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Properties

Label 1792.1.bt.a
Level $1792$
Weight $1$
Character orbit 1792.bt
Analytic conductor $0.894$
Analytic rank $0$
Dimension $32$
Projective image $D_{64}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,1,Mod(13,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(64))
 
chi = DirichletCharacter(H, H._module([0, 47, 32]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1792.bt (of order \(64\), degree \(32\), 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: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(32\)
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: \(D_{64}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{64} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{64}^{3} q^{2} + \zeta_{64}^{6} q^{4} - \zeta_{64}^{7} q^{7} + \zeta_{64}^{9} q^{8} - \zeta_{64}^{17} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{64}^{3} q^{2} + \zeta_{64}^{6} q^{4} - \zeta_{64}^{7} q^{7} + \zeta_{64}^{9} q^{8} - \zeta_{64}^{17} q^{9} + ( - \zeta_{64}^{18} + \zeta_{64}^{5}) q^{11} - \zeta_{64}^{10} q^{14} + \zeta_{64}^{12} q^{16} - \zeta_{64}^{20} q^{18} + ( - \zeta_{64}^{21} + \zeta_{64}^{8}) q^{22} + ( - \zeta_{64}^{30} - \zeta_{64}^{28}) q^{23} + \zeta_{64}^{27} q^{25} - \zeta_{64}^{13} q^{28} + (\zeta_{64}^{31} + \zeta_{64}^{26}) q^{29} + \zeta_{64}^{15} q^{32} - \zeta_{64}^{23} q^{36} + (\zeta_{64}^{22} + \zeta_{64}^{13}) q^{37} + ( - \zeta_{64}^{11} + \zeta_{64}^{4}) q^{43} + ( - \zeta_{64}^{24} + \zeta_{64}^{11}) q^{44} + ( - \zeta_{64}^{31} + \zeta_{64}) q^{46} + \zeta_{64}^{14} q^{49} + \zeta_{64}^{30} q^{50} + (\zeta_{64}^{20} - \zeta_{64}^{19}) q^{53} - \zeta_{64}^{16} q^{56} + (\zeta_{64}^{29} - \zeta_{64}^{2}) q^{58} + \zeta_{64}^{24} q^{63} + \zeta_{64}^{18} q^{64} + ( - \zeta_{64}^{25} - \zeta_{64}^{8}) q^{67} + (\zeta_{64}^{29} - \zeta_{64}) q^{71} - \zeta_{64}^{26} q^{72} + (\zeta_{64}^{25} + \zeta_{64}^{16}) q^{74} + (\zeta_{64}^{25} - \zeta_{64}^{12}) q^{77} + (\zeta_{64}^{19} - \zeta_{64}^{9}) q^{79} - \zeta_{64}^{2} q^{81} + ( - \zeta_{64}^{14} + \zeta_{64}^{7}) q^{86} + ( - \zeta_{64}^{27} + \zeta_{64}^{14}) q^{88} + (\zeta_{64}^{4} + \zeta_{64}^{2}) q^{92} + \zeta_{64}^{17} q^{98} + ( - \zeta_{64}^{22} - \zeta_{64}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{64}^{27}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.956940 − 0.290285i
−0.956940 − 0.290285i
0.773010 − 0.634393i
0.995185 − 0.0980171i
0.471397 − 0.881921i
−0.881921 + 0.471397i
0.0980171 − 0.995185i
0.634393 − 0.773010i
−0.290285 − 0.956940i
−0.290285 + 0.956940i
−0.634393 − 0.773010i
−0.0980171 − 0.995185i
−0.881921 − 0.471397i
0.471397 + 0.881921i
−0.995185 − 0.0980171i
−0.773010 − 0.634393i
−0.956940 + 0.290285i
0.956940 + 0.290285i
−0.773010 + 0.634393i
−0.995185 + 0.0980171i
0.634393 − 0.773010i 0 −0.195090 − 0.980785i 0 0 0.471397 + 0.881921i −0.881921 − 0.471397i −0.290285 − 0.956940i 0
69.1 −0.634393 − 0.773010i 0 −0.195090 + 0.980785i 0 0 −0.471397 + 0.881921i 0.881921 − 0.471397i 0.290285 − 0.956940i 0
125.1 −0.471397 − 0.881921i 0 −0.555570 + 0.831470i 0 0 −0.0980171 − 0.995185i 0.995185 + 0.0980171i −0.634393 − 0.773010i 0
181.1 0.956940 − 0.290285i 0 0.831470 − 0.555570i 0 0 −0.773010 + 0.634393i 0.634393 − 0.773010i 0.0980171 + 0.995185i 0
237.1 −0.995185 + 0.0980171i 0 0.980785 − 0.195090i 0 0 −0.290285 + 0.956940i −0.956940 + 0.290285i −0.881921 − 0.471397i 0
293.1 −0.0980171 + 0.995185i 0 −0.980785 − 0.195090i 0 0 −0.956940 + 0.290285i 0.290285 − 0.956940i −0.471397 − 0.881921i 0
349.1 −0.290285 + 0.956940i 0 −0.831470 − 0.555570i 0 0 0.634393 − 0.773010i 0.773010 − 0.634393i −0.995185 − 0.0980171i 0
