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Properties

Label 375.3.k.a
Level $375$
Weight $3$
Character orbit 375.k
Analytic conductor $10.218$
Analytic rank $0$
Dimension $80$
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] = [375,3,Mod(7,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.k (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: \(10.2180099135\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 75)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} + 4 q^{7} + 12 q^{8} + 24 q^{12} - 32 q^{13} + 80 q^{16} + 100 q^{17} + 48 q^{18} - 100 q^{19} + 100 q^{22} + 96 q^{23} - 40 q^{26} - 196 q^{28} + 200 q^{29} - 636 q^{32} - 216 q^{33} + 100 q^{34} - 120 q^{36} + 184 q^{37} + 564 q^{38} + 160 q^{41} + 12 q^{42} + 472 q^{43} - 700 q^{44} + 288 q^{47} + 48 q^{48} - 620 q^{52} - 304 q^{53} - 72 q^{57} - 1272 q^{58} + 800 q^{59} - 240 q^{61} - 1212 q^{62} + 12 q^{63} + 100 q^{64} + 80 q^{67} - 104 q^{68} - 36 q^{72} + 116 q^{73} + 88 q^{77} + 120 q^{78} + 200 q^{79} + 180 q^{81} + 168 q^{82} + 1264 q^{83} - 1200 q^{84} + 876 q^{87} + 212 q^{88} - 1500 q^{89} + 1504 q^{92} + 648 q^{93} - 200 q^{94} + 60 q^{96} + 260 q^{97} + 92 q^{98}+O(q^{100}) \) 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 −2.93190 + 1.49388i 1.71073 − 0.270952i 4.01322 − 5.52372i 0 −4.61091 + 3.35002i −4.34391 − 4.34391i −1.45557 + 9.19012i 2.85317 − 0.927051i 0
7.2 −2.73870 + 1.39544i −1.71073 + 0.270952i 3.20210 − 4.40731i 0 4.30707 − 3.12927i 0.250082 + 0.250082i −0.696119 + 4.39513i 2.85317 − 0.927051i 0
7.3 −1.53105 + 0.780110i −1.71073 + 0.270952i −0.615594 + 0.847292i 0 2.40784 − 1.74940i 3.26334 + 3.26334i 1.35675 − 8.56621i 2.85317 − 0.927051i 0
7.4 −0.510204 + 0.259962i 1.71073 − 0.270952i −2.15841 + 2.97080i 0 −0.802382 + 0.582965i −6.06980 − 6.06980i 0.687243 − 4.33908i 2.85317 − 0.927051i 0
7.5 −0.250128 + 0.127447i 1.71073 − 0.270952i −2.30482 + 3.17231i 0 −0.393368 + 0.285799i 6.72381 + 6.72381i 0.347860 − 2.19630i 2.85317 − 0.927051i 0
7.6 0.186256 − 0.0949024i −1.71073 + 0.270952i −2.32546 + 3.20072i 0 −0.292920 + 0.212819i −5.11319 − 5.11319i −0.260180 + 1.64271i 2.85317 − 0.927051i 0
7.7 1.53414 − 0.781681i 1.71073 − 0.270952i −0.608595 + 0.837659i 0 2.41269 − 1.75292i 1.91431 + 1.91431i −1.35628 + 8.56322i 2.85317 − 0.927051i 0
7.8 2.16137 − 1.10127i −1.71073 + 0.270952i 1.10757 − 1.52445i 0 −3.39912 + 2.46960i −6.20191 − 6.20191i −0.802843 + 5.06895i 2.85317 − 0.927051i 0
7.9 3.18220 − 1.62141i −1.71073 + 0.270952i 5.14628 − 7.08325i 0 −5.00455 + 3.63602i 6.93420 + 6.93420i 2.65683 − 16.7746i 2.85317 − 0.927051i 0
