Newspace parameters
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.o (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 | − | 1.00000i | −1.67328 | − | 0.447368i | −1.00000 | 0.866025 | + | 0.500000i | −0.447368 | + | 1.67328i | −1.32225 | − | 2.29021i | 1.00000i | 2.59972 | + | 1.49714i | 0.500000 | − | 0.866025i | |||||
161.2 | − | 1.00000i | −1.14810 | + | 1.29687i | −1.00000 | 0.866025 | + | 0.500000i | 1.29687 | + | 1.14810i | 0.742621 | + | 1.28626i | 1.00000i | −0.363755 | − | 2.97787i | 0.500000 | − | 0.866025i | |||||
161.3 | − | 1.00000i | −0.556373 | + | 1.64026i | −1.00000 | 0.866025 | + | 0.500000i | 1.64026 | + | 0.556373i | −1.24695 | − | 2.15979i | 1.00000i | −2.38090 | − | 1.82519i | 0.500000 | − | 0.866025i | |||||
161.4 | − | 1.00000i | −0.474351 | − | 1.66583i | −1.00000 | 0.866025 | + | 0.500000i | −1.66583 | + | 0.474351i | 1.75266 | + | 3.03570i | 1.00000i | −2.54998 | + | 1.58038i | 0.500000 | − | 0.866025i | |||||
161.5 | − | 1.00000i | −0.151443 | − | 1.72542i | −1.00000 | 0.866025 | + | 0.500000i | −1.72542 | + | 0.151443i | −1.64767 | − | 2.85385i | 1.00000i | −2.95413 | + | 0.522605i | 0.500000 | − | 0.866025i | |||||
161.6 | − | 1.00000i | 0.433517 | + | 1.67692i | −1.00000 | 0.866025 | + | 0.500000i | 1.67692 | − | 0.433517i | −1.74230 | − | 3.01775i | 1.00000i | −2.62413 | + | 1.45395i | 0.500000 | − | 0.866025i | |||||
161.7 | − | 1.00000i | 0.500302 | − | 1.65822i | −1.00000 | 0.866025 | + | 0.500000i | −1.65822 | − | 0.500302i | −1.28701 | − | 2.22916i | 1.00000i | −2.49940 | − | 1.65922i | 0.500000 | − | 0.866025i | |||||
161.8 | − | 1.00000i | 1.29636 | + | 1.14867i | −1.00000 | 0.866025 | + | 0.500000i | 1.14867 | − | 1.29636i | 1.71736 | + | 2.97455i | 1.00000i | 0.361105 | + | 2.97819i | 0.500000 | − | 0.866025i | |||||
161.9 | − | 1.00000i | 1.62650 | − | 0.595404i | −1.00000 | 0.866025 | + | 0.500000i | −0.595404 | − | 1.62650i | −0.581871 | − | 1.00783i | 1.00000i | 2.29099 | − | 1.93685i | 0.500000 | − | 0.866025i | |||||
161.10 | − | 1.00000i | 1.64686 | − | 0.536508i | −1.00000 | 0.866025 | + | 0.500000i | −0.536508 | − | 1.64686i | 0.615418 | + | 1.06593i | 1.00000i | 2.42432 | − | 1.76711i | 0.500000 | − | 0.866025i | |||||
161.11 | 1.00000i | −1.69869 | − | 0.338296i | −1.00000 | −0.866025 | − | 0.500000i | 0.338296 | − | 1.69869i | −1.24695 | − | 2.15979i | − | 1.00000i | 2.77111 | + | 1.14932i | 0.500000 | − | 0.866025i | |||||
161.12 | 1.00000i | −1.69717 | + | 0.345843i | −1.00000 | −0.866025 | − | 0.500000i | −0.345843 | − | 1.69717i | 0.742621 | + | 1.28626i | − | 1.00000i | 2.76078 | − | 1.17391i | 0.500000 | − | 0.866025i | |||||
