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Properties

Label 931.2.bj.a
Level $931$
Weight $2$
Character orbit 931.bj
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(117,931)]
 
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([15, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.bj (of order \(18\), degree \(6\), not 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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} + 9 q^{3} - 3 q^{4} + 9 q^{5} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} + 9 q^{3} - 3 q^{4} + 9 q^{5} - 18 q^{8} - 3 q^{9} + 9 q^{10} - 12 q^{11} - 6 q^{12} + 30 q^{13} + 9 q^{15} - 15 q^{16} - 18 q^{17} + 36 q^{18} - 12 q^{19} - 3 q^{23} + 36 q^{24} - 27 q^{25} - 12 q^{27} - 6 q^{29} + 3 q^{30} + 9 q^{31} + 60 q^{32} + 9 q^{33} + 36 q^{34} + 27 q^{36} - 36 q^{37} - 18 q^{38} + 12 q^{39} - 9 q^{40} - 54 q^{41} + 12 q^{43} + 18 q^{44} + 27 q^{45} - 45 q^{47} - 63 q^{48} - 63 q^{50} - 3 q^{51} - 57 q^{52} + 27 q^{53} + 9 q^{54} + 45 q^{55} - 54 q^{57} + 30 q^{58} - 36 q^{59} - 78 q^{60} + 42 q^{61} + 45 q^{62} - 36 q^{64} + 45 q^{65} - 9 q^{66} + 30 q^{67} + 9 q^{68} - 6 q^{71} - 6 q^{72} - 60 q^{73} + 9 q^{74} + 21 q^{75} - 54 q^{76} + 3 q^{78} + 27 q^{79} + 45 q^{80} + 24 q^{81} + 9 q^{82} - 36 q^{83} - 48 q^{85} - 48 q^{86} - 9 q^{88} + 9 q^{89} + 18 q^{90} + 48 q^{92} - 3 q^{93} - 90 q^{94} - 75 q^{95} - 63 q^{96} + 27 q^{97} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −0.736235 − 2.02279i −0.719095 − 0.603392i −2.01754 + 1.69292i −0.606188 + 0.722427i −0.691112 + 1.89882i 0 1.18138 + 0.682070i −0.367929 − 2.08663i 1.90761 + 0.694315i
117.2 −0.587321 − 1.61365i 2.22906 + 1.87040i −0.726837 + 0.609889i 2.43273 − 2.89922i 1.70900 − 4.69545i 0 −1.56326 − 0.902551i 0.949353 + 5.38405i −6.10712 − 2.22281i
117.3 −0.578027 − 1.58812i −1.65197 − 1.38617i −0.655906 + 0.550371i −0.486861 + 0.580218i −1.24651 + 3.42476i 0 −1.67405 − 0.966511i 0.286602 + 1.62540i 1.20287 + 0.437809i
117.4 −0.436734 − 1.19992i 1.30218 + 1.09266i 0.283028 − 0.237489i −1.54412 + 1.84020i 0.742394 − 2.03971i 0 −2.62027 − 1.51281i −0.0191728 − 0.108734i 2.88246 + 1.04913i
117.5 −0.115959 − 0.318595i −2.21309 − 1.85700i 1.44403 − 1.21169i 1.47942 − 1.76310i −0.335004 + 0.920417i 0 −1.14072 − 0.658598i 0.928362 + 5.26501i −0.733267 − 0.266887i
117.6 −0.0315715 − 0.0867419i 1.08833 + 0.913216i 1.52556 − 1.28010i −0.251137 + 0.299294i 0.0448540 − 0.123235i 0 −0.319086 − 0.184224i −0.170449 − 0.966664i 0.0338901 + 0.0123350i
117.7 0.180665 + 0.496373i −0.319296 − 0.267922i 1.31834 − 1.10622i 1.46488 − 1.74578i 0.0753033 − 0.206894i 0 1.70220 + 0.982764i −0.490776 − 2.78333i 1.13121 + 0.411727i
117.8 0.504612 + 1.38641i 1.96840 + 1.65169i −0.135411 + 0.113624i −0.974518 + 1.16139i −1.29664 + 3.56248i 0 2.32959 + 1.34499i 0.625601 + 3.54796i −2.10191 − 0.765033i
117.9 0.527515 + 1.44933i −0.946314 − 0.794052i −0.290211 + 0.243516i −1.11372 + 1.32728i 0.651652 − 1.79040i 0 2.16540 + 1.25020i −0.255952 − 1.45158i −2.51118 − 0.913996i
