Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(58,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("775.58");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 775.bz (of order \(20\), degree \(8\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(6.18840615665\) |
Analytic rank: | \(0\) |
Dimension: | \(624\) |
Relative dimension: | \(78\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
58.1 | −0.432453 | + | 2.73040i | −0.813284 | − | 0.414389i | −5.36597 | − | 1.74351i | 1.39279 | + | 1.74932i | 1.48316 | − | 2.04139i | 0.597123 | − | 3.77009i | 4.57095 | − | 8.97100i | −1.27364 | − | 1.75302i | −5.37868 | + | 3.04637i |
58.2 | −0.427920 | + | 2.70178i | −2.21257 | − | 1.12736i | −5.21439 | − | 1.69426i | −0.783635 | − | 2.09426i | 3.99268 | − | 5.49545i | −0.194778 | + | 1.22978i | 4.32511 | − | 8.48851i | 1.86116 | + | 2.56166i | 5.99356 | − | 1.22103i |
58.3 | −0.420856 | + | 2.65718i | 1.36758 | + | 0.696816i | −4.98137 | − | 1.61855i | −2.22970 | + | 0.168664i | −2.42712 | + | 3.34064i | 0.172196 | − | 1.08720i | 3.95447 | − | 7.76108i | −0.378638 | − | 0.521151i | 0.490211 | − | 5.99569i |
58.4 | −0.417323 | + | 2.63487i | 2.25350 | + | 1.14822i | −4.86628 | − | 1.58115i | 1.91087 | + | 1.16128i | −3.96584 | + | 5.45851i | −0.787011 | + | 4.96899i | 3.77471 | − | 7.40828i | 1.99651 | + | 2.74796i | −3.85727 | + | 4.55027i |
58.5 | −0.407008 | + | 2.56975i | 0.965093 | + | 0.491739i | −4.53583 | − | 1.47378i | 0.800701 | − | 2.08779i | −1.65645 | + | 2.27990i | −0.0975664 | + | 0.616010i | 3.27099 | − | 6.41969i | −1.07376 | − | 1.47790i | 5.03921 | + | 2.90735i |
58.6 | −0.385823 | + | 2.43599i | −0.635361 | − | 0.323733i | −3.88308 | − | 1.26169i | −1.16565 | + | 1.90821i | 1.03375 | − | 1.42283i | −0.334318 | + | 2.11080i | 2.33224 | − | 4.57727i | −1.46447 | − | 2.01568i | −4.19864 | − | 3.57574i |
58.7 | −0.372726 | + | 2.35330i | 1.04430 | + | 0.532096i | −3.49698 | − | 1.13624i | 2.12831 | − | 0.685784i | −1.64142 | + | 2.25922i | 0.575570 | − | 3.63401i | 1.81394 | − | 3.56005i | −0.955926 | − | 1.31572i | 0.820578 | + | 5.26416i |
58.8 | −0.371953 | + | 2.34842i | −1.05680 | − | 0.538468i | −3.47462 | − | 1.12897i | 2.05961 | − | 0.870635i | 1.65763 | − | 2.28153i | −0.551526 | + | 3.48220i | 1.78480 | − | 3.50286i | −0.936471 | − | 1.28894i | 1.27854 | + | 5.16067i |
58.9 | −0.359324 | + | 2.26869i | −3.05823 | − | 1.55825i | −3.11571 | − | 1.01235i | −1.52232 | + | 1.63785i | 4.63407 | − | 6.37825i | 0.553964 | − | 3.49759i | 1.33066 | − | 2.61157i | 5.16129 | + | 7.10391i | −3.16875 | − | 4.04219i |
58.10 | −0.354386 | + | 2.23750i | 2.89940 | + | 1.47732i | −2.97872 | − | 0.967845i | −1.95864 | − | 1.07876i | −4.33301 | + | 5.96388i | −0.115111 | + | 0.726782i | 1.16424 | − | 2.28494i | 4.46070 | + | 6.13962i | 3.10785 | − | 4.00016i |
58.11 | −0.337588 | + | 2.13145i | 2.05236 | + | 1.04573i | −2.52700 | − | 0.821071i | −0.701956 | + | 2.12303i | −2.92177 | + | 4.02147i | 0.190417 | − | 1.20224i | 0.643716 | − | 1.26336i | 1.35527 | + | 1.86537i | −4.28816 | − | 2.21289i |
