Newspace parameters
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.q (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(53.8889634572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
| Defining polynomial: | \(x^{8} - x^{7} + 98 x^{6} + 83 x^{5} + 9122 x^{4} - 91 x^{3} + 28567 x^{2} + 2058 x + 86436\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 21) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \(x^{8} - x^{7} + 98 x^{6} + 83 x^{5} + 9122 x^{4} - 91 x^{3} + 28567 x^{2} + 2058 x + 86436\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 141627885 \nu - 307508418 \)\()/ 9888988410 \) |
| \(\beta_{2}\) | \(=\) | \((\)\( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 39697581525 \nu - 10196496828 \)\()/ 4944494205 \) |
| \(\beta_{3}\) | \(=\) | \((\)\(65603 \nu^{7} - 25662916 \nu^{6} + 83553680 \nu^{5} - 2627557456 \nu^{4} + 3363632432 \nu^{3} - 228707310580 \nu^{2} + 464946681171 \nu - 714835153872\)\()/ 9888988410 \) |
| \(\beta_{4}\) | \(=\) | \((\)\( 731039 \nu^{7} - 7994302 \nu^{6} + 26027960 \nu^{5} - 628584958 \nu^{4} + 1047811304 \nu^{3} - 71245033510 \nu^{2} - 451136527737 \nu - 222679608984 \)\()/ 9888988410 \) |
| \(\beta_{5}\) | \(=\) | \((\)\( 1437521 \nu^{7} - 1642579 \nu^{6} + 136763060 \nu^{5} + 99854873 \nu^{4} + 13093975028 \nu^{3} - 2811264295 \nu^{2} + 41000921157 \nu + 2771341902 \)\()/ 4944494205 \) |
| \(\beta_{6}\) | \(=\) | \((\)\( 484709 \nu^{7} - 433105 \nu^{6} + 45997385 \nu^{5} + 45127907 \nu^{4} + 4285048727 \nu^{3} + 153033125 \nu^{2} + 442204518 \nu - 5171325642 \)\()/ 282542526 \) |
| \(\beta_{7}\) | \(=\) | \((\)\( 2434409 \nu^{7} - 2482177 \nu^{6} + 231017885 \nu^{5} + 226651007 \nu^{4} + 21466747139 \nu^{3} + 768595625 \nu^{2} + 2220933918 \nu + 102356221716 \)\()/ 1412712630 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{2} - 2 \beta_{1} + 2\)\()/8\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 98 \beta_{1} + 1\)\()/2\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-16 \beta_{6} + 95 \beta_{5} - 95 \beta_{2} - 542\)\()/8\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-101 \beta_{4} - 97 \beta_{3} - 22 \beta_{2} + 9050 \beta_{1} - 9050\)\()/2\) |
| \(\nu^{5}\) | \(=\) | \((\)\(-360 \beta_{7} + 1928 \beta_{6} - 9009 \beta_{5} - 1928 \beta_{4} - 360 \beta_{3} + 85442 \beta_{1} - 360\)\()/8\) |
| \(\nu^{6}\) | \(=\) | \((\)\(-9205 \beta_{7} + 9957 \beta_{6} - 4220 \beta_{5} + 4220 \beta_{2} + 860245\)\()/2\) |
| \(\nu^{7}\) | \(=\) | \((\)\(219312 \beta_{4} + 69024 \beta_{3} + 862207 \beta_{2} - 11352926 \beta_{1} + 11352926\)\()/8\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | 4.50000 | − | 7.79423i | 0 | −29.1836 | − | 50.5475i | 0 | 21.4366 | + | 127.857i | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||||||||||||||||||||
| 193.2 | 0 | 4.50000 | − | 7.79423i | 0 | −11.8764 | − | 20.5705i | 0 | −30.6840 | − | 125.958i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 4.50000 | − | 7.79423i | 0 | −11.0358 | − | 19.1146i | 0 | −126.882 | − | 26.6059i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 4.50000 | − | 7.79423i | 0 | 52.0958 | + | 90.2327i | 0 | 7.12980 | − | 129.446i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
