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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(26,567)] 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("567.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,16,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 26.15
Character \(\chi\) \(=\) 567.26
Dual form 567.2.s.g.458.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.24330 - 1.29517i) q^{2} +(2.35494 - 4.07887i) q^{4} -3.71400 q^{5} +(2.37028 - 1.17549i) q^{7} -7.01949i q^{8} +(-8.33163 + 4.81027i) q^{10} -3.36536i q^{11} +(-0.132449 + 0.0764695i) q^{13} +(3.79480 - 5.70689i) q^{14} +(-4.38157 - 7.58910i) q^{16} +(-0.989830 - 1.71444i) q^{17} +(0.839544 + 0.484711i) q^{19} +(-8.74624 + 15.1489i) q^{20} +(-4.35872 - 7.54952i) q^{22} +3.57073i q^{23} +8.79383 q^{25} +(-0.198082 + 0.343088i) q^{26} +(0.787200 - 12.4363i) q^{28} +(1.99809 + 1.15360i) q^{29} +(3.68784 + 2.12918i) q^{31} +(-7.50025 - 4.33027i) q^{32} +(-4.44097 - 2.56400i) q^{34} +(-8.80323 + 4.36577i) q^{35} +(1.45972 - 2.52831i) q^{37} +2.51113 q^{38} +26.0704i q^{40} +(4.95144 + 8.57615i) q^{41} +(-4.85872 + 8.41554i) q^{43} +(-13.7269 - 7.92520i) q^{44} +(4.62471 + 8.01023i) q^{46} +(-2.17774 - 3.77196i) q^{47} +(4.23646 - 5.57247i) q^{49} +(19.7272 - 11.3895i) q^{50} +0.720323i q^{52} +(5.00850 - 2.89166i) q^{53} +12.4990i q^{55} +(-8.25132 - 16.6382i) q^{56} +5.97643 q^{58} +(0.543251 - 0.940938i) q^{59} +(9.55242 - 5.51510i) q^{61} +11.0306 q^{62} -4.90749 q^{64} +(0.491917 - 0.284008i) q^{65} +(-2.12714 + 3.68431i) q^{67} -9.32394 q^{68} +(-14.0939 + 21.1954i) q^{70} +12.2311i q^{71} +(-9.75901 + 5.63436i) q^{73} -7.56236i q^{74} +(3.95414 - 2.28293i) q^{76} +(-3.95594 - 7.97685i) q^{77} +(-1.31193 - 2.27232i) q^{79} +(16.2732 + 28.1860i) q^{80} +(22.2152 + 12.8259i) q^{82} +(7.94920 - 13.7684i) q^{83} +(3.67623 + 6.36742i) q^{85} +25.1715i q^{86} -23.6231 q^{88} +(-0.743409 + 1.28762i) q^{89} +(-0.224053 + 0.336946i) q^{91} +(14.5646 + 8.40885i) q^{92} +(-9.77066 - 5.64110i) q^{94} +(-3.11807 - 1.80022i) q^{95} +(11.5109 + 6.64580i) q^{97} +(2.28635 - 17.9877i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} + 4 q^{7} - 12 q^{13} - 28 q^{16} - 12 q^{22} + 32 q^{25} - 16 q^{28} + 48 q^{31} - 4 q^{37} - 28 q^{43} + 12 q^{46} - 16 q^{49} - 72 q^{58} - 12 q^{61} - 80 q^{64} - 20 q^{67} - 60 q^{70}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24330 − 1.29517i 1.58625 − 0.915824i 0.592337 − 0.805690i \(-0.298205\pi\)
0.993917 − 0.110134i \(-0.0351280\pi\)
\(3\) 0 0
\(4\) 2.35494 − 4.07887i 1.17747 − 2.03943i
\(5\) −3.71400 −1.66095 −0.830477 − 0.557053i \(-0.811932\pi\)
−0.830477 + 0.557053i \(0.811932\pi\)
\(6\) 0 0
\(7\) 2.37028 − 1.17549i 0.895882 − 0.444292i
