Newspace parameters
Level: | \( N \) | \(=\) | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 990.bh (of order \(20\), degree \(8\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.90518980011\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 110) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 | −0.156434 | − | 0.987688i | 0 | −0.951057 | + | 0.309017i | −2.14909 | − | 0.617590i | 0 | −0.00697389 | + | 0.0136870i | 0.453990 | + | 0.891007i | 0 | −0.273795 | + | 2.21924i | ||||||
73.2 | −0.156434 | − | 0.987688i | 0 | −0.951057 | + | 0.309017i | 1.90161 | − | 1.17638i | 0 | −0.692077 | + | 1.35828i | 0.453990 | + | 0.891007i | 0 | −1.45937 | − | 1.69418i | ||||||
73.3 | −0.156434 | − | 0.987688i | 0 | −0.951057 | + | 0.309017i | 2.03838 | + | 0.919234i | 0 | −1.55752 | + | 3.05681i | 0.453990 | + | 0.891007i | 0 | 0.589044 | − | 2.15709i | ||||||
73.4 | 0.156434 | + | 0.987688i | 0 | −0.951057 | + | 0.309017i | −1.55051 | − | 1.61119i | 0 | −1.46937 | + | 2.88379i | −0.453990 | − | 0.891007i | 0 | 1.34880 | − | 1.78346i | ||||||
73.5 | 0.156434 | + | 0.987688i | 0 | −0.951057 | + | 0.309017i | −0.923903 | − | 2.03627i | 0 | 1.49255 | − | 2.92930i | −0.453990 | − | 0.891007i | 0 | 1.86667 | − | 1.23107i | ||||||
73.6 | 0.156434 | + | 0.987688i | 0 | −0.951057 | + | 0.309017i | 0.507931 | + | 2.17761i | 0 | −1.71860 | + | 3.37294i | −0.453990 | − | 0.891007i | 0 | −2.07135 | + | 0.842332i | ||||||
127.1 | −0.891007 | − | 0.453990i | 0 | 0.587785 | + | 0.809017i | −1.60893 | + | 1.55285i | 0 | −0.0165943 | + | 0.104772i | −0.156434 | − | 0.987688i | 0 | 2.13855 | − | 0.653156i | ||||||
127.2 | −0.891007 | − | 0.453990i | 0 | 0.587785 | + | 0.809017i | −0.537022 | − | 2.17062i | 0 | −0.516545 | + | 3.26134i | −0.156434 | − | 0.987688i | 0 | −0.506953 | + | 2.17784i | ||||||
127.3 | −0.891007 | − | 0.453990i | 0 | 0.587785 | + | 0.809017i | 2.22860 | − | 0.182641i | 0 | 0.119289 | − | 0.753160i | −0.156434 | − | 0.987688i | 0 | −2.06861 | − | 0.849027i | ||||||
127.4 | 0.891007 | + | 0.453990i | 0 | 0.587785 | + | 0.809017i | −2.18748 | + | 0.463629i | 0 | 0.620489 | − | 3.91761i | 0.156434 | + | 0.987688i | 0 | −2.15954 | − | 0.579997i | ||||||
127.5 | 0.891007 | + | 0.453990i | 0 | 0.587785 | + | 0.809017i | −0.667967 | − | 2.13397i | 0 | −0.0611439 | + | 0.386047i | 0.156434 | + | 0.987688i | 0 | 0.373639 | − | 2.20463i | ||||||
127.6 | 0.891007 | + | 0.453990i | 0 | 0.587785 | + | 0.809017i | 1.87069 | − | 1.22496i | 0 | −0.466963 | + | 2.94829i | 0.156434 | + | 0.987688i | 0 | 2.22292 | − | 0.242173i | ||||||
217.1 | −0.156434 | + | 0.987688i | 0 | −0.951057 | − | 0.309017i | −2.14909 | + | 0.617590i | 0 | −0.00697389 | − | 0.0136870i | 0.453990 | − | 0.891007i | 0 | −0.273795 | − | 2.21924i | ||||||
217.2 | −0.156434 | + | 0.987688i | 0 | −0.951057 | − | 0.309017i | 1.90161 | + | 1.17638i | 0 | −0.692077 | − | 1.35828i | 0.453990 | − | 0.891007i | 0 | −1.45937 | + | 1.69418i | ||||||
