Properties

Label 9.15.d.a
Level $9$
Weight $15$
Character orbit 9.d
Analytic conductor $11.190$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,15,Mod(2,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 9.d (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: \(11.1896071337\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 3 q^{2} - 2199 q^{3} + 98303 q^{4} - 107994 q^{5} + 24957 q^{6} - 146330 q^{7} - 5316525 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 3 q^{2} - 2199 q^{3} + 98303 q^{4} - 107994 q^{5} + 24957 q^{6} - 146330 q^{7} - 5316525 q^{9} - 32772 q^{10} - 15978711 q^{11} - 53795046 q^{12} - 23281436 q^{13} - 349442850 q^{14} + 207365400 q^{15} - 671072257 q^{16} - 2272246668 q^{18} + 195706222 q^{19} + 2822935854 q^{20} + 1958992050 q^{21} - 204235521 q^{22} - 6101133672 q^{23} + 10931998527 q^{24} + 14614789727 q^{25} - 34016586282 q^{27} - 9052979204 q^{28} + 85191770166 q^{29} + 91857763818 q^{30} + 15357554728 q^{31} - 237242922057 q^{32} + 76352726421 q^{33} - 16566014943 q^{34} - 570464313633 q^{36} - 77721425444 q^{37} + 476314869819 q^{38} + 353897273784 q^{39} - 88855733850 q^{40} - 938882938233 q^{41} + 700091392140 q^{42} - 29709800063 q^{43} - 84148700094 q^{45} + 1553142786144 q^{46} - 1272741434580 q^{47} + 803069876301 q^{48} - 945184029255 q^{49} + 3573994524699 q^{50} - 1993755292875 q^{51} + 1825489374856 q^{52} + 6488294941323 q^{54} - 3772940931372 q^{55} - 7790759725422 q^{56} - 10739943123909 q^{57} + 3747043240302 q^{58} + 5874788534451 q^{59} + 1848130753878 q^{60} - 4867968901544 q^{61} + 5766639947244 q^{63} + 1211169309434 q^{64} - 22156719464856 q^{65} - 6375059860230 q^{66} - 8251061484125 q^{67} + 30687017243265 q^{68} + 23068377735552 q^{69} + 6871822387932 q^{70} - 46243200240621 q^{72} + 16120119648922 q^{73} + 8549243621166 q^{74} + 1090840096221 q^{75} + 14422972781521 q^{76} + 13919363311398 q^{77} + 109811947190436 q^{78} - 10740904544582 q^{79} - 140975607018501 q^{81} - 67076318933382 q^{82} - 11295125984532 q^{83} + 30827942753826 q^{84} + 10590724937466 q^{85} + 97019495740221 q^{86} + 151937485500582 q^{87} - 26136212294541 q^{88} - 392755154255226 q^{90} + 70870424994080 q^{91} - 474274297000284 q^{92} - 124466752375728 q^{93} - 46678164316620 q^{94} + 440317551118746 q^{95} + 823174447870656 q^{96} + 7621755375583 q^{97} - 195117933369168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −193.294 111.598i 1404.39 + 1676.50i 16716.4 + 28953.6i −20060.8 + 11582.1i −84364.2 480785.i 361557. 626235.i 3.80522e6i −838368. + 4.70892e6i 5.17016e6
2.2 −165.573 95.5937i 919.037 1984.52i 10084.3 + 17466.5i 5072.07 2928.36i −341876. + 240730.i −596946. + 1.03394e6i 723570.i −3.09371e6 3.64771e6i −1.11973e6
2.3 −164.198 94.7998i −2133.26 481.824i 9782.02 + 16943.0i 93735.9 54118.4i 304601. + 281348.i 584546. 1.01246e6i 602934.i 4.31866e6 + 2.05572e6i −2.05217e7
2.4 −122.622 70.7960i −1808.59 + 1229.62i 1832.14 + 3173.36i −101737. + 58738.1i 308826. 22738.4i −525106. + 909510.i 1.80101e6i 1.75902e6 4.44777e6i 1.66337e7
