Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [196,2,Mod(27,196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([7, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("196.27");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 196.j (of order \(14\), 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: | \(1.56506787962\) |
Analytic rank: | \(0\) |
Dimension: | \(156\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 | −1.41412 | + | 0.0163478i | −0.0687858 | + | 0.301370i | 1.99947 | − | 0.0462354i | −0.644334 | − | 0.147065i | 0.0923445 | − | 0.427298i | 1.15895 | − | 2.37841i | −2.82673 | + | 0.0980691i | 2.61681 | + | 1.26019i | 0.913570 | + | 0.197434i |
27.2 | −1.41194 | + | 0.0802101i | 0.320649 | − | 1.40485i | 1.98713 | − | 0.226503i | 3.61677 | + | 0.825505i | −0.340052 | + | 2.00928i | −1.15779 | + | 2.37897i | −2.78754 | + | 0.479196i | 0.832108 | + | 0.400722i | −5.17287 | − | 0.875459i |
27.3 | −1.34277 | + | 0.443800i | −0.570354 | + | 2.49888i | 1.60608 | − | 1.19185i | −1.90324 | − | 0.434402i | −0.343147 | − | 3.60856i | −2.60517 | − | 0.461609i | −1.62767 | + | 2.31316i | −3.21621 | − | 1.54884i | 2.74841 | − | 0.261353i |
27.4 | −1.26947 | − | 0.623262i | −0.743013 | + | 3.25535i | 1.22309 | + | 1.58242i | 2.48779 | + | 0.567822i | 2.97216 | − | 3.66947i | 2.27356 | + | 1.35311i | −0.566410 | − | 2.77113i | −7.34233 | − | 3.53588i | −2.80426 | − | 2.27137i |
27.5 | −1.23433 | − | 0.690243i | −0.0211920 | + | 0.0928483i | 1.04713 | + | 1.70397i | −3.74506 | − | 0.854785i | 0.0902457 | − | 0.0999775i | 1.51239 | + | 2.17087i | −0.116345 | − | 2.82603i | 2.69473 | + | 1.29772i | 4.03262 | + | 3.64008i |
27.6 | −1.18418 | + | 0.773118i | 0.570354 | − | 2.49888i | 0.804576 | − | 1.83103i | −1.90324 | − | 0.434402i | 1.25653 | + | 3.40009i | 2.60517 | + | 0.461609i | 0.462835 | + | 2.79030i | −3.21621 | − | 1.54884i | 2.58963 | − | 0.957018i |
27.7 | −1.04449 | − | 0.953440i | 0.451180 | − | 1.97675i | 0.181906 | + | 1.99171i | 1.77951 | + | 0.406161i | −2.35596 | + | 1.63452i | 1.22122 | − | 2.34705i | 1.70898 | − | 2.25375i | −1.00107 | − | 0.482089i | −1.47142 | − | 2.12088i |
27.8 | −0.943039 | + | 1.05389i | −0.320649 | + | 1.40485i | −0.221354 | − | 1.98771i | 3.61677 | + | 0.825505i | −1.17817 | − | 1.66276i | 1.15779 | − | 2.37897i | 2.30357 | + | 1.64121i | 0.832108 | + | 0.400722i | −4.28075 | + | 3.03318i |
27.9 | −0.894470 | + | 1.09541i | 0.0687858 | − | 0.301370i | −0.399847 | − | 1.95962i | −0.644334 | − | 0.147065i | 0.268597 | + | 0.344915i | −1.15895 | + | 2.37841i | 2.50424 | + | 1.31483i | 2.61681 | + | 1.26019i | 0.737435 | − | 0.574265i |
27.10 | −0.530578 | − | 1.31091i | −0.325950 | + | 1.42808i | −1.43697 | + | 1.39108i | 0.906872 | + | 0.206988i | 2.04503 | − | 0.330416i | −2.40631 | − | 1.09985i | 2.58601 | + | 1.14567i | 0.769737 | + | 0.370686i | −0.209824 | − | 1.29865i |
