Properties

 Label 121.3.d.f Level $121$ Weight $3$ Character orbit 121.d Analytic conductor $3.297$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,3,Mod(40,121)]

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([7]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("121.40");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 121.d (of order $$10$$, 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: $$3.29701119876$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{3} + 24 q^{4} + 4 q^{5} - 16 q^{9}+O(q^{10})$$ 32 * q + 4 * q^3 + 24 * q^4 + 4 * q^5 - 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{3} + 24 q^{4} + 4 q^{5} - 16 q^{9} + 208 q^{12} + 4 q^{14} + 68 q^{15} + 24 q^{16} - 52 q^{20} - 48 q^{23} + 16 q^{25} - 168 q^{26} - 104 q^{27} + 116 q^{31} - 720 q^{34} - 152 q^{36} + 4 q^{37} + 132 q^{38} + 176 q^{42} - 304 q^{45} + 244 q^{47} - 172 q^{48} - 88 q^{49} - 268 q^{53} - 48 q^{56} - 88 q^{58} + 56 q^{59} + 100 q^{60} + 40 q^{64} + 1136 q^{67} - 264 q^{69} + 188 q^{70} - 272 q^{71} + 96 q^{75} + 720 q^{78} + 356 q^{80} + 272 q^{81} + 180 q^{82} - 336 q^{86} - 96 q^{89} - 140 q^{91} + 156 q^{92} + 256 q^{93} - 152 q^{97}+O(q^{100})$$ 32 * q + 4 * q^3 + 24 * q^4 + 4 * q^5 - 16 * q^9 + 208 * q^12 + 4 * q^14 + 68 * q^15 + 24 * q^16 - 52 * q^20 - 48 * q^23 + 16 * q^25 - 168 * q^26 - 104 * q^27 + 116 * q^31 - 720 * q^34 - 152 * q^36 + 4 * q^37 + 132 * q^38 + 176 * q^42 - 304 * q^45 + 244 * q^47 - 172 * q^48 - 88 * q^49 - 268 * q^53 - 48 * q^56 - 88 * q^58 + 56 * q^59 + 100 * q^60 + 40 * q^64 + 1136 * q^67 - 264 * q^69 + 188 * q^70 - 272 * q^71 + 96 * q^75 + 720 * q^78 + 356 * q^80 + 272 * q^81 + 180 * q^82 - 336 * q^86 - 96 * q^89 - 140 * q^91 + 156 * q^92 + 256 * q^93 - 152 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
40.1 −2.10207 + 2.89326i 0.0368220 0.113327i −2.71615 8.35945i 2.78683 2.02475i 0.250480 + 0.344756i 3.69341 1.20006i 16.2907 + 5.29316i 7.26967 + 5.28172i 12.3192i
40.2 −1.69909 + 2.33860i −1.19042 + 3.66375i −1.34607 4.14278i 1.59197 1.15663i −6.54541 9.00898i −9.28903 + 3.01819i 0.978641 + 0.317980i −4.72480 3.43277i 5.68820i
40.3 −1.27082 + 1.74913i −0.881072 + 2.71166i −0.208417 0.641443i −6.18158 + 4.49118i −3.62337 4.98715i 5.75919 1.87127i −6.83809 2.22183i 0.704334 + 0.511729i 16.5199i
40.4 −0.867839 + 1.19448i 1.41664 4.35997i 0.562434 + 1.73099i 3.42082 2.48537i 3.97848 + 5.47590i 2.09516 0.680759i −8.17251 2.65541i −9.72133 7.06296i 6.24301i
40.5 0.867839 1.19448i 1.41664 4.35997i 0.562434 + 1.73099i 3.42082 2.48537i −3.97848 5.47590i −2.09516 + 0.680759i 8.17251 + 2.65541i −9.72133 7.06296i 6.24301i
40.6 1.27082 1.74913i −0.881072 + 2.71166i −0.208417 0.641443i −6.18158 + 4.49118i 3.62337 + 4.98715i −5.75919 + 1.87127i 6.83809 + 2.22183i 0.704334 + 0.511729i 16.5199i
40.7 1.69909 2.33860i −1.19042 + 3.66375i −1.34607 4.14278i 1.59197 1.15663i 6.54541 + 9.00898i 9.28903 3.01819i −0.978641 0.317980i −4.72480 3.43277i 5.68820i
