Newspace parameters
Level: | \( N \) | \(=\) | \( 125 = 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 125.e (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.998130025266\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
Coefficient field: | 8.0.58140625.2 |
Defining polynomial: |
\( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 25) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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
\( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
\(\nu^{4}\) | \(=\) |
\( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
|
\(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \)
|
\(\nu^{6}\) | \(=\) |
\( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \)
|
\(\nu^{7}\) | \(=\) |
\( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
24.1 |
|
−0.174207 | − | 0.0566033i | 0.865190 | − | 1.19083i | −1.59089 | − | 1.15585i | 0 | −0.218127 | + | 0.158479i | − | 3.26086i | 0.427051 | + | 0.587785i | 0.257524 | + | 0.792578i | 0 | |||||||||||||||||||||||||||||
24.2 | 1.98322 | + | 0.644389i | −1.29224 | + | 1.77862i | 1.89991 | + | 1.38036i | 0 | −3.70892 | + | 2.69469i | − | 0.992398i | 0.427051 | + | 0.587785i | −0.566541 | − | 1.74363i | 0 | ||||||||||||||||||||||||||||||
49.1 | −0.666375 | + | 0.917186i | 2.47539 | − | 0.804303i | 0.220859 | + | 0.679734i | 0 | −0.911842 | + | 2.80636i | − | 0.407162i | −2.92705 | − | 0.951057i | 3.05361 | − | 2.21858i | 0 | ||||||||||||||||||||||||||||||
49.2 | 1.35736 | − | 1.86824i | 0.451659 | − | 0.146753i | −1.02988 | − | 3.16963i | 0 | 0.338893 | − | 1.04301i | 3.03582i | −2.92705 | − | 0.951057i | −2.24459 | + | 1.63079i | 0 | |||||||||||||||||||||||||||||||
74.1 | −0.666375 | − | 0.917186i | 2.47539 | + | 0.804303i | 0.220859 | − | 0.679734i | 0 | −0.911842 | − | 2.80636i | 0.407162i | −2.92705 | + | 0.951057i | 3.05361 | + | 2.21858i | 0 | |||||||||||||||||||||||||||||||
74.2 | 1.35736 | + | 1.86824i | 0.451659 | + | 0.146753i | −1.02988 | + | 3.16963i | 0 | 0.338893 | + | 1.04301i | − | 3.03582i | −2.92705 | + | 0.951057i | −2.24459 | − | 1.63079i | 0 | ||||||||||||||||||||||||||||||
99.1 | −0.174207 | + | 0.0566033i | 0.865190 | + | 1.19083i | −1.59089 | + | 1.15585i | 0 | −0.218127 | − | 0.158479i | 3.26086i | 0.427051 | − | 0.587785i | 0.257524 | − | 0.792578i | 0 | |||||||||||||||||||||||||||||||
99.2 | 1.98322 | − | 0.644389i | −1.29224 | − | 1.77862i | 1.89991 | − | 1.38036i | 0 | −3.70892 | − | 2.69469i | 0.992398i | 0.427051 | − | 0.587785i | −0.566541 | + | 1.74363i | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 125.2.e.b | 8 | |
5.b | even | 2 | 1 | 25.2.e.a | ✓ | 8 | |
5.c | odd | 4 | 2 | 125.2.d.b | 16 | ||
15.d | odd | 2 | 1 | 225.2.m.a | 8 | ||
20.d | odd | 2 | 1 | 400.2.y.c | 8 | ||
25.d | even | 5 | 1 | 25.2.e.a | ✓ | 8 | |
25.d | even | 5 | 1 | 625.2.b.c | 8 | ||
25.d | even | 5 | 1 | 625.2.e.a | 8 | ||
25.d | even | 5 | 1 | 625.2.e.i | 8 | ||
25.e | even | 10 | 1 | inner | 125.2.e.b | 8 | |
