Properties

Label 931.2.w.b
Level $931$
Weight $2$
Character orbit 931.w
Analytic conductor $7.434$
Analytic rank $0$
Dimension $30$
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] = [931,2,Mod(99,931)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.w (of order \(9\), 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: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} - 6 q^{4} - 3 q^{5} + 9 q^{6} + 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} - 6 q^{4} - 3 q^{5} + 9 q^{6} + 9 q^{8} - 3 q^{9} + 36 q^{10} - 18 q^{12} - 30 q^{15} + 18 q^{16} - 12 q^{17} - 30 q^{18} - 6 q^{19} + 30 q^{20} + 12 q^{22} - 15 q^{23} - 45 q^{24} + 21 q^{25} + 45 q^{26} - 27 q^{27} + 21 q^{30} - 12 q^{31} - 75 q^{32} - 18 q^{33} + 42 q^{34} - 21 q^{36} - 36 q^{37} - 57 q^{38} - 24 q^{39} + 39 q^{40} - 3 q^{41} - 12 q^{43} + 72 q^{44} - 27 q^{45} + 6 q^{46} - 27 q^{47} - 18 q^{48} - 3 q^{50} - 27 q^{51} + 15 q^{52} - 15 q^{53} + 57 q^{54} - 33 q^{55} - 12 q^{57} - 60 q^{58} + 84 q^{59} + 75 q^{60} - 42 q^{61} - 96 q^{62} + 3 q^{64} + 45 q^{65} - 99 q^{66} + 39 q^{67} - 51 q^{68} - 15 q^{69} + 30 q^{71} - 117 q^{72} - 15 q^{73} + 24 q^{74} + 144 q^{75} - 84 q^{76} + 6 q^{78} + 12 q^{79} + 165 q^{80} - 15 q^{81} - 3 q^{82} - 12 q^{83} + 42 q^{85} - 48 q^{86} - 24 q^{87} + 36 q^{88} + 66 q^{89} - 180 q^{90} - 75 q^{92} + 30 q^{93} + 42 q^{94} - 15 q^{95} - 66 q^{96} + 63 q^{97} + 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −1.80311 0.656278i −0.100636 0.570737i 1.28842 + 1.08111i 0.131215 0.110102i −0.193104 + 1.09515i 0 0.305182 + 0.528591i 2.50346 0.911187i −0.308853 + 0.112413i
99.2 −1.65002 0.600557i 0.488762 + 2.77191i 0.829796 + 0.696282i −2.66117 + 2.23299i 0.858223 4.86723i 0 0.804890 + 1.39411i −4.62552 + 1.68355i 5.73201 2.08628i
99.3 0.712406 + 0.259295i 0.218901 + 1.24145i −1.09180 0.916129i −0.752493 + 0.631417i −0.165955 + 0.941177i 0 −1.29838 2.24887i 1.32580 0.482551i −0.699804 + 0.254708i
99.4 1.29036 + 0.469651i −0.457545 2.59487i −0.0876440 0.0735420i −0.327974 + 0.275202i 0.628286 3.56318i 0 −1.45172 2.51445i −3.70490 + 1.34847i −0.552451 + 0.201076i
99.5 2.39006 + 0.869910i 0.524166 + 2.97269i 3.42354 + 2.87269i 2.17073 1.82146i −1.33319 + 7.56088i 0 3.14003 + 5.43870i −5.74307 + 2.09031i 6.77267 2.46505i
442.1 −1.80311 + 0.656278i −0.100636 + 0.570737i 1.28842 1.08111i 0.131215 + 0.110102i −0.193104 1.09515i 0 0.305182 0.528591i 2.50346 + 0.911187i −0.308853 0.112413i
442.2 −1.65002 + 0.600557i 0.488762 2.77191i 0.829796 0.696282i −2.66117 2.23299i 0.858223 + 4.86723i 0 0.804890 1.39411i −4.62552 1.68355i 5.73201 + 2.08628i
442.3 0.712406 0.259295i 0.218901 1.24145i −1.09180 + 0.916129i −0.752493 0.631417i −0.165955 0.941177i 0 −1.29838 + 2.24887i 1.32580 + 0.482551i −0.699804 0.254708i
442.4 1.29036 0.469651i −0.457545 + 2.59487i −0.0876440 + 0.0735420i −0.327974 0.275202i 0.628286 + 3.56318i 0 −1.45172 + 2.51445i −3.70490 1.34847i −0.552451 0.201076i
442.5 2.39006 0.869910i 0.524166 2.97269i 3.42354 2.87269i 2.17073 + 1.82146i −1.33319 7.56088i 0 3.14003 5.43870i −5.74307 2.09031i 6.77267 + 2.46505i