40.8 2.10207 2.89326i 0.0368220 0.113327i −2.71615 8.35945i 2.78683 2.02475i −0.250480 0.344756i −3.69341 + 1.20006i −16.2907 5.29316i 7.26967 + 5.28172i 12.3192i
94.1 −3.40123 + 1.10513i −0.0964013 0.0700396i 7.11098 5.16643i −1.06447 + 3.27611i 0.405285 + 0.131685i −2.28265 3.14180i −10.0682 + 13.8577i −2.77677 8.54600i 12.3192i
94.2 −2.74919 + 0.893266i 3.11657 + 2.26432i 3.52406 2.56038i −0.608078 + 1.87147i −10.5907 3.44113i 5.74094 + 7.90172i −0.604833 + 0.832482i 1.80471 + 5.55434i 5.68820i
94.3 −2.05623 + 0.668110i 2.30668 + 1.67590i 0.545644 0.396433i 2.36116 7.26689i −5.86274 1.90492i −3.55937 4.89906i 4.22617 5.81683i −0.269032 0.827995i 16.5199i
94.4 −1.40419 + 0.456250i −3.70881 2.69461i −1.47247 + 1.06981i −1.30664 + 4.02142i 6.43731 + 2.09161i −1.29488 1.78225i 5.05089 6.95196i 3.71322 + 11.4281i 6.24301i
94.5 1.40419 0.456250i −3.70881 2.69461i −1.47247 + 1.06981i −1.30664 + 4.02142i −6.43731 2.09161i 1.29488 + 1.78225i −5.05089 + 6.95196i 3.71322 + 11.4281i 6.24301i
94.6 2.05623 0.668110i 2.30668 + 1.67590i 0.545644 0.396433i 2.36116 7.26689i 5.86274 + 1.90492i 3.55937 + 4.89906i −4.22617 + 5.81683i −0.269032 0.827995i 16.5199i
94.7 2.74919 0.893266i 3.11657 + 2.26432i 3.52406 2.56038i −0.608078 + 1.87147i 10.5907 + 3.44113i −5.74094 7.90172i 0.604833 0.832482i 1.80471 + 5.55434i 5.68820i
94.8 3.40123 1.10513i −0.0964013 0.0700396i 7.11098 5.16643i −1.06447 + 3.27611i −0.405285 0.131685i 2.28265 + 3.14180i 10.0682 13.8577i −2.77677 8.54600i 12.3192i
112.1 −3.40123 1.10513i −0.0964013 + 0.0700396i 7.11098 + 5.16643i −1.06447 3.27611i 0.405285 0.131685i −2.28265 + 3.14180i −10.0682 13.8577i −2.77677 + 8.54600i 12.3192i
112.2 −2.74919 0.893266i 3.11657 2.26432i 3.52406 + 2.56038i −0.608078 1.87147i −10.5907 + 3.44113i 5.74094 7.90172i −0.604833 0.832482i 1.80471 5.55434i 5.68820i
112.3 −2.05623 0.668110i 2.30668 1.67590i 0.545644 + 0.396433i 2.36116 + 7.26689i −5.86274 + 1.90492i −3.55937 + 4.89906i 4.22617 + 5.81683i −0.269032 + 0.827995i 16.5199i
112.4 −1.40419 0.456250i −3.70881 + 2.69461i −1.47247 1.06981i −1.30664 4.02142i 6.43731 2.09161i −1.29488 + 1.78225i 5.05089 + 6.95196i 3.71322 11.4281i 6.24301i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.8 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 121.3.d.f 32
11.b odd 2 1 inner 121.3.d.f 32
11.c even 5 1 121.3.b.c 8
11.c even 5 3 inner 121.3.d.f 32
11.d odd 10 1 121.3.b.c 8
11.d odd 10 3 inner 121.3.d.f 32
33.f even 10 1 1089.3.c.k 8
33.h odd 10 1 1089.3.c.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.3.b.c 8 11.c even 5 1
121.3.b.c 8 11.d odd 10 1
121.3.d.f 32 1.a even 1 1 trivial
121.3.d.f 32 11.b odd 2 1 inner
121.3.d.f 32 11.c even 5 3 inner
121.3.d.f 32 11.d odd 10 3 inner
1089.3.c.k 8 33.f even 10 1
1089.3.c.k 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - 28 T_{2}^{30} + 522 T_{2}^{28} - 8228 T_{2}^{26} + 119075 T_{2}^{24} - 1257500 T_{2}^{22} + 11244036 T_{2}^{20} - 89290816 T_{2}^{18} + 616642741 T_{2}^{16} - 3161731584 T_{2}^{14} + \cdots + 1406408618241$$ acting on $$S_{3}^{\mathrm{new}}(121, [\chi])$$.