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Properties

Label 869.6.a.d
Level $869$
Weight $6$
Character orbit 869.a
Self dual yes
Analytic conductor $139.374$
Analytic rank $0$
Dimension $86$
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] = [869,6,Mod(1,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 869.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: \(139.373539417\)
Analytic rank: \(0\)
Dimension: \(86\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 86 q + 4 q^{2} + 78 q^{3} + 1530 q^{4} + 169 q^{5} - 78 q^{6} + 490 q^{7} - 192 q^{8} + 7632 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 86 q + 4 q^{2} + 78 q^{3} + 1530 q^{4} + 169 q^{5} - 78 q^{6} + 490 q^{7} - 192 q^{8} + 7632 q^{9} + 1019 q^{10} - 10406 q^{11} + 3369 q^{12} + 2992 q^{13} + 143 q^{14} + 3202 q^{15} + 29438 q^{16} - 4043 q^{17} + 2679 q^{18} + 16115 q^{19} + 8151 q^{20} + 909 q^{21} - 484 q^{22} - 403 q^{23} - 18745 q^{24} + 65143 q^{25} + 11635 q^{26} + 8841 q^{27} + 19998 q^{28} - 9797 q^{29} - 7753 q^{30} + 34557 q^{31} - 6195 q^{32} - 9438 q^{33} + 55362 q^{34} + 26255 q^{35} + 174892 q^{36} + 5322 q^{37} + 34138 q^{38} + 20246 q^{39} + 44862 q^{40} - 29897 q^{41} + 32393 q^{42} + 36010 q^{43} - 185130 q^{44} + 141885 q^{45} + 144330 q^{46} + 32330 q^{47} + 153596 q^{48} + 338884 q^{49} + 9642 q^{50} + 58559 q^{51} + 301121 q^{52} + 38487 q^{53} + 25966 q^{54} - 20449 q^{55} + 72298 q^{56} + 67302 q^{57} + 277501 q^{58} + 180602 q^{59} + 361842 q^{60} + 170221 q^{61} - 52231 q^{62} + 228776 q^{63} + 633510 q^{64} - 47719 q^{65} + 9438 q^{66} + 307210 q^{67} + 26244 q^{68} + 135307 q^{69} + 92715 q^{70} + 213698 q^{71} + 372144 q^{72} + 94636 q^{73} + 298200 q^{74} + 253758 q^{75} + 531922 q^{76} - 59290 q^{77} + 427545 q^{78} + 536726 q^{79} + 699157 q^{80} + 753290 q^{81} + 619268 q^{82} + 282139 q^{83} + 39834 q^{84} - 52876 q^{85} - 203149 q^{86} - 130307 q^{87} + 23232 q^{88} + 10087 q^{89} + 159005 q^{90} + 887428 q^{91} - 156182 q^{92} + 107249 q^{93} + 473622 q^{94} - 132719 q^{95} - 308216 q^{96} + 75639 q^{97} - 66857 q^{98} - 923472 q^{99}+O(q^{100}) \) 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 −11.1701 10.5725 92.7712 88.4661 −118.096 −179.562 −678.820 −131.222 −988.175
1.2 −11.1509 14.5590 92.3430 −52.3398 −162.346 −19.7989 −672.881 −31.0360 583.637
1.3 −10.8276 −5.15236 85.2374 4.05757 55.7878 31.6682 −576.435 −216.453 −43.9339
1.4 −10.6893 −24.5157 82.2601 21.9898 262.055 170.934 −537.243 358.020 −235.055
1.5 −10.6628 30.2864 81.6963 104.710 −322.939 194.693 −529.904 674.263 −1116.51
1.6 −10.3982 −19.1344 76.1231 −104.924 198.964 −149.636 −458.801 123.125 1091.02
1.7 −10.1750 8.69047 71.5298 −21.7832 −88.4252 161.242 −402.214 −167.476 221.643
1.8 −9.90984 3.67660 66.2050 40.5424 −36.4345 −91.9081 −338.966 −229.483 −401.769
1.9 −9.49507 19.4656 58.1564 −83.8413 −184.827 226.228 −248.357 135.909 796.079
1.10 −9.44124 23.1215 57.1371 60.1768 −218.296 −108.921 −237.325 291.604 −568.144
1.11 −9.07637 9.88361 50.3806 −68.9403 −89.7073 −62.9023 −166.829 −145.314 625.728
1.12 −8.97422 −16.0069 48.5366 −96.7799 143.649 135.373 −148.403 13.2203 868.524
1.13 −8.65061 −10.5727 42.8331 14.7597 91.4601 −214.226 −93.7126 −131.219 −127.681
1.14 −8.56686 −25.8206 41.3912 19.3215 221.201 75.4205 −80.4527 423.702 −165.524
1.15 −8.14980 1.38451 34.4192 −11.2484 −11.2835 91.3738 −19.7162 −241.083 91.6725
1.16 −7.97020 −22.2602 31.5242 57.0497 177.418 244.329 3.79251 252.516 −454.698
1.17 −7.89095 −22.4524 30.2672 −9.69329 177.171 −158.775 13.6738 261.112 76.4893
1.18 −7.82149 29.4915 29.1758 −66.8495 −230.667 −215.399 22.0896 626.747 522.863
1.19 −7.72997 28.1962 27.7524 −41.7744 −217.956 −0.504551 32.8339 552.025 322.914
1.20 −6.99253 8.40897 16.8954 −0.431638 −58.8000 6.01587 105.619 −172.289 3.01824
See all 86 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.86
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(79\) \(-1\)

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 869.6.a.d ✓ 86
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.6.a.d ✓ 86 1.a even 1 1 trivial