# Properties

 Label 405.4.e.d 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 15) 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( -20 + 20 \zeta_{6} ) q^{7} -21 q^{8} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( -20 + 20 \zeta_{6} ) q^{7} -21 q^{8} -15 q^{10} + ( 24 - 24 \zeta_{6} ) q^{11} -74 \zeta_{6} q^{13} -60 \zeta_{6} q^{14} + ( 71 - 71 \zeta_{6} ) q^{16} + 54 q^{17} -124 q^{19} + ( 5 - 5 \zeta_{6} ) q^{20} + 72 \zeta_{6} q^{22} + 120 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 222 q^{26} + 20 q^{28} + ( 78 - 78 \zeta_{6} ) q^{29} -200 \zeta_{6} q^{31} + 45 \zeta_{6} q^{32} + ( -162 + 162 \zeta_{6} ) q^{34} -100 q^{35} -70 q^{37} + ( 372 - 372 \zeta_{6} ) q^{38} -105 \zeta_{6} q^{40} -330 \zeta_{6} q^{41} + ( -92 + 92 \zeta_{6} ) q^{43} -24 q^{44} -360 q^{46} + ( 24 - 24 \zeta_{6} ) q^{47} -57 \zeta_{6} q^{49} -75 \zeta_{6} q^{50} + ( -74 + 74 \zeta_{6} ) q^{52} + 450 q^{53} + 120 q^{55} + ( 420 - 420 \zeta_{6} ) q^{56} + 234 \zeta_{6} q^{58} -24 \zeta_{6} q^{59} + ( 322 - 322 \zeta_{6} ) q^{61} + 600 q^{62} + 433 q^{64} + ( 370 - 370 \zeta_{6} ) q^{65} + 196 \zeta_{6} q^{67} -54 \zeta_{6} q^{68} + ( 300 - 300 \zeta_{6} ) q^{70} -288 q^{71} -430 q^{73} + ( 210 - 210 \zeta_{6} ) q^{74} + 124 \zeta_{6} q^{76} + 480 \zeta_{6} q^{77} + ( 520 - 520 \zeta_{6} ) q^{79} + 355 q^{80} + 990 q^{82} + ( -156 + 156 \zeta_{6} ) q^{83} + 270 \zeta_{6} q^{85} -276 \zeta_{6} q^{86} + ( -504 + 504 \zeta_{6} ) q^{88} + 1026 q^{89} + 1480 q^{91} + ( 120 - 120 \zeta_{6} ) q^{92} + 72 \zeta_{6} q^{94} -620 \zeta_{6} q^{95} + ( 286 - 286 \zeta_{6} ) q^{97} + 171 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8} + O(q^{10})$$ $$2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8} - 30 q^{10} + 24 q^{11} - 74 q^{13} - 60 q^{14} + 71 q^{16} + 108 q^{17} - 248 q^{19} + 5 q^{20} + 72 q^{22} + 120 q^{23} - 25 q^{25} + 444 q^{26} + 40 q^{28} + 78 q^{29} - 200 q^{31} + 45 q^{32} - 162 q^{34} - 200 q^{35} - 140 q^{37} + 372 q^{38} - 105 q^{40} - 330 q^{41} - 92 q^{43} - 48 q^{44} - 720 q^{46} + 24 q^{47} - 57 q^{49} - 75 q^{50} - 74 q^{52} + 900 q^{53} + 240 q^{55} + 420 q^{56} + 234 q^{58} - 24 q^{59} + 322 q^{61} + 1200 q^{62} + 866 q^{64} + 370 q^{65} + 196 q^{67} - 54 q^{68} + 300 q^{70} - 576 q^{71} - 860 q^{73} + 210 q^{74} + 124 q^{76} + 480 q^{77} + 520 q^{79} + 710 q^{80} + 1980 q^{82} - 156 q^{83} + 270 q^{85} - 276 q^{86} - 504 q^{88} + 2052 q^{89} + 2960 q^{91} + 120 q^{92} + 72 q^{94} - 620 q^{95} + 286 q^{97} + 342 q^{98} + O(q^{100})$$

## 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
−1.50000 + 2.59808i 0 −0.500000 0.866025i 2.50000 + 4.33013i 0 −10.0000 + 17.3205i −21.0000 0 −15.0000
271.1 −1.50000 2.59808i 0 −0.500000 + 0.866025i 2.50000 4.33013i 0 −10.0000 17.3205i −21.0000 0 −15.0000
 $$n$$: 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.d 2
3.b odd 2 1 405.4.e.k 2
9.c even 3 1 15.4.a.b 1
9.c even 3 1 inner 405.4.e.d 2
9.d odd 6 1 45.4.a.b 1
9.d odd 6 1 405.4.e.k 2
36.f odd 6 1 240.4.a.f 1
36.h even 6 1 720.4.a.r 1
45.h odd 6 1 225.4.a.g 1
45.j even 6 1 75.4.a.a 1
45.k odd 12 2 75.4.b.a 2
45.l even 12 2 225.4.b.d 2
63.l odd 6 1 735.4.a.i 1
63.o even 6 1 2205.4.a.c 1
72.n even 6 1 960.4.a.bi 1
72.p odd 6 1 960.4.a.l 1
99.h odd 6 1 1815.4.a.a 1
180.p odd 6 1 1200.4.a.o 1
180.x even 12 2 1200.4.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 9.c even 3 1
45.4.a.b 1 9.d odd 6 1
75.4.a.a 1 45.j even 6 1
75.4.b.a 2 45.k odd 12 2
225.4.a.g 1 45.h odd 6 1
225.4.b.d 2 45.l even 12 2
240.4.a.f 1 36.f odd 6 1
405.4.e.d 2 1.a even 1 1 trivial
405.4.e.d 2 9.c even 3 1 inner
405.4.e.k 2 3.b odd 2 1
405.4.e.k 2 9.d odd 6 1
720.4.a.r 1 36.h even 6 1
735.4.a.i 1 63.l odd 6 1
960.4.a.l 1 72.p odd 6 1
960.4.a.bi 1 72.n even 6 1
1200.4.a.o 1 180.p odd 6 1
1200.4.f.m 2 180.x even 12 2
1815.4.a.a 1 99.h odd 6 1
2205.4.a.c 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 $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} + 3 T_{2} + 9$$ $$T_{7}^{2} + 20 T_{7} + 400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 - 5 T + T^{2}$$
$7$ $$400 + 20 T + T^{2}$$
$11$ $$576 - 24 T + T^{2}$$
$13$ $$5476 + 74 T + T^{2}$$
$17$ $$( -54 + T )^{2}$$
$19$ $$( 124 + T )^{2}$$
$23$ $$14400 - 120 T + T^{2}$$
$29$ $$6084 - 78 T + T^{2}$$
$31$ $$40000 + 200 T + T^{2}$$
$37$ $$( 70 + T )^{2}$$
$41$ $$108900 + 330 T + T^{2}$$
$43$ $$8464 + 92 T + T^{2}$$
$47$ $$576 - 24 T + T^{2}$$
$53$ $$( -450 + T )^{2}$$
$59$ $$576 + 24 T + T^{2}$$
$61$ $$103684 - 322 T + T^{2}$$
$67$ $$38416 - 196 T + T^{2}$$
$71$ $$( 288 + T )^{2}$$
$73$ $$( 430 + T )^{2}$$
$79$ $$270400 - 520 T + T^{2}$$
$83$ $$24336 + 156 T + T^{2}$$
$89$ $$( -1026 + T )^{2}$$
$97$ $$81796 - 286 T + T^{2}$$