Properties

Label 1911.1.cv.a
Level $1911$
Weight $1$
Character orbit 1911.cv
Analytic conductor $0.954$
Analytic rank $0$
Dimension $12$
Projective image $D_{28}$
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.cv (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953713239142\)
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: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

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

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(\zeta_{28}^{7}\) \(\zeta_{28}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
−0.974928 0.222521i
0.433884 + 0.900969i
0.974928 0.222521i
−0.781831 0.623490i
−0.781831 + 0.623490i
0.974928 + 0.222521i
0.433884 0.900969i
−0.974928 + 0.222521i
0.781831 0.623490i
−0.433884 0.900969i
−0.433884 + 0.900969i
0.781831 + 0.623490i
0 0.781831 + 0.623490i −0.433884 + 0.900969i 0 0 −0.222521 + 0.974928i 0 0.222521 + 0.974928i 0
125.1 0 0.974928 + 0.222521i 0.781831 + 0.623490i 0 0 −0.900969 + 0.433884i 0 0.900969 + 0.433884i 0
356.1 0 −0.781831 + 0.623490i 0.433884 + 0.900969i 0 0 −0.222521 0.974928i 0 0.222521 0.974928i 0
398.1 0 −0.433884 + 0.900969i 0.974928 0.222521i 0 0 0.623490 0.781831i 0 −0.623490 0.781831i 0
629.1 0 −0.433884 0.900969i 0.974928 + 0.222521i 0 0 0.623490 + 0.781831i 0 −0.623490 + 0.781831i 0
671.1 0 −0.781831 0.623490i 0.433884 0.900969i 0 0 −0.222521 + 0.974928i 0 0.222521 + 0.974928i 0
902.1 0 0.974928 0.222521i 0.781831 0.623490i 0 0 −0.900969 0.433884i 0 0.900969 0.433884i 0
944.1 0 0.781831 0.623490i −0.433884 0.900969i 0 0 −0.222521 0.974928i 0 0.222521 0.974928i 0
1217.1 0 0.433884 + 0.900969i −0.974928 0.222521i 0 0 0.623490 + 0.781831i 0 −0.623490 + 0.781831i 0
1448.1 0 −0.974928 0.222521i −0.781831 0.623490i 0 0 −0.900969 + 0.433884i 0 0.900969 + 0.433884i 0
1490.1 0 −0.974928 + 0.222521i −0.781831 + 0.623490i 0 0 −0.900969 0.433884i 0 0.900969 0.433884i 0
1721.1 0 0.433884 0.900969i −0.974928 + 0.222521i 0 0 0.623490 0.781831i 0 −0.623490 0.781831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1721.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
637.bn even 28 1 inner
1911.cv odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.cv.a 12
3.b odd 2 1 CM 1911.1.cv.a 12
13.d odd 4 1 1911.1.cv.b yes 12
39.f even 4 1 1911.1.cv.b yes 12
49.f odd 14 1 1911.1.cv.b yes 12
147.k even 14 1 1911.1.cv.b yes 12
637.bn even 28 1 inner 1911.1.cv.a 12
1911.cv odd 28 1 inner 1911.1.cv.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.cv.a 12 1.a even 1 1 trivial
1911.1.cv.a 12 3.b odd 2 1 CM
1911.1.cv.a 12 637.bn even 28 1 inner
1911.1.cv.a 12 1911.cv odd 28 1 inner
1911.1.cv.b yes 12 13.d odd 4 1
1911.1.cv.b yes 12 39.f even 4 1
1911.1.cv.b yes 12 49.f odd 14 1
1911.1.cv.b yes 12 147.k even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{19}^{12} - \cdots\) acting on \(S_{1}^{\mathrm{new}}(1911, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$11$ \( T^{12} \)
$13$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( 1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( 1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} \)
$37$ \( 1 - 8 T + 11 T^{2} + 28 T^{3} - 23 T^{4} - 12 T^{5} + 41 T^{6} - 34 T^{7} + 31 T^{8} - 14 T^{9} + 9 T^{10} - 2 T^{11} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( ( 7 + 14 T + 7 T^{2} + T^{6} )^{2} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 - 9 T^{2} + 25 T^{4} - T^{6} + 9 T^{8} + 3 T^{10} + T^{12} \)
$67$ \( 1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( 1 - 6 T + 4 T^{2} + 14 T^{3} + 61 T^{4} + 54 T^{5} + 48 T^{6} + 20 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12} \)
$79$ \( ( -7 + 14 T^{2} - 7 T^{4} + T^{6} )^{2} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( 1 + 8 T + 32 T^{2} + 42 T^{3} + 26 T^{4} - 2 T^{5} + 34 T^{6} + 34 T^{7} + 17 T^{8} + 2 T^{10} + 2 T^{11} + T^{12} \)
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