Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [495,2,Mod(164,495)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(495, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("495.164");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 495.r (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: | \(3.95259490005\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(64\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
164.1 | −2.35894 | − | 1.36194i | −0.491830 | − | 1.66075i | 2.70974 | + | 4.69341i | 1.99447 | + | 1.01099i | −1.10164 | + | 4.58746i | −0.885979 | + | 1.53456i | − | 9.31422i | −2.51621 | + | 1.63362i | −3.32794 | − | 5.10120i | |
164.2 | −2.35894 | − | 1.36194i | 0.491830 | + | 1.66075i | 2.70974 | + | 4.69341i | 0.121696 | − | 2.23275i | 1.10164 | − | 4.58746i | −0.885979 | + | 1.53456i | − | 9.31422i | −2.51621 | + | 1.63362i | −3.32794 | + | 5.10120i | |
164.3 | −2.22820 | − | 1.28645i | −1.41739 | + | 0.995490i | 2.30993 | + | 4.00091i | −2.12513 | + | 0.695571i | 4.43889 | − | 0.394744i | −0.0241406 | + | 0.0418127i | − | 6.74064i | 1.01800 | − | 2.82200i | 5.63004 | + | 1.18401i | |
164.4 | −2.22820 | − | 1.28645i | 1.41739 | − | 0.995490i | 2.30993 | + | 4.00091i | −1.66495 | + | 1.49263i | −4.43889 | + | 0.394744i | −0.0241406 | + | 0.0418127i | − | 6.74064i | 1.01800 | − | 2.82200i | 5.63004 | − | 1.18401i | |
164.5 | −2.16245 | − | 1.24849i | −1.60498 | + | 0.651194i | 2.11746 | + | 3.66754i | 2.12949 | + | 0.682097i | 4.28369 | + | 0.595622i | 2.33858 | − | 4.05054i | − | 5.58053i | 2.15189 | − | 2.09030i | −3.75333 | − | 4.13365i | |
164.6 | −2.16245 | − | 1.24849i | 1.60498 | − | 0.651194i | 2.11746 | + | 3.66754i | 0.474033 | − | 2.18524i | −4.28369 | − | 0.595622i | 2.33858 | − | 4.05054i | − | 5.58053i | 2.15189 | − | 2.09030i | −3.75333 | + | 4.13365i | |
164.7 | −2.03042 | − | 1.17226i | −0.0364388 | + | 1.73167i | 1.74841 | + | 3.02833i | 0.752578 | + | 2.10562i | 2.10396 | − | 3.47330i | −1.67901 | + | 2.90812i | − | 3.50932i | −2.99734 | − | 0.126200i | 0.940291 | − | 5.15751i | |
164.8 | −2.03042 | − | 1.17226i | 0.0364388 | − | 1.73167i | 1.74841 | + | 3.02833i | −1.44723 | − | 1.70456i | −2.10396 | + | 3.47330i | −1.67901 | + | 2.90812i | − | 3.50932i | −2.99734 | − | 0.126200i | 0.940291 | + | 5.15751i | |
164.9 | −1.84053 | − | 1.06263i | −1.00779 | − | 1.40867i | 1.25838 | + | 2.17957i | −1.05343 | + | 1.97238i | 0.357974 | + | 3.66362i | 1.55655 | − | 2.69602i | − | 1.09824i | −0.968712 | + | 2.83930i | 4.03479 | − | 2.51082i | |
164.10 | −1.84053 | − | 1.06263i | 1.00779 | + | 1.40867i | 1.25838 | + | 2.17957i | −2.23485 | − | 0.0738892i | −0.357974 | − | 3.66362i | 1.55655 | − | 2.69602i | − | 1.09824i | −0.968712 | + | 2.83930i | 4.03479 | + | 2.51082i | |
164.11 | −1.70741 | − | 0.985775i | −1.59259 | − | 0.680924i | 0.943504 | + | 1.63420i | −1.79646 | − | 1.33143i | 2.04797 | + | 2.73255i | 0.0538350 | − | 0.0932449i | 0.222770i | 2.07268 | + | 2.16887i | 1.75481 | + | 4.04422i | ||
164.12 | −1.70741 | − | 0.985775i | 1.59259 | + | 0.680924i | 0.943504 | + | 1.63420i | 0.254824 | + | 2.22150i | −2.04797 | − | 2.73255i | 0.0538350 | − | 0.0932449i | 0.222770i | 2.07268 | + | 2.16887i | 1.75481 | − | 4.04422i | ||
