# Properties

 Label 96.4.f.b Level $96$ Weight $4$ Character orbit 96.f Analytic conductor $5.664$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,4,Mod(47,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("96.47");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 96.f (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: $$5.66418336055$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096$$ x^8 - 10*x^6 + 120*x^4 - 640*x^2 + 4096 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} + \beta_{4} q^{5} - \beta_{6} q^{7} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 9) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 + b4 * q^5 - b6 * q^7 + (-3*b3 + 3*b2 + 3*b1 - 9) * q^9 $$q + (\beta_{2} + 1) q^{3} + \beta_{4} q^{5} - \beta_{6} q^{7} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 9) q^{9} + (\beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + (\beta_{7} - \beta_{6}) q^{13} + (\beta_{7} - \beta_{5} + 2 \beta_{4}) q^{15} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{17} + ( - 11 \beta_{3} + 11 \beta_{2} - 34) q^{19} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4}) q^{21} + (2 \beta_{5} + 4 \beta_{4}) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} + 35) q^{25} + (9 \beta_{3} - 12 \beta_{2} + 18 \beta_1 + 51) q^{27} + (4 \beta_{5} - 3 \beta_{4}) q^{29} + ( - 2 \beta_{7} + \beta_{6}) q^{31} + (9 \beta_{3} - \beta_{2} + 9 \beta_1 - 28) q^{33} + ( - 2 \beta_{3} - 2 \beta_{2} - 42 \beta_1) q^{35} + ( - \beta_{7} + 9 \beta_{6}) q^{37} + ( - \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - 14 \beta_{4}) q^{39} + ( - 28 \beta_{3} - 28 \beta_{2} - 32 \beta_1) q^{41} + (39 \beta_{3} - 39 \beta_{2} + 58) q^{43} + (12 \beta_{6} - 6 \beta_{5} - 3 \beta_{4}) q^{45} + (8 \beta_{5} - 24 \beta_{4}) q^{47} + (46 \beta_{3} - 46 \beta_{2} - 73) q^{49} + ( - 24 \beta_{3} + 16 \beta_{2} + 6 \beta_1 - 152) q^{51} + (8 \beta_{5} - 7 \beta_{4}) q^{53} + (2 \beta_{7} + 8 \beta_{6}) q^{55} + ( - 33 \beta_{3} - 23 \beta_{2} + 33 \beta_1 + 142) q^{57} + ( - 11 \beta_{3} - 11 \beta_{2} + 8 \beta_1) q^{59} + ( - 7 \beta_{7} - 17 \beta_{6}) q^{61} + ( - 6 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 36 \beta_{4}) q^{63} + (76 \beta_{3} + 76 \beta_{2} - 36 \beta_1) q^{65} + ( - 9 \beta_{3} + 9 \beta_{2} + 178) q^{67} + (8 \beta_{7} - 12 \beta_{6} - 2 \beta_{5} - 8 \beta_{4}) q^{69} + (2 \beta_{5} + 44 \beta_{4}) q^{71} + ( - 148 \beta_{3} + 148 \beta_{2} - 