Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,4,Mod(3,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([21, 10]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.x (of order \(60\), degree \(16\), 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: | \(10.3253342510\) |
Analytic rank: | \(0\) |
Dimension: | \(928\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{60})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −2.98654 | + | 4.59886i | −1.16114 | + | 3.02487i | −8.97624 | − | 20.1610i | 9.73544 | + | 5.49738i | −10.4432 | − | 14.3738i | 7.57257 | + | 16.9014i | 76.1974 | + | 12.0685i | 12.2633 | + | 11.0419i | −54.3569 | + | 28.3538i |
3.2 | −2.93952 | + | 4.52647i | 2.08359 | − | 5.42794i | −8.59424 | − | 19.3030i | 5.69869 | − | 9.61899i | 18.4447 | + | 25.3869i | −18.5027 | − | 0.805399i | 69.9913 | + | 11.0855i | −5.05630 | − | 4.55272i | 26.7886 | + | 54.0702i |
3.3 | −2.78034 | + | 4.28135i | 0.882639 | − | 2.29935i | −7.34575 | − | 16.4988i | −10.5665 | + | 3.65353i | 7.39030 | + | 10.1719i | −6.43718 | − | 17.3656i | 50.7243 | + | 8.03394i | 15.5569 | + | 14.0075i | 13.7365 | − | 55.3971i |
3.4 | −2.76823 | + | 4.26271i | −3.25777 | + | 8.48679i | −7.25365 | − | 16.2920i | −10.0021 | + | 4.99589i | −27.1584 | − | 37.3803i | 14.5748 | − | 11.4270i | 49.3668 | + | 7.81893i | −41.3476 | − | 37.2295i | 6.39202 | − | 56.4656i |
3.5 | −2.71831 | + | 4.18582i | 3.28166 | − | 8.54901i | −6.87804 | − | 15.4483i | −6.26616 | + | 9.25933i | 26.8641 | + | 36.9752i | 12.3275 | + | 13.8214i | 43.9240 | + | 6.95687i | −42.2513 | − | 38.0433i | −21.7246 | − | 51.3987i |
3.6 | −2.62864 | + | 4.04775i | 0.173128 | − | 0.451014i | −6.22064 | − | 13.9718i | −6.75712 | − | 8.90737i | 1.37050 | + | 1.88633i | 16.2876 | + | 8.81554i | 34.7703 | + | 5.50708i | 19.8915 | + | 17.9104i | 53.8168 | − | 3.93686i |
3.7 | −2.46147 | + | 3.79033i | −1.42743 | + | 3.71858i | −5.05389 | − | 11.3512i | 2.23871 | + | 10.9539i | −10.5811 | − | 14.5636i | −18.1267 | − | 3.79783i | 19.7544 | + | 3.12879i | 8.27464 | + | 7.45052i | −47.0295 | − | 18.4773i |
3.8 | −2.44758 | + | 3.76895i | −2.50628 | + | 6.52907i | −4.96040 | − | 11.1412i | −4.07235 | − | 10.4123i | −18.4734 | − | 25.4265i | −15.2656 | + | 10.4863i | 18.6227 | + | 2.94955i | −16.2825 | − | 14.6608i | 49.2108 | + | 10.1365i |
3.9 | −2.38969 | + | 3.67979i | 1.74074 | − | 4.53478i | −4.57638 | − | 10.2787i | 10.9198 | + | 2.39967i | 12.5272 | + | 17.2422i | 14.0149 | − | 12.1072i | 14.0907 | + | 2.23174i | 2.53089 | + | 2.27882i | −34.9251 | + | 34.4481i |
3.10 | −2.26197 | + | 3.48313i | −1.55172 | + | 4.04237i | −3.76181 | − | 8.44916i | 8.63834 | − | 7.09782i | −10.5702 | − | 14.5486i | 3.60941 | − | 18.1651i | 5.12243 | + | 0.811313i | 6.13196 | + | 5.52124i | 5.18297 | + | 46.1436i |
3.11 | −1.98207 | + | 3.05213i | 0.456252 | − | 1.18858i | −2.13297 | − | 4.79073i | −9.54840 | + | 5.81619i | 2.72336 | + | 3.74839i | −7.21114 | + | 17.0587i | −9.90591 | − | 1.56894i | 18.8604 | + | 16.9819i | 1.17388 | − | 40.6711i |
