# Properties

 Label 648.2.bd.a Level $648$ Weight $2$ Character orbit 648.bd Analytic conductor $5.174$ Analytic rank $0$ Dimension $1908$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 648.bd (of order $$54$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.17430605098$$ Analytic rank: $$0$$ Dimension: $$1908$$ Relative dimension: $$106$$ over $$\Q(\zeta_{54})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1908q - 18q^{2} - 18q^{4} - 18q^{6} - 36q^{7} - 18q^{8} - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1908q - 18q^{2} - 18q^{4} - 18q^{6} - 36q^{7} - 18q^{8} - 36q^{9} - 18q^{10} - 18q^{12} - 18q^{14} - 36q^{15} - 18q^{16} - 36q^{17} - 18q^{18} - 18q^{20} - 18q^{22} - 36q^{23} - 18q^{24} - 36q^{25} - 9q^{26} - 9q^{28} - 18q^{30} - 36q^{31} - 18q^{32} - 36q^{33} - 18q^{34} - 18q^{36} - 18q^{38} - 36q^{39} - 18q^{40} - 36q^{41} + 27q^{42} - 90q^{44} - 18q^{46} - 36q^{47} - 117q^{48} - 36q^{49} + 99q^{50} - 18q^{52} + 108q^{54} - 18q^{55} - 144q^{56} - 36q^{57} - 18q^{58} - 135q^{60} + 81q^{62} - 36q^{63} - 18q^{64} - 36q^{65} + 54q^{66} - 63q^{68} - 18q^{70} - 36q^{71} - 18q^{72} - 36q^{73} - 18q^{74} - 18q^{76} - 45q^{78} - 36q^{79} - 36q^{80} - 36q^{81} - 36q^{82} - 45q^{84} - 18q^{86} - 36q^{87} - 18q^{88} - 36q^{89} - 81q^{90} - 108q^{92} - 18q^{94} - 36q^{95} - 135q^{96} - 36q^{97} - 189q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.41420 0.00635899i −1.68145 0.415592i 1.99992 + 0.0179858i 2.71787 0.813676i 2.37527 + 0.598422i −1.20371 2.79050i −2.82817 0.0381529i 2.65457 + 1.39760i −3.84878 + 1.13342i
13.2 −1.41298 + 0.0591140i 1.07290 1.35974i 1.99301 0.167054i −2.54435 + 0.761729i −1.43560 + 1.98471i −0.0791644 0.183524i −2.80620 + 0.353858i −0.697789 2.91772i 3.55008 1.22671i
13.3 −1.41277 0.0639239i 0.429774 + 1.67788i 1.99183 + 0.180619i −4.11411 + 1.23168i −0.499914 2.39793i −1.24557 2.88757i −2.80244 0.382498i −2.63059 + 1.44222i 5.89101 1.47709i
13.4 −1.40484 0.162553i −0.0416517 1.73155i 1.94715 + 0.456721i 1.45600 0.435896i −0.222954 + 2.43932i 1.13129 + 2.62263i −2.66120 0.958136i −2.99653 + 0.144244i −2.11630 + 0.375689i
13.5 −1.40382 + 0.171153i −1.43648 + 0.967741i 1.94141 0.480537i 3.87465 1.15999i 1.85093 1.60439i 1.40145 + 3.24893i −2.64315 + 1.00687i 1.12695 2.78028i −5.24077 + 2.29158i
13.6 −1.40381 + 0.171199i 1.73135 0.0491944i 1.94138 0.480663i 0.878928 0.263134i −2.42207 + 0.365466i −1.82006 4.21937i −2.64305 + 1.00712i 2.99516 0.170346i −1.18880 + 0.519862i
13.7 −1.39070 + 0.256830i 1.07027 + 1.36181i 1.86808 0.714346i 1.22125 0.365619i −1.83818 1.61898i 0.532782 + 1.23513i −2.41446 + 1.47322i −0.709034 + 2.91501i −1.60449 + 0.822120i
13.8 −1.39047 + 0.258070i −1.54866 0.775654i 1.86680 0.717676i −2.80707 + 0.840380i 2.35354 + 0.678857i 0.791894 + 1.83582i −2.41051 + 1.47967i 1.79672 + 2.40245i 3.68626 1.89294i
