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Properties

Label 110.2.f.b.87.2
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 87.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 110.87
Dual form 110.2.f.b.43.2

$q$-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} +(0.292893 - 0.292893i) q^{3} -1.00000i q^{4} +(0.707107 - 2.12132i) q^{5} -0.414214i 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} +(-0.292893 - 0.292893i) q^{12} +(3.00000 + 3.00000i) q^{13} +3.00000i q^{14} +(-0.414214 - 0.828427i) q^{15} -1.00000 q^{16} +(-0.878680 + 0.878680i) q^{17} +(2.00000 + 2.00000i) q^{18} -3.00000 q^{19} +(-2.12132 - 0.707107i) q^{20} +1.24264i q^{21} +(-1.12132 - 3.12132i) q^{22} +(-5.82843 + 5.82843i) q^{23} -0.414214 q^{24} +(-4.00000 - 3.00000i) q^{25} +4.24264 q^{26} +(1.70711 + 1.70711i) q^{27} +(2.12132 + 2.12132i) q^{28} +7.24264 q^{29} +(-0.878680 - 0.292893i) q^{30} +1.24264 q^{31} +(-0.707107 + 0.707107i) q^{32} +(-0.464466 - 1.29289i) q^{33} +1.24264i q^{34} +(3.00000 + 6.00000i) q^{35} +2.82843 q^{36} +(-4.12132 - 4.12132i) q^{37} +(-2.12132 + 2.12132i) q^{38} +1.75736 q^{39} +(-2.00000 + 1.00000i) q^{40} -10.2426i q^{41} +(0.878680 + 0.878680i) q^{42} +(-7.24264 - 7.24264i) q^{43} +(-3.00000 - 1.41421i) q^{44} +(6.00000 + 2.00000i) q^{45} +8.24264i q^{46} +(1.58579 + 1.58579i) q^{47} +(-0.292893 + 0.292893i) q^{48} -2.00000i q^{49} +(-4.94975 + 0.707107i) q^{50} +0.514719i q^{51} +(3.00000 - 3.00000i) q^{52} +(2.46447 - 2.46447i) q^{53} +2.41421 q^{54} +(-5.36396 - 5.12132i) q^{55} +3.00000 q^{56} +(-0.878680 + 0.878680i) q^{57} +(5.12132 - 5.12132i) q^{58} -1.41421i q^{59} +(-0.828427 + 0.414214i) q^{60} +1.24264i q^{61} +(0.878680 - 0.878680i) q^{62} +(-6.00000 - 6.00000i) q^{63} +1.00000i q^{64} +(8.48528 - 4.24264i) q^{65} +(-1.24264 - 0.585786i) q^{66} +(-4.00000 - 4.00000i) q^{67} +(0.878680 + 0.878680i) q^{68} +3.41421i q^{69} +(6.36396 + 2.12132i) q^{70} +7.24264 q^{71} +(2.00000 - 2.00000i) q^{72} +(6.00000 + 6.00000i) q^{73} -5.82843 q^{74} +(-2.05025 + 0.292893i) q^{75} +3.00000i q^{76} +(3.36396 + 9.36396i) q^{77} +(1.24264 - 1.24264i) q^{78} +1.75736 q^{79} +(-0.707107 + 2.12132i) q^{80} -7.48528 q^{81} +(-7.24264 - 7.24264i) q^{82} +(1.24264 + 1.24264i) q^{83} +1.24264 q^{84} +(1.24264 + 2.48528i) q^{85} -10.2426 q^{86} +(2.12132 - 2.12132i) q^{87} +(-3.12132 + 1.12132i) q^{88} -11.4853i q^{89} +(5.65685 - 2.82843i) q^{90} -12.7279 q^{91} +(5.82843 + 5.82843i) q^{92} +(0.363961 - 0.363961i) q^{93} +2.24264 q^{94} +(-2.12132 + 6.36396i) q^{95} +0.414214i q^{96} +(6.24264 + 6.24264i) 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
\(3\) 0.292893 − 0.292893i 0.169102 − 0.169102i −0.617483 − 0.786585i \(-0.711847\pi\)
0.786585 + 0.617483i \(0.211847\pi\)
\(4\) − 1.00000i − 0.500000i
\(5\) 0.707107 − 2.12132i 0.316228 − 0.948683i
\(6\) − 0.414214i − 0.169102i
\(7\) −2.12132 + 2.12132i −0.801784 + 0.801784i −0.983374 − 0.181591i \(-0.941875\pi\)
