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Properties

Label 110.2.a.d.1.1
Level $110$
Weight $2$
Character 110.1
Self dual yes
Analytic conductor $0.878$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [110,2,Mod(1,110)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("110.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 110.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.878354422234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 110.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.37228 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.37228 q^{6} +3.37228 q^{7} -1.00000 q^{8} +8.37228 q^{9} -1.00000 q^{10} -1.00000 q^{11} -3.37228 q^{12} +2.00000 q^{13} -3.37228 q^{14} -3.37228 q^{15} +1.00000 q^{16} +1.37228 q^{17} -8.37228 q^{18} +0.627719 q^{19} +1.00000 q^{20} -11.3723 q^{21} +1.00000 q^{22} +2.74456 q^{23} +3.37228 q^{24} +1.00000 q^{25} -2.00000 q^{26} -18.1168 q^{27} +3.37228 q^{28} +1.37228 q^{29} +3.37228 q^{30} +3.37228 q^{31} -1.00000 q^{32} +3.37228 q^{33} -1.37228 q^{34} +3.37228 q^{35} +8.37228 q^{36} +9.37228 q^{37} -0.627719 q^{38} -6.74456 q^{39} -1.00000 q^{40} -11.4891 q^{41} +11.3723 q^{42} -4.00000 q^{43} -1.00000 q^{44} +8.37228 q^{45} -2.74456 q^{46} +2.74456 q^{47} -3.37228 q^{48} +4.37228 q^{49} -1.00000 q^{50} -4.62772 q^{51} +2.00000 q^{52} -4.11684 q^{53} +18.1168 q^{54} -1.00000 q^{55} -3.37228 q^{56} -2.11684 q^{57} -1.37228 q^{58} -2.74456 q^{59} -3.37228 q^{60} -5.37228 q^{61} -3.37228 q^{62} +28.2337 q^{63} +1.00000 q^{64} +2.00000 q^{65} -3.37228 q^{66} +8.00000 q^{67} +1.37228 q^{68} -9.25544 q^{69} -3.37228 q^{70} +10.1168 q^{71} -8.37228 q^{72} -15.4891 q^{73} -9.37228 q^{74} -3.37228 q^{75} +0.627719 q^{76} -3.37228 q^{77} +6.74456 q^{78} -1.25544 q^{79} +1.00000 q^{80} +35.9783 q^{81} +11.4891 q^{82} -2.74456 q^{83} -11.3723 q^{84} +1.37228 q^{85} +4.00000 q^{86} -4.62772 q^{87} +1.00000 q^{88} -1.37228 q^{89} -8.37228 q^{90} +6.74456 q^{91} +2.74456 q^{92} -11.3723 q^{93} -2.74456 q^{94} +0.627719 q^{95} +3.37228 q^{96} -12.7446 q^{97} -4.37228 q^{98} -8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} - 2 q^{8} + 11 q^{9} - 2 q^{10} - 2 q^{11} - q^{12} + 4 q^{13} - q^{14} - q^{15} + 2 q^{16} - 3 q^{17} - 11 q^{18} + 7 q^{19} + 2 q^{20} - 17 q^{21}+ \cdots - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −3.37228 −1.94699 −0.973494 − 0.228714i \(-0.926548\pi\)
−0.973494 + 0.228714i \(0.926548\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 3.37228 1.37673
\(7\) 3.37228 1.27460 0.637301 − 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) −1.00000 −0.353553
\(9\) 8.37228 2.79076
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) −3.37228 −0.973494
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −3.37228 −0.901280
\(15\) −3.37228 −0.870719
\(16\) 1.00000 0.250000
\(17\) 1.37228 0.332827 0.166414 − 0.986056i \(-0.446781\pi\)
0.166414 + 0.986056i \(0.446781\pi\)
\(18\) −8.37228 −1.97337
\(19\) 0.627719 0.144009 0.0720043 − 0.997404i \(-0.477060\pi\)
0.0720043 + 0.997404i \(0.477060\pi\)
\(20\) 1.00000 0.223607
\(21\) −11.3723 −2.48164
\(22\) 1.00000 0.213201
\(23\) 2.74456 0.572281 0.286140 − 0.958188i \(-0.407628\pi\)
0.286140 + 0.958188i \(0.407628\pi\)
\(24\) 3.37228 0.688364
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −18.1168 −3.48659
\(28\) 3.37228 0.637301
\(29\) 1.37228 0.254826 0.127413 − 0.991850i \(-0.459333\pi\)
0.127413 + 0.991850i \(0.459333\pi\)
\(30\) 3.37228 0.615692
\(31\) 3.37228 0.605680 0.302840 − 0.953041i \(-0.402065\pi\)
0.302840 + 0.953041i \(0.402065\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.37228 0.587039
\(34\) −1.37228 −0.235344
\(35\) 3.37228 0.570020
