Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,2,Mod(11,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([24, 20]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.q (of order \(15\), 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: | \(1.39738203537\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −2.26854 | − | 1.01002i | −1.03919 | − | 1.15413i | 2.78788 | + | 3.09625i | 0.598087 | − | 2.15460i | 1.19174 | + | 3.66780i | 1.63141 | − | 2.08291i | −1.66242 | − | 5.11641i | 0.0614698 | − | 0.584846i | −3.53297 | + | 4.28371i |
11.2 | −2.25839 | − | 1.00550i | 0.712831 | + | 0.791679i | 2.75102 | + | 3.05531i | −2.23601 | − | 0.0162832i | −0.813815 | − | 2.50467i | −2.33148 | + | 1.25068i | −1.61290 | − | 4.96398i | 0.194958 | − | 1.85490i | 5.03340 | + | 2.28508i |
11.3 | −2.16145 | − | 0.962341i | 2.09781 | + | 2.32985i | 2.40752 | + | 2.67382i | 2.08640 | + | 0.804329i | −2.29220 | − | 7.05468i | −0.0714197 | − | 2.64479i | −1.16834 | − | 3.59577i | −0.713827 | + | 6.79161i | −3.73561 | − | 3.74634i |
11.4 | −1.94331 | − | 0.865215i | −1.55527 | − | 1.72730i | 1.68958 | + | 1.87647i | 0.566195 | + | 2.16320i | 1.52788 | + | 4.70232i | −2.64421 | + | 0.0903750i | −0.345128 | − | 1.06219i | −0.251123 | + | 2.38928i | 0.771342 | − | 4.69363i |
11.5 | −1.40506 | − | 0.625574i | 0.893605 | + | 0.992448i | 0.244597 | + | 0.271652i | −1.97083 | + | 1.05633i | −0.634720 | − | 1.95347i | 2.64452 | + | 0.0807455i | 0.776821 | + | 2.39081i | 0.127161 | − | 1.20985i | 3.42995 | − | 0.251317i |
11.6 | −1.26136 | − | 0.561596i | −0.564524 | − | 0.626968i | −0.0626100 | − | 0.0695354i | 2.11758 | + | 0.718244i | 0.359969 | + | 1.10787i | 2.36555 | + | 1.18498i | 0.893265 | + | 2.74919i | 0.239184 | − | 2.27569i | −2.26767 | − | 2.09519i |
11.7 | −0.853279 | − | 0.379904i | −1.30628 | − | 1.45077i | −0.754503 | − | 0.837961i | −1.24736 | − | 1.85583i | 0.563466 | + | 1.73417i | −0.508729 | + | 2.59638i | 0.902719 | + | 2.77828i | −0.0847812 | + | 0.806639i | 0.359309 | + | 2.05742i |
11.8 | −0.677204 | − | 0.301511i | 0.992974 | + | 1.10281i | −0.970565 | − | 1.07792i | −0.0581969 | − | 2.23531i | −0.339937 | − | 1.04622i | −1.08929 | − | 2.41111i | 0.790409 | + | 2.43263i | 0.0833944 | − | 0.793444i | −0.634559 | + | 1.53131i |
11.9 | −0.428333 | − | 0.190706i | 1.73082 | + | 1.92227i | −1.19116 | − | 1.32292i | 0.335942 | + | 2.21069i | −0.374778 | − | 1.15345i | −1.07014 | + | 2.41967i | 0.547701 | + | 1.68565i | −0.385798 | + | 3.67063i | 0.277696 | − | 1.01098i |
11.10 | 0.189312 | + | 0.0842871i | −0.609162 | − | 0.676543i | −1.30953 | − | 1.45438i | −1.65641 | + | 1.50211i | −0.0582978 | − | 0.179422i | −2.20881 | − | 1.45642i | −0.253398 | − | 0.779878i | 0.226953 | − | 2.15932i | −0.440185 | + | 0.144753i |
11.11 | 0.529445 | + | 0.235724i | −0.229691 | − | 0.255097i | −1.11352 | − | 1.23668i | 1.94658 | + | 1.10038i | −0.0614760 | − | 0.189204i | 1.74489 | − | 1.98881i | −0.656210 | − | 2.01961i | 0.301269 | − | 2.86638i | 0.771220 | + | 1.04145i |
11.12 | 0.669019 | + | 0.297867i | 1.50997 | + | 1.67699i | −0.979399 | − | 1.08773i | 1.33545 | − | 1.79348i | 0.510679 | + | 1.57171i | 1.21370 | + | 2.35094i | −0.783844 | − | 2.41242i | −0.218706 | + | 2.08085i | 1.42766 | − | 0.802086i |
