Newspace parameters
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.j (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
115.1 | −1.99592 | − | 0.127650i | −1.83110 | − | 1.83110i | 3.96741 | + | 0.509561i | −1.67465 | + | 1.67465i | 3.42099 | + | 3.88848i | −2.41883 | − | 2.41883i | −7.85360 | − | 1.52349i | − | 2.29414i | 3.55625 | − | 3.12871i | |
115.2 | −1.95592 | + | 0.417606i | 0.749243 | + | 0.749243i | 3.65121 | − | 1.63360i | 3.60984 | − | 3.60984i | −1.77834 | − | 1.15257i | −3.85896 | − | 3.85896i | −6.45926 | + | 4.71996i | − | 7.87727i | −5.55305 | + | 8.56803i | |
115.3 | −1.93784 | − | 0.494748i | 2.46525 | + | 2.46525i | 3.51045 | + | 1.91749i | −2.34604 | + | 2.34604i | −3.55759 | − | 5.99695i | 6.17762 | + | 6.17762i | −5.85402 | − | 5.45257i | 3.15495i | 5.70695 | − | 3.38555i | ||
115.4 | −1.88068 | − | 0.680460i | −2.79799 | − | 2.79799i | 3.07395 | + | 2.55946i | 5.84946 | − | 5.84946i | 3.35822 | + | 7.16606i | 9.27837 | + | 9.27837i | −4.03952 | − | 6.90524i | 6.65749i | −14.9813 | + | 7.02067i | ||
115.5 | −1.76140 | + | 0.947342i | −1.95177 | − | 1.95177i | 2.20509 | − | 3.33730i | −5.56104 | + | 5.56104i | 5.28684 | + | 1.58886i | 4.14904 | + | 4.14904i | −0.722483 | + | 7.96731i | − | 1.38120i | 4.52703 | − | 15.0634i | |
115.6 | −1.67184 | + | 1.09770i | 2.87011 | + | 2.87011i | 1.59010 | − | 3.67037i | −0.478405 | + | 0.478405i | −7.94889 | − | 1.64783i | 5.13465 | + | 5.13465i | 1.37059 | + | 7.88172i | 7.47505i | 0.274669 | − | 1.32496i | ||
115.7 | −1.64296 | − | 1.14047i | 3.48624 | + | 3.48624i | 1.39866 | + | 3.74750i | 5.68473 | − | 5.68473i | −1.75181 | − | 9.70371i | −3.01446 | − | 3.01446i | 1.97596 | − | 7.75213i | 15.3077i | −15.8231 | + | 2.85655i | ||
115.8 | −1.55587 | − | 1.25669i | 0.0765415 | + | 0.0765415i | 0.841444 | + | 3.91050i | −1.52277 | + | 1.52277i | −0.0228992 | − | 0.215278i | −3.30548 | − | 3.30548i | 3.60512 | − | 7.14165i | − | 8.98828i | 4.28288 | − | 0.455573i | |
115.9 | −1.54442 | + | 1.27074i | −4.05827 | − | 4.05827i | 0.770451 | − | 3.92510i | 3.34858 | − | 3.34858i | 11.4247 | + | 1.11067i | −7.12993 | − | 7.12993i | 3.79787 | + | 7.04103i | 23.9391i | −0.916437 | + | 9.42676i | ||
115.10 | −1.17987 | − | 1.61490i | −3.41961 | − | 3.41961i | −1.21583 | + | 3.81074i | −2.66341 | + | 2.66341i | −1.48766 | + | 9.55703i | −1.35907 | − | 1.35907i | 7.58850 | − | 2.53271i | 14.3875i | 7.44363 | + | 1.15869i | ||
115.11 | −1.10596 | + | 1.66639i | 1.14630 | + | 1.14630i | −1.55370 | − | 3.68592i | −4.78622 | + | 4.78622i | −3.17793 | + | 0.642415i | −7.03698 | − | 7.03698i | 7.86051 | + | 1.48742i | − | 6.37201i | −2.68233 | − | 13.2691i | |
115.12 | −0.860518 | + | 1.80541i | 0.887294 | + | 0.887294i | −2.51902 | − | 3.10718i | 5.25581 | − | 5.25581i | −2.36546 | + | 0.838398i | −0.502888 | − | 0.502888i | 7.77739 | − | 1.87408i | − | 7.42542i | 4.96617 | + | 14.0116i | |
