# Properties

 Label 690.2.d.a 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} + \beta_{3} q^{5} + q^{6} -4 \beta_{1} q^{7} + \beta_{1} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} + \beta_{3} q^{5} + q^{6} -4 \beta_{1} q^{7} + \beta_{1} q^{8} - q^{9} -\beta_{2} q^{10} -\beta_{1} q^{12} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} -4 q^{14} + \beta_{2} q^{15} + q^{16} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{17} + \beta_{1} q^{18} -2 q^{19} -\beta_{3} q^{20} + 4 q^{21} -\beta_{1} q^{23} - q^{24} + 5 q^{25} + ( 2 - 2 \beta_{3} ) q^{26} -\beta_{1} q^{27} + 4 \beta_{1} q^{28} + ( -4 + 2 \beta_{3} ) q^{29} + \beta_{3} q^{30} -\beta_{1} q^{32} + ( 2 - 2 \beta_{3} ) q^{34} -4 \beta_{2} q^{35} + q^{36} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{37} + 2 \beta_{1} q^{38} + ( -2 + 2 \beta_{3} ) q^{39} + \beta_{2} q^{40} + ( 2 + 4 \beta_{3} ) q^{41} -4 \beta_{1} q^{42} -\beta_{3} q^{45} - q^{46} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{47} + \beta_{1} q^{48} -9 q^{49} -5 \beta_{1} q^{50} + ( -2 + 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{52} -4 \beta_{2} q^{53} - q^{54} + 4 q^{56} -2 \beta_{1} q^{57} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{58} + 6 q^{59} -\beta_{2} q^{60} + ( 4 - 2 \beta_{3} ) q^{61} + 4 \beta_{1} q^{63} - q^{64} + ( -10 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{68} + q^{69} -4 \beta_{3} q^{70} + ( -12 + 2 \beta_{3} ) q^{71} -\beta_{1} q^{72} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 4 - 2 \beta_{3} ) q^{74} + 5 \beta_{1} q^{75} + 2 q^{76} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -8 - 2 \beta_{3} ) q^{79} + \beta_{3} q^{80} + q^{81} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{82} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{83} -4 q^{84} + ( -10 \beta_{1} + 2 \beta_{2} ) q^{85} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 12 + 2 \beta_{3} ) q^{89} + \beta_{2} q^{90} + ( 8 - 8 \beta_{3} ) q^{91} + \beta_{1} q^{92} + ( 4 - 4 \beta_{3} ) q^{94} -2 \beta_{3} q^{95} + q^{96} + 6 \beta_{2} q^{97} + 9 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} - 16q^{14} + 4q^{16} - 8q^{19} + 16q^{21} - 4q^{24} + 20q^{25} + 8q^{26} - 16q^{29} + 8q^{34} + 4q^{36} - 8q^{39} + 8q^{41} - 4q^{46} - 36q^{49} - 8q^{51} - 4q^{54} + 16q^{56} + 24q^{59} + 16q^{61} - 4q^{64} + 4q^{69} - 48q^{71} + 16q^{74} + 8q^{76} - 32q^{79} + 4q^{81} - 16q^{84} + 48q^{89} + 32q^{91} + 16q^{94} + 4q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2 \beta_{1}$$

## 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
 − 1.61803i 0.618034i 1.61803i − 0.618034i
1.00000i 1.00000i −1.00000 −2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
139.2 1.00000i 1.00000i −1.00000 2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
139.3 1.00000i 1.00000i −1.00000 −2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
139.4 1.00000i 1.00000i −1.00000 2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
 $$n$$: 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.a 4
3.b odd 2 1 2070.2.d.b 4
5.b even 2 1 inner 690.2.d.a 4
5.c odd 4 1 3450.2.a.bc 2
5.c odd 4 1 3450.2.a.bn 2
15.d odd 2 1 2070.2.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.a 4 1.a even 1 1 trivial
690.2.d.a 4 5.b even 2 1 inner
2070.2.d.b 4 3.b odd 2 1
2070.2.d.b 4 15.d odd 2 1
3450.2.a.bc 2 5.c odd 4 1
3450.2.a.bn 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 $$S_{2}^{\mathrm{new}}(690, [\chi])$$:

 $$T_{7}^{2} + 16$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$256 + 48 T^{2} + T^{4}$$
$17$ $$256 + 48 T^{2} + T^{4}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -4 + 8 T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$16 + 72 T^{2} + T^{4}$$
$41$ $$( -76 - 4 T + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$4096 + 192 T^{2} + T^{4}$$
$53$ $$( 80 + T^{2} )^{2}$$
$59$ $$( -6 + T )^{4}$$
$61$ $$( -4 - 8 T + T^{2} )^{2}$$
$67$ $$4096 + 192 T^{2} + T^{4}$$
$71$ $$( 124 + 24 T + T^{2} )^{2}$$
$73$ $$4096 + 192 T^{2} + T^{4}$$
$79$ $$( 44 + 16 T + T^{2} )^{2}$$
$83$ $$256 + 112 T^{2} + T^{4}$$
$89$ $$( 124 - 24 T + T^{2} )^{2}$$
$97$ $$( 180 + T^{2} )^{2}$$