Properties

Label 9075.2.a.df
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + 4 x + 1\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{3} ) q^{2} + q^{3} + \beta_{1} q^{4} + ( -\beta_{2} - \beta_{3} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{3} ) q^{2} + q^{3} + \beta_{1} q^{4} + ( -\beta_{2} - \beta_{3} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{8} + q^{9} + \beta_{1} q^{12} + ( 4 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{13} + ( -3 + \beta_{1} - 3 \beta_{3} ) q^{14} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{16} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -\beta_{2} - \beta_{3} ) q^{18} + ( -2 - 3 \beta_{1} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{23} + ( -1 + \beta_{2} ) q^{24} + ( -4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{26} + q^{27} + ( 1 + \beta_{1} ) q^{28} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{29} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} + ( 2 + 2 \beta_{2} + 5 \beta_{3} ) q^{32} + ( 2 + 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{34} + \beta_{1} q^{36} + ( -5 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{37} + ( 3 + 5 \beta_{2} + 8 \beta_{3} ) q^{38} + ( 4 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{39} + ( -5 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{41} + ( -3 + \beta_{1} - 3 \beta_{3} ) q^{42} + ( 1 - 3 \beta_{1} - 5 \beta_{2} ) q^{43} + ( -4 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{46} + ( 3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{48} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{49} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{51} + ( \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{52} + ( -4 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{53} + ( -\beta_{2} - \beta_{3} ) q^{54} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{56} + ( -2 - 3 \beta_{1} ) q^{57} + ( 5 - \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{58} + ( 3 - 5 \beta_{2} - \beta_{3} ) q^{59} + ( -9 + \beta_{1} + \beta_{3} ) q^{61} + ( -7 - \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{62} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{64} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 + \beta_{1} - 4 \beta_{3} ) q^{68} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{69} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{71} + ( -1 + \beta_{2} ) q^{72} + ( 2 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{73} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{74} + ( -6 - 2 \beta_{1} - 3 \beta_{2} ) q^{76} + ( -4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{78} + ( -7 + 2 \beta_{2} - 3 \beta_{3} ) q^{79} + q^{81} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} ) q^{82} + ( -2 - 9 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{83} + ( 1 + \beta_{1} ) q^{84} + ( 13 + 2 \beta_{2} + 10 \beta_{3} ) q^{86} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{87} + ( -5 - 8 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( -6 + \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{92} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{93} + ( -5 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{94} + ( 2 + 2 \beta_{2} + 5 \beta_{3} ) q^{96} + ( 10 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{97} + ( 7 - 2 \beta_{1} + \beta_{2} + 7 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + q^{12} + 7 q^{13} - 5 