491.1 −1.59663 1.33973i 2.73420 0.995166i 0.407048 + 2.30848i −0.0434172 + 0.246231i −5.69875 2.07417i 0 0.358588 0.621093i 4.18734 3.51360i 0.399204 0.334972i
491.2 −1.30141 1.09201i −2.86843 + 1.04402i 0.153879 + 0.872693i −0.224994 + 1.27600i 4.87308 + 1.77366i 0 −0.946138 + 1.63876i 4.83975 4.06104i 1.68622 1.41491i
491.3 0.220093 + 0.184680i −2.07321 + 0.754587i −0.332962 1.88832i 0.608956 3.45356i −0.595656 0.216801i 0 0.562764 0.974735i 1.43067 1.20047i 0.771831 0.647643i
491.4 0.467268 + 0.392084i −0.141115 + 0.0513615i −0.282687 1.60320i −0.519693 + 2.94733i −0.0860764 0.0313292i 0 1.10647 1.91647i −2.28086 + 1.91387i −1.39844 + 1.17343i
491.5 1.44463 + 1.21219i 1.90886 0.694769i 0.270260 + 1.53272i 0.445192 2.52481i 3.59980 + 1.31022i 0 0.418312 0.724538i 0.862920 0.724076i 3.70369 3.10776i
785.1 −1.59663 + 1.33973i 2.73420 + 0.995166i 0.407048 2.30848i −0.0434172 0.246231i −5.69875 + 2.07417i 0 0.358588 + 0.621093i 4.18734 + 3.51360i 0.399204 + 0.334972i
785.2 −1.30141 + 1.09201i −2.86843 1.04402i 0.153879 0.872693i −0.224994 1.27600i 4.87308 1.77366i 0 −0.946138 1.63876i 4.83975 + 4.06104i 1.68622 + 1.41491i
785.3 0.220093 0.184680i −2.07321 0.754587i −0.332962 + 1.88832i 0.608956 + 3.45356i −0.595656 + 0.216801i 0 0.562764 + 0.974735i 1.43067 + 1.20047i 0.771831 + 0.647643i
785.4 0.467268 0.392084i −0.141115 0.0513615i −0.282687 + 1.60320i −0.519693 2.94733i −0.0860764 + 0.0313292i 0 1.10647 + 1.91647i −2.28086 1.91387i −1.39844 1.17343i
785.5 1.44463 1.21219i 1.90886 + 0.694769i 0.270260 1.53272i 0.445192 + 2.52481i 3.59980 1.31022i 0 0.418312 + 0.724538i 0.862920 + 0.724076i 3.70369 + 3.10776i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.w.b 30
7.b odd 2 1 133.2.v.b 30
7.c even 3 1 931.2.v.e 30
7.c even 3 1 931.2.x.d 30
7.d odd 6 1 931.2.v.d 30
7.d odd 6 1 931.2.x.e 30
19.e even 9 1 inner 931.2.w.b 30
133.u even 9 1 931.2.x.d 30
133.w even 9 1 931.2.v.e 30
133.x odd 18 1 931.2.x.e 30
133.y odd 18 1 133.2.v.b 30
133.y odd 18 1 2527.2.a.r 15
133.z odd 18 1 931.2.v.d 30
133.ba even 18 1 2527.2.a.s 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.v.b 30 7.b odd 2 1
133.2.v.b 30 133.y odd 18 1
931.2.v.d 30 7.d odd 6 1
931.2.v.d 30 133.z odd 18 1
931.2.v.e 30 7.c even 3 1
931.2.v.e 30 133.w even 9 1
931.2.w.b 30 1.a even 1 1 trivial
931.2.w.b 30 19.e even 9 1 inner
931.2.x.d 30 7.c even 3 1
931.2.x.d 30 133.u even 9 1
931.2.x.e 30 7.d odd 6 1
931.2.x.e 30 133.x odd 18 1
2527.2.a.r 15 133.y odd 18 1
2527.2.a.s 15 133.ba even 18 1

Hecke kernels

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

\( T_{2}^{30} + 3 T_{2}^{28} - 7 T_{2}^{27} - 18 T_{2}^{26} - 6 T_{2}^{25} + 136 T_{2}^{24} + 135 T_{2}^{23} + 867 T_{2}^{22} + 877 T_{2}^{21} - 1023 T_{2}^{20} - 117 T_{2}^{19} + 13145 T_{2}^{18} + 5592 T_{2}^{17} + 20736 T_{2}^{16} + \cdots + 11449 \) Copy content Toggle raw display
\( T_{3}^{30} - 3 T_{3}^{29} + 6 T_{3}^{28} + 12 T_{3}^{27} - 102 T_{3}^{26} + 258 T_{3}^{25} + 423 T_{3}^{24} - 2601 T_{3}^{23} + 8712 T_{3}^{22} - 14465 T_{3}^{21} - 7476 T_{3}^{20} + 105963 T_{3}^{19} - 34109 T_{3}^{18} + 156342 T_{3}^{17} + \cdots + 26569 \) Copy content Toggle raw display