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Properties

Label 931.2.be.b
Level $931$
Weight $2$
Character orbit 931.be
Analytic conductor $7.434$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(48,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.48");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.be (of order \(18\), degree \(6\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{2} + 6 q^{4} - 18 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{2} + 6 q^{4} - 18 q^{8} + 6 q^{9} + 27 q^{10} + 6 q^{11} + 3 q^{12} + 30 q^{13} + 9 q^{15} + 12 q^{16} + 9 q^{17} - 12 q^{19} + 24 q^{23} - 27 q^{24} - 9 q^{26} - 12 q^{27} - 6 q^{29} + 3 q^{30} + 9 q^{31} - 30 q^{32} + 36 q^{34} + 27 q^{36} + 27 q^{38} - 24 q^{39} - 9 q^{40} - 54 q^{41} + 12 q^{43} + 45 q^{44} - 27 q^{45} - 72 q^{46} + 54 q^{47} - 63 q^{48} - 63 q^{50} + 6 q^{51} - 48 q^{52} - 18 q^{54} + 45 q^{55} - 54 q^{57} - 60 q^{58} + 36 q^{59} - 6 q^{60} + 6 q^{61} + 45 q^{62} - 36 q^{64} - 45 q^{65} - 144 q^{66} - 51 q^{67} - 9 q^{68} - 6 q^{71} + 12 q^{72} + 66 q^{73} + 90 q^{74} - 42 q^{75} - 54 q^{76} + 3 q^{78} - 36 q^{79} - 72 q^{80} - 39 q^{81} + 45 q^{82} - 36 q^{83} - 48 q^{85} - 66 q^{86} + 9 q^{87} + 9 q^{88} + 18 q^{90} + 48 q^{92} + 87 q^{93} + 180 q^{94} + 78 q^{95} + 27 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
48.1 −2.36394 + 0.416827i −0.183442 + 0.153926i 3.53510 − 1.28667i −0.320710 + 0.881145i 0.369487 − 0.440337i 0 −3.66282 + 2.11473i −0.510987 + 2.89795i 0.390856 − 2.21666i
48.2 −2.27811 + 0.401692i 1.89663 − 1.59146i 3.14903 − 1.14615i 0.265932 − 0.730641i −3.68144 + 4.38737i 0 −2.70677 + 1.56275i 0.543507 − 3.08238i −0.312328 + 1.77130i
48.3 −1.25371 + 0.221064i −1.53410 + 1.28726i −0.356454 + 0.129739i −0.814183 + 2.23695i 1.63876 − 1.95299i 0 2.62321 − 1.51451i 0.175473 − 0.995155i 0.526245 − 2.98448i
48.4 −0.736767 + 0.129912i 0.400026 − 0.335662i −1.35344 + 0.492611i 0.554316 − 1.52297i −0.251120 + 0.299273i 0 2.22898 − 1.28690i −0.473592 + 2.68588i −0.210550 + 1.19409i
48.5 −0.635026 + 0.111972i 1.39334 − 1.16915i −1.48867 + 0.541830i −1.23911 + 3.40443i −0.753896 + 0.898458i 0 2.00154 − 1.15559i 0.0535396 − 0.303638i 0.405667 − 2.30065i
48.6 0.572377 − 0.100926i −1.27610 + 1.07077i −1.56196 + 0.568505i 0.102345 − 0.281190i −0.622340 + 0.741676i 0 −1.84333 + 1.06425i −0.0390756 + 0.221608i 0.0302006 − 0.171276i
48.7 0.662111 − 0.116748i 2.26756 − 1.90271i −1.45462 + 0.529440i 0.333702 − 0.916840i 1.27924 − 1.52454i 0 −2.06581 + 1.19270i 1.00058 − 5.67458i 0.113909 − 0.646009i
48.8 1.21113 − 0.213555i −0.125493 + 0.105301i −0.458160 + 0.166757i 0.295476 − 0.811814i −0.129500 + 0.154333i 0 −2.64937 + 1.52962i −0.516284 + 2.92799i 0.184493 − 1.04631i
48.9 1.95081 − 0.343981i −2.10377 + 1.76528i 1.80796 − 0.658045i −0.766026 + 2.10464i −3.49685 + 4.16738i 0 −0.130386 + 0.0752786i 0.788722 − 4.47306i −0.770417 + 4.36925i
48.10 2.32891 − 0.410650i 1.93710 − 1.62542i 3.37581 − 1.22869i −0.362921 + 0.997117i 3.84386 − 4.58094i 0 3.26138 − 1.88296i 0.589425 − 3.34280i −0.435745 + 2.47123i
