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Properties

Label 8049.2.a.c
Level $8049$
Weight $2$
Character orbit 8049.a
Self dual yes
Analytic conductor $64.272$
Analytic rank $0$
Dimension $119$
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: \(119\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9} - 10 q^{10} + 56 q^{11} - 137 q^{12} - 37 q^{13} + 31 q^{14} - 17 q^{15} + 173 q^{16} + 17 q^{17} + 11 q^{18} + 16 q^{19} + 61 q^{20} - 10 q^{21} - 3 q^{22} + 76 q^{23} - 33 q^{24} + 134 q^{25} + 47 q^{26} - 119 q^{27} - q^{28} + 47 q^{29} + 10 q^{30} + 51 q^{31} + 87 q^{32} - 56 q^{33} + 13 q^{34} + 58 q^{35} + 137 q^{36} - 67 q^{37} + 35 q^{38} + 37 q^{39} - 40 q^{40} + 47 q^{41} - 31 q^{42} + 12 q^{43} + 148 q^{44} + 17 q^{45} + 26 q^{46} + 107 q^{47} - 173 q^{48} + 163 q^{49} + 76 q^{50} - 17 q^{51} - 57 q^{52} + 64 q^{53} - 11 q^{54} + 71 q^{55} + 91 q^{56} - 16 q^{57} + 12 q^{58} + 98 q^{59} - 61 q^{60} - 50 q^{61} + 40 q^{62} + 10 q^{63} + 245 q^{64} + 40 q^{65} + 3 q^{66} + 12 q^{67} + 75 q^{68} - 76 q^{69} - 9 q^{70} + 194 q^{71} + 33 q^{72} - 79 q^{73} + 72 q^{74} - 134 q^{75} + 12 q^{76} + 71 q^{77} - 47 q^{78} + 127 q^{79} + 148 q^{80} + 119 q^{81} - 54 q^{82} + 77 q^{83} + q^{84} - 25 q^{85} + 142 q^{86} - 47 q^{87} + q^{88} + 93 q^{89} - 10 q^{90} + 61 q^{91} + 156 q^{92} - 51 q^{93} + 16 q^{94} + 138 q^{95} - 87 q^{96} - 110 q^{97} + 96 q^{98} + 56 q^{99}+O(q^{100}) \) 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 −2.76674 −1.00000 5.65483 −0.357353 2.76674 −2.67071 −10.1119 1.00000 0.988701
1.2 −2.73168 −1.00000 5.46207 −0.892345 2.73168 −1.57860 −9.45726 1.00000 2.43760
1.3 −2.70718 −1.00000 5.32884 −1.26891 2.70718 −3.95115 −9.01180 1.00000 3.43519
1.4 −2.67169 −1.00000 5.13794 4.23533 2.67169 3.62248 −8.38362 1.00000 −11.3155
1.5 −2.65333 −1.00000 5.04016 2.95577 2.65333 2.19565 −8.06655 1.00000 −7.84263
1.6 −2.62888 −1.00000 4.91100 3.06865 2.62888 −2.79906 −7.65266 1.00000 −8.06710
1.7 −2.61270 −1.00000 4.82618 1.51274 2.61270 3.06130 −7.38394 1.00000 −3.95232
1.8 −2.60392 −1.00000 4.78038 3.66153 2.60392 −3.52018 −7.23988 1.00000 −9.53431
1.9 −2.46430 −1.00000 4.07278 −0.321627 2.46430 1.46780 −5.10794 1.00000 0.792584
1.10 −2.39135 −1.00000 3.71855 −1.51157 2.39135 2.32148 −4.10966 1.00000 3.61469
1.11 −2.34737 −1.00000 3.51014 −3.81233 2.34737 −0.357303 −3.54486 1.00000 8.94895
1.12 −2.34205 −1.00000 3.48520 0.568796 2.34205 −0.877328 −3.47840 1.00000 −1.33215
1.13 −2.28998 −1.00000 3.24400 0.0628594 2.28998 −3.99468 −2.84873 1.00000 −0.143947
1.14 −2.26557 −1.00000 3.13283 0.270785 2.26557 1.54301 −2.56650 1.00000 −0.613482
1.15 −2.25432 −1.00000 3.08196 −2.79786 2.25432 −0.509718 −2.43909 1.00000 6.30728
1.16 −2.24285 −1.00000 3.03036 −3.65739 2.24285 −4.78600 −2.31094 1.00000 8.20296
1.17 −2.22986 −1.00000 2.97225 2.09159 2.22986 2.90452 −2.16799 1.00000 −4.66395
1.18 −2.22780 −1.00000 2.96308 4.08366 2.22780 2.32778 −2.14554 1.00000 −9.09757
1.19 −2.21831 −1.00000 2.92089 1.70107 2.21831 0.0884207 −2.04282 1.00000 −3.77350
1.20 −2.08408 −1.00000 2.34338 −1.55135 2.08408 4.26949 −0.715634 1.00000 3.23314
See next 80 embeddings (of 119 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.119
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.c ✓ 119
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8049.2.a.c ✓ 119 1.a even 1 1 trivial