Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,2,Mod(43,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([12, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("342.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 342.w (of order \(9\), degree \(6\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(2.73088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(54\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −0.173648 | + | 0.984808i | −1.61970 | + | 0.613651i | −0.939693 | − | 0.342020i | −0.648788 | + | 3.67946i | −0.323070 | − | 1.70165i | −3.85338 | 0.500000 | − | 0.866025i | 2.24687 | − | 1.98786i | −3.51090 | − | 1.27786i | ||
43.2 | −0.173648 | + | 0.984808i | −1.52941 | + | 0.812965i | −0.939693 | − | 0.342020i | 0.272361 | − | 1.54463i | −0.535036 | − | 1.64734i | 2.24074 | 0.500000 | − | 0.866025i | 1.67817 | − | 2.48671i | 1.47387 | + | 0.536446i | ||
43.3 | −0.173648 | + | 0.984808i | −1.30786 | − | 1.13556i | −0.939693 | − | 0.342020i | −0.124608 | + | 0.706685i | 1.34542 | − | 1.09080i | 1.23071 | 0.500000 | − | 0.866025i | 0.421002 | + | 2.97031i | −0.674311 | − | 0.245429i | ||
43.4 | −0.173648 | + | 0.984808i | −0.224346 | − | 1.71746i | −0.939693 | − | 0.342020i | −0.440347 | + | 2.49733i | 1.73033 | + | 0.0772962i | −1.32843 | 0.500000 | − | 0.866025i | −2.89934 | + | 0.770610i | −2.38293 | − | 0.867314i | ||
43.5 | −0.173648 | + | 0.984808i | −0.0104059 | + | 1.73202i | −0.939693 | − | 0.342020i | −0.0597169 | + | 0.338671i | −1.70390 | − | 0.311010i | −2.70759 | 0.500000 | − | 0.866025i | −2.99978 | − | 0.0360466i | −0.323156 | − | 0.117619i | ||
43.6 | −0.173648 | + | 0.984808i | 0.792978 | − | 1.53987i | −0.939693 | − | 0.342020i | 0.336337 | − | 1.90746i | 1.37877 | + | 1.04833i | −4.43581 | 0.500000 | − | 0.866025i | −1.74237 | − | 2.44216i | 1.82008 | + | 0.662455i | ||
43.7 | −0.173648 | + | 0.984808i | 1.00018 | + | 1.41409i | −0.939693 | − | 0.342020i | −0.232829 | + | 1.32044i | −1.56628 | + | 0.739433i | 3.81041 | 0.500000 | − | 0.866025i | −0.999276 | + | 2.82868i | −1.25995 | − | 0.458584i | ||
43.8 | −0.173648 | + | 0.984808i | 1.39081 | − | 1.03230i | −0.939693 | − | 0.342020i | 0.183383 | − | 1.04002i | 0.775102 | + | 1.54894i | 2.95323 | 0.500000 | − | 0.866025i | 0.868725 | − | 2.87147i | 0.992372 | + | 0.361194i | ||
43.9 | −0.173648 | + | 0.984808i | 1.50775 | + | 0.852462i | −0.939693 | − | 0.342020i | 0.714207 | − | 4.05047i | −1.10133 | + | 1.33681i | −0.377794 | 0.500000 | − | 0.866025i | 1.54662 | + | 2.57060i | 3.86492 | + | 1.40671i | ||
85.1 | −0.766044 | + | 0.642788i | −1.73088 | − | 0.0636567i | 0.173648 | − | 0.984808i | 0.451077 | − | 0.378498i | 1.36685 | − | 1.06382i | −1.49179 | 0.500000 | + | 0.866025i | 2.99190 | + | 0.220364i | −0.102251 | + | 0.579893i | ||
85.2 | −0.766044 | + | 0.642788i | −1.13333 | − | 1.30979i | 0.173648 | − | 0.984808i | 1.81802 | − | 1.52550i | 1.71010 | + | 0.274870i | 1.50605 | 0.500000 | + | 0.866025i | −0.431123 | + | 2.96886i | −0.412111 | + | 2.33720i | ||
