# Properties

 Label 48.7.e.b Level $48$ Weight $7$ Character orbit 48.e 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.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0425960138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -21 + \beta ) q^{3} + 10 \beta q^{5} -2 q^{7} + ( 153 - 42 \beta ) q^{9} +O(q^{10})$$ $$q + ( -21 + \beta ) q^{3} + 10 \beta q^{5} -2 q^{7} + ( 153 - 42 \beta ) q^{9} -2 \beta q^{11} -2950 q^{13} + ( -2880 - 210 \beta ) q^{15} -264 \beta q^{17} -5258 q^{19} + ( 42 - 2 \beta ) q^{21} -604 \beta q^{23} -13175 q^{25} + ( 8883 + 1035 \beta ) q^{27} -130 \beta q^{29} -22898 q^{31} + ( 576 + 42 \beta ) q^{33} -20 \beta q^{35} + 34058 q^{37} + ( 61950 - 2950 \beta ) q^{39} + 988 \beta q^{41} + 6406 q^{43} + ( 120960 + 1530 \beta ) q^{45} + 10600 \beta q^{47} -117645 q^{49} + ( 76032 + 5544 \beta ) q^{51} + 11346 \beta q^{53} + 5760 q^{55} + ( 110418 - 5258 \beta ) q^{57} -19258 \beta q^{59} -62566 q^{61} + ( -306 + 84 \beta ) q^{63} -29500 \beta q^{65} -438698 q^{67} + ( 173952 + 12684 \beta ) q^{69} -4020 \beta q^{71} -730510 q^{73} + ( 276675 - 13175 \beta ) q^{75} + 4 \beta q^{77} -340562 q^{79} + ( -484623 - 12852 \beta ) q^{81} + 29242 \beta q^{83} + 760320 q^{85} + ( 37440 + 2730 \beta ) q^{87} + 22788 \beta q^{89} + 5900 q^{91} + ( 480858 - 22898 \beta ) q^{93} -52580 \beta q^{95} -281086 q^{97} + ( -24192 - 306 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 42 q^{3} - 4 q^{7} + 306 q^{9} + O(q^{10})$$ $$2 q - 42 q^{3} - 4 q^{7} + 306 q^{9} - 5900 q^{13} - 5760 q^{15} - 10516 q^{19} + 84 q^{21} - 26350 q^{25} + 17766 q^{27} - 45796 q^{31} + 1152 q^{33} + 68116 q^{37} + 123900 q^{39} + 12812 q^{43} + 241920 q^{45} - 235290 q^{49} + 152064 q^{51} + 11520 q^{55} + 220836 q^{57} - 125132 q^{61} - 612 q^{63} - 877396 q^{67} + 347904 q^{69} - 1461020 q^{73} + 553350 q^{75} - 681124 q^{79} - 969246 q^{81} + 1520640 q^{85} + 74880 q^{87} + 11800 q^{91} + 961716 q^{93} - 562172 q^{97} - 48384 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 −21.0000 16.9706i 0 169.706i 0 −2.00000 0 153.000 + 712.764i 0
17.2 0 −21.0000 + 16.9706i 0 169.706i 0 −2.00000 0 153.000 712.764i 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 48.7.e.b 2
3.b odd 2 1 inner 48.7.e.b 2
4.b odd 2 1 6.7.b.a 2
8.b even 2 1 192.7.e.f 2
8.d odd 2 1 192.7.e.c 2
12.b even 2 1 6.7.b.a 2
20.d odd 2 1 150.7.d.a 2
20.e even 4 2 150.7.b.a 4
24.f even 2 1 192.7.e.c 2
24.h odd 2 1 192.7.e.f 2
28.d even 2 1 294.7.b.a 2
36.f odd 6 2 162.7.d.b 4
36.h even 6 2 162.7.d.b 4
60.h even 2 1 150.7.d.a 2
60.l odd 4 2 150.7.b.a 4
84.h odd 2 1 294.7.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 4.b odd 2 1
6.7.b.a 2 12.b even 2 1
48.7.e.b 2 1.a even 1 1 trivial
48.7.e.b 2 3.b odd 2 1 inner
150.7.b.a 4 20.e even 4 2
150.7.b.a 4 60.l odd 4 2
150.7.d.a 2 20.d odd 2 1
150.7.d.a 2 60.h even 2 1
162.7.d.b 4 36.f odd 6 2
162.7.d.b 4 36.h even 6 2
192.7.e.c 2 8.d odd 2 1
192.7.e.c 2 24.f even 2 1
192.7.e.f 2 8.b even 2 1
192.7.e.f 2 24.h odd 2 1
294.7.b.a 2 28.d even 2 1
294.7.b.a 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 28800$$ acting on $$S_{7}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$729 + 42 T + T^{2}$$
$5$ $$28800 + T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$1152 + T^{2}$$
$13$ $$( 2950 + T )^{2}$$
$17$ $$20072448 + T^{2}$$
$19$ $$( 5258 + T )^{2}$$
$23$ $$105067008 + T^{2}$$
$29$ $$4867200 + T^{2}$$
$31$ $$( 22898 + T )^{2}$$
$37$ $$( -34058 + T )^{2}$$
$41$ $$281129472 + T^{2}$$
$43$ $$( -6406 + T )^{2}$$
$47$ $$32359680000 + T^{2}$$
$53$ $$37074734208 + T^{2}$$
$59$ $$106810722432 + T^{2}$$
$61$ $$( 62566 + T )^{2}$$
$67$ $$( 438698 + T )^{2}$$
$71$ $$4654195200 + T^{2}$$
$73$ $$( 730510 + T )^{2}$$
$79$ $$( 340562 + T )^{2}$$
$83$ $$246267234432 + T^{2}$$
$89$ $$149556367872 + T^{2}$$
$97$ $$( 281086 + T )^{2}$$