# Properties

 Label 48.15.e.d Level $48$ Weight $15$ Character orbit 48.e 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(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, 15, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 15);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$15$$ 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: $$59.6779047129$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} + 23x^{2} - 22x + 51$$ x^4 - 2*x^3 + 23*x^2 - 22*x + 51 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{16}\cdot 3^{7}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1 + 819) q^{3} + ( - 19 \beta_{3} - 38 \beta_{2} - 49 \beta_1) q^{5} + ( - 259 \beta_{3} + 148 \beta_{2} + 413518) q^{7} + (1053 \beta_{3} - 1170 \beta_{2} + \cdots - 538407) q^{9}+O(q^{10})$$ q + (-b2 - b1 + 819) * q^3 + (-19*b3 - 38*b2 - 49*b1) * q^5 + (-259*b3 + 148*b2 + 413518) * q^7 + (1053*b3 - 1170*b2 + 4743*b1 - 538407) * q^9 $$q + ( - \beta_{2} - \beta_1 + 819) q^{3} + ( - 19 \beta_{3} - 38 \beta_{2} - 49 \beta_1) q^{5} + ( - 259 \beta_{3} + 148 \beta_{2} + 413518) q^{7} + (1053 \beta_{3} - 1170 \beta_{2} + \cdots - 538407) q^{9}+ \cdots + (26205950331 \beta_{3} + \cdots + 39602152594368) q^{99}+O(q^{100})$$ q + (-b2 - b1 + 819) * q^3 + (-19*b3 - 38*b2 - 49*b1) * q^5 + (-259*b3 + 148*b2 + 413518) * q^7 + (1053*b3 - 1170*b2 + 4743*b1 - 538407) * q^9 + (6415*b3 + 12830*b2 - 17386*b1) * q^11 + (1820*b3 - 1040*b2 - 62053030) * q^13 + (25785*b3 - 28554*b2 + 115734*b1 - 130044960) * q^15 + (8956*b3 + 17912*b2 - 377256*b1) * q^17 + (361963*b3 - 206836*b2 + 313016902) * q^19 + (-350649*b3 - 569362*b2 - 2538391*b1 - 628041078) * q^21 + (746066*b3 + 1492132*b2 - 7576436*b1) * q^23 + (1524180*b3 - 870960*b2 - 252624215) * q^25 + (-270459*b3 - 47223*b2 + 17198001*b1 + 4711196763) * q^27 + (761687*b3 + 1523374*b2 + 19683877*b1) * q^29 + (6262305*b3 - 3578460*b2 - 12136121378) * q^31 + (5951907*b3 + 16155282*b2 - 70121355*b1 + 31712047776) * q^33 + (-14641162*b3 - 29282324*b2 - 124068472*b1) * q^35 + (27151684*b3 - 15515248*b2 - 57340274902) * q^37 + (2464020*b3 + 63148150*b2 + 76984570*b1 - 44028317970) * q^39 + (-28498306*b3 - 56996612*b2 + 81870938*b1) * q^41 + (957243*b3 - 546996*b2 + 118780954006) * q^43 + (-6505839*b3 + 115835130*b2 + 536995683*b1 + 19589532480) * q^45 + (14994420*b3 + 29988840*b2 + 1107981440*b1) * q^47 + (-214202324*b3 + 122401328*b2 + 136604468595) * q^49 + (160798284*b3 + 90327240*b2 - 420876396*b1 - 82597408128) * q^51 + (-19035919*b3 - 38071838*b2 - 1779317541*b1) * q^53 + (-81393732*b3 + 46510704*b2 + 1548779595840) * q^55 + (490046193*b3 - 95218594*b2 + 2656579259*b1 + 1607380500978) * q^57 + (136521107*b3 + 273042214*b2 - 1501312982*b1) * q^59 + (1206734676*b3 - 689562672*b2 - 2002126733446) * q^61 + (1099860363*b3 + 486456696*b2 + 414198540*b1 - 3389590546146) * q^63 + (1226681170*b3 + 2453362340*b2 + 3770046670*b1) * q^65 + (1960855743*b3 - 1120488996*b2 + 1426624547302) * q^67 + (3091726314*b3 + 2945315580*b2 - 13237442706*b1 + 1691719480512) * q^69 + (1176185318*b3 + 2352370636*b2 - 1464781812*b1) * q^71 + (882648144*b3 - 504370368*b2 + 7033949980850) * q^73 + (2063521980*b3 + 1169745095*b2 + 12757214675*b1 + 5482072134315) * q^75 + (1086705106*b3 + 2173410212*b2 + 30028135238*b1) * q^77 + (707806897*b3 - 404461084*b2 + 20073013180798) * q^79 + (-7670399490*b3 - 8285513652*b2 + 10112960646*b1 + 8858907955377) * q^81 + (10471726701*b3 + 20943453402*b2 + 18833137322*b1) * q^83 + (4393257456*b3 - 2510432832*b2 - 4051155029760) * q^85 + (-8688524805*b3 - 2257453758*b2 + 11573730618*b1 + 11582168050080) * q^87 + (17315815378*b3 + 34631630756*b2 - 18724551558*b1) * q^89 + (16824337530*b3 - 9613907160*b2 - 30184258517140) * q^91 + (8478266355*b3 + 15904239758*b2 + 63512966213*b1 + 13434444757818) * q^93 + (3534041102*b3 + 7068082204*b2 + 129735390932*b1) * q^95 + (13872970076*b3 - 7927411472*b2 + 44561041643714) * q^97 + (26205950331*b3 - 4048290306*b2 - 173709411894*b1 + 39602152594368) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3276 q^{3} + 1654072 q^{7} - 2153628 q^{9}+O(q^{10})$$ 4 * q + 3276 * q^3 + 1654072 * q^7 - 2153628 * q^9 $$4 q + 3276 q^{3} + 1654072 q^{7} - 2153628 q^{9} - 248212120 q^{13} - 520179840 q^{15} + 1252067608 q^{19} - 2512164312 q^{21} - 1010496860 q^{25} + 18844787052 q^{27} - 48544485512 q^{31} + 126848191104 q^{33} - 229361099608 q^{37} - 176113271880 q^{39} + 475123816024 q^{43} + 78358129920 q^{45} + 546417874380 q^{49} - 330389632512 q^{51} + 6195118383360 q^{55} + 6429522003912 q^{57} - 8008506933784 q^{61} - 13558362184584 q^{63} + 5706498189208 q^{67} + 6766877922048 q^{69} + 28135799923400 q^{73} + 21928288537260 q^{75} + 80292052723192 q^{79} + 35435631821508 q^{81} - 16204620119040 q^{85} + 46328672200320 q^{87} - 120737034068560 q^{91} + 53737779031272 q^{93} + 178244166574856 q^{97} + 158408610377472 q^{99}+O(q^{100})$$ 4 * q + 3276 * q^3 + 1654072 * q^7 - 2153628 * q^9 - 248212120 * q^13 - 520179840 * q^15 + 1252067608 * q^19 - 2512164312 * q^21 - 1010496860 * q^25 + 18844787052 * q^27 - 48544485512 * q^31 + 126848191104 * q^33 - 229361099608 * q^37 - 176113271880 * q^39 + 475123816024 * q^43 + 78358129920 * q^45 + 546417874380 * q^49 - 330389632512 * q^51 + 6195118383360 * q^55 + 6429522003912 * q^57 - 8008506933784 * q^61 - 13558362184584 * q^63 + 5706498189208 * q^67 + 6766877922048 * q^69 + 28135799923400 * q^73 + 21928288537260 * q^75 + 80292052723192 * q^79 + 35435631821508 * q^81 - 16204620119040 * q^85 + 46328672200320 * q^87 - 120737034068560 * q^91 + 53737779031272 * q^93 + 178244166574856 * q^97 + 158408610377472 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 23x^{2} - 22x + 51$$ :

