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: \( x^{2} - x - 14 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4368))\):

\( T_{5}^{2} + T_{5} - 14 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} + 7T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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