2.5 −47.7787 27.5850i 2180.12 173.279i −6670.13 11553.0i 20922.0 12079.3i −108943. 51859.8i 105744. 183153.i 1.63989e6i 4.72292e6 755540.i −1.33284e6
2.6 −40.5755 23.4263i −594.691 2104.59i −7094.42 12287.9i −113204. + 65358.5i −25172.9 + 99326.3i 742609. 1.28624e6i 1.43242e6i −4.07565e6 + 2.50316e6i 6.12443e6
2.7 −36.6279 21.1471i 13.1197 + 2186.96i −7297.60 12639.8i 53737.7 31025.5i 45767.4 80381.2i −26704.6 + 46253.7i 1.31024e6i −4.78262e6 + 57384.4i −2.62440e6
2.8 25.4810 + 14.7114i −1531.84 1560.91i −7759.15 13439.2i 68095.1 39314.7i −16069.4 62309.0i −576494. + 998517.i 938657.i −89920.6 + 4.78212e6i 2.31351e6
2.9 104.538 + 60.3552i −1916.47 + 1053.62i −906.505 1570.11i −10010.1 + 5779.31i −263936. 5524.99i 308433. 534222.i 2.19657e6i 2.56273e6 4.03847e6i −1.39525e6
2.10 104.562 + 60.3690i 1696.28 + 1380.43i −903.169 1564.33i −116056. + 67005.0i 94031.7 + 246744.i −391422. + 677963.i 2.19626e6i 971781. + 4.68321e6i −1.61801e7
2.11 129.548 + 74.7948i 1398.48 1681.44i 2996.54 + 5190.15i 31950.1 18446.4i 306934. 113228.i 153772. 266342.i 1.55438e6i −871480. 4.70290e6i 5.51879e6
2.12 195.306 + 112.760i 840.821 + 2018.91i 17237.5 + 29856.3i 103308. 59645.1i −63434.5 + 489115.i 163246. 282751.i 4.07989e6i −3.36901e6 + 3.39508e6i 2.69023e7
2.13 209.734 + 121.090i −1566.90 1525.71i 21133.6 + 36604.4i −69749.9 + 40270.1i −143885. 509729.i −376400. + 651944.i 6.26837e6i 127396. + 4.78127e6i −1.95052e7
5.1 −193.294 + 111.598i 1404.39 1676.50i 16716.4 28953.6i −20060.8 11582.1i −84364.2 + 480785.i 361557. + 626235.i 3.80522e6i −838368. 4.70892e6i 5.17016e6
5.2 −165.573 + 95.5937i 919.037 + 1984.52i 10084.3 17466.5i 5072.07 + 2928.36i −341876. 240730.i −596946. 1.03394e6i 723570.i −3.09371e6 + 3.64771e6i −1.11973e6
5.3 −164.198 + 94.7998i −2133.26 + 481.824i 9782.02 16943.0i 93735.9 + 54118.4i 304601. 281348.i 584546. + 1.01246e6i 602934.i 4.31866e6 2.05572e6i −2.05217e7
5.4 −122.622 + 70.7960i −1808.59 1229.62i 1832.14 3173.36i −101737. 58738.1i 308826. + 22738.4i −525106. 909510.i 1.80101e6i 1.75902e6 + 4.44777e6i 1.66337e7
5.5 −47.7787 + 27.5850i 2180.12 + 173.279i −6670.13 + 11553.0i 20922.0 + 12079.3i −108943. + 51859.8i 105744. + 183153.i 1.63989e6i 4.72292e6 + 755540.i −1.33284e6
5.6 −40.5755 + 23.4263i −594.691 + 2104.59i −7094.42 + 12287.9i −113204. 65358.5i −25172.9 99326.3i 742609. + 1.28624e6i 1.43242e6i −4.07565e6 2.50316e6i 6.12443e6
5.7 −36.6279 + 21.1471i 13.1197 2186.96i −7297.60 + 12639.8i 53737.7 + 31025.5i 45767.4 + 80381.2i −26704.6 46253.7i 1.31024e6i −4.78262e6 57384.4i −2.62440e6
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.15.d.a 26
3.b odd 2 1 27.15.d.a 26
9.c even 3 1 27.15.d.a 26
9.c even 3 1 81.15.b.a 26
9.d odd 6 1 inner 9.15.d.a 26
9.d odd 6 1 81.15.b.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.15.d.a 26 1.a even 1 1 trivial
9.15.d.a 26 9.d odd 6 1 inner
27.15.d.a 26 3.b odd 2 1
27.15.d.a 26 9.c even 3 1
81.15.b.a 26 9.c even 3 1
81.15.b.a 26 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(9, [\chi])\).