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Properties

Label 4007.2.a.b
Level $4007$
Weight $2$
Character orbit 4007.a
Self dual yes
Analytic conductor $31.996$
Analytic rank $0$
Dimension $195$
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] = [4007,2,Mod(1,4007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4007, 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("4007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4007 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4007.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: \(31.9960560899\)
Analytic rank: \(0\)
Dimension: \(195\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 195 q + 14 q^{2} + 22 q^{3} + 220 q^{4} + 14 q^{5} + 13 q^{6} + 48 q^{7} + 39 q^{8} + 245 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 195 q + 14 q^{2} + 22 q^{3} + 220 q^{4} + 14 q^{5} + 13 q^{6} + 48 q^{7} + 39 q^{8} + 245 q^{9} + 40 q^{10} + 13 q^{11} + 57 q^{12} + 97 q^{13} + 9 q^{14} + 23 q^{15} + 270 q^{16} + 66 q^{17} + 60 q^{18} + 33 q^{19} + 24 q^{20} + 27 q^{21} + 127 q^{22} + 42 q^{23} + 33 q^{24} + 357 q^{25} + 8 q^{26} + 79 q^{27} + 131 q^{28} + 57 q^{29} + 51 q^{30} + 41 q^{31} + 67 q^{32} + 74 q^{33} + 26 q^{34} + 14 q^{35} + 279 q^{36} + 133 q^{37} + 7 q^{38} + 28 q^{39} + 97 q^{40} + 64 q^{41} - q^{42} + 123 q^{43} + 21 q^{44} + 40 q^{45} + 84 q^{46} + 14 q^{47} + 122 q^{48} + 335 q^{49} + 35 q^{50} + 37 q^{51} + 220 q^{52} + 83 q^{53} + 21 q^{54} + 47 q^{55} + q^{56} + 235 q^{57} + 138 q^{58} + 18 q^{59} - 4 q^{60} + 81 q^{61} + 39 q^{62} + 102 q^{63} + 343 q^{64} + 165 q^{65} - 54 q^{66} + 147 q^{67} + 74 q^{68} - 2 q^{69} + 13 q^{70} + 31 q^{71} + 112 q^{72} + 300 q^{73} + 5 q^{74} + 84 q^{75} + 64 q^{76} + 67 q^{77} + 61 q^{78} + 144 q^{79} - 8 q^{80} + 359 q^{81} + 85 q^{82} + 26 q^{83} + 12 q^{84} + 201 q^{85} - 2 q^{86} + 80 q^{87} + 347 q^{88} + 29 q^{89} + 62 q^{90} + 70 q^{91} + 79 q^{92} + 76 q^{93} + 72 q^{94} + 71 q^{95} + 31 q^{96} + 264 q^{97} + 30 q^{98} + 19 q^{99}+O(q^{100}) \) 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 −2.80768 0.834495 5.88309 4.09159 −2.34300 3.13441 −10.9025 −2.30362 −11.4879
1.2 −2.77582 −0.198171 5.70518 −0.442867 0.550088 0.719726 −10.2849 −2.96073 1.22932
1.3 −2.77052 −1.23937 5.67576 −3.30641 3.43369 0.994092 −10.1837 −1.46397 9.16047
1.4 −2.74088 3.29634 5.51241 −3.80383 −9.03487 −0.926403 −9.62710 7.86587 10.4258
1.5 −2.73169 2.16378 5.46210 −1.55160 −5.91076 4.97440 −9.45738 1.68193 4.23848
1.6 −2.71610 −2.73922 5.37720 −0.158886 7.44001 4.34515 −9.17282 4.50335 0.431551
1.7 −2.69070 −2.66626 5.23985 2.66315 7.17410 −3.38912 −8.71746 4.10894 −7.16574
1.8 −2.63607 1.85738 4.94884 −0.474491 −4.89618 −2.15797 −7.77333 0.449870 1.25079
1.9 −2.60343 1.83156 4.77782 −3.48115 −4.76834 −4.50968 −7.23185 0.354625 9.06293
1.10 −2.59764 −1.72984 4.74771 1.17522 4.49349 2.46697 −7.13754 −0.00766352 −3.05279
1.11 −2.59160 −1.40196 4.71637 −3.26317 3.63331 −1.83814 −7.03972 −1.03451 8.45682
1.12 −2.58803 0.217741 4.69789 −3.50376 −0.563520 3.48169 −6.98221 −2.95259 9.06783
1.13 −2.55145 −0.786154 4.50991 3.16082 2.00584 −3.53434 −6.40393 −2.38196 −8.06468
1.14 −2.54911 2.95187 4.49797 1.50040 −7.52466 0.00462218 −6.36761 5.71356 −3.82468
1.15 −2.50867 2.54308 4.29344 3.25074 −6.37975 2.35749 −5.75350 3.46723 −8.15505
1.16 −2.47249 0.496095 4.11323 2.63577 −1.22659 3.34809 −5.22495 −2.75389 −6.51692
1.17 −2.43669 0.943453 3.93744 −0.444210 −2.29890 −0.206330 −4.72093 −2.10990 1.08240
1.18 −2.42566 0.883893 3.88381 −3.90163 −2.14402 3.11093 −4.56949 −2.21873 9.46401
1.19 −2.41162 −0.257694 3.81590 2.26401 0.621460 −3.03056 −4.37927 −2.93359 −5.45992
1.20 −2.39027 −2.93715 3.71341 −2.06612 7.02059 −0.622431 −4.09551 5.62685 4.93859
See next 80 embeddings (of 195 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.195
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4007\) \(-1\)

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 4007.2.a.b ✓ 195
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4007.2.a.b ✓ 195 1.a even 1 1 trivial