# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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