Newspace parameters
Level: | \( N \) | \(=\) | \( 152 = 2^{3} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 152.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.21372611072\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | 12.0.319794774016000000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{11} + 2\nu^{7} + 4\nu^{5} + 16\nu^{3} ) / 32 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{11} - 2\nu^{7} - 4\nu^{5} + 16\nu^{3} ) / 32 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{6} + 2\nu^{4} - 2\nu^{2} ) / 4 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{11} - 2\nu^{7} + 12\nu^{5} - 16\nu^{3} ) / 32 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{11} + 6\nu^{7} - 4\nu^{5} ) / 32 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{8} + 2\nu^{4} - 4\nu^{2} + 8 ) / 8 \) |
\(\beta_{9}\) | \(=\) | \( ( -\nu^{11} + 4\nu^{9} - 2\nu^{7} + 4\nu^{5} + 32\nu ) / 32 \) |
\(\beta_{10}\) | \(=\) | \( ( -\nu^{10} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} + 16 ) / 16 \) |
\(\beta_{11}\) | \(=\) | \( ( \nu^{10} - 2\nu^{8} + 2\nu^{6} + 8\nu^{2} - 32 ) / 16 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{4} + \beta_{3} \) |
\(\nu^{4}\) | \(=\) | \( \beta_{11} + \beta_{10} + \beta_{8} + \beta_{5} + \beta_{2} \) |
\(\nu^{5}\) | \(=\) | \( 2\beta_{6} + 2\beta_{3} \) |
\(\nu^{6}\) | \(=\) | \( 2\beta_{11} + 2\beta_{10} + 2\beta_{8} - 2\beta_{5} \) |
\(\nu^{7}\) | \(=\) | \( 4\beta_{7} - 2\beta_{4} + 2\beta_{3} \) |
\(\nu^{8}\) | \(=\) | \( -2\beta_{11} - 2\beta_{10} + 6\beta_{8} - 2\beta_{5} + 2\beta_{2} - 8 \) |
\(\nu^{9}\) | \(=\) | \( 8\beta_{9} - 4\beta_{6} - 4\beta_{4} - 8\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( 8\beta_{11} - 8\beta_{10} + 8\beta_{8} - 4\beta_{2} + 16 \) |
\(\nu^{11}\) | \(=\) | \( -8\beta_{7} - 8\beta_{6} - 12\beta_{4} + 4\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).
\(n\) | \(39\) | \(77\) | \(97\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
75.1 |
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−1.37364 | − | 0.336338i | 2.97320i | 1.77375 | + | 0.924013i | 3.04222i | 1.00000 | − | 4.08409i | − | 2.23607i | −2.12571 | − | 1.86584i | −5.83991 | 1.02321 | − | 4.17890i | |||||||||||||||||||||||||||||||||||||||||||
75.2 | −1.37364 | + | 0.336338i | − | 2.97320i | 1.77375 | − | 0.924013i | − | 3.04222i | 1.00000 | + | 4.08409i | 2.23607i | −2.12571 | + | 1.86584i | −5.83991 | 1.02321 | + | 4.17890i | |||||||||||||||||||||||||||||||||||||||||||
75.3 | −1.17117 | − | 0.792696i | 1.26152i | 0.743268 | + | 1.85676i | − | 3.51876i | 1.00000 | − | 1.47745i | 2.23607i | 0.601353 | − | 2.76376i | 1.40857 | −2.78930 | + | 4.12105i | ||||||||||||||||||||||||||||||||||||||||||||
