Newspace parameters
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 8.0.1223810289.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 225) |
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
\( x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
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\(\beta_{2}\) | \(=\) |
\( ( -76\nu^{7} - 32\nu^{6} - 361\nu^{5} - 722\nu^{4} - 3496\nu^{3} - 1691\nu^{2} - 741\nu - 2934 ) / 6165 \)
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\(\beta_{3}\) | \(=\) |
\( ( -12\nu^{7} + 31\nu^{6} - 57\nu^{5} - 114\nu^{4} + 133\nu^{3} - 267\nu^{2} - 117\nu - 968 ) / 685 \)
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\(\beta_{4}\) | \(=\) |
\( ( -263\nu^{7} + 1079\nu^{6} - 3818\nu^{5} + 4694\nu^{4} - 10043\nu^{3} + 9047\nu^{2} - 29793\nu + 2988 ) / 6165 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -326\nu^{7} + 728\nu^{6} - 2576\nu^{5} + 1013\nu^{4} - 6776\nu^{3} + 6104\nu^{2} - 10371\nu + 2016 ) / 6165 \)
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\(\beta_{6}\) | \(=\) |
\( ( 156\nu^{7} - 403\nu^{6} + 1426\nu^{5} - 1258\nu^{4} + 3751\nu^{3} - 3379\nu^{2} + 4261\nu - 1116 ) / 2055 \)
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\(\beta_{7}\) | \(=\) |
\( ( 488\nu^{7} - 119\nu^{6} + 2318\nu^{5} + 4636\nu^{4} + 10118\nu^{3} + 10858\nu^{2} + 4758\nu + 17217 ) / 6165 \)
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\(\nu\) | \(=\) |
\( \beta_1 \)
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\(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - \beta_{3} - \beta_{2} + \beta _1 - 2 \)
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\(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{3} - 5\beta_{2} - 1 \)
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\(\nu^{4}\) | \(=\) |
\( -6\beta_{6} - 7\beta_{5} - 2\beta_{4} - 9\beta_1 \)
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\(\nu^{5}\) | \(=\) |
\( 8\beta_{7} - 11\beta_{6} - 8\beta_{5} - 8\beta_{4} + 11\beta_{3} + 30\beta_{2} - 30\beta _1 + 8 \)
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\(\nu^{6}\) | \(=\) |
\( 19\beta_{7} + 38\beta_{3} + 68\beta_{2} + 33 \)
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\(\nu^{7}\) | \(=\) |
\( 87\beta_{6} + 60\beta_{5} + 57\beta_{4} + 196\beta_1 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) |
\(\chi(n)\) | \(-1 + \beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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226.1 |
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−0.816862 | − | 1.41485i | 0 | −0.334526 | + | 0.579416i | 0 | 0 | 0.252674 | + | 0.437645i | −2.17440 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
226.2 | −0.236627 | − | 0.409850i | 0 | 0.888015 | − | 1.53809i | 0 | 0 | 1.28153 | + | 2.21967i | −1.78702 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
226.3 | 0.736627 | + | 1.27588i | 0 | −0.0852394 | + | 0.147639i | 0 | 0 | −1.93291 | − | 3.34791i | 2.69535 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
226.4 | 1.31686 | + | 2.28087i | 0 | −2.46825 | + | 4.27513i | 0 | 0 | 0.898714 | + | 1.55662i | −7.73393 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
451.1 | −0.816862 | + | 1.41485i | 0 | −0.334526 | − | 0.579416i | 0 | 0 | 0.252674 | − | 0.437645i | −2.17440 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
451.2 | −0.236627 | + | 0.409850i | 0 | 0.888015 | + | 1.53809i | 0 | 0 | 1.28153 | − | 2.21967i | −1.78702 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
