# Properties

 Label 405.2.e.h Level $405$ Weight $2$ Character orbit 405.e Analytic conductor $3.234$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 + z * q^5 + (-3*z + 3) * q^7 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + 2 q^{10} + ( - 2 \zeta_{6} + 2) q^{11} + 5 \zeta_{6} q^{13} - 6 \zeta_{6} q^{14} + ( - 4 \zeta_{6} + 4) q^{16} - 8 q^{17} + q^{19} + ( - 2 \zeta_{6} + 2) q^{20} - 4 \zeta_{6} q^{22} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 10 q^{26} - 6 q^{28} + (2 \zeta_{6} - 2) q^{29} - 8 \zeta_{6} q^{32} + (16 \zeta_{6} - 16) q^{34} + 3 q^{35} + 5 q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + 10 \zeta_{6} q^{41} + (4 \zeta_{6} - 4) q^{43} - 4 q^{44} - 12 q^{46} + (4 \zeta_{6} - 4) q^{47} - 2 \zeta_{6} q^{49} + 2 \zeta_{6} q^{50} + ( - 10 \zeta_{6} + 10) q^{52} - 2 q^{53} + 2 q^{55} + 4 \zeta_{6} q^{58} + 8 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} - 8 q^{64} + (5 \zeta_{6} - 5) q^{65} + 9 \zeta_{6} q^{67} + 16 \zeta_{6} q^{68} + ( - 6 \zeta_{6} + 6) q^{70} + 2 q^{71} - 5 q^{73} + ( - 10 \zeta_{6} + 10) q^{74} - 2 \zeta_{6} q^{76} - 6 \zeta_{6} q^{77} + ( - 3 \zeta_{6} + 3) q^{79} + 4 q^{80} + 20 q^{82} + (6 \zeta_{6} - 6) q^{83} - 8 \zeta_{6} q^{85} + 8 \zeta_{6} q^{86} - 12 q^{89} + 15 q^{91} + (12 \zeta_{6} - 12) q^{92} + 8 \zeta_{6} q^{94} + \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} - 4 q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 2*z * q^4 + z * q^5 + (-3*z + 3) * q^7 + 2 * q^10 + (-2*z + 2) * q^11 + 5*z * q^13 - 6*z * q^14 + (-4*z + 4) * q^16 - 8 * q^17 + q^19 + (-2*z + 2) * q^20 - 4*z * q^22 - 6*z * q^23 + (z - 1) * q^25 + 10 * q^26 - 6 * q^28 + (2*z - 2) * q^29 - 8*z * q^32 + (16*z - 16) * q^34 + 3 * q^35 + 5 * q^37 + (-2*z + 2) * q^38 + 10*z * q^41 + (4*z - 4) * q^43 - 4 * q^44 - 12 * q^46 + (4*z - 4) * q^47 - 2*z * q^49 + 2*z * q^50 + (-10*z + 10) * q^52 - 2 * q^53 + 2 * q^55 + 4*z * q^58 + 8*z * q^59 + (7*z - 7) * q^61 - 8 * q^64 + (5*z - 5) * q^65 + 9*z * q^67 + 16*z * q^68 + (-6*z + 6) * q^70 + 2 * q^71 - 5 * q^73 + (-10*z + 10) * q^74 - 2*z * q^76 - 6*z * q^77 + (-3*z + 3) * q^79 + 4 * q^80 + 20 * q^82 + (6*z - 6) * q^83 - 8*z * q^85 + 8*z * q^86 - 12 * q^89 + 15 * q^91 + (12*z - 12) * q^92 + 8*z * q^94 + z * q^95 + (-13*z + 13) * q^97 - 4 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} + q^{5} + 3 q^{7}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 3 * q^7 $$2 q + 2 q^{2} - 2 q^{4} + q^{5} + 3 q^{7} + 4 q^{10} + 2 q^{11} + 5 q^{13} - 6 q^{14} + 4 q^{16} - 16 q^{17} + 2 q^{19} + 2 q^{20} - 4 q^{22} - 6 q^{23} - q^{25} + 20 q^{26} - 12 q^{28} - 2 q^{29} - 8 q^{32} - 16 q^{34} + 6 q^{35} + 10 q^{37} + 2 q^{38} + 10 q^{41} - 4 q^{43} - 8 q^{44} - 24 q^{46} - 4 q^{47} - 2 q^{49} + 2 q^{50} + 10 q^{52} - 4 q^{53} + 4 q^{55} + 4 q^{58} + 8 q^{59} - 7 q^{61} - 16 q^{64} - 5 q^{65} + 