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Properties

Label 125.5.f.c
Level $125$
Weight $5$
Character orbit 125.f
Analytic conductor $12.921$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [125,5,Mod(7,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([17]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 125.f (of order \(20\), degree \(8\), 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: \(12.9212453855\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 8 q^{2} - 2 q^{3} - 10 q^{4} - 6 q^{6} - 42 q^{7} - 50 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 8 q^{2} - 2 q^{3} - 10 q^{4} - 6 q^{6} - 42 q^{7} - 50 q^{8} - 10 q^{9} - 6 q^{11} + 178 q^{12} + 288 q^{13} - 10 q^{14} + 762 q^{16} - 1042 q^{17} + 28 q^{18} - 1310 q^{19} - 6 q^{21} + 3226 q^{22} + 578 q^{23} - 976 q^{26} - 5120 q^{27} - 10318 q^{28} + 3290 q^{29} - 6 q^{31} + 1298 q^{32} + 2806 q^{33} - 8010 q^{34} - 2758 q^{36} - 4692 q^{37} - 14480 q^{38} + 8790 q^{39} + 1434 q^{41} + 23886 q^{42} + 14958 q^{43} - 22060 q^{44} - 6 q^{46} + 12158 q^{47} + 14168 q^{48} - 16 q^{51} - 29372 q^{52} - 22852 q^{53} + 53940 q^{54} - 1030 q^{56} - 34150 q^{57} - 23000 q^{58} + 23240 q^{59} + 2634 q^{61} + 13716 q^{62} + 42428 q^{63} - 66060 q^{64} + 5178 q^{66} + 25558 q^{67} + 71982 q^{68} - 39560 q^{69} + 7974 q^{71} + 37860 q^{72} + 4228 q^{73} - 1040 q^{76} - 34374 q^{77} - 51844 q^{78} + 26590 q^{79} - 14368 q^{81} - 145614 q^{82} - 99832 q^{83} + 149990 q^{84} - 6 q^{86} + 31320 q^{87} + 95490 q^{88} - 93760 q^{89} - 6 q^{91} + 199958 q^{92} + 134396 q^{93} - 130410 q^{94} + 17694 q^{96} + 40108 q^{97} + 25052 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −6.39421 + 3.25801i 1.72535 − 0.273268i 20.8667 − 28.7206i 0 −10.1419 + 7.36854i −61.1323 − 61.1323i −21.8920 + 138.221i −74.1334 + 24.0874i 0
7.2 −4.46923 + 2.27719i −1.27414 + 0.201805i 5.38386 − 7.41025i 0 5.23489 − 3.80337i 24.5824 + 24.5824i 5.36744 − 33.8887i −75.4529 + 24.5161i 0
7.3 −2.97729 + 1.51700i −15.7767 + 2.49879i −2.84163 + 3.91117i 0 43.1811 − 31.3729i 44.2767 + 44.2767i 10.8907 − 68.7610i 165.625 − 53.8149i 0
7.4 −2.11781 + 1.07908i 16.3370 − 2.58752i −6.08387 + 8.37373i 0 −31.8064 + 23.1087i 2.05469 + 2.05469i 9.79775 − 61.8606i 183.166 − 59.5143i 0
7.5 0.207300 − 0.105625i 4.56103 − 0.722396i −9.37275 + 12.9005i 0 0.869199 − 0.631510i −11.1285 − 11.1285i −1.16270 + 7.34097i −56.7545 + 18.4406i 0
7.6 1.54995 − 0.789739i −11.3554 + 1.79852i −7.62591 + 10.4962i 0 −16.1800 + 11.7554i −58.8078 − 58.8078i −7.88456 + 49.7811i 48.6753 − 15.8156i 0
7.7 3.69958 − 1.88503i −0.916650 + 0.145183i 0.728966 − 1.00334i 0 −3.11754 + 2.26503i 37.9739 + 37.9739i −9.58703 + 60.5301i −76.2164 + 24.7642i 0
