# Properties

 Label 320.2.bf.a Level $320$ Weight $2$ Character orbit 320.bf Analytic conductor $2.555$ Analytic rank $0$ Dimension $368$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 320.bf (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.55521286468$$ Analytic rank: $$0$$ Dimension: $$368$$ Relative dimension: $$46$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$368q - 16q^{4} - 8q^{5} - 16q^{6} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$368q - 16q^{4} - 8q^{5} - 16q^{6} - 16q^{9} - 8q^{10} - 16q^{11} - 16q^{14} - 8q^{15} - 16q^{16} - 16q^{19} - 8q^{20} - 16q^{21} - 16q^{24} - 8q^{25} - 96q^{26} - 16q^{29} + 72q^{30} - 16q^{34} - 8q^{35} - 176q^{36} - 16q^{39} + 32q^{40} - 16q^{41} - 16q^{44} - 8q^{45} - 16q^{46} - 16q^{49} + 16q^{50} + 16q^{51} + 48q^{54} - 72q^{55} - 112q^{56} - 80q^{59} + 88q^{60} - 16q^{61} + 80q^{64} - 16q^{65} - 144q^{66} - 16q^{69} + 88q^{70} + 48q^{71} + 80q^{74} - 72q^{75} - 80q^{76} - 48q^{79} + 16q^{80} - 16q^{81} - 128q^{84} - 8q^{85} - 16q^{86} - 16q^{89} - 8q^{90} - 16q^{91} - 192q^{94} - 16q^{96} - 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.41315 + 0.0548766i 0.462030 2.32278i 1.99398 0.155097i −1.42528 1.72296i −0.525451 + 3.30779i 0.330301 + 0.797417i −2.80927 + 0.328598i −2.41021 0.998340i 2.10868 + 2.35658i
29.2 −1.40092 0.193431i −0.612243 + 3.07795i 1.92517 + 0.541963i −2.22253 + 0.245694i 1.45308 4.19355i 1.41327 + 3.41194i −2.59218 1.13164i −6.32731 2.62086i 3.16112 + 0.0857073i
29.3 −1.38805 + 0.270776i −0.246966 + 1.24158i 1.85336 0.751702i 1.74404 + 1.39940i 0.00660982 1.79025i 0.414009 + 0.999505i −2.36901 + 1.54525i 1.29111 + 0.534796i −2.79974 1.47019i
29.4 −1.34695 + 0.430954i 0.329632 1.65717i 1.62856 1.16095i −1.38798 + 1.75314i 0.270166 + 2.37419i 0.267328 + 0.645386i −1.69327 + 2.26557i 0.134078 + 0.0555368i 1.11402 2.95955i
29.5 −1.33571 0.464641i 0.488587 2.45629i 1.56822 + 1.24125i 0.609350 + 2.15144i −1.79390 + 3.05386i −0.627512 1.51495i −1.51794 2.38660i −3.02301 1.25217i 0.185734 3.15682i
29.6 −1.33383 0.469997i 0.0916179 0.460594i 1.55821 + 1.25379i 1.74536 1.39776i −0.338681 + 0.571295i 1.00872 + 2.43526i −1.48910 2.40470i 2.56789 + 1.06365i −2.98495 + 1.04406i
29.7 −1.26165 0.638935i −0.547765 + 2.75380i 1.18353 + 1.61222i 1.52882 1.63178i 2.45059 3.12435i −1.54675 3.73418i −0.463089 2.79026i −4.51173 1.86882i −2.97144 + 1.08192i
29.8 −1.20469 + 0.740758i 0.318063 1.59901i 0.902555 1.78477i 2.13894 0.651887i 0.801313 + 2.16192i −1.98639 4.79556i 0.234782 + 2.81867i 0.315964 + 0.130877i −2.09386 + 2.36975i
29.9 −1.15919 0.810109i −0.212827 + 1.06995i 0.687447 + 1.87814i −0.607268 + 2.15203i 1.11349 1.06787i −0.679902 1.64143i 0.724616 2.73403i 1.67213 + 0.692620i 2.44732 2.00266i
