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Properties

Label 1386.2.e.e
Level $1386$
Weight $2$
Character orbit 1386.e
Analytic conductor $11.067$
Analytic rank $0$
Dimension $8$
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] = [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: 8.0.6679465984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 462)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{4} + \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{4} + \beta_1) q^{7} + \beta_1 q^{8} + (\beta_{2} + 1) q^{10} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 4 q^{10} + 8 q^{11} + 8 q^{14} + 8 q^{16} + 12 q^{17} + 4 q^{19} + 4 q^{22} + 8 q^{23} - 16 q^{25} - 8 q^{35} + 16 q^{37} - 4 q^{40} + 20 q^{41} - 8 q^{44} - 12 q^{49} - 40 q^{55} - 8 q^{56} + 56 q^{61} - 20 q^{62} - 8 q^{64} + 16 q^{67} - 12 q^{68} - 12 q^{70} - 8 q^{71} - 4 q^{73} - 4 q^{76} + 20 q^{77} - 4 q^{83} + 24 q^{86} - 4 q^{88} + 20 q^{91} - 8 q^{92} - 20 q^{94} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 55\nu^{3} + 46\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 11\nu^{5} + 33\nu^{4} - 28\nu^{3} + 96\nu^{2} + 8\nu + 60 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 3\nu^{6} + 11\nu^{5} + 33\nu^{4} + 28\nu^{3} + 96\nu^{2} - 8\nu + 60 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 11\nu^{5} + 39\nu^{4} - 28\nu^{3} + 138\nu^{2} - 4\nu + 96 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 3\nu^{6} + 11\nu^{5} + 39\nu^{4} + 28\nu^{3} + 138\nu^{2} + 4\nu + 96 ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 7\beta_{6} + 5\beta_{5} - 5\beta_{4} + 2\beta_{3} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} + 9\beta_{6} - 9\beta_{5} - 9\beta_{4} - 14\beta_{2} + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 45\beta_{7} - 45\beta_{6} - 31\beta_{5} + 31\beta_{4} - 18\beta_{3} + 26\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} - 67\beta_{6} + 71\beta_{5} + 71\beta_{4} + 90\beta_{2} - 178 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -291\beta_{7} + 291\beta_{6} + 205\beta_{5} - 205\beta_{4} + 142\beta_{3} - 230\beta_1 ) / 2 \) 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
− 2.20392i
2.62511i
− 1.25051i
0.829319i
− 0.829319i
1.25051i
− 2.62511i
2.20392i
− 1.00000i 0 −1.00000 − 3.06118i 0 2.37330 + 1.16938i 1.00000i 0 −3.06118
307.2 − 1.00000i 0 −1.00000 − 0.266088i 0 −1.34926 + 2.27585i 1.00000i 0 −0.266088
307.3 − 1.00000i 0 −1.00000 1.18572i 0 −1.74138 − 1.99189i 1.00000i 0 1.18572
307.4 − 1.00000i 0 −1.00000 4.14155i 0 0.717333 + 2.54665i 1.00000i 0 4.14155
307.5 1.00000i 0 −1.00000 − 4.14155i 0 0.717333 − 2.54665i − 1.00000i 0 4.14155
307.6 1.00000i 0 −1.00000 − 1.18572i 0 −1.74138 + 1.99189i − 1.00000i 0 1.18572
307.7 1.00000i 0 −1.00000 0.266088i 0 −1.34926 − 2.27585i − 1.00000i 0 −0.266088
307.8 1.00000i 0 −1.00000 3.06118i 0 2.37330 − 1.16938i − 1.00000i 0 −3.06118
\(n\): 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
77.b even 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.e 8
3.b odd 2 1 462.2.e.a ✓ 8
7.b odd 2 1 1386.2.e.a 8
11.b odd 2 1 1386.2.e.a 8
12.b even 2 1 3696.2.q.b 8
21.c even 2 1 462.2.e.b yes 8
33.d even 2 1 462.2.e.b yes 8
77.b even 2 1 inner 1386.2.e.e 8
84.h odd 2 1 3696.2.q.c 8
132.d odd 2 1 3696.2.q.c 8
231.h odd 2 1 462.2.e.a ✓ 8
924.n even 2 1 3696.2.q.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.e.a ✓ 8 3.b odd 2 1
462.2.e.a ✓ 8 231.h odd 2 1
462.2.e.b yes 8 21.c even 2 1
462.2.e.b yes 8 33.d even 2 1
1386.2.e.a 8 7.b odd 2 1
1386.2.e.a 8 11.b odd 2 1
1386.2.e.e 8 1.a even 1 1 trivial
1386.2.e.e 8 77.b even 2 1 inner
3696.2.q.b 8 12.b even 2 1
3696.2.q.b 8 924.n even 2 1
3696.2.q.c 8 84.h odd 2 1
3696.2.q.c 8 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 200T_{5}^{4} + 240T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 26T_{13}^{2} + 56T_{13} - 16 \) 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
$5$ \( T^{8} + 28 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} - 26 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 6 T^{3} - 10 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 58 T^{2} + \cdots - 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + \cdots - 424)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 56 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$31$ \( T^{8} + 112 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + \cdots + 432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 10 T^{3} + \cdots - 656)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 144 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + \cdots + 1507984 \) Copy content Toggle raw display
$53$ \( (T^{4} - 182 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 272 T^{6} + \cdots + 11943936 \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{3} + \cdots - 3232)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots + 2624)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{3} + \cdots - 1416)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} - 22 T^{2} + 64)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 300 T^{6} + \cdots + 20502784 \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} + \cdots + 4296)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 200 T^{6} + \cdots + 147456 \) Copy content Toggle raw display
$97$ \( T^{8} + 136 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
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