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Properties

Label 880.4.b.b.529.2
Level $880$
Weight $4$
Character 880.529
Analytic conductor $51.922$
Analytic rank $1$
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] = [880,4,Mod(529,880)] 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("880.529"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 880.b (of order \(2\), degree \(1\), not 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: \(51.9216808051\)
Analytic rank: \(1\)
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: no (minimal twist has level 220)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 529.2
Root \(0.500000 - 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 880.529
Dual form 880.4.b.b.529.1

$q$-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 \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(-0.732944\pi\)
0.668220 − 0.743964i \(-0.267056\pi\)
\(14\) 0 0
\(15\) −47.5000 − 10.8972i −0.817630 − 0.187577i
\(16\) 0 0
\(17\) − 26.1534i − 0.373125i −0.982443 − 0.186563i \(-0.940265\pi\)
0.982443 − 0.186563i \(-0.0597347\pi\)
\(18\) 0 0
\(19\) −68.0000 −0.821067 −0.410533 − 0.911846i \(-0.634657\pi\)
−0.410533 + 0.911846i \(0.634657\pi\)
\(20\) 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 \(-0.669227\pi\)
−0.506950 + 0.861975i \(0.669227\pi\)
\(32\) 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 \(-0.876726\pi\)
0.925942 − 0.377667i \(-0.123274\pi\)
\(38\) 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 \(-0.800134\pi\)
0.809265 − 0.587444i \(-0.199866\pi\)
\(54\) 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 \(-0.550431\pi\)
−0.157771 + 0.987476i \(0.550431\pi\)
\(60\) 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 \(-0.832690\pi\)
−0.865013 + 0.501749i \(0.832690\pi\)
\(72\) 0 0
\(73\) − 331.276i − 0.531136i −0.964092 − 0.265568i \(-0.914440\pi\)
0.964092 − 0.265568i \(-0.0855596\pi\)
\(74\) 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 \(-0.450387\pi\)
0.155234 + 0.987878i \(0.450387\pi\)
\(80\) 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 \(0.132310\pi\)
−0.914849 + 0.403796i \(0.867690\pi\)
\(98\) 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 880.4.b.b.529.2 2
4.3 odd 2 220.4.b.a.89.1 ✓ 2
5.4 even 2 inner 880.4.b.b.529.1 2
12.11 even 2 1980.4.c.a.1189.1 2
20.3 even 4 1100.4.a.f.1.2 2
20.7 even 4 1100.4.a.f.1.1 2
20.19 odd 2 220.4.b.a.89.2 yes 2
60.59 even 2 1980.4.c.a.1189.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.4.b.a.89.1 ✓ 2 4.3 odd 2
220.4.b.a.89.2 yes 2 20.19 odd 2
880.4.b.b.529.1 2 5.4 even 2 inner
880.4.b.b.529.2 2 1.1 even 1 trivial
1100.4.a.f.1.1 2 20.7 even 4
1100.4.a.f.1.2 2 20.3 even 4
1980.4.c.a.1189.1 2 12.11 even 2
1980.4.c.a.1189.2 2 60.59 even 2