Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(236,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.236");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.be (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: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
236.1 | −2.27381 | + | 1.31279i | −0.933716 | + | 1.45883i | 2.44681 | − | 4.23800i | −1.00000 | 0.207969 | − | 4.54286i | −2.64220 | − | 0.136993i | 7.59741i | −1.25635 | − | 2.72426i | 2.27381 | − | 1.31279i | ||||
236.2 | −2.03337 | + | 1.17397i | 1.49299 | + | 0.878053i | 1.75639 | − | 3.04216i | −1.00000 | −4.06660 | − | 0.0326841i | 0.135526 | − | 2.64228i | 3.55190i | 1.45805 | + | 2.62185i | 2.03337 | − | 1.17397i | ||||
236.3 | −1.91788 | + | 1.10729i | 0.302150 | − | 1.70549i | 1.45217 | − | 2.51523i | −1.00000 | 1.30898 | + | 3.60549i | 2.53521 | − | 0.756781i | 2.00271i | −2.81741 | − | 1.03063i | 1.91788 | − | 1.10729i | ||||
236.4 | −1.67628 | + | 0.967799i | −1.71487 | + | 0.243339i | 0.873269 | − | 1.51255i | −1.00000 | 2.63910 | − | 2.06755i | 2.28906 | + | 1.32673i | − | 0.490599i | 2.88157 | − | 0.834589i | 1.67628 | − | 0.967799i | |||
236.5 | −1.07088 | + | 0.618275i | 1.48704 | − | 0.888095i | −0.235471 | + | 0.407847i | −1.00000 | −1.04336 | + | 1.87045i | −2.10494 | − | 1.60288i | − | 3.05545i | 1.42257 | − | 2.64127i | 1.07088 | − | 0.618275i | |||
236.6 | −0.948743 | + | 0.547757i | −0.237931 | + | 1.71563i | −0.399925 | + | 0.692690i | −1.00000 | −0.714013 | − | 1.75802i | 1.38314 | + | 2.25542i | − | 3.06727i | −2.88678 | − | 0.816404i | 0.948743 | − | 0.547757i | |||
236.7 | −0.454062 | + | 0.262153i | 0.322156 | − | 1.70183i | −0.862552 | + | 1.49398i | −1.00000 | 0.299860 | + | 0.857190i | −1.09499 | + | 2.40853i | − | 1.95309i | −2.79243 | − | 1.09651i | 0.454062 | − | 0.262153i | |||
236.8 | −0.0900587 | + | 0.0519954i | −1.13092 | − | 1.31188i | −0.994593 | + | 1.72269i | −1.00000 | 0.170061 | + | 0.0593436i | 2.34312 | − | 1.22873i | − | 0.414839i | −0.442054 | + | 2.96725i | 0.0900587 | − | 0.0519954i | |||
236.9 | 0.343770 | − | 0.198476i | −1.65197 | + | 0.520559i | −0.921215 | + | 1.59559i | −1.00000 | −0.464581 | + | 0.506829i | −0.324456 | − | 2.62578i | 1.52526i | 2.45804 | − | 1.71990i | −0.343770 | + | 0.198476i | ||||
236.10 | 0.403396 | − | 0.232901i | 0.625194 | + | 1.61528i | −0.891514 | + | 1.54415i | −1.00000 | 0.628402 | + | 0.505990i | −2.59662 | − | 0.507526i | 1.76214i | −2.21827 | + | 2.01973i | −0.403396 | + | 0.232901i | ||||
236.11 | 0.559417 | − | 0.322980i | 1.73059 | + | 0.0712304i | −0.791368 | + | 1.37069i | −1.00000 | 0.991125 | − | 0.519096i | 1.24717 | + | 2.33336i | 2.31430i | 2.98985 | + | 0.246540i | −0.559417 | + | 0.322980i | ||||
