# Properties

 Label 96.4.o.a Level $96$ Weight $4$ Character orbit 96.o Analytic conductor $5.664$ Analytic rank $0$ Dimension $184$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,4,Mod(11,96)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 5, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("96.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.66418336055$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$184 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10})$$ 184 * q - 4 * q^3 - 8 * q^4 - 4 * q^6 - 8 * q^7 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$184 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9} - 128 q^{10} - 4 q^{12} - 8 q^{13} - 8 q^{15} + 472 q^{16} - 4 q^{18} - 8 q^{19} - 4 q^{21} - 440 q^{22} + 496 q^{24} - 8 q^{25} + 260 q^{27} - 8 q^{28} - 1132 q^{30} - 8 q^{33} - 72 q^{34} + 576 q^{36} - 8 q^{37} - 604 q^{39} + 280 q^{40} + 896 q^{42} - 8 q^{43} - 4 q^{45} - 1448 q^{46} - 2448 q^{48} - 112 q^{51} + 784 q^{52} - 224 q^{54} + 280 q^{55} - 4 q^{57} - 368 q^{58} + 2792 q^{60} + 1816 q^{61} - 3032 q^{64} - 3268 q^{66} - 3272 q^{67} - 4 q^{69} - 1016 q^{70} + 1376 q^{72} - 8 q^{73} + 496 q^{75} + 5200 q^{76} + 3316 q^{78} + 5648 q^{79} - 3488 q^{82} + 2408 q^{84} - 1008 q^{85} - 1292 q^{87} + 1552 q^{88} + 2816 q^{90} - 3608 q^{91} + 104 q^{93} - 160 q^{94} + 280 q^{96} - 16 q^{97} - 5316 q^{99}+O(q^{100})$$ 184 * q - 4 * q^3 - 8 * q^4 - 4 * q^6 - 8 * q^7 - 4 * q^9 - 128 * q^10 - 4 * q^12 - 8 * q^13 - 8 * q^15 + 472 * q^16 - 4 * q^18 - 8 * q^19 - 4 * q^21 - 440 * q^22 + 496 * q^24 - 8 * q^25 + 260 * q^27 - 8 * q^28 - 1132 * q^30 - 8 * q^33 - 72 * q^34 + 576 * q^36 - 8 * q^37 - 604 * q^39 + 280 * q^40 + 896 * q^42 - 8 * q^43 - 4 * q^45 - 1448 * q^46 - 2448 * q^48 - 112 * q^51 + 784 * q^52 - 224 * q^54 + 280 * q^55 - 4 * q^57 - 368 * q^58 + 2792 * q^60 + 1816 * q^61 - 3032 * q^64 - 3268 * q^66 - 3272 * q^67 - 4 * q^69 - 1016 * q^70 + 1376 * q^72 - 8 * q^73 + 496 * q^75 + 5200 * q^76 + 3316 * q^78 + 5648 * q^79 - 3488 * q^82 + 2408 * q^84 - 1008 * q^85 - 1292 * q^87 + 1552 * q^88 + 2816 * q^90 - 3608 * q^91 + 104 * q^93 - 160 * q^94 + 280 * q^96 - 16 * q^97 - 5316 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.82816 + 0.0386855i −2.52365 + 4.54216i 7.99701 0.218818i 9.95230 + 4.12238i 6.96158 12.9436i 12.7553 12.7553i −22.6084 + 0.928221i −14.2624 22.9256i −28.3062 11.2737i
11.2 −2.78850 0.473589i −5.15727 + 0.634507i 7.55143 + 2.64120i −7.83732 3.24632i 14.6815 + 0.673103i −17.0610 + 17.0610i −19.8063 10.9412i 26.1948 6.54464i 20.3169 + 12.7640i
11.3 −2.77431 + 0.550619i 0.760131 5.14025i 7.39364 3.05518i 14.9748 + 6.20278i 0.721477 + 14.6792i −0.838016 + 0.838016i −18.8300 + 12.5471i −25.8444 7.81454i −44.9602 8.96303i
11.4 −2.70969 + 0.810909i 2.11809 + 4.74486i 6.68485 4.39463i −8.30675 3.44077i −9.58702 11.1395i −7.65148 + 7.65148i −14.5502 + 17.3289i −18.0274 + 20.1001i 25.2989 + 2.58740i
11.5 −2.63988 + 1.01541i 4.93812 1.61709i 5.93788 5.36111i −2.01136 0.833133i −11.3940 + 9.28313i 14.9578 14.9578i −10.2315 + 20.1821i 21.7701 15.9708i 6.15571 + 0.157010i
