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Properties

Label 8029.2.a.g
Level $8029$
Weight $2$
Character orbit 8029.a
Self dual yes
Analytic conductor $64.112$
Analytic rank $0$
Dimension $70$
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] = [8029,2,Mod(1,8029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8029, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8029 = 7 \cdot 31 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8029.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: \(64.1118877829\)
Analytic rank: \(0\)
Dimension: \(70\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 70 q + 5 q^{2} + 22 q^{3} + 71 q^{4} + 24 q^{5} + 9 q^{6} + 70 q^{7} + 9 q^{8} + 78 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q + 5 q^{2} + 22 q^{3} + 71 q^{4} + 24 q^{5} + 9 q^{6} + 70 q^{7} + 9 q^{8} + 78 q^{9} + 4 q^{10} + 61 q^{11} + 49 q^{12} + 28 q^{13} + 5 q^{14} + 22 q^{15} + 73 q^{16} + 37 q^{17} + 8 q^{18} + 23 q^{19} + 45 q^{20} + 22 q^{21} - 10 q^{22} + 26 q^{23} + 3 q^{24} + 66 q^{25} + 57 q^{26} + 76 q^{27} + 71 q^{28} + 38 q^{29} - 14 q^{30} + 70 q^{31} - 2 q^{32} + 44 q^{33} + 34 q^{34} + 24 q^{35} + 46 q^{36} + 70 q^{37} + 21 q^{38} + 10 q^{39} + 13 q^{40} + 71 q^{41} + 9 q^{42} + 30 q^{43} + 108 q^{44} + 13 q^{45} - 14 q^{46} + 78 q^{47} + 85 q^{48} + 70 q^{49} - 12 q^{50} + 21 q^{51} + 23 q^{52} + 47 q^{53} + 17 q^{54} + 5 q^{55} + 9 q^{56} + 9 q^{57} + 8 q^{58} + 109 q^{59} - q^{60} + 41 q^{61} + 5 q^{62} + 78 q^{63} + 29 q^{64} + 36 q^{65} + 5 q^{66} + 23 q^{67} + 47 q^{68} + 8 q^{69} + 4 q^{70} + 99 q^{71} + 8 q^{72} + 33 q^{73} + 5 q^{74} + 94 q^{75} - 19 q^{76} + 61 q^{77} + 37 q^{78} + 52 q^{79} + 78 q^{80} + 102 q^{81} + 118 q^{83} + 49 q^{84} - 21 q^{85} + 74 q^{86} + 11 q^{87} - 21 q^{88} + 86 q^{89} - 7 q^{90} + 28 q^{91} + 14 q^{92} + 22 q^{93} + 35 q^{94} + 24 q^{95} - 40 q^{96} + 9 q^{97} + 5 q^{98} + 92 q^{99}+O(q^{100}) \) 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 −2.75348 −0.719252 5.58165 0.633986 1.98045 1.00000 −9.86202 −2.48268 −1.74567
1.2 −2.70672 2.12991 5.32631 −2.23028 −5.76507 1.00000 −9.00339 1.53652 6.03675
1.3 −2.69962 2.88270 5.28797 3.78908 −7.78221 1.00000 −8.87628 5.30996 −10.2291
1.4 −2.54550 −1.20009 4.47956 2.75807 3.05483 1.00000 −6.31172 −1.55978 −7.02066
1.5 −2.49220 −0.243085 4.21105 3.85659 0.605816 1.00000 −5.51038 −2.94091 −9.61139
1.6 −2.46404 −1.44903 4.07148 −2.15186 3.57045 1.00000 −5.10420 −0.900323 5.30226
1.7 −2.35192 3.14960 3.53152 −3.21971 −7.40760 1.00000 −3.60200 6.91997 7.57250
1.8 −2.30530 0.494804 3.31440 1.43730 −1.14067 1.00000 −3.03008 −2.75517 −3.31341
1.9 −2.19859 1.85317 2.83381 −0.955285 −4.07436 1.00000 −1.83320 0.434225 2.10028
1.10 −2.18873 2.37642 2.79052 1.52040 −5.20134 1.00000 −1.73024 2.64738 −3.32774
1.11 −2.01064 −3.19206 2.04266 −1.79732 6.41808 1.00000 −0.0857692 7.18928 3.61376
1.12 −2.00414 −1.87169 2.01657 −1.61289 3.75112 1.00000 −0.0331998 0.503210 3.23245
1.13 −1.96118 1.44792 1.84625 −1.74883 −2.83964 1.00000 0.301540 −0.903529 3.42978
1.14 −1.91642 −2.33665 1.67267 3.67008 4.47800 1.00000 0.627304 2.45993 −7.03342
1.15 −1.91066 −1.87934 1.65063 −2.89304 3.59078 1.00000 0.667519 0.531906 5.52763
1.16 −1.74189 1.96861 1.03416 1.87277 −3.42909 1.00000 1.68238 0.875422 −3.26215
1.17 −1.48293 −0.919958 0.199089 1.22196 1.36424 1.00000 2.67063 −2.15368 −1.81208
1.18 −1.43732 3.17240 0.0658921 3.13251 −4.55975 1.00000 2.77993 7.06410 −4.50243
1.19 −1.39384 1.56439 −0.0572072 4.02287 −2.18052 1.00000 2.86742 −0.552671 −5.60723
1.20 −1.35296 0.0580579 −0.169502 −2.28030 −0.0785499 1.00000 2.93525 −2.99663 3.08516
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.70
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(31\) \(-1\)
\(37\) \(-1\)

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 8029.2.a.g ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8029.2.a.g ✓ 70 1.a even 1 1 trivial