0.295718	−0.147859	+	0.989008i	$0.547238\pi$
$240$	0	0
$241$	−25.9022	−1.66851	−0.834253	−	0.551383i	$-0.814100\pi$
$241$	−25.9022	−1.66851	−0.834253	+	0.551383i	$0.814100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.487260	0.0311299
$246$	0	0
$247$	21.4835	1.36696
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.803360	−0.0507076	−0.0253538	−	0.999679i	$-0.508071\pi$
$251$	−0.803360	−0.0507076	−0.0253538	+	0.999679i	$0.508071\pi$
$252$	0	0
$253$	−5.97718	−0.375782
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.20486	0.387049	0.193524	−	0.981095i	$-0.438008\pi$
$257$	6.20486	0.387049	0.193524	+	0.981095i	$0.438008\pi$
$258$	0	0
$259$	−19.6406	−1.22040
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.8348	1.28473	0.642366	−	0.766398i	$-0.277953\pi$
$263$	20.8348	1.28473	0.642366	+	0.766398i	$0.277953\pi$
$264$	0	0
$265$	4.04839	0.248691
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.8374	−1.39242	−0.696210	−	0.717838i	$-0.745132\pi$
$269$	−22.8374	−1.39242	−0.696210	+	0.717838i	$0.745132\pi$
$270$	0	0
$271$	−28.5632	−1.73509	−0.867547	−	0.497356i	$-0.834304\pi$
$271$	−28.5632	−1.73509	−0.867547	+	0.497356i	$0.834304\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.97718	−0.360438
$276$	0	0
$277$	−26.2565	−1.57760	−0.788801	−	0.614648i	$-0.789298\pi$
$277$	−26.2565	−1.57760	−0.788801	+	0.614648i	$0.789298\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.27463	−0.195348	−0.0976740	−	0.995218i	$-0.531140\pi$
$281$	−3.27463	−0.195348	−0.0976740	+	0.995218i	$0.531140\pi$
$282$	0	0
$283$	−8.87870	−0.527784	−0.263892	−	0.964552i	$-0.585006\pi$
$283$	−8.87870	−0.527784	−0.263892	+	0.964552i	$0.585006\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.8950	1.64659
$288$	0	0
$289$	31.2394	1.83761
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.3646	−0.956029	−0.478014	−	0.878352i	$-0.658643\pi$
$293$	−16.3646	−0.956029	−0.478014	+	0.878352i	$0.658643\pi$
$294$	0	0
$295$	8.13797	0.473811
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.40168	−0.312387
$300$	0	0
$301$	−3.89784	−0.224668
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.34516	−0.191543
$306$	0	0
$307$	−21.0803	−1.20312	−0.601558	−	0.798829i	$-0.705453\pi$
$307$	−21.0803	−1.20312	−0.601558	+	0.798829i	$0.705453\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.9877	1.41692	0.708460	−	0.705751i	$-0.249390\pi$
$311$	24.9877	1.41692	0.708460	+	0.705751i	$0.249390\pi$
$312$	0	0
$313$	−26.8602	−1.51823	−0.759114	−	0.650957i	$-0.774368\pi$
$313$	−26.8602	−1.51823	−0.759114	+	0.650957i	$0.774368\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.0456	1.40670	0.703350	−	0.710843i	$-0.251687\pi$
$317$	25.0456	1.40670	0.703350	+	0.710843i	$0.251687\pi$
$318$	0	0
$319$	−55.5723	−3.11145
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.6233	1.53700
$324$	0	0
$325$	−5.40168	−0.299631
$326$	0	0
$327$	0	0
$328$	0	0
$329$	28.5086	1.57173
$330$	0	0
$331$	−13.4803	−0.740947	−0.370473	−	0.928843i	$-0.620804\pi$
$331$	−13.4803	−0.740947	−0.370473	+	0.928843i	$0.620804\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.70525	−0.0931675
$336$	0	0
$337$	24.3478	1.32631	0.663155	−	0.748482i	$-0.269217\pi$
