Newspace parameters
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.p (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.50917259820\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( 81\nu^{11} - 531\nu^{9} + 3481\nu^{7} - 3627\nu^{5} + 1782\nu^{3} + 76298\nu ) / 21995 \) |
\(\beta_{2}\) | \(=\) | \( ( 117\nu^{10} - 767\nu^{8} + 7472\nu^{6} - 27234\nu^{4} + 90554\nu^{2} - 60864 ) / 21995 \) |
\(\beta_{3}\) | \(=\) | \( ( -1298\nu^{10} + 10953\nu^{8} - 71803\nu^{6} + 202311\nu^{4} - 490451\nu^{2} + 240966 ) / 197955 \) |
\(\beta_{4}\) | \(=\) | \( ( 461\nu^{10} - 5466\nu^{8} + 28501\nu^{6} - 98847\nu^{4} + 142112\nu^{2} - 186237 ) / 65985 \) |
\(\beta_{5}\) | \(=\) | \( ( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} - 438921\nu ) / 197955 \) |
\(\beta_{6}\) | \(=\) | \( ( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} + 154944\nu ) / 197955 \) |
\(\beta_{7}\) | \(=\) | \( ( -288\nu^{10} + 1888\nu^{8} - 9933\nu^{6} + 12896\nu^{4} - 6336\nu^{2} - 68439 ) / 21995 \) |
\(\beta_{8}\) | \(=\) | \( ( -288\nu^{11} + 1888\nu^{9} - 9933\nu^{7} + 12896\nu^{5} - 6336\nu^{3} - 90434\nu ) / 21995 \) |
\(\beta_{9}\) | \(=\) | \( ( 1138\nu^{10} - 12348\nu^{8} + 80948\nu^{6} - 273351\nu^{4} + 552916\nu^{2} - 271656 ) / 65985 \) |
\(\beta_{10}\) | \(=\) | \( ( 4712\nu^{11} - 47997\nu^{9} + 314647\nu^{7} - 1022364\nu^{5} + 2149199\nu^{3} - 1055934\nu ) / 197955 \) |
\(\beta_{11}\) | \(=\) | \( ( -5192\nu^{11} + 43812\nu^{9} - 287212\nu^{7} + 809244\nu^{5} - 1763849\nu^{3} + 172044\nu ) / 197955 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} - \beta_{5} ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -2\beta_{9} + 2\beta_{7} + \beta_{4} - 9\beta_{3} - \beta_{2} + 9 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{11} + 8\beta_{6} + 4\beta_{5} ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( -\beta_{9} + 2\beta_{7} - 2\beta_{4} - 12\beta_{3} - 4\beta_{2} \) |
\(\nu^{5}\) | \(=\) | \( ( 18\beta_{11} + 3\beta_{10} + 17\beta_{6} + 34\beta_{5} + 18\beta_1 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( 32\beta_{9} - 5\beta_{7} - 64\beta_{4} - 32\beta_{2} - 153 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( 27\beta_{8} - 74\beta_{6} + 74\beta_{5} + 96\beta_1 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( 51\beta_{9} - 51\beta_{7} - 55\beta_{4} + 222\beta_{3} + 55\beta_{2} - 222 \) |
\(\nu^{9}\) | \(=\) | \( ( -495\beta_{11} - 177\beta_{10} + 177\beta_{8} - 656\beta_{6} - 328\beta_{5} ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( -191\beta_{9} - 835\beta_{7} + 835\beta_{4} + 2952\beta_{3} + 1670\beta_{2} ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( -2505\beta_{11} - 1026\beta_{10} - 1477\beta_{6} - 2954\beta_{5} - 2505\beta_1 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(136\) |
