Newspace parameters
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bg (of order \(15\), degree \(8\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | −0.809017 | + | 0.587785i | −0.104528 | + | 0.994522i | 0.309017 | − | 0.951057i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −3.47616 | + | 0.738880i | 0.309017 | + | 0.951057i | −0.978148 | − | 0.207912i | −0.104528 | − | 0.994522i |
121.2 | −0.809017 | + | 0.587785i | −0.104528 | + | 0.994522i | 0.309017 | − | 0.951057i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.813060 | + | 0.172821i | 0.309017 | + | 0.951057i | −0.978148 | − | 0.207912i | −0.104528 | − | 0.994522i |
121.3 | −0.809017 | + | 0.587785i | −0.104528 | + | 0.994522i | 0.309017 | − | 0.951057i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 2.66379 | − | 0.566206i | 0.309017 | + | 0.951057i | −0.978148 | − | 0.207912i | −0.104528 | − | 0.994522i |
361.1 | 0.309017 | + | 0.951057i | 0.669131 | + | 0.743145i | −0.809017 | + | 0.587785i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.272387 | − | 2.59159i | −0.809017 | − | 0.587785i | −0.104528 | + | 0.994522i | 0.669131 | − | 0.743145i |
361.2 | 0.309017 | + | 0.951057i | 0.669131 | + | 0.743145i | −0.809017 | + | 0.587785i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.258263 | − | 2.45720i | −0.809017 | − | 0.587785i | −0.104528 | + | 0.994522i | 0.669131 | − | 0.743145i |
361.3 | 0.309017 | + | 0.951057i | 0.669131 | + | 0.743145i | −0.809017 | + | 0.587785i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.408047 | + | 3.88231i | −0.809017 | − | 0.587785i | −0.104528 | + | 0.994522i | 0.669131 | − | 0.743145i |
391.1 | −0.809017 | − | 0.587785i | 0.913545 | + | 0.406737i | 0.309017 | + | 0.951057i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.348368 | + | 0.386902i | 0.309017 | − | 0.951057i | 0.669131 | + | 0.743145i | 0.913545 | − | 0.406737i |
391.2 | −0.809017 | − | 0.587785i | 0.913545 | + | 0.406737i | 0.309017 | + | 0.951057i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.955933 | − | 1.06167i | 0.309017 | − | 0.951057i | 0.669131 | + | 0.743145i | 0.913545 | − | 0.406737i |
391.3 | −0.809017 | − | 0.587785i | 0.913545 | + | 0.406737i | 0.309017 | + | 0.951057i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 2.70884 | − | 3.00848i | 0.309017 | − | 0.951057i | 0.669131 | + | 0.743145i | 0.913545 | − | 0.406737i |
421.1 | −0.809017 | + | 0.587785i | 0.913545 | − | 0.406737i | 0.309017 | − | 0.951057i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.348368 | − | 0.386902i | 0.309017 | + | 0.951057i | 0.669131 | − | 0.743145i | 0.913545 | + | 0.406737i |
421.2 | −0.809017 | + | 0.587785i | 0.913545 | − | 0.406737i | 0.309017 | − | 0.951057i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.955933 | + | 1.06167i | 0.309017 | + | 0.951057i | 0.669131 | − | 0.743145i | 0.913545 | + | 0.406737i |
421.3 | −0.809017 | + | 0.587785i | 0.913545 | − | 0.406737i | 0.309017 | − | 0.951057i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 2.70884 | + | 3.00848i | 0.309017 | + | 0.951057i | 0.669131 | − | 0.743145i | 0.913545 | + | 0.406737i |
541.1 | 0.309017 | − | 0.951057i | 0.669131 | − | 0.743145i | −0.809017 | − | 0.587785i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.272387 | + | 2.59159i | −0.809017 | + | 0.587785i | −0.104528 | − | 0.994522i | 0.669131 | + | 0.743145i |
541.2 | 0.309017 | − | 0.951057i | 0.669131 | − | 0.743145i | −0.809017 | − | 0.587785i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.258263 | + | 2.45720i | −0.809017 | + | 0.587785i | −0.104528 | − | 0.994522i | 0.669131 | + | 0.743145i |
541.3 | 0.309017 | − | 0.951057i | 0.669131 | − | 0.743145i | −0.809017 | − | 0.587785i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.408047 | − | 3.88231i | −0.809017 | + | 0.587785i | −0.104528 | − | 0.994522i | 0.669131 | + | 0.743145i |
661.1 | −0.809017 | − | 0.587785i | −0.104528 | − | 0.994522i | 0.309017 | + | 0.951057i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −3.47616 | − | 0.738880i | 0.309017 | − | 0.951057i | −0.978148 | + | 0.207912i | −0.104528 | + | 0.994522i |
661.2 | −0.809017 | − | 0.587785i | −0.104528 | − | 0.994522i | 0.309017 | + | 0.951057i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.813060 | − | 0.172821i | 0.309017 | − | 0.951057i | −0.978148 | + | 0.207912i | −0.104528 | + | 0.994522i |
661.3 | −0.809017 | − | 0.587785i | −0.104528 | − | 0.994522i | 0.309017 | + | 0.951057i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 2.66379 | + | 0.566206i | 0.309017 | − | 0.951057i | −0.978148 | + | 0.207912i | −0.104528 | + | 0.994522i |
691.1 | 0.309017 | − | 0.951057i | −0.978148 | − | 0.207912i | −0.809017 | − | 0.587785i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −2.05288 | + | 0.914002i | −0.809017 | + | 0.587785i | 0.913545 | + | 0.406737i | −0.978148 | + | 0.207912i |
691.2 | 0.309017 | − | 0.951057i | −0.978148 | − | 0.207912i | −0.809017 | − | 0.587785i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.302260 | − | 0.134575i | −0.809017 | + | 0.587785i | 0.913545 | + | 0.406737i | −0.978148 | + | 0.207912i |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.g | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.bg.g | ✓ | 24 |
31.g | even | 15 | 1 | inner | 930.2.bg.g | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bg.g | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
930.2.bg.g | ✓ | 24 | 31.g | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{24} - 9 T_{7}^{23} + 30 T_{7}^{22} + 4 T_{7}^{21} - 421 T_{7}^{20} + 2484 T_{7}^{19} - 4220 T_{7}^{18} - 6559 T_{7}^{17} + 82621 T_{7}^{16} - 146085 T_{7}^{15} + 391610 T_{7}^{14} - 255930 T_{7}^{13} - 3950800 T_{7}^{12} + \cdots + 5382400 \)
acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).