Properties

Label 9450.2.a.eg
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1890)
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.

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

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.eg 2
3.b odd 2 1 9450.2.a.ew 2
5.b even 2 1 9450.2.a.en 2
5.c odd 4 2 1890.2.g.o 4
15.d odd 2 1 9450.2.a.ef 2
15.e even 4 2 1890.2.g.p yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.g.o 4 5.c odd 4 2
1890.2.g.p yes 4 15.e even 4 2
9450.2.a.ef 2 15.d odd 2 1
9450.2.a.eg 2 1.a even 1 1 trivial
9450.2.a.en 2 5.b even 2 1
9450.2.a.ew 2 3.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11}^{2} + 4 T_{11} - 6 \)
\( T_{13}^{2} + 2 T_{13} - 9 \)
\( T_{17} - 5 \)
\( T_{19}^{2} + 8 T_{19} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 + 4 T + 16 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 2 T + 17 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 5 T + 17 T^{2} )^{2} \)
$19$ \( 1 + 8 T + 44 T^{2} + 152 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 45 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 2 T + 49 T^{2} + 58 T^{3} + 841 T^{4} \)
$31$ \( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 12 T + 100 T^{2} + 444 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 8 T + 58 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 7 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 2 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 6 T + 25 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 - 7 T + 59 T^{2} )^{2} \)
$61$ \( 1 + 4 T + 86 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 18 T + 175 T^{2} + 1206 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 6 T + 61 T^{2} - 426 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 20 T + 248 T^{2} - 1580 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 4 T + 130 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 18 T + 219 T^{2} - 1602 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 4 T + 188 T^{2} - 388 T^{3} + 9409 T^{4} \)
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