LMFDB - Embedded newform 8280.2.a.y.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.00000 q^{7} -2.24264 q^{11} -5.41421 q^{13} +4.41421 q^{17} -7.07107 q^{19} +1.00000 q^{23} +1.00000 q^{25} -1.24264 q^{29} -10.6569 q^{31} +1.00000 q^{35} -3.00000 q^{37} +1.58579 q^{41} +2.00000 q^{43} -5.41421 q^{47} -6.00000 q^{49} +6.41421 q^{53} +2.24264 q^{55} +12.0711 q^{59} +1.07107 q^{61} +5.41421 q^{65} +10.6569 q^{67} -2.07107 q^{71} +15.5563 q^{73} +2.24264 q^{77} +5.65685 q^{79} -16.0711 q^{83} -4.41421 q^{85} +10.8284 q^{89} +5.41421 q^{91} +7.07107 q^{95} -1.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{23} + 2 q^{25} + 6 q^{29} - 10 q^{31} + 2 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 8 q^{47} - 12 q^{49} + 10 q^{53} - 4 q^{55} + 10 q^{59} - 12 q^{61} + 8 q^{65} + 10 q^{67} + 10 q^{71} - 4 q^{77} - 18 q^{83} - 6 q^{85} + 16 q^{89} + 8 q^{91} - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^23 + 2 * q^25 + 6 * q^29 - 10 * q^31 + 2 * q^35 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 8 * q^47 - 12 * q^49 + 10 * q^53 - 4 * q^55 + 10 * q^59 - 12 * q^61 + 8 * q^65 + 10 * q^67 + 10 * q^71 - 4 * q^77 - 18 * q^83 - 6 * q^85 + 16 * q^89 + 8 * q^91 - 8 * 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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.24264	−0.676182	−0.338091	−	0.941113i	$-0.609781\pi$
$11$	−2.24264	−0.676182	−0.338091	+	0.941113i	$0.609781\pi$
$12$	0	0
$13$	−5.41421	−1.50163	−0.750816	−	0.660511i	$-0.770340\pi$
$13$	−5.41421	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.41421	1.07060	0.535302	−	0.844661i	$-0.320198\pi$
$17$	4.41421	1.07060	0.535302	+	0.844661i	$0.320198\pi$
$18$	0	0
$19$	−7.07107	−1.62221	−0.811107	−	0.584898i	$-0.801135\pi$
$19$	−7.07107	−1.62221	−0.811107	+	0.584898i	$0.801135\pi$
$20$	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
$29$	−1.24264	−0.230753	−0.115376	−	0.993322i	$-0.536807\pi$
$29$	−1.24264	−0.230753	−0.115376	+	0.993322i	$0.536807\pi$
$30$	0	0
$31$	−10.6569	−1.91403	−0.957014	−	0.290043i	$-0.906331\pi$
$31$	−10.6569	−1.91403	−0.957014	+	0.290043i	$0.906331\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.58579	0.247658	0.123829	−	0.992304i	$-0.460483\pi$
$41$	1.58579	0.247658	0.123829	+	0.992304i	$0.460483\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.41421	−0.789744	−0.394872	−	0.918736i	$-0.629211\pi$
$47$	−5.41421	−0.789744	−0.394872	+	0.918736i	$0.629211\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.41421	0.881060	0.440530	−	0.897738i	$-0.354791\pi$
$53$	6.41421	0.881060	0.440530	+	0.897738i	$0.354791\pi$
$54$	0	0
$55$	2.24264	0.302398
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0711	1.57152	0.785760	−	0.618532i	$-0.212272\pi$
$59$	12.0711	1.57152	0.785760	+	0.618532i	$0.212272\pi$
$60$	0	0
$61$	1.07107	0.137136	0.0685681	−	0.997646i	$-0.478157\pi$
$61$	1.07107	0.137136	0.0685681	+	0.997646i	$0.478157\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.41421	0.671551
$66$	0	0
$67$	10.6569	1.30194	0.650971	−	0.759103i	$-0.274362\pi$
$67$	10.6569	1.30194	0.650971	+	0.759103i	$0.274362\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.07107	−0.245791	−0.122895	−	0.992420i	$-0.539218\pi$
$71$	−2.07107	−0.245791	−0.122895	+	0.992420i	$0.539218\pi$
$72$	0	0
$73$	15.5563	1.82073	0.910366	−	0.413803i	$-0.135800\pi$
$73$	15.5563	1.82073	0.910366	+	0.413803i	$0.135800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.24264	0.255573
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.0711	−1.76403	−0.882014	−	0.471222i	$-0.843813\pi$
$83$	−16.0711	−1.76403	−0.882014	+	0.471222i	$0.843813\pi$
$84$	0	0
$85$	−4.41421	−0.478789
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.8284	1.14781	0.573905	−	0.818922i	$-0.305428\pi$
$89$	10.8284	1.14781	0.573905	+	0.818922i	$0.305428\pi$
$90$	0	0
$91$	5.41421	0.567564
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.07107	0.725476
$96$	0	0
$97$	−1.17157	−0.118955	−0.0594776	−	0.998230i	$-0.518943\pi$
$97$	−1.17157	−0.118955	−0.0594776	+	0.998230i	$0.518943\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.75736	−0.274368	−0.137184	−	0.990546i	$-0.543805\pi$