25.e | even | 10 | 1 | 625.2.b.c | 8 | ||
25.e | even | 10 | 1 | 625.2.e.a | 8 | ||
25.e | even | 10 | 1 | 625.2.e.i | 8 | ||
25.f | odd | 20 | 2 | 125.2.d.b | 16 | ||
25.f | odd | 20 | 2 | 625.2.a.f | 8 | ||
25.f | odd | 20 | 4 | 625.2.d.o | 16 | ||
75.j | odd | 10 | 1 | 225.2.m.a | 8 | ||
75.l | even | 20 | 2 | 5625.2.a.x | 8 | ||
100.j | odd | 10 | 1 | 400.2.y.c | 8 | ||
100.l | even | 20 | 2 | 10000.2.a.bj | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.2.e.a | ✓ | 8 | 5.b | even | 2 | 1 | |
25.2.e.a | ✓ | 8 | 25.d | even | 5 | 1 | |
125.2.d.b | 16 | 5.c | odd | 4 | 2 | ||
125.2.d.b | 16 | 25.f | odd | 20 | 2 | ||
125.2.e.b | 8 | 1.a | even | 1 | 1 | trivial | |
125.2.e.b | 8 | 25.e | even | 10 | 1 | inner | |
225.2.m.a | 8 | 15.d | odd | 2 | 1 | ||
225.2.m.a | 8 | 75.j | odd | 10 | 1 | ||
400.2.y.c | 8 | 20.d | odd | 2 | 1 | ||
400.2.y.c | 8 | 100.j | odd | 10 | 1 | ||
625.2.a.f | 8 | 25.f | odd | 20 | 2 | ||
625.2.b.c | 8 | 25.d | even | 5 | 1 | ||
625.2.b.c | 8 | 25.e | even | 10 | 1 | ||
625.2.d.o | 16 | 25.f | odd | 20 | 4 | ||
625.2.e.a | 8 | 25.d | even | 5 | 1 | ||
625.2.e.a | 8 | 25.e | even | 10 | 1 | ||
625.2.e.i | 8 | 25.d | even | 5 | 1 | ||
625.2.e.i | 8 | 25.e | even | 10 | 1 | ||
5625.2.a.x | 8 | 75.l | even | 20 | 2 | ||
10000.2.a.bj | 8 | 100.l | even | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 5T_{2}^{7} + 11T_{2}^{6} - 10T_{2}^{5} + T_{2}^{4} - 10T_{2}^{3} + 26T_{2}^{2} + 10T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 5 T^{7} + 11 T^{6} - 10 T^{5} + \cdots + 1 \)
$3$
\( T^{8} - 5 T^{7} + 9 T^{6} - 15 T^{5} + \cdots + 16 \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 21 T^{6} + 121 T^{4} + \cdots + 16 \)
$11$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{2} \)
$13$
\( T^{8} - 5 T^{7} + 4 T^{6} + 5 T^{5} + \cdots + 1 \)
$17$
\( T^{8} - 10 T^{7} + 56 T^{6} + \cdots + 1936 \)
$19$
\( T^{8} + 5 T^{7} + 30 T^{6} + 40 T^{5} + \cdots + 400 \)
$23$
\( T^{8} + 5 T^{7} - T^{6} + 15 T^{5} + \cdots + 256 \)
$29$
\( T^{8} + 5 T^{7} + 30 T^{6} + \cdots + 483025 \)
$31$
\( T^{8} + 9 T^{7} + 117 T^{6} + \cdots + 1936 \)
$37$
\( T^{8} + 30 T^{7} + 406 T^{6} + \cdots + 116281 \)
$41$
\( T^{8} + 4 T^{7} + 52 T^{6} + \cdots + 13456 \)
$43$
\( T^{8} + 129 T^{6} + 4421 T^{4} + \cdots + 246016 \)
$47$
\( T^{8} + 16 T^{6} - 615 T^{5} + \cdots + 65536 \)
$53$
\( T^{8} - 10 T^{7} - 6 T^{6} + \cdots + 8755681 \)
$59$
\( T^{8} + 15 T^{5} + 5635 T^{4} + \cdots + 4080400 \)
$61$
\( T^{8} + 9 T^{7} - 43 T^{6} + \cdots + 116281 \)
$67$
\( T^{8} + 20 T^{7} + 116 T^{6} + \cdots + 246016 \)
$71$
\( T^{8} - 6 T^{7} + 142 T^{6} + \cdots + 24245776 \)
$73$
\( T^{8} + 15 T^{7} + 49 T^{6} - 120 T^{5} + \cdots + 1 \)
$79$
\( T^{8} - 15 T^{7} + 100 T^{6} + \cdots + 33408400 \)
$83$
\( T^{8} - 45 T^{7} + 949 T^{6} + \cdots + 99856 \)
$89$
\( T^{8} + 25 T^{7} + 520 T^{6} + \cdots + 1392400 \)
$97$
\( T^{8} - 60 T^{7} + \cdots + 301334881 \)
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