Newspace parameters
Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 105.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.838429221223\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
\(n\) | \(22\) | \(31\) | \(71\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{12}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 |
|
−0.366025 | + | 0.633975i | 0.500000 | + | 0.866025i | 0.732051 | + | 1.26795i | −0.500000 | + | 0.866025i | −0.732051 | −0.866025 | − | 2.50000i | −2.53590 | −0.500000 | + | 0.866025i | −0.366025 | − | 0.633975i | ||||||||||||||||
16.2 | 1.36603 | − | 2.36603i | 0.500000 | + | 0.866025i | −2.73205 | − | 4.73205i | −0.500000 | + | 0.866025i | 2.73205 | 0.866025 | + | 2.50000i | −9.46410 | −0.500000 | + | 0.866025i | 1.36603 | + | 2.36603i | |||||||||||||||||
46.1 | −0.366025 | − | 0.633975i | 0.500000 | − | 0.866025i | 0.732051 | − | 1.26795i | −0.500000 | − | 0.866025i | −0.732051 | −0.866025 | + | 2.50000i | −2.53590 | −0.500000 | − | 0.866025i | −0.366025 | + | 0.633975i | |||||||||||||||||
46.2 | 1.36603 | + | 2.36603i | 0.500000 | − | 0.866025i | −2.73205 | + | 4.73205i | −0.500000 | − | 0.866025i | 2.73205 | 0.866025 | − | 2.50000i | −9.46410 | −0.500000 | − | 0.866025i | 1.36603 | − | 2.36603i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 105.2.i.d | ✓ | 4 |
3.b | odd | 2 | 1 | 315.2.j.c | 4 | ||
4.b | odd | 2 | 1 | 1680.2.bg.o | 4 | ||
5.b | even | 2 | 1 | 525.2.i.f | 4 | ||
5.c | odd | 4 | 1 | 525.2.r.a | 4 | ||
5.c | odd | 4 | 1 | 525.2.r.f | 4 | ||
7.b | odd | 2 | 1 | 735.2.i.l | 4 | ||
7.c | even | 3 | 1 | inner | 105.2.i.d | ✓ | 4 |
7.c | even | 3 | 1 | 735.2.a.g | 2 | ||
7.d | odd | 6 | 1 | 735.2.a.h | 2 | ||
7.d | odd | 6 | 1 | 735.2.i.l | 4 | ||
21.g | even | 6 | 1 | 2205.2.a.ba | 2 | ||
21.h | odd | 6 | 1 | 315.2.j.c | 4 | ||
21.h | odd | 6 | 1 | 2205.2.a.z | 2 | ||
28.g | odd | 6 | 1 | 1680.2.bg.o | 4 | ||
35.i | odd | 6 | 1 | 3675.2.a.be | 2 | ||
35.j | even | 6 | 1 | 525.2.i.f | 4 | ||
35.j | even | 6 | 1 | 3675.2.a.bg | 2 | ||
35.l | odd | 12 | 1 | 525.2.r.a | 4 | ||
35.l | odd | 12 | 1 | 525.2.r.f | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.2.i.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
105.2.i.d | ✓ | 4 | 7.c | even | 3 | 1 | inner |
315.2.j.c | 4 | 3.b | odd | 2 | 1 | ||
315.2.j.c | 4 | 21.h | odd | 6 | 1 | ||
525.2.i.f | 4 | 5.b | even | 2 | 1 | ||
525.2.i.f | 4 | 35.j | even | 6 | 1 | ||
525.2.r.a | 4 | 5.c | odd | 4 | 1 | ||
525.2.r.a | 4 | 35.l | odd | 12 | 1 | ||
525.2.r.f | 4 | 5.c | odd | 4 | 1 | ||
525.2.r.f | 4 | 35.l | odd | 12 | 1 | ||
735.2.a.g | 2 | 7.c | even | 3 | 1 | ||
735.2.a.h | 2 | 7.d | odd | 6 | 1 | ||
735.2.i.l | 4 | 7.b | odd | 2 | 1 | ||
735.2.i.l | 4 | 7.d | odd | 6 | 1 | ||
1680.2.bg.o | 4 | 4.b | odd | 2 | 1 | ||
1680.2.bg.o | 4 | 28.g | odd | 6 | 1 | ||
2205.2.a.z | 2 | 21.h | odd | 6 | 1 | ||
2205.2.a.ba | 2 | 21.g | even | 6 | 1 | ||
3675.2.a.be | 2 | 35.i | odd | 6 | 1 | ||
3675.2.a.bg | 2 | 35.j | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \)
$3$
\( (T^{2} - T + 1)^{2} \)
$5$
\( (T^{2} + T + 1)^{2} \)
$7$
\( T^{4} + 11T^{2} + 49 \)
$11$
\( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \)
$13$
\( (T^{2} - 8 T + 13)^{2} \)
$17$
\( T^{4} + 10 T^{3} + 78 T^{2} + \cdots + 484 \)
$19$
\( T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121 \)
$23$
\( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \)
$29$
\( (T^{2} - 2 T - 26)^{2} \)
$31$
\( T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9 \)
$37$
\( T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529 \)
$41$
\( (T^{2} - 2 T - 2)^{2} \)
$43$
\( (T^{2} + 4 T - 23)^{2} \)
$47$
\( (T^{2} + 2 T + 4)^{2} \)
$53$
\( T^{4} + 4 T^{3} + 120 T^{2} + \cdots + 10816 \)
$59$
\( T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4 \)
$61$
\( (T^{2} + 4 T + 16)^{2} \)
$67$
\( T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521 \)
$71$
\( (T^{2} - 2 T - 26)^{2} \)
$73$
\( T^{4} + 8 T^{3} + 123 T^{2} + \cdots + 3481 \)
$79$
\( T^{4} + 6 T^{3} + 135 T^{2} + \cdots + 9801 \)
$83$
\( (T^{2} - 6 T - 138)^{2} \)
$89$
\( T^{4} + 6 T^{3} + 174 T^{2} + \cdots + 19044 \)
$97$
\( (T^{2} - 16 T + 16)^{2} \)
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