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Properties

Label 968.2.q.a
Level $968$
Weight $2$
Character orbit 968.q
Analytic conductor $7.730$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(89,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.q (of order \(11\), degree \(10\), 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: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{3} + 3 q^{5} + 2 q^{7} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{3} + 3 q^{5} + 2 q^{7} + 144 q^{9} + q^{11} - q^{13} - 6 q^{15} + 8 q^{17} - 5 q^{19} - 4 q^{21} - 35 q^{23} - 19 q^{25} + 34 q^{27} + 9 q^{29} + 50 q^{31} - 3 q^{33} - 10 q^{35} + 32 q^{37} - 95 q^{39} - 6 q^{41} - 3 q^{43} + 65 q^{45} + 21 q^{47} - 14 q^{49} + 26 q^{51} + q^{53} + 63 q^{55} - 14 q^{57} + 34 q^{59} - 32 q^{61} + 10 q^{63} - 64 q^{65} + q^{67} + 4 q^{69} - 59 q^{71} + 28 q^{73} + 11 q^{75} + 20 q^{77} + 31 q^{79} + 104 q^{81} + q^{83} - 16 q^{85} + 51 q^{87} + 54 q^{89} + 40 q^{91} - 22 q^{93} - 21 q^{95} - 36 q^{97} + 83 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 0 −3.12596 0 1.55728 + 3.40997i 0 3.49187 − 2.24409i 0 6.77164 0
89.2 0 −2.77374 0 −1.33499 − 2.92322i 0 −1.07461 + 0.690609i 0 4.69362 0
89.3 0 −2.15195 0 0.430924 + 0.943593i 0 −2.82025 + 1.81247i 0 1.63089 0
89.4 0 −1.82346 0 −0.501378 − 1.09787i 0 4.07724 − 2.62028i 0 0.325007 0
89.5 0 −1.78143 0 −1.14323 − 2.50333i 0 1.25386 − 0.805806i 0 0.173510 0
89.6 0 −1.54404 0 0.897061 + 1.96429i 0 −2.81967 + 1.81209i 0 −0.615929 0
89.7 0 −0.919162 0 1.12537 + 2.46422i 0 0.805194 − 0.517467i 0 −2.15514 0
89.8 0 0.00335716 0 −1.15442 − 2.52784i 0 −2.47255 + 1.58901i 0 −2.99999 0
89.9 0 0.422850 0 0.402236 + 0.880774i 0 0.804292 − 0.516887i 0 −2.82120 0
89.10 0 0.589185 0 −0.356472 − 0.780565i 0 1.61709 − 1.03924i 0 −2.65286 0
89.11 0 1.08937 0 −0.789179 − 1.72806i 0 −1.68913 + 1.08554i 0 −1.81327 0
89.12 0 2.11359 0 1.18261 + 2.58955i 0 −3.07168 + 1.97405i 0 1.46725 0
89.13 0 2.18720 0 1.77057 + 3.87701i 0 0.676491 − 0.434755i 0 1.78386 0
89.14 0 2.39908 0 −0.491823 − 1.07694i 0 2.35987 − 1.51659i 0 2.75559 0
89.15 0 2.92661 0 0.506339 + 1.10873i 0 1.31057 − 0.842250i 0 5.56503 0
89.16 0 3.30749 0 −1.71913 − 3.76436i 0 −2.33034 + 1.49762i 0 7.93952 0
177.1 0 −3.28476 0 −1.61124 + 1.85947i 0 −1.70970 + 3.74371i 0 7.78967 0
177.2 0 −2.76994 0 0.503901 − 0.581533i 0 −0.142278 + 0.311545i 0 4.67257 0
177.3 0 −2.54322 0 1.12706 − 1.30070i 0 1.64410 − 3.60007i 0 3.46796 0
177.4 0 −2.13452 0 −1.18617 + 1.36892i 0 0.626091 − 1.37095i 0 1.55616 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.q.a ✓ 160
121.e even 11 1 inner 968.2.q.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.q.a ✓ 160 1.a even 1 1 trivial
968.2.q.a ✓ 160 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} + 4 T_{3}^{79} - 148 T_{3}^{78} - 611 T_{3}^{77} + 10405 T_{3}^{76} + 44520 T_{3}^{75} + \cdots + 442821632 \) acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\). Copy content Toggle raw display