Properties

Label 4368.2.a.be
Level $4368$
Weight $2$
Character orbit 4368.a
Self dual yes
Analytic conductor $34.879$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + (\beta + 1) q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta + 1) q^{5} + q^{7} + q^{9} + (\beta - 1) q^{11} + q^{13} + ( - \beta - 1) q^{15} + ( - 3 \beta + 1) q^{17} + (3 \beta - 3) q^{19} - q^{21} + (\beta + 3) q^{23} + 3 \beta q^{25} - q^{27} + (3 \beta - 1) q^{29} + ( - 4 \beta + 4) q^{31} + ( - \beta + 1) q^{33} + (\beta + 1) q^{35} + ( - \beta - 5) q^{37} - q^{39} + ( - 2 \beta + 4) q^{41} + ( - \beta + 9) q^{43} + (\beta + 1) q^{45} + q^{49} + (3 \beta - 1) q^{51} + (4 \beta + 2) q^{53} + (\beta + 3) q^{55} + ( - 3 \beta + 3) q^{57} + ( - 4 \beta + 8) q^{59} + (3 \beta - 1) q^{61} + q^{63} + (\beta + 1) q^{65} + (2 \beta + 2) q^{67} + ( - \beta - 3) q^{69} - 8 q^{71} + ( - \beta - 1) q^{73} - 3 \beta q^{75} + (\beta - 1) q^{77} + (2 \beta + 6) q^{79} + q^{81} + (2 \beta - 14) q^{83} + ( - 5 \beta - 11) q^{85} + ( - 3 \beta + 1) q^{87} + 10 q^{89} + q^{91} + (4 \beta - 4) q^{93} + (3 \beta + 9) q^{95} + (8 \beta - 6) q^{97} + (\beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{15} - q^{17} - 3 q^{19} - 2 q^{21} + 7 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + 4 q^{31} + q^{33} + 3 q^{35} - 11 q^{37} - 2 q^{39} + 6 q^{41} + 17 q^{43} + 3 q^{45} + 2 q^{49} + q^{51} + 8 q^{53} + 7 q^{55} + 3 q^{57} + 12 q^{59} + q^{61} + 2 q^{63} + 3 q^{65} + 6 q^{67} - 7 q^{69} - 16 q^{71} - 3 q^{73} - 3 q^{75} - q^{77} + 14 q^{79} + 2 q^{81} - 26 q^{83} - 27 q^{85} - q^{87} + 20 q^{89} + 2 q^{91} - 4 q^{93} + 21 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
0 −1.00000 0 −0.561553 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 3.56155 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.be 2
4.b odd 2 1 546.2.a.j 2
12.b even 2 1 1638.2.a.u 2
28.d even 2 1 3822.2.a.bo 2
52.b odd 2 1 7098.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.j 2 4.b odd 2 1
1638.2.a.u 2 12.b even 2 1
3822.2.a.bo 2 28.d even 2 1
4368.2.a.be 2 1.a even 1 1 trivial
7098.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4368))\):

\( T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + T_{17} - 38 \) 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
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 17T + 68 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$61$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 26T + 152 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 268 \) Copy content Toggle raw display
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