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Properties

Label 59.14.a.b
Level $59$
Weight $14$
Character orbit 59.a
Self dual yes
Analytic conductor $63.266$
Analytic rank $0$
Dimension $35$
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: \(0\)
Dimension: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 63 q^{2} + 1458 q^{3} + 159743 q^{4} + 83378 q^{5} + 179366 q^{6} + 506762 q^{7} + 802809 q^{8} + 20814655 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q + 63 q^{2} + 1458 q^{3} + 159743 q^{4} + 83378 q^{5} + 179366 q^{6} + 506762 q^{7} + 802809 q^{8} + 20814655 q^{9} + 4772878 q^{10} + 10681834 q^{11} + 17915904 q^{12} + 44226440 q^{13} + 23728315 q^{14} + 203161175 q^{15} + 713058395 q^{16} - 198140645 q^{17} - 444942563 q^{18} + 809708762 q^{19} + 1849250492 q^{20} + 2385704495 q^{21} + 3313436917 q^{22} + 2206285272 q^{23} + 6790575156 q^{24} + 13016148077 q^{25} + 2694797991 q^{26} - 328278723 q^{27} + 1205137819 q^{28} + 3288782830 q^{29} - 19910071134 q^{30} + 981269400 q^{31} - 33409389319 q^{32} - 7911913598 q^{33} - 18609365403 q^{34} - 14067473881 q^{35} + 616120745 q^{36} + 36998144098 q^{37} - 84767303624 q^{38} - 12103277434 q^{39} + 6372217380 q^{40} + 20813698362 q^{41} - 147801423308 q^{42} + 94982956504 q^{43} + 22433398317 q^{44} + 132283375385 q^{45} + 241035138952 q^{46} - 7851236892 q^{47} + 501219024824 q^{48} + 718731987087 q^{49} + 473515847721 q^{50} + 440513366100 q^{51} + 801131090067 q^{52} + 326180862358 q^{53} + 884027766682 q^{54} + 819540399780 q^{55} + 1345932266057 q^{56} + 1661701566645 q^{57} + 1141277035202 q^{58} + 1476318677435 q^{59} + 5489622249218 q^{60} + 2296237018642 q^{61} + 4897460632572 q^{62} + 1490589296264 q^{63} + 6756824777903 q^{64} + 272122696104 q^{65} + 3229398343430 q^{66} - 460496416016 q^{67} + 222299339639 q^{68} + 2574891260924 q^{69} + 443093980858 q^{70} + 2635964894451 q^{71} + 7327602382759 q^{72} + 2765330319916 q^{73} + 7036676106883 q^{74} + 9114485505008 q^{75} + 20518171606306 q^{76} + 6895221903902 q^{77} + 25905875270986 q^{78} + 11635861231274 q^{79} + 35697025844376 q^{80} + 22941811701275 q^{81} + 12071119258061 q^{82} + 18499268196066 q^{83} + 47461536290584 q^{84} + 22216649482768 q^{85} + 38565978561003 q^{86} + 41714519349083 q^{87} + 42810563939151 q^{88} + 26100717697774 q^{89} + 63276939340518 q^{90} + 48415917253570 q^{91} + 53069360689820 q^{92} + 45938541501550 q^{93} + 49075065328342 q^{94} + 58795276476446 q^{95} + 95410873672132 q^{96} + 12621624962006 q^{97} + 42600357054028 q^{98} + 58636214702872 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 −179.892 961.746 24169.3 54934.5 −173011. 423569. −2.87419e6 −669367. −9.88230e6
1.2 −178.857 −1769.54 23797.7 −18829.8 316494. −348044. −2.79119e6 1.53694e6 3.36784e6
1.3 −156.966 89.9401 16446.2 41348.1 −14117.5 −529832. −1.29562e6 −1.58623e6 −6.49023e6
1.4 −150.743 −177.425 14531.4 15163.4 26745.6 −5353.43 −955626. −1.56284e6 −2.28578e6
1.5 −148.124 690.970 13748.8 −44831.0 −102349. 166511. −823093. −1.11688e6 6.64056e6
1.6 −126.745 2457.88 7872.42 41719.8 −311525. 140898. 40505.4 4.44683e6 −5.28780e6
1.7 −120.514 −2309.10 6331.66 −30115.2 278279. −383621. 224197. 3.73760e6 3.62931e6
1.8 −106.334 −1973.07 3114.93 −55720.1 209804. 393713. 539866. 2.29866e6 5.92495e6
1.9 −105.298 1465.27 2895.70 −40531.6 −154290. 580027. 557691. 552690. 4.26790e6
1.10 −93.5750 −715.291 564.289 24949.4 66933.4 9558.27 713763. −1.08268e6 −2.33465e6
1.11 −79.8727 1044.09 −1812.35 −16559.3 −83394.1 −261364. 799075. −504204. 1.32264e6
1.12 −66.4572 −891.820 −3775.44 −23945.4 59267.9 85271.8 795323. −798980. 1.59134e6
1.13 −62.8305 1175.30 −4244.33 65587.6 −73844.9 −260148. 781381. −212983. −4.12090e6
1.14 −61.5078 2362.88 −4408.78 −52520.3 −145336. −77351.1 775047. 3.98889e6 3.23041e6
1.15 −38.1436 −2114.94 −6737.07 34872.8 80671.5 −112871. 569448. 2.87865e6 −1.33017e6
1.16 −31.8283 −2202.43 −7178.96 17835.3 70099.7 452914. 489232. 3.25638e6 −567668.
1.17 −17.9012 2177.42 −7871.55 44148.8 −38978.4 357686. 287557. 3.14681e6 −790318.
1.18 −5.68524 −132.051 −8159.68 23423.0 750.739 −419839. 92963.2 −1.57689e6 −133165.
1.19 17.9022 −197.865 −7871.51 12906.8 −3542.23 510539. −287573. −1.55517e6 231060.
1.20 35.8192 1090.57 −6908.98 −59365.1 39063.3 −220063. −540905. −404985. −2.12641e6
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
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.b ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.14.a.b ✓ 35 1.a even 1 1 trivial