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Properties

Label 65.3.g.a
Level $65$
Weight $3$
Character orbit 65.g
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(34,65)]
 
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([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.34");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 65.g (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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{5} + 12 q^{6} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{5} + 12 q^{6} - 56 q^{9} - 8 q^{11} - 24 q^{14} + 44 q^{15} + 24 q^{16} - 60 q^{19} + 8 q^{20} + 28 q^{21} + 64 q^{24} - 72 q^{26} - 40 q^{29} + 68 q^{31} - 140 q^{34} - 164 q^{35} + 208 q^{39} + 292 q^{40} - 332 q^{41} + 356 q^{44} - 40 q^{45} - 236 q^{46} + 68 q^{50} + 228 q^{54} - 152 q^{55} + 108 q^{59} + 184 q^{60} + 672 q^{61} - 268 q^{65} + 80 q^{66} - 732 q^{70} + 368 q^{71} - 400 q^{74} - 704 q^{76} - 464 q^{79} + 220 q^{80} + 144 q^{81} - 540 q^{84} - 236 q^{85} + 352 q^{86} + 624 q^{89} - 20 q^{91} - 456 q^{94} - 300 q^{96} - 1164 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.57147 + 2.57147i − 0.530912i − 9.22493i 1.15926 − 4.86376i 1.36523 + 1.36523i −5.08660 − 5.08660i 13.4358 + 13.4358i 8.71813 9.52600 + 15.4880i
34.2 −2.12780 + 2.12780i − 3.33709i − 5.05509i 0.983580 + 4.90230i 7.10066 + 7.10066i 8.57772 + 8.57772i 2.24502 + 2.24502i −2.13614 −12.5240 − 8.33827i
34.3 −1.64592 + 1.64592i 4.98115i − 1.41808i 4.98106 + 0.434772i −8.19855 − 8.19855i 0.0863113 + 0.0863113i −4.24962 − 4.24962i −15.8118 −8.91401 + 7.48281i
34.4 −1.35950 + 1.35950i 2.30697i 0.303507i −4.94346 − 0.749794i −3.13633 − 3.13633i 0.997639 + 0.997639i −5.85063 − 5.85063i 3.67790 7.73999 − 5.70130i
34.5 −1.10974 + 1.10974i − 4.98039i 1.53696i −4.19573 − 2.71953i 5.52693 + 5.52693i −4.09605 − 4.09605i −6.14458 − 6.14458i −15.8043 7.67413 − 1.63820i
34.6 −0.266801 + 0.266801i − 1.28209i 3.85763i 4.70456 − 1.69326i 0.342062 + 0.342062i 3.28160 + 3.28160i −2.09642 − 2.09642i 7.35624 −0.803415 + 1.70694i
34.7 0.266801 − 0.266801i 1.28209i 3.85763i −1.69326 + 4.70456i 0.342062 + 0.342062i −3.28160 − 3.28160i 2.09642 + 2.09642i 7.35624 0.803415 + 1.70694i
34.8 1.10974 − 1.10974i 4.98039i 1.53696i −2.71953 − 4.19573i 5.52693 + 5.52693i 4.09605 + 4.09605i 6.14458 + 6.14458i −15.8043 −7.67413 − 1.63820i
34.9 1.35950 − 1.35950i − 2.30697i 0.303507i −0.749794 − 4.94346i −3.13633 − 3.13633i −0.997639 − 0.997639i 5.85063 + 5.85063i 3.67790 −7.73999 − 5.70130i
34.10 1.64592 − 1.64592i − 4.98115i − 1.41808i 0.434772 + 4.98106i −8.19855 − 8.19855i −0.0863113 − 0.0863113i 4.24962 + 4.24962i −15.8118 8.91401 + 7.48281i
34.11 2.12780 − 2.12780i 3.33709i − 5.05509i 4.90230 + 0.983580i 7.10066 + 7.10066i −8.57772 − 8.57772i −2.24502 − 2.24502i −2.13614 12.5240 − 8.33827i
34.12 2.57147 − 2.57147i 0.530912i − 9.22493i −4.86376 + 1.15926i 1.36523 + 1.36523i 5.08660 + 5.08660i −13.4358 − 13.4358i 8.71813 −9.52600 + 15.4880i
44.1 −2.57147 − 2.57147i 0.530912i 9.22493i 1.15926 + 4.86376i 1.36523 − 1.36523i −5.08660 + 5.08660i 13.4358 − 13.4358i 8.71813 9.52600 − 15.4880i
44.2 −2.12780 − 2.12780i 3.33709i 5.05509i 0.983580 − 4.90230i 7.10066 − 7.10066i 8.57772 − 8.57772i 2.24502 − 2.24502i −2.13614 −12.5240 + 8.33827i
44.3 −1.64592 − 1.64592i − 4.98115i 1.41808i 4.98106 − 0.434772i −8.19855 + 8.19855i 0.0863113 − 0.0863113i −4.24962 + 4.24962i −15.8118 −8.91401 − 7.48281i
44.4 −1.35950 − 1.35950i − 2.30697i − 0.303507i −4.94346 + 0.749794i −3.13633 + 3.13633i 0.997639 − 0.997639i −5.85063 + 5.85063i 3.67790 7.73999 + 5.70130i
44.5 −1.10974 − 1.10974i 4.98039i − 1.53696i −4.19573 + 2.71953i 5.52693 − 5.52693i −4.09605 + 4.09605i −6.14458 + 6.14458i −15.8043 7.67413 + 1.63820i
44.6 −0.266801 − 0.266801i 1.28209i − 3.85763i 4.70456 + 1.69326i 0.342062 − 0.342062i 3.28160 − 3.28160i −2.09642 + 2.09642i 7.35624 −0.803415 − 1.70694i
44.7 0.266801 + 0.266801i − 1.28209i − 3.85763i −1.69326 − 4.70456i 0.342062 − 0.342062i −3.28160 + 3.28160i 2.09642 − 2.09642i 7.35624 0.803415 − 1.70694i
44.8 1.10974 + 1.10974i − 4.98039i − 1.53696i −2.71953 + 4.19573i 5.52693 − 5.52693i 4.09605 − 4.09605i 6.14458 − 6.14458i −15.8043 −7.67413 + 1.63820i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.d odd 4 1 inner
65.g 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.g.a ✓ 24
5.b even 2 1 inner 65.3.g.a ✓ 24
5.c odd 4 2 325.3.j.e 24
13.d odd 4 1 inner 65.3.g.a ✓ 24
65.f even 4 1 325.3.j.e 24
65.g odd 4 1 inner 65.3.g.a ✓ 24
65.k even 4 1 325.3.j.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.g.a ✓ 24 1.a even 1 1 trivial
65.3.g.a ✓ 24 5.b even 2 1 inner
65.3.g.a ✓ 24 13.d odd 4 1 inner
65.3.g.a ✓ 24 65.g odd 4 1 inner
325.3.j.e 24 5.c odd 4 2
325.3.j.e 24 65.f even 4 1
325.3.j.e 24 65.k even 4 1

Hecke kernels

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