Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,2,Mod(67,980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.x (of order \(12\), degree \(4\), 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: | \(7.82533939809\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 140) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.41262 | + | 0.0671791i | −2.38471 | + | 0.638980i | 1.99097 | − | 0.189797i | 0.525600 | − | 2.17342i | 3.32575 | − | 1.06284i | 0 | −2.79973 | + | 0.401862i | 2.68045 | − | 1.54756i | −0.596463 | + | 3.10552i | ||
67.2 | −1.40543 | + | 0.157385i | −1.08292 | + | 0.290169i | 1.95046 | − | 0.442386i | −1.68711 | + | 1.46754i | 1.47630 | − | 0.578247i | 0 | −2.67161 | + | 0.928715i | −1.50955 | + | 0.871538i | 2.14014 | − | 2.32805i | ||
67.3 | −1.34754 | + | 0.429119i | 2.71477 | − | 0.727420i | 1.63171 | − | 1.15651i | 2.01971 | + | 0.959579i | −3.34610 | + | 2.14518i | 0 | −1.70252 | + | 2.25864i | 4.24276 | − | 2.44956i | −3.13340 | − | 0.426375i | ||
67.4 | −1.24267 | − | 0.675113i | 0.402205 | − | 0.107770i | 1.08845 | + | 1.67788i | 2.22809 | − | 0.188711i | −0.572564 | − | 0.137611i | 0 | −0.219816 | − | 2.81987i | −2.44792 | + | 1.41331i | −2.89618 | − | 1.26971i | ||
67.5 | −1.05431 | + | 0.942570i | 1.20413 | − | 0.322645i | 0.223125 | − | 1.98751i | −2.09885 | − | 0.771256i | −0.965404 | + | 1.47514i | 0 | 1.63813 | + | 2.30576i | −1.25225 | + | 0.722989i | 2.93979 | − | 1.16517i | ||
67.6 | −0.809319 | − | 1.15974i | 0.551941 | − | 0.147892i | −0.690004 | + | 1.87720i | −1.08510 | − | 1.95513i | −0.618214 | − | 0.520418i | 0 | 2.73551 | − | 0.719030i | −2.31531 | + | 1.33674i | −1.38926 | + | 2.84077i | ||
67.7 | −0.674418 | − | 1.24304i | −2.47915 | + | 0.664287i | −1.09032 | + | 1.67666i | −1.93364 | + | 1.12296i | 2.49773 | + | 2.63369i | 0 | 2.81950 | + | 0.224544i | 3.10685 | − | 1.79374i | 2.69997 | + | 1.64626i | ||
67.8 | −0.667698 | + | 1.24667i | −2.02821 | + | 0.543458i | −1.10836 | − | 1.66480i | 0.518800 | + | 2.17505i | 0.676724 | − | 2.89137i | 0 | 2.81549 | − | 0.270171i | 1.22023 | − | 0.704499i | −3.05797 | − | 0.805507i | ||
67.9 | −0.212826 | − | 1.39811i | 0.807254 | − | 0.216303i | −1.90941 | + | 0.595107i | 0.780454 | + | 2.09545i | −0.474220 | − | 1.08259i | 0 | 1.23840 | + | 2.54291i | −1.99320 | + | 1.15078i | 2.76356 | − | 1.53712i | ||
67.10 | −0.0450897 | + | 1.41349i | 2.02821 | − | 0.543458i | −1.99593 | − | 0.127468i | 0.518800 | + | 2.17505i | 0.676724 | + | 2.89137i | 0 | 0.270171 | − | 2.81549i | 1.22023 | − | 0.704499i | −3.09782 | + | 0.635249i | ||
67.11 | 0.441772 | + | 1.34344i | −1.20413 | + | 0.322645i | −1.60968 | + | 1.18699i | −2.09885 | − | 0.771256i | −0.965404 | − | 1.47514i | 0 | −2.30576 | − | 1.63813i | −1.25225 | + | 0.722989i | 0.108927 | − | 3.16040i | ||
