Properties

Label 405.2.e.a
Level $405$
Weight $2$
Character orbit 405.e
Analytic conductor $3.234$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(1\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + 2 q^{10} + (5 \zeta_{6} - 5) q^{11} - 4 \zeta_{6} q^{13} + ( - 4 \zeta_{6} + 4) q^{16} - 4 q^{17} - 5 q^{19} + (2 \zeta_{6} - 2) q^{20} - 10 \zeta_{6} q^{22} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 8 q^{26} + ( - 5 \zeta_{6} + 5) q^{29} + 9 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( - 8 \zeta_{6} + 8) q^{34} - 10 q^{37} + ( - 10 \zeta_{6} + 10) q^{38} - 7 \zeta_{6} q^{41} + ( - 2 \zeta_{6} + 2) q^{43} + 10 q^{44} + 12 q^{46} + (2 \zeta_{6} - 2) q^{47} + 7 \zeta_{6} q^{49} - 2 \zeta_{6} q^{50} + (8 \zeta_{6} - 8) q^{52} + 8 q^{53} + 5 q^{55} + 10 \zeta_{6} q^{58} + \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} - 18 q^{62} - 8 q^{64} + (4 \zeta_{6} - 4) q^{65} - 6 \zeta_{6} q^{67} + 8 \zeta_{6} q^{68} + q^{71} - 8 q^{73} + ( - 20 \zeta_{6} + 20) q^{74} + 10 \zeta_{6} q^{76} + (12 \zeta_{6} - 12) q^{79} - 4 q^{80} + 14 q^{82} + (6 \zeta_{6} - 6) q^{83} + 4 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} - 9 q^{89} + (12 \zeta_{6} - 12) q^{92} - 4 \zeta_{6} q^{94} + 5 \zeta_{6} q^{95} + (14 \zeta_{6} - 14) q^{97} - 14 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{4} - q^{5} + 4 q^{10} - 5 q^{11} - 4 q^{13} + 4 q^{16} - 8 q^{17} - 10 q^{19} - 2 q^{20} - 10 q^{22} - 6 q^{23} - q^{25} + 16 q^{26} + 5 q^{29} + 9 q^{31} + 8 q^{32} + 8 q^{34} - 20 q^{37} + 10 q^{38} - 7 q^{41} + 2 q^{43} + 20 q^{44} + 24 q^{46} - 2 q^{47} + 7 q^{49} - 2 q^{50} - 8 q^{52} + 16 q^{53} + 10 q^{55} + 10 q^{58} + q^{59} + 2 q^{61} - 36 q^{62} - 16 q^{64} - 4 q^{65} - 6 q^{67} + 8 q^{68} + 2 q^{71} - 16 q^{73} + 20 q^{74} + 10 q^{76} - 12 q^{79} - 8 q^{80} + 28 q^{82} - 6 q^{83} + 4 q^{85} + 4 q^{86} - 18 q^{89} - 12 q^{92} - 4 q^{94} + 5 q^{95} - 14 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
−1.00000 + 1.73205i 0 −1.00000 1.73205i −0.500000 0.866025i 0 0 0 0 2.00000
271.1 −1.00000 1.73205i 0 −1.00000 + 1.73205i −0.500000 + 0.866025i 0 0 0 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a 2
3.b odd 2 1 405.2.e.g 2
9.c even 3 1 405.2.a.f yes 1
9.c even 3 1 inner 405.2.e.a 2
9.d odd 6 1 405.2.a.a 1
9.d odd 6 1 405.2.e.g 2
36.f odd 6 1 6480.2.a.r 1
36.h even 6 1 6480.2.a.f 1
45.h odd 6 1 2025.2.a.f 1
45.j even 6 1 2025.2.a.a 1
45.k odd 12 2 2025.2.b.b 2
45.l even 12 2 2025.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.a 1 9.d odd 6 1
405.2.a.f yes 1 9.c even 3 1
405.2.e.a 2 1.a even 1 1 trivial
405.2.e.a 2 9.c even 3 1 inner
405.2.e.g 2 3.b odd 2 1
405.2.e.g 2 9.d odd 6 1
2025.2.a.a 1 45.j even 6 1
2025.2.a.f 1 45.h odd 6 1
2025.2.b.a 2 45.l even 12 2
2025.2.b.b 2 45.k odd 12 2
6480.2.a.f 1 36.h even 6 1
6480.2.a.r 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 \(S_{2}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 5T_{11} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) 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
$11$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T + 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$31$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( (T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( (T + 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
show more
show less