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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 785.4
Character \(\chi\) \(=\) 888.785
Dual form 888.2.br.a.569.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.64415 + 0.544774i) q^{3} +(-0.557275 - 2.07978i) q^{5} +(1.90625 - 3.30173i) q^{7} +(2.40644 - 1.79138i) q^{9} -2.70521 q^{11} +(0.737048 - 0.197491i) q^{13} +(2.04925 + 3.11587i) q^{15} +(-0.554152 - 0.148484i) q^{17} +(-2.06065 + 0.552149i) q^{19} +(-1.33547 + 6.46700i) q^{21} +(-4.13412 + 4.13412i) q^{23} +(0.315207 - 0.181985i) q^{25} +(-2.98065 + 4.25625i) q^{27} +(-1.48612 - 1.48612i) q^{29} +(0.369408 - 0.369408i) q^{31} +(4.44777 - 1.47373i) q^{33} +(-7.92916 - 2.12461i) q^{35} +(-1.29198 - 5.94397i) q^{37} +(-1.10423 + 0.726229i) q^{39} +(-1.11956 + 1.93913i) q^{41} +(-5.18129 - 5.18129i) q^{43} +(-5.06671 - 4.00658i) q^{45} +1.34429i q^{47} +(-3.76760 - 6.52568i) q^{49} +(0.991997 - 0.0577567i) q^{51} +(-0.415147 + 0.239685i) q^{53} +(1.50755 + 5.62624i) q^{55} +(3.08721 - 2.03040i) q^{57} +(-12.4806 - 3.34417i) q^{59} +(0.788533 + 2.94285i) q^{61} +(-1.32735 - 11.3602i) q^{63} +(-0.821477 - 1.42284i) q^{65} +(-2.47617 - 1.42962i) q^{67} +(4.54494 - 9.04926i) q^{69} +(5.44141 + 3.14160i) q^{71} -14.3331i q^{73} +(-0.419107 + 0.470927i) q^{75} +(-5.15682 + 8.93187i) q^{77} +(-10.2736 + 2.75281i) q^{79} +(2.58194 - 8.62169i) q^{81} +(-4.79474 + 2.76824i) q^{83} +1.23526i q^{85} +(3.25300 + 1.63380i) q^{87} +(3.12906 - 11.6778i) q^{89} +(0.752937 - 2.81000i) q^{91} +(-0.406117 + 0.808604i) q^{93} +(2.29669 + 3.97799i) q^{95} +(0.215187 + 0.215187i) q^{97} +(-6.50994 + 4.84605i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.64415 + 0.544774i −0.949249 + 0.314525i
\(4\) 0 0
\(5\) −0.557275 − 2.07978i −0.249221 − 0.930105i −0.971215 − 0.238205i \(-0.923441\pi\)
0.721994 − 0.691899i \(-0.243226\pi\)
\(6\) 0 0
\(7\) 1.90625 − 3.30173i 0.720496 − 1.24794i −0.240305 − 0.970697i \(-0.577248\pi\)
0.960801 − 0.277238i \(-0.0894191\pi\)
\(8\) 0 0
\(9\) 2.40644 − 1.79138i 0.802148 − 0.597125i
\(10\) 0 0
\(11\) −2.70521 −0.815652 −0.407826 − 0.913060i \(-0.633713\pi\)
−0.407826 + 0.913060i \(0.633713\pi\)
\(12\) 0 0
\(13\) 0.737048 − 0.197491i 0.204420 − 0.0547743i −0.155156 − 0.987890i \(-0.549588\pi\)
0.359576 + 0.933116i \(0.382921\pi\)
\(14\) 0 0
\(15\) 2.04925 + 3.11587i 0.529114 + 0.804515i
\(16\) 0 0
\(17\) −0.554152 − 0.148484i −0.134401 − 0.0360128i 0.190991 − 0.981592i \(-0.438830\pi\)
−0.325393 + 0.945579i \(0.605496\pi\)
\(18\) 0 0
\(19\) −2.06065 + 0.552149i −0.472745 + 0.126672i −0.487322 − 0.873222i \(-0.662026\pi\)
0.0145776 + 0.999894i \(0.495360\pi\)
\(20\) 0 0
\(21\) −1.33547 + 6.46700i −0.291423 + 1.41122i
\(22\) 0 0
\(23\) −4.13412 + 4.13412i −0.862023 + 0.862023i −0.991573 − 0.129550i \(-0.958647\pi\)
0.129550 + 0.991573i \(0.458647\pi\)
\(24\) 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 \(-0.312515\pi\)
−0.831496 + 0.555531i \(0.812515\pi\)
\(30\) 0 0
\(31\) 0.369408 − 0.369408i 0.0663476 − 0.0663476i −0.673154 − 0.739502i \(-0.735061\pi\)
