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Properties

Label 1205.2.d.c
Level $1205$
Weight $2$
Character orbit 1205.d
Analytic conductor $9.622$
Analytic rank $0$
Dimension $42$
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] = [1205,2,Mod(481,1205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1205.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1205 = 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1205.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: \(9.62197344356\)
Analytic rank: \(0\)
Dimension: \(42\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 6 q^{3} + 48 q^{4} - 42 q^{5} - 2 q^{6} + 6 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 6 q^{3} + 48 q^{4} - 42 q^{5} - 2 q^{6} + 6 q^{8} + 60 q^{9} - 8 q^{12} + 6 q^{15} + 52 q^{16} + 6 q^{18} - 48 q^{20} - 8 q^{24} + 42 q^{25} - 12 q^{27} - 20 q^{29} + 2 q^{30} - 8 q^{32} + 82 q^{36} - 6 q^{40} - 18 q^{41} - 60 q^{45} - 12 q^{47} - 44 q^{48} - 20 q^{49} - 28 q^{53} - 2 q^{54} - 40 q^{58} + 12 q^{59} + 8 q^{60} + 6 q^{61} + 50 q^{64} - 2 q^{67} + 28 q^{72} - 6 q^{75} - 14 q^{77} - 24 q^{79} - 52 q^{80} + 98 q^{81} + 38 q^{82} + 2 q^{83} - 20 q^{87} - 6 q^{90} - 48 q^{91} + 28 q^{94} + 16 q^{96} + 10 q^{97} + 20 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1 −2.65266 −2.55371 5.03660 −1.00000 6.77412 − 3.09402i −8.05508 3.52143 2.65266
481.2 −2.65266 −2.55371 5.03660 −1.00000 6.77412 3.09402i −8.05508 3.52143 2.65266
481.3 −2.59746 0.184723 4.74680 −1.00000 −0.479810 − 1.63057i −7.13471 −2.96588 2.59746
481.4 −2.59746 0.184723 4.74680 −1.00000 −0.479810 1.63057i −7.13471 −2.96588 2.59746
481.5 −2.43203 2.71572 3.91476 −1.00000 −6.60471 0.796450i −4.65676 4.37513 2.43203
481.6 −2.43203 2.71572 3.91476 −1.00000 −6.60471 − 0.796450i −4.65676 4.37513 2.43203
481.7 −1.96118 1.54545 1.84621 −1.00000 −3.03090 − 3.56501i 0.301608 −0.611577 1.96118
481.8 −1.96118 1.54545 1.84621 −1.00000 −3.03090 3.56501i 0.301608 −0.611577 1.96118
481.9 −1.77159 −1.65886 1.13852 −1.00000 2.93882 1.22826i 1.52619 −0.248169 1.77159
481.10 −1.77159 −1.65886 1.13852 −1.00000 2.93882 − 1.22826i 1.52619 −0.248169 1.77159
481.11 −1.43869 −3.24335 0.0698400 −1.00000 4.66619 − 2.49639i 2.77691 7.51934 1.43869
481.12 −1.43869 −3.24335 0.0698400 −1.00000 4.66619 2.49639i 2.77691 7.51934 1.43869
481.13 −1.42616 −0.123830 0.0339350 −1.00000 0.176601 2.22466i 2.80393 −2.98467 1.42616
481.14 −1.42616 −0.123830 0.0339350 −1.00000 0.176601 − 2.22466i 2.80393 −2.98467 1.42616
481.15 −1.14954 3.06063 −0.678558 −1.00000 −3.51831 − 4.38979i 3.07911 6.36744 1.14954
481.16 −1.14954 3.06063 −0.678558 −1.00000 −3.51831 4.38979i 3.07911 6.36744 1.14954
481.17 −0.563416 −1.90443 −1.68256 −1.00000 1.07299 − 5.09769i 2.07481 0.626872 0.563416
481.18 −0.563416 −1.90443 −1.68256 −1.00000 1.07299 5.09769i 2.07481 0.626872 0.563416
481.19 −0.410869 0.874016 −1.83119 −1.00000 −0.359106 − 2.07132i 1.57411 −2.23610 0.410869
481.20 −0.410869 0.874016 −1.83119 −1.00000 −0.359106 2.07132i 1.57411 −2.23610 0.410869
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.42
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1205.2.d.c ✓ 42
241.b even 2 1 inner 1205.2.d.c ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.d.c ✓ 42 1.a even 1 1 trivial
1205.2.d.c ✓ 42 241.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 33 T_{2}^{19} - T_{2}^{18} + 460 T_{2}^{17} + 29 T_{2}^{16} - 3532 T_{2}^{15} - 346 T_{2}^{14} + \cdots - 27 \) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\). Copy content Toggle raw display