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Properties

Label 65.3.p.a
Level $65$
Weight $3$
Character orbit 65.p
Analytic conductor $1.771$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,3,Mod(6,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.6");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 65.p (of order \(12\), degree \(4\), 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: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9} - 12 q^{11} - 12 q^{13} + 48 q^{14} + 20 q^{15} + 128 q^{16} + 60 q^{17} - 136 q^{18} + 68 q^{19} - 80 q^{20} - 48 q^{21} - 48 q^{22} - 48 q^{23} - 56 q^{24} - 84 q^{26} + 24 q^{27} - 16 q^{28} + 28 q^{29} + 240 q^{30} + 128 q^{31} - 408 q^{32} + 136 q^{33} - 28 q^{34} + 40 q^{35} + 300 q^{36} + 56 q^{37} - 88 q^{39} + 68 q^{41} - 320 q^{42} - 372 q^{43} - 240 q^{44} - 40 q^{45} + 260 q^{46} + 152 q^{47} + 424 q^{48} - 132 q^{49} + 372 q^{52} - 288 q^{53} + 152 q^{54} - 40 q^{55} - 288 q^{56} + 252 q^{57} + 492 q^{58} + 492 q^{59} - 160 q^{60} - 100 q^{61} + 120 q^{62} + 844 q^{63} + 120 q^{65} - 456 q^{66} - 20 q^{67} + 72 q^{68} - 576 q^{69} + 120 q^{70} - 132 q^{71} - 780 q^{72} - 424 q^{73} - 160 q^{74} - 60 q^{75} - 992 q^{76} - 60 q^{78} - 248 q^{79} - 480 q^{80} + 600 q^{82} + 112 q^{83} + 1100 q^{84} - 120 q^{85} + 852 q^{86} - 160 q^{87} - 1188 q^{88} + 168 q^{89} - 160 q^{91} + 1192 q^{92} - 1008 q^{93} + 328 q^{94} + 120 q^{95} + 124 q^{96} - 1008 q^{97} - 636 q^{98} - 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −1.00018 − 3.73273i 1.63472 − 2.83142i −9.46878 + 5.46680i −1.58114 + 1.58114i −12.2039 − 3.27004i 2.30767 − 8.61234i 18.9464 + 18.9464i −0.844632 − 1.46295i 7.48338 + 4.32053i
6.2 −0.947644 − 3.53665i −2.34119 + 4.05506i −8.14580 + 4.70298i 1.58114 − 1.58114i 16.5600 + 4.43723i −2.82701 + 10.5506i 13.9961 + 13.9961i −6.46235 − 11.1931i −7.09030 − 4.09359i
6.3 −0.590063 − 2.20214i 1.05817 − 1.83281i −1.03716 + 0.598805i 1.58114 − 1.58114i −4.66050 − 1.24878i −0.314408 + 1.17338i −4.51768 − 4.51768i 2.26053 + 3.91536i −4.41487 − 2.54892i
6.4 −0.305985 − 1.14195i 2.30106 − 3.98555i 2.25367 − 1.30116i −1.58114 + 1.58114i −5.25539 − 1.40818i −3.05692 + 11.4086i −5.51932 − 5.51932i −6.08971 − 10.5477i 2.28939 + 1.32178i
6.5 0.0928988 + 0.346703i −0.295143 + 0.511202i 3.35253 − 1.93558i 1.58114 − 1.58114i −0.204654 − 0.0548368i −0.0877254 + 0.327396i 1.99774 + 1.99774i 4.32578 + 7.49247i 0.695072 + 0.401300i
6.6 0.107580 + 0.401493i −2.08519 + 3.61165i 3.31448 − 1.91361i −1.58114 + 1.58114i −1.67438 − 0.448649i −2.68917 + 10.0361i 2.30053 + 2.30053i −4.19603 − 7.26773i −0.804916 − 0.464718i
6.7 0.312366 + 1.16577i 1.19180 − 2.06426i 2.20266 − 1.27171i −1.58114 + 1.58114i 2.77872 + 0.744556i 2.00836 − 7.49531i 5.58415 + 5.58415i 1.65922 + 2.87385i −2.33713 − 1.34934i
6.8 0.715386 + 2.66986i −2.11281 + 3.65949i −3.15226 + 1.81996i 1.58114 − 1.58114i −11.2818 − 3.02294i 0.943772 − 3.52221i 0.703773 + 0.703773i −4.42790 − 7.66935i 5.35254 + 3.09029i
