↑

Properties

Label 180.3.u.a
Level $180$
Weight $3$
Character orbit 180.u
Analytic conductor $4.905$
Analytic rank $0$
Dimension $48$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,3,Mod(13,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.13"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 4, 9])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} + 12 q^{11} + 22 q^{15} + 36 q^{17} + 52 q^{21} - 12 q^{23} - 6 q^{25} + 46 q^{27} - 14 q^{33} + 72 q^{35} - 84 q^{37} + 72 q^{41} + 110 q^{45} - 78 q^{47} - 4 q^{51} - 312 q^{53} + 132 q^{55}+ \cdots - 90 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −2.93271 + 0.631853i 0 −4.98610 + 0.372540i 0 5.98052 + 1.60248i 0 8.20152 − 3.70608i 0
13.2 0 −2.69602 − 1.31586i 0 4.02568 + 2.96545i 0 −4.41867 − 1.18398i 0 5.53700 + 7.09518i 0
13.3 0 −2.08827 − 2.15386i 0 −2.95142 − 4.03598i 0 −4.02302 − 1.07796i 0 −0.278258 + 8.99570i 0
13.4 0 −1.80382 + 2.39713i 0 1.85957 + 4.64134i 0 3.33405 + 0.893356i 0 −2.49245 − 8.64799i 0
13.5 0 −1.70695 + 2.46705i 0 2.79014 − 4.14911i 0 −10.5029 − 2.81425i 0 −3.17264 − 8.42226i 0
13.6 0 −0.412115 − 2.97156i 0 −3.05431 + 3.95868i 0 −2.46810 − 0.661325i 0 −8.66032 + 2.44925i 0
13.7 0 0.165444 − 2.99543i 0 4.61203 − 1.93111i 0 9.89538 + 2.65146i 0 −8.94526 − 0.991152i 0
13.8 0 0.594985 + 2.94041i 0 −1.82257 − 4.65599i 0 12.3966 + 3.32166i 0 −8.29199 + 3.49900i 0
13.9 0 1.55644 + 2.56466i 0 −4.88163 + 1.08150i 0 −10.5922 − 2.83818i 0 −4.15500 + 7.98348i 0
13.10 0 2.49269 − 1.66927i 0 −0.806137 − 4.93459i 0 −4.69726 − 1.25863i 0 3.42704 − 8.32198i 0
13.11 0 2.62769 + 1.44749i 0 4.94178 + 0.760767i 0 0.709316 + 0.190061i 0 4.80952 + 7.60713i 0
13.12 0 2.83660 − 0.976571i 0 −2.32510 + 4.42650i 0 4.38636 + 1.17532i 0 7.09262 − 5.54029i 0
97.1 0 −2.93271 − 0.631853i 0 −4.98610 − 0.372540i 0 5.98052 − 1.60248i 0 8.20152 + 3.70608i 0
97.2 0 −2.69602 + 1.31586i 0 4.02568 − 2.96545i 0 −4.41867 + 1.18398i 0 5.53700 − 7.09518i 0
97.3 0 −2.08827 + 2.15386i 0 −2.95142 + 4.03598i 0 −4.02302 + 1.07796i 0 −0.278258 − 8.99570i 0
97.4 0 −1.80382 − 2.39713i 0 1.85957 − 4.64134i 0 3.33405 − 0.893356i 0 −2.49245 + 8.64799i 0
97.5 0 −1.70695 − 2.46705i 0 2.79014 + 4.14911i 0 −10.5029 + 2.81425i 0 −3.17264 + 8.42226i 0
97.6 0 −0.412115 + 2.97156i 0 −3.05431 − 3.95868i 0 −2.46810 + 0.661325i 0 −8.66032 − 2.44925i 0
97.7 0 0.165444 + 2.99543i 0 4.61203 + 1.93111i 0 9.89538 − 2.65146i 0 −8.94526 + 0.991152i 0
97.8 0 0.594985 − 2.94041i 0 −1.82257 + 4.65599i 0 12.3966 − 3.32166i 0 −8.29199 − 3.49900i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.c even 3 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.u.a ✓ 48
3.b odd 2 1 540.3.v.a 48
5.c odd 4 1 inner 180.3.u.a ✓ 48
9.c even 3 1 inner 180.3.u.a ✓ 48
9.c even 3 1 1620.3.l.f 24
9.d odd 6 1 540.3.v.a 48
9.d odd 6 1 1620.3.l.g 24
15.e even 4 1 540.3.v.a 48
45.k odd 12 1 inner 180.3.u.a ✓ 48
45.k odd 12 1 1620.3.l.f 24
45.l even 12 1 540.3.v.a 48
45.l even 12 1 1620.3.l.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.u.a ✓ 48 1.a even 1 1 trivial
180.3.u.a ✓ 48 5.c odd 4 1 inner
180.3.u.a ✓ 48 9.c even 3 1 inner
180.3.u.a ✓ 48 45.k odd 12 1 inner
540.3.v.a 48 3.b odd 2 1
540.3.v.a 48 9.d odd 6 1
540.3.v.a 48 15.e even 4 1
540.3.v.a 48 45.l even 12 1
1620.3.l.f 24 9.c even 3 1
1620.3.l.f 24 45.k odd 12 1
1620.3.l.g 24 9.d odd 6 1
1620.3.l.g 24 45.l even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(180, [\chi])\).