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Properties

Label 1900.1.j.a
Level $1900$
Weight $1$
Character orbit 1900.j
Analytic conductor $0.948$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,1,Mod(607,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.607");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1900.j (of order \(4\), degree \(2\), 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: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) 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 8.0.4170272000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{2} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{3} - \zeta_{16}^{2} q^{4} + ( - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{6} - \zeta_{16}^{7} q^{8} + ( - \zeta_{16}^{6} + \cdots - \zeta_{16}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{5} q^{2} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{3} - \zeta_{16}^{2} q^{4} + ( - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{6} - \zeta_{16}^{7} q^{8} + ( - \zeta_{16}^{6} + \cdots - \zeta_{16}^{2}) q^{9} + \cdots + ( - \zeta_{16}^{6} - \zeta_{16}^{4} + \cdots - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{19} - 8 q^{26} - 8 q^{36} + 16 q^{39} - 8 q^{44} + 8 q^{54} + 8 q^{66} + 8 q^{74} - 8 q^{81} + 8 q^{96} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-\zeta_{16}^{4}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
−0.382683 − 0.923880i
0.923880 − 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 − 0.382683i
0.382683 − 0.923880i
−0.923880 + 0.382683i 1.30656 − 1.30656i 0.707107 − 0.707107i 0 −0.707107 + 1.70711i 0 −0.382683 + 0.923880i − 2.41421i 0
607.2 −0.382683 − 0.923880i −0.541196 + 0.541196i −0.707107 + 0.707107i 0 0.707107 + 0.292893i 0 0.923880 + 0.382683i 0.414214i 0
607.3 0.382683 + 0.923880i 0.541196 − 0.541196i −0.707107 + 0.707107i 0 0.707107 + 0.292893i 0 −0.923880 − 0.382683i 0.414214i 0
607.4 0.923880 − 0.382683i −1.30656 + 1.30656i 0.707107 − 0.707107i 0 −0.707107 + 1.70711i 0 0.382683 − 0.923880i − 2.41421i 0
1443.1 −0.923880 − 0.382683i 1.30656 + 1.30656i 0.707107 + 0.707107i 0 −0.707107 − 1.70711i 0 −0.382683 − 0.923880i 2.41421i 0
1443.2 −0.382683 + 0.923880i −0.541196 − 0.541196i −0.707107 − 0.707107i 0 0.707107 − 0.292893i 0 0.923880 − 0.382683i − 0.414214i 0
1443.3 0.382683 − 0.923880i 0.541196 + 0.541196i −0.707107 − 0.707107i 0 0.707107 − 0.292893i 0 −0.923880 + 0.382683i − 0.414214i 0
1443.4 0.923880 + 0.382683i −1.30656 − 1.30656i 0.707107 + 0.707107i 0 −0.707107 − 1.70711i 0 0.382683 + 0.923880i 2.41421i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner
20.e even 4 2 inner
380.j odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.j.a ✓ 8
4.b odd 2 1 1900.1.j.b yes 8
5.b even 2 1 inner 1900.1.j.a ✓ 8
5.c odd 4 2 1900.1.j.b yes 8
19.b odd 2 1 inner 1900.1.j.a ✓ 8
20.d odd 2 1 1900.1.j.b yes 8
20.e even 4 2 inner 1900.1.j.a ✓ 8
76.d even 2 1 1900.1.j.b yes 8
95.d odd 2 1 CM 1900.1.j.a ✓ 8
95.g even 4 2 1900.1.j.b yes 8
380.d even 2 1 1900.1.j.b yes 8
380.j odd 4 2 inner 1900.1.j.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.j.a ✓ 8 1.a even 1 1 trivial
1900.1.j.a ✓ 8 5.b even 2 1 inner
1900.1.j.a ✓ 8 19.b odd 2 1 inner
1900.1.j.a ✓ 8 20.e even 4 2 inner
1900.1.j.a ✓ 8 95.d odd 2 1 CM
1900.1.j.a ✓ 8 380.j odd 4 2 inner
1900.1.j.b yes 8 4.b odd 2 1
1900.1.j.b yes 8 5.c odd 4 2
1900.1.j.b yes 8 20.d odd 2 1
1900.1.j.b yes 8 76.d even 2 1
1900.1.j.b yes 8 95.g even 4 2
1900.1.j.b yes 8 380.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{199} - 2 \) acting on \(S_{1}^{\mathrm{new}}(1900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) 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
$37$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
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