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Properties

Label 110.2.f.b.87.1
Level $110$
Weight $2$
Character 110.87
Analytic conductor $0.878$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(43,110)] 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("110.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 110.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.878354422234\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 87.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 110.87
Dual form 110.2.f.b.43.1

$q$-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+(-0.707107 + 0.707107i) q^{2} +(1.70711 - 1.70711i) q^{3} -1.00000i q^{4} +(-0.707107 + 2.12132i) q^{5} +2.41421i q^{6} +(2.12132 - 2.12132i) q^{7} +(0.707107 + 0.707107i) q^{8} -2.82843i q^{9} +(-1.00000 - 2.00000i) q^{10} +(-1.41421 - 3.00000i) q^{11} +(-1.70711 - 1.70711i) q^{12} +(3.00000 + 3.00000i) q^{13} +3.00000i q^{14} +(2.41421 + 4.82843i) q^{15} -1.00000 q^{16} +(-5.12132 + 5.12132i) q^{17} +(2.00000 + 2.00000i) q^{18} -3.00000 q^{19} +(2.12132 + 0.707107i) q^{20} -7.24264i q^{21} +(3.12132 + 1.12132i) q^{22} +(-0.171573 + 0.171573i) q^{23} +2.41421 q^{24} +(-4.00000 - 3.00000i) q^{25} -4.24264 q^{26} +(0.292893 + 0.292893i) q^{27} +(-2.12132 - 2.12132i) q^{28} -1.24264 q^{29} +(-5.12132 - 1.70711i) q^{30} -7.24264 q^{31} +(0.707107 - 0.707107i) q^{32} +(-7.53553 - 2.70711i) q^{33} -7.24264i q^{34} +(3.00000 + 6.00000i) q^{35} -2.82843 q^{36} +(0.121320 + 0.121320i) q^{37} +(2.12132 - 2.12132i) q^{38} +10.2426 q^{39} +(-2.00000 + 1.00000i) q^{40} -1.75736i q^{41} +(5.12132 + 5.12132i) q^{42} +(1.24264 + 1.24264i) q^{43} +(-3.00000 + 1.41421i) q^{44} +(6.00000 + 2.00000i) q^{45} -0.242641i q^{46} +(4.41421 + 4.41421i) q^{47} +(-1.70711 + 1.70711i) q^{48} -2.00000i q^{49} +(4.94975 - 0.707107i) q^{50} +17.4853i q^{51} +(3.00000 - 3.00000i) q^{52} +(9.53553 - 9.53553i) q^{53} -0.414214 q^{54} +(7.36396 - 0.878680i) q^{55} +3.00000 q^{56} +(-5.12132 + 5.12132i) q^{57} +(0.878680 - 0.878680i) q^{58} +1.41421i q^{59} +(4.82843 - 2.41421i) q^{60} -7.24264i q^{61} +(5.12132 - 5.12132i) q^{62} +(-6.00000 - 6.00000i) q^{63} +1.00000i q^{64} +(-8.48528 + 4.24264i) q^{65} +(7.24264 - 3.41421i) q^{66} +(-4.00000 - 4.00000i) q^{67} +(5.12132 + 5.12132i) q^{68} +0.585786i q^{69} +(-6.36396 - 2.12132i) q^{70} -1.24264 q^{71} +(2.00000 - 2.00000i) q^{72} +(6.00000 + 6.00000i) q^{73} -0.171573 q^{74} +(-11.9497 + 1.70711i) q^{75} +3.00000i q^{76} +(-9.36396 - 3.36396i) q^{77} +(-7.24264 + 7.24264i) q^{78} +10.2426 q^{79} +(0.707107 - 2.12132i) q^{80} +9.48528 q^{81} +(1.24264 + 1.24264i) q^{82} +(-7.24264 - 7.24264i) q^{83} -7.24264 q^{84} +(-7.24264 - 14.4853i) q^{85} -1.75736 q^{86} +(-2.12132 + 2.12132i) q^{87} +(1.12132 - 3.12132i) q^{88} +5.48528i q^{89} +(-5.65685 + 2.82843i) q^{90} +12.7279 q^{91} +(0.171573 + 0.171573i) q^{92} +(-12.3640 + 12.3640i) q^{93} -6.24264 q^{94} +(2.12132 - 6.36396i) q^{95} -2.41421i q^{96} +(-2.24264 - 2.24264i) q^{97} +(1.41421 + 1.41421i) q^{98} +(-8.48528 + 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{10} - 4 q^{12} + 12 q^{13} + 4 q^{15} - 4 q^{16} - 12 q^{17} + 8 q^{18} - 12 q^{19} + 4 q^{22} - 12 q^{23} + 4 q^{24} - 16 q^{25} + 4 q^{27} + 12 q^{29} - 12 q^{30} - 12 q^{31} - 16 q^{33}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.500000 + 0.500000i
