# Properties

 Label 405.4.e.e Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} -24 q^{8} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} -24 q^{8} -10 q^{10} + ( 10 - 10 \zeta_{6} ) q^{11} + 80 \zeta_{6} q^{13} + ( 16 - 16 \zeta_{6} ) q^{16} -7 q^{17} -113 q^{19} + ( -20 + 20 \zeta_{6} ) q^{20} + 20 \zeta_{6} q^{22} -81 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -160 q^{26} + ( -220 + 220 \zeta_{6} ) q^{29} + 189 \zeta_{6} q^{31} -160 \zeta_{6} q^{32} + ( 14 - 14 \zeta_{6} ) q^{34} + 170 q^{37} + ( 226 - 226 \zeta_{6} ) q^{38} -120 \zeta_{6} q^{40} -130 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + 40 q^{44} + 162 q^{46} + ( 160 - 160 \zeta_{6} ) q^{47} + 343 \zeta_{6} q^{49} -50 \zeta_{6} q^{50} + ( -320 + 320 \zeta_{6} ) q^{52} -631 q^{53} + 50 q^{55} -440 \zeta_{6} q^{58} -560 \zeta_{6} q^{59} + ( -229 + 229 \zeta_{6} ) q^{61} -378 q^{62} + 448 q^{64} + ( -400 + 400 \zeta_{6} ) q^{65} -750 \zeta_{6} q^{67} -28 \zeta_{6} q^{68} -890 q^{71} -890 q^{73} + ( -340 + 340 \zeta_{6} ) q^{74} -452 \zeta_{6} q^{76} + ( 27 - 27 \zeta_{6} ) q^{79} + 80 q^{80} + 260 q^{82} + ( 429 - 429 \zeta_{6} ) q^{83} -35 \zeta_{6} q^{85} -20 \zeta_{6} q^{86} + ( -240 + 240 \zeta_{6} ) q^{88} + 750 q^{89} + ( 324 - 324 \zeta_{6} ) q^{92} + 320 \zeta_{6} q^{94} -565 \zeta_{6} q^{95} + ( 1480 - 1480 \zeta_{6} ) q^{97} -686 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 48 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 48 q^{8} - 20 q^{10} + 10 q^{11} + 80 q^{13} + 16 q^{16} - 14 q^{17} - 226 q^{19} - 20 q^{20} + 20 q^{22} - 81 q^{23} - 25 q^{25} - 320 q^{26} - 220 q^{29} + 189 q^{31} - 160 q^{32} + 14 q^{34} + 340 q^{37} + 226 q^{38} - 120 q^{40} - 130 q^{41} - 10 q^{43} + 80 q^{44} + 324 q^{46} + 160 q^{47} + 343 q^{49} - 50 q^{50} - 320 q^{52} - 1262 q^{53} + 100 q^{55} - 440 q^{58} - 560 q^{59} - 229 q^{61} - 756 q^{62} + 896 q^{64} - 400 q^{65} - 750 q^{67} - 28 q^{68} - 1780 q^{71} - 1780 q^{73} - 340 q^{74} - 452 q^{76} + 27 q^{79} + 160 q^{80} + 520 q^{82} + 429 q^{83} - 35 q^{85} - 20 q^{86} - 240 q^{88} + 1500 q^{89} + 324 q^{92} + 320 q^{94} - 565 q^{95} + 1480 q^{97} - 1372 q^{98} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i 0 2.00000 + 3.46410i 2.50000 + 4.33013i 0 0 −24.0000 0 −10.0000
271.1 −1.00000 1.73205i 0 2.00000 3.46410i 2.50000 4.33013i 0 0 −24.0000 0 −10.0000
 $$n$$: 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
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 405.4.e.e 2
3.b odd 2 1 405.4.e.j 2
9.c even 3 1 135.4.a.d yes 1
9.c even 3 1 inner 405.4.e.e 2
9.d odd 6 1 135.4.a.a 1
9.d odd 6 1 405.4.e.j 2
36.f odd 6 1 2160.4.a.d 1
36.h even 6 1 2160.4.a.n 1
45.h odd 6 1 675.4.a.i 1
45.j even 6 1 675.4.a.b 1
45.k odd 12 2 675.4.b.c 2
45.l even 12 2 675.4.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 9.d odd 6 1
135.4.a.d yes 1 9.c even 3 1
405.4.e.e 2 1.a even 1 1 trivial
405.4.e.e 2 9.c even 3 1 inner
405.4.e.j 2 3.b odd 2 1
405.4.e.j 2 9.d odd 6 1
675.4.a.b 1 45.j even 6 1
675.4.a.i 1 45.h odd 6 1
675.4.b.c 2 45.k odd 12 2
675.4.b.d 2 45.l even 12 2
2160.4.a.d 1 36.f odd 6 1
2160.4.a.n 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} + 2 T_{2} + 4$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 - 5 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$100 - 10 T + T^{2}$$
$13$ $$6400 - 80 T + T^{2}$$
$17$ $$( 7 + T )^{2}$$
$19$ $$( 113 + T )^{2}$$
$23$ $$6561 + 81 T + T^{2}$$
$29$ $$48400 + 220 T + T^{2}$$
$31$ $$35721 - 189 T + T^{2}$$
$37$ $$( -170 + T )^{2}$$
$41$ $$16900 + 130 T + T^{2}$$
$43$ $$100 + 10 T + T^{2}$$
$47$ $$25600 - 160 T + T^{2}$$
$53$ $$( 631 + T )^{2}$$
$59$ $$313600 + 560 T + T^{2}$$
$61$ $$52441 + 229 T + T^{2}$$
$67$ $$562500 + 750 T + T^{2}$$
$71$ $$( 890 + T )^{2}$$
$73$ $$( 890 + T )^{2}$$
$79$ $$729 - 27 T + T^{2}$$
$83$ $$184041 - 429 T + T^{2}$$
$89$ $$( -750 + T )^{2}$$
$97$ $$2190400 - 1480 T + T^{2}$$