7.8 5.42141 − 2.76234i 11.3884 − 1.80375i 12.3565 − 17.0073i 0 56.7588 − 41.2377i 9.79605 + 9.79605i 4.78031 − 30.1817i 49.4076 − 16.0535i 0
7.9 6.66808 − 3.39756i −11.5382 + 1.82747i 23.5154 − 32.3661i 0 −70.7286 + 51.3874i −32.8567 − 32.8567i 28.1051 − 177.449i 52.7545 − 17.1410i 0
18.1 −6.39421 − 3.25801i 1.72535 + 0.273268i 20.8667 + 28.7206i 0 −10.1419 − 7.36854i −61.1323 + 61.1323i −21.8920 − 138.221i −74.1334 − 24.0874i 0
18.2 −4.46923 − 2.27719i −1.27414 − 0.201805i 5.38386 + 7.41025i 0 5.23489 + 3.80337i 24.5824 − 24.5824i 5.36744 + 33.8887i −75.4529 − 24.5161i 0
18.3 −2.97729 − 1.51700i −15.7767 − 2.49879i −2.84163 − 3.91117i 0 43.1811 + 31.3729i 44.2767 − 44.2767i 10.8907 + 68.7610i 165.625 + 53.8149i 0
18.4 −2.11781 − 1.07908i 16.3370 + 2.58752i −6.08387 − 8.37373i 0 −31.8064 − 23.1087i 2.05469 − 2.05469i 9.79775 + 61.8606i 183.166 + 59.5143i 0
18.5 0.207300 + 0.105625i 4.56103 + 0.722396i −9.37275 − 12.9005i 0 0.869199 + 0.631510i −11.1285 + 11.1285i −1.16270 − 7.34097i −56.7545 − 18.4406i 0
18.6 1.54995 + 0.789739i −11.3554 − 1.79852i −7.62591 − 10.4962i 0 −16.1800 − 11.7554i −58.8078 + 58.8078i −7.88456 − 49.7811i 48.6753 + 15.8156i 0
18.7 3.69958 + 1.88503i −0.916650 − 0.145183i 0.728966 + 1.00334i 0 −3.11754 − 2.26503i 37.9739 − 37.9739i −9.58703 − 60.5301i −76.2164 − 24.7642i 0
18.8 5.42141 + 2.76234i 11.3884 + 1.80375i 12.3565 + 17.0073i 0 56.7588 + 41.2377i 9.79605 − 9.79605i 4.78031 + 30.1817i 49.4076 + 16.0535i 0
18.9 6.66808 + 3.39756i −11.5382 − 1.82747i 23.5154 + 32.3661i 0 −70.7286 − 51.3874i −32.8567 + 32.8567i 28.1051 + 177.449i 52.7545 + 17.1410i 0
32.1 −6.53352 − 1.03481i 7.66178 + 15.0371i 26.3992 + 8.57763i 0 −34.4979 − 106.174i −5.89577 − 5.89577i −69.3001 − 35.3102i −119.800 + 164.891i 0
32.2 −6.19314 − 0.980897i −1.94321 − 3.81377i 22.1759 + 7.20540i 0 8.29367 + 25.5253i 2.13863 + 2.13863i −40.8803 − 20.8296i 36.8419 − 50.7085i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.5.f.c 72
5.b even 2 1 25.5.f.a ✓ 72
5.c odd 4 1 125.5.f.a 72
5.c odd 4 1 125.5.f.b 72
25.d even 5 1 125.5.f.a 72
25.e even 10 1 125.5.f.b 72
25.f odd 20 1 25.5.f.a ✓ 72
25.f odd 20 1 inner 125.5.f.c 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.f.a ✓ 72 5.b even 2 1
25.5.f.a ✓ 72 25.f odd 20 1
125.5.f.a 72 5.c odd 4 1
125.5.f.a 72 25.d even 5 1
125.5.f.b 72 5.c odd 4 1
125.5.f.b 72 25.e even 10 1
125.5.f.c 72 1.a even 1 1 trivial
125.5.f.c 72 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 8 T_{2}^{71} + 37 T_{2}^{70} - 130 T_{2}^{69} - 3650 T_{2}^{68} + 31268 T_{2}^{67} + \cdots + 13\!\cdots\!76 \) acting on \(S_{5}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display