# Properties

 Label 405.4.e.g 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 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + 7 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( 24 - 24 \zeta_{6} ) q^{7} -15 q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + 7 \zeta_{6} q^{4} -5 \zeta_{6} q^{5} + ( 24 - 24 \zeta_{6} ) q^{7} -15 q^{8} + 5 q^{10} + ( -52 + 52 \zeta_{6} ) q^{11} -22 \zeta_{6} q^{13} + 24 \zeta_{6} q^{14} + ( -41 + 41 \zeta_{6} ) q^{16} -14 q^{17} -20 q^{19} + ( 35 - 35 \zeta_{6} ) q^{20} -52 \zeta_{6} q^{22} + 168 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 22 q^{26} + 168 q^{28} + ( -230 + 230 \zeta_{6} ) q^{29} + 288 \zeta_{6} q^{31} -161 \zeta_{6} q^{32} + ( 14 - 14 \zeta_{6} ) q^{34} -120 q^{35} -34 q^{37} + ( 20 - 20 \zeta_{6} ) q^{38} + 75 \zeta_{6} q^{40} -122 \zeta_{6} q^{41} + ( 188 - 188 \zeta_{6} ) q^{43} -364 q^{44} -168 q^{46} + ( -256 + 256 \zeta_{6} ) q^{47} -233 \zeta_{6} q^{49} -25 \zeta_{6} q^{50} + ( 154 - 154 \zeta_{6} ) q^{52} -338 q^{53} + 260 q^{55} + ( -360 + 360 \zeta_{6} ) q^{56} -230 \zeta_{6} q^{58} -100 \zeta_{6} q^{59} + ( -742 + 742 \zeta_{6} ) q^{61} -288 q^{62} -167 q^{64} + ( -110 + 110 \zeta_{6} ) q^{65} + 84 \zeta_{6} q^{67} -98 \zeta_{6} q^{68} + ( 120 - 120 \zeta_{6} ) q^{70} -328 q^{71} -38 q^{73} + ( 34 - 34 \zeta_{6} ) q^{74} -140 \zeta_{6} q^{76} + 1248 \zeta_{6} q^{77} + ( 240 - 240 \zeta_{6} ) q^{79} + 205 q^{80} + 122 q^{82} + ( -1212 + 1212 \zeta_{6} ) q^{83} + 70 \zeta_{6} q^{85} + 188 \zeta_{6} q^{86} + ( 780 - 780 \zeta_{6} ) q^{88} + 330 q^{89} -528 q^{91} + ( -1176 + 1176 \zeta_{6} ) q^{92} -256 \zeta_{6} q^{94} + 100 \zeta_{6} q^{95} + ( -866 + 866 \zeta_{6} ) q^{97} + 233 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8} + O(q^{10})$$ $$2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8} + 10 q^{10} - 52 q^{11} - 22 q^{13} + 24 q^{14} - 41 q^{16} - 28 q^{17} - 40 q^{19} + 35 q^{20} - 52 q^{22} + 168 q^{23} - 25 q^{25} + 44 q^{26} + 336 q^{28} - 230 q^{29} + 288 q^{31} - 161 q^{32} + 14 q^{34} - 240 q^{35} - 68 q^{37} + 20 q^{38} + 75 q^{40} - 122 q^{41} + 188 q^{43} - 728 q^{44} - 336 q^{46} - 256 q^{47} - 233 q^{49} - 25 q^{50} + 154 q^{52} - 676 q^{53} + 520 q^{55} - 360 q^{56} - 230 q^{58} - 100 q^{59} - 742 q^{61} - 576 q^{62} - 334 q^{64} - 110 q^{65} + 84 q^{67} - 98 q^{68} + 120 q^{70} - 656 q^{71} - 76 q^{73} + 34 q^{74} - 140 q^{76} + 1248 q^{77} + 240 q^{79} + 410 q^{80} + 244 q^{82} - 1212 q^{83} + 70 q^{85} + 188 q^{86} + 780 q^{88} + 660 q^{89} - 1056 q^{91} - 1176 q^{92} - 256 q^{94} + 100 q^{95} - 866 q^{97} + 466 q^{98} + O(q^{100})$$

## 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 12.0000 20.7846i −15.0000 0 5.00000
271.1 −0.500000 0.866025i 0 3.50000 6.06218i −2.50000 + 4.33013i 0 12.0000 + 20.7846i −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.g 2
3.b odd 2 1 405.4.e.i 2
9.c even 3 1 15.4.a.a 1
9.c even 3 1 inner 405.4.e.g 2
9.d odd 6 1 45.4.a.c 1
9.d odd 6 1 405.4.e.i 2
36.f odd 6 1 240.4.a.e 1
36.h even 6 1 720.4.a.n 1
45.h odd 6 1 225.4.a.f 1
45.j even 6 1 75.4.a.b 1
45.k odd 12 2 75.4.b.b 2
45.l even 12 2 225.4.b.e 2
63.l odd 6 1 735.4.a.e 1
63.o even 6 1 2205.4.a.l 1
72.n even 6 1 960.4.a.b 1
72.p odd 6 1 960.4.a.ba 1
99.h odd 6 1 1815.4.a.e 1
180.p odd 6 1 1200.4.a.t 1
180.x even 12 2 1200.4.f.b 2

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

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{7}^{2} - 24 T_{7} + 576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 + 5 T + T^{2}$$
$7$ $$576 - 24 T + T^{2}$$
$11$ $$2704 + 52 T + T^{2}$$
$13$ $$484 + 22 T + T^{2}$$
$17$ $$( 14 + T )^{2}$$
$19$ $$( 20 + T )^{2}$$
$23$ $$28224 - 168 T + T^{2}$$
$29$ $$52900 + 230 T + T^{2}$$
$31$ $$82944 - 288 T + T^{2}$$
$37$ $$( 34 + T )^{2}$$
$41$ $$14884 + 122 T + T^{2}$$
$43$ $$35344 - 188 T + T^{2}$$
$47$ $$65536 + 256 T + T^{2}$$
$53$ $$( 338 + T )^{2}$$
$59$ $$10000 + 100 T + T^{2}$$
$61$ $$550564 + 742 T + T^{2}$$
$67$ $$7056 - 84 T + T^{2}$$
$71$ $$( 328 + T )^{2}$$
$73$ $$( 38 + T )^{2}$$
$79$ $$57600 - 240 T + T^{2}$$
$83$ $$1468944 + 1212 T + T^{2}$$
$89$ $$( -330 + T )^{2}$$
$97$ $$749956 + 866 T + T^{2}$$