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Properties

Label 528.2.y.b.97.1
Level $528$
Weight $2$
Character 528.97
Analytic conductor $4.216$
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] = [528,2,Mod(49,528)] 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("528.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.y (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-1,0,-3] 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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 97.1
Root \(-0.309017 - 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 528.97
Dual form 528.2.y.b.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.809017 + 0.587785i) q^{3} +(-0.190983 - 0.587785i) q^{5} +(-0.809017 - 0.587785i) q^{7} +(0.309017 - 0.951057i) q^{9} +(3.30902 + 0.224514i) q^{11} +(0.0729490 - 0.224514i) q^{13} +(0.500000 + 0.363271i) q^{15} +(-0.354102 - 1.08981i) q^{17} +(4.73607 - 3.44095i) q^{19} +1.00000 q^{21} -0.236068 q^{23} +(3.73607 - 2.71441i) q^{25} +(0.309017 + 0.951057i) q^{27} +(4.85410 + 3.52671i) q^{29} +(1.88197 - 5.79210i) q^{31} +(-2.80902 + 1.76336i) q^{33} +(-0.190983 + 0.587785i) q^{35} +(5.04508 + 3.66547i) q^{37} +(0.0729490 + 0.224514i) q^{39} +(-0.190983 + 0.138757i) q^{41} +6.70820 q^{43} -0.618034 q^{45} +(-8.16312 + 5.93085i) q^{47} +(-1.85410 - 5.70634i) q^{49} +(0.927051 + 0.673542i) q^{51} +(-0.118034 + 0.363271i) q^{53} +(-0.500000 - 1.98787i) q^{55} +(-1.80902 + 5.56758i) q^{57} +(5.97214 + 4.33901i) q^{59} +(-3.57295 - 10.9964i) q^{61} +(-0.809017 + 0.587785i) q^{63} -0.145898 q^{65} -1.85410 q^{67} +(0.190983 - 0.138757i) q^{69} +(-3.19098 - 9.82084i) q^{71} +(4.61803 + 3.35520i) q^{73} +(-1.42705 + 4.39201i) q^{75} +(-2.54508 - 2.12663i) q^{77} +(-3.39919 + 10.4616i) q^{79} +(-0.809017 - 0.587785i) q^{81} +(-0.454915 - 1.40008i) q^{83} +(-0.572949 + 0.416272i) q^{85} -6.00000 q^{87} -8.23607 q^{89} +(-0.190983 + 0.138757i) q^{91} +(1.88197 + 5.79210i) q^{93} +(-2.92705 - 2.12663i) q^{95} +(2.42705 - 7.46969i) q^{97} +(1.23607 - 3.07768i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 3 q^{5} - q^{7} - q^{9} + 11 q^{11} + 7 q^{13} + 2 q^{15} + 12 q^{17} + 10 q^{19} + 4 q^{21} + 8 q^{23} + 6 q^{25} - q^{27} + 6 q^{29} + 12 q^{31} - 9 q^{33} - 3 q^{35} + 9 q^{37} + 7 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(1\) \(1\)

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
\(3\) −0.809017 + 0.587785i −0.467086 + 0.339358i
\(4\) 0 0
\(5\) −0.190983 − 0.587785i −0.0854102 − 0.262866i 0.899226 − 0.437485i \(-0.144131\pi\)
−0.984636 + 0.174619i \(0.944131\pi\)
\(6\) 0 0
\(7\) −0.809017 − 0.587785i −0.305780 − 0.222162i 0.424304 − 0.905520i \(-0.360519\pi\)
−0.730084 + 0.683358i \(0.760519\pi\)
\(8\) 0 0
\(9\) 0.309017 − 0.951057i 0.103006 − 0.317019i
\(10\) 0 0
\(11\) 3.30902 + 0.224514i 0.997706 + 0.0676935i
\(12\) 0 0
\(13\) 0.0729490 − 0.224514i 0.0202324 − 0.0622690i −0.940431 − 0.339986i \(-0.889578\pi\)
0.960663 + 0.277717i \(0.0895777\pi\)
\(14\) 0 0
