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Properties

Label 968.2.o.i
Level $968$
Weight $2$
Character orbit 968.o
Analytic conductor $7.730$
Analytic rank $0$
Dimension $40$
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(245,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.245");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.o (of order \(10\), degree \(4\), not 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: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 88)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 5 q^{2} - q^{4} + 7 q^{6} + 10 q^{7} + 5 q^{8} + 20 q^{10} - 6 q^{12} + 2 q^{14} - 18 q^{15} + 15 q^{16} + 6 q^{17} + 20 q^{18} + 8 q^{20} - 8 q^{23} - 25 q^{24} - 4 q^{25} - 10 q^{26} - 32 q^{28}+ \cdots + 144 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −1.37413 + 0.334310i −1.22337 + 0.397498i 1.77647 − 0.918772i 0.929834 − 1.27981i 1.54819 − 0.955200i 0.577320 − 1.77681i −2.13395 + 1.85641i −1.08841 + 0.790780i −0.849862 + 2.06948i
245.2 −1.21055 − 0.731134i −0.582243 + 0.189182i 0.930886 + 1.77016i −0.858340 + 1.18140i 0.843155 + 0.196682i −1.36837 + 4.21140i 0.167332 − 2.82347i −2.12383 + 1.54306i 1.90283 − 0.802592i
245.3 −0.742578 + 1.20357i 1.22337 − 0.397498i −0.897156 − 1.78749i −0.929834 + 1.27981i −0.430033 + 1.76759i 0.577320 − 1.77681i 2.81757 + 0.247559i −1.08841 + 0.790780i −0.849862 − 2.06948i
245.4 −0.210425 − 1.39847i −2.32812 + 0.756451i −1.91144 + 0.588548i −0.117836 + 0.162187i 1.54777 + 3.09663i 0.725001 − 2.23132i 1.22528 + 2.54925i 2.42086 − 1.75886i 0.251609 + 0.130662i
245.5 0.121361 − 1.40900i 2.62285 − 0.852217i −1.97054 − 0.341993i −1.70795 + 2.35079i −0.882460 − 3.79902i −0.730368 + 2.24784i −0.721013 + 2.73498i 3.72604 − 2.70713i 3.10498 + 2.69179i
245.6 0.321268 + 1.37724i 0.582243 − 0.189182i −1.79357 + 0.884925i 0.858340 − 1.18140i 0.447605 + 0.741110i −1.36837 + 4.21140i −1.79497 − 2.18588i −2.12383 + 1.54306i 1.90283 + 0.802592i
245.7 1.00414 − 0.995839i 0.826127 − 0.268425i 0.0166095 − 1.99993i 2.15963 − 2.97247i 0.562242 − 1.09223i 0.369362 − 1.13678i −1.97493 − 2.02476i −1.81662 + 1.31985i −0.791527 − 5.13543i
245.8 1.25740 − 0.647266i −0.826127 + 0.268425i 1.16209 − 1.62774i −2.15963 + 2.97247i −0.865027 + 0.872241i 0.369362 − 1.13678i 0.407629 − 2.79890i −1.81662 + 1.31985i −0.791527 + 5.13543i
245.9 1.26500 + 0.632278i 2.32812 − 0.756451i 1.20045 + 1.59966i 0.117836 − 0.162187i 3.42336 + 0.515106i 0.725001 − 2.23132i 0.507138 + 2.78259i 2.42086 − 1.75886i 0.251609 − 0.130662i
245.10 1.37754 + 0.319983i −2.62285 + 0.852217i 1.79522 + 0.881578i 1.70795 − 2.35079i −3.88578 + 0.334692i −0.730368 + 2.24784i 2.19090 + 1.78885i 3.72604 − 2.70713i 3.10498 − 2.69179i
269.1 −1.40939 + 0.116729i 0.317655 − 0.437215i 1.97275 − 0.329033i −2.75892 − 0.896426i −0.396664 + 0.653285i 2.10709 − 1.53089i −2.74196 + 0.694011i 0.836799 + 2.57540i 3.99302 + 0.941367i
269.2 −1.19831 + 0.751039i −1.67918 + 2.31119i 0.871881 − 1.79995i 2.48027 + 0.805887i 0.276378 − 4.03065i 0.158860 − 0.115419i 0.307053 + 2.81171i −1.59492 − 4.90866i −3.57737 + 0.897076i
269.3 −0.725319 − 1.21405i 1.10501 − 1.52092i −0.947824 + 1.76114i −0.468972 − 0.152378i −2.64796 − 0.238387i 0.141392 − 0.102727i 2.82559 − 0.126687i −0.165095 − 0.508111i 0.155160 + 0.679877i
269.4 −0.548639 + 1.30345i 0.188809 − 0.259874i −1.39799 − 1.43025i −1.22432 − 0.397805i 0.235146 + 0.388681i −2.67516 + 1.94362i 2.63126 − 1.03752i 0.895166 + 2.75504i 1.19023 − 1.37759i
