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Properties

Label 574.2.f.b
Level $574$
Weight $2$
Character orbit 574.f
Analytic conductor $4.583$
Analytic rank $0$
Dimension $24$
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(155,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.f (of order \(4\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 4 q^{11} - 12 q^{13} + 12 q^{15} + 24 q^{16} + 8 q^{17} + 32 q^{18} + 16 q^{19} - 4 q^{22} + 8 q^{23} - 48 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{30} + 8 q^{31} - 8 q^{34} - 8 q^{35} + 24 q^{37} - 16 q^{38} + 4 q^{41} + 8 q^{42} + 4 q^{44} + 8 q^{45} - 4 q^{47} + 8 q^{51} + 12 q^{52} - 16 q^{53} + 4 q^{55} - 8 q^{57} + 12 q^{58} + 8 q^{59} - 12 q^{60} - 24 q^{64} - 12 q^{65} + 16 q^{66} - 8 q^{67} - 8 q^{68} - 48 q^{69} + 8 q^{70} - 16 q^{71} - 32 q^{72} + 16 q^{75} - 16 q^{76} + 72 q^{78} - 8 q^{79} - 24 q^{81} - 24 q^{82} - 8 q^{83} + 12 q^{85} - 8 q^{86} + 4 q^{88} + 36 q^{89} - 8 q^{92} + 4 q^{94} - 28 q^{95} + 36 q^{97} + 24 q^{98} + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 1.00000i −2.23561 + 2.23561i −1.00000 3.12689i −2.23561 − 2.23561i −0.707107 + 0.707107i − 1.00000i − 6.99588i −3.12689
155.2 1.00000i −1.79721 + 1.79721i −1.00000 − 3.90955i −1.79721 − 1.79721i 0.707107 − 0.707107i − 1.00000i − 3.45993i 3.90955
155.3 1.00000i −1.71114 + 1.71114i −1.00000 − 2.15578i −1.71114 − 1.71114i −0.707107 + 0.707107i − 1.00000i − 2.85598i 2.15578
155.4 1.00000i −1.20036 + 1.20036i −1.00000 1.77947i −1.20036 − 1.20036i 0.707107 − 0.707107i − 1.00000i 0.118277i −1.77947
155.5 1.00000i −0.715262 + 0.715262i −1.00000 − 2.56013i −0.715262 − 0.715262i −0.707107 + 0.707107i − 1.00000i 1.97680i 2.56013
155.6 1.00000i 0.0200226 − 0.0200226i −1.00000 0.349701i 0.0200226 + 0.0200226i 0.707107 − 0.707107i − 1.00000i 2.99920i −0.349701
155.7 1.00000i 0.343442 − 0.343442i −1.00000 − 1.44236i 0.343442 + 0.343442i 0.707107 − 0.707107i − 1.00000i 2.76409i 1.44236
155.8 1.00000i 0.413728 − 0.413728i −1.00000 4.21553i 0.413728 + 0.413728i −0.707107 + 0.707107i − 1.00000i 2.65766i −4.21553
155.9 1.00000i 0.983825 − 0.983825i −1.00000 0.548928i 0.983825 + 0.983825i −0.707107 + 0.707107i − 1.00000i 1.06418i −0.548928
155.10 1.00000i 1.85024 − 1.85024i −1.00000 − 0.347015i 1.85024 + 1.85024i −0.707107 + 0.707107i − 1.00000i − 3.84678i 0.347015
155.11 1.00000i 1.93364 − 1.93364i −1.00000 − 3.27393i 1.93364 + 1.93364i 0.707107 − 0.707107i − 1.00000i − 4.47796i 3.27393
155.12 1.00000i 2.11467 − 2.11467i −1.00000 3.66824i 2.11467 + 2.11467i 0.707107 − 0.707107i − 1.00000i − 5.94369i −3.66824
337.1 − 1.00000i −2.23561 − 2.23561i −1.00000 − 3.12689i −2.23561 + 2.23561i −0.707107 − 0.707107i 1.00000i 6.99588i −3.12689
337.2 − 1.00000i −1.79721 − 1.79721i −1.00000 3.90955i −1.79721 + 1.79721i 0.707107 + 0.707107i 1.00000i 3.45993i 3.90955
337.3 − 1.00000i −1.71114 − 1.71114i −1.00000 2.15578i −1.71114 + 1.71114i −0.707107 − 0.707107i 1.00000i 2.85598i 2.15578
337.4 − 1.00000i −1.20036 − 1.20036i −1.00000 − 1.77947i −1.20036 + 1.20036i 0.707107 + 0.707107i 1.00000i − 0.118277i −1.77947
337.5 − 1.00000i −0.715262 − 0.715262i −1.00000 2.56013i −0.715262 + 0.715262i −0.707107 − 0.707107i 1.00000i − 1.97680i 2.56013
337.6 − 1.00000i 0.0200226 + 0.0200226i −1.00000 − 0.349701i 0.0200226 − 0.0200226i 0.707107 + 0.707107i 1.00000i − 2.99920i −0.349701
337.7 − 1.00000i 0.343442 + 0.343442i −1.00000 1.44236i 0.343442 − 0.343442i 0.707107 + 0.707107i 1.00000i − 2.76409i 1.44236
337.8 − 1.00000i 0.413728 + 0.413728i −1.00000 − 4.21553i 0.413728 − 0.413728i −0.707107 − 0.707107i 1.00000i − 2.65766i −4.21553
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.f.b ✓ 24
41.c even 4 1 inner 574.2.f.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.f.b ✓ 24 1.a even 1 1 trivial
574.2.f.b ✓ 24 41.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 186 T_{3}^{20} + 48 T_{3}^{17} + 11121 T_{3}^{16} + 864 T_{3}^{15} - 2208 T_{3}^{13} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display