Properties

Label 723.1.bc.a
Level $723$
Weight $1$
Character orbit 723.bc
Analytic conductor $0.361$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 723 = 3 \cdot 241 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 723.bc (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.360824004134\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{40}^{13} q^{3} + \zeta_{40}^{10} q^{4} + ( -\zeta_{40} - \zeta_{40}^{2} ) q^{7} -\zeta_{40}^{6} q^{9} +O(q^{10})\) \( q + \zeta_{40}^{13} q^{3} + \zeta_{40}^{10} q^{4} + ( -\zeta_{40} - \zeta_{40}^{2} ) q^{7} -\zeta_{40}^{6} q^{9} -\zeta_{40}^{3} q^{12} + ( \zeta_{40}^{6} + \zeta_{40}^{15} ) q^{13} - q^{16} + ( \zeta_{40}^{17} - \zeta_{40}^{18} ) q^{19} + ( -\zeta_{40}^{14} - \zeta_{40}^{15} ) q^{21} + \zeta_{40}^{14} q^{25} -\zeta_{40}^{19} q^{27} + ( -\zeta_{40}^{11} - \zeta_{40}^{12} ) q^{28} + ( -\zeta_{40}^{4} - \zeta_{40}^{9} ) q^{31} -\zeta_{40}^{16} q^{36} + ( \zeta_{40} + \zeta_{40}^{18} ) q^{37} + ( -\zeta_{40}^{8} + \zeta_{40}^{19} ) q^{39} + ( \zeta_{40}^{5} + \zeta_{40}^{12} ) q^{43} -\zeta_{40}^{13} q^{48} + ( \zeta_{40}^{2} + \zeta_{40}^{3} + \zeta_{40}^{4} ) q^{49} + ( -\zeta_{40}^{5} + \zeta_{40}^{16} ) q^{52} + ( -\zeta_{40}^{10} + \zeta_{40}^{11} ) q^{57} + ( \zeta_{40}^{9} - \zeta_{40}^{17} ) q^{61} + ( \zeta_{40}^{7} + \zeta_{40}^{8} ) q^{63} -\zeta_{40}^{10} q^{64} + ( \zeta_{40}^{7} + \zeta_{40}^{11} ) q^{67} + ( \zeta_{40}^{3} + \zeta_{40}^{16} ) q^{73} -\zeta_{40}^{7} q^{75} + ( -\zeta_{40}^{7} + \zeta_{40}^{8} ) q^{76} + ( -\zeta_{40}^{5} + \zeta_{40}^{9} ) q^{79} + \zeta_{40}^{12} q^{81} + ( \zeta_{40}^{4} + \zeta_{40}^{5} ) q^{84} + ( -\zeta_{40}^{7} - \zeta_{40}^{8} - \zeta_{40}^{16} - \zeta_{40}^{17} ) q^{91} + ( \zeta_{40}^{2} - \zeta_{40}^{17} ) q^{93} + ( -\zeta_{40}^{13} + \zeta_{40}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{16} - 4q^{28} - 4q^{31} + 4q^{36} + 4q^{39} + 4q^{43} + 4q^{49} - 4q^{52} - 4q^{63} - 4q^{73} - 4q^{76} + 4q^{81} + 4q^{84} + 8q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/723\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(242\)
\(\chi(n)\) \(\zeta_{40}^{3}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.987688 0.156434i
−0.453990 0.891007i
0.453990 0.891007i
0.891007 + 0.453990i
−0.891007 0.453990i
−0.453990 + 0.891007i
0.453990 + 0.891007i
0.987688 + 0.156434i
−0.156434 0.987688i
0.156434 0.987688i
−0.987688 + 0.156434i
0.891007 0.453990i
−0.891007 + 0.453990i
0.987688 0.156434i
−0.156434 + 0.987688i
0.156434 + 0.987688i
0 0.453990 0.891007i 1.00000i 0 0 0.0366318 0.152583i 0 −0.587785 0.809017i 0
41.1 0 0.156434 0.987688i 1.00000i 0 0 1.04178 + 0.0819895i 0 −0.951057 0.309017i 0
47.1 0 −0.156434 0.987688i 1.00000i 0 0 0.133795 + 1.70002i 0 −0.951057 + 0.309017i 0
116.1 0 0.987688 0.156434i 1.00000i 0 0 −1.47879 1.26301i 0 0.951057 0.309017i 0
125.1 0 −0.987688 + 0.156434i 1.00000i 0 0 0.303221 0.355026i 0 0.951057 0.309017i 0
194.1 0 0.156434 + 0.987688i 1.00000i 0 0 1.04178 0.0819895i 0 −0.951057 + 0.309017i 0
200.1 0 −0.156434 + 0.987688i 1.00000i 0 0 0.133795 1.70002i 0 −0.951057 0.309017i 0