6.9 0.729421 + 2.72224i 2.39711 − 4.15192i −3.41442 + 1.97132i 1.58114 − 1.58114i 13.0510 + 3.49701i −1.27402 + 4.75472i 0.114321 + 0.114321i −6.99232 − 12.1111i 5.45755 + 3.15092i
6.10 0.886220 + 3.30742i −0.0164891 + 0.0285599i −6.68953 + 3.86220i −1.58114 + 1.58114i −0.109072 − 0.0292259i 0.185607 − 0.692695i −9.01754 − 9.01754i 4.49946 + 7.79329i −6.63073 − 3.82825i
11.1 −1.00018 + 3.73273i 1.63472 + 2.83142i −9.46878 − 5.46680i −1.58114 − 1.58114i −12.2039 + 3.27004i 2.30767 + 8.61234i 18.9464 − 18.9464i −0.844632 + 1.46295i 7.48338 − 4.32053i
11.2 −0.947644 + 3.53665i −2.34119 − 4.05506i −8.14580 − 4.70298i 1.58114 + 1.58114i 16.5600 − 4.43723i −2.82701 − 10.5506i 13.9961 − 13.9961i −6.46235 + 11.1931i −7.09030 + 4.09359i
11.3 −0.590063 + 2.20214i 1.05817 + 1.83281i −1.03716 − 0.598805i 1.58114 + 1.58114i −4.66050 + 1.24878i −0.314408 − 1.17338i −4.51768 + 4.51768i 2.26053 − 3.91536i −4.41487 + 2.54892i
11.4 −0.305985 + 1.14195i 2.30106 + 3.98555i 2.25367 + 1.30116i −1.58114 − 1.58114i −5.25539 + 1.40818i −3.05692 − 11.4086i −5.51932 + 5.51932i −6.08971 + 10.5477i 2.28939 − 1.32178i
11.5 0.0928988 − 0.346703i −0.295143 − 0.511202i 3.35253 + 1.93558i 1.58114 + 1.58114i −0.204654 + 0.0548368i −0.0877254 − 0.327396i 1.99774 − 1.99774i 4.32578 − 7.49247i 0.695072 − 0.401300i
11.6 0.107580 − 0.401493i −2.08519 − 3.61165i 3.31448 + 1.91361i −1.58114 − 1.58114i −1.67438 + 0.448649i −2.68917 − 10.0361i 2.30053 − 2.30053i −4.19603 + 7.26773i −0.804916 + 0.464718i
11.7 0.312366 − 1.16577i 1.19180 + 2.06426i 2.20266 + 1.27171i −1.58114 − 1.58114i 2.77872 − 0.744556i 2.00836 + 7.49531i 5.58415 − 5.58415i 1.65922 − 2.87385i −2.33713 + 1.34934i
11.8 0.715386 − 2.66986i −2.11281 − 3.65949i −3.15226 − 1.81996i 1.58114 + 1.58114i −11.2818 + 3.02294i 0.943772 + 3.52221i 0.703773 − 0.703773i −4.42790 + 7.66935i 5.35254 − 3.09029i
11.9 0.729421 − 2.72224i 2.39711 + 4.15192i −3.41442 − 1.97132i 1.58114 + 1.58114i 13.0510 − 3.49701i −1.27402 − 4.75472i 0.114321 − 0.114321i −6.99232 + 12.1111i 5.45755 − 3.15092i
11.10 0.886220 − 3.30742i −0.0164891 − 0.0285599i −6.68953 − 3.86220i −1.58114 − 1.58114i −0.109072 + 0.0292259i 0.185607 + 0.692695i −9.01754 + 9.01754i 4.49946 − 7.79329i −6.63073 + 3.82825i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.3.p.a ✓ 40
5.b even 2 1 325.3.t.d 40
5.c odd 4 1 325.3.w.e 40
5.c odd 4 1 325.3.w.f 40
13.f odd 12 1 inner 65.3.p.a ✓ 40
65.o even 12 1 325.3.w.e 40
65.s odd 12 1 325.3.t.d 40
65.t even 12 1 325.3.w.f 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.p.a ✓ 40 1.a even 1 1 trivial
65.3.p.a ✓ 40 13.f odd 12 1 inner
325.3.t.d 40 5.b even 2 1
325.3.t.d 40 65.s odd 12 1
325.3.w.e 40 5.c odd 4 1
325.3.w.e 40 65.o even 12 1
325.3.w.f 40 5.c odd 4 1
325.3.w.f 40 65.t even 12 1

Hecke kernels

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