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Properties

Label 8049.2.a.d
Level $8049$
Weight $2$
Character orbit 8049.a
Self dual yes
Analytic conductor $64.272$
Analytic rank $0$
Dimension $129$
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: \(0\)
Dimension: \(129\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 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 −2.82087 1.00000 5.95730 1.28710 −2.82087 2.58338 −11.1630 1.00000 −3.63075
1.2 −2.77720 1.00000 5.71284 2.12496 −2.77720 −3.21138 −10.3113 1.00000 −5.90145
1.3 −2.72806 1.00000 5.44232 −3.98613 −2.72806 −2.69921 −9.39088 1.00000 10.8744
1.4 −2.70395 1.00000 5.31133 1.76368 −2.70395 −4.33108 −8.95365 1.00000 −4.76891
1.5 −2.70347 1.00000 5.30878 −2.52602 −2.70347 3.06715 −8.94519 1.00000 6.82903
1.6 −2.67215 1.00000 5.14039 −2.79285 −2.67215 3.90889 −8.39161 1.00000 7.46293
1.7 −2.65892 1.00000 5.06986 3.58637 −2.65892 4.24083 −8.16251 1.00000 −9.53588
1.8 −2.65190 1.00000 5.03256 −2.74220 −2.65190 0.184989 −8.04204 1.00000 7.27204
1.9 −2.62578 1.00000 4.89474 −0.512843 −2.62578 1.48852 −7.60096 1.00000 1.34662
1.10 −2.58791 1.00000 4.69728 1.64365 −2.58791 −4.45906 −6.98031 1.00000 −4.25362
1.11 −2.57732 1.00000 4.64257 −1.35723 −2.57732 3.16714 −6.81075 1.00000 3.49801
1.12 −2.46621 1.00000 4.08220 1.94142 −2.46621 −0.674366 −5.13515 1.00000 −4.78795
1.13 −2.42396 1.00000 3.87556 −3.49875 −2.42396 3.65352 −4.54627 1.00000 8.48081
1.14 −2.31100 1.00000 3.34070 3.34170 −2.31100 4.45674 −3.09836 1.00000 −7.72265
1.15 −2.31049 1.00000 3.33834 3.79120 −2.31049 −2.69218 −3.09222 1.00000 −8.75950
1.16 −2.30919 1.00000 3.33237 −0.223778 −2.30919 3.94364 −3.07670 1.00000 0.516746
1.17 −2.21866 1.00000 2.92244 −3.24553 −2.21866 −2.52068 −2.04658 1.00000 7.20072
1.18 −2.20805 1.00000 2.87549 −1.04969 −2.20805 −2.20196 −1.93312 1.00000 2.31777
1.19 −2.16763 1.00000 2.69860 2.25786 −2.16763 0.531131 −1.51431 1.00000 −4.89419
1.20 −2.16635 1.00000 2.69306 3.84293 −2.16635 1.73846 −1.50141 1.00000 −8.32512
See next 80 embeddings (of 129 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.129
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.d ✓ 129
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8049.2.a.d ✓ 129 1.a even 1 1 trivial