Properties

Label 1386.2.a.m
Level $1386$
Weight $2$
Character orbit 1386.a
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.00000 0 1.00000 −3.23607 0 1.00000 −1.00000 0 3.23607
1.2 −1.00000 0 1.00000 1.23607 0 1.00000 −1.00000 0 −1.23607
\(n\): 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.m 2
3.b odd 2 1 154.2.a.d 2
7.b odd 2 1 9702.2.a.cu 2
12.b even 2 1 1232.2.a.p 2
15.d odd 2 1 3850.2.a.bj 2
15.e even 4 2 3850.2.c.q 4
21.c even 2 1 1078.2.a.w 2
21.g even 6 2 1078.2.e.n 4
21.h odd 6 2 1078.2.e.q 4
24.f even 2 1 4928.2.a.bk 2
24.h odd 2 1 4928.2.a.bt 2
33.d even 2 1 1694.2.a.l 2
84.h odd 2 1 8624.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 3.b odd 2 1
1078.2.a.w 2 21.c even 2 1
1078.2.e.n 4 21.g even 6 2
1078.2.e.q 4 21.h odd 6 2
1232.2.a.p 2 12.b even 2 1
1386.2.a.m 2 1.a even 1 1 trivial
1694.2.a.l 2 33.d even 2 1
3850.2.a.bj 2 15.d odd 2 1
3850.2.c.q 4 15.e even 4 2
4928.2.a.bk 2 24.f even 2 1
4928.2.a.bt 2 24.h odd 2 1
8624.2.a.bf 2 84.h odd 2 1
9702.2.a.cu 2 7.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)
\( T_{17}^{2} - 4 T_{17} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -4 + 2 T + T^{2} \)
$17$ \( -16 - 4 T + T^{2} \)
$19$ \( 20 + 10 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( -76 + 4 T + T^{2} \)
$41$ \( -16 + 4 T + T^{2} \)
$43$ \( 16 + 12 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( 20 + 10 T + T^{2} \)
$61$ \( 4 + 6 T + T^{2} \)
$67$ \( -176 + 4 T + T^{2} \)
$71$ \( -16 + 4 T + T^{2} \)
$73$ \( -64 - 8 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -124 - 2 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 44 - 16 T + T^{2} \)
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