67.12 | 0.883367 | − | 1.10438i | −0.807254 | + | 0.216303i | −0.439327 | − | 1.95115i | 0.780454 | + | 2.09545i | −0.474220 | + | 1.08259i | 0 | −2.54291 | − | 1.23840i | −1.99320 | + | 1.15078i | 3.00360 | + | 0.989126i | ||
67.13 | 0.952442 | + | 1.04540i | −2.71477 | + | 0.727420i | −0.185707 | + | 1.99136i | 2.01971 | + | 0.959579i | −3.34610 | − | 2.14518i | 0 | −2.25864 | + | 1.70252i | 4.24276 | − | 2.44956i | 0.920513 | + | 3.02534i | ||
67.14 | 1.13844 | + | 0.839014i | 1.08292 | − | 0.290169i | 0.592112 | + | 1.91034i | −1.68711 | + | 1.46754i | 1.47630 | + | 0.578247i | 0 | −0.928715 | + | 2.67161i | −1.50955 | + | 0.871538i | −3.15196 | + | 0.255205i | ||
67.15 | 1.18977 | + | 0.764487i | 2.38471 | − | 0.638980i | 0.831118 | + | 1.81913i | 0.525600 | − | 2.17342i | 3.32575 | + | 1.06284i | 0 | −0.401862 | + | 2.79973i | 2.68045 | − | 1.54756i | 2.28689 | − | 2.18406i | ||
67.16 | 1.20559 | − | 0.739299i | 2.47915 | − | 0.664287i | 0.906874 | − | 1.78258i | −1.93364 | + | 1.12296i | 2.49773 | − | 2.63369i | 0 | −0.224544 | − | 2.81950i | 3.10685 | − | 1.79374i | −1.50097 | + | 2.78336i | ||
67.17 | 1.28076 | − | 0.599707i | −0.551941 | + | 0.147892i | 1.28070 | − | 1.53616i | −1.08510 | − | 1.95513i | −0.618214 | + | 0.520418i | 0 | 0.719030 | − | 2.73551i | −2.31531 | + | 1.33674i | −2.56227 | − | 1.85332i | ||
67.18 | 1.41374 | + | 0.0366689i | −0.402205 | + | 0.107770i | 1.99731 | + | 0.103680i | 2.22809 | − | 0.188711i | −0.572564 | + | 0.137611i | 0 | 2.81987 | + | 0.219816i | −2.44792 | + | 1.41331i | 3.15686 | − | 0.185086i | ||
263.1 | −1.39811 | + | 0.212826i | 0.216303 | + | 0.807254i | 1.90941 | − | 0.595107i | 1.42448 | + | 1.72362i | −0.474220 | − | 1.08259i | 0 | −2.54291 | + | 1.23840i | 1.99320 | − | 1.15078i | −2.35841 | − | 2.10663i | ||
263.2 | −1.24304 | + | 0.674418i | −0.664287 | − | 2.47915i | 1.09032 | − | 1.67666i | 1.93933 | − | 1.11310i | 2.49773 | + | 2.63369i | 0 | −0.224544 | + | 2.81950i | −3.10685 | + | 1.79374i | −1.65998 | + | 2.69156i | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
7.c | even | 3 | 1 | inner |
20.e | even | 4 | 1 | inner |
28.g | odd | 6 | 1 | inner |
35.l | odd | 12 | 1 | inner |
140.w | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.2.x.m | 72 | |
4.b | odd | 2 | 1 | inner | 980.2.x.m | 72 | |
5.c | odd | 4 | 1 | inner | 980.2.x.m | 72 | |
7.b | odd | 2 | 1 | 140.2.w.b | ✓ | 72 | |
7.c | even | 3 | 1 | 980.2.k.j | 36 | ||
7.c | even | 3 | 1 | inner | 980.2.x.m | 72 | |
7.d | odd | 6 | 1 | 140.2.w.b | ✓ | 72 | |
7.d | odd | 6 | 1 | 980.2.k.k | 36 | ||
20.e | even | 4 | 1 | inner | 980.2.x.m | 72 | |
28.d | even | 2 | 1 | 140.2.w.b | ✓ | 72 | |
28.f | even | 6 | 1 | 140.2.w.b | ✓ | 72 | |
28.f | even | 6 | 1 | 980.2.k.k | 36 | ||
28.g | odd | 6 | 1 | 980.2.k.j | 36 | ||
28.g | odd | 6 | 1 | inner | 980.2.x.m | 72 | |
35.c | odd | 2 | 1 | 700.2.be.e | 72 | ||
35.f | even | 4 | 1 | 140.2.w.b | ✓ | 72 | |
35.f | even | 4 | 1 | 700.2.be.e | 72 | ||