0.739502 + 0.673154i \(0.235061\pi\)
\(32\) 0 0
\(33\) 4.44777 − 1.47373i 0.774257 − 0.256543i
\(34\) 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
\(39\) −1.10423 + 0.726229i −0.176818 + 0.116290i
\(40\) 0 0
\(41\) −1.11956 + 1.93913i −0.174845 + 0.302841i −0.940108 − 0.340877i \(-0.889276\pi\)
0.765262 + 0.643719i \(0.222609\pi\)
\(42\) 0 0
\(43\) −5.18129 − 5.18129i −0.790140 − 0.790140i 0.191377 − 0.981517i \(-0.438705\pi\)
−0.981517 + 0.191377i \(0.938705\pi\)
\(44\) 0 0
\(45\) −5.06671 − 4.00658i −0.755301 − 0.597265i
\(46\) 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
\(51\) 0.991997 − 0.0577567i 0.138907 − 0.00808756i
\(52\) 0 0
\(53\) −0.415147 + 0.239685i −0.0570248 + 0.0329233i −0.528241 − 0.849094i \(-0.677148\pi\)
0.471217 + 0.882018i \(0.343815\pi\)
\(54\) 0 0
\(55\) 1.50755 + 5.62624i 0.203277 + 0.758642i
\(56\) 0 0
\(57\) 3.08721 − 2.03040i 0.408911 − 0.268933i
\(58\) 0 0
\(59\) −12.4806 − 3.34417i −1.62484 − 0.435374i −0.672419 − 0.740171i \(-0.734745\pi\)
−0.952417 + 0.304797i \(0.901411\pi\)
\(60\) 0 0
\(61\) 0.788533 + 2.94285i 0.100961 + 0.376793i 0.997856 − 0.0654520i \(-0.0208489\pi\)
−0.896894 + 0.442245i \(0.854182\pi\)
\(62\) 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 0.341059 − 0.940042i \(-0.389214\pi\)
−0.643571 + 0.765387i \(0.722548\pi\)
\(68\) 0 0
\(69\) 4.54494 − 9.04926i 0.547147 − 1.08940i
\(70\) 0 0
\(71\) 5.44141 + 3.14160i 0.645777 + 0.372839i 0.786836 − 0.617162i \(-0.211718\pi\)
−0.141060 + 0.990001i \(0.545051\pi\)
\(72\) 0 0
\(73\) − 14.3331i − 1.67756i −0.544469 − 0.838781i \(-0.683269\pi\)
0.544469 − 0.838781i \(-0.316731\pi\)
\(74\) 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 −0.785317 − 0.619094i \(-0.787500\pi\)
−0.370557 + 0.928810i \(0.620833\pi\)
\(80\) 0 0
\(81\) 2.58194 − 8.62169i 0.286882 − 0.957966i
\(82\) 0 0
\(83\) −4.79474 + 2.76824i −0.526291 + 0.303854i −0.739505 − 0.673151i \(-0.764940\pi\)
0.213214 + 0.977006i \(0.431607\pi\)
\(84\) 0 0
\(85\) 1.23526i 0.133983i
\(86\) 0 0
\(87\) 3.25300 + 1.63380i 0.348758 + 0.175162i
\(88\) 0 0
\(89\) 3.12906 − 11.6778i 0.331680 − 1.23785i −0.575744 − 0.817630i \(-0.695287\pi\)
0.907424 − 0.420217i \(-0.138046\pi\)
\(90\) 0 0
\(91\) 0.752937 − 2.81000i 0.0789293 − 0.294568i
\(92\) 0 0
\(93\) −0.406117 + 0.808604i −0.0421124 + 0.0838483i
\(94\) 0 0
\(95\) 2.29669 + 3.97799i 0.235636 + 0.408133i
\(96\) 0 0
\(97\) 0.215187 + 0.215187i 0.0218489 + 0.0218489i 0.717947 − 0.696098i \(-0.245082\pi\)
−0.696098 + 0.717947i \(0.745082\pi\)
\(98\) 0 0
\(99\) −6.50994 + 4.84605i −0.654274 + 0.487047i
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.785.4 yes 152
3.2 odd 2 inner 888.2.br.a.785.17 yes 152
37.14 odd 12 inner 888.2.br.a.569.17 yes 152
111.14 even 12 inner 888.2.br.a.569.4 ✓ 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.4 ✓ 152 111.14 even 12 inner
888.2.br.a.569.17 yes 152 37.14 odd 12 inner
888.2.br.a.785.4 yes 152 1.1 even 1 trivial
888.2.br.a.785.17 yes 152 3.2 odd 2 inner