Newspace parameters
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(124.954010194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{649})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 325x^{2} + 26244 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 25) |
| 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
\( x^{4} + 325x^{2} + 26244 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 487\nu ) / 162 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{3} - 815\nu ) / 162 \)
|
| \(\beta_{3}\) | \(=\) |
\( 10\nu^{2} + 1625 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 1625 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -487\beta_{2} - 815\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 45.4755i | 0 | 0 | 0 | − | 1369.97i | 0 | 118.981 | 0 | ||||||||||||||||||||||||||||
| 49.2 | 0 | − | 5.47548i | 0 | 0 | 0 | − | 769.970i | 0 | 2157.02 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 0 | 5.47548i | 0 | 0 | 0 | 769.970i | 0 | 2157.02 | 0 | |||||||||||||||||||||||||||||||
| 49.4 | 0 | 45.4755i | 0 | 0 | 0 | 1369.97i | 0 | 118.981 | 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 | 400.8.c.s | 4 | |
| 4.b | odd | 2 | 1 | 25.8.b.b | 4 | ||
| 5.b | even | 2 | 1 | inner | 400.8.c.s | 4 | |
| 5.c | odd | 4 | 1 | 400.8.a.v | 2 | ||
| 5.c | odd | 4 | 1 | 400.8.a.bd | 2 | ||
| 12.b | even | 2 | 1 | 225.8.b.l | 4 | ||
| 20.d | odd | 2 | 1 | 25.8.b.b | 4 | ||
| 20.e | even | 4 | 1 | 25.8.a.c | ✓ | 2 | |
| 20.e | even | 4 | 1 | 25.8.a.e | yes | 2 | |
| 60.h | even | 2 | 1 | 225.8.b.l | 4 | ||
| 60.l | odd | 4 | 1 | 225.8.a.k | 2 | ||
| 60.l | odd | 4 | 1 | 225.8.a.v | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.8.a.c | ✓ | 2 | 20.e | even | 4 | 1 | |
| 25.8.a.e | yes | 2 | 20.e | even | 4 | 1 | |
| 25.8.b.b | 4 | 4.b | odd | 2 | 1 | ||
| 25.8.b.b | 4 | 20.d | odd | 2 | 1 | ||
| 225.8.a.k | 2 | 60.l | odd | 4 | 1 | ||
| 225.8.a.v | 2 | 60.l | odd | 4 | 1 | ||
| 225.8.b.l | 4 | 12.b | even | 2 | 1 | ||
| 225.8.b.l | 4 | 60.h | even | 2 | 1 | ||
| 400.8.a.v | 2 | 5.c | odd | 4 | 1 | ||
| 400.8.a.bd | 2 | 5.c | odd | 4 | 1 | ||
| 400.8.c.s | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.8.c.s | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2098T_{3}^{2} + 62001 \)
acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 2098 T^{2} + 62001 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 1112678986896 \)
|
| $11$ |
\( (T^{2} + 4344 T - 5423041)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 26\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots + 47\!\cdots\!41 \)
|
| $19$ |
\( (T^{2} - 18200 T - 117098225)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 52\!\cdots\!36 \)
|
| $29$ |
\( (T^{2} + 55800 T - 27320953600)^{2} \)
|
| $31$ |
\( (T^{2} - 301776 T + 19481626044)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 52\!\cdots\!56 \)
|
| $41$ |
\( (T^{2} + 108486 T + 2293303049)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 59\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 96\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} - 2067600 T - 174052062400)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 8129212445516)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 73\!\cdots\!41 \)
|
| $71$ |
\( (T^{2} + \cdots + 5183381635664)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 50\!\cdots\!61 \)
|
| $79$ |
\( (T^{2} + \cdots + 11646468081900)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!41 \)
|
| $89$ |
\( (T^{2} + \cdots - 22736713427775)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 50\!\cdots\!76 \)
|
| show more | |
| show less |