115.13 | −0.775308 | + | 1.84361i | −2.20075 | − | 2.20075i | −2.79779 | − | 2.85873i | −0.154410 | + | 0.154410i | 5.76358 | − | 2.35106i | 6.51226 | + | 6.51226i | 7.43954 | − | 2.94164i | 0.686594i | −0.164957 | − | 0.404387i | ||
115.14 | −0.678905 | − | 1.88125i | 3.22659 | + | 3.22659i | −3.07817 | + | 2.55438i | −4.60664 | + | 4.60664i | 3.87946 | − | 8.26055i | −3.51075 | − | 3.51075i | 6.89520 | + | 4.05662i | 11.8217i | 11.7937 | + | 5.53875i | ||
115.15 | −0.603225 | − | 1.90686i | 0.735236 | + | 0.735236i | −3.27224 | + | 2.30053i | 0.444064 | − | 0.444064i | 0.958480 | − | 1.84551i | 8.56417 | + | 8.56417i | 6.36070 | + | 4.85196i | − | 7.91886i | −1.11464 | − | 0.578897i | |
115.16 | −0.462601 | − | 1.94576i | −1.38331 | − | 1.38331i | −3.57200 | + | 1.80023i | 5.77712 | − | 5.77712i | −2.05167 | + | 3.33151i | −3.98488 | − | 3.98488i | 5.15523 | + | 6.11748i | − | 5.17291i | −13.9134 | − | 8.56842i | |
115.17 | 0.462601 | − | 1.94576i | −1.38331 | − | 1.38331i | −3.57200 | − | 1.80023i | −5.77712 | + | 5.77712i | −3.33151 | + | 2.05167i | 3.98488 | + | 3.98488i | −5.15523 | + | 6.11748i | − | 5.17291i | 8.56842 | + | 13.9134i | |
115.18 | 0.603225 | − | 1.90686i | 0.735236 | + | 0.735236i | −3.27224 | − | 2.30053i | −0.444064 | + | 0.444064i | 1.84551 | − | 0.958480i | −8.56417 | − | 8.56417i | −6.36070 | + | 4.85196i | − | 7.91886i | 0.578897 | + | 1.11464i | |
115.19 | 0.678905 | − | 1.88125i | 3.22659 | + | 3.22659i | −3.07817 | − | 2.55438i | 4.60664 | − | 4.60664i | 8.26055 | − | 3.87946i | 3.51075 | + | 3.51075i | −6.89520 | + | 4.05662i | 11.8217i | −5.53875 | − | 11.7937i | ||
115.20 | 0.775308 | + | 1.84361i | −2.20075 | − | 2.20075i | −2.79779 | + | 2.85873i | 0.154410 | − | 0.154410i | 2.35106 | − | 5.76358i | −6.51226 | − | 6.51226i | −7.43954 | − | 2.94164i | 0.686594i | 0.404387 | + | 0.164957i | ||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
17.c | even | 4 | 1 | inner |
136.j | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.j.b | ✓ | 64 |
4.b | odd | 2 | 1 | 544.3.n.b | 64 | ||
8.b | even | 2 | 1 | 544.3.n.b | 64 | ||
8.d | odd | 2 | 1 | inner | 136.3.j.b | ✓ | 64 |
17.c | even | 4 | 1 | inner | 136.3.j.b | ✓ | 64 |
68.f | odd | 4 | 1 | 544.3.n.b | 64 | ||
136.i | even | 4 | 1 | 544.3.n.b | 64 | ||
136.j | odd | 4 | 1 | inner | 136.3.j.b | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.j.b | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
136.3.j.b | ✓ | 64 | 8.d | odd | 2 | 1 | inner |
136.3.j.b | ✓ | 64 | 17.c | even | 4 | 1 | inner |
136.3.j.b | ✓ | 64 | 136.j | odd | 4 | 1 | inner |
544.3.n.b | 64 | 4.b | odd | 2 | 1 | ||
544.3.n.b | 64 | 8.b | even | 2 | 1 | ||
544.3.n.b | 64 | 68.f | odd | 4 | 1 | ||
544.3.n.b | 64 | 136.i | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{32} + 4 T_{3}^{31} + 8 T_{3}^{30} - 40 T_{3}^{29} + 1580 T_{3}^{28} + 5584 T_{3}^{27} + 10496 T_{3}^{26} - 44704 T_{3}^{25} + 859760 T_{3}^{24} + 2747968 T_{3}^{23} + 4908800 T_{3}^{22} - 17074624 T_{3}^{21} + \cdots + 136273199104 \)
acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\).