# Properties

 Label 1260.2.c.c Level $1260$ Weight $2$ Character orbit 1260.c Analytic conductor $10.061$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{7} + ( -2 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{7} + ( -2 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{8} + \beta_{1} q^{10} + ( \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} -2 \beta_{2} q^{13} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{14} + ( -2 - 3 \beta_{4} + \beta_{6} + \beta_{7} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{6} + \beta_{7} ) q^{20} + ( 4 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{22} + ( \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{23} - q^{25} -2 \beta_{1} q^{26} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{28} + 2 q^{29} + ( 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{31} + ( 2 + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{32} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{34} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{35} + 2 q^{37} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{38} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{40} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{3} - 4 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 4 + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{44} + ( 6 + 3 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{46} + ( 6 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{47} + ( 4 + \beta_{1} + 5 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{49} -\beta_{4} q^{50} + ( 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{52} -2 q^{53} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{55} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{56} + 2 \beta_{4} q^{58} + ( -6 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} -2 \beta_{2} q^{61} + ( -4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{62} + ( 2 + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{64} + 2 q^{65} + ( -\beta_{3} - 8 \beta_{4} - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{67} + ( 8 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{68} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{70} + ( -3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{71} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{73} + 2 \beta_{4} q^{74} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{76} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{77} + ( -\beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{79} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{80} + ( 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{82} + ( -\beta_{6} + \beta_{7} ) q^{83} + ( 2 - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{85} + ( 2 + \beta_{3} + 4 \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{86} + ( 4 - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{88} + 12 \beta_{2} q^{89} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{91} + ( 4 + 2 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} ) q^{92} + ( -6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{94} + ( \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{95} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{97} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + 2q^{4} - 14q^{8} + O(q^{10})$$ $$8q - 2q^{2} + 2q^{4} - 14q^{8} + 6q^{14} - 14q^{16} + 28q^{22} - 8q^{25} + 14q^{28} + 16q^{29} + 18q^{32} + 16q^{37} + 20q^{44} + 44q^{46} + 36q^{49} + 2q^{50} - 16q^{53} - 2q^{56} - 4q^{58} + 2q^{64} + 16q^{65} + 4q^{70} - 4q^{74} - 24q^{77} + 24q^{85} + 20q^{86} + 36q^{88} + 12q^{92} - 26q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3}$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} - 2 \nu^{2} - 8$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 3 \nu^{5} - \nu^{4} + 2 \nu^{3} - 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{6} - 3 \nu^{5} - \nu^{4} - 2 \nu^{3} - 6 \nu^{2} - 8$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{6} - 3 \beta_{4} - 2$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{3} - 10 \beta_{2} - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
