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Properties

Label 880.2.m.a
Level $880$
Weight $2$
Character orbit 880.m
Analytic conductor $7.027$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,2,Mod(879,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 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("880.879"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-3,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta - 1) q^{5} - 2 q^{9} + (2 \beta - 1) q^{11} + (\beta + 1) q^{15} + 9 q^{23} + (3 \beta - 2) q^{25} + 5 q^{27} + (6 \beta - 3) q^{31} + ( - 2 \beta + 1) q^{33} + ( - 6 \beta + 3) q^{37}+ \cdots + ( - 4 \beta + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 3 q^{5} - 4 q^{9} + 3 q^{15} + 18 q^{23} - q^{25} + 10 q^{27} + 6 q^{45} + 24 q^{47} + 14 q^{49} + 11 q^{55} - 26 q^{67} - 18 q^{69} + q^{75} + 2 q^{81} + 18 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
879.1
0.500000 + 1.65831i
0.500000 − 1.65831i
0 −1.00000 0 −1.50000 − 1.65831i 0 0 0 −2.00000 0
879.2 0 −1.00000 0 −1.50000 + 1.65831i 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
20.d odd 2 1 inner
220.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.m.a ✓ 2
4.b odd 2 1 880.2.m.b yes 2
5.b even 2 1 880.2.m.b yes 2
11.b odd 2 1 CM 880.2.m.a ✓ 2
20.d odd 2 1 inner 880.2.m.a ✓ 2
44.c even 2 1 880.2.m.b yes 2
55.d odd 2 1 880.2.m.b yes 2
220.g even 2 1 inner 880.2.m.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.m.a ✓ 2 1.a even 1 1 trivial
880.2.m.a ✓ 2 11.b odd 2 1 CM
880.2.m.a ✓ 2 20.d odd 2 1 inner
880.2.m.a ✓ 2 220.g even 2 1 inner
880.2.m.b yes 2 4.b odd 2 1
880.2.m.b yes 2 5.b even 2 1
880.2.m.b yes 2 44.c even 2 1
880.2.m.b yes 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). 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
$5$ \( T^{2} + 3T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 11 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 99 \) Copy content Toggle raw display
$37$ \( T^{2} + 99 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 176 \) Copy content Toggle raw display
$59$ \( T^{2} + 11 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 13)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 275 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 99 \) Copy content Toggle raw display
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