Newspace parameters
Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 354.e (of order \(29\), degree \(28\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.82670423155\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{29})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{29}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | 0.561187 | + | 0.827689i | −0.0541389 | + | 0.998533i | −0.370138 | + | 0.928977i | −2.31715 | + | 1.07203i | −0.856857 | + | 0.515554i | 0.0770120 | − | 0.277372i | −0.976621 | + | 0.214970i | −0.994138 | − | 0.108119i | −2.18766 | − | 1.31627i |
7.2 | 0.561187 | + | 0.827689i | −0.0541389 | + | 0.998533i | −0.370138 | + | 0.928977i | 2.80275 | − | 1.29669i | −0.856857 | + | 0.515554i | −0.846851 | + | 3.05008i | −0.976621 | + | 0.214970i | −0.994138 | − | 0.108119i | 2.64613 | + | 1.59212i |
19.1 | −0.468408 | + | 0.883512i | 0.725995 | − | 0.687699i | −0.561187 | − | 0.827689i | −2.36756 | + | 0.521140i | 0.267528 | + | 0.963550i | 2.60217 | − | 1.97812i | 0.994138 | − | 0.108119i | 0.0541389 | − | 0.998533i | 0.648553 | − | 2.33588i |
19.2 | −0.468408 | + | 0.883512i | 0.725995 | − | 0.687699i | −0.561187 | − | 0.827689i | 3.92252 | − | 0.863413i | 0.267528 | + | 0.963550i | −0.604650 | + | 0.459643i | 0.994138 | − | 0.108119i | 0.0541389 | − | 0.998533i | −1.07451 | + | 3.87003i |
25.1 | −0.796093 | − | 0.605174i | −0.468408 | − | 0.883512i | 0.267528 | + | 0.963550i | −1.39125 | − | 1.31786i | −0.161782 | + | 0.986827i | 1.52711 | − | 1.79785i | 0.370138 | − | 0.928977i | −0.561187 | + | 0.827689i | 0.310029 | + | 1.89109i |
25.2 | −0.796093 | − | 0.605174i | −0.468408 | − | 0.883512i | 0.267528 | + | 0.963550i | 0.451252 | + | 0.427449i | −0.161782 | + | 0.986827i | −1.39072 | + | 1.63728i | 0.370138 | − | 0.928977i | −0.561187 | + | 0.827689i | −0.100558 | − | 0.613375i |
49.1 | 0.370138 | − | 0.928977i | 0.994138 | + | 0.108119i | −0.725995 | − | 0.687699i | −2.09789 | + | 2.46983i | 0.468408 | − | 0.883512i | 3.22015 | + | 1.93750i | −0.907575 | + | 0.419889i | 0.976621 | + | 0.214970i | 1.51790 | + | 2.86307i |
49.2 | 0.370138 | − | 0.928977i | 0.994138 | + | 0.108119i | −0.725995 | − | 0.687699i | 0.988453 | − | 1.16370i | 0.468408 | − | 0.883512i | −2.12914 | − | 1.28106i | −0.907575 | + | 0.419889i | 0.976621 | + | 0.214970i | −0.715183 | − | 1.34898i |
79.1 | −0.907575 | − | 0.419889i | 0.947653 | − | 0.319302i | 0.647386 | + | 0.762162i | −1.91571 | + | 1.15265i | −0.994138 | − | 0.108119i | 0.0631741 | + | 1.16518i | −0.267528 | − | 0.963550i | 0.796093 | − | 0.605174i | 2.22264 | − | 0.241727i |
79.2 | −0.907575 | − | 0.419889i | 0.947653 | − | 0.319302i | 0.647386 | + | 0.762162i | 1.82294 | − | 1.09682i | −0.994138 | − | 0.108119i | −0.192354 | − | 3.54777i | −0.267528 | − | 0.963550i | 0.796093 | − | 0.605174i | −2.11500 | + | 0.230020i |
85.1 | −0.796093 | + | 0.605174i | −0.468408 | + | 0.883512i | 0.267528 | − | 0.963550i | −1.39125 | + | 1.31786i | −0.161782 | − | 0.986827i | 1.52711 | + | 1.79785i | 0.370138 | + | 0.928977i | −0.561187 | − | 0.827689i | 0.310029 | − | 1.89109i |