q^{14} - 9 q^{16} + 8 q^{17} + q^{18} - 11 q^{19} - 5 q^{23} - 3 q^{24} - 12 q^{26} + 4 q^{27} + 5 q^{28} - 17 q^{29} - 5 q^{31} + 17 q^{34} + q^{36} - 15 q^{37} + q^{38} + 7 q^{39} - 10 q^{41} - 5 q^{42} - 4 q^{43} - 15 q^{46} + 8 q^{47} - 9 q^{48} - 8 q^{49} + 8 q^{51} - 7 q^{52} - 10 q^{53} + q^{54} + 10 q^{56} - 11 q^{57} + 7 q^{58} + 9 q^{59} - 37 q^{61} - 20 q^{62} - 7 q^{64} - 3 q^{67} + 17 q^{68} - 5 q^{69} + 13 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} - 29 q^{76} - 12 q^{78} - 20 q^{79} + 4 q^{81} - 5 q^{82} - 17 q^{83} + 5 q^{84} + 34 q^{86} - 17 q^{87} - 24 q^{89} - 20 q^{91} - 10 q^{92} - 5 q^{93} - 23 q^{94} + 32 q^{97} + 13 q^{98} + O(q^{100}) \)

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
1.82709
−1.95630
−0.209057
1.33826
−1.95630 1.00000 1.82709 0 −1.95630 1.54732 0.338261 1.00000 0
1.2 −0.209057 1.00000 −1.95630 0 −0.209057 0.488830 0.827091 1.00000 0
1.3 1.33826 1.00000 −0.209057 0 1.33826 −3.78339 −2.95630 1.00000 0
1.4 1.82709 1.00000 1.33826 0 1.82709 1.74724 −1.20906 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.df 4
5.b even 2 1 1815.2.a.q 4
11.b odd 2 1 9075.2.a.co 4
11.d odd 10 2 825.2.n.j 8
15.d odd 2 1 5445.2.a.bq 4
55.d odd 2 1 1815.2.a.u 4
55.h odd 10 2 165.2.m.c 8
55.l even 20 4 825.2.bx.e 16
165.d even 2 1 5445.2.a.bj 4
165.r even 10 2 495.2.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.c 8 55.h odd 10 2
495.2.n.c 8 165.r even 10 2
825.2.n.j 8 11.d odd 10 2
825.2.bx.e 16 55.l even 20 4
1815.2.a.q 4 5.b even 2 1
1815.2.a.u 4 55.d odd 2 1
5445.2.a.bj 4 165.d even 2 1
5445.2.a.bq 4 15.d odd 2 1
9075.2.a.co 4 11.b odd 2 1
9075.2.a.df 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{4} - T_{2}^{3} - 4 T_{2}^{2} + 4 T_{2} + 1 \)
\( T_{7}^{4} - 10 T_{7}^{2} + 15 T_{7} - 5 \)
\( T_{13}^{4} - 7 T_{13}^{3} - 16 T_{13}^{2} + 202 T_{13} - 359 \)
\( T_{17}^{4} - 8 T_{17}^{3} - 6 T_{17}^{2} + 88 T_{17} - 59 \)
\( T_{19}^{4} + 11 T_{19}^{3} + 6 T_{19}^{2} - 184 T_{19} - 239 \)
\( T_{23}^{4} + 5 T_{23}^{3} - 25 T_{23}^{2} + 25 T_{23} - 5 \)
\( T_{37}^{4} + 15 T_{37}^{3} + 65 T_{37}^{2} + 45 T_{37} - 155 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T - 4 T^{2} - T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( -5 + 15 T - 10 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -359 + 202 T - 16 T^{2} - 7 T^{3} + T^{4} \)
$17$ \( -59 + 88 T - 6 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( -239 - 184 T + 6 T^{2} + 11 T^{3} + T^{4} \)
$23$ \( -5 + 25 T - 25 T^{2} + 5 T^{3} + T^{4} \)
$29$ \( -419 + 38 T + 84 T^{2} + 17 T^{3} + T^{4} \)
$31$ \( -5 - 235 T - 45 T^{2} + 5 T^{3} + T^{4} \)
$37$ \( -155 + 45 T + 65 T^{2} + 15 T^{3} + T^{4} \)
$41$ \( -5 - 145 T - 10 T^{2} + 10 T^{3} + T^{4} \)
$43$ \( -569 - 541 T - 124 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( 121 + 143 T - 16 T^{2} - 8 T^{3} + T^{4} \)
$53$ \( 25 - 25 T - 10 T^{2} + 10 T^{3} + T^{4} \)
$59$ \( 1341 + 261 T - 69 T^{2} - 9 T^{3} + T^{4} \)
$61$ \( 6571 + 2998 T + 504 T^{2} + 37 T^{3} + T^{4} \)
$67$ \( 31 + 72 T - 46 T^{2} + 3 T^{3} + T^{4} \)
$71$ \( -389 + 83 T + 39 T^{2} - 13 T^{3} + T^{4} \)
$73$ \( -4205 + 1320 T - 40 T^{2} - 15 T^{3} + T^{4} \)
$79$ \( -305 + 10 T + 95 T^{2} + 20 T^{3} + T^{4} \)
$83$ \( -14669 - 5347 T - 261 T^{2} + 17 T^{3} + T^{4} \)
$89$ \( -31869 - 6066 T - 129 T^{2} + 24 T^{3} + T^{4} \)
$97$ \( 61 - 463 T + 284 T^{2} - 32 T^{3} + T^{4} \)
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