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Properties

Label 1386.2.e.d
Level $1386$
Weight $2$
Character orbit 1386.e
Analytic conductor $11.067$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(307,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1386.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.e (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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + ( - \beta_{7} + \beta_{2}) q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + ( - \beta_{7} + \beta_{2}) q^{7} - \beta_1 q^{8} - \beta_{7} q^{10} + ( - 2 \beta_{5} + \beta_{4} - \beta_1) q^{11} - \beta_{7} q^{13} + ( - \beta_{5} + 2 \beta_{4}) q^{14} + q^{16} + 2 \beta_{3} q^{17} + 2 \beta_{7} q^{19} + \beta_{4} q^{20} + ( - \beta_{7} - 2 \beta_{2} + 1) q^{22} + (2 \beta_{5} - \beta_{4}) q^{23} + 3 q^{25} + \beta_{4} q^{26} + (\beta_{7} - \beta_{2}) q^{28} - 6 \beta_1 q^{29} + 4 \beta_{6} q^{31} + \beta_1 q^{32} + 2 \beta_{6} q^{34} + (\beta_{3} - 3 \beta_1) q^{35} + 2 q^{37} - 2 \beta_{4} q^{38} + \beta_{7} q^{40} + 4 \beta_{3} q^{41} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{43} + (2 \beta_{5} - \beta_{4} + \beta_1) q^{44} + (\beta_{7} + 2 \beta_{2}) q^{46} - 5 \beta_{4} q^{47} + (3 \beta_{6} + 2) q^{49} + 3 \beta_1 q^{50} + \beta_{7} q^{52} + ( - 2 \beta_{5} + \beta_{4}) q^{53} + (\beta_{7} + 2 \beta_{6}) q^{55} + (\beta_{5} - 2 \beta_{4}) q^{56} + 6 q^{58} - 2 \beta_{4} q^{59} - 9 \beta_{7} q^{61} - 4 \beta_{3} q^{62} - q^{64} - 2 \beta_1 q^{65} + 12 q^{67} - 2 \beta_{3} q^{68} + (\beta_{6} + 3) q^{70} + ( - 2 \beta_{5} + \beta_{4}) q^{71} + 2 \beta_1 q^{74} - 2 \beta_{7} q^{76} + (\beta_{5} - 2 \beta_{4} + \cdots - 5 \beta_1) q^{77}+ \cdots + ( - 3 \beta_{3} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{22} + 24 q^{25} + 16 q^{37} + 16 q^{49} + 48 q^{58} - 8 q^{64} + 96 q^{67} + 24 q^{70} - 8 q^{88} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - \nu^{5} + 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} - \beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{7} - 5\beta_{5} + 8\beta_{4} - 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} + 13\beta_{5} + 8\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.14412 − 1.14412i
−0.437016 + 0.437016i
0.437016 − 0.437016i
−1.14412 + 1.14412i
−1.14412 − 1.14412i
0.437016 + 0.437016i
−0.437016 − 0.437016i
1.14412 + 1.14412i
− 1.00000i 0 −1.00000 − 1.41421i 0 −2.12132 − 1.58114i 1.00000i 0 −1.41421
307.2 − 1.00000i 0 −1.00000 − 1.41421i 0 −2.12132 + 1.58114i 1.00000i 0 −1.41421
307.3 − 1.00000i 0 −1.00000 1.41421i 0 2.12132 − 1.58114i 1.00000i 0 1.41421
307.4 − 1.00000i 0 −1.00000 1.41421i 0 2.12132 + 1.58114i 1.00000i 0 1.41421
307.5 1.00000i 0 −1.00000 − 1.41421i 0 2.12132 − 1.58114i − 1.00000i 0 1.41421
307.6 1.00000i 0 −1.00000 − 1.41421i 0 2.12132 + 1.58114i − 1.00000i 0 1.41421
307.7 1.00000i 0 −1.00000 1.41421i 0 −2.12132 − 1.58114i − 1.00000i 0 −1.41421
307.8 1.00000i 0 −1.00000 1.41421i 0 −2.12132 + 1.58114i − 1.00000i 0 −1.41421
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
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
11.b odd 2 1 inner
21.c even 2 1 inner
33.d even 2 1 inner
77.b even 2 1 inner
231.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.e.d ✓ 8
3.b odd 2 1 inner 1386.2.e.d ✓ 8
7.b odd 2 1 inner 1386.2.e.d ✓ 8
11.b odd 2 1 inner 1386.2.e.d ✓ 8
21.c even 2 1 inner 1386.2.e.d ✓ 8
33.d even 2 1 inner 1386.2.e.d ✓ 8
77.b even 2 1 inner 1386.2.e.d ✓ 8
231.h odd 2 1 inner 1386.2.e.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.e.d ✓ 8 1.a even 1 1 trivial
1386.2.e.d ✓ 8 3.b odd 2 1 inner
1386.2.e.d ✓ 8 7.b odd 2 1 inner
1386.2.e.d ✓ 8 11.b odd 2 1 inner
1386.2.e.d ✓ 8 21.c even 2 1 inner
1386.2.e.d ✓ 8 33.d even 2 1 inner
1386.2.e.d ✓ 8 77.b even 2 1 inner
1386.2.e.d ✓ 8 231.h 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{2} + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$37$ \( (T - 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$67$ \( (T - 12)^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
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