85.2 | −0.796093 | + | 0.605174i | −0.468408 | + | 0.883512i | 0.267528 | − | 0.963550i | 0.451252 | − | 0.427449i | −0.161782 | − | 0.986827i | −1.39072 | − | 1.63728i | 0.370138 | + | 0.928977i | −0.561187 | − | 0.827689i | −0.100558 | + | 0.613375i |
121.1 | −0.907575 | + | 0.419889i | 0.947653 | + | 0.319302i | 0.647386 | − | 0.762162i | −1.91571 | − | 1.15265i | −0.994138 | + | 0.108119i | 0.0631741 | − | 1.16518i | −0.267528 | + | 0.963550i | 0.796093 | + | 0.605174i | 2.22264 | + | 0.241727i |
121.2 | −0.907575 | + | 0.419889i | 0.947653 | + | 0.319302i | 0.647386 | − | 0.762162i | 1.82294 | + | 1.09682i | −0.994138 | + | 0.108119i | −0.192354 | + | 3.54777i | −0.267528 | + | 0.963550i | 0.796093 | + | 0.605174i | −2.11500 | − | 0.230020i |
127.1 | −0.647386 | + | 0.762162i | −0.796093 | − | 0.605174i | −0.161782 | − | 0.986827i | −1.07211 | − | 2.02222i | 0.976621 | − | 0.214970i | −1.17973 | − | 0.128303i | 0.856857 | + | 0.515554i | 0.267528 | + | 0.963550i | 2.23533 | + | 0.492034i |
127.2 | −0.647386 | + | 0.762162i | −0.796093 | − | 0.605174i | −0.161782 | − | 0.986827i | 0.140789 | + | 0.265557i | 0.976621 | − | 0.214970i | −0.694075 | − | 0.0754852i | 0.856857 | + | 0.515554i | 0.267528 | + | 0.963550i | −0.293542 | − | 0.0646136i |
133.1 | 0.994138 | − | 0.108119i | −0.647386 | − | 0.762162i | 0.976621 | − | 0.214970i | −2.85024 | + | 2.16670i | −0.725995 | − | 0.687699i | 1.29781 | + | 3.25725i | 0.947653 | − | 0.319302i | −0.161782 | + | 0.986827i | −2.59927 | + | 2.46216i |
133.2 | 0.994138 | − | 0.108119i | −0.647386 | − | 0.762162i | 0.976621 | − | 0.214970i | 2.26091 | − | 1.71870i | −0.725995 | − | 0.687699i | 0.732533 | + | 1.83852i | 0.947653 | − | 0.319302i | −0.161782 | + | 0.986827i | 2.06183 | − | 1.95307i |
139.1 | 0.856857 | + | 0.515554i | 0.370138 | − | 0.928977i | 0.468408 | + | 0.883512i | −1.41393 | − | 0.153774i | 0.796093 | − | 0.605174i | 2.98026 | + | 1.00417i | −0.0541389 | + | 0.998533i | −0.725995 | − | 0.687699i | −1.13226 | − | 0.860721i |
139.2 | 0.856857 | + | 0.515554i | 0.370138 | − | 0.928977i | 0.468408 | + | 0.883512i | 3.29813 | + | 0.358693i | 0.796093 | − | 0.605174i | −0.183400 | − | 0.0617948i | −0.0541389 | + | 0.998533i | −0.725995 | − | 0.687699i | 2.64110 | + | 2.00771i |
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.c | even | 29 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 354.2.e.b | ✓ | 56 |
59.c | even | 29 | 1 | inner | 354.2.e.b | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
354.2.e.b | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
354.2.e.b | ✓ | 56 | 59.c | even | 29 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{56} + 2 T_{5}^{55} - 25 T_{5}^{54} - 166 T_{5}^{53} + 74 T_{5}^{52} + 3193 T_{5}^{51} + 9373 T_{5}^{50} - 19592 T_{5}^{49} - 170496 T_{5}^{48} - 248192 T_{5}^{47} + 1133735 T_{5}^{46} + 5256384 T_{5}^{45} + \cdots + 8873553280201 \)
acting on \(S_{2}^{\mathrm{new}}(354, [\chi])\).