Newspace parameters
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.h (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(20.2605127644\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 28) |
| 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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).
| \(n\) | \(99\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 117.1 |
|
0 | 6.00000 | − | 3.46410i | 0 | −18.0000 | − | 10.3923i | 0 | 0 | 0 | −16.5000 | + | 28.5788i | 0 | ||||||||||||||||||
| 129.1 | 0 | 6.00000 | + | 3.46410i | 0 | −18.0000 | + | 10.3923i | 0 | 0 | 0 | −16.5000 | − | 28.5788i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 196.5.h.b | 2 | |
| 7.b | odd | 2 | 1 | 196.5.h.a | 2 | ||
| 7.c | even | 3 | 1 | 28.5.b.a | ✓ | 2 | |
| 7.c | even | 3 | 1 | 196.5.h.a | 2 | ||
| 7.d | odd | 6 | 1 | 28.5.b.a | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 196.5.h.b | 2 | |
| 21.g | even | 6 | 1 | 252.5.d.a | 2 | ||
| 21.h | odd | 6 | 1 | 252.5.d.a | 2 | ||
| 28.f | even | 6 | 1 | 112.5.c.b | 2 | ||
| 28.g | odd | 6 | 1 | 112.5.c.b | 2 | ||
| 35.i | odd | 6 | 1 | 700.5.d.a | 2 | ||
| 35.j | even | 6 | 1 | 700.5.d.a | 2 | ||
| 35.k | even | 12 | 2 | 700.5.h.a | 4 | ||
| 35.l | odd | 12 | 2 | 700.5.h.a | 4 | ||
| 56.j | odd | 6 | 1 | 448.5.c.c | 2 | ||
| 56.k | odd | 6 | 1 | 448.5.c.d | 2 | ||
| 56.m | even | 6 | 1 | 448.5.c.d | 2 | ||
| 56.p | even | 6 | 1 | 448.5.c.c | 2 | ||
| 84.j | odd | 6 | 1 | 1008.5.f.c | 2 | ||
| 84.n | even | 6 | 1 | 1008.5.f.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.5.b.a | ✓ | 2 | 7.c | even | 3 | 1 | |
| 28.5.b.a | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 112.5.c.b | 2 | 28.f | even | 6 | 1 | ||
| 112.5.c.b | 2 | 28.g | odd | 6 | 1 | ||
| 196.5.h.a | 2 | 7.b | odd | 2 | 1 | ||
| 196.5.h.a | 2 | 7.c | even | 3 | 1 | ||
| 196.5.h.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 196.5.h.b | 2 | 7.d | odd | 6 | 1 | inner | |
| 252.5.d.a | 2 | 21.g | even | 6 | 1 | ||
| 252.5.d.a | 2 | 21.h | odd | 6 | 1 | ||
| 448.5.c.c | 2 | 56.j | odd | 6 | 1 | ||
| 448.5.c.c | 2 | 56.p | even | 6 | 1 | ||
| 448.5.c.d | 2 | 56.k | odd | 6 | 1 | ||
| 448.5.c.d | 2 | 56.m | even | 6 | 1 | ||
| 700.5.d.a | 2 | 35.i | odd | 6 | 1 | ||
| 700.5.d.a | 2 | 35.j | even | 6 | 1 | ||
| 700.5.h.a | 4 | 35.k | even | 12 | 2 | ||
| 700.5.h.a | 4 | 35.l | odd | 12 | 2 | ||
| 1008.5.f.c | 2 | 84.j | odd | 6 | 1 | ||
| 1008.5.f.c | 2 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 12T_{3} + 48 \)
acting on \(S_{5}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 12T + 48 \)
|
| $5$ |
\( T^{2} + 36T + 432 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 18T + 324 \)
|
| $13$ |
\( T^{2} + 17328 \)
|
| $17$ |
\( T^{2} + 720T + 172800 \)
|
| $19$ |
\( T^{2} + 156T + 8112 \)
|
| $23$ |
\( T^{2} + 738T + 544644 \)
|
| $29$ |
\( (T + 846)^{2} \)
|
| $31$ |
\( T^{2} - 2016 T + 1354752 \)
|
| $37$ |
\( T^{2} + 2386 T + 5692996 \)
|
| $41$ |
\( T^{2} + 2904768 \)
|
| $43$ |
\( (T + 2510)^{2} \)
|
| $47$ |
\( T^{2} - 5904 T + 11619072 \)
|
| $53$ |
\( T^{2} - 270T + 72900 \)
|
| $59$ |
\( T^{2} + 5436 T + 9850032 \)
|
| $61$ |
\( T^{2} + 11244 T + 42142512 \)
|
| $67$ |
\( T^{2} + 2450 T + 6002500 \)
|
| $71$ |
\( (T + 3150)^{2} \)
|
| $73$ |
\( T^{2} - 408T + 55488 \)
|
| $79$ |
\( T^{2} - 3982 T + 15856324 \)
|
| $83$ |
\( T^{2} + 25090992 \)
|
| $89$ |
\( T^{2} - 13176 T + 57868992 \)
|
| $97$ |
\( T^{2} + 158297088 \)
|
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