14) q^{73} + ( - 6 \beta_{3} + 37 \beta_{2} + 6 \beta_1 + 67) q^{75} + ( - 8 \beta_{5} + 26 \beta_{4}) q^{77} + (14 \beta_{7} - 7 \beta_{6}) q^{79} + (144 \beta_{3} + 18 \beta_1 + 9) q^{81} + (21 \beta_{3} + 21 \beta_{2} + 62 \beta_1) q^{83} + (8 \beta_{7} - 8 \beta_{6}) q^{85} + (5 \beta_{7} - 24 \beta_{6} + 7 \beta_{5} - 38 \beta_{4}) q^{87} + ( - 112 \beta_{3} - 112 \beta_{2} + 106 \beta_1) q^{89} + ( - 116 \beta_{3} + 116 \beta_{2} - 512) q^{91} + (\beta_{7} - 9 \beta_{6} + 8 \beta_{5} + 29 \beta_{4}) q^{93} + ( - 22 \beta_{5} - 12 \beta_{4}) q^{95} + (90 \beta_{3} - 90 \beta_{2} - 394) q^{97} + (57 \beta_{3} - 39 \beta_{2} + 24 \beta_1 - 216) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^3 + b4 * q^5 - b6 * q^7 + (-3*b3 + 3*b2 + 3*b1 - 9) * q^9 + (b3 + b2 + 2*b1) * q^11 + (b7 - b6) * q^13 + (b7 - b5 + 2*b4) * q^15 + (4*b3 + 4*b2 - 2*b1) * q^17 + (-11*b3 + 11*b2 - 34) * q^19 + (-b7 - 3*b6 - 2*b5 + b4) * q^21 + (2*b5 + 4*b4) * q^23 + (-2*b3 + 2*b2 + 35) * q^25 + (9*b3 - 12*b2 + 18*b1 + 51) * q^27 + (4*b5 - 3*b4) * q^29 + (-2*b7 + b6) * q^31 + (9*b3 - b2 + 9*b1 - 28) * q^33 + (-2*b3 - 2*b2 - 42*b1) * q^35 + (-b7 + 9*b6) * q^37 + (-b7 + 3*b6 - 5*b5 - 14*b4) * q^39 + (-28*b3 - 28*b2 - 32*b1) * q^41 + (39*b3 - 39*b2 + 58) * q^43 + (12*b6 - 6*b5 - 3*b4) * q^45 + (8*b5 - 24*b4) * q^47 + (46*b3 - 46*b2 - 73) * q^49 + (-24*b3 + 16*b2 + 6*b1 - 152) * q^51 + (8*b5 - 7*b4) * q^53 + (2*b7 + 8*b6) * q^55 + (-33*b3 - 23*b2 + 33*b1 + 142) * q^57 + (-11*b3 - 11*b2 + 8*b1) * q^59 + (-7*b7 - 17*b6) * q^61 + (-6*b7 - 3*b6 - 6*b5 + 36*b4) * q^63 + (76*b3 + 76*b2 - 36*b1) * q^65 + (-9*b3 + 9*b2 + 178) * q^67 + (8*b7 - 12*b6 - 2*b5 - 8*b4) * q^69 + (2*b5 + 44*b4) * q^71 + (-148*b3 + 148*b2 - 14) * q^73 + (-6*b3 + 37*b2 + 6*b1 + 67) * q^75 + (-8*b5 + 26*b4) * q^77 + (14*b7 - 7*b6) * q^79 + (144*b3 + 18*b1 + 9) * q^81 + (21*b3 + 21*b2 + 62*b1) * q^83 + (8*b7 - 8*b6) * q^85 + (5*b7 - 24*b6 + 7*b5 - 38*b4) * q^87 + (-112*b3 - 112*b2 + 106*b1) * q^89 + (-116*b3 + 116*b2 - 512) * q^91 + (b7 - 9*b6 + 8*b5 + 29*b4) * q^93 + (-22*b5 - 12*b4) * q^95 + (90*b3 - 90*b2 - 394) * q^97 + (57*b3 - 39*b2 + 24*b1 - 216) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{3} - 48 q^{9}+O(q^{10})$$ 8 * q + 12 * q^3 - 48 * q^9 $$8 q + 12 q^{3} - 48 q^{9} - 184 q^{19} + 296 q^{25} + 324 q^{27} - 264 q^{33} + 152 q^{43} - 952 q^{49} - 1056 q^{51} + 1176 q^{57} + 1496 q^{67} + 1072 q^{73} + 708 q^{75} - 504 q^{81} - 3168 q^{91} - 3872 