Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(95,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 17, 12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.95");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.ci (of order \(18\), degree \(6\), 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: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(840\) |
Relative dimension: | \(140\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
95.1 | −1.41419 | + | 0.00832720i | 1.09084 | − | 1.34539i | 1.99986 | − | 0.0235525i | 2.13665 | + | 0.376749i | −1.53144 | + | 1.91172i | 2.48173 | + | 0.917059i | −2.82799 | + | 0.0499609i | −0.620154 | − | 2.93520i | −3.02476 | − | 0.515002i |
95.2 | −1.41419 | + | 0.00841960i | −1.72851 | + | 0.110651i | 1.99986 | − | 0.0238138i | 0.00617965 | + | 0.00108964i | 2.44351 | − | 0.171035i | 2.62747 | + | 0.310501i | −2.82798 | + | 0.0505152i | 2.97551 | − | 0.382524i | −0.00874837 | − | 0.00148893i |
95.3 | −1.41401 | − | 0.0242190i | −0.531095 | + | 1.64862i | 1.99883 | + | 0.0684917i | 1.58913 | + | 0.280207i | 0.790900 | − | 2.31829i | −2.09916 | − | 1.61045i | −2.82469 | − | 0.145257i | −2.43588 | − | 1.75115i | −2.24025 | − | 0.434701i |
95.4 | −1.41331 | + | 0.0506390i | −0.250798 | − | 1.71380i | 1.99487 | − | 0.143137i | −1.97895 | − | 0.348942i | 0.441239 | + | 2.40942i | −1.91178 | + | 1.82896i | −2.81212 | + | 0.303315i | −2.87420 | + | 0.859633i | 2.81453 | + | 0.392950i |
95.5 | −1.40530 | + | 0.158515i | 1.04908 | + | 1.37820i | 1.94975 | − | 0.445523i | 0.0136217 | + | 0.00240188i | −1.69274 | − | 1.77049i | −2.34832 | − | 1.21877i | −2.66936 | + | 0.935159i | −0.798872 | + | 2.89168i | −0.0195234 | − | 0.00121611i |
95.6 | −1.40027 | − | 0.198086i | 0.186948 | + | 1.72193i | 1.92152 | + | 0.554748i | −2.73494 | − | 0.482245i | 0.0793128 | − | 2.44821i | 0.265748 | + | 2.63237i | −2.58077 | − | 1.15742i | −2.93010 | + | 0.643822i | 3.73414 | + | 1.21703i |
95.7 | −1.39717 | + | 0.218922i | −0.617768 | − | 1.61814i | 1.90415 | − | 0.611740i | 3.83680 | + | 0.676532i | 1.21737 | + | 2.12556i | 1.64737 | − | 2.07031i | −2.52649 | + | 1.27156i | −2.23673 | + | 1.99926i | −5.50876 | − | 0.105268i |
95.8 | −1.39122 | − | 0.253995i | 1.51307 | + | 0.842975i | 1.87097 | + | 0.706724i | 3.16384 | + | 0.557871i | −1.89090 | − | 1.55707i | 0.270632 | + | 2.63187i | −2.42343 | − | 1.45842i | 1.57879 | + | 2.55097i | −4.25990 | − | 1.57972i |
95.9 | −1.38746 | + | 0.273772i | 0.796278 | − | 1.53816i | 1.85010 | − | 0.759696i | −0.188974 | − | 0.0333212i | −0.683700 | + | 2.35214i | −1.65353 | − | 2.06539i | −2.35896 | + | 1.56055i | −1.73188 | − | 2.44961i | 0.271317 | − | 0.00550392i |
95.10 | −1.37324 | − | 0.337961i | −1.07725 | + | 1.35630i | 1.77156 | + | 0.928203i | −0.713698 | − | 0.125844i | 1.93769 | − | 1.49845i | 2.30108 | − | 1.30576i | −2.11908 | − | 1.87336i | −0.679077 | − | 2.92213i | 0.937547 | + | 0.414016i |
95.11 | −1.37052 | − | 0.348809i | −1.72044 | − | 0.200188i | 1.75666 | + | 0.956102i | −4.20241 | − | 0.740998i | 2.28808 | + | 0.874469i | −1.31632 | − | 2.29506i | −2.07405 | − | 1.92310i | 2.91985 | + | 0.688824i | 5.50103 | + | 2.48139i |
