Properties

 Label 192.5.g.b Level $192$ Weight $5$ Character orbit 192.g Analytic conductor $19.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,5,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (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: $$19.8470329121$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \beta q^{3} + 42 q^{5} - 44 \beta q^{7} - 27 q^{9} +O(q^{10})$$ q - 3*b * q^3 + 42 * q^5 - 44*b * q^7 - 27 * q^9 $$q - 3 \beta q^{3} + 42 q^{5} - 44 \beta q^{7} - 27 q^{9} - 12 \beta q^{11} + 182 q^{13} - 126 \beta q^{15} - 246 q^{17} - 68 \beta q^{19} - 396 q^{21} + 432 \beta q^{23} + 1139 q^{25} + 81 \beta q^{27} - 78 q^{29} - 852 \beta q^{31} - 108 q^{33} - 1848 \beta q^{35} - 530 q^{37} - 546 \beta q^{39} - 918 q^{41} - 492 \beta q^{43} - 1134 q^{45} - 2184 \beta q^{47} - 3407 q^{49} + 738 \beta q^{51} + 4626 q^{53} - 504 \beta q^{55} - 612 q^{57} - 132 \beta q^{59} - 1346 q^{61} + 1188 \beta q^{63} + 7644 q^{65} + 628 \beta q^{67} + 3888 q^{69} + 1056 \beta q^{71} - 926 q^{73} - 3417 \beta q^{75} - 1584 q^{77} + 2540 \beta q^{79} + 729 q^{81} + 6924 \beta q^{83} - 10332 q^{85} + 234 \beta q^{87} + 11586 q^{89} - 8008 \beta q^{91} - 7668 q^{93} - 2856 \beta q^{95} - 13118 q^{97} + 324 \beta q^{99} +O(q^{100})$$ q - 3*b * q^3 + 42 * q^5 - 44*b * q^7 - 27 * q^9 - 12*b * q^11 + 182 * q^13 - 126*b * q^15 - 246 * q^17 - 68*b * q^19 - 396 * q^21 + 432*b * q^23 + 1139 * q^25 + 81*b * q^27 - 78 * q^29 - 852*b * q^31 - 108 * q^33 - 1848*b * q^35 - 530 * q^37 - 546*b * q^39 - 918 * q^41 - 492*b * q^43 - 1134 * q^45 - 2184*b * q^47 - 3407 * q^49 + 738*b * q^51 + 4626 * q^53 - 504*b * q^55 - 612 * q^57 - 132*b * q^59 - 1346 * q^61 + 1188*b * q^63 + 7644 * q^65 + 628*b * q^67 + 3888 * q^69 + 1056*b * q^71 - 926 * q^73 - 3417*b * q^75 - 1584 * q^77 + 2540*b * q^79 + 729 * q^81 + 6924*b * q^83 - 10332 * q^85 + 234*b * q^87 + 11586 * q^89 - 8008*b * q^91 - 7668 * q^93 - 2856*b * q^95 - 13118 * q^97 + 324*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 84 q^{5} - 54 q^{9}+O(q^{10})$$ 2 * q + 84 * q^5 - 54 * q^9 $$2 q + 84 q^{5} - 54 q^{9} + 364 q^{13} - 492 q^{17} - 792 q^{21} + 2278 q^{25} - 156 q^{29} - 216 q^{33} - 1060 q^{37} - 1836 q^{41} - 2268 q^{45} - 6814 q^{49} + 9252 q^{53} - 1224 q^{57} - 2692 q^{61} + 15288 q^{65} + 7776 q^{69} - 1852 q^{73} - 3168 q^{77} + 1458 q^{81} - 20664 q^{85} + 23172 q^{89} - 15336 q^{93} - 26236 q^{97}+O(q^{100})$$ 2 * q + 84 * q^5 - 54 * q^9 + 364 * q^13 - 492 * q^17 - 792 * q^21 + 2278 * q^25 - 156 * q^29 - 216 * q^33 - 1060 * q^37 - 1836 * q^41 - 2268 * q^45 - 6814 * q^49 + 9252 * q^53 - 1224 * q^57 - 2692 * q^61 + 15288 * q^65 + 7776 * q^69 - 1852 * q^73 - 3168 * q^77 + 1458 * q^81 - 20664 * q^85 + 23172 * q^89 - 15336 * q^93 - 26236 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 5.19615i 0 42.0000 0 76.2102i 0 −27.0000 0
127.2 0 5.19615i 0 42.0000 0 76.2102i 0 −27.0000 0
 $$n$$: e.g. 2-40 or 990-1000 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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.5.g.b 2
3.b odd 2 1 576.5.g.d 2
4.b odd 2 1 inner 192.5.g.b 2
8.b even 2 1 48.5.g.a 2
8.d odd 2 1 48.5.g.a 2
12.b even 2 1 576.5.g.d 2
16.e even 4 2 768.5.b.c 4
16.f odd 4 2 768.5.b.c 4
24.f even 2 1 144.5.g.f 2
24.h odd 2 1 144.5.g.f 2
40.e odd 2 1 1200.5.e.b 2
40.f even 2 1 1200.5.e.b 2
40.i odd 4 2 1200.5.j.b 4
40.k even 4 2 1200.5.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.g.a 2 8.b even 2 1
48.5.g.a 2 8.d odd 2 1
144.5.g.f 2 24.f even 2 1
144.5.g.f 2 24.h odd 2 1
192.5.g.b 2 1.a even 1 1 trivial
192.5.g.b 2 4.b odd 2 1 inner
576.5.g.d 2 3.b odd 2 1
576.5.g.d 2 12.b even 2 1
768.5.b.c 4 16.e even 4 2
768.5.b.c 4 16.f odd 4 2
1200.5.e.b 2 40.e odd 2 1
1200.5.e.b 2 40.f even 2 1
1200.5.j.b 4 40.i odd 4 2
1200.5.j.b 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 42$$ acting on $$S_{5}^{\mathrm{new}}(192, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 27$$
$5$ $$(T - 42)^{2}$$
$7$ $$T^{2} + 5808$$
$11$ $$T^{2} + 432$$
$13$ $$(T - 182)^{2}$$
$17$ $$(T + 246)^{2}$$
$19$ $$T^{2} + 13872$$
$23$ $$T^{2} + 559872$$
$29$ $$(T + 78)^{2}$$
$31$ $$T^{2} + 2177712$$
$37$ $$(T + 530)^{2}$$
$41$ $$(T + 918)^{2}$$
$43$ $$T^{2} + 726192$$
$47$ $$T^{2} + 14309568$$
$53$ $$(T - 4626)^{2}$$
$59$ $$T^{2} + 52272$$
$61$ $$(T + 1346)^{2}$$
$67$ $$T^{2} + 1183152$$
$71$ $$T^{2} + 3345408$$
$73$ $$(T + 926)^{2}$$
$79$ $$T^{2} + 19354800$$
$83$ $$T^{2} + 143825328$$
$89$ $$(T - 11586)^{2}$$
$97$ $$(T + 13118)^{2}$$
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