Properties

Label 804.2.ba.a
Level 804
Weight 2
Character orbit 804.ba
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.ba (of order \(66\) and degree \(20\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{33} - \zeta_{33}^{12} ) q^{3} + ( 1 - \zeta_{33} + \zeta_{33}^{3} - \zeta_{33}^{4} + 2 \zeta_{33}^{6} - \zeta_{33}^{7} - 2 \zeta_{33}^{9} - \zeta_{33}^{10} + \zeta_{33}^{11} - \zeta_{33}^{13} + \zeta_{33}^{14} - \zeta_{33}^{16} + 4 \zeta_{33}^{17} - \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{33} - \zeta_{33}^{12} ) q^{3} + ( 1 - \zeta_{33} + \zeta_{33}^{3} - \zeta_{33}^{4} + 2 \zeta_{33}^{6} - \zeta_{33}^{7} - 2 \zeta_{33}^{9} - \zeta_{33}^{10} + \zeta_{33}^{11} - \zeta_{33}^{13} + \zeta_{33}^{14} - \zeta_{33}^{16} + 4 \zeta_{33}^{17} - \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} + ( -3 \zeta_{33}^{3} + 4 \zeta_{33}^{7} + \zeta_{33}^{14} + \zeta_{33}^{18} ) q^{13} + ( -3 + 3 \zeta_{33}^{2} - 3 \zeta_{33}^{3} + 3 \zeta_{33}^{5} - 3 \zeta_{33}^{6} - 2 \zeta_{33}^{8} - 3 \zeta_{33}^{9} + 5 \zeta_{33}^{10} - 3 \zeta_{33}^{12} + 3 \zeta_{33}^{13} - 3 \zeta_{33}^{15} + 3 \zeta_{33}^{16} - 3 \zeta_{33}^{18} + \zeta_{33}^{19} ) q^{19} + ( -4 + 4 \zeta_{33}^{2} - 4 \zeta_{33}^{3} + 4 \zeta_{33}^{5} - 4 \zeta_{33}^{6} + \zeta_{33}^{7} + 4 \zeta_{33}^{8} - 4 \zeta_{33}^{9} + 5 \zeta_{33}^{10} - 4 \zeta_{33}^{12} + 4 \zeta_{33}^{13} - 4 \zeta_{33}^{15} + 4 \zeta_{33}^{16} - 8 \zeta_{33}^{18} + 4 \zeta_{33}^{19} ) q^{21} + ( 5 \zeta_{33}^{5} + 5 \zeta_{33}^{16} ) q^{25} + ( -3 \zeta_{33}^{3} - 6 \zeta_{33}^{14} ) q^{27} + ( 6 - 6 \zeta_{33} + 12 \zeta_{33}^{3} - 6 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 6 \zeta_{33}^{7} + \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 6 \zeta_{33}^{11} - 6 \zeta_{33}^{13} + 11 \zeta_{33}^{14} - 6 \zeta_{33}^{16} + 6 \zeta_{33}^{17} - 6 \zeta_{33}^{19} ) q^{31} + ( 7 \zeta_{33}^{5} + 4 \zeta_{33}^{6} + 4 \zeta_{33}^{16} + 7 \zeta_{33}^{17} ) q^{37} + ( 7 \zeta_{33}^{4} - 7 \zeta_{33}^{8} + 2 \zeta_{33}^{15} - 5 \zeta_{33}^{19} ) q^{39} + ( -\zeta_{33} + 7 \zeta_{33}^{2} + 6 \zeta_{33}^{12} + 6 \zeta_{33}^{13} ) q^{43} + ( 3 \zeta_{33} + 7 \zeta_{33}^{4} - 5 \zeta_{33}^{7} - 5 \zeta_{33}^{12} + 7 \zeta_{33}^{15} + 3 \zeta_{33}^{18} ) q^{49} + ( 1 + 7 \zeta_{33} - 7 \zeta_{33}^{3} + 7 \zeta_{33}^{4} - 7 \zeta_{33}^{6} + 7 \zeta_{33}^{7} + \zeta_{33}^{9} + 7 \zeta_{33}^{10} - 6 \zeta_{33}^{11} + 7 \zeta_{33}^{13} - 7 \zeta_{33}^{14} + 7 \zeta_{33}^{16} - 7 \zeta_{33}^{17} + 7 \zeta_{33}^{19} ) q^{57} + ( 4 \zeta_{33}^{2} + 4 \zeta_{33}^{4} + 9 \zeta_{33}^{13} - 5 \zeta_{33}^{15} ) q^{61} + ( 9 - 6 \zeta_{33}^{8} + 3 \zeta_{33}^{11} + 3 \zeta_{33}^{19} ) q^{63} + ( 2 \zeta_{33}^{5} + 9 \zeta_{33}^{16} ) q^{67} + ( -9 + 9 \zeta_{33}^{6} - \zeta_{33}^{11} + 8 \zeta_{33}^{17} ) q^{73} + ( -5 \zeta_{33}^{6} - 10 \zeta_{33}^{17} ) q^{75} + ( -3 - 7 \zeta_{33}^{2} + 7 \zeta_{33}^{3} - 7 \zeta_{33}^{5} + 7 \zeta_{33}^{6} - 7 \zeta_{33}^{8} + 7 \zeta_{33}^{9} - 10 \zeta_{33}^{10} - 3 \zeta_{33}^{11} + 7 \zeta_{33}^{12} - 7 \zeta_{33}^{13} + 7 \zeta_{33}^{15} - 7 \zeta_{33}^{16} + 7 \zeta_{33}^{18} - 7 \zeta_{33}^{19} ) q^{79} + 9 \zeta_{33}^{15} q^{81} + ( 11 - 12 \zeta_{33} - 10 \zeta_{33}^{2} + 11 \zeta_{33}^{3} - 11 \zeta_{33}^{4} + 6 \zeta_{33}^{5} + 11 \zeta_{33}^{6} - 11 \zeta_{33}^{7} + 5 \zeta_{33}^{9} - 11 \zeta_{33}^{10} + 11 \zeta_{33}^{11} + 9 \zeta_{33}^{12} - 20 \zeta_{33}^{13} + 11 \zeta_{33}^{14} - 16 \zeta_{33}^{16} + 11 \zeta_{33}^{17} - 11 \zeta_{33}^{19} ) q^{91} + ( -11 + 11 \zeta_{33}^{2} - 11 \zeta_{33}^{3} - 7 \zeta_{33}^{4} + 11 \zeta_{33}^{5} - 11 \zeta_{33}^{6} + 11 \zeta_{33}^{8} - 11 \zeta_{33}^{9} + 4 \zeta_{33}^{10} - 11 \zeta_{33}^{12} + 11 \zeta_{33}^{13} - 22 \zeta_{33}^{15} + 11 \zeta_{33}^{16} - 11 \zeta_{33}^{18} + 11 \zeta_{33}^{19} ) q^{93} + ( 3 + 3 \zeta_{33} - 3 \zeta_{33}^{2} + 3 \zeta_{33}^{3} - 3 \zeta_{33}^{5} + 3 \zeta_{33}^{6} - 3 \zeta_{33}^{8} + 3 \zeta_{33}^{9} + 8 \zeta_{33}^{10} - 5 \zeta_{33}^{12} - 3 \zeta_{33}^{13} + 3 \zeta_{33}^{15} - 3 \zeta_{33}^{16} + 3 \zeta_{33}^{18} - 3 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−0.888835 0.458227i
0.723734 + 0.690079i
−0.995472 + 0.0950560i
−0.995472 0.0950560i
0.0475819 0.998867i
0.928368 0.371662i
−0.786053 + 0.618159i
−0.786053 0.618159i
−0.327068 0.945001i
0.580057 + 0.814576i
−0.888835 + 0.458227i
0.580057 0.814576i
0.981929 0.189251i
0.928368 + 0.371662i
0.723734 0.690079i
0.0475819 + 0.998867i
0.235759 + 0.971812i
−0.327068 + 0.945001i
0.981929 + 0.189251i
0.235759 0.971812i
0 0.936417 + 1.45709i 0 0 0 −0.502989 5.26754i 0 −1.24625 + 2.72890i 0
101.1 0 −0.487975 1.66189i 0 0 0 1.00786 1.95498i 0 −2.52376 + 1.62192i 0
113.1 0 1.57553 + 0.719520i 0 0 0 3.27565 + 1.13371i 0 1.96458 + 2.26725i 0
185.1 0 1.57553 0.719520i 0 0 0 3.27565 1.13371i 0 1.96458 2.26725i 0
197.1 0 −0.936417 + 1.45709i 0 0 0 −0.614593 0.437650i 0 −1.24625 2.72890i 0
221.1 0 −1.71442 0.246497i 0 0 0 4.93365 1.19689i 0 2.87848 + 0.845198i 0
233.1 0 1.71442 0.246497i 0 0 0 0.518680 0.543976i 0 2.87848 0.845198i 0
245.1 0 1.71442 + 0.246497i 0 0 0 0.518680 + 0.543976i 0 2.87848 + 0.845198i 0
281.1 0 1.30900 + 1.13425i 0 0 0 1.65416 4.13190i 0 0.426945 + 2.96946i 0
329.1 0 −1.57553 0.719520i 0 0 0 −0.842599 4.37182i 0 1.96458 + 2.26725i 0
353.1 0 0.936417 1.45709i 0 0 0 −0.502989 + 5.26754i 0 −1.24625 2.72890i 0
413.1 0 −1.57553 + 0.719520i 0 0 0 −0.842599 + 4.37182i 0 1.96458 2.26725i 0
497.1 0 −1.30900 + 1.13425i 0 0 0 −1.35800 + 1.72684i 0 0.426945 2.96946i 0
593.1 0 −1.71442 + 0.246497i 0 0 0 4.93365 + 1.19689i 0 2.87848 0.845198i 0
605.1 0 −0.487975 + 1.66189i 0 0 0 1.00786 + 1.95498i 0 −2.52376 1.62192i 0
653.1 0 −0.936417 1.45709i 0 0 0 −0.614593 + 0.437650i 0 −1.24625 + 2.72890i 0
677.1 0 0.487975 1.66189i 0 0 0 −5.07182 0.241600i 0 −2.52376 1.62192i 0
701.1 0 1.30900 1.13425i 0 0 0 1.65416 + 4.13190i 0 0.426945 2.96946i 0
749.1 0 −1.30900 1.13425i 0 0 0 −1.35800 1.72684i 0 0.426945 + 2.96946i 0
785.1 0 0.487975 + 1.66189i 0 0 0 −5.07182 + 0.241600i 0 −2.52376 + 1.62192i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 785.1
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.h Odd 1 yes
201.p Even 1 yes

Hecke kernels

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