Show commands: Magma / Pari/GP / SageMath

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 569.3
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.68742 - 0.390639i) q^{3} +(0.623820 - 2.32813i) q^{5} +(0.882605 + 1.52872i) q^{7} +(2.69480 + 1.31835i) q^{9} +6.17618 q^{11} +(-2.69137 - 0.721150i) q^{13} +(-1.96211 + 3.68485i) q^{15} +(-4.97259 + 1.33240i) q^{17} +(0.561027 + 0.150327i) q^{19} +(-0.892152 - 2.92437i) q^{21} +(4.84488 + 4.84488i) q^{23} +(-0.700908 - 0.404669i) q^{25} +(-4.03228 - 3.27731i) q^{27} +(3.17615 - 3.17615i) q^{29} +(3.25762 + 3.25762i) q^{31} +(-10.4218 - 2.41266i) q^{33} +(4.10964 - 1.10117i) q^{35} +(2.96611 - 5.31058i) q^{37} +(4.25977 + 2.26824i) q^{39} +(-1.88735 - 3.26899i) q^{41} +(1.31134 - 1.31134i) q^{43} +(4.75036 - 5.45144i) q^{45} -9.96870i q^{47} +(1.94202 - 3.36367i) q^{49} +(8.91136 - 0.305838i) q^{51} +(6.88835 + 3.97699i) q^{53} +(3.85283 - 14.3790i) q^{55} +(-0.887968 - 0.472824i) q^{57} +(10.8224 - 2.89984i) q^{59} +(0.699062 - 2.60894i) q^{61} +(0.363065 + 5.28317i) q^{63} +(-3.35786 + 5.81598i) q^{65} +(-12.8155 + 7.39903i) q^{67} +(-6.28276 - 10.0680i) q^{69} +(6.50876 - 3.75783i) q^{71} -12.2386i q^{73} +(1.02465 + 0.956651i) q^{75} +(5.45113 + 9.44163i) q^{77} +(-9.05095 - 2.42520i) q^{79} +(5.52392 + 7.10537i) q^{81} +(0.219584 + 0.126777i) q^{83} +12.4080i q^{85} +(-6.60024 + 4.11878i) q^{87} +(1.58722 + 5.92360i) q^{89} +(-1.27298 - 4.75083i) q^{91} +(-4.22444 - 6.76955i) q^{93} +(0.699961 - 1.21237i) q^{95} +(-10.9153 + 10.9153i) q^{97} +(16.6436 + 8.14236i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −1.68742 − 0.390639i −0.974235 − 0.225536i
\(4\) 0 0
\(5\) 0.623820 − 2.32813i 0.278981 − 1.04117i −0.674145 − 0.738599i \(-0.735488\pi\)
0.953126 − 0.302572i \(-0.0978454\pi\)
\(6\) 0 0
\(7\) 0.882605 + 1.52872i 0.333593 + 0.577800i 0.983214 − 0.182458i \(-0.0584055\pi\)
−0.649620 + 0.760259i \(0.725072\pi\)
\(8\) 0 0
\(9\) 2.69480 + 1.31835i 0.898267 + 0.439449i
\(10\) 0 0
\(11\) 6.17618 1.86219 0.931095 − 0.364777i \(-0.118855\pi\)
0.931095 + 0.364777i \(0.118855\pi\)
\(12\) 0 0
\(13\) −2.69137 − 0.721150i −0.746451 − 0.200011i −0.134508 − 0.990913i \(-0.542945\pi\)
−0.611943 + 0.790902i \(0.709612\pi\)
\(14\) 0 0
\(15\) −1.96211 + 3.68485i −0.506614 + 0.951425i
\(16\) 0 0
\(17\) −4.97259 + 1.33240i −1.20603 + 0.323155i −0.805203 − 0.592999i \(-0.797944\pi\)
−0.400827 + 0.916154i \(0.631277\pi\)
\(18\) 0 0
\(19\) 0.561027 + 0.150327i 0.128709 + 0.0344873i 0.322598 − 0.946536i \(-0.395444\pi\)
−0.193890 + 0.981023i \(0.562110\pi\)
\(20\) 0 0
\(21\) −0.892152 − 2.92437i −0.194684 − 0.638150i
\(22\) 0 0
\(23\) 4.84488 + 4.84488i 1.01023 + 1.01023i 0.999947 + 0.0102791i \(0.00327200\pi\)
0.0102791 + 0.999947i \(0.496728\pi\)
\(24\) 0 0
\(25\) −0.700908 − 0.404669i −0.140182 − 0.0809338i
\(26\) 0 0
\(27\) −4.03228 − 3.27731i −0.776012 − 0.630718i
\(28\) 0 0
\(29\) 3.17615 − 3.17615i 0.589796 − 0.589796i −0.347780 − 0.937576i \(-0.613064\pi\)
0.937576 + 0.347780i \(0.113064\pi\)
\(30\) 0 0
\(31\) 3.25762 + 3.25762i 0.585086 + 0.585086i 0.936297 − 0.351210i \(-0.114230\pi\)
−0.351210 + 0.936297i \(0.614230\pi\)
\(32\) 0 0
\(33\) −10.4218 − 2.41266i −1.81421 − 0.419990i
\(34\) 0 0
\(35\) 4.10964 − 1.10117i 0.694655 − 0.186132i
