Newspace parameters
Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 161.i (of order \(11\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.28559147254\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −0.188704 | + | 1.31247i | 0.554762 | + | 1.21476i | 0.232022 | + | 0.0681278i | 0.605110 | + | 0.698334i | −1.69902 | + | 0.498877i | −0.841254 | − | 0.540641i | −1.23485 | + | 2.70395i | 0.796701 | − | 0.919442i | −1.03073 | + | 0.662409i |
8.2 | −0.0827955 | + | 0.575856i | −0.801904 | − | 1.75592i | 1.59423 | + | 0.468109i | −2.45011 | − | 2.82758i | 1.07755 | − | 0.316398i | −0.841254 | − | 0.540641i | −0.884916 | + | 1.93770i | −0.475639 | + | 0.548917i | 1.83114 | − | 1.17680i |
8.3 | 0.0810785 | − | 0.563913i | 1.06791 | + | 2.33839i | 1.60756 | + | 0.472023i | −0.715681 | − | 0.825940i | 1.40523 | − | 0.412614i | −0.841254 | − | 0.540641i | 0.869852 | − | 1.90471i | −2.36306 | + | 2.72711i | −0.523785 | + | 0.336616i |
8.4 | 0.222230 | − | 1.54565i | −1.15077 | − | 2.51984i | −0.420652 | − | 0.123514i | −0.138046 | − | 0.159314i | −4.15052 | + | 1.21870i | −0.841254 | − | 0.540641i | 1.01298 | − | 2.21813i | −3.06074 | + | 3.53228i | −0.276921 | + | 0.177967i |
8.5 | 0.252821 | − | 1.75841i | 0.198117 | + | 0.433815i | −1.10909 | − | 0.325658i | 2.02744 | + | 2.33979i | 0.812911 | − | 0.238692i | −0.841254 | − | 0.540641i | 0.622919 | − | 1.36400i | 1.81564 | − | 2.09536i | 4.62688 | − | 2.97352i |
29.1 | −1.13379 | + | 2.48266i | −1.50202 | − | 0.441033i | −3.56837 | − | 4.11812i | 1.57859 | − | 1.01450i | 2.79791 | − | 3.22896i | 0.142315 | − | 0.989821i | 9.03219 | − | 2.65209i | −0.462204 | − | 0.297041i | 0.728860 | + | 5.06933i |
29.2 | −0.813943 | + | 1.78229i | 1.29442 | + | 0.380077i | −1.20432 | − | 1.38986i | −3.05799 | + | 1.96525i | −1.73099 | + | 1.99767i | 0.142315 | − | 0.989821i | −0.302592 | + | 0.0888490i | −0.992685 | − | 0.637960i | −1.01361 | − | 7.04982i |
29.3 | −0.282831 | + | 0.619314i | 0.745240 | + | 0.218822i | 1.00616 | + | 1.16118i | 2.35286 | − | 1.51209i | −0.346297 | + | 0.399648i | 0.142315 | − | 0.989821i | −2.31023 | + | 0.678345i | −2.01626 | − | 1.29577i | 0.270997 | + | 1.88483i |
29.4 | 0.490936 | − | 1.07500i | 1.94080 | + | 0.569869i | 0.395116 | + | 0.455989i | −2.35800 | + | 1.51540i | 1.56541 | − | 1.80658i | 0.142315 | − | 0.989821i | 2.95201 | − | 0.866789i | 0.918175 | + | 0.590075i | 0.471422 | + | 3.27881i |
29.5 | 0.908799 | − | 1.98999i | −0.286353 | − | 0.0840808i | −1.82443 | − | 2.10551i | 0.768064 | − | 0.493605i | −0.427557 | + | 0.493427i | 0.142315 | − | 0.989821i | −1.64984 | + | 0.484437i | −2.44883 | − | 1.57377i | −0.284254 | − | 1.97703i |
36.1 | −1.59835 | − | 1.84459i | −1.86860 | − | 1.20088i | −0.563170 | + | 3.91694i | −0.732261 | + | 1.60343i | 0.771545 | + | 5.36621i | 0.959493 | + | 0.281733i | 4.01871 | − | 2.58267i | 0.803315 | + | 1.75901i | 4.12807 | − | 1.21211i |
