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Properties

Label 6033.2.a.e
Level $6033$
Weight $2$
Character orbit 6033.a
Self dual yes
Analytic conductor $48.174$
Analytic rank $0$
Dimension $97$
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] = [6033,2,Mod(1,6033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6033, 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("6033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6033 = 3 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6033.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: \(48.1737475394\)
Analytic rank: \(0\)
Dimension: \(97\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 97 q + 12 q^{2} + 97 q^{3} + 120 q^{4} + 6 q^{5} + 12 q^{6} + 50 q^{7} + 30 q^{8} + 97 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 97 q + 12 q^{2} + 97 q^{3} + 120 q^{4} + 6 q^{5} + 12 q^{6} + 50 q^{7} + 30 q^{8} + 97 q^{9} + 35 q^{10} + 18 q^{11} + 120 q^{12} + 67 q^{13} - q^{14} + 6 q^{15} + 158 q^{16} + 25 q^{17} + 12 q^{18} + 51 q^{19} + 10 q^{20} + 50 q^{21} + 39 q^{22} + 87 q^{23} + 30 q^{24} + 149 q^{25} + 14 q^{26} + 97 q^{27} + 83 q^{28} + 23 q^{29} + 35 q^{30} + 72 q^{31} + 57 q^{32} + 18 q^{33} + 28 q^{34} + 45 q^{35} + 120 q^{36} + 72 q^{37} + 3 q^{38} + 67 q^{39} + 90 q^{40} + 5 q^{41} - q^{42} + 122 q^{43} + 11 q^{44} + 6 q^{45} + 56 q^{46} + 49 q^{47} + 158 q^{48} + 167 q^{49} + 13 q^{50} + 25 q^{51} + 128 q^{52} + 30 q^{53} + 12 q^{54} + 120 q^{55} - 21 q^{56} + 51 q^{57} + 37 q^{58} + 2 q^{59} + 10 q^{60} + 158 q^{61} + 17 q^{62} + 50 q^{63} + 212 q^{64} + q^{65} + 39 q^{66} + 77 q^{67} + 56 q^{68} + 87 q^{69} + 9 q^{70} + 38 q^{71} + 30 q^{72} + 82 q^{73} - 6 q^{74} + 149 q^{75} + 93 q^{76} + 49 q^{77} + 14 q^{78} + 134 q^{79} - 25 q^{80} + 97 q^{81} + 53 q^{82} + 69 q^{83} + 83 q^{84} + 72 q^{85} + 23 q^{87} + 107 q^{88} + 35 q^{90} + 84 q^{91} + 108 q^{92} + 72 q^{93} + 65 q^{94} + 89 q^{95} + 57 q^{96} + 65 q^{97} + 18 q^{99}+O(q^{100}) \) 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 −2.77569 1.00000 5.70444 −4.08383 −2.77569 2.32651 −10.2824 1.00000 11.3354
1.2 −2.72701 1.00000 5.43660 −1.98915 −2.72701 −1.33626 −9.37164 1.00000 5.42444
1.3 −2.72505 1.00000 5.42589 −0.505520 −2.72505 −2.64822 −9.33570 1.00000 1.37757
1.4 −2.69536 1.00000 5.26496 2.52315 −2.69536 0.986697 −8.80025 1.00000 −6.80079
1.5 −2.65791 1.00000 5.06448 3.14919 −2.65791 3.30721 −8.14510 1.00000 −8.37025
1.6 −2.63042 1.00000 4.91910 −0.782993 −2.63042 4.49130 −7.67844 1.00000 2.05960
1.7 −2.53542 1.00000 4.42834 −3.51748 −2.53542 −3.36027 −6.15684 1.00000 8.91826
1.8 −2.50001 1.00000 4.25007 −1.07876 −2.50001 3.84793 −5.62521 1.00000 2.69691
1.9 −2.41345 1.00000 3.82475 3.09815 −2.41345 4.57422 −4.40394 1.00000 −7.47724
1.10 −2.38207 1.00000 3.67426 −3.40207 −2.38207 2.20870 −3.98822 1.00000 8.10396
1.11 −2.37614 1.00000 3.64603 3.12222 −2.37614 −0.986181 −3.91118 1.00000 −7.41883
1.12 −2.30219 1.00000 3.30008 −3.13763 −2.30219 1.81298 −2.99304 1.00000 7.22342
1.13 −2.19685 1.00000 2.82615 3.79567 −2.19685 −1.66772 −1.81492 1.00000 −8.33851
1.14 −2.11811 1.00000 2.48641 0.996414 −2.11811 −4.64745 −1.03027 1.00000 −2.11052
1.15 −2.06099 1.00000 2.24767 −0.122069 −2.06099 −4.45814 −0.510454 1.00000 0.251582
1.16 −2.05972 1.00000 2.24246 −1.93322 −2.05972 2.16498 −0.499401 1.00000 3.98191
1.17 −1.96725 1.00000 1.87006 −1.28401 −1.96725 2.13094 0.255618 1.00000 2.52597
1.18 −1.95525 1.00000 1.82299 1.82748 −1.95525 −0.999221 0.346105 1.00000 −3.57318
1.19 −1.92742 1.00000 1.71496 −2.34894 −1.92742 −3.43132 0.549392 1.00000 4.52741
1.20 −1.88583 1.00000 1.55635 −4.37058 −1.88583 3.62055 0.836652 1.00000 8.24216
See all 97 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.97
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2011\) \(1\)

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 6033.2.a.e ✓ 97
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6033.2.a.e ✓ 97 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{97} - 12 T_{2}^{96} - 85 T_{2}^{95} + 1570 T_{2}^{94} + 1783 T_{2}^{93} - 97687 T_{2}^{92} + \cdots - 3597258368 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\). Copy content Toggle raw display