Properties

Label 405.2.j.c
Level $405$
Weight $2$
Character orbit 405.j
Analytic conductor $3.234$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{3} q^{8} + 5 q^{10} + ( 1 - \beta_{2} ) q^{16} + 2 \beta_{3} q^{17} -4 q^{19} + 3 \beta_{1} q^{20} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} -8 \beta_{2} q^{31} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{32} + ( -10 + 10 \beta_{2} ) q^{34} -4 \beta_{1} q^{38} + 5 \beta_{2} q^{40} -20 q^{46} + 4 \beta_{1} q^{47} -7 \beta_{2} q^{49} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{50} + 2 \beta_{3} q^{53} + ( -2 + 2 \beta_{2} ) q^{61} -8 \beta_{3} q^{62} + 13 q^{64} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{68} -12 \beta_{2} q^{76} + ( 16 - 16 \beta_{2} ) q^{79} -\beta_{3} q^{80} -8 \beta_{1} q^{83} + 10 \beta_{2} q^{85} -12 \beta_{1} q^{92} + 20 \beta_{2} q^{94} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{95} -7 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + O(q^{10}) \) \( 4q + 6q^{4} + 20q^{10} + 2q^{16} - 16q^{19} + 10q^{25} - 16q^{31} - 20q^{34} + 10q^{40} - 80q^{46} - 14q^{49} - 4q^{61} + 52q^{64} - 24q^{76} + 32q^{79} + 20q^{85} + 40q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i 0 1.50000 2.59808i −1.93649 1.11803i 0 0 2.23607i 0 5.00000
109.2 1.93649 1.11803i 0 1.50000 2.59808i 1.93649 + 1.11803i 0 0 2.23607i 0 5.00000
379.1 −1.93649 1.11803i 0 1.50000 + 2.59808i −1.93649 + 1.11803i 0 0 2.23607i 0 5.00000
379.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 1.93649 1.11803i 0 0 2.23607i 0 5.00000
\(n\): 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 \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.j.c 4
3.b odd 2 1 inner 405.2.j.c 4
5.b even 2 1 inner 405.2.j.c 4
9.c even 3 1 45.2.b.a 2
9.c even 3 1 inner 405.2.j.c 4
9.d odd 6 1 45.2.b.a 2
9.d odd 6 1 inner 405.2.j.c 4
15.d odd 2 1 CM 405.2.j.c 4
36.f odd 6 1 720.2.f.d 2
36.h even 6 1 720.2.f.d 2
45.h odd 6 1 45.2.b.a 2
45.h odd 6 1 inner 405.2.j.c 4
45.j even 6 1 45.2.b.a 2
45.j even 6 1 inner 405.2.j.c 4
45.k odd 12 2 225.2.a.f 2
45.l even 12 2 225.2.a.f 2
63.l odd 6 1 2205.2.d.a 2
63.o even 6 1 2205.2.d.a 2
72.j odd 6 1 2880.2.f.k 2
72.l even 6 1 2880.2.f.j 2
72.n even 6 1 2880.2.f.k 2
72.p odd 6 1 2880.2.f.j 2
180.n even 6 1 720.2.f.d 2
180.p odd 6 1 720.2.f.d 2
180.v odd 12 2 3600.2.a.bs 2
180.x even 12 2 3600.2.a.bs 2
315.z even 6 1 2205.2.d.a 2
315.bg odd 6 1 2205.2.d.a 2
360.z odd 6 1 2880.2.f.j 2
360.bd even 6 1 2880.2.f.j 2
360.bh odd 6 1 2880.2.f.k 2
360.bk even 6 1 2880.2.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 9.c even 3 1
45.2.b.a 2 9.d odd 6 1
45.2.b.a 2 45.h odd 6 1
45.2.b.a 2 45.j even 6 1
225.2.a.f 2 45.k odd 12 2
225.2.a.f 2 45.l even 12 2
405.2.j.c 4 1.a even 1 1 trivial
405.2.j.c 4 3.b odd 2 1 inner
405.2.j.c 4 5.b even 2 1 inner
405.2.j.c 4 9.c even 3 1 inner
405.2.j.c 4 9.d odd 6 1 inner
405.2.j.c 4 15.d odd 2 1 CM
405.2.j.c 4 45.h odd 6 1 inner
405.2.j.c 4 45.j even 6 1 inner
720.2.f.d 2 36.f odd 6 1
720.2.f.d 2 36.h even 6 1
720.2.f.d 2 180.n even 6 1
720.2.f.d 2 180.p odd 6 1
2205.2.d.a 2 63.l odd 6 1
2205.2.d.a 2 63.o even 6 1
2205.2.d.a 2 315.z even 6 1
2205.2.d.a 2 315.bg odd 6 1
2880.2.f.j 2 72.l even 6 1
2880.2.f.j 2 72.p odd 6 1
2880.2.f.j 2 360.z odd 6 1
2880.2.f.j 2 360.bd even 6 1
2880.2.f.k 2 72.j odd 6 1
2880.2.f.k 2 72.n even 6 1
2880.2.f.k 2 360.bh odd 6 1
2880.2.f.k 2 360.bk even 6 1
3600.2.a.bs 2 180.v odd 12 2
3600.2.a.bs 2 180.x even 12 2

Hecke kernels

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

\( T_{2}^{4} - 5 T_{2}^{2} + 25 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 - 5 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 5 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( ( 4 + T )^{4} \)
$23$ \( 6400 - 80 T^{2} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 64 + 8 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( 6400 - 80 T^{2} + T^{4} \)
$53$ \( ( 20 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 4 + 2 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 256 - 16 T + T^{2} )^{2} \)
$83$ \( 102400 - 320 T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
show more
show less