Properties

Label 1900.1.j.b
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
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

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

\(f(q)\) \(=\) \( q + \zeta_{16}^{3} q^{2} + ( \zeta_{16} + \zeta_{16}^{3} ) q^{3} + \zeta_{16}^{6} q^{4} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{6} -\zeta_{16} q^{8} + ( \zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{9} +O(q^{10})\) \( q + \zeta_{16}^{3} q^{2} + ( \zeta_{16} + \zeta_{16}^{3} ) q^{3} + \zeta_{16}^{6} q^{4} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{6} -\zeta_{16} q^{8} + ( \zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{9} + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{11} + ( -\zeta_{16} + \zeta_{16}^{7} ) q^{12} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{13} -\zeta_{16}^{4} q^{16} + ( -\zeta_{16} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{18} + q^{19} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{22} + ( -\zeta_{16}^{2} - \zeta_{16}^{4} ) q^{24} + ( -1 - \zeta_{16}^{2} ) q^{26} + ( -\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{27} -\zeta_{16}^{7} q^{32} + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{33} + ( -1 - \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{36} + ( \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{37} + \zeta_{16}^{3} q^{38} + ( -2 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{39} + ( 1 + \zeta_{16}^{4} ) q^{44} + ( -\zeta_{16}^{5} - \zeta_{16}^{7} ) q^{48} -\zeta_{16}^{4} q^{49} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{52} + ( \zeta_{16} - \zeta_{16}^{3} ) q^{53} + ( -1 - \zeta_{16}^{2} - \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{54} + ( \zeta_{16} + \zeta_{16}^{3} ) q^{57} + ( \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{61} + \zeta_{16}^{2} q^{64} + ( 1 + \zeta_{16}^{2} + \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{66} + ( \zeta_{16} - \zeta_{16}^{3} ) q^{67} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{72} + ( -1 + \zeta_{16}^{2} ) q^{74} + \zeta_{16}^{6} q^{76} + ( -\zeta_{16} - 2 \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{78} + ( -1 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{81} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{88} + ( 1 + \zeta_{16}^{2} ) q^{96} + ( \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{97} -\zeta_{16}^{7} q^{98} + ( 2 + \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{19} - 8q^{26} - 8q^{36} - 16q^{39} + 8q^{44} - 8q^{54} + 8q^{66} - 8q^{74} - 8q^{81} + 8q^{96} + 16q^{99} + O(q^{100}) \)

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.

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 −0.541196 + 0.541196i 0.707107 + 0.707107i 0 0.707107 0.292893i 0 −0.382683 0.923880i 0.414214i 0
607.2 −0.382683 + 0.923880i −1.30656 + 1.30656i −0.707107 0.707107i 0 −0.707107 1.70711i 0 0.923880 0.382683i 2.41421i 0
607.3 0.382683 0.923880i 1.30656 1.30656i −0.707107 0.707107i 0 −0.707107 1.70711i 0 −0.923880 + 0.382683i 2.41421i 0
607.4 0.923880 + 0.382683i 0.541196 0.541196i 0.707107 + 0.707107i 0 0.707107 0.292893i 0 0.382683 + 0.923880i 0.414214i 0
1443.1 −0.923880 + 0.382683i −0.541196 0.541196i 0.707107 0.707107i 0 0.707107 + 0.292893i 0 −0.382683 + 0.923880i 0.414214i 0
1443.2 −0.382683 0.923880i −1.30656 1.30656i −0.707107 + 0.707107i 0 −0.707107 + 1.70711i 0 0.923880 + 0.382683i 2.41421i 0
1443.3 0.382683 + 0.923880i 1.30656 + 1.30656i −0.707107 + 0.707107i 0 −0.707107 + 1.70711i 0 −0.923880 0.382683i 2.41421i 0
1443.4 0.923880 0.382683i 0.541196 + 0.541196i 0.707107 0.707107i 0 0.707107 + 0.292893i 0 0.382683 0.923880i 0.414214i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1443.4
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.b yes 8
4.b odd 2 1 1900.1.j.a 8
5.b even 2 1 inner 1900.1.j.b yes 8
5.c odd 4 2 1900.1.j.a 8
19.b odd 2 1 inner 1900.1.j.b yes 8
20.d odd 2 1 1900.1.j.a 8
20.e even 4 2 inner 1900.1.j.b yes 8
76.d even 2 1 1900.1.j.a 8
95.d odd 2 1 CM 1900.1.j.b yes 8
95.g even 4 2 1900.1.j.a 8
380.d even 2 1 1900.1.j.a 8
380.j odd 4 2 inner 1900.1.j.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.j.a 8 4.b odd 2 1
1900.1.j.a 8 5.c odd 4 2
1900.1.j.a 8 20.d odd 2 1
1900.1.j.a 8 76.d even 2 1
1900.1.j.a 8 95.g even 4 2
1900.1.j.a 8 380.d even 2 1
1900.1.j.b yes 8 1.a even 1 1 trivial
1900.1.j.b yes 8 5.b even 2 1 inner
1900.1.j.b yes 8 19.b odd 2 1 inner
1900.1.j.b yes 8 20.e even 4 2 inner
1900.1.j.b yes 8 95.d odd 2 1 CM
1900.1.j.b yes 8 380.j odd 4 2 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{8} \)
$3$ \( 4 + 12 T^{4} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 2 + T^{2} )^{4} \)
$13$ \( 4 + 12 T^{4} + T^{8} \)
$17$ \( T^{8} \)
$19$ \( ( -1 + T )^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 4 + 12 T^{4} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( 4 + 12 T^{4} + T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( -2 + T^{2} )^{4} \)
$67$ \( 4 + 12 T^{4} + T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( 4 + 12 T^{4} + T^{8} \)
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