Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,4,Mod(23,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.23");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.e (of order \(6\), degree \(2\), not 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: | \(9.97132279097\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −4.70109 | + | 2.71418i | −0.837548 | − | 1.45068i | 10.7335 | − | 18.5910i | − | 7.70909i | 7.87478 | + | 4.54651i | 13.0120 | + | 7.51249i | 73.1038i | 12.0970 | − | 20.9527i | 20.9238 | + | 36.2412i | |||
23.2 | −4.19172 | + | 2.42009i | 3.09831 | + | 5.36643i | 7.71365 | − | 13.3604i | 15.2399i | −25.9745 | − | 14.9964i | 3.73794 | + | 2.15810i | 35.9495i | −5.69903 | + | 9.87102i | −36.8818 | − | 63.8811i | ||||
23.3 | −4.17905 | + | 2.41278i | −2.22176 | − | 3.84820i | 7.64299 | − | 13.2380i | 12.7712i | 18.5697 | + | 10.7212i | −22.6787 | − | 13.0936i | 35.1589i | 3.62756 | − | 6.28312i | −30.8140 | − | 53.3715i | ||||
23.4 | −3.32067 | + | 1.91719i | 0.139581 | + | 0.241762i | 3.35124 | − | 5.80452i | 11.3710i | −0.927008 | − | 0.535209i | 26.9007 | + | 15.5311i | − | 4.97517i | 13.4610 | − | 23.3152i | −21.8004 | − | 37.7594i | |||
23.5 | −2.73781 | + | 1.58068i | −3.54442 | − | 6.13911i | 0.997073 | − | 1.72698i | − | 13.6039i | 19.4079 | + | 11.2051i | 12.4114 | + | 7.16574i | − | 18.9866i | −11.6258 | + | 20.1364i | 21.5034 | + | 37.2450i | ||
23.6 | −1.92949 | + | 1.11399i | 4.87434 | + | 8.44260i | −1.51803 | + | 2.62931i | 8.20685i | −18.8100 | − | 10.8600i | −7.23560 | − | 4.17747i | − | 24.5882i | −34.0183 | + | 58.9214i | −9.14239 | − | 15.8351i | |||
23.7 | −1.49617 | + | 0.863817i | −3.44796 | − | 5.97204i | −2.50764 | + | 4.34336i | 20.8281i | 10.3175 | + | 5.95681i | 6.55206 | + | 3.78283i | − | 22.4856i | −10.2768 | + | 17.8000i | −17.9916 | − | 31.1624i | |||
23.8 | −0.337850 | + | 0.195058i | −1.80483 | − | 3.12606i | −3.92391 | + | 6.79640i | − | 7.52136i | 1.21953 | + | 0.704093i | 16.9261 | + | 9.77228i | − | 6.18247i | 6.98515 | − | 12.0986i | 1.46710 | + | 2.54109i | ||
23.9 | −0.129088 | + | 0.0745292i | 3.24429 | + | 5.61927i | −3.98889 | + | 6.90896i | 10.2526i | −0.837600 | − | 0.483588i | 25.6987 | + | 14.8372i | − | 2.38162i | −7.55083 | + | 13.0784i | −0.764114 | − | 1.32348i | |||
23.10 | 0.129088 | − | 0.0745292i | 3.24429 | + | 5.61927i | −3.98889 | + | 6.90896i | − | 10.2526i | 0.837600 | + | 0.483588i | −25.6987 | − | 14.8372i | 2.38162i | −7.55083 | + | 13.0784i | −0.764114 | − | 1.32348i | |||
23.11 | 0.337850 | − | 0.195058i | −1.80483 | − | 3.12606i | −3.92391 | + | 6.79640i | 7.52136i | −1.21953 | − | 0.704093i | −16.9261 | − | 9.77228i | 6.18247i | 6.98515 | − | 12.0986i | 1.46710 | + | 2.54109i | ||||
23.12 | 1.49617 | − | 0.863817i | −3.44796 | − | 5.97204i | −2.50764 | + | 4.34336i | − | 20.8281i | −10.3175 | − | 5.95681i | −6.55206 | − | 3.78283i | 22.4856i | −10.2768 | + | 17.8000i | −17.9916 | − | 31.1624i | |||
23.13 | 1.92949 | − | 1.11399i | 4.87434 | + | 8.44260i | −1.51803 | + | 2.62931i | − | 8.20685i | 18.8100 | + | 10.8600i | 7.23560 | + | 4.17747i | 24.5882i | −34.0183 | + | 58.9214i | −9.14239 | − | 15.8351i | |||
