Newspace parameters
Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 688.bg (of order \(21\), degree \(12\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.49370765906\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{21})\) |
Twist minimal: | no (minimal twist has level 86) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | 0.200549 | − | 0.0618612i | 0 | 0.961158 | − | 2.44899i | 0 | −1.57428 | + | 2.72674i | 0 | −2.44232 | + | 1.66515i | 0 | ||||||||||
17.2 | 0 | 2.92230 | − | 0.901411i | 0 | −0.758409 | + | 1.93239i | 0 | 1.24220 | − | 2.15155i | 0 | 5.24859 | − | 3.57843i | 0 | ||||||||||
81.1 | 0 | 0.200549 | + | 0.0618612i | 0 | 0.961158 | + | 2.44899i | 0 | −1.57428 | − | 2.72674i | 0 | −2.44232 | − | 1.66515i | 0 | ||||||||||
81.2 | 0 | 2.92230 | + | 0.901411i | 0 | −0.758409 | − | 1.93239i | 0 | 1.24220 | + | 2.15155i | 0 | 5.24859 | + | 3.57843i | 0 | ||||||||||
225.1 | 0 | −1.67079 | − | 1.55027i | 0 | 0.0705384 | − | 0.0106319i | 0 | −0.174372 | − | 0.302021i | 0 | 0.164022 | + | 2.18872i | 0 | ||||||||||
225.2 | 0 | 1.75084 | + | 1.62455i | 0 | −0.619298 | + | 0.0933442i | 0 | 1.53091 | + | 2.65162i | 0 | 0.202115 | + | 2.69704i | 0 | ||||||||||
273.1 | 0 | −0.145090 | + | 0.369683i | 0 | −0.0922707 | − | 1.23127i | 0 | 0.695839 | − | 1.20523i | 0 | 2.08354 | + | 1.93324i | 0 | ||||||||||
273.2 | 0 | 1.03020 | − | 2.62491i | 0 | 0.260188 | + | 3.47196i | 0 | −1.96215 | + | 3.39854i | 0 | −3.62970 | − | 3.36787i | 0 | ||||||||||
289.1 | 0 | −1.83026 | + | 1.24785i | 0 | −0.518590 | − | 0.481181i | 0 | 0.925176 | + | 1.60245i | 0 | 0.696697 | − | 1.77516i | 0 | ||||||||||
289.2 | 0 | 0.676467 | − | 0.461207i | 0 | 1.10645 | + | 1.02664i | 0 | −1.97394 | − | 3.41896i | 0 | −0.851128 | + | 2.16864i | 0 | ||||||||||
353.1 | 0 | −1.37320 | − | 0.206976i | 0 | −1.49011 | − | 1.01594i | 0 | −0.0705203 | + | 0.122145i | 0 | −1.02388 | − | 0.315826i | 0 | ||||||||||
353.2 | 0 | 1.65565 | + | 0.249549i | 0 | 3.34665 | + | 2.28171i | 0 | 0.158382 | − | 0.274326i | 0 | −0.187822 | − | 0.0579354i | 0 | ||||||||||
369.1 | 0 | −1.83026 | − | 1.24785i | 0 | −0.518590 | + | 0.481181i | 0 | 0.925176 | − | 1.60245i | 0 | 0.696697 | + | 1.77516i | 0 | ||||||||||
369.2 | 0 | 0.676467 | + | 0.461207i | 0 | 1.10645 | − | 1.02664i | 0 | −1.97394 | + | 3.41896i | 0 | −0.851128 | − | 2.16864i | 0 | ||||||||||
401.1 | 0 | −0.215103 | − | 2.87035i | 0 | −3.08306 | + | 0.950998i | 0 | −1.19592 | + | 2.07139i | 0 | −5.22617 | + | 0.787718i | 0 | ||||||||||
401.2 | 0 | −0.00157347 | − | 0.0209965i | 0 | 2.31675 | − | 0.714623i | 0 | 0.898666 | − | 1.55654i | 0 | 2.96605 | − | 0.447061i | 0 | ||||||||||
497.1 | 0 | −1.37320 | + | 0.206976i | 0 | −1.49011 | + | 1.01594i | 0 | −0.0705203 | − | 0.122145i | 0 | −1.02388 | + | 0.315826i | 0 | ||||||||||
497.2 | 0 | 1.65565 | − | 0.249549i | 0 | 3.34665 | − | 2.28171i | 0 | 0.158382 | + | 0.274326i | 0 | −0.187822 | + | 0.0579354i | 0 | ||||||||||
513.1 | 0 | −0.215103 | + | 2.87035i | 0 | −3.08306 | − | 0.950998i | 0 | −1.19592 | − | 2.07139i | 0 | −5.22617 | − | 0.787718i | 0 | ||||||||||
513.2 | 0 | −0.00157347 | + | 0.0209965i | 0 | 2.31675 | + | 0.714623i | 0 | 0.898666 | + | 1.55654i | 0 | 2.96605 | + | 0.447061i | 0 | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.g | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 688.2.bg.b | 24 | |
4.b | odd | 2 | 1 | 86.2.g.b | ✓ | 24 | |
12.b | even | 2 | 1 | 774.2.z.f | 24 | ||
43.g | even | 21 | 1 | inner | 688.2.bg.b | 24 | |
172.o | odd | 42 | 1 | 86.2.g.b | ✓ | 24 | |
172.o | odd | 42 | 1 | 3698.2.a.t | 12 | ||
172.p | even | 42 | 1 | 3698.2.a.s | 12 | ||
516.bb | even | 42 | 1 | 774.2.z.f | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
86.2.g.b | ✓ | 24 | 4.b | odd | 2 | 1 | |
86.2.g.b | ✓ | 24 | 172.o | odd | 42 | 1 | |
688.2.bg.b | 24 | 1.a | even | 1 | 1 | trivial | |
688.2.bg.b | 24 | 43.g | even | 21 | 1 | inner | |
774.2.z.f | 24 | 12.b | even | 2 | 1 | ||
774.2.z.f | 24 | 516.bb | even | 42 | 1 | ||
3698.2.a.s | 12 | 172.p | even | 42 | 1 | ||
3698.2.a.t | 12 | 172.o | odd | 42 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{24} - 6 T_{3}^{23} + 17 T_{3}^{22} - 34 T_{3}^{21} + 43 T_{3}^{20} - 60 T_{3}^{19} + 422 T_{3}^{18} - 1532 T_{3}^{17} + 4016 T_{3}^{16} - 2618 T_{3}^{15} - 11961 T_{3}^{14} + 15786 T_{3}^{13} + 94040 T_{3}^{12} - 179484 T_{3}^{11} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\).