Properties

Label 1900.1.w.a
Level $1900$
Weight $1$
Character orbit 1900.w
Analytic conductor $0.948$
Analytic rank $0$
Dimension $8$
Projective image $D_{30}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1900.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \(x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{30}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{30} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{30}^{8} q^{5} + ( \zeta_{30}^{2} + \zeta_{30}^{13} ) q^{7} + \zeta_{30}^{3} q^{9} +O(q^{10})\) \( q + \zeta_{30}^{8} q^{5} + ( \zeta_{30}^{2} + \zeta_{30}^{13} ) q^{7} + \zeta_{30}^{3} q^{9} + ( -\zeta_{30}^{10} - \zeta_{30}^{14} ) q^{11} + ( -\zeta_{30}^{2} + \zeta_{30}^{10} ) q^{17} + \zeta_{30}^{6} q^{19} + ( -1 - \zeta_{30}^{9} ) q^{23} -\zeta_{30} q^{25} + ( -\zeta_{30}^{6} + \zeta_{30}^{10} ) q^{35} + ( \zeta_{30} + \zeta_{30}^{14} ) q^{43} + \zeta_{30}^{11} q^{45} + ( \zeta_{30}^{4} - \zeta_{30}^{14} ) q^{47} + ( -1 + \zeta_{30}^{4} - \zeta_{30}^{11} ) q^{49} + ( \zeta_{30}^{3} + \zeta_{30}^{7} ) q^{55} + ( \zeta_{30}^{11} + \zeta_{30}^{13} ) q^{61} + ( -\zeta_{30} + \zeta_{30}^{5} ) q^{63} + ( -\zeta_{30}^{2} - \zeta_{30}^{7} ) q^{73} + ( \zeta_{30} + \zeta_{30}^{8} ) q^{77} + \zeta_{30}^{6} q^{81} + ( 1 - \zeta_{30}^{12} ) q^{83} + ( -\zeta_{30}^{3} - \zeta_{30}^{10} ) q^{85} + \zeta_{30}^{14} q^{95} + ( \zeta_{30}^{2} - \zeta_{30}^{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{5} + 2q^{9} + O(q^{10}) \) \( 8q + q^{5} + 2q^{9} + 3q^{11} - 5q^{17} - 2q^{19} - 10q^{23} + q^{25} - 2q^{35} - q^{45} - 6q^{49} + q^{55} - 2q^{61} + 5q^{63} - 2q^{81} + 10q^{83} + 2q^{85} + q^{95} + 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(\zeta_{30}^{9}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
−0.978148 0.207912i
0.669131 0.743145i
0.913545 0.406737i
−0.104528 + 0.994522i
0.913545 + 0.406737i
−0.104528 0.994522i
−0.978148 + 0.207912i
0.669131 + 0.743145i
0 0 0 −0.104528 + 0.994522i 0 0.813473i 0 0.809017 + 0.587785i 0
189.2 0 0 0 0.913545 0.406737i 0 1.98904i 0 0.809017 + 0.587785i 0
569.1 0 0 0 −0.978148 + 0.207912i 0 1.48629i 0 −0.309017 + 0.951057i 0
569.2 0 0 0 0.669131 + 0.743145i 0 0.415823i 0 −0.309017 + 0.951057i 0
1329.1 0 0 0 −0.978148 0.207912i 0 1.48629i 0 −0.309017 0.951057i 0
1329.2 0 0 0 0.669131 0.743145i 0 0.415823i 0 −0.309017 0.951057i 0
1709.1 0 0 0 −0.104528 0.994522i 0 0.813473i 0 0.809017 0.587785i 0
1709.2 0 0 0 0.913545 + 0.406737i 0 1.98904i 0 0.809017 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1709.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
25.e even 10 1 inner
475.m odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.w.a 8
19.b odd 2 1 CM 1900.1.w.a 8
25.e even 10 1 inner 1900.1.w.a 8
475.m odd 10 1 inner 1900.1.w.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.w.a 8 1.a even 1 1 trivial
1900.1.w.a 8 19.b odd 2 1 CM
1900.1.w.a 8 25.e even 10 1 inner
1900.1.w.a 8 475.m odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
$7$ \( 1 + 8 T^{2} + 14 T^{4} + 7 T^{6} + T^{8} \)
$11$ \( 1 + 7 T + 18 T^{2} - 11 T^{3} + 15 T^{4} - 11 T^{5} + 8 T^{6} - 3 T^{7} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( 1 + 5 T + 8 T^{2} + 5 T^{3} + 9 T^{4} + 15 T^{5} + 12 T^{6} + 5 T^{7} + T^{8} \)
$19$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$23$ \( ( 5 + 10 T + 10 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( 1 + 8 T^{2} + 14 T^{4} + 7 T^{6} + T^{8} \)
$47$ \( 81 - 27 T^{2} + 9 T^{4} - 3 T^{6} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( 1 - 3 T + 23 T^{2} - T^{3} - T^{5} + 3 T^{6} + 2 T^{7} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 81 - 27 T^{2} + 9 T^{4} - 3 T^{6} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 5 - 10 T + 10 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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