Properties

Label 825.4.a.c
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{2} - 3q^{3} + q^{4} + 9q^{6} - 7q^{7} + 21q^{8} + 9q^{9} + O(q^{10}) \) \( q - 3q^{2} - 3q^{3} + q^{4} + 9q^{6} - 7q^{7} + 21q^{8} + 9q^{9} + 11q^{11} - 3q^{12} - 16q^{13} + 21q^{14} - 71q^{16} + 21q^{17} - 27q^{18} + 125q^{19} + 21q^{21} - 33q^{22} - 81q^{23} - 63q^{24} + 48q^{26} - 27q^{27} - 7q^{28} + 186q^{29} - 58q^{31} + 45q^{32} - 33q^{33} - 63q^{34} + 9q^{36} - 253q^{37} - 375q^{38} + 48q^{39} + 63q^{41} - 63q^{42} - 100q^{43} + 11q^{44} + 243q^{46} - 219q^{47} + 213q^{48} - 294q^{49} - 63q^{51} - 16q^{52} - 192q^{53} + 81q^{54} - 147q^{56} - 375q^{57} - 558q^{58} + 249q^{59} - 64q^{61} + 174q^{62} - 63q^{63} + 433q^{64} + 99q^{66} + 272q^{67} + 21q^{68} + 243q^{69} - 645q^{71} + 189q^{72} - 112q^{73} + 759q^{74} + 125q^{76} - 77q^{77} - 144q^{78} + 509q^{79} + 81q^{81} - 189q^{82} - 1254q^{83} + 21q^{84} + 300q^{86} - 558q^{87} + 231q^{88} + 756q^{89} + 112q^{91} - 81q^{92} + 174q^{93} + 657q^{94} - 135q^{96} + 839q^{97} + 882q^{98} + 99q^{99} + O(q^{100}) \)

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
−3.00000 −3.00000 1.00000 0 9.00000 −7.00000 21.0000 9.00000 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 825.4.a.c 1
3.b odd 2 1 2475.4.a.i 1
5.b even 2 1 825.4.a.g yes 1
5.c odd 4 2 825.4.c.d 2
15.d odd 2 1 2475.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.c 1 1.a even 1 1 trivial
825.4.a.g yes 1 5.b even 2 1
825.4.c.d 2 5.c odd 4 2
2475.4.a.d 1 15.d odd 2 1
2475.4.a.i 1 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2} + 3 \)
\( T_{7} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( 7 + T \)
$11$ \( -11 + T \)
$13$ \( 16 + T \)
$17$ \( -21 + T \)
$19$ \( -125 + T \)
$23$ \( 81 + T \)
$29$ \( -186 + T \)
$31$ \( 58 + T \)
$37$ \( 253 + T \)
$41$ \( -63 + T \)
$43$ \( 100 + T \)
$47$ \( 219 + T \)
$53$ \( 192 + T \)
$59$ \( -249 + T \)
$61$ \( 64 + T \)
$67$ \( -272 + T \)
$71$ \( 645 + T \)
$73$ \( 112 + T \)
$79$ \( -509 + T \)
$83$ \( 1254 + T \)
$89$ \( -756 + T \)
$97$ \( -839 + T \)
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