Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(23,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 9, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bv (of order \(12\), degree \(4\), 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: | \(176\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.86946 | − | 1.86946i | −0.784062 | − | 1.54442i | 4.98975i | −0.167803 | − | 2.22976i | −1.42147 | + | 4.35301i | −2.39984 | − | 1.11389i | 5.58921 | − | 5.58921i | −1.77049 | + | 2.42185i | −3.85475 | + | 4.48215i | ||
23.2 | −1.77797 | − | 1.77797i | 0.265860 | − | 1.71153i | 4.32238i | −0.447954 | + | 2.19074i | −3.51574 | + | 2.57035i | 1.37680 | + | 2.25930i | 4.12912 | − | 4.12912i | −2.85864 | − | 0.910053i | 4.69153 | − | 3.09862i | ||
23.3 | −1.76616 | − | 1.76616i | −1.54103 | + | 0.790710i | 4.23867i | 1.75658 | + | 1.38362i | 4.11824 | + | 1.32519i | −2.15527 | + | 1.53454i | 3.95386 | − | 3.95386i | 1.74955 | − | 2.43702i | −0.658708 | − | 5.54612i | ||
23.4 | −1.76155 | − | 1.76155i | 1.72928 | + | 0.0979388i | 4.20612i | −2.22663 | + | 0.205270i | −2.87369 | − | 3.21874i | 0.0218295 | − | 2.64566i | 3.88619 | − | 3.88619i | 2.98082 | + | 0.338727i | 4.28391 | + | 3.56072i | ||
23.5 | −1.69531 | − | 1.69531i | 1.72524 | − | 0.153414i | 3.74812i | 1.48276 | − | 1.67374i | −3.18490 | − | 2.66473i | 1.46770 | + | 2.20133i | 2.96360 | − | 2.96360i | 2.95293 | − | 0.529352i | −5.35124 | + | 0.323772i | ||
23.6 | −1.57216 | − | 1.57216i | 0.440043 | + | 1.67522i | 2.94339i | 0.402568 | + | 2.19953i | 1.94190 | − | 3.32554i | 2.54364 | − | 0.727930i | 1.48316 | − | 1.48316i | −2.61272 | + | 1.47434i | 2.82512 | − | 4.09092i | ||
23.7 | −1.41417 | − | 1.41417i | −1.46976 | − | 0.916412i | 1.99973i | 2.20169 | + | 0.390582i | 0.782522 | + | 3.37444i | 1.74936 | − | 1.98488i | −0.000387571 | 0 | 0.000387571i | 1.32038 | + | 2.69381i | −2.56121 | − | 3.66590i | ||
23.8 | −1.33173 | − | 1.33173i | −1.22217 | + | 1.22732i | 1.54698i | −2.16580 | + | 0.556151i | 3.26204 | − | 0.00685671i | −1.84223 | − | 1.89899i | −0.603293 | + | 0.603293i | −0.0126118 | − | 2.99997i | 3.62489 | + | 2.14361i | ||
23.9 | −1.29695 | − | 1.29695i | −1.63045 | − | 0.584483i | 1.36418i | −2.10879 | − | 0.743641i | 1.35657 | + | 2.87267i | 1.22224 | + | 2.34651i | −0.824634 | + | 0.824634i | 2.31676 | + | 1.90595i | 1.77054 | + | 3.69947i | ||
23.10 | −1.14944 | − | 1.14944i | 0.768701 | + | 1.55213i | 0.642437i | −0.945461 | − | 2.02635i | 0.900504 | − | 2.66766i | −1.70140 | + | 2.02614i | −1.56044 | + | 1.56044i | −1.81820 | + | 2.38624i | −1.24242 | + | 3.41593i | ||
23.11 | −1.14308 | − | 1.14308i | 1.67242 | + | 0.450568i | 0.613246i | 0.197501 | + | 2.22733i | −1.39667 | − | 2.42674i | −2.27793 | + | 1.34575i | −1.58517 | + | 1.58517i | 2.59398 | + | 1.50708i | 2.32025 | − | 2.77176i | ||
23.12 | −1.08313 | − | 1.08313i | 1.10330 | − | 1.33519i | 0.346328i | 2.23510 | − | 0.0656858i | −2.64119 | + | 0.251168i | −2.63546 | − | 0.233113i | −1.79114 | + | 1.79114i | −0.565465 | − | 2.94623i | −2.49205 | − | 2.34975i | ||
23.13 | −1.02458 | − | 1.02458i | 0.918977 | − | 1.46816i | 0.0995230i | −0.891651 | − | 2.05060i | −2.44581 | + | 0.562677i | 2.22810 | − | 1.42674i | −1.94719 | + | 1.94719i | −1.31096 | − | 2.69840i | −1.18743 | + | 3.01457i | ||
