Newspace parameters
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(9.97132279097\) |
Analytic rank: | \(0\) |
Dimension: | \(9\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
Defining polynomial: |
\( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 13^{2} \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 6901 \nu^{8} - 12396 \nu^{7} + 501446 \nu^{6} + 728955 \nu^{5} - 11530440 \nu^{4} - 12738943 \nu^{3} + 91159455 \nu^{2} + 52058000 \nu - 149293824 ) / 8071424 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 3843 \nu^{8} + 11372 \nu^{7} + 195178 \nu^{6} - 384275 \nu^{5} - 3297016 \nu^{4} + 4265751 \nu^{3} + 16793257 \nu^{2} - 22791952 \nu - 6050816 ) / 4035712 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 595 \nu^{8} + 2737 \nu^{7} + 19422 \nu^{6} - 72705 \nu^{5} - 171803 \nu^{4} + 485407 \nu^{3} + 252488 \nu^{2} - 1604391 \nu - 646464 ) / 504464 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 23843 \nu^{8} + 103372 \nu^{7} + 1059978 \nu^{6} - 3803155 \nu^{5} - 14845688 \nu^{4} + 38793527 \nu^{3} + 67400873 \nu^{2} - 97179408 \nu - 92826880 ) / 8071424 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 32107 \nu^{8} + 90940 \nu^{7} + 1548474 \nu^{6} - 2570427 \nu^{5} - 23878504 \nu^{4} + 12527327 \nu^{3} + 111900273 \nu^{2} + 12391680 \nu - 122506496 ) / 8071424 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 53515 \nu^{8} + 88524 \nu^{7} + 2837370 \nu^{6} - 2210427 \nu^{5} - 46218904 \nu^{4} + 3078399 \nu^{3} + 217315329 \nu^{2} + 51740944 \nu - 202361088 ) / 8071424 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 27975 \nu^{8} - 97156 \nu^{7} - 1304226 \nu^{6} + 3186791 \nu^{5} + 19362096 \nu^{4} - 25660427 \nu^{3} - 85614861 \nu^{2} + 34322440 \nu + 59238144 ) / 4035712 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{6} + \beta_{5} + 2\beta _1 + 12 \)
|
\(\nu^{3}\) | \(=\) |
\( 3\beta_{8} + 3\beta_{6} + 2\beta_{5} + 4\beta_{3} - \beta_{2} + 23\beta _1 + 12 \)
|
\(\nu^{4}\) | \(=\) |
\( 32\beta_{8} + 37\beta_{6} + 27\beta_{5} - 5\beta_{4} + 8\beta_{3} - 8\beta_{2} + 77\beta _1 + 260 \)
|
\(\nu^{5}\) | \(=\) |
\( 128 \beta_{8} - 10 \beta_{7} + 166 \beta_{6} + 75 \beta_{5} - 18 \beta_{4} + 138 \beta_{3} - 45 \beta_{2} + 636 \beta _1 + 604 \)
|
\(\nu^{6}\) | \(=\) |
\( 1004 \beta_{8} - 46 \beta_{7} + 1315 \beta_{6} + 748 \beta_{5} - 299 \beta_{4} + 490 \beta_{3} - 339 \beta_{2} + 2746 \beta _1 + 6752 \)
|
\(\nu^{7}\) | \(=\) |
\( 4677 \beta_{8} - 644 \beta_{7} + 6896 \beta_{6} + 2684 \beta_{5} - 1299 \beta_{4} + 4588 \beta_{3} - 1762 \beta_{2} + 19224 \beta _1 + 23360 \)
|
\(\nu^{8}\) | \(=\) |
\( 32278 \beta_{8} - 3242 \beta_{7} + 46550 \beta_{6} + 21858 \beta_{5} - 12940 \beta_{4} + 21190 \beta_{3} - 12178 \beta_{2} + 95033 \beta _1 + 192772 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−4.84018 | −6.19662 | 15.4273 | 15.2399 | 29.9927 | 4.31620 | −35.9495 | 11.3981 | −73.7636 | |||||||||||||||||||||||||||||||||||||||||||||
1.2 | −3.83438 | −0.279163 | 6.70249 | 11.3710 | 1.07042 | 31.0623 | 4.97517 | −26.9221 | −43.6008 | ||||||||||||||||||||||||||||||||||||||||||||||
1.3 | −1.72763 | 6.89591 | −5.01528 | 20.8281 | −11.9136 | 7.56566 | 22.4856 | 20.5536 | −35.9833 | ||||||||||||||||||||||||||||||||||||||||||||||
