Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,4,Mod(4,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 41 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 41.f (of order \(10\), degree \(4\), 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: | \(2.41907831024\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −4.41144 | + | 3.20510i | − | 1.48791i | 6.71601 | − | 20.6698i | 0.515160 | − | 1.58550i | 4.76888 | + | 6.56381i | 3.18727 | − | 4.38691i | 23.1412 | + | 71.2212i | 24.7861 | 2.80908 | + | 8.64547i | |||
4.2 | −2.78864 | + | 2.02606i | 7.54001i | 1.19943 | − | 3.69147i | 0.380962 | − | 1.17248i | −15.2765 | − | 21.0264i | −1.94929 | + | 2.68297i | −4.38694 | − | 13.5016i | −29.8517 | 1.31316 | + | 4.04148i | ||||
4.3 | −2.43695 | + | 1.77055i | − | 6.59197i | 0.331746 | − | 1.02101i | −4.60057 | + | 14.1591i | 11.6714 | + | 16.0643i | −15.8644 | + | 21.8355i | −6.44735 | − | 19.8429i | −16.4541 | −13.8580 | − | 42.6505i | |||
4.4 | −1.73189 | + | 1.25830i | − | 4.21774i | −1.05598 | + | 3.24998i | 4.47618 | − | 13.7763i | 5.30716 | + | 7.30468i | 9.95533 | − | 13.7023i | −7.55278 | − | 23.2451i | 9.21068 | 9.58234 | + | 29.4914i | |||
4.5 | 0.355319 | − | 0.258154i | 4.28641i | −2.41253 | + | 7.42500i | 1.11581 | − | 3.43412i | 1.10655 | + | 1.52304i | −14.0326 | + | 19.3142i | 2.14533 | + | 6.60266i | 8.62667 | −0.490063 | − | 1.50826i | ||||
4.6 | 0.454809 | − | 0.330438i | 1.60812i | −2.37447 | + | 7.30788i | −5.11607 | + | 15.7456i | 0.531384 | + | 0.731388i | 13.6716 | − | 18.8173i | 2.72464 | + | 8.38558i | 24.4139 | 2.87612 | + | 8.85179i | ||||
4.7 | 2.08916 | − | 1.51786i | − | 9.41969i | −0.411459 | + | 1.26634i | −1.61945 | + | 4.98415i | −14.2978 | − | 19.6792i | 7.26325 | − | 9.99701i | 7.44643 | + | 22.9178i | −61.7306 | 4.18197 | + | 12.8708i | |||
4.8 | 2.80536 | − | 2.03822i | − | 1.54098i | 1.24361 | − | 3.82743i | 3.68903 | − | 11.3537i | −3.14085 | − | 4.32301i | −3.25858 | + | 4.48505i | 4.26007 | + | 13.1112i | 24.6254 | −12.7922 | − | 39.3702i | |||
4.9 | 3.25753 | − | 2.36673i | 9.81535i | 2.53793 | − | 7.81093i | 1.19307 | − | 3.67189i | 23.2303 | + | 31.9738i | 12.9368 | − | 17.8060i | −0.264905 | − | 0.815295i | −69.3412 | −4.80392 | − | 14.7849i | ||||
4.10 | 4.33380 | − | 3.14869i | − | 1.16718i | 6.39542 | − | 19.6831i | −5.31529 | + | 16.3588i | −3.67507 | − | 5.05830i | −12.6004 | + | 17.3430i | −21.0165 | − | 64.6822i | 25.6377 | 28.4733 | + | 87.6317i | |||
23.1 | −1.69608 | − | 5.22000i | − | 6.95975i | −17.8996 | + | 13.0048i | 12.4188 | − | 9.02282i | −36.3299 | + | 11.8043i | 15.1090 | + | 4.90923i | 62.7209 | + | 45.5694i | −21.4382 | −68.1625 | − | 49.5229i | |||
23.2 | −1.49988 | − | 4.61614i | 4.75851i | −12.5870 | + | 9.14499i | −13.9422 | + | 10.1296i | 21.9660 | − | 7.13718i | −21.8801 | − | 7.10927i | 29.6797 | + | 21.5636i | 4.35654 | 67.6714 | + | 49.1662i | ||||
23.3 | −0.925466 | − | 2.84829i | 6.53368i | −0.784140 | + | 0.569711i | 9.17905 | − | 6.66897i | 18.6098 | − | 6.04670i | 23.6307 | + | 7.67807i | −17.0348 | − | 12.3765i | −15.6889 | −27.4901 | − | 19.9727i | ||||
23.4 | −0.814530 | − | 2.50687i | − | 2.19754i | 0.851213 | − | 0.618443i | 4.32167 | − | 3.13988i | −5.50894 | + | 1.78996i | −24.2069 | − | 7.86529i | −19.3034 | − | 14.0248i | 22.1708 | −11.3914 | − | 8.27632i | |||
23.5 | −0.625259 | − | 1.92435i | − | 8.16596i | 3.15996 | − | 2.29585i | −13.3127 | + | 9.67225i | −15.7142 | + | 5.10585i | 17.8022 | + | 5.78429i | −19.4894 | − | 14.1599i | −39.6830 | 26.9367 | + | 19.5707i | |||
23.6 | 0.239215 | + | 0.736229i | 6.90238i | 5.98733 | − | 4.35005i | −13.1428 | + | 9.54883i | −5.08173 | + | 1.65116i | 5.83421 | + | 1.89565i | 9.64508 | + | 7.00756i | −20.6429 | −10.1741 | − | 7.39191i | ||||
23.7 | 0.283232 | + | 0.871700i | − | 1.56069i | 5.79250 | − | 4.20849i | 3.14119 | − | 2.28221i | 1.36045 | − | 0.442038i | 0.892763 | + | 0.290076i | 11.2413 | + | 8.16726i | 24.5643 | 2.87909 | + | 2.09178i | |||
23.8 | 1.06811 | + | 3.28729i | − | 9.65587i | −3.19332 | + | 2.32008i | 6.29911 | − | 4.57657i | 31.7417 | − | 10.3135i | 1.36476 | + | 0.443437i | 11.3331 | + | 8.23400i | −66.2357 | 21.7726 | + | 15.8188i | |||
23.9 | 1.09430 | + | 3.36790i | 8.13837i | −3.67312 | + | 2.66868i | 15.1333 | − | 10.9950i | −27.4092 | + | 8.90579i | −27.8693 | − | 9.05530i | 9.91193 | + | 7.20144i | −39.2330 | 53.5903 | + | 38.9357i | ||||
23.10 | 1.44931 | + | 4.46051i | 0.304759i | −11.3236 | + | 8.22705i | −5.31421 | + | 3.86100i | −1.35938 | + | 0.441690i | 7.51362 | + | 2.44132i | −22.7535 | − | 16.5314i | 26.9071 | −24.9240 | − | 18.1083i | ||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 41.4.f.a | ✓ | 40 |
41.f | even | 10 | 1 | inner | 41.4.f.a | ✓ | 40 |
41.g | even | 20 | 2 | 1681.4.a.l | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
41.4.f.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
41.4.f.a | ✓ | 40 | 41.f | even | 10 | 1 | inner |
1681.4.a.l | 40 | 41.g | even | 20 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(41, [\chi])\).