Newspace parameters
Level: | \( N \) | \(=\) | \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4004.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(31.9721009693\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2157.1 | 0 | −3.27485 | 0 | − | 2.17871i | 0 | 1.00000i | 0 | 7.72466 | 0 | |||||||||||||||||
2157.2 | 0 | −3.27485 | 0 | 2.17871i | 0 | − | 1.00000i | 0 | 7.72466 | 0 | |||||||||||||||||
2157.3 | 0 | −3.10315 | 0 | 1.71558i | 0 | 1.00000i | 0 | 6.62954 | 0 | ||||||||||||||||||
2157.4 | 0 | −3.10315 | 0 | − | 1.71558i | 0 | − | 1.00000i | 0 | 6.62954 | 0 | ||||||||||||||||
2157.5 | 0 | −2.75761 | 0 | 1.98236i | 0 | 1.00000i | 0 | 4.60441 | 0 | ||||||||||||||||||
2157.6 | 0 | −2.75761 | 0 | − | 1.98236i | 0 | − | 1.00000i | 0 | 4.60441 | 0 | ||||||||||||||||
2157.7 | 0 | −2.10949 | 0 | − | 3.39046i | 0 | 1.00000i | 0 | 1.44995 | 0 | |||||||||||||||||
2157.8 | 0 | −2.10949 | 0 | 3.39046i | 0 | − | 1.00000i | 0 | 1.44995 | 0 | |||||||||||||||||
2157.9 | 0 | −1.64159 | 0 | − | 4.02832i | 0 | − | 1.00000i | 0 | −0.305195 | 0 | ||||||||||||||||
2157.10 | 0 | −1.64159 | 0 | 4.02832i | 0 | 1.00000i | 0 | −0.305195 | 0 | ||||||||||||||||||
2157.11 | 0 | −1.08723 | 0 | 3.34996i | 0 | 1.00000i | 0 | −1.81792 | 0 | ||||||||||||||||||
2157.12 | 0 | −1.08723 | 0 | − | 3.34996i | 0 | − | 1.00000i | 0 | −1.81792 | 0 | ||||||||||||||||
2157.13 | 0 | −1.07570 | 0 | − | 3.26058i | 0 | 1.00000i | 0 | −1.84286 | 0 | |||||||||||||||||
2157.14 | 0 | −1.07570 | 0 | 3.26058i | 0 | − | 1.00000i | 0 | −1.84286 | 0 | |||||||||||||||||
2157.15 | 0 | −0.819032 | 0 | 1.69723i | 0 | 1.00000i | 0 | −2.32919 | 0 | ||||||||||||||||||
2157.16 | 0 | −0.819032 | 0 | − | 1.69723i | 0 | − | 1.00000i | 0 | −2.32919 | 0 | ||||||||||||||||
2157.17 | 0 | −0.722868 | 0 | 0.383791i | 0 | − | 1.00000i | 0 | −2.47746 | 0 | |||||||||||||||||
2157.18 | 0 | −0.722868 | 0 | − | 0.383791i | 0 | 1.00000i | 0 | −2.47746 | 0 | |||||||||||||||||
2157.19 | 0 | −0.189991 | 0 | − | 1.01340i | 0 | − | 1.00000i | 0 | −2.96390 | 0 | ||||||||||||||||
2157.20 | 0 | −0.189991 | 0 | 1.01340i | 0 | 1.00000i | 0 | −2.96390 | 0 | ||||||||||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4004.2.m.c | ✓ | 36 |
13.b | even | 2 | 1 | inner | 4004.2.m.c | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4004.2.m.c | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
4004.2.m.c | ✓ | 36 | 13.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{18} + 2 T_{3}^{17} - 35 T_{3}^{16} - 70 T_{3}^{15} + 474 T_{3}^{14} + 954 T_{3}^{13} - 3141 T_{3}^{12} - 6480 T_{3}^{11} + 10562 T_{3}^{10} + 23362 T_{3}^{9} - 16538 T_{3}^{8} - 44114 T_{3}^{7} + 7822 T_{3}^{6} + 40824 T_{3}^{5} + \cdots + 324 \)
acting on \(S_{2}^{\mathrm{new}}(4004, [\chi])\).