Properties

Label 270.4.m.a
Level $270$
Weight $4$
Character orbit 270.m
Analytic conductor $15.931$
Analytic rank $0$
Dimension $72$
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] = [270,4,Mod(17,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.m (of order \(12\), degree \(4\), not 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: \(15.9305157015\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 48 q^{11} + 576 q^{16} + 192 q^{20} + 312 q^{23} + 288 q^{25} - 288 q^{37} + 72 q^{38} + 2772 q^{41} + 1008 q^{46} + 3444 q^{47} + 672 q^{50} - 1584 q^{55} - 1056 q^{56} - 504 q^{58} - 36 q^{61} - 696 q^{65} - 1224 q^{67} - 1152 q^{68} - 9744 q^{77} + 3744 q^{82} + 2820 q^{83} + 1656 q^{85} + 4176 q^{86} - 2016 q^{91} + 1248 q^{92} + 3540 q^{95} - 1512 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.93185 0.517638i 0 3.46410 + 2.00000i 8.83318 6.85382i 0 −6.42552 + 23.9804i −5.65685 5.65685i 0 −20.6122 + 8.66817i
17.2 −1.93185 0.517638i 0 3.46410 + 2.00000i −3.66689 + 10.5619i 0 4.59296 17.1411i −5.65685 5.65685i 0 12.5511 18.5059i
17.3 −1.93185 0.517638i 0 3.46410 + 2.00000i 11.1581 + 0.704786i 0 −3.01094 + 11.2370i −5.65685 5.65685i 0 −21.1910 7.13740i
17.4 −1.93185 0.517638i 0 3.46410 + 2.00000i −10.4719 3.91650i 0 −1.06330 + 3.96829i −5.65685 5.65685i 0 18.2029 + 12.9868i
17.5 −1.93185 0.517638i 0 3.46410 + 2.00000i 0.302381 + 11.1763i 0 2.28208 8.51682i −5.65685 5.65685i 0 5.20110 21.7474i
17.6 −1.93185 0.517638i 0 3.46410 + 2.00000i 0.744935 11.1555i 0 −3.80534 + 14.2017i −5.65685 5.65685i 0 −7.21361 + 21.1652i
17.7 −1.93185 0.517638i 0 3.46410 + 2.00000i −10.4745 + 3.90957i 0 −4.71196 + 17.5853i −5.65685 5.65685i 0 22.2589 2.13070i
17.8 −1.93185 0.517638i 0 3.46410 + 2.00000i −4.14145 10.3850i 0 8.50201 31.7299i −5.65685 5.65685i 0 2.62500 + 22.2061i
17.9 −1.93185 0.517638i 0 3.46410 + 2.00000i 11.1803 0.0416934i 0 8.57118 31.9881i −5.65685 5.65685i 0 −21.6202 5.70678i
17.10 1.93185 + 0.517638i 0 3.46410 + 2.00000i 11.1621 + 0.639246i 0 6.82827 25.4835i 5.65685 + 5.65685i 0 21.2325 + 7.01283i
17.11 1.93185 + 0.517638i 0 3.46410 + 2.00000i −8.43508 + 7.33821i 0 2.96302 11.0581i 5.65685 + 5.65685i 0 −20.0939 + 9.81002i
17.12 1.93185 + 0.517638i 0 3.46410 + 2.00000i 4.23604 + 10.3468i 0 2.69511 10.0583i 5.65685 + 5.65685i 0 2.82751 + 22.1812i
17.13 1.93185 + 0.517638i 0 3.46410 + 2.00000i −7.32550 8.44612i 0 1.57990 5.89626i 5.65685 + 5.65685i 0 −9.77975 20.1086i
17.14 1.93185 + 0.517638i 0 3.46410 + 2.00000i 5.40078 9.78936i 0 0.447095 1.66858i 5.65685 + 5.65685i 0 15.5008 16.1159i
17.15 1.93185 + 0.517638i 0 3.46410 + 2.00000i −10.3242 4.29078i 0 0.319128 1.19100i 5.65685 + 5.65685i 0 −17.7238 13.6334i
17.16 1.93185 + 0.517638i 0 3.46410 + 2.00000i 10.7798 2.96568i 0 −5.36661 + 20.0285i 5.65685 + 5.65685i 0 22.3602 0.149197i
17.17 1.93185 + 0.517638i 0 3.46410 + 2.00000i −6.55934 9.05401i 0 −7.99699 + 29.8452i 5.65685 + 5.65685i 0 −7.98497 20.8864i
17.18 1.93185 + 0.517638i 0 3.46410 + 2.00000i 4.52953 + 10.2217i 0 −6.40009 + 23.8855i 5.65685 + 5.65685i 0 3.45924 + 22.0915i
143.1 −1.93185 + 0.517638i 0 3.46410 2.00000i 8.83318 + 6.85382i 0 −6.42552 23.9804i −5.65685 + 5.65685i 0 −20.6122 8.66817i
143.2 −1.93185 + 0.517638i 0 3.46410 2.00000i −3.66689 10.5619i 0 4.59296 + 17.1411i −5.65685 + 5.65685i 0 12.5511 + 18.5059i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.4.m.a 72
3.b odd 2 1 90.4.l.a 72
5.c odd 4 1 inner 270.4.m.a 72
9.c even 3 1 90.4.l.a 72
9.d odd 6 1 inner 270.4.m.a 72
15.e even 4 1 90.4.l.a 72
45.k odd 12 1 90.4.l.a 72
45.l even 12 1 inner 270.4.m.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.l.a 72 3.b odd 2 1
90.4.l.a 72 9.c even 3 1
90.4.l.a 72 15.e even 4 1
90.4.l.a 72 45.k odd 12 1
270.4.m.a 72 1.a even 1 1 trivial
270.4.m.a 72 5.c odd 4 1 inner
270.4.m.a 72 9.d odd 6 1 inner
270.4.m.a 72 45.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(270, [\chi])\).