↑

Properties

Label 578.2.f.b.35.10
Level $578$
Weight $2$
Character 578.35
Analytic conductor $4.615$
Analytic rank $0$
Dimension $224$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [578,2,Mod(35,578)] 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("578.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.f (of order \(17\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [224] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label 35.10
Character \(\chi\) \(=\) 578.35
Dual form 578.2.f.b.545.10

$q$-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+(0.739009 + 0.673696i) q^{2} +(0.263973 + 0.927767i) q^{3} +(0.0922684 + 0.995734i) q^{4} +(0.148902 + 0.197178i) q^{5} +(-0.429955 + 0.863466i) q^{6} +(2.02926 - 1.25646i) q^{7} +(-0.602635 + 0.798017i) q^{8} +(1.75958 - 1.08949i) q^{9} +(-0.0227982 + 0.246031i) q^{10} +(0.558927 + 6.03178i) q^{11} +(-0.899453 + 0.348450i) q^{12} +(3.42464 - 4.53495i) q^{13} +(2.34611 + 0.438565i) q^{14} +(-0.143630 + 0.190196i) q^{15} +(-0.982973 + 0.183750i) q^{16} +(-3.91504 + 1.29322i) q^{17} +(2.03433 + 0.380282i) q^{18} +(0.811924 - 0.740166i) q^{19} +(-0.182598 + 0.166460i) q^{20} +(1.70138 + 1.55101i) q^{21} +(-3.65053 + 4.83409i) q^{22} +(-6.56300 + 4.06364i) q^{23} +(-0.899453 - 0.348450i) q^{24} +(1.35161 - 4.75041i) q^{25} +(5.58601 - 1.04421i) q^{26} +(3.61379 + 3.29441i) q^{27} +(1.43834 + 1.90467i) q^{28} +(0.791554 + 8.54223i) q^{29} +(-0.234278 + 0.0437941i) q^{30} +(1.31429 - 1.74041i) q^{31} +(-0.850217 - 0.526432i) q^{32} +(-5.44855 + 2.11078i) q^{33} +(-3.76449 - 1.68185i) q^{34} +(0.549909 + 0.213036i) q^{35} +(1.24719 + 1.65155i) q^{36} +(-0.809899 - 0.313756i) q^{37} +1.09867 q^{38} +(5.11139 + 1.98016i) q^{39} -0.247085 q^{40} +(-2.52782 - 8.88438i) q^{41} +(0.212424 + 2.29242i) q^{42} +(-1.89747 - 0.354698i) q^{43} +(-5.95448 + 1.11309i) q^{44} +(0.476829 + 0.184724i) q^{45} +(-7.58777 - 1.41840i) q^{46} +(-3.85994 - 2