Properties

Label 405.1.h.a
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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: $C_6\times S_3$
Artin field: Galois closure of 12.0.6053445140625.2

$q$-expansion

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

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

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.a 2
3.b odd 2 1 405.1.h.b 2
5.b even 2 1 405.1.h.b 2
5.c odd 4 2 2025.1.j.c 4
9.c even 3 1 135.1.d.b yes 1
9.c even 3 1 inner 405.1.h.a 2
9.d odd 6 1 135.1.d.a 1
9.d odd 6 1 405.1.h.b 2
15.d odd 2 1 CM 405.1.h.a 2
15.e even 4 2 2025.1.j.c 4
27.e even 9 6 3645.1.n.e 6
27.f odd 18 6 3645.1.n.d 6
36.f odd 6 1 2160.1.c.a 1
36.h even 6 1 2160.1.c.b 1
45.h odd 6 1 135.1.d.b yes 1
45.h odd 6 1 inner 405.1.h.a 2
45.j even 6 1 135.1.d.a 1
45.j even 6 1 405.1.h.b 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.e 6
135.p even 18 6 3645.1.n.d 6
180.n even 6 1 2160.1.c.a 1
180.p odd 6 1 2160.1.c.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 9.d odd 6 1
135.1.d.a 1 45.j even 6 1
135.1.d.b yes 1 9.c even 3 1
135.1.d.b yes 1 45.h odd 6 1
405.1.h.a 2 1.a even 1 1 trivial
405.1.h.a 2 9.c even 3 1 inner
405.1.h.a 2 15.d odd 2 1 CM
405.1.h.a 2 45.h odd 6 1 inner
405.1.h.b 2 3.b odd 2 1
405.1.h.b 2 5.b even 2 1
405.1.h.b 2 9.d odd 6 1
405.1.h.b 2 45.j even 6 1
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.f odd 6 1
2160.1.c.a 1 180.n even 6 1
2160.1.c.b 1 36.h even 6 1
2160.1.c.b 1 180.p odd 6 1
3645.1.n.d 6 27.f odd 18 6
3645.1.n.d 6 135.p even 18 6
3645.1.n.e 6 27.e even 9 6
3645.1.n.e 6 135.n odd 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 \(S_{1}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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