# Properties

 Label 48.13.g.b Level $48$ Weight $13$ Character orbit 48.g Analytic conductor $43.872$ 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] = [48,13,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, 13, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$13$$ 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: $$43.8717032293$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-2803})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 700x^{2} - 701x + 491401$$ x^4 - x^3 - 700*x^2 - 701*x + 491401 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{11}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{2} + 5598) q^{5} + ( - 5 \beta_{3} - 63 \beta_1) q^{7} - 177147 q^{9}+O(q^{10})$$ q + b1 * q^3 + (b2 + 5598) * q^5 + (-5*b3 - 63*b1) * q^7 - 177147 * q^9 $$q + \beta_1 q^{3} + (\beta_{2} + 5598) q^{5} + ( - 5 \beta_{3} - 63 \beta_1) q^{7} - 177147 q^{9} + ( - 198 \beta_{3} + 2442 \beta_1) q^{11} + (18 \beta_{2} + 2226490) q^{13} + ( - 729 \beta_{3} + 5679 \beta_1) q^{15} + ( - 178 \beta_{2} + 18136890) q^{17} + (2914 \beta_{3} - 47286 \beta_1) q^{19} + ( - 1215 \beta_{2} + 11258676) q^{21} + (9270 \beta_{3} + 102426 \beta_1) q^{23} + (11196 \beta_{2} + 179527283) q^{25} - 177147 \beta_1 q^{27} + ( - 18985 \beta_{2} - 182102778) q^{29} + ( - 90205 \beta_{3} + 1330785 \beta_1) q^{31} + ( - 48114 \beta_{2} - 428695740) q^{33} + (18342 \beta_{3} + 2333058 \beta_1) q^{35} + (196380 \beta_{2} - 681143470) q^{37} + ( - 13122 \beta_{3} + 2227948 \beta_1) q^{39} + ( - 148350 \beta_{2} - 1791030150) q^{41} + (750430 \beta_{3} + 8008326 \beta_1) q^{43} + ( - 177147 \beta_{2} - 991668906) q^{45} + ( - 485910 \beta_{3} + 25132542 \beta_1) q^{47} + (154440 \beta_{2} + 9856316593) q^{49} + (129762 \beta_{3} + 18122472 \beta_1) q^{51} + (131871 \beta_{2} + 8154817830) q^{53} + ( - 2872584 \beta_{3} + 120425184 \beta_1) q^{55} + (708102 \beta_{2} + 8319216780) q^{57} + (746568 \beta_{3} + 135469668 \beta_1) q^{59} + ( - 2482560 \beta_{2} - 32684060446) q^{61} + (885735 \beta_{3} + 11160261 \beta_1) q^{63} + (2327254 \beta_{2} + 19525836492) q^{65} + (5716680 \beta_{3} + 179960508 \beta_1) q^{67} + (2252610 \beta_{2} - 18326920032) q^{69} + (6276078 \beta_{3} + 358260018 \beta_1) q^{71} + ( - 6082488 \beta_{2} - 42624362270) q^{73} + ( - 8161884 \beta_{3} + 180434159 \beta_1) q^{75} + (117612 \beta_{2} - 102223004400) q^{77} + ( - 4352441 \beta_{3} + 240436989 \beta_1) q^{79} + 31381059609 q^{81} + ( - 28062630 \beta_{3} + 138076386 \beta_1) q^{83} + (17140446 \beta_{2} + 31695516108) q^{85} + (13840065 \beta_{3} - 183640563 \beta_1) q^{87} + ( - 7633820 \beta_{2} + 503055597474) q^{89} + ( - 10298474 \beta_{3} - 91925694 \beta_1) q^{91} + ( - 21919815 \beta_{2} - 233969065380) q^{93} + (50548032 \beta_{3} - 1836755832 \beta_1) q^{95} + (19661076 \beta_{2} + 372854702050) q^{97} + (35075106 \beta_{3} - 432592974 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b2 + 5598) * q^5 + (-5*b3 - 63*b1) * q^7 - 177147 * q^9 + (-198*b3 + 2442*b1) * q^11 + (18*b2 + 2226490) * q^13 + (-729*b3 + 5679*b1) * q^15 + (-178*b2 + 18136890) * q^17 + (2914*b3 - 47286*b1) * q^19 + (-1215*b2 + 11258676) * q^21 + (9270*b3 + 102426*b1) * q^23 + (11196*b2 + 179527283) * q^25 - 177147*b1 * q^27 + (-18985*b2 - 182102778) * q^29 + (-90205*b3 + 1330785*b1) * q^31 + (-48114*b2 - 428695740) * q^33 + (18342*b3 + 2333058*b1) * q^35 + (196380*b2 - 681143470) * q^37 + (-13122*b3 + 2227948*b1) * q^39 + (-148350*b2 - 1791030150) * q^41 + (750430*b3 + 8008326*b1) * q^43 + (-177147*b2 - 991668906) * q^45 + (-485910*b3 + 25132542*b1) * q^47 + (154440*b2 + 9856316593) * q^49 + (129762*b3 + 18122472*b1) * q^51 + (131871*b2 + 8154817830) * q^53 + (-2872584*b3 + 120425184*b1) * q^55 + (708102*b2 + 8319216780) * q^57 + (746568*b3 + 135469668*b1) * q^59 + (-2482560*b2 - 32684060446) * q^61 + (885735*b3 + 11160261*b1) * q^63 + (2327254*b2 + 19525836492) * q^65 + (5716680*b3 + 179960508*b1) * q^67 + (2252610*b2 - 18326920032) * q^69 + (6276078*b3 + 358260018*b1) * q^71 + (-6082488*b2 - 42624362270) * q^73 + (-8161884*b3 + 180434159*b1) * q^75 + (117612*b2 - 102223004400) * q^77 + (-4352441*b3 + 240436989*b1) * q^79 + 31381059609 * q^81 + (-28062630*b3 + 138076386*b1) * q^83 + (17140446*b2 + 31695516108) * q^85 + (13840065*b3 - 183640563*b1) * q^87 + (-7633820*b2 + 503055597474) * q^89 + (-10298474*b3 - 91925694*b1) * q^91 + (-21919815*b2 - 233969065380) * q^93 + (50548032*b3 - 1836755832*b1) * q^95 + (19661076*b2 + 372854702050) * q^97 + (35075106*b3 - 432592974*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 22392 q^{5} - 708588 q^{9}+O(q^{10})$$ 4 * q + 22392 * q^5 - 708588 * q^9 $$4 q + 22392 q^{5} - 708588 q^{9} + 8905960 q^{13} + 72547560 q^{17} + 45034704 q^{21} + 718109132 q^{25} - 728411112 q^{29} - 1714782960 q^{33} - 2724573880 q^{37} - 7164120600 q^{41} - 3966675624 q^{45} + 39425266372 q^{49} + 32619271320 q^{53} + 33276867120 q^{57} - 130736241784 q^{61} + 78103345968 q^{65} - 73307680128 q^{69} - 170497449080 q^{73} - 408892017600 q^{77} + 125524238436 q^{81} + 126782064432 q^{85} + 2012222389896 q^{89} - 935876261520 q^{93} + 1491418808200 q^{97}+O(q^{100})$$ 4 * q + 22392 * q^5 - 708588 * q^9 + 8905960 * q^13 + 72547560 * q^17 + 45034704 * q^21 + 718109132 * q^25 - 728411112 * q^29 - 1714782960 * q^33 - 2724573880 * q^37 - 7164120600 * q^41 - 3966675624 * q^45 + 39425266372 * q^49 + 32619271320 * q^53 + 33276867120 * q^57 - 130736241784 * q^61 + 78103345968 * q^65 - 73307680128 * q^69 - 170497449080 * q^73 - 408892017600 * q^77 + 125524238436 * q^81 + 126782064432 * q^85 + 2012222389896 * q^89 - 935876261520 * q^93 + 1491418808200 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 700x^{2} - 701x + 491401$$ :

