Properties

Label 285.2.a.c
Level $285$
Weight $2$
Character orbit 285.a
Self dual yes
Analytic conductor $2.276$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} - q^{16} + 2 q^{17} + q^{18} - q^{19} - q^{20} - 4 q^{21} + 4 q^{22} - 4 q^{23} + 3 q^{24} + q^{25} + 2 q^{26} - q^{27} - 4 q^{28} - 2 q^{29} - q^{30} + 5 q^{32} - 4 q^{33} + 2 q^{34} + 4 q^{35} - q^{36} - 6 q^{37} - q^{38} - 2 q^{39} - 3 q^{40} - 6 q^{41} - 4 q^{42} + 8 q^{43} - 4 q^{44} + q^{45} - 4 q^{46} - 12 q^{47} + q^{48} + 9 q^{49} + q^{50} - 2 q^{51} - 2 q^{52} - 14 q^{53} - q^{54} + 4 q^{55} - 12 q^{56} + q^{57} - 2 q^{58} + 4 q^{59} + q^{60} + 14 q^{61} + 4 q^{63} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 4 q^{67} - 2 q^{68} + 4 q^{69} + 4 q^{70} - 3 q^{72} - 14 q^{73} - 6 q^{74} - q^{75} + q^{76} + 16 q^{77} - 2 q^{78} + 16 q^{79} - q^{80} + q^{81} - 6 q^{82} + 4 q^{84} + 2 q^{85} + 8 q^{86} + 2 q^{87} - 12 q^{88} - 6 q^{89} + q^{90} + 8 q^{91} + 4 q^{92} - 12 q^{94} - q^{95} - 5 q^{96} - 10 q^{97} + 9 q^{98} + 4 q^{99}+O(q^{100}) \) 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
1.00000 −1.00000 −1.00000 1.00000 −1.00000 4.00000 −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.c 1
3.b odd 2 1 855.2.a.a 1
4.b odd 2 1 4560.2.a.w 1
5.b even 2 1 1425.2.a.c 1
5.c odd 4 2 1425.2.c.f 2
15.d odd 2 1 4275.2.a.j 1
19.b odd 2 1 5415.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.c 1 1.a even 1 1 trivial
855.2.a.a 1 3.b odd 2 1
1425.2.a.c 1 5.b even 2 1
1425.2.c.f 2 5.c odd 4 2
4275.2.a.j 1 15.d odd 2 1
4560.2.a.w 1 4.b odd 2 1
5415.2.a.e 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(285))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 14 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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