Newspace parameters
Level: | \( N \) | \(=\) | \( 286 = 2 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 286.h (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.28372149781\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{5})\) |
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: | \( x^{12} + 15x^{10} + 80x^{8} + 180x^{6} + 160x^{4} + 55x^{2} + 5 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \( x^{12} + 15x^{10} + 80x^{8} + 180x^{6} + 160x^{4} + 55x^{2} + 5 \) :
\(\beta_{1}\) | \(=\) | \( ( 3 \nu^{11} + 5 \nu^{10} + 44 \nu^{9} + 72 \nu^{8} + 224 \nu^{7} + 356 \nu^{6} + 452 \nu^{5} + 680 \nu^{4} + 292 \nu^{3} + 380 \nu^{2} + 45 \nu + 47 ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( 5 \nu^{11} - \nu^{10} + 72 \nu^{9} - 16 \nu^{8} + 356 \nu^{7} - 88 \nu^{6} + 680 \nu^{5} - 188 \nu^{4} + 380 \nu^{3} - 120 \nu^{2} + 47 \nu - 15 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( - 3 \nu^{11} + 5 \nu^{10} - 44 \nu^{9} + 72 \nu^{8} - 224 \nu^{7} + 356 \nu^{6} - 452 \nu^{5} + 680 \nu^{4} - 292 \nu^{3} + 380 \nu^{2} - 45 \nu + 47 ) / 8 \) |
\(\beta_{4}\) | \(=\) | \( ( 5 \nu^{11} + 5 \nu^{10} + 72 \nu^{9} + 72 \nu^{8} + 356 \nu^{7} + 356 \nu^{6} + 680 \nu^{5} + 676 \nu^{4} + 380 \nu^{3} + 360 \nu^{2} + 43 \nu + 35 ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{10} - 43\nu^{8} - 211\nu^{6} - 396\nu^{4} - \nu^{3} - 206\nu^{2} - 5\nu - 19 ) / 2 \) |
\(\beta_{6}\) | \(=\) | \( ( 5 \nu^{11} - 5 \nu^{10} + 72 \nu^{9} - 72 \nu^{8} + 356 \nu^{7} - 356 \nu^{6} + 680 \nu^{5} - 676 \nu^{4} + 380 \nu^{3} - 360 \nu^{2} + 43 \nu - 35 ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( ( -7\nu^{10} - 100\nu^{8} - 490\nu^{6} - 924\nu^{4} - 496\nu^{2} - 2\nu - 45 ) / 4 \) |
\(\beta_{8}\) | \(=\) | \( ( - \nu^{11} + 6 \nu^{10} - 14 \nu^{9} + 86 \nu^{8} - 66 \nu^{7} + 424 \nu^{6} - 112 \nu^{5} + 810 \nu^{4} - 30 \nu^{3} + 454 \nu^{2} + 17 \nu + 50 ) / 4 \) |
\(\beta_{9}\) | \(=\) | \( ( 7 \nu^{11} - 5 \nu^{10} + 100 \nu^{9} - 72 \nu^{8} + 488 \nu^{7} - 356 \nu^{6} + 904 \nu^{5} - 680 \nu^{4} + 444 \nu^{3} - 380 \nu^{2} + 25 \nu - 43 ) / 8 \) |
\(\beta_{10}\) | \(=\) | \( ( 7 \nu^{11} - 5 \nu^{10} + 100 \nu^{9} - 72 \nu^{8} + 488 \nu^{7} - 356 \nu^{6} + 904 \nu^{5} - 680 \nu^{4} + 444 \nu^{3} - 380 \nu^{2} + 33 \nu - 43 ) / 8 \) |
\(\beta_{11}\) | \(=\) | \( ( 11 \nu^{11} + \nu^{10} + 158 \nu^{9} + 14 \nu^{8} + 780 \nu^{7} + 66 \nu^{6} + 1490 \nu^{5} + 114 \nu^{4} + 834 \nu^{3} + 44 \nu^{2} + 95 \nu + 1 ) / 4 \) |
\(\nu\) | \(=\) | \( \beta_{10} - \beta_{9} \) |
\(\nu^{2}\) | \(=\) | \( \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) |
\(\nu^{3}\) | \(=\) | \( -4\beta_{10} + 5\beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - 1 \) |
\(\nu^{4}\) | \(=\) | \( -5\beta_{7} + 6\beta_{6} - \beta_{4} - 4\beta_{3} - 5\beta_{2} - 4\beta _1 + 12 \) |
\(\nu^{5}\) | \(=\) | \( 20\beta_{10} - 28\beta_{9} - 6\beta_{8} - 6\beta_{7} + 2\beta_{6} + 6\beta_{5} + 8\beta_{4} - 2\beta _1 + 7 \) |
\(\nu^{6}\) | \(=\) | \( \beta_{10} + \beta_{9} + 2 \beta_{8} + 24 \beta_{7} - 34 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} + 16 \beta_{3} + 24 \beta_{2} + 16 \beta _1 - 52 \) |
\(\nu^{7}\) | \(=\) | \( 2 \beta_{11} - 104 \beta_{10} + 154 \beta_{9} + 34 \beta_{8} + 33 \beta_{7} - 21 \beta_{6} - 32 \beta_{5} - 54 \beta_{4} - \beta_{3} - \beta_{2} + 17 \beta _1 - 41 \) |
