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Properties

Label 6014.2.a.k
Level $6014$
Weight $2$
Character orbit 6014.a
Self dual yes
Analytic conductor $48.022$
Analytic rank $0$
Dimension $37$
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] = [6014,2,Mod(1,6014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6014, 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("6014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6014 = 2 \cdot 31 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6014.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: \(48.0220317756\)
Analytic rank: \(0\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 37 q^{2} + 9 q^{3} + 37 q^{4} + 9 q^{5} + 9 q^{6} + 19 q^{7} + 37 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 37 q^{2} + 9 q^{3} + 37 q^{4} + 9 q^{5} + 9 q^{6} + 19 q^{7} + 37 q^{8} + 52 q^{9} + 9 q^{10} + 5 q^{11} + 9 q^{12} + 16 q^{13} + 19 q^{14} + 22 q^{15} + 37 q^{16} + 3 q^{17} + 52 q^{18} + 36 q^{19} + 9 q^{20} + 6 q^{21} + 5 q^{22} + 11 q^{23} + 9 q^{24} + 58 q^{25} + 16 q^{26} + 24 q^{27} + 19 q^{28} + 5 q^{29} + 22 q^{30} + 37 q^{31} + 37 q^{32} + q^{33} + 3 q^{34} + 28 q^{35} + 52 q^{36} + 21 q^{37} + 36 q^{38} + 38 q^{39} + 9 q^{40} + 21 q^{41} + 6 q^{42} + 14 q^{43} + 5 q^{44} + 55 q^{45} + 11 q^{46} + 59 q^{47} + 9 q^{48} + 82 q^{49} + 58 q^{50} + 46 q^{51} + 16 q^{52} + 8 q^{53} + 24 q^{54} + 25 q^{55} + 19 q^{56} + 5 q^{58} + 41 q^{59} + 22 q^{60} + 16 q^{61} + 37 q^{62} + 23 q^{63} + 37 q^{64} - 46 q^{65} + q^{66} + 45 q^{67} + 3 q^{68} + 68 q^{69} + 28 q^{70} + 55 q^{71} + 52 q^{72} + 29 q^{73} + 21 q^{74} - 12 q^{75} + 36 q^{76} + 30 q^{77} + 38 q^{78} + 25 q^{79} + 9 q^{80} + 73 q^{81} + 21 q^{82} + 70 q^{83} + 6 q^{84} - 21 q^{85} + 14 q^{86} + 37 q^{87} + 5 q^{88} + 55 q^{90} + 18 q^{91} + 11 q^{92} + 9 q^{93} + 59 q^{94} - 9 q^{95} + 9 q^{96} + 37 q^{97} + 82 q^{98} + 33 q^{99}+O(q^{100}) \) 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 1.00000 −3.24519 1.00000 −3.01148 −3.24519 −2.55038 1.00000 7.53129 −3.01148
1.2 1.00000 −3.11031 1.00000 2.99950 −3.11031 2.46329 1.00000 6.67403 2.99950
1.3 1.00000 −2.85895 1.00000 2.60012 −2.85895 −2.89016 1.00000 5.17357 2.60012
1.4 1.00000 −2.75773 1.00000 2.81004 −2.75773 5.04887 1.00000 4.60506 2.81004
1.5 1.00000 −2.56862 1.00000 −3.99872 −2.56862 2.48908 1.00000 3.59779 −3.99872
1.6 1.00000 −2.38480 1.00000 1.29081 −2.38480 −1.08194 1.00000 2.68726 1.29081
1.7 1.00000 −2.26430 1.00000 −0.734715 −2.26430 2.48512 1.00000 2.12704 −0.734715
1.8 1.00000 −2.11580 1.00000 −1.69302 −2.11580 −4.07217 1.00000 1.47662 −1.69302
1.9 1.00000 −1.95043 1.00000 −1.96504 −1.95043 0.849460 1.00000 0.804176 −1.96504
1.10 1.00000 −1.81353 1.00000 0.592127 −1.81353 0.674162 1.00000 0.288888 0.592127
1.11 1.00000 −1.30157 1.00000 3.71801 −1.30157 4.36912 1.00000 −1.30591 3.71801
1.12 1.00000 −0.850163 1.00000 1.27366 −0.850163 −2.84006 1.00000 −2.27722 1.27366
1.13 1.00000 −0.835304 1.00000 −3.09750 −0.835304 4.05347 1.00000 −2.30227 −3.09750
1.14 1.00000 −0.749812 1.00000 −0.301670 −0.749812 −4.79556 1.00000 −2.43778 −0.301670
1.15 1.00000 −0.237489 1.00000 4.10563 −0.237489 4.01630 1.00000 −2.94360 4.10563
1.16 1.00000 −0.177510 1.00000 −1.23284 −0.177510 −3.05433 1.00000 −2.96849 −1.23284
1.17 1.00000 −0.0432689 1.00000 −1.25379 −0.0432689 4.04669 1.00000 −2.99813 −1.25379
1.18 1.00000 0.242040 1.00000 −4.38700 0.242040 3.95726 1.00000 −2.94142 −4.38700
1.19 1.00000 0.277622 1.00000 −0.233184 0.277622 −0.765476 1.00000 −2.92293 −0.233184
1.20 1.00000 0.452250 1.00000 −3.58205 0.452250 −1.19090 1.00000 −2.79547 −3.58205
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(31\) \(-1\)
\(97\) \(-1\)

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 6014.2.a.k ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6014.2.a.k ✓ 37 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{37} - 9 T_{3}^{36} - 41 T_{3}^{35} + 586 T_{3}^{34} + 226 T_{3}^{33} - 16855 T_{3}^{32} + \cdots + 36928 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\). Copy content Toggle raw display