Properties

Label 605.2.a.b
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.83094932229\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + q^{5} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} + q^{5} + 3q^{8} - 3q^{9} - q^{10} - 2q^{13} - q^{16} - 6q^{17} + 3q^{18} + 4q^{19} - q^{20} + 4q^{23} + q^{25} + 2q^{26} - 6q^{29} - 8q^{31} - 5q^{32} + 6q^{34} + 3q^{36} - 2q^{37} - 4q^{38} + 3q^{40} - 2q^{41} - 4q^{43} - 3q^{45} - 4q^{46} - 12q^{47} - 7q^{49} - q^{50} + 2q^{52} - 2q^{53} + 6q^{58} + 4q^{59} + 10q^{61} + 8q^{62} + 7q^{64} - 2q^{65} - 16q^{67} + 6q^{68} + 8q^{71} - 9q^{72} - 14q^{73} + 2q^{74} - 4q^{76} - 8q^{79} - q^{80} + 9q^{81} + 2q^{82} + 4q^{83} - 6q^{85} + 4q^{86} + 10q^{89} + 3q^{90} - 4q^{92} + 12q^{94} + 4q^{95} + 10q^{97} + 7q^{98} + O(q^{100}) \)

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 0 −1.00000 1.00000 0 0 3.00000 −3.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)

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 605.2.a.b 1
3.b odd 2 1 5445.2.a.i 1
4.b odd 2 1 9680.2.a.r 1
5.b even 2 1 3025.2.a.f 1
11.b odd 2 1 55.2.a.a 1
11.c even 5 4 605.2.g.c 4
11.d odd 10 4 605.2.g.a 4
33.d even 2 1 495.2.a.a 1
44.c even 2 1 880.2.a.h 1
55.d odd 2 1 275.2.a.a 1
55.e even 4 2 275.2.b.b 2
77.b even 2 1 2695.2.a.c 1
88.b odd 2 1 3520.2.a.p 1
88.g even 2 1 3520.2.a.n 1
132.d odd 2 1 7920.2.a.i 1
143.d odd 2 1 9295.2.a.b 1
165.d even 2 1 2475.2.a.i 1
165.l odd 4 2 2475.2.c.f 2
220.g even 2 1 4400.2.a.p 1
220.i odd 4 2 4400.2.b.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 11.b odd 2 1
275.2.a.a 1 55.d odd 2 1
275.2.b.b 2 55.e even 4 2
495.2.a.a 1 33.d even 2 1
605.2.a.b 1 1.a even 1 1 trivial
605.2.g.a 4 11.d odd 10 4
605.2.g.c 4 11.c even 5 4
880.2.a.h 1 44.c even 2 1
2475.2.a.i 1 165.d even 2 1
2475.2.c.f 2 165.l odd 4 2
2695.2.a.c 1 77.b even 2 1
3025.2.a.f 1 5.b even 2 1
3520.2.a.n 1 88.g even 2 1
3520.2.a.p 1 88.b odd 2 1
4400.2.a.p 1 220.g even 2 1
4400.2.b.n 2 220.i odd 4 2
5445.2.a.i 1 3.b odd 2 1
7920.2.a.i 1 132.d odd 2 1
9295.2.a.b 1 143.d odd 2 1
9680.2.a.r 1 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2} + 1 \)
\( T_{3} \)

Hecke characteristic polynomials

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