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Properties

Label 87.2.f.b.41.1
Level $87$
Weight $2$
Character 87.41
Analytic conductor $0.695$
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] = [87,2,Mod(17,87)] 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("87.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.694698497585\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 41.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 87.41
Dual form 87.2.f.b.17.1

$q$-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+(-0.366025 + 0.366025i) q^{2} +(1.50000 - 0.866025i) q^{3} +1.73205i q^{4} -0.267949 q^{5} +(-0.232051 + 0.866025i) q^{6} +0.732051 q^{7} +(-1.36603 - 1.36603i) q^{8} +(1.50000 - 2.59808i) q^{9} +(0.0980762 - 0.0980762i) q^{10} +(-1.36603 + 1.36603i) q^{11} +(1.50000 + 2.59808i) q^{12} -2.26795i q^{13} +(-0.267949 + 0.267949i) q^{14} +(-0.401924 + 0.232051i) q^{15} -2.46410 q^{16} +(-2.73205 + 2.73205i) q^{17} +(0.401924 + 1.50000i) q^{18} +(-1.73205 - 1.73205i) q^{19} -0.464102i q^{20} +(1.09808 - 0.633975i) q^{21} -1.00000i q^{22} -5.46410i q^{23} +(-3.23205 - 0.866025i) q^{24} -4.92820 q^{25} +(0.830127 + 0.830127i) q^{26} -5.19615i q^{27} +1.26795i q^{28} +(-2.00000 + 5.00000i) q^{29} +(0.0621778 - 0.232051i) q^{30} +(2.36603 + 2.36603i) q^{31} +(3.63397 - 3.63397i) q^{32} +(-0.866025 + 3.23205i) q^{33} -2.00000i q^{34} -0.196152 q^{35} +(4.50000 + 2.59808i) q^{36} +(3.46410 - 3.46410i) q^{37} +1.26795 q^{38} +(-1.96410 - 3.40192i) q^{39} +(0.366025 + 0.366025i) q^{40} +(5.19615 + 5.19615i) q^{41} +(-0.169873 + 0.633975i) q^{42} +(8.56218 + 8.56218i) q^{43} +(-2.36603 - 2.36603i) q^{44} +(-0.401924 + 0.696152i) q^{45} +(2.00000 + 2.00000i) q^{46} +(7.83013 + 7.83013i) q^{47} +(-3.69615 + 2.13397i) q^{48} -6.46410 q^{49} +(1.80385 - 1.80385i) q^{50} +(-1.73205 + 6.46410i) q^{51} +3.92820 q^{52} -9.92820i q^{53} +(1.90192 + 1.90192i) q^{54} +(0.366025 - 0.366025i) q^{55} +(-1.00000 - 1.00000i) q^{56} +(-4.09808 - 1.09808i) q^{57} +(-1.09808 - 2.56218i) q^{58} -7.46410i q^{59} +(-0.401924 - 0.696152i) q^{60} +(-2.00000 - 2.00000i) q^{61} -1.73205 q^{62} +(1.09808 - 1.90192i) q^{63} -2.26795i q^{64} +0.607695i q^{65} +(-0.866025 - 1.50000i) q^{66} +4.92820i q^{67} +(-4.73205 - 4.73205i) q^{68} +(-4.73205 - 8.19615i) q^{69} +(0.0717968 - 0.0717968i) q^{70} +13.2679 q^{71} +(-5.59808 + 1.50000i) q^{72} +(-4.00000 + 4.00000i) q^{73} +2.53590i q^{74} +(-7.39230 + 4.26795i) q^{75} +(3.00000 - 3.00000i) q^{76} +(-1.00000 + 1.00000i) q^{77} +(1.96410 + 0.526279i) q^{78} +(-6.83013 - 6.83013i) q^{79} +0.660254 q^{80} +(-4.50000 - 7.79423i) q^{81} -3.80385 q^{82} +13.8564i q^{83} +(1.09808 + 1.90192i) q^{84} +(0.732051 - 0.732051i) q^{85} -6.26795 q^{86} +(1.33013 + 9.23205i) q^{87} +3.73205 q^{88} +(-5.00000 + 5.00000i) q^{89} +(-0.107695 - 0.401924i) q^{90} -1.66025i q^{91} +9.46410 q^{92} +(5.59808 + 1.50000i) q^{93} -5.73205 q^{94} +(0.464102 + 0.464102i) q^{95} +(2.30385 - 8.59808i) q^{96} +(-4.26795 + 4.26795i) q^{97} +(2.36603 - 2.36603i) q^{98} +(1.50000 + 5.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{3} - 8 q^{5} + 6 q^{6} - 4 q^{7} - 2 q^{8} + 6 q^{9} - 10 q^{10} - 2 q^{11} + 6 q^{12} - 8 q^{14} - 12 q^{15} + 4 q^{16} - 4 q^{17} + 12 q^{18} - 6 q^{21} - 6 q^{24} + 8 q^{25} - 14 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.366025 + 0.366025i −0.258819 + 0.258819i −0.824574 − 0.565755i \(-0.808585\pi\)
