Properties

Label 1911.1.w.e
Level $1911$
Weight $1$
Character orbit 1911.w
Analytic conductor $0.954$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -39
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.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.2
Defining polynomial: \(x^{8} + 4 x^{6} + 14 x^{4} + 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.90724673403.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{5} q^{3} + ( \beta_{5} - \beta_{6} ) q^{4} + \beta_{1} q^{5} + \beta_{7} q^{6} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{8} + ( -1 - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + \beta_{5} q^{3} + ( \beta_{5} - \beta_{6} ) q^{4} + \beta_{1} q^{5} + \beta_{7} q^{6} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{8} + ( -1 - \beta_{5} ) q^{9} + \beta_{6} q^{10} + ( \beta_{3} + \beta_{7} ) q^{11} + ( -1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{12} - q^{13} + ( -\beta_{1} + \beta_{4} ) q^{15} + ( -1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{16} + ( -\beta_{3} - \beta_{7} ) q^{18} -\beta_{7} q^{20} + ( -2 + \beta_{2} ) q^{22} + ( -\beta_{3} + \beta_{4} - \beta_{7} ) q^{24} + ( \beta_{5} + \beta_{6} ) q^{25} -\beta_{3} q^{26} + q^{27} + ( -\beta_{2} - \beta_{6} ) q^{30} + ( -\beta_{3} - \beta_{7} ) q^{32} -\beta_{3} q^{33} + ( 1 - \beta_{2} ) q^{36} -\beta_{5} q^{39} + ( 2 + 2 \beta_{5} ) q^{40} + ( \beta_{1} - \beta_{4} ) q^{41} + ( \beta_{1} - 2 \beta_{3} ) q^{44} -\beta_{4} q^{45} -\beta_{1} q^{47} + ( 1 - \beta_{2} ) q^{48} + ( -\beta_{1} + \beta_{4} ) q^{50} + ( -\beta_{5} + \beta_{6} ) q^{52} + \beta_{3} q^{54} -\beta_{2} q^{55} + ( \beta_{3} + \beta_{7} ) q^{59} + ( \beta_{3} + \beta_{7} ) q^{60} + q^{64} -\beta_{1} q^{65} + ( -2 \beta_{5} + \beta_{6} ) q^{66} + ( \beta_{1} - \beta_{4} ) q^{71} + ( -\beta_{1} + \beta_{3} ) q^{72} + ( -1 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{75} -\beta_{7} q^{78} + ( -\beta_{2} - \beta_{6} ) q^{79} + ( \beta_{3} + \beta_{7} ) q^{80} + \beta_{5} q^{81} + ( \beta_{2} + \beta_{6} ) q^{82} + \beta_{7} q^{83} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{88} + \beta_{3} q^{89} + \beta_{2} q^{90} -\beta_{6} q^{94} + \beta_{3} q^{96} -\beta_{7} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9} + O(q^{10}) \) \( 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9} - 4 q^{12} - 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} + 8 q^{27} + 8 q^{36} + 4 q^{39} + 8 q^{40} + 8 q^{48} + 4 q^{52} + 8 q^{64} + 8 q^{66} - 4 q^{75} - 4 q^{81} + 8 q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 4 x^{6} + 14 x^{4} + 8 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 20 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 34 \nu \)\()/14\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} + 7 \nu^{5} + 28 \nu^{3} + 16 \nu \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{6} + 7 \nu^{4} + 28 \nu^{2} + 2 \)\()/14\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{6} - 7 \nu^{4} - 21 \nu^{2} - 2 \)\()/7\)
\(\beta_{7}\)\(=\)\((\)\( -6 \nu^{7} - 21 \nu^{5} - 70 \nu^{3} - 6 \nu \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{5}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 3 \beta_{4} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-4 \beta_{6} - 6 \beta_{5} - 4 \beta_{2} - 6\)
\(\nu^{5}\)\(=\)\(-4 \beta_{7} - 10 \beta_{4} - 4 \beta_{3}\)
\(\nu^{6}\)\(=\)\(14 \beta_{2} + 20\)
\(\nu^{7}\)\(=\)\(14 \beta_{3} + 34 \beta_{1}\)

