Newspace parameters
Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 209.j (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.66887340224\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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.
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −2.38547 | − | 0.868238i | 0.291201 | + | 1.65148i | 3.40452 | + | 2.85673i | 2.29004 | − | 1.92157i | 0.739230 | − | 4.19238i | −0.775424 | + | 1.34307i | −3.10248 | − | 5.37365i | 0.176489 | − | 0.0642366i | −7.13118 | + | 2.59554i |
23.2 | −1.96733 | − | 0.716050i | 0.502296 | + | 2.84866i | 1.82557 | + | 1.53184i | −2.76559 | + | 2.32060i | 1.05160 | − | 5.96393i | −0.132423 | + | 0.229364i | −0.401042 | − | 0.694626i | −5.04350 | + | 1.83568i | 7.10249 | − | 2.58510i |
23.3 | −1.69844 | − | 0.618183i | −0.255473 | − | 1.44886i | 0.970468 | + | 0.814320i | 1.59794 | − | 1.34083i | −0.461753 | + | 2.61873i | 1.29928 | − | 2.25041i | 0.662558 | + | 1.14758i | 0.785155 | − | 0.285773i | −3.54289 | + | 1.28951i |
23.4 | −0.0793104 | − | 0.0288666i | 0.115804 | + | 0.656759i | −1.52663 | − | 1.28100i | 1.81015 | − | 1.51890i | 0.00977392 | − | 0.0554307i | −0.266530 | + | 0.461644i | 0.168500 | + | 0.291851i | 2.40116 | − | 0.873949i | −0.187409 | + | 0.0682115i |
23.5 | 0.248801 | + | 0.0905562i | −0.369765 | − | 2.09704i | −1.47839 | − | 1.24051i | −2.83976 | + | 2.38284i | 0.0979021 | − | 0.555231i | −1.58658 | + | 2.74803i | −0.520257 | − | 0.901111i | −1.44178 | + | 0.524764i | −0.922316 | + | 0.335696i |
23.6 | 1.68033 | + | 0.611589i | 0.505415 | + | 2.86635i | 0.917370 | + | 0.769764i | 0.525977 | − | 0.441347i | −0.903766 | + | 5.12551i | 1.17848 | − | 2.04120i | −0.717465 | − | 1.24269i | −5.14143 | + | 1.87133i | 1.15374 | − | 0.419925i |
23.7 | 1.99568 | + | 0.726369i | 0.0787632 | + | 0.446688i | 1.92305 | + | 1.61363i | −0.260317 | + | 0.218432i | −0.167274 | + | 0.948659i | −1.83015 | + | 3.16990i | 0.541945 | + | 0.938677i | 2.62575 | − | 0.955695i | −0.678172 | + | 0.246834i |
100.1 | −2.38547 | + | 0.868238i | 0.291201 | − | 1.65148i | 3.40452 | − | 2.85673i | 2.29004 | + | 1.92157i | 0.739230 | + | 4.19238i | −0.775424 | − | 1.34307i | −3.10248 | + | 5.37365i | 0.176489 | + | 0.0642366i | −7.13118 | − | 2.59554i |
100.2 | −1.96733 | + | 0.716050i | 0.502296 | − | 2.84866i | 1.82557 | − | 1.53184i | −2.76559 | − | 2.32060i | 1.05160 | + | 5.96393i | −0.132423 | − | 0.229364i | −0.401042 | + | 0.694626i | −5.04350 | − | 1.83568i | 7.10249 | + | 2.58510i |
100.3 | −1.69844 | + | 0.618183i | −0.255473 | + | 1.44886i | 0.970468 | − | 0.814320i | 1.59794 | + | 1.34083i | −0.461753 | − | 2.61873i | 1.29928 | + | 2.25041i | 0.662558 | − | 1.14758i | 0.785155 | + | 0.285773i | −3.54289 | − | 1.28951i |
100.4 | −0.0793104 | + | 0.0288666i | 0.115804 | − | 0.656759i | −1.52663 | + | 1.28100i | 1.81015 | + | 1.51890i | 0.00977392 | + | 0.0554307i | −0.266530 | − | 0.461644i | 0.168500 | − | 0.291851i | 2.40116 | + | 0.873949i | −0.187409 | − | 0.0682115i |
