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Properties

Label 968.2.q.b
Level $968$
Weight $2$
Character orbit 968.q
Analytic conductor $7.730$
Analytic rank $0$
Dimension $170$
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: \(170\)
Relative dimension: \(17\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 170 q + 10 q^{3} - 3 q^{5} + 2 q^{7} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 170 q + 10 q^{3} - 3 q^{5} + 2 q^{7} + 180 q^{9} + 2 q^{11} + 3 q^{13} + 10 q^{15} - 6 q^{17} + 9 q^{19} + 16 q^{21} + 23 q^{23} - 18 q^{25} - 26 q^{27} + q^{29} - 38 q^{31} + q^{33} - 10 q^{35} - 24 q^{37} + 79 q^{39} - 4 q^{41} + 3 q^{43} - 37 q^{45} - 25 q^{47} - 39 q^{49} + 9 q^{51} + 17 q^{53} + 69 q^{55} + 63 q^{57} - 28 q^{59} + 34 q^{61} + 22 q^{63} + 84 q^{65} - 19 q^{67} - 12 q^{69} + 15 q^{71} - 46 q^{73} + 23 q^{75} + 108 q^{77} + 47 q^{79} + 226 q^{81} - 9 q^{83} - 8 q^{85} - 91 q^{87} - 100 q^{89} - 16 q^{91} - 6 q^{93} + 45 q^{95} + 18 q^{97} - 74 q^{99}+O(q^{100}) \) 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 −2.92790 0 −1.58795 − 3.47712i 0 −4.12052 + 2.64810i 0 5.57262 0
89.2 0 −2.74456 0 1.17344 + 2.56948i 0 0.111432 − 0.0716129i 0 4.53259 0
89.3 0 −2.73957 0 −0.663430 − 1.45271i 0 3.33076 − 2.14055i 0 4.50527 0
89.4 0 −2.73222 0 0.309747 + 0.678251i 0 −0.883657 + 0.567892i 0 4.46504 0
89.5 0 −1.05598 0 −0.502316 − 1.09992i 0 −0.0815169 + 0.0523877i 0 −1.88492 0
89.6 0 −1.03194 0 −0.0505481 − 0.110685i 0 −3.31366 + 2.12956i 0 −1.93511 0
89.7 0 −0.999400 0 0.729872 + 1.59820i 0 2.00590 − 1.28912i 0 −2.00120 0
89.8 0 −0.168615 0 −1.69959 − 3.72159i 0 1.53704 − 0.987794i 0 −2.97157 0
89.9 0 0.354731 0 1.80265 + 3.94726i 0 −3.77453 + 2.42574i 0 −2.87417 0
89.10 0 0.828920 0 0.388590 + 0.850894i 0 3.45207 − 2.21851i 0 −2.31289 0
89.11 0 0.899266 0 0.440967 + 0.965584i 0 −1.93013 + 1.24042i 0 −2.19132 0
89.12 0 1.24375 0 1.26032 + 2.75972i 0 1.83915 − 1.18195i 0 −1.45308 0
89.13 0 1.27915 0 −1.43170 − 3.13498i 0 −1.95916 + 1.25907i 0 −1.36377 0
89.14 0 2.00219 0 −0.684988 − 1.49991i 0 −0.643255 + 0.413395i 0 1.00878 0
89.15 0 2.63121 0 −0.954536 − 2.09014i 0 4.07705 − 2.62016i 0 3.92325 0
89.16 0 3.05084 0 1.06220 + 2.32589i 0 0.403770 − 0.259487i 0 6.30765 0
89.17 0 3.11012 0 0.0255015 + 0.0558405i 0 −3.53400 + 2.27116i 0 6.67282 0
177.1 0 −3.35746 0 0.195173 − 0.225241i 0 0.0423050 − 0.0926350i 0 8.27254 0
177.2 0 −2.74058 0 2.88694 − 3.33170i 0 0.227828 − 0.498874i 0 4.51077 0
177.3 0 −2.20507 0 −2.59828 + 2.99857i 0 1.82594 − 3.99824i 0 1.86234 0
See next 80 embeddings (of 170 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.17
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.b ✓ 170
121.e even 11 1 inner 968.2.q.b ✓ 170
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.q.b ✓ 170 1.a even 1 1 trivial
968.2.q.b ✓ 170 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{85} - 5 T_{3}^{84} - 160 T_{3}^{83} + 836 T_{3}^{82} + 12163 T_{3}^{81} - 66862 T_{3}^{80} + \cdots - 289676288 \) acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\). Copy content Toggle raw display