\(36\) 8.37228 1.39538
\(37\) 9.37228 1.54079 0.770397 − 0.637565i \(-0.220058\pi\)
0.770397 + 0.637565i \(0.220058\pi\)
\(38\) −0.627719 −0.101829
\(39\) −6.74456 −1.07999
\(40\) −1.00000 −0.158114
\(41\) −11.4891 −1.79430 −0.897150 − 0.441726i \(-0.854366\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 11.3723 1.75478
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 8.37228 1.24807
\(46\) −2.74456 −0.404664
\(47\) 2.74456 0.400336 0.200168 − 0.979762i \(-0.435851\pi\)
0.200168 + 0.979762i \(0.435851\pi\)
\(48\) −3.37228 −0.486747
\(49\) 4.37228 0.624612
\(50\) −1.00000 −0.141421
\(51\) −4.62772 −0.648010
\(52\) 2.00000 0.277350
\(53\) −4.11684 −0.565492 −0.282746 − 0.959195i \(-0.591245\pi\)
−0.282746 + 0.959195i \(0.591245\pi\)
\(54\) 18.1168 2.46539
\(55\) −1.00000 −0.134840
\(56\) −3.37228 −0.450640
\(57\) −2.11684 −0.280383
\(58\) −1.37228 −0.180189
\(59\) −2.74456 −0.357312 −0.178656 − 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) −3.37228 −0.435360
\(61\) −5.37228 −0.687850 −0.343925 − 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) −3.37228 −0.428280
\(63\) 28.2337 3.55711
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −3.37228 −0.415099
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 1.37228 0.166414
\(69\) −9.25544 −1.11422
\(70\) −3.37228 −0.403065
\(71\) 10.1168 1.20065 0.600324 − 0.799757i \(-0.295038\pi\)
0.600324 + 0.799757i \(0.295038\pi\)
\(72\) −8.37228 −0.986683
\(73\) −15.4891 −1.81286 −0.906432 − 0.422351i \(-0.861205\pi\)
−0.906432 + 0.422351i \(0.861205\pi\)
\(74\) −9.37228 −1.08951
\(75\) −3.37228 −0.389398
\(76\) 0.627719 0.0720043
\(77\) −3.37228 −0.384307
\(78\) 6.74456 0.763671
\(79\) −1.25544 −0.141248 −0.0706239 − 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) 1.00000 0.111803
\(81\) 35.9783 3.99758
\(82\) 11.4891 1.26876
\(83\) −2.74456 −0.301255 −0.150627 − 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(84\) −11.3723 −1.24082
\(85\) 1.37228 0.148845
\(86\) 4.00000 0.431331
\(87\) −4.62772 −0.496144
\(88\) 1.00000 0.106600
\(89\) −1.37228 −0.145462 −0.0727308 − 0.997352i \(-0.523171\pi\)
−0.0727308 + 0.997352i \(0.523171\pi\)
\(90\) −8.37228 −0.882516
\(91\) 6.74456 0.707022
\(92\) 2.74456 0.286140
\(93\) −11.3723 −1.17925
\(94\) −2.74456 −0.283080
\(95\) 0.627719 0.0644026
\(96\) 3.37228 0.344182
\(97\) −12.7446 −1.29401 −0.647007 − 0.762484i \(-0.723980\pi\)
−0.647007 + 0.762484i \(0.723980\pi\)
\(98\) −4.37228 −0.441667
\(99\) −8.37228 −0.841446
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 110.2.a.d.1.1 ✓ 2
3.2 odd 2 990.2.a.m.1.2 2
4.3 odd 2 880.2.a.n.1.2 2
5.2 odd 4 550.2.b.f.199.2 4
5.3 odd 4 550.2.b.f.199.3 4
5.4 even 2 550.2.a.n.1.2 2
7.6 odd 2 5390.2.a.bp.1.2 2
8.3 odd 2 3520.2.a.bj.1.1 2
8.5 even 2 3520.2.a.bq.1.2 2
11.10 odd 2 1210.2.a.r.1.1 2
12.11 even 2 7920.2.a.bq.1.1 2
15.2 even 4 4950.2.c.bc.199.4 4
15.8 even 4 4950.2.c.bc.199.1 4
15.14 odd 2 4950.2.a.bw.1.1 2
20.3 even 4 4400.2.b.p.4049.4 4
20.7 even 4 4400.2.b.p.4049.1 4
20.19 odd 2 4400.2.a.bl.1.1 2
44.43 even 2 9680.2.a.bt.1.2 2
55.54 odd 2 6050.2.a.cb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 ✓ 2 1.1 even 1 trivial
550.2.a.n.1.2 2 5.4 even 2
550.2.b.f.199.2 4 5.2 odd 4
550.2.b.f.199.3 4 5.3 odd 4
880.2.a.n.1.2 2 4.3 odd 2
990.2.a.m.1.2 2 3.2 odd 2
1210.2.a.r.1.1 2 11.10 odd 2
3520.2.a.bj.1.1 2 8.3 odd 2
3520.2.a.bq.1.2 2 8.5 even 2
4400.2.a.bl.1.1 2 20.19 odd 2
4400.2.b.p.4049.1 4 20.7 even 4
4400.2.b.p.4049.4 4 20.3 even 4
4950.2.a.bw.1.1 2 15.14 odd 2
4950.2.c.bc.199.1 4 15.8 even 4
4950.2.c.bc.199.4 4 15.2 even 4
5390.2.a.bp.1.2 2 7.6 odd 2
6050.2.a.cb.1.2 2 55.54 odd 2
7920.2.a.bq.1.1 2 12.11 even 2
9680.2.a.bt.1.2 2 44.43 even 2