# Properties

 Label 48.5.l.a Level 48 Weight 5 Character orbit 48.l Analytic conductor 4.962 Analytic rank 0 Dimension 32 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 48.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.96175822802$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{4} + 180q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{4} + 180q^{8} + 296q^{10} - 192q^{11} + 360q^{12} - 156q^{14} + 352q^{16} - 324q^{18} + 704q^{19} - 1200q^{20} - 1568q^{22} - 2304q^{23} + 1188q^{24} + 2700q^{26} + 4680q^{28} - 1728q^{29} + 1512q^{30} - 3360q^{32} - 9312q^{34} - 5184q^{35} - 756q^{36} + 3648q^{37} - 5880q^{38} + 5232q^{40} + 4500q^{42} + 1088q^{43} + 18840q^{44} + 680q^{46} + 2160q^{48} + 10976q^{49} - 25884q^{50} - 4032q^{51} - 25584q^{52} + 960q^{53} + 972q^{54} + 11776q^{55} + 15456q^{56} + 12624q^{58} + 13056q^{59} + 7992q^{60} + 3776q^{61} + 21852q^{62} - 8664q^{64} + 4032q^{65} - 8856q^{66} - 896q^{67} - 17280q^{68} - 9792q^{69} - 18240q^{70} - 39936q^{71} + 4860q^{72} + 24204q^{74} - 1152q^{75} + 16776q^{76} + 9408q^{77} - 3780q^{78} - 14232q^{80} - 23328q^{81} - 33800q^{82} + 24000q^{83} - 11448q^{84} - 11200q^{85} - 1200q^{86} - 11424q^{88} + 4104q^{90} + 30528q^{91} - 11664q^{92} - 8040q^{94} + 10080q^{96} + 52968q^{98} - 5184q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −3.98297 0.368670i −3.67423 + 3.67423i 15.7282 + 2.93680i −9.21870 + 9.21870i 15.9890 13.2798i 17.0639 −61.5622 17.4957i 27.0000i 40.1165 33.3192i
19.2 −3.94725 0.647479i 3.67423 3.67423i 15.1615 + 5.11152i −21.2428 + 21.2428i −16.8821 + 12.1241i 35.6927 −56.5368 29.9932i 27.0000i 97.6049 70.0964i
19.3 −3.15376 + 2.46044i −3.67423 + 3.67423i 3.89246 15.5193i 18.4189 18.4189i 2.54743 20.6279i 4.95460 25.9084 + 58.5214i 27.0000i −12.7702 + 103.407i
19.4 −3.09682 + 2.53174i 3.67423 3.67423i 3.18056 15.6807i −7.37831 + 7.37831i −2.07622 + 20.6807i −75.1984 29.8498 + 56.6126i 27.0000i 4.16931 41.5293i
19.5 −2.82809 2.82877i 3.67423 3.67423i −0.00384203 + 16.0000i 21.6601 21.6601i −20.7846 0.00249547i 15.5177 45.2711 45.2385i 27.0000i −122.528 0.0147111i
19.6 −1.62589 3.65465i −3.67423 + 3.67423i −10.7130 + 11.8841i −2.10268 + 2.10268i 19.4020 + 7.45415i 84.8276 60.8504 + 19.8299i 27.0000i 11.1033 + 4.26583i
19.7 −0.922521 + 3.89217i 3.67423 3.67423i −14.2979 7.18121i 17.5812 17.5812i 10.9112 + 17.6903i 50.9945 41.1406 49.0250i 27.0000i 52.2099 + 84.6479i
19.8 −0.827018 3.91357i 3.67423 3.67423i −14.6321 + 6.47318i −25.9147 + 25.9147i −17.4180 11.3407i −48.6273 37.4343 + 51.9103i 27.0000i 122.851 + 79.9872i