75.4 | −1.17117 | + | 0.792696i | − | 1.26152i | 0.743268 | − | 1.85676i | 3.51876i | 1.00000 | + | 1.47745i | − | 2.23607i | 0.601353 | + | 2.76376i | 1.40857 | −2.78930 | − | 4.12105i | |||||||||||||||||||||||||||||||||||||||||||
75.5 | −0.491416 | − | 1.32609i | 0.754098i | −1.51702 | + | 1.30332i | 2.08884i | 1.00000 | − | 0.370575i | 2.23607i | 2.47381 | + | 1.37123i | 2.43134 | 2.76999 | − | 1.02649i | |||||||||||||||||||||||||||||||||||||||||||||
75.6 | −0.491416 | + | 1.32609i | − | 0.754098i | −1.51702 | − | 1.30332i | − | 2.08884i | 1.00000 | + | 0.370575i | − | 2.23607i | 2.47381 | − | 1.37123i | 2.43134 | 2.76999 | + | 1.02649i | ||||||||||||||||||||||||||||||||||||||||||
75.7 | 0.491416 | − | 1.32609i | 0.754098i | −1.51702 | − | 1.30332i | − | 2.08884i | 1.00000 | + | 0.370575i | − | 2.23607i | −2.47381 | + | 1.37123i | 2.43134 | −2.76999 | − | 1.02649i | |||||||||||||||||||||||||||||||||||||||||||
75.8 | 0.491416 | + | 1.32609i | − | 0.754098i | −1.51702 | + | 1.30332i | 2.08884i | 1.00000 | − | 0.370575i | 2.23607i | −2.47381 | − | 1.37123i | 2.43134 | −2.76999 | + | 1.02649i | ||||||||||||||||||||||||||||||||||||||||||||
75.9 | 1.17117 | − | 0.792696i | 1.26152i | 0.743268 | − | 1.85676i | 3.51876i | 1.00000 | + | 1.47745i | − | 2.23607i | −0.601353 | − | 2.76376i | 1.40857 | 2.78930 | + | 4.12105i | ||||||||||||||||||||||||||||||||||||||||||||
75.10 | 1.17117 | + | 0.792696i | − | 1.26152i | 0.743268 | + | 1.85676i | − | 3.51876i | 1.00000 | − | 1.47745i | 2.23607i | −0.601353 | + | 2.76376i | 1.40857 | 2.78930 | − | 4.12105i | |||||||||||||||||||||||||||||||||||||||||||
75.11 | 1.37364 | − | 0.336338i | 2.97320i | 1.77375 | − | 0.924013i | − | 3.04222i | 1.00000 | + | 4.08409i | 2.23607i | 2.12571 | − | 1.86584i | −5.83991 | −1.02321 | − | 4.17890i | ||||||||||||||||||||||||||||||||||||||||||||
75.12 | 1.37364 | + | 0.336338i | − | 2.97320i | 1.77375 | + | 0.924013i | 3.04222i | 1.00000 | − | 4.08409i | − | 2.23607i | 2.12571 | + | 1.86584i | −5.83991 | −1.02321 | + | 4.17890i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
152.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 152.2.b.c | ✓ | 12 |
3.b | odd | 2 | 1 | 1368.2.e.e | 12 | ||
4.b | odd | 2 | 1 | 608.2.b.c | 12 | ||
8.b | even | 2 | 1 | 608.2.b.c | 12 | ||
8.d | odd | 2 | 1 | inner | 152.2.b.c | ✓ | 12 |
12.b | even | 2 | 1 | 5472.2.e.e | 12 | ||
19.b | odd | 2 | 1 | inner | 152.2.b.c | ✓ | 12 |
24.f | even | 2 | 1 | 1368.2.e.e | 12 | ||
24.h | odd | 2 | 1 | 5472.2.e.e | 12 | ||
57.d | even | 2 | 1 | 1368.2.e.e | 12 | ||
