Properties

Label 405.2.e.d
Level $405$
Weight $2$
Character orbit 405.e
Analytic conductor $3.234$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + (3 \zeta_{6} - 3) q^{11} + 4 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} + 6 q^{17} - q^{19} + ( - 2 \zeta_{6} + 2) q^{20} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 4 q^{28} + (9 \zeta_{6} - 9) q^{29} + \zeta_{6} q^{31} + 2 q^{35} + 8 q^{37} + 3 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 6 q^{44} + ( - 12 \zeta_{6} + 12) q^{47} + 3 \zeta_{6} q^{49} + (8 \zeta_{6} - 8) q^{52} - 6 q^{53} + 3 q^{55} + 3 \zeta_{6} q^{59} + ( - 10 \zeta_{6} + 10) q^{61} - 8 q^{64} + ( - 4 \zeta_{6} + 4) q^{65} - 14 \zeta_{6} q^{67} + 12 \zeta_{6} q^{68} + 3 q^{71} + 2 q^{73} - 2 \zeta_{6} q^{76} - 6 \zeta_{6} q^{77} + ( - 16 \zeta_{6} + 16) q^{79} + 4 q^{80} + (12 \zeta_{6} - 12) q^{83} - 6 \zeta_{6} q^{85} - 15 q^{89} - 8 q^{91} + ( - 12 \zeta_{6} + 12) q^{92} + \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - q^{5} - 2 q^{7} - 3 q^{11} + 4 q^{13} - 4 q^{16} + 12 q^{17} - 2 q^{19} + 2 q^{20} - 6 q^{23} - q^{25} - 8 q^{28} - 9 q^{29} + q^{31} + 4 q^{35} + 16 q^{37} + 3 q^{41} + 4 q^{43} - 12 q^{44} + 12 q^{47} + 3 q^{49} - 8 q^{52} - 12 q^{53} + 6 q^{55} + 3 q^{59} + 10 q^{61} - 16 q^{64} + 4 q^{65} - 14 q^{67} + 12 q^{68} + 6 q^{71} + 4 q^{73} - 2 q^{76} - 6 q^{77} + 16 q^{79} + 8 q^{80} - 12 q^{83} - 6 q^{85} - 30 q^{89} - 16 q^{91} + 12 q^{92} + q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 1.00000 + 1.73205i −0.500000 0.866025i 0 −1.00000 + 1.73205i 0 0 0
271.1 0 0 1.00000 1.73205i −0.500000 + 0.866025i 0 −1.00000 1.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.e.d 2
3.b odd 2 1 405.2.e.e 2
9.c even 3 1 405.2.a.d yes 1
9.c even 3 1 inner 405.2.e.d 2
9.d odd 6 1 405.2.a.c 1
9.d odd 6 1 405.2.e.e 2
36.f odd 6 1 6480.2.a.o 1
36.h even 6 1 6480.2.a.c 1
45.h odd 6 1 2025.2.a.c 1
45.j even 6 1 2025.2.a.d 1
45.k odd 12 2 2025.2.b.f 2
45.l even 12 2 2025.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.c 1 9.d odd 6 1
405.2.a.d yes 1 9.c even 3 1
405.2.e.d 2 1.a even 1 1 trivial
405.2.e.d 2 9.c even 3 1 inner
405.2.e.e 2 3.b odd 2 1
405.2.e.e 2 9.d odd 6 1
2025.2.a.c 1 45.h odd 6 1
2025.2.a.d 1 45.j even 6 1
2025.2.b.e 2 45.l even 12 2
2025.2.b.f 2 45.k odd 12 2
6480.2.a.c 1 36.h even 6 1
6480.2.a.o 1 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
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