19.9 −0.149438 + 3.99721i −3.67423 + 3.67423i −15.9553 1.19467i −0.419719 + 0.419719i −14.1376 15.2357i −40.4181 7.15966 63.5983i 27.0000i −1.61498 1.74043i
19.10 0.896210 3.89831i −3.67423 + 3.67423i −14.3936 6.98741i 30.1272 30.1272i 11.0304 + 17.6162i −80.6454 −40.1388 + 49.8486i 27.0000i −90.4449 144.446i
19.11 2.05307 3.43291i −3.67423 + 3.67423i −7.56978 14.0960i −34.5052 + 34.5052i 5.06985 + 20.1568i −15.7115 −63.9318 2.95383i 27.0000i 47.6116 + 189.295i
19.12 2.69875 2.95241i 3.67423 3.67423i −1.43350 15.9357i 5.17564 5.17564i −0.932024 20.7637i 6.77541 −50.9173 38.7741i 27.0000i −1.31288 29.2484i
19.13 3.43325 + 2.05251i −3.67423 + 3.67423i 7.57441 + 14.0936i −15.8310 + 15.8310i −20.1560 + 5.07316i −14.0988 −2.92234 + 63.9332i 27.0000i −86.8452 + 21.8585i
19.14 3.75428 1.38036i −3.67423 + 3.67423i 12.1892 10.3645i 13.5312 13.5312i −8.72233 + 18.8659i 44.0276 31.4549 55.7368i 27.0000i 32.1220 69.4780i
19.15 3.79804 + 1.25496i 3.67423 3.67423i 12.8502 + 9.53276i 27.4836 27.4836i 18.5659 9.34386i −74.9802 36.8422 + 52.3322i 27.0000i 138.874 69.8928i
19.16 3.90016 + 0.888110i 3.67423 3.67423i 14.4225 + 6.92754i −17.3646 + 17.3646i 17.5932 11.0670i 89.8255 50.0978 + 39.8273i 27.0000i −83.1466 + 52.3032i
43.1 −3.98297 + 0.368670i −3.67423 3.67423i 15.7282 2.93680i −9.21870 9.21870i 15.9890 + 13.2798i 17.0639 −61.5622 + 17.4957i 27.0000i 40.1165 + 33.3192i
43.2 −3.94725 + 0.647479i 3.67423 + 3.67423i 15.1615 5.11152i −21.2428 21.2428i −16.8821 12.1241i 35.6927 −56.5368 + 29.9932i 27.0000i 97.6049 + 70.0964i
43.3 −3.15376 2.46044i −3.67423 3.67423i 3.89246 + 15.5193i 18.4189 + 18.4189i 2.54743 + 20.6279i 4.95460 25.9084 58.5214i 27.0000i −12.7702 103.407i
43.4 −3.09682 2.53174i 3.67423 + 3.67423i 3.18056 + 15.6807i −7.37831 7.37831i −2.07622 20.6807i −75.1984 29.8498 56.6126i 27.0000i 4.16931 + 41.5293i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.5.l.a 32
3.b odd 2 1 144.5.m.c 32
4.b odd 2 1 192.5.l.a 32
8.b even 2 1 384.5.l.b 32
8.d odd 2 1 384.5.l.a 32
12.b even 2 1 576.5.m.b 32
16.e even 4 1 192.5.l.a 32
16.e even 4 1 384.5.l.a 32
16.f odd 4 1 inner 48.5.l.a 32
16.f odd 4 1 384.5.l.b 32
48.i odd 4 1 576.5.m.b 32
48.k even 4 1 144.5.m.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.l.a 32 1.a even 1 1 trivial
48.5.l.a 32 16.f odd 4 1 inner
144.5.m.c 32 3.b odd 2 1
144.5.m.c 32 48.k even 4 1
192.5.l.a 32 4.b odd 2 1
192.5.l.a 32 16.e even 4 1
384.5.l.a 32 8.d odd 2 1
384.5.l.a 32 16.e even 4 1
384.5.l.b 32 8.b even 2 1
384.5.l.b 32 16.f odd 4 1
576.5.m.b 32 12.b even 2 1
576.5.m.b 32 48.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database