# Properties

 Label 48.11.g.c Level $48$ Weight $11$ Character orbit 48.g Analytic conductor $30.497$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,11,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (of order $$2$$, degree $$1$$, 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: $$30.4971481283$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{2545})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 637x^{2} + 636x + 404496$$ x^4 - x^3 + 637*x^2 + 636*x + 404496 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ Twist minimal: yes 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 81 \beta_1 q^{3} + ( - \beta_{2} + 750) q^{5} + (13 \beta_{3} + 428 \beta_1) q^{7} - 19683 q^{9}+O(q^{10})$$ q + 81*b1 * q^3 + (-b2 + 750) * q^5 + (13*b3 + 428*b1) * q^7 - 19683 * q^9 $$q + 81 \beta_1 q^{3} + ( - \beta_{2} + 750) q^{5} + (13 \beta_{3} + 428 \beta_1) q^{7} - 19683 q^{9} + ( - 82 \beta_{3} + 74148 \beta_1) q^{11} + (294 \beta_{2} + 216034) q^{13} + (81 \beta_{3} + 60750 \beta_1) q^{15} + ( - 1766 \beta_{2} - 13782) q^{17} + (646 \beta_{3} + 939628 \beta_1) q^{19} + (3159 \beta_{2} - 104004) q^{21} + ( - 1926 \beta_{3} - 325296 \beta_1) q^{23} + ( - 1500 \beta_{2} - 7737205) q^{25} - 1594323 \beta_1 q^{27} + (7297 \beta_{2} - 20197578) q^{29} + (3581 \beta_{3} + 4476852 \beta_1) q^{31} + ( - 19926 \beta_{2} - 18017964) q^{33} + (10178 \beta_{3} + 19377960 \beta_1) q^{35} + ( - 17604 \beta_{2} - 53127910) q^{37} + ( - 23814 \beta_{3} + 17498754 \beta_1) q^{39} + (63270 \beta_{2} - 136113318) q^{41} + ( - 28406 \beta_{3} + 93464964 \beta_1) q^{43} + (19683 \beta_{2} - 14762250) q^{45} + (25238 \beta_{3} + 145810248 \beta_1) q^{47} + (33384 \beta_{2} - 461295743) q^{49} + (143046 \beta_{3} - 1116342 \beta_1) q^{51} + ( - 366207 \beta_{2} - 222201018) q^{53} + (12648 \beta_{3} - 64594440 \beta_1) q^{55} + (156978 \beta_{2} - 228329604) q^{57} + ( - 323912 \beta_{3} + 38534412 \beta_1) q^{59} + (461112 \beta_{2} - 52083814) q^{61} + ( - 255879 \beta_{3} - 8424324 \beta_1) q^{63} + (4466 \beta_{2} - 268954980) q^{65} + (194808 \beta_{3} - 204681404 \beta_1) q^{67} + ( - 468018 \beta_{2} + 79046928) q^{69} + (1097858 \beta_{3} + 324580704 \beta_1) q^{71} + ( - 1409784 \beta_{2} + 1220417362) q^{73} + (121500 \beta_{3} - 626713605 \beta_1) q^{75} + (2786484 \beta_{2} + 4592806128) q^{77} + ( - 863351 \beta_{3} - 110548940 \beta_1) q^{79} + 387420489 q^{81} + ( - 1745378 \beta_{3} + 209882172 \beta_1) q^{83} + ( - 1310718 \beta_{2} + 2578478220) q^{85} + ( - 591057 \beta_{3} - 1636003818 \beta_1) q^{87} + ( - 2291668 \beta_{2} + 5385095634) q^{89} + (2682610 \beta_{3} - 5510283688 \beta_1) q^{91} + (870183 \beta_{2} - 1087875036) q^{93} + (1424128 \beta_{3} + 1651705320 \beta_1) q^{95} + (7560444 \beta_{2} + 3741442114) q^{97} + (1614006 \beta_{3} - 1459455084 \beta_1) q^{99}+O(q^{100})$$ q + 81*b1 * q^3 + (-b2 + 750) * q^5 + (13*b3 + 428*b1) * q^7 - 19683 * q^9 + (-82*b3 + 74148*b1) * q^11 + (294*b2 + 216034) * q^13 + (81*b3 + 60750*b1) * q^15 + (-1766*b2 - 13782) * q^17 + (646*b3 + 939628*b1) * q^19 + (3159*b2 - 104004) * q^21 + (-1926*b3 - 325296*b1) * q^23 + (-1500*b2 - 7737205) * q^25 - 1594323*b1 * q^27 + (7297*b2 - 20197578) * q^29 + (3581*b3 + 4476852*b1) * q^31 + (-19926*b2 - 18017964) * q^33 + (10178*b3 + 19377960*b1) * q^35 + (-17604*b2 - 53127910) * q^37 + (-23814*b3 + 17498754*b1) * q^39 + (63270*b2 - 136113318) * q^41 + (-28406*b3 + 93464964*b1) * q^43 + (19683*b2 - 14762250) * q^45 + (25238*b3 + 145810248*b1) * q^47 + (33384*b2 - 461295743) * q^49 + (143046*b3 - 1116342*b1) * q^51 + (-366207*b2 - 222201018) * q^53 + (12648*b3 - 64594440*b1) * q^55 + (156978*b2 - 228329604) * q^57 + (-323912*b3 + 38534412*b1) * q^59 + (461112*b2 - 52083814) * q^61 + (-255879*b3 - 8424324*b1) * q^63 + (4466*b2 - 268954980) * q^65 + (194808*b3 - 204681404*b1) * q^67 + (-468018*b2 + 79046928) * q^69 + (1097858*b3 + 324580704*b1) * q^71 + (-1409784*b2 + 1220417362) * q^73 + (121500*b3 - 626713605*b1) * q^75 + (2786484*b2 + 4592806128) * q^77 + (-863351*b3 - 110548940*b1) * q^79 + 387420489 * q^81 + (-1745378*b3 + 209882172*b1) * q^83 + (-1310718*b2 + 2578478220) * q^85 + (-591057*b3 - 1636003818*b1) * q^87 + (-2291668*b2 + 5385095634) * q^89 + (2682610*b3 - 5510283688*b1) * q^91 + (870183*b2 - 1087875036) * q^93 + (1424128*b3 + 1651705320*b1) * q^95 + (7560444*b2 + 3741442114) * q^97 + (1614006*b3 - 1459455084*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3000 q^{5} - 78732 q^{9}+O(q^{10})$$ 4 * q + 3000 * q^5 - 78732 * q^9 $$4 q + 3000 q^{5} - 78732 q^{9} + 864136 q^{13} - 55128 q^{17} - 416016 q^{21} - 30948820 q^{25} - 80790312 q^{29} - 72071856 q^{33} - 212511640 q^{37} - 544453272 q^{41} - 59049000 q^{45} - 1845182972 q^{49} - 888804072 q^{53} - 913318416 q^{57} - 208335256 q^{61} - 1075819920 q^{65} + 316187712 q^{69} + 4881669448 q^{73} + 18371224512 q^{77} + 1549681956 q^{81} + 10313912880 q^{85} + 21540382536 q^{89} - 4351500144 q^{93} + 14965768456 q^{97}+O(q^{100})$$ 4 * q + 3000 * q^5 - 78732 * q^9 + 864136 * q^13 - 55128 * q^17 - 416016 * q^21 - 30948820 * q^25 - 80790312 * q^29 - 72071856 * q^33 - 212511640 * q^37 - 544453272 * q^41 - 59049000 * q^45 - 1845182972 * q^49 - 888804072 * q^53 - 913318416 * q^57 - 208335256 * q^61 - 1075819920 * q^65 + 316187712 * q^69 + 4881669448 * q^73 + 18371224512 * q^77 + 1549681956 * q^81 + 10313912880 * q^85 + 21540382536 * q^89 - 4351500144 * q^93 + 14965768456 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 637x^{2} + 636x + 404496$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 637\nu^{2} - 637\nu + 201930 ) / 202566$$ (-v^3 + 637*v^2 - 637*v + 201930) / 202566 $$\beta_{2}$$ $$=$$ $$( 48\nu^{3} + 45816 ) / 637$$ (48*v^3 + 45816) / 637 $$\beta_{3}$$ $$=$$ $$( 4\nu^{3} - 4\nu^{2} + 5092\nu + 1272 ) / 53$$ (4*v^3 - 4*v^2 + 5092*v + 1272) / 53
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96$$ (b3 - b2 + 24*b1 + 24) / 96 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 30552\beta _1 - 30552 ) / 96$$ (b3 + b2 + 30552*b1 - 30552) / 96 $$\nu^{3}$$ $$=$$ $$( 637\beta_{2} - 45816 ) / 48$$ (637*b2 - 45816) / 48

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −12.3620 + 21.4116i 12.8620 − 22.2776i −12.3620 − 21.4116i 12.8620 + 22.2776i
0 140.296i 0 −460.752 0 26520.8i 0 −19683.0 0
31.2 0 140.296i 0 1960.75 0 28003.4i 0 −19683.0 0
31.3 0 140.296i 0 −460.752 0 26520.8i 0 −19683.0 0
31.4 0 140.296i 0 1960.75 0 28003.4i 0 −19683.0 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 48.11.g.c 4
3.b odd 2 1 144.11.g.e 4
4.b odd 2 1 inner 48.11.g.c 4
8.b even 2 1 192.11.g.b 4
8.d odd 2 1 192.11.g.b 4
12.b even 2 1 144.11.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.11.g.c 4 1.a even 1 1 trivial
48.11.g.c 4 4.b odd 2 1 inner
144.11.g.e 4 3.b odd 2 1
144.11.g.e 4 12.b even 2 1
192.11.g.b 4 8.b even 2 1
192.11.g.b 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 1500T_{5} - 903420$$ acting on $$S_{11}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 19683)^{2}$$
$5$ $$(T^{2} - 1500 T - 903420)^{2}$$
$7$ $$T^{4} + \cdots + 55\!\cdots\!44$$
$11$ $$T^{4} + \cdots + 17\!\cdots\!84$$
$13$ $$(T^{2} - 432068 T - 80037571964)^{2}$$
$17$ $$(T^{2} + \cdots - 4571656851996)^{2}$$
$19$ $$T^{4} + \cdots + 66\!\cdots\!64$$
$23$ $$T^{4} + \cdots + 25\!\cdots\!44$$
$29$ $$(T^{2} + \cdots + 329887474368804)^{2}$$
$31$ $$T^{4} + \cdots + 13\!\cdots\!04$$
$37$ $$(T^{2} + \cdots + 23\!\cdots\!80)^{2}$$
$41$ $$(T^{2} + \cdots + 12\!\cdots\!24)^{2}$$
$43$ $$T^{4} + \cdots + 51\!\cdots\!84$$
$47$ $$T^{4} + \cdots + 37\!\cdots\!84$$
$53$ $$(T^{2} + \cdots - 14\!\cdots\!56)^{2}$$
$59$ $$T^{4} + \cdots + 20\!\cdots\!64$$
$61$ $$(T^{2} + \cdots - 30\!\cdots\!84)^{2}$$
$67$ $$T^{4} + \cdots + 16\!\cdots\!64$$
$71$ $$T^{4} + \cdots + 24\!\cdots\!64$$
$73$ $$(T^{2} + \cdots - 14\!\cdots\!76)^{2}$$
$79$ $$T^{4} + \cdots + 10\!\cdots\!00$$
$83$ $$T^{4} + \cdots + 17\!\cdots\!44$$
$89$ $$(T^{2} + \cdots + 21\!\cdots\!76)^{2}$$
$97$ $$(T^{2} + \cdots - 69\!\cdots\!24)^{2}$$