Properties

Label 1900.1.bb.a
Level $1900$
Weight $1$
Character orbit 1900.bb
Analytic conductor $0.948$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
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.bb (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: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\)

$q$-expansion

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

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

Character values

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

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(\zeta_{30}^{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
−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 1.82709 0 −0.809017 + 0.587785i 0
341.2 0 0 0 0.913545 + 0.406737i 0 −0.209057 0 −0.809017 + 0.587785i 0
721.1 0 0 0 −0.978148 0.207912i 0 1.33826 0 0.309017 + 0.951057i 0
721.2 0 0 0 0.669131 0.743145i 0 −1.95630 0 0.309017 + 0.951057i 0
1481.1 0 0 0 −0.978148 + 0.207912i 0 1.33826 0 0.309017 0.951057i 0
1481.2 0 0 0 0.669131 + 0.743145i 0 −1.95630 0 0.309017 0.951057i 0
1861.1 0 0 0 −0.104528 + 0.994522i 0 1.82709 0 −0.809017 0.587785i 0
1861.2 0 0 0 0.913545 0.406737i 0 −0.209057 0 −0.809017 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1861.2
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.d even 5 1 inner
475.o 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.bb.a 8
19.b odd 2 1 CM 1900.1.bb.a 8
25.d even 5 1 inner 1900.1.bb.a 8
475.o odd 10 1 inner 1900.1.bb.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.bb.a 8 1.a even 1 1 trivial
1900.1.bb.a 8 19.b odd 2 1 CM
1900.1.bb.a 8 25.d even 5 1 inner
1900.1.bb.a 8 475.o odd 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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