Newspace parameters
Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 552.j (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.40774219157\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
323.1 | −1.39813 | − | 0.212673i | 1.67830 | + | 0.428159i | 1.90954 | + | 0.594688i | 1.58050 | −2.25542 | − | 0.955550i | − | 2.65755i | −2.54331 | − | 1.23756i | 2.63336 | + | 1.43716i | −2.20975 | − | 0.336130i | |||
323.2 | −1.39813 | + | 0.212673i | 1.67830 | − | 0.428159i | 1.90954 | − | 0.594688i | 1.58050 | −2.25542 | + | 0.955550i | 2.65755i | −2.54331 | + | 1.23756i | 2.63336 | − | 1.43716i | −2.20975 | + | 0.336130i | ||||
323.3 | −1.29483 | − | 0.568707i | −1.25835 | + | 1.19019i | 1.35315 | + | 1.47275i | −0.156681 | 2.30622 | − | 0.825452i | 3.17007i | −0.914523 | − | 2.67650i | 0.166904 | − | 2.99535i | 0.202875 | + | 0.0891057i | ||||
323.4 | −1.29483 | + | 0.568707i | −1.25835 | − | 1.19019i | 1.35315 | − | 1.47275i | −0.156681 | 2.30622 | + | 0.825452i | − | 3.17007i | −0.914523 | + | 2.67650i | 0.166904 | + | 2.99535i | 0.202875 | − | 0.0891057i | |||
323.5 | −1.29433 | − | 0.569844i | 0.611727 | + | 1.62043i | 1.35056 | + | 1.47513i | −2.61830 | 0.131618 | − | 2.44595i | − | 0.280990i | −0.907465 | − | 2.67890i | −2.25158 | + | 1.98252i | 3.38893 | + | 1.49202i | |||
323.6 | −1.29433 | + | 0.569844i | 0.611727 | − | 1.62043i | 1.35056 | − | 1.47513i | −2.61830 | 0.131618 | + | 2.44595i | 0.280990i | −0.907465 | + | 2.67890i | −2.25158 | − | 1.98252i | 3.38893 | − | 1.49202i | ||||
323.7 | −1.28262 | − | 0.595732i | −1.03555 | − | 1.38840i | 1.29021 | + | 1.52819i | 1.31940 | 0.501098 | + | 2.39769i | 1.36534i | −0.744447 | − | 2.72870i | −0.855283 | + | 2.87550i | −1.69228 | − | 0.786008i | ||||
323.8 | −1.28262 | + | 0.595732i | −1.03555 | + | 1.38840i | 1.29021 | − | 1.52819i | 1.31940 | 0.501098 | − | 2.39769i | − | 1.36534i | −0.744447 | + | 2.72870i | −0.855283 | − | 2.87550i | −1.69228 | + | 0.786008i | |||
323.9 | −1.03749 | − | 0.961053i | 1.07871 | − | 1.35513i | 0.152756 | + | 1.99416i | 4.03392 | −2.42150 | + | 0.369231i | 0.282887i | 1.75801 | − | 2.21572i | −0.672762 | − | 2.92359i | −4.18514 | − | 3.87681i | ||||
323.10 | −1.03749 | + | 0.961053i | 1.07871 | + | 1.35513i | 0.152756 | − | 1.99416i | 4.03392 | −2.42150 | − | 0.369231i | − | 0.282887i | 1.75801 | + | 2.21572i | −0.672762 | + | 2.92359i | −4.18514 | + | 3.87681i | |||
323.11 | −1.03694 | − | 0.961640i | −0.303431 | − | 1.70527i | 0.150496 | + | 1.99433i | −3.63397 | −1.32521 | + | 2.06005i | 2.53044i | 1.76177 | − | 2.21273i | −2.81586 | + | 1.03486i | 3.76822 | + | 3.49457i | ||||
323.12 | −1.03694 | + | 0.961640i | −0.303431 | + | 1.70527i | 0.150496 | − | 1.99433i | −3.63397 | −1.32521 | − | 2.06005i | − | 2.53044i | 1.76177 | + | 2.21273i | −2.81586 | − | 1.03486i | 3.76822 | − | 3.49457i | |||
