Newspace parameters
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.m (of order \(15\), degree \(8\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.790518980011\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{15})\) |
Defining polynomial: |
\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) |
\(\chi(n)\) | \(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\) | \(-1 - \zeta_{15}^{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 |
|
0.273659 | − | 2.60369i | −1.28716 | + | 1.15897i | −4.74803 | − | 1.00922i | −0.338261 | − | 3.21834i | 2.66535 | + | 3.66854i | −0.669131 | + | 0.743145i | −2.30902 | + | 7.10642i | 0.313585 | − | 2.98357i | −8.47214 | ||||||||||||||||||||||||||
16.1 | 0.373619 | + | 0.0794152i | 1.72256 | + | 0.181049i | −1.69381 | − | 0.754131i | 1.20906 | − | 0.256993i | 0.629204 | + | 0.204441i | 0.104528 | + | 0.994522i | −1.19098 | − | 0.865300i | 2.93444 | + | 0.623735i | 0.472136 | |||||||||||||||||||||||||||
25.1 | 0.273659 | + | 2.60369i | −1.28716 | − | 1.15897i | −4.74803 | + | 1.00922i | −0.338261 | + | 3.21834i | 2.66535 | − | 3.66854i | −0.669131 | − | 0.743145i | −2.30902 | − | 7.10642i | 0.313585 | + | 2.98357i | −8.47214 | |||||||||||||||||||||||||||
31.1 | 0.373619 | − | 0.0794152i | 1.72256 | − | 0.181049i | −1.69381 | + | 0.754131i | 1.20906 | + | 0.256993i | 0.629204 | − | 0.204441i | 0.104528 | − | 0.994522i | −1.19098 | + | 0.865300i | 2.93444 | − | 0.623735i | 0.472136 | |||||||||||||||||||||||||||
49.1 | −0.255585 | + | 0.283856i | 0.704489 | + | 1.58231i | 0.193806 | + | 1.84395i | −0.827091 | − | 0.918578i | −0.629204 | − | 0.204441i | −0.913545 | − | 0.406737i | −1.19098 | − | 0.865300i | −2.00739 | + | 2.22943i | 0.472136 | |||||||||||||||||||||||||||
58.1 | −2.39169 | − | 1.06485i | 0.360114 | − | 1.69420i | 3.24803 | + | 3.60730i | 2.95630 | − | 1.31623i | −2.66535 | + | 3.66854i | 0.978148 | − | 0.207912i | −2.30902 | − | 7.10642i | −2.74064 | − | 1.22021i | −8.47214 | |||||||||||||||||||||||||||
70.1 | −2.39169 | + | 1.06485i | 0.360114 | + | 1.69420i | 3.24803 | − | 3.60730i | 2.95630 | + | 1.31623i | −2.66535 | − | 3.66854i | 0.978148 | + | 0.207912i | −2.30902 | + | 7.10642i | −2.74064 | + | 1.22021i | −8.47214 | |||||||||||||||||||||||||||
97.1 | −0.255585 | − | 0.283856i | 0.704489 | − | 1.58231i | 0.193806 | − | 1.84395i | −0.827091 | + | 0.918578i | −0.629204 | + | 0.204441i | −0.913545 | + | 0.406737i | −1.19098 | + | 0.865300i | −2.00739 | − | 2.22943i | 0.472136 | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
99.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.2.m.a | ✓ | 8 |
3.b | odd | 2 | 1 | 297.2.n.a | 8 | ||
9.c | even | 3 | 1 | inner | 99.2.m.a | ✓ | 8 |
9.c | even | 3 | 1 | 891.2.f.b | 4 | ||
9.d | odd | 6 | 1 | 297.2.n.a | 8 | ||
9.d | odd | 6 | 1 | 891.2.f.a | 4 | ||
11.c | even | 5 | 1 | inner | 99.2.m.a | ✓ | 8 |
11.c | even | 5 | 1 | 1089.2.e.g | 4 | ||
11.d | odd | 10 | 1 | 1089.2.e.d | 4 | ||
33.h | odd | 10 | 1 | 297.2.n.a | 8 | ||
