Properties

Label 945.2.a.a
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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} + q^{5} + q^{7} + ( -1 - 4 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} + q^{5} + q^{7} + ( -1 - 4 \beta ) q^{8} + ( -1 - \beta ) q^{10} + ( -3 + 2 \beta ) q^{11} + ( -2 + \beta ) q^{13} + ( -1 - \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} + ( -1 - 3 \beta ) q^{17} + ( -2 - \beta ) q^{19} + 3 \beta q^{20} + ( 1 - \beta ) q^{22} + ( -4 - 3 \beta ) q^{23} + q^{25} + q^{26} + 3 \beta q^{28} + ( -4 + 3 \beta ) q^{29} + ( -1 - 4 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( 4 + 7 \beta ) q^{34} + q^{35} + ( 3 - 8 \beta ) q^{37} + ( 3 + 4 \beta ) q^{38} + ( -1 - 4 \beta ) q^{40} + ( -7 + 9 \beta ) q^{41} + ( -1 + 2 \beta ) q^{43} + ( 6 - 3 \beta ) q^{44} + ( 7 + 10 \beta ) q^{46} + ( -3 + 4 \beta ) q^{47} + q^{49} + ( -1 - \beta ) q^{50} + ( 3 - 3 \beta ) q^{52} + ( -5 + 5 \beta ) q^{53} + ( -3 + 2 \beta ) q^{55} + ( -1 - 4 \beta ) q^{56} + ( 1 - 2 \beta ) q^{58} + ( -1 + 8 \beta ) q^{59} + ( -2 - 3 \beta ) q^{61} + ( 5 + 9 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( -2 + \beta ) q^{65} + ( -7 - 3 \beta ) q^{67} + ( -9 - 12 \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( -6 - 5 \beta ) q^{71} + ( -3 + 8 \beta ) q^{73} + ( 5 + 13 \beta ) q^{74} + ( -3 - 9 \beta ) q^{76} + ( -3 + 2 \beta ) q^{77} + ( -3 + 11 \beta ) q^{79} + ( 5 + 3 \beta ) q^{80} + ( -2 - 11 \beta ) q^{82} + ( -11 + 2 \beta ) q^{83} + ( -1 - 3 \beta ) q^{85} + ( -1 - 3 \beta ) q^{86} + ( -5 + 2 \beta ) q^{88} + ( 5 - 6 \beta ) q^{89} + ( -2 + \beta ) q^{91} + ( -9 - 21 \beta ) q^{92} + ( -1 - 5 \beta ) q^{94} + ( -2 - \beta ) q^{95} + ( 5 - 5 \beta ) q^{97} + ( -1 - \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 3q^{4} + 2q^{5} + 2q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - 3q^{2} + 3q^{4} + 2q^{5} + 2q^{7} - 6q^{8} - 3q^{10} - 4q^{11} - 3q^{13} - 3q^{14} + 13q^{16} - 5q^{17} - 5q^{19} + 3q^{20} + q^{22} - 11q^{23} + 2q^{25} + 2q^{26} + 3q^{28} - 5q^{29} - 6q^{31} - 15q^{32} + 15q^{34} + 2q^{35} - 2q^{37} + 10q^{38} - 6q^{40} - 5q^{41} + 9q^{44} + 24q^{46} - 2q^{47} + 2q^{49} - 3q^{50} + 3q^{52} - 5q^{53} - 4q^{55} - 6q^{56} + 6q^{59} - 7q^{61} + 19q^{62} + 4q^{64} - 3q^{65} - 17q^{67} - 30q^{68} - 3q^{70} - 17q^{71} + 2q^{73} + 23q^{74} - 15q^{76} - 4q^{77} + 5q^{79} + 13q^{80} - 15q^{82} - 20q^{83} - 5q^{85} - 5q^{86} - 8q^{88} + 4q^{89} - 3q^{91} - 39q^{92} - 7q^{94} - 5q^{95} + 5q^{97} - 3q^{98} + O(q^{100}) \)

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
1.61803
−0.618034
−2.61803 0 4.85410 1.00000 0 1.00000 −7.47214 0 −2.61803
1.2 −0.381966 0 −1.85410 1.00000 0 1.00000 1.47214 0 −0.381966
\(n\): 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.a 2
3.b odd 2 1 945.2.a.l yes 2
5.b even 2 1 4725.2.a.bh 2
7.b odd 2 1 6615.2.a.k 2
15.d odd 2 1 4725.2.a.u 2
21.c even 2 1 6615.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.a 2 1.a even 1 1 trivial
945.2.a.l yes 2 3.b odd 2 1
4725.2.a.u 2 15.d odd 2 1
4725.2.a.bh 2 5.b even 2 1
6615.2.a.k 2 7.b odd 2 1
6615.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

\( T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{11}^{2} + 4 T_{11} - 1 \)