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Properties

Label 968.2.w.b
Level $968$
Weight $2$
Character orbit 968.w
Analytic conductor $7.730$
Analytic rank $0$
Dimension $1280$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(43,968)]
 
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([11, 11, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.w (of order \(22\), 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: \(1280\)
Relative dimension: \(128\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 11 q^{2} - 36 q^{3} - 11 q^{4} - 33 q^{6} - 11 q^{8} + 1156 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 11 q^{2} - 36 q^{3} - 11 q^{4} - 33 q^{6} - 11 q^{8} + 1156 q^{9} - 11 q^{10} - 30 q^{11} - 8 q^{12} - 7 q^{14} + 17 q^{16} - 22 q^{17} - 66 q^{18} - 22 q^{19} - 7 q^{20} - 19 q^{22} - 132 q^{24} + 114 q^{25} - 31 q^{26} - 168 q^{27} - 11 q^{28} - 44 q^{30} - 11 q^{32} - 96 q^{33} - 31 q^{34} - 22 q^{35} - 82 q^{36} - 13 q^{38} - 22 q^{41} - 30 q^{42} - 22 q^{43} - 8 q^{44} - 11 q^{46} - 58 q^{48} - 150 q^{49} - 33 q^{50} - 110 q^{51} + 143 q^{52} + 22 q^{54} + 96 q^{56} - 154 q^{57} + 40 q^{58} - 30 q^{59} - 112 q^{60} - 66 q^{62} - 23 q^{64} - 22 q^{65} - 180 q^{66} - 30 q^{67} - 11 q^{68} + 37 q^{70} - 132 q^{72} - 22 q^{73} - 11 q^{74} - 104 q^{75} + 84 q^{78} + 108 q^{80} + 768 q^{81} + 7 q^{82} - 22 q^{83} - 44 q^{84} - 61 q^{86} + 93 q^{88} + 18 q^{89} - 143 q^{90} - 38 q^{91} + 45 q^{92} - 44 q^{96} - 14 q^{97} - 99 q^{98} - 254 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41360 − 0.0415774i 2.70505 1.99654 + 0.117548i 0.673360 + 1.04777i −3.82387 − 0.112469i 3.98082 + 1.16887i −2.81743 − 0.249177i 4.31731 −0.908300 − 1.50912i
43.2 −1.41281 − 0.0630277i −3.44485 1.99206 + 0.178092i 0.343474 + 0.534455i 4.86692 + 0.217121i 3.53550 + 1.03812i −2.80317 − 0.377165i 8.86701 −0.451577 − 0.776731i
43.3 −1.40899 − 0.121400i 1.28249 1.97052 + 0.342105i 0.430300 + 0.669560i −1.80703 − 0.155695i −2.24754 − 0.659938i −2.73492 − 0.721245i −1.35521 −0.525005 − 0.995644i
43.4 −1.40694 − 0.143200i −1.54105 1.95899 + 0.402949i 2.15085 + 3.34679i 2.16817 + 0.220678i −3.48890 − 1.02443i −2.69849 − 0.847454i −0.625166 −2.54687 − 5.01675i
43.5 −1.40460 − 0.164660i −0.408048 1.94577 + 0.462561i −0.373091 − 0.580541i 0.573143 + 0.0671891i −1.68281 − 0.494117i −2.65686 − 0.970101i −2.83350 0.428450 + 0.876858i
43.6 −1.40068 + 0.195151i −2.95440 1.92383 − 0.546689i 0.295078 + 0.459150i 4.13819 − 0.576553i −4.06840 − 1.19459i −2.58800 + 1.14118i 5.72850 −0.502914 − 0.585539i
43.7 −1.39824 + 0.211942i −0.188163 1.91016 − 0.592694i 0.990266 + 1.54088i 0.263098 − 0.0398798i 3.50383 + 1.02882i −2.54525 + 1.23357i −2.96459 −1.71121 − 1.94465i
43.8 −1.38443 − 0.288724i 1.27953 1.83328 + 0.799436i −1.78188 − 2.77266i −1.77142 − 0.369432i −0.359576 − 0.105581i −2.30722 − 1.63607i −1.36280 1.66635 + 4.35302i
