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Properties

Label 2011.4.a.b
Level $2011$
Weight $4$
Character orbit 2011.a
Self dual yes
Analytic conductor $118.653$
Analytic rank $0$
Dimension $258$
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] = [2011,4,Mod(1,2011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2011.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2011 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2011.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: \(118.652841022\)
Analytic rank: \(0\)
Dimension: \(258\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 258 q + 28 q^{2} + 39 q^{3} + 1072 q^{4} + 249 q^{5} + 131 q^{6} + 64 q^{7} + 318 q^{8} + 2579 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 258 q + 28 q^{2} + 39 q^{3} + 1072 q^{4} + 249 q^{5} + 131 q^{6} + 64 q^{7} + 318 q^{8} + 2579 q^{9} + 297 q^{10} + 375 q^{11} + 507 q^{12} + 547 q^{13} + 841 q^{14} + 286 q^{15} + 4608 q^{16} + 1106 q^{17} + 740 q^{18} + 639 q^{19} + 2002 q^{20} + 1646 q^{21} + 288 q^{22} + 872 q^{23} + 1549 q^{24} + 7147 q^{25} + 1744 q^{26} + 1332 q^{27} + 598 q^{28} + 4915 q^{29} + 995 q^{30} + 1534 q^{31} + 2572 q^{32} + 1680 q^{33} + 993 q^{34} + 2192 q^{35} + 12373 q^{36} + 1495 q^{37} + 2535 q^{38} + 2516 q^{39} + 3041 q^{40} + 7118 q^{41} + 954 q^{42} + 1375 q^{43} + 4352 q^{44} + 6813 q^{45} + 1619 q^{46} + 2558 q^{47} + 4895 q^{48} + 15630 q^{49} + 4722 q^{50} + 3266 q^{51} + 3643 q^{52} + 4305 q^{53} + 4112 q^{54} + 2036 q^{55} + 9348 q^{56} + 2070 q^{57} + 1838 q^{58} + 7695 q^{59} + 4334 q^{60} + 8889 q^{61} + 6237 q^{62} + 3950 q^{63} + 21214 q^{64} + 5420 q^{65} + 4129 q^{66} + 2067 q^{67} + 11494 q^{68} + 14366 q^{69} + 3790 q^{70} + 5930 q^{71} + 8479 q^{72} + 5244 q^{73} + 9908 q^{74} + 4457 q^{75} + 6987 q^{76} + 12456 q^{77} + 3946 q^{78} + 5738 q^{79} + 15496 q^{80} + 28262 q^{81} + 4697 q^{82} + 5665 q^{83} + 13217 q^{84} + 7380 q^{85} + 8095 q^{86} + 6904 q^{87} + 5293 q^{88} + 10582 q^{89} + 14750 q^{90} + 3988 q^{91} + 10120 q^{92} + 4700 q^{93} + 13045 q^{94} + 12058 q^{95} + 18160 q^{96} + 5528 q^{97} + 12344 q^{98} + 8693 q^{99}+O(q^{100}) \) 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 −5.62746 −5.53739 23.6683 20.2935 31.1614 −28.4423 −88.1726 3.66269 −114.201
1.2 −5.48488 −4.60773 22.0839 −9.02063 25.2728 5.90719 −77.2483 −5.76883 49.4770
1.3 −5.46009 −4.78706 21.8126 4.54049 26.1378 −8.04989 −75.4183 −4.08410 −24.7915
1.4 −5.41804 5.24242 21.3552 18.0667 −28.4037 24.8695 −72.3591 0.483000 −97.8862
1.5 −5.40978 8.31528 21.2657 −9.65958 −44.9839 −15.6382 −71.7647 42.1439 52.2562
1.6 −5.39885 0.630647 21.1475 15.5449 −3.40477 0.664127 −70.9816 −26.6023 −83.9244
1.7 −5.39184 −2.98173 21.0719 5.01166 16.0770 25.2254 −70.4818 −18.1093 −27.0221
1.8 −5.37830 1.68358 20.9261 −20.5458 −9.05479 −5.85907 −69.5203 −24.1656 110.501
1.9 −5.35006 −10.1939 20.6231 4.88167 54.5380 −15.1252 −67.5343 76.9160 −26.1172
1.10 −5.33549 0.327025 20.4674 1.25949 −1.74484 −15.7384 −66.5198 −26.8931 −6.72000
1.11 −5.31589 2.72612 20.2587 16.2353 −14.4918 −3.11533 −65.1661 −19.5683 −86.3054
1.12 −5.27577 10.2516 19.8338 0.307468 −54.0851 5.56584 −62.4322 78.0954 −1.62213
1.13 −5.13351 −4.94736 18.3529 3.47195 25.3973 −5.13679 −53.1468 −2.52364 −17.8233
1.14 −5.07113 5.82339 17.7164 −7.59587 −29.5312 −6.98609 −49.2730 6.91186 38.5197
1.15 −5.03752 −0.438570 17.3766 10.8997 2.20931 −24.5011 −47.2349 −26.8077 −54.9072
1.16 −5.02385 2.42981 17.2390 −11.9617 −12.2070 22.8596 −46.4156 −21.0960 60.0935
1.17 −5.01366 −8.68657 17.1368 −16.3961 43.5516 −34.1999 −45.8090 48.4566 82.2046
1.18 −5.00470 7.22228 17.0470 1.35333 −36.1453 25.7323 −45.2773 25.1614 −6.77303
1.19 −5.00102 −9.66440 17.0102 −7.09530 48.3319 27.4900 −45.0601 66.4007 35.4838
1.20 −4.95174 −7.07181 16.5197 −11.8064 35.0178 −29.2151 −42.1876 23.0105 58.4625
See next 80 embeddings (of 258 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.258
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2011\) \(1\)

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 2011.4.a.b ✓ 258
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2011.4.a.b ✓ 258 1.a even 1 1 trivial