Properties

Label 1386.2.e.b
Level $1386$
Weight $2$
Character orbit 1386.e
Analytic conductor $11.067$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: 8.0.12745506816.1
Defining polynomial: \(x^{8} + 23 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{4} -\beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} + \beta_{7} ) q^{7} + \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{4} -\beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} + \beta_{7} ) q^{7} + \beta_{1} q^{8} + \beta_{3} q^{10} + ( -1 - \beta_{1} - \beta_{5} ) q^{11} -\beta_{7} q^{13} + ( -1 + \beta_{2} + \beta_{6} ) q^{14} + q^{16} + ( \beta_{3} - \beta_{7} ) q^{17} + ( -\beta_{3} - 2 \beta_{7} ) q^{19} + \beta_{2} q^{20} -\beta_{4} q^{22} -4 q^{23} + ( 1 - \beta_{4} + \beta_{5} ) q^{25} -\beta_{6} q^{26} + ( \beta_{1} - \beta_{3} - \beta_{7} ) q^{28} + ( -2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{29} + ( -3 \beta_{2} - \beta_{6} ) q^{31} -\beta_{1} q^{32} + ( \beta_{2} - \beta_{6} ) q^{34} + ( 4 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( -2 + \beta_{4} - \beta_{5} ) q^{37} + ( -\beta_{2} - 2 \beta_{6} ) q^{38} -\beta_{3} q^{40} + ( 3 \beta_{3} - \beta_{7} ) q^{41} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{43} + ( 1 + \beta_{1} + \beta_{5} ) q^{44} + 4 \beta_{1} q^{46} + ( 3 \beta_{2} - \beta_{6} ) q^{47} + ( 5 + 2 \beta_{2} + 2 \beta_{6} ) q^{49} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{50} + \beta_{7} q^{52} + ( 6 + \beta_{4} - \beta_{5} ) q^{53} + ( -2 \beta_{2} + 3 \beta_{3} - \beta_{6} + \beta_{7} ) q^{55} + ( 1 - \beta_{2} - \beta_{6} ) q^{56} + ( -4 + \beta_{4} - \beta_{5} ) q^{58} -\beta_{6} q^{59} -3 \beta_{7} q^{61} + ( 3 \beta_{3} + \beta_{7} ) q^{62} - q^{64} + 2 \beta_{1} q^{65} + ( -2 - \beta_{4} + \beta_{5} ) q^{67} + ( -\beta_{3} + \beta_{7} ) q^{68} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{70} -2 q^{71} + ( 5 \beta_{3} - \beta_{7} ) q^{73} + ( -\beta_{4} - \beta_{5} ) q^{74} + ( \beta_{3} + 2 \beta_{7} ) q^{76} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{77} + 4 \beta_{1} q^{79} -\beta_{2} q^{80} + ( 3 \beta_{2} - \beta_{6} ) q^{82} + ( -5 \beta_{3} - 4 \beta_{7} ) q^{83} + ( 8 \beta_{1} + \beta_{4} + \beta_{5} ) q^{85} + ( -4 + 2 \beta_{4} - 2 \beta_{5} ) q^{86} + \beta_{4} q^{88} + ( -4 \beta_{2} - 4 \beta_{6} ) q^{89} + ( -4 + \beta_{4} - \beta_{5} - \beta_{6} ) q^{91} + 4 q^{92} + ( -3 \beta_{3} + \beta_{7} ) q^{94} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{95} + ( -4 \beta_{2} - 6 \beta_{6} ) q^{97} + ( -5 \beta_{1} - 2 \beta_{3} - 2 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} - 4q^{11} - 8q^{14} + 8q^{16} - 4q^{22} - 32q^{23} - 8q^{37} + 4q^{44} + 40q^{49} + 56q^{53} + 8q^{56} - 24q^{58} - 8q^{64} - 24q^{67} + 24q^{70} - 16q^{71} - 4q^{77} - 16q^{86} + 4q^{88} - 24q^{91} + 32q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 23 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 24 \nu^{2} \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 24 \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 24 \nu^{3} - 5 \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{6} - \nu^{4} - 67 \nu^{2} - 9 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} + \nu^{4} - 67 \nu^{2} + 9 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{7} - \nu^{5} - 115 \nu^{3} - 24 \nu \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + \nu^{5} - 115 \nu^{3} + 24 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 6 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + 5 \beta_{3} + 5 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{5} - 5 \beta_{4} - 18\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - 5 \beta_{6} + 24 \beta_{3} - 24 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(-12 \beta_{5} - 12 \beta_{4} - 67 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(-24 \beta_{7} - 24 \beta_{6} - 115 \beta_{3} - 115 \beta_{2}\)\()/2\)

