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Properties

Label 945.2.cf.a
Level $945$
Weight $2$
Character orbit 945.cf
Analytic conductor $7.546$
Analytic rank $0$
Dimension $144$
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(8,945)]
 
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([2, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cf (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: \(144\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 12 q^{11} + 72 q^{16} + 48 q^{20} + 24 q^{23} - 12 q^{25} + 60 q^{32} - 24 q^{37} - 72 q^{38} - 48 q^{41} - 48 q^{46} - 12 q^{47} - 24 q^{55} + 72 q^{65} - 12 q^{67} - 24 q^{76} - 96 q^{82} - 120 q^{83} - 48 q^{85} + 144 q^{86} - 48 q^{88} + 24 q^{91} - 156 q^{92} - 120 q^{95} - 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.58356 + 0.692264i 0 4.46352 − 2.57701i 2.16034 − 0.577011i 0 −0.258819 − 0.965926i −5.96521 + 5.96521i 0 −5.18193 + 2.98627i
8.2 −2.56723 + 0.687887i 0 4.38543 − 2.53193i 0.798496 + 2.08864i 0 0.258819 + 0.965926i −5.75804 + 5.75804i 0 −3.48667 − 4.81274i
8.3 −2.45082 + 0.656695i 0 3.84322 − 2.21888i −0.100229 − 2.23382i 0 0.258819 + 0.965926i −4.37366 + 4.37366i 0 1.71258 + 5.40887i
8.4 −2.22790 + 0.596964i 0 2.87511 − 1.65995i −0.702276 + 2.12292i 0 −0.258819 − 0.965926i −2.15266 + 2.15266i 0 0.297292 − 5.14889i
8.5 −2.14161 + 0.573842i 0 2.52513 − 1.45789i 1.18982 − 1.89323i 0 0.258819 + 0.965926i −1.43571 + 1.43571i 0 −1.46171 + 4.73733i
8.6 −1.97607 + 0.529487i 0 1.89246 − 1.09261i −2.18471 − 0.476494i 0 −0.258819 − 0.965926i −0.267935 + 0.267935i 0 4.56944 − 0.215188i
8.7 −1.94896 + 0.522223i 0 1.79368 − 1.03558i 1.64429 + 1.51536i 0 −0.258819 − 0.965926i −0.101539 + 0.101539i 0 −3.99601 − 2.09469i
8.8 −1.89278 + 0.507169i 0 1.59335 − 0.919920i −1.79424 − 1.33442i 0 −0.258819 − 0.965926i 0.221919 − 0.221919i 0 4.07289 + 1.61579i
8.9 −1.76646 + 0.473323i 0 1.16431 − 0.672216i −0.0404174 + 2.23570i 0 0.258819 + 0.965926i 0.847742 − 0.847742i 0 −0.986813 − 3.96842i
8.10 −1.39473 + 0.373716i 0 0.0735484 − 0.0424632i 2.07304 − 0.838156i 0 0.258819 + 0.965926i 1.95531 − 1.95531i 0 −2.57809 + 1.94373i
8.11 −1.35450 + 0.362938i 0 −0.0290960 + 0.0167986i 1.91443 − 1.15541i 0 −0.258819 − 0.965926i 2.01644 − 2.01644i 0 −2.17376 + 2.25982i
8.12 −1.29498 + 0.346990i 0 −0.175469 + 0.101307i −2.04598 + 0.902198i 0 0.258819 + 0.965926i 2.08807 − 2.08807i 0 2.33646 − 1.87827i
8.13 −0.984152 + 0.263703i 0 −0.833034 + 0.480953i −2.23090 + 0.151881i 0 0.258819 + 0.965926i 2.13390 − 2.13390i 0 2.15550 − 0.737770i
8.14 −0.556131 + 0.149015i 0 −1.44497 + 0.834256i −0.251163 + 2.22192i 0 −0.258819 − 0.965926i 1.49351 − 1.49351i 0 −0.191420 − 1.27311i
8.15 −0.480556 + 0.128764i 0 −1.51770 + 0.876243i 0.0596385 − 2.23527i 0 −0.258819 − 0.965926i 1.32009 − 1.32009i 0 0.259164 + 1.08185i
8.16 −0.408324 + 0.109410i 0 −1.57729 + 0.910651i 0.732580 + 2.11266i 0 0.258819 + 0.965926i 1.14224 − 1.14224i 0 −0.530276 − 0.782497i
8.17 −0.323660 + 0.0867243i 0 −1.63482 + 0.943862i 1.44390 − 1.70738i 0 0.258819 + 0.965926i 0.921139 − 0.921139i 0 −0.319262 + 0.677831i
8.18 −0.213960 + 0.0573303i 0 −1.68956 + 0.975467i −2.00023 + 0.999546i 0 −0.258819 − 0.965926i 0.618832 − 0.618832i 0 0.370664 − 0.328536i
8.19 0.0906199 − 0.0242815i 0 −1.72443 + 0.995599i 2.11748 + 0.718524i 0 −0.258819 − 0.965926i −0.264770 + 0.264770i 0 0.209333 + 0.0136969i
8.20 0.205139 − 0.0549668i 0 −1.69299 + 0.977448i −1.19958 − 1.88706i 0 0.258819 + 0.965926i −0.593915 + 0.593915i 0 −0.349807 − 0.321173i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l 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.cf.a 144
3.b odd 2 1 315.2.cc.a ✓ 144
5.c odd 4 1 inner 945.2.cf.a 144
9.c even 3 1 315.2.cc.a ✓ 144
9.d odd 6 1 inner 945.2.cf.a 144
15.e even 4 1 315.2.cc.a ✓ 144
45.k odd 12 1 315.2.cc.a ✓ 144
45.l even 12 1 inner 945.2.cf.a 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.cc.a ✓ 144 3.b odd 2 1
315.2.cc.a ✓ 144 9.c even 3 1
315.2.cc.a ✓ 144 15.e even 4 1
315.2.cc.a ✓ 144 45.k odd 12 1
945.2.cf.a 144 1.a even 1 1 trivial
945.2.cf.a 144 5.c odd 4 1 inner
945.2.cf.a 144 9.d odd 6 1 inner
945.2.cf.a 144 45.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).