Properties

Label 375.3.d.e
Level $375$
Weight $3$
Character orbit 375.d
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(374,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.374");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.d (of order \(2\), degree \(1\), 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: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 68 q^{4} + 12 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 68 q^{4} + 12 q^{6} + 20 q^{9} + 116 q^{16} + 40 q^{19} + 112 q^{21} + 96 q^{24} - 16 q^{31} - 64 q^{34} - 56 q^{36} - 8 q^{39} - 416 q^{46} - 468 q^{49} - 208 q^{51} - 472 q^{54} - 276 q^{61} - 136 q^{64} - 340 q^{66} - 352 q^{69} + 148 q^{76} + 580 q^{79} - 268 q^{81} + 952 q^{84} + 444 q^{91} + 936 q^{94} + 1456 q^{96} + 220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1 −3.76125 −1.36465 2.67165i 10.1470 0 5.13278 + 10.0488i 11.2763i −23.1204 −5.27548 + 7.29173i 0
374.2 −3.76125 −1.36465 + 2.67165i 10.1470 0 5.13278 10.0488i 11.2763i −23.1204 −5.27548 7.29173i 0
374.3 −3.38842 −2.11620 2.12643i 7.48139 0 7.17057 + 7.20524i 6.70269i −11.7964 −0.0434208 + 8.99990i 0
374.4 −3.38842 −2.11620 + 2.12643i 7.48139 0 7.17057 7.20524i 6.70269i −11.7964 −0.0434208 8.99990i 0
374.5 −2.98144 2.72527 1.25416i 4.88897 0 −8.12522 + 3.73919i 12.5530i −2.65041 5.85419 6.83583i 0
374.6 −2.98144 2.72527 + 1.25416i 4.88897 0 −8.12522 3.73919i 12.5530i −2.65041 5.85419 + 6.83583i 0
374.7 −2.76294 2.51808 1.63072i 3.63386 0 −6.95733 + 4.50559i 1.45760i 1.01162 3.68150 8.21259i 0
374.8 −2.76294 2.51808 + 1.63072i 3.63386 0 −6.95733 4.50559i 1.45760i 1.01162 3.68150 + 8.21259i 0
374.9 −1.84620 −2.91022 0.728424i −0.591546 0 5.37285 + 1.34482i 11.6440i 8.47691 7.93880 + 4.23975i 0
374.10 −1.84620 −2.91022 + 0.728424i −0.591546 0 5.37285 1.34482i 11.6440i 8.47691 7.93880 4.23975i 0
374.11 −1.67701 −0.0488427 2.99960i −1.18764 0 0.0819097 + 5.03036i 0.986092i 8.69972 −8.99523 + 0.293018i 0
374.12 −1.67701 −0.0488427 + 2.99960i −1.18764 0 0.0819097 5.03036i 0.986092i 8.69972 −8.99523 0.293018i 0
374.13 −0.654766 1.42077 2.64224i −3.57128 0 −0.930269 + 1.73005i 1.39150i 4.95742 −4.96285 7.50800i 0
374.14 −0.654766 1.42077 + 2.64224i −3.57128 0 −0.930269 1.73005i 1.39150i 4.95742 −4.96285 + 7.50800i 0
374.15 −0.446371 −2.81092 1.04821i −3.80075 0 1.25471 + 0.467892i 6.22297i 3.48203 6.80250 + 5.89288i 0
374.16 −0.446371 −2.81092 + 1.04821i −3.80075 0 1.25471 0.467892i 6.22297i 3.48203 6.80250 5.89288i 0
374.17 0.446371 2.81092 1.04821i −3.80075 0 1.25471 0.467892i 6.22297i −3.48203 6.80250 5.89288i 0
374.18 0.446371 2.81092 + 1.04821i −3.80075 0 1.25471 + 0.467892i 6.22297i −3.48203 6.80250 + 5.89288i 0
374.19 0.654766 −1.42077 2.64224i −3.57128 0 −0.930269 1.73005i 1.39150i −4.95742 −4.96285 + 7.50800i 0
374.20 0.654766 −1.42077 + 2.64224i −3.57128 0 −0.930269 + 1.73005i 1.39150i −4.95742 −4.96285 7.50800i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.3.d.e 32
3.b odd 2 1 inner 375.3.d.e 32
5.b even 2 1 inner 375.3.d.e 32
5.c odd 4 2 375.3.c.c 32
15.d odd 2 1 inner 375.3.d.e 32
15.e even 4 2 375.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.3.c.c 32 5.c odd 4 2
375.3.c.c 32 15.e even 4 2
375.3.d.e 32 1.a even 1 1 trivial
375.3.d.e 32 3.b odd 2 1 inner
375.3.d.e 32 5.b even 2 1 inner
375.3.d.e 32 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 49 T_{2}^{14} + 956 T_{2}^{12} - 9479 T_{2}^{10} + 50466 T_{2}^{8} - 139855 T_{2}^{6} + 179160 T_{2}^{4} - 75825 T_{2}^{2} + 9025 \) acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display