# Properties

 Label 2156.2.i.c Level $2156$ Weight $2$ Character orbit 2156.i Analytic conductor $17.216$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2156 = 2^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2156.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.2157466758$$ 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 44) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - 3*z * q^5 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + 4 q^{13} - 3 q^{15} + ( - 6 \zeta_{6} + 6) q^{17} + 8 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 5 q^{27} + ( - 5 \zeta_{6} + 5) q^{31} - \zeta_{6} q^{33} + \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} - 10 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} - 6 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} - 3 q^{55} + 8 q^{57} + ( - 3 \zeta_{6} + 3) q^{59} - 4 \zeta_{6} q^{61} - 12 \zeta_{6} q^{65} + ( - \zeta_{6} + 1) q^{67} + 3 q^{69} + 15 q^{71} + (4 \zeta_{6} - 4) q^{73} + 4 \zeta_{6} q^{75} - 2 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 6 q^{83} - 18 q^{85} - 9 \zeta_{6} q^{89} - 5 \zeta_{6} q^{93} + ( - 24 \zeta_{6} + 24) q^{95} + 7 q^{97} + 2 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 - 3*z * q^5 + 2*z * q^9 + (-z + 1) * q^11 + 4 * q^13 - 3 * q^15 + (-6*z + 6) * q^17 + 8*z * q^19 + 3*z * q^23 + (4*z - 4) * q^25 + 5 * q^27 + (-5*z + 5) * q^31 - z * q^33 + z * q^37 + (-4*z + 4) * q^39 - 10 * q^43 + (-6*z + 6) * q^45 - 6*z * q^51 + (-6*z + 6) * q^53 - 3 * q^55 + 8 * q^57 + (-3*z + 3) * q^59 - 4*z * q^61 - 12*z * q^65 + (-z + 1) * q^67 + 3 * q^69 + 15 * q^71 + (4*z - 4) * q^73 + 4*z * q^75 - 2*z * q^79 + (z - 1) * q^81 - 6 * q^83 - 18 * q^85 - 9*z * q^89 - 5*z * q^93 + (-24*z + 24) * q^95 + 7 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^9 $$2 q + q^{3} - 3 q^{5} + 2 q^{9} + q^{11} + 8 q^{13} - 6 q^{15} + 6 q^{17} + 8 q^{19} + 3 q^{23} - 4 q^{25} + 10 q^{27} + 5 q^{31} - q^{33} + q^{37} + 4 q^{39} - 20 q^{43} + 6 q^{45} - 6 q^{51} + 6 q^{53} - 6 q^{55} + 16 q^{57} + 3 q^{59} - 4 q^{61} - 12 q^{65} + q^{67} + 6 q^{69} + 30 q^{71} - 4 q^{73} + 4 q^{75} - 2 q^{79} - q^{81} - 12 q^{83} - 36 q^{85} - 9 q^{89} - 5 q^{93} + 24 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^9 + q^11 + 8 * q^13 - 6 * q^15 + 6 * q^17 + 8 * q^19 + 3 * q^23 - 4 * q^25 + 10 * q^27 + 5 * q^31 - q^33 + q^37 + 4 * q^39 - 20 * q^43 + 6 * q^45 - 6 * q^51 + 6 * q^53 - 6 * q^55 + 16 * q^57 + 3 * q^59 - 4 * q^61 - 12 * q^65 + q^67 + 6 * q^69 + 30 * q^71 - 4 * q^73 + 4 * q^75 - 2 * q^79 - q^81 - 12 * q^83 - 36 * q^85 - 9 * q^89 - 5 * q^93 + 24 * q^95 + 14 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1079$$ $$1277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 0 0 0 1.00000 1.73205i 0
1145.1 0 0.500000 0.866025i 0 −1.50000 2.59808i 0 0 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.2.i.c 2
7.b odd 2 1 2156.2.i.b 2
7.c even 3 1 2156.2.a.a 1
7.c even 3 1 inner 2156.2.i.c 2
7.d odd 6 1 44.2.a.a 1
7.d odd 6 1 2156.2.i.b 2
21.g even 6 1 396.2.a.c 1
28.f even 6 1 176.2.a.a 1
28.g odd 6 1 8624.2.a.w 1
35.i odd 6 1 1100.2.a.b 1
35.k even 12 2 1100.2.b.c 2
56.j odd 6 1 704.2.a.f 1
56.m even 6 1 704.2.a.i 1
63.i even 6 1 3564.2.i.a 2
63.k odd 6 1 3564.2.i.j 2
63.s even 6 1 3564.2.i.a 2
63.t odd 6 1 3564.2.i.j 2
77.i even 6 1 484.2.a.a 1
77.n even 30 4 484.2.e.b 4
77.p odd 30 4 484.2.e.a 4
84.j odd 6 1 1584.2.a.p 1
91.s odd 6 1 7436.2.a.d 1
105.p even 6 1 9900.2.a.h 1
105.w odd 12 2 9900.2.c.g 2
112.v even 12 2 2816.2.c.k 2
112.x odd 12 2 2816.2.c.e 2
140.s even 6 1 4400.2.a.v 1
140.x odd 12 2 4400.2.b.k 2
168.ba even 6 1 6336.2.a.j 1
168.be odd 6 1 6336.2.a.i 1
231.k odd 6 1 4356.2.a.j 1
308.m odd 6 1 1936.2.a.c 1
616.s even 6 1 7744.2.a.m 1
616.z odd 6 1 7744.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 7.d odd 6 1
176.2.a.a 1 28.f even 6 1
396.2.a.c 1 21.g even 6 1
484.2.a.a 1 77.i even 6 1
484.2.e.a 4 77.p odd 30 4
484.2.e.b 4 77.n even 30 4
704.2.a.f 1 56.j odd 6 1
704.2.a.i 1 56.m even 6 1
1100.2.a.b 1 35.i odd 6 1
1100.2.b.c 2 35.k even 12 2
1584.2.a.p 1 84.j odd 6 1
1936.2.a.c 1 308.m odd 6 1
2156.2.a.a 1 7.c even 3 1
2156.2.i.b 2 7.b odd 2 1
2156.2.i.b 2 7.d odd 6 1
2156.2.i.c 2 1.a even 1 1 trivial
2156.2.i.c 2 7.c even 3 1 inner
2816.2.c.e 2 112.x odd 12 2
2816.2.c.k 2 112.v even 12 2
3564.2.i.a 2 63.i even 6 1
3564.2.i.a 2 63.s even 6 1
3564.2.i.j 2 63.k odd 6 1
3564.2.i.j 2 63.t odd 6 1
4356.2.a.j 1 231.k odd 6 1
4400.2.a.v 1 140.s even 6 1
4400.2.b.k 2 140.x odd 12 2
6336.2.a.i 1 168.be odd 6 1
6336.2.a.j 1 168.ba even 6 1
7436.2.a.d 1 91.s odd 6 1
7744.2.a.m 1 616.s even 6 1
7744.2.a.bc 1 616.z odd 6 1
8624.2.a.w 1 28.g odd 6 1
9900.2.a.h 1 105.p even 6 1
9900.2.c.g 2 105.w odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2156, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 8T + 64$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 5T + 25$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} - T + 1$$
$71$ $$(T - 15)^{2}$$
$73$ $$T^{2} + 4T + 16$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 9T + 81$$
$97$ $$(T - 7)^{2}$$