Properties

 Label 48.6.c.b Level 48 Weight 6 Character orbit 48.c Analytic conductor 7.698 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.69842335102$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 9 - 18 \zeta_{6} ) q^{3} + ( 62 - 124 \zeta_{6} ) q^{7} -243 q^{9} +O(q^{10})$$ $$q + ( 9 - 18 \zeta_{6} ) q^{3} + ( 62 - 124 \zeta_{6} ) q^{7} -243 q^{9} -1202 q^{13} + ( 1618 - 3236 \zeta_{6} ) q^{19} -1674 q^{21} + 3125 q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} + ( -1626 + 3252 \zeta_{6} ) q^{31} + 16550 q^{37} + ( -10818 + 21636 \zeta_{6} ) q^{39} + ( 13866 - 27732 \zeta_{6} ) q^{43} + 5275 q^{49} -43686 q^{57} -38626 q^{61} + ( -15066 + 30132 \zeta_{6} ) q^{63} + ( 37138 - 74276 \zeta_{6} ) q^{67} + 1450 q^{73} + ( 28125 - 56250 \zeta_{6} ) q^{75} + ( -27050 + 54100 \zeta_{6} ) q^{79} + 59049 q^{81} + ( -74524 + 149048 \zeta_{6} ) q^{91} + 43902 q^{93} + 134386 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 486q^{9} + O(q^{10})$$ $$2q - 486q^{9} - 2404q^{13} - 3348q^{21} + 6250q^{25} + 33100q^{37} + 10550q^{49} - 87372q^{57} - 77252q^{61} + 2900q^{73} + 118098q^{81} + 87804q^{93} + 268772q^{97} + O(q^{100})$$

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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 15.5885i 0 0 0 107.387i 0 −243.000 0
47.2 0 15.5885i 0 0 0 107.387i 0 −243.000 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 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.6.c.b 2
3.b odd 2 1 CM 48.6.c.b 2
4.b odd 2 1 inner 48.6.c.b 2
8.b even 2 1 192.6.c.b 2
8.d odd 2 1 192.6.c.b 2
12.b even 2 1 inner 48.6.c.b 2
24.f even 2 1 192.6.c.b 2
24.h odd 2 1 192.6.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.c.b 2 1.a even 1 1 trivial
48.6.c.b 2 3.b odd 2 1 CM
48.6.c.b 2 4.b odd 2 1 inner
48.6.c.b 2 12.b even 2 1 inner
192.6.c.b 2 8.b even 2 1
192.6.c.b 2 8.d odd 2 1
192.6.c.b 2 24.f even 2 1
192.6.c.b 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(48, [\chi])$$:

 $$T_{5}$$ $$T_{11}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 243 T^{2}$$
$5$ $$( 1 - 3125 T^{2} )^{2}$$
$7$ $$( 1 - 236 T + 16807 T^{2} )( 1 + 236 T + 16807 T^{2} )$$
$11$ $$( 1 + 161051 T^{2} )^{2}$$
$13$ $$( 1 + 1202 T + 371293 T^{2} )^{2}$$
$17$ $$( 1 - 1419857 T^{2} )^{2}$$
$19$ $$( 1 - 1432 T + 2476099 T^{2} )( 1 + 1432 T + 2476099 T^{2} )$$
$23$ $$( 1 + 6436343 T^{2} )^{2}$$
$29$ $$( 1 - 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 10324 T + 28629151 T^{2} )( 1 + 10324 T + 28629151 T^{2} )$$
$37$ $$( 1 - 16550 T + 69343957 T^{2} )^{2}$$
$41$ $$( 1 - 115856201 T^{2} )^{2}$$
$43$ $$( 1 - 3352 T + 147008443 T^{2} )( 1 + 3352 T + 147008443 T^{2} )$$
$47$ $$( 1 + 229345007 T^{2} )^{2}$$
$53$ $$( 1 - 418195493 T^{2} )^{2}$$
$59$ $$( 1 + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 38626 T + 844596301 T^{2} )^{2}$$
$67$ $$( 1 - 35536 T + 1350125107 T^{2} )( 1 + 35536 T + 1350125107 T^{2} )$$
$71$ $$( 1 + 1804229351 T^{2} )^{2}$$
$73$ $$( 1 - 1450 T + 2073071593 T^{2} )^{2}$$
$79$ $$( 1 - 100564 T + 3077056399 T^{2} )( 1 + 100564 T + 3077056399 T^{2} )$$
$83$ $$( 1 + 3939040643 T^{2} )^{2}$$
$89$ $$( 1 - 5584059449 T^{2} )^{2}$$
$97$ $$( 1 - 134386 T + 8587340257 T^{2} )^{2}$$