Newspace parameters
Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 465.bg (of order \(15\), degree \(8\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.71304369399\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 | −0.791724 | + | 2.43667i | 0.669131 | − | 0.743145i | −3.69252 | − | 2.68277i | 0.500000 | − | 0.866025i | 1.28104 | + | 2.21882i | −0.323047 | + | 3.07359i | 5.31499 | − | 3.86156i | −0.104528 | − | 0.994522i | 1.71436 | + | 1.90399i |
76.2 | −0.547008 | + | 1.68352i | 0.669131 | − | 0.743145i | −0.916984 | − | 0.666228i | 0.500000 | − | 0.866025i | 0.885078 | + | 1.53300i | 0.289063 | − | 2.75025i | −1.24097 | + | 0.901616i | −0.104528 | − | 0.994522i | 1.18447 | + | 1.31548i |
76.3 | −0.0126485 | + | 0.0389281i | 0.669131 | − | 0.743145i | 1.61668 | + | 1.17459i | 0.500000 | − | 0.866025i | 0.0204657 | + | 0.0354476i | −0.375369 | + | 3.57140i | −0.132401 | + | 0.0961952i | −0.104528 | − | 0.994522i | 0.0273885 | + | 0.0304180i |
76.4 | 0.197542 | − | 0.607972i | 0.669131 | − | 0.743145i | 1.28743 | + | 0.935370i | 0.500000 | − | 0.866025i | −0.319630 | − | 0.553615i | 0.264793 | − | 2.51934i | 1.85734 | − | 1.34944i | −0.104528 | − | 0.994522i | −0.427748 | − | 0.475063i |
76.5 | 0.599664 | − | 1.84558i | 0.669131 | − | 0.743145i | −1.42852 | − | 1.03788i | 0.500000 | − | 0.866025i | −0.970277 | − | 1.68057i | 0.0159459 | − | 0.151715i | 0.367763 | − | 0.267195i | −0.104528 | − | 0.994522i | −1.29848 | − | 1.44211i |
76.6 | 0.863191 | − | 2.65663i | 0.669131 | − | 0.743145i | −4.69455 | − | 3.41079i | 0.500000 | − | 0.866025i | −1.39667 | − | 2.41911i | 0.217892 | − | 2.07310i | −8.59378 | + | 6.24374i | −0.104528 | − | 0.994522i | −1.86911 | − | 2.07586i |
121.1 | −2.17121 | + | 1.57748i | −0.104528 | + | 0.994522i | 1.60769 | − | 4.94796i | 0.500000 | − | 0.866025i | −1.34188 | − | 2.32421i | −3.36862 | + | 0.716022i | 2.65600 | + | 8.17434i | −0.978148 | − | 0.207912i | 0.280530 | + | 2.66906i |
121.2 | −1.42167 | + | 1.03290i | −0.104528 | + | 0.994522i | 0.336215 | − | 1.03476i | 0.500000 | − | 0.866025i | −0.878637 | − | 1.52184i | 1.53605 | − | 0.326498i | −0.495233 | − | 1.52417i | −0.978148 | − | 0.207912i | 0.183685 | + | 1.74765i |
121.3 | −0.491640 | + | 0.357197i | −0.104528 | + | 0.994522i | −0.503914 | + | 1.55089i | 0.500000 | − | 0.866025i | −0.303850 | − | 0.526284i | −3.62443 | + | 0.770396i | −0.681808 | − | 2.09839i | −0.978148 | − | 0.207912i | 0.0635220 | + | 0.604371i |
121.4 | 0.278085 | − | 0.202041i | −0.104528 | + | 0.994522i | −0.581523 | + | 1.78974i | 0.500000 | − | 0.866025i | 0.171866 | + | 0.297681i | 2.80882 | − | 0.597033i | 0.412326 | + | 1.26901i | −0.978148 | − | 0.207912i | −0.0359298 | − | 0.341849i |
121.5 | 0.943990 | − | 0.685849i | −0.104528 | + | 0.994522i | −0.197306 | + | 0.607244i | 0.500000 | − | 0.866025i | 0.583418 | + | 1.01051i | −4.63922 | + | 0.986096i | 0.951367 | + | 2.92801i | −0.978148 | − | 0.207912i | −0.121968 | − | 1.16044i |