100.5 | 0.248801 | − | 0.0905562i | −0.369765 | + | 2.09704i | −1.47839 | + | 1.24051i | −2.83976 | − | 2.38284i | 0.0979021 | + | 0.555231i | −1.58658 | − | 2.74803i | −0.520257 | + | 0.901111i | −1.44178 | − | 0.524764i | −0.922316 | − | 0.335696i |
100.6 | 1.68033 | − | 0.611589i | 0.505415 | − | 2.86635i | 0.917370 | − | 0.769764i | 0.525977 | + | 0.441347i | −0.903766 | − | 5.12551i | 1.17848 | + | 2.04120i | −0.717465 | + | 1.24269i | −5.14143 | − | 1.87133i | 1.15374 | + | 0.419925i |
100.7 | 1.99568 | − | 0.726369i | 0.0787632 | − | 0.446688i | 1.92305 | − | 1.61363i | −0.260317 | − | 0.218432i | −0.167274 | − | 0.948659i | −1.83015 | − | 3.16990i | 0.541945 | − | 0.938677i | 2.62575 | + | 0.955695i | −0.678172 | − | 0.246834i |
111.1 | −1.96241 | − | 1.64666i | −2.67281 | + | 0.972823i | 0.792280 | + | 4.49324i | 0.366121 | − | 2.07637i | 6.84707 | + | 2.49213i | 1.57957 | + | 2.73590i | 3.28232 | − | 5.68514i | 3.89939 | − | 3.27198i | −4.13756 | + | 3.47183i |
111.2 | −1.28771 | − | 1.08051i | −1.02863 | + | 0.374390i | 0.143380 | + | 0.813151i | −0.0882338 | + | 0.500399i | 1.72910 | + | 0.629342i | 0.0395283 | + | 0.0684651i | −0.986992 | + | 1.70952i | −1.38023 | + | 1.15815i | 0.654306 | − | 0.549028i |
111.3 | −0.368271 | − | 0.309016i | 2.42559 | − | 0.882842i | −0.307164 | − | 1.74201i | 0.229238 | − | 1.30007i | −1.16609 | − | 0.424421i | 1.88709 | + | 3.26854i | −0.905934 | + | 1.56912i | 2.80593 | − | 2.35446i | −0.486166 | + | 0.407942i |
111.4 | −0.0229450 | − | 0.0192531i | −1.38899 | + | 0.505551i | −0.347141 | − | 1.96873i | −0.508658 | + | 2.88474i | 0.0416038 | + | 0.0151425i | −2.21460 | − | 3.83580i | −0.0598916 | + | 0.103735i | −0.624422 | + | 0.523953i | 0.0672114 | − | 0.0563971i |
111.5 | 0.938345 | + | 0.787365i | −1.27599 | + | 0.464422i | −0.0867486 | − | 0.491976i | −0.392586 | + | 2.22647i | −1.56299 | − | 0.568881i | 2.23878 | + | 3.87769i | 1.53089 | − | 2.65157i | −0.885675 | + | 0.743170i | −2.12142 | + | 1.78009i |
111.6 | 0.948012 | + | 0.795477i | −2.09808 | + | 0.763637i | −0.0813525 | − | 0.461373i | 0.604339 | − | 3.42738i | −2.59646 | − | 0.945033i | −0.971037 | − | 1.68188i | 1.52743 | − | 2.64559i | 1.52065 | − | 1.27598i | 3.29932 | − | 2.76846i |
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 209.2.j.b | ✓ | 42 |
19.e | even | 9 | 1 | inner | 209.2.j.b | ✓ | 42 |
19.e | even | 9 | 1 | 3971.2.a.t | 21 | ||
19.f | odd | 18 | 1 | 3971.2.a.s | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
209.2.j.b | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
209.2.j.b | ✓ | 42 | 19.e | even | 9 | 1 | inner |
3971.2.a.s | 21 | 19.f | odd | 18 | 1 | ||
3971.2.a.t | 21 | 19.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{42} + 3 T_{2}^{41} + 3 T_{2}^{40} + 13 T_{2}^{39} + 33 T_{2}^{38} - 54 T_{2}^{37} + 98 T_{2}^{36} + 960 T_{2}^{35} + 435 T_{2}^{34} - 1098 T_{2}^{33} + 4728 T_{2}^{32} - 7194 T_{2}^{31} - 11964 T_{2}^{30} + 101247 T_{2}^{29} + 174462 T_{2}^{28} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\).