23.14 | −0.902350 | − | 0.902350i | −0.320197 | − | 1.70220i | − | 0.371529i | −0.953884 | + | 2.02240i | −1.24705 | + | 1.82491i | −0.594335 | − | 2.57813i | −2.13995 | + | 2.13995i | −2.79495 | + | 1.09008i | 2.68565 | − | 0.964176i | |
23.15 | −0.710963 | − | 0.710963i | −0.818385 | + | 1.52651i | − | 0.989064i | 2.18368 | − | 0.481173i | 1.66714 | − | 0.503454i | 1.29429 | + | 2.30756i | −2.12511 | + | 2.12511i | −1.66049 | − | 2.49855i | −1.89461 | − | 1.21042i | |
23.16 | −0.481194 | − | 0.481194i | 1.08354 | + | 1.35127i | − | 1.53691i | −2.19786 | + | 0.411594i | 0.128828 | − | 1.17162i | 2.63650 | + | 0.221076i | −1.70194 | + | 1.70194i | −0.651863 | + | 2.92832i | 1.25565 | + | 0.859540i | |
23.17 | −0.477311 | − | 0.477311i | −1.72825 | + | 0.114750i | − | 1.54435i | 0.814847 | − | 2.08231i | 0.879681 | + | 0.770138i | −2.51050 | − | 0.835101i | −1.69176 | + | 1.69176i | 2.97366 | − | 0.396634i | −1.38284 | + | 0.604974i | |
23.18 | −0.421918 | − | 0.421918i | 1.58110 | + | 0.707197i | − | 1.64397i | 2.16393 | − | 0.563383i | −0.368715 | − | 0.965474i | 1.26919 | − | 2.32145i | −1.53746 | + | 1.53746i | 1.99974 | + | 2.23630i | −1.15070 | − | 0.675301i | |
23.19 | −0.391477 | − | 0.391477i | −1.67401 | + | 0.444638i | − | 1.69349i | −0.676575 | + | 2.13125i | 0.829400 | + | 0.481269i | 2.43542 | − | 1.03378i | −1.44592 | + | 1.44592i | 2.60459 | − | 1.48865i | 1.09920 | − | 0.569474i | |
23.20 | −0.0820754 | − | 0.0820754i | −0.222382 | − | 1.71772i | − | 1.98653i | −2.22963 | − | 0.169605i | −0.122730 | + | 0.159234i | −1.67271 | + | 2.04989i | −0.327196 | + | 0.327196i | −2.90109 | + | 0.763977i | 0.169077 | + | 0.196918i | |
See next 80 embeddings (of 176 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
63.j | odd | 6 | 1 | inner |
315.bv | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.bv.a | ✓ | 176 |
3.b | odd | 2 | 1 | 945.2.by.a | 176 | ||
5.c | odd | 4 | 1 | inner | 315.2.bv.a | ✓ | 176 |
7.c | even | 3 | 1 | 315.2.bx.a | yes | 176 | |
9.c | even | 3 | 1 | 945.2.ca.a | 176 | ||
9.d | odd | 6 | 1 | 315.2.bx.a | yes | 176 | |
15.e | even | 4 | 1 | 945.2.by.a | 176 | ||
21.h | odd | 6 | 1 | 945.2.ca.a | 176 | ||
35.l | odd | 12 | 1 | 315.2.bx.a | yes | 176 | |
45.k | odd | 12 | 1 | 945.2.ca.a | 176 | ||
45.l | even | 12 | 1 | 315.2.bx.a | yes | 176 | |
63.h | even | 3 | 1 | 945.2.by.a | 176 | ||
63.j | odd | 6 | 1 | inner | 315.2.bv.a | ✓ | 176 |
105.x | even | 12 | 1 | 945.2.ca.a | 176 | ||
315.bt | odd | 12 | 1 | 945.2.by.a | 176 | ||
315.bv | even | 12 | 1 | inner | 315.2.bv.a | ✓ | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bv.a | ✓ | 176 | 1.a | even | 1 | 1 | trivial |
315.2.bv.a | ✓ | 176 | 5.c | odd | 4 | 1 | inner |
315.2.bv.a | ✓ | 176 | 63.j | odd | 6 | 1 | inner |
315.2.bv.a | ✓ | 176 | 315.bv | even | 12 | 1 | inner |
315.2.bx.a | yes | 176 | 7.c | even | 3 | 1 | |
315.2.bx.a | yes | 176 | 9.d | odd | 6 | 1 | |
315.2.bx.a | yes | 176 | 35.l | odd | 12 | 1 | |
315.2.bx.a | yes | 176 | 45.l | even | 12 | 1 | |
945.2.by.a | 176 | 3.b | odd | 2 | 1 | ||
945.2.by.a | 176 | 15.e | even | 4 | 1 | ||
945.2.by.a | 176 | 63.h | even | 3 | 1 | ||
945.2.by.a | 176 | 315.bt | odd | 12 | 1 | ||
945.2.ca.a | 176 | 9.c | even | 3 | 1 | ||
945.2.ca.a | 176 | 21.h | odd | 6 | 1 | ||
945.2.ca.a | 176 | 45.k | odd | 12 | 1 | ||
945.2.ca.a | 176 | 105.x | even | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).