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Properties

Label 945.2.cj.e
Level $945$
Weight $2$
Character orbit 945.cj
Analytic conductor $7.546$
Analytic rank $0$
Dimension $160$
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] = [945,2,Mod(208,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([8, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cj (of order \(12\), degree \(4\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{2} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{2} + 6 q^{7} + 16 q^{8} - 24 q^{10} - 32 q^{11} + 76 q^{16} + 6 q^{17} + 60 q^{20} + 8 q^{22} + 16 q^{23} - 4 q^{25} + 36 q^{26} + 22 q^{28} + 48 q^{31} + 6 q^{32} + 36 q^{35} - 4 q^{37} - 12 q^{41} - 4 q^{43} - 16 q^{46} + 54 q^{47} + 44 q^{50} - 8 q^{53} + 92 q^{56} - 56 q^{58} - 24 q^{61} - 62 q^{65} + 12 q^{67} + 2 q^{70} + 40 q^{71} + 36 q^{73} - 96 q^{76} + 110 q^{77} - 36 q^{80} - 66 q^{82} - 138 q^{83} - 20 q^{85} - 32 q^{86} - 92 q^{88} - 48 q^{91} + 26 q^{92} + 94 q^{95} - 48 q^{97} - 102 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 −2.58237 − 0.691944i 0 4.45780 + 2.57371i 1.90273 − 1.17458i 0 −2.52895 − 0.777435i −5.94998 − 5.94998i 0 −5.72629 + 1.71661i
208.2 −2.57185 − 0.689126i 0 4.40748 + 2.54466i −0.846517 + 2.06964i 0 −0.442917 − 2.60841i −5.81636 − 5.81636i 0 3.60336 − 4.73945i
208.3 −2.48857 − 0.666809i 0 4.01627 + 2.31880i 2.17104 − 0.535318i 0 1.50397 + 2.17671i −4.80505 − 4.80505i 0 −5.75974 − 0.115499i
208.4 −2.31102 − 0.619235i 0 3.22530 + 1.86213i 0.200784 + 2.22704i 0 −2.38619 + 1.14284i −2.91707 − 2.91707i 0 0.915043 − 5.27105i
208.5 −2.12342 − 0.568970i 0 2.45315 + 1.41633i −0.987763 − 2.00607i 0 2.61450 + 0.405477i −1.29433 − 1.29433i 0 0.956046 + 4.82175i
208.6 −2.09295 − 0.560803i 0 2.33387 + 1.34746i −1.64778 − 1.51156i 0 −1.30912 + 2.29917i −1.06472 − 1.06472i 0 2.60102 + 4.08770i
208.7 −2.03275 − 0.544675i 0 2.10336 + 1.21438i −2.13753 + 0.656469i 0 2.62752 − 0.310030i −0.638021 − 0.638021i 0 4.70264 − 0.170180i
208.8 −1.99388 − 0.534260i 0 1.95809 + 1.13050i 0.931996 − 2.03258i 0 0.0413570 − 2.64543i −0.380972 − 0.380972i 0 −2.94422 + 3.55480i
208.9 −1.65968 − 0.444711i 0 0.824732 + 0.476159i 1.87165 + 1.22348i 0 2.22637 − 1.42943i 1.27291 + 1.27291i 0 −2.56226 − 2.86294i
208.10 −1.46948 − 0.393746i 0 0.272285 + 0.157204i 1.38104 − 1.75862i 0 1.74483 + 1.98886i 1.81325 + 1.81325i 0 −2.72186 + 2.04047i
208.11 −1.40755 − 0.377151i 0 0.106895 + 0.0617160i 1.85463 + 1.24914i 0 −2.04950 − 1.67318i 1.93361 + 1.93361i 0 −2.13936 − 2.45770i
208.12 −1.22833 − 0.329131i 0 −0.331572 − 0.191433i −1.83985 + 1.27081i 0 −2.63515 − 0.236659i 2.14268 + 2.14268i 0 2.67822 − 0.955424i
