# Properties

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

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## 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: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ 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 primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{4} q^{3} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{5} + ( -2 + \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{4} q^{3} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{5} + ( -2 + \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{7} - q^{9} + ( -2 \zeta_{16} - 2 \zeta_{16}^{7} ) q^{11} + ( 2 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{13} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{15} + ( -2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} ) q^{17} + ( 2 \zeta_{16} - 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{19} + ( \zeta_{16} + \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{21} -4 q^{23} + ( -1 + 4 \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{25} -\zeta_{16}^{4} q^{27} + ( -3 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{29} + ( 2 + 3 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{31} + ( 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{33} + ( 2 \zeta_{16} - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{4} + 4 \zeta_{16}^{5} - 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{35} + ( -\zeta_{16}^{2} - 4 \zeta_{16}^{3} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{37} + ( -2 + 2 \zeta_{16} - \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{39} + ( -4 \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{41} + ( 2 \zeta_{16} - 2 \zeta_{16}^{2} - 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{43} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{45} + ( 4 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{47} + ( 1 - 4 \zeta_{16} + 2 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{49} + ( -2 \zeta_{16} - 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{51} + ( \zeta_{16} + \zeta_{16}^{2} + 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{53} + ( -4 + 2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} ) q^{57} + ( -4 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 4 \zeta_{16}^{5} - 4 \zeta_{16}^{6} ) q^{59} + ( -4 \zeta_{16} - \zeta_{16}^{2} + 6 \zeta_{16}^{4} - \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{61} + ( 2 - \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{63} + ( -2 + 4 \zeta_{16} - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 6 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{65} + ( -4 \zeta_{16} - 4 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 4 \zeta_{16}^{5} - 4 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{67} -4 \zeta_{16}^{4} q^{69} + ( -4 - 2 \zeta_{16} + 4 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{71} + ( 2 - 4 \zeta_{16} - 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{73} + ( -2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - \zeta_{16}^{4} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} ) q^{75} + ( 4 \zeta_{16} - 4 \zeta_{16}^{4} + 4 \zeta_{16}^{7} ) q^{77} + ( 6 - 3 \zeta_{16} - 4 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{79} + q^{81} + ( -2 \zeta_{16} - 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{83} + ( 2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{85} + ( -\zeta_{16} - \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{87} + ( 2 + 4 \zeta_{16} - 4 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{89} + ( -2 \zeta_{16} - 6 \zeta_{16}^{2} + 8 \zeta_{16}^{3} - 4 \zeta_{16}^{4} + 8 \zeta_{16}^{5} - 6 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{91} + ( -\zeta_{16} + 3 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 3 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{93} + ( 8 - 6 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{95} + ( -4 + 4 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - 4 \zeta_{16}^{6} ) q^{97} + ( 2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{7} - 8q^{9} + O(q^{10})$$ $$8q - 16q^{7} - 8q^{9} - 32q^{23} - 8q^{25} + 16q^{31} - 16q^{39} + 32q^{47} + 8q^{49} - 32q^{55} + 16q^{63} - 16q^{65} - 32q^{71} + 16q^{73} + 48q^{79} + 8q^{81} + 16q^{89} + 64q^{95} - 32q^{97} + O(q^{100})$$

