Newspace parameters
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{7} + 8x^{6} - 3x^{5} + 50x^{4} - 27x^{3} + 53x^{2} + 20x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - x^{7} + 8x^{6} - 3x^{5} + 50x^{4} - 27x^{3} + 53x^{2} + 20x + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 689\nu^{7} - 7573\nu^{6} + 14556\nu^{5} - 51899\nu^{4} + 65994\nu^{3} - 301543\nu^{2} + 311369\nu - 57732 ) / 75828 \) |
\(\beta_{3}\) | \(=\) | \( ( -329\nu^{7} + 21\nu^{6} - 2209\nu^{5} - 1081\nu^{4} - 17462\nu^{3} - 3431\nu^{2} - 1692\nu - 8164 ) / 18957 \) |
\(\beta_{4}\) | \(=\) | \( ( 1653 \nu^{7} + 3659 \nu^{6} + 10196 \nu^{5} + 37929 \nu^{4} + 67606 \nu^{3} + 217641 \nu^{2} - 19483 \nu + 208424 ) / 75828 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2041 \nu^{7} + 3357 \nu^{6} - 16412 \nu^{5} + 14959 \nu^{4} - 97726 \nu^{3} + 124955 \nu^{2} - 94449 \nu - 34052 ) / 75828 \) |
\(\beta_{6}\) | \(=\) | \( ( 1871\nu^{7} - 1195\nu^{6} + 17076\nu^{5} - 4685\nu^{4} + 112020\nu^{3} - 34651\nu^{2} + 187457\nu - 33126 ) / 37914 \) |
\(\beta_{7}\) | \(=\) | \( ( 6159 \nu^{7} - 5771 \nu^{6} + 42256 \nu^{5} - 12261 \nu^{4} + 255062 \nu^{3} - 136173 \nu^{2} + 59659 \nu + 161284 ) / 75828 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{7} + 4\beta_{5} - \beta_{3} + \beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{7} - \beta_{6} - 7\beta_{3} + \beta_{2} + \beta _1 - 1 \) |
\(\nu^{4}\) | \(=\) | \( -8\beta_{7} - 7\beta_{6} - 25\beta_{5} + 8\beta_{4} - 8\beta_{3} + \beta_{2} + 7\beta _1 - 25 \) |
\(\nu^{5}\) | \(=\) | \( -2\beta_{7} + 8\beta_{6} - 12\beta_{5} + 8\beta_{4} + 41\beta_{3} - 2\beta_{2} - 39\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 10\beta_{7} + 57\beta_{6} - 47\beta_{4} + 120\beta_{3} - 57\beta_{2} - 57\beta _1 + 166 \) |
\(\nu^{7}\) | \(=\) | \( 83\beta_{7} + 26\beta_{6} + 121\beta_{5} - 83\beta_{4} + 83\beta_{3} - 57\beta_{2} + 177\beta _1 + 121 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(176\) | \(451\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
151.1 |
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−1.21868 | − | 2.11082i | −0.500000 | + | 0.866025i | −1.97036 | + | 3.41277i | 0 | 2.43736 | 0.970361 | − | 2.46138i | 4.73024 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||
151.2 | −0.247087 | − | 0.427967i | −0.500000 | + | 0.866025i | 0.877896 | − | 1.52056i | 0 | 0.494173 | −1.87790 | + | 1.86373i | −1.85601 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
151.3 | 0.614340 | + | 1.06407i | −0.500000 | + | 0.866025i | 0.245174 | − | 0.424653i | 0 | −1.22868 | −1.24517 | − | 2.33443i | 3.05984 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
