Newspace parameters
Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 112.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.894324502638\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | 12.0.20138089353117696.1 |
Defining polynomial: |
\( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{10} - 3\nu^{8} - 2\nu^{7} + 2\nu^{6} + 4\nu^{5} + 2\nu^{4} + 8\nu^{3} + 8\nu^{2} - 16\nu - 48 ) / 16 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - \nu^{11} + 10 \nu^{10} + 15 \nu^{9} + 12 \nu^{8} - 10 \nu^{7} - 32 \nu^{6} - 34 \nu^{5} - 52 \nu^{4} - 32 \nu^{3} + 144 \nu^{2} + 272 \nu + 96 ) / 128 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{11} - 3\nu^{9} - 2\nu^{8} + 2\nu^{7} + 4\nu^{6} + 2\nu^{5} + 8\nu^{4} + 8\nu^{3} - 16\nu^{2} - 48\nu ) / 32 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + 64 \nu^{3} + 112 \nu^{2} + 48 \nu + 32 ) / 128 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} - 12 \nu^{4} - 96 \nu^{3} - 144 \nu^{2} - 80 \nu + 160 ) / 64 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} - 12 \nu^{4} - 96 \nu^{3} - 144 \nu^{2} + 48 \nu + 160 ) / 64 \)
|
\(\beta_{7}\) | \(=\) |
\( ( \nu^{11} + \nu^{10} - 3 \nu^{9} - 5 \nu^{8} - 4 \nu^{7} + 2 \nu^{6} + 6 \nu^{5} + 10 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 48 \nu - 64 ) / 16 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 13 \nu^{11} + 14 \nu^{10} - 19 \nu^{9} - 44 \nu^{8} - 46 \nu^{7} + 26 \nu^{5} + 196 \nu^{4} + 256 \nu^{3} + 112 \nu^{2} - 336 \nu - 480 ) / 128 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 7 \nu^{11} - 14 \nu^{10} + 9 \nu^{9} + 32 \nu^{8} + 34 \nu^{7} + 8 \nu^{6} + 2 \nu^{5} - 52 \nu^{4} - 192 \nu^{3} - 144 \nu^{2} + 176 \nu + 352 ) / 64 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 15 \nu^{11} + 26 \nu^{10} - 17 \nu^{9} - 68 \nu^{8} - 58 \nu^{7} - 32 \nu^{6} + 30 \nu^{5} + 172 \nu^{4} + 384 \nu^{3} + 272 \nu^{2} - 368 \nu - 672 ) / 128 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 7 \nu^{11} - 8 \nu^{10} + 9 \nu^{9} + 30 \nu^{8} + 22 \nu^{7} + 4 \nu^{6} - 6 \nu^{5} - 72 \nu^{4} - 144 \nu^{3} - 96 \nu^{2} + 208 \nu + 256 ) / 32 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 2\beta_{9} + 2\beta_{7} - \beta_{6} - \beta_{5} + 2 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( 2\beta_{11} + 2\beta_{10} - \beta_{6} - \beta_{5} + 2\beta_{4} - 4\beta_{2} + 2 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 2\beta_{9} + 2\beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 2 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 2\beta_{11} + 4\beta_{10} + 2\beta_{9} + 2\beta_{7} + \beta_{6} + \beta_{5} + 4\beta_{4} + 2\beta _1 + 2 ) / 2 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 6 \beta _1 + 10 ) / 2 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{11} + 8 \beta_{10} + 10 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} + 12 \beta_{3} - 4 \beta_{2} + 2 \beta _1 - 2 ) / 2 \)
|
\(\nu^{8}\) | \(=\) |
\( ( 10 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 8 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta _1 - 6 ) / 2 \)
|
\(\nu^{9}\) | \(=\) |
\( ( 2 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + 20 \beta_{4} - 4 \beta_{3} - 12 \beta_{2} + 14 \beta _1 - 2 ) / 2 \)
|
\(\nu^{10}\) | \(=\) |
\( ( 6 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} - \beta_{6} + 3 \beta_{5} - 12 \beta_{4} + 28 \beta_{3} + 20 \beta_{2} + 10 \beta _1 + 10 ) / 2 \)
