Properties

Label 804.2.s.a
Level 804
Weight 2
Character orbit 804.s
Analytic conductor 6.420
Analytic rank 0
Dimension 20
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.s (of order \(22\) and degree \(10\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{33}^{3} + 2 \zeta_{33}^{14} ) q^{3} + ( -3 \zeta_{33}^{5} + 2 \zeta_{33}^{7} - \zeta_{33}^{16} - \zeta_{33}^{18} ) q^{7} -3 \zeta_{33}^{6} q^{9} +O(q^{10})\) \( q + ( \zeta_{33}^{3} + 2 \zeta_{33}^{14} ) q^{3} + ( -3 \zeta_{33}^{5} + 2 \zeta_{33}^{7} - \zeta_{33}^{16} - \zeta_{33}^{18} ) q^{7} -3 \zeta_{33}^{6} q^{9} + ( 3 - 4 \zeta_{33} + \zeta_{33}^{2} + 3 \zeta_{33}^{3} - 4 \zeta_{33}^{4} + \zeta_{33}^{5} + 3 \zeta_{33}^{6} - 4 \zeta_{33}^{7} + \zeta_{33}^{8} + 2 \zeta_{33}^{9} + 4 \zeta_{33}^{11} - \zeta_{33}^{12} - 3 \zeta_{33}^{13} + 4 \zeta_{33}^{14} - \zeta_{33}^{15} - 3 \zeta_{33}^{16} + 4 \zeta_{33}^{17} - \zeta_{33}^{18} - 3 \zeta_{33}^{19} ) q^{13} + ( 3 \zeta_{33}^{2} + 5 \zeta_{33}^{8} + 5 \zeta_{33}^{13} + 3 \zeta_{33}^{19} ) q^{19} + ( -5 + 5 \zeta_{33}^{2} - 5 \zeta_{33}^{3} + 5 \zeta_{33}^{5} - 5 \zeta_{33}^{6} + 4 \zeta_{33}^{8} - 5 \zeta_{33}^{9} + 4 \zeta_{33}^{10} - 5 \zeta_{33}^{12} + 5 \zeta_{33}^{13} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} - 5 \zeta_{33}^{18} ) q^{21} -5 \zeta_{33}^{15} q^{25} + ( 6 - 6 \zeta_{33} + 6 \zeta_{33}^{3} - 6 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 6 \zeta_{33}^{7} + 3 \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 6 \zeta_{33}^{11} - 6 \zeta_{33}^{13} + 6 \zeta_{33}^{14} - 6 \zeta_{33}^{16} + 6 \zeta_{33}^{17} - 6 \zeta_{33}^{19} ) q^{27} + ( 6 - 6 \zeta_{33} + 6 \zeta_{33}^{3} - 6 \zeta_{33}^{4} - \zeta_{33}^{5} + 6 \zeta_{33}^{6} - 6 \zeta_{33}^{7} + 5 \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 6 \zeta_{33}^{11} - 6 \zeta_{33}^{13} + 6 \zeta_{33}^{14} - \zeta_{33}^{16} + 6 \zeta_{33}^{17} - 6 \zeta_{33}^{19} ) q^{31} + ( -4 \zeta_{33}^{4} + 4 \zeta_{33}^{7} + 3 \zeta_{33}^{15} + 7 \zeta_{33}^{18} ) q^{37} + ( -2 \zeta_{33} - 7 \zeta_{33}^{2} + 5 \zeta_{33}^{12} - 5 \zeta_{33}^{13} ) q^{39} + ( -\zeta_{33}^{3} + 7 \zeta_{33}^{6} + 6 \zeta_{33}^{14} + 6 \zeta_{33}^{17} ) q^{43} + ( -5 + 5 \zeta_{33}^{2} + 5 \zeta_{33}^{5} - 5 \zeta_{33}^{6} + 5 \zeta_{33}^{8} - 5 \zeta_{33}^{9} + 8 \zeta_{33}^{10} - 12 \zeta_{33}^{12} + 5 \zeta_{33}^{13} + 8 \zeta_{33}^{14} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} - 5 \zeta_{33}^{18} + 5 \zeta_{33}^{19} ) q^{49} + ( -7 - 7 \zeta_{33}^{5} - 8 \zeta_{33}^{11} + \zeta_{33}^{16} ) q^{57} + ( 4 \zeta_{33} + 5 \zeta_{33}^{6} - 5 \zeta_{33}^{12} - 4 \zeta_{33}^{17} ) q^{61} + ( -3 - 3 \zeta_{33}^{2} + 6 \zeta_{33}^{11} - 9 \zeta_{33}^{13} ) q^{63} + ( 2 \zeta_{33}^{4} + 9 \zeta_{33}^{15} ) q^{67} + ( 1 - 8 \zeta_{33}^{7} - 8 \zeta_{33}^{11} + \zeta_{33}^{18} ) q^{73} + ( 