# Properties

 Label 48.6.j.a Level 48 Weight 6 Character orbit 48.j Analytic conductor 7.698 Analytic rank 0 Dimension 40 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.69842335102$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 44q^{4} - 492q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 44q^{4} - 492q^{8} + 968q^{10} + 1208q^{11} - 792q^{12} + 324q^{14} - 1800q^{15} - 384q^{16} - 324q^{18} + 2360q^{19} - 7600q^{20} - 1024q^{22} + 5220q^{24} - 12980q^{26} + 10472q^{28} - 8144q^{29} - 1368q^{30} + 23064q^{31} + 44160q^{32} + 4864q^{34} - 4776q^{35} - 1620q^{36} + 21296q^{37} - 72088q^{38} - 90864q^{40} - 14220q^{42} + 31416q^{43} + 43768q^{44} + 90664q^{46} + 44784q^{48} - 96040q^{49} + 106212q^{50} + 10440q^{51} - 116432q^{52} - 49456q^{53} - 14580q^{54} - 158592q^{56} - 111472q^{58} + 28960q^{59} + 49464q^{60} + 48080q^{61} + 191100q^{62} + 31752q^{63} + 369032q^{64} - 55376q^{65} + 63720q^{66} + 101488q^{67} - 122336q^{68} - 22320q^{69} - 460192q^{70} - 28836q^{72} - 94996q^{74} - 96624q^{75} + 211304q^{76} - 14896q^{77} + 200124q^{78} - 71960q^{79} + 623368q^{80} - 262440q^{81} + 33880q^{82} - 126440q^{83} - 116856q^{84} + 132400q^{85} - 581168q^{86} - 466400q^{88} - 46008q^{90} + 429496q^{91} - 12752q^{92} + 524120q^{94} - 76848q^{95} + 51840q^{96} + 477736q^{98} + 97848q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −5.65510 + 0.140725i −6.36396 6.36396i 31.9604 1.59163i 24.8890 24.8890i 36.8844 + 35.0933i 146.252i −180.515 + 13.4984i 81.0000i −137.247 + 144.252i
13.2 −5.59211 0.853399i 6.36396 + 6.36396i 30.5434 + 9.54460i 16.4445 16.4445i −30.1570 41.0190i 18.3733i −162.657 79.4402i 81.0000i −105.993 + 77.9257i
13.3 −5.12822 2.38775i −6.36396 6.36396i 20.5973 + 24.4898i 12.6306 12.6306i 17.4402 + 47.8313i 213.506i −47.1519 174.770i 81.0000i −94.9313 + 34.6138i
13.4 −4.89728 + 2.83137i −6.36396 6.36396i 15.9667 27.7320i −71.9915 + 71.9915i 49.1848 + 13.1474i 158.840i 0.326398 + 181.019i 81.0000i 148.728 556.397i
13.5 −4.79989 + 2.99350i 6.36396 + 6.36396i 14.0779 28.7369i −26.7417 + 26.7417i −49.5968 11.4958i 39.6806i 18.4514 + 180.076i 81.0000i 48.3060 208.408i
13.6 −3.74446 4.24017i 6.36396 + 6.36396i −3.95807 + 31.7543i −41.3210 + 41.3210i 3.15468 50.8139i 215.468i 149.464 102.120i 81.0000i 329.933 + 20.4833i
13.7 −3.04548 + 4.76708i −6.36396 6.36396i −13.4501 29.0361i 25.2736 25.2736i 49.7188 10.9562i 124.502i 179.379 + 24.3109i 81.0000i 43.5111 + 197.451i
13.8 −2.72907 4.95502i −6.36396 6.36396i −17.1044 + 27.0451i −7.19648 + 7.19648i −14.1659 + 48.9012i 96.5221i 180.688 + 10.9446i 81.0000i 55.2984 + 16.0190i
