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Properties

Label 2178.2.b.h
Level $2178$
Weight $2$
Character orbit 2178.b
Analytic conductor $17.391$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2178,2,Mod(2177,2178)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2178.2177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2178.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,4,0,0,0,4,0,0,0,0,0,0,0,4,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.3914175602\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{5} + q^{8} + \beta_1 q^{10} - \beta_{3} q^{13} + q^{16} + 5 q^{17} + 4 \beta_1 q^{19} + \beta_1 q^{20} - 4 \beta_{3} q^{23} + (\beta_{2} + 3) q^{25} - \beta_{3} q^{26}+ \cdots + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} + 20 q^{17} + 12 q^{25} - 16 q^{29} + 4 q^{32} + 20 q^{34} + 12 q^{37} + 28 q^{49} + 12 q^{50} - 16 q^{58} + 4 q^{64} + 4 q^{65} + 32 q^{67} + 20 q^{68} + 12 q^{74}+ \cdots + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times\).

\(n\) \(1333\) \(1937\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
− 1.93185i
− 0.517638i
0.517638i
1.93185i
1.00000 0 1.00000 − 1.93185i 0 0 1.00000 0 − 1.93185i
2177.2 1.00000 0 1.00000 − 0.517638i 0 0 1.00000 0 − 0.517638i
2177.3 1.00000 0 1.00000 0.517638i 0 0 1.00000 0 0.517638i
2177.4 1.00000 0 1.00000 1.93185i 0 0 1.00000 0 1.93185i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2178.2.b.h yes 4
3.b odd 2 1 2178.2.b.g ✓ 4
11.b odd 2 1 2178.2.b.g ✓ 4
33.d even 2 1 inner 2178.2.b.h yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2178.2.b.g ✓ 4 3.b odd 2 1
2178.2.b.g ✓ 4 11.b odd 2 1
2178.2.b.h yes 4 1.a even 1 1 trivial
2178.2.b.h yes 4 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2178, [\chi])\):

\( T_{5}^{4} + 4T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{17} - 5 \) Copy content Toggle raw display
\( T_{29}^{2} + 8T_{29} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T - 5)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 64T^{2} + 256 \) Copy content Toggle raw display
$23$ \( T^{4} + 64T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 11)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 39)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 196T^{2} + 529 \) Copy content Toggle raw display
$59$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 84T^{2} + 36 \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 256T^{2} + 4096 \) Copy content Toggle raw display
$83$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 388 T^{2} + 27889 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T - 183)^{2} \) Copy content Toggle raw display
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