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Properties

Label 1456.2.r.j
Level $1456$
Weight $2$
Character orbit 1456.r
Analytic conductor $11.626$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(417,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), degree \(2\), 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (2 \beta_1 + 3) q^{7} + 2 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (2 \beta_1 + 3) q^{7} + 2 \beta_1 q^{9} + ( - 3 \beta_1 - 3) q^{11} - q^{13} + 5 q^{15} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{17} - 3 \beta_1 q^{19} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{21} + (\beta_{3} + \beta_{2} + 6 \beta_1) q^{23} + \beta_{3} q^{27} - 2 \beta_{3} q^{29} + (5 \beta_1 + 5) q^{31} + (3 \beta_{3} + 3 \beta_{2}) q^{33} + (3 \beta_{3} + \beta_{2}) q^{35} + (3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{37} + \beta_{2} q^{39} + 2 \beta_{3} q^{41} + 8 q^{43} - 2 \beta_{2} q^{45} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{47} + (8 \beta_1 + 5) q^{49} + ( - 3 \beta_{3} - 3 \beta_{2} + 10 \beta_1) q^{51} + (2 \beta_{2} + 3 \beta_1 + 3) q^{53} - 3 \beta_{3} q^{55} + 3 \beta_{3} q^{57} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{59} + 3 \beta_1 q^{61} + (2 \beta_1 - 4) q^{63} + ( - \beta_{3} - \beta_{2}) q^{65} + ( - 3 \beta_1 - 3) q^{67} + ( - 6 \beta_{3} + 5) q^{69} - 4 \beta_{3} q^{71} + (3 \beta_{2} - 4 \beta_1 - 4) q^{73} + ( - 9 \beta_1 - 3) q^{77} + (3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{79} + (11 \beta_1 + 11) q^{81} + (3 \beta_{3} + 10) q^{85} + ( - 10 \beta_1 - 10) q^{87} + (\beta_{3} + \beta_{2}) q^{89} + ( - 2 \beta_1 - 3) q^{91} + ( - 5 \beta_{3} - 5 \beta_{2}) q^{93} + 3 \beta_{2} q^{95} + ( - 6 \beta_{3} - 4) q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 4 q^{9} - 6 q^{11} - 4 q^{13} + 20 q^{15} + 6 q^{17} + 6 q^{19} - 12 q^{23} + 10 q^{31} + 4 q^{37} + 32 q^{43} - 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} + 6 q^{59} - 6 q^{61} - 20 q^{63} - 6 q^{67} + 20 q^{69} - 8 q^{73} + 6 q^{77} + 8 q^{79} + 22 q^{81} + 40 q^{85} - 20 q^{87} - 8 q^{91} - 16 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
0.809017 − 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 − 0.535233i
0 −1.11803 + 1.93649i 0 −1.11803 − 1.93649i 0 2.00000 − 1.73205i 0 −1.00000 − 1.73205i 0
417.2 0 1.11803 − 1.93649i 0 1.11803 + 1.93649i 0 2.00000 − 1.73205i 0 −1.00000 − 1.73205i 0
625.1 0 −1.11803 − 1.93649i 0 −1.11803 + 1.93649i 0 2.00000 + 1.73205i 0 −1.00000 + 1.73205i 0
625.2 0 1.11803 + 1.93649i 0 1.11803 − 1.93649i 0 2.00000 + 1.73205i 0 −1.00000 + 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.r.j 4
4.b odd 2 1 91.2.e.b ✓ 4
7.c even 3 1 inner 1456.2.r.j 4
12.b even 2 1 819.2.j.c 4
28.d even 2 1 637.2.e.h 4
28.f even 6 1 637.2.a.e 2
28.f even 6 1 637.2.e.h 4
28.g odd 6 1 91.2.e.b ✓ 4
28.g odd 6 1 637.2.a.f 2
52.b odd 2 1 1183.2.e.d 4
84.j odd 6 1 5733.2.a.w 2
84.n even 6 1 819.2.j.c 4
84.n even 6 1 5733.2.a.v 2
364.x even 6 1 8281.2.a.ba 2
364.bl odd 6 1 1183.2.e.d 4
364.bl odd 6 1 8281.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b ✓ 4 4.b odd 2 1
91.2.e.b ✓ 4 28.g odd 6 1
637.2.a.e 2 28.f even 6 1
637.2.a.f 2 28.g odd 6 1
637.2.e.h 4 28.d even 2 1
637.2.e.h 4 28.f even 6 1
819.2.j.c 4 12.b even 2 1
819.2.j.c 4 84.n even 6 1
1183.2.e.d 4 52.b odd 2 1
1183.2.e.d 4 364.bl odd 6 1
1456.2.r.j 4 1.a even 1 1 trivial
1456.2.r.j 4 7.c even 3 1 inner
5733.2.a.v 2 84.n even 6 1
5733.2.a.w 2 84.j odd 6 1
8281.2.a.z 2 364.bl odd 6 1
8281.2.a.ba 2 364.x even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3}^{4} + 5T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{5}^{4} + 5T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$41$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$43$ \( (T - 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 164)^{2} \) Copy content Toggle raw display
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