Properties

 Label 460.2.g.a Level $460$ Weight $2$ Character orbit 460.g Analytic conductor $3.673$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.g (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{2} q^{3} + 2 q^{4} + \beta_{3} q^{5} + 2 q^{6} + ( -2 \beta_{1} - \beta_{2} ) q^{7} -2 \beta_{2} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{2} q^{3} + 2 q^{4} + \beta_{3} q^{5} + 2 q^{6} + ( -2 \beta_{1} - \beta_{2} ) q^{7} -2 \beta_{2} q^{8} - q^{9} + ( 2 \beta_{1} + \beta_{2} ) q^{10} -2 \beta_{2} q^{12} -2 \beta_{3} q^{14} + ( 2 \beta_{1} + \beta_{2} ) q^{15} + 4 q^{16} + \beta_{2} q^{18} + 2 \beta_{3} q^{20} -2 \beta_{3} q^{21} + ( 3 \beta_{1} + \beta_{2} ) q^{23} + 4 q^{24} -5 q^{25} + 4 \beta_{2} q^{27} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{28} -6 q^{29} + 2 \beta_{3} q^{30} -4 \beta_{2} q^{32} -5 \beta_{2} q^{35} -2 q^{36} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{40} -12 q^{41} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{42} + ( -2 \beta_{1} - \beta_{2} ) q^{43} -\beta_{3} q^{45} + ( 1 + 3 \beta_{3} ) q^{46} -7 \beta_{2} q^{47} -4 \beta_{2} q^{48} -3 q^{49} + 5 \beta_{2} q^{50} -8 q^{54} -4 \beta_{3} q^{56} + 6 \beta_{2} q^{58} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{60} -6 \beta_{3} q^{61} + ( 2 \beta_{1} + \beta_{2} ) q^{63} + 8 q^{64} + ( -10 \beta_{1} - 5 \beta_{2} ) q^{67} + ( 1 + 3 \beta_{3} ) q^{69} + 10 q^{70} + 2 \beta_{2} q^{72} + 5 \beta_{2} q^{75} + 4 \beta_{3} q^{80} -5 q^{81} + 12 \beta_{2} q^{82} + ( 6 \beta_{1} + 3 \beta_{2} ) q^{83} -4 \beta_{3} q^{84} -2 \beta_{3} q^{86} + 6 \beta_{2} q^{87} + 8 \beta_{3} q^{89} + ( -2 \beta_{1} - \beta_{2} ) q^{90} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{92} + 14 q^{94} + 8 q^{96} + 3 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + 8q^{6} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{4} + 8q^{6} - 4q^{9} + 16q^{16} + 16q^{24} - 20q^{25} - 24q^{29} - 8q^{36} - 48q^{41} + 4q^{46} - 12q^{49} - 32q^{54} + 32q^{64} + 4q^{69} + 40q^{70} - 20q^{81} + 56q^{94} + 32q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 9$$:

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$277$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1
 −0.707107 + 1.58114i −0.707107 − 1.58114i 0.707107 − 1.58114i 0.707107 + 1.58114i
−1.41421 −1.41421 2.00000 2.23607i 2.00000 3.16228i −2.82843 −1.00000 3.16228i
459.2 −1.41421 −1.41421 2.00000 2.23607i 2.00000 3.16228i −2.82843 −1.00000 3.16228i
459.3 1.41421 1.41421 2.00000 2.23607i 2.00000 3.16228i 2.82843 −1.00000 3.16228i
459.4 1.41421 1.41421 2.00000 2.23607i 2.00000 3.16228i 2.82843 −1.00000 3.16228i
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c odd 2 1 inner
460.g even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.g.a 4
4.b odd 2 1 inner 460.2.g.a 4
5.b even 2 1 inner 460.2.g.a 4
20.d odd 2 1 CM 460.2.g.a 4
23.b odd 2 1 inner 460.2.g.a 4
92.b even 2 1 inner 460.2.g.a 4
115.c odd 2 1 inner 460.2.g.a 4
460.g even 2 1 inner 460.2.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.g.a 4 1.a even 1 1 trivial
460.2.g.a 4 4.b odd 2 1 inner
460.2.g.a 4 5.b even 2 1 inner
460.2.g.a 4 20.d odd 2 1 CM
460.2.g.a 4 23.b odd 2 1 inner
460.2.g.a 4 92.b even 2 1 inner
460.2.g.a 4 115.c odd 2 1 inner
460.2.g.a 4 460.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( 10 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$529 + 44 T^{2} + T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 12 + T )^{4}$$
$43$ $$( 10 + T^{2} )^{2}$$
$47$ $$( -98 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( 250 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 90 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$