0.181591 + 0.983374i \(0.441875\pi\)
\(8\) −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
\(12\) −0.292893 − 0.292893i −0.0845510 − 0.0845510i
\(13\) 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
\(15\) −0.414214 − 0.828427i −0.106949 − 0.213899i
\(16\) −1.00000 −0.250000
\(17\) −0.878680 + 0.878680i −0.213111 + 0.213111i −0.805588 − 0.592477i \(-0.798150\pi\)
0.592477 + 0.805588i \(0.298150\pi\)
\(18\) 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
\(21\) 1.24264i 0.271166i
\(22\) −1.12132 − 3.12132i −0.239066 − 0.665468i
\(23\) −5.82843 + 5.82843i −1.21531 + 1.21531i −0.246055 + 0.969256i \(0.579134\pi\)
−0.969256 + 0.246055i \(0.920866\pi\)
\(24\) −0.414214 −0.0845510
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) 4.24264 0.832050
\(27\) 1.70711 + 1.70711i 0.328533 + 0.328533i
\(28\) 2.12132 + 2.12132i 0.400892 + 0.400892i
\(29\) 7.24264 1.34492 0.672462 − 0.740131i \(-0.265237\pi\)
0.672462 + 0.740131i \(0.265237\pi\)
\(30\) −0.878680 − 0.292893i −0.160424 − 0.0534747i
\(31\) 1.24264 0.223185 0.111592 − 0.993754i \(-0.464405\pi\)
0.111592 + 0.993754i \(0.464405\pi\)
\(32\) −0.707107 + 0.707107i −0.125000 + 0.125000i
\(33\) −0.464466 − 1.29289i −0.0808532 − 0.225064i
\(34\) 1.24264i 0.213111i
\(35\) 3.00000 + 6.00000i 0.507093 + 1.01419i
\(36\) 2.82843 0.471405
\(37\) −4.12132 − 4.12132i −0.677541 − 0.677541i 0.281902 − 0.959443i \(-0.409035\pi\)
−0.959443 + 0.281902i \(0.909035\pi\)
\(38\) −2.12132 + 2.12132i −0.344124 + 0.344124i
\(39\) 1.75736 0.281403
\(40\) −2.00000 + 1.00000i −0.316228 + 0.158114i
\(41\) − 10.2426i − 1.59963i −0.600245 − 0.799816i \(-0.704930\pi\)
0.600245 − 0.799816i \(-0.295070\pi\)
\(42\) 0.878680 + 0.878680i 0.135583 + 0.135583i
\(43\) −7.24264 − 7.24264i −1.10449 − 1.10449i −0.993862 − 0.110631i \(-0.964713\pi\)
−0.110631 − 0.993862i \(-0.535287\pi\)
\(44\) −3.00000 − 1.41421i −0.452267 − 0.213201i
\(45\) 6.00000 + 2.00000i 0.894427 + 0.298142i
\(46\) 8.24264i 1.21531i
\(47\) 1.58579 + 1.58579i 0.231311 + 0.231311i 0.813240 − 0.581929i \(-0.197702\pi\)
−0.581929 + 0.813240i \(0.697702\pi\)
\(48\) −0.292893 + 0.292893i −0.0422755 + 0.0422755i
\(49\) − 2.00000i − 0.285714i
\(50\) −4.94975 + 0.707107i −0.700000 + 0.100000i
\(51\) 0.514719i 0.0720750i
\(52\) 3.00000 − 3.00000i 0.416025 − 0.416025i
\(53\) 2.46447 − 2.46447i 0.338520 − 0.338520i −0.517290 − 0.855810i \(-0.673059\pi\)
0.855810 + 0.517290i \(0.173059\pi\)
\(54\) 2.41421 0.328533
\(55\) −5.36396 − 5.12132i −0.723276 − 0.690559i
\(56\) 3.00000 0.400892
\(57\) −0.878680 + 0.878680i −0.116384 + 0.116384i
\(58\) 5.12132 − 5.12132i 0.672462 − 0.672462i
\(59\) − 1.41421i − 0.184115i −0.995754 − 0.0920575i \(-0.970656\pi\)
0.995754 − 0.0920575i \(-0.0293443\pi\)
\(60\) −0.828427 + 0.414214i −0.106949 + 0.0534747i
\(61\) 1.24264i 0.159104i 0.996831 + 0.0795519i \(0.0253489\pi\)
−0.996831 + 0.0795519i \(0.974651\pi\)
\(62\) 0.878680 − 0.878680i 0.111592 − 0.111592i
