Properties

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

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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.

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

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 0.707107 2.12132i −1.00000 + 1.41421i 3.00000 + 3.00000i 0.707107 + 0.707107i 2.82843 1.00000i 1.00000 + 2.00000i
47.2 0.707107 0.707107i 0.292893 1.70711i 1.00000i −0.707107 + 2.12132i −1.00000 1.41421i 3.00000 + 3.00000i −0.707107 0.707107i −2.82843 1.00000i 1.00000 + 2.00000i
323.1 −0.707107 0.707107i 1.70711 + 0.292893i 1.00000i 0.707107 + 2.12132i −1.00000 1.41421i 3.00000 3.00000i 0.707107 0.707107i 2.82843 + 1.00000i 1.00000 2.00000i
323.2 0.707107 + 0.707107i 0.292893 + 1.70711i 1.00000i −0.707107 2.12132i −1.00000 + 1.41421i 3.00000 3.00000i −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.b 4
3.b odd 2 1 inner 690.2.i.b 4
5.c odd 4 1 inner 690.2.i.b 4
15.e even 4 1 inner 690.2.i.b 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 6 T_{7} + 18$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$25 + 8 T^{2} + T^{4}$$
$7$ $$( 18 - 6 T + T^{2} )^{2}$$
$11$ $$( 18 + T^{2} )^{2}$$
$13$ $$( 8 - 4 T + T^{2} )^{2}$$
$17$ $$1296 + T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$1 + T^{4}$$
$29$ $$( -50 + T^{2} )^{2}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( 72 - 12 T + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$20736 + T^{4}$$
$53$ $$256 + T^{4}$$
$59$ $$( -2 + T^{2} )^{2}$$
$61$ $$( 10 + T )^{4}$$
$67$ $$( 200 + 20 T + T^{2} )^{2}$$
$71$ $$( 32 + T^{2} )^{2}$$
$73$ $$( 2 + 2 T + T^{2} )^{2}$$
$79$ $$( 196 + T^{2} )^{2}$$
$83$ $$20736 + T^{4}$$
$89$ $$( -128 + T^{2} )^{2}$$
$97$ $$( 18 + 6 T + T^{2} )^{2}$$
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