\(36\) 0 0
\(37\) 2.96611 − 5.31058i 0.487625 − 0.873053i
\(38\) 0 0
\(39\) 4.25977 + 2.26824i 0.682109 + 0.363209i
\(40\) 0 0
\(41\) −1.88735 − 3.26899i −0.294755 − 0.510530i 0.680173 − 0.733052i \(-0.261905\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(42\) 0 0
\(43\) 1.31134 − 1.31134i 0.199978 − 0.199978i −0.600013 − 0.799991i \(-0.704838\pi\)
0.799991 + 0.600013i \(0.204838\pi\)
\(44\) 0 0
\(45\) 4.75036 − 5.45144i 0.708141 − 0.812652i
\(46\) 0 0
\(47\) − 9.96870i − 1.45408i −0.686593 − 0.727042i \(-0.740894\pi\)
0.686593 − 0.727042i \(-0.259106\pi\)
\(48\) 0 0
\(49\) 1.94202 − 3.36367i 0.277431 − 0.480525i
\(50\) 0 0
\(51\) 8.91136 − 0.305838i 1.24784 − 0.0428259i
\(52\) 0 0
\(53\) 6.88835 + 3.97699i 0.946188 + 0.546282i 0.891895 − 0.452243i \(-0.149376\pi\)
0.0542932 + 0.998525i \(0.482709\pi\)
\(54\) 0 0
\(55\) 3.85283 − 14.3790i 0.519515 − 1.93886i
\(56\) 0 0
\(57\) −0.887968 − 0.472824i −0.117614 − 0.0626271i
\(58\) 0 0
\(59\) 10.8224 − 2.89984i 1.40895 − 0.377527i 0.527400 − 0.849617i \(-0.323167\pi\)
0.881551 + 0.472090i \(0.156500\pi\)
\(60\) 0 0
\(61\) 0.699062 − 2.60894i 0.0895057 − 0.334040i −0.906624 − 0.421941i \(-0.861349\pi\)
0.996129 + 0.0879005i \(0.0280158\pi\)
\(62\) 0 0
\(63\) 0.363065 + 5.28317i 0.0457419 + 0.665617i
\(64\) 0 0
\(65\) −3.35786 + 5.81598i −0.416491 + 0.721384i
\(66\) 0 0
\(67\) −12.8155 + 7.39903i −1.56566 + 0.903935i −0.568996 + 0.822340i \(0.692668\pi\)
−0.996666 + 0.0815949i \(0.973999\pi\)
\(68\) 0 0
\(69\) −6.28276 − 10.0680i −0.756356 − 1.21204i
\(70\) 0 0
\(71\) 6.50876 − 3.75783i 0.772448 − 0.445973i −0.0612994 − 0.998119i \(-0.519524\pi\)
0.833747 + 0.552147i \(0.186191\pi\)
\(72\) 0 0
\(73\) − 12.2386i − 1.43242i −0.697884 − 0.716211i \(-0.745875\pi\)
0.697884 − 0.716211i \(-0.254125\pi\)
\(74\) 0 0
\(75\) 1.02465 + 0.956651i 0.118316 + 0.110464i
\(76\) 0 0
\(77\) 5.45113 + 9.44163i 0.621214 + 1.07597i
\(78\) 0 0
\(79\) −9.05095 − 2.42520i −1.01831 − 0.272856i −0.289213 − 0.957265i \(-0.593394\pi\)
−0.729098 + 0.684409i \(0.760060\pi\)
\(80\) 0 0
\(81\) 5.52392 + 7.10537i 0.613769 + 0.789486i
\(82\) 0 0
\(83\) 0.219584 + 0.126777i 0.0241025 + 0.0139156i 0.512003 − 0.858984i \(-0.328904\pi\)
−0.487900 + 0.872899i \(0.662237\pi\)
\(84\) 0 0
\(85\) 12.4080i 1.34584i
\(86\) 0 0
\(87\) −6.60024 + 4.11878i −0.707620 + 0.441580i
\(88\) 0 0
\(89\) 1.58722 + 5.92360i 0.168245 + 0.627900i 0.997604 + 0.0691834i \(0.0220394\pi\)
−0.829359 + 0.558717i \(0.811294\pi\)
\(90\) 0 0
\(91\) −1.27298 − 4.75083i −0.133445 − 0.498022i
\(92\) 0 0
\(93\) −4.22444 − 6.76955i −0.438054 − 0.701970i
\(94\) 0 0
\(95\) 0.699961 − 1.21237i 0.0718144 − 0.124386i
\(96\) 0 0
\(97\) −10.9153 + 10.9153i −1.10828 + 1.10828i −0.114901 + 0.993377i \(0.536655\pi\)
−0.993377 + 0.114901i \(0.963345\pi\)
\(98\) 0 0
\(99\) 16.6436 + 8.14236i 1.67274 + 0.818338i
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.569.3 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.16 yes 152
37.8 odd 12 inner 888.2.br.a.785.16 yes 152
111.8 even 12 inner 888.2.br.a.785.3 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.3 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.16 yes 152 3.2 odd 2 inner
888.2.br.a.785.3 yes 152 111.8 even 12 inner
888.2.br.a.785.16 yes 152 37.8 odd 12 inner