Properties

Label 9450.2.a.em
Level 9450
Weight 2
Character orbit 9450.a
Self dual yes
Analytic conductor 75.459
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.16228
−3.16228
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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9450.2.a.em yes 2
3.b odd 2 1 9450.2.a.ee 2
5.b even 2 1 9450.2.a.eh yes 2
15.d odd 2 1 9450.2.a.ex yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9450.2.a.ee 2 3.b odd 2 1
9450.2.a.eh yes 2 5.b even 2 1
9450.2.a.em yes 2 1.a even 1 1 trivial
9450.2.a.ex yes 2 15.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

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

\( T_{11} + 2 \)
\( T_{13}^{2} + 2 T_{13} - 9 \)
\( T_{17}^{2} - 2 T_{17} - 39 \)
\( T_{19} - 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{2} \)
$13$ \( 1 + 2 T + 17 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 2 T - 5 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 3 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 4 T + 40 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 10 T + 73 T^{2} + 290 T^{3} + 841 T^{4} \)
$31$ \( 1 + 22 T^{2} + 961 T^{4} \)
$37$ \( 1 - 4 T - 12 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 4 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 76 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 2 T + 5 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 6 T + 105 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 114 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 14 T + 161 T^{2} + 854 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 12 T + 80 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 8 T + 118 T^{2} - 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 4 T + 110 T^{2} + 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 10 T + 93 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 156 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 5 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 4 T + 108 T^{2} - 388 T^{3} + 9409 T^{4} \)
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