Properties

Label 285.2.a.a
Level $285$
Weight $2$
Character orbit 285.a
Self dual yes
Analytic conductor $2.276$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.27573645761\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + 3q^{8} + q^{9} + q^{10} - 6q^{11} - q^{12} + 2q^{14} - q^{15} - q^{16} - 6q^{17} - q^{18} + q^{19} + q^{20} - 2q^{21} + 6q^{22} - 8q^{23} + 3q^{24} + q^{25} + q^{27} + 2q^{28} + 4q^{29} + q^{30} - 5q^{32} - 6q^{33} + 6q^{34} + 2q^{35} - q^{36} + 4q^{37} - q^{38} - 3q^{40} + 2q^{42} - 2q^{43} + 6q^{44} - q^{45} + 8q^{46} - 8q^{47} - q^{48} - 3q^{49} - q^{50} - 6q^{51} + 2q^{53} - q^{54} + 6q^{55} - 6q^{56} + q^{57} - 4q^{58} + 12q^{59} + q^{60} + 2q^{61} - 2q^{63} + 7q^{64} + 6q^{66} - 8q^{67} + 6q^{68} - 8q^{69} - 2q^{70} + 16q^{71} + 3q^{72} + 14q^{73} - 4q^{74} + q^{75} - q^{76} + 12q^{77} + 8q^{79} + q^{80} + q^{81} + 2q^{84} + 6q^{85} + 2q^{86} + 4q^{87} - 18q^{88} + q^{90} + 8q^{92} + 8q^{94} - q^{95} - 5q^{96} - 12q^{97} + 3q^{98} - 6q^{99} + O(q^{100}) \)

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 −1.00000 −1.00000 −2.00000 3.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 285.2.a.a 1
3.b odd 2 1 855.2.a.c 1
4.b odd 2 1 4560.2.a.h 1
5.b even 2 1 1425.2.a.g 1
5.c odd 4 2 1425.2.c.c 2
15.d odd 2 1 4275.2.a.h 1
19.b odd 2 1 5415.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.a 1 1.a even 1 1 trivial
855.2.a.c 1 3.b odd 2 1
1425.2.a.g 1 5.b even 2 1
1425.2.c.c 2 5.c odd 4 2
4275.2.a.h 1 15.d odd 2 1
4560.2.a.h 1 4.b odd 2 1
5415.2.a.h 1 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(285))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( 2 + T \)
$11$ \( 6 + T \)
$13$ \( T \)
$17$ \( 6 + T \)
$19$ \( -1 + T \)
$23$ \( 8 + T \)
$29$ \( -4 + T \)
$31$ \( T \)
$37$ \( -4 + T \)
$41$ \( T \)
$43$ \( 2 + T \)
$47$ \( 8 + T \)
$53$ \( -2 + T \)
$59$ \( -12 + T \)
$61$ \( -2 + T \)
$67$ \( 8 + T \)
$71$ \( -16 + T \)
$73$ \( -14 + T \)
$79$ \( -8 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 12 + T \)
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