Newspace parameters
Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 195.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.55708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | 12.0.2593100598870016.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{5} \) |
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 \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{10} - 4\nu^{8} - \nu^{6} + 8\nu^{4} - 12\nu^{2} - 32 ) / 80 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{11} + 4\nu^{9} + \nu^{7} - 8\nu^{5} + 12\nu^{3} + 112\nu ) / 80 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{11} - 4\nu^{9} - \nu^{7} + 8\nu^{5} - 12\nu^{3} + 48\nu ) / 80 \) |
\(\beta_{4}\) | \(=\) | \( ( -3\nu^{11} + 8\nu^{9} - 43\nu^{7} + 44\nu^{5} - 76\nu^{3} + 144\nu ) / 160 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{11} + \nu^{7} + 4\nu^{5} + 12\nu^{3} ) / 32 \) |
\(\beta_{6}\) | \(=\) | \( ( - 2 \nu^{11} - 3 \nu^{10} + 2 \nu^{9} + 8 \nu^{8} - 2 \nu^{7} - 3 \nu^{6} + 26 \nu^{5} + 44 \nu^{4} - 64 \nu^{3} - 36 \nu^{2} + 56 \nu + 64 ) / 80 \) |
\(\beta_{7}\) | \(=\) | \( ( -7\nu^{11} + 32\nu^{9} - 47\nu^{7} + 116\nu^{5} - 284\nu^{3} + 496\nu ) / 160 \) |
\(\beta_{8}\) | \(=\) | \( ( - 2 \nu^{11} + 5 \nu^{10} + 2 \nu^{9} - 2 \nu^{7} + 5 \nu^{6} + 26 \nu^{5} + 20 \nu^{4} - 64 \nu^{3} - 20 \nu^{2} + 56 \nu + 80 ) / 80 \) |
\(\beta_{9}\) | \(=\) | \( ( 2 \nu^{11} + 5 \nu^{10} - 2 \nu^{9} - 20 \nu^{8} + 2 \nu^{7} + 45 \nu^{6} - 26 \nu^{5} - 80 \nu^{4} + 64 \nu^{3} + 100 \nu^{2} - 56 \nu - 240 ) / 80 \) |
\(\beta_{10}\) | \(=\) | \( ( -3\nu^{11} + 8\nu^{9} - 13\nu^{7} + 24\nu^{5} - 46\nu^{3} + 84\nu ) / 40 \) |
\(\beta_{11}\) | \(=\) | \( ( \nu^{11} + 6 \nu^{10} - \nu^{9} - 16 \nu^{8} + \nu^{7} + 26 \nu^{6} - 13 \nu^{5} - 48 \nu^{4} + 32 \nu^{3} + 132 \nu^{2} - 28 \nu - 168 ) / 40 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} - \beta _1 + 1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 2\beta_{10} - 2\beta_{7} + 2\beta_{5} + \beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{11} - \beta_{9} + \beta_{8} - 2\beta_{7} + 3\beta_{6} + 2\beta_{4} - 2\beta_{3} + 2\beta_{2} + 3\beta _1 - 1 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 2\beta_{10} + 6\beta_{5} - 2\beta_{4} + 3\beta_{3} - 3\beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( -\beta_{11} + 5\beta_{9} + \beta_{8} - 2\beta_{7} + 7\beta_{6} + 2\beta_{4} - 2\beta_{3} + 2\beta_{2} - 3\beta _1 + 3 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 2\beta_{10} + 2\beta_{7} + 2\beta_{5} - 12\beta_{4} + 3\beta_{3} - \beta_{2} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( -\beta_{11} + \beta_{9} - \beta_{8} - 2\beta_{7} + 5\beta_{6} + 2\beta_{4} - 2\beta_{3} + 2\beta_{2} - 27\beta _1 - 15 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( 6\beta_{10} + 2\beta_{5} - 6\beta_{4} - 19\beta_{3} + 3\beta_{2} ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( \beta_{11} - 5 \beta_{9} + 23 \beta_{8} - 6 \beta_{7} - 15 \beta_{6} + 6 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} - 13 \beta _1 - 27 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( -34\beta_{10} + 22\beta_{7} + 14\beta_{5} + 20\beta_{4} - 27\beta_{3} + \beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).
