Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [343,2,Mod(30,343)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([32]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("343.30");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 343 = 7^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 343.g (of order \(21\), degree \(12\), 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: | \(2.73886878933\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{21})\) |
Twist minimal: | no (minimal twist has level 49) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
30.1 | −2.09732 | + | 1.42993i | 1.40598 | − | 1.30456i | 1.62338 | − | 4.13631i | 2.38971 | + | 0.737129i | −1.08336 | + | 4.74653i | 0 | 1.38019 | + | 6.04701i | 0.0507148 | − | 0.676742i | −6.06605 | + | 1.87113i | ||
30.2 | −0.968018 | + | 0.659983i | −0.630335 | + | 0.584866i | −0.229202 | + | 0.583997i | −3.03447 | − | 0.936009i | 0.224174 | − | 0.982171i | 0 | −0.684966 | − | 3.00103i | −0.168936 | + | 2.25429i | 3.55517 | − | 1.09662i | ||
30.3 | 0.174575 | − | 0.119023i | −1.58372 | + | 1.46948i | −0.714372 | + | 1.82019i | 3.75050 | + | 1.15688i | −0.101576 | + | 0.445032i | 0 | 0.185965 | + | 0.814768i | 0.124614 | − | 1.66286i | 0.792437 | − | 0.244435i | ||
30.4 | 1.52543 | − | 1.04002i | 1.14975 | − | 1.06682i | 0.514606 | − | 1.31120i | 1.32986 | + | 0.410207i | 0.644358 | − | 2.82312i | 0 | 0.242977 | + | 1.06455i | −0.0403518 | + | 0.538458i | 2.45523 | − | 0.757337i | ||
67.1 | −2.50546 | + | 0.772833i | 0.539916 | + | 1.37568i | 4.02760 | − | 2.74597i | −2.04183 | − | 0.307757i | −2.41591 | − | 3.02946i | 0 | −4.69930 | + | 5.89274i | 0.598158 | − | 0.555009i | 5.35358 | − | 0.806922i | ||
67.2 | −1.16438 | + | 0.359164i | −0.671120 | − | 1.70999i | −0.425690 | + | 0.290231i | 0.478452 | + | 0.0721150i | 1.39561 | + | 1.75004i | 0 | 1.91089 | − | 2.39618i | −0.274497 | + | 0.254696i | −0.583002 | + | 0.0878735i | ||
67.3 | 0.681025 | − | 0.210069i | 0.953694 | + | 2.42997i | −1.23281 | + | 0.840516i | −2.65874 | − | 0.400741i | 1.15995 | + | 1.45453i | 0 | −1.55172 | + | 1.94579i | −2.79608 | + | 2.59438i | −1.89485 | + | 0.285603i | ||
67.4 | 1.16258 | − | 0.358609i | −0.190201 | − | 0.484624i | −0.429482 | + | 0.292816i | 2.56601 | + | 0.386764i | −0.394914 | − | 0.495207i | 0 | −1.91142 | + | 2.39684i | 2.00047 | − | 1.85617i | 3.12189 | − | 0.470550i | ||
79.1 | −2.25512 | + | 0.339905i | −0.489949 | − | 0.334042i | 3.05889 | − | 0.943543i | 0.260520 | − | 3.47640i | 1.21844 | + | 0.586769i | 0 | −2.46797 | + | 1.18851i | −0.967557 | − | 2.46529i | 0.594140 | + | 7.92825i | ||
79.2 | −1.23528 | + | 0.186189i | 2.40355 | + | 1.63871i | −0.419885 | + | 0.129517i | −0.186621 | + | 2.49028i | −3.27417 | − | 1.57676i | 0 | 2.74561 | − | 1.32222i | 1.99564 | + | 5.08481i | −0.233134 | − | 3.11095i | ||
79.3 | −0.251257 | + | 0.0378709i | −1.64786 | − | 1.12349i | −1.84945 | + | 0.570480i | −0.0348203 | + | 0.464645i | 0.456585 | + | 0.219880i | 0 | 0.900946 | − | 0.433873i | 0.357192 | + | 0.910110i | −0.00884768 | − | 0.118064i | ||
79.4 | 1.78609 | − | 0.269210i | 1.92585 | + | 1.31302i | 1.20650 | − | 0.372155i | 0.264371 | − | 3.52778i | 3.79321 | + | 1.82671i | 0 | −1.20005 | + | 0.577915i | 0.888840 | + | 2.26473i | −0.477524 | − | 6.37211i | ||
116.1 | −0.151065 | − | 2.01582i | −1.33832 | − | 0.412818i | −2.06305 | + | 0.310954i | 1.66779 | − | 1.54748i | −0.629993 | + | 2.76018i | 0 | 0.0388416 | + | 0.170176i | −0.858025 | − | 0.584992i | −3.37138 | − | 3.12819i | ||
116.2 | −0.118164 | − | 1.57679i | 3.07373 | + | 0.948121i | −0.494650 | + | 0.0745565i | −0.291074 | + | 0.270077i | 1.13178 | − | 4.95867i | 0 | −0.527696 | − | 2.31199i | 6.07018 | + | 4.13858i | 0.460250 | + | 0.427049i | ||
