Newspace parameters
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.u (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.726638658394\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 12.0.2346760387617129.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
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 \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800 ) / 224 \) |
\(\beta_{3}\) | \(=\) | \( ( - 9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256 ) / 224 \) |
\(\beta_{4}\) | \(=\) | \( ( - 11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288 ) / 224 \) |
\(\beta_{5}\) | \(=\) | \( ( - 13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544 ) / 224 \) |
\(\beta_{6}\) | \(=\) | \( ( 8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464 ) / 112 \) |
\(\beta_{7}\) | \(=\) | \( ( 13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664 ) / 224 \) |
\(\beta_{8}\) | \(=\) | \( ( 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88 ) / 28 \) |
\(\beta_{9}\) | \(=\) | \( ( 3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48 ) / 28 \) |
\(\beta_{10}\) | \(=\) | \( ( - 15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464 ) / 112 \) |
\(\beta_{11}\) | \(=\) | \( ( - 17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056 ) / 112 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} \) |
\(\nu^{4}\) | \(=\) | \( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta _1 - 1 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{10} + 2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - \beta_1 \) |
\(\nu^{6}\) | \(=\) | \( - 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta _1 - 6 \) |
\(\nu^{7}\) | \(=\) | \( - \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -4\beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 2\beta_{2} + 3\beta _1 - 6 \) |
\(\nu^{9}\) | \(=\) | \( 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta _1 + 4 \) |
\(\nu^{10}\) | \(=\) | \( 5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta _1 + 1 \) |
\(\nu^{11}\) | \(=\) | \( - 2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta _1 - 5 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).
\(n\) | \(15\) | \(66\) |
\(\chi(n)\) | \(-\beta_{4}\) | \(-1 - \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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30.1 |
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−2.24179 | + | 1.29430i | 0.518466 | 2.35043 | − | 4.07106i | 1.39608 | + | 0.806027i | −1.16229 | + | 0.671051i | 2.62954 | + | 0.292422i | 6.99143i | −2.73119 | −4.17296 | ||||||||||||||||||||||||||||||||||||||||||||
30.2 | −1.19430 | + | 0.689527i | 2.88120 | −0.0491037 | + | 0.0850501i | 0.697972 | + | 0.402974i | −3.44101 | + | 1.98667i | −2.25549 | − | 1.38302i | − | 2.89354i | 5.30133 | −1.11145 | ||||||||||||||||||||||||||||||||||||||||||||
