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Properties

Label 79.8.a.b
Level $79$
Weight $8$
Character orbit 79.a
Self dual yes
Analytic conductor $24.678$
Analytic rank $0$
Dimension $25$
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] = [79,8,Mod(1,79)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, 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("79.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 79.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: \(24.6784170132\)
Analytic rank: \(0\)
Dimension: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 8 q^{2} + 107 q^{3} + 1728 q^{4} + 999 q^{5} + 1113 q^{6} + 883 q^{7} + 2334 q^{8} + 24630 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 8 q^{2} + 107 q^{3} + 1728 q^{4} + 999 q^{5} + 1113 q^{6} + 883 q^{7} + 2334 q^{8} + 24630 q^{9} + 9125 q^{10} + 7750 q^{11} + 15367 q^{12} + 2889 q^{13} + 19333 q^{14} + 33191 q^{15} + 142884 q^{16} + 78798 q^{17} - 3299 q^{18} + 30540 q^{19} - 12216 q^{20} - 132083 q^{21} - 284791 q^{22} + 164186 q^{23} + 128679 q^{24} + 508126 q^{25} + 505560 q^{26} + 607295 q^{27} + 536123 q^{28} + 638580 q^{29} + 2043027 q^{30} + 910298 q^{31} + 1629475 q^{32} + 598836 q^{33} + 1077090 q^{34} + 1581093 q^{35} + 4005767 q^{36} + 1211146 q^{37} + 1478978 q^{38} + 1344825 q^{39} + 3812582 q^{40} + 2816906 q^{41} + 4045603 q^{42} - 137148 q^{43} + 2136404 q^{44} + 2911464 q^{45} + 1092854 q^{46} + 2190469 q^{47} + 5884593 q^{48} + 6517102 q^{49} + 1975400 q^{50} + 751758 q^{51} - 201731 q^{52} + 4108064 q^{53} + 4068807 q^{54} + 868206 q^{55} + 1949841 q^{56} + 1601962 q^{57} - 2185482 q^{58} + 4346931 q^{59} - 4960425 q^{60} + 4437224 q^{61} - 1283135 q^{62} - 2298764 q^{63} + 6180676 q^{64} + 3307347 q^{65} - 13646860 q^{66} - 4982464 q^{67} + 5261328 q^{68} + 4906500 q^{69} - 20329017 q^{70} + 9728355 q^{71} - 32705097 q^{72} + 1445696 q^{73} - 1636246 q^{74} - 4906804 q^{75} - 9170621 q^{76} + 4420424 q^{77} - 9115731 q^{78} - 12325975 q^{79} - 18733905 q^{80} + 509769 q^{81} - 34706430 q^{82} - 22164078 q^{83} - 98610835 q^{84} - 43977968 q^{85} - 29164330 q^{86} - 57086618 q^{87} - 66593968 q^{88} + 20252339 q^{89} - 34814996 q^{90} - 25616983 q^{91} - 12368519 q^{92} - 35295820 q^{93} - 53030651 q^{94} - 18689280 q^{95} - 35962515 q^{96} + 15021627 q^{97} - 46370127 q^{98} + 5324944 q^{99}+O(q^{100}) \) 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 −21.2793 74.0188 324.811 −413.030 −1575.07 −1438.99 −4188.00 3291.79 8789.01
1.2 −20.1021 −81.6324 276.096 108.869 1640.99 1538.37 −2977.05 4476.85 −2188.50
1.3 −18.1336 −33.2751 200.828 −253.217 603.397 −67.6239 −1320.64 −1079.77 4591.74
1.4 −17.3981 79.4640 174.694 241.731 −1382.52 334.534 −812.386 4127.52 −4205.66
1.5 −17.0433 34.4011 162.475 −177.928 −586.310 87.6173 −587.576 −1003.56 3032.49
1.6 −14.2316 −74.8098 74.5388 395.145 1064.66 −669.744 760.839 3409.50 −5623.55
1.7 −13.5954 −42.8316 56.8345 78.6147 582.312 964.929 967.523 −352.452 −1068.80
1.8 −10.3985 −5.32354 −19.8720 436.140 55.3567 1538.77 1537.64 −2158.66 −4535.18
1.9 −9.53941 21.1568 −36.9996 −173.139 −201.823 −1786.48 1574.00 −1739.39 1651.64
1.10 −5.08366 55.6713 −102.156 260.915 −283.014 778.923 1170.04 912.298 −1326.41
1.11 −1.21112 −23.4568 −126.533 −160.207 28.4090 111.335 308.271 −1636.78 194.031
1.12 0.223229 −37.9960 −127.950 530.811 −8.48180 −1314.17 −57.1354 −743.301 118.492
1.13 0.779058 14.4867 −127.393 −535.379 11.2860 −397.866 −198.966 −1977.14 −417.091
1.14 0.876271 81.7023 −127.232 200.557 71.5933 −1593.69 −223.653 4488.26 175.742
1.15 4.43059 54.0622 −108.370 −154.136 239.527 1565.63 −1047.26 735.719 −682.911
1.16 7.03424 −63.7134 −78.5195 −161.566 −448.175 −1543.46 −1452.71 1872.40 −1136.50
1.17 8.71791 −29.3587 −51.9980 −180.532 −255.947 −300.254 −1569.21 −1325.07 −1573.86
1.18 10.7098 89.0191 −13.3010 480.043 953.373 990.409 −1513.30 5737.40 5141.15
1.19 11.1268 −45.0964 −4.19403 460.083 −501.779 776.179 −1470.90 −153.316 5119.26
1.20 13.4012 −76.8889 51.5933 −482.927 −1030.41 509.449 −1023.95 3724.90 −6471.82
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(79\) \(1\)

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 79.8.a.b ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.8.a.b ✓ 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 8 T_{2}^{24} - 2432 T_{2}^{23} + 18166 T_{2}^{22} + 2562247 T_{2}^{21} + \cdots + 52\!\cdots\!72 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(79))\). Copy content Toggle raw display