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Properties

Label 180.4.k.f
Level $180$
Weight $4$
Character orbit 180.k
Analytic conductor $10.620$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,4,Mod(127,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.127");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.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: \(10.6203438010\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 60)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 24 q^{5} - 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 24 q^{5} - 84 q^{8} + 128 q^{10} - 412 q^{13} - 180 q^{16} - 20 q^{17} - 52 q^{20} - 436 q^{22} + 132 q^{25} - 704 q^{26} + 508 q^{28} - 340 q^{32} + 508 q^{37} + 1792 q^{38} - 2696 q^{40} + 1696 q^{41} + 2584 q^{46} - 832 q^{50} + 504 q^{52} - 1772 q^{53} + 512 q^{56} - 1060 q^{58} + 2096 q^{61} + 472 q^{62} - 28 q^{65} - 5872 q^{68} + 2956 q^{70} - 3348 q^{73} - 3480 q^{76} + 384 q^{77} + 4828 q^{80} - 928 q^{82} - 476 q^{85} + 3616 q^{86} + 380 q^{88} - 472 q^{92} + 8300 q^{97} - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 −2.79539 + 0.431034i 0 7.62842 − 2.40982i −2.54602 + 10.8866i 0 −9.77420 + 9.77420i −20.2857 + 10.0245i 0 2.42464 − 31.5297i
127.2 −2.66431 + 0.949450i 0 6.19709 − 5.05926i 10.6252 + 3.47923i 0 24.7270 − 24.7270i −11.7074 + 19.3633i 0 −31.6122 + 0.818373i
127.3 −2.62464 − 1.05416i 0 5.77750 + 5.53358i −10.9830 − 2.09135i 0 5.27814 − 5.27814i −9.33060 − 20.6141i 0 26.6218 + 17.0669i
127.4 −1.83686 + 2.15080i 0 −1.25186 − 7.90145i −8.47773 − 7.28890i 0 13.7652 − 13.7652i 19.2939 + 11.8214i 0 31.2494 − 4.84515i
127.5 −1.72961 + 2.23795i 0 −2.01687 − 7.74159i −7.18431 + 8.56655i 0 −13.0358 + 13.0358i 20.8137 + 8.87628i 0 −6.74547 − 30.8950i
127.6 −1.05416 − 2.62464i 0 −5.77750 + 5.53358i −10.9830 − 2.09135i 0 −5.27814 + 5.27814i 20.6141 + 9.33060i 0 6.08878 + 31.0311i
127.7 0.129446 + 2.82546i 0 −7.96649 + 0.731491i 8.68710 − 7.03806i 0 −4.43008 + 4.43008i −3.09803 − 22.4143i 0 21.0103 + 23.6340i
127.8 0.431034 − 2.79539i 0 −7.62842 − 2.40982i −2.54602 + 10.8866i 0 9.77420 − 9.77420i −10.0245 + 20.2857i 0 29.3348 + 11.8096i
127.9 0.949450 − 2.66431i 0 −6.19709 − 5.05926i 10.6252 + 3.47923i 0 −24.7270 + 24.7270i −19.3633 + 11.7074i 0 19.3578 − 25.0055i
127.10 1.79487 + 2.18597i 0 −1.55691 + 7.84704i 3.87875 + 10.4860i 0 17.0336 − 17.0336i −19.9478 + 10.6810i 0 −15.9601 + 27.2997i
127.11 2.15080 − 1.83686i 0 1.25186 − 7.90145i −8.47773 − 7.28890i 0 −13.7652 + 13.7652i −11.8214 − 19.2939i 0 −31.6226 − 0.104521i
127.12 2.18597 + 1.79487i 0 1.55691 + 7.84704i 3.87875 + 10.4860i 0 −17.0336 + 17.0336i −10.6810 + 19.9478i 0 −10.3421 + 29.8838i
127.13 2.23795 − 1.72961i 0 2.01687 − 7.74159i −7.18431 + 8.56655i 0 13.0358 − 13.0358i −8.87628 − 20.8137i 0 −1.26133 + 31.5976i
127.14 2.82546 + 0.129446i 0 7.96649 + 0.731491i 8.68710 − 7.03806i 0 4.43008 − 4.43008i 22.4143 + 3.09803i 0 25.4561 − 18.7613i
163.1 −2.79539 − 0.431034i 0 7.62842 + 2.40982i −2.54602 − 10.8866i 0 −9.77420 − 9.77420i −20.2857 − 10.0245i 0 2.42464 + 31.5297i
163.2 −2.66431 − 0.949450i 0 6.19709 + 5.05926i 10.6252 − 3.47923i 0 24.7270 + 24.7270i −11.7074 − 19.3633i 0 −31.6122 − 0.818373i
163.3 −2.62464 + 1.05416i 0 5.77750 − 5.53358i −10.9830 + 2.09135i 0 5.27814 + 5.27814i −9.33060 + 20.6141i 0 26.6218 − 17.0669i
163.4 −1.83686 − 2.15080i 0 −1.25186 + 7.90145i −8.47773 + 7.28890i 0 13.7652 + 13.7652i 19.2939 − 11.8214i 0 31.2494 + 4.84515i
163.5 −1.72961 − 2.23795i 0 −2.01687 + 7.74159i −7.18431 − 8.56655i 0 −13.0358 − 13.0358i 20.8137 − 8.87628i 0 −6.74547 + 30.8950i
163.6 −1.05416 + 2.62464i 0 −5.77750 − 5.53358i −10.9830 + 2.09135i 0 −5.27814 − 5.27814i 20.6141 − 9.33060i 0 6.08878 − 31.0311i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.4.k.f 28
3.b odd 2 1 60.4.j.b ✓ 28
4.b odd 2 1 inner 180.4.k.f 28
5.c odd 4 1 inner 180.4.k.f 28
12.b even 2 1 60.4.j.b ✓ 28
15.e even 4 1 60.4.j.b ✓ 28
20.e even 4 1 inner 180.4.k.f 28
60.l odd 4 1 60.4.j.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.j.b ✓ 28 3.b odd 2 1
60.4.j.b ✓ 28 12.b even 2 1
60.4.j.b ✓ 28 15.e even 4 1
60.4.j.b ✓ 28 60.l odd 4 1
180.4.k.f 28 1.a even 1 1 trivial
180.4.k.f 28 4.b odd 2 1 inner
180.4.k.f 28 5.c odd 4 1 inner
180.4.k.f 28 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{28} + 2132368 T_{7}^{24} + 1081090099808 T_{7}^{20} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
\( T_{13}^{14} + 206 T_{13}^{13} + 21218 T_{13}^{12} + 1094656 T_{13}^{11} + 43430660 T_{13}^{10} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
\( T_{17}^{14} + 10 T_{17}^{13} + 50 T_{17}^{12} - 287200 T_{17}^{11} + 171569348 T_{17}^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display