# Properties

 Label 48.5.e.c Level 48 Weight 5 Character orbit 48.e Analytic conductor 4.962 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.96175822802$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{13})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}\cdot 3$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( -6 - \beta_{1} + 4 \beta_{2} ) q^{7} + ( 25 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( -6 - \beta_{1} + 4 \beta_{2} ) q^{7} + ( 25 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{9} + ( -3 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{11} + ( -62 - 4 \beta_{1} + 16 \beta_{2} ) q^{13} + ( 136 - 5 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{15} + ( 4 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{17} + ( -206 + \beta_{1} - 4 \beta_{2} ) q^{19} + ( 306 - 3 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} ) q^{21} + ( 14 \beta_{1} + 28 \beta_{2} + 16 \beta_{3} ) q^{23} + ( -511 + 20 \beta_{1} - 80 \beta_{2} ) q^{25} + ( 385 + 23 \beta_{1} + 23 \beta_{2} + 12 \beta_{3} ) q^{27} + ( -11 \beta_{1} - 22 \beta_{2} - 3 \beta_{3} ) q^{29} + ( -950 + 11 \beta_{1} - 44 \beta_{2} ) q^{31} + ( 632 + 41 \beta_{1} - 18 \beta_{2} - 15 \beta_{3} ) q^{33} + ( -14 \beta_{1} - 28 \beta_{2} - 60 \beta_{3} ) q^{35} + ( -702 - 28 \beta_{1} + 112 \beta_{2} ) q^{37} + ( 1186 - 12 \beta_{1} - 62 \beta_{2} + 36 \beta_{3} ) q^{39} + ( 50 \beta_{1} + 100 \beta_{2} + 66 \beta_{3} ) q^{41} + ( 242 - 7 \beta_{1} + 28 \beta_{2} ) q^{43} + ( 688 - 85 \beta_{1} + 110 \beta_{2} - 69 \beta_{3} ) q^{45} + ( -20 \beta_{1} - 40 \beta_{2} - 24 \beta_{3} ) q^{47} + ( -493 + 12 \beta_{1} - 48 \beta_{2} ) q^{49} + ( -928 - 76 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} ) q^{51} + ( -101 \beta_{1} - 202 \beta_{2} + 59 \beta_{3} ) q^{53} + ( -1168 - 52 \beta_{1} + 208 \beta_{2} ) q^{55} + ( -518 + 3 \beta_{1} - 206 \beta_{2} - 9 \beta_{3} ) q^{57} + ( 57 \beta_{1} + 114 \beta_{2} + 112 \beta_{3} ) q^{59} + ( 2146 - 44 \beta_{1} + 176 \beta_{2} ) q^{61} + ( 1098 + 81 \beta_{1} + 288 \beta_{2} ) q^{63} + ( -18 \beta_{1} - 36 \beta_{2} - 202 \beta_{3} ) q^{65} + ( 3778 + 5 \beta_{1} - 20 \beta_{2} ) q^{67} + ( -1840 + 86 \beta_{1} + 84 \beta_{2} + 174 \beta_{3} ) q^{69} + ( -102 \beta_{1} - 204 \beta_{2} - 136 \beta_{3} ) q^{71} + ( 1378 + 48 \beta_{1} - 192 \beta_{2} ) q^{73} + ( -6751 + 60 \beta_{1} - 511 \beta_{2} - 180 \beta_{3} ) q^{75} + ( 182 \beta_{1} + 364 \beta_{2} - 138 \beta_{3} ) q^{77} + ( 266 + 139 \beta_{1} - 556 \beta_{2} ) q^{79} + ( -3647 + 50 \beta_{1} + 500 \beta_{2} + 174 \beta_{3} ) q^{81} + ( 47 \beta_{1} + 94 \beta_{2} + 308 \beta_{3} ) q^{83} + ( 704 + 112 \beta_{1} - 448 \beta_{2} ) q^{85} + ( 1752 + 9 \beta_{1} - 66 \beta_{2} - 108 \beta_{3} ) q^{87} + ( 30 \beta_{1} + 60 \beta_{2} + 258 \beta_{3} ) q^{89} + ( 7860 + 86 \beta_{1} - 344 \beta_{2} ) q^{91} + ( -4382 + 33 \beta_{1} - 950 \beta_{2} - 99 \beta_{3} ) q^{93} + ( 226 \beta_{1} + 452 \beta_{2} + 272 \beta_{3} ) q^{95} + ( 8114 - 100 \beta_{1} + 400 \beta_{2} ) q^{97} + ( -9136 - 143 \beta_{1} + 778 \beta_{2} + 24 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 24q^{7} + 100q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 24q^{7} + 100q^{9} - 248q^{13} + 544q^{15} - 824q^{19} + 1224q^{21} - 2044q^{25} + 1540q^{27} - 3800q^{31} + 2528q^{33} - 2808q^{37} + 4744q^{39} + 968q^{43} + 2752q^{45} - 1972q^{49} - 3712q^{51} - 4672q^{55} - 2072q^{57} + 8584q^{61} + 4392q^{63} + 15112q^{67} - 7360q^{69} + 5512q^{73} - 27004q^{75} + 1064q^{79} - 14588q^{81} + 2816q^{85} + 7008q^{87} + 31440q^{91} - 17528q^{93} + 32456q^{97} - 36544q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - x^{2} + 2 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$8 \nu^{3} + 72 \nu^{2} - 224 \nu - 12$$$$)/21$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{3} + 36 \nu^{2} + 28 \nu - 48$$$$)/21$$ $$\beta_{3}$$ $$=$$ $$($$$$32 \nu^{3} - 48 \nu^{2} + 112 \nu - 48$$$$)/21$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{3} + 8 \beta_{2} - 2 \beta_{1} + 24$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{2} + \beta_{1} + 12$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$15 \beta_{3} + 4 \beta_{2} + 8 \beta_{1} + 48$$$$)/24$$

## 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
 −1.30278 + 1.41421i −1.30278 − 1.41421i 2.30278 − 1.41421i 2.30278 + 1.41421i
0 −6.21110 6.51323i 0 16.4520i 0 −49.2666 0 −3.84441 + 80.9087i 0
17.2 0 −6.21110 + 6.51323i 0 16.4520i 0 −49.2666 0 −3.84441 80.9087i 0
17.3 0 8.21110 3.68481i 0 44.7363i 0 37.2666 0 53.8444 60.5126i 0
17.4 0 8.21110 + 3.68481i 0 44.7363i 0 37.2666 0 53.8444 + 60.5126i 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 48.5.e.c 4
3.b odd 2 1 inner 48.5.e.c 4
4.b odd 2 1 24.5.e.a 4
8.b even 2 1 192.5.e.e 4
8.d odd 2 1 192.5.e.f 4
12.b even 2 1 24.5.e.a 4
20.d odd 2 1 600.5.l.a 4
20.e even 4 2 600.5.c.a 8
24.f even 2 1 192.5.e.f 4
24.h odd 2 1 192.5.e.e 4
36.f odd 6 2 648.5.m.e 8
36.h even 6 2 648.5.m.e 8
60.h even 2 1 600.5.l.a 4
60.l odd 4 2 600.5.c.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.e.a 4 4.b odd 2 1
24.5.e.a 4 12.b even 2 1
48.5.e.c 4 1.a even 1 1 trivial
48.5.e.c 4 3.b odd 2 1 inner
192.5.e.e 4 8.b even 2 1
