# Properties

 Label 48.12.a.f Level $48$ Weight $12$ Character orbit 48.a Self dual yes Analytic conductor $36.880$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 48.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.8804726669$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 243 q^{3} - 5370 q^{5} + 27760 q^{7} + 59049 q^{9}+O(q^{10})$$ q + 243 * q^3 - 5370 * q^5 + 27760 * q^7 + 59049 * q^9 $$q + 243 q^{3} - 5370 q^{5} + 27760 q^{7} + 59049 q^{9} - 637836 q^{11} + 766214 q^{13} - 1304910 q^{15} + 3084354 q^{17} + 19511404 q^{19} + 6745680 q^{21} - 15312360 q^{23} - 19991225 q^{25} + 14348907 q^{27} + 10751262 q^{29} + 50937400 q^{31} - 154994148 q^{33} - 149071200 q^{35} + 664740830 q^{37} + 186190002 q^{39} + 898833450 q^{41} + 957947188 q^{43} - 317093130 q^{45} + 1555741344 q^{47} - 1206709143 q^{49} + 749498022 q^{51} + 3792417030 q^{53} + 3425179320 q^{55} + 4741271172 q^{57} - 555306924 q^{59} + 4950420998 q^{61} + 1639200240 q^{63} - 4114569180 q^{65} - 5292399284 q^{67} - 3720903480 q^{69} + 14831086248 q^{71} + 13971005210 q^{73} - 4857867675 q^{75} - 17706327360 q^{77} - 3720542360 q^{79} + 3486784401 q^{81} - 8768454036 q^{83} - 16562980980 q^{85} + 2612556666 q^{87} - 25472769174 q^{89} + 21270100640 q^{91} + 12377788200 q^{93} - 104776239480 q^{95} - 39092494846 q^{97} - 37663577964 q^{99}+O(q^{100})$$ q + 243 * q^3 - 5370 * q^5 + 27760 * q^7 + 59049 * q^9 - 637836 * q^11 + 766214 * q^13 - 1304910 * q^15 + 3084354 * q^17 + 19511404 * q^19 + 6745680 * q^21 - 15312360 * q^23 - 19991225 * q^25 + 14348907 * q^27 + 10751262 * q^29 + 50937400 * q^31 - 154994148 * q^33 - 149071200 * q^35 + 664740830 * q^37 + 186190002 * q^39 + 898833450 * q^41 + 957947188 * q^43 - 317093130 * q^45 + 1555741344 * q^47 - 1206709143 * q^49 + 749498022 * q^51 + 3792417030 * q^53 + 3425179320 * q^55 + 4741271172 * q^57 - 555306924 * q^59 + 4950420998 * q^61 + 1639200240 * q^63 - 4114569180 * q^65 - 5292399284 * q^67 - 3720903480 * q^69 + 14831086248 * q^71 + 13971005210 * q^73 - 4857867675 * q^75 - 17706327360 * q^77 - 3720542360 * q^79 + 3486784401 * q^81 - 8768454036 * q^83 - 16562980980 * q^85 + 2612556666 * q^87 - 25472769174 * q^89 + 21270100640 * q^91 + 12377788200 * q^93 - 104776239480 * q^95 - 39092494846 * q^97 - 37663577964 * q^99

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 243.000 0 −5370.00 0 27760.0 0 59049.0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.12.a.f 1
3.b odd 2 1 144.12.a.l 1
4.b odd 2 1 3.12.a.a 1
8.b even 2 1 192.12.a.g 1
8.d odd 2 1 192.12.a.q 1
12.b even 2 1 9.12.a.a 1
20.d odd 2 1 75.12.a.a 1
20.e even 4 2 75.12.b.a 2
28.d even 2 1 147.12.a.c 1
36.f odd 6 2 81.12.c.a 2
36.h even 6 2 81.12.c.e 2
60.h even 2 1 225.12.a.f 1
60.l odd 4 2 225.12.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 4.b odd 2 1
9.12.a.a 1 12.b even 2 1
48.12.a.f 1 1.a even 1 1 trivial
75.12.a.a 1 20.d odd 2 1
75.12.b.a 2 20.e even 4 2
81.12.c.a 2 36.f odd 6 2
81.12.c.e 2 36.h even 6 2
144.12.a.l 1 3.b odd 2 1
147.12.a.c 1 28.d even 2 1
192.12.a.g 1 8.b even 2 1
192.12.a.q 1 8.d odd 2 1
225.12.a.f 1 60.h even 2 1
225.12.b.a 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 5370$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 243$$
$5$ $$T + 5370$$
$7$ $$T - 27760$$
$11$ $$T + 637836$$
$13$ $$T - 766214$$
$17$ $$T - 3084354$$
$19$ $$T - 19511404$$
$23$ $$T + 15312360$$
$29$ $$T - 10751262$$
$31$ $$T - 50937400$$
$37$ $$T - 664740830$$
$41$ $$T - 898833450$$
$43$ $$T - 957947188$$
$47$ $$T - 1555741344$$
$53$ $$T - 3792417030$$
$59$ $$T + 555306924$$
$61$ $$T - 4950420998$$
$67$ $$T + 5292399284$$
$71$ $$T - 14831086248$$
$73$ $$T - 13971005210$$
$79$ $$T + 3720542360$$
$83$ $$T + 8768454036$$
$89$ $$T + 25472769174$$
$97$ $$T + 39092494846$$