# Properties

 Label 48.13.e.c Level $48$ Weight $13$ Character orbit 48.e 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(17,48)]

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([0, 0, 1]))

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (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: $$43.8717032293$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{1009})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 499x^{2} + 500x + 64518$$ x^4 - 2*x^3 - 499*x^2 + 500*x + 64518 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 6) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 195) q^{3} + ( - 10 \beta_{3} - \beta_{2} - 18 \beta_1) q^{5} + (68 \beta_{3} - 85 \beta_{2} + \cdots - 38270) q^{7}+ \cdots + ( - 390 \beta_{3} + 39 \beta_{2} + \cdots - 382743) q^{9}+O(q^{10})$$ q + (b3 - 195) * q^3 + (-10*b3 - b2 - 18*b1) * q^5 + (68*b3 - 85*b2 - 17*b1 - 38270) * q^7 + (-390*b3 + 39*b2 - 462*b1 - 382743) * q^9 $$q + (\beta_{3} - 195) q^{3} + ( - 10 \beta_{3} - \beta_{2} - 18 \beta_1) q^{5} + (68 \beta_{3} - 85 \beta_{2} + \cdots - 38270) q^{7}+ \cdots + ( - 640701270 \beta_{3} + \cdots + 5864210352) q^{99}+O(q^{100})$$ q + (b3 - 195) * q^3 + (-10*b3 - b2 - 18*b1) * q^5 + (68*b3 - 85*b2 - 17*b1 - 38270) * q^7 + (-390*b3 + 39*b2 - 462*b1 - 382743) * q^9 + (1210*b3 + 121*b2 - 1463*b1) * q^11 + (2000*b3 - 2500*b2 - 500*b1 + 1813250) * q^13 + (2106*b3 + 3207*b2 + 6927*b1 + 4403448) * q^15 + (-30200*b3 - 3020*b2 + 27596*b1) * q^17 + (11740*b3 - 14675*b2 - 2935*b1 + 30067018) * q^19 + (-38270*b3 + 17901*b2 - 212058*b1 + 40808082) * q^21 + (-67300*b3 - 6730*b2 - 312946*b1) * q^23 + (51312*b3 - 64140*b2 - 12828*b1 + 108901441) * q^25 + (-312777*b3 + 67995*b2 + 380043*b1 + 196235325) * q^27 + (722450*b3 + 72245*b2 + 727970*b1) * q^29 + (172500*b3 - 215625*b2 - 43125*b1 - 682931918) * q^31 + (-254826*b3 + 311025*b2 + 68442*b1 - 641872440) * q^33 + (2563460*b3 + 256346*b2 + 3316278*b1) * q^35 + (-41552*b3 + 51940*b2 + 10388*b1 - 3820030) * q^37 + (1813250*b3 + 526500*b2 - 6237000*b1 + 627164250) * q^39 + (2641460*b3 + 264146*b2 - 2043948*b1) * q^41 + (-2208516*b3 + 2760645*b2 + 552129*b1 - 407279990) * q^43 + (3492486*b3 - 1700037*b2 + 4277430*b1 - 3894169392) * q^45 + (10095000*b3 + 1009500*b2 + 9199700*b1) * q^47 + (-5204720*b3 + 6505900*b2 + 1301180*b1 + 18234412275) * q^49 + (6360120*b3 - 6050412*b2 + 512496*b1 + 15753159072) * q^51 + (34839950*b3 + 3483995*b2 + 371974*b1) * q^53 + (-616176*b3 + 770220*b2 + 154044*b1 - 1571449968) * q^55 + (30067018*b3 + 3090555*b2 - 36611190*b1 - 106077750) * q^57 + (28803410*b3 + 2880341*b2 - 98111743*b1) * q^59 + (9953520*b3 - 12441900*b2 - 2488380*b1 + 11369266274) * q^61 + (45478176*b3 + 36698565*b2 + 109417857*b1 - 11361862350) * q^63 + (46007500*b3 + 4600750*b2 + 44638500*b1) * q^65 + (2148492*b3 - 2685615*b2 - 537123*b1 + 53358402490) * q^67 + (14173380*b3 + 58409862*b2 + 94378404*b1 + 23890231728) * q^69 + (240115700*b3 + 24011570*b2 + 162983170*b1) * q^71 + (-26838720*b3 + 33548400*b2 + 6709680*b1 - 63595906270) * q^73 + (108901441*b3 + 13507884*b2 - 160016472*b1 + 3926289693) * q^75 + (165190300*b3 + 16519030*b2 - 200279156*b1) * q^77 + (-31255340*b3 + 39069175*b2 + 7813835*b1 - 77145039758) * q^79 + (246619620*b3 - 94753854*b2 + 197761392*b1 + 54753914817) * q^81 + (-314302050*b3 - 31430205*b2 + 94573883*b1) * q^83 + (29077824*b3 - 36347280*b2 - 7269456*b1 - 4711013568) * q^85 + (-152147970*b3 - 121781235*b2 - 357903555*b1 - 335272823640) * q^87 + (20792140*b3 + 2079214*b2 - 68967472*b1) * q^89 + (46761000*b3 - 58451250*b2 - 11690250*b1 + 830933586500) * q^91 + (-682931918*b3 + 45410625*b2 - 537941250*b1 + 217761239010) * q^93 + (75831620*b3 + 7583162*b2 - 87590334*b1) * q^95 + (-175913776*b3 + 219892220*b2 + 43978444*b1 + 319057118930) * q^97 + (-640701270*b3 - 66933603*b2 + 777166929*b1 + 5864210352) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 780 q^{3} - 153080 q^{7} - 1530972 q^{9}+O(q^{10})$$ 4 * q - 780 * q^3 - 153080 * q^7 - 1530972 * q^9 $$4 q - 780 q^{3} - 153080 q^{7} - 1530972 q^{9} + 7253000 q^{13} + 17613792 q^{15} + 120268072 q^{19} + 163232328 q^{21} + 435605764 q^{25} + 784941300 q^{27} - 2731727672 q^{31} - 2567489760 q^{33} - 15280120 q^{37} + 2508657000 q^{39} - 1629119960 q^{43} - 15576677568 q^{45} + 72937649100 q^{49} + 63012636288 q^{51} - 6285799872 q^{55} - 424311000 q^{57} + 45477065096 q^{61} - 45447449400 q^{63} + 213433609960 q^{67} + 95560926912 q^{69} - 254383625080 q^{73} + 15705158772 q^{75} - 308580159032 q^{79} + 219015659268 q^{81} - 18844054272 q^{85} - 1341091294560 q^{87} + 3323734346000 q^{91} + 871044956040 q^{93} + 1276228475720 q^{97} + 23456841408 q^{99}+O(q^{100})$$ 4 * q - 780 * q^3 - 153080 * q^7 - 1530972 * q^9 + 7253000 * q^13 + 17613792 * q^15 + 120268072 * q^19 + 163232328 * q^21 + 435605764 * q^25 + 784941300 * q^27 - 2731727672 * q^31 - 2567489760 * q^33 - 15280120 * q^37 + 2508657000 * q^39 - 1629119960 * q^43 - 15576677568 * q^45 + 72937649100 * q^49 + 63012636288 * q^51 - 6285799872 * q^55 - 424311000 * q^57 + 45477065096 * q^61 - 45447449400 * q^63 + 213433609960 * q^67 + 95560926912 * q^69 - 254383625080 * q^73 + 15705158772 * q^75 - 308580159032 * q^79 + 219015659268 * q^81 - 18844054272 * q^85 - 1341091294560 * q^87 + 3323734346000 * q^91 + 871044956040 * q^93 + 1276228475720 * q^97 + 23456841408 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 499x^{2} + 500x + 64518$$ :

