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Properties

Label 2496.1.eh.a
Level $2496$
Weight $1$
Character orbit 2496.eh
Analytic conductor $1.246$
Analytic rank $0$
Dimension $32$
Projective image $D_{32}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,1,Mod(77,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 15, 8, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.77");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2496.eh (of order \(16\), degree \(8\), 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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{64})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{64}^{27} q^{2} - \zeta_{64}^{26} q^{3} - \zeta_{64}^{22} q^{4} + ( - \zeta_{64}^{19} + \zeta_{64}^{9}) q^{5} - \zeta_{64}^{21} q^{6} - \zeta_{64}^{17} q^{8} - \zeta_{64}^{20} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{64}^{27} q^{2} - \zeta_{64}^{26} q^{3} - \zeta_{64}^{22} q^{4} + ( - \zeta_{64}^{19} + \zeta_{64}^{9}) q^{5} - \zeta_{64}^{21} q^{6} - \zeta_{64}^{17} q^{8} - \zeta_{64}^{20} q^{9} + ( - \zeta_{64}^{14} + \zeta_{64}^{4}) q^{10} + (\zeta_{64}^{23} + \zeta_{64}^{21}) q^{11} - \zeta_{64}^{16} q^{12} - \zeta_{64}^{2} q^{13} + ( - \zeta_{64}^{13} + \zeta_{64}^{3}) q^{15} - \zeta_{64}^{12} q^{16} - \zeta_{64}^{15} q^{18} + ( - \zeta_{64}^{31} - \zeta_{64}^{9}) q^{20} + (\zeta_{64}^{18} + \zeta_{64}^{16}) q^{22} - \zeta_{64}^{11} q^{24} + ( - \zeta_{64}^{28} + \cdots - \zeta_{64}^{6}) q^{25} + \cdots + (\zeta_{64}^{11} + \zeta_{64}^{9}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 32 q^{55} - 32 q^{75}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{64}^{28}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
−0.995185 − 0.0980171i
−0.0980171 + 0.995185i
0.0980171 − 0.995185i
0.995185 + 0.0980171i
−0.995185 + 0.0980171i
−0.0980171 − 0.995185i
0.0980171 + 0.995185i
0.995185 − 0.0980171i
−0.290285 − 0.956940i
−0.956940 + 0.290285i
0.956940 − 0.290285i
0.290285 + 0.956940i
0.881921 − 0.471397i
−0.471397 − 0.881921i
0.471397 + 0.881921i
−0.881921 + 0.471397i
0.773010 − 0.634393i
0.634393 + 0.773010i
−0.634393 − 0.773010i
−0.773010 + 0.634393i
−0.881921 + 0.471397i 0.831470 − 0.555570i 0.555570 − 0.831470i −0.924678 + 0.183930i −0.471397 + 0.881921i 0 −0.0980171 + 0.995185i 0.382683 − 0.923880i 0.728789 − 0.598102i
77.2 −0.471397 − 0.881921i −0.831470 + 0.555570i −0.555570 + 0.831470i −1.72995 + 0.344109i 0.881921 + 0.471397i 0 0.995185 + 0.0980171i 0.382683 − 0.923880i 1.11897 + 1.36347i
77.3 0.471397 + 0.881921i −0.831470 + 0.555570i −0.555570 + 0.831470i 1.72995 − 0.344109i −0.881921 − 0.471397i 0 −0.995185 − 0.0980171i 0.382683 − 0.923880i 1.11897 + 1.36347i
77.4 0.881921 − 0.471397i 0.831470 − 0.555570i 0.555570 − 0.831470i 0.924678 − 0.183930i 0.471397 − 0.881921i 0 0.0980171 − 0.995185i 0.382683 − 0.923880i 0.728789 − 0.598102i
389.1 −0.881921 − 0.471397i 0.831470 + 0.555570i 0.555570 + 0.831470i −0.924678 − 0.183930i −0.471397 − 0.881921i 0 −0.0980171 − 0.995185i 0.382683 + 0.923880i 0.728789 + 0.598102i
389.2 −0.471397 + 0.881921i −0.831470 − 0.555570i −0.555570 − 0.831470i −1.72995 − 0.344109i 0.881921 − 0.471397i 0 0.995185 − 0.0980171i 0.382683 + 0.923880i 1.11897 − 1.36347i
389.3 0.471397 − 0.881921i −0.831470 − 0.555570i −0.555570 − 0.831470i 1.72995 + 0.344109i −0.881921 + 0.471397i 0 −0.995185 + 0.0980171i 0.382683 + 0.923880i 1.11897 − 1.36347i
389.4 0.881921 + 0.471397i 0.831470 + 0.555570i 0.555570 + 0.831470i 0.924678 + 0.183930i 0.471397 + 0.881921i 0 0.0980171 + 0.995185i 0.382683 + 0.923880i 0.728789 + 0.598102i
701.1 −0.995185 + 0.0980171i 0.195090 − 0.980785i 0.980785 − 0.195090i 0.162997 + 0.108911i −0.0980171 + 0.995185i 0 −0.956940 + 0.290285i −0.923880 − 0.382683i −0.172887 − 0.0924099i
701.2 −0.0980171 − 0.995185i −0.195090 + 0.980785i −0.980785 + 0.195090i 1.65493 + 1.10579i 0.995185 + 0.0980171i 0 0.290285 + 0.956940i −0.923880 − 0.382683i 0.938254 − 1.75535i
701.3 0.0980171 + 0.995185i −0.195090 + 0.980785i −0.980785 + 0.195090i −1.65493 − 1.10579i −0.995185 − 0.0980171i 0 −0.290285 − 0.956940i −0.923880 − 0.382683i 0.938254 − 1.75535i
