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Properties

Label 931.2.w.g
Level $931$
Weight $2$
Character orbit 931.w
Analytic conductor $7.434$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

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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: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 36 q^{8} - 24 q^{15} + 24 q^{16} - 120 q^{18} - 24 q^{22} - 36 q^{23} - 84 q^{25} - 12 q^{29} - 84 q^{30} + 60 q^{32} - 48 q^{37} - 120 q^{44} + 72 q^{46} + 84 q^{50} + 24 q^{51} + 48 q^{53} + 36 q^{57} - 144 q^{58} + 144 q^{60} - 60 q^{64} + 96 q^{65} + 12 q^{67} - 24 q^{71} - 84 q^{72} - 108 q^{74} - 24 q^{78} + 60 q^{81} - 24 q^{85} + 72 q^{86} + 72 q^{88} + 60 q^{92} + 96 q^{93} + 60 q^{95} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −2.47611 − 0.901230i −0.226269 − 1.28324i 3.78681 + 3.17751i 1.04434 − 0.876307i −0.596223 + 3.38135i 0 −3.87786 − 6.71665i 1.22358 − 0.445348i −3.37566 + 1.22864i
99.2 −2.47611 − 0.901230i 0.226269 + 1.28324i 3.78681 + 3.17751i −1.04434 + 0.876307i 0.596223 − 3.38135i 0 −3.87786 − 6.71665i 1.22358 − 0.445348i 3.37566 − 1.22864i
99.3 −1.61978 − 0.589551i −0.204211 − 1.15814i 0.744021 + 0.624308i −1.63998 + 1.37611i −0.352005 + 1.99632i 0 0.886644 + 1.53571i 1.51949 − 0.553051i 3.46769 − 1.26214i
99.4 −1.61978 − 0.589551i 0.204211 + 1.15814i 0.744021 + 0.624308i 1.63998 − 1.37611i 0.352005 − 1.99632i 0 0.886644 + 1.53571i 1.51949 − 0.553051i −3.46769 + 1.26214i
99.5 −1.36303 − 0.496103i −0.319735 − 1.81330i 0.0796485 + 0.0668331i −2.19025 + 1.83783i −0.463778 + 2.63021i 0 1.37510 + 2.38174i −0.366765 + 0.133492i 3.89713 − 1.41844i
99.6 −1.36303 − 0.496103i 0.319735 + 1.81330i 0.0796485 + 0.0668331i 2.19025 − 1.83783i 0.463778 − 2.63021i 0 1.37510 + 2.38174i −0.366765 + 0.133492i −3.89713 + 1.41844i
99.7 −0.426103 − 0.155089i −0.569593 − 3.23032i −1.37458 − 1.15341i −0.0264316 + 0.0221787i −0.258282 + 1.46479i 0 0.860281 + 1.49005i −7.29148 + 2.65388i 0.0147023 − 0.00535118i
99.8 −0.426103 − 0.155089i 0.569593 + 3.23032i −1.37458 − 1.15341i 0.0264316 − 0.0221787i 0.258282 − 1.46479i 0 0.860281 + 1.49005i −7.29148 + 2.65388i −0.0147023 + 0.00535118i
99.9 −0.128216 − 0.0466668i −0.477606 − 2.70864i −1.51783 − 1.27361i 1.81215 − 1.52057i −0.0651667 + 0.369579i 0 0.271619 + 0.470458i −4.28953 + 1.56126i −0.303306 + 0.110394i
99.10 −0.128216 − 0.0466668i 0.477606 + 2.70864i −1.51783 − 1.27361i −1.81215 + 1.52057i 0.0651667 − 0.369579i 0 0.271619 + 0.470458i −4.28953 + 1.56126i 0.303306 − 0.110394i
99.11 0.289634 + 0.105418i −0.115620 − 0.655713i −1.45931 − 1.22451i 1.74076 − 1.46067i 0.0356367 − 0.202106i 0 −0.601804 − 1.04236i 2.40249 − 0.874433i 0.658165 − 0.239553i
