Newspace parameters
Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 288.w (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: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
35.1 | −1.39224 | − | 0.248345i | 0 | 1.87665 | + | 0.691511i | 4.01952 | − | 1.66494i | 0 | 2.72280 | + | 2.72280i | −2.44101 | − | 1.42881i | 0 | −6.00960 | + | 1.31976i | ||||||
35.2 | −1.33776 | − | 0.458699i | 0 | 1.57919 | + | 1.22726i | −2.70076 | + | 1.11869i | 0 | −1.06647 | − | 1.06647i | −1.54963 | − | 2.36614i | 0 | 4.12611 | − | 0.257702i | ||||||
35.3 | −1.19734 | + | 0.752583i | 0 | 0.867238 | − | 1.80219i | −0.0913223 | + | 0.0378270i | 0 | −3.05457 | − | 3.05457i | 0.317923 | + | 2.81050i | 0 | 0.0808758 | − | 0.114019i | ||||||
35.4 | −0.345448 | − | 1.37137i | 0 | −1.76133 | + | 0.947475i | −1.64859 | + | 0.682869i | 0 | 2.51270 | + | 2.51270i | 1.90779 | + | 2.08814i | 0 | 1.50597 | + | 2.02494i | ||||||
35.5 | 0.349793 | + | 1.37027i | 0 | −1.75529 | + | 0.958624i | −2.97412 | + | 1.23192i | 0 | 0.237717 | + | 0.237717i | −1.92756 | − | 2.06990i | 0 | −2.72839 | − | 3.64443i | ||||||
35.6 | 0.430512 | − | 1.34709i | 0 | −1.62932 | − | 1.15988i | 1.78959 | − | 0.741273i | 0 | −2.33709 | − | 2.33709i | −2.26391 | + | 1.69550i | 0 | −0.228123 | − | 2.72987i | ||||||
35.7 | 1.16775 | + | 0.797726i | 0 | 0.727265 | + | 1.86308i | 1.65625 | − | 0.686041i | 0 | −0.456585 | − | 0.456585i | −0.636970 | + | 2.75577i | 0 | 2.48135 | + | 0.520112i | ||||||
35.8 | 1.32473 | − | 0.495070i | 0 | 1.50981 | − | 1.31167i | −0.0505677 | + | 0.0209458i | 0 | 1.44150 | + | 1.44150i | 1.35072 | − | 2.48506i | 0 | −0.0566189 | + | 0.0527821i | ||||||
107.1 | −1.39224 | + | 0.248345i | 0 | 1.87665 | − | 0.691511i | 4.01952 | + | 1.66494i | 0 | 2.72280 | − | 2.72280i | −2.44101 | + | 1.42881i | 0 | −6.00960 | − | 1.31976i | ||||||
107.2 | −1.33776 | + | 0.458699i | 0 | 1.57919 | − | 1.22726i | −2.70076 | − | 1.11869i | 0 | −1.06647 | + | 1.06647i | −1.54963 | + | 2.36614i | 0 | 4.12611 | + | 0.257702i | ||||||
107.3 | −1.19734 | − | 0.752583i | 0 | 0.867238 | + | 1.80219i | −0.0913223 | − | 0.0378270i | 0 | −3.05457 | + | 3.05457i | 0.317923 | − | 2.81050i | 0 | 0.0808758 | + | 0.114019i | ||||||
107.4 | −0.345448 | + | 1.37137i | 0 | −1.76133 | − | 0.947475i | −1.64859 | − | 0.682869i | 0 | 2.51270 | − | 2.51270i | 1.90779 | − | 2.08814i | 0 | 1.50597 | − | 2.02494i | ||||||
107.5 | 0.349793 | − | 1.37027i | 0 | −1.75529 | − | 0.958624i | −2.97412 | − | 1.23192i | 0 | 0.237717 | − | 0.237717i | −1.92756 | + | 2.06990i | 0 | −2.72839 | + | 3.64443i | ||||||
107.6 | 0.430512 | + | 1.34709i | 0 | −1.62932 | + | 1.15988i | 1.78959 | + | 0.741273i | 0 | −2.33709 | + | 2.33709i | −2.26391 | − | 1.69550i | 0 | −0.228123 | + | 2.72987i | ||||||
107.7 | 1.16775 | − | 0.797726i | 0 | 0.727265 | − | 1.86308i | 1.65625 | + | 0.686041i | 0 | −0.456585 | + | 0.456585i | −0.636970 | − | 2.75577i | 0 | 2.48135 | − | 0.520112i | ||||||
107.8 | 1.32473 | + | 0.495070i | 0 | 1.50981 | + | 1.31167i | −0.0505677 | − | 0.0209458i | 0 | 1.44150 | − | 1.44150i | 1.35072 | + | 2.48506i | 0 | −0.0566189 | − | 0.0527821i | ||||||
179.1 | −1.40684 | + | 0.144194i | 0 | 1.95842 | − | 0.405715i | 0.366958 | + | 0.885915i | 0 | 1.21471 | + | 1.21471i | −2.69668 | + | 0.853169i | 0 | −0.643996 | − | 1.19343i | ||||||
179.2 | −1.04926 | − | 0.948179i | 0 | 0.201912 | + | 1.98978i | −1.39555 | − | 3.36915i | 0 | −1.05755 | − | 1.05755i | 1.67481 | − | 2.27926i | 0 | −1.73026 | + | 4.85836i | ||||||
179.3 | −0.850468 | + | 1.12991i | 0 | −0.553410 | − | 1.92191i | 0.352978 | + | 0.852163i | 0 | −3.43393 | − | 3.43393i | 2.64225 | + | 1.00922i | 0 | −1.26307 | − | 0.325903i | ||||||
179.4 | −0.337906 | − | 1.37325i | 0 | −1.77164 | + | 0.928061i | 1.32268 | + | 3.19322i | 0 | −2.32913 | − | 2.32913i | 1.87311 | + | 2.11931i | 0 | 3.93816 | − | 2.89538i | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
96.o | 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.w.a | ✓ | 32 |
3.b | odd | 2 | 1 | 288.2.w.b | yes | 32 | |
4.b | odd | 2 | 1 | 1152.2.w.b | 32 | ||
12.b | even | 2 | 1 | 1152.2.w.a | 32 | ||
32.g | even | 8 | 1 | 1152.2.w.a | 32 | ||
32.h | odd | 8 | 1 | 288.2.w.b | yes | 32 | |
96.o | even | 8 | 1 | inner | 288.2.w.a | ✓ | 32 |
96.p | odd | 8 | 1 | 1152.2.w.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
288.2.w.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
288.2.w.a | ✓ | 32 | 96.o | even | 8 | 1 | inner |
288.2.w.b | yes | 32 | 3.b | odd | 2 | 1 | |
288.2.w.b | yes | 32 | 32.h | odd | 8 | 1 | |
1152.2.w.a | 32 | 12.b | even | 2 | 1 | ||
1152.2.w.a | 32 | 32.g | even | 8 | 1 | ||
1152.2.w.b | 32 | 4.b | odd | 2 | 1 | ||
1152.2.w.b | 32 | 96.p | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{32} - 16 T_{5}^{29} + 224 T_{5}^{27} + 896 T_{5}^{26} + 6080 T_{5}^{25} + 48512 T_{5}^{24} + 87168 T_{5}^{23} + 188416 T_{5}^{22} + 181376 T_{5}^{21} + 155648 T_{5}^{20} - 380672 T_{5}^{19} - 6281728 T_{5}^{18} + \cdots + 4096 \)
acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).