0.565755 + 0.824574i \(0.308585\pi\)
\(3\) 1.50000 − 0.866025i 0.866025 − 0.500000i
\(4\) 1.73205i 0.866025i
\(5\) −0.267949 −0.119831 −0.0599153 − 0.998203i \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) −0.232051 + 0.866025i −0.0947343 + 0.353553i
\(7\) 0.732051 0.276689 0.138345 − 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) −1.36603 − 1.36603i −0.482963 − 0.482963i
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) 0.0980762 − 0.0980762i 0.0310144 − 0.0310144i
\(11\) −1.36603 + 1.36603i −0.411872 + 0.411872i −0.882390 − 0.470518i \(-0.844067\pi\)
0.470518 + 0.882390i \(0.344067\pi\)
\(12\) 1.50000 + 2.59808i 0.433013 + 0.750000i
\(13\) − 2.26795i − 0.629016i −0.949255 − 0.314508i \(-0.898160\pi\)
0.949255 − 0.314508i \(-0.101840\pi\)
\(14\) −0.267949 + 0.267949i −0.0716124 + 0.0716124i
\(15\) −0.401924 + 0.232051i −0.103776 + 0.0599153i
\(16\) −2.46410 −0.616025
\(17\) −2.73205 + 2.73205i −0.662620 + 0.662620i −0.955997 − 0.293377i \(-0.905221\pi\)
0.293377 + 0.955997i \(0.405221\pi\)
\(18\) 0.401924 + 1.50000i 0.0947343 + 0.353553i
\(19\) −1.73205 − 1.73205i −0.397360 − 0.397360i 0.479941 − 0.877301i \(-0.340658\pi\)
−0.877301 + 0.479941i \(0.840658\pi\)
\(20\) − 0.464102i − 0.103776i
\(21\) 1.09808 − 0.633975i 0.239620 − 0.138345i
\(22\) − 1.00000i − 0.213201i
\(23\) − 5.46410i − 1.13934i −0.821872 − 0.569672i \(-0.807070\pi\)
0.821872 − 0.569672i \(-0.192930\pi\)
\(24\) −3.23205 − 0.866025i −0.659740 − 0.176777i
\(25\) −4.92820 −0.985641
\(26\) 0.830127 + 0.830127i 0.162801 + 0.162801i
\(27\) − 5.19615i − 1.00000i
\(28\) 1.26795i 0.239620i
\(29\) −2.00000 + 5.00000i −0.371391 + 0.928477i
\(30\) 0.0621778 − 0.232051i 0.0113521 − 0.0423665i
\(31\) 2.36603 + 2.36603i 0.424951 + 0.424951i 0.886904 − 0.461953i \(-0.152851\pi\)
−0.461953 + 0.886904i \(0.652851\pi\)
\(32\) 3.63397 − 3.63397i 0.642402 − 0.642402i
\(33\) −0.866025 + 3.23205i −0.150756 + 0.562628i
\(34\) − 2.00000i − 0.342997i
\(35\) −0.196152 −0.0331558
\(36\) 4.50000 + 2.59808i 0.750000 + 0.433013i
\(37\) 3.46410 − 3.46410i 0.569495 − 0.569495i −0.362492 − 0.931987i \(-0.618074\pi\)
0.931987 + 0.362492i \(0.118074\pi\)
\(38\) 1.26795 0.205689
\(39\) −1.96410 − 3.40192i −0.314508 − 0.544744i
\(40\) 0.366025 + 0.366025i 0.0578737 + 0.0578737i
\(41\) 5.19615 + 5.19615i 0.811503 + 0.811503i 0.984859 − 0.173356i \(-0.0554613\pi\)
−0.173356 + 0.984859i \(0.555461\pi\)
\(42\) −0.169873 + 0.633975i −0.0262120 + 0.0978244i
\(43\) 8.56218 + 8.56218i 1.30572 + 1.30572i 0.924473 + 0.381246i \(0.124505\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −2.36603 − 2.36603i −0.356692 − 0.356692i
\(45\) −0.401924 + 0.696152i −0.0599153 + 0.103776i
\(46\) 2.00000 + 2.00000i 0.294884 + 0.294884i
\(47\) 7.83013 + 7.83013i 1.14214 + 1.14214i 0.988058 + 0.154084i \(0.0492425\pi\)
0.154084 + 0.988058i \(0.450757\pi\)
\(48\) −3.69615 + 2.13397i −0.533494 + 0.308013i
\(49\) −6.46410 −0.923443
