↑

Properties

Label 1260.2.di.b
Level $1260$
Weight $2$
Character orbit 1260.di
Analytic conductor $10.061$
Analytic rank $0$
Dimension $92$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(709,1260)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.709");
 
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.di (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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q - 4 q^{5} + 2 q^{9} - 4 q^{11} - 23 q^{15} + 4 q^{19} + 8 q^{21} + 12 q^{25} - 8 q^{29} + 12 q^{31} - 21 q^{35} - 34 q^{39} + 24 q^{41} - 23 q^{45} + 34 q^{49} - 44 q^{51} + 4 q^{55} - 4 q^{59} - 10 q^{61} + 13 q^{65} - 6 q^{69} - 4 q^{71} - 17 q^{75} - 2 q^{79} - 2 q^{81} + 16 q^{85} - 40 q^{89} - 32 q^{91} + 10 q^{95} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
709.1 0 −1.71565 − 0.237766i 0 2.08150 − 0.816920i 0 −0.154199 − 2.64125i 0 2.88693 + 0.815848i 0
709.2 0 −1.67985 − 0.422033i 0 −0.921186 − 2.03750i 0 1.15665 + 2.37953i 0 2.64378 + 1.41790i 0
709.3 0 −1.66900 + 0.463072i 0 −2.15579 − 0.593784i 0 −2.40273 − 1.10764i 0 2.57113 − 1.54574i 0
709.4 0 −1.65903 − 0.497601i 0 −1.65727 + 1.50115i 0 2.33742 − 1.23955i 0 2.50479 + 1.65107i 0
709.5 0 −1.65280 − 0.517930i 0 1.49086 + 1.66654i 0 2.21097 + 1.45314i 0 2.46350 + 1.71207i 0
709.6 0 −1.64502 − 0.542144i 0 −1.29076 + 1.82591i 0 −1.78612 + 1.95186i 0 2.41216 + 1.78367i 0
709.7 0 −1.59361 + 0.678541i 0 −2.20754 − 0.356034i 0 1.76824 − 1.96808i 0 2.07916 − 2.16265i 0
709.8 0 −1.56222 + 0.747986i 0 2.16859 − 0.545185i 0 −0.107179 + 2.64358i 0 1.88103 − 2.33703i 0
709.9 0 −1.54644 + 0.780076i 0 0.485091 + 2.18282i 0 −2.53880 − 0.744650i 0 1.78296 − 2.41268i 0
709.10 0 −1.39714 + 1.02373i 0 0.584373 − 2.15836i 0 2.64457 − 0.0790527i 0 0.903973 − 2.86057i 0
709.11 0 −1.36411 − 1.06734i 0 0.0788554 − 2.23468i 0 −1.96682 + 1.76963i 0 0.721577 + 2.91193i 0
709.12 0 −1.15843 + 1.28765i 0 −1.61444 + 1.54712i 0 1.28049 + 2.31524i 0 −0.316075 − 2.98330i 0
709.13 0 −1.11697 − 1.32378i 0 −1.58114 − 1.58114i 0 1.59586 − 2.11027i 0 −0.504768 + 2.95723i 0
709.14 0 −1.05276 − 1.37539i 0 2.22471 − 0.225123i 0 −2.53929 + 0.742978i 0 −0.783408 + 2.89591i 0
709.15 0 −0.892594 + 1.48434i 0 −0.197849 − 2.22730i 0 −0.786252 − 2.52622i 0 −1.40655 − 2.64983i 0
709.16 0 −0.767556 + 1.55269i 0 2.23397 + 0.0969190i 0 −2.64193 + 0.142234i 0 −1.82171 − 2.38356i 0
709.17 0 −0.666363 − 1.59874i 0 1.67132 + 1.48549i 0 2.46160 + 0.969809i 0 −2.11192 + 2.13068i 0
709.18 0 −0.610376 + 1.62094i 0 0.319725 + 2.21309i 0 2.27413 + 1.35216i 0 −2.25488 − 1.97876i 0
709.19 0 −0.597316 − 1.62580i 0 0.908804 − 2.04306i 0 2.18198 − 1.49631i 0 −2.28643 + 1.94223i 0
709.20 0 −0.561007 + 1.63868i 0 −1.39803 − 1.74514i 0 −1.49402 + 2.18355i 0 −2.37054 − 1.83862i 0
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 709.46
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{23} + T_{11}^{22} - 143 T_{11}^{21} - 188 T_{11}^{20} + 8596 T_{11}^{19} + 14113 T_{11}^{18} + \cdots + 15875136 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display