# Properties

 Label 48.13.g.a Level $48$ Weight $13$ Character orbit 48.g Analytic conductor $43.872$ Analytic rank $0$ Dimension $4$ 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: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224$$ x^4 - x^3 + 4333*x^2 + 4332*x + 18766224 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.

 $$f(q)$$ $$=$$ $$q - 243 \beta_1 q^{3} + ( - 5 \beta_{2} - 5490) q^{5} + ( - 3 \beta_{3} - 5372 \beta_1) q^{7} - 177147 q^{9}+O(q^{10})$$ q - 243*b1 * q^3 + (-5*b2 - 5490) * q^5 + (-3*b3 - 5372*b1) * q^7 - 177147 * q^9 $$q - 243 \beta_1 q^{3} + ( - 5 \beta_{2} - 5490) q^{5} + ( - 3 \beta_{3} - 5372 \beta_1) q^{7} - 177147 q^{9} + (374 \beta_{3} + 93060 \beta_1) q^{11} + ( - 2250 \beta_{2} - 1452518) q^{13} + ( - 1215 \beta_{3} + 1334070 \beta_1) q^{15} + (4826 \beta_{2} - 8470566) q^{17} + (4974 \beta_{3} + 16087436 \beta_1) q^{19} + (2187 \beta_{2} - 3916188) q^{21} + ( - 21702 \beta_{3} + 55966464 \beta_1) q^{23} + (54900 \beta_{2} + 35537075) q^{25} + 43046721 \beta_1 q^{27} + ( - 119443 \beta_{2} + 10645974) q^{29} + (123861 \beta_{3} - 208303380 \beta_1) q^{31} + ( - 272646 \beta_{2} + 67840740) q^{33} + ( - 10390 \beta_{3} - 120230280 \beta_1) q^{35} + (372708 \beta_{2} + 2350716434) q^{37} + ( - 546750 \beta_{3} + 352961874 \beta_1) q^{39} + (370134 \beta_{2} + 4492220634) q^{41} + (996882 \beta_{3} + 341094852 \beta_1) q^{43} + (885735 \beta_{2} + 972537030) q^{45} + ( - 1486618 \beta_{3} + 6216878808 \beta_1) q^{47} + (96696 \beta_{2} + 13485211441) q^{49} + (1172718 \beta_{3} + 2058347538 \beta_1) q^{51} + ( - 4280187 \beta_{2} + 15136739190) q^{53} + ( - 1587960 \beta_{3} + 18154513080 \beta_1) q^{55} + ( - 3626046 \beta_{2} + 11727740844) q^{57} + (7263544 \beta_{3} + 15536372220 \beta_1) q^{59} + ( - 1069776 \beta_{2} + 42071800418) q^{61} + (531441 \beta_{3} + 951633684 \beta_1) q^{63} + (19615090 \beta_{2} + 120266243820) q^{65} + ( - 19551816 \beta_{3} + 28609418996 \beta_1) q^{67} + (15820758 \beta_{2} + 40799552256) q^{69} + ( - 3537406 \beta_{3} + 12886833744 \beta_1) q^{71} + ( - 30749832 \beta_{2} + 197256657250) q^{73} + (13340700 \beta_{3} - 8635509225 \beta_1) q^{75} + ( - 6864924 \beta_{2} + 35097497424) q^{77} + (38237649 \beta_{3} + 5869754828 \beta_1) q^{79} + 31381059609 q^{81} + ( - 17505770 \beta_{3} + 77882293404 \beta_1) q^{83} + (15858090 \beta_{2} - 194350284180) q^{85} + ( - 29024649 \beta_{3} - 2586971682 \beta_1) q^{87} + ( - 99275060 \beta_{2} - 159667615134) q^{89} + ( - 7729446 \beta_{3} - 59572225304 \beta_1) q^{91} + ( - 90294669 \beta_{2} - 151853164020) q^{93} + (53129920 \beta_{3} + 159919980840 \beta_1) q^{95} + (338673564 \beta_{2} - 140885510750) q^{97} + ( - 66252978 \beta_{3} - 16485299820 \beta_1) q^{99}+O(q^{100})$$ q - 243*b1 * q^3 + (-5*b2 - 5490) * q^5 + (-3*b3 - 5372*b1) * q^7 - 177147 * q^9 + (374*b3 + 93060*b1) * q^11 + (-2250*b2 - 1452518) * q^13 + (-1215*b3 + 1334070*b1) * q^15 + (4826*b2 - 8470566) * q^17 + (4974*b3 + 16087436*b1) * q^19 + (2187*b2 - 3916188) * q^21 + (-21702*b3 + 55966464*b1) * q^23 + (54900*b2 + 35537075) * q^25 + 43046721*b1 * q^27 + (-119443*b2 + 10645974) * q^29 + (123861*b3 - 208303380*b1) * q^31 + (-272646*b2 + 67840740) * q^33 + (-10390*b3 - 120230280*b1) * q^35 + (372708*b2 + 2350716434) * q^37 + (-546750*b3 + 352961874*b1) * q^39 + (370134*b2 + 4492220634) * q^41 + (996882*b3 + 341094852*b1) * q^43 + (885735*b2 + 972537030) * q^45 + (-1486618*b3 + 6216878808*b1) * q^47 + (96696*b2 + 13485211441) * q^49 + (1172718*b3 + 2058347538*b1) * q^51 + (-4280187*b2 + 15136739190) * q^53 + (-1587960*b3 + 18154513080*b1) * q^55 + (-3626046*b2 + 11727740844) * q^57 + (7263544*b3 + 15536372220*b1) * q^59 + (-1069776*b2 + 42071800418) * q^61 + (531441*b3 + 951633684*b1) * q^63 + (19615090*b2 + 120266243820) * q^65 + (-19551816*b3 + 28609418996*b1) * q^67 + (15820758*b2 + 40799552256) * q^69 + (-3537406*b3 + 12886833744*b1) * q^71 + (-30749832*b2 + 197256657250) * q^73 + (13340700*b3 - 8635509225*b1) * q^75 + (-6864924*b2 + 35097497424) * q^77 + (38237649*b3 + 5869754828*b1) * q^79 + 31381059609 * q^81 + (-17505770*b3 + 77882293404*b1) * q^83 + (15858090*b2 - 194350284180) * q^85 + (-29024649*b3 - 2586971682*b1) * q^87 + (-99275060*b2 - 159667615134) * q^89 + (-7729446*b3 - 59572225304*b1) * q^91 + (-90294669*b2 - 151853164020) * q^93 + (53129920*b3 + 159919980840*b1) * q^95 + (338673564*b2 - 140885510750) * q^97 + (-66252978*b3 - 16485299820*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10})$$ 4 * q - 21960 * q^5 - 708588 * q^9 $$4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100})$$ 4 * q - 21960 * q^5 - 708588 * q^9 - 5810072 * q^13 - 33882264 * q^17 - 15664752 * q^21 + 142148300 * q^25 + 42583896 * q^29 + 271362960 * q^33 + 9402865736 * q^37 + 17968882536 * q^41 + 3890148120 * q^45 + 53940845764 * q^49 + 60546956760 * q^53 + 46910963376 * q^57 + 168287201672 * q^61 + 481064975280 * q^65 + 163198209024 * q^69 + 789026629000 * q^73 + 140389989696 * q^77 + 125524238436 * q^81 - 777401136720 * q^85 - 638670460536 * q^89 - 607412656080 * q^93 - 563542043000 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 4333\nu^{2} - 4333\nu + 9380946 ) / 9385278$$ (-v^3 + 4333*v^2 - 4333*v + 9380946) / 9385278 $$\beta_{2}$$ $$=$$ $$( 48\nu^{3} + 311928 ) / 4333$$ (48*v^3 + 311928) / 4333 $$\beta_{3}$$ $$=$$ $$( 4\nu^{3} - 4\nu^{2} + 34660\nu + 8664 ) / 361$$ (4*v^3 - 4*v^2 + 34660*v + 8664) / 361
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96$$ (b3 - b2 + 24*b1 + 24) / 96 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 207960\beta _1 - 207960 ) / 96$$ (b3 + b2 + 207960*b1 - 207960) / 96 $$\nu^{3}$$ $$=$$ $$( 4333\beta_{2} - 311928 ) / 48$$ (4333*b2 - 311928) / 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
 −32.6599 − 56.5686i 33.1599 + 57.4347i −32.6599 + 56.5686i 33.1599 − 57.4347i
0 420.888i 0 −21286.8 0 7111.90i 0 −177147. 0
31.2 0 420.888i 0 10306.8 0 25721.1i 0 −177147. 0
31.3 0 420.888i 0 −21286.8 0 7111.90i 0 −177147. 0
31.4 0 420.888i 0 10306.8 0 25721.1i 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.a 4
3.b odd 2 1 144.13.g.h 4
4.b odd 2 1 inner 48.13.g.a 4
8.b even 2 1 192.13.g.c 4
8.d odd 2 1 192.13.g.c 4
12.b even 2 1 144.13.g.h 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 10980T_{5} - 219397500$$ acting on $$S_{13}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 177147)^{2}$$
$5$ $$(T^{2} + 10980 T - 219397500)^{2}$$
$7$ $$T^{4} + \cdots + 33\!\cdots\!36$$
$11$ $$T^{4} + \cdots + 17\!\cdots\!44$$
$13$ $$(T^{2} + \cdots - 48421555459676)^{2}$$
$17$ $$(T^{2} + \cdots - 160721494694748)^{2}$$
$19$ $$T^{4} + \cdots + 12\!\cdots\!76$$
$23$ $$T^{4} + \cdots + 22\!\cdots\!00$$
$29$ $$(T^{2} + \cdots - 14\!\cdots\!20)^{2}$$
$31$ $$T^{4} + \cdots + 10\!\cdots\!04$$
$37$ $$(T^{2} + \cdots + 41\!\cdots\!00)^{2}$$
$41$ $$(T^{2} + \cdots + 18\!\cdots\!32)^{2}$$
$43$ $$T^{4} + \cdots + 86\!\cdots\!76$$
$47$ $$T^{4} + \cdots + 24\!\cdots\!16$$
$53$ $$(T^{2} + \cdots + 46\!\cdots\!24)^{2}$$
$59$ $$T^{4} + \cdots + 73\!\cdots\!24$$
$61$ $$(T^{2} + \cdots + 17\!\cdots\!20)^{2}$$
$67$ $$T^{4} + \cdots + 80\!\cdots\!76$$
$71$ $$T^{4} + \cdots + 15\!\cdots\!76$$
$73$ $$(T^{2} + \cdots + 29\!\cdots\!04)^{2}$$
$79$ $$T^{4} + \cdots + 19\!\cdots\!00$$
$83$ $$T^{4} + \cdots + 81\!\cdots\!04$$
$89$ $$(T^{2} + \cdots - 72\!\cdots\!44)^{2}$$
$97$ $$(T^{2} + \cdots - 11\!\cdots\!84)^{2}$$