Properties

Label 308.2.a.c
Level 308
Weight 2
Character orbit 308.a
Self dual Yes
Analytic conductor 2.459
Analytic rank 0
Dimension 3
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 308.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + q^{11} + ( 4 + \beta_{2} ) q^{13} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{15} + ( 2 \beta_{1} - \beta_{2} ) q^{17} -2 \beta_{1} q^{19} -\beta_{1} q^{21} + ( -2 - \beta_{1} + \beta_{2} ) q^{23} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{25} + ( -2 - \beta_{1} - \beta_{2} ) q^{27} + ( 2 - 2 \beta_{2} ) q^{29} + ( -4 + 3 \beta_{1} ) q^{31} -\beta_{1} q^{33} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( 2 + \beta_{1} + \beta_{2} ) q^{37} + ( 2 - 6 \beta_{1} ) q^{39} + ( -2 \beta_{1} + \beta_{2} ) q^{41} + ( -2 + 2 \beta_{1} ) q^{43} + ( -10 - 4 \beta_{1} ) q^{45} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( -10 - 2 \beta_{2} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -\beta_{1} - \beta_{2} ) q^{55} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( -8 + \beta_{1} ) q^{59} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{61} + ( 1 + \beta_{1} + \beta_{2} ) q^{63} + ( -8 - 4 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} ) q^{67} + ( 6 + \beta_{1} + \beta_{2} ) q^{69} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{71} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{73} + ( -14 - 6 \beta_{1} - 3 \beta_{2} ) q^{75} + q^{77} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -1 + 2 \beta_{1} - 2 \beta_{2} ) q^{81} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 4 - 6 \beta_{1} - 4 \beta_{2} ) q^{85} + ( -4 + 2 \beta_{1} ) q^{87} + ( -8 + 5 \beta_{1} + \beta_{2} ) q^{89} + ( 4 + \beta_{2} ) q^{91} + ( -12 + \beta_{1} - 3 \beta_{2} ) q^{93} + ( 4 + 6 \beta_{1} + 2 \beta_{2} ) q^{95} + ( 4 + 3 \beta_{1} + \beta_{2} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} - q^{5} + 3q^{7} + 4q^{9} + O(q^{10}) \) \( 3q - q^{3} - q^{5} + 3q^{7} + 4q^{9} + 3q^{11} + 12q^{13} + 9q^{15} + 2q^{17} - 2q^{19} - q^{21} - 7q^{23} + 18q^{25} - 7q^{27} + 6q^{29} - 9q^{31} - q^{33} - q^{35} + 7q^{37} - 2q^{41} - 4q^{43} - 34q^{45} - 4q^{47} + 3q^{49} - 30q^{51} + 2q^{53} - q^{55} + 26q^{57} - 23q^{59} + 14q^{61} + 4q^{63} - 28q^{65} - 5q^{67} + 19q^{69} - 5q^{71} - 14q^{73} - 48q^{75} + 3q^{77} + 14q^{79} - q^{81} + 4q^{83} + 6q^{85} - 10q^{87} - 19q^{89} + 12q^{91} - 35q^{93} + 18q^{95} + 15q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)

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
2.85577
0.321637
−2.17741
0 −2.85577 0 −4.15544 0 1.00000 0 5.15544 0
1.2 0 −0.321637 0 3.89655 0 1.00000 0 −2.89655 0
1.3 0 2.17741 0 −0.741113 0 1.00000 0 1.74111 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{3} + T_{3}^{2} - 6 T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(308))\).