# Properties

 Label 690.2.d.d Level $690$ Weight $2$ Character orbit 690.d Analytic conductor $5.510$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{3} q^{3} - q^{4} -\beta_{4} q^{5} - q^{6} + 2 \beta_{3} q^{7} + \beta_{3} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{3} q^{3} - q^{4} -\beta_{4} q^{5} - q^{6} + 2 \beta_{3} q^{7} + \beta_{3} q^{8} - q^{9} -\beta_{1} q^{10} + ( \beta_{4} + \beta_{5} ) q^{11} + \beta_{3} q^{12} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{13} + 2 q^{14} -\beta_{1} q^{15} + q^{16} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{17} + \beta_{3} q^{18} + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{19} + \beta_{4} q^{20} + 2 q^{21} + ( \beta_{1} + \beta_{2} ) q^{22} -\beta_{3} q^{23} + q^{24} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{25} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{26} + \beta_{3} q^{27} -2 \beta_{3} q^{28} + 2 q^{29} + \beta_{4} q^{30} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{31} -\beta_{3} q^{32} + ( \beta_{1} + \beta_{2} ) q^{33} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{34} + 2 \beta_{1} q^{35} + q^{36} + ( -\beta_{1} - \beta_{2} + 6 \beta_{3} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{38} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{39} + \beta_{1} q^{40} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{41} -2 \beta_{3} q^{42} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{43} + ( -\beta_{4} - \beta_{5} ) q^{44} + \beta_{4} q^{45} - q^{46} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} -\beta_{3} q^{48} + 3 q^{49} + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{50} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{51} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{53} + q^{54} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{55} -2 q^{56} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{57} -2 \beta_{3} q^{58} + ( 4 + \beta_{1} - \beta_{2} - 3 \beta_{4} - 3 \beta_{5} ) q^{59} + \beta_{1} q^{60} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{62} -2 \beta_{3} q^{63} - q^{64} + ( -6 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{65} + ( -\beta_{4} - \beta_{5} ) q^{66} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{68} - q^{69} -2 \beta_{4} q^{70} + ( -4 - \beta_{1} + \beta_{2} - 3 \beta_{4} - 3 \beta_{5} ) q^{71} -\beta_{3} q^{72} + ( -\beta_{1} - \beta_{2} - 10 \beta_{3} - \beta_{4} + \beta_{5} ) q^{73} + ( 6 + \beta_{4} + \beta_{5} ) q^{74} + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{75} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{76} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{78} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{4} + 3 \beta_{5} ) q^{79} -\beta_{4} q^{80} + q^{81} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{82} + ( -5 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{83} -2 q^{84} + ( -6 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{85} + ( -2 - \beta_{4} - \beta_{5} ) q^{86} -2 \beta_{3} q^{87} + ( -\beta_{1} - \beta_{2} ) q^{88} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{89} + \beta_{1} q^{90} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{91} + \beta_{3} q^{92} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{93} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{94} + ( -7 - 2 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{95} - q^{96} + ( 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} -3 \beta_{3} q^{98} + ( -\beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} - 6q^{6} - 6q^{9} + O(q^{10})$$ $$6q - 6q^{4} - 6q^{6} - 6q^{9} + 2q^{10} + 12q^{14} + 2q^{15} + 6q^{16} - 8q^{19} + 12q^{21} + 6q^{24} - 10q^{25} - 8q^{26} + 12q^{29} + 16q^{31} + 