$337$	24.3478	1.32631	0.663155	+	0.748482i	$0.269217\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.1818	1.14706
$342$	0	0
$343$	17.8207	0.962228
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.15100	0.276520	0.138260	−	0.990396i	$-0.455849\pi$
$347$	5.15100	0.276520	0.138260	+	0.990396i	$0.455849\pi$
$348$	0	0
$349$	23.0646	1.23462	0.617310	−	0.786720i	$-0.288222\pi$
$349$	23.0646	1.23462	0.617310	+	0.786720i	$0.288222\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.62321	0.139619	0.0698097	−	0.997560i	$-0.477761\pi$
$353$	2.62321	0.139619	0.0698097	+	0.997560i	$0.477761\pi$
$354$	0	0
$355$	−7.56247	−0.401374
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.36987	−0.0722991	−0.0361495	−	0.999346i	$-0.511509\pi$
$359$	−1.36987	−0.0722991	−0.0361495	+	0.999346i	$0.511509\pi$
$360$	0	0
$361$	−3.18204	−0.167476
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.85722	0.411266
$366$	0	0
$367$	−8.01055	−0.418147	−0.209074	−	0.977900i	$-0.567045\pi$
$367$	−8.01055	−0.418147	−0.209074	+	0.977900i	$0.567045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.0775	−0.575118
$372$	0	0
$373$	−7.80815	−0.404291	−0.202145	−	0.979356i	$-0.564791\pi$
$373$	−7.80815	−0.404291	−0.202145	+	0.979356i	$0.564791\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−50.2217	−2.58655
$378$	0	0
$379$	−23.8126	−1.22317	−0.611585	−	0.791179i	$-0.709468\pi$
$379$	−23.8126	−1.22317	−0.611585	+	0.791179i	$0.709468\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.4730	0.841732	0.420866	−	0.907123i	$-0.361726\pi$
$383$	16.4730	0.841732	0.420866	+	0.907123i	$0.361726\pi$
$384$	0	0
$385$	16.3553	0.833542
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.71444	−0.289734	−0.144867	−	0.989451i	$-0.546275\pi$
$389$	−5.71444	−0.289734	−0.144867	+	0.989451i	$0.546275\pi$
$390$	0	0
$391$	−6.94546	−0.351247
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.36798	−0.0688305
$396$	0	0
$397$	−14.6525	−0.735388	−0.367694	−	0.929947i	$-0.619853\pi$
$397$	−14.6525	−0.735388	−0.367694	+	0.929947i	$0.619853\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.2851	1.06293	0.531465	−	0.847080i	$-0.321642\pi$
$401$	21.2851	1.06293	0.531465	+	0.847080i	$0.321642\pi$
$402$	0	0
$403$	19.1424	0.953549
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−42.9031	−2.12663
$408$	0	0
$409$	31.1908	1.54228	0.771142	−	0.636663i	$-0.219686\pi$
$409$	31.1908	1.54228	0.771142	+	0.636663i	$0.219686\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−22.2678	−1.09573
$414$	0	0
$415$	13.9709	0.685807
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.4950	1.78290	0.891448	−	0.453123i	$-0.149690\pi$
$419$	36.4950	1.78290	0.891448	+	0.453123i	$0.149690\pi$
$420$	0	0
$421$	−2.94307	−0.143437	−0.0717183	−	0.997425i	$-0.522848\pi$
$421$	−2.94307	−0.143437	−0.0717183	+	0.997425i	$0.522848\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.94546	−0.336904
$426$	0	0
$427$	9.15331	0.442960
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.53529	0.170289	0.0851444	−	0.996369i	$-0.472865\pi$
$431$	3.53529	0.170289	0.0851444	+	0.996369i	$0.472865\pi$
$432$	0	0
$433$	38.5181	1.85106	0.925532	−	0.378671i	$-0.123619\pi$
