Properties

Label 1000.2.o.a
Level $1000$
Weight $2$
Character orbit 1000.o
Analytic conductor $7.985$
Analytic rank $0$
Dimension $112$
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] = [1000,2,Mod(149,1000)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 5 q^{2} - 3 q^{4} + q^{6} - 10 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 5 q^{2} - 3 q^{4} + q^{6} - 10 q^{8} - 30 q^{9} + 5 q^{12} - 3 q^{14} - 15 q^{16} + 10 q^{17} + 30 q^{22} + 10 q^{23} - 16 q^{24} - 14 q^{26} - 15 q^{28} - 18 q^{31} + 10 q^{33} + 9 q^{34} + 41 q^{36} - 45 q^{38} - 10 q^{39} - 10 q^{41} - 75 q^{42} - 32 q^{44} + 13 q^{46} + 10 q^{47} + 70 q^{48} - 80 q^{49} + 100 q^{52} + 43 q^{54} + 36 q^{56} + 30 q^{58} - 20 q^{62} - 60 q^{63} - 36 q^{64} + 40 q^{66} + 22 q^{71} + 65 q^{72} + 10 q^{73} + 4 q^{74} - 36 q^{76} + 55 q^{78} + 14 q^{79} - 6 q^{81} + 78 q^{84} - 59 q^{86} + 10 q^{87} - 110 q^{88} + 24 q^{89} - 90 q^{92} + 45 q^{94} + 46 q^{96} + 50 q^{97} - 90 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −1.37152 + 0.344862i 2.07611 + 1.50838i 1.76214 0.945971i 0 −3.36760 1.35280i 2.17256i −2.09058 + 1.90511i 1.10796 + 3.40994i 0
149.2 −1.36533 + 0.368612i 0.831817 + 0.604350i 1.72825 1.00655i 0 −1.35847 0.518519i 1.74290i −1.98860 + 2.01133i −0.600371 1.84775i 0
149.3 −1.36479 0.370599i 0.457893 + 0.332679i 1.72531 + 1.01158i 0 −0.501638 0.623732i 1.85821i −1.97980 2.02000i −0.828060 2.54851i 0
149.4 −1.32197 0.502384i −0.457893 0.332679i 1.49522 + 1.32827i 0 0.438189 + 0.669830i 1.85821i −1.30934 2.50712i −0.828060 2.54851i 0
149.5 −1.26861 + 0.625006i −2.03813 1.48079i 1.21874 1.58578i 0 3.51109 + 0.604697i 3.58786i −0.554981 + 2.77344i 1.03418 + 3.18289i 0
149.6 −0.938193 + 1.05820i −1.09282 0.793983i −0.239588 1.98560i 0 1.86547 0.411520i 0.296885i 2.32595 + 1.60934i −0.363197 1.11780i 0
149.7 −0.906879 1.08516i −2.07611 1.50838i −0.355140 + 1.96822i 0 0.245946 + 3.62082i 2.17256i 2.45790 1.39955i 1.10796 + 3.40994i 0
149.8 −0.887910 1.10073i −0.831817 0.604350i −0.423232 + 1.95471i 0 0.0733494 + 1.45222i 1.74290i 2.52740 1.26974i −0.600371 1.84775i 0
149.9 −0.806832 + 1.16147i 1.06117 + 0.770982i −0.698045 1.87423i 0 −1.75166 + 0.610464i 2.31589i 2.74007 + 0.701425i −0.395392 1.21689i 0
149.10 −0.658957 1.25131i 2.03813 + 1.48079i −1.13155 + 1.64912i 0 0.509884 3.52610i 3.58786i 2.80920 + 0.329223i 1.03418 + 3.18289i 0
149.11 −0.314378 + 1.37883i −0.0988780 0.0718391i −1.80233 0.866946i 0 0.130139 0.113751i 4.12326i 1.76198 2.21256i −0.922435 2.83896i 0
149.12 −0.137018 1.40756i 1.09282 + 0.793983i −1.96245 + 0.385721i 0 0.967842 1.64700i 0.296885i 0.811817 + 2.70942i −0.363197 1.11780i 0
149.13 −0.135750 + 1.40768i −2.54275 1.84742i −1.96314 0.382186i 0 2.94576 3.32861i 1.17589i 0.804493 2.71160i 2.12559 + 6.54190i 0
149.14 0.0299570 1.41390i −1.06117 0.770982i −1.99821 0.0847122i 0 −1.12188 + 1.47728i 2.31589i −0.179635 + 2.82272i −0.395392 1.21689i 0
149.15 0.238643 + 1.39393i 1.39188 + 1.01126i −1.88610 + 0.665306i 0 −1.07746 + 2.18152i 4.42380i −1.37750 2.47032i −0.0123697 0.0380700i 0
149.16 0.338435 + 1.37312i 2.58107 + 1.87526i −1.77092 + 0.929423i 0 −1.70143 + 4.17877i 2.97769i −1.87555 2.11714i 2.21828 + 6.82716i 0
149.17 0.532208 + 1.31025i −0.235999 0.171463i −1.43351 + 1.39465i 0 0.0990593 0.400472i 0.234809i −2.59027 1.13601i −0.900755 2.77224i 0
149.18 0.556118 1.30028i 0.0988780 + 0.0718391i −1.38147 1.44622i 0 0.148399 0.0886183i 4.12326i −2.64875 + 0.992028i −0.922435 2.83896i 0
149.19 0.717591 1.21863i 2.54275 + 1.84742i −0.970125 1.74896i 0 4.07598 1.77299i 1.17589i −2.82749 0.0728129i 2.12559 + 6.54190i 0
149.20 0.908199 + 1.08405i −2.01785 1.46605i −0.350349 + 1.96907i 0 −0.243326 3.51892i 0.110917i −2.45277 + 1.40851i 0.995348 + 3.06337i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.28
Significant digits:
Format:

Inner twists

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

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