811.1
 −0.599676 + 1.28078i 0.599676 − 1.28078i 0.599676 + 1.28078i −0.599676 − 1.28078i −1.17915 − 0.780776i 1.17915 + 0.780776i 1.17915 − 0.780776i −1.17915 + 0.780776i
−1.28078 0.599676i 0 1.28078 + 1.53610i 1.00000i 0 2.60399 0.468213i −0.719224 2.73546i 0 −0.599676 + 1.28078i
811.2 −1.28078 0.599676i 0 1.28078 + 1.53610i 1.00000i 0 −2.60399 0.468213i −0.719224 2.73546i 0 0.599676 1.28078i
811.3 −1.28078 + 0.599676i 0 1.28078 1.53610i 1.00000i 0 −2.60399 + 0.468213i −0.719224 + 2.73546i 0 0.599676 + 1.28078i
811.4 −1.28078 + 0.599676i 0 1.28078 1.53610i 1.00000i 0 2.60399 + 0.468213i −0.719224 + 2.73546i 0 −0.599676 1.28078i
811.5 0.780776 1.17915i 0 −0.780776 1.84130i 1.00000i 0 −2.17238 + 1.51022i −2.78078 0.516994i 0 −1.17915 0.780776i
811.6 0.780776 1.17915i 0 −0.780776 1.84130i 1.00000i 0 2.17238 + 1.51022i −2.78078 0.516994i 0 1.17915 + 0.780776i
811.7 0.780776 + 1.17915i 0 −0.780776 + 1.84130i 1.00000i 0 2.17238 1.51022i −2.78078 + 0.516994i 0 1.17915 0.780776i
811.8 0.780776 + 1.17915i 0 −0.780776 + 1.84130i 1.00000i 0 −2.17238 1.51022i −2.78078 + 0.516994i 0 −1.17915 + 0.780776i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.c.c 8
3.b odd 2 1 140.2.g.c 8
4.b odd 2 1 inner 1260.2.c.c 8
7.b odd 2 1 inner 1260.2.c.c 8
12.b even 2 1 140.2.g.c 8
15.d odd 2 1 700.2.g.j 8
15.e even 4 1 700.2.c.i 8
15.e even 4 1 700.2.c.j 8
21.c even 2 1 140.2.g.c 8
21.g even 6 2 980.2.o.e 16
21.h odd 6 2 980.2.o.e 16
24.f even 2 1 2240.2.k.e 8
24.h odd 2 1 2240.2.k.e 8
28.d even 2 1 inner 1260.2.c.c 8
60.h even 2 1 700.2.g.j 8
60.l odd 4 1 700.2.c.i 8
60.l odd 4 1 700.2.c.j 8
84.h odd 2 1 140.2.g.c 8
84.j odd 6 2 980.2.o.e 16
84.n even 6 2 980.2.o.e 16
105.g even 2 1 700.2.g.j 8
105.k odd 4 1 700.2.c.i 8
105.k odd 4 1 700.2.c.j 8
168.e odd 2 1 2240.2.k.e 8
168.i even 2 1 2240.2.k.e 8
420.o odd 2 1 700.2.g.j 8
420.w even 4 1 700.2.c.i 8
420.w even 4 1 700.2.c.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 3.b odd 2 1
140.2.g.c 8 12.b even 2 1
140.2.g.c 8 21.c even 2 1
140.2.g.c 8 84.h odd 2 1
700.2.c.i 8 15.e even 4 1
700.2.c.i 8 60.l odd 4 1
700.2.c.i 8 105.k odd 4 1
700.2.c.i 8 420.w even 4 1
700.2.c.j 8 15.e even 4 1
700.2.c.j 8 60.l odd 4 1
700.2.c.j 8 105.k odd 4 1
700.2.c.j 8 420.w even 4 1
700.2.g.j 8 15.d odd 2 1
700.2.g.j 8 60.h even 2 1
700.2.g.j 8 105.g even 2 1
700.2.g.j 8 420.o odd 2 1
980.2.o.e 16 21.g even 6 2
980.2.o.e 16 21.h odd 6 2
980.2.o.e 16 84.j odd 6 2
980.2.o.e 16 84.n even 6 2
1260.2.c.c 8 1.a even 1 1 trivial
1260.2.c.c 8 4.b odd 2 1 inner
1260.2.c.c 8 7.b odd 2 1 inner
1260.2.c.c 8 28.d even 2 1 inner
2240.2.k.e 8 24.f even 2 1
2240.2.k.e 8 24.h odd 2 1
2240.2.k.e 8 168.e odd 2 1
2240.2.k.e 8 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{4} + 28 T_{11}^{2} + 128$$ $$T_{19}^{4} - 28 T_{19}^{2} + 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$2401 - 882 T^{2} + 162 T^{4} - 18 T^{6} + T^{8}$$
$11$ $$( 128 + 28 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + T^{2} )^{4}$$
$17$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 128 - 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 1352 + 74 T^{2} + T^{4} )^{2}$$
$29$ $$( -2 + T )^{8}$$
$31$ $$( 512 - 56 T^{2} + T^{4} )^{2}$$
$37$ $$( -2 + T )^{8}$$
$41$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$43$ $$( 8 + 58 T^{2} + T^{4} )^{2}$$
$47$ $$( 2592 - 126 T^{2} + T^{4} )^{2}$$
$53$ $$( 2 + T )^{8}$$
$59$ $$( 512 - 124 T^{2} + T^{4} )^{2}$$
$61$ $$( 4 + T^{2} )^{4}$$
$67$ $$( 2888 + 218 T^{2} + T^{4} )^{2}$$
$71$ $$( 2048 + 92 T^{2} + T^{4} )^{2}$$
$73$ $$( 20736 + 324 T^{2} + T^{4} )^{2}$$
$79$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$83$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$89$ $$( 144 + T^{2} )^{4}$$
$97$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$