Properties

Label 945.2.a.f
Level 945
Weight 2
Character orbit 945.a
Self dual yes
Analytic conductor 7.546
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 945.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.54586299101\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 1 + \beta ) q^{4} - q^{5} + q^{7} -3 q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 1 + \beta ) q^{4} - q^{5} + q^{7} -3 q^{8} + \beta q^{10} -3 q^{11} + ( -1 + \beta ) q^{13} -\beta q^{14} + ( -2 + \beta ) q^{16} + \beta q^{17} + ( -1 - \beta ) q^{19} + ( -1 - \beta ) q^{20} + 3 \beta q^{22} + ( -3 + \beta ) q^{23} + q^{25} -3 q^{26} + ( 1 + \beta ) q^{28} + ( -3 - \beta ) q^{29} + ( -7 + 2 \beta ) q^{31} + ( 3 + \beta ) q^{32} + ( -3 - \beta ) q^{34} - q^{35} + ( -1 + 2 \beta ) q^{37} + ( 3 + 2 \beta ) q^{38} + 3 q^{40} + 3 \beta q^{41} + ( 5 - 4 \beta ) q^{43} + ( -3 - 3 \beta ) q^{44} + ( -3 + 2 \beta ) q^{46} + ( -3 - 2 \beta ) q^{47} + q^{49} -\beta q^{50} + ( 2 + \beta ) q^{52} + ( -6 + 5 \beta ) q^{53} + 3 q^{55} -3 q^{56} + ( 3 + 4 \beta ) q^{58} + ( 3 - 4 \beta ) q^{59} + ( 5 - 7 \beta ) q^{61} + ( -6 + 5 \beta ) q^{62} + ( 1 - 6 \beta ) q^{64} + ( 1 - \beta ) q^{65} + ( 2 + 5 \beta ) q^{67} + ( 3 + 2 \beta ) q^{68} + \beta q^{70} + ( -9 + 3 \beta ) q^{71} + ( -1 - 4 \beta ) q^{73} + ( -6 - \beta ) q^{74} + ( -4 - 3 \beta ) q^{76} -3 q^{77} + ( -10 - 3 \beta ) q^{79} + ( 2 - \beta ) q^{80} + ( -9 - 3 \beta ) q^{82} + ( 3 + 4 \beta ) q^{83} -\beta q^{85} + ( 12 - \beta ) q^{86} + 9 q^{88} + ( 3 - 6 \beta ) q^{89} + ( -1 + \beta ) q^{91} -\beta q^{92} + ( 6 + 5 \beta ) q^{94} + ( 1 + \beta ) q^{95} + ( 2 - 5 \beta ) q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 3q^{4} - 2q^{5} + 2q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - q^{2} + 3q^{4} - 2q^{5} + 2q^{7} - 6q^{8} + q^{10} - 6q^{11} - q^{13} - q^{14} - 3q^{16} + q^{17} - 3q^{19} - 3q^{20} + 3q^{22} - 5q^{23} + 2q^{25} - 6q^{26} + 3q^{28} - 7q^{29} - 12q^{31} + 7q^{32} - 7q^{34} - 2q^{35} + 8q^{38} + 6q^{40} + 3q^{41} + 6q^{43} - 9q^{44} - 4q^{46} - 8q^{47} + 2q^{49} - q^{50} + 5q^{52} - 7q^{53} + 6q^{55} - 6q^{56} + 10q^{58} + 2q^{59} + 3q^{61} - 7q^{62} - 4q^{64} + q^{65} + 9q^{67} + 8q^{68} + q^{70} - 15q^{71} - 6q^{73} - 13q^{74} - 11q^{76} - 6q^{77} - 23q^{79} + 3q^{80} - 21q^{82} + 10q^{83} - q^{85} + 23q^{86} + 18q^{88} - q^{91} - q^{92} + 17q^{94} + 3q^{95} - q^{97} - q^{98} + O(q^{100}) \)

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
2.30278
−1.30278
−2.30278 0 3.30278 −1.00000 0 1.00000 −3.00000 0 2.30278
1.2 1.30278 0 −0.302776 −1.00000 0 1.00000 −3.00000 0 −1.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 945.2.a.f 2
3.b odd 2 1 945.2.a.j yes 2
5.b even 2 1 4725.2.a.bf 2
7.b odd 2 1 6615.2.a.q 2
15.d odd 2 1 4725.2.a.z 2
21.c even 2 1 6615.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.f 2 1.a even 1 1 trivial
945.2.a.j yes 2 3.b odd 2 1
4725.2.a.z 2 15.d odd 2 1
4725.2.a.bf 2 5.b even 2 1
6615.2.a.q 2 7.b odd 2 1
6615.2.a.u 2 21.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

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