Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.u (of order \(5\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.19401608085\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 5 \) |
Twist minimal: | no (minimal twist has level 50) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -64\nu^{7} + 206\nu^{6} - 318\nu^{5} + 399\nu^{4} - 732\nu^{3} - 126\nu^{2} + 2175\nu + 235 ) / 1355 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -420\nu^{7} + 776\nu^{6} - 698\nu^{5} + 1924\nu^{4} - 2297\nu^{3} - 5129\nu^{2} - 1055\nu + 10265 ) / 1355 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 504\nu^{7} - 877\nu^{6} + 946\nu^{5} - 2363\nu^{4} + 2919\nu^{3} + 5125\nu^{2} + 3705\nu - 11505 ) / 1355 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -613\nu^{7} + 1321\nu^{6} - 1513\nu^{5} + 3394\nu^{4} - 4352\nu^{3} - 6178\nu^{2} - 530\nu + 12985 ) / 1355 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -714\nu^{7} + 1265\nu^{6} - 1295\nu^{5} + 3325\nu^{4} - 3390\nu^{3} - 8367\nu^{2} - 2200\nu + 14605 ) / 1355 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -1020\nu^{7} + 1962\nu^{6} - 2121\nu^{5} + 5563\nu^{4} - 6314\nu^{3} - 10443\nu^{2} - 3530\nu + 21445 ) / 1355 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 5 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} - 3\beta_{4} + 2\beta_{3} + 11\beta_{2} + 6\beta _1 - 4 ) / 5 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + 16\beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{3} + 17\beta_{2} - 3\beta _1 + 2 ) / 5 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 13\beta_{7} - 3\beta_{6} - 6\beta_{5} + 13\beta_{4} - 7\beta_{3} - 6\beta_{2} - 6\beta _1 + 14 ) / 5 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 16\beta_{7} - 6\beta_{6} - 22\beta_{5} + 6\beta_{4} + 51\beta_{3} + 43\beta_{2} + 8\beta _1 - 67 ) / 5 \)
|
\(\nu^{6}\) | \(=\) |
\( ( -23\beta_{7} + 63\beta_{6} + 46\beta_{5} + 12\beta_{4} + 52\beta_{3} + 156\beta_{2} - 34\beta _1 - 144 ) / 5 \)
|
\(\nu^{7}\) | \(=\) |
\( ( -31\beta_{7} - 9\beta_{6} + 137\beta_{5} + 84\beta_{4} - 31\beta_{3} - 33\beta_{2} - 168\beta _1 + 62 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(177\) | \(351\) |
\(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
81.1 |
|
0 | −1.09089 | + | 0.792578i | 0 | −2.15743 | − | 0.587785i | 0 | 0.833366 | 0 | −0.365190 | + | 1.12394i | 0 | ||||||||||||||||||||||||||||||||||||
81.2 | 0 | 2.39991 | − | 1.74363i | 0 | 2.15743 | − | 0.587785i | 0 | −1.83337 | 0 | 1.79224 | − | 5.51595i | 0 | |||||||||||||||||||||||||||||||||||||
161.1 | 0 | −0.529876 | + | 1.63079i | 0 | 2.02373 | + | 0.951057i | 0 | 2.77447 | 0 | 0.0483405 | + | 0.0351215i | 0 | |||||||||||||||||||||||||||||||||||||
161.2 | 0 | 0.720859 | − | 2.21858i | 0 | −2.02373 | + | 0.951057i | 0 | −3.77447 | 0 | −1.97539 | − | 1.43521i | 0 | |||||||||||||||||||||||||||||||||||||
241.1 | 0 | −0.529876 | − | 1.63079i | 0 | 2.02373 | − | 0.951057i | 0 | 2.77447 | 0 | 0.0483405 | − | 0.0351215i | 0 | |||||||||||||||||||||||||||||||||||||
241.2 | 0 | 0.720859 | + | 2.21858i | 0 | −2.02373 | − | 0.951057i | 0 | −3.77447 | 0 | −1.97539 | + | 1.43521i | 0 | |||||||||||||||||||||||||||||||||||||