323.13 | −0.866210 | − | 1.11789i | −1.39323 | + | 1.02904i | −0.499360 | + | 1.93666i | −3.31842 | 2.35718 | + | 0.666110i | − | 4.93974i | 2.59752 | − | 1.11932i | 0.882155 | − | 2.86737i | 2.87445 | + | 3.70963i | |||
323.14 | −0.866210 | + | 1.11789i | −1.39323 | − | 1.02904i | −0.499360 | − | 1.93666i | −3.31842 | 2.35718 | − | 0.666110i | 4.93974i | 2.59752 | + | 1.11932i | 0.882155 | + | 2.86737i | 2.87445 | − | 3.70963i | ||||
323.15 | −0.610595 | − | 1.27561i | 0.0769352 | + | 1.73034i | −1.25435 | + | 1.55776i | 2.14134 | 2.16026 | − | 1.15468i | 2.49021i | 2.75299 | + | 0.648892i | −2.98816 | + | 0.266248i | −1.30749 | − | 2.73151i | ||||
323.16 | −0.610595 | + | 1.27561i | 0.0769352 | − | 1.73034i | −1.25435 | − | 1.55776i | 2.14134 | 2.16026 | + | 1.15468i | − | 2.49021i | 2.75299 | − | 0.648892i | −2.98816 | − | 0.266248i | −1.30749 | + | 2.73151i | |||
323.17 | −0.581428 | − | 1.28916i | 1.42468 | − | 0.985030i | −1.32388 | + | 1.49911i | −1.37188 | −2.09821 | − | 1.26392i | − | 0.424283i | 2.70234 | + | 0.835077i | 1.05943 | − | 2.80671i | 0.797651 | + | 1.76858i | |||
323.18 | −0.581428 | + | 1.28916i | 1.42468 | + | 0.985030i | −1.32388 | − | 1.49911i | −1.37188 | −2.09821 | + | 1.26392i | 0.424283i | 2.70234 | − | 0.835077i | 1.05943 | + | 2.80671i | 0.797651 | − | 1.76858i | ||||
323.19 | −0.484105 | − | 1.32877i | −1.59465 | − | 0.676097i | −1.53128 | + | 1.28653i | 1.67234 | −0.126404 | + | 2.44623i | − | 2.28745i | 2.45082 | + | 1.41191i | 2.08579 | + | 2.15627i | −0.809588 | − | 2.22216i | |||
323.20 | −0.484105 | + | 1.32877i | −1.59465 | + | 0.676097i | −1.53128 | − | 1.28653i | 1.67234 | −0.126404 | − | 2.44623i | 2.28745i | 2.45082 | − | 1.41191i | 2.08579 | − | 2.15627i | −0.809588 | + | 2.22216i | ||||
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 552.2.j.c | ✓ | 42 |
3.b | odd | 2 | 1 | 552.2.j.d | yes | 42 | |
4.b | odd | 2 | 1 | 2208.2.j.c | 42 | ||
8.b | even | 2 | 1 | 2208.2.j.d | 42 | ||
8.d | odd | 2 | 1 | 552.2.j.d | yes | 42 | |
12.b | even | 2 | 1 | 2208.2.j.d | 42 | ||
24.f | even | 2 | 1 | inner | 552.2.j.c | ✓ | 42 |
24.h | odd | 2 | 1 | 2208.2.j.c | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
552.2.j.c | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
552.2.j.c | ✓ | 42 | 24.f | even | 2 | 1 | inner |
552.2.j.d | yes | 42 | 3.b | odd | 2 | 1 | |
552.2.j.d | yes | 42 | 8.d | odd | 2 | 1 | |
2208.2.j.c | 42 | 4.b | odd | 2 | 1 | ||
2208.2.j.c | 42 | 24.h | odd | 2 | 1 | ||
2208.2.j.d | 42 | 8.b | even | 2 | 1 | ||
2208.2.j.d | 42 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{21} + 4 T_{5}^{20} - 50 T_{5}^{19} - 192 T_{5}^{18} + 1064 T_{5}^{17} + 3716 T_{5}^{16} - 13112 T_{5}^{15} - 38152 T_{5}^{14} + 105136 T_{5}^{13} + 226032 T_{5}^{12} - 566000 T_{5}^{11} - 770480 T_{5}^{10} + \cdots - 24576 \)
acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\).