# Properties

 Label 48.4.j.a Level $48$ Weight $4$ Character orbit 48.j Analytic conductor $2.832$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(13,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("48.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.j (of order $$4$$, degree $$2$$, 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: $$2.83209168028$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 20 q^{4} + 84 q^{8}+O(q^{10})$$ 24 * q - 20 * q^4 + 84 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 20 q^{4} + 84 q^{8} + 72 q^{10} - 40 q^{11} - 24 q^{12} - 348 q^{14} + 120 q^{15} - 192 q^{16} - 36 q^{18} + 24 q^{19} + 80 q^{20} + 704 q^{22} + 228 q^{24} - 20 q^{26} - 344 q^{28} + 400 q^{29} - 408 q^{30} - 744 q^{31} - 960 q^{32} - 704 q^{34} - 456 q^{35} + 108 q^{36} + 16 q^{37} + 1256 q^{38} + 1744 q^{40} + 660 q^{42} + 1240 q^{43} - 200 q^{44} - 1432 q^{46} - 528 q^{48} - 1176 q^{49} + 708 q^{50} + 744 q^{51} + 1008 q^{52} + 752 q^{53} + 108 q^{54} + 1344 q^{56} + 1936 q^{58} - 1376 q^{59} - 1224 q^{60} - 912 q^{61} - 996 q^{62} - 504 q^{63} - 56 q^{64} + 976 q^{65} - 1368 q^{66} - 2256 q^{67} - 1568 q^{68} - 528 q^{69} - 1760 q^{70} - 612 q^{72} - 2740 q^{74} + 1104 q^{75} - 1880 q^{76} + 1904 q^{77} + 1692 q^{78} + 5992 q^{79} + 712 q^{80} - 1944 q^{81} - 40 q^{82} + 2680 q^{83} + 1800 q^{84} - 240 q^{85} - 1712 q^{86} - 3936 q^{88} + 648 q^{90} - 3496 q^{91} + 5296 q^{92} + 5272 q^{94} - 7728 q^{95} + 2880 q^{96} + 6760 q^{98} - 360 q^{99}+O(q^{100})$$ 24 * q - 20 * q^4 + 84 * q^8 + 72 * q^10 - 40 * q^11 - 24 * q^12 - 348 * q^14 + 120 * q^15 - 192 * q^16 - 36 * q^18 + 24 * q^19 + 80 * q^20 + 704 * q^22 + 228 * q^24 - 20 * q^26 - 344 * q^28 + 400 * q^29 - 408 * q^30 - 744 * q^31 - 960 * q^32 - 704 * q^34 - 456 * q^35 + 108 * q^36 + 16 * q^37 + 1256 * q^38 + 1744 * q^40 + 660 * q^42 + 1240 * q^43 - 200 * q^44 - 1432 * q^46 - 528 * q^48 - 1176 * q^49 + 708 * q^50 + 744 * q^51 + 1008 * q^52 + 752 * q^53 + 108 * q^54 + 1344 * q^56 + 1936 * q^58 - 1376 * q^59 - 1224 * q^60 - 912 * q^61 - 996 * q^62 - 504 * q^63 - 56 * q^64 + 976 * q^65 - 1368 * q^66 - 2256 * q^67 - 1568 * q^68 - 528 * q^69 - 1760 * q^70 - 612 * q^72 - 2740 * q^74 + 1104 * q^75 - 1880 * q^76 + 1904 * q^77 + 1692 * q^78 + 5992 * q^79 + 712 * q^80 - 1944 * q^81 - 40 * q^82 + 2680 * q^83 + 1800 * q^84 - 240 * q^85 - 1712 * q^86 - 3936 * q^88 + 648 * q^90 - 3496 * q^91 + 5296 * q^92 + 5272 * q^94 - 7728 * q^95 + 2880 * q^96 + 6760 * q^98 - 360 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −2.77551 + 0.544550i −2.12132 2.12132i 7.40693 3.02281i −3.72414 + 3.72414i 7.04291 + 4.73259i 20.2675i −18.9120 + 12.4233i 9.00000i 8.30842 12.3644i
13.2 −2.24080 + 1.72593i 2.12132 + 2.12132i 2.04234 7.73491i 14.6111 14.6111i −8.41470 1.09220i 26.8889i 8.77342 + 20.8573i 9.00000i −7.52282 + 57.9584i
13.3 −1.92738 2.07008i 2.12132 + 2.12132i −0.570442 + 7.97964i −7.29121 + 7.29121i 0.302715 8.47988i 22.1610i 17.6179 14.1989i 9.00000i 29.1463 + 1.04047i
13.4 −1.40656 + 2.45389i −2.12132 2.12132i −4.04315 6.90311i 3.22588 3.22588i 8.18926 2.22171i 24.6080i 22.6264 0.211795i 9.00000i 3.37855 + 12.4534i
