Properties

Label 2368.2.a.ba
Level $2368$
Weight $2$
Character orbit 2368.a
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + \beta q^{5} + (2 \beta - 2) q^{7} + (3 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + \beta q^{5} + (2 \beta - 2) q^{7} + (3 \beta + 1) q^{9} - \beta q^{11} + ( - \beta + 1) q^{13} + (2 \beta + 3) q^{15} - 6 q^{17} + 2 q^{19} + (2 \beta + 4) q^{21} + ( - 3 \beta + 3) q^{23} + (\beta - 2) q^{25} + (4 \beta + 7) q^{27} + (3 \beta - 3) q^{29} + (\beta - 2) q^{31} + ( - 2 \beta - 3) q^{33} + 6 q^{35} - q^{37} + ( - \beta - 2) q^{39} + (3 \beta + 3) q^{41} + (2 \beta - 4) q^{43} + (4 \beta + 9) q^{45} - 2 \beta q^{47} + ( - 4 \beta + 9) q^{49} + ( - 6 \beta - 6) q^{51} + 6 q^{53} + ( - \beta - 3) q^{55} + (2 \beta + 2) q^{57} + (2 \beta + 6) q^{59} + ( - 5 \beta + 4) q^{61} + (2 \beta + 16) q^{63} - 3 q^{65} + ( - 5 \beta + 8) q^{67} + ( - 3 \beta - 6) q^{69} - 6 q^{71} + ( - \beta - 10) q^{73} + q^{75} - 6 q^{77} + ( - 7 \beta + 7) q^{79} + (6 \beta + 16) q^{81} + ( - 4 \beta + 12) q^{83} - 6 \beta q^{85} + (3 \beta + 6) q^{87} - 4 \beta q^{89} + (2 \beta - 8) q^{91} + q^{93} + 2 \beta q^{95} + ( - 8 \beta + 2) q^{97} + ( - 4 \beta - 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9} - q^{11} + q^{13} + 8 q^{15} - 12 q^{17} + 4 q^{19} + 10 q^{21} + 3 q^{23} - 3 q^{25} + 18 q^{27} - 3 q^{29} - 3 q^{31} - 8 q^{33} + 12 q^{35} - 2 q^{37} - 5 q^{39} + 9 q^{41} - 6 q^{43} + 22 q^{45} - 2 q^{47} + 14 q^{49} - 18 q^{51} + 12 q^{53} - 7 q^{55} + 6 q^{57} + 14 q^{59} + 3 q^{61} + 34 q^{63} - 6 q^{65} + 11 q^{67} - 15 q^{69} - 12 q^{71} - 21 q^{73} + 2 q^{75} - 12 q^{77} + 7 q^{79} + 38 q^{81} + 20 q^{83} - 6 q^{85} + 15 q^{87} - 4 q^{89} - 14 q^{91} + 2 q^{93} + 2 q^{95} - 4 q^{97} - 22 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
0 −0.302776 0 −1.30278 0 −4.60555 0 −2.90833 0
1.2 0 3.30278 0 2.30278 0 2.60555 0 7.90833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(37\) \(1\)

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 2368.2.a.ba 2
4.b odd 2 1 2368.2.a.s 2
8.b even 2 1 592.2.a.f 2
8.d odd 2 1 74.2.a.a 2
24.f even 2 1 666.2.a.j 2
24.h odd 2 1 5328.2.a.bf 2
40.e odd 2 1 1850.2.a.u 2
40.k even 4 2 1850.2.b.i 4
56.e even 2 1 3626.2.a.a 2
88.g even 2 1 8954.2.a.p 2
296.h odd 2 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 8.d odd 2 1
592.2.a.f 2 8.b even 2 1
666.2.a.j 2 24.f even 2 1
1850.2.a.u 2 40.e odd 2 1
1850.2.b.i 4 40.k even 4 2
2368.2.a.s 2 4.b odd 2 1
2368.2.a.ba 2 1.a even 1 1 trivial
2738.2.a.l 2 296.h odd 2 1
3626.2.a.a 2 56.e even 2 1
5328.2.a.bf 2 24.h odd 2 1
8954.2.a.p 2 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2368))\):

\( T_{3}^{2} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T - 79 \) Copy content Toggle raw display
$67$ \( T^{2} - 11T - 51 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 21T + 107 \) Copy content Toggle raw display
$79$ \( T^{2} - 7T - 147 \) Copy content Toggle raw display
$83$ \( T^{2} - 20T + 48 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 204 \) Copy content Toggle raw display
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