 $$\beta_{1}$$ $$=$$ $$( -256\nu^{3} + 384\nu^{2} + 62848\nu - 31488 ) / 339$$ (-256*v^3 + 384*v^2 + 62848*v - 31488) / 339 $$\beta_{2}$$ $$=$$ $$( -8\nu^{3} - 4056\nu^{2} + 46712\nu + 995676 ) / 339$$ (-8*v^3 - 4056*v^2 + 46712*v + 995676) / 339 $$\beta_{3}$$ $$=$$ $$( 34\nu^{3} - 5136\nu^{2} - 7330\nu + 1277466 ) / 339$$ (34*v^3 - 5136*v^2 - 7330*v + 1277466) / 339
 $$\nu$$ $$=$$ $$( -64\beta_{3} + 80\beta_{2} - 11\beta _1 + 5184 ) / 10368$$ (-64*b3 + 80*b2 - 11*b1 + 5184) / 10368 $$\nu^{2}$$ $$=$$ $$( -352\beta_{3} + 8\beta_{2} - 47\beta _1 + 1298592 ) / 5184$$ (-352*b3 + 8*b2 - 47*b1 + 1298592) / 5184 $$\nu^{3}$$ $$=$$ $$( -16768\beta_{3} + 19664\beta_{2} - 16571\beta _1 + 3893184 ) / 10368$$ (-16768*b3 + 19664*b2 - 16571*b1 + 3893184) / 10368

