# Properties

 Label 48.11.e.d Level $48$ Weight $11$ Character orbit 48.e 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(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, 11, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$11$$ 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: $$30.4971481283$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{85})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 37x^{2} + 38x + 531$$ x^4 - 2*x^3 - 37*x^2 + 38*x + 531 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{14}\cdot 3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 21) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} + \cdots + 11278) q^{7}+ \cdots + (7 \beta_{3} - 62 \beta_{2} + \cdots + 39753) q^{9}+O(q^{10})$$ q + (-b1 - 21) * q^3 + (-b3 - 2*b2 + 4*b1) * q^5 + (-b3 + 5*b2 - 43*b1 + 11278) * q^7 + (7*b3 - 62*b2 - 20*b1 + 39753) * q^9 $$q + ( - \beta_1 - 21) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} + \cdots + 11278) q^{7}+ \cdots + ( - 2000271 \beta_{3} + \cdots + 656728128) q^{99}+O(q^{100})$$ q + (-b1 - 21) * q^3 + (-b3 - 2*b2 + 4*b1) * q^5 + (-b3 + 5*b2 - 43*b1 + 11278) * q^7 + (7*b3 - 62*b2 - 20*b1 + 39753) * q^9 + (-35*b3 - 189*b2 + 21*b1) * q^11 + (20*b3 - 100*b2 + 860*b1 + 68810) * q^13 + (35*b3 + 1877*b2 + 743*b1 + 295200) * q^15 + (-236*b3 + 2860*b2 + 4276*b1) * q^17 + (97*b3 - 485*b2 + 4171*b1 + 392182) * q^19 + (189*b3 - 1674*b2 - 12952*b1 + 2407002) * q^21 + (-826*b3 - 4590*b2 + 366*b1) * q^23 + (-420*b3 + 2100*b2 - 18060*b1 - 8433095) * q^25 + (1599*b3 - 5727*b2 - 39306*b1 - 8654877) * q^27 + (-2947*b3 - 34494*b2 - 16812*b1) * q^29 + (-3045*b3 + 15225*b2 - 130935*b1 + 5446462) * q^31 + (5033*b3 + 69146*b2 + 29456*b1 + 6493536) * q^33 + (-9598*b3 + 71914*b2 + 129502*b1) * q^35 + (-5684*b3 + 28420*b2 - 244412*b1 - 17753542) * q^37 + (-3780*b3 + 33480*b2 - 35330*b1 - 54321810) * q^39 + (-13414*b3 + 158308*b2 + 238792*b1) * q^41 + (-6783*b3 + 33915*b2 - 291669*b1 + 117672166) * q^43 + (-52581*b3 - 175386*b2 - 411492*b1 + 78079680) * q^45 + (14460*b3 - 159940*b2 - 246700*b1) * q^47 + (-22556*b3 + 112780*b2 - 969908*b1 - 12514605) * q^49 + (-98364*b3 + 346344*b2 + 78720*b1 + 177144192) * q^51 + (59339*b3 + 437750*b2 + 81716*b1) * q^53 + (-37548*b3 + 187740*b2 - 1614564*b1 - 675339840) * q^55 + (-18333*b3 + 162378*b2 - 229804*b1 - 264688302) * q^57 + (23273*b3 + 739875*b2 + 600237*b1) * q^59 + (19644*b3 - 98220*b2 + 844692*b1 - 296009686) * q^61 + (130057*b3 - 924173*b2 - 2903825*b1 + 226251774) * q^63 + (-102410*b3 - 2027020*b2 - 1412560*b1) * q^65 + (49917*b3 - 249585*b2 + 2146431*b1 + 74341462) * q^67 + (122926*b3 + 1635604*b2 + 698920*b1 + 149067072) * q^69 + (-213598*b3 - 