# Properties

 Label 1470.2.g.j Level $1470$ Weight $2$ Character orbit 1470.g Analytic conductor $11.738$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1698758656.6 Defining polynomial: $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} -\beta_{5} q^{3} - q^{4} + ( -1 + \beta_{7} ) q^{5} - q^{6} + \beta_{5} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} -\beta_{5} q^{3} - q^{4} + ( -1 + \beta_{7} ) q^{5} - q^{6} + \beta_{5} q^{8} - q^{9} + \beta_{4} q^{10} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + \beta_{5} q^{12} + ( 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{13} + \beta_{4} q^{15} + q^{16} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + \beta_{5} q^{18} + ( 2 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} + ( 1 - \beta_{7} ) q^{20} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{22} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{23} + q^{24} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{25} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{26} + \beta_{5} q^{27} + ( 4 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 1 - \beta_{7} ) q^{30} + ( -4 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{31} -\beta_{5} q^{32} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{33} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + q^{36} + ( 3 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{38} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{39} -\beta_{4} q^{40} + ( -2 - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{44} + ( 1 - \beta_{7} ) q^{45} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{46} + ( -\beta_{1} - \beta_{2} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} -\beta_{5} q^{48} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{50} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{51} + ( -2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{52} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} + q^{54} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{55} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{57} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{58} + ( 6 + \beta_{1} - 4 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{59} -\beta_{4} q^{60} + ( -6 + 3 \beta_{6} + 3 \beta_{7} ) q^{61} + ( \beta_{2} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{62} - q^{64} + ( 4 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{65} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{66} + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{67} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{68} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{69} + ( -2 + \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{71} -\beta_{5} q^{72} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 2 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{74} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{75} + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{76} + ( -2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{78} + ( 4 + 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{79} + ( -1 + \beta_{7} ) q^{80} + q^{81} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{82} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{83} + ( 4 - \beta_{2} + 4 \beta_{3} - \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{85} + ( 2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{86} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{87} + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{88} + ( 10 + 4 \beta_{3} + \beta_{6} + \beta_{7} ) q^{89} -\beta_{4} q^{90} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{92} + ( \beta_{2} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{93} + ( 2 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{94} + ( 2 - \beta_{1} - 6 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{95} - q^{96} + ( \beta_{1} - 4 \beta_{2} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{9} + O(q^{10})$$ $$8q - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{9} + 8q^{16} + 24q^{19} + 4q^{20} + 8q^{24} + 4q^{25} + 16q^{29} + 4q^{30} - 32q^{31} + 8q^{34} + 8q^{36} - 24q^{41} + 4q^{45} + 8q^{46} - 4q^{50} + 8q^{51} + 8q^{54} + 40q^{59} - 24q^{61} - 8q^{64} + 28q^{65} + 8q^{69} - 40q^{71} + 16q^{74} - 4q^{75} - 24q^{76} + 16q^{79} - 4q^{80} + 8q^{81} + 28q^{85} + 8q^{86} + 88q^{89} + 24q^{94} + 24q^{95} - 8q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 10 \nu^{5} - 20 \nu^{4} - 15 \nu^{3} - 2 \nu^{2} - 184 \nu + 80$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 18 \nu^{5} - 89 \nu^{3} - 104 \nu$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 14 \nu^{4} + 37 \nu^{2} - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + 10 \nu^{5} + 20 \nu^{4} - 15 \nu^{3} + 2 \nu^{2} - 184 \nu - 80$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} + 46 \nu^{5} + 179 \nu^{3} + 168 \nu$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 6 \nu^{6} - 10 \nu^{5} + 92 \nu^{4} + 15 \nu^{3} + 358 \nu^{2} + 120 \nu + 336$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 6 \nu^{6} + 10 \nu^{5} + 92 \nu^{4} - 15 \nu^{3} + 358 \nu^{2} - 120 \nu + 336$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{4} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{4} - 4 \beta_{3} - \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{7} + 9 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 8 \beta_{2} + 5 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} - 17 \beta_{4} + 44 \beta_{3} + 17 \beta_{1} + 74$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$81 \beta_{7} - 81 \beta_{6} - 104 \beta_{5} - 37 \beta_{4} - 112 \beta_{2} - 37 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$89 \beta_{7} + 89 \beta_{6} + 201 \beta_{4} - 436 \beta_{3} - 201 \beta_{1} - 650$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-761 \beta_{7} + 761 \beta_{6} + 1160 \beta_{5} + 325 \beta_{4} + 1240 \beta_{2} + 325 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 − 1.69230i − 3.16053i 2.16053i 0.692297i 1.69230i 3.16053i − 2.16053i − 0.692297i
1.00000i 1.00000i −1.00000 −2.18183 + 0.489528i −1.00000 0 1.00000i −1.00000 0.489528 + 2.18183i
589.2 1.00000i 1.00000i −1.00000 −1.63280 1.52773i −1.00000 0 1.00000i −1.00000 −1.52773 + 1.63280i
589.3 1.00000i 1.00000i −1.00000 −0.0743018 + 2.23483i −1.00000 0 1.00000i −1.00000 2.23483 + 0.0743018i
589.4 1.00000i 1.00000i −1.00000 1.88893 1.19663i −1.00000 0 1.00000i −1.00000 −1.19663 1.88893i
589.5 1.00000i 1.00000i −1.00000 −2.18183 0.489528i −1.00000 0 1.00000i −1.00000 0.489528 2.18183i
589.6 1.00000i 1.00000i −1.00000 −1.63280 + 1.52773i −1.00000 0 1.00000i −1.00000 −1.52773 1.63280i
589.7 1.00000i 1.00000i −1.00000 −0.0743018 2.23483i −1.00000 0 1.00000i −1.00000 2.23483 0.0743018i
589.8 1.00000i 1.00000i −1.00000 1.88893 + 1.19663i −1.00000 0 1.00000i −1.00000 −1.19663 + 1.88893i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 589.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1470.2.g.j 8
5.b even 2 1 inner 1470.2.g.j 8
5.c odd 4 1 7350.2.a.ds 4
5.c odd 4 1 7350.2.a.dt 4
7.b odd 2 1 1470.2.g.k yes 8
7.c even 3 2 1470.2.n.l 16
7.d odd 6 2 1470.2.n.k 16
35.c odd 2 1 1470.2.g.k yes 8
35.f even 4 1 7350.2.a.dr 4
35.f even 4 1 7350.2.a.du 4
35.i odd 6 2 1470.2.n.k 16
35.j even 6 2 1470.2.n.l 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 1.a even 1 1 trivial
1470.2.g.j 8 5.b even 2 1 inner
1470.2.g.k yes 8 7.b odd 2 1
1470.2.g.k yes 8 35.c odd 2 1
1470.2.n.k 16 7.d odd 6 2
1470.2.n.k 16 35.i odd 6 2
1470.2.n.l 16 7.c even 3 2
1470.2.n.l 16 35.j even 6 2
7350.2.a.dr 4 35.f even 4 1
7350.2.a.ds 4 5.c odd 4 1
7350.2.a.dt 4 5.c odd 4 1
7350.2.a.du 4 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{4} - 18 T_{11}^{2} - 16 T_{11} + 32$$ $$T_{17}^{8} + 80 T_{17}^{6} + 1336 T_{17}^{4} + 8000 T_{17}^{2} + 15376$$ $$T_{19}^{4} - 12 T_{19}^{3} + 12 T_{19}^{2} + 224 T_{19} - 448$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$625 + 500 T + 150 T^{2} + 20 T^{3} + 2 T^{4} + 4 T^{5} + 6 T^{6} + 4 T^{7} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 32 - 16 T - 18 T^{2} + T^{4} )^{2}$$
$13$ $$256 + 1472 T^{2} + 644 T^{4} + 52 T^{6} + T^{8}$$
$17$ $$15376 + 8000 T^{2} + 1336 T^{4} + 80 T^{6} + T^{8}$$
$19$ $$( -448 + 224 T + 12 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$23$ $$16384 + 9216 T^{2} + 1552 T^{4} + 88 T^{6} + T^{8}$$
$29$ $$( 1568 + 336 T - 66 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$( 64 + 128 T + 78 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$37$ $$40000 + 31200 T^{2} + 4084 T^{4} + 148 T^{6} + T^{8}$$
$41$ $$( -376 - 120 T + 26 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$43$ $$541696 + 297344 T^{2} + 16516 T^{4} + 236 T^{6} + T^{8}$$
$47$ $$1048576 + 200704 T^{2} + 11140 T^{4} + 204 T^{6} + T^{8}$$
$53$ $$4096 + 11776 T^{2} + 5392 T^{4} + 184 T^{6} + T^{8}$$
$59$ $$( -3008 + 736 T + 52 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$61$ $$( -648 - 1512 T - 126 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$67$ $$1024 + 1152 T^{2} + 388 T^{4} + 44 T^{6} + T^{8}$$
$71$ $$( 2944 - 1472 T - 36 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$73$ $$61504 + 244640 T^{2} + 19764 T^{4} + 316 T^{6} + T^{8}$$
$79$ $$( -3584 + 1792 T - 176 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$83$ $$16384 + 26624 T^{2} + 6976 T^{4} + 240 T^{6} + T^{8}$$
$89$ $$( 5048 - 3688 T + 658 T^{2} - 44 T^{3} + T^{4} )^{2}$$
$97$ $$107661376 + 4479328 T^{2} + 66804 T^{4} + 428 T^{6} + T^{8}$$