$433$	38.5181	1.85106	0.925532	+	0.378671i	$0.123619\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.97718	−0.190254
$438$	0	0
$439$	−13.3043	−0.634980	−0.317490	−	0.948262i	$-0.602840\pi$
$439$	−13.3043	−0.634980	−0.317490	+	0.948262i	$0.602840\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.1259	1.00372	0.501862	−	0.864948i	$-0.332649\pi$
$443$	21.1259	1.00372	0.501862	+	0.864948i	$0.332649\pi$
$444$	0	0
$445$	−6.49805	−0.308037
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.13932	0.100961	0.0504805	−	0.998725i	$-0.483925\pi$
$449$	2.13932	0.100961	0.0504805	+	0.998725i	$0.483925\pi$
$450$	0	0
$451$	60.9343	2.86928
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.7805	0.692922
$456$	0	0
$457$	0.0350873	0.00164131	0.000820657	−	1.00000i	$-0.499739\pi$
$457$	0.0350873	0.00164131	0.000820657		1.00000i	$0.499739\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.2381	−1.40833	−0.704164	−	0.710037i	$-0.748678\pi$
$461$	−30.2381	−1.40833	−0.704164	+	0.710037i	$0.748678\pi$
$462$	0	0
$463$	−4.31412	−0.200494	−0.100247	−	0.994963i	$-0.531963\pi$
$463$	−4.31412	−0.200494	−0.100247	+	0.994963i	$0.531963\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.7680	0.914753	0.457376	−	0.889273i	$-0.348789\pi$
$467$	19.7680	0.914753	0.457376	+	0.889273i	$0.348789\pi$
$468$	0	0
$469$	4.66604	0.215458
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.51449	−0.391497
$474$	0	0
$475$	−3.97718	−0.182486
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.5883	0.849320	0.424660	−	0.905353i	$-0.360394\pi$
$479$	18.5883	0.849320	0.424660	+	0.905353i	$0.360394\pi$
$480$	0	0
$481$	−38.7723	−1.76786
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.8769	−0.630118
$486$	0	0
$487$	15.1337	0.685775	0.342888	−	0.939376i	$-0.388595\pi$
$487$	15.1337	0.685775	0.342888	+	0.939376i	$0.388595\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.0762	1.31219	0.656095	−	0.754678i	$-0.272207\pi$
$491$	29.0762	1.31219	0.656095	+	0.754678i	$0.272207\pi$
$492$	0	0
$493$	−64.5748	−2.90830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.6931	0.928211
$498$	0	0
$499$	13.4060	0.600133	0.300067	−	0.953918i	$-0.402991\pi$
$499$	13.4060	0.600133	0.300067	+	0.953918i	$0.402991\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.2143	−0.589197	−0.294598	−	0.955621i	$-0.595186\pi$
$503$	−13.2143	−0.589197	−0.294598	+	0.955621i	$0.595186\pi$
$504$	0	0
$505$	18.4293	0.820091
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.41682	0.373069	0.186534	−	0.982448i	$-0.440274\pi$
$509$	8.41682	0.373069	0.186534	+	0.982448i	$0.440274\pi$
$510$	0	0
$511$	−21.4996	−0.951086
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.29130	0.365358
$516$	0	0
$517$	62.2745	2.73883
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.03908	−0.176955	−0.0884777	−	0.996078i	$-0.528200\pi$
$521$	−4.03908	−0.176955	−0.0884777	+	0.996078i	$0.528200\pi$
$522$	0	0
$523$	8.85699	0.387289	0.193645	−	0.981072i	$-0.437969\pi$
$523$	8.85699	0.387289	0.193645	+	0.981072i	$0.437969\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.6132	1.07217
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	55.0674	2.38523
$534$	0	0
$535$	−0.583632	−0.0252326
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.91244	−0.125448
