Newspace parameters
Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 288.v (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.29969157821\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{8})\) |
Twist minimal: | no (minimal twist has level 96) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.40823 | + | 0.129991i | 0 | 1.96620 | − | 0.366115i | 2.51374 | + | 1.04122i | 0 | 2.01027 | + | 2.01027i | −2.72127 | + | 0.771162i | 0 | −3.67526 | − | 1.13951i | ||||||
37.2 | −1.34827 | − | 0.426820i | 0 | 1.63565 | + | 1.15093i | 1.46213 | + | 0.605634i | 0 | −3.54889 | − | 3.54889i | −1.71405 | − | 2.24989i | 0 | −1.71285 | − | 1.44062i | ||||||
37.3 | −0.884039 | − | 1.10385i | 0 | −0.436951 | + | 1.95168i | −2.14986 | − | 0.890503i | 0 | −1.10001 | − | 1.10001i | 2.54064 | − | 1.24304i | 0 | 0.917585 | + | 3.16036i | ||||||
37.4 | −0.605567 | + | 1.27800i | 0 | −1.26658 | − | 1.54783i | −1.60930 | − | 0.666593i | 0 | −0.589445 | − | 0.589445i | 2.74513 | − | 0.681373i | 0 | 1.82644 | − | 1.65302i | ||||||
37.5 | −0.333592 | − | 1.37431i | 0 | −1.77743 | + | 0.916914i | 1.20409 | + | 0.498752i | 0 | 2.59422 | + | 2.59422i | 1.85306 | + | 2.13686i | 0 | 0.283762 | − | 1.82117i | ||||||
37.6 | 0.603367 | + | 1.27904i | 0 | −1.27190 | + | 1.54346i | −3.68816 | − | 1.52768i | 0 | −1.63704 | − | 1.63704i | −2.74158 | − | 0.695531i | 0 | −0.271341 | − | 5.63906i | ||||||
37.7 | 1.26685 | − | 0.628571i | 0 | 1.20980 | − | 1.59260i | 3.09318 | + | 1.28124i | 0 | −1.73503 | − | 1.73503i | 0.531562 | − | 2.77803i | 0 | 4.72394 | − | 0.321153i | ||||||
37.8 | 1.29526 | − | 0.567706i | 0 | 1.35542 | − | 1.47066i | −0.825824 | − | 0.342068i | 0 | 1.17750 | + | 1.17750i | 0.920723 | − | 2.67437i | 0 | −1.26385 | + | 0.0257578i | ||||||
109.1 | −1.40823 | − | 0.129991i | 0 | 1.96620 | + | 0.366115i | 2.51374 | − | 1.04122i | 0 | 2.01027 | − | 2.01027i | −2.72127 | − | 0.771162i | 0 | −3.67526 | + | 1.13951i | ||||||
109.2 | −1.34827 | + | 0.426820i | 0 | 1.63565 | − | 1.15093i | 1.46213 | − | 0.605634i | 0 | −3.54889 | + | 3.54889i | −1.71405 | + | 2.24989i | 0 | −1.71285 | + | 1.44062i | ||||||
109.3 | −0.884039 | + | 1.10385i | 0 | −0.436951 | − | 1.95168i | −2.14986 | + | 0.890503i | 0 | −1.10001 | + | 1.10001i | 2.54064 | + | 1.24304i | 0 | 0.917585 | − | 3.16036i | ||||||
109.4 | −0.605567 | − | 1.27800i | 0 | −1.26658 | + | 1.54783i | −1.60930 | + | 0.666593i | 0 | −0.589445 | + | 0.589445i | 2.74513 | + | 0.681373i | 0 | 1.82644 | + | 1.65302i | ||||||
109.5 | −0.333592 | + | 1.37431i | 0 | −1.77743 | − | 0.916914i | 1.20409 | − | 0.498752i | 0 | 2.59422 | − | 2.59422i | 1.85306 | − | 2.13686i | 0 | 0.283762 | + | 1.82117i | ||||||
