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Properties

Label 8017.2.a.a
Level $8017$
Weight $2$
Character orbit 8017.a
Self dual yes
Analytic conductor $64.016$
Analytic rank $1$
Dimension $327$
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: \(1\)
Dimension: \(327\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 327 q - 23 q^{2} - 48 q^{3} + 315 q^{4} - 55 q^{5} - 38 q^{6} - 87 q^{7} - 69 q^{8} + 303 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 327 q - 23 q^{2} - 48 q^{3} + 315 q^{4} - 55 q^{5} - 38 q^{6} - 87 q^{7} - 69 q^{8} + 303 q^{9} - 48 q^{10} - 70 q^{11} - 120 q^{12} - 53 q^{13} - 52 q^{14} - 77 q^{15} + 295 q^{16} - 164 q^{17} - 58 q^{18} - 47 q^{19} - 153 q^{20} - 39 q^{21} - 68 q^{22} - 256 q^{23} - 107 q^{24} + 288 q^{25} - 95 q^{26} - 189 q^{27} - 167 q^{28} - 99 q^{29} - 81 q^{30} - 71 q^{31} - 146 q^{32} - 95 q^{33} - 40 q^{34} - 192 q^{35} + 261 q^{36} - 54 q^{37} - 179 q^{38} - 115 q^{39} - 121 q^{40} - 111 q^{41} - 62 q^{42} - 110 q^{43} - 157 q^{44} - 137 q^{45} - 11 q^{46} - 324 q^{47} - 236 q^{48} + 296 q^{49} - 73 q^{50} - 88 q^{51} - 138 q^{52} - 170 q^{53} - 127 q^{54} - 151 q^{55} - 151 q^{56} - 106 q^{57} - 81 q^{58} - 123 q^{59} - 83 q^{60} - 62 q^{61} - 287 q^{62} - 400 q^{63} + 263 q^{64} - 143 q^{65} - 64 q^{66} - 95 q^{67} - 442 q^{68} - 22 q^{69} - 26 q^{70} - 210 q^{71} - 129 q^{72} - 121 q^{73} - 159 q^{74} - 194 q^{75} - 86 q^{76} - 178 q^{77} - 68 q^{78} - 145 q^{79} - 338 q^{80} + 259 q^{81} - 103 q^{82} - 418 q^{83} - 102 q^{84} - 40 q^{85} - 89 q^{86} - 372 q^{87} - 186 q^{88} - 100 q^{89} - 150 q^{90} - 69 q^{91} - 458 q^{92} - 81 q^{93} - 46 q^{94} - 377 q^{95} - 190 q^{96} - 87 q^{97} - 147 q^{98} - 171 q^{99}+O(q^{100}) \) 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 −2.79695 −1.72914 5.82292 −0.405996 4.83631 −2.26625 −10.6925 −0.0100790 1.13555
1.2 −2.79413 −2.90420 5.80714 −2.69949 8.11470 −0.505392 −10.6376 5.43438 7.54271
1.3 −2.77418 0.313969 5.69607 4.29926 −0.871005 −0.431093 −10.2536 −2.90142 −11.9269
1.4 −2.76749 2.84439 5.65899 −0.717749 −7.87182 −1.28590 −10.1262 5.09057 1.98636
1.5 −2.76002 −0.752350 5.61769 −2.62302 2.07650 4.59409 −9.98488 −2.43397 7.23958
1.6 −2.74897 0.108150 5.55686 1.28185 −0.297303 2.70833 −9.77771 −2.98830 −3.52377
1.7 −2.74759 −3.21743 5.54926 2.43668 8.84019 −3.59070 −9.75191 7.35187 −6.69501
1.8 −2.73986 −0.187585 5.50685 −3.57442 0.513957 1.37739 −9.60828 −2.96481 9.79342
1.9 −2.73007 2.35314 5.45331 −1.11386 −6.42426 −0.749989 −9.42779 2.53728 3.04092
1.10 −2.72447 0.869183 5.42273 2.47434 −2.36806 −1.68212 −9.32512 −2.24452 −6.74127
1.11 −2.72133 1.64758 5.40564 −2.87444 −4.48362 −4.41970 −9.26787 −0.285473 7.82231
1.12 −2.70756 −2.54523 5.33088 0.896657 6.89137 3.68403 −9.01856 3.47821 −2.42775
1.13 −2.67797 1.71362 5.17153 −1.95790 −4.58901 3.43960 −8.49326 −0.0635223 5.24319
1.14 −2.67394 −1.93873 5.14997 −1.50753 5.18405 −1.18909 −8.42282 0.758668 4.03106
1.15 −2.65464 2.85639 5.04709 2.05834 −7.58268 0.740875 −8.08892 5.15896 −5.46413
1.16 −2.60838 2.59159 4.80367 −3.67050 −6.75986 1.01472 −7.31304 3.71634 9.57408
1.17 −2.59947 −1.49239 4.75723 4.02767 3.87942 −3.51002 −7.16732 −0.772773 −10.4698
1.18 −2.59616 −1.46619 4.74007 −0.558786 3.80647 −4.63820 −7.11368 −0.850285 1.45070
1.19 −2.58656 2.73680 4.69031 2.52723 −7.07890 −4.80679 −6.95866 4.49006 −6.53684
1.20 −2.58337 1.72136 4.67379 0.176863 −4.44690 3.62607 −6.90738 −0.0369272 −0.456901
See next 80 embeddings (of 327 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.327
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.a ✓ 327
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8017.2.a.a ✓ 327 1.a even 1 1 trivial