# Properties

 Label 77.2.n.a Level $77$ Weight $2$ Character orbit 77.n Analytic conductor $0.615$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 77.n (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$6$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 5q^{2} - 9q^{3} - 9q^{4} - 15q^{5} - 5q^{7} - 11q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 5q^{2} - 9q^{3} - 9q^{4} - 15q^{5} - 5q^{7} - 11q^{9} - q^{11} - 12q^{12} - 8q^{14} - 27q^{16} + 15q^{17} + 20q^{18} - 15q^{19} - 76q^{22} + 10q^{23} + 75q^{24} + q^{25} + 27q^{26} - 40q^{28} - 40q^{29} + 25q^{30} + 9q^{31} + 42q^{33} + 5q^{35} - 38q^{36} - q^{37} + 33q^{38} - 45q^{39} + 75q^{40} + 64q^{42} + 30q^{44} - 84q^{45} - 20q^{46} + 3q^{47} + 59q^{49} + 30q^{50} + 55q^{51} - 15q^{52} - 3q^{53} - 8q^{56} + 60q^{57} + 46q^{58} - 3q^{59} - 15q^{60} - 30q^{61} - 40q^{63} + 12q^{64} - 93q^{66} + 44q^{67} - 75q^{68} - 27q^{70} + 20q^{71} - 60q^{72} - 60q^{73} + 45q^{74} - 57q^{75} + 92q^{78} - 70q^{79} - 75q^{80} - 29q^{81} - 129q^{82} - 125q^{84} + 10q^{85} - 62q^{86} + 19q^{88} + 6q^{89} - 12q^{91} + 30q^{92} - 92q^{93} + 105q^{94} + 30q^{95} + 75q^{96} - 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −1.95877 1.76368i 0.650787 + 1.46169i 0.517138 + 4.92024i 0.446428 + 2.10028i 1.30322 4.01090i −1.30623 + 2.30082i 4.56624 6.28489i 0.294372 0.326934i 2.82977 4.90131i
17.2 −1.33360 1.20078i −1.17499 2.63908i 0.127565 + 1.21370i 0.418270 + 1.96780i −1.60198 + 4.93040i −1.17364 2.37120i −0.822345 + 1.13186i −3.57672 + 3.97235i 1.80510 3.12652i
17.3 −1.11268 1.00186i 0.245339 + 0.551041i 0.0252713 + 0.240440i −0.491349 2.31161i 0.279082 0.858925i 2.00635 1.72470i −1.54736 + 2.12976i 1.76394 1.95905i −1.76920 + 3.06434i
17.4 −0.0202070 0.0181945i −0.500742 1.12469i −0.208980 1.98831i −0.240558 1.13174i −0.0103446 + 0.0318373i −0.296416 + 2.62909i −0.0639186 + 0.0879764i 0.993218 1.10308i −0.0157304 + 0.0272459i
17.5 0.386517 + 0.348022i 0.460952 + 1.03532i −0.180780 1.72001i 0.678628 + 3.19269i −0.182146 + 0.560588i −1.97073 1.76528i 1.14015 1.56929i 1.14799 1.27497i −0.848825 + 1.47021i
17.6 1.75157 + 1.57712i −0.821231 1.84451i 0.371633 + 3.53585i −0.00617973 0.0290734i 1.47058 4.52598i −2.21185 + 1.45180i −2.15474 + 2.96574i −0.720422 + 0.800109i 0.0350280 0.0606703i
19.1 −2.06490 + 0.217030i −0.138863 0.125032i 2.26043 0.480470i −1.04356 2.34387i 0.313874 + 0.228043i 2.62996 0.288618i −0.613987 + 0.199497i −0.309936 2.94884i 2.66354 + 4.61339i
19.2 −1.30277 + 0.136927i 2.02980 + 1.82764i −0.277833 + 0.0590552i 0.166444 + 0.373838i −2.89461 2.10306i −2.50634 + 0.847492i 2.84553 0.924570i 0.466233 + 4.43591i −0.268026 0.464235i
19.3 −0.476751 + 0.0501086i −1.66384 1.49813i −1.73151 + 0.368045i −0.547175 1.22898i 0.868306 + 0.630861i −1.63770 2.07796i 1.71889 0.558501i 0.210388 + 2.00171i 0.322449 + 0.558497i
19.4 1.11135 0.116808i 1.15989 + 1.04437i −0.734836 + 0.156194i −1.25246 2.81307i 1.41104 + 1.02518i 1.16071 + 2.37755i −2.92398 + 0.950058i −0.0589482 0.560855i −1.72051 2.98001i
19.5 1.39661 0.146790i 0.151995 + 0.136856i −0.0273118 + 0.00580530i 0.618266 + 1.38865i 0.232367 + 0.168824i −1.57500 2.12588i −2.70844 + 0.880026i −0.309213 2.94196i 1.06732 + 1.84865i
19.6 2.05902 0.216412i −2.33449 2.10199i 2.23645 0.475371i 0.483200 + 1.08528i −5.26167 3.82283i 2.02173 + 1.70664i 0.563952 0.183239i 0.717924 + 6.83059i 1.22979 + 2.13006i