76.d | even | 2 | 1 | 608.2.b.c | 12 | ||
152.b | even | 2 | 1 | inner | 152.2.b.c | ✓ | 12 |
152.g | odd | 2 | 1 | 608.2.b.c | 12 | ||
228.b | odd | 2 | 1 | 5472.2.e.e | 12 | ||
456.l | odd | 2 | 1 | 1368.2.e.e | 12 | ||
456.p | even | 2 | 1 | 5472.2.e.e | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
152.2.b.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
152.2.b.c | ✓ | 12 | 8.d | odd | 2 | 1 | inner |
152.2.b.c | ✓ | 12 | 19.b | odd | 2 | 1 | inner |
152.2.b.c | ✓ | 12 | 152.b | even | 2 | 1 | inner |
608.2.b.c | 12 | 4.b | odd | 2 | 1 | ||
608.2.b.c | 12 | 8.b | even | 2 | 1 | ||
608.2.b.c | 12 | 76.d | even | 2 | 1 | ||
608.2.b.c | 12 | 152.g | odd | 2 | 1 | ||
1368.2.e.e | 12 | 3.b | odd | 2 | 1 | ||
1368.2.e.e | 12 | 24.f | even | 2 | 1 | ||
1368.2.e.e | 12 | 57.d | even | 2 | 1 | ||
1368.2.e.e | 12 | 456.l | odd | 2 | 1 | ||
5472.2.e.e | 12 | 12.b | even | 2 | 1 | ||
5472.2.e.e | 12 | 24.h | odd | 2 | 1 | ||
5472.2.e.e | 12 | 228.b | odd | 2 | 1 | ||
5472.2.e.e | 12 | 456.p | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 11T_{3}^{4} + 20T_{3}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(152, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 2 T^{10} + 2 T^{8} + 8 T^{4} + \cdots + 64 \)
$3$
\( (T^{6} + 11 T^{4} + 20 T^{2} + 8)^{2} \)
$5$
\( (T^{6} + 26 T^{4} + 209 T^{2} + 500)^{2} \)
$7$
\( (T^{2} + 5)^{6} \)
$11$
\( (T^{3} - 7 T - 4)^{4} \)
$13$
\( (T^{6} - 51 T^{4} + 464 T^{2} - 640)^{2} \)
$17$
\( (T^{3} - 3 T^{2} - 25 T + 59)^{4} \)
$19$
\( (T^{6} + 12 T^{5} + 77 T^{4} + 376 T^{3} + \cdots + 6859)^{2} \)
$23$
\( (T^{6} + 69 T^{4} + 1184 T^{2} + 320)^{2} \)
$29$
\( (T^{6} - 71 T^{4} + 244 T^{2} - 40)^{2} \)
$31$
\( (T^{6} - 44 T^{4} + 504 T^{2} - 640)^{2} \)
$37$
\( (T^{6} - 84 T^{4} + 1784 T^{2} + \cdots - 2560)^{2} \)
$41$
\( (T^{6} + 136 T^{4} + 5640 T^{2} + \cdots + 70688)^{2} \)
$43$
\( (T^{3} + 6 T^{2} - 51 T - 10)^{4} \)
$47$
\( (T^{6} + 106 T^{4} + 1609 T^{2} + 80)^{2} \)
$53$
\( (T^{6} - 191 T^{4} + 8724 T^{2} + \cdots - 112360)^{2} \)
$59$
\( (T^{6} + 103 T^{4} + 2656 T^{2} + \cdots + 12800)^{2} \)
$61$
\( (T^{6} + 214 T^{4} + 10809 T^{2} + \cdots + 50000)^{2} \)
$67$
\( (T^{6} + 155 T^{4} + 6452 T^{2} + \cdots + 75272)^{2} \)
$71$
\( (T^{6} - 476 T^{4} + 72384 T^{2} + \cdots - 3504640)^{2} \)
$73$
\( (T^{3} - T^{2} - 65 T + 97)^{4} \)
$79$
\( (T^{6} - 424 T^{4} + 51464 T^{2} + \cdots - 1697440)^{2} \)
$83$
\( (T - 2)^{12} \)
$89$
\( (T^{6} + 232 T^{4} + 13960 T^{2} + \cdots + 231200)^{2} \)
$97$
\( (T^{6} + 316 T^{4} + 1920 T^{2} + \cdots + 2048)^{2} \)
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