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Properties

Label 6041.2.a.e
Level $6041$
Weight $2$
Character orbit 6041.a
Self dual yes
Analytic conductor $48.238$
Analytic rank $0$
Dimension $112$
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: \(112\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 3 q^{2} + 14 q^{3} + 131 q^{4} + 13 q^{5} + 18 q^{6} - 112 q^{7} - 9 q^{8} + 116 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 3 q^{2} + 14 q^{3} + 131 q^{4} + 13 q^{5} + 18 q^{6} - 112 q^{7} - 9 q^{8} + 116 q^{9} + 32 q^{10} + 14 q^{11} + 36 q^{12} + 22 q^{13} + 3 q^{14} + 19 q^{15} + 169 q^{16} + 11 q^{17} - 18 q^{18} + 52 q^{19} + 40 q^{20} - 14 q^{21} + 16 q^{22} + 38 q^{23} + 64 q^{24} + 99 q^{25} + 45 q^{26} + 65 q^{27} - 131 q^{28} + 10 q^{29} + q^{30} + 133 q^{31} - 26 q^{32} + 27 q^{33} + 52 q^{34} - 13 q^{35} + 183 q^{36} - 13 q^{37} + 20 q^{38} + 74 q^{39} + 92 q^{40} + 25 q^{41} - 18 q^{42} - 11 q^{43} + 16 q^{44} + 63 q^{45} + 28 q^{46} + 71 q^{47} + 70 q^{48} + 112 q^{49} + 5 q^{50} + 57 q^{51} + 79 q^{52} - 10 q^{53} + 75 q^{54} + 146 q^{55} + 9 q^{56} - 83 q^{57} - 19 q^{58} + 56 q^{59} - 3 q^{60} + 80 q^{61} + 42 q^{62} - 116 q^{63} + 263 q^{64} - 26 q^{65} + 48 q^{66} + 29 q^{67} + 57 q^{68} + 56 q^{69} - 32 q^{70} + 100 q^{71} - 62 q^{72} + 73 q^{73} + 24 q^{74} + 89 q^{75} + 155 q^{76} - 14 q^{77} + 33 q^{78} + 140 q^{79} + 80 q^{80} + 120 q^{81} + 114 q^{82} + 36 q^{83} - 36 q^{84} - 2 q^{85} + 12 q^{86} + 96 q^{87} + 29 q^{88} + 47 q^{89} + 52 q^{90} - 22 q^{91} + 81 q^{92} - 10 q^{93} + 127 q^{94} + 96 q^{95} + 175 q^{96} + 80 q^{97} - 3 q^{98} + 74 q^{99}+O(q^{100}) \) 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 −2.81373 −3.44120 5.91707 1.30649 9.68259 −1.00000 −11.0216 8.84183 −3.67611
1.2 −2.80869 2.13107 5.88877 −3.52544 −5.98552 −1.00000 −10.9224 1.54145 9.90189
1.3 −2.77254 −2.14289 5.68701 −0.156494 5.94125 −1.00000 −10.2224 1.59196 0.433887
1.4 −2.76309 2.60057 5.63466 −1.29315 −7.18561 −1.00000 −10.0429 3.76297 3.57310
1.5 −2.71246 −0.244067 5.35744 −2.10613 0.662022 −1.00000 −9.10692 −2.94043 5.71281
1.6 −2.71151 1.27956 5.35230 1.05589 −3.46954 −1.00000 −9.08979 −1.36273 −2.86306
1.7 −2.67889 −2.12545 5.17645 −3.74850 5.69384 −1.00000 −8.50937 1.51753 10.0418
1.8 −2.57911 −0.702301 4.65183 2.85376 1.81131 −1.00000 −6.83937 −2.50677 −7.36016
1.9 −2.52776 3.15958 4.38957 3.22941 −7.98667 −1.00000 −6.04026 6.98297 −8.16316
1.10 −2.46248 −0.0194781 4.06380 −1.44141 0.0479644 −1.00000 −5.08207 −2.99962 3.54944
1.11 −2.45673 3.01946 4.03553 −1.39061 −7.41801 −1.00000 −5.00075 6.11715 3.41635
1.12 −2.43414 −1.84866 3.92503 3.83434 4.49989 −1.00000 −4.68578 0.417546 −9.33330
1.13 −2.41077 2.26439 3.81183 4.38346 −5.45893 −1.00000 −4.36791 2.12745 −10.5675
1.14 −2.37296 0.810716 3.63095 0.528258 −1.92380 −1.00000 −3.87017 −2.34274 −1.25354
1.15 −2.27713 −2.93009 3.18534 −2.32769 6.67221 −1.00000 −2.69918 5.58544 5.30045
1.16 −2.25493 0.275101 3.08473 2.79647 −0.620335 −1.00000 −2.44598 −2.92432 −6.30585
1.17 −2.22590 −2.92856 2.95465 1.01796 6.51870 −1.00000 −2.12496 5.57647 −2.26588
1.18 −2.13001 −1.53283 2.53693 −2.08639 3.26494 −1.00000 −1.14367 −0.650428 4.44403
1.19 −2.10602 −0.719393 2.43533 1.34496 1.51506 −1.00000 −0.916817 −2.48247 −2.83252
1.20 −2.06364 −0.217047 2.25861 3.48472 0.447908 −1.00000 −0.533675 −2.95289 −7.19121
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.112
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.e ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6041.2.a.e ✓ 112 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} + 3 T_{2}^{111} - 173 T_{2}^{110} - 521 T_{2}^{109} + 14516 T_{2}^{108} + 43895 T_{2}^{107} + \cdots + 24980734323 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\). Copy content Toggle raw display