3.12 | −1.97283 | + | 3.03788i | 2.51318 | − | 6.54707i | −2.08281 | − | 4.67806i | 9.50586 | + | 5.88546i | 14.9312 | + | 20.5510i | −16.0333 | + | 9.27000i | −10.3009 | − | 1.63150i | −16.4831 | − | 14.8414i | −36.6327 | + | 17.2667i |
3.13 | −1.96169 | + | 3.02073i | 3.41104 | − | 8.88607i | −2.02272 | − | 4.54311i | −3.76740 | − | 10.5265i | 20.1510 | + | 27.7355i | 11.4301 | − | 14.5724i | −10.7683 | − | 1.70553i | −47.2621 | − | 42.5550i | 39.1881 | + | 9.26933i |
3.14 | −1.78858 | + | 2.75417i | −3.22364 | + | 8.39787i | −1.13255 | − | 2.54375i | 11.1404 | − | 0.944160i | −17.3634 | − | 23.8987i | 13.6862 | + | 12.4775i | −16.9168 | − | 2.67935i | −40.0675 | − | 36.0769i | −17.3251 | + | 32.3713i |
3.15 | −1.67008 | + | 2.57170i | 1.53287 | − | 3.99327i | −0.570578 | − | 1.28154i | 4.62112 | − | 10.1806i | 7.70948 | + | 10.6112i | 5.95919 | + | 17.5353i | −19.9805 | − | 3.16461i | 6.46837 | + | 5.82415i | 18.4639 | + | 28.8866i |
3.16 | −1.55621 | + | 2.39635i | 1.14091 | − | 2.97218i | −0.0668230 | − | 0.150087i | −9.96831 | − | 5.06288i | 5.34689 | + | 7.35936i | −12.1399 | − | 13.9866i | −22.1135 | − | 3.50243i | 12.5327 | + | 11.2845i | 27.6452 | − | 16.0087i |
3.17 | −1.55612 | + | 2.39621i | −0.0369693 | + | 0.0963083i | −0.0664387 | − | 0.149224i | 1.07427 | + | 11.1286i | −0.173247 | − | 0.238454i | 18.4311 | − | 1.81536i | −22.1149 | − | 3.50265i | 20.0570 | + | 18.0594i | −28.3382 | − | 14.7433i |
3.18 | −1.25359 | + | 1.93036i | −1.53819 | + | 4.00713i | 1.09910 | + | 2.46861i | −10.5630 | − | 3.66383i | −5.80693 | − | 7.99256i | 13.9174 | − | 12.2190i | −24.3299 | − | 3.85348i | 6.37387 | + | 5.73906i | 20.3141 | − | 15.7974i |
3.19 | −1.25167 | + | 1.92741i | −3.02258 | + | 7.87409i | 1.10568 | + | 2.48340i | 2.53086 | + | 10.8901i | −11.3933 | − | 15.6815i | −8.19908 | − | 16.6065i | −24.3295 | − | 3.85341i | −32.8005 | − | 29.5337i | −24.1575 | − | 8.75286i |
3.20 | −1.13801 | + | 1.75238i | −2.76584 | + | 7.20527i | 1.47812 | + | 3.31992i | −10.3242 | + | 4.29074i | −9.47881 | − | 13.0465i | −2.38757 | + | 18.3657i | −24.0099 | − | 3.80279i | −24.2011 | − | 21.7908i | 4.23004 | − | 22.9748i |
See next 80 embeddings (of 928 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
25.f | odd | 20 | 1 | inner |
175.x | even | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.4.x.a | ✓ | 928 |
7.d | odd | 6 | 1 | inner | 175.4.x.a | ✓ | 928 |
25.f | odd | 20 | 1 | inner | 175.4.x.a | ✓ | 928 |
175.x | even | 60 | 1 | inner | 175.4.x.a | ✓ | 928 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
175.4.x.a | ✓ | 928 | 1.a | even | 1 | 1 | trivial |
175.4.x.a | ✓ | 928 | 7.d | odd | 6 | 1 | inner |
175.4.x.a | ✓ | 928 | 25.f | odd | 20 | 1 | inner |
175.4.x.a | ✓ | 928 | 175.x | even | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(175, [\chi])\).