164.13 | −1.62560 | − | 0.938540i | −0.155415 | + | 1.72506i | 0.761715 | + | 1.31933i | 2.19111 | − | 0.446145i | 1.87168 | − | 2.65840i | 1.01834 | − | 1.76382i | 0.894560i | −2.95169 | − | 0.536200i | −3.98059 | − | 1.33119i | ||
164.14 | −1.62560 | − | 0.938540i | 0.155415 | − | 1.72506i | 0.761715 | + | 1.31933i | 1.48193 | − | 1.67448i | −1.87168 | + | 2.65840i | 1.01834 | − | 1.76382i | 0.894560i | −2.95169 | − | 0.536200i | −3.98059 | + | 1.33119i | ||
164.15 | −1.58655 | − | 0.915995i | −1.69477 | − | 0.357410i | 0.678093 | + | 1.17449i | 0.985404 | + | 2.00723i | 2.36146 | + | 2.11945i | −1.65812 | + | 2.87195i | 1.17946i | 2.74452 | + | 1.21146i | 0.275221 | − | 4.08720i | ||
164.16 | −1.58655 | − | 0.915995i | 1.69477 | + | 0.357410i | 0.678093 | + | 1.17449i | −1.24561 | − | 1.85700i | −2.36146 | − | 2.11945i | −1.65812 | + | 2.87195i | 1.17946i | 2.74452 | + | 1.21146i | 0.275221 | + | 4.08720i | ||
164.17 | −1.29115 | − | 0.745448i | −1.09243 | + | 1.34410i | 0.111386 | + | 0.192926i | 0.565909 | − | 2.16327i | 2.41245 | − | 0.921083i | −0.573487 | + | 0.993309i | 2.64966i | −0.613184 | − | 2.93667i | −2.34328 | + | 2.37126i | ||
164.18 | −1.29115 | − | 0.745448i | 1.09243 | − | 1.34410i | 0.111386 | + | 0.192926i | 2.15640 | + | 0.591544i | −2.41245 | + | 0.921083i | −0.573487 | + | 0.993309i | 2.64966i | −0.613184 | − | 2.93667i | −2.34328 | − | 2.37126i | ||
164.19 | −1.12460 | − | 0.649288i | −1.13838 | + | 1.30540i | −0.156849 | − | 0.271671i | −2.22900 | + | 0.177667i | 2.12781 | − | 0.728919i | −1.61390 | + | 2.79536i | 3.00452i | −0.408162 | − | 2.97210i | 2.62209 | + | 1.24746i | ||
164.20 | −1.12460 | − | 0.649288i | 1.13838 | − | 1.30540i | −0.156849 | − | 0.271671i | −1.26836 | + | 1.84154i | −2.12781 | + | 0.728919i | −1.61390 | + | 2.79536i | 3.00452i | −0.408162 | − | 2.97210i | 2.62209 | − | 1.24746i | ||
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
11.b | odd | 2 | 1 | inner |
45.h | odd | 6 | 1 | inner |
55.d | odd | 2 | 1 | inner |
99.g | even | 6 | 1 | inner |
495.r | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 495.2.r.c | ✓ | 128 |
5.b | even | 2 | 1 | inner | 495.2.r.c | ✓ | 128 |
9.d | odd | 6 | 1 | inner | 495.2.r.c | ✓ | 128 |
11.b | odd | 2 | 1 | inner | 495.2.r.c | ✓ | 128 |
45.h | odd | 6 | 1 | inner | 495.2.r.c | ✓ | 128 |
55.d | odd | 2 | 1 | inner | 495.2.r.c | ✓ | 128 |
99.g | even | 6 | 1 | inner | 495.2.r.c | ✓ | 128 |
495.r | even | 6 | 1 | inner | 495.2.r.c | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
495.2.r.c | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
495.2.r.c | ✓ | 128 | 5.b | even | 2 | 1 | inner |
495.2.r.c | ✓ | 128 | 9.d | odd | 6 | 1 | inner |
495.2.r.c | ✓ | 128 | 11.b | odd | 2 | 1 | inner |
495.2.r.c | ✓ | 128 | 45.h | odd | 6 | 1 | inner |
495.2.r.c | ✓ | 128 | 55.d | odd | 2 | 1 | inner |
495.2.r.c | ✓ | 128 | 99.g | even | 6 | 1 | inner |
495.2.r.c | ✓ | 128 | 495.r | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):
\( T_{2}^{64} - 49 T_{2}^{62} + 1320 T_{2}^{60} - 24557 T_{2}^{58} + 348458 T_{2}^{56} - 3963045 T_{2}^{54} + 37266104 T_{2}^{52} - 295506029 T_{2}^{50} + 2003338167 T_{2}^{48} - 11720212940 T_{2}^{46} + \cdots + 227889216 \) |
\( T_{23}^{64} + 303 T_{23}^{62} + 53520 T_{23}^{60} + 6281307 T_{23}^{58} + 546857901 T_{23}^{56} + 36402572421 T_{23}^{54} + 1912157107605 T_{23}^{52} + 79918113532788 T_{23}^{50} + \cdots + 34\!\cdots\!16 \) |