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Properties

Label 1912.1.x.a
Level $1912$
Weight $1$
Character orbit 1912.x
Analytic conductor $0.954$
Analytic rank $0$
Dimension $16$
Projective image $D_{34}$
RM discriminant 8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1912,1,Mod(501,1912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1912, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([0, 17, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1912.501");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1912 = 2^{3} \cdot 239 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1912.x (of order \(34\), degree \(16\), 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: \(0.954212304154\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{34}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{34} - \cdots)\)

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{34}^{6} q^{2} + \zeta_{34}^{12} q^{4} + (\zeta_{34}^{7} + \zeta_{34}^{4}) q^{7} - \zeta_{34} q^{8} - \zeta_{34}^{15} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{34}^{6} q^{2} + \zeta_{34}^{12} q^{4} + (\zeta_{34}^{7} + \zeta_{34}^{4}) q^{7} - \zeta_{34} q^{8} - \zeta_{34}^{15} q^{9} + (\zeta_{34}^{13} + \zeta_{34}^{10}) q^{14} - \zeta_{34}^{7} q^{16} + ( - \zeta_{34}^{8} + \zeta_{34}^{3}) q^{17} + \zeta_{34}^{4} q^{18} + ( - \zeta_{34}^{16} + \zeta_{34}^{14}) q^{23} - \zeta_{34}^{5} q^{25} + (\zeta_{34}^{16} - \zeta_{34}^{2}) q^{28} + (\zeta_{34}^{9} - \zeta_{34}^{2}) q^{31} - \zeta_{34}^{13} q^{32} + ( - \zeta_{34}^{14} + \zeta_{34}^{9}) q^{34} + \zeta_{34}^{10} q^{36} + (\zeta_{34}^{12} - \zeta_{34}^{6}) q^{41} + (\zeta_{34}^{5} - \zeta_{34}^{3}) q^{46} + ( - \zeta_{34}^{16} - \zeta_{34}^{3}) q^{47} + (\zeta_{34}^{14} + \cdots + \zeta_{34}^{8}) q^{49} + \cdots + (\zeta_{34}^{14} - \zeta_{34}^{3} - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - q^{4} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - q^{4} - q^{8} - q^{9} - q^{16} + 2 q^{17} - q^{18} - q^{25} + 2 q^{31} - q^{32} + 2 q^{34} - q^{36} - q^{49} - q^{50} + 2 q^{62} - q^{64} + 2 q^{68} + 2 q^{71} - q^{72} - q^{81} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1912\mathbb{Z}\right)^\times\).

\(n\) \(479\) \(957\) \(1441\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{34}^{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
501.1
0.273663 + 0.961826i
−0.445738 − 0.895163i
−0.0922684 + 0.995734i
−0.932472 − 0.361242i
0.982973 + 0.183750i
−0.739009 − 0.673696i
−0.0922684 − 0.995734i
−0.932472 + 0.361242i
0.602635 − 0.798017i
0.982973 − 0.183750i
−0.445738 + 0.895163i
0.273663 − 0.961826i
−0.739009 + 0.673696i
0.850217 + 0.526432i
0.602635 + 0.798017i
0.850217 − 0.526432i
0.0922684 + 0.995734i 0 −0.982973 + 0.183750i 0 0 −0.486734 − 0.533922i −0.273663 − 0.961826i −0.850217 − 0.526432i 0
589.1 0.932472 + 0.361242i 0 0.739009 + 0.673696i 0 0 −0.365931 − 1.95756i 0.445738 + 0.895163i −0.602635 − 0.798017i 0
677.1 −0.850217 − 0.526432i 0 0.445738 + 0.895163i 0 0 1.53511 − 0.436776i 0.0922684 − 0.995734i −0.982973 + 0.183750i 0
885.1 −0.602635 + 0.798017i 0 −0.273663 − 0.961826i 0 0 0.942485 + 0.469302i 0.932472 + 0.361242i 0.739009 − 0.673696i 0
1029.1 0.445738 + 0.895163i 0 −0.602635 + 0.798017i 0 0 1.01267 + 1.63552i −0.982973 − 0.183750i 0.932472 − 0.361242i 0
1173.1 −0.273663 − 0.961826i 0 −0.850217 + 0.526432i 0 0 −1.42871 + 1.07891i 0.739009 + 0.673696i 0.0922684 − 0.995734i 0
1189.1 −0.850217 + 0.526432i 0 0.445738 − 0.895163i 0 0 1.53511 + 0.436776i 0.0922684 + 0.995734i −0.982973 − 0.183750i 0
1333.1 −0.602635 − 0.798017i 0 −0.273663 + 0.961826i 0 0 0.942485 − 0.469302i 0.932472 − 0.361242i 0.739009 + 0.673696i 0
1541.1 0.739009 + 0.673696i 0 0.0922684 + 0.995734i 0 0 0.132756 + 0.342683i −0.602635 + 0.798017i −0.273663 + 0.961826i 0
1637.1 0.445738 − 0.895163i 0 −0.602635 − 0.798017i 0 0 1.01267 − 1.63552i −0.982973 + 0.183750i 0.932472 + 0.361242i 0
1701.1 0.932472 − 0.361242i 0 0.739009 − 0.673696i 0 0 −0.365931 + 1.95756i 0.445738 − 0.895163i −0.602635 + 0.798017i 0
1725.1 0.0922684 − 0.995734i 0 −0.982973 − 0.183750i 0 0 −0.486734 + 0.533922i −0.273663 + 0.961826i −0.850217 + 0.526432i 0
1749.1 −0.273663 + 0.961826i 0 −0.850217 − 0.526432i 0 0 −1.42871 − 1.07891i 0.739009 − 0.673696i 0.0922684 + 0.995734i 0
1837.1 −0.982973 − 0.183750i 0 0.932472 + 0.361242i 0 0 −1.34164 + 0.124322i −0.850217 − 0.526432i 0.445738 − 0.895163i 0
1845.1 0.739009 − 0.673696i 0 0.0922684 − 0.995734i 0 0 0.132756 − 0.342683i −0.602635 − 0.798017i −0.273663 − 0.961826i 0
1861.1 −0.982973 + 0.183750i 0 0.932472 − 0.361242i 0 0 −1.34164 − 0.124322i −0.850217 + 0.526432i 0.445738 + 0.895163i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
239.f odd 34 1 inner
1912.x odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1912.1.x.a ✓ 16
8.b even 2 1 RM 1912.1.x.a ✓ 16
239.f odd 34 1 inner 1912.1.x.a ✓ 16
1912.x odd 34 1 inner 1912.1.x.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1912.1.x.a ✓ 16 1.a even 1 1 trivial
1912.1.x.a ✓ 16 8.b even 2 1 RM
1912.1.x.a ✓ 16 239.f odd 34 1 inner
1912.1.x.a ✓ 16 1912.x odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1912, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 51 T^{9} + \cdots + 17 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} - 51 T^{9} + \cdots + 17 \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} + 17 T^{7} + \cdots + 17 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} + 17 T^{12} + \cdots + 17 \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{16} + 17 T^{10} + \cdots + 17 \) Copy content Toggle raw display
$79$ \( T^{16} + 17 T^{11} + \cdots + 17 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} + 34 T^{11} + \cdots + 17 \) Copy content Toggle raw display
$97$ \( T^{16} - 51 T^{9} + \cdots + 17 \) Copy content Toggle raw display
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