Properties

Label 6005.2.a.g
Level $6005$
Weight $2$
Character orbit 6005.a
Self dual yes
Analytic conductor $47.950$
Analytic rank $0$
Dimension $113$
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] = [6005,2,Mod(1,6005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6005, 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("6005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6005.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: \(47.9501664138\)
Analytic rank: \(0\)
Dimension: \(113\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 113 q - 3 q^{2} + 6 q^{3} + 141 q^{4} - 113 q^{5} + 7 q^{6} + 7 q^{7} - 12 q^{8} + 141 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 113 q - 3 q^{2} + 6 q^{3} + 141 q^{4} - 113 q^{5} + 7 q^{6} + 7 q^{7} - 12 q^{8} + 141 q^{9} + 3 q^{10} + 38 q^{11} - 4 q^{12} + 17 q^{13} + 23 q^{14} - 6 q^{15} + 193 q^{16} - 11 q^{17} - 3 q^{18} + 76 q^{19} - 141 q^{20} + 19 q^{21} + 41 q^{22} - 28 q^{23} + 29 q^{24} + 113 q^{25} + 21 q^{26} + 18 q^{27} + 29 q^{28} + 24 q^{29} - 7 q^{30} + 59 q^{31} - 22 q^{32} + 3 q^{33} + 55 q^{34} - 7 q^{35} + 232 q^{36} + 41 q^{37} - 6 q^{38} + 55 q^{39} + 12 q^{40} + 24 q^{41} + 17 q^{42} + 136 q^{43} + 85 q^{44} - 141 q^{45} + 84 q^{46} - 91 q^{47} - 19 q^{48} + 198 q^{49} - 3 q^{50} + 97 q^{51} + 45 q^{52} + 9 q^{53} + 54 q^{54} - 38 q^{55} + 98 q^{56} + 22 q^{57} + 69 q^{58} + 59 q^{59} + 4 q^{60} + 51 q^{61} - 30 q^{62} - 22 q^{63} + 298 q^{64} - 17 q^{65} + 76 q^{66} + 201 q^{67} - 34 q^{68} + 42 q^{69} - 23 q^{70} + 69 q^{71} - 7 q^{72} + 30 q^{73} + 35 q^{74} + 6 q^{75} + 170 q^{76} - 37 q^{77} - 11 q^{78} + 143 q^{79} - 193 q^{80} + 197 q^{81} + 55 q^{82} - 15 q^{83} + 83 q^{84} + 11 q^{85} + 78 q^{86} - 51 q^{87} + 113 q^{88} + 53 q^{89} + 3 q^{90} + 217 q^{91} - 40 q^{92} + 36 q^{93} + 81 q^{94} - 76 q^{95} + 66 q^{96} + 63 q^{97} - 62 q^{98} + 184 q^{99}+O(q^{100}) \) 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 −2.81643 −3.03025 5.93229 −1.00000 8.53449 1.00907 −11.0750 6.18240 2.81643
1.2 −2.80393 2.29330 5.86204 −1.00000 −6.43026 2.82627 −10.8289 2.25923 2.80393
1.3 −2.78685 −1.57697 5.76652 −1.00000 4.39478 −0.289151 −10.4967 −0.513163 2.78685
1.4 −2.77794 1.80495 5.71693 −1.00000 −5.01404 −4.50639 −10.3254 0.257853 2.77794
1.5 −2.68650 2.64233 5.21726 −1.00000 −7.09862 −4.41903 −8.64317 3.98193 2.68650
1.6 −2.67625 1.85244 5.16230 −1.00000 −4.95758 4.71146 −8.46311 0.431523 2.67625
1.7 −2.66122 −2.12417 5.08210 −1.00000 5.65288 −3.49276 −8.20215 1.51209 2.66122
1.8 −2.62680 −1.30643 4.90009 −1.00000 3.43174 −4.40647 −7.61796 −1.29324 2.62680
1.9 −2.61532 0.118458 4.83989 −1.00000 −0.309805 −0.279288 −7.42721 −2.98597 2.61532
1.10 −2.57539 −2.35580 4.63261 −1.00000 6.06709 3.76299 −6.78000 2.54978 2.57539
1.11 −2.56745 −0.556289 4.59182 −1.00000 1.42824 3.15883 −6.65437 −2.69054 2.56745
1.12 −2.49553 −3.37079 4.22769 −1.00000 8.41191 −2.54255 −5.55927 8.36220 2.49553
1.13 −2.48400 0.156242 4.17024 −1.00000 −0.388105 1.49577 −5.39087 −2.97559 2.48400
1.14 −2.47572 3.00665 4.12920 −1.00000 −7.44362 1.05496 −5.27130 6.03992 2.47572
1.15 −2.24487 −2.47232 3.03945 −1.00000 5.55004 −1.67086 −2.33343 3.11237 2.24487
1.16 −2.21162 0.635696 2.89127 −1.00000 −1.40592 −2.43179 −1.97116 −2.59589 2.21162
1.17 −2.18171 2.09909 2.75987 −1.00000 −4.57961 −2.68153 −1.65781 1.40617 2.18171
1.18 −2.16728 2.60896 2.69711 −1.00000 −5.65435 4.04826 −1.51083 3.80667 2.16728
1.19 −2.09533 −3.35381 2.39043 −1.00000 7.02736 3.72693 −0.818074 8.24805 2.09533
1.20 −2.08489 −0.988264 2.34676 −1.00000 2.06042 4.76657 −0.722948 −2.02333 2.08489
See next 80 embeddings (of 113 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.113
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(1201\) \(-1\)

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 6005.2.a.g 113
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6005.2.a.g 113 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{113} + 3 T_{2}^{112} - 179 T_{2}^{111} - 538 T_{2}^{110} + 15552 T_{2}^{109} + 46831 T_{2}^{108} - 873916 T_{2}^{107} - 2636588 T_{2}^{106} + 35708941 T_{2}^{105} + 107939281 T_{2}^{104} - 1130959591 T_{2}^{103} + \cdots - 28158256848 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\). Copy content Toggle raw display