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Properties

Label 276.2.a.b
Level $276$
Weight $2$
Character orbit 276.a
Self dual yes
Analytic conductor $2.204$
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] = [276,2,Mod(1,276)] 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("276.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(276, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta + 2) q^{5} + \beta q^{7} + q^{9} - 4 \beta q^{11} - 4 \beta q^{13} + (\beta + 2) q^{15} + (3 \beta + 2) q^{17} + (3 \beta - 4) q^{19} + \beta q^{21} + q^{23} + (4 \beta + 1) q^{25} + \cdots - 4 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} - 10 q^{49}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 1.00000 0 0.585786 0 −1.41421 0 1.00000 0
1.2 0 1.00000 0 3.41421 0 1.41421 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(23\) \( -1 \)

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 276.2.a.b ✓ 2
3.b odd 2 1 828.2.a.e 2
4.b odd 2 1 1104.2.a.l 2
5.b even 2 1 6900.2.a.m 2
5.c odd 4 2 6900.2.f.l 4
8.b even 2 1 4416.2.a.bc 2
8.d odd 2 1 4416.2.a.bi 2
12.b even 2 1 3312.2.a.s 2
23.b odd 2 1 6348.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.a.b ✓ 2 1.a even 1 1 trivial
828.2.a.e 2 3.b odd 2 1
1104.2.a.l 2 4.b odd 2 1
3312.2.a.s 2 12.b even 2 1
4416.2.a.bc 2 8.b even 2 1
4416.2.a.bi 2 8.d odd 2 1
6348.2.a.h 2 23.b odd 2 1
6900.2.a.m 2 5.b even 2 1
6900.2.f.l 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(276))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 32 \) Copy content Toggle raw display
$13$ \( T^{2} - 32 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 2 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 34 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$67$ \( T^{2} + 24T + 126 \) Copy content Toggle raw display
$71$ \( T^{2} - 128 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 50 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 14 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 196 \) Copy content Toggle raw display
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