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Properties

Label 375.2.l.c
Level $375$
Weight $2$
Character orbit 375.l
Analytic conductor $2.994$
Analytic rank $0$
Dimension $64$
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] = [375,2,Mod(32,375)]
 
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([10, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.l (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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 10 q^{3} - 20 q^{4} - 6 q^{6} + 20 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 10 q^{3} - 20 q^{4} - 6 q^{6} + 20 q^{7} - 10 q^{9} + 10 q^{12} + 20 q^{13} - 8 q^{16} + 10 q^{18} - 6 q^{21} - 20 q^{22} + 10 q^{27} - 40 q^{28} - 12 q^{31} + 10 q^{33} + 20 q^{34} - 22 q^{36} + 20 q^{37} + 30 q^{39} - 90 q^{42} + 20 q^{43} - 12 q^{46} - 100 q^{48} - 16 q^{51} - 20 q^{52} + 120 q^{54} - 70 q^{57} + 20 q^{58} - 12 q^{61} + 20 q^{63} - 100 q^{64} - 30 q^{66} + 60 q^{67} - 80 q^{69} + 150 q^{72} + 60 q^{73} - 64 q^{76} + 80 q^{78} - 60 q^{79} + 14 q^{81} + 60 q^{82} - 130 q^{84} + 60 q^{87} - 20 q^{88} - 12 q^{91} + 20 q^{93} + 260 q^{94} + 42 q^{96} - 120 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.400841 + 2.53081i −1.67651 − 0.435117i −4.34220 − 1.41087i 0 1.77321 − 4.06850i −1.29796 − 1.29796i 2.98460 − 5.85760i 2.62135 + 1.45895i 0
32.2 −0.338090 + 2.13462i 1.71899 − 0.212337i −2.54018 − 0.825354i 0 −0.127915 + 3.74117i 0.0545283 + 0.0545283i 0.658272 − 1.29193i 2.90983 − 0.730008i 0
32.3 −0.184928 + 1.16759i 0.224509 − 1.71744i 0.573046 + 0.186194i 0 1.96375 + 0.579736i 3.04094 + 3.04094i −1.39673 + 2.74125i −2.89919 − 0.771160i 0
32.4 −0.0809972 + 0.511396i −0.181767 + 1.72249i 1.64715 + 0.535191i 0 −0.866150 − 0.232472i −0.155466 − 0.155466i −0.877235 + 1.72167i −2.93392 − 0.626184i 0
32.5 0.0809972 − 0.511396i 1.50036 + 0.865399i 1.64715 + 0.535191i 0 0.564087 − 0.697184i −0.155466 − 0.155466i 0.877235 − 1.72167i 1.50217 + 2.59682i 0
32.6 0.184928 − 1.16759i −1.52140 − 0.827854i 0.573046 + 0.186194i 0 −1.24794 + 1.62328i 3.04094 + 3.04094i 1.39673 − 2.74125i 1.62932 + 2.51899i 0
32.7 0.338090 − 2.13462i −1.18218 + 1.26588i −2.54018 − 0.825354i 0 2.30249 + 2.95148i 0.0545283 + 0.0545283i −0.658272 + 1.29193i −0.204907 − 2.99299i 0
32.8 0.400841 − 2.53081i 0.633409 − 1.61208i −4.34220 − 1.41087i 0 −3.82596 − 2.24922i −1.29796 − 1.29796i −2.98460 + 5.85760i −2.19759 − 2.04221i 0
107.1 −2.24299 − 1.14286i 1.19130 + 1.25730i 2.54931 + 3.50882i 0 −1.23517 − 4.18160i 1.80298 + 1.80298i −0.920375 − 5.81102i −0.161596 + 2.99564i 0
107.2 −1.33917 − 0.682341i 0.231310 − 1.71654i 0.152214 + 0.209505i 0 −1.48103 + 2.14090i −1.98452 − 1.98452i 0.409350 + 2.58454i −2.89299 − 0.794104i 0