208.13 −0.943703 − 0.252864i 0 −0.905417 − 0.522743i −0.290604 − 2.21710i 0 −1.85665 + 1.88490i 2.10394 + 2.10394i 0 −0.286383 + 2.16577i
208.14 −0.907846 − 0.243257i 0 −0.967040 − 0.558321i −0.369705 + 2.20529i 0 0.311704 + 2.62733i 2.07129 + 2.07129i 0 0.872087 − 1.91213i
208.15 −0.867363 − 0.232409i 0 −1.03375 − 0.596834i −2.17152 − 0.533393i 0 0.983677 − 2.45609i 2.02783 + 2.02783i 0 1.75953 + 0.967326i
208.16 −0.772339 − 0.206948i 0 −1.17837 − 0.680332i −0.657057 + 2.13735i 0 0.914904 − 2.48253i 1.90009 + 1.90009i 0 0.949791 − 1.51478i
208.17 −0.455617 − 0.122082i 0 −1.53937 − 0.888755i 2.19185 − 0.442509i 0 0.0667159 − 2.64491i 1.25993 + 1.25993i 0 −1.05266 − 0.0659706i
208.18 −0.423314 − 0.113427i 0 −1.56572 − 0.903970i 1.75491 − 1.38574i 0 2.44885 + 1.00156i 1.18003 + 1.18003i 0 −0.900059 + 0.387551i
208.19 −0.193604 − 0.0518760i 0 −1.69726 − 0.979913i −2.23570 + 0.0407506i 0 2.25666 + 1.38111i 0.561218 + 0.561218i 0 0.434954 + 0.108090i
208.20 −0.148370 − 0.0397556i 0 −1.71162 − 0.988203i 2.21576 + 0.300657i 0 −2.32574 + 1.26133i 0.431895 + 0.431895i 0 −0.316800 − 0.132697i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.k odd 6 1 inner
315.cg even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.cj.e 160
3.b odd 2 1 315.2.cg.e yes 160
5.c odd 4 1 inner 945.2.cj.e 160
7.d odd 6 1 945.2.bv.e 160
9.c even 3 1 945.2.bv.e 160
9.d odd 6 1 315.2.bs.e ✓ 160
15.e even 4 1 315.2.cg.e yes 160
21.g even 6 1 315.2.bs.e ✓ 160
35.k even 12 1 945.2.bv.e 160
45.k odd 12 1 945.2.bv.e 160
45.l even 12 1 315.2.bs.e ✓ 160
63.k odd 6 1 inner 945.2.cj.e 160
63.s even 6 1 315.2.cg.e yes 160
105.w odd 12 1 315.2.bs.e ✓ 160
315.bw odd 12 1 315.2.cg.e yes 160
315.cg even 12 1 inner 945.2.cj.e 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.e ✓ 160 9.d odd 6 1
315.2.bs.e ✓ 160 21.g even 6 1
315.2.bs.e ✓ 160 45.l even 12 1
315.2.bs.e ✓ 160 105.w odd 12 1
315.2.cg.e yes 160 3.b odd 2 1
315.2.cg.e yes 160 15.e even 4 1
315.2.cg.e yes 160 63.s even 6 1
315.2.cg.e yes 160 315.bw odd 12 1
945.2.bv.e 160 7.d odd 6 1
945.2.bv.e 160 9.c even 3 1
945.2.bv.e 160 35.k even 12 1
945.2.bv.e 160 45.k odd 12 1
945.2.cj.e 160 1.a even 1 1 trivial
945.2.cj.e 160 5.c odd 4 1 inner
945.2.cj.e 160 63.k odd 6 1 inner
945.2.cj.e 160 315.cg even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{160} - 2 T_{2}^{159} + 2 T_{2}^{158} - 12 T_{2}^{157} - 277 T_{2}^{156} + 596 T_{2}^{155} + \cdots + 294499921 \) Copy content Toggle raw display
\( T_{11}^{40} + 8 T_{11}^{39} - 194 T_{11}^{38} - 1652 T_{11}^{37} + 16475 T_{11}^{36} + \cdots + 4204122386796 \) Copy content Toggle raw display