# Properties

 Label 3072.2.d.g Level $3072$ Weight $2$ Character orbit 3072.d Analytic conductor $24.530$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 1536) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{6} q^{5} + ( \beta_{2} - \beta_{7} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{6} q^{5} + ( \beta_{2} - \beta_{7} ) q^{7} - q^{9} + ( -2 \beta_{1} - \beta_{5} ) q^{11} -\beta_{3} q^{13} + \beta_{7} q^{15} + ( -2 - \beta_{4} ) q^{17} + ( -2 \beta_{1} - \beta_{5} ) q^{19} + ( \beta_{3} - \beta_{6} ) q^{21} -4 \beta_{2} q^{23} -5 q^{25} -\beta_{1} q^{27} + ( 2 \beta_{3} + \beta_{6} ) q^{29} + ( -5 \beta_{2} + \beta_{7} ) q^{31} + ( 2 - \beta_{4} ) q^{33} + ( 10 \beta_{1} - \beta_{5} ) q^{35} + ( 3 \beta_{3} + 2 \beta_{6} ) q^{37} + \beta_{2} q^{39} + ( -2 - \beta_{4} ) q^{41} + ( -6 \beta_{1} + \beta_{5} ) q^{43} + \beta_{6} q^{45} + ( 5 + 2 \beta_{4} ) q^{49} + ( -2 \beta_{1} + \beta_{5} ) q^{51} + ( 6 \beta_{3} + \beta_{6} ) q^{53} + ( -10 \beta_{2} - 2 \beta_{7} ) q^{55} + ( 2 - \beta_{4} ) q^{57} -2 \beta_{5} q^{59} + ( 3 \beta_{3} - 2 \beta_{6} ) q^{61} + ( -\beta_{2} + \beta_{7} ) q^{63} + \beta_{4} q^{65} + 12 \beta_{1} q^{67} -4 \beta_{3} q^{69} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{71} + ( 6 + 2 \beta_{4} ) q^{73} -5 \beta_{1} q^{75} + 8 \beta_{3} q^{77} + ( -3 \beta_{2} - \beta_{7} ) q^{79} + q^{81} + ( -2 \beta_{1} - \beta_{5} ) q^{83} + ( -10 \beta_{3} + 2 \beta_{6} ) q^{85} + ( -2 \beta_{2} - \beta_{7} ) q^{87} + 10 q^{89} + ( -2 \beta_{1} + \beta_{5} ) q^{91} + ( -5 \beta_{3} + \beta_{6} ) q^{93} + ( -10 \beta_{2} - 2 \beta_{7} ) q^{95} + ( -4 + 2 \beta_{4} ) q^{97} + ( 2 \beta_{1} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{9} + O(q^{10})$$ $$8q - 8q^{9} - 16q^{17} - 40q^{25} + 16q^{33} - 16q^{41} + 40q^{49} + 16q^{57} + 48q^{73} + 8q^{81} + 80q^{89} - 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{4} + 14$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$-2 \nu^{6} - 12 \nu^{2}$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 6 \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 2 \beta_{3} - 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{4} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 5 \beta_{6} - 11 \beta_{3} - 11 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{5} - 9 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} - 13 \beta_{6} - 29 \beta_{3} + 29 \beta_{2}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1537.1
 −1.14412 + 1.14412i −0.437016 + 0.437016i 0.437016 − 0.437016i 1.14412 − 1.14412i 0.437016 + 0.437016i 1.14412 + 1.14412i −1.14412 − 1.14412i −0.437016 − 0.437016i
