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Properties

Label 675.3.i.c
Level $675$
Weight $3$
Character orbit 675.i
Analytic conductor $18.392$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,3,Mod(224,675)] 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("675.224"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,-32,0,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3924178443\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 45)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 36 q^{11} - 108 q^{14} - 64 q^{16} + 104 q^{19} - 108 q^{29} + 64 q^{31} - 108 q^{34} - 288 q^{41} - 216 q^{46} + 108 q^{49} + 36 q^{56} - 972 q^{59} + 124 q^{61} - 512 q^{64} + 1080 q^{74}+ \cdots + 300 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
224.1 −1.86827 + 3.23594i 0 −4.98088 − 8.62715i 0 0 6.28840 + 3.63061i 22.2764 0 0
224.2 −1.63532 + 2.83245i 0 −3.34853 − 5.79983i 0 0 −5.48940 − 3.16931i 8.82112 0 0
224.3 −1.54563 + 2.67710i 0 −2.77793 − 4.81151i 0 0 −1.91037 − 1.10296i 4.80953 0 0
224.4 −1.41438 + 2.44978i 0 −2.00096 − 3.46576i 0 0 6.88441 + 3.97472i 0.00541780 0 0
224.5 −0.916957 + 1.58822i 0 0.318381 + 0.551452i 0 0 −5.60142 − 3.23398i −8.50342 0 0
224.6 −0.696021 + 1.20554i 0 1.03111 + 1.78593i 0 0 7.63842 + 4.41004i −8.43886 0 0
224.7 −0.346451 + 0.600071i 0 1.75994 + 3.04831i 0 0 10.6367 + 6.14112i −5.21055 0 0
224.8 −0.0238054 + 0.0412321i 0 1.99887 + 3.46214i 0 0 3.30399 + 1.90756i −0.380778 0 0
224.9 0.0238054 − 0.0412321i 0 1.99887 + 3.46214i 0 0 −3.30399 − 1.90756i 0.380778 0 0
224.10 0.346451 − 0.600071i 0 1.75994 + 3.04831i 0 0 −10.6367 − 6.14112i 5.21055 0 0
224.11 0.696021 − 1.20554i 0 1.03111 + 1.78593i 0 0 −7.63842 − 4.41004i 8.43886 0 0
224.12 0.916957 − 1.58822i 0 0.318381 + 0.551452i 0 0 5.60142 + 3.23398i 8.50342 0 0
224.13 1.41438 − 2.44978i 0 −2.00096 − 3.46576i 0 0 −6.88441 − 3.97472i −0.00541780 0 0
224.14 1.54563 − 2.67710i 0 −2.77793 − 4.81151i 0 0 1.91037 + 1.10296i −4.80953 0 0
224.15 1.63532 − 2.83245i 0 −3.34853 − 5.79983i 0 0 5.48940 + 3.16931i −8.82112 0 0
224.16 1.86827 − 3.23594i 0 −4.98088 − 8.62715i 0 0 −6.28840 − 3.63061i −22.2764 0 0
449.1 −1.86827 − 3.23594i 0 −4.98088 + 8.62715i 0 0 6.28840 − 3.63061i 22.2764 0 0
449.2 −1.63532 − 2.83245i 0 −3.34853 + 5.79983i 0 0 −5.48940 + 3.16931i 8.82112 0 0
449.3 −1.54563 − 2.67710i 0 −2.77793 + 4.81151i 0 0 −1.91037 + 1.10296i 4.80953 0 0
449.4 −1.41438 − 2.44978i 0 −2.00096 + 3.46576i 0 0 6.88441 − 3.97472i 0.00541780 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 224.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.3.i.c 32
3.b odd 2 1 225.3.i.b 32
5.b even 2 1 inner 675.3.i.c 32
5.c odd 4 1 135.3.i.a 16
5.c odd 4 1 675.3.j.b 16
9.c even 3 1 225.3.i.b 32
9.d odd 6 1 inner 675.3.i.c 32
15.d odd 2 1 225.3.i.b 32
15.e even 4 1 45.3.i.a ✓ 16
15.e even 4 1 225.3.j.b 16
20.e even 4 1 2160.3.bs.c 16
45.h odd 6 1 inner 675.3.i.c 32
45.j even 6 1 225.3.i.b 32
45.k odd 12 1 45.3.i.a ✓ 16
45.k odd 12 1 225.3.j.b 16
45.k odd 12 1 405.3.c.a 16
45.l even 12 1 135.3.i.a 16
45.l even 12 1 405.3.c.a 16
45.l even 12 1 675.3.j.b 16
60.l odd 4 1 720.3.bs.c 16
180.v odd 12 1 2160.3.bs.c 16
180.x even 12 1 720.3.bs.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.i.a ✓ 16 15.e even 4 1
45.3.i.a ✓ 16 45.k odd 12 1
135.3.i.a 16 5.c odd 4 1
135.3.i.a 16 45.l even 12 1
225.3.i.b 32 3.b odd 2 1
225.3.i.b 32 9.c even 3 1
225.3.i.b 32 15.d odd 2 1
225.3.i.b 32 45.j even 6 1
225.3.j.b 16 15.e even 4 1
225.3.j.b 16 45.k odd 12 1
405.3.c.a 16 45.k odd 12 1
405.3.c.a 16 45.l even 12 1
675.3.i.c 32 1.a even 1 1 trivial
675.3.i.c 32 5.b even 2 1 inner
675.3.i.c 32 9.d odd 6 1 inner
675.3.i.c 32 45.h odd 6 1 inner
675.3.j.b 16 5.c odd 4 1
675.3.j.b 16 45.l even 12 1
720.3.bs.c 16 60.l odd 4 1
720.3.bs.c 16 180.x even 12 1
2160.3.bs.c 16 20.e even 4 1
2160.3.bs.c 16 180.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 48 T_{2}^{30} + 1392 T_{2}^{28} + 26368 T_{2}^{26} + 370350 T_{2}^{24} + 3861288 T_{2}^{22} + \cdots + 6561 \) acting on \(S_{3}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display