Properties

Label 804.2.c.b
Level 804
Weight 2
Character orbit 804.c
Analytic conductor 6.420
Analytic rank 0
Dimension 128
CM No

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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: \(128\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128q + 4q^{4} - 6q^{6} - 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q + 4q^{4} - 6q^{6} - 4q^{9} - 4q^{10} + 4q^{12} - 8q^{13} - 4q^{16} + 18q^{18} - 8q^{21} - 8q^{22} - 24q^{24} - 136q^{25} + 14q^{30} - 32q^{33} + 34q^{36} - 48q^{37} + 16q^{40} - 28q^{42} - 16q^{45} - 28q^{46} - 22q^{48} - 152q^{49} + 8q^{52} - 16q^{54} - 32q^{57} - 20q^{58} - 14q^{60} + 8q^{61} + 16q^{64} + 14q^{66} + 56q^{69} - 4q^{70} - 8q^{72} - 48q^{73} - 36q^{76} + 40q^{78} + 44q^{81} + 60q^{82} + 46q^{84} + 64q^{85} - 28q^{88} - 14q^{90} + 32q^{93} - 8q^{96} + 64q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1 −1.41002 0.108878i 1.62789 + 0.591579i 1.97629 + 0.307041i 3.04949i −2.23094 1.01138i 0.278695i −2.75317 0.648108i 2.30007 + 1.92605i −0.332024 + 4.29983i
671.2 −1.41002 + 0.108878i 1.62789 0.591579i 1.97629 0.307041i 3.04949i −2.23094 + 1.01138i 0.278695i −2.75317 + 0.648108i 2.30007 1.92605i −0.332024 4.29983i
671.3 −1.40995 0.109793i −0.420694 1.68018i 1.97589 + 0.309604i 0.933302i 0.408683 + 2.41516i 1.91511i −2.75191 0.653464i −2.64603 + 1.41369i −0.102470 + 1.31591i
671.4 −1.40995 + 0.109793i −0.420694 + 1.68018i 1.97589 0.309604i 0.933302i 0.408683 2.41516i 1.91511i −2.75191 + 0.653464i −2.64603 1.41369i −0.102470 1.31591i
671.5 −1.40536 0.157993i 1.10632 + 1.33268i 1.95008 + 0.444075i 1.25971i −1.34423 2.04769i 4.86310i −2.67040 0.932185i −0.552098 + 2.94876i 0.199026 1.77035i
671.6 −1.40536 + 0.157993i 1.10632 1.33268i 1.95008 0.444075i 1.25971i −1.34423 + 2.04769i 4.86310i −2.67040 + 0.932185i −0.552098 2.94876i 0.199026 + 1.77035i
671.7 −1.39093 0.255554i 1.65649 0.506011i 1.86938 + 0.710916i 2.60371i −2.43338 + 0.280506i 2.89039i −2.41851 1.46656i 2.48790 1.67640i −0.665387 + 3.62158i
671.8 −1.39093 + 0.255554i 1.65649 + 0.506011i 1.86938 0.710916i 2.60371i −2.43338 0.280506i 2.89039i −2.41851 + 1.46656i 2.48790 + 1.67640i −0.665387 3.62158i
671.9 −1.38271 0.296856i −1.71448 + 0.246081i 1.82375 + 0.820930i 2.85244i 2.44367 + 0.168697i 2.12311i −2.27802 1.67650i 2.87889 0.843802i −0.846765 + 3.94409i
671.10 −1.38271 + 0.296856i −1.71448 0.246081i 1.82375 0.820930i 2.85244i 2.44367 0.168697i 2.12311i −2.27802 + 1.67650i 2.87889 + 0.843802i −0.846765 3.94409i
671.11 −1.37515 0.330103i 0.355099 + 1.69526i 1.78206 + 0.907880i 0.0453328i 0.0712965 2.44845i 2.82199i −2.15091 1.83673i −2.74781 + 1.20397i 0.0149645 0.0623394i
671.12 −1.37515 + 0.330103i 0.355099 1.69526i 1.78206 0.907880i 0.0453328i 0.0712965 + 2.44845i 2.82199i −2.15091 + 1.83673i −2.74781 1.20397i 0.0149645 + 0.0623394i
671.13 −1.36593 0.366392i −0.973809 1.43237i 1.73151 + 1.00093i 2.42154i 0.805342 + 2.31331i 1.75439i −1.99839 2.00161i −1.10339 + 2.78972i 0.887230 3.30764i
671.14 −1.36593 + 0.366392i −0.973809 + 1.43237i 1.73151 1.00093i 2.42154i 0.805342 2.31331i 1.75439i −1.99839 + 2.00161i −1.10339 2.78972i 0.887230 + 3.30764i
671.15 −1.28652 0.587251i 1.29382 1.15153i 1.31027 + 1.51102i 1.10955i −2.34077 + 0.721672i 0.262541i −0.798342 2.71342i 0.347946 2.97975i 0.651585 1.42746i
671.16 −1.28652 + 0.587251i 1.29382 + 1.15153i 1.31027 1.51102i 1.10955i −2.34077 0.721672i 0.262541i −0.798342 + 2.71342i 0.347946 + 2.97975i 0.651585 + 1.42746i
671.17 −1.27950 0.602386i −1.48757 + 0.887204i 1.27426 + 1.54151i 3.40681i 2.43779 0.239089i 4.07118i −0.701842 2.73997i 1.42574 2.63956i 2.05221 4.35902i
671.18 −1.27950 + 0.602386i −1.48757 0.887204i 1.27426 1.54151i 3.40681i 2.43779 + 0.239089i 4.07118i −0.701842 + 2.73997i 1.42574 + 2.63956i 2.05221 + 4.35902i
671.19 −1.27439 0.613127i −1.63926 0.559297i 1.24815 + 1.56273i 0.350980i 1.74615 + 1.71784i 2.98214i −0.632485 2.75680i 2.37437 + 1.83367i −0.215195 + 0.447286i
671.20 −1.27439 + 0.613127i −1.63926 + 0.559297i 1.24815 1.56273i 0.350980i 1.74615 1.71784i 2.98214i −0.632485 + 2.75680i 2.37437 1.83367i −0.215195 0.447286i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 671.128
Significant digits:
Format:

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 \(T_{5}^{64} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).