Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.n (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(23.6007640023\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.1731891456.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{22} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 776 ) / 65 \) |
\(\beta_{2}\) | \(=\) | \( ( -9\nu^{7} + 65\nu^{5} - 377\nu^{3} + 256\nu ) / 832 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} - 25\nu^{5} + 145\nu^{3} - 544\nu ) / 40 \) |
\(\beta_{4}\) | \(=\) | \( ( -117\nu^{7} + 128\nu^{6} + 845\nu^{5} - 6565\nu^{3} + 3328\nu + 19008 ) / 4160 \) |
\(\beta_{5}\) | \(=\) | \( ( -117\nu^{7} - 128\nu^{6} + 845\nu^{5} - 6565\nu^{3} + 3328\nu - 19008 ) / 4160 \) |
\(\beta_{6}\) | \(=\) | \( ( -17\nu^{7} + 104\nu^{6} + 169\nu^{5} - 936\nu^{4} - 1313\nu^{3} + 5096\nu^{2} + 5152\nu - 7488 ) / 104 \) |
\(\beta_{7}\) | \(=\) | \( ( -17\nu^{7} - 104\nu^{6} + 169\nu^{5} + 936\nu^{4} - 1313\nu^{3} - 5096\nu^{2} + 5152\nu + 7488 ) / 104 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{6} - 8\beta_{5} - 8\beta_{4} + 2\beta_{3} + 16\beta_{2} ) / 64 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} - \beta_{6} + 8\beta_{5} - 8\beta_{4} + 18\beta _1 + 144 ) / 64 \) |
\(\nu^{3}\) | \(=\) | \( ( -5\beta_{5} - 5\beta_{4} + 26\beta_{2} ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( 9\beta_{7} - 9\beta_{6} - 72\beta_{5} + 72\beta_{4} + 98\beta _1 - 784 ) / 64 \) |
\(\nu^{5}\) | \(=\) | \( ( -29\beta_{7} - 29\beta_{6} - 232\beta_{5} - 232\beta_{4} - 202\beta_{3} + 1616\beta_{2} ) / 64 \) |
\(\nu^{6}\) | \(=\) | \( ( -65\beta_{5} + 65\beta_{4} - 594 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( -181\beta_{7} - 181\beta_{6} + 1448\beta_{5} + 1448\beta_{4} - 1402\beta_{3} - 11216\beta_{2} ) / 64 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(177\) | \(351\) |
\(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 |
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0 | −4.12311 | + | 4.12311i | 0 | 0 | 0 | −4.12311 | − | 4.12311i | 0 | − | 7.00000i | 0 | |||||||||||||||||||||||||||||||||||||
143.2 | 0 | −4.12311 | + | 4.12311i | 0 | 0 | 0 | −4.12311 | − | 4.12311i | 0 | − | 7.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
143.3 | 0 | 4.12311 | − | 4.12311i | 0 | 0 | 0 | 4.12311 | + | 4.12311i | 0 | − | 7.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
143.4 | 0 | 4.12311 | − | 4.12311i | 0 | 0 | 0 | 4.12311 | + | 4.12311i | 0 | − | 7.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
207.1 | 0 | −4.12311 | − | 4.12311i | 0 | 0 | 0 | −4.12311 | + | 4.12311i | 0 | 7.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
207.2 | 0 | −4.12311 | − | 4.12311i | 0 | 0 | 0 | −4.12311 | + | 4.12311i | 0 | 7.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
207.3 | 0 | 4.12311 | + | 4.12311i | 0 | 0 | 0 | 4.12311 | − | 4.12311i | 0 | 7.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
207.4 | 0 | 4.12311 | + | 4.12311i | 0 | 0 | 0 | 4.12311 | − | 4.12311i | 0 | 7.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
20.d | odd | 2 | 1 | inner |
20.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.4.n.d | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 400.4.n.d | ✓ | 8 |
5.b | even | 2 | 1 | inner | 400.4.n.d | ✓ | 8 |
5.c | odd | 4 | 2 | inner | 400.4.n.d | ✓ | 8 |
20.d | odd | 2 | 1 | inner | 400.4.n.d | ✓ | 8 |
20.e | even | 4 | 2 | inner | 400.4.n.d | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
400.4.n.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
400.4.n.d | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
400.4.n.d | ✓ | 8 | 5.b | even | 2 | 1 | inner |
400.4.n.d | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
400.4.n.d | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
400.4.n.d | ✓ | 8 | 20.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 1156 \)
acting on \(S_{4}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 1156)^{2} \)
$5$
\( T^{8} \)
$7$
\( (T^{4} + 1156)^{2} \)
$11$
\( (T^{2} + 192)^{4} \)
$13$
\( (T^{4} + 42614784)^{2} \)
$17$
\( (T^{4} + 42614784)^{2} \)
$19$
\( (T^{2} - 9408)^{4} \)
$23$
\( (T^{4} + 224820036)^{2} \)
$29$
\( (T^{2} + 30276)^{4} \)
$31$
\( (T^{2} + 37632)^{4} \)
$37$
\( T^{8} \)
$41$
\( (T - 252)^{8} \)
$43$
\( (T^{4} + 6664109956)^{2} \)
$47$
\( (T^{4} + 26203191876)^{2} \)
$53$
\( (T^{4} + 102318096384)^{2} \)
$59$
\( (T^{2} - 762048)^{4} \)
$61$
\( (T - 56)^{8} \)
$67$
\( (T^{4} + 40636915396)^{2} \)
$71$
\( (T^{2} + 150528)^{4} \)
$73$
\( (T^{4} + 102318096384)^{2} \)
$79$
\( (T^{2} - 480000)^{4} \)
$83$
\( (T^{4} + 216621361476)^{2} \)
$89$
\( (T^{2} + 1764)^{4} \)
$97$
\( (T^{4} + 1217120845824)^{2} \)
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