269.5 −0.126804 − 1.40852i −1.10501 + 1.52092i −1.96784 + 0.357211i 0.468972 + 0.152378i 2.28236 + 1.36357i 0.141392 − 0.102727i 0.752668 + 2.72644i −0.165095 − 0.508111i 0.155160 − 0.679877i
269.6 0.301096 + 1.38179i 1.50170 − 2.06692i −1.81868 + 0.832103i 2.56349 + 0.832928i 3.30820 + 1.45270i 3.19487 − 2.32121i −1.69739 − 2.26249i −1.08998 − 3.35462i −0.379074 + 3.79299i
269.7 0.568603 + 1.29487i −1.50170 + 2.06692i −1.35338 + 1.47254i −2.56349 − 0.832928i −3.53026 − 0.769256i 3.19487 − 2.32121i −2.67628 − 0.915165i −1.08998 − 3.35462i −0.379074 − 3.79299i
269.8 1.20883 − 0.733982i −0.317655 + 0.437215i 0.922541 − 1.77452i 2.75892 + 0.896426i −0.0630835 + 0.761672i 2.10709 − 1.53089i −0.187268 − 2.82222i 0.836799 + 2.57540i 3.99302 − 0.941367i
269.9 1.21001 + 0.732035i −0.188809 + 0.259874i 0.928249 + 1.77154i 1.22432 + 0.397805i −0.418698 + 0.176235i −2.67516 + 1.94362i −0.173638 + 2.82309i 0.895166 + 2.75504i 1.19023 + 1.37759i
269.10 1.41090 − 0.0967440i 1.67918 − 2.31119i 1.98128 − 0.272992i −2.48027 − 0.805887i 2.14556 − 3.42332i 0.158860 − 0.115419i 2.76898 − 0.576842i −1.59492 − 4.90866i −3.57737 − 0.897076i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
11.c even 5 1 inner
88.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.o.i 40
8.b even 2 1 inner 968.2.o.i 40
11.b odd 2 1 88.2.o.a ✓ 40
11.c even 5 1 968.2.c.i 20
11.c even 5 2 968.2.o.d 40
11.c even 5 1 inner 968.2.o.i 40
11.d odd 10 1 88.2.o.a ✓ 40
11.d odd 10 1 968.2.c.h 20
11.d odd 10 2 968.2.o.j 40
33.d even 2 1 792.2.br.b 40
33.f even 10 1 792.2.br.b 40
44.c even 2 1 352.2.w.a 40
44.g even 10 1 352.2.w.a 40
44.g even 10 1 3872.2.c.h 20
44.h odd 10 1 3872.2.c.i 20
88.b odd 2 1 88.2.o.a ✓ 40
88.g even 2 1 352.2.w.a 40
88.k even 10 1 352.2.w.a 40
88.k even 10 1 3872.2.c.h 20
88.l odd 10 1 3872.2.c.i 20
88.o even 10 1 968.2.c.i 20
88.o even 10 2 968.2.o.d 40
88.o even 10 1 inner 968.2.o.i 40
88.p odd 10 1 88.2.o.a ✓ 40
88.p odd 10 1 968.2.c.h 20
88.p odd 10 2 968.2.o.j 40
264.m even 2 1 792.2.br.b 40
264.u even 10 1 792.2.br.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.o.a ✓ 40 11.b odd 2 1
88.2.o.a ✓ 40 11.d odd 10 1
88.2.o.a ✓ 40 88.b odd 2 1
88.2.o.a ✓ 40 88.p odd 10 1
352.2.w.a 40 44.c even 2 1
352.2.w.a 40 44.g even 10 1
352.2.w.a 40 88.g even 2 1
352.2.w.a 40 88.k even 10 1
792.2.br.b 40 33.d even 2 1
792.2.br.b 40 33.f even 10 1
792.2.br.b 40 264.m even 2 1
792.2.br.b 40 264.u even 10 1
968.2.c.h 20 11.d odd 10 1
968.2.c.h 20 88.p odd 10 1
968.2.c.i 20 11.c even 5 1
968.2.c.i 20 88.o even 10 1
968.2.o.d 40 11.c even 5 2
968.2.o.d 40 88.o even 10 2
968.2.o.i 40 1.a even 1 1 trivial
968.2.o.i 40 8.b even 2 1 inner
968.2.o.i 40 11.c even 5 1 inner
968.2.o.i 40 88.o even 10 1 inner
968.2.o.j 40 11.d odd 10 2
968.2.o.j 40 88.p odd 10 2
3872.2.c.h 20 44.g even 10 1
3872.2.c.h 20 88.k even 10 1
3872.2.c.i 20 44.h odd 10 1
3872.2.c.i 20 88.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{40} - 15 T_{3}^{38} + 181 T_{3}^{36} - 1921 T_{3}^{34} + 18015 T_{3}^{32} - 120568 T_{3}^{30} + \cdots + 14641 \) Copy content Toggle raw display
\( T_{5}^{40} - 23 T_{5}^{38} + 425 T_{5}^{36} - 7117 T_{5}^{34} + 97237 T_{5}^{32} - 998258 T_{5}^{30} + \cdots + 16777216 \) Copy content Toggle raw display
\( T_{7}^{20} - 5 T_{7}^{19} + 31 T_{7}^{18} - 147 T_{7}^{17} + 691 T_{7}^{16} - 1836 T_{7}^{15} + \cdots + 4096 \) Copy content Toggle raw display