Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [352,2,Mod(19,352)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(352, base_ring=CyclotomicField(40))
chi = DirichletCharacter(H, H._module([20, 35, 12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("352.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 352 = 2^{5} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 352.bc (of order \(40\), degree \(16\), 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: | \(2.81073415115\) |
Analytic rank: | \(0\) |
Dimension: | \(736\) |
Relative dimension: | \(46\) over \(\Q(\zeta_{40})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{40}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.41245 | + | 0.0705991i | 0.150757 | − | 1.91555i | 1.99003 | − | 0.199436i | −2.63276 | − | 0.632071i | −0.0777007 | + | 2.71626i | 2.34464 | + | 4.60161i | −2.79674 | + | 0.422187i | −0.683545 | − | 0.108263i | 3.76327 | + | 0.706898i |
19.2 | −1.39865 | − | 0.209265i | −0.134573 | + | 1.70991i | 1.91242 | + | 0.585374i | 0.794943 | + | 0.190849i | 0.546042 | − | 2.36339i | 1.17216 | + | 2.30049i | −2.55229 | − | 1.21893i | 0.0573940 | + | 0.00909032i | −1.07190 | − | 0.433283i |
19.3 | −1.39496 | − | 0.232560i | 0.00366593 | − | 0.0465800i | 1.89183 | + | 0.648825i | −0.0528412 | − | 0.0126860i | −0.0159465 | + | 0.0641247i | −1.30784 | − | 2.56679i | −2.48814 | − | 1.34505i | 2.96091 | + | 0.468962i | 0.0707611 | + | 0.0299853i |
19.4 | −1.31946 | + | 0.508948i | −0.158585 | + | 2.01502i | 1.48194 | − | 1.34307i | −3.93727 | − | 0.945256i | −0.816293 | − | 2.73944i | −0.0456497 | − | 0.0895925i | −1.27181 | + | 2.52636i | −1.07208 | − | 0.169801i | 5.67616 | − | 0.756643i |
19.5 | −1.30861 | + | 0.536227i | 0.141964 | − | 1.80382i | 1.42492 | − | 1.40342i | −1.17636 | − | 0.282420i | 0.781484 | + | 2.43663i | −0.850228 | − | 1.66867i | −1.11211 | + | 2.60062i | −0.270561 | − | 0.0428526i | 1.69084 | − | 0.261221i |
19.6 | −1.29988 | − | 0.557061i | 0.118949 | − | 1.51139i | 1.37937 | + | 1.44822i | 3.60772 | + | 0.866136i | −0.996557 | + | 1.89836i | −0.662879 | − | 1.30097i | −0.986257 | − | 2.65090i | 0.692913 | + | 0.109747i | −4.20710 | − | 3.13559i |
19.7 | −1.27293 | + | 0.616165i | 0.0557342 | − | 0.708171i | 1.24068 | − | 1.56867i | 2.69247 | + | 0.646406i | 0.365405 | + | 0.935790i | 0.404991 | + | 0.794840i | −0.612737 | + | 2.76126i | 2.46467 | + | 0.390365i | −3.82561 | + | 0.836183i |
19.8 | −1.19190 | + | 0.761162i | −0.217455 | + | 2.76302i | 0.841265 | − | 1.81446i | 1.21827 | + | 0.292481i | −1.84392 | − | 3.45877i | 0.587203 | + | 1.15245i | 0.378394 | + | 2.80300i | −4.62394 | − | 0.732361i | −1.67469 | + | 0.578692i |
19.9 | −1.17122 | − | 0.792623i | −0.0864834 | + | 1.09888i | 0.743499 | + | 1.85667i | −1.85905 | − | 0.446319i | 0.972285 | − | 1.21847i | 0.571540 | + | 1.12171i | 0.600838 | − | 2.76387i | 1.76302 | + | 0.279234i | 1.82359 | + | 1.99626i |
19.10 | −1.11604 | − | 0.868590i | −0.259149 | + | 3.29279i | 0.491101 | + | 1.93877i | −1.78730 | − | 0.429093i | 3.14931 | − | 3.44980i | −2.10638 | − | 4.13400i | 1.13591 | − | 2.59031i | −7.81227 | − | 1.23734i | 1.62200 | + | 2.03132i |