30.3 | −0.156598 | + | 0.0904119i | −1.82601 | −0.983651 | + | 1.70373i | 2.32670 | + | 1.34332i | 0.285950 | − | 0.165093i | −0.393717 | + | 2.61629i | − | 0.717383i | 0.334323 | −0.485809 | ||||||||||||||||||||||||||||||||||||||||||||
30.4 | 0.433001 | − | 0.249993i | 0.849601 | −0.875007 | + | 1.51556i | 0.902810 | + | 0.521238i | 0.367878 | − | 0.212395i | 1.52469 | − | 2.16225i | 1.87496i | −2.27818 | 0.521224 | |||||||||||||||||||||||||||||||||||||||||||||
30.5 | 1.16500 | − | 0.672613i | 2.05010 | −0.0951832 | + | 0.164862i | −3.08979 | − | 1.78389i | 2.38837 | − | 1.37893i | −2.09638 | + | 1.61406i | 2.94654i | 1.20292 | −4.79947 | |||||||||||||||||||||||||||||||||||||||||||||
30.6 | 1.99469 | − | 1.15163i | −1.47336 | 1.65252 | − | 2.86225i | −0.733776 | − | 0.423646i | −2.93889 | + | 1.69677i | 2.09135 | + | 1.62057i | − | 3.00585i | −0.829208 | −1.95154 | ||||||||||||||||||||||||||||||||||||||||||||
88.1 | −2.24179 | − | 1.29430i | 0.518466 | 2.35043 | + | 4.07106i | 1.39608 | − | 0.806027i | −1.16229 | − | 0.671051i | 2.62954 | − | 0.292422i | − | 6.99143i | −2.73119 | −4.17296 | ||||||||||||||||||||||||||||||||||||||||||||
88.2 | −1.19430 | − | 0.689527i | 2.88120 | −0.0491037 | − | 0.0850501i | 0.697972 | − | 0.402974i | −3.44101 | − | 1.98667i | −2.25549 | + | 1.38302i | 2.89354i | 5.30133 | −1.11145 | |||||||||||||||||||||||||||||||||||||||||||||
88.3 | −0.156598 | − | 0.0904119i | −1.82601 | −0.983651 | − | 1.70373i | 2.32670 | − | 1.34332i | 0.285950 | + | 0.165093i | −0.393717 | − | 2.61629i | 0.717383i | 0.334323 | −0.485809 | |||||||||||||||||||||||||||||||||||||||||||||
88.4 | 0.433001 | + | 0.249993i | 0.849601 | −0.875007 | − | 1.51556i | 0.902810 | − | 0.521238i | 0.367878 | + | 0.212395i | 1.52469 | + | 2.16225i | − | 1.87496i | −2.27818 | 0.521224 | ||||||||||||||||||||||||||||||||||||||||||||
88.5 | 1.16500 | + | 0.672613i | 2.05010 | −0.0951832 | − | 0.164862i | −3.08979 | + | 1.78389i | 2.38837 | + | 1.37893i | −2.09638 | − | 1.61406i | − | 2.94654i | 1.20292 | −4.79947 | ||||||||||||||||||||||||||||||||||||||||||||
88.6 | 1.99469 | + | 1.15163i | −1.47336 | 1.65252 | + | 2.86225i | −0.733776 | + | 0.423646i | −2.93889 | − | 1.69677i | 2.09135 | − | 1.62057i | 3.00585i | −0.829208 | −1.95154 | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.u | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.2.u.b | yes | 12 |
3.b | odd | 2 | 1 | 819.2.do.e | 12 | ||
7.b | odd | 2 | 1 | 637.2.u.g | 12 | ||
7.c | even | 3 | 1 | 91.2.k.b | ✓ | 12 | |
7.c | even | 3 | 1 | 637.2.q.g | 12 | ||
7.d | odd | 6 | 1 | 637.2.k.i | 12 | ||
7.d | odd | 6 | 1 | 637.2.q.i | 12 | ||
13.e | even | 6 | 1 | 91.2.k.b | ✓ | 12 | |
13.f | odd | 12 | 2 | 1183.2.e.j | 24 | ||
21.h | odd | 6 | 1 | 819.2.bm.f | 12 | ||
39.h | odd | 6 | 1 | 819.2.bm.f | 12 | ||
91.k | even | 6 | 1 | 637.2.q.g | 12 | ||
91.l | odd | 6 | 1 | 637.2.q.i | 12 | ||
