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Properties

Label 59.12.a.b
Level $59$
Weight $12$
Character orbit 59.a
Self dual yes
Analytic conductor $45.332$
Analytic rank $0$
Dimension $29$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(45.3322476530\)
Analytic rank: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q + 87 q^{2} + 36287 q^{4} + 6690 q^{5} + 21398 q^{6} + 85024 q^{7} + 187569 q^{8} + 2007973 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q + 87 q^{2} + 36287 q^{4} + 6690 q^{5} + 21398 q^{6} + 85024 q^{7} + 187569 q^{8} + 2007973 q^{9} + 950478 q^{10} + 622704 q^{11} + 5036266 q^{13} + 1730731 q^{14} - 5581744 q^{15} + 42790923 q^{16} + 23955578 q^{17} + 77439937 q^{18} + 12995184 q^{19} + 61465412 q^{20} + 77514608 q^{21} - 20501155 q^{22} + 7192184 q^{23} - 239360040 q^{24} + 313830783 q^{25} - 291189929 q^{26} - 181905432 q^{27} - 85268861 q^{28} - 32083126 q^{29} - 843420210 q^{30} + 334980904 q^{31} + 349440049 q^{32} + 684594688 q^{33} - 75966355 q^{34} - 156198184 q^{35} + 3407655761 q^{36} + 1791899762 q^{37} + 2569471048 q^{38} + 2067942032 q^{39} + 4438361712 q^{40} + 1278928138 q^{41} + 8300684368 q^{42} + 1473495312 q^{43} + 9147708849 q^{44} + 8686456418 q^{45} + 12009504888 q^{46} + 3367607192 q^{47} + 9917882072 q^{48} + 17632480421 q^{49} + 21850285297 q^{50} + 5677188456 q^{51} + 30523549023 q^{52} + 13556687426 q^{53} + 23217241582 q^{54} + 15901887728 q^{55} + 19840334257 q^{56} + 29191984824 q^{57} + 20284257022 q^{58} - 20732804671 q^{59} + 3824012210 q^{60} + 10774565854 q^{61} + 60973163100 q^{62} + 4013673308 q^{63} + 78667105399 q^{64} + 38200101988 q^{65} + 119508354062 q^{66} + 66911001404 q^{67} + 103233410647 q^{68} + 91883982128 q^{69} + 114891102466 q^{70} + 53717062896 q^{71} + 264540864763 q^{72} + 63777079114 q^{73} + 135769100867 q^{74} + 195304750556 q^{75} + 111450883698 q^{76} + 169933076580 q^{77} + 195949301338 q^{78} + 78656177360 q^{79} + 141321896752 q^{80} + 155954224853 q^{81} + 88268324465 q^{82} + 80666324040 q^{83} + 296626327288 q^{84} + 263882363292 q^{85} - 5704588981 q^{86} - 112463561776 q^{87} - 69746064345 q^{88} - 7454227014 q^{89} - 108560813034 q^{90} - 35449720212 q^{91} + 32056943980 q^{92} - 159915457016 q^{93} - 152381806226 q^{94} - 442358549364 q^{95} - 666216593600 q^{96} + 76602531466 q^{97} - 471058976984 q^{98} - 239033021840 q^{99}+O(q^{100}) \) 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 −87.1755 51.2929 5551.57 −8102.37 −4471.48 77101.4 −305425. −174516. 706328.
1.2 −83.5879 478.204 4938.93 2593.14 −39972.1 −41705.5 −241647. 51532.2 −216755.
1.3 −82.6823 −53.9577 4788.36 −1376.86 4461.34 −59108.3 −226579. −174236. 113842.
1.4 −71.7670 502.446 3102.51 13046.1 −36059.0 −12415.0 −75678.8 75304.7 −936283.
1.5 −66.3555 −768.568 2355.05 1599.91 50998.7 −35413.1 −20374.7 413550. −106163.
1.6 −60.2851 −556.447 1586.29 −6810.91 33545.4 49449.9 27834.4 132486. 410596.
1.7 −52.6379 −101.397 722.747 1306.93 5337.34 −69646.2 69758.5 −166866. −68794.0
1.8 −48.9701 207.557 350.073 −4497.54 −10164.1 −15653.3 83147.7 −134067. 220245.
1.9 −41.1818 −427.819 −352.061 1603.68 17618.4 67267.2 98838.8 5882.19 −66042.5
1.10 −38.1697 621.247 −591.075 3222.21 −23712.8 62852.4 100733. 208800. −122991.
1.11 −30.2749 −779.585 −1131.43 3374.86 23601.9 9974.13 96256.9 430606. −102173.
1.12 −23.1854 206.292 −1510.44 −13038.5 −4782.97 74436.9 82503.9 −134591. 302302.
1.13 −12.1507 −360.424 −1900.36 12255.1 4379.41 38038.4 47975.4 −47241.7 −148909.
1.14 −0.722438 446.814 −2047.48 −10481.4 −322.795 −77427.5 2958.73 22495.6 7572.15
1.15 9.55619 770.772 −1956.68 −5978.51 7365.64 30094.4 −38269.5 416943. −57131.8
1.16 11.5595 −276.373 −1914.38 −380.116 −3194.74 −43803.2 −45803.2 −100765. −4393.96
1.17 20.9426 180.797 −1609.41 6840.56 3786.35 −67266.6 −76595.5 −144459. 143259.
1.18 31.9034 −227.344 −1030.18 −3902.06 −7253.03 9078.58 −98204.2 −125462. −124489.
1.19 36.2796 701.365 −731.788 12160.6 25445.3 11534.4 −100850. 314766. 441181.
1.20 44.6553 −55.0108 −53.9055 −12501.5 −2456.52 −12083.7 −93861.2 −174121. −558260.
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 59.12.a.b ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.12.a.b ✓ 29 1.a even 1 1 trivial