↑

Properties

Label 110.2.a.d.1.2
Level $110$
Weight $2$
Character 110.1
Self dual yes
Analytic conductor $0.878$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(-2.37228\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.37228 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.37228 q^{6} -2.37228 q^{7} -1.00000 q^{8} +2.62772 q^{9} -1.00000 q^{10} -1.00000 q^{11} +2.37228 q^{12} +2.00000 q^{13} +2.37228 q^{14} +2.37228 q^{15} +1.00000 q^{16} -4.37228 q^{17} -2.62772 q^{18} +6.37228 q^{19} +1.00000 q^{20} -5.62772 q^{21} +1.00000 q^{22} -8.74456 q^{23} -2.37228 q^{24} +1.00000 q^{25} -2.00000 q^{26} -0.883156 q^{27} -2.37228 q^{28} -4.37228 q^{29} -2.37228 q^{30} -2.37228 q^{31} -1.00000 q^{32} -2.37228 q^{33} +4.37228 q^{34} -2.37228 q^{35} +2.62772 q^{36} +3.62772 q^{37} -6.37228 q^{38} +4.74456 q^{39} -1.00000 q^{40} +11.4891 q^{41} +5.62772 q^{42} -4.00000 q^{43} -1.00000 q^{44} +2.62772 q^{45} +8.74456 q^{46} -8.74456 q^{47} +2.37228 q^{48} -1.37228 q^{49} -1.00000 q^{50} -10.3723 q^{51} +2.00000 q^{52} +13.1168 q^{53} +0.883156 q^{54} -1.00000 q^{55} +2.37228 q^{56} +15.1168 q^{57} +4.37228 q^{58} +8.74456 q^{59} +2.37228 q^{60} +0.372281 q^{61} +2.37228 q^{62} -6.23369 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.37228 q^{66} +8.00000 q^{67} -4.37228 q^{68} -20.7446 q^{69} +2.37228 q^{70} -7.11684 q^{71} -2.62772 q^{72} +7.48913 q^{73} -3.62772 q^{74} +2.37228 q^{75} +6.37228 q^{76} +2.37228 q^{77} -4.74456 q^{78} -12.7446 q^{79} +1.00000 q^{80} -9.97825 q^{81} -11.4891 q^{82} +8.74456 q^{83} -5.62772 q^{84} -4.37228 q^{85} +4.00000 q^{86} -10.3723 q^{87} +1.00000 q^{88} +4.37228 q^{89} -2.62772 q^{90} -4.74456 q^{91} -8.74456 q^{92} -5.62772 q^{93} +8.74456 q^{94} +6.37228 q^{95} -2.37228 q^{96} -1.25544 q^{97} +1.37228 q^{98} -2.62772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} - 2 q^{8} + 11 q^{9} - 2 q^{10} - 2 q^{11} - q^{12} + 4 q^{13} - q^{14} - q^{15} + 2 q^{16} - 3 q^{17} - 11 q^{18} + 7 q^{19} + 2 q^{20} - 17 q^{21}+ \cdots - 11 q^{99}+O(q^{100}) \) 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
\(3\) 2.37228 1.36964 0.684819 − 0.728714i \(-0.259881\pi\)
0.684819 + 0.728714i \(0.259881\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.37228 −0.968480
\(7\) −2.37228 −0.896638 −0.448319 − 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.62772 0.875906
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 2.37228 0.684819
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.37228 0.634019
\(15\) 2.37228 0.612520
\(16\) 1.00000 0.250000
\(17\) −4.37228 −1.06043 −0.530217 − 0.847862i \(-0.677890\pi\)
−0.530217 + 0.847862i \(0.677890\pi\)
\(18\) −2.62772 −0.619359
\(19\) 6.37228 1.46190 0.730951 − 0.682430i \(-0.239077\pi\)
0.730951 + 0.682430i \(0.239077\pi\)
\(20\) 1.00000 0.223607
\(21\) −5.62772 −1.22807
\(22\) 1.00000 0.213201
\(23\) −8.74456 −1.82337 −0.911684 − 0.410893i \(-0.865217\pi\)
−0.911684 + 0.410893i \(0.865217\pi\)
\(24\) −2.37228 −0.484240
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −0.883156 −0.169963
\(28\) −2.37228 −0.448319
\(29\) −4.37228 −0.811912 −0.405956 − 0.913893i \(-0.633061\pi\)
−0.405956 + 0.913893i \(0.633061\pi\)
\(30\) −2.37228 −0.433117
\(31\) −2.37228 −0.426074 −0.213037 − 0.977044i \(-0.568336\pi\)
−0.213037 + 0.977044i \(0.568336\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.37228 −0.412961
\(34\) 4.37228 0.749840
\(35\) −2.37228 −0.400989
\(36\) 2.62772 0.437953
\(37\) 3.62772 0.596393 0.298197 − 0.954504i \(-0.403615\pi\)
0.298197 + 0.954504i \(0.403615\pi\)
\(38\) −6.37228 −1.03372