13.5 −0.987020 + 2.65062i 2.12132 + 2.12132i −6.05158 5.23243i −11.8955 + 11.8955i −7.71660 + 3.52903i 0.485059i 19.8422 10.8759i 9.00000i −19.7893 43.2714i
13.6 −0.716137 2.73627i −2.12132 2.12132i −6.97430 + 3.91908i −11.7719 + 11.7719i −4.28534 + 7.32365i 14.7089i 15.7182 + 16.2769i 9.00000i 40.6415 + 23.7808i
13.7 −0.220074 2.81985i 2.12132 + 2.12132i −7.90313 + 1.24115i 10.2951 10.2951i 5.51496 6.44866i 32.8369i 5.23914 + 22.0125i 9.00000i −31.2964 26.7650i
13.8 0.954009 + 2.66268i −2.12132 2.12132i −6.17974 + 5.08044i −8.83384 + 8.83384i 3.62464 7.67216i 29.4760i −19.4231 11.6079i 9.00000i −31.9493 15.0941i
13.9 1.94824 2.05046i −2.12132 2.12132i −0.408732 7.98955i 2.24191 2.24191i −8.48251 + 0.216833i 9.00196i −17.1785 14.7275i 9.00000i −0.229160 8.96471i
13.10 2.07099 + 1.92640i 2.12132 + 2.12132i 0.577966 + 7.97909i 0.644922 0.644922i 0.306713 + 8.47974i 7.13926i −14.1740 + 17.6380i 9.00000i 2.57800 0.0932465i
13.11 2.59717 1.12013i 2.12132 + 2.12132i 5.49064 5.81833i 0.706564 0.706564i 7.88559 + 3.13329i 4.44122i 7.74288 21.2614i 9.00000i 1.04363 2.62651i
13.12 2.70307 + 0.832707i −2.12132 2.12132i 6.61320 + 4.50173i 11.7911 11.7911i −3.96764 7.50052i 12.5754i 14.1273 + 17.6754i 9.00000i 41.6906 22.0536i
37.1 −2.77551 0.544550i −2.12132 + 2.12132i 7.40693 + 3.02281i −3.72414 3.72414i 7.04291 4.73259i 20.2675i −18.9120 12.4233i 9.00000i 8.30842 + 12.3644i
37.2 −2.24080 1.72593i 2.12132 2.12132i 2.04234 + 7.73491i 14.6111 + 14.6111i −8.41470 + 1.09220i 26.8889i 8.77342 20.8573i 9.00000i −7.52282 57.9584i
37.3 −1.92738 + 2.07008i 2.12132 2.12132i −0.570442 7.97964i −7.29121 7.29121i 0.302715 + 8.47988i 22.1610i 17.6179 + 14.1989i 9.00000i 29.1463 1.04047i
37.4 −1.40656 2.45389i −2.12132 + 2.12132i −4.04315 + 6.90311i 3.22588 + 3.22588i 8.18926 + 2.22171i 24.6080i 22.6264 + 0.211795i 9.00000i 3.37855 12.4534i
37.5 −0.987020 2.65062i 2.12132 2.12132i −6.05158 + 5.23243i −11.8955 11.8955i −7.71660 3.52903i 0.485059i 19.8422 + 10.8759i 9.00000i −19.7893 + 43.2714i
37.6 −0.716137 + 2.73627i −2.12132 + 2.12132i −6.97430 3.91908i −11.7719 11.7719i −4.28534 7.32365i 14.7089i 15.7182 16.2769i 9.00000i 40.6415 23.7808i
37.7 −0.220074 + 2.81985i 2.12132 2.12132i −7.90313 1.24115i 10.2951 + 10.2951i 5.51496 + 6.44866i 32.8369i 5.23914 22.0125i 9.00000i −31.2964 + 26.7650i
37.8 0.954009 2.66268i −2.12132 + 2.12132i −6.17974 5.08044i −8.83384 8.83384i 3.62464 + 7.67216i 29.4760i −19.4231 + 11.6079i 9.00000i −31.9493 + 15.0941i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.4.j.a 24
3.b odd 2 1 144.4.k.b 24
4.b odd 2 1 192.4.j.a 24
8.b even 2 1 384.4.j.b 24
8.d odd 2 1 384.4.j.a 24
12.b even 2 1 576.4.k.b 24
16.e even 4 1 inner 48.4.j.a 24
16.e even 4 1 384.4.j.b 24
16.f odd 4 1 192.4.j.a 24
16.f odd 4 1 384.4.j.a 24
48.i odd 4 1 144.4.k.b 24
48.k even 4 1 576.4.k.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.j.a 24 1.a even 1 1 trivial
48.4.j.a 24 16.e even 4 1 inner
144.4.k.b 24 3.b odd 2 1
144.4.k.b 24 48.i odd 4 1
192.4.j.a 24 4.b odd 2 1
192.4.j.a 24 16.f odd 4 1
384.4.j.a 24 8.d odd 2 1
384.4.j.a 24 16.f odd 4 1
384.4.j.b 24 8.b even 2 1
384.4.j.b 24 16.e even 4 1
576.4.k.b 24 12.b even 2 1
576.4.k.b 24 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(48, [\chi])$$.