Properties

Label 825.2.a.a
Level $825$
Weight $2$
Character orbit 825.a
Self dual yes
Analytic conductor $6.588$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{11} - q^{12} + 2 q^{13} + 4 q^{14} - q^{16} + 2 q^{17} - q^{18} - 4 q^{21} - q^{22} - 8 q^{23} + 3 q^{24} - 2 q^{26} + q^{27} + 4 q^{28} - 6 q^{29} - 8 q^{31} - 5 q^{32} + q^{33} - 2 q^{34} - q^{36} - 6 q^{37} + 2 q^{39} - 2 q^{41} + 4 q^{42} - q^{44} + 8 q^{46} - 8 q^{47} - q^{48} + 9 q^{49} + 2 q^{51} - 2 q^{52} - 6 q^{53} - q^{54} - 12 q^{56} + 6 q^{58} - 4 q^{59} + 6 q^{61} + 8 q^{62} - 4 q^{63} + 7 q^{64} - q^{66} + 4 q^{67} - 2 q^{68} - 8 q^{69} + 3 q^{72} + 14 q^{73} + 6 q^{74} - 4 q^{77} - 2 q^{78} - 4 q^{79} + q^{81} + 2 q^{82} - 12 q^{83} + 4 q^{84} - 6 q^{87} + 3 q^{88} - 6 q^{89} - 8 q^{91} + 8 q^{92} - 8 q^{93} + 8 q^{94} - 5 q^{96} - 2 q^{97} - 9 q^{98} + q^{99}+O(q^{100}) \) 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 1.00000 −1.00000 0 −1.00000 −4.00000 3.00000 1.00000 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 825.2.a.a 1
3.b odd 2 1 2475.2.a.g 1
5.b even 2 1 33.2.a.a 1
5.c odd 4 2 825.2.c.a 2
11.b odd 2 1 9075.2.a.q 1
15.d odd 2 1 99.2.a.b 1
15.e even 4 2 2475.2.c.d 2
20.d odd 2 1 528.2.a.g 1
35.c odd 2 1 1617.2.a.j 1
40.e odd 2 1 2112.2.a.j 1
40.f even 2 1 2112.2.a.bb 1
45.h odd 6 2 891.2.e.g 2
45.j even 6 2 891.2.e.e 2
55.d odd 2 1 363.2.a.b 1
55.h odd 10 4 363.2.e.g 4
55.j even 10 4 363.2.e.e 4
60.h even 2 1 1584.2.a.o 1
65.d even 2 1 5577.2.a.a 1
85.c even 2 1 9537.2.a.m 1
105.g even 2 1 4851.2.a.b 1
120.i odd 2 1 6336.2.a.x 1
120.m even 2 1 6336.2.a.n 1
165.d even 2 1 1089.2.a.j 1
220.g even 2 1 5808.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 5.b even 2 1
99.2.a.b 1 15.d odd 2 1
363.2.a.b 1 55.d odd 2 1
363.2.e.e 4 55.j even 10 4
363.2.e.g 4 55.h odd 10 4
528.2.a.g 1 20.d odd 2 1
825.2.a.a 1 1.a even 1 1 trivial
825.2.c.a 2 5.c odd 4 2
891.2.e.e 2 45.j even 6 2
891.2.e.g 2 45.h odd 6 2
1089.2.a.j 1 165.d even 2 1
1584.2.a.o 1 60.h even 2 1
1617.2.a.j 1 35.c odd 2 1
2112.2.a.j 1 40.e odd 2 1
2112.2.a.bb 1 40.f even 2 1
2475.2.a.g 1 3.b odd 2 1
2475.2.c.d 2 15.e even 4 2
4851.2.a.b 1 105.g even 2 1
5577.2.a.a 1 65.d even 2 1
5808.2.a.t 1 220.g even 2 1
6336.2.a.n 1 120.m even 2 1
6336.2.a.x 1 120.i odd 2 1
9075.2.a.q 1 11.b odd 2 1
9537.2.a.m 1 85.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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