Properties

Label 804.2.o.a
Level 804
Weight 2
Character orbit 804.o
Analytic conductor 6.420
Analytic rank 1
Dimension 2
CM disc. -3
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.o (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -4 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -4 + 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -3 - 3 \zeta_{6} ) q^{13} + ( 8 - 8 \zeta_{6} ) q^{19} -6 \zeta_{6} q^{21} -5 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 - \zeta_{6} ) q^{31} + ( -10 + 10 \zeta_{6} ) q^{37} + ( 9 - 9 \zeta_{6} ) q^{39} + ( -1 + 2 \zeta_{6} ) q^{43} + ( 5 - 5 \zeta_{6} ) q^{49} + ( 8 + 8 \zeta_{6} ) q^{57} + ( -9 - 9 \zeta_{6} ) q^{61} + ( 12 - 6 \zeta_{6} ) q^{63} + ( -2 - 7 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{73} + ( 5 - 10 \zeta_{6} ) q^{75} + ( -14 + 7 \zeta_{6} ) q^{79} + 9 q^{81} + 18 q^{91} + 3 \zeta_{6} q^{93} + ( -11 - 11 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 6q^{7} - 6q^{9} - 9q^{13} + 8q^{19} - 6q^{21} - 10q^{25} + 3q^{31} - 10q^{37} + 9q^{39} + 5q^{49} + 24q^{57} - 27q^{61} + 18q^{63} - 11q^{67} - 7q^{73} - 21q^{79} + 18q^{81} + 36q^{91} + 3q^{93} - 33q^{97} + O(q^{100}) \)

Character Values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
365.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 0 0 −3.00000 1.73205i 0 −3.00000 0
641.1 0 1.73205i 0 0 0 −3.00000 + 1.73205i 0 −3.00000 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 CM by \(\Q(\sqrt{-3}) \) yes
67.d Odd 1 yes
201.f Even 1 yes

Hecke kernels

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