$101$	−2.75736	−0.274368	−0.137184	+	0.990546i	$0.543805\pi$
$102$	0	0
$103$	−10.4853	−1.03315	−0.516573	−	0.856243i	$-0.672792\pi$
$103$	−10.4853	−1.03315	−0.516573	+	0.856243i	$0.672792\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.24264	−0.313478	−0.156739	−	0.987640i	$-0.550098\pi$
$107$	−3.24264	−0.313478	−0.156739	+	0.987640i	$0.550098\pi$
$108$	0	0
$109$	−18.3848	−1.76094	−0.880471	−	0.474100i	$-0.842774\pi$
$109$	−18.3848	−1.76094	−0.880471	+	0.474100i	$0.842774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.757359	0.0712464	0.0356232	−	0.999365i	$-0.488658\pi$
$113$	0.757359	0.0712464	0.0356232	+	0.999365i	$0.488658\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.41421	−0.404650
$120$	0	0
$121$	−5.97056	−0.542778
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.4142	1.01285	0.506424	−	0.862285i	$-0.330967\pi$
$127$	11.4142	1.01285	0.506424	+	0.862285i	$0.330967\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	7.07107	0.613139
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.485281	0.0414604	0.0207302	−	0.999785i	$-0.493401\pi$
$137$	0.485281	0.0414604	0.0207302	+	0.999785i	$0.493401\pi$
$138$	0	0
$139$	2.51472	0.213296	0.106648	−	0.994297i	$-0.465988\pi$
$139$	2.51472	0.213296	0.106648	+	0.994297i	$0.465988\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.1421	1.01538
$144$	0	0
$145$	1.24264	0.103196
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.5858	0.867221	0.433611	−	0.901100i	$-0.357239\pi$
$149$	10.5858	0.867221	0.433611	+	0.901100i	$0.357239\pi$
$150$	0	0
$151$	−0.343146	−0.0279248	−0.0139624	−	0.999903i	$-0.504445\pi$
$151$	−0.343146	−0.0279248	−0.0139624	+	0.999903i	$0.504445\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.6569	0.855979
$156$	0	0
$157$	1.48528	0.118538	0.0592692	−	0.998242i	$-0.481123\pi$
$157$	1.48528	0.118538	0.0592692	+	0.998242i	$0.481123\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	7.65685	0.599731	0.299866	−	0.953981i	$-0.403058\pi$
$163$	7.65685	0.599731	0.299866	+	0.953981i	$0.403058\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.0711	0.856705	0.428352	−	0.903612i	$-0.359094\pi$
$167$	11.0711	0.856705	0.428352	+	0.903612i	$0.359094\pi$
$168$	0	0
$169$	16.3137	1.25490
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3137	0.860165	0.430083	−	0.902790i	$-0.358484\pi$
$173$	11.3137	0.860165	0.430083	+	0.902790i	$0.358484\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.3431	0.922570	0.461285	−	0.887252i	$-0.347389\pi$
$179$	12.3431	0.922570	0.461285	+	0.887252i	$0.347389\pi$
$180$	0	0
$181$	−6.14214	−0.456541	−0.228271	−	0.973598i	$-0.573307\pi$
$181$	−6.14214	−0.456541	−0.228271	+	0.973598i	$0.573307\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	−9.89949	−0.723923
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.8995	1.00573	0.502866	−	0.864364i	$-0.332279\pi$
$191$	13.8995	1.00573	0.502866	+	0.864364i	$0.332279\pi$
$192$	0	0
$193$	7.51472	0.540921	0.270461	−	0.962731i	$-0.412824\pi$
$193$	7.51472	0.540921	0.270461	+	0.962731i	$0.412824\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.7990	−0.983137	−0.491569	−	0.870839i	$-0.663576\pi$
$197$	−13.7990	−0.983137	−0.491569	+	0.870839i	$0.663576\pi$
$198$	0	0
$199$	23.4558	1.66274	0.831370	−	0.555719i	$-0.187557\pi$
$199$	23.4558	1.66274	0.831370	+	0.555719i	$0.187557\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.24264	0.0872163
$204$	0	0
$205$	−1.58579	−0.110756
$206$	0	0
$207$	0	0
$208$	0	0
$209$	15.8579	1.09691
$210$	0	0
$211$	10.1716	0.700240	0.350120	−	0.936705i	$-0.386141\pi$
$211$	10.1716	0.700240	0.350120	+	0.936705i	$0.386141\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	10.6569	0.723434
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−23.8995	−1.60765
$222$	0	0
$223$	−26.9706	−1.80608	−0.903041	−	0.429554i	$-0.858671\pi$
$223$	−26.9706	−1.80608	−0.903041	+	0.429554i	$0.858671\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	6.82843	0.451235	0.225618	−	0.974216i	$-0.427560\pi$
$229$	6.82843	0.451235	0.225618	+	0.974216i	$0.427560\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.3431	1.20170	0.600850	−	0.799362i	$-0.294829\pi$
