Properties

Label 405.1.h.b
Level $405$
Weight $1$
Character orbit 405.h
Analytic conductor $0.202$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 405.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.202121330116\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.135.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.2460375.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{5} + q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{5} + q^{8} - q^{10} + \zeta_{6} q^{16} - q^{17} - q^{19} -\zeta_{6}^{2} q^{23} -\zeta_{6} q^{25} -\zeta_{6}^{2} q^{31} -\zeta_{6} q^{34} -\zeta_{6} q^{38} + \zeta_{6}^{2} q^{40} + q^{46} -2 \zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} -\zeta_{6}^{2} q^{50} - q^{53} + \zeta_{6} q^{61} + q^{62} + q^{64} + \zeta_{6} q^{79} - q^{80} + \zeta_{6} q^{83} -\zeta_{6}^{2} q^{85} -2 \zeta_{6}^{2} q^{94} -\zeta_{6}^{2} q^{95} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{5} + 2q^{8} - 2q^{10} + q^{16} - 2q^{17} - 2q^{19} + q^{23} - q^{25} + q^{31} - q^{34} - q^{38} - q^{40} + 2q^{46} - 2q^{47} - q^{49} + q^{50} - 2q^{53} + q^{61} + 2q^{62} + 2q^{64} + q^{79} - 2q^{80} + q^{83} + q^{85} + 2q^{94} + q^{95} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0 −0.500000 + 0.866025i 0 0 1.00000 0 −1.00000
269.1 0.500000 0.866025i 0 0 −0.500000 0.866025i 0 0 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
9.c even 3 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.1.h.b 2
3.b odd 2 1 405.1.h.a 2
5.b even 2 1 405.1.h.a 2
5.c odd 4 2 2025.1.j.c 4
9.c even 3 1 135.1.d.a 1
9.c even 3 1 inner 405.1.h.b 2
9.d odd 6 1 135.1.d.b yes 1
9.d odd 6 1 405.1.h.a 2
15.d odd 2 1 CM 405.1.h.b 2
15.e even 4 2 2025.1.j.c 4
27.e even 9 6 3645.1.n.d 6
27.f odd 18 6 3645.1.n.e 6
36.f odd 6 1 2160.1.c.b 1
36.h even 6 1 2160.1.c.a 1
45.h odd 6 1 135.1.d.a 1
45.h odd 6 1 inner 405.1.h.b 2
45.j even 6 1 135.1.d.b yes 1
45.j even 6 1 405.1.h.a 2
45.k odd 12 2 675.1.c.c 2
45.k odd 12 2 2025.1.j.c 4
45.l even 12 2 675.1.c.c 2
45.l even 12 2 2025.1.j.c 4
135.n odd 18 6 3645.1.n.d 6
135.p even 18 6 3645.1.n.e 6
180.n even 6 1 2160.1.c.b 1
180.p odd 6 1 2160.1.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 9.c even 3 1
135.1.d.a 1 45.h odd 6 1
135.1.d.b yes 1 9.d odd 6 1
135.1.d.b yes 1 45.j even 6 1
405.1.h.a 2 3.b odd 2 1
405.1.h.a 2 5.b even 2 1
405.1.h.a 2 9.d odd 6 1
405.1.h.a 2 45.j even 6 1
405.1.h.b 2 1.a even 1 1 trivial
405.1.h.b 2 9.c even 3 1 inner
405.1.h.b 2 15.d odd 2 1 CM
405.1.h.b 2 45.h odd 6 1 inner
675.1.c.c 2 45.k odd 12 2
675.1.c.c 2 45.l even 12 2
2025.1.j.c 4 5.c odd 4 2
2025.1.j.c 4 15.e even 4 2
2025.1.j.c 4 45.k odd 12 2
2025.1.j.c 4 45.l even 12 2
2160.1.c.a 1 36.h even 6 1
2160.1.c.a 1 180.p odd 6 1
2160.1.c.b 1 36.f odd 6 1
2160.1.c.b 1 180.n even 6 1
3645.1.n.d 6 27.e even 9 6
3645.1.n.d 6 135.n odd 18 6
3645.1.n.e 6 27.f odd 18 6
3645.1.n.e 6 135.p even 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(405, [\chi])\).

Hecke characteristic polynomials

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