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Properties

Label 220.4.b.a.89.2
Level $220$
Weight $4$
Character 220.89
Analytic conductor $12.980$
Analytic rank $0$
Dimension $2$
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] = [220,4,Mod(89,220)] 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("220.89"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 220.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.9804202013\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 89.2
Root \(0.500000 - 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 220.89
Dual form 220.4.b.a.89.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.35890i q^{3} +(-2.50000 - 10.8972i) q^{5} +8.71780i q^{7} +8.00000 q^{9} -11.0000 q^{11} +69.7424i q^{13} +(47.5000 - 10.8972i) q^{15} +26.1534i q^{17} +68.0000 q^{19} -38.0000 q^{21} +117.690i q^{23} +(-112.500 + 54.4862i) q^{25} +152.561i q^{27} -260.000 q^{29} +175.000 q^{31} -47.9479i q^{33} +(95.0000 - 21.7945i) q^{35} +169.997i q^{37} -304.000 q^{39} -380.000 q^{41} +305.123i q^{43} +(-20.0000 - 87.1780i) q^{45} -305.123i q^{47} +267.000 q^{49} -114.000 q^{51} +453.325i q^{53} +(27.5000 + 119.870i) q^{55} +296.405i q^{57} +143.000 q^{59} +676.000 q^{61} +69.7424i q^{63} +(760.000 - 174.356i) q^{65} -527.427i q^{67} -513.000 q^{69} +1035.00 q^{71} +331.276i q^{73} +(-237.500 - 490.376i) q^{75} -95.8958i q^{77} -218.000 q^{79} -449.000 q^{81} -758.448i q^{83} +(285.000 - 65.3835i) q^{85} -1133.31i q^{87} -1279.00 q^{89} -608.000 q^{91} +762.807i q^{93} +(-170.000 - 741.013i) q^{95} -771.525i q^{97} -88.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{5} + 16 q^{9} - 22 q^{11} + 95 q^{15} + 136 q^{19} - 76 q^{21} - 225 q^{25} - 520 q^{29} + 350 q^{31} + 190 q^{35} - 608 q^{39} - 760 q^{41} - 40 q^{45} + 534 q^{49} - 228 q^{51} + 55 q^{55}+ \cdots - 176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 0
\(3\) 4.35890i 0.838870i 0.907785 + 0.419435i \(0.137772\pi\)
−0.907785 + 0.419435i \(0.862228\pi\)
\(4\) 0 0
\(5\) −2.50000 − 10.8972i −0.223607 − 0.974679i
\(6\) 0 0
\(7\) 8.71780i 0.470717i 0.971909 + 0.235358i \(0.0756264\pi\)
−0.971909 + 0.235358i \(0.924374\pi\)
\(8\) 0 0
\(9\) 8.00000 0.296296
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 69.7424i 1.48793i 0.668220 + 0.743964i \(0.267056\pi\)
−0.668220 + 0.743964i \(0.732944\pi\)
\(14\) 0 0
\(15\) 47.5000 − 10.8972i 0.817630 − 0.187577i
\(16\) 0 0
\(17\) 26.1534i 0.373125i 0.982443 + 0.186563i \(0.0597347\pi\)
−0.982443 + 0.186563i \(0.940265\pi\)
\(18\) 0 0
\(19\) 68.0000 0.821067 0.410533 − 0.911846i \(-0.365343\pi\)
0.410533 + 0.911846i \(0.365343\pi\)
\(20\) 0 0
\(21\) −38.0000 −0.394870
\(22\) 0 0
\(23\) 117.690i 1.06696i 0.845812 + 0.533481i \(0.179116\pi\)
−0.845812 + 0.533481i \(0.820884\pi\)
\(24\) 0 0
\(25\) −112.500 + 54.4862i −0.900000 + 0.435890i
\(26\) 0 0
\(27\) 152.561i 1.08742i
\(28\) 0 0
\(29\) −260.000 −1.66485 −0.832427 − 0.554134i \(-0.813049\pi\)
−0.832427 + 0.554134i \(0.813049\pi\)
\(30\) 0 0
\(31\) 175.000 1.01390 0.506950 − 0.861975i \(-0.330773\pi\)
0.506950 + 0.861975i \(0.330773\pi\)
