Newspace parameters
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.s (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.5774358654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} + 230 x^{14} - 12 x^{13} + 20791 x^{12} - 4768 x^{11} + 945164 x^{10} - 389856 x^{9} + \cdots + 5722038847 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{38}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 56) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 230 x^{14} - 12 x^{13} + 20791 x^{12} - 4768 x^{11} + 945164 x^{10} - 389856 x^{9} + \cdots + 5722038847 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 74\!\cdots\!18 \nu^{15} + \cdots + 60\!\cdots\!88 ) / 32\!\cdots\!04 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 15\!\cdots\!32 \nu^{15} + \cdots + 68\!\cdots\!63 ) / 55\!\cdots\!08 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 78\!\cdots\!32 \nu^{15} + \cdots + 23\!\cdots\!08 ) / 58\!\cdots\!84 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18\!\cdots\!42 \nu^{15} + \cdots + 54\!\cdots\!26 ) / 11\!\cdots\!68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 28\!\cdots\!22 \nu^{15} + \cdots + 17\!\cdots\!87 ) / 11\!\cdots\!68 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 61\!\cdots\!88 \nu^{15} + \cdots + 56\!\cdots\!52 ) / 19\!\cdots\!28 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 76\!\cdots\!06 \nu^{15} + \cdots - 12\!\cdots\!74 ) / 11\!\cdots\!68 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!94 \nu^{15} + \cdots + 60\!\cdots\!52 ) / 11\!\cdots\!68 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 36\!\cdots\!00 \nu^{15} + \cdots + 22\!\cdots\!87 ) / 39\!\cdots\!56 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 12\!\cdots\!58 \nu^{15} + \cdots - 28\!\cdots\!42 ) / 11\!\cdots\!68 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\!\cdots\!66 \nu^{15} + \cdots - 89\!\cdots\!27 ) / 11\!\cdots\!68 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 43\!\cdots\!92 \nu^{15} + \cdots - 47\!\cdots\!11 ) / 29\!\cdots\!92 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 78\!\cdots\!74 \nu^{15} + \cdots - 14\!\cdots\!95 ) / 37\!\cdots\!28 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 37\!\cdots\!42 \nu^{15} + \cdots + 18\!\cdots\!91 ) / 16\!\cdots\!24 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 49\!\cdots\!80 \nu^{15} + \cdots + 23\!\cdots\!07 ) / 17\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( ( - 6 \beta_{15} - 8 \beta_{14} - 10 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} - 2 \beta_{10} + \cdots + 56 ) / 224 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 18 \beta_{15} - \beta_{14} - 19 \beta_{13} - 10 \beta_{12} - 2 \beta_{11} + 14 \beta_{10} + \cdots - 3220 ) / 112 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 174 \beta_{15} + 348 \beta_{14} + 522 \beta_{13} + 232 \beta_{12} - 284 \beta_{11} - 70 \beta_{10} + \cdots - 7056 ) / 224 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 753 \beta_{15} - 169 \beta_{14} + 666 \beta_{13} + 463 \beta_{12} + 121 \beta_{11} - 727 \beta_{10} + \cdots + 72086 ) / 56 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2891 \beta_{15} - 9750 \beta_{14} - 15237 \beta_{13} - 9036 \beta_{12} + 9212 \beta_{11} + \cdots + 370972 ) / 112 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 110580 \beta_{15} + 45231 \beta_{14} - 99041 \beta_{13} - 78232 \beta_{12} - 21844 \beta_{11} + \cdots - 7071064 ) / 