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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Newspace parameters

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

Self dual: no
Analytic conductor: \(0.345852053755\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
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

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

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

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.

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 629.4
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

$p$ $F_p(T)$
$2$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$11$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$13$ \( T^{16} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( ( 1 + 12 T^{2} + 19 T^{4} + 8 T^{6} + T^{8} )^{2} \)
$29$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( ( 25 + 10 T^{4} + 5 T^{6} + T^{8} )^{2} \)
$41$ \( T^{16} \)
$43$ \( ( 1 + 3 T^{2} + T^{4} )^{4} \)
$47$ \( T^{16} \)
$53$ \( 1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( T^{16} \)
$67$ \( ( -1 + T + T^{2} )^{8} \)
$71$ \( 1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( ( 5 + 10 T + 10 T^{2} + 5 T^{3} + T^{4} )^{4} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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