Newspace parameters
Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 28.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.49074695476\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + 94817024 x^{9} + 1652635052 x^{8} - 4373672096 x^{7} + \cdots + 224266997486896 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{46} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + 94817024 x^{9} + 1652635052 x^{8} - 4373672096 x^{7} + \cdots + 224266997486896 \) :
\(\beta_{1}\) | \(=\) | \( ( 17\!\cdots\!41 \nu^{15} + \cdots + 46\!\cdots\!92 ) / 71\!\cdots\!16 \) |
\(\beta_{2}\) | \(=\) | \( ( - 93\!\cdots\!21 \nu^{15} + \cdots - 62\!\cdots\!76 ) / 35\!\cdots\!80 \) |
\(\beta_{3}\) | \(=\) | \( ( - 93\!\cdots\!21 \nu^{15} + \cdots - 58\!\cdots\!96 ) / 35\!\cdots\!80 \) |
\(\beta_{4}\) | \(=\) | \( ( - 32\!\cdots\!19 \nu^{15} + \cdots - 18\!\cdots\!04 ) / 17\!\cdots\!40 \) |
\(\beta_{5}\) | \(=\) | \( ( 22\!\cdots\!87 \nu^{15} + \cdots + 85\!\cdots\!28 ) / 71\!\cdots\!16 \) |
\(\beta_{6}\) | \(=\) | \( ( - 17\!\cdots\!27 \nu^{15} + \cdots - 55\!\cdots\!52 ) / 35\!\cdots\!80 \) |
\(\beta_{7}\) | \(=\) | \( ( - 17\!\cdots\!19 \nu^{15} + \cdots - 95\!\cdots\!04 ) / 35\!\cdots\!80 \) |
\(\beta_{8}\) | \(=\) | \( ( - 11\!\cdots\!11 \nu^{15} + \cdots - 25\!\cdots\!96 ) / 21\!\cdots\!80 \) |
\(\beta_{9}\) | \(=\) | \( ( - 20\!\cdots\!64 \nu^{15} + \cdots - 11\!\cdots\!68 ) / 37\!\cdots\!36 \) |
\(\beta_{10}\) | \(=\) | \( ( - 28\!\cdots\!37 \nu^{15} + \cdots - 82\!\cdots\!52 ) / 35\!\cdots\!80 \) |
\(\beta_{11}\) | \(=\) | \( ( 33\!\cdots\!53 \nu^{15} + \cdots + 35\!\cdots\!68 ) / 30\!\cdots\!40 \) |
\(\beta_{12}\) | \(=\) | \( ( 61\!\cdots\!75 \nu^{15} + \cdots + 86\!\cdots\!52 ) / 35\!\cdots\!08 \) |
\(\beta_{13}\) | \(=\) | \( ( - 63\!\cdots\!42 \nu^{15} + \cdots - 17\!\cdots\!28 ) / 21\!\cdots\!48 \) |
\(\beta_{14}\) | \(=\) | \( ( - 67\!\cdots\!07 \nu^{15} + \cdots - 51\!\cdots\!72 ) / 21\!\cdots\!80 \) |
\(\beta_{15}\) | \(=\) | \( ( 92\!\cdots\!09 \nu^{15} + \cdots + 63\!\cdots\!04 ) / 21\!\cdots\!80 \) |
\(\nu\) | \(=\) | \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( - \beta_{12} - \beta_{10} + 2 \beta_{7} - \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} - 4 \beta _1 + 331 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( - 5 \beta_{15} + 2 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} + 4 \beta_{11} - 14 \beta_{10} + 4 \beta_{9} + 7 \beta_{8} - 35 \beta_{7} + 48 \beta_{6} - 46 \beta_{5} + 26 \beta_{4} - 1008 \beta_{3} + 575 \beta_{2} + 56 \beta _1 - 473 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( 64 \beta_{15} - 16 \beta_{14} + 