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Properties

Label 375.3.f.a
Level $375$
Weight $3$
Character orbit 375.f
Analytic conductor $10.218$
Analytic rank $0$
Dimension $32$
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] = [375,3,Mod(193,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.f (of order \(4\), degree \(2\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 36 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 36 q^{6} + 48 q^{11} + 100 q^{16} + 48 q^{21} - 416 q^{26} - 408 q^{31} + 156 q^{36} + 312 q^{41} + 632 q^{46} + 336 q^{51} - 960 q^{56} + 120 q^{61} - 48 q^{66} + 304 q^{71} + 256 q^{76} - 288 q^{81} + 608 q^{86} - 608 q^{91} + 288 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 −2.60613 + 2.60613i 1.22474 + 1.22474i − 9.58383i 0 −6.38369 −5.99807 + 5.99807i 14.5522 + 14.5522i 3.00000i 0
193.2 −2.21125 + 2.21125i −1.22474 − 1.22474i − 5.77922i 0 5.41643 −4.82250 + 4.82250i 3.93430 + 3.93430i 3.00000i 0
193.3 −1.77058 + 1.77058i 1.22474 + 1.22474i − 2.26992i 0 −4.33702 7.43730 − 7.43730i −3.06325 − 3.06325i 3.00000i 0
193.4 −1.59267 + 1.59267i 1.22474 + 1.22474i − 1.07319i 0 −3.90123 6.23907 − 6.23907i −4.66144 − 4.66144i 3.00000i 0
193.5 −1.51578 + 1.51578i 1.22474 + 1.22474i − 0.595188i 0 −3.71289 −7.75402 + 7.75402i −5.16095 − 5.16095i 3.00000i 0
193.6 −1.49321 + 1.49321i −1.22474 − 1.22474i − 0.459347i 0 3.65760 −1.71066 + 1.71066i −5.28693 − 5.28693i 3.00000i 0
193.7 −0.607306 + 0.607306i −1.22474 − 1.22474i 3.26236i 0 1.48759 2.05628 − 2.05628i −4.41047 − 4.41047i 3.00000i 0
193.8 −0.500834 + 0.500834i 1.22474 + 1.22474i 3.49833i 0 −1.22679 0.497825 − 0.497825i −3.75542 − 3.75542i 3.00000i 0
193.9 0.500834 − 0.500834i −1.22474 − 1.22474i 3.49833i 0 −1.22679 −0.497825 + 0.497825i 3.75542 + 3.75542i 3.00000i 0
193.10 0.607306 − 0.607306i 1.22474 + 1.22474i 3.26236i 0 1.48759 −2.05628 + 2.05628i 4.41047 + 4.41047i 3.00000i 0
193.11 1.49321 − 1.49321i 1.22474 + 1.22474i − 0.459347i 0 3.65760 1.71066 − 1.71066i 5.28693 + 5.28693i 3.00000i 0
193.12 1.51578 − 1.51578i −1.22474 − 1.22474i − 0.595188i 0 −3.71289 7.75402 − 7.75402i 5.16095 + 5.16095i 3.00000i 0
193.13 1.59267 − 1.59267i −1.22474 − 1.22474i − 1.07319i 0 −3.90123 −6.23907 + 6.23907i 4.66144 + 4.66144i 3.00000i 0
193.14 1.77058 − 1.77058i −1.22474 − 1.22474i − 2.26992i 0 −4.33702 −7.43730 + 7.43730i 3.06325 + 3.06325i 3.00000i 0
193.15 2.21125 − 2.21125i 1.22474 + 1.22474i − 5.77922i 0 5.41643 4.82250 − 4.82250i −3.93430 − 3.93430i 3.00000i 0
193.16 2.60613 − 2.60613i −1.22474 − 1.22474i − 9.58383i 0 −6.38369 5.99807 − 5.99807i −14.5522 − 14.5522i 3.00000i 0
307.1 −2.60613 − 2.60613i 1.22474 − 1.22474i 9.58383i 0 −6.38369 −5.99807 − 5.99807i 14.5522 − 14.5522i − 3.00000i 0
307.2 −2.21125 − 2.21125i −1.22474 + 1.22474i 5.77922i 0 5.41643 −4.82250 − 4.82250i 3.93430 − 3.93430i − 3.00000i 0
307.3 −1.77058 − 1.77058i 1.22474 − 1.22474i 2.26992i 0 −4.33702 7.43730 + 7.43730i −3.06325 + 3.06325i − 3.00000i 0
307.4 −1.59267 − 1.59267i 1.22474 − 1.22474i 1.07319i 0 −3.90123 6.23907 + 6.23907i −4.66144 + 4.66144i − 3.00000i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.3.f.a ✓ 32
5.b even 2 1 inner 375.3.f.a ✓ 32
5.c odd 4 2 inner 375.3.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.3.f.a ✓ 32 1.a even 1 1 trivial
375.3.f.a ✓ 32 5.b even 2 1 inner
375.3.f.a ✓ 32 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 387 T_{2}^{28} + 51763 T_{2}^{24} + 3129489 T_{2}^{20} + 94492320 T_{2}^{16} + \cdots + 1026625681 \) acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display