# Properties

 Label 48.5.i.a Level 48 Weight 5 Character orbit 48.i Analytic conductor 4.962 Analytic rank 0 Dimension 60 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.96175822802$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60q - 2q^{3} - 4q^{4} - 68q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q - 2q^{3} - 4q^{4} - 68q^{6} - 104q^{10} - 32q^{12} - 4q^{13} - 4q^{15} - 976q^{16} + 516q^{18} + 700q^{19} + 160q^{21} + 792q^{22} - 824q^{24} + 1822q^{27} - 840q^{28} + 132q^{30} - 8q^{31} - 4q^{33} - 1048q^{34} - 956q^{36} - 4q^{37} - 1896q^{40} + 760q^{42} + 4476q^{43} - 1252q^{45} + 6904q^{46} - 4112q^{48} - 12356q^{49} - 6720q^{51} - 2688q^{52} + 6088q^{54} + 4472q^{58} + 3648q^{60} + 3772q^{61} + 9600q^{63} + 16856q^{64} + 10804q^{66} - 15108q^{67} - 164q^{69} + 984q^{70} + 7752q^{72} - 13598q^{75} - 29752q^{76} - 52q^{78} + 25080q^{79} - 4q^{81} - 29696q^{82} - 36560q^{84} + 13696q^{85} + 22752q^{88} - 37968q^{90} - 6336q^{91} + 8572q^{93} - 1680q^{94} - 3160q^{96} - 8q^{97} + 10172q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −3.99998 + 0.0120968i −4.29675 + 7.90809i 15.9997 0.0967736i 21.9903 21.9903i 17.0913 31.6842i 77.1285i −63.9974 + 0.580637i −44.0758 67.9582i −87.6947 + 88.2267i
5.2 −3.98827 + 0.306169i 7.27370 5.30031i 15.8125 2.44216i 2.97955 2.97955i −27.3866 + 23.3660i 5.93984i −62.3168 + 14.5813i 24.8134 77.1058i −10.9710 + 12.7955i
5.3 −3.84751 1.09391i 5.82159 + 6.86360i 13.6067 + 8.41770i −20.5976 + 20.5976i −14.8904 32.7761i 44.8073i −43.1437 47.2718i −13.2181 + 79.9142i 101.781 56.7174i
5.4 −3.69713 1.52683i −8.89463 1.37314i 11.3376 + 11.2898i −13.4157 + 13.4157i 30.7881 + 18.6573i 4.18833i −24.6791 59.0503i 77.2290 + 24.4272i 70.0832 29.1162i
5.5 −3.47271 + 1.98501i −4.74405 7.64814i 8.11943 13.7868i −3.34416 + 3.34416i 31.6564 + 17.1428i 6.36882i −0.829500 + 63.9946i −35.9881 + 72.5662i 4.97509 18.2515i
5.6 −3.16452 2.44659i −1.45703 8.88128i 4.02839 + 15.4846i 30.0724 30.0724i −17.1180 + 31.6697i 37.1392i 25.1365 58.8571i −76.7541 + 25.8807i −168.740 + 21.5899i
5.7 −3.01530 + 2.62831i −5.72284 + 6.94616i 2.18401 15.8502i −22.2177 + 22.2177i −1.00059 35.9861i 25.7845i 35.0739 + 53.5334i −15.4983 79.5035i 8.59799 125.388i
5.8 −2.98404 + 2.66374i 6.71421 + 5.99328i 1.80902 15.8974i 27.3100 27.3100i −36.0000 0.000682234i 70.5035i 36.9483 + 52.2573i 9.16120 + 80.4803i −8.74755 + 154.241i
5.9 −2.23278 3.31884i 8.02371 + 4.07678i −6.02942 + 14.8205i 12.9156 12.9156i −4.38496 35.7319i 84.6751i 62.6491 13.0801i 47.7598 + 65.4217i −71.7023 14.0272i
5.10 −2.12409 3.38943i 3.23855 8.39713i −6.97648 + 14.3989i −31.4183 + 31.4183i −35.3405 + 6.85944i 26.1649i 63.6228 6.93833i −60.0236 54.3890i 173.225 + 39.7548i
5.11 −2.02902 3.44718i −3.29492 + 8.37517i −7.76615 + 13.9888i 8.78364 8.78364i 35.5562 5.63519i 70.1022i 63.9797 1.61227i −59.2870 55.1911i −48.1010 12.4566i
5.12 −1.96807 + 3.48234i 8.97524 0.667113i −8.25342 13.7070i −17.7419 + 17.7419i −15.3408 + 32.5678i 66.4694i 63.9757 1.76496i 80.1099 11.9750i −26.8661 96.7008i
5.13 −1.13167 + 3.83658i −8.98791 0.466380i −13.4387 8.68346i 23.5639 23.5639i 11.9606 33.9550i 30.4899i 48.5229 41.7317i 80.5650 + 8.38356i 63.7382 + 117.071i
5.14 −0.431020 + 3.97671i 2.29223 8.70320i −15.6284 3.42808i −3.74626 + 3.74626i 33.6221 + 12.8668i 71.9695i 20.3687 60.6722i −70.4914 39.8994i −13.2831 16.5125i
5.15 −0.312407 3.98778i −8.85016 1.63546i −15.8048 + 2.49162i 2.85930 2.85930i −3.75701 + 35.8034i 27.2580i 14.8736 + 62.2477i 75.6505 + 28.9482i −12.2955 10.5090i
5.16 0.312407 + 3.98778i 1.63546 + 8.85016i −15.8048 + 2.49162i −2.85930 + 2.85930i −34.7816 + 9.28671i 27.2580i −14.8736 62.2477i −75.6505 + 28.9482i −12.2955 10.5090i
5.17 0.431020 3.97671i 8.70320 2.29223i −15.6284 3.42808i 3.74626 3.74626i −5.36427 35.5981i 71.9695i −20.3687 + 60.6722i 70.4914 39.8994i −13.2831 16.5125i
5.18 1.13167 3.83658i 0.466380 + 8.98791i −13.4387 8.68346i −23.5639 + 23.5639i 35.0106 + 8.38202i 30.4899i −48.5229 + 41.7317i −80.5650 + 8.38356i 63.7382 + 117.071i
5.19 1.96807 3.48234i 0.667113 8.97524i −8.25342 13.7070i 17.7419 17.7419i −29.9419 19.9870i 66.4694i −63.9757 + 1.76496i −80.1099 11.9750i −26.8661 96.7008i
5.20 2.02902 + 3.44718i −8.37517 + 3.29492i −7.76615 + 13.9888i −8.78364 + 8.78364i −28.3516 22.1853i 70.1022i −63.9797 + 1.61227i 59.2870 55.1911i −48.1010 12.4566i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.30 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
16.e even 4 1 inner
48.i 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.i.a 60
3.b odd 2 1 inner 48.5.i.a 60
4.b odd 2 1 192.5.i.a 60
12.b even 2 1 192.5.i.a 60
16.e even 4 1 inner 48.5.i.a 60
16.f odd 4 1 192.5.i.a 60
48.i odd 4 1 inner 48.5.i.a 60
48.k even 4 1 192.5.i.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.i.a 60 1.a even 1 1 trivial
48.5.i.a 60 3.b odd 2 1 inner
48.5.i.a 60 16.e even 4 1 inner
48.5.i.a 60 48.i odd 4 1 inner
192.5.i.a 60 4.b odd 2 1
192.5.i.a 60 12.b even 2 1
192.5.i.a 60 16.f odd 4 1
192.5.i.a 60 48.k even 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