↑

Properties

Label 6010.2.a.j
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $0$
Dimension $33$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6010,2,Mod(1,6010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6010 = 2 \cdot 5 \cdot 601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6010.a (trivial)

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: yes
Analytic conductor: \(47.9900916148\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 33 q + 33 q^{2} + 6 q^{3} + 33 q^{4} + 33 q^{5} + 6 q^{6} + 4 q^{7} + 33 q^{8} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q + 33 q^{2} + 6 q^{3} + 33 q^{4} + 33 q^{5} + 6 q^{6} + 4 q^{7} + 33 q^{8} + 49 q^{9} + 33 q^{10} + 12 q^{11} + 6 q^{12} + 20 q^{13} + 4 q^{14} + 6 q^{15} + 33 q^{16} + 33 q^{17} + 49 q^{18} + 17 q^{19} + 33 q^{20} + 26 q^{21} + 12 q^{22} + 7 q^{23} + 6 q^{24} + 33 q^{25} + 20 q^{26} + 21 q^{27} + 4 q^{28} + 33 q^{29} + 6 q^{30} + 35 q^{31} + 33 q^{32} + 25 q^{33} + 33 q^{34} + 4 q^{35} + 49 q^{36} + 16 q^{37} + 17 q^{38} + 22 q^{39} + 33 q^{40} + 39 q^{41} + 26 q^{42} - 3 q^{43} + 12 q^{44} + 49 q^{45} + 7 q^{46} + 19 q^{47} + 6 q^{48} + 69 q^{49} + 33 q^{50} + 21 q^{51} + 20 q^{52} + 41 q^{53} + 21 q^{54} + 12 q^{55} + 4 q^{56} + 33 q^{58} + 18 q^{59} + 6 q^{60} + 30 q^{61} + 35 q^{62} - 15 q^{63} + 33 q^{64} + 20 q^{65} + 25 q^{66} - 9 q^{67} + 33 q^{68} + 23 q^{69} + 4 q^{70} + 36 q^{71} + 49 q^{72} + 35 q^{73} + 16 q^{74} + 6 q^{75} + 17 q^{76} + 26 q^{77} + 22 q^{78} + 32 q^{79} + 33 q^{80} + 53 q^{81} + 39 q^{82} + 24 q^{83} + 26 q^{84} + 33 q^{85} - 3 q^{86} + 12 q^{87} + 12 q^{88} + 40 q^{89} + 49 q^{90} + 5 q^{91} + 7 q^{92} + 18 q^{93} + 19 q^{94} + 17 q^{95} + 6 q^{96} + 39 q^{97} + 69 q^{98} + 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 1.00000 −3.20038 1.00000 1.00000 −3.20038 −3.69470 1.00000 7.24241 1.00000
1.2 1.00000 −3.13566 1.00000 1.00000 −3.13566 3.51903 1.00000 6.83234 1.00000
1.3 1.00000 −2.97405 1.00000 1.00000 −2.97405 −4.80622 1.00000 5.84500 1.00000
1.4 1.00000 −2.94052 1.00000 1.00000 −2.94052 1.02542 1.00000 5.64669 1.00000
1.5 1.00000 −2.70935 1.00000 1.00000 −2.70935 2.38883 1.00000 4.34059 1.00000
1.6 1.00000 −2.12592 1.00000 1.00000 −2.12592 −2.80754 1.00000 1.51952 1.00000
1.7 1.00000 −1.86907 1.00000 1.00000 −1.86907 −0.450739 1.00000 0.493433 1.00000
1.8 1.00000 −1.57998 1.00000 1.00000 −1.57998 −4.21564 1.00000 −0.503648 1.00000
1.9 1.00000 −1.55579 1.00000 1.00000 −1.55579 2.74607 1.00000 −0.579528 1.00000
1.10 1.00000 −1.52452 1.00000 1.00000 −1.52452 3.34647 1.00000 −0.675836 1.00000
1.11 1.00000 −1.48456 1.00000 1.00000 −1.48456 −3.64737 1.00000 −0.796082 1.00000
1.12 1.00000 −0.800428 1.00000 1.00000 −0.800428 2.44913 1.00000 −2.35931 1.00000
1.13 1.00000 −0.635616 1.00000 1.00000 −0.635616 −1.32070 1.00000 −2.59599 1.00000
1.14 1.00000 −0.552487 1.00000 1.00000 −0.552487 2.28822 1.00000 −2.69476 1.00000
1.15 1.00000 −0.489009 1.00000 1.00000 −0.489009 −2.03673 1.00000 −2.76087 1.00000
1.16 1.00000 −0.361555 1.00000 1.00000 −0.361555 4.32690 1.00000 −2.86928 1.00000
1.17 1.00000 0.162499 1.00000 1.00000 0.162499 −4.01770 1.00000 −2.97359 1.00000
1.18 1.00000 0.351716 1.00000 1.00000 0.351716 −0.0941576 1.00000 −2.87630 1.00000
1.19 1.00000 0.649976 1.00000 1.00000 0.649976 1.95740 1.00000 −2.57753 1.00000
1.20 1.00000 0.828443 1.00000 1.00000 0.828443 4.24055 1.00000 −2.31368 1.00000
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(601\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6010.2.a.j ✓ 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.j ✓ 33 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{33} - 6 T_{3}^{32} - 56 T_{3}^{31} + 389 T_{3}^{30} + 1302 T_{3}^{29} - 11223 T_{3}^{28} + \cdots - 657664 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display