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 = [4,0,1,0,1] 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 264)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 289.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 528.289
Dual form 528.2.y.h.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.309017 - 0.951057i) q^{3} +(-0.309017 - 0.224514i) q^{5} +(1.30902 - 4.02874i) q^{7} +(-0.809017 + 0.587785i) q^{9} +(-3.04508 - 1.31433i) q^{11} +(-1.80902 + 1.31433i) q^{13} +(-0.118034 + 0.363271i) q^{15} +(1.50000 + 1.08981i) q^{17} +(-0.118034 - 0.363271i) q^{19} -4.23607 q^{21} -6.23607 q^{23} +(-1.50000 - 4.61653i) q^{25} +(0.809017 + 0.587785i) q^{27} +(0.145898 - 0.449028i) q^{29} +(6.97214 - 5.06555i) q^{31} +(-0.309017 + 3.30220i) q^{33} +(-1.30902 + 0.951057i) q^{35} +(1.16312 - 3.57971i) q^{37} +(1.80902 + 1.31433i) q^{39} +(-1.54508 - 4.75528i) q^{41} -11.4721 q^{43} +0.381966 q^{45} +(-0.0450850 - 0.138757i) q^{47} +(-8.85410 - 6.43288i) q^{49} +(0.572949 - 1.76336i) q^{51} +(7.73607 - 5.62058i) q^{53} +(0.645898 + 1.08981i) q^{55} +(-0.309017 + 0.224514i) q^{57} +(-4.50000 + 13.8496i) q^{59} +(5.54508 + 4.02874i) q^{61} +(1.30902 + 4.02874i) q^{63} +0.854102 q^{65} +4.56231 q^{67} +(1.92705 + 5.93085i) q^{69} +(10.1631 + 7.38394i) q^{71} +(1.14590 - 3.52671i) q^{73} +(-3.92705 + 2.85317i) q^{75} +(-9.28115 + 10.5474i) q^{77} +(11.5172 - 8.36775i) q^{79} +(0.309017 - 0.951057i) q^{81} +(-1.57295 - 1.14281i) q^{83} +(-0.218847 - 0.673542i) q^{85} -0.472136 q^{87} +3.47214 q^{89} +(2.92705 + 9.00854i) q^{91} +(-6.97214 - 5.06555i) q^{93} +(-0.0450850 + 0.138757i) q^{95} +(5.54508 - 4.02874i) q^{97} +(3.23607 - 0.726543i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + q^{5} + 3 q^{7} - q^{9} - q^{11} - 5 q^{13} + 4 q^{15} + 6 q^{17} + 4 q^{19} - 8 q^{21} - 16 q^{23} - 6 q^{25} + q^{27} + 14 q^{29} + 10 q^{31} + q^{33} - 3 q^{35} - 11 q^{37} + 5 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(1\) \(1\)

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
\(3\) −0.309017 − 0.951057i −0.178411 − 0.549093i
\(4\) 0 0
\(5\) −0.309017 − 0.224514i −0.138197 − 0.100406i 0.516539 − 0.856264i \(-0.327220\pi\)
−0.654736 + 0.755858i \(0.727220\pi\)
\(6\) 0 0
\(7\) 1.30902 − 4.02874i 0.494762 − 1.52272i −0.322566 − 0.946547i \(-0.604545\pi\)
0.817327 − 0.576173i \(-0.195455\pi\)
\(8\) 0 0
\(9\) −0.809017 + 0.587785i −0.269672 + 0.195928i
\(10\) 0 0
