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Properties

Label 624.4.bc.d
Level $624$
Weight $4$
Character orbit 624.bc
Analytic conductor $36.817$
Analytic rank $0$
Dimension $28$
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(31,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bc (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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 3.00000i 0 −13.9138 + 13.9138i 0 3.70358 − 3.70358i 0 −9.00000 0
31.2 0 3.00000i 0 −11.5719 + 11.5719i 0 25.2676 − 25.2676i 0 −9.00000 0
31.3 0 3.00000i 0 −8.48307 + 8.48307i 0 −5.05973 + 5.05973i 0 −9.00000 0
31.4 0 3.00000i 0 −6.77018 + 6.77018i 0 −21.6794 + 21.6794i 0 −9.00000 0
31.5 0 3.00000i 0 −6.10630 + 6.10630i 0 −23.8576 + 23.8576i 0 −9.00000 0
31.6 0 3.00000i 0 −2.84589 + 2.84589i 0 9.35646 − 9.35646i 0 −9.00000 0
31.7 0 3.00000i 0 −1.88620 + 1.88620i 0 −9.36349 + 9.36349i 0 −9.00000 0
31.8 0 3.00000i 0 −0.273393 + 0.273393i 0 21.0333 − 21.0333i 0 −9.00000 0
31.9 0 3.00000i 0 5.36394 − 5.36394i 0 11.4083 − 11.4083i 0 −9.00000 0
31.10 0 3.00000i 0 7.68384 − 7.68384i 0 −9.93516 + 9.93516i 0 −9.00000 0
31.11 0 3.00000i 0 7.87509 − 7.87509i 0 15.6298 − 15.6298i 0 −9.00000 0
31.12 0 3.00000i 0 8.84364 − 8.84364i 0 3.20396 − 3.20396i 0 −9.00000 0
31.13 0 3.00000i 0 10.3664 − 10.3664i 0 −6.70327 + 6.70327i 0 −9.00000 0
31.14 0 3.00000i 0 13.7178 − 13.7178i 0 −9.00439 + 9.00439i 0 −9.00000 0
463.1 0 − 3.00000i 0 −13.9138 − 13.9138i 0 3.70358 + 3.70358i 0 −9.00000 0
463.2 0 − 3.00000i 0 −11.5719 − 11.5719i 0 25.2676 + 25.2676i 0 −9.00000 0
463.3 0 − 3.00000i 0 −8.48307 − 8.48307i 0 −5.05973 − 5.05973i 0 −9.00000 0
463.4 0 − 3.00000i 0 −6.77018 − 6.77018i 0 −21.6794 − 21.6794i 0 −9.00000 0
463.5 0 − 3.00000i 0 −6.10630 − 6.10630i 0 −23.8576 − 23.8576i 0 −9.00000 0
463.6 0 − 3.00000i 0 −2.84589 − 2.84589i 0 9.35646 + 9.35646i 0 −9.00000 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bc.d yes 28
4.b odd 2 1 624.4.bc.c ✓ 28
13.d odd 4 1 624.4.bc.c ✓ 28
52.f even 4 1 inner 624.4.bc.d yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.bc.c ✓ 28 4.b odd 2 1
624.4.bc.c ✓ 28 13.d odd 4 1
624.4.bc.d yes 28 1.a even 1 1 trivial
624.4.bc.d yes 28 52.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{28} - 4 T_{5}^{27} + 8 T_{5}^{26} + 72 T_{5}^{25} + 250468 T_{5}^{24} - 1122080 T_{5}^{23} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
\( T_{7}^{28} - 8 T_{7}^{27} + 32 T_{7}^{26} + 6464 T_{7}^{25} + 2466704 T_{7}^{24} - 15378944 T_{7}^{23} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display