↑

Properties

Label 1000.2.v.h
Level $1000$
Weight $2$
Character orbit 1000.v
Analytic conductor $7.985$
Analytic rank $0$
Dimension $208$
Inner twists $4$

Related objects

Downloads

Learn more

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.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{2} + 16 q^{3} - 10 q^{4} - 26 q^{6} + 34 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{2} + 16 q^{3} - 10 q^{4} - 26 q^{6} + 34 q^{8} - 20 q^{9} - 4 q^{11} + 2 q^{12} - 30 q^{14} + 18 q^{16} + 32 q^{17} + 20 q^{18} + 42 q^{22} - 8 q^{26} + 28 q^{27} - 30 q^{28} - 22 q^{32} - 36 q^{33} + 30 q^{34} - 38 q^{36} + 56 q^{38} - 44 q^{41} + 70 q^{42} + 8 q^{43} + 60 q^{44} + 2 q^{46} - 104 q^{48} + 8 q^{51} - 60 q^{52} - 140 q^{54} + 2 q^{56} - 92 q^{57} + 70 q^{58} - 40 q^{59} + 120 q^{62} + 20 q^{64} - 2 q^{66} + 12 q^{67} + 46 q^{68} - 60 q^{72} + 16 q^{73} - 24 q^{76} + 20 q^{78} + 16 q^{81} + 22 q^{82} + 96 q^{83} - 10 q^{84} - 26 q^{86} - 134 q^{88} - 100 q^{89} + 4 q^{91} - 70 q^{92} + 150 q^{94} - 26 q^{96} - 24 q^{97} - 206 q^{98}+O(q^{100}) \) 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 −1.41029 + 0.105290i 0.469813 − 0.922060i 1.97783 − 0.296978i 0 −0.565488 + 1.34984i 1.43598 + 1.43598i −2.75804 + 0.627071i 1.13389 + 1.56066i 0
43.2 −1.37380 + 0.335666i 0.469813 − 0.922060i 1.77466 − 0.922278i 0 −0.335925 + 1.42443i −1.43598 − 1.43598i −2.12845 + 1.86272i 1.13389 + 1.56066i 0
43.3 −1.24495 − 0.670892i −0.442138 + 0.867744i 1.09981 + 1.67046i 0 1.13260 − 0.783673i 2.37646 + 2.37646i −0.248513 − 2.81749i 1.20586 + 1.65973i 0
43.4 −1.24068 − 0.678763i 0.984272 − 1.93174i 1.07856 + 1.68425i 0 −2.53236 + 1.72858i 0.651921 + 0.651921i −0.194937 − 2.82170i −0.999483 − 1.37567i 0
43.5 −1.15383 − 0.817729i −0.969400 + 1.90256i 0.662637 + 1.88704i 0 2.67430 − 1.40251i −0.557029 − 0.557029i 0.778517 − 2.71917i −0.916623 − 1.26162i 0
43.6 −0.976702 + 1.02277i −0.442138 + 0.867744i −0.0921055 − 1.99788i 0 −0.455663 − 1.29973i −2.37646 − 2.37646i 2.13332 + 1.85713i 1.20586 + 1.65973i 0
43.7 −0.970205 + 1.02893i 0.984272 − 1.93174i −0.117405 − 1.99655i 0 1.03269 + 2.88694i −0.651921 − 0.651921i 2.16822 + 1.81626i −0.999483 − 1.37567i 0
43.8 −0.860423 − 1.12235i 1.15761 − 2.27193i −0.519346 + 1.93139i 0 −3.54593 + 0.655579i −2.32000 − 2.32000i 2.61456 − 1.07893i −2.05826 − 2.83295i 0
43.9 −0.844663 + 1.13426i −0.969400 + 1.90256i −0.573088 − 1.91613i 0 −1.33917 − 2.70657i 0.557029 + 0.557029i 2.65746 + 0.968457i −0.916623 − 1.26162i 0
43.10 −0.471485 + 1.33330i 1.15761 − 2.27193i −1.55540 − 1.25727i 0 2.48338 + 2.61462i 2.32000 + 2.32000i 2.40967 − 1.48105i −2.05826 − 2.83295i 0
43.11 −0.470539 − 1.33364i −0.453931 + 0.890890i −1.55719 + 1.25506i 0 1.40172 + 0.186182i −2.45165 − 2.45165i 2.40651 + 1.48617i 1.17572 + 1.61825i 0