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
1.54779 + 1.54779i
0.323042 + 0.323042i
−0.323042 0.323042i
−1.54779 1.54779i
−1.54779 + 1.54779i
−0.323042 + 0.323042i
0.323042 0.323042i
1.54779 1.54779i
1.00000i 0 −1.00000 3.09557i 0 −2.44949 1.00000i 1.00000i 0 −3.09557
307.2 1.00000i 0 −1.00000 0.646084i 0 2.44949 1.00000i 1.00000i 0 −0.646084
307.3 1.00000i 0 −1.00000 0.646084i 0 −2.44949 1.00000i 1.00000i 0 0.646084
307.4 1.00000i 0 −1.00000 3.09557i 0 2.44949 1.00000i 1.00000i 0 3.09557
307.5 1.00000i 0 −1.00000 3.09557i 0 2.44949 + 1.00000i 1.00000i 0 3.09557
307.6 1.00000i 0 −1.00000 0.646084i 0 −2.44949 + 1.00000i 1.00000i 0 0.646084
307.7 1.00000i 0 −1.00000 0.646084i 0 2.44949 + 1.00000i 1.00000i 0 −0.646084
307.8 1.00000i 0 −1.00000 3.09557i 0 −2.44949 + 1.00000i 1.00000i 0 −3.09557
\(n\): 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
7.b odd 2 1 inner
11.b odd 2 1 inner
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.b 8
3.b odd 2 1 154.2.c.a 8
7.b odd 2 1 inner 1386.2.e.b 8
11.b odd 2 1 inner 1386.2.e.b 8
12.b even 2 1 1232.2.e.e 8
21.c even 2 1 154.2.c.a 8
21.g even 6 2 1078.2.i.b 16
21.h odd 6 2 1078.2.i.b 16
33.d even 2 1 154.2.c.a 8
77.b even 2 1 inner 1386.2.e.b 8
84.h odd 2 1 1232.2.e.e 8
132.d odd 2 1 1232.2.e.e 8
231.h odd 2 1 154.2.c.a 8
231.k odd 6 2 1078.2.i.b 16
231.l even 6 2 1078.2.i.b 16
924.n even 2 1 1232.2.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.c.a 8 3.b odd 2 1
154.2.c.a 8 21.c even 2 1
154.2.c.a 8 33.d even 2 1
154.2.c.a 8 231.h odd 2 1
1078.2.i.b 16 21.g even 6 2
1078.2.i.b 16 21.h odd 6 2
1078.2.i.b 16 231.k odd 6 2
1078.2.i.b 16 231.l even 6 2
1232.2.e.e 8 12.b even 2 1
1232.2.e.e 8 84.h odd 2 1
1232.2.e.e 8 132.d odd 2 1
1232.2.e.e 8 924.n even 2 1
1386.2.e.b 8 1.a even 1 1 trivial
1386.2.e.b 8 7.b odd 2 1 inner
1386.2.e.b 8 11.b odd 2 1 inner
1386.2.e.b 8 77.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 10 T_{5}^{2} + 4 \)
\( T_{13}^{4} - 10 T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( T^{8} \)
$5$ \( ( 4 + 10 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 - 10 T^{2} + T^{4} )^{2} \)
$11$ \( ( 121 + 22 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( ( 4 - 10 T^{2} + T^{4} )^{2} \)
$17$ \( ( -14 + T^{2} )^{4} \)
$19$ \( ( 100 - 34 T^{2} + T^{4} )^{2} \)
$23$ \( ( 4 + T )^{8} \)
$29$ \( ( 144 + 60 T^{2} + T^{4} )^{2} \)
$31$ \( ( 100 + 76 T^{2} + T^{4} )^{2} \)
$37$ \( ( -20 + 2 T + T^{2} )^{4} \)
$41$ \( ( 2500 - 124 T^{2} + T^{4} )^{2} \)
$43$ \( ( 6400 + 176 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2500 + 124 T^{2} + T^{4} )^{2} \)
$53$ \( ( 28 - 14 T + T^{2} )^{4} \)
$59$ \( ( 4 + 10 T^{2} + T^{4} )^{2} \)
$61$ \( ( 324 - 90 T^{2} + T^{4} )^{2} \)
$67$ \( ( -12 + 6 T + T^{2} )^{4} \)
$71$ \( ( 2 + T )^{8} \)
$73$ \( ( 10404 - 300 T^{2} + T^{4} )^{2} \)
$79$ \( ( 16 + T^{2} )^{4} \)
$83$ \( ( 13924 - 250 T^{2} + T^{4} )^{2} \)
$89$ \( ( 96 + T^{2} )^{4} \)
$97$ \( ( 18496 + 328 T^{2} + T^{4} )^{2} \)
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