Newspace parameters
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(64.1535279252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1595208.1 |
| Defining polynomial: | \(x^{4} - x^{3} - 20 x^{2} + 33 x - 3\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \(x^{4} - x^{3} - 20 x^{2} + 33 x - 3\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( -2 \nu^{3} + 40 \nu - 29 \) |
| \(\beta_{2}\) | \(=\) | \((\)\( 22 \nu^{3} + 40 \nu^{2} - 240 \nu - 141 \)\()/3\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -8 \nu^{3} + 40 \nu^{2} + 120 \nu - 516 \)\()/3\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(-\beta_{3} + \beta_{2} + 5 \beta_{1} + 20\)\()/80\) |
| \(\nu^{2}\) | \(=\) | \((\)\(5 \beta_{3} + \beta_{2} - 3 \beta_{1} + 820\)\()/80\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-\beta_{3} + \beta_{2} + 3 \beta_{1} - 38\)\()/4\) |
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 |
|
0 | −24.1383 | 0 | 0 | 0 | 179.876 | 0 | 339.657 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.49000 | 0 | 0 | 0 | −188.968 | 0 | −212.860 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.69449 | 0 | 0 | 0 | 10.2635 | 0 | −220.962 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 28.9338 | 0 | 0 | 0 | 146.828 | 0 | 594.165 | 0 | |||||||||||||||||||||||||||||||
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 | 400.6.a.ba | 4 | |
| 4.b | odd | 2 | 1 | 200.6.a.j | 4 | ||
| 5.b | even | 2 | 1 | 400.6.a.z | 4 | ||
| 5.c | odd | 4 | 2 | 80.6.c.d | 8 | ||
| 15.e | even | 4 | 2 | 720.6.f.n | 8 | ||
| 20.d | odd | 2 | 1 | 200.6.a.k | 4 | ||
| 20.e | even | 4 | 2 | 40.6.c.a | ✓ | 8 | |
| 40.i | odd | 4 | 2 | 320.6.c.i | 8 | ||
| 40.k | even | 4 | 2 | 320.6.c.j | 8 | ||
| 60.l | odd | 4 | 2 | 360.6.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.6.c.a | ✓ | 8 | 20.e | even | 4 | 2 | |
| 80.6.c.d | 8 | 5.c | odd | 4 | 2 | ||
| 200.6.a.j | 4 | 4.b | odd | 2 | 1 | ||
| 200.6.a.k | 4 | 20.d | odd | 2 | 1 | ||
| 320.6.c.i | 8 | 40.i | odd | 4 | 2 | ||
| 320.6.c.j | 8 | 40.k | even | 4 | 2 | ||
| 360.6.f.b | 8 | 60.l | odd | 4 | 2 | ||
| 400.6.a.z | 4 | 5.b | even | 2 | 1 | ||
| 400.6.a.ba | 4 | 1.a | even | 1 | 1 | trivial | |
| 720.6.f.n | 8 | 15.e | even | 4 | 2 | ||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4 T_{3}^{3} - 728 T_{3}^{2} - 432 T_{3} + 18000 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | 1 |
| $3$ | \( 1 - 4 T + 244 T^{2} - 3348 T^{3} + 18486 T^{4} - 813564 T^{5} + 14407956 T^{6} - 57395628 T^{7} + 3486784401 T^{8} \) |
| $5$ | 1 |
| $7$ | \( 1 - 148 T + 33316 T^{2} - 2108948 T^{3} + 503710694 T^{4} - 35445089036 T^{5} + 9410945395684 T^{6} - 702639103471564 T^{7} + 79792266297612001 T^{8} \) |
| $11$ | \( 1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 8239191359696 T^{5} + 4570277964394604 T^{6} - 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \) |
| $13$ | \( 1 + 440 T + 886036 T^{2} + 564452872 T^{3} + 392923998710 T^{4} + 209577400203496 T^{5} + 122147586683920564 T^{6} + 22521792926199933080 T^{7} + \)\(19\!\cdots\!01\)\( T^{8} \) |
| $17$ | \( 1 - 672 T + 3356228 T^{2} - 3540729312 T^{3} + 5520858237894 T^{4} - 5027329298748384 T^{5} + 6766135176516146372 T^{6} - \)\(19\!\cdots\!96\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \) |