13.9 −1.76330 + 5.37502i 6.36396 + 6.36396i −25.7816 18.9555i 47.6164 47.6164i −45.4279 + 22.9848i 212.678i 147.347 105.152i 81.0000i 171.977 + 339.901i
13.10 0.0662295 5.65647i 6.36396 + 6.36396i −31.9912 0.749250i −23.0892 + 23.0892i 36.4190 35.5761i 103.844i −6.35687 + 180.908i 81.0000i 129.074 + 132.132i
13.11 1.74038 5.38248i −6.36396 6.36396i −25.9421 18.7352i 63.5721 63.5721i −45.3296 + 23.1781i 39.2344i −145.991 + 107.026i 81.0000i −231.535 452.815i
13.12 1.92664 + 5.31865i 6.36396 + 6.36396i −24.5761 + 20.4942i 31.1853 31.1853i −21.5866 + 46.1087i 246.402i −156.351 91.2269i 81.0000i 225.946 + 105.781i
13.13 2.58832 + 5.02997i −6.36396 6.36396i −18.6012 + 26.0383i 74.2576 74.2576i 15.5386 48.4825i 68.3939i −179.118 26.1682i 81.0000i 565.716 + 181.311i
13.14 2.70388 + 4.96881i 6.36396 + 6.36396i −17.3781 + 26.8701i −55.5954 + 55.5954i −14.4139 + 48.8287i 156.212i −180.501 13.6951i 81.0000i −426.566 125.920i
13.15 3.19025 4.67143i −6.36396 6.36396i −11.6446 29.8061i −59.0275 + 59.0275i −50.0315 + 9.42617i 85.1278i −176.386 40.6922i 81.0000i 87.4303 + 464.056i
13.16 3.53705 4.41467i 6.36396 + 6.36396i −6.97862 31.2298i 37.1934 37.1934i 50.6044 5.58517i 5.83317i −162.553 79.6528i 81.0000i −32.6419 295.752i
13.17 4.81279 + 2.97273i −6.36396 6.36396i 14.3258 + 28.6142i −30.6986 + 30.6986i −11.7101 49.5467i 92.2922i −16.1150 + 180.301i 81.0000i −239.005 + 56.4873i
13.18 5.54514 + 1.11868i 6.36396 + 6.36396i 29.4971 + 12.4065i 42.8171 42.8171i 28.1698 + 42.4083i 71.3138i 149.687 + 101.793i 81.0000i 285.325 189.528i
13.19 5.58787 0.880730i −6.36396 6.36396i 30.4486 9.84281i 3.64660 3.64660i −41.1659 29.9561i 162.720i 161.474 81.8174i 81.0000i 17.1651 23.5884i
13.20 5.65636 0.0744172i 6.36396 + 6.36396i 31.9889 0.841862i −63.8648 + 63.8648i 36.4705 + 35.5233i 155.205i 180.878 7.14241i 81.0000i −356.490 + 365.995i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.20 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.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.6.j.a 40
3.b odd 2 1 144.6.k.c 40
4.b odd 2 1 192.6.j.a 40
8.b even 2 1 384.6.j.a 40
8.d odd 2 1 384.6.j.b 40
12.b even 2 1 576.6.k.c 40
16.e even 4 1 inner 48.6.j.a 40
16.e even 4 1 384.6.j.a 40
16.f odd 4 1 192.6.j.a 40
16.f odd 4 1 384.6.j.b 40
48.i odd 4 1 144.6.k.c 40
48.k even 4 1 576.6.k.c 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.j.a 40 1.a even 1 1 trivial
48.6.j.a 40 16.e even 4 1 inner
144.6.k.c 40 3.b odd 2 1
144.6.k.c 40 48.i odd 4 1
192.6.j.a 40 4.b odd 2 1
192.6.j.a 40 16.f odd 4 1
384.6.j.a 40 8.b even 2 1
384.6.j.a 40 16.e even 4 1
384.6.j.b 40 8.d odd 2 1
384.6.j.b 40 16.f odd 4 1
576.6.k.c 40 12.b even 2 1
576.6.k.c 40 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database