Newspace parameters
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.44030560092\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.359712057600.22 |
| Defining polynomial: |
\( x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{22} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{7} + 130\nu^{5} + 572\nu^{3} + 420\nu ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 64\nu^{7} + 1040\nu^{5} + 5008\nu^{3} + 7248\nu ) / 135 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{7} + 148\nu^{5} + 788\nu^{3} + 996\nu ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -64\nu^{7} - 30\nu^{6} - 1040\nu^{5} - 420\nu^{4} - 4468\nu^{3} - 1470\nu^{2} - 3468\nu - 765 ) / 135 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} - 56\nu^{4} - 196\nu^{2} - 102 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16\nu^{6} + 80\nu^{4} - 1088\nu^{2} - 3048 ) / 45 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\nu^{6} + 320\nu^{4} + 1552\nu^{2} + 672 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} - 4\beta_{4} + \beta_{2} - 8\beta_1 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} - 2\beta_{6} - 4\beta_{5} - 136 ) / 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -18\beta_{5} + 36\beta_{4} + \beta_{2} + 56\beta_1 ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13\beta_{7} + 16\beta_{6} + 44\beta_{5} + 1000 ) / 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 38\beta_{5} - 76\beta_{4} + 4\beta_{3} - 11\beta_{2} - 116\beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -133\beta_{7} - 126\beta_{6} - 492\beta_{5} - 8152 ) / 32 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -1288\beta_{5} + 2576\beta_{4} - 260\beta_{3} + 591\beta_{2} + 4064\beta_1 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 129.1 |
|
0 | − | 8.57295i | 0 | −0.582576 | − | 11.1652i | 0 | − | 22.1403i | 0 | −46.4955 | 0 | ||||||||||||||||||||||||||||||||||||||
| 129.2 | 0 | − | 8.57295i | 0 | −0.582576 | + | 11.1652i | 0 | − | 22.1403i | 0 | −46.4955 | 0 | |||||||||||||||||||||||||||||||||||||||
| 129.3 | 0 | − | 4.30169i | 0 | 8.58258 | − | 7.16515i | 0 | 28.3162i | 0 | 8.49545 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 129.4 | 0 | − | 4.30169i | 0 | 8.58258 | + | 7.16515i | 0 | 28.3162i | 0 | 8.49545 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 129.5 | 0 | 4.30169i | 0 | 8.58258 | − | 7.16515i | 0 | − | 28.3162i | 0 | 8.49545 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 129.6 | 0 | 4.30169i | 0 | 8.58258 | + | 7.16515i | 0 | − | 28.3162i | 0 | 8.49545 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 129.7 | 0 | 8.57295i | 0 | −0.582576 | − | 11.1652i | 0 | 22.1403i | 0 | −46.4955 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 129.8 | 0 | 8.57295i | 0 | −0.582576 | + | 11.1652i | 0 | 22.1403i | 0 | −46.4955 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 20.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 160.4.c.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1440.4.f.k | 8 | ||
| 4.b | odd | 2 | 1 | inner | 160.4.c.d | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 160.4.c.d | ✓ | 8 |
