Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.e (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.94706303019\) |
| 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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(\zeta_{10}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
3.61803 | + | 2.62866i | 0.927051 | − | 2.85317i | 3.70820 | + | 11.4127i | −4.69098 | + | 3.40820i | 10.8541 | − | 7.88597i | −4.72542 | − | 14.5434i | −5.52786 | + | 17.0130i | −7.28115 | − | 5.29007i | −25.9311 | ||||||||||||||
| 16.1 | 1.38197 | + | 4.25325i | −2.42705 | − | 1.76336i | −9.70820 | + | 7.05342i | −5.80902 | + | 17.8783i | 4.14590 | − | 12.7598i | 23.2254 | − | 16.8743i | −14.4721 | − | 10.5146i | 2.78115 | + | 8.55951i | −84.0689 | |||||||||||||||
| 25.1 | 3.61803 | − | 2.62866i | 0.927051 | + | 2.85317i | 3.70820 | − | 11.4127i | −4.69098 | − | 3.40820i | 10.8541 | + | 7.88597i | −4.72542 | + | 14.5434i | −5.52786 | − | 17.0130i | −7.28115 | + | 5.29007i | −25.9311 | |||||||||||||||
| 31.1 | 1.38197 | − | 4.25325i | −2.42705 | + | 1.76336i | −9.70820 | − | 7.05342i | −5.80902 | − | 17.8783i | 4.14590 | + | 12.7598i | 23.2254 | + | 16.8743i | −14.4721 | + | 10.5146i | 2.78115 | − | 8.55951i | −84.0689 | |||||||||||||||
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 | 33.4.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 99.4.f.a | 4 | ||
| 11.c | even | 5 | 1 | inner | 33.4.e.a | ✓ | 4 |
| 11.c | even | 5 | 1 | 363.4.a.n | 2 | ||
| 11.d | odd | 10 | 1 | 363.4.a.o | 2 | ||
| 33.f | even | 10 | 1 | 1089.4.a.q | 2 | ||
| 33.h | odd | 10 | 1 | 99.4.f.a | 4 | ||
| 33.h | odd | 10 | 1 | 1089.4.a.p | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.4.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 33.4.e.a | ✓ | 4 | 11.c | even | 5 | 1 | inner |
| 99.4.f.a | 4 | 3.b | odd | 2 | 1 | ||
| 99.4.f.a | 4 | 33.h | odd | 10 | 1 | ||
| 363.4.a.n | 2 | 11.c | even | 5 | 1 | ||
| 363.4.a.o | 2 | 11.d | odd | 10 | 1 | ||
| 1089.4.a.p | 2 | 33.h | odd | 10 | 1 | ||
| 1089.4.a.q | 2 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 10T_{2}^{3} + 60T_{2}^{2} - 200T_{2} + 400 \)
acting on \(S_{4}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400 \)
$3$
\( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \)
$5$
\( T^{4} + 21 T^{3} + 496 T^{2} + \cdots + 11881 \)
$7$
\( T^{4} - 37 T^{3} + 619 T^{2} + \cdots + 192721 \)
$11$
\( T^{4} - 41 T^{3} + 1881 T^{2} + \cdots + 1771561 \)
$13$
\( T^{4} + 77 T^{3} + 2299 T^{2} + \cdots + 14641 \)
$17$
\( T^{4} - 192 T^{3} + 14634 T^{2} + \cdots + 2595321 \)
$19$
\( T^{4} + 52 T^{3} + 11254 T^{2} + \cdots + 1274641 \)
$23$
\( (T^{2} + 74 T - 1051)^{2} \)
$29$
\( T^{4} - 414 T^{3} + \cdots + 1740892176 \)
$31$
\( T^{4} + 198 T^{3} + \cdots + 54479161 \)
$37$
\( T^{4} - 201 T^{3} + \cdots + 4216034761 \)
$41$
\( T^{4} + 129 T^{3} + 6271 T^{2} + \cdots + 241081 \)
$43$
\( (T^{2} + 66 T - 7731)^{2} \)
$47$
\( T^{4} + 35 T^{3} + \cdots + 674700625 \)
$53$
\( T^{4} - 188 T^{3} + \cdots + 10042561 \)
$59$
\( T^{4} + 1320 T^{3} + \cdots + 47173668025 \)
$61$
\( T^{4} - 1275 T^{3} + \cdots + 55792802025 \)
$67$
\( (T^{2} - 75 T - 391995)^{2} \)
$71$
\( T^{4} - 117 T^{3} + \cdots + 53332821721 \)
$73$
\( T^{4} + 982 T^{3} + \cdots + 8360542096 \)
$79$
\( T^{4} + 1469 T^{3} + \cdots + 285379255681 \)
$83$
\( T^{4} + 967 T^{3} + \cdots + 53935882081 \)
$89$
\( (T^{2} + 2712 T + 1809091)^{2} \)
$97$
\( T^{4} - 2391 T^{3} + \cdots + 932420053161 \)
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