$233$	18.3431	1.20170	0.600850	+	0.799362i	$0.294829\pi$
$234$	0	0
$235$	5.41421	0.353184
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.07107	0.392705	0.196352	−	0.980533i	$-0.437090\pi$
$239$	6.07107	0.392705	0.196352	+	0.980533i	$0.437090\pi$
$240$	0	0
$241$	3.75736	0.242033	0.121016	−	0.992651i	$-0.461385\pi$
$241$	3.75736	0.242033	0.121016	+	0.992651i	$0.461385\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	38.2843	2.43597
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.34315	−0.274137	−0.137068	−	0.990562i	$-0.543768\pi$
$251$	−4.34315	−0.274137	−0.137068	+	0.990562i	$0.543768\pi$
$252$	0	0
$253$	−2.24264	−0.140994
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.7279	0.793946	0.396973	−	0.917830i	$-0.370061\pi$
$257$	12.7279	0.793946	0.396973	+	0.917830i	$0.370061\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−29.7279	−1.83310	−0.916551	−	0.399918i	$-0.869039\pi$
$263$	−29.7279	−1.83310	−0.916551	+	0.399918i	$0.869039\pi$
$264$	0	0
$265$	−6.41421	−0.394022
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.4142	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$269$	18.4142	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$270$	0	0
$271$	23.9706	1.45611	0.728054	−	0.685520i	$-0.240425\pi$
$271$	23.9706	1.45611	0.728054	+	0.685520i	$0.240425\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.24264	−0.135236
$276$	0	0
$277$	−11.6569	−0.700392	−0.350196	−	0.936676i	$-0.613885\pi$
$277$	−11.6569	−0.700392	−0.350196	+	0.936676i	$0.613885\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.07107	−0.541135	−0.270567	−	0.962701i	$-0.587211\pi$
$281$	−9.07107	−0.541135	−0.270567	+	0.962701i	$0.587211\pi$
$282$	0	0
$283$	−24.4558	−1.45375	−0.726875	−	0.686770i	$-0.759028\pi$
$283$	−24.4558	−1.45375	−0.726875	+	0.686770i	$0.759028\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.58579	−0.0936060
$288$	0	0
$289$	2.48528	0.146193
$290$	0	0
$291$	0	0
$292$	0	0
$293$	33.0416	1.93031	0.965156	−	0.261674i	$-0.0842745\pi$
$293$	33.0416	1.93031	0.965156	+	0.261674i	$0.0842745\pi$
$294$	0	0
$295$	−12.0711	−0.702805
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.41421	−0.313112
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.07107	−0.0613292
$306$	0	0
$307$	−22.3848	−1.27757	−0.638783	−	0.769387i	$-0.720562\pi$
$307$	−22.3848	−1.27757	−0.638783	+	0.769387i	$0.720562\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.9706	1.18913	0.594566	−	0.804047i	$-0.297324\pi$
$311$	20.9706	1.18913	0.594566	+	0.804047i	$0.297324\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.55635	−0.199744	−0.0998722	−	0.995000i	$-0.531843\pi$
$317$	−3.55635	−0.199744	−0.0998722	+	0.995000i	$0.531843\pi$
$318$	0	0
$319$	2.78680	0.156031
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−31.2132	−1.73675
$324$	0	0
$325$	−5.41421	−0.300327
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.41421	0.298495
$330$	0	0
$331$	30.7990	1.69287	0.846433	−	0.532496i	$-0.178746\pi$
$331$	30.7990	1.69287	0.846433	+	0.532496i	$0.178746\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.6569	−0.582246
$336$	0	0
$337$	7.65685	0.417095	0.208548	−	0.978012i	$-0.433126\pi$
$337$	7.65685	0.417095	0.208548	+	0.978012i	$0.433126\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.8995	1.29423
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.1716	−0.707087	−0.353544	−	0.935418i	$-0.615023\pi$
$347$	−13.1716	−0.707087	−0.353544	+	0.935418i	$0.615023\pi$
$348$	0	0
$349$	−11.8284	−0.633161	−0.316581	−	0.948566i	$-0.602535\pi$
$349$	−11.8284	−0.633161	−0.316581	+	0.948566i	$0.602535\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.7574	0.838680	0.419340	−	0.907829i	$-0.362262\pi$
$353$	15.7574	0.838680	0.419340	+	0.907829i	$0.362262\pi$
$354$	0	0
$355$	2.07107	0.109921
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.7574	0.620530	0.310265	−	0.950650i	$-0.399582\pi$
$359$	11.7574	0.620530	0.310265	+	0.950650i	$0.399582\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.5563	−0.814257
$366$	0	0
$367$	−19.1421	−0.999211	−0.499606	−	0.866253i	$-0.666522\pi$