451.3 | 0.736627 | − | 1.27588i | 0 | −0.0852394 | − | 0.147639i | 0 | 0 | −1.93291 | + | 3.34791i | 2.69535 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
451.4 | 1.31686 | − | 2.28087i | 0 | −2.46825 | − | 4.27513i | 0 | 0 | 0.898714 | − | 1.55662i | −7.73393 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.e.e | 8 | |
3.b | odd | 2 | 1 | 225.2.e.c | ✓ | 8 | |
5.b | even | 2 | 1 | 675.2.e.c | 8 | ||
5.c | odd | 4 | 2 | 675.2.k.c | 16 | ||
9.c | even | 3 | 1 | inner | 675.2.e.e | 8 | |
9.c | even | 3 | 1 | 2025.2.a.p | 4 | ||
9.d | odd | 6 | 1 | 225.2.e.c | ✓ | 8 | |
9.d | odd | 6 | 1 | 2025.2.a.y | 4 | ||
15.d | odd | 2 | 1 | 225.2.e.e | yes | 8 | |
15.e | even | 4 | 2 | 225.2.k.c | 16 | ||
45.h | odd | 6 | 1 | 225.2.e.e | yes | 8 | |
45.h | odd | 6 | 1 | 2025.2.a.q | 4 | ||
45.j | even | 6 | 1 | 675.2.e.c | 8 | ||
45.j | even | 6 | 1 | 2025.2.a.z | 4 | ||
45.k | odd | 12 | 2 | 675.2.k.c | 16 | ||
45.k | odd | 12 | 2 | 2025.2.b.o | 8 | ||
45.l | even | 12 | 2 | 225.2.k.c | 16 | ||
45.l | even | 12 | 2 | 2025.2.b.n | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.2.e.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
225.2.e.c | ✓ | 8 | 9.d | odd | 6 | 1 | |
225.2.e.e | yes | 8 | 15.d | odd | 2 | 1 | |
225.2.e.e | yes | 8 | 45.h | odd | 6 | 1 | |
225.2.k.c | 16 | 15.e | even | 4 | 2 | ||
225.2.k.c | 16 | 45.l | even | 12 | 2 | ||
675.2.e.c | 8 | 5.b | even | 2 | 1 | ||
675.2.e.c | 8 | 45.j | even | 6 | 1 | ||
675.2.e.e | 8 | 1.a | even | 1 | 1 | trivial | |
675.2.e.e | 8 | 9.c | even | 3 | 1 | inner | |
675.2.k.c | 16 | 5.c | odd | 4 | 2 | ||
675.2.k.c | 16 | 45.k | odd | 12 | 2 | ||
2025.2.a.p | 4 | 9.c | even | 3 | 1 | ||
2025.2.a.q | 4 | 45.h | odd | 6 | 1 | ||
2025.2.a.y | 4 | 9.d | odd | 6 | 1 | ||
2025.2.a.z | 4 | 45.j | even | 6 | 1 | ||
2025.2.b.n | 8 | 45.l | even | 12 | 2 | ||
2025.2.b.o | 8 | 45.k | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{7} + 8T_{2}^{6} - 2T_{2}^{5} + 23T_{2}^{4} - 8T_{2}^{3} + 37T_{2}^{2} + 15T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 2 T^{7} + 8 T^{6} - 2 T^{5} + \cdots + 9 \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - T^{7} + 13 T^{6} - 36 T^{5} + \cdots + 81 \)
$11$
\( T^{8} + T^{7} + 26 T^{6} - 107 T^{5} + \cdots + 81 \)
$13$
\( T^{8} + 2 T^{7} + 34 T^{6} + \cdots + 11449 \)
$17$
\( (T^{4} + 11 T^{3} + 20 T^{2} - 110 T - 303)^{2} \)
$19$
\( (T^{4} - 2 T^{3} - 27 T^{2} + 80 T - 25)^{2} \)
$23$
\( T^{8} - 15 T^{7} + 168 T^{6} + \cdots + 59049 \)
$29$
\( T^{8} - T^{7} + 41 T^{6} - 244 T^{5} + \cdots + 16641 \)
$31$
\( T^{8} - 4 T^{7} + 58 T^{6} + \cdots + 59049 \)
$37$
\( (T^{4} + T^{3} - 99 T^{2} - 503 T - 647)^{2} \)
$41$
\( T^{8} + 5 T^{7} + 50 T^{6} + \cdots + 42849 \)
$43$
\( T^{8} - 10 T^{7} + 148 T^{6} + \cdots + 452929 \)
$47$
\( T^{8} - 20 T^{7} + 293 T^{6} + \cdots + 145161 \)
$53$
\( (T^{4} + 20 T^{3} + 86 T^{2} - 179 T - 471)^{2} \)
$59$
\( T^{8} - 17 T^{7} + 287 T^{6} + \cdots + 5349969 \)
$61$
\( T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1 \)
$67$
\( T^{8} + 17 T^{7} + 253 T^{6} + \cdots + 59049 \)
$71$
\( (T^{4} - 8 T^{3} - 40 T^{2} + 263 T + 381)^{2} \)
$73$
\( (T^{4} - 2 T^{3} - 96 T^{2} - 241 T + 113)^{2} \)
$79$
\( T^{8} - 7 T^{7} + 82 T^{6} + \cdots + 42849 \)
$83$
\( T^{8} - 30 T^{7} + 612 T^{6} + \cdots + 531441 \)
$89$
\( (T^{4} - 9 T^{3} - 99 T^{2} + 405 T + 2025)^{2} \)
$97$
\( T^{8} - 19 T^{7} + 280 T^{6} + \cdots + 908209 \)
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