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Properties

Label 930.4.d.d
Level $930$
Weight $4$
Character orbit 930.d
Analytic conductor $54.872$
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] = [930,4,Mod(559,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.559");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 930.d (of order \(2\), degree \(1\), 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: \(54.8717763053\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 96 q^{4} - 22 q^{5} - 144 q^{6} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 96 q^{4} - 22 q^{5} - 144 q^{6} - 216 q^{9} + 44 q^{10} - 50 q^{11} - 52 q^{14} + 66 q^{15} + 384 q^{16} - 410 q^{19} + 88 q^{20} - 78 q^{21} + 576 q^{24} - 8 q^{25} - 80 q^{26} + 188 q^{29} + 132 q^{30} - 744 q^{31} + 424 q^{34} + 2 q^{35} + 864 q^{36} - 120 q^{39} - 176 q^{40} + 152 q^{41} + 200 q^{44} + 198 q^{45} - 668 q^{46} - 4702 q^{49} + 1728 q^{50} + 636 q^{51} + 1296 q^{54} + 832 q^{55} + 208 q^{56} - 4412 q^{59} - 264 q^{60} - 524 q^{61} - 1536 q^{64} + 1856 q^{65} + 300 q^{66} - 1002 q^{69} + 2108 q^{70} + 1114 q^{71} - 2392 q^{74} + 2592 q^{75} + 1640 q^{76} - 2894 q^{79} - 352 q^{80} + 1944 q^{81} + 312 q^{84} + 4922 q^{85} + 1060 q^{86} - 3870 q^{89} - 396 q^{90} - 2424 q^{91} + 968 q^{94} + 1720 q^{95} - 2304 q^{96} + 450 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1 − 2.00000i − 3.00000i −4.00000 −11.1015 + 1.32502i −6.00000 − 20.6417i 8.00000i −9.00000 2.65004 + 22.2031i
559.2 − 2.00000i − 3.00000i −4.00000 −8.99906 − 6.63452i −6.00000 − 34.9388i 8.00000i −9.00000 −13.2690 + 17.9981i
559.3 − 2.00000i − 3.00000i −4.00000 −8.93720 − 6.71762i −6.00000 2.36380i 8.00000i −9.00000 −13.4352 + 17.8744i
559.4 − 2.00000i − 3.00000i −4.00000 −7.91656 + 7.89481i −6.00000 − 17.1903i 8.00000i −9.00000 15.7896 + 15.8331i
559.5 − 2.00000i − 3.00000i −4.00000 −7.27003 − 8.49392i −6.00000 6.72354i 8.00000i −9.00000 −16.9878 + 14.5401i
559.6 − 2.00000i − 3.00000i −4.00000 −3.10506 + 10.7405i −6.00000 14.5634i 8.00000i −9.00000 21.4810 + 6.21012i
559.7 − 2.00000i − 3.00000i −4.00000 −1.55943 − 11.0711i −6.00000 16.8419i 8.00000i −9.00000 −22.1421 + 3.11886i
559.8 − 2.00000i − 3.00000i −4.00000 1.45233 + 11.0856i −6.00000 33.3634i 8.00000i −9.00000 22.1712 − 2.90465i
559.9 − 2.00000i − 3.00000i −4.00000 8.11245 + 7.69338i −6.00000 − 22.7714i 8.00000i −9.00000 15.3868 − 16.2249i
559.10 − 2.00000i − 3.00000i −4.00000 8.84400 + 6.83986i −6.00000 − 34.6053i 8.00000i −9.00000 13.6797 − 17.6880i
559.11 − 2.00000i − 3.00000i −4.00000 9.27878 − 6.23732i −6.00000 11.3524i 8.00000i −9.00000 −12.4746 − 18.5576i
559.12 − 2.00000i − 3.00000i −4.00000 10.2013 + 4.57524i −6.00000 31.9390i 8.00000i −9.00000 9.15049 − 20.4027i
559.13 2.00000i 3.00000i −4.00000 −11.1015 − 1.32502i −6.00000 20.6417i − 8.00000i −9.00000 2.65004 − 22.2031i
559.14 2.00000i 3.00000i −4.00000 −8.99906 + 6.63452i −6.00000 34.9388i − 8.00000i −9.00000 −13.2690 − 17.9981i
559.15 2.00000i 3.00000i −4.00000 −8.93720 + 6.71762i −6.00000 − 2.36380i − 8.00000i −9.00000 −13.4352 − 17.8744i
559.16 2.00000i 3.00000i −4.00000 −7.91656 − 7.89481i −6.00000 17.1903i − 8.00000i −9.00000 15.7896 − 15.8331i
559.17 2.00000i 3.00000i −4.00000 −7.27003 + 8.49392i −6.00000 − 6.72354i − 8.00000i −9.00000 −16.9878 − 14.5401i
559.18 2.00000i 3.00000i −4.00000 −3.10506 − 10.7405i −6.00000 − 14.5634i − 8.00000i −9.00000 21.4810 − 6.21012i
559.19 2.00000i 3.00000i −4.00000 −1.55943 + 11.0711i −6.00000 − 16.8419i − 8.00000i −9.00000 −22.1421 − 3.11886i
559.20 2.00000i 3.00000i −4.00000 1.45233 − 11.0856i −6.00000 − 33.3634i − 8.00000i −9.00000 22.1712 + 2.90465i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.4.d.d ✓ 24
5.b even 2 1 inner 930.4.d.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.4.d.d ✓ 24 1.a even 1 1 trivial
930.4.d.d ✓ 24 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 6467 T_{7}^{22} + 17968179 T_{7}^{20} + 28130488513 T_{7}^{18} + 27398394644008 T_{7}^{16} + \cdots + 21\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display