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Properties

Label 3009.2.a.j
Level $3009$
Weight $2$
Character orbit 3009.a
Self dual yes
Analytic conductor $24.027$
Analytic rank $0$
Dimension $24$
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: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 24 q^{3} + 34 q^{4} - 3 q^{5} + 2 q^{6} + 19 q^{7} + 3 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 24 q^{3} + 34 q^{4} - 3 q^{5} + 2 q^{6} + 19 q^{7} + 3 q^{8} + 24 q^{9} + 10 q^{10} + 11 q^{11} + 34 q^{12} + 21 q^{13} + 9 q^{14} - 3 q^{15} + 50 q^{16} - 24 q^{17} + 2 q^{18} + 13 q^{19} + q^{20} + 19 q^{21} + 5 q^{22} + 18 q^{23} + 3 q^{24} + 39 q^{25} - 7 q^{26} + 24 q^{27} + 40 q^{28} - 10 q^{29} + 10 q^{30} + 47 q^{31} + 18 q^{32} + 11 q^{33} - 2 q^{34} + 5 q^{35} + 34 q^{36} + 54 q^{37} + 5 q^{38} + 21 q^{39} + 29 q^{40} + 9 q^{42} + 16 q^{43} + 7 q^{44} - 3 q^{45} - 8 q^{46} + 10 q^{47} + 50 q^{48} + 55 q^{49} + 21 q^{50} - 24 q^{51} + 68 q^{52} + 6 q^{53} + 2 q^{54} + 15 q^{55} - 5 q^{56} + 13 q^{57} + q^{58} - 24 q^{59} + q^{60} + 42 q^{61} - 7 q^{62} + 19 q^{63} + 53 q^{64} - 9 q^{65} + 5 q^{66} + 28 q^{67} - 34 q^{68} + 18 q^{69} - 20 q^{70} + 54 q^{71} + 3 q^{72} + 33 q^{73} - 17 q^{74} + 39 q^{75} - 56 q^{76} - 23 q^{77} - 7 q^{78} + 35 q^{79} + 3 q^{80} + 24 q^{81} - 12 q^{82} - 17 q^{83} + 40 q^{84} + 3 q^{85} + 36 q^{86} - 10 q^{87} + 47 q^{88} + 15 q^{89} + 10 q^{90} + 74 q^{91} + 6 q^{92} + 47 q^{93} + 9 q^{94} + 12 q^{95} + 18 q^{96} + 52 q^{97} - 48 q^{98} + 11 q^{99}+O(q^{100}) \) 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 −2.75777 1.00000 5.60528 1.77452 −2.75777 3.92758 −9.94254 1.00000 −4.89371
1.2 −2.64834 1.00000 5.01370 −1.72747 −2.64834 −1.00744 −7.98130 1.00000 4.57492
1.3 −2.44925 1.00000 3.99884 −4.02919 −2.44925 −0.238385 −4.89567 1.00000 9.86850
1.4 −2.13124 1.00000 2.54217 2.58726 −2.13124 2.60195 −1.15550 1.00000 −5.51407
1.5 −2.10017 1.00000 2.41071 −2.30678 −2.10017 4.51480 −0.862553 1.00000 4.84464
1.6 −1.95642 1.00000 1.82758 2.67470 −1.95642 −3.83979 0.337329 1.00000 −5.23284
1.7 −1.64279 1.00000 0.698755 −1.75290 −1.64279 −4.95142 2.13767 1.00000 2.87965
1.8 −1.12409 1.00000 −0.736419 1.51768 −1.12409 −0.757398 3.07598 1.00000 −1.70601
1.9 −0.969837 1.00000 −1.05942 −2.73388 −0.969837 4.83240 2.96713 1.00000 2.65141
1.10 −0.780404 1.00000 −1.39097 −0.0767861 −0.780404 2.72552 2.64633 1.00000 0.0599241
1.11 −0.113340 1.00000 −1.98715 4.03693 −0.113340 1.78945 0.451904 1.00000 −0.457545
1.12 0.0458562 1.00000 −1.99790 −1.34672 0.0458562 −3.65590 −0.183328 1.00000 −0.0617554
1.13 0.412985 1.00000 −1.82944 −1.14252 0.412985 2.74893 −1.58150 1.00000 −0.471845
1.14 0.749114 1.00000 −1.43883 −2.56478 0.749114 0.240543 −2.57608 1.00000 −1.92132
1.15 0.898521 1.00000 −1.19266 −4.37170 0.898521 −1.93786 −2.86867 1.00000 −3.92807
1.16 0.945621 1.00000 −1.10580 2.83482 0.945621 1.11494 −2.93691 1.00000 2.68066
1.17 1.22092 1.00000 −0.509348 2.54973 1.22092 4.41434 −3.06372 1.00000 3.11302
1.18 1.84785 1.00000 1.41454 −2.08675 1.84785 −3.05385 −1.08184 1.00000 −3.85599
1.19 1.91019 1.00000 1.64881 0.480421 1.91019 3.75225 −0.670838 1.00000 0.917693
1.20 2.34339 1.00000 3.49147 4.22060 2.34339 −0.391610 3.49511 1.00000 9.89050
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
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.j ✓ 24
3.b odd 2 1 9027.2.a.t 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3009.2.a.j ✓ 24 1.a even 1 1 trivial
9027.2.a.t 24 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\):

\( T_{2}^{24} - 2 T_{2}^{23} - 39 T_{2}^{22} + 77 T_{2}^{21} + 655 T_{2}^{20} - 1277 T_{2}^{19} + \cdots - 132 \) Copy content Toggle raw display
\( T_{5}^{24} + 3 T_{5}^{23} - 75 T_{5}^{22} - 212 T_{5}^{21} + 2387 T_{5}^{20} + 6312 T_{5}^{19} + \cdots - 351888 \) Copy content Toggle raw display