Properties

Label 9075.2.a.v
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: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} + (\beta - 1) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} + (\beta - 1) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9} + ( - 2 \beta + 1) q^{12} + 4 \beta q^{13} - 2 q^{14} + 3 q^{16} + ( - 2 \beta - 4) q^{17} + (\beta - 1) q^{18} + ( - 2 \beta + 4) q^{19} + ( - 2 \beta - 2) q^{21} + 4 q^{23} + (\beta - 3) q^{24} + ( - 4 \beta + 8) q^{26} + q^{27} + (2 \beta + 6) q^{28} + ( - 2 \beta + 2) q^{29} + (\beta + 3) q^{32} - 2 \beta q^{34} + ( - 2 \beta + 1) q^{36} + (4 \beta - 6) q^{37} + (6 \beta - 8) q^{38} + 4 \beta q^{39} + (2 \beta - 2) q^{41} - 2 q^{42} + (2 \beta - 6) q^{43} + (4 \beta - 4) q^{46} + 4 q^{47} + 3 q^{48} + (8 \beta + 5) q^{49} + ( - 2 \beta - 4) q^{51} + (4 \beta - 16) q^{52} + (8 \beta + 2) q^{53} + (\beta - 1) q^{54} + (4 \beta + 2) q^{56} + ( - 2 \beta + 4) q^{57} + (4 \beta - 6) q^{58} - 4 q^{59} + ( - 4 \beta + 6) q^{61} + ( - 2 \beta - 2) q^{63} + (2 \beta - 7) q^{64} - 4 \beta q^{67} + (6 \beta + 4) q^{68} + 4 q^{69} + (4 \beta + 8) q^{71} + (\beta - 3) q^{72} - 8 \beta q^{73} + ( - 10 \beta + 14) q^{74} + ( - 10 \beta + 12) q^{76} + ( - 4 \beta + 8) q^{78} + 6 \beta q^{79} + q^{81} + ( - 4 \beta + 6) q^{82} - 10 q^{83} + (2 \beta + 6) q^{84} + ( - 8 \beta + 10) q^{86} + ( - 2 \beta + 2) q^{87} + ( - 4 \beta - 2) q^{89} + ( - 8 \beta - 16) q^{91} + ( - 8 \beta + 4) q^{92} + (4 \beta - 4) q^{94} + (\beta + 3) q^{96} + (4 \beta - 6) q^{97} + ( - 3 \beta + 11) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{18} + 8 q^{19} - 4 q^{21} + 8 q^{23} - 6 q^{24} + 16 q^{26} + 2 q^{27} + 12 q^{28} + 4 q^{29} + 6 q^{32} + 2 q^{36} - 12 q^{37} - 16 q^{38} - 4 q^{41} - 4 q^{42} - 12 q^{43} - 8 q^{46} + 8 q^{47} + 6 q^{48} + 10 q^{49} - 8 q^{51} - 32 q^{52} + 4 q^{53} - 2 q^{54} + 4 q^{56} + 8 q^{57} - 12 q^{58} - 8 q^{59} + 12 q^{61} - 4 q^{63} - 14 q^{64} + 8 q^{68} + 8 q^{69} + 16 q^{71} - 6 q^{72} + 28 q^{74} + 24 q^{76} + 16 q^{78} + 2 q^{81} + 12 q^{82} - 20 q^{83} + 12 q^{84} + 20 q^{86} + 4 q^{87} - 4 q^{89} - 32 q^{91} + 8 q^{92} - 8 q^{94} + 6 q^{96} - 12 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 0.828427 −4.41421 1.00000 0
1.2 0.414214 1.00000 −1.82843 0 0.414214 −4.82843 −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.v 2
5.b even 2 1 1815.2.a.k 2
11.b odd 2 1 825.2.a.g 2
15.d odd 2 1 5445.2.a.m 2
33.d even 2 1 2475.2.a.m 2
55.d odd 2 1 165.2.a.a 2
55.e even 4 2 825.2.c.e 4
165.d even 2 1 495.2.a.d 2
165.l odd 4 2 2475.2.c.m 4
220.g even 2 1 2640.2.a.bb 2
385.h even 2 1 8085.2.a.ba 2
660.g odd 2 1 7920.2.a.cg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 55.d odd 2 1
495.2.a.d 2 165.d even 2 1
825.2.a.g 2 11.b odd 2 1
825.2.c.e 4 55.e even 4 2
1815.2.a.k 2 5.b even 2 1
2475.2.a.m 2 33.d even 2 1
2475.2.c.m 4 165.l odd 4 2
2640.2.a.bb 2 220.g even 2 1
5445.2.a.m 2 15.d odd 2 1
7920.2.a.cg 2 660.g odd 2 1
8085.2.a.ba 2 385.h even 2 1
9075.2.a.v 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 32 \) Copy content Toggle raw display
\( T_{17}^{2} + 8T_{17} + 8 \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 8 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display
\( T_{37}^{2} + 12T_{37} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 32 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 128 \) Copy content Toggle raw display
$79$ \( T^{2} - 72 \) Copy content Toggle raw display
$83$ \( (T + 10)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
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