Properties

Label 825.2.f.b
Level $825$
Weight $2$
Character orbit 825.f
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -2 q^{4} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -2 q^{4} + ( 3 + \beta_{2} ) q^{9} + ( 1 + 2 \beta_{2} ) q^{11} + 2 \beta_{1} q^{12} + 4 q^{16} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{23} + ( -3 \beta_{1} - \beta_{3} ) q^{27} -5 q^{31} + ( -\beta_{1} - 2 \beta_{3} ) q^{33} + ( -6 - 2 \beta_{2} ) q^{36} + ( -5 \beta_{1} + \beta_{3} ) q^{37} + ( -2 - 4 \beta_{2} ) q^{44} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{47} -4 \beta_{1} q^{48} + 7 q^{49} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -1 - 2 \beta_{2} ) q^{59} -8 q^{64} + ( 5 \beta_{1} - \beta_{3} ) q^{67} + 9 \beta_{2} q^{69} + ( -5 - 10 \beta_{2} ) q^{71} + ( 6 + 5 \beta_{2} ) q^{81} + ( -5 - 10 \beta_{2} ) q^{89} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{92} + 5 \beta_{1} q^{93} + ( 5 \beta_{1} - \beta_{3} ) q^{97} + ( -3 + 5 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 10 q^{9} + O(q^{10}) \) \( 4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
626.1
1.65831 + 0.500000i
1.65831 0.500000i
−1.65831 + 0.500000i
−1.65831 0.500000i
0 −1.65831 0.500000i −2.00000 0 0 0 0 2.50000 + 1.65831i 0
626.2 0 −1.65831 + 0.500000i −2.00000 0 0 0 0 2.50000 1.65831i 0
626.3 0 1.65831 0.500000i −2.00000 0 0 0 0 2.50000 1.65831i 0
626.4 0 1.65831 + 0.500000i −2.00000 0 0 0 0 2.50000 + 1.65831i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.f.b 4
3.b odd 2 1 inner 825.2.f.b 4
5.b even 2 1 inner 825.2.f.b 4
5.c odd 4 1 165.2.d.a 2
5.c odd 4 1 165.2.d.b yes 2
11.b odd 2 1 CM 825.2.f.b 4
15.d odd 2 1 inner 825.2.f.b 4
15.e even 4 1 165.2.d.a 2
15.e even 4 1 165.2.d.b yes 2
33.d even 2 1 inner 825.2.f.b 4
55.d odd 2 1 inner 825.2.f.b 4
55.e even 4 1 165.2.d.a 2
55.e even 4 1 165.2.d.b yes 2
165.d even 2 1 inner 825.2.f.b 4
165.l odd 4 1 165.2.d.a 2
165.l odd 4 1 165.2.d.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.a 2 5.c odd 4 1
165.2.d.a 2 15.e even 4 1
165.2.d.a 2 55.e even 4 1
165.2.d.a 2 165.l odd 4 1
165.2.d.b yes 2 5.c odd 4 1
165.2.d.b yes 2 15.e even 4 1
165.2.d.b yes 2 55.e even 4 1
165.2.d.b yes 2 165.l odd 4 1
825.2.f.b 4 1.a even 1 1 trivial
825.2.f.b 4 3.b odd 2 1 inner
825.2.f.b 4 5.b even 2 1 inner
825.2.f.b 4 11.b odd 2 1 CM
825.2.f.b 4 15.d odd 2 1 inner
825.2.f.b 4 33.d even 2 1 inner
825.2.f.b 4 55.d odd 2 1 inner
825.2.f.b 4 165.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2} \)
\( T_{23}^{2} + 81 \)
\( T_{37}^{2} - 99 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 5 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 11 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 81 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 5 + T )^{4} \)
$37$ \( ( -99 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 144 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 11 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -99 + T^{2} )^{2} \)
$71$ \( ( 275 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 275 + T^{2} )^{2} \)
$97$ \( ( -99 + T^{2} )^{2} \)
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