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Properties

Label 125.3.f.a
Level $125$
Weight $3$
Character orbit 125.f
Analytic conductor $3.406$
Analytic rank $0$
Dimension $32$
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] = [125,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
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: \(3.40600330450\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{4} - 6 q^{6} - 10 q^{7} + 30 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{4} - 6 q^{6} - 10 q^{7} + 30 q^{8} + 10 q^{9} - 6 q^{11} + 80 q^{12} + 10 q^{14} + 2 q^{16} + 140 q^{18} - 90 q^{19} - 6 q^{21} - 150 q^{22} - 160 q^{23} + 4 q^{26} + 210 q^{27} + 160 q^{28} + 110 q^{29} - 6 q^{31} - 290 q^{32} + 50 q^{33} + 260 q^{34} - 58 q^{36} + 150 q^{37} - 630 q^{38} - 390 q^{39} - 86 q^{41} - 120 q^{42} + 230 q^{43} - 340 q^{44} - 6 q^{46} - 150 q^{47} - 400 q^{48} - 16 q^{51} + 700 q^{52} + 10 q^{53} + 660 q^{54} - 70 q^{56} - 650 q^{57} + 120 q^{58} + 260 q^{59} + 114 q^{61} + 120 q^{62} + 40 q^{63} - 340 q^{64} + 138 q^{66} + 330 q^{67} + 710 q^{68} - 340 q^{69} - 66 q^{71} - 290 q^{72} - 320 q^{73} - 80 q^{76} - 210 q^{77} + 590 q^{78} + 210 q^{79} + 62 q^{81} + 30 q^{82} + 30 q^{83} + 10 q^{84} - 6 q^{86} + 200 q^{87} - 30 q^{88} + 10 q^{89} - 6 q^{91} + 270 q^{92} + 520 q^{93} - 790 q^{94} + 174 q^{96} - 90 q^{97} + 170 q^{98}+O(q^{100}) \) 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 −2.54321 + 1.29583i −2.29349 + 0.363254i 2.43759 − 3.35505i 0 5.36211 − 3.89580i −3.40272 − 3.40272i −0.0656701 + 0.414625i −3.43135 + 1.11491i 0
7.2 0.508965 − 0.259330i 5.29495 − 0.838638i −2.15935 + 2.97209i 0 2.47746 − 1.79998i 1.66138 + 1.66138i −0.685716 + 4.32944i 18.7737 − 6.09994i 0
7.3 0.776265 − 0.395527i −1.87175 + 0.296456i −1.90500 + 2.62200i 0 −1.33572 + 0.970456i −5.60844 − 5.60844i −0.986866 + 6.23083i −5.14394 + 1.67137i 0
7.4 3.32707 − 1.69523i −0.541921 + 0.0858318i 5.84445 − 8.04419i 0 −1.65750 + 1.20425i 1.68463 + 1.68463i 3.47161 − 21.9189i −8.27320 + 2.68812i 0
18.1 −2.54321 − 1.29583i −2.29349 − 0.363254i 2.43759 + 3.35505i 0 5.36211 + 3.89580i −3.40272 + 3.40272i −0.0656701 − 0.414625i −3.43135 − 1.11491i 0
18.2 0.508965 + 0.259330i 5.29495 + 0.838638i −2.15935 − 2.97209i 0 2.47746 + 1.79998i 1.66138 − 1.66138i −0.685716 − 4.32944i 18.7737 + 6.09994i 0
18.3 0.776265 + 0.395527i −1.87175 − 0.296456i −1.90500 − 2.62200i 0 −1.33572 − 0.970456i −5.60844 + 5.60844i −0.986866 − 6.23083i −5.14394 − 1.67137i 0
18.4 3.32707 + 1.69523i −0.541921 − 0.0858318i 5.84445 + 8.04419i 0 −1.65750 − 1.20425i 1.68463 − 1.68463i 3.47161 + 21.9189i −8.27320 − 2.68812i 0
32.1 −3.24491 − 0.513943i −1.43193 − 2.81033i 6.46108 + 2.09933i 0 3.20215 + 9.85519i 7.51823 + 7.51823i −8.17758 − 4.16668i −0.557438 + 0.767248i 0
