Properties

Label 825.6.a.a
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 9 q^{3} - 31 q^{4} + 9 q^{6} + 26 q^{7} + 63 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 9 q^{3} - 31 q^{4} + 9 q^{6} + 26 q^{7} + 63 q^{8} + 81 q^{9} + 121 q^{11} + 279 q^{12} + 692 q^{13} - 26 q^{14} + 929 q^{16} + 1442 q^{17} - 81 q^{18} + 2160 q^{19} - 234 q^{21} - 121 q^{22} + 1582 q^{23} - 567 q^{24} - 692 q^{26} - 729 q^{27} - 806 q^{28} - 5526 q^{29} + 4792 q^{31} - 2945 q^{32} - 1089 q^{33} - 1442 q^{34} - 2511 q^{36} + 10194 q^{37} - 2160 q^{38} - 6228 q^{39} - 10622 q^{41} + 234 q^{42} - 8580 q^{43} - 3751 q^{44} - 1582 q^{46} + 2362 q^{47} - 8361 q^{48} - 16131 q^{49} - 12978 q^{51} - 21452 q^{52} + 30804 q^{53} + 729 q^{54} + 1638 q^{56} - 19440 q^{57} + 5526 q^{58} + 6416 q^{59} + 42096 q^{61} - 4792 q^{62} + 2106 q^{63} - 26783 q^{64} + 1089 q^{66} + 28444 q^{67} - 44702 q^{68} - 14238 q^{69} + 45690 q^{71} + 5103 q^{72} + 18374 q^{73} - 10194 q^{74} - 66960 q^{76} + 3146 q^{77} + 6228 q^{78} - 105214 q^{79} + 6561 q^{81} + 10622 q^{82} - 62292 q^{83} + 7254 q^{84} + 8580 q^{86} + 49734 q^{87} + 7623 q^{88} - 72246 q^{89} + 17992 q^{91} - 49042 q^{92} - 43128 q^{93} - 2362 q^{94} + 26505 q^{96} - 79262 q^{97} + 16131 q^{98} + 9801 q^{99}+O(q^{100}) \) 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
0
−1.00000 −9.00000 −31.0000 0 9.00000 26.0000 63.0000 81.0000 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 825.6.a.a 1
5.b even 2 1 33.6.a.b 1
15.d odd 2 1 99.6.a.a 1
20.d odd 2 1 528.6.a.a 1
55.d odd 2 1 363.6.a.b 1
165.d even 2 1 1089.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.b 1 5.b even 2 1
99.6.a.a 1 15.d odd 2 1
363.6.a.b 1 55.d odd 2 1
528.6.a.a 1 20.d odd 2 1
825.6.a.a 1 1.a even 1 1 trivial
1089.6.a.h 1 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 26 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T - 692 \) Copy content Toggle raw display
$17$ \( T - 1442 \) Copy content Toggle raw display
$19$ \( T - 2160 \) Copy content Toggle raw display
$23$ \( T - 1582 \) Copy content Toggle raw display
$29$ \( T + 5526 \) Copy content Toggle raw display
$31$ \( T - 4792 \) Copy content Toggle raw display
$37$ \( T - 10194 \) Copy content Toggle raw display
$41$ \( T + 10622 \) Copy content Toggle raw display
$43$ \( T + 8580 \) Copy content Toggle raw display
$47$ \( T - 2362 \) Copy content Toggle raw display
$53$ \( T - 30804 \) Copy content Toggle raw display
$59$ \( T - 6416 \) Copy content Toggle raw display
$61$ \( T - 42096 \) Copy content Toggle raw display
$67$ \( T - 28444 \) Copy content Toggle raw display
$71$ \( T - 45690 \) Copy content Toggle raw display
$73$ \( T - 18374 \) Copy content Toggle raw display
$79$ \( T + 105214 \) Copy content Toggle raw display
$83$ \( T + 62292 \) Copy content Toggle raw display
$89$ \( T + 72246 \) Copy content Toggle raw display
$97$ \( T + 79262 \) Copy content Toggle raw display
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