# Properties

 Label 177.5.c.a Level $177$ Weight $5$ Character orbit 177.c Analytic conductor $18.296$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 177.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.2964834658$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 320q^{4} + 80q^{7} + 1080q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 320q^{4} + 80q^{7} + 1080q^{9} + 360q^{12} + 144q^{15} + 3944q^{16} - 528q^{17} + 444q^{19} + 444q^{20} + 1304q^{22} + 4880q^{25} - 1452q^{26} - 1160q^{28} - 996q^{29} + 10320q^{35} - 8640q^{36} - 5196q^{41} - 10476q^{46} + 576q^{48} + 5104q^{49} + 936q^{51} - 2184q^{53} - 2520q^{57} - 11736q^{59} - 11448q^{60} + 15240q^{62} + 2160q^{63} - 81012q^{64} + 17352q^{66} + 29568q^{68} - 5964q^{71} + 14376q^{74} - 2736q^{75} + 3480q^{76} + 37692q^{78} + 19020q^{79} + 33096q^{80} + 29160q^{81} + 25128q^{84} + 20220q^{85} - 65880q^{86} + 1512q^{87} - 14932q^{88} - 17864q^{94} + 11004q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
58.1 7.81880i 5.19615 −45.1337 21.3172 40.6277i −30.7266 227.790i 27.0000 166.675i
58.2 7.77798i −5.19615 −44.4970 28.5836 40.4156i 72.7001 221.650i 27.0000 222.323i
58.3 7.65331i −5.19615 −42.5731 −38.1687 39.7678i −35.1454 203.372i 27.0000 292.117i
58.4 6.76718i 5.19615 −29.7948 6.77685 35.1633i 45.8846 93.3518i 27.0000 45.8602i
58.5 6.70986i 5.19615 −29.0222 −41.0870 34.8655i −6.70931 87.3773i 27.0000 275.688i
58.6 6.10324i −5.19615 −21.2495 −12.8499 31.7133i 4.61608 32.0389i 27.0000 78.4262i
58.7 5.77154i −5.19615 −17.3107 34.3452 29.9898i −39.2727 7.56484i 27.0000 198.225i
58.8 4.96663i 5.19615 −8.66741 41.3974 25.8074i 1.08721 36.4182i 27.0000 205.606i
58.9 4.85825i −5.19615 −7.60257 −12.5087 25.2442i 50.4755 40.7968i 27.0000 60.7705i
58.10 4.64719i 5.19615 −5.59638 −0.691812 24.1475i −76.1022 48.3476i 27.0000 3.21498i
58.11 4.10319i −5.19615 −0.836195 −39.6276 21.3208i −85.8400 62.2200i 27.0000 162.600i
58.12 4.05068i −5.19615 −0.408022 −16.4107 21.0480i 47.3137 63.1581i 27.0000 66.4744i
58.13 3.44422i 5.19615 4.13738 −16.2205 17.8967i 92.6602 69.3575i 27.0000 55.8670i
58.14 2.43119i 5.19615 10.0893 25.9319 12.6328i −17.4595 63.4281i 27.0000 63.0454i
58.15 2.33963i −5.19615 10.5261 30.0439 12.1571i −41.7287 62.0613i 27.0000 70.2916i
58.16 2.31914i 5.19615 10.6216 −38.6371 12.0506i 9.48964 61.7392i 27.0000 89.6048i
58.17 2.15560i −5.19615 11.3534 30.9385 11.2008i 73.3307 58.9629i 27.0000 66.6909i
58.18 1.07792i 5.19615 14.8381 26.1400 5.60104i 49.1298 33.2410i 27.0000 28.1769i
58.19 0.850217i −5.19615 15.2771 −11.2738 4.41785i −26.4494 26.5923i 27.0000 9.58520i
58.20 0.389117i 5.19615 15.8486 −17.9988 2.02191i −47.2538 12.3928i 27.0000 7.00365i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 58.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.5.c.a 40
3.b odd 2 1 531.5.c.d 40
59.b odd 2 1 inner 177.5.c.a 40
177.d even 2 1 531.5.c.d 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.5.c.a 40 1.a even 1 1 trivial
177.5.c.a 40 59.b odd 2 1 inner
531.5.c.d 40 3.b odd 2 1
531.5.c.d 40 177.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(177, [\chi])$$.