Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(92,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.92");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.w (of order \(7\), degree \(6\), 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: | \(5.08647060876\) |
Analytic rank: | \(0\) |
Dimension: | \(174\) |
Relative dimension: | \(29\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
92.1 | −1.68715 | + | 2.11563i | −0.820369 | − | 0.395069i | −1.18434 | − | 5.18892i | −0.379531 | − | 0.182773i | 2.21991 | − | 1.06905i | −1.63169 | − | 2.08269i | 8.09995 | + | 3.90073i | −1.35354 | − | 1.69729i | 1.02701 | − | 0.494580i |
92.2 | −1.57084 | + | 1.96978i | 2.34650 | + | 1.13001i | −0.967425 | − | 4.23856i | 3.79256 | + | 1.82640i | −5.91185 | + | 2.84700i | 0.107940 | − | 2.64355i | 5.32882 | + | 2.56623i | 2.35865 | + | 2.95765i | −9.55512 | + | 4.60150i |
92.3 | −1.44058 | + | 1.80643i | −0.133830 | − | 0.0644489i | −0.742877 | − | 3.25476i | −3.89600 | − | 1.87621i | 0.309215 | − | 0.148910i | −1.21227 | + | 2.35168i | 2.78627 | + | 1.34180i | −1.85671 | − | 2.32824i | 9.00174 | − | 4.33501i |
92.4 | −1.42465 | + | 1.78645i | −2.04852 | − | 0.986513i | −0.716746 | − | 3.14027i | 0.342127 | + | 0.164760i | 4.68077 | − | 2.25414i | 2.57961 | − | 0.587895i | 2.51370 | + | 1.21053i | 1.35274 | + | 1.69628i | −0.781746 | + | 0.376469i |
92.5 | −1.18201 | + | 1.48220i | −0.0940579 | − | 0.0452959i | −0.354712 | − | 1.55409i | 1.30132 | + | 0.626680i | 0.178315 | − | 0.0858720i | −2.43875 | + | 1.02591i | −0.693369 | − | 0.333909i | −1.86367 | − | 2.33697i | −2.46703 | + | 1.18806i |
92.6 | −1.14796 | + | 1.43949i | 1.04469 | + | 0.503095i | −0.309290 | − | 1.35509i | −0.563090 | − | 0.271170i | −1.92345 | + | 0.926286i | 1.79557 | − | 1.94318i | −1.01200 | − | 0.487355i | −1.03220 | − | 1.29434i | 1.03675 | − | 0.499271i |
92.7 | −1.12298 | + | 1.40817i | −2.74049 | − | 1.31975i | −0.276823 | − | 1.21284i | −2.53396 | − | 1.22029i | 4.93595 | − | 2.37703i | −1.05285 | − | 2.42724i | −1.22675 | − | 0.590772i | 3.89808 | + | 4.88803i | 4.56397 | − | 2.19789i |
92.8 | −1.10143 | + | 1.38115i | 3.05365 | + | 1.47056i | −0.249384 | − | 1.09262i | 0.775588 | + | 0.373504i | −5.39445 | + | 2.59783i | 0.705731 | + | 2.54989i | −1.39947 | − | 0.673949i | 5.29177 | + | 6.63567i | −1.37012 | + | 0.659815i |
92.9 | −0.807700 | + | 1.01282i | −2.74330 | − | 1.32111i | 0.0716088 | + | 0.313739i | 2.02464 | + | 0.975017i | 3.55382 | − | 1.71143i | 0.327305 | + | 2.62543i | −2.70992 | − | 1.30503i | 3.90993 | + | 4.90289i | −2.62283 | + | 1.26309i |
92.10 | −0.487175 | + | 0.610898i | 1.39678 | + | 0.672652i | 0.309185 | + | 1.35463i | −2.44805 | − | 1.17892i | −1.09140 | + | 0.525588i | −2.52492 | − | 0.790430i | −2.38614 | − | 1.14911i | −0.371947 | − | 0.466407i | 1.91283 | − | 0.921169i |
