Properties

Label 804.2.i.a
Level 804
Weight 2
Character orbit 804.i
Analytic conductor 6.420
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.i (of order \(3\) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0 −2.00000 0 1.50000 2.59808i 0 1.00000 0
565.1 0 −1.00000 0 −2.00000 0 1.50000 + 2.59808i 0 1.00000 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
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\):

\( T_{5} + 2 \)
\( T_{7}^{2} - 3 T_{7} + 9 \)