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Properties

Label 756.2.bp.a
Level $756$
Weight $2$
Character orbit 756.bp
Analytic conductor $6.037$
Analytic rank $0$
Dimension $144$
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] = [756,2,Mod(193,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bp (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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{9} - 12 q^{11} - 12 q^{15} - 24 q^{17} - 3 q^{21} + 15 q^{23} + 6 q^{29} + 18 q^{33} + 18 q^{35} + 18 q^{39} - 12 q^{41} + 6 q^{45} + 18 q^{47} + 36 q^{49} + 18 q^{51} + 15 q^{53} + 3 q^{57} + 30 q^{59} + 18 q^{61} + 3 q^{63} + 9 q^{65} + 30 q^{69} - 12 q^{71} - 36 q^{73} + 102 q^{75} + 69 q^{77} + 18 q^{79} + 12 q^{81} - 36 q^{85} + 78 q^{87} - 72 q^{89} - 18 q^{91} - 60 q^{93} + 42 q^{95} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 0 −1.72645 + 0.139144i 0 −0.604458 + 0.220005i 0 −2.43449 − 1.03598i 0 2.96128 − 0.480450i 0
193.2 0 −1.72611 − 0.143355i 0 1.14483 − 0.416685i 0 −1.12940 + 2.39259i 0 2.95890 + 0.494894i 0
193.3 0 −1.68394 + 0.405388i 0 1.57900 − 0.574710i 0 2.62692 + 0.315138i 0 2.67132 − 1.36530i 0
193.4 0 −1.67981 − 0.422185i 0 −3.86396 + 1.40636i 0 0.267801 − 2.63216i 0 2.64352 + 1.41838i 0
193.5 0 −1.35655 − 1.07693i 0 2.04264 − 0.743461i 0 0.979055 − 2.45794i 0 0.680440 + 2.92181i 0
193.6 0 −1.22275 + 1.22673i 0 −1.75075 + 0.637221i 0 2.59849 + 0.497853i 0 −0.00975629 − 2.99998i 0
193.7 0 −0.985812 + 1.42414i 0 −1.13435 + 0.412871i 0 −2.60567 − 0.458792i 0 −1.05635 − 2.80787i 0
193.8 0 −0.844572 − 1.51218i 0 −0.569103 + 0.207137i 0 2.00271 + 1.72892i 0 −1.57340 + 2.55430i 0
193.9 0 −0.699375 + 1.58457i 0 3.09474 − 1.12639i 0 −0.555998 + 2.58667i 0 −2.02175 − 2.21642i 0
193.10 0 −0.636453 − 1.61088i 0 −2.33777 + 0.850878i 0 −2.09399 + 1.61716i 0 −2.18985 + 2.05050i 0
193.11 0 −0.513768 + 1.65410i 0 −0.633570 + 0.230601i 0 1.95184 − 1.78614i 0 −2.47208 − 1.69965i 0
193.12 0 0.128228 − 1.72730i 0 3.80842 − 1.38615i 0 −2.63217 − 0.267690i 0 −2.96712 − 0.442975i 0
193.13 0 0.156536 − 1.72496i 0 −1.43093 + 0.520817i 0 −0.946337 − 2.47072i 0 −2.95099 − 0.540038i 0
193.14 0 0.378367 + 1.69022i 0 −3.32039 + 1.20852i 0 0.693513 + 2.55324i 0 −2.71368 + 1.27905i 0
193.15 0 0.557871 − 1.63975i 0 2.46711 − 0.897956i 0 1.73437 + 1.99799i 0 −2.37756 − 1.82954i 0
193.16 0 0.726734 + 1.57221i 0 1.04532 − 0.380466i 0 −2.56389 + 0.653064i 0 −1.94371 + 2.28516i 0
193.17 0 0.963882 + 1.43907i 0 −0.981872 + 0.357372i 0 0.459325 − 2.60557i 0 −1.14186 + 2.77419i 0
193.18 0 1.12192 − 1.31958i 0 −3.46735 + 1.26201i 0 2.63004 − 0.287949i 0 −0.482607 − 2.96093i 0
193.19 0 1.17989 + 1.26802i 0 3.75808 − 1.36783i 0 2.59104 − 0.535278i 0 −0.215726 + 2.99223i 0
193.20 0 1.44372 − 0.956902i 0 −2.40619 + 0.875781i 0 −1.62201 + 2.09024i 0 1.16868 − 2.76300i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.u even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bp.a ✓ 144
7.c even 3 1 756.2.bq.a yes 144
27.e even 9 1 756.2.bq.a yes 144
189.u even 9 1 inner 756.2.bp.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bp.a ✓ 144 1.a even 1 1 trivial
756.2.bp.a ✓ 144 189.u even 9 1 inner
756.2.bq.a yes 144 7.c even 3 1
756.2.bq.a yes 144 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).