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Properties

Label 6014.2.a.j
Level $6014$
Weight $2$
Character orbit 6014.a
Self dual yes
Analytic conductor $48.022$
Analytic rank $0$
Dimension $32$
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: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{2} - 2 q^{3} + 32 q^{4} + 2 q^{6} + 5 q^{7} - 32 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{2} - 2 q^{3} + 32 q^{4} + 2 q^{6} + 5 q^{7} - 32 q^{8} + 30 q^{9} - 4 q^{11} - 2 q^{12} + 10 q^{13} - 5 q^{14} - q^{15} + 32 q^{16} + 14 q^{17} - 30 q^{18} + 33 q^{19} + 4 q^{22} - 2 q^{23} + 2 q^{24} + 46 q^{25} - 10 q^{26} - 5 q^{27} + 5 q^{28} - q^{29} + q^{30} - 32 q^{31} - 32 q^{32} + 32 q^{33} - 14 q^{34} + 8 q^{35} + 30 q^{36} + 31 q^{37} - 33 q^{38} + 4 q^{39} + 31 q^{41} + 15 q^{43} - 4 q^{44} + q^{45} + 2 q^{46} - 14 q^{47} - 2 q^{48} + 75 q^{49} - 46 q^{50} + 27 q^{51} + 10 q^{52} - 31 q^{53} + 5 q^{54} + 14 q^{55} - 5 q^{56} + 51 q^{57} + q^{58} - 8 q^{59} - q^{60} + 24 q^{61} + 32 q^{62} + 23 q^{63} + 32 q^{64} + 20 q^{65} - 32 q^{66} + 17 q^{67} + 14 q^{68} - 31 q^{69} - 8 q^{70} - 31 q^{71} - 30 q^{72} + 19 q^{73} - 31 q^{74} - 40 q^{75} + 33 q^{76} + 8 q^{77} - 4 q^{78} + 39 q^{79} + 116 q^{81} - 31 q^{82} - 6 q^{83} + 56 q^{85} - 15 q^{86} - 17 q^{87} + 4 q^{88} + 8 q^{89} - q^{90} + 34 q^{91} - 2 q^{92} + 2 q^{93} + 14 q^{94} - 22 q^{95} + 2 q^{96} + 32 q^{97} - 75 q^{98} - 27 q^{99}+O(q^{100}) \) 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 −3.39762 1.00000 2.43962 3.39762 3.88412 −1.00000 8.54381 −2.43962
1.2 −1.00000 −3.38837 1.00000 −3.95025 3.38837 −2.93892 −1.00000 8.48103 3.95025
1.3 −1.00000 −2.82503 1.00000 −0.629658 2.82503 −0.109398 −1.00000 4.98077 0.629658
1.4 −1.00000 −2.75899 1.00000 2.79935 2.75899 −3.68185 −1.00000 4.61205 −2.79935
1.5 −1.00000 −2.60322 1.00000 0.943880 2.60322 3.39783 −1.00000 3.77677 −0.943880
1.6 −1.00000 −2.29039 1.00000 −1.87953 2.29039 0.391521 −1.00000 2.24587 1.87953
1.7 −1.00000 −2.05301 1.00000 −2.11820 2.05301 −1.54348 −1.00000 1.21485 2.11820
1.8 −1.00000 −1.88640 1.00000 4.44357 1.88640 −1.18572 −1.00000 0.558507 −4.44357
1.9 −1.00000 −1.64406 1.00000 −4.02843 1.64406 2.75493 −1.00000 −0.297060 4.02843
1.10 −1.00000 −1.52309 1.00000 −0.0721797 1.52309 3.89998 −1.00000 −0.680188 0.0721797
1.11 −1.00000 −1.01996 1.00000 −1.74888 1.01996 2.14281 −1.00000 −1.95968 1.74888
1.12 −1.00000 −0.848161 1.00000 4.13342 0.848161 1.58594 −1.00000 −2.28062 −4.13342
1.13 −1.00000 −0.676893 1.00000 −1.03316 0.676893 −4.68414 −1.00000 −2.54182 1.03316
1.14 −1.00000 −0.475406 1.00000 1.98672 0.475406 3.40549 −1.00000 −2.77399 −1.98672
1.15 −1.00000 −0.416033 1.00000 0.917555 0.416033 −0.743221 −1.00000 −2.82692 −0.917555
1.16 −1.00000 −0.0623199 1.00000 1.97455 0.0623199 −0.185636 −1.00000 −2.99612 −1.97455
1.17 −1.00000 0.00816480 1.00000 4.13053 −0.00816480 −5.05598 −1.00000 −2.99993 −4.13053
1.18 −1.00000 0.284664 1.00000 −2.60924 −0.284664 −1.90185 −1.00000 −2.91897 2.60924
1.19 −1.00000 0.502069 1.00000 0.295460 −0.502069 −2.57922 −1.00000 −2.74793 −0.295460
1.20 −1.00000 0.519470 1.00000 −3.34361 −0.519470 4.82750 −1.00000 −2.73015 3.34361
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
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.j ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6014.2.a.j ✓ 32 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 2 T_{3}^{31} - 61 T_{3}^{30} - 119 T_{3}^{29} + 1630 T_{3}^{28} + 3088 T_{3}^{27} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\). Copy content Toggle raw display