# Properties

 Label 32.4.g.a Level $32$ Weight $4$ Character orbit 32.g Analytic conductor $1.888$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 32.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.88806112018$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$11$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{9} + 116q^{10} - 4q^{11} - 52q^{12} - 4q^{13} - 212q^{14} - 304q^{16} - 184q^{18} - 4q^{19} + 76q^{20} - 4q^{21} + 192q^{22} + 324q^{23} - 48q^{24} - 4q^{25} + 16q^{26} - 268q^{27} + 376q^{28} - 4q^{29} + 1188q^{30} - 752q^{31} + 616q^{32} - 8q^{33} + 528q^{34} - 460q^{35} + 1456q^{36} - 4q^{37} + 980q^{38} + 596q^{39} - 536q^{40} - 4q^{41} - 2264q^{42} + 804q^{43} - 2044q^{44} + 104q^{45} - 1444q^{46} - 2448q^{48} - 3564q^{50} - 1384q^{51} - 2524q^{52} + 748q^{53} - 1088q^{54} - 292q^{55} + 1192q^{56} - 4q^{57} + 3200q^{58} + 1372q^{59} + 5752q^{60} - 1828q^{61} + 3384q^{62} + 2512q^{63} + 4952q^{64} - 8q^{65} + 5996q^{66} + 2036q^{67} + 2768q^{68} - 1060q^{69} + 1400q^{70} + 220q^{71} - 1708q^{72} - 4q^{73} - 3476q^{74} - 1712q^{75} - 5124q^{76} + 1900q^{77} - 11916q^{78} - 10312q^{80} - 6404q^{82} + 2436q^{83} - 6560q^{84} + 496q^{85} - 928q^{86} - 1292q^{87} + 1248q^{88} - 4q^{89} + 7400q^{90} - 3604q^{91} + 10152q^{92} - 112q^{93} + 12840q^{94} - 6088q^{95} + 17792q^{96} - 8q^{97} + 11224q^{98} - 5424q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −2.82381 0.161516i −0.477813 1.15354i 7.94783 + 0.912182i −16.3468 6.77105i 1.16294 + 3.33456i −18.0222 18.0222i −22.2958 3.85953i 17.9895 17.9895i 45.0666 + 21.7604i
5.2 −2.51802 + 1.28824i −0.998206 2.40988i 4.68088 6.48763i 17.4005 + 7.20752i 5.61801 + 4.78221i 4.37099 + 4.37099i −3.42892 + 22.3661i 14.2808 14.2808i −53.0999 + 4.26731i
5.3 −2.10117 1.89344i 1.94138 + 4.68690i 0.829801 + 7.95685i 4.93338 + 2.04347i 4.79518 13.5238i 14.0755 + 14.0755i 13.3222 18.2898i 0.893826 0.893826i −6.49667 13.6347i
5.4 −1.75970 + 2.21438i 3.54796 + 8.56554i −1.80692 7.79327i −7.55322 3.12865i −25.2107 7.21625i 7.16166 + 7.16166i 20.4369 + 9.71261i −41.6886 + 41.6886i 20.2194 11.2202i
5.5 −0.719102 + 2.73549i −3.21198 7.75440i −6.96578 3.93419i −13.6472 5.65283i 23.5218 3.21012i 9.07689 + 9.07689i 15.7710 16.2257i −30.7220 + 30.7220i 25.2770 33.2666i
5.6 −0.582457 2.76780i −1.90169 4.59109i −7.32149 + 3.22425i 0.188811 + 0.0782080i −11.5996 + 7.93763i −11.4103 11.4103i 13.1886 + 18.3865i 1.63019 1.63019i 0.106490 0.568145i
5.7 1.17521 + 2.57272i 0.729459 + 1.76107i −5.23776 + 6.04698i 4.29822 + 1.78038i −3.67347 + 3.94632i 1.47807 + 1.47807i −21.7126 6.36880i 16.5226 16.5226i 0.470898 + 13.1504i
5.8 1.49948 2.39824i 2.92731 + 7.06715i −3.50313 7.19223i 13.5234 + 5.60159i 21.3382 + 3.57665i −23.0737 23.0737i −22.5016 2.38325i −22.2836 + 22.2836i 33.7121 24.0330i