\(15\) 0.500000 + 0.363271i 0.129099 + 0.0937962i
\(16\) 0 0
\(17\) −0.354102 − 1.08981i −0.0858823 − 0.264319i 0.898888 − 0.438178i \(-0.144376\pi\)
−0.984770 + 0.173860i \(0.944376\pi\)
\(18\) 0 0
\(19\) 4.73607 − 3.44095i 1.08653 − 0.789409i 0.107719 − 0.994181i \(-0.465645\pi\)
0.978810 + 0.204772i \(0.0656454\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −0.236068 −0.0492236 −0.0246118 − 0.999697i \(-0.507835\pi\)
−0.0246118 + 0.999697i \(0.507835\pi\)
\(24\) 0 0
\(25\) 3.73607 − 2.71441i 0.747214 − 0.542882i
\(26\) 0 0
\(27\) 0.309017 + 0.951057i 0.0594703 + 0.183031i
\(28\) 0 0
\(29\) 4.85410 + 3.52671i 0.901384 + 0.654894i 0.938821 − 0.344405i \(-0.111919\pi\)
−0.0374370 + 0.999299i \(0.511919\pi\)
\(30\) 0 0
\(31\) 1.88197 − 5.79210i 0.338011 − 1.04029i −0.627209 − 0.778851i \(-0.715803\pi\)
0.965220 − 0.261440i \(-0.0841973\pi\)
\(32\) 0 0
\(33\) −2.80902 + 1.76336i −0.488987 + 0.306961i
\(34\) 0 0
\(35\) −0.190983 + 0.587785i −0.0322820 + 0.0993538i
\(36\) 0 0
\(37\) 5.04508 + 3.66547i 0.829407 + 0.602599i 0.919391 − 0.393344i \(-0.128682\pi\)
−0.0899846 + 0.995943i \(0.528682\pi\)
\(38\) 0 0
\(39\) 0.0729490 + 0.224514i 0.0116812 + 0.0359510i
\(40\) 0 0
\(41\) −0.190983 + 0.138757i −0.0298265 + 0.0216702i −0.602599 − 0.798044i \(-0.705868\pi\)
0.572772 + 0.819715i \(0.305868\pi\)
\(42\) 0 0
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0 0
\(47\) −8.16312 + 5.93085i −1.19071 + 0.865104i −0.993339 − 0.115224i \(-0.963241\pi\)
−0.197374 + 0.980328i \(0.563241\pi\)
\(48\) 0 0
\(49\) −1.85410 − 5.70634i −0.264872 − 0.815191i
\(50\) 0 0
\(51\) 0.927051 + 0.673542i 0.129813 + 0.0943147i
\(52\) 0 0
\(53\) −0.118034 + 0.363271i −0.0162132 + 0.0498991i −0.958836 − 0.283961i \(-0.908351\pi\)
0.942623 + 0.333860i \(0.108351\pi\)
\(54\) 0 0
\(55\) −0.500000 − 1.98787i −0.0674200 − 0.268044i
\(56\) 0 0
\(57\) −1.80902 + 5.56758i −0.239610 + 0.737444i
\(58\) 0 0
\(59\) 5.97214 + 4.33901i 0.777506 + 0.564891i 0.904229 − 0.427047i \(-0.140446\pi\)
−0.126724 + 0.991938i \(0.540446\pi\)
\(60\) 0 0
\(61\) −3.57295 − 10.9964i −0.457469 − 1.40795i −0.868212 − 0.496194i \(-0.834730\pi\)
0.410742 − 0.911751i \(-0.365270\pi\)
\(62\) 0 0
\(63\) −0.809017 + 0.587785i −0.101927 + 0.0740540i
\(64\) 0 0
\(65\) −0.145898 −0.0180964
\(66\) 0 0
\(67\) −1.85410 −0.226515 −0.113257 − 0.993566i \(-0.536128\pi\)
−0.113257 + 0.993566i \(0.536128\pi\)
\(68\) 0 0
\(69\) 0.190983 − 0.138757i 0.0229917 − 0.0167044i
\(70\) 0 0
\(71\) −3.19098 − 9.82084i −0.378700 − 1.16552i −0.940948 − 0.338550i \(-0.890063\pi\)
0.562248 − 0.826968i \(-0.309937\pi\)
\(72\) 0 0
\(73\) 4.61803 + 3.35520i 0.540500 + 0.392696i 0.824271 − 0.566196i \(-0.191585\pi\)
−0.283771 + 0.958892i \(0.591585\pi\)
\(74\) 0 0
\(75\) −1.42705 + 4.39201i −0.164782 + 0.507146i
\(76\) 0 0
\(77\) −2.54508 − 2.12663i −0.290039 − 0.242352i
\(78\) 0 0
\(79\) −3.39919 + 10.4616i −0.382438 + 1.17702i 0.555883 + 0.831260i \(0.312380\pi\)
−0.938322 + 0.345764i \(0.887620\pi\)
\(80\) 0 0
