Properties

 Label 270.3.b.c Level $270$ Weight $3$ Character orbit 270.b Analytic conductor $7.357$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$270 = 2 \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 270.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.35696713773$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + 2 q^{4} + 5 \zeta_{8} q^{5} + 5 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{8}+O(q^{10})$$ q + (-z^3 + z) * q^2 + 2 * q^4 + 5*z * q^5 + 5*z^2 * q^7 + (-2*z^3 + 2*z) * q^8 $$q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + 2 q^{4} + 5 \zeta_{8} q^{5} + 5 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{8} + (5 \zeta_{8}^{2} + 5) q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + 9 \zeta_{8}^{2} q^{13} + (5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{14} + 4 q^{16} + (8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{17} + 21 q^{19} + 10 \zeta_{8} q^{20} - 2 \zeta_{8}^{2} q^{22} + (\zeta_{8}^{3} - \zeta_{8}) q^{23} + 25 \zeta_{8}^{2} q^{25} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{26} + 10 \zeta_{8}^{2} q^{28} + ( - 27 \zeta_{8}^{3} - 27 \zeta_{8}) q^{29} + 40 q^{31} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{32} - 16 q^{34} + 25 \zeta_{8}^{3} q^{35} - 25 \zeta_{8}^{2} q^{37} + ( - 21 \zeta_{8}^{3} + 21 \zeta_{8}) q^{38} + (10 \zeta_{8}^{2} + 10) q^{40} + ( - 37 \zeta_{8}^{3} - 37 \zeta_{8}) q^{41} + 64 \zeta_{8}^{2} q^{43} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{44} - 2 q^{46} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{47} + 24 q^{49} + (25 \zeta_{8}^{3} + 25 \zeta_{8}) q^{50} + 18 \zeta_{8}^{2} q^{52} + (51 \zeta_{8}^{3} - 51 \zeta_{8}) q^{53} + ( - 5 \zeta_{8}^{2} + 5) q^{55} + (10 \zeta_{8}^{3} + 10 \zeta_{8}) q^{56} - 54 \zeta_{8}^{2} q^{58} + ( - 64 \zeta_{8}^{3} - 64 \zeta_{8}) q^{59} - 97 q^{61} + ( - 40 \zeta_{8}^{3} + 40 \zeta_{8}) q^{62} + 8 q^{64} + 45 \zeta_{8}^{3} q^{65} - 131 \zeta_{8}^{2} q^{67} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{68} + (25 \zeta_{8}^{2} - 25) q^{70} + (63 \zeta_{8}^{3} + 63 \zeta_{8}) q^{71} - 17 \zeta_{8}^{2} q^{73} + ( - 25 \zeta_{8}^{3} - 25 \zeta_{8}) q^{74} + 42 q^{76} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{77} - 117 q^{79} + 20 \zeta_{8} q^{80} - 74 \zeta_{8}^{2} q^{82} + ( - 41 \zeta_{8}^{3} + 41 \zeta_{8}) q^{83} + ( - 40 \zeta_{8}^{2} - 40) q^{85} + (64 \zeta_{8}^{3} + 64 \zeta_{8}) q^{86} - 4 \zeta_{8}^{2} q^{88} + ( - 104 \zeta_{8}^{3} - 104 \zeta_{8}) q^{89} - 45 q^{91} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{92} - 32 q^{94} + 105 \zeta_{8} q^{95} - 41 \zeta_{8}^{2} q^{97} + ( - 24 \zeta_{8}^{3} + 24 \zeta_{8}) q^{98} +O(q^{100})$$ q + (-z^3 + z) * q^2 + 2 * q^4 + 5*z * q^5 + 5*z^2 * q^7 + (-2*z^3 + 2*z) * q^8 + (5*z^2 + 5) * q^10 + (-z^3 - z) * q^11 + 9*z^2 * q^13 + (5*z^3 + 5*z) * q^14 + 4 * q^16 + (8*z^3 - 8*z) * q^17 + 21 * q^19 + 10*z * q^20 - 2*z^2 * q^22 + (z^3 - z) * q^23 + 25*z^2 * q^25 + (9*z^3 + 9*z) * q^26 + 10*z^2 * q^28 + (-27*z^3 - 27*z) * q^29 + 40 * q^31 + (-4*z^3 + 4*z) * q^32 - 16 * q^34 + 25*z^3 * q^35 - 25*z^2 * q^37 + (-21*z^3 + 21*z) * q^38 + (10*z^2 + 10) * q^40 + (-37*z^3 - 37*z) * q^41 + 64*z^2 * q^43 + (-2*z^3 - 2*z) * q^44 - 2 * q^46 + (16*z^3 - 16*z) * q^47 + 24 * q^49 + (25*z^3 + 25*z) * q^50 + 18*z^2 * q^52 + (51*z^3 - 51*z) * q^53 + (-5*z^2 + 5) * q^55 + (10*z^3 + 10*z) * q^56 - 54*z^2 * q^58 + (-64*z^3 - 64*z) * q^59 - 97 * q^61 + (-40*z^3 + 40*z) * q^62 + 8 * q^64 + 45*z^3 * q^65 - 131*z^2 * q^67 + (16*z^3 - 16*z) * q^68 + (25*z^2 - 25) * q^70 + (63*z^3 + 63*z) * q^71 - 17*z^2 * q^73 + (-25*z^3 - 25*z) * q^74 + 42 * q^76 + (-5*z^3 + 5*z) * q^77 - 117 * q^79 + 20*z * q^80 - 74*z^2 * q^82 + (-41*z^3 + 41*z) * q^83 + (-40*z^2 - 40) * q^85 + (64*z^3 + 64*z) * q^86 - 4*z^2 * q^88 + (-104*z^3 - 104*z) * q^89 - 45 * q^91 + (2*z^3 - 2*z) * q^92 - 32 * q^94 + 105*z * q^95 - 41*z^2 * q^97 + (-24*z^3 + 24*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4}+O(q^{10})$$ 4 * q + 8 * q^4 $$4 q + 8 q^{4} + 20 q^{10} + 16 q^{16} + 84 q^{19} + 160 q^{31} - 64 q^{34} + 40 q^{40} - 8 q^{46} + 96 q^{49} + 20 q^{55} - 388 q^{61} + 32 q^{64} - 100 q^{70} + 168 q^{76} - 468 q^{79} - 160 q^{85} - 180 q^{91} - 128 q^{94}+O(q^{100})$$ 4 * q + 8 * q^4 + 20 * q^10 + 16 * q^16 + 84 * q^19 + 160 * q^31 - 64 * q^34 + 40 * q^40 - 8 * q^46 + 96 * q^49 + 20 * q^55 - 388 * q^61 + 32 * q^64 - 100 * q^70 + 168 * q^76 - 468 * q^79 - 160 * q^85 - 180 * q^91 - 128 * q^94