236.12 | 1.36556 | − | 0.788404i | −1.67710 | + | 0.432827i | 0.243162 | − | 0.421168i | −1.00000 | −1.94893 | + | 1.91328i | 0.569408 | + | 2.58375i | 2.38678i | 2.62532 | − | 1.45179i | −1.36556 | + | 0.788404i | ||||
236.13 | 1.48159 | − | 0.855399i | 1.02526 | − | 1.39601i | 0.463416 | − | 0.802660i | −1.00000 | 0.324870 | − | 2.94533i | 1.20122 | − | 2.35734i | 1.83597i | −0.897694 | − | 2.86254i | −1.48159 | + | 0.855399i | ||||
236.14 | 2.01305 | − | 1.16223i | 0.708464 | + | 1.58053i | 1.70157 | − | 2.94720i | −1.00000 | 3.26312 | + | 2.35828i | 2.64011 | − | 0.172663i | − | 3.26155i | −1.99616 | + | 2.23950i | −2.01305 | + | 1.16223i | |||
236.15 | 2.11687 | − | 1.22217i | 1.73076 | + | 0.0669518i | 1.98742 | − | 3.44231i | −1.00000 | 3.74561 | − | 1.97356i | −2.50358 | + | 0.855621i | − | 4.82720i | 2.99103 | + | 0.231754i | −2.11687 | + | 1.22217i | |||
236.16 | 2.18143 | − | 1.25945i | −1.57808 | − | 0.713898i | 2.17243 | − | 3.76276i | −1.00000 | −4.34160 | + | 0.430199i | −2.57718 | − | 0.598455i | − | 5.90648i | 1.98070 | + | 2.25318i | −2.18143 | + | 1.25945i | |||
311.1 | −2.27381 | − | 1.31279i | −0.933716 | − | 1.45883i | 2.44681 | + | 4.23800i | −1.00000 | 0.207969 | + | 4.54286i | −2.64220 | + | 0.136993i | − | 7.59741i | −1.25635 | + | 2.72426i | 2.27381 | + | 1.31279i | |||
311.2 | −2.03337 | − | 1.17397i | 1.49299 | − | 0.878053i | 1.75639 | + | 3.04216i | −1.00000 | −4.06660 | + | 0.0326841i | 0.135526 | + | 2.64228i | − | 3.55190i | 1.45805 | − | 2.62185i | 2.03337 | + | 1.17397i | |||
311.3 | −1.91788 | − | 1.10729i | 0.302150 | + | 1.70549i | 1.45217 | + | 2.51523i | −1.00000 | 1.30898 | − | 3.60549i | 2.53521 | + | 0.756781i | − | 2.00271i | −2.81741 | + | 1.03063i | 1.91788 | + | 1.10729i | |||
311.4 | −1.67628 | − | 0.967799i | −1.71487 | − | 0.243339i | 0.873269 | + | 1.51255i | −1.00000 | 2.63910 | + | 2.06755i | 2.28906 | − | 1.32673i | 0.490599i | 2.88157 | + | 0.834589i | 1.67628 | + | 0.967799i | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.be.c | yes | 32 |
3.b | odd | 2 | 1 | 945.2.be.c | 32 | ||
7.d | odd | 6 | 1 | 315.2.t.c | ✓ | 32 | |
9.c | even | 3 | 1 | 945.2.t.c | 32 | ||
9.d | odd | 6 | 1 | 315.2.t.c | ✓ | 32 | |
21.g | even | 6 | 1 | 945.2.t.c | 32 | ||
63.k | odd | 6 | 1 | 945.2.be.c | 32 | ||
63.s | even | 6 | 1 | inner | 315.2.be.c | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.t.c | ✓ | 32 | 7.d | odd | 6 | 1 | |
315.2.t.c | ✓ | 32 | 9.d | odd | 6 | 1 | |
315.2.be.c | yes | 32 | 1.a | even | 1 | 1 | trivial |
315.2.be.c | yes | 32 | 63.s | even | 6 | 1 | inner |
945.2.t.c | 32 | 9.c | even | 3 | 1 | ||
945.2.t.c | 32 | 21.g | even | 6 | 1 | ||
945.2.be.c | 32 | 3.b | odd | 2 | 1 | ||
945.2.be.c | 32 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 24 T_{2}^{30} + 348 T_{2}^{28} - 3298 T_{2}^{26} - 6 T_{2}^{25} + 23178 T_{2}^{24} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).