Character values

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

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
0.382683 + 0.662827i
−0.923880 1.60021i
0.923880 + 1.60021i
−0.382683 0.662827i
0.382683 0.662827i
−0.923880 + 1.60021i
0.923880 1.60021i
−0.382683 + 0.662827i
−0.923880 1.60021i −0.500000 + 0.866025i −1.20711 + 2.09077i 0.382683 + 0.662827i 1.84776 0 2.61313 −0.500000 0.866025i 0.707107 1.22474i
116.2 −0.382683 0.662827i −0.500000 + 0.866025i 0.207107 0.358719i −0.923880 1.60021i 0.765367 0 −1.08239 −0.500000 0.866025i −0.707107 + 1.22474i
116.3 0.382683 + 0.662827i −0.500000 + 0.866025i 0.207107 0.358719i 0.923880 + 1.60021i −0.765367 0 1.08239 −0.500000 0.866025i −0.707107 + 1.22474i
116.4 0.923880 + 1.60021i −0.500000 + 0.866025i −1.20711 + 2.09077i −0.382683 0.662827i −1.84776 0 −2.61313 −0.500000 0.866025i 0.707107 1.22474i
1598.1 −0.923880 + 1.60021i −0.500000 0.866025i −1.20711 2.09077i 0.382683 0.662827i 1.84776 0 2.61313 −0.500000 + 0.866025i 0.707107 + 1.22474i
1598.2 −0.382683 + 0.662827i −0.500000 0.866025i 0.207107 + 0.358719i −0.923880 + 1.60021i 0.765367 0 −1.08239 −0.500000 + 0.866025i −0.707107 1.22474i
1598.3 0.382683 0.662827i −0.500000 0.866025i 0.207107 + 0.358719i 0.923880 1.60021i −0.765367 0 1.08239 −0.500000 + 0.866025i −0.707107 1.22474i
1598.4 0.923880 1.60021i −0.500000 0.866025i −1.20711 2.09077i −0.382683 + 0.662827i −1.84776 0 −2.61313 −0.500000 + 0.866025i 0.707107 + 1.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1598.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
13.b even 2 1 inner
21.h odd 6 1 inner
91.r even 6 1 inner
273.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.w.e 8
3.b odd 2 1 inner 1911.1.w.e 8
7.b odd 2 1 1911.1.w.f 8
7.c even 3 1 1911.1.h.e yes 4
7.c even 3 1 inner 1911.1.w.e 8
7.d odd 6 1 1911.1.h.d 4
7.d odd 6 1 1911.1.w.f 8
13.b even 2 1 inner 1911.1.w.e 8
21.c even 2 1 1911.1.w.f 8
21.g even 6 1 1911.1.h.d 4
21.g even 6 1 1911.1.w.f 8
21.h odd 6 1 1911.1.h.e yes 4
21.h odd 6 1 inner 1911.1.w.e 8
39.d odd 2 1 CM 1911.1.w.e 8
91.b odd 2 1 1911.1.w.f 8
91.r even 6 1 1911.1.h.e yes 4
91.r even 6 1 inner 1911.1.w.e 8
91.s odd 6 1 1911.1.h.d 4
91.s odd 6 1 1911.1.w.f 8
273.g even 2 1 1911.1.w.f 8
273.w odd 6 1 1911.1.h.e yes 4
273.w odd 6 1 inner 1911.1.w.e 8
273.ba even 6 1 1911.1.h.d 4
273.ba even 6 1 1911.1.w.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.h.d 4 7.d odd 6 1
1911.1.h.d 4 21.g even 6 1
1911.1.h.d 4 91.s odd 6 1
1911.1.h.d 4 273.ba even 6 1
1911.1.h.e yes 4 7.c even 3 1
1911.1.h.e yes 4 21.h odd 6 1
1911.1.h.e yes 4 91.r even 6 1
1911.1.h.e yes 4 273.w odd 6 1
1911.1.w.e 8 1.a even 1 1 trivial
1911.1.w.e 8 3.b odd 2 1 inner
1911.1.w.e 8 7.c even 3 1 inner
1911.1.w.e 8 13.b even 2 1 inner
1911.1.w.e 8 21.h odd 6 1 inner
1911.1.w.e 8 39.d odd 2 1 CM
1911.1.w.e 8 91.r even 6 1 inner
1911.1.w.e 8 273.w odd 6 1 inner
1911.1.w.f 8 7.b odd 2 1
1911.1.w.f 8 7.d odd 6 1
1911.1.w.f 8 21.c even 2 1
1911.1.w.f 8 21.g even 6 1
1911.1.w.f 8 91.b odd 2 1
1911.1.w.f 8 91.s odd 6 1
1911.1.w.f 8 273.g even 2 1
1911.1.w.f 8 273.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1911, [\chi])\):

\( T_{2}^{8} + 4 T_{2}^{6} + 14 T_{2}^{4} + 8 T_{2}^{2} + 4 \)
\( T_{61} \)
\( T_{199}^{2} - 2 T_{199} + 4 \)

Hecke characteristic polynomials

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