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Properties

Label 6007.2.a.c
Level $6007$
Weight $2$
Character orbit 6007.a
Self dual yes
Analytic conductor $47.966$
Analytic rank $0$
Dimension $261$
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] = [6007,2,Mod(1,6007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6007, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6007 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6007.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: \(47.9661364942\)
Analytic rank: \(0\)
Dimension: \(261\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 261 q + 26 q^{2} + 25 q^{3} + 274 q^{4} + 66 q^{5} + 25 q^{6} + 37 q^{7} + 72 q^{8} + 310 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 261 q + 26 q^{2} + 25 q^{3} + 274 q^{4} + 66 q^{5} + 25 q^{6} + 37 q^{7} + 72 q^{8} + 310 q^{9} + 35 q^{10} + 32 q^{11} + 51 q^{12} + 60 q^{13} + 55 q^{14} + 16 q^{15} + 288 q^{16} + 270 q^{17} + 45 q^{18} + 34 q^{19} + 157 q^{20} + 27 q^{21} + 38 q^{22} + 116 q^{23} + 48 q^{24} + 335 q^{25} + 46 q^{26} + 73 q^{27} + 70 q^{28} + 99 q^{29} + 33 q^{30} + 33 q^{31} + 150 q^{32} + 172 q^{33} + 24 q^{34} + 114 q^{35} + 339 q^{36} + 36 q^{37} + 112 q^{38} + 30 q^{39} + 106 q^{40} + 209 q^{41} + 64 q^{42} + 64 q^{43} + 65 q^{44} + 153 q^{45} + 135 q^{47} + 87 q^{48} + 332 q^{49} + 82 q^{50} + 52 q^{51} + 102 q^{52} + 163 q^{53} + 52 q^{54} + 56 q^{55} + 134 q^{56} + 181 q^{57} + q^{58} + 89 q^{59} - 43 q^{60} + 112 q^{61} + 228 q^{62} + 130 q^{63} + 268 q^{64} + 248 q^{65} + 5 q^{66} + 42 q^{67} + 453 q^{68} + 51 q^{69} - 22 q^{70} + 98 q^{71} + 113 q^{72} + 206 q^{73} + 81 q^{74} + 29 q^{75} + 62 q^{76} + 185 q^{77} - 25 q^{78} + 29 q^{79} + 258 q^{80} + 393 q^{81} + 79 q^{82} + 265 q^{83} - 25 q^{84} + 84 q^{85} + 36 q^{86} + 131 q^{87} + 24 q^{88} + 195 q^{89} + 89 q^{90} - 18 q^{91} + 261 q^{92} + 52 q^{93} + 3 q^{94} + 104 q^{95} + 92 q^{96} + 213 q^{97} + 156 q^{98} + 47 q^{99}+O(q^{100}) \) 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 −2.82040 1.97504 5.95465 −2.55663 −5.57039 −3.03915 −11.1537 0.900772 7.21072
1.2 −2.81023 −2.20286 5.89742 4.33627 6.19056 0.601928 −10.9527 1.85260 −12.1859
1.3 −2.72871 0.0576599 5.44587 −1.32868 −0.157337 0.160488 −9.40280 −2.99668 3.62560
1.4 −2.68856 2.47306 5.22837 0.0319975 −6.64899 3.92320 −8.67966 3.11605 −0.0860273
1.5 −2.66623 −2.56917 5.10878 0.348811 6.85001 3.60714 −8.28872 3.60066 −0.930012
1.6 −2.65529 2.71142 5.05057 3.32212 −7.19960 −2.23518 −8.10014 4.35179 −8.82118
1.7 −2.63601 0.145552 4.94856 0.530894 −0.383678 0.464377 −7.77243 −2.97881 −1.39944
1.8 −2.63092 0.625555 4.92176 3.53160 −1.64579 2.86224 −7.68693 −2.60868 −9.29136
1.9 −2.57859 −1.33205 4.64914 0.797435 3.43482 −0.447865 −6.83106 −1.22564 −2.05626
1.10 −2.56111 −3.17696 4.55927 0.542831 8.13653 −2.10285 −6.55456 7.09306 −1.39025
1.11 −2.55274 −1.51769 4.51649 −0.0618695 3.87427 −0.267248 −6.42394 −0.696618 0.157937
1.12 −2.53967 2.55882 4.44992 2.30710 −6.49855 −1.02038 −6.22199 3.54756 −5.85928
1.13 −2.53843 −2.17354 4.44361 −2.13295 5.51736 −4.01617 −6.20293 1.72426 5.41433
1.14 −2.52735 0.935367 4.38748 −0.964544 −2.36400 −4.26765 −6.03399 −2.12509 2.43774
1.15 −2.52396 2.90244 4.37039 −2.11388 −7.32566 −3.46277 −5.98279 5.42417 5.33537
1.16 −2.49977 0.0230289 4.24887 3.35515 −0.0575671 0.906816 −5.62166 −2.99947 −8.38713
1.17 −2.47861 −3.39135 4.14353 2.04119 8.40584 3.41777 −5.31299 8.50123 −5.05931
1.18 −2.47715 2.24713 4.13629 −0.500546 −5.56650 2.06873 −5.29193 2.04961 1.23993
1.19 −2.47082 3.41011 4.10496 1.44488 −8.42576 2.75509 −5.20098 8.62882 −3.57005
1.20 −2.46886 −1.31806 4.09526 2.84222 3.25411 −1.17521 −5.17289 −1.26271 −7.01703
See next 80 embeddings (of 261 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.261
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(6007\) \(-1\)

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 6007.2.a.c ✓ 261
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6007.2.a.c ✓ 261 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{261} - 26 T_{2}^{260} - 60 T_{2}^{259} + 7360 T_{2}^{258} - 35761 T_{2}^{257} + \cdots + 8552084955431 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6007))\). Copy content Toggle raw display