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Properties

Label 624.4.bv.j
Level $624$
Weight $4$
Character orbit 624.bv
Analytic conductor $36.817$
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] = [624,4,Mod(49,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 36 q^{3} - 42 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 36 q^{3} - 42 q^{7} - 108 q^{9} - 42 q^{11} - 66 q^{13} + 36 q^{15} + 114 q^{17} - 102 q^{19} + 130 q^{23} - 824 q^{25} + 648 q^{27} - 144 q^{29} + 126 q^{33} - 218 q^{35} + 438 q^{37} + 54 q^{39} - 786 q^{41} + 710 q^{43} - 108 q^{45} + 856 q^{49} - 684 q^{51} + 464 q^{53} - 322 q^{55} + 936 q^{59} - 738 q^{61} + 378 q^{63} + 70 q^{65} + 30 q^{67} + 390 q^{69} + 2202 q^{71} + 1236 q^{75} + 880 q^{77} + 496 q^{79} - 972 q^{81} + 90 q^{85} - 432 q^{87} - 4380 q^{89} - 312 q^{91} + 72 q^{93} + 158 q^{95} + 4980 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.50000 − 2.59808i 0 − 21.1153i 0 −21.9582 − 12.6776i 0 −4.50000 + 7.79423i 0
49.2 0 −1.50000 − 2.59808i 0 − 14.7219i 0 10.8587 + 6.26929i 0 −4.50000 + 7.79423i 0
49.3 0 −1.50000 − 2.59808i 0 − 12.7872i 0 −0.0260449 − 0.0150370i 0 −4.50000 + 7.79423i 0
49.4 0 −1.50000 − 2.59808i 0 − 4.15670i 0 −11.4016 − 6.58274i 0 −4.50000 + 7.79423i 0
49.5 0 −1.50000 − 2.59808i 0 − 3.71979i 0 8.41491 + 4.85835i 0 −4.50000 + 7.79423i 0
49.6 0 −1.50000 − 2.59808i 0 − 0.713259i 0 16.2795 + 9.39900i 0 −4.50000 + 7.79423i 0
49.7 0 −1.50000 − 2.59808i 0 0.0822940i 0 14.7638 + 8.52389i 0 −4.50000 + 7.79423i 0
49.8 0 −1.50000 − 2.59808i 0 1.14918i 0 −28.9607 − 16.7205i 0 −4.50000 + 7.79423i 0
49.9 0 −1.50000 − 2.59808i 0 9.59964i 0 −22.8747 − 13.2067i 0 −4.50000 + 7.79423i 0
49.10 0 −1.50000 − 2.59808i 0 16.1421i 0 14.8729 + 8.58685i 0 −4.50000 + 7.79423i 0
49.11 0 −1.50000 − 2.59808i 0 16.4107i 0 20.2700 + 11.7029i 0 −4.50000 + 7.79423i 0
49.12 0 −1.50000 − 2.59808i 0 20.7584i 0 −21.2386 − 12.2621i 0 −4.50000 + 7.79423i 0
433.1 0 −1.50000 + 2.59808i 0 − 20.7584i 0 −21.2386 + 12.2621i 0 −4.50000 − 7.79423i 0
433.2 0 −1.50000 + 2.59808i 0 − 16.4107i 0 20.2700 − 11.7029i 0 −4.50000 − 7.79423i 0
433.3 0 −1.50000 + 2.59808i 0 − 16.1421i 0 14.8729 − 8.58685i 0 −4.50000 − 7.79423i 0
433.4 0 −1.50000 + 2.59808i 0 − 9.59964i 0 −22.8747 + 13.2067i 0 −4.50000 − 7.79423i 0
433.5 0 −1.50000 + 2.59808i 0 − 1.14918i 0 −28.9607 + 16.7205i 0 −4.50000 − 7.79423i 0
433.6 0 −1.50000 + 2.59808i 0 − 0.0822940i 0 14.7638 − 8.52389i 0 −4.50000 − 7.79423i 0
433.7 0 −1.50000 + 2.59808i 0 0.713259i 0 16.2795 − 9.39900i 0 −4.50000 − 7.79423i 0
433.8 0 −1.50000 + 2.59808i 0 3.71979i 0 8.41491 − 4.85835i 0 −4.50000 − 7.79423i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bv.j 24
4.b odd 2 1 312.4.bf.b ✓ 24
13.e even 6 1 inner 624.4.bv.j 24
52.i odd 6 1 312.4.bf.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.bf.b ✓ 24 4.b odd 2 1
312.4.bf.b ✓ 24 52.i odd 6 1
624.4.bv.j 24 1.a even 1 1 trivial
624.4.bv.j 24 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 1912 T_{5}^{22} + 1524076 T_{5}^{20} + 657843128 T_{5}^{18} + 166969402870 T_{5}^{16} + \cdots + 47\!\cdots\!44 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display