↑

Properties

Label 224.2.u.b
Level $224$
Weight $2$
Character orbit 224.u
Analytic conductor $1.789$
Analytic rank $0$
Dimension $40$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,2,Mod(29,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.u (of order \(8\), degree \(4\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{2} + 4 q^{3} + 8 q^{5} + 12 q^{6} - 8 q^{8} + 8 q^{9} - 16 q^{10} + 12 q^{11} - 36 q^{12} + 8 q^{16} + 4 q^{19} - 4 q^{21} - 52 q^{22} + 16 q^{23} - 8 q^{24} - 16 q^{25} + 12 q^{26} - 32 q^{27}+ \cdots - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.39530 − 0.230541i −3.03470 − 1.25702i 1.89370 + 0.643346i 0.551446 + 1.33131i 3.94452 + 2.45353i −0.707107 + 0.707107i −2.49396 − 1.33423i 5.50802 + 5.50802i −0.462509 − 1.98470i
29.2 −1.32480 + 0.494880i 0.0422763 + 0.0175114i 1.51019 − 1.31123i 1.39396 + 3.36532i −0.0646737 − 0.00227744i −0.707107 + 0.707107i −1.35179 + 2.48448i −2.11984 − 2.11984i −3.51215 − 3.76853i
29.3 −0.533524 + 1.30971i −0.910708 − 0.377228i −1.43070 − 1.39753i −0.227513 − 0.549265i 0.979945 − 0.991507i −0.707107 + 0.707107i 2.59368 − 1.12820i −1.43423 − 1.43423i 0.840764 − 0.00493097i
29.4 −0.420598 − 1.35022i 2.28471 + 0.946358i −1.64619 + 1.13580i 0.446012 + 1.07677i 0.316848 − 3.48290i −0.707107 + 0.707107i 2.22597 + 1.74501i 2.20299 + 2.20299i 1.26628 − 1.05510i
29.5 0.146191 + 1.40664i 2.41040 + 0.998422i −1.95726 + 0.411276i 0.0734640 + 0.177358i −1.05204 + 3.53652i −0.707107 + 0.707107i −0.864649 − 2.69302i 2.69188 + 2.69188i −0.238738 + 0.129265i
29.6 0.270256 − 1.38815i 0.350600 + 0.145223i −1.85392 − 0.750313i −1.27761 − 3.08441i 0.296344 − 0.447438i −0.707107 + 0.707107i −1.54258 + 2.37075i −2.01949 − 2.01949i −4.62691 + 0.939927i
29.7 0.461048 + 1.33695i −1.66728 − 0.690611i −1.57487 + 1.23280i 0.923458 + 2.22943i 0.154616 − 2.54748i −0.707107 + 0.707107i −2.37428 − 1.53715i 0.181564 + 0.181564i −2.55487 + 2.26249i
29.8 1.16944 − 0.795236i 1.29558 + 0.536647i 0.735199 − 1.85997i 0.937867 + 2.26421i 1.94187 − 0.402713i −0.707107 + 0.707107i −0.619339 − 2.75979i −0.730784 − 0.730784i 2.89737 + 1.90204i
29.9 1.27165 + 0.618795i 2.04641 + 0.847650i 1.23419 + 1.57378i −1.31439 − 3.17322i 2.07779 + 2.34422i −0.707107 + 0.707107i 0.595604 + 2.76501i 1.34795 + 1.34795i 0.292129 − 4.84857i
29.10 1.35563 − 0.402828i −1.81729 − 0.752744i 1.67546 − 1.09217i −0.920909 − 2.22327i −2.76679 − 0.288389i −0.707107 + 0.707107i 1.83135 − 2.15550i 0.614581 + 0.614581i −2.14401 − 2.64296i
85.1 −1.39530 + 0.230541i −3.03470 + 1.25702i 1.89370 − 0.643346i 0.551446 − 1.33131i 3.94452 − 2.45353i −0.707107 − 0.707107i −2.49396 + 1.33423i 5.50802 − 5.50802i −0.462509 + 1.98470i
85.2 −1.32480 − 0.494880i 0.0422763 − 0.0175114i 1.51019 + 1.31123i 1.39396 − 3.36532i −0.0646737 + 0.00227744i −0.707107 − 0.707107i −1.35179 − 2.48448i −2.11984 + 2.11984i −3.51215 + 3.76853i
85.3 −0.533524 − 1.30971i −0.910708 + 0.377228i −1.43070 + 1.39753i −0.227513 + 0.549265i 0.979945 + 0.991507i −0.707107 − 0.707107i 2.59368 + 1.12820i −1.43423 + 1.43423i 0.840764 + 0.00493097i
85.4 −0.420598 + 1.35022i 2.28471 − 0.946358i −1.64619 − 1.13580i 0.446012 − 1.07677i 0.316848 + 3.48290i −0.707107 − 0.707107i 2.22597 − 1.74501i 2.20299 − 2.20299i 1.26628 + 1.05510i
85.5 0.146191 − 1.40664i 2.41040 − 0.998422i −1.95726 − 0.411276i 0.0734640 − 0.177358i −1.05204 − 3.53652i −0.707107 − 0.707107i −0.864649 + 2.69302i 2.69188 − 2.69188i −0.238738 − 0.129265i
85.6 0.270256 + 1.38815i 0.350600 − 0.145223i −1.85392 + 0.750313i −1.27761 + 3.08441i 0.296344 + 0.447438i −0.707107 − 0.707107i −1.54258 − 2.37075i −2.01949 + 2.01949i −4.62691 − 0.939927i
85.7 0.461048 − 1.33695i −1.66728 + 0.690611i −1.57487 − 1.23280i 0.923458 − 2.22943i 0.154616 + 2.54748i −0.707107 − 0.707107i −2.37428 + 1.53715i 0.181564 − 0.181564i −2.55487 − 2.26249i
85.8 1.16944 + 0.795236i 1.29558 − 0.536647i 0.735199 + 1.85997i 0.937867 − 2.26421i 1.94187 + 0.402713i −0.707107 − 0.707107i −0.619339 + 2.75979i −0.730784 + 0.730784i 2.89737 − 1.90204i
85.9 1.27165 − 0.618795i 2.04641 − 0.847650i 1.23419 − 1.57378i −1.31439 + 3.17322i 2.07779 − 2.34422i −0.707107 − 0.707107i 0.595604 − 2.76501i 1.34795 − 1.34795i 0.292129 + 4.84857i
85.10 1.35563 + 0.402828i −1.81729 + 0.752744i 1.67546 + 1.09217i −0.920909 + 2.22327i −2.76679 + 0.288389i −0.707107 − 0.707107i 1.83135 + 2.15550i 0.614581 − 0.614581i −2.14401 + 2.64296i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.u.b ✓ 40
4.b odd 2 1 896.2.u.b 40
32.g even 8 1 inner 224.2.u.b ✓ 40
32.h odd 8 1 896.2.u.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.b ✓ 40 1.a even 1 1 trivial
224.2.u.b ✓ 40 32.g even 8 1 inner
896.2.u.b 40 4.b odd 2 1
896.2.u.b 40 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 4 T_{3}^{39} + 4 T_{3}^{38} + 24 T_{3}^{37} - 88 T_{3}^{36} - 40 T_{3}^{35} + 872 T_{3}^{34} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display