Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [672,2,Mod(155,672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(672, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 1, 4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("672.155");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 672 = 2^{5} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 672.bs (of order \(8\), degree \(4\), 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: | \(5.36594701583\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
155.1 | −1.41004 | + | 0.108548i | −0.206243 | − | 1.71973i | 1.97643 | − | 0.306113i | −0.448541 | + | 1.08287i | 0.477484 | + | 2.40250i | −0.707107 | + | 0.707107i | −2.75363 | + | 0.646170i | −2.91493 | + | 0.709364i | 0.514919 | − | 1.57559i |
155.2 | −1.40912 | − | 0.119931i | 1.40370 | − | 1.01471i | 1.97123 | + | 0.337995i | 1.13465 | − | 2.73929i | −2.09967 | + | 1.26150i | −0.707107 | + | 0.707107i | −2.73717 | − | 0.712688i | 0.940735 | − | 2.84869i | −1.92738 | + | 3.72390i |
155.3 | −1.38036 | + | 0.307600i | 0.475550 | + | 1.66549i | 1.81076 | − | 0.849195i | −1.65624 | + | 3.99852i | −1.16873 | − | 2.15269i | −0.707107 | + | 0.707107i | −2.23829 | + | 1.72918i | −2.54770 | + | 1.58405i | 1.05626 | − | 6.02883i |
155.4 | −1.34893 | + | 0.424720i | −1.25748 | + | 1.19111i | 1.63923 | − | 1.14583i | 0.367345 | − | 0.886850i | 1.19037 | − | 2.14080i | −0.707107 | + | 0.707107i | −1.72454 | + | 2.24186i | 0.162531 | − | 2.99559i | −0.118861 | + | 1.35232i |
155.5 | −1.34768 | − | 0.428658i | 0.151828 | + | 1.72538i | 1.63251 | + | 1.15539i | 0.849301 | − | 2.05039i | 0.534982 | − | 2.39035i | −0.707107 | + | 0.707107i | −1.70483 | − | 2.25689i | −2.95390 | + | 0.523925i | −2.02351 | + | 2.39922i |
155.6 | −1.28283 | − | 0.595271i | −1.60604 | + | 0.648559i | 1.29131 | + | 1.52726i | −0.327082 | + | 0.789647i | 2.44635 | + | 0.124039i | −0.707107 | + | 0.707107i | −0.747392 | − | 2.72789i | 2.15874 | − | 2.08323i | 0.889645 | − | 0.818280i |
155.7 | −1.26204 | − | 0.638158i | 1.63116 | + | 0.582516i | 1.18551 | + | 1.61077i | −0.102034 | + | 0.246332i | −1.68685 | − | 1.77610i | −0.707107 | + | 0.707107i | −0.468240 | − | 2.78940i | 2.32135 | + | 1.90035i | 0.285970 | − | 0.245768i |
155.8 | −1.22233 | + | 0.711266i | 1.73195 | + | 0.0187876i | 0.988202 | − | 1.73881i | −0.345711 | + | 0.834621i | −2.13038 | + | 1.20891i | −0.707107 | + | 0.707107i | 0.0288427 | + | 2.82828i | 2.99929 | + | 0.0650782i | −0.171063 | − | 1.26608i |
155.9 | −1.21431 | − | 0.724880i | −1.31477 | − | 1.12755i | 0.949098 | + | 1.76046i | 0.733476 | − | 1.77077i | 0.779205 | + | 2.32225i | −0.707107 | + | 0.707107i | 0.123621 | − | 2.82572i | 0.457262 | + | 2.96495i | −2.17426 | + | 1.61858i |
155.10 | −1.07981 | + | 0.913242i | 1.37333 | − | 1.05544i | 0.331980 | − | 1.97225i | −0.295998 | + | 0.714603i | −0.519067 | + | 2.39386i | −0.707107 | + | 0.707107i | 1.44267 | + | 2.43284i | 0.772089 | − | 2.89894i | −0.332983 | − | 1.04195i |
155.11 | −0.958529 | + | 1.03982i | −0.183529 | + | 1.72230i | −0.162445 | − | 1.99339i | 0.468472 | − | 1.13099i | −1.61496 | − | 1.84171i | −0.707107 | + | 0.707107i | 2.22847 | + | 1.74181i | −2.93263 | − | 0.632184i | 0.726982 | + | 1.57121i |
