# Properties

 Label 405.4.e.m 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: 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.

 $$f(q)$$ $$=$$ $$q + ( 5 - 5 \zeta_{6} ) q^{2} -17 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( -9 + 9 \zeta_{6} ) q^{7} -45 q^{8} +O(q^{10})$$ $$q + ( 5 - 5 \zeta_{6} ) q^{2} -17 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( -9 + 9 \zeta_{6} ) q^{7} -45 q^{8} -25 q^{10} + ( 8 - 8 \zeta_{6} ) q^{11} -43 \zeta_{6} q^{13} + 45 \zeta_{6} q^{14} + ( -89 + 89 \zeta_{6} ) q^{16} -122 q^{17} -59 q^{19} + ( -85 + 85 \zeta_{6} ) q^{20} -40 \zeta_{6} q^{22} + 213 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -215 q^{26} + 153 q^{28} + ( -224 + 224 \zeta_{6} ) q^{29} + 36 \zeta_{6} q^{31} + 85 \zeta_{6} q^{32} + ( -610 + 610 \zeta_{6} ) q^{34} + 45 q^{35} + 206 q^{37} + ( -295 + 295 \zeta_{6} ) q^{38} + 225 \zeta_{6} q^{40} -413 \zeta_{6} q^{41} + ( 392 - 392 \zeta_{6} ) q^{43} -136 q^{44} + 1065 q^{46} + ( 311 - 311 \zeta_{6} ) q^{47} + 262 \zeta_{6} q^{49} + 125 \zeta_{6} q^{50} + ( -731 + 731 \zeta_{6} ) q^{52} -377 q^{53} -40 q^{55} + ( 405 - 405 \zeta_{6} ) q^{56} + 1120 \zeta_{6} q^{58} -337 \zeta_{6} q^{59} + ( -40 + 40 \zeta_{6} ) q^{61} + 180 q^{62} -287 q^{64} + ( -215 + 215 \zeta_{6} ) q^{65} -348 \zeta_{6} q^{67} + 2074 \zeta_{6} q^{68} + ( 225 - 225 \zeta_{6} ) q^{70} + 62 q^{71} -1214 q^{73} + ( 1030 - 1030 \zeta_{6} ) q^{74} + 1003 \zeta_{6} q^{76} + 72 \zeta_{6} q^{77} + ( 294 - 294 \zeta_{6} ) q^{79} + 445 q^{80} -2065 q^{82} + ( -534 + 534 \zeta_{6} ) q^{83} + 610 \zeta_{6} q^{85} -1960 \zeta_{6} q^{86} + ( -360 + 360 \zeta_{6} ) q^{88} -810 q^{89} + 387 q^{91} + ( 3621 - 3621 \zeta_{6} ) q^{92} -1555 \zeta_{6} q^{94} + 295 \zeta_{6} q^{95} + ( 928 - 928 \zeta_{6} ) q^{97} + 1310 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} - 17 q^{4} - 5 q^{5} - 9 q^{7} - 90 q^{8} + O(q^{10})$$ $$2 q + 5 q^{2} - 17 q^{4} - 5 q^{5} - 9 q^{7} - 90 q^{8} - 50 q^{10} + 8 q^{11} - 43 q^{13} + 45 q^{14} - 89 q^{16} - 244 q^{17} - 118 q^{19} - 85 q^{20} - 40 q^{22} + 213 q^{23} - 25 q^{25} - 430 q^{26} + 306 q^{28} - 224 q^{29} + 36 q^{31} + 85 q^{32} - 610 q^{34} + 90 q^{35} + 412 q^{37} - 295 q^{38} + 225 q^{40} - 413 q^{41} + 392 q^{43} - 272 q^{44} + 2130 q^{46} + 311 q^{47} + 262 q^{49} + 125 q^{50} - 731 q^{52} - 754 q^{53} - 80 q^{55} + 405 q^{56} + 1120 q^{58} - 337 q^{59} - 40 q^{61} + 360 q^{62} - 574 q^{64} - 215 q^{65} - 348 q^{67} + 2074 q^{68} + 225 q^{70} + 124 q^{71} - 2428 q^{73} + 1030 q^{74} + 1003 q^{76} + 72 q^{77} + 294 q^{79} + 890 q^{80} - 4130 q^{82} - 534 q^{83} + 610 q^{85} - 1960 q^{86} - 360 q^{88} - 1620 q^{89} + 774 q^{91} + 3621 q^{92} - 1555 q^{94} + 295 q^{95} + 928 q^{97} + 2620 q^{98} + O(q^{100})$$

## 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 −4.50000 + 7.79423i −45.0000 0 −25.0000
271.1 2.50000 + 4.33013i 0 −8.50000 + 14.7224i −2.50000 + 4.33013i 0 −4.50000 7.79423i −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.m 2
3.b odd 2 1 405.4.e.a 2
9.c even 3 1 405.4.a.a 1
9.c even 3 1 inner 405.4.e.m 2
9.d odd 6 1 405.4.a.b yes 1
9.d odd 6 1 405.4.e.a 2
45.h odd 6 1 2025.4.a.a 1
45.j even 6 1 2025.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.a 1 9.c even 3 1
405.4.a.b yes 1 9.d odd 6 1
405.4.e.a 2 3.b odd 2 1
405.4.e.a 2 9.d odd 6 1
405.4.e.m 2 1.a even 1 1 trivial
405.4.e.m 2 9.c even 3 1 inner
2025.4.a.a 1 45.h odd 6 1
2025.4.a.f 1 45.j even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 5 T_{2} + 25$$ $$T_{7}^{2} + 9 T_{7} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 - 5 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 + 5 T + T^{2}$$
$7$ $$81 + 9 T + T^{2}$$
$11$ $$64 - 8 T + T^{2}$$
$13$ $$1849 + 43 T + T^{2}$$
$17$ $$( 122 + T )^{2}$$
$19$ $$( 59 + T )^{2}$$
$23$ $$45369 - 213 T + T^{2}$$
$29$ $$50176 + 224 T + T^{2}$$
$31$ $$1296 - 36 T + T^{2}$$
$37$ $$( -206 + T )^{2}$$
$41$ $$170569 + 413 T + T^{2}$$
$43$ $$153664 - 392 T + T^{2}$$
$47$ $$96721 - 311 T + T^{2}$$
$53$ $$( 377 + T )^{2}$$
$59$ $$113569 + 337 T + T^{2}$$
$61$ $$1600 + 40 T + T^{2}$$
$67$ $$121104 + 348 T + T^{2}$$
$71$ $$( -62 + T )^{2}$$
$73$ $$( 1214 + T )^{2}$$
$79$ $$86436 - 294 T + T^{2}$$
$83$ $$285156 + 534 T + T^{2}$$
$89$ $$( 810 + T )^{2}$$
$97$ $$861184 - 928 T + T^{2}$$