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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(473,888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("888.473"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.br (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 569.17
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.17

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.350286 + 1.69626i) q^{3} +(0.557275 - 2.07978i) q^{5} +(1.90625 + 3.30173i) q^{7} +(-2.75460 - 1.18835i) q^{9} +2.70521 q^{11} +(0.737048 + 0.197491i) q^{13} +(3.33264 + 1.67380i) q^{15} +(0.554152 - 0.148484i) q^{17} +(-2.06065 - 0.552149i) q^{19} +(-6.26832 + 2.07695i) q^{21} +(4.13412 + 4.13412i) q^{23} +(0.315207 + 0.181985i) q^{25} +(2.98065 - 4.25625i) q^{27} +(1.48612 - 1.48612i) q^{29} +(0.369408 + 0.369408i) q^{31} +(-0.947598 + 4.58874i) q^{33} +(7.92916 - 2.12461i) q^{35} +(-1.29198 + 5.94397i) q^{37} +(-0.593175 + 1.18105i) q^{39} +(1.11956 + 1.93913i) q^{41} +(-5.18129 + 5.18129i) q^{43} +(-4.00658 + 5.06671i) q^{45} +1.34429i q^{47} +(-3.76760 + 6.52568i) q^{49} +(0.0577567 + 0.991997i) q^{51} +(0.415147 + 0.239685i) q^{53} +(1.50755 - 5.62624i) q^{55} +(1.65840 - 3.30198i) q^{57} +(12.4806 - 3.34417i) q^{59} +(0.788533 - 2.94285i) q^{61} +(-1.32735 - 11.3602i) q^{63} +(0.821477 - 1.42284i) q^{65} +(-2.47617 + 1.42962i) q^{67} +(-8.46067 + 5.56442i) q^{69} +(-5.44141 + 3.14160i) q^{71} +14.3331i q^{73} +(-0.419107 + 0.470927i) q^{75} +(5.15682 + 8.93187i) q^{77} +(-10.2736 - 2.75281i) q^{79} +(6.17563 + 6.54687i) q^{81} +(4.79474 + 2.76824i) q^{83} -1.23526i q^{85} +(2.00028 + 3.04141i) q^{87} +(-3.12906 - 11.6778i) q^{89} +(0.752937 + 2.81000i) q^{91} +(-0.756010 + 0.497213i) q^{93} +(-2.29669 + 3.97799i) q^{95} +(0.215187 - 0.215187i) q^{97} +(-7.45177 - 3.21475i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.350286 + 1.69626i −0.202238 + 0.979336i
\(4\) 0 0
\(5\) 0.557275 − 2.07978i 0.249221 − 0.930105i −0.721994 − 0.691899i \(-0.756774\pi\)
0.971215 − 0.238205i \(-0.0765592\pi\)
\(6\) 0 0
\(7\) 1.90625 + 3.30173i 0.720496 + 1.24794i 0.960801 + 0.277238i \(0.0894191\pi\)
−0.240305 + 0.970697i \(0.577248\pi\)
\(8\) 0 0
\(9\) −2.75460 − 1.18835i −0.918200 − 0.396118i
\(10\) 0 0
\(11\) 2.70521 0.815652 0.407826 − 0.913060i \(-0.366287\pi\)
0.407826 + 0.913060i \(0.366287\pi\)
\(12\) 0 0
\(13\) 0.737048 + 0.197491i 0.204420 + 0.0547743i 0.359576 − 0.933116i \(-0.382921\pi\)
−0.155156 + 0.987890i \(0.549588\pi\)
\(14\) 0 0
\(15\) 3.33264 + 1.67380i 0.860484 + 0.432173i
\(16\) 0 0
\(17\) 0.554152 − 0.148484i 0.134401 − 0.0360128i −0.190991 − 0.981592i \(-0.561170\pi\)
0.325393 + 0.945579i \(0.394504\pi\)
\(18\) 0 0
\(19\) −2.06065 − 0.552149i −0.472745 − 0.126672i 0.0145776 − 0.999894i \(-0.495360\pi\)
−0.487322 + 0.873222i \(0.662026\pi\)
\(20\) 0 0
\(21\) −6.26832 + 2.07695i −1.36786 + 0.453228i
\(22\) 0 0
\(23\) 4.13412 + 4.13412i 0.862023 + 0.862023i 0.991573 − 0.129550i \(-0.0413532\pi\)
−0.129550 + 0.991573i \(0.541353\pi\)
\(24\) 0 0
\(25\) 0.315207 + 0.181985i 0.0630415 + 0.0363970i
\(26\) 0 0
\(27\) 2.98065 − 4.25625i 0.573627 − 0.819116i
\(28\) 0 0
\(29\) 1.48612 − 1.48612i 0.275965 − 0.275965i −0.555531 − 0.831496i \(-0.687485\pi\)
0.831496 + 0.555531i \(0.187485\pi\)
\(30\) 0 0
\(31\) 0.369408 + 0.369408i 0.0663476 + 0.0663476i 0.739502 − 0.673154i \(-0.235061\pi\)
−0.673154 + 0.739502i \(0.735061\pi\)
