Properties

 Label 48.15.g.b 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: $$x^{4} - x^{3} + 3669553x^{2} + 3669552x + 13465611880704$$ x^4 - x^3 + 3669553*x^2 + 3669552*x + 13465611880704 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.

 $$f(q)$$ $$=$$ $$q + 729 \beta_1 q^{3} + (\beta_{2} + 25278) q^{5} + ( - 5 \beta_{3} - 414484 \beta_1) q^{7} - 1594323 q^{9}+O(q^{10})$$ q + 729*b1 * q^3 + (b2 + 25278) * q^5 + (-5*b3 - 414484*b1) * q^7 - 1594323 * q^9 $$q + 729 \beta_1 q^{3} + (\beta_{2} + 25278) q^{5} + ( - 5 \beta_{3} - 414484 \beta_1) q^{7} - 1594323 q^{9} + ( - 110 \beta_{3} - 4984188 \beta_1) q^{11} + ( - 390 \beta_{2} - 28411214) q^{13} + ( - 729 \beta_{3} + 18427662 \beta_1) q^{15} + ( - 4330 \beta_{2} - 55535766) q^{17} + ( - 230 \beta_{3} + 419025484 \beta_1) q^{19} + ( - 10935 \beta_{2} + 906476508) q^{21} + (10710 \beta_{3} + 1191600720 \beta_1) q^{23} + (50556 \beta_{2} + 2990110043) q^{25} - 1162261467 \beta_1 q^{27} + (209615 \beta_{2} - 4383176730) q^{29} + ( - 88645 \beta_{3} + 3525228468 \beta_1) q^{31} + ( - 240570 \beta_{2} + 10900419156) q^{33} + (288094 \beta_{3} + 31795915368 \beta_1) q^{35} + ( - 929340 \beta_{2} + 53126740106) q^{37} + (284310 \beta_{3} - 20711775006 \beta_1) q^{39} + (85770 \beta_{2} - 268267911174) q^{41} + ( - 3102410 \beta_{3} - 10594113948 \beta_1) q^{43} + ( - 1594323 \beta_{2} - 40301296794) q^{45} + (2829370 \beta_{3} - 229834692024 \beta_1) q^{47} + (12434520 \beta_{2} - 471266514719) q^{49} + (3156570 \beta_{3} - 40485573414 \beta_1) q^{51} + ( - 8100705 \beta_{2} + 317712640662) q^{53} + (2203608 \beta_{3} + 804021017976 \beta_1) q^{55} + ( - 503010 \beta_{2} - 916408733508) q^{57} + ( - 8795320 \beta_{3} + 1027208495148 \beta_1) q^{59} + ( - 2053560 \beta_{2} - 2672800762390) q^{61} + (7971615 \beta_{3} + 660821374332 \beta_1) q^{63} + ( - 38269634 \beta_{2} - 4015491537252) q^{65} + ( - 32229240 \beta_{3} + 2656900077412 \beta_1) q^{67} + (23422770 \beta_{2} - 2606030774640) q^{69} + (9841870 \beta_{3} + 2890058549088 \beta_1) q^{71} + ( - 46820040 \beta_{2} - 10233190147598) q^{73} + ( - 36855324 \beta_{3} + 2179790221347 \beta_1) q^{75} + (211542540 \beta_{2} - 20147768370576) q^{77} + (192261295 \beta_{3} + 2150637439156 \beta_1) q^{79} + 2541865828329 q^{81} + (8978690 \beta_{3} + 15341394888348 \beta_1) q^{83} + ( - 164989506 \beta_{2} - 38012460595668) q^{85} + ( - 152809335 \beta_{3} - 3195335836170 \beta_1) q^{87} + (63654580 \beta_{2} - 75235880686926) q^{89} + ( - 19592690 \beta_{3} - 4710570725224 \beta_1) q^{91} + ( - 193866615 \beta_{2} - 7709674659516) q^{93} + ( - 424839424 \beta_{3} + 12536695312872 \beta_1) q^{95} + ( - 101383260 \beta_{2} - 85085855560766) q^{97} + (175375530 \beta_{3} + 7946405564724 \beta_1) q^{99}+O(q^{100})$$ q + 729*b1 * q^3 + (b2 + 25278) * q^5 + (-5*b3 - 414484*b1) * q^7 - 1594323 * q^9 + (-110*b3 - 4984188*b1) * q^11 + (-390*b2 - 28411214) * q^13 + (-729*b3 + 18427662*b1) * q^15 + (-4330*b2 - 55535766) * q^17 + (-230*b3 + 419025484*b1) * q^19 + (-10935*b2 + 906476508) * q^21 + (10710*b3 + 1191600720*b1) * q^23 + (50556*b2 + 2990110043) * q^25 - 1162261467*b1 * q^27 + (209615*b2 - 4383176730) * q^29 + (-88645*b3 + 3525228468*b1) * q^31 + (-240570*b2 + 10900419156) * q^33 + (288094*b3 + 31795915368*b1) * q^35 + (-929340*b2 + 53126740106) * q^37 + (284310*b3 - 20711775006*b1) * q^39 + (85770*b2 - 268267911174) * q^41 + (-3102410*b3 - 10594113948*b1) * q^43 + (-1594323*b2 - 40301296794) * q^45 + (2829370*b3 - 229834692024*b1) * q^47 + (12434520*b2 - 471266514719) * q^49 + (3156570*b3 - 40485573414*b1) * q^51 + (-8100705*b2 + 317712640662) * q^53 + (2203608*b3 + 804021017976*b1) * q^55 + (-503010*b2 - 916408733508) * q^57 + (-8795320*b3 + 1027208495148*b1) * q^59 + (-2053560*b2 - 2672800762390) * q^61 + (7971615*b3 + 660821374332*b1) * q^63 + (-38269634*b2 - 4015491537252) * q^65 + (-32229240*b3 + 2656900077412*b1) * q^67 + (23422770*b2 - 2606030774640) * q^69 + (9841870*b3 + 2890058549088*b1) * q^71 + (-46820040*b2 - 10233190147598) * q^73 + (-36855324*b3 + 2179790221347*b1) * q^75 + (211542540*b2 - 20147768370576) * q^77 + (192261295*b3 + 2150637439156*b1) * q^79 + 2541865828329 * q^81 + (8978690*b3 + 15341394888348*b1) * q^83 + (-164989506*b2 - 38012460595668) * q^85 + (-152809335*b3 - 3195335836170*b1) * q^87 + (63654580*b2 - 75235880686926) * q^89 + (-19592690*b3 - 4710570725224*b1) * q^91 + (-193866615*b2 - 7709674659516) * q^93 + (-424839424*b3 + 12536695312872*b1) * q^95 + (-101383260*b2 - 85085855560766) * q^97 + (175375530*b3 + 7946405564724*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 101112 q^{5} - 6377292 q^{9}+O(q^{10})$$ 4 * q + 101112 * q^5 - 6377292 * q^9 $$4 q + 101112 q^{5} - 6377292 q^{9} - 113644856 q^{13} - 222143064 q^{17} + 3625906032 q^{21} + 11960440172 q^{25} - 17532706920 q^{29} + 43601676624 q^{33} + 212506960424 q^{37} - 1073071644696 q^{41} - 161205187176 q^{45} - 1885066058876 q^{49} + 1270850562648 q^{53} - 3665634934032 q^{57} - 10691203049560 q^{61} - 16061966149008 q^{65} - 10424123098560 q^{69} - 40932760590392 q^{73} - 80591073482304 q^{77} + 10167463313316 q^{81} - 152049842382672 q^{85} - 300943522747704 q^{89} - 30838698638064 q^{93} - 340343422243064 q^{97}+O(q^{100})$$ 4 * q + 101112 * q^5 - 6377292 * q^9 - 113644856 * q^13 - 222143064 * q^17 + 3625906032 * q^21 + 11960440172 * q^25 - 17532706920 * q^29 + 43601676624 * q^33 + 212506960424 * q^37 - 1073071644696 * q^41 - 161205187176 * q^45 - 1885066058876 * q^49 + 