\(32\) 0 0
\(33\) − 47.9479i − 0.252929i
\(34\) 0 0
\(35\) 95.0000 − 21.7945i 0.458798 − 0.105255i
\(36\) 0 0
\(37\) 169.997i 0.755334i 0.925942 + 0.377667i \(0.123274\pi\)
−0.925942 + 0.377667i \(0.876726\pi\)
\(38\) 0 0
\(39\) −304.000 −1.24818
\(40\) 0 0
\(41\) −380.000 −1.44746 −0.723732 − 0.690081i \(-0.757575\pi\)
−0.723732 + 0.690081i \(0.757575\pi\)
\(42\) 0 0
\(43\) 305.123i 1.08211i 0.840987 + 0.541056i \(0.181975\pi\)
−0.840987 + 0.541056i \(0.818025\pi\)
\(44\) 0 0
\(45\) −20.0000 − 87.1780i −0.0662539 − 0.288794i
\(46\) 0 0
\(47\) − 305.123i − 0.946952i −0.880807 − 0.473476i \(-0.842999\pi\)
0.880807 − 0.473476i \(-0.157001\pi\)
\(48\) 0 0
\(49\) 267.000 0.778426
\(50\) 0 0
\(51\) −114.000 −0.313004
\(52\) 0 0
\(53\) 453.325i 1.17489i 0.809265 + 0.587444i \(0.199866\pi\)
−0.809265 + 0.587444i \(0.800134\pi\)
\(54\) 0 0
\(55\) 27.5000 + 119.870i 0.0674200 + 0.293877i
\(56\) 0 0
\(57\) 296.405i 0.688769i
\(58\) 0 0
\(59\) 143.000 0.315543 0.157771 − 0.987476i \(-0.449569\pi\)
0.157771 + 0.987476i \(0.449569\pi\)
\(60\) 0 0
\(61\) 676.000 1.41890 0.709450 − 0.704756i \(-0.248943\pi\)
0.709450 + 0.704756i \(0.248943\pi\)
\(62\) 0 0
\(63\) 69.7424i 0.139472i
\(64\) 0 0
\(65\) 760.000 − 174.356i 1.45025 − 0.332711i
\(66\) 0 0
\(67\) − 527.427i − 0.961723i −0.876797 − 0.480861i \(-0.840324\pi\)
0.876797 − 0.480861i \(-0.159676\pi\)
\(68\) 0 0
\(69\) −513.000 −0.895043
\(70\) 0 0
\(71\) 1035.00 1.73003 0.865013 − 0.501749i \(-0.167310\pi\)
0.865013 + 0.501749i \(0.167310\pi\)
\(72\) 0 0
\(73\) 331.276i 0.531136i 0.964092 + 0.265568i \(0.0855596\pi\)
−0.964092 + 0.265568i \(0.914440\pi\)
\(74\) 0 0
\(75\) −237.500 − 490.376i −0.365655 − 0.754983i
\(76\) 0 0
\(77\) − 95.8958i − 0.141926i
\(78\) 0 0
\(79\) −218.000 −0.310467 −0.155234 − 0.987878i \(-0.549613\pi\)
−0.155234 + 0.987878i \(0.549613\pi\)
\(80\) 0 0
\(81\) −449.000 −0.615912
\(82\) 0 0
\(83\) − 758.448i − 1.00302i −0.865152 − 0.501509i \(-0.832778\pi\)
0.865152 − 0.501509i \(-0.167222\pi\)
\(84\) 0 0
\(85\) 285.000 − 65.3835i 0.363678 − 0.0834333i
\(86\) 0 0
\(87\) − 1133.31i − 1.39660i
\(88\) 0 0
\(89\) −1279.00 −1.52330 −0.761650 − 0.647988i \(-0.775610\pi\)
−0.761650 + 0.647988i \(0.775610\pi\)
\(90\) 0 0
\(91\) −608.000 −0.700393
\(92\) 0 0
\(93\) 762.807i 0.850532i
\(94\) 0 0
\(95\) −170.000 − 741.013i −0.183596 − 0.800277i
\(96\) 0 0
\(97\) − 771.525i − 0.807593i −0.914849 − 0.403796i \(-0.867690\pi\)
0.914849 − 0.403796i \(-0.132310\pi\)
\(98\) 0 0
\(99\) −88.0000 −0.0893367
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 220.4.b.a.89.2 yes 2
3.2 odd 2 1980.4.c.a.1189.2 2
4.3 odd 2 880.4.b.b.529.1 2
5.2 odd 4 1100.4.a.f.1.2 2
5.3 odd 4 1100.4.a.f.1.1 2
5.4 even 2 inner 220.4.b.a.89.1 ✓ 2
15.14 odd 2 1980.4.c.a.1189.1 2
20.19 odd 2 880.4.b.b.529.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.4.b.a.89.1 ✓ 2 5.4 even 2 inner
220.4.b.a.89.2 yes 2 1.1 even 1 trivial
880.4.b.b.529.1 2 4.3 odd 2
880.4.b.b.529.2 2 20.19 odd 2
1100.4.a.f.1.1 2 5.3 odd 4
1100.4.a.f.1.2 2 5.2 odd 4
1980.4.c.a.1189.1 2 15.14 odd 2
1980.4.c.a.1189.2 2 3.2 odd 2