# Properties

 Label 405.3.h.a Level $405$ Weight $3$ Character orbit 405.h Analytic conductor $11.035$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 405.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0354507066$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + 3 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} - 7 q^{8} +O(q^{10})$$ q + (z - 1) * q^2 + 3*z * q^4 - 5*z * q^5 - 7 * q^8 $$q + (\zeta_{6} - 1) q^{2} + 3 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} - 7 q^{8} + 5 q^{10} + (5 \zeta_{6} - 5) q^{16} - 14 q^{17} - 22 q^{19} + ( - 15 \zeta_{6} + 15) q^{20} - 34 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 2 \zeta_{6} q^{31} - 33 \zeta_{6} q^{32} + ( - 14 \zeta_{6} + 14) q^{34} + ( - 22 \zeta_{6} + 22) q^{38} + 35 \zeta_{6} q^{40} + 34 q^{46} + ( - 14 \zeta_{6} + 14) q^{47} - 49 \zeta_{6} q^{49} - 25 \zeta_{6} q^{50} - 86 q^{53} + ( - 118 \zeta_{6} + 118) q^{61} + 2 q^{62} + 13 q^{64} - 42 \zeta_{6} q^{68} - 66 \zeta_{6} q^{76} + (98 \zeta_{6} - 98) q^{79} + 25 q^{80} + (154 \zeta_{6} - 154) q^{83} + 70 \zeta_{6} q^{85} + ( - 102 \zeta_{6} + 102) q^{92} + 14 \zeta_{6} q^{94} + 110 \zeta_{6} q^{95} + 49 q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + 3*z * q^4 - 5*z * q^5 - 7 * q^8 + 5 * q^10 + (5*z - 5) * q^16 - 14 * q^17 - 22 * q^19 + (-15*z + 15) * q^20 - 34*z * q^23 + (25*z - 25) * q^25 - 2*z * q^31 - 33*z * q^32 + (-14*z + 14) * q^34 + (-22*z + 22) * q^38 + 35*z * q^40 + 34 * q^46 + (-14*z + 14) * q^47 - 49*z * q^49 - 25*z * q^50 - 86 * q^53 + (-118*z + 118) * q^61 + 2 * q^62 + 13 * q^64 - 42*z * q^68 - 66*z * q^76 + (98*z - 98) * q^79 + 25 * q^80 + (154*z - 154) * q^83 + 70*z * q^85 + (-102*z + 102) * q^92 + 14*z * q^94 + 110*z * q^95 + 49 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} - 5 q^{5} - 14 q^{8}+O(q^{10})$$ 2 * q - q^2 + 3 * q^4 - 5 * q^5 - 14 * q^8 $$2 q - q^{2} + 3 q^{4} - 5 q^{5} - 14 q^{8} + 10 q^{10} - 5 q^{16} - 28 q^{17} - 44 q^{19} + 15 q^{20} - 34 q^{23} - 25 q^{25} - 2 q^{31} - 33 q^{32} + 14 q^{34} + 22 q^{38} + 35 q^{40} + 68 q^{46} + 14 q^{47} - 49 q^{49} - 25 q^{50} - 172 q^{53} + 118 q^{61} + 4 q^{62} + 26 q^{64} - 42 q^{68} - 66 q^{76} - 98 q^{79} + 50 q^{80} - 154 q^{83} + 70 q^{85} + 102 q^{92} + 14 q^{94} + 110 q^{95} + 98 q^{98}+O(q^{100})$$ 2 * q - q^2 + 3 * q^4 - 5 * q^5 - 14 * q^8 + 10 * q^10 - 5 * q^16 - 28 * q^17 - 44 * q^19 + 15 * q^20 - 34 * q^23 - 25 * q^25 - 2 * q^31 - 33 * q^32 + 14 * q^34 + 22 * q^38 + 35 * q^40 + 68 * q^46 + 14 * q^47 - 49 * q^49 - 25 * q^50 - 172 * q^53 + 118 * q^61 + 4 * q^62 + 26 * q^64 - 42 * q^68 - 66 * q^76 - 98 * q^79 + 50 * q^80 - 154 * q^83 + 70 * q^85 + 102 * q^92 + 14 * q^94 + 110 * q^95 + 98 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
134.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 1.50000 2.59808i −2.50000 + 4.33013i 0 0 −7.00000 0 5.00000
269.1 −0.500000 + 0.866025i 0 1.50000 + 2.59808i −2.50000 4.33013i 0 0 −7.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
9.c even 3 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.h.a 2
3.b odd 2 1 405.3.h.b 2
5.b even 2 1 405.3.h.b 2
9.c even 3 1 15.3.d.b yes 1
9.c even 3 1 inner 405.3.h.a 2
9.d odd 6 1 15.3.d.a 1
9.d odd 6 1 405.3.h.b 2
15.d odd 2 1 CM 405.3.h.a 2
36.f odd 6 1 240.3.c.b 1
36.h even 6 1 240.3.c.a 1
45.h odd 6 1 15.3.d.b yes 1
45.h odd 6 1 inner 405.3.h.a 2
45.j even 6 1 15.3.d.a 1
45.j even 6 1 405.3.h.b 2
45.k odd 12 2 75.3.c.d 2
45.l even 12 2 75.3.c.d 2
72.j odd 6 1 960.3.c.b 1
72.l even 6 1 960.3.c.d 1
72.n even 6 1 960.3.c.c 1
72.p odd 6 1 960.3.c.a 1
180.n even 6 1 240.3.c.b 1
180.p odd 6 1 240.3.c.a 1
180.v odd 12 2 1200.3.l.l 2
180.x even 12 2 1200.3.l.l 2
360.z odd 6 1 960.3.c.d 1
360.bd even 6 1 960.3.c.a 1
360.bh odd 6 1 960.3.c.c 1
360.bk even 6 1 960.3.c.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 9.d odd 6 1
15.3.d.a 1 45.j even 6 1
15.3.d.b yes 1 9.c even 3 1
15.3.d.b yes 1 45.h odd 6 1
75.3.c.d 2 45.k odd 12 2
75.3.c.d 2 45.l even 12 2
240.3.c.a 1 36.h even 6 1
240.3.c.a 1 180.p odd 6 1
240.3.c.b 1 36.f odd 6 1
240.3.c.b 1 180.n even 6 1
405.3.h.a 2 1.a even 1 1 trivial
405.3.h.a 2 9.c even 3 1 inner
405.3.h.a 2 15.d odd 2 1 CM
405.3.h.a 2 45.h odd 6 1 inner
405.3.h.b 2 3.b odd 2 1
405.3.h.b 2 5.b even 2 1
405.3.h.b 2 9.d odd 6 1
405.3.h.b 2 45.j even 6 1
960.3.c.a 1 72.p odd 6 1
960.3.c.a 1 360.bd even 6 1
960.3.c.b 1 72.j odd 6 1
960.3.c.b 1 360.bk even 6 1
960.3.c.c 1 72.n even 6 1
960.3.c.c 1 360.bh odd 6 1
960.3.c.d 1 72.l even 6 1
960.3.c.d 1 360.z odd 6 1
1200.3.l.l 2 180.v odd 12 2
1200.3.l.l 2 180.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T + 14)^{2}$$
$19$ $$(T + 22)^{2}$$
$23$ $$T^{2} + 34T + 1156$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 14T + 196$$
$53$ $$(T + 86)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 118T + 13924$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 98T + 9604$$
$83$ $$T^{2} + 154T + 23716$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$