# Properties

 Label 1216.2.i.d Level $1216$ Weight $2$ Character orbit 1216.i Analytic conductor $9.710$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + 4 q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (z - 1) * q^3 + 4 * q^7 + 2*z * q^9 $$q + (\zeta_{6} - 1) q^{3} + 4 q^{7} + 2 \zeta_{6} q^{9} + 3 q^{11} + 2 \zeta_{6} q^{13} + ( - 6 \zeta_{6} + 6) q^{17} + ( - 3 \zeta_{6} - 2) q^{19} + (4 \zeta_{6} - 4) q^{21} - 6 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} - 5 q^{27} - 2 q^{31} + (3 \zeta_{6} - 3) q^{33} + 10 q^{37} - 2 q^{39} + (9 \zeta_{6} - 9) q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + 9 q^{49} + 6 \zeta_{6} q^{51} + 6 \zeta_{6} q^{53} + ( - 2 \zeta_{6} + 5) q^{57} + ( - 9 \zeta_{6} + 9) q^{59} - 4 \zeta_{6} q^{61} + 8 \zeta_{6} q^{63} + 7 \zeta_{6} q^{67} + 6 q^{69} + (6 \zeta_{6} - 6) q^{71} + ( - \zeta_{6} + 1) q^{73} - 5 q^{75} + 12 q^{77} + (4 \zeta_{6} - 4) q^{79} + (\zeta_{6} - 1) q^{81} + 3 q^{83} - 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( - 2 \zeta_{6} + 2) q^{93} + (17 \zeta_{6} - 17) q^{97} + 6 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + 4 * q^7 + 2*z * q^9 + 3 * q^11 + 2*z * q^13 + (-6*z + 6) * q^17 + (-3*z - 2) * q^19 + (4*z - 4) * q^21 - 6*z * q^23 + 5*z * q^25 - 5 * q^27 - 2 * q^31 + (3*z - 3) * q^33 + 10 * q^37 - 2 * q^39 + (9*z - 9) * q^41 + (-4*z + 4) * q^43 + 9 * q^49 + 6*z * q^51 + 6*z * q^53 + (-2*z + 5) * q^57 + (-9*z + 9) * q^59 - 4*z * q^61 + 8*z * q^63 + 7*z * q^67 + 6 * q^69 + (6*z - 6) * q^71 + (-z + 1) * q^73 - 5 * q^75 + 12 * q^77 + (4*z - 4) * q^79 + (z - 1) * q^81 + 3 * q^83 - 6*z * q^89 + 8*z * q^91 + (-2*z + 2) * q^93 + (17*z - 17) * q^97 + 6*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^3 + 8 * q^7 + 2 * q^9 $$2 q - q^{3} + 8 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} + 6 q^{17} - 7 q^{19} - 4 q^{21} - 6 q^{23} + 5 q^{25} - 10 q^{27} - 4 q^{31} - 3 q^{33} + 20 q^{37} - 4 q^{39} - 9 q^{41} + 4 q^{43} + 18 q^{49} + 6 q^{51} + 6 q^{53} + 8 q^{57} + 9 q^{59} - 4 q^{61} + 8 q^{63} + 7 q^{67} + 12 q^{69} - 6 q^{71} + q^{73} - 10 q^{75} + 24 q^{77} - 4 q^{79} - q^{81} + 6 q^{83} - 6 q^{89} + 8 q^{91} + 2 q^{93} - 17 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - q^3 + 8 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^13 + 6 * q^17 - 7 * q^19 - 4 * q^21 - 6 * q^23 + 5 * q^25 - 10 * q^27 - 4 * q^31 - 3 * q^33 + 20 * q^37 - 4 * q^39 - 9 * q^41 + 4 * q^43 + 18 * q^49 + 6 * q^51 + 6 * q^53 + 8 * q^57 + 9 * q^59 - 4 * q^61 + 8 * q^63 + 7 * q^67 + 12 * q^69 - 6 * q^71 + q^73 - 10 * q^75 + 24 * q^77 - 4 * q^79 - q^81 + 6 * q^83 - 6 * q^89 + 8 * q^91 + 2 * q^93 - 17 * q^97 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 0 0 0 4.00000 0 1.00000 1.73205i 0
961.1 0 −0.500000 + 0.866025i 0 0 0 4.00000 0 1.00000 + 1.73205i 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.d 2
4.b odd 2 1 1216.2.i.h 2
8.b even 2 1 304.2.i.c 2
8.d odd 2 1 38.2.c.a 2
19.c even 3 1 inner 1216.2.i.d 2
24.f even 2 1 342.2.g.b 2
24.h odd 2 1 2736.2.s.m 2
40.e odd 2 1 950.2.e.d 2
40.k even 4 2 950.2.j.e 4
76.g odd 6 1 1216.2.i.h 2
152.b even 2 1 722.2.c.b 2
152.k odd 6 1 38.2.c.a 2
152.k odd 6 1 722.2.a.c 1
152.l odd 6 1 5776.2.a.n 1
152.o even 6 1 722.2.a.d 1
152.o even 6 1 722.2.c.b 2
152.p even 6 1 304.2.i.c 2
152.p even 6 1 5776.2.a.g 1
152.u odd 18 6 722.2.e.j 6
152.v even 18 6 722.2.e.i 6
456.s odd 6 1 6498.2.a.e 1
456.u even 6 1 342.2.g.b 2
456.u even 6 1 6498.2.a.s 1
456.x odd 6 1 2736.2.s.m 2
760.bm odd 6 1 950.2.e.d 2
760.bw even 12 2 950.2.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 8.d odd 2 1
38.2.c.a 2 152.k odd 6 1
304.2.i.c 2 8.b even 2 1
304.2.i.c 2 152.p even 6 1
342.2.g.b 2 24.f even 2 1
342.2.g.b 2 456.u even 6 1
722.2.a.c 1 152.k odd 6 1
722.2.a.d 1 152.o even 6 1
722.2.c.b 2 152.b even 2 1
722.2.c.b 2 152.o even 6 1
722.2.e.i 6 152.v even 18 6
722.2.e.j 6 152.u odd 18 6
950.2.e.d 2 40.e odd 2 1
950.2.e.d 2 760.bm odd 6 1
950.2.j.e 4 40.k even 4 2
950.2.j.e 4 760.bw even 12 2
1216.2.i.d 2 1.a even 1 1 trivial
1216.2.i.d 2 19.c even 3 1 inner
1216.2.i.h 2 4.b odd 2 1
1216.2.i.h 2 76.g odd 6 1
2736.2.s.m 2 24.h odd 2 1
2736.2.s.m 2 456.x odd 6 1
5776.2.a.g 1 152.p even 6 1
5776.2.a.n 1 152.l odd 6 1
6498.2.a.e 1 456.s odd 6 1
6498.2.a.s 1 456.u even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{5}$$ T5 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 4)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$T^{2} - T + 1$$
$79$ $$T^{2} + 4T + 16$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} + 17T + 289$$