236.1 0 −0.453990 + 0.891007i 1.00000i 0 0 −1.93874 0.465451i 0 −0.587785 0.809017i 0
302.1 0 −0.891007 + 0.453990i 1.00000i 0 0 1.10749 + 0.678671i 0 0.587785 0.809017i 0
320.1 0 0.891007 + 0.453990i 1.00000i 0 0 0.794622 + 1.29671i 0 0.587785 + 0.809017i 0
434.1 0 0.453990 + 0.891007i 1.00000i 0 0 0.0366318 + 0.152583i 0 −0.587785 + 0.809017i 0
455.1 0 0.987688 + 0.156434i 1.00000i 0 0 −1.47879 + 1.26301i 0 0.951057 + 0.309017i 0
509.1 0 −0.987688 0.156434i 1.00000i 0 0 0.303221 + 0.355026i 0 0.951057 + 0.309017i 0
530.1 0 −0.453990 0.891007i 1.00000i 0 0 −1.93874 + 0.465451i 0 −0.587785 + 0.809017i 0
644.1 0 −0.891007 0.453990i 1.00000i 0 0 1.10749 0.678671i 0 0.587785 + 0.809017i 0
662.1 0 0.891007 0.453990i 1.00000i 0 0 0.794622 1.29671i 0 0.587785 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 662.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
241.o even 40 1 inner
723.bc odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 723.1.bc.a 16
3.b odd 2 1 CM 723.1.bc.a 16
241.o even 40 1 inner 723.1.bc.a 16
723.bc odd 40 1 inner 723.1.bc.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.1.bc.a 16 1.a even 1 1 trivial
723.1.bc.a 16 3.b odd 2 1 CM
723.1.bc.a 16 241.o even 40 1 inner
723.1.bc.a 16 723.bc odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(723, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1 - 8 T + 68 T^{2} - 256 T^{3} + 557 T^{4} - 628 T^{5} + 298 T^{6} + 32 T^{7} - 100 T^{8} + 32 T^{9} + 16 T^{10} - 16 T^{11} + 2 T^{12} + 4 T^{13} - 2 T^{14} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 - 12 T + 68 T^{2} - 144 T^{3} + 222 T^{4} - 92 T^{5} + 118 T^{6} - 72 T^{7} - 80 T^{8} + 8 T^{9} + 6 T^{10} + 16 T^{11} + 7 T^{12} - 4 T^{13} - 2 T^{14} + T^{16} \)
$17$ \( T^{16} \)
$19$ \( 1 - 8 T + 58 T^{2} - 136 T^{3} + 82 T^{4} + 52 T^{5} + 28 T^{6} + 72 T^{7} + 75 T^{8} + 52 T^{9} + 16 T^{10} - 16 T^{11} + 2 T^{12} + 4 T^{13} - 2 T^{14} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( 16 + 32 T + 16 T^{2} - 32 T^{3} - 56 T^{4} - 112 T^{5} - 128 T^{6} - 16 T^{7} + 156 T^{8} + 160 T^{9} + 112 T^{10} + 64 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$37$ \( 1 - 12 T + 58 T^{2} - 104 T^{3} + 237 T^{4} - 112 T^{5} - 42 T^{6} + 208 T^{7} - 80 T^{8} + 8 T^{9} + 86 T^{10} - 24 T^{11} + 2 T^{12} + 16 T^{13} - 2 T^{14} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( 1 - 12 T + 76 T^{2} - 188 T^{3} + 174 T^{4} + 72 T^{5} - 158 T^{6} + 156 T^{7} - 64 T^{8} - 60 T^{9} + 82 T^{10} - 64 T^{11} + 39 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( T^{16} \)
$61$ \( 625 + 125 T^{4} + 150 T^{8} - 20 T^{12} + T^{16} \)
$67$ \( 625 + 125 T^{4} + 150 T^{8} - 20 T^{12} + T^{16} \)
$71$ \( T^{16} \)
$73$ \( 1 - 8 T - 4 T^{2} + 68 T^{3} + 269 T^{4} + 328 T^{5} + 382 T^{6} + 284 T^{7} + 156 T^{8} + 60 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$79$ \( 1 + 4 T^{4} + 46 T^{8} - 11 T^{12} + T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( 1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16} \)
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