Properties

Label 791.1.x.a
Level $791$
Weight $1$
Character orbit 791.x
Analytic conductor $0.395$
Analytic rank $0$
Dimension $12$
Projective image $D_{28}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 791 = 7 \cdot 113 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 791.x (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.394760424993\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{28})\)
Defining polynomial: \(x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{28}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{28}^{3} + \zeta_{28}^{7} ) q^{2} + ( -1 + \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{4} + \zeta_{28}^{8} q^{7} + ( \zeta_{28}^{3} - \zeta_{28}^{7} - \zeta_{28}^{9} + \zeta_{28}^{13} ) q^{8} -\zeta_{28}^{9} q^{9} +O(q^{10})\) \( q + ( -\zeta_{28}^{3} + \zeta_{28}^{7} ) q^{2} + ( -1 + \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{4} + \zeta_{28}^{8} q^{7} + ( \zeta_{28}^{3} - \zeta_{28}^{7} - \zeta_{28}^{9} + \zeta_{28}^{13} ) q^{8} -\zeta_{28}^{9} q^{9} + ( -\zeta_{28}^{10} - \zeta_{28}^{12} ) q^{11} + ( -\zeta_{28} - \zeta_{28}^{11} ) q^{14} + ( 1 + \zeta_{28}^{2} - \zeta_{28}^{6} + \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{16} + ( \zeta_{28}^{2} + \zeta_{28}^{12} ) q^{18} + ( -\zeta_{28} + \zeta_{28}^{3} + \zeta_{28}^{5} + \zeta_{28}^{13} ) q^{22} + ( \zeta_{28}^{8} + \zeta_{28}^{11} ) q^{23} + \zeta_{28}^{5} q^{25} + ( -1 + \zeta_{28}^{4} - \zeta_{28}^{8} ) q^{28} + ( \zeta_{28}^{4} - \zeta_{28}^{13} ) q^{29} + ( \zeta_{28} - \zeta_{28}^{3} - \zeta_{28}^{5} + \zeta_{28}^{7} + \zeta_{28}^{9} - \zeta_{28}^{13} ) q^{32} + ( \zeta_{28} - \zeta_{28}^{5} + \zeta_{28}^{9} ) q^{36} + ( -\zeta_{28}^{6} + \zeta_{28}^{9} ) q^{37} + ( -\zeta_{28}^{4} - \zeta_{28}^{13} ) q^{43} + ( \zeta_{28}^{2} + \zeta_{28}^{4} - \zeta_{28}^{6} - \zeta_{28}^{8} + \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{44} + ( 1 - \zeta_{28} - \zeta_{28}^{4} - \zeta_{28}^{11} ) q^{46} -\zeta_{28}^{2} q^{49} + ( -\zeta_{28}^{8} + \zeta_{28}^{12} ) q^{50} + ( -\zeta_{28}^{8} - \zeta_{28}^{12} ) q^{53} + ( \zeta_{28} + \zeta_{28}^{3} - \zeta_{28}^{7} + \zeta_{28}^{11} ) q^{56} + ( -\zeta_{28}^{2} + \zeta_{28}^{6} - \zeta_{28}^{7} + \zeta_{28}^{11} ) q^{58} + \zeta_{28}^{3} q^{63} + ( -1 - \zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{6} + \zeta_{28}^{8} - \zeta_{28}^{10} - \zeta_{28}^{12} ) q^{64} + ( \zeta_{28}^{2} + \zeta_{28}^{13} ) q^{67} + ( \zeta_{28}^{10} - \zeta_{28}^{11} ) q^{71} + ( -\zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{8} - \zeta_{28}^{12} ) q^{72} + ( -\zeta_{28}^{2} + \zeta_{28}^{9} - \zeta_{28}^{12} - \zeta_{28}^{13} ) q^{74} + ( \zeta_{28}^{4} + \zeta_{28}^{6} ) q^{77} + ( -\zeta_{28}^{3} + \zeta_{28}^{10} ) q^{79} -\zeta_{28}^{4} q^{81} + ( -\zeta_{28}^{2} + \zeta_{28}^{6} + \zeta_{28}^{7} - \zeta_{28}^{11} ) q^{86} + ( \zeta_{28} - \zeta_{28}^{3} - 2 \zeta_{28}^{5} - \zeta_{28}^{7} + \zeta_{28}^{9} + \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{88} + ( -1 - \zeta_{28}^{3} + \zeta_{28}^{4} + \zeta_{28}^{7} - \zeta_{28}^{8} - \zeta_{28}^{11} ) q^{92} + ( \zeta_{28}^{5} - \zeta_{28}^{9} ) q^{98} + ( -\zeta_{28}^{5} - \zeta_{28}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{4} - 2q^{7} + O(q^{10}) \) \( 12q - 12q^{4} - 2q^{7} + 12q^{16} - 2q^{23} - 12q^{28} - 2q^{29} - 2q^{37} + 2q^{43} + 14q^{46} - 2q^{49} + 4q^{53} - 12q^{64} + 2q^{67} + 2q^{71} + 2q^{79} + 2q^{81} - 12q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/791\mathbb{Z}\right)^\times\).