1.4 | −0.390115 | 3.60967 | −7.84781 | −7.52136 | −1.40819 | 19.5446 | 6.18247 | −13.9703 | 2.93420 | ||||||||||||||||||||||||||||||||||||||||||||||
1.5 | 0.149058 | −6.48858 | −7.97778 | −10.2526 | −0.967177 | −29.6743 | −2.38162 | 15.1017 | −1.52823 | ||||||||||||||||||||||||||||||||||||||||||||||
1.6 | 2.22799 | −9.74867 | −3.03607 | −8.20685 | −21.7199 | 8.35495 | −24.5882 | 68.0366 | −18.2848 | ||||||||||||||||||||||||||||||||||||||||||||||
1.7 | 3.16135 | 7.08883 | 1.99415 | 13.6039 | 22.4103 | −14.3315 | −18.9866 | 23.2516 | 43.0068 | ||||||||||||||||||||||||||||||||||||||||||||||
1.8 | 4.82555 | 4.44352 | 15.2860 | −12.7712 | 21.4425 | 26.1871 | 35.1589 | −7.25513 | −61.6281 | ||||||||||||||||||||||||||||||||||||||||||||||
1.9 | 5.42835 | 1.67510 | 21.4670 | 7.70909 | 9.09301 | −15.0250 | 73.1038 | −24.1941 | 41.8477 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.4.a.l | yes | 9 |
3.b | odd | 2 | 1 | 1521.4.a.bg | 9 | ||
13.b | even | 2 | 1 | 169.4.a.k | ✓ | 9 | |
13.c | even | 3 | 2 | 169.4.c.k | 18 | ||
13.d | odd | 4 | 2 | 169.4.b.g | 18 | ||
13.e | even | 6 | 2 | 169.4.c.l | 18 | ||
13.f | odd | 12 | 4 | 169.4.e.h | 36 | ||
39.d | odd | 2 | 1 | 1521.4.a.bh | 9 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
169.4.a.k | ✓ | 9 | 13.b | even | 2 | 1 | |
169.4.a.l | yes | 9 | 1.a | even | 1 | 1 | trivial |
169.4.b.g | 18 | 13.d | odd | 4 | 2 | ||
169.4.c.k | 18 | 13.c | even | 3 | 2 | ||
169.4.c.l | 18 | 13.e | even | 6 | 2 | ||
169.4.e.h | 36 | 13.f | odd | 12 | 4 | ||
1521.4.a.bg | 9 | 3.b | odd | 2 | 1 | ||
1521.4.a.bh | 9 | 39.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 5T_{2}^{8} - 42T_{2}^{7} + 205T_{2}^{6} + 486T_{2}^{5} - 2310T_{2}^{4} - 1257T_{2}^{3} + 5898T_{2}^{2} + 1464T_{2} - 344 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{9} - 5 T^{8} - 42 T^{7} + 205 T^{6} + \cdots - 344 \)
$3$
\( T^{9} - T^{8} - 154 T^{7} + \cdots - 143717 \)
$5$
\( T^{9} - 30 T^{8} + \cdots - 3059376152 \)
$7$
\( T^{9} - 38 T^{8} + \cdots + 27715644424 \)
$11$
\( T^{9} - 181 T^{8} + \cdots - 276199564381 \)
$13$
\( T^{9} \)
$17$
\( T^{9} + 55 T^{8} + \cdots + 5572934105557 \)
$19$
\( T^{9} + \cdots - 865058822963419 \)
$23$
\( T^{9} + 204 T^{8} + \cdots + 68\!\cdots\!92 \)
$29$
\( T^{9} - 280 T^{8} + \cdots + 56\!\cdots\!96 \)
$31$
\( T^{9} - 706 T^{8} + \cdots + 30\!\cdots\!56 \)
$37$
\( T^{9} - 298 T^{8} + \cdots + 31\!\cdots\!48 \)
$41$
\( T^{9} - 1201 T^{8} + \cdots - 61\!\cdots\!53 \)
$43$
\( T^{9} + 533 T^{8} + \cdots - 35\!\cdots\!77 \)
$47$
\( T^{9} - 956 T^{8} + \cdots - 11\!\cdots\!84 \)
$53$
\( T^{9} + 278 T^{8} + \cdots - 12\!\cdots\!76 \)
$59$
\( T^{9} - 1377 T^{8} + \cdots - 27\!\cdots\!23 \)
$61$
\( T^{9} + 136 T^{8} + \cdots - 27\!\cdots\!32 \)
$67$
\( T^{9} + 931 T^{8} + \cdots - 32\!\cdots\!99 \)
$71$
\( T^{9} - 2046 T^{8} + \cdots - 44\!\cdots\!72 \)
$73$
\( T^{9} + 45 T^{8} + \cdots + 31\!\cdots\!21 \)
$79$
\( T^{9} - 412 T^{8} + \cdots + 63\!\cdots\!88 \)
$83$
\( T^{9} - 3709 T^{8} + \cdots - 14\!\cdots\!61 \)
$89$
\( T^{9} - 1663 T^{8} + \cdots - 43\!\cdots\!23 \)
$97$
\( T^{9} + 1087 T^{8} + \cdots - 20\!\cdots\!89 \)
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