Properties

Label 6041.2.a.f
Level $6041$
Weight $2$
Character orbit 6041.a
Self dual yes
Analytic conductor $48.238$
Analytic rank $0$
Dimension $132$
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] = [6041,2,Mod(1,6041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6041, 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("6041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6041 = 7 \cdot 863 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6041.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: \(48.2376278611\)
Analytic rank: \(0\)
Dimension: \(132\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9} + 22 q^{10} + 32 q^{11} + 30 q^{12} + 16 q^{13} + 8 q^{14} + 59 q^{15} + 254 q^{16} + 11 q^{17} + 33 q^{18} + 40 q^{19} + 4 q^{20} + 10 q^{21} + 66 q^{22} + 62 q^{23} + 36 q^{24} + 235 q^{25} + 25 q^{26} + 37 q^{27} + 174 q^{28} + 28 q^{29} + 45 q^{30} + 121 q^{31} + 53 q^{32} - 13 q^{33} + 44 q^{34} + 11 q^{35} + 274 q^{36} + 61 q^{37} - 28 q^{38} + 114 q^{39} + 32 q^{40} - q^{41} + 16 q^{42} + 105 q^{43} + 54 q^{44} + 29 q^{45} + 104 q^{46} + 33 q^{47} + 16 q^{48} + 132 q^{49} - 14 q^{50} + 53 q^{51} - 11 q^{52} + 48 q^{53} + 11 q^{54} + 118 q^{55} + 30 q^{56} + 93 q^{57} + 87 q^{58} + 12 q^{59} + 41 q^{60} + 54 q^{61} - 28 q^{62} + 178 q^{63} + 376 q^{64} + 22 q^{65} + 6 q^{66} + 123 q^{67} - 47 q^{68} + 58 q^{69} + 22 q^{70} + 108 q^{71} + 97 q^{72} + q^{73} - 10 q^{74} + 23 q^{75} + 71 q^{76} + 32 q^{77} + 5 q^{78} + 204 q^{79} - 10 q^{80} + 296 q^{81} + 80 q^{82} - 10 q^{83} + 30 q^{84} + 94 q^{85} + 48 q^{86} + 4 q^{87} + 155 q^{88} + q^{89} - 66 q^{90} + 16 q^{91} + 49 q^{92} + 90 q^{93} + 79 q^{94} + 100 q^{95} + q^{96} + 18 q^{97} + 8 q^{98} + 96 q^{99}+O(q^{100}) \) 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 −2.81389 0.225620 5.91799 −1.24518 −0.634869 1.00000 −11.0248 −2.94910 3.50380
1.2 −2.78355 3.33951 5.74813 2.68755 −9.29568 1.00000 −10.4331 8.15233 −7.48091
1.3 −2.77966 −1.97929 5.72652 −3.34292 5.50177 1.00000 −10.3585 0.917605 9.29218
1.4 −2.75252 −2.26942 5.57639 4.03697 6.24664 1.00000 −9.84409 2.15028 −11.1119
1.5 −2.71866 −0.898623 5.39111 3.58669 2.44305 1.00000 −9.21926 −2.19248 −9.75100
1.6 −2.65744 2.58260 5.06196 1.09780 −6.86308 1.00000 −8.13696 3.66980 −2.91732
1.7 −2.62362 −1.27143 4.88340 1.41870 3.33576 1.00000 −7.56495 −1.38346 −3.72213
1.8 −2.60157 0.915852 4.76817 −4.25711 −2.38265 1.00000 −7.20159 −2.16122 11.0752
1.9 −2.58774 −2.55272 4.69637 −0.926438 6.60577 1.00000 −6.97750 3.51640 2.39738
1.10 −2.58389 1.40748 4.67649 −2.16822 −3.63678 1.00000 −6.91577 −1.01899 5.60245
1.11 −2.57785 −0.937633 4.64529 −3.87896 2.41707 1.00000 −6.81916 −2.12084 9.99936
1.12 −2.56875 −3.19690 4.59849 −1.91049 8.21205 1.00000 −6.67489 7.22017 4.90758
1.13 −2.49223 0.173328 4.21120 2.29751 −0.431973 1.00000 −5.51083 −2.96996 −5.72592
1.14 −2.47641 1.36965 4.13259 −1.36117 −3.39181 1.00000 −5.28115 −1.12406 3.37081
1.15 −2.46575 3.35396 4.07990 −4.25898 −8.27001 1.00000 −5.12851 8.24904 10.5016
1.16 −2.32322 2.41590 3.39733 0.249961 −5.61265 1.00000 −3.24629 2.83657 −0.580712
1.17 −2.26422 −1.83534 3.12671 1.84374 4.15563 1.00000 −2.55113 0.368489 −4.17465
1.18 −2.25429 −3.03067 3.08183 3.72305 6.83200 1.00000 −2.43875 6.18494 −8.39284
1.19 −2.23793 1.90424 3.00834 4.23440 −4.26157 1.00000 −2.25659 0.626144 −9.47631
1.20 −2.22730 1.19588 2.96085 2.18062 −2.66358 1.00000 −2.14009 −1.56986 −4.85688
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.132
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(863\) \(1\)

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 6041.2.a.f 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6041.2.a.f 132 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} - 8 T_{2}^{131} - 187 T_{2}^{130} + 1646 T_{2}^{129} + 16720 T_{2}^{128} - 164947 T_{2}^{127} - 946079 T_{2}^{126} + 10728739 T_{2}^{125} + 37663821 T_{2}^{124} - 509286443 T_{2}^{123} - 1101579171 T_{2}^{122} + \cdots + 57534003 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\). Copy content Toggle raw display