Properties

Label 1815.1.v.a
Level $1815$
Weight $1$
Character orbit 1815.v
Analytic conductor $0.906$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -11
Inner twists $16$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1815.v (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.905802997929\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.12375.1
Artin image: $C_{10}\times C_4\wr C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{80} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{20}^{7} q^{3} + \zeta_{20}^{3} q^{4} + \zeta_{20} q^{5} -\zeta_{20}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{20}^{7} q^{3} + \zeta_{20}^{3} q^{4} + \zeta_{20} q^{5} -\zeta_{20}^{4} q^{9} + q^{12} -\zeta_{20}^{8} q^{15} + \zeta_{20}^{6} q^{16} + \zeta_{20}^{4} q^{20} + ( -1 - \zeta_{20}^{5} ) q^{23} + \zeta_{20}^{2} q^{25} -\zeta_{20} q^{27} -\zeta_{20}^{7} q^{36} + ( -\zeta_{20}^{3} + \zeta_{20}^{8} ) q^{37} -\zeta_{20}^{5} q^{45} + ( \zeta_{20}^{2} - \zeta_{20}^{7} ) q^{47} + \zeta_{20}^{3} q^{48} + \zeta_{20} q^{49} + ( \zeta_{20}^{4} + \zeta_{20}^{9} ) q^{53} + 2 \zeta_{20}^{8} q^{59} + \zeta_{20} q^{60} + \zeta_{20}^{9} q^{64} + ( -1 - \zeta_{20}^{5} ) q^{67} + ( -\zeta_{20}^{2} + \zeta_{20}^{7} ) q^{69} -\zeta_{20}^{9} q^{75} + \zeta_{20}^{7} q^{80} + \zeta_{20}^{8} q^{81} + ( -\zeta_{20}^{3} - \zeta_{20}^{8} ) q^{92} + ( -\zeta_{20}^{4} - \zeta_{20}^{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{9} + 8 q^{12} + 2 q^{15} + 2 q^{16} - 2 q^{20} - 8 q^{23} + 2 q^{25} - 2 q^{37} + 2 q^{47} - 2 q^{53} - 4 q^{59} - 8 q^{67} - 2 q^{69} - 2 q^{81} + 2 q^{92} + 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(\zeta_{20}^{5}\) \(-1\) \(-\zeta_{20}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1
0.951057 0.309017i
−0.951057 0.309017i
0.587785 0.809017i
−0.587785 0.809017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 + 0.809017i
0.951057 + 0.309017i
0 0.587785 + 0.809017i 0.587785 0.809017i 0.951057 0.309017i 0 0 0 −0.309017 + 0.951057i 0
578.1 0 −0.587785 + 0.809017i −0.587785 0.809017i −0.951057 0.309017i 0 0 0 −0.309017 0.951057i 0
602.1 0 −0.951057 + 0.309017i −0.951057 0.309017i 0.587785 0.809017i 0 0 0 0.809017 0.587785i 0
887.1 0 0.951057 + 0.309017i 0.951057 0.309017i −0.587785 0.809017i 0 0 0 0.809017 + 0.587785i 0
1322.1 0 −0.587785 0.809017i −0.587785 + 0.809017i −0.951057 + 0.309017i 0 0 0 −0.309017 + 0.951057i 0
1328.1 0 0.951057 0.309017i 0.951057 + 0.309017i −0.587785 + 0.809017i 0 0 0 0.809017 0.587785i 0
1613.1 0 −0.951057 0.309017i −0.951057 + 0.309017i 0.587785 + 0.809017i 0 0 0 0.809017 + 0.587785i 0
1667.1 0 0.587785 0.809017i 0.587785 + 0.809017i 0.951057 + 0.309017i 0 0 0 −0.309017 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1667.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
11.c even 5 3 inner
11.d odd 10 3 inner
15.e even 4 1 inner
165.l odd 4 1 inner
165.u odd 20 3 inner
165.v even 20 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.1.v.a 8
3.b odd 2 1 1815.1.v.b 8
