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Properties

Label 1936.4.a.ba
Level $1936$
Weight $4$
Character orbit 1936.a
Self dual yes
Analytic conductor $114.228$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1936,4,Mod(1,1936)] 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("1936.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,10,0,10,0,0] 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: \(114.227697771\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{26}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 121)
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 \(\beta = 4\sqrt{26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{3} + 5 q^{5} + \beta q^{7} - 2 q^{9} - 3 \beta q^{13} + 25 q^{15} + \beta q^{17} - 5 \beta q^{19} + 5 \beta q^{21} - 35 q^{23} - 100 q^{25} - 145 q^{27} + 10 \beta q^{29} - 15 q^{31} + 5 \beta q^{35} + \cdots + 785 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{3} + 10 q^{5} - 4 q^{9} + 50 q^{15} - 70 q^{23} - 200 q^{25} - 290 q^{27} - 30 q^{31} - 530 q^{37} - 20 q^{45} - 760 q^{47} + 146 q^{49} + 1020 q^{53} - 42 q^{59} - 1170 q^{67} - 350 q^{69}+ \cdots + 1570 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.09902
5.09902
0 5.00000 0 5.00000 0 −20.3961 0 −2.00000 0
1.2 0 5.00000 0 5.00000 0 20.3961 0 −2.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1936.4.a.ba 2
4.b odd 2 1 121.4.a.d ✓ 2
11.b odd 2 1 inner 1936.4.a.ba 2
12.b even 2 1 1089.4.a.r 2
44.c even 2 1 121.4.a.d ✓ 2
44.g even 10 4 121.4.c.e 8
44.h odd 10 4 121.4.c.e 8
132.d odd 2 1 1089.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.d ✓ 2 4.b odd 2 1
121.4.a.d ✓ 2 44.c even 2 1
121.4.c.e 8 44.g even 10 4
121.4.c.e 8 44.h odd 10 4
1089.4.a.r 2 12.b even 2 1
1089.4.a.r 2 132.d odd 2 1
1936.4.a.ba 2 1.a even 1 1 trivial
1936.4.a.ba 2 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1936))\):

\( T_{3} - 5 \) Copy content Toggle raw display
\( T_{5} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} - 416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 5)^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 416 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3744 \) Copy content Toggle raw display
$17$ \( T^{2} - 416 \) Copy content Toggle raw display
$19$ \( T^{2} - 10400 \) Copy content Toggle raw display
$23$ \( (T + 35)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 41600 \) Copy content Toggle raw display
$31$ \( (T + 15)^{2} \) Copy content Toggle raw display
$37$ \( (T + 265)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10400 \) Copy content Toggle raw display
$43$ \( T^{2} - 201344 \) Copy content Toggle raw display
$47$ \( (T + 380)^{2} \) Copy content Toggle raw display
$53$ \( (T - 510)^{2} \) Copy content Toggle raw display
$59$ \( (T + 21)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 41600 \) Copy content Toggle raw display
$67$ \( (T + 585)^{2} \) Copy content Toggle raw display
$71$ \( (T + 313)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 220064 \) Copy content Toggle raw display
$79$ \( T^{2} - 374400 \) Copy content Toggle raw display
$83$ \( T^{2} - 425984 \) Copy content Toggle raw display
$89$ \( (T + 185)^{2} \) Copy content Toggle raw display
$97$ \( (T - 785)^{2} \) Copy content Toggle raw display
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