\(n\) \(227\) \(568\)
\(\chi(n)\) \(-1\) \(-\zeta_{28}^{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
−0.433884 0.900969i
0.781831 0.623490i
0.781831 + 0.623490i
−0.781831 0.623490i
−0.781831 + 0.623490i
0.433884 + 0.900969i
−0.974928 0.222521i
0.433884 0.900969i
−0.974928 + 0.222521i
0.974928 0.222521i
−0.433884 + 0.900969i
0.974928 + 0.222521i
−0.974928 1.22252i 0 −0.321552 + 1.40881i 0 0 −0.900969 + 0.433884i 0.626980 0.301938i −0.781831 0.623490i 0
258.1 0.433884 + 1.90097i 0 −2.52446 + 1.21572i 0 0 0.623490 + 0.781831i −2.19064 2.74698i −0.974928 0.222521i 0
279.1 0.433884 1.90097i 0 −2.52446 1.21572i 0 0 0.623490 0.781831i −2.19064 + 2.74698i −0.974928 + 0.222521i 0
286.1 −0.433884 + 1.90097i 0 −2.52446 1.21572i 0 0 0.623490 0.781831i 2.19064 2.74698i 0.974928 0.222521i 0
307.1 −0.433884 1.90097i 0 −2.52446 + 1.21572i 0 0 0.623490 + 0.781831i 2.19064 + 2.74698i 0.974928 + 0.222521i 0
454.1 0.974928 + 1.22252i 0 −0.321552 + 1.40881i 0 0 −0.900969 + 0.433884i −0.626980 + 0.301938i 0.781831 + 0.623490i 0
573.1 0.781831 0.376510i 0 −0.153989 + 0.193096i 0 0 −0.222521 + 0.974928i −0.240787 + 1.05496i −0.433884 + 0.900969i 0
622.1 0.974928 1.22252i 0 −0.321552 1.40881i 0 0 −0.900969 0.433884i −0.626980 0.301938i 0.781831 0.623490i 0
664.1 0.781831 + 0.376510i 0 −0.153989 0.193096i 0 0 −0.222521 0.974928i −0.240787 1.05496i −0.433884 0.900969i 0
692.1 −0.781831 0.376510i 0 −0.153989 0.193096i 0 0 −0.222521 0.974928i 0.240787 + 1.05496i 0.433884 + 0.900969i 0
734.1 −0.974928 + 1.22252i 0 −0.321552 1.40881i 0 0 −0.900969 0.433884i 0.626980 + 0.301938i −0.781831 + 0.623490i 0
783.1 −0.781831 + 0.376510i 0 −0.153989 + 0.193096i 0 0 −0.222521 + 0.974928i 0.240787 1.05496i 0.433884 0.900969i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 783.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
113.h even 28 1 inner
791.x odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 791.1.x.a 12
7.b odd 2 1 CM 791.1.x.a 12
113.h even 28 1 inner 791.1.x.a 12
791.x odd 28 1 inner 791.1.x.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
791.1.x.a 12 1.a even 1 1 trivial
791.1.x.a 12 7.b odd 2 1 CM
791.1.x.a 12 113.h even 28 1 inner
791.1.x.a 12 791.x odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(791, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$11$ \( ( 7 + 14 T + 7 T^{2} + T^{6} )^{2} \)
$13$ \( T^{12} \)
$17$ \( T^{12} \)
$19$ \( T^{12} \)
$23$ \( 1 - 6 T + 32 T^{2} - 70 T^{3} + 61 T^{4} - 2 T^{5} - 8 T^{6} - 8 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12} \)
$29$ \( 1 + 8 T + 39 T^{2} + 84 T^{3} + 82 T^{4} + 40 T^{5} - T^{6} - 22 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12} \)
$31$ \( T^{12} \)
$37$ \( 1 + 8 T + 39 T^{2} + 84 T^{3} + 82 T^{4} + 40 T^{5} - T^{6} - 22 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 - 8 T + 39 T^{2} - 84 T^{3} + 82 T^{4} - 40 T^{5} - T^{6} + 22 T^{7} - 4 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( ( 1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$59$ \( T^{12} \)
$61$ \( T^{12} \)
$67$ \( 1 + 6 T + 11 T^{2} - 14 T^{3} + 26 T^{4} - 68 T^{5} + 55 T^{6} - 20 T^{7} + 24 T^{8} - 14 T^{9} + 2 T^{10} - 2 T^{11} + T^{12} \)
$71$ \( 1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} \)
$73$ \( T^{12} \)
$79$ \( 64 - 64 T + 32 T^{2} - 16 T^{4} + 16 T^{5} - 8 T^{6} + 8 T^{7} - 4 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( T^{12} \)
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