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Properties

Label 1000.2.t.a
Level $1000$
Weight $2$
Character orbit 1000.t
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(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: \(112\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 3 q^{2} - 3 q^{4} - 7 q^{6} + 16 q^{7} + 12 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 3 q^{2} - 3 q^{4} - 7 q^{6} + 16 q^{7} + 12 q^{8} + 18 q^{9} + q^{12} - 15 q^{14} + 9 q^{16} + 10 q^{17} + 24 q^{18} - 16 q^{22} + 6 q^{23} - 28 q^{24} - 2 q^{26} + 37 q^{28} + 6 q^{31} + 18 q^{32} + 18 q^{33} + 17 q^{34} - 31 q^{36} + 5 q^{38} - 34 q^{39} - 2 q^{41} - 19 q^{42} - 32 q^{44} - 19 q^{46} + 30 q^{47} + 46 q^{48} + 48 q^{49} + 53 q^{54} - 42 q^{56} + 28 q^{57} - 66 q^{58} - 4 q^{62} + 60 q^{63} - 36 q^{64} - 70 q^{66} + 34 q^{68} - 34 q^{71} - 77 q^{72} + 26 q^{73} - 4 q^{74} + 4 q^{76} - 99 q^{78} + 14 q^{79} - 30 q^{81} - 38 q^{82} + 24 q^{84} + 53 q^{86} - 38 q^{87} - 16 q^{88} + 24 q^{89} - 36 q^{92} + 73 q^{94} - 72 q^{96} + 34 q^{97} - 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.40431 − 0.167096i 1.75086 − 2.40986i 1.94416 + 0.469307i 0 −2.86143 + 3.09162i 3.41885 −2.65178 − 0.983912i −1.81484 − 5.58549i 0
101.2 −1.39044 + 0.258214i −0.910283 + 1.25290i 1.86665 − 0.718062i 0 0.942180 − 1.97713i 1.01594 −2.41005 + 1.48042i 0.185915 + 0.572186i 0
101.3 −1.37198 − 0.343029i −1.07287 + 1.47668i 1.76466 + 0.941259i 0 1.97850 − 1.65795i 2.31401 −2.09820 − 1.89672i −0.102477 − 0.315391i 0
101.4 −1.31976 − 0.508170i 1.00642 − 1.38522i 1.48353 + 1.34132i 0 −2.03216 + 1.31672i −0.719899 −1.27628 − 2.52411i 0.0211043 + 0.0649523i 0
101.5 −1.14927 + 0.824121i 1.13822 − 1.56662i 0.641649 − 1.89428i 0 −0.0170347 + 2.73850i −3.40673 0.823685 + 2.70584i −0.231711 − 0.713133i 0
101.6 −1.03905 + 0.959357i −0.239455 + 0.329581i 0.159268 − 1.99365i 0 −0.0673796 − 0.572176i 2.90112 1.74713 + 2.22430i 0.875766 + 2.69533i 0
101.7 −0.990232 − 1.00967i 0.260015 − 0.357881i −0.0388822 + 1.99962i 0 −0.618818 + 0.0918539i −1.14468 2.05747 − 1.94083i 0.866581 + 2.66706i 0
101.8 −0.809643 + 1.15952i −0.801761 + 1.10353i −0.688956 − 1.87759i 0 −0.630420 − 1.82312i −1.98026 2.73490 + 0.721321i 0.352095 + 1.08364i 0
101.9 −0.774941 − 1.18299i −1.68097 + 2.31366i −0.798934 + 1.83350i 0 4.03970 + 0.195626i −4.93157 2.78813 − 0.475718i −1.60031 − 4.92524i 0
101.10 −0.633630 − 1.26432i −0.289798 + 0.398873i −1.19703 + 1.60223i 0 0.687929 + 0.113661i 4.76281 2.78420 + 0.498210i 0.851934 + 2.62198i 0
101.11 −0.611345 + 1.27525i 1.91233 − 2.63210i −1.25251 − 1.55923i 0 2.18748 + 4.04782i 1.36581 2.75413 − 0.644035i −2.34388 − 7.21373i 0
101.12 −0.486021 − 1.32808i 1.35783 − 1.86889i −1.52757 + 1.29094i 0 −3.14196 − 0.894979i −0.146324 2.45690 + 1.40130i −0.722003 − 2.22210i 0
101.13 −0.254983 + 1.39104i −1.91233 + 2.63210i −1.86997 − 0.709382i 0 −3.17374 − 3.33127i 1.36581 1.46359 − 2.42031i −2.34388 − 7.21373i 0
101.14 −0.0265316 + 1.41396i 0.801761 − 1.10353i −1.99859 − 0.0750294i 0 1.53908 + 1.16294i −1.98026 0.159115 − 2.82395i 0.352095 + 1.08364i 0
101.15 0.0758714 − 1.41218i −1.52910 + 2.10462i −1.98849 − 0.214288i 0 2.85609 + 2.31904i 1.53499 −0.453481 + 2.79184i −1.16425 − 3.58319i 0
101.16 0.145252 − 1.40673i 0.0135478 − 0.0186469i −1.95780 − 0.408663i 0 −0.0242634 − 0.0217667i −2.98406 −0.859257 + 2.69475i 0.926887 + 2.85266i 0
101.17 0.276717 + 1.38688i 0.239455 − 0.329581i −1.84686 + 0.767544i 0 0.523350 + 0.240894i 2.90112 −1.57554 − 2.34897i 0.875766 + 2.69533i 0
101.18 0.445374 + 1.34225i −1.13822 + 1.56662i −1.60328 + 1.19561i 0 −2.60973 − 0.830042i −3.40673 −2.31887 − 1.61952i −0.231711 − 0.713133i 0
101.19 0.709346 − 1.22345i −0.0135478 + 0.0186469i −0.993656 − 1.73570i 0 0.0132035 + 0.0298022i −2.98406 −2.82838 − 0.0155221i 0.926887 + 2.85266i 0
101.20 0.768676 − 1.18707i 1.52910 − 2.10462i −0.818276 − 1.82495i 0 −1.32296 − 3.43292i 1.53499 −2.79533 − 0.431439i −1.16425 − 3.58319i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.28
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} - 51 T_{3}^{110} + 1497 T_{3}^{108} - 33193 T_{3}^{106} + 616897 T_{3}^{104} + \cdots + 959512576 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display