Newspace parameters
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.97132279097\) |
Analytic rank: | \(0\) |
Dimension: | \(18\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
Defining polynomial: |
\( x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + 490851 x^{10} - 318018 x^{9} + 3391249 x^{8} - 990441 x^{7} + \cdots + 16777216 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 13^{4} \) |
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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + 490851 x^{10} - 318018 x^{9} + 3391249 x^{8} - 990441 x^{7} + \cdots + 16777216 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 71\!\cdots\!85 \nu^{17} + \cdots + 26\!\cdots\!56 ) / 32\!\cdots\!52 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 12\!\cdots\!53 \nu^{17} + \cdots + 61\!\cdots\!76 ) / 25\!\cdots\!16 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!41 \nu^{17} + \cdots + 25\!\cdots\!24 ) / 20\!\cdots\!16 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!51 \nu^{17} + \cdots - 56\!\cdots\!96 ) / 16\!\cdots\!28 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 17\!\cdots\!07 \nu^{17} + \cdots + 13\!\cdots\!44 ) / 20\!\cdots\!16 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 28\!\cdots\!09 \nu^{17} + \cdots + 74\!\cdots\!32 ) / 31\!\cdots\!64 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 12\!\cdots\!01 \nu^{17} + \cdots + 90\!\cdots\!72 ) / 10\!\cdots\!08 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 11\!\cdots\!13 \nu^{17} + \cdots - 33\!\cdots\!56 ) / 81\!\cdots\!64 \)
|
\(\beta_{10}\) | \(=\) |
\( ( - 38\!\cdots\!47 \nu^{17} + \cdots - 14\!\cdots\!88 ) / 20\!\cdots\!16 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 28\!\cdots\!69 \nu^{17} + \cdots + 11\!\cdots\!84 ) / 12\!\cdots\!56 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 54\!\cdots\!69 \nu^{17} + \cdots + 17\!\cdots\!56 ) / 20\!\cdots\!16 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 25\!\cdots\!97 \nu^{17} + \cdots - 63\!\cdots\!36 ) / 81\!\cdots\!64 \)
|
\(\beta_{14}\) | \(=\) |
\( ( 82\!\cdots\!31 \nu^{17} + \cdots + 18\!\cdots\!88 ) / 20\!\cdots\!16 \)
|
\(\beta_{15}\) | \(=\) |
\( ( - 65\!\cdots\!29 \nu^{17} + \cdots + 10\!\cdots\!20 ) / 12\!\cdots\!56 \)
|
\(\beta_{16}\) | \(=\) |
\( ( 11\!\cdots\!21 \nu^{17} + \cdots - 19\!\cdots\!44 ) / 16\!\cdots\!28 \)
|
\(\beta_{17}\) | \(=\) |
\( ( 12\!\cdots\!63 \nu^{17} + \cdots - 14\!\cdots\!88 ) / 16\!\cdots\!28 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{17} + \beta_{16} - \beta_{13} + \beta_{10} + \beta_{6} - 12\beta_{3} - 2\beta_{2} + 2\beta_1 \)
|
\(\nu^{3}\) | \(=\) |
\( -2\beta_{17} + 3\beta_{16} - 3\beta_{13} + 2\beta_{12} - \beta_{11} - 4\beta_{9} + 4\beta_{7} - 23\beta_{2} - 12 \)
|
\(\nu^{4}\) | \(=\) |
\( -8\beta_{14} + 27\beta_{12} - 37\beta_{10} + 8\beta_{7} - 32\beta_{6} - 5\beta_{4} + 260\beta_{3} - 77\beta _1 - 260 \)
|
\(\nu^{5}\) | \(=\) |
\( 75 \beta_{17} - 128 \beta_{16} + 10 \beta_{15} - 45 \beta_{14} + 166 \beta_{13} + 45 \beta_{11} - 166 \beta_{10} + 138 \beta_{9} - 10 \beta_{8} - 128 \beta_{6} - 18 \beta_{5} - 18 \beta_{4} + 604 \beta_{3} + 636 \beta_{2} - 636 \beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( 748 \beta_{17} - 1004 \beta_{16} + 46 \beta_{15} + 1315 \beta_{13} - 748 \beta_{12} + 339 \beta_{11} + 490 \beta_{9} - 490 \beta_{7} - 299 \beta_{5} + 2746 \beta_{2} + 6752 \)
