↑

Properties

Label 2669.2.a.d
Level $2669$
Weight $2$
Character orbit 2669.a
Self dual yes
Analytic conductor $21.312$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2669,2,Mod(1,2669)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2669, 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("2669.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2669 = 17 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2669.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: \(21.3120722995\)
Analytic rank: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{2} + 8 q^{3} + 75 q^{4} + 6 q^{5} + 8 q^{6} + 10 q^{7} + 12 q^{8} + 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{2} + 8 q^{3} + 75 q^{4} + 6 q^{5} + 8 q^{6} + 10 q^{7} + 12 q^{8} + 74 q^{9} + 25 q^{10} - 2 q^{11} + 13 q^{12} + 20 q^{13} + 7 q^{14} + 30 q^{15} + 101 q^{16} - 60 q^{17} + 26 q^{18} + 76 q^{19} + 28 q^{20} + 14 q^{21} + 9 q^{22} - 8 q^{23} + 31 q^{24} + 92 q^{25} + 16 q^{26} + 29 q^{27} + 29 q^{28} - 17 q^{29} + 21 q^{30} + 65 q^{31} + 17 q^{32} + 4 q^{33} - 3 q^{34} + 40 q^{35} + 97 q^{36} + 18 q^{37} + 46 q^{38} + 21 q^{39} + 50 q^{40} + 5 q^{41} + 11 q^{42} + 73 q^{43} + q^{44} + 35 q^{45} - 28 q^{46} + 11 q^{47} + 48 q^{48} + 130 q^{49} + 25 q^{50} - 8 q^{51} + 55 q^{52} + 17 q^{53} + 16 q^{54} + 51 q^{55} + 50 q^{56} - 15 q^{57} - 36 q^{58} + 61 q^{59} + 101 q^{60} + 62 q^{61} + 2 q^{62} - 18 q^{63} + 94 q^{64} - 25 q^{65} - 69 q^{66} + 92 q^{67} - 75 q^{68} + 51 q^{69} + 26 q^{70} + 4 q^{71} + 44 q^{72} + 28 q^{73} - 47 q^{74} + 35 q^{75} + 97 q^{76} + 24 q^{77} + q^{78} + 73 q^{79} + 11 q^{80} + 68 q^{81} + 56 q^{82} + 28 q^{83} + 82 q^{84} - 6 q^{85} + 10 q^{86} + 57 q^{87} + 35 q^{88} + 33 q^{89} + 103 q^{90} + 161 q^{91} - 100 q^{92} + 4 q^{93} + 136 q^{94} - 29 q^{95} + q^{96} + 50 q^{97} - 18 q^{98} + 5 q^{99}+O(q^{100}) \) 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 −2.79045 1.65585 5.78660 2.75943 −4.62057 2.88169 −10.5663 −0.258159 −7.70004
1.2 −2.74794 0.913711 5.55117 −0.250676 −2.51082 −4.51892 −9.75840 −2.16513 0.688842
1.3 −2.58887 −2.04548 4.70224 −2.25718 5.29548 4.39137 −6.99576 1.18399 5.84354
1.4 −2.56296 −2.90354 4.56875 −3.86807 7.44166 −2.48411 −6.58361 5.43056 9.91369
1.5 −2.51896 −1.82563 4.34514 0.948921 4.59869 −4.23350 −5.90730 0.332934 −2.39029
1.6 −2.49207 0.183037 4.21040 1.48314 −0.456141 −1.05623 −5.50847 −2.96650 −3.69608
1.7 −2.48107 2.13821 4.15569 −3.58244 −5.30505 0.835816 −5.34841 1.57195 8.88828
1.8 −2.22131 3.18653 2.93424 −0.373371 −7.07829 5.00245 −2.07524 7.15398 0.829374
1.9 −2.16892 3.14125 2.70422 2.47833 −6.81312 −0.221826 −1.52739 6.86745 −5.37529
1.10 −2.15070 1.41747 2.62552 −2.04223 −3.04855 −1.85317 −1.34532 −0.990787 4.39223
1.11 −2.13403 −2.46016 2.55407 3.49540 5.25004 4.63778 −1.18239 3.05238 −7.45928
1.12 −2.03270 −1.08329 2.13188 −1.14909 2.20200 2.79071 −0.268074 −1.82649 2.33575
1.13 −1.99538 −2.84714 1.98156 1.81977 5.68113 −2.02350 0.0367928 5.10619 −3.63115
1.14 −1.77693 0.000323701 0 1.15749 2.89437 −0.000575195 0 −3.09817 1.49708 −3.00000 −5.14311
1.15 −1.71612 −0.593452 0.945076 −4.31407 1.01844 3.92958 1.81038 −2.64781 7.40347
1.16 −1.62976 1.35125 0.656113 −1.35560 −2.20221 0.933459 2.19021 −1.17413 2.20930
1.17 −1.60407 −0.666881 0.573031 3.23815 1.06972 1.09196 2.28895 −2.55527 −5.19420
1.18 −1.17168 −2.89772 −0.627168 −1.66886 3.39520 −2.50404 3.07820 5.39678 1.95537
1.19 −1.05628 0.198712 −0.884277 −2.30277 −0.209895 −2.31788 3.04660 −2.96051 2.43236
1.20 −1.01512 1.32983 −0.969531 −1.00143 −1.34993 −1.54838 3.01443 −1.23156 1.01657
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.60
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(157\) \(-1\)

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 2669.2.a.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2669.2.a.d ✓ 60 1.a even 1 1 trivial