# Properties

 Label 48.15.g.a Level $48$ Weight $15$ Character orbit 48.g Analytic conductor $59.678$ 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] = [48,15,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, 15, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 15);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$15$$ 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: $$59.6779047129$$ 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: $$x^{4} - x^{3} + 16897x^{2} + 16896x + 285474816$$ x^4 - x^3 + 16897*x^2 + 16896*x + 285474816 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{9}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + (7 \beta_{2} + 4590) q^{5} + ( - 17 \beta_{3} - 354 \beta_1) q^{7} - 1594323 q^{9}+O(q^{10})$$ q - b1 * q^3 + (7*b2 + 4590) * q^5 + (-17*b3 - 354*b1) * q^7 - 1594323 * q^9 $$q - \beta_1 q^{3} + (7 \beta_{2} + 4590) q^{5} + ( - 17 \beta_{3} - 354 \beta_1) q^{7} - 1594323 q^{9} + (618 \beta_{3} + 2872 \beta_1) q^{11} + ( - 13338 \beta_{2} - 39597038) q^{13} + ( - 1701 \beta_{3} - 5388 \beta_1) q^{15} + (80378 \beta_{2} + 65624202) q^{17} + (36466 \beta_{3} + 222432 \beta_1) q^{19} + ( - 111537 \beta_{2} - 551675124) q^{21} + ( - 106818 \beta_{3} - 1129660 \beta_1) q^{23} + (64260 \beta_{2} - 4174928485) q^{25} + 1594323 \beta_1 q^{27} + ( - 1206967 \beta_{2} - 14985012042) q^{29} + (43631 \beta_{3} + 6621078 \beta_1) q^{31} + (4054698 \beta_{2} + 4116660084) q^{33} + ( - 666618 \beta_{3} - 20964964 \beta_1) q^{35} + ( - 7502004 \beta_{2} + 3681968330) q^{37} + (3241134 \beta_{3} + 41117570 \beta_1) q^{39} + (18965670 \beta_{2} + 111582587610) q^{41} + (102238 \beta_{3} + 83448528 \beta_1) q^{43} + ( - 11160261 \beta_{2} - 7317942570) q^{45} + ( - 2052654 \beta_{3} + 164683460 \beta_1) q^{47} + ( - 77189112 \beta_{2} + 183567183457) q^{49} + ( - 19531854 \beta_{3} - 74787294 \beta_1) q^{51} + (47863929 \beta_{2} + 209252792934) q^{53} + (7228728 \beta_{3} + 708274584 \beta_1) q^{55} + (239253426 \beta_{2} + 327353562972) q^{57} + (28471272 \beta_{3} + 2578572004 \beta_1) q^{59} + ( - 39111768 \beta_{2} - 957183231766) q^{61} + (27103491 \beta_{3} + 564390342 \beta_1) q^{63} + ( - 338400686 \beta_{2} - 3816391683780) q^{65} + ( - 72809688 \beta_{3} + 4833709164 \beta_1) q^{67} + ( - 700832898 \beta_{2} - 1721147969808) q^{69} + (120972054 \beta_{3} + 8842669124 \beta_1) q^{71} + (538384104 \beta_{2} - 2240021401742) q^{73} + ( - 15615180 \beta_{3} + 4167602845 \beta_1) q^{75} + (1691022420 \beta_{2} + 12467132686512) q^{77} + ( - 58878701 \beta_{3} + 22433757486 \beta_1) q^{79} + 2541865828329 q^{81} + ( - 896010918 \beta_{3} + 18369881600 \beta_1) q^{83} + (828304434 \beta_{2} + 22204438715340) q^{85} + (293292981 \beta_{3} + 15122606280 \beta_1) q^{87} + ( - 7693399124 \beta_{2} - 19824177873294) q^{89} + (1794662038 \beta_{3} + 50868538188 \beta_1) q^{91} + (286262991 \beta_{2} + 10523502959220) q^{93} + (516635904 \beta_{3} + 42078162392 \beta_1) q^{95} + (8416384380 \beta_{2} - 20423490292094) q^{97} + ( - 985291614 \beta_{3} - 4578895656 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 + (7*b2 + 4590) * q^5 + (-17*b3 - 354*b1) * q^7 - 1594323 * q^9 + (618*b3 + 2872*b1) * q^11 + (-13338*b2 - 39597038) * q^13 + (-1701*b3 - 5388*b1) * q^15 + (80378*b2 + 65624202) * q^17 + (36466*b3 + 222432*b1) * q^19 + (-111537*b2 - 551675124) * q^21 + (-106818*b3 - 1129660*b1) * q^23 + (64260*b2 - 4174928485) * q^25 + 1594323*b1 * q^27 + (-1206967*b2 - 14985012042) * q^29 + (43631*b3 + 6621078*b1) * q^31 + (4054698*b2 + 4116660084) * q^33 + (-666618*b3 - 20964964*b1) * q^35 + (-7502004*b2 + 3681968330) * q^37 + (3241134*b3 + 41117570*b1) * q^39 + (18965670*b2 + 111582587610) * q^41 + (102238*b3 + 83448528*b1) * q^43 + (-11160261*b2 - 7317942570) * q^45 + (-2052654*b3 + 164683460*b1) * q^47 + (-77189112*b2 + 183567183457) * q^49 + (-19531854*b3 - 74787294*b1) * q^51 + (47863929*b2 + 209252792934) * q^53 + (7228728*b3 + 708274584*b1) * q^55 + (239253426*b2 + 327353562972) * q^57 + (28471272*b3 + 2578572004*b1) * q^59 + (-39111768*b2 - 957183231766) * q^61 + (27103491*b3 + 564390342*b1) * q^63 + (-338400686*b2 - 3816391683780) * q^65 + (-72809688*b3 + 4833709164*b1) * q^67 + (-700832898*b2 - 1721147969808) * q^69 + (120972054*b3 + 8842669124*b1) * q^71 + (538384104*b2 - 2240021401742) * q^73 + (-15615180*b3 + 4167602845*b1) * q^75 + (1691022420*b2 + 12467132686512) * q^77 + (-58878701*b3 + 22433757486*b1) * q^79 + 2541865828329 * q^81 + (-896010918*b3 + 18369881600*b1) * q^83 + (828304434*b2 + 22204438715340) * q^85 + (293292981*b3 + 15122606280*b1) * q^87 + (-7693399124*b2 - 19824177873294) * q^89 + (1794662038*b3 + 50868538188*b1) * q^91 + (286262991*b2 + 10523502959220) * q^93 + (516635904*b3 + 42078162392*b1) * q^95 + (8416384380*b2 - 20423490292094) * q^97 + (-985291614*b3 - 4578895656*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 18360 q^{5} - 6377292 q^{9}+O(q^{10})$$ 4 * q + 18360 * q^5 - 6377292 * q^9 $$4 q + 18360 q^{5} - 6377292 q^{9} - 158388152 q^{13} + 262496808 q^{17} - 2206700496 q^{21} - 16699713940 q^{25} - 59940048168 q^{29} + 16466640336 q^{33} + 14727873320 q^{37} + 446330350440 q^{41} - 29271770280 q^{45} + 734268733828 q^{49} + 837011171736 q^{53} + 1309414251888 q^{57} - 3828732927064 q^{61} - 15265566735120 q^{65} - 6884591879232 q^{69} - 8960085606968 q^{73} + 49868530746048 q^{77} + 10167463313316 q^{81} + 88817754861360 q^{85} - 79296711493176 q^{89} + 42094011836880 q^{93} - 81693961168376 q^{97}+O(q^{100})$$ 4 * q + 18360 * q^5 - 6377292 * q^9 - 158388152 * q^13 + 262496808 * q^17 - 2206700496 * q^21 - 16699713940 * q^25 - 59940048168 * q^29 + 16466640336 * q^33 + 14727873320 * q^37 + 446330350440 * q^41 - 29271770280 * q^45 + 734268733828 * q^49 + 837011171736 * q^53 + 1309414251888 * q^57 - 3828732927064 * q^61 - 15265566735120 * q^65 - 6884591879232 * q^69 - 8960085606968 * q^73 + 49868530746048 * q^77 + 10167463313316 * q^81 + 88817754861360 * q^85 - 79296711493176 * q^89 + 42094011836880 * q^93 - 81693961168376 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 16897x^{2} + 16896x + 285474816$$ :

