Newspace parameters
Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 84.m (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(34.2198032451\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
Defining polynomial: |
\( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + \cdots + 45\!\cdots\!96 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{20}\cdot 3^{10}\cdot 7^{4} \) |
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
\( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + \cdots + 45\!\cdots\!96 \)
:
\(\beta_{1}\) | \(=\) |
\( ( - 15\!\cdots\!43 \nu^{11} + \cdots - 14\!\cdots\!20 ) / 14\!\cdots\!72 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 34\!\cdots\!83 \nu^{11} + \cdots + 41\!\cdots\!04 ) / 20\!\cdots\!24 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 42\!\cdots\!13 \nu^{11} + \cdots + 17\!\cdots\!08 ) / 20\!\cdots\!24 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 49\!\cdots\!23 \nu^{11} + \cdots - 86\!\cdots\!88 ) / 20\!\cdots\!24 \)
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\(\beta_{5}\) | \(=\) |
\( ( 92\!\cdots\!02 \nu^{11} + \cdots + 10\!\cdots\!68 ) / 20\!\cdots\!24 \)
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\(\beta_{6}\) | \(=\) |
\( ( 15\!\cdots\!68 \nu^{11} + \cdots + 78\!\cdots\!92 ) / 28\!\cdots\!32 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 18\!\cdots\!39 \nu^{11} + \cdots - 44\!\cdots\!40 ) / 20\!\cdots\!24 \)
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\(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!45 \nu^{11} + \cdots - 20\!\cdots\!48 ) / 50\!\cdots\!56 \)
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\(\beta_{9}\) | \(=\) |
\( ( 14\!\cdots\!29 \nu^{11} + \cdots - 79\!\cdots\!00 ) / 50\!\cdots\!56 \)
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\(\beta_{10}\) | \(=\) |
\( ( 17\!\cdots\!71 \nu^{11} + \cdots + 25\!\cdots\!72 ) / 20\!\cdots\!24 \)
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\(\beta_{11}\) | \(=\) |
\( ( 25\!\cdots\!43 \nu^{11} + \cdots - 99\!\cdots\!36 ) / 20\!\cdots\!24 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{9} + 2\beta_{8} + 11\beta_{5} - 2\beta_{4} - 13\beta_{3} + 5\beta_{2} + 89\beta _1 - 4 ) / 168 \)
|
\(\nu^{2}\) | \(=\) |
\( ( - 49 \beta_{11} - 98 \beta_{10} + 379 \beta_{9} + \beta_{8} - 357 \beta_{6} + 460 \beta_{5} - 3002 \beta_{4} - 3851 \beta_{3} + 6992 \beta_{2} + 8294560 \beta _1 - 8295551 ) / 168 \)
|
\(\nu^{3}\) | \(=\) |
\( ( 35084 \beta_{11} - 35084 \beta_{10} + 789666 \beta_{9} - 376181 \beta_{8} - 275352 \beta_{7} - 5422393 \beta_{5} - 4661151 \beta_{4} + 5080849 \beta_{3} + 6316864 \beta_{2} + \cdots - 5863038373 ) / 504 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 53732714 \beta_{11} + 26866357 \beta_{10} + 42409788 \beta_{9} - 238180922 \beta_{8} - 173747049 \beta_{7} + 173747049 \beta_{6} - 2066402749 \beta_{5} + \cdots + 602856833 ) / 504 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 8924207908 \beta_{11} + 17848415816 \beta_{10} - 91153716975 \beta_{9} - 55734876919 \beta_{8} + 65807686308 \beta_{6} + 94177321174 \beta_{5} + \cdots + 13\!\cdots\!03 ) / 504 \)
|
\(\nu^{6}\) | \(=\) |
\( ( - 4764138914563 \beta_{11} + 4764138914563 \beta_{10} - 56310552509697 \beta_{9} + 33032234588515 \beta_{8} + 31905883209375 \beta_{7} + \cdots + 66\!\cdots\!34 ) / 504 \)
|
\(\nu^{7}\) | \(=\) |
\( ( - 12\!\cdots\!96 \beta_{11} - 637399972567148 \beta_{10} + \cdots - 18\!\cdots\!84 ) / 168 \)
|
\(\nu^{8}\) | \(=\) |
\( ( - 29\!\cdots\!67 \beta_{11} + \cdots - 42\!\cdots\!61 ) / 168 \)
|
\(\nu^{9}\) | \(=\) |
\( ( 38\!\cdots\!56 \beta_{11} + \cdots - 56\!\cdots\!67 ) / 504 \)
|
\(\nu^{10}\) | \(=\) |
\( ( 11\!\cdots\!10 \beta_{11} + \cdots + 16\!\cdots\!09 ) / 168 \)
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\(\nu^{11}\) | \(=\) |
\( ( 76\!\cdots\!72 \beta_{11} + \cdots + 11\!\cdots\!37 ) / 504 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(43\) | \(73\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
