Newspace parameters
Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 624.x (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.98266508613\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | −1.41187 | + | 0.0813196i | −0.707107 | − | 0.707107i | 1.98677 | − | 0.229626i | 0.430937 | − | 0.430937i | 1.05585 | + | 0.940844i | 0.332860i | −2.78640 | + | 0.485767i | 1.00000i | −0.573385 | + | 0.643473i | ||||
157.2 | −1.37720 | − | 0.321439i | 0.707107 | + | 0.707107i | 1.79335 | + | 0.885370i | −0.616150 | + | 0.616150i | −0.746535 | − | 1.20112i | 2.54951i | −2.18521 | − | 1.79578i | 1.00000i | 1.04661 | − | 0.650506i | ||||
157.3 | −1.28294 | + | 0.595032i | 0.707107 | + | 0.707107i | 1.29187 | − | 1.52678i | −2.23094 | + | 2.23094i | −1.32793 | − | 0.486425i | 0.469035i | −0.748916 | + | 2.72748i | 1.00000i | 1.53469 | − | 4.18965i | ||||
157.4 | −1.14904 | − | 0.824448i | −0.707107 | − | 0.707107i | 0.640572 | + | 1.89464i | 1.44135 | − | 1.44135i | 0.229519 | + | 1.39546i | 2.43406i | 0.825993 | − | 2.70513i | 1.00000i | −2.84448 | + | 0.467847i | ||||
157.5 | −1.09850 | + | 0.890678i | −0.707107 | − | 0.707107i | 0.413387 | − | 1.95681i | 1.72155 | − | 1.72155i | 1.40656 | + | 0.146950i | − | 3.99681i | 1.28878 | + | 2.51774i | 1.00000i | −0.357769 | + | 3.42446i | |||
157.6 | −0.862181 | − | 1.12100i | 0.707107 | + | 0.707107i | −0.513289 | + | 1.93301i | 2.38716 | − | 2.38716i | 0.183014 | − | 1.40232i | − | 2.28372i | 2.60946 | − | 1.09121i | 1.00000i | −4.73418 | − | 0.617847i | |||
157.7 | −0.856472 | + | 1.12537i | −0.707107 | − | 0.707107i | −0.532911 | − | 1.92769i | −2.33677 | + | 2.33677i | 1.40137 | − | 0.190139i | − | 1.80314i | 2.62579 | + | 1.05130i | 1.00000i | −0.628351 | − | 4.63111i | |||
157.8 | −0.620977 | − | 1.27059i | −0.707107 | − | 0.707107i | −1.22878 | + | 1.57801i | −2.40184 | + | 2.40184i | −0.459343 | + | 1.33754i | − | 0.488115i | 2.76804 | + | 0.581357i | 1.00000i | 4.54323 | + | 1.56026i | |||
157.9 | −0.434856 | + | 1.34570i | 0.707107 | + | 0.707107i | −1.62180 | − | 1.17037i | −0.929776 | + | 0.929776i | −1.25904 | + | 0.644062i | − | 3.22395i | 2.28021 | − | 1.67351i | 1.00000i | −0.846879 | − | 1.65552i | |||
157.10 | −0.159779 | + | 1.40516i | −0.707107 | − | 0.707107i | −1.94894 | − | 0.449029i | −0.624160 | + | 0.624160i | 1.10658 | − | 0.880617i | − | 0.519233i | 0.942356 | − | 2.66683i | 1.00000i | −0.777317 | − | 0.976772i | |||
157.11 | −0.0467634 | + | 1.41344i | 0.707107 | + | 0.707107i | −1.99563 | − | 0.132195i | 2.42463 | − | 2.42463i | −1.03252 | + | 0.966386i | 3.56880i | 0.280172 | − | 2.81452i | 1.00000i | 3.31369 | + | 3.54046i | ||||
