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Properties

Label 6035.2.a.g
Level $6035$
Weight $2$
Character orbit 6035.a
Self dual yes
Analytic conductor $48.190$
Analytic rank $0$
Dimension $58$
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: \(58\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 58 q + q^{2} + 6 q^{3} + 69 q^{4} + 58 q^{5} + 10 q^{6} + 13 q^{7} - 3 q^{8} + 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 58 q + q^{2} + 6 q^{3} + 69 q^{4} + 58 q^{5} + 10 q^{6} + 13 q^{7} - 3 q^{8} + 84 q^{9} + q^{10} + 28 q^{11} + 18 q^{12} + 37 q^{13} + 28 q^{14} + 6 q^{15} + 83 q^{16} + 58 q^{17} - 12 q^{18} + 19 q^{19} + 69 q^{20} + 31 q^{21} + 13 q^{22} + 14 q^{23} + 13 q^{24} + 58 q^{25} + 18 q^{26} + 9 q^{27} + 8 q^{28} + 60 q^{29} + 10 q^{30} + 39 q^{31} - 30 q^{32} + 13 q^{33} + q^{34} + 13 q^{35} + 113 q^{36} + 60 q^{37} - q^{38} + 41 q^{39} - 3 q^{40} + 65 q^{41} - 30 q^{42} + 17 q^{43} + 69 q^{44} + 84 q^{45} + 24 q^{46} + 16 q^{47} + 14 q^{48} + 117 q^{49} + q^{50} + 6 q^{51} + 61 q^{52} + 5 q^{53} + 24 q^{54} + 28 q^{55} + 105 q^{56} + 8 q^{57} - 34 q^{58} + 22 q^{59} + 18 q^{60} + 113 q^{61} - 19 q^{62} + 8 q^{63} + 89 q^{64} + 37 q^{65} - 37 q^{66} + 19 q^{67} + 69 q^{68} + 75 q^{69} + 28 q^{70} + 58 q^{71} - 17 q^{72} + 49 q^{73} + 29 q^{74} + 6 q^{75} - 6 q^{76} + 17 q^{77} - 12 q^{78} + 7 q^{79} + 83 q^{80} + 134 q^{81} + 7 q^{82} - 12 q^{83} - 18 q^{84} + 58 q^{85} + 23 q^{86} - 36 q^{87} - 33 q^{88} + 52 q^{89} - 12 q^{90} + 31 q^{91} + 80 q^{92} - 37 q^{93} + 4 q^{94} + 19 q^{95} - 35 q^{96} + 26 q^{97} - 33 q^{98} + 57 q^{99}+O(q^{100}) \) 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 −2.80026 1.99842 5.84147 1.00000 −5.59610 −4.62775 −10.7571 0.993679 −2.80026
1.2 −2.75680 2.57899 5.59996 1.00000 −7.10976 3.29093 −9.92438 3.65117 −2.75680
1.3 −2.70439 −2.73763 5.31370 1.00000 7.40360 −4.83847 −8.96152 4.49461 −2.70439
1.4 −2.68796 −2.40870 5.22515 1.00000 6.47449 1.33386 −8.66909 2.80181 −2.68796
1.5 −2.49259 −0.124755 4.21303 1.00000 0.310963 1.75866 −5.51618 −2.98444 −2.49259
1.6 −2.45480 −1.45794 4.02604 1.00000 3.57896 −2.38415 −4.97353 −0.874405 −2.45480
1.7 −2.40526 1.86065 3.78528 1.00000 −4.47536 2.19847 −4.29406 0.462037 −2.40526
1.8 −2.20983 3.43880 2.88333 1.00000 −7.59915 −1.16489 −1.95201 8.82536 −2.20983
1.9 −2.17363 0.623249 2.72466 1.00000 −1.35471 −4.14588 −1.57514 −2.61156 −2.17363
1.10 −2.15077 −3.16911 2.62583 1.00000 6.81604 −1.54855 −1.34601 7.04327 −2.15077
1.11 −2.12400 −1.09989 2.51137 1.00000 2.33617 5.15845 −1.08615 −1.79024 −2.12400
1.12 −2.00836 0.177134 2.03352 1.00000 −0.355749 −1.65376 −0.0673116 −2.96862 −2.00836
1.13 −1.83919 1.25269 1.38262 1.00000 −2.30393 1.52576 1.13547 −1.43078 −1.83919
1.14 −1.67839 3.22950 0.816985 1.00000 −5.42036 0.934275 1.98556 7.42970 −1.67839
1.15 −1.56061 −2.33055 0.435498 1.00000 3.63707 3.61265 2.44157 2.43146 −1.56061
1.16 −1.54425 −3.31836 0.384710 1.00000 5.12438 3.26635 2.49441 8.01151 −1.54425
1.17 −1.51455 1.68592 0.293855 1.00000 −2.55340 −0.258342 2.58404 −0.157685 −1.51455
1.18 −1.30337 −0.487797 −0.301228 1.00000 0.635780 −1.77737 2.99935 −2.76205 −1.30337
1.19 −1.22809 1.74628 −0.491793 1.00000 −2.14460 4.49008 3.06015 0.0495068 −1.22809
1.20 −1.22218 −2.16829 −0.506288 1.00000 2.65003 −1.52570 3.06312 1.70146 −1.22218
See all 58 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.58
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.g ✓ 58
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6035.2.a.g ✓ 58 1.a even 1 1 trivial