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

\(f(q)\) \(=\) \( q + 2 q^{2} + 9 q^{3} - 28 q^{4} + 18 q^{6} - 148 q^{7} - 120 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 9 q^{3} - 28 q^{4} + 18 q^{6} - 148 q^{7} - 120 q^{8} + 81 q^{9} + 121 q^{11} - 252 q^{12} - 574 q^{13} - 296 q^{14} + 656 q^{16} + 722 q^{17} + 162 q^{18} + 2160 q^{19} - 1332 q^{21} + 242 q^{22} + 2536 q^{23} - 1080 q^{24} - 1148 q^{26} + 729 q^{27} + 4144 q^{28} + 4650 q^{29} + 5032 q^{31} + 5152 q^{32} + 1089 q^{33} + 1444 q^{34} - 2268 q^{36} - 8118 q^{37} + 4320 q^{38} - 5166 q^{39} - 5138 q^{41} - 2664 q^{42} - 8304 q^{43} - 3388 q^{44} + 5072 q^{46} - 24728 q^{47} + 5904 q^{48} + 5097 q^{49} + 6498 q^{51} + 16072 q^{52} + 28746 q^{53} + 1458 q^{54} + 17760 q^{56} + 19440 q^{57} + 9300 q^{58} - 5860 q^{59} - 53658 q^{61} + 10064 q^{62} - 11988 q^{63} - 10688 q^{64} + 2178 q^{66} - 30908 q^{67} - 20216 q^{68} + 22824 q^{69} - 69648 q^{71} - 9720 q^{72} + 18446 q^{73} - 16236 q^{74} - 60480 q^{76} - 17908 q^{77} - 10332 q^{78} - 25300 q^{79} + 6561 q^{81} - 10276 q^{82} + 17556 q^{83} + 37296 q^{84} - 16608 q^{86} + 41850 q^{87} - 14520 q^{88} + 132570 q^{89} + 84952 q^{91} - 71008 q^{92} + 45288 q^{93} - 49456 q^{94} + 46368 q^{96} - 70658 q^{97} + 10194 q^{98} + 9801 q^{99}+O(q^{100}) \) 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
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

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