Properties

Label 945.2.a.d
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.

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

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
1.61803
−0.618034
−1.61803 0 0.618034 1.00000 0 −1.00000 2.23607 0 −1.61803
1.2 0.618034 0 −1.61803 1.00000 0 −1.00000 −2.23607 0 0.618034
\(n\): 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.d 2
3.b odd 2 1 945.2.a.i yes 2
5.b even 2 1 4725.2.a.bc 2
7.b odd 2 1 6615.2.a.m 2
15.d odd 2 1 4725.2.a.x 2
21.c even 2 1 6615.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.d 2 1.a even 1 1 trivial
945.2.a.i yes 2 3.b odd 2 1
4725.2.a.x 2 15.d odd 2 1
4725.2.a.bc 2 5.b even 2 1
6615.2.a.m 2 7.b odd 2 1
6615.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 19 \)