$367$	−19.1421	−0.999211	−0.499606	+	0.866253i	$0.666522\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.41421	−0.333009
$372$	0	0
$373$	−24.9706	−1.29293	−0.646463	−	0.762945i	$-0.723753\pi$
$373$	−24.9706	−1.29293	−0.646463	+	0.762945i	$0.723753\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.72792	0.346506
$378$	0	0
$379$	−17.6569	−0.906972	−0.453486	−	0.891263i	$-0.649820\pi$
$379$	−17.6569	−0.906972	−0.453486	+	0.891263i	$0.649820\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.0711	−1.33217	−0.666085	−	0.745876i	$-0.732031\pi$
$383$	−26.0711	−1.33217	−0.666085	+	0.745876i	$0.732031\pi$
$384$	0	0
$385$	−2.24264	−0.114296
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.1421	1.83248	0.916240	−	0.400631i	$-0.131209\pi$
$389$	36.1421	1.83248	0.916240	+	0.400631i	$0.131209\pi$
$390$	0	0
$391$	4.41421	0.223236
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.65685	−0.284627
$396$	0	0
$397$	−16.0000	−0.803017	−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000	−0.803017	−0.401508	+	0.915855i	$0.631514\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.4853	0.723360	0.361680	−	0.932302i	$-0.382203\pi$
$401$	14.4853	0.723360	0.361680	+	0.932302i	$0.382203\pi$
$402$	0	0
$403$	57.6985	2.87417
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.72792	0.333491
$408$	0	0
$409$	−28.4558	−1.40705	−0.703525	−	0.710670i	$-0.748392\pi$
$409$	−28.4558	−1.40705	−0.703525	+	0.710670i	$0.748392\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−12.0711	−0.593978
$414$	0	0
$415$	16.0711	0.788898
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.3848	−1.77751	−0.888756	−	0.458380i	$-0.848430\pi$
$419$	−36.3848	−1.77751	−0.888756	+	0.458380i	$0.848430\pi$
$420$	0	0
$421$	14.8701	0.724722	0.362361	−	0.932038i	$-0.381971\pi$
$421$	14.8701	0.724722	0.362361	+	0.932038i	$0.381971\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.41421	0.214121
$426$	0	0
$427$	−1.07107	−0.0518326
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.17157	0.249106	0.124553	−	0.992213i	$-0.460250\pi$
$431$	5.17157	0.249106	0.124553	+	0.992213i	$0.460250\pi$
$432$	0	0
$433$	24.1716	1.16161	0.580806	−	0.814042i	$-0.302738\pi$
$433$	24.1716	1.16161	0.580806	+	0.814042i	$0.302738\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.07107	−0.338255
$438$	0	0
$439$	−2.20101	−0.105048	−0.0525242	−	0.998620i	$-0.516727\pi$
$439$	−2.20101	−0.105048	−0.0525242	+	0.998620i	$0.516727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.0711	0.716048	0.358024	−	0.933712i	$-0.383451\pi$
$443$	15.0711	0.716048	0.358024	+	0.933712i	$0.383451\pi$
$444$	0	0
$445$	−10.8284	−0.513317
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.7279	1.49733	0.748667	−	0.662947i	$-0.230694\pi$
$449$	31.7279	1.49733	0.748667	+	0.662947i	$0.230694\pi$
$450$	0	0
$451$	−3.55635	−0.167462
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.41421	−0.253822
$456$	0	0
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.4853	−0.860945	−0.430473	−	0.902604i	$-0.641653\pi$
$461$	−18.4853	−0.860945	−0.430473	+	0.902604i	$0.641653\pi$
$462$	0	0
$463$	28.0416	1.30321	0.651603	−	0.758561i	$-0.274097\pi$
$463$	28.0416	1.30321	0.651603	+	0.758561i	$0.274097\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.4142	0.481912	0.240956	−	0.970536i	$-0.422539\pi$
$467$	10.4142	0.481912	0.240956	+	0.970536i	$0.422539\pi$
$468$	0	0
$469$	−10.6569	−0.492088
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.48528	−0.206233
$474$	0	0
$475$	−7.07107	−0.324443
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.89949	0.452319	0.226160	−	0.974090i	$-0.427383\pi$
$479$	9.89949	0.452319	0.226160	+	0.974090i	$0.427383\pi$
$480$	0	0
$481$	16.2426	0.740601
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.17157	0.0531984
$486$	0	0
$487$	5.07107	0.229792	0.114896	−	0.993378i	$-0.463347\pi$
$487$	5.07107	0.229792	0.114896	+	0.993378i	$0.463347\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.8995	−1.12370	−0.561849	−	0.827240i	$-0.689910\pi$
$491$	−24.8995	−1.12370	−0.561849	+	0.827240i	$0.689910\pi$
$492$	0	0
$493$	−5.48528	−0.247045
