Properties

Label 6046.2.a.e
Level $6046$
Weight $2$
Character orbit 6046.a
Self dual yes
Analytic conductor $48.278$
Analytic rank $1$
Dimension $56$
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] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(1\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.35147 1.00000 0.434868 −3.35147 3.30614 1.00000 8.23237 0.434868
1.2 1.00000 −3.31079 1.00000 −3.14984 −3.31079 −0.324393 1.00000 7.96133 −3.14984
1.3 1.00000 −3.25917 1.00000 −0.219000 −3.25917 −4.63324 1.00000 7.62218 −0.219000
1.4 1.00000 −3.18186 1.00000 3.25833 −3.18186 0.730955 1.00000 7.12426 3.25833
1.5 1.00000 −2.93762 1.00000 1.81603 −2.93762 −0.742842 1.00000 5.62960 1.81603
1.6 1.00000 −2.89764 1.00000 3.79029 −2.89764 −4.56649 1.00000 5.39633 3.79029
1.7 1.00000 −2.64315 1.00000 −1.33957 −2.64315 3.06951 1.00000 3.98626 −1.33957
1.8 1.00000 −2.45102 1.00000 0.781132 −2.45102 −0.0575638 1.00000 3.00749 0.781132
1.9 1.00000 −2.44881 1.00000 2.87933 −2.44881 −3.14262 1.00000 2.99666 2.87933
1.10 1.00000 −2.43348 1.00000 −1.73446 −2.43348 −1.67668 1.00000 2.92185 −1.73446
1.11 1.00000 −2.39212 1.00000 0.456785 −2.39212 2.48835 1.00000 2.72224 0.456785
1.12 1.00000 −2.37532 1.00000 −4.29598 −2.37532 −4.13959 1.00000 2.64215 −4.29598
1.13 1.00000 −2.14327 1.00000 4.37018 −2.14327 −1.18558 1.00000 1.59362 4.37018
1.14 1.00000 −2.05550 1.00000 −1.29697 −2.05550 0.414264 1.00000 1.22507 −1.29697
1.15 1.00000 −1.97472 1.00000 −3.51408 −1.97472 4.87260 1.00000 0.899523 −3.51408
1.16 1.00000 −1.79217 1.00000 −3.46015 −1.79217 −4.09544 1.00000 0.211861 −3.46015
1.17 1.00000 −1.73760 1.00000 −0.720137 −1.73760 −3.09490 1.00000 0.0192639 −0.720137
1.18 1.00000 −1.70174 1.00000 −3.77151 −1.70174 2.90105 1.00000 −0.104070 −3.77151
1.19 1.00000 −1.47034 1.00000 2.17542 −1.47034 −0.538935 1.00000 −0.838111 2.17542
1.20 1.00000 −1.44356 1.00000 1.04385 −1.44356 3.20508 1.00000 −0.916133 1.04385
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.56
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3023\) \(-1\)

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 6046.2.a.e 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6046.2.a.e 56 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):

\( T_{3}^{56} + 18 T_{3}^{55} + 61 T_{3}^{54} - 789 T_{3}^{53} - 6192 T_{3}^{52} + 7239 T_{3}^{51} + 197391 T_{3}^{50} + 285595 T_{3}^{49} - 3381976 T_{3}^{48} - 10742780 T_{3}^{47} + 33897576 T_{3}^{46} + \cdots + 1185404 \) Copy content Toggle raw display
\( T_{11}^{56} + 53 T_{11}^{55} + 1068 T_{11}^{54} + 7444 T_{11}^{53} - 66251 T_{11}^{52} - 1553959 T_{11}^{51} - 6873704 T_{11}^{50} + 72326295 T_{11}^{49} + 904636698 T_{11}^{48} + 859433184 T_{11}^{47} + \cdots + 97\!\cdots\!21 \) Copy content Toggle raw display