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Properties

Label 1476.2.d.c
Level $1476$
Weight $2$
Character orbit 1476.d
Analytic conductor $11.786$
Analytic rank $0$
Dimension $8$
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] = [1476,2,Mod(1475,1476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1476, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1476.1475");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1476.d (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: \(11.7859193383\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{4} - 2) q^{4} - 2 \beta_{6} q^{5} + ( - \beta_{7} - \beta_1) q^{7} + ( - \beta_{4} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{4} - 2) q^{4} - 2 \beta_{6} q^{5} + ( - \beta_{7} - \beta_1) q^{7} + ( - \beta_{4} - 2) q^{8} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{10} + ( - 2 \beta_{7} + 2 \beta_1) q^{11} + ( - 2 \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{13}+ \cdots + (3 \beta_{6} - 3 \beta_{4} + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 12 q^{4} - 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 12 q^{4} - 20 q^{8} + 4 q^{16} - 32 q^{23} - 24 q^{25} + 44 q^{32} - 16 q^{37} - 16 q^{46} - 24 q^{49} - 12 q^{50} + 16 q^{59} + 48 q^{61} + 56 q^{62} + 36 q^{64} - 48 q^{73} - 8 q^{74} - 28 q^{82} + 96 q^{83} - 56 q^{86} + 48 q^{92} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 6\nu^{3} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} + 6\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1475.1
0.254137 + 1.39119i
−0.254137 − 1.39119i
1.39119 − 0.254137i
−1.39119 + 0.254137i
1.39119 + 0.254137i
−1.39119 − 0.254137i
0.254137 − 1.39119i
−0.254137 + 1.39119i
0.500000 − 1.32288i 0 −1.50000 − 1.32288i − 2.82843i 0 −0.508274 −2.50000 + 1.32288i 0 −3.74166 − 1.41421i
1475.2 0.500000 − 1.32288i 0 −1.50000 − 1.32288i − 2.82843i 0 0.508274 −2.50000 + 1.32288i 0 −3.74166 − 1.41421i
1475.3 0.500000 − 1.32288i 0 −1.50000 − 1.32288i 2.82843i 0 −2.78238 −2.50000 + 1.32288i 0 3.74166 + 1.41421i
1475.4 0.500000 − 1.32288i 0 −1.50000 − 1.32288i 2.82843i 0 2.78238 −2.50000 + 1.32288i 0 3.74166 + 1.41421i
1475.5 0.500000 + 1.32288i 0 −1.50000 + 1.32288i − 2.82843i 0 −2.78238 −2.50000 − 1.32288i 0 3.74166 − 1.41421i
1475.6 0.500000 + 1.32288i 0 −1.50000 + 1.32288i − 2.82843i 0 2.78238 −2.50000 − 1.32288i 0 3.74166 − 1.41421i
1475.7 0.500000 + 1.32288i 0 −1.50000 + 1.32288i 2.82843i 0 −0.508274 −2.50000 − 1.32288i 0 −3.74166 + 1.41421i
1475.8 0.500000 + 1.32288i 0 −1.50000 + 1.32288i 2.82843i 0 0.508274 −2.50000 − 1.32288i 0 −3.74166 + 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1475.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
41.b even 2 1 inner
492.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.2.d.c yes 8
3.b odd 2 1 1476.2.d.b ✓ 8
4.b odd 2 1 1476.2.d.b ✓ 8
12.b even 2 1 inner 1476.2.d.c yes 8
41.b even 2 1 inner 1476.2.d.c yes 8
123.b odd 2 1 1476.2.d.b ✓ 8
164.d odd 2 1 1476.2.d.b ✓ 8
492.d even 2 1 inner 1476.2.d.c yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.2.d.b ✓ 8 3.b odd 2 1
1476.2.d.b ✓ 8 4.b odd 2 1
1476.2.d.b ✓ 8 123.b odd 2 1
1476.2.d.b ✓ 8 164.d odd 2 1
1476.2.d.c yes 8 1.a even 1 1 trivial
1476.2.d.c yes 8 12.b even 2 1 inner
1476.2.d.c yes 8 41.b even 2 1 inner
1476.2.d.c yes 8 492.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1476, [\chi])\):

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{2} + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 242)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 64 T^{2} + 800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 32 T^{2} + 32)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 64 T^{2} + 338)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 72 T^{2} + 400)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 92 T^{6} + \cdots + 2825761 \) Copy content Toggle raw display
$43$ \( (T^{4} + 92 T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 128 T^{2} + 50)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 56 T^{2} + 98)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 52)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T - 20)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 256 T^{2} + 12800)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 296 T^{2} + 21218)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 22)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 288 T^{2} + 19602)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 24 T + 130)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 256 T^{2} + 5408)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 32 T^{2} + 32)^{2} \) Copy content Toggle raw display
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