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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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: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
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

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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{4} -\beta_{4} q^{5} + ( \beta_{2} - \beta_{7} ) q^{7} -\beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{4} -\beta_{4} q^{5} + ( \beta_{2} - \beta_{7} ) q^{7} -\beta_{1} q^{8} -\beta_{7} q^{10} + ( -\beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{11} -\beta_{7} q^{13} + ( 2 \beta_{4} - \beta_{5} ) q^{14} + q^{16} + 2 \beta_{3} q^{17} + 2 \beta_{7} q^{19} + \beta_{4} q^{20} + ( 1 - 2 \beta_{2} - \beta_{7} ) q^{22} + ( -\beta_{4} + 2 \beta_{5} ) q^{23} + 3 q^{25} + \beta_{4} q^{26} + ( -\beta_{2} + \beta_{7} ) q^{28} -6 \beta_{1} q^{29} + 4 \beta_{6} q^{31} + \beta_{1} q^{32} + 2 \beta_{6} q^{34} + ( -3 \beta_{1} + \beta_{3} ) q^{35} + 2 q^{37} -2 \beta_{4} q^{38} + \beta_{7} q^{40} + 4 \beta_{3} q^{41} + ( -4 \beta_{2} - 2 \beta_{7} ) q^{43} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{44} + ( 2 \beta_{2} + \beta_{7} ) q^{46} -5 \beta_{4} q^{47} + ( 2 + 3 \beta_{6} ) q^{49} + 3 \beta_{1} q^{50} + \beta_{7} q^{52} + ( \beta_{4} - 2 \beta_{5} ) q^{53} + ( 2 \beta_{6} + \beta_{7} ) q^{55} + ( -2 \beta_{4} + \beta_{5} ) q^{56} + 6 q^{58} -2 \beta_{4} q^{59} -9 \beta_{7} q^{61} -4 \beta_{3} q^{62} - q^{64} -2 \beta_{1} q^{65} + 12 q^{67} -2 \beta_{3} q^{68} + ( 3 + \beta_{6} ) q^{70} + ( \beta_{4} - 2 \beta_{5} ) q^{71} + 2 \beta_{1} q^{74} -2 \beta_{7} q^{76} + ( -5 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{77} + ( 2 \beta_{2} + \beta_{7} ) q^{79} -\beta_{4} q^{80} + 4 \beta_{6} q^{82} + 2 \beta_{3} q^{83} + ( -4 \beta_{2} - 2 \beta_{7} ) q^{85} + ( -2 \beta_{4} + 4 \beta_{5} ) q^{86} + ( -1 + 2 \beta_{2} + \beta_{7} ) q^{88} + 12 \beta_{4} q^{89} + ( 3 + \beta_{6} ) q^{91} + ( \beta_{4} - 2 \beta_{5} ) q^{92} -5 \beta_{7} q^{94} + 4 \beta_{1} q^{95} + 2 \beta_{6} q^{97} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{16} + 8q^{22} + 24q^{25} + 16q^{37} + 16q^{49} + 48q^{58} - 8q^{64} + 96q^{67} + 24q^{70} - 8q^{88} + 24q^{91} + O(q^{100}) \)

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

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

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
1.14412 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

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

Hecke characteristic polynomials

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