27.11 | −0.521710 | − | 1.31447i | 0.555098 | − | 2.43205i | −1.45564 | + | 1.37154i | −2.13916 | − | 0.488249i | −3.48644 | + | 0.539164i | −2.09149 | + | 1.62039i | 2.56226 | + | 1.19784i | −2.90380 | − | 1.39840i | 0.474234 | + | 3.06657i |
27.12 | −0.304214 | + | 1.38111i | 0.743013 | − | 3.25535i | −1.81491 | − | 0.840303i | 2.48779 | + | 0.567822i | 4.26995 | + | 2.01650i | −2.27356 | − | 1.35311i | 1.71267 | − | 2.25095i | −7.34233 | − | 3.53588i | −1.54104 | + | 3.26316i |
27.13 | −0.229937 | + | 1.39540i | 0.0211920 | − | 0.0928483i | −1.89426 | − | 0.641706i | −3.74506 | − | 0.854785i | 0.124687 | + | 0.0509205i | −1.51239 | − | 2.17087i | 1.33099 | − | 2.49569i | 2.69473 | + | 1.29772i | 2.05389 | − | 5.02929i |
27.14 | −0.119871 | − | 1.40912i | −0.148687 | + | 0.651440i | −1.97126 | + | 0.337826i | 2.00983 | + | 0.458730i | 0.935782 | + | 0.131430i | 2.05174 | + | 1.67044i | 0.712336 | + | 2.73726i | 2.30064 | + | 1.10793i | 0.405488 | − | 2.88709i |
27.15 | 0.0942021 | + | 1.41107i | −0.451180 | + | 1.97675i | −1.98225 | + | 0.265852i | 1.77951 | + | 0.406161i | −2.83184 | − | 0.450434i | −1.22122 | + | 2.34705i | −0.561869 | − | 2.77206i | −1.00107 | − | 0.482089i | −0.405490 | + | 2.54928i |
27.16 | 0.454626 | − | 1.33915i | −0.636184 | + | 2.78731i | −1.58663 | − | 1.21762i | −3.28016 | − | 0.748676i | 3.44339 | + | 2.11913i | −1.09871 | + | 2.40683i | −2.35190 | + | 1.57117i | −4.66144 | − | 2.24483i | −2.49384 | + | 4.05225i |
27.17 | 0.584470 | − | 1.28779i | 0.166766 | − | 0.730650i | −1.31679 | − | 1.50534i | −1.97974 | − | 0.451863i | −0.843452 | − | 0.641802i | −1.05109 | − | 2.42800i | −2.70819 | + | 0.815917i | 2.19687 | + | 1.05796i | −1.73900 | + | 2.28538i |
27.18 | 0.658802 | − | 1.25139i | 0.637509 | − | 2.79311i | −1.13196 | − | 1.64884i | 1.25203 | + | 0.285768i | −3.07528 | − | 2.63788i | 1.52974 | + | 2.15868i | −2.80908 | + | 0.330264i | −4.69213 | − | 2.25961i | 1.18245 | − | 1.37852i |
27.19 | 0.694101 | + | 1.23216i | 0.325950 | − | 1.42808i | −1.03645 | + | 1.71049i | 0.906872 | + | 0.206988i | 1.98587 | − | 0.589609i | 2.40631 | + | 1.09985i | −2.82700 | − | 0.0898155i | 0.769737 | + | 0.370686i | 0.374419 | + | 1.26108i |
27.20 | 0.702410 | + | 1.22744i | −0.555098 | + | 2.43205i | −1.01324 | + | 1.72434i | −2.13916 | − | 0.488249i | −3.37511 | + | 1.02694i | 2.09149 | − | 1.62039i | −2.82824 | − | 0.0325071i | −2.90380 | − | 1.39840i | −0.903267 | − | 2.96865i |
See next 80 embeddings (of 156 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
49.f | odd | 14 | 1 | inner |
196.j | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 196.2.j.a | ✓ | 156 |
4.b | odd | 2 | 1 | inner | 196.2.j.a | ✓ | 156 |
49.f | odd | 14 | 1 | inner | 196.2.j.a | ✓ | 156 |
196.j | even | 14 | 1 | inner | 196.2.j.a | ✓ | 156 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
196.2.j.a | ✓ | 156 | 1.a | even | 1 | 1 | trivial |
196.2.j.a | ✓ | 156 | 4.b | odd | 2 | 1 | inner |
196.2.j.a | ✓ | 156 | 49.f | odd | 14 | 1 | inner |
196.2.j.a | ✓ | 156 | 196.j | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(196, [\chi])\).