Properties

Label 6047.2.a.a
Level $6047$
Weight $2$
Character orbit 6047.a
Self dual yes
Analytic conductor $48.286$
Analytic rank $1$
Dimension $217$
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] = [6047,2,Mod(1,6047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6047, 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("6047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6047 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6047.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: \(48.2855381023\)
Analytic rank: \(1\)
Dimension: \(217\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 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 −2.79459 −1.21209 5.80974 −0.914621 3.38731 −4.48023 −10.6467 −1.53083 2.55599
1.2 −2.78935 0.603668 5.78045 3.11422 −1.68384 0.146622 −10.5450 −2.63558 −8.68663
1.3 −2.77438 −3.26218 5.69718 2.18536 9.05052 2.86186 −10.2574 7.64181 −6.06301
1.4 −2.74777 2.83106 5.55022 −0.268060 −7.77911 0.663291 −9.75518 5.01493 0.736566
1.5 −2.70650 0.217632 5.32513 0.669687 −0.589021 −0.581450 −8.99946 −2.95264 −1.81251
1.6 −2.67338 −1.23035 5.14695 0.526879 3.28919 −0.672464 −8.41297 −1.48624 −1.40855
1.7 −2.65722 0.109936 5.06084 0.753739 −0.292126 3.98656 −8.13334 −2.98791 −2.00285
1.8 −2.65451 −2.99510 5.04641 −2.08040 7.95052 −1.11556 −8.08670 5.97065 5.52243
1.9 −2.63907 −2.61338 4.96471 2.16033 6.89691 −4.93155 −7.82409 3.82978 −5.70128
1.10 −2.62503 1.59617 4.89080 2.83370 −4.19001 −4.08027 −7.58846 −0.452230 −7.43856
1.11 −2.59272 3.24215 4.72220 −1.12445 −8.40598 −0.119273 −7.05789 7.51153 2.91540
1.12 −2.57504 1.64822 4.63081 −3.44910 −4.24422 −0.596125 −6.77442 −0.283374 8.88154
1.13 −2.54610 −2.31345 4.48264 −3.02578 5.89027 −2.15804 −6.32106 2.35204 7.70396
1.14 −2.50842 0.797647 4.29216 −3.82675 −2.00083 −2.73995 −5.74970 −2.36376 9.59909
1.15 −2.50438 2.01082 4.27191 3.19046 −5.03585 0.790850 −5.68973 1.04339 −7.99013
1.16 −2.47362 −1.15677 4.11880 −1.98795 2.86140 3.96077 −5.24112 −1.66189 4.91743
1.17 −2.46440 −0.383920 4.07326 2.75346 0.946133 −1.51513 −5.10935 −2.85261 −6.78563
1.18 −2.46358 2.70933 4.06925 0.207418 −6.67465 −2.27877 −5.09776 4.34044 −0.510992
1.19 −2.45739 0.0299595 4.03876 −2.30057 −0.0736221 −3.67314 −5.01004 −2.99910 5.65341
1.20 −2.44753 −2.68992 3.99041 −1.41647 6.58366 −0.890122 −4.87158 4.23566 3.46686
See next 80 embeddings (of 217 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.217
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(6047\) \(1\)

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 6047.2.a.a 217
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6047.2.a.a 217 1.a even 1 1 trivial