\(\chi(n)\) | \(-1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 |
|
−2.15715 | − | 1.24543i | 0 | 2.10220 | + | 3.64112i | −0.617942 | + | 1.07031i | 0 | −1.53189 | − | 2.15715i | − | 5.49086i | 0 | 2.66599 | − | 1.53921i | |||||||||||||||||||||||||||||||||||||||||||
26.2 | −1.58850 | − | 0.917122i | 0 | 0.682224 | + | 1.18165i | 1.90412 | − | 3.29804i | 0 | 2.11581 | + | 1.58850i | 1.16576i | 0 | −6.04940 | + | 3.49262i | |||||||||||||||||||||||||||||||||||||||||||||
26.3 | −0.568650 | − | 0.328310i | 0 | −0.784425 | − | 1.35866i | −1.65604 | + | 2.86834i | 0 | −2.58392 | + | 0.568650i | 2.34338i | 0 | 1.88341 | − | 1.08739i | |||||||||||||||||||||||||||||||||||||||||||||
26.4 | 0.568650 | + | 0.328310i | 0 | −0.784425 | − | 1.35866i | 1.65604 | − | 2.86834i | 0 | −2.58392 | + | 0.568650i | − | 2.34338i | 0 | 1.88341 | − | 1.08739i | ||||||||||||||||||||||||||||||||||||||||||||
26.5 | 1.58850 | + | 0.917122i | 0 | 0.682224 | + | 1.18165i | −1.90412 | + | 3.29804i | 0 | 2.11581 | + | 1.58850i | − | 1.16576i | 0 | −6.04940 | + | 3.49262i | ||||||||||||||||||||||||||||||||||||||||||||
26.6 | 2.15715 | + | 1.24543i | 0 | 2.10220 | + | 3.64112i | 0.617942 | − | 1.07031i | 0 | −1.53189 | − | 2.15715i | 5.49086i | 0 | 2.66599 | − | 1.53921i | |||||||||||||||||||||||||||||||||||||||||||||
80.1 | −2.15715 | + | 1.24543i | 0 | 2.10220 | − | 3.64112i | −0.617942 | − | 1.07031i | 0 | −1.53189 | + | 2.15715i | 5.49086i | 0 | 2.66599 | + | 1.53921i | |||||||||||||||||||||||||||||||||||||||||||||
80.2 | −1.58850 | + | 0.917122i | 0 | 0.682224 | − | 1.18165i | 1.90412 | + | 3.29804i | 0 | 2.11581 | − | 1.58850i | − | 1.16576i | 0 | −6.04940 | − | 3.49262i | ||||||||||||||||||||||||||||||||||||||||||||
80.3 | −0.568650 | + | 0.328310i | 0 | −0.784425 | + | 1.35866i | −1.65604 | − | 2.86834i | 0 | −2.58392 | − | 0.568650i | − | 2.34338i | 0 | 1.88341 | + | 1.08739i | ||||||||||||||||||||||||||||||||||||||||||||
80.4 | 0.568650 | − | 0.328310i | 0 | −0.784425 | + | 1.35866i | 1.65604 | + | 2.86834i | 0 | −2.58392 | − | 0.568650i | 2.34338i | 0 | 1.88341 | + | 1.08739i | |||||||||||||||||||||||||||||||||||||||||||||
80.5 | 1.58850 | − | 0.917122i | 0 | 0.682224 | − | 1.18165i | −1.90412 | − | 3.29804i | 0 | 2.11581 | − | 1.58850i | 1.16576i | 0 | −6.04940 | − | 3.49262i | |||||||||||||||||||||||||||||||||||||||||||||
80.6 | 2.15715 | − | 1.24543i | 0 | 2.10220 | − | 3.64112i | 0.617942 | + | 1.07031i | 0 | −1.53189 | + | 2.15715i | − | 5.49086i | 0 | 2.66599 | + | 1.53921i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 189.2.p.d | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 189.2.p.d | ✓ | 12 |
7.c | even | 3 | 1 | 1323.2.c.d | 12 | ||
7.d | odd | 6 | 1 | inner | 189.2.p.d | ✓ | 12 |
7.d | odd | 6 | 1 | 1323.2.c.d | 12 | ||
9.c | even | 3 | 1 | 567.2.i.f | 12 | ||
9.c | even | 3 | 1 | 567.2.s.f | 12 | ||
