Properties

Label 574.2.r.c
Level $574$
Weight $2$
Character orbit 574.r
Analytic conductor $4.583$
Analytic rank $0$
Dimension $40$
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] = [574,2,Mod(9,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.r (of order \(12\), degree \(4\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} + 20 q^{4} - 8 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} + 20 q^{4} - 8 q^{6} - 12 q^{7} + 12 q^{10} - 4 q^{12} - 28 q^{13} + 12 q^{14} + 20 q^{15} - 20 q^{16} + 14 q^{17} - 16 q^{18} + 24 q^{23} - 4 q^{24} - 20 q^{25} + 14 q^{26} + 4 q^{27} + 44 q^{29} + 10 q^{30} - 8 q^{31} + 28 q^{34} - 2 q^{35} - 16 q^{37} - 12 q^{40} + 4 q^{41} - 64 q^{42} - 36 q^{45} + 2 q^{47} - 8 q^{48} + 28 q^{51} - 14 q^{52} + 20 q^{53} + 2 q^{54} - 44 q^{55} + 12 q^{56} - 40 q^{57} - 22 q^{58} - 8 q^{59} + 10 q^{60} + 24 q^{63} - 40 q^{64} + 36 q^{65} - 48 q^{66} + 6 q^{67} - 14 q^{68} - 36 q^{69} + 8 q^{70} + 40 q^{71} + 16 q^{72} + 36 q^{75} - 26 q^{79} + 84 q^{81} - 22 q^{82} - 24 q^{83} + 164 q^{85} - 28 q^{86} + 16 q^{89} + 48 q^{92} + 46 q^{93} - 2 q^{94} - 18 q^{95} + 4 q^{96} + 8 q^{97} + 24 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0.866025 + 0.500000i −0.855646 + 3.19332i 0.500000 + 0.866025i 0.112148 + 0.0647489i −2.33767 + 2.33767i 1.76095 + 1.97460i 1.00000i −6.86706 3.96470i 0.0647489 + 0.112148i
9.2 0.866025 + 0.500000i −0.593109 + 2.21351i 0.500000 + 0.866025i −2.00711 1.15881i −1.62040 + 1.62040i −2.64158 0.148454i 1.00000i −1.94978 1.12571i −1.15881 2.00711i
9.3 0.866025 + 0.500000i −0.552179 + 2.06076i 0.500000 + 0.866025i 1.83592 + 1.05997i −1.50858 + 1.50858i 0.869194 2.49890i 1.00000i −1.34375 0.775813i 1.05997 + 1.83592i
9.4 0.866025 + 0.500000i −0.321998 + 1.20171i 0.500000 + 0.866025i 2.36946 + 1.36801i −0.879715 + 0.879715i 2.54753 + 0.714193i 1.00000i 1.25765 + 0.726102i 1.36801 + 2.36946i
9.5 0.866025 + 0.500000i −0.111957 + 0.417828i 0.500000 + 0.866025i 0.681525 + 0.393479i −0.305871 + 0.305871i −2.17302 1.50929i 1.00000i 2.43603 + 1.40644i 0.393479 + 0.681525i
9.6 0.866025 + 0.500000i −0.0822802 + 0.307074i 0.500000 + 0.866025i −0.999657 0.577152i −0.224794 + 0.224794i 2.64547 0.0385485i 1.00000i 2.51055 + 1.44947i −0.577152 0.999657i
9.7 0.866025 + 0.500000i 0.200396 0.747888i 0.500000 + 0.866025i 1.99851 + 1.15384i 0.547492 0.547492i −1.57271 + 2.12757i 1.00000i 2.07890 + 1.20025i 1.15384 + 1.99851i
9.8 0.866025 + 0.500000i 0.320619 1.19657i 0.500000 + 0.866025i −1.97710 1.14148i 0.875948 0.875948i −1.58481 2.11858i 1.00000i 1.26910 + 0.732715i −1.14148 1.97710i
9.9 0.866025 + 0.500000i 0.417822 1.55933i 0.500000 + 0.866025i 0.463562 + 0.267638i 1.14151 1.14151i 1.34738 2.27697i 1.00000i 0.341135 + 0.196954i 0.267638 + 0.463562i
