Newspace parameters
Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 33.e (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.263506326670\) |
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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
\(n\) | \(13\) | \(23\) |
\(\chi(n)\) | \(\zeta_{10}^{2}\) | \(1\) |
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 |
|
−1.30902 | − | 0.951057i | 0.309017 | − | 0.951057i | 0.190983 | + | 0.587785i | 0.309017 | − | 0.224514i | −1.30902 | + | 0.951057i | 0.927051 | + | 2.85317i | −0.690983 | + | 2.12663i | −0.809017 | − | 0.587785i | −0.618034 | ||||||||||||||
16.1 | −0.190983 | − | 0.587785i | −0.809017 | − | 0.587785i | 1.30902 | − | 0.951057i | −0.809017 | + | 2.48990i | −0.190983 | + | 0.587785i | −2.42705 | + | 1.76336i | −1.80902 | − | 1.31433i | 0.309017 | + | 0.951057i | 1.61803 | |||||||||||||||
25.1 | −1.30902 | + | 0.951057i | 0.309017 | + | 0.951057i | 0.190983 | − | 0.587785i | 0.309017 | + | 0.224514i | −1.30902 | − | 0.951057i | 0.927051 | − | 2.85317i | −0.690983 | − | 2.12663i | −0.809017 | + | 0.587785i | −0.618034 | |||||||||||||||
31.1 | −0.190983 | + | 0.587785i | −0.809017 | + | 0.587785i | 1.30902 | + | 0.951057i | −0.809017 | − | 2.48990i | −0.190983 | − | 0.587785i | −2.42705 | − | 1.76336i | −1.80902 | + | 1.31433i | 0.309017 | − | 0.951057i | 1.61803 | |||||||||||||||
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 | 33.2.e.a | ✓ | 4 |
3.b | odd | 2 | 1 | 99.2.f.b | 4 | ||
4.b | odd | 2 | 1 | 528.2.y.f | 4 | ||
5.b | even | 2 | 1 | 825.2.n.f | 4 | ||
5.c | odd | 4 | 2 | 825.2.bx.b | 8 | ||
9.c | even | 3 | 2 | 891.2.n.d | 8 | ||
9.d | odd | 6 | 2 | 891.2.n.a | 8 | ||
11.b | odd | 2 | 1 | 363.2.e.j | 4 | ||
11.c | even | 5 | 1 | inner | 33.2.e.a | ✓ | 4 |
11.c | even | 5 | 1 | 363.2.a.h | 2 | ||
11.c | even | 5 | 2 | 363.2.e.h | 4 | ||
11.d | odd | 10 | 1 | 363.2.a.e | 2 | ||
11.d | odd | 10 | 2 | 363.2.e.c | 4 | ||
11.d | odd | 10 | 1 | 363.2.e.j | 4 | ||
33.f | even | 10 | 1 | 1089.2.a.s | 2 | ||
33.h | odd | 10 | 1 | 99.2.f.b | 4 | ||
33.h | odd | 10 | 1 | 1089.2.a.m | 2 | ||
44.g | even | 10 | 1 | 5808.2.a.bm | 2 | ||
44.h | odd | 10 | 1 | 528.2.y.f | 4 | ||
44.h | odd | 10 | 1 | 5808.2.a.bl | 2 | ||
55.h | odd | 10 | 1 | 9075.2.a.bv | 2 | ||
55.j | even | 10 | 1 | 825.2.n.f | 4 | ||
55.j | even | 10 | 1 | 9075.2.a.x | 2 | ||
55.k | odd | 20 | 2 | 825.2.bx.b | 8 | ||
99.m | even | 15 | 2 | 891.2.n.d | 8 | ||
99.n | odd | 30 | 2 | 891.2.n.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.2.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
33.2.e.a | ✓ | 4 | 11.c | even | 5 | 1 | inner |
99.2.f.b | 4 | 3.b | odd | 2 | 1 | ||
99.2.f.b | 4 | 33.h | odd | 10 | 1 | ||
363.2.a.e | 2 | 11.d | odd | 10 | 1 | ||
363.2.a.h | 2 | 11.c | even | 5 | 1 | ||
363.2.e.c | 4 | 11.d | odd | 10 | 2 | ||
363.2.e.h | 4 | 11.c | even | 5 | 2 | ||
363.2.e.j | 4 | 11.b | odd | 2 | 1 | ||
363.2.e.j | 4 | 11.d | odd | 10 | 1 | ||
528.2.y.f | 4 | 4.b | odd | 2 | 1 | ||
528.2.y.f | 4 | 44.h | odd | 10 | 1 | ||
825.2.n.f | 4 | 5.b | even | 2 | 1 | ||
825.2.n.f | 4 | 55.j | even | 10 | 1 | ||
825.2.bx.b | 8 | 5.c | odd | 4 | 2 | ||
825.2.bx.b | 8 | 55.k | odd | 20 | 2 | ||
891.2.n.a | 8 | 9.d | odd | 6 | 2 | ||
891.2.n.a | 8 | 99.n | odd | 30 | 2 | ||
891.2.n.d | 8 | 9.c | even | 3 | 2 | ||
891.2.n.d | 8 | 99.m | even | 15 | 2 | ||
1089.2.a.m | 2 | 33.h | odd | 10 | 1 | ||
1089.2.a.s | 2 | 33.f | even | 10 | 1 | ||
5808.2.a.bl | 2 | 44.h | odd | 10 | 1 | ||
5808.2.a.bm | 2 | 44.g | even | 10 | 1 | ||
9075.2.a.x | 2 | 55.j | even | 10 | 1 | ||
9075.2.a.bv | 2 | 55.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \)
$3$
\( T^{4} + T^{3} + T^{2} + T + 1 \)
$5$
\( T^{4} + T^{3} + 6 T^{2} - 4 T + 1 \)
$7$
\( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \)
$11$
\( T^{4} - 9 T^{3} + 41 T^{2} - 99 T + 121 \)
$13$
\( T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121 \)
$17$
\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \)
$19$
\( T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 \)
$23$
\( (T^{2} + 2 T - 19)^{2} \)
$29$
\( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \)
$31$
\( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \)
$37$
\( T^{4} + 3 T^{3} + 19 T^{2} + 7 T + 1 \)
$41$
\( T^{4} - 23 T^{3} + 249 T^{2} + \cdots + 5041 \)
$43$
\( (T^{2} - 8 T + 11)^{2} \)
$47$
\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \)
$53$
\( T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 \)
$59$
\( T^{4} + 20 T^{3} + 190 T^{2} + \cdots + 3025 \)
$61$
\( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \)
$67$
\( (T^{2} - T - 101)^{2} \)
$71$
\( T^{4} + 27 T^{3} + 324 T^{2} + \cdots + 6561 \)
$73$
\( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \)
$79$
\( T^{4} - 5 T^{3} + 85 T^{2} + 75 T + 25 \)
$83$
\( T^{4} - 21 T^{3} + 171 T^{2} + \cdots + 81 \)
$89$
\( (T^{2} - 10 T + 5)^{2} \)
$97$
\( T^{4} + 33 T^{3} + 634 T^{2} + \cdots + 44521 \)
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