109.6 | 0.603367 | − | 1.27904i | 0 | −1.27190 | − | 1.54346i | −3.68816 | + | 1.52768i | 0 | −1.63704 | + | 1.63704i | −2.74158 | + | 0.695531i | 0 | −0.271341 | + | 5.63906i | ||||||
109.7 | 1.26685 | + | 0.628571i | 0 | 1.20980 | + | 1.59260i | 3.09318 | − | 1.28124i | 0 | −1.73503 | + | 1.73503i | 0.531562 | + | 2.77803i | 0 | 4.72394 | + | 0.321153i | ||||||
109.8 | 1.29526 | + | 0.567706i | 0 | 1.35542 | + | 1.47066i | −0.825824 | + | 0.342068i | 0 | 1.17750 | − | 1.17750i | 0.920723 | + | 2.67437i | 0 | −1.26385 | − | 0.0257578i | ||||||
181.1 | −1.34416 | − | 0.439595i | 0 | 1.61351 | + | 1.18177i | −0.184062 | + | 0.444366i | 0 | −0.134531 | − | 0.134531i | −1.64931 | − | 2.29777i | 0 | 0.442749 | − | 0.516384i | ||||||
181.2 | −0.890433 | − | 1.09869i | 0 | −0.414259 | + | 1.95663i | 0.00259461 | − | 0.00626394i | 0 | −2.41880 | − | 2.41880i | 2.51860 | − | 1.28710i | 0 | −0.00919248 | + | 0.00272694i | ||||||
181.3 | −0.525864 | + | 1.31281i | 0 | −1.44693 | − | 1.38072i | 0.155637 | − | 0.375742i | 0 | 0.709092 | + | 0.709092i | 2.57351 | − | 1.17348i | 0 | 0.411433 | + | 0.401911i | ||||||
181.4 | 0.126318 | + | 1.40856i | 0 | −1.96809 | + | 0.355853i | −1.36206 | + | 3.28830i | 0 | −2.73097 | − | 2.73097i | −0.749846 | − | 2.72722i | 0 | −4.80383 | − | 1.50317i | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 288.2.v.d | 32 | |
3.b | odd | 2 | 1 | 96.2.n.a | ✓ | 32 | |
4.b | odd | 2 | 1 | 1152.2.v.c | 32 | ||
12.b | even | 2 | 1 | 384.2.n.a | 32 | ||
24.f | even | 2 | 1 | 768.2.n.b | 32 | ||
24.h | odd | 2 | 1 | 768.2.n.a | 32 | ||
32.g | even | 8 | 1 | inner | 288.2.v.d | 32 | |
32.h | odd | 8 | 1 | 1152.2.v.c | 32 | ||
96.o | even | 8 | 1 | 384.2.n.a | 32 | ||
96.o | even | 8 | 1 | 768.2.n.b | 32 | ||
96.p | odd | 8 | 1 | 96.2.n.a | ✓ | 32 | |
96.p | odd | 8 | 1 | 768.2.n.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.2.n.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
96.2.n.a | ✓ | 32 | 96.p | odd | 8 | 1 | |
288.2.v.d | 32 | 1.a | even | 1 | 1 | trivial | |
288.2.v.d | 32 | 32.g | even | 8 | 1 | inner | |
384.2.n.a | 32 | 12.b | even | 2 | 1 | ||
384.2.n.a | 32 | 96.o | even | 8 | 1 | ||
768.2.n.a | 32 | 24.h | odd | 2 | 1 | ||
768.2.n.a | 32 | 96.p | odd | 8 | 1 | ||
768.2.n.b | 32 | 24.f | even | 2 | 1 | ||
768.2.n.b | 32 | 96.o | even | 8 | 1 | ||
1152.2.v.c | 32 | 4.b | odd | 2 | 1 | ||
1152.2.v.c | 32 | 32.h | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{32} + 32 T_{5}^{29} - 384 T_{5}^{27} + 160 T_{5}^{26} - 1408 T_{5}^{25} + 34560 T_{5}^{24} - 29696 T_{5}^{23} + 1152 T_{5}^{22} + 692992 T_{5}^{21} - 298496 T_{5}^{20} - 6861824 T_{5}^{19} - 4331776 T_{5}^{18} + \cdots + 4096 \)
acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).