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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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: 8.0.6679465984.1
Defining polynomial: \(x^{8} + 14 x^{6} + 61 x^{4} + 88 x^{2} + 36\)
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

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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + \beta_{1} q^{8} + ( 1 + \beta_{2} ) q^{10} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{11} + ( \beta_{6} + \beta_{7} ) q^{13} + ( 1 + \beta_{7} ) q^{14} + q^{16} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{17} + ( 1 + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} ) q^{22} + ( -2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{23} + ( -3 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{25} + ( -\beta_{4} + \beta_{5} ) q^{26} + ( -\beta_{1} - \beta_{4} ) q^{28} + ( -\beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( -3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{31} -\beta_{1} q^{32} + ( -\beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{34} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{35} + ( 2 - 2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{38} + ( -1 - \beta_{2} ) q^{40} + ( 3 + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{44} + ( 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{46} + ( -\beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{47} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{49} + ( 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{50} + ( -\beta_{6} - \beta_{7} ) q^{52} + ( -3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -6 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( -1 - \beta_{7} ) q^{56} + ( \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{58} + ( -4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 6 - 2 \beta_{2} - \beta_{6} - \beta_{7} ) q^{61} + ( -3 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{62} - q^{64} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{65} + ( 2 - \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{68} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{70} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{74} + ( -1 - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{76} + ( 1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{77} + ( -6 \beta_{1} - 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{79} + ( \beta_{1} + \beta_{3} ) q^{80} + ( -3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{82} + ( 1 + 3 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -4 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{85} + ( 4 + 2 \beta_{2} ) q^{86} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} ) q^{88} + ( 6 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( 3 + 7 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{91} + ( 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{92} + ( -1 + 3 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{94} + ( 2 \beta_{1} + 4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{95} + ( -\beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{97} + ( 3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{10} + 8q^{11} + 8q^{14} + 8q^{16} + 12q^{17} + 4q^{19} + 4q^{22} + 8q^{23} - 16q^{25} - 8q^{35} + 16q^{37} - 4q^{40} + 20q^{41} - 8q^{44} - 12q^{49} - 40q^{55} - 8q^{56} + 56q^{61} - 20q^{62} - 8q^{64} + 16q^{67} - 12q^{68} - 12q^{70} - 8q^{71} - 4q^{73} - 4q^{76} + 20q^{77} - 4q^{83} + 24q^{86} - 4q^{88} + 20q^{91} - 8q^{92} - 20q^{94} + 20q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 14 \nu^{5} + 55 \nu^{3} + 46 \nu \)\()/12\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 9 \nu^{2} + 12 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 12 \nu^{5} + 41 \nu^{3} + 38 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} - 11 \nu^{5} + 33 \nu^{4} - 28 \nu^{3} + 96 \nu^{2} + 8 \nu + 60 \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{6} + 11 \nu^{5} + 33 \nu^{4} + 28 \nu^{3} + 96 \nu^{2} - 8 \nu + 60 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} - 11 \nu^{5} + 39 \nu^{4} - 28 \nu^{3} + 138 \nu^{2} - 4 \nu + 96 \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{6} + 11 \nu^{5} + 39 \nu^{4} + 28 \nu^{3} + 138 \nu^{2} + 4 \nu + 96 \)\()/12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{7} + 7 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{7} + 9 \beta_{6} - 9 \beta_{5} - 9 \beta_{4} - 14 \beta_{2} + 30\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(45 \beta_{7} - 45 \beta_{6} - 31 \beta_{5} + 31 \beta_{4} - 18 \beta_{3} + 26 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-67 \beta_{7} - 67 \beta_{6} + 71 \beta_{5} + 71 \beta_{4} + 90 \beta_{2} - 178\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-291 \beta_{7} + 291 \beta_{6} + 205 \beta_{5} - 205 \beta_{4} + 142 \beta_{3} - 230 \beta_{1}\)\()/2\)

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

\( T_{5}^{8} + 28 T_{5}^{6} + 200 T_{5}^{4} + 240 T_{5}^{2} + 16 \)
\( T_{13}^{4} - 26 T_{13}^{2} + 56 T_{13} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( T^{8} \)
$5$ \( 16 + 240 T^{2} + 200 T^{4} + 28 T^{6} + T^{8} \)
$7$ \( 2401 + 294 T^{2} - 112 T^{3} + 50 T^{4} - 16 T^{5} + 6 T^{6} + T^{8} \)
$11$ \( 14641 - 10648 T + 4840 T^{2} - 1848 T^{3} + 606 T^{4} - 168 T^{5} + 40 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( ( -16 + 56 T - 26 T^{2} + T^{4} )^{2} \)
$17$ \( ( -24 + 80 T - 10 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$19$ \( ( -40 + 208 T - 58 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$23$ \( ( -424 + 288 T - 46 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$29$ \( 9216 + 5632 T^{2} + 976 T^{4} + 56 T^{6} + T^{8} \)
$31$ \( 6400 + 13504 T^{2} + 2436 T^{4} + 112 T^{6} + T^{8} \)
$37$ \( ( 432 + 448 T - 64 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$41$ \( ( -656 + 384 T - 22 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$43$ \( 73984 + 43264 T^{2} + 4448 T^{4} + 144 T^{6} + T^{8} \)
$47$ \( 1507984 + 423440 T^{2} + 16872 T^{4} + 228 T^{6} + T^{8} \)
$53$ \( ( 512 + 64 T - 182 T^{2} + T^{4} )^{2} \)
$59$ \( 11943936 + 993280 T^{2} + 25920 T^{4} + 272 T^{6} + T^{8} \)
$61$ \( ( -3232 + 96 T + 206 T^{2} - 28 T^{3} + T^{4} )^{2} \)
$67$ \( ( 2624 + 1248 T - 172 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$71$ \( ( -1416 - 1360 T - 222 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$73$ \( ( 64 - 22 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$79$ \( 20502784 + 1359104 T^{2} + 31524 T^{4} + 300 T^{6} + T^{8} \)
$83$ \( ( 4296 - 176 T - 146 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$89$ \( 147456 + 77824 T^{2} + 10512 T^{4} + 200 T^{6} + T^{8} \)
$97$ \( 1024 + 47616 T^{2} + 4688 T^{4} + 136 T^{6} + T^{8} \)
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