155.12 | −0.938001 | − | 1.05837i | 0.196518 | − | 1.72087i | −0.240309 | + | 1.98551i | −0.272376 | + | 0.657573i | −2.00565 | + | 1.40618i | −0.707107 | + | 0.707107i | 2.32682 | − | 1.60807i | −2.92276 | − | 0.676364i | 0.951446 | − | 0.328529i |
155.13 | −0.891321 | + | 1.09797i | −1.72725 | − | 0.128875i | −0.411094 | − | 1.95729i | 1.42904 | − | 3.45001i | 1.68104 | − | 1.78161i | −0.707107 | + | 0.707107i | 2.51548 | + | 1.29321i | 2.96678 | + | 0.445200i | 2.51429 | + | 4.64412i |
155.14 | −0.868232 | − | 1.11632i | −1.73050 | + | 0.0733671i | −0.492347 | + | 1.93845i | −1.63009 | + | 3.93540i | 1.58437 | + | 1.86809i | −0.707107 | + | 0.707107i | 2.59141 | − | 1.13341i | 2.98923 | − | 0.253923i | 5.80847 | − | 1.59713i |
155.15 | −0.838438 | + | 1.13887i | −0.303955 | − | 1.70517i | −0.594043 | − | 1.90974i | −1.38677 | + | 3.34796i | 2.19681 | + | 1.08352i | −0.707107 | + | 0.707107i | 2.67301 | + | 0.924664i | −2.81522 | + | 1.03659i | −2.65017 | − | 4.38641i |
155.16 | −0.669636 | − | 1.24563i | 1.65765 | + | 0.502191i | −1.10318 | + | 1.66823i | 1.25795 | − | 3.03696i | −0.484479 | − | 2.40110i | −0.707107 | + | 0.707107i | 2.81672 | + | 0.257038i | 2.49561 | + | 1.66491i | −4.62529 | + | 0.466720i |
155.17 | −0.651302 | − | 1.25531i | −0.203847 | + | 1.72001i | −1.15161 | + | 1.63517i | −0.577425 | + | 1.39403i | 2.29192 | − | 0.864357i | −0.707107 | + | 0.707107i | 2.80270 | + | 0.380642i | −2.91689 | − | 0.701238i | 2.12602 | − | 0.183084i |
155.18 | −0.550610 | + | 1.30262i | 0.864233 | − | 1.50103i | −1.39366 | − | 1.43447i | 1.15904 | − | 2.79817i | 1.47943 | + | 1.95225i | −0.707107 | + | 0.707107i | 2.63594 | − | 1.02558i | −1.50620 | − | 2.59449i | 3.00678 | + | 3.05049i |
155.19 | −0.422431 | + | 1.34965i | 1.42128 | + | 0.989935i | −1.64310 | − | 1.14027i | −0.956738 | + | 2.30977i | −1.93646 | + | 1.50004i | −0.707107 | + | 0.707107i | 2.23306 | − | 1.73593i | 1.04006 | + | 2.81394i | −2.71322 | − | 2.26698i |
155.20 | −0.393886 | − | 1.35825i | 1.34373 | − | 1.09288i | −1.68971 | + | 1.06999i | −0.833504 | + | 2.01226i | −2.01369 | − | 1.39465i | −0.707107 | + | 0.707107i | 2.11888 | + | 1.87360i | 0.611211 | − | 2.93708i | 3.06146 | + | 0.339511i |
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
32.h | odd | 8 | 1 | inner |
96.o | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 672.2.bs.a | ✓ | 192 |
3.b | odd | 2 | 1 | inner | 672.2.bs.a | ✓ | 192 |
32.h | odd | 8 | 1 | inner | 672.2.bs.a | ✓ | 192 |
96.o | even | 8 | 1 | inner | 672.2.bs.a | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
672.2.bs.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
672.2.bs.a | ✓ | 192 | 3.b | odd | 2 | 1 | inner |
672.2.bs.a | ✓ | 192 | 32.h | odd | 8 | 1 | inner |
672.2.bs.a | ✓ | 192 | 96.o | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{192} - 1552 T_{5}^{186} + 320464 T_{5}^{184} - 436352 T_{5}^{182} + 1204352 T_{5}^{180} - 373140512 T_{5}^{178} + 40026662656 T_{5}^{176} - 104422601024 T_{5}^{174} + \cdots + 19\!\cdots\!36 \)
acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).