|
\(\nu^{7}\) | \(=\) |
\( 1762 \beta_{14} - 2684 \beta_{12} + 6896 \beta_{10} + 644 \beta_{8} - 4588 \beta_{7} + 4677 \beta_{6} + 1299 \beta_{4} - 23360 \beta_{3} + 19224 \beta _1 + 23360 \)
|
\(\nu^{8}\) | \(=\) |
\( - 21858 \beta_{17} + 32278 \beta_{16} - 3242 \beta_{15} + 12178 \beta_{14} - 46550 \beta_{13} - 12178 \beta_{11} + 46550 \beta_{10} - 21190 \beta_{9} + 3242 \beta_{8} + 32278 \beta_{6} + 12940 \beta_{5} + \cdots + 95033 \beta_1 \)
|
\(\nu^{9}\) | \(=\) |
\( - 93585 \beta_{17} + 164491 \beta_{16} - 29122 \beta_{15} - 260223 \beta_{13} + 93585 \beta_{12} - 65220 \beta_{11} - 155738 \beta_{9} + 155738 \beta_{7} + 63424 \beta_{5} - 609880 \beta_{2} + \cdots - 831272 \)
|
\(\nu^{10}\) | \(=\) |
\( - 423443 \beta_{14} + 667892 \beta_{12} - 1637813 \beta_{10} - 155970 \beta_{8} + 812858 \beta_{7} - 1058555 \beta_{6} - 500294 \beta_{4} + 5853008 \beta_{3} - 3244441 \beta _1 - 5853008 \)
|
\(\nu^{11}\) | \(=\) |
\( 3212681 \beta_{17} - 5697380 \beta_{16} + 1156558 \beta_{15} - 2341040 \beta_{14} + 9404329 \beta_{13} + 2341040 \beta_{11} - 9404329 \beta_{10} + 5347886 \beta_{9} - 1156558 \beta_{8} + \cdots - 19896385 \beta_1 \)
|
\(\nu^{12}\) | \(=\) |
\( 21117643 \beta_{17} - 35204052 \beta_{16} + 6439744 \beta_{15} + 57218756 \beta_{13} - 21117643 \beta_{12} + 14595321 \beta_{11} + 29575380 \beta_{9} - 29575380 \beta_{7} + \cdots + 185118648 \)
|
\(\nu^{13}\) | \(=\) |
\( 82492865 \beta_{14} - 109440828 \beta_{12} + 332264719 \beta_{10} + 43215104 \beta_{8} - 184233596 \beta_{7} + 195850226 \beta_{6} + 101502237 \beta_{4} + \cdots + 974986692 \)
|
\(\nu^{14}\) | \(=\) |
\( - 684437956 \beta_{17} + 1181688405 \beta_{16} - 246219578 \beta_{15} + 501405476 \beta_{14} - 1986451992 \beta_{13} - 501405476 \beta_{11} + 1986451992 \beta_{10} + \cdots + 3738644514 \beta_1 \)
|
\(\nu^{15}\) | \(=\) |
\( - 3716482924 \beta_{17} + 6705745868 \beta_{16} - 1562310104 \beta_{15} - 11588149060 \beta_{13} + 3716482924 \beta_{12} - 2873366536 \beta_{11} + \cdots - 33051801880 \)
|
\(\nu^{16}\) | \(=\) |
\( - 17193582024 \beta_{14} + 22567596581 \beta_{12} - 68620566389 \beta_{10} - 9017389320 \beta_{8} + 36572897116 \beta_{7} - 39902478713 \beta_{6} + \cdots - 198745293060 \)
|
\(\nu^{17}\) | \(=\) |
\( 126112059954 \beta_{17} - 229119790391 \beta_{16} + 55380969056 \beta_{15} - 99346555769 \beta_{14} + 400991725275 \beta_{13} + 99346555769 \beta_{11} + \cdots - 745830358799 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 |
|
−2.42009 | + | 4.19172i | 3.09831 | − | 5.36643i | −7.71365 | − | 13.3604i | −15.2399 | 14.9964 | + | 25.9745i | 2.15810 | + | 3.73794i | 35.9495 | −5.69903 | − | 9.87102i | 36.8818 | − | 63.8811i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.2 | −1.91719 | + | 3.32067i | 0.139581 | − | 0.241762i | −3.35124 | − | 5.80452i | −11.3710 | 0.535209 | + | 0.927008i | 15.5311 | + | 26.9007i | −4.97517 | 13.4610 | + | 23.3152i | 21.8004 | − | 37.7594i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.3 | −0.863817 | + | 1.49617i | −3.44796 | + | 5.97204i | 2.50764 | + | 4.34336i | −20.8281 | −5.95681 | − | 10.3175i | 3.78283 | + | 6.55206i | −22.4856 | −10.2768 | − | 17.8000i | 17.9916 | − | 31.1624i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.4 | −0.195058 | + | 0.337850i | −1.80483 | + | 3.12606i | 3.92391 | + | 6.79640i | 7.52136 | −0.704093 | − | 1.21953i | 9.77228 | + | 16.9261i | −6.18247 | 6.98515 | + | 12.0986i | −1.46710 | + | 2.54109i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.5 | 0.0745292 | − | 0.129088i | 3.24429 | − | 5.61927i | 3.98889 | + | 6.90896i | 10.2526 | −0.483588 | − | 0.837600i | −14.8372 | − | 25.6987i | 2.38162 | −7.55083 | − | 13.0784i | 0.764114 | − | 1.32348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.6 | 1.11399 | − | 1.92949i | 4.87434 | − | 8.44260i | 1.51803 | + | 2.62931i | 8.20685 | −10.8600 | − | 18.8100i | 4.17747 | + | 7.23560i | 24.5882 | −34.0183 | − | 58.9214i | 9.14239 | − | 15.8351i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.7 | 1.58068 | − | 2.73781i | −3.54442 | + | 6.13911i | −0.997073 | − | 1.72698i | −13.6039 | 11.2051 | + | 19.4079i | −7.16574 | − | 12.4114i | 18.9866 | −11.6258 | − | 20.1364i | −21.5034 | + | 37.2450i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.8 | 2.41278 | − | 4.17905i | −2.22176 | + | 3.84820i | −7.64299 | − | 13.2380i | 12.7712 | 10.7212 | + | 18.5697i | 13.0936 | + | 22.6787i | −35.1589 | 3.62756 | + | 6.28312i | 30.8140 | − | 53.3715i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
22.9 | 2.71418 | − | 4.70109i | −0.837548 | + | 1.45068i | −10.7335 | − | 18.5910i | −7.70909 | 4.54651 | + | 7.87478i | −7.51249 | − | 13.0120i | −73.1038 | 12.0970 | + | 20.9527i | −20.9238 | + | 36.2412i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.1 | −2.42009 | − | 4.19172i | 3.09831 | + | 5.36643i | −7.71365 | + | 13.3604i | −15.2399 | 14.9964 | − | 25.9745i | 2.15810 | − | 3.73794i | 35.9495 | −5.69903 | + | 9.87102i | 36.8818 | + | 63.8811i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.2 | −1.91719 | − | 3.32067i | 0.139581 | + | 0.241762i | −3.35124 | + | 5.80452i | −11.3710 | 0.535209 | − | 0.927008i | 15.5311 | − | 26.9007i | −4.97517 | 13.4610 | − | 23.3152i | 21.8004 | + | 37.7594i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.3 | −0.863817 | − | 1.49617i | −3.44796 | − | 5.97204i | 2.50764 | − | 4.34336i | −20.8281 | −5.95681 | + | 10.3175i | 3.78283 | − | 6.55206i | −22.4856 | −10.2768 | + | 17.8000i | 17.9916 | + | 31.1624i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.4 | −0.195058 | − | 0.337850i | −1.80483 | − | 3.12606i | 3.92391 | − | 6.79640i | 7.52136 | −0.704093 | + | 1.21953i | 9.77228 | − | 16.9261i | −6.18247 | 6.98515 | − | 12.0986i | −1.46710 | − | 2.54109i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.5 | 0.0745292 | + | 0.129088i | 3.24429 | + | 5.61927i | 3.98889 | − | 6.90896i | 10.2526 | −0.483588 | + | 0.837600i | −14.8372 | + | 25.6987i | 2.38162 | −7.55083 | + | 13.0784i | 0.764114 | + | 1.32348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.6 | 1.11399 | + | 1.92949i | 4.87434 | + | 8.44260i | 1.51803 | − | 2.62931i | 8.20685 | −10.8600 | + | 18.8100i | 4.17747 | − | 7.23560i | 24.5882 | −34.0183 | + | 58.9214i | 9.14239 | + | 15.8351i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.7 | 1.58068 | + | 2.73781i | −3.54442 | − | 6.13911i | −0.997073 | + | 1.72698i | −13.6039 | 11.2051 | − | 19.4079i | −7.16574 | + | 12.4114i | 18.9866 | −11.6258 | + | 20.1364i | −21.5034 | − | 