Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,5,Mod(2,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.2");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 25.f (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: | \(2.58424907710\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(9\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −7.04610 | − | 1.11599i | −6.53588 | − | 12.8274i | 33.1852 | + | 10.7825i | −24.8595 | + | 2.64690i | 31.7372 | + | 97.6770i | 31.4475 | + | 31.4475i | −120.091 | − | 61.1893i | −74.2134 | + | 102.146i | 178.116 | + | 9.09265i |
2.2 | −5.62994 | − | 0.891695i | 2.94614 | + | 5.78212i | 15.6842 | + | 5.09611i | 0.584324 | − | 24.9932i | −11.4307 | − | 35.1800i | −65.0193 | − | 65.0193i | −2.49547 | − | 1.27151i | 22.8574 | − | 31.4606i | −25.5760 | + | 140.189i |
2.3 | −4.54749 | − | 0.720251i | 0.590325 | + | 1.15858i | 4.94395 | + | 1.60639i | 22.6193 | + | 10.6475i | −1.85003 | − | 5.69380i | 49.8348 | + | 49.8348i | 44.3120 | + | 22.5781i | 46.6168 | − | 64.1625i | −95.1920 | − | 64.7107i |
2.4 | −1.28906 | − | 0.204167i | 7.17656 | + | 14.0848i | −13.5969 | − | 4.41790i | −24.2814 | + | 5.95099i | −6.37537 | − | 19.6214i | 39.9737 | + | 39.9737i | 35.2313 | + | 17.9513i | −99.2677 | + | 136.630i | 32.5152 | − | 2.71373i |
2.5 | −0.977279 | − | 0.154786i | −2.72438 | − | 5.34689i | −14.2858 | − | 4.64173i | −12.9766 | + | 21.3684i | 1.83485 | + | 5.64710i | −52.4931 | − | 52.4931i | 27.3486 | + | 13.9348i | 26.4436 | − | 36.3965i | 15.9893 | − | 18.8743i |
2.6 | 1.02878 | + | 0.162942i | −4.17666 | − | 8.19716i | −14.1851 | − | 4.60901i | 1.67139 | − | 24.9441i | −2.96119 | − | 9.11361i | 26.8728 | + | 26.8728i | −28.6915 | − | 14.6190i | −2.13832 | + | 2.94315i | 5.78393 | − | 25.3896i |
2.7 | 3.78337 | + | 0.599226i | 3.54623 | + | 6.95987i | −1.26212 | − | 0.410089i | 23.2986 | + | 9.06514i | 9.24615 | + | 28.4567i | −13.0105 | − | 13.0105i | −59.1377 | − | 30.1322i | 11.7465 | − | 16.1677i | 82.7149 | + | 48.2579i |
2.8 | 6.19314 | + | 0.980897i | 1.94321 | + | 3.81377i | 22.1759 | + | 7.20540i | −22.2324 | − | 11.4333i | 8.29367 | + | 25.5253i | −2.13863 | − | 2.13863i | 40.8803 | + | 20.8296i | 36.8419 | − | 50.7085i | −126.474 | − | 92.6155i |
2.9 | 6.53352 | + | 1.03481i | −7.66178 | − | 15.0371i | 26.3992 | + | 8.57763i | 14.4844 | + | 20.3765i | −34.4979 | − | 106.174i | 5.89577 | + | 5.89577i | 69.3001 | + | 35.3102i | −119.800 | + | 164.891i | 73.5482 | + | 148.119i |
3.1 | −3.10514 | − | 6.09417i | 2.14164 | + | 13.5218i | −18.0925 | + | 24.9022i | −10.3349 | + | 22.7638i | 75.7541 | − | 55.0385i | 9.33601 | − | 9.33601i | 99.8507 | + | 15.8148i | −101.217 | + | 32.8873i | 170.818 | − | 7.70230i |
3.2 | −2.65089 | − | 5.20267i | −2.30541 | − | 14.5558i | −10.6360 | + | 14.6392i | 13.2120 | + | 21.2237i | −69.6176 | + | 50.5802i | 28.1685 | − | 28.1685i | 12.0823 | + | 1.91365i | −129.521 | + | 42.0839i | 75.3962 | − | 124.999i |