157.12 | 0.366592 | − | 1.36587i | 0.707107 | + | 0.707107i | −1.73122 | − | 1.00144i | −0.366436 | + | 0.366436i | 1.22504 | − | 0.706599i | 3.68431i | −2.00249 | + | 1.99751i | 1.00000i | 0.366172 | + | 0.634837i | ||||
157.13 | 0.528198 | + | 1.31187i | 0.707107 | + | 0.707107i | −1.44201 | + | 1.38586i | −1.06983 | + | 1.06983i | −0.554141 | + | 1.30113i | 0.473561i | −2.57973 | − | 1.15973i | 1.00000i | −1.96857 | − | 0.838400i | ||||
157.14 | 0.707327 | − | 1.22462i | −0.707107 | − | 0.707107i | −0.999377 | − | 1.73241i | −1.67831 | + | 1.67831i | −1.36609 | + | 0.365780i | 4.31119i | −2.82843 | − | 0.00152607i | 1.00000i | 0.868176 | + | 3.24241i | ||||
157.15 | 1.09789 | − | 0.891425i | 0.707107 | + | 0.707107i | 0.410724 | − | 1.95737i | 0.927746 | − | 0.927746i | 1.40666 | + | 0.145993i | − | 3.33268i | −1.29392 | − | 2.51511i | 1.00000i | 0.191547 | − | 1.84558i | |||
157.16 | 1.21541 | − | 0.723027i | −0.707107 | − | 0.707107i | 0.954464 | − | 1.75755i | 0.790012 | − | 0.790012i | −1.37069 | − | 0.348170i | − | 1.56424i | −0.110691 | − | 2.82626i | 1.00000i | 0.388992 | − | 1.53139i | |||
157.17 | 1.26768 | + | 0.626887i | −0.707107 | − | 0.707107i | 1.21402 | + | 1.58938i | 2.76434 | − | 2.76434i | −0.453109 | − | 1.33966i | 1.16682i | 0.542630 | + | 2.77589i | 1.00000i | 5.23722 | − | 1.77137i | ||||
157.18 | 1.32653 | + | 0.490219i | 0.707107 | + | 0.707107i | 1.51937 | + | 1.30058i | 1.28803 | − | 1.28803i | 0.591362 | + | 1.28464i | − | 2.72177i | 1.37792 | + | 2.47009i | 1.00000i | 2.34003 | − | 1.07720i | |||
157.19 | 1.39184 | + | 0.250586i | 0.707107 | + | 0.707107i | 1.87441 | + | 0.697549i | −0.400225 | + | 0.400225i | 0.806986 | + | 1.16137i | 2.81691i | 2.43408 | + | 1.44057i | 1.00000i | −0.657339 | + | 0.456757i | ||||
157.20 | 1.39911 | + | 0.206159i | −0.707107 | − | 0.707107i | 1.91500 | + | 0.576877i | −1.52131 | + | 1.52131i | −0.843541 | − | 1.13509i | 2.12662i | 2.56036 | + | 1.20191i | 1.00000i | −2.44210 | + | 1.81484i | ||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 624.2.x.a | ✓ | 40 |
4.b | odd | 2 | 1 | 2496.2.x.a | 40 | ||
16.e | even | 4 | 1 | inner | 624.2.x.a | ✓ | 40 |
16.f | odd | 4 | 1 | 2496.2.x.a | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
624.2.x.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
624.2.x.a | ✓ | 40 | 16.e | even | 4 | 1 | inner |
2496.2.x.a | 40 | 4.b | odd | 2 | 1 | ||
2496.2.x.a | 40 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{40} + 16 T_{5}^{37} + 492 T_{5}^{36} + 192 T_{5}^{35} + 128 T_{5}^{34} + 5456 T_{5}^{33} + 90236 T_{5}^{32} + 64032 T_{5}^{31} + 42752 T_{5}^{30} + 637568 T_{5}^{29} + 7521440 T_{5}^{28} + 7145216 T_{5}^{27} + \cdots + 879952896 \)
acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\).