Properties

Label 1000.2.t.b
Level $1000$
Weight $2$
Character orbit 1000.t
Analytic conductor $7.985$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(101,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.t (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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 6 q^{4} + 2 q^{6} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 6 q^{4} + 2 q^{6} + 60 q^{9} + 6 q^{14} - 30 q^{16} + 32 q^{24} - 28 q^{26} - 36 q^{31} - 18 q^{34} + 82 q^{36} + 20 q^{39} - 20 q^{41} + 64 q^{44} + 26 q^{46} + 160 q^{49} - 86 q^{54} + 72 q^{56} + 72 q^{64} + 80 q^{66} + 44 q^{71} - 8 q^{74} - 72 q^{76} - 28 q^{79} - 12 q^{81} - 156 q^{84} - 118 q^{86} - 48 q^{89} - 90 q^{94} + 92 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.41390 0.0299570i 0.770982 1.06117i 1.99821 + 0.0847122i 0 −1.12188 + 1.47728i −2.31589 −2.82272 0.179635i 0.395392 + 1.21689i 0
101.2 −1.40768 0.135750i −1.84742 + 2.54275i 1.96314 + 0.382186i 0 2.94576 3.32861i −1.17589 −2.71160 0.804493i −2.12559 6.54190i 0
101.3 −1.40756 + 0.137018i −0.793983 + 1.09282i 1.96245 0.385721i 0 0.967842 1.64700i 0.296885 −2.70942 + 0.811817i 0.363197 + 1.11780i 0
101.4 −1.39393 + 0.238643i 1.01126 1.39188i 1.88610 0.665306i 0 −1.07746 + 2.18152i 4.42380 −2.47032 + 1.37750i 0.0123697 + 0.0380700i 0
101.5 −1.37883 0.314378i −0.0718391 + 0.0988780i 1.80233 + 0.866946i 0 0.130139 0.113751i −4.12326 −2.21256 1.76198i 0.922435 + 2.83896i 0
101.6 −1.37312 + 0.338435i 1.87526 2.58107i 1.77092 0.929423i 0 −1.70143 + 4.17877i −2.97769 −2.11714 + 1.87555i −2.21828 6.82716i 0
101.7 −1.31025 + 0.532208i −0.171463 + 0.235999i 1.43351 1.39465i 0 0.0990593 0.400472i 0.234809 −1.13601 + 2.59027i 0.900755 + 2.77224i 0
101.8 −1.30028 0.556118i −0.0718391 + 0.0988780i 1.38147 + 1.44622i 0 0.148399 0.0886183i 4.12326 −0.992028 2.64875i 0.922435 + 2.83896i 0
101.9 −1.25131 + 0.658957i −1.48079 + 2.03813i 1.13155 1.64912i 0 0.509884 3.52610i −3.58786 −0.329223 + 2.80920i −1.03418 3.18289i 0
101.10 −1.21863 0.717591i −1.84742 + 2.54275i 0.970125 + 1.74896i 0 4.07598 1.77299i 1.17589 0.0728129 2.82749i −2.12559 6.54190i 0
101.11 −1.16147 0.806832i 0.770982 1.06117i 0.698045 + 1.87423i 0 −1.75166 + 0.610464i 2.31589 0.701425 2.74007i 0.395392 + 1.21689i 0
101.12 −1.10073 + 0.887910i 0.604350 0.831817i 0.423232 1.95471i 0 0.0733494 + 1.45222i 1.74290 1.26974 + 2.52740i 0.600371 + 1.84775i 0
101.13 −1.08516 + 0.906879i 1.50838 2.07611i 0.355140 1.96822i 0 0.245946 + 3.62082i 2.17256 1.39955 + 2.45790i −1.10796 3.40994i 0
101.14 −1.08405 + 0.908199i −1.46605 + 2.01785i 0.350349 1.96907i 0 −0.243326 3.51892i 0.110917 1.40851 + 2.45277i −0.995348 3.06337i 0
