Properties

Label 555.2.c.c
Level $555$
Weight $2$
Character orbit 555.c
Analytic conductor $4.432$
Analytic rank $0$
Dimension $26$
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] = [555,2,Mod(334,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.334");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.c (of order \(2\), degree \(1\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 32 q^{4} - 2 q^{5} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 32 q^{4} - 2 q^{5} - 26 q^{9} + 20 q^{11} - 16 q^{14} + 44 q^{16} - 28 q^{19} + 4 q^{20} + 16 q^{21} + 10 q^{25} + 32 q^{26} - 46 q^{29} - 4 q^{30} + 32 q^{31} + 20 q^{34} - 12 q^{35} + 32 q^{36} - 22 q^{39} - 36 q^{40} + 56 q^{41} - 20 q^{44} + 2 q^{45} + 36 q^{46} - 74 q^{49} - 60 q^{50} + 4 q^{51} - 8 q^{55} + 72 q^{56} + 10 q^{59} - 24 q^{60} + 36 q^{61} - 32 q^{64} - 36 q^{65} + 28 q^{66} + 4 q^{69} - 92 q^{70} + 36 q^{71} + 116 q^{76} - 56 q^{79} - 96 q^{80} + 26 q^{81} - 8 q^{84} - 24 q^{85} + 108 q^{86} - 58 q^{89} + 60 q^{91} - 4 q^{94} - 4 q^{95} - 20 q^{96} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
334.1 2.77939i 1.00000i −5.72501 −2.22462 + 0.225945i −2.77939 0.0319162i 10.3533i −1.00000 0.627991 + 6.18310i
334.2 2.55509i 1.00000i −4.52851 2.13933 0.650593i 2.55509 2.23683i 6.46057i −1.00000 −1.66233 5.46619i
334.3 2.40245i 1.00000i −3.77179 −1.28557 + 1.82956i 2.40245 3.30864i 4.25663i −1.00000 4.39544 + 3.08853i
334.4 2.36117i 1.00000i −3.57513 1.66180 + 1.49613i −2.36117 3.86072i 3.71916i −1.00000 3.53262 3.92381i
334.5 2.32415i 1.00000i −3.40166 −2.03612 0.924245i 2.32415 3.18077i 3.25767i −1.00000 −2.14808 + 4.73224i
334.6 1.82554i 1.00000i −1.33258 1.97176 1.05459i −1.82554 4.15424i 1.21840i −1.00000 −1.92520 3.59951i
334.7 1.59785i 1.00000i −0.553137 −1.43895 + 1.71155i −1.59785 4.47204i 2.31188i −1.00000 2.73481 + 2.29923i
334.8 1.46114i 1.00000i −0.134925 1.43182 1.71753i 1.46114 3.04927i 2.72513i −1.00000 −2.50954 2.09208i
334.9 1.33085i 1.00000i 0.228850 1.74423 1.39916i −1.33085 3.76351i 2.96625i −1.00000 −1.86207 2.32130i
334.10 0.919689i 1.00000i 1.15417 −0.388610 2.20204i 0.919689 0.259618i 2.90086i −1.00000 −2.02519 + 0.357401i
334.11 0.550826i 1.00000i 1.69659 −0.808939 + 2.08461i 0.550826 2.51774i 2.03618i −1.00000 1.14826 + 0.445584i
334.12 0.214651i 1.00000i 1.95393 −2.20564 0.367653i −0.214651 4.41925i 0.848712i −1.00000 −0.0789169 + 0.473441i
334.13 0.103900i 1.00000i 1.98920 0.439510 2.19245i −0.103900 0.609052i 0.414478i −1.00000 −0.227795 0.0456650i
334.14 0.103900i 1.00000i 1.98920 0.439510 + 2.19245i −0.103900 0.609052i 0.414478i −1.00000 −0.227795 + 0.0456650i
334.15 0.214651i 1.00000i 1.95393 −2.20564 + 0.367653i −0.214651 4.41925i 0.848712i −1.00000 −0.0789169 0.473441i
334.16 0.550826i 1.00000i 1.69659 −0.808939 2.08461i 0.550826 2.51774i 2.03618i −1.00000 1.14826 0.445584i
334.17 0.919689i 1.00000i 1.15417 −0.388610 + 2.20204i 0.919689 0.259618i 2.90086i −1.00000 −2.02519 0.357401i
334.18 1.33085i 1.00000i 0.228850 1.74423 + 1.39916i −1.33085 3.76351i 2.96625i −1.00000 −1.86207 + 2.32130i
334.19 1.46114i 1.00000i −0.134925 1.43182 + 1.71753i 1.46114 3.04927i 2.72513i −1.00000 −2.50954 + 2.09208i
334.20 1.59785i 1.00000i −0.553137 −1.43895 1.71155i −1.59785 4.47204i 2.31188i −1.00000 2.73481 2.29923i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 334.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.c.c 26
3.b odd 2 1 1665.2.c.f 26
5.b even 2 1 inner 555.2.c.c 26
5.c odd 4 1 2775.2.a.bg 13
5.c odd 4 1 2775.2.a.bh 13
15.d odd 2 1 1665.2.c.f 26
15.e even 4 1 8325.2.a.cu 13
15.e even 4 1 8325.2.a.cv 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.c 26 1.a even 1 1 trivial
555.2.c.c 26 5.b even 2 1 inner
1665.2.c.f 26 3.b odd 2 1
1665.2.c.f 26 15.d odd 2 1
2775.2.a.bg 13 5.c odd 4 1
2775.2.a.bh 13 5.c odd 4 1
8325.2.a.cu 13 15.e even 4 1
8325.2.a.cv 13 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 42 T_{2}^{24} + 771 T_{2}^{22} + 8130 T_{2}^{20} + 54435 T_{2}^{18} + 241528 T_{2}^{16} + 719422 T_{2}^{14} + 1425812 T_{2}^{12} + 1821683 T_{2}^{10} + 1407938 T_{2}^{8} + 583595 T_{2}^{6} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display