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Properties

Label 869.4.a.b
Level $869$
Weight $4$
Character orbit 869.a
Self dual yes
Analytic conductor $51.273$
Analytic rank $1$
Dimension $46$
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] = [869,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(51.2726597950\)
Analytic rank: \(1\)
Dimension: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 8 q^{2} - 24 q^{3} + 146 q^{4} - 30 q^{5} - 90 q^{6} - 98 q^{7} - 96 q^{8} + 342 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 8 q^{2} - 24 q^{3} + 146 q^{4} - 30 q^{5} - 90 q^{6} - 98 q^{7} - 96 q^{8} + 342 q^{9} - 105 q^{10} + 506 q^{11} - 255 q^{12} - 216 q^{13} - 105 q^{14} - 326 q^{15} + 350 q^{16} - 370 q^{17} - 381 q^{18} - 980 q^{19} - 545 q^{20} - 450 q^{21} - 88 q^{22} + 62 q^{23} - 105 q^{24} + 1010 q^{25} - 537 q^{26} - 330 q^{27} - 762 q^{28} - 866 q^{29} + 1023 q^{30} - 1032 q^{31} - 351 q^{32} - 264 q^{33} - 586 q^{34} - 522 q^{35} + 1092 q^{36} - 744 q^{37} + 850 q^{38} - 796 q^{39} - 954 q^{40} - 2174 q^{41} - 163 q^{42} - 1946 q^{43} + 1606 q^{44} - 388 q^{45} + 358 q^{46} - 598 q^{47} - 1732 q^{48} + 822 q^{49} - 1486 q^{50} - 2206 q^{51} - 871 q^{52} - 962 q^{53} - 4262 q^{54} - 330 q^{55} - 654 q^{56} - 2244 q^{57} + 33 q^{58} - 2920 q^{59} - 3078 q^{60} - 4032 q^{61} - 2099 q^{62} - 2834 q^{63} + 430 q^{64} - 3844 q^{65} - 990 q^{66} - 3268 q^{67} - 2732 q^{68} - 4030 q^{69} - 3129 q^{70} - 1288 q^{71} - 3480 q^{72} - 2768 q^{73} - 3596 q^{74} - 5568 q^{75} - 7078 q^{76} - 1078 q^{77} + 909 q^{78} - 3634 q^{79} - 6427 q^{80} + 2650 q^{81} - 1572 q^{82} - 3098 q^{83} - 2334 q^{84} - 2186 q^{85} - 2901 q^{86} - 1286 q^{87} - 1056 q^{88} - 1830 q^{89} - 2163 q^{90} - 8238 q^{91} + 1370 q^{92} - 2808 q^{93} - 8770 q^{94} - 4690 q^{95} - 2696 q^{96} + 404 q^{97} + 1479 q^{98} + 3762 q^{99}+O(q^{100}) \) 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 −5.39524 −4.60478 21.1086 18.3703 24.8439 −25.5254 −70.7243 −5.79597 −99.1124
1.2 −5.17795 −2.76087 18.8112 −21.6022 14.2956 −17.3663 −55.9796 −19.3776 111.855
1.3 −5.10485 10.1252 18.0595 −5.73458 −51.6874 7.36085 −51.3523 75.5188 29.2742
1.4 −5.05749 −8.05694 17.5782 −0.993903 40.7479 9.60442 −48.4415 37.9143 5.02665
1.5 −5.05709 1.18058 17.5742 6.91397 −5.97031 −6.79506 −48.4177 −25.6062 −34.9646
1.6 −4.67577 3.98873 13.8628 −10.2532 −18.6504 13.5140 −27.4132 −11.0900 47.9417
1.7 −4.21294 1.78948 9.74887 −6.11226 −7.53898 21.3581 −7.36788 −23.7978 25.7506
1.8 −3.76167 −9.18096 6.15019 11.2522 34.5358 19.6297 6.95839 57.2899 −42.3271
1.9 −3.58161 −4.95489 4.82792 −12.0020 17.7465 −3.20489 11.3611 −2.44902 42.9863
1.10 −3.57579 7.47687 4.78631 13.2885 −26.7357 −24.2462 11.4915 28.9036 −47.5170
1.11 −3.53379 8.82477 4.48766 −17.4220 −31.1849 −30.6541 12.4119 50.8766 61.5657
1.12 −3.36096 −1.78339 3.29603 8.74790 5.99390 9.15714 15.8098 −23.8195 −29.4013
1.13 −3.19692 −8.44385 2.22032 −0.736856 26.9944 23.2114 18.4772 44.2987 2.35567
1.14 −3.03557 3.21644 1.21468 17.3380 −9.76371 7.13602 20.5973 −16.6545 −52.6308
1.15 −2.35063 −4.75422 −2.47456 7.16552 11.1754 −35.2162 24.6218 −4.39739 −16.8434
1.16 −2.25107 7.98152 −2.93267 −5.93888 −17.9670 −6.27149 24.6102 36.7047 13.3689
1.17 −2.00614 5.97556 −3.97539 −0.115361 −11.9878 35.1724 24.0243 8.70737 0.231430
1.18 −1.91628 1.92296 −4.32785 −10.9585 −3.68494 −15.3683 23.6237 −23.3022 20.9996
1.19 −1.83063 −8.13967 −4.64878 18.0678 14.9007 −10.0321 23.1553 39.2542 −33.0756
1.20 −0.938237 −0.332553 −7.11971 6.48024 0.312014 16.5876 14.1859 −26.8894 −6.08000
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
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 869.4.a.b ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.4.a.b ✓ 46 1.a even 1 1 trivial