Properties

Label 4368.2.a.bh
Level $4368$
Weight $2$
Character orbit 4368.a
Self dual yes
Analytic conductor $34.879$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta q^{5} - q^{7} + q^{9} + ( -2 + \beta ) q^{11} - q^{13} -\beta q^{15} + ( -4 + \beta ) q^{17} + ( -2 + \beta ) q^{19} - q^{21} + ( 2 - \beta ) q^{23} + ( 9 + \beta ) q^{25} + q^{27} + ( 4 + \beta ) q^{29} -8 q^{31} + ( -2 + \beta ) q^{33} + \beta q^{35} -\beta q^{37} - q^{39} + ( -2 + 2 \beta ) q^{41} + ( -2 + \beta ) q^{43} -\beta q^{45} + q^{49} + ( -4 + \beta ) q^{51} + 10 q^{53} + ( -14 + \beta ) q^{55} + ( -2 + \beta ) q^{57} -8 q^{59} + ( -8 - \beta ) q^{61} - q^{63} + \beta q^{65} + ( -4 - 2 \beta ) q^{67} + ( 2 - \beta ) q^{69} -3 \beta q^{73} + ( 9 + \beta ) q^{75} + ( 2 - \beta ) q^{77} + ( -4 - 2 \beta ) q^{79} + q^{81} + ( -4 + 2 \beta ) q^{83} + ( -14 + 3 \beta ) q^{85} + ( 4 + \beta ) q^{87} -14 q^{89} + q^{91} -8 q^{93} + ( -14 + \beta ) q^{95} + ( -2 + 4 \beta ) q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - q^{15} - 7 q^{17} - 3 q^{19} - 2 q^{21} + 3 q^{23} + 19 q^{25} + 2 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} - q^{37} - 2 q^{39} - 2 q^{41} - 3 q^{43} - q^{45} + 2 q^{49} - 7 q^{51} + 20 q^{53} - 27 q^{55} - 3 q^{57} - 16 q^{59} - 17 q^{61} - 2 q^{63} + q^{65} - 10 q^{67} + 3 q^{69} - 3 q^{73} + 19 q^{75} + 3 q^{77} - 10 q^{79} + 2 q^{81} - 6 q^{83} - 25 q^{85} + 9 q^{87} - 28 q^{89} + 2 q^{91} - 16 q^{93} - 27 q^{95} - 3 q^{99} + O(q^{100}) \)

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
4.27492
−3.27492
0 1.00000 0 −4.27492 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 3.27492 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 4368.2.a.bh 2
4.b odd 2 1 546.2.a.h 2
12.b even 2 1 1638.2.a.y 2
28.d even 2 1 3822.2.a.bm 2
52.b odd 2 1 7098.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.h 2 4.b odd 2 1
1638.2.a.y 2 12.b even 2 1
3822.2.a.bm 2 28.d even 2 1
4368.2.a.bh 2 1.a even 1 1 trivial
7098.2.a.bu 2 52.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4368))\):

\( T_{5}^{2} + T_{5} - 14 \)
\( T_{11}^{2} + 3 T_{11} - 12 \)
\( T_{17}^{2} + 7 T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -14 + T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -12 + 3 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -2 + 7 T + T^{2} \)
$19$ \( -12 + 3 T + T^{2} \)
$23$ \( -12 - 3 T + T^{2} \)
$29$ \( 6 - 9 T + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( -14 + T + T^{2} \)
$41$ \( -56 + 2 T + T^{2} \)
$43$ \( -12 + 3 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( 58 + 17 T + T^{2} \)
$67$ \( -32 + 10 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -126 + 3 T + T^{2} \)
$79$ \( -32 + 10 T + T^{2} \)
$83$ \( -48 + 6 T + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( -228 + T^{2} \)
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