Properties

Label 1575.2.a.y
Level 1575
Weight 2
Character orbit 1575.a
Self dual yes
Analytic conductor 12.576
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.174928.1
Defining polynomial: \(x^{4} - 9 x^{2} + 13\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} - q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} - q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} -\beta_{3} q^{11} + ( 2 + 2 \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( 6 + 3 \beta_{2} ) q^{16} + 2 \beta_{1} q^{17} + ( 2 - 2 \beta_{2} ) q^{19} + ( -2 - 3 \beta_{2} ) q^{22} -\beta_{3} q^{23} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{26} + ( -3 - \beta_{2} ) q^{28} + ( -2 \beta_{1} + \beta_{3} ) q^{29} + 6 q^{31} + ( 5 \beta_{1} + \beta_{3} ) q^{32} + ( 10 + 2 \beta_{2} ) q^{34} + 3 q^{37} -2 \beta_{3} q^{38} + 4 \beta_{1} q^{41} + ( -5 - 2 \beta_{2} ) q^{43} + ( -5 \beta_{1} - \beta_{3} ) q^{44} + ( -2 - 3 \beta_{2} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{47} + q^{49} + ( 20 + 6 \beta_{2} ) q^{52} -4 \beta_{1} q^{53} + ( -2 \beta_{1} - \beta_{3} ) q^{56} + ( -8 + \beta_{2} ) q^{58} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + 6 \beta_{1} q^{62} + ( 15 + 2 \beta_{2} ) q^{64} + ( -5 - 2 \beta_{2} ) q^{67} + ( 8 \beta_{1} + 2 \beta_{3} ) q^{68} + ( 4 \beta_{1} + \beta_{3} ) q^{71} + 2 \beta_{2} q^{73} + 3 \beta_{1} q^{74} + ( -8 - 2 \beta_{2} ) q^{76} + \beta_{3} q^{77} + ( -1 - 2 \beta_{2} ) q^{79} + ( 20 + 4 \beta_{2} ) q^{82} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{83} + ( -7 \beta_{1} - 2 \beta_{3} ) q^{86} + ( -23 - 2 \beta_{2} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{2} ) q^{91} + ( -5 \beta_{1} - \beta_{3} ) q^{92} + ( -14 - 8 \beta_{2} ) q^{94} + ( -2 - 4 \beta_{2} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{4} - 4q^{7} + O(q^{10}) \) \( 4q + 10q^{4} - 4q^{7} + 4q^{13} + 18q^{16} + 12q^{19} - 2q^{22} - 10q^{28} + 24q^{31} + 36q^{34} + 12q^{37} - 16q^{43} - 2q^{46} + 4q^{49} + 68q^{52} - 34q^{58} + 56q^{64} - 16q^{67} - 4q^{73} - 28q^{76} + 72q^{82} - 88q^{88} - 4q^{91} - 40q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 13\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)

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
−2.68190
−1.34440
1.34440
2.68190
−2.68190 0 5.19258 0 0 −1.00000 −8.56218 0 0
1.2 −1.34440 0 −0.192582 0 0 −1.00000 2.94771 0 0
1.3 1.34440 0 −0.192582 0 0 −1.00000 −2.94771 0 0
1.4 2.68190 0 5.19258 0 0 −1.00000 8.56218 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.a.y 4
3.b odd 2 1 inner 1575.2.a.y 4
5.b even 2 1 1575.2.a.z yes 4
5.c odd 4 2 1575.2.d.l 8
15.d odd 2 1 1575.2.a.z yes 4
15.e even 4 2 1575.2.d.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.a.y 4 1.a even 1 1 trivial
1575.2.a.y 4 3.b odd 2 1 inner
1575.2.a.z yes 4 5.b even 2 1
1575.2.a.z yes 4 15.d odd 2 1
1575.2.d.l 8 5.c odd 4 2
1575.2.d.l 8 15.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{4} - 9 T_{2}^{2} + 13 \)
\( T_{11}^{4} - 42 T_{11}^{2} + 325 \)
\( T_{13}^{2} - 2 T_{13} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - 4 T^{6} + 16 T^{8} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 + 2 T^{2} + 127 T^{4} + 242 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 2 T - 2 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 + 32 T^{2} + 718 T^{4} + 9248 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 50 T^{2} + 1567 T^{4} + 26450 T^{6} + 279841 T^{8} \)
$29$ \( 1 + 42 T^{2} + 1079 T^{4} + 35322 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 6 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 3 T + 37 T^{2} )^{4} \)
$41$ \( 1 + 20 T^{2} + 1606 T^{4} + 33620 T^{6} + 2825761 T^{8} \)
$43$ \( ( 1 + 8 T + 73 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 24 T^{2} + 3518 T^{4} - 53016 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 68 T^{2} + 4918 T^{4} + 191012 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 24 T^{2} + 6062 T^{4} + 83544 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 8 T + 121 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 + 90 T^{2} + 2711 T^{4} + 453690 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 + 2 T + 118 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 129 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 36 T^{2} - 2602 T^{4} + 248004 T^{6} + 47458321 T^{8} \)
$89$ \( 1 + 160 T^{2} + 12846 T^{4} + 1267360 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 + 78 T^{2} + 9409 T^{4} )^{2} \)
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