# Properties

 Label 616.2.q.a Level $616$ Weight $2$ Character orbit 616.q Analytic conductor $4.919$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$616 = 2^{3} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 616.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.91878476451$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - 2*z * q^5 + (z + 2) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} + 2 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + 3 q^{13} - 2 q^{15} + ( - 2 \zeta_{6} + 2) q^{17} - 4 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} + 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + 5 q^{27} - 7 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} - \zeta_{6} q^{33} + ( - 6 \zeta_{6} + 2) q^{35} - 12 \zeta_{6} q^{37} + ( - 3 \zeta_{6} + 3) q^{39} - 8 q^{41} + 8 q^{43} + ( - 4 \zeta_{6} + 4) q^{45} + 10 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} - 2 \zeta_{6} q^{51} + (14 \zeta_{6} - 14) q^{53} - 2 q^{55} - 4 q^{57} + ( - 9 \zeta_{6} + 9) q^{59} + 5 \zeta_{6} q^{61} + (6 \zeta_{6} - 2) q^{63} - 6 \zeta_{6} q^{65} + ( - 3 \zeta_{6} + 3) q^{67} + 4 q^{69} - 6 q^{71} + (4 \zeta_{6} - 4) q^{73} - \zeta_{6} q^{75} + ( - 2 \zeta_{6} + 3) q^{77} + 17 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 6 q^{83} - 4 q^{85} + (7 \zeta_{6} - 7) q^{87} + 2 \zeta_{6} q^{89} + (3 \zeta_{6} + 6) q^{91} - 8 \zeta_{6} q^{93} + (8 \zeta_{6} - 8) q^{95} + 7 q^{97} + 2 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 - 2*z * q^5 + (z + 2) * q^7 + 2*z * q^9 + (-z + 1) * q^11 + 3 * q^13 - 2 * q^15 + (-2*z + 2) * q^17 - 4*z * q^19 + (-2*z + 3) * q^21 + 4*z * q^23 + (-z + 1) * q^25 + 5 * q^27 - 7 * q^29 + (-8*z + 8) * q^31 - z * q^33 + (-6*z + 2) * q^35 - 12*z * q^37 + (-3*z + 3) * q^39 - 8 * q^41 + 8 * q^43 + (-4*z + 4) * q^45 + 10*z * q^47 + (5*z + 3) * q^49 - 2*z * q^51 + (14*z - 14) * q^53 - 2 * q^55 - 4 * q^57 + (-9*z + 9) * q^59 + 5*z * q^61 + (6*z - 2) * q^63 - 6*z * q^65 + (-3*z + 3) * q^67 + 4 * q^69 - 6 * q^71 + (4*z - 4) * q^73 - z * q^75 + (-2*z + 3) * q^77 + 17*z * q^79 + (z - 1) * q^81 - 6 * q^83 - 4 * q^85 + (7*z - 7) * q^87 + 2*z * q^89 + (3*z + 6) * q^91 - 8*z * q^93 + (8*z - 8) * q^95 + 7 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 $$2 q + q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + q^{11} + 6 q^{13} - 4 q^{15} + 2 q^{17} - 4 q^{19} + 4 q^{21} + 4 q^{23} + q^{25} + 10 q^{27} - 14 q^{29} + 8 q^{31} - q^{33} - 2 q^{35} - 12 q^{37} + 3 q^{39} - 16 q^{41} + 16 q^{43} + 4 q^{45} + 10 q^{47} + 11 q^{49} - 2 q^{51} - 14 q^{53} - 4 q^{55} - 8 q^{57} + 9 q^{59} + 5 q^{61} + 2 q^{63} - 6 q^{65} + 3 q^{67} + 8 q^{69} - 12 q^{71} - 4 q^{73} - q^{75} + 4 q^{77} + 17 q^{79} - q^{81} - 12 q^{83} - 8 q^{85} - 7 q^{87} + 2 q^{89} + 15 q^{91} - 8 q^{93} - 8 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + q^11 + 6 * q^13 - 4 * q^15 + 2 * q^17 - 4 * q^19 + 4 * q^21 + 4 * q^23 + q^25 + 10 * q^27 - 14 * q^29 + 8 * q^31 - q^33 - 2 * q^35 - 12 * q^37 + 3 * q^39 - 16 * q^41 + 16 * q^43 + 4 * q^45 + 10 * q^47 + 11 * q^49 - 2 * q^51 - 14 * q^53 - 4 * q^55 - 8 * q^57 + 9 * q^59 + 5 * q^61 + 2 * q^63 - 6 * q^65 + 3 * q^67 + 8 * q^69 - 12 * q^71 - 4 * q^73 - q^75 + 4 * q^77 + 17 * q^79 - q^81 - 12 * q^83 - 8 * q^85 - 7 * q^87 + 2 * q^89 + 15 * q^91 - 8 * q^93 - 8 * q^95 + 14 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/616\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$309$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 0 1.00000 1.73205i 0
529.1 0 0.500000 0.866025i 0 −1.00000 1.73205i 0 2.50000 + 0.866025i 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 616.2.q.a 2
4.b odd 2 1 1232.2.q.b 2
7.c even 3 1 inner 616.2.q.a 2
7.c even 3 1 4312.2.a.d 1
7.d odd 6 1 4312.2.a.g 1
28.f even 6 1 8624.2.a.i 1
28.g odd 6 1 1232.2.q.b 2
28.g odd 6 1 8624.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.a 2 1.a even 1 1 trivial
616.2.q.a 2 7.c even 3 1 inner
1232.2.q.b 2 4.b odd 2 1
1232.2.q.b 2 28.g odd 6 1
4312.2.a.d 1 7.c even 3 1
4312.2.a.g 1 7.d odd 6 1
8624.2.a.i 1 28.f even 6 1
8624.2.a.v 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(616, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T + 7)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} + 12T + 144$$
$41$ $$(T + 8)^{2}$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} - 10T + 100$$
$53$ $$T^{2} + 14T + 196$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 4T + 16$$
$79$ $$T^{2} - 17T + 289$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 2T + 4$$
$97$ $$(T - 7)^{2}$$