# 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: $$x^{2} - x + 1$$ 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.

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

## 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 $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} + 3T_{2} + 9$$ T2^2 + 3*T2 + 9 $$T_{7}^{2} + 20T_{7} + 400$$ T7^2 + 20*T7 + 400

## Hecke characteristic polynomials

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