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$217$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 0 2.00000 −3.53553 3.53553i 0 5.00000i −2.82843 0 5.00000 + 5.00000i
269.2 −1.41421 0 2.00000 −3.53553 + 3.53553i 0 5.00000i −2.82843 0 5.00000 5.00000i
269.3 1.41421 0 2.00000 3.53553 3.53553i 0 5.00000i 2.82843 0 5.00000 5.00000i
269.4 1.41421 0 2.00000 3.53553 + 3.53553i 0 5.00000i 2.82843 0 5.00000 + 5.00000i
 $$n$$: 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.b.c 4
3.b odd 2 1 inner 270.3.b.c 4
4.b odd 2 1 2160.3.c.i 4
5.b even 2 1 inner 270.3.b.c 4
5.c odd 4 1 1350.3.d.f 2
5.c odd 4 1 1350.3.d.g 2
9.c even 3 2 810.3.j.e 8
9.d odd 6 2 810.3.j.e 8
12.b even 2 1 2160.3.c.i 4
15.d odd 2 1 inner 270.3.b.c 4
15.e even 4 1 1350.3.d.f 2
15.e even 4 1 1350.3.d.g 2
20.d odd 2 1 2160.3.c.i 4
45.h odd 6 2 810.3.j.e 8
45.j even 6 2 810.3.j.e 8
60.h even 2 1 2160.3.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 1.a even 1 1 trivial
270.3.b.c 4 3.b odd 2 1 inner
270.3.b.c 4 5.b even 2 1 inner
270.3.b.c 4 15.d odd 2 1 inner
810.3.j.e 8 9.c even 3 2
810.3.j.e 8 9.d odd 6 2
810.3.j.e 8 45.h odd 6 2
810.3.j.e 8 45.j even 6 2
1350.3.d.f 2 5.c odd 4 1
1350.3.d.f 2 15.e even 4 1
1350.3.d.g 2 5.c odd 4 1
1350.3.d.g 2 15.e even 4 1
2160.3.c.i 4 4.b odd 2 1
2160.3.c.i 4 12.b even 2 1
2160.3.c.i 4 20.d odd 2 1
2160.3.c.i 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(270, [\chi])$$:

 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{17}^{2} - 128$$ T17^2 - 128

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 625$$
$7$ $$(T^{2} + 25)^{2}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$(T^{2} + 81)^{2}$$
$17$ $$(T^{2} - 128)^{2}$$
$19$ $$(T - 21)^{4}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$(T^{2} + 1458)^{2}$$
$31$ $$(T - 40)^{4}$$
$37$ $$(T^{2} + 625)^{2}$$
$41$ $$(T^{2} + 2738)^{2}$$
$43$ $$(T^{2} + 4096)^{2}$$
$47$ $$(T^{2} - 512)^{2}$$
$53$ $$(T^{2} - 5202)^{2}$$
$59$ $$(T^{2} + 8192)^{2}$$
$61$ $$(T + 97)^{4}$$
$67$ $$(T^{2} + 17161)^{2}$$
$71$ $$(T^{2} + 7938)^{2}$$
$73$ $$(T^{2} + 289)^{2}$$
$79$ $$(T + 117)^{4}$$
$83$ $$(T^{2} - 3362)^{2}$$
$89$ $$(T^{2} + 21632)^{2}$$
$97$ $$(T^{2} + 1681)^{2}$$