# Properties

 Label 405.4.e.h 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + 7 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( 6 - 6 \zeta_{6} ) q^{7} + 15 q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + 7 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( 6 - 6 \zeta_{6} ) q^{7} + 15 q^{8} + 5 q^{10} + ( -47 + 47 \zeta_{6} ) q^{11} + 5 \zeta_{6} q^{13} -6 \zeta_{6} q^{14} + ( -41 + 41 \zeta_{6} ) q^{16} + 131 q^{17} -56 q^{19} + ( -35 + 35 \zeta_{6} ) q^{20} + 47 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 5 q^{26} + 42 q^{28} + ( -157 + 157 \zeta_{6} ) q^{29} -225 \zeta_{6} q^{31} + 161 \zeta_{6} q^{32} + ( 131 - 131 \zeta_{6} ) q^{34} + 30 q^{35} -70 q^{37} + ( -56 + 56 \zeta_{6} ) q^{38} + 75 \zeta_{6} q^{40} + 140 \zeta_{6} q^{41} + ( -397 + 397 \zeta_{6} ) q^{43} -329 q^{44} + 3 q^{46} + ( -347 + 347 \zeta_{6} ) q^{47} + 307 \zeta_{6} q^{49} + 25 \zeta_{6} q^{50} + ( -35 + 35 \zeta_{6} ) q^{52} -4 q^{53} -235 q^{55} + ( 90 - 90 \zeta_{6} ) q^{56} + 157 \zeta_{6} q^{58} + 748 \zeta_{6} q^{59} + ( 338 - 338 \zeta_{6} ) q^{61} -225 q^{62} -167 q^{64} + ( -25 + 25 \zeta_{6} ) q^{65} -492 \zeta_{6} q^{67} + 917 \zeta_{6} q^{68} + ( 30 - 30 \zeta_{6} ) q^{70} -32 q^{71} + 970 q^{73} + ( -70 + 70 \zeta_{6} ) q^{74} -392 \zeta_{6} q^{76} + 282 \zeta_{6} q^{77} + ( 1257 - 1257 \zeta_{6} ) q^{79} -205 q^{80} + 140 q^{82} + ( -102 + 102 \zeta_{6} ) q^{83} + 655 \zeta_{6} q^{85} + 397 \zeta_{6} q^{86} + ( -705 + 705 \zeta_{6} ) q^{88} + 1488 q^{89} + 30 q^{91} + ( -21 + 21 \zeta_{6} ) q^{92} + 347 \zeta_{6} q^{94} -280 \zeta_{6} q^{95} + ( -974 + 974 \zeta_{6} ) q^{97} + 307 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 7 q^{4} + 5 q^{5} + 6 q^{7} + 30 q^{8} + O(q^{10})$$ $$2 q + q^{2} + 7 q^{4} + 5 q^{5} + 6 q^{7} + 30 q^{8} + 10 q^{10} - 47 q^{11} + 5 q^{13} - 6 q^{14} - 41 q^{16} + 262 q^{17} - 112 q^{19} - 35 q^{20} + 47 q^{22} + 3 q^{23} - 25 q^{25} + 10 q^{26} + 84 q^{28} - 157 q^{29} - 225 q^{31} + 161 q^{32} + 131 q^{34} + 60 q^{35} - 140 q^{37} - 56 q^{38} + 75 q^{40} + 140 q^{41} - 397 q^{43} - 658 q^{44} + 6 q^{46} - 347 q^{47} + 307 q^{49} + 25 q^{50} - 35 q^{52} - 8 q^{53} - 470 q^{55} + 90 q^{56} + 157 q^{58} + 748 q^{59} + 338 q^{61} - 450 q^{62} - 334 q^{64} - 25 q^{65} - 492 q^{67} + 917 q^{68} + 30 q^{70} - 64 q^{71} + 1940 q^{73} - 70 q^{74} - 392 q^{76} + 282 q^{77} + 1257 q^{79} - 410 q^{80} + 280 q^{82} - 102 q^{83} + 655 q^{85} + 397 q^{86} - 705 q^{88} + 2976 q^{89} + 60 q^{91} - 21 q^{92} + 347 q^{94} - 280 q^{95} - 974 q^{97} + 614 q^{98} + O(q^{100})$$

## 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
0.500000 0.866025i 0 3.50000 + 6.06218i 2.50000 + 4.33013i 0 3.00000 5.19615i 15.0000 0 5.00000
271.1 0.500000 + 0.866025i 0 3.50000 6.06218i 2.50000 4.33013i 0 3.00000 + 5.19615i 15.0000 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
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.h 2
3.b odd 2 1 405.4.e.f 2
9.c even 3 1 135.4.a.b 1
9.c even 3 1 inner 405.4.e.h 2
9.d odd 6 1 135.4.a.c yes 1
9.d odd 6 1 405.4.e.f 2
36.f odd 6 1 2160.4.a.f 1
36.h even 6 1 2160.4.a.p 1
45.h odd 6 1 675.4.a.c 1
45.j even 6 1 675.4.a.h 1
45.k odd 12 2 675.4.b.f 2
45.l even 12 2 675.4.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.b 1 9.c even 3 1
135.4.a.c yes 1 9.d odd 6 1
405.4.e.f 2 3.b odd 2 1
405.4.e.f 2 9.d odd 6 1
405.4.e.h 2 1.a even 1 1 trivial
405.4.e.h 2 9.c even 3 1 inner
675.4.a.c 1 45.h odd 6 1
675.4.a.h 1 45.j even 6 1
675.4.b.e 2 45.l even 12 2
675.4.b.f 2 45.k odd 12 2
2160.4.a.f 1 36.f odd 6 1
2160.4.a.p 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 $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{7}^{2} - 6 T_{7} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 - 5 T + T^{2}$$
$7$ $$36 - 6 T + T^{2}$$
$11$ $$2209 + 47 T + T^{2}$$
$13$ $$25 - 5 T + T^{2}$$
$17$ $$( -131 + T )^{2}$$
$19$ $$( 56 + T )^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$24649 + 157 T + T^{2}$$
$31$ $$50625 + 225 T + T^{2}$$
$37$ $$( 70 + T )^{2}$$
$41$ $$19600 - 140 T + T^{2}$$
$43$ $$157609 + 397 T + T^{2}$$
$47$ $$120409 + 347 T + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$559504 - 748 T + T^{2}$$
$61$ $$114244 - 338 T + T^{2}$$
$67$ $$242064 + 492 T + T^{2}$$
$71$ $$( 32 + T )^{2}$$
$73$ $$( -970 + T )^{2}$$
$79$ $$1580049 - 1257 T + T^{2}$$
$83$ $$10404 + 102 T + T^{2}$$
$89$ $$( -1488 + T )^{2}$$
$97$ $$948676 + 974 T + T^{2}$$