q^{97} - 2112 q^{99}+O(q^{100})$$ 8 * q + 12 * q^3 - 48 * q^9 - 184 * q^19 + 296 * q^25 + 324 * q^27 - 264 * q^33 + 152 * q^43 - 952 * q^49 - 1056 * q^51 + 1176 * q^57 + 1496 * q^67 + 1072 * q^73 + 708 * q^75 - 504 * q^81 - 3168 * q^91 - 3872 * q^97 - 2112 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 10\nu^{5} - 120\nu^{3} + 128\nu ) / 256$$ (-v^7 + 10*v^5 - 120*v^3 + 128*v) / 256 $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 4\nu^{5} - 10\nu^{4} - 8\nu^{3} + 56\nu^{2} + 160\nu - 256 ) / 128$$ (v^6 + 4*v^5 - 10*v^4 - 8*v^3 + 56*v^2 + 160*v - 256) / 128 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 10\nu^{4} - 8\nu^{3} - 56\nu^{2} + 160\nu + 256 ) / 128$$ (-v^6 + 4*v^5 + 10*v^4 - 8*v^3 - 56*v^2 + 160*v + 256) / 128 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} + 14\nu^{5} - 72\nu^{3} + 256\nu ) / 256$$ (-3*v^7 + 14*v^5 - 72*v^3 + 256*v) / 256 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 10\nu^{5} - 120\nu^{3} + 1152\nu ) / 64$$ (-v^7 + 10*v^5 - 120*v^3 + 1152*v) / 64 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 22\nu^{4} + 136\nu^{2} - 1280 ) / 64$$ (-v^6 - 22*v^4 + 136*v^2 - 1280) / 64 $$\beta_{7}$$ $$=$$ $$( 5\nu^{6} - 18\nu^{4} + 600\nu^{2} - 1280 ) / 64$$ (5*v^6 - 18*v^4 + 600*v^2 - 1280) / 64
 $$\nu$$ $$=$$ $$( \beta_{5} - 4\beta_1 ) / 16$$ (b5 - 4*b1) / 16 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 4\beta_{3} - 4\beta_{2} + 24 ) / 8$$ (b7 + b6 + 4*b3 - 4*b2 + 24) / 8 $$\nu^{3}$$ $$=$$ $$( -\beta_{5} + 8\beta_{4} + 8\beta_{3} + 8\beta_{2} - 20\beta_1 ) / 8$$ (-b5 + 8*b4 + 8*b3 + 8*b2 - 20*b1) / 8 $$\nu^{4}$$ $$=$$ $$( 3\beta_{7} - 5\beta_{6} + 20\beta_{3} - 20\beta_{2} - 120 ) / 4$$ (3*b7 - 5*b6 + 20*b3 - 20*b2 - 120) / 4 $$\nu^{5}$$ $$=$$ $$( -11\beta_{5} + 8\beta_{4} + 72\beta_{3} + 72\beta_{2} + 20\beta_1 ) / 4$$ (-11*b5 + 8*b4 + 72*b3 + 72*b2 + 20*b1) / 4 $$\nu^{6}$$ $$=$$ $$( \beta_{7} - 39\beta_{6} - 84\beta_{3} + 84\beta_{2} - 424 ) / 2$$ (b7 - 39*b6 - 84*b3 + 84*b2 - 424) / 2 $$\nu^{7}$$ $$=$$ $$( -9\beta_{5} - 200\beta_{4} + 120\beta_{3} + 120\beta_{2} + 124\beta_1 ) / 2$$ (-9*b5 - 200*b4 + 120*b3 + 120*b2 + 124*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −2.58576 − 1.14624i 2.58576 − 1.14624i −2.58576 + 1.14624i 2.58576 + 1.14624i 1.95291 + 2.04601i −1.95291 + 2.04601i 1.95291 − 2.04601i −1.95291 − 2.04601i
0 −1.37228 5.01167i 0 −12.2683 0 14.0624i 0 −23.2337 + 13.7548i 0
47.2 0 −1.37228 5.01167i 0 12.2683 0 14.0624i 0 −23.2337 + 13.7548i 0
