Properties

Label 2415.2.a.j
Level $2415$
Weight $2$
Character orbit 2415.a
Self dual yes
Analytic conductor $19.284$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2838720881\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + q^{4} - q^{5} + \beta q^{6} - q^{7} -\beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + q^{4} - q^{5} + \beta q^{6} - q^{7} -\beta q^{8} + q^{9} -\beta q^{10} + ( 1 + \beta ) q^{11} + q^{12} + ( -3 - \beta ) q^{13} -\beta q^{14} - q^{15} -5 q^{16} + ( -4 - 2 \beta ) q^{17} + \beta q^{18} -2 \beta q^{19} - q^{20} - q^{21} + ( 3 + \beta ) q^{22} + q^{23} -\beta q^{24} + q^{25} + ( -3 - 3 \beta ) q^{26} + q^{27} - q^{28} -\beta q^{30} + ( -3 + \beta ) q^{31} -3 \beta q^{32} + ( 1 + \beta ) q^{33} + ( -6 - 4 \beta ) q^{34} + q^{35} + q^{36} + ( -3 + \beta ) q^{37} -6 q^{38} + ( -3 - \beta ) q^{39} + \beta q^{40} -2 \beta q^{41} -\beta q^{42} + 4 \beta q^{43} + ( 1 + \beta ) q^{44} - q^{45} + \beta q^{46} + ( 2 - 6 \beta ) q^{47} -5 q^{48} + q^{49} + \beta q^{50} + ( -4 - 2 \beta ) q^{51} + ( -3 - \beta ) q^{52} + ( -1 + \beta ) q^{53} + \beta q^{54} + ( -1 - \beta ) q^{55} + \beta q^{56} -2 \beta q^{57} + ( -3 + 3 \beta ) q^{59} - q^{60} + 4 q^{61} + ( 3 - 3 \beta ) q^{62} - q^{63} + q^{64} + ( 3 + \beta ) q^{65} + ( 3 + \beta ) q^{66} + ( -8 - 4 \beta ) q^{67} + ( -4 - 2 \beta ) q^{68} + q^{69} + \beta q^{70} -6 q^{71} -\beta q^{72} + ( -5 + 5 \beta ) q^{73} + ( 3 - 3 \beta ) q^{74} + q^{75} -2 \beta q^{76} + ( -1 - \beta ) q^{77} + ( -3 - 3 \beta ) q^{78} + ( 5 + 3 \beta ) q^{79} + 5 q^{80} + q^{81} -6 q^{82} -2 q^{83} - q^{84} + ( 4 + 2 \beta ) q^{85} + 12 q^{86} + ( -3 - \beta ) q^{88} + ( 10 - 2 \beta ) q^{89} -\beta q^{90} + ( 3 + \beta ) q^{91} + q^{92} + ( -3 + \beta ) q^{93} + ( -18 + 2 \beta ) q^{94} + 2 \beta q^{95} -3 \beta q^{96} + ( -4 + 4 \beta ) q^{97} + \beta q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{15} - 10 q^{16} - 8 q^{17} - 2 q^{20} - 2 q^{21} + 6 q^{22} + 2 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 2 q^{28} - 6 q^{31} + 2 q^{33} - 12 q^{34} + 2 q^{35} + 2 q^{36} - 6 q^{37} - 12 q^{38} - 6 q^{39} + 2 q^{44} - 2 q^{45} + 4 q^{47} - 10 q^{48} + 2 q^{49} - 8 q^{51} - 6 q^{52} - 2 q^{53} - 2 q^{55} - 6 q^{59} - 2 q^{60} + 8 q^{61} + 6 q^{62} - 2 q^{63} + 2 q^{64} + 6 q^{65} + 6 q^{66} - 16 q^{67} - 8 q^{68} + 2 q^{69} - 12 q^{71} - 10 q^{73} + 6 q^{74} + 2 q^{75} - 2 q^{77} - 6 q^{78} + 10 q^{79} + 10 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{83} - 2 q^{84} + 8 q^{85} + 24 q^{86} - 6 q^{88} + 20 q^{89} + 6 q^{91} + 2 q^{92} - 6 q^{93} - 36 q^{94} - 8 q^{97} + 2 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 1.00000 1.00000 −1.00000 −1.73205 −1.00000 1.73205 1.00000 1.73205
1.2 1.73205 1.00000 1.00000 −1.00000 1.73205 −1.00000 −1.73205 1.00000 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

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 2415.2.a.j 2
3.b odd 2 1 7245.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.j 2 1.a even 1 1 trivial
7245.2.a.w 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{2} - 3 \)
\( T_{11}^{2} - 2 T_{11} - 2 \)
\( T_{13}^{2} + 6 T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -2 - 2 T + T^{2} \)
$13$ \( 6 + 6 T + T^{2} \)
$17$ \( 4 + 8 T + T^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( 6 + 6 T + T^{2} \)
$37$ \( 6 + 6 T + T^{2} \)
$41$ \( -12 + T^{2} \)
$43$ \( -48 + T^{2} \)
$47$ \( -104 - 4 T + T^{2} \)
$53$ \( -2 + 2 T + T^{2} \)
$59$ \( -18 + 6 T + T^{2} \)
$61$ \( ( -4 + T )^{2} \)
$67$ \( 16 + 16 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( -50 + 10 T + T^{2} \)
$79$ \( -2 - 10 T + T^{2} \)
$83$ \( ( 2 + T )^{2} \)
$89$ \( 88 - 20 T + T^{2} \)
$97$ \( -32 + 8 T + T^{2} \)
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