5.c odd 4 1 1815.1.v.b 8
11.b odd 2 1 CM 1815.1.v.a 8
11.c even 5 1 165.1.l.b yes 2
11.c even 5 3 inner 1815.1.v.a 8
11.d odd 10 1 165.1.l.b yes 2
11.d odd 10 3 inner 1815.1.v.a 8
15.e even 4 1 inner 1815.1.v.a 8
33.d even 2 1 1815.1.v.b 8
33.f even 10 1 165.1.l.a 2
33.f even 10 3 1815.1.v.b 8
33.h odd 10 1 165.1.l.a 2
33.h odd 10 3 1815.1.v.b 8
44.g even 10 1 2640.1.ch.a 2
44.h odd 10 1 2640.1.ch.a 2
55.e even 4 1 1815.1.v.b 8
55.h odd 10 1 825.1.l.a 2
55.j even 10 1 825.1.l.a 2
55.k odd 20 1 165.1.l.a 2
55.k odd 20 1 825.1.l.b 2
55.k odd 20 3 1815.1.v.b 8
55.l even 20 1 165.1.l.a 2
55.l even 20 1 825.1.l.b 2
55.l even 20 3 1815.1.v.b 8
132.n odd 10 1 2640.1.ch.b 2
132.o even 10 1 2640.1.ch.b 2
165.l odd 4 1 inner 1815.1.v.a 8
165.o odd 10 1 825.1.l.b 2
165.r even 10 1 825.1.l.b 2
165.u odd 20 1 165.1.l.b yes 2
165.u odd 20 1 825.1.l.a 2
165.u odd 20 3 inner 1815.1.v.a 8
165.v even 20 1 165.1.l.b yes 2
165.v even 20 1 825.1.l.a 2
165.v even 20 3 inner 1815.1.v.a 8
220.v even 20 1 2640.1.ch.b 2
220.w odd 20 1 2640.1.ch.b 2
660.bp odd 20 1 2640.1.ch.a 2
660.bv even 20 1 2640.1.ch.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.1.l.a 2 33.f even 10 1
165.1.l.a 2 33.h odd 10 1
165.1.l.a 2 55.k odd 20 1
165.1.l.a 2 55.l even 20 1
165.1.l.b yes 2 11.c even 5 1
165.1.l.b yes 2 11.d odd 10 1
165.1.l.b yes 2 165.u odd 20 1
165.1.l.b yes 2 165.v even 20 1
825.1.l.a 2 55.h odd 10 1
825.1.l.a 2 55.j even 10 1
825.1.l.a 2 165.u odd 20 1
825.1.l.a 2 165.v even 20 1
825.1.l.b 2 55.k odd 20 1
825.1.l.b 2 55.l even 20 1
825.1.l.b 2 165.o odd 10 1
825.1.l.b 2 165.r even 10 1
1815.1.v.a 8 1.a even 1 1 trivial
1815.1.v.a 8 11.b odd 2 1 CM
1815.1.v.a 8 11.c even 5 3 inner
1815.1.v.a 8 11.d odd 10 3 inner
1815.1.v.a 8 15.e even 4 1 inner
1815.1.v.a 8 165.l odd 4 1 inner
1815.1.v.a 8 165.u odd 20 3 inner
1815.1.v.a 8 165.v even 20 3 inner
1815.1.v.b 8 3.b odd 2 1
1815.1.v.b 8 5.c odd 4 1
1815.1.v.b 8 33.d even 2 1
1815.1.v.b 8 33.f even 10 3
1815.1.v.b 8 33.h odd 10 3
1815.1.v.b 8 55.e even 4 1
1815.1.v.b 8 55.k odd 20 3
1815.1.v.b 8 55.l even 20 3
2640.1.ch.a 2 44.g even 10 1
2640.1.ch.a 2 44.h odd 10 1
2640.1.ch.a 2 660.bp odd 20 1
2640.1.ch.a 2 660.bv even 20 1
2640.1.ch.b 2 132.n odd 10 1
2640.1.ch.b 2 132.o even 10 1
2640.1.ch.b 2 220.v even 20 1
2640.1.ch.b 2 220.w odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 2 T_{23} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1815, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$5$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 2 + 2 T + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 16 + 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( 16 - 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$53$ \( 16 + 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$59$ \( ( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( ( 2 + 2 T + T^{2} )^{4} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( 16 - 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
show more
show less