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Properties

Label 200.2.v.c
Level $200$
Weight $2$
Character orbit 200.v
Analytic conductor $1.597$
Analytic rank $0$
Dimension $208$
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] = [200,2,Mod(3,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.v (of order \(20\), degree \(8\), 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: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q + 2 q^{2} - 16 q^{3} - 10 q^{4} - 26 q^{6} - 34 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q + 2 q^{2} - 16 q^{3} - 10 q^{4} - 26 q^{6} - 34 q^{8} - 20 q^{9} + 10 q^{10} - 4 q^{11} - 2 q^{12} - 30 q^{14} + 18 q^{16} - 32 q^{17} - 20 q^{18} - 10 q^{20} - 42 q^{22} - 40 q^{25} - 8 q^{26} - 28 q^{27} + 30 q^{28} - 10 q^{30} + 22 q^{32} + 36 q^{33} + 30 q^{34} - 100 q^{35} - 38 q^{36} - 56 q^{38} + 20 q^{40} - 44 q^{41} - 70 q^{42} - 8 q^{43} + 60 q^{44} + 2 q^{46} + 104 q^{48} - 30 q^{50} + 8 q^{51} + 60 q^{52} - 140 q^{54} + 2 q^{56} + 92 q^{57} - 70 q^{58} - 40 q^{59} + 10 q^{60} - 120 q^{62} + 20 q^{64} - 40 q^{65} - 2 q^{66} - 12 q^{67} - 46 q^{68} + 30 q^{70} + 60 q^{72} - 16 q^{73} + 80 q^{75} - 24 q^{76} - 20 q^{78} + 30 q^{80} + 16 q^{81} - 22 q^{82} - 96 q^{83} - 10 q^{84} - 26 q^{86} + 134 q^{88} - 100 q^{89} + 130 q^{90} + 4 q^{91} + 70 q^{92} + 150 q^{94} - 26 q^{96} + 24 q^{97} + 206 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.40750 + 0.137639i −0.284156 − 1.79409i 1.96211 − 0.387455i 0.834343 − 2.07458i 0.646887 + 2.48607i 2.57003 + 2.57003i −2.70834 + 0.815406i −0.284845 + 0.0925516i −0.888795 + 3.03481i
3.2 −1.37847 − 0.315925i 0.194262 + 1.22652i 1.80038 + 0.870989i −0.558511 − 2.16519i 0.119704 − 1.75210i −2.92162 − 2.92162i −2.20661 − 1.76942i 1.38655 − 0.450516i 0.0858538 + 3.16111i
3.3 −1.37526 − 0.329632i −0.390689 − 2.46671i 1.78269 + 0.906659i −1.86583 + 1.23235i −0.275807 + 3.52116i −1.88452 − 1.88452i −2.15280 − 1.83452i −3.07886 + 1.00038i 2.97222 − 1.07977i
3.4 −1.33775 + 0.458729i 0.0641871 + 0.405261i 1.57913 − 1.22733i 1.19321 + 1.89109i −0.271771 − 0.512693i −0.723968 − 0.723968i −1.54947 + 2.36625i 2.69305 − 0.875026i −2.46372 − 1.98244i
3.5 −1.32153 + 0.503554i 0.415545 + 2.62365i 1.49287 − 1.33092i −2.16296 − 0.567092i −1.87030 − 3.25797i 2.50442 + 2.50442i −1.30267 + 2.51059i −3.85767 + 1.25343i 3.14397 − 0.339742i
3.6 −1.12548 − 0.856327i −0.204230 − 1.28946i 0.533408 + 1.92756i 1.97621 + 1.04623i −0.874342 + 1.62615i 1.02699 + 1.02699i 1.05028 − 2.62620i 1.23217 − 0.400358i −1.32827 − 2.86979i
3.7 −0.906307 − 1.08564i −0.000586028 − 0.00370003i −0.357214 + 1.96784i −2.23596 − 0.0217085i −0.00348577 + 0.00398958i 2.79054 + 2.79054i 2.46011 − 1.39566i 2.85316 − 0.927047i 2.00290 + 2.44712i
3.8 −0.827548 + 1.14681i −0.527096 − 3.32795i −0.630327 − 1.89807i 2.14301 + 0.638361i 4.25271 + 2.14957i −2.04742 − 2.04742i 2.69835 + 0.847885i −7.94427 + 2.58125i −2.50552 + 1.92934i
3.9 −0.747524 + 1.20050i 0.0906054 + 0.572060i −0.882416 − 1.79481i 1.33570 − 1.79329i −0.754489 − 0.318856i 0.578731 + 0.578731i 2.81430 + 0.282321i 2.53413 − 0.823388i 1.15439 + 2.94404i
3.10 −0.416394 + 1.35152i 0.464977 + 2.93575i −1.65323 − 1.12553i 0.317692 + 2.21338i −4.16135 − 0.594000i 0.353934 + 0.353934i 2.20958 − 1.76572i −5.54925 + 1.80306i −3.12373 − 0.492271i
