# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.18939904.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + (\beta_{6} + \beta_{4}) q^{5} + ( - \beta_1 + 1) q^{7} - q^{9}+O(q^{10})$$ q + b4 * q^3 + (b6 + b4) * q^5 + (-b1 + 1) * q^7 - q^9 $$q + \beta_{4} q^{3} + (\beta_{6} + \beta_{4}) q^{5} + ( - \beta_1 + 1) q^{7} - q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{11} + (\beta_{7} + \beta_{6} - 2 \beta_{4}) q^{13} + ( - \beta_{3} - 1) q^{15} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{19} + (\beta_{7} + \beta_{4}) q^{21} + 2 \beta_{2} q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 1) q^{25} - \beta_{4} q^{27} + (\beta_{6} + 2 \beta_{5} - 3 \beta_{4}) q^{29} + ( - \beta_1 - 3) q^{31} + (\beta_{3} + \beta_{2} - \beta_1) q^{33} + (\beta_{7} + \beta_{6} + 3 \beta_{5}) q^{35} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + 4 \beta_{4}) q^{37} + ( - \beta_{3} + \beta_1 + 2) q^{39} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{41} + (\beta_{7} + 3 \beta_{6} + \beta_{5}) q^{43} + ( - \beta_{6} - \beta_{4}) q^{45} + 2 \beta_{2} q^{47} + (4 \beta_{2} - 2 \beta_1 + 1) q^{49} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{51} + (\beta_{6} + 2 \beta_{5} + 5 \beta_{4}) q^{53} + ( - 4 \beta_{2} + 4) q^{55} + (\beta_{3} + \beta_{2} + \beta_1) q^{57} + 4 \beta_{5} q^{59} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - 4 \beta_{4}) q^{61} + (\beta_1 - 1) q^{63} + (\beta_{3} + 5 \beta_{2} + \beta_1 - 2) q^{65} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4}) q^{67} - 2 \beta_{5} q^{69} + (2 \beta_1 - 2) q^{71} + (4 \beta_{2} + 2 \beta_1 + 2) q^{73} + ( - 2 \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{75} + ( - 2 \beta_{7} + 2 \beta_{5} - 6 \beta_{4}) q^{77} + (4 \beta_{3} - 4 \beta_{2} - \beta_1 - 3) q^{79} + q^{81} + ( - \beta_{7} + 3 \beta_{6} + 5 \beta_{5}) q^{83} + ( - 2 \beta_{7} - 2 \beta_{5} + 6 \beta_{4}) q^{85} + ( - \beta_{3} + 2 \beta_{2} + 3) q^{87} + (2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 4 \beta_{4}) q^{91} + (\beta_{7} - 3 \beta_{4}) q^{93} + (2 \beta_{2} + 2 \beta_1 + 6) q^{95} + ( - 2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{97} + (\beta_{7} + \beta_{6} - \beta_{5}) q^{99}+O(q^{100})$$ q + b4 * q^3 + (b6 + b4) * q^5 + (-b1 + 1) * q^7 - q^9 + (-b7 - b6 + b5) * q^11 + (b7 + b6 - 2*b4) * q^13 + (-b3 - 1) * q^15 + (b3 + b2 + b1) * q^17 + (b7 - b6 + b5) * q^19 + (b7 + b4) * q^21 + 2*b2 * q^23 + (-2*b3 + 2*b2 - 1) * q^25 - b4 * q^27 + (b6 + 2*b5 - 3*b4) * q^29 + (-b1 - 3) * q^31 + (b3 + b2 - b1) * q^33 + (b7 + b6 + 3*b5) * q^35 + (b7 - b6 + 2*b5 + 4*b4) * q^37 + (-b3 + b1 + 2) * q^39 + (-b3 + 3*b2 - b1) * q^41 + (b7 + 3*b6 + b5) * q^43 + (-b6 - b4) * q^45 + 2*b2 * q^47 + (4*b2 - 2*b1 + 1) * q^49 + (-b7 + b6 - b5) * q^51 + (b6 + 2*b5 + 5*b4) * q^53 + (-4*b2 + 4) * q^55 + (b3 + b2 + b1) * q^57 + 4*b5 * q^59 + (b7 - b6 - 2*b5 - 4*b4) * q^61 + (b1 - 1) * q^63 + (b3 + 5*b2 + b1 - 2) * q^65 + (-2*b7 - 2*b6 - 2*b5 - 4*b4) * q^67 - 2*b5 * q^69 + (2*b1 - 2) * q^71 + (4*b2 + 2*b1 + 2) * q^73 + (-2*b6 - 2*b5 - b4) * q^75 + (-2*b7 + 2*b5 - 6*b4) * q^77 + (4*b3 - 4*b2 - b1 - 3) * q^79 + q^81 + (-b7 + 3*b6 + 5*b5) * q^83 + (-2*b7 - 2*b5 + 6*b4) * q^85 + (-b3 + 2*b2 + 3) * q^87 + (2*b3 + 6*b2 - 2*b1 + 2) * q^89 + (-b7 + b6 - b5 + 4*b4) * q^91 + (b7 - 3*b4) * q^93 + (2*b2 + 2*b1 + 6) * q^95 + (-2*b3 + 6*b2 + 2*b1) * q^97 + (b7 + b6 - b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{7} - 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^7 - 8 * q^9 $$8 q + 8 q^{7} - 8 q^{9} - 8 q^{15} - 8 q^{25} - 24 q^{31} + 16 q^{39} + 8 q^{49} + 32 q^{55} - 8 q^{63} - 16 q^{65} - 16 q^{71} + 16 q^{73} - 24 q^{79} + 8 q^{81} + 24 q^{87} + 16 q^{89} + 48 q^{95}+O(q^{100})$$ 8 * q + 8 * q^7 - 8 * q^9 - 8 * q^15 - 8 * q^25 - 24 * q^31 + 16 * q^39 + 8 * q^49 + 32 * q^55 - 8 * q^63 - 16 * q^65 - 16 * q^71 + 16 * q^73 - 24 * q^79 + 8 * q^81 + 24 * q^87 + 16 * q^89 + 48 * q^95

