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Properties

Label 180.5.f.i
Level $180$
Weight $5$
Character orbit 180.f
Analytic conductor $18.607$
Analytic rank $0$
Dimension $24$
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] = [180,5,Mod(19,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 180.f (of order \(2\), degree \(1\), 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: \(18.6065933551\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 14 q^{4} + 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 14 q^{4} + 24 q^{5} + 274 q^{10} + 36 q^{14} + 594 q^{16} + 12 q^{20} + 1208 q^{25} + 2868 q^{26} + 1680 q^{29} + 3076 q^{34} - 7222 q^{40} + 4848 q^{41} + 3828 q^{44} - 15280 q^{46} + 5416 q^{49} - 14472 q^{50} - 32172 q^{56} + 2896 q^{61} - 18298 q^{64} + 2688 q^{65} + 27608 q^{70} - 31836 q^{74} + 50136 q^{76} + 27348 q^{80} - 15680 q^{85} + 58152 q^{86} + 38544 q^{89} + 4808 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.90764 − 0.854601i 0 14.5393 + 6.67895i 9.11612 − 23.2787i 0 10.5267 −51.1066 − 38.5243i 0 −55.5165 + 83.1740i
19.2 −3.90764 + 0.854601i 0 14.5393 − 6.67895i 9.11612 + 23.2787i 0 10.5267 −51.1066 + 38.5243i 0 −55.5165 − 83.1740i
19.3 −3.76887 − 1.34002i 0 12.4087 + 10.1007i −11.9809 + 21.9421i 0 63.7886 −33.2317 − 54.6960i 0 74.5573 − 66.6424i
19.4 −3.76887 + 1.34002i 0 12.4087 − 10.1007i −11.9809 − 21.9421i 0 63.7886 −33.2317 + 54.6960i 0 74.5573 + 66.6424i
19.5 −3.44337 − 2.03548i 0 7.71364 + 14.0178i 20.5121 − 14.2918i 0 −51.0440 1.97210 − 63.9696i 0 −99.7214 + 7.45996i
19.6 −3.44337 + 2.03548i 0 7.71364 − 14.0178i 20.5121 + 14.2918i 0 −51.0440 1.97210 + 63.9696i 0 −99.7214 − 7.45996i
19.7 −2.59896 − 3.04063i 0 −2.49086 + 15.8049i −24.9405 − 1.72355i 0 2.65040 54.5305 − 33.5025i 0 59.5786 + 80.3143i
19.8 −2.59896 + 3.04063i 0 −2.49086 − 15.8049i −24.9405 + 1.72355i 0 2.65040 54.5305 + 33.5025i 0 59.5786 − 80.3143i
19.9 −0.988963 − 3.87582i 0 −14.0439 + 7.66608i −11.6321 + 22.1290i 0 −66.7450 43.6012 + 46.8501i 0 97.2718 + 23.1992i
19.10 −0.988963 + 3.87582i 0 −14.0439 − 7.66608i −11.6321 − 22.1290i 0 −66.7450 43.6012 − 46.8501i 0 97.2718 − 23.1992i
19.11 −0.828581 − 3.91324i 0 −14.6269 + 6.48488i 24.9254 + 1.93048i 0 −67.1774 37.4965 + 51.8654i 0 −13.0983 − 99.1385i
19.12 −0.828581 + 3.91324i 0 −14.6269 − 6.48488i 24.9254 − 1.93048i 0 −67.1774 37.4965 − 51.8654i 0 −13.0983 + 99.1385i
19.13 0.828581 − 3.91324i 0 −14.6269 − 6.48488i 24.9254 + 1.93048i 0 67.1774 −37.4965 + 51.8654i 0 28.2071 − 95.9394i
19.14 0.828581 + 3.91324i 0 −14.6269 + 6.48488i 24.9254 − 1.93048i 0 67.1774 −37.4965 − 51.8654i 0 28.2071 + 95.9394i
19.15 0.988963 − 3.87582i 0 −14.0439 − 7.66608i −11.6321 + 22.1290i 0 66.7450 −43.6012 + 46.8501i 0 74.2643 + 66.9688i
19.16 0.988963 + 3.87582i 0 −14.0439 + 7.66608i −11.6321 − 22.1290i 0 66.7450 −43.6012 − 46.8501i 0 74.2643 − 66.9688i
19.17 2.59896 − 3.04063i 0 −2.49086 − 15.8049i −24.9405 − 1.72355i 0 −2.65040 −54.5305 − 33.5025i 0 −70.0600 + 71.3554i
19.18 2.59896 + 3.04063i 0 −2.49086 + 15.8049i −24.9405 + 1.72355i 0 −2.65040 −54.5305 + 33.5025i 0 −70.0600 − 71.3554i
19.19 3.44337 − 2.03548i 0 7.71364 − 14.0178i 20.5121 − 14.2918i 0 51.0440 −1.97210 − 63.9696i 0 41.5401 − 90.9638i
19.20 3.44337 + 2.03548i 0 7.71364 + 14.0178i 20.5121 + 14.2918i 0 51.0440 −1.97210 + 63.9696i 0 41.5401 + 90.9638i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.5.f.i 24
3.b odd 2 1 60.5.f.a ✓ 24
4.b odd 2 1 inner 180.5.f.i 24
5.b even 2 1 inner 180.5.f.i 24
12.b even 2 1 60.5.f.a ✓ 24
15.d odd 2 1 60.5.f.a ✓ 24
15.e even 4 2 300.5.c.e 24
20.d odd 2 1 inner 180.5.f.i 24
24.f even 2 1 960.5.j.d 24
24.h odd 2 1 960.5.j.d 24
60.h even 2 1 60.5.f.a ✓ 24
60.l odd 4 2 300.5.c.e 24
120.i odd 2 1 960.5.j.d 24
120.m even 2 1 960.5.j.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.f.a ✓ 24 3.b odd 2 1
60.5.f.a ✓ 24 12.b even 2 1
60.5.f.a ✓ 24 15.d odd 2 1
60.5.f.a ✓ 24 60.h even 2 1
180.5.f.i 24 1.a even 1 1 trivial
180.5.f.i 24 4.b odd 2 1 inner
180.5.f.i 24 5.b even 2 1 inner
180.5.f.i 24 20.d odd 2 1 inner
300.5.c.e 24 15.e even 4 2
300.5.c.e 24 60.l odd 4 2
960.5.j.d 24 24.f even 2 1
960.5.j.d 24 24.h odd 2 1
960.5.j.d 24 120.i odd 2 1
960.5.j.d 24 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{12} - 15760 T_{7}^{10} + 92404320 T_{7}^{8} - 239939919616 T_{7}^{6} + 240221796962560 T_{7}^{4} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
\( T_{13}^{12} + 218544 T_{13}^{10} + 18382787808 T_{13}^{8} + 747903076611840 T_{13}^{6} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
\( T_{23}^{12} - 1770688 T_{23}^{10} + 990424991232 T_{23}^{8} + \cdots + 70\!\cdots\!84 \) Copy content Toggle raw display
\( T_{29}^{6} - 420 T_{29}^{5} - 2195660 T_{29}^{4} + 1045759008 T_{29}^{3} + 1193995478080 T_{29}^{2} + \cdots - 34\!\cdots\!64 \) Copy content Toggle raw display