116.3 | 0.0658689 | + | 0.878959i | 0.269714 | + | 0.0831957i | 1.20943 | − | 0.182293i | 1.49036 | − | 1.38285i | −0.0553598 | + | 0.242547i | 0 | 0.632162 | + | 2.76968i | −2.41289 | − | 1.64508i | 1.31364 | + | 1.21888i | ||
116.4 | 0.192191 | + | 2.56461i | 0.776689 | + | 0.239577i | −4.56263 | + | 0.687706i | −1.80872 | + | 1.67825i | −0.465149 | + | 2.03795i | 0 | −1.49604 | − | 6.55456i | −1.93287 | − | 1.31781i | −4.65167 | − | 4.31612i | ||
128.1 | −2.50546 | − | 0.772833i | 0.539916 | − | 1.37568i | 4.02760 | + | 2.74597i | −2.04183 | + | 0.307757i | −2.41591 | + | 3.02946i | 0 | −4.69930 | − | 5.89274i | 0.598158 | + | 0.555009i | 5.35358 | + | 0.806922i | ||
128.2 | −1.16438 | − | 0.359164i | −0.671120 | + | 1.70999i | −0.425690 | − | 0.290231i | 0.478452 | − | 0.0721150i | 1.39561 | − | 1.75004i | 0 | 1.91089 | + | 2.39618i | −0.274497 | − | 0.254696i | −0.583002 | − | 0.0878735i | ||
128.3 | 0.681025 | + | 0.210069i | 0.953694 | − | 2.42997i | −1.23281 | − | 0.840516i | −2.65874 | + | 0.400741i | 1.15995 | − | 1.45453i | 0 | −1.55172 | − | 1.94579i | −2.79608 | − | 2.59438i | −1.89485 | − | 0.285603i | ||
128.4 | 1.16258 | + | 0.358609i | −0.190201 | + | 0.484624i | −0.429482 | − | 0.292816i | 2.56601 | − | 0.386764i | −0.394914 | + | 0.495207i | 0 | −1.91142 | − | 2.39684i | 2.00047 | + | 1.85617i | 3.12189 | + | 0.470550i | ||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.g | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 343.2.g.g | 48 | |
7.b | odd | 2 | 1 | 49.2.g.a | ✓ | 48 | |
7.c | even | 3 | 1 | 343.2.e.c | 48 | ||
7.c | even | 3 | 1 | 343.2.g.h | 48 | ||
7.d | odd | 6 | 1 | 343.2.e.d | 48 | ||
7.d | odd | 6 | 1 | 343.2.g.i | 48 | ||
21.c | even | 2 | 1 | 441.2.bb.d | 48 | ||
28.d | even | 2 | 1 | 784.2.bg.c | 48 | ||
49.e | even | 7 | 1 | 343.2.g.h | 48 | ||
49.f | odd | 14 | 1 | 343.2.g.i | 48 | ||
49.g | even | 21 | 1 | 343.2.e.c | 48 | ||
49.g | even | 21 | 1 | inner | 343.2.g.g | 48 | |
49.g | even | 21 | 1 | 2401.2.a.i | 24 | ||
49.h | odd | 42 | 1 | 49.2.g.a | ✓ | 48 | |
49.h | odd | 42 | 1 | 343.2.e.d | 48 | ||
49.h | odd | 42 | 1 | 2401.2.a.h | 24 | ||
147.o | even | 42 | 1 | 441.2.bb.d | 48 | ||
196.p | even | 42 | 1 | 784.2.bg.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
49.2.g.a | ✓ | 48 | 7.b | odd | 2 | 1 | |
49.2.g.a | ✓ | 48 | 49.h | odd | 42 | 1 | |
343.2.e.c | 48 | 7.c | even | 3 | 1 | ||
343.2.e.c | 48 | 49.g | even | 21 | 1 | ||
343.2.e.d | 48 | 7.d | odd | 6 | 1 | ||
343.2.e.d | 48 | 49.h | odd | 42 | 1 | ||
343.2.g.g | 48 | 1.a | even | 1 | 1 | trivial | |
343.2.g.g | 48 | 49.g | even | 21 | 1 | inner | |
343.2.g.h | 48 | 7.c | even | 3 | 1 | ||
343.2.g.h | 48 | 49.e | even | 7 | 1 | ||
343.2.g.i | 48 | 7.d | odd | 6 | 1 | ||
343.2.g.i | 48 | 49.f | odd | 14 | 1 | ||
441.2.bb.d | 48 | 21.c | even | 2 | 1 | ||
441.2.bb.d | 48 | 147.o | even | 42 | 1 | ||
784.2.bg.c | 48 | 28.d | even | 2 | 1 | ||
784.2.bg.c | 48 | 196.p | even | 42 | 1 | ||
2401.2.a.h | 24 | 49.h | odd | 42 | 1 | ||
2401.2.a.i | 24 | 49.g | even | 21 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(343, [\chi])\):
\( T_{2}^{48} + 13 T_{2}^{47} + 85 T_{2}^{46} + 382 T_{2}^{45} + 1339 T_{2}^{44} + 3829 T_{2}^{43} + 9018 T_{2}^{42} + 17506 T_{2}^{41} + 27882 T_{2}^{40} + 36353 T_{2}^{39} + 41812 T_{2}^{38} + 54238 T_{2}^{37} + 79166 T_{2}^{36} + \cdots + 729 \) |
\( T_{3}^{48} - 14 T_{3}^{47} + 89 T_{3}^{46} - 329 T_{3}^{45} + 679 T_{3}^{44} + 35 T_{3}^{43} - 6061 T_{3}^{42} + 24122 T_{3}^{41} - 49506 T_{3}^{40} + 37289 T_{3}^{39} + 91315 T_{3}^{38} - 314237 T_{3}^{37} + 455848 T_{3}^{36} + \cdots + 4439449 \) |