Newspace parameters
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.e (of order \(5\), degree \(4\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.89856959337\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 124.1 |
|
−1.61803 | + | 1.17557i | −0.309017 | − | 0.951057i | 0.618034 | − | 1.90211i | −3.23607 | − | 2.35114i | 1.61803 | + | 1.17557i | −0.309017 | + | 0.951057i | 0 | −0.809017 | + | 0.587785i | 8.00000 | ||||||||||||||||
| 130.1 | 0.618034 | − | 1.90211i | 0.809017 | − | 0.587785i | −1.61803 | − | 1.17557i | 1.23607 | + | 3.80423i | −0.618034 | − | 1.90211i | 0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 8.00000 | |||||||||||||||||
| 148.1 | 0.618034 | + | 1.90211i | 0.809017 | + | 0.587785i | −1.61803 | + | 1.17557i | 1.23607 | − | 3.80423i | −0.618034 | + | 1.90211i | 0.809017 | − | 0.587785i | 0 | 0.309017 | + | 0.951057i | 8.00000 | |||||||||||||||||
| 202.1 | −1.61803 | − | 1.17557i | −0.309017 | + | 0.951057i | 0.618034 | + | 1.90211i | −3.23607 | + | 2.35114i | 1.61803 | − | 1.17557i | −0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | 8.00000 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.2.e.d | 4 | |
| 11.b | odd | 2 | 1 | 363.2.e.i | 4 | ||
| 11.c | even | 5 | 1 | 363.2.a.c | yes | 1 | |
| 11.c | even | 5 | 3 | inner | 363.2.e.d | 4 | |
| 11.d | odd | 10 | 1 | 363.2.a.a | ✓ | 1 | |
| 11.d | odd | 10 | 3 | 363.2.e.i | 4 | ||
| 33.f | even | 10 | 1 | 1089.2.a.k | 1 | ||
| 33.h | odd | 10 | 1 | 1089.2.a.a | 1 | ||
| 44.g | even | 10 | 1 | 5808.2.a.bh | 1 | ||
| 44.h | odd | 10 | 1 | 5808.2.a.bi | 1 | ||
| 55.h | odd | 10 | 1 | 9075.2.a.t | 1 | ||
| 55.j | even | 10 | 1 | 9075.2.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.2.a.a | ✓ | 1 | 11.d | odd | 10 | 1 | |
| 363.2.a.c | yes | 1 | 11.c | even | 5 | 1 | |
| 363.2.e.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 363.2.e.d | 4 | 11.c | even | 5 | 3 | inner | |
| 363.2.e.i | 4 | 11.b | odd | 2 | 1 | ||
| 363.2.e.i | 4 | 11.d | odd | 10 | 3 | ||
| 1089.2.a.a | 1 | 33.h | odd | 10 | 1 | ||
| 1089.2.a.k | 1 | 33.f | even | 10 | 1 | ||
| 5808.2.a.bh | 1 | 44.g | even | 10 | 1 | ||
| 5808.2.a.bi | 1 | 44.h | odd | 10 | 1 | ||
| 9075.2.a.b | 1 | 55.j | even | 10 | 1 | ||
| 9075.2.a.t | 1 | 55.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 8T_{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$3$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$5$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$7$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$17$
\( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \)
$19$
\( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \)
$23$
\( (T - 2)^{4} \)
$29$
\( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$31$
\( T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625 \)
$37$
\( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \)
$41$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$43$
\( (T + 12)^{4} \)
$47$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$53$
\( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$59$
\( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \)
$61$
\( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \)
$67$
\( (T + 1)^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641 \)
$79$
\( T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641 \)
$83$
\( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$89$
\( (T - 12)^{4} \)
$97$
\( T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625 \)
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