Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -2 q^{4} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{6} -2 \zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -2 q^{4} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{6} -2 \zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + 2 q^{10} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{12} + 2 q^{13} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{14} + ( \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{15} + 4 q^{16} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{17} + ( -4 + \zeta_{8} + \zeta_{8}^{3} ) q^{18} -4 \zeta_{8}^{2} q^{19} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{20} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + 8 \zeta_{8}^{2} q^{22} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{23} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{24} + 3 q^{25} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{26} + ( -\zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + 4 \zeta_{8}^{2} q^{28} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{30} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( 8 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{33} + 8 q^{34} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( -2 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{36} + 8 q^{37} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{38} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} -4 q^{40} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{42} + 4 \zeta_{8}^{2} q^{43} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{44} + ( 4 - \zeta_{8} - \zeta_{8}^{3} ) q^{45} -10 \zeta_{8}^{2} q^{46} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{47} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{48} + 3 q^{49} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{50} + ( 4 \zeta_{8} - 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{51} -4 q^{52} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{53} + ( -5 \zeta_{8} - 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{54} -8 \zeta_{8}^{2} q^{55} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{56} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{57} -8 q^{58} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{60} -6 q^{61} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} -8 q^{64} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{65} + ( -8 + 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{66} -\zeta_{8}^{2} q^{67} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{68} + ( -10 - 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{69} -4 \zeta_{8}^{2} q^{70} + ( -11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{71} + ( 8 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{72} + 10 q^{73} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{74} + ( 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{75} + 8 \zeta_{8}^{2} q^{76} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{77} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{78} -8 \zeta_{8}^{2} q^{79} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{80} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} -2 q^{82} + ( 11 \zeta_{8} - 11 \zeta_{8}^{3} ) q^{83} + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{84} -8 q^{85} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{86} + ( -4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} -16 \zeta_{8}^{2} q^{88} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{89} + ( 2 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{90} -4 \zeta_{8}^{2} q^{91} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{92} -10 \zeta_{8}^{2} q^{94} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} + ( -4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{96} -6 q^{97} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{98} + ( 4 \zeta_{8} + 16 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 8q^{4} + 4q^{9} + 8q^{10} + 8q^{13} + 16q^{16} - 16q^{18} + 8q^{21} + 12q^{25} + 32q^{33} + 32q^{34} - 8q^{36} + 32q^{37} - 16q^{40} + 16q^{42} + 16q^{45} + 12q^{49} - 16q^{52} + 16q^{57} - 32q^{58} - 24q^{61} - 32q^{64} - 32q^{66} - 40q^{69} + 32q^{72} + 40q^{73} - 28q^{81} - 8q^{82} - 16q^{84} - 32q^{85} + 8q^{90} - 24q^{97} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
1.41421i −1.41421 + 1.00000i −2.00000 1.41421i 1.41421 + 2.00000i 2.00000i 2.82843i 1.00000 2.82843i 2.00000
671.2 1.41421i 1.41421 1.00000i −2.00000 1.41421i −1.41421 2.00000i 2.00000i 2.82843i 1.00000 2.82843i 2.00000
671.3 1.41421i −1.41421 1.00000i −2.00000 1.41421i 1.41421 2.00000i 2.00000i 2.82843i 1.00000 + 2.82843i 2.00000
671.4 1.41421i 1.41421 + 1.00000i −2.00000 1.41421i −1.41421 + 2.00000i 2.00000i 2.82843i 1.00000 + 2.82843i 2.00000
\(n\): 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
4.b Odd 1 yes
12.b Even 1 yes

Hecke kernels

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