\(50\) 1.80385 − 1.80385i 0.255103 − 0.255103i
\(51\) −1.73205 + 6.46410i −0.242536 + 0.905155i
\(52\) 3.92820 0.544744
\(53\) − 9.92820i − 1.36374i −0.731472 − 0.681872i \(-0.761166\pi\)
0.731472 − 0.681872i \(-0.238834\pi\)
\(54\) 1.90192 + 1.90192i 0.258819 + 0.258819i
\(55\) 0.366025 − 0.366025i 0.0493549 − 0.0493549i
\(56\) −1.00000 − 1.00000i −0.133631 − 0.133631i
\(57\) −4.09808 − 1.09808i −0.542803 − 0.145444i
\(58\) −1.09808 − 2.56218i −0.144184 − 0.336430i
\(59\) − 7.46410i − 0.971743i −0.874030 − 0.485872i \(-0.838502\pi\)
0.874030 − 0.485872i \(-0.161498\pi\)
\(60\) −0.401924 − 0.696152i −0.0518881 − 0.0898729i
\(61\) −2.00000 − 2.00000i −0.256074 − 0.256074i 0.567381 − 0.823455i \(-0.307957\pi\)
−0.823455 + 0.567381i \(0.807957\pi\)
\(62\) −1.73205 −0.219971
\(63\) 1.09808 − 1.90192i 0.138345 − 0.239620i
\(64\) − 2.26795i − 0.283494i
\(65\) 0.607695i 0.0753753i
\(66\) −0.866025 − 1.50000i −0.106600 − 0.184637i
\(67\) 4.92820i 0.602076i 0.953612 + 0.301038i \(0.0973331\pi\)
−0.953612 + 0.301038i \(0.902667\pi\)
\(68\) −4.73205 − 4.73205i −0.573845 − 0.573845i
\(69\) −4.73205 − 8.19615i −0.569672 − 0.986701i
\(70\) 0.0717968 − 0.0717968i 0.00858136 − 0.00858136i
\(71\) 13.2679 1.57462 0.787308 − 0.616560i \(-0.211474\pi\)
0.787308 + 0.616560i \(0.211474\pi\)
\(72\) −5.59808 + 1.50000i −0.659740 + 0.176777i
\(73\) −4.00000 + 4.00000i −0.468165 + 0.468165i −0.901319 − 0.433155i \(-0.857400\pi\)
0.433155 + 0.901319i \(0.357400\pi\)
\(74\) 2.53590i 0.294792i
\(75\) −7.39230 + 4.26795i −0.853590 + 0.492820i
\(76\) 3.00000 − 3.00000i 0.344124 − 0.344124i
\(77\) −1.00000 + 1.00000i −0.113961 + 0.113961i
\(78\) 1.96410 + 0.526279i 0.222391 + 0.0595894i
\(79\) −6.83013 − 6.83013i −0.768449 − 0.768449i 0.209384 − 0.977833i \(-0.432854\pi\)
−0.977833 + 0.209384i \(0.932854\pi\)
\(80\) 0.660254 0.0738186
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) −3.80385 −0.420065
\(83\) 13.8564i 1.52094i 0.649374 + 0.760469i \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 1.09808 + 1.90192i 0.119810 + 0.207517i
\(85\) 0.732051 − 0.732051i 0.0794021 − 0.0794021i
\(86\) −6.26795 −0.675890
\(87\) 1.33013 + 9.23205i 0.142605 + 0.989780i
\(88\) 3.73205 0.397838
\(89\) −5.00000 + 5.00000i −0.529999 + 0.529999i −0.920572 − 0.390573i \(-0.872277\pi\)
0.390573 + 0.920572i \(0.372277\pi\)
\(90\) −0.107695 − 0.401924i −0.0113521 − 0.0423665i
\(91\) − 1.66025i − 0.174042i
\(92\) 9.46410 0.986701
\(93\) 5.59808 + 1.50000i 0.580493 + 0.155543i
\(94\) −5.73205 −0.591216
\(95\) 0.464102 + 0.464102i 0.0476158 + 0.0476158i
\(96\) 2.30385 − 8.59808i 0.235135 − 0.877537i
\(97\) −4.26795 + 4.26795i −0.433345 + 0.433345i −0.889764 − 0.456420i \(-0.849131\pi\)
0.456420 + 0.889764i \(0.349131\pi\)
\(98\) 2.36603 − 2.36603i 0.239005 − 0.239005i
\(99\) 1.50000 + 5.59808i 0.150756 + 0.562628i
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 87.2.f.b.41.1 yes 4
3.2 odd 2 87.2.f.a.41.2 yes 4
29.17 odd 4 87.2.f.a.17.2 ✓ 4
87.17 even 4 inner 87.2.f.b.17.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.f.a.17.2 ✓ 4 29.17 odd 4
87.2.f.a.41.2 yes 4 3.2 odd 2
87.2.f.b.17.1 yes 4 87.17 even 4 inner
87.2.f.b.41.1 yes 4 1.1 even 1 trivial