Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [211,4,Mod(58,211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([12]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("211.58");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 211 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 211.f (of order \(7\), 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: | \(12.4494030112\) |
Analytic rank: | \(0\) |
Dimension: | \(312\) |
Relative dimension: | \(52\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | −4.97605 | + | 2.39634i | −1.94388 | + | 0.936125i | 14.0307 | − | 17.5940i | −2.40114 | − | 1.15633i | 7.42959 | − | 9.31641i | −15.4643 | − | 7.44721i | −17.8246 | + | 78.0948i | −13.9319 | + | 17.4700i | 14.7192 | ||
58.2 | −4.90121 | + | 2.36030i | 4.10047 | − | 1.97468i | 13.4629 | − | 16.8819i | 14.3224 | + | 6.89730i | −15.4364 | + | 19.3566i | 24.7699 | + | 11.9285i | −16.4541 | + | 72.0900i | −3.91977 | + | 4.91524i | −86.4767 | ||
58.3 | −4.48223 | + | 2.15853i | 7.21184 | − | 3.47304i | 10.4432 | − | 13.0954i | 1.78326 | + | 0.858773i | −24.8285 | + | 31.1339i | −22.0474 | − | 10.6174i | −9.68599 | + | 42.4371i | 23.1145 | − | 28.9846i | −9.84667 | ||
58.4 | −4.27434 | + | 2.05841i | −7.67440 | + | 3.69580i | 9.04499 | − | 11.3421i | −12.8775 | − | 6.20149i | 25.1955 | − | 31.5942i | −1.72606 | − | 0.831228i | −6.86931 | + | 30.0964i | 28.4033 | − | 35.6166i | 67.8082 | ||
58.5 | −4.26590 | + | 2.05435i | −6.92250 | + | 3.33370i | 8.98962 | − | 11.2726i | 12.9045 | + | 6.21448i | 22.6821 | − | 28.4425i | 20.2172 | + | 9.73607i | −6.76219 | + | 29.6271i | 19.9733 | − | 25.0457i | −67.8160 | ||
58.6 | −4.23580 | + | 2.03986i | 0.891304 | − | 0.429229i | 8.79311 | − | 11.0262i | −14.0955 | − | 6.78802i | −2.89982 | + | 3.63626i | 16.2686 | + | 7.83456i | −6.38475 | + | 27.9734i | −16.2240 | + | 20.3443i | 73.5522 | ||
58.7 | −4.21502 | + | 2.02985i | 6.96432 | − | 3.35384i | 8.65822 | − | 10.8571i | −14.7232 | − | 7.09032i | −22.5470 | + | 28.2730i | 6.24622 | + | 3.00802i | −6.12821 | + | 26.8495i | 20.4193 | − | 25.6050i | 76.4509 | ||
58.8 | −4.11840 | + | 1.98331i | −6.02527 | + | 2.90162i | 8.03973 | − | 10.0815i | 13.2398 | + | 6.37594i | 19.0596 | − | 23.9000i | −24.3098 | − | 11.7070i | −4.97871 | + | 21.8132i | 11.0503 | − | 13.8566i | −67.1721 | ||
58.9 | −3.98998 | + | 1.92147i | 0.271323 | − | 0.130662i | 7.23994 | − | 9.07860i | 3.61502 | + | 1.74090i | −0.831507 | + | 1.04268i | 4.01486 | + | 1.93345i | −3.55938 | + | 15.5947i | −16.7777 | + | 21.0385i | −17.7689 | ||
58.10 | −3.21096 | + | 1.54632i | 3.63743 | − | 1.75169i | 2.93123 | − | 3.67565i | 17.1840 | + | 8.27538i | −8.97095 | + | 11.2492i | −17.4875 | − | 8.42155i | 2.61598 | − | 11.4613i | −6.67179 | + | 8.36616i | −67.9734 | ||
