Properties

Label 2960.1.ez.a
Level $2960$
Weight $1$
Character orbit 2960.ez
Analytic conductor $1.477$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2960,1,Mod(1099,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 6, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2960.1099");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2960.ez (of order \(12\), degree \(4\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.14018560.7

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{7} q^{2} + \zeta_{24}^{5} q^{3} - \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} + q^{6} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{7} + \zeta_{24}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{7} q^{2} + \zeta_{24}^{5} q^{3} - \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} + q^{6} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{7} + \zeta_{24}^{9} q^{8} + q^{10} + ( - \zeta_{24}^{6} + 1) q^{11} - \zeta_{24}^{7} q^{12} - \zeta_{24}^{11} q^{13} + (\zeta_{24}^{6} - 1) q^{14} + \zeta_{24}^{10} q^{15} + \zeta_{24}^{4} q^{16} + (\zeta_{24}^{7} - \zeta_{24}) q^{17} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{19} - \zeta_{24}^{7} q^{20} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{21} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{22} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{23} - \zeta_{24}^{2} q^{24} + \zeta_{24}^{10} q^{25} - \zeta_{24}^{6} q^{26} + \zeta_{24}^{3} q^{27} + (\zeta_{24}^{7} + \zeta_{24}) q^{28} + \zeta_{24}^{5} q^{30} + \zeta_{24}^{6} q^{31} - \zeta_{24}^{11} q^{32} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{33} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{34} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{35} - \zeta_{24}^{5} q^{37} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{38} + \zeta_{24}^{4} q^{39} - \zeta_{24}^{2} q^{40} + \zeta_{24}^{2} q^{41} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{42} + \zeta_{24}^{3} q^{43} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{44} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{46} + \zeta_{24}^{9} q^{48} + \zeta_{24}^{4} q^{49} + \zeta_{24}^{5} q^{50} + ( - \zeta_{24}^{6} - 1) q^{51} - \zeta_{24} q^{52} + \zeta_{24} q^{53} - \zeta_{24}^{10} q^{54} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{55} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{56} + (\zeta_{24}^{7} + \zeta_{24}) q^{57} + q^{60} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{61} + \zeta_{24} q^{62} - \zeta_{24}^{6} q^{64} + \zeta_{24}^{4} q^{65} + ( - \zeta_{24}^{6} + 1) q^{66} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{68} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{69} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{70} - \zeta_{24}^{8} q^{71} - q^{74} - \zeta_{24}^{3} q^{75} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{76} + (2 \zeta_{24}^{11} + \zeta_{24}^{5}) q^{77} - \zeta_{24}^{11} q^{78} - \zeta_{24}^{2} q^{79} + \zeta_{24}^{9} q^{80} + \zeta_{24}^{8} q^{81} - \zeta_{24}^{9} q^{82} + (\zeta_{24}^{6} - 1) q^{84} + ( - \zeta_{24}^{6} - 1) q^{85} - \zeta_{24}^{10} q^{86} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{88} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{91} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{92} + \zeta_{24}^{11} q^{93} + (\zeta_{24}^{7} + \zeta_{24}) q^{95} + \zeta_{24}^{4} q^{96} - \zeta_{24}^{11} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{6} + 8 q^{10} + 8 q^{11} - 8 q^{14} + 4 q^{16} + 4 q^{19} - 4 q^{21} - 4 q^{34} - 4 q^{35} + 4 q^{39} - 4 q^{44} - 4 q^{46} + 4 q^{49} - 8 q^{51} + 4 q^{56} + 8 q^{60} - 4 q^{61} + 4 q^{65} + 8 q^{66} + 4 q^{69} + 8 q^{71} - 8 q^{74} - 4 q^{76} - 4 q^{81} - 8 q^{84} - 8 q^{85} - 4 q^{91} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-1\) \(\zeta_{24}^{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1099.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i −0.258819 0.965926i −0.866025 0.500000i −0.258819 0.965926i 1.00000 1.22474 + 0.707107i 0.707107 0.707107i 0 1.00000
1099.2 0.258819 0.965926i 0.258819 + 0.965926i −0.866025 0.500000i 0.258819 + 0.965926i 1.00000 −1.22474 0.707107i −0.707107 + 0.707107i 0 1.00000
1379.1 −0.258819 0.965926i −0.258819 + 0.965926i −0.866025 + 0.500000i −0.258819 + 0.965926i 1.00000 1.22474 0.707107i 0.707107 + 0.707107i 0 1.00000
1379.2 0.258819 + 0.965926i 0.258819 0.965926i −0.866025 + 0.500000i 0.258819 0.965926i 1.00000 −1.22474 + 0.707107i −0.707107 0.707107i 0 1.00000
2579.1 −0.965926 0.258819i −0.965926 + 0.258819i 0.866025 + 0.500000i −0.965926 + 0.258819i 1.00000 1.22474 + 0.707107i −0.707107 0.707107i 0 1.00000
2579.2 0.965926 + 0.258819i 0.965926 0.258819i 0.866025 + 0.500000i 0.965926 0.258819i 1.00000 −1.22474 0.707107i 0.707107 + 0.707107i 0 1.00000
2859.1 −0.965926 + 0.258819i −0.965926 0.258819i 0.866025 0.500000i −0.965926 0.258819i 1.00000 1.22474 0.707107i −0.707107 + 0.707107i 0 1.00000
2859.2 0.965926 0.258819i 0.965926 + 0.258819i 0.866025 0.500000i 0.965926 + 0.258819i 1.00000 −1.22474 + 0.707107i 0.707107 0.707107i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1099.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.f odd 4 1 inner
37.c even 3 1 inner
80.k odd 4 1 inner
185.n even 6 1 inner
592.bi odd 12 1 inner
2960.ez odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.ez.a 8
5.b even 2 1 inner 2960.1.ez.a 8
16.f odd 4 1 inner 2960.1.ez.a 8
37.c even 3 1 inner 2960.1.ez.a 8
80.k odd 4 1 inner 2960.1.ez.a 8
185.n even 6 1 inner 2960.1.ez.a 8
592.bi odd 12 1 inner 2960.1.ez.a 8
2960.ez odd 12 1 inner 2960.1.ez.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.ez.a 8 1.a even 1 1 trivial
2960.1.ez.a 8 5.b even 2 1 inner
2960.1.ez.a 8 16.f odd 4 1 inner
2960.1.ez.a 8 37.c even 3 1 inner
2960.1.ez.a 8 80.k odd 4 1 inner
2960.1.ez.a 8 185.n even 6 1 inner
2960.1.ez.a 8 592.bi odd 12 1 inner
2960.1.ez.a 8 2960.ez odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2960, [\chi])\).

Hecke characteristic polynomials

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