|
\(\nu^{11}\) | \(=\) |
\( ( - 6 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 5 \beta_{5} + 52 \beta_{4} + 20 \beta_{3} + 12 \beta_{2} + 6 \beta _1 - 58 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
\(n\) | \(15\) | \(17\) | \(85\) |
\(\chi(n)\) | \(1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 |
|
−1.27801 | + | 0.605558i | 1.39123 | + | 1.39123i | 1.26660 | − | 1.54781i | 2.16478 | − | 2.16478i | −2.62048 | − | 0.935533i | − | 1.00000i | −0.681431 | + | 2.74511i | 0.871066i | −1.45570 | + | 4.07750i | |||||||||||||||||||||||||||||||||||||||
29.2 | −0.411286 | − | 1.35309i | 2.21570 | + | 2.21570i | −1.66169 | + | 1.11301i | −0.393125 | + | 0.393125i | 2.08675 | − | 3.90932i | − | 1.00000i | 2.18943 | + | 1.79064i | 6.81864i | 0.693620 | + | 0.370246i | ||||||||||||||||||||||||||||||||||||||||
29.3 | 0.163666 | + | 1.40471i | −2.05500 | − | 2.05500i | −1.94643 | + | 0.459808i | 2.72766 | − | 2.72766i | 2.55034 | − | 3.22301i | − | 1.00000i | −0.964462 | − | 2.65891i | 5.44602i | 4.27801 | + | 3.38515i | ||||||||||||||||||||||||||||||||||||||||
29.4 | 0.312504 | − | 1.37925i | −0.599978 | − | 0.599978i | −1.80468 | − | 0.862045i | 0.974969 | − | 0.974969i | −1.01502 | + | 0.640026i | − | 1.00000i | −1.75295 | + | 2.21972i | − | 2.28005i | −1.04005 | − | 1.64941i | |||||||||||||||||||||||||||||||||||||||
29.5 | 0.857418 | + | 1.12465i | 0.416854 | + | 0.416854i | −0.529667 | + | 1.92859i | −1.13169 | + | 1.13169i | −0.111396 | + | 0.826233i | − | 1.00000i | −2.62313 | + | 1.05792i | − | 2.65247i | −2.24309 | − | 0.302422i | |||||||||||||||||||||||||||||||||||||||
29.6 | 1.35570 | − | 0.402577i | 0.631188 | + | 0.631188i | 1.67586 | − | 1.09155i | −2.34259 | + | 2.34259i | 1.10981 | + | 0.601602i | − | 1.00000i | 1.83254 | − | 2.15448i | − | 2.20320i | −2.23279 | + | 4.11894i | |||||||||||||||||||||||||||||||||||||||
85.1 | −1.27801 | − | 0.605558i | 1.39123 | − | 1.39123i | 1.26660 | + | 1.54781i | 2.16478 | + | 2.16478i | −2.62048 | + | 0.935533i | 1.00000i | −0.681431 | − | 2.74511i | − | 0.871066i | −1.45570 | − | 4.07750i | ||||||||||||||||||||||||||||||||||||||||
85.2 | −0.411286 | + | 1.35309i | 2.21570 | − | 2.21570i | −1.66169 | − | 1.11301i | −0.393125 | − | 0.393125i | 2.08675 | + | 3.90932i | 1.00000i | 2.18943 | − | 1.79064i | − | 6.81864i | 0.693620 | − | 0.370246i | ||||||||||||||||||||||||||||||||||||||||
85.3 | 0.163666 | − | 1.40471i | −2.05500 | + | 2.05500i | −1.94643 | − | 0.459808i | 2.72766 | + | 2.72766i | 2.55034 | + | 3.22301i | 1.00000i | −0.964462 | + | 2.65891i | − | 5.44602i | 4.27801 | − | 3.38515i | ||||||||||||||||||||||||||||||||||||||||
85.4 | 0.312504 | + | 1.37925i | −0.599978 | + | 0.599978i | −1.80468 | + | 0.862045i | 0.974969 | + | 0.974969i | −1.01502 | − | 0.640026i | 1.00000i | −1.75295 | − | 2.21972i | 2.28005i | −1.04005 | + | 1.64941i | |||||||||||||||||||||||||||||||||||||||||
85.5 | 0.857418 | − | 1.12465i | 0.416854 | − | 0.416854i | −0.529667 | − | 1.92859i | −1.13169 | − | 1.13169i | −0.111396 | − | 0.826233i | 1.00000i | −2.62313 | − | 1.05792i | 2.65247i | −2.24309 | + | 0.302422i | |||||||||||||||||||||||||||||||||||||||||
