# Properties

 Label 96.3.m.a Level $96$ Weight $3$ Character orbit 96.m Analytic conductor $2.616$ Analytic rank $0$ Dimension $64$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 96.m (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.61581053786$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 40q^{10} + 48q^{12} + 32q^{14} - 8q^{16} - 24q^{18} - 160q^{20} - 184q^{22} + 128q^{23} - 72q^{24} - 200q^{26} - 120q^{28} + 40q^{32} + 120q^{34} - 192q^{35} + 280q^{38} + 584q^{40} - 192q^{43} + 104q^{44} + 32q^{46} - 312q^{50} - 192q^{51} - 424q^{52} + 320q^{53} - 72q^{54} - 256q^{55} - 392q^{56} - 352q^{58} - 256q^{59} - 144q^{60} + 64q^{61} - 48q^{62} + 408q^{64} + 144q^{66} + 64q^{67} + 856q^{68} - 192q^{69} + 984q^{70} + 512q^{71} + 1056q^{74} + 384q^{75} + 296q^{76} - 448q^{77} + 360q^{78} + 512q^{79} + 328q^{80} - 760q^{82} - 448q^{86} - 1072q^{88} + 192q^{91} - 784q^{92} - 480q^{94} + 600q^{96} + 272q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.97063 0.341475i −1.60021 0.662827i 3.76679 + 1.34584i −2.29382 + 0.950132i 2.92708 + 1.85262i 3.26583 + 3.26583i −6.96339 3.93843i 2.12132 + 2.12132i 4.84473 1.08908i
19.2 −1.92437 0.544810i 1.60021 + 0.662827i 3.40636 + 2.09683i −8.76568 + 3.63086i −2.71827 2.14733i −4.20494 4.20494i −5.41272 5.89088i 2.12132 + 2.12132i 18.8465 2.21148i
19.3 −1.84914 + 0.762031i 1.60021 + 0.662827i 2.83862 2.81820i 6.06733 2.51317i −3.46410 0.00625081i −9.25703 9.25703i −3.10143 + 7.37436i 2.12132 + 2.12132i −9.30421 + 9.27069i
19.4 −1.49565 1.32779i −1.60021 0.662827i 0.473939 + 3.97182i 5.66092 2.34483i 1.51325 + 3.11610i −8.77775 8.77775i 4.56491 6.56975i 2.12132 + 2.12132i −11.5802 4.00947i
19.5 −1.46158 + 1.36521i 1.60021 + 0.662827i 0.272417 3.99071i −1.99171 + 0.824993i −3.24372 + 1.21584i 8.28172 + 8.28172i 5.04999 + 6.20464i 2.12132 + 2.12132i 1.78475 3.92489i
19.6 −1.19507 1.60369i 1.60021 + 0.662827i −1.14363 + 3.83303i 2.44597 1.01315i −0.849386 3.35835i 3.89082 + 3.89082i 7.51370 2.74670i 2.12132 + 2.12132i −4.54788 2.71179i
19.7 −0.795895 + 1.83482i −1.60021 0.662827i −2.73310 2.92064i 6.54910 2.71273i 2.48976 2.40854i 1.95699 + 1.95699i 7.53410 2.69022i 2.12132 + 2.12132i −0.235042 + 14.1754i
19.8 −0.712728 1.86869i −1.60021 0.662827i −2.98404 + 2.66374i −4.75081 + 1.96785i −0.0981101 + 3.46271i 5.89664 + 5.89664i 7.10452 + 3.67773i 2.12132 + 2.12132i 7.06335 + 7.47527i
19.9 0.408652 + 1.95781i −1.60021 0.662827i −3.66601 + 1.60012i −4.62667 + 1.91643i 0.643760 3.40376i −2.45923 2.45923i −4.63085 6.52344i 2.12132 + 2.12132i −5.64269 8.27497i
19.10 0.729010 + 1.86240i 1.60021 + 0.662827i −2.93709 + 2.71542i 8.33737 3.45345i −0.0678852 + 3.46344i 1.00716 + 1.00716i −7.19837 3.49048i 2.12132 + 2.12132i 12.5098 + 13.0099i
19.11 0.948970 1.76053i −1.60021 0.662827i −2.19891 3.34138i −4.89089 + 2.02587i −2.68547 + 2.18820i −6.40097 6.40097i −7.96928 + 0.700377i 2.12132 + 2.12132i −1.07471 + 10.5330i
19.12 1.02518 + 1.71727i 1.60021 + 0.662827i −1.89801 + 3.52102i −7.00900 + 2.90322i 0.502250 + 3.42750i 2.35957 + 2.35957i −7.99233 + 0.350291i 2.12132 + 2.12132i −12.1711 9.05999i
19.13 1.32170 1.50104i 1.60021 + 0.662827i −0.506212 3.96784i 2.17599 0.901324i 3.10992 1.52591i 0.825541 + 0.825541i −6.62493 4.48446i 2.12132 + 2.12132i 1.52309 4.45751i
19.14 1.64362 + 1.13952i −1.60021 0.662827i 1.40297 + 3.74589i 0.838363 0.347262i −1.87482 2.91291i 8.29065 + 8.29065i −1.96258 + 7.75553i 2.12132 + 2.12132i 1.77366 + 0.384569i
19.15 1.92938 0.526758i −1.60021 0.662827i 3.44505 2.03264i 3.51382 1.45547i −3.43656 0.435928i −1.77216 1.77216i 5.57613 5.73645i 2.12132 + 2.12132i 6.01282 4.65909i
19.16 1.98432 + 0.249937i 1.60021 + 0.662827i 3.87506 + 0.991913i −1.26028 + 0.522023i 3.00966 + 1.71521i −2.90283 2.90283i 7.44145 + 2.93680i 2.12132 + 2.12132i −2.63126 + 0.720872i
43.1 −1.99199 0.178809i 0.662827 + 1.60021i 3.93605 + 0.712374i −1.28772 + 3.10884i −1.03421 3.30612i 0.299204 0.299204i −7.71320 2.12285i −2.12132 + 2.12132i 3.12102 5.96252i
43.2 −1.99038 0.195968i −0.662827 1.60021i 3.92319 + 0.780100i −0.862642 + 2.08260i 1.00569 + 3.31491i 6.00123 6.00123i −7.65575 2.32151i −2.12132 + 2.12132i 2.12510 3.97611i
43.3 −1.48044 + 1.34472i −0.662827 1.60021i 0.383430 3.98158i −0.872612 + 2.10667i 3.13312 + 1.47770i −6.43807 + 6.43807i 4.78648 + 6.41012i −2.12132 + 2.12132i −1.54104 4.29223i
43.4 −1.37677 1.45069i −0.662827 1.60021i −0.209027 + 3.99453i −1.48548 + 3.58626i −1.40885 + 3.16467i −7.09685 + 7.09685i 6.08263 5.19631i −2.12132 + 2.12132i 7.24772 2.78247i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.3.m.a 64
3.b odd 2 1 288.3.u.b 64
4.b odd 2 1 384.3.m.a 64
32.g even 8 1 384.3.m.a 64
32.h odd 8 1 inner 96.3.m.a 64
96.o even 8 1 288.3.u.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.m.a 64 1.a even 1 1 trivial
96.3.m.a 64 32.h odd 8 1 inner
288.3.u.b 64 3.b odd 2 1
288.3.u.b 64 96.o even 8 1
384.3.m.a 64 4.b odd 2 1
384.3.m.a 64 32.g even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(96, [\chi])$$.