Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.l (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
121.1 | −2.69765 | −0.273945 | − | 1.71025i | 5.27730 | −0.500000 | − | 0.866025i | 0.739006 | + | 4.61365i | −0.230272 | − | 2.63571i | −8.84100 | −2.84991 | + | 0.937027i | 1.34882 | + | 2.33623i | ||||||
121.2 | −2.65219 | 1.69529 | + | 0.354953i | 5.03414 | −0.500000 | − | 0.866025i | −4.49624 | − | 0.941405i | −1.76710 | + | 1.96910i | −8.04712 | 2.74802 | + | 1.20350i | 1.32610 | + | 2.29687i | ||||||
121.3 | −2.35938 | −0.431498 | + | 1.67744i | 3.56668 | −0.500000 | − | 0.866025i | 1.01807 | − | 3.95772i | 2.33019 | + | 1.25307i | −3.69640 | −2.62762 | − | 1.44762i | 1.17969 | + | 2.04328i | ||||||
121.4 | −1.69039 | −1.59654 | − | 0.671613i | 0.857411 | −0.500000 | − | 0.866025i | 2.69877 | + | 1.13529i | 2.40616 | − | 1.10017i | 1.93142 | 2.09787 | + | 2.14451i | 0.845194 | + | 1.46392i | ||||||
121.5 | −1.58441 | 1.09319 | − | 1.34348i | 0.510363 | −0.500000 | − | 0.866025i | −1.73206 | + | 2.12863i | −1.17061 | + | 2.37269i | 2.36020 | −0.609890 | − | 2.93735i | 0.792206 | + | 1.37214i | ||||||
121.6 | −1.42579 | −1.59516 | + | 0.674891i | 0.0328702 | −0.500000 | − | 0.866025i | 2.27435 | − | 0.962251i | −1.98240 | − | 1.75217i | 2.80471 | 2.08904 | − | 2.15311i | 0.712894 | + | 1.23477i | ||||||
121.7 | −1.17766 | 1.55400 | + | 0.764908i | −0.613115 | −0.500000 | − | 0.866025i | −1.83009 | − | 0.900802i | −1.48383 | − | 2.19049i | 3.07736 | 1.82983 | + | 2.37733i | 0.588830 | + | 1.01988i | ||||||
121.8 | −0.390993 | 0.919343 | + | 1.46793i | −1.84712 | −0.500000 | − | 0.866025i | −0.359457 | − | 0.573950i | 2.26118 | + | 1.37370i | 1.50420 | −1.30962 | + | 2.69906i | 0.195497 | + | 0.338610i | ||||||
121.9 | −0.259663 | −1.21719 | − | 1.23225i | −1.93258 | −0.500000 | − | 0.866025i | 0.316059 | + | 0.319971i | −0.593390 | + | 2.57835i | 1.02114 | −0.0368964 | + | 2.99977i | 0.129832 | + | 0.224875i | ||||||
121.10 | 0.0255806 | 0.676102 | − | 1.59464i | −1.99935 | −0.500000 | − | 0.866025i | 0.0172951 | − | 0.0407920i | −2.37158 | − | 1.17286i | −0.102306 | −2.08577 | − | 2.15628i | −0.0127903 | − | 0.0221535i | ||||||
121.11 | 0.699049 | −1.56820 | + | 0.735367i | −1.51133 | −0.500000 | − | 0.866025i | −1.09625 | + | 0.514057i | 2.62064 | − | 0.363625i | −2.45459 | 1.91847 | − | 2.30640i | −0.349525 | − | 0.605394i | ||||||
121.12 | 0.831231 | −0.611026 | + | 1.62069i | −1.30905 | −0.500000 | − | 0.866025i | −0.507903 | + | 1.34717i | −2.57526 | − | 0.606656i | −2.75059 | −2.25330 | − | 1.98057i | −0.415616 | − | 0.719867i | ||||||
