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Properties

Label 1911.1.w.f
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1911,1,Mod(116,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.116");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

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: no
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 14x^{4} + 8x^{2} + 4 \) Copy content Toggle raw display
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

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

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{5} q^{3} + ( - \beta_{6} + \beta_{5}) q^{4} - \beta_1 q^{5} - \beta_{7} q^{6} + (\beta_{7} - \beta_{4} + \beta_1) q^{8} + ( - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{5} q^{3} + ( - \beta_{6} + \beta_{5}) q^{4} - \beta_1 q^{5} - \beta_{7} q^{6} + (\beta_{7} - \beta_{4} + \beta_1) q^{8} + ( - \beta_{5} - 1) q^{9} - \beta_{6} q^{10} + (\beta_{7} + \beta_{3}) q^{11} + ( - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{12} + q^{13} + (\beta_{4} - \beta_1) q^{15} + (\beta_{6} - \beta_{5} + \beta_{2} - 1) q^{16} + ( - \beta_{7} - \beta_{3}) q^{18} + \beta_{7} q^{20} + (\beta_{2} - 2) q^{22} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{24} + (\beta_{6} + \beta_{5}) q^{25} + \beta_{3} q^{26} - q^{27} + ( - \beta_{6} - \beta_{2}) q^{30} + ( - \beta_{7} - \beta_{3}) q^{32} + \beta_{3} q^{33} + ( - \beta_{2} + 1) q^{36} - \beta_{5} q^{39} + ( - 2 \beta_{5} - 2) q^{40} + (\beta_{4} - \beta_1) q^{41} + ( - 2 \beta_{3} + \beta_1) q^{44} + \beta_{4} q^{45} + \beta_1 q^{47} + (\beta_{2} - 1) q^{48} + (\beta_{4} - \beta_1) q^{50} + ( - \beta_{6} + \beta_{5}) q^{52} - \beta_{3} q^{54} + \beta_{2} q^{55} + ( - \beta_{7} - \beta_{3}) q^{59} + (\beta_{7} + \beta_{3}) q^{60} + q^{64} - \beta_1 q^{65} + ( - \beta_{6} + 2 \beta_{5}) q^{66} + ( - \beta_{4} + \beta_1) q^{71} + (\beta_{3} - \beta_1) q^{72} + (\beta_{6} + \beta_{5} + \beta_{2} + 1) q^{75} - \beta_{7} q^{78} + ( - \beta_{6} - \beta_{2}) q^{79} + ( - \beta_{7} - \beta_{3}) q^{80} + \beta_{5} q^{81} + ( - \beta_{6} - \beta_{2}) q^{82} - \beta_{7} q^{83} + (2 \beta_{6} - 2 \beta_{5}) q^{88} - \beta_{3} q^{89} - \beta_{2} q^{90} + \beta_{6} q^{94} - \beta_{3} q^{96} - \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{4} - 4 q^{9} + 4 q^{12} + 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} - 8 q^{27} + 8 q^{36} + 4 q^{39} - 8 q^{40} - 8 q^{48} - 4 q^{52} + 8 q^{64} - 8 q^{66} + 4 q^{75} - 4 q^{81} + 8 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 34\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + 7\nu^{5} + 28\nu^{3} + 16\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{6} + 7\nu^{4} + 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} - 7\nu^{4} - 21\nu^{2} - 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{7} - 21\nu^{5} - 70\nu^{3} - 6\nu ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{6} - 6\beta_{5} - 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 10\beta_{4} - 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{3} + 34\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 116.4
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.f 8
3.b odd 2 1 inner 1911.1.w.f 8
7.b odd 2 1 1911.1.w.e 8
7.c even 3 1 1911.1.h.d ✓ 4
7.c even 3 1 inner 1911.1.w.f 8
7.d odd 6 1 1911.1.h.e yes 4
7.d odd 6 1 1911.1.w.e 8
13.b even 2 1 inner 1911.1.w.f 8
21.c even 2 1 1911.1.w.e 8
21.g even 6 1 1911.1.h.e yes 4
21.g even 6 1 1911.1.w.e 8
21.h odd 6 1 1911.1.h.d ✓ 4
21.h odd 6 1 inner 1911.1.w.f 8
39.d odd 2 1 CM 1911.1.w.f 8
91.b odd 2 1 1911.1.w.e 8
91.r even 6 1 1911.1.h.d ✓ 4
91.r even 6 1 inner 1911.1.w.f 8
91.s odd 6 1 1911.1.h.e yes 4
91.s odd 6 1 1911.1.w.e 8
273.g even 2 1 1911.1.w.e 8
273.w odd 6 1 1911.1.h.d ✓ 4
273.w odd 6 1 inner 1911.1.w.f 8
273.ba even 6 1 1911.1.h.e yes 4
273.ba even 6 1 1911.1.w.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.h.d ✓ 4 7.c even 3 1
1911.1.h.d ✓ 4 21.h odd 6 1
1911.1.h.d ✓ 4 91.r even 6 1
1911.1.h.d ✓ 4 273.w odd 6 1
1911.1.h.e yes 4 7.d odd 6 1
1911.1.h.e yes 4 21.g even 6 1
1911.1.h.e yes 4 91.s odd 6 1
1911.1.h.e yes 4 273.ba even 6 1
1911.1.w.e 8 7.b odd 2 1
1911.1.w.e 8 7.d odd 6 1
1911.1.w.e 8 21.c even 2 1
1911.1.w.e 8 21.g even 6 1
1911.1.w.e 8 91.b odd 2 1
1911.1.w.e 8 91.s odd 6 1
1911.1.w.e 8 273.g even 2 1
1911.1.w.e 8 273.ba even 6 1
1911.1.w.f 8 1.a even 1 1 trivial
1911.1.w.f 8 3.b odd 2 1 inner
1911.1.w.f 8 7.c even 3 1 inner
1911.1.w.f 8 13.b even 2 1 inner
1911.1.w.f 8 21.h odd 6 1 inner
1911.1.w.f 8 39.d odd 2 1 CM
1911.1.w.f 8 91.r even 6 1 inner
1911.1.w.f 8 273.w odd 6 1 inner

Hecke kernels

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

\( T_{2}^{8} + 4T_{2}^{6} + 14T_{2}^{4} + 8T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{61} \) Copy content Toggle raw display
\( T_{199}^{2} + 2T_{199} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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