Newspace parameters
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.350464.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | no (minimal twist has level 95) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) :
\(\beta_{1}\) | \(=\) | \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) |
\(\beta_{2}\) | \(=\) | \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 7\nu^{2} - 32\nu + 13 ) / 23 \) |
\(\beta_{3}\) | \(=\) | \( ( -6\nu^{5} + 25\nu^{4} - 24\nu^{3} - 6\nu^{2} + 12\nu + 67 ) / 23 \) |
\(\beta_{4}\) | \(=\) | \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) |
\(\beta_{5}\) | \(=\) | \( ( 21\nu^{5} - 30\nu^{4} + 15\nu^{3} + 67\nu^{2} + 96\nu - 39 ) / 23 \) |
\(\nu\) | \(=\) | \( ( -\beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 3\beta_{2} ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -2\beta_{4} + \beta_{3} + 4\beta_{2} - 3\beta _1 - 4 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{3} - 3\beta _1 - 7 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{5} + 6\beta_{4} + 5\beta_{3} - 17\beta_{2} - 11\beta _1 - 18 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
324.1 |
|
− | 2.17009i | − | 1.70928i | −2.70928 | 0 | −3.70928 | − | 1.07838i | 1.53919i | 0.0783777 | 0 | |||||||||||||||||||||||||||||||||
324.2 | − | 1.48119i | − | 0.806063i | −0.193937 | 0 | −1.19394 | 3.35026i | − | 2.67513i | 2.35026 | 0 | ||||||||||||||||||||||||||||||||||
324.3 | − | 0.311108i | 2.90321i | 1.90321 | 0 | 0.903212 | 4.42864i | − | 1.21432i | −5.42864 | 0 | |||||||||||||||||||||||||||||||||||
324.4 | 0.311108i | − | 2.90321i | 1.90321 | 0 | 0.903212 | − | 4.42864i | 1.21432i | −5.42864 | 0 | |||||||||||||||||||||||||||||||||||
324.5 | 1.48119i | 0.806063i | −0.193937 | 0 | −1.19394 | − | 3.35026i | 2.67513i | 2.35026 | 0 | ||||||||||||||||||||||||||||||||||||
324.6 | 2.17009i | 1.70928i | −2.70928 | 0 | −3.70928 | 1.07838i | − | 1.53919i | 0.0783777 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.b.d | 6 | |
5.b | even | 2 | 1 | inner | 475.2.b.d | 6 | |
5.c | odd | 4 | 1 | 95.2.a.a | ✓ | 3 | |
5.c | odd | 4 | 1 | 475.2.a.f | 3 | ||
15.e | even | 4 | 1 | 855.2.a.i | 3 | ||
15.e | even | 4 | 1 | 4275.2.a.bk | 3 | ||
20.e | even | 4 | 1 | 1520.2.a.p | 3 | ||
20.e | even | 4 | 1 | 7600.2.a.bx | 3 | ||
35.f | even | 4 | 1 | 4655.2.a.u | 3 | ||
40.i | odd | 4 | 1 | 6080.2.a.bo | 3 | ||
40.k | even | 4 | 1 | 6080.2.a.by | 3 | ||
95.g | even | 4 | 1 | 1805.2.a.f | 3 | ||
95.g | even | 4 | 1 | 9025.2.a.bb | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.2.a.a | ✓ | 3 | 5.c | odd | 4 | 1 | |
475.2.a.f | 3 | 5.c | odd | 4 | 1 | ||
475.2.b.d | 6 | 1.a | even | 1 | 1 | trivial | |
475.2.b.d | 6 | 5.b | even | 2 | 1 | inner | |
855.2.a.i | 3 | 15.e | even | 4 | 1 | ||
1520.2.a.p | 3 | 20.e | even | 4 | 1 | ||
1805.2.a.f | 3 | 95.g | even | 4 | 1 | ||
4275.2.a.bk | 3 | 15.e | even | 4 | 1 | ||
4655.2.a.u | 3 | 35.f | even | 4 | 1 | ||
6080.2.a.bo | 3 | 40.i | odd | 4 | 1 | ||
6080.2.a.by | 3 | 40.k | even | 4 | 1 | ||
7600.2.a.bx | 3 | 20.e | even | 4 | 1 | ||
9025.2.a.bb | 3 | 95.g | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 7 T^{4} + 11 T^{2} + 1 \)
$3$
\( T^{6} + 12 T^{4} + 32 T^{2} + 16 \)
$5$
\( T^{6} \)
$7$
\( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \)
$11$
\( (T^{3} + 8 T^{2} + 8 T - 16)^{2} \)
$13$
\( T^{6} + 40 T^{4} + 80 T^{2} + 16 \)
$17$
\( T^{6} + 76 T^{4} + 1712 T^{2} + \cdots + 10816 \)
$19$
\( (T - 1)^{6} \)
$23$
\( T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 256 \)
$29$
\( (T^{3} - 10 T^{2} + 12 T + 40)^{2} \)
$31$
\( (T^{3} - 4 T^{2} - 48 T + 64)^{2} \)
$37$
\( T^{6} + 152 T^{4} + 5616 T^{2} + \cdots + 59536 \)
$41$
\( (T^{3} + 2 T^{2} - 36 T - 104)^{2} \)
$43$
\( T^{6} + 304 T^{4} + 25472 T^{2} + \cdots + 350464 \)
$47$
\( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \)
$53$
\( T^{6} + 104 T^{4} + 2832 T^{2} + \cdots + 8464 \)
$59$
\( (T^{3} - 20 T^{2} + 112 T - 160)^{2} \)
$61$
\( (T^{3} + 2 T^{2} - 84 T + 232)^{2} \)
$67$
\( T^{6} + 156 T^{4} + 5312 T^{2} + \cdots + 13456 \)
$71$
\( (T^{3} + 4 T^{2} - 80 T - 64)^{2} \)
$73$
\( T^{6} + 44 T^{4} + 432 T^{2} + \cdots + 64 \)
$79$
\( (T^{3} - 192 T + 160)^{2} \)
$83$
\( T^{6} + 368 T^{4} + 38976 T^{2} + \cdots + 1149184 \)
$89$
\( (T^{3} + 2 T^{2} - 132 T - 680)^{2} \)
$97$
\( T^{6} + 520 T^{4} + 73520 T^{2} + \cdots + 3055504 \)
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