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Properties

Label 1260.2.t.d
Level $1260$
Weight $2$
Character orbit 1260.t
Analytic conductor $10.061$
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] = [1260,2,Mod(961,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.t (of order \(3\), 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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - q^{3} + 26 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - q^{3} + 26 q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} + 7 q^{13} - q^{15} + 8 q^{17} - 9 q^{19} - 11 q^{21} + 6 q^{23} + 26 q^{25} - 7 q^{27} - 7 q^{29} - q^{31} - q^{33} + 3 q^{35} + 10 q^{37} + 7 q^{39} + 13 q^{41} - 8 q^{43} + 3 q^{45} + 6 q^{47} + 11 q^{49} - 6 q^{51} + 17 q^{53} - 2 q^{55} - 16 q^{57} - 8 q^{59} + 10 q^{61} + 41 q^{63} + 7 q^{65} + 24 q^{67} + 54 q^{69} + 4 q^{71} - 17 q^{73} - q^{75} - 11 q^{77} + 19 q^{79} - 21 q^{81} + 9 q^{83} + 8 q^{85} + 26 q^{87} + q^{89} - 2 q^{91} + 35 q^{93} - 9 q^{95} + 9 q^{97} + 33 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1 0 −1.73092 + 0.0625133i 0 1.00000 0 2.63998 − 0.174676i 0 2.99218 − 0.216411i 0
961.2 0 −1.68638 − 0.395134i 0 1.00000 0 −1.81728 − 1.92289i 0 2.68774 + 1.33269i 0
961.3 0 −1.42078 + 0.990642i 0 1.00000 0 −1.60522 + 2.10316i 0 1.03726 − 2.81498i 0
961.4 0 −1.10956 + 1.32999i 0 1.00000 0 2.61266 + 0.417155i 0 −0.537734 − 2.95141i 0
961.5 0 −0.930887 − 1.46063i 0 1.00000 0 −2.18169 + 1.49674i 0 −1.26690 + 2.71937i 0
961.6 0 −0.890787 − 1.48543i 0 1.00000 0 1.80204 − 1.93717i 0 −1.41300 + 2.64640i 0
961.7 0 0.287054 − 1.70810i 0 1.00000 0 0.538095 + 2.59045i 0 −2.83520 − 0.980632i 0
961.8 0 0.411737 + 1.68240i 0 1.00000 0 0.375786 + 2.61893i 0 −2.66095 + 1.38541i 0
961.9 0 0.715946 − 1.57716i 0 1.00000 0 0.120551 − 2.64300i 0 −1.97484 − 2.25832i 0
961.10 0 1.19904 − 1.24992i 0 1.00000 0 −2.29230 + 1.32113i 0 −0.124621 − 2.99741i 0
961.11 0 1.33148 + 1.10778i 0 1.00000 0 −2.61971 − 0.370327i 0 0.545654 + 2.94996i 0
961.12 0 1.64674 + 0.536888i 0 1.00000 0 1.57985 − 2.12228i 0 2.42350 + 1.76823i 0
961.13 0 1.67733 − 0.431913i 0 1.00000 0 2.34723 + 1.22087i 0 2.62690 − 1.44892i 0
1201.1 0 −1.73092 − 0.0625133i 0 1.00000 0 2.63998 + 0.174676i 0 2.99218 + 0.216411i 0
1201.2 0 −1.68638 + 0.395134i 0 1.00000 0 −1.81728 + 1.92289i 0 2.68774 − 1.33269i 0
1201.3 0 −1.42078 − 0.990642i 0 1.00000 0 −1.60522 − 2.10316i 0 1.03726 + 2.81498i 0
1201.4 0 −1.10956 − 1.32999i 0 1.00000 0 2.61266 − 0.417155i 0 −0.537734 + 2.95141i 0
1201.5 0 −0.930887 + 1.46063i 0 1.00000 0 −2.18169 − 1.49674i 0 −1.26690 − 2.71937i 0
1201.6 0 −0.890787 + 1.48543i 0 1.00000 0 1.80204 + 1.93717i 0 −1.41300 − 2.64640i 0
1201.7 0 0.287054 + 1.70810i 0 1.00000 0 0.538095 − 2.59045i 0 −2.83520 + 0.980632i 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.t.d yes 26
3.b odd 2 1 3780.2.t.d 26
7.c even 3 1 1260.2.q.d ✓ 26
9.c even 3 1 1260.2.q.d ✓ 26
9.d odd 6 1 3780.2.q.d 26
21.h odd 6 1 3780.2.q.d 26
63.g even 3 1 inner 1260.2.t.d yes 26
63.n odd 6 1 3780.2.t.d 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.q.d ✓ 26 7.c even 3 1
1260.2.q.d ✓ 26 9.c even 3 1
1260.2.t.d yes 26 1.a even 1 1 trivial
1260.2.t.d yes 26 63.g even 3 1 inner
3780.2.q.d 26 9.d odd 6 1
3780.2.q.d 26 21.h odd 6 1
3780.2.t.d 26 3.b odd 2 1
3780.2.t.d 26 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{13} + T_{11}^{12} - 66 T_{11}^{11} - 36 T_{11}^{10} + 1395 T_{11}^{9} - 39 T_{11}^{8} + \cdots - 15552 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display