Properties

Label 6034.2.a.m
Level $6034$
Weight $2$
Character orbit 6034.a
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $21$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 21 q^{2} - 6 q^{3} + 21 q^{4} - 11 q^{5} - 6 q^{6} + 21 q^{7} + 21 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 21 q^{2} - 6 q^{3} + 21 q^{4} - 11 q^{5} - 6 q^{6} + 21 q^{7} + 21 q^{8} + 5 q^{9} - 11 q^{10} - 34 q^{11} - 6 q^{12} - 19 q^{13} + 21 q^{14} - 24 q^{15} + 21 q^{16} - 17 q^{17} + 5 q^{18} - 15 q^{19} - 11 q^{20} - 6 q^{21} - 34 q^{22} - 32 q^{23} - 6 q^{24} + 6 q^{25} - 19 q^{26} - 3 q^{27} + 21 q^{28} - 46 q^{29} - 24 q^{30} + 7 q^{31} + 21 q^{32} - 13 q^{33} - 17 q^{34} - 11 q^{35} + 5 q^{36} - 34 q^{37} - 15 q^{38} - 25 q^{39} - 11 q^{40} - 27 q^{41} - 6 q^{42} - 47 q^{43} - 34 q^{44} - 13 q^{45} - 32 q^{46} - 7 q^{47} - 6 q^{48} + 21 q^{49} + 6 q^{50} - 29 q^{51} - 19 q^{52} - 57 q^{53} - 3 q^{54} + 17 q^{55} + 21 q^{56} - 28 q^{57} - 46 q^{58} - 30 q^{59} - 24 q^{60} - 17 q^{61} + 7 q^{62} + 5 q^{63} + 21 q^{64} - 40 q^{65} - 13 q^{66} - 38 q^{67} - 17 q^{68} - 13 q^{69} - 11 q^{70} - 66 q^{71} + 5 q^{72} - 15 q^{73} - 34 q^{74} + 15 q^{75} - 15 q^{76} - 34 q^{77} - 25 q^{78} - 17 q^{79} - 11 q^{80} - 11 q^{81} - 27 q^{82} - 19 q^{83} - 6 q^{84} - 28 q^{85} - 47 q^{86} + 45 q^{87} - 34 q^{88} - 39 q^{89} - 13 q^{90} - 19 q^{91} - 32 q^{92} - 25 q^{93} - 7 q^{94} - 35 q^{95} - 6 q^{96} + 21 q^{98} - 52 q^{99}+O(q^{100}) \) 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 1.00000 −3.34739 1.00000 1.24845 −3.34739 1.00000 1.00000 8.20504 1.24845
1.2 1.00000 −2.48749 1.00000 1.32783 −2.48749 1.00000 1.00000 3.18763 1.32783
1.3 1.00000 −2.30616 1.00000 −2.85291 −2.30616 1.00000 1.00000 2.31835 −2.85291
1.4 1.00000 −2.23818 1.00000 3.73239 −2.23818 1.00000 1.00000 2.00943 3.73239
1.5 1.00000 −2.23488 1.00000 0.812592 −2.23488 1.00000 1.00000 1.99468 0.812592
1.6 1.00000 −1.63595 1.00000 −2.93756 −1.63595 1.00000 1.00000 −0.323656 −2.93756
1.7 1.00000 −1.49715 1.00000 −0.207716 −1.49715 1.00000 1.00000 −0.758538 −0.207716
1.8 1.00000 −1.27918 1.00000 −0.808180 −1.27918 1.00000 1.00000 −1.36370 −0.808180
1.9 1.00000 −1.16337 1.00000 −3.30128 −1.16337 1.00000 1.00000 −1.64657 −3.30128
1.10 1.00000 −0.950934 1.00000 −2.70877 −0.950934 1.00000 1.00000 −2.09572 −2.70877
1.11 1.00000 −0.235507 1.00000 1.96765 −0.235507 1.00000 1.00000 −2.94454 1.96765
1.12 1.00000 −0.00804327 1.00000 3.67628 −0.00804327 1.00000 1.00000 −2.99994 3.67628
1.13 1.00000 0.402295 1.00000 1.29038 0.402295 1.00000 1.00000 −2.83816 1.29038
1.14 1.00000 0.584190 1.00000 0.414785 0.584190 1.00000 1.00000 −2.65872 0.414785
1.15 1.00000 0.800681 1.00000 −2.16386 0.800681 1.00000 1.00000 −2.35891 −2.16386
1.16 1.00000 0.893081 1.00000 0.353208 0.893081 1.00000 1.00000 −2.20241 0.353208
1.17 1.00000 1.34513 1.00000 −2.31517 1.34513 1.00000 1.00000 −1.19062 −2.31517
1.18 1.00000 1.87377 1.00000 −1.03478 1.87377 1.00000 1.00000 0.510997 −1.03478
1.19 1.00000 2.14514 1.00000 −0.309854 2.14514 1.00000 1.00000 1.60160 −0.309854
1.20 1.00000 2.28514 1.00000 −4.30862 2.28514 1.00000 1.00000 2.22188 −4.30862
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(431\) \(1\)

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 6034.2.a.m 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.m 21 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3}^{21} + 6 T_{3}^{20} - 16 T_{3}^{19} - 155 T_{3}^{18} - 5 T_{3}^{17} + 1546 T_{3}^{16} + 1479 T_{3}^{15} - 7657 T_{3}^{14} - 11444 T_{3}^{13} + 19951 T_{3}^{12} + 39484 T_{3}^{11} - 26535 T_{3}^{10} - 71161 T_{3}^{9} + 16140 T_{3}^{8} + \cdots - 4 \) Copy content Toggle raw display
\( T_{5}^{21} + 11 T_{5}^{20} + 5 T_{5}^{19} - 350 T_{5}^{18} - 1077 T_{5}^{17} + 3184 T_{5}^{16} + 17851 T_{5}^{15} - 1753 T_{5}^{14} - 113339 T_{5}^{13} - 86999 T_{5}^{12} + 353928 T_{5}^{11} + 407587 T_{5}^{10} - 607364 T_{5}^{9} + \cdots - 1721 \) Copy content Toggle raw display
\( T_{11}^{21} + 34 T_{11}^{20} + 462 T_{11}^{19} + 2886 T_{11}^{18} + 3617 T_{11}^{17} - 60918 T_{11}^{16} - 353290 T_{11}^{15} - 312972 T_{11}^{14} + 3485592 T_{11}^{13} + 11786108 T_{11}^{12} - 2134972 T_{11}^{11} + \cdots - 206981567 \) Copy content Toggle raw display