Newspace parameters
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.90019100057\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). 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 | − | 1.00000i | 0 | 0 | 0 | 26.0000i | 0 | 26.0000 | 0 | |||||||||||||||||||||||
| 49.2 | 0 | 1.00000i | 0 | 0 | 0 | − | 26.0000i | 0 | 26.0000 | 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 | 100.4.c.b | 2 | |
| 3.b | odd | 2 | 1 | 900.4.d.a | 2 | ||
| 4.b | odd | 2 | 1 | 400.4.c.l | 2 | ||
| 5.b | even | 2 | 1 | inner | 100.4.c.b | 2 | |
| 5.c | odd | 4 | 1 | 100.4.a.b | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 100.4.a.c | yes | 1 | |
| 15.d | odd | 2 | 1 | 900.4.d.a | 2 | ||
| 15.e | even | 4 | 1 | 900.4.a.c | 1 | ||
| 15.e | even | 4 | 1 | 900.4.a.p | 1 | ||
| 20.d | odd | 2 | 1 | 400.4.c.l | 2 | ||
| 20.e | even | 4 | 1 | 400.4.a.i | 1 | ||
| 20.e | even | 4 | 1 | 400.4.a.l | 1 | ||
| 40.i | odd | 4 | 1 | 1600.4.a.x | 1 | ||
| 40.i | odd | 4 | 1 | 1600.4.a.bc | 1 | ||
| 40.k | even | 4 | 1 | 1600.4.a.y | 1 | ||
| 40.k | even | 4 | 1 | 1600.4.a.bd | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.4.a.b | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 100.4.a.c | yes | 1 | 5.c | odd | 4 | 1 | |
| 100.4.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 100.4.c.b | 2 | 5.b | even | 2 | 1 | inner | |
| 400.4.a.i | 1 | 20.e | even | 4 | 1 | ||
| 400.4.a.l | 1 | 20.e | even | 4 | 1 | ||
| 400.4.c.l | 2 | 4.b | odd | 2 | 1 | ||
| 400.4.c.l | 2 | 20.d | odd | 2 | 1 | ||
| 900.4.a.c | 1 | 15.e | even | 4 | 1 | ||
| 900.4.a.p | 1 | 15.e | even | 4 | 1 | ||
| 900.4.d.a | 2 | 3.b | odd | 2 | 1 | ||
| 900.4.d.a | 2 | 15.d | odd | 2 | 1 | ||
| 1600.4.a.x | 1 | 40.i | odd | 4 | 1 | ||
| 1600.4.a.y | 1 | 40.k | even | 4 | 1 | ||
| 1600.4.a.bc | 1 | 40.i | odd | 4 | 1 | ||
| 1600.4.a.bd | 1 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 676 \)
$11$
\( (T - 45)^{2} \)
$13$
\( T^{2} + 1936 \)
$17$
\( T^{2} + 13689 \)
$19$
\( (T - 91)^{2} \)
$23$
\( T^{2} + 324 \)
$29$
\( (T + 144)^{2} \)
$31$
\( (T - 26)^{2} \)
$37$
\( T^{2} + 45796 \)
$41$
\( (T + 459)^{2} \)
$43$
\( T^{2} + 211600 \)
$47$
\( T^{2} + 219024 \)
$53$
\( T^{2} + 311364 \)
$59$
\( (T - 72)^{2} \)
$61$
\( (T + 118)^{2} \)
$67$
\( T^{2} + 63001 \)
$71$
\( (T - 108)^{2} \)
$73$
\( T^{2} + 89401 \)
$79$
\( (T - 898)^{2} \)
$83$
\( T^{2} + 859329 \)
$89$
\( (T + 351)^{2} \)
$97$
\( T^{2} + 148996 \)
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