\(63\) −6.00000 − 6.00000i −0.755929 − 0.755929i
\(64\) 1.00000i 0.125000i
\(65\) 8.48528 − 4.24264i 1.05247 − 0.526235i
\(66\) −1.24264 − 0.585786i −0.152958 − 0.0721053i
\(67\) −4.00000 − 4.00000i −0.488678 − 0.488678i 0.419211 − 0.907889i \(-0.362307\pi\)
−0.907889 + 0.419211i \(0.862307\pi\)
\(68\) 0.878680 + 0.878680i 0.106556 + 0.106556i
\(69\) 3.41421i 0.411023i
\(70\) 6.36396 + 2.12132i 0.760639 + 0.253546i
\(71\) 7.24264 0.859543 0.429772 − 0.902938i \(-0.358594\pi\)
0.429772 + 0.902938i \(0.358594\pi\)
\(72\) 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 \(0.584595\pi\)
\(74\) −5.82843 −0.677541
\(75\) −2.05025 + 0.292893i −0.236743 + 0.0338204i
\(76\) 3.00000i 0.344124i
\(77\) 3.36396 + 9.36396i 0.383359 + 1.06712i
\(78\) 1.24264 − 1.24264i 0.140701 − 0.140701i
\(79\) 1.75736 0.197718 0.0988592 − 0.995101i \(-0.468481\pi\)
0.0988592 + 0.995101i \(0.468481\pi\)
\(80\) −0.707107 + 2.12132i −0.0790569 + 0.237171i
\(81\) −7.48528 −0.831698
\(82\) −7.24264 − 7.24264i −0.799816 − 0.799816i
\(83\) 1.24264 + 1.24264i 0.136398 + 0.136398i 0.772009 − 0.635612i \(-0.219252\pi\)
−0.635612 + 0.772009i \(0.719252\pi\)
\(84\) 1.24264 0.135583
\(85\) 1.24264 + 2.48528i 0.134783 + 0.269567i
\(86\) −10.2426 −1.10449
\(87\) 2.12132 − 2.12132i 0.227429 − 0.227429i
\(88\) −3.12132 + 1.12132i −0.332734 + 0.119533i
\(89\) − 11.4853i − 1.21744i −0.793386 − 0.608719i \(-0.791684\pi\)
0.793386 − 0.608719i \(-0.208316\pi\)
\(90\) 5.65685 − 2.82843i 0.596285 − 0.298142i
\(91\) −12.7279 −1.33425
\(92\) 5.82843 + 5.82843i 0.607656 + 0.607656i
\(93\) 0.363961 − 0.363961i 0.0377410 − 0.0377410i
\(94\) 2.24264 0.231311
\(95\) −2.12132 + 6.36396i −0.217643 + 0.652929i
\(96\) 0.414214i 0.0422755i
\(97\) 6.24264 + 6.24264i 0.633844 + 0.633844i 0.949030 − 0.315186i \(-0.102067\pi\)
−0.315186 + 0.949030i \(0.602067\pi\)
\(98\) −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.87.2 yes 4
3.2 odd 2 990.2.m.c.307.1 4
4.3 odd 2 880.2.bd.c.417.2 4
5.2 odd 4 550.2.f.b.43.2 4
5.3 odd 4 110.2.f.c.43.1 yes 4
5.4 even 2 550.2.f.a.307.1 4
11.10 odd 2 110.2.f.c.87.1 yes 4
15.8 even 4 990.2.m.d.703.2 4
20.3 even 4 880.2.bd.b.593.2 4
33.32 even 2 990.2.m.d.307.2 4
44.43 even 2 880.2.bd.b.417.2 4
55.32 even 4 550.2.f.a.43.1 4
55.43 even 4 inner 110.2.f.b.43.2 ✓ 4
55.54 odd 2 550.2.f.b.307.2 4
165.98 odd 4 990.2.m.c.703.1 4
220.43 odd 4 880.2.bd.c.593.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.f.b.43.2 ✓ 4 55.43 even 4 inner
110.2.f.b.87.2 yes 4 1.1 even 1 trivial
110.2.f.c.43.1 yes 4 5.3 odd 4
110.2.f.c.87.1 yes 4 11.10 odd 2
550.2.f.a.43.1 4 55.32 even 4
550.2.f.a.307.1 4 5.4 even 2
550.2.f.b.43.2 4 5.2 odd 4
550.2.f.b.307.2 4 55.54 odd 2
880.2.bd.b.417.2 4 44.43 even 2
880.2.bd.b.593.2 4 20.3 even 4
880.2.bd.c.417.2 4 4.3 odd 2
880.2.bd.c.593.2 4 220.43 odd 4
990.2.m.c.307.1 4 3.2 odd 2
990.2.m.c.703.1 4 165.98 odd 4
990.2.m.d.307.2 4 33.32 even 2
990.2.m.d.703.2 4 15.8 even 4