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Properties

Label 6034.2.a.r
Level $6034$
Weight $2$
Character orbit 6034.a
Self dual yes
Analytic conductor $48.182$
Analytic rank $0$
Dimension $31$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(0\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 31 q^{2} + 7 q^{3} + 31 q^{4} + 15 q^{5} + 7 q^{6} - 31 q^{7} + 31 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 31 q^{2} + 7 q^{3} + 31 q^{4} + 15 q^{5} + 7 q^{6} - 31 q^{7} + 31 q^{8} + 42 q^{9} + 15 q^{10} + 12 q^{11} + 7 q^{12} + 26 q^{13} - 31 q^{14} + 6 q^{15} + 31 q^{16} + 33 q^{17} + 42 q^{18} + 34 q^{19} + 15 q^{20} - 7 q^{21} + 12 q^{22} - 14 q^{23} + 7 q^{24} + 58 q^{25} + 26 q^{26} + 28 q^{27} - 31 q^{28} + 11 q^{29} + 6 q^{30} + 19 q^{31} + 31 q^{32} + 43 q^{33} + 33 q^{34} - 15 q^{35} + 42 q^{36} + 2 q^{37} + 34 q^{38} - 16 q^{39} + 15 q^{40} + 53 q^{41} - 7 q^{42} + 22 q^{43} + 12 q^{44} + 43 q^{45} - 14 q^{46} + 27 q^{47} + 7 q^{48} + 31 q^{49} + 58 q^{50} + 17 q^{51} + 26 q^{52} + 11 q^{53} + 28 q^{54} + 19 q^{55} - 31 q^{56} + 45 q^{57} + 11 q^{58} + 54 q^{59} + 6 q^{60} + 41 q^{61} + 19 q^{62} - 42 q^{63} + 31 q^{64} + 30 q^{65} + 43 q^{66} + 13 q^{67} + 33 q^{68} + 17 q^{69} - 15 q^{70} + 43 q^{71} + 42 q^{72} + 42 q^{73} + 2 q^{74} + 62 q^{75} + 34 q^{76} - 12 q^{77} - 16 q^{78} - 12 q^{79} + 15 q^{80} + 63 q^{81} + 53 q^{82} + 35 q^{83} - 7 q^{84} + 16 q^{85} + 22 q^{86} - 4 q^{87} + 12 q^{88} + 115 q^{89} + 43 q^{90} - 26 q^{91} - 14 q^{92} + q^{93} + 27 q^{94} - 13 q^{95} + 7 q^{96} + 32 q^{97} + 31 q^{98} + 34 q^{99}+O(q^{100}) \) 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 −3.29198 1.00000 −1.58360 −3.29198 −1.00000 1.00000 7.83715 −1.58360
1.2 1.00000 −3.19812 1.00000 3.31602 −3.19812 −1.00000 1.00000 7.22798 3.31602
1.3 1.00000 −2.74816 1.00000 3.57738 −2.74816 −1.00000 1.00000 4.55238 3.57738
1.4 1.00000 −2.34364 1.00000 −2.37141 −2.34364 −1.00000 1.00000 2.49264 −2.37141
1.5 1.00000 −2.29462 1.00000 1.21852 −2.29462 −1.00000 1.00000 2.26529 1.21852
1.6 1.00000 −2.15124 1.00000 −0.116889 −2.15124 −1.00000 1.00000 1.62782 −0.116889
1.7 1.00000 −1.78656 1.00000 −1.18496 −1.78656 −1.00000 1.00000 0.191811 −1.18496
1.8 1.00000 −1.76695 1.00000 2.46864 −1.76695 −1.00000 1.00000 0.122112 2.46864
1.9 1.00000 −1.69299 1.00000 −2.70073 −1.69299 −1.00000 1.00000 −0.133772 −2.70073
1.10 1.00000 −1.40540 1.00000 2.72818 −1.40540 −1.00000 1.00000 −1.02486 2.72818
1.11 1.00000 −0.820659 1.00000 4.15538 −0.820659 −1.00000 1.00000 −2.32652 4.15538
1.12 1.00000 −0.585502 1.00000 −3.38680 −0.585502 −1.00000 1.00000 −2.65719 −3.38680
1.13 1.00000 −0.307546 1.00000 −1.88818 −0.307546 −1.00000 1.00000 −2.90542 −1.88818
1.14 1.00000 −0.217225 1.00000 0.607550 −0.217225 −1.00000 1.00000 −2.95281 0.607550
1.15 1.00000 −0.118197 1.00000 −0.813929 −0.118197 −1.00000 1.00000 −2.98603 −0.813929
1.16 1.00000 0.239534 1.00000 −1.50535 0.239534 −1.00000 1.00000 −2.94262 −1.50535
1.17 1.00000 0.630504 1.00000 3.47508 0.630504 −1.00000 1.00000 −2.60246 3.47508
1.18 1.00000 0.718722 1.00000 3.07705 0.718722 −1.00000 1.00000 −2.48344 3.07705
1.19 1.00000 0.747767 1.00000 −1.63599 0.747767 −1.00000 1.00000 −2.44084 −1.63599
1.20 1.00000 1.15174 1.00000 2.73654 1.15174 −1.00000 1.00000 −1.67348 2.73654
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(431\) \(1\)

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 6034.2.a.r ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.r ✓ 31 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3}^{31} - 7 T_{3}^{30} - 43 T_{3}^{29} + 392 T_{3}^{28} + 638 T_{3}^{27} - 9610 T_{3}^{26} + \cdots + 21632 \) Copy content Toggle raw display
\( T_{5}^{31} - 15 T_{5}^{30} + 6 T_{5}^{29} + 943 T_{5}^{28} - 3427 T_{5}^{27} - 23833 T_{5}^{26} + \cdots + 639529984 \) Copy content Toggle raw display
\( T_{11}^{31} - 12 T_{11}^{30} - 139 T_{11}^{29} + 2078 T_{11}^{28} + 7412 T_{11}^{27} + \cdots + 17135117778944 \) Copy content Toggle raw display