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Properties

Label 964.1.bh.a
Level $964$
Weight $1$
Character orbit 964.bh
Analytic conductor $0.481$
Analytic rank $0$
Dimension $16$
Projective image $D_{60}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [964,1,Mod(83,964)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(964, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([30, 37]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("964.83");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 964 = 2^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 964.bh (of order \(60\), degree \(16\), 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: \(0.481098672178\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{60}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{60}^{5} q^{2} + \zeta_{60}^{10} q^{4} + (\zeta_{60}^{23} - \zeta_{60}) q^{5} - \zeta_{60}^{15} q^{8} + \zeta_{60}^{26} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{60}^{5} q^{2} + \zeta_{60}^{10} q^{4} + (\zeta_{60}^{23} - \zeta_{60}) q^{5} - \zeta_{60}^{15} q^{8} + \zeta_{60}^{26} q^{9} + ( - \zeta_{60}^{28} + \zeta_{60}^{6}) q^{10} + ( - \zeta_{60}^{29} - \zeta_{60}^{12}) q^{13} + \zeta_{60}^{20} q^{16} + ( - \zeta_{60}^{23} - \zeta_{60}^{10}) q^{17} + \zeta_{60} q^{18} + ( - \zeta_{60}^{11} - \zeta_{60}^{3}) q^{20} + ( - \zeta_{60}^{24} + \cdots + \zeta_{60}^{2}) q^{25} + \cdots + \zeta_{60}^{18} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 2 q^{9} + 2 q^{10} + 4 q^{13} - 8 q^{16} - 8 q^{17} - 2 q^{26} + 2 q^{34} - 4 q^{36} + 2 q^{37} + 4 q^{40} + 2 q^{52} + 4 q^{53} - 16 q^{64} - 18 q^{65} + 8 q^{68} + 4 q^{73} - 2 q^{74} + 2 q^{81} - 2 q^{85} - 4 q^{89} - 2 q^{90} - 2 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/964\mathbb{Z}\right)^\times\).

\(n\) \(483\) \(489\)
\(\chi(n)\) \(-1\) \(-\zeta_{60}^{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
−0.406737 + 0.913545i
0.743145 − 0.669131i
−0.994522 + 0.104528i
−0.406737 − 0.913545i
0.207912 + 0.978148i
−0.207912 − 0.978148i
0.406737 + 0.913545i
0.994522 − 0.104528i
−0.743145 + 0.669131i
0.406737 − 0.913545i
−0.743145 − 0.669131i
−0.207912 + 0.978148i
−0.994522 − 0.104528i
0.994522 + 0.104528i
0.207912 − 0.978148i
0.743145 + 0.669131i
0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.198825 + 0.0646021i 0 0 1.00000i 0.104528 + 0.994522i 0.139886 + 0.155360i
107.1 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.14988 + 1.58268i 0 0 − 1.00000i 0.978148 − 0.207912i −0.204489 + 1.94558i
123.1 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.73767 + 0.564602i 0 0 − 1.00000i −0.913545 − 0.406737i 1.78716 − 0.379874i
151.1 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.198825 − 0.0646021i 0 0 − 1.00000i 0.104528 − 0.994522i 0.139886 − 0.155360i
159.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.786610 − 1.08268i 0 0 − 1.00000i −0.669131 − 0.743145i −1.22256 + 0.544320i
323.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.786610 + 1.08268i 0 0 1.00000i −0.669131 − 0.743145i −1.22256 + 0.544320i
331.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.198825 + 0.0646021i 0 0 1.00000i 0.104528 − 0.994522i 0.139886 − 0.155360i
359.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.73767 − 0.564602i 0 0 1.00000i −0.913545 − 0.406737i 1.78716 − 0.379874i
375.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.14988 − 1.58268i 0 0 1.00000i 0.978148 − 0.207912i −0.204489 + 1.94558i
399.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.198825 − 0.0646021i 0 0 − 1.00000i 0.104528 + 0.994522i 0.139886 + 0.155360i
491.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.14988 + 1.58268i 0 0 − 1.00000i 0.978148 + 0.207912i −0.204489 − 1.94558i
579.1 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.786610 − 1.08268i 0 0 − 1.00000i −0.669131 + 0.743145i −1.22256 − 0.544320i
627.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.73767 − 0.564602i 0 0 1.00000i −0.913545 + 0.406737i 1.78716 + 0.379874i
819.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.73767 + 0.564602i 0 0 − 1.00000i −0.913545 + 0.406737i 1.78716 + 0.379874i
867.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.786610 + 1.08268i 0 0 1.00000i −0.669131 + 0.743145i −1.22256 − 0.544320i
955.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.14988 − 1.58268i 0 0 1.00000i 0.978148 + 0.207912i −0.204489 − 1.94558i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
241.q even 60 1 inner
964.bh odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 964.1.bh.a ✓ 16
4.b odd 2 1 CM 964.1.bh.a ✓ 16
241.q even 60 1 inner 964.1.bh.a ✓ 16
964.bh odd 60 1 inner 964.1.bh.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
964.1.bh.a ✓ 16 1.a even 1 1 trivial
964.1.bh.a ✓ 16 4.b odd 2 1 CM
964.1.bh.a ✓ 16 241.q even 60 1 inner
964.1.bh.a ✓ 16 964.bh odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(964, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} - 10 T^{5} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{8} - 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} - 2 T^{7} + 6 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
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