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Properties

Label 2646.2.d.e
Level $2646$
Weight $2$
Character orbit 2646.d
Analytic conductor $21.128$
Analytic rank $0$
Dimension $16$
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] = [2646,2,Mod(2645,2646)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2646, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2646.2645");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.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: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + (\beta_{12} + \beta_{9}) q^{5} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} + (\beta_{12} + \beta_{9}) q^{5} + \beta_1 q^{8} + ( - \beta_{14} - \beta_{2}) q^{10} + ( - \beta_{15} - 2 \beta_1) q^{11} + (\beta_{6} + \beta_{3} + \beta_{2}) q^{13} + q^{16} + ( - 3 \beta_{8} - \beta_{5}) q^{17} + (\beta_{14} + \beta_{6} - 2 \beta_{3}) q^{19} + ( - \beta_{12} - \beta_{9}) q^{20} + ( - \beta_{7} - 2) q^{22} + ( - \beta_{15} - \beta_{13} + \cdots - 3 \beta_1) q^{23}+ \cdots + ( - 2 \beta_{14} - 3 \beta_{6} + \cdots + 7 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{48}^{8} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{48}^{10} - \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{48}^{12} + 2\zeta_{48}^{4} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{5} - \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{48}^{15} - \zeta_{48}^{9} + 2\zeta_{48}^{7} + 2\zeta_{48} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -2\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( \zeta_{48}^{13} - 2\zeta_{48}^{11} + \zeta_{48}^{5} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( \zeta_{48}^{15} - \zeta_{48}^{9} - 2\zeta_{48}^{7} + 2\zeta_{48} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( 2\zeta_{48}^{14} + \zeta_{48}^{10} - \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( \zeta_{48}^{13} + 2\zeta_{48}^{11} + \zeta_{48}^{5} - \zeta_{48}^{3} \) Copy content Toggle raw display

Display \(\zeta_{48}^j\) in terms of \(\beta_i\)

\(\zeta_{48}\)\(=\) \( ( \beta_{13} + \beta_{8} + \beta_{7} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{14} + \beta_{11} + \beta_{9} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( \beta_{12} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( ( \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( \beta_{15} + \beta_{12} + \beta_{10} + \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( ( \beta_{11} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( -\beta_{13} - \beta_{8} + \beta_{7} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( ( \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( \beta_{14} - \beta_{11} + \beta_{9} - \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( \beta_{15} + \beta_{12} - \beta_{10} - \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( \beta_{15} - \beta_{12} + \beta_{10} - \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( \beta_{14} + \beta_{11} - \beta_{9} - \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( -\beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{48}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2645.1
0.608761 − 0.793353i
0.130526 − 0.991445i
−0.130526 + 0.991445i
−0.608761 + 0.793353i
−0.991445 − 0.130526i
0.793353 + 0.608761i
−0.793353 − 0.608761i
0.991445 + 0.130526i
0.608761 + 0.793353i
0.130526 + 0.991445i
−0.130526 − 0.991445i
−0.608761 − 0.793353i
−0.991445 + 0.130526i
0.793353 − 0.608761i
−0.793353 + 0.608761i
0.991445 − 0.130526i
− 1.00000i 0 −1.00000 −4.29725 0 0 1.00000i 0 4.29725i
2645.2 − 1.00000i 0 −1.00000 −3.21486 0 0 1.00000i 0 3.21486i
2645.3 − 1.00000i 0 −1.00000 −1.68412 0 0 1.00000i 0 1.68412i
2645.4 − 1.00000i 0 −1.00000 −0.601731 0 0 1.00000i 0 0.601731i
2645.5 − 1.00000i 0 −1.00000 0.601731 0 0 1.00000i 0 − 0.601731i
2645.6 − 1.00000i 0 −1.00000 1.68412 0 0 1.00000i 0 − 1.68412i
2645.7 − 1.00000i 0 −1.00000 3.21486 0 0 1.00000i 0 − 3.21486i
2645.8 − 1.00000i 0 −1.00000 4.29725 0 0 1.00000i 0 − 4.29725i
2645.9 1.00000i 0 −1.00000 −4.29725 0 0 − 1.00000i 0 − 4.29725i
2645.10 1.00000i 0 −1.00000 −3.21486 0 0 − 1.00000i 0 − 3.21486i
2645.11 1.00000i 0 −1.00000 −1.68412 0 0 − 1.00000i 0 − 1.68412i
2645.12 1.00000i 0 −1.00000 −0.601731 0 0 − 1.00000i 0 − 0.601731i
2645.13 1.00000i 0 −1.00000 0.601731 0 0 − 1.00000i 0 0.601731i
2645.14 1.00000i 0 −1.00000 1.68412 0 0 − 1.00000i 0 1.68412i
2645.15 1.00000i 0 −1.00000 3.21486 0 0 − 1.00000i 0 3.21486i
2645.16 1.00000i 0 −1.00000 4.29725 0 0 − 1.00000i 0 4.29725i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2645.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.d.e ✓ 16
3.b odd 2 1 inner 2646.2.d.e ✓ 16
7.b odd 2 1 inner 2646.2.d.e ✓ 16
21.c even 2 1 inner 2646.2.d.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2646.2.d.e ✓ 16 1.a even 1 1 trivial
2646.2.d.e ✓ 16 3.b odd 2 1 inner
2646.2.d.e ✓ 16 7.b odd 2 1 inner
2646.2.d.e ✓ 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 32T_{5}^{6} + 284T_{5}^{4} - 640T_{5}^{2} + 196 \) acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 32 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 40 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 28 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 84 T^{6} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 116 T^{6} + \cdots + 388129)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 60 T^{6} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 76 T^{6} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + \cdots + 3064)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 152 T^{6} + \cdots + 1817104)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots + 4567)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 152 T^{6} + \cdots + 8836)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 196 T^{6} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 268 T^{6} + \cdots + 9186961)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 304 T^{6} + \cdots + 430336)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 5047)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 60 T^{6} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 488 T^{6} + \cdots + 23059204)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 160 T^{2} + \cdots + 382)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 328 T^{6} + \cdots + 868624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 148 T^{6} + \cdots + 187489)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 752 T^{6} + \cdots + 1015314496)^{2} \) Copy content Toggle raw display
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