Newspace parameters
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(16.0383819813\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2 \) |
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
\(n\) | \(51\) | \(77\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | − | 12.0000i | 0 | 0 | 0 | 88.0000i | 0 | 99.0000 | 0 | |||||||||||||||||||||||
49.2 | 0 | 12.0000i | 0 | 0 | 0 | − | 88.0000i | 0 | 99.0000 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 100.6.c.b | 2 | |
3.b | odd | 2 | 1 | 900.6.d.a | 2 | ||
4.b | odd | 2 | 1 | 400.6.c.f | 2 | ||
5.b | even | 2 | 1 | inner | 100.6.c.b | 2 | |
5.c | odd | 4 | 1 | 4.6.a.a | ✓ | 1 | |
5.c | odd | 4 | 1 | 100.6.a.b | 1 | ||
15.d | odd | 2 | 1 | 900.6.d.a | 2 | ||
15.e | even | 4 | 1 | 36.6.a.a | 1 | ||
15.e | even | 4 | 1 | 900.6.a.h | 1 | ||
20.d | odd | 2 | 1 | 400.6.c.f | 2 | ||
20.e | even | 4 | 1 | 16.6.a.b | 1 | ||
20.e | even | 4 | 1 | 400.6.a.d | 1 | ||
35.f | even | 4 | 1 | 196.6.a.e | 1 | ||
35.k | even | 12 | 2 | 196.6.e.d | 2 | ||
35.l | odd | 12 | 2 | 196.6.e.g | 2 | ||
40.i | odd | 4 | 1 | 64.6.a.f | 1 | ||
40.k | even | 4 | 1 | 64.6.a.b | 1 | ||
45.k | odd | 12 | 2 | 324.6.e.a | 2 | ||
45.l | even | 12 | 2 | 324.6.e.d | 2 | ||
55.e | even | 4 | 1 | 484.6.a.a | 1 | ||
60.l | odd | 4 | 1 | 144.6.a.c | 1 | ||
65.f | even | 4 | 1 | 676.6.d.a | 2 | ||
65.h | odd | 4 | 1 | 676.6.a.a | 1 | ||
65.k | even | 4 | 1 | 676.6.d.a | 2 | ||
80.i | odd | 4 | 1 | 256.6.b.g | 2 | ||
80.j | even | 4 | 1 | 256.6.b.c | 2 | ||
80.s | even | 4 | 1 | 256.6.b.c | 2 | ||
80.t | odd | 4 | 1 | 256.6.b.g | 2 | ||
120.q | odd | 4 | 1 | 576.6.a.bd | 1 | ||
120.w | even | 4 | 1 | 576.6.a.bc | 1 | ||
140.j | odd | 4 | 1 | 784.6.a.d | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.6.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
16.6.a.b | 1 | 20.e | even | 4 | 1 | ||
36.6.a.a | 1 | 15.e | even | 4 | 1 | ||
64.6.a.b | 1 | 40.k | even | 4 | 1 | ||
64.6.a.f | 1 | 40.i | odd | 4 | 1 | ||
100.6.a.b | 1 | 5.c | odd | 4 | 1 | ||
100.6.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
100.6.c.b | 2 | 5.b | even | 2 | 1 | inner | |
144.6.a.c | 1 | 60.l | odd | 4 | 1 | ||
196.6.a.e | 1 | 35.f | even | 4 | 1 | ||
196.6.e.d | 2 | 35.k | even | 12 | 2 | ||
196.6.e.g | 2 | 35.l | odd | 12 | 2 | ||
256.6.b.c | 2 | 80.j | even | 4 | 1 | ||
256.6.b.c | 2 | 80.s | even | 4 | 1 | ||
256.6.b.g | 2 | 80.i | odd | 4 | 1 | ||
256.6.b.g | 2 | 80.t | odd | 4 | 1 | ||
324.6.e.a | 2 | 45.k | odd | 12 | 2 | ||
324.6.e.d | 2 | 45.l | even | 12 | 2 | ||
400.6.a.d | 1 | 20.e | even | 4 | 1 | ||
400.6.c.f | 2 | 4.b | odd | 2 | 1 | ||
400.6.c.f | 2 | 20.d | odd | 2 | 1 | ||
484.6.a.a | 1 | 55.e | even | 4 | 1 | ||
576.6.a.bc | 1 | 120.w | even | 4 | 1 | ||
576.6.a.bd | 1 | 120.q | odd | 4 | 1 | ||
676.6.a.a | 1 | 65.h | odd | 4 | 1 | ||
676.6.d.a | 2 | 65.f | even | 4 | 1 | ||
676.6.d.a | 2 | 65.k | even | 4 | 1 | ||
784.6.a.d | 1 | 140.j | odd | 4 | 1 | ||
900.6.a.h | 1 | 15.e | even | 4 | 1 | ||
900.6.d.a | 2 | 3.b | odd | 2 | 1 | ||
900.6.d.a | 2 | 15.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 144 \)
acting on \(S_{6}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 144 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 7744 \)
$11$
\( (T - 540)^{2} \)
$13$
\( T^{2} + 174724 \)
$17$
\( T^{2} + 352836 \)
$19$
\( (T + 836)^{2} \)
$23$
\( T^{2} + 16842816 \)
$29$
\( (T - 594)^{2} \)
$31$
\( (T - 4256)^{2} \)
$37$
\( T^{2} + 88804 \)
$41$
\( (T - 17226)^{2} \)
$43$
\( T^{2} + 146410000 \)
$47$
\( T^{2} + 1679616 \)
$53$
\( T^{2} + 380016036 \)
$59$
\( (T - 7668)^{2} \)
$61$
\( (T + 34738)^{2} \)
$67$
\( T^{2} + 475763344 \)
$71$
\( (T + 46872)^{2} \)
$73$
\( T^{2} + 4564623844 \)
$79$
\( (T - 76912)^{2} \)
$83$
\( T^{2} + 4585456656 \)
$89$
\( (T + 29754)^{2} \)
$97$
\( T^{2} + 14981270404 \)
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