Properties

Label 1911.1.cc.a
Level $1911$
Weight $1$
Character orbit 1911.cc
Analytic conductor $0.954$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1911.cc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.968877.1
Artin image: $C_6\times C_4\wr C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-expansion

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

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

Character values

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

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{3}\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
668.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 −0.866025 0.500000i 0.866025 + 0.500000i 0 0 0 0 0.500000 + 0.866025i 0
1097.1 0 0.866025 0.500000i −0.866025 + 0.500000i 0 0 0 0 0.500000 0.866025i 0
1256.1 0 0.866025 + 0.500000i −0.866025 0.500000i 0 0 0 0 0.500000 + 0.866025i 0
1685.1 0 −0.866025 + 0.500000i 0.866025 0.500000i 0 0 0 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner
91.i even 4 1 inner
91.bb even 12 1 inner
273.o odd 4 1 inner
273.cb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.cc.a 4
3.b odd 2 1 CM 1911.1.cc.a 4
7.b odd 2 1 1911.1.cc.b 4
7.c even 3 1 273.1.o.b yes 2
7.c even 3 1 inner 1911.1.cc.a 4
7.d odd 6 1 273.1.o.a 2
7.d odd 6 1 1911.1.cc.b 4
13.d odd 4 1 1911.1.cc.b 4
21.c even 2 1 1911.1.cc.b 4
21.g even 6 1 273.1.o.a 2
21.g even 6 1 1911.1.cc.b 4
21.h odd 6 1 273.1.o.b yes 2
21.h odd 6 1 inner 1911.1.cc.a 4
39.f even 4 1 1911.1.cc.b 4
91.g even 3 1 3549.1.ca.a 4
91.h even 3 1 3549.1.ca.a 4
91.i even 4 1 inner 1911.1.cc.a 4
91.k even 6 1 3549.1.ca.d 4
91.l odd 6 1 3549.1.ca.c 4
91.m odd 6 1 3549.1.ca.b 4
91.p odd 6 1 3549.1.ca.c 4
91.r even 6 1 3549.1.o.a 2
91.s odd 6 1 3549.1.o.b 2
91.u even 6 1 3549.1.ca.d 4
91.v odd 6 1 3549.1.ca.b 4
91.w even 12 1 3549.1.ca.a 4
91.w even 12 1 3549.1.ca.d 4
91.x odd 12 1 3549.1.ca.b 4
91.x odd 12 1 3549.1.ca.c 4
91.z odd 12 1 273.1.o.a 2
91.z odd 12 1 1911.1.cc.b 4
91.z odd 12 1 3549.1.o.b 2
91.ba even 12 1 3549.1.ca.a 4
91.ba even 12 1 3549.1.ca.d 4
91.bb even 12 1 273.1.o.b yes 2
91.bb even 12 1 inner 1911.1.cc.a 4
91.bb even 12 1 3549.1.o.a 2
91.bd odd 12 1 3549.1.ca.b 4
91.bd odd 12 1 3549.1.ca.c 4
273.o odd 4 1 inner 1911.1.cc.a 4
273.r even 6 1 3549.1.ca.b 4
273.s odd 6 1 3549.1.ca.a 4
273.w odd 6 1 3549.1.o.a 2
273.x odd 6 1 3549.1.ca.d 4
273.y even 6 1 3549.1.ca.c 4
273.ba even 6 1 3549.1.o.b 2
273.bf even 6 1 3549.1.ca.b 4
273.bm odd 6 1 3549.1.ca.a 4
273.bp odd 6 1 3549.1.ca.d 4
273.br even 6 1 3549.1.ca.c 4
273.bs odd 12 1 3549.1.ca.a 4
273.bs odd 12 1 3549.1.ca.d 4
273.bv even 12 1 3549.1.ca.b 4
