Properties

Label 804.2.i.e
Level 804
Weight 2
Character orbit 804.i
Analytic conductor 6.420
Analytic rank 0
Dimension 8
CM No
Inner twists 2

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{5} -\beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{5} -\beta_{3} q^{7} + q^{9} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + ( -2 \beta_{1} + \beta_{7} ) q^{13} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{15} -\beta_{1} q^{17} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} -\beta_{3} q^{21} + ( 3 + 3 \beta_{4} ) q^{23} + ( 3 + 3 \beta_{2} - \beta_{5} ) q^{25} + q^{27} + ( \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{29} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{31} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{33} + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{35} + ( 4 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{37} + ( -2 \beta_{1} + \beta_{7} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{41} + ( 1 + \beta_{6} ) q^{43} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{45} + ( -4 \beta_{4} - \beta_{6} + \beta_{7} ) q^{47} + ( 2 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{49} -\beta_{1} q^{51} + ( -1 + 2 \beta_{2} - 4 \beta_{5} + 3 \beta_{6} ) q^{53} + ( -4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} + 4 \beta_{7} ) q^{55} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{57} + ( \beta_{5} + \beta_{6} ) q^{59} + ( 4 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{61} -\beta_{3} q^{63} + ( -7 + 6 \beta_{1} + \beta_{3} - 7 \beta_{4} - \beta_{5} ) q^{65} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{67} + ( 3 + 3 \beta_{4} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -4 + 3 \beta_{1} - 4 \beta_{4} + 2 \beta_{7} ) q^{73} + ( 3 + 3 \beta_{2} - \beta_{5} ) q^{75} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{77} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} - \beta_{6} + \beta_{7} ) q^{79} + q^{81} + ( -4 + \beta_{1} - 4 \beta_{4} - 2 \beta_{7} ) q^{83} + ( -2 + 3 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{85} + ( \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{87} + ( -1 - 2 \beta_{2} + \beta_{5} - 4 \beta_{6} ) q^{89} + ( -4 + \beta_{2} + 3 \beta_{5} - \beta_{6} ) q^{91} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{93} + ( -2 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{95} + ( 8 - 2 \beta_{1} + 3 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{97} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} - 6q^{5} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} - 6q^{5} + 8q^{9} + 3q^{11} - 2q^{13} - 6q^{15} - q^{17} + 4q^{19} + 12q^{23} + 18q^{25} + 8q^{27} - 9q^{29} - q^{31} + 3q^{33} - 9q^{35} + 14q^{37} - 2q^{39} + 10q^{41} + 8q^{43} - 6q^{45} + 16q^{47} + 6q^{49} - q^{51} - 12q^{53} + 4q^{55} + 4q^{57} + 4q^{61} - 22q^{65} + 20q^{67} + 12q^{69} + 6q^{71} - 13q^{73} + 18q^{75} - 5q^{77} - 18q^{79} + 8q^{81} - 15q^{83} - 5q^{85} - 9q^{87} - 4q^{89} - 34q^{91} - q^{93} - 6q^{95} + 30q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 8 x^{6} - 9 x^{5} + 54 x^{4} - 50 x^{3} + 85 x^{2} + 24 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 315 \nu^{7} + 154 \nu^{6} + 2025 \nu^{5} + 180 \nu^{4} + 17278 \nu^{3} + 4905 \nu^{2} + 1485 \nu + 9771 \)\()/32593\)
\(\beta_{3}\)\(=\)\((\)\( 1867 \nu^{7} - 8503 \nu^{6} + 7346 \nu^{5} - 59463 \nu^{4} + 97026 \nu^{3} - 292202 \nu^{2} + 55363 \nu + 13524 \)\()/97779\)
\(\beta_{4}\)\(=\)\((\)\( -3257 \nu^{7} + 4202 \nu^{6} - 25594 \nu^{5} + 35388 \nu^{4} - 175338 \nu^{3} + 214684 \nu^{2} - 262130 \nu - 73713 \)\()/97779\)
\(\beta_{5}\)\(=\)\((\)\( -1106 \nu^{7} - 1265 \nu^{6} - 7110 \nu^{5} - 632 \nu^{4} - 33142 \nu^{3} - 17222 \nu^{2} - 5214 \nu + 13496 \)\()/32593\)
\(\beta_{6}\)\(=\)\((\)\( -1890 \nu^{7} - 924 \nu^{6} - 12150 \nu^{5} - 1080 \nu^{4} - 71075 \nu^{3} - 29430 \nu^{2} - 8910 \nu - 123812 \)\()/32593\)
\(\beta_{7}\)\(=\)\((\)\( 6436 \nu^{7} - 10615 \nu^{6} + 64655 \nu^{5} - 84789 \nu^{4} + 442935 \nu^{3} - 542330 \nu^{2} + 1068661 \nu - 60795 \)\()/97779\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + 4 \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 6 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-6 \beta_{7} + 7 \beta_{5} - 23 \beta_{4} - 7 \beta_{3} + 9 \beta_{1} - 23\)
\(\nu^{5}\)\(=\)\(9 \beta_{7} - 9 \beta_{6} + 23 \beta_{4} + \beta_{3} - 38 \beta_{2} - 38 \beta_{1}\)
\(\nu^{6}\)\(=\)\(38 \beta_{6} - 45 \beta_{5} + 70 \beta_{2} + 142\)
\(\nu^{7}\)\(=\)\(-70 \beta_{7} + 18 \beta_{5} - 197 \beta_{4} - 18 \beta_{3} + 250 \beta_{1} - 197\)

Character Values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.31931 2.28512i
1.13858 + 1.97209i
0.830946 + 1.43924i
−0.150216 0.260181i
−1.31931 + 2.28512i
1.13858 1.97209i
0.830946 1.43924i
−0.150216 + 0.260181i
0 1.00000 0 −3.96236 0 −0.107790 + 0.186699i 0 1.00000 0
37.2 0 1.00000 0 −2.18550 0 1.80395 3.12454i 0 1.00000 0
37.3 0 1.00000 0 0.238118 0 −1.47881 + 2.56138i 0 1.00000 0
37.4 0 1.00000 0 2.90974 0 −0.217351 + 0.376462i 0 1.00000 0
565.1 0 1.00000 0 −3.96236 0 −0.107790 0.186699i 0 1.00000 0
565.2 0 1.00000 0 −2.18550 0 1.80395 + 3.12454i 0 1.00000 0
565.3 0 1.00000 0 0.238118 0 −1.47881 2.56138i 0 1.00000 0
565.4 0 1.00000 0 2.90974 0 −0.217351 0.376462i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 565.4
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 \(S_{2}^{\mathrm{new}}(804, [\chi])\):

\( T_{5}^{4} + 3 T_{5}^{3} - 10 T_{5}^{2} - 23 T_{5} + 6 \)
\( T_{7}^{8} + 11 T_{7}^{6} + 14 T_{7}^{5} + 122 T_{7}^{4} + 77 T_{7}^{3} + 38 T_{7}^{2} + 7 T_{7} + 1 \)