101.15 −1.05820 0.938193i −0.793983 + 1.09282i 0.239588 + 1.98560i 0 1.86547 0.411520i −0.296885 1.60934 2.32595i 0.363197 + 1.11780i 0
101.16 −0.987445 1.01240i 1.01126 1.39188i −0.0499066 + 1.99938i 0 −2.40770 + 0.350605i −4.42380 2.07345 1.92375i 0.0123697 + 0.0380700i 0
101.17 −0.911952 1.08090i 1.87526 2.58107i −0.336688 + 1.97146i 0 −4.50002 + 0.326846i 2.97769 2.43799 1.43395i −2.21828 6.82716i 0
101.18 −0.747190 1.20071i −0.171463 + 0.235999i −0.883414 + 1.79432i 0 0.411483 + 0.0295418i −0.234809 2.81454 0.279973i 0.900755 + 2.77224i 0
101.19 −0.747170 + 1.20072i 0.893213 1.22940i −0.883474 1.79429i 0 0.808790 + 1.99107i −3.99458 2.81455 + 0.279830i 0.213451 + 0.656936i 0
101.20 −0.625006 1.26861i −1.48079 + 2.03813i −1.21874 + 1.58578i 0 3.51109 + 0.604697i 3.58786 2.77344 + 0.554981i −1.03418 3.18289i 0
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner
40.f even 2 1 inner
200.o even 10 1 inner
200.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.t.b 224
5.b even 2 1 inner 1000.2.t.b 224
5.c odd 4 1 200.2.o.a 112
5.c odd 4 1 1000.2.o.a 112
8.b even 2 1 inner 1000.2.t.b 224
20.e even 4 1 800.2.be.a 112
25.d even 5 1 inner 1000.2.t.b 224
25.e even 10 1 inner 1000.2.t.b 224
25.f odd 20 1 200.2.o.a 112
25.f odd 20 1 1000.2.o.a 112
40.f even 2 1 inner 1000.2.t.b 224
40.i odd 4 1 200.2.o.a 112
40.i odd 4 1 1000.2.o.a 112
40.k even 4 1 800.2.be.a 112
100.l even 20 1 800.2.be.a 112
200.o even 10 1 inner 1000.2.t.b 224
200.t even 10 1 inner 1000.2.t.b 224
200.v even 20 1 800.2.be.a 112
200.x odd 20 1 200.2.o.a 112
200.x odd 20 1 1000.2.o.a 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.o.a 112 5.c odd 4 1
200.2.o.a 112 25.f odd 20 1
200.2.o.a 112 40.i odd 4 1
200.2.o.a 112 200.x odd 20 1
800.2.be.a 112 20.e even 4 1
800.2.be.a 112 40.k even 4 1
800.2.be.a 112 100.l even 20 1
800.2.be.a 112 200.v even 20 1
1000.2.o.a 112 5.c odd 4 1
1000.2.o.a 112 25.f odd 20 1
1000.2.o.a 112 40.i odd 4 1
1000.2.o.a 112 200.x odd 20 1
1000.2.t.b 224 1.a even 1 1 trivial
1000.2.t.b 224 5.b even 2 1 inner
1000.2.t.b 224 8.b even 2 1 inner
1000.2.t.b 224 25.d even 5 1 inner
1000.2.t.b 224 25.e even 10 1 inner
1000.2.t.b 224 40.f even 2 1 inner
1000.2.t.b 224 200.o even 10 1 inner
1000.2.t.b 224 200.t even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} - 57 T_{3}^{110} + 1797 T_{3}^{108} - 41631 T_{3}^{106} + 794837 T_{3}^{104} - 13058078 T_{3}^{102} + 188634219 T_{3}^{100} - 2435797056 T_{3}^{98} + 28488245305 T_{3}^{96} + \cdots + 70338682617856 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display