405.1 −0.881921 − 0.471397i 0 0.555570 + 0.831470i 0 0 −0.995185 − 0.0980171i −0.0980171 − 0.995185i 0.773010 + 0.634393i 0
461.1 0.773010 + 0.634393i 0 0.195090 + 0.980785i 0 0 −0.881921 + 0.471397i −0.471397 + 0.881921i −0.956940 + 0.290285i 0
517.1 0.773010 − 0.634393i 0 0.195090 − 0.980785i 0 0 −0.881921 − 0.471397i −0.471397 − 0.881921i −0.956940 − 0.290285i 0
573.1 0.881921 − 0.471397i 0 0.555570 − 0.831470i 0 0 0.995185 − 0.0980171i 0.0980171 − 0.995185i −0.773010 + 0.634393i 0
629.1 0.290285 + 0.956940i 0 −0.831470 + 0.555570i 0 0 −0.634393 − 0.773010i −0.773010 − 0.634393i 0.995185 − 0.0980171i 0
685.1 −0.0980171 − 0.995185i 0 −0.980785 + 0.195090i 0 0 −0.956940 − 0.290285i 0.290285 + 0.956940i −0.471397 + 0.881921i 0
741.1 −0.995185 − 0.0980171i 0 0.980785 + 0.195090i 0 0 −0.290285 − 0.956940i −0.956940 − 0.290285i −0.881921 + 0.471397i 0
797.1 −0.956940 − 0.290285i 0 0.831470 + 0.555570i 0 0 0.773010 + 0.634393i −0.634393 − 0.773010i −0.0980171 + 0.995185i 0
853.1 0.471397 − 0.881921i 0 −0.555570 − 0.831470i 0 0 0.0980171 − 0.995185i −0.995185 + 0.0980171i 0.634393 − 0.773010i 0
909.1 −0.634393 + 0.773010i 0 −0.195090 − 0.980785i 0 0 −0.471397 − 0.881921i 0.881921 + 0.471397i 0.290285 + 0.956940i 0
965.1 0.634393 + 0.773010i 0 −0.195090 + 0.980785i 0 0 0.471397 − 0.881921i −0.881921 + 0.471397i −0.290285 + 0.956940i 0
1021.1 0.471397 + 0.881921i 0 −0.555570 + 0.831470i 0 0 0.0980171 + 0.995185i −0.995185 − 0.0980171i 0.634393 + 0.773010i 0
1077.1 −0.956940 + 0.290285i 0 0.831470 − 0.555570i 0 0 0.773010 − 0.634393i −0.634393 + 0.773010i −0.0980171 − 0.995185i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
256.m even 64 1 inner
1792.bt odd 64 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.1.bt.a ✓ 32
7.b odd 2 1 CM 1792.1.bt.a ✓ 32
256.m even 64 1 inner 1792.1.bt.a ✓ 32
1792.bt odd 64 1 inner 1792.1.bt.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.1.bt.a ✓ 32 1.a even 1 1 trivial
1792.1.bt.a ✓ 32 7.b odd 2 1 CM
1792.1.bt.a ✓ 32 256.m even 64 1 inner
1792.1.bt.a ✓ 32 1792.bt odd 64 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1792, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{32} + 1 \) Copy content Toggle raw display
$3$ \( T^{32} \) Copy content Toggle raw display
$5$ \( T^{32} \) Copy content Toggle raw display
$7$ \( T^{32} + 1 \) Copy content Toggle raw display
$11$ \( T^{32} + 64 T^{27} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{32} \) Copy content Toggle raw display
$17$ \( T^{32} \) Copy content Toggle raw display
$19$ \( T^{32} \) Copy content Toggle raw display
$23$ \( (T^{16} + 2 T^{8} + 64 T^{7} + \cdots + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{32} - 96 T^{25} + \cdots + 2 \) Copy content Toggle raw display
$31$ \( T^{32} \) Copy content Toggle raw display
$37$ \( T^{32} + 80 T^{26} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{32} \) Copy content Toggle raw display
$43$ \( T^{32} + 4 T^{24} + \cdots + 2 \) Copy content Toggle raw display
$47$ \( T^{32} \) Copy content Toggle raw display
$53$ \( T^{32} + 32 T^{27} + \cdots + 2 \) Copy content Toggle raw display
$59$ \( T^{32} \) Copy content Toggle raw display
$61$ \( T^{32} \) Copy content Toggle raw display
$67$ \( T^{32} + 8 T^{28} + \cdots + 2 \) Copy content Toggle raw display
$71$ \( T^{32} - 32 T^{28} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{32} \) Copy content Toggle raw display
$79$ \( T^{32} + 32 T^{26} + \cdots + 4 \) Copy content Toggle raw display
$83$ \( T^{32} \) Copy content Toggle raw display
$89$ \( T^{32} \) Copy content Toggle raw display
$97$ \( T^{32} \) Copy content Toggle raw display
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