Properties

Label 400.3.bg.c
Level $400$
Weight $3$
Character orbit 400.bg
Analytic conductor $10.899$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,3,Mod(17,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.bg (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: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 25)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{3} - 10 q^{5} + 10 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{3} - 10 q^{5} + 10 q^{7} - 10 q^{9} + 6 q^{11} - 10 q^{13} + 10 q^{15} + 60 q^{17} - 90 q^{19} - 6 q^{21} - 10 q^{23} - 40 q^{25} + 100 q^{27} - 110 q^{29} + 6 q^{31} - 190 q^{33} + 120 q^{35} + 50 q^{37} - 390 q^{39} - 86 q^{41} - 230 q^{43} + 310 q^{45} - 70 q^{47} + 16 q^{51} - 190 q^{53} + 250 q^{55} - 650 q^{57} + 260 q^{59} + 114 q^{61} + 20 q^{63} + 360 q^{65} - 270 q^{67} + 340 q^{69} + 66 q^{71} + 30 q^{73} + 90 q^{75} - 250 q^{77} + 210 q^{79} + 62 q^{81} + 600 q^{85} - 300 q^{87} - 10 q^{89} + 6 q^{91} + 520 q^{93} - 310 q^{95} + 270 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −0.838638 + 5.29495i 0 −3.73307 + 3.32629i 0 −1.66138 1.66138i 0 −18.7737 6.09994i 0
17.2 0 0.0858318 0.541921i 0 2.26962 + 4.45520i 0 −1.68463 1.68463i 0 8.27320 + 2.68812i 0
17.3 0 0.296456 1.87175i 0 1.22928 4.84653i 0 5.60844 + 5.60844i 0 5.14394 + 1.67137i 0
17.4 0 0.363254 2.29349i 0 −4.45624 + 2.26758i 0 3.40272 + 3.40272i 0 3.43135 + 1.11491i 0
33.1 0 −3.42034 0.541729i 0 4.00059 2.99921i 0 8.06323 8.06323i 0 2.84577 + 0.924645i 0
33.2 0 −0.872241 0.138149i 0 −2.66494 + 4.23062i 0 −1.62783 + 1.62783i 0 −7.81779 2.54015i 0
33.3 0 3.57679 + 0.566508i 0 −4.45026 2.27929i 0 −6.54971 + 6.54971i 0 3.91299 + 1.27141i 0
33.4 0 4.42692 + 0.701156i 0 1.95091 4.60369i 0 4.77540 4.77540i 0 10.5465 + 3.42677i 0
97.1 0 −3.42034 + 0.541729i 0 4.00059 + 2.99921i 0 8.06323 + 8.06323i 0 2.84577 0.924645i 0
97.2 0 −0.872241 + 0.138149i 0 −2.66494 4.23062i 0 −1.62783 1.62783i 0 −7.81779 + 2.54015i 0
97.3 0 3.57679 0.566508i 0 −4.45026 + 2.27929i 0 −6.54971 6.54971i 0 3.91299 1.27141i 0
97.4 0 4.42692 0.701156i 0 1.95091 + 4.60369i 0 4.77540 + 4.77540i 0 10.5465 3.42677i 0
113.1 0 −1.72787 + 3.39113i 0 2.36408 + 4.40581i 0 2.38950 2.38950i 0 −3.22416 4.43767i 0
113.2 0 −1.61679 + 3.17313i 0 −0.872190 4.92334i 0 0.574149 0.574149i 0 −2.16466 2.97940i 0
113.3 0 0.665351 1.30583i 0 −3.20727 3.83580i 0 −3.62927 + 3.62927i 0 4.02758 + 5.54349i 0
113.4 0 2.19472 4.30737i 0 4.99561 0.209511i 0 3.57009 3.57009i 0 −8.44662 11.6258i 0
177.1 0 −1.72787 3.39113i 0 2.36408 4.40581i 0 2.38950 + 2.38950i 0 −3.22416 + 4.43767i 0
177.2 0 −1.61679 3.17313i 0 −0.872190 + 4.92334i 0 0.574149 + 0.574149i 0 −2.16466 + 2.97940i 0
177.3 0 0.665351 + 1.30583i 0 −3.20727 + 3.83580i 0 −3.62927 3.62927i 0 4.02758 5.54349i 0
177.4 0 2.19472 + 4.30737i 0 4.99561 + 0.209511i 0 3.57009 + 3.57009i 0 −8.44662 + 11.6258i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.bg.c 32
4.b odd 2 1 25.3.f.a 32
12.b even 2 1 225.3.r.a 32
20.d odd 2 1 125.3.f.c 32
20.e even 4 1 125.3.f.a 32
20.e even 4 1 125.3.f.b 32
25.f odd 20 1 inner 400.3.bg.c 32
100.h odd 10 1 125.3.f.b 32
100.j odd 10 1 125.3.f.a 32
100.l even 20 1 25.3.f.a 32
100.l even 20 1 125.3.f.c 32
300.u odd 20 1 225.3.r.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.3.f.a 32 4.b odd 2 1
25.3.f.a 32 100.l even 20 1
125.3.f.a 32 20.e even 4 1
125.3.f.a 32 100.j odd 10 1
125.3.f.b 32 20.e even 4 1
125.3.f.b 32 100.h odd 10 1
125.3.f.c 32 20.d odd 2 1
125.3.f.c 32 100.l even 20 1
225.3.r.a 32 12.b even 2 1
225.3.r.a 32 300.u odd 20 1
400.3.bg.c 32 1.a even 1 1 trivial
400.3.bg.c 32 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 10 T_{3}^{31} + 55 T_{3}^{30} - 310 T_{3}^{29} + 1192 T_{3}^{28} - 1960 T_{3}^{27} - 2380 T_{3}^{26} + 30930 T_{3}^{25} - 86732 T_{3}^{24} + 603820 T_{3}^{23} - 3704615 T_{3}^{22} - 6537080 T_{3}^{21} + \cdots + 40837943056 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display