117.10 0.785212 + 2.15735i 1.58192 + 1.32739i −2.50552 + 2.10238i 2.15552 − 2.56885i −1.62150 + 4.45504i 0 −2.52649 − 1.45867i 0.219567 + 1.24523i 7.23445 + 2.63313i
117.11 0.927536 + 2.54838i −0.646481 − 0.542462i −4.10185 + 3.44186i −1.05601 + 1.25851i 0.782767 − 2.15063i 0 −7.87857 − 4.54870i −0.397272 − 2.25304i −4.18664 − 1.52381i
129.1 −2.42618 + 0.427801i −1.81066 − 0.659027i 3.82394 − 1.39180i 0.965340 − 2.65225i 4.67492 + 0.824314i 0 −4.41506 + 2.54904i 0.546042 + 0.458184i −1.20745 + 6.84780i
129.2 −1.99215 + 0.351269i 0.662476 + 0.241122i 1.96587 − 0.715519i −0.0536790 + 0.147482i −1.40445 − 0.247642i 0 −0.161239 + 0.0930914i −1.91740 − 1.60889i 0.0551306 − 0.312661i
129.3 −1.84540 + 0.325394i 2.57736 + 0.938083i 1.42023 − 0.516922i 0.854536 − 2.34782i −5.06151 − 0.892480i 0 0.792944 − 0.457807i 3.46466 + 2.90719i −0.812995 + 4.61072i
129.4 −0.928599 + 0.163737i 0.187541 + 0.0682594i −1.04390 + 0.379948i −0.803754 + 2.20830i −0.185327 − 0.0326782i 0 2.54034 − 1.46667i −2.26762 − 1.90276i 0.384786 − 2.18223i
129.5 −0.697238 + 0.122942i −2.27698 − 0.828751i −1.40836 + 0.512601i 0.581955 − 1.59891i 1.68948 + 0.297901i 0 2.14522 − 1.23854i 2.19966 + 1.84573i −0.209188 + 1.18637i
129.6 −0.0226089 + 0.00398656i 1.47792 + 0.537920i −1.87889 + 0.683860i −0.513048 + 1.40959i −0.0355586 − 0.00626995i 0 0.0795172 − 0.0459093i −0.403236 − 0.338355i 0.00598004 − 0.0339145i
129.7 0.397723 − 0.0701294i 1.42173 + 0.517468i −1.72612 + 0.628256i 1.40124 − 3.84988i 0.601745 + 0.106104i 0 −1.34196 + 0.774783i −0.544588 − 0.456963i 0.287317 − 1.62946i
129.8 1.19145 − 0.210084i −1.79234 − 0.652360i −0.503976 + 0.183432i 0.404136 − 1.11035i −2.27253 − 0.400709i 0 −2.65741 + 1.53425i 0.488789 + 0.410143i 0.248238 − 1.40783i
129.9 1.20110 − 0.211786i −1.44898 − 0.527386i −0.481603 + 0.175289i −1.20912 + 3.32203i −1.85206 − 0.326568i 0 −2.65378 + 1.53216i −0.476721 − 0.400016i −0.748713 + 4.24616i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.bf 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.bj.a 66
7.b odd 2 1 133.2.bf.a yes 66
7.c even 3 1 133.2.bb.a ✓ 66
7.c even 3 1 931.2.be.b 66
7.d odd 6 1 931.2.be.a 66
7.d odd 6 1 931.2.bf.a 66
19.f odd 18 1 931.2.bf.a 66
133.ba even 18 1 133.2.bb.a ✓ 66
133.bb even 18 1 931.2.be.b 66
133.bd odd 18 1 931.2.be.a 66
133.be odd 18 1 133.2.bf.a yes 66
133.bf even 18 1 inner 931.2.bj.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.ba even 18 1
133.2.bf.a yes 66 7.b odd 2 1
133.2.bf.a yes 66 133.be odd 18 1
931.2.be.a 66 7.d odd 6 1
931.2.be.a 66 133.bd odd 18 1
931.2.be.b 66 7.c even 3 1
931.2.be.b 66 133.bb even 18 1
931.2.bf.a 66 7.d odd 6 1
931.2.bf.a 66 19.f odd 18 1
931.2.bj.a 66 1.a even 1 1 trivial
931.2.bj.a 66 133.bf even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 3 T_{2}^{65} + 6 T_{2}^{64} + 15 T_{2}^{63} + 42 T_{2}^{62} + 24 T_{2}^{61} - 395 T_{2}^{60} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\). Copy content Toggle raw display