Properties

Label 9075.2.a.u
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} + ( - \beta - 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} + ( - \beta - 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} + q^{9} + 3 \beta q^{12} + (2 \beta - 3) q^{13} + (\beta + 1) q^{14} + (3 \beta + 5) q^{16} + (3 \beta - 6) q^{17} + ( - \beta - 1) q^{18} + ( - 3 \beta - 1) q^{19} - q^{21} + ( - 2 \beta + 3) q^{23} + ( - 4 \beta - 1) q^{24} + ( - \beta + 1) q^{26} + q^{27} - 3 \beta q^{28} + 6 q^{29} + ( - 5 \beta + 2) q^{31} + ( - 3 \beta - 6) q^{32} + 3 q^{34} + 3 \beta q^{36} + (2 \beta + 3) q^{37} + (7 \beta + 4) q^{38} + (2 \beta - 3) q^{39} + ( - 2 \beta + 3) q^{41} + (\beta + 1) q^{42} + ( - 6 \beta + 3) q^{43} + (\beta - 1) q^{46} + (5 \beta + 2) q^{47} + (3 \beta + 5) q^{48} - 6 q^{49} + (3 \beta - 6) q^{51} + ( - 3 \beta + 6) q^{52} + ( - \beta + 2) q^{53} + ( - \beta - 1) q^{54} + (4 \beta + 1) q^{56} + ( - 3 \beta - 1) q^{57} + ( - 6 \beta - 6) q^{58} + ( - \beta + 9) q^{59} + (9 \beta - 3) q^{61} + (8 \beta + 3) q^{62} - q^{63} + (6 \beta - 1) q^{64} + ( - 3 \beta + 3) q^{67} + ( - 9 \beta + 9) q^{68} + ( - 2 \beta + 3) q^{69} + (7 \beta - 1) q^{71} + ( - 4 \beta - 1) q^{72} + ( - 6 \beta + 4) q^{73} + ( - 7 \beta - 5) q^{74} + ( - 12 \beta - 9) q^{76} + ( - \beta + 1) q^{78} - 11 q^{79} + q^{81} + (\beta - 1) q^{82} + (4 \beta - 5) q^{83} - 3 \beta q^{84} + (9 \beta + 3) q^{86} + 6 q^{87} + ( - 2 \beta - 5) q^{89} + ( - 2 \beta + 3) q^{91} + (3 \beta - 6) q^{92} + ( - 5 \beta + 2) q^{93} + ( - 12 \beta - 7) q^{94} + ( - 3 \beta - 6) q^{96} + ( - 3 \beta - 3) q^{97} + (6 \beta + 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} + 3 q^{12} - 4 q^{13} + 3 q^{14} + 13 q^{16} - 9 q^{17} - 3 q^{18} - 5 q^{19} - 2 q^{21} + 4 q^{23} - 6 q^{24} + q^{26} + 2 q^{27} - 3 q^{28} + 12 q^{29} - q^{31} - 15 q^{32} + 6 q^{34} + 3 q^{36} + 8 q^{37} + 15 q^{38} - 4 q^{39} + 4 q^{41} + 3 q^{42} - q^{46} + 9 q^{47} + 13 q^{48} - 12 q^{49} - 9 q^{51} + 9 q^{52} + 3 q^{53} - 3 q^{54} + 6 q^{56} - 5 q^{57} - 18 q^{58} + 17 q^{59} + 3 q^{61} + 14 q^{62} - 2 q^{63} + 4 q^{64} + 3 q^{67} + 9 q^{68} + 4 q^{69} + 5 q^{71} - 6 q^{72} + 2 q^{73} - 17 q^{74} - 30 q^{76} + q^{78} - 22 q^{79} + 2 q^{81} - q^{82} - 6 q^{83} - 3 q^{84} + 15 q^{86} + 12 q^{87} - 12 q^{89} + 4 q^{91} - 9 q^{92} - q^{93} - 26 q^{94} - 15 q^{96} - 9 q^{97} + 18 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 1.00000 4.85410 0 −2.61803 −1.00000 −7.47214 1.00000 0
1.2 −0.381966 1.00000 −1.85410 0 −0.381966 −1.00000 1.47214 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.u 2
5.b even 2 1 363.2.a.i 2
11.b odd 2 1 9075.2.a.cb 2
11.d odd 10 2 825.2.n.c 4
15.d odd 2 1 1089.2.a.l 2
20.d odd 2 1 5808.2.a.ci 2
55.d odd 2 1 363.2.a.d 2
55.h odd 10 2 33.2.e.b 4
55.h odd 10 2 363.2.e.k 4
55.j even 10 2 363.2.e.b 4
55.j even 10 2 363.2.e.f 4
55.l even 20 4 825.2.bx.d 8
165.d even 2 1 1089.2.a.t 2
165.r even 10 2 99.2.f.a 4
220.g even 2 1 5808.2.a.cj 2
220.o even 10 2 528.2.y.b 4
495.bo even 30 4 891.2.n.b 8
495.br odd 30 4 891.2.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 55.h odd 10 2
99.2.f.a 4 165.r even 10 2
363.2.a.d 2 55.d odd 2 1
363.2.a.i 2 5.b even 2 1
363.2.e.b 4 55.j even 10 2
363.2.e.f 4 55.j even 10 2
363.2.e.k 4 55.h odd 10 2
528.2.y.b 4 220.o even 10 2
825.2.n.c 4 11.d odd 10 2
825.2.bx.d 8 55.l even 20 4
891.2.n.b 8 495.bo even 30 4
891.2.n.c 8 495.br odd 30 4
1089.2.a.l 2 15.d odd 2 1
1089.2.a.t 2 165.d even 2 1
5808.2.a.ci 2 20.d odd 2 1
5808.2.a.cj 2 220.g even 2 1
9075.2.a.u 2 1.a even 1 1 trivial
9075.2.a.cb 2 11.b odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 9T_{17} + 9 \) Copy content Toggle raw display
\( T_{19}^{2} + 5T_{19} - 5 \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 1 \) Copy content Toggle raw display
\( T_{37}^{2} - 8T_{37} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 11 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 45 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 71 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T - 99 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T - 55 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
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