\(81\) −0.809017 − 0.587785i −0.0898908 − 0.0653095i
\(82\) 0 0
\(83\) −0.454915 − 1.40008i −0.0499334 − 0.153679i 0.922981 − 0.384846i \(-0.125746\pi\)
−0.972914 + 0.231167i \(0.925746\pi\)
\(84\) 0 0
\(85\) −0.572949 + 0.416272i −0.0621450 + 0.0451510i
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −8.23607 −0.873021 −0.436511 − 0.899699i \(-0.643786\pi\)
−0.436511 + 0.899699i \(0.643786\pi\)
\(90\) 0 0
\(91\) −0.190983 + 0.138757i −0.0200205 + 0.0145457i
\(92\) 0 0
\(93\) 1.88197 + 5.79210i 0.195151 + 0.600612i
\(94\) 0 0
\(95\) −2.92705 − 2.12663i −0.300309 − 0.218187i
\(96\) 0 0
\(97\) 2.42705 − 7.46969i 0.246430 − 0.758433i −0.748968 − 0.662606i \(-0.769451\pi\)
0.995398 − 0.0958268i \(-0.0305495\pi\)
\(98\) 0 0
\(99\) 1.23607 − 3.07768i 0.124230 − 0.309319i
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 528.2.y.b.97.1 4
4.3 odd 2 33.2.e.b.31.1 yes 4
11.4 even 5 5808.2.a.cj.1.1 2
11.5 even 5 inner 528.2.y.b.49.1 4
11.7 odd 10 5808.2.a.ci.1.1 2
12.11 even 2 99.2.f.a.64.1 4
20.3 even 4 825.2.bx.d.724.2 8
20.7 even 4 825.2.bx.d.724.1 8
20.19 odd 2 825.2.n.c.526.1 4
36.7 odd 6 891.2.n.c.757.1 8
36.11 even 6 891.2.n.b.757.1 8
36.23 even 6 891.2.n.b.460.1 8
36.31 odd 6 891.2.n.c.460.1 8
44.3 odd 10 363.2.e.k.202.1 4
44.7 even 10 363.2.a.i.1.2 2
44.15 odd 10 363.2.a.d.1.1 2
44.19 even 10 363.2.e.b.202.1 4
44.27 odd 10 33.2.e.b.16.1 ✓ 4
44.31 odd 10 363.2.e.k.124.1 4
44.35 even 10 363.2.e.b.124.1 4
44.39 even 10 363.2.e.f.148.1 4
44.43 even 2 363.2.e.f.130.1 4
132.59 even 10 1089.2.a.t.1.2 2
132.71 even 10 99.2.f.a.82.1 4
132.95 odd 10 1089.2.a.l.1.1 2
220.27 even 20 825.2.bx.d.49.2 8
220.59 odd 10 9075.2.a.cb.1.2 2
220.139 even 10 9075.2.a.u.1.1 2
220.159 odd 10 825.2.n.c.676.1 4
220.203 even 20 825.2.bx.d.49.1 8
396.115 odd 30 891.2.n.c.676.1 8
396.203 even 30 891.2.n.b.379.1 8
396.247 odd 30 891.2.n.c.379.1 8
396.335 even 30 891.2.n.b.676.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 44.27 odd 10
33.2.e.b.31.1 yes 4 4.3 odd 2
99.2.f.a.64.1 4 12.11 even 2
99.2.f.a.82.1 4 132.71 even 10
363.2.a.d.1.1 2 44.15 odd 10
363.2.a.i.1.2 2 44.7 even 10
363.2.e.b.124.1 4 44.35 even 10
363.2.e.b.202.1 4 44.19 even 10
363.2.e.f.130.1 4 44.43 even 2
363.2.e.f.148.1 4 44.39 even 10
363.2.e.k.124.1 4 44.31 odd 10
363.2.e.k.202.1 4 44.3 odd 10
528.2.y.b.49.1 4 11.5 even 5 inner
528.2.y.b.97.1 4 1.1 even 1 trivial
825.2.n.c.526.1 4 20.19 odd 2
825.2.n.c.676.1 4 220.159 odd 10
825.2.bx.d.49.1 8 220.203 even 20
825.2.bx.d.49.2 8 220.27 even 20
825.2.bx.d.724.1 8 20.7 even 4
825.2.bx.d.724.2 8 20.3 even 4
891.2.n.b.379.1 8 396.203 even 30
891.2.n.b.460.1 8 36.23 even 6
891.2.n.b.676.1 8 396.335 even 30
891.2.n.b.757.1 8 36.11 even 6
891.2.n.c.379.1 8 396.247 odd 30
891.2.n.c.460.1 8 36.31 odd 6
891.2.n.c.676.1 8 396.115 odd 30
891.2.n.c.757.1 8 36.7 odd 6
1089.2.a.l.1.1 2 132.95 odd 10
1089.2.a.t.1.2 2 132.59 even 10
5808.2.a.ci.1.1 2 11.7 odd 10
5808.2.a.cj.1.1 2 11.4 even 5
9075.2.a.u.1.1 2 220.139 even 10
9075.2.a.cb.1.2 2 220.59 odd 10