Properties

Label 9075.2.a.w
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} + ( -1 + \beta ) q^{6} + ( 1 + \beta ) q^{7} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} + ( -1 + \beta ) q^{6} + ( 1 + \beta ) q^{7} + ( -3 + \beta ) q^{8} + q^{9} + ( 1 - 2 \beta ) q^{12} -2 \beta q^{13} + q^{14} + 3 q^{16} + ( -1 + \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( -5 + \beta ) q^{19} + ( 1 + \beta ) q^{21} + q^{23} + ( -3 + \beta ) q^{24} + ( -4 + 2 \beta ) q^{26} + q^{27} + ( -3 - \beta ) q^{28} + ( -4 - 2 \beta ) q^{29} + 6 \beta q^{31} + ( 3 + \beta ) q^{32} + ( 3 - 2 \beta ) q^{34} + ( 1 - 2 \beta ) q^{36} + ( -3 - 2 \beta ) q^{37} + ( 7 - 6 \beta ) q^{38} -2 \beta q^{39} + ( 1 - 7 \beta ) q^{41} + q^{42} + ( 6 - 4 \beta ) q^{43} + ( -1 + \beta ) q^{46} + ( 1 + 6 \beta ) q^{47} + 3 q^{48} + ( -4 + 2 \beta ) q^{49} + ( -1 + \beta ) q^{51} + ( 8 - 2 \beta ) q^{52} + ( 2 - 4 \beta ) q^{53} + ( -1 + \beta ) q^{54} + ( -1 - 2 \beta ) q^{56} + ( -5 + \beta ) q^{57} -2 \beta q^{58} + 11 q^{59} + ( -6 + 2 \beta ) q^{61} + ( 12 - 6 \beta ) q^{62} + ( 1 + \beta ) q^{63} + ( -7 + 2 \beta ) q^{64} + ( -6 - 4 \beta ) q^{67} + ( -5 + 3 \beta ) q^{68} + q^{69} + ( 5 - 2 \beta ) q^{71} + ( -3 + \beta ) q^{72} + ( 6 - 2 \beta ) q^{73} + ( -1 - \beta ) q^{74} + ( -9 + 11 \beta ) q^{76} + ( -4 + 2 \beta ) q^{78} + ( -9 + 3 \beta ) q^{79} + q^{81} + ( -15 + 8 \beta ) q^{82} + ( -4 - 6 \beta ) q^{83} + ( -3 - \beta ) q^{84} + ( -14 + 10 \beta ) q^{86} + ( -4 - 2 \beta ) q^{87} + ( -2 - 4 \beta ) q^{89} + ( -4 - 2 \beta ) q^{91} + ( 1 - 2 \beta ) q^{92} + 6 \beta q^{93} + ( 11 - 5 \beta ) q^{94} + ( 3 + \beta ) q^{96} + ( 3 - 2 \beta ) q^{97} + ( 8 - 6 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 6 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{14} + 6 q^{16} - 2 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{21} + 2 q^{23} - 6 q^{24} - 8 q^{26} + 2 q^{27} - 6 q^{28} - 8 q^{29} + 6 q^{32} + 6 q^{34} + 2 q^{36} - 6 q^{37} + 14 q^{38} + 2 q^{41} + 2 q^{42} + 12 q^{43} - 2 q^{46} + 2 q^{47} + 6 q^{48} - 8 q^{49} - 2 q^{51} + 16 q^{52} + 4 q^{53} - 2 q^{54} - 2 q^{56} - 10 q^{57} + 22 q^{59} - 12 q^{61} + 24 q^{62} + 2 q^{63} - 14 q^{64} - 12 q^{67} - 10 q^{68} + 2 q^{69} + 10 q^{71} - 6 q^{72} + 12 q^{73} - 2 q^{74} - 18 q^{76} - 8 q^{78} - 18 q^{79} + 2 q^{81} - 30 q^{82} - 8 q^{83} - 6 q^{84} - 28 q^{86} - 8 q^{87} - 4 q^{89} - 8 q^{91} + 2 q^{92} + 22 q^{94} + 6 q^{96} + 6 q^{97} + 16 q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 1.00000 3.82843 0 −2.41421 −0.414214 −4.41421 1.00000 0
1.2 0.414214 1.00000 −1.82843 0 0.414214 2.41421 −1.58579 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 9075.2.a.w 2
5.b even 2 1 9075.2.a.ca 2
11.b odd 2 1 825.2.a.f yes 2
33.d even 2 1 2475.2.a.l 2
55.d odd 2 1 825.2.a.d 2
55.e even 4 2 825.2.c.d 4
165.d even 2 1 2475.2.a.w 2
165.l odd 4 2 2475.2.c.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 55.d odd 2 1
825.2.a.f yes 2 11.b odd 2 1
825.2.c.d 4 55.e even 4 2
2475.2.a.l 2 33.d even 2 1
2475.2.a.w 2 165.d even 2 1
2475.2.c.o 4 165.l odd 4 2
9075.2.a.w 2 1.a even 1 1 trivial
9075.2.a.ca 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{7}^{2} - 2 T_{7} - 1 \)
\( T_{13}^{2} - 8 \)
\( T_{17}^{2} + 2 T_{17} - 1 \)
\( T_{19}^{2} + 10 T_{19} + 23 \)
\( T_{23} - 1 \)
\( T_{37}^{2} + 6 T_{37} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -1 - 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( -1 + 2 T + T^{2} \)
$19$ \( 23 + 10 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 8 + 8 T + T^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( 1 + 6 T + T^{2} \)
$41$ \( -97 - 2 T + T^{2} \)
$43$ \( 4 - 12 T + T^{2} \)
$47$ \( -71 - 2 T + T^{2} \)
$53$ \( -28 - 4 T + T^{2} \)
$59$ \( ( -11 + T )^{2} \)
$61$ \( 28 + 12 T + T^{2} \)
$67$ \( 4 + 12 T + T^{2} \)
$71$ \( 17 - 10 T + T^{2} \)
$73$ \( 28 - 12 T + T^{2} \)
$79$ \( 63 + 18 T + T^{2} \)
$83$ \( -56 + 8 T + T^{2} \)
$89$ \( -28 + 4 T + T^{2} \)
$97$ \( 1 - 6 T + T^{2} \)
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