↑

Properties

Label 714.2.e.a
Level $714$
Weight $2$
Character orbit 714.e
Analytic conductor $5.701$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(713,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.713");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.e (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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{4} - 8 q^{9} - 16 q^{15} + 48 q^{16} + 8 q^{18} + 4 q^{21} - 40 q^{25} + 24 q^{30} + 8 q^{36} - 20 q^{42} - 56 q^{43} - 56 q^{49} - 24 q^{51} + 16 q^{60} - 48 q^{64} + 40 q^{67} - 32 q^{70} - 8 q^{72} - 4 q^{84} + 120 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
713.1 − 1.00000i −1.71879 + 0.213902i −1.00000 − 2.59445i 0.213902 + 1.71879i 1.78990 − 1.94840i 1.00000i 2.90849 − 0.735308i −2.59445
713.2 − 1.00000i −1.71879 + 0.213902i −1.00000 − 2.59445i 0.213902 + 1.71879i 1.78990 + 1.94840i 1.00000i 2.90849 − 0.735308i −2.59445
713.3 − 1.00000i −1.56194 − 0.748555i −1.00000 0.299697i −0.748555 + 1.56194i −1.80493 − 1.93448i 1.00000i 1.87933 + 2.33840i 0.299697
713.4 − 1.00000i −1.56194 − 0.748555i −1.00000 0.299697i −0.748555 + 1.56194i −1.80493 + 1.93448i 1.00000i 1.87933 + 2.33840i 0.299697
713.5 − 1.00000i −1.36662 + 1.06412i −1.00000 2.53130i 1.06412 + 1.36662i −1.83455 − 1.90641i 1.00000i 0.735308 − 2.90849i 2.53130
713.6 − 1.00000i −1.36662 + 1.06412i −1.00000 2.53130i 1.06412 + 1.36662i −1.83455 + 1.90641i 1.00000i 0.735308 − 2.90849i 2.53130
713.7 − 1.00000i −0.792756 − 1.53998i −1.00000 1.21309i −1.53998 + 0.792756i 2.62530 − 0.328330i 1.00000i −1.74308 + 2.44166i 1.21309
713.8 − 1.00000i −0.792756 − 1.53998i −1.00000 1.21309i −1.53998 + 0.792756i 2.62530 + 0.328330i 1.00000i −1.74308 + 2.44166i 1.21309
713.9 − 1.00000i −0.575153 + 1.63377i −1.00000 − 2.55256i 1.63377 + 0.575153i 0.211918 − 2.63725i 1.00000i −2.33840 − 1.87933i −2.55256
713.10 − 1.00000i −0.575153 + 1.63377i −1.00000 − 2.55256i 1.63377 + 0.575153i 0.211918 + 2.63725i 1.00000i −2.33840 − 1.87933i −2.55256
713.11 − 1.00000i −0.528367 − 1.64949i −1.00000 − 3.71273i −1.64949 + 0.528367i −0.857786 − 2.50284i 1.00000i −2.44166 + 1.74308i −3.71273
713.12 − 1.00000i −0.528367 − 1.64949i −1.00000 − 3.71273i −1.64949 + 0.528367i −0.857786 + 2.50284i 1.00000i −2.44166 + 1.74308i −3.71273
713.13 − 1.00000i 0.528367 + 1.64949i −1.00000 3.71273i 1.64949 − 0.528367i 0.857786 − 2.50284i 1.00000i −2.44166 + 1.74308i 3.71273
713.14 − 1.00000i 0.528367 + 1.64949i −1.00000 3.71273i 1.64949 − 0.528367i 0.857786 + 2.50284i 1.00000i −2.44166 + 1.74308i 3.71273
713.15 − 1.00000i 0.575153 − 1.63377i −1.00000 2.55256i −1.63377 − 0.575153i −0.211918 − 2.63725i 1.00000i −2.33840 − 1.87933i 2.55256
713.16 − 1.00000i 0.575153 − 1.63377i −1.00000 2.55256i −1.63377 − 0.575153i −0.211918 + 2.63725i 1.00000i −2.33840 − 1.87933i 2.55256
713.17 − 1.00000i 0.792756 + 1.53998i −1.00000 − 1.21309i 1.53998 − 0.792756i −2.62530 − 0.328330i 1.00000i −1.74308 + 2.44166i −1.21309
713.18 − 1.00000i 0.792756 + 1.53998i −1.00000 − 1.21309i 1.53998 − 0.792756i −2.62530 + 0.328330i 1.00000i −1.74308 + 2.44166i −1.21309
713.19 − 1.00000i 1.36662 − 1.06412i −1.00000 − 2.53130i −1.06412 − 1.36662i 1.83455 − 1.90641i 1.00000i 0.735308 − 2.90849i −2.53130
713.20 − 1.00000i 1.36662 − 1.06412i −1.00000 − 2.53130i −1.06412 − 1.36662i 1.83455 + 1.90641i 1.00000i 0.735308 − 2.90849i −2.53130
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 713.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
17.b even 2 1 inner
21.c even 2 1 inner
51.c odd 2 1 inner
119.d odd 2 1 inner
357.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.e.a ✓ 48
3.b odd 2 1 inner 714.2.e.a ✓ 48
7.b odd 2 1 inner 714.2.e.a ✓ 48
17.b even 2 1 inner 714.2.e.a ✓ 48
21.c even 2 1 inner 714.2.e.a ✓ 48
51.c odd 2 1 inner 714.2.e.a ✓ 48
119.d odd 2 1 inner 714.2.e.a ✓ 48
357.c even 2 1 inner 714.2.e.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.e.a ✓ 48 1.a even 1 1 trivial
714.2.e.a ✓ 48 3.b odd 2 1 inner
714.2.e.a ✓ 48 7.b odd 2 1 inner
714.2.e.a ✓ 48 17.b even 2 1 inner
714.2.e.a ✓ 48 21.c even 2 1 inner
714.2.e.a ✓ 48 51.c odd 2 1 inner
714.2.e.a ✓ 48 119.d odd 2 1 inner
714.2.e.a ✓ 48 357.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(714, [\chi])\).