24 \beta_{13} - 389 \beta_{12} + 20 \beta_{11} - 197 \beta_{10} + 440 \beta_{9} - 60 \beta_{8} + 1060 \beta_{7} - 597 \beta_{6} + 2329 \beta_{5} - 944 \beta_{4} + 9383 \beta_{3} - 3100 \beta_{2} + \cdots + 94421 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( - 3276 \beta_{15} + 746 \beta_{14} - 3564 \beta_{13} + 461 \beta_{12} + 732 \beta_{11} - 16051 \beta_{10} - 6750 \beta_{9} + 478 \beta_{8} - 24426 \beta_{7} + 26825 \beta_{6} - 49645 \beta_{5} + \cdots - 1040817 ) / 16 \) |
\(\nu^{6}\) | \(=\) | \( ( 27408 \beta_{15} - 1286 \beta_{14} + 3516 \beta_{13} - 58644 \beta_{12} + 9842 \beta_{11} + 53762 \beta_{10} + 173322 \beta_{9} - 16200 \beta_{8} + 272964 \beta_{7} - 175832 \beta_{6} + \cdots + 16536506 ) / 8 \) |
\(\nu^{7}\) | \(=\) | \( ( - 994386 \beta_{15} + 97504 \beta_{14} - 466370 \beta_{13} - 173379 \beta_{12} - 143694 \beta_{11} - 6313423 \beta_{10} - 4132090 \beta_{9} + 48896 \beta_{8} - 7326348 \beta_{7} + \cdots - 325364495 ) / 16 \) |
\(\nu^{8}\) | \(=\) | \( ( 4458492 \beta_{15} + 125016 \beta_{14} - 387120 \beta_{13} - 2508560 \beta_{12} + 1968504 \beta_{11} + 20295682 \beta_{10} + 28558992 \beta_{9} - 1708260 \beta_{8} + \cdots + 1522190365 ) / 4 \) |
\(\nu^{9}\) | \(=\) | \( ( - 146781779 \beta_{15} + 47246 \beta_{14} + 2493160 \beta_{13} - 81824902 \beta_{12} - 58256618 \beta_{11} - 1086989293 \beta_{10} - 823255652 \beta_{9} + \cdots - 36505630337 ) / 8 \) |
\(\nu^{10}\) | \(=\) | \( ( 2607280116 \beta_{15} + 201519766 \beta_{14} - 826376492 \beta_{13} + 1073823469 \beta_{12} + 1422038670 \beta_{11} + 16817530523 \beta_{10} + 17618120222 \beta_{9} + \cdots + 518623160875 ) / 8 \) |
\(\nu^{11}\) | \(=\) | \( ( - 83490538352 \beta_{15} - 6391578346 \beta_{14} + 32231694568 \beta_{13} - 86420623275 \beta_{12} - 48369260454 \beta_{11} - 698198497995 \beta_{10} + \cdots - 12370219291491 ) / 16 \) |
\(\nu^{12}\) | \(=\) | \( ( 358178757498 \beta_{15} + 44814331584 \beta_{14} - 202874410600 \beta_{13} + 414987622535 \beta_{12} + 237626189708 \beta_{11} + 2877019856637 \beta_{10} + \cdots + 31729488258250 ) / 4 \) |
\(\nu^{13}\) | \(=\) | \( ( - 11339647088759 \beta_{15} - 1612797452186 \beta_{14} + 7911692216368 \beta_{13} - 17904084161658 \beta_{12} - 8276600843780 \beta_{11} + \cdots - 494319932192243 ) / 8 \) |
\(\nu^{14}\) | \(=\) | \( ( 187342485511976 \beta_{15} + 33290093313390 \beta_{14} - 157186917116908 \beta_{13} + 343507344114547 \beta_{12} + 148796550614158 \beta_{11} + \cdots - 31\!\cdots\!19 ) / 8 \) |
\(\nu^{15}\) | \(=\) | \( ( - 58\!\cdots\!12 \beta_{15} + \cdots + 39\!\cdots\!47 ) / 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).
