# Properties

 Label 1260.2.d.a Level $1260$ Weight $2$ Character orbit 1260.d Analytic conductor $10.061$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Defining polynomial: $$x^{4} + 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q - q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -\beta_{1} q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} -2 \beta_{3} q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} -\beta_{2} q^{23} + q^{25} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -\beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{3} ) q^{35} + ( 6 - 2 \beta_{3} ) q^{37} + ( 4 - 2 \beta_{3} ) q^{41} + ( 4 - 2 \beta_{3} ) q^{43} + ( 2 - 2 \beta_{3} ) q^{47} + ( 3 - 2 \beta_{2} ) q^{49} + ( -4 \beta_{1} - \beta_{2} ) q^{53} + \beta_{1} q^{55} + ( -6 + 2 \beta_{3} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{65} + ( -\beta_{1} - 2 \beta_{2} ) q^{71} + ( -5 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -2 - \beta_{2} ) q^{77} + 4 \beta_{3} q^{79} + ( 10 - 2 \beta_{3} ) q^{83} + 2 \beta_{3} q^{85} + ( -2 - 4 \beta_{3} ) q^{89} + ( 2 - 5 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} + \beta_{2} ) q^{95} + ( 7 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + O(q^{10})$$ $$4q - 4q^{5} + 4q^{25} + 24q^{37} + 16q^{41} + 16q^{43} + 8q^{47} + 12q^{49} - 24q^{59} - 8q^{77} + 40q^{83} - 8q^{89} + 8q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} + 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 − 2.28825i 2.28825i 0.874032i − 0.874032i
0 0 0 −1.00000 0 −2.23607 1.41421i 0 0 0
881.2 0 0 0 −1.00000 0 −2.23607 + 1.41421i 0 0 0
881.3 0 0 0 −1.00000 0 2.23607 1.41421i 0 0 0
881.4 0 0 0 −1.00000 0 2.23607 + 1.41421i 0 0 0
 $$n$$: 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
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.d.a 4
3.b odd 2 1 1260.2.d.b yes 4
4.b odd 2 1 5040.2.f.b 4
5.b even 2 1 6300.2.d.b 4
5.c odd 4 2 6300.2.f.c 8
7.b odd 2 1 1260.2.d.b yes 4
12.b even 2 1 5040.2.f.d 4
15.d odd 2 1 6300.2.d.a 4
15.e even 4 2 6300.2.f.a 8
21.c even 2 1 inner 1260.2.d.a 4
28.d even 2 1 5040.2.f.d 4
35.c odd 2 1 6300.2.d.a 4
35.f even 4 2 6300.2.f.a 8
84.h odd 2 1 5040.2.f.b 4
105.g even 2 1 6300.2.d.b 4
105.k odd 4 2 6300.2.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.d.a 4 1.a even 1 1 trivial
1260.2.d.a 4 21.c even 2 1 inner
1260.2.d.b yes 4 3.b odd 2 1
1260.2.d.b yes 4 7.b odd 2 1
5040.2.f.b 4 4.b odd 2 1
5040.2.f.b 4 84.h odd 2 1
5040.2.f.d 4 12.b even 2 1
5040.2.f.d 4 28.d even 2 1
6300.2.d.a 4 15.d odd 2 1
6300.2.d.a 4 35.c odd 2 1
6300.2.d.b 4 5.b even 2 1
6300.2.d.b 4 105.g even 2 1
6300.2.f.a 8 15.e even 4 2
6300.2.f.a 8 35.f even 4 2
6300.2.f.c 8 5.c odd 4 2
6300.2.f.c 8 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{41}^{2} - 8 T_{41} - 4$$ acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$49 - 6 T^{2} + T^{4}$$
$11$ $$( 2 + T^{2} )^{2}$$
$13$ $$64 + 24 T^{2} + T^{4}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$64 + 24 T^{2} + T^{4}$$
$23$ $$( 10 + T^{2} )^{2}$$
$29$ $$484 + 116 T^{2} + T^{4}$$
$31$ $$64 + 24 T^{2} + T^{4}$$
$37$ $$( 16 - 12 T + T^{2} )^{2}$$
$41$ $$( -4 - 8 T + T^{2} )^{2}$$
$43$ $$( -4 - 8 T + T^{2} )^{2}$$
$47$ $$( -16 - 4 T + T^{2} )^{2}$$
$53$ $$484 + 84 T^{2} + T^{4}$$
$59$ $$( 16 + 12 T + T^{2} )^{2}$$
$61$ $$1024 + 96 T^{2} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$1444 + 84 T^{2} + T^{4}$$
$73$ $$1600 + 280 T^{2} + T^{4}$$
$79$ $$( -80 + T^{2} )^{2}$$
$83$ $$( 80 - 20 T + T^{2} )^{2}$$
$89$ $$( -76 + 4 T + T^{2} )^{2}$$
$97$ $$64 + 376 T^{2} + T^{4}$$