Newspace parameters
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.k (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{10} - 2x^{9} + 15x^{8} + 14x^{7} + 110x^{6} + 36x^{5} + 233x^{4} + 164x^{3} + 345x^{2} + 76x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \( x^{10} - 2x^{9} + 15x^{8} + 14x^{7} + 110x^{6} + 36x^{5} + 233x^{4} + 164x^{3} + 345x^{2} + 76x + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 503 \nu^{9} - 2241 \nu^{8} + 8466 \nu^{7} - 67528 \nu^{6} + 19422 \nu^{5} - 156870 \nu^{4} + 1003571 \nu^{3} - 301041 \nu^{2} - 66732 \nu + 438544 ) / 2044008 \) |
\(\beta_{3}\) | \(=\) | \( ( - 3064 \nu^{9} + 9207 \nu^{8} - 34782 \nu^{7} - 28346 \nu^{6} - 79794 \nu^{5} + 644490 \nu^{4} + 579550 \nu^{3} + 1236807 \nu^{2} + 274164 \nu + 3620228 ) / 1022004 \) |
\(\beta_{4}\) | \(=\) | \( ( - 3977 \nu^{9} - 10833 \nu^{8} - 12699 \nu^{7} - 334006 \nu^{6} - 625302 \nu^{5} - 1723536 \nu^{4} - 478621 \nu^{3} - 3196425 \nu^{2} - 3391749 \nu - 3118196 ) / 1022004 \) |
\(\beta_{5}\) | \(=\) | \( ( - 27409 \nu^{9} + 54315 \nu^{8} - 413376 \nu^{7} - 375260 \nu^{6} - 3082518 \nu^{5} - 967302 \nu^{4} - 6543167 \nu^{3} - 3491505 \nu^{2} - 9757146 \nu - 2149816 ) / 2044008 \) |
\(\beta_{6}\) | \(=\) | \( ( - 14545 \nu^{9} + 29941 \nu^{8} - 216152 \nu^{7} - 200758 \nu^{6} - 1521222 \nu^{5} - 440214 \nu^{4} - 2887781 \nu^{3} - 2349679 \nu^{2} - 3963994 \nu - 1074620 ) / 340668 \) |
\(\beta_{7}\) | \(=\) | \( ( 45928 \nu^{9} - 96426 \nu^{8} + 695481 \nu^{7} + 561428 \nu^{6} + 5037264 \nu^{5} + 744876 \nu^{4} + 10649492 \nu^{3} + 5556402 \nu^{2} + 16896855 \nu + 165664 ) / 1022004 \) |
\(\beta_{8}\) | \(=\) | \( ( 97465 \nu^{9} - 178473 \nu^{8} + 1399728 \nu^{7} + 1662752 \nu^{6} + 10555542 \nu^{5} + 4653846 \nu^{4} + 20509619 \nu^{3} + 17738031 \nu^{2} + \cdots + 5377120 ) / 2044008 \) |
\(\beta_{9}\) | \(=\) | \( ( 51344 \nu^{9} - 117222 \nu^{8} + 805587 \nu^{7} + 484042 \nu^{6} + 5520312 \nu^{5} + 367938 \nu^{4} + 11604142 \nu^{3} + 5109630 \nu^{2} + 14145267 \nu + 742892 ) / 1022004 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{9} + 4\beta_{5} - \beta_{4} + 2\beta_{2} + 2\beta_1 \) |
\(\nu^{3}\) | \(=\) | \( -3\beta_{8} - 3\beta_{6} - 3\beta_{4} - 2\beta_{3} + 11\beta_{2} - 6 \) |
\(\nu^{4}\) | \(=\) | \( -16\beta_{9} - 16\beta_{8} + 6\beta_{7} - 18\beta_{6} - 40\beta_{5} + 2\beta_{4} - 37\beta _1 - 40 \) |
\(\nu^{5}\) | \(=\) | \( - 65 \beta_{9} + 4 \beta_{8} + 34 \beta_{7} - 4 \beta_{6} - 126 \beta_{5} + 65 \beta_{4} + 34 \beta_{3} - 168 \beta_{2} - 168 \beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -30\beta_{9} + 297\beta_{8} + 267\beta_{6} + 267\beta_{4} + 126\beta_{3} - 657\beta_{2} + 592 \) |
