Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [828,2,Mod(47,828)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("828.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 828.o (of order \(6\), degree \(2\), 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: | \(6.61161328736\) |
Analytic rank: | \(0\) |
Dimension: | \(132\) |
Relative dimension: | \(66\) 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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −1.41421 | − | 0.00214998i | −1.34750 | − | 1.08823i | 1.99999 | + | 0.00608105i | −1.10823 | + | 0.639837i | 1.90331 | + | 1.54189i | −3.10783 | − | 1.79431i | −2.82840 | − | 0.0128998i | 0.631500 | + | 2.93278i | 1.56865 | − | 0.902482i |
47.2 | −1.40722 | + | 0.140428i | −1.25401 | + | 1.19476i | 1.96056 | − | 0.395228i | −0.532014 | + | 0.307158i | 1.59689 | − | 1.85740i | 1.07017 | + | 0.617862i | −2.70345 | + | 0.831492i | 0.145075 | − | 2.99649i | 0.705529 | − | 0.506950i |
47.3 | −1.39754 | − | 0.216507i | 1.65212 | + | 0.520096i | 1.90625 | + | 0.605156i | 2.17762 | − | 1.25725i | −2.19630 | − | 1.08455i | 3.49734 | + | 2.01919i | −2.53304 | − | 1.25845i | 2.45900 | + | 1.71852i | −3.31551 | + | 1.28559i |
47.4 | −1.35893 | + | 0.391560i | −1.70231 | − | 0.319612i | 1.69336 | − | 1.06420i | −0.811777 | + | 0.468680i | 2.43846 | − | 0.232226i | 4.54437 | + | 2.62369i | −1.88445 | + | 2.10923i | 2.79570 | + | 1.08816i | 0.919629 | − | 0.954761i |
47.5 | −1.35549 | − | 0.403287i | −1.22500 | + | 1.22449i | 1.67472 | + | 1.09331i | 2.26529 | − | 1.30786i | 2.15430 | − | 1.16576i | 0.222415 | + | 0.128411i | −1.82915 | − | 2.15736i | 0.00123454 | − | 3.00000i | −3.59802 | + | 0.859237i |
47.6 | −1.35188 | + | 0.415239i | 0.255232 | − | 1.71314i | 1.65515 | − | 1.12271i | 2.99840 | − | 1.73112i | 0.366321 | + | 2.42194i | 1.15385 | + | 0.666173i | −1.77138 | + | 2.20505i | −2.86971 | − | 0.874496i | −3.33464 | + | 3.58532i |
47.7 | −1.35126 | − | 0.417239i | 0.0496784 | + | 1.73134i | 1.65182 | + | 1.12760i | −3.55240 | + | 2.05098i | 0.655254 | − | 2.36022i | 3.04028 | + | 1.75531i | −1.76157 | − | 2.21289i | −2.99506 | + | 0.172020i | 5.65597 | − | 1.28921i |
47.8 | −1.32104 | + | 0.504830i | 1.42989 | − | 0.977450i | 1.49029 | − | 1.33380i | −1.85603 | + | 1.07158i | −1.39550 | + | 2.01310i | −3.04212 | − | 1.75637i | −1.29539 | + | 2.51435i | 1.08918 | − | 2.79530i | 1.91093 | − | 2.35258i |
47.9 | −1.30862 | − | 0.536209i | 1.72401 | + | 0.166709i | 1.42496 | + | 1.40339i | −2.66178 | + | 1.53678i | −2.16668 | − | 1.14259i | −2.35100 | − | 1.35735i | −1.11222 | − | 2.60057i | 2.94442 | + | 0.574816i | 4.30729 | − | 0.583786i |
47.10 | −1.25744 | + | 0.647177i | −0.0751077 | − | 1.73042i | 1.16232 | − | 1.62758i | −3.62509 | + | 2.09295i | 1.21433 | + | 2.12730i | 1.00949 | + | 0.582830i | −0.408228 | + | 2.79881i | −2.98872 | + | 0.259936i | 3.20384 | − | 4.97784i |
47.11 | −1.25736 | − | 0.647332i | −0.895324 | − | 1.48270i | 1.16192 | + | 1.62786i | 2.77988 | − | 1.60497i | 0.165950 | + | 2.44386i | 0.136435 | + | 0.0787710i | −0.407193 | − | 2.79896i | −1.39679 | + | 2.65499i | −4.53427 | + | 0.218519i |
