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Properties

Label 465.2.g.e
Level $465$
Weight $2$
Character orbit 465.g
Analytic conductor $3.713$
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] = [465,2,Mod(464,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.464");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.g (of order \(2\), degree \(1\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{5} q^{3} + q^{4} + ( - \beta_{2} - \beta_1) q^{5} - \beta_{4} q^{6} + \beta_{3} q^{7} - \beta_{2} q^{8} + ( - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{5} q^{3} + q^{4} + ( - \beta_{2} - \beta_1) q^{5} - \beta_{4} q^{6} + \beta_{3} q^{7} - \beta_{2} q^{8} + ( - 2 \beta_1 + 1) q^{9} + (\beta_{3} - 3) q^{10} - \beta_{5} q^{12} + (\beta_{6} - 2 \beta_{5}) q^{13} - 3 \beta_1 q^{14} + ( - \beta_{6} + \beta_{4}) q^{15} - 5 q^{16} + (2 \beta_{6} + 2 \beta_{5}) q^{17} + (2 \beta_{3} + \beta_{2}) q^{18} + 4 q^{19} + ( - \beta_{2} - \beta_1) q^{20} + \beta_{7} q^{21} + \beta_{4} q^{24} + ( - 2 \beta_{3} + 1) q^{25} + ( - \beta_{7} - 2 \beta_{4}) q^{26} + ( - 2 \beta_{6} - \beta_{5}) q^{27} + \beta_{3} q^{28} + ( - \beta_{7} - 2 \beta_{4}) q^{29} + (\beta_{7} + 3 \beta_{5}) q^{30} + ( - \beta_{7} + \beta_{4} + 2) q^{31} - 3 \beta_{2} q^{32} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{34} + ( - 2 \beta_{2} + 3 \beta_1) q^{35} + ( - 2 \beta_1 + 1) q^{36} + ( - \beta_{6} + 2 \beta_{5}) q^{37} + 4 \beta_{2} q^{38} + ( - 3 \beta_1 + 6) q^{39} + ( - \beta_{3} + 3) q^{40} + 2 \beta_1 q^{41} - 3 \beta_{6} q^{42} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{45} - 6 \beta_{2} q^{47} + 5 \beta_{5} q^{48} + q^{49} + (\beta_{2} + 6 \beta_1) q^{50} + (6 \beta_1 + 6) q^{51} + (\beta_{6} - 2 \beta_{5}) q^{52} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{53} + (2 \beta_{7} - \beta_{4}) q^{54} + 3 \beta_1 q^{56} - 4 \beta_{5} q^{57} + (3 \beta_{6} - 6 \beta_{5}) q^{58} - 7 \beta_1 q^{59} + ( - \beta_{6} + \beta_{4}) q^{60} + (3 \beta_{6} + 3 \beta_{5} + 2 \beta_{2}) q^{62} + (\beta_{3} - 4 \beta_{2}) q^{63} + q^{64} + (\beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_{4}) q^{65}+ \cdots + \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 8 q^{9} - 24 q^{10} - 40 q^{16} + 32 q^{19} + 8 q^{25} + 16 q^{31} + 8 q^{36} + 48 q^{39} + 24 q^{40} - 32 q^{45} + 8 q^{49} + 48 q^{51} + 8 q^{64} - 48 q^{70} + 32 q^{76} - 56 q^{81} - 24 q^{90} - 144 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} + 3\beta_{3} + 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 3\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{7} + \beta_{4} + 3 ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{4} + 3\beta_{3} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} - 3\beta_1 ) / 12 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
464.1
−0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.258819 − 0.965926i
0.965926 + 0.258819i
0.965926 − 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i
−1.73205 −1.41421 − 1.00000i 1.00000 1.73205 + 1.41421i 2.44949 + 1.73205i − 2.44949i 1.73205 1.00000 + 2.82843i −3.00000 − 2.44949i
464.2 −1.73205 −1.41421 + 1.00000i 1.00000 1.73205 − 1.41421i 2.44949 − 1.73205i 2.44949i 1.73205 1.00000 − 2.82843i −3.00000 + 2.44949i
464.3 −1.73205 1.41421 − 1.00000i 1.00000 1.73205 − 1.41421i −2.44949 + 1.73205i 2.44949i 1.73205 1.00000 − 2.82843i −3.00000 + 2.44949i
464.4 −1.73205 1.41421 + 1.00000i 1.00000 1.73205 + 1.41421i −2.44949 − 1.73205i − 2.44949i 1.73205 1.00000 + 2.82843i −3.00000 − 2.44949i
464.5 1.73205 −1.41421 − 1.00000i 1.00000 −1.73205 + 1.41421i −2.44949 − 1.73205i 2.44949i −1.73205 1.00000 + 2.82843i −3.00000 + 2.44949i
464.6 1.73205 −1.41421 + 1.00000i 1.00000 −1.73205 − 1.41421i −2.44949 + 1.73205i − 2.44949i −1.73205 1.00000 − 2.82843i −3.00000 − 2.44949i
464.7 1.73205 1.41421 − 1.00000i 1.00000 −1.73205 − 1.41421i 2.44949 − 1.73205i − 2.44949i −1.73205 1.00000 − 2.82843i −3.00000 − 2.44949i
464.8 1.73205 1.41421 + 1.00000i 1.00000 −1.73205 + 1.41421i 2.44949 + 1.73205i 2.44949i −1.73205 1.00000 + 2.82843i −3.00000 + 2.44949i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 464.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner
155.c odd 2 1 inner
465.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.g.e ✓ 8
3.b odd 2 1 inner 465.2.g.e ✓ 8
5.b even 2 1 inner 465.2.g.e ✓ 8
15.d odd 2 1 inner 465.2.g.e ✓ 8
31.b odd 2 1 inner 465.2.g.e ✓ 8
93.c even 2 1 inner 465.2.g.e ✓ 8
155.c odd 2 1 inner 465.2.g.e ✓ 8
465.g even 2 1 inner 465.2.g.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.g.e ✓ 8 1.a even 1 1 trivial
465.2.g.e ✓ 8 3.b odd 2 1 inner
465.2.g.e ✓ 8 5.b even 2 1 inner
465.2.g.e ✓ 8 15.d odd 2 1 inner
465.2.g.e ✓ 8 31.b odd 2 1 inner
465.2.g.e ✓ 8 93.c even 2 1 inner
465.2.g.e ✓ 8 155.c odd 2 1 inner
465.2.g.e ✓ 8 465.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{37}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$19$ \( (T - 4)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 31)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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