\(8\) − 7.01949i − 2.48176i
\(9\) 0 0
\(10\) −8.33163 + 4.81027i −2.63469 + 1.52114i
\(11\) − 3.36536i − 1.01469i −0.861742 − 0.507347i \(-0.830626\pi\)
0.861742 − 0.507347i \(-0.169374\pi\)
\(12\) 0 0
\(13\) −0.132449 + 0.0764695i −0.0367348 + 0.0212088i −0.518255 − 0.855226i \(-0.673418\pi\)
0.481520 + 0.876435i \(0.340085\pi\)
\(14\) 3.79480 − 5.70689i 1.01420 − 1.52523i
\(15\) 0 0
\(16\) −4.38157 − 7.58910i −1.09539 − 1.89728i
\(17\) −0.989830 − 1.71444i −0.240069 − 0.415812i 0.720665 − 0.693284i \(-0.243837\pi\)
−0.960734 + 0.277472i \(0.910503\pi\)
\(18\) 0 0
\(19\) 0.839544 + 0.484711i 0.192605 + 0.111200i 0.593201 − 0.805054i \(-0.297864\pi\)
−0.400597 + 0.916254i \(0.631197\pi\)
\(20\) −8.74624 + 15.1489i −1.95572 + 3.38740i
\(21\) 0 0
\(22\) −4.35872 − 7.54952i −0.929281 − 1.60956i
\(23\) 3.57073i 0.744550i 0.928123 + 0.372275i \(0.121422\pi\)
−0.928123 + 0.372275i \(0.878578\pi\)
\(24\) 0 0
\(25\) 8.79383 1.75877
\(26\) −0.198082 + 0.343088i −0.0388471 + 0.0672852i
\(27\) 0 0
\(28\) 0.787200 − 12.4363i 0.148767 − 2.35023i
\(29\) 1.99809 + 1.15360i 0.371036 + 0.214218i 0.673911 − 0.738812i \(-0.264613\pi\)
−0.302875 + 0.953030i \(0.597946\pi\)
\(30\) 0 0
\(31\) 3.68784 + 2.12918i 0.662356 + 0.382412i 0.793174 − 0.608995i \(-0.208427\pi\)
−0.130818 + 0.991406i \(0.541760\pi\)
\(32\) −7.50025 − 4.33027i −1.32587 − 0.765491i
\(33\) 0 0
\(34\) −4.44097 − 2.56400i −0.761621 − 0.439722i
\(35\) −8.80323 + 4.36577i −1.48802 + 0.737949i
\(36\) 0 0
\(37\) 1.45972 − 2.52831i 0.239977 − 0.415652i −0.720731 − 0.693215i \(-0.756194\pi\)
0.960707 + 0.277563i \(0.0895268\pi\)
\(38\) 2.51113 0.407360
\(39\) 0 0
\(40\) 26.0704i 4.12210i
\(41\) 4.95144 + 8.57615i 0.773286 + 1.33937i 0.935753 + 0.352656i \(0.114721\pi\)
−0.162468 + 0.986714i \(0.551945\pi\)
\(42\) 0 0
\(43\) −4.85872 + 8.41554i −0.740947 + 1.28336i 0.211117 + 0.977461i \(0.432290\pi\)
−0.952064 + 0.305897i \(0.901044\pi\)
\(44\) −13.7269 − 7.92520i −2.06940 − 1.19477i
\(45\) 0 0
\(46\) 4.62471 + 8.01023i 0.681876 + 1.18104i
\(47\) −2.17774 − 3.77196i −0.317656 − 0.550197i 0.662342 − 0.749201i \(-0.269562\pi\)
−0.979999 + 0.199005i \(0.936229\pi\)
\(48\) 0 0
\(49\) 4.23646 − 5.57247i 0.605208 − 0.796067i
\(50\) 19.7272 − 11.3895i 2.78985 − 1.61072i
\(51\) 0 0
\(52\) 0.720323i 0.0998908i
\(53\) 5.00850 − 2.89166i 0.687970 − 0.397200i −0.114881 − 0.993379i \(-0.536649\pi\)
0.802851 + 0.596179i \(0.203315\pi\)
\(54\) 0 0
\(55\) 12.4990i 1.68536i
\(56\) −8.25132 − 16.6382i −1.10263 − 2.22337i
\(57\) 0 0
\(58\) 5.97643 0.784744
\(59\) 0.543251 − 0.940938i 0.0707252 − 0.122500i −0.828494 − 0.559998i \(-0.810802\pi\)
0.899219 + 0.437498i \(0.144135\pi\)
\(60\) 0 0
