Newspace parameters
Level: | \( N \) | \(=\) | \( 6039 = 3^{2} \cdot 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6039.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(48.2216577807\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
Twist minimal: | no (minimal twist has level 671) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.68161 | 0 | 5.19104 | 3.56342 | 0 | 3.18744 | −8.55713 | 0 | −9.55570 | ||||||||||||||||||
1.2 | −2.67686 | 0 | 5.16560 | −1.61394 | 0 | −4.48909 | −8.47390 | 0 | 4.32029 | ||||||||||||||||||
1.3 | −2.54333 | 0 | 4.46852 | 0.268217 | 0 | 2.27426 | −6.27827 | 0 | −0.682163 | ||||||||||||||||||
1.4 | −2.16302 | 0 | 2.67865 | −3.42546 | 0 | 1.52570 | −1.46794 | 0 | 7.40934 | ||||||||||||||||||
1.5 | −2.08860 | 0 | 2.36225 | 0.0546248 | 0 | −2.53368 | −0.756597 | 0 | −0.114089 | ||||||||||||||||||
1.6 | −1.61525 | 0 | 0.609041 | −2.50050 | 0 | 4.06392 | 2.24675 | 0 | 4.03894 | ||||||||||||||||||
1.7 | −1.29392 | 0 | −0.325780 | 3.15298 | 0 | 5.03078 | 3.00937 | 0 | −4.07969 | ||||||||||||||||||
1.8 | −0.942064 | 0 | −1.11252 | −4.16220 | 0 | −0.914635 | 2.93219 | 0 | 3.92106 | ||||||||||||||||||
1.9 | −0.731918 | 0 | −1.46430 | −2.76447 | 0 | −1.34613 | 2.53558 | 0 | 2.02337 | ||||||||||||||||||
1.10 | −0.543169 | 0 | −1.70497 | 0.382346 | 0 | −4.84691 | 2.01242 | 0 | −0.207678 | ||||||||||||||||||
1.11 | −0.472572 | 0 | −1.77668 | 3.20957 | 0 | 4.06194 | 1.78475 | 0 | −1.51675 | ||||||||||||||||||
1.12 | 0.404310 | 0 | −1.83653 | −0.0975766 | 0 | 1.38056 | −1.55115 | 0 | −0.0394512 | ||||||||||||||||||
1.13 | 0.634118 | 0 | −1.59789 | 1.64555 | 0 | −3.41198 | −2.28149 | 0 | 1.04347 | ||||||||||||||||||
1.14 | 1.29453 | 0 | −0.324186 | −2.94198 | 0 | −0.962580 | −3.00873 | 0 | −3.80849 | ||||||||||||||||||
1.15 | 1.45442 | 0 | 0.115325 | −3.82594 | 0 | 3.26042 | −2.74110 | 0 | −5.56450 | ||||||||||||||||||
1.16 | 1.47389 | 0 | 0.172365 | −0.593969 | 0 | 3.31761 | −2.69374 | 0 | −0.875448 | ||||||||||||||||||
1.17 | 2.19302 | 0 | 2.80936 | 3.87665 | 0 | 3.34814 | 1.77494 | 0 | 8.50160 | ||||||||||||||||||
1.18 | 2.40042 | 0 | 3.76203 | 3.90458 | 0 | −3.05635 | 4.22962 | 0 | 9.37264 | ||||||||||||||||||
1.19 | 2.55546 | 0 | 4.53040 | −4.10969 | 0 | −4.64385 | 6.46634 | 0 | −10.5022 | ||||||||||||||||||
1.20 | 2.60374 | 0 | 4.77947 | −0.851432 | 0 | 3.04314 | 7.23702 | 0 | −2.21691 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(11\) | \(-1\) |
\(61\) | \(-1\) |
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 | 6039.2.a.l | 21 | |
3.b | odd | 2 | 1 | 671.2.a.d | ✓ | 21 | |
33.d | even | 2 | 1 | 7381.2.a.j | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.a.d | ✓ | 21 | 3.b | odd | 2 | 1 | |
6039.2.a.l | 21 | 1.a | even | 1 | 1 | trivial | |
7381.2.a.j | 21 | 33.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{21} - 37 T_{2}^{19} - 2 T_{2}^{18} + 582 T_{2}^{17} + 65 T_{2}^{16} - 5074 T_{2}^{15} - 876 T_{2}^{14} + 26808 T_{2}^{13} + 6340 T_{2}^{12} - 88228 T_{2}^{11} - 26710 T_{2}^{10} + 179236 T_{2}^{9} + 66498 T_{2}^{8} + \cdots + 2082 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).