\(3\) 1.70711 − 1.70711i 0.985599 − 0.985599i −0.0142992 − 0.999898i \(-0.504552\pi\)
0.999898 + 0.0142992i \(0.00455173\pi\)
\(4\) − 1.00000i − 0.500000i
\(5\) −0.707107 + 2.12132i −0.316228 + 0.948683i
\(6\) 2.41421i 0.985599i
\(7\) 2.12132 − 2.12132i 0.801784 − 0.801784i −0.181591 − 0.983374i \(-0.558125\pi\)
0.983374 + 0.181591i \(0.0581245\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) − 2.82843i − 0.942809i
\(10\) −1.00000 − 2.00000i −0.316228 − 0.632456i
\(11\) −1.41421 − 3.00000i −0.426401 − 0.904534i
\(12\) −1.70711 − 1.70711i −0.492799 − 0.492799i
\(13\) 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 − 0.155747i \(-0.0497784\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 3.00000i 0.801784i
\(15\) 2.41421 + 4.82843i 0.623347 + 1.24669i
\(16\) −1.00000 −0.250000
\(17\) −5.12132 + 5.12132i −1.24210 + 1.24210i −0.282975 + 0.959127i \(0.591322\pi\)
−0.959127 + 0.282975i \(0.908678\pi\)
\(18\) 2.00000 + 2.00000i 0.471405 + 0.471405i
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 2.12132 + 0.707107i 0.474342 + 0.158114i
\(21\) − 7.24264i − 1.58047i
\(22\) 3.12132 + 1.12132i 0.665468 + 0.239066i
\(23\) −0.171573 + 0.171573i −0.0357754 + 0.0357754i −0.724768 − 0.688993i \(-0.758053\pi\)
0.688993 + 0.724768i \(0.258053\pi\)
\(24\) 2.41421 0.492799
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) −4.24264 −0.832050
\(27\) 0.292893 + 0.292893i 0.0563673 + 0.0563673i
\(28\) −2.12132 − 2.12132i −0.400892 − 0.400892i
\(29\) −1.24264 −0.230753 −0.115376 − 0.993322i \(-0.536807\pi\)
−0.115376 + 0.993322i \(0.536807\pi\)
\(30\) −5.12132 − 1.70711i −0.935021 − 0.311674i
\(31\) −7.24264 −1.30082 −0.650408 − 0.759585i \(-0.725402\pi\)
−0.650408 + 0.759585i \(0.725402\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) −7.53553 − 2.70711i −1.31177 − 0.471247i
\(34\) − 7.24264i − 1.24210i
\(35\) 3.00000 + 6.00000i 0.507093 + 1.01419i
\(36\) −2.82843 −0.471405
\(37\) 0.121320 + 0.121320i 0.0199449 + 0.0199449i 0.717009 − 0.697064i \(-0.245511\pi\)
−0.697064 + 0.717009i \(0.745511\pi\)
\(38\) 2.12132 − 2.12132i 0.344124 − 0.344124i
\(39\) 10.2426 1.64014
\(40\) −2.00000 + 1.00000i −0.316228 + 0.158114i
\(41\) − 1.75736i − 0.274453i −0.990540 − 0.137227i \(-0.956181\pi\)
0.990540 − 0.137227i \(-0.0438189\pi\)
\(42\) 5.12132 + 5.12132i 0.790237 + 0.790237i
\(43\) 1.24264 + 1.24264i 0.189501 + 0.189501i 0.795480 − 0.605979i \(-0.207219\pi\)
−0.605979 + 0.795480i \(0.707219\pi\)
\(44\) −3.00000 + 1.41421i −0.452267 + 0.213201i
\(45\) 6.00000 + 2.00000i 0.894427 + 0.298142i
\(46\) − 0.242641i − 0.0357754i
\(47\) 4.41421 + 4.41421i 0.643879 + 0.643879i 0.951507 − 0.307628i \(-0.0995351\pi\)
−0.307628 + 0.951507i \(0.599535\pi\)
\(48\) −1.70711 + 1.70711i −0.246400 + 0.246400i
\(49\) − 2.00000i − 0.285714i
\(50\) 4.94975 − 0.707107i 0.700000 − 0.100000i
\(51\) 17.4853i 2.44843i
\(52\) 3.00000 − 3.00000i 0.416025 − 0.416025i
\(53\) 9.53553 − 9.53553i 1.30981 − 1.30981i 0.388254 − 0.921552i \(-0.373078\pi\)
0.921552 − 0.388254i \(-0.126922\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 7.36396 − 0.878680i 0.992956 − 0.118481i
\(56\) 3.00000 0.400892
\(57\) −5.12132 + 5.12132i −0.678335 + 0.678335i
\(58\) 0.878680 − 0.878680i 0.115376 − 0.115376i
\(59\) 1.41421i 0.184115i 0.995754 + 0.0920575i \(0.0293443\pi\)
