# Properties

 Label 1260.2.k.d Level $1260$ Weight $2$ Character orbit 1260.k Analytic conductor $10.061$ Analytic rank $0$ Dimension $2$ 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.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i + 2) q^{5} - i q^{7} +O(q^{10})$$ q + (-i + 2) * q^5 - i * q^7 $$q + ( - i + 2) q^{5} - i q^{7} + 4 q^{11} - 6 i q^{13} + 2 i q^{17} - 6 q^{19} - 2 i q^{23} + ( - 4 i + 3) q^{25} + 6 q^{29} - 2 q^{31} + ( - 2 i - 1) q^{35} + 4 i q^{37} - 8 q^{41} - 4 i q^{43} + 4 i q^{47} - q^{49} - 6 i q^{53} + ( - 4 i + 8) q^{55} + 4 q^{59} + 14 q^{61} + ( - 12 i - 6) q^{65} - 4 i q^{67} - 10 i q^{73} - 4 i q^{77} + 16 i q^{83} + (4 i + 2) q^{85} + 8 q^{89} - 6 q^{91} + (6 i - 12) q^{95} - 10 i q^{97} +O(q^{100})$$ q + (-i + 2) * q^5 - i * q^7 + 4 * q^11 - 6*i * q^13 + 2*i * q^17 - 6 * q^19 - 2*i * q^23 + (-4*i + 3) * q^25 + 6 * q^29 - 2 * q^31 + (-2*i - 1) * q^35 + 4*i * q^37 - 8 * q^41 - 4*i * q^43 + 4*i * q^47 - q^49 - 6*i * q^53 + (-4*i + 8) * q^55 + 4 * q^59 + 14 * q^61 + (-12*i - 6) * q^65 - 4*i * q^67 - 10*i * q^73 - 4*i * q^77 + 16*i * q^83 + (4*i + 2) * q^85 + 8 * q^89 - 6 * q^91 + (6*i - 12) * q^95 - 10*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} + 8 q^{11} - 12 q^{19} + 6 q^{25} + 12 q^{29} - 4 q^{31} - 2 q^{35} - 16 q^{41} - 2 q^{49} + 16 q^{55} + 8 q^{59} + 28 q^{61} - 12 q^{65} + 4 q^{85} + 16 q^{89} - 12 q^{91} - 24 q^{95}+O(q^{100})$$ 2 * q + 4 * q^5 + 8 * q^11 - 12 * q^19 + 6 * q^25 + 12 * q^29 - 4 * q^31 - 2 * q^35 - 16 * q^41 - 2 * q^49 + 16 * q^55 + 8 * q^59 + 28 * q^61 - 12 * q^65 + 4 * q^85 + 16 * q^89 - 12 * q^91 - 24 * q^95

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1
 1.00000i − 1.00000i
0 0 0 2.00000 1.00000i 0 1.00000i 0 0 0
1009.2 0 0 0 2.00000 + 1.00000i 0 1.00000i 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
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 1260.2.k.d 2
3.b odd 2 1 420.2.k.a 2
4.b odd 2 1 5040.2.t.o 2
5.b even 2 1 inner 1260.2.k.d 2
5.c odd 4 1 6300.2.a.n 1
5.c odd 4 1 6300.2.a.bc 1
12.b even 2 1 1680.2.t.a 2
15.d odd 2 1 420.2.k.a 2
15.e even 4 1 2100.2.a.e 1
15.e even 4 1 2100.2.a.j 1
20.d odd 2 1 5040.2.t.o 2
21.c even 2 1 2940.2.k.d 2
21.g even 6 2 2940.2.bb.c 4
21.h odd 6 2 2940.2.bb.h 4
60.h even 2 1 1680.2.t.a 2
60.l odd 4 1 8400.2.a.bh 1
60.l odd 4 1 8400.2.a.cd 1
105.g even 2 1 2940.2.k.d 2
105.o odd 6 2 2940.2.bb.h 4
105.p even 6 2 2940.2.bb.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.a 2 3.b odd 2 1
420.2.k.a 2 15.d odd 2 1
1260.2.k.d 2 1.a even 1 1 trivial
1260.2.k.d 2 5.b even 2 1 inner
1680.2.t.a 2 12.b even 2 1
1680.2.t.a 2 60.h even 2 1
2100.2.a.e 1 15.e even 4 1
2100.2.a.j 1 15.e even 4 1
2940.2.k.d 2 21.c even 2 1
2940.2.k.d 2 105.g even 2 1
2940.2.bb.c 4 21.g even 6 2
2940.2.bb.c 4 105.p even 6 2
2940.2.bb.h 4 21.h odd 6 2
2940.2.bb.h 4 105.o odd 6 2
5040.2.t.o 2 4.b odd 2 1
5040.2.t.o 2 20.d odd 2 1
6300.2.a.n 1 5.c odd 4 1
6300.2.a.bc 1 5.c odd 4 1
8400.2.a.bh 1 60.l odd 4 1
8400.2.a.cd 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 4$$ acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 100$$