Newspace parameters
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(37.2687615464\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
| Defining polynomial: |
\( x^{16} + 485774x^{12} + 87183614355x^{8} + 6839940225440174x^{4} + 198392288899684017121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{36} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 485774x^{12} + 87183614355x^{8} + 6839940225440174x^{4} + 198392288899684017121 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -346\nu^{12} - 125696265\nu^{8} - 14422204736895\nu^{4} - 515812550322387299 ) / 142209872480800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 110729 \nu^{14} - 39704089485 \nu^{10} + \cdots + 13\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4593352202400 \nu^{14} + \cdots + 12\!\cdots\!09 ) / 70\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 62\!\cdots\!13 \nu^{14} + \cdots - 21\!\cdots\!33 ) / 18\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 63\!\cdots\!13 \nu^{14} + \cdots + 23\!\cdots\!59 ) / 18\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17\!\cdots\!25 \nu^{14} + \cdots + 10\!\cdots\!71 ) / 56\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 35\!\cdots\!65 \nu^{14} + \cdots - 21\!\cdots\!52 ) / 11\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 65\!\cdots\!76 \nu^{15} + \cdots + 25\!\cdots\!43 \nu ) / 67\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 27\!\cdots\!47 \nu^{15} + \cdots + 17\!\cdots\!96 \nu ) / 42\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 55\!\cdots\!91 \nu^{15} + \cdots - 10\!\cdots\!83 \nu ) / 22\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 62\!\cdots\!45 \nu^{15} + \cdots + 37\!\cdots\!03 \nu ) / 33\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 13\!\cdots\!76 \nu^{15} + \cdots - 10\!\cdots\!43 \nu ) / 67\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 60\!\cdots\!15 \nu^{15} + \cdots - 21\!\cdots\!41 \nu ) / 22\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 20\!\cdots\!96 \nu^{15} + \cdots + 21\!\cdots\!71 \nu ) / 67\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 29\!\cdots\!00 \nu^{15} + \cdots - 21\!\cdots\!36 \nu ) / 33\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{14} + 6\beta_{13} + 2\beta_{12} - 3\beta_{11} + 12\beta_{10} - 38\beta_{9} - 2\beta_{8} ) / 162 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - 154\beta_{3} - 2594\beta_{2} - 77\beta _1 + 1297 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 72 \beta_{15} - 1154 \beta_{14} - 2436 \beta_{13} - 625 \beta_{12} - 483 \beta_{11} - 750 \beta_{10} + 19567 \beta_{9} + 3457 \beta_{8} ) / 81 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 96\beta_{7} - 96\beta_{6} + 2127\beta_{5} - 2127\beta_{4} + 1078\beta_{3} + 11611\beta _1 - 6557901 ) / 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 99504 \beta_{15} - 159152 \beta_{14} - 328566 \beta_{13} + 119672 \beta_{12} + 413043 \beta_{11} - 1950684 \beta_{10} + 27933562 \beta_{9} - 6706136 \beta_{8} ) / 162 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 365024 \beta_{7} - 365024 \beta_{6} - 406936 \beta_{5} - 406936 \beta_{4} + 9562710 \beta_{3} + 466711184 \beta_{2} + 4723171 \beta _1 - 233355592 ) / 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 51666624 \beta_{15} + 402203792 \beta_{14} + 772529316 \beta_{13} + 200636200 \beta_{12} + 102715737 \beta_{11} + 50431602 \beta_{10} - 9802759642 \beta_{9} - 5095485928 \beta_{8} ) / 162 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 69354528 \beta_{7} + 69354528 \beta_{6} - 553988649 \beta_{5} + 553988649 \beta_{4} - 269272570 \beta_{3} - 2800707517 \beta _1 + 831688852677 ) / 54 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 11945304648 \beta_{15} - 11077715194 \beta_{14} - 1834011510 \beta_{13} - 37653432149 \beta_{12} - 28946255865 \beta_{11} + 151620937404 \beta_{10} + \cdots + 1483960759397 \beta_{8} ) / 81 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 155759031134 \beta_{7} + 155759031134 \beta_{6} + 259402944703 \beta_{5} + 259402944703 \beta_{4} - 2472136817594 \beta_{3} - 187985751037982 \beta_{2} + \cdots + 93992875518991 ) / 54 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 10374491746224 \beta_{15} - 67426468882732 \beta_{14} - 118152875712756 \beta_{13} - 26352036126998 \beta_{12} - 11382858372297 \beta_{11} + \cdots + 16\!\cdots\!30 \beta_{8} ) / 162 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 3532308996000 \beta_{7} - 3532308996000 \beta_{6} + 18766028206320 \beta_{5} - 18766028206320 \beta_{4} + 8814749330240 \beta_{3} + \cdots - 18\!\cdots\!81 ) / 9 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 43\!\cdots\!60 \beta_{15} + \cdots - 79\!\cdots\!02 \beta_{8} ) / 162 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 29\!\cdots\!38 \beta_{7} + \cdots - 16\!\cdots\!53 ) / 54 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 88\!\cdots\!08 \beta_{15} + \cdots - 18\!