112 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 47810 \beta_{15} + 1202280 \beta_{14} + 1748478 \beta_{13} + 1347460 \beta_{12} - 1239476 \beta_{11} + \cdots - 77480312 ) / 224 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3973859 \beta_{15} - 2026919 \beta_{14} + 3847558 \beta_{13} + 3285441 \beta_{12} + 685843 \beta_{11} + \cdots + 172074462 ) / 56 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 23839322 \beta_{15} - 80315620 \beta_{14} - 93145314 \beta_{13} - 93800968 \beta_{12} + \cdots + 7544495952 ) / 224 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 565989130 \beta_{15} + 312301235 \beta_{14} - 608573355 \beta_{13} - 549097646 \beta_{12} + \cdots - 14670000100 ) / 112 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 1921862131 \beta_{15} + 2883521894 \beta_{14} + 2103456147 \beta_{13} + 3038716164 \beta_{12} + \cdots - 341851756428 ) / 112 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 19842082136 \beta_{15} - 11031004420 \beta_{14} + 24056395784 \beta_{13} + 22691634048 \beta_{12} + \cdots + 174876161432 ) / 56 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 440367412574 \beta_{15} - 437513919800 \beta_{14} - 109866301534 \beta_{13} - 358151156972 \beta_{12} + \cdots + 58428587870072 ) / 224 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 2709863699570 \beta_{15} + 1453825641077 \beta_{14} - 3758450786613 \beta_{13} + \cdots + 27902704716124 ) / 112 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 44001238176086 \beta_{15} + 34300333800636 \beta_{14} - 7033981710402 \beta_{13} + \cdots - 47\!\cdots\!56 ) / 224 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −12.7833 | − | 7.38047i | 0 | 34.7085 | − | 20.0390i | 0 | 30.0415 | + | 38.7106i | 0 | 68.4426 | + | 118.546i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −9.24784 | − | 5.33924i | 0 | 7.20435 | − | 4.15943i | 0 | −41.8806 | − | 25.4364i | 0 | 16.5150 | + | 28.6049i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | −7.86567 | − | 4.54125i | 0 | −38.9762 | + | 22.5029i | 0 | 24.9652 | − | 42.1632i | 0 | 0.745853 | + | 1.29185i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | −0.465735 | − | 0.268892i | 0 | −25.0164 | + | 14.4432i | 0 | 24.2904 | + | 42.5555i | 0 | −40.3554 | − | 69.8976i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 2.39239 | + | 1.38125i | 0 | −0.113987 | + | 0.0658102i | 0 | −37.6924 | + | 31.3095i | 0 | −36.6843 | − | 63.5391i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 3.70610 | + | 2.13972i | 0 | 25.8222 | − | 14.9084i | 0 | 45.6687 | − | 17.7587i | 0 | −31.3432 | − | 54.2881i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 9.77134 | + | 5.64149i | 0 | 12.8670 | − | 7.42875i | 0 | 30.6268 | − | 38.2491i | 0 | 23.1527 | + | 40.1017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 14.4928 | + | 8.36740i | 0 | −16.4954 | + | 9.52362i | 0 | −48.0196 | − | 9.75278i | 0 | 99.5267 | + | 172.385i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.1 | 0 | −12.7833 | + | 7.38047i | 0 | 34.7085 | + | 20.0390i | 0 | 30.0415 | − | 38.7106i | 0 | 68.4426 | − | 118.546i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | −9.24784 | + | 5.33924i | 0 | 7.20435 | + | 4.15943i | 0 | −41.8806 | + | 25.4364i | 0 | 16.5150 | − | 28.6049i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | −7.86567 | + | 4.54125i | 0 | −38.9762 | − | 22.5029i | 0 | 24.9652 | + | 42.1632i | 0 | 0.745853 | − | 