Properties

Label 306.2.a.b
Level $306$
Weight $2$
Character orbit 306.a
Self dual yes
Analytic conductor $2.443$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 306.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.44342230185\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 102)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2q^{5} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + 2q^{5} - q^{8} - 2q^{10} + 4q^{11} - 2q^{13} + q^{16} - q^{17} + 4q^{19} + 2q^{20} - 4q^{22} - q^{25} + 2q^{26} + 10q^{29} + 8q^{31} - q^{32} + q^{34} - 2q^{37} - 4q^{38} - 2q^{40} - 10q^{41} + 12q^{43} + 4q^{44} - 7q^{49} + q^{50} - 2q^{52} - 6q^{53} + 8q^{55} - 10q^{58} - 12q^{59} - 10q^{61} - 8q^{62} + q^{64} - 4q^{65} - 12q^{67} - q^{68} + 10q^{73} + 2q^{74} + 4q^{76} - 8q^{79} + 2q^{80} + 10q^{82} - 4q^{83} - 2q^{85} - 12q^{86} - 4q^{88} + 6q^{89} + 8q^{95} - 14q^{97} + 7q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(1\)

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 306.2.a.b 1
3.b odd 2 1 102.2.a.c 1
4.b odd 2 1 2448.2.a.p 1
5.b even 2 1 7650.2.a.ca 1
8.b even 2 1 9792.2.a.k 1
8.d odd 2 1 9792.2.a.l 1
12.b even 2 1 816.2.a.b 1
15.d odd 2 1 2550.2.a.c 1
15.e even 4 2 2550.2.d.m 2
17.b even 2 1 5202.2.a.c 1
21.c even 2 1 4998.2.a.be 1
24.f even 2 1 3264.2.a.bc 1
24.h odd 2 1 3264.2.a.m 1
51.c odd 2 1 1734.2.a.j 1
51.f odd 4 2 1734.2.b.b 2
51.g odd 8 4 1734.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 3.b odd 2 1
306.2.a.b 1 1.a even 1 1 trivial
816.2.a.b 1 12.b even 2 1
1734.2.a.j 1 51.c odd 2 1
1734.2.b.b 2 51.f odd 4 2
1734.2.f.e 4 51.g odd 8 4
2448.2.a.p 1 4.b odd 2 1
2550.2.a.c 1 15.d odd 2 1
2550.2.d.m 2 15.e even 4 2
3264.2.a.m 1 24.h odd 2 1
3264.2.a.bc 1 24.f even 2 1
4998.2.a.be 1 21.c even 2 1
5202.2.a.c 1 17.b even 2 1
7650.2.a.ca 1 5.b even 2 1
9792.2.a.k 1 8.b even 2 1
9792.2.a.l 1 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(306))\):

\( T_{5} - 2 \)
\( T_{7} \)
\( T_{23} \)

Hecke characteristic polynomials

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