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Properties

Label 538.8.a.d
Level $538$
Weight $8$
Character orbit 538.a
Self dual yes
Analytic conductor $168.063$
Analytic rank $0$
Dimension $43$
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] = [538,8,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(168.063143710\)
Analytic rank: \(0\)
Dimension: \(43\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 43 q + 344 q^{2} + 123 q^{3} + 2752 q^{4} + 1249 q^{5} + 984 q^{6} + 2292 q^{7} + 22016 q^{8} + 38034 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 43 q + 344 q^{2} + 123 q^{3} + 2752 q^{4} + 1249 q^{5} + 984 q^{6} + 2292 q^{7} + 22016 q^{8} + 38034 q^{9} + 9992 q^{10} + 9236 q^{11} + 7872 q^{12} + 20734 q^{13} + 18336 q^{14} + 45412 q^{15} + 176128 q^{16} + 73650 q^{17} + 304272 q^{18} + 29172 q^{19} + 79936 q^{20} + 121999 q^{21} + 73888 q^{22} + 292888 q^{23} + 62976 q^{24} + 922452 q^{25} + 165872 q^{26} + 361146 q^{27} + 146688 q^{28} + 564477 q^{29} + 363296 q^{30} + 430638 q^{31} + 1409024 q^{32} + 443001 q^{33} + 589200 q^{34} + 933371 q^{35} + 2434176 q^{36} + 1358862 q^{37} + 233376 q^{38} + 1743349 q^{39} + 639488 q^{40} + 1869846 q^{41} + 975992 q^{42} + 1398712 q^{43} + 591104 q^{44} + 3495454 q^{45} + 2343104 q^{46} + 2157317 q^{47} + 503808 q^{48} + 7543237 q^{49} + 7379616 q^{50} + 230971 q^{51} + 1326976 q^{52} + 5925672 q^{53} + 2889168 q^{54} + 1107773 q^{55} + 1173504 q^{56} + 9117130 q^{57} + 4515816 q^{58} + 2165139 q^{59} + 2906368 q^{60} + 5047648 q^{61} + 3445104 q^{62} + 5691679 q^{63} + 11272192 q^{64} + 12361273 q^{65} + 3544008 q^{66} + 4415992 q^{67} + 4713600 q^{68} - 7538249 q^{69} + 7466968 q^{70} + 6241518 q^{71} + 19473408 q^{72} + 3473436 q^{73} + 10870896 q^{74} - 5305402 q^{75} + 1867008 q^{76} - 3658634 q^{77} + 13946792 q^{78} + 15661969 q^{79} + 5115904 q^{80} + 38608119 q^{81} + 14958768 q^{82} + 12126950 q^{83} + 7807936 q^{84} + 9453663 q^{85} + 11189696 q^{86} + 41966326 q^{87} + 4728832 q^{88} + 29273561 q^{89} + 27963632 q^{90} + 35476972 q^{91} + 18744832 q^{92} + 44382745 q^{93} + 17258536 q^{94} + 62134118 q^{95} + 4030464 q^{96} + 58118870 q^{97} + 60345896 q^{98} + 81437781 q^{99}+O(q^{100}) \) 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 8.00000 −86.9882 64.0000 −404.737 −695.905 −311.659 512.000 5379.94 −3237.89
1.2 8.00000 −85.3227 64.0000 −39.4502 −682.582 −1573.59 512.000 5092.96 −315.602
1.3 8.00000 −83.8072 64.0000 491.541 −670.458 −774.992 512.000 4836.65 3932.33
1.4 8.00000 −78.2289 64.0000 −435.660 −625.831 1442.05 512.000 3932.76 −3485.28
1.5 8.00000 −69.4454 64.0000 377.122 −555.563 −302.336 512.000 2635.66 3016.97
1.6 8.00000 −68.6733 64.0000 −92.2308 −549.387 −211.395 512.000 2529.03 −737.847
1.7 8.00000 −61.4104 64.0000 414.437 −491.283 1630.21 512.000 1584.24 3315.49
1.8 8.00000 −61.0758 64.0000 308.233 −488.606 −1086.67 512.000 1543.25 2465.87
1.9 8.00000 −56.8922 64.0000 −193.797 −455.137 485.218 512.000 1049.72 −1550.38
1.10 8.00000 −53.6923 64.0000 −214.599 −429.538 380.530 512.000 695.859 −1716.79
1.11 8.00000 −52.6428 64.0000 173.825 −421.143 701.621 512.000 584.267 1390.60
1.12 8.00000 −47.1657 64.0000 395.078 −377.326 1404.80 512.000 37.6026 3160.63
1.13 8.00000 −42.5136 64.0000 −385.072 −340.109 1083.70 512.000 −379.592 −3080.57
1.14 8.00000 −32.2549 64.0000 −211.678 −258.039 −876.962 512.000 −1146.62 −1693.43
1.15 8.00000 −30.6882 64.0000 83.2051 −245.506 52.0613 512.000 −1245.23 665.641
1.16 8.00000 −20.8703 64.0000 −401.676 −166.963 184.574 512.000 −1751.43 −3213.41
1.17 8.00000 −20.0835 64.0000 112.777 −160.668 −1635.26 512.000 −1783.65 902.217
1.18 8.00000 −16.9346 64.0000 −19.7537 −135.477 920.984 512.000 −1900.22 −158.030
1.19 8.00000 −15.5956 64.0000 406.109 −124.765 774.181 512.000 −1943.78 3248.87
1.20 8.00000 2.36922 64.0000 414.136 18.9538 −1162.79 512.000 −2181.39 3313.08
See all 43 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.43
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(269\) \( -1 \)

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 538.8.a.d ✓ 43
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.8.a.d ✓ 43 1.a even 1 1 trivial