91.p | odd | 6 | 1 | 637.2.u.g | 12 | ||
91.t | odd | 6 | 1 | 637.2.k.i | 12 | ||
91.u | even | 6 | 1 | inner | 91.2.u.b | yes | 12 |
91.w | even | 12 | 2 | 8281.2.a.co | 12 | ||
91.x | odd | 12 | 2 | 1183.2.e.j | 24 | ||
91.bd | odd | 12 | 2 | 8281.2.a.cp | 12 | ||
273.x | odd | 6 | 1 | 819.2.do.e | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.k.b | ✓ | 12 | 7.c | even | 3 | 1 | |
91.2.k.b | ✓ | 12 | 13.e | even | 6 | 1 | |
91.2.u.b | yes | 12 | 1.a | even | 1 | 1 | trivial |
91.2.u.b | yes | 12 | 91.u | even | 6 | 1 | inner |
637.2.k.i | 12 | 7.d | odd | 6 | 1 | ||
637.2.k.i | 12 | 91.t | odd | 6 | 1 | ||
637.2.q.g | 12 | 7.c | even | 3 | 1 | ||
637.2.q.g | 12 | 91.k | even | 6 | 1 | ||
637.2.q.i | 12 | 7.d | odd | 6 | 1 | ||
637.2.q.i | 12 | 91.l | odd | 6 | 1 | ||
637.2.u.g | 12 | 7.b | odd | 2 | 1 | ||
637.2.u.g | 12 | 91.p | odd | 6 | 1 | ||
819.2.bm.f | 12 | 21.h | odd | 6 | 1 | ||
819.2.bm.f | 12 | 39.h | odd | 6 | 1 | ||
819.2.do.e | 12 | 3.b | odd | 2 | 1 | ||
819.2.do.e | 12 | 273.x | odd | 6 | 1 | ||
1183.2.e.j | 24 | 13.f | odd | 12 | 2 | ||
1183.2.e.j | 24 | 91.x | odd | 12 | 2 | ||
8281.2.a.co | 12 | 91.w | even | 12 | 2 | ||
8281.2.a.cp | 12 | 91.bd | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 8T_{2}^{10} + 52T_{2}^{8} - 18T_{2}^{7} - 91T_{2}^{6} + 36T_{2}^{5} + 130T_{2}^{4} - 72T_{2}^{3} + 6T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 8 T^{10} + 52 T^{8} - 18 T^{7} + \cdots + 1 \)
$3$
\( (T^{6} - 3 T^{5} - 5 T^{4} + 16 T^{3} + 4 T^{2} + \cdots + 7)^{2} \)
$5$
\( T^{12} - 3 T^{11} - 8 T^{10} + 33 T^{9} + \cdots + 121 \)
$7$
\( T^{12} - 3 T^{11} + 3 T^{9} + \cdots + 117649 \)
$11$
\( T^{12} + 62 T^{10} + 1355 T^{8} + \cdots + 85849 \)
$13$
\( T^{12} + 2 T^{11} - 18 T^{10} + \cdots + 4826809 \)
$17$
\( T^{12} - 17 T^{11} + 193 T^{10} + \cdots + 361 \)
$19$
\( T^{12} + 79 T^{10} + 1984 T^{8} + \cdots + 1 \)
$23$
\( T^{12} - 3 T^{11} + 59 T^{10} + \cdots + 628849 \)
$29$
\( T^{12} + T^{11} + 87 T^{10} + \cdots + 16072081 \)
$31$
\( T^{12} + 18 T^{11} + \cdots + 241274089 \)
$37$
\( T^{12} - 15 T^{11} + 39 T^{10} + \cdots + 123201 \)
$41$
\( T^{12} + 6 T^{11} - 159 T^{10} + \cdots + 389707081 \)
$43$
\( T^{12} - 11 T^{11} + \cdots + 418898089 \)
$47$
\( T^{12} - 15 T^{11} + 92 T^{10} + \cdots + 121 \)
$53$
\( T^{12} + 8 T^{11} + 102 T^{10} + \cdots + 289 \)
$59$
\( T^{12} - 27 T^{11} + \cdots + 35582408689 \)
$61$
\( (T^{6} + 5 T^{5} - 75 T^{4} - 354 T^{3} + \cdots + 1777)^{2} \)
$67$
\( T^{12} + 439 T^{10} + \cdots + 5708255809 \)
$71$
\( T^{12} - 30 T^{11} + \cdots + 639230089 \)
$73$
\( T^{12} + 42 T^{11} + \cdots + 484396081 \)
$79$
\( T^{12} + 35 T^{11} + \cdots + 65086724641 \)
$83$
\( T^{12} + 463 T^{10} + \cdots + 402363481 \)
$89$
\( T^{12} - 48 T^{11} + \cdots + 145033849 \)
$97$
\( T^{12} + 3 T^{11} - 176 T^{10} + \cdots + 1681 \)
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