43.9 −1.37277 − 0.339843i −1.51622 1.76901 + 0.933055i −1.59494 − 2.48177i 2.08142 + 0.515275i 4.91856 + 1.44422i −2.11136 − 1.88206i −0.701088 1.34607 + 3.94893i
43.10 −1.37221 + 0.342096i −0.588382 1.76594 − 0.938859i −1.85549 − 2.88721i 0.807386 − 0.201283i 0.881832 + 0.258929i −2.10207 + 1.89244i −2.65381 3.53384 + 3.32711i
43.11 −1.37170 − 0.344160i 1.63145 1.76311 + 0.944167i 0.295508 + 0.459819i −2.23786 − 0.561481i 2.29624 + 0.674238i −2.09351 − 1.90190i −0.338355 −0.247096 − 0.732435i
43.12 −1.36741 + 0.360821i 1.94141 1.73962 − 0.986780i 2.33676 + 3.63607i −2.65470 + 0.700502i −2.87990 − 0.845616i −2.02272 + 1.97702i 0.769076 −4.50728 − 4.12884i
43.13 −1.36220 + 0.380000i 2.63753 1.71120 − 1.03527i −1.93418 − 3.00964i −3.59285 + 1.00226i 3.35285 + 0.984484i −1.93760 + 2.06051i 3.95655 3.77841 + 3.36475i
43.14 −1.36213 + 0.380275i −0.850297 1.71078 − 1.03597i −0.343822 − 0.534997i 1.15821 − 0.323347i −1.37431 − 0.403534i −1.93635 + 2.06168i −2.27699 0.671775 + 0.597987i
43.15 −1.35029 − 0.420369i 2.73563 1.64658 + 1.13524i −1.94857 − 3.03204i −3.69390 − 1.14997i −4.44475 − 1.30510i −1.74615 − 2.22508i 4.48366 1.35657 + 4.91326i
43.16 −1.32297 + 0.499755i 2.90369 1.50049 − 1.32232i −0.508957 − 0.791953i −3.84149 + 1.45113i −2.20694 − 0.648016i −1.32426 + 2.49926i 5.43143 1.06912 + 0.793375i
43.17 −1.31805 + 0.512591i −2.16063 1.47450 − 1.35124i −0.990395 − 1.54108i 2.84782 − 1.10752i 1.20757 + 0.354573i −1.25083 + 2.53681i 1.66833 2.09533 + 1.52356i
43.18 −1.30025 + 0.556203i 0.444915 1.38128 − 1.44640i 1.52864 + 2.37861i −0.578499 + 0.247463i 2.72072 + 0.798875i −0.991508 + 2.64895i −2.80205 −3.31060 − 2.24255i
43.19 −1.30023 − 0.556231i −2.37243 1.38121 + 1.44646i −1.55004 − 2.41190i 3.08471 + 1.31962i −1.64159 − 0.482015i −0.991337 − 2.64901i 2.62841 0.673835 + 3.99822i
43.20 −1.29198 − 0.575148i −1.96086 1.33841 + 1.48616i −0.0129030 − 0.0200774i 2.53338 + 1.12778i −0.541303 − 0.158941i −0.874433 − 2.68986i 0.844963 0.00512285 + 0.0333607i
See next 80 embeddings (of 1280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.128
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
121.f odd 22 1 inner
968.w even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.w.b ✓ 1280
8.d odd 2 1 inner 968.2.w.b ✓ 1280
121.f odd 22 1 inner 968.2.w.b ✓ 1280
968.w even 22 1 inner 968.2.w.b ✓ 1280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.w.b ✓ 1280 1.a even 1 1 trivial
968.2.w.b ✓ 1280 8.d odd 2 1 inner
968.2.w.b ✓ 1280 121.f odd 22 1 inner
968.2.w.b ✓ 1280 968.w even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{320} + 9 T_{3}^{319} - 584 T_{3}^{318} - 5467 T_{3}^{317} + 168131 T_{3}^{316} + \cdots - 25\!\cdots\!68 \) acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\). Copy content Toggle raw display