Properties

Label 1386.2.e.c
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} -\beta_{5} q^{5} + \beta_{1} q^{7} -\beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} - q^{4} -\beta_{5} q^{5} + \beta_{1} q^{7} -\beta_{2} q^{8} -\beta_{3} q^{10} + ( 2 \beta_{2} + \beta_{4} ) q^{11} + 2 \beta_{3} q^{13} -\beta_{4} q^{14} + q^{16} + \beta_{7} q^{17} -\beta_{3} q^{19} + \beta_{5} q^{20} + ( -2 + \beta_{1} ) q^{22} + 2 \beta_{4} q^{23} -3 q^{25} -2 \beta_{5} q^{26} -\beta_{1} q^{28} + 6 \beta_{2} q^{29} -\beta_{6} q^{31} + \beta_{2} q^{32} + \beta_{6} q^{34} -\beta_{7} q^{35} -10 q^{37} + \beta_{5} q^{38} + \beta_{3} q^{40} -\beta_{7} q^{41} + 2 \beta_{1} q^{43} + ( -2 \beta_{2} - \beta_{4} ) q^{44} + 2 \beta_{1} q^{46} + \beta_{5} q^{47} -7 q^{49} -3 \beta_{2} q^{50} -2 \beta_{3} q^{52} + 4 \beta_{4} q^{53} + ( -2 \beta_{3} + \beta_{6} ) q^{55} + \beta_{4} q^{56} -6 q^{58} + 4 \beta_{5} q^{59} + \beta_{7} q^{62} - q^{64} + 16 \beta_{2} q^{65} -12 q^{67} -\beta_{7} q^{68} -\beta_{6} q^{70} -2 \beta_{4} q^{71} + 3 \beta_{3} q^{73} -10 \beta_{2} q^{74} + \beta_{3} q^{76} + ( 7 \beta_{2} - 2 \beta_{4} ) q^{77} + 2 \beta_{1} q^{79} -\beta_{5} q^{80} -\beta_{6} q^{82} + \beta_{7} q^{83} + 8 \beta_{1} q^{85} -2 \beta_{4} q^{86} + ( 2 - \beta_{1} ) q^{88} + 2 \beta_{6} q^{91} -2 \beta_{4} q^{92} + \beta_{3} q^{94} -8 \beta_{2} q^{95} -2 \beta_{6} q^{97} -7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{16} - 16q^{22} - 24q^{25} - 80q^{37} - 56q^{49} - 48q^{58} - 8q^{64} - 96q^{67} + 16q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{4} + 1 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{2} \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} - 7 \nu^{3} + 4 \nu \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu \)\()/12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 3 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - 5 \beta_{5} + 5 \beta_{3}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - 11 \beta_{5} - 11 \beta_{3}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{4} + 9 \beta_{2}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} + 7 \beta_{6} + 13 \beta_{5} - 13 \beta_{3}\)\()/8\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.28897 0.581861i
−0.581861 + 1.28897i
−1.28897 + 0.581861i
0.581861 1.28897i
0.581861 + 1.28897i
−1.28897 0.581861i
−0.581861 1.28897i
1.28897 + 0.581861i
1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
307.2 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
307.3 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.4 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.5 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.6 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.7 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
307.8 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
\(n\): 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.c 8
3.b odd 2 1 inner 1386.2.e.c 8
7.b odd 2 1 inner 1386.2.e.c 8
11.b odd 2 1 inner 1386.2.e.c 8
21.c even 2 1 inner 1386.2.e.c 8
33.d even 2 1 inner 1386.2.e.c 8
77.b even 2 1 inner 1386.2.e.c 8
231.h odd 2 1 inner 1386.2.e.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.e.c 8 1.a even 1 1 trivial
1386.2.e.c 8 3.b odd 2 1 inner
1386.2.e.c 8 7.b odd 2 1 inner
1386.2.e.c 8 11.b odd 2 1 inner
1386.2.e.c 8 21.c even 2 1 inner
1386.2.e.c 8 33.d even 2 1 inner
1386.2.e.c 8 77.b even 2 1 inner
1386.2.e.c 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{2} + 8 \)
\( T_{13}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( T^{8} \)
$5$ \( ( 8 + T^{2} )^{4} \)
$7$ \( ( 7 + T^{2} )^{4} \)
$11$ \( ( 121 - 6 T^{2} + T^{4} )^{2} \)
$13$ \( ( -32 + T^{2} )^{4} \)
$17$ \( ( -56 + T^{2} )^{4} \)
$19$ \( ( -8 + T^{2} )^{4} \)
$23$ \( ( -28 + T^{2} )^{4} \)
$29$ \( ( 36 + T^{2} )^{4} \)
$31$ \( ( 56 + T^{2} )^{4} \)
$37$ \( ( 10 + T )^{8} \)
$41$ \( ( -56 + T^{2} )^{4} \)
$43$ \( ( 28 + T^{2} )^{4} \)
$47$ \( ( 8 + T^{2} )^{4} \)
$53$ \( ( -112 + T^{2} )^{4} \)
$59$ \( ( 128 + T^{2} )^{4} \)
$61$ \( T^{8} \)
$67$ \( ( 12 + T )^{8} \)
$71$ \( ( -28 + T^{2} )^{4} \)
$73$ \( ( -72 + T^{2} )^{4} \)
$79$ \( ( 28 + T^{2} )^{4} \)
$83$ \( ( -56 + T^{2} )^{4} \)
$89$ \( T^{8} \)
$97$ \( ( 224 + T^{2} )^{4} \)
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