Properties

Label 825.6.a.b
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 9q^{3} - 28q^{4} + 18q^{6} - 148q^{7} - 120q^{8} + 81q^{9} + O(q^{10}) \) \( q + 2q^{2} + 9q^{3} - 28q^{4} + 18q^{6} - 148q^{7} - 120q^{8} + 81q^{9} + 121q^{11} - 252q^{12} - 574q^{13} - 296q^{14} + 656q^{16} + 722q^{17} + 162q^{18} + 2160q^{19} - 1332q^{21} + 242q^{22} + 2536q^{23} - 1080q^{24} - 1148q^{26} + 729q^{27} + 4144q^{28} + 4650q^{29} + 5032q^{31} + 5152q^{32} + 1089q^{33} + 1444q^{34} - 2268q^{36} - 8118q^{37} + 4320q^{38} - 5166q^{39} - 5138q^{41} - 2664q^{42} - 8304q^{43} - 3388q^{44} + 5072q^{46} - 24728q^{47} + 5904q^{48} + 5097q^{49} + 6498q^{51} + 16072q^{52} + 28746q^{53} + 1458q^{54} + 17760q^{56} + 19440q^{57} + 9300q^{58} - 5860q^{59} - 53658q^{61} + 10064q^{62} - 11988q^{63} - 10688q^{64} + 2178q^{66} - 30908q^{67} - 20216q^{68} + 22824q^{69} - 69648q^{71} - 9720q^{72} + 18446q^{73} - 16236q^{74} - 60480q^{76} - 17908q^{77} - 10332q^{78} - 25300q^{79} + 6561q^{81} - 10276q^{82} + 17556q^{83} + 37296q^{84} - 16608q^{86} + 41850q^{87} - 14520q^{88} + 132570q^{89} + 84952q^{91} - 71008q^{92} + 45288q^{93} - 49456q^{94} + 46368q^{96} - 70658q^{97} + 10194q^{98} + 9801q^{99} + O(q^{100}) \)

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
0
2.00000 9.00000 −28.0000 0 18.0000 −148.000 −120.000 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.b 1
5.b even 2 1 33.6.a.a 1
15.d odd 2 1 99.6.a.b 1
20.d odd 2 1 528.6.a.i 1
55.d odd 2 1 363.6.a.c 1
165.d even 2 1 1089.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 5.b even 2 1
99.6.a.b 1 15.d odd 2 1
363.6.a.c 1 55.d odd 2 1
528.6.a.i 1 20.d odd 2 1
825.6.a.b 1 1.a even 1 1 trivial
1089.6.a.d 1 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -9 + T \)
$5$ \( T \)
$7$ \( 148 + T \)
$11$ \( -121 + T \)
$13$ \( 574 + T \)
$17$ \( -722 + T \)
$19$ \( -2160 + T \)
$23$ \( -2536 + T \)
$29$ \( -4650 + T \)
$31$ \( -5032 + T \)
$37$ \( 8118 + T \)
$41$ \( 5138 + T \)
$43$ \( 8304 + T \)
$47$ \( 24728 + T \)
$53$ \( -28746 + T \)
$59$ \( 5860 + T \)
$61$ \( 53658 + T \)
$67$ \( 30908 + T \)
$71$ \( 69648 + T \)
$73$ \( -18446 + T \)
$79$ \( 25300 + T \)
$83$ \( -17556 + T \)
$89$ \( -132570 + T \)
$97$ \( 70658 + T \)
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