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Properties

Label 224.2.u.c
Level $224$
Weight $2$
Character orbit 224.u
Analytic conductor $1.789$
Analytic rank $0$
Dimension $52$
Inner twists $2$

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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: \(52\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 20 q^{6} - 8 q^{10} + 12 q^{12} - 12 q^{16} - 20 q^{18} + 20 q^{22} - 20 q^{23} - 8 q^{24} + 20 q^{26} - 24 q^{27} - 8 q^{28} + 20 q^{30} + 60 q^{32} - 48 q^{33} + 48 q^{34} + 8 q^{36} - 60 q^{38}+ \cdots - 64 q^{99}+O(q^{100}) \) 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 −1.33683 + 0.461403i −0.825255 − 0.341832i 1.57421 − 1.23363i −0.175588 − 0.423906i 1.26095 + 0.0761946i 0.707107 − 0.707107i −1.53525 + 2.37550i −1.55712 − 1.55712i 0.430322 + 0.485673i
29.2 −1.24749 − 0.666168i 2.18713 + 0.905938i 1.11244 + 1.66207i 0.797931 + 1.92638i −2.12490 − 2.58714i 0.707107 − 0.707107i −0.280536 − 2.81448i 1.84149 + 1.84149i 0.287882 − 2.93468i
29.3 −1.03545 + 0.963248i 2.47594 + 1.02557i 0.144306 − 1.99479i −1.23251 − 2.97553i −3.55158 + 1.32302i 0.707107 − 0.707107i 1.77205 + 2.20450i 2.95715 + 2.95715i 4.14237 + 1.89380i
29.4 −0.926451 − 1.06850i −1.34443 − 0.556881i −0.283377 + 1.97982i 0.598728 + 1.44546i 0.650522 + 1.95244i 0.707107 − 0.707107i 2.37797 − 1.53142i −0.623944 − 0.623944i 0.989776 − 1.97888i
29.5 −0.742328 + 1.20372i −2.57348 − 1.06597i −0.897897 − 1.78712i 0.224493 + 0.541974i 3.19350 − 2.30646i 0.707107 − 0.707107i 2.81773 + 0.245807i 3.36518 + 3.36518i −0.819034 − 0.132095i
29.6 −0.729813 + 1.21135i 1.74965 + 0.724727i −0.934745 − 1.76812i 1.44353 + 3.48498i −2.15481 + 1.59052i 0.707107 − 0.707107i 2.82401 + 0.158094i 0.414709 + 0.414709i −5.27504 − 0.794768i
29.7 −0.0564658 − 1.41309i −2.95511 − 1.22405i −1.99362 + 0.159582i −1.21151 − 2.92484i −1.56282 + 4.24494i 0.707107 − 0.707107i 0.338075 + 2.80815i 5.11306 + 5.11306i −4.06464 + 1.87712i
29.8 0.171231 + 1.40381i −0.380077 − 0.157433i −1.94136 + 0.480752i −1.58477 − 3.82598i 0.155925 − 0.560513i 0.707107 − 0.707107i −1.00731 − 2.64298i −2.00165 − 2.00165i 5.09958 − 2.87984i
29.9 0.631975 − 1.26515i 2.91760 + 1.20851i −1.20122 − 1.59909i −0.629460 − 1.51965i 3.37279 − 2.92745i 0.707107 − 0.707107i −2.78223 + 0.509137i 4.93055 + 4.93055i −2.32039 − 0.164019i
29.10 0.863963 − 1.11963i −0.301186 − 0.124755i −0.507135 − 1.93464i −0.107860 − 0.260396i −0.399893 + 0.229432i 0.707107 − 0.707107i −2.60422 − 1.10365i −2.04617 − 2.04617i −0.384733 − 0.104210i
29.11 0.875361 + 1.11074i 0.999166 + 0.413868i −0.467486 + 1.94460i 0.523077 + 1.26282i 0.414931 + 1.47210i 0.707107 − 0.707107i −2.56916 + 1.18297i −1.29427 − 1.29427i −0.944783 + 1.68643i
29.12 1.41113 − 0.0933551i −2.44686 − 1.01352i 1.98257 − 0.263472i 1.68815 + 4.07556i −3.54745 − 1.20178i 0.707107 − 0.707107i 2.77306 − 0.556876i 2.83858 + 2.83858i 2.76268 + 5.59355i
29.13 1.41405 − 0.0212810i 0.496926 + 0.205834i 1.99909 − 0.0601851i −0.334218 − 0.806875i 0.707060 + 0.280485i 0.707107 − 0.707107i 2.82555 − 0.127648i −1.91675 − 1.91675i −0.489774 − 1.13385i
85.1 −1.33683 − 0.461403i −0.825255 + 0.341832i 1.57421 + 1.23363i −0.175588 + 0.423906i 1.26095 − 0.0761946i 0.707107 + 0.707107i −1.53525 − 2.37550i −1.55712 + 1.55712i 0.430322 − 0.485673i
85.2 −1.24749 + 0.666168i 2.18713 − 0.905938i 1.11244 − 1.66207i 0.797931 − 1.92638i −2.12490 + 2.58714i 0.707107 + 0.707107i −0.280536 + 2.81448i 1.84149 − 1.84149i 0.287882 + 2.93468i
85.3 −1.03545 − 0.963248i 2.47594 − 1.02557i 0.144306 + 1.99479i −1.23251 + 2.97553i −3.55158 − 1.32302i 0.707107 + 0.707107i 1.77205 − 2.20450i 2.95715 − 2.95715i 4.14237 − 1.89380i
85.4 −0.926451 + 1.06850i −1.34443 + 0.556881i −0.283377 − 1.97982i 0.598728 − 1.44546i 0.650522 − 1.95244i 0.707107 + 0.707107i 2.37797 + 1.53142i −0.623944 + 0.623944i 0.989776 + 1.97888i
85.5 −0.742328 − 1.20372i −2.57348 + 1.06597i −0.897897 + 1.78712i 0.224493 − 0.541974i 3.19350 + 2.30646i 0.707107 + 0.707107i 2.81773 − 0.245807i 3.36518 − 3.36518i −0.819034 + 0.132095i
85.6 −0.729813 − 1.21135i 1.74965 − 0.724727i −0.934745 + 1.76812i 1.44353 − 3.48498i −2.15481 − 1.59052i 0.707107 + 0.707107i 2.82401 − 0.158094i 0.414709 − 0.414709i −5.27504 + 0.794768i
85.7 −0.0564658 + 1.41309i −2.95511 + 1.22405i −1.99362 − 0.159582i −1.21151 + 2.92484i −1.56282 − 4.24494i 0.707107 + 0.707107i 0.338075 − 2.80815i 5.11306 − 5.11306i −4.06464 − 1.87712i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.13
Significant digits:
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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.c ✓ 52
4.b odd 2 1 896.2.u.c 52
32.g even 8 1 inner 224.2.u.c ✓ 52
32.h odd 8 1 896.2.u.c 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.c ✓ 52 1.a even 1 1 trivial
224.2.u.c ✓ 52 32.g even 8 1 inner
896.2.u.c 52 4.b odd 2 1
896.2.u.c 52 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{52} + 8 T_{3}^{49} - 72 T_{3}^{47} - 40 T_{3}^{46} - 88 T_{3}^{45} + 16440 T_{3}^{44} + \cdots + 151519232 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display