107.3 −0.710801 − 0.362171i −1.53227 − 0.807549i −0.801501 − 1.10317i 0 0.796670 + 1.12895i 2.94096 + 2.94096i 0.419762 + 2.65027i 1.69573 + 2.47477i 0
107.4 −0.685506 − 0.349283i 0.559289 + 1.63927i −0.827650 − 1.13916i 0 0.189171 − 1.31908i −1.53819 − 1.53819i 0.410179 + 2.58977i −2.37439 + 1.83365i 0
107.5 0.685506 + 0.349283i −1.03848 + 1.38621i −0.827650 − 1.13916i 0 −1.19606 + 0.587530i −1.53819 − 1.53819i −0.410179 − 2.58977i −0.843130 − 2.87909i 0
107.6 0.710801 + 0.362171i 1.70683 − 0.294526i −0.801501 − 1.10317i 0 1.31988 + 0.408813i 2.94096 + 2.94096i −0.419762 − 2.65027i 2.82651 − 1.00541i 0
107.7 1.33917 + 0.682341i 0.310450 − 1.70400i 0.152214 + 0.209505i 0 1.57845 − 2.07011i −1.98452 − 1.98452i −0.409350 − 2.58454i −2.80724 − 1.05801i 0
107.8 2.24299 + 1.14286i −1.52152 + 0.827629i 2.54931 + 3.50882i 0 −4.35863 + 0.117475i 1.80298 + 1.80298i 0.920375 + 5.81102i 1.63006 − 2.51851i 0
143.1 −1.12503 + 2.20799i 1.73197 − 0.0162612i −2.43396 − 3.35006i 0 −1.91261 + 3.84248i 1.56473 − 1.56473i 5.24001 − 0.829937i 2.99947 − 0.0563281i 0
143.2 −0.888111 + 1.74302i −0.667932 − 1.59808i −1.07379 − 1.47795i 0 3.37868 + 0.255059i 0.551254 − 0.551254i −0.334566 + 0.0529901i −2.10773 + 2.13482i 0
143.3 −0.674175 + 1.32314i −0.399927 + 1.68525i −0.120624 − 0.166024i 0 −1.96020 − 1.66531i 1.72218 − 1.72218i −2.63243 + 0.416937i −2.68012 − 1.34795i 0
143.4 −0.0231951 + 0.0455229i 1.39036 − 1.03290i 1.17404 + 1.61592i 0 0.0147712 + 0.0872517i −1.44136 + 1.44136i −0.201718 + 0.0319491i 0.866221 − 2.87222i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.l.c 64
3.b odd 2 1 inner 375.2.l.c 64
5.b even 2 1 75.2.l.a ✓ 64
5.c odd 4 1 375.2.l.a 64
5.c odd 4 1 375.2.l.b 64
15.d odd 2 1 75.2.l.a ✓ 64
15.e even 4 1 375.2.l.a 64
15.e even 4 1 375.2.l.b 64
25.d even 5 1 375.2.l.b 64
25.e even 10 1 375.2.l.a 64
25.f odd 20 1 75.2.l.a ✓ 64
25.f odd 20 1 inner 375.2.l.c 64
75.h odd 10 1 375.2.l.a 64
75.j odd 10 1 375.2.l.b 64
75.l even 20 1 75.2.l.a ✓ 64
75.l even 20 1 inner 375.2.l.c 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.l.a ✓ 64 5.b even 2 1
75.2.l.a ✓ 64 15.d odd 2 1
75.2.l.a ✓ 64 25.f odd 20 1
75.2.l.a ✓ 64 75.l even 20 1
375.2.l.a 64 5.c odd 4 1
375.2.l.a 64 15.e even 4 1
375.2.l.a 64 25.e even 10 1
375.2.l.a 64 75.h odd 10 1
375.2.l.b 64 5.c odd 4 1
375.2.l.b 64 15.e even 4 1
375.2.l.b 64 25.d even 5 1
375.2.l.b 64 75.j odd 10 1
375.2.l.c 64 1.a even 1 1 trivial
375.2.l.c 64 3.b odd 2 1 inner
375.2.l.c 64 25.f odd 20 1 inner
375.2.l.c 64 75.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\):

\( T_{2}^{64} + 10 T_{2}^{62} + 2 T_{2}^{60} - 230 T_{2}^{58} + 898 T_{2}^{56} + 13480 T_{2}^{54} + \cdots + 625 \) Copy content Toggle raw display
\( T_{7}^{32} - 10 T_{7}^{31} + 50 T_{7}^{30} - 90 T_{7}^{29} + 231 T_{7}^{28} - 1920 T_{7}^{27} + \cdots + 6250000 \) Copy content Toggle raw display