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Properties

Label 869.2.a.h
Level $869$
Weight $2$
Character orbit 869.a
Self dual yes
Analytic conductor $6.939$
Analytic rank $0$
Dimension $21$
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] = [869,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 7 q^{3} + 32 q^{4} + 6 q^{5} + 4 q^{6} + 10 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 7 q^{3} + 32 q^{4} + 6 q^{5} + 4 q^{6} + 10 q^{7} + 30 q^{9} + 3 q^{10} - 21 q^{11} + 11 q^{12} + 10 q^{13} + 9 q^{14} + 19 q^{15} + 50 q^{16} - 15 q^{17} + q^{18} + 51 q^{19} + 13 q^{20} + 17 q^{21} + 6 q^{23} - 7 q^{24} + 33 q^{25} + 7 q^{26} + 22 q^{27} + 22 q^{28} - 13 q^{29} - 39 q^{30} + 46 q^{31} + 5 q^{32} - 7 q^{33} + 22 q^{34} - 11 q^{35} + 22 q^{36} + 15 q^{37} - 2 q^{38} - 14 q^{39} - 6 q^{40} - 41 q^{41} - 7 q^{42} + 40 q^{43} - 32 q^{44} - 3 q^{45} - 24 q^{46} + 8 q^{47} + 34 q^{48} + 55 q^{49} + 18 q^{50} + 31 q^{51} - 39 q^{52} + 3 q^{53} + 14 q^{54} - 6 q^{55} + 16 q^{56} + 8 q^{57} - 73 q^{58} + 33 q^{59} - 18 q^{60} + 9 q^{61} - 5 q^{62} + 40 q^{63} + 78 q^{64} - 21 q^{65} - 4 q^{66} + 33 q^{67} - 62 q^{68} + 6 q^{69} + 35 q^{70} - 3 q^{71} - 42 q^{72} + 24 q^{73} - 54 q^{74} + 82 q^{75} + 82 q^{76} - 10 q^{77} - 15 q^{78} + 21 q^{79} + 21 q^{80} + 41 q^{81} - 50 q^{82} + 47 q^{83} + 60 q^{84} - 28 q^{85} - 3 q^{86} + 25 q^{87} + 10 q^{89} - 89 q^{90} + 76 q^{91} + 4 q^{92} + 8 q^{93} + 36 q^{94} + 31 q^{95} - 116 q^{96} + 16 q^{97} - 11 q^{98} - 30 q^{99}+O(q^{100}) \) 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 −2.68141 3.08359 5.18996 1.27314 −8.26836 2.12553 −8.55358 6.50850 −3.41380
1.2 −2.67412 −1.30236 5.15090 2.08090 3.48267 −5.04648 −8.42588 −1.30385 −5.56457
1.3 −2.61714 0.00861090 4.84940 −3.44651 −0.0225359 1.04136 −7.45726 −2.99993 9.01998
1.4 −2.17525 1.86276 2.73170 4.04480 −4.05196 −0.369185 −1.59163 0.469860 −8.79845
1.5 −1.99791 −3.33785 1.99165 −1.02821 6.66873 3.38384 0.0166730 8.14123 2.05427
1.6 −1.86333 0.250270 1.47201 −1.43937 −0.466337 4.42491 0.983819 −2.93736 2.68202
1.7 −1.20949 −2.13907 −0.537135 −1.02611 2.58718 −0.387737 3.06864 1.57562 1.24107
1.8 −1.11826 −0.308986 −0.749503 0.903648 0.345526 −4.24675 3.07465 −2.90453 −1.01051
1.9 −0.887101 2.88793 −1.21305 1.69947 −2.56188 4.17957 2.85030 5.34012 −1.50760
1.10 −0.448370 2.45414 −1.79896 −3.35700 −1.10036 −2.59630 1.70334 3.02280 1.50518
1.11 0.142620 −0.160249 −1.97966 3.18339 −0.0228548 0.767383 −0.567580 −2.97432 0.454016
1.12 0.538928 −1.16559 −1.70956 −2.76839 −0.628171 1.27340 −1.99918 −1.64139 −1.49196
1.13 0.588983 3.01193 −1.65310 4.19109 1.77398 −2.16045 −2.15161 6.07175 2.46848
1.14 1.12939 −2.85627 −0.724484 −2.65356 −3.22584 −4.28474 −3.07700 5.15830 −2.99690
1.15 1.38759 1.96275 −0.0745955 −0.689694 2.72350 3.69585 −2.87869 0.852403 −0.957012
1.16 1.63701 −2.27056 0.679810 3.62218 −3.71693 4.67022 −2.16117 2.15544 5.92956
1.17 1.96551 2.72665 1.86323 1.39322 5.35927 −0.118929 −0.268818 4.43464 2.73840
1.18 2.42496 2.92168 3.88042 −4.07091 7.08494 2.89348 4.55993 5.53619 −9.87178
1.19 2.45750 0.143506 4.03932 1.16348 0.352668 4.27613 5.01164 −2.97941 2.85926
1.20 2.62155 1.05843 4.87253 2.96395 2.77472 −3.13739 7.53049 −1.87973 7.77013
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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 869.2.a.h ✓ 21
3.b odd 2 1 7821.2.a.r 21
11.b odd 2 1 9559.2.a.l 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.a.h ✓ 21 1.a even 1 1 trivial
7821.2.a.r 21 3.b odd 2 1
9559.2.a.l 21 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(869))\):

\( T_{2}^{21} - 37 T_{2}^{19} + 582 T_{2}^{17} - T_{2}^{16} - 5078 T_{2}^{15} + 28 T_{2}^{14} + 26906 T_{2}^{13} + \cdots - 810 \) Copy content Toggle raw display
\( T_{3}^{21} - 7 T_{3}^{20} - 22 T_{3}^{19} + 247 T_{3}^{18} + 19 T_{3}^{17} - 3474 T_{3}^{16} + 3273 T_{3}^{15} + \cdots + 4 \) Copy content Toggle raw display