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: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8} + \beta_{3} q^{10} + ( - \beta_{5} - \beta_1 - 1) q^{11} - \beta_{7} q^{13} + (\beta_{6} + \beta_{2} - 1) q^{14} + q^{16} + ( - \beta_{7} + \beta_{3}) q^{17} + ( - 2 \beta_{7} - \beta_{3}) q^{19} + \beta_{2} q^{20} - \beta_{4} q^{22} - 4 q^{23} + (\beta_{5} - \beta_{4} + 1) q^{25} - \beta_{6} q^{26} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{28} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{29} + ( - \beta_{6} - 3 \beta_{2}) q^{31} - \beta_1 q^{32} + ( - \beta_{6} + \beta_{2}) q^{34} + (\beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_1) q^{35} + ( - \beta_{5} + \beta_{4} - 2) q^{37} + ( - 2 \beta_{6} - \beta_{2}) q^{38} - \beta_{3} q^{40} + ( - \beta_{7} + 3 \beta_{3}) q^{41} + (2 \beta_{5} + 2 \beta_{4}) q^{43} + (\beta_{5} + \beta_1 + 1) q^{44} + 4 \beta_1 q^{46} + ( - \beta_{6} + 3 \beta_{2}) q^{47} + (2 \beta_{6} + 2 \beta_{2} + 5) q^{49} + (\beta_{5} + \beta_{4} + \beta_1) q^{50} + \beta_{7} q^{52} + ( - \beta_{5} + \beta_{4} + 6) q^{53} + (\beta_{7} - \beta_{6} + 3 \beta_{3} - 2 \beta_{2}) q^{55} + ( - \beta_{6} - \beta_{2} + 1) q^{56} + ( - \beta_{5} + \beta_{4} - 4) q^{58} - \beta_{6} q^{59} - 3 \beta_{7} q^{61} + (\beta_{7} + 3 \beta_{3}) q^{62} - q^{64} + 2 \beta_1 q^{65} + (\beta_{5} - \beta_{4} - 2) q^{67} + (\beta_{7} - \beta_{3}) q^{68} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{70} - 2 q^{71} + ( - \beta_{7} + 5 \beta_{3}) q^{73} + ( - \beta_{5} - \beta_{4}) q^{74} + (2 \beta_{7} + \beta_{3}) q^{76} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{77} + 4 \beta_1 q^{79} - \beta_{2} q^{80} + ( - \beta_{6} + 3 \beta_{2}) q^{82} + ( - 4 \beta_{7} - 5 \beta_{3}) q^{83} + (\beta_{5} + \beta_{4} + 8 \beta_1) q^{85} + ( - 2 \beta_{5} + 2 \beta_{4} - 4) q^{86} + \beta_{4} q^{88} + ( - 4 \beta_{6} - 4 \beta_{2}) q^{89} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 4) q^{91} + 4 q^{92} + (\beta_{7} - 3 \beta_{3}) q^{94} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{95} + ( - 6 \beta_{6} - 4 \beta_{2}) q^{97} + ( - 2 \beta_{7} - 2 \beta_{3} - 5 \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 \( 8 q - 8 q^{4} - 4 q^{11} - 8 q^{14} + 8 q^{16} - 4 q^{22} - 32 q^{23} - 8 q^{37} + 4 q^{44} + 40 q^{49} + 56 q^{53} + 8 q^{56} - 24 q^{58} - 8 q^{64} - 24 q^{67} + 24 q^{70} - 16 q^{71} - 4 q^{77} - 16 q^{86} + 4 q^{88} - 24 q^{91} + 32 q^{92}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 24\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 24\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 24\nu^{3} - 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} - \nu^{4} - 67\nu^{2} - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + \nu^{4} - 67\nu^{2} + 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - \nu^{5} - 115\nu^{3} - 24\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + \nu^{5} - 115\nu^{3} + 24\nu ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + 5\beta_{3} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{5} - 5\beta_{4} - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 5\beta_{6} + 24\beta_{3} - 24\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{5} - 12\beta_{4} - 67\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -24\beta_{7} - 24\beta_{6} - 115\beta_{3} - 115\beta_{2} ) / 2 \) Copy content Toggle raw display

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

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