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Properties

Label 360.2.q.e
Level $360$
Weight $2$
Character orbit 360.q
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
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] = [360,2,Mod(121,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("360.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.q (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_{2}) q^{7} + (\beta_{7} - \beta_{3} - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_{2}) q^{7} + (\beta_{7} - \beta_{3} - \beta_{2} - \beta_1) q^{9} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + q^{7} - q^{11} - 4 q^{13} + 3 q^{15} + 10 q^{17} + 2 q^{19} - 7 q^{23} - 4 q^{25} - 18 q^{27} - 7 q^{29} + 2 q^{31} - 3 q^{33} - 2 q^{35} + 12 q^{37} - 6 q^{39} - 12 q^{41} + 11 q^{43} - 3 q^{45} - 7 q^{47} - 3 q^{49} + 39 q^{51} + 24 q^{53} + 2 q^{55} + 27 q^{57} - 11 q^{59} - 19 q^{61} - 33 q^{63} - 4 q^{65} + 10 q^{67} - 9 q^{69} + 24 q^{71} + 18 q^{73} - 3 q^{75} - 32 q^{77} + 24 q^{79} - 12 q^{81} - 23 q^{83} - 5 q^{85} + 24 q^{87} + 42 q^{89} + 28 q^{91} + 18 q^{93} - q^{95} - q^{97} - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 7\nu^{3} - 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - \nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 5\nu^{4} + 2\nu^{3} - 2\nu^{2} + 10\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} + 9\nu^{4} + 5\nu^{3} + 22\nu^{2} + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 4\nu^{4} + 31\nu^{3} + 22\nu^{2} + 30\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 29\nu^{3} - 20\nu^{2} + 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 11\nu^{5} + 9\nu^{4} - 32\nu^{3} + 24\nu^{2} - 10\nu + 16 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} + 3\beta_{4} - 4\beta_{3} - 10\beta_{2} + 2\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{7} - 18\beta_{6} + 7\beta_{5} - 11\beta_{4} + 4\beta_{3} + 35 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{7} - 7\beta_{5} - 11\beta_{4} + 18\beta_{3} + 50\beta_{2} - 16\beta _1 + 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{7} + 32\beta_{6} - 13\beta_{5} + 21\beta_{4} - 4\beta_{3} + 2\beta _1 - 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39\beta_{7} + 6\beta_{6} + 45\beta_{5} + 47\beta_{4} - 80\beta_{3} - 250\beta_{2} + 128\beta _1 - 101 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.385731i
− 2.06288i
2.33086i
1.07834i
− 0.385731i
2.06288i
− 2.33086i
− 1.07834i
0 −1.73065 + 0.0696054i 0 −0.500000 + 0.866025i 0 0.165947 + 0.287429i 0 2.99031 − 0.240925i 0
121.2 0 −0.574618 − 1.63396i 0 −0.500000 + 0.866025i 0 2.28651 + 3.96035i 0 −2.33963 + 1.87780i 0
121.3 0 1.05903 − 1.37057i 0 −0.500000 + 0.866025i 0 −1.51859 − 2.63027i 0 −0.756906 − 2.90295i 0
121.4 0 1.24624 + 1.20287i 0 −0.500000 + 0.866025i 0 −0.433868 − 0.751481i 0 0.106223 + 2.99812i 0
241.1 0 −1.73065 − 0.0696054i 0 −0.500000 − 0.866025i 0 0.165947 − 0.287429i 0 2.99031 + 0.240925i 0
241.2 0 −0.574618 + 1.63396i 0 −0.500000 − 0.866025i 0 2.28651 − 3.96035i 0 −2.33963 − 1.87780i 0
241.3 0 1.05903 + 1.37057i 0 −0.500000 − 0.866025i 0 −1.51859 + 2.63027i 0 −0.756906 + 2.90295i 0
241.4 0 1.24624 − 1.20287i 0 −0.500000 − 0.866025i 0 −0.433868 + 0.751481i 0 0.106223 − 2.99812i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.q.e ✓ 8
3.b odd 2 1 1080.2.q.e 8
4.b odd 2 1 720.2.q.l 8
9.c even 3 1 inner 360.2.q.e ✓ 8
9.c even 3 1 3240.2.a.u 4
9.d odd 6 1 1080.2.q.e 8
9.d odd 6 1 3240.2.a.s 4
12.b even 2 1 2160.2.q.l 8
36.f odd 6 1 720.2.q.l 8
36.f odd 6 1 6480.2.a.cb 4
36.h even 6 1 2160.2.q.l 8
36.h even 6 1 6480.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.e ✓ 8 1.a even 1 1 trivial
360.2.q.e ✓ 8 9.c even 3 1 inner
720.2.q.l 8 4.b odd 2 1
720.2.q.l 8 36.f odd 6 1
1080.2.q.e 8 3.b odd 2 1
1080.2.q.e 8 9.d odd 6 1
2160.2.q.l 8 12.b even 2 1
2160.2.q.l 8 36.h even 6 1
3240.2.a.s 4 9.d odd 6 1
3240.2.a.u 4 9.c even 3 1
6480.2.a.bz 4 36.h even 6 1
6480.2.a.cb 4 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - T_{7}^{7} + 16T_{7}^{6} + 29T_{7}^{5} + 214T_{7}^{4} + 113T_{7}^{3} + 109T_{7}^{2} - 28T_{7} + 16 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + \cdots + 190096 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{4} - 5 T^{3} + \cdots + 172)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} - 84 T^{2} + \cdots + 1348)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 7 T^{7} + \cdots + 3844 \) Copy content Toggle raw display
$29$ \( T^{8} + 7 T^{7} + \cdots + 42436 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} + \cdots + 179776 \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 5602689 \) Copy content Toggle raw display
$43$ \( T^{8} - 11 T^{7} + \cdots + 44944 \) Copy content Toggle raw display
$47$ \( T^{8} + 7 T^{7} + \cdots + 1149184 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 11 T^{7} + \cdots + 222784 \) Copy content Toggle raw display
$61$ \( T^{8} + 19 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$67$ \( T^{8} - 10 T^{7} + \cdots + 2111209 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 9 T^{3} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 1434288384 \) Copy content Toggle raw display
$83$ \( T^{8} + 23 T^{7} + \cdots + 27836176 \) Copy content Toggle raw display
$89$ \( (T^{4} - 21 T^{3} + \cdots - 7506)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + \cdots + 1459264 \) Copy content Toggle raw display
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