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Properties

Label 1088.2.o.c
Level $1088$
Weight $2$
Character orbit 1088.o
Analytic conductor $8.688$
Analytic rank $0$
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] = [1088,2,Mod(769,1088)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1088, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) 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("1088.769"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.o (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-4,0,-4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{3} + ( - 2 i - 2) q^{5} + (2 i - 2) q^{7} - i q^{9} + ( - i + 1) q^{11} - 2 q^{13} + 4 i q^{15} + (i - 4) q^{17} + 4 i q^{19} + 4 q^{21} + ( - 4 i + 4) q^{23} + 3 i q^{25} + (4 i - 4) q^{27} + \cdots + ( - i - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{11} - 4 q^{13} - 8 q^{17} + 8 q^{21} + 8 q^{23} - 8 q^{27} - 12 q^{29} + 12 q^{31} - 4 q^{33} + 16 q^{35} + 16 q^{37} + 4 q^{39} + 2 q^{41} - 4 q^{45} + 10 q^{51}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
− 1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 −2.00000 + 2.00000i 0 −2.00000 − 2.00000i 0 1.00000i 0
897.1 0 −1.00000 − 1.00000i 0 −2.00000 − 2.00000i 0 −2.00000 + 2.00000i 0 − 1.00000i 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
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.o.c 2
4.b odd 2 1 1088.2.o.m 2
8.b even 2 1 272.2.o.e 2
8.d odd 2 1 136.2.k.b ✓ 2
17.c even 4 1 inner 1088.2.o.c 2
24.f even 2 1 1224.2.w.a 2
24.h odd 2 1 2448.2.be.a 2
68.f odd 4 1 1088.2.o.m 2
136.i even 4 1 272.2.o.e 2
136.j odd 4 1 136.2.k.b ✓ 2
136.o even 8 2 4624.2.a.o 2
136.p odd 8 2 2312.2.a.j 2
136.p odd 8 2 2312.2.b.d 2
408.q even 4 1 1224.2.w.a 2
408.t odd 4 1 2448.2.be.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.k.b ✓ 2 8.d odd 2 1
136.2.k.b ✓ 2 136.j odd 4 1
272.2.o.e 2 8.b even 2 1
272.2.o.e 2 136.i even 4 1
1088.2.o.c 2 1.a even 1 1 trivial
1088.2.o.c 2 17.c even 4 1 inner
1088.2.o.m 2 4.b odd 2 1
1088.2.o.m 2 68.f odd 4 1
1224.2.w.a 2 24.f even 2 1
1224.2.w.a 2 408.q even 4 1
2312.2.a.j 2 136.p odd 8 2
2312.2.b.d 2 136.p odd 8 2
2448.2.be.a 2 24.h odd 2 1
2448.2.be.a 2 408.t odd 4 1
4624.2.a.o 2 136.o even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1088, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} + 4T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$37$ \( T^{2} - 16T + 128 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 196 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$67$ \( (T - 6)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
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