| $19$ | \( 1 - 688 T + 5308396 T^{2} - 6058136368 T^{3} + 15069081422710 T^{4} - 15000545402668432 T^{5} + 32546127598645797196 T^{6} - \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \) |
| $23$ | \( 1 - 4492 T + 27426980 T^{2} - 79192657228 T^{3} + 264562508116390 T^{4} - 509711105000837204 T^{5} + \)\(11\!\cdots\!20\)\( T^{6} - \)\(11\!\cdots\!44\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \) |
| $29$ | \( 1 + 2936 T + 58625996 T^{2} + 77951973928 T^{3} + 1466411094282230 T^{4} + 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} + \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \) |
| $31$ | \( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} + \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \) |
| $37$ | \( 1 + 8792 T + 164536948 T^{2} + 653354012200 T^{3} + 11523689002604438 T^{4} + 45306152527774275400 T^{5} + \)\(79\!\cdots\!52\)\( T^{6} + \)\(29\!\cdots\!56\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \) |
| $41$ | \( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \) |
| $43$ | \( 1 - 48276 T + 1306891924 T^{2} - 24102642735684 T^{3} + 333606713965182294 T^{4} - \)\(35\!\cdots\!12\)\( T^{5} + \)\(28\!\cdots\!76\)\( T^{6} - \)\(15\!\cdots\!32\)\( T^{7} + \)\(46\!\cdots\!01\)\( T^{8} \) |
| $47$ | \( 1 - 14724 T + 590074244 T^{2} - 7755349482468 T^{3} + 177012391425566214 T^{4} - \)\(17\!\cdots\!76\)\( T^{5} + \)\(31\!\cdots\!56\)\( T^{6} - \)\(17\!\cdots\!32\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \) |
| $53$ | \( 1 + 84296 T + 4144600628 T^{2} + 135070817073080 T^{3} + 3234464962764166486 T^{4} + \)\(56\!\cdots\!40\)\( T^{5} + \)\(72\!\cdots\!72\)\( T^{6} + \)\(61\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \) |
| $59$ | \( 1 - 45840 T + 3064286732 T^{2} - 94721285480976 T^{3} + 3348109683185502486 T^{4} - \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \) |
| $61$ | \( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \) |
| $67$ | \( 1 - 72700 T + 7283604532 T^{2} - 314621051775212 T^{3} + 16093744308113184182 T^{4} - \)\(42\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \) |
| $71$ | \( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} - \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} \) |
| $73$ | \( 1 + 133072 T + 13424418340 T^{2} + 846379227250928 T^{3} + 45561026057566462310 T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(57\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \) |
| $79$ | \( 1 - 21632 T + 7152876604 T^{2} - 332616618908288 T^{3} + 24121899620797566790 T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} - \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \) |
| $83$ | \( 1 - 74660 T + 6167006708 T^{2} - 554766838355444 T^{3} + 42874466663031633142 T^{4} - \)\(21\!\cdots\!92\)\( T^{5} + \)\(95\!\cdots\!92\)\( T^{6} - \)\(45\!\cdots\!20\)\( T^{7} + \)\(24\!\cdots\!01\)\( T^{8} \) |
| $89$ | \( 1 - 20952 T + 16118164796 T^{2} - 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} - \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \) |
| $97$ | \( 1 + 59456 T + 24399465604 T^{2} + 684112850575552 T^{3} + \)\(25\!\cdots\!74\)\( T^{4} + \)\(58\!\cdots\!64\)\( T^{5} + \)\(17\!\cdots\!96\)\( T^{6} + \)\(37\!\cdots\!08\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \) |
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