Newspace parameters
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.h (of order \(58\), degree \(28\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.82290067918\) |
Analytic rank: | \(0\) |
Dimension: | \(1064\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{58})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{58}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.23800 | + | 3.67424i | 2.23781 | − | 1.99805i | −8.78304 | − | 6.67669i | 2.27487 | + | 0.906393i | 4.57091 | + | 10.6958i | 0.939802 | − | 0.434799i | 22.5687 | − | 15.3019i | 1.01559 | − | 8.94251i | −6.14659 | + | 7.23632i |
5.2 | −1.21095 | + | 3.59398i | −2.84830 | + | 0.941900i | −8.26594 | − | 6.28360i | 1.37514 | + | 0.547907i | 0.0639885 | − | 11.3773i | 8.82700 | − | 4.08381i | 20.0368 | − | 13.5853i | 7.22565 | − | 5.36563i | −3.63441 | + | 4.27875i |
5.3 | −1.15362 | + | 3.42383i | 0.846084 | + | 2.87822i | −7.20737 | − | 5.47890i | −6.99239 | − | 2.78602i | −10.8306 | − | 0.423531i | 0.665705 | − | 0.307988i | 15.1117 | − | 10.2460i | −7.56829 | + | 4.87043i | 17.6054 | − | 20.7267i |
5.4 | −1.00783 | + | 2.99114i | −2.99774 | − | 0.116447i | −4.74683 | − | 3.60844i | −2.80678 | − | 1.11832i | 3.36953 | − | 8.84930i | −12.2974 | + | 5.68939i | 5.12741 | − | 3.47647i | 8.97288 | + | 0.698155i | 6.17382 | − | 7.26839i |
5.5 | −0.999960 | + | 2.96778i | −1.56678 | − | 2.55836i | −4.62340 | − | 3.51462i | 4.69371 | + | 1.87015i | 9.15935 | − | 2.09161i | −2.98209 | + | 1.37966i | 4.68548 | − | 3.17683i | −4.09038 | + | 8.01678i | −10.2437 | + | 12.0598i |
5.6 | −0.964344 | + | 2.86207i | −0.263179 | + | 2.98843i | −4.07712 | − | 3.09934i | 5.65262 | + | 2.25221i | −8.29931 | − | 3.63511i | −4.32408 | + | 2.00053i | 2.80324 | − | 1.90065i | −8.86147 | − | 1.57299i | −11.8970 | + | 14.0063i |
5.7 | −0.912087 | + | 2.70698i | 2.96240 | − | 0.473468i | −3.31146 | − | 2.51731i | −6.32194 | − | 2.51889i | −1.42030 | + | 8.45101i | −6.08665 | + | 2.81599i | 0.377434 | − | 0.255907i | 8.55166 | − | 2.80521i | 12.5848 | − | 14.8159i |
5.8 | −0.892372 | + | 2.64847i | 0.713097 | − | 2.91402i | −3.03367 | − | 2.30614i | −2.87136 | − | 1.14405i | 7.08133 | + | 4.48900i | 2.33765 | − | 1.08151i | −0.437890 | + | 0.296897i | −7.98299 | − | 4.15595i | 5.59230 | − | 6.58377i |
5.9 | −0.869361 | + | 2.58017i | 2.78047 | + | 1.12649i | −2.71713 | − | 2.06551i | 3.05628 | + | 1.21773i | −5.32377 | + | 6.19477i | 9.38908 | − | 4.34385i | −1.32265 | + | 0.896782i | 6.46204 | + | 6.26435i | −5.79898 | + | 6.82709i |
5.10 | −0.792761 | + | 2.35283i | −2.47356 | − | 1.69749i | −1.72297 | − | 1.30976i | −7.18639 | − | 2.86332i | 5.95485 | − | 4.47417i | 10.9013 | − | 5.04348i | −3.77238 | + | 2.55774i | 3.23704 | + | 8.39771i | 12.4340 | − | 14.6384i |
5.11 | −0.561287 | + | 1.66584i | 2.68858 | − | 1.33099i | 0.724387 | + | 0.550665i | 7.94437 | + | 3.16532i | 0.708151 | + | 5.22582i | −11.0072 | + | 5.09247i | −7.14376 | + | 4.84359i | 5.45694 | − | 7.15694i | −9.73200 | + | 11.4574i |