$540$	0	0
$541$	−28.1566	−1.21055	−0.605274	−	0.796017i	$-0.706936\pi$
$541$	−28.1566	−1.21055	−0.605274	+	0.796017i	$0.706936\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.37064	−0.187218
$546$	0	0
$547$	−19.4765	−0.832756	−0.416378	−	0.909191i	$-0.636701\pi$
$547$	−19.4765	−0.832756	−0.416378	+	0.909191i	$0.636701\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−36.9775	−1.57529
$552$	0	0
$553$	3.74318	0.159176
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.88185	0.291593	0.145797	−	0.989315i	$-0.453425\pi$
$557$	6.88185	0.291593	0.145797	+	0.989315i	$0.453425\pi$
$558$	0	0
$559$	−7.69469	−0.325451
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.30665	−0.392229	−0.196114	−	0.980581i	$-0.562832\pi$
$563$	−9.30665	−0.392229	−0.196114	+	0.980581i	$0.562832\pi$
$564$	0	0
$565$	11.4504	0.481723
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.5779	−0.778827	−0.389414	−	0.921063i	$-0.627322\pi$
$569$	−18.5779	−0.778827	−0.389414	+	0.921063i	$0.627322\pi$
$570$	0	0
$571$	39.5512	1.65517	0.827583	−	0.561343i	$-0.189715\pi$
$571$	39.5512	1.65517	0.827583	+	0.561343i	$0.189715\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−13.1923	−0.549204	−0.274602	−	0.961558i	$-0.588546\pi$
$577$	−13.1923	−0.549204	−0.274602	+	0.961558i	$0.588546\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−38.2285	−1.58598
$582$	0	0
$583$	−24.1979	−1.00218
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.7066	0.441908	0.220954	−	0.975284i	$-0.429083\pi$
$587$	10.7066	0.441908	0.220954	+	0.975284i	$0.429083\pi$
$588$	0	0
$589$	14.0943	0.580744
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.98410	−0.163607	−0.0818036	−	0.996648i	$-0.526068\pi$
$593$	−3.98410	−0.163607	−0.0818036	+	0.996648i	$0.526068\pi$
$594$	0	0
$595$	19.0048	0.779119
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−25.9056	−1.05847	−0.529237	−	0.848474i	$-0.677522\pi$
$599$	−25.9056	−1.05847	−0.529237	+	0.848474i	$0.677522\pi$
$600$	0	0
$601$	12.3935	0.505542	0.252771	−	0.967526i	$-0.418658\pi$
$601$	12.3935	0.505542	0.252771	+	0.967526i	$0.418658\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.7267	1.00528
$606$	0	0
$607$	−22.4768	−0.912304	−0.456152	−	0.889902i	$-0.650773\pi$
$607$	−22.4768	−0.912304	−0.456152	+	0.889902i	$0.650773\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	56.2785	2.27679
$612$	0	0
$613$	−23.9995	−0.969332	−0.484666	−	0.874699i	$-0.661059\pi$
$613$	−23.9995	−0.969332	−0.484666	+	0.874699i	$0.661059\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.7574	1.23825	0.619123	−	0.785294i	$-0.287488\pi$
$617$	30.7574	1.23825	0.619123	+	0.785294i	$0.287488\pi$
$618$	0	0
$619$	−34.4693	−1.38544	−0.692719	−	0.721207i	$-0.743588\pi$
$619$	−34.4693	−1.38544	−0.692719	+	0.721207i	$0.743588\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.7805	0.712362
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−49.8532	−1.98778
$630$	0	0
$631$	−8.22555	−0.327454	−0.163727	−	0.986506i	$-0.552352\pi$
$631$	−8.22555	−0.327454	−0.163727	+	0.986506i	$0.552352\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.9710	0.673475
$636$	0	0
$637$	−2.63202	−0.104284
