# Properties

 Label 405.4.e.n 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + ( 5 - 5 \zeta_{6} ) q^{2} -17 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( 30 - 30 \zeta_{6} ) q^{7} -45 q^{8} +O(q^{10})$$ $$q + ( 5 - 5 \zeta_{6} ) q^{2} -17 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( 30 - 30 \zeta_{6} ) q^{7} -45 q^{8} -25 q^{10} + ( 50 - 50 \zeta_{6} ) q^{11} + 20 \zeta_{6} q^{13} -150 \zeta_{6} q^{14} + ( -89 + 89 \zeta_{6} ) q^{16} + 10 q^{17} -44 q^{19} + ( -85 + 85 \zeta_{6} ) q^{20} -250 \zeta_{6} q^{22} + 120 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 100 q^{26} -510 q^{28} + ( -50 + 50 \zeta_{6} ) q^{29} -108 \zeta_{6} q^{31} + 85 \zeta_{6} q^{32} + ( 50 - 50 \zeta_{6} ) q^{34} -150 q^{35} -40 q^{37} + ( -220 + 220 \zeta_{6} ) q^{38} + 225 \zeta_{6} q^{40} + 400 \zeta_{6} q^{41} + ( -280 + 280 \zeta_{6} ) q^{43} -850 q^{44} + 600 q^{46} + ( -280 + 280 \zeta_{6} ) q^{47} -557 \zeta_{6} q^{49} + 125 \zeta_{6} q^{50} + ( 340 - 340 \zeta_{6} ) q^{52} + 610 q^{53} -250 q^{55} + ( -1350 + 1350 \zeta_{6} ) q^{56} + 250 \zeta_{6} q^{58} + 50 \zeta_{6} q^{59} + ( 518 - 518 \zeta_{6} ) q^{61} -540 q^{62} -287 q^{64} + ( 100 - 100 \zeta_{6} ) q^{65} + 180 \zeta_{6} q^{67} -170 \zeta_{6} q^{68} + ( -750 + 750 \zeta_{6} ) q^{70} -700 q^{71} -410 q^{73} + ( -200 + 200 \zeta_{6} ) q^{74} + 748 \zeta_{6} q^{76} -1500 \zeta_{6} q^{77} + ( 516 - 516 \zeta_{6} ) q^{79} + 445 q^{80} + 2000 q^{82} + ( 660 - 660 \zeta_{6} ) q^{83} -50 \zeta_{6} q^{85} + 1400 \zeta_{6} q^{86} + ( -2250 + 2250 \zeta_{6} ) q^{88} + 1500 q^{89} + 600 q^{91} + ( 2040 - 2040 \zeta_{6} ) q^{92} + 1400 \zeta_{6} q^{94} + 220 \zeta_{6} q^{95} + ( 1630 - 1630 \zeta_{6} ) q^{97} -2785 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} - 17 q^{4} - 5 q^{5} + 30 q^{7} - 90 q^{8} + O(q^{10})$$ $$2 q + 5 q^{2} - 17 q^{4} - 5 q^{5} + 30 q^{7} - 90 q^{8} - 50 q^{10} + 50 q^{11} + 20 q^{13} - 150 q^{14} - 89 q^{16} + 20 q^{17} - 88 q^{19} - 85 q^{20} - 250 q^{22} + 120 q^{23} - 25 q^{25} + 200 q^{26} - 1020 q^{28} - 50 q^{29} - 108 q^{31} + 85 q^{32} + 50 q^{34} - 300 q^{35} - 80 q^{37} - 220 q^{38} + 225 q^{40} + 400 q^{41} - 280 q^{43} - 1700 q^{44} + 1200 q^{46} - 280 q^{47} - 557 q^{49} + 125 q^{50} + 340 q^{52} + 1220 q^{53} - 500 q^{55} - 1350 q^{56} + 250 q^{58} + 50 q^{59} + 518 q^{61} - 1080 q^{62} - 574 q^{64} + 100 q^{65} + 180 q^{67} - 170 q^{68} - 750 q^{70} - 1400 q^{71} - 820 q^{73} - 200 q^{74} + 748 q^{76} - 1500 q^{77} + 516 q^{79} + 890 q^{80} + 4000 q^{82} + 660 q^{83} - 50 q^{85} + 1400 q^{86} - 2250 q^{88} + 3000 q^{89} + 1200 q^{91} + 2040 q^{92} + 1400 q^{94} + 220 q^{95} + 1630 q^{97} - 5570 q^{98} + O(q^{100})$$

## 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
2.50000 4.33013i 0 −8.50000 14.7224i −2.50000 4.33013i 0 15.0000 25.9808i −45.0000 0 −25.0000
271.1 2.50000 + 4.33013i 0 −8.50000 + 14.7224i −2.50000 + 4.33013i 0 15.0000 + 25.9808i −45.0000 0 −25.0000
 $$n$$: 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.n 2
3.b odd 2 1 405.4.e.b 2
9.c even 3 1 45.4.a.a 1
9.c even 3 1 inner 405.4.e.n 2
9.d odd 6 1 45.4.a.e yes 1
9.d odd 6 1 405.4.e.b 2
36.f odd 6 1 720.4.a.bc 1
36.h even 6 1 720.4.a.o 1
45.h odd 6 1 225.4.a.a 1
45.j even 6 1 225.4.a.h 1
45.k odd 12 2 225.4.b.a 2
45.l even 12 2 225.4.b.b 2
63.l odd 6 1 2205.4.a.a 1
63.o even 6 1 2205.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 9.c even 3 1
45.4.a.e yes 1 9.d odd 6 1
225.4.a.a 1 45.h odd 6 1
225.4.a.h 1 45.j even 6 1
225.4.b.a 2 45.k odd 12 2
225.4.b.b 2 45.l even 12 2
405.4.e.b 2 3.b odd 2 1
405.4.e.b 2 9.d odd 6 1
405.4.e.n 2 1.a even 1 1 trivial
405.4.e.n 2 9.c even 3 1 inner
720.4.a.o 1 36.h even 6 1
720.4.a.bc 1 36.f odd 6 1
2205.4.a.a 1 63.l odd 6 1
2205.4.a.t 1 63.o even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 5 T_{2} + 25$$ $$T_{7}^{2} - 30 T_{7} + 900$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 - 5 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 + 5 T + T^{2}$$
$7$ $$900 - 30 T + T^{2}$$
$11$ $$2500 - 50 T + T^{2}$$
$13$ $$400 - 20 T + T^{2}$$
$17$ $$( -10 + T )^{2}$$
$19$ $$( 44 + T )^{2}$$
$23$ $$14400 - 120 T + T^{2}$$
$29$ $$2500 + 50 T + T^{2}$$
$31$ $$11664 + 108 T + T^{2}$$
$37$ $$( 40 + T )^{2}$$
$41$ $$160000 - 400 T + T^{2}$$
$43$ $$78400 + 280 T + T^{2}$$
$47$ $$78400 + 280 T + T^{2}$$
$53$ $$( -610 + T )^{2}$$
$59$ $$2500 - 50 T + T^{2}$$
$61$ $$268324 - 518 T + T^{2}$$
$67$ $$32400 - 180 T + T^{2}$$
$71$ $$( 700 + T )^{2}$$
$73$ $$( 410 + T )^{2}$$
$79$ $$266256 - 516 T + T^{2}$$
$83$ $$435600 - 660 T + T^{2}$$
$89$ $$( -1500 + T )^{2}$$
$97$ $$2656900 - 1630 T + T^{2}$$