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Properties

Label 618.2.d.a
Level $618$
Weight $2$
Character orbit 618.d
Analytic conductor $4.935$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [618,2,Mod(617,618)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(618, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("618.617");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.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: \(4.93475484492\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{4} + 16 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{4} + 16 q^{7} + 8 q^{9} - 4 q^{13} - 16 q^{15} + 36 q^{16} + 12 q^{19} + 48 q^{25} - 16 q^{28} - 4 q^{30} - 4 q^{33} - 8 q^{36} - 40 q^{46} + 44 q^{49} + 4 q^{52} + 48 q^{55} - 28 q^{58} + 16 q^{60} - 20 q^{61} - 64 q^{63} - 36 q^{64} - 12 q^{76} - 40 q^{79} - 16 q^{82} - 72 q^{91} + 68 q^{93} + 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
617.1 − 1.00000i −1.71485 + 0.243496i −1.00000 1.75068 0.243496 + 1.71485i −3.43952 1.00000i 2.88142 − 0.835119i − 1.75068i
617.2 − 1.00000i −1.68990 + 0.379791i −1.00000 −0.460542 0.379791 + 1.68990i 2.41687 1.00000i 2.71152 − 1.28362i 0.460542i
617.3 − 1.00000i −1.62094 − 0.610375i −1.00000 −2.75094 −0.610375 + 1.62094i −3.46895 1.00000i 2.25489 + 1.97876i 2.75094i
617.4 − 1.00000i −1.57327 − 0.724446i −1.00000 3.90722 −0.724446 + 1.57327i 0.121232 1.00000i 1.95035 + 2.27950i − 3.90722i
617.5 − 1.00000i −1.15729 − 1.28867i −1.00000 −1.27559 −1.28867 + 1.15729i 4.86590 1.00000i −0.321341 + 2.98274i 1.27559i
617.6 − 1.00000i −0.954161 + 1.44554i −1.00000 −0.996429 1.44554 + 0.954161i 0.677088 1.00000i −1.17915 − 2.75855i 0.996429i
617.7 − 1.00000i −0.948854 + 1.44903i −1.00000 4.03047 1.44903 + 0.948854i 3.65770 1.00000i −1.19935 − 2.74983i − 4.03047i
617.8 − 1.00000i −0.533567 − 1.64782i −1.00000 0.191227 −1.64782 + 0.533567i −2.20699 1.00000i −2.43061 + 1.75844i − 0.191227i
617.9 − 1.00000i −0.407604 + 1.68341i −1.00000 −3.46233 1.68341 + 0.407604i 1.37666 1.00000i −2.66772 − 1.37233i 3.46233i
617.10 − 1.00000i 0.407604 − 1.68341i −1.00000 3.46233 −1.68341 − 0.407604i 1.37666 1.00000i −2.66772 − 1.37233i − 3.46233i
617.11 − 1.00000i 0.533567 + 1.64782i −1.00000 −0.191227 1.64782 − 0.533567i −2.20699 1.00000i −2.43061 + 1.75844i 0.191227i
617.12 − 1.00000i 0.948854 − 1.44903i −1.00000 −4.03047 −1.44903 − 0.948854i 3.65770 1.00000i −1.19935 − 2.74983i 4.03047i
617.13 − 1.00000i 0.954161 − 1.44554i −1.00000 0.996429 −1.44554 − 0.954161i 0.677088 1.00000i −1.17915 − 2.75855i − 0.996429i
617.14 − 1.00000i 1.15729 + 1.28867i −1.00000 1.27559 1.28867 − 1.15729i 4.86590 1.00000i −0.321341 + 2.98274i − 1.27559i
617.15 − 1.00000i 1.57327 + 0.724446i −1.00000 −3.90722 0.724446 − 1.57327i 0.121232 1.00000i 1.95035 + 2.27950i 3.90722i
617.16 − 1.00000i 1.62094 + 0.610375i −1.00000 2.75094 0.610375 − 1.62094i −3.46895 1.00000i 2.25489 + 1.97876i − 2.75094i
617.17 − 1.00000i 1.68990 − 0.379791i −1.00000 0.460542 −0.379791 − 1.68990i 2.41687 1.00000i 2.71152 − 1.28362i − 0.460542i
617.18 − 1.00000i 1.71485 − 0.243496i −1.00000 −1.75068 −0.243496 − 1.71485i −3.43952 1.00000i 2.88142 − 0.835119i 1.75068i
617.19 1.00000i −1.71485 − 0.243496i −1.00000 1.75068 0.243496 − 1.71485i −3.43952 − 1.00000i 2.88142 + 0.835119i 1.75068i
617.20 1.00000i −1.68990 − 0.379791i −1.00000 −0.460542 0.379791 − 1.68990i 2.41687 − 1.00000i 2.71152 + 1.28362i − 0.460542i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 617.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.b odd 2 1 inner
309.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 618.2.d.a ✓ 36
3.b odd 2 1 inner 618.2.d.a ✓ 36
103.b odd 2 1 inner 618.2.d.a ✓ 36
309.c even 2 1 inner 618.2.d.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.d.a ✓ 36 1.a even 1 1 trivial
618.2.d.a ✓ 36 3.b odd 2 1 inner
618.2.d.a ✓ 36 103.b odd 2 1 inner
618.2.d.a ✓ 36 309.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(618, [\chi])\).