Newspace parameters
Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 840.bz (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.70743376979\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.39955 | + | 0.203137i | 0.500000 | + | 0.866025i | 1.91747 | − | 0.568599i | −0.835606 | + | 2.07407i | −0.875696 | − | 1.11048i | −2.10664 | + | 1.60064i | −2.56809 | + | 1.18529i | −0.500000 | + | 0.866025i | 0.748152 | − | 3.07250i |
19.2 | −1.39446 | − | 0.235554i | 0.500000 | + | 0.866025i | 1.88903 | + | 0.656942i | −0.911079 | + | 2.04204i | −0.493233 | − | 1.32541i | 2.41279 | − | 1.08556i | −2.47943 | − | 1.36105i | −0.500000 | + | 0.866025i | 1.75147 | − | 2.63293i |
19.3 | −1.39442 | − | 0.235769i | 0.500000 | + | 0.866025i | 1.88883 | + | 0.657523i | 1.85007 | − | 1.25588i | −0.493029 | − | 1.32549i | 2.44325 | + | 1.01516i | −2.47880 | − | 1.36219i | −0.500000 | + | 0.866025i | −2.87588 | + | 1.31503i |
19.4 | −1.38940 | − | 0.263748i | 0.500000 | + | 0.866025i | 1.86087 | + | 0.732904i | 0.519269 | − | 2.17494i | −0.466288 | − | 1.33513i | −1.58536 | − | 2.11817i | −2.39220 | − | 1.50910i | −0.500000 | + | 0.866025i | −1.29511 | + | 2.88491i |
19.5 | −1.38671 | + | 0.277563i | 0.500000 | + | 0.866025i | 1.84592 | − | 0.769797i | −1.58066 | − | 1.58162i | −0.933730 | − | 1.06214i | 0.458574 | − | 2.60571i | −2.34608 | + | 1.57984i | −0.500000 | + | 0.866025i | 2.63091 | + | 1.75451i |
19.6 | −1.34731 | + | 0.429821i | 0.500000 | + | 0.866025i | 1.63051 | − | 1.15821i | 2.17016 | − | 0.538885i | −1.04589 | − | 0.951897i | −2.41852 | + | 1.07274i | −1.69898 | + | 2.26130i | −0.500000 | + | 0.866025i | −2.69226 | + | 1.65883i |
19.7 | −1.32928 | + | 0.482718i | 0.500000 | + | 0.866025i | 1.53397 | − | 1.28333i | 0.805895 | + | 2.08579i | −1.08269 | − | 0.909830i | −0.514752 | − | 2.59519i | −1.41958 | + | 2.44638i | −0.500000 | + | 0.866025i | −2.07811 | − | 2.38358i |
19.8 | −1.24841 | − | 0.664442i | 0.500000 | + | 0.866025i | 1.11703 | + | 1.65899i | 1.77230 | + | 1.36343i | −0.0487790 | − | 1.41337i | −1.63954 | − | 2.07652i | −0.292210 | − | 2.81329i | −0.500000 | + | 0.866025i | −1.30663 | − | 2.87971i |
19.9 | −1.19963 | − | 0.748930i | 0.500000 | + | 0.866025i | 0.878208 | + | 1.79687i | −1.77230 | − | 1.36343i | 0.0487790 | − | 1.41337i | 1.63954 | + | 2.07652i | 0.292210 | − | 2.81329i | −0.500000 | + | 0.866025i | 1.10498 | + | 2.96294i |
19.10 | −1.11818 | + | 0.865837i | 0.500000 | + | 0.866025i | 0.500651 | − | 1.93632i | −2.23603 | − | 0.0125931i | −1.30893 | − | 0.535453i | 0.842207 | + | 2.50812i | 1.11672 | + | 2.59864i | −0.500000 | + | 0.866025i | 2.51119 | − | 1.92196i |
19.11 | −1.09295 | + | 0.897480i | 0.500000 | + | 0.866025i | 0.389060 | − | 1.96179i | 1.75772 | + | 1.38219i | −1.32371 | − | 0.497779i | 2.38660 | + | 1.14199i | 1.33545 | + | 2.49331i | −0.500000 | + | 0.866025i | −3.16157 | + | 0.0668608i |
19.12 | −0.923113 | − | 1.07138i | 0.500000 | + | 0.866025i | −0.295723 | + | 1.97802i | −0.519269 | + | 2.17494i | 0.466288 | − | 1.33513i | 1.58536 | + | 2.11817i | 2.39220 | − | 1.50910i | −0.500000 | + | 0.866025i | 2.80954 | − | 1.45138i |
