Newspace parameters
Level: | \( N \) | \(=\) | \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 3100.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(84.4688819517\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
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: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{5}\cdot 5^{2} \) |
Twist minimal: | no (minimal twist has level 124) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( -2\zeta_{12}^{3} + 4\zeta_{12} \) |
\(\beta_{2}\) | \(=\) | \( 10\zeta_{12}^{3} \) |
\(\beta_{3}\) | \(=\) | \( 20\zeta_{12}^{2} - 10 \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{2} + 5\beta_1 ) / 20 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( ( \beta_{3} + 10 ) / 20 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( ( \beta_{2} ) / 10 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times\).
\(n\) | \(1551\) | \(1801\) | \(2977\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1549.1 |
|
0 | −3.46410 | 0 | 0 | 0 | − | 10.0000i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||
1549.2 | 0 | −3.46410 | 0 | 0 | 0 | 10.0000i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||
1549.3 | 0 | 3.46410 | 0 | 0 | 0 | − | 10.0000i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||
1549.4 | 0 | 3.46410 | 0 | 0 | 0 | 10.0000i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
31.b | odd | 2 | 1 | inner |
155.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3100.3.f.a | 4 | |
5.b | even | 2 | 1 | inner | 3100.3.f.a | 4 | |
5.c | odd | 4 | 1 | 124.3.c.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 3100.3.d.a | 2 | ||
15.e | even | 4 | 1 | 1116.3.h.c | 2 | ||
20.e | even | 4 | 1 | 496.3.e.a | 2 | ||
31.b | odd | 2 | 1 | inner | 3100.3.f.a | 4 | |
155.c | odd | 2 | 1 | inner | 3100.3.f.a | 4 | |
155.f | even | 4 | 1 | 124.3.c.a | ✓ | 2 | |
155.f | even | 4 | 1 | 3100.3.d.a | 2 | ||
465.m | odd | 4 | 1 | 1116.3.h.c | 2 | ||
620.m | odd | 4 | 1 | 496.3.e.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
124.3.c.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
124.3.c.a | ✓ | 2 | 155.f | even | 4 | 1 | |
496.3.e.a | 2 | 20.e | even | 4 | 1 | ||
496.3.e.a | 2 | 620.m | odd | 4 | 1 | ||
1116.3.h.c | 2 | 15.e | even | 4 | 1 | ||
1116.3.h.c | 2 | 465.m | odd | 4 | 1 | ||
3100.3.d.a | 2 | 5.c | odd | 4 | 1 | ||
3100.3.d.a | 2 | 155.f | even | 4 | 1 | ||
3100.3.f.a | 4 | 1.a | even | 1 | 1 | trivial | |
3100.3.f.a | 4 | 5.b | even | 2 | 1 | inner | |
3100.3.f.a | 4 | 31.b | odd | 2 | 1 | inner | |
3100.3.f.a | 4 | 155.c | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 12 \)
acting on \(S_{3}^{\mathrm{new}}(3100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 12)^{2} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 100)^{2} \)
$11$
\( (T^{2} + 300)^{2} \)
$13$
\( (T^{2} - 300)^{2} \)
$17$
\( (T^{2} - 192)^{2} \)
$19$
\( (T - 14)^{4} \)
$23$
\( (T^{2} - 1200)^{2} \)
$29$
\( (T^{2} + 300)^{2} \)
$31$
\( (T + 31)^{4} \)
$37$
\( (T^{2} - 12)^{2} \)
$41$
\( (T - 54)^{4} \)
$43$
\( (T^{2} - 5292)^{2} \)
$47$
\( (T^{2} + 900)^{2} \)
$53$
\( (T^{2} - 972)^{2} \)
$59$
\( (T - 6)^{4} \)
$61$
\( (T^{2} + 300)^{2} \)
$67$
\( (T^{2} + 12100)^{2} \)
$71$
\( (T - 66)^{4} \)
$73$
\( (T^{2} - 8112)^{2} \)
$79$
\( (T^{2} + 4800)^{2} \)
$83$
\( (T^{2} - 1452)^{2} \)
$89$
\( (T^{2} + 1200)^{2} \)
$97$
\( (T^{2} + 12100)^{2} \)
show more
show less