Properties

Label 270.2.k.a
Level $270$
Weight $2$
Character orbit 270.k
Analytic conductor $2.156$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,2,Mod(31,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.k (of order \(9\), degree \(6\), 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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{3} - \zeta_{18}^{5} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} + \zeta_{18}^{2}) q^{6} + ( - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{7} + \zeta_{18}^{3} q^{8} - 3 \zeta_{18}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{3} - \zeta_{18}^{5} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} + \zeta_{18}^{2}) q^{6} + ( - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{7} + \zeta_{18}^{3} q^{8} - 3 \zeta_{18}^{2} q^{9} + ( - \zeta_{18}^{3} + 1) q^{10} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \cdots - 1) q^{11} + \cdots + ( - 3 \zeta_{18}^{5} + \cdots - 3 \zeta_{18}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 9 q^{12} + 6 q^{13} + 12 q^{14} + 6 q^{17} + 18 q^{18} - 6 q^{19} - 18 q^{21} + 3 q^{22} + 18 q^{23} - 27 q^{27} - 6 q^{29} - 18 q^{31} - 9 q^{33} + 3 q^{34} + 12 q^{37} + 18 q^{38} + 3 q^{41} + 9 q^{43} - 12 q^{44} - 6 q^{47} - 12 q^{49} + 6 q^{52} + 24 q^{55} + 12 q^{56} - 27 q^{57} - 30 q^{58} - 30 q^{59} - 24 q^{61} + 6 q^{62} - 3 q^{64} - 12 q^{65} - 9 q^{67} + 6 q^{68} - 6 q^{70} - 18 q^{71} + 18 q^{73} - 18 q^{76} + 30 q^{77} + 12 q^{79} + 6 q^{80} - 48 q^{82} + 24 q^{83} - 18 q^{84} - 3 q^{85} - 9 q^{86} + 36 q^{87} - 6 q^{88} - 21 q^{89} + 12 q^{91} - 18 q^{93} + 6 q^{94} + 9 q^{95} - 9 q^{97} + 3 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-\zeta_{18}^{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
−0.766044 0.642788i
0.939693 + 0.342020i −1.70574 + 0.300767i 0.766044 + 0.642788i 0.173648 0.984808i −1.70574 0.300767i 2.87939 2.41609i 0.500000 + 0.866025i 2.81908 1.02606i 0.500000 0.866025i
61.1 0.939693 0.342020i −1.70574 0.300767i 0.766044 0.642788i 0.173648 + 0.984808i −1.70574 + 0.300767i 2.87939 + 2.41609i 0.500000 0.866025i 2.81908 + 1.02606i 0.500000 + 0.866025i
121.1 −0.173648 0.984808i 1.11334 + 1.32683i −0.939693 + 0.342020i 0.766044 + 0.642788i 1.11334 1.32683i 0.652704 + 0.237565i 0.500000 + 0.866025i −0.520945 + 2.95442i 0.500000 0.866025i
151.1 −0.766044 0.642788i 0.592396 + 1.62760i 0.173648 + 0.984808i −0.939693 0.342020i 0.592396 1.62760i −0.532089 + 3.01763i 0.500000 0.866025i −2.29813 + 1.92836i 0.500000 + 0.866025i
211.1 −0.766044 + 0.642788i 0.592396 1.62760i 0.173648 0.984808i −0.939693 + 0.342020i 0.592396 + 1.62760i −0.532089 3.01763i 0.500000 + 0.866025i −2.29813 1.92836i 0.500000 0.866025i
241.1 −0.173648 + 0.984808i 1.11334 1.32683i −0.939693 0.342020i 0.766044 0.642788i 1.11334 + 1.32683i 0.652704 0.237565i 0.500000 0.866025i −0.520945 2.95442i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.k.a 6
3.b odd 2 1 810.2.k.a 6
27.e even 9 1 inner 270.2.k.a 6
27.e even 9 1 7290.2.a.c 3
27.f odd 18 1 810.2.k.a 6
27.f odd 18 1 7290.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.k.a 6 1.a even 1 1 trivial
270.2.k.a 6 27.e even 9 1 inner
810.2.k.a 6 3.b odd 2 1
810.2.k.a 6 27.f odd 18 1
7290.2.a.c 3 27.e even 9 1
7290.2.a.e 3 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 6T_{7}^{5} + 24T_{7}^{4} - 64T_{7}^{3} + 192T_{7}^{2} - 192T_{7} + 64 \) acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{6} - 18 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + \cdots + 207936 \) Copy content Toggle raw display
$31$ \( T^{6} + 18 T^{5} + \cdots + 18496 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + \cdots + 239121 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( (T^{3} - 36 T - 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 30 T^{5} + \cdots + 288369 \) Copy content Toggle raw display
$61$ \( T^{6} + 24 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 811801 \) Copy content Toggle raw display
$71$ \( T^{6} + 18 T^{5} + \cdots + 1498176 \) Copy content Toggle raw display
$73$ \( T^{6} - 18 T^{5} + \cdots + 128881 \) Copy content Toggle raw display
$79$ \( T^{6} - 12 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{6} - 24 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$89$ \( T^{6} + 21 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} + 9 T^{5} + \cdots + 3996001 \) Copy content Toggle raw display
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