85.3 | −0.766044 | + | 0.642788i | −1.01451 | + | 1.40384i | 0.173648 | − | 0.984808i | 1.60183 | − | 1.34410i | −0.125210 | − | 1.72752i | −0.685971 | 0.500000 | + | 0.866025i | −0.941533 | − | 2.84842i | −0.363106 | + | 2.05928i | ||
85.4 | −0.766044 | + | 0.642788i | −0.916667 | + | 1.46960i | 0.173648 | − | 0.984808i | −2.20943 | + | 1.85393i | −0.242432 | − | 1.71500i | 1.38926 | 0.500000 | + | 0.866025i | −1.31944 | − | 2.69427i | 0.500836 | − | 2.84038i | ||
85.5 | −0.766044 | + | 0.642788i | −0.379894 | − | 1.68988i | 0.173648 | − | 0.984808i | −2.05175 | + | 1.72162i | 1.37725 | + | 1.05033i | 0.931774 | 0.500000 | + | 0.866025i | −2.71136 | + | 1.28395i | 0.465093 | − | 2.63767i | ||
85.6 | −0.766044 | + | 0.642788i | 0.705559 | + | 1.58183i | 0.173648 | − | 0.984808i | 0.287887 | − | 0.241566i | −1.55727 | − | 0.758228i | −3.26780 | 0.500000 | + | 0.866025i | −2.00437 | + | 2.23215i | −0.0652586 | + | 0.370100i | ||
85.7 | −0.766044 | + | 0.642788i | 1.10517 | − | 1.33364i | 0.173648 | − | 0.984808i | −0.765245 | + | 0.642117i | 0.0106356 | + | 1.73202i | −5.16688 | 0.500000 | + | 0.866025i | −0.557188 | − | 2.94780i | 0.173467 | − | 0.983780i | ||
85.8 | −0.766044 | + | 0.642788i | 1.67515 | − | 0.440302i | 0.173648 | − | 0.984808i | −1.49025 | + | 1.25047i | −1.00022 | + | 1.41406i | 2.53958 | 0.500000 | + | 0.866025i | 2.61227 | − | 1.47515i | 0.337812 | − | 1.91582i | ||
85.9 | −0.766044 | + | 0.642788i | 1.68940 | + | 0.382000i | 0.173648 | − | 0.984808i | 2.35785 | − | 1.97847i | −1.53970 | + | 0.793297i | −1.63361 | 0.500000 | + | 0.866025i | 2.70815 | + | 1.29070i | −0.534481 | + | 3.03119i | ||
139.1 | 0.939693 | + | 0.342020i | −1.65424 | − | 0.513314i | 0.766044 | + | 0.642788i | −1.17751 | − | 0.428579i | −1.37891 | − | 1.04814i | 2.38962 | 0.500000 | + | 0.866025i | 2.47302 | + | 1.69829i | −0.959915 | − | 0.805465i | ||
139.2 | 0.939693 | + | 0.342020i | −1.63454 | − | 0.572959i | 0.766044 | + | 0.642788i | −1.12406 | − | 0.409125i | −1.34000 | − | 1.09745i | −3.90541 | 0.500000 | + | 0.866025i | 2.34344 | + | 1.87305i | −0.916343 | − | 0.768903i | ||
See all 54 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.w | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 342.2.w.a | yes | 54 |
9.c | even | 3 | 1 | 342.2.v.a | ✓ | 54 | |
19.e | even | 9 | 1 | 342.2.v.a | ✓ | 54 | |
171.w | even | 9 | 1 | inner | 342.2.w.a | yes | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
342.2.v.a | ✓ | 54 | 9.c | even | 3 | 1 | |
342.2.v.a | ✓ | 54 | 19.e | even | 9 | 1 | |
342.2.w.a | yes | 54 | 1.a | even | 1 | 1 | trivial |
342.2.w.a | yes | 54 | 171.w | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{54} + 3 T_{5}^{52} + 6 T_{5}^{51} - 114 T_{5}^{50} - 105 T_{5}^{49} + 2256 T_{5}^{48} + 246 T_{5}^{47} + 22224 T_{5}^{46} - 33923 T_{5}^{45} - 137706 T_{5}^{44} - 84216 T_{5}^{43} + 4974390 T_{5}^{42} + \cdots + 387420489 \)
acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\).