92.11 | −0.440322 | + | 0.552147i | −1.61690 | − | 0.778657i | 0.334060 | + | 1.46361i | 2.37681 | + | 1.14461i | 1.14189 | − | 0.549905i | 2.31524 | − | 1.28050i | −2.22779 | − | 1.07285i | 0.137583 | + | 0.172524i | −1.67856 | + | 0.808350i |
92.12 | −0.347630 | + | 0.435914i | −0.809113 | − | 0.389648i | 0.375867 | + | 1.64678i | −1.77186 | − | 0.853282i | 0.451125 | − | 0.217250i | 2.01500 | + | 1.71458i | −1.85320 | − | 0.892452i | −1.36763 | − | 1.71495i | 0.987907 | − | 0.475751i |
92.13 | −0.165915 | + | 0.208051i | 2.56994 | + | 1.23762i | 0.429285 | + | 1.88082i | 0.602735 | + | 0.290262i | −0.683878 | + | 0.329338i | 1.20359 | − | 2.35614i | −0.942039 | − | 0.453662i | 3.20241 | + | 4.01570i | −0.160392 | + | 0.0772407i |
92.14 | −0.118827 | + | 0.149004i | 1.11947 | + | 0.539107i | 0.436959 | + | 1.91444i | 0.156007 | + | 0.0751292i | −0.213352 | + | 0.102745i | 0.0412747 | + | 2.64543i | −0.680602 | − | 0.327760i | −0.907899 | − | 1.13847i | −0.0297324 | + | 0.0143184i |
92.15 | 0.153433 | − | 0.192399i | −1.67003 | − | 0.804242i | 0.431566 | + | 1.89082i | 0.217813 | + | 0.104893i | −0.410972 | + | 0.197914i | −2.59813 | − | 0.499745i | 0.873441 | + | 0.420627i | 0.271713 | + | 0.340717i | 0.0536009 | − | 0.0258129i |
92.16 | 0.252803 | − | 0.317004i | 1.07472 | + | 0.517556i | 0.408459 | + | 1.78958i | 3.73621 | + | 1.79926i | 0.435758 | − | 0.209850i | −1.42709 | + | 2.22787i | 1.40118 | + | 0.674774i | −0.983320 | − | 1.23304i | 1.51490 | − | 0.729536i |
92.17 | 0.311250 | − | 0.390295i | −1.89343 | − | 0.911826i | 0.389588 | + | 1.70690i | −2.41457 | − | 1.16280i | −0.945210 | + | 0.455189i | 0.718518 | − | 2.54632i | 1.68699 | + | 0.812413i | 0.883165 | + | 1.10745i | −1.20537 | + | 0.580476i |
92.18 | 0.529450 | − | 0.663910i | 2.82603 | + | 1.36094i | 0.284583 | + | 1.24684i | −2.32207 | − | 1.11825i | 2.39979 | − | 1.15568i | −1.55674 | + | 2.13929i | 2.50862 | + | 1.20809i | 4.26380 | + | 5.34664i | −1.97184 | + | 0.949586i |
92.19 | 0.708170 | − | 0.888017i | 0.627836 | + | 0.302350i | 0.157972 | + | 0.692122i | −3.97243 | − | 1.91302i | 0.713107 | − | 0.343414i | 2.60413 | + | 0.467470i | 2.77316 | + | 1.33548i | −1.56771 | − | 1.96584i | −4.51195 | + | 2.17284i |
92.20 | 0.743539 | − | 0.932368i | 1.30142 | + | 0.626732i | 0.128582 | + | 0.563353i | 0.955511 | + | 0.460150i | 1.55200 | − | 0.747405i | 2.35959 | − | 1.19679i | 2.76975 | + | 1.33384i | −0.569562 | − | 0.714208i | 1.13949 | − | 0.548748i |
See next 80 embeddings (of 174 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.e | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.w.b | ✓ | 174 |
49.e | even | 7 | 1 | inner | 637.2.w.b | ✓ | 174 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
637.2.w.b | ✓ | 174 | 1.a | even | 1 | 1 | trivial |
637.2.w.b | ✓ | 174 | 49.e | even | 7 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{174} + 3 T_{2}^{173} + 49 T_{2}^{172} + 147 T_{2}^{171} + 1338 T_{2}^{170} + \cdots + 84\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\).