# Properties

 Label 230.5.f.a Level $230$ Weight $5$ Character orbit 230.f Analytic conductor $23.775$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 230.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.7750915093$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 88q^{2} + 24q^{5} - 80q^{7} + 704q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 88q^{2} + 24q^{5} - 80q^{7} + 704q^{8} - 184q^{10} + 8q^{11} + 20q^{13} + 396q^{15} - 2816q^{16} + 1080q^{17} - 2648q^{18} + 544q^{20} - 3096q^{21} - 16q^{22} - 1884q^{25} - 80q^{26} - 3828q^{27} + 640q^{28} - 2520q^{30} - 1580q^{31} + 5632q^{32} + 3644q^{33} + 8208q^{35} + 10592q^{36} + 3104q^{37} - 4064q^{38} - 704q^{40} + 4124q^{41} + 6192q^{42} - 960q^{43} - 11316q^{45} + 2424q^{47} + 7832q^{50} + 14840q^{51} + 160q^{52} - 3116q^{53} - 2572q^{55} - 2560q^{56} - 9408q^{57} - 3928q^{58} + 6912q^{60} + 19136q^{61} + 3160q^{62} + 4564q^{63} - 9220q^{65} - 14576q^{66} - 5152q^{67} - 8640q^{68} - 23672q^{70} + 7900q^{71} - 21184q^{72} + 16424q^{73} + 24156q^{75} + 16256q^{76} - 27012q^{77} - 1808q^{78} - 1536q^{80} - 116684q^{81} - 8248q^{82} + 11184q^{83} - 14620q^{85} + 3840q^{86} + 8312q^{87} + 128q^{88} + 14544q^{90} + 10296q^{91} - 7488q^{93} + 19536q^{95} + 41292q^{97} + 51024q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −2.00000 2.00000i −11.8046 + 11.8046i 8.00000i −2.32340 24.8918i 47.2184 62.0285 + 62.0285i 16.0000 16.0000i 197.698i −45.1368 + 54.4304i
47.2 −2.00000 2.00000i −10.5012 + 10.5012i 8.00000i 10.9451 + 22.4768i 42.0050 −20.3725 20.3725i 16.0000 16.0000i 139.552i 23.0634 66.8437i
47.3 −2.00000 2.00000i −10.3237 + 10.3237i 8.00000i −22.1733 + 11.5475i 41.2946 6.42895 + 6.42895i 16.0000 16.0000i 132.156i 67.4416 + 21.2517i
47.4 −2.00000 2.00000i −9.43718 + 9.43718i 8.00000i −10.0800 22.8778i 37.7487 −36.9388 36.9388i 16.0000 16.0000i 97.1208i −25.5957 + 65.9156i
47.5 −2.00000 2.00000i −6.81573 + 6.81573i 8.00000i 24.3590 5.62491i 27.2629 −23.8398 23.8398i 16.0000 16.0000i 11.9084i −59.9678 37.4682i
47.6 −2.00000 2.00000i −6.79892 + 6.79892i 8.00000i 24.9298 1.87172i 27.1957 46.9898 + 46.9898i 16.0000 16.0000i 11.4506i −53.6031 46.1162i
47.7 −2.00000 2.00000i −5.15211 + 5.15211i 8.00000i −17.4200 + 17.9317i 20.6085 −7.30291 7.30291i 16.0000 16.0000i 27.9115i 70.7033 1.02335i
47.8 −2.00000 2.00000i −2.34537 + 2.34537i 8.00000i −23.4912 8.55343i 9.38149 −59.8209 59.8209i 16.0000 16.0000i 69.9985i 29.8756 + 64.0894i
47.9 −2.00000 2.00000i −2.20663 + 2.20663i 8.00000i −2.16754 24.9059i 8.82650 −2.30518 2.30518i 16.0000 16.0000i 71.2616i −45.4766 + 54.1468i
47.10 −2.00000 2.00000i −1.91399 + 1.91399i 8.00000i −2.47793 + 24.8769i 7.65597 57.2690 + 57.2690i 16.0000 16.0000i 73.6733i 54.7096 44.7979i
47.11 −2.00000 2.00000i −1.72340 + 1.72340i 8.00000i 4.34818 + 24.6190i 6.89361 −48.9296 48.9296i 16.0000 16.0000i 75.0598i 40.5416 57.9343i
47.12 −2.00000 2.00000i −0.665037 + 0.665037i 8.00000i 13.6244 20.9613i 2.66015 18.0918 + 18.0918i 16.0000 16.0000i 80.1155i −69.1714 + 14.6738i
47.13 −2.00000 2.00000i −0.545965 + 0.545965i 8.00000i −21.0759 13.4464i 2.18386 44.0481 + 44.0481i 16.0000 16.0000i 80.4038i 15.2589 + 69.0447i
47.14 −2.00000 2.00000i 3.09361 3.09361i 8.00000i 15.7881 + 19.3839i −12.3744 −30.9151 30.9151i 16.0000 16.0000i 61.8591i 7.19166 70.3440i
47.15 −2.00000 2.00000i 4.47696 4.47696i 8.00000i −20.9409 + 13.6558i −17.9078 −15.4160 15.4160i 16.0000 16.0000i 40.9136i 69.1933 + 14.5701i
47.16 −2.00000 2.00000i 6.39678 6.39678i 8.00000i 22.1785 + 11.5375i −25.5871 18.6881 + 18.6881i 16.0000 16.0000i 0.837704i −21.2821 67.4320i
47.17 −2.00000 2.00000i 6.64524 6.64524i 8.00000i −21.5847 12.6135i −26.5810 27.2438 + 27.2438i 16.0000 16.0000i 7.31852i 17.9425 + 68.3964i
47.18 −2.00000 2.00000i 6.67173 6.67173i 8.00000i 5.03883 24.4869i −26.6869 −43.7544 43.7544i 16.0000 16.0000i 8.02391i −59.0515 + 38.8962i
47.19 −2.00000 2.00000i 7.00362 7.00362i 8.00000i 23.3343 8.97266i −28.0145 −20.2451 20.2451i 16.0000 16.0000i 17.1014i −64.6140 28.7233i
47.20 −2.00000 2.00000i 11.2589 11.2589i 8.00000i 19.3240 15.8613i −45.0357 67.6527 + 67.6527i 16.0000 16.0000i 172.526i −70.3707 6.92536i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 93.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.5.f.a 44
5.c odd 4 1 inner 230.5.f.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.5.f.a 44 1.a even 1 1 trivial
230.5.f.a 44 5.c odd 4 1 inner