Properties

Label 825.2.a.g
Level $825$
Weight $2$
Character orbit 825.a
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + q^{3} + ( 1 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + ( 3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + q^{3} + ( 1 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + ( 3 + \beta ) q^{8} + q^{9} - q^{11} + ( 1 + 2 \beta ) q^{12} + 4 \beta q^{13} -2 q^{14} + 3 q^{16} + ( 4 - 2 \beta ) q^{17} + ( 1 + \beta ) q^{18} + ( -4 - 2 \beta ) q^{19} + ( 2 - 2 \beta ) q^{21} + ( -1 - \beta ) q^{22} + 4 q^{23} + ( 3 + \beta ) q^{24} + ( 8 + 4 \beta ) q^{26} + q^{27} + ( -6 + 2 \beta ) q^{28} + ( -2 - 2 \beta ) q^{29} + ( -3 + \beta ) q^{32} - q^{33} + 2 \beta q^{34} + ( 1 + 2 \beta ) q^{36} + ( -6 - 4 \beta ) q^{37} + ( -8 - 6 \beta ) q^{38} + 4 \beta q^{39} + ( 2 + 2 \beta ) q^{41} -2 q^{42} + ( 6 + 2 \beta ) q^{43} + ( -1 - 2 \beta ) q^{44} + ( 4 + 4 \beta ) q^{46} + 4 q^{47} + 3 q^{48} + ( 5 - 8 \beta ) q^{49} + ( 4 - 2 \beta ) q^{51} + ( 16 + 4 \beta ) q^{52} + ( 2 - 8 \beta ) q^{53} + ( 1 + \beta ) q^{54} + ( 2 - 4 \beta ) q^{56} + ( -4 - 2 \beta ) q^{57} + ( -6 - 4 \beta ) q^{58} -4 q^{59} + ( -6 - 4 \beta ) q^{61} + ( 2 - 2 \beta ) q^{63} + ( -7 - 2 \beta ) q^{64} + ( -1 - \beta ) q^{66} + 4 \beta q^{67} + ( -4 + 6 \beta ) q^{68} + 4 q^{69} + ( 8 - 4 \beta ) q^{71} + ( 3 + \beta ) q^{72} -8 \beta q^{73} + ( -14 - 10 \beta ) q^{74} + ( -12 - 10 \beta ) q^{76} + ( -2 + 2 \beta ) q^{77} + ( 8 + 4 \beta ) q^{78} + 6 \beta q^{79} + q^{81} + ( 6 + 4 \beta ) q^{82} + 10 q^{83} + ( -6 + 2 \beta ) q^{84} + ( 10 + 8 \beta ) q^{86} + ( -2 - 2 \beta ) q^{87} + ( -3 - \beta ) q^{88} + ( -2 + 4 \beta ) q^{89} + ( -16 + 8 \beta ) q^{91} + ( 4 + 8 \beta ) q^{92} + ( 4 + 4 \beta ) q^{94} + ( -3 + \beta ) q^{96} + ( -6 - 4 \beta ) q^{97} + ( -11 - 3 \beta ) q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} - 2q^{11} + 2q^{12} - 4q^{14} + 6q^{16} + 8q^{17} + 2q^{18} - 8q^{19} + 4q^{21} - 2q^{22} + 8q^{23} + 6q^{24} + 16q^{26} + 2q^{27} - 12q^{28} - 4q^{29} - 6q^{32} - 2q^{33} + 2q^{36} - 12q^{37} - 16q^{38} + 4q^{41} - 4q^{42} + 12q^{43} - 2q^{44} + 8q^{46} + 8q^{47} + 6q^{48} + 10q^{49} + 8q^{51} + 32q^{52} + 4q^{53} + 2q^{54} + 4q^{56} - 8q^{57} - 12q^{58} - 8q^{59} - 12q^{61} + 4q^{63} - 14q^{64} - 2q^{66} - 8q^{68} + 8q^{69} + 16q^{71} + 6q^{72} - 28q^{74} - 24q^{76} - 4q^{77} + 16q^{78} + 2q^{81} + 12q^{82} + 20q^{83} - 12q^{84} + 20q^{86} - 4q^{87} - 6q^{88} - 4q^{89} - 32q^{91} + 8q^{92} + 8q^{94} - 6q^{96} - 12q^{97} - 22q^{98} - 2q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−0.414214 1.00000 −1.82843 0 −0.414214 4.82843 1.58579 1.00000 0
1.2 2.41421 1.00000 3.82843 0 2.41421 −0.828427 4.41421 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.g 2
3.b odd 2 1 2475.2.a.m 2
5.b even 2 1 165.2.a.a 2
5.c odd 4 2 825.2.c.e 4
11.b odd 2 1 9075.2.a.v 2
15.d odd 2 1 495.2.a.d 2
15.e even 4 2 2475.2.c.m 4
20.d odd 2 1 2640.2.a.bb 2
35.c odd 2 1 8085.2.a.ba 2
55.d odd 2 1 1815.2.a.k 2
60.h even 2 1 7920.2.a.cg 2
165.d even 2 1 5445.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 5.b even 2 1
495.2.a.d 2 15.d odd 2 1
825.2.a.g 2 1.a even 1 1 trivial
825.2.c.e 4 5.c odd 4 2
1815.2.a.k 2 55.d odd 2 1
2475.2.a.m 2 3.b odd 2 1
2475.2.c.m 4 15.e even 4 2
2640.2.a.bb 2 20.d odd 2 1
5445.2.a.m 2 165.d even 2 1
7920.2.a.cg 2 60.h even 2 1
8085.2.a.ba 2 35.c odd 2 1
9075.2.a.v 2 11.b odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{7}^{2} - 4 T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 - 4 T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( 8 + 8 T + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( -4 + 4 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + 12 T + T^{2} \)
$41$ \( -4 - 4 T + T^{2} \)
$43$ \( 28 - 12 T + T^{2} \)
$47$ \( ( -4 + T )^{2} \)
$53$ \( -124 - 4 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( 4 + 12 T + T^{2} \)
$67$ \( -32 + T^{2} \)
$71$ \( 32 - 16 T + T^{2} \)
$73$ \( -128 + T^{2} \)
$79$ \( -72 + T^{2} \)
$83$ \( ( -10 + T )^{2} \)
$89$ \( -28 + 4 T + T^{2} \)
$97$ \( 4 + 12 T + T^{2} \)
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