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Properties

Label 4019.2.a.a
Level $4019$
Weight $2$
Character orbit 4019.a
Self dual yes
Analytic conductor $32.092$
Analytic rank $1$
Dimension $149$
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: \(1\)
Dimension: \(149\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 149 q - 8 q^{2} - 12 q^{3} + 124 q^{4} - 36 q^{5} - 45 q^{6} - 32 q^{7} - 21 q^{8} + 115 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 149 q - 8 q^{2} - 12 q^{3} + 124 q^{4} - 36 q^{5} - 45 q^{6} - 32 q^{7} - 21 q^{8} + 115 q^{9} - 58 q^{10} - 33 q^{11} - 33 q^{12} - 107 q^{13} - 28 q^{14} - 24 q^{15} + 74 q^{16} - 39 q^{17} - 33 q^{18} - 93 q^{19} - 63 q^{20} - 113 q^{21} - 38 q^{22} - 11 q^{23} - 130 q^{24} + 85 q^{25} - 33 q^{26} - 30 q^{27} - 94 q^{28} - 85 q^{29} - 16 q^{30} - 129 q^{31} - 35 q^{32} - 64 q^{33} - 78 q^{34} - 27 q^{35} + 79 q^{36} - 135 q^{37} - 11 q^{38} - 73 q^{39} - 146 q^{40} - 101 q^{41} + 4 q^{42} - 55 q^{43} - 82 q^{44} - 168 q^{45} - 113 q^{46} - 40 q^{47} - 65 q^{48} + 27 q^{49} - 5 q^{50} - 49 q^{51} - 177 q^{52} - 32 q^{53} - 155 q^{54} - 128 q^{55} - 44 q^{56} - 47 q^{57} - 46 q^{58} - 53 q^{59} - 11 q^{60} - 347 q^{61} - 11 q^{62} - 73 q^{63} + q^{64} - 31 q^{65} - 37 q^{66} - 40 q^{67} - 80 q^{68} - 175 q^{69} - 61 q^{70} - 31 q^{71} - 68 q^{72} - 193 q^{73} - 33 q^{74} - 56 q^{75} - 248 q^{76} - 84 q^{77} + 40 q^{78} - 111 q^{79} - 54 q^{80} + 49 q^{81} - 74 q^{82} - 24 q^{83} - 159 q^{84} - 258 q^{85} - q^{86} - 66 q^{87} - 97 q^{88} - 76 q^{89} - 75 q^{90} - 134 q^{91} + 31 q^{92} - 97 q^{93} - 111 q^{94} - 14 q^{95} - 216 q^{96} - 140 q^{97} - 13 q^{98} - 116 q^{99}+O(q^{100}) \) 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 −2.78495 3.42897 5.75594 1.05066 −9.54951 −0.902468 −10.4601 8.75783 −2.92604
1.2 −2.73191 −1.70544 5.46336 2.32218 4.65913 1.19146 −9.46160 −0.0914604 −6.34400
1.3 −2.70774 1.77949 5.33184 1.68400 −4.81839 −4.00894 −9.02175 0.166582 −4.55984
1.4 −2.68954 1.48905 5.23361 2.84645 −4.00487 −1.97182 −8.69693 −0.782716 −7.65564
1.5 −2.62051 −0.192294 4.86706 −1.97989 0.503908 −4.52578 −7.51315 −2.96302 5.18832
1.6 −2.59547 1.08147 4.73647 0.749609 −2.80691 2.15869 −7.10242 −1.83043 −1.94559
1.7 −2.58406 1.99817 4.67739 −2.81422 −5.16341 −2.01468 −6.91855 0.992695 7.27214
1.8 −2.57089 0.190612 4.60945 −1.40475 −0.490042 1.52623 −6.70860 −2.96367 3.61144
1.9 −2.51991 −1.75801 4.34994 3.61499 4.43004 3.76242 −5.92164 0.0906134 −9.10944
1.10 −2.46575 −2.43164 4.07992 2.59428 5.99581 −0.656341 −5.12858 2.91285 −6.39684
1.11 −2.45850 −1.74268 4.04423 −0.525405 4.28438 −0.912805 −5.02575 0.0369224 1.29171
1.12 −2.45807 −2.34751 4.04212 −0.597755 5.77034 −0.214441 −5.01969 2.51079 1.46933
1.13 −2.42490 −1.02548 3.88013 −4.00476 2.48668 −1.16451 −4.55913 −1.94839 9.71114
1.14 −2.40917 −3.09144 3.80411 −1.16894 7.44782 2.97947 −4.34641 6.55703 2.81617
1.15 −2.36640 0.537231 3.59984 −0.969490 −1.27130 3.20883 −3.78587 −2.71138 2.29420
1.16 −2.35135 2.82086 3.52886 0.184417 −6.63284 2.16509 −3.59489 4.95726 −0.433630
1.17 −2.30650 −1.73394 3.31993 −0.392279 3.99934 −3.42422 −3.04443 0.00656301 0.904791
1.18 −2.27033 2.56855 3.15439 −3.84625 −5.83145 2.23399 −2.62085 3.59745 8.73225
1.19 −2.23088 1.97807 2.97683 1.53013 −4.41285 −1.35303 −2.17920 0.912770 −3.41354
1.20 −2.21437 0.176199 2.90346 3.35713 −0.390170 0.00990688 −2.00059 −2.96895 −7.43394
See next 80 embeddings (of 149 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.149
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.a ✓ 149
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4019.2.a.a ✓ 149 1.a even 1 1 trivial