Properties

Label 804.2.a.e
Level 804
Weight 2
Character orbit 804.a
Self dual Yes
Analytic conductor 6.420
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1076.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( 1 - \beta_{1} - \beta_{2} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( 2 + \beta_{1} - \beta_{2} ) q^{21} + ( 3 - \beta_{1} ) q^{23} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + 8 q^{29} + ( 1 - 2 \beta_{2} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} ) q^{33} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{35} + ( 6 - \beta_{1} + \beta_{2} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} ) q^{39} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{41} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{47} + ( 8 - 2 \beta_{2} ) q^{49} + ( -1 + \beta_{1} + \beta_{2} ) q^{51} + 3 \beta_{2} q^{53} + ( -7 - 5 \beta_{1} - \beta_{2} ) q^{55} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} + ( 4 - 3 \beta_{2} ) q^{59} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{61} + ( -2 - \beta_{1} + \beta_{2} ) q^{63} + ( 5 + 3 \beta_{1} + \beta_{2} ) q^{65} - q^{67} + ( -3 + \beta_{1} ) q^{69} + ( 5 - 5 \beta_{1} - \beta_{2} ) q^{71} + ( -7 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{75} + ( 1 - 5 \beta_{1} - 3 \beta_{2} ) q^{77} + ( -6 - 2 \beta_{2} ) q^{79} + q^{81} + ( 6 - 3 \beta_{2} ) q^{83} + ( 5 + 3 \beta_{1} + \beta_{2} ) q^{85} -8 q^{87} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{89} + ( -5 + 3 \beta_{1} + 5 \beta_{2} ) q^{91} + ( -1 + 2 \beta_{2} ) q^{93} + ( -3 + \beta_{1} - \beta_{2} ) q^{95} + ( 5 + \beta_{1} - \beta_{2} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 3q^{5} - 5q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 3q^{5} - 5q^{7} + 3q^{9} + 4q^{11} + 2q^{13} + 3q^{15} + 2q^{17} - 4q^{19} + 5q^{21} + 9q^{23} + 4q^{25} - 3q^{27} + 24q^{29} + q^{31} - 4q^{33} + 19q^{35} + 19q^{37} - 2q^{39} + 13q^{41} - q^{43} - 3q^{45} + 2q^{47} + 22q^{49} - 2q^{51} + 3q^{53} - 22q^{55} + 4q^{57} + 9q^{59} - 5q^{63} + 16q^{65} - 3q^{67} - 9q^{69} + 14q^{71} - 23q^{73} - 4q^{75} - 20q^{79} + 3q^{81} + 15q^{83} + 16q^{85} - 24q^{87} + 12q^{89} - 10q^{91} - q^{93} - 10q^{95} + 14q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 8 x - 6\):

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

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
3.14743
−0.818558
−2.32887
0 −1.00000 0 −4.14743 0 −3.38854 0 1.00000 0
1.2 0 −1.00000 0 −0.181442 0 −4.69285 0 1.00000 0
1.3 0 −1.00000 0 1.32887 0 3.08139 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(67\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{3} + 3 T_{5}^{2} - 5 T_{5} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\).