# Properties

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

# 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: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} + ( \beta_{4} + \beta_{6} ) q^{5} + ( -1 + \beta_{1} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} + ( \beta_{4} + \beta_{6} ) q^{5} + ( -1 + \beta_{1} ) q^{7} - q^{9} + ( -\beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( 1 + \beta_{3} ) q^{15} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -\beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{4} + \beta_{7} ) q^{21} -2 \beta_{2} q^{23} + ( -1 + 2 \beta_{2} - 2 \beta_{3} ) q^{25} + \beta_{4} q^{27} + ( -3 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{29} + ( 3 + \beta_{1} ) q^{31} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( 4 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -2 - \beta_{1} + \beta_{3} ) q^{39} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{41} + ( -\beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{43} + ( -\beta_{4} - \beta_{6} ) q^{45} -2 \beta_{2} q^{47} + ( 1 - 2 \beta_{1} + 4 \beta_{2} ) q^{49} + ( \beta_{5} - \beta_{6} + \beta_{7} ) q^{51} + ( 5 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{53} + ( -4 + 4 \beta_{2} ) q^{55} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} -4 \beta_{5} q^{59} + ( -4 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{61} + ( 1 - \beta_{1} ) q^{63} + ( -2 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{65} + ( 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{67} -2 \beta_{5} q^{69} + ( 2 - 2 \beta_{1} ) q^{71} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{73} + ( \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{75} + ( -6 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{77} + ( 3 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{79} + q^{81} + ( -5 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{83} + ( 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{85} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{87} + ( 2 - 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -4 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( -3 \beta_{4} + \beta_{7} ) q^{93} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{97} + ( \beta_{5} - \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} - 8q^{9} + O(q^{10})$$ $$8q - 8q^{7} - 8q^{9} + 8q^{15} - 8q^{25} + 24q^{31} - 16q^{39} + 8q^{49} - 32q^{55} + 8q^{63} - 16q^{65} + 16q^{71} + 16q^{73} + 24q^{79} + 8q^{81} - 24q^{87} + 16q^{89} - 48q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} - 7 \nu^{2} + 6 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} + 19 \nu^{2} - 12 \nu + 4$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} - 11 \nu^{4} + 17 \nu^{3} - 24 \nu^{2} + 16 \nu - 5$$ $$\beta_{4}$$ $$=$$ $$-8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31$$ $$\beta_{5}$$ $$=$$ $$10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 166 \nu - 42$$ $$\beta_{6}$$ $$=$$ $$10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 168 \nu - 43$$ $$\beta_{7}$$ $$=$$ $$-28 \nu^{7} + 98 \nu^{6} - 342 \nu^{5} + 610 \nu^{4} - 890 \nu^{3} + 774 \nu^{2} - 440 \nu + 109$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - 5 \beta_{6} + 7 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} - 10$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} - 6 \beta_{6} + 8 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} + 11 \beta_{6} - 13 \beta_{5} - 20 \beta_{4} - 25 \beta_{3} - 25 \beta_{2} + 15 \beta_{1} + 52$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$10 \beta_{7} + 32 \beta_{6} - 40 \beta_{5} - 45 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} - \beta_{1} - 6$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{7} + 11 \beta_{6} - 19 \beta_{5} + 133 \beta_{3} + 147 \beta_{2} - 63 \beta_{1} - 236$$$$)/4$$