48.11 2.48191 − 0.437628i −0.199970 + 0.167794i 4.08898 − 1.48827i 1.43024 − 3.92955i −0.422875 + 0.503963i 0 5.13205 − 2.96299i −0.509112 + 2.88732i 1.83004 − 10.3787i
97.1 −2.36394 − 0.416827i −0.183442 − 0.153926i 3.53510 + 1.28667i −0.320710 − 0.881145i 0.369487 + 0.440337i 0 −3.66282 − 2.11473i −0.510987 − 2.89795i 0.390856 + 2.21666i
97.2 −2.27811 − 0.401692i 1.89663 + 1.59146i 3.14903 + 1.14615i 0.265932 + 0.730641i −3.68144 − 4.38737i 0 −2.70677 − 1.56275i 0.543507 + 3.08238i −0.312328 − 1.77130i
97.3 −1.25371 − 0.221064i −1.53410 − 1.28726i −0.356454 − 0.129739i −0.814183 − 2.23695i 1.63876 + 1.95299i 0 2.62321 + 1.51451i 0.175473 + 0.995155i 0.526245 + 2.98448i
97.4 −0.736767 − 0.129912i 0.400026 + 0.335662i −1.35344 − 0.492611i 0.554316 + 1.52297i −0.251120 − 0.299273i 0 2.22898 + 1.28690i −0.473592 − 2.68588i −0.210550 − 1.19409i
97.5 −0.635026 − 0.111972i 1.39334 + 1.16915i −1.48867 − 0.541830i −1.23911 − 3.40443i −0.753896 − 0.898458i 0 2.00154 + 1.15559i 0.0535396 + 0.303638i 0.405667 + 2.30065i
97.6 0.572377 + 0.100926i −1.27610 − 1.07077i −1.56196 − 0.568505i 0.102345 + 0.281190i −0.622340 − 0.741676i 0 −1.84333 − 1.06425i −0.0390756 − 0.221608i 0.0302006 + 0.171276i
97.7 0.662111 + 0.116748i 2.26756 + 1.90271i −1.45462 − 0.529440i 0.333702 + 0.916840i 1.27924 + 1.52454i 0 −2.06581 − 1.19270i 1.00058 + 5.67458i 0.113909 + 0.646009i
97.8 1.21113 + 0.213555i −0.125493 − 0.105301i −0.458160 − 0.166757i 0.295476 + 0.811814i −0.129500 − 0.154333i 0 −2.64937 − 1.52962i −0.516284 − 2.92799i 0.184493 + 1.04631i
97.9 1.95081 + 0.343981i −2.10377 − 1.76528i 1.80796 + 0.658045i −0.766026 − 2.10464i −3.49685 − 4.16738i 0 −0.130386 − 0.0752786i 0.788722 + 4.47306i −0.770417 − 4.36925i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 48.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.ba even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.be.b 66
7.b odd 2 1 931.2.be.a 66
7.c even 3 1 133.2.bb.a ✓ 66
7.c even 3 1 931.2.bj.a 66
7.d odd 6 1 133.2.bf.a yes 66
7.d odd 6 1 931.2.bf.a 66
19.f odd 18 1 931.2.be.a 66
133.ba even 18 1 inner 931.2.be.b 66
133.bb even 18 1 931.2.bj.a 66
133.bd odd 18 1 133.2.bf.a yes 66
133.be odd 18 1 931.2.bf.a 66
133.bf even 18 1 133.2.bb.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.bb.a ✓ 66 7.c even 3 1
133.2.bb.a ✓ 66 133.bf even 18 1
133.2.bf.a yes 66 7.d odd 6 1
133.2.bf.a yes 66 133.bd odd 18 1
931.2.be.a 66 7.b odd 2 1
931.2.be.a 66 19.f odd 18 1
931.2.be.b 66 1.a even 1 1 trivial
931.2.be.b 66 133.ba even 18 1 inner
931.2.bf.a 66 7.d odd 6 1
931.2.bf.a 66 133.be odd 18 1
931.2.bj.a 66 7.c even 3 1
931.2.bj.a 66 133.bb even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{66} - 6 T_{2}^{65} + 15 T_{2}^{64} - 12 T_{2}^{63} - 39 T_{2}^{62} + 150 T_{2}^{61} - 584 T_{2}^{60} + \cdots + 243 \) Copy content Toggle raw display
\( T_{3}^{66} - 3 T_{3}^{64} + 10 T_{3}^{63} + 21 T_{3}^{62} + 123 T_{3}^{61} + 789 T_{3}^{60} + \cdots + 11229201 \) Copy content Toggle raw display