\(11\) −3.04508 − 1.31433i −0.918128 − 0.396285i
\(12\) 0 0
\(13\) −1.80902 + 1.31433i −0.501731 + 0.364529i −0.809678 − 0.586875i \(-0.800358\pi\)
0.307947 + 0.951404i \(0.400358\pi\)
\(14\) 0 0
\(15\) −0.118034 + 0.363271i −0.0304762 + 0.0937962i
\(16\) 0 0
\(17\) 1.50000 + 1.08981i 0.363803 + 0.264319i 0.754637 − 0.656143i \(-0.227813\pi\)
−0.390833 + 0.920461i \(0.627813\pi\)
\(18\) 0 0
\(19\) −0.118034 − 0.363271i −0.0270789 − 0.0833401i 0.936604 − 0.350390i \(-0.113951\pi\)
−0.963683 + 0.267050i \(0.913951\pi\)
\(20\) 0 0
\(21\) −4.23607 −0.924386
\(22\) 0 0
\(23\) −6.23607 −1.30031 −0.650155 − 0.759802i \(-0.725296\pi\)
−0.650155 + 0.759802i \(0.725296\pi\)
\(24\) 0 0
\(25\) −1.50000 − 4.61653i −0.300000 − 0.923305i
\(26\) 0 0
\(27\) 0.809017 + 0.587785i 0.155695 + 0.113119i
\(28\) 0 0
\(29\) 0.145898 − 0.449028i 0.0270926 − 0.0833824i −0.936596 − 0.350411i \(-0.886042\pi\)
0.963689 + 0.267029i \(0.0860419\pi\)
\(30\) 0 0
\(31\) 6.97214 − 5.06555i 1.25223 − 0.909800i 0.253883 − 0.967235i \(-0.418292\pi\)
0.998349 + 0.0574346i \(0.0182921\pi\)
\(32\) 0 0
\(33\) −0.309017 + 3.30220i −0.0537930 + 0.574839i
\(34\) 0 0
\(35\) −1.30902 + 0.951057i −0.221264 + 0.160758i
\(36\) 0 0
\(37\) 1.16312 − 3.57971i 0.191216 − 0.588501i −0.808784 − 0.588105i \(-0.799874\pi\)
1.00000 0.000395703i \(-0.000125956\pi\)
\(38\) 0 0
\(39\) 1.80902 + 1.31433i 0.289675 + 0.210461i
\(40\) 0 0
\(41\) −1.54508 − 4.75528i −0.241302 − 0.742650i −0.996223 − 0.0868346i \(-0.972325\pi\)
0.754921 − 0.655816i \(-0.227675\pi\)
\(42\) 0 0
\(43\) −11.4721 −1.74948 −0.874742 − 0.484589i \(-0.838969\pi\)
−0.874742 + 0.484589i \(0.838969\pi\)
\(44\) 0 0
\(45\) 0.381966 0.0569401
\(46\) 0 0
\(47\) −0.0450850 − 0.138757i −0.00657632 − 0.0202398i 0.947715 − 0.319119i \(-0.103387\pi\)
−0.954291 + 0.298880i \(0.903387\pi\)
\(48\) 0 0
\(49\) −8.85410 − 6.43288i −1.26487 − 0.918983i
\(50\) 0 0
\(51\) 0.572949 − 1.76336i 0.0802289 − 0.246919i
\(52\) 0 0
\(53\) 7.73607 − 5.62058i 1.06263 − 0.772046i 0.0880574 − 0.996115i \(-0.471934\pi\)
0.974573 + 0.224069i \(0.0719341\pi\)
\(54\) 0 0
\(55\) 0.645898 + 1.08981i 0.0870929 + 0.146950i
\(56\) 0 0
\(57\) −0.309017 + 0.224514i −0.0409303 + 0.0297376i
\(58\) 0 0
\(59\) −4.50000 + 13.8496i −0.585850 + 1.80306i 0.00997934 + 0.999950i \(0.496823\pi\)
−0.595829 + 0.803111i \(0.703177\pi\)
\(60\) 0 0
\(61\) 5.54508 + 4.02874i 0.709975 + 0.515827i 0.883166 − 0.469061i \(-0.155408\pi\)
−0.173190 + 0.984888i \(0.555408\pi\)
