Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 q^{5} - q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 2 q^{5} - q^{7} + 3 q^{9} - q^{11} + ( 2 - \beta ) q^{13} + 2 \beta q^{15} + ( 2 - \beta ) q^{17} -2 \beta q^{19} -\beta q^{21} + ( 4 - 2 \beta ) q^{23} - q^{25} + ( -2 + 2 \beta ) q^{29} + ( 4 + \beta ) q^{31} -\beta q^{33} -2 q^{35} + 4 q^{37} + ( -6 + 2 \beta ) q^{39} + ( -2 - 3 \beta ) q^{41} + ( -2 + 2 \beta ) q^{43} + 6 q^{45} + ( -4 - \beta ) q^{47} + q^{49} + ( -6 + 2 \beta ) q^{51} + 4 \beta q^{53} -2 q^{55} -12 q^{57} + 3 \beta q^{59} + ( -2 + 3 \beta ) q^{61} -3 q^{63} + ( 4 - 2 \beta ) q^{65} -6 \beta q^{67} + ( -12 + 4 \beta ) q^{69} + ( -8 - 2 \beta ) q^{71} + ( 10 + \beta ) q^{73} -\beta q^{75} + q^{77} + ( -6 - 2 \beta ) q^{79} -9 q^{81} + ( -12 - 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} + ( 12 - 2 \beta ) q^{87} + 6 q^{89} + ( -2 + \beta ) q^{91} + ( 6 + 4 \beta ) q^{93} -4 \beta q^{95} + ( 10 - 2 \beta ) q^{97} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{7} + 6q^{9} - 2q^{11} + 4q^{13} + 4q^{17} + 8q^{23} - 2q^{25} - 4q^{29} + 8q^{31} - 4q^{35} + 8q^{37} - 12q^{39} - 4q^{41} - 4q^{43} + 12q^{45} - 8q^{47} + 2q^{49} - 12q^{51} - 4q^{55} - 24q^{57} - 4q^{61} - 6q^{63} + 8q^{65} - 24q^{69} - 16q^{71} + 20q^{73} + 2q^{77} - 12q^{79} - 18q^{81} - 24q^{83} + 8q^{85} + 24q^{87} + 12q^{89} - 4q^{91} + 12q^{93} + 20q^{97} - 6q^{99} + O(q^{100}) \)

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
−2.44949
2.44949
0 −2.44949 0 2.00000 0 −1.00000 0 3.00000 0
1.2 0 2.44949 0 2.00000 0 −1.00000 0 3.00000 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 \( T_{3}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(308))\).