Properties

Label 1911.1.cj.b
Level $1911$
Weight $1$
Character orbit 1911.cj
Analytic conductor $0.954$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1911.cj (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{14}^{3} q^{3} -\zeta_{14}^{2} q^{4} + \zeta_{14} q^{7} + \zeta_{14}^{6} q^{9} +O(q^{10})\) \( q + \zeta_{14}^{3} q^{3} -\zeta_{14}^{2} q^{4} + \zeta_{14} q^{7} + \zeta_{14}^{6} q^{9} -\zeta_{14}^{5} q^{12} + \zeta_{14}^{5} q^{13} + \zeta_{14}^{4} q^{16} + ( \zeta_{14} + \zeta_{14}^{6} ) q^{19} + \zeta_{14}^{4} q^{21} -\zeta_{14}^{6} q^{25} -\zeta_{14}^{2} q^{27} -\zeta_{14}^{3} q^{28} + ( \zeta_{14}^{2} + \zeta_{14}^{5} ) q^{31} + \zeta_{14} q^{36} + ( 1 + \zeta_{14}^{3} ) q^{37} -\zeta_{14} q^{39} + ( -\zeta_{14}^{3} - \zeta_{14}^{5} ) q^{43} - q^{48} + \zeta_{14}^{2} q^{49} + q^{52} + ( -\zeta_{14}^{2} + \zeta_{14}^{4} ) q^{57} + ( -1 + \zeta_{14}^{3} ) q^{61} - q^{63} -\zeta_{14}^{6} q^{64} + ( \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{67} + ( -\zeta_{14} - \zeta_{14}^{4} ) q^{73} + \zeta_{14}^{2} q^{75} + ( \zeta_{14} - \zeta_{14}^{3} ) q^{76} + ( -\zeta_{14}^{2} + \zeta_{14}^{5} ) q^{79} -\zeta_{14}^{5} q^{81} -\zeta_{14}^{6} q^{84} + \zeta_{14}^{6} q^{91} + ( -\zeta_{14} + \zeta_{14}^{5} ) q^{93} + ( -\zeta_{14} - \zeta_{14}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{4} + q^{7} - q^{9} + O(q^{10}) \) \( 6 q + q^{3} + q^{4} + q^{7} - q^{9} - q^{12} + q^{13} - q^{16} - q^{21} + q^{25} + q^{27} - q^{28} + q^{36} + 7 q^{37} - q^{39} - 2 q^{43} - 6 q^{48} - q^{49} + 6 q^{52} - 5 q^{61} - 6 q^{63} + q^{64} - q^{75} + 2 q^{79} - q^{81} + q^{84} - q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{14}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−0.623490 0.781831i
0.222521 + 0.974928i
0.222521 0.974928i
−0.623490 + 0.781831i
0.900969 0.433884i
0.900969 + 0.433884i
0 0.900969 0.433884i 0.222521 0.974928i 0 0 −0.623490 0.781831i 0 0.623490 0.781831i 0
428.1 0 −0.623490 0.781831i 0.900969 0.433884i 0 0 0.222521 + 0.974928i 0 −0.222521 + 0.974928i 0
701.1 0 −0.623490 + 0.781831i 0.900969 + 0.433884i 0 0 0.222521 0.974928i 0 −0.222521 0.974928i 0
974.1 0 0.900969 + 0.433884i 0.222521 + 0.974928i 0 0 −0.623490 + 0.781831i 0 0.623490 + 0.781831i 0
1247.1 0 0.222521 0.974928i −0.623490 + 0.781831i 0 0 0.900969 0.433884i 0 −0.900969 0.433884i 0
1793.1 0 0.222521 + 0.974928i −0.623490 0.781831i 0 0 0.900969 + 0.433884i 0 −0.900969 + 0.433884i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.1
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
637.bg even 14 1 inner
1911.cj odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.cj.b yes 6
3.b odd 2 1 CM 1911.1.cj.b yes 6
13.b even 2 1 1911.1.cj.a 6
39.d odd 2 1 1911.1.cj.a 6
49.e even 7 1 1911.1.cj.a 6
147.l odd 14 1 1911.1.cj.a 6
637.bg even 14 1 inner 1911.1.cj.b yes 6
1911.cj odd 14 1 inner 1911.1.cj.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.cj.a 6 13.b even 2 1
1911.1.cj.a 6 39.d odd 2 1
1911.1.cj.a 6 49.e even 7 1
1911.1.cj.a 6 147.l odd 14 1
1911.1.cj.b yes 6 1.a even 1 1 trivial
1911.1.cj.b yes 6 3.b odd 2 1 CM
1911.1.cj.b yes 6 637.bg even 14 1 inner
1911.1.cj.b yes 6 1911.cj odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{6} - 7 T_{37}^{5} + 21 T_{37}^{4} - 35 T_{37}^{3} + 35 T_{37}^{2} - 21 T_{37} + 7 \) acting on \(S_{1}^{\mathrm{new}}(1911, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
$37$ \( 7 - 21 T + 35 T^{2} - 35 T^{3} + 21 T^{4} - 7 T^{5} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( 1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( T^{6} \)
$61$ \( 1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6} \)
$67$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 7 + 14 T + 7 T^{2} + T^{6} \)
$79$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
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