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Properties

Label 1000.2.v.i
Level $1000$
Weight $2$
Character orbit 1000.v
Analytic conductor $7.985$
Analytic rank $0$
Dimension $208$
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(43,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.v (of order \(20\), degree \(8\), 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: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 208 q + 8 q^{2} - 4 q^{3} + 10 q^{4} - 26 q^{6} - 46 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q + 8 q^{2} - 4 q^{3} + 10 q^{4} - 26 q^{6} - 46 q^{8} + 20 q^{9} - 4 q^{11} + 32 q^{12} + 30 q^{14} + 18 q^{16} + 12 q^{17} + 20 q^{18} - 38 q^{22} - 8 q^{26} + 8 q^{27} + 40 q^{28} - 22 q^{32} - 16 q^{33} - 30 q^{34} - 38 q^{36} - 74 q^{38} - 44 q^{41} - 60 q^{42} + 8 q^{43} - 60 q^{44} + 2 q^{46} + 16 q^{48} + 8 q^{51} - 100 q^{52} + 140 q^{54} + 2 q^{56} - 92 q^{57} - 90 q^{58} + 40 q^{59} + 40 q^{62} - 20 q^{64} - 2 q^{66} + 72 q^{67} + 46 q^{68} + 150 q^{72} - 4 q^{73} - 24 q^{76} + 110 q^{78} + 16 q^{81} + 22 q^{82} - 44 q^{83} + 10 q^{84} - 26 q^{86} + 126 q^{88} + 100 q^{89} + 4 q^{91} + 110 q^{92} - 150 q^{94} - 26 q^{96} + 76 q^{97} + 34 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41134 − 0.0900914i −0.946277 + 1.85717i 1.98377 + 0.254299i 0 1.50283 − 2.53585i −2.34478 − 2.34478i −2.77686 − 0.537623i −0.790294 − 1.08775i 0
43.2 −1.40100 + 0.192882i 1.21293 − 2.38052i 1.92559 − 0.540456i 0 −1.24016 + 3.56906i −3.29288 − 3.29288i −2.59351 + 1.12859i −2.43230 − 3.34777i 0
43.3 −1.39203 + 0.249490i 1.21293 − 2.38052i 1.87551 − 0.694597i 0 −1.09453 + 3.61637i 3.29288 + 3.29288i −2.43747 + 1.43482i −2.43230 − 3.34777i 0
43.4 −1.37771 − 0.319257i −0.175452 + 0.344344i 1.79615 + 0.879685i 0 0.351656 − 0.418391i 1.98413 + 1.98413i −2.19372 − 1.78538i 1.67557 + 2.30622i 0
43.5 −1.31443 + 0.521810i −0.946277 + 1.85717i 1.45543 − 1.37176i 0 0.274718 − 2.93489i 2.34478 + 2.34478i −1.19725 + 2.56254i −0.790294 − 1.08775i 0
43.6 −1.21162 + 0.729366i −0.175452 + 0.344344i 0.936050 − 1.76743i 0 −0.0385716 − 0.545184i −1.98413 − 1.98413i 0.154967 + 2.82418i 1.67557 + 2.30622i 0
43.7 −1.04689 − 0.950803i −1.35709 + 2.66344i 0.191948 + 1.99077i 0 3.95313 − 1.49800i −2.33956 − 2.33956i 1.69188 − 2.26661i −3.48887 − 4.80202i 0
43.8 −0.701835 + 1.22777i −1.35709 + 2.66344i −1.01485 − 1.72339i 0 −2.31765 − 3.53550i 2.33956 + 2.33956i 2.82819 − 0.0364771i −3.48887 − 4.80202i 0
43.9 −0.576765 − 1.29126i 0.202676 − 0.397774i −1.33468 + 1.48950i 0 −0.630525 − 0.0322845i 0.724575 + 0.724575i 2.69313 + 0.864326i 1.64621 + 2.26581i 0
43.10 −0.451717 − 1.34013i −0.585698 + 1.14950i −1.59190 + 1.21072i 0 1.80505 + 0.265665i 1.03289 + 1.03289i 2.34161 + 1.58646i 0.785054 + 1.08053i 0
43.11 −0.149516 + 1.40629i 0.202676 − 0.397774i −1.95529 − 0.420525i 0 0.529082 + 0.344495i −0.724575 − 0.724575i 0.883727 − 2.68682i 1.64621 + 2.26581i 0
43.12 −0.0154853 + 1.41413i −0.585698 + 1.14950i −1.99952 − 0.0437964i 0 −1.61647 − 0.846053i −1.03289 − 1.03289i 0.0928969 − 2.82690i 0.785054 + 1.08053i 0
