Properties

Label 65.3.h.a
Level $65$
Weight $3$
Character orbit 65.h
Analytic conductor $1.771$
Analytic rank $0$
Dimension $24$
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] = [65,3,Mod(12,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.12");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 65.h (of order \(4\), degree \(2\), 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: \(1.77112171834\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} + 16 q^{10} + 72 q^{12} - 36 q^{13} - 104 q^{16} - 48 q^{17} + 8 q^{22} - 104 q^{23} - 88 q^{25} + 88 q^{26} + 56 q^{27} - 24 q^{30} - 64 q^{35} + 256 q^{36} + 124 q^{38} - 368 q^{40} + 216 q^{42} + 8 q^{43} + 196 q^{48} - 296 q^{51} + 16 q^{52} + 220 q^{53} + 332 q^{55} + 584 q^{56} - 8 q^{61} - 596 q^{62} + 420 q^{65} - 360 q^{66} - 640 q^{68} - 184 q^{75} + 388 q^{77} - 636 q^{78} - 224 q^{81} - 1004 q^{82} - 52 q^{87} + 780 q^{88} + 452 q^{90} - 512 q^{91} + 812 q^{92} - 136 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.66395 + 2.66395i −2.12513 + 2.12513i 10.1932i −4.54019 + 2.09444i 11.3225i 2.95114 2.95114i 16.4984 + 16.4984i 0.0323960i 6.51533 17.6743i
12.2 −2.14254 + 2.14254i 1.85789 1.85789i 5.18097i 1.93662 4.60972i 7.96121i 5.84982 5.84982i 2.53029 + 2.53029i 2.09649i 5.72722 + 14.0258i
12.3 −1.78036 + 1.78036i 0.433767 0.433767i 2.33933i 3.33534 + 3.72498i 1.54452i −5.63522 + 5.63522i −2.95658 2.95658i 8.62369i −12.5699 0.693690i
12.4 −1.30775 + 1.30775i −3.70495 + 3.70495i 0.579580i 1.70825 4.69914i 9.69031i −2.51452 + 2.51452i −5.98895 5.98895i 18.4534i 3.91133 + 8.37926i
12.5 −0.474292 + 0.474292i −0.839739 + 0.839739i 3.55009i −4.99852 + 0.121682i 0.796563i −1.35251 + 1.35251i −3.58095 3.58095i 7.58968i 2.31304 2.42847i
12.6 −0.456148 + 0.456148i 3.37817 3.37817i 3.58386i −0.779969 + 4.93879i 3.08189i 7.82065 7.82065i −3.45936 3.45936i 13.8241i −1.89704 2.60860i
12.7 0.456148 0.456148i 3.37817 3.37817i 3.58386i 0.779969 4.93879i 3.08189i −7.82065 + 7.82065i 3.45936 + 3.45936i 13.8241i −1.89704 2.60860i
12.8 0.474292 0.474292i −0.839739 + 0.839739i 3.55009i 4.99852 0.121682i 0.796563i 1.35251 1.35251i 3.58095 + 3.58095i 7.58968i 2.31304 2.42847i
12.9 1.30775 1.30775i −3.70495 + 3.70495i 0.579580i −1.70825 + 4.69914i 9.69031i 2.51452 2.51452i 5.98895 + 5.98895i 18.4534i 3.91133 + 8.37926i
12.10 1.78036 1.78036i 0.433767 0.433767i 2.33933i −3.33534 3.72498i 1.54452i 5.63522 5.63522i 2.95658 + 2.95658i 8.62369i −12.5699 0.693690i
12.11 2.14254 2.14254i 1.85789 1.85789i 5.18097i −1.93662 + 4.60972i 7.96121i −5.84982 + 5.84982i −2.53029 2.53029i 2.09649i 5.72722 + 14.0258i
12.12 2.66395 2.66395i −2.12513 + 2.12513i 10.1932i 4.54019 2.09444i 11.3225i −2.95114 + 2.95114i −16.4984 16.4984i 0.0323960i 6.51533 17.6743i
38.1 −2.66395 2.66395i −2.12513 2.12513i 10.1932i −4.54019 2.09444i 11.3225i 2.95114 + 2.95114i 16.4984 16.4984i 0.0323960i 6.51533 + 17.6743i
38.2 −2.14254 2.14254i 1.85789 + 1.85789i 5.18097i 1.93662 + 4.60972i 7.96121i 5.84982 + 5.84982i 2.53029 2.53029i 2.09649i 5.72722 14.0258i
38.3 −1.78036 1.78036i 0.433767 + 0.433767i 2.33933i 3.33534 3.72498i 1.54452i −5.63522 5.63522i −2.95658 + 2.95658i 8.62369i −12.5699 + 0.693690i
38.4 −1.30775 1.30775i −3.70495 3.70495i 0.579580i 1.70825 + 4.69914i 9.69031i −2.51452 2.51452i −5.98895 + 5.98895i 18.4534i 3.91133 8.37926i
38.5 −0.474292 0.474292i −0.839739 0.839739i 3.55009i −4.99852 0.121682i 0.796563i −1.35251 1.35251i −3.58095 + 3.58095i 7.58968i 2.31304 + 2.42847i
38.6 −0.456148 0.456148i 3.37817 + 3.37817i 3.58386i −0.779969 4.93879i 3.08189i 7.82065 + 7.82065i −3.45936 + 3.45936i 13.8241i −1.89704 + 2.60860i
38.7 0.456148 + 0.456148i 3.37817 + 3.37817i 3.58386i 0.779969 + 4.93879i 3.08189i −7.82065 7.82065i 3.45936 3.45936i 13.8241i −1.89704 + 2.60860i
38.8 0.474292 + 0.474292i −0.839739 0.839739i 3.55009i 4.99852 + 0.121682i 0.796563i 1.35251 + 1.35251i 3.58095 3.58095i 7.58968i 2.31304 + 2.42847i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
13.b even 2 1 inner
65.h odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.3.h.a 24
5.b even 2 1 325.3.h.b 24
5.c odd 4 1 inner 65.3.h.a 24
5.c odd 4 1 325.3.h.b 24
13.b even 2 1 inner 65.3.h.a 24
65.d even 2 1 325.3.h.b 24
65.h odd 4 1 inner 65.3.h.a 24
65.h odd 4 1 325.3.h.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.h.a 24 1.a even 1 1 trivial
65.3.h.a 24 5.c odd 4 1 inner
65.3.h.a 24 13.b even 2 1 inner
65.3.h.a 24 65.h odd 4 1 inner
325.3.h.b 24 5.b even 2 1
325.3.h.b 24 5.c odd 4 1
325.3.h.b 24 65.d even 2 1
325.3.h.b 24 65.h odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(65, [\chi])\).