Newspace parameters
Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 49.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.2726500974\) |
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: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 7) |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
\(n\) | \(3\) |
\(\chi(n)\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−4.50000 | + | 7.79423i | 0 | −8.50000 | − | 14.7224i | 0 | 0 | 0 | −423.000 | −364.500 | + | 631.333i | 0 | ||||||||||||||||||
31.1 | −4.50000 | − | 7.79423i | 0 | −8.50000 | + | 14.7224i | 0 | 0 | 0 | −423.000 | −364.500 | − | 631.333i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
7.c | even | 3 | 1 | inner |
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 | 49.7.d.a | 2 | |
7.b | odd | 2 | 1 | CM | 49.7.d.a | 2 | |
7.c | even | 3 | 1 | 7.7.b.a | ✓ | 1 | |
7.c | even | 3 | 1 | inner | 49.7.d.a | 2 | |
7.d | odd | 6 | 1 | 7.7.b.a | ✓ | 1 | |
7.d | odd | 6 | 1 | inner | 49.7.d.a | 2 | |
21.g | even | 6 | 1 | 63.7.d.a | 1 | ||
21.h | odd | 6 | 1 | 63.7.d.a | 1 | ||
28.f | even | 6 | 1 | 112.7.c.a | 1 | ||
28.g | odd | 6 | 1 | 112.7.c.a | 1 | ||
35.i | odd | 6 | 1 | 175.7.d.a | 1 | ||
35.j | even | 6 | 1 | 175.7.d.a | 1 | ||
35.k | even | 12 | 2 | 175.7.c.a | 2 | ||
35.l | odd | 12 | 2 | 175.7.c.a | 2 | ||
56.j | odd | 6 | 1 | 448.7.c.a | 1 | ||
56.k | odd | 6 | 1 | 448.7.c.b | 1 | ||
56.m | even | 6 | 1 | 448.7.c.b | 1 | ||
56.p | even | 6 | 1 | 448.7.c.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
7.7.b.a | ✓ | 1 | 7.c | even | 3 | 1 | |
7.7.b.a | ✓ | 1 | 7.d | odd | 6 | 1 | |
49.7.d.a | 2 | 1.a | even | 1 | 1 | trivial | |
49.7.d.a | 2 | 7.b | odd | 2 | 1 | CM | |
49.7.d.a | 2 | 7.c | even | 3 | 1 | inner | |
49.7.d.a | 2 | 7.d | odd | 6 | 1 | inner | |
63.7.d.a | 1 | 21.g | even | 6 | 1 | ||
63.7.d.a | 1 | 21.h | odd | 6 | 1 | ||
112.7.c.a | 1 | 28.f | even | 6 | 1 | ||
112.7.c.a | 1 | 28.g | odd | 6 | 1 | ||
175.7.c.a | 2 | 35.k | even | 12 | 2 | ||
175.7.c.a | 2 | 35.l | odd | 12 | 2 | ||
175.7.d.a | 1 | 35.i | odd | 6 | 1 | ||
175.7.d.a | 1 | 35.j | even | 6 | 1 | ||
448.7.c.a | 1 | 56.j | odd | 6 | 1 | ||
448.7.c.a | 1 | 56.p | even | 6 | 1 | ||
448.7.c.b | 1 | 56.k | odd | 6 | 1 | ||
448.7.c.b | 1 | 56.m | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 9T_{2} + 81 \)
acting on \(S_{7}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 9T + 81 \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 1962 T + 3849444 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} - 22734 T + 516834756 \)
$29$
\( (T + 21222)^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} + 101194 T + 10240225636 \)
$41$
\( T^{2} \)
$43$
\( (T + 126614)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} + 50346 T + 2534719716 \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} - 53926 T + 2908013476 \)
$71$
\( (T + 242478)^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} + 929378 T + 863743466884 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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