# Properties

 Label 690.2.i.a Level $690$ Weight $2$ Character orbit 690.i Analytic conductor $5.510$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + ( 1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{2} + ( 1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} + ( 1 + 2 \zeta_{8}^{2} ) q^{10} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( -1 - \zeta_{8} + \zeta_{8}^{2} ) q^{12} + ( 2 + 2 \zeta_{8}^{2} ) q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} + ( -2 + 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{15} - q^{16} -2 \zeta_{8} q^{17} + ( -2 - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{18} + 4 \zeta_{8}^{2} q^{19} + ( \zeta_{8} + 2 \zeta_{8}^{3} ) q^{20} + ( -2 - \zeta_{8} - \zeta_{8}^{3} ) q^{21} + ( 1 - \zeta_{8}^{2} ) q^{22} -\zeta_{8}^{3} q^{23} + ( -\zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{24} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{26} + ( 1 - 5 \zeta_{8} - \zeta_{8}^{2} ) q^{27} + ( -1 - \zeta_{8}^{2} ) q^{28} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + ( -1 - 2 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{30} + 2 q^{31} -\zeta_{8} q^{32} + ( 1 + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{33} -2 \zeta_{8}^{2} q^{34} + ( -\zeta_{8} + 3 \zeta_{8}^{3} ) q^{35} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{36} + ( 6 - 6 \zeta_{8}^{2} ) q^{37} + 4 \zeta_{8}^{3} q^{38} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{39} + ( -2 + \zeta_{8}^{2} ) q^{40} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{41} + ( 1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{42} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{44} + ( -6 + \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{45} + q^{46} + 4 \zeta_{8} q^{47} + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{48} + 5 \zeta_{8}^{2} q^{49} + ( 4 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{50} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + ( -2 + 2 \zeta_{8}^{2} ) q^{52} -12 \zeta_{8}^{3} q^{53} + ( \zeta_{8} - 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{54} + ( 1 - 3 \zeta_{8}^{2} ) q^{55} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{56} + ( -4 - 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{57} + ( -3 - 3 \zeta_{8}^{2} ) q^{58} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -1 - \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{60} -2 q^{61} + 2 \zeta_{8} q^{62} + ( -1 - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} -\zeta_{8}^{2} q^{64} + ( 6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{65} + ( 2 + \zeta_{8} + \zeta_{8}^{3} ) q^{66} + ( -2 + 2 \zeta_{8}^{2} ) q^{67} -2 \zeta_{8}^{3} q^{68} + ( \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{69} + ( -3 - \zeta_{8}^{2} ) q^{70} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( 2 - \zeta_{8} - 2 \zeta_{8}^{2} ) q^{72} + ( -1 - \zeta_{8}^{2} ) q^{73} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} + ( 1 - 3 \zeta_{8} + 7 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{75} -4 q^{76} + 2 \zeta_{8} q^{77} + ( -2 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{78} + 2 \zeta_{8}^{2} q^{79} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( 4 - 4 \zeta_{8}^{2} ) q^{82} + 12 \zeta_{8}^{3} q^{83} + ( \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{84} + ( -2 - 4 \zeta_{8}^{2} ) q^{85} + ( 3 - 6 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{87} + ( 1 + \zeta_{8}^{2} ) q^{88} + ( -2 - 6 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{90} -4 q^{91} + \zeta_{8} q^{92} + ( 2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{93} + 4 \zeta_{8}^{2} q^{94} + ( 4 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{96} + ( 9 - 9 \zeta_{8}^{2} ) q^{97} + 5 \zeta_{8}^{3} q^{98} + ( \zeta_{8} + 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 4q^{6} - 4q^{7} + O(q^{10})$$ $$4q + 4q^{3} - 4q^{6} - 4q^{7} + 4q^{10} - 4q^{12} + 8q^{13} - 8q^{15} - 4q^{16} - 8q^{18} - 8q^{21} + 4q^{22} + 16q^{25} + 4q^{27} - 4q^{28} - 4q^{30} + 8q^{31} + 4q^{33} - 4q^{36} + 24q^{37} - 8q^{40} + 4q^{42} - 24q^{45} + 4q^{46} - 4q^{48} + 8q^{51} - 8q^{52} + 4q^{55} - 16q^{57} - 12q^{58} - 4q^{60} - 8q^{61} - 4q^{63} + 8q^{66} - 8q^{67} - 12q^{70} + 8q^{72} - 4q^{73} + 4q^{75} - 16q^{76} - 8q^{78} + 28q^{81} + 16q^{82} - 8q^{85} + 12q^{87} + 4q^{88} - 8q^{90} - 16q^{91} + 8q^{93} + 4q^{96} + 36q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$461$$ $$511$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i 1.70711 0.292893i 1.00000i −2.12132 + 0.707107i −1.00000 + 1.41421i −1.00000 1.00000i 0.707107 + 0.707107i 2.82843 1.00000i 1.00000 2.00000i
47.2 0.707107 0.707107i 0.292893 1.70711i 1.00000i 2.12132 0.707107i −1.00000 1.41421i −1.00000 1.00000i −0.707107 0.707107i −2.82843 1.00000i 1.00000 2.00000i
323.1 −0.707107 0.707107i 1.70711 + 0.292893i 1.00000i −2.12132 0.707107i −1.00000 1.41421i −1.00000 + 1.00000i 0.707107 0.707107i 2.82843 + 1.00000i 1.00000 + 2.00000i
323.2 0.707107 + 0.707107i 0.292893 + 1.70711i 1.00000i 2.12132 + 0.707107i −1.00000 + 1.41421i −1.00000 + 1.00000i −0.707107 + 0.707107i −2.82843 + 1.00000i 1.00000 + 2.00000i
 $$n$$: 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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.i.a 4
3.b odd 2 1 inner 690.2.i.a 4
5.c odd 4 1 inner 690.2.i.a 4
15.e even 4 1 inner 690.2.i.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.i.a 4 1.a even 1 1 trivial
690.2.i.a 4 3.b odd 2 1 inner
690.2.i.a 4 5.c odd 4 1 inner
690.2.i.a 4 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 2 T_{7} + 2$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$( 2 + 2 T + T^{2} )^{2}$$
$11$ $$( 2 + T^{2} )^{2}$$
$13$ $$( 8 - 4 T + T^{2} )^{2}$$
$17$ $$16 + T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$1 + T^{4}$$
$29$ $$( -18 + T^{2} )^{2}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( 72 - 12 T + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$256 + T^{4}$$
$53$ $$20736 + T^{4}$$
$59$ $$( -2 + T^{2} )^{2}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$( 8 + 4 T + T^{2} )^{2}$$
$71$ $$( 32 + T^{2} )^{2}$$
$73$ $$( 2 + 2 T + T^{2} )^{2}$$
$79$ $$( 4 + T^{2} )^{2}$$
$83$ $$20736 + T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 162 - 18 T + T^{2} )^{2}$$