\(\nu^{8}\) | \(=\) | \( - 8 \beta_{10} - 13 \beta_{9} - 21 \beta_{8} - 117 \beta_{7} + 192 \beta_{6} - 21 \beta_{5} - 49 \beta_{4} - 66 \beta_{3} - 122 \beta_{2} - 66 \beta _1 + 235 \) |
\(\nu^{9}\) | \(=\) | \( - 26 \beta_{11} + 548 \beta_{10} - 838 \beta_{9} - 192 \beta_{8} - 179 \beta_{7} + 160 \beta_{6} + 166 \beta_{5} + 339 \beta_{4} + 8 \beta_{3} + 13 \beta_{2} - 106 \beta _1 + 228 \) |
\(\nu^{10}\) | \(=\) | \( 44 \beta_{10} + 116 \beta_{9} + 160 \beta_{8} + 580 \beta_{7} - 1084 \beta_{6} + 160 \beta_{5} + 272 \beta_{4} + 280 \beta_{3} + 652 \beta_{2} + 280 \beta _1 - 1095 \) |
\(\nu^{11}\) | \(=\) | \( 232 \beta_{11} - 2911 \beta_{10} + 4539 \beta_{9} + 1084 \beta_{8} + 968 \beta_{7} - 1080 \beta_{6} - 852 \beta_{5} - 2048 \beta_{4} - 44 \beta_{3} - 116 \beta_{2} + 588 \beta _1 - 1240 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/286\mathbb{Z}\right)^\times\).
\(n\) | \(67\) | \(79\) |
\(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 |
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0.309017 | + | 0.951057i | −1.24555 | − | 0.904942i | −0.809017 | + | 0.587785i | 0.590186 | − | 1.81641i | 0.475756 | − | 1.46423i | −1.16309 | + | 0.845036i | −0.809017 | − | 0.587785i | −0.194587 | − | 0.598877i | 1.90988 | ||||||||||||||||||||||||||||||||||||||
27.2 | 0.309017 | + | 0.951057i | 0.905613 | + | 0.657966i | −0.809017 | + | 0.587785i | −0.576732 | + | 1.77500i | −0.345913 | + | 1.06461i | −3.05121 | + | 2.21683i | −0.809017 | − | 0.587785i | −0.539836 | − | 1.66144i | −1.86634 | |||||||||||||||||||||||||||||||||||||||
27.3 | 0.309017 | + | 0.951057i | 2.45797 | + | 1.78582i | −0.809017 | + | 0.587785i | 1.29556 | − | 3.98733i | −0.938860 | + | 2.88951i | −0.0217689 | + | 0.0158160i | −0.809017 | − | 0.587785i | 1.92541 | + | 5.92579i | 4.19253 | |||||||||||||||||||||||||||||||||||||||
53.1 | 0.309017 | − | 0.951057i | −1.24555 | + | 0.904942i | −0.809017 | − | 0.587785i | 0.590186 | + | 1.81641i | 0.475756 | + | 1.46423i | −1.16309 | − | 0.845036i | −0.809017 | + | 0.587785i | −0.194587 | + | 0.598877i | 1.90988 | |||||||||||||||||||||||||||||||||||||||
53.2 | 0.309017 | − | 0.951057i | 0.905613 | − | 0.657966i | −0.809017 | − | 0.587785i | −0.576732 | − | 1.77500i | −0.345913 | − | 1.06461i | −3.05121 | − | 2.21683i | −0.809017 | + | 0.587785i | −0.539836 | + | 1.66144i | −1.86634 | |||||||||||||||||||||||||||||||||||||||
53.3 | 0.309017 | − | 0.951057i | 2.45797 | − | 1.78582i | −0.809017 | − | 0.587785i | 1.29556 | + | 3.98733i | −0.938860 | − | 2.88951i | −0.0217689 | − | 0.0158160i | −0.809017 | + | 0.587785i | 1.92541 | − | 5.92579i | 4.19253 | |||||||||||||||||||||||||||||||||||||||
157.1 | −0.809017 | + | 0.587785i | −0.558621 | − | 1.71926i | 0.309017 | − | 0.951057i | 0.436733 | + | 0.317305i | 1.46249 | + | 1.06256i | −0.151882 | + | 0.467445i | 0.309017 | + | 0.951057i | −0.216737 | + | 0.157469i | −0.539832 | |||||||||||||||||||||||||||||||||||||||
157.2 | −0.809017 | + | 0.587785i | −0.0778392 | − | 0.239564i | 0.309017 | − | 0.951057i | 1.77049 | + | 1.28633i | 0.203786 | + | 0.148059i | −0.976187 | + | 3.00440i | 0.309017 | + | 0.951057i | 2.37572 | − | 1.72606i | −2.18844 | |||||||||||||||||||||||||||||||||||||||
