Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.2164278413\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
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: | \( x^{8} - 2x^{7} - 6x^{6} + 12x^{5} + 96x^{3} - 384x^{2} - 1024x + 4096 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{16} \) |
Twist minimal: | no (minimal twist has level 56) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{8} - 2x^{7} - 6x^{6} + 12x^{5} + 96x^{3} - 384x^{2} - 1024x + 4096 \) :
\(\beta_{1}\) | \(=\) | \( ( 3\nu^{7} + 2\nu^{6} - 2\nu^{5} - 12\nu^{4} + 32\nu^{3} + 288\nu^{2} - 640\nu - 3072 ) / 512 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{7} + 6\nu^{6} - 18\nu^{5} - 4\nu^{4} + 272\nu^{3} - 800\nu^{2} + 1024\nu - 512 ) / 256 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{7} + 6\nu^{6} + 18\nu^{5} + 28\nu^{4} - 128\nu^{3} - 672\nu^{2} + 384\nu + 3072 ) / 512 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} - 2\nu^{5} + 16\nu^{4} - 24\nu^{3} + 64\nu - 1024 ) / 128 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} + 10\nu^{6} - 14\nu^{5} - 92\nu^{4} - 80\nu^{3} - 224\nu^{2} + 2560\nu - 2816 ) / 256 \) |
\(\beta_{6}\) | \(=\) | \( ( -5\nu^{7} - 2\nu^{6} + 38\nu^{5} + 44\nu^{4} + 80\nu^{3} - 160\nu^{2} + 1024\nu + 5888 ) / 256 \) |
\(\beta_{7}\) | \(=\) | \( ( 3\nu^{7} + 14\nu^{6} - 42\nu^{5} + 12\nu^{4} + 16\nu^{3} + 736\nu^{2} + 512\nu - 9216 ) / 256 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{3} - 2\beta _1 + 4 ) / 16 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + \beta_{6} - \beta_{4} - 2\beta_{3} - \beta_{2} + 15 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{6} - \beta_{5} - 2\beta_{4} - 6\beta_{3} + 4\beta_{2} + 10\beta _1 + 8 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( 3\beta_{7} + 3\beta_{6} - 4\beta_{5} + 13\beta_{4} + 2\beta_{3} + \beta_{2} - 8\beta _1 + 41 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -8\beta_{7} + 17\beta_{6} + 5\beta_{5} + 2\beta_{4} + 14\beta_{3} - 4\beta_{2} + 78\beta _1 - 232 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 17\beta_{7} + 5\beta_{6} - 25\beta_{4} + 62\beta_{3} + 3\beta_{2} + 80\beta _1 + 411 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -32\beta_{7} + 3\beta_{6} + 23\beta_{5} + 126\beta_{4} - 22\beta_{3} + 12\beta_{2} + 218\beta _1 + 1504 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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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113.1 |
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0 | − | 8.87222i | 0 | 1.57179i | 0 | 7.00000 | 0 | −51.7163 | 0 | |||||||||||||||||||||||||||||||||||||||||
113.2 | 0 | − | 5.24924i | 0 | 9.85277i | 0 | 7.00000 | 0 | −0.554549 | 0 | ||||||||||||||||||||||||||||||||||||||||||
113.3 | 0 | − | 3.82805i | 0 | 2.56837i | 0 | 7.00000 | 0 | 12.3460 | 0 | ||||||||||||||||||||||||||||||||||||||||||
113.4 | 0 | − | 2.25282i | 0 | − | 20.1954i | 0 | 7.00000 | 0 | 21.9248 | 0 | |||||||||||||||||||||||||||||||||||||||||
