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Properties

Label 930.4.d.e
Level $930$
Weight $4$
Character orbit 930.d
Analytic conductor $54.872$
Analytic rank $0$
Dimension $26$
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: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 104 q^{4} - 2 q^{5} + 156 q^{6} - 234 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 104 q^{4} - 2 q^{5} + 156 q^{6} - 234 q^{9} + 8 q^{10} + 62 q^{11} - 60 q^{14} - 12 q^{15} + 416 q^{16} - 466 q^{19} + 8 q^{20} + 90 q^{21} - 624 q^{24} - 42 q^{25} + 152 q^{26} - 260 q^{29} - 12 q^{30} + 806 q^{31} - 512 q^{34} + 246 q^{35} + 936 q^{36} - 228 q^{39} - 32 q^{40} + 940 q^{41} - 248 q^{44} + 18 q^{45} + 1228 q^{46} - 2120 q^{49} + 320 q^{50} + 768 q^{51} - 1404 q^{54} + 724 q^{55} + 240 q^{56} - 1520 q^{59} + 48 q^{60} + 2544 q^{61} - 1664 q^{64} + 568 q^{65} + 372 q^{66} - 1842 q^{69} + 388 q^{70} + 690 q^{71} - 1264 q^{74} - 480 q^{75} + 1864 q^{76} - 3778 q^{79} - 32 q^{80} + 2106 q^{81} - 360 q^{84} - 2918 q^{85} + 604 q^{86} - 4206 q^{89} - 72 q^{90} + 6312 q^{91} - 3000 q^{94} + 672 q^{95} + 2496 q^{96} - 558 q^{99}+O(q^{100}) \) 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 −11.1660 − 0.566870i 6.00000 23.5000i 8.00000i −9.00000 −1.13374 + 22.3319i
559.2 − 2.00000i 3.00000i −4.00000 −9.48168 − 5.92433i 6.00000 − 19.7694i 8.00000i −9.00000 −11.8487 + 18.9634i
559.3 − 2.00000i 3.00000i −4.00000 −9.29234 + 6.21711i 6.00000 − 23.5649i 8.00000i −9.00000 12.4342 + 18.5847i
559.4 − 2.00000i 3.00000i −4.00000 −8.33412 + 7.45269i 6.00000 11.4090i 8.00000i −9.00000 14.9054 + 16.6682i
559.5 − 2.00000i 3.00000i −4.00000 −6.85061 − 8.83568i 6.00000 − 1.91123i 8.00000i −9.00000 −17.6714 + 13.7012i
559.6 − 2.00000i 3.00000i −4.00000 −1.55756 − 11.0713i 6.00000 15.5721i 8.00000i −9.00000 −22.1426 + 3.11511i
559.7 − 2.00000i 3.00000i −4.00000 0.0618835 + 11.1802i 6.00000 − 31.9571i 8.00000i −9.00000 22.3603 − 0.123767i
559.8 − 2.00000i 3.00000i −4.00000 3.67028 + 10.5607i 6.00000 9.02574i 8.00000i −9.00000 21.1215 − 7.34056i
559.9 − 2.00000i 3.00000i −4.00000 5.48941 − 9.73993i 6.00000 − 23.8960i 8.00000i −9.00000 −19.4799 − 10.9788i
559.10 − 2.00000i 3.00000i −4.00000 7.44048 − 8.34501i 6.00000 23.9119i 8.00000i −9.00000 −16.6900 − 14.8810i
559.11 − 2.00000i 3.00000i −4.00000 8.92226 + 6.73745i 6.00000 − 2.69865i 8.00000i −9.00000 13.4749 − 17.8445i
559.12 − 2.00000i 3.00000i −4.00000 9.12220 + 6.46417i 6.00000 28.9290i 8.00000i −9.00000 12.9283 − 18.2444i
559.13 − 2.00000i 3.00000i −4.00000 10.9757 − 2.12917i 6.00000 − 23.5504i 8.00000i −9.00000 −4.25834 − 21.9515i
559.14 2.00000i − 3.00000i −4.00000 −11.1660 + 0.566870i 6.00000 − 23.5000i − 8.00000i −9.00000 −1.13374 − 22.3319i
559.15 2.00000i − 3.00000i −4.00000 −9.48168 + 5.92433i 6.00000 19.7694i − 8.00000i −9.00000 −11.8487 − 18.9634i
559.16 2.00000i − 3.00000i −4.00000 −9.29234 − 6.21711i 6.00000 23.5649i − 8.00000i −9.00000 12.4342 − 18.5847i
559.17 2.00000i − 3.00000i −4.00000 −8.33412 − 7.45269i 6.00000 − 11.4090i − 8.00000i −9.00000 14.9054 − 16.6682i
559.18 2.00000i − 3.00000i −4.00000 −6.85061 + 8.83568i 6.00000 1.91123i − 8.00000i −9.00000 −17.6714 − 13.7012i
559.19 2.00000i − 3.00000i −4.00000 −1.55756 + 11.0713i 6.00000 − 15.5721i − 8.00000i −9.00000 −22.1426 − 3.11511i
559.20 2.00000i − 3.00000i −4.00000 0.0618835 − 11.1802i 6.00000 31.9571i − 8.00000i −9.00000 22.3603 + 0.123767i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.26
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.e ✓ 26
5.b even 2 1 inner 930.4.d.e ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.4.d.e ✓ 26 1.a even 1 1 trivial
930.4.d.e ✓ 26 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{26} + 5519 T_{7}^{24} + 13453439 T_{7}^{22} + 19084914461 T_{7}^{20} + 17461826627500 T_{7}^{18} + \cdots + 12\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display