Newspace parameters
Level: | \( N \) | = | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | = | \( 2 \) |
Character orbit: | \([\chi]\) | = | 403.k (of order \(5\) and degree \(4\)) |
Newform invariants
Self dual: | No |
Analytic conductor: | \(3.21797120146\) |
Analytic rank: | \(1\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{10})\) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).
\(n\) | \(249\) | \(313\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
66.1 |
|
−0.500000 | − | 1.53884i | −0.809017 | + | 2.48990i | −0.500000 | + | 0.363271i | −0.381966 | 4.23607 | −2.30902 | + | 1.67760i | −1.80902 | − | 1.31433i | −3.11803 | − | 2.26538i | 0.190983 | + | 0.587785i | ||||||||||||||||
157.1 | −0.500000 | − | 0.363271i | 0.309017 | − | 0.224514i | −0.500000 | − | 1.53884i | −2.61803 | −0.236068 | −1.19098 | − | 3.66547i | −0.690983 | + | 2.12663i | −0.881966 | + | 2.71441i | 1.30902 | + | 0.951057i | |||||||||||||||||
287.1 | −0.500000 | + | 1.53884i | −0.809017 | − | 2.48990i | −0.500000 | − | 0.363271i | −0.381966 | 4.23607 | −2.30902 | − | 1.67760i | −1.80902 | + | 1.31433i | −3.11803 | + | 2.26538i | 0.190983 | − | 0.587785i | |||||||||||||||||
326.1 | −0.500000 | + | 0.363271i | 0.309017 | + | 0.224514i | −0.500000 | + | 1.53884i | −2.61803 | −0.236068 | −1.19098 | + | 3.66547i | −0.690983 | − | 2.12663i | −0.881966 | − | 2.71441i | 1.30902 | − | 0.951057i |
Inner twists
Char. orbit | Parity | Mult. | Self Twist | Proved |
---|---|---|---|---|
1.a | Even | 1 | trivial | yes |
31.d | Even | 1 | yes |
Hecke kernels
This newform can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\).