Properties

Label 945.2.a.h
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: \( 1 \)
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.

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

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
−0.618034
1.61803
−0.618034 0 −1.61803 −1.00000 0 −1.00000 2.23607 0 0.618034
1.2 1.61803 0 0.618034 −1.00000 0 −1.00000 −2.23607 0 −1.61803
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

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