803026*b2 + 478562*b1) * q^71 + (-46224*b3 + 231120*b2 - 1987632*b1 + 1633567250) * q^73 + (79380*b3 - 703080*b2 + 7730015*b1 + 1287507795) * q^75 + (-153146*b3 + 1340556*b2 + 2259432*b1) * q^77 + (160363*b3 - 801815*b2 + 6895609*b1 - 49820642) * q^79 + (397530*b3 - 4229568*b2 + 6742260*b1 + 364741137) * q^81 + (388359*b3 - 651583*b2 - 2981737*b1) * q^83 + (540624*b3 - 2703120*b2 + 23246832*b1 - 3220128000) * q^85 + (1018345*b3 + 6360919*b2 + 3019021*b1 - 52567200) * q^87 + (587062*b3 + 5538400*b2 + 2016028*b1) * q^89 + (156750*b3 - 783750*b2 + 6740250*b1 - 2079308020) * q^91 + (575505*b3 - 5097330*b2 - 10543792*b1 + 7936117098) * q^93 + (-555142*b3 - 9947954*b2 - 6617102*b1) * q^95 + (-79756*b3 + 398780*b2 - 3429508*b1 - 9794088766) * q^97 + (-2000271*b3 - 11748765*b2 - 12335547*b1 + 656728128) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9}+O(q^{10})$$ 4 * q - 84 * q^3 + 45112 * q^7 + 159012 * q^9 $$4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9} + 275240 q^{13} + 1180800 q^{15} + 1568728 q^{19} + 9628008 q^{21} - 33732380 q^{25} - 34619508 q^{27} + 21785848 q^{31} + 25974144 q^{33} - 71014168 q^{37} - 217287240 q^{39} + 470688664 q^{43} + 312318720 q^{45} - 50058420 q^{49} + 708576768 q^{51} - 2701359360 q^{55} - 1058753208 q^{57} - 1184038744 q^{61} + 905007096 q^{63} + 297365848 q^{67} + 596268288 q^{69} + 6534269000 q^{73} + 5150031180 q^{75} - 199282568 q^{79} + 1458964548 q^{81} - 12880512000 q^{85} - 210268800 q^{87} - 8317232080 q^{91} + 31744468392 q^{93} - 39176355064 q^{97} + 2626912512 q^{99}+O(q^{100})$$ 4 * q - 84 * q^3 + 45112 * q^7 + 159012 * q^9 + 275240 * q^13 + 1180800 * q^15 + 1568728 * q^19 + 9628008 * q^21 - 33732380 * q^25 - 34619508 * q^27 + 21785848 * q^31 + 25974144 * q^33 - 71014168 * q^37 - 217287240 * q^39 + 470688664 * q^43 + 312318720 * q^45 - 50058420 * q^49 + 708576768 * q^51 - 2701359360 * q^55 - 1058753208 * q^57 - 1184038744 * q^61 + 905007096 * q^63 + 297365848 * q^67 + 596268288 * q^69 + 6534269000 * q^73 + 5150031180 * q^75 - 199282568 * q^79 + 1458964548 * q^81 - 12880512000 * q^85 - 210268800 * q^87 - 8317232080 * q^91 + 31744468392 * q^93 - 39176355064 * q^97 + 2626912512 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 37x^{2} + 38x + 531$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{3} - 180\nu^{2} + 1732\nu + 2760 ) / 31$$ (-4*v^3 - 180*v^2 + 1732*v + 2760) / 31 $$\beta_{2}$$ $$=$$ $$( 260\nu^{3} - 204\nu^{2} - 5444\nu - 840 ) / 31$$ (260*v^3 - 204*v^2 - 5444*v - 840) / 31 $$\beta_{3}$$ $$=$$ $$( -256\nu^{3} + 9312\nu^{2} + 3712\nu - 176016 ) / 31$$ (-256*v^3 + 9312*v^2 + 3712*v - 176016) / 31
