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Properties

Label 71.10.a.a
Level $71$
Weight $10$
Character orbit 71.a
Self dual yes
Analytic conductor $36.568$
Analytic rank $1$
Dimension $23$
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] = [71,10,Mod(1,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 71.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: \(36.5675443676\)
Analytic rank: \(1\)
Dimension: \(23\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q - 89 q^{3} + 4778 q^{4} - 3297 q^{5} - 7194 q^{6} - 16808 q^{7} + 107630 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q - 89 q^{3} + 4778 q^{4} - 3297 q^{5} - 7194 q^{6} - 16808 q^{7} + 107630 q^{9} - 77330 q^{10} - 120006 q^{11} - 89088 q^{12} - 332244 q^{13} - 185088 q^{14} - 120528 q^{15} + 475298 q^{16} - 664854 q^{17} - 2623418 q^{18} - 2644545 q^{19} - 2673318 q^{20} - 1382784 q^{21} + 472690 q^{22} + 1060548 q^{23} + 3961146 q^{24} + 5272226 q^{25} + 5226526 q^{26} + 2389456 q^{27} - 4191426 q^{28} - 3040535 q^{29} - 5189324 q^{30} - 17219076 q^{31} - 14033440 q^{32} - 38661740 q^{33} - 40608446 q^{34} - 21166336 q^{35} - 52672148 q^{36} - 31470349 q^{37} - 67861104 q^{38} - 62228304 q^{39} - 146582126 q^{40} - 83781198 q^{41} - 170224082 q^{42} - 28402627 q^{43} - 234961370 q^{44} - 161931259 q^{45} - 88418480 q^{46} - 134283630 q^{47} - 296392336 q^{48} - 154160071 q^{49} - 211014684 q^{50} - 171710166 q^{51} - 329759324 q^{52} - 109027914 q^{53} - 592490382 q^{54} - 170947268 q^{55} - 406289904 q^{56} - 251834810 q^{57} - 321745818 q^{58} - 275264470 q^{59} - 657942858 q^{60} - 494204822 q^{61} - 427665380 q^{62} - 246129318 q^{63} - 477294142 q^{64} - 251178198 q^{65} - 271497042 q^{66} - 386359706 q^{67} - 141602446 q^{68} - 884855448 q^{69} - 393028542 q^{70} - 584468663 q^{71} - 926278080 q^{72} - 1404580743 q^{73} - 759812804 q^{74} - 1388106150 q^{75} - 1583505004 q^{76} - 770535522 q^{77} + 144309568 q^{78} - 832628317 q^{79} - 1006206658 q^{80} + 107366123 q^{81} - 259806114 q^{82} - 304290445 q^{83} + 647806000 q^{84} + 144255690 q^{85} + 1601734094 q^{86} + 3016497569 q^{87} + 270205346 q^{88} - 186810541 q^{89} + 5095873966 q^{90} + 1399930116 q^{91} + 4951239130 q^{92} + 2176102714 q^{93} + 755320388 q^{94} + 4981736848 q^{95} + 5553718210 q^{96} + 901409370 q^{97} + 7491072512 q^{98} + 4590908072 q^{99}+O(q^{100}) \) 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 −42.7100 −155.568 1312.14 1490.00 6644.31 −947.723 −34174.1 4518.41 −63637.9
1.2 −37.0063 54.2611 857.463 981.973 −2008.00 7593.19 −12784.3 −16738.7 −36339.2
1.3 −36.9739 −149.480 855.068 −2151.88 5526.84 −4750.80 −12684.6 2661.16 79563.2
1.4 −34.3259 270.307 666.267 −825.754 −9278.54 3375.82 −5295.36 53383.0 28344.8
1.5 −29.2011 −41.4090 340.704 2109.02 1209.19 −3658.60 5002.02 −17968.3 −61585.6
1.6 −28.7699 160.613 315.706 −2039.39 −4620.82 1090.79 5647.37 6113.58 58673.1
1.7 −17.6322 −65.1130 −201.107 −1203.92 1148.08 −337.662 12573.6 −15443.3 21227.8
1.8 −15.0899 194.495 −284.295 1965.55 −2934.91 −7288.43 12016.0 18145.3 −29660.0
1.9 −10.3508 230.496 −404.861 −998.774 −2385.82 −4562.58 9490.24 33445.6 10338.1
1.10 −9.43874 −190.922 −422.910 680.126 1802.06 −9665.61 8824.37 16768.0 −6419.53
1.11 −7.42582 −231.758 −456.857 −2654.96 1720.99 −2455.28 7194.56 34028.6 19715.2
1.12 −1.10214 50.1839 −510.785 1619.54 −55.3095 2861.03 1127.25 −17164.6 −1784.95
1.13 2.95140 26.6415 −503.289 −39.8458 78.6299 6117.24 −2996.53 −18973.2 −117.601
1.14 3.85845 −201.520 −497.112 −802.464 −777.553 8229.19 −3893.61 20927.2 −3096.26
1.15 17.2836 141.369 −213.277 709.327 2443.37 −65.9924 −12535.4 302.257 12259.7
1.16 19.4351 −253.652 −134.277 1310.07 −4929.75 −454.949 −12560.5 44656.3 25461.4
1.17 19.8852 186.421 −116.578 −1012.99 3707.03 −1006.54 −12499.4 15069.9 −20143.5
1.18 28.0969 −30.6154 277.438 1963.96 −860.199 −5458.23 −6590.47 −18745.7 55181.4
1.19 31.3039 39.0235 467.935 −1003.34 1221.59 6330.93 −1379.40 −18160.2 −31408.6
1.20 32.3734 142.521 536.040 −986.027 4613.89 −11750.5 778.260 629.149 −31921.1
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(71\) \(1\)

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 71.10.a.a ✓ 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.10.a.a ✓ 23 1.a even 1 1 trivial