95.12 | −1.36638 | + | 0.364685i | 1.60153 | − | 0.659617i | 1.73401 | − | 0.996600i | −2.53354 | − | 0.446731i | −1.94776 | + | 1.48535i | 2.18735 | + | 1.48846i | −2.00588 | + | 1.99411i | 2.12981 | − | 2.11280i | 3.62470 | − | 0.313538i |
95.13 | −1.35923 | − | 0.390498i | −1.12398 | − | 1.31783i | 1.69502 | + | 1.06155i | −0.140742 | − | 0.0248166i | 1.01314 | + | 2.23015i | 1.14582 | + | 2.38476i | −1.88939 | − | 2.10480i | −0.473344 | + | 2.96242i | 0.181610 | + | 0.0886910i |
95.14 | −1.35369 | + | 0.409293i | −1.06364 | + | 1.36699i | 1.66496 | − | 1.10811i | 4.32435 | + | 0.762500i | 0.880342 | − | 2.28583i | −0.0657043 | + | 2.64494i | −1.80029 | + | 2.18150i | −0.737329 | − | 2.90798i | −6.16592 | + | 0.737740i |
95.15 | −1.34821 | − | 0.427004i | 1.72966 | − | 0.0910470i | 1.63533 | + | 1.15138i | −3.12490 | − | 0.551004i | −2.37082 | − | 0.615820i | −2.64376 | − | 0.102727i | −1.71313 | − | 2.25060i | 2.98342 | − | 0.314960i | 3.97773 | + | 2.07721i |
95.16 | −1.34200 | + | 0.446123i | −1.17175 | − | 1.27554i | 1.60195 | − | 1.19740i | −2.14959 | − | 0.379031i | 2.14154 | + | 1.18903i | 1.07635 | − | 2.41691i | −1.61563 | + | 2.32158i | −0.253990 | + | 2.98923i | 3.05386 | − | 0.450322i |
95.17 | −1.33299 | − | 0.472387i | −1.55434 | − | 0.764214i | 1.55370 | + | 1.25937i | 3.05802 | + | 0.539212i | 1.71091 | + | 1.75294i | −2.62079 | − | 0.362599i | −1.47615 | − | 2.41267i | 1.83195 | + | 2.37570i | −3.82158 | − | 2.16333i |
95.18 | −1.32216 | − | 0.501896i | 1.68643 | − | 0.394900i | 1.49620 | + | 1.32717i | 0.496060 | + | 0.0874688i | −2.42793 | − | 0.324294i | 1.26084 | − | 2.32600i | −1.31211 | − | 2.50567i | 2.68811 | − | 1.33194i | −0.611969 | − | 0.364618i |
95.19 | −1.32113 | − | 0.504600i | 0.893683 | + | 1.48369i | 1.49076 | + | 1.33328i | 3.15348 | + | 0.556044i | −0.431999 | − | 2.41109i | 1.10140 | − | 2.40560i | −1.29670 | − | 2.51367i | −1.40266 | + | 2.65189i | −3.88557 | − | 2.32585i |
95.20 | −1.30138 | + | 0.553549i | −1.64933 | − | 0.528879i | 1.38717 | − | 1.44075i | 0.746991 | + | 0.131715i | 2.43916 | − | 0.224713i | −2.50378 | + | 0.855036i | −1.00770 | + | 2.64283i | 2.44057 | + | 1.74459i | −1.04503 | + | 0.242086i |
See next 80 embeddings (of 840 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
189.bc | odd | 18 | 1 | inner |
756.ci | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 756.2.ci.a | yes | 840 |
4.b | odd | 2 | 1 | inner | 756.2.ci.a | yes | 840 |
7.c | even | 3 | 1 | 756.2.bs.a | ✓ | 840 | |
27.f | odd | 18 | 1 | 756.2.bs.a | ✓ | 840 | |
28.g | odd | 6 | 1 | 756.2.bs.a | ✓ | 840 | |
108.l | even | 18 | 1 | 756.2.bs.a | ✓ | 840 | |
189.bc | odd | 18 | 1 | inner | 756.2.ci.a | yes | 840 |
756.ci | even | 18 | 1 | inner | 756.2.ci.a | yes | 840 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
756.2.bs.a | ✓ | 840 | 7.c | even | 3 | 1 | |
756.2.bs.a | ✓ | 840 | 27.f | odd | 18 | 1 | |
756.2.bs.a | ✓ | 840 | 28.g | odd | 6 | 1 | |
756.2.bs.a | ✓ | 840 | 108.l | even | 18 | 1 | |
756.2.ci.a | yes | 840 | 1.a | even | 1 | 1 | trivial |
756.2.ci.a | yes | 840 | 4.b | odd | 2 | 1 | inner |
756.2.ci.a | yes | 840 | 189.bc | odd | 18 | 1 | inner |
756.2.ci.a | yes | 840 | 756.ci | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).