Properties

Label 270.11.h.a
Level $270$
Weight $11$
Character orbit 270.h
Analytic conductor $171.546$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,11,Mod(71,270)] 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("270.71"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 270.h (of order \(6\), degree \(2\), not 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: \(171.546458222\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 14843.5 + 25709.7i 11585.2i 0 31622.8
71.2 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 12405.3 + 21486.6i 11585.2i 0 31622.8
71.3 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 9067.39 + 15705.2i 11585.2i 0 31622.8
71.4 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 8980.55 + 15554.8i 11585.2i 0 −31622.8
71.5 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 −3790.77 − 6565.80i 11585.2i 0 31622.8
71.6 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 694.085 + 1202.19i 11585.2i 0 −31622.8
71.7 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 −1609.34 − 2787.46i 11585.2i 0 −31622.8
71.8 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 1479.49 + 2562.55i 11585.2i 0 −31622.8
71.9 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 2304.47 + 3991.46i 11585.2i 0 −31622.8
71.10 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 −2915.84 − 5050.38i 11585.2i 0 31622.8
71.11 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 3432.10 + 5944.57i 11585.2i 0 31622.8
71.12 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 −5291.82 − 9165.70i 11585.2i 0 31622.8
71.13 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 6861.95 + 11885.2i 11585.2i 0 −31622.8
71.14 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 −7045.91 − 12203.9i 11585.2i 0 −31622.8
71.15 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 −7055.28 − 12220.1i 11585.2i 0 31622.8
71.16 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 −7235.43 − 12532.1i 11585.2i 0 31622.8
71.17 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 10395.6 + 18005.7i 11585.2i 0 −31622.8
71.18 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 −13044.1 − 22593.0i 11585.2i 0 −31622.8
71.19 −19.5959 + 11.3137i 0 256.000 − 443.405i 1210.31 + 698.771i 0 −13360.0 − 23140.2i 11585.2i 0 −31622.8
71.20 −19.5959 + 11.3137i 0 256.000 − 443.405i −1210.31 − 698.771i 0 −16355.6 − 28328.7i 11585.2i 0 31622.8
See all 80 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 71.40
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.11.h.a 80
3.b odd 2 1 90.11.h.a ✓ 80
9.c even 3 1 90.11.h.a ✓ 80
9.d odd 6 1 inner 270.11.h.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.11.h.a ✓ 80 3.b odd 2 1
90.11.h.a ✓ 80 9.c even 3 1
270.11.h.a 80 1.a even 1 1 trivial
270.11.h.a 80 9.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(270, [\chi])\).