Newspace parameters
Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 49.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.06512819111\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{22})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 22x^{2} + 484 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 7) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} + 22x^{2} + 484 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 22 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 22 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 22\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 22\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
\(n\) | \(3\) |
\(\chi(n)\) | \(1 + \beta_{2}\) |
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 |
|
−3.34521 | + | 5.79407i | −8.53562 | + | 4.92804i | −14.3808 | − | 24.9083i | −6.57125 | − | 3.79391i | − | 65.9413i | 0 | 85.3808 | 8.07125 | − | 13.9798i | 43.9644 | − | 25.3828i | |||||||||||||||||
19.2 | 1.34521 | − | 2.32997i | 5.53562 | − | 3.19599i | 4.38083 | + | 7.58782i | 21.5712 | + | 12.4542i | − | 17.1971i | 0 | 66.6192 | −20.0712 | + | 34.7644i | 58.0356 | − | 33.5069i | ||||||||||||||||||
31.1 | −3.34521 | − | 5.79407i | −8.53562 | − | 4.92804i | −14.3808 | + | 24.9083i | −6.57125 | + | 3.79391i | 65.9413i | 0 | 85.3808 | 8.07125 | + | 13.9798i | 43.9644 | + | 25.3828i | |||||||||||||||||||
31.2 | 1.34521 | + | 2.32997i | 5.53562 | + | 3.19599i | 4.38083 | − | 7.58782i | 21.5712 | − | 12.4542i | 17.1971i | 0 | 66.6192 | −20.0712 | − | 34.7644i | 58.0356 | + | 33.5069i | |||||||||||||||||||
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 | 49.5.d.b | 4 | |
7.b | odd | 2 | 1 | 7.5.d.a | ✓ | 4 | |
7.c | even | 3 | 1 | 7.5.d.a | ✓ | 4 | |
7.c | even | 3 | 1 | 49.5.b.a | 4 | ||
7.d | odd | 6 | 1 | 49.5.b.a | 4 | ||
7.d | odd | 6 | 1 | inner | 49.5.d.b | 4 | |
21.c | even | 2 | 1 | 63.5.m.d | 4 | ||
21.g | even | 6 | 1 | 441.5.d.d | 4 | ||
21.h | odd | 6 | 1 | 63.5.m.d | 4 | ||
21.h | odd | 6 | 1 | 441.5.d.d | 4 | ||
28.d | even | 2 | 1 | 112.5.s.a | 4 | ||
28.f | even | 6 | 1 | 784.5.c.c | 4 | ||
28.g | odd | 6 | 1 | 112.5.s.a | 4 | ||
28.g | odd | 6 | 1 | 784.5.c.c | 4 | ||
35.c | odd | 2 | 1 | 175.5.i.a | 4 | ||
35.f | even | 4 | 2 | 175.5.j.a | 8 | ||
35.j | even | 6 | 1 | 175.5.i.a | 4 | ||
35.l | odd | 12 | 2 | 175.5.j.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
7.5.d.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
7.5.d.a | ✓ | 4 | 7.c | even | 3 | 1 | |
49.5.b.a | 4 | 7.c | even | 3 | 1 | ||
49.5.b.a | 4 | 7.d | odd | 6 | 1 | ||
49.5.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
49.5.d.b | 4 | 7.d | odd | 6 | 1 | inner | |
63.5.m.d | 4 | 21.c | even | 2 | 1 | ||
63.5.m.d | 4 | 21.h | odd | 6 | 1 | ||
112.5.s.a | 4 | 28.d | even | 2 | 1 | ||
112.5.s.a | 4 | 28.g | odd | 6 | 1 | ||
175.5.i.a | 4 | 35.c | odd | 2 | 1 | ||
175.5.i.a | 4 | 35.j | even | 6 | 1 | ||
175.5.j.a | 8 | 35.f | even | 4 | 2 | ||
175.5.j.a | 8 | 35.l | odd | 12 | 2 | ||
441.5.d.d | 4 | 21.g | even | 6 | 1 | ||
441.5.d.d | 4 | 21.h | odd | 6 | 1 | ||
784.5.c.c | 4 | 28.f | even | 6 | 1 | ||
784.5.c.c | 4 | 28.g | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} + 34T_{2}^{2} - 72T_{2} + 324 \)
acting on \(S_{5}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 4 T^{3} + 34 T^{2} - 72 T + 324 \)
$3$
\( T^{4} + 6 T^{3} - 51 T^{2} + \cdots + 3969 \)
$5$
\( T^{4} - 30 T^{3} + 111 T^{2} + \cdots + 35721 \)
$7$
\( T^{4} \)
$11$
\( T^{4} + 58 T^{3} + 8881 T^{2} + \cdots + 30437289 \)
$13$
\( T^{4} + 55272 T^{2} + \cdots + 3111696 \)
$17$
\( T^{4} - 246 T^{3} + \cdots + 5076990009 \)
$19$
\( T^{4} + 642 T^{3} + \cdots + 3136784049 \)
$23$
\( T^{4} - 290 T^{3} + \cdots + 254625849 \)
$29$
\( (T^{2} + 1088 T + 188136)^{2} \)
$31$
\( T^{4} + 3618 T^{3} + \cdots + 1071889573041 \)
$37$
\( T^{4} + 270 T^{3} + \cdots + 48279954529 \)
$41$
\( T^{4} + 4053672 T^{2} + \cdots + 3218392944144 \)
$43$
\( (T^{2} - 1236 T - 2313076)^{2} \)
$47$
\( T^{4} - 1542 T^{3} + \cdots + 4451285577249 \)
$53$
\( T^{4} + 4510 T^{3} + \cdots + 6460874330625 \)
$59$
\( T^{4} + \cdots + 163173619767249 \)
$61$
\( T^{4} - 282 T^{3} + \cdots + 14401820070729 \)
$67$
\( T^{4} + 1318 T^{3} + \cdots + 151890252361 \)
$71$
\( (T^{2} + 5204 T + 5524236)^{2} \)
$73$
\( T^{4} + 5214 T^{3} + \cdots + 6851102086521 \)
$79$
\( T^{4} + \cdots + 188584385155609 \)
$83$
\( T^{4} + \cdots + 115179601694976 \)
$89$
\( T^{4} + 33990 T^{3} + \cdots + 75\!\cdots\!81 \)
$97$
\( T^{4} + 217069608 T^{2} + \cdots + 11\!\cdots\!84 \)
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