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Properties

Label 616.2.bs.a
Level $616$
Weight $2$
Character orbit 616.bs
Analytic conductor $4.919$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [616,2,Mod(41,616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(616, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("616.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.bs (of order \(10\), 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: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{9} + 4 q^{11} + 12 q^{15} - 24 q^{23} + 36 q^{25} + 20 q^{29} - 30 q^{35} + 16 q^{37} - 60 q^{39} - 30 q^{49} + 60 q^{51} - 80 q^{57} + 40 q^{63} + 64 q^{67} - 24 q^{71} + 50 q^{77} - 8 q^{81}+ \cdots - 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −3.27750 + 1.06492i 0 −0.994198 + 1.36840i 0 1.65506 − 2.06417i 0 7.18090 − 5.21723i 0
41.2 0 −2.71747 + 0.882959i 0 0.894789 − 1.23157i 0 −2.11752 + 1.58622i 0 4.17797 − 3.03547i 0
41.3 0 −2.67340 + 0.868639i 0 −1.17661 + 1.61946i 0 −2.63412 − 0.247838i 0 3.96546 − 2.88107i 0
41.4 0 −2.31908 + 0.753515i 0 2.02191 − 2.78292i 0 1.25317 − 2.33015i 0 2.38330 − 1.73157i 0
41.5 0 −1.82637 + 0.593422i 0 −0.261077 + 0.359341i 0 2.60311 − 0.473070i 0 0.556410 − 0.404255i 0
41.6 0 −1.74058 + 0.565549i 0 −2.00195 + 2.75545i 0 0.810582 + 2.51852i 0 0.282723 − 0.205410i 0
41.7 0 −1.52261 + 0.494727i 0 −1.49894 + 2.06311i 0 1.29160 + 2.30906i 0 −0.353456 + 0.256801i 0
41.8 0 −1.19972 + 0.389811i 0 1.72755 − 2.37777i 0 1.75602 + 1.97899i 0 −1.13969 + 0.828031i 0
41.9 0 −1.13683 + 0.369378i 0 0.421636 − 0.580332i 0 0.160066 − 2.64090i 0 −1.27111 + 0.923516i 0
41.10 0 −0.811253 + 0.263592i 0 1.07118 − 1.47435i 0 −2.00771 − 1.72311i 0 −1.83840 + 1.33568i 0
41.11 0 −0.796503 + 0.258800i 0 1.93911 − 2.66896i 0 −2.30970 + 1.29046i 0 −1.85961 + 1.35109i 0
41.12 0 −0.238236 + 0.0774075i 0 −0.542378 + 0.746520i 0 −0.962116 − 2.46462i 0 −2.37629 + 1.72647i 0
41.13 0 0.238236 − 0.0774075i 0 0.542378 − 0.746520i 0 −2.22703 + 1.42840i 0 −2.37629 + 1.72647i 0
41.14 0 0.796503 − 0.258800i 0 −1.93911 + 2.66896i 0 −1.11007 − 2.40161i 0 −1.85961 + 1.35109i 0
41.15 0 0.811253 − 0.263592i 0 −1.07118 + 1.47435i 0 −2.63709 + 0.213920i 0 −1.83840 + 1.33568i 0
41.16 0 1.13683 − 0.369378i 0 −0.421636 + 0.580332i 0 −1.42279 + 2.23062i 0 −1.27111 + 0.923516i 0
41.17 0 1.19972 − 0.389811i 0 −1.72755 + 2.37777i 0 2.58387 − 0.568876i 0 −1.13969 + 0.828031i 0
41.18 0 1.52261 − 0.494727i 0 1.49894 − 2.06311i 0 2.40216 − 1.10889i 0 −0.353456 + 0.256801i 0
41.19 0 1.74058 − 0.565549i 0 2.00195 − 2.75545i 0 2.13612 − 1.56108i 0 0.282723 − 0.205410i 0
41.20 0 1.82637 − 0.593422i 0 0.261077 − 0.359341i 0 1.82790 + 1.91279i 0 0.556410 − 0.404255i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 616.2.bs.a ✓ 96
7.b odd 2 1 inner 616.2.bs.a ✓ 96
11.d odd 10 1 inner 616.2.bs.a ✓ 96
77.l even 10 1 inner 616.2.bs.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.bs.a ✓ 96 1.a even 1 1 trivial
616.2.bs.a ✓ 96 7.b odd 2 1 inner
616.2.bs.a ✓ 96 11.d odd 10 1 inner
616.2.bs.a ✓ 96 77.l even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(616, [\chi])\).