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.5166	1.60031	0.800154	−	0.599795i	$-0.204751\pi$
$641$	40.5166	1.60031	0.800154	+	0.599795i	$0.204751\pi$
$642$	0	0
$643$	0.890308	0.0351103	0.0175552	−	0.999846i	$-0.494412\pi$
$643$	0.890308	0.0351103	0.0175552	+	0.999846i	$0.494412\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.0168	1.21940	0.609699	−	0.792633i	$-0.291290\pi$
$647$	31.0168	1.21940	0.609699	+	0.792633i	$0.291290\pi$
$648$	0	0
$649$	−48.6421	−1.90937
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.8735	−0.464645	−0.232323	−	0.972639i	$-0.574633\pi$
$653$	−11.8735	−0.464645	−0.232323	+	0.972639i	$0.574633\pi$
$654$	0	0
$655$	−18.3727	−0.717881
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.0551	−1.28764	−0.643822	−	0.765175i	$-0.722652\pi$
$659$	−33.0551	−1.28764	−0.643822	+	0.765175i	$0.722652\pi$
$660$	0	0
$661$	−4.71253	−0.183296	−0.0916482	−	0.995791i	$-0.529214\pi$
$661$	−4.71253	−0.183296	−0.0916482	+	0.995791i	$0.529214\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.8827	0.422013
$666$	0	0
$667$	9.29741	0.359997
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.9946	0.771884
$672$	0	0
$673$	−13.0906	−0.504605	−0.252303	−	0.967648i	$-0.581188\pi$
$673$	−13.0906	−0.504605	−0.252303	+	0.967648i	$0.581188\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.915188	0.0351735	0.0175868	−	0.999845i	$-0.494402\pi$
$677$	0.915188	0.0351735	0.0175868	+	0.999845i	$0.494402\pi$
$678$	0	0
$679$	37.9712	1.45720
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.1856	−1.15502	−0.577511	−	0.816383i	$-0.695976\pi$
$683$	−30.1856	−1.15502	−0.577511	+	0.816383i	$0.695976\pi$
$684$	0	0
$685$	7.74607	0.295962
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−21.8681	−0.833108
$690$	0	0
$691$	−38.8018	−1.47609	−0.738044	−	0.674752i	$-0.764250\pi$
$691$	−38.8018	−1.47609	−0.738044	+	0.674752i	$0.764250\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.2906	−0.390345
$696$	0	0
$697$	70.8054	2.68195
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.6348	0.552749	0.276375	−	0.961050i	$-0.410867\pi$
$701$	14.6348	0.552749	0.276375	+	0.961050i	$0.410867\pi$
$702$	0	0
$703$	−28.5475	−1.07669
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−50.4277	−1.89653
$708$	0	0
$709$	−6.89807	−0.259062	−0.129531	−	0.991575i	$-0.541347\pi$
$709$	−6.89807	−0.259062	−0.129531	+	0.991575i	$0.541347\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.54378	−0.132716
$714$	0	0
$715$	32.2868	1.20746
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.9504	−0.818610	−0.409305	−	0.912398i	$-0.634229\pi$
$719$	−21.9504	−0.818610	−0.409305	+	0.912398i	$0.634229\pi$
$720$	0	0
$721$	−22.6874	−0.844922
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.29741	0.345297
$726$	0	0
$727$	39.8620	1.47840	0.739200	−	0.673486i	$-0.235204\pi$
$727$	39.8620	1.47840	0.739200	+	0.673486i	$0.235204\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.89381	−0.365936
$732$	0	0
$733$	50.6101	1.86933	0.934663	−	0.355536i	$-0.115701\pi$
$733$	50.6101	1.86933	0.934663	+	0.355536i	$0.115701\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.1926	0.375448
$738$	0	0
$739$	11.1158	0.408901	0.204451	−	0.978877i	$-0.434459\pi$