\cdots\!53 \beta_{8} ) / 81 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | −132.984 | − | 76.7783i | 0 | −336.136 | − | 582.205i | 181.019i | 0 | 868.648 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.2 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | −74.2039 | − | 42.8417i | 0 | 282.029 | + | 488.488i | 181.019i | 0 | 484.698 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.3 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | 117.830 | + | 68.0293i | 0 | 93.5265 | + | 161.993i | 181.019i | 0 | −769.663 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.4 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | 148.146 | + | 85.5319i | 0 | −280.419 | − | 485.701i | 181.019i | 0 | −967.683 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.5 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | −148.146 | − | 85.5319i | 0 | −280.419 | − | 485.701i | − | 181.019i | 0 | −967.683 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.6 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | −117.830 | − | 68.0293i | 0 | 93.5265 | + | 161.993i | − | 181.019i | 0 | −769.663 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.7 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | 74.2039 | + | 42.8417i | 0 | 282.029 | + | 488.488i | − | 181.019i | 0 | 484.698 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.8 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | 132.984 | + | 76.7783i | 0 | −336.136 | − | 582.205i | − | 181.019i | 0 | 868.648 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.1 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | −132.984 | + | 76.7783i | 0 | −336.136 | + | 582.205i | − | 181.019i | 0 | 868.648 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.2 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | −74.2039 | + | 42.8417i | 0 | 282.029 | − | 488.488i | − | 181.019i | 0 | 484.698 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.3 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | 117.830 | − | 68.0293i | 0 | 93.5265 | − | 161.993i | − | 181.019i | 0 | −769.663 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.4 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | 148.146 | − | 85.5319i | 0 | −280.419 | + | 485.701i | − | 181.019i | 0 | −967.683 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.5 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | −148.146 | + | 85.5319i | 0 | −280.419 | + | 485.701i | 181.019i | 0 | −967.683 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.6 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | −117.830 | + | 68.0293i | 0 | 93.5265 | − | 161.993i | 181.019i | 0 | −769.663 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.7 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | 74.2039 | − | 42.8417i | 0 | 282.029 | − | 488.488i | 181.019i | 0 | 484.698 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.8 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | 132.984 | − | 76.7783i | 0 | −336.136 | + | 582.205i | 181.019i | 0 | 868.648 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.7.d.g | 16 | |
| 3.b | odd | 2 | 1 | inner | 162.7.d.g | 16 | |
| 9.c | even | 3 | 1 | 162.7.b.b | ✓ | 8 | |
| 9.c | even | 3 | 1 | inner | 162.7.d.g | 16 | |
| 9.d | odd | 6 | 1 | 162.7.b.b | ✓ | 8 | |
| 9.d | odd | 6 | 1 | inner | 162.7.d.g | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.7.b.b | ✓ | 8 | 9.c | even | 3 | 1 | |
| 162.7.b.b | ✓ | 8 | 9.d | odd | 6 | 1 | |
| 162.7.d.g | 16 | 1.a | even | 1 | 1 | trivial | |
| 162.7.d.g | 16 | 3.b | odd | 2 | 1 | inner | |
| 162.7.d.g | 16 | 9.c | even | 3 | 1 | inner | |
| 162.7.d.g | 16 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 78696 T_{5}^{14} + 4000981230 T_{5}^{12} - 122466234531456 T_{5}^{10} + \cdots + 87\!\cdots\!25 \)
acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 32 T^{2} + 1024)^{4} \)
$3$
\( T^{16} \)
$5$
\( T^{16} - 78696 T^{14} + \cdots + 87\!\cdots\!25 \)
$7$
\( (T^{8} + 482 T^{7} + \cdots + 15\!\cdots\!16)^{2} \)
$11$
\( T^{16} - 12104748 T^{14} + \cdots + 51\!\cdots\!96 \)
$13$
\( (T^{8} + 2270 T^{7} + \cdots + 10\!\cdots\!21)^{2} \)
$17$
\( (T^{8} + 51916752 T^{6} + \cdots + 89\!\cdots\!25)^{2} \)
$19$
\( (T^{4} + 11842 T^{3} + \cdots - 41\!\cdots\!44)^{4} \)
$23$
\( T^{16} - 582093972 T^{14} + \cdots + 20\!\cdots\!00 \)
$29$
\( T^{16} - 1686224736 T^{14} + \cdots + 12\!\cdots\!01 \)
$31$
\( (T^{8} + 38528 T^{7} + \cdots + 62\!\cdots\!36)^{2} \)
$37$
\( (T^{4} + 5674 T^{3} + \cdots + 68\!\cdots\!73)^{4} \)
$41$
\( T^{16} - 13639103856 T^{14} + \cdots + 13\!\cdots\!96 \)
$43$
\( (T^{8} - 113302 T^{7} + \cdots + 91\!\cdots\!04)^{2} \)
$47$
\( T^{16} - 40347462384 T^{14} + \cdots + 10\!\cdots\!76 \)
$53$
\( (T^{8} + 79583256720 T^{6} + \cdots + 57\!\cdots\!16)^{2} \)
$59$
\( T^{16} - 268690196304 T^{14} + \cdots + 64\!\cdots\!76 \)
$61$
\( (T^{8} - 163738 T^{7} + \cdots + 69\!\cdots\!49)^{2} \)
$67$
\( (T^{8} + 856646 T^{7} + \cdots + 86\!\cdots\!96)^{2} \)
$71$
\( (T^{8} + 812781263700 T^{6} + \cdots + 41\!\cdots\!04)^{2} \)
$73$
\( (T^{4} + 1094608 T^{3} + \cdots - 40\!\cdots\!71)^{4} \)
$79$
\( (T^{8} - 663442 T^{7} + \cdots + 66\!\cdots\!16)^{2} \)
$83$
\( T^{16} - 705097881360 T^{14} + \cdots + 13\!\cdots\!76 \)
$89$
\( (T^{8} + 2921642775864 T^{6} + \cdots + 10\!\cdots\!61)^{2} \)
$97$
\( (T^{8} - 1100032 T^{7} + \cdots + 52\!\cdots\!24)^{2} \)
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