Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,2,Mod(19,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 11, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("460.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 460.o (of order \(22\), degree \(10\), 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: | \(3.67311849298\) |
Analytic rank: | \(0\) |
Dimension: | \(640\) |
Relative dimension: | \(64\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.41365 | − | 0.0397971i | 2.67003 | − | 0.783991i | 1.99683 | + | 0.112519i | 1.97695 | + | 1.04482i | −3.80569 | + | 1.00203i | 0.790909 | − | 0.113716i | −2.81835 | − | 0.238531i | 3.99064 | − | 2.56463i | −2.75315 | − | 1.55569i |
19.2 | −1.41287 | + | 0.0615925i | 2.08651 | − | 0.612653i | 1.99241 | − | 0.174045i | −0.546640 | − | 2.16822i | −2.91023 | + | 0.994114i | −1.11477 | + | 0.160279i | −2.80430 | + | 0.368620i | 1.45440 | − | 0.934688i | 0.905878 | + | 3.02975i |
19.3 | −1.41242 | − | 0.0712068i | −1.44577 | + | 0.424517i | 1.98986 | + | 0.201148i | −0.632143 | − | 2.14485i | 2.07226 | − | 0.496647i | 4.69086 | − | 0.674443i | −2.79619 | − | 0.425797i | −0.613721 | + | 0.394415i | 0.740123 | + | 3.07445i |
19.4 | −1.40349 | + | 0.173829i | −0.723511 | + | 0.212442i | 1.93957 | − | 0.487934i | 1.78262 | − | 1.34991i | 0.978512 | − | 0.423927i | −1.71228 | + | 0.246189i | −2.63735 | + | 1.02196i | −2.04542 | + | 1.31451i | −2.26724 | + | 2.20446i |
19.5 | −1.40029 | + | 0.197940i | 1.48939 | − | 0.437323i | 1.92164 | − | 0.554348i | −2.20194 | + | 0.389205i | −1.99901 | + | 0.907190i | −0.230848 | + | 0.0331909i | −2.58113 | + | 1.15662i | −0.496740 | + | 0.319236i | 3.00632 | − | 0.980851i |
19.6 | −1.39440 | + | 0.235922i | −2.94595 | + | 0.865009i | 1.88868 | − | 0.657939i | −1.40285 | + | 1.74127i | 3.90375 | − | 1.90118i | 1.84920 | − | 0.265875i | −2.47835 | + | 1.36301i | 5.40662 | − | 3.47462i | 1.54533 | − | 2.75898i |
19.7 | −1.34217 | − | 0.445613i | −0.610817 | + | 0.179352i | 1.60286 | + | 1.19618i | −1.58117 | + | 1.58110i | 0.899744 | + | 0.0314662i | 0.165059 | − | 0.0237318i | −1.61828 | − | 2.31973i | −2.18283 | + | 1.40282i | 2.82677 | − | 1.41752i |
19.8 | −1.33042 | + | 0.479557i | −0.820997 | + | 0.241066i | 1.54005 | − | 1.27603i | 0.912951 | + | 2.04121i | 0.976668 | − | 0.714435i | −3.61150 | + | 0.519256i | −1.43699 | + | 2.43620i | −1.90784 | + | 1.22609i | −2.19348 | − | 2.27786i |
19.9 | −1.27975 | − | 0.601871i | −2.51556 | + | 0.738636i | 1.27550 | + | 1.54049i | 2.08492 | + | 0.808149i | 3.66385 | + | 0.568779i | −0.0121874 | + | 0.00175228i | −0.705145 | − | 2.73912i | 3.25872 | − | 2.09425i | −2.18177 | − | 2.28908i |
19.10 | −1.27640 | − | 0.608942i | 1.81640 | − | 0.533344i | 1.25838 | + | 1.55450i | 0.889872 | + | 2.05137i | −2.64323 | − | 0.425324i | −4.44634 | + | 0.639287i | −0.659592 | − | 2.75044i | 0.491100 | − | 0.315611i | 0.113335 | − | 3.16025i |
19.11 | −1.23367 | + | 0.691424i | 0.820997 | − | 0.241066i | 1.04387 | − | 1.70597i | 0.912951 | + | 2.04121i | −0.846157 | + | 0.865052i | 3.61150 | − | 0.519256i | −0.108234 | + | 2.82636i | −1.90784 | + | 1.22609i | −2.53761 | − | 1.88693i |
19.12 | −1.21441 | − | 0.724711i | −0.381027 | + | 0.111880i | 0.949588 | + | 1.76019i | −1.73128 | − | 1.41516i | 0.543804 | + | 0.140267i | −2.86375 | + | 0.411745i | 0.122443 | − | 2.82578i | −2.39110 | + | 1.53666i | 1.07690 | + | 2.97326i |