$739$	11.1158	0.408901	0.204451	+	0.978877i	$0.434459\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.8523	1.49873	0.749363	−	0.662159i	$-0.230360\pi$
$743$	40.8523	1.49873	0.749363	+	0.662159i	$0.230360\pi$
$744$	0	0
$745$	17.0647	0.625204
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.59698	0.0583525
$750$	0	0
$751$	−5.21900	−0.190444	−0.0952221	−	0.995456i	$-0.530356\pi$
$751$	−5.21900	−0.190444	−0.0952221	+	0.995456i	$0.530356\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.9141	−0.579173
$756$	0	0
$757$	3.27998	0.119213	0.0596064	−	0.998222i	$-0.481015\pi$
$757$	3.27998	0.119213	0.0596064	+	0.998222i	$0.481015\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.5191	0.925066	0.462533	−	0.886602i	$-0.346941\pi$
$761$	25.5191	0.925066	0.462533	+	0.886602i	$0.346941\pi$
$762$	0	0
$763$	11.9593	0.432956
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−43.9587	−1.58726
$768$	0	0
$769$	34.5086	1.24441	0.622205	−	0.782854i	$-0.286237\pi$
$769$	34.5086	1.24441	0.622205	+	0.782854i	$0.286237\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−50.8067	−1.82739	−0.913695	−	0.406401i	$-0.866784\pi$
$773$	−50.8067	−1.82739	−0.913695	+	0.406401i	$0.866784\pi$
$774$	0	0
$775$	−3.54378	−0.127296
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.5453	1.45269
$780$	0	0
$781$	45.2022	1.61746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.5482	0.876164
$786$	0	0
$787$	−28.9087	−1.03048	−0.515242	−	0.857045i	$-0.672298\pi$
$787$	−28.9087	−1.03048	−0.515242	+	0.857045i	$0.672298\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−31.3317	−1.11403
$792$	0	0
$793$	18.0695	0.641666
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.7056	1.47729	0.738644	−	0.674095i	$-0.235466\pi$
$797$	41.7056	1.47729	0.738644	+	0.674095i	$0.235466\pi$
$798$	0	0
$799$	72.3627	2.56001
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−46.9640	−1.65732
$804$	0	0
$805$	−2.73629	−0.0964414
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.2397	1.37959	0.689797	−	0.724002i	$-0.257700\pi$
$809$	39.2397	1.37959	0.689797	+	0.724002i	$0.257700\pi$
$810$	0	0
$811$	32.2340	1.13189	0.565945	−	0.824443i	$-0.308511\pi$
$811$	32.2340	1.13189	0.565945	+	0.824443i	$0.308511\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.71580	−0.270273
$816$	0	0
$817$	−5.66549	−0.198210
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.1197	−0.911583	−0.455792	−	0.890087i	$-0.650644\pi$
$821$	−26.1197	−0.911583	−0.455792	+	0.890087i	$0.650644\pi$
$822$	0	0
$823$	14.6734	0.511481	0.255740	−	0.966745i	$-0.417681\pi$
$823$	14.6734	0.511481	0.255740	+	0.966745i	$0.417681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.3498	−0.498990	−0.249495	−	0.968376i	$-0.580265\pi$
$827$	−14.3498	−0.498990	−0.249495	+	0.968376i	$0.580265\pi$
$828$	0	0
$829$	−14.9407	−0.518911	−0.259456	−	0.965755i	$-0.583543\pi$
$829$	−14.9407	−0.518911	−0.259456	+	0.965755i	$0.583543\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.38424	−0.117257
$834$	0	0
$835$	0.0793410	0.00274571
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38.1982	−1.31875	−0.659375	−	0.751814i	$-0.729179\pi$
$839$	−38.1982	−1.31875	−0.659375	+	0.751814i	$0.729179\pi$
$840$	0	0
$841$	57.4419	1.98076