701.4 0.995185 − 0.0980171i 0.195090 − 0.980785i 0.980785 − 0.195090i −0.162997 − 0.108911i 0.0980171 − 0.995185i 0 0.956940 − 0.290285i −0.923880 − 0.382683i −0.172887 − 0.0924099i
1013.1 −0.773010 + 0.634393i −0.980785 + 0.195090i 0.195090 − 0.980785i 0.704900 + 1.05496i 0.634393 − 0.773010i 0 0.471397 + 0.881921i 0.923880 − 0.382683i −1.21415 − 0.368309i
1013.2 −0.634393 − 0.773010i 0.980785 − 0.195090i −0.195090 + 0.980785i 0.858923 + 1.28547i −0.773010 − 0.634393i 0 0.881921 − 0.471397i 0.923880 − 0.382683i 0.448786 − 1.47945i
1013.3 0.634393 + 0.773010i 0.980785 − 0.195090i −0.195090 + 0.980785i −0.858923 − 1.28547i 0.773010 + 0.634393i 0 −0.881921 + 0.471397i 0.923880 − 0.382683i 0.448786 − 1.47945i
1013.4 0.773010 − 0.634393i −0.980785 + 0.195090i 0.195090 − 0.980785i −0.704900 − 1.05496i −0.634393 + 0.773010i 0 −0.471397 − 0.881921i 0.923880 − 0.382683i −1.21415 − 0.368309i
1325.1 −0.956940 − 0.290285i −0.555570 − 0.831470i 0.831470 + 0.555570i 0.113263 + 0.569414i 0.290285 + 0.956940i 0 −0.634393 − 0.773010i −0.382683 + 0.923880i 0.0569057 − 0.577774i
1325.2 −0.290285 + 0.956940i 0.555570 + 0.831470i −0.831470 − 0.555570i 0.373380 + 1.87711i −0.956940 + 0.290285i 0 0.773010 − 0.634393i −0.382683 + 0.923880i −1.90466 − 0.187593i
1325.3 0.290285 − 0.956940i 0.555570 + 0.831470i −0.831470 − 0.555570i −0.373380 − 1.87711i 0.956940 − 0.290285i 0 −0.773010 + 0.634393i −0.382683 + 0.923880i −1.90466 − 0.187593i
1325.4 0.956940 + 0.290285i −0.555570 − 0.831470i 0.831470 + 0.555570i −0.113263 − 0.569414i −0.290285 − 0.956940i 0 0.634393 + 0.773010i −0.382683 + 0.923880i 0.0569057 − 0.577774i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
64.i even 16 1 inner
192.q odd 16 1 inner
832.cb even 16 1 inner
2496.eh odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.eh.a ✓ 32
3.b odd 2 1 inner 2496.1.eh.a ✓ 32
13.b even 2 1 inner 2496.1.eh.a ✓ 32
39.d odd 2 1 CM 2496.1.eh.a ✓ 32
64.i even 16 1 inner 2496.1.eh.a ✓ 32
192.q odd 16 1 inner 2496.1.eh.a ✓ 32
832.cb even 16 1 inner 2496.1.eh.a ✓ 32
2496.eh odd 16 1 inner 2496.1.eh.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2496.1.eh.a ✓ 32 1.a even 1 1 trivial
2496.1.eh.a ✓ 32 3.b odd 2 1 inner
2496.1.eh.a ✓ 32 13.b even 2 1 inner
2496.1.eh.a ✓ 32 39.d odd 2 1 CM
2496.1.eh.a ✓ 32 64.i even 16 1 inner
2496.1.eh.a ✓ 32 192.q odd 16 1 inner
2496.1.eh.a ✓ 32 832.cb even 16 1 inner
2496.1.eh.a ✓ 32 2496.eh odd 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2496, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{32} + 1 \) Copy content Toggle raw display
$3$ \( (T^{16} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{32} + 32 T^{26} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{32} \) Copy content Toggle raw display
$11$ \( T^{32} - 32 T^{26} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{16} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{32} \) Copy content Toggle raw display
$19$ \( T^{32} \) Copy content Toggle raw display
$23$ \( T^{32} \) Copy content Toggle raw display
$29$ \( T^{32} \) Copy content Toggle raw display
$31$ \( T^{32} \) Copy content Toggle raw display
$37$ \( T^{32} \) Copy content Toggle raw display
$41$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$43$ \( (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{32} + 48 T^{28} + \cdots + 4 \) Copy content Toggle raw display
$53$ \( T^{32} \) Copy content Toggle raw display
$59$ \( T^{32} + 32 T^{26} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{16} + 16 T^{12} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{32} \) Copy content Toggle raw display
$71$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$73$ \( T^{32} \) Copy content Toggle raw display
$79$ \( (T^{16} + 24 T^{12} + \cdots + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{32} + 32 T^{26} + \cdots + 4 \) Copy content Toggle raw display
$89$ \( T^{32} + 280 T^{24} + \cdots + 4 \) Copy content Toggle raw display
$97$ \( T^{32} \) Copy content Toggle raw display
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