5.12 | −0.551931 | + | 1.63807i | −1.84525 | + | 2.36538i | 0.805720 | + | 0.612492i | −3.83005 | − | 1.52603i | −2.85622 | − | 4.32819i | −0.768529 | + | 0.355559i | −7.17084 | + | 4.86195i | −2.19007 | − | 8.72947i | 4.61367 | − | 5.43163i |
5.13 | −0.528130 | + | 1.56743i | −2.80203 | + | 1.07173i | 1.00644 | + | 0.765078i | 6.70916 | + | 2.67318i | −0.200026 | − | 4.95802i | 6.64296 | − | 3.07336i | −7.20679 | + | 4.88632i | 6.70279 | − | 6.00604i | −7.73334 | + | 9.10439i |
5.14 | −0.386111 | + | 1.14594i | 1.94700 | + | 2.28237i | 2.02029 | + | 1.53578i | −1.21804 | − | 0.485310i | −3.36720 | + | 1.34988i | −3.89579 | + | 1.80238i | −6.54344 | + | 4.43656i | −1.41841 | + | 8.88753i | 1.02643 | − | 1.20841i |
5.15 | −0.322077 | + | 0.955892i | −0.589478 | − | 2.94152i | 2.37438 | + | 1.80495i | 5.46847 | + | 2.17884i | 3.00163 | + | 0.383919i | 3.90866 | − | 1.80834i | −5.82961 | + | 3.95258i | −8.30503 | + | 3.46792i | −3.84400 | + | 4.52551i |
5.16 | −0.225119 | + | 0.668129i | 0.0215190 | − | 2.99992i | 2.78865 | + | 2.11988i | −5.48264 | − | 2.18448i | 1.99949 | + | 0.689717i | −11.1057 | + | 5.13804i | −4.37833 | + | 2.96858i | −8.99907 | − | 0.129111i | 2.69377 | − | 3.17135i |
5.17 | −0.221328 | + | 0.656877i | −2.72451 | − | 1.25582i | 2.80187 | + | 2.12993i | −2.12339 | − | 0.846038i | 1.42792 | − | 1.51172i | −1.48778 | + | 0.688319i | −4.31412 | + | 2.92505i | 5.84586 | + | 6.84295i | 1.02571 | − | 1.20756i |
5.18 | −0.192085 | + | 0.570088i | 2.47232 | − | 1.69930i | 2.89627 | + | 2.20169i | 1.33026 | + | 0.530023i | 0.493857 | + | 1.73585i | 7.94838 | − | 3.67731i | −3.80316 | + | 2.57861i | 3.22474 | − | 8.40244i | −0.557682 | + | 0.656554i |
5.19 | −0.0718854 | + | 0.213348i | 2.81983 | − | 1.02398i | 3.14402 | + | 2.39002i | −7.60675 | − | 3.03081i | 0.0157589 | + | 0.675216i | 4.35627 | − | 2.01543i | −1.48128 | + | 1.00433i | 6.90294 | − | 5.77490i | 1.19343 | − | 1.40502i |
5.20 | 0.0718854 | − | 0.213348i | 0.416138 | + | 2.97100i | 3.14402 | + | 2.39002i | 7.60675 | + | 3.03081i | 0.663771 | + | 0.124789i | 4.35627 | − | 2.01543i | 1.48128 | − | 1.00433i | −8.65366 | + | 2.47269i | 1.19343 | − | 1.40502i |
See next 80 embeddings (of 1064 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
59.c | even | 29 | 1 | inner |
177.h | odd | 58 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.3.h.a | ✓ | 1064 |
3.b | odd | 2 | 1 | inner | 177.3.h.a | ✓ | 1064 |
59.c | even | 29 | 1 | inner | 177.3.h.a | ✓ | 1064 |
177.h | odd | 58 | 1 | inner | 177.3.h.a | ✓ | 1064 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.3.h.a | ✓ | 1064 | 1.a | even | 1 | 1 | trivial |
177.3.h.a | ✓ | 1064 | 3.b | odd | 2 | 1 | inner |
177.3.h.a | ✓ | 1064 | 59.c | even | 29 | 1 | inner |
177.3.h.a | ✓ | 1064 | 177.h | odd | 58 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).