3.3 | −2.14039 | − | 4.20075i | −0.349755 | − | 2.20827i | −3.66045 | + | 5.03818i | −17.0385 | − | 18.2946i | −8.52776 | + | 6.19578i | −46.9663 | + | 46.9663i | −45.5061 | − | 7.20746i | 72.2815 | − | 23.4857i | −40.3819 | + | 110.732i |
3.4 | −1.10795 | − | 2.17447i | 0.976269 | + | 6.16392i | 5.90379 | − | 8.12588i | 23.7728 | − | 7.73659i | 12.3216 | − | 8.95217i | 23.7798 | − | 23.7798i | −62.7773 | − | 9.94294i | 39.9948 | − | 12.9951i | −43.1620 | − | 43.1215i |
3.5 | 0.312556 | + | 0.613425i | −1.24611 | − | 7.86762i | 9.12596 | − | 12.5608i | −24.9872 | + | 0.799781i | 4.43671 | − | 3.22346i | 47.0743 | − | 47.0743i | 21.4373 | + | 3.39533i | 16.6890 | − | 5.42258i | −8.30050 | − | 15.0778i |
3.6 | 0.958433 | + | 1.88103i | 1.41659 | + | 8.94397i | 6.78488 | − | 9.33859i | −5.24264 | + | 24.4441i | −15.4662 | + | 11.2368i | −27.3860 | + | 27.3860i | 57.4312 | + | 9.09622i | −0.952289 | + | 0.309417i | −51.0049 | + | 13.5665i |
3.7 | 1.47174 | + | 2.88846i | −2.00120 | − | 12.6351i | 3.22739 | − | 4.44212i | 22.9144 | − | 9.99657i | 33.5507 | − | 24.3760i | −45.6970 | + | 45.6970i | 68.8109 | + | 10.8986i | −78.6049 | + | 25.5403i | 62.5988 | + | 51.4749i |
3.8 | 2.50963 | + | 4.92542i | 1.48300 | + | 9.36327i | −8.55700 | + | 11.7777i | −1.80390 | − | 24.9348i | −42.3963 | + | 30.8027i | 0.684509 | − | 0.684509i | 7.87292 | + | 1.24695i | −8.43590 | + | 2.74099i | 118.288 | − | 71.4622i |
3.9 | 3.33979 | + | 6.55470i | −0.874157 | − | 5.51921i | −22.4054 | + | 30.8383i | 2.16102 | + | 24.9064i | 33.2573 | − | 24.1628i | 37.7565 | − | 37.7565i | −160.710 | − | 25.4540i | 47.3380 | − | 15.3811i | −156.037 | + | 97.3470i |
8.1 | −6.66808 | − | 3.39756i | 11.5382 | + | 1.82747i | 23.5154 | + | 32.3661i | −4.94350 | − | 24.5064i | −70.7286 | − | 51.3874i | 32.8567 | − | 32.8567i | −28.1051 | − | 177.449i | 52.7545 | + | 17.1410i | −50.2981 | + | 180.206i |
8.2 | −5.42141 | − | 2.76234i | −11.3884 | − | 1.80375i | 12.3565 | + | 17.0073i | 24.3635 | + | 5.60550i | 56.7588 | + | 41.2377i | −9.79605 | + | 9.79605i | −4.78031 | − | 30.1817i | 49.4076 | + | 16.0535i | −116.600 | − | 97.6900i |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 25.5.f.a | ✓ | 72 |
5.b | even | 2 | 1 | 125.5.f.c | 72 | ||
5.c | odd | 4 | 1 | 125.5.f.a | 72 | ||
5.c | odd | 4 | 1 | 125.5.f.b | 72 | ||
25.d | even | 5 | 1 | 125.5.f.b | 72 | ||
25.e | even | 10 | 1 | 125.5.f.a | 72 | ||
25.f | odd | 20 | 1 | inner | 25.5.f.a | ✓ | 72 |
25.f | odd | 20 | 1 | 125.5.f.c | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.5.f.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
25.5.f.a | ✓ | 72 | 25.f | odd | 20 | 1 | inner |
125.5.f.a | 72 | 5.c | odd | 4 | 1 | ||
125.5.f.a | 72 | 25.e | even | 10 | 1 | ||
125.5.f.b | 72 | 5.c | odd | 4 | 1 | ||
125.5.f.b | 72 | 25.d | even | 5 | 1 | ||
125.5.f.c | 72 | 5.b | even | 2 | 1 | ||
125.5.f.c | 72 | 25.f | odd | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(25, [\chi])\).