13.9 −1.35488 0.405333i 1.49899 + 0.867767i 1.67141 + 1.09836i −1.99050 + 0.595916i −1.67922 1.78331i 1.60061 + 3.71063i −1.81936 2.16562i 1.49396 + 2.60155i 2.93843 0.000580680i
13.10 −1.35470 0.405938i −1.35025 + 1.08482i 1.67043 + 1.09985i −1.24137 + 0.371641i 2.26955 0.921488i 1.17620 + 2.72673i −1.81646 2.16806i 0.646340 2.92955i 1.83254 0.000455190i
13.11 −1.34876 0.425256i 1.03611 + 1.38798i 1.63832 + 1.14714i 3.18189 0.952596i −0.807215 2.31266i −0.256454 0.594528i −1.72187 2.24392i −0.852968 + 2.87619i −4.69671 0.0682924i
13.12 −1.33846 + 0.456643i 1.66450 0.479002i 1.58296 1.22240i −0.183674 + 0.0549883i −2.00913 + 1.40121i 1.23663 + 2.86682i −1.56053 + 2.35897i 2.54112 1.59460i 0.220730 0.157473i
13.13 −1.33558 + 0.464990i −0.266503 + 1.71143i 1.56757 1.24207i 0.902416 0.270166i −0.439858 2.40967i −0.0371225 0.0860596i −1.51607 + 2.38779i −2.85795 0.912201i −1.07963 + 0.780443i
13.14 −1.31801 0.512678i −1.65507 + 0.510644i 1.47432 + 1.35143i −1.81662 + 0.543860i 2.44320 + 0.175479i −0.762450 1.76756i −1.25033 2.53706i 2.47849 1.69030i 2.67315 + 0.214525i
13.15 −1.28811 0.583755i −0.393780 + 1.68669i 1.31846 + 1.50388i 1.38808 0.415563i 1.49185 1.94278i −1.33275 3.08966i −0.820426 2.70682i −2.68988 1.32837i −2.03059 0.275005i
13.16 −1.24166 + 0.676962i −1.38930 + 1.03433i 1.08345 1.68111i −0.453349 + 0.135724i 1.02484 2.22479i −1.03892 2.40849i −0.207223 + 2.82083i 0.860326 2.87399i 0.471026 0.475423i
13.17 −1.23745 0.684630i 0.393780 1.68669i 1.06256 + 1.69439i −1.38808 + 0.415563i −1.64204 + 1.81761i −1.33275 3.08966i −0.154840 2.82419i −2.68988 1.32837i 2.00218 + 0.436081i
13.18 −1.22371 + 0.708889i 0.742903 1.56464i 0.994952 1.73496i 2.98234 0.892854i 0.200055 + 2.44131i −0.865364 2.00614i 0.0123558 + 2.82840i −1.89619 2.32475i −3.01660 + 3.20675i
13.19 −1.21220 + 0.728408i −0.572218 1.63480i 0.938845 1.76595i 1.28376 0.384332i 1.88444 + 1.56489i 0.764986 + 1.77344i 0.148264 + 2.82454i −2.34513 + 1.87092i −1.27622 + 1.40099i
13.20 −1.19829 0.751060i 1.65507 0.510644i 0.871818 + 1.79998i 1.81662 0.543860i −2.36678 0.631152i −0.762450 1.76756i 0.307200 2.81169i 2.47849 1.69030i −2.58531 0.712685i
See next 80 embeddings (of 1908 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 637.106 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
81.g even 27 1 inner
648.bd even 54 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.bd.a 1908
8.b even 2 1 inner 648.2.bd.a 1908
81.g even 27 1 inner 648.2.bd.a 1908
648.bd even 54 1 inner 648.2.bd.a 1908

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.bd.a 1908 1.a even 1 1 trivial
648.2.bd.a 1908 8.b even 2 1 inner
648.2.bd.a 1908 81.g even 27 1 inner
648.2.bd.a 1908 648.bd even 54 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(648, [\chi])$$.