Properties

Label 2205.2.a.d
Level $2205$
Weight $2$
Character orbit 2205.a
Self dual yes
Analytic conductor $17.607$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{4} - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - q^{5} + q^{13} + 4 q^{16} + 6 q^{17} - 5 q^{19} + 2 q^{20} - 6 q^{23} + q^{25} + 6 q^{29} - 5 q^{31} - 7 q^{37} + 12 q^{41} - q^{43} + 6 q^{47} - 2 q^{52} - 6 q^{59} - 2 q^{61} - 8 q^{64} - q^{65} - 7 q^{67} - 12 q^{68} - 12 q^{71} - 11 q^{73} + 10 q^{76} - 13 q^{79} - 4 q^{80} - 12 q^{83} - 6 q^{85} + 6 q^{89} + 12 q^{92} + 5 q^{95} + 10 q^{97}+O(q^{100}) \) 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
0 0 −2.00000 −1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 2205.2.a.d 1
3.b odd 2 1 735.2.a.d 1
7.b odd 2 1 2205.2.a.f 1
7.d odd 6 2 315.2.j.b 2
15.d odd 2 1 3675.2.a.i 1
21.c even 2 1 735.2.a.e 1
21.g even 6 2 105.2.i.a 2
21.h odd 6 2 735.2.i.c 2
84.j odd 6 2 1680.2.bg.m 2
105.g even 2 1 3675.2.a.h 1
105.p even 6 2 525.2.i.c 2
105.w odd 12 4 525.2.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.a 2 21.g even 6 2
315.2.j.b 2 7.d odd 6 2
525.2.i.c 2 105.p even 6 2
525.2.r.b 4 105.w odd 12 4
735.2.a.d 1 3.b odd 2 1
735.2.a.e 1 21.c even 2 1
735.2.i.c 2 21.h odd 6 2
1680.2.bg.m 2 84.j odd 6 2
2205.2.a.d 1 1.a even 1 1 trivial
2205.2.a.f 1 7.b odd 2 1
3675.2.a.h 1 105.g even 2 1
3675.2.a.i 1 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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