\(\nu^{7}\) | \(=\) | \( 1080\beta_{9} + 1080\beta_{8} - 564\beta_{7} + 1176\beta_{6} + 2280\beta_{5} - 96\beta_{4} + 2797\beta _1 + 2280 \) |
\(\nu^{8}\) | \(=\) | \( 5005 \beta_{9} - 468 \beta_{8} - 2256 \beta_{7} + 468 \beta_{6} + 9784 \beta_{5} - 5005 \beta_{4} - 2256 \beta_{3} + 11402 \beta_{2} + 11402 \beta_1 \) |
\(\nu^{9}\) | \(=\) | \( 1788\beta_{9} - 20451\beta_{8} - 18663\beta_{6} - 18663\beta_{4} - 9542\beta_{3} + 47603\beta_{2} - 39726 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
\(n\) | \(157\) | \(365\) | \(379\) |
\(\chi(n)\) | \(\beta_{5}\) | \(1\) | \(-1 - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
373.1 |
|
0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −2.07085 | + | 3.58682i | −0.500000 | − | 0.866025i | 2.11344 | + | 1.59166i | −1.00000 | 1.00000 | −4.14170 | ||||||||||||||||||||||||||||||||||||||
373.2 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.769836 | + | 1.33339i | −0.500000 | − | 0.866025i | −0.131875 | − | 2.64246i | −1.00000 | 1.00000 | −1.53967 | |||||||||||||||||||||||||||||||||||||||
373.3 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 0.114009 | − | 0.197470i | −0.500000 | − | 0.866025i | 0.848534 | − | 2.50599i | −1.00000 | 1.00000 | 0.228019 | |||||||||||||||||||||||||||||||||||||||
373.4 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 0.623307 | − | 1.07960i | −0.500000 | − | 0.866025i | −2.27938 | + | 1.34329i | −1.00000 | 1.00000 | 1.24661 | |||||||||||||||||||||||||||||||||||||||
373.5 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 1.10337 | − | 1.91109i | −0.500000 | − | 0.866025i | 1.44928 | + | 2.21350i | −1.00000 | 1.00000 | 2.20674 | |||||||||||||||||||||||||||||||||||||||
445.1 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −2.07085 | − | 3.58682i | −0.500000 | + | 0.866025i | 2.11344 | − | 1.59166i | −1.00000 | 1.00000 | −4.14170 | |||||||||||||||||||||||||||||||||||||||
445.2 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.769836 | − | 1.33339i | −0.500000 | + | 0.866025i | −0.131875 | + | 2.64246i | −1.00000 | 1.00000 | −1.53967 | |||||||||||||||||||||||||||||||||||||||
445.3 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 0.114009 | + | 0.197470i | −0.500000 | + | 0.866025i | 0.848534 | + | 2.50599i | −1.00000 | 1.00000 | 0.228019 | |||||||||||||||||||||||||||||||||||||||
445.4 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 0.623307 | + | 1.07960i | −0.500000 | + | 0.866025i | −2.27938 | − | 1.34329i | −1.00000 | 1.00000 | 1.24661 | |||||||||||||||||||||||||||||||||||||||
