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Properties

Label 3021.2.a.j
Level $3021$
Weight $2$
Character orbit 3021.a
Self dual yes
Analytic conductor $24.123$
Analytic rank $0$
Dimension $24$
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] = [3021,2,Mod(1,3021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3021 = 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3021.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: \(24.1228064506\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} + 24 q^{3} + 28 q^{4} + 13 q^{5} + 6 q^{6} + 4 q^{7} + 18 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} + 24 q^{3} + 28 q^{4} + 13 q^{5} + 6 q^{6} + 4 q^{7} + 18 q^{8} + 24 q^{9} - 2 q^{10} + 21 q^{11} + 28 q^{12} + 18 q^{14} + 13 q^{15} + 32 q^{16} + 18 q^{17} + 6 q^{18} + 24 q^{19} + 28 q^{20} + 4 q^{21} - q^{22} + 21 q^{23} + 18 q^{24} + 25 q^{25} + 5 q^{26} + 24 q^{27} + 18 q^{28} + 8 q^{29} - 2 q^{30} + 17 q^{31} + 37 q^{32} + 21 q^{33} + 2 q^{34} + 20 q^{35} + 28 q^{36} - 6 q^{37} + 6 q^{38} + 3 q^{40} + 11 q^{41} + 18 q^{42} + 6 q^{43} + 13 q^{44} + 13 q^{45} - 3 q^{46} + 13 q^{47} + 32 q^{48} + 32 q^{49} - 22 q^{50} + 18 q^{51} + 9 q^{52} + 24 q^{53} + 6 q^{54} + 25 q^{55} + 39 q^{56} + 24 q^{57} + 5 q^{58} + 60 q^{59} + 28 q^{60} + 17 q^{61} + 4 q^{63} + 20 q^{64} + 33 q^{65} - q^{66} - 6 q^{67} + 14 q^{68} + 21 q^{69} - 17 q^{70} + 47 q^{71} + 18 q^{72} + 2 q^{73} + 13 q^{74} + 25 q^{75} + 28 q^{76} + 29 q^{77} + 5 q^{78} - 6 q^{79} + 67 q^{80} + 24 q^{81} - 7 q^{82} + 32 q^{83} + 18 q^{84} + 14 q^{85} + 16 q^{86} + 8 q^{87} - 16 q^{88} + 69 q^{89} - 2 q^{90} + 10 q^{91} - 6 q^{92} + 17 q^{93} - 46 q^{94} + 13 q^{95} + 37 q^{96} - 14 q^{97} - 16 q^{98} + 21 q^{99}+O(q^{100}) \) 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 −2.52065 1.00000 4.35367 4.15755 −2.52065 −1.28919 −5.93276 1.00000 −10.4797
1.2 −2.47609 1.00000 4.13104 1.54744 −2.47609 3.43138 −5.27666 1.00000 −3.83161
1.3 −2.23096 1.00000 2.97720 −2.23626 −2.23096 −4.03952 −2.18010 1.00000 4.98901
1.4 −2.10266 1.00000 2.42118 0.973583 −2.10266 −2.02936 −0.885597 1.00000 −2.04711
1.5 −1.88674 1.00000 1.55980 −1.54145 −1.88674 1.20385 0.830551 1.00000 2.90833
1.6 −1.15471 1.00000 −0.666634 −0.971901 −1.15471 0.580389 3.07920 1.00000 1.12227
1.7 −1.14489 1.00000 −0.689217 3.81195 −1.14489 4.89385 3.07887 1.00000 −4.36428
1.8 −0.933171 1.00000 −1.12919 −1.17001 −0.933171 −4.34579 2.92007 1.00000 1.09182
1.9 −0.897660 1.00000 −1.19421 3.60923 −0.897660 −2.88935 2.86731 1.00000 −3.23986
1.10 −0.564915 1.00000 −1.68087 −2.70419 −0.564915 2.51805 2.07938 1.00000 1.52764
1.11 −0.0748135 1.00000 −1.99440 2.31955 −0.0748135 0.783009 0.298835 1.00000 −0.173534
1.12 0.205832 1.00000 −1.95763 1.57621 0.205832 2.92257 −0.814608 1.00000 0.324434
1.13 0.731654 1.00000 −1.46468 −0.224965 0.731654 −0.743430 −2.53495 1.00000 −0.164596
1.14 0.752610 1.00000 −1.43358 −4.04716 0.752610 −0.848504 −2.58414 1.00000 −3.04594
1.15 0.932223 1.00000 −1.13096 3.22734 0.932223 −5.02860 −2.91875 1.00000 3.00860
1.16 1.21599 1.00000 −0.521380 3.11346 1.21599 2.82986 −3.06596 1.00000 3.78592
1.17 1.55337 1.00000 0.412964 −2.70501 1.55337 −4.05408 −2.46526 1.00000 −4.20188
1.18 1.79063 1.00000 1.20635 −0.766944 1.79063 4.19951 −1.42113 1.00000 −1.37331
1.19 2.19579 1.00000 2.82149 2.59973 2.19579 0.988223 1.80383 1.00000 5.70847
1.20 2.26554 1.00000 3.13268 −2.65408 2.26554 3.33634 2.56612 1.00000 −6.01293
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)
\(53\) \(-1\)

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 3021.2.a.j ✓ 24
3.b odd 2 1 9063.2.a.o 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3021.2.a.j ✓ 24 1.a even 1 1 trivial
9063.2.a.o 24 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 6 T_{2}^{23} - 20 T_{2}^{22} + 178 T_{2}^{21} + 78 T_{2}^{20} - 2221 T_{2}^{19} + \cdots - 204 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3021))\). Copy content Toggle raw display