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Properties

Label 8017.2.a.b
Level $8017$
Weight $2$
Character orbit 8017.a
Self dual yes
Analytic conductor $64.016$
Analytic rank $0$
Dimension $340$
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] = [8017,2,Mod(1,8017)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8017, 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("8017.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8017 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8017.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: \(64.0160673005\)
Analytic rank: \(0\)
Dimension: \(340\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 340 q + 20 q^{2} + 44 q^{3} + 350 q^{4} + 53 q^{5} + 34 q^{6} + 81 q^{7} + 54 q^{8} + 360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 340 q + 20 q^{2} + 44 q^{3} + 350 q^{4} + 53 q^{5} + 34 q^{6} + 81 q^{7} + 54 q^{8} + 360 q^{9} + 36 q^{10} + 70 q^{11} + 92 q^{12} + 45 q^{13} + 44 q^{14} + 71 q^{15} + 362 q^{16} + 162 q^{17} + 41 q^{18} + 49 q^{19} + 147 q^{20} + 41 q^{21} + 32 q^{22} + 244 q^{23} + 85 q^{24} + 355 q^{25} + 83 q^{26} + 155 q^{27} + 129 q^{28} + 91 q^{29} + 51 q^{30} + 65 q^{31} + 113 q^{32} + 73 q^{33} + 26 q^{34} + 200 q^{35} + 380 q^{36} + 28 q^{37} + 171 q^{38} + 117 q^{39} + 95 q^{40} + 115 q^{41} + 42 q^{42} + 98 q^{43} + 139 q^{44} + 127 q^{45} + 29 q^{46} + 312 q^{47} + 168 q^{48} + 365 q^{49} + 64 q^{50} + 72 q^{51} + 100 q^{52} + 154 q^{53} + 89 q^{54} + 161 q^{55} + 89 q^{56} + 82 q^{57} + 29 q^{58} + 149 q^{59} + 93 q^{60} + 70 q^{61} + 257 q^{62} + 376 q^{63} + 346 q^{64} + 125 q^{65} + 48 q^{66} + 65 q^{67} + 464 q^{68} + 58 q^{69} - 54 q^{70} + 216 q^{71} + 90 q^{72} + 93 q^{73} + 147 q^{74} + 162 q^{75} + 64 q^{76} + 190 q^{77} + 12 q^{78} + 139 q^{79} + 274 q^{80} + 376 q^{81} + 59 q^{82} + 402 q^{83} + 10 q^{84} + 32 q^{85} + 53 q^{86} + 364 q^{87} + 42 q^{88} + 114 q^{89} + 126 q^{90} + 43 q^{91} + 422 q^{92} + 47 q^{93} + 2 q^{94} + 347 q^{95} + 146 q^{96} + 47 q^{97} + 96 q^{98} + 129 q^{99}+O(q^{100}) \) 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 −2.82489 2.49518 5.98003 3.36028 −7.04861 4.80660 −11.2432 3.22590 −9.49245
1.2 −2.81482 0.685798 5.92319 0.844371 −1.93040 −4.31326 −11.0431 −2.52968 −2.37675
1.3 −2.75702 −3.03076 5.60115 −1.17209 8.35587 3.32948 −9.92845 6.18552 3.23147
1.4 −2.75627 0.354012 5.59704 −0.448330 −0.975755 −0.00120082 −9.91443 −2.87468 1.23572
1.5 −2.72150 1.69330 5.40656 −4.24685 −4.60831 −1.07458 −9.27094 −0.132735 11.5578
1.6 −2.70245 −0.906232 5.30323 2.51115 2.44905 2.28815 −8.92682 −2.17874 −6.78625
1.7 −2.68673 2.91370 5.21852 −0.199710 −7.82834 −2.77102 −8.64730 5.48967 0.536568
1.8 −2.67670 −1.97751 5.16470 −3.95415 5.29320 −4.51333 −8.47093 0.910563 10.5841
1.9 −2.67123 −2.04696 5.13545 2.79534 5.46788 −0.784684 −8.37550 1.19003 −7.46698
1.10 −2.65743 −2.52364 5.06196 4.19446 6.70640 4.50856 −8.13696 3.36874 −11.1465
1.11 −2.64379 0.908830 4.98964 −0.356677 −2.40276 −1.07237 −7.90399 −2.17403 0.942979
1.12 −2.62968 −1.21671 4.91523 1.08723 3.19957 0.844404 −7.66614 −1.51960 −2.85908
1.13 −2.61837 −0.666893 4.85585 −0.00270488 1.74617 0.156538 −7.47768 −2.55525 0.00708236
1.14 −2.61686 2.24685 4.84797 1.56834 −5.87971 3.28305 −7.45273 2.04835 −4.10414
1.15 −2.60921 3.19659 4.80800 −1.37562 −8.34060 −1.63991 −7.32666 7.21821 3.58928
1.16 −2.60414 2.31571 4.78154 3.50289 −6.03042 −2.38601 −7.24351 2.36250 −9.12202
1.17 −2.59088 −0.557344 4.71265 −1.39623 1.44401 −3.59677 −7.02815 −2.68937 3.61746
1.18 −2.56223 −1.92311 4.56505 0.295481 4.92747 2.92407 −6.57225 0.698367 −0.757092
1.19 −2.52841 −2.54349 4.39287 2.51878 6.43098 −1.43237 −6.05015 3.46932 −6.36851
1.20 −2.50135 −2.46945 4.25678 1.14683 6.17696 −1.51997 −5.64500 3.09817 −2.86862
See next 80 embeddings (of 340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.340
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(8017\) \(-1\)

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 8017.2.a.b ✓ 340
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8017.2.a.b ✓ 340 1.a even 1 1 trivial