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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{6} - 4q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{4} - q^{6} - 4q^{7} + 3q^{8} + q^{9} + q^{11} - q^{12} + 2q^{13} + 4q^{14} - q^{16} + 2q^{17} - q^{18} - 4q^{21} - q^{22} - 8q^{23} + 3q^{24} - 2q^{26} + q^{27} + 4q^{28} - 6q^{29} - 8q^{31} - 5q^{32} + q^{33} - 2q^{34} - q^{36} - 6q^{37} + 2q^{39} - 2q^{41} + 4q^{42} - q^{44} + 8q^{46} - 8q^{47} - q^{48} + 9q^{49} + 2q^{51} - 2q^{52} - 6q^{53} - q^{54} - 12q^{56} + 6q^{58} - 4q^{59} + 6q^{61} + 8q^{62} - 4q^{63} + 7q^{64} - q^{66} + 4q^{67} - 2q^{68} - 8q^{69} + 3q^{72} + 14q^{73} + 6q^{74} - 4q^{77} - 2q^{78} - 4q^{79} + q^{81} + 2q^{82} - 12q^{83} + 4q^{84} - 6q^{87} + 3q^{88} - 6q^{89} - 8q^{91} + 8q^{92} - 8q^{93} + 8q^{94} - 5q^{96} - 2q^{97} - 9q^{98} + q^{99} + O(q^{100}) \)

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
−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:

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

Hecke kernels

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

\( T_{2} + 1 \)
\( T_{7} + 4 \)

Hecke characteristic polynomials

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