1.29185i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | −0.465735 | + | 0.268892i | 0 | −25.0164 | − | 14.4432i | 0 | 24.2904 | − | 42.5555i | 0 | −40.3554 | + | 69.8976i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 2.39239 | − | 1.38125i | 0 | −0.113987 | − | 0.0658102i | 0 | −37.6924 | − | 31.3095i | 0 | −36.6843 | + | 63.5391i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | 3.70610 | − | 2.13972i | 0 | 25.8222 | + | 14.9084i | 0 | 45.6687 | + | 17.7587i | 0 | −31.3432 | + | 54.2881i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 9.77134 | − | 5.64149i | 0 | 12.8670 | + | 7.42875i | 0 | 30.6268 | + | 38.2491i | 0 | 23.1527 | − | 40.1017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 14.4928 | − | 8.36740i | 0 | −16.4954 | − | 9.52362i | 0 | −48.0196 | + | 9.75278i | 0 | 99.5267 | − | 172.385i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.5.s.d | 16 | |
| 4.b | odd | 2 | 1 | 56.5.o.a | ✓ | 16 | |
| 7.c | even | 3 | 1 | 784.5.c.j | 16 | ||
| 7.d | odd | 6 | 1 | inner | 112.5.s.d | 16 | |
| 7.d | odd | 6 | 1 | 784.5.c.j | 16 | ||
| 12.b | even | 2 | 1 | 504.5.by.b | 16 | ||
| 28.d | even | 2 | 1 | 392.5.o.a | 16 | ||
| 28.f | even | 6 | 1 | 56.5.o.a | ✓ | 16 | |
| 28.f | even | 6 | 1 | 392.5.c.c | 16 | ||
| 28.g | odd | 6 | 1 | 392.5.c.c | 16 | ||
| 28.g | odd | 6 | 1 | 392.5.o.a | 16 | ||
| 84.j | odd | 6 | 1 | 504.5.by.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.5.o.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 56.5.o.a | ✓ | 16 | 28.f | even | 6 | 1 | |
| 112.5.s.d | 16 | 1.a | even | 1 | 1 | trivial | |
| 112.5.s.d | 16 | 7.d | odd | 6 | 1 | inner | |
| 392.5.c.c | 16 | 28.f | even | 6 | 1 | ||
| 392.5.c.c | 16 | 28.g | odd | 6 | 1 | ||
| 392.5.o.a | 16 | 28.d | even | 2 | 1 | ||
| 392.5.o.a | 16 | 28.g | odd | 6 | 1 | ||
| 504.5.by.b | 16 | 12.b | even | 2 | 1 | ||
| 504.5.by.b | 16 | 84.j | odd | 6 | 1 | ||
| 784.5.c.j | 16 | 7.c | even | 3 | 1 | ||
| 784.5.c.j | 16 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 424 T_{3}^{14} + 130466 T_{3}^{12} + 206304 T_{3}^{11} - 17768480 T_{3}^{10} + \cdots + 2953506904929 \)
acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} + \cdots + 2953506904929 \)
|
| $5$ |
\( T^{16} + \cdots + 23\!\cdots\!61 \)
|
| $7$ |
\( T^{16} + \cdots + 11\!\cdots\!01 \)
|
| $11$ |
\( T^{16} + \cdots + 43\!\cdots\!29 \)
|
| $13$ |
\( T^{16} + \cdots + 56\!\cdots\!44 \)
|
| $17$ |
\( T^{16} + \cdots + 71\!\cdots\!61 \)
|
| $19$ |
\( T^{16} + \cdots + 42\!\cdots\!61 \)
|
| $23$ |
\( T^{16} + \cdots + 24\!\cdots\!09 \)
|
| $29$ |
\( (T^{8} + \cdots + 16\!\cdots\!64)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 18\!\cdots\!01 \)
|
| $37$ |
\( T^{16} + \cdots + 76\!\cdots\!49 \)
|
| $41$ |
\( T^{16} + \cdots + 12\!\cdots\!76 \)
|
| $43$ |
\( (T^{8} + \cdots - 39\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 60\!\cdots\!49 \)
|
| $53$ |
\( T^{16} + \cdots + 18\!\cdots\!21 \)
|
| $59$ |
\( T^{16} + \cdots + 58\!\cdots\!29 \)
|
| $61$ |
\( T^{16} + \cdots + 31\!\cdots\!09 \)
|
| $67$ |
\( T^{16} + \cdots + 11\!\cdots\!09 \)
|
| $71$ |
\( (T^{8} + \cdots - 52\!\cdots\!88)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 24\!\cdots\!29 \)
|
| $79$ |
\( T^{16} + \cdots + 96\!\cdots\!41 \)
|
| $83$ |
\( T^{16} + \cdots + 15\!\cdots\!16 \)
|
| $89$ |
\( T^{16} + \cdots + 42\!\cdots\!89 \)
|
| $97$ |
\( T^{16} + \cdots + 95\!\cdots\!36 \)
|
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