Properties

Label 164.2.a.a
Level 164
Weight 2
Character orbit 164.a
Self dual Yes
Analytic conductor 1.310
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 164.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + 2 \beta_{1} q^{13} + ( -1 - 2 \beta_{1} + 3 \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} + ( 2 - \beta_{1} - \beta_{2} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{21} + ( -2 - 2 \beta_{2} ) q^{23} + ( 3 - 2 \beta_{1} ) q^{25} + ( -5 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -8 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{33} + ( -7 + 4 \beta_{1} - \beta_{3} ) q^{35} + ( 4 - \beta_{2} - \beta_{3} ) q^{37} + ( -6 + 2 \beta_{1} ) q^{39} - q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( 6 + 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( 4 + 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{55} + ( -4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{57} + ( 2 + 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + ( -6 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{63} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 7 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( -6 + 2 \beta_{3} ) q^{69} + ( -4 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{71} + ( 4 - \beta_{2} + 3 \beta_{3} ) q^{73} + ( 6 - 5 \beta_{1} + 3 \beta_{2} ) q^{75} + ( 2 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{77} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{79} + ( 7 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{81} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 12 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{87} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 12 - 2 \beta_{1} + 6 \beta_{3} ) q^{91} + ( -2 - 4 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 7 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{95} + ( 2 - 4 \beta_{3} ) q^{97} + ( 16 + 7 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{5} + 12q^{9} + 4q^{11} - 10q^{15} - 4q^{17} + 6q^{19} - 12q^{23} + 12q^{25} - 10q^{27} - 4q^{29} - 8q^{31} - 20q^{33} - 26q^{35} + 16q^{37} - 24q^{39} - 4q^{41} + 4q^{43} + 4q^{45} - 6q^{47} + 16q^{49} - 4q^{51} - 16q^{53} - 2q^{55} + 4q^{57} + 12q^{59} + 24q^{61} - 10q^{63} + 4q^{65} + 28q^{67} - 28q^{69} - 2q^{71} + 8q^{73} + 30q^{75} + 8q^{77} - 18q^{79} + 28q^{81} - 12q^{83} + 32q^{85} + 44q^{87} + 4q^{89} + 36q^{91} - 28q^{93} + 14q^{95} + 16q^{97} + 58q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} - 6 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 7 \nu - 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu^{2} + 7 \nu - 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 7 \beta_{1} + 3\)

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
0.707500
3.31526
−2.46810
−1.55466
0 −3.24028 0 2.56613 0 −0.858626 0 7.49944 0
1.2 0 0.0950939 0 1.17025 0 3.14501 0 −2.99096 0
1.3 0 2.21551 0 3.59669 0 −5.06479 0 1.90849 0
1.4 0 2.92968 0 −3.33307 0 2.77840 0 5.58303 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
\(2\) \(-1\)
\(41\) \(1\)

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(\Gamma_0(164))\).