↑

Properties

Label 714.2.k.c
Level $714$
Weight $2$
Character orbit 714.k
Analytic conductor $5.701$
Analytic rank $0$
Dimension $40$
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] = [714,2,Mod(251,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.k (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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{2} + 40 q^{4} + 8 q^{7} - 40 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{2} + 40 q^{4} + 8 q^{7} - 40 q^{8} - 24 q^{9} - 8 q^{11} - 8 q^{14} + 8 q^{15} + 40 q^{16} + 24 q^{18} - 12 q^{21} + 8 q^{22} + 8 q^{28} - 8 q^{29} - 8 q^{30} - 40 q^{32} - 24 q^{36} + 40 q^{37} - 24 q^{39} + 12 q^{42} - 8 q^{44} + 52 q^{51} + 24 q^{53} - 8 q^{56} - 56 q^{57} + 8 q^{58} + 8 q^{60} + 28 q^{63} + 40 q^{64} - 8 q^{65} - 24 q^{67} + 72 q^{71} + 24 q^{72} - 40 q^{74} + 24 q^{78} + 8 q^{81} - 12 q^{84} + 16 q^{85} + 8 q^{88} - 48 q^{91} + 8 q^{93} + 144 q^{95} - 56 q^{99}+O(q^{100}) \) 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 −1.00000 −1.73133 + 0.0499674i 1.00000 2.07041 + 2.07041i 1.73133 − 0.0499674i 1.23373 − 2.34049i −1.00000 2.99501 − 0.173020i −2.07041 − 2.07041i
251.2 −1.00000 −1.60451 − 0.652352i 1.00000 −2.04094 − 2.04094i 1.60451 + 0.652352i 2.59952 + 0.492452i −1.00000 2.14887 + 2.09340i 2.04094 + 2.04094i
251.3 −1.00000 −1.48324 + 0.894430i 1.00000 −0.246237 − 0.246237i 1.48324 − 0.894430i 1.89364 + 1.84774i −1.00000 1.39999 − 2.65330i 0.246237 + 0.246237i
251.4 −1.00000 −1.21132 − 1.23803i 1.00000 0.726631 + 0.726631i 1.21132 + 1.23803i −1.05596 + 2.42589i −1.00000 −0.0654297 + 2.99929i −0.726631 − 0.726631i
251.5 −1.00000 −1.13621 + 1.30730i 1.00000 0.144485 + 0.144485i 1.13621 − 1.30730i 1.73170 − 2.00030i −1.00000 −0.418051 − 2.97073i −0.144485 − 0.144485i
251.6 −1.00000 −0.809073 + 1.53147i 1.00000 −2.61434 − 2.61434i 0.809073 − 1.53147i −0.815162 − 2.51704i −1.00000 −1.69080 − 2.47814i 2.61434 + 2.61434i
251.7 −1.00000 −0.717532 − 1.57644i 1.00000 −0.707291 − 0.707291i 0.717532 + 1.57644i 0.496577 − 2.59873i −1.00000 −1.97030 + 2.26228i 0.707291 + 0.707291i
251.8 −1.00000 −0.384637 + 1.68880i 1.00000 0.149319 + 0.149319i 0.384637 − 1.68880i −2.64395 − 0.0975963i −1.00000 −2.70411 − 1.29915i −0.149319 − 0.149319i
251.9 −1.00000 −0.307000 − 1.70463i 1.00000 2.84926 + 2.84926i 0.307000 + 1.70463i −2.27494 − 1.35079i −1.00000 −2.81150 + 1.04664i −2.84926 − 2.84926i
251.10 −1.00000 −0.241160 + 1.71518i 1.00000 2.90910 + 2.90910i 0.241160 − 1.71518i −0.774038 + 2.52999i −1.00000 −2.88368 − 0.827264i −2.90910 − 2.90910i
251.11 −1.00000 0.241160 − 1.71518i 1.00000 −2.90910 − 2.90910i −0.241160 + 1.71518i −2.52999 + 0.774038i −1.00000 −2.88368 − 0.827264i 2.90910 + 2.90910i
251.12 −1.00000 0.307000 + 1.70463i 1.00000 −2.84926 − 2.84926i −0.307000 − 1.70463i 1.35079 + 2.27494i −1.00000 −2.81150 + 1.04664i 2.84926 + 2.84926i
251.13 −1.00000 0.384637 − 1.68880i 1.00000 −0.149319 − 0.149319i −0.384637 + 1.68880i 0.0975963 + 2.64395i −1.00000 −2.70411 − 1.29915i 0.149319 + 0.149319i
251.14 −1.00000 0.717532 + 1.57644i 1.00000 0.707291 + 0.707291i −0.717532 − 1.57644i 2.59873 − 0.496577i −1.00000 −1.97030 + 2.26228i −0.707291 − 0.707291i
251.15 −1.00000 0.809073 − 1.53147i 1.00000 2.61434 + 2.61434i −0.809073 + 1.53147i 2.51704 + 0.815162i −1.00000 −1.69080 − 2.47814i −2.61434 − 2.61434i
251.16 −1.00000 1.13621 − 1.30730i 1.00000 −0.144485 − 0.144485i −1.13621 + 1.30730i 2.00030 − 1.73170i −1.00000 −0.418051 − 2.97073i 0.144485 + 0.144485i
251.17 −1.00000 1.21132 + 1.23803i 1.00000 −0.726631 − 0.726631i −1.21132 − 1.23803i −2.42589 + 1.05596i −1.00000 −0.0654297 + 2.99929i 0.726631 + 0.726631i
251.18 −1.00000 1.48324 − 0.894430i 1.00000 0.246237 + 0.246237i −1.48324 + 0.894430i −1.84774 − 1.89364i −1.00000 1.39999 − 2.65330i −0.246237 − 0.246237i
251.19 −1.00000 1.60451 + 0.652352i 1.00000 2.04094 + 2.04094i −1.60451 − 0.652352i −0.492452 − 2.59952i −1.00000 2.14887 + 2.09340i −2.04094 − 2.04094i
251.20 −1.00000 1.73133 − 0.0499674i 1.00000 −2.07041 − 2.07041i −1.73133 + 0.0499674i 2.34049 − 1.23373i −1.00000 2.99501 − 0.173020i 2.07041 + 2.07041i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
51.f odd 4 1 inner
357.l even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.k.c ✓ 40
3.b odd 2 1 714.2.k.d yes 40
7.b odd 2 1 inner 714.2.k.c ✓ 40
17.c even 4 1 714.2.k.d yes 40
21.c even 2 1 714.2.k.d yes 40
51.f odd 4 1 inner 714.2.k.c ✓ 40
119.f odd 4 1 714.2.k.d yes 40
357.l even 4 1 inner 714.2.k.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.k.c ✓ 40 1.a even 1 1 trivial
714.2.k.c ✓ 40 7.b odd 2 1 inner
714.2.k.c ✓ 40 51.f odd 4 1 inner
714.2.k.c ✓ 40 357.l even 4 1 inner
714.2.k.d yes 40 3.b odd 2 1
714.2.k.d yes 40 17.c even 4 1
714.2.k.d yes 40 21.c even 2 1
714.2.k.d yes 40 119.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):

\( T_{5}^{40} + 882 T_{5}^{36} + 290609 T_{5}^{32} + 43970400 T_{5}^{28} + 3019132256 T_{5}^{24} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{11}^{20} + 4 T_{11}^{19} + 8 T_{11}^{18} - 48 T_{11}^{17} + 1265 T_{11}^{16} + 4016 T_{11}^{15} + \cdots + 760384 \) Copy content Toggle raw display