Properties

Label 6021.2.a.r
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $0$
Dimension $35$
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] = [6021,2,Mod(1,6021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6021, 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("6021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6021.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: \(48.0779270570\)
Analytic rank: \(0\)
Dimension: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8} - 7 q^{10} + 34 q^{11} + 2 q^{13} + 18 q^{14} + 42 q^{16} + 20 q^{17} - 14 q^{19} + 27 q^{20} + 8 q^{22} + 8 q^{23} + 27 q^{25} + 28 q^{26} - 4 q^{28} + 23 q^{29} + 12 q^{31} + 29 q^{32} - 21 q^{34} + 41 q^{35} - 8 q^{37} + 18 q^{38} + 16 q^{40} + 50 q^{41} + 2 q^{43} + 83 q^{44} - 5 q^{46} + 21 q^{47} + 43 q^{49} + 39 q^{50} + 6 q^{52} + 37 q^{53} + 20 q^{55} + 33 q^{56} - 32 q^{58} + 81 q^{59} - 6 q^{61} + 26 q^{62} - q^{64} + 29 q^{65} + 12 q^{67} + 55 q^{68} + 50 q^{70} + 43 q^{71} - 20 q^{73} + 48 q^{74} - 15 q^{76} + 29 q^{77} + 28 q^{79} + 88 q^{80} - 6 q^{82} + 64 q^{83} - 67 q^{85} + 41 q^{86} + 10 q^{88} + 50 q^{89} + 2 q^{91} + 32 q^{92} + 15 q^{94} + 25 q^{95} + 9 q^{97} + 74 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −2.73710 0 5.49174 0.00733490 0 −3.12271 −9.55725 0 −0.0200764
1.2 −2.45047 0 4.00478 1.31775 0 −2.98449 −4.91265 0 −3.22911
1.3 −2.31533 0 3.36077 2.18639 0 4.16397 −3.15063 0 −5.06222
1.4 −2.30052 0 3.29241 0.673783 0 1.52445 −2.97323 0 −1.55005
1.5 −2.13504 0 2.55839 2.06664 0 −0.593247 −1.19218 0 −4.41235
1.6 −2.10383 0 2.42609 2.21279 0 −3.17552 −0.896424 0 −4.65533
1.7 −2.08857 0 2.36212 −1.86886 0 −1.80880 −0.756310 0 3.90323
1.8 −1.74554 0 1.04691 −3.98518 0 −0.618122 1.66366 0 6.95629
1.9 −1.41065 0 −0.0100678 3.64343 0 2.96382 2.83550 0 −5.13961
1.10 −1.33519 0 −0.217276 0.108401 0 4.69056 2.96048 0 −0.144736
1.11 −0.982520 0 −1.03465 −0.593429 0 1.54053 2.98161 0 0.583056
1.12 −0.941194 0 −1.11415 2.01775 0 −3.14622 2.93102 0 −1.89910
1.13 −0.627854 0 −1.60580 −3.06245 0 0.00430391 2.26392 0 1.92277
1.14 −0.563437 0 −1.68254 3.58037 0 0.250597 2.07488 0 −2.01732
1.15 −0.544885 0 −1.70310 −3.52229 0 −1.92545 2.01776 0 1.91924
1.16 −0.384611 0 −1.85207 −1.68485 0 −1.18684 1.48155 0 0.648011
1.17 −0.0961107 0 −1.99076 2.13805 0 −1.72518 0.383555 0 −0.205489
1.18 0.145112 0 −1.97894 4.00700 0 −4.35864 −0.577391 0 0.581462
1.19 0.378423 0 −1.85680 0.673616 0 3.71590 −1.45950 0 0.254912
1.20 0.424438 0 −1.81985 −1.02884 0 −4.02930 −1.62129 0 −0.436677
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(223\) \(-1\)

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 6021.2.a.r yes 35
3.b odd 2 1 6021.2.a.q 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.q 35 3.b odd 2 1
6021.2.a.r yes 35 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{35} - 4 T_{2}^{34} - 45 T_{2}^{33} + 191 T_{2}^{32} + 898 T_{2}^{31} - 4104 T_{2}^{30} - 10447 T_{2}^{29} + 52472 T_{2}^{28} + 78189 T_{2}^{27} - 444821 T_{2}^{26} - 390260 T_{2}^{25} + 2637421 T_{2}^{24} + 1288333 T_{2}^{23} + \cdots + 534 \) Copy content Toggle raw display
\( T_{5}^{35} - 10 T_{5}^{34} - 51 T_{5}^{33} + 805 T_{5}^{32} + 348 T_{5}^{31} - 28343 T_{5}^{30} + 35781 T_{5}^{29} + 572902 T_{5}^{28} - 1316105 T_{5}^{27} - 7307228 T_{5}^{26} + 23445918 T_{5}^{25} + \cdots + 4984161 \) Copy content Toggle raw display