321.1 | 0 | −1.09089 | − | 0.792578i | 0 | −2.15743 | + | 0.587785i | 0 | 0.833366 | 0 | −0.365190 | − | 1.12394i | 0 | |||||||||||||||||||||||||||||||||||||
321.2 | 0 | 2.39991 | + | 1.74363i | 0 | 2.15743 | + | 0.587785i | 0 | −1.83337 | 0 | 1.79224 | + | 5.51595i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.2.u.d | 8 | |
4.b | odd | 2 | 1 | 50.2.d.b | ✓ | 8 | |
12.b | even | 2 | 1 | 450.2.h.e | 8 | ||
20.d | odd | 2 | 1 | 250.2.d.d | 8 | ||
20.e | even | 4 | 2 | 250.2.e.c | 16 | ||
25.d | even | 5 | 1 | inner | 400.2.u.d | 8 | |
25.d | even | 5 | 1 | 10000.2.a.t | 4 | ||
25.e | even | 10 | 1 | 10000.2.a.x | 4 | ||
100.h | odd | 10 | 1 | 250.2.d.d | 8 | ||
100.h | odd | 10 | 1 | 1250.2.a.f | 4 | ||
100.j | odd | 10 | 1 | 50.2.d.b | ✓ | 8 | |
100.j | odd | 10 | 1 | 1250.2.a.l | 4 | ||
100.l | even | 20 | 2 | 250.2.e.c | 16 | ||
100.l | even | 20 | 2 | 1250.2.b.e | 8 | ||
300.n | even | 10 | 1 | 450.2.h.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
50.2.d.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
50.2.d.b | ✓ | 8 | 100.j | odd | 10 | 1 | |
250.2.d.d | 8 | 20.d | odd | 2 | 1 | ||
250.2.d.d | 8 | 100.h | odd | 10 | 1 | ||
250.2.e.c | 16 | 20.e | even | 4 | 2 | ||
250.2.e.c | 16 | 100.l | even | 20 | 2 | ||
400.2.u.d | 8 | 1.a | even | 1 | 1 | trivial | |
400.2.u.d | 8 | 25.d | even | 5 | 1 | inner | |
450.2.h.e | 8 | 12.b | even | 2 | 1 | ||
450.2.h.e | 8 | 300.n | even | 10 | 1 | ||
1250.2.a.f | 4 | 100.h | odd | 10 | 1 | ||
1250.2.a.l | 4 | 100.j | odd | 10 | 1 | ||
1250.2.b.e | 8 | 100.l | even | 20 | 2 | ||
10000.2.a.t | 4 | 25.d | even | 5 | 1 | ||
10000.2.a.x | 4 | 25.e | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 3T_{3}^{7} + 8T_{3}^{6} - 6T_{3}^{5} + 25T_{3}^{4} + 24T_{3}^{3} + 128T_{3}^{2} + 192T_{3} + 256 \)
acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - 3 T^{7} + 8 T^{6} - 6 T^{5} + \cdots + 256 \)
$5$
\( T^{8} - 15 T^{6} + 105 T^{4} + \cdots + 625 \)
$7$
\( (T^{4} + 2 T^{3} - 11 T^{2} - 12 T + 16)^{2} \)
$11$
\( T^{8} + T^{7} - 3 T^{6} - 7 T^{5} + \cdots + 256 \)
$13$
\( T^{8} + 13 T^{7} + 93 T^{6} + \cdots + 39601 \)
$17$
\( T^{8} + 11 T^{7} + 72 T^{6} + \cdots + 11881 \)
$19$
\( T^{8} + 20 T^{7} + 190 T^{6} + \cdots + 6400 \)
$23$
\( T^{8} - 3 T^{7} + 63 T^{6} + 49 T^{5} + \cdots + 256 \)
$29$
\( T^{8} + 15 T^{7} + 105 T^{6} + \cdots + 25 \)
$31$
\( T^{8} - 9 T^{7} + 87 T^{6} + \cdots + 1597696 \)
$37$
\( T^{8} + 6 T^{7} + 47 T^{6} + \cdots + 5041 \)
$41$
\( T^{8} + 9 T^{7} + 27 T^{6} + \cdots + 7921 \)
$43$
\( (T^{4} + 6 T^{3} - 39 T^{2} - 64 T + 176)^{2} \)
$47$
\( T^{8} - T^{7} + 17 T^{6} - 33 T^{5} + \cdots + 256 \)
$53$
\( T^{8} - 7 T^{7} + 108 T^{6} + \cdots + 65536 \)
$59$
\( T^{8} + 10 T^{7} + 260 T^{6} + \cdots + 102400 \)
$61$
\( T^{8} - 6 T^{7} - 18 T^{6} + \cdots + 2920681 \)
$67$
\( T^{8} - 11 T^{7} + 237 T^{6} + \cdots + 891136 \)
$71$
\( T^{8} - 9 T^{7} + 217 T^{6} + \cdots + 4096 \)
$73$
\( T^{8} + 8 T^{7} + 208 T^{6} + \cdots + 1175056 \)
$79$
\( T^{8} - 10 T^{7} + 120 T^{6} + \cdots + 102400 \)
$83$
\( T^{8} + 27 T^{7} + 403 T^{6} + \cdots + 13424896 \)
$89$
\( T^{8} + 15 T^{7} + 70 T^{6} + \cdots + 9610000 \)
$97$
\( (T^{4} + 18 T^{3} + 124 T^{2} + 7 T + 1)^{2} \)
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