43.13 0.0870490 − 1.41153i −0.778908 + 1.52869i −1.98484 − 0.245745i 0 2.09000 + 1.23252i −1.70575 − 1.70575i −0.519656 + 2.78028i 0.0331518 + 0.0456295i 0
43.14 0.388899 − 1.35969i 1.26923 − 2.49100i −1.69751 − 1.05756i 0 −2.89338 − 2.69450i 0.761148 + 0.761148i −2.09812 + 1.89681i −2.83077 − 3.89622i 0
43.15 0.499091 − 1.32322i 0.525223 − 1.03081i −1.50182 − 1.32081i 0 −1.10185 − 1.20945i 2.75302 + 2.75302i −2.49727 + 1.32803i 0.976649 + 1.34424i 0
43.16 0.518976 + 1.31555i −0.778908 + 1.52869i −1.46133 + 1.36547i 0 −2.41530 − 0.231335i 1.70575 + 1.70575i −2.55474 − 1.21380i 0.0331518 + 0.0456295i 0
43.17 0.790033 + 1.17297i 1.26923 − 2.49100i −0.751697 + 1.85336i 0 3.92458 − 0.479210i −0.761148 − 0.761148i −2.76780 + 0.582501i −2.83077 − 3.89622i 0
43.18 0.883561 + 1.10423i 0.525223 − 1.03081i −0.438642 + 1.95131i 0 1.60231 − 0.330815i −2.75302 − 2.75302i −2.54225 + 1.23974i 0.976649 + 1.34424i 0
43.19 0.941284 − 1.05545i −0.124160 + 0.243677i −0.227968 − 1.98697i 0 0.140320 + 0.360414i −3.34252 − 3.34252i −2.31173 − 1.62969i 1.71939 + 2.36654i 0
43.20 1.03432 − 0.964455i −1.42165 + 2.79015i 0.139652 − 1.99512i 0 1.22052 + 4.25704i 0.509777 + 0.509777i −1.77976 − 2.19829i −4.00048 − 5.50618i 0
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
25.f odd 20 1 inner
200.v even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.v.i 208
5.b even 2 1 1000.2.v.g 208
5.c odd 4 1 200.2.v.c ✓ 208
5.c odd 4 1 1000.2.v.h 208
8.d odd 2 1 inner 1000.2.v.i 208
20.e even 4 1 800.2.bp.c 208
25.d even 5 1 1000.2.v.h 208
25.e even 10 1 200.2.v.c ✓ 208
25.f odd 20 1 1000.2.v.g 208
25.f odd 20 1 inner 1000.2.v.i 208
40.e odd 2 1 1000.2.v.g 208
40.i odd 4 1 800.2.bp.c 208
40.k even 4 1 200.2.v.c ✓ 208
40.k even 4 1 1000.2.v.h 208
100.h odd 10 1 800.2.bp.c 208
200.n odd 10 1 1000.2.v.h 208
200.o even 10 1 800.2.bp.c 208
200.s odd 10 1 200.2.v.c ✓ 208
200.v even 20 1 1000.2.v.g 208
200.v even 20 1 inner 1000.2.v.i 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.v.c ✓ 208 5.c odd 4 1
200.2.v.c ✓ 208 25.e even 10 1
200.2.v.c ✓ 208 40.k even 4 1
200.2.v.c ✓ 208 200.s odd 10 1
800.2.bp.c 208 20.e even 4 1
800.2.bp.c 208 40.i odd 4 1
800.2.bp.c 208 100.h odd 10 1
800.2.bp.c 208 200.o even 10 1
1000.2.v.g 208 5.b even 2 1
1000.2.v.g 208 25.f odd 20 1
1000.2.v.g 208 40.e odd 2 1
1000.2.v.g 208 200.v even 20 1
1000.2.v.h 208 5.c odd 4 1
1000.2.v.h 208 25.d even 5 1
1000.2.v.h 208 40.k even 4 1
1000.2.v.h 208 200.n odd 10 1
1000.2.v.i 208 1.a even 1 1 trivial
1000.2.v.i 208 8.d odd 2 1 inner
1000.2.v.i 208 25.f odd 20 1 inner
1000.2.v.i 208 200.v even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3}^{104} + 2 T_{3}^{103} - 3 T_{3}^{102} - 34 T_{3}^{101} - 189 T_{3}^{100} - 208 T_{3}^{99} + \cdots + 22278400 \) Copy content Toggle raw display
\( T_{7}^{208} + 5336 T_{7}^{204} + 13443436 T_{7}^{200} + 21300445696 T_{7}^{196} + 23851936544246 T_{7}^{192} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{208} + 10 T_{13}^{206} - 4969 T_{13}^{204} - 72140 T_{13}^{202} + 13245701 T_{13}^{200} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display