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.1781	0.556545
$846$	0	0
$847$	−67.6593	−2.32480
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.17782	0.246052
$852$	0	0
$853$	−13.0969	−0.448430	−0.224215	−	0.974540i	$-0.571982\pi$
$853$	−13.0969	−0.448430	−0.224215	+	0.974540i	$0.571982\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.6533	−1.52533	−0.762664	−	0.646795i	$-0.776109\pi$
$857$	−44.6533	−1.52533	−0.762664	+	0.646795i	$0.776109\pi$
$858$	0	0
$859$	−23.6964	−0.808512	−0.404256	−	0.914646i	$-0.632470\pi$
$859$	−23.6964	−0.808512	−0.404256	+	0.914646i	$0.632470\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.5825	−0.972961	−0.486480	−	0.873692i	$-0.661719\pi$
$863$	−28.5825	−0.972961	−0.486480	+	0.873692i	$0.661719\pi$
$864$	0	0
$865$	−19.1590	−0.651425
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.17666	0.277374
$870$	0	0
$871$	9.21119	0.312109
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.73629	−0.0925033
$876$	0	0
$877$	29.4797	0.995457	0.497729	−	0.867333i	$-0.334168\pi$
$877$	29.4797	0.995457	0.497729	+	0.867333i	$0.334168\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.1078	−0.677450	−0.338725	−	0.940885i	$-0.609996\pi$
$881$	−20.1078	−0.677450	−0.338725	+	0.940885i	$0.609996\pi$
$882$	0	0
$883$	−50.7789	−1.70885	−0.854424	−	0.519577i	$-0.826090\pi$
$883$	−50.7789	−1.70885	−0.854424	+	0.519577i	$0.826090\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.365439	0.0122703	0.00613513	−	0.999981i	$-0.498047\pi$
$887$	0.365439	0.0122703	0.00613513	+	0.999981i	$0.498047\pi$
$888$	0	0
$889$	−46.4376	−1.55747
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41.4371	1.38664
$894$	0	0
$895$	4.68009	0.156438
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−32.9480	−1.09888
$900$	0	0
$901$	−28.1179	−0.936744
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.46476	0.248137
$906$	0	0
$907$	16.9221	0.561890	0.280945	−	0.959724i	$-0.409352\pi$
$907$	16.9221	0.561890	0.280945	+	0.959724i	$0.409352\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.0149	0.398072	0.199036	−	0.979992i	$-0.436219\pi$
$911$	12.0149	0.398072	0.199036	+	0.979992i	$0.436219\pi$
$912$	0	0
$913$	−83.5068	−2.76367
$914$	0	0
$915$	0	0
$916$	0	0
$917$	50.2730	1.66016
$918$	0	0
$919$	−12.1340	−0.400263	−0.200131	−	0.979769i	$-0.564137\pi$
$919$	−12.1340	−0.400263	−0.200131	+	0.979769i	$0.564137\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.8500	1.34459
$924$	0	0
$925$	7.17782	0.236005
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.70399	0.0559060	0.0279530	−	0.999609i	$-0.491101\pi$
$929$	1.70399	0.0559060	0.0279530	+	0.999609i	$0.491101\pi$
$930$	0	0
$931$	−1.93792	−0.0635127
$932$	0	0
$933$	0	0
$934$	0	0
$935$	41.5143	1.35766
$936$	0	0
$937$	20.1572	0.658505	0.329253	−	0.944242i	$-0.393203\pi$
$937$	20.1572	0.658505	0.329253	+	0.944242i	$0.393203\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.934801	0.0304736	0.0152368	−	0.999884i	$-0.495150\pi$
$941$	0.934801	0.0304736	0.0152368	+	0.999884i	$0.495150\pi$
$942$	0	0
$943$	−10.1945	−0.331978
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.0830101	−0.00269747	−0.00134873	−	0.999999i	$-0.500429\pi$