217.3 | −0.156434 | + | 0.987688i | 0 | −0.951057 | − | 0.309017i | 2.03838 | − | 0.919234i | 0 | −1.55752 | − | 3.05681i | 0.453990 | − | 0.891007i | 0 | 0.589044 | + | 2.15709i | ||||||
217.4 | 0.156434 | − | 0.987688i | 0 | −0.951057 | − | 0.309017i | −1.55051 | + | 1.61119i | 0 | −1.46937 | − | 2.88379i | −0.453990 | + | 0.891007i | 0 | 1.34880 | + | 1.78346i | ||||||
217.5 | 0.156434 | − | 0.987688i | 0 | −0.951057 | − | 0.309017i | −0.923903 | + | 2.03627i | 0 | 1.49255 | + | 2.92930i | −0.453990 | + | 0.891007i | 0 | 1.86667 | + | 1.23107i | ||||||
217.6 | 0.156434 | − | 0.987688i | 0 | −0.951057 | − | 0.309017i | 0.507931 | − | 2.17761i | 0 | −1.71860 | − | 3.37294i | −0.453990 | + | 0.891007i | 0 | −2.07135 | − | 0.842332i | ||||||
343.1 | −0.891007 | + | 0.453990i | 0 | 0.587785 | − | 0.809017i | −1.60893 | − | 1.55285i | 0 | −0.0165943 | − | 0.104772i | −0.156434 | + | 0.987688i | 0 | 2.13855 | + | 0.653156i | ||||||
343.2 | −0.891007 | + | 0.453990i | 0 | 0.587785 | − | 0.809017i | −0.537022 | + | 2.17062i | 0 | −0.516545 | − | 3.26134i | −0.156434 | + | 0.987688i | 0 | −0.506953 | − | 2.17784i | ||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.d | odd | 10 | 1 | inner |
55.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 990.2.bh.c | 48 | |
3.b | odd | 2 | 1 | 110.2.k.a | ✓ | 48 | |
5.c | odd | 4 | 1 | inner | 990.2.bh.c | 48 | |
11.d | odd | 10 | 1 | inner | 990.2.bh.c | 48 | |
12.b | even | 2 | 1 | 880.2.cm.c | 48 | ||
15.d | odd | 2 | 1 | 550.2.bh.b | 48 | ||
15.e | even | 4 | 1 | 110.2.k.a | ✓ | 48 | |
15.e | even | 4 | 1 | 550.2.bh.b | 48 | ||
33.f | even | 10 | 1 | 110.2.k.a | ✓ | 48 | |
55.l | even | 20 | 1 | inner | 990.2.bh.c | 48 | |
60.l | odd | 4 | 1 | 880.2.cm.c | 48 | ||
132.n | odd | 10 | 1 | 880.2.cm.c | 48 | ||
165.r | even | 10 | 1 | 550.2.bh.b | 48 | ||
165.u | odd | 20 | 1 | 110.2.k.a | ✓ | 48 | |
165.u | odd | 20 | 1 | 550.2.bh.b | 48 | ||
660.bv | even | 20 | 1 | 880.2.cm.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
110.2.k.a | ✓ | 48 | 3.b | odd | 2 | 1 | |
110.2.k.a | ✓ | 48 | 15.e | even | 4 | 1 | |
110.2.k.a | ✓ | 48 | 33.f | even | 10 | 1 | |
110.2.k.a | ✓ | 48 | 165.u | odd | 20 | 1 | |
550.2.bh.b | 48 | 15.d | odd | 2 | 1 | ||
550.2.bh.b | 48 | 15.e | even | 4 | 1 | ||
550.2.bh.b | 48 | 165.r | even | 10 | 1 | ||
550.2.bh.b | 48 | 165.u | odd | 20 | 1 | ||
880.2.cm.c | 48 | 12.b | even | 2 | 1 | ||
880.2.cm.c | 48 | 60.l | odd | 4 | 1 | ||
880.2.cm.c | 48 | 132.n | odd | 10 | 1 | ||
880.2.cm.c | 48 | 660.bv | even | 20 | 1 | ||
990.2.bh.c | 48 | 1.a | even | 1 | 1 | trivial | |
990.2.bh.c | 48 | 5.c | odd | 4 | 1 | inner | |
990.2.bh.c | 48 | 11.d | odd | 10 | 1 | inner | |
990.2.bh.c | 48 | 55.l | even | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{48} + 20 T_{7}^{47} + 200 T_{7}^{46} + 1280 T_{7}^{45} + 5417 T_{7}^{44} + 12340 T_{7}^{43} - 17400 T_{7}^{42} - 307780 T_{7}^{41} - 1496254 T_{7}^{40} - 3777740 T_{7}^{39} + 131200 T_{7}^{38} + 47252320 T_{7}^{37} + \cdots + 256 \)
acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\).