\(32\) 0 0
\(33\) −0.947598 + 4.58874i −0.164956 + 0.798798i
\(34\) 0 0
\(35\) 7.92916 − 2.12461i 1.34027 − 0.359125i
\(36\) 0 0
\(37\) −1.29198 + 5.94397i −0.212400 + 0.977183i
\(38\) 0 0
\(39\) −0.593175 + 1.18105i −0.0949840 + 0.189119i
\(40\) 0 0
\(41\) 1.11956 + 1.93913i 0.174845 + 0.302841i 0.940108 − 0.340877i \(-0.110724\pi\)
−0.765262 + 0.643719i \(0.777391\pi\)
\(42\) 0 0
\(43\) −5.18129 + 5.18129i −0.790140 + 0.790140i −0.981517 − 0.191377i \(-0.938705\pi\)
0.191377 + 0.981517i \(0.438705\pi\)
\(44\) 0 0
\(45\) −4.00658 + 5.06671i −0.597265 + 0.755301i
\(46\) 0 0
\(47\) 1.34429i 0.196084i 0.995182 + 0.0980421i \(0.0312580\pi\)
−0.995182 + 0.0980421i \(0.968742\pi\)
\(48\) 0 0
\(49\) −3.76760 + 6.52568i −0.538229 + 0.932239i
\(50\) 0 0
\(51\) 0.0577567 + 0.991997i 0.00808756 + 0.138907i
\(52\) 0 0
\(53\) 0.415147 + 0.239685i 0.0570248 + 0.0329233i 0.528241 − 0.849094i \(-0.322852\pi\)
−0.471217 + 0.882018i \(0.656185\pi\)
\(54\) 0 0
\(55\) 1.50755 − 5.62624i 0.203277 − 0.758642i
\(56\) 0 0
\(57\) 1.65840 − 3.30198i 0.219661 − 0.437358i
\(58\) 0 0
\(59\) 12.4806 − 3.34417i 1.62484 − 0.435374i 0.672419 − 0.740171i \(-0.265255\pi\)
0.952417 + 0.304797i \(0.0985887\pi\)
\(60\) 0 0
\(61\) 0.788533 − 2.94285i 0.100961 − 0.376793i −0.896894 − 0.442245i \(-0.854182\pi\)
0.997856 + 0.0654520i \(0.0208489\pi\)
\(62\) 0 0
\(63\) −1.32735 − 11.3602i −0.167230 − 1.43126i
\(64\) 0 0
\(65\) 0.821477 − 1.42284i 0.101892 − 0.176481i
\(66\) 0 0
\(67\) −2.47617 + 1.42962i −0.302512 + 0.174655i −0.643571 − 0.765387i \(-0.722548\pi\)
0.341059 + 0.940042i \(0.389214\pi\)
\(68\) 0 0
\(69\) −8.46067 + 5.56442i −1.01854 + 0.669877i
\(70\) 0 0
\(71\) −5.44141 + 3.14160i −0.645777 + 0.372839i −0.786836 − 0.617162i \(-0.788282\pi\)
0.141060 + 0.990001i \(0.454949\pi\)
\(72\) 0 0
\(73\) 14.3331i 1.67756i 0.544469 + 0.838781i \(0.316731\pi\)
−0.544469 + 0.838781i \(0.683269\pi\)
\(74\) 0 0
\(75\) −0.419107 + 0.470927i −0.0483943 + 0.0543780i
\(76\) 0 0
\(77\) 5.15682 + 8.93187i 0.587674 + 1.01788i
\(78\) 0 0
\(79\) −10.2736 − 2.75281i −1.15587 − 0.309716i −0.370557 − 0.928810i \(-0.620833\pi\)
−0.785317 + 0.619094i \(0.787500\pi\)
\(80\) 0 0
\(81\) 6.17563 + 6.54687i 0.686181 + 0.727430i
\(82\) 0 0
\(83\) 4.79474 + 2.76824i 0.526291 + 0.303854i 0.739505 − 0.673151i \(-0.235060\pi\)
−0.213214 + 0.977006i \(0.568393\pi\)
\(84\) 0 0
\(85\) − 1.23526i − 0.133983i
\(86\) 0 0
\(87\) 2.00028 + 3.04141i 0.214452 + 0.326074i
\(88\) 0 0
\(89\) −3.12906 − 11.6778i −0.331680 − 1.23785i −0.907424 − 0.420217i \(-0.861954\pi\)
0.575744 − 0.817630i \(-0.304713\pi\)
\(90\) 0 0
\(91\) 0.752937 + 2.81000i 0.0789293 + 0.294568i
\(92\) 0 0
\(93\) −0.756010 + 0.497213i −0.0783946 + 0.0515586i
\(94\) 0 0
\(95\) −2.29669 + 3.97799i −0.235636 + 0.408133i
\(96\) 0 0
\(97\) 0.215187 − 0.215187i 0.0218489 − 0.0218489i −0.696098 − 0.717947i \(-0.745082\pi\)
0.717947 + 0.696098i \(0.245082\pi\)
\(98\) 0 0
\(99\) −7.45177 − 3.21475i −0.748931 − 0.323094i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 888.2.br.a.569.17 yes 152
3.2 odd 2 inner 888.2.br.a.569.4 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.4 yes 152
111.8 even 12 inner 888.2.br.a.785.17 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.4 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.17 yes 152 1.1 even 1 trivial
888.2.br.a.785.4 yes 152 37.8 odd 12 inner
888.2.br.a.785.17 yes 152 111.8 even 12 inner