# Properties

 Label 177.6.d.b Level $177$ Weight $6$ Character orbit 177.d Analytic conductor $28.388$ Analytic rank $0$ Dimension $92$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 177.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.3879361069$$ Analytic rank: $$0$$ Dimension: $$92$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$92q + 22q^{3} + 1724q^{4} - 80q^{7} + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$92q + 22q^{3} + 1724q^{4} - 80q^{7} + 2q^{9} - 1244q^{12} + 1116q^{15} + 14724q^{16} + 1784q^{19} + 6388q^{21} - 8140q^{22} - 48208q^{25} - 6458q^{27} - 19092q^{28} - 20832q^{36} - 134984q^{45} + 51180q^{46} + 61720q^{48} + 174556q^{49} + 8332q^{51} + 236784q^{57} + 375208q^{60} - 429890q^{63} + 561472q^{64} - 11596q^{66} + 169948q^{75} + 111488q^{76} + 356264q^{78} + 180260q^{79} + 79554q^{81} + 269308q^{84} + 111028q^{85} - 318764q^{87} - 1242976q^{88} - 513608q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
176.1 −10.9379 4.58836 + 14.8979i 87.6366 82.7497i −50.1868 162.951i 211.698 −608.545 −200.894 + 136.714i 905.104i
176.2 −10.9379 4.58836 14.8979i 87.6366 82.7497i −50.1868 + 162.951i 211.698 −608.545 −200.894 136.714i 905.104i
176.3 −10.7490 9.69295 + 12.2085i 83.5412 33.1454i −104.190 131.229i −174.539 −554.017 −55.0934 + 236.672i 356.281i
176.4 −10.7490 9.69295 12.2085i 83.5412 33.1454i −104.190 + 131.229i −174.539 −554.017 −55.0934 236.672i 356.281i
176.5 −10.3821 −12.6016 9.17604i 75.7873 9.35226i 130.831 + 95.2663i 67.1723 −454.603 74.6006 + 231.266i 97.0958i
176.6 −10.3821 −12.6016 + 9.17604i 75.7873 9.35226i 130.831 95.2663i 67.1723 −454.603 74.6006 231.266i 97.0958i
176.7 −9.95283 −15.5523 1.06177i 67.0588 97.9414i 154.789 + 10.5677i −155.750 −348.934 240.745 + 33.0259i 974.794i
176.8 −9.95283 −15.5523 + 1.06177i 67.0588 97.9414i 154.789 10.5677i −155.750 −348.934 240.745 33.0259i 974.794i
176.9 −9.63720 15.3454 2.74211i 60.8757 48.0995i −147.887 + 26.4263i 55.2643 −278.281 227.962 84.1575i 463.545i
176.10 −9.63720 15.3454 + 2.74211i 60.8757 48.0995i −147.887 26.4263i 55.2643 −278.281 227.962 + 84.1575i 463.545i
176.11 −9.16184 −4.65820 + 14.8762i 51.9393 80.4460i 42.6777 136.293i −176.825 −182.680 −199.602 138.593i 737.033i
176.12 −9.16184 −4.65820 14.8762i 51.9393 80.4460i 42.6777 + 136.293i −176.825 −182.680 −199.602 + 138.593i 737.033i
176.13 −8.73168 1.48415 15.5176i 44.2422 95.2512i −12.9591 + 135.495i 144.410 −106.895 −238.595 46.0609i 831.703i
176.14 −8.73168 1.48415 + 15.5176i 44.2422 95.2512i −12.9591 135.495i 144.410 −106.895 −238.595 + 46.0609i 831.703i
176.15 −8.62331 −4.57233 + 14.9028i 42.3615 15.6511i 39.4286 128.512i −96.1896 −89.3504 −201.188 136.281i 134.964i
176.16 −8.62331 −4.57233 14.9028i 42.3615 15.6511i 39.4286 + 128.512i −96.1896 −89.3504 −201.188 + 136.281i 134.964i
176.17 −7.56324 11.2364 + 10.8048i 25.2027 51.3528i −84.9838 81.7190i 90.7178 51.4099 9.51431 + 242.814i 388.394i
176.18 −7.56324 11.2364 10.8048i 25.2027 51.3528i −84.9838 + 81.7190i 90.7178 51.4099 9.51431 242.814i 388.394i
176.19 −7.43533 −12.1962 9.70834i 23.2842 26.1059i 90.6831 + 72.1847i 171.796 64.8050 54.4964 + 236.810i 194.106i
176.20 −7.43533 −12.1962 + 9.70834i 23.2842 26.1059i 90.6831 72.1847i 171.796 64.8050 54.4964 236.810i 194.106i
See all 92 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 176.92 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
59.b odd 2 1 inner
177.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.6.d.b 92
3.b odd 2 1 inner 177.6.d.b 92
59.b odd 2 1 inner 177.6.d.b 92
177.d even 2 1 inner 177.6.d.b 92

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.d.b 92 1.a even 1 1 trivial
177.6.d.b 92 3.b odd 2 1 inner
177.6.d.b 92 59.b odd 2 1 inner
177.6.d.b 92 177.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$86\!\cdots\!83$$$$T_{2}^{36} +$$$$11\!\cdots\!84$$$$T_{2}^{34} -$$$$11\!\cdots\!61$$$$T_{2}^{32} +$$$$96\!\cdots\!75$$$$T_{2}^{30} -$$$$63\!\cdots\!14$$$$T_{2}^{28} +$$$$33\!\cdots\!84$$$$T_{2}^{26} -$$$$14\!\cdots\!60$$$$T_{2}^{24} +$$$$49\!\cdots\!72$$$$T_{2}^{22} -$$$$13\!\cdots\!68$$$$T_{2}^{20} +$$$$30\!\cdots\!28$$$$T_{2}^{18} -$$$$53\!\cdots\!84$$$$T_{2}^{16} +$$$$72\!\cdots\!96$$$$T_{2}^{14} -$$$$74\!\cdots\!80$$$$T_{2}^{12} +$$$$57\!\cdots\!44$$$$T_{2}^{10} -$$$$31\!\cdots\!00$$$$T_{2}^{8} +$$$$11\!\cdots\!40$$$$T_{2}^{6} -$$$$28\!\cdots\!72$$$$T_{2}^{4} +$$$$38\!\cdots\!44$$$$T_{2}^{2} -$$$$21\!\cdots\!32$$">$$T_{2}^{46} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(177, [\chi])$$.