273.bv even 12 1 3549.1.ca.c 4
273.bw even 12 1 3549.1.ca.b 4
273.bw even 12 1 3549.1.ca.c 4
273.cb odd 12 1 273.1.o.b yes 2
273.cb odd 12 1 inner 1911.1.cc.a 4
273.cb odd 12 1 3549.1.o.a 2
273.cd even 12 1 273.1.o.a 2
273.cd even 12 1 1911.1.cc.b 4
273.cd even 12 1 3549.1.o.b 2
273.ch odd 12 1 3549.1.ca.a 4
273.ch odd 12 1 3549.1.ca.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.o.a 2 7.d odd 6 1
273.1.o.a 2 21.g even 6 1
273.1.o.a 2 91.z odd 12 1
273.1.o.a 2 273.cd even 12 1
273.1.o.b yes 2 7.c even 3 1
273.1.o.b yes 2 21.h odd 6 1
273.1.o.b yes 2 91.bb even 12 1
273.1.o.b yes 2 273.cb odd 12 1
1911.1.cc.a 4 1.a even 1 1 trivial
1911.1.cc.a 4 3.b odd 2 1 CM
1911.1.cc.a 4 7.c even 3 1 inner
1911.1.cc.a 4 21.h odd 6 1 inner
1911.1.cc.a 4 91.i even 4 1 inner
1911.1.cc.a 4 91.bb even 12 1 inner
1911.1.cc.a 4 273.o odd 4 1 inner
1911.1.cc.a 4 273.cb odd 12 1 inner
1911.1.cc.b 4 7.b odd 2 1
1911.1.cc.b 4 7.d odd 6 1
1911.1.cc.b 4 13.d odd 4 1
1911.1.cc.b 4 21.c even 2 1
1911.1.cc.b 4 21.g even 6 1
1911.1.cc.b 4 39.f even 4 1
1911.1.cc.b 4 91.z odd 12 1
1911.1.cc.b 4 273.cd even 12 1
3549.1.o.a 2 91.r even 6 1
3549.1.o.a 2 91.bb even 12 1
3549.1.o.a 2 273.w odd 6 1
3549.1.o.a 2 273.cb odd 12 1
3549.1.o.b 2 91.s odd 6 1
3549.1.o.b 2 91.z odd 12 1
3549.1.o.b 2 273.ba even 6 1
3549.1.o.b 2 273.cd even 12 1
3549.1.ca.a 4 91.g even 3 1
3549.1.ca.a 4 91.h even 3 1
3549.1.ca.a 4 91.w even 12 1
3549.1.ca.a 4 91.ba even 12 1
3549.1.ca.a 4 273.s odd 6 1
3549.1.ca.a 4 273.bm odd 6 1
3549.1.ca.a 4 273.bs odd 12 1
3549.1.ca.a 4 273.ch odd 12 1
3549.1.ca.b 4 91.m odd 6 1
3549.1.ca.b 4 91.v odd 6 1
3549.1.ca.b 4 91.x odd 12 1
3549.1.ca.b 4 91.bd odd 12 1
3549.1.ca.b 4 273.r even 6 1
3549.1.ca.b 4 273.bf even 6 1
3549.1.ca.b 4 273.bv even 12 1
3549.1.ca.b 4 273.bw even 12 1
3549.1.ca.c 4 91.l odd 6 1
3549.1.ca.c 4 91.p odd 6 1
3549.1.ca.c 4 91.x odd 12 1
3549.1.ca.c 4 91.bd odd 12 1
3549.1.ca.c 4 273.y even 6 1
3549.1.ca.c 4 273.br even 6 1
3549.1.ca.c 4 273.bv even 12 1
3549.1.ca.c 4 273.bw even 12 1
3549.1.ca.d 4 91.k even 6 1
3549.1.ca.d 4 91.u even 6 1
3549.1.ca.d 4 91.w even 12 1
3549.1.ca.d 4 91.ba even 12 1
3549.1.ca.d 4 273.x odd 6 1
3549.1.ca.d 4 273.bp odd 6 1
3549.1.ca.d 4 273.bs odd 12 1
3549.1.ca.d 4 273.ch odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{4} - 2 T_{19}^{3} + 2 T_{19}^{2} - 4 T_{19} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1911, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( 16 - 4 T^{2} + T^{4} \)
$67$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 2 - 2 T + T^{2} )^{2} \)
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