# Properties

 Label 48.7.g.a Level 48 Weight 7 Character orbit 48.g Analytic conductor 11.043 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0425960138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 9 - 18 \zeta_{6} ) q^{3} -90 q^{5} + ( -108 + 216 \zeta_{6} ) q^{7} -243 q^{9} +O(q^{10})$$ $$q + ( 9 - 18 \zeta_{6} ) q^{3} -90 q^{5} + ( -108 + 216 \zeta_{6} ) q^{7} -243 q^{9} + ( -972 + 1944 \zeta_{6} ) q^{11} + 1762 q^{13} + ( -810 + 1620 \zeta_{6} ) q^{15} -1638 q^{17} + ( -7236 + 14472 \zeta_{6} ) q^{19} + 2916 q^{21} + ( -7776 + 15552 \zeta_{6} ) q^{23} -7525 q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} -16002 q^{29} + ( 9180 - 18360 \zeta_{6} ) q^{31} + 26244 q^{33} + ( 9720 - 19440 \zeta_{6} ) q^{35} + 61130 q^{37} + ( 15858 - 31716 \zeta_{6} ) q^{39} -98550 q^{41} + ( 28404 - 56808 \zeta_{6} ) q^{43} + 21870 q^{45} + ( 103032 - 206064 \zeta_{6} ) q^{47} + 82657 q^{49} + ( -14742 + 29484 \zeta_{6} ) q^{51} -275346 q^{53} + ( 87480 - 174960 \zeta_{6} ) q^{55} + 195372 q^{57} + ( -146772 + 293544 \zeta_{6} ) q^{59} + 106634 q^{61} + ( 26244 - 52488 \zeta_{6} ) q^{63} -158580 q^{65} + ( -318492 + 636984 \zeta_{6} ) q^{67} + 209952 q^{69} + ( -42768 + 85536 \zeta_{6} ) q^{71} + 100978 q^{73} + ( -67725 + 135450 \zeta_{6} ) q^{75} -314928 q^{77} + ( 44604 - 89208 \zeta_{6} ) q^{79} + 59049 q^{81} + ( -372276 + 744552 \zeta_{6} ) q^{83} + 147420 q^{85} + ( -144018 + 288036 \zeta_{6} ) q^{87} -819054 q^{89} + ( -190296 + 380592 \zeta_{6} ) q^{91} -247860 q^{93} + ( 651240 - 1302480 \zeta_{6} ) q^{95} + 557026 q^{97} + ( 236196 - 472392 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 180q^{5} - 486q^{9} + O(q^{10})$$ $$2q - 180q^{5} - 486q^{9} + 3524q^{13} - 3276q^{17} + 5832q^{21} - 15050q^{25} - 32004q^{29} + 52488q^{33} + 122260q^{37} - 197100q^{41} + 43740q^{45} + 165314q^{49} - 550692q^{53} + 390744q^{57} + 213268q^{61} - 317160q^{65} + 419904q^{69} + 201956q^{73} - 629856q^{77} + 118098q^{81} + 294840q^{85} - 1638108q^{89} - 495720q^{93} + 1114052q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 15.5885i 0 −90.0000 0 187.061i 0 −243.000 0
31.2 0 15.5885i 0 −90.0000 0 187.061i 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
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 48.7.g.a 2
3.b odd 2 1 144.7.g.e 2
4.b odd 2 1 inner 48.7.g.a 2
8.b even 2 1 192.7.g.c 2
8.d odd 2 1 192.7.g.c 2
12.b even 2 1 144.7.g.e 2
16.e even 4 2 768.7.b.c 4
16.f odd 4 2 768.7.b.c 4
24.f even 2 1 576.7.g.e 2
24.h odd 2 1 576.7.g.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.g.a 2 1.a even 1 1 trivial
48.7.g.a 2 4.b odd 2 1 inner
144.7.g.e 2 3.b odd 2 1
144.7.g.e 2 12.b even 2 1
192.7.g.c 2 8.b even 2 1
192.7.g.c 2 8.d odd 2 1
576.7.g.e 2 24.f even 2 1
576.7.g.e 2 24.h odd 2 1
768.7.b.c 4 16.e even 4 2
768.7.b.c 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 90$$ acting on $$S_{7}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 243 T^{2}$$
$5$ $$( 1 + 90 T + 15625 T^{2} )^{2}$$
$7$ $$1 - 200306 T^{2} + 13841287201 T^{4}$$
$11$ $$1 - 708770 T^{2} + 3138428376721 T^{4}$$
$13$ $$( 1 - 1762 T + 4826809 T^{2} )^{2}$$
$17$ $$( 1 + 1638 T + 24137569 T^{2} )^{2}$$
$19$ $$1 + 62987326 T^{2} + 2213314919066161 T^{4}$$
$23$ $$1 - 114673250 T^{2} + 21914624432020321 T^{4}$$
$29$ $$( 1 + 16002 T + 594823321 T^{2} )^{2}$$
$31$ $$1 - 1522190162 T^{2} + 787662783788549761 T^{4}$$
$37$ $$( 1 - 61130 T + 2565726409 T^{2} )^{2}$$
$41$ $$( 1 + 98550 T + 4750104241 T^{2} )^{2}$$
$43$ $$1 - 10222364450 T^{2} + 39959630797262576401 T^{4}$$
$47$ $$1 + 10288348414 T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$
$53$ $$( 1 + 275346 T + 22164361129 T^{2} )^{2}$$
$59$ $$1 - 19735007330 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$
$61$ $$( 1 - 106634 T + 51520374361 T^{2} )^{2}$$
$67$ $$1 + 123394697854 T^{2} +$$$$81\!\cdots\!61$$$$T^{4}$$
$71$ $$1 - 250713262370 T^{2} +$$$$16\!\cdots\!41$$$$T^{4}$$
$73$ $$( 1 - 100978 T + 151334226289 T^{2} )^{2}$$
$79$ $$1 - 480206360594 T^{2} +$$$$59\!\cdots\!41$$$$T^{4}$$
$83$ $$1 - 238112486210 T^{2} +$$$$10\!\cdots\!61$$$$T^{4}$$
$89$ $$( 1 + 819054 T + 496981290961 T^{2} )^{2}$$
$97$ $$( 1 - 557026 T + 832972004929 T^{2} )^{2}$$