 $$\beta_{1}$$ $$=$$ $$( -243\nu^{3} + 4105971\nu^{2} - 4105971\nu + 34683137280 ) / 47581952$$ (-243*v^3 + 4105971*v^2 - 4105971*v + 34683137280) / 47581952 $$\beta_{2}$$ $$=$$ $$( -48\nu^{3} - 1216536 ) / 16897$$ (-48*v^3 - 1216536) / 16897 $$\beta_{3}$$ $$=$$ $$( 202821\nu^{3} - 1165893\nu^{2} + 6852966981\nu - 6422600448 ) / 23790976$$ (202821*v^3 - 1165893*v^2 + 6852966981*v - 6422600448) / 23790976
 $$\nu$$ $$=$$ $$( 81\beta_{3} + 243\beta_{2} + 46\beta _1 + 5832 ) / 23328$$ (81*b3 + 243*b2 + 46*b1 + 5832) / 23328 $$\nu^{2}$$ $$=$$ $$( 81\beta_{3} - 243\beta_{2} + 270382\beta _1 - 197080776 ) / 23328$$ (81*b3 - 243*b2 + 270382*b1 - 197080776) / 23328 $$\nu^{3}$$ $$=$$ $$( -16897\beta_{2} - 1216536 ) / 48$$ (-16897*b2 - 1216536) / 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
 −64.7428 − 112.138i 65.2428 + 113.004i −64.7428 + 112.138i 65.2428 − 113.004i
0 1262.67i 0 −39085.2 0 114233.i 0 −1.59432e6 0
31.2 0 1262.67i 0 48265.2 0 988060.i 0 −1.59432e6 0
31.3 0 1262.67i 0 −39085.2 0 114233.i 0 −1.59432e6 0
31.4 0 1262.67i 0 48265.2 0 988060.i 0 −1.59432e6 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 48.15.g.a 4
3.b odd 2 1 144.15.g.e 4
4.b odd 2 1 inner 48.15.g.a 4
12.b even 2 1 144.15.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.15.g.a 4 1.a even 1 1 trivial
48.15.g.a 4 4.b odd 2 1 inner
144.15.g.e 4 3.b odd 2 1
144.15.g.e 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 9180T_{5} - 1886450940$$ acting on $$S_{15}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1594323)^{2}$$
$5$ $$(T^{2} - 9180 T - 1886450940)^{2}$$
$7$ $$T^{4} + \cdots + 12\!\cdots\!24$$
$11$ $$T^{4} + \cdots + 15\!\cdots\!64$$
$13$ $$(T^{2} + \cdots - 53\!\cdots\!96)^{2}$$
$17$ $$(T^{2} + \cdots - 24\!\cdots\!36)^{2}$$
$19$ $$T^{4} + \cdots + 17\!\cdots\!44$$
$23$ $$T^{4} + \cdots + 10\!\cdots\!44$$
$29$ $$(T^{2} + \cdots + 16\!\cdots\!24)^{2}$$
$31$ $$T^{4} + \cdots + 45\!\cdots\!00$$
$37$ $$(T^{2} + \cdots - 21\!\cdots\!60)^{2}$$
$41$ $$(T^{2} + \cdots - 15\!\cdots\!00)^{2}$$
$43$ $$T^{4} + \cdots + 12\!\cdots\!44$$
$47$ $$T^{4} + \cdots + 15\!\cdots\!84$$
$53$ $$(T^{2} + \cdots - 45\!\cdots\!04)^{2}$$
$59$ $$T^{4} + \cdots + 92\!\cdots\!44$$
$61$ $$(T^{2} + \cdots + 85\!\cdots\!16)^{2}$$
$67$ $$T^{4} + \cdots + 10\!\cdots\!24$$
$71$ $$T^{4} + \cdots + 11\!\cdots\!24$$
$73$ $$(T^{2} + \cdots - 62\!\cdots\!96)^{2}$$
$79$ $$T^{4} + \cdots + 64\!\cdots\!24$$
$83$ $$T^{4} + \cdots + 78\!\cdots\!84$$
$89$ $$(T^{2} + \cdots - 19\!\cdots\!24)^{2}$$
$97$ $$(T^{2} + \cdots - 23\!\cdots\!64)^{2}$$