Newspace parameters
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.bj (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.01223013094\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
103.1 | −1.41404 | − | 0.0221477i | 0.799650 | + | 1.53641i | 1.99902 | + | 0.0626353i | −2.43207 | + | 1.40416i | −1.09671 | − | 2.19026i | −2.50476 | − | 0.852161i | −2.82531 | − | 0.132842i | −1.72112 | + | 2.45718i | 3.47014 | − | 1.93167i |
103.2 | −1.41404 | + | 0.0221477i | −0.799650 | − | 1.53641i | 1.99902 | − | 0.0626353i | −2.43207 | + | 1.40416i | 1.16477 | + | 2.15484i | 2.50476 | + | 0.852161i | −2.82531 | + | 0.132842i | −1.72112 | + | 2.45718i | 3.40794 | − | 2.03940i |
103.3 | −1.40719 | − | 0.140755i | 1.46735 | − | 0.920267i | 1.96038 | + | 0.396138i | 0.259273 | − | 0.149691i | −2.19437 | + | 1.08846i | 0.0899789 | − | 2.64422i | −2.70287 | − | 0.833374i | 1.30622 | − | 2.70070i | −0.385916 | + | 0.174150i |
103.4 | −1.40719 | + | 0.140755i | −1.46735 | + | 0.920267i | 1.96038 | − | 0.396138i | 0.259273 | − | 0.149691i | 1.93531 | − | 1.50153i | −0.0899789 | + | 2.64422i | −2.70287 | + | 0.833374i | 1.30622 | − | 2.70070i | −0.343777 | + | 0.247138i |
103.5 | −1.33100 | − | 0.477940i | 1.43852 | + | 0.964710i | 1.54315 | + | 1.27228i | 3.01564 | − | 1.74108i | −1.45360 | − | 1.97156i | 1.90425 | + | 1.83680i | −1.44586 | − | 2.43094i | 1.13867 | + | 2.77551i | −4.84596 | + | 0.876089i |
103.6 | −1.33100 | + | 0.477940i | −1.43852 | − | 0.964710i | 1.54315 | − | 1.27228i | 3.01564 | − | 1.74108i | 2.37575 | + | 0.596507i | −1.90425 | − | 1.83680i | −1.44586 | + | 2.43094i | 1.13867 | + | 2.77551i | −3.18169 | + | 3.75868i |
103.7 | −1.16385 | − | 0.803396i | −0.477968 | − | 1.66480i | 0.709111 | + | 1.87007i | 0.121150 | − | 0.0699460i | −0.781205 | + | 2.32158i | −2.50334 | + | 0.856310i | 0.677105 | − | 2.74618i | −2.54309 | + | 1.59144i | −0.197195 | − | 0.0159245i |
103.8 | −1.16385 | + | 0.803396i | 0.477968 | + | 1.66480i | 0.709111 | − | 1.87007i | 0.121150 | − | 0.0699460i | −1.89378 | − | 1.55358i | 2.50334 | − | 0.856310i | 0.677105 | + | 2.74618i | −2.54309 | + | 1.59144i | −0.0848066 | + | 0.178738i |
103.9 | −1.15944 | − | 0.809751i | −0.630196 | + | 1.61334i | 0.688607 | + | 1.87772i | 2.22429 | − | 1.28420i | 2.03708 | − | 1.36027i | −2.01495 | − | 1.71464i | 0.722083 | − | 2.73470i | −2.20571 | − | 2.03344i | −3.61882 | − | 0.312174i |
103.10 | −1.15944 | + | 0.809751i | 0.630196 | − | 1.61334i | 0.688607 | − | 1.87772i | 2.22429 | − | 1.28420i | 0.575725 | + | 2.38087i | 2.01495 | + | 1.71464i | 0.722083 | + | 2.73470i | −2.20571 | − | 2.03344i | −1.53906 | + | 3.29008i |
103.11 | −1.05304 | − | 0.943979i | 1.67837 | − | 0.427864i | 0.217806 | + | 1.98810i | −2.49760 | + | 1.44199i | −2.17130 | − | 1.13379i | 0.0846904 | + | 2.64440i | 1.64737 | − | 2.29917i | 2.63387 | − | 1.43623i | 3.99129 | + | 0.839202i |
103.12 | −1.05304 | + | 0.943979i | −1.67837 | + | 0.427864i | 0.217806 | − | 1.98810i | −2.49760 | + | 1.44199i | 1.36351 | − | 2.03491i | −0.0846904 | − | 2.64440i | 1.64737 | + | 2.29917i | 2.63387 | − | 1.43623i | 1.26887 | − | 3.87616i |
103.13 | −0.615190 | − | 1.27340i | −1.73108 | − | 0.0580485i | −1.24308 | + | 1.56676i | 1.66094 | − | 0.958945i | 0.991022 | + | 2.24006i | 0.708998 | + | 2.54898i | 2.75984 | + | 0.619084i | 2.99326 | + | 0.200973i | −2.24291 | − | 1.52511i |
103.14 | −0.615190 | + | 1.27340i | 1.73108 | + | 0.0580485i | −1.24308 | − | 1.56676i | 1.66094 | − | 0.958945i | −1.13886 | + | 2.16864i | −0.708998 | − | 2.54898i | 2.75984 | − | 0.619084i | 2.99326 | + | 0.200973i | 0.199325 | + | 2.70497i |