19.13 | −0.922863 | + | 1.07160i | 0.500000 | + | 0.866025i | −0.296648 | − | 1.97788i | −2.09232 | + | 0.788801i | −1.38946 | − | 0.263423i | 0.635767 | − | 2.56823i | 2.39326 | + | 1.50742i | −0.500000 | + | 0.866025i | 1.08564 | − | 2.97008i |
19.14 | −0.901393 | − | 1.08972i | 0.500000 | + | 0.866025i | −0.374981 | + | 1.96453i | −1.85007 | + | 1.25588i | 0.493029 | − | 1.32549i | −2.44325 | − | 1.01516i | 2.47880 | − | 1.36219i | −0.500000 | + | 0.866025i | 3.03620 | + | 0.884025i |
19.15 | −0.901225 | − | 1.08986i | 0.500000 | + | 0.866025i | −0.375586 | + | 1.96442i | 0.911079 | − | 2.04204i | 0.493233 | − | 1.32541i | −2.41279 | + | 1.08556i | 2.47943 | − | 1.36105i | −0.500000 | + | 0.866025i | −3.04663 | + | 0.847392i |
19.16 | −0.808138 | + | 1.16057i | 0.500000 | + | 0.866025i | −0.693826 | − | 1.87579i | −0.750153 | − | 2.10648i | −1.40915 | − | 0.119585i | −2.62822 | − | 0.304041i | 2.73769 | + | 0.710669i | −0.500000 | + | 0.866025i | 3.05094 | + | 0.831727i |
19.17 | −0.687198 | + | 1.23603i | 0.500000 | + | 0.866025i | −1.05552 | − | 1.69879i | 0.760135 | − | 2.10290i | −1.41403 | + | 0.0228815i | 2.38144 | − | 1.15271i | 2.82510 | − | 0.137241i | −0.500000 | + | 0.866025i | 2.07688 | + | 2.38466i |
19.18 | −0.523853 | − | 1.31361i | 0.500000 | + | 0.866025i | −1.45116 | + | 1.37628i | 0.835606 | − | 2.07407i | 0.875696 | − | 1.11048i | 2.10664 | − | 1.60064i | 2.56809 | + | 1.18529i | −0.500000 | + | 0.866025i | −3.16226 | − | 0.0111565i |
19.19 | −0.452977 | − | 1.33971i | 0.500000 | + | 0.866025i | −1.58962 | + | 1.21371i | 1.58066 | + | 1.58162i | 0.933730 | − | 1.06214i | −0.458574 | + | 2.60571i | 2.34608 | + | 1.57984i | −0.500000 | + | 0.866025i | 1.40290 | − | 2.83406i |
19.20 | −0.301421 | − | 1.38172i | 0.500000 | + | 0.866025i | −1.81829 | + | 0.832957i | −2.17016 | + | 0.538885i | 1.04589 | − | 0.951897i | 2.41852 | − | 1.07274i | 1.69898 | + | 2.26130i | −0.500000 | + | 0.866025i | 1.39872 | + | 2.83612i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
35.i | odd | 6 | 1 | inner |
280.ba | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 840.2.bz.b | yes | 96 |
5.b | even | 2 | 1 | 840.2.bz.a | ✓ | 96 | |
7.d | odd | 6 | 1 | 840.2.bz.a | ✓ | 96 | |
8.d | odd | 2 | 1 | inner | 840.2.bz.b | yes | 96 |
35.i | odd | 6 | 1 | inner | 840.2.bz.b | yes | 96 |
40.e | odd | 2 | 1 | 840.2.bz.a | ✓ | 96 | |
56.m | even | 6 | 1 | 840.2.bz.a | ✓ | 96 | |
280.ba | even | 6 | 1 | inner | 840.2.bz.b | yes | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.2.bz.a | ✓ | 96 | 5.b | even | 2 | 1 | |
840.2.bz.a | ✓ | 96 | 7.d | odd | 6 | 1 | |
840.2.bz.a | ✓ | 96 | 40.e | odd | 2 | 1 | |
840.2.bz.a | ✓ | 96 | 56.m | even | 6 | 1 | |
840.2.bz.b | yes | 96 | 1.a | even | 1 | 1 | trivial |
840.2.bz.b | yes | 96 | 8.d | odd | 2 | 1 | inner |
840.2.bz.b | yes | 96 | 35.i | odd | 6 | 1 | inner |
840.2.bz.b | yes | 96 | 280.ba | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17}^{48} + 216 T_{17}^{46} - 176 T_{17}^{45} + 26976 T_{17}^{44} - 34492 T_{17}^{43} + 2274984 T_{17}^{42} - 3838688 T_{17}^{41} + 143959340 T_{17}^{40} - 284664672 T_{17}^{39} + 7089264040 T_{17}^{38} + \cdots + 10\!\cdots\!24 \)
acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).