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.07107	0.0929001
$498$	0	0
$499$	−9.82843	−0.439981	−0.219990	−	0.975502i	$-0.570603\pi$
$499$	−9.82843	−0.439981	−0.219990	+	0.975502i	$0.570603\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.6985	−1.10125	−0.550626	−	0.834752i	$-0.685611\pi$
$503$	−24.6985	−1.10125	−0.550626	+	0.834752i	$0.685611\pi$
$504$	0	0
$505$	2.75736	0.122701
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.7990	−0.700278	−0.350139	−	0.936698i	$-0.613866\pi$
$509$	−15.7990	−0.700278	−0.350139	+	0.936698i	$0.613866\pi$
$510$	0	0
$511$	−15.5563	−0.688172
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.4853	0.462037
$516$	0	0
$517$	12.1421	0.534011
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.07107	0.222168	0.111084	−	0.993811i	$-0.464568\pi$
$521$	5.07107	0.222168	0.111084	+	0.993811i	$0.464568\pi$
$522$	0	0
$523$	30.3431	1.32681	0.663407	−	0.748259i	$-0.269110\pi$
$523$	30.3431	1.32681	0.663407	+	0.748259i	$0.269110\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−47.0416	−2.04917
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.58579	−0.371892
$534$	0	0
$535$	3.24264	0.140192
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.4558	0.579584
$540$	0	0
$541$	44.4853	1.91257	0.956286	−	0.292434i	$-0.0944651\pi$
$541$	44.4853	1.91257	0.956286	+	0.292434i	$0.0944651\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.3848	0.787517
$546$	0	0
$547$	−8.68629	−0.371399	−0.185700	−	0.982607i	$-0.559455\pi$
$547$	−8.68629	−0.371399	−0.185700	+	0.982607i	$0.559455\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.78680	0.374330
$552$	0	0
$553$	−5.65685	−0.240554
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.5858	−1.08410	−0.542052	−	0.840345i	$-0.682353\pi$
$557$	−25.5858	−1.08410	−0.542052	+	0.840345i	$0.682353\pi$
$558$	0	0
$559$	−10.8284	−0.457994
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.24264	0.136661	0.0683305	−	0.997663i	$-0.478233\pi$
$563$	3.24264	0.136661	0.0683305	+	0.997663i	$0.478233\pi$
$564$	0	0
$565$	−0.757359	−0.0318623
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.14214	−0.173647	−0.0868237	−	0.996224i	$-0.527672\pi$
$569$	−4.14214	−0.173647	−0.0868237	+	0.996224i	$0.527672\pi$
$570$	0	0
$571$	30.7279	1.28592	0.642962	−	0.765898i	$-0.277705\pi$
$571$	30.7279	1.28592	0.642962	+	0.765898i	$0.277705\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−21.9411	−0.913421	−0.456711	−	0.889615i	$-0.650972\pi$
$577$	−21.9411	−0.913421	−0.456711	+	0.889615i	$0.650972\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.0711	0.666740
$582$	0	0
$583$	−14.3848	−0.595757
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	75.3553	3.10496
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.8701	1.10342	0.551711	−	0.834036i	$-0.313975\pi$
$593$	26.8701	1.10342	0.551711	+	0.834036i	$0.313975\pi$
$594$	0	0
$595$	4.41421	0.180965
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.48528	0.346699	0.173350	−	0.984860i	$-0.444541\pi$
$599$	8.48528	0.346699	0.173350	+	0.984860i	$0.444541\pi$
$600$	0	0
$601$	0.313708	0.0127964	0.00639822	−	0.999980i	$-0.497963\pi$
$601$	0.313708	0.0127964	0.00639822	+	0.999980i	$0.497963\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.97056	0.242738
$606$	0	0
$607$	−14.0416	−0.569932	−0.284966	−	0.958538i	$-0.591982\pi$
$607$	−14.0416	−0.569932	−0.284966	+	0.958538i	$0.591982\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.3137	1.18591
$612$	0	0
$613$	48.6274	1.96404	0.982021	−	0.188769i	$-0.0604499\pi$
$613$	48.6274	1.96404	0.982021	+	0.188769i	$0.0604499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.8995	−1.16345	−0.581725	−	0.813386i	$-0.697622\pi$
$617$	−28.8995	−1.16345	−0.581725	+	0.813386i	$0.697622\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.8284	−0.433832
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.2426	−0.528019
$630$	0	0
$631$	−31.8995	−1.26990	−0.634949	−	0.772554i	$-0.718979\pi$
$631$	−31.8995	−1.26990	−0.634949	+	0.772554i	$0.718979\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.4142	−0.452959
$636$	0	0
$637$	32.4853	1.28711
