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: | \(\Q(i, \sqrt{46})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 529 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{4} + 529 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{2} ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 184\nu ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -16\nu^{3} + 368\nu ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 23\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -23\beta_{3} + 46\beta_{2} ) / 32 \)
|
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 | − | 56.2586i | 0 | 0 | 0 | 1344.40i | 0 | −978.035 | 0 | |||||||||||||||||||||||||||||
| 49.2 | 0 | − | 52.2586i | 0 | 0 | 0 | 500.397i | 0 | −543.965 | 0 | ||||||||||||||||||||||||||||||
| 49.3 | 0 | 52.2586i | 0 | 0 | 0 | − | 500.397i | 0 | −543.965 | 0 | ||||||||||||||||||||||||||||||
| 49.4 | 0 | 56.2586i | 0 | 0 | 0 | − | 1344.40i | 0 | −978.035 | 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.q | 4 | |
| 4.b | odd | 2 | 1 | 200.8.c.h | 4 | ||
| 5.b | even | 2 | 1 | inner | 400.8.c.q | 4 | |
| 5.c | odd | 4 | 1 | 80.8.a.h | 2 | ||
| 5.c | odd | 4 | 1 | 400.8.a.z | 2 | ||
| 20.d | odd | 2 | 1 | 200.8.c.h | 4 | ||
| 20.e | even | 4 | 1 | 40.8.a.b | ✓ | 2 | |
| 20.e | even | 4 | 1 | 200.8.a.m | 2 | ||
| 40.i | odd | 4 | 1 | 320.8.a.q | 2 | ||
| 40.k | even | 4 | 1 | 320.8.a.o | 2 | ||
| 60.l | odd | 4 | 1 | 360.8.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.8.a.b | ✓ | 2 | 20.e | even | 4 | 1 | |
| 80.8.a.h | 2 | 5.c | odd | 4 | 1 | ||
| 200.8.a.m | 2 | 20.e | even | 4 | 1 | ||
| 200.8.c.h | 4 | 4.b | odd | 2 | 1 | ||
| 200.8.c.h | 4 | 20.d | odd | 2 | 1 | ||
| 320.8.a.o | 2 | 40.k | even | 4 | 1 | ||
| 320.8.a.q | 2 | 40.i | odd | 4 | 1 | ||
| 360.8.a.l | 2 | 60.l | odd | 4 | 1 | ||
| 400.8.a.z | 2 | 5.c | odd | 4 | 1 | ||
| 400.8.c.q | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.8.c.q | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 5896T_{3}^{2} + 8643600 \)
acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 5896 T^{2} + 8643600 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 452568343824 \)
|
| $11$ |
\( (T^{2} + 4640 T + 1131264)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 15\!\cdots\!04 \)
|
| $17$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $19$ |
\( (T^{2} - 39720 T + 29646224)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 60\!\cdots\!24 \)
|
| $29$ |
\( (T^{2} + 186940 T - 4295905404)^{2} \)
|
| $31$ |
\( (T^{2} - 327128 T + 11743009680)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 55\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + 162964 T - 1010797212)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 38\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 22\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 44\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + \cdots - 5676215308272)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 2532614905500)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 94\!\cdots\!64 \)
|
| $71$ |
\( (T^{2} - 1071112 T + 285545583120)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 83\!\cdots\!04 \)
|
| $79$ |
\( (T^{2} + \cdots - 12100786389440)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 79\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} + \cdots - 12795653008796)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 19\!\cdots\!16 \)
|
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