192.5.e.e 4 24.h odd 2 1
192.5.e.f 4 8.d odd 2 1
192.5.e.f 4 24.f even 2 1
600.5.c.a 8 20.e even 4 2
600.5.c.a 8 60.l odd 4 2
600.5.l.a 4 20.d odd 2 1
600.5.l.a 4 60.h even 2 1
648.5.m.e 8 36.f odd 6 2
648.5.m.e 8 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 2272 T_{5}^{2} + 541696$$ acting on $$S_{5}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T - 42 T^{2} - 324 T^{3} + 6561 T^{4}$$
$5$ $$1 - 228 T^{2} + 45446 T^{4} - 89062500 T^{6} + 152587890625 T^{8}$$
$7$ $$( 1 + 12 T + 2966 T^{2} + 28812 T^{3} + 5764801 T^{4} )^{2}$$
$11$ $$1 - 14820 T^{2} + 30482054 T^{4} - 3176798616420 T^{6} + 45949729863572161 T^{8}$$
$13$ $$( 1 + 124 T + 31014 T^{2} + 3541564 T^{3} + 815730721 T^{4} )^{2}$$
$17$ $$1 - 211716 T^{2} + 22389387014 T^{4} - 1476879462378756 T^{6} + 48661191875666868481 T^{8}$$
$19$ $$( 1 + 412 T + 301206 T^{2} + 53692252 T^{3} + 16983563041 T^{4} )^{2}$$
$23$ $$1 - 634116 T^{2} + 213728847494 T^{4} - 49658248742446596 T^{6} +$$$$61\!\cdots\!61$$$$T^{8}$$
$29$ $$1 - 2601316 T^{2} + 2691910499334 T^{4} - 1301298997978056676 T^{6} +$$$$25\!\cdots\!21$$$$T^{8}$$
$31$ $$( 1 + 1900 T + 2523030 T^{2} + 1754689900 T^{3} + 852891037441 T^{4} )^{2}$$
$37$ $$( 1 + 1404 T + 2773478 T^{2} + 2631322044 T^{3} + 3512479453921 T^{4} )^{2}$$
$41$ $$1 - 4336900 T^{2} + 9972871940742 T^{4} - 34629822226174864900 T^{6} +$$$$63\!\cdots\!41$$$$T^{8}$$
$43$ $$( 1 - 484 T + 6804438 T^{2} - 1654699684 T^{3} + 11688200277601 T^{4} )^{2}$$
$47$ $$1 - 18491140 T^{2} + 132894384489222 T^{4} -$$$$44\!\cdots\!40$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 2855652 T^{2} - 56957106011002 T^{4} -$$$$17\!\cdots\!72$$$$T^{6} +$$$$38\!\cdots\!21$$$$T^{8}$$
$59$ $$1 - 33977316 T^{2} + 531124679369606 T^{4} -$$$$49\!\cdots\!36$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8}$$
$61$ $$( 1 - 4292 T + 28672806 T^{2} - 59426349572 T^{3} + 191707312997281 T^{4} )^{2}$$
$67$ $$( 1 - 7556 T + 54528726 T^{2} - 152261870276 T^{3} + 406067677556641 T^{4} )^{2}$$
$71$ $$1 - 72386052 T^{2} + 2410878712841606 T^{4} -$$$$46\!\cdots\!72$$$$T^{6} +$$$$41\!\cdots\!21$$$$T^{8}$$
$73$ $$( 1 - 2756 T + 54382278 T^{2} - 78265552196 T^{3} + 806460091894081 T^{4} )^{2}$$
$79$ $$( 1 - 532 T + 41802006 T^{2} - 20721443092 T^{3} + 1517108809906561 T^{4} )^{2}$$
$83$ $$1 - 99356772 T^{2} + 6615483960569606 T^{4} -$$$$22\!\cdots\!52$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$1 - 186937348 T^{2} + 16504528715125638 T^{4} -$$$$73\!\cdots\!88$$$$T^{6} +$$$$15\!\cdots\!61$$$$T^{8}$$
$97$ $$( 1 - 16228 T + 224175558 T^{2} - 1436653172068 T^{3} + 7837433594376961 T^{4} )^{2}$$