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 16.3824 + 1.41421i 16.3824 − 1.41421i −15.3824 − 1.41421i −15.3824 + 1.41421i
0 −385.589 618.678i 0 16348.2i 0 −213230. 0 −234084. + 477110.i 0
17.2 0 −385.589 + 618.678i 0 16348.2i 0 −213230. 0 −234084. 477110.i 0
17.3 0 −4.41144 728.987i 0 1793.38i 0 136690. 0 −531402. + 6431.76i 0
17.4 0 −4.41144 + 728.987i 0 1793.38i 0 136690. 0 −531402. 6431.76i 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
3.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.e.c 4
3.b odd 2 1 inner 48.13.e.c 4
4.b odd 2 1 6.13.b.a 4
8.b even 2 1 192.13.e.h 4
8.d odd 2 1 192.13.e.e 4
12.b even 2 1 6.13.b.a 4
20.d odd 2 1 150.13.d.a 4
20.e even 4 2 150.13.b.a 8
24.f even 2 1 192.13.e.e 4
24.h odd 2 1 192.13.e.h 4
36.f odd 6 2 162.13.d.d 8
36.h even 6 2 162.13.d.d 8
60.h even 2 1 150.13.d.a 4
60.l odd 4 2 150.13.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.13.b.a 4 4.b odd 2 1
6.13.b.a 4 12.b even 2 1
48.13.e.c 4 1.a even 1 1 trivial
48.13.e.c 4 3.b odd 2 1 inner
150.13.b.a 8 20.e even 4 2
150.13.b.a 8 60.l odd 4 2
150.13.d.a 4 20.d odd 2 1
150.13.d.a 4 60.h even 2 1
162.13.d.d 8 36.f odd 6 2
162.13.d.d 8 36.h even 6 2
192.13.e.e 4 8.d odd 2 1
192.13.e.e 4 24.f even 2 1
192.13.e.h 4 8.b even 2 1
192.13.e.h 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 270478368T_{5}^{2} + 859568578560000$$ acting on $$S_{13}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + \cdots + 282429536481$$
$5$ $$T^{4} + \cdots + 859568578560000$$
$7$ $$(T^{2} + 76540 T - 29146513676)^{2}$$
$11$ $$T^{4} + \cdots + 22\!\cdots\!84$$
$13$ $$(T^{2} + \cdots - 23192320437500)^{2}$$
$17$ $$T^{4} + \cdots + 36\!\cdots\!04$$
$19$ $$(T^{2} + \cdots - 8399894140076)^{2}$$
$23$ $$T^{4} + \cdots + 61\!\cdots\!04$$
$29$ $$T^{4} + \cdots + 24\!\cdots\!00$$
$31$ $$(T^{2} + \cdots + 26\!\cdots\!24)^{2}$$
$37$ $$(T^{2} + \cdots - 11\!\cdots\!96)^{2}$$
$41$ $$T^{4} + \cdots + 40\!\cdots\!84$$
$43$ $$(T^{2} + \cdots - 32\!\cdots\!44)^{2}$$
$47$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 40\!\cdots\!84$$
$59$ $$T^{4} + \cdots + 68\!\cdots\!04$$
$61$ $$(T^{2} + \cdots - 52\!\cdots\!24)^{2}$$
$67$ $$(T^{2} + \cdots + 28\!\cdots\!64)^{2}$$
$71$ $$T^{4} + \cdots + 64\!\cdots\!00$$
$73$ $$(T^{2} + \cdots - 72\!\cdots\!00)^{2}$$
$79$ $$(T^{2} + \cdots - 51\!\cdots\!36)^{2}$$
$83$ $$T^{4} + \cdots + 22\!\cdots\!44$$
$89$ $$T^{4} + \cdots + 16\!\cdots\!24$$
$97$ $$(T^{2} + \cdots - 10\!\cdots\!24)^{2}$$