Newspace parameters
Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 725.e (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.78915414654\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Relative dimension: | \(13\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 145) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | − | 2.26693i | 2.65401 | −3.13899 | 0 | − | 6.01647i | 2.59753 | + | 2.59753i | 2.58201i | 4.04378 | 0 | ||||||||||||||
157.2 | − | 2.23019i | −1.25170 | −2.97373 | 0 | 2.79153i | −0.483409 | − | 0.483409i | 2.17160i | −1.43324 | 0 | |||||||||||||||
157.3 | − | 1.41066i | −2.58872 | 0.0100302 | 0 | 3.65181i | 0.510628 | + | 0.510628i | − | 2.83547i | 3.70148 | 0 | ||||||||||||||
157.4 | − | 1.26373i | 0.913274 | 0.402981 | 0 | − | 1.15413i | −2.03055 | − | 2.03055i | − | 3.03672i | −2.16593 | 0 | |||||||||||||
157.5 | − | 0.895351i | 2.11245 | 1.19835 | 0 | − | 1.89138i | −1.38465 | − | 1.38465i | − | 2.86364i | 1.46244 | 0 | |||||||||||||
157.6 | − | 0.222351i | −1.02589 | 1.95056 | 0 | 0.228107i | 2.35964 | + | 2.35964i | − | 0.878412i | −1.94756 | 0 | ||||||||||||||
157.7 | 0.342532i | 2.64611 | 1.88267 | 0 | 0.906377i | 1.55474 | + | 1.55474i | 1.32994i | 4.00189 | 0 | ||||||||||||||||
157.8 | 0.839004i | −0.711801 | 1.29607 | 0 | − | 0.597203i | −1.13987 | − | 1.13987i | 2.76542i | −2.49334 | 0 | |||||||||||||||
157.9 | 1.36192i | 0.228160 | 0.145179 | 0 | 0.310736i | −3.45046 | − | 3.45046i | 2.92156i | −2.94794 | 0 | ||||||||||||||||
157.10 | 1.77873i | 1.38965 | −1.16389 | 0 | 2.47181i | 3.41296 | + | 3.41296i | 1.48721i | −1.06888 | 0 | ||||||||||||||||
157.11 | 1.82099i | −2.59340 | −1.31599 | 0 | − | 4.72256i | 0.820621 | + | 0.820621i | 1.24557i | 3.72575 | 0 | |||||||||||||||
157.12 | 2.41122i | 1.85973 | −3.81399 | 0 | 4.48422i | 0.291676 | + | 0.291676i | − | 4.37394i | 0.458586 | 0 | |||||||||||||||
157.13 | 2.73482i | −1.63186 | −5.47925 | 0 | − | 4.46285i | −1.05887 | − | 1.05887i | − | 9.51511i | −0.337028 | 0 | ||||||||||||||
568.1 | − | 2.73482i | −1.63186 | −5.47925 | 0 | 4.46285i | −1.05887 | + | 1.05887i | 9.51511i | −0.337028 | 0 | |||||||||||||||
568.2 | − | 2.41122i | 1.85973 | −3.81399 | 0 | − | 4.48422i | 0.291676 | − | 0.291676i | 4.37394i | 0.458586 | 0 | ||||||||||||||
568.3 | − | 1.82099i | −2.59340 | −1.31599 | 0 | 4.72256i | 0.820621 | − | 0.820621i | − | 1.24557i | 3.72575 | 0 | ||||||||||||||
568.4 | − | 1.77873i | 1.38965 | −1.16389 | 0 | − | 2.47181i | 3.41296 | − | 3.41296i | − | 1.48721i | −1.06888 | 0 | |||||||||||||
568.5 | − | 1.36192i | 0.228160 | 0.145179 | 0 | − | 0.310736i | −3.45046 | + | 3.45046i | − | 2.92156i | −2.94794 | 0 | |||||||||||||
568.6 | − | 0.839004i | −0.711801 | 1.29607 | 0 | 0.597203i | −1.13987 | + | 1.13987i | − | 2.76542i | −2.49334 | 0 | ||||||||||||||
568.7 | − | 0.342532i | 2.64611 | 1.88267 | 0 | − | 0.906377i | 1.55474 | − | 1.55474i | − | 1.32994i | 4.00189 | 0 | |||||||||||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
145.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 725.2.e.c | 26 | |
5.b | even | 2 | 1 | 145.2.e.a | ✓ | 26 | |
5.c | odd | 4 | 1 | 145.2.j.a | yes | 26 | |
5.c | odd | 4 | 1 | 725.2.j.c | 26 | ||
29.c | odd | 4 | 1 | 725.2.j.c | 26 | ||
145.e | even | 4 | 1 | inner | 725.2.e.c | 26 | |
145.f | odd | 4 | 1 | 145.2.j.a | yes | 26 | |
145.j | even | 4 | 1 | 145.2.e.a | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
145.2.e.a | ✓ | 26 | 5.b | even | 2 | 1 | |
145.2.e.a | ✓ | 26 | 145.j | even | 4 | 1 | |
145.2.j.a | yes | 26 | 5.c | odd | 4 | 1 | |
145.2.j.a | yes | 26 | 145.f | odd | 4 | 1 | |
725.2.e.c | 26 | 1.a | even | 1 | 1 | trivial | |
725.2.e.c | 26 | 145.e | even | 4 | 1 | inner | |
725.2.j.c | 26 | 5.c | odd | 4 | 1 | ||
725.2.j.c | 26 | 29.c | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{26} + 37 T_{2}^{24} + 598 T_{2}^{22} + 5562 T_{2}^{20} + 33015 T_{2}^{18} + 131099 T_{2}^{16} + 354976 T_{2}^{14} + 655536 T_{2}^{12} + 811627 T_{2}^{10} + 648415 T_{2}^{8} + 311274 T_{2}^{6} + 78422 T_{2}^{4} + \cdots + 225 \)
acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\).