19.11 | −0.938875 | − | 1.05760i | 0.233178 | − | 2.96281i | −0.237028 | + | 1.98590i | −1.45205 | − | 0.348606i | −3.35239 | + | 2.53510i | −0.603345 | − | 1.18413i | 2.32283 | − | 1.61384i | −5.76080 | − | 0.912421i | 0.994606 | + | 1.86298i |
19.12 | −0.892958 | − | 1.09664i | 0.121306 | − | 1.54134i | −0.405253 | + | 1.95851i | 0.448720 | + | 0.107728i | −1.79862 | + | 1.24332i | 1.16515 | + | 2.28673i | 2.50966 | − | 1.30445i | 0.602046 | + | 0.0953547i | −0.282549 | − | 0.588282i |
19.13 | −0.873487 | + | 1.11221i | 0.221517 | − | 2.81464i | −0.474042 | − | 1.94301i | −1.54483 | − | 0.370880i | 2.93699 | + | 2.70493i | −1.34669 | − | 2.64302i | 2.57511 | + | 1.16996i | −4.91008 | − | 0.777680i | 1.76188 | − | 1.39422i |
19.14 | −0.823127 | − | 1.14998i | −0.176573 | + | 2.24358i | −0.644925 | + | 1.89316i | 3.36360 | + | 0.807530i | 2.72542 | − | 1.64369i | 0.830538 | + | 1.63002i | 2.70796 | − | 0.816660i | −2.03939 | − | 0.323008i | −1.84003 | − | 4.53279i |
19.15 | −0.821434 | + | 1.15119i | −0.0829010 | + | 1.05336i | −0.650493 | − | 1.89126i | 2.88799 | + | 0.693345i | −1.14452 | − | 0.960699i | −1.79651 | − | 3.52584i | 2.71154 | + | 0.804701i | 1.86038 | + | 0.294654i | −3.17047 | + | 2.75510i |
19.16 | −0.730601 | + | 1.21088i | 0.222334 | − | 2.82502i | −0.932444 | − | 1.76934i | 2.89506 | + | 0.695043i | 3.25831 | + | 2.33318i | 2.11218 | + | 4.14539i | 2.82369 | + | 0.163605i | −4.96824 | − | 0.786892i | −2.95675 | + | 2.99776i |
19.17 | −0.618838 | + | 1.27163i | 0.00409177 | − | 0.0519908i | −1.23408 | − | 1.57387i | −2.91327 | − | 0.699414i | 0.0635809 | + | 0.0373771i | −0.210704 | − | 0.413530i | 2.76507 | − | 0.595320i | 2.96038 | + | 0.468878i | 2.69224 | − | 3.27177i |
19.18 | −0.528312 | − | 1.31183i | −0.00496317 | + | 0.0630631i | −1.44177 | + | 1.38611i | 0.438953 | + | 0.105383i | 0.0853499 | − | 0.0268061i | −1.74049 | − | 3.41591i | 2.58004 | + | 1.15906i | 2.95911 | + | 0.468677i | −0.0936596 | − | 0.631506i |
19.19 | −0.367801 | − | 1.36555i | −0.0569089 | + | 0.723097i | −1.72945 | + | 1.00450i | −2.99198 | − | 0.718310i | 1.00835 | − | 0.188244i | −0.251406 | − | 0.493413i | 2.00778 | + | 1.99219i | 2.44343 | + | 0.387002i | 0.119564 | + | 4.34988i |
19.20 | −0.362021 | + | 1.36709i | −0.202587 | + | 2.57412i | −1.73788 | − | 0.989833i | −1.06398 | − | 0.255438i | −3.44571 | − | 1.20884i | −1.31451 | − | 2.57988i | 1.98234 | − | 2.01750i | −3.62197 | − | 0.573664i | 0.734389 | − | 1.36208i |
See next 80 embeddings (of 736 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
32.h | odd | 8 | 1 | inner |
352.bc | even | 40 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 352.2.bc.a | ✓ | 736 |
11.d | odd | 10 | 1 | inner | 352.2.bc.a | ✓ | 736 |
32.h | odd | 8 | 1 | inner | 352.2.bc.a | ✓ | 736 |
352.bc | even | 40 | 1 | inner | 352.2.bc.a | ✓ | 736 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
352.2.bc.a | ✓ | 736 | 1.a | even | 1 | 1 | trivial |
352.2.bc.a | ✓ | 736 | 11.d | odd | 10 | 1 | inner |
352.2.bc.a | ✓ | 736 | 32.h | odd | 8 | 1 | inner |
352.2.bc.a | ✓ | 736 | 352.bc | even | 40 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).