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Properties

Label 6007.2.a.b
Level $6007$
Weight $2$
Character orbit 6007.a
Self dual yes
Analytic conductor $47.966$
Analytic rank $1$
Dimension $237$
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: \(1\)
Dimension: \(237\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 237 q - 26 q^{2} - 24 q^{3} + 226 q^{4} - 67 q^{5} - 30 q^{6} - 37 q^{7} - 75 q^{8} + 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 237 q - 26 q^{2} - 24 q^{3} + 226 q^{4} - 67 q^{5} - 30 q^{6} - 37 q^{7} - 75 q^{8} + 189 q^{9} - 39 q^{10} - 38 q^{11} - 67 q^{12} - 52 q^{13} - 54 q^{14} - 24 q^{15} + 208 q^{16} - 255 q^{17} - 71 q^{18} - 24 q^{19} - 154 q^{20} - 60 q^{21} - 39 q^{22} - 118 q^{23} - 85 q^{24} + 170 q^{25} - 61 q^{26} - 87 q^{27} - 99 q^{28} - 87 q^{29} - 30 q^{30} - 28 q^{31} - 156 q^{32} - 173 q^{33} - 4 q^{34} - 113 q^{35} + 152 q^{36} - 49 q^{37} - 145 q^{38} - 49 q^{39} - 91 q^{40} - 197 q^{41} - 61 q^{42} - 63 q^{43} - 106 q^{44} - 181 q^{45} - 2 q^{46} - 119 q^{47} - 142 q^{48} + 150 q^{49} - 89 q^{50} - 40 q^{51} - 97 q^{52} - 190 q^{53} - 97 q^{54} - 55 q^{55} - 154 q^{56} - 202 q^{57} - 27 q^{58} - 86 q^{59} - 48 q^{60} - 96 q^{61} - 239 q^{62} - 149 q^{63} + 183 q^{64} - 259 q^{65} - 72 q^{66} - 28 q^{67} - 482 q^{68} - 83 q^{69} + 20 q^{70} - 63 q^{71} - 193 q^{72} - 206 q^{73} - 132 q^{74} - 89 q^{75} - 11 q^{76} - 179 q^{77} - 58 q^{78} - 32 q^{79} - 320 q^{80} + 57 q^{81} - 77 q^{82} - 245 q^{83} - 133 q^{84} + q^{85} - 39 q^{86} - 179 q^{87} - 104 q^{88} - 227 q^{89} - 146 q^{90} - 36 q^{91} - 315 q^{92} - 87 q^{93} - 48 q^{94} - 111 q^{95} - 134 q^{96} - 221 q^{97} - 161 q^{98} - 68 q^{99}+O(q^{100}) \) 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 −2.79278 2.70470 5.79960 2.04099 −7.55362 −2.59733 −10.6114 4.31540 −5.70003
1.2 −2.78154 3.16840 5.73694 −2.96782 −8.81302 0.536102 −10.3944 7.03875 8.25511
1.3 −2.77308 −0.650314 5.68998 0.444469 1.80337 −4.52369 −10.2326 −2.57709 −1.23255
1.4 −2.73398 −2.84716 5.47464 −1.36635 7.78406 −2.19683 −9.49960 5.10630 3.73558
1.5 −2.72886 −1.67732 5.44669 −2.00878 4.57718 3.72534 −9.40555 −0.186597 5.48170
1.6 −2.72815 1.45530 5.44281 2.25925 −3.97027 3.02620 −9.39250 −0.882113 −6.16358
1.7 −2.72222 −1.56731 5.41047 −3.83172 4.26655 −2.76166 −9.28405 −0.543552 10.4308
1.8 −2.71930 −0.422286 5.39458 −1.71313 1.14832 0.813338 −9.23087 −2.82167 4.65850
1.9 −2.71184 0.480935 5.35409 −1.73647 −1.30422 3.65301 −9.09575 −2.76870 4.70903
1.10 −2.70797 −0.668026 5.33310 2.30781 1.80899 −3.86069 −9.02593 −2.55374 −6.24947
1.11 −2.69134 −2.16445 5.24332 1.36308 5.82528 5.10442 −8.72889 1.68484 −3.66852
1.12 −2.68501 −2.77509 5.20930 3.15364 7.45116 −1.25017 −8.61702 4.70113 −8.46756
1.13 −2.66512 −3.23472 5.10287 −3.53010 8.62093 1.66538 −8.26953 7.46344 9.40814
1.14 −2.63584 2.53685 4.94764 0.682671 −6.68672 1.00934 −7.76949 3.43560 −1.79941
1.15 −2.62005 0.737084 4.86469 −2.98180 −1.93120 −0.252259 −7.50563 −2.45671 7.81248
1.16 −2.61224 1.23083 4.82380 3.92824 −3.21523 −2.33975 −7.37643 −1.48505 −10.2615
1.17 −2.54661 1.71939 4.48524 −4.31092 −4.37861 4.46436 −6.32895 −0.0437112 10.9783
1.18 −2.54016 0.887378 4.45240 −3.77266 −2.25408 −4.31374 −6.22948 −2.21256 9.58314
1.19 −2.52945 −1.63373 4.39812 1.74511 4.13244 0.157946 −6.06591 −0.330925 −4.41416
1.20 −2.52132 1.70278 4.35706 −2.82870 −4.29327 −1.18756 −5.94292 −0.100525 7.13206
See next 80 embeddings (of 237 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.237
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.b ✓ 237
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6007.2.a.b ✓ 237 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{237} + 26 T_{2}^{236} - 12 T_{2}^{235} - 6111 T_{2}^{234} - 37323 T_{2}^{233} + \cdots - 10\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6007))\). Copy content Toggle raw display