Properties

Label 804.2.g.b
Level 804
Weight 2
Character orbit 804.g
Analytic conductor 6.420
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 2 \beta_{1} - \beta_{2} ) q^{5} + \beta_{2} q^{7} + ( 2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 2 \beta_{1} - \beta_{2} ) q^{5} + \beta_{2} q^{7} + ( 2 + \beta_{3} ) q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{11} -3 \beta_{2} q^{13} + ( -5 - \beta_{3} ) q^{15} + 3 \beta_{3} q^{17} -3 q^{19} + ( 1 - \beta_{3} ) q^{21} + 3 \beta_{3} q^{23} + 5 q^{25} + ( -\beta_{1} - 3 \beta_{2} ) q^{27} -\beta_{3} q^{29} + 3 \beta_{2} q^{31} + ( -5 - \beta_{3} ) q^{33} + 2 \beta_{3} q^{35} + 5 q^{37} + ( -3 + 3 \beta_{3} ) q^{39} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{41} -8 \beta_{2} q^{43} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{45} -\beta_{3} q^{47} + 5 q^{49} + ( 3 \beta_{1} - 9 \beta_{2} ) q^{51} + 10 q^{55} + 3 \beta_{1} q^{57} -\beta_{3} q^{59} + 2 \beta_{2} q^{61} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{63} -6 \beta_{3} q^{65} + ( -7 + 3 \beta_{2} ) q^{67} + ( 3 \beta_{1} - 9 \beta_{2} ) q^{69} -6 \beta_{3} q^{71} + 5 q^{73} -5 \beta_{1} q^{75} + 2 \beta_{3} q^{77} + 10 \beta_{2} q^{79} + ( -1 + 4 \beta_{3} ) q^{81} + 2 \beta_{3} q^{83} + 15 \beta_{2} q^{85} + ( -\beta_{1} + 3 \beta_{2} ) q^{87} + \beta_{3} q^{89} + 6 q^{91} + ( 3 - 3 \beta_{3} ) q^{93} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{95} -6 \beta_{2} q^{97} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{9} - 20q^{15} - 12q^{19} + 4q^{21} + 20q^{25} - 20q^{33} + 20q^{37} - 12q^{39} + 20q^{49} + 40q^{55} - 28q^{67} + 20q^{73} - 4q^{81} + 24q^{91} + 12q^{93} + O(q^{100}) \)

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

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
−1.58114 0.707107i
0 −1.58114 0.707107i 0 3.16228 0 1.41421i 0 2.00000 + 2.23607i 0
401.2 0 −1.58114 + 0.707107i 0 3.16228 0 1.41421i 0 2.00000 2.23607i 0
401.3 0 1.58114 0.707107i 0 −3.16228 0 1.41421i 0 2.00000 2.23607i 0
401.4 0 1.58114 + 0.707107i 0 −3.16228 0 1.41421i 0 2.00000 + 2.23607i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
67.b Odd 1 yes
201.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{2} - 10 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).