 $$\beta_{1}$$ $$=$$ $$( 243\nu^{3} + 170100\nu^{2} - 170100\nu - 59790393 ) / 245350$$ (243*v^3 + 170100*v^2 - 170100*v - 59790393) / 245350 $$\beta_{2}$$ $$=$$ $$( -432\nu^{3} + 432\nu^{2} + 605232\nu + 151416 ) / 701$$ (-432*v^3 + 432*v^2 + 605232*v + 151416) / 701 $$\beta_{3}$$ $$=$$ $$( -21627\nu^{3} + 2700\nu^{2} - 2700\nu + 21784977 ) / 35050$$ (-21627*v^3 + 2700*v^2 - 2700*v + 21784977) / 35050
 $$\nu$$ $$=$$ $$( -9\beta_{3} + 9\beta_{2} - 7\beta _1 + 1944 ) / 7776$$ (-9*b3 + 9*b2 - 7*b1 + 1944) / 7776 $$\nu^{2}$$ $$=$$ $$( 9\beta_{3} + 9\beta_{2} + 11207\beta _1 + 2723544 ) / 7776$$ (9*b3 + 9*b2 + 11207*b1 + 2723544) / 7776 $$\nu^{3}$$ $$=$$ $$( -1575\beta_{3} + 175\beta _1 + 1021572 ) / 972$$ (-1575*b3 + 175*b1 + 1021572) / 972

## 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
 −22.6751 + 13.6689i 23.1751 − 12.8028i −22.6751 − 13.6689i 23.1751 + 12.8028i
0 420.888i 0 −14209.3 0 83928.6i 0 −177147. 0
31.2 0 420.888i 0 25405.3 0 30429.0i 0 −177147. 0
31.3 0 420.888i 0 −14209.3 0 83928.6i 0 −177147. 0
31.4 0 420.888i 0 25405.3 0 30429.0i 0 −177147. 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 48.13.g.b 4
3.b odd 2 1 144.13.g.f 4
4.b odd 2 1 inner 48.13.g.b 4
8.b even 2 1 192.13.g.b 4
8.d odd 2 1 192.13.g.b 4
12.b even 2 1 144.13.g.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.13.g.b 4 1.a even 1 1 trivial
48.13.g.b 4 4.b odd 2 1 inner
144.13.g.f 4 3.b odd 2 1
144.13.g.f 4 12.b even 2 1
192.13.g.b 4 8.b even 2 1
192.13.g.b 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 11196T_{5} - 360992700$$ acting on $$S_{13}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 177147)^{2}$$
$5$ $$(T^{2} - 11196 T - 360992700)^{2}$$
$7$ $$T^{4} + \cdots + 65\!\cdots\!64$$
$11$ $$T^{4} + \cdots + 16\!\cdots\!84$$
$13$ $$(T^{2} + \cdots + 4830142701604)^{2}$$
$17$ $$(T^{2} + \cdots + 316516185520164)^{2}$$
$19$ $$T^{4} + \cdots + 51\!\cdots\!84$$
$23$ $$T^{4} + \cdots + 87\!\cdots\!64$$
$29$ $$(T^{2} + \cdots - 10\!\cdots\!16)^{2}$$
$31$ $$T^{4} + \cdots + 57\!\cdots\!00$$
$37$ $$(T^{2} + \cdots - 14\!\cdots\!00)^{2}$$
$41$ $$(T^{2} + \cdots - 54\!\cdots\!00)^{2}$$
$43$ $$T^{4} + \cdots + 38\!\cdots\!04$$
$47$ $$T^{4} + \cdots + 64\!\cdots\!44$$
$53$ $$(T^{2} + \cdots + 59\!\cdots\!36)^{2}$$
$59$ $$T^{4} + \cdots + 10\!\cdots\!24$$
$61$ $$(T^{2} + \cdots - 13\!\cdots\!84)^{2}$$
$67$ $$T^{4} + \cdots + 22\!\cdots\!04$$
$71$ $$T^{4} + \cdots + 31\!\cdots\!44$$
$73$ $$(T^{2} + \cdots - 12\!\cdots\!76)^{2}$$
$79$ $$T^{4} + \cdots + 59\!\cdots\!64$$
$83$ $$T^{4} + \cdots + 99\!\cdots\!24$$
$89$ $$(T^{2} + \cdots + 23\!\cdots\!76)^{2}$$
$97$ $$(T^{2} + \cdots - 12\!\cdots\!04)^{2}$$