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Properties

Label 869.4.a.a
Level $869$
Weight $4$
Character orbit 869.a
Self dual yes
Analytic conductor $51.273$
Analytic rank $1$
Dimension $44$
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: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{2} - 16 q^{3} + 154 q^{4} - 32 q^{5} + 20 q^{6} - 70 q^{7} + 12 q^{8} + 298 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{2} - 16 q^{3} + 154 q^{4} - 32 q^{5} + 20 q^{6} - 70 q^{7} + 12 q^{8} + 298 q^{9} - 85 q^{10} - 484 q^{11} - 127 q^{12} - 180 q^{13} - 165 q^{14} - 72 q^{15} + 398 q^{16} + 208 q^{17} - 255 q^{18} - 824 q^{19} - 417 q^{20} - 190 q^{21} + 22 q^{22} - 316 q^{23} - 855 q^{24} + 290 q^{25} - 401 q^{26} - 868 q^{27} - 804 q^{28} + 184 q^{29} - 939 q^{30} - 918 q^{31} - 21 q^{32} + 176 q^{33} - 2178 q^{34} - 622 q^{35} - 436 q^{36} - 102 q^{37} - 1642 q^{38} - 340 q^{39} - 714 q^{40} + 472 q^{41} - 275 q^{42} - 942 q^{43} - 1694 q^{44} - 2284 q^{45} - 1586 q^{46} - 1008 q^{47} - 2524 q^{48} + 916 q^{49} - 1230 q^{50} - 1462 q^{51} - 2719 q^{52} + 442 q^{53} + 712 q^{54} + 352 q^{55} - 2462 q^{56} + 984 q^{57} - 2109 q^{58} - 3640 q^{59} - 2134 q^{60} - 2290 q^{61} - 101 q^{62} - 2034 q^{63} - 734 q^{64} + 1940 q^{65} - 220 q^{66} - 3854 q^{67} + 1386 q^{68} - 1664 q^{69} - 1773 q^{70} - 1284 q^{71} - 1632 q^{72} - 2568 q^{73} + 1368 q^{74} - 4382 q^{75} - 5734 q^{76} + 770 q^{77} - 4037 q^{78} + 3476 q^{79} - 3611 q^{80} + 208 q^{81} - 2556 q^{82} - 3894 q^{83} - 4178 q^{84} - 4206 q^{85} - 3657 q^{86} - 6806 q^{87} - 132 q^{88} - 1452 q^{89} - 807 q^{90} - 7886 q^{91} - 4214 q^{92} + 1166 q^{93} - 1862 q^{94} - 1322 q^{95} - 5832 q^{96} - 7142 q^{97} - 1881 q^{98} - 3278 q^{99}+O(q^{100}) \) 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 −5.47497 −3.04539 21.9753 −2.25535 16.6734 −6.16113 −76.5146 −17.7256 12.3480
1.2 −5.24070 −0.0832829 19.4650 7.99165 0.436461 −5.18961 −60.0845 −26.9931 −41.8819
1.3 −4.99617 4.62015 16.9617 10.2491 −23.0831 20.2604 −44.7740 −5.65417 −51.2061
1.4 −4.85492 7.16572 15.5703 −16.7916 −34.7890 −11.8981 −36.7531 24.3476 81.5221
1.5 −4.60078 −9.89755 13.1672 −14.4195 45.5365 25.1213 −23.7733 70.9616 66.3410
1.6 −4.10784 −6.26323 8.87435 −8.78630 25.7284 −7.64885 −3.59169 12.2281 36.0927
1.7 −4.10296 7.95647 8.83430 0.788247 −32.6451 −1.45994 −3.42309 36.3055 −3.23415
1.8 −3.98435 −9.46239 7.87507 10.6807 37.7015 −22.8344 0.497775 62.5369 −42.5558
1.9 −3.93374 1.84551 7.47431 19.6845 −7.25977 −25.9798 2.06792 −23.5941 −77.4338
1.10 −3.72162 0.0741499 5.85042 −12.7920 −0.275957 −24.3075 7.99991 −26.9945 47.6069
1.11 −3.38865 −0.749382 3.48292 −6.34450 2.53939 33.5056 15.3068 −26.4384 21.4993
1.12 −3.09878 7.32878 1.60246 0.147521 −22.7103 8.43888 19.8246 26.7111 −0.457135
1.13 −2.69986 −5.19514 −0.710752 7.76167 14.0262 29.8877 23.5178 −0.0104900 −20.9554
1.14 −2.37823 −0.665481 −2.34404 −20.9360 1.58266 −2.74623 24.6005 −26.5571 49.7904
1.15 −2.14714 −1.35316 −3.38977 10.1025 2.90544 −17.2045 24.4555 −25.1689 −21.6915
1.16 −2.10598 −8.46983 −3.56483 16.5702 17.8373 17.9216 24.3553 44.7381 −34.8967
1.17 −1.76212 3.61728 −4.89493 9.63609 −6.37409 12.0564 22.7224 −13.9153 −16.9800
1.18 −1.32115 8.31167 −6.25457 5.06567 −10.9809 −24.0415 18.8324 42.0838 −6.69249
1.19 −1.20929 −6.95499 −6.53763 −15.7720 8.41057 −35.6927 17.5801 21.3719 19.0729
1.20 −0.804718 4.27644 −7.35243 −12.0828 −3.44133 20.1016 12.3544 −8.71202 9.72326
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
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.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.4.a.a ✓ 44 1.a even 1 1 trivial