Properties

Label 2310.2.a.z
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + \beta q^{13} + q^{14} + q^{15} + q^{16} + \beta q^{17} - q^{18} + q^{20} - q^{21} - q^{22} + ( 2 + \beta ) q^{23} - q^{24} + q^{25} -\beta q^{26} + q^{27} - q^{28} + ( 2 - 2 \beta ) q^{29} - q^{30} + ( 4 - 2 \beta ) q^{31} - q^{32} + q^{33} -\beta q^{34} - q^{35} + q^{36} -\beta q^{37} + \beta q^{39} - q^{40} + 2 q^{41} + q^{42} + q^{44} + q^{45} + ( -2 - \beta ) q^{46} -2 \beta q^{47} + q^{48} + q^{49} - q^{50} + \beta q^{51} + \beta q^{52} + 6 q^{53} - q^{54} + q^{55} + q^{56} + ( -2 + 2 \beta ) q^{58} + 8 q^{59} + q^{60} + ( 2 + 2 \beta ) q^{61} + ( -4 + 2 \beta ) q^{62} - q^{63} + q^{64} + \beta q^{65} - q^{66} + ( -2 + \beta ) q^{67} + \beta q^{68} + ( 2 + \beta ) q^{69} + q^{70} + ( 8 + 2 \beta ) q^{71} - q^{72} + ( 2 - 4 \beta ) q^{73} + \beta q^{74} + q^{75} - q^{77} -\beta q^{78} -4 q^{79} + q^{80} + q^{81} -2 q^{82} -4 \beta q^{83} - q^{84} + \beta q^{85} + ( 2 - 2 \beta ) q^{87} - q^{88} + ( 12 + \beta ) q^{89} - q^{90} -\beta q^{91} + ( 2 + \beta ) q^{92} + ( 4 - 2 \beta ) q^{93} + 2 \beta q^{94} - q^{96} -10 q^{97} - q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 8 q^{31} - 2 q^{32} + 2 q^{33} - 2 q^{35} + 2 q^{36} - 2 q^{40} + 4 q^{41} + 2 q^{42} + 2 q^{44} + 2 q^{45} - 4 q^{46} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 4 q^{58} + 16 q^{59} + 2 q^{60} + 4 q^{61} - 8 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} - 4 q^{67} + 4 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} + 4 q^{73} + 2 q^{75} - 2 q^{77} - 8 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} - 2 q^{84} + 4 q^{87} - 2 q^{88} + 24 q^{89} - 2 q^{90} + 4 q^{92} + 8 q^{93} - 2 q^{96} - 20 q^{97} - 2 q^{98} + 2 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−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.z 2
3.b odd 2 1 6930.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.z 2 1.a even 1 1 trivial
6930.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2310))\):

\( T_{13}^{2} - 12 \)
\( T_{17}^{2} - 12 \)
\( T_{19} \)
\( T_{23}^{2} - 4 T_{23} - 8 \)
\( T_{29}^{2} - 4 T_{29} - 44 \)
\( T_{31}^{2} - 8 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -8 - 4 T + T^{2} \)
$29$ \( -44 - 4 T + T^{2} \)
$31$ \( -32 - 8 T + T^{2} \)
$37$ \( -12 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( -8 + 4 T + T^{2} \)
$71$ \( 16 - 16 T + T^{2} \)
$73$ \( -188 - 4 T + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -192 + T^{2} \)
$89$ \( 132 - 24 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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