Properties

 Label 121.4.c.j Level $121$ Weight $4$ Character orbit 121.c Analytic conductor $7.139$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(3,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8]))

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

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

chi := DirichletCharacter("121.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.c (of order $$5$$, degree $$4$$, not 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: $$7.13923111069$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 8 q^{3} - 20 q^{4} - 14 q^{5} - 150 q^{9}+O(q^{10})$$ 24 * q - 8 * q^3 - 20 * q^4 - 14 * q^5 - 150 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 8 q^{3} - 20 q^{4} - 14 q^{5} - 150 q^{9} + 1072 q^{12} + 196 q^{14} - 200 q^{15} + 92 q^{16} - 576 q^{20} + 2048 q^{23} - 160 q^{25} - 60 q^{26} - 344 q^{27} + 16 q^{31} - 896 q^{34} + 364 q^{36} - 370 q^{37} + 484 q^{38} + 1408 q^{42} - 6952 q^{45} - 1448 q^{47} + 2292 q^{48} - 158 q^{49} + 822 q^{53} - 4624 q^{56} + 1628 q^{58} - 1464 q^{59} + 252 q^{60} + 2396 q^{64} - 4864 q^{67} - 2432 q^{69} + 3452 q^{70} - 1096 q^{71} + 744 q^{75} + 7056 q^{78} + 1136 q^{80} - 2670 q^{81} + 1336 q^{82} - 6112 q^{86} + 1384 q^{89} - 216 q^{91} - 4260 q^{92} - 400 q^{93} - 1734 q^{97}+O(q^{100})$$ 24 * q - 8 * q^3 - 20 * q^4 - 14 * q^5 - 150 * q^9 + 1072 * q^12 + 196 * q^14 - 200 * q^15 + 92 * q^16 - 576 * q^20 + 2048 * q^23 - 160 * q^25 - 60 * q^26 - 344 * q^27 + 16 * q^31 - 896 * q^34 + 364 * q^36 - 370 * q^37 + 484 * q^38 + 1408 * q^42 - 6952 * q^45 - 1448 * q^47 + 2292 * q^48 - 158 * q^49 + 822 * q^53 - 4624 * q^56 + 1628 * q^58 - 1464 * q^59 + 252 * q^60 + 2396 * q^64 - 4864 * q^67 - 2432 * q^69 + 3452 * q^70 - 1096 * q^71 + 744 * q^75 + 7056 * q^78 + 1136 * q^80 - 2670 * q^81 + 1336 * q^82 - 6112 * q^86 + 1384 * q^89 - 216 * q^91 - 4260 * q^92 - 400 * q^93 - 1734 * q^97

