Newspace parameters
Level: | \( N \) | \(=\) | \( 272 = 2^{4} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 272.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.41146319060\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
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: | \( x^{12} - 3 x^{11} + 25 x^{10} - 16 x^{9} + 294 x^{8} - 156 x^{7} + 1960 x^{6} + 2136 x^{5} + 2980 x^{4} + 976 x^{3} + 528 x^{2} - 64 x + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{22} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{12} - 3 x^{11} + 25 x^{10} - 16 x^{9} + 294 x^{8} - 156 x^{7} + 1960 x^{6} + 2136 x^{5} + 2980 x^{4} + 976 x^{3} + 528 x^{2} - 64 x + 64 \) :
\(\beta_{1}\) | \(=\) | \( ( - 154524933 \nu^{11} + 478470931 \nu^{10} - 3898157409 \nu^{9} + 2843759544 \nu^{8} - 45535904778 \nu^{7} + 29009461680 \nu^{6} + \cdots - 30127498968 ) / 39666231960 \) |
\(\beta_{2}\) | \(=\) | \( ( 854388159 \nu^{11} - 1561421493 \nu^{10} + 17825980707 \nu^{9} + 12893673988 \nu^{8} + 222884377654 \nu^{7} + 165035138660 \nu^{6} + \cdots + 89227289184 ) / 158664927840 \) |
\(\beta_{3}\) | \(=\) | \( ( 1472487891 \nu^{11} - 3475305217 \nu^{10} + 33418610343 \nu^{9} + 1518635812 \nu^{8} + 405027996766 \nu^{7} + 48997291940 \nu^{6} + \cdots + 51072357216 ) / 158664927840 \) |
\(\beta_{4}\) | \(=\) | \( ( 2357301861 \nu^{11} - 1165878247 \nu^{10} + 38217782753 \nu^{9} + 118201104972 \nu^{8} + 528185910946 \nu^{7} + \cdots + 613533279136 ) / 158664927840 \) |
\(\beta_{5}\) | \(=\) | \( ( - 116341408 \nu^{11} + 373742662 \nu^{10} - 2949227696 \nu^{9} + 2346295329 \nu^{8} - 33658113765 \nu^{7} + 23952776007 \nu^{6} + \cdots + 78430328582 ) / 5949934794 \) |
\(\beta_{6}\) | \(=\) | \( ( - 1075350649 \nu^{11} + 5442385203 \nu^{10} - 34670140997 \nu^{9} + 75858352572 \nu^{8} - 378966455394 \nu^{7} + \cdots + 276973569376 ) / 39666231960 \) |
\(\beta_{7}\) | \(=\) | \( ( 149042703 \nu^{11} - 292603176 \nu^{10} + 3247596644 \nu^{9} + 1513474161 \nu^{8} + 40974795138 \nu^{7} + 22285243110 \nu^{6} + \cdots + 27351173848 ) / 4958278995 \) |
\(\beta_{8}\) | \(=\) | \( ( 1117537 \nu^{11} - 3584989 \nu^{10} + 28327961 \nu^{9} - 22460841 \nu^{8} + 323543127 \nu^{7} - 229980045 \nu^{6} + 2129151100 \nu^{5} + \cdots - 63777578 ) / 33691590 \) |
\(\beta_{9}\) | \(=\) | \( ( - 5343614297 \nu^{11} + 16067870099 \nu^{10} - 135092448541 \nu^{9} + 90458903436 \nu^{8} - 1604931624562 \nu^{7} + \cdots - 148594773792 ) / 79332463920 \) |