10 \zeta_{33}^{7} + 5 \zeta_{33}^{18} ) q^{75} + ( 7 + 7 \zeta_{33}^{8} + 10 \zeta_{33}^{11} - 3 \zeta_{33}^{19} ) q^{79} + 9 \zeta_{33}^{12} q^{81} + ( -9 \zeta_{33}^{3} + 6 \zeta_{33}^{4} + 11 \zeta_{33}^{5} + \zeta_{33}^{6} - 10 \zeta_{33}^{14} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} + 10 \zeta_{33}^{17} ) q^{91} + ( -4 \zeta_{33} - 11 \zeta_{33}^{8} + 7 \zeta_{33}^{12} - 7 \zeta_{33}^{19} ) q^{93} + ( 3 \zeta_{33}^{3} + 3 \zeta_{33}^{8} - 8 \zeta_{33}^{14} + 11 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character Values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(-\zeta_{33}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.786053 + 0.618159i
0.928368 + 0.371662i
0.981929 + 0.189251i
−0.327068 0.945001i
−0.995472 0.0950560i
0.580057 0.814576i
0.0475819 0.998867i
−0.888835 + 0.458227i
−0.786053 0.618159i
0.928368 0.371662i
0.723734 0.690079i
0.235759 + 0.971812i
0.723734 + 0.690079i
0.235759 0.971812i
−0.995472 + 0.0950560i
0.580057 + 0.814576i
0.0475819 + 0.998867i
−0.888835 0.458227i
0.981929 0.189251i
−0.327068 + 0.945001i
0 −1.57553 + 0.719520i 0 0 0 −3.36481 2.91562i 0 1.96458 2.26725i 0
5.2 0 1.57553 0.719520i 0 0 0 −2.61965 2.26994i 0 1.96458 2.26725i 0
53.1 0 −0.936417 + 1.45709i 0 0 0 0.686312 0.313428i 0 −1.24625 2.72890i 0
53.2 0 0.936417 1.45709i 0 0 0 4.81332 2.19817i 0 −1.24625 2.72890i 0
125.1 0 −0.487975 + 1.66189i 0 0 0 1.18913 1.85033i 0 −2.52376 1.62192i 0
125.2 0 0.487975 1.66189i 0 0 0 2.74514 4.27152i 0 −2.52376 1.62192i 0
137.1 0 −1.71442 0.246497i 0 0 0 −1.43029 + 4.87111i 0 2.87848 + 0.845198i 0
137.2 0 1.71442 + 0.246497i 0 0 0 0.211757 0.721178i 0 2.87848 + 0.845198i 0
161.1 0 −1.57553 0.719520i 0 0 0 −3.36481 + 2.91562i 0 1.96458 + 2.26725i 0
161.2 0 1.57553 + 0.719520i 0 0 0 −2.61965 + 2.26994i 0 1.96458 + 2.26725i 0
209.1 0 −1.30900 + 1.13425i 0 0 0 2.17449 + 0.312644i 0 0.426945 2.96946i 0
209.2 0 1.30900 1.13425i 0 0 0 −4.40541 0.633402i 0 0.426945 2.96946i 0
377.1 0 −1.30900 1.13425i 0 0 0 2.17449 0.312644i 0 0.426945 + 2.96946i 0
377.2 0 1.30900 + 1.13425i 0 0 0 −4.40541 + 0.633402i 0 0.426945 + 2.96946i 0
521.1 0 −0.487975 1.66189i 0 0 0 1.18913 + 1.85033i 0 −2.52376 + 1.62192i 0
521.2 0 0.487975 + 1.66189i 0 0 0 2.74514 + 4.27152i 0 −2.52376 + 1.62192i 0
581.1 0 −1.71442 + 0.246497i 0 0 0 −1.43029 4.87111i 0 2.87848 0.845198i 0
581.2 0 1.71442 0.246497i 0 0 0 0.211757 + 0.721178i 0 2.87848 0.845198i 0
713.1 0 −0.936417 1.45709i 0 0 0 0.686312 + 0.313428i 0 −1.24625 + 2.72890i 0
713.2 0 0.936417 + 1.45709i 0 0 0 4.81332 + 2.19817i 0 −1.24625 + 2.72890i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 713.2
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.f Odd 1 yes
201.j Even 1 yes

Hecke kernels

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