9.10 0.866025 + 0.500000i 0.846280 3.15836i 0.500000 + 0.866025i 2.71889 + 1.56975i 2.31208 2.31208i −2.46634 0.957682i 1.00000i −6.66098 3.84572i 1.56975 + 2.71889i
319.1 0.866025 0.500000i −0.855646 3.19332i 0.500000 0.866025i 0.112148 0.0647489i −2.33767 2.33767i 1.76095 1.97460i 1.00000i −6.86706 + 3.96470i 0.0647489 0.112148i
319.2 0.866025 0.500000i −0.593109 2.21351i 0.500000 0.866025i −2.00711 + 1.15881i −1.62040 1.62040i −2.64158 + 0.148454i 1.00000i −1.94978 + 1.12571i −1.15881 + 2.00711i
319.3 0.866025 0.500000i −0.552179 2.06076i 0.500000 0.866025i 1.83592 1.05997i −1.50858 1.50858i 0.869194 + 2.49890i 1.00000i −1.34375 + 0.775813i 1.05997 1.83592i
319.4 0.866025 0.500000i −0.321998 1.20171i 0.500000 0.866025i 2.36946 1.36801i −0.879715 0.879715i 2.54753 0.714193i 1.00000i 1.25765 0.726102i 1.36801 2.36946i
319.5 0.866025 0.500000i −0.111957 0.417828i 0.500000 0.866025i 0.681525 0.393479i −0.305871 0.305871i −2.17302 + 1.50929i 1.00000i 2.43603 1.40644i 0.393479 0.681525i
319.6 0.866025 0.500000i −0.0822802 0.307074i 0.500000 0.866025i −0.999657 + 0.577152i −0.224794 0.224794i 2.64547 + 0.0385485i 1.00000i 2.51055 1.44947i −0.577152 + 0.999657i
319.7 0.866025 0.500000i 0.200396 + 0.747888i 0.500000 0.866025i 1.99851 1.15384i 0.547492 + 0.547492i −1.57271 2.12757i 1.00000i 2.07890 1.20025i 1.15384 1.99851i
319.8 0.866025 0.500000i 0.320619 + 1.19657i 0.500000 0.866025i −1.97710 + 1.14148i 0.875948 + 0.875948i −1.58481 + 2.11858i 1.00000i 1.26910 0.732715i −1.14148 + 1.97710i
319.9 0.866025 0.500000i 0.417822 + 1.55933i 0.500000 0.866025i 0.463562 0.267638i 1.14151 + 1.14151i 1.34738 + 2.27697i 1.00000i 0.341135 0.196954i 0.267638 0.463562i
319.10 0.866025 0.500000i 0.846280 + 3.15836i 0.500000 0.866025i 2.71889 1.56975i 2.31208 + 2.31208i −2.46634 + 0.957682i 1.00000i −6.66098 + 3.84572i 1.56975 2.71889i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.c even 4 1 inner
287.r even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.r.c 40
7.c even 3 1 inner 574.2.r.c 40
41.c even 4 1 inner 574.2.r.c 40
287.r even 12 1 inner 574.2.r.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.r.c 40 1.a even 1 1 trivial
574.2.r.c 40 7.c even 3 1 inner
574.2.r.c 40 41.c even 4 1 inner
574.2.r.c 40 287.r even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 4 T_{3}^{39} + 8 T_{3}^{38} - 28 T_{3}^{37} - 67 T_{3}^{36} + 474 T_{3}^{35} - 968 T_{3}^{34} + 3426 T_{3}^{33} + 7278 T_{3}^{32} - 54078 T_{3}^{31} + 109642 T_{3}^{30} - 386712 T_{3}^{29} + 684529 T_{3}^{28} + \cdots + 38416 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display