Properties

Label 2310.2.a.x
Level $2310$
Weight $2$
Character orbit 2310.a
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.4454428669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - q^{11} - q^{12} + ( 1 + \beta ) q^{13} + q^{14} - q^{15} + q^{16} + ( -1 - \beta ) q^{17} - q^{18} + q^{20} + q^{21} + q^{22} + ( -1 + \beta ) q^{23} + q^{24} + q^{25} + ( -1 - \beta ) q^{26} - q^{27} - q^{28} + 2 q^{29} + q^{30} - q^{32} + q^{33} + ( 1 + \beta ) q^{34} - q^{35} + q^{36} + ( 3 - \beta ) q^{37} + ( -1 - \beta ) q^{39} - q^{40} + 6 q^{41} - q^{42} + 4 q^{43} - q^{44} + q^{45} + ( 1 - \beta ) q^{46} -4 q^{47} - q^{48} + q^{49} - q^{50} + ( 1 + \beta ) q^{51} + ( 1 + \beta ) q^{52} -6 q^{53} + q^{54} - q^{55} + q^{56} -2 q^{58} + 4 q^{59} - q^{60} + 6 q^{61} - q^{63} + q^{64} + ( 1 + \beta ) q^{65} - q^{66} + ( -7 - \beta ) q^{67} + ( -1 - \beta ) q^{68} + ( 1 - \beta ) q^{69} + q^{70} - q^{72} + 6 q^{73} + ( -3 + \beta ) q^{74} - q^{75} + q^{77} + ( 1 + \beta ) q^{78} + 12 q^{79} + q^{80} + q^{81} -6 q^{82} -4 q^{83} + q^{84} + ( -1 - \beta ) q^{85} -4 q^{86} -2 q^{87} + q^{88} + ( 1 + \beta ) q^{89} - q^{90} + ( -1 - \beta ) q^{91} + ( -1 + \beta ) q^{92} + 4 q^{94} + q^{96} + ( 4 + 2 \beta ) q^{97} - q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{28} + 4 q^{29} + 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} - 2 q^{35} + 2 q^{36} + 6 q^{37} - 2 q^{39} - 2 q^{40} + 12 q^{41} - 2 q^{42} + 8 q^{43} - 2 q^{44} + 2 q^{45} + 2 q^{46} - 8 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 2 q^{51} + 2 q^{52} - 12 q^{53} + 2 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{58} + 8 q^{59} - 2 q^{60} + 12 q^{61} - 2 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} - 14 q^{67} - 2 q^{68} + 2 q^{69} + 2 q^{70} - 2 q^{72} + 12 q^{73} - 6 q^{74} - 2 q^{75} + 2 q^{77} + 2 q^{78} + 24 q^{79} + 2 q^{80} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 2 q^{84} - 2 q^{85} - 8 q^{86} - 4 q^{87} + 2 q^{88} + 2 q^{89} - 2 q^{90} - 2 q^{91} - 2 q^{92} + 8 q^{94} + 2 q^{96} + 8 q^{97} - 2 q^{98} - 2 q^{99} + O(q^{100}) \)

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
−2.37228
3.37228
−1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 2310.2.a.x 2
3.b odd 2 1 6930.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.x 2 1.a even 1 1 trivial
6930.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2310))\):

\( T_{13}^{2} - 2 T_{13} - 32 \)
\( T_{17}^{2} + 2 T_{17} - 32 \)
\( T_{19} \)
\( T_{23}^{2} + 2 T_{23} - 32 \)
\( T_{29} - 2 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 - 2 T + T^{2} \)
$17$ \( -32 + 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( -24 - 6 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 16 + 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( -32 - 2 T + T^{2} \)
$97$ \( -116 - 8 T + T^{2} \)
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