| 5.c | odd | 4 | 1 | 800.4.a.y | 4 | ||
| 5.c | odd | 4 | 1 | 800.4.a.z | 4 | ||
| 8.b | even | 2 | 1 | 320.4.c.j | 8 | ||
| 8.d | odd | 2 | 1 | 320.4.c.j | 8 | ||
| 12.b | even | 2 | 1 | 1440.4.f.k | 8 | ||
| 15.d | odd | 2 | 1 | 1440.4.f.k | 8 | ||
| 20.d | odd | 2 | 1 | inner | 160.4.c.d | ✓ | 8 |
| 20.e | even | 4 | 1 | 800.4.a.y | 4 | ||
| 20.e | even | 4 | 1 | 800.4.a.z | 4 | ||
| 40.e | odd | 2 | 1 | 320.4.c.j | 8 | ||
| 40.f | even | 2 | 1 | 320.4.c.j | 8 | ||
| 40.i | odd | 4 | 1 | 1600.4.a.cu | 4 | ||
| 40.i | odd | 4 | 1 | 1600.4.a.cv | 4 | ||
| 40.k | even | 4 | 1 | 1600.4.a.cu | 4 | ||
| 40.k | even | 4 | 1 | 1600.4.a.cv | 4 | ||
| 60.h | even | 2 | 1 | 1440.4.f.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.4.c.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 160.4.c.d | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 160.4.c.d | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 160.4.c.d | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
| 320.4.c.j | 8 | 8.b | even | 2 | 1 | ||
| 320.4.c.j | 8 | 8.d | odd | 2 | 1 | ||
| 320.4.c.j | 8 | 40.e | odd | 2 | 1 | ||
| 320.4.c.j | 8 | 40.f | even | 2 | 1 | ||
| 800.4.a.y | 4 | 5.c | odd | 4 | 1 | ||
| 800.4.a.y | 4 | 20.e | even | 4 | 1 | ||
| 800.4.a.z | 4 | 5.c | odd | 4 | 1 | ||
| 800.4.a.z | 4 | 20.e | even | 4 | 1 | ||
| 1440.4.f.k | 8 | 3.b | odd | 2 | 1 | ||
| 1440.4.f.k | 8 | 12.b | even | 2 | 1 | ||
| 1440.4.f.k | 8 | 15.d | odd | 2 | 1 | ||
| 1440.4.f.k | 8 | 60.h | even | 2 | 1 | ||
| 1600.4.a.cu | 4 | 40.i | odd | 4 | 1 | ||
| 1600.4.a.cu | 4 | 40.k | even | 4 | 1 | ||
| 1600.4.a.cv | 4 | 40.i | odd | 4 | 1 | ||
| 1600.4.a.cv | 4 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 92T_{3}^{2} + 1360 \)
acting on \(S_{4}^{\mathrm{new}}(160, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 92 T^{2} + 1360)^{2} \)
$5$
\( (T^{4} - 16 T^{3} + 230 T^{2} + \cdots + 15625)^{2} \)
$7$
\( (T^{4} + 1292 T^{2} + 393040)^{2} \)
$11$
\( (T^{4} - 4992 T^{2} + 3133440)^{2} \)
$13$
\( (T^{4} + 6080 T^{2} + 5607424)^{2} \)
$17$
\( (T^{2} + 5376)^{4} \)
$19$
\( (T^{4} - 30080 T^{2} + \cdots + 217600000)^{2} \)
$23$
\( (T^{4} + 11852 T^{2} + 2514640)^{2} \)
$29$
\( (T^{2} - 156 T - 15420)^{4} \)
$31$
\( (T^{4} - 23552 T^{2} + 89128960)^{2} \)
$37$
\( (T^{4} + 42176 T^{2} + \cdots + 140185600)^{2} \)
$41$
\( (T^{2} + 48 T - 13620)^{4} \)
$43$
\( (T^{4} + 251708 T^{2} + \cdots + 57154000)^{2} \)
$47$
\( (T^{4} + 211052 T^{2} + \cdots + 3485953360)^{2} \)
$53$
\( (T^{4} + 151488 T^{2} + \cdots + 5693607936)^{2} \)
$59$
\( (T^{4} - 313728 T^{2} + \cdots + 4289679360)^{2} \)
$61$
\( (T^{2} - 528 T + 8460)^{4} \)
$67$
\( (T^{4} + 388572 T^{2} + \cdots + 27064708560)^{2} \)
$71$
\( (T^{4} - 46592 T^{2} + \cdots + 272957440)^{2} \)
$73$
\( (T^{4} + 584448 T^{2} + \cdots + 1090584576)^{2} \)
$79$
\( (T^{4} - 1215488 T^{2} + \cdots + 196885872640)^{2} \)
$83$
\( (T^{4} + 1986972 T^{2} + \cdots + 739915650000)^{2} \)
$89$
\( (T^{2} - 20 T - 537500)^{4} \)
$97$
\( (T^{2} + 1290496)^{4} \)
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