Newspace parameters
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(23.0054083620\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{15})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 7x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | no (minimal twist has level 4) |
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 in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} - 3\nu ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 11\nu ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( 8\nu^{2} - 28 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + \beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 28 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{2} + 11\beta_1 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
\(n\) | \(51\) | \(77\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
99.1 |
|
−7.74597 | − | 2.00000i | −30.9839 | 56.0000 | + | 30.9839i | 0 | 240.000 | + | 61.9677i | −309.839 | −371.806 | − | 352.000i | 231.000 | 0 | ||||||||||||||||||||||
99.2 | −7.74597 | + | 2.00000i | −30.9839 | 56.0000 | − | 30.9839i | 0 | 240.000 | − | 61.9677i | −309.839 | −371.806 | + | 352.000i | 231.000 | 0 | |||||||||||||||||||||||
99.3 | 7.74597 | − | 2.00000i | 30.9839 | 56.0000 | − | 30.9839i | 0 | 240.000 | − | 61.9677i | 309.839 | 371.806 | − | 352.000i | 231.000 | 0 | |||||||||||||||||||||||
99.4 | 7.74597 | + | 2.00000i | 30.9839 | 56.0000 | + | 30.9839i | 0 | 240.000 | + | 61.9677i | 309.839 | 371.806 | + | 352.000i | 231.000 | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 100.7.d.a | 4 | |
4.b | odd | 2 | 1 | inner | 100.7.d.a | 4 | |
5.b | even | 2 | 1 | inner | 100.7.d.a | 4 | |
5.c | odd | 4 | 1 | 4.7.b.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 100.7.b.c | 2 | ||
15.e | even | 4 | 1 | 36.7.d.c | 2 | ||
20.d | odd | 2 | 1 | inner | 100.7.d.a | 4 | |
20.e | even | 4 | 1 | 4.7.b.a | ✓ | 2 | |
20.e | even | 4 | 1 | 100.7.b.c | 2 | ||
40.i | odd | 4 | 1 | 64.7.c.c | 2 | ||
40.k | even | 4 | 1 | 64.7.c.c | 2 | ||
60.l | odd | 4 | 1 | 36.7.d.c | 2 | ||
80.i | odd | 4 | 1 | 256.7.d.f | 4 | ||
80.j | even | 4 | 1 | 256.7.d.f | 4 | ||
80.s | even | 4 | 1 | 256.7.d.f | 4 | ||
80.t | odd | 4 | 1 | 256.7.d.f | 4 | ||
120.q | odd | 4 | 1 | 576.7.g.h | 2 | ||
120.w | even | 4 | 1 | 576.7.g.h | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.7.b.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
4.7.b.a | ✓ | 2 | 20.e | even | 4 | 1 | |
36.7.d.c | 2 | 15.e | even | 4 | 1 | ||
36.7.d.c | 2 | 60.l | odd | 4 | 1 | ||
64.7.c.c | 2 | 40.i | odd | 4 | 1 | ||
64.7.c.c | 2 | 40.k | even | 4 | 1 | ||
100.7.b.c | 2 | 5.c | odd | 4 | 1 | ||
100.7.b.c | 2 | 20.e | even | 4 | 1 | ||
100.7.d.a | 4 | 1.a | even | 1 | 1 | trivial | |
100.7.d.a | 4 | 4.b | odd | 2 | 1 | inner | |
100.7.d.a | 4 | 5.b | even | 2 | 1 | inner | |
100.7.d.a | 4 | 20.d | odd | 2 | 1 | inner | |
256.7.d.f | 4 | 80.i | odd | 4 | 1 | ||
256.7.d.f | 4 | 80.j | even | 4 | 1 | ||
256.7.d.f | 4 | 80.s | even | 4 | 1 | ||
256.7.d.f | 4 | 80.t | odd | 4 | 1 | ||
576.7.g.h | 2 | 120.q | odd | 4 | 1 | ||
576.7.g.h | 2 | 120.w | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 960 \)
acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 112T^{2} + 4096 \)
$3$
\( (T^{2} - 960)^{2} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - 96000)^{2} \)
$11$
\( (T^{2} + 922560)^{2} \)
$13$
\( (T^{2} + 2149156)^{2} \)
$17$
\( (T^{2} + 22714756)^{2} \)
$19$
\( (T^{2} + 56687040)^{2} \)
$23$
\( (T^{2} - 109674240)^{2} \)
$29$
\( (T + 25498)^{4} \)
$31$
\( (T^{2} + 1754787840)^{2} \)
$37$
\( (T^{2} + 3976036)^{2} \)
$41$
\( (T - 29362)^{4} \)
$43$
\( (T^{2} - 463704000)^{2} \)
$47$
\( (T^{2} - 57154560)^{2} \)
$53$
\( (T^{2} + 37192665316)^{2} \)
$59$
\( (T^{2} + 6149722560)^{2} \)
$61$
\( (T + 10918)^{4} \)
$67$
\( (T^{2} - 155350887360)^{2} \)
$71$
\( (T^{2} + 283280336640)^{2} \)
$73$
\( (T^{2} + 83304967876)^{2} \)
$79$
\( (T^{2} + 96538352640)^{2} \)
$83$
\( (T^{2} - 41678324160)^{2} \)
$89$
\( (T + 310738)^{4} \)
$97$
\( (T^{2} + 2123099611396)^{2} \)
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