Properties

Label 804.2.g.c
Level 804
Weight 2
Character orbit 804.g
Analytic conductor 6.420
Analytic rank 0
Dimension 16
CM No
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.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{12} q^{5} + \beta_{10} q^{7} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{12} q^{5} + \beta_{10} q^{7} + \beta_{2} q^{9} + ( \beta_{1} + \beta_{12} - \beta_{15} ) q^{11} + \beta_{4} q^{13} + ( 1 - \beta_{11} - \beta_{14} ) q^{15} -\beta_{14} q^{17} + ( 2 + \beta_{7} ) q^{19} + ( -1 + \beta_{2} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{21} + ( \beta_{8} + \beta_{14} ) q^{23} + ( \beta_{5} + \beta_{7} - \beta_{11} ) q^{25} + ( \beta_{3} - \beta_{4} ) q^{27} + ( \beta_{2} + \beta_{5} - \beta_{9} + \beta_{11} ) q^{29} + ( \beta_{1} - \beta_{4} + \beta_{6} - \beta_{10} ) q^{31} + ( 1 + \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{33} + ( \beta_{5} - \beta_{8} + \beta_{11} ) q^{35} + ( -1 + \beta_{7} ) q^{37} + ( -\beta_{5} - \beta_{8} + \beta_{9} - \beta_{14} ) q^{39} + ( 2 \beta_{1} - \beta_{6} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{41} + ( -\beta_{1} + \beta_{4} + \beta_{6} + \beta_{10} - \beta_{13} ) q^{43} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{10} + \beta_{15} ) q^{45} + ( \beta_{2} - \beta_{9} + \beta_{14} ) q^{47} + ( \beta_{2} + 2 \beta_{7} + \beta_{9} ) q^{49} + ( -\beta_{6} + \beta_{12} + \beta_{15} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{6} - 2 \beta_{12} + \beta_{15} ) q^{53} + ( -2 - \beta_{2} - \beta_{7} - \beta_{9} ) q^{55} + ( 3 \beta_{1} - \beta_{3} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{57} + ( \beta_{2} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{14} ) q^{59} + ( 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{10} ) q^{61} + ( -\beta_{1} - \beta_{6} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{63} + ( \beta_{2} - \beta_{9} - \beta_{14} ) q^{65} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{13} ) q^{67} + ( -\beta_{4} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{69} + ( -\beta_{2} - \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{14} ) q^{71} + ( -1 - \beta_{2} + \beta_{5} - \beta_{9} - \beta_{11} ) q^{73} + ( \beta_{4} - 2 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{75} + ( -\beta_{2} + 2 \beta_{8} + \beta_{9} - \beta_{14} ) q^{77} + ( -3 \beta_{1} + 2 \beta_{4} - \beta_{6} + 2 \beta_{10} - \beta_{13} ) q^{79} + ( 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{14} ) q^{81} + ( -\beta_{2} - 3 \beta_{8} + \beta_{9} ) q^{83} + ( 2 \beta_{1} + \beta_{4} + \beta_{13} ) q^{85} + ( -\beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{87} + ( \beta_{2} - \beta_{5} - 4 \beta_{8} - \beta_{9} - \beta_{11} - 3 \beta_{14} ) q^{89} + \beta_{7} q^{91} + ( -2 + 3 \beta_{8} - \beta_{11} + 2 \beta_{14} ) q^{93} + ( -\beta_{1} - 2 \beta_{3} - 3 \beta_{12} + \beta_{13} - \beta_{15} ) q^{95} + ( -\beta_{1} - 3 \beta_{4} + \beta_{6} + 2 \beta_{10} - \beta_{13} ) q^{97} + ( 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - \beta_{10} + 2 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{9} + O(q^{10}) \) \( 16q + 2q^{9} + 18q^{15} + 28q^{19} - 16q^{21} + 14q^{33} - 20q^{37} - 4q^{49} - 32q^{55} + 4q^{67} - 16q^{73} + 6q^{81} - 4q^{91} - 30q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{14} - x^{12} - 27 x^{10} + 88 x^{8} - 243 x^{6} - 81 x^{4} - 729 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + 14 \nu^{13} - 43 \nu^{11} + 228 \nu^{9} + 196 \nu^{7} + 105 \nu^{5} + 2646 \nu^{3} + 5589 \nu \)\()/11664\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} + 14 \nu^{13} - 43 \nu^{11} + 228 \nu^{9} + 196 \nu^{7} + 105 \nu^{5} - 9018 \nu^{3} + 5589 \nu \)\()/11664\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{14} + 52 \nu^{12} - 131 \nu^{10} + 300 \nu^{8} - 412 \nu^{6} + 5703 \nu^{4} - 14580 \nu^{2} + 7533 \)\()/7776\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} - 38 \nu^{13} + 67 \nu^{11} - 204 \nu^{9} + 452 \nu^{7} - 2217 \nu^{5} + 9018 \nu^{3} - 3645 \nu \)\()/5832\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{14} + 19 \nu^{12} - 44 \nu^{10} + 36 \nu^{8} - 304 \nu^{6} + 855 \nu^{4} - 4725 \nu^{2} + 972 \)\()/1944\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{14} - 32 \nu^{12} - 179 \nu^{10} + 156 \nu^{8} - 220 \nu^{6} + 339 \nu^{4} - 7560 \nu^{2} + 50301 \)\()/7776\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{14} + \nu^{12} + \nu^{10} + 27 \nu^{8} - 88 \nu^{6} + 243 \nu^{4} + 81 \nu^{2} + 729 \)\()/729\)
\(\beta_{10}\)\(=\)\((\)\( 4 \nu^{15} - 13 \nu^{13} + 86 \nu^{11} - 180 \nu^{9} + 514 \nu^{7} - 1764 \nu^{5} + 4617 \nu^{3} - 13122 \nu \)\()/4374\)
\(\beta_{11}\)\(=\)\((\)\( 53 \nu^{14} - 44 \nu^{12} + 343 \nu^{10} - 1116 \nu^{8} + 1100 \nu^{6} - 9171 \nu^{4} - 2916 \nu^{2} - 47385 \)\()/23328\)
\(\beta_{12}\)\(=\)\((\)\( 95 \nu^{15} - 212 \nu^{13} - 59 \nu^{11} - 180 \nu^{9} + 2852 \nu^{7} - 24633 \nu^{5} - 12636 \nu^{3} + 91125 \nu \)\()/69984\)
\(\beta_{13}\)\(=\)\((\)\( 3 \nu^{15} + 10 \nu^{13} - 25 \nu^{11} - 4 \nu^{9} + 84 \nu^{7} + 91 \nu^{5} - 3222 \nu^{3} - 729 \nu \)\()/1944\)
\(\beta_{14}\)\(=\)\((\)\( -23 \nu^{14} + 14 \nu^{12} + 113 \nu^{10} - 180 \nu^{8} - 404 \nu^{6} + 1881 \nu^{4} + 14094 \nu^{2} - 34263 \)\()/5832\)
\(\beta_{15}\)\(=\)\((\)\( 169 \nu^{15} - 412 \nu^{13} - 493 \nu^{11} - 108 \nu^{9} + 7420 \nu^{7} - 24543 \nu^{5} - 48276 \nu^{3} + 276291 \nu \)\()/69984\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{14} + \beta_{11} + \beta_{8} - \beta_{7} + 2 \beta_{5}\)
\(\nu^{5}\)\(=\)\(3 \beta_{15} - \beta_{13} - 5 \beta_{12} + \beta_{10} - 2 \beta_{6} - 2 \beta_{4} - 3 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-\beta_{14} - 4 \beta_{11} - 6 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 3 \beta_{5} - 2 \beta_{2} + 10\)