99.12 0.289634 + 0.105418i 0.115620 + 0.655713i −1.45931 − 1.22451i −1.74076 + 1.46067i −0.0356367 + 0.202106i 0 −0.601804 − 1.04236i 2.40249 − 0.874433i −0.658165 + 0.239553i
99.13 1.07173 + 0.390077i −0.162643 − 0.922396i −0.535649 − 0.449463i 0.877160 − 0.736025i 0.185496 − 1.05200i 0 −1.53925 − 2.66606i 1.99472 − 0.726018i 1.22718 − 0.446658i
99.14 1.07173 + 0.390077i 0.162643 + 0.922396i −0.535649 − 0.449463i −0.877160 + 0.736025i −0.185496 + 1.05200i 0 −1.53925 − 2.66606i 1.99472 − 0.726018i −1.22718 + 0.446658i
99.15 1.78770 + 0.650670i −0.349643 − 1.98292i 1.24042 + 1.04083i −2.63819 + 2.21370i 0.665172 − 3.77238i 0 −0.362175 − 0.627305i −0.990655 + 0.360569i −6.15668 + 2.24085i
99.16 1.78770 + 0.650670i 0.349643 + 1.98292i 1.24042 + 1.04083i 2.63819 − 2.21370i −0.665172 + 3.77238i 0 −0.362175 − 0.627305i −0.990655 + 0.360569i 6.15668 − 2.24085i
99.17 2.34965 + 0.855204i −0.362435 − 2.05547i 3.25741 + 2.73330i 2.89634 − 2.43032i 0.906250 − 5.13960i 0 2.81582 + 4.87715i −1.27452 + 0.463886i 8.88381 − 3.23344i
99.18 2.34965 + 0.855204i 0.362435 + 2.05547i 3.25741 + 2.73330i −2.89634 + 2.43032i −0.906250 + 5.13960i 0 2.81582 + 4.87715i −1.27452 + 0.463886i −8.88381 + 3.23344i
99.19 2.39390 + 0.871309i −0.406256 − 2.30399i 3.43950 + 2.88608i 0.877784 − 0.736548i 1.03495 − 5.86951i 0 3.17162 + 5.49341i −2.32426 + 0.845961i 2.74309 − 0.998403i
99.20 2.39390 + 0.871309i 0.406256 + 2.30399i 3.43950 + 2.88608i −0.877784 + 0.736548i −1.03495 + 5.86951i 0 3.17162 + 5.49341i −2.32426 + 0.845961i −2.74309 + 0.998403i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.e even 9 1 inner
133.y odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.w.g ✓ 120
7.b odd 2 1 inner 931.2.w.g ✓ 120
7.c even 3 1 931.2.v.i 120
7.c even 3 1 931.2.x.i 120
7.d odd 6 1 931.2.v.i 120
7.d odd 6 1 931.2.x.i 120
19.e even 9 1 inner 931.2.w.g ✓ 120
133.u even 9 1 931.2.x.i 120
133.w even 9 1 931.2.v.i 120
133.x odd 18 1 931.2.x.i 120
133.y odd 18 1 inner 931.2.w.g ✓ 120
133.z odd 18 1 931.2.v.i 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.v.i 120 7.c even 3 1
931.2.v.i 120 7.d odd 6 1
931.2.v.i 120 133.w even 9 1
931.2.v.i 120 133.z odd 18 1
931.2.w.g ✓ 120 1.a even 1 1 trivial
931.2.w.g ✓ 120 7.b odd 2 1 inner
931.2.w.g ✓ 120 19.e even 9 1 inner
931.2.w.g ✓ 120 133.y odd 18 1 inner
931.2.x.i 120 7.c even 3 1
931.2.x.i 120 7.d odd 6 1
931.2.x.i 120 133.u even 9 1
931.2.x.i 120 133.x odd 18 1

Hecke kernels

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

\( T_{2}^{60} - 14 T_{2}^{57} - 3 T_{2}^{56} - 6 T_{2}^{55} + 533 T_{2}^{54} - 30 T_{2}^{53} + 282 T_{2}^{52} + \cdots + 15625 \) Copy content Toggle raw display
\( T_{3}^{120} - 15 T_{3}^{116} + 2104 T_{3}^{114} + 4668 T_{3}^{112} + 9954 T_{3}^{110} + \cdots + 38\!\cdots\!21 \) Copy content Toggle raw display