Newspace parameters
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.k (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.14984578911\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.18939904.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 48) |
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
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4 \)
|
\(\beta_{2}\) | \(=\) |
\( -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 42\nu^{4} - 59\nu^{3} + 48\nu^{2} - 24\nu + 4 \)
|
\(\beta_{3}\) | \(=\) |
\( -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 \)
|
\(\beta_{4}\) | \(=\) |
\( -7\nu^{7} + 25\nu^{6} - 87\nu^{5} + 158\nu^{4} - 231\nu^{3} + 206\nu^{2} - 118\nu + 31 \)
|
\(\beta_{5}\) | \(=\) |
\( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \)
|
\(\beta_{6}\) | \(=\) |
\( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 257\nu^{3} - 224\nu^{2} + 130\nu - 32 \)
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\(\beta_{7}\) | \(=\) |
\( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} + \beta_{6} + 5\beta_{5} - 3\beta_{4} + 5\beta_{3} - 3\beta_{2} + 4\beta _1 - 5 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 12\beta_{7} - 11\beta_{6} + 11\beta_{5} - 5\beta_{4} - \beta_{3} - 9\beta_{2} + 2\beta _1 + 5 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 6\beta_{7} - 17\beta_{6} - 13\beta_{5} + 13\beta_{4} - 23\beta_{3} + 3\beta_{2} - 18\beta _1 + 21 ) / 2 \)
|
\(\nu^{6}\) | \(=\) |
\( ( -46\beta_{7} + 29\beta_{6} - 67\beta_{5} + 37\beta_{4} - 15\beta_{3} + 47\beta_{2} - 24\beta _1 - 1 ) / 2 \)
|
\(\nu^{7}\) | \(=\) |
\( ( -76\beta_{7} + 105\beta_{6} - 7\beta_{5} - 31\beta_{4} + 91\beta_{3} + 39\beta_{2} + 68\beta _1 - 83 ) / 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(65\) | \(127\) |
\(\chi(n)\) | \(\beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
−0.874559 | + | 1.11137i | 0 | −0.470294 | − | 1.94392i | 0.334904 | + | 0.334904i | 0 | 4.55765i | 2.57172 | + | 1.17740i | 0 | −0.665096 | + | 0.0793096i | ||||||||||||||||||||||||||||||||
37.2 | −0.635665 | − | 1.26330i | 0 | −1.19186 | + | 1.60607i | 2.68554 | + | 2.68554i | 0 | 2.15894i | 2.78658 | + | 0.484753i | 0 | 1.68554 | − | 5.09976i | |||||||||||||||||||||||||||||||||
37.3 | 0.167452 | − | 1.40426i | 0 | −1.94392 | − | 0.470294i | −1.74912 | − | 1.74912i | 0 | − | 2.55765i | −0.985930 | + | 2.65103i | 0 | −2.74912 | + | 2.16333i | ||||||||||||||||||||||||||||||||
37.4 | 1.34277 | − | 0.443806i | 0 | 1.60607 | − | 1.19186i | −1.27133 | − | 1.27133i | 0 | − | 0.158942i | 1.62764 | − | 2.31318i | 0 | −2.27133 | − | 1.14288i | ||||||||||||||||||||||||||||||||
109.1 | −0.874559 | − | 1.11137i | 0 | −0.470294 | + | 1.94392i | 0.334904 | − | 0.334904i | 0 | − | 4.55765i | 2.57172 | − | 1.17740i | 0 | −0.665096 | − | 0.0793096i | ||||||||||||||||||||||||||||||||
109.2 | −0.635665 | + | 1.26330i | 0 | −1.19186 | − | 1.60607i | 2.68554 | − | 2.68554i | 0 | − | 2.15894i | 2.78658 | − | 0.484753i | 0 | 1.68554 | + | 5.09976i | ||||||||||||||||||||||||||||||||
109.3 | 0.167452 | + | 1.40426i | 0 | −1.94392 | + | 0.470294i | −1.74912 | + | 1.74912i | 0 | 2.55765i | −0.985930 | − | 2.65103i | 0 | −2.74912 | − | 2.16333i | |||||||||||||||||||||||||||||||||
109.4 | 1.34277 | + | 0.443806i | 0 | 1.60607 | + | 1.19186i | −1.27133 | + | 1.27133i | 0 | 0.158942i | 1.62764 | + | 2.31318i | 0 | −2.27133 | + | 1.14288i | |||||||||||||||||||||||||||||||||
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 | 144.2.k.b | 8 | |
3.b | odd | 2 | 1 | 48.2.j.a | ✓ | 8 | |
4.b | odd | 2 | 1 | 576.2.k.b | 8 | ||
8.b | even | 2 | 1 | 1152.2.k.c | 8 | ||
8.d | odd | 2 | 1 | 1152.2.k.f | 8 | ||