32.2 −1.97355 − 0.312579i 2.04864 + 4.02069i −0.00704800 − 0.00229003i 0 −2.78631 − 8.57538i 3.91191 + 3.91191i 7.13464 + 3.63528i −6.67894 + 9.19277i 0
32.3 0.589367 + 0.0933465i −0.107372 − 0.210730i −3.46559 − 1.12604i 0 −0.0436108 − 0.134220i −7.64532 − 7.64532i −4.06409 − 2.07076i 5.25719 − 7.23590i 0
32.4 2.92327 + 0.463000i 0.441718 + 0.866921i 4.52691 + 1.47088i 0 0.889877 + 2.73876i 4.44588 + 4.44588i 2.00389 + 1.02103i 4.73363 − 6.51528i 0
43.1 −3.24491 + 0.513943i −1.43193 + 2.81033i 6.46108 − 2.09933i 0 3.20215 − 9.85519i 7.51823 − 7.51823i −8.17758 + 4.16668i −0.557438 − 0.767248i 0
43.2 −1.97355 + 0.312579i 2.04864 − 4.02069i −0.00704800 + 0.00229003i 0 −2.78631 + 8.57538i 3.91191 − 3.91191i 7.13464 − 3.63528i −6.67894 − 9.19277i 0
43.3 0.589367 − 0.0933465i −0.107372 + 0.210730i −3.46559 + 1.12604i 0 −0.0436108 + 0.134220i −7.64532 + 7.64532i −4.06409 + 2.07076i 5.25719 + 7.23590i 0
43.4 2.92327 − 0.463000i 0.441718 − 0.866921i 4.52691 − 1.47088i 0 0.889877 − 2.73876i 4.44588 − 4.44588i 2.00389 − 1.02103i 4.73363 + 6.51528i 0
82.1 −0.566108 − 3.57427i −3.17313 − 1.61679i −8.65068 + 2.81078i 0 −3.98250 + 12.2569i −0.574149 − 0.574149i 8.37205 + 16.4311i 2.16466 + 2.97940i 0
82.2 −0.295731 − 1.86717i 4.30737 + 2.19472i 0.405347 − 0.131705i 0 2.82409 − 8.69166i −3.57009 − 3.57009i −3.79877 − 7.45551i 8.44662 + 11.6258i 0
82.3 0.0455490 + 0.287585i −3.39113 − 1.72787i 3.72360 − 1.20987i 0 0.342446 − 1.05394i −2.38950 − 2.38950i 1.04630 + 2.05348i 3.22416 + 4.43767i 0
82.4 0.286042 + 1.80600i 1.30583 + 0.665351i 0.624420 − 0.202886i 0 −0.828102 + 2.54863i 3.62927 + 3.62927i 3.86553 + 7.58652i −4.02758 − 5.54349i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
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 125.3.f.a 32
5.b even 2 1 125.3.f.b 32
5.c odd 4 1 25.3.f.a ✓ 32
5.c odd 4 1 125.3.f.c 32
15.e even 4 1 225.3.r.a 32
20.e even 4 1 400.3.bg.c 32
25.d even 5 1 25.3.f.a ✓ 32
25.e even 10 1 125.3.f.c 32
25.f odd 20 1 inner 125.3.f.a 32
25.f odd 20 1 125.3.f.b 32
75.j odd 10 1 225.3.r.a 32
100.j odd 10 1 400.3.bg.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.3.f.a ✓ 32 5.c odd 4 1
25.3.f.a ✓ 32 25.d even 5 1
125.3.f.a 32 1.a even 1 1 trivial
125.3.f.a 32 25.f odd 20 1 inner
125.3.f.b 32 5.b even 2 1
125.3.f.b 32 25.f odd 20 1
125.3.f.c 32 5.c odd 4 1
125.3.f.c 32 25.e even 10 1
225.3.r.a 32 15.e even 4 1
225.3.r.a 32 75.j odd 10 1
400.3.bg.c 32 20.e even 4 1
400.3.bg.c 32 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 5 T_{2}^{30} - 10 T_{2}^{29} - 88 T_{2}^{28} + 460 T_{2}^{27} + 1255 T_{2}^{26} + \cdots + 12952801 \) acting on \(S_{3}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display