Newspace parameters
| Level: | \( N \) | \(=\) | \( 182 = 2 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 182.m (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.45327731679\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(157\) |
| \(\chi(n)\) | \(\zeta_{12}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−0.866025 | − | 0.500000i | 1.36603 | − | 2.36603i | 0.500000 | + | 0.866025i | − | 1.00000i | −2.36603 | + | 1.36603i | 0.866025 | − | 0.500000i | − | 1.00000i | −2.23205 | − | 3.86603i | −0.500000 | + | 0.866025i | ||||||||||||||
| 43.2 | 0.866025 | + | 0.500000i | −0.366025 | + | 0.633975i | 0.500000 | + | 0.866025i | 1.00000i | −0.633975 | + | 0.366025i | −0.866025 | + | 0.500000i | 1.00000i | 1.23205 | + | 2.13397i | −0.500000 | + | 0.866025i | |||||||||||||||||
| 127.1 | −0.866025 | + | 0.500000i | 1.36603 | + | 2.36603i | 0.500000 | − | 0.866025i | 1.00000i | −2.36603 | − | 1.36603i | 0.866025 | + | 0.500000i | 1.00000i | −2.23205 | + | 3.86603i | −0.500000 | − | 0.866025i | |||||||||||||||||
| 127.2 | 0.866025 | − | 0.500000i | −0.366025 | − | 0.633975i | 0.500000 | − | 0.866025i | − | 1.00000i | −0.633975 | − | 0.366025i | −0.866025 | − | 0.500000i | − | 1.00000i | 1.23205 | − | 2.13397i | −0.500000 | − | 0.866025i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 182.2.m.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1638.2.bj.c | 4 | ||
| 4.b | odd | 2 | 1 | 1456.2.cc.b | 4 | ||
| 7.b | odd | 2 | 1 | 1274.2.m.a | 4 | ||
| 7.c | even | 3 | 1 | 1274.2.o.b | 4 | ||
| 7.c | even | 3 | 1 | 1274.2.v.a | 4 | ||
| 7.d | odd | 6 | 1 | 1274.2.o.a | 4 | ||
| 7.d | odd | 6 | 1 | 1274.2.v.b | 4 | ||
| 13.c | even | 3 | 1 | 2366.2.d.k | 4 | ||
| 13.e | even | 6 | 1 | inner | 182.2.m.a | ✓ | 4 |
| 13.e | even | 6 | 1 | 2366.2.d.k | 4 | ||
| 13.f | odd | 12 | 1 | 2366.2.a.q | 2 | ||
| 13.f | odd | 12 | 1 | 2366.2.a.s | 2 | ||
| 39.h | odd | 6 | 1 | 1638.2.bj.c | 4 | ||
| 52.i | odd | 6 | 1 | 1456.2.cc.b | 4 | ||
| 91.k | even | 6 | 1 | 1274.2.v.a | 4 | ||
| 91.l | odd | 6 | 1 | 1274.2.v.b | 4 | ||
| 91.p | odd | 6 | 1 | 1274.2.o.a | 4 | ||
| 91.t | odd | 6 | 1 | 1274.2.m.a | 4 | ||
| 91.u | even | 6 | 1 | 1274.2.o.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.2.m.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 182.2.m.a | ✓ | 4 | 13.e | even | 6 | 1 | inner |
| 1274.2.m.a | 4 | 7.b | odd | 2 | 1 | ||
| 1274.2.m.a | 4 | 91.t | odd | 6 | 1 | ||
| 1274.2.o.a | 4 | 7.d | odd | 6 | 1 | ||
| 1274.2.o.a | 4 | 91.p | odd | 6 | 1 | ||
| 1274.2.o.b | 4 | 7.c | even | 3 | 1 | ||
| 1274.2.o.b | 4 | 91.u | even | 6 | 1 | ||
| 1274.2.v.a | 4 | 7.c | even | 3 | 1 | ||
| 1274.2.v.a | 4 | 91.k | even | 6 | 1 | ||
| 1274.2.v.b | 4 | 7.d | odd | 6 | 1 | ||
| 1274.2.v.b | 4 | 91.l | odd | 6 | 1 | ||
| 1456.2.cc.b | 4 | 4.b | odd | 2 | 1 | ||
| 1456.2.cc.b | 4 | 52.i | odd | 6 | 1 | ||
| 1638.2.bj.c | 4 | 3.b | odd | 2 | 1 | ||
| 1638.2.bj.c | 4 | 39.h | odd | 6 | 1 | ||
| 2366.2.a.q | 2 | 13.f | odd | 12 | 1 | ||
| 2366.2.a.s | 2 | 13.f | odd | 12 | 1 | ||
| 2366.2.d.k | 4 | 13.c | even | 3 | 1 | ||
| 2366.2.d.k | 4 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(182, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{2} + 1 \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
|
| $5$ |
\( (T^{2} + 1)^{2} \)
|
| $7$ |
\( T^{4} - T^{2} + 1 \)
|
| $11$ |
\( T^{4} - 6 T^{3} + \cdots + 4 \)
|
| $13$ |
\( T^{4} - T^{2} + 169 \)
|
| $17$ |
\( T^{4} + 8 T^{3} + \cdots + 169 \)
|
| $19$ |
\( T^{4} + 12 T^{3} + \cdots + 64 \)
|
| $23$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $29$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $31$ |
\( T^{4} + 104T^{2} + 2116 \)
|
| $37$ |
\( (T^{2} - 9 T + 27)^{2} \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 121 \)
|
| $43$ |
\( T^{4} + 14 T^{3} + \cdots + 484 \)
|
| $47$ |
\( T^{4} + 128T^{2} + 1024 \)
|
| $53$ |
\( (T^{2} - 10 T + 13)^{2} \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 6084 \)
|
| $61$ |
\( T^{4} - 20 T^{3} + \cdots + 9409 \)
|
| $67$ |
\( T^{4} - 30 T^{3} + \cdots + 4356 \)
|
| $71$ |
\( (T^{2} + 24 T + 192)^{2} \)
|
| $73$ |
\( T^{4} + 218 T^{2} + 11449 \)
|
| $79$ |
\( (T^{2} + 18 T + 54)^{2} \)
|
| $83$ |
\( T^{4} + 216T^{2} + 2916 \)
|
| $89$ |
\( T^{4} + 12 T^{3} + \cdots + 576 \)
|
| $97$ |
\( T^{4} + 12 T^{3} + \cdots + 64 \)
|
| show more | |
| show less |