11.13 | 0.956389 | + | 0.425812i | −1.22328 | − | 1.35859i | −0.604896 | − | 0.671805i | 1.72038 | − | 1.42838i | −0.591430 | − | 1.82023i | −2.07580 | + | 1.64044i | −0.939473 | − | 2.89140i | −0.0357700 | + | 0.340329i | 2.25358 | − | 0.633531i |
11.14 | 1.37473 | + | 0.612069i | 2.04628 | + | 2.27262i | 0.176990 | + | 0.196568i | −2.23021 | + | 0.161703i | 1.42208 | + | 4.37671i | 0.781664 | − | 2.52765i | −0.807034 | − | 2.48380i | −0.663974 | + | 6.31729i | −3.16491 | − | 1.14275i |
11.15 | 1.53521 | + | 0.683522i | −1.65355 | − | 1.83645i | 0.551421 | + | 0.612415i | −1.53440 | − | 1.62653i | −1.28330 | − | 3.94959i | 2.14197 | − | 1.55304i | −0.610656 | − | 1.87941i | −0.324749 | + | 3.08978i | −1.24387 | − | 3.54587i |
11.16 | 1.86868 | + | 0.831989i | −0.100275 | − | 0.111367i | 1.46149 | + | 1.62315i | −0.993637 | + | 2.00317i | −0.0947260 | − | 0.291537i | 1.42653 | + | 2.22823i | 0.116408 | + | 0.358267i | 0.311238 | − | 2.96123i | −3.52340 | + | 2.91658i |
11.17 | 2.21264 | + | 0.985130i | 0.483361 | + | 0.536826i | 2.58702 | + | 2.87318i | −1.04758 | − | 1.97550i | 0.540658 | + | 1.66398i | −2.59815 | + | 0.499637i | 1.39679 | + | 4.29888i | 0.259040 | − | 2.46460i | −0.371785 | − | 5.40306i |
11.18 | 2.44336 | + | 1.08785i | −1.77288 | − | 1.96899i | 3.44830 | + | 3.82973i | 1.66349 | + | 1.49426i | −2.18982 | − | 6.73956i | −1.54320 | − | 2.14908i | 2.60627 | + | 8.02126i | −0.420206 | + | 3.99799i | 2.43896 | + | 5.46063i |
16.1 | −2.26854 | + | 1.01002i | −1.03919 | + | 1.15413i | 2.78788 | − | 3.09625i | 0.598087 | + | 2.15460i | 1.19174 | − | 3.66780i | 1.63141 | + | 2.08291i | −1.66242 | + | 5.11641i | 0.0614698 | + | 0.584846i | −3.53297 | − | 4.28371i |
16.2 | −2.25839 | + | 1.00550i | 0.712831 | − | 0.791679i | 2.75102 | − | 3.05531i | −2.23601 | + | 0.0162832i | −0.813815 | + | 2.50467i | −2.33148 | − | 1.25068i | −1.61290 | + | 4.96398i | 0.194958 | + | 1.85490i | 5.03340 | − | 2.28508i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
25.d | even | 5 | 1 | inner |
175.q | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.2.q.a | ✓ | 144 |
5.b | even | 2 | 1 | 875.2.q.a | 144 | ||
5.c | odd | 4 | 2 | 875.2.u.b | 288 | ||
7.c | even | 3 | 1 | inner | 175.2.q.a | ✓ | 144 |
25.d | even | 5 | 1 | inner | 175.2.q.a | ✓ | 144 |
25.e | even | 10 | 1 | 875.2.q.a | 144 | ||
25.f | odd | 20 | 2 | 875.2.u.b | 288 | ||
35.j | even | 6 | 1 | 875.2.q.a | 144 | ||
35.l | odd | 12 | 2 | 875.2.u.b | 288 | ||
175.q | even | 15 | 1 | inner | 175.2.q.a | ✓ | 144 |
175.t | even | 30 | 1 | 875.2.q.a | 144 | ||
175.w | odd | 60 | 2 | 875.2.u.b | 288 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
175.2.q.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
175.2.q.a | ✓ | 144 | 7.c | even | 3 | 1 | inner |
175.2.q.a | ✓ | 144 | 25.d | even | 5 | 1 | inner |
175.2.q.a | ✓ | 144 | 175.q | even | 15 | 1 | inner |
875.2.q.a | 144 | 5.b | even | 2 | 1 | ||
875.2.q.a | 144 | 25.e | even | 10 | 1 | ||
875.2.q.a | 144 | 35.j | even | 6 | 1 | ||
875.2.q.a | 144 | 175.t | even | 30 | 1 | ||
875.2.u.b | 288 | 5.c | odd | 4 | 2 | ||
875.2.u.b | 288 | 25.f | odd | 20 | 2 | ||
875.2.u.b | 288 | 35.l | odd | 12 | 2 | ||
875.2.u.b | 288 | 175.w | odd | 60 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(175, [\chi])\).