\(39\) 4.74456 0.759738
\(40\) −1.00000 −0.158114
\(41\) 11.4891 1.79430 0.897150 − 0.441726i \(-0.145634\pi\)
0.897150 + 0.441726i \(0.145634\pi\)
\(42\) 5.62772 0.868376
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 2.62772 0.391717
\(46\) 8.74456 1.28932
\(47\) −8.74456 −1.27553 −0.637763 − 0.770233i \(-0.720140\pi\)
−0.637763 + 0.770233i \(0.720140\pi\)
\(48\) 2.37228 0.342409
\(49\) −1.37228 −0.196040
\(50\) −1.00000 −0.141421
\(51\) −10.3723 −1.45241
\(52\) 2.00000 0.277350
\(53\) 13.1168 1.80174 0.900869 − 0.434092i \(-0.142931\pi\)
0.900869 + 0.434092i \(0.142931\pi\)
\(54\) 0.883156 0.120182
\(55\) −1.00000 −0.134840
\(56\) 2.37228 0.317009
\(57\) 15.1168 2.00227
\(58\) 4.37228 0.574109
\(59\) 8.74456 1.13845 0.569223 − 0.822183i \(-0.307244\pi\)
0.569223 + 0.822183i \(0.307244\pi\)
\(60\) 2.37228 0.306260
\(61\) 0.372281 0.0476657 0.0238329 − 0.999716i \(-0.492413\pi\)
0.0238329 + 0.999716i \(0.492413\pi\)
\(62\) 2.37228 0.301280
\(63\) −6.23369 −0.785371
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 2.37228 0.292008
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −4.37228 −0.530217
\(69\) −20.7446 −2.49735
\(70\) 2.37228 0.283542
\(71\) −7.11684 −0.844614 −0.422307 − 0.906453i \(-0.638780\pi\)
−0.422307 + 0.906453i \(0.638780\pi\)
\(72\) −2.62772 −0.309680
\(73\) 7.48913 0.876536 0.438268 − 0.898844i \(-0.355592\pi\)
0.438268 + 0.898844i \(0.355592\pi\)
\(74\) −3.62772 −0.421714
\(75\) 2.37228 0.273927
\(76\) 6.37228 0.730951
\(77\) 2.37228 0.270347
\(78\) −4.74456 −0.537216
\(79\) −12.7446 −1.43388 −0.716938 − 0.697137i \(-0.754457\pi\)
−0.716938 + 0.697137i \(0.754457\pi\)
\(80\) 1.00000 0.111803
\(81\) −9.97825 −1.10869
\(82\) −11.4891 −1.26876
\(83\) 8.74456 0.959840 0.479920 − 0.877312i \(-0.340666\pi\)
0.479920 + 0.877312i \(0.340666\pi\)
\(84\) −5.62772 −0.614034
\(85\) −4.37228 −0.474240
\(86\) 4.00000 0.431331
\(87\) −10.3723 −1.11203
\(88\) 1.00000 0.106600
\(89\) 4.37228 0.463461 0.231730 − 0.972780i \(-0.425561\pi\)
0.231730 + 0.972780i \(0.425561\pi\)
\(90\) −2.62772 −0.276986
\(91\) −4.74456 −0.497365
\(92\) −8.74456 −0.911684
\(93\) −5.62772 −0.583567
\(94\) 8.74456 0.901933
\(95\) 6.37228 0.653782
\(96\) −2.37228 −0.242120
\(97\) −1.25544 −0.127470 −0.0637352 − 0.997967i \(-0.520301\pi\)
−0.0637352 + 0.997967i \(0.520301\pi\)
\(98\) 1.37228 0.138621
\(99\) −2.62772 −0.264096
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.1.2 ✓ 2
3.2 odd 2 990.2.a.m.1.1 2
4.3 odd 2 880.2.a.n.1.1 2
5.2 odd 4 550.2.b.f.199.1 4
5.3 odd 4 550.2.b.f.199.4 4
5.4 even 2 550.2.a.n.1.1 2
7.6 odd 2 5390.2.a.bp.1.1 2
8.3 odd 2 3520.2.a.bj.1.2 2
8.5 even 2 3520.2.a.bq.1.1 2
11.10 odd 2 1210.2.a.r.1.2 2
12.11 even 2 7920.2.a.bq.1.2 2
15.2 even 4 4950.2.c.bc.199.3 4
15.8 even 4 4950.2.c.bc.199.2 4
15.14 odd 2 4950.2.a.bw.1.2 2
20.3 even 4 4400.2.b.p.4049.2 4
20.7 even 4 4400.2.b.p.4049.3 4
20.19 odd 2 4400.2.a.bl.1.2 2
44.43 even 2 9680.2.a.bt.1.1 2
55.54 odd 2 6050.2.a.cb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.2 ✓ 2 1.1 even 1 trivial
550.2.a.n.1.1 2 5.4 even 2
550.2.b.f.199.1 4 5.2 odd 4
550.2.b.f.199.4 4 5.3 odd 4
880.2.a.n.1.1 2 4.3 odd 2
990.2.a.m.1.1 2 3.2 odd 2
1210.2.a.r.1.2 2 11.10 odd 2
3520.2.a.bj.1.2 2 8.3 odd 2
3520.2.a.bq.1.1 2 8.5 even 2
4400.2.a.bl.1.2 2 20.19 odd 2
4400.2.b.p.4049.2 4 20.3 even 4
4400.2.b.p.4049.3 4 20.7 even 4
4950.2.a.bw.1.2 2 15.14 odd 2
4950.2.c.bc.199.2 4 15.8 even 4
4950.2.c.bc.199.3 4 15.2 even 4
5390.2.a.bp.1.1 2 7.6 odd 2
6050.2.a.cb.1.1 2 55.54 odd 2
7920.2.a.bq.1.2 2 12.11 even 2
9680.2.a.bt.1.1 2 44.43 even 2