Properties

 Label 48.4.k.a Level $48$ Weight $4$ Character orbit 48.k Analytic conductor $2.832$ Analytic rank $0$ Dimension $44$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(11,48)]

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([2, 1, 2]))

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

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

chi := DirichletCharacter("48.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.k (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: $$44$$ Relative dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 2 q^{3} - 4 q^{4} + 28 q^{6} - 8 q^{7}+O(q^{10})$$ 44 * q - 2 * q^3 - 4 * q^4 + 28 * q^6 - 8 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 2 q^{3} - 4 q^{4} + 28 q^{6} - 8 q^{7} + 56 q^{10} - 80 q^{12} - 4 q^{13} - 112 q^{16} + 52 q^{18} + 20 q^{19} - 56 q^{21} - 40 q^{22} - 120 q^{24} - 134 q^{27} - 296 q^{28} - 332 q^{30} - 4 q^{33} + 520 q^{34} - 604 q^{36} - 4 q^{37} + 596 q^{39} + 632 q^{40} + 696 q^{42} - 436 q^{43} - 252 q^{45} + 664 q^{46} + 1200 q^{48} + 972 q^{49} - 648 q^{51} + 320 q^{52} + 1592 q^{54} + 280 q^{55} - 424 q^{58} + 800 q^{60} - 916 q^{61} - 2056 q^{64} - 668 q^{66} - 1636 q^{67} + 52 q^{69} - 5192 q^{70} - 3704 q^{72} + 1454 q^{75} - 568 q^{76} - 4932 q^{78} - 4 q^{81} + 768 q^{82} - 2096 q^{84} + 736 q^{85} + 1284 q^{87} + 8864 q^{88} + 2672 q^{90} + 424 q^{91} - 2084 q^{93} + 5616 q^{94} + 8008 q^{96} - 8 q^{97} + 1196 q^{99}+O(q^{100})$$ 44 * q - 2 * q^3 - 4 * q^4 + 28 * q^6 - 8 * q^7 + 56 * q^10 - 80 * q^12 - 4 * q^13 - 112 * q^16 + 52 * q^18 + 20 * q^19 - 56 * q^21 - 40 * q^22 - 120 * q^24 - 134 * q^27 - 296 * q^28 - 332 * q^30 - 4 * q^33 + 520 * q^34 - 604 * q^36 - 4 * q^37 + 596 * q^39 + 632 * q^40 + 696 * q^42 - 436 * q^43 - 252 * q^45 + 664 * q^46 + 1200 * q^48 + 972 * q^49 - 648 * q^51 + 320 * q^52 + 1592 * q^54 + 280 * q^55 - 424 * q^58 + 800 * q^60 - 916 * q^61 - 2056 * q^64 - 668 * q^66 - 1636 * q^67 + 52 * q^69 - 5192 * q^70 - 3704 * q^72 + 1454 * q^75 - 568 * q^76 - 4932 * q^78 - 4 * q^81 + 768 * q^82 - 2096 * q^84 + 736 * q^85 + 1284 * q^87 + 8864 * q^88 + 2672 * q^90 + 424 * q^91 - 2084 * q^93 + 5616 * q^94 + 8008 * q^96 - 8 * q^97 + 1196 * 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 −2.81978 0.221045i −2.71261 + 4.43190i 7.90228 + 1.24659i 3.17566 3.17566i 8.62861 11.8974i −32.3513 −22.0071 5.26187i −12.2835 24.0440i −9.65662 + 8.25269i
11.2 −2.60843 + 1.09367i 4.68788 + 2.24138i 5.60778 5.70550i 2.69043 2.69043i −14.6793 0.719504i 10.6336 −8.38758 + 21.0154i 16.9524 + 21.0147i −4.07535 + 9.96021i
11.3 −2.56930 1.18266i −4.81196 1.96089i 5.20263 + 6.07722i −6.30133 + 6.30133i 10.0443 + 10.7290i 24.6728 −6.17986 21.7672i 19.3098 + 18.8714i 23.6423 8.73769i
11.4 −2.53336 1.25780i 3.76578 3.58035i 4.83587 + 6.37294i 4.71515 4.71515i −14.0435 + 4.33371i 4.67595 −4.23510 22.2275i 1.36225 26.9656i −17.8759 + 6.01447i
11.5 −2.44754 + 1.41758i 0.563550 5.16550i 3.98092 6.93918i −13.1633 + 13.1633i 5.94321 + 13.4417i −13.2717 0.0933841 + 22.6272i −26.3648 5.82204i 13.5576 50.8776i
11.6 −1.97996 + 2.01984i −5.19561 0.0749974i −0.159526 7.99841i 5.37662 5.37662i 10.4386 10.3458i 14.8575 16.4714 + 15.5143i 26.9888 + 0.779314i 0.214439 + 21.5054i
11.7 −1.70353 2.25788i 3.86039 + 3.47813i −2.19600 + 7.69270i −13.5794 + 13.5794i 1.27693 14.6414i −19.7355 21.1101 8.14640i 2.80518 + 26.8539i 53.7934 + 7.52774i