61.1 |
|
0 | 40.5000 | − | 23.3827i | 0 | −939.615 | − | 542.487i | 0 | 1451.57 | + | 1912.52i | 0 | 1093.50 | − | 1894.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
61.2 | 0 | 40.5000 | − | 23.3827i | 0 | −225.043 | − | 129.928i | 0 | 597.275 | − | 2325.52i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
61.3 | 0 | 40.5000 | − | 23.3827i | 0 | −203.366 | − | 117.413i | 0 | 1130.34 | − | 2118.29i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
61.4 | 0 | 40.5000 | − | 23.3827i | 0 | −133.903 | − | 77.3091i | 0 | −2234.88 | + | 877.558i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
61.5 | 0 | 40.5000 | − | 23.3827i | 0 | 632.551 | + | 365.204i | 0 | 984.437 | + | 2189.91i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
61.6 | 0 | 40.5000 | − | 23.3827i | 0 | 1011.88 | + | 584.207i | 0 | −1829.74 | − | 1554.62i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
73.1 | 0 | 40.5000 | + | 23.3827i | 0 | −939.615 | + | 542.487i | 0 | 1451.57 | − | 1912.52i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
73.2 | 0 | 40.5000 | + | 23.3827i | 0 | −225.043 | + | 129.928i | 0 | 597.275 | + | 2325.52i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
73.3 | 0 | 40.5000 | + | 23.3827i | 0 | −203.366 | + | 117.413i | 0 | 1130.34 | + | 2118.29i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
73.4 | 0 | 40.5000 | + | 23.3827i | 0 | −133.903 | + | 77.3091i | 0 | −2234.88 | − | 877.558i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
73.5 | 0 | 40.5000 | + | 23.3827i | 0 | 632.551 | − | 365.204i | 0 | 984.437 | − | 2189.91i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
73.6 | 0 | 40.5000 | + | 23.3827i | 0 | 1011.88 | − | 584.207i | 0 | −1829.74 | + | 1554.62i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 84.9.m.b | ✓ | 12 |
3.b | odd | 2 | 1 | 252.9.z.d | 12 | ||
7.d | odd | 6 | 1 | inner | 84.9.m.b | ✓ | 12 |
21.g | even | 6 | 1 | 252.9.z.d | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
84.9.m.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
84.9.m.b | ✓ | 12 | 7.d | odd | 6 | 1 | inner |
252.9.z.d | 12 | 3.b | odd | 2 | 1 | ||
252.9.z.d | 12 | 21.g | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 285 T_{5}^{11} - 1570602 T_{5}^{10} + 455337945 T_{5}^{9} + 2118409775334 T_{5}^{8} - 311260236586983 T_{5}^{7} + \cdots + 76\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( (T^{2} - 81 T + 2187)^{6} \)
$5$
\( T^{12} - 285 T^{11} + \cdots + 76\!\cdots\!00 \)
$7$
\( T^{12} - 198 T^{11} + \cdots + 36\!\cdots\!01 \)
$11$
\( T^{12} + 17919 T^{11} + \cdots + 16\!\cdots\!00 \)
$13$
\( T^{12} + 5107913481 T^{10} + \cdots + 21\!\cdots\!64 \)
$17$
\( T^{12} + 205782 T^{11} + \cdots + 10\!\cdots\!00 \)
$19$
\( T^{12} - 74313 T^{11} + \cdots + 83\!\cdots\!96 \)
$23$
\( T^{12} + 62832 T^{11} + \cdots + 63\!\cdots\!64 \)
$29$
\( (T^{6} + 287727 T^{5} + \cdots + 23\!\cdots\!00)^{2} \)
$31$
\( T^{12} - 1442952 T^{11} + \cdots + 15\!\cdots\!81 \)
$37$
\( T^{12} + 2058621 T^{11} + \cdots + 40\!\cdots\!84 \)
$41$
\( T^{12} + 44050323101928 T^{10} + \cdots + 15\!\cdots\!64 \)
$43$
\( (T^{6} - 3860661 T^{5} + \cdots + 51\!\cdots\!00)^{2} \)
$47$
\( T^{12} - 12088194 T^{11} + \cdots + 18\!\cdots\!00 \)
$53$
\( T^{12} + 5506743 T^{11} + \cdots + 11\!\cdots\!56 \)
$59$
\( T^{12} - 7511901 T^{11} + \cdots + 26\!\cdots\!24 \)
$61$
\( T^{12} + 37215576 T^{11} + \cdots + 93\!\cdots\!00 \)
$67$
\( T^{12} + 36824553 T^{11} + \cdots + 65\!\cdots\!44 \)
$71$
\( (T^{6} + 15005778 T^{5} + \cdots - 52\!\cdots\!20)^{2} \)
$73$
\( T^{12} - 95080185 T^{11} + \cdots + 94\!\cdots\!00 \)
$79$
\( T^{12} - 8514456 T^{11} + \cdots + 47\!\cdots\!25 \)
$83$
\( T^{12} + \cdots + 78\!\cdots\!04 \)
$89$
\( T^{12} - 83038554 T^{11} + \cdots + 11\!\cdots\!96 \)
$97$
\( T^{12} + \cdots + 53\!\cdots\!00 \)
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