113.5 | 0 | 2.25282i | 0 | 20.1954i | 0 | 7.00000 | 0 | 21.9248 | 0 | |||||||||||||||||||||||||||||||||||||||||||
113.6 | 0 | 3.82805i | 0 | − | 2.56837i | 0 | 7.00000 | 0 | 12.3460 | 0 | ||||||||||||||||||||||||||||||||||||||||||
113.7 | 0 | 5.24924i | 0 | − | 9.85277i | 0 | 7.00000 | 0 | −0.554549 | 0 | ||||||||||||||||||||||||||||||||||||||||||
113.8 | 0 | 8.87222i | 0 | − | 1.57179i | 0 | 7.00000 | 0 | −51.7163 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.4.b.a | 8 | |
3.b | odd | 2 | 1 | 2016.4.c.a | 8 | ||
4.b | odd | 2 | 1 | 56.4.b.a | ✓ | 8 | |
8.b | even | 2 | 1 | inner | 224.4.b.a | 8 | |
8.d | odd | 2 | 1 | 56.4.b.a | ✓ | 8 | |
12.b | even | 2 | 1 | 504.4.c.a | 8 | ||
16.e | even | 4 | 2 | 1792.4.a.u | 8 | ||
16.f | odd | 4 | 2 | 1792.4.a.w | 8 | ||
24.f | even | 2 | 1 | 504.4.c.a | 8 | ||
24.h | odd | 2 | 1 | 2016.4.c.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.4.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
56.4.b.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
224.4.b.a | 8 | 1.a | even | 1 | 1 | trivial | |
224.4.b.a | 8 | 8.b | even | 2 | 1 | inner | |
504.4.c.a | 8 | 12.b | even | 2 | 1 | ||
504.4.c.a | 8 | 24.f | even | 2 | 1 | ||
1792.4.a.u | 8 | 16.e | even | 4 | 2 | ||
1792.4.a.w | 8 | 16.f | odd | 4 | 2 | ||
2016.4.c.a | 8 | 3.b | odd | 2 | 1 | ||
2016.4.c.a | 8 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 126T_{3}^{6} + 4340T_{3}^{4} + 50696T_{3}^{2} + 161312 \)
acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + 126 T^{6} + 4340 T^{4} + \cdots + 161312 \)
$5$
\( T^{8} + 514 T^{6} + 44188 T^{4} + \cdots + 645248 \)
$7$
\( (T - 7)^{8} \)
$11$
\( T^{8} + 3908 T^{6} + \cdots + 223860787200 \)
$13$
\( T^{8} + 4162 T^{6} + \cdots + 8031006848 \)
$17$
\( (T^{4} + 72 T^{3} - 5384 T^{2} + \cdots - 1875824)^{2} \)
$19$
\( T^{8} + 20958 T^{6} + \cdots + 61589268512 \)
$23$
\( (T^{4} + 130 T^{3} - 19220 T^{2} + \cdots - 85969984)^{2} \)
$29$
\( T^{8} + 157048 T^{6} + \cdots + 60\!\cdots\!32 \)
$31$
\( (T^{4} - 132 T^{3} - 64720 T^{2} + \cdots + 793521152)^{2} \)
$37$
\( T^{8} + 182136 T^{6} + \cdots + 64\!\cdots\!08 \)
$41$
\( (T^{4} - 356 T^{3} - 28784 T^{2} + \cdots - 1202210128)^{2} \)
$43$
\( T^{8} + 361156 T^{6} + \cdots + 18\!\cdots\!88 \)
$47$
\( (T^{4} - 381312 T^{2} + \cdots + 12485394432)^{2} \)
$53$
\( T^{8} + 1194896 T^{6} + \cdots + 31\!\cdots\!92 \)
$59$
\( T^{8} + 851214 T^{6} + \cdots + 13\!\cdots\!12 \)
$61$
\( T^{8} + 1109282 T^{6} + \cdots + 20\!\cdots\!00 \)
$67$
\( T^{8} + 1033628 T^{6} + \cdots + 13\!\cdots\!48 \)
$71$
\( (T^{4} + 656 T^{3} + \cdots + 31205572608)^{2} \)
$73$
\( (T^{4} - 696 T^{3} + \cdots - 78638316912)^{2} \)
$79$
\( (T^{4} - 304 T^{3} + \cdots - 117180334080)^{2} \)
$83$
\( T^{8} + 2480782 T^{6} + \cdots + 49\!\cdots\!68 \)
$89$
\( (T^{4} - 924 T^{3} + 45920 T^{2} + \cdots + 3549530480)^{2} \)
$97$
\( (T^{4} + 1100 T^{3} + \cdots + 2140769108912)^{2} \)
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