Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(70,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([2, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.70");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.h (of order \(5\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(236\) |
Relative dimension: | \(59\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
70.1 | −0.869346 | − | 2.67557i | 2.15515 | − | 1.56581i | −4.78489 | + | 3.47642i | 0.984705 | + | 0.715430i | −6.06301 | − | 4.40504i | 3.79531 | + | 2.75745i | 8.90919 | + | 6.47291i | 1.26587 | − | 3.89595i | 1.05813 | − | 3.25660i |
70.2 | −0.853051 | − | 2.62542i | −0.947545 | + | 0.688432i | −4.54710 | + | 3.30366i | −3.22100 | − | 2.34019i | 2.61573 | + | 1.90044i | −2.13293 | − | 1.54966i | 8.08578 | + | 5.87467i | −0.503148 | + | 1.54853i | −3.39631 | + | 10.4528i |
70.3 | −0.845205 | − | 2.60127i | 0.160163 | − | 0.116365i | −4.43422 | + | 3.22165i | 1.41160 | + | 1.02559i | −0.438068 | − | 0.318275i | −2.29738 | − | 1.66914i | 7.70268 | + | 5.59632i | −0.914940 | + | 2.81589i | 1.47474 | − | 4.53879i |
70.4 | −0.791807 | − | 2.43693i | −2.09599 | + | 1.52283i | −3.69364 | + | 2.68359i | 3.32471 | + | 2.41555i | 5.37064 | + | 3.90200i | 0.868100 | + | 0.630711i | 5.31841 | + | 3.86405i | 1.14713 | − | 3.53049i | 3.25399 | − | 10.0147i |
70.5 | −0.748831 | − | 2.30466i | −1.16304 | + | 0.844998i | −3.13269 | + | 2.27603i | 0.674317 | + | 0.489920i | 2.81836 | + | 2.04766i | 1.46363 | + | 1.06339i | 3.67042 | + | 2.66672i | −0.288411 | + | 0.887638i | 0.624151 | − | 1.92094i |
70.6 | −0.723495 | − | 2.22669i | 0.360320 | − | 0.261787i | −2.81666 | + | 2.04643i | −1.15632 | − | 0.840113i | −0.843609 | − | 0.612918i | 1.47092 | + | 1.06869i | 2.80633 | + | 2.03892i | −0.865753 | + | 2.66452i | −1.03408 | + | 3.18258i |
70.7 | −0.711166 | − | 2.18874i | 1.70599 | − | 1.23947i | −2.66681 | + | 1.93755i | 0.461971 | + | 0.335641i | −3.92613 | − | 2.85250i | −1.71777 | − | 1.24803i | 2.41364 | + | 1.75361i | 0.447050 | − | 1.37588i | 0.406095 | − | 1.24983i |
70.8 | −0.694635 | − | 2.13787i | 0.233103 | − | 0.169359i | −2.46992 | + | 1.79450i | −2.36051 | − | 1.71501i | −0.523988 | − | 0.380700i | 3.46385 | + | 2.51663i | 1.91494 | + | 1.39129i | −0.901397 | + | 2.77421i | −2.02678 | + | 6.23777i |
70.9 | −0.667832 | − | 2.05538i | −2.48560 | + | 1.80590i | −2.16054 | + | 1.56972i | −0.411576 | − | 0.299027i | 5.37176 | + | 3.90281i | −3.21015 | − | 2.33231i | 1.17243 | + | 0.851823i | 1.98991 | − | 6.12431i | −0.339750 | + | 1.04564i |
70.10 | −0.665082 | − | 2.04691i | 0.235143 | − | 0.170841i | −2.12948 | + | 1.54716i | 2.09182 | + | 1.51980i | −0.506086 | − | 0.367693i | −0.996170 | − | 0.723760i | 1.10077 | + | 0.799754i | −0.900946 | + | 2.77283i | 1.71966 | − | 5.29257i |
70.11 | −0.625954 | − | 1.92649i | −2.12502 | + | 1.54392i | −1.70151 | + | 1.23622i | −1.76588 | − | 1.28299i | 4.30450 | + | 3.12740i | 1.77767 | + | 1.29155i | 0.169083 | + | 0.122846i | 1.20497 | − | 3.70852i | −1.36630 | + | 4.20504i |
