Properties

Label 360.2.bd.a
Level $360$
Weight $2$
Character orbit 360.bd
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(59,360)]
 
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([3, 3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bd (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1) q^{2} + (\beta_{4} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + (\beta_{6} + \beta_{5} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_1) q^{6} + ( - 2 \beta_{7} + \beta_{6} - \beta_1) q^{7} + 2 \beta_{7} q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1) q^{2} + (\beta_{4} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + (\beta_{6} + \beta_{5} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_1) q^{6} + ( - 2 \beta_{7} + \beta_{6} - \beta_1) q^{7} + 2 \beta_{7} q^{8} - 3 \beta_{2} q^{9} + (\beta_{4} - 2 \beta_{3} - 1) q^{10} + (3 \beta_{2} + 3) q^{11} + 2 \beta_{4} q^{12} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{13} + ( - 2 \beta_{2} + 4) q^{14} + (2 \beta_{7} - 4 \beta_{6} - \beta_{5} + 3 \beta_1) q^{15} + (4 \beta_{2} - 4) q^{16} + 3 \beta_{4} q^{17} - 3 \beta_{7} q^{18} - 5 q^{19} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5}) q^{20} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_1) q^{21} + (3 \beta_{7} + 3 \beta_{6} - 3 \beta_1) q^{22} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{23} + ( - 2 \beta_{6} - 2 \beta_1) q^{24} + (\beta_{4} + \beta_{3} + 4 \beta_{2} - 4) q^{25} + ( - 4 \beta_{2} + 2) q^{26} - 3 \beta_{4} q^{27} + ( - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_1) q^{28} + (4 \beta_{7} - 6 \beta_{6} - 8 \beta_{5} + 2 \beta_1) q^{29} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{30} + (3 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 3 \beta_1) q^{31} + (4 \beta_{7} - 4 \beta_{6} + 4 \beta_1) q^{32} + (6 \beta_{4} - 3 \beta_{3}) q^{33} + ( - 3 \beta_{6} - 3 \beta_1) q^{34} + ( - 3 \beta_{4} + 2 \beta_{2} - 1) q^{35} + ( - 6 \beta_{2} + 6) q^{36} + ( - 5 \beta_{6} + 5 \beta_1) q^{38} + (\beta_{7} - 2 \beta_{5} + 2 \beta_1) q^{39} + (4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{40} + ( - \beta_{2} + 2) q^{41} + (2 \beta_{4} - 4 \beta_{3}) q^{42} + ( - 3 \beta_{4} - 3 \beta_{3}) q^{43} + (12 \beta_{2} - 6) q^{44} + (3 \beta_{7} - 3 \beta_{6} - 6 \beta_{5}) q^{45} + 4 q^{46} + (5 \beta_{6} - 5 \beta_1) q^{47} + 4 \beta_{3} q^{48} + \beta_{2} q^{49} + (5 \beta_{7} - 7 \beta_{6} - 2 \beta_{5} + 3 \beta_1) q^{50} + ( - 9 \beta_{2} + 9) q^{51} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{52} + \beta_{7} q^{53} + (3 \beta_{6} + 3 \beta_1) q^{54} + ( - 3 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} + 3 \beta_1) q^{55} + (4 \beta_{2} + 4) q^{56} + ( - 5 \beta_{4} + 5 \beta_{3}) q^{57} + ( - 8 \beta_{4} + 4 \beta_{3}) q^{58} + (\beta_{2} - 2) q^{59} + ( - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 4 \beta_1) q^{60} + ( - 3 \beta_{6} - 3 \beta_1) q^{61} - 6 \beta_{4} q^{62} + (3 \beta_{7} - 6 \beta_{6} + 6 \beta_1) q^{63} - 8 q^{64} + ( - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 1) q^{65} + ( - 3 \beta_{7} + 6 \beta_{5} - 6 \beta_1) q^{66} + (2 \beta_{4} - \beta_{3}) q^{67} + 6 \beta_{3} q^{68} + ( - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5}) q^{69} + (2 \beta_{7} + 2 \beta_{6} + 4 \beta_1) q^{70} + (\beta_{7} - 2 \beta_{5} + 2 \beta_1) q^{71} + ( - 6 \beta_{7} + 6 \beta_{6} - 6 \beta_1) q^{72} + ( - \beta_{4} + 2 \beta_{3}) q^{73} + (4 \beta_{3} - 3 \beta_{2} + 6) q^{75} - 10 \beta_{2} q^{76} + ( - 9 \beta_{7} + 9 \beta_{6} - 9 \beta_1) q^{77} + ( - 2 \beta_{4} - 2 \beta_{3}) q^{78} + ( - 3 \beta_{6} - 3 \beta_1) q^{79} + ( - 4 \beta_{7} + 4 \beta_{5} - 4 \beta_1) q^{80} + (9 \beta_{2} - 9) q^{81} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{82} + ( - 4 \beta_{7} + 6 \beta_{6} + 8 \beta_{5} - 2 \beta_1) q^{84} + ( - 3 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} + 6 \beta_1) q^{85} + ( - 3 \beta_{7} + 9 \beta_{6} + 6 \beta_{5} + 3 \beta_1) q^{86} + (6 \beta_{7} + 6 \beta_{6} - 6 \beta_1) q^{87} + (12 \beta_{7} - 6 \beta_{6} + 6 \beta_1) q^{88} + ( - 12 \beta_{2} + 6) q^{89} + ( - 6 \beta_{4} + 3 \beta_{3} + 3 \beta_{2}) q^{90} - 6 q^{91} + (4 \beta_{6} - 4 \beta_1) q^{92} + 9 \beta_{7} q^{93} + 10 \beta_{2} q^{94} + ( - 5 \beta_{6} - 5 \beta_{5} - 5 \beta_1) q^{95} + (4 \beta_{7} - 8 \beta_{6} - 8 \beta_{5}) q^{96} + (\beta_{4} + \beta_{3}) q^{97} + \beta_{7} q^{98} + ( - 18 \beta_{2} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 12 q^{9} - 8 q^{10} + 36 q^{11} + 24 q^{14} - 16 q^{16} - 40 q^{19} - 16 q^{25} - 36 q^{30} + 24 q^{36} - 8 q^{40} + 12 q^{41} + 32 q^{46} + 4 q^{49} + 36 q^{51} + 48 q^{56} - 12 q^{59} - 64 q^{64} + 12 q^{65} + 36 q^{75} - 40 q^{76} - 36 q^{81} + 12 q^{90} - 48 q^{91} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{5} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.258819 + 0.965926i
0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
−1.22474 + 0.707107i −0.866025 1.50000i 1.00000 1.73205i −0.448288 + 2.19067i 2.12132 + 1.22474i −1.22474 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 3.00000i
59.2 −1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 1.73205i 1.67303 1.48356i −2.12132 1.22474i −1.22474 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 + 3.00000i
59.3 1.22474 0.707107i −0.866025 1.50000i 1.00000 1.73205i 0.448288 2.19067i −2.12132 1.22474i 1.22474 + 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 3.00000i
59.4 1.22474 0.707107i 0.866025 + 1.50000i 1.00000 1.73205i −1.67303 + 1.48356i 2.12132 + 1.22474i 1.22474 + 2.12132i 2.82843i −1.50000 + 2.59808i −1.00000 + 3.00000i
299.1 −1.22474 0.707107i −0.866025 + 1.50000i 1.00000 + 1.73205i −0.448288 2.19067i 2.12132 1.22474i −1.22474 + 2.12132i 2.82843i −1.50000 2.59808i −1.00000 + 3.00000i
299.2 −1.22474 0.707107i 0.866025 1.50000i 1.00000 + 1.73205i 1.67303 + 1.48356i −2.12132 + 1.22474i −1.22474 + 2.12132i 2.82843i −1.50000 2.59808i −1.00000 3.00000i
299.3 1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 + 1.73205i 0.448288 + 2.19067i −2.12132 + 1.22474i 1.22474 2.12132i 2.82843i −1.50000 2.59808i −1.00000 + 3.00000i
299.4 1.22474 + 0.707107i 0.866025 1.50000i 1.00000 + 1.73205i −1.67303 1.48356i 2.12132 1.22474i 1.22474 2.12132i 2.82843i −1.50000 2.59808i −1.00000 3.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
9.d odd 6 1 inner
40.e odd 2 1 inner
45.h odd 6 1 inner
72.l even 6 1 inner
360.bd even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bd.a 8
5.b even 2 1 inner 360.2.bd.a 8
8.d odd 2 1 inner 360.2.bd.a 8
9.d odd 6 1 inner 360.2.bd.a 8
40.e odd 2 1 inner 360.2.bd.a 8
45.h odd 6 1 inner 360.2.bd.a 8
72.l even 6 1 inner 360.2.bd.a 8
360.bd even 6 1 inner 360.2.bd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bd.a 8 1.a even 1 1 trivial
360.2.bd.a 8 5.b even 2 1 inner
360.2.bd.a 8 8.d odd 2 1 inner
360.2.bd.a 8 9.d odd 6 1 inner
360.2.bd.a 8 40.e odd 2 1 inner
360.2.bd.a 8 45.h odd 6 1 inner
360.2.bd.a 8 72.l even 6 1 inner
360.2.bd.a 8 360.bd even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 6T_{7}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{6} + 39 T^{4} + 200 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 9 T + 27)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$19$ \( (T + 5)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
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