# Properties

 Label 1470.2.m.a Level $1470$ Weight $2$ Character orbit 1470.m 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{16}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.923880 − 0.382683i −0.923880 + 0.382683i −0.382683 − 0.923880i 0.382683 + 0.923880i 0.923880 + 0.382683i −0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i
−0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.00000 2.00000i 1.00000i 0 0.707107 0.707107i 1.00000i −0.707107 + 2.12132i
97.2 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.00000 2.00000i 1.00000i 0 0.707107 0.707107i 1.00000i −0.707107 + 2.12132i
97.3 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.00000 2.00000i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 0.707107 2.12132i
97.4 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.00000 2.00000i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 0.707107 2.12132i
1273.1 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.00000 + 2.00000i 1.00000i 0 0.707107 + 0.707107i 1.00000i −0.707107 2.12132i
1273.2 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.00000 + 2.00000i 1.00000i 0 0.707107 + 0.707107i 1.00000i −0.707107 2.12132i
1273.3 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.00000 + 2.00000i 1.00000i 0 −0.707107 0.707107i 1.00000i 0.707107 + 2.12132i
1273.4 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.00000 + 2.00000i 1.00000i 0 −0.707107 0.707107i 1.00000i 0.707107 + 2.12132i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1273.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.m.a 8
5.c odd 4 1 1470.2.m.b yes 8
7.b odd 2 1 1470.2.m.b yes 8
35.f even 4 1 inner 1470.2.m.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.m.a 8 1.a even 1 1 trivial
1470.2.m.a 8 35.f even 4 1 inner
1470.2.m.b yes 8 5.c odd 4 1
1470.2.m.b yes 8 7.b odd 2 1

## Hecke kernels

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

 $$T_{11}^{4} - 24 T_{11}^{2} - 8 T_{11} + 62$$ $$T_{13}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$( 5 + 2 T + T^{2} )^{4}$$
$7$ $$T^{8}$$
$11$ $$( 62 - 8 T - 24 T^{2} + T^{4} )^{2}$$
$13$ $$99856 - 45504 T + 10368 T^{2} + 1504 T^{3} + 152 T^{4} - 80 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$17$ $$3844 - 3968 T + 2048 T^{2} + 16 T^{3} - 60 T^{4} + 32 T^{6} + 8 T^{7} + T^{8}$$
$19$ $$( 2 - 4 T + T^{2} )^{4}$$
$23$ $$( 18 + 6 T + T^{2} )^{4}$$
$29$ $$64516 + 39440 T^{2} + 4116 T^{4} + 136 T^{6} + T^{8}$$
$31$ $$3844 + 7104 T^{2} + 2044 T^{4} + 128 T^{6} + T^{8}$$
$37$ $$454276 - 442144 T + 215168 T^{2} - 47376 T^{3} + 5444 T^{4} - 144 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$41$ $$565504 + 191232 T^{2} + 11296 T^{4} + 208 T^{6} + T^{8}$$
$43$ $$18645124 - 3730752 T + 373248 T^{2} - 44864 T^{3} + 20300 T^{4} - 4320 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$47$ $$8655364 + 4142336 T + 991232 T^{2} + 120288 T^{3} + 8588 T^{4} + 576 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$53$ $$312299584 + 79736064 T + 10179072 T^{2} + 637696 T^{3} + 35600 T^{4} + 4000 T^{5} + 512 T^{6} + 32 T^{7} + T^{8}$$
$59$ $$( 124 - 160 T - 20 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$61$ $$984064 + 194560 T^{2} + 11200 T^{4} + 192 T^{6} + T^{8}$$
$67$ $$12630916 - 4207936 T + 700928 T^{2} + 80480 T^{3} + 6348 T^{4} - 672 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$71$ $$( 3844 + 992 T - 68 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$73$ $$3717184 + 2529536 T + 860672 T^{2} + 114816 T^{3} + 7952 T^{4} + 288 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$79$ $$565504 + 191232 T^{2} + 11296 T^{4} + 208 T^{6} + T^{8}$$
$83$ $$3268864 - 694272 T + 73728 T^{2} + 89088 T^{3} + 50208 T^{4} + 384 T^{5} + T^{8}$$
$89$ $$( -2168 + 384 T + 280 T^{2} + 32 T^{3} + T^{4} )^{2}$$
$97$ $$764854336 + 222132992 T + 32256512 T^{2} + 2041600 T^{3} + 76048 T^{4} + 3424 T^{5} + 512 T^{6} + 32 T^{7} + T^{8}$$