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Properties

Label 6035.2.a.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{2} - 4 q^{3} + 23 q^{4} + 36 q^{5} - 2 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{2} - 4 q^{3} + 23 q^{4} + 36 q^{5} - 2 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9} - q^{10} - 22 q^{11} - 14 q^{12} - 15 q^{13} - 28 q^{14} - 4 q^{15} + q^{16} - 36 q^{17} - 12 q^{18} - 23 q^{19} + 23 q^{20} - 21 q^{21} + 2 q^{23} - 13 q^{24} + 36 q^{25} - 18 q^{26} - 13 q^{27} - 20 q^{28} - 4 q^{29} - 2 q^{30} - 43 q^{31} - 2 q^{32} - 19 q^{33} + q^{34} - 7 q^{35} - 35 q^{36} - 30 q^{37} - 11 q^{38} - 20 q^{39} - 3 q^{40} - 39 q^{41} + 2 q^{42} - 7 q^{43} - 45 q^{44} + 10 q^{45} - 52 q^{46} - 12 q^{47} - 12 q^{48} - 15 q^{49} - q^{50} + 4 q^{51} - 19 q^{52} - 31 q^{53} + 48 q^{54} - 22 q^{55} - 30 q^{56} + 18 q^{57} - 12 q^{58} - 66 q^{59} - 14 q^{60} - 93 q^{61} - 7 q^{62} - 22 q^{63} - 41 q^{64} - 15 q^{65} - 21 q^{66} - 19 q^{67} - 23 q^{68} - 73 q^{69} - 28 q^{70} + 36 q^{71} - q^{72} - 47 q^{73} - 27 q^{74} - 4 q^{75} - 56 q^{76} - 9 q^{77} - 78 q^{78} - 21 q^{79} + q^{80} - 40 q^{81} - 15 q^{82} - 8 q^{83} - 54 q^{84} - 36 q^{85} - 17 q^{86} - 32 q^{87} - 13 q^{88} - 62 q^{89} - 12 q^{90} - 33 q^{91} + 42 q^{92} - 24 q^{93} - 40 q^{94} - 23 q^{95} + 21 q^{96} - 60 q^{97} + 11 q^{98} - 65 q^{99}+O(q^{100}) \) 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 −2.66525 0.180914 5.10356 1.00000 −0.482180 −2.30037 −8.27176 −2.96727 −2.66525
1.2 −2.40320 1.20051 3.77538 1.00000 −2.88508 −1.77185 −4.26659 −1.55876 −2.40320
1.3 −2.39029 −2.40790 3.71350 1.00000 5.75558 4.16969 −4.09576 2.79797 −2.39029
1.4 −2.31945 −2.05009 3.37984 1.00000 4.75509 0.529758 −3.20047 1.20289 −2.31945
1.5 −2.29064 1.47549 3.24701 1.00000 −3.37981 2.13493 −2.85645 −0.822934 −2.29064
1.6 −1.85242 0.130041 1.43148 1.00000 −0.240891 2.44217 1.05315 −2.98309 −1.85242
1.7 −1.71589 −3.02116 0.944266 1.00000 5.18397 −0.410823 1.81152 6.12740 −1.71589
1.8 −1.63901 0.960813 0.686340 1.00000 −1.57478 4.58027 2.15310 −2.07684 −1.63901
1.9 −1.62869 2.31397 0.652617 1.00000 −3.76873 −4.05639 2.19446 2.35446 −1.62869
1.10 −1.59880 −0.663250 0.556157 1.00000 1.06040 −3.10735 2.30841 −2.56010 −1.59880
1.11 −1.59058 2.74131 0.529934 1.00000 −4.36027 0.195561 2.33825 4.51480 −1.59058
1.12 −1.02277 −0.691517 −0.953933 1.00000 0.707265 1.61582 3.02121 −2.52180 −1.02277
1.13 −0.861525 −2.77636 −1.25777 1.00000 2.39191 0.623554 2.80665 4.70820 −0.861525
1.14 −0.644629 1.56810 −1.58445 1.00000 −1.01085 2.57685 2.31064 −0.541047 −0.644629
1.15 −0.619688 −2.94210 −1.61599 1.00000 1.82319 −1.69113 2.24078 5.65598 −0.619688
1.16 −0.381657 −0.970824 −1.85434 1.00000 0.370521 −3.53323 1.47103 −2.05750 −0.381657
1.17 −0.163606 −0.433822 −1.97323 1.00000 0.0709760 2.35725 0.650046 −2.81180 −0.163606
1.18 −0.134450 −0.934655 −1.98192 1.00000 0.125664 −1.20552 0.535370 −2.12642 −0.134450
1.19 −0.0473223 3.03750 −1.99776 1.00000 −0.143741 −2.32686 0.189183 6.22638 −0.0473223
1.20 0.111835 1.47233 −1.98749 1.00000 0.164659 −0.779752 −0.445943 −0.832238 0.111835
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.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6035.2.a.b ✓ 36 1.a even 1 1 trivial