19.13 | −1.19551 | − | 0.755479i | 1.52020 | − | 0.446372i | 0.858502 | + | 1.80637i | 1.91991 | − | 1.14628i | −2.15465 | − | 0.614838i | 4.13344 | − | 0.594300i | 0.338324 | − | 2.80812i | −0.411991 | + | 0.264771i | −3.16126 | − | 0.0800530i |
19.14 | −1.09143 | + | 0.899318i | 2.94595 | − | 0.865009i | 0.382455 | − | 1.96309i | −1.40285 | + | 1.74127i | −2.43739 | + | 3.59345i | −1.84920 | + | 0.265875i | 1.34802 | + | 2.48653i | 5.40662 | − | 3.47462i | −0.0348340 | − | 3.16209i |
19.15 | −1.09031 | − | 0.900680i | 2.64511 | − | 0.776673i | 0.377553 | + | 1.96404i | −2.22046 | + | 0.263774i | −3.58352 | − | 1.53558i | 2.67954 | − | 0.385260i | 1.35732 | − | 2.48147i | 3.86960 | − | 2.48684i | 2.65856 | + | 1.71232i |
19.16 | −1.06659 | + | 0.928648i | −1.48939 | + | 0.437323i | 0.275227 | − | 1.98097i | −2.20194 | + | 0.389205i | 1.18245 | − | 1.84956i | 0.230848 | − | 0.0331909i | 1.54607 | + | 2.36847i | −0.496740 | + | 0.319236i | 1.98713 | − | 2.45994i |
19.17 | −1.05046 | + | 0.946853i | 0.723511 | − | 0.212442i | 0.206938 | − | 1.98927i | 1.78262 | − | 1.34991i | −0.558869 | + | 0.908221i | 1.71228 | − | 0.246189i | 1.66616 | + | 2.28559i | −2.04542 | + | 1.31451i | −0.594407 | + | 3.10591i |
19.18 | −0.971783 | + | 1.02744i | −2.08651 | + | 0.612653i | −0.111277 | − | 1.99690i | −0.546640 | − | 2.16822i | 1.39816 | − | 2.73913i | 1.11477 | − | 0.160279i | 2.15984 | + | 1.82622i | 1.45440 | − | 0.934688i | 2.75894 | + | 1.54540i |
19.19 | −0.925935 | − | 1.06895i | −0.832655 | + | 0.244489i | −0.285289 | + | 1.97955i | 0.527840 | + | 2.17287i | 1.03233 | + | 0.663681i | 3.68197 | − | 0.529388i | 2.38019 | − | 1.52797i | −1.89022 | + | 1.21477i | 1.83394 | − | 2.57617i |
19.20 | −0.921514 | − | 1.07276i | −2.25339 | + | 0.661656i | −0.301623 | + | 1.97713i | 0.592624 | − | 2.15611i | 2.78633 | + | 1.80762i | −0.146802 | + | 0.0211070i | 2.39893 | − | 1.49838i | 2.11623 | − | 1.36002i | −2.85909 | + | 1.35114i |
See next 80 embeddings (of 640 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
92.h | even | 22 | 1 | inner |
115.i | odd | 22 | 1 | inner |
460.o | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 460.2.o.b | ✓ | 640 |
4.b | odd | 2 | 1 | inner | 460.2.o.b | ✓ | 640 |
5.b | even | 2 | 1 | inner | 460.2.o.b | ✓ | 640 |
20.d | odd | 2 | 1 | inner | 460.2.o.b | ✓ | 640 |
23.d | odd | 22 | 1 | inner | 460.2.o.b | ✓ | 640 |
92.h | even | 22 | 1 | inner | 460.2.o.b | ✓ | 640 |
115.i | odd | 22 | 1 | inner | 460.2.o.b | ✓ | 640 |
460.o | even | 22 | 1 | inner | 460.2.o.b | ✓ | 640 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
460.2.o.b | ✓ | 640 | 1.a | even | 1 | 1 | trivial |
460.2.o.b | ✓ | 640 | 4.b | odd | 2 | 1 | inner |
460.2.o.b | ✓ | 640 | 5.b | even | 2 | 1 | inner |
460.2.o.b | ✓ | 640 | 20.d | odd | 2 | 1 | inner |
460.2.o.b | ✓ | 640 | 23.d | odd | 22 | 1 | inner |
460.2.o.b | ✓ | 640 | 92.h | even | 22 | 1 | inner |
460.2.o.b | ✓ | 640 | 115.i | odd | 22 | 1 | inner |
460.2.o.b | ✓ | 640 | 460.o | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{320} + 73 T_{3}^{318} + 2903 T_{3}^{316} + 82945 T_{3}^{314} + 1899638 T_{3}^{312} + \cdots + 58\!\cdots\!04 \) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).