Properties

Label 825.2.k.d
Level $825$
Weight $2$
Character orbit 825.k
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{1} q^{3} + \beta_{2} q^{4} -3 \beta_{2} q^{6} + 2 \beta_{3} q^{7} -\beta_{3} q^{8} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{1} q^{3} + \beta_{2} q^{4} -3 \beta_{2} q^{6} + 2 \beta_{3} q^{7} -\beta_{3} q^{8} + 3 \beta_{2} q^{9} -\beta_{2} q^{11} -\beta_{3} q^{12} + 2 \beta_{1} q^{13} -6 q^{14} + 5 q^{16} + 4 \beta_{1} q^{17} + 3 \beta_{3} q^{18} -2 \beta_{2} q^{19} + 6 q^{21} -\beta_{3} q^{22} + 4 \beta_{3} q^{23} -3 q^{24} + 6 \beta_{2} q^{26} -3 \beta_{3} q^{27} -2 \beta_{1} q^{28} + 6 q^{29} + 4 q^{31} + 3 \beta_{1} q^{32} + \beta_{3} q^{33} + 12 \beta_{2} q^{34} -3 q^{36} -2 \beta_{3} q^{38} -6 \beta_{2} q^{39} + 6 \beta_{2} q^{41} + 6 \beta_{1} q^{42} + 6 \beta_{1} q^{43} + q^{44} -12 q^{46} -4 \beta_{1} q^{47} -5 \beta_{1} q^{48} -5 \beta_{2} q^{49} -12 \beta_{2} q^{51} + 2 \beta_{3} q^{52} + 4 \beta_{3} q^{53} + 9 q^{54} + 6 \beta_{2} q^{56} + 2 \beta_{3} q^{57} + 6 \beta_{1} q^{58} + 2 q^{61} + 4 \beta_{1} q^{62} -6 \beta_{1} q^{63} -\beta_{2} q^{64} -3 q^{66} + 2 \beta_{3} q^{67} + 4 \beta_{3} q^{68} + 12 q^{69} -12 \beta_{2} q^{71} + 3 \beta_{1} q^{72} -2 \beta_{1} q^{73} + 2 q^{76} + 2 \beta_{1} q^{77} -6 \beta_{3} q^{78} -10 \beta_{2} q^{79} -9 q^{81} + 6 \beta_{3} q^{82} + 6 \beta_{3} q^{83} + 6 \beta_{2} q^{84} + 18 \beta_{2} q^{86} -6 \beta_{1} q^{87} -\beta_{1} q^{88} -12 q^{89} -12 q^{91} -4 \beta_{1} q^{92} -4 \beta_{1} q^{93} -12 \beta_{2} q^{94} -9 \beta_{2} q^{96} -8 \beta_{3} q^{97} -5 \beta_{3} q^{98} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 24 q^{14} + 20 q^{16} + 24 q^{21} - 12 q^{24} + 24 q^{29} + 16 q^{31} - 12 q^{36} + 4 q^{44} - 48 q^{46} + 36 q^{54} + 8 q^{61} - 12 q^{66} + 48 q^{69} + 8 q^{76} - 36 q^{81} - 48 q^{89} - 48 q^{91} + 12 q^{99} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
518.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i 1.22474 + 1.22474i 1.00000i 0 3.00000i 2.44949 2.44949i −1.22474 + 1.22474i 3.00000i 0
518.2 1.22474 + 1.22474i −1.22474 1.22474i 1.00000i 0 3.00000i −2.44949 + 2.44949i 1.22474 1.22474i 3.00000i 0
782.1 −1.22474 + 1.22474i 1.22474 1.22474i 1.00000i 0 3.00000i 2.44949 + 2.44949i −1.22474 1.22474i 3.00000i 0
782.2 1.22474 1.22474i −1.22474 + 1.22474i 1.00000i 0 3.00000i −2.44949 2.44949i 1.22474 + 1.22474i 3.00000i 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
5.b even 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.k.d 4
3.b odd 2 1 825.2.k.e yes 4
5.b even 2 1 inner 825.2.k.d 4
5.c odd 4 2 825.2.k.e yes 4
15.d odd 2 1 825.2.k.e yes 4
15.e even 4 2 inner 825.2.k.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.k.d 4 1.a even 1 1 trivial
825.2.k.d 4 5.b even 2 1 inner
825.2.k.d 4 15.e even 4 2 inner
825.2.k.e yes 4 3.b odd 2 1
825.2.k.e yes 4 5.c odd 4 2
825.2.k.e yes 4 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{4} + 9 \)
\( T_{7}^{4} + 144 \)
\( T_{13}^{4} + 144 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 144 + T^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( 144 + T^{4} \)
$17$ \( 2304 + T^{4} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( 2304 + T^{4} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 36 + T^{2} )^{2} \)
$43$ \( 11664 + T^{4} \)
$47$ \( 2304 + T^{4} \)
$53$ \( 2304 + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( 144 + T^{4} \)
$71$ \( ( 144 + T^{2} )^{2} \)
$73$ \( 144 + T^{4} \)
$79$ \( ( 100 + T^{2} )^{2} \)
$83$ \( 11664 + T^{4} \)
$89$ \( ( 12 + T )^{4} \)
$97$ \( 36864 + T^{4} \)
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