70.12 | −0.588103 | − | 1.81000i | 2.66323 | − | 1.93495i | −1.31219 | + | 0.953359i | 2.95435 | + | 2.14646i | −5.06851 | − | 3.68249i | −2.27051 | − | 1.64962i | −0.582072 | − | 0.422900i | 2.42172 | − | 7.45327i | 2.14762 | − | 6.60970i |
70.13 | −0.549055 | − | 1.68982i | −1.21899 | + | 0.885650i | −0.935991 | + | 0.680037i | 0.281486 | + | 0.204512i | 2.16588 | + | 1.57360i | 0.593936 | + | 0.431520i | −1.21184 | − | 0.880454i | −0.225485 | + | 0.693970i | 0.191036 | − | 0.587949i |
70.14 | −0.536577 | − | 1.65141i | 2.11318 | − | 1.53531i | −0.821217 | + | 0.596649i | 1.72377 | + | 1.25239i | −3.66932 | − | 2.66592i | 2.10792 | + | 1.53149i | −1.38359 | − | 1.00524i | 1.18128 | − | 3.63561i | 1.14328 | − | 3.51865i |
70.15 | −0.536210 | − | 1.65028i | −0.389829 | + | 0.283228i | −0.817882 | + | 0.594226i | −2.94146 | − | 2.13710i | 0.676436 | + | 0.491460i | −3.08149 | − | 2.23883i | −1.38843 | − | 1.00876i | −0.855302 | + | 2.63235i | −1.94958 | + | 6.00018i |
70.16 | −0.447883 | − | 1.37844i | 1.22213 | − | 0.887928i | −0.0814705 | + | 0.0591918i | −1.35535 | − | 0.984717i | −1.77133 | − | 1.28695i | −0.969669 | − | 0.704506i | −2.22706 | − | 1.61806i | −0.221870 | + | 0.682846i | −0.750339 | + | 2.30931i |
70.17 | −0.428672 | − | 1.31932i | −0.369969 | + | 0.268798i | 0.0612004 | − | 0.0444647i | 2.08561 | + | 1.51529i | 0.513225 | + | 0.372880i | −3.54951 | − | 2.57887i | −2.32945 | − | 1.69245i | −0.862426 | + | 2.65428i | 1.10510 | − | 3.40114i |
70.18 | −0.427484 | − | 1.31566i | 1.72167 | − | 1.25087i | 0.0698112 | − | 0.0507208i | −0.553752 | − | 0.402324i | −2.38171 | − | 1.73041i | 2.59238 | + | 1.88347i | −2.33491 | − | 1.69641i | 0.472432 | − | 1.45400i | −0.292602 | + | 0.900537i |
70.19 | −0.418710 | − | 1.28866i | −1.45739 | + | 1.05885i | 0.132718 | − | 0.0964256i | 2.39305 | + | 1.73865i | 1.97472 | + | 1.43472i | 3.32854 | + | 2.41832i | −2.37222 | − | 1.72352i | 0.0757533 | − | 0.233145i | 1.23853 | − | 3.81180i |
70.20 | −0.417408 | − | 1.28465i | 2.67326 | − | 1.94224i | 0.141939 | − | 0.103125i | −3.12493 | − | 2.27039i | −3.61094 | − | 2.62350i | 1.24958 | + | 0.907872i | −2.37730 | − | 1.72721i | 2.44699 | − | 7.53107i | −1.61229 | + | 4.96212i |
See next 80 embeddings (of 236 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
671.h | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.h.b | ✓ | 236 |
11.c | even | 5 | 1 | 671.2.m.b | yes | 236 | |
61.e | even | 5 | 1 | 671.2.m.b | yes | 236 | |
671.h | even | 5 | 1 | inner | 671.2.h.b | ✓ | 236 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.h.b | ✓ | 236 | 1.a | even | 1 | 1 | trivial |
671.2.h.b | ✓ | 236 | 671.h | even | 5 | 1 | inner |
671.2.m.b | yes | 236 | 11.c | even | 5 | 1 | |
671.2.m.b | yes | 236 | 61.e | even | 5 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{236} - 2 T_{2}^{235} + 92 T_{2}^{234} - 188 T_{2}^{233} + 4455 T_{2}^{232} - 9182 T_{2}^{231} + \cdots + 102191881 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\).