# Properties

 Label 48.17.e.b Level $48$ Weight $17$ Character orbit 48.e Analytic conductor $77.916$ 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] = [48,17,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, 17, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 17);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$17$$ 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: $$77.9157810512$$ 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: $$x^{4} + 3814x^{2} + 2981440$$ x^4 + 3814*x^2 + 2981440 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{9}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 3) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 513) q^{3} + (15 \beta_{3} - 45 \beta_{2} + 10 \beta_1) q^{5} + ( - 217 \beta_{3} - 651 \beta_{2} + \cdots + 785386) q^{7}+ \cdots + ( - 729 \beta_{3} + 135 \beta_{2} + \cdots + 4654665) q^{9}+O(q^{10})$$ q + (-b2 + 513) * q^3 + (15*b3 - 45*b2 + 10*b1) * q^5 + (-217*b3 - 651*b2 + 217*b1 + 785386) * q^7 + (-729*b3 + 135*b2 - 18954*b1 + 4654665) * q^9 $$q + ( - \beta_{2} + 513) q^{3} + (15 \beta_{3} - 45 \beta_{2} + 10 \beta_1) q^{5} + ( - 217 \beta_{3} - 651 \beta_{2} + \cdots + 785386) q^{7}+ \cdots + (374876825841 \beta_{3} + \cdots - 47\!\cdots\!40) q^{99}+O(q^{100})$$ q + (-b2 + 513) * q^3 + (15*b3 - 45*b2 + 10*b1) * q^5 + (-217*b3 - 651*b2 + 217*b1 + 785386) * q^7 + (-729*b3 + 135*b2 - 18954*b1 + 4654665) * q^9 + (7341*b3 - 22023*b2 - 78257*b1) * q^11 + (10972*b3 + 32916*b2 - 10972*b1 - 395182606) * q^13 + (-49005*b3 + 10275*b2 - 782055*b1 - 1707489720) * q^15 + (-189948*b3 + 569844*b2 - 1257852*b1) * q^17 + (642849*b3 + 1928547*b2 - 642849*b1 + 14029279090) * q^19 + (-474579*b3 - 29575*b2 - 12339054*b1 + 31113859266) * q^21 + (-269922*b3 + 809766*b2 - 20053294*b1) * q^23 + (-2677100*b3 - 8031300*b2 + 2677100*b1 - 2018512175) * q^25 + (12852999*b3 - 6926958*b2 - 53242515*b1 + 79495916913) * q^27 + (-19387467*b3 + 58162401*b2 - 5958586*b1) * q^29 + (1406435*b3 + 4219305*b2 - 1406435*b1 - 617945289062) * q^31 + (29898801*b3 - 8940783*b2 - 618470802*b1 - 902674487880) * q^33 + (-32370030*b3 + 97110090*b2 - 1046733590*b1) * q^35 + (-10171068*b3 - 30513204*b2 + 10171068*b1 + 92640803474) * q^37 + (23995764*b3 + 356967130*b2 + 623889864*b1 - 1755542556846) * q^39 + (117316386*b3 - 351949158*b2 - 711527972*b1) * q^41 + (-337207927*b3 - 1011623781*b2 + 337207927*b1 + 7016255515666) * q^43 + (546017355*b3 + 1636632135*b2 - 2161302750*b1 + 4698831610800) * q^45 + (-279906036*b3 + 839718108*b2 - 5881833140*b1) * q^47 + (-340857524*b3 - 1022572572*b2 + 340857524*b1 + 7369565634291) * q^49 + (1353590676*b3 - 320159340*b2 + 6696310176*b1 + 20710393805664) * q^51 + (-1217891637*b3 + 3653674911*b2 + 7506036882*b1) * q^53 + (771928300*b3 + 2315784900*b2 - 771928300*b1 - 72563413309200) * q^55 + (1405910763*b3 - 16268322157*b2 + 36553679838*b1 - 83782277122086) * q^57 + (723473769*b3 - 2170421307*b2 + 88594469927*b1) * q^59 + (-688871180*b3 - 2066613540*b2 + 688871180*b1 + 90567448270802) * q^61 + (8281674009*b3 - 32517007929*b2 - 36887214105*b1 + 66674590946586) * q^63 + (-3695375970*b3 + 11086127910*b2 + 49370448140*b1) * q^65 + (5788376453*b3 + 17365129359*b2 - 5788376453*b1 + 6693601340866) * q^67 + (13759763382*b3 - 3523618698*b2 - 42268012596*b1 + 14705793322704) * q^69 + (-14167417398*b3 + 42502252194*b2 - 146168753154*b1) * q^71 + (33109695792*b3 + 99329087376*b2 - 33109695792*b1 + 79407221884994) * q^73 + (-5854817700*b3 + 11342851475*b2 - 152225260200*b1 + 377841466256625) * q^75 + (22266543222*b3 - 66799629666*b2 - 796917081884*b1) * q^77 + (39564406019*b3 + 118693218057*b2 - 39564406019*b1 - 1198504291296230) * q^79 + (21122653986*b3 - 101573452278*b2 - 245797839792*b1 - 1611048213013599) * q^81 + (65301927471*b3 - 195905782413*b2 - 1047121580195*b1) * q^83 + (62226401520*b3 + 186679204560*b2 - 62226401520*b1 + 1999998523143360) * q^85 + (58824632673*b3 - 12110061039*b2 + 1030554088299*b1 + 2212542398810520) * q^87 + (100743455694*b3 - 302230367082*b2 + 2235826543152*b1) * q^89 + (94371880694*b3 + 283115642082*b2 - 94371880694*b1 - 2332134557914252) * q^91 + (3075873345*b3 + 613046675957*b2 + 79972706970*b1 - 516051849190446) * q^93 + (341233243890*b3 - 1023699731670*b2 + 3264442503550*b1) * q^95 + (-66747558308*b3 - 200242674924*b2 + 66747558308*b1 - 3458448750650446) * q^97 + (374876825841*b3 + 763573764357*b2 - 1838065223817*b1 - 4735794274334640) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2052 q^{3} + 3141544 q^{7} + 18618660 q^{9}+O(q^{10})$$ 4 * q + 2052 * q^3 + 3141544 * q^7 + 18618660 * q^9 $$4 q + 2052 q^{3} + 3141544 q^{7} + 18618660 q^{9} - 1580730424 q^{13} - 6829958880 q^{15} + 56117116360 q^{19} + 124455437064 q^{21} - 8074048700 q^{25} + 317983667652 q^{27} - 2471781156248 q^{31} - 3610697951520 q^{33} + 370563213896 q^{37} - 7022170227384 q^{39} + 28065022062664 q^{43} + 18795326443200 q^{45} + 29478262537164 q^{49} + 82841575222656 q^{51} - 290253653236800 q^{55} - 335129108488344 q^{57} + 362269793083208 q^{61} + 266698363786344 q^{63} + 26774405363464 q^{67} + 58823173290816 q^{69} + 317628887539976 q^{73} + 15\!\cdots\!00 q^{75}+ \cdots - 18\!\cdots\!60 q^{99}+O(q^{100})$$ 4 * q + 2052 * q^3 + 3141544 * q^7 + 18618660 * q^9 - 1580730424 * q^13 - 6829958880 * q^15 + 56117116360 * q^19 + 124455437064 * q^21 - 8074048700 * q^25 + 317983667652 * q^27 - 2471781156248 * q^31 - 3610697951520 * q^33 + 370563213896 * q^37 - 7022170227384 * q^39 + 28065022062664 * q^43 + 18795326443200 * q^45 + 29478262537164 * q^49 + 82841575222656 * q^51 - 290253653236800 * q^55 - 335129108488344 * q^57 + 362269793083208 * q^61 + 266698363786344 * q^63 + 26774405363464 * q^67 + 58823173290816 * q^69 + 317628887539976 * q^73 + 1511365865026500 * q^75 - 4794017165184920 * q^79 - 6444192852054396 * q^81 + 7999994092573440 * q^85 + 8850169595242080 * q^87 - 9328538231657008 * q^91 - 2064207396761784 * q^93 - 13833795002601784 * q^97 - 18943177097338560 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3814x^{2} + 2981440$$ :

