Properties

Label 804.2.i.d
Level 804
Weight 2
Character orbit 804.i
Analytic conductor 6.420
Analytic rank 0
Dimension 8
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: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{6} q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{6} q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} + q^{9} + 2 \beta_{2} q^{11} + ( \beta_{3} + \beta_{4} ) q^{13} -\beta_{6} q^{15} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( \beta_{1} - \beta_{6} ) q^{21} + \beta_{5} q^{23} + ( \beta_{3} + \beta_{6} ) q^{25} - q^{27} + ( \beta_{1} - \beta_{6} - \beta_{7} ) q^{29} + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{31} -2 \beta_{2} q^{33} + ( -\beta_{1} + 5 \beta_{2} - \beta_{4} + \beta_{6} ) q^{35} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{37} + ( -\beta_{3} - \beta_{4} ) q^{39} + ( 3 \beta_{2} - \beta_{4} - \beta_{7} ) q^{41} + ( -\beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{43} + \beta_{6} q^{45} + ( 3 \beta_{2} - \beta_{4} - \beta_{7} ) q^{47} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{49} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{51} + ( 5 - \beta_{3} ) q^{53} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{57} + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( 5 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( -\beta_{1} + \beta_{6} ) q^{63} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{65} + ( -5 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{67} -\beta_{5} q^{69} + ( -4 \beta_{1} + 2 \beta_{2} + 4 \beta_{6} - \beta_{7} ) q^{71} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( -\beta_{3} - \beta_{6} ) q^{75} -2 \beta_{1} q^{77} + ( 4 \beta_{1} + 4 \beta_{2} + \beta_{4} - 4 \beta_{6} ) q^{79} + q^{81} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{85} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{87} + ( -2 - \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( -2 + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{91} + ( -2 \beta_{1} - 3 \beta_{2} + 2 \beta_{6} + \beta_{7} ) q^{93} + ( -3 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{95} + ( 5 + \beta_{1} - 5 \beta_{2} - \beta_{5} ) q^{97} + 2 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + 2q^{5} + q^{7} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{3} + 2q^{5} + q^{7} + 8q^{9} + 8q^{11} - 2q^{15} + 4q^{17} - 5q^{19} - q^{21} + q^{23} + 2q^{25} - 8q^{27} - 2q^{29} + 9q^{31} - 8q^{33} + 21q^{35} - 5q^{37} + 11q^{41} + 6q^{43} + 2q^{45} + 11q^{47} + 7q^{49} - 4q^{51} + 40q^{53} + 2q^{55} + 5q^{57} + 28q^{59} + 20q^{61} + q^{63} - 4q^{65} - 33q^{67} - q^{69} + 11q^{71} - 7q^{73} - 2q^{75} - 2q^{77} + 12q^{79} + 8q^{81} - 4q^{83} + 5q^{85} + 2q^{87} - 16q^{89} - 8q^{91} - 9q^{93} - 18q^{95} + 20q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 11 x^{6} + 4 x^{5} + 91 x^{4} - 6 x^{3} + 129 x^{2} + 36 x + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 47 \nu^{7} - 199 \nu^{6} + 597 \nu^{5} - 1256 \nu^{4} + 2985 \nu^{3} - 14726 \nu^{2} + 3555 \nu - 1044 \)\()/18060\)
\(\beta_{3}\)\(=\)\((\)\( 20 \nu^{7} + 37 \nu^{6} + 190 \nu^{5} + 170 \nu^{4} + 2756 \nu^{3} + 330 \nu^{2} + 360 \nu - 15411 \)\()/4515\)
\(\beta_{4}\)\(=\)\((\)\( 83 \nu^{7} - 915 \nu^{6} + 1541 \nu^{5} - 7572 \nu^{4} + 481 \nu^{3} - 58078 \nu^{2} - 3021 \nu - 11988 \)\()/18060\)
\(\beta_{5}\)\(=\)\((\)\( 67 \nu^{7} + 139 \nu^{6} - 417 \nu^{5} + 3128 \nu^{4} - 2085 \nu^{3} + 10286 \nu^{2} - 62907 \nu + 13344 \)\()/9030\)
\(\beta_{6}\)\(=\)\((\)\( -38 \nu^{7} + 20 \nu^{6} - 361 \nu^{5} - 323 \nu^{4} - 3611 \nu^{3} - 627 \nu^{2} - 684 \nu - 1692 \)\()/4515\)
\(\beta_{7}\)\(=\)\((\)\( 577 \nu^{7} - 573 \nu^{6} + 6535 \nu^{5} + 2346 \nu^{4} + 52541 \nu^{3} + 340 \nu^{2} + 74499 \nu + 20844 \)\()/9030\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{4} - 5 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - 9 \beta_{6} - \beta_{5} - \beta_{3} - 3\)
\(\nu^{4}\)\(=\)\(-\beta_{5} - 11 \beta_{4} - 11 \beta_{3} + 41 \beta_{2} - 16 \beta_{1} - 41\)
\(\nu^{5}\)\(=\)\(11 \beta_{7} + 91 \beta_{6} - 18 \beta_{4} + 56 \beta_{2} - 91 \beta_{1}\)
\(\nu^{6}\)\(=\)\(18 \beta_{7} + 212 \beta_{6} + 18 \beta_{5} + 113 \beta_{3} + 397\)
\(\nu^{7}\)\(=\)\(113 \beta_{5} + 248 \beta_{4} + 248 \beta_{3} - 798 \beta_{2} + 966 \beta_{1} + 798\)

Character Values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
−1.30177 2.25473i
−0.531167 0.920008i
0.641129 + 1.11047i
1.69181 + 2.93030i
−1.30177 + 2.25473i
−0.531167 + 0.920008i
0.641129 1.11047i
1.69181 2.93030i
0 −1.00000 0 −2.60354 0 −1.30177 + 2.25473i 0 1.00000 0
37.2 0 −1.00000 0 −1.06233 0 −0.531167 + 0.920008i 0 1.00000 0
37.3 0 −1.00000 0 1.28226 0 0.641129 1.11047i 0 1.00000 0
37.4 0 −1.00000 0 3.38361 0 1.69181 2.93030i 0 1.00000 0
565.1 0 −1.00000 0 −2.60354 0 −1.30177 2.25473i 0 1.00000 0
565.2 0 −1.00000 0 −1.06233 0 −0.531167 0.920008i 0 1.00000 0
565.3 0 −1.00000 0 1.28226 0 0.641129 + 1.11047i 0 1.00000 0
565.4 0 −1.00000 0 3.38361 0 1.69181 + 2.93030i 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 \(S_{2}^{\mathrm{new}}(804, [\chi])\):

\( T_{5}^{4} - T_{5}^{3} - 10 T_{5}^{2} + 3 T_{5} + 12 \)
\(T_{7}^{8} - \cdots\)