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Properties

Label 832.4.a.u
Level $832$
Weight $4$
Character orbit 832.a
Self dual yes
Analytic conductor $49.090$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [832,4,Mod(1,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 832.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3,0,3,0,25] 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: yes
Analytic conductor: \(49.0895891248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} + ( - 3 \beta + 3) q^{5} + ( - \beta + 13) q^{7} + (3 \beta - 8) q^{9} + ( - 4 \beta - 26) q^{11} + 13 q^{13} + (3 \beta + 51) q^{15} + ( - 19 \beta + 3) q^{17} + ( - 8 \beta - 58) q^{19}+ \cdots + ( - 58 \beta - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 3 q^{5} + 25 q^{7} - 13 q^{9} - 56 q^{11} + 26 q^{13} + 105 q^{15} - 13 q^{17} - 124 q^{19} - q^{21} + 172 q^{23} + 83 q^{25} - 9 q^{27} - 196 q^{29} + 78 q^{31} + 230 q^{33} + 147 q^{35}+ \cdots - 74 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.77200
−3.77200
0 −5.77200 0 −11.3160 0 8.22800 0 6.31601 0
1.2 0 2.77200 0 14.3160 0 16.7720 0 −19.3160 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.4.a.u 2
4.b odd 2 1 832.4.a.y 2
8.b even 2 1 208.4.a.j 2
8.d odd 2 1 104.4.a.c ✓ 2
24.f even 2 1 936.4.a.e 2
24.h odd 2 1 1872.4.a.bc 2
104.h odd 2 1 1352.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.4.a.c ✓ 2 8.d odd 2 1
208.4.a.j 2 8.b even 2 1
832.4.a.u 2 1.a even 1 1 trivial
832.4.a.y 2 4.b odd 2 1
936.4.a.e 2 24.f even 2 1
1352.4.a.f 2 104.h odd 2 1
1872.4.a.bc 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(832))\):

\( T_{3}^{2} + 3T_{3} - 16 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} - 162 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 16 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 162 \) Copy content Toggle raw display
$7$ \( T^{2} - 25T + 138 \) Copy content Toggle raw display
$11$ \( T^{2} + 56T + 492 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 13T - 6546 \) Copy content Toggle raw display
$19$ \( T^{2} + 124T + 2676 \) Copy content Toggle raw display
$23$ \( T^{2} - 172T + 4768 \) Copy content Toggle raw display
$29$ \( T^{2} + 196T + 4932 \) Copy content Toggle raw display
$31$ \( T^{2} - 78T - 19576 \) Copy content Toggle raw display
$37$ \( T^{2} + 161T - 65954 \) Copy content Toggle raw display
$41$ \( T^{2} - 234T - 97344 \) Copy content Toggle raw display
$43$ \( T^{2} - 135T - 146572 \) Copy content Toggle raw display
$47$ \( T^{2} - 237T + 13586 \) Copy content Toggle raw display
$53$ \( T^{2} + 666T + 109064 \) Copy content Toggle raw display
$59$ \( T^{2} - 136T - 353076 \) Copy content Toggle raw display
$61$ \( T^{2} - 146T + 3504 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 74748 \) Copy content Toggle raw display
$71$ \( T^{2} - 563T + 71194 \) Copy content Toggle raw display
$73$ \( T^{2} + 1480 T + 533292 \) Copy content Toggle raw display
$79$ \( T^{2} - 896T + 190192 \) Copy content Toggle raw display
$83$ \( T^{2} - 1902 T + 903744 \) Copy content Toggle raw display
$89$ \( T^{2} + 272 T - 1068036 \) Copy content Toggle raw display
$97$ \( T^{2} + 2160 T + 521372 \) Copy content Toggle raw display
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