Properties

Label 9450.2.a.er
Level 9450
Weight 2
Character orbit 9450.a
Self dual Yes
Analytic conductor 75.459
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 9450.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(75.4586299101\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
1.00000 0 1.00000 0 0 −1.00000 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −1.00000 1.00000 0 0
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\(T_{11}^{2} \) \(\mathstrut -\mathstrut 3 \)
\(T_{13}^{2} \) \(\mathstrut -\mathstrut 4 T_{13} \) \(\mathstrut -\mathstrut 8 \)
\(T_{17} \) \(\mathstrut -\mathstrut 4 \)
\(T_{19}^{2} \) \(\mathstrut -\mathstrut 8 T_{19} \) \(\mathstrut +\mathstrut 13 \)