Properties

Label 2669.2.a.b
Level $2669$
Weight $2$
Character orbit 2669.a
Self dual yes
Analytic conductor $21.312$
Analytic rank $1$
Dimension $45$
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] = [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: \(1\)
Dimension: \(45\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 45 q - 2 q^{2} - 20 q^{3} + 34 q^{4} - 10 q^{5} - 14 q^{6} - 20 q^{7} - 9 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 45 q - 2 q^{2} - 20 q^{3} + 34 q^{4} - 10 q^{5} - 14 q^{6} - 20 q^{7} - 9 q^{8} + 39 q^{9} - 21 q^{10} - 26 q^{11} - 23 q^{12} - 6 q^{13} - 7 q^{14} - 10 q^{15} + 20 q^{16} + 45 q^{17} - 3 q^{18} - 56 q^{19} - 26 q^{20} + 2 q^{21} - 23 q^{22} - 38 q^{23} - 35 q^{24} + 27 q^{25} - 10 q^{26} - 71 q^{27} - 29 q^{28} - 29 q^{29} + 9 q^{30} - 57 q^{31} + 4 q^{33} - 2 q^{34} - 16 q^{35} + 44 q^{36} - 14 q^{37} - 6 q^{38} - 25 q^{39} - 48 q^{40} - 23 q^{41} - q^{42} - 43 q^{43} - 29 q^{44} - 63 q^{45} - 42 q^{46} + 11 q^{47} - 6 q^{48} - 9 q^{49} + 14 q^{50} - 20 q^{51} - 27 q^{52} + 7 q^{53} + 10 q^{54} - 41 q^{55} - 14 q^{56} - 5 q^{57} - 58 q^{58} - 59 q^{59} + q^{60} - 40 q^{61} - 34 q^{62} - 56 q^{63} - 67 q^{64} - 3 q^{65} - 53 q^{66} - 44 q^{67} + 34 q^{68} - 17 q^{69} + 14 q^{70} - 18 q^{71} - 25 q^{72} - 2 q^{73} - 5 q^{74} - 85 q^{75} - 123 q^{76} - 4 q^{77} + 33 q^{78} - 119 q^{79} - 17 q^{80} + 21 q^{81} - 6 q^{82} - 32 q^{83} + 54 q^{84} - 10 q^{85} - 14 q^{86} - 3 q^{87} - 33 q^{88} - 25 q^{89} - 23 q^{90} - 177 q^{91} - 62 q^{92} + 36 q^{93} - 64 q^{94} - 47 q^{95} - 153 q^{96} - 82 q^{97} + 13 q^{98} - 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.57997 2.27677 4.65627 0.454801 −5.87400 −2.24616 −6.85310 2.18367 −1.17338
1.2 −2.57201 2.33809 4.61521 0.369809 −6.01357 2.27226 −6.72634 2.46665 −0.951150
1.3 −2.52566 −1.38618 4.37894 3.81505 3.50102 −1.76068 −6.00839 −1.07850 −9.63552
1.4 −2.50488 0.477138 4.27442 0.361901 −1.19517 −0.370992 −5.69715 −2.77234 −0.906519
1.5 −2.33974 −2.03142 3.47436 −3.05646 4.75298 0.309944 −3.44962 1.12665 7.15130
1.6 −2.29985 −1.57945 3.28932 −1.13990 3.63250 −3.55948 −2.96525 −0.505346 2.62159
1.7 −2.25978 −3.22925 3.10659 −1.10091 7.29738 2.65511 −2.50065 7.42805 2.48780
1.8 −1.96618 1.61430 1.86585 −2.80677 −3.17400 3.78870 0.263762 −0.394028 5.51860
1.9 −1.83672 2.17585 1.37355 −2.57891 −3.99644 −3.01876 1.15062 1.73434 4.73675
1.10 −1.77149 −2.81155 1.13817 3.99941 4.98063 −2.62351 1.52672 4.90481 −7.08491
1.11 −1.62128 −0.288989 0.628561 2.60755 0.468532 0.210946 2.22349 −2.91649 −4.22758
1.12 −1.52876 −3.31671 0.337100 −0.791880 5.07045 −3.85375 2.54217 8.00058 1.21059
1.13 −1.45876 1.14095 0.127991 1.62587 −1.66437 −2.45935 2.73082 −1.69823 −2.37176
1.14 −1.42791 0.606117 0.0389142 1.41385 −0.865478 1.73370 2.80025 −2.63262 −2.01884
1.15 −1.37659 −2.26906 −0.104998 −0.453399 3.12356 4.69088 2.89772 2.14862 0.624145
1.16 −0.864732 0.578982 −1.25224 −4.36543 −0.500664 −1.93357 2.81231 −2.66478 3.77492
1.17 −0.810938 −0.989402 −1.34238 −2.55175 0.802344 −1.16091 2.71046 −2.02108 2.06931
1.18 −0.440764 −0.189246 −1.80573 1.67241 0.0834127 −4.38089 1.67743 −2.96419 −0.737139
1.19 −0.356197 −0.420410 −1.87312 2.53359 0.149749 3.26999 1.37959 −2.82326 −0.902455
1.20 −0.325803 2.31804 −1.89385 −0.307738 −0.755226 −0.806220 1.26863 2.37332 0.100262
See all 45 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.45
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.b 45
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2669.2.a.b 45 1.a even 1 1 trivial