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Properties

Label 8049.2.a.a
Level $8049$
Weight $2$
Character orbit 8049.a
Self dual yes
Analytic conductor $64.272$
Analytic rank $1$
Dimension $95$
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] = [8049,2,Mod(1,8049)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8049, 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("8049.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8049 = 3 \cdot 2683 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8049.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: \(64.2715885869\)
Analytic rank: \(1\)
Dimension: \(95\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 95 q - 9 q^{2} + 95 q^{3} + 65 q^{4} - 15 q^{5} - 9 q^{6} - 36 q^{7} - 27 q^{8} + 95 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 95 q - 9 q^{2} + 95 q^{3} + 65 q^{4} - 15 q^{5} - 9 q^{6} - 36 q^{7} - 27 q^{8} + 95 q^{9} - 36 q^{10} - 48 q^{11} + 65 q^{12} - 73 q^{13} - 17 q^{14} - 15 q^{15} + 13 q^{16} - 9 q^{17} - 9 q^{18} - 66 q^{19} - 35 q^{20} - 36 q^{21} - 37 q^{22} - 58 q^{23} - 27 q^{24} + 24 q^{25} - 25 q^{26} + 95 q^{27} - 75 q^{28} - 31 q^{29} - 36 q^{30} - 129 q^{31} - 53 q^{32} - 48 q^{33} - 61 q^{34} - 38 q^{35} + 65 q^{36} - 127 q^{37} + q^{38} - 73 q^{39} - 74 q^{40} - 31 q^{41} - 17 q^{42} - 62 q^{43} - 76 q^{44} - 15 q^{45} - 60 q^{46} - 75 q^{47} + 13 q^{48} + 5 q^{49} - 30 q^{50} - 9 q^{51} - 137 q^{52} - 28 q^{53} - 9 q^{54} - 117 q^{55} - 23 q^{56} - 66 q^{57} - 90 q^{58} - 60 q^{59} - 35 q^{60} - 96 q^{61} + 10 q^{62} - 36 q^{63} - 75 q^{64} - 28 q^{65} - 37 q^{66} - 116 q^{67} + 3 q^{68} - 58 q^{69} - 73 q^{70} - 144 q^{71} - 27 q^{72} - 121 q^{73} - 16 q^{74} + 24 q^{75} - 118 q^{76} - 3 q^{77} - 25 q^{78} - 135 q^{79} - 36 q^{80} + 95 q^{81} - 102 q^{82} - 21 q^{83} - 75 q^{84} - 129 q^{85} - 46 q^{86} - 31 q^{87} - 77 q^{88} - 63 q^{89} - 36 q^{90} - 123 q^{91} - 42 q^{92} - 129 q^{93} - 44 q^{94} - 80 q^{95} - 53 q^{96} - 144 q^{97} + 10 q^{98} - 48 q^{99}+O(q^{100}) \) 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 −2.76182 1.00000 5.62766 1.63171 −2.76182 −0.505997 −10.0189 1.00000 −4.50650
1.2 −2.67709 1.00000 5.16680 1.53691 −2.67709 1.80566 −8.47779 1.00000 −4.11445
1.3 −2.56720 1.00000 4.59052 −3.13252 −2.56720 −0.347168 −6.65039 1.00000 8.04180
1.4 −2.55942 1.00000 4.55064 −0.0700614 −2.55942 −1.78635 −6.52816 1.00000 0.179317
1.5 −2.50719 1.00000 4.28600 −3.32561 −2.50719 −4.48246 −5.73145 1.00000 8.33794
1.6 −2.50349 1.00000 4.26747 3.53618 −2.50349 −0.351578 −5.67658 1.00000 −8.85279
1.7 −2.49336 1.00000 4.21683 −1.71250 −2.49336 −4.18090 −5.52734 1.00000 4.26986
1.8 −2.43614 1.00000 3.93477 0.978315 −2.43614 −0.911749 −4.71336 1.00000 −2.38331
1.9 −2.41240 1.00000 3.81966 4.15952 −2.41240 −1.40645 −4.38975 1.00000 −10.0344
1.10 −2.40517 1.00000 3.78485 0.506302 −2.40517 −0.320355 −4.29287 1.00000 −1.21774
1.11 −2.35419 1.00000 3.54222 −0.888451 −2.35419 0.154132 −3.63067 1.00000 2.09158
1.12 −2.32680 1.00000 3.41399 0.167407 −2.32680 5.19602 −3.29007 1.00000 −0.389523
1.13 −2.21332 1.00000 2.89880 −0.906600 −2.21332 −0.601026 −1.98934 1.00000 2.00660
1.14 −2.12258 1.00000 2.50534 −0.383524 −2.12258 2.66366 −1.07263 1.00000 0.814060
1.15 −1.98224 1.00000 1.92929 1.71154 −1.98224 −4.33478 0.140165 1.00000 −3.39268
1.16 −1.98008 1.00000 1.92072 −2.93482 −1.98008 2.42063 0.156981 1.00000 5.81119
1.17 −1.91515 1.00000 1.66780 2.10726 −1.91515 1.73510 0.636208 1.00000 −4.03573
1.18 −1.86344 1.00000 1.47242 −2.78098 −1.86344 2.31504 0.983118 1.00000 5.18220
1.19 −1.80117 1.00000 1.24420 2.31270 −1.80117 −0.312870 1.36132 1.00000 −4.16555
1.20 −1.77386 1.00000 1.14660 −0.218845 −1.77386 −4.33927 1.51382 1.00000 0.388201
See all 95 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.95
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2683\) \(-1\)

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 8049.2.a.a ✓ 95
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8049.2.a.a ✓ 95 1.a even 1 1 trivial