85.6 | 1.35570 | + | 0.402577i | 0.631188 | − | 0.631188i | 1.67586 | + | 1.09155i | −2.34259 | − | 2.34259i | 1.10981 | − | 0.601602i | 1.00000i | 1.83254 | + | 2.15448i | 2.20320i | −2.23279 | − | 4.11894i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 112.2.m.d | ✓ | 12 |
4.b | odd | 2 | 1 | 448.2.m.d | 12 | ||
7.b | odd | 2 | 1 | 784.2.m.h | 12 | ||
7.c | even | 3 | 2 | 784.2.x.l | 24 | ||
7.d | odd | 6 | 2 | 784.2.x.m | 24 | ||
8.b | even | 2 | 1 | 896.2.m.g | 12 | ||
8.d | odd | 2 | 1 | 896.2.m.h | 12 | ||
16.e | even | 4 | 1 | inner | 112.2.m.d | ✓ | 12 |
16.e | even | 4 | 1 | 896.2.m.g | 12 | ||
16.f | odd | 4 | 1 | 448.2.m.d | 12 | ||
16.f | odd | 4 | 1 | 896.2.m.h | 12 | ||
32.g | even | 8 | 2 | 7168.2.a.bj | 12 | ||
32.h | odd | 8 | 2 | 7168.2.a.bi | 12 | ||
112.l | odd | 4 | 1 | 784.2.m.h | 12 | ||
112.w | even | 12 | 2 | 784.2.x.l | 24 | ||
112.x | odd | 12 | 2 | 784.2.x.m | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.2.m.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
112.2.m.d | ✓ | 12 | 16.e | even | 4 | 1 | inner |
448.2.m.d | 12 | 4.b | odd | 2 | 1 | ||
448.2.m.d | 12 | 16.f | odd | 4 | 1 | ||
784.2.m.h | 12 | 7.b | odd | 2 | 1 | ||
784.2.m.h | 12 | 112.l | odd | 4 | 1 | ||
784.2.x.l | 24 | 7.c | even | 3 | 2 | ||
784.2.x.l | 24 | 112.w | even | 12 | 2 | ||
784.2.x.m | 24 | 7.d | odd | 6 | 2 | ||
784.2.x.m | 24 | 112.x | odd | 12 | 2 | ||
896.2.m.g | 12 | 8.b | even | 2 | 1 | ||
896.2.m.g | 12 | 16.e | even | 4 | 1 | ||
896.2.m.h | 12 | 8.d | odd | 2 | 1 | ||
896.2.m.h | 12 | 16.f | odd | 4 | 1 | ||
7168.2.a.bi | 12 | 32.h | odd | 8 | 2 | ||
7168.2.a.bj | 12 | 32.g | even | 8 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 76 T_{3}^{8} - 288 T_{3}^{7} + 552 T_{3}^{6} - 376 T_{3}^{5} + 164 T_{3}^{4} - 144 T_{3}^{3} + 288 T_{3}^{2} - 192 T_{3} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 2 T^{11} + 5 T^{10} - 6 T^{9} + \cdots + 64 \)
$3$
\( T^{12} - 4 T^{11} + 8 T^{10} - 4 T^{9} + \cdots + 64 \)
$5$
\( T^{12} - 4 T^{11} + 8 T^{10} + 12 T^{9} + \cdots + 2304 \)
$7$
\( (T^{2} + 1)^{6} \)
$11$
\( T^{12} + 32 T^{9} + 740 T^{8} + \cdots + 541696 \)
$13$
\( T^{12} + 20 T^{9} + 1724 T^{8} + \cdots + 3211264 \)
$17$
\( (T^{6} + 4 T^{5} - 44 T^{4} - 200 T^{3} + \cdots - 96)^{2} \)
$19$
\( T^{12} - 76 T^{9} + 2444 T^{8} + \cdots + 2849344 \)
$23$
\( T^{12} + 136 T^{10} + \cdots + 10137856 \)
$29$
\( T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 8620096 \)
$31$
\( (T^{6} + 4 T^{5} - 16 T^{4} - 72 T^{3} + \cdots + 64)^{2} \)
$37$
\( T^{12} + 20 T^{11} + 200 T^{10} + \cdots + 5053504 \)
$41$
\( T^{12} + 120 T^{10} + 4704 T^{8} + \cdots + 25600 \)
$43$
\( T^{12} - 16 T^{11} + 128 T^{10} + \cdots + 23040000 \)
$47$
\( (T^{6} - 8 T^{5} - 104 T^{4} + 1032 T^{3} + \cdots + 19776)^{2} \)
$53$
\( T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 78400 \)
$59$
\( T^{12} + 16 T^{11} + \cdots + 119596096 \)
$61$
\( T^{12} + 20 T^{11} + 200 T^{10} + \cdots + 256 \)
$67$
\( T^{12} - 24 T^{11} + 288 T^{10} + \cdots + 3686400 \)
$71$
\( T^{12} + 432 T^{10} + 60960 T^{8} + \cdots + 6553600 \)
$73$
\( T^{12} + 616 T^{10} + \cdots + 3050573824 \)
$79$
\( (T^{6} - 12 T^{5} - 44 T^{4} + 576 T^{3} + \cdots - 2240)^{2} \)
$83$
\( T^{12} + 20 T^{11} + \cdots + 320093429824 \)
$89$
\( T^{12} + 552 T^{10} + \cdots + 35476475904 \)
$97$
\( (T^{6} - 24 T^{5} - 60 T^{4} + 2568 T^{3} + \cdots - 39008)^{2} \)
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