23.14 | 2.73781 | − | 1.58068i | −3.54442 | − | 6.13911i | 0.997073 | − | 1.72698i | 13.6039i | −19.4079 | − | 11.2051i | −12.4114 | − | 7.16574i | 18.9866i | −11.6258 | + | 20.1364i | 21.5034 | + | 37.2450i | ||||
23.15 | 3.32067 | − | 1.91719i | 0.139581 | + | 0.241762i | 3.35124 | − | 5.80452i | − | 11.3710i | 0.927008 | + | 0.535209i | −26.9007 | − | 15.5311i | 4.97517i | 13.4610 | − | 23.3152i | −21.8004 | − | 37.7594i | |||
23.16 | 4.17905 | − | 2.41278i | −2.22176 | − | 3.84820i | 7.64299 | − | 13.2380i | − | 12.7712i | −18.5697 | − | 10.7212i | 22.6787 | + | 13.0936i | − | 35.1589i | 3.62756 | − | 6.28312i | −30.8140 | − | 53.3715i | ||
23.17 | 4.19172 | − | 2.42009i | 3.09831 | + | 5.36643i | 7.71365 | − | 13.3604i | − | 15.2399i | 25.9745 | + | 14.9964i | −3.73794 | − | 2.15810i | − | 35.9495i | −5.69903 | + | 9.87102i | −36.8818 | − | 63.8811i | ||
23.18 | 4.70109 | − | 2.71418i | −0.837548 | − | 1.45068i | 10.7335 | − | 18.5910i | 7.70909i | −7.87478 | − | 4.54651i | −13.0120 | − | 7.51249i | − | 73.1038i | 12.0970 | − | 20.9527i | 20.9238 | + | 36.2412i | |||
147.1 | −4.70109 | − | 2.71418i | −0.837548 | + | 1.45068i | 10.7335 | + | 18.5910i | 7.70909i | 7.87478 | − | 4.54651i | 13.0120 | − | 7.51249i | − | 73.1038i | 12.0970 | + | 20.9527i | 20.9238 | − | 36.2412i | |||
147.2 | −4.19172 | − | 2.42009i | 3.09831 | − | 5.36643i | 7.71365 | + | 13.3604i | − | 15.2399i | −25.9745 | + | 14.9964i | 3.73794 | − | 2.15810i | − | 35.9495i | −5.69903 | − | 9.87102i | −36.8818 | + | 63.8811i | ||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.4.e.h | 36 | |
13.b | even | 2 | 1 | inner | 169.4.e.h | 36 | |
13.c | even | 3 | 1 | 169.4.b.g | 18 | ||
13.c | even | 3 | 1 | inner | 169.4.e.h | 36 | |
13.d | odd | 4 | 1 | 169.4.c.k | 18 | ||
13.d | odd | 4 | 1 | 169.4.c.l | 18 | ||
13.e | even | 6 | 1 | 169.4.b.g | 18 | ||
13.e | even | 6 | 1 | inner | 169.4.e.h | 36 | |
13.f | odd | 12 | 1 | 169.4.a.k | ✓ | 9 | |
13.f | odd | 12 | 1 | 169.4.a.l | yes | 9 | |
13.f | odd | 12 | 1 | 169.4.c.k | 18 | ||
13.f | odd | 12 | 1 | 169.4.c.l | 18 | ||
39.k | even | 12 | 1 | 1521.4.a.bg | 9 | ||
39.k | even | 12 | 1 | 1521.4.a.bh | 9 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
169.4.a.k | ✓ | 9 | 13.f | odd | 12 | 1 | |
169.4.a.l | yes | 9 | 13.f | odd | 12 | 1 | |
169.4.b.g | 18 | 13.c | even | 3 | 1 | ||
169.4.b.g | 18 | 13.e | even | 6 | 1 | ||
169.4.c.k | 18 | 13.d | odd | 4 | 1 | ||
169.4.c.k | 18 | 13.f | odd | 12 | 1 | ||
169.4.c.l | 18 | 13.d | odd | 4 | 1 | ||
169.4.c.l | 18 | 13.f | odd | 12 | 1 | ||
169.4.e.h | 36 | 1.a | even | 1 | 1 | trivial | |
169.4.e.h | 36 | 13.b | even | 2 | 1 | inner | |
169.4.e.h | 36 | 13.c | even | 3 | 1 | inner | |
169.4.e.h | 36 | 13.e | even | 6 | 1 | inner | |
1521.4.a.bg | 9 | 39.k | even | 12 | 1 | ||
1521.4.a.bh | 9 | 39.k | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{36} - 109 T_{2}^{34} + 7095 T_{2}^{32} - 304748 T_{2}^{30} + 9732537 T_{2}^{28} - 233733762 T_{2}^{26} + 4367945071 T_{2}^{24} - 62734700289 T_{2}^{22} + 700620402282 T_{2}^{20} + \cdots + 14003408896 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).