Properties

Label 945.2.a.k
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: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−0.414214 0 −1.82843 1.00000 0 −1.00000 1.58579 0 −0.414214
1.2 2.41421 0 3.82843 1.00000 0 −1.00000 4.41421 0 2.41421
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

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