Newspace parameters
Level: | \( N \) | \(=\) | \( 345 = 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 345.m (of order \(11\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.75483886973\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −1.71238 | + | 1.97619i | 0.841254 | − | 0.540641i | −0.688461 | − | 4.78835i | −0.415415 | − | 0.909632i | −0.372136 | + | 2.58826i | −3.22191 | + | 0.946039i | 6.24206 | + | 4.01153i | 0.415415 | − | 0.909632i | 2.50896 | + | 0.736696i |
16.2 | −1.32781 | + | 1.53237i | 0.841254 | − | 0.540641i | −0.300461 | − | 2.08975i | −0.415415 | − | 0.909632i | −0.288560 | + | 2.00698i | 3.59545 | − | 1.05572i | 0.189750 | + | 0.121945i | 0.415415 | − | 0.909632i | 1.94549 | + | 0.571247i |
16.3 | −0.615779 | + | 0.710647i | 0.841254 | − | 0.540641i | 0.158794 | + | 1.10444i | −0.415415 | − | 0.909632i | −0.133822 | + | 0.930750i | 0.864280 | − | 0.253775i | −2.46475 | − | 1.58400i | 0.415415 | − | 0.909632i | 0.902231 | + | 0.264919i |
16.4 | 0.707484 | − | 0.816480i | 0.841254 | − | 0.540641i | 0.118524 | + | 0.824351i | −0.415415 | − | 0.909632i | 0.153751 | − | 1.06936i | 2.39967 | − | 0.704607i | 2.57463 | + | 1.65461i | 0.415415 | − | 0.909632i | −1.03660 | − | 0.304372i |
16.5 | 1.45237 | − | 1.67613i | 0.841254 | − | 0.540641i | −0.415386 | − | 2.88907i | −0.415415 | − | 0.909632i | 0.315630 | − | 2.19526i | −1.14822 | + | 0.337148i | −1.71422 | − | 1.10166i | 0.415415 | − | 0.909632i | −2.12799 | − | 0.624835i |
31.1 | −0.387876 | + | 2.69773i | 0.415415 | + | 0.909632i | −5.20834 | − | 1.52931i | 0.654861 | + | 0.755750i | −2.61507 | + | 0.767855i | −4.11911 | − | 2.64719i | 3.88144 | − | 8.49917i | −0.654861 | + | 0.755750i | −2.29282 | + | 1.47350i |
31.2 | −0.290073 | + | 2.01750i | 0.415415 | + | 0.909632i | −2.06718 | − | 0.606979i | 0.654861 | + | 0.755750i | −1.95568 | + | 0.574241i | 3.11544 | + | 2.00217i | 0.130777 | − | 0.286362i | −0.654861 | + | 0.755750i | −1.71468 | + | 1.10196i |
31.3 | −0.124133 | + | 0.863362i | 0.415415 | + | 0.909632i | 1.18900 | + | 0.349122i | 0.654861 | + | 0.755750i | −0.836908 | + | 0.245738i | −2.78196 | − | 1.78785i | −1.17370 | + | 2.57004i | −0.654861 | + | 0.755750i | −0.733775 | + | 0.471568i |
31.4 | 0.0510170 | − | 0.354831i | 0.415415 | + | 0.909632i | 1.79568 | + | 0.527260i | 0.654861 | + | 0.755750i | 0.343959 | − | 0.100995i | 2.04891 | + | 1.31675i | 0.576535 | − | 1.26244i | −0.654861 | + | 0.755750i | 0.301573 | − | 0.193809i |
31.5 | 0.193334 | − | 1.34467i | 0.415415 | + | 0.909632i | 0.148229 | + | 0.0435239i | 0.654861 | + | 0.755750i | 1.30347 | − | 0.382733i | −0.958335 | − | 0.615885i | 1.21586 | − | 2.66237i | −0.654861 | + | 0.755750i | 1.14284 | − | 0.734459i |
121.1 | −0.831671 | + | 1.82111i | −0.959493 | − | 0.281733i | −1.31503 | − | 1.51762i | −0.841254 | + | 0.540641i | 1.31105 | − | 1.51303i | 0.450906 | − | 3.13612i | 0.0155624 | − | 0.00456953i | 0.841254 | + | 0.540641i | −0.284918 | − | 1.98165i |
