Newspace parameters
Level: | \( N \) | = | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | = | \( 2 \) |
Character orbit: | \([\chi]\) | = | 403.c (of order \(2\) and degree \(1\)) |
Newform invariants
Self dual: | No |
Analytic conductor: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
311.1 | − | 2.71381i | 0.265781 | −5.36475 | 0.954130i | − | 0.721280i | 4.28452i | 9.13130i | −2.92936 | 2.58933 | ||||||||||||||||
311.2 | − | 2.67507i | −1.67043 | −5.15599 | − | 2.86564i | 4.46851i | − | 5.21058i | 8.44248i | −0.209666 | −7.66579 | |||||||||||||||
311.3 | − | 2.45635i | 2.80039 | −4.03363 | 3.48535i | − | 6.87871i | − | 3.54003i | 4.99531i | 4.84216 | 8.56123 | |||||||||||||||
311.4 | − | 2.44691i | 2.35527 | −3.98738 | − | 2.67528i | − | 5.76313i | 0.955539i | 4.86294i | 2.54728 | −6.54617 | |||||||||||||||
311.5 | − | 2.21678i | −2.59178 | −2.91410 | 1.00571i | 5.74540i | 1.82096i | 2.02635i | 3.71732 | 2.22943 | |||||||||||||||||
311.6 | − | 1.99449i | −1.58843 | −1.97798 | − | 2.66789i | 3.16810i | 1.73927i | − | 0.0439183i | −0.476895 | −5.32107 | |||||||||||||||
311.7 | − | 1.88981i | −0.327950 | −1.57140 | 1.75790i | 0.619765i | − | 0.652358i | − | 0.809976i | −2.89245 | 3.32211 | |||||||||||||||
311.8 | − | 1.48947i | 2.99747 | −0.218531 | 0.692562i | − | 4.46466i | 1.15260i | − | 2.65345i | 5.98485 | 1.03155 | |||||||||||||||
311.9 | − | 1.44935i | 0.0382575 | −0.100611 | 3.85977i | − | 0.0554484i | 1.66255i | − | 2.75288i | −2.99854 | 5.59415 | |||||||||||||||
311.10 | − | 1.44015i | −2.79139 | −0.0740342 | 2.61757i | 4.02002i | − | 4.99660i | − | 2.77368i | 4.79185 | 3.76969 | |||||||||||||||
311.11 | − | 1.40506i | 1.52886 | 0.0258025 | − | 1.01618i | − | 2.14814i | − | 2.90958i | − | 2.84638i | −0.662589 | −1.42779 | |||||||||||||
311.12 | − | 1.01245i | −1.05136 | 0.974950 | − | 3.24438i | 1.06445i | − | 1.79572i | − | 3.01198i | −1.89463 | −3.28477 | ||||||||||||||
311.13 | − | 0.937316i | 1.68575 | 1.12144 | 0.290760i | − | 1.58008i | 3.85549i | − | 2.92577i | −0.158249 | 0.272534 | |||||||||||||||
311.14 | − | 0.624841i | −2.78388 | 1.60957 | − | 0.619834i | 1.73948i | 2.35216i | − | 2.25541i | 4.74999 | −0.387298 | |||||||||||||||
311.15 | − | 0.475674i | 0.345654 | 1.77373 | − | 0.161594i | − | 0.164419i | − | 4.07996i | − | 1.79507i | −2.88052 | −0.0768661 | |||||||||||||
311.16 | − | 0.327248i | −1.21221 | 1.89291 | − | 2.01769i | 0.396693i | 0.699230i | − | 1.27395i | −1.53055 | −0.660284 | |||||||||||||||
311.17 | 0.327248i | −1.21221 | 1.89291 | 2.01769i | − | 0.396693i | − | 0.699230i | 1.27395i | −1.53055 | −0.660284 | ||||||||||||||||
311.18 | 0.475674i | 0.345654 | 1.77373 | 0.161594i | 0.164419i | 4.07996i | 1.79507i | −2.88052 | −0.0768661 | ||||||||||||||||||
311.19 | 0.624841i | −2.78388 | 1.60957 | 0.619834i | − | 1.73948i | − | 2.35216i | 2.25541i | 4.74999 | −0.387298 | ||||||||||||||||
311.20 | 0.937316i | 1.68575 | 1.12144 | − | 0.290760i | 1.58008i | − | 3.85549i | 2.92577i | −0.158249 | 0.272534 | ||||||||||||||||
See all 32 embeddings |
Inner twists
This newform does not have CM; other inner twists have not been computed.
Hecke kernels
This newform can be constructed as the kernel of the linear operator \(T_{2}^{32} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\).