Properties

Label 245.2.a.b
Level $245$
Weight $2$
Character orbit 245.a
Self dual yes
Analytic conductor $1.956$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.95633484952\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 2q^{4} - q^{5} - 6q^{6} + 6q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 2q^{4} - q^{5} - 6q^{6} + 6q^{9} + 2q^{10} + q^{11} + 6q^{12} + 3q^{13} - 3q^{15} - 4q^{16} - 3q^{17} - 12q^{18} + 6q^{19} - 2q^{20} - 2q^{22} - 4q^{23} + q^{25} - 6q^{26} + 9q^{27} - q^{29} + 6q^{30} + 6q^{31} + 8q^{32} + 3q^{33} + 6q^{34} + 12q^{36} - 12q^{38} + 9q^{39} + 6q^{41} - 6q^{43} + 2q^{44} - 6q^{45} + 8q^{46} - 9q^{47} - 12q^{48} - 2q^{50} - 9q^{51} + 6q^{52} - 10q^{53} - 18q^{54} - q^{55} + 18q^{57} + 2q^{58} - 6q^{59} - 6q^{60} - 12q^{62} - 8q^{64} - 3q^{65} - 6q^{66} - 14q^{67} - 6q^{68} - 12q^{69} - 8q^{71} + 6q^{73} + 3q^{75} + 12q^{76} - 18q^{78} - q^{79} + 4q^{80} + 9q^{81} - 12q^{82} + 12q^{83} + 3q^{85} + 12q^{86} - 3q^{87} + 12q^{89} + 12q^{90} - 8q^{92} + 18q^{93} + 18q^{94} - 6q^{95} + 24q^{96} - 15q^{97} + 6q^{99} + O(q^{100}) \)

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
−2.00000 3.00000 2.00000 −1.00000 −6.00000 0 0 6.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 245.2.a.b yes 1
3.b odd 2 1 2205.2.a.l 1
4.b odd 2 1 3920.2.a.a 1
5.b even 2 1 1225.2.a.h 1
5.c odd 4 2 1225.2.b.a 2
7.b odd 2 1 245.2.a.a 1
7.c even 3 2 245.2.e.c 2
7.d odd 6 2 245.2.e.d 2
21.c even 2 1 2205.2.a.j 1
28.d even 2 1 3920.2.a.bj 1
35.c odd 2 1 1225.2.a.j 1
35.f even 4 2 1225.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 7.b odd 2 1
245.2.a.b yes 1 1.a even 1 1 trivial
245.2.e.c 2 7.c even 3 2
245.2.e.d 2 7.d odd 6 2
1225.2.a.h 1 5.b even 2 1
1225.2.a.j 1 35.c odd 2 1
1225.2.b.a 2 5.c odd 4 2
1225.2.b.b 2 35.f even 4 2
2205.2.a.j 1 21.c even 2 1
2205.2.a.l 1 3.b odd 2 1
3920.2.a.a 1 4.b odd 2 1
3920.2.a.bj 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(245))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( -3 + T \)
$17$ \( 3 + T \)
$19$ \( -6 + T \)
$23$ \( 4 + T \)
$29$ \( 1 + T \)
$31$ \( -6 + T \)
$37$ \( T \)
$41$ \( -6 + T \)
$43$ \( 6 + T \)
$47$ \( 9 + T \)
$53$ \( 10 + T \)
$59$ \( 6 + T \)
$61$ \( T \)
$67$ \( 14 + T \)
$71$ \( 8 + T \)
$73$ \( -6 + T \)
$79$ \( 1 + T \)
$83$ \( -12 + T \)
$89$ \( -12 + T \)
$97$ \( 15 + T \)
show more
show less