47.12 | −1.24577 | − | 0.669373i | 0.632882 | − | 1.61228i | 1.10388 | + | 1.66777i | 0.439521 | − | 0.253757i | −1.86764 | + | 1.58490i | −2.35304 | − | 1.35853i | −0.258822 | − | 2.81656i | −2.19892 | − | 2.04077i | −0.717399 | + | 0.0219199i |
47.13 | −1.22156 | + | 0.712591i | −0.317113 | + | 1.70277i | 0.984428 | − | 1.74095i | 2.33026 | − | 1.34538i | −0.826008 | − | 2.30602i | −3.25138 | − | 1.87718i | 0.0380433 | + | 2.82817i | −2.79888 | − | 1.07994i | −1.88786 | + | 3.30399i |
47.14 | −1.17797 | + | 0.782556i | 1.48590 | + | 0.890002i | 0.775213 | − | 1.84365i | 1.46262 | − | 0.844445i | −2.44682 | + | 0.114406i | −0.558634 | − | 0.322527i | 0.529584 | + | 2.77841i | 1.41579 | + | 2.64491i | −1.06209 | + | 2.13931i |
47.15 | −1.17633 | + | 0.785014i | −1.35464 | − | 1.07932i | 0.767507 | − | 1.84687i | 2.67514 | − | 1.54449i | 2.44079 | + | 0.206227i | −1.19116 | − | 0.687714i | 0.546978 | + | 2.77503i | 0.670120 | + | 2.92420i | −1.93440 | + | 3.91685i |
47.16 | −1.14034 | − | 0.836430i | 1.64820 | − | 0.532379i | 0.600769 | + | 1.90764i | −0.636131 | + | 0.367271i | −2.32482 | − | 0.771512i | 2.19630 | + | 1.26803i | 0.910522 | − | 2.67786i | 2.43315 | − | 1.75494i | 1.03260 | + | 0.113265i |
47.17 | −1.00627 | − | 0.993688i | 1.01905 | + | 1.40055i | 0.0251665 | + | 1.99984i | −1.87865 | + | 1.08464i | 0.366266 | − | 2.42195i | −1.45711 | − | 0.841265i | 1.96190 | − | 2.03739i | −0.923070 | + | 2.85446i | 2.96823 | + | 0.775351i |
47.18 | −0.863315 | + | 1.12013i | −1.71951 | − | 0.208062i | −0.509373 | − | 1.93405i | −1.75788 | + | 1.01492i | 1.71753 | − | 1.74645i | −4.07349 | − | 2.35183i | 2.60613 | + | 1.09913i | 2.91342 | + | 0.715530i | 0.380774 | − | 2.84525i |
47.19 | −0.852594 | + | 1.12831i | 1.71309 | + | 0.255573i | −0.546167 | − | 1.92398i | −2.82290 | + | 1.62980i | −1.74894 | + | 1.71500i | 3.09894 | + | 1.78918i | 2.63651 | + | 1.02413i | 2.86936 | + | 0.875640i | 0.567866 | − | 4.57467i |
47.20 | −0.828510 | − | 1.14611i | −1.40140 | + | 1.01788i | −0.627142 | + | 1.89913i | −2.23183 | + | 1.28855i | 2.32768 | + | 0.762840i | −2.37288 | − | 1.36998i | 2.69621 | − | 0.854674i | 0.927852 | − | 2.85291i | 3.32591 | + | 1.49035i |
See next 80 embeddings (of 132 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
36.h | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 828.2.o.a | ✓ | 132 |
4.b | odd | 2 | 1 | 828.2.o.b | yes | 132 | |
9.d | odd | 6 | 1 | 828.2.o.b | yes | 132 | |
36.h | even | 6 | 1 | inner | 828.2.o.a | ✓ | 132 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
828.2.o.a | ✓ | 132 | 1.a | even | 1 | 1 | trivial |
828.2.o.a | ✓ | 132 | 36.h | even | 6 | 1 | inner |
828.2.o.b | yes | 132 | 4.b | odd | 2 | 1 | |
828.2.o.b | yes | 132 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{132} - 264 T_{7}^{130} + 36816 T_{7}^{128} + 732 T_{7}^{127} - 3538956 T_{7}^{126} - 177732 T_{7}^{125} + 260801214 T_{7}^{124} + 22853088 T_{7}^{123} - 15593339592 T_{7}^{122} + \cdots + 37\!\cdots\!76 \)
acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\).