Newspace parameters
Level: | \( N \) | \(=\) | \( 2214 = 2 \cdot 3^{3} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2214.i (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(17.6788790075\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 738) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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.
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
901.1 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.84171 | + | 3.18994i | 0 | −3.70331 | + | 2.13810i | −1.00000 | 0 | −3.68343 | ||||||||||
901.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.84171 | + | 3.18994i | 0 | 3.70331 | − | 2.13810i | −1.00000 | 0 | −3.68343 | ||||||||||
901.3 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.55925 | + | 2.70071i | 0 | 0.0882889 | − | 0.0509736i | −1.00000 | 0 | −3.11851 | ||||||||||
901.4 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.55925 | + | 2.70071i | 0 | −0.0882889 | + | 0.0509736i | −1.00000 | 0 | −3.11851 | ||||||||||
901.5 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.13494 | + | 1.96578i | 0 | −4.05054 | + | 2.33858i | −1.00000 | 0 | −2.26989 | ||||||||||
901.6 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.13494 | + | 1.96578i | 0 | 4.05054 | − | 2.33858i | −1.00000 | 0 | −2.26989 | ||||||||||
901.7 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.714167 | + | 1.23697i | 0 | −2.06322 | + | 1.19120i | −1.00000 | 0 | −1.42833 | ||||||||||
901.8 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.714167 | + | 1.23697i | 0 | 2.06322 | − | 1.19120i | −1.00000 | 0 | −1.42833 | ||||||||||
901.9 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.290704 | + | 0.503513i | 0 | −1.28705 | + | 0.743077i | −1.00000 | 0 | −0.581407 | ||||||||||
901.10 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.290704 | + | 0.503513i | 0 | 1.28705 | − | 0.743077i | −1.00000 | 0 | −0.581407 | ||||||||||
901.11 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.0761831 | − | 0.131953i | 0 | 1.94692 | − | 1.12405i | −1.00000 | 0 | 0.152366 | ||||||||||
901.12 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.0761831 | − | 0.131953i | 0 | −1.94692 | + | 1.12405i | −1.00000 | 0 | 0.152366 | ||||||||||
901.13 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.18467 | − | 2.05191i | 0 | −4.22182 | + | 2.43747i | −1.00000 | 0 | 2.36934 | ||||||||||
901.14 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.18467 | − | 2.05191i | 0 | 4.22182 | − | 2.43747i | −1.00000 | 0 | 2.36934 | ||||||||||
901.15 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.58544 | − | 2.74607i | 0 | 1.86289 | − | 1.07554i | −1.00000 | 0 | 3.17089 | ||||||||||
901.16 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.58544 | − | 2.74607i | 0 | −1.86289 | + | 1.07554i | −1.00000 | 0 | 3.17089 | ||||||||||
901.17 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.69449 | − | 2.93494i | 0 | 0.610625 | − | 0.352544i | −1.00000 | 0 | 3.38898 | ||||||||||
901.18 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.69449 | − | 2.93494i | 0 | −0.610625 | + | 0.352544i | −1.00000 | 0 | 3.38898 | ||||||||||
1639.1 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.84171 | − | 3.18994i | 0 | −3.70331 | − | 2.13810i | −1.00000 | 0 | −3.68343 | ||||||||||
1639.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.84171 | − | 3.18994i | 0 | 3.70331 | + | 2.13810i | −1.00000 | 0 | −3.68343 | ||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
41.b | even | 2 | 1 | inner |
369.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2214.2.i.b | 36 | |
3.b | odd | 2 | 1 | 738.2.i.b | ✓ | 36 | |
9.c | even | 3 | 1 | inner | 2214.2.i.b | 36 | |
9.d | odd | 6 | 1 | 738.2.i.b | ✓ | 36 | |
41.b | even | 2 | 1 | inner | 2214.2.i.b | 36 | |
123.b | odd | 2 | 1 | 738.2.i.b | ✓ | 36 | |
369.i | even | 6 | 1 | inner | 2214.2.i.b | 36 | |
369.k | odd | 6 | 1 | 738.2.i.b | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
738.2.i.b | ✓ | 36 | 3.b | odd | 2 | 1 | |
738.2.i.b | ✓ | 36 | 9.d | odd | 6 | 1 | |
738.2.i.b | ✓ | 36 | 123.b | odd | 2 | 1 | |
738.2.i.b | ✓ | 36 | 369.k | odd | 6 | 1 | |
2214.2.i.b | 36 | 1.a | even | 1 | 1 | trivial | |
2214.2.i.b | 36 | 9.c | even | 3 | 1 | inner | |
2214.2.i.b | 36 | 41.b | even | 2 | 1 | inner | |
2214.2.i.b | 36 | 369.i | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{18} + 2 T_{5}^{17} + 31 T_{5}^{16} + 52 T_{5}^{15} + 608 T_{5}^{14} + 979 T_{5}^{13} + \cdots + 7056 \)
acting on \(S_{2}^{\mathrm{new}}(2214, [\chi])\).