\(n\) | \(15\) | \(17\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 |
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−4.36116 | − | 3.60282i | −28.2107 | 6.03935 | + | 31.4249i | − | 57.4402i | 123.031 | + | 101.638i | 15.0189 | + | 128.769i | 86.8799 | − | 158.808i | 552.846 | −206.947 | + | 250.506i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
27.2 | −4.36116 | − | 3.60282i | 28.2107 | 6.03935 | + | 31.4249i | 57.4402i | −123.031 | − | 101.638i | −15.0189 | + | 128.769i | 86.8799 | − | 158.808i | 552.846 | 206.947 | − | 250.506i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.3 | −4.36116 | + | 3.60282i | −28.2107 | 6.03935 | − | 31.4249i | 57.4402i | 123.031 | − | 101.638i | 15.0189 | − | 128.769i | 86.8799 | + | 158.808i | 552.846 | −206.947 | − | 250.506i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.4 | −4.36116 | + | 3.60282i | 28.2107 | 6.03935 | − | 31.4249i | − | 57.4402i | −123.031 | + | 101.638i | −15.0189 | − | 128.769i | 86.8799 | + | 158.808i | 552.846 | 206.947 | + | 250.506i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.5 | −1.70662 | − | 5.39328i | −6.44707 | −26.1749 | + | 18.4085i | 76.3023i | 11.0027 | + | 34.7708i | 120.438 | − | 47.9752i | 143.953 | + | 109.752i | −201.435 | 411.519 | − | 130.219i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.6 | −1.70662 | − | 5.39328i | 6.44707 | −26.1749 | + | 18.4085i | − | 76.3023i | −11.0027 | − | 34.7708i | −120.438 | − | 47.9752i | 143.953 | + | 109.752i | −201.435 | −411.519 | + | 130.219i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.7 | −1.70662 | + | 5.39328i | −6.44707 | −26.1749 | − | 18.4085i | − | 76.3023i | 11.0027 | − | 34.7708i | 120.438 | + | 47.9752i | 143.953 | − | 109.752i | −201.435 | 411.519 | + | 130.219i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.8 | −1.70662 | + | 5.39328i | 6.44707 | −26.1749 | − | 18.4085i | 76.3023i | −11.0027 | + | 34.7708i | −120.438 | + | 47.9752i | 143.953 | − | 109.752i | −201.435 | −411.519 | − | 130.219i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.9 | 2.70662 | − | 4.96731i | −17.0283 | −17.3484 | − | 26.8893i | 28.3847i | −46.0892 | + | 84.5850i | −117.282 | + | 55.2436i | −180.523 | + | 13.3961i | 46.9637 | 140.996 | + | 76.8266i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.10 | 2.70662 | − | 4.96731i | 17.0283 | −17.3484 | − | 26.8893i | − | 28.3847i | 46.0892 | − | 84.5850i | 117.282 | + | 55.2436i | −180.523 | + | 13.3961i | 46.9637 | −140.996 | − | 76.8266i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.11 | 2.70662 | + | 4.96731i | −17.0283 | −17.3484 | + | 26.8893i | − | 28.3847i | −46.0892 | − | 84.5850i | −117.282 | − | 55.2436i | −180.523 | − | 13.3961i | 46.9637 | 140.996 | − | 76.8266i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.12 | 2.70662 | + | 4.96731i | 17.0283 | −17.3484 | + | 26.8893i | 28.3847i | 46.0892 | + | 84.5850i | 117.282 | − | 55.2436i | −180.523 | − | 13.3961i | 46.9637 | −140.996 | + | 76.8266i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.13 | 5.36116 | − | 1.80500i | −15.7679 | 25.4840 | − | 19.3537i | − | 70.8301i | −84.5340 | + | 28.4610i | 78.9551 | − | 102.826i | 101.690 | − | 149.757i | 5.62572 | −127.848 | − | 379.731i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.14 | 5.36116 | − | 1.80500i | 15.7679 | 25.4840 | − | 19.3537i | 70.8301i | 84.5340 | − | 28.4610i | −78.9551 | − | 102.826i | 101.690 | − | 149.757i | 5.62572 | 127.848 | + | 379.731i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.15 | 5.36116 | + | 1.80500i | −15.7679 | 25.4840 | + | 19.3537i | 70.8301i | −84.5340 | − | 28.4610i | 78.9551 | + | 102.826i | 101.690 | + | 149.757i | 5.62572 | −127.848 | + | 379.731i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