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.0711	0.832257	0.416129	−	0.909306i	$-0.363387\pi$
$641$	21.0711	0.832257	0.416129	+	0.909306i	$0.363387\pi$
$642$	0	0
$643$	31.8284	1.25519	0.627595	−	0.778540i	$-0.284039\pi$
$643$	31.8284	1.25519	0.627595	+	0.778540i	$0.284039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.7279	1.67981	0.839904	−	0.542735i	$-0.182611\pi$
$647$	42.7279	1.67981	0.839904	+	0.542735i	$0.182611\pi$
$648$	0	0
$649$	−27.0711	−1.06263
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−49.8406	−1.95041	−0.975207	−	0.221294i	$-0.928972\pi$
$653$	−49.8406	−1.95041	−0.975207	+	0.221294i	$0.928972\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	35.6985	1.39062	0.695308	−	0.718712i	$-0.255268\pi$
$659$	35.6985	1.39062	0.695308	+	0.718712i	$0.255268\pi$
$660$	0	0
$661$	18.4853	0.718994	0.359497	−	0.933146i	$-0.382948\pi$
$661$	18.4853	0.718994	0.359497	+	0.933146i	$0.382948\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.07107	−0.274204
$666$	0	0
$667$	−1.24264	−0.0481152
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.40202	−0.0927290
$672$	0	0
$673$	32.8701	1.26705	0.633524	−	0.773723i	$-0.281608\pi$
$673$	32.8701	1.26705	0.633524	+	0.773723i	$0.281608\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.89949	0.342035	0.171018	−	0.985268i	$-0.445294\pi$
$677$	8.89949	0.342035	0.171018	+	0.985268i	$0.445294\pi$
$678$	0	0
$679$	1.17157	0.0449608
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.7279	−1.71147	−0.855733	−	0.517417i	$-0.826893\pi$
$683$	−44.7279	−1.71147	−0.855733	+	0.517417i	$0.826893\pi$
$684$	0	0
$685$	−0.485281	−0.0185416
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−34.7279	−1.32303
$690$	0	0
$691$	−0.627417	−0.0238681	−0.0119340	−	0.999929i	$-0.503799\pi$
$691$	−0.627417	−0.0238681	−0.0119340	+	0.999929i	$0.503799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.51472	−0.0953887
$696$	0	0
$697$	7.00000	0.265144
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.6985	1.12170	0.560848	−	0.827919i	$-0.310475\pi$
$701$	29.6985	1.12170	0.560848	+	0.827919i	$0.310475\pi$
$702$	0	0
$703$	21.2132	0.800071
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.75736	0.103701
$708$	0	0
$709$	37.0711	1.39223	0.696117	−	0.717929i	$-0.254910\pi$
$709$	37.0711	1.39223	0.696117	+	0.717929i	$0.254910\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.6569	−0.399102
$714$	0	0
$715$	−12.1421	−0.454090
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.414214	−0.0154476	−0.00772378	−	0.999970i	$-0.502459\pi$
$719$	−0.414214	−0.0154476	−0.00772378	+	0.999970i	$0.502459\pi$
$720$	0	0
$721$	10.4853	0.390492
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.24264	−0.0461505
$726$	0	0
$727$	−9.14214	−0.339063	−0.169532	−	0.985525i	$-0.554225\pi$
$727$	−9.14214	−0.339063	−0.169532	+	0.985525i	$0.554225\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.82843	0.326531
$732$	0	0
$733$	−24.1716	−0.892797	−0.446399	−	0.894834i	$-0.647294\pi$
$733$	−24.1716	−0.892797	−0.446399	+	0.894834i	$0.647294\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.8995	−0.880349
$738$	0	0
$739$	−3.14214	−0.115585	−0.0577927	−	0.998329i	$-0.518406\pi$
$739$	−3.14214	−0.115585	−0.0577927	+	0.998329i	$0.518406\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.6274	0.976865	0.488433	−	0.872602i	$-0.337569\pi$
$743$	26.6274	0.976865	0.488433	+	0.872602i	$0.337569\pi$
$744$	0	0
$745$	−10.5858	−0.387833
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.24264	0.118484
$750$	0	0
$751$	−36.7279	−1.34022	−0.670110	−	0.742261i	$-0.733753\pi$
$751$	−36.7279	−1.34022	−0.670110	+	0.742261i	$0.733753\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.343146	0.0124884
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.1005	0.402393	0.201196	−	0.979551i	$-0.435517\pi$
$761$	11.1005	0.402393	0.201196	+	0.979551i	$0.435517\pi$
$762$	0	0
$763$	18.3848	0.665574
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−65.3553	−2.35984
$768$	0	0
$769$	−16.5858	−0.598099	−0.299049	−	0.954238i	$-0.596670\pi$
$769$	−16.5858	−0.598099	−0.299049	+	0.954238i	$0.596670\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.1127	1.33485	0.667425	−	0.744677i	$-0.267396\pi$
