Properties

Label 704.2.e.a
Level $704$
Weight $2$
Character orbit 704.e
Analytic conductor $5.621$
Analytic rank $1$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,2,Mod(703,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.62146830230\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 176)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 3 q^{5} - 8 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 3 q^{5} - 8 q^{9} + \beta q^{11} + 3 \beta q^{15} + \beta q^{23} + 4 q^{25} + 5 \beta q^{27} - 3 \beta q^{31} + 11 q^{33} - 7 q^{37} + 24 q^{45} + 2 \beta q^{47} - 7 q^{49} - 6 q^{53} - 3 \beta q^{55} - \beta q^{59} + 3 \beta q^{67} + 11 q^{69} + 5 \beta q^{71} - 4 \beta q^{75} + 31 q^{81} - 9 q^{89} - 33 q^{93} - 17 q^{97} - 8 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{5} - 16 q^{9} + 8 q^{25} + 22 q^{33} - 14 q^{37} + 48 q^{45} - 14 q^{49} - 12 q^{53} + 22 q^{69} + 62 q^{81} - 18 q^{89} - 66 q^{93} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(321\) \(639\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 + 1.65831i
0.500000 1.65831i
0 3.31662i 0 −3.00000 0 0 0 −8.00000 0
703.2 0 3.31662i 0 −3.00000 0 0 0 −8.00000 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 \(\Q(\sqrt{-11}) \)
4.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.2.e.a 2
4.b odd 2 1 inner 704.2.e.a 2
8.b even 2 1 176.2.e.a 2
8.d odd 2 1 176.2.e.a 2
11.b odd 2 1 CM 704.2.e.a 2
16.e even 4 2 2816.2.g.a 4
16.f odd 4 2 2816.2.g.a 4
24.f even 2 1 1584.2.o.a 2
24.h odd 2 1 1584.2.o.a 2
44.c even 2 1 inner 704.2.e.a 2
88.b odd 2 1 176.2.e.a 2
88.g even 2 1 176.2.e.a 2
176.i even 4 2 2816.2.g.a 4
176.l odd 4 2 2816.2.g.a 4
264.m even 2 1 1584.2.o.a 2
264.p odd 2 1 1584.2.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.e.a 2 8.b even 2 1
176.2.e.a 2 8.d odd 2 1
176.2.e.a 2 88.b odd 2 1
176.2.e.a 2 88.g even 2 1
704.2.e.a 2 1.a even 1 1 trivial
704.2.e.a 2 4.b odd 2 1 inner
704.2.e.a 2 11.b odd 2 1 CM
704.2.e.a 2 44.c even 2 1 inner
1584.2.o.a 2 24.f even 2 1
1584.2.o.a 2 24.h odd 2 1
1584.2.o.a 2 264.m even 2 1
1584.2.o.a 2 264.p odd 2 1
2816.2.g.a 4 16.e even 4 2
2816.2.g.a 4 16.f odd 4 2
2816.2.g.a 4 176.i even 4 2
2816.2.g.a 4 176.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 11 \) acting on \(S_{2}^{\mathrm{new}}(704, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 11 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) 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
$23$ \( T^{2} + 11 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 99 \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 44 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 11 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 99 \) 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
$97$ \( (T + 17)^{2} \) Copy content Toggle raw display
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