43.12 −0.210653 − 1.39844i 0.965749 − 1.89539i −1.91125 + 0.589170i 0 −2.85402 − 0.951269i 2.48252 + 2.48252i 1.22653 + 2.54865i −0.896471 − 1.23389i 0
43.13 −0.129781 − 1.40825i −1.45055 + 2.84686i −1.96631 + 0.365526i 0 4.19734 + 1.67326i 0.769903 + 0.769903i 0.769940 + 2.72162i −4.23718 − 5.83198i 0
43.14 −0.0353919 + 1.41377i −0.453931 + 0.890890i −1.99749 − 0.100072i 0 −1.24345 − 0.673285i 2.45165 + 2.45165i 0.212174 − 2.82046i 1.17572 + 1.61825i 0
43.15 0.231798 + 1.39509i 0.965749 − 1.89539i −1.89254 + 0.646756i 0 2.86809 + 0.907957i −2.48252 − 2.48252i −1.34097 − 2.49034i −0.896471 − 1.23389i 0
43.16 0.311743 + 1.37943i −1.45055 + 2.84686i −1.80563 + 0.860054i 0 −4.37924 − 1.11344i −0.769903 − 0.769903i −1.74927 − 2.22262i −4.23718 − 5.83198i 0
43.17 0.425811 − 1.34859i −0.0516414 + 0.101352i −1.63737 − 1.14848i 0 0.114692 + 0.112800i −1.00143 − 1.00143i −2.24604 + 1.71910i 1.75575 + 2.41658i 0
43.18 0.821706 + 1.15100i −0.0516414 + 0.101352i −0.649598 + 1.89157i 0 −0.159090 + 0.0238423i 1.00143 + 1.00143i −2.71097 + 0.806625i 1.75575 + 2.41658i 0
43.19 0.840342 − 1.13746i 0.526974 − 1.03425i −0.587651 − 1.91172i 0 −0.733578 − 1.46853i −1.02781 − 1.02781i −2.66834 − 0.938065i 0.971395 + 1.33701i 0
43.20 1.15071 + 0.822113i 0.526974 − 1.03425i 0.648260 + 1.89202i 0 1.45666 − 0.756882i 1.02781 + 1.02781i −0.809500 + 2.71011i 0.971395 + 1.33701i 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.h 208
5.b even 2 1 200.2.v.c ✓ 208
5.c odd 4 1 1000.2.v.g 208
5.c odd 4 1 1000.2.v.i 208
8.d odd 2 1 inner 1000.2.v.h 208
20.d odd 2 1 800.2.bp.c 208
25.d even 5 1 1000.2.v.i 208
25.e even 10 1 1000.2.v.g 208
25.f odd 20 1 200.2.v.c ✓ 208
25.f odd 20 1 inner 1000.2.v.h 208
40.e odd 2 1 200.2.v.c ✓ 208
40.f even 2 1 800.2.bp.c 208
40.k even 4 1 1000.2.v.g 208
40.k even 4 1 1000.2.v.i 208
100.l even 20 1 800.2.bp.c 208
200.n odd 10 1 1000.2.v.i 208
200.s odd 10 1 1000.2.v.g 208
200.v even 20 1 200.2.v.c ✓ 208
200.v even 20 1 inner 1000.2.v.h 208
200.x odd 20 1 800.2.bp.c 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.v.c ✓ 208 5.b even 2 1
200.2.v.c ✓ 208 25.f odd 20 1
200.2.v.c ✓ 208 40.e odd 2 1
200.2.v.c ✓ 208 200.v even 20 1
800.2.bp.c 208 20.d odd 2 1
800.2.bp.c 208 40.f even 2 1
800.2.bp.c 208 100.l even 20 1
800.2.bp.c 208 200.x odd 20 1
1000.2.v.g 208 5.c odd 4 1
1000.2.v.g 208 25.e even 10 1
1000.2.v.g 208 40.k even 4 1
1000.2.v.g 208 200.s odd 10 1
1000.2.v.h 208 1.a even 1 1 trivial
1000.2.v.h 208 8.d odd 2 1 inner
1000.2.v.h 208 25.f odd 20 1 inner
1000.2.v.h 208 200.v even 20 1 inner
1000.2.v.i 208 5.c odd 4 1
1000.2.v.i 208 25.d even 5 1
1000.2.v.i 208 40.k even 4 1
1000.2.v.i 208 200.n odd 10 1

Hecke kernels

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

\( T_{3}^{104} - 8 T_{3}^{103} + 37 T_{3}^{102} - 114 T_{3}^{101} + 111 T_{3}^{100} + 732 T_{3}^{99} + \cdots + 22278400 \) 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