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Properties

Label 3009.2.a.k
Level $3009$
Weight $2$
Character orbit 3009.a
Self dual yes
Analytic conductor $24.027$
Analytic rank $0$
Dimension $26$
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] = [3009,2,Mod(1,3009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3009 = 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3009.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: \(24.0269859682\)
Analytic rank: \(0\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 3 q^{2} + 26 q^{3} + 35 q^{4} + 8 q^{5} + 3 q^{6} + 13 q^{7} + 6 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 3 q^{2} + 26 q^{3} + 35 q^{4} + 8 q^{5} + 3 q^{6} + 13 q^{7} + 6 q^{8} + 26 q^{9} - 8 q^{10} + 10 q^{11} + 35 q^{12} + 24 q^{13} - 3 q^{14} + 8 q^{15} + 45 q^{16} + 26 q^{17} + 3 q^{18} + 24 q^{19} + 11 q^{20} + 13 q^{21} + 23 q^{22} + 13 q^{23} + 6 q^{24} + 42 q^{25} + 9 q^{26} + 26 q^{27} + 14 q^{28} + 26 q^{29} - 8 q^{30} + 37 q^{31} - 5 q^{32} + 10 q^{33} + 3 q^{34} - 5 q^{35} + 35 q^{36} + 28 q^{37} - 29 q^{38} + 24 q^{39} - 33 q^{40} + 7 q^{41} - 3 q^{42} + 21 q^{43} + 7 q^{44} + 8 q^{45} + 40 q^{46} + 22 q^{47} + 45 q^{48} + 41 q^{49} - 12 q^{50} + 26 q^{51} + 26 q^{52} + 4 q^{53} + 3 q^{54} + 20 q^{55} - 5 q^{56} + 24 q^{57} + 31 q^{58} + 26 q^{59} + 11 q^{60} - 21 q^{62} + 13 q^{63} + 88 q^{64} + 6 q^{65} + 23 q^{66} + 38 q^{67} + 35 q^{68} + 13 q^{69} + 46 q^{70} + 62 q^{71} + 6 q^{72} + 27 q^{73} + 11 q^{74} + 42 q^{75} + 98 q^{76} - 25 q^{77} + 9 q^{78} + 49 q^{79} - 21 q^{80} + 26 q^{81} + 30 q^{82} + 9 q^{83} + 14 q^{84} + 8 q^{85} - 18 q^{86} + 26 q^{87} - 17 q^{88} - 27 q^{89} - 8 q^{90} - 12 q^{91} + 26 q^{92} + 37 q^{93} - 19 q^{94} - q^{95} - 5 q^{96} + 46 q^{97} - 35 q^{98} + 10 q^{99}+O(q^{100}) \) 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 −2.76818 1.00000 5.66279 3.65865 −2.76818 −2.79644 −10.1393 1.00000 −10.1278
1.2 −2.70471 1.00000 5.31544 −2.42158 −2.70471 4.39333 −8.96730 1.00000 6.54967
1.3 −2.39971 1.00000 3.75861 2.52908 −2.39971 2.94364 −4.22017 1.00000 −6.06906
1.4 −2.27048 1.00000 3.15509 −1.29680 −2.27048 −3.28058 −2.62261 1.00000 2.94436
1.5 −2.09676 1.00000 2.39640 −1.58178 −2.09676 1.76445 −0.831151 1.00000 3.31662
1.6 −1.69649 1.00000 0.878089 3.95187 −1.69649 −0.497632 1.90331 1.00000 −6.70433
1.7 −1.43158 1.00000 0.0494328 2.88482 −1.43158 5.00829 2.79240 1.00000 −4.12987
1.8 −1.42345 1.00000 0.0261965 0.403729 −1.42345 −3.03614 2.80960 1.00000 −0.574686
1.9 −0.930214 1.00000 −1.13470 −3.65300 −0.930214 0.804964 2.91594 1.00000 3.39807
1.10 −0.922404 1.00000 −1.14917 1.55008 −0.922404 −1.97606 2.90481 1.00000 −1.42980
1.11 −0.847327 1.00000 −1.28204 −3.53879 −0.847327 2.30103 2.78096 1.00000 2.99851
1.12 −0.371721 1.00000 −1.86182 1.89339 −0.371721 3.60506 1.43552 1.00000 −0.703814
1.13 0.173969 1.00000 −1.96973 −0.0737264 0.173969 −1.51403 −0.690610 1.00000 −0.0128261
1.14 0.322353 1.00000 −1.89609 3.19886 0.322353 0.538718 −1.25592 1.00000 1.03116
1.15 0.666373 1.00000 −1.55595 −0.790051 0.666373 4.52650 −2.36959 1.00000 −0.526469
1.16 1.05658 1.00000 −0.883638 −1.73246 1.05658 −5.07262 −3.04680 1.00000 −1.83048
1.17 1.09580 1.00000 −0.799218 2.76597 1.09580 −3.59144 −3.06739 1.00000 3.03096
1.18 1.42776 1.00000 0.0385038 −3.91181 1.42776 2.75596 −2.80055 1.00000 −5.58513
1.19 1.58307 1.00000 0.506106 3.69251 1.58307 2.90397 −2.36494 1.00000 5.84550
1.20 1.82595 1.00000 1.33410 −2.34287 1.82595 0.812091 −1.21590 1.00000 −4.27797
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(-1\)

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 3009.2.a.k ✓ 26
3.b odd 2 1 9027.2.a.u 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3009.2.a.k ✓ 26 1.a even 1 1 trivial
9027.2.a.u 26 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\):

\( T_{2}^{26} - 3 T_{2}^{25} - 39 T_{2}^{24} + 120 T_{2}^{23} + 656 T_{2}^{22} - 2084 T_{2}^{21} + \cdots + 3008 \) Copy content Toggle raw display
\( T_{5}^{26} - 8 T_{5}^{25} - 54 T_{5}^{24} + 579 T_{5}^{23} + 851 T_{5}^{22} - 17707 T_{5}^{21} + \cdots - 1205216 \) Copy content Toggle raw display