↑

Properties

Label 750.3.d.b
Level $750$
Weight $3$
Character orbit 750.d
Analytic conductor $20.436$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [750,3,Mod(251,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.251");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 750.d (of order \(2\), degree \(1\), 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: \(20.4360198270\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 8 q^{6} - 4 q^{9} + 128 q^{16} + 16 q^{19} + 124 q^{21} - 16 q^{24} + 96 q^{31} + 32 q^{34} + 8 q^{36} - 100 q^{39} - 112 q^{46} + 160 q^{49} - 4 q^{51} - 40 q^{54} + 128 q^{61} - 256 q^{64} + 144 q^{66} + 264 q^{69} - 32 q^{76} - 296 q^{79} - 596 q^{81} - 248 q^{84} - 600 q^{91} - 400 q^{94} + 32 q^{96} + 724 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 − 1.41421i −3.00000 + 0.00277587i −2.00000 0 0.00392567 + 4.24264i −5.72578 2.82843i 8.99998 − 0.0166552i 0
251.2 − 1.41421i 3.00000 + 0.00277587i −2.00000 0 0.00392567 − 4.24264i 5.72578 2.82843i 8.99998 + 0.0166552i 0
251.3 1.41421i −3.00000 − 0.00277587i −2.00000 0 0.00392567 − 4.24264i −5.72578 − 2.82843i 8.99998 + 0.0166552i 0
251.4 1.41421i 3.00000 − 0.00277587i −2.00000 0 0.00392567 + 4.24264i 5.72578 − 2.82843i 8.99998 − 0.0166552i 0
251.5 − 1.41421i −1.34568 + 2.68126i −2.00000 0 3.79187 + 1.90308i −6.63242 2.82843i −5.37828 − 7.21624i 0
251.6 − 1.41421i 1.34568 + 2.68126i −2.00000 0 3.79187 − 1.90308i 6.63242 2.82843i −5.37828 + 7.21624i 0
251.7 1.41421i −1.34568 − 2.68126i −2.00000 0 3.79187 − 1.90308i −6.63242 − 2.82843i −5.37828 + 7.21624i 0
251.8 1.41421i 1.34568 − 2.68126i −2.00000 0 3.79187 + 1.90308i 6.63242 − 2.82843i −5.37828 − 7.21624i 0
251.9 − 1.41421i −2.61270 + 1.47438i −2.00000 0 2.08508 + 3.69492i 7.39353 2.82843i 4.65243 − 7.70421i 0
251.10 − 1.41421i 2.61270 + 1.47438i −2.00000 0 2.08508 − 3.69492i −7.39353 2.82843i 4.65243 + 7.70421i 0
251.11 1.41421i −2.61270 − 1.47438i −2.00000 0 2.08508 − 3.69492i 7.39353 − 2.82843i 4.65243 + 7.70421i 0
251.12 1.41421i 2.61270 − 1.47438i −2.00000 0 2.08508 + 3.69492i −7.39353 − 2.82843i 4.65243 − 7.70421i 0
251.13 − 1.41421i −2.67900 + 1.35016i −2.00000 0 1.90941 + 3.78869i 0.207413 2.82843i 5.35413 − 7.23417i 0
251.14 − 1.41421i 2.67900 + 1.35016i −2.00000 0 1.90941 − 3.78869i −0.207413 2.82843i 5.35413 + 7.23417i 0
251.15 1.41421i −2.67900 − 1.35016i −2.00000 0 1.90941 − 3.78869i 0.207413 − 2.82843i 5.35413 + 7.23417i 0
251.16 1.41421i 2.67900 − 1.35016i −2.00000 0 1.90941 + 3.78869i −0.207413 − 2.82843i 5.35413 − 7.23417i 0
251.17 − 1.41421i −2.10322 − 2.13927i −2.00000 0 −3.02538 + 2.97440i −4.95030 2.82843i −0.152952 + 8.99870i 0
251.18 − 1.41421i 2.10322 − 2.13927i −2.00000 0 −3.02538 − 2.97440i 4.95030 2.82843i −0.152952 − 8.99870i 0
251.19 1.41421i −2.10322 + 2.13927i −2.00000 0 −3.02538 − 2.97440i −4.95030 − 2.82843i −0.152952 − 8.99870i 0
251.20 1.41421i 2.10322 + 2.13927i −2.00000 0 −3.02538 + 2.97440i 4.95030 − 2.82843i −0.152952 + 8.99870i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.32
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 750.3.d.b ✓ 32
3.b odd 2 1 inner 750.3.d.b ✓ 32
5.b even 2 1 inner 750.3.d.b ✓ 32
5.c odd 4 1 750.3.b.a 16
5.c odd 4 1 750.3.b.b 16
15.d odd 2 1 inner 750.3.d.b ✓ 32
15.e even 4 1 750.3.b.a 16
15.e even 4 1 750.3.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.3.b.a 16 5.c odd 4 1
750.3.b.a 16 15.e even 4 1
750.3.b.b 16 5.c odd 4 1
750.3.b.b 16 15.e even 4 1
750.3.d.b ✓ 32 1.a even 1 1 trivial
750.3.d.b ✓ 32 3.b odd 2 1 inner
750.3.d.b ✓ 32 5.b even 2 1 inner
750.3.d.b ✓ 32 15.d odd 2 1 inner