35.i | odd | 6 | 1 | 700.2.be.e | 72 | ||
35.k | even | 12 | 1 | 140.2.w.b | ✓ | 72 | |
35.k | even | 12 | 1 | 700.2.be.e | 72 | ||
35.k | even | 12 | 1 | 980.2.k.k | 36 | ||
35.l | odd | 12 | 1 | 980.2.k.j | 36 | ||
35.l | odd | 12 | 1 | inner | 980.2.x.m | 72 | |
140.c | even | 2 | 1 | 700.2.be.e | 72 | ||
140.j | odd | 4 | 1 | 140.2.w.b | ✓ | 72 | |
140.j | odd | 4 | 1 | 700.2.be.e | 72 | ||
140.s | even | 6 | 1 | 700.2.be.e | 72 | ||
140.w | even | 12 | 1 | 980.2.k.j | 36 | ||
140.w | even | 12 | 1 | inner | 980.2.x.m | 72 | |
140.x | odd | 12 | 1 | 140.2.w.b | ✓ | 72 | |
140.x | odd | 12 | 1 | 700.2.be.e | 72 | ||
140.x | odd | 12 | 1 | 980.2.k.k | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
140.2.w.b | ✓ | 72 | 7.b | odd | 2 | 1 | |
140.2.w.b | ✓ | 72 | 7.d | odd | 6 | 1 | |
140.2.w.b | ✓ | 72 | 28.d | even | 2 | 1 | |
140.2.w.b | ✓ | 72 | 28.f | even | 6 | 1 | |
140.2.w.b | ✓ | 72 | 35.f | even | 4 | 1 | |
140.2.w.b | ✓ | 72 | 35.k | even | 12 | 1 | |
140.2.w.b | ✓ | 72 | 140.j | odd | 4 | 1 | |
140.2.w.b | ✓ | 72 | 140.x | odd | 12 | 1 | |
700.2.be.e | 72 | 35.c | odd | 2 | 1 | ||
700.2.be.e | 72 | 35.f | even | 4 | 1 | ||
700.2.be.e | 72 | 35.i | odd | 6 | 1 | ||
700.2.be.e | 72 | 35.k | even | 12 | 1 | ||
700.2.be.e | 72 | 140.c | even | 2 | 1 | ||
700.2.be.e | 72 | 140.j | odd | 4 | 1 | ||
700.2.be.e | 72 | 140.s | even | 6 | 1 | ||
700.2.be.e | 72 | 140.x | odd | 12 | 1 | ||
980.2.k.j | 36 | 7.c | even | 3 | 1 | ||
980.2.k.j | 36 | 28.g | odd | 6 | 1 | ||
980.2.k.j | 36 | 35.l | odd | 12 | 1 | ||
980.2.k.j | 36 | 140.w | even | 12 | 1 | ||
980.2.k.k | 36 | 7.d | odd | 6 | 1 | ||
980.2.k.k | 36 | 28.f | even | 6 | 1 | ||
980.2.k.k | 36 | 35.k | even | 12 | 1 | ||
980.2.k.k | 36 | 140.x | odd | 12 | 1 | ||
980.2.x.m | 72 | 1.a | even | 1 | 1 | trivial | |
980.2.x.m | 72 | 4.b | odd | 2 | 1 | inner | |
980.2.x.m | 72 | 5.c | odd | 4 | 1 | inner | |
980.2.x.m | 72 | 7.c | even | 3 | 1 | inner | |
980.2.x.m | 72 | 20.e | even | 4 | 1 | inner | |
980.2.x.m | 72 | 28.g | odd | 6 | 1 | inner | |
980.2.x.m | 72 | 35.l | odd | 12 | 1 | inner | |
980.2.x.m | 72 | 140.w | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):
\( T_{3}^{72} - 167 T_{3}^{68} + 17716 T_{3}^{64} - 1150565 T_{3}^{60} + 54628057 T_{3}^{56} - 1771981650 T_{3}^{52} + 42135437541 T_{3}^{48} - 631596499705 T_{3}^{44} + 6449394079565 T_{3}^{40} + \cdots + 136048896 \) |
\( T_{11}^{36} - 86 T_{11}^{34} + 4450 T_{11}^{32} - 150556 T_{11}^{30} + 3773207 T_{11}^{28} - 70383174 T_{11}^{26} + 1014933178 T_{11}^{24} - 11131220640 T_{11}^{22} + 94141574089 T_{11}^{20} + \cdots + 84332160000 \) |
\( T_{13}^{18} - 36 T_{13}^{15} + 1009 T_{13}^{14} - 1068 T_{13}^{13} + 648 T_{13}^{12} - 5320 T_{13}^{11} + 187328 T_{13}^{10} - 191280 T_{13}^{9} + 108000 T_{13}^{8} - 527296 T_{13}^{7} + 7172032 T_{13}^{6} - 10425920 T_{13}^{5} + \cdots + 12800 \) |