# Properties

 Label 688.2.i.d Level $688$ Weight $2$ Character orbit 688.i Analytic conductor $5.494$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 688.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.49370765906$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 3 \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 3 \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + 5 \zeta_{6} q^{13} -\zeta_{6} q^{15} -3 \zeta_{6} q^{17} + ( 1 - \zeta_{6} ) q^{19} + 3 q^{21} + ( -7 + 7 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + 5 q^{27} -3 \zeta_{6} q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + 3 q^{35} + ( 9 - 9 \zeta_{6} ) q^{37} + 5 q^{39} -10 q^{41} + ( 7 - 6 \zeta_{6} ) q^{43} + 2 q^{45} + 8 q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} -3 q^{51} + ( 5 - 5 \zeta_{6} ) q^{53} -\zeta_{6} q^{57} -12 q^{59} + 13 \zeta_{6} q^{61} + ( -6 + 6 \zeta_{6} ) q^{63} + 5 q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + 7 \zeta_{6} q^{69} -\zeta_{6} q^{71} -11 \zeta_{6} q^{73} + 4 q^{75} -5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + ( 9 - 9 \zeta_{6} ) q^{83} -3 q^{85} -3 q^{87} + ( 1 - \zeta_{6} ) q^{89} + ( -15 + 15 \zeta_{6} ) q^{91} -5 \zeta_{6} q^{93} -\zeta_{6} q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + 5 q^{13} - q^{15} - 3 q^{17} + q^{19} + 6 q^{21} - 7 q^{23} + 4 q^{25} + 10 q^{27} - 3 q^{29} + 5 q^{31} + 6 q^{35} + 9 q^{37} + 10 q^{39} - 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 2 q^{49} - 6 q^{51} + 5 q^{53} - q^{57} - 24 q^{59} + 13 q^{61} - 6 q^{63} + 10 q^{65} - 3 q^{67} + 7 q^{69} - q^{71} - 11 q^{73} + 8 q^{75} - 5 q^{79} - q^{81} + 9 q^{83} - 6 q^{85} - 6 q^{87} + q^{89} - 15 q^{91} - 5 q^{93} - q^{95} - 4 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$$.

 $$n$$ $$431$$ $$433$$ $$517$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 1.00000 + 1.73205i 0
337.1 0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.50000 2.59808i 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
43.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.i.d 2
4.b odd 2 1 43.2.c.a 2
12.b even 2 1 387.2.h.a 2
43.c even 3 1 inner 688.2.i.d 2
172.f even 6 1 1849.2.a.a 1
172.g odd 6 1 43.2.c.a 2
172.g odd 6 1 1849.2.a.c 1
516.p even 6 1 387.2.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.a 2 4.b odd 2 1
43.2.c.a 2 172.g odd 6 1
387.2.h.a 2 12.b even 2 1
387.2.h.a 2 516.p even 6 1
688.2.i.d 2 1.a even 1 1 trivial
688.2.i.d 2 43.c even 3 1 inner
1849.2.a.a 1 172.f even 6 1
1849.2.a.c 1 172.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(688, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{2} - T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 - 5 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$49 + 7 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$81 - 9 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$43 - 8 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$25 - 5 T + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$1 + T + T^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$1 - T + T^{2}$$
$97$ $$( 2 + T )^{2}$$