\(\nu^{7}\)\(=\)\(4 \beta_{15} + 3 \beta_{13} - 12 \beta_{12} + 2 \beta_{10} + 6 \beta_{6} + 9 \beta_{4} + 5 \beta_{3} + 4 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-14 \beta_{14} - 10 \beta_{11} + 13 \beta_{9} - 12 \beta_{8} - 4 \beta_{7} - 2 \beta_{5} + 6 \beta_{2} - 34\)
\(\nu^{9}\)\(=\)\(4 \beta_{15} - 17 \beta_{13} + 4 \beta_{12} + 6 \beta_{10} - 15 \beta_{6} + 16 \beta_{4} + 20 \beta_{3} - 36 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-6 \beta_{14} + 22 \beta_{11} + 34 \beta_{9} - 24 \beta_{8} - 16 \beta_{7} + 10 \beta_{5} - 30 \beta_{2} + 129\)
\(\nu^{11}\)\(=\)\(8 \beta_{15} - 50 \beta_{13} - 8 \beta_{12} + 62 \beta_{10} - 72 \beta_{6} + 32 \beta_{4} - 36 \beta_{3} + 103 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-54 \beta_{14} - 22 \beta_{11} - 34 \beta_{9} - 124 \beta_{8} + 68 \beta_{7} - 6 \beta_{5} + 165 \beta_{2} + 446\)
\(\nu^{13}\)\(=\)\(-144 \beta_{15} + 102 \beta_{13} + 112 \beta_{12} + 34 \beta_{10} + 8 \beta_{6} - 19 \beta_{4} + 119 \beta_{3} + 520 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-107 \beta_{14} + 325 \beta_{11} + 150 \beta_{9} + 35 \beta_{8} - 35 \beta_{7} + 172 \beta_{5} + 554 \beta_{2} - 494\)
\(\nu^{15}\)\(=\)\(349 \beta_{15} - 185 \beta_{13} + 53 \beta_{12} + 325 \beta_{10} - 754 \beta_{6} - 914 \beta_{4} + 264 \beta_{3} - 701 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−1.72824 0.114788i
−1.72824 + 0.114788i
−1.49351 0.877170i
−1.49351 + 0.877170i
−0.760738 1.55605i
−0.760738 + 1.55605i
−0.673707 1.59566i
−0.673707 + 1.59566i
0.673707 1.59566i
0.673707 + 1.59566i
0.760738 1.55605i
0.760738 + 1.55605i
1.49351 0.877170i
1.49351 + 0.877170i
1.72824 0.114788i
1.72824 + 0.114788i
0 −1.72824 0.114788i 0 −2.40477 0 3.48008i 0 2.97365 + 0.396761i 0
401.2 0 −1.72824 + 0.114788i 0 −2.40477 0 3.48008i 0 2.97365 0.396761i 0
401.3 0 −1.49351 0.877170i 0 1.18740 0 0.405325i 0 1.46115 + 2.62012i 0
401.4 0 −1.49351 + 0.877170i 0 1.18740 0 0.405325i 0 1.46115 2.62012i 0
401.5 0 −0.760738 1.55605i 0 −3.48890 0 1.71955i 0 −1.84255 + 2.36749i 0
401.6 0 −0.760738 + 1.55605i 0 −3.48890 0 1.71955i 0 −1.84255 2.36749i 0
401.7 0 −0.673707 1.59566i 0 0.796732 0 3.71051i 0 −2.09224 + 2.15001i 0
401.8 0 −0.673707 + 1.59566i 0 0.796732 0 3.71051i 0 −2.09224 2.15001i 0
401.9 0 0.673707 1.59566i 0 −0.796732 0 3.71051i 0 −2.09224 2.15001i 0
401.10 0 0.673707 + 1.59566i 0 −0.796732 0 3.71051i 0 −2.09224 + 2.15001i 0
401.11 0 0.760738 1.55605i 0 3.48890 0 1.71955i 0 −1.84255 2.36749i 0
401.12 0 0.760738 + 1.55605i 0 3.48890 0 1.71955i 0 −1.84255 + 2.36749i 0
401.13 0 1.49351 0.877170i 0 −1.18740 0 0.405325i 0 1.46115 2.62012i 0
401.14 0 1.49351 + 0.877170i 0 −1.18740 0 0.405325i 0 1.46115 + 2.62012i 0
401.15 0 1.72824 0.114788i 0 2.40477 0 3.48008i 0 2.97365 0.396761i 0
401.16 0 1.72824 + 0.114788i 0 2.40477 0 3.48008i 0 2.97365 + 0.396761i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
67.b Odd 1 yes
201.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{8} - 20 T_{5}^{6} + 108 T_{5}^{4} - 160 T_{5}^{2} + 63 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).