$773$	37.1127	1.33485	0.667425	+	0.744677i	$0.267396\pi$
$774$	0	0
$775$	−10.6569	−0.382806
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.2132	−0.401755
$780$	0	0
$781$	4.64466	0.166199
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.48528	−0.0530120
$786$	0	0
$787$	45.7696	1.63151	0.815754	−	0.578399i	$-0.196322\pi$
$787$	45.7696	1.63151	0.815754	+	0.578399i	$0.196322\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.757359	−0.0269286
$792$	0	0
$793$	−5.79899	−0.205928
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.5269	0.975053	0.487527	−	0.873108i	$-0.337899\pi$
$797$	27.5269	0.975053	0.487527	+	0.873108i	$0.337899\pi$
$798$	0	0
$799$	−23.8995	−0.845503
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.8873	−1.23115
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.58579	−0.196386	−0.0981929	−	0.995167i	$-0.531306\pi$
$809$	−5.58579	−0.196386	−0.0981929	+	0.995167i	$0.531306\pi$
$810$	0	0
$811$	−24.1716	−0.848779	−0.424389	−	0.905480i	$-0.639511\pi$
$811$	−24.1716	−0.848779	−0.424389	+	0.905480i	$0.639511\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.65685	−0.268208
$816$	0	0
$817$	−14.1421	−0.494771
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.1127	−1.36504	−0.682521	−	0.730866i	$-0.739117\pi$
$821$	−39.1127	−1.36504	−0.682521	+	0.730866i	$0.739117\pi$
$822$	0	0
$823$	−38.6274	−1.34647	−0.673234	−	0.739430i	$-0.735095\pi$
$823$	−38.6274	−1.34647	−0.673234	+	0.739430i	$0.735095\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.0416	1.07942	0.539712	−	0.841850i	$-0.318533\pi$
$827$	31.0416	1.07942	0.539712	+	0.841850i	$0.318533\pi$
$828$	0	0
$829$	−1.20101	−0.0417128	−0.0208564	−	0.999782i	$-0.506639\pi$
$829$	−1.20101	−0.0417128	−0.0208564	+	0.999782i	$0.506639\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.4853	−0.917661
$834$	0	0
$835$	−11.0711	−0.383130
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.3431	1.39280	0.696400	−	0.717654i	$-0.254784\pi$
$839$	40.3431	1.39280	0.696400	+	0.717654i	$0.254784\pi$
$840$	0	0
$841$	−27.4558	−0.946753
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−16.3137	−0.561209
$846$	0	0
$847$	5.97056	0.205151
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	−6.48528	−0.222052	−0.111026	−	0.993818i	$-0.535414\pi$
$853$	−6.48528	−0.222052	−0.111026	+	0.993818i	$0.535414\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.3431	0.694909	0.347454	−	0.937697i	$-0.387046\pi$
$857$	20.3431	0.694909	0.347454	+	0.937697i	$0.387046\pi$
$858$	0	0
$859$	34.4558	1.17562	0.587809	−	0.809000i	$-0.299991\pi$
$859$	34.4558	1.17562	0.587809	+	0.809000i	$0.299991\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.85786	0.267485	0.133742	−	0.991016i	$-0.457301\pi$
$863$	7.85786	0.267485	0.133742	+	0.991016i	$0.457301\pi$
$864$	0	0
$865$	−11.3137	−0.384678
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.6863	−0.430353
$870$	0	0
$871$	−57.6985	−1.95504
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	12.8284	0.433185	0.216593	−	0.976262i	$-0.430506\pi$
$877$	12.8284	0.433185	0.216593	+	0.976262i	$0.430506\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25.4142	−0.856227	−0.428113	−	0.903725i	$-0.640822\pi$
$881$	−25.4142	−0.856227	−0.428113	+	0.903725i	$0.640822\pi$
$882$	0	0
$883$	16.3848	0.551392	0.275696	−	0.961245i	$-0.411092\pi$
$883$	16.3848	0.551392	0.275696	+	0.961245i	$0.411092\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.48528	−0.0834476	−0.0417238	−	0.999129i	$-0.513285\pi$
$887$	−2.48528	−0.0834476	−0.0417238	+	0.999129i	$0.513285\pi$
$888$	0	0
$889$	−11.4142	−0.382820
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38.2843	1.28113
$894$	0	0
$895$	−12.3431	−0.412586
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.2426	0.441667
$900$	0	0
$901$	28.3137	0.943266
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.14214	0.204171
$906$	0	0
$907$	1.97056	0.0654315	0.0327157	−	0.999465i	$-0.489584\pi$
$907$	1.97056	0.0654315	0.0327157	+	0.999465i	$0.489584\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.97056	−0.164682	−0.0823410	−	0.996604i	$-0.526240\pi$