7.10 3.41817 − 1.74164i 1.71073 − 0.270952i 6.29941 − 8.67040i 0 5.37565 − 3.90564i 0.908106 + 0.908106i 4.03119 − 25.4519i 2.85317 − 0.927051i 0
43.1 −3.75152 + 0.594183i 0.786335 − 1.54327i 9.91663 − 3.22211i 0 −2.03297 + 6.25683i 5.76987 − 5.76987i −21.7507 + 11.0826i −1.76336 − 2.42705i 0
43.2 −2.92729 + 0.463638i −0.786335 + 1.54327i 4.54985 − 1.47834i 0 1.58631 − 4.88217i −2.66748 + 2.66748i −2.07035 + 1.05489i −1.76336 − 2.42705i 0
43.3 −2.52458 + 0.399854i −0.786335 + 1.54327i 2.40940 − 0.782861i 0 1.36808 − 4.21053i 7.99303 − 7.99303i 3.34014 − 1.70189i −1.76336 − 2.42705i 0
43.4 −2.47604 + 0.392165i 0.786335 − 1.54327i 2.17273 − 0.705963i 0 −1.34178 + 4.12956i −3.46290 + 3.46290i 3.83175 − 1.95238i −1.76336 − 2.42705i 0
43.5 −0.367346 + 0.0581819i 0.786335 − 1.54327i −3.67267 + 1.19332i 0 −0.199067 + 0.612664i 1.57845 − 1.57845i 2.60526 − 1.32745i −1.76336 − 2.42705i 0
43.6 0.247004 − 0.0391215i −0.786335 + 1.54327i −3.74475 + 1.21674i 0 −0.133852 + 0.411956i −8.31728 + 8.31728i −1.76867 + 0.901180i −1.76336 − 2.42705i 0
43.7 1.20062 − 0.190159i −0.786335 + 1.54327i −2.39891 + 0.779452i 0 −0.650620 + 2.00240i 8.35473 − 8.35473i −7.06431 + 3.59945i −1.76336 − 2.42705i 0
43.8 1.75107 − 0.277342i 0.786335 − 1.54327i −0.814895 + 0.264776i 0 0.948914 − 2.92046i −5.49856 + 5.49856i −7.67216 + 3.90916i −1.76336 − 2.42705i 0
43.9 2.60745 − 0.412980i −0.786335 + 1.54327i 2.82402 − 0.917581i 0 −1.41299 + 4.34874i −5.98338 + 5.98338i −2.42430 + 1.23524i −1.76336 − 2.42705i 0
43.10 3.44703 − 0.545956i 0.786335 − 1.54327i 7.77971 − 2.52778i 0 1.86796 − 5.74899i 0.992761 − 0.992761i 12.9984 − 6.62301i −1.76336 − 2.42705i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
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 375.3.k.a 80
5.b even 2 1 75.3.k.a ✓ 80
5.c odd 4 1 375.3.k.b 80
5.c odd 4 1 375.3.k.c 80
15.d odd 2 1 225.3.r.b 80
25.d even 5 1 375.3.k.b 80
25.e even 10 1 375.3.k.c 80
25.f odd 20 1 75.3.k.a ✓ 80
25.f odd 20 1 inner 375.3.k.a 80
75.l even 20 1 225.3.r.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.k.a ✓ 80 5.b even 2 1
75.3.k.a ✓ 80 25.f odd 20 1
225.3.r.b 80 15.d odd 2 1
225.3.r.b 80 75.l even 20 1
375.3.k.a 80 1.a even 1 1 trivial
375.3.k.a 80 25.f odd 20 1 inner
375.3.k.b 80 5.c odd 4 1
375.3.k.b 80 25.d even 5 1
375.3.k.c 80 5.c odd 4 1
375.3.k.c 80 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 4 T_{2}^{79} + 8 T_{2}^{78} - 4 T_{2}^{77} - 348 T_{2}^{76} - 580 T_{2}^{75} + \cdots + 87\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display