\(\beta_{10}\) | \(=\) | \( ( 413871677 \nu^{11} - 1342367993 \nu^{10} + 10504360891 \nu^{9} - 8541536301 \nu^{8} + 119140931715 \nu^{7} - 87675314073 \nu^{6} + \cdots + 38648292212 ) / 5949934794 \) |
\(\beta_{11}\) | \(=\) | \( ( 1723724581 \nu^{11} - 5364724411 \nu^{10} + 43520193917 \nu^{9} - 32138337348 \nu^{8} + 506683696134 \nu^{7} - 327933428520 \nu^{6} + \cdots - 382439746496 ) / 23799739176 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 27 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( -2\beta_{11} - \beta_{10} + 4\beta_{8} + \beta_{5} - 26\beta _1 - 51 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( - 2 \beta_{11} + 7 \beta_{10} - 22 \beta_{9} - 8 \beta_{8} - 33 \beta_{7} + 5 \beta_{6} + 17 \beta_{5} + 40 \beta_{4} - 78 \beta_{3} - 42 \beta_{2} - 66 \beta _1 - 367 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( 34 \beta_{11} + 27 \beta_{10} + 2 \beta_{9} - 108 \beta_{8} - 17 \beta_{7} - 61 \beta_{6} - 35 \beta_{5} + 148 \beta_{4} - 442 \beta_{3} + 166 \beta_{2} + 370 \beta _1 + 909 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( 70\beta_{11} - 5\beta_{10} - 184\beta_{8} - 287\beta_{5} + 1086\beta _1 + 5417 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( 574 \beta_{11} + 591 \beta_{10} - 62 \beta_{9} - 2320 \beta_{8} + 265 \beta_{7} + 1165 \beta_{6} - 811 \beta_{5} - 3052 \beta_{4} + 7226 \beta_{3} - 422 \beta_{2} + 5542 \beta _1 + 15893 ) / 8 \) |
\(\nu^{8}\) | \(=\) | \( ( - 1622 \beta_{11} - 1219 \beta_{10} + 2298 \beta_{9} + 6864 \beta_{8} + 4385 \beta_{7} + 2841 \beta_{6} + 4919 \beta_{5} - 14484 \beta_{4} + 25698 \beta_{3} + 14250 \beta_{2} - 18158 \beta _1 - 83785 ) / 8 \) |
\(\nu^{9}\) | \(=\) | \( ( -9838\beta_{11} - 11771\beta_{10} + 45656\beta_{8} + 16391\beta_{5} - 86614\beta _1 - 277049 ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( - 32782 \beta_{11} - 35923 \beta_{10} - 21050 \beta_{9} + 160952 \beta_{8} - 52209 \beta_{7} - 68705 \beta_{6} + 85135 \beta_{5} + 263996 \beta_{4} - 456826 \beta_{3} - 258698 \beta_{2} + \cdots - 1344609 ) / 8 \) |
\(\nu^{11}\) | \(=\) | \( ( 170270 \beta_{11} + 223075 \beta_{10} + 32934 \beta_{9} - 858536 \beta_{8} - 63481 \beta_{7} - 393345 \beta_{6} - 311199 \beta_{5} + 1083996 \beta_{4} - 2055754 \beta_{3} - 660506 \beta_{2} + \cdots + 4827633 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).
