Properties

Label 1260.2.by.b
Level $1260$
Weight $2$
Character orbit 1260.by
Analytic conductor $10.061$
Analytic rank $0$
Dimension $92$
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] = [1260,2,Mod(529,1260)]
 
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, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.529");
 
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.by (of order \(6\), 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: \(92\)
Relative dimension: \(46\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 2 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 2 q^{5} + 8 q^{9} + 2 q^{11} - 23 q^{15} + 4 q^{19} - 16 q^{21} - 6 q^{25} - 8 q^{29} - 24 q^{31} - 21 q^{35} + 8 q^{39} + 24 q^{41} + 22 q^{45} - 32 q^{49} + 58 q^{51} + 4 q^{55} + 8 q^{59} + 20 q^{61} - 26 q^{65} - 6 q^{69} - 4 q^{71} + 7 q^{75} + 4 q^{79} + 40 q^{81} + 16 q^{85} - 40 q^{89} - 32 q^{91} - 20 q^{95} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0 −1.73178 0.0308374i 0 −1.82997 1.28499i 0 2.58090 0.582198i 0 2.99810 + 0.106807i 0
529.2 0 −1.72845 + 0.111624i 0 −1.03305 + 1.98313i 0 1.19778 + 2.35909i 0 2.97508 0.385872i 0
529.3 0 −1.70896 + 0.281868i 0 1.75673 + 1.38344i 0 −2.30807 1.29337i 0 2.84110 0.963403i 0
529.4 0 −1.69964 + 0.333494i 0 −0.812314 2.08330i 0 −1.14400 + 2.38564i 0 2.77756 1.13364i 0
529.5 0 −1.69435 0.359407i 0 2.14707 0.624586i 0 −2.64530 + 0.0486853i 0 2.74165 + 1.21792i 0
529.6 0 −1.61264 + 0.631969i 0 0.196400 + 2.22743i 0 2.20782 1.45793i 0 2.20123 2.03828i 0
529.7 0 −1.58514 0.698092i 0 −2.16138 0.573097i 0 −1.25382 2.32979i 0 2.02534 + 2.21315i 0
529.8 0 −1.52923 + 0.813293i 0 1.98915 1.02141i 0 2.53981 0.741175i 0 1.67711 2.48743i 0
529.9 0 −1.44879 0.949220i 0 1.64783 + 1.51151i 0 1.91428 + 1.82634i 0 1.19796 + 2.75043i 0
529.10 0 −1.42888 0.978925i 0 −1.55644 + 1.60546i 0 −2.23582 + 1.41461i 0 1.08341 + 2.79754i 0
529.11 0 −1.38444 1.04083i 0 0.795436 2.08980i 0 0.820288 2.51538i 0 0.833332 + 2.88194i 0
529.12 0 −1.24055 + 1.20874i 0 0.639876 2.14256i 0 −1.73123 2.00071i 0 0.0779117 2.99899i 0
529.13 0 −1.23553 1.21386i 0 0.563662 2.16386i 0 2.16061 + 1.52701i 0 0.0530816 + 2.99953i 0
529.14 0 −1.10932 + 1.33019i 0 1.31494 + 1.80858i 0 −0.204856 + 2.63781i 0 −0.538805 2.95122i 0
529.15 0 −1.05137 + 1.37646i 0 −2.12213 + 0.704663i 0 2.07068 + 1.64690i 0 −0.789262 2.89432i 0
529.16 0 −0.664746 + 1.59941i 0 −0.917391 + 2.03921i 0 −0.626206 2.57058i 0 −2.11622 2.12640i 0
529.17 0 −0.651915 1.60468i 0 −1.74822 + 1.39417i 0 2.36449 1.18709i 0 −2.15001 + 2.09223i 0
529.18 0 −0.587940 + 1.62921i 0 2.15987 0.578741i 0 −1.02962 + 2.43719i 0 −2.30865 1.91576i 0
529.19 0 −0.474433 1.66581i 0 −1.30394 1.81652i 0 −2.63906 + 0.188075i 0 −2.54983 + 1.58063i 0
529.20 0 −0.398582 1.68557i 0 2.12867 0.684661i 0 −0.0952265 2.64404i 0 −2.68227 + 1.34367i 0
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.h even 3 1 inner
315.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.by.b 92
3.b odd 2 1 3780.2.by.b 92
5.b even 2 1 inner 1260.2.by.b 92
7.c even 3 1 1260.2.di.b yes 92
9.c even 3 1 1260.2.di.b yes 92
9.d odd 6 1 3780.2.di.b 92
15.d odd 2 1 3780.2.by.b 92
21.h odd 6 1 3780.2.di.b 92
35.j even 6 1 1260.2.di.b yes 92
45.h odd 6 1 3780.2.di.b 92
45.j even 6 1 1260.2.di.b yes 92
63.h even 3 1 inner 1260.2.by.b 92
63.j odd 6 1 3780.2.by.b 92
105.o odd 6 1 3780.2.di.b 92
315.r even 6 1 inner 1260.2.by.b 92
315.br odd 6 1 3780.2.by.b 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.by.b 92 1.a even 1 1 trivial
1260.2.by.b 92 5.b even 2 1 inner
1260.2.by.b 92 63.h even 3 1 inner
1260.2.by.b 92 315.r even 6 1 inner
1260.2.di.b yes 92 7.c even 3 1
1260.2.di.b yes 92 9.c even 3 1
1260.2.di.b yes 92 35.j even 6 1
1260.2.di.b yes 92 45.j even 6 1
3780.2.by.b 92 3.b odd 2 1
3780.2.by.b 92 15.d odd 2 1
3780.2.by.b 92 63.j odd 6 1
3780.2.by.b 92 315.br odd 6 1
3780.2.di.b 92 9.d odd 6 1
3780.2.di.b 92 21.h odd 6 1
3780.2.di.b 92 45.h odd 6 1
3780.2.di.b 92 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{46} - T_{11}^{45} + 144 T_{11}^{44} - 233 T_{11}^{43} + 12041 T_{11}^{42} - 23805 T_{11}^{41} + 686061 T_{11}^{40} - 1575159 T_{11}^{39} + 29596896 T_{11}^{38} - 74834889 T_{11}^{37} + \cdots + 252019943018496 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display