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Properties

Label 8029.2.a.a
Level $8029$
Weight $2$
Character orbit 8029.a
Self dual yes
Analytic conductor $64.112$
Analytic rank $1$
Dimension $64$
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: \(1\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 7 q^{2} - 12 q^{3} + 57 q^{4} - 4 q^{5} - 9 q^{6} - 64 q^{7} - 21 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 7 q^{2} - 12 q^{3} + 57 q^{4} - 4 q^{5} - 9 q^{6} - 64 q^{7} - 21 q^{8} + 52 q^{9} + 4 q^{10} - 53 q^{11} - 11 q^{12} + 12 q^{13} + 7 q^{14} - 18 q^{15} + 55 q^{16} + 3 q^{17} + 2 q^{18} - 19 q^{19} - 13 q^{20} + 12 q^{21} - 6 q^{22} - 26 q^{23} - 27 q^{24} + 42 q^{25} - 39 q^{26} - 54 q^{27} - 57 q^{28} - 48 q^{29} - 38 q^{30} + 64 q^{31} - 44 q^{32} + 16 q^{33} - 8 q^{34} + 4 q^{35} + 20 q^{36} + 64 q^{37} - 15 q^{38} - 34 q^{39} + 31 q^{40} + 11 q^{41} + 9 q^{42} + 4 q^{43} - 114 q^{44} - 13 q^{45} - 32 q^{46} - 38 q^{47} + 5 q^{48} + 64 q^{49} - 44 q^{50} - 43 q^{51} + 35 q^{52} - 73 q^{53} + 17 q^{54} + 3 q^{55} + 21 q^{56} - 9 q^{57} + 20 q^{58} - 81 q^{59} + 3 q^{60} - 13 q^{61} - 7 q^{62} - 52 q^{63} + 27 q^{64} - 52 q^{65} - 53 q^{66} - 49 q^{67} - 3 q^{68} + 6 q^{69} - 4 q^{70} - 121 q^{71} - 96 q^{72} - 11 q^{73} - 7 q^{74} - 62 q^{75} - 29 q^{76} + 53 q^{77} - 23 q^{78} - 50 q^{79} - 56 q^{80} + 52 q^{81} + 48 q^{82} - 58 q^{83} + 11 q^{84} - 37 q^{85} - 142 q^{86} - 31 q^{87} + 39 q^{88} - 82 q^{89} + 137 q^{90} - 12 q^{91} - 66 q^{92} - 12 q^{93} - 23 q^{94} - 72 q^{95} + 2 q^{96} + 5 q^{97} - 7 q^{98} - 162 q^{99}+O(q^{100}) \) 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 −2.71962 0.182939 5.39631 4.05552 −0.497525 −1.00000 −9.23666 −2.96653 −11.0295
1.2 −2.67561 −3.38938 5.15887 −3.66710 9.06865 −1.00000 −8.45189 8.48790 9.81173
1.3 −2.65229 −1.92510 5.03463 −0.257069 5.10593 −1.00000 −8.04870 0.706024 0.681820
1.4 −2.63980 1.35549 4.96856 −3.80390 −3.57823 −1.00000 −7.83643 −1.16265 10.0415
1.5 −2.62214 1.29194 4.87563 −1.48824 −3.38764 −1.00000 −7.54030 −1.33090 3.90239
1.6 −2.51699 3.06733 4.33524 1.56860 −7.72044 −1.00000 −5.87777 6.40851 −3.94816
1.7 −2.46747 1.33466 4.08843 0.919105 −3.29323 −1.00000 −5.15314 −1.21869 −2.26787
1.8 −2.36785 −1.08907 3.60673 0.795724 2.57877 −1.00000 −3.80449 −1.81392 −1.88416
1.9 −2.34317 2.62506 3.49043 −2.59450 −6.15096 −1.00000 −3.49233 3.89095 6.07934
1.10 −2.17774 −0.753648 2.74253 1.70729 1.64124 −1.00000 −1.61704 −2.43202 −3.71803
1.11 −2.06088 −1.93212 2.24722 3.93468 3.98186 −1.00000 −0.509490 0.733082 −8.10889
1.12 −2.03869 0.0741722 2.15627 −0.776530 −0.151214 −1.00000 −0.318587 −2.99450 1.58311
1.13 −2.01575 −2.63523 2.06323 −0.573452 5.31195 −1.00000 −0.127463 3.94442 1.15593
1.14 −1.76035 2.23881 1.09883 1.43818 −3.94108 −1.00000 1.58638 2.01227 −2.53170
1.15 −1.72862 0.450561 0.988128 0.439298 −0.778850 −1.00000 1.74914 −2.79699 −0.759379
1.16 −1.72682 −1.84166 0.981900 −3.68492 3.18021 −1.00000 1.75807 0.391700 6.36318
1.17 −1.68849 −0.385117 0.850999 −2.67060 0.650267 −1.00000 1.94008 −2.85168 4.50928
1.18 −1.51927 2.10112 0.308194 2.68002 −3.19218 −1.00000 2.57032 1.41471 −4.07168
1.19 −1.38730 1.99496 −0.0753958 −3.29078 −2.76761 −1.00000 2.87920 0.979877 4.56530
1.20 −1.33398 −1.10643 −0.220500 −1.23549 1.47596 −1.00000 2.96210 −1.77581 1.64812
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
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.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8029.2.a.a ✓ 64 1.a even 1 1 trivial