−0.995754 + 0.0920575i \(0.970656\pi\)
\(60\) 4.82843 − 2.41421i 0.623347 − 0.311674i
\(61\) − 7.24264i − 0.927325i −0.886012 − 0.463663i \(-0.846535\pi\)
0.886012 − 0.463663i \(-0.153465\pi\)
\(62\) 5.12132 − 5.12132i 0.650408 − 0.650408i
\(63\) −6.00000 − 6.00000i −0.755929 − 0.755929i
\(64\) 1.00000i 0.125000i
\(65\) −8.48528 + 4.24264i −1.05247 + 0.526235i
\(66\) 7.24264 − 3.41421i 0.891507 − 0.420261i
\(67\) −4.00000 − 4.00000i −0.488678 − 0.488678i 0.419211 − 0.907889i \(-0.362307\pi\)
−0.907889 + 0.419211i \(0.862307\pi\)
\(68\) 5.12132 + 5.12132i 0.621051 + 0.621051i
\(69\) 0.585786i 0.0705204i
\(70\) −6.36396 − 2.12132i −0.760639 − 0.253546i
\(71\) −1.24264 −0.147474 −0.0737372 − 0.997278i \(-0.523493\pi\)
−0.0737372 + 0.997278i \(0.523493\pi\)
\(72\) 2.00000 − 2.00000i 0.235702 − 0.235702i
\(73\) 6.00000 + 6.00000i 0.702247 + 0.702247i 0.964892 − 0.262646i \(-0.0845950\pi\)
−0.262646 + 0.964892i \(0.584595\pi\)
\(74\) −0.171573 −0.0199449
\(75\) −11.9497 + 1.70711i −1.37984 + 0.197120i
\(76\) 3.00000i 0.344124i
\(77\) −9.36396 − 3.36396i −1.06712 − 0.383359i
\(78\) −7.24264 + 7.24264i −0.820068 + 0.820068i
\(79\) 10.2426 1.15239 0.576194 − 0.817313i \(-0.304537\pi\)
0.576194 + 0.817313i \(0.304537\pi\)
\(80\) 0.707107 − 2.12132i 0.0790569 − 0.237171i
\(81\) 9.48528 1.05392
\(82\) 1.24264 + 1.24264i 0.137227 + 0.137227i
\(83\) −7.24264 − 7.24264i −0.794983 − 0.794983i 0.187317 − 0.982300i \(-0.440021\pi\)
−0.982300 + 0.187317i \(0.940021\pi\)
\(84\) −7.24264 −0.790237
\(85\) −7.24264 − 14.4853i −0.785575 − 1.57115i
\(86\) −1.75736 −0.189501
\(87\) −2.12132 + 2.12132i −0.227429 + 0.227429i
\(88\) 1.12132 − 3.12132i 0.119533 − 0.332734i
\(89\) 5.48528i 0.581439i 0.956808 + 0.290719i \(0.0938946\pi\)
−0.956808 + 0.290719i \(0.906105\pi\)
\(90\) −5.65685 + 2.82843i −0.596285 + 0.298142i
\(91\) 12.7279 1.33425
\(92\) 0.171573 + 0.171573i 0.0178877 + 0.0178877i
\(93\) −12.3640 + 12.3640i −1.28208 + 1.28208i
\(94\) −6.24264 −0.643879
\(95\) 2.12132 − 6.36396i 0.217643 − 0.652929i
\(96\) − 2.41421i − 0.246400i
\(97\) −2.24264 − 2.24264i −0.227706 − 0.227706i 0.584028 − 0.811734i \(-0.301476\pi\)
−0.811734 + 0.584028i \(0.801476\pi\)
\(98\) 1.41421 + 1.41421i 0.142857 + 0.142857i
\(99\) −8.48528 + 4.00000i −0.852803 + 0.402015i
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.f.b.87.1 yes 4
3.2 odd 2 990.2.m.c.307.2 4
4.3 odd 2 880.2.bd.c.417.1 4
5.2 odd 4 550.2.f.b.43.1 4
5.3 odd 4 110.2.f.c.43.2 yes 4
5.4 even 2 550.2.f.a.307.2 4
11.10 odd 2 110.2.f.c.87.2 yes 4
15.8 even 4 990.2.m.d.703.1 4
20.3 even 4 880.2.bd.b.593.1 4
33.32 even 2 990.2.m.d.307.1 4
44.43 even 2 880.2.bd.b.417.1 4
55.32 even 4 550.2.f.a.43.2 4
55.43 even 4 inner 110.2.f.b.43.1 ✓ 4
55.54 odd 2 550.2.f.b.307.1 4
165.98 odd 4 990.2.m.c.703.2 4
220.43 odd 4 880.2.bd.c.593.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.f.b.43.1 ✓ 4 55.43 even 4 inner
110.2.f.b.87.1 yes 4 1.1 even 1 trivial
110.2.f.c.43.2 yes 4 5.3 odd 4
110.2.f.c.87.2 yes 4 11.10 odd 2
550.2.f.a.43.2 4 55.32 even 4
550.2.f.a.307.2 4 5.4 even 2
550.2.f.b.43.1 4 5.2 odd 4
550.2.f.b.307.1 4 55.54 odd 2
880.2.bd.b.417.1 4 44.43 even 2
880.2.bd.b.593.1 4 20.3 even 4
880.2.bd.c.417.1 4 4.3 odd 2
880.2.bd.c.593.1 4 220.43 odd 4
990.2.m.c.307.2 4 3.2 odd 2
990.2.m.c.703.2 4 165.98 odd 4
990.2.m.d.307.1 4 33.32 even 2
990.2.m.d.703.1 4 15.8 even 4