161.13 | 1.00000i | −1.23550 | − | 1.21390i | −1.00000 | −0.866025 | − | 0.500000i | 1.21390 | − | 1.23550i | −1.74230 | − | 3.01775i | − | 1.00000i | 0.0529076 | + | 2.99953i | 0.500000 | − | 0.866025i | |||||
161.14 | 1.00000i | −0.449207 | + | 1.67279i | −1.00000 | −0.866025 | − | 0.500000i | −1.67279 | − | 0.449207i | −1.32225 | − | 2.29021i | − | 1.00000i | −2.59643 | − | 1.50285i | 0.500000 | − | 0.866025i | |||||
161.15 | 1.00000i | −0.346599 | − | 1.69702i | −1.00000 | −0.866025 | − | 0.500000i | 1.69702 | − | 0.346599i | 1.71736 | + | 2.97455i | − | 1.00000i | −2.75974 | + | 1.17637i | 0.500000 | − | 0.866025i | |||||
161.16 | 1.00000i | 1.20548 | + | 1.24372i | −1.00000 | −0.866025 | − | 0.500000i | −1.24372 | + | 1.20548i | 1.75266 | + | 3.03570i | − | 1.00000i | −0.0936563 | + | 2.99854i | 0.500000 | − | 0.866025i | |||||
161.17 | 1.00000i | 1.28806 | − | 1.15797i | −1.00000 | −0.866025 | − | 0.500000i | 1.15797 | + | 1.28806i | 0.615418 | + | 1.06593i | − | 1.00000i | 0.318204 | − | 2.98308i | 0.500000 | − | 0.866025i | |||||
161.18 | 1.00000i | 1.32888 | − | 1.11089i | −1.00000 | −0.866025 | − | 0.500000i | 1.11089 | + | 1.32888i | −0.581871 | − | 1.00783i | − | 1.00000i | 0.531863 | − | 2.95248i | 0.500000 | − | 0.866025i | |||||
161.19 | 1.00000i | 1.41853 | + | 0.993862i | −1.00000 | −0.866025 | − | 0.500000i | −0.993862 | + | 1.41853i | −1.64767 | − | 2.85385i | − | 1.00000i | 1.02448 | + | 2.81965i | 0.500000 | − | 0.866025i | |||||
161.20 | 1.00000i | 1.68621 | + | 0.395836i | −1.00000 | −0.866025 | − | 0.500000i | −0.395836 | + | 1.68621i | −1.28701 | − | 2.22916i | − | 1.00000i | 2.68663 | + | 1.33493i | 0.500000 | − | 0.866025i | |||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
31.e | odd | 6 | 1 | inner |
93.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.o.e | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 930.2.o.e | ✓ | 40 |
31.e | odd | 6 | 1 | inner | 930.2.o.e | ✓ | 40 |
93.g | even | 6 | 1 | inner | 930.2.o.e | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.o.e | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
930.2.o.e | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
930.2.o.e | ✓ | 40 | 31.e | odd | 6 | 1 | inner |
930.2.o.e | ✓ | 40 | 93.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):
\(T_{7}^{20} + \cdots\) |
\(15\!\cdots\!68\)\( T_{11}^{22} + \)\(65\!\cdots\!05\)\( T_{11}^{20} + \)\(14\!\cdots\!04\)\( T_{11}^{18} + \)\(23\!\cdots\!21\)\( T_{11}^{16} + \)\(26\!\cdots\!74\)\( T_{11}^{14} + \)\(21\!\cdots\!62\)\( T_{11}^{12} + \)\(13\!\cdots\!26\)\( T_{11}^{10} + \)\(59\!\cdots\!06\)\( T_{11}^{8} + \)\(17\!\cdots\!44\)\( T_{11}^{6} + 332005777128 T_{11}^{4} + 10332663156 T_{11}^{2} + 276922881 \)">\(T_{11}^{40} + \cdots\) |