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Properties

Label 4021.2.a.b
Level $4021$
Weight $2$
Character orbit 4021.a
Self dual yes
Analytic conductor $32.108$
Analytic rank $1$
Dimension $151$
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] = [4021,2,Mod(1,4021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4021.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: \(32.1078466528\)
Analytic rank: \(1\)
Dimension: \(151\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9} - 7 q^{10} - 126 q^{11} - 47 q^{12} - 10 q^{13} - 58 q^{14} - 50 q^{15} + 74 q^{16} - 39 q^{17} - 47 q^{18} - 48 q^{19} - 64 q^{20} - 50 q^{21} - 7 q^{22} - 91 q^{23} - 44 q^{24} + 102 q^{25} - 94 q^{26} - 92 q^{27} - 27 q^{28} - 79 q^{29} - 12 q^{30} - 33 q^{31} - 102 q^{32} - 7 q^{33} - 17 q^{34} - 183 q^{35} - 6 q^{36} - 24 q^{37} - 77 q^{38} - 81 q^{39} - 37 q^{40} - 62 q^{41} + 4 q^{42} - 74 q^{43} - 209 q^{44} - 35 q^{45} - 37 q^{46} - 127 q^{47} - 59 q^{48} + 75 q^{49} - 79 q^{50} - 138 q^{51} - 12 q^{52} - 104 q^{53} - 43 q^{54} - 49 q^{55} - 167 q^{56} - 19 q^{57} - 32 q^{58} - 207 q^{59} - 94 q^{60} - 57 q^{61} - 69 q^{62} - 49 q^{63} - 4 q^{64} - 78 q^{65} - 51 q^{66} - 92 q^{67} - 80 q^{68} - 84 q^{69} + 5 q^{70} - 217 q^{71} - 79 q^{72} + 2 q^{73} - 113 q^{74} - 173 q^{75} - 70 q^{76} - 85 q^{77} - 46 q^{78} - 70 q^{79} - 95 q^{80} - 49 q^{81} - 12 q^{82} - 109 q^{83} - 114 q^{84} - 25 q^{85} - 136 q^{86} - 60 q^{87} - 2 q^{88} - 144 q^{89} - 55 q^{90} - 62 q^{91} - 175 q^{92} - 42 q^{93} - 15 q^{94} - 189 q^{95} - 64 q^{96} - 24 q^{97} - 37 q^{98} - 293 q^{99}+O(q^{100}) \) 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 −2.75476 1.58414 5.58872 −2.10175 −4.36394 2.25410 −9.88608 −0.490495 5.78981
1.2 −2.72951 0.628920 5.45022 1.83150 −1.71664 −2.53691 −9.41739 −2.60446 −4.99910
1.3 −2.70449 −0.271185 5.31424 −1.63010 0.733417 −2.12965 −8.96332 −2.92646 4.40858
1.4 −2.69273 1.85515 5.25082 1.44838 −4.99542 4.11859 −8.75359 0.441579 −3.90011
1.5 −2.69045 −2.62936 5.23855 −3.01308 7.07417 1.15272 −8.71316 3.91353 8.10656
1.6 −2.66021 −2.13808 5.07674 4.04432 5.68776 2.33919 −8.18478 1.57140 −10.7588
1.7 −2.64544 −1.40778 4.99834 −2.43306 3.72420 1.18791 −7.93191 −1.01815 6.43651
1.8 −2.64112 2.33893 4.97551 −1.70298 −6.17740 0.267464 −7.85867 2.47061 4.49778
1.9 −2.55186 −0.0699266 4.51199 −3.83680 0.178443 5.05477 −6.41025 −2.99511 9.79098
1.10 −2.54734 1.98068 4.48893 0.894542 −5.04546 −4.01929 −6.34013 0.923096 −2.27870
1.11 −2.52668 −3.06344 4.38409 0.824998 7.74031 3.92448 −6.02383 6.38463 −2.08450
1.12 −2.49227 −2.01078 4.21142 3.04731 5.01140 −1.48654 −5.51145 1.04322 −7.59472
1.13 −2.43685 −1.07161 3.93825 2.02983 2.61136 3.59143 −4.72324 −1.85165 −4.94641
1.14 −2.41856 −1.61942 3.84946 −1.02567 3.91668 2.10305 −4.47303 −0.377473 2.48065
1.15 −2.38084 2.66837 3.66838 −4.16670 −6.35294 0.560262 −3.97213 4.12018 9.92023
1.16 −2.37064 −0.181119 3.61993 1.77155 0.429369 −0.953276 −3.84026 −2.96720 −4.19971
1.17 −2.36898 2.54130 3.61208 1.21451 −6.02031 −1.53479 −3.81900 3.45822 −2.87715
1.18 −2.33695 0.0283872 3.46135 −2.49189 −0.0663397 4.25940 −3.41512 −2.99919 5.82343
1.19 −2.23472 −1.90948 2.99397 0.847533 4.26716 −4.08817 −2.22124 0.646133 −1.89400
1.20 −2.23064 −3.03788 2.97574 2.24920 6.77641 −3.34201 −2.17651 6.22873 −5.01714
See next 80 embeddings (of 151 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.151
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4021\) \(1\)

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 4021.2.a.b ✓ 151
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4021.2.a.b ✓ 151 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{151} + 16 T_{2}^{150} - 83 T_{2}^{149} - 2656 T_{2}^{148} - 1920 T_{2}^{147} + 209918 T_{2}^{146} + \cdots - 1920453 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4021))\). Copy content Toggle raw display