121.6 | 2.05342 | − | 1.49190i | −0.104528 | + | 0.994522i | 1.37275 | − | 4.22490i | 0.500000 | − | 0.866025i | 1.26909 | + | 2.19812i | 1.56122 | − | 0.331847i | −1.91561 | − | 5.89563i | −0.978148 | − | 0.207912i | −0.265311 | − | 2.52427i |
196.1 | −2.17121 | − | 1.57748i | −0.104528 | − | 0.994522i | 1.60769 | + | 4.94796i | 0.500000 | + | 0.866025i | −1.34188 | + | 2.32421i | −3.36862 | − | 0.716022i | 2.65600 | − | 8.17434i | −0.978148 | + | 0.207912i | 0.280530 | − | 2.66906i |
196.2 | −1.42167 | − | 1.03290i | −0.104528 | − | 0.994522i | 0.336215 | + | 1.03476i | 0.500000 | + | 0.866025i | −0.878637 | + | 1.52184i | 1.53605 | + | 0.326498i | −0.495233 | + | 1.52417i | −0.978148 | + | 0.207912i | 0.183685 | − | 1.74765i |
196.3 | −0.491640 | − | 0.357197i | −0.104528 | − | 0.994522i | −0.503914 | − | 1.55089i | 0.500000 | + | 0.866025i | −0.303850 | + | 0.526284i | −3.62443 | − | 0.770396i | −0.681808 | + | 2.09839i | −0.978148 | + | 0.207912i | 0.0635220 | − | 0.604371i |
196.4 | 0.278085 | + | 0.202041i | −0.104528 | − | 0.994522i | −0.581523 | − | 1.78974i | 0.500000 | + | 0.866025i | 0.171866 | − | 0.297681i | 2.80882 | + | 0.597033i | 0.412326 | − | 1.26901i | −0.978148 | + | 0.207912i | −0.0359298 | + | 0.341849i |
196.5 | 0.943990 | + | 0.685849i | −0.104528 | − | 0.994522i | −0.197306 | − | 0.607244i | 0.500000 | + | 0.866025i | 0.583418 | − | 1.01051i | −4.63922 | − | 0.986096i | 0.951367 | − | 2.92801i | −0.978148 | + | 0.207912i | −0.121968 | + | 1.16044i |
196.6 | 2.05342 | + | 1.49190i | −0.104528 | − | 0.994522i | 1.37275 | + | 4.22490i | 0.500000 | + | 0.866025i | 1.26909 | − | 2.19812i | 1.56122 | + | 0.331847i | −1.91561 | + | 5.89563i | −0.978148 | + | 0.207912i | −0.265311 | + | 2.52427i |
226.1 | −0.652567 | + | 2.00839i | −0.978148 | − | 0.207912i | −1.98977 | − | 1.44565i | 0.500000 | + | 0.866025i | 1.05588 | − | 1.82883i | 2.63550 | − | 1.17340i | 0.785009 | − | 0.570342i | 0.913545 | + | 0.406737i | −2.06560 | + | 0.439058i |
226.2 | −0.403889 | + | 1.24304i | −0.978148 | − | 0.207912i | 0.236004 | + | 0.171467i | 0.500000 | + | 0.866025i | 0.653506 | − | 1.13191i | −2.27510 | + | 1.01294i | −2.42325 | + | 1.76060i | 0.913545 | + | 0.406737i | −1.27845 | + | 0.271743i |
See all 48 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 | 465.2.bg.c | ✓ | 48 |
31.g | even | 15 | 1 | inner | 465.2.bg.c | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
465.2.bg.c | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
465.2.bg.c | ✓ | 48 | 31.g | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{48} + 2 T_{2}^{47} + 19 T_{2}^{46} + 40 T_{2}^{45} + 249 T_{2}^{44} + 440 T_{2}^{43} + 2538 T_{2}^{42} + 4226 T_{2}^{41} + 22794 T_{2}^{40} + 36407 T_{2}^{39} + 166979 T_{2}^{38} + 248093 T_{2}^{37} + 1056614 T_{2}^{36} + \cdots + 2025 \)
acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\).