Properties

Label 825.2.a.n
Level $825$
Weight $2$
Character orbit 825.a
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{2} - \beta_1 + 3) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{2} - \beta_1 + 3) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9} - q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{12} + ( - \beta_{2} + 3 \beta_1 + 1) q^{13} + (3 \beta_{2} - \beta_1 + 1) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + (3 \beta_{2} + \beta_1 + 1) q^{17} + (\beta_{2} + 1) q^{18} + (2 \beta_{2} + 4 \beta_1 - 2) q^{19} + (\beta_{2} + \beta_1 - 3) q^{21} + ( - \beta_{2} - 1) q^{22} + 4 q^{23} + (3 \beta_1 - 4) q^{24} + (\beta_{2} + 7 \beta_1 - 5) q^{26} - q^{27} + (3 \beta_{2} - 3 \beta_1 + 5) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{29} + (2 \beta_1 + 2) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + q^{33} + (\beta_{2} - \beta_1 + 9) q^{34} + (\beta_{2} - \beta_1 + 2) q^{36} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{2} + 6 \beta_1) q^{38} + (\beta_{2} - 3 \beta_1 - 1) q^{39} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 3 \beta_{2} + \beta_1 - 1) q^{42} + ( - \beta_{2} - \beta_1 + 3) q^{43} + ( - \beta_{2} + \beta_1 - 2) q^{44} + (4 \beta_{2} + 4) q^{46} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{2} + 4 \beta_1 - 3) q^{48} + ( - 6 \beta_{2} - 4 \beta_1 + 5) q^{49} + ( - 3 \beta_{2} - \beta_1 - 1) q^{51} + ( - 3 \beta_{2} + 7 \beta_1 - 11) q^{52} + (2 \beta_{2} + 4 \beta_1 - 4) q^{53} + ( - \beta_{2} - 1) q^{54} + ( - \beta_{2} - 7 \beta_1 + 15) q^{56} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{58} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{59} + ( - 4 \beta_1 + 2) q^{61} + (2 \beta_{2} + 4 \beta_1) q^{62} + ( - \beta_{2} - \beta_1 + 3) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + (\beta_{2} + 1) q^{66} + (4 \beta_{2} + 4) q^{67} + (3 \beta_{2} - 5 \beta_1 + 11) q^{68} - 4 q^{69} + ( - 2 \beta_{2} - 4) q^{71} + ( - 3 \beta_1 + 4) q^{72} + ( - 3 \beta_{2} - 3 \beta_1 + 7) q^{73} + (2 \beta_{2} - 2 \beta_1 - 2) q^{74} + ( - 4 \beta_{2} + 6 \beta_1 - 8) q^{76} + (\beta_{2} + \beta_1 - 3) q^{77} + ( - \beta_{2} - 7 \beta_1 + 5) q^{78} + (2 \beta_{2} - 2) q^{79} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{82} + (\beta_{2} + \beta_1 - 1) q^{83} + ( - 3 \beta_{2} + 3 \beta_1 - 5) q^{84} + (3 \beta_{2} - \beta_1 + 1) q^{86} + (2 \beta_{2} + 2 \beta_1 + 2) q^{87} + (3 \beta_1 - 4) q^{88} + 2 \beta_1 q^{89} + ( - 8 \beta_{2} + 2 \beta_1 + 2) q^{91} + (4 \beta_{2} - 4 \beta_1 + 8) q^{92} + ( - 2 \beta_1 - 2) q^{93} + (2 \beta_{2} + 6 \beta_1 - 6) q^{94} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{96} + (4 \beta_1 - 4) q^{97} + (5 \beta_{2} - 2 \beta_1 - 9) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 2 q^{19} - 8 q^{21} - 3 q^{22} + 12 q^{23} - 9 q^{24} - 8 q^{26} - 3 q^{27} + 12 q^{28} - 8 q^{29} + 8 q^{31} + 11 q^{32} + 3 q^{33} + 26 q^{34} + 5 q^{36} + 4 q^{37} + 6 q^{38} - 6 q^{39} - 8 q^{41} - 2 q^{42} + 8 q^{43} - 5 q^{44} + 12 q^{46} + 8 q^{47} - 5 q^{48} + 11 q^{49} - 4 q^{51} - 26 q^{52} - 8 q^{53} - 3 q^{54} + 38 q^{56} + 2 q^{57} - 20 q^{58} - 8 q^{59} + 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} + 3 q^{66} + 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} - 8 q^{74} - 18 q^{76} - 8 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 20 q^{82} - 2 q^{83} - 12 q^{84} + 2 q^{86} + 8 q^{87} - 9 q^{88} + 2 q^{89} + 8 q^{91} + 20 q^{92} - 8 q^{93} - 12 q^{94} - 11 q^{96} - 8 q^{97} - 29 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
0.311108
2.17009
−1.48119
−1.21432 −1.00000 −0.525428 0 1.21432 4.90321 3.06668 1.00000 0
1.2 1.53919 −1.00000 0.369102 0 −1.53919 0.290725 −2.51026 1.00000 0
1.3 2.67513 −1.00000 5.15633 0 −2.67513 2.80606 8.44358 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.n 3
3.b odd 2 1 2475.2.a.y 3
5.b even 2 1 825.2.a.h 3
5.c odd 4 2 165.2.c.a 6
11.b odd 2 1 9075.2.a.cc 3
15.d odd 2 1 2475.2.a.be 3
15.e even 4 2 495.2.c.d 6
20.e even 4 2 2640.2.d.i 6
55.d odd 2 1 9075.2.a.ck 3
55.e even 4 2 1815.2.c.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.a 6 5.c odd 4 2
495.2.c.d 6 15.e even 4 2
825.2.a.h 3 5.b even 2 1
825.2.a.n 3 1.a even 1 1 trivial
1815.2.c.d 6 55.e even 4 2
2475.2.a.y 3 3.b odd 2 1
2475.2.a.be 3 15.d odd 2 1
2640.2.d.i 6 20.e even 4 2
9075.2.a.cc 3 11.b odd 2 1
9075.2.a.ck 3 55.d odd 2 1

Hecke kernels

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

\( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{3} - 8T_{7}^{2} + 16T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 8 T^{2} + 16 T - 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} - 28 T + 148 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 28 T + 116 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 52 T - 184 \) Copy content Toggle raw display
$23$ \( (T - 4)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 8T^{2} - 32 \) Copy content Toggle raw display
$31$ \( T^{3} - 8 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$41$ \( T^{3} + 8T^{2} - 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} + 16 T - 4 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 32 T - 272 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 64 T + 80 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 52 T + 40 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} - 16 T + 320 \) Copy content Toggle raw display
$71$ \( T^{3} + 12 T^{2} + 32 T - 16 \) Copy content Toggle raw display
$73$ \( T^{3} - 18 T^{2} + 60 T + 92 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$83$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 32 T - 128 \) Copy content Toggle raw display
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