| 289.1 | 0 | 4.50000 | + | 7.79423i | 0 | −29.1836 | + | 50.5475i | 0 | 21.4366 | − | 127.857i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
| 289.2 | 0 | 4.50000 | + | 7.79423i | 0 | −11.8764 | + | 20.5705i | 0 | −30.6840 | + | 125.958i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
| 289.3 | 0 | 4.50000 | + | 7.79423i | 0 | −11.0358 | + | 19.1146i | 0 | −126.882 | + | 26.6059i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
| 289.4 | 0 | 4.50000 | + | 7.79423i | 0 | 52.0958 | − | 90.2327i | 0 | 7.12980 | + | 129.446i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.6.q.j | 8 | |
| 4.b | odd | 2 | 1 | 21.6.e.c | ✓ | 8 | |
| 7.c | even | 3 | 1 | inner | 336.6.q.j | 8 | |
| 12.b | even | 2 | 1 | 63.6.e.e | 8 | ||
| 28.d | even | 2 | 1 | 147.6.e.o | 8 | ||
| 28.f | even | 6 | 1 | 147.6.a.l | 4 | ||
| 28.f | even | 6 | 1 | 147.6.e.o | 8 | ||
| 28.g | odd | 6 | 1 | 21.6.e.c | ✓ | 8 | |
| 28.g | odd | 6 | 1 | 147.6.a.m | 4 | ||
| 84.j | odd | 6 | 1 | 441.6.a.v | 4 | ||
| 84.n | even | 6 | 1 | 63.6.e.e | 8 | ||
| 84.n | even | 6 | 1 | 441.6.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.6.e.c | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 21.6.e.c | ✓ | 8 | 28.g | odd | 6 | 1 | |
| 63.6.e.e | 8 | 12.b | even | 2 | 1 | ||
| 63.6.e.e | 8 | 84.n | even | 6 | 1 | ||
| 147.6.a.l | 4 | 28.f | even | 6 | 1 | ||
| 147.6.a.m | 4 | 28.g | odd | 6 | 1 | ||
| 147.6.e.o | 8 | 28.d | even | 2 | 1 | ||
| 147.6.e.o | 8 | 28.f | even | 6 | 1 | ||
| 336.6.q.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 336.6.q.j | 8 | 7.c | even | 3 | 1 | inner | |
| 441.6.a.v | 4 | 84.j | odd | 6 | 1 | ||
| 441.6.a.w | 4 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 7657 T_{5}^{6} + 605400 T_{5}^{5} + 61817893 T_{5}^{4} + 2317773900 T_{5}^{3} + 67214905692 T_{5}^{2} + 965081458800 T_{5} + \)\(10\!\cdots\!36\)\( \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( T^{8} \) |
| $3$ | \( ( 81 - 9 T + T^{2} )^{4} \) |
| $5$ | \( 10164899803536 + 965081458800 T + 67214905692 T^{2} + 2317773900 T^{3} + 61817893 T^{4} + 605400 T^{5} + 7657 T^{6} + T^{8} \) |
| $7$ | \( 79792266297612001 + 1224870869565294 T + 18476141086592 T^{2} + 206920120008 T^{3} + 1643194665 T^{4} + 12311544 T^{5} + 65408 T^{6} + 258 T^{7} + T^{8} \) |
| $11$ | \( 2829568482592751376 - 169576701600014928 T + 9761813367494172 T^{2} - 25381944601332 T^{3} + 99024901837 T^{4} - 105799218 T^{5} + 399967 T^{6} - 402 T^{7} + T^{8} \) |
| $13$ | \( ( 149501563456 + 515112852 T - 1148423 T^{2} - 462 T^{3} + T^{4} )^{2} \) |
| $17$ | \( \)\(25\!\cdots\!24\)\( + 44837110146970681344 T + 720903316973027328 T^{2} + 1454766301642752 T^{3} + 2840324474880 T^{4} + 1349604864 T^{5} + 1670928 T^{6} + 276 T^{7} + T^{8} \) |
| $19$ | \( \)\(54\!\cdots\!76\)\( + \)\(58\!\cdots\!80\)\( T + 44221112569626422096 T^{2} + 2834297273515140 T^{3} + 27788839301865 T^{4} + 1411756650 T^{5} + 6156971 T^{6} - 510 T^{7} + T^{8} \) |
