↑

Properties

Label 399.2.ck.a
Level $399$
Weight $2$
Character orbit 399.ck
Analytic conductor $3.186$
Analytic rank $0$
Dimension $78$
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] = [399,2,Mod(10,399)] 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("399.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 3, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.ck (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [78] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.18603104065\)
Analytic rank: \(0\)
Dimension: \(78\)
Relative dimension: \(13\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 78 q - 6 q^{7} + 12 q^{10} + 6 q^{11} + 72 q^{12} + 3 q^{13} - 15 q^{14} + 18 q^{19} + 9 q^{21} - 12 q^{22} + 30 q^{23} - 9 q^{24} + 18 q^{25} + 39 q^{27} - 9 q^{28} - 6 q^{29} - 18 q^{31} - 51 q^{32} - 12 q^{33}+ \cdots - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.74715 − 2.08217i −0.173648 − 0.984808i −0.935610 + 5.30611i −3.54879 + 0.625747i −1.74715 + 2.08217i −0.498430 + 2.59838i 7.97502 − 4.60438i −0.939693 + 0.342020i 7.50318 + 6.29591i
10.2 −1.37475 − 1.63837i −0.173648 − 0.984808i −0.447004 + 2.53509i 1.82948 − 0.322586i −1.37475 + 1.63837i 2.54314 + 0.729693i 1.06352 − 0.614025i −0.939693 + 0.342020i −3.04360 − 2.55388i
10.3 −1.12502 − 1.34075i −0.173648 − 0.984808i −0.184639 + 1.04714i −0.946358 + 0.166869i −1.12502 + 1.34075i 2.38220 − 1.15114i −1.41980 + 0.819723i −0.939693 + 0.342020i 1.28841 + 1.08110i
10.4 −0.868231 − 1.03472i −0.173648 − 0.984808i 0.0304812 − 0.172868i 2.82413 − 0.497969i −0.868231 + 1.03472i −2.62808 + 0.305313i −2.54486 + 1.46928i −0.939693 + 0.342020i −2.96725 − 2.48982i
10.5 −0.763835 − 0.910304i −0.173648 − 0.984808i 0.102088 − 0.578971i −3.42867 + 0.604567i −0.763835 + 0.910304i −2.05773 − 1.66305i −2.66324 + 1.53762i −0.939693 + 0.342020i 3.16928 + 2.65934i
10.6 −0.376037 − 0.448144i −0.173648 − 0.984808i 0.287868 − 1.63258i −2.56641 + 0.452528i −0.376037 + 0.448144i 0.436486 + 2.60950i −1.85315 + 1.06991i −0.939693 + 0.342020i 1.16786 + 0.979954i
10.7 −0.137026 − 0.163301i −0.173648 − 0.984808i 0.339405 − 1.92486i 2.98443 − 0.526235i −0.137026 + 0.163301i −0.181283 − 2.63953i −0.730069 + 0.421505i −0.939693 + 0.342020i −0.494879 − 0.415253i
10.8 0.416536 + 0.496408i −0.173648 − 0.984808i 0.274377 − 1.55607i 2.05441 − 0.362248i 0.416536 − 0.496408i 2.34609 + 1.22306i 2.00913 − 1.15997i −0.939693 + 0.342020i 1.03556 + 0.868936i
10.9 0.623350 + 0.742879i −0.173648 − 0.984808i 0.183992 − 1.04347i −1.88728 + 0.332779i 0.623350 − 0.742879i −2.31983 − 1.27216i 2.56954 − 1.48352i −0.939693 + 0.342020i −1.42365 − 1.19459i
10.10 1.09739 + 1.30782i −0.173648 − 0.984808i −0.158826 + 0.900749i −2.41276 + 0.425435i 1.09739 − 1.30782i 1.71138 + 2.01772i 1.60471 − 0.926477i −0.939693 + 0.342020i −3.20413 − 2.68858i
10.11 1.23327 + 1.46975i −0.173648 − 0.984808i −0.291923 + 1.65558i 1.06585 − 0.187938i 1.23327 − 1.46975i 0.911736 − 2.48369i 0.529852 − 0.305910i −0.939693 + 0.342020i 1.59070 + 1.33475i
10.12 1.24334 + 1.48175i −0.173648 − 0.984808i −0.302405 + 1.71502i 3.34147 − 0.589192i 1.24334 − 1.48175i −2.31720 + 1.27695i 0.433057 − 0.250025i −0.939693 + 0.342020i 5.02762 + 4.21867i
10.13 1.77818 + 2.11915i −0.173648 − 0.984808i −0.981581 + 5.56683i −1.01522 + 0.179011i 1.77818 − 2.11915i −2.26817 + 1.36213i −8.75089 + 5.05233i −0.939693 + 0.342020i −2.18459 − 1.83309i
40.1 −1.74715 + 2.08217i −0.173648 + 0.984808i −0.935610 − 5.30611i −3.54879 − 0.625747i −1.74715 − 2.08217i −0.498430 − 2.59838i 7.97502 + 4.60438i −0.939693 − 0.342020i 7.50318 − 6.29591i
40.2 −1.37475 + 1.63837i −0.173648 + 0.984808i −0.447004 − 2.53509i 1.82948 + 0.322586i −1.37475 − 1.63837i 2.54314 − 0.729693i 1.06352 + 0.614025i −0.939693 − 0.342020i −3.04360 + 2.55388i
40.3 −1.12502 + 1.34075i −0.173648 + 0.984808i −0.184639 − 1.04714i −0.946358 − 0.166869i −1.12502 − 1.34075i 2.38220 + 1.15114i −1.41980 − 0.819723i −0.939693 − 0.342020i 1.28841 − 1.08110i
40.4 −0.868231 + 1.03472i −0.173648 + 0.984808i 0.0304812 + 0.172868i 2.82413 + 0.497969i −0.868231 − 1.03472i −2.62808 − 0.305313i −2.54486 − 1.46928i −0.939693 − 0.342020i −2.96725 + 2.48982i
40.5 −0.763835 + 0.910304i −0.173648 + 0.984808i 0.102088 + 0.578971i −3.42867 − 0.604567i −0.763835 − 0.910304i −2.05773 + 1.66305i −2.66324 − 1.53762i −0.939693 − 0.342020i 3.16928 − 2.65934i
40.6 −0.376037 + 0.448144i −0.173648 + 0.984808i 0.287868 + 1.63258i −2.56641 − 0.452528i −0.376037 − 0.448144i 0.436486 − 2.60950i −1.85315 − 1.06991i −0.939693 − 0.342020i 1.16786 − 0.979954i
40.7 −0.137026 + 0.163301i −0.173648 + 0.984808i 0.339405 + 1.92486i 2.98443 + 0.526235i −0.137026 − 0.163301i −0.181283 + 2.63953i −0.730069 − 0.421505i −0.939693 − 0.342020i −0.494879 + 0.415253i
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.bf even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 399.2.ck.a yes 78
7.d odd 6 1 399.2.cd.a ✓ 78
19.f odd 18 1 399.2.cd.a ✓ 78
133.bf even 18 1 inner 399.2.ck.a yes 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.cd.a ✓ 78 7.d odd 6 1
399.2.cd.a ✓ 78 19.f odd 18 1
399.2.ck.a yes 78 1.a even 1 1 trivial
399.2.ck.a yes 78 133.bf even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{78} + 39 T_{2}^{73} - 529 T_{2}^{72} - 87 T_{2}^{71} - 342 T_{2}^{70} + 3873 T_{2}^{68} + \cdots + 623808 \) acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\). Copy content Toggle raw display