Properties

Label 690.2.w.a
Level $690$
Weight $2$
Character orbit 690.w
Analytic conductor $5.510$
Analytic rank $0$
Dimension $240$
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] = [690,2,Mod(7,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([0, 11, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.w (of order \(44\), degree \(20\), 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: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 24 q^{6} + 44 q^{10} - 16 q^{13} + 24 q^{16} + 44 q^{21} + 72 q^{23} + 16 q^{25} + 44 q^{28} - 16 q^{31} - 44 q^{33} - 24 q^{36} + 44 q^{37} + 88 q^{43} - 8 q^{46} + 48 q^{47} + 8 q^{50} - 16 q^{52} + 56 q^{55} + 44 q^{57} + 16 q^{58} + 88 q^{61} + 8 q^{62} + 88 q^{65} - 132 q^{67} + 56 q^{70} - 64 q^{71} + 16 q^{73} - 32 q^{75} - 16 q^{77} - 16 q^{78} + 24 q^{81} - 24 q^{82} + 92 q^{85} - 16 q^{87} - 44 q^{88} + 116 q^{92} - 80 q^{93} + 20 q^{95} + 24 q^{96} - 88 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −2.20400 + 0.377364i −0.142315 + 0.989821i 2.34384 4.29243i −0.977147 + 0.212565i −0.909632 0.415415i 2.17146 0.533633i
7.2 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −2.11721 0.719325i −0.142315 + 0.989821i −2.05801 + 3.76897i −0.977147 + 0.212565i −0.909632 0.415415i 2.16313 + 0.566452i
7.3 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −0.721297 + 2.11654i −0.142315 + 0.989821i −0.913900 + 1.67368i −0.977147 + 0.212565i −0.909632 0.415415i 0.568467 2.16260i
7.4 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −0.160701 2.23029i −0.142315 + 0.989821i 1.09028 1.99670i −0.977147 + 0.212565i −0.909632 0.415415i 0.319399 + 2.21314i
7.5 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −0.0140096 + 2.23602i −0.142315 + 0.989821i −0.151445 + 0.277351i −0.977147 + 0.212565i −0.909632 0.415415i −0.145542 2.23133i
7.6 −0.997452 + 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 2.23599 + 0.0191781i −0.142315 + 0.989821i 0.197874 0.362379i −0.977147 + 0.212565i −0.909632 0.415415i −2.23166 + 0.140384i
7.7 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −2.14628 0.627266i −0.142315 + 0.989821i −0.236933 + 0.433910i 0.977147 0.212565i −0.909632 0.415415i −2.18556 0.472554i
7.8 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.79544 + 1.33281i −0.142315 + 0.989821i 2.35788 4.31814i 0.977147 0.212565i −0.909632 0.415415i −1.69579 + 1.45750i
7.9 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.32705 1.79970i −0.142315 + 0.989821i 1.34666 2.46622i 0.977147 0.212565i −0.909632 0.415415i −1.45206 1.70045i
7.10 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −0.244725 + 2.22264i −0.142315 + 0.989821i −1.14160 + 2.09069i 0.977147 0.212565i −0.909632 0.415415i −0.0855405 + 2.23443i
7.11 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.24397 1.85810i −0.142315 + 0.989821i 0.294639 0.539591i 0.977147 0.212565i −0.909632 0.415415i 1.10824 1.94211i
7.12 0.997452 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.68213 + 1.47324i −0.142315 + 0.989821i −0.663233 + 1.21462i 0.977147 0.212565i −0.909632 0.415415i 1.78294 + 1.34949i
37.1 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −2.15875 0.582912i 0.841254 + 0.540641i 0.0886059 + 0.237562i −0.0713392 0.997452i 0.989821 + 0.142315i 1.61533 + 1.54619i
37.2 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −1.27260 1.83861i 0.841254 + 0.540641i −1.45896 3.91162i −0.0713392 0.997452i 0.989821 + 0.142315i 0.235778 + 2.22360i
37.3 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −0.00639501 + 2.23606i 0.841254 + 0.540641i −1.30166 3.48989i −0.0713392 0.997452i 0.989821 + 0.142315i 1.07724 1.95948i
37.4 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i 0.565910 + 2.16327i 0.841254 + 0.540641i 1.25538 + 3.36582i −0.0713392 0.997452i 0.989821 + 0.142315i 0.540059 2.16987i
37.5 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i 2.11884 0.714499i 0.841254 + 0.540641i 0.271075 + 0.726780i −0.0713392 0.997452i 0.989821 + 0.142315i −2.20209 0.388352i
37.6 −0.877679 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i 2.22914 0.175901i 0.841254 + 0.540641i −0.0297310 0.0797118i −0.0713392 0.997452i 0.989821 + 0.142315i −2.04077 0.913927i
37.7 0.877679 + 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −1.94930 + 1.09554i 0.841254 + 0.540641i 0.266002 + 0.713180i 0.0713392 + 0.997452i 0.989821 + 0.142315i −2.23590 + 0.0273353i
37.8 0.877679 + 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −0.516511 2.17560i 0.841254 + 0.540641i −1.41675 3.79846i 0.0713392 + 0.997452i 0.989821 + 0.142315i 0.589321 2.15701i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.w.a 240
5.c odd 4 1 inner 690.2.w.a 240
23.d odd 22 1 inner 690.2.w.a 240
115.l even 44 1 inner 690.2.w.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.w.a 240 1.a even 1 1 trivial
690.2.w.a 240 5.c odd 4 1 inner
690.2.w.a 240 23.d odd 22 1 inner
690.2.w.a 240 115.l even 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{240} + 88 T_{7}^{237} - 352 T_{7}^{236} + 924 T_{7}^{235} + 3872 T_{7}^{234} - 163460 T_{7}^{233} + 542862 T_{7}^{232} + 125576 T_{7}^{231} - 12594648 T_{7}^{230} + 90828496 T_{7}^{229} + \cdots + 19\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display