9 q^{67} + 16 q^{68} + 6 q^{70} + 4 q^{71} - 10 q^{73} + 10 q^{74} - 2 q^{76} - 6 q^{77} + 3 q^{79} + 8 q^{80} + 40 q^{82} - 6 q^{83} - 8 q^{85} + 8 q^{86} - 24 q^{89} + 30 q^{91} - 12 q^{92} + 8 q^{94} + q^{95} + 13 q^{97} - 8 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 3 * q^7 + 4 * q^10 + 2 * q^11 + 5 * q^13 - 6 * q^14 + 4 * q^16 - 16 * q^17 + 2 * q^19 + 2 * q^20 - 4 * q^22 - 6 * q^23 - q^25 + 20 * q^26 - 12 * q^28 - 2 * q^29 - 8 * q^32 - 16 * q^34 + 6 * q^35 + 10 * q^37 + 2 * q^38 + 10 * q^41 - 4 * q^43 - 8 * q^44 - 24 * q^46 - 4 * q^47 - 2 * q^49 + 2 * q^50 + 10 * q^52 - 4 * q^53 + 4 * q^55 + 4 * q^58 + 8 * q^59 - 7 * q^61 - 16 * q^64 - 5 * q^65 + 9 * q^67 + 16 * q^68 + 6 * q^70 + 4 * q^71 - 10 * q^73 + 10 * q^74 - 2 * q^76 - 6 * q^77 + 3 * q^79 + 8 * q^80 + 40 * q^82 - 6 * q^83 - 8 * q^85 + 8 * q^86 - 24 * q^89 + 30 * q^91 - 12 * q^92 + 8 * q^94 + q^95 + 13 * q^97 - 8 * 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
1.00000 1.73205i 0 −1.00000 1.73205i 0.500000 + 0.866025i 0 1.50000 2.59808i 0 0 2.00000
271.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i 0.500000 0.866025i 0 1.50000 + 2.59808i 0 0 2.00000
 $$n$$: 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 405.2.e.h 2
3.b odd 2 1 405.2.e.b 2
9.c even 3 1 135.2.a.a 1
9.c even 3 1 inner 405.2.e.h 2
9.d odd 6 1 135.2.a.b yes 1
9.d odd 6 1 405.2.e.b 2
36.f odd 6 1 2160.2.a.j 1
36.h even 6 1 2160.2.a.v 1
45.h odd 6 1 675.2.a.a 1
45.j even 6 1 675.2.a.i 1
45.k odd 12 2 675.2.b.a 2
45.l even 12 2 675.2.b.b 2
63.l odd 6 1 6615.2.a.a 1
63.o even 6 1 6615.2.a.j 1
72.j odd 6 1 8640.2.a.c 1
72.l even 6 1 8640.2.a.bb 1
72.n even 6 1 8640.2.a.bh 1
72.p odd 6 1 8640.2.a.ce 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.a 1 9.c even 3 1
135.2.a.b yes 1 9.d odd 6 1
405.2.e.b 2 3.b odd 2 1
405.2.e.b 2 9.d odd 6 1
405.2.e.h 2 1.a even 1 1 trivial
405.2.e.h 2 9.c even 3 1 inner
675.2.a.a 1 45.h odd 6 1
675.2.a.i 1 45.j even 6 1
675.2.b.a 2 45.k odd 12 2
675.2.b.b 2 45.l even 12 2
2160.2.a.j 1 36.f odd 6 1
2160.2.a.v 1 36.h even 6 1
6615.2.a.a 1 63.l odd 6 1
6615.2.a.j 1 63.o even 6 1
8640.2.a.c 1 72.j odd 6 1
8640.2.a.bb 1 72.l even 6 1
8640.2.a.bh 1 72.n even 6 1
8640.2.a.ce 1 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{7}^{2} - 3T_{7} + 9$$ T7^2 - 3*T7 + 9 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 3T + 9$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T + 8)^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 2T + 4$$
$31$ $$T^{2}$$
$37$ $$(T - 5)^{2}$$
$41$ $$T^{2} - 10T + 100$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$T^{2} + 4T + 16$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$(T - 2)^{2}$$
$73$ $$(T + 5)^{2}$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$T^{2} + 6T + 36$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2} - 13T + 169$$