Properties

Label 945.2.a.l
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.

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

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
−0.618034
1.61803
0.381966 0 −1.85410 −1.00000 0 1.00000 −1.47214 0 −0.381966
1.2 2.61803 0 4.85410 −1.00000 0 1.00000 7.47214 0 −2.61803
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

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