Properties

Label 225.3.i.b
Level $225$
Weight $3$
Character orbit 225.i
Analytic conductor $6.131$
Analytic rank $0$
Dimension $32$
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] = [225,3,Mod(74,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.74");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.i (of order \(6\), degree \(2\), 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: \(6.13080594811\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} - 44 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} - 44 q^{6} - 16 q^{9} - 36 q^{11} + 108 q^{14} - 64 q^{16} + 104 q^{19} + 144 q^{21} - 216 q^{24} + 108 q^{29} + 64 q^{31} - 108 q^{34} - 172 q^{36} - 320 q^{39} + 288 q^{41} - 216 q^{46} + 108 q^{49} - 212 q^{51} + 632 q^{54} - 36 q^{56} + 972 q^{59} + 124 q^{61} - 512 q^{64} + 416 q^{66} - 180 q^{69} - 1080 q^{74} - 212 q^{76} + 80 q^{79} - 224 q^{81} - 1428 q^{84} + 108 q^{86} + 272 q^{91} + 300 q^{94} + 332 q^{96} + 1648 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −1.86827 + 3.23594i −1.45927 + 2.62117i −4.98088 8.62715i 0 −5.75562 9.61919i −6.28840 3.63061i 22.2764 −4.74103 7.65001i 0
74.2 −1.63532 + 2.83245i 1.96733 2.26486i −3.34853 5.79983i 0 3.19790 + 9.27615i 5.48940 + 3.16931i 8.82112 −1.25919 8.91148i 0
74.3 −1.54563 + 2.67710i 1.01971 + 2.82138i −2.77793 4.81151i 0 −9.12922 1.63094i 1.91037 + 1.10296i 4.80953 −6.92040 + 5.75396i 0
74.4 −1.41438 + 2.44978i −2.99795 0.110987i −2.00096 3.46576i 0 4.51214 7.18734i −6.88441 3.97472i 0.00541780 8.97536 + 0.665467i 0
74.5 −0.916957 + 1.58822i 2.88508 0.822398i 0.318381 + 0.551452i 0 −1.33934 + 5.33623i 5.60142 + 3.23398i −8.50342 7.64732 4.74536i 0
74.6 −0.696021 + 1.20554i 1.88791 + 2.33148i 1.03111 + 1.78593i 0 −4.12473 + 0.653207i −7.63842 4.41004i −8.43886 −1.87156 + 8.80325i 0
74.7 −0.346451 + 0.600071i 0.450312 2.96601i 1.75994 + 3.04831i 0 1.62381 + 1.29780i −10.6367 6.14112i −5.21055 −8.59444 2.67126i 0
74.8 −0.0238054 + 0.0412321i 2.42528 1.76580i 1.99887 + 3.46214i 0 0.0150729 + 0.142035i −3.30399 1.90756i −0.380778 2.76393 8.56508i 0
74.9 0.0238054 0.0412321i −2.42528 + 1.76580i 1.99887 + 3.46214i 0 0.0150729 + 0.142035i 3.30399 + 1.90756i 0.380778 2.76393 8.56508i 0
74.10 0.346451 0.600071i −0.450312 + 2.96601i 1.75994 + 3.04831i 0 1.62381 + 1.29780i 10.6367 + 6.14112i 5.21055 −8.59444 2.67126i 0
74.11 0.696021 1.20554i −1.88791 2.33148i 1.03111 + 1.78593i 0 −4.12473 + 0.653207i 7.63842 + 4.41004i 8.43886 −1.87156 + 8.80325i 0
74.12 0.916957 1.58822i −2.88508 + 0.822398i 0.318381 + 0.551452i 0 −1.33934 + 5.33623i −5.60142 3.23398i 8.50342 7.64732 4.74536i 0
74.13 1.41438 2.44978i 2.99795 + 0.110987i −2.00096 3.46576i 0 4.51214 7.18734i 6.88441 + 3.97472i −0.00541780 8.97536 + 0.665467i 0
74.14 1.54563 2.67710i −1.01971 2.82138i −2.77793 4.81151i 0 −9.12922 1.63094i −1.91037 1.10296i −4.80953 −6.92040 + 5.75396i 0
74.15 1.63532 2.83245i −1.96733 + 2.26486i −3.34853 5.79983i 0 3.19790 + 9.27615i −5.48940 3.16931i −8.82112 −1.25919 8.91148i 0
74.16 1.86827 3.23594i 1.45927 2.62117i −4.98088 8.62715i 0 −5.75562 9.61919i 6.28840 + 3.63061i −22.2764 −4.74103 7.65001i 0
149.1 −1.86827 3.23594i −1.45927 2.62117i −4.98088 + 8.62715i 0 −5.75562 + 9.61919i −6.28840 + 3.63061i 22.2764 −4.74103 + 7.65001i 0
149.2 −1.63532 2.83245i 1.96733 + 2.26486i −3.34853 + 5.79983i 0 3.19790 9.27615i 5.48940 3.16931i 8.82112 −1.25919 + 8.91148i 0
149.3 −1.54563 2.67710i 1.01971 2.82138i −2.77793 + 4.81151i 0 −9.12922 + 1.63094i 1.91037 1.10296i 4.80953 −6.92040 5.75396i 0
149.4 −1.41438 2.44978i −2.99795 + 0.110987i −2.00096 + 3.46576i 0 4.51214 + 7.18734i −6.88441 + 3.97472i 0.00541780 8.97536 0.665467i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.16
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 48 T_{2}^{30} + 1392 T_{2}^{28} + 26368 T_{2}^{26} + 370350 T_{2}^{24} + 3861288 T_{2}^{22} + 30996744 T_{2}^{20} + 186329952 T_{2}^{18} + 848102067 T_{2}^{16} + 2735420800 T_{2}^{14} + \cdots + 6561 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display