| $23$ | \( \)\(90\!\cdots\!76\)\( + \)\(51\!\cdots\!20\)\( T + \)\(27\!\cdots\!44\)\( T^{2} + 163456653366558720 T^{3} + 165861153884160 T^{4} - 83386679040 T^{5} + 40488144 T^{6} - 6900 T^{7} + T^{8} \) |
| $29$ | \( ( 408027025117872 + 62527747272 T - 52650397 T^{2} - 540 T^{3} + T^{4} )^{2} \) |
| $31$ | \( \)\(75\!\cdots\!09\)\( - \)\(50\!\cdots\!98\)\( T + \)\(49\!\cdots\!52\)\( T^{2} - 73339378003949028 T^{3} + 602809801419817 T^{4} + 3691164188 T^{5} + 58801968 T^{6} + 6410 T^{7} + T^{8} \) |
| $37$ | \( \)\(25\!\cdots\!36\)\( + \)\(75\!\cdots\!80\)\( T + \)\(20\!\cdots\!08\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + 30068709801011305 T^{4} + 2279577843610 T^{5} + 279418707 T^{6} + 15250 T^{7} + T^{8} \) |
| $41$ | \( ( -1856858915261952 - 1101575496480 T - 192741244 T^{2} - 4308 T^{3} + T^{4} )^{2} \) |
| $43$ | \( ( -991662745581932 + 199921376588 T + 199961493 T^{2} + 29198 T^{3} + T^{4} )^{2} \) |
| $47$ | \( \)\(73\!\cdots\!36\)\( + \)\(12\!\cdots\!48\)\( T + \)\(15\!\cdots\!28\)\( T^{2} + 5876954381324343168 T^{3} + 3512848312989072 T^{4} + 852661119456 T^{5} + 176153856 T^{6} + 15060 T^{7} + T^{8} \) |
| $53$ | \( \)\(72\!\cdots\!84\)\( + \)\(68\!\cdots\!40\)\( T + \)\(60\!\cdots\!12\)\( T^{2} + \)\(42\!\cdots\!28\)\( T^{3} + 366485627229994953 T^{4} + 9260410268772 T^{5} + 685378293 T^{6} + 13692 T^{7} + T^{8} \) |
| $59$ | \( \)\(66\!\cdots\!96\)\( - \)\(11\!\cdots\!60\)\( T + \)\(69\!\cdots\!76\)\( T^{2} - 92739830872274379000 T^{3} + 54097578476747217 T^{4} - 7480212648750 T^{5} + 1024872459 T^{6} - 34830 T^{7} + T^{8} \) |
| $61$ | \( \)\(32\!\cdots\!76\)\( + \)\(60\!\cdots\!60\)\( T + \)\(20\!\cdots\!24\)\( T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + 283845325040842512 T^{4} - 3902598174816 T^{5} + 561368672 T^{6} - 5364 T^{7} + T^{8} \) |
| $67$ | \( \)\(30\!\cdots\!56\)\( + \)\(27\!\cdots\!00\)\( T + \)\(15\!\cdots\!44\)\( T^{2} - \)\(20\!\cdots\!92\)\( T^{3} + 7519978747196869197 T^{4} - 26942404854654 T^{5} + 2881922927 T^{6} + 5994 T^{7} + T^{8} \) |
| $71$ | \( ( 21932335650275568 - 16377596837712 T + 1521744768 T^{2} + 89268 T^{3} + T^{4} )^{2} \) |
| $73$ | \( \)\(14\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T + \)\(45\!\cdots\!00\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + 1466139740779982221 T^{4} + 62299772588518 T^{5} + 3193750863 T^{6} + 59638 T^{7} + T^{8} \) |
| $79$ | \( \)\(27\!\cdots\!81\)\( + \)\(24\!\cdots\!06\)\( T + \)\(84\!\cdots\!28\)\( T^{2} - \)\(70\!\cdots\!88\)\( T^{3} + 13482185316532207897 T^{4} - 196348017119564 T^{5} + 5723264752 T^{6} + 44062 T^{7} + T^{8} \) |
| $83$ | \( ( -41533908097096407132 + 738604511000820 T + 7607249829 T^{2} - 208446 T^{3} + T^{4} )^{2} \) |
| $89$ | \( \)\(22\!\cdots\!24\)\( - \)\(88\!\cdots\!72\)\( T + \)\(26\!\cdots\!72\)\( T^{2} - \)\(38\!\cdots\!52\)\( T^{3} + \)\(47\!\cdots\!12\)\( T^{4} - 2429594520301248 T^{5} + 22815563908 T^{6} - 77520 T^{7} + T^{8} \) |
| $97$ | \( ( -11638556269792123644 - 99054118022220 T + 9271508101 T^{2} + 188630 T^{3} + T^{4} )^{2} \) |
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