Properties

Label 804.2.o.d
Level 804
Weight 2
Character orbit 804.o
Analytic conductor 6.420
Analytic rank 0
Dimension 36
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.o (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 4q^{9} - 36q^{13} + 18q^{15} + 16q^{21} + 76q^{25} + 6q^{31} + 4q^{33} + 42q^{37} - 21q^{39} + 2q^{49} + 18q^{51} + 20q^{55} + 18q^{57} - 24q^{61} - 12q^{63} - 8q^{67} + 3q^{69} + 14q^{73} + 72q^{79} - 12q^{81} - 18q^{85} - 21q^{87} - 68q^{91} + 9q^{93} - 48q^{97} + 15q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
365.1 0 −1.72171 0.188968i 0 3.09374 0 −1.38610 0.800262i 0 2.92858 + 0.650697i 0
365.2 0 −1.71192 + 0.263286i 0 −0.454473 0 −0.768091 0.443457i 0 2.86136 0.901449i 0
365.3 0 −1.53706 + 0.798403i 0 −3.20386 0 2.52764 + 1.45934i 0 1.72511 2.45439i 0
365.4 0 −1.41646 0.996822i 0 0.835859 0 2.58614 + 1.49311i 0 1.01269 + 2.82391i 0
365.5 0 −1.34511 1.09118i 0 −3.88149 0 −3.35972 1.93974i 0 0.618641 + 2.93552i 0
365.6 0 −0.870385 + 1.49747i 0 −2.31311 0 −0.381576 0.220303i 0 −1.48486 2.60676i 0
365.7 0 −0.869231 + 1.49814i 0 2.66956 0 0.767031 + 0.442845i 0 −1.48888 2.60447i 0
365.8 0 −0.536215 + 1.64696i 0 1.65208 0 −3.18562 1.83922i 0 −2.42495 1.76625i 0
365.9 0 −0.355178 1.69524i 0 −3.60368 0 3.20029 + 1.84769i 0 −2.74770 + 1.20423i 0
365.10 0 0.355178 1.69524i 0 3.60368 0 3.20029 + 1.84769i 0 −2.74770 1.20423i 0
365.11 0 0.536215 + 1.64696i 0 −1.65208 0 −3.18562 1.83922i 0 −2.42495 + 1.76625i 0
365.12 0 0.869231 + 1.49814i 0 −2.66956 0 0.767031 + 0.442845i 0 −1.48888 + 2.60447i 0
365.13 0 0.870385 + 1.49747i 0 2.31311 0 −0.381576 0.220303i 0 −1.48486 + 2.60676i 0
365.14 0 1.34511 1.09118i 0 3.88149 0 −3.35972 1.93974i 0 0.618641 2.93552i 0
365.15 0 1.41646 0.996822i 0 −0.835859 0 2.58614 + 1.49311i 0 1.01269 2.82391i 0
365.16 0 1.53706 + 0.798403i 0 3.20386 0 2.52764 + 1.45934i 0 1.72511 + 2.45439i 0
365.17 0 1.71192 + 0.263286i 0 0.454473 0 −0.768091 0.443457i 0 2.86136 + 0.901449i 0
365.18 0 1.72171 0.188968i 0 −3.09374 0 −1.38610 0.800262i 0 2.92858 0.650697i 0
641.1 0 −1.72171 + 0.188968i 0 3.09374 0 −1.38610 + 0.800262i 0 2.92858 0.650697i 0
641.2 0 −1.71192 0.263286i 0 −0.454473 0 −0.768091 + 0.443457i 0 2.86136 + 0.901449i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.18
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 \(T_{5}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).