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Properties

Label 59.14.a.a
Level $59$
Weight $14$
Character orbit 59.a
Self dual yes
Analytic conductor $63.266$
Analytic rank $1$
Dimension $29$
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] = [59,14,Mod(1,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 59.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: \(63.2662480816\)
Analytic rank: \(1\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 129 q^{2} + 98303 q^{4} - 41622 q^{5} - 193882 q^{6} - 669728 q^{7} - 1556487 q^{8} + 11248717 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - 129 q^{2} + 98303 q^{4} - 41622 q^{5} - 193882 q^{6} - 669728 q^{7} - 1556487 q^{8} + 11248717 q^{9} - 5227122 q^{10} - 7033776 q^{11} - 61963358 q^{13} - 21448901 q^{14} - 196556368 q^{15} + 406737043 q^{16} + 200994074 q^{17} + 6765579 q^{18} - 273375664 q^{19} - 1779481060 q^{20} - 2572011472 q^{21} - 4005875855 q^{22} - 719116792 q^{23} - 6296894550 q^{24} + 4529392999 q^{25} - 2125121489 q^{26} - 1233345096 q^{27} - 6135898397 q^{28} - 4899903454 q^{29} + 18107739780 q^{30} - 3659110376 q^{31} + 308923537 q^{32} + 2543994496 q^{33} + 1628246065 q^{34} + 27250670792 q^{35} + 132819166321 q^{36} - 31065281094 q^{37} + 46666711756 q^{38} - 53660669712 q^{39} + 11007153978 q^{40} - 8965773542 q^{41} + 61098087454 q^{42} - 73075727056 q^{43} - 93941401389 q^{44} - 172047916294 q^{45} - 153638353992 q^{46} - 126208822744 q^{47} + 153946572864 q^{48} + 58056076309 q^{49} - 420271687559 q^{50} - 356110768904 q^{51} - 1073886597511 q^{52} - 513098743318 q^{53} - 1077589129912 q^{54} - 480310362608 q^{55} - 933792123383 q^{56} - 857931045768 q^{57} - 802135690156 q^{58} - 1223235475589 q^{59} - 4926444586986 q^{60} - 2952177421946 q^{61} - 3898201613700 q^{62} - 1322707856604 q^{63} - 539003555949 q^{64} - 864809865500 q^{65} - 6001613857658 q^{66} - 543573734988 q^{67} + 1692259274719 q^{68} - 337480683728 q^{69} - 7408826965054 q^{70} - 3473464909104 q^{71} - 11669414031711 q^{72} - 4078986322982 q^{73} - 8136700217557 q^{74} - 2150153669900 q^{75} - 7492210804102 q^{76} - 11040669113100 q^{77} - 15044366058294 q^{78} - 10779125966160 q^{79} - 29424215142548 q^{80} - 13773188111387 q^{81} - 13319755484705 q^{82} - 20373347950440 q^{83} - 50341326289768 q^{84} - 36953338322292 q^{85} - 10672791242821 q^{86} - 30300626777488 q^{87} - 43025727091365 q^{88} - 33097577228886 q^{89} - 46091208899758 q^{90} - 26038247542172 q^{91} - 36040952030600 q^{92} - 44252999555352 q^{93} - 17835139070014 q^{94} - 47165611332684 q^{95} - 83595321433830 q^{96} - 37052800032182 q^{97} - 65781867303518 q^{98} - 46904533654640 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 −178.990 1981.36 23845.5 −48658.3 −354645. −263558. −2.80183e6 2.33148e6 8.70937e6
1.2 −157.045 −1913.87 16471.2 16159.5 300564. 493320. −1.30020e6 2.06858e6 −2.53778e6
1.3 −154.172 −658.796 15577.1 −35797.2 101568. 123605. −1.13858e6 −1.16031e6 5.51893e6
1.4 −145.363 1648.80 12938.4 3654.32 −239674. 128719. −689944. 1.12421e6 −531202.
1.5 −134.425 −1823.15 9878.14 62238.5 245078. −111285. −226660. 1.72957e6 −8.36642e6
1.6 −129.346 1706.06 8538.29 9203.97 −220672. −449232. −44790.7 1.31632e6 −1.19049e6
1.7 −107.030 35.1967 3263.33 −63050.4 −3767.09 −566995. 527514. −1.59308e6 6.74825e6
1.8 −100.856 −1438.29 1980.02 2360.69 145061. −240451. 626518. 474364. −238091.
1.9 −97.2926 −223.163 1273.85 35927.6 21712.2 568700. 673085. −1.54452e6 −3.49549e6
1.10 −91.0387 791.748 96.0440 45402.1 −72079.7 118122. 737045. −967458. −4.13335e6
1.11 −46.2512 998.167 −6052.83 −12552.5 −46166.4 113726. 658840. −597985. 580570.
1.12 −22.6399 −931.301 −7679.43 −54445.8 21084.6 259178. 359328. −727001. 1.23265e6
1.13 −21.5362 −986.018 −7728.19 13722.1 21235.1 −543189. 342861. −622091. −295523.
1.14 −6.75449 162.334 −8146.38 −44144.0 −1096.48 151956. 110357. −1.56797e6 298170.
1.15 −4.87123 1912.70 −8168.27 6333.31 −9317.21 −428028. 79694.6 2.06411e6 −30851.0
1.16 −1.38336 −910.252 −8190.09 63666.4 1259.21 60553.3 22662.4 −765765. −88073.7
1.17 7.38110 1757.01 −8137.52 9369.74 12968.7 335457. −120530. 1.49276e6 69159.0
1.18 17.2151 −2033.00 −7895.64 −41107.2 −34998.4 −288076. −276951. 2.53877e6 −707667.
1.19 63.5883 −1480.00 −4148.52 39600.4 −94110.5 118742. −784713. 596065. 2.51812e6
1.20 76.2835 227.253 −2372.82 50036.4 17335.6 −163679. −805922. −1.54268e6 3.81695e6
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(59\) \(1\)

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 59.14.a.a ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.14.a.a ✓ 29 1.a even 1 1 trivial