9.d | odd | 6 | 1 | 567.2.i.f | 12 | ||
9.d | odd | 6 | 1 | 567.2.s.f | 12 | ||
21.g | even | 6 | 1 | inner | 189.2.p.d | ✓ | 12 |
21.g | even | 6 | 1 | 1323.2.c.d | 12 | ||
21.h | odd | 6 | 1 | 1323.2.c.d | 12 | ||
63.i | even | 6 | 1 | 567.2.s.f | 12 | ||
63.k | odd | 6 | 1 | 567.2.i.f | 12 | ||
63.s | even | 6 | 1 | 567.2.i.f | 12 | ||
63.t | odd | 6 | 1 | 567.2.s.f | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.p.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
189.2.p.d | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
189.2.p.d | ✓ | 12 | 7.d | odd | 6 | 1 | inner |
189.2.p.d | ✓ | 12 | 21.g | even | 6 | 1 | inner |
567.2.i.f | 12 | 9.c | even | 3 | 1 | ||
567.2.i.f | 12 | 9.d | odd | 6 | 1 | ||
567.2.i.f | 12 | 63.k | odd | 6 | 1 | ||
567.2.i.f | 12 | 63.s | even | 6 | 1 | ||
567.2.s.f | 12 | 9.c | even | 3 | 1 | ||
567.2.s.f | 12 | 9.d | odd | 6 | 1 | ||
567.2.s.f | 12 | 63.i | even | 6 | 1 | ||
567.2.s.f | 12 | 63.t | odd | 6 | 1 | ||
1323.2.c.d | 12 | 7.c | even | 3 | 1 | ||
1323.2.c.d | 12 | 7.d | odd | 6 | 1 | ||
1323.2.c.d | 12 | 21.g | even | 6 | 1 | ||
1323.2.c.d | 12 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 10T_{2}^{10} + 75T_{2}^{8} - 232T_{2}^{6} + 535T_{2}^{4} - 225T_{2}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 10 T^{10} + 75 T^{8} - 232 T^{6} + \cdots + 81 \)
$3$
\( T^{12} \)
$5$
\( T^{12} + 27 T^{10} + 531 T^{8} + \cdots + 59049 \)
$7$
\( (T^{6} + 4 T^{5} + 2 T^{4} - 11 T^{3} + \cdots + 343)^{2} \)
$11$
\( T^{12} - 37 T^{10} + 1155 T^{8} + \cdots + 50625 \)
$13$
\( (T^{6} + 42 T^{4} + 369 T^{2} + 675)^{2} \)
$17$
\( T^{12} + 30 T^{10} + 675 T^{8} + \cdots + 59049 \)
$19$
\( (T^{6} + 3 T^{5} - 21 T^{4} - 72 T^{3} + \cdots + 243)^{2} \)
$23$
\( T^{12} - 94 T^{10} + 8319 T^{8} + \cdots + 531441 \)
$29$
\( (T^{6} + 37 T^{4} + 322 T^{2} + 81)^{2} \)
$31$
\( (T^{6} + 6 T^{5} - 51 T^{4} - 378 T^{3} + \cdots + 64827)^{2} \)
$37$
\( (T^{6} - 4 T^{5} + 35 T^{4} - 58 T^{3} + \cdots + 4489)^{2} \)
$41$
\( (T^{6} - 114 T^{4} + 1953 T^{2} + \cdots - 243)^{2} \)
$43$
\( (T^{3} - 5 T^{2} - 16 T - 1)^{4} \)
$47$
\( T^{12} + 135 T^{10} + \cdots + 387420489 \)
$53$
\( T^{12} - 118 T^{10} + 12207 T^{8} + \cdots + 81 \)
$59$
\( T^{12} + 63 T^{10} + 3231 T^{8} + \cdots + 4782969 \)
$61$
\( (T^{6} - 9 T^{5} - 24 T^{4} + 459 T^{3} + \cdots + 49923)^{2} \)
$67$
\( (T^{6} - 18 T^{5} + 309 T^{4} + \cdots + 458329)^{2} \)
$71$
\( (T^{6} + 85 T^{4} + 2290 T^{2} + \cdots + 19881)^{2} \)
$73$
\( (T^{6} + 21 T^{5} + 123 T^{4} + \cdots + 177147)^{2} \)
$79$
\( (T^{6} + 18 T^{5} + 309 T^{4} + 284 T^{3} + \cdots + 49)^{2} \)
$83$
\( (T^{6} - 159 T^{4} + 5814 T^{2} + \cdots - 6075)^{2} \)
$89$
\( T^{12} + 201 T^{10} + \cdots + 7695324729 \)
$97$
\( (T^{6} + 204 T^{4} + 3744 T^{2} + \cdots + 1728)^{2} \)
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