## 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
 −0.923880 + 0.382683i 0.382683 + 0.923880i 0.923880 − 0.382683i −0.382683 − 0.923880i −0.382683 + 0.923880i 0.923880 + 0.382683i 0.382683 − 0.923880i −0.923880 − 0.382683i
0 1.00000i 0 4.02734i 0 −4.61313 0 −1.00000 0
1537.2 0 1.00000i 0 0.331821i 0 −3.08239 0 −1.00000 0
1537.3 0 1.00000i 0 1.19891i 0 0.613126 0 −1.00000 0
1537.4 0 1.00000i 0 2.49661i 0 −0.917608 0 −1.00000 0
1537.5 0 1.00000i 0 2.49661i 0 −0.917608 0 −1.00000 0
1537.6 0 1.00000i 0 1.19891i 0 0.613126 0 −1.00000 0
1537.7 0 1.00000i 0 0.331821i 0 −3.08239 0 −1.00000 0
1537.8 0 1.00000i 0 4.02734i 0 −4.61313 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
8.b even 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.e 8
4.b odd 2 1 3072.2.d.j 8
8.b even 2 1 inner 3072.2.d.e 8
8.d odd 2 1 3072.2.d.j 8
16.e even 4 1 3072.2.a.m 4
16.e even 4 1 3072.2.a.s 4
16.f odd 4 1 3072.2.a.j 4
16.f odd 4 1 3072.2.a.p 4
32.g even 8 2 1536.2.j.e 8
32.g even 8 2 1536.2.j.j yes 8
32.h odd 8 2 1536.2.j.f yes 8
32.h odd 8 2 1536.2.j.i yes 8
48.i odd 4 1 9216.2.a.bl 4
48.i odd 4 1 9216.2.a.bm 4
48.k even 4 1 9216.2.a.z 4
48.k even 4 1 9216.2.a.ba 4
96.o even 8 2 4608.2.k.bc 8
96.o even 8 2 4608.2.k.bj 8
96.p odd 8 2 4608.2.k.be 8
96.p odd 8 2 4608.2.k.bh 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.e 8 32.g even 8 2
1536.2.j.f yes 8 32.h odd 8 2
1536.2.j.i yes 8 32.h odd 8 2
1536.2.j.j yes 8 32.g even 8 2
3072.2.a.j 4 16.f odd 4 1
3072.2.a.m 4 16.e even 4 1
3072.2.a.p 4 16.f odd 4 1
3072.2.a.s 4 16.e even 4 1
3072.2.d.e 8 1.a even 1 1 trivial
3072.2.d.e 8 8.b even 2 1 inner
3072.2.d.j 8 4.b odd 2 1
3072.2.d.j 8 8.d odd 2 1
4608.2.k.bc 8 96.o even 8 2
4608.2.k.be 8 96.p odd 8 2
4608.2.k.bh 8 96.p odd 8 2
4608.2.k.bj 8 96.o even 8 2
9216.2.a.z 4 48.k even 4 1
9216.2.a.ba 4 48.k even 4 1
9216.2.a.bl 4 48.i odd 4 1
9216.2.a.bm 4 48.i odd 4 1

## Hecke kernels

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

 $$T_{5}^{8} + 24 T_{5}^{6} + 136 T_{5}^{4} + 160 T_{5}^{2} + 16$$ $$T_{7}^{4} + 8 T_{7}^{3} + 16 T_{7}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$16 + 160 T^{2} + 136 T^{4} + 24 T^{6} + T^{8}$$
$7$ $$( -8 + 16 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$11$ $$( 32 + 16 T^{2} + T^{4} )^{2}$$
$13$ $$35344 + 26464 T^{2} + 2584 T^{4} + 88 T^{6} + T^{8}$$
$17$ $$( -32 - 64 T - 32 T^{2} + T^{4} )^{2}$$
$19$ $$1024 + 2048 T^{2} + 960 T^{4} + 64 T^{6} + T^{8}$$
$23$ $$( 4 + T )^{8}$$
$29$ $$4624 + 6560 T^{2} + 1800 T^{4} + 88 T^{6} + T^{8}$$
$31$ $$( 248 + 128 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$37$ $$150544 + 46688 T^{2} + 4504 T^{4} + 152 T^{6} + T^{8}$$
$41$ $$( 992 - 64 T - 96 T^{2} + T^{4} )^{2}$$
$43$ $$984064 + 194560 T^{2} + 11200 T^{4} + 192 T^{6} + T^{8}$$
$47$ $$( -16 - 8 T + T^{2} )^{4}$$
$53$ $$35344 + 114592 T^{2} + 12040 T^{4} + 216 T^{6} + T^{8}$$
$59$ $$18939904 + 1359872 T^{2} + 33280 T^{4} + 320 T^{6} + T^{8}$$
$61$ $$16 + 29792 T^{2} + 13464 T^{4} + 280 T^{6} + T^{8}$$
$67$ $$62980096 + 3784704 T^{2} + 67072 T^{4} + 448 T^{6} + T^{8}$$
$71$ $$( -2176 - 768 T + 16 T^{3} + T^{4} )^{2}$$
$73$ $$( 1552 + 32 T - 120 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$79$ $$( -7688 + 1344 T + 80 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$83$ $$295936 + 103424 T^{2} + 7488 T^{4} + 160 T^{6} + T^{8}$$
$89$ $$( 272 - 288 T - 168 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$97$ $$( -256 - 256 T - 32 T^{2} + 16 T^{3} + T^{4} )^{2}$$
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