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Properties

Label 6046.2.a.g
Level $6046$
Weight $2$
Character orbit 6046.a
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $69$
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: \(0\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) 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 −3.31246 1.00000 3.61669 3.31246 −0.0193368 −1.00000 7.97240 −3.61669
1.2 −1.00000 −3.30557 1.00000 −1.09873 3.30557 3.10435 −1.00000 7.92681 1.09873
1.3 −1.00000 −3.30292 1.00000 −2.02929 3.30292 −1.40073 −1.00000 7.90925 2.02929
1.4 −1.00000 −3.27820 1.00000 0.761195 3.27820 3.33846 −1.00000 7.74657 −0.761195
1.5 −1.00000 −3.16263 1.00000 1.22760 3.16263 −4.76545 −1.00000 7.00223 −1.22760
1.6 −1.00000 −3.15165 1.00000 1.27267 3.15165 −3.14596 −1.00000 6.93292 −1.27267
1.7 −1.00000 −3.13336 1.00000 −4.20234 3.13336 −4.60058 −1.00000 6.81797 4.20234
1.8 −1.00000 −2.76254 1.00000 2.17667 2.76254 −4.35539 −1.00000 4.63161 −2.17667
1.9 −1.00000 −2.73247 1.00000 −4.08673 2.73247 −1.35525 −1.00000 4.46639 4.08673
1.10 −1.00000 −2.42918 1.00000 1.49357 2.42918 3.38071 −1.00000 2.90091 −1.49357
1.11 −1.00000 −2.36493 1.00000 −1.05232 2.36493 −2.21269 −1.00000 2.59290 1.05232
1.12 −1.00000 −2.32306 1.00000 −1.40989 2.32306 −1.24456 −1.00000 2.39659 1.40989
1.13 −1.00000 −2.30469 1.00000 4.02559 2.30469 −3.39321 −1.00000 2.31160 −4.02559
1.14 −1.00000 −2.29528 1.00000 2.67586 2.29528 −0.797143 −1.00000 2.26829 −2.67586
1.15 −1.00000 −2.17249 1.00000 −2.48902 2.17249 3.29516 −1.00000 1.71973 2.48902
1.16 −1.00000 −2.10025 1.00000 4.19656 2.10025 2.72635 −1.00000 1.41104 −4.19656
1.17 −1.00000 −1.93704 1.00000 −1.86801 1.93704 4.01320 −1.00000 0.752114 1.86801
1.18 −1.00000 −1.82817 1.00000 −4.13260 1.82817 2.43979 −1.00000 0.342212 4.13260
1.19 −1.00000 −1.76075 1.00000 0.351086 1.76075 −1.33291 −1.00000 0.100238 −0.351086
1.20 −1.00000 −1.66158 1.00000 −1.21846 1.66158 −1.75845 −1.00000 −0.239146 1.21846
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
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.g ✓ 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6046.2.a.g ✓ 69 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):

\( T_{3}^{69} - 153 T_{3}^{67} - 5 T_{3}^{66} + 11120 T_{3}^{65} + 729 T_{3}^{64} - 510967 T_{3}^{63} + \cdots - 4657414144 \) Copy content Toggle raw display
\( T_{11}^{69} - 42 T_{11}^{68} + 442 T_{11}^{67} + 5724 T_{11}^{66} - 149564 T_{11}^{65} + \cdots - 19\!\cdots\!50 \) Copy content Toggle raw display