Properties

Label 9450.2.a.ec
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: \(\Q(\sqrt{7}) \)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.64575
2.64575
−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.ec 2
3.b odd 2 1 9450.2.a.eq yes 2
5.b even 2 1 9450.2.a.et yes 2
15.d odd 2 1 9450.2.a.ej yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9450.2.a.ec 2 1.a even 1 1 trivial
9450.2.a.ej yes 2 15.d odd 2 1
9450.2.a.eq yes 2 3.b odd 2 1
9450.2.a.et yes 2 5.b even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11}^{2} - 28 \)
\( T_{13}^{2} - 4 T_{13} - 3 \)
\( T_{17} - 1 \)
\( T_{19}^{2} + 2 T_{19} - 27 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ \( 1 - 4 T + 23 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - T + 17 T^{2} )^{2} \)
$19$ \( 1 + 2 T + 11 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( 1 - 2 T + 40 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( 1 + 16 T + 115 T^{2} + 464 T^{3} + 841 T^{4} \)
$31$ \( 1 + 4 T + 38 T^{2} + 124 T^{3} + 961 T^{4} \)
$37$ \( 1 - 14 T + 116 T^{2} - 518 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 4 T + 58 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 6 T + 88 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 91 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 99 T^{2} + 2809 T^{4} \)
$59$ \( 1 + 16 T + 154 T^{2} + 944 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 115 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 6 T + 80 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 150 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 118 T^{2} + 5329 T^{4} \)
$79$ \( 1 + 16 T + 159 T^{2} + 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 2 T + 160 T^{2} + 166 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + T + 89 T^{2} )^{2} \)
$97$ \( 1 - 10 T + 156 T^{2} - 970 T^{3} + 9409 T^{4} \)
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