29.10 −1.14869 + 0.824933i −0.336544 + 1.69192i 0.638972 1.89518i −1.00832 1.99582i −1.00914 2.22112i −0.601901 1.45312i 0.829417 + 2.70408i 0.0223079 + 0.00924022i 2.80466 + 1.46078i
29.11 −0.970636 + 1.02853i −0.0730884 + 0.367440i −0.115730 1.99665i −2.22727 + 0.198194i −0.306979 0.431824i 1.26909 + 3.06387i 2.16594 + 1.81899i 2.64197 + 1.09434i 1.95802 2.48318i
29.12 −0.963775 1.03496i −0.0117336 + 0.0589886i −0.142276 + 1.99493i −2.09609 0.778715i 0.0723593 0.0447080i 0.480951 + 1.16112i 2.20179 1.77542i 2.76830 + 1.14667i 1.21422 + 2.91987i
29.13 −0.942250 + 1.05459i −0.630723 + 3.17086i −0.224328 1.98738i 0.0178005 + 2.23600i −2.74966 3.65290i −1.29895 3.13595i 2.30725 + 1.63603i −6.88490 2.85182i −2.37484 2.08810i
29.14 −0.878389 1.10835i 0.572105 2.87617i −0.456865 + 1.94712i 1.03679 1.98118i −3.69032 + 1.89230i −0.591145 1.42715i 2.55939 1.20396i −5.17340 2.14289i −3.10654 + 0.591119i
29.15 −0.713777 + 1.22087i 0.446234 2.24337i −0.981046 1.74286i 1.63244 + 1.52812i 2.42035 + 2.14606i 1.38403 + 3.34135i 2.82805 + 0.0462814i −2.06195 0.854087i −3.03083 + 0.902259i
29.16 −0.635601 + 1.26333i 0.627776 3.15604i −1.19202 1.60595i −1.59499 1.56716i 3.58812 + 2.79908i −0.393699 0.950472i 2.78650 0.485175i −6.79487 2.81453i 2.99363 1.01891i
29.17 −0.628783 1.26674i 0.368711 1.85364i −1.20926 + 1.59301i 0.834497 + 2.07452i −2.57992 + 0.698473i 1.47604 + 3.56349i 2.77829 + 0.530167i −0.528383 0.218863i 2.10316 2.36151i
29.18 −0.561336 + 1.29804i −0.279704 + 1.40617i −1.36980 1.45727i 2.09531 0.780827i −1.66825 1.15240i 0.431062 + 1.04068i 2.66051 0.960039i 0.872568 + 0.361429i −0.162628 + 3.15809i
29.19 −0.458870 + 1.33770i 0.110253 0.554278i −1.57888 1.22766i −1.78619 + 1.34519i 0.690865 + 0.401826i −1.43120 3.45521i 2.36674 1.54873i 2.47657 + 1.02583i −0.979827 3.00665i
29.20 −0.424621 1.34896i −0.454817 + 2.28652i −1.63939 + 1.14559i −0.548285 2.16781i 3.27755 0.357373i 0.363715 + 0.878084i 2.24148 + 1.72504i −2.24966 0.931842i −2.69147 + 1.66011i
See next 80 embeddings (of 368 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 309.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
64.i even 16 1 inner
320.bf even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.bf.a 368
5.b even 2 1 inner 320.2.bf.a 368
64.i even 16 1 inner 320.2.bf.a 368
320.bf even 16 1 inner 320.2.bf.a 368

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.bf.a 368 1.a even 1 1 trivial
320.2.bf.a 368 5.b even 2 1 inner
320.2.bf.a 368 64.i even 16 1 inner
320.2.bf.a 368 320.bf even 16 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(320, [\chi])$$.