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Properties

Label 252.4.w.a
Level $252$
Weight $4$
Character orbit 252.w
Analytic conductor $14.868$
Analytic rank $0$
Dimension $48$
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] = [252,4,Mod(5,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.w (of order \(6\), 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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{7} + 12 q^{9} + 12 q^{11} - 36 q^{13} + 66 q^{15} - 72 q^{17} + 24 q^{21} + 30 q^{23} - 600 q^{25} - 396 q^{27} + 42 q^{29} + 390 q^{35} + 84 q^{37} - 840 q^{39} - 618 q^{41} - 42 q^{43} + 366 q^{45} + 396 q^{47} + 318 q^{49} - 738 q^{51} - 1620 q^{53} + 492 q^{57} + 1500 q^{59} + 672 q^{63} - 588 q^{67} - 924 q^{69} + 564 q^{75} - 2472 q^{77} + 1608 q^{79} + 2592 q^{81} - 360 q^{85} + 2640 q^{87} + 1722 q^{89} + 540 q^{91} + 660 q^{93} - 792 q^{97} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −5.18021 + 0.406745i 0 −8.68596 − 15.0445i 0 6.03770 − 17.5085i 0 26.6691 − 4.21405i 0
5.2 0 −5.15302 + 0.668133i 0 9.18977 + 15.9171i 0 18.3240 − 2.68897i 0 26.1072 − 6.88581i 0
5.3 0 −4.97073 − 1.51388i 0 −3.24590 − 5.62207i 0 −7.45627 + 16.9530i 0 22.4163 + 15.0502i 0
5.4 0 −4.76836 − 2.06463i 0 9.23798 + 16.0007i 0 −13.5371 − 12.6391i 0 18.4746 + 19.6898i 0
5.5 0 −4.59055 + 2.43451i 0 1.72890 + 2.99454i 0 −17.7636 + 5.23980i 0 15.1463 − 22.3515i 0
5.6 0 −4.38438 − 2.78877i 0 −1.97944 − 3.42849i 0 15.7046 + 9.81667i 0 11.4456 + 24.4540i 0
5.7 0 −3.44710 + 3.88813i 0 −0.950243 − 1.64587i 0 14.2951 + 11.7749i 0 −3.23505 − 26.8055i 0
5.8 0 −2.54719 − 4.52900i 0 −3.81512 − 6.60797i 0 −15.6850 − 9.84781i 0 −14.0236 + 23.0725i 0
5.9 0 −1.96476 + 4.81038i 0 2.75454 + 4.77100i 0 −0.994545 − 18.4935i 0 −19.2795 − 18.9024i 0
5.10 0 −1.35688 − 5.01586i 0 3.33893 + 5.78320i 0 14.9872 − 10.8804i 0 −23.3178 + 13.6118i 0
5.11 0 −0.808952 + 5.13280i 0 −10.5226 − 18.2257i 0 −18.5183 + 0.266265i 0 −25.6912 − 8.30437i 0
5.12 0 −0.452892 − 5.17638i 0 6.54401 + 11.3346i 0 1.26985 + 18.4767i 0 −26.5898 + 4.68868i 0
5.13 0 0.569877 + 5.16481i 0 8.28296 + 14.3465i 0 −3.56863 + 18.1732i 0 −26.3505 + 5.88661i 0
5.14 0 0.973978 + 5.10405i 0 −6.29153 − 10.8972i 0 18.2954 + 2.87731i 0 −25.1027 + 9.94247i 0
5.15 0 1.05398 − 5.08813i 0 −9.99264 − 17.3078i 0 −0.138706 + 18.5197i 0 −24.7782 − 10.7256i 0
5.16 0 3.03995 − 4.21411i 0 5.20294 + 9.01176i 0 −16.5030 + 8.40550i 0 −8.51746 − 25.6213i 0
5.17 0 3.04499 − 4.21047i 0 −4.90701 − 8.49919i 0 13.1134 − 13.0781i 0 −8.45605 − 25.6417i 0
5.18 0 3.58371 + 3.76258i 0 −1.13691 − 1.96919i 0 −14.5181 + 11.4989i 0 −1.31403 + 26.9680i 0
5.19 0 3.79518 + 3.54917i 0 5.15592 + 8.93032i 0 −5.91622 − 17.5499i 0 1.80675 + 26.9395i 0
5.20 0 3.80953 + 3.53377i 0 −3.77250 − 6.53416i 0 14.8160 − 11.1124i 0 2.02498 + 26.9240i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.w.a ✓ 48
3.b odd 2 1 756.4.w.a 48
7.d odd 6 1 252.4.bm.a yes 48
9.c even 3 1 756.4.bm.a 48
9.d odd 6 1 252.4.bm.a yes 48
21.g even 6 1 756.4.bm.a 48
63.i even 6 1 inner 252.4.w.a ✓ 48
63.t odd 6 1 756.4.w.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.w.a ✓ 48 1.a even 1 1 trivial
252.4.w.a ✓ 48 63.i even 6 1 inner
252.4.bm.a yes 48 7.d odd 6 1
252.4.bm.a yes 48 9.d odd 6 1
756.4.w.a 48 3.b odd 2 1
756.4.w.a 48 63.t odd 6 1
756.4.bm.a 48 9.c even 3 1
756.4.bm.a 48 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(252, [\chi])\).