24.1 −0.548010 2.57818i 1.59126 0.167248i −4.51962 + 2.01227i −1.59568 1.43676i −1.30322 4.01090i 2.59186 + 0.531305i 4.56624 + 6.28489i −0.430319 + 0.0914672i −2.82977 + 4.90131i
24.2 −0.373106 1.75533i −2.87300 + 0.301965i −1.11488 + 0.496374i −1.49503 1.34613i 1.60198 + 4.93040i −1.89247 + 1.84893i −0.822345 1.13186i 5.22852 1.11136i −1.80510 + 3.12652i
24.3 −0.311296 1.46453i 0.599886 0.0630505i −0.220863 + 0.0983344i 1.75624 + 1.58133i −0.279082 0.858925i −2.26028 1.37519i −1.54736 2.12976i −2.57856 + 0.548089i 1.76920 3.06434i
24.4 −0.00565338 0.0265970i −1.22438 + 0.128687i 1.82642 0.813173i 0.859835 + 0.774198i 0.0103446 + 0.0318373i 2.59202 0.530526i −0.0639186 0.0879764i −1.45190 + 0.308612i 0.0157304 0.0272459i
24.5 0.108137 + 0.508745i 1.12709 0.118461i 1.57996 0.703445i −2.42564 2.18406i 0.182146 + 0.560588i −1.06990 + 2.41978i 1.14015 + 1.56929i −1.67816 + 0.356703i 0.848825 1.47021i
24.6 0.490042 + 2.30547i −2.00801 + 0.211051i −3.24795 + 1.44608i 0.0220884 + 0.0198885i −1.47058 4.52598i 2.06424 + 1.65497i −2.15474 2.96574i 1.05313 0.223849i −0.0350280 + 0.0606703i
40.1 −0.842093 1.89137i 0.653128 + 3.07273i −1.52991 + 1.69913i 1.18148 0.124179i 5.26167 3.82283i 2.63875 + 0.192354i 0.563952 + 0.183239i −6.27443 + 2.79356i −1.22979 2.13006i
40.2 −0.571183 1.28290i −0.0425239 0.200059i 0.0186834 0.0207501i 1.51174 0.158890i −0.232367 + 0.168824i −2.52376 0.794113i −2.70844 0.880026i 2.70242 1.20320i −1.06732 1.84865i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.n.a 48
3.b odd 2 1 693.2.cg.a 48
7.b odd 2 1 539.2.s.d 48
7.c even 3 1 539.2.m.a 48
7.c even 3 1 539.2.s.d 48
7.d odd 6 1 inner 77.2.n.a 48
7.d odd 6 1 539.2.m.a 48
11.b odd 2 1 847.2.r.c 48
11.c even 5 1 847.2.i.b 48
11.c even 5 1 847.2.r.a 48
11.c even 5 1 847.2.r.c 48
11.c even 5 1 847.2.r.d 48
11.d odd 10 1 inner 77.2.n.a 48
11.d odd 10 1 847.2.i.b 48
11.d odd 10 1 847.2.r.a 48
11.d odd 10 1 847.2.r.d 48
21.g even 6 1 693.2.cg.a 48
33.f even 10 1 693.2.cg.a 48
77.i even 6 1 847.2.r.c 48
77.l even 10 1 539.2.s.d 48
77.n even 30 1 inner 77.2.n.a 48
77.n even 30 1 539.2.m.a 48
77.n even 30 1 847.2.i.b 48
77.n even 30 1 847.2.r.a 48
77.n even 30 1 847.2.r.d 48
77.o odd 30 1 539.2.m.a 48
77.o odd 30 1 539.2.s.d 48
77.p odd 30 1 847.2.i.b 48
77.p odd 30 1 847.2.r.a 48
77.p odd 30 1 847.2.r.c 48
77.p odd 30 1 847.2.r.d 48
231.bf odd 30 1 693.2.cg.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.n.a 48 1.a even 1 1 trivial
77.2.n.a 48 7.d odd 6 1 inner
77.2.n.a 48 11.d odd 10 1 inner
77.2.n.a 48 77.n even 30 1 inner
539.2.m.a 48 7.c even 3 1
539.2.m.a 48 7.d odd 6 1
539.2.m.a 48 77.n even 30 1
539.2.m.a 48 77.o odd 30 1
539.2.s.d 48 7.b odd 2 1
539.2.s.d 48 7.c even 3 1
539.2.s.d 48 77.l even 10 1
539.2.s.d 48 77.o odd 30 1
693.2.cg.a 48 3.b odd 2 1
693.2.cg.a 48 21.g even 6 1
693.2.cg.a 48 33.f even 10 1
693.2.cg.a 48 231.bf odd 30 1
847.2.i.b 48 11.c even 5 1
847.2.i.b 48 11.d odd 10 1
847.2.i.b 48 77.n even 30 1
847.2.i.b 48 77.p odd 30 1
847.2.r.a 48 11.c even 5 1
847.2.r.a 48 11.d odd 10 1
847.2.r.a 48 77.n even 30 1
847.2.r.a 48 77.p odd 30 1
847.2.r.c 48 11.b odd 2 1
847.2.r.c 48 11.c even 5 1
847.2.r.c 48 77.i even 6 1
847.2.r.c 48 77.p odd 30 1
847.2.r.d 48 11.c even 5 1
847.2.r.d 48 11.d odd 10 1
847.2.r.d 48 77.n even 30 1
847.2.r.d 48 77.p odd 30 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(77, [\chi])$$.