Properties

Label 4019.2.a.b
Level $4019$
Weight $2$
Character orbit 4019.a
Self dual yes
Analytic conductor $32.092$
Analytic rank $0$
Dimension $186$
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] = [4019,2,Mod(1,4019)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4019, 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("4019.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4019.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: \(32.0918765724\)
Analytic rank: \(0\)
Dimension: \(186\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 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 −2.80607 −1.31751 5.87400 −0.399238 3.69703 −2.36747 −10.8707 −1.26416 1.12029
1.2 −2.79148 −3.39321 5.79236 3.31167 9.47208 −0.0588258 −10.5863 8.51387 −9.24445
1.3 −2.78390 2.40470 5.75012 −2.23828 −6.69445 −0.0567746 −10.4400 2.78257 6.23116
1.4 −2.77200 −0.494167 5.68400 0.905775 1.36983 1.77801 −10.2121 −2.75580 −2.51081
1.5 −2.72152 −2.92759 5.40665 −3.58721 7.96750 −4.23646 −9.27126 5.57081 9.76265
1.6 −2.71067 −2.48956 5.34775 −1.96358 6.74838 3.14635 −9.07467 3.19790 5.32263
1.7 −2.69011 0.353602 5.23670 −0.999886 −0.951229 3.25121 −8.70707 −2.87497 2.68981
1.8 −2.67918 1.79491 5.17799 3.40719 −4.80888 4.24264 −8.51439 0.221695 −9.12846
1.9 −2.62160 −1.74677 4.87277 −3.59409 4.57932 5.00470 −7.53124 0.0512031 9.42226
1.10 −2.60285 −0.581840 4.77483 3.23074 1.51444 −2.22687 −7.22248 −2.66146 −8.40914
1.11 −2.59341 1.69325 4.72575 −3.22826 −4.39129 −0.750732 −7.06898 −0.132895 8.37219
1.12 −2.56992 −0.512548 4.60446 −4.01178 1.31721 0.174790 −6.69325 −2.73729 10.3099
1.13 −2.56492 −2.35383 4.57881 1.72936 6.03739 −3.13382 −6.61443 2.54053 −4.43566
1.14 −2.55030 3.02512 4.50404 −1.05624 −7.71496 5.05580 −6.38607 6.15132 2.69374
1.15 −2.52937 0.978887 4.39770 −0.161932 −2.47596 −2.77531 −6.06468 −2.04178 0.409586
1.16 −2.43886 −0.628245 3.94805 3.87591 1.53220 −4.09962 −4.75103 −2.60531 −9.45281
1.17 −2.43453 2.49356 3.92696 3.69129 −6.07065 0.601351 −4.69125 3.21782 −8.98658
1.18 −2.40616 −0.556574 3.78961 1.68267 1.33921 2.28423 −4.30608 −2.69023 −4.04878
1.19 −2.39579 3.13150 3.73980 0.529965 −7.50242 −1.46029 −4.16820 6.80631 −1.26968
1.20 −2.34165 −0.437788 3.48332 0.358726 1.02515 3.39349 −3.47342 −2.80834 −0.840010
See next 80 embeddings (of 186 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.186
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4019\) \(-1\)

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 4019.2.a.b 186
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4019.2.a.b 186 1.a even 1 1 trivial