99.m | even | 15 | 1 | inner | 99.2.m.a | ✓ | 8 |
99.m | even | 15 | 1 | 891.2.f.b | 4 | ||
99.m | even | 15 | 1 | 1089.2.e.g | 4 | ||
99.m | even | 15 | 1 | 9801.2.a.n | 2 | ||
99.n | odd | 30 | 1 | 297.2.n.a | 8 | ||
99.n | odd | 30 | 1 | 891.2.f.a | 4 | ||
99.n | odd | 30 | 1 | 9801.2.a.bb | 2 | ||
99.o | odd | 30 | 1 | 1089.2.e.d | 4 | ||
99.o | odd | 30 | 1 | 9801.2.a.bc | 2 | ||
99.p | even | 30 | 1 | 9801.2.a.m | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.2.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
99.2.m.a | ✓ | 8 | 9.c | even | 3 | 1 | inner |
99.2.m.a | ✓ | 8 | 11.c | even | 5 | 1 | inner |
99.2.m.a | ✓ | 8 | 99.m | even | 15 | 1 | inner |
297.2.n.a | 8 | 3.b | odd | 2 | 1 | ||
297.2.n.a | 8 | 9.d | odd | 6 | 1 | ||
297.2.n.a | 8 | 33.h | odd | 10 | 1 | ||
297.2.n.a | 8 | 99.n | odd | 30 | 1 | ||
891.2.f.a | 4 | 9.d | odd | 6 | 1 | ||
891.2.f.a | 4 | 99.n | odd | 30 | 1 | ||
891.2.f.b | 4 | 9.c | even | 3 | 1 | ||
891.2.f.b | 4 | 99.m | even | 15 | 1 | ||
1089.2.e.d | 4 | 11.d | odd | 10 | 1 | ||
1089.2.e.d | 4 | 99.o | odd | 30 | 1 | ||
1089.2.e.g | 4 | 11.c | even | 5 | 1 | ||
1089.2.e.g | 4 | 99.m | even | 15 | 1 | ||
9801.2.a.m | 2 | 99.p | even | 30 | 1 | ||
9801.2.a.n | 2 | 99.m | even | 15 | 1 | ||
9801.2.a.bb | 2 | 99.n | odd | 30 | 1 | ||
9801.2.a.bc | 2 | 99.o | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4T_{2}^{7} + 10T_{2}^{6} + 26T_{2}^{5} + 39T_{2}^{4} - 14T_{2}^{3} - 5T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 4 T^{7} + 10 T^{6} + 26 T^{5} + \cdots + 1 \)
$3$
\( T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + \cdots + 81 \)
$5$
\( T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256 \)
$7$
\( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \)
$11$
\( T^{8} - T^{7} - 20 T^{6} + T^{5} + \cdots + 14641 \)
$13$
\( T^{8} - 8 T^{7} - 128 T^{5} + \cdots + 65536 \)
$17$
\( (T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81)^{2} \)
$19$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$23$
\( (T^{4} + 7 T^{3} + 38 T^{2} + 77 T + 121)^{2} \)
$29$
\( T^{8} + 9 T^{7} + 45 T^{6} + \cdots + 6561 \)
$31$
\( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \)
$37$
\( (T^{4} - 12 T^{3} + 94 T^{2} - 403 T + 961)^{2} \)
$41$
\( T^{8} - 3 T^{7} - 100 T^{6} + \cdots + 707281 \)
$43$
\( (T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2} \)
$47$
\( T^{8} - 23 T^{7} + 280 T^{6} + \cdots + 25411681 \)
$53$
\( (T^{4} - 14 T^{3} + 96 T^{2} - 319 T + 841)^{2} \)
$59$
\( T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 \)
$61$
\( T^{8} - 90 T^{6} + 1350 T^{5} + \cdots + 4100625 \)
$67$
\( (T^{2} + 6 T + 36)^{4} \)
$71$
\( (T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2} \)
$73$
\( (T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1)^{2} \)
$79$
\( T^{8} + 22 T^{7} + \cdots + 104060401 \)
$83$
\( T^{8} + 17 T^{7} + \cdots + 373301041 \)
$89$
\( (T^{2} + 18 T + 76)^{4} \)
$97$
\( T^{8} - 6 T^{7} + 216 T^{5} + \cdots + 1679616 \)
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