\(62\) 0 0
\(63\) 1.30902 + 4.02874i 0.164921 + 0.507574i
\(64\) 0 0
\(65\) 0.854102 0.105938
\(66\) 0 0
\(67\) 4.56231 0.557374 0.278687 − 0.960382i \(-0.410101\pi\)
0.278687 + 0.960382i \(0.410101\pi\)
\(68\) 0 0
\(69\) 1.92705 + 5.93085i 0.231990 + 0.713991i
\(70\) 0 0
\(71\) 10.1631 + 7.38394i 1.20614 + 0.876312i 0.994875 − 0.101116i \(-0.0322415\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(72\) 0 0
\(73\) 1.14590 − 3.52671i 0.134117 − 0.412770i −0.861334 − 0.508038i \(-0.830371\pi\)
0.995452 + 0.0952680i \(0.0303708\pi\)
\(74\) 0 0
\(75\) −3.92705 + 2.85317i −0.453457 + 0.329456i
\(76\) 0 0
\(77\) −9.28115 + 10.5474i −1.05769 + 1.20199i
\(78\) 0 0
\(79\) 11.5172 − 8.36775i 1.29579 − 0.941446i 0.295884 − 0.955224i \(-0.404386\pi\)
0.999905 + 0.0137785i \(0.00438597\pi\)
\(80\) 0 0
\(81\) 0.309017 − 0.951057i 0.0343352 − 0.105673i
\(82\) 0 0
\(83\) −1.57295 − 1.14281i −0.172654 − 0.125440i 0.498103 − 0.867118i \(-0.334030\pi\)
−0.670756 + 0.741678i \(0.734030\pi\)
\(84\) 0 0
\(85\) −0.218847 − 0.673542i −0.0237373 − 0.0730559i
\(86\) 0 0
\(87\) −0.472136 −0.0506183
\(88\) 0 0
\(89\) 3.47214 0.368046 0.184023 − 0.982922i \(-0.441088\pi\)
0.184023 + 0.982922i \(0.441088\pi\)
\(90\) 0 0
\(91\) 2.92705 + 9.00854i 0.306838 + 0.944351i
\(92\) 0 0
\(93\) −6.97214 − 5.06555i −0.722977 − 0.525273i
\(94\) 0 0
\(95\) −0.0450850 + 0.138757i −0.00462562 + 0.0142362i
\(96\) 0 0
\(97\) 5.54508 − 4.02874i 0.563018 − 0.409057i −0.269544 − 0.962988i \(-0.586873\pi\)
0.832562 + 0.553931i \(0.186873\pi\)
\(98\) 0 0
\(99\) 3.23607 − 0.726543i 0.325237 − 0.0730203i
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.h.289.1 4
4.3 odd 2 264.2.q.b.25.1 ✓ 4
11.2 odd 10 5808.2.a.bv.1.1 2
11.4 even 5 inner 528.2.y.h.433.1 4
11.9 even 5 5808.2.a.bw.1.1 2
12.11 even 2 792.2.r.d.289.1 4
44.15 odd 10 264.2.q.b.169.1 yes 4
44.31 odd 10 2904.2.a.z.1.1 2
44.35 even 10 2904.2.a.ba.1.1 2
132.35 odd 10 8712.2.a.bc.1.2 2
132.59 even 10 792.2.r.d.433.1 4
132.119 even 10 8712.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
264.2.q.b.25.1 ✓ 4 4.3 odd 2
264.2.q.b.169.1 yes 4 44.15 odd 10
528.2.y.h.289.1 4 1.1 even 1 trivial
528.2.y.h.433.1 4 11.4 even 5 inner
792.2.r.d.289.1 4 12.11 even 2
792.2.r.d.433.1 4 132.59 even 10
2904.2.a.z.1.1 2 44.31 odd 10
2904.2.a.ba.1.1 2 44.35 even 10
5808.2.a.bv.1.1 2 11.2 odd 10
5808.2.a.bw.1.1 2 11.9 even 5
8712.2.a.ba.1.2 2 132.119 even 10
8712.2.a.bc.1.2 2 132.35 odd 10