36.2 | −0.737191 | − | 0.850763i | 1.61822 | + | 1.03996i | 0.104281 | − | 0.725293i | 0.719526 | − | 1.57554i | −0.308170 | − | 2.14337i | 0.959493 | + | 0.281733i | −2.58796 | + | 1.66318i | 0.290854 | + | 0.636882i | −1.87084 | + | 0.549328i |
36.3 | 0.515215 | + | 0.594590i | −1.14233 | − | 0.734130i | 0.196539 | − | 1.36696i | 0.0858162 | − | 0.187911i | −0.152039 | − | 1.05745i | 0.959493 | + | 0.281733i | 2.23776 | − | 1.43812i | −0.480277 | − | 1.05166i | 0.155944 | − | 0.0457893i |
36.4 | 1.34132 | + | 1.54796i | 0.455720 | + | 0.292874i | −0.312428 | + | 2.17298i | 0.0741704 | − | 0.162411i | 0.157908 | + | 1.09828i | 0.959493 | + | 0.281733i | −0.336572 | + | 0.216302i | −1.12434 | − | 2.46196i | 0.350892 | − | 0.103031i |
36.5 | 1.78872 | + | 2.06430i | −2.35987 | − | 1.51660i | −0.777161 | + | 5.40527i | −1.03647 | + | 2.26955i | −1.09045 | − | 7.58425i | 0.959493 | + | 0.281733i | −7.95252 | + | 5.11077i | 2.02268 | + | 4.42905i | −6.53898 | + | 1.92002i |
50.1 | −1.13379 | − | 2.48266i | −1.50202 | + | 0.441033i | −3.56837 | + | 4.11812i | 1.57859 | + | 1.01450i | 2.79791 | + | 3.22896i | 0.142315 | + | 0.989821i | 9.03219 | + | 2.65209i | −0.462204 | + | 0.297041i | 0.728860 | − | 5.06933i |
50.2 | −0.813943 | − | 1.78229i | 1.29442 | − | 0.380077i | −1.20432 | + | 1.38986i | −3.05799 | − | 1.96525i | −1.73099 | − | 1.99767i | 0.142315 | + | 0.989821i | −0.302592 | − | 0.0888490i | −0.992685 | + | 0.637960i | −1.01361 | + | 7.04982i |
50.3 | −0.282831 | − | 0.619314i | 0.745240 | − | 0.218822i | 1.00616 | − | 1.16118i | 2.35286 | + | 1.51209i | −0.346297 | − | 0.399648i | 0.142315 | + | 0.989821i | −2.31023 | − | 0.678345i | −2.01626 | + | 1.29577i | 0.270997 | − | 1.88483i |
50.4 | 0.490936 | + | 1.07500i | 1.94080 | − | 0.569869i | 0.395116 | − | 0.455989i | −2.35800 | − | 1.51540i | 1.56541 | + | 1.80658i | 0.142315 | + | 0.989821i | 2.95201 | + | 0.866789i | 0.918175 | − | 0.590075i | 0.471422 | − | 3.27881i |
50.5 | 0.908799 | + | 1.98999i | −0.286353 | + | 0.0840808i | −1.82443 | + | 2.10551i | 0.768064 | + | 0.493605i | −0.427557 | − | 0.493427i | 0.142315 | + | 0.989821i | −1.64984 | − | 0.484437i | −2.44883 | + | 1.57377i | −0.284254 | + | 1.97703i |
See all 50 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 161.2.i.a | ✓ | 50 |
23.c | even | 11 | 1 | inner | 161.2.i.a | ✓ | 50 |
23.c | even | 11 | 1 | 3703.2.a.q | 25 | ||
23.d | odd | 22 | 1 | 3703.2.a.r | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
161.2.i.a | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
161.2.i.a | ✓ | 50 | 23.c | even | 11 | 1 | inner |
3703.2.a.q | 25 | 23.c | even | 11 | 1 | ||
3703.2.a.r | 25 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{50} - 2 T_{2}^{49} + 9 T_{2}^{48} - 21 T_{2}^{47} + 113 T_{2}^{46} - 195 T_{2}^{45} + 554 T_{2}^{44} - 1316 T_{2}^{43} + 4277 T_{2}^{42} - 7850 T_{2}^{41} + 17228 T_{2}^{40} - 30695 T_{2}^{39} + 89086 T_{2}^{38} - 139147 T_{2}^{37} + \cdots + 279841 \)
acting on \(S_{2}^{\mathrm{new}}(161, [\chi])\).