58.11 | −3.20031 | + | 1.54119i | 0.0460954 | − | 0.0221984i | 2.87883 | − | 3.60993i | 7.24370 | + | 3.48838i | −0.113308 | + | 0.142084i | −7.06865 | − | 3.40408i | 2.67375 | − | 11.7144i | −16.8326 | + | 21.1074i | −28.5584 | ||
58.12 | −3.08366 | + | 1.48501i | −4.08815 | + | 1.96875i | 2.31577 | − | 2.90388i | −0.273383 | − | 0.131654i | 9.68284 | − | 12.1419i | 17.9048 | + | 8.62251i | 3.26406 | − | 14.3008i | −3.99722 | + | 5.01235i | 1.03853 | ||
58.13 | −3.06711 | + | 1.47704i | 7.77421 | − | 3.74386i | 2.23761 | − | 2.80587i | 4.61133 | + | 2.22070i | −18.3145 | + | 22.9657i | 22.9736 | + | 11.0635i | 3.34152 | − | 14.6401i | 29.5876 | − | 37.1016i | −17.4235 | ||
58.14 | −2.84004 | + | 1.36769i | −2.86373 | + | 1.37910i | 1.20732 | − | 1.51394i | −12.4872 | − | 6.01350i | 6.24692 | − | 7.83339i | −27.9663 | − | 13.4678i | 4.25320 | − | 18.6345i | −10.5352 | + | 13.2107i | 43.6886 | ||
58.15 | −2.59971 | + | 1.25195i | 4.51364 | − | 2.17366i | 0.203164 | − | 0.254760i | −11.4836 | − | 5.53021i | −9.01283 | + | 11.3017i | −23.1270 | − | 11.1374i | 4.92737 | − | 21.5882i | −1.18602 | + | 1.48723i | 36.7775 | ||
58.16 | −2.08199 | + | 1.00264i | 5.67503 | − | 2.73295i | −1.65849 | + | 2.07969i | −2.24841 | − | 1.08278i | −9.07523 | + | 11.3800i | −0.706202 | − | 0.340089i | 5.48149 | − | 24.0160i | 7.90271 | − | 9.90969i | 5.76681 | ||
58.17 | −2.03194 | + | 0.978531i | −7.76198 | + | 3.73797i | −1.81666 | + | 2.27801i | −5.73829 | − | 2.76341i | 12.1142 | − | 15.1907i | −2.97528 | − | 1.43282i | 5.47701 | − | 23.9964i | 29.4417 | − | 36.9187i | 14.3640 | ||
58.18 | −1.90830 | + | 0.918988i | −5.61985 | + | 2.70637i | −2.19085 | + | 2.74724i | −8.82929 | − | 4.25196i | 8.23722 | − | 10.3291i | 26.0543 | + | 12.5471i | 5.42661 | − | 23.7755i | 7.42397 | − | 9.30937i | 20.7564 | ||
58.19 | −1.73917 | + | 0.837542i | −0.342599 | + | 0.164987i | −2.66467 | + | 3.34139i | 16.3636 | + | 7.88027i | 0.457655 | − | 0.573881i | 19.5807 | + | 9.42957i | 5.27209 | − | 23.0985i | −16.7441 | + | 20.9964i | −35.0591 | ||
58.20 | −1.60260 | + | 0.771774i | −7.73273 | + | 3.72389i | −3.01521 | + | 3.78096i | 10.4825 | + | 5.04811i | 9.51851 | − | 11.9358i | −10.0207 | − | 4.82572i | 5.08063 | − | 22.2597i | 29.0935 | − | 36.4822i | −20.6953 | ||
See next 80 embeddings (of 312 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
211.f | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 211.4.f.a | ✓ | 312 |
211.f | even | 7 | 1 | inner | 211.4.f.a | ✓ | 312 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
211.4.f.a | ✓ | 312 | 1.a | even | 1 | 1 | trivial |
211.4.f.a | ✓ | 312 | 211.f | even | 7 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(211, [\chi])\).