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

 $$\beta_{1}$$ $$=$$ $$-\nu^{4} + 2\nu^{3} - 7\nu^{2} + 6\nu - 5$$ -v^4 + 2*v^3 - 7*v^2 + 6*v - 5 $$\beta_{2}$$ $$=$$ $$\nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4$$ v^6 - 3*v^5 + 10*v^4 - 15*v^3 + 19*v^2 - 12*v + 4 $$\beta_{3}$$ $$=$$ $$-\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5$$ -v^6 + 3*v^5 - 11*v^4 + 17*v^3 - 24*v^2 + 16*v - 5 $$\beta_{4}$$ $$=$$ $$-8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31$$ -8*v^7 + 28*v^6 - 98*v^5 + 175*v^4 - 256*v^3 + 223*v^2 - 126*v + 31 $$\beta_{5}$$ $$=$$ $$10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42$$ 10*v^7 - 35*v^6 + 123*v^5 - 220*v^4 + 325*v^3 - 285*v^2 + 166*v - 42 $$\beta_{6}$$ $$=$$ $$10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43$$ 10*v^7 - 35*v^6 + 123*v^5 - 220*v^4 + 325*v^3 - 285*v^2 + 168*v - 43 $$\beta_{7}$$ $$=$$ $$-28\nu^{7} + 98\nu^{6} - 342\nu^{5} + 610\nu^{4} - 890\nu^{3} + 774\nu^{2} - 440\nu + 109$$ -28*v^7 + 98*v^6 - 342*v^5 + 610*v^4 - 890*v^3 + 774*v^2 - 440*v + 109
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} + 1 ) / 2$$ (b6 - b5 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 3 ) / 2$$ (b6 - b5 + b3 + b2 - b1 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} - 5\beta_{6} + 7\beta_{5} + 6\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta _1 - 10 ) / 4$$ (-b7 - 5*b6 + 7*b5 + 6*b4 + 3*b3 + 3*b2 - 3*b1 - 10) / 4 $$\nu^{4}$$ $$=$$ $$( -\beta_{7} - 6\beta_{6} + 8\beta_{5} + 6\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 7 ) / 2$$ (-b7 - 6*b6 + 8*b5 + 6*b4 - 4*b3 - 4*b2 + 2*b1 + 7) / 2 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} + 11\beta_{6} - 13\beta_{5} - 20\beta_{4} - 25\beta_{3} - 25\beta_{2} + 15\beta _1 + 52 ) / 4$$ (5*b7 + 11*b6 - 13*b5 - 20*b4 - 25*b3 - 25*b2 + 15*b1 + 52) / 4 $$\nu^{6}$$ $$=$$ $$( 10\beta_{7} + 32\beta_{6} - 40\beta_{5} - 45\beta_{4} + 6\beta_{3} + 8\beta_{2} - \beta _1 - 6 ) / 2$$ (10*b7 + 32*b6 - 40*b5 - 45*b4 + 6*b3 + 8*b2 - b1 - 6) / 2 $$\nu^{7}$$ $$=$$ $$( -3\beta_{7} + 11\beta_{6} - 19\beta_{5} + 133\beta_{3} + 147\beta_{2} - 63\beta _1 - 236 ) / 4$$ (-3*b7 + 11*b6 - 19*b5 + 133*b3 + 147*b2 - 63*b1 - 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

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

 $$T_{5}^{8} + 24T_{5}^{6} + 160T_{5}^{4} + 320T_{5}^{2} + 64$$ T5^8 + 24*T5^6 + 160*T5^4 + 320*T5^2 + 64 $$T_{7}^{4} - 4T_{7}^{3} - 8T_{7}^{2} + 24T_{7} + 4$$ T7^4 - 4*T7^3 - 8*T7^2 + 24*T7 + 4

## Hecke characteristic polynomials

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