\(n\) | \(106\) | \(131\) | \(157\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 |
|
−2.51912 | − | 1.00000i | 4.34596 | 1.45804 | − | 1.69532i | 2.51912i | 1.26849 | −5.90976 | −1.00000 | −3.67298 | + | 4.27072i | |||||||||||||||||||||||||||||||||||||||||||||||||
64.2 | −2.51912 | 1.00000i | 4.34596 | 1.45804 | + | 1.69532i | − | 2.51912i | 1.26849 | −5.90976 | −1.00000 | −3.67298 | − | 4.27072i | ||||||||||||||||||||||||||||||||||||||||||||||||||
64.3 | −1.61036 | − | 1.00000i | 0.593272 | 1.11567 | + | 1.93785i | 1.61036i | −4.86509 | 2.26534 | −1.00000 | −1.79664 | − | 3.12065i | ||||||||||||||||||||||||||||||||||||||||||||||||||
64.4 | −1.61036 | 1.00000i | 0.593272 | 1.11567 | − | 1.93785i | − | 1.61036i | −4.86509 | 2.26534 | −1.00000 | −1.79664 | + | 3.12065i | ||||||||||||||||||||||||||||||||||||||||||||||||||
64.5 | −0.246506 | − | 1.00000i | −1.93923 | 2.15160 | + | 0.608775i | 0.246506i | 2.59264 | 0.971044 | −1.00000 | −0.530383 | − | 0.150067i | ||||||||||||||||||||||||||||||||||||||||||||||||||
64.6 | −0.246506 | 1.00000i | −1.93923 | 2.15160 | − | 0.608775i | − | 0.246506i | 2.59264 | 0.971044 | −1.00000 | −0.530383 | + | 0.150067i | ||||||||||||||||||||||||||||||||||||||||||||||||||
64.7 | 0.246506 | − | 1.00000i | −1.93923 | −2.15160 | − | 0.608775i | − | 0.246506i | −2.59264 | −0.971044 | −1.00000 | −0.530383 | − | 0.150067i | |||||||||||||||||||||||||||||||||||||||||||||||||
64.8 | 0.246506 | 1.00000i | −1.93923 | −2.15160 | + | 0.608775i | 0.246506i | −2.59264 | −0.971044 | −1.00000 | −0.530383 | + | 0.150067i | |||||||||||||||||||||||||||||||||||||||||||||||||||
64.9 | 1.61036 | − | 1.00000i | 0.593272 | −1.11567 | − | 1.93785i | − | 1.61036i | 4.86509 | −2.26534 | −1.00000 | −1.79664 | − | 3.12065i | |||||||||||||||||||||||||||||||||||||||||||||||||
64.10 | 1.61036 | 1.00000i | 0.593272 | −1.11567 | + | 1.93785i | 1.61036i | 4.86509 | −2.26534 | −1.00000 | −1.79664 | + | 3.12065i | |||||||||||||||||||||||||||||||||||||||||||||||||||
64.11 | 2.51912 | − | 1.00000i | 4.34596 | −1.45804 | + | 1.69532i | − | 2.51912i | −1.26849 | 5.90976 | −1.00000 | −3.67298 | + | 4.27072i | |||||||||||||||||||||||||||||||||||||||||||||||||
64.12 | 2.51912 | 1.00000i | 4.34596 | −1.45804 | − | 1.69532i | 2.51912i | −1.26849 | 5.90976 | −1.00000 | −3.67298 | − | 4.27072i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
65.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 195.2.h.c | ✓ | 12 |
3.b | odd | 2 | 1 | 585.2.h.g | 12 | ||
4.b | odd | 2 | 1 | 3120.2.r.n | 12 | ||
5.b | even | 2 | 1 | inner | 195.2.h.c | ✓ | 12 |
5.c | odd | 4 | 1 | 975.2.b.j | 6 | ||
5.c | odd | 4 | 1 | 975.2.b.l | 6 | ||
13.b | even | 2 | 1 | inner | 195.2.h.c | ✓ | 12 |