445.5 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 1.10337 | + | 1.91109i | −0.500000 | + | 0.866025i | 1.44928 | − | 2.21350i | −1.00000 | 1.00000 | 2.20674 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.k.e | yes | 10 |
3.b | odd | 2 | 1 | 1638.2.p.j | 10 | ||
7.c | even | 3 | 1 | 546.2.j.e | ✓ | 10 | |
13.c | even | 3 | 1 | 546.2.j.e | ✓ | 10 | |
21.h | odd | 6 | 1 | 1638.2.m.k | 10 | ||
39.i | odd | 6 | 1 | 1638.2.m.k | 10 | ||
91.g | even | 3 | 1 | inner | 546.2.k.e | yes | 10 |
273.bm | odd | 6 | 1 | 1638.2.p.j | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.j.e | ✓ | 10 | 7.c | even | 3 | 1 | |
546.2.j.e | ✓ | 10 | 13.c | even | 3 | 1 | |
546.2.k.e | yes | 10 | 1.a | even | 1 | 1 | trivial |
546.2.k.e | yes | 10 | 91.g | even | 3 | 1 | inner |
1638.2.m.k | 10 | 21.h | odd | 6 | 1 | ||
1638.2.m.k | 10 | 39.i | odd | 6 | 1 | ||
1638.2.p.j | 10 | 3.b | odd | 2 | 1 | ||
1638.2.p.j | 10 | 273.bm | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 2 T_{5}^{9} + 15 T_{5}^{8} - 14 T_{5}^{7} + 110 T_{5}^{6} - 36 T_{5}^{5} + 233 T_{5}^{4} - 164 T_{5}^{3} + 345 T_{5}^{2} - 76 T_{5} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T + 1)^{5} \)
$3$
\( (T + 1)^{10} \)
$5$
\( T^{10} + 2 T^{9} + 15 T^{8} - 14 T^{7} + \cdots + 16 \)
$7$
\( T^{10} - 4 T^{9} + 18 T^{8} + \cdots + 16807 \)
$11$
\( (T^{5} + 6 T^{4} - 12 T^{3} - 65 T^{2} + \cdots + 30)^{2} \)
$13$
\( T^{10} + 4 T^{9} + 24 T^{8} + \cdots + 371293 \)
$17$
\( T^{10} - 4 T^{9} + 36 T^{8} - 62 T^{7} + \cdots + 784 \)
$19$
\( (T^{5} + 3 T^{4} - 69 T^{3} - 109 T^{2} + \cdots - 111)^{2} \)
$23$
\( T^{10} - 6 T^{9} + 108 T^{8} + \cdots + 129600 \)
$29$
\( T^{10} + 60 T^{8} - 54 T^{7} + \cdots + 236196 \)
$31$
\( T^{10} + 10 T^{9} + 126 T^{8} + \cdots + 1254400 \)
$37$
\( T^{10} - T^{9} + 39 T^{8} + \cdots + 319225 \)
$41$
\( T^{10} + 4 T^{9} + 48 T^{8} + \cdots + 770884 \)
$43$
\( T^{10} - 3 T^{9} + 111 T^{8} + \cdots + 308025 \)
$47$
\( T^{10} + 15 T^{9} + \cdots + 271854144 \)
$53$
\( T^{10} + 17 T^{9} + 279 T^{8} + \cdots + 256 \)
$59$
\( T^{10} - 2 T^{9} + 171 T^{8} + \cdots + 1607824 \)
$61$
\( (T^{5} + 11 T^{4} - 8 T^{3} - 400 T^{2} + \cdots + 80)^{2} \)
$67$
\( (T^{5} - T^{4} - 137 T^{3} + 199 T^{2} + \cdots - 20)^{2} \)
$71$
\( T^{10} - 18 T^{9} + 315 T^{8} + \cdots + 202500 \)
$73$
\( T^{10} - 12 T^{9} + \cdots + 37941975369 \)
$79$
\( T^{10} + 4 T^{9} + 87 T^{8} + \cdots + 817216 \)
$83$
\( (T^{5} - 60 T^{3} - 27 T^{2} + 405 T + 486)^{2} \)
$89$
\( T^{10} - 7 T^{9} + \cdots + 205435562500 \)
$97$
\( T^{10} + 6 T^{9} + 279 T^{8} + \cdots + 1734489 \)
show more
show less