$911$	−4.97056	−0.164682	−0.0823410	+	0.996604i	$0.526240\pi$
$912$	0	0
$913$	36.0416	1.19280
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.65685	0.186806
$918$	0	0
$919$	29.9411	0.987667	0.493833	−	0.869557i	$-0.335595\pi$
$919$	29.9411	0.987667	0.493833	+	0.869557i	$0.335595\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.2132	0.369087
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.6152	0.676364	0.338182	−	0.941081i	$-0.390188\pi$
$929$	20.6152	0.676364	0.338182	+	0.941081i	$0.390188\pi$
$930$	0	0
$931$	42.4264	1.39047
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.89949	0.323748
$936$	0	0
$937$	45.1716	1.47569	0.737845	−	0.674970i	$-0.235843\pi$
$937$	45.1716	1.47569	0.737845	+	0.674970i	$0.235843\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−35.5563	−1.15910	−0.579552	−	0.814935i	$-0.696772\pi$
$941$	−35.5563	−1.15910	−0.579552	+	0.814935i	$0.696772\pi$
$942$	0	0
$943$	1.58579	0.0516403
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.45584	0.0473086	0.0236543	−	0.999720i	$-0.492470\pi$
$947$	1.45584	0.0473086	0.0236543	+	0.999720i	$0.492470\pi$
$948$	0	0
$949$	−84.2254	−2.73407
$950$	0	0
$951$	0	0
$952$	0	0
$953$	57.3137	1.85657	0.928287	−	0.371866i	$-0.121282\pi$
$953$	57.3137	1.85657	0.928287	+	0.371866i	$0.121282\pi$
$954$	0	0
$955$	−13.8995	−0.449777
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.485281	−0.0156706
$960$	0	0
$961$	82.5685	2.66350
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.51472	−0.241907
$966$	0	0
$967$	−34.5269	−1.11031	−0.555155	−	0.831747i	$-0.687341\pi$
$967$	−34.5269	−1.11031	−0.555155	+	0.831747i	$0.687341\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53.4558	−1.71548	−0.857740	−	0.514084i	$-0.828132\pi$
$971$	−53.4558	−1.71548	−0.857740	+	0.514084i	$0.828132\pi$
$972$	0	0
$973$	−2.51472	−0.0806182
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.6985	−1.30206	−0.651030	−	0.759052i	$-0.725663\pi$
$977$	−40.6985	−1.30206	−0.651030	+	0.759052i	$0.725663\pi$
$978$	0	0
$979$	−24.2843	−0.776129
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−0.757359	−0.0241560	−0.0120780	−	0.999927i	$-0.503845\pi$
$983$	−0.757359	−0.0241560	−0.0120780	+	0.999927i	$0.503845\pi$
$984$	0	0
$985$	13.7990	0.439672
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	−24.1716	−0.767835	−0.383918	−	0.923367i	$-0.625425\pi$
$991$	−24.1716	−0.767835	−0.383918	+	0.923367i	$0.625425\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−23.4558	−0.743600
$996$	0	0
$997$	−40.0833	−1.26945	−0.634725	−	0.772738i	$-0.718887\pi$
$997$	−40.0833	−1.26945	−0.634725	+	0.772738i	$0.718887\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.y.1.1		2
3.2	odd	2		2760.2.a.q.1.2	✓	2
12.11	even	2		5520.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.q.1.2	✓	2	3.2	odd	2
5520.2.a.bk.1.1		2	12.11	even	2
8280.2.a.y.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.00000 q^{7} -2.24264 q^{11} -5.41421 q^{13} +4.41421 q^{17} -7.07107 q^{19} +1.00000 q^{23} +1.00000 q^{25} -1.24264 q^{29} -10.6569 q^{31} +1.00000 q^{35} -3.00000 q^{37} +1.58579 q^{41} +2.00000 q^{43} -5.41421 q^{47} -6.00000 q^{49} +6.41421 q^{53} +2.24264 q^{55} +12.0711 q^{59} +1.07107 q^{61} +5.41421 q^{65} +10.6569 q^{67} -2.07107 q^{71} +15.5563 q^{73} +2.24264 q^{77} +5.65685 q^{79} -16.0711 q^{83} -4.41421 q^{85} +10.8284 q^{89} +5.41421 q^{91} +7.07107 q^{95} -1.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{23} + 2 q^{25} + 6 q^{29} - 10 q^{31} + 2 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 8 q^{47} - 12 q^{49} + 10 q^{53} - 4 q^{55} + 10 q^{59} - 12 q^{61} + 8 q^{65} + 10 q^{67} + 10 q^{71} - 4 q^{77} - 18 q^{83} - 6 q^{85} + 16 q^{89} + 8 q^{91} - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^23 + 2 * q^25 + 6 * q^29 - 10 * q^31 + 2 * q^35 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 8 * q^47 - 12 * q^49 + 10 * q^53 - 4 * q^55 + 10 * q^59 - 12 * q^61 + 8 * q^65 + 10 * q^67 + 10 * q^71 - 4 * q^77 - 18 * q^83 - 6 * q^85 + 16 * q^89 + 8 * q^91 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more