Newspace parameters
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(71.8519512762\) |
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: | \( x^{8} + 208x^{6} + 15517x^{4} + 287808x^{2} + 1830609 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{20}\cdot 7^{5} \) |
Twist minimal: | no (minimal twist has level 112) |
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 \( x^{8} + 208x^{6} + 15517x^{4} + 287808x^{2} + 1830609 \) :
\(\beta_{1}\) | \(=\) | \( ( 196\nu^{6} + 27552\nu^{4} + 567504\nu^{2} - 45923360 ) / 154009 \) |
\(\beta_{2}\) | \(=\) | \( ( -4040\nu^{6} - 831924\nu^{4} - 56404144\nu^{2} - 570933138 ) / 462027 \) |
\(\beta_{3}\) | \(=\) | \( ( 1222\nu^{7} + 303786\nu^{5} + 27739628\nu^{3} + 809257932\nu ) / 18943107 \) |
\(\beta_{4}\) | \(=\) | \( ( -93146\nu^{7} - 18153962\nu^{5} - 1214256788\nu^{3} - 11248398780\nu ) / 625122531 \) |
\(\beta_{5}\) | \(=\) | \( ( - 40870 \nu^{7} + 406802 \nu^{6} - 8341306 \nu^{5} + 77029898 \nu^{4} - 552201520 \nu^{3} + 5186588396 \nu^{2} - 5108756376 \nu + 52462110921 ) / 208374177 \) |
\(\beta_{6}\) | \(=\) | \( ( 77668\nu^{7} + 12942020\nu^{5} + 713838720\nu^{3} + 17813518128\nu ) / 208374177 \) |
\(\beta_{7}\) | \(=\) | \( ( 286090 \nu^{7} + 2847614 \nu^{6} + 58389142 \nu^{5} + 539209286 \nu^{4} + 3865410640 \nu^{3} + 36306118772 \nu^{2} + 35761294632 \nu + 367234776447 ) / 208374177 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + 7\beta_{6} - 7\beta_{5} + 42\beta_{4} + 14\beta_{3} ) / 784 \) |
\(\nu^{2}\) | \(=\) | \( ( 9\beta_{7} + 63\beta_{5} + 21\beta_{2} - 49\beta _1 - 20384 ) / 392 \) |
\(\nu^{3}\) | \(=\) | \( ( -317\beta_{7} - 525\beta_{6} + 2219\beta_{5} - 6846\beta_{4} + 714\beta_{3} ) / 784 \) |
\(\nu^{4}\) | \(=\) | \( ( -762\beta_{7} - 5334\beta_{5} - 2121\beta_{2} + 1792\beta _1 + 599270 ) / 196 \) |
\(\nu^{5}\) | \(=\) | \( ( 23483\beta_{7} + 11949\beta_{6} - 164381\beta_{5} + 453264\beta_{4} - 21672\beta_{3} ) / 392 \) |
\(\nu^{6}\) | \(=\) | \( ( 94086\beta_{7} + 658602\beta_{5} + 267750\beta_{2} - 26957\beta _1 - 8806672 ) / 196 \) |
\(\nu^{7}\) | \(=\) | \( ( -367279\beta_{7} + 95781\beta_{6} + 2570953\beta_{5} - 6983508\beta_{4} - 182196\beta_{3} ) / 56 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
447.1 |
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0 | −26.7378 | 0 | − | 100.497i | 0 | 89.7065 | − | 93.5935i | 0 | 471.912 | 0 | |||||||||||||||||||||||||||||||||||||||
447.2 | 0 | −26.7378 | 0 | 100.497i | 0 | 89.7065 | + | 93.5935i | 0 | 471.912 | 0 | |||||||||||||||||||||||||||||||||||||||||
447.3 | 0 | −8.31193 | 0 | − | 20.3058i | 0 | −99.9236 | + | 82.5970i | 0 | −173.912 | 0 | ||||||||||||||||||||||||||||||||||||||||
447.4 | 0 | −8.31193 | 0 | 20.3058i | 0 | −99.9236 | − | 82.5970i | 0 | −173.912 | 0 | |||||||||||||||||||||||||||||||||||||||||
447.5 | 0 | 8.31193 | 0 | − | 20.3058i | 0 | 99.9236 | − | 82.5970i | 0 | −173.912 | 0 | ||||||||||||||||||||||||||||||||||||||||
447.6 | 0 | 8.31193 | 0 | 20.3058i | 0 | 99.9236 | + | 82.5970i | 0 | −173.912 | 0 | |||||||||||||||||||||||||||||||||||||||||
