# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{2} -\zeta_{8}^{2} q^{3} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + q^{6} -2 \zeta_{8}^{2} q^{7} -\zeta_{8}^{2} q^{8} - q^{9} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{2} -\zeta_{8}^{2} q^{3} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + q^{6} -2 \zeta_{8}^{2} q^{7} -\zeta_{8}^{2} q^{8} - q^{9} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{10} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{11} + \zeta_{8}^{2} q^{12} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} + 2 q^{14} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{15} + q^{16} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} -\zeta_{8}^{2} q^{18} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{20} -2 q^{21} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{22} + \zeta_{8}^{2} q^{23} - q^{24} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + \zeta_{8}^{2} q^{27} + 2 \zeta_{8}^{2} q^{28} + ( 2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{30} + 6 q^{31} + \zeta_{8}^{2} q^{32} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{33} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{34} + ( 2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{35} + q^{36} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{37} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{38} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{39} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{40} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} -2 \zeta_{8}^{2} q^{42} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{43} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{44} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{45} - q^{46} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} -\zeta_{8}^{2} q^{48} + 3 q^{49} + ( -3 - 4 \zeta_{8}^{2} ) q^{50} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{52} + ( -\zeta_{8} - 12 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{53} - q^{54} + ( 3 + 9 \zeta_{8}^{2} ) q^{55} -2 q^{56} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{57} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{58} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{59} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{60} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{61} + 6 \zeta_{8}^{2} q^{62} + 2 \zeta_{8}^{2} q^{63} - q^{64} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{65} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{66} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{67} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{68} + q^{69} + ( 4 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{70} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + \zeta_{8}^{2} q^{72} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{73} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{74} + ( 3 + 4 \zeta_{8}^{2} ) q^{75} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{77} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{78} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{79} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} + q^{81} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{82} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{83} + 2 q^{84} + ( 6 - 4 \zeta_{8} - 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{85} + ( 6 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{86} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{87} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{88} + ( -12 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{89} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{90} + ( -4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} -\zeta_{8}^{2} q^{92} -6 \zeta_{8}^{2} q^{93} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{94} + ( 3 - 4 \zeta_{8} + 9 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{95} + q^{96} -6 \zeta_{8}^{2} q^{97} + 3 \zeta_{8}^{2} q^{98} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + 8q^{14} + 4q^{16} - 8q^{19} - 8q^{21} - 4q^{24} - 16q^{25} + 8q^{26} + 8q^{29} + 24q^{31} - 16q^{34} + 4q^{36} - 8q^{39} - 16q^{41} - 4q^{46} + 12q^{49} - 12q^{50} + 16q^{51} - 4q^{54} + 12q^{55} - 8q^{56} - 8q^{61} - 4q^{64} + 24q^{65} + 4q^{69} - 8q^{74} + 12q^{75} + 8q^{76} - 8q^{79} + 4q^{81} + 8q^{84} + 24q^{85} + 24q^{86} - 48q^{89} - 16q^{91} - 8q^{94} + 12q^{95} + 4q^{96} + O(q^{100})$$

## Character values

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

 $$n$$ $$277$$ $$461$$ $$511$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i 1.00000i −1.00000 −0.707107 + 2.12132i 1.00000 2.00000i 1.00000i −1.00000 2.12132 + 0.707107i
139.2 1.00000i 1.00000i −1.00000 0.707107 2.12132i 1.00000 2.00000i 1.00000i −1.00000 −2.12132 0.707107i
139.3 1.00000i 1.00000i −1.00000 −0.707107 2.12132i 1.00000 2.00000i 1.00000i −1.00000 2.12132 0.707107i
139.4 1.00000i 1.00000i −1.00000 0.707107 + 2.12132i 1.00000 2.00000i 1.00000i −1.00000 −2.12132 + 0.707107i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.d.b 4
3.b odd 2 1 2070.2.d.a 4
5.b even 2 1 inner 690.2.d.b 4
5.c odd 4 1 3450.2.a.bg 2
5.c odd 4 1 3450.2.a.bk 2
15.d odd 2 1 2070.2.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.b 4 1.a even 1 1 trivial
690.2.d.b 4 5.b even 2 1 inner
2070.2.d.a 4 3.b odd 2 1
2070.2.d.a 4 15.d odd 2 1
3450.2.a.bg 2 5.c odd 4 1
3450.2.a.bk 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$25 + 8 T^{2} + T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( -18 + T^{2} )^{2}$$
$13$ $$16 + 24 T^{2} + T^{4}$$
$17$ $$64 + 48 T^{2} + T^{4}$$
$19$ $$( -14 + 4 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -28 - 4 T + T^{2} )^{2}$$
$31$ $$( -6 + T )^{4}$$
$37$ $$4 + 12 T^{2} + T^{4}$$
$41$ $$( 8 + 8 T + T^{2} )^{2}$$
$43$ $$324 + 108 T^{2} + T^{4}$$
$47$ $$16 + 24 T^{2} + T^{4}$$
$53$ $$20164 + 292 T^{2} + T^{4}$$
$59$ $$( -72 + T^{2} )^{2}$$
$61$ $$( 2 + 4 T + T^{2} )^{2}$$
$67$ $$4 + 12 T^{2} + T^{4}$$
$71$ $$( -32 + T^{2} )^{2}$$
$73$ $$784 + 72 T^{2} + T^{4}$$
$79$ $$( -28 + 4 T + T^{2} )^{2}$$
$83$ $$( 2 + T^{2} )^{2}$$
$89$ $$( 136 + 24 T + T^{2} )^{2}$$
$97$ $$( 36 + T^{2} )^{2}$$