Properties

Label 756.2.be.b
Level $756$
Weight $2$
Character orbit 756.be
Analytic conductor $6.037$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(107,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.be (of order \(6\), degree \(2\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + 2 \beta_1 q^{5} + (2 \beta_{2} + 1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + 2 \beta_1 q^{5} + (2 \beta_{2} + 1) q^{7} + 2 \beta_{3} q^{8} + 4 \beta_{2} q^{10} + ( - \beta_{3} - \beta_1) q^{11} - q^{13} + (2 \beta_{3} + \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} + ( - \beta_{3} + \beta_1) q^{17} + ( - 4 \beta_{2} - 4) q^{19} + 4 \beta_{3} q^{20} + ( - 4 \beta_{2} + 2) q^{22} + ( - 2 \beta_{3} + \beta_1) q^{23} + 3 \beta_{2} q^{25} - \beta_1 q^{26} + (6 \beta_{2} - 4) q^{28} - 5 \beta_{3} q^{29} + ( - 3 \beta_{2} + 6) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + 2 q^{34} + (4 \beta_{3} + 2 \beta_1) q^{35} + (5 \beta_{2} - 5) q^{37} + ( - 4 \beta_{3} - 4 \beta_1) q^{38} + (8 \beta_{2} - 8) q^{40} - 5 \beta_{3} q^{41} + (10 \beta_{2} - 5) q^{43} + ( - 4 \beta_{3} + 2 \beta_1) q^{44} + ( - 2 \beta_{2} + 4) q^{46} + (6 \beta_{3} - 3 \beta_1) q^{47} + (8 \beta_{2} - 3) q^{49} + 3 \beta_{3} q^{50} - 2 \beta_{2} q^{52} + ( - 7 \beta_{3} + 7 \beta_1) q^{53} + ( - 8 \beta_{2} + 4) q^{55} + (6 \beta_{3} - 4 \beta_1) q^{56} + ( - 10 \beta_{2} + 10) q^{58} + ( - 3 \beta_{3} - 3 \beta_1) q^{59} + (5 \beta_{2} - 5) q^{61} + ( - 3 \beta_{3} + 6 \beta_1) q^{62} - 8 q^{64} - 2 \beta_1 q^{65} + (3 \beta_{2} - 6) q^{67} + 2 \beta_1 q^{68} + (12 \beta_{2} - 8) q^{70} + ( - 6 \beta_{3} + 12 \beta_1) q^{71} + 4 \beta_{2} q^{73} + (5 \beta_{3} - 5 \beta_1) q^{74} + ( - 16 \beta_{2} + 8) q^{76} + ( - 5 \beta_{3} + \beta_1) q^{77} + (9 \beta_{2} + 9) q^{79} + (8 \beta_{3} - 8 \beta_1) q^{80} + ( - 10 \beta_{2} + 10) q^{82} + (\beta_{3} - 2 \beta_1) q^{83} + 4 q^{85} + (10 \beta_{3} - 5 \beta_1) q^{86} + ( - 4 \beta_{2} + 8) q^{88} - 10 \beta_1 q^{89} + ( - 2 \beta_{2} - 1) q^{91} + ( - 2 \beta_{3} + 4 \beta_1) q^{92} + (6 \beta_{2} - 12) q^{94} + ( - 8 \beta_{3} - 8 \beta_1) q^{95} + 5 q^{97} + (8 \beta_{3} - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 8 q^{7} + 8 q^{10} - 4 q^{13} - 8 q^{16} - 24 q^{19} + 6 q^{25} - 4 q^{28} + 18 q^{31} + 8 q^{34} - 10 q^{37} - 16 q^{40} + 12 q^{46} + 4 q^{49} - 4 q^{52} + 20 q^{58} - 10 q^{61} - 32 q^{64} - 18 q^{67} - 8 q^{70} + 8 q^{73} + 54 q^{79} + 20 q^{82} + 16 q^{85} + 24 q^{88} - 8 q^{91} - 36 q^{94} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i −2.44949 1.41421i 0 2.00000 + 1.73205i 2.82843i 0 2.00000 + 3.46410i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.44949 + 1.41421i 0 2.00000 + 1.73205i 2.82843i 0 2.00000 + 3.46410i
431.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −2.44949 + 1.41421i 0 2.00000 1.73205i 2.82843i 0 2.00000 3.46410i
431.2 1.22474 0.707107i 0 1.00000 1.73205i 2.44949 1.41421i 0 2.00000 1.73205i 2.82843i 0 2.00000 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.be.b yes 4
3.b odd 2 1 inner 756.2.be.b yes 4
4.b odd 2 1 756.2.be.a 4
7.c even 3 1 756.2.be.a 4
12.b even 2 1 756.2.be.a 4
21.h odd 6 1 756.2.be.a 4
28.g odd 6 1 inner 756.2.be.b yes 4
84.n even 6 1 inner 756.2.be.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.be.a 4 4.b odd 2 1
756.2.be.a 4 7.c even 3 1
756.2.be.a 4 12.b even 2 1
756.2.be.a 4 21.h odd 6 1
756.2.be.b yes 4 1.a even 1 1 trivial
756.2.be.b yes 4 3.b odd 2 1 inner
756.2.be.b yes 4 28.g odd 6 1 inner
756.2.be.b yes 4 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{4} - 8T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{2} + 12T_{19} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$53$ \( T^{4} - 98T^{2} + 9604 \) Copy content Toggle raw display
$59$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 27 T + 243)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 200 T^{2} + 40000 \) Copy content Toggle raw display
$97$ \( (T - 5)^{4} \) Copy content Toggle raw display
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