\(n\) | \(69\) | \(239\) | \(241\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
239.1 |
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0 | − | 4.83337i | 0 | −8.57307 | 0 | − | 8.54360i | 0 | −14.3615 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
239.2 | 0 | − | 4.31458i | 0 | 4.73683 | 0 | − | 6.12094i | 0 | −9.61562 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
239.3 | 0 | − | 4.14258i | 0 | −3.83246 | 0 | 13.9064i | 0 | −8.16093 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.4 | 0 | − | 2.14999i | 0 | −1.41418 | 0 | 7.84997i | 0 | 4.37755 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.5 | 0 | − | 1.99259i | 0 | 6.36975 | 0 | − | 6.28119i | 0 | 5.02960 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
239.6 | 0 | − | 0.518786i | 0 | −3.28687 | 0 | − | 4.36794i | 0 | 8.73086 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
239.7 | 0 | 0.518786i | 0 | −3.28687 | 0 | 4.36794i | 0 | 8.73086 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.8 | 0 | 1.99259i | 0 | 6.36975 | 0 | 6.28119i | 0 | 5.02960 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.9 | 0 | 2.14999i | 0 | −1.41418 | 0 | − | 7.84997i | 0 | 4.37755 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.10 | 0 | 4.14258i | 0 | −3.83246 | 0 | − | 13.9064i | 0 | −8.16093 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.11 | 0 | 4.31458i | 0 | 4.73683 | 0 | 6.12094i | 0 | −9.61562 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
239.12 | 0 | 4.83337i | 0 | −8.57307 | 0 | 8.54360i | 0 | −14.3615 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 272.3.d.b | ✓ | 12 |
3.b | odd | 2 | 1 | 2448.3.k.f | 12 | ||
4.b | odd | 2 | 1 | inner | 272.3.d.b | ✓ | 12 |
8.b | even | 2 | 1 | 1088.3.d.b | 12 | ||
8.d | odd | 2 | 1 | 1088.3.d.b | 12 | ||
12.b | even | 2 | 1 | 2448.3.k.f | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
272.3.d.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
272.3.d.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
1088.3.d.b | 12 | 8.b | even | 2 | 1 | ||
1088.3.d.b | 12 | 8.d | odd | 2 | 1 | ||
2448.3.k.f | 12 | 3.b | odd | 2 | 1 | ||
2448.3.k.f | 12 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 68T_{3}^{10} + 1700T_{3}^{8} + 18928T_{3}^{6} + 90304T_{3}^{4} + 159936T_{3}^{2} + 36864 \)
acting on \(S_{3}^{\mathrm{new}}(272, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} + 68 T^{10} + 1700 T^{8} + \cdots + 36864 \)
$5$
\( (T^{6} + 6 T^{5} - 64 T^{4} - 336 T^{3} + \cdots + 4608)^{2} \)
$7$
\( T^{12} + 424 T^{10} + \cdots + 24531077376 \)
$11$
\( T^{12} + 1076 T^{10} + \cdots + 895794103296 \)
$13$
\( (T^{6} - 16 T^{5} - 488 T^{4} + \cdots - 377600)^{2} \)
$17$
\( (T^{2} - 17)^{6} \)
$19$
\( T^{12} + \cdots + 137783273324544 \)
$23$
\( T^{12} + 4840 T^{10} + \cdots + 28\!\cdots\!24 \)
$29$
\( (T^{6} + 46 T^{5} - 2080 T^{4} + \cdots + 380420352)^{2} \)
$31$
\( T^{12} + 4120 T^{10} + \cdots + 1647793264896 \)
$37$
\( (T^{6} + 58 T^{5} - 4424 T^{4} + \cdots + 1464176512)^{2} \)
$41$
\( (T^{6} - 16 T^{5} - 6748 T^{4} + \cdots - 617041728)^{2} \)
$43$
\( T^{12} + 10384 T^{10} + \cdots + 11\!\cdots\!64 \)
$47$
\( T^{12} + 13184 T^{10} + \cdots + 96\!\cdots\!04 \)
$53$
\( (T^{6} - 6988 T^{4} + \cdots + 4699279296)^{2} \)
$59$
\( T^{12} + 11792 T^{10} + \cdots + 14\!\cdots\!44 \)
$61$
\( (T^{6} - 38 T^{5} - 7544 T^{4} + \cdots - 4119920768)^{2} \)
$67$
\( T^{12} + 35440 T^{10} + \cdots + 47\!\cdots\!36 \)
$71$
\( T^{12} + 46520 T^{10} + \cdots + 11\!\cdots\!36 \)
$73$
\( (T^{6} - 92 T^{5} + \cdots - 285567998400)^{2} \)
$79$
\( T^{12} + 48904 T^{10} + \cdots + 78\!\cdots\!84 \)
$83$
\( T^{12} + 40208 T^{10} + \cdots + 24\!\cdots\!04 \)
$89$
\( (T^{6} - 36 T^{5} - 28336 T^{4} + \cdots - 48635889408)^{2} \)
$97$
\( (T^{6} - 72 T^{5} - 30268 T^{4} + \cdots + 48604036288)^{2} \)
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