151.4 | 1.35143 | + | 2.34074i | −0.500000 | + | 0.866025i | −2.65271 | + | 4.59463i | 0 | −2.70285 | 1.65271 | + | 2.06605i | −8.93406 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.1 | −1.21868 | + | 2.11082i | −0.500000 | − | 0.866025i | −1.97036 | − | 3.41277i | 0 | 2.43736 | 0.970361 | + | 2.46138i | 4.73024 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.2 | −0.247087 | + | 0.427967i | −0.500000 | − | 0.866025i | 0.877896 | + | 1.52056i | 0 | 0.494173 | −1.87790 | − | 1.86373i | −1.85601 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.3 | 0.614340 | − | 1.06407i | −0.500000 | − | 0.866025i | 0.245174 | + | 0.424653i | 0 | −1.22868 | −1.24517 | + | 2.33443i | 3.05984 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.4 | 1.35143 | − | 2.34074i | −0.500000 | − | 0.866025i | −2.65271 | − | 4.59463i | 0 | −2.70285 | 1.65271 | − | 2.06605i | −8.93406 | −0.500000 | + | 0.866025i | 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 | 525.2.i.j | yes | 8 |
5.b | even | 2 | 1 | 525.2.i.i | ✓ | 8 | |
5.c | odd | 4 | 2 | 525.2.r.h | 16 | ||
7.c | even | 3 | 1 | inner | 525.2.i.j | yes | 8 |
7.c | even | 3 | 1 | 3675.2.a.br | 4 | ||
7.d | odd | 6 | 1 | 3675.2.a.bq | 4 | ||
35.i | odd | 6 | 1 | 3675.2.a.bx | 4 | ||
35.j | even | 6 | 1 | 525.2.i.i | ✓ | 8 | |
35.j | even | 6 | 1 | 3675.2.a.bw | 4 | ||
35.l | odd | 12 | 2 | 525.2.r.h | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.i.i | ✓ | 8 | 5.b | even | 2 | 1 | |
525.2.i.i | ✓ | 8 | 35.j | even | 6 | 1 | |
525.2.i.j | yes | 8 | 1.a | even | 1 | 1 | trivial |
525.2.i.j | yes | 8 | 7.c | even | 3 | 1 | inner |
525.2.r.h | 16 | 5.c | odd | 4 | 2 | ||
525.2.r.h | 16 | 35.l | odd | 12 | 2 | ||
3675.2.a.bq | 4 | 7.d | odd | 6 | 1 | ||
3675.2.a.br | 4 | 7.c | even | 3 | 1 | ||
3675.2.a.bw | 4 | 35.j | even | 6 | 1 | ||
3675.2.a.bx | 4 | 35.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} + 8T_{2}^{6} - 3T_{2}^{5} + 50T_{2}^{4} - 27T_{2}^{3} + 53T_{2}^{2} + 20T_{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{7} + 8 T^{6} - 3 T^{5} + \cdots + 16 \)
$3$
\( (T^{2} + T + 1)^{4} \)
$5$
\( T^{8} \)
$7$
\( T^{8} + T^{7} + 11 T^{6} + 12 T^{5} + \cdots + 2401 \)
$11$
\( T^{8} + 8 T^{7} + 72 T^{6} + \cdots + 107584 \)
$13$
\( (T^{4} - 7 T^{3} - 19 T^{2} + 179 T - 194)^{2} \)
$17$
\( T^{8} + 6 T^{7} + 56 T^{6} + \cdots + 1600 \)
$19$
\( T^{8} + 3 T^{7} + 46 T^{6} + \cdots + 38416 \)
$23$
\( T^{8} - 2 T^{7} + 72 T^{6} + \cdots + 921600 \)
$29$
\( (T - 4)^{8} \)
$31$
\( T^{8} + 9 T^{7} + 137 T^{6} + \cdots + 589824 \)
$37$
\( T^{8} - 8 T^{7} + 126 T^{6} + \cdots + 597529 \)
$41$
\( (T^{4} - 4 T^{3} - 60 T^{2} + 236 T - 200)^{2} \)
$43$
\( (T^{4} + 5 T^{3} - 52 T^{2} + 32 T + 160)^{2} \)
$47$
\( T^{8} - 6 T^{7} + 100 T^{6} + \cdots + 732736 \)
$53$
\( T^{8} + 6 T^{7} + 112 T^{6} + \cdots + 1236544 \)
$59$
\( T^{8} + 10 T^{7} + 140 T^{6} + \cdots + 64 \)
$61$
\( (T^{2} - 5 T + 25)^{4} \)
$67$
\( T^{8} - T^{7} + 18 T^{6} - T^{5} + \cdots + 3600 \)
$71$
\( (T^{4} - 22 T^{3} - 32 T^{2} + 2880 T - 12736)^{2} \)
$73$
\( T^{8} - 4 T^{7} + 138 T^{6} + \cdots + 2455489 \)
$79$
\( T^{8} + 8 T^{7} + 134 T^{6} + \cdots + 23409 \)
$83$
\( (T^{4} + 2 T^{3} - 112 T^{2} + 396 T - 312)^{2} \)
$89$
\( T^{8} - 10 T^{7} + 96 T^{6} + \cdots + 576 \)
$97$
\( (T^{4} - 12 T^{3} - 58 T^{2} + 196 T - 79)^{2} \)
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