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Properties

Label 6035.2.a.f
Level $6035$
Weight $2$
Character orbit 6035.a
Self dual yes
Analytic conductor $48.190$
Analytic rank $0$
Dimension $49$
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: \(0\)
Dimension: \(49\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 49 q + 3 q^{2} + 6 q^{3} + 55 q^{4} - 49 q^{5} - 6 q^{6} + 11 q^{7} + 9 q^{8} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 49 q + 3 q^{2} + 6 q^{3} + 55 q^{4} - 49 q^{5} - 6 q^{6} + 11 q^{7} + 9 q^{8} + 43 q^{9} - 3 q^{10} - 10 q^{11} + 2 q^{12} + 43 q^{13} - 16 q^{14} - 6 q^{15} + 63 q^{16} - 49 q^{17} + 8 q^{18} + 15 q^{19} - 55 q^{20} + 19 q^{21} - 2 q^{23} - 3 q^{24} + 49 q^{25} + 22 q^{26} + 27 q^{27} + 32 q^{28} - 38 q^{29} + 6 q^{30} + 11 q^{31} + 16 q^{32} + 51 q^{33} - 3 q^{34} - 11 q^{35} + 83 q^{36} + 54 q^{37} + 39 q^{38} - 25 q^{39} - 9 q^{40} - 29 q^{41} + 52 q^{42} + 29 q^{43} - 47 q^{44} - 43 q^{45} + 10 q^{46} + 60 q^{47} + 16 q^{48} + 82 q^{49} + 3 q^{50} - 6 q^{51} + 113 q^{52} + 41 q^{53} + 28 q^{54} + 10 q^{55} - 16 q^{56} + 18 q^{57} + 14 q^{58} + 8 q^{59} - 2 q^{60} + 65 q^{61} + 37 q^{62} - 4 q^{63} + 77 q^{64} - 43 q^{65} + 27 q^{66} + 65 q^{67} - 55 q^{68} + 23 q^{69} + 16 q^{70} + 49 q^{71} + 121 q^{72} + 59 q^{73} - 53 q^{74} + 6 q^{75} + 50 q^{76} + 41 q^{77} + 26 q^{78} + 11 q^{79} - 63 q^{80} + 41 q^{81} + 77 q^{82} + 16 q^{83} + 30 q^{84} + 49 q^{85} - q^{86} + 8 q^{87} + 11 q^{88} - 4 q^{89} - 8 q^{90} + 47 q^{91} + 60 q^{92} + 69 q^{93} + 66 q^{94} - 15 q^{95} - 19 q^{96} + 76 q^{97} + 31 q^{98} - 39 q^{99}+O(q^{100}) \) 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 −2.74491 −1.52760 5.53451 −1.00000 4.19313 1.91085 −9.70189 −0.666425 2.74491
1.2 −2.66790 1.47521 5.11768 −1.00000 −3.93572 2.11715 −8.31766 −0.823747 2.66790
1.3 −2.60831 −0.558654 4.80328 −1.00000 1.45714 2.40232 −7.31182 −2.68791 2.60831
1.4 −2.60338 2.30923 4.77761 −1.00000 −6.01182 −4.03482 −7.23119 2.33255 2.60338
1.5 −2.37029 −1.04857 3.61829 −1.00000 2.48541 0.282680 −3.83582 −1.90050 2.37029
1.6 −2.35541 2.89986 3.54798 −1.00000 −6.83036 3.81816 −3.64613 5.40916 2.35541
1.7 −2.10648 −2.79941 2.43726 −1.00000 5.89690 −4.08251 −0.921083 4.83668 2.10648
1.8 −2.06915 −2.99869 2.28138 −1.00000 6.20473 3.60194 −0.582222 5.99212 2.06915
1.9 −2.00954 1.06318 2.03827 −1.00000 −2.13651 0.504155 −0.0769013 −1.86965 2.00954
1.10 −1.85441 1.98520 1.43882 −1.00000 −3.68137 −2.79102 1.04065 0.941017 1.85441
1.11 −1.76458 2.45692 1.11374 −1.00000 −4.33542 0.398587 1.56388 3.03644 1.76458
1.12 −1.68607 0.433728 0.842832 −1.00000 −0.731295 5.25109 1.95107 −2.81188 1.68607
1.13 −1.67974 −2.02457 0.821521 −1.00000 3.40075 −2.28769 1.97954 1.09890 1.67974
1.14 −1.31158 −2.44528 −0.279754 −1.00000 3.20719 2.74652 2.99008 2.97941 1.31158
1.15 −1.26937 0.563612 −0.388690 −1.00000 −0.715434 3.94272 3.03214 −2.68234 1.26937
1.16 −1.26707 3.30982 −0.394528 −1.00000 −4.19378 −2.69759 3.03404 7.95489 1.26707
1.17 −1.08459 −0.444340 −0.823667 −1.00000 0.481926 −0.817730 3.06252 −2.80256 1.08459
1.18 −0.946634 1.66092 −1.10388 −1.00000 −1.57229 −4.44729 2.93824 −0.241339 0.946634
1.19 −0.807272 0.370983 −1.34831 −1.00000 −0.299484 1.58315 2.70300 −2.86237 0.807272
1.20 −0.377033 2.92202 −1.85785 −1.00000 −1.10170 4.80817 1.45453 5.53818 0.377033
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
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.f ✓ 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6035.2.a.f ✓ 49 1.a even 1 1 trivial