 $$\beta_{1}$$ $$=$$ $$( 256\nu^{3} - 384\nu^{2} + 3968\nu - 1920 ) / 9$$ (256*v^3 - 384*v^2 + 3968*v - 1920) / 9 $$\beta_{2}$$ $$=$$ $$( -280\nu^{3} - 876\nu^{2} - 7580\nu - 9888 ) / 9$$ (-280*v^3 - 876*v^2 - 7580*v - 9888) / 9 $$\beta_{3}$$ $$=$$ $$( -160\nu^{3} + 2832\nu^{2} - 7664\nu + 31008 ) / 9$$ (-160*v^3 + 2832*v^2 - 7664*v + 31008) / 9
 $$\nu$$ $$=$$ $$( -16\beta_{3} - 32\beta_{2} - 45\beta _1 + 10368 ) / 20736$$ (-16*b3 - 32*b2 - 45*b1 + 10368) / 20736 $$\nu^{2}$$ $$=$$ $$( 40\beta_{3} - 64\beta_{2} - 45\beta _1 - 217728 ) / 20736$$ (40*b3 - 64*b2 - 45*b1 - 217728) / 20736 $$\nu^{3}$$ $$=$$ $$( 308\beta_{3} + 400\beta_{2} + 1359\beta _1 - 331776 ) / 20736$$ (308*b3 + 400*b2 + 1359*b1 - 331776) / 20736

## 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
 0.5 − 4.37225i 0.5 + 4.37225i 0.5 − 1.54383i 0.5 + 1.54383i
0 −385.790 2152.70i 0 105253.i 0 1.21591e6 0 −4.48530e6 + 1.66099e6i 0
17.2 0 −385.790 + 2152.70i 0 105253.i 0 1.21591e6 0 −4.48530e6 1.66099e6i 0
17.3 0 2023.79 829.000i 0 40425.0i 0 −388872. 0 3.40849e6 3.35545e6i 0
17.4 0 2023.79 + 829.000i 0 40425.0i 0 −388872. 0 3.40849e6 + 3.35545e6i 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 48.15.e.d 4
3.b odd 2 1 inner 48.15.e.d 4
4.b odd 2 1 6.15.b.a 4
12.b even 2 1 6.15.b.a 4
20.d odd 2 1 150.15.d.a 4
20.e even 4 2 150.15.b.a 8
60.h even 2 1 150.15.d.a 4
60.l odd 4 2 150.15.b.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12712279680T_{5}^{2} + 18103614409876377600$$ acting on $$S_{15}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + \cdots + 22876792454961$$
$5$ $$T^{4} + \cdots + 18\!\cdots\!00$$
$7$ $$(T^{2} - 827036 T - 472833268796)^{2}$$
$11$ $$T^{4} + \cdots + 30\!\cdots\!44$$
$13$ $$(T^{2} + \cdots + 38\!\cdots\!00)^{2}$$
$17$ $$T^{4} + \cdots + 15\!\cdots\!44$$
$19$ $$(T^{2} + \cdots - 11\!\cdots\!76)^{2}$$
$23$ $$T^{4} + \cdots + 47\!\cdots\!64$$
$29$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$31$ $$(T^{2} + \cdots - 22\!\cdots\!16)^{2}$$
$37$ $$(T^{2} + \cdots - 37\!\cdots\!16)^{2}$$
$41$ $$T^{4} + \cdots + 11\!\cdots\!64$$
$43$ $$(T^{2} + \cdots + 14\!\cdots\!56)^{2}$$
$47$ $$T^{4} + \cdots + 13\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 88\!\cdots\!24$$
$59$ $$T^{4} + \cdots + 10\!\cdots\!64$$
$61$ $$(T^{2} + \cdots - 99\!\cdots\!04)^{2}$$
$67$ $$(T^{2} + \cdots - 34\!\cdots\!76)^{2}$$
$71$ $$T^{4} + \cdots + 40\!\cdots\!00$$
$73$ $$(T^{2} + \cdots + 41\!\cdots\!80)^{2}$$
$79$ $$(T^{2} + \cdots + 39\!\cdots\!24)^{2}$$
$83$ $$T^{4} + \cdots + 20\!\cdots\!04$$
$89$ $$T^{4} + \cdots + 19\!\cdots\!64$$
$97$ $$(T^{2} + \cdots + 13\!\cdots\!76)^{2}$$