37.2450i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.8 | 2.41278 | + | 4.17905i | −2.22176 | − | 3.84820i | −7.64299 | + | 13.2380i | 12.7712 | 10.7212 | − | 18.5697i | 13.0936 | − | 22.6787i | −35.1589 | 3.62756 | − | 6.28312i | 30.8140 | + | 53.3715i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.9 | 2.71418 | + | 4.70109i | −0.837548 | − | 1.45068i | −10.7335 | + | 18.5910i | −7.70909 | 4.54651 | − | 7.87478i | −7.51249 | + | 13.0120i | −73.1038 | 12.0970 | − | 20.9527i | −20.9238 | − | 36.2412i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.4.c.l | 18 | |
13.b | even | 2 | 1 | 169.4.c.k | 18 | ||
13.c | even | 3 | 1 | 169.4.a.k | ✓ | 9 | |
13.c | even | 3 | 1 | inner | 169.4.c.l | 18 | |
13.d | odd | 4 | 2 | 169.4.e.h | 36 | ||
13.e | even | 6 | 1 | 169.4.a.l | yes | 9 | |
13.e | even | 6 | 1 | 169.4.c.k | 18 | ||
13.f | odd | 12 | 2 | 169.4.b.g | 18 | ||
13.f | odd | 12 | 2 | 169.4.e.h | 36 | ||
39.h | odd | 6 | 1 | 1521.4.a.bg | 9 | ||
39.i | odd | 6 | 1 | 1521.4.a.bh | 9 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
169.4.a.k | ✓ | 9 | 13.c | even | 3 | 1 | |
169.4.a.l | yes | 9 | 13.e | even | 6 | 1 | |
169.4.b.g | 18 | 13.f | odd | 12 | 2 | ||
169.4.c.k | 18 | 13.b | even | 2 | 1 | ||
169.4.c.k | 18 | 13.e | even | 6 | 1 | ||
169.4.c.l | 18 | 1.a | even | 1 | 1 | trivial | |
169.4.c.l | 18 | 13.c | even | 3 | 1 | inner | |
169.4.e.h | 36 | 13.d | odd | 4 | 2 | ||
169.4.e.h | 36 | 13.f | odd | 12 | 2 | ||
1521.4.a.bg | 9 | 39.h | odd | 6 | 1 | ||
1521.4.a.bh | 9 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - 5 T_{2}^{17} + 67 T_{2}^{16} - 200 T_{2}^{15} + 2303 T_{2}^{14} - 6060 T_{2}^{13} + 48373 T_{2}^{12} - 82227 T_{2}^{11} + 596508 T_{2}^{10} - 861638 T_{2}^{9} + 4613216 T_{2}^{8} - 2514618 T_{2}^{7} + \cdots + 118336 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{18} - 5 T^{17} + 67 T^{16} + \cdots + 118336 \)
$3$
\( T^{18} + T^{17} + 155 T^{16} + \cdots + 20654576089 \)
$5$
\( (T^{9} + 30 T^{8} - 266 T^{7} + \cdots + 3059376152)^{2} \)
$7$
\( T^{18} - 38 T^{17} + \cdots + 76\!\cdots\!76 \)
$11$
\( T^{18} - 181 T^{17} + \cdots + 76\!\cdots\!61 \)
$13$
\( T^{18} \)
$17$
\( T^{18} - 55 T^{17} + \cdots + 31\!\cdots\!49 \)
$19$
\( T^{18} - 161 T^{17} + \cdots + 74\!\cdots\!61 \)
$23$
\( T^{18} - 204 T^{17} + \cdots + 46\!\cdots\!64 \)
$29$
\( T^{18} + 280 T^{17} + \cdots + 31\!\cdots\!16 \)
$31$
\( (T^{9} + 706 T^{8} + \cdots - 30\!\cdots\!56)^{2} \)
$37$
\( T^{18} - 298 T^{17} + \cdots + 96\!\cdots\!04 \)
$41$
\( T^{18} - 1201 T^{17} + \cdots + 38\!\cdots\!09 \)
$43$
\( T^{18} - 533 T^{17} + \cdots + 12\!\cdots\!29 \)
$47$
\( (T^{9} + 956 T^{8} + \cdots + 11\!\cdots\!84)^{2} \)
$53$
\( (T^{9} + 278 T^{8} + \cdots - 12\!\cdots\!76)^{2} \)
$59$
\( T^{18} - 1377 T^{17} + \cdots + 75\!\cdots\!29 \)
$61$
\( T^{18} - 136 T^{17} + \cdots + 75\!\cdots\!24 \)
$67$
\( T^{18} + 931 T^{17} + \cdots + 10\!\cdots\!01 \)
$71$
\( T^{18} - 2046 T^{17} + \cdots + 19\!\cdots\!84 \)
$73$
\( (T^{9} - 45 T^{8} + \cdots - 31\!\cdots\!21)^{2} \)
$79$
\( (T^{9} - 412 T^{8} + \cdots + 63\!\cdots\!88)^{2} \)
$83$
\( (T^{9} + 3709 T^{8} + \cdots + 14\!\cdots\!61)^{2} \)
$89$
\( T^{18} - 1663 T^{17} + \cdots + 19\!\cdots\!29 \)
$97$
\( T^{18} + 1087 T^{17} + \cdots + 43\!\cdots\!21 \)
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