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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −3.62629 + 2.63465i 0.719300 + 2.21378i 3.73643 11.4995i −15.2540 11.0827i −8.44091 6.13268i −5.80624 + 17.8698i 5.66698 + 17.4412i 17.4600 12.6855i 84.5143
3.2 −2.93750 + 2.13422i 2.92726 + 9.00917i 1.60188 4.93009i 1.70313 + 1.23740i −27.8264 20.2170i −3.03234 + 9.33259i −3.15984 9.72499i −50.7528 + 36.8741i −7.64382
3.3 −0.688785 + 0.500431i −2.41049 7.41872i −2.24814 + 6.91907i 7.88774 + 5.73078i 5.37287 + 3.90362i 7.93077 24.4084i −4.01877 12.3685i −27.3835 + 19.8953i −8.30082
3.4 0.688785 0.500431i −2.41049 7.41872i −2.24814 + 6.91907i 7.88774 + 5.73078i −5.37287 3.90362i −7.93077 + 24.4084i 4.01877 + 12.3685i −27.3835 + 19.8953i 8.30082
3.5 2.93750 2.13422i 2.92726 + 9.00917i 1.60188 4.93009i 1.70313 + 1.23740i 27.8264 + 20.2170i 3.03234 9.33259i 3.15984 + 9.72499i −50.7528 + 36.8741i 7.64382
3.6 3.62629 2.63465i 0.719300 + 2.21378i 3.73643 11.4995i −15.2540 11.0827i 8.44091 + 6.13268i 5.80624 17.8698i −5.66698 17.4412i 17.4600 12.6855i −84.5143
9.1 −1.38512 + 4.26295i −1.88315 + 1.36819i −9.78210 7.10711i 5.82651 + 17.9321i −3.22414 9.92289i −15.2009 11.0441i 14.8364 10.7792i −6.66914 + 20.5255i −84.5143
9.2 −1.12203 + 3.45324i −7.66366 + 5.56797i −4.19379 3.04697i −0.650538 2.00215i −10.6287 32.7119i −7.93878 5.76786i −8.27257 + 6.01037i 19.3859 59.6635i 7.64382
9.3 −0.263092 + 0.809715i 6.31074 4.58502i 5.88572 + 4.27622i −3.01285 9.27260i 2.05225 + 6.31618i 20.7630 + 15.0852i −10.5213 + 7.64416i 10.4596 32.1912i 8.30082
9.4 0.263092 0.809715i 6.31074 4.58502i 5.88572 + 4.27622i −3.01285 9.27260i −2.05225 6.31618i −20.7630 15.0852i 10.5213 7.64416i 10.4596 32.1912i −8.30082
9.5 1.12203 3.45324i −7.66366 + 5.56797i −4.19379 3.04697i −0.650538 2.00215i 10.6287 + 32.7119i 7.93878 + 5.76786i 8.27257 6.01037i 19.3859 59.6635i −7.64382
9.6 1.38512 4.26295i −1.88315 + 1.36819i −9.78210 7.10711i 5.82651 + 17.9321i 3.22414 + 9.92289i 15.2009 + 11.0441i −14.8364 + 10.7792i −6.66914 + 20.5255i 84.5143
27.1 −1.38512 4.26295i −1.88315 1.36819i −9.78210 + 7.10711i 5.82651 17.9321i −3.22414 + 9.92289i −15.2009 + 11.0441i 14.8364 + 10.7792i −6.66914 20.5255i −84.5143
27.2 −1.12203 3.45324i −7.66366 5.56797i −4.19379 + 3.04697i −0.650538 + 2.00215i −10.6287 + 32.7119i −7.93878 + 5.76786i −8.27257 6.01037i 19.3859 + 59.6635i 7.64382
27.3 −0.263092 0.809715i 6.31074 + 4.58502i 5.88572 4.27622i −3.01285 + 9.27260i 2.05225 6.31618i 20.7630 15.0852i −10.5213 7.64416i 10.4596 + 32.1912i 8.30082
27.4 0.263092 + 0.809715i 6.31074 + 4.58502i 5.88572 4.27622i −3.01285 + 9.27260i −2.05225 + 6.31618i −20.7630 + 15.0852i 10.5213 + 7.64416i 10.4596 + 32.1912i −8.30082
27.5 1.12203 + 3.45324i −7.66366 5.56797i −4.19379 + 3.04697i −0.650538 + 2.00215i 10.6287 32.7119i 7.93878 5.76786i 8.27257 + 6.01037i 19.3859 + 59.6635i −7.64382
27.6 1.38512 + 4.26295i −1.88315 1.36819i −9.78210 + 7.10711i 5.82651 17.9321i 3.22414 9.92289i 15.2009 11.0441i −14.8364 10.7792i −6.66914 20.5255i 84.5143
81.1 −3.62629 2.63465i 0.719300 2.21378i 3.73643 + 11.4995i −15.2540 + 11.0827i −8.44091 + 6.13268i −5.80624 17.8698i 5.66698 17.4412i 17.4600 + 12.6855i 84.5143
81.2 −2.93750 2.13422i 2.92726 9.00917i 1.60188 + 4.93009i 1.70313 1.23740i −27.8264 + 20.2170i −3.03234 9.33259i −3.15984 + 9.72499i −50.7528 36.8741i −7.64382
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.j 24
11.b odd 2 1 inner 121.4.c.j 24
11.c even 5 1 121.4.a.h 6
11.c even 5 3 inner 121.4.c.j 24
11.d odd 10 1 121.4.a.h 6
11.d odd 10 3 inner 121.4.c.j 24
33.f even 10 1 1089.4.a.bj 6
33.h odd 10 1 1089.4.a.bj 6
44.g even 10 1 1936.4.a.bq 6
44.h odd 10 1 1936.4.a.bq 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.h 6 11.c even 5 1
121.4.a.h 6 11.d odd 10 1
121.4.c.j 24 1.a even 1 1 trivial
121.4.c.j 24 11.b odd 2 1 inner
121.4.c.j 24 11.c even 5 3 inner
121.4.c.j 24 11.d odd 10 3 inner
1089.4.a.bj 6 33.f even 10 1
1089.4.a.bj 6 33.h odd 10 1
1936.4.a.bq 6 44.g even 10 1
1936.4.a.bq 6 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + 34 T_{2}^{22} + 867 T_{2}^{20} + 19844 T_{2}^{18} + 430661 T_{2}^{16} + 5401988 T_{2}^{14} + 63016611 T_{2}^{12} + 664077154 T_{2}^{10} + 5404004353 T_{2}^{8} + 3917039808 T_{2}^{6} + \cdots + 1358954496$$ acting on $$S_{4}^{\mathrm{new}}(121, [\chi])$$.