Properties

Label 882.2.y.b
Level $882$
Weight $2$
Character orbit 882.y
Analytic conductor $7.043$
Analytic rank $0$
Dimension $336$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(193,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([14, 22])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [336,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9} + 2 q^{10} - q^{12} + 21 q^{13} - 3 q^{14} - 7 q^{15} + 28 q^{16} + 7 q^{17} - 3 q^{18} - 34 q^{19} + 2 q^{20}+ \cdots - 33 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 −0.988831 + 0.149042i −1.72151 + 0.190770i 0.955573 − 0.294755i 0.620475 − 0.298805i 1.67385 − 0.445217i 1.17782 − 2.36912i −0.900969 + 0.433884i 2.92721 − 0.656826i −0.569010 + 0.387944i
193.2 −0.988831 + 0.149042i −1.72099 − 0.195461i 0.955573 − 0.294755i 3.37211 − 1.62392i 1.73090 − 0.0632214i −2.07177 + 1.64553i −0.900969 + 0.433884i 2.92359 + 0.672773i −3.09242 + 2.10837i
193.3 −0.988831 + 0.149042i −1.71571 + 0.237388i 0.955573 − 0.294755i −2.20928 + 1.06393i 1.66116 − 0.490449i −2.48278 + 0.914229i −0.900969 + 0.433884i 2.88729 − 0.814576i 2.02603 − 1.38132i
193.4 −0.988831 + 0.149042i −1.34126 + 1.09591i 0.955573 − 0.294755i −3.04871 + 1.46818i 1.16294 − 1.28358i 2.45568 − 0.984711i −0.900969 + 0.433884i 0.597949 − 2.93981i 2.79584 − 1.90617i
193.5 −0.988831 + 0.149042i −1.30288 − 1.14127i 0.955573 − 0.294755i 0.319989 − 0.154098i 1.45842 + 0.934344i −2.34313 − 1.22872i −0.900969 + 0.433884i 0.394983 + 2.97388i −0.293447 + 0.200069i
193.6 −0.988831 + 0.149042i −1.29686 + 1.14811i 0.955573 − 0.294755i −2.13890 + 1.03004i 1.11125 − 1.32858i −0.709438 + 2.54886i −0.900969 + 0.433884i 0.363666 − 2.97788i 1.96149 − 1.33732i
193.7 −0.988831 + 0.149042i −1.25599 + 1.19269i 0.955573 − 0.294755i 2.27765 − 1.09686i 1.06420 − 1.36656i 0.746060 + 2.53838i −0.900969 + 0.433884i 0.154997 − 2.99599i −2.08873 + 1.42407i
193.8 −0.988831 + 0.149042i −1.22909 − 1.22038i 0.955573 − 0.294755i −2.38065 + 1.14646i 1.39725 + 1.02356i 0.584552 + 2.58037i −0.900969 + 0.433884i 0.0213443 + 2.99992i 2.18319 − 1.48847i
193.9 −0.988831 + 0.149042i −1.10763 + 1.33160i 0.955573 − 0.294755i 0.803117 − 0.386761i 0.896791 − 1.48181i −1.54497 − 2.14781i −0.900969 + 0.433884i −0.546322 − 2.94984i −0.736504 + 0.502140i
193.10 −0.988831 + 0.149042i −0.791704 − 1.54052i 0.955573 − 0.294755i −1.63609 + 0.787900i 1.01246 + 1.40532i 1.21299 − 2.35131i −0.900969 + 0.433884i −1.74641 + 2.43927i 1.50039 − 1.02295i
193.11 −0.988831 + 0.149042i −0.706030 − 1.58162i 0.955573 − 0.294755i 2.82729 − 1.36155i 0.933872 + 1.45873i 2.29182 + 1.32196i −0.900969 + 0.433884i −2.00304 + 2.23334i −2.59278 + 1.76773i
193.12 −0.988831 + 0.149042i −0.273342 + 1.71035i 0.955573 − 0.294755i 1.42958 − 0.688448i 0.0153749 − 1.73198i −2.59729 − 0.504055i −0.900969 + 0.433884i −2.85057 − 0.935018i −1.31100 + 0.893827i
193.13 −0.988831 + 0.149042i −0.104521 + 1.72889i 0.955573 − 0.294755i −2.53873 + 1.22259i −0.154324 − 1.72516i 2.51659 − 0.816574i −0.900969 + 0.433884i −2.97815 − 0.361413i 2.32815 − 1.58731i
193.14 −0.988831 + 0.149042i 0.154519 − 1.72514i 0.955573 − 0.294755i 3.43321 − 1.65335i 0.104326 + 1.72891i −2.32033 − 1.27124i −0.900969 + 0.433884i −2.95225 − 0.533135i −3.14845 + 2.14658i
193.15 −0.988831 + 0.149042i 0.182160 − 1.72245i 0.955573 − 0.294755i 0.639573 − 0.308002i 0.0765919 + 1.73036i −1.13819 + 2.38842i −0.900969 + 0.433884i −2.93364 − 0.627521i −0.586524 + 0.399885i
193.16 −0.988831 + 0.149042i 0.345681 + 1.69720i 0.955573 − 0.294755i 3.39561 − 1.63524i −0.594776 − 1.62673i 2.19792 − 1.47280i −0.900969 + 0.433884i −2.76101 + 1.17338i −3.11397 + 2.12307i
193.17 −0.988831 + 0.149042i 0.518149 + 1.65273i 0.955573 − 0.294755i −0.369132 + 0.177764i −0.758689 − 1.55705i 0.562698 + 2.58522i −0.900969 + 0.433884i −2.46304 + 1.71272i 0.338514 − 0.230795i
193.18 −0.988831 + 0.149042i 0.524634 − 1.65068i 0.955573 − 0.294755i −1.80411 + 0.868814i −0.272753 + 1.71044i −1.17055 − 2.37272i −0.900969 + 0.433884i −2.44952 − 1.73201i 1.65447 − 1.12800i
193.19 −0.988831 + 0.149042i 0.812898 − 1.52944i 0.955573 − 0.294755i 1.13402 − 0.546117i −0.575866 + 1.63352i 2.59724 + 0.504333i −0.900969 + 0.433884i −1.67839 − 2.48656i −1.03996 + 0.709035i
193.20 −0.988831 + 0.149042i 1.06408 + 1.36665i 0.955573 − 0.294755i −3.40275 + 1.63868i −1.25588 − 1.19279i −1.90345 − 1.83763i −0.900969 + 0.433884i −0.735474 + 2.90845i 3.12052 − 2.12753i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 193.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.z even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.y.b ✓ 336
9.c even 3 1 882.2.bb.a yes 336
49.g even 21 1 882.2.bb.a yes 336
441.z even 21 1 inner 882.2.y.b ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.y.b ✓ 336 1.a even 1 1 trivial
882.2.y.b ✓ 336 441.z even 21 1 inner
882.2.bb.a yes 336 9.c even 3 1
882.2.bb.a yes 336 49.g even 21 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{336} + 4 T_{5}^{335} + 180 T_{5}^{334} + 732 T_{5}^{333} + 17223 T_{5}^{332} + \cdots + 10\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display