Properties

Label 693.1.bp.a
Level $693$
Weight $1$
Character orbit 693.bp
Analytic conductor $0.346$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,1,Mod(62,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.62");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 693.bp (of order \(10\), 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: \(0.345852053755\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{40}^{7} + \zeta_{40}) q^{2} + (\zeta_{40}^{14} + \zeta_{40}^{8} + \zeta_{40}^{2}) q^{4} + \zeta_{40}^{18} q^{7} + (\zeta_{40}^{15} + \zeta_{40}^{9} + \zeta_{40}^{3} - \zeta_{40}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{40}^{7} + \zeta_{40}) q^{2} + (\zeta_{40}^{14} + \zeta_{40}^{8} + \zeta_{40}^{2}) q^{4} + \zeta_{40}^{18} q^{7} + (\zeta_{40}^{15} + \zeta_{40}^{9} + \zeta_{40}^{3} - \zeta_{40}) q^{8} - \zeta_{40}^{11} q^{11} + (\zeta_{40}^{19} - \zeta_{40}^{5}) q^{14} + (\zeta_{40}^{16} + \zeta_{40}^{10} - \zeta_{40}^{8} + \zeta_{40}^{4} + \zeta_{40}^{2}) q^{16} + ( - \zeta_{40}^{18} - \zeta_{40}^{12}) q^{22} + ( - \zeta_{40}^{17} - \zeta_{40}^{3}) q^{23} + \zeta_{40}^{12} q^{25} + ( - \zeta_{40}^{12} - \zeta_{40}^{6} - 1) q^{28} + ( - \zeta_{40}^{9} - \zeta_{40}^{7}) q^{29} + (\zeta_{40}^{17} - \zeta_{40}^{15} + \zeta_{40}^{11} - \zeta_{40}^{9} + \zeta_{40}^{5} + \zeta_{40}^{3}) q^{32} + (\zeta_{40}^{10} + \zeta_{40}^{6}) q^{37} + ( - \zeta_{40}^{14} - \zeta_{40}^{6}) q^{43} + ( - \zeta_{40}^{19} - \zeta_{40}^{13} + \zeta_{40}^{5}) q^{44} + ( - \zeta_{40}^{18} - \zeta_{40}^{10} + \zeta_{40}^{4}) q^{46} - \zeta_{40}^{16} q^{49} + (\zeta_{40}^{19} + \zeta_{40}^{13}) q^{50} + (\zeta_{40}^{15} + \zeta_{40}^{13}) q^{53} + ( - \zeta_{40}^{19} - \zeta_{40}^{13} - \zeta_{40}^{7} - \zeta_{40}) q^{56} + ( - \zeta_{40}^{16} - \zeta_{40}^{14} - \zeta_{40}^{10} - \zeta_{40}^{8}) q^{58} + (\zeta_{40}^{18} - \zeta_{40}^{16} + \zeta_{40}^{12} - \zeta_{40}^{10} + \zeta_{40}^{6} + \zeta_{40}^{4} + \zeta_{40}^{2}) q^{64} + (\zeta_{40}^{16} - \zeta_{40}^{4}) q^{67} + (\zeta_{40}^{17} - \zeta_{40}^{15}) q^{71} + (\zeta_{40}^{17} + \zeta_{40}^{13} + \zeta_{40}^{11} + \zeta_{40}^{7}) q^{74} + \zeta_{40}^{9} q^{77} + (\zeta_{40}^{8} - 1) q^{79} + ( - \zeta_{40}^{15} - \zeta_{40}^{13} - \zeta_{40}^{7} + \zeta_{40}) q^{86} + ( - \zeta_{40}^{14} + \zeta_{40}^{12} + \zeta_{40}^{6} + 1) q^{88} + ( - \zeta_{40}^{19} - \zeta_{40}^{17} + \zeta_{40}^{11} + \zeta_{40}^{5}) q^{92} + ( - \zeta_{40}^{17} + \zeta_{40}^{3}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 4 q^{16} - 4 q^{22} + 4 q^{25} - 20 q^{28} + 4 q^{49} + 8 q^{58} + 4 q^{64} - 8 q^{67} - 20 q^{79} + 20 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{40}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
−0.987688 0.156434i
0.156434 0.987688i
−0.156434 + 0.987688i
0.987688 + 0.156434i
−0.453990 0.891007i
0.891007 0.453990i
−0.891007 + 0.453990i
0.453990 + 0.891007i
−0.987688 + 0.156434i
0.156434 + 0.987688i
−0.156434 0.987688i
0.987688 0.156434i
