Newspace parameters
Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 196.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(31.4352286833\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 4) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
165.1 |
|
0 | 6.00000 | − | 10.3923i | 0 | −27.0000 | − | 46.7654i | 0 | 0 | 0 | 49.5000 | + | 85.7365i | 0 | ||||||||||||||||||
177.1 | 0 | 6.00000 | + | 10.3923i | 0 | −27.0000 | + | 46.7654i | 0 | 0 | 0 | 49.5000 | − | 85.7365i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 196.6.e.g | 2 | |
7.b | odd | 2 | 1 | 196.6.e.d | 2 | ||
7.c | even | 3 | 1 | 4.6.a.a | ✓ | 1 | |
7.c | even | 3 | 1 | inner | 196.6.e.g | 2 | |
7.d | odd | 6 | 1 | 196.6.a.e | 1 | ||
7.d | odd | 6 | 1 | 196.6.e.d | 2 | ||
21.h | odd | 6 | 1 | 36.6.a.a | 1 | ||
28.f | even | 6 | 1 | 784.6.a.d | 1 | ||
28.g | odd | 6 | 1 | 16.6.a.b | 1 | ||
35.j | even | 6 | 1 | 100.6.a.b | 1 | ||
35.l | odd | 12 | 2 | 100.6.c.b | 2 | ||
56.k | odd | 6 | 1 | 64.6.a.b | 1 | ||
56.p | even | 6 | 1 | 64.6.a.f | 1 | ||
63.g | even | 3 | 1 | 324.6.e.a | 2 | ||
63.h | even | 3 | 1 | 324.6.e.a | 2 | ||
63.j | odd | 6 | 1 | 324.6.e.d | 2 | ||
63.n | odd | 6 | 1 | 324.6.e.d | 2 | ||
77.h | odd | 6 | 1 | 484.6.a.a | 1 | ||
84.n | even | 6 | 1 | 144.6.a.c | 1 | ||
91.r | even | 6 | 1 | 676.6.a.a | 1 | ||
91.z | odd | 12 | 2 | 676.6.d.a | 2 | ||
105.o | odd | 6 | 1 | 900.6.a.h | 1 | ||
105.x | even | 12 | 2 | 900.6.d.a | 2 | ||
112.u | odd | 12 | 2 | 256.6.b.c | 2 | ||
112.w | even | 12 | 2 | 256.6.b.g | 2 | ||
140.p | odd | 6 | 1 | 400.6.a.d | 1 | ||
140.w | even | 12 | 2 | 400.6.c.f | 2 | ||
168.s | odd | 6 | 1 | 576.6.a.bc | 1 | ||
168.v | even | 6 | 1 | 576.6.a.bd | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.6.a.a | ✓ | 1 | 7.c | even | 3 | 1 | |
16.6.a.b | 1 | 28.g | odd | 6 | 1 | ||
36.6.a.a | 1 | 21.h | odd | 6 | 1 | ||
64.6.a.b | 1 | 56.k | odd | 6 | 1 | ||
64.6.a.f | 1 | 56.p | even | 6 | 1 | ||
100.6.a.b | 1 | 35.j | even | 6 | 1 | ||
100.6.c.b | 2 | 35.l | odd | 12 | 2 | ||
144.6.a.c | 1 | 84.n | even | 6 | 1 | ||
196.6.a.e | 1 | 7.d | odd | 6 | 1 | ||
196.6.e.d | 2 | 7.b | odd | 2 | 1 | ||
196.6.e.d | 2 | 7.d | odd | 6 | 1 | ||
196.6.e.g | 2 | 1.a | even | 1 | 1 | trivial | |
196.6.e.g | 2 | 7.c | even | 3 | 1 | inner | |
256.6.b.c | 2 | 112.u | odd | 12 | 2 | ||
256.6.b.g | 2 | 112.w | even | 12 | 2 | ||
324.6.e.a | 2 | 63.g | even | 3 | 1 | ||
324.6.e.a | 2 | 63.h | even | 3 | 1 | ||
324.6.e.d | 2 | 63.j | odd | 6 | 1 | ||
324.6.e.d | 2 | 63.n | odd | 6 | 1 | ||
400.6.a.d | 1 | 140.p | odd | 6 | 1 | ||
400.6.c.f | 2 | 140.w | even | 12 | 2 | ||
484.6.a.a | 1 | 77.h | odd | 6 | 1 | ||
576.6.a.bc | 1 | 168.s | odd | 6 | 1 | ||
576.6.a.bd | 1 | 168.v | even | 6 | 1 | ||
676.6.a.a | 1 | 91.r | even | 6 | 1 | ||
676.6.d.a | 2 | 91.z | odd | 12 | 2 | ||
784.6.a.d | 1 | 28.f | even | 6 | 1 | ||
900.6.a.h | 1 | 105.o | odd | 6 | 1 | ||
900.6.d.a | 2 | 105.x | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 12T_{3} + 144 \)
acting on \(S_{6}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 12T + 144 \)
$5$
\( T^{2} + 54T + 2916 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 540T + 291600 \)
$13$
\( (T + 418)^{2} \)
$17$
\( T^{2} + 594T + 352836 \)
$19$
\( T^{2} + 836T + 698896 \)
$23$
\( T^{2} - 4104 T + 16842816 \)
$29$
\( (T + 594)^{2} \)
$31$
\( T^{2} + 4256 T + 18113536 \)
$37$
\( T^{2} - 298T + 88804 \)
$41$
\( (T - 17226)^{2} \)
$43$
\( (T + 12100)^{2} \)
$47$
\( T^{2} - 1296 T + 1679616 \)
$53$
\( T^{2} + 19494 T + 380016036 \)
$59$
\( T^{2} - 7668 T + 58798224 \)
$61$
\( T^{2} - 34738 T + 1206728644 \)
$67$
\( T^{2} + 21812 T + 475763344 \)
$71$
\( (T + 46872)^{2} \)
$73$
\( T^{2} + 67562 T + 4564623844 \)
$79$
\( T^{2} - 76912 T + 5915455744 \)
$83$
\( (T - 67716)^{2} \)
$89$
\( T^{2} + 29754 T + 885300516 \)
$97$
\( (T + 122398)^{2} \)
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