27.16 | 5.36116 | + | 1.80500i | 15.7679 | 25.4840 | + | 19.3537i | − | 70.8301i | 84.5340 | + | 28.4610i | −78.9551 | + | 102.826i | 101.690 | + | 149.757i | 5.62572 | 127.848 | − | 379.731i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 28.6.d.b | ✓ | 16 |
3.b | odd | 2 | 1 | 252.6.b.d | 16 | ||
4.b | odd | 2 | 1 | inner | 28.6.d.b | ✓ | 16 |
7.b | odd | 2 | 1 | inner | 28.6.d.b | ✓ | 16 |
8.b | even | 2 | 1 | 448.6.f.d | 16 | ||
8.d | odd | 2 | 1 | 448.6.f.d | 16 | ||
12.b | even | 2 | 1 | 252.6.b.d | 16 | ||
21.c | even | 2 | 1 | 252.6.b.d | 16 | ||
28.d | even | 2 | 1 | inner | 28.6.d.b | ✓ | 16 |
56.e | even | 2 | 1 | 448.6.f.d | 16 | ||
56.h | odd | 2 | 1 | 448.6.f.d | 16 | ||
84.h | odd | 2 | 1 | 252.6.b.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
28.6.d.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
28.6.d.b | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
28.6.d.b | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
28.6.d.b | ✓ | 16 | 28.d | even | 2 | 1 | inner |
252.6.b.d | 16 | 3.b | odd | 2 | 1 | ||
252.6.b.d | 16 | 12.b | even | 2 | 1 | ||
252.6.b.d | 16 | 21.c | even | 2 | 1 | ||
252.6.b.d | 16 | 84.h | odd | 2 | 1 | ||
448.6.f.d | 16 | 8.b | even | 2 | 1 | ||
448.6.f.d | 16 | 8.d | odd | 2 | 1 | ||
448.6.f.d | 16 | 56.e | even | 2 | 1 | ||
448.6.f.d | 16 | 56.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 1376T_{3}^{6} + 556192T_{3}^{4} - 78187008T_{3}^{2} + 2384750592 \)
acting on \(S_{6}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} - 4 T^{7} + 20 T^{6} + \cdots + 1048576)^{2} \)
$3$
\( (T^{8} - 1376 T^{6} + \cdots + 2384750592)^{2} \)
$5$
\( (T^{8} + 14944 T^{6} + \cdots + 77644413167616)^{2} \)
$7$
\( T^{16} - 4424 T^{14} + \cdots + 63\!\cdots\!01 \)
$11$
\( (T^{8} + 289072 T^{6} + \cdots + 16\!\cdots\!48)^{2} \)
$13$
\( (T^{8} + 1669600 T^{6} + \cdots + 34\!\cdots\!24)^{2} \)
$17$
\( (T^{8} + 7386624 T^{6} + \cdots + 18\!\cdots\!36)^{2} \)
$19$
\( (T^{8} - 14532320 T^{6} + \cdots + 43\!\cdots\!08)^{2} \)
$23$
\( (T^{8} + 9935536 T^{6} + \cdots + 27\!\cdots\!08)^{2} \)
$29$
\( (T^{4} - 6648 T^{3} + \cdots - 47450817355248)^{4} \)
$31$
\( (T^{8} - 101160448 T^{6} + \cdots + 19\!\cdots\!72)^{2} \)
$37$
\( (T^{4} + 6568 T^{3} + \cdots - 39\!\cdots\!84)^{4} \)
$41$
\( (T^{8} + 721188736 T^{6} + \cdots + 18\!\cdots\!96)^{2} \)
$43$
\( (T^{8} + 313622576 T^{6} + \cdots + 22\!\cdots\!88)^{2} \)
$47$
\( (T^{8} - 633042432 T^{6} + \cdots + 51\!\cdots\!28)^{2} \)
$53$
\( (T^{4} + 10472 T^{3} + \cdots + 85\!\cdots\!76)^{4} \)
$59$
\( (T^{8} - 534584288 T^{6} + \cdots + 16\!\cdots\!08)^{2} \)
$61$
\( (T^{8} + 2196041824 T^{6} + \cdots + 75\!\cdots\!56)^{2} \)
$67$
\( (T^{8} + 9454705072 T^{6} + \cdots + 51\!\cdots\!92)^{2} \)
$71$
\( (T^{8} + 6241223408 T^{6} + \cdots + 50\!\cdots\!88)^{2} \)
$73$
\( (T^{8} + 7874786432 T^{6} + \cdots + 22\!\cdots\!04)^{2} \)
$79$
\( (T^{8} + 12613036912 T^{6} + \cdots + 63\!\cdots\!72)^{2} \)
$83$
\( (T^{8} - 11901939552 T^{6} + \cdots + 62\!\cdots\!92)^{2} \)
$89$
\( (T^{8} + 18068143744 T^{6} + \cdots + 12\!\cdots\!96)^{2} \)
$97$
\( (T^{8} + 35259705856 T^{6} + \cdots + 31\!\cdots\!84)^{2} \)
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