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Properties

Label 574.2.n.a
Level $574$
Weight $2$
Character orbit 574.n
Analytic conductor $4.583$
Analytic rank $0$
Dimension $40$
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] = [574,2,Mod(113,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.n (of order \(10\), 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 \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 10 q^{2} - 10 q^{4} - 8 q^{5} - 10 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 10 q^{2} - 10 q^{4} - 8 q^{5} - 10 q^{8} - 24 q^{9} - 8 q^{10} - 10 q^{11} - 20 q^{15} - 10 q^{16} + 10 q^{17} + 6 q^{18} - 10 q^{19} + 12 q^{20} - 10 q^{25} + 20 q^{26} + 20 q^{30} - 18 q^{31} + 40 q^{32} - 30 q^{33} - 30 q^{34} - 10 q^{35} + 6 q^{36} + 10 q^{37} - 24 q^{39} - 8 q^{40} + 22 q^{41} - 16 q^{43} + 58 q^{45} - 10 q^{46} + 30 q^{47} + 10 q^{49} + 60 q^{50} - 34 q^{51} - 20 q^{52} + 10 q^{53} - 38 q^{57} + 22 q^{59} + 10 q^{60} + 50 q^{61} + 22 q^{62} - 10 q^{64} - 50 q^{65} + 40 q^{66} + 40 q^{69} - 30 q^{71} + 6 q^{72} - 40 q^{73} + 10 q^{74} - 10 q^{75} + 10 q^{76} + 12 q^{77} + 26 q^{78} + 12 q^{80} + 16 q^{81} + 2 q^{82} - 52 q^{83} + 14 q^{86} - 22 q^{87} + 20 q^{89} + 58 q^{90} - 16 q^{91} - 10 q^{92} - 70 q^{93} - 20 q^{94} - 80 q^{95} + 60 q^{97} + 10 q^{98} + 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 −0.809017 − 0.587785i − 2.68579i 0.309017 + 0.951057i −0.472602 − 1.45452i −1.57867 + 2.17285i 0.587785 + 0.809017i 0.309017 − 0.951057i −4.21348 −0.472602 + 1.45452i
113.2 −0.809017 − 0.587785i − 2.29227i 0.309017 + 0.951057i −1.10040 − 3.38669i −1.34737 + 1.85449i −0.587785 − 0.809017i 0.309017 − 0.951057i −2.25452 −1.10040 + 3.38669i
113.3 −0.809017 − 0.587785i − 1.38973i 0.309017 + 0.951057i 0.279490 + 0.860181i −0.816864 + 1.12432i 0.587785 + 0.809017i 0.309017 − 0.951057i 1.06864 0.279490 − 0.860181i
113.4 −0.809017 − 0.587785i − 0.923650i 0.309017 + 0.951057i 1.28915 + 3.96760i −0.542908 + 0.747248i −0.587785 − 0.809017i 0.309017 − 0.951057i 2.14687 1.28915 − 3.96760i
113.5 −0.809017 − 0.587785i − 0.523795i 0.309017 + 0.951057i −0.0503515 − 0.154966i −0.307879 + 0.423759i −0.587785 − 0.809017i 0.309017 − 0.951057i 2.72564 −0.0503515 + 0.154966i
113.6 −0.809017 − 0.587785i − 0.327982i 0.309017 + 0.951057i −1.16097 − 3.57309i −0.192783 + 0.265343i 0.587785 + 0.809017i 0.309017 − 0.951057i 2.89243 −1.16097 + 3.57309i
113.7 −0.809017 − 0.587785i 1.00668i 0.309017 + 0.951057i 0.370724 + 1.14097i 0.591713 − 0.814423i 0.587785 + 0.809017i 0.309017 − 0.951057i 1.98659 0.370724 − 1.14097i
113.8 −0.809017 − 0.587785i 1.22915i 0.309017 + 0.951057i −0.955396 − 2.94041i 0.722478 − 0.994406i −0.587785 − 0.809017i 0.309017 − 0.951057i 1.48918 −0.955396 + 2.94041i
113.9 −0.809017 − 0.587785i 2.51057i 0.309017 + 0.951057i −0.546271 − 1.68125i 1.47567 − 2.03109i −0.587785 − 0.809017i 0.309017 − 0.951057i −3.30294 −0.546271 + 1.68125i
113.10 −0.809017 − 0.587785i 3.39682i 0.309017 + 0.951057i 0.346626 + 1.06680i 1.99660 − 2.74809i 0.587785 + 0.809017i 0.309017 − 0.951057i −8.53841 0.346626 − 1.06680i
127.1 −0.809017 + 0.587785i − 3.39682i 0.309017 − 0.951057i 0.346626 − 1.06680i 1.99660 + 2.74809i 0.587785 − 0.809017i 0.309017 + 0.951057i −8.53841 0.346626 + 1.06680i
127.2 −0.809017 + 0.587785i − 2.51057i 0.309017 − 0.951057i −0.546271 + 1.68125i 1.47567 + 2.03109i −0.587785 + 0.809017i 0.309017 + 0.951057i −3.30294 −0.546271 − 1.68125i
127.3 −0.809017 + 0.587785i − 1.22915i 0.309017 − 0.951057i −0.955396 + 2.94041i 0.722478 + 0.994406i −0.587785 + 0.809017i 0.309017 + 0.951057i 1.48918 −0.955396 − 2.94041i
127.4 −0.809017 + 0.587785i − 1.00668i 0.309017 − 0.951057i 0.370724 − 1.14097i 0.591713 + 0.814423i 0.587785 − 0.809017i 0.309017 + 0.951057i 1.98659 0.370724 + 1.14097i
127.5 −0.809017 + 0.587785i 0.327982i 0.309017 − 0.951057i −1.16097 + 3.57309i −0.192783 − 0.265343i 0.587785 − 0.809017i 0.309017 + 0.951057i 2.89243 −1.16097 − 3.57309i
127.6 −0.809017 + 0.587785i 0.523795i 0.309017 − 0.951057i −0.0503515 + 0.154966i −0.307879 − 0.423759i −0.587785 + 0.809017i 0.309017 + 0.951057i 2.72564 −0.0503515 − 0.154966i
127.7 −0.809017 + 0.587785i 0.923650i 0.309017 − 0.951057i 1.28915 − 3.96760i −0.542908 − 0.747248i −0.587785 + 0.809017i 0.309017 + 0.951057i 2.14687 1.28915 + 3.96760i
127.8 −0.809017 + 0.587785i 1.38973i 0.309017 − 0.951057i 0.279490 − 0.860181i −0.816864 − 1.12432i 0.587785 − 0.809017i 0.309017 + 0.951057i 1.06864 0.279490 + 0.860181i
127.9 −0.809017 + 0.587785i 2.29227i 0.309017 − 0.951057i −1.10040 + 3.38669i −1.34737 − 1.85449i −0.587785 + 0.809017i 0.309017 + 0.951057i −2.25452 −1.10040 − 3.38669i
127.10 −0.809017 + 0.587785i 2.68579i 0.309017 − 0.951057i −0.472602 + 1.45452i −1.57867 − 2.17285i 0.587785 − 0.809017i 0.309017 + 0.951057i −4.21348 −0.472602 − 1.45452i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.n.a ✓ 40
41.f even 10 1 inner 574.2.n.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.n.a ✓ 40 1.a even 1 1 trivial
574.2.n.a ✓ 40 41.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 72 T_{3}^{38} + 2354 T_{3}^{36} + 46332 T_{3}^{34} + 613523 T_{3}^{32} + 5785636 T_{3}^{30} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display