Newspace parameters
| Level: | \( N \) | \(=\) | \( 66 = 2 \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 66.e (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.527012653340\) |
| 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: | \(\Z[a_1, a_2]\) |
| 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 \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(\zeta_{10}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −1.11803 | − | 0.812299i | 0.809017 | + | 0.587785i | −0.500000 | + | 1.53884i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −1.38197 | ||||||||||||||
| 31.1 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 1.11803 | + | 3.44095i | −0.309017 | − | 0.951057i | −0.500000 | − | 0.363271i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | −3.61803 | |||||||||||||||
| 37.1 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −1.11803 | + | 0.812299i | 0.809017 | − | 0.587785i | −0.500000 | − | 1.53884i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −1.38197 | |||||||||||||||
| 49.1 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 1.11803 | − | 3.44095i | −0.309017 | + | 0.951057i | −0.500000 | + | 0.363271i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | −3.61803 | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 66.2.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 198.2.f.a | 4 | ||
| 4.b | odd | 2 | 1 | 528.2.y.g | 4 | ||
| 11.b | odd | 2 | 1 | 726.2.e.c | 4 | ||
| 11.c | even | 5 | 1 | inner | 66.2.e.b | ✓ | 4 |
| 11.c | even | 5 | 1 | 726.2.a.k | 2 | ||
| 11.c | even | 5 | 2 | 726.2.e.j | 4 | ||
| 11.d | odd | 10 | 1 | 726.2.a.m | 2 | ||
| 11.d | odd | 10 | 2 | 726.2.e.a | 4 | ||
| 11.d | odd | 10 | 1 | 726.2.e.c | 4 | ||
| 33.f | even | 10 | 1 | 2178.2.a.o | 2 | ||
| 33.h | odd | 10 | 1 | 198.2.f.a | 4 | ||
| 33.h | odd | 10 | 1 | 2178.2.a.v | 2 | ||
| 44.g | even | 10 | 1 | 5808.2.a.by | 2 | ||
| 44.h | odd | 10 | 1 | 528.2.y.g | 4 | ||
| 44.h | odd | 10 | 1 | 5808.2.a.bz | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 66.2.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 66.2.e.b | ✓ | 4 | 11.c | even | 5 | 1 | inner |
| 198.2.f.a | 4 | 3.b | odd | 2 | 1 | ||
| 198.2.f.a | 4 | 33.h | odd | 10 | 1 | ||
| 528.2.y.g | 4 | 4.b | odd | 2 | 1 | ||
| 528.2.y.g | 4 | 44.h | odd | 10 | 1 | ||
| 726.2.a.k | 2 | 11.c | even | 5 | 1 | ||
| 726.2.a.m | 2 | 11.d | odd | 10 | 1 | ||
| 726.2.e.a | 4 | 11.d | odd | 10 | 2 | ||
| 726.2.e.c | 4 | 11.b | odd | 2 | 1 | ||
| 726.2.e.c | 4 | 11.d | odd | 10 | 1 | ||
| 726.2.e.j | 4 | 11.c | even | 5 | 2 | ||
| 2178.2.a.o | 2 | 33.f | even | 10 | 1 | ||
| 2178.2.a.v | 2 | 33.h | odd | 10 | 1 | ||
| 5808.2.a.by | 2 | 44.g | even | 10 | 1 | ||
| 5808.2.a.bz | 2 | 44.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 10T_{5}^{2} + 25T_{5} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(66, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$3$
\( T^{4} + T^{3} + T^{2} + T + 1 \)
$5$
\( T^{4} + 10 T^{2} + 25 T + 25 \)
$7$
\( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \)
$11$
\( T^{4} + 4 T^{3} + 6 T^{2} + 44 T + 121 \)
$13$
\( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \)
$17$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$19$
\( T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936 \)
$23$
\( (T^{2} - 2 T - 44)^{2} \)
$29$
\( T^{4} + 11 T^{3} + 46 T^{2} - 4 T + 1 \)
$31$
\( T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361 \)
$37$
\( T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400 \)
$41$
\( T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936 \)
$43$
\( (T^{2} + 14 T + 44)^{2} \)
$47$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$53$
\( T^{4} + 5 T^{3} + 10 T^{2} + 25 \)
$59$
\( T^{4} + 20 T^{3} + 190 T^{2} + \cdots + 3025 \)
$61$
\( T^{4} + 18 T^{3} + 144 T^{2} + \cdots + 1296 \)
$67$
\( (T^{2} + 8 T - 64)^{2} \)
$71$
\( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$73$
\( T^{4} + 27 T^{3} + 324 T^{2} + \cdots + 6561 \)
$79$
\( T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1 \)
$83$
\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \)
$89$
\( (T^{2} - 18 T + 76)^{2} \)
$97$
\( T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 \)
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