157.3 | −0.809017 | + | 0.587785i | 0.518426 | + | 1.59555i | 0.309017 | − | 0.951057i | −2.01624 | − | 1.46488i | −1.35726 | − | 0.986104i | 1.36414 | − | 4.19838i | 0.309017 | + | 0.951057i | 0.150035 | − | 0.109007i | 2.49221 | |||||||||||||||||||||||||||||||||||||||
235.1 | −0.809017 | − | 0.587785i | −0.558621 | + | 1.71926i | 0.309017 | + | 0.951057i | 0.436733 | − | 0.317305i | 1.46249 | − | 1.06256i | −0.151882 | − | 0.467445i | 0.309017 | − | 0.951057i | −0.216737 | − | 0.157469i | −0.539832 | |||||||||||||||||||||||||||||||||||||||
235.2 | −0.809017 | − | 0.587785i | −0.0778392 | + | 0.239564i | 0.309017 | + | 0.951057i | 1.77049 | − | 1.28633i | 0.203786 | − | 0.148059i | −0.976187 | − | 3.00440i | 0.309017 | − | 0.951057i | 2.37572 | + | 1.72606i | −2.18844 | |||||||||||||||||||||||||||||||||||||||
235.3 | −0.809017 | − | 0.587785i | 0.518426 | − | 1.59555i | 0.309017 | + | 0.951057i | −2.01624 | + | 1.46488i | −1.35726 | + | 0.986104i | 1.36414 | + | 4.19838i | 0.309017 | − | 0.951057i | 0.150035 | + | 0.109007i | 2.49221 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 286.2.h.c | ✓ | 12 |
11.c | even | 5 | 1 | inner | 286.2.h.c | ✓ | 12 |
11.c | even | 5 | 1 | 3146.2.a.bg | 6 | ||
11.d | odd | 10 | 1 | 3146.2.a.be | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
286.2.h.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
286.2.h.c | ✓ | 12 | 11.c | even | 5 | 1 | inner |
3146.2.a.be | 6 | 11.d | odd | 10 | 1 | ||
3146.2.a.bg | 6 | 11.c | even | 5 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 4 T_{3}^{11} + 9 T_{3}^{10} - 10 T_{3}^{9} + 35 T_{3}^{8} - 14 T_{3}^{7} + 71 T_{3}^{6} - 26 T_{3}^{5} + 145 T_{3}^{4} - 220 T_{3}^{3} + 224 T_{3}^{2} + 24 T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(286, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{3} \)
$3$
\( T^{12} - 4 T^{11} + 9 T^{10} - 10 T^{9} + \cdots + 16 \)
$5$
\( T^{12} - 3 T^{11} + 21 T^{10} + \cdots + 1936 \)
$7$
\( T^{12} + 8 T^{11} + 51 T^{10} + 255 T^{9} + \cdots + 1 \)
$11$
\( T^{12} + 2 T^{11} + 16 T^{10} + \cdots + 1771561 \)
$13$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \)
$17$
\( T^{12} + T^{11} + 9 T^{10} + 55 T^{9} + \cdots + 3481 \)
$19$
\( T^{12} + 3 T^{11} + 81 T^{10} + \cdots + 25411681 \)
$23$
\( (T^{6} - 16 T^{5} + 50 T^{4} + 400 T^{3} + \cdots - 6064)^{2} \)
$29$
\( T^{12} + 31 T^{11} + 519 T^{10} + \cdots + 2193361 \)
$31$
\( T^{12} - 3 T^{11} + 16 T^{10} + \cdots + 19321 \)
$37$
\( T^{12} - 16 T^{11} + 209 T^{10} + \cdots + 18800896 \)
$41$
\( T^{12} + 10 T^{11} + \cdots + 403206400 \)
$43$
\( (T^{6} - 30 T^{5} + 210 T^{4} + \cdots + 34880)^{2} \)
$47$
\( T^{12} - 11 T^{11} + \cdots + 5032341721 \)
$53$
\( T^{12} + 11 T^{11} + 64 T^{10} + \cdots + 477481 \)
$59$
\( T^{12} - 5 T^{11} + \cdots + 9205443025 \)
$61$
\( T^{12} + 16 T^{11} + 209 T^{10} + \cdots + 11881 \)
$67$
\( (T^{6} - 18 T^{5} - 25 T^{4} + 1170 T^{3} + \cdots + 14384)^{2} \)
$71$
\( T^{12} - 15 T^{10} - 95 T^{9} + \cdots + 6477025 \)
$73$
\( T^{12} + 120 T^{10} - 1280 T^{9} + \cdots + 102400 \)
$79$
\( T^{12} + 18 T^{11} + 321 T^{10} + \cdots + 80209936 \)
$83$
\( T^{12} - 15 T^{11} + \cdots + 5507822265625 \)
$89$
\( (T^{6} - 6 T^{5} - 120 T^{4} + 525 T^{3} + \cdots + 176)^{2} \)
$97$
\( T^{12} - 13 T^{11} + \cdots + 3508903696 \)
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