1270850562648 * q^53 - 3665634934032 * q^57 - 10691203049560 * q^61 - 16061966149008 * q^65 - 10424123098560 * q^69 - 40932760590392 * q^73 - 80591073482304 * q^77 + 10167463313316 * q^81 - 152049842382672 * q^85 - 300943522747704 * q^89 - 30838698638064 * q^93 - 340343422243064 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 3669553x^{2} + 3669552x + 13465611880704$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 3669553\nu^{2} - 3669553\nu + 6732804105576 ) / 6732807775128$$ (-v^3 + 3669553*v^2 - 3669553*v + 6732804105576) / 6732807775128 $$\beta_{2}$$ $$=$$ $$( 48\nu^{3} + 264207768 ) / 3669553$$ (48*v^3 + 264207768) / 3669553 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} + 7339105\nu + 1834776 ) / 76449$$ (v^3 - v^2 + 7339105*v + 1834776) / 76449
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96$$ (b3 - b2 + 24*b1 + 24) / 96 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 176138520\beta _1 - 176138520 ) / 96$$ (b3 + b2 + 176138520*b1 - 176138520) / 96 $$\nu^{3}$$ $$=$$ $$( 3669553\beta_{2} - 264207768 ) / 48$$ (3669553*b2 - 264207768) / 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
 958.054 − 1659.40i −957.554 + 1658.53i 958.054 + 1659.40i −957.554 − 1658.53i
0 1262.67i 0 −66671.2 0 1.51421e6i 0 −1.59432e6 0
31.2 0 1262.67i 0 117227. 0 78395.8i 0 −1.59432e6 0
31.3 0 1262.67i 0 −66671.2 0 1.51421e6i 0 −1.59432e6 0
31.4 0 1262.67i 0 117227. 0 78395.8i 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.b 4
3.b odd 2 1 144.15.g.d 4
4.b odd 2 1 inner 48.15.g.b 4
12.b even 2 1 144.15.g.d 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 50556T_{5} - 7815671100$$ acting on $$S_{15}^{\mathrm{new}}(48, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1594323)^{2}$$
$5$ $$(T^{2} - 50556 T - 7815671100)^{2}$$
$7$ $$T^{4} + \cdots + 14\!\cdots\!24$$
$11$ $$T^{4} + \cdots + 53\!\cdots\!24$$
$13$ $$(T^{2} + \cdots - 478754938252604)^{2}$$
$17$ $$(T^{2} + \cdots - 15\!\cdots\!44)^{2}$$
$19$ $$T^{4} + \cdots + 27\!\cdots\!24$$
$23$ $$T^{4} + \cdots + 18\!\cdots\!00$$
$29$ $$(T^{2} + \cdots - 35\!\cdots\!00)^{2}$$
$31$ $$T^{4} + \cdots + 26\!\cdots\!84$$
$37$ $$(T^{2} + \cdots - 44\!\cdots\!64)^{2}$$
$41$ $$(T^{2} + \cdots + 71\!\cdots\!76)^{2}$$
$43$ $$T^{4} + \cdots + 59\!\cdots\!44$$
$47$ $$T^{4} + \cdots + 19\!\cdots\!84$$
$53$ $$(T^{2} + \cdots - 45\!\cdots\!56)^{2}$$
$59$ $$T^{4} + \cdots + 14\!\cdots\!44$$
$61$ $$(T^{2} + \cdots + 71\!\cdots\!00)^{2}$$
$67$ $$T^{4} + \cdots + 26\!\cdots\!24$$
$71$ $$T^{4} + \cdots + 51\!\cdots\!24$$
$73$ $$(T^{2} + \cdots + 86\!\cdots\!04)^{2}$$
$79$ $$T^{4} + \cdots + 85\!\cdots\!64$$
$83$ $$T^{4} + \cdots + 49\!\cdots\!44$$
$89$ $$(T^{2} + \cdots + 56\!\cdots\!76)^{2}$$
$97$ $$(T^{2} + \cdots + 71\!\cdots\!56)^{2}$$