Properties

Label 104.2.b.b
Level $104$
Weight $2$
Character orbit 104.b
Analytic conductor $0.830$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [104,2,Mod(53,104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("104.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.b (of order \(2\), degree \(1\), 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: \(0.830444181021\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + 2 \beta_{2} q^{3} + (\beta_{3} + \beta_1) q^{4} + (2 \beta_{3} - 2 \beta_1 + 2) q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{7} + (2 \beta_{2} + 2) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + 2 \beta_{2} q^{3} + (\beta_{3} + \beta_1) q^{4} + (2 \beta_{3} - 2 \beta_1 + 2) q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{7} + (2 \beta_{2} + 2) q^{8} - q^{9} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{10} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{11} + ( - 2 \beta_{3} + 2 \beta_1 - 4) q^{12} + \beta_{2} q^{13} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{14} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{15} + ( - 2 \beta_{3} + 2 \beta_1) q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{17}+ \cdots + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{6} - 12 q^{7} + 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{6} - 12 q^{7} + 8 q^{8} - 4 q^{9} + 12 q^{10} - 8 q^{12} - 12 q^{14} + 8 q^{16} + 8 q^{17} - 2 q^{18} - 12 q^{22} + 16 q^{23} - 16 q^{24} - 28 q^{25} + 2 q^{26} - 12 q^{28} + 24 q^{30} - 20 q^{31} - 8 q^{32} + 24 q^{33} - 8 q^{34} + 4 q^{38} - 8 q^{39} + 8 q^{41} + 12 q^{44} + 8 q^{46} + 20 q^{47} + 20 q^{49} - 14 q^{50} - 4 q^{52} + 8 q^{54} + 24 q^{55} - 24 q^{56} + 8 q^{57} + 4 q^{58} + 48 q^{60} - 4 q^{62} + 12 q^{63} - 24 q^{68} - 24 q^{70} - 12 q^{71} - 8 q^{72} - 16 q^{73} - 28 q^{74} + 4 q^{76} - 4 q^{78} + 8 q^{79} + 48 q^{80} - 44 q^{81} - 20 q^{82} + 24 q^{84} + 4 q^{86} - 16 q^{87} + 24 q^{88} - 12 q^{90} + 4 q^{94} - 24 q^{95} - 16 q^{96} - 16 q^{97} + 46 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.366025 1.36603i 2.00000i −1.73205 + 1.00000i 3.46410i 2.73205 0.732051i −1.26795 2.00000 + 2.00000i −1.00000 4.73205 1.26795i
53.2 −0.366025 + 1.36603i 2.00000i −1.73205 1.00000i 3.46410i 2.73205 + 0.732051i −1.26795 2.00000 2.00000i −1.00000 4.73205 + 1.26795i
53.3 1.36603 0.366025i 2.00000i 1.73205 1.00000i 3.46410i −0.732051 2.73205i −4.73205 2.00000 2.00000i −1.00000 1.26795 + 4.73205i
53.4 1.36603 + 0.366025i 2.00000i 1.73205 + 1.00000i 3.46410i −0.732051 + 2.73205i −4.73205 2.00000 + 2.00000i −1.00000 1.26795 4.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.2.b.b 4
3.b odd 2 1 936.2.g.b 4
4.b odd 2 1 416.2.b.b 4
8.b even 2 1 inner 104.2.b.b 4
8.d odd 2 1 416.2.b.b 4
12.b even 2 1 3744.2.g.b 4
16.e even 4 1 3328.2.a.n 2
16.e even 4 1 3328.2.a.bd 2
16.f odd 4 1 3328.2.a.m 2
16.f odd 4 1 3328.2.a.bc 2
24.f even 2 1 3744.2.g.b 4
24.h odd 2 1 936.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.b.b 4 1.a even 1 1 trivial
104.2.b.b 4 8.b even 2 1 inner
416.2.b.b 4 4.b odd 2 1
416.2.b.b 4 8.d odd 2 1
936.2.g.b 4 3.b odd 2 1
936.2.g.b 4 24.h odd 2 1
3328.2.a.m 2 16.f odd 4 1
3328.2.a.n 2 16.e even 4 1
3328.2.a.bc 2 16.f odd 4 1
3328.2.a.bd 2 16.e even 4 1
3744.2.g.b 4 12.b even 2 1
3744.2.g.b 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 22)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} - 10 T + 22)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$61$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$89$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 92)^{2} \) Copy content Toggle raw display
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