121.13 | 1.50060 | −1.11516 | − | 1.32530i | 0.251799 | −0.500000 | − | 0.866025i | −1.67341 | − | 1.98874i | 0.0793460 | − | 2.64456i | −2.62335 | −0.512840 | + | 2.95584i | −0.750300 | − | 1.29956i | ||||||
121.14 | 1.85256 | 1.72871 | − | 0.107594i | 1.43197 | −0.500000 | − | 0.866025i | 3.20253 | − | 0.199324i | 0.370146 | − | 2.61973i | −1.05231 | 2.97685 | − | 0.371996i | −0.926279 | − | 1.60436i | ||||||
121.15 | 1.89985 | 0.960253 | − | 1.44150i | 1.60945 | −0.500000 | − | 0.866025i | 1.82434 | − | 2.73864i | 1.72052 | + | 2.00992i | −0.741992 | −1.15583 | − | 2.76840i | −0.949927 | − | 1.64532i | ||||||
121.16 | 2.08660 | 0.878572 | + | 1.49269i | 2.35391 | −0.500000 | − | 0.866025i | 1.83323 | + | 3.11464i | −0.122839 | + | 2.64290i | 0.738471 | −1.45622 | + | 2.62286i | −1.04330 | − | 1.80705i | ||||||
121.17 | 2.58565 | −1.01529 | + | 1.40328i | 4.68556 | −0.500000 | − | 0.866025i | −2.62518 | + | 3.62838i | 2.41754 | − | 1.07494i | 6.94392 | −0.938372 | − | 2.84947i | −1.29282 | − | 2.23923i | ||||||
121.18 | 2.75701 | −0.581455 | − | 1.63154i | 5.60109 | −0.500000 | − | 0.866025i | −1.60307 | − | 4.49816i | −2.40845 | + | 1.09515i | 9.92824 | −2.32382 | + | 1.89733i | −1.37850 | − | 2.38764i | ||||||
151.1 | −2.69765 | −0.273945 | + | 1.71025i | 5.27730 | −0.500000 | + | 0.866025i | 0.739006 | − | 4.61365i | −0.230272 | + | 2.63571i | −8.84100 | −2.84991 | − | 0.937027i | 1.34882 | − | 2.33623i | ||||||
151.2 | −2.65219 | 1.69529 | − | 0.354953i | 5.03414 | −0.500000 | + | 0.866025i | −4.49624 | + | 0.941405i | −1.76710 | − | 1.96910i | −8.04712 | 2.74802 | − | 1.20350i | 1.32610 | − | 2.29687i | ||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.l.c | yes | 36 |
3.b | odd | 2 | 1 | 945.2.l.c | 36 | ||
7.c | even | 3 | 1 | 315.2.k.c | ✓ | 36 | |
9.c | even | 3 | 1 | 315.2.k.c | ✓ | 36 | |
9.d | odd | 6 | 1 | 945.2.k.c | 36 | ||
21.h | odd | 6 | 1 | 945.2.k.c | 36 | ||
63.h | even | 3 | 1 | inner | 315.2.l.c | yes | 36 |
63.j | odd | 6 | 1 | 945.2.l.c | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.k.c | ✓ | 36 | 7.c | even | 3 | 1 | |
315.2.k.c | ✓ | 36 | 9.c | even | 3 | 1 | |
315.2.l.c | yes | 36 | 1.a | even | 1 | 1 | trivial |
315.2.l.c | yes | 36 | 63.h | even | 3 | 1 | inner |
945.2.k.c | 36 | 9.d | odd | 6 | 1 | ||
945.2.k.c | 36 | 21.h | odd | 6 | 1 | ||
945.2.l.c | 36 | 3.b | odd | 2 | 1 | ||
945.2.l.c | 36 | 63.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - 29 T_{2}^{16} + 344 T_{2}^{14} + 2 T_{2}^{13} - 2159 T_{2}^{12} - 42 T_{2}^{11} + 7749 T_{2}^{10} + 312 T_{2}^{9} - 16013 T_{2}^{8} - 1003 T_{2}^{7} + 18068 T_{2}^{6} + 1417 T_{2}^{5} - 9642 T_{2}^{4} - 839 T_{2}^{3} + \cdots - 9 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).