Properties

Label 1216.2.b.e
Level $1216$
Weight $2$
Character orbit 1216.b
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM discriminant -19
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(607,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, 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("1216.607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_{6} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + \beta_{6} q^{7} + 3 q^{9} + (\beta_{5} - 2 \beta_1) q^{11} + ( - \beta_{4} - 3) q^{17} + (2 \beta_{5} + \beta_1) q^{19} + \beta_{7} q^{23} - \beta_{4} q^{25} + ( - 3 \beta_{5} + 4 \beta_1) q^{35} + (3 \beta_{5} + 2 \beta_1) q^{43} + 3 \beta_{3} q^{45} + (2 \beta_{7} + \beta_{6}) q^{47} + ( - 3 \beta_{4} - 8) q^{49} + ( - \beta_{7} + 3 \beta_{6}) q^{55} + ( - \beta_{3} + 2 \beta_{2}) q^{61} + 3 \beta_{6} q^{63} + ( - 3 \beta_{4} + 7) q^{73} + (10 \beta_{3} - \beta_{2}) q^{77} + 9 q^{81} + (4 \beta_{5} + 2 \beta_1) q^{83} + ( - 6 \beta_{3} + \beta_{2}) q^{85} + ( - 2 \beta_{7} + \beta_{6}) q^{95} + (3 \beta_{5} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 28 q^{17} - 4 q^{25} - 76 q^{49} + 44 q^{73} + 72 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 16\nu^{5} + 44\nu^{3} + 175\nu ) / 500 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{6} - 196\nu^{4} - 1764\nu^{2} - 4975 ) / 700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{6} + 28\nu^{4} + 252\nu^{2} + 325 ) / 700 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 31\nu^{2} + 125 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 4\nu^{5} + 36\nu^{3} + 45\nu ) / 100 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{7} - 182\nu^{5} - 938\nu^{3} - 5525\nu ) / 1750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\nu^{7} + 112\nu^{5} + 308\nu^{3} + 1250\nu ) / 875 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - 4\beta_{6} - 4\beta_{5} - 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{4} + \beta_{3} - 5\beta_{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 16\beta_{5} + 8\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36\beta_{4} - 47\beta_{3} + 11\beta_{2} - 80 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 67\beta_{7} + 44\beta_{6} - 44\beta_{5} + 180\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49\beta_{3} + 7\beta_{2} + 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 281\beta_{7} + 4\beta_{6} + 4\beta_{5} - 1116\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
−0.656712 + 2.13746i
0.656712 2.13746i
1.52274 + 1.63746i
−1.52274 1.63746i
−1.52274 + 1.63746i
1.52274 1.63746i
0.656712 + 2.13746i
−0.656712 2.13746i
0 0 0 3.04547i 0 5.27492i 0 3.00000 0
607.2 0 0 0 3.04547i 0 5.27492i 0 3.00000 0
607.3 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.4 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.5 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.6 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.7 0 0 0 3.04547i 0 5.27492i 0 3.00000 0
607.8 0 0 0 3.04547i 0 5.27492i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner
152.g odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.b.e 8
4.b odd 2 1 inner 1216.2.b.e 8
8.b even 2 1 inner 1216.2.b.e 8
8.d odd 2 1 inner 1216.2.b.e 8
19.b odd 2 1 CM 1216.2.b.e 8
76.d even 2 1 inner 1216.2.b.e 8
152.b even 2 1 inner 1216.2.b.e 8
152.g odd 2 1 inner 1216.2.b.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.b.e 8 1.a even 1 1 trivial
1216.2.b.e 8 4.b odd 2 1 inner
1216.2.b.e 8 8.b even 2 1 inner
1216.2.b.e 8 8.d odd 2 1 inner
1216.2.b.e 8 19.b odd 2 1 CM
1216.2.b.e 8 76.d even 2 1 inner
1216.2.b.e 8 152.b even 2 1 inner
1216.2.b.e 8 152.g odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 11T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 11 T^{2} + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 33 T^{2} + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 47 T^{2} + 196)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 7 T - 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 87 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 113 T^{2} + 784)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 347 T^{2} + 26896)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 11 T - 98)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} - 76)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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