Properties

Label 1911.1.h.e
Level $1911$
Weight $1$
Character orbit 1911.h
Self dual yes
Analytic conductor $0.954$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -39
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1520.1
1.84776
0.765367
−0.765367
−1.84776
−1.84776 1.00000 2.41421 0.765367 −1.84776 0 −2.61313 1.00000 −1.41421
1520.2 −0.765367 1.00000 −0.414214 −1.84776 −0.765367 0 1.08239 1.00000 1.41421
1520.3 0.765367 1.00000 −0.414214 1.84776 0.765367 0 −1.08239 1.00000 1.41421
1520.4 1.84776 1.00000 2.41421 −0.765367 1.84776 0 2.61313 1.00000 −1.41421
\(n\): e.g. 2-40 or 990-1000
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
13.b even 2 1 inner

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

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