47.3 0 −1.37228 + 5.01167i 0 −12.2683 0 14.0624i 0 −23.2337 13.7548i 0
47.4 0 −1.37228 + 5.01167i 0 12.2683 0 14.0624i 0 −23.2337 13.7548i 0
47.5 0 4.37228 2.80770i 0 −13.1715 0 26.9490i 0 11.2337 24.5521i 0
47.6 0 4.37228 2.80770i 0 13.1715 0 26.9490i 0 11.2337 24.5521i 0
47.7 0 4.37228 + 2.80770i 0 −13.1715 0 26.9490i 0 11.2337 + 24.5521i 0
47.8 0 4.37228 + 2.80770i 0 13.1715 0 26.9490i 0 11.2337 + 24.5521i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.4.f.b 8
3.b odd 2 1 inner 96.4.f.b 8
4.b odd 2 1 24.4.f.b 8
8.b even 2 1 24.4.f.b 8
8.d odd 2 1 inner 96.4.f.b 8
12.b even 2 1 24.4.f.b 8
16.e even 4 2 768.4.c.v 16
16.f odd 4 2 768.4.c.v 16
24.f even 2 1 inner 96.4.f.b 8
24.h odd 2 1 24.4.f.b 8
48.i odd 4 2 768.4.c.v 16
48.k even 4 2 768.4.c.v 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.b 8 4.b odd 2 1
24.4.f.b 8 8.b even 2 1
24.4.f.b 8 12.b even 2 1
24.4.f.b 8 24.h odd 2 1
96.4.f.b 8 1.a even 1 1 trivial
96.4.f.b 8 3.b odd 2 1 inner
96.4.f.b 8 8.d odd 2 1 inner
96.4.f.b 8 24.f even 2 1 inner
768.4.c.v 16 16.e even 4 2
768.4.c.v 16 16.f odd 4 2
768.4.c.v 16 48.i odd 4 2
768.4.c.v 16 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 324T_{5}^{2} + 26112$$ acting on $$S_{4}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - 6 T^{3} + 30 T^{2} - 162 T + 729)^{2}$$
$5$ $$(T^{4} - 324 T^{2} + 26112)^{2}$$
$7$ $$(T^{4} + 924 T^{2} + 143616)^{2}$$
$11$ $$(T^{4} + 484 T^{2} + 352)^{2}$$
$13$ $$(T^{4} + 5808 T^{2} + 574464)^{2}$$
$17$ $$(T^{4} + 2464 T^{2} + 90112)^{2}$$
$19$ $$(T^{2} + 46 T - 3464)^{4}$$
$23$ $$(T^{4} - 17472 T^{2} + 1671168)^{2}$$
$29$ $$(T^{4} - 43620 T^{2} + \cdots + 448108032)^{2}$$
$31$ $$(T^{4} + 20988 T^{2} + 41505024)^{2}$$
$37$ $$(T^{4} + 77616 T^{2} + \cdots + 1494180864)^{2}$$
$41$ $$(T^{4} + 193600 T^{2} + \cdots + 3151126528)^{2}$$
$43$ $$(T^{2} - 38 T - 49832)^{4}$$
$47$ $$(T^{4} - 321792 T^{2} + \cdots + 427819008)^{2}$$
$53$ $$(T^{4} - 177156 T^{2} + \cdots + 7033554432)^{2}$$
$59$ $$(T^{4} + 21604 T^{2} + 296032)^{2}$$
$61$ $$(T^{4} + 550704 T^{2} + \cdots + 18820015104)^{2}$$
$67$ $$(T^{2} - 374 T + 32296)^{4}$$
$71$ $$(T^{4} - 654912 T^{2} + \cdots + 103614087168)^{2}$$
$73$ $$(T^{2} - 268 T - 704876)^{4}$$
$79$ $$(T^{4} + 1028412 T^{2} + \cdots + 99653562624)^{2}$$
$83$ $$(T^{4} + 396484 T^{2} + \cdots + 2128431712)^{2}$$
$89$ $$(T^{4} + 2644576 T^{2} + \cdots + 147293673472)^{2}$$
$97$ $$(T^{2} + 968 T - 33044)^{4}$$