Properties

Label 945.2.a.e
Level 945
Weight 2
Character orbit 945.a
Self dual Yes
Analytic conductor 7.546
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

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

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.

Atkin-Lehner signs

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

Hecke kernels

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

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