−0.453990 + 0.891007i
0.891007 + 0.453990i
−0.891007 0.453990i
0.453990 0.891007i
−1.44168 1.04744i 0 0.672288 + 2.06909i 0 0 −0.951057 + 0.309017i 0.647354 1.99235i 0 0
62.2 −0.734572 0.533698i 0 −0.0542543 0.166977i 0 0 0.951057 0.309017i −0.329843 + 1.01515i 0 0
62.3 0.734572 + 0.533698i 0 −0.0542543 0.166977i 0 0 0.951057 0.309017i 0.329843 1.01515i 0 0
62.4 1.44168 + 1.04744i 0 0.672288 + 2.06909i 0 0 −0.951057 + 0.309017i −0.647354 + 1.99235i 0 0
314.1 −0.610425 1.87869i 0 −2.34786 + 1.70582i 0 0 0.587785 + 0.809017i 3.03979 + 2.20854i 0 0
314.2 −0.0966818 0.297556i 0 0.729825 0.530249i 0 0 −0.587785 0.809017i −0.481456 0.349798i 0 0
314.3 0.0966818 + 0.297556i 0 0.729825 0.530249i 0 0 −0.587785 0.809017i 0.481456 + 0.349798i 0 0
314.4 0.610425 + 1.87869i 0 −2.34786 + 1.70582i 0 0 0.587785 + 0.809017i −3.03979 2.20854i 0 0
503.1 −1.44168 + 1.04744i 0 0.672288 2.06909i 0 0 −0.951057 0.309017i 0.647354 + 1.99235i 0 0
503.2 −0.734572 + 0.533698i 0 −0.0542543 + 0.166977i 0 0 0.951057 + 0.309017i −0.329843 1.01515i 0 0
503.3 0.734572 0.533698i 0 −0.0542543 + 0.166977i 0 0 0.951057 + 0.309017i 0.329843 + 1.01515i 0 0
503.4 1.44168 1.04744i 0 0.672288 2.06909i 0 0 −0.951057 0.309017i −0.647354 1.99235i 0 0
629.1 −0.610425 + 1.87869i 0 −2.34786 1.70582i 0 0 0.587785 0.809017i 3.03979 2.20854i 0 0
629.2 −0.0966818 + 0.297556i 0 0.729825 + 0.530249i 0 0 −0.587785 + 0.809017i −0.481456 + 0.349798i 0 0
629.3 0.0966818 0.297556i 0 0.729825 + 0.530249i 0 0 −0.587785 + 0.809017i 0.481456 0.349798i 0 0
629.4 0.610425 1.87869i 0 −2.34786 1.70582i 0 0 0.587785 0.809017i −3.03979 + 2.20854i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
11.d odd 10 1 inner
21.c even 2 1 inner
33.f even 10 1 inner
77.l even 10 1 inner
231.r odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.1.bp.a 16
3.b odd 2 1 inner 693.1.bp.a 16
7.b odd 2 1 CM 693.1.bp.a 16
11.d odd 10 1 inner 693.1.bp.a 16
21.c even 2 1 inner 693.1.bp.a 16
33.f even 10 1 inner 693.1.bp.a 16
77.l even 10 1 inner 693.1.bp.a 16
231.r odd 10 1 inner 693.1.bp.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.1.bp.a 16 1.a even 1 1 trivial
693.1.bp.a 16 3.b odd 2 1 inner
693.1.bp.a 16 7.b odd 2 1 CM
693.1.bp.a 16 11.d odd 10 1 inner
693.1.bp.a 16 21.c even 2 1 inner
693.1.bp.a 16 33.f even 10 1 inner
693.1.bp.a 16 77.l even 10 1 inner
693.1.bp.a 16 231.r odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(693, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{14} + 12 T^{12} + 32 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - T^{12} + T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} + 8 T^{6} + 19 T^{4} + 12 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 4 T^{14} + 12 T^{12} + 32 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} - 4 T^{14} + 17 T^{12} - 72 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} - 4 T^{14} + 17 T^{12} - 72 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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