Properties

Label 804.2.bd.a
Level 804
Weight 2
Character orbit 804.bd
Analytic conductor 6.420
Analytic rank 0
Dimension 2640
CM No

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.bd (of order \(66\) and degree \(20\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(2640\)
Relative dimension: \(132\) over \(\Q(\zeta_{66})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2640q - 42q^{4} - 19q^{6} - 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 2640q - 42q^{4} - 19q^{6} - 32q^{9} - 46q^{10} - 24q^{12} - 88q^{13} - 26q^{16} - 23q^{18} - 128q^{22} - 22q^{24} + 160q^{25} + 44q^{28} - 18q^{30} - 44q^{33} - 6q^{34} - 35q^{36} - 48q^{37} - 92q^{40} - 10q^{42} + 40q^{45} + 46q^{46} - 11q^{48} - 216q^{49} - 76q^{52} + 129q^{54} - 42q^{57} - 56q^{58} - 90q^{60} - 80q^{61} - 48q^{64} - 74q^{66} - 4q^{69} + 96q^{70} - 20q^{72} - 216q^{73} + 76q^{76} - 66q^{78} - 16q^{81} + 20q^{82} - 55q^{84} - 104q^{85} - 30q^{88} - 64q^{90} - 192q^{93} - 32q^{94} - 205q^{96} - 64q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.41230 + 0.0734794i −0.651270 1.60494i 1.98920 0.207550i 1.42405 0.650344i 1.03772 + 2.21881i −2.49089 2.61237i −2.79411 + 0.439290i −2.15170 + 2.09050i −1.96341 + 1.02312i
23.2 −1.41084 + 0.0976039i 0.479660 1.66431i 1.98095 0.275407i 2.88456 1.31734i −0.514280 + 2.39489i 1.85789 + 1.94850i −2.76792 + 0.581904i −2.53985 1.59660i −3.94108 + 2.14010i
23.3 −1.41021 + 0.106353i 0.881313 + 1.49107i 1.97738 0.299961i −1.21127 + 0.553168i −1.40142 2.00899i −1.08361 1.13646i −2.75661 + 0.633309i −1.44658 + 2.62820i 1.64931 0.908906i
23.4 −1.40914 + 0.119671i −1.73118 + 0.0550183i 1.97136 0.337268i 2.63175 1.20188i 2.43289 0.284701i −0.694593 0.728468i −2.73756 + 0.711173i 2.99395 0.190493i −3.56468 + 2.00856i
23.5 −1.39933 + 0.204660i −1.70511 + 0.304307i 1.91623 0.572774i −2.29127 + 1.04639i 2.32372 0.774794i 0.382733 + 0.401398i −2.56420 + 1.19367i 2.81479 1.03775i 2.99208 1.93317i
23.6 −1.39917 + 0.205706i −0.594255 + 1.62692i 1.91537 0.575637i −1.29680 + 0.592230i 0.496799 2.39858i 3.42631 + 3.59341i −2.56152 + 1.19942i −2.29372 1.93361i 1.69263 1.09539i
23.7 −1.39851 0.210199i 1.49247 + 0.878932i 1.91163 + 0.587929i 1.84970 0.844728i −1.90248 1.54291i −2.22048 2.32877i −2.54985 1.22405i 1.45496 + 2.62357i −2.76437 + 0.792551i
23.8 −1.39805 0.213188i −0.861586 + 1.50255i 1.90910 + 0.596096i −2.23249 + 1.01954i 1.52487 1.91697i −3.06753 3.21714i −2.54194 1.24037i −1.51534 2.58916i 3.33849 0.949437i
23.9 −1.39424 0.236847i 1.65796 + 0.501176i 1.88781 + 0.660444i 1.24254 0.567451i −2.19289 1.09144i 2.47543 + 2.59615i −2.47563 1.36794i 2.49765 + 1.66186i −1.86680 + 0.496869i
23.10 −1.38414 0.290119i 1.40173 1.01742i 1.83166 + 0.803128i −0.809462 + 0.369669i −2.23536 + 1.00158i −0.477128 0.500397i −2.30227 1.64304i 0.929699 2.85231i 1.22765 0.276832i
23.11 −1.35898 0.391381i −0.812487 1.52966i 1.69364 + 1.06376i 0.160397 0.0732507i 0.505471 + 2.39677i 0.691946 + 0.725692i −1.88529 2.10848i −1.67973 + 2.48566i −0.246644 + 0.0367698i
23.12 −1.33789 + 0.458299i 0.594255 1.62692i 1.57992 1.22631i −1.29680 + 0.592230i −0.0494359 + 2.44899i −3.42631 3.59341i −1.55175 + 2.36475i −2.29372 1.93361i 1.46357 1.38666i
23.13 −1.33755 + 0.459299i 1.70511 0.304307i 1.57809 1.22867i −2.29127 + 1.04639i −2.14090 + 1.19018i −0.382733 0.401398i −1.54645 + 2.36823i 2.81479 1.03775i 2.58409 2.45198i
23.14 −1.33646 0.462455i 0.536519 1.64686i 1.57227 + 1.23611i −3.31157 + 1.51234i −1.47864 + 1.95285i 0.404012 + 0.423716i −1.52964 2.37912i −2.42429 1.76714i 5.12518 0.489740i
23.15 −1.32151 0.503605i −0.807567 + 1.53227i 1.49276 + 1.33104i 1.52869 0.698128i 1.83886 1.61820i 1.04475 + 1.09570i −1.30238 2.51074i −1.69567 2.47481i −2.37175 + 0.152726i
23.16 −1.30733 + 0.539338i 1.73118 0.0550183i 1.41823 1.41019i 2.63175 1.20188i −2.23355 + 1.00562i 0.694593 + 0.728468i −1.09353 + 2.60848i 2.99395 0.190493i −2.79235 + 2.99066i
23.17 −1.30218 + 0.551664i −0.881313 1.49107i 1.39133 1.43673i −1.21127 + 0.553168i 1.97020 + 1.45545i 1.08361 + 1.13646i −1.01917 + 2.63843i −1.44658 + 2.62820i 1.27212 1.38854i
23.18 −1.29873 + 0.559731i −0.479660 + 1.66431i 1.37340 1.45388i 2.88456 1.31734i −0.308618 2.42997i −1.85789 1.94850i −0.969897 + 2.65693i −2.53985 1.59660i −3.00892 + 3.32544i
23.19 −1.28898 + 0.581844i 0.651270 + 1.60494i 1.32292 1.49996i 1.42405 0.650344i −1.77330 1.68980i 2.49089 + 2.61237i −0.832461 + 2.70315i −2.15170 + 2.09050i −1.45717 + 1.66685i
23.20 −1.28787 0.584297i −1.68798 0.388227i 1.31719 + 1.50499i −0.837417 + 0.382435i 1.94705 + 1.48627i 3.05562 + 3.20464i −0.817004 2.70786i 2.69856 + 1.31064i 1.30194 0.00322483i
See next 80 embeddings (of 2640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 791.132
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(804, [\chi])\).