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Properties

Label 1911.1.h.d
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1911,1,Mod(1520,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.1520");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + \beta_{2} q^{10} + \beta_1 q^{11} + ( - \beta_{2} - 1) q^{12} + q^{13} + \beta_{3} q^{15} + (\beta_{2} + 1) q^{16} - \beta_1 q^{18} - \beta_1 q^{20} + ( - \beta_{2} - 2) q^{22} + (\beta_{3} + \beta_1) q^{24} + ( - \beta_{2} + 1) q^{25} - \beta_1 q^{26} - q^{27} - \beta_{2} q^{30} - \beta_1 q^{32} - \beta_1 q^{33} + (\beta_{2} + 1) q^{36} - q^{39} + 2 q^{40} + \beta_{3} q^{41} + (\beta_{3} + 2 \beta_1) q^{44} - \beta_{3} q^{45} + \beta_{3} q^{47} + ( - \beta_{2} - 1) q^{48} + \beta_{3} q^{50} + (\beta_{2} + 1) q^{52} + \beta_1 q^{54} - \beta_{2} q^{55} - \beta_1 q^{59} + \beta_1 q^{60} + q^{64} - \beta_{3} q^{65} + (\beta_{2} + 2) q^{66} - \beta_{3} q^{71} + ( - \beta_{3} - \beta_1) q^{72} + (\beta_{2} - 1) q^{75} + \beta_1 q^{78} - \beta_{2} q^{79} - \beta_1 q^{80} + q^{81} - \beta_{2} q^{82} + \beta_1 q^{83} + ( - 2 \beta_{2} - 2) q^{88} + \beta_1 q^{89} + \beta_{2} q^{90} - \beta_{2} q^{94} + \beta_1 q^{96} + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{9} - 4 q^{12} + 4 q^{13} + 4 q^{16} - 8 q^{22} + 4 q^{25} - 4 q^{27} + 4 q^{36} - 4 q^{39} + 8 q^{40} - 4 q^{48} + 4 q^{52} + 4 q^{64} + 8 q^{66} - 4 q^{75} + 4 q^{81} - 8 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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.d ✓ 4
3.b odd 2 1 inner 1911.1.h.d ✓ 4
7.b odd 2 1 1911.1.h.e yes 4
7.c even 3 2 1911.1.w.f 8
7.d odd 6 2 1911.1.w.e 8
13.b even 2 1 inner 1911.1.h.d ✓ 4
21.c even 2 1 1911.1.h.e yes 4
21.g even 6 2 1911.1.w.e 8
21.h odd 6 2 1911.1.w.f 8
39.d odd 2 1 CM 1911.1.h.d ✓ 4
91.b odd 2 1 1911.1.h.e yes 4
91.r even 6 2 1911.1.w.f 8
91.s odd 6 2 1911.1.w.e 8
273.g even 2 1 1911.1.h.e yes 4
273.w odd 6 2 1911.1.w.f 8
273.ba even 6 2 1911.1.w.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.h.d ✓ 4 1.a even 1 1 trivial
1911.1.h.d ✓ 4 3.b odd 2 1 inner
1911.1.h.d ✓ 4 13.b even 2 1 inner
1911.1.h.d ✓ 4 39.d odd 2 1 CM
1911.1.h.e yes 4 7.b odd 2 1
1911.1.h.e yes 4 21.c even 2 1
1911.1.h.e yes 4 91.b odd 2 1
1911.1.h.e yes 4 273.g even 2 1
1911.1.w.e 8 7.d odd 6 2
1911.1.w.e 8 21.g even 6 2
1911.1.w.e 8 91.s odd 6 2
1911.1.w.e 8 273.ba even 6 2
1911.1.w.f 8 7.c even 3 2
1911.1.w.f 8 21.h odd 6 2
1911.1.w.f 8 91.r even 6 2
1911.1.w.f 8 273.w odd 6 2

Hecke kernels

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

\( T_{2}^{4} - 4T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{61} \) Copy content Toggle raw display
\( T_{199} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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