Properties

Label 1386.2.a.n
Level $1386$
Weight $2$
Character orbit 1386.a
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.16228
3.16228
−1.00000 0 1.00000 −3.16228 0 −1.00000 −1.00000 0 3.16228
1.2 −1.00000 0 1.00000 3.16228 0 −1.00000 −1.00000 0 −3.16228
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.a.n 2
3.b odd 2 1 1386.2.a.o yes 2
7.b odd 2 1 9702.2.a.cn 2
21.c even 2 1 9702.2.a.dc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.a.n 2 1.a even 1 1 trivial
1386.2.a.o yes 2 3.b odd 2 1
9702.2.a.cn 2 7.b odd 2 1
9702.2.a.dc 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5}^{2} - 10 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + 8 T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -10 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 6 + 8 T + T^{2} \)
$19$ \( -6 - 4 T + T^{2} \)
$23$ \( -40 + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( 26 - 12 T + T^{2} \)
$37$ \( -36 - 4 T + T^{2} \)
$41$ \( 6 - 8 T + T^{2} \)
$43$ \( -40 + T^{2} \)
$47$ \( -86 + 4 T + T^{2} \)
$53$ \( -36 - 4 T + T^{2} \)
$59$ \( 24 - 16 T + T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( -36 + 4 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( -74 - 8 T + T^{2} \)
$79$ \( -36 - 4 T + T^{2} \)
$83$ \( -6 - 4 T + T^{2} \)
$89$ \( 60 + 20 T + T^{2} \)
$97$ \( -156 - 4 T + T^{2} \)
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