58.12 | −0.335900 | + | 2.12079i | −2.21885 | − | 1.13056i | −2.48281 | − | 0.806715i | 2.18571 | − | 0.471884i | 3.14300 | − | 4.32596i | 0.188721 | − | 1.19154i | 0.595211 | − | 1.16817i | 1.88177 | + | 2.59004i | 0.266588 | + | 4.79394i |
58.13 | −0.333655 | + | 2.10662i | 1.00524 | + | 0.512194i | −2.42439 | − | 0.787732i | −0.617699 | − | 2.14906i | −1.41440 | + | 1.94675i | 0.165232 | − | 1.04323i | 0.531750 | − | 1.04362i | −1.01520 | − | 1.39730i | 4.73334 | − | 0.584210i |
58.14 | −0.320924 | + | 2.02624i | −1.93450 | − | 0.985675i | −2.10053 | − | 0.682503i | −2.16149 | + | 0.572676i | 2.61804 | − | 3.60342i | −0.468266 | + | 2.95651i | 0.194306 | − | 0.381347i | 1.00736 | + | 1.38652i | −0.466702 | − | 4.56348i |
58.15 | −0.302579 | + | 1.91041i | −1.04392 | − | 0.531905i | −1.65598 | − | 0.538061i | 1.28497 | + | 1.82999i | 1.33202 | − | 1.83337i | 0.225002 | − | 1.42060i | −0.227254 | + | 0.446011i | −0.956506 | − | 1.31652i | −3.88482 | + | 1.90110i |
58.16 | −0.292237 | + | 1.84511i | 0.500677 | + | 0.255108i | −1.41693 | − | 0.460389i | −0.00183075 | + | 2.23607i | −0.617020 | + | 0.849255i | −0.226119 | + | 1.42766i | −0.432661 | + | 0.849145i | −1.57776 | − | 2.17160i | −4.12527 | − | 0.656840i |
58.17 | −0.277660 | + | 1.75307i | −1.36666 | − | 0.696350i | −1.09406 | − | 0.355482i | 0.647198 | − | 2.14036i | 1.60022 | − | 2.20252i | 0.563366 | − | 3.55695i | −0.684635 | + | 1.34367i | −0.380488 | − | 0.523697i | 3.57251 | + | 1.72888i |
58.18 | −0.276615 | + | 1.74648i | 1.93521 | + | 0.986041i | −1.07156 | − | 0.348170i | 1.93192 | + | 1.12592i | −2.25741 | + | 3.10705i | 0.0127590 | − | 0.0805573i | −0.701056 | + | 1.37590i | 1.00942 | + | 1.38935i | −2.50079 | + | 3.06261i |
58.19 | −0.270232 | + | 1.70618i | 0.232154 | + | 0.118288i | −0.935906 | − | 0.304094i | −1.45606 | − | 1.69703i | −0.264556 | + | 0.364130i | −0.629780 | + | 3.97627i | −0.796737 | + | 1.56368i | −1.72345 | − | 2.37213i | 3.28891 | − | 2.02570i |
58.20 | −0.263622 | + | 1.66444i | 0.760114 | + | 0.387297i | −0.798767 | − | 0.259535i | −2.20524 | + | 0.369994i | −0.845018 | + | 1.16307i | 0.814632 | − | 5.14338i | −0.887568 | + | 1.74195i | −1.33558 | − | 1.83827i | −0.0344834 | − | 3.76805i |
See next 80 embeddings (of 624 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
775.bz | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 775.2.bz.a | yes | 624 |
25.f | odd | 20 | 1 | 775.2.br.a | ✓ | 624 | |
31.f | odd | 10 | 1 | 775.2.br.a | ✓ | 624 | |
775.bz | even | 20 | 1 | inner | 775.2.bz.a | yes | 624 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
775.2.br.a | ✓ | 624 | 25.f | odd | 20 | 1 | |
775.2.br.a | ✓ | 624 | 31.f | odd | 10 | 1 | |
775.2.bz.a | yes | 624 | 1.a | even | 1 | 1 | trivial |
775.2.bz.a | yes | 624 | 775.bz | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).