4q^{34} - 4q^{35} + 6q^{36} - 8q^{39} - 2q^{40} - 4q^{41} - 6q^{46} + 18q^{49} - 4q^{50} + 4q^{51} + 6q^{54} - 20q^{55} - 12q^{56} + 20q^{59} - 2q^{60} + 20q^{61} - 6q^{64} - 32q^{65} - 6q^{69} - 20q^{71} + 36q^{74} - 4q^{75} + 8q^{76} + 8q^{79} + 6q^{81} - 12q^{84} - 36q^{85} - 12q^{86} + 32q^{89} - 2q^{90} + 16q^{91} + 8q^{94} - 44q^{95} - 6q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{5} + 25 \nu^{4} + 10 \nu^{3} - 4 \nu^{2} - 121 \nu + 323$$$$)/121$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{5} - 27 \nu^{4} - 35 \nu^{3} + 14 \nu^{2} - 121 \nu - 223$$$$)/121$$ $$\beta_{3}$$ $$=$$ $$($$$$-25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258$$$$)/242$$ $$\beta_{4}$$ $$=$$ $$($$$$-65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1573 \nu + 574$$$$)/242$$ $$\beta_{5}$$ $$=$$ $$($$$$75 \nu^{5} + 30 \nu^{4} + 12 \nu^{3} - 392 \nu^{2} + 1573 \nu - 774$$$$)/242$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} - \beta_{4} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 5 \beta_{1} + 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{5} - 9 \beta_{4} - 5 \beta_{2} + 5 \beta_{1} - 30$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$25 \beta_{5} + 29 \beta_{4} - 32 \beta_{3} + 29 \beta_{2} + 25 \beta_{1} + 32$$$$)/2$$

## 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
 −1.75233 + 1.75233i 0.432320 − 0.432320i 1.32001 − 1.32001i −1.75233 − 1.75233i 0.432320 + 0.432320i 1.32001 + 1.32001i
1.00000i 1.00000i −1.00000 −1.75233 + 1.38900i −1.00000 2.00000i 1.00000i −1.00000 1.38900 + 1.75233i
139.2 1.00000i 1.00000i −1.00000 0.432320 2.19388i −1.00000 2.00000i 1.00000i −1.00000 −2.19388 0.432320i
139.3 1.00000i 1.00000i −1.00000 1.32001 + 1.80487i −1.00000 2.00000i 1.00000i −1.00000 1.80487 1.32001i
139.4 1.00000i 1.00000i −1.00000 −1.75233 1.38900i −1.00000 2.00000i 1.00000i −1.00000 1.38900 1.75233i
139.5 1.00000i 1.00000i −1.00000 0.432320 + 2.19388i −1.00000 2.00000i 1.00000i −1.00000 −2.19388 + 0.432320i
139.6 1.00000i 1.00000i −1.00000 1.32001 1.80487i −1.00000 2.00000i 1.00000i −1.00000 1.80487 + 1.32001i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.6 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.d 6
3.b odd 2 1 2070.2.d.d 6
5.b even 2 1 inner 690.2.d.d 6
5.c odd 4 1 3450.2.a.bq 3
5.c odd 4 1 3450.2.a.br 3
15.d odd 2 1 2070.2.d.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.d 6 1.a even 1 1 trivial
690.2.d.d 6 5.b even 2 1 inner
2070.2.d.d 6 3.b odd 2 1
2070.2.d.d 6 15.d odd 2 1
3450.2.a.bq 3 5.c odd 4 1
3450.2.a.br 3 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 $$S_{2}^{\mathrm{new}}(690, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11}^{3} - 10 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$125 + 25 T^{2} + 8 T^{3} + 5 T^{4} + T^{6}$$
$7$ $$( 4 + T^{2} )^{3}$$
$11$ $$( -8 - 10 T + T^{3} )^{2}$$
$13$ $$4096 + 912 T^{2} + 56 T^{4} + T^{6}$$
$17$ $$256 + 512 T^{2} + 52 T^{4} + T^{6}$$
$19$ $$( -232 - 62 T + 4 T^{2} + T^{3} )^{2}$$
$23$ $$( 1 + T^{2} )^{3}$$
$29$ $$( -2 + T )^{6}$$
$31$ $$( 80 - 12 T - 8 T^{2} + T^{3} )^{2}$$
$37$ $$26896 + 3700 T^{2} + 128 T^{4} + T^{6}$$
$41$ $$( 16 - 24 T + 2 T^{2} + T^{3} )^{2}$$
$43$ $$16 + 52 T^{2} + 32 T^{4} + T^{6}$$
$47$ $$4096 + 912 T^{2} + 56 T^{4} + T^{6}$$
$53$ $$16 + 308 T^{2} + 168 T^{4} + T^{6}$$
$59$ $$( 848 - 88 T - 10 T^{2} + T^{3} )^{2}$$
$61$ $$( 284 - 46 T - 10 T^{2} + T^{3} )^{2}$$
$67$ $$80656 + 7796 T^{2} + 192 T^{4} + T^{6}$$
$71$ $$( -512 - 64 T + 10 T^{2} + T^{3} )^{2}$$
$73$ $$262144 + 23312 T^{2} + 328 T^{4} + T^{6}$$
$79$ $$( 352 - 92 T - 4 T^{2} + T^{3} )^{2}$$
$83$ $$3748096 + 87268 T^{2} + 548 T^{4} + T^{6}$$
$89$ $$( 512 + 8 T - 16 T^{2} + T^{3} )^{2}$$
$97$ $$7744 + 7472 T^{2} + 364 T^{4} + T^{6}$$