Newspace parameters
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(180.262542657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{2305})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 1153x^{2} + 331776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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
\( x^{4} + 1153x^{2} + 331776 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 577\nu ) / 576 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} + 8645\nu ) / 576 \)
|
| \(\beta_{3}\) | \(=\) |
\( 10\nu^{2} + 5765 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 5765 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -577\beta_{2} + 8645\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
− | 16.0000i | − | 233.052i | −256.000 | 0 | −3728.83 | 2401.00i | 4096.00i | −34630.3 | 0 | ||||||||||||||||||||||||||||
| 99.2 | − | 16.0000i | 247.052i | −256.000 | 0 | 3952.83 | 2401.00i | 4096.00i | −41351.7 | 0 | ||||||||||||||||||||||||||||||
| 99.3 | 16.0000i | − | 247.052i | −256.000 | 0 | 3952.83 | − | 2401.00i | − | 4096.00i | −41351.7 | 0 | ||||||||||||||||||||||||||||
| 99.4 | 16.0000i | 233.052i | −256.000 | 0 | −3728.83 | − | 2401.00i | − | 4096.00i | −34630.3 | 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 | 350.10.c.j | 4 | |
| 5.b | even | 2 | 1 | inner | 350.10.c.j | 4 | |
| 5.c | odd | 4 | 1 | 14.10.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 350.10.a.j | 2 | ||
| 15.e | even | 4 | 1 | 126.10.a.o | 2 | ||
| 20.e | even | 4 | 1 | 112.10.a.c | 2 | ||
| 35.f | even | 4 | 1 | 98.10.a.e | 2 | ||
| 35.k | even | 12 | 2 | 98.10.c.h | 4 | ||
| 35.l | odd | 12 | 2 | 98.10.c.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.10.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 98.10.a.e | 2 | 35.f | even | 4 | 1 | ||
| 98.10.c.h | 4 | 35.k | even | 12 | 2 | ||
| 98.10.c.j | 4 | 35.l | odd | 12 | 2 | ||
| 112.10.a.c | 2 | 20.e | even | 4 | 1 | ||
| 126.10.a.o | 2 | 15.e | even | 4 | 1 | ||
| 350.10.a.j | 2 | 5.c | odd | 4 | 1 | ||
| 350.10.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 350.10.c.j | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 115348T_{3}^{2} + 3314995776 \)
acting on \(S_{10}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 256)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 3314995776 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 5764801)^{2} \)
|
| $11$ |
\( (T^{2} - 44940 T - 2036361600)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 57\!\cdots\!36 \)
|
| $17$ |
\( T^{4} + \cdots + 33\!\cdots\!56 \)
|
| $19$ |
\( (T^{2} + 508774 T - 352577596856)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 31904129519604)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 2367849772544)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 36\!\cdots\!36 \)
|
| $41$ |
\( (T^{2} + \cdots - 527816477266884)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 73\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 24\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + \cdots + 15\!\cdots\!36)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 11\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 45\!\cdots\!16 \)
|
| $71$ |
\( (T^{2} + \cdots - 85\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 70\!\cdots\!36 \)
|
| $79$ |
\( (T^{2} + \cdots + 68\!\cdots\!44)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 58\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + \cdots + 50\!\cdots\!76)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
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