\(61\) 9.55242 − 5.51510i 1.22306 − 0.706136i 0.257493 − 0.966280i \(-0.417104\pi\)
0.965570 + 0.260145i \(0.0837702\pi\)
\(62\) 11.0306 1.40089
\(63\) 0 0
\(64\) −4.90749 −0.613436
\(65\) 0.491917 − 0.284008i 0.0610147 − 0.0352269i
\(66\) 0 0
\(67\) −2.12714 + 3.68431i −0.259871 + 0.450111i −0.966207 − 0.257766i \(-0.917014\pi\)
0.706336 + 0.707877i \(0.250347\pi\)
\(68\) −9.32394 −1.13069
\(69\) 0 0
\(70\) −14.0939 + 21.1954i −1.68454 + 2.53334i
\(71\) 12.2311i 1.45157i 0.687922 + 0.725784i \(0.258523\pi\)
−0.687922 + 0.725784i \(0.741477\pi\)
\(72\) 0 0
\(73\) −9.75901 + 5.63436i −1.14221 + 0.659452i −0.946976 − 0.321305i \(-0.895878\pi\)
−0.195229 + 0.980758i \(0.562545\pi\)
\(74\) − 7.56236i − 0.879106i
\(75\) 0 0
\(76\) 3.95414 − 2.28293i 0.453571 − 0.261869i
\(77\) −3.95594 − 7.97685i −0.450821 − 0.909046i
\(78\) 0 0
\(79\) −1.31193 − 2.27232i −0.147603 − 0.255657i 0.782738 − 0.622352i \(-0.213823\pi\)
−0.930341 + 0.366695i \(0.880489\pi\)
\(80\) 16.2732 + 28.1860i 1.81940 + 3.15129i
\(81\) 0 0
\(82\) 22.2152 + 12.8259i 2.45325 + 1.41639i
\(83\) 7.94920 − 13.7684i 0.872538 − 1.51128i 0.0131751 − 0.999913i \(-0.495806\pi\)
0.859363 − 0.511367i \(-0.170861\pi\)
\(84\) 0 0
\(85\) 3.67623 + 6.36742i 0.398743 + 0.690644i
\(86\) 25.1715i 2.71431i
\(87\) 0 0
\(88\) −23.6231 −2.51823
\(89\) −0.743409 + 1.28762i −0.0788012 + 0.136488i −0.902733 − 0.430201i \(-0.858443\pi\)
0.823932 + 0.566689i \(0.191776\pi\)
\(90\) 0 0
\(91\) −0.224053 + 0.336946i −0.0234871 + 0.0353216i
\(92\) 14.5646 + 8.40885i 1.51846 + 0.876683i
\(93\) 0 0
\(94\) −9.77066 − 5.64110i −1.00777 − 0.581835i
\(95\) −3.11807 − 1.80022i −0.319907 − 0.184698i
\(96\) 0 0
\(97\) 11.5109 + 6.64580i 1.16875 + 0.674779i 0.953385 − 0.301755i \(-0.0975725\pi\)
0.215365 + 0.976534i \(0.430906\pi\)
\(98\) 2.28635 − 17.9877i 0.230957 − 1.81703i
\(99\) 0 0
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 567.2.s.g.26.15 32
3.2 odd 2 inner 567.2.s.g.26.2 32
7.3 odd 6 567.2.i.g.269.15 32
9.2 odd 6 567.2.p.e.404.15 yes 32
9.4 even 3 567.2.i.g.215.15 32
9.5 odd 6 567.2.i.g.215.2 32
9.7 even 3 567.2.p.e.404.2 yes 32
21.17 even 6 567.2.i.g.269.2 32
63.31 odd 6 inner 567.2.s.g.458.2 32
63.38 even 6 567.2.p.e.80.2 ✓ 32
63.52 odd 6 567.2.p.e.80.15 yes 32
63.59 even 6 inner 567.2.s.g.458.15 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.2 32 9.5 odd 6
567.2.i.g.215.15 32 9.4 even 3
567.2.i.g.269.2 32 21.17 even 6
567.2.i.g.269.15 32 7.3 odd 6
567.2.p.e.80.2 ✓ 32 63.38 even 6
567.2.p.e.80.15 yes 32 63.52 odd 6
567.2.p.e.404.2 yes 32 9.7 even 3
567.2.p.e.404.15 yes 32 9.2 odd 6
567.2.s.g.26.2 32 3.2 odd 2 inner
567.2.s.g.26.15 32 1.1 even 1 trivial
567.2.s.g.458.2 32 63.31 odd 6 inner
567.2.s.g.458.15 32 63.59 even 6 inner