11.6 −2.63756 1.02140i 4.56009 + 2.49110i 5.91346 + 5.38804i 19.2155 + 7.95931i −9.48309 11.2281i −16.0945 + 16.0945i −10.0938 20.2513i 14.5888 + 22.7193i −42.5523 40.6200i
11.7 −2.63034 1.03987i 2.61562 4.48982i 5.83734 + 5.47042i −9.81381 4.06501i −11.5488 + 9.08984i −9.78975 + 9.78975i −9.66562 20.4591i −13.3170 23.4874i 21.5865 + 20.8974i
11.8 −2.56841 1.18459i −3.89517 3.43913i 5.19349 + 6.08504i 3.04783 + 1.26245i 5.93045 + 13.4473i 23.6832 23.6832i −6.13073 21.7811i 3.34477 + 26.7920i −6.33258 6.85292i
11.9 −2.42314 + 1.45889i −2.62715 4.48309i 3.74326 7.07022i −17.7025 7.33262i 12.9063 + 7.03043i 4.60267 4.60267i 1.24423 + 22.5932i −13.1961 + 23.5555i 53.5932 8.05807i
11.10 −2.36663 1.54889i 4.20779 + 3.04868i 3.20188 + 7.33130i −10.9726 4.54499i −5.23622 13.7325i 15.8650 15.8650i 3.77771 22.3098i 8.41105 + 25.6565i 18.9284 + 27.7517i
11.11 −2.30818 + 1.63471i −5.17000 0.520689i 2.65541 7.54644i 11.2426 + 4.65685i 12.7845 7.24963i −7.06175 + 7.06175i 6.20710 + 21.7594i 26.4578 + 5.38392i −33.5626 + 7.62963i
11.12 −1.84929 2.14012i −1.83843 + 4.86006i −1.16026 + 7.91542i 0.237684 + 0.0984520i 13.8009 5.05319i −2.73691 + 2.73691i 19.0856 12.1548i −20.2403 17.8698i −0.228847 0.690740i
11.13 −1.73518 + 2.23364i 5.18129 0.392730i −1.97827 7.75154i 0.597865 + 0.247644i −8.11328 + 12.2546i −22.9337 + 22.9337i 20.7468 + 9.03162i 26.6915 4.06969i −1.59055 + 0.905706i
11.14 −1.54625 + 2.36836i −3.37123 + 3.95408i −3.21823 7.32414i −8.07473 3.34466i −4.15191 14.0983i 4.83421 4.83421i 22.3224 + 3.70302i −4.26955 26.6603i 20.4069 13.9522i
11.15 −1.50364 2.39563i −2.78340 4.38779i −3.47811 + 7.20435i 8.14957 + 3.37566i −6.32628 + 13.2657i −18.6414 + 18.6414i 22.4888 2.50049i −11.5053 + 24.4260i −4.16719 24.5992i
11.16 −1.45436 + 2.42587i 3.62905 + 3.71887i −3.76965 7.05618i 15.2979 + 6.33660i −14.2994 + 3.39502i 23.1014 23.1014i 22.5998 + 1.11758i −0.659935 + 26.9919i −37.6205 + 27.8950i
11.17 −1.36045 2.47975i 4.50648 2.58681i −4.29836 + 6.74716i 8.09459 + 3.35289i −12.5455 7.65575i 8.54901 8.54901i 22.5790 + 1.47971i 13.6168 23.3149i −2.69794 24.6340i
11.18 −1.17156 + 2.57439i 0.498392 5.17220i −5.25492 6.03207i 3.82709 + 1.58523i 12.7313 + 7.34257i −9.56512 + 9.56512i 21.6853 6.46128i −26.5032 5.15557i −8.56464 + 7.99521i
11.19 −0.913528 2.67684i −5.17736 + 0.441578i −6.33093 + 4.89074i −15.3452 6.35617i 5.91169 + 13.4556i 7.04435 7.04435i 18.8752 + 12.4791i 26.6100 4.57242i −2.99622 + 46.8831i
11.20 −0.404190 2.79940i 5.09775 + 1.00644i −7.67326 + 2.26298i −16.8122 6.96383i 0.756973 14.6774i −21.0906 + 21.0906i 9.43644 + 20.5658i 24.9741 + 10.2612i −12.6992 + 49.8787i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.4.o.a 184
3.b odd 2 1 inner 96.4.o.a 184
32.h odd 8 1 inner 96.4.o.a 184
96.o even 8 1 inner 96.4.o.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.o.a 184 1.a even 1 1 trivial
96.4.o.a 184 3.b odd 2 1 inner
96.4.o.a 184 32.h odd 8 1 inner
96.4.o.a 184 96.o even 8 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(96, [\chi])$$.