Properties

Label 804.2.q.a
Level 804
Weight 2
Character orbit 804.q
Analytic conductor 6.420
Analytic rank 0
Dimension 60
CM No

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.q (of order \(11\) and degree \(10\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60q - 6q^{3} - 2q^{5} - 2q^{7} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 6q^{3} - 2q^{5} - 2q^{7} - 6q^{9} + 7q^{11} - 2q^{13} + 9q^{15} - 19q^{17} + 2q^{19} - 2q^{21} + 4q^{23} + 16q^{25} - 6q^{27} + 16q^{29} - 28q^{31} - 4q^{33} + 28q^{35} + 2q^{37} - 2q^{39} + 32q^{41} + 19q^{43} - 2q^{45} + 2q^{47} - 70q^{49} - 19q^{51} + 31q^{53} - 5q^{55} + 13q^{57} + 59q^{59} + 32q^{61} + 9q^{63} + 28q^{65} + 7q^{67} + 4q^{69} + 16q^{71} + 19q^{73} + 16q^{75} - 46q^{77} + 48q^{79} - 6q^{81} + 60q^{83} - 66q^{85} + 5q^{87} - 22q^{89} + 24q^{91} + 5q^{93} + 103q^{95} - 46q^{97} - 4q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −0.654861 + 0.755750i 0 −2.85861 + 1.83711i 0 −0.661793 4.60287i 0 −0.142315 0.989821i 0
25.2 0 −0.654861 + 0.755750i 0 −2.63012 + 1.69027i 0 0.473818 + 3.29548i 0 −0.142315 0.989821i 0
25.3 0 −0.654861 + 0.755750i 0 0.361898 0.232578i 0 −0.272618 1.89610i 0 −0.142315 0.989821i 0
25.4 0 −0.654861 + 0.755750i 0 0.412663 0.265202i 0 0.411742 + 2.86373i 0 −0.142315 0.989821i 0
25.5 0 −0.654861 + 0.755750i 0 1.20911 0.777049i 0 −0.250069 1.73927i 0 −0.142315 0.989821i 0
25.6 0 −0.654861 + 0.755750i 0 3.26560 2.09868i 0 0.572019 + 3.97848i 0 −0.142315 0.989821i 0
193.1 0 −0.654861 0.755750i 0 −2.85861 1.83711i 0 −0.661793 + 4.60287i 0 −0.142315 + 0.989821i 0
193.2 0 −0.654861 0.755750i 0 −2.63012 1.69027i 0 0.473818 3.29548i 0 −0.142315 + 0.989821i 0
193.3 0 −0.654861 0.755750i 0 0.361898 + 0.232578i 0 −0.272618 + 1.89610i 0 −0.142315 + 0.989821i 0
193.4 0 −0.654861 0.755750i 0 0.412663 + 0.265202i 0 0.411742 2.86373i 0 −0.142315 + 0.989821i 0
193.5 0 −0.654861 0.755750i 0 1.20911 + 0.777049i 0 −0.250069 + 1.73927i 0 −0.142315 + 0.989821i 0
193.6 0 −0.654861 0.755750i 0 3.26560 + 2.09868i 0 0.572019 3.97848i 0 −0.142315 + 0.989821i 0
241.1 0 0.415415 0.909632i 0 −3.88811 1.14165i 0 1.58324 + 1.82715i 0 −0.654861 0.755750i 0
241.2 0 0.415415 0.909632i 0 −1.26159 0.370436i 0 −2.70108 3.11721i 0 −0.654861 0.755750i 0
241.3 0 0.415415 0.909632i 0 −0.925675 0.271803i 0 1.06841 + 1.23301i 0 −0.654861 0.755750i 0
241.4 0 0.415415 0.909632i 0 0.346311 + 0.101686i 0 −0.316787 0.365591i 0 −0.654861 0.755750i 0
241.5 0 0.415415 0.909632i 0 2.96527 + 0.870683i 0 2.08291 + 2.40380i 0 −0.654861 0.755750i 0
241.6 0 0.415415 0.909632i 0 4.02046 + 1.18051i 0 −1.53030 1.76606i 0 −0.654861 0.755750i 0
265.1 0 −0.142315 + 0.989821i 0 −1.58453 3.46964i 0 −2.06192 0.605433i 0 −0.959493 0.281733i 0
265.2 0 −0.142315 + 0.989821i 0 −1.15584 2.53094i 0 2.78570 + 0.817955i 0 −0.959493 0.281733i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.6
Significant digits:
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Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{60} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).