# Properties

 Label 230.5.f.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 88q^{2} + 24q^{5} - 80q^{7} - 704q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 88q^{2} + 24q^{5} - 80q^{7} - 704q^{8} - 136q^{10} - 632q^{11} + 500q^{13} + 12q^{15} - 2816q^{16} + 120q^{17} + 2648q^{18} - 736q^{20} - 856q^{21} - 1264q^{22} + 2052q^{25} + 2000q^{26} + 588q^{27} + 640q^{28} - 1704q^{30} + 3220q^{31} - 5632q^{32} - 2884q^{33} + 720q^{35} + 10592q^{36} + 1856q^{37} - 1696q^{38} - 1856q^{40} - 5156q^{41} - 1712q^{42} + 960q^{43} + 460q^{45} - 1224q^{47} + 4456q^{50} + 3640q^{51} + 4000q^{52} + 13556q^{53} + 12980q^{55} + 2560q^{56} - 3072q^{57} - 3112q^{58} - 6912q^{60} - 4864q^{61} + 6440q^{62} - 13484q^{63} - 4100q^{65} - 11536q^{66} - 1888q^{67} - 960q^{68} + 8824q^{70} - 1060q^{71} + 21184q^{72} + 16616q^{73} - 11044q^{75} - 6784q^{76} + 8892q^{77} - 23152q^{78} - 1536q^{80} - 4044q^{81} - 10312q^{82} - 27024q^{83} - 11068q^{85} + 3840q^{86} - 8392q^{87} + 10112q^{88} - 7184q^{90} - 54024q^{91} + 22528q^{93} - 25968q^{95} - 3252q^{97} - 27984q^{98} + O(q^{100})$$

## 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 −11.2624 + 11.2624i 8.00000i −23.2954 9.07330i −45.0495 2.86636 + 2.86636i −16.0000 + 16.0000i 172.682i −28.4442 64.7374i
47.2 2.00000 + 2.00000i −11.0455 + 11.0455i 8.00000i 3.15467 + 24.8002i −44.1818 −2.46420 2.46420i −16.0000 + 16.0000i 163.004i −43.2910 + 55.9097i
47.3 2.00000 + 2.00000i −9.95475 + 9.95475i 8.00000i 24.5881 + 4.51957i −39.8190 −57.5399 57.5399i −16.0000 + 16.0000i 117.194i 40.1370 + 58.2153i
47.4 2.00000 + 2.00000i −9.27707 + 9.27707i 8.00000i 23.6357 8.14565i −37.1083 15.7230 + 15.7230i −16.0000 + 16.0000i 91.1281i 63.5628 + 30.9802i
47.5 2.00000 + 2.00000i −8.96878 + 8.96878i 8.00000i 1.96751 24.9225i −35.8751 53.1255 + 53.1255i −16.0000 + 16.0000i 79.8782i 53.7799 45.9099i
47.6 2.00000 + 2.00000i −5.57799 + 5.57799i 8.00000i −24.4867 5.04022i −22.3120 −57.8690 57.8690i −16.0000 + 16.0000i 18.7720i −38.8929 59.0537i
47.7 2.00000 + 2.00000i −4.66373 + 4.66373i 8.00000i −12.4209 + 21.6961i −18.6549 −24.9944 24.9944i −16.0000 + 16.0000i 37.4993i −68.2341 + 18.5503i
47.8 2.00000 + 2.00000i −4.44879 + 4.44879i 8.00000i 18.7807 + 16.5011i −17.7952 10.7782 + 10.7782i −16.0000 + 16.0000i 41.4165i 4.55931 + 70.5635i
47.9 2.00000 + 2.00000i −4.05030 + 4.05030i 8.00000i −24.3623 + 5.61047i −16.2012 52.2645 + 52.2645i −16.0000 + 16.0000i 48.1901i −59.9456 37.5037i
47.10 2.00000 + 2.00000i −3.68907 + 3.68907i 8.00000i 6.02861 24.2622i −14.7563 −4.81567 4.81567i −16.0000 + 16.0000i 53.7815i 60.5817 36.4673i
47.11 2.00000 + 2.00000i −0.310392 + 0.310392i 8.00000i 6.21241 + 24.2158i −1.24157 40.7961 + 40.7961i −16.0000 + 16.0000i 80.8073i −36.0068 + 60.8565i
47.12 2.00000 + 2.00000i 1.43653 1.43653i 8.00000i 23.6990 7.95979i 5.74612 −17.1725 17.1725i −16.0000 + 16.0000i 76.8728i 63.3175 + 31.4784i
47.13 2.00000 + 2.00000i 2.64936 2.64936i 8.00000i 22.4224 + 11.0560i 10.5974 −62.5314 62.5314i −16.0000 + 16.0000i 66.9618i 22.7329 + 66.9568i
47.14 2.00000 + 2.00000i 3.59966 3.59966i 8.00000i −10.4188 22.7255i 14.3986 7.45573 + 7.45573i −16.0000 + 16.0000i 55.0849i 24.6134 66.2886i
47.15 2.00000 + 2.00000i 4.87410 4.87410i 8.00000i 23.2684 9.14227i 19.4964 49.4466 + 49.4466i −16.0000 + 16.0000i 33.4862i 64.8214 + 28.2523i
47.16 2.00000 + 2.00000i 5.56222 5.56222i 8.00000i −17.4021 + 17.9490i 22.2489 −33.3744 33.3744i −16.0000 + 16.0000i 19.1234i −70.7022 + 1.09393i
47.17 2.00000 + 2.00000i 7.12415 7.12415i 8.00000i −12.4603 + 21.6735i 28.4966 −34.8776 34.8776i −16.0000 + 16.0000i 20.5070i −68.2676 + 18.4264i
47.18 2.00000 + 2.00000i 7.50341 7.50341i 8.00000i −23.5955 8.26153i 30.0136 38.1189 + 38.1189i −16.0000 + 16.0000i 31.6022i −30.6679 63.7140i
47.19 2.00000 + 2.00000i 7.61108 7.61108i 8.00000i −1.36901 + 24.9625i 30.4443 45.0835 + 45.0835i −16.0000 + 16.0000i 34.8570i −52.6630 + 47.1870i
47.20 2.00000 + 2.00000i 9.67771 9.67771i 8.00000i 14.1269 20.6259i 38.7108 −43.7928 43.7928i −16.0000 + 16.0000i 106.316i 69.5058 12.9980i
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.b 44
5.c odd 4 1 inner 230.5.f.b 44

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