Newspace parameters
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.84118909057\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.12261951429820416.4 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 62x^{6} + 1113x^{4} + 5786x^{2} + 5776 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{3} \) |
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 62x^{6} + 1113x^{4} + 5786x^{2} + 5776 \) :
\(\beta_{1}\) | \(=\) | \( ( -17\nu^{6} - 1018\nu^{4} - 15948\nu^{2} - 42252 ) / 4631 \) |
\(\beta_{2}\) | \(=\) | \( ( -2\nu^{6} - 95\nu^{4} - 1059\nu^{2} - 2321 ) / 421 \) |
\(\beta_{3}\) | \(=\) | \( ( -2\nu^{6} - 95\nu^{4} - 638\nu^{2} + 4415 ) / 421 \) |
\(\beta_{4}\) | \(=\) | \( ( -36\nu^{7} - 2973\nu^{5} - 79265\nu^{3} - 644650\nu ) / 87989 \) |
\(\beta_{5}\) | \(=\) | \( ( 25\nu^{7} + 1398\nu^{5} + 20605\nu^{3} + 128158\nu ) / 31996 \) |
\(\beta_{6}\) | \(=\) | \( ( 25\nu^{7} + 1398\nu^{5} + 20605\nu^{3} + 64166\nu ) / 31996 \) |
\(\beta_{7}\) | \(=\) | \( ( 223\nu^{7} + 13750\nu^{5} + 228591\nu^{3} + 754098\nu ) / 31996 \) |
\(\nu\) | \(=\) | \( ( -\beta_{6} + \beta_{5} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} - \beta_{2} - 16 \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{7} + 47\beta_{6} - 26\beta_{5} - 11\beta_{4} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( -33\beta_{3} + 50\beta_{2} - 22\beta _1 + 421 \) |
\(\nu^{5}\) | \(=\) | \( ( 155\beta_{7} - 1949\beta_{6} + 768\beta_{5} + 385\beta_{4} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 1038\beta_{3} - 2056\beta_{2} + 1045\beta _1 - 12686 \) |
\(\nu^{7}\) | \(=\) | \( ( -6195\beta_{7} + 75377\beta_{6} - 24084\beta_{5} - 12463\beta_{4} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
98.1 |
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−5.00866 | 0 | 17.0866 | − | 7.07107i | 0 | 10.4716i | −45.5118 | 0 | 35.4165i | |||||||||||||||||||||||||||||||||||||||||
98.2 | −5.00866 | 0 | 17.0866 | 7.07107i | 0 | − | 10.4716i | −45.5118 | 0 | − | 35.4165i | |||||||||||||||||||||||||||||||||||||||||
98.3 | −1.38325 | 0 | −6.08663 | − | 7.07107i | 0 | 14.2249i | 19.4853 | 0 | 9.78103i | ||||||||||||||||||||||||||||||||||||||||||
98.4 | −1.38325 | 0 | −6.08663 | 7.07107i | 0 | − | 14.2249i | 19.4853 | 0 | − | 9.78103i | |||||||||||||||||||||||||||||||||||||||||
98.5 | 1.38325 | 0 | −6.08663 | − | 7.07107i | 0 | − | 14.2249i | −19.4853 | 0 | − | 9.78103i | ||||||||||||||||||||||||||||||||||||||||
98.6 | 1.38325 | 0 | −6.08663 | 7.07107i | 0 | 14.2249i | −19.4853 | 0 | 9.78103i | |||||||||||||||||||||||||||||||||||||||||||
98.7 | 5.00866 | 0 | 17.0866 | − | 7.07107i | 0 | − | 10.4716i | 45.5118 | 0 | − | 35.4165i | ||||||||||||||||||||||||||||||||||||||||
98.8 | 5.00866 | 0 | 17.0866 | 7.07107i | 0 | 10.4716i | 45.5118 | 0 | 35.4165i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
33.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.4.d.c | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 99.4.d.c | ✓ | 8 |
4.b | odd | 2 | 1 | 1584.4.b.g | 8 | ||
11.b | odd | 2 | 1 | inner | 99.4.d.c | ✓ | 8 |
12.b | even | 2 | 1 | 1584.4.b.g | 8 | ||
33.d | even | 2 | 1 | inner | 99.4.d.c | ✓ | 8 |
44.c | even | 2 | 1 | 1584.4.b.g | 8 | ||
132.d | odd | 2 | 1 | 1584.4.b.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.4.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
99.4.d.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
99.4.d.c | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
99.4.d.c | ✓ | 8 | 33.d | even | 2 | 1 | inner |
1584.4.b.g | 8 | 4.b | odd | 2 | 1 | ||
1584.4.b.g | 8 | 12.b | even | 2 | 1 | ||
1584.4.b.g | 8 | 44.c | even | 2 | 1 | ||
1584.4.b.g | 8 | 132.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 27T_{2}^{2} + 48 \)
acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 27 T^{2} + 48)^{2} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 50)^{4} \)
$7$
\( (T^{4} + 312 T^{2} + 22188)^{2} \)
$11$
\( T^{8} - 676 T^{6} + \cdots + 3138428376721 \)
$13$
\( (T^{4} + 8088 T^{2} + 11808768)^{2} \)
$17$
\( (T^{4} - 13572 T^{2} + 3345408)^{2} \)
$19$
\( (T^{4} + 9000 T^{2} + 8167500)^{2} \)
$23$
\( (T^{4} + 58600 T^{2} + \cdots + 595360000)^{2} \)
$29$
\( (T^{4} - 28500 T^{2} + \cdots + 201720000)^{2} \)
$31$
\( (T^{2} + 20 T - 19232)^{4} \)
$37$
\( (T^{2} + 140 T - 48800)^{4} \)
$41$
\( (T^{4} - 112500 T^{2} + \cdots + 2453880000)^{2} \)
$43$
\( (T^{4} + 77688 T^{2} + \cdots + 738151788)^{2} \)
$47$
\( (T^{2} + 57800)^{4} \)
$53$
\( (T^{4} + 214900 T^{2} + \cdots + 11524022500)^{2} \)
$59$
\( (T^{4} + 507016 T^{2} + \cdots + 52804363264)^{2} \)
$61$
\( (T^{4} + 530400 T^{2} + \cdots + 58968120000)^{2} \)
$67$
\( (T - 260)^{8} \)
$71$
\( (T^{4} + 452944 T^{2} + \cdots + 136235584)^{2} \)
$73$
\( (T^{4} + 1936008 T^{2} + \cdots + 816399986688)^{2} \)
$79$
\( (T^{4} + 1811400 T^{2} + \cdots + 811044007500)^{2} \)
$83$
\( (T^{4} - 316368 T^{2} + \cdots + 9548295168)^{2} \)
$89$
\( (T^{4} + 1418452 T^{2} + \cdots + 176199076)^{2} \)
$97$
\( (T^{2} - 2200 T + 726700)^{4} \)
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