121.2 | −0.168180 | + | 0.368263i | −0.959493 | − | 0.281733i | 1.20239 | + | 1.38763i | −0.841254 | + | 0.540641i | 0.265120 | − | 0.305964i | 0.179033 | − | 1.24520i | −1.49013 | + | 0.437542i | 0.841254 | + | 0.540641i | −0.0576160 | − | 0.400728i |
121.3 | 0.582364 | − | 1.27520i | −0.959493 | − | 0.281733i | 0.0227365 | + | 0.0262393i | −0.841254 | + | 0.540641i | −0.918039 | + | 1.05947i | −0.746335 | + | 5.19088i | 2.73690 | − | 0.803626i | 0.841254 | + | 0.540641i | 0.199509 | + | 1.38762i |
121.4 | 0.627141 | − | 1.37325i | −0.959493 | − | 0.281733i | −0.182780 | − | 0.210939i | −0.841254 | + | 0.540641i | −0.988626 | + | 1.14093i | 0.326039 | − | 2.26765i | 2.49274 | − | 0.731935i | 0.841254 | + | 0.540641i | 0.214849 | + | 1.49431i |
121.5 | 1.16525 | − | 2.55155i | −0.959493 | − | 0.281733i | −3.84287 | − | 4.43491i | −0.841254 | + | 0.540641i | −1.83691 | + | 2.11991i | 0.000825631 | − | 0.00574239i | −10.4110 | + | 3.05695i | 0.841254 | + | 0.540641i | 0.399198 | + | 2.77649i |
151.1 | −1.71238 | − | 1.97619i | 0.841254 | + | 0.540641i | −0.688461 | + | 4.78835i | −0.415415 | + | 0.909632i | −0.372136 | − | 2.58826i | −3.22191 | − | 0.946039i | 6.24206 | − | 4.01153i | 0.415415 | + | 0.909632i | 2.50896 | − | 0.736696i |
151.2 | −1.32781 | − | 1.53237i | 0.841254 | + | 0.540641i | −0.300461 | + | 2.08975i | −0.415415 | + | 0.909632i | −0.288560 | − | 2.00698i | 3.59545 | + | 1.05572i | 0.189750 | − | 0.121945i | 0.415415 | + | 0.909632i | 1.94549 | − | 0.571247i |
151.3 | −0.615779 | − | 0.710647i | 0.841254 | + | 0.540641i | 0.158794 | − | 1.10444i | −0.415415 | + | 0.909632i | −0.133822 | − | 0.930750i | 0.864280 | + | 0.253775i | −2.46475 | + | 1.58400i | 0.415415 | + | 0.909632i | 0.902231 | − | 0.264919i |
151.4 | 0.707484 | + | 0.816480i | 0.841254 | + | 0.540641i | 0.118524 | − | 0.824351i | −0.415415 | + | 0.909632i | 0.153751 | + | 1.06936i | 2.39967 | + | 0.704607i | 2.57463 | − | 1.65461i | 0.415415 | + | 0.909632i | −1.03660 | + | 0.304372i |
151.5 | 1.45237 | + | 1.67613i | 0.841254 | + | 0.540641i | −0.415386 | + | 2.88907i | −0.415415 | + | 0.909632i | 0.315630 | + | 2.19526i | −1.14822 | − | 0.337148i | −1.71422 | + | 1.10166i | 0.415415 | + | 0.909632i | −2.12799 | + | 0.624835i |
See all 50 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 345.2.m.c | ✓ | 50 |
23.c | even | 11 | 1 | inner | 345.2.m.c | ✓ | 50 |
23.c | even | 11 | 1 | 7935.2.a.bv | 25 | ||
23.d | odd | 22 | 1 | 7935.2.a.bw | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
345.2.m.c | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
345.2.m.c | ✓ | 50 | 23.c | even | 11 | 1 | inner |
7935.2.a.bv | 25 | 23.c | even | 11 | 1 | ||
7935.2.a.bw | 25 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{50} + 7 T_{2}^{48} + 54 T_{2}^{46} + 11 T_{2}^{45} + 325 T_{2}^{44} + 77 T_{2}^{43} + 2314 T_{2}^{42} + 3201 T_{2}^{41} + 14050 T_{2}^{40} - 20548 T_{2}^{39} + 86116 T_{2}^{38} - 25168 T_{2}^{37} + 686053 T_{2}^{36} + \cdots + 64009 \)
acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\).