Properties

Label 945.2.a.n
Level 945
Weight 2
Character orbit 945.a
Self dual Yes
Analytic conductor 7.546
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: 4.4.144344.1
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \(+ q^{5}\) \(+ q^{7}\) \( + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \(+ q^{5}\) \(+ q^{7}\) \( + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{8} \) \( + \beta_{1} q^{10} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{11} \) \( + ( 1 - \beta_{1} - \beta_{2} ) q^{13} \) \( + \beta_{1} q^{14} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{16} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} \) \( + ( 1 - \beta_{3} ) q^{19} \) \( + ( 2 + \beta_{2} ) q^{20} \) \( + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} \) \(+ q^{25}\) \( + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{26} \) \( + ( 2 + \beta_{2} ) q^{28} \) \( + ( -2 - 2 \beta_{2} ) q^{29} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} \) \( + ( 4 + 4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{32} \) \( + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{34} \) \(+ q^{35}\) \( + ( 2 + 2 \beta_{1} ) q^{37} \) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{38} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{40} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} \) \( + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} \) \( + ( -9 + 3 \beta_{1} - 3 \beta_{2} ) q^{44} \) \( + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{46} \) \( + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} \) \(+ q^{49}\) \( + \beta_{1} q^{50} \) \( + ( -7 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{52} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{55} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{56} \) \( + ( -2 - 6 \beta_{1} - 2 \beta_{3} ) q^{58} \) \( -2 \beta_{1} q^{59} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} \) \( + ( -5 + \beta_{2} ) q^{62} \) \( + ( 11 + 8 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{64} \) \( + ( 1 - \beta_{1} - \beta_{2} ) q^{65} \) \( + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{67} \) \( + ( 5 - 7 \beta_{1} - \beta_{3} ) q^{68} \) \( + \beta_{1} q^{70} \) \( + ( -2 - 4 \beta_{1} + 4 \beta_{2} ) q^{71} \) \( + ( -3 + \beta_{1} + \beta_{2} ) q^{73} \) \( + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{74} \) \( + ( -4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{76} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{77} \) \( + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{80} \) \( + ( 5 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{82} \) \( + ( 7 - 2 \beta_{1} - 2 \beta_{2} ) q^{83} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{3} ) q^{85} \) \( + ( -5 + 3 \beta_{1} + \beta_{2} ) q^{86} \) \( + ( 3 - 9 \beta_{1} + \beta_{2} - \beta_{3} ) q^{88} \) \( + ( -1 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} \) \( + ( 1 - \beta_{1} - \beta_{2} ) q^{91} \) \( + ( 5 - 7 \beta_{1} - \beta_{3} ) q^{92} \) \( + ( -11 + 7 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{94} \) \( + ( 1 - \beta_{3} ) q^{95} \) \( + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} \) \( + \beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 9q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 9q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 19q^{16} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 9q^{20} \) \(\mathstrut +\mathstrut 10q^{22} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 22q^{26} \) \(\mathstrut +\mathstrut 9q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 23q^{32} \) \(\mathstrut -\mathstrut 13q^{34} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut q^{38} \) \(\mathstrut +\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 18q^{43} \) \(\mathstrut -\mathstrut 36q^{44} \) \(\mathstrut -\mathstrut 13q^{46} \) \(\mathstrut +\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut -\mathstrut 34q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 6q^{56} \) \(\mathstrut -\mathstrut 14q^{58} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 8q^{61} \) \(\mathstrut -\mathstrut 19q^{62} \) \(\mathstrut +\mathstrut 54q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 10q^{67} \) \(\mathstrut +\mathstrut 13q^{68} \) \(\mathstrut +\mathstrut q^{70} \) \(\mathstrut -\mathstrut 8q^{71} \) \(\mathstrut -\mathstrut 10q^{73} \) \(\mathstrut +\mathstrut 36q^{74} \) \(\mathstrut -\mathstrut 5q^{76} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 12q^{79} \) \(\mathstrut +\mathstrut 19q^{80} \) \(\mathstrut +\mathstrut 24q^{82} \) \(\mathstrut +\mathstrut 24q^{83} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut +\mathstrut 13q^{92} \) \(\mathstrut -\mathstrut 38q^{94} \) \(\mathstrut +\mathstrut 4q^{95} \) \(\mathstrut +\mathstrut 10q^{97} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(8\) \(x^{2}\mathstrut +\mathstrut \) \(5\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)

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
−2.46608
−0.857589
1.51533
2.80834
−2.46608 0 4.08154 1.00000 0 1.00000 −5.13325 0 −2.46608
1.2 −0.857589 0 −1.26454 1.00000 0 1.00000 2.79964 0 −0.857589
1.3 1.51533 0 0.296215 1.00000 0 1.00000 −2.58179 0 1.51533
1.4 2.80834 0 5.88678 1.00000 0 1.00000 10.9154 0 2.80834
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

\(T_{2}^{4} \) \(\mathstrut -\mathstrut T_{2}^{3} \) \(\mathstrut -\mathstrut 8 T_{2}^{2} \) \(\mathstrut +\mathstrut 5 T_{2} \) \(\mathstrut +\mathstrut 9 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 4 T_{11}^{3} \) \(\mathstrut -\mathstrut 13 T_{11}^{2} \) \(\mathstrut -\mathstrut 18 T_{11} \) \(\mathstrut +\mathstrut 36 \)