## 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
 0.5 − 2.10607i 0.5 − 0.0297061i 0.5 + 1.44392i 0.5 + 0.691860i 0.5 − 0.691860i 0.5 − 1.44392i 0.5 + 0.0297061i 0.5 + 2.10607i
0 1.00000i 0 1.79793i 0 0.158942 0 −1.00000 0
1537.2 0 1.00000i 0 0.473626i 0 −4.55765 0 −1.00000 0
1537.3 0 1.00000i 0 2.47363i 0 2.55765 0 −1.00000 0
1537.4 0 1.00000i 0 3.79793i 0 −2.15894 0 −1.00000 0
1537.5 0 1.00000i 0 3.79793i 0 −2.15894 0 −1.00000 0
1537.6 0 1.00000i 0 2.47363i 0 2.55765 0 −1.00000 0
1537.7 0 1.00000i 0 0.473626i 0 −4.55765 0 −1.00000 0
1537.8 0 1.00000i 0 1.79793i 0 0.158942 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.f 8
4.b odd 2 1 3072.2.d.i 8
8.b even 2 1 inner 3072.2.d.f 8
8.d odd 2 1 3072.2.d.i 8
16.e even 4 1 3072.2.a.i 4
16.e even 4 1 3072.2.a.t 4
16.f odd 4 1 3072.2.a.n 4
16.f odd 4 1 3072.2.a.o 4
32.g even 8 2 48.2.j.a 8
32.g even 8 2 384.2.j.b 8
32.h odd 8 2 192.2.j.a 8
32.h odd 8 2 384.2.j.a 8
48.i odd 4 1 9216.2.a.y 4
48.i odd 4 1 9216.2.a.bo 4
48.k even 4 1 9216.2.a.x 4
48.k even 4 1 9216.2.a.bn 4
96.o even 8 2 576.2.k.b 8
96.o even 8 2 1152.2.k.f 8
96.p odd 8 2 144.2.k.b 8
96.p odd 8 2 1152.2.k.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 32.g even 8 2
144.2.k.b 8 96.p odd 8 2
192.2.j.a 8 32.h odd 8 2
384.2.j.a 8 32.h odd 8 2
384.2.j.b 8 32.g even 8 2
576.2.k.b 8 96.o even 8 2
1152.2.k.c 8 96.p odd 8 2
1152.2.k.f 8 96.o even 8 2
3072.2.a.i 4 16.e even 4 1
3072.2.a.n 4 16.f odd 4 1
3072.2.a.o 4 16.f odd 4 1
3072.2.a.t 4 16.e even 4 1
3072.2.d.f 8 1.a even 1 1 trivial
3072.2.d.f 8 8.b even 2 1 inner
3072.2.d.i 8 4.b odd 2 1
3072.2.d.i 8 8.d odd 2 1
9216.2.a.x 4 48.k even 4 1
9216.2.a.y 4 48.i odd 4 1
9216.2.a.bn 4 48.k even 4 1
9216.2.a.bo 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 $$S_{2}^{\mathrm{new}}(3072, [\chi])$$:

 $$T_{5}^{8} + 24 T_{5}^{6} + 160 T_{5}^{4} + 320 T_{5}^{2} + 64$$ $$T_{7}^{4} + 4 T_{7}^{3} - 8 T_{7}^{2} - 24 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$64 + 320 T^{2} + 160 T^{4} + 24 T^{6} + T^{8}$$
$7$ $$( 4 - 24 T - 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$1024 + 2560 T^{2} + 640 T^{4} + 48 T^{6} + T^{8}$$
$13$ $$16 + 2272 T^{2} + 792 T^{4} + 56 T^{6} + T^{8}$$
$17$ $$( 16 - 64 T - 32 T^{2} + T^{4} )^{2}$$
$19$ $$256 + 5120 T^{2} + 1056 T^{4} + 64 T^{6} + T^{8}$$
$23$ $$( -8 + T^{2} )^{4}$$
$29$ $$61504 + 20288 T^{2} + 2208 T^{4} + 88 T^{6} + T^{8}$$
$31$ $$( -28 - 24 T + 40 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$37$ $$1106704 + 159584 T^{2} + 7768 T^{4} + 152 T^{6} + T^{8}$$
$41$ $$( -112 - 192 T - 64 T^{2} + T^{4} )^{2}$$
$43$ $$12544 + 44032 T^{2} + 8992 T^{4} + 192 T^{6} + T^{8}$$
$47$ $$( -8 + T^{2} )^{4}$$
$53$ $$18496 + 40256 T^{2} + 4768 T^{4} + 152 T^{6} + T^{8}$$
$59$ $$( 32 + T^{2} )^{4}$$
$61$ $$1106704 + 159584 T^{2} + 7768 T^{4} + 152 T^{6} + T^{8}$$
$67$ $$65536 + 65536 T^{2} + 8704 T^{4} + 256 T^{6} + T^{8}$$
$71$ $$( 64 + 192 T - 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$73$ $$( 64 - 64 T - 96 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$79$ $$( -10108 + 2888 T - 168 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$83$ $$1024 + 39424 T^{2} + 46720 T^{4} + 432 T^{6} + T^{8}$$
$89$ $$( -1904 + 1632 T - 200 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$97$ $$( 512 + 768 T - 224 T^{2} + T^{4} )^{2}$$