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Properties

Label 930.2.j.h
Level $930$
Weight $2$
Character orbit 930.j
Analytic conductor $7.426$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [930,2,Mod(497,930)] 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("930.497"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,-4,0,0,8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 8 q^{6} - 8 q^{7} + 4 q^{12} + 24 q^{13} - 12 q^{15} - 40 q^{16} - 56 q^{21} + 16 q^{22} - 48 q^{25} + 8 q^{27} - 8 q^{28} + 4 q^{30} - 40 q^{31} - 20 q^{33} - 8 q^{37} - 8 q^{40} + 12 q^{42}+ \cdots + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
497.1 −0.707107 + 0.707107i −1.73016 + 0.0809240i − 1.00000i −0.662050 − 2.13581i 1.16619 − 1.28063i 2.89525 + 2.89525i 0.707107 + 0.707107i 2.98690 − 0.280023i 1.97839 + 1.04211i
497.2 −0.707107 + 0.707107i −1.56098 + 0.750564i − 1.00000i 0.256652 + 2.22129i 0.573050 − 1.63451i 0.463267 + 0.463267i 0.707107 + 0.707107i 1.87331 − 2.34323i −1.75217 − 1.38921i
497.3 −0.707107 + 0.707107i −1.50338 + 0.860137i − 1.00000i 2.19538 − 0.424609i 0.454844 − 1.67126i −0.358790 − 0.358790i 0.707107 + 0.707107i 1.52033 − 2.58623i −1.25213 + 1.85261i
497.4 −0.707107 + 0.707107i −1.30199 − 1.14229i − 1.00000i 1.36562 − 1.77061i 1.72837 − 0.112923i −0.0800893 − 0.0800893i 0.707107 + 0.707107i 0.390345 + 2.97450i 0.286372 + 2.21765i
497.5 −0.707107 + 0.707107i −0.404727 − 1.68410i − 1.00000i −2.01952 + 0.959968i 1.47702 + 0.904654i −2.95632 − 2.95632i 0.707107 + 0.707107i −2.67239 + 1.36320i 0.749217 − 2.10682i
497.6 −0.707107 + 0.707107i 0.0792035 − 1.73024i − 1.00000i −1.29721 + 1.82133i 1.16746 + 1.27947i 2.17556 + 2.17556i 0.707107 + 0.707107i −2.98745 − 0.274082i −0.370615 − 2.20514i
497.7 −0.707107 + 0.707107i 0.0980849 + 1.72927i − 1.00000i −0.278761 + 2.21862i −1.29214 − 1.15342i −1.59354 − 1.59354i 0.707107 + 0.707107i −2.98076 + 0.339231i −1.37169 − 1.76592i
497.8 −0.707107 + 0.707107i 0.877144 + 1.49353i − 1.00000i −1.98408 − 1.03122i −1.67632 − 0.435847i 2.39520 + 2.39520i 0.707107 + 0.707107i −1.46124 + 2.62007i 2.13214 − 0.673776i
497.9 −0.707107 + 0.707107i 1.33303 − 1.10591i − 1.00000i −0.368595 − 2.20548i −0.160600 + 1.72459i −2.86656 − 2.86656i 0.707107 + 0.707107i 0.553937 − 2.94842i 1.82015 + 1.29887i
497.10 −0.707107 + 0.707107i 1.69956 + 0.333903i − 1.00000i 1.37834 + 1.76073i −1.43788 + 0.965667i −2.07397 − 2.07397i 0.707107 + 0.707107i 2.77702 + 1.13498i −2.21966 − 0.270390i
497.11 0.707107 − 0.707107i −1.72927 − 0.0980849i − 1.00000i 0.278761 − 2.21862i −1.29214 + 1.15342i −1.59354 − 1.59354i −0.707107 − 0.707107i 2.98076 + 0.339231i −1.37169 − 1.76592i
497.12 0.707107 − 0.707107i −1.49353 − 0.877144i − 1.00000i 1.98408 + 1.03122i −1.67632 + 0.435847i 2.39520 + 2.39520i −0.707107 − 0.707107i 1.46124 + 2.62007i 2.13214 − 0.673776i
497.13 0.707107 − 0.707107i −0.860137 + 1.50338i − 1.00000i −2.19538 + 0.424609i 0.454844 + 1.67126i −0.358790 − 0.358790i −0.707107 − 0.707107i −1.52033 − 2.58623i −1.25213 + 1.85261i
497.14 0.707107 − 0.707107i −0.750564 + 1.56098i − 1.00000i −0.256652 − 2.22129i 0.573050 + 1.63451i 0.463267 + 0.463267i −0.707107 − 0.707107i −1.87331 − 2.34323i −1.75217 − 1.38921i
497.15 0.707107 − 0.707107i −0.333903 − 1.69956i − 1.00000i −1.37834 − 1.76073i −1.43788 − 0.965667i −2.07397 − 2.07397i −0.707107 − 0.707107i −2.77702 + 1.13498i −2.21966 − 0.270390i
497.16 0.707107 − 0.707107i −0.0809240 + 1.73016i − 1.00000i 0.662050 + 2.13581i 1.16619 + 1.28063i 2.89525 + 2.89525i −0.707107 − 0.707107i −2.98690 − 0.280023i 1.97839 + 1.04211i
497.17 0.707107 − 0.707107i 1.10591 − 1.33303i − 1.00000i 0.368595 + 2.20548i −0.160600 − 1.72459i −2.86656 − 2.86656i −0.707107 − 0.707107i −0.553937 − 2.94842i 1.82015 + 1.29887i
497.18 0.707107 − 0.707107i 1.14229 + 1.30199i − 1.00000i −1.36562 + 1.77061i 1.72837 + 0.112923i −0.0800893 − 0.0800893i −0.707107 − 0.707107i −0.390345 + 2.97450i 0.286372 + 2.21765i
497.19 0.707107 − 0.707107i 1.68410 + 0.404727i − 1.00000i 2.01952 − 0.959968i 1.47702 − 0.904654i −2.95632 − 2.95632i −0.707107 − 0.707107i 2.67239 + 1.36320i 0.749217 − 2.10682i
497.20 0.707107 − 0.707107i 1.73024 − 0.0792035i − 1.00000i 1.29721 − 1.82133i 1.16746 − 1.27947i 2.17556 + 2.17556i −0.707107 − 0.707107i 2.98745 − 0.274082i −0.370615 − 2.20514i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 497.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.h ✓ 40
3.b odd 2 1 inner 930.2.j.h ✓ 40
5.c odd 4 1 inner 930.2.j.h ✓ 40
15.e even 4 1 inner 930.2.j.h ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.h ✓ 40 1.a even 1 1 trivial
930.2.j.h ✓ 40 3.b odd 2 1 inner
930.2.j.h ✓ 40 5.c odd 4 1 inner
930.2.j.h ✓ 40 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} - 8 T_{7}^{17} + 525 T_{7}^{16} + 1996 T_{7}^{15} + \cdots + 32400 \) Copy content Toggle raw display
\( T_{11}^{20} + 90 T_{11}^{18} + 3281 T_{11}^{16} + 62320 T_{11}^{14} + 659288 T_{11}^{12} + 3839388 T_{11}^{10} + \cdots + 36 \) Copy content Toggle raw display
\( T_{17}^{40} + 4496 T_{17}^{36} + 7932976 T_{17}^{32} + 6919396576 T_{17}^{28} + 3060402118496 T_{17}^{24} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display