 $$\nu$$ $$=$$ $$( 4\beta_{3} + 7\beta_{2} + 199\beta _1 + 5184 ) / 10368$$ (4*b3 + 7*b2 + 199*b1 + 5184) / 10368 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta _1 + 5616 ) / 288$$ (b3 + b2 + b1 + 5616) / 288 $$\nu^{3}$$ $$=$$ $$( 112\beta_{3} + 1411\beta_{2} + 4195\beta _1 + 300672 ) / 10368$$ (112*b3 + 1411*b2 + 4195*b1 + 300672) / 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
 5.10977 − 1.41421i 5.10977 + 1.41421i −4.10977 + 1.41421i −4.10977 − 1.41421i
0 −242.269 18.8335i 0 4818.41i 0 −670.530 0 58339.6 + 9125.53i 0
17.2 0 −242.269 + 18.8335i 0 4818.41i 0 −670.530 0 58339.6 9125.53i 0
17.3 0 200.269 137.627i 0 3630.47i 0 23226.5 0 21166.4 55125.0i 0
17.4 0 200.269 + 137.627i 0 3630.47i 0 23226.5 0 21166.4 + 55125.0i 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 48.11.e.d 4
3.b odd 2 1 inner 48.11.e.d 4
4.b odd 2 1 6.11.b.a 4
8.b even 2 1 192.11.e.h 4
8.d odd 2 1 192.11.e.g 4
12.b even 2 1 6.11.b.a 4
20.d odd 2 1 150.11.d.a 4
20.e even 4 2 150.11.b.a 8
24.f even 2 1 192.11.e.g 4
24.h odd 2 1 192.11.e.h 4
36.f odd 6 2 162.11.d.d 8
36.h even 6 2 162.11.d.d 8
60.h even 2 1 150.11.d.a 4
60.l odd 4 2 150.11.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 4.b odd 2 1
6.11.b.a 4 12.b even 2 1
48.11.e.d 4 1.a even 1 1 trivial
48.11.e.d 4 3.b odd 2 1 inner
150.11.b.a 8 20.e even 4 2
150.11.b.a 8 60.l odd 4 2
150.11.d.a 4 20.d odd 2 1
150.11.d.a 4 60.h even 2 1
162.11.d.d 8 36.f odd 6 2
162.11.d.d 8 36.h even 6 2
192.11.e.g 4 8.d odd 2 1
192.11.e.g 4 24.f even 2 1
192.11.e.h 4 8.b even 2 1
192.11.e.h 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 36397440T_{5}^{2} + 306009247334400$$ acting on $$S_{11}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + \cdots + 3486784401$$
$5$ $$T^{4} + \cdots + 306009247334400$$
$7$ $$(T^{2} - 22556 T - 15574076)^{2}$$
$11$ $$T^{4} + \cdots + 21\!\cdots\!24$$
$13$ $$(T^{2} - 137620 T - 52372127900)^{2}$$
$17$ $$T^{4} + \cdots + 32\!\cdots\!84$$
$19$ $$(T^{2} + \cdots - 1189491369116)^{2}$$
$23$ $$T^{4} + \cdots + 61\!\cdots\!24$$
$29$ $$T^{4} + \cdots + 20\!\cdots\!00$$
$31$ $$(T^{2} + \cdots - 12\!\cdots\!56)^{2}$$
$37$ $$(T^{2} + \cdots - 42\!\cdots\!96)^{2}$$
$41$ $$T^{4} + \cdots + 28\!\cdots\!04$$
$43$ $$(T^{2} + \cdots + 72\!\cdots\!16)^{2}$$
$47$ $$T^{4} + \cdots + 26\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 37\!\cdots\!04$$
$59$ $$T^{4} + \cdots + 20\!\cdots\!04$$
$61$ $$(T^{2} + \cdots + 32\!\cdots\!36)^{2}$$
$67$ $$(T^{2} + \cdots - 35\!\cdots\!96)^{2}$$
$71$ $$T^{4} + \cdots + 49\!\cdots\!00$$
$73$ $$(T^{2} + \cdots + 23\!\cdots\!40)^{2}$$
$79$ $$(T^{2} + \cdots - 36\!\cdots\!76)^{2}$$
$83$ $$T^{4} + \cdots + 57\!\cdots\!44$$
$89$ $$T^{4} + \cdots + 15\!\cdots\!84$$
$97$ $$(T^{2} + \cdots + 95\!\cdots\!96)^{2}$$