5.9 2.01560 1.98428i −1.20185 2.90153i 0.125267 7.99902i −3.98512 1.65069i −8.17991 3.46351i 22.4050 + 22.4050i −15.6198 16.3714i 12.1174 12.1174i −11.3078 + 4.58047i
5.10 2.70658 + 0.821250i −3.28810 7.93817i 6.65110 + 4.44555i 11.2895 + 4.67626i −2.38026 24.1856i −11.8490 11.8490i 14.3508 + 17.4944i −33.1111 + 33.1111i 26.7154 + 21.9281i
5.11 2.81450 + 0.280314i 1.64064 + 3.96085i 7.84285 + 1.57789i −11.8087 4.89132i 3.50730 + 11.6077i −5.11236 5.11236i 21.6314 + 6.63943i 6.09524 6.09524i −31.8645 17.0768i
13.1 −2.82381 + 0.161516i −0.477813 + 1.15354i 7.94783 0.912182i −16.3468 + 6.77105i 1.16294 3.33456i −18.0222 + 18.0222i −22.2958 + 3.85953i 17.9895 + 17.9895i 45.0666 21.7604i
13.2 −2.51802 1.28824i −0.998206 + 2.40988i 4.68088 + 6.48763i 17.4005 7.20752i 5.61801 4.78221i 4.37099 4.37099i −3.42892 22.3661i 14.2808 + 14.2808i −53.0999 4.26731i
13.3 −2.10117 + 1.89344i 1.94138 4.68690i 0.829801 7.95685i 4.93338 2.04347i 4.79518 + 13.5238i 14.0755 14.0755i 13.3222 + 18.2898i 0.893826 + 0.893826i −6.49667 + 13.6347i
13.4 −1.75970 2.21438i 3.54796 8.56554i −1.80692 + 7.79327i −7.55322 + 3.12865i −25.2107 + 7.21625i 7.16166 7.16166i 20.4369 9.71261i −41.6886 41.6886i 20.2194 + 11.2202i
13.5 −0.719102 2.73549i −3.21198 + 7.75440i −6.96578 + 3.93419i −13.6472 + 5.65283i 23.5218 + 3.21012i 9.07689 9.07689i 15.7710 + 16.2257i −30.7220 30.7220i 25.2770 + 33.2666i
13.6 −0.582457 + 2.76780i −1.90169 + 4.59109i −7.32149 3.22425i 0.188811 0.0782080i −11.5996 7.93763i −11.4103 + 11.4103i 13.1886 18.3865i 1.63019 + 1.63019i 0.106490 + 0.568145i
13.7 1.17521 2.57272i 0.729459 1.76107i −5.23776 6.04698i 4.29822 1.78038i −3.67347 3.94632i 1.47807 1.47807i −21.7126 + 6.36880i 16.5226 + 16.5226i 0.470898 13.1504i
13.8 1.49948 + 2.39824i 2.92731 7.06715i −3.50313 + 7.19223i 13.5234 5.60159i 21.3382 3.57665i −23.0737 + 23.0737i −22.5016 + 2.38325i −22.2836 22.2836i 33.7121 + 24.0330i
13.9 2.01560 + 1.98428i −1.20185 + 2.90153i 0.125267 + 7.99902i −3.98512 + 1.65069i −8.17991 + 3.46351i 22.4050 22.4050i −15.6198 + 16.3714i 12.1174 + 12.1174i −11.3078 4.58047i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.11 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.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.4.g.a 44
4.b odd 2 1 128.4.g.a 44
8.b even 2 1 256.4.g.b 44
8.d odd 2 1 256.4.g.a 44
32.g even 8 1 inner 32.4.g.a 44
32.g even 8 1 256.4.g.b 44
32.h odd 8 1 128.4.g.a 44
32.h odd 8 1 256.4.g.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.g.a 44 1.a even 1 1 trivial
32.4.g.a 44 32.g even 8 1 inner
128.4.g.a 44 4.b odd 2 1
128.4.g.a 44 32.h odd 8 1
256.4.g.a 44 8.d odd 2 1
256.4.g.a 44 32.h odd 8 1
256.4.g.b 44 8.b even 2 1
256.4.g.b 44 32.g even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database