Properties

Label 6019.2.a.b
Level $6019$
Weight $2$
Character orbit 6019.a
Self dual yes
Analytic conductor $48.062$
Analytic rank $1$
Dimension $101$
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] = [6019,2,Mod(1,6019)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6019, 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("6019.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6019 = 13 \cdot 463 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6019.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: \(48.0619569766\)
Analytic rank: \(1\)
Dimension: \(101\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 101 q - 8 q^{2} - 13 q^{3} + 86 q^{4} - 43 q^{5} - 10 q^{6} - q^{7} - 24 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 101 q - 8 q^{2} - 13 q^{3} + 86 q^{4} - 43 q^{5} - 10 q^{6} - q^{7} - 24 q^{8} + 52 q^{9} - 19 q^{10} - 42 q^{11} - 28 q^{12} + 101 q^{13} - 45 q^{14} - 15 q^{15} + 48 q^{16} - 83 q^{17} - 4 q^{18} - 18 q^{19} - 51 q^{20} - 50 q^{21} - 20 q^{22} - 64 q^{23} - 23 q^{24} + 46 q^{25} - 8 q^{26} - 37 q^{27} - 11 q^{28} - 117 q^{29} - 28 q^{30} - 10 q^{31} - 36 q^{32} - 20 q^{33} - 10 q^{34} - 53 q^{35} - 16 q^{36} - 27 q^{37} - 68 q^{38} - 13 q^{39} - 42 q^{40} - 60 q^{41} - 31 q^{42} - 16 q^{43} - 89 q^{44} - 56 q^{45} + 5 q^{46} - 23 q^{47} - 37 q^{48} + 48 q^{49} - 30 q^{50} - 68 q^{51} + 86 q^{52} - 189 q^{53} - 23 q^{54} + 3 q^{55} - 106 q^{56} - 25 q^{57} - 82 q^{59} + 6 q^{60} - 68 q^{61} - 57 q^{62} + 3 q^{63} - 2 q^{64} - 43 q^{65} - 40 q^{66} - 13 q^{67} - 138 q^{68} - 92 q^{69} + 18 q^{70} - 39 q^{71} - 20 q^{72} + 19 q^{73} - 88 q^{74} - 21 q^{75} - 53 q^{76} - 147 q^{77} - 10 q^{78} - 19 q^{79} - 104 q^{80} - 55 q^{81} + 27 q^{82} - 49 q^{83} - 59 q^{84} - 27 q^{85} - 99 q^{86} - 33 q^{87} - 41 q^{88} - 70 q^{89} - 49 q^{90} - q^{91} - 111 q^{92} - 84 q^{93} + 4 q^{94} - 82 q^{95} - 7 q^{96} + 25 q^{97} - 37 q^{98} - 41 q^{99}+O(q^{100}) \) 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 −2.77092 −0.383319 5.67797 −1.00861 1.06214 0.495501 −10.1914 −2.85307 2.79476
1.2 −2.75200 0.205353 5.57349 −1.15382 −0.565131 4.79173 −9.83422 −2.95783 3.17530
1.3 −2.66557 −0.463578 5.10529 3.99027 1.23570 −1.40213 −8.27737 −2.78510 −10.6364
1.4 −2.61088 −2.81902 4.81671 −3.25148 7.36014 3.68170 −7.35411 4.94689 8.48923
1.5 −2.59744 2.60815 4.74669 −0.0717768 −6.77450 0.808846 −7.13437 3.80242 0.186436
1.6 −2.53156 1.58470 4.40878 2.52136 −4.01175 −1.30202 −6.09798 −0.488741 −6.38296
1.7 −2.51903 1.84872 4.34553 −3.50196 −4.65699 1.30916 −5.90847 0.417768 8.82155
1.8 −2.47893 −2.41040 4.14507 2.56792 5.97521 2.73390 −5.31747 2.81004 −6.36569
1.9 −2.44907 −1.63543 3.99796 −2.35876 4.00528 1.29431 −4.89316 −0.325384 5.77678
1.10 −2.35123 −2.62167 3.52828 −1.32904 6.16416 −2.62369 −3.59334 3.87318 3.12489
1.11 −2.33786 −0.380371 3.46560 −3.85216 0.889254 −3.60381 −3.42637 −2.85532 9.00581
1.12 −2.32242 2.86947 3.39365 0.571930 −6.66413 −0.979167 −3.23665 5.23387 −1.32826
1.13 −2.31630 0.457851 3.36526 2.03378 −1.06052 1.32033 −3.16236 −2.79037 −4.71086
1.14 −2.22409 −1.49293 2.94657 −0.404702 3.32042 −4.24821 −2.10525 −0.771146 0.900094
1.15 −2.18475 1.46102 2.77312 −4.12132 −3.19197 2.89283 −1.68907 −0.865408 9.00405
1.16 −2.11661 2.26026 2.48003 −0.342197 −4.78408 3.30081 −1.01604 2.10876 0.724298
1.17 −2.06993 −2.73222 2.28462 1.87227 5.65552 −2.93607 −0.589152 4.46505 −3.87547
1.18 −2.04146 1.76541 2.16756 −1.03624 −3.60401 −3.97509 −0.342069 0.116656 2.11544
1.19 −2.03639 −1.20777 2.14690 3.16451 2.45949 3.67207 −0.299141 −1.54130 −6.44419
1.20 −1.88082 −1.45946 1.53747 1.38464 2.74498 −0.844871 0.869931 −0.869978 −2.60424
See next 80 embeddings (of 101 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.101
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(463\) \(-1\)

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 6019.2.a.b 101
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6019.2.a.b 101 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{101} + 8 T_{2}^{100} - 112 T_{2}^{99} - 1048 T_{2}^{98} + 5737 T_{2}^{97} + 66172 T_{2}^{96} - 172069 T_{2}^{95} - 2682375 T_{2}^{94} + 3044183 T_{2}^{93} + 78445571 T_{2}^{92} - 17915741 T_{2}^{91} + \cdots - 25589025 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6019))\). Copy content Toggle raw display