447.7 | 0 | 26.7378 | 0 | − | 100.497i | 0 | −89.7065 | + | 93.5935i | 0 | 471.912 | 0 | ||||||||||||||||||||||||||||||||||||||||
447.8 | 0 | 26.7378 | 0 | 100.497i | 0 | −89.7065 | − | 93.5935i | 0 | 471.912 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 448.6.f.b | 8 | |
4.b | odd | 2 | 1 | inner | 448.6.f.b | 8 | |
7.b | odd | 2 | 1 | inner | 448.6.f.b | 8 | |
8.b | even | 2 | 1 | 112.6.f.a | ✓ | 8 | |
8.d | odd | 2 | 1 | 112.6.f.a | ✓ | 8 | |
24.f | even | 2 | 1 | 1008.6.b.g | 8 | ||
24.h | odd | 2 | 1 | 1008.6.b.g | 8 | ||
28.d | even | 2 | 1 | inner | 448.6.f.b | 8 | |
56.e | even | 2 | 1 | 112.6.f.a | ✓ | 8 | |
56.h | odd | 2 | 1 | 112.6.f.a | ✓ | 8 | |
168.e | odd | 2 | 1 | 1008.6.b.g | 8 | ||
168.i | even | 2 | 1 | 1008.6.b.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.6.f.a | ✓ | 8 | 8.b | even | 2 | 1 | |
112.6.f.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
112.6.f.a | ✓ | 8 | 56.e | even | 2 | 1 | |
112.6.f.a | ✓ | 8 | 56.h | odd | 2 | 1 | |
448.6.f.b | 8 | 1.a | even | 1 | 1 | trivial | |
448.6.f.b | 8 | 4.b | odd | 2 | 1 | inner | |
448.6.f.b | 8 | 7.b | odd | 2 | 1 | inner | |
448.6.f.b | 8 | 28.d | even | 2 | 1 | inner | |
1008.6.b.g | 8 | 24.f | even | 2 | 1 | ||
1008.6.b.g | 8 | 24.h | odd | 2 | 1 | ||
1008.6.b.g | 8 | 168.e | odd | 2 | 1 | ||
1008.6.b.g | 8 | 168.i | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 784T_{3}^{2} + 49392 \)
acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} - 784 T^{2} + 49392)^{2} \)
$5$
\( (T^{4} + 10512 T^{2} + 4164336)^{2} \)
$7$
\( T^{8} - 4900 T^{6} + \cdots + 79\!\cdots\!01 \)
$11$
\( (T^{4} + 405720 T^{2} + \cdots + 41092118928)^{2} \)
$13$
\( (T^{4} + 1587792 T^{2} + \cdots + 532363517424)^{2} \)
$17$
\( (T^{4} + 2502912 T^{2} + \cdots + 1219231678464)^{2} \)
$19$
\( (T^{4} - 2939216 T^{2} + \cdots + 390652133872)^{2} \)
$23$
\( (T^{4} + 20548248 T^{2} + \cdots + 2172546584208)^{2} \)
$29$
\( (T^{2} - 2436 T - 2270268)^{4} \)
$31$
\( (T^{4} - 76004096 T^{2} + \cdots + 10\!\cdots\!72)^{2} \)
$37$
\( (T^{2} + 4844 T - 27918044)^{4} \)
$41$
\( (T^{4} + 171938112 T^{2} + \cdots + 19\!\cdots\!36)^{2} \)
$43$
\( (T^{4} + 298206552 T^{2} + \cdots + 19\!\cdots\!28)^{2} \)
$47$
\( (T^{4} - 1042594560 T^{2} + \cdots + 27\!\cdots\!72)^{2} \)
$53$
\( (T^{2} + 25548 T - 212204124)^{4} \)
$59$
\( (T^{4} - 688136400 T^{2} + \cdots + 43\!\cdots\!68)^{2} \)
$61$
\( (T^{4} + 823393296 T^{2} + \cdots + 57\!\cdots\!04)^{2} \)
$67$
\( (T^{4} + 5888524824 T^{2} + \cdots + 61\!\cdots\!72)^{2} \)
$71$
\( (T^{4} + 3593673720 T^{2} + \cdots + 28\!\cdots\!28)^{2} \)
$73$
\( (T^{4} + 2931878592 T^{2} + \cdots + 12\!\cdots\!16)^{2} \)
$79$
\( (T^{4} + 4035162936 T^{2} + \cdots + 36\!\cdots\!72)^{2} \)
$83$
\( (T^{4} - 11503940112 T^{2} + \cdots + 67\!\cdots\!28)^{2} \)
$89$
\( (T^{4} + 4366945728 T^{2} + \cdots + 27\!\cdots\!96)^{2} \)
$97$
\( (T^{4} + 2413035264 T^{2} + \cdots + 93\!\cdots\!24)^{2} \)
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