Properties

Label 6035.2.a.a
Level $6035$
Weight $2$
Character orbit 6035.a
Self dual yes
Analytic conductor $48.190$
Analytic rank $1$
Dimension $36$
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] = [6035,2,Mod(1,6035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6035, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6035.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6035 = 5 \cdot 17 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6035.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: \(48.1897176198\)
Analytic rank: \(1\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 8 q^{3} + 23 q^{4} + 36 q^{5} - 10 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 8 q^{3} + 23 q^{4} + 36 q^{5} - 10 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9} - 3 q^{10} - 20 q^{11} - 8 q^{12} - 29 q^{13} - 12 q^{14} - 8 q^{15} + q^{16} + 36 q^{17} - 8 q^{18} - 19 q^{19} + 23 q^{20} - 19 q^{21} - 10 q^{22} - 10 q^{23} - 23 q^{24} + 36 q^{25} - 32 q^{26} - 23 q^{27} - 20 q^{28} - 52 q^{29} - 10 q^{30} - 15 q^{31} - 16 q^{32} - 19 q^{33} - 3 q^{34} - 7 q^{35} + 9 q^{36} - 52 q^{37} + 7 q^{38} - 10 q^{39} - 9 q^{40} - 51 q^{41} - 2 q^{42} - 13 q^{43} - 27 q^{44} + 10 q^{45} + 12 q^{46} - 24 q^{47} + 12 q^{48} - 15 q^{49} - 3 q^{50} - 8 q^{51} - 49 q^{52} - 13 q^{53} - 48 q^{54} - 20 q^{55} - 12 q^{56} - 20 q^{57} - 20 q^{58} - 14 q^{59} - 8 q^{60} - 75 q^{61} - 7 q^{62} + 16 q^{63} - 41 q^{64} - 29 q^{65} - q^{66} - 5 q^{67} + 23 q^{68} - 37 q^{69} - 12 q^{70} - 36 q^{71} - 23 q^{72} - 21 q^{73} + q^{74} - 8 q^{75} - 40 q^{76} - 31 q^{77} + 84 q^{78} - 49 q^{79} + q^{80} - 56 q^{81} - 51 q^{82} + 6 q^{83} + 10 q^{84} + 36 q^{85} - 41 q^{86} - 4 q^{87} - 21 q^{88} - 78 q^{89} - 8 q^{90} - 25 q^{91} - 24 q^{92} - 36 q^{93} + 6 q^{94} - 19 q^{95} - 71 q^{96} - 48 q^{97} + 51 q^{98} - 17 q^{99}+O(q^{100}) \) 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 −2.62226 −1.54085 4.87626 1.00000 4.04052 0.0125838 −7.54231 −0.625776 −2.62226
1.2 −2.61440 2.49037 4.83510 1.00000 −6.51084 2.00109 −7.41211 3.20197 −2.61440
1.3 −2.37331 2.40096 3.63259 1.00000 −5.69821 −2.59145 −3.87465 2.76459 −2.37331
1.4 −2.27701 0.848689 3.18476 1.00000 −1.93247 1.85809 −2.69770 −2.27973 −2.27701
1.5 −2.15542 −0.292006 2.64584 1.00000 0.629396 −5.08250 −1.39207 −2.91473 −2.15542
1.6 −2.13074 −1.60683 2.54004 1.00000 3.42373 2.82688 −1.15069 −0.418101 −2.13074
1.7 −2.04946 −2.65729 2.20029 1.00000 5.44602 −0.551924 −0.410497 4.06121 −2.04946
1.8 −1.90835 −2.27887 1.64180 1.00000 4.34888 −0.428939 0.683573 2.19324 −1.90835
1.9 −1.70160 0.687277 0.895431 1.00000 −1.16947 −0.626542 1.87953 −2.52765 −1.70160
1.10 −1.14290 −1.19489 −0.693776 1.00000 1.36564 −1.41810 3.07872 −1.57223 −1.14290
1.11 −1.12696 2.34856 −0.729961 1.00000 −2.64673 −2.68480 3.07656 2.51573 −1.12696
1.12 −1.12140 0.383871 −0.742456 1.00000 −0.430474 1.34550 3.07540 −2.85264 −1.12140
1.13 −1.03393 2.68090 −0.930989 1.00000 −2.77186 1.41622 3.03044 4.18720 −1.03393
1.14 −0.904707 −2.66665 −1.18150 1.00000 2.41254 3.83492 2.87833 4.11101 −0.904707
1.15 −0.890323 −0.682987 −1.20732 1.00000 0.608079 2.96744 2.85556 −2.53353 −0.890323
1.16 −0.599472 −1.15414 −1.64063 1.00000 0.691877 −3.77052 2.18246 −1.66795 −0.599472
1.17 −0.256013 0.196473 −1.93446 1.00000 −0.0502997 2.99337 1.00727 −2.96140 −0.256013
1.18 −0.174528 1.77073 −1.96954 1.00000 −0.309043 −0.0296109 0.692797 0.135502 −0.174528
1.19 0.0925953 0.181897 −1.99143 1.00000 0.0168428 −0.294707 −0.369587 −2.96691 0.0925953
1.20 0.139816 −2.98888 −1.98045 1.00000 −0.417894 2.58191 −0.556532 5.93339 0.139816
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(17\) \(-1\)
\(71\) \(1\)

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 6035.2.a.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6035.2.a.a 36 1.a even 1 1 trivial