 $$\beta_{1}$$ $$=$$ $$48\nu$$ 48*v $$\beta_{2}$$ $$=$$ $$( -3\nu^{3} + 132\nu^{2} - 6558\nu + 251724 ) / 22$$ (-3*v^3 + 132*v^2 - 6558*v + 251724) / 22 $$\beta_{3}$$ $$=$$ $$( 9\nu^{3} + 396\nu^{2} + 20730\nu + 755172 ) / 22$$ (9*v^3 + 396*v^2 + 20730*v + 755172) / 22
 $$\nu$$ $$=$$ $$( \beta_1 ) / 48$$ (b1) / 48 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3\beta_{2} - \beta _1 - 68652 ) / 36$$ (b3 + 3*b2 - b1 - 68652) / 36 $$\nu^{3}$$ $$=$$ $$( 88\beta_{3} - 264\beta_{2} - 3367\beta_1 ) / 72$$ (88*b3 - 264*b2 - 3367*b1) / 72

## 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
 − 33.1293i 33.1293i 52.1196i − 52.1196i
0 −4343.70 4917.21i 0 482304.i 0 −5.53804e6 0 −5.31128e6 + 4.27178e7i 0
17.2 0 −4343.70 + 4917.21i 0 482304.i 0 −5.53804e6 0 −5.31128e6 4.27178e7i 0
17.3 0 5369.70 3770.02i 0 276758.i 0 7.10881e6 0 1.46206e7 4.04878e7i 0
17.4 0 5369.70 + 3770.02i 0 276758.i 0 7.10881e6 0 1.46206e7 + 4.04878e7i 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 48.17.e.b 4
3.b odd 2 1 inner 48.17.e.b 4
4.b odd 2 1 3.17.b.a 4
12.b even 2 1 3.17.b.a 4
20.d odd 2 1 75.17.c.d 4
20.e even 4 2 75.17.d.b 8
60.h even 2 1 75.17.c.d 4
60.l odd 4 2 75.17.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.17.b.a 4 4.b odd 2 1
3.17.b.a 4 12.b even 2 1
48.17.e.b 4 1.a even 1 1 trivial
48.17.e.b 4 3.b odd 2 1 inner
75.17.c.d 4 20.d odd 2 1
75.17.c.d 4 60.h even 2 1
75.17.d.b 8 20.e even 4 2
75.17.d.b 8 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 309212805600T_{5}^{2} + 17817390680833497600000$$ acting on $$S_{17}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + \cdots + 18\!\cdots\!41$$
$5$ $$T^{4} + \cdots + 17\!\cdots\!00$$
$7$ $$(T^{2} + \cdots - 39368833865900)^{2}$$
$11$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$13$ $$(T^{2} + \cdots + 53\!\cdots\!60)^{2}$$
$17$ $$T^{4} + \cdots + 28\!\cdots\!60$$
$19$ $$(T^{2} + \cdots - 15\!\cdots\!64)^{2}$$
$23$ $$T^{4} + \cdots + 32\!\cdots\!40$$
$29$ $$T^{4} + \cdots + 52\!\cdots\!00$$
$31$ $$(T^{2} + \cdots + 38\!\cdots\!44)^{2}$$
$37$ $$(T^{2} + \cdots - 79\!\cdots\!60)^{2}$$
$41$ $$T^{4} + \cdots + 10\!\cdots\!00$$
$43$ $$(T^{2} + \cdots - 47\!\cdots\!00)^{2}$$
$47$ $$T^{4} + \cdots + 26\!\cdots\!60$$
$53$ $$T^{4} + \cdots + 12\!\cdots\!60$$
$59$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$61$ $$(T^{2} + \cdots + 77\!\cdots\!04)^{2}$$
$67$ $$(T^{2} + \cdots - 28\!\cdots\!20)^{2}$$
$71$ $$T^{4} + \cdots + 29\!\cdots\!00$$
$73$ $$(T^{2} + \cdots - 92\!\cdots\!60)^{2}$$
$79$ $$(T^{2} + \cdots + 10\!\cdots\!96)^{2}$$
$83$ $$T^{4} + \cdots + 20\!\cdots\!60$$
$89$ $$T^{4} + \cdots + 56\!\cdots\!00$$
$97$ $$(T^{2} + \cdots + 81\!\cdots\!20)^{2}$$