15.d | odd | 2 | 1 | 585.2.h.g | 12 | ||
20.d | odd | 2 | 1 | 3120.2.r.n | 12 | ||
39.d | odd | 2 | 1 | 585.2.h.g | 12 | ||
52.b | odd | 2 | 1 | 3120.2.r.n | 12 | ||
65.d | even | 2 | 1 | inner | 195.2.h.c | ✓ | 12 |
65.h | odd | 4 | 1 | 975.2.b.j | 6 | ||
65.h | odd | 4 | 1 | 975.2.b.l | 6 | ||
195.e | odd | 2 | 1 | 585.2.h.g | 12 | ||
260.g | odd | 2 | 1 | 3120.2.r.n | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.h.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
195.2.h.c | ✓ | 12 | 5.b | even | 2 | 1 | inner |
195.2.h.c | ✓ | 12 | 13.b | even | 2 | 1 | inner |
195.2.h.c | ✓ | 12 | 65.d | even | 2 | 1 | inner |
585.2.h.g | 12 | 3.b | odd | 2 | 1 | ||
585.2.h.g | 12 | 15.d | odd | 2 | 1 | ||
585.2.h.g | 12 | 39.d | odd | 2 | 1 | ||
585.2.h.g | 12 | 195.e | odd | 2 | 1 | ||
975.2.b.j | 6 | 5.c | odd | 4 | 1 | ||
975.2.b.j | 6 | 65.h | odd | 4 | 1 | ||
975.2.b.l | 6 | 5.c | odd | 4 | 1 | ||
975.2.b.l | 6 | 65.h | odd | 4 | 1 | ||
3120.2.r.n | 12 | 4.b | odd | 2 | 1 | ||
3120.2.r.n | 12 | 20.d | odd | 2 | 1 | ||
3120.2.r.n | 12 | 52.b | odd | 2 | 1 | ||
3120.2.r.n | 12 | 260.g | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 9T_{2}^{4} + 17T_{2}^{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} - 9 T^{4} + 17 T^{2} - 1)^{2} \)
$3$
\( (T^{2} + 1)^{6} \)
$5$
\( T^{12} - 2 T^{10} + 27 T^{8} + \cdots + 15625 \)
$7$
\( (T^{6} - 32 T^{4} + 208 T^{2} - 256)^{2} \)
$11$
\( (T^{6} + 20 T^{4} + 84 T^{2} + 64)^{2} \)
$13$
\( T^{12} + 26 T^{10} + 343 T^{8} + \cdots + 4826809 \)
$17$
\( (T^{6} + 88 T^{4} + 2192 T^{2} + \cdots + 16384)^{2} \)
$19$
\( (T^{6} + 64 T^{4} + 1232 T^{2} + \cdots + 6400)^{2} \)
$23$
\( (T^{2} + 16)^{6} \)
$29$
\( (T^{3} + 6 T^{2} - 28 T - 104)^{4} \)
$31$
\( (T^{6} + 160 T^{4} + 6224 T^{2} + \cdots + 43264)^{2} \)
$37$
\( (T^{6} - 132 T^{4} + 2304 T^{2} + \cdots - 256)^{2} \)
$41$
\( (T^{6} + 76 T^{4} + 1460 T^{2} + \cdots + 1024)^{2} \)
$43$
\( (T^{6} + 72 T^{4} + 656 T^{2} + 256)^{2} \)
$47$
\( (T^{6} - 20 T^{4} + 84 T^{2} - 64)^{2} \)
$53$
\( (T^{6} + 104 T^{4} + 3216 T^{2} + \cdots + 25600)^{2} \)
$59$
\( (T^{6} + 164 T^{4} + 8500 T^{2} + \cdots + 141376)^{2} \)
$61$
\( (T^{3} - 2 T^{2} - 96 T - 256)^{4} \)
$67$
\( (T^{6} - 240 T^{4} + 10064 T^{2} + \cdots - 1024)^{2} \)
$71$
\( (T^{6} + 468 T^{4} + 70484 T^{2} + \cdots + 3356224)^{2} \)
$73$
\( (T^{6} - 52 T^{4} + 512 T^{2} - 1024)^{2} \)
$79$
\( (T^{3} + 16 T^{2} + 52 T + 32)^{4} \)
$83$
\( (T^{6} - 228 T^{4} + 9428 T^{2} + \cdots - 12544)^{2} \)
$89$
\( (T^{6} + 140 T^{4} + 5044 T^{2} + \cdots + 16384)^{2} \)
$97$
\( (T^{6} - 340 T^{4} + 31232 T^{2} + \cdots - 640000)^{2} \)
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