$947$	−0.0830101	−0.00269747	−0.00134873	+	0.999999i	$0.500429\pi$
$948$	0	0
$949$	−42.4422	−1.37773
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.0754	0.877059	0.438530	−	0.898717i	$-0.355499\pi$
$953$	27.0754	0.877059	0.438530	+	0.898717i	$0.355499\pi$
$954$	0	0
$955$	26.7805	0.866598
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21.1955	−0.684437
$960$	0	0
$961$	−18.4416	−0.594891
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.8196	0.412678
$966$	0	0
$967$	2.18529	0.0702743	0.0351371	−	0.999383i	$-0.488813\pi$
$967$	2.18529	0.0702743	0.0351371	+	0.999383i	$0.488813\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.0519	0.322580	0.161290	−	0.986907i	$-0.448435\pi$
$971$	10.0519	0.322580	0.161290	+	0.986907i	$0.448435\pi$
$972$	0	0
$973$	28.1581	0.902706
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.22903	0.103306	0.0516529	−	0.998665i	$-0.483551\pi$
$977$	3.22903	0.103306	0.0516529	+	0.998665i	$0.483551\pi$
$978$	0	0
$979$	38.8400	1.24133
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.6353	1.26417	0.632085	−	0.774899i	$-0.282199\pi$
$983$	39.6353	1.26417	0.632085	+	0.774899i	$0.282199\pi$
$984$	0	0
$985$	2.60577	0.0830268
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.42450	0.0452965
$990$	0	0
$991$	5.77615	0.183485	0.0917427	−	0.995783i	$-0.470756\pi$
$991$	5.77615	0.183485	0.0917427	+	0.995783i	$0.470756\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12.5686	−0.398451
$996$	0	0
$997$	57.5757	1.82344	0.911721	−	0.410810i	$-0.134754\pi$
$997$	57.5757	1.82344	0.911721	+	0.410810i	$0.134754\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bu.1.2	yes	6
3.2	odd	2		8280.2.a.bt.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bt.1.2	✓	6	3.2	odd	2
8280.2.a.bu.1.2	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.73629 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.73629 q^{7} -5.97718 q^{11} -5.40168 q^{13} -6.94546 q^{17} -3.97718 q^{19} +1.00000 q^{23} +1.00000 q^{25} +9.29741 q^{29} -3.54378 q^{31} -2.73629 q^{35} +7.17782 q^{37} -10.1945 q^{41} +1.42450 q^{43} -10.4187 q^{47} +0.487260 q^{49} +4.04839 q^{53} -5.97718 q^{55} +8.13797 q^{59} -3.34516 q^{61} -5.40168 q^{65} -1.70525 q^{67} -7.56247 q^{71} +7.85722 q^{73} +16.3553 q^{77} -1.36798 q^{79} +13.9709 q^{83} -6.94546 q^{85} -6.49805 q^{89} +14.7805 q^{91} -3.97718 q^{95} -13.8769 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 2 q^{7}+O(q^{10}) \) 6 * q + 6 * q^5 - 2 * q^7	\( 6 q + 6 q^{5} - 2 q^{7} - 4 q^{11} + 4 q^{17} + 8 q^{19} + 6 q^{23} + 6 q^{25} + 18 q^{29} - 8 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 2 q^{59} - 8 q^{61} - 2 q^{67} + 2 q^{71} + 8 q^{73} + 16 q^{77} - 28 q^{79} + 24 q^{83} + 4 q^{85} + 4 q^{89} - 8 q^{91} + 8 q^{95} + 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^5 - 2 * q^7 - 4 * q^11 + 4 * q^17 + 8 * q^19 + 6 * q^23 + 6 * q^25 + 18 * q^29 - 8 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^41 + 8 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 2 * q^59 - 8 * q^61 - 2 * q^67 + 2 * q^71 + 8 * q^73 + 16 * q^77 - 28 * q^79 + 24 * q^83 + 4 * q^85 + 4 * q^89 - 8 * q^91 + 8 * q^95 + 24 * q^97

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