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Properties

Label 4010.2.a.o
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, 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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} + 2 q^{3} + 22 q^{4} - 22 q^{5} + 2 q^{6} + 13 q^{7} + 22 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} + 2 q^{3} + 22 q^{4} - 22 q^{5} + 2 q^{6} + 13 q^{7} + 22 q^{8} + 32 q^{9} - 22 q^{10} - 3 q^{11} + 2 q^{12} + 6 q^{13} + 13 q^{14} - 2 q^{15} + 22 q^{16} + 17 q^{17} + 32 q^{18} + 13 q^{19} - 22 q^{20} + 16 q^{21} - 3 q^{22} + 19 q^{23} + 2 q^{24} + 22 q^{25} + 6 q^{26} + 14 q^{27} + 13 q^{28} + 14 q^{29} - 2 q^{30} + 13 q^{31} + 22 q^{32} + 12 q^{33} + 17 q^{34} - 13 q^{35} + 32 q^{36} + 35 q^{37} + 13 q^{38} + 30 q^{39} - 22 q^{40} - 5 q^{41} + 16 q^{42} + 19 q^{43} - 3 q^{44} - 32 q^{45} + 19 q^{46} + 29 q^{47} + 2 q^{48} + 61 q^{49} + 22 q^{50} + q^{51} + 6 q^{52} + 29 q^{53} + 14 q^{54} + 3 q^{55} + 13 q^{56} + 33 q^{57} + 14 q^{58} - 4 q^{59} - 2 q^{60} + 20 q^{61} + 13 q^{62} + 50 q^{63} + 22 q^{64} - 6 q^{65} + 12 q^{66} + 48 q^{67} + 17 q^{68} + 19 q^{69} - 13 q^{70} + 2 q^{71} + 32 q^{72} + 16 q^{73} + 35 q^{74} + 2 q^{75} + 13 q^{76} + 53 q^{77} + 30 q^{78} + 29 q^{79} - 22 q^{80} + 54 q^{81} - 5 q^{82} + 13 q^{83} + 16 q^{84} - 17 q^{85} + 19 q^{86} + 56 q^{87} - 3 q^{88} + 20 q^{89} - 32 q^{90} + 42 q^{91} + 19 q^{92} + 50 q^{93} + 29 q^{94} - 13 q^{95} + 2 q^{96} + 36 q^{97} + 61 q^{98} - q^{99}+O(q^{100}) \) 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 −3.30817 1.00000 −1.00000 −3.30817 −2.76226 1.00000 7.94397 −1.00000
1.2 1.00000 −2.90289 1.00000 −1.00000 −2.90289 4.43766 1.00000 5.42676 −1.00000
1.3 1.00000 −2.88307 1.00000 −1.00000 −2.88307 −0.867210 1.00000 5.31208 −1.00000
1.4 1.00000 −2.52966 1.00000 −1.00000 −2.52966 2.69506 1.00000 3.39917 −1.00000
1.5 1.00000 −1.85051 1.00000 −1.00000 −1.85051 −3.61831 1.00000 0.424381 −1.00000
1.6 1.00000 −1.60919 1.00000 −1.00000 −1.60919 3.90377 1.00000 −0.410498 −1.00000
1.7 1.00000 −1.34023 1.00000 −1.00000 −1.34023 0.629438 1.00000 −1.20379 −1.00000
1.8 1.00000 −1.31697 1.00000 −1.00000 −1.31697 −3.01403 1.00000 −1.26559 −1.00000
1.9 1.00000 −1.18460 1.00000 −1.00000 −1.18460 4.68706 1.00000 −1.59673 −1.00000
1.10 1.00000 −0.310341 1.00000 −1.00000 −0.310341 1.61370 1.00000 −2.90369 −1.00000
1.11 1.00000 −0.165649 1.00000 −1.00000 −0.165649 −1.98987 1.00000 −2.97256 −1.00000
1.12 1.00000 0.417458 1.00000 −1.00000 0.417458 −4.20952 1.00000 −2.82573 −1.00000
1.13 1.00000 0.529822 1.00000 −1.00000 0.529822 −4.65586 1.00000 −2.71929 −1.00000
1.14 1.00000 0.642301 1.00000 −1.00000 0.642301 4.88072 1.00000 −2.58745 −1.00000
1.15 1.00000 1.74418 1.00000 −1.00000 1.74418 3.19985 1.00000 0.0421511 −1.00000
1.16 1.00000 1.77730 1.00000 −1.00000 1.77730 1.31771 1.00000 0.158778 −1.00000
1.17 1.00000 1.82747 1.00000 −1.00000 1.82747 0.568203 1.00000 0.339635 −1.00000
1.18 1.00000 2.45072 1.00000 −1.00000 2.45072 4.38820 1.00000 3.00604 −1.00000
1.19 1.00000 2.51296 1.00000 −1.00000 2.51296 −2.03791 1.00000 3.31499 −1.00000
1.20 1.00000 3.00389 1.00000 −1.00000 3.00389 −1.21989 1.00000 6.02333 −1.00000
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(401\) \(1\)

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 4010.2.a.o ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.o ✓ 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{22} - 2 T_{3}^{21} - 47 T_{3}^{20} + 88 T_{3}^{19} + 936 T_{3}^{18} - 1612 T_{3}^{17} + \cdots - 3520 \) Copy content Toggle raw display
\( T_{7}^{22} - 13 T_{7}^{21} - 23 T_{7}^{20} + 947 T_{7}^{19} - 1876 T_{7}^{18} - 26849 T_{7}^{17} + \cdots - 182495808 \) Copy content Toggle raw display
\( T_{11}^{22} + 3 T_{11}^{21} - 152 T_{11}^{20} - 380 T_{11}^{19} + 9909 T_{11}^{18} + 18631 T_{11}^{17} + \cdots - 13105691136 \) Copy content Toggle raw display