0 1.00000i 0 3.16228i 0 1.74806 0 −1.00000 0
1537.2 0 1.00000i 0 3.16228i 0 4.57649 0 −1.00000 0
1537.3 0 1.00000i 0 3.16228i 0 −4.57649 0 −1.00000 0
1537.4 0 1.00000i 0 3.16228i 0 −1.74806 0 −1.00000 0
1537.5 0 1.00000i 0 3.16228i 0 −4.57649 0 −1.00000 0
1537.6 0 1.00000i 0 3.16228i 0 −1.74806 0 −1.00000 0
1537.7 0 1.00000i 0 3.16228i 0 1.74806 0 −1.00000 0
1537.8 0 1.00000i 0 3.16228i 0 4.57649 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1537.8 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.d.g 8
4.b odd 2 1 inner 3072.2.d.g 8
8.b even 2 1 inner 3072.2.d.g 8
8.d odd 2 1 inner 3072.2.d.g 8
16.e even 4 1 3072.2.a.k 4
16.e even 4 1 3072.2.a.q 4
16.f odd 4 1 3072.2.a.k 4
16.f odd 4 1 3072.2.a.q 4
32.g even 8 2 1536.2.j.g 8
32.g even 8 2 1536.2.j.h yes 8
32.h odd 8 2 1536.2.j.g 8
32.h odd 8 2 1536.2.j.h yes 8
48.i odd 4 1 9216.2.a.bd 4
48.i odd 4 1 9216.2.a.bj 4
48.k even 4 1 9216.2.a.bd 4
48.k even 4 1 9216.2.a.bj 4
96.o even 8 2 4608.2.k.bf 8
96.o even 8 2 4608.2.k.bg 8
96.p odd 8 2 4608.2.k.bf 8
96.p odd 8 2 4608.2.k.bg 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.g 8 32.g even 8 2
1536.2.j.g 8 32.h odd 8 2
1536.2.j.h yes 8 32.g even 8 2
1536.2.j.h yes 8 32.h odd 8 2
3072.2.a.k 4 16.e even 4 1
3072.2.a.k 4 16.f odd 4 1
3072.2.a.q 4 16.e even 4 1
3072.2.a.q 4 16.f odd 4 1
3072.2.d.g 8 1.a even 1 1 trivial
3072.2.d.g 8 4.b odd 2 1 inner
3072.2.d.g 8 8.b even 2 1 inner
3072.2.d.g 8 8.d odd 2 1 inner
4608.2.k.bf 8 96.o even 8 2
4608.2.k.bf 8 96.p odd 8 2
4608.2.k.bg 8 96.o even 8 2
4608.2.k.bg 8 96.p odd 8 2
9216.2.a.bd 4 48.i odd 4 1
9216.2.a.bd 4 48.k even 4 1
9216.2.a.bj 4 48.i odd 4 1
9216.2.a.bj 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3072, [\chi])$$:

 $$T_{5}^{2} + 10$$ $$T_{7}^{4} - 24 T_{7}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$( 10 + T^{2} )^{4}$$
$7$ $$( 64 - 24 T^{2} + T^{4} )^{2}$$
$11$ $$( 256 + 48 T^{2} + T^{4} )^{2}$$
$13$ $$( 2 + T^{2} )^{4}$$
$17$ $$( -16 + 4 T + T^{2} )^{4}$$
$19$ $$( 256 + 48 T^{2} + T^{4} )^{2}$$
$23$ $$( -32 + T^{2} )^{4}$$
$29$ $$( 4 + 36 T^{2} + T^{4} )^{2}$$
$31$ $$( 1600 - 120 T^{2} + T^{4} )^{2}$$
$37$ $$( 484 + 116 T^{2} + T^{4} )^{2}$$
$41$ $$( -16 + 4 T + T^{2} )^{4}$$
$43$ $$( 256 + 112 T^{2} + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$( 3844 + 164 T^{2} + T^{4} )^{2}$$
$59$ $$( 80 + T^{2} )^{4}$$
$61$ $$( 484 + 116 T^{2} + T^{4} )^{2}$$
$67$ $$( 144 + T^{2} )^{4}$$
$71$ $$( 1024 - 96 T^{2} + T^{4} )^{2}$$
$73$ $$( -44 - 12 T + T^{2} )^{4}$$
$79$ $$( 64 - 56 T^{2} + T^{4} )^{2}$$
$83$ $$( 256 + 48 T^{2} + T^{4} )^{2}$$
$89$ $$( -10 + T )^{8}$$
$97$ $$( -64 + 8 T + T^{2} )^{4}$$