11.8 −1.13198 2.59203i −1.13522 + 5.07063i −5.43724 + 5.86826i 11.2665 11.2665i 14.4283 2.79732i 30.2121 21.3655 + 7.45073i −24.4225 11.5126i −41.9567 16.4497i
11.9 −0.775594 2.72001i −2.65327 4.46768i −6.79691 + 4.21925i 5.27809 5.27809i −10.0943 + 10.6820i −22.9284 16.7480 + 15.2152i −12.9203 + 23.7079i −18.4501 10.2628i
11.10 −0.763869 + 2.72333i −0.151014 + 5.19396i −6.83301 4.16053i −4.66675 + 4.66675i −14.0295 4.37877i 0.405799 16.5500 15.4304i −26.9544 1.56872i −9.14429 16.2739i
11.11 −0.668281 + 2.74835i 2.28317 4.66767i −7.10680 3.67333i 11.5146 11.5146i 11.3026 + 9.39425i 0.829117 14.8449 17.0771i −16.5743 21.3142i 23.9512 + 39.3412i
11.12 0.668281 2.74835i −4.66767 + 2.28317i −7.10680 3.67333i −11.5146 + 11.5146i 3.15562 + 14.3542i 0.829117 −14.8449 + 17.0771i 16.5743 21.3142i 23.9512 + 39.3412i
11.13 0.763869 2.72333i 5.19396 0.151014i −6.83301 4.16053i 4.66675 4.66675i 3.55625 14.2602i 0.405799 −16.5500 + 15.4304i 26.9544 1.56872i −9.14429 16.2739i
11.14 0.775594 + 2.72001i −4.46768 2.65327i −6.79691 + 4.21925i −5.27809 + 5.27809i 3.75181 14.2100i −22.9284 −16.7480 15.2152i 12.9203 + 23.7079i −18.4501 10.2628i
11.15 1.13198 + 2.59203i 5.07063 1.13522i −5.43724 + 5.86826i −11.2665 + 11.2665i 8.68239 + 11.8582i 30.2121 −21.3655 7.45073i 24.4225 11.5126i −41.9567 16.4497i
11.16 1.70353 + 2.25788i 3.47813 + 3.86039i −2.19600 + 7.69270i 13.5794 13.5794i −2.79119 + 14.4295i −19.7355 −21.1101 + 8.14640i −2.80518 + 26.8539i 53.7934 + 7.52774i
11.17 1.97996 2.01984i −0.0749974 5.19561i −0.159526 7.99841i −5.37662 + 5.37662i −10.6428 10.1356i 14.8575 −16.4714 15.5143i −26.9888 + 0.779314i 0.214439 + 21.5054i
11.18 2.44754 1.41758i −5.16550 + 0.563550i 3.98092 6.93918i 13.1633 13.1633i −11.8439 + 8.70183i −13.2717 −0.0933841 22.6272i 26.3648 5.82204i 13.5576 50.8776i
11.19 2.53336 + 1.25780i −3.58035 + 3.76578i 4.83587 + 6.37294i −4.71515 + 4.71515i −13.8069 + 5.03673i 4.67595 4.23510 + 22.2275i −1.36225 26.9656i −17.8759 + 6.01447i
11.20 2.56930 + 1.18266i −1.96089 4.81196i 5.20263 + 6.07722i 6.30133 6.30133i 0.652791 14.6824i 24.6728 6.17986 + 21.7672i −19.3098 + 18.8714i 23.6423 8.73769i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.22 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
16.f odd 4 1 inner
48.k 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.k.a 44
3.b odd 2 1 inner 48.4.k.a 44
4.b odd 2 1 192.4.k.a 44
8.b even 2 1 384.4.k.b 44
8.d odd 2 1 384.4.k.a 44
12.b even 2 1 192.4.k.a 44
16.e even 4 1 192.4.k.a 44
16.e even 4 1 384.4.k.a 44
16.f odd 4 1 inner 48.4.k.a 44
16.f odd 4 1 384.4.k.b 44
24.f even 2 1 384.4.k.a 44
24.h odd 2 1 384.4.k.b 44
48.i odd 4 1 192.4.k.a 44
48.i odd 4 1 384.4.k.a 44
48.k even 4 1 inner 48.4.k.a 44
48.k even 4 1 384.4.k.b 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.k.a 44 1.a even 1 1 trivial
48.4.k.a 44 3.b odd 2 1 inner
48.4.k.a 44 16.f odd 4 1 inner
48.4.k.a 44 48.k even 4 1 inner
192.4.k.a 44 4.b odd 2 1
192.4.k.a 44 12.b even 2 1
192.4.k.a 44 16.e even 4 1
192.4.k.a 44 48.i odd 4 1
384.4.k.a 44 8.d odd 2 1
384.4.k.a 44 16.e even 4 1
384.4.k.a 44 24.f even 2 1
384.4.k.a 44 48.i odd 4 1
384.4.k.b 44 8.b even 2 1
384.4.k.b 44 16.f odd 4 1
384.4.k.b 44 24.h odd 2 1
384.4.k.b 44 48.k even 4 1

Hecke kernels

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