103.15 | −0.412858 | − | 1.35261i | 1.24372 | + | 1.20547i | −1.65910 | + | 1.11687i | 0.514884 | − | 0.297269i | 1.11705 | − | 2.17995i | 1.74201 | − | 1.99133i | 2.19566 | + | 1.78300i | 0.0936760 | + | 2.99854i | −0.614662 | − | 0.573707i |
103.16 | −0.412858 | + | 1.35261i | −1.24372 | − | 1.20547i | −1.65910 | − | 1.11687i | 0.514884 | − | 0.297269i | 2.14401 | − | 1.18458i | −1.74201 | + | 1.99133i | 2.19566 | − | 1.78300i | 0.0936760 | + | 2.99854i | 0.189514 | + | 0.819166i |
103.17 | −0.379891 | − | 1.36223i | −1.39755 | − | 1.02316i | −1.71137 | + | 1.03500i | −2.34739 | + | 1.35527i | −0.862862 | + | 2.29248i | −1.09995 | − | 2.40626i | 2.06005 | + | 1.93809i | 0.906301 | + | 2.85983i | 2.73794 | + | 2.68284i |
103.18 | −0.379891 | + | 1.36223i | 1.39755 | + | 1.02316i | −1.71137 | − | 1.03500i | −2.34739 | + | 1.35527i | −1.92470 | + | 1.51511i | 1.09995 | + | 2.40626i | 2.06005 | − | 1.93809i | 0.906301 | + | 2.85983i | −0.954438 | − | 3.71255i |
103.19 | −0.171292 | − | 1.40380i | 1.50942 | − | 0.849495i | −1.94132 | + | 0.480920i | 3.50837 | − | 2.02556i | −1.45107 | − | 1.97342i | −2.35939 | + | 1.19719i | 1.00765 | + | 2.64285i | 1.55672 | − | 2.56450i | −3.44444 | − | 4.57810i |
103.20 | −0.171292 | + | 1.40380i | −1.50942 | + | 0.849495i | −1.94132 | − | 0.480920i | 3.50837 | − | 2.02556i | −0.933971 | − | 2.26444i | 2.35939 | − | 1.19719i | 1.00765 | − | 2.64285i | 1.55672 | − | 2.56450i | 2.24253 | + | 5.27202i |
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
63.t | odd | 6 | 1 | inner |
252.bj | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.2.bj.b | yes | 84 |
3.b | odd | 2 | 1 | 756.2.bj.b | 84 | ||
4.b | odd | 2 | 1 | inner | 252.2.bj.b | yes | 84 |
7.d | odd | 6 | 1 | 252.2.n.b | ✓ | 84 | |
9.c | even | 3 | 1 | 252.2.n.b | ✓ | 84 | |
9.d | odd | 6 | 1 | 756.2.n.b | 84 | ||
12.b | even | 2 | 1 | 756.2.bj.b | 84 | ||
21.g | even | 6 | 1 | 756.2.n.b | 84 | ||
28.f | even | 6 | 1 | 252.2.n.b | ✓ | 84 | |
36.f | odd | 6 | 1 | 252.2.n.b | ✓ | 84 | |
36.h | even | 6 | 1 | 756.2.n.b | 84 | ||
63.i | even | 6 | 1 | 756.2.bj.b | 84 | ||
63.t | odd | 6 | 1 | inner | 252.2.bj.b | yes | 84 |
84.j | odd | 6 | 1 | 756.2.n.b | 84 | ||
252.r | odd | 6 | 1 | 756.2.bj.b | 84 | ||
252.bj | even | 6 | 1 | inner | 252.2.bj.b | yes | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.n.b | ✓ | 84 | 7.d | odd | 6 | 1 | |
252.2.n.b | ✓ | 84 | 9.c | even | 3 | 1 | |
252.2.n.b | ✓ | 84 | 28.f | even | 6 | 1 | |
252.2.n.b | ✓ | 84 | 36.f | odd | 6 | 1 | |
252.2.bj.b | yes | 84 | 1.a | even | 1 | 1 | trivial |
252.2.bj.b | yes | 84 | 4.b | odd | 2 | 1 | inner |
252.2.bj.b | yes | 84 | 63.t | odd | 6 | 1 | inner |
252.2.bj.b | yes | 84 | 252.bj | even | 6 | 1 | inner |
756.2.n.b | 84 | 9.d | odd | 6 | 1 | ||
756.2.n.b | 84 | 21.g | even | 6 | 1 | ||
756.2.n.b | 84 | 36.h | even | 6 | 1 | ||
756.2.n.b | 84 | 84.j | odd | 6 | 1 | ||
756.2.bj.b | 84 | 3.b | odd | 2 | 1 | ||
756.2.bj.b | 84 | 12.b | even | 2 | 1 | ||
756.2.bj.b | 84 | 63.i | even | 6 | 1 | ||
756.2.bj.b | 84 | 252.r | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{42} - 3 T_{5}^{41} - 52 T_{5}^{40} + 165 T_{5}^{39} + 1632 T_{5}^{38} - 5289 T_{5}^{37} - 33705 T_{5}^{36} + 112518 T_{5}^{35} + 518868 T_{5}^{34} - 1777332 T_{5}^{33} - 6029808 T_{5}^{32} + 21439284 T_{5}^{31} + \cdots + 2187 \)
acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).