12.b | even | 2 | 1 | 192.2.j.a | 8 | ||
16.e | even | 4 | 1 | inner | 144.2.k.b | 8 | |
16.e | even | 4 | 1 | 1152.2.k.c | 8 | ||
16.f | odd | 4 | 1 | 576.2.k.b | 8 | ||
16.f | odd | 4 | 1 | 1152.2.k.f | 8 | ||
24.f | even | 2 | 1 | 384.2.j.a | 8 | ||
24.h | odd | 2 | 1 | 384.2.j.b | 8 | ||
32.g | even | 8 | 1 | 9216.2.a.y | 4 | ||
32.g | even | 8 | 1 | 9216.2.a.bo | 4 | ||
32.h | odd | 8 | 1 | 9216.2.a.x | 4 | ||
32.h | odd | 8 | 1 | 9216.2.a.bn | 4 | ||
48.i | odd | 4 | 1 | 48.2.j.a | ✓ | 8 | |
48.i | odd | 4 | 1 | 384.2.j.b | 8 | ||
48.k | even | 4 | 1 | 192.2.j.a | 8 | ||
48.k | even | 4 | 1 | 384.2.j.a | 8 | ||
96.o | even | 8 | 1 | 3072.2.a.n | 4 | ||
96.o | even | 8 | 1 | 3072.2.a.o | 4 | ||
96.o | even | 8 | 2 | 3072.2.d.i | 8 | ||
96.p | odd | 8 | 1 | 3072.2.a.i | 4 | ||
96.p | odd | 8 | 1 | 3072.2.a.t | 4 | ||
96.p | odd | 8 | 2 | 3072.2.d.f | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.2.j.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
48.2.j.a | ✓ | 8 | 48.i | odd | 4 | 1 | |
144.2.k.b | 8 | 1.a | even | 1 | 1 | trivial | |
144.2.k.b | 8 | 16.e | even | 4 | 1 | inner | |
192.2.j.a | 8 | 12.b | even | 2 | 1 | ||
192.2.j.a | 8 | 48.k | even | 4 | 1 | ||
384.2.j.a | 8 | 24.f | even | 2 | 1 | ||
384.2.j.a | 8 | 48.k | even | 4 | 1 | ||
384.2.j.b | 8 | 24.h | odd | 2 | 1 | ||
384.2.j.b | 8 | 48.i | odd | 4 | 1 | ||
576.2.k.b | 8 | 4.b | odd | 2 | 1 | ||
576.2.k.b | 8 | 16.f | odd | 4 | 1 | ||
1152.2.k.c | 8 | 8.b | even | 2 | 1 | ||
1152.2.k.c | 8 | 16.e | even | 4 | 1 | ||
1152.2.k.f | 8 | 8.d | odd | 2 | 1 | ||
1152.2.k.f | 8 | 16.f | odd | 4 | 1 | ||
3072.2.a.i | 4 | 96.p | odd | 8 | 1 | ||
3072.2.a.n | 4 | 96.o | even | 8 | 1 | ||
3072.2.a.o | 4 | 96.o | even | 8 | 1 | ||
3072.2.a.t | 4 | 96.p | odd | 8 | 1 | ||
3072.2.d.f | 8 | 96.p | odd | 8 | 2 | ||
3072.2.d.i | 8 | 96.o | even | 8 | 2 | ||
9216.2.a.x | 4 | 32.h | odd | 8 | 1 | ||
9216.2.a.y | 4 | 32.g | even | 8 | 1 | ||
9216.2.a.bn | 4 | 32.h | odd | 8 | 1 | ||
9216.2.a.bo | 4 | 32.g | even | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 16T_{5}^{5} + 128T_{5}^{4} + 192T_{5}^{3} + 128T_{5}^{2} - 128T_{5} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 2 T^{6} - 4 T^{5} + 2 T^{4} + \cdots + 16 \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 16 T^{5} + 128 T^{4} + \cdots + 64 \)
$7$
\( T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16 \)
$11$
\( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1024 \)
$13$
\( T^{8} - 64 T^{5} + 776 T^{4} + \cdots + 16 \)
$17$
\( (T^{4} - 32 T^{2} - 64 T + 16)^{2} \)
$19$
\( T^{8} + 8 T^{7} + 32 T^{6} - 32 T^{5} + \cdots + 256 \)
$23$
\( (T^{2} + 8)^{4} \)
$29$
\( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 61504 \)
$31$
\( (T^{4} - 12 T^{3} + 40 T^{2} - 24 T - 28)^{2} \)
$37$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704 \)
$41$
\( T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544 \)
$43$
\( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 12544 \)
$47$
\( (T^{2} - 8)^{4} \)
$53$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 18496 \)
$59$
\( (T^{2} + 8 T + 32)^{4} \)
$61$
\( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704 \)
$67$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 65536 \)
$71$
\( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \)
$73$
\( T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096 \)
$79$
\( (T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2} \)
$83$
\( T^{8} - 40 T^{7} + 800 T^{6} + \cdots + 1024 \)
$89$
\( T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216 \)
$97$
\( (T^{4} - 224 T^{2} + 768 T + 512)^{2} \)
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