3.11 −0.345584 − 1.37134i −0.000586028 − 0.00370003i −1.76114 + 0.947827i 2.23596 + 0.0217085i −0.00487148 + 0.00208232i −2.79054 − 2.79054i 1.90842 + 2.08757i 2.85316 − 0.927047i −0.742944 − 3.07377i
3.12 −0.316332 + 1.37838i −0.240037 − 1.51554i −1.79987 − 0.872053i −1.41053 + 1.73505i 2.16492 + 0.148550i 3.08323 + 3.08323i 1.77138 − 2.20505i 0.613940 − 0.199481i −1.94537 − 2.49310i
3.13 −0.0312429 − 1.41387i −0.204230 − 1.28946i −1.99805 + 0.0883468i −1.97621 − 1.04623i −1.81674 + 0.329041i −1.02699 − 1.02699i 0.187336 + 2.82222i 1.23217 − 0.400358i −1.41749 + 2.82679i
3.14 0.237717 + 1.39409i −0.290912 − 1.83675i −1.88698 + 0.662799i −1.05082 − 1.97377i 2.49144 − 0.842184i −0.966817 − 0.966817i −1.37257 − 2.47306i −0.435834 + 0.141611i 2.50182 − 1.93414i
3.15 0.533157 + 1.30986i 0.238378 + 1.50506i −1.43149 + 1.39673i 2.20632 − 0.363530i −1.84433 + 1.11468i 0.599487 + 0.599487i −2.59273 − 1.13038i 0.644782 − 0.209502i 1.65249 + 2.69616i
3.16 0.541681 − 1.30636i −0.390689 − 2.46671i −1.41316 − 1.41526i 1.86583 − 1.23235i −3.43405 − 0.825789i 1.88452 + 1.88452i −2.61433 + 1.07948i −3.07886 + 1.00038i −0.599212 − 3.10499i
3.17 0.554658 − 1.30091i 0.194262 + 1.22652i −1.38471 − 1.44312i 0.558511 + 2.16519i 1.70334 + 0.427585i 2.92162 + 2.92162i −2.64540 + 1.00094i 1.38655 − 0.450516i 3.12649 + 0.474374i
3.18 0.746320 + 1.20125i 0.238378 + 1.50506i −0.886011 + 1.79304i −2.20632 + 0.363530i −1.63005 + 1.40961i −0.599487 − 0.599487i −2.81514 + 0.273858i 0.644782 − 0.209502i −2.08331 − 2.37903i
3.19 0.938660 − 1.05779i −0.284156 − 1.79409i −0.237834 − 1.98581i −0.834343 + 2.07458i −2.16449 − 1.38346i −2.57003 − 2.57003i −2.32381 − 1.61242i −0.284845 + 0.0925516i 1.41130 + 2.82988i
3.20 0.988117 + 1.01174i −0.290912 − 1.83675i −0.0472506 + 1.99944i 1.05082 + 1.97377i 1.57086 − 2.10925i 0.966817 + 0.966817i −2.06961 + 1.92788i −0.435834 + 0.141611i −0.958623 + 3.01348i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.26
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 200.2.v.c ✓ 208
4.b odd 2 1 800.2.bp.c 208
5.b even 2 1 1000.2.v.h 208
5.c odd 4 1 1000.2.v.g 208
5.c odd 4 1 1000.2.v.i 208
8.b even 2 1 800.2.bp.c 208
8.d odd 2 1 inner 200.2.v.c ✓ 208
25.d even 5 1 1000.2.v.g 208
25.e even 10 1 1000.2.v.i 208
25.f odd 20 1 inner 200.2.v.c ✓ 208
25.f odd 20 1 1000.2.v.h 208
40.e odd 2 1 1000.2.v.h 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.g 208
200.s odd 10 1 1000.2.v.i 208
200.v even 20 1 inner 200.2.v.c ✓ 208
200.v even 20 1 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 1.a even 1 1 trivial
200.2.v.c ✓ 208 8.d odd 2 1 inner
200.2.v.c ✓ 208 25.f odd 20 1 inner
200.2.v.c ✓ 208 200.v even 20 1 inner
800.2.bp.c 208 4.b odd 2 1
800.2.bp.c 208 8.b 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.d even 5 1
1000.2.v.g 208 40.k even 4 1
1000.2.v.g 208 200.n odd 10 1
1000.2.v.h 208 5.b even 2 1
1000.2.v.h 208 25.f odd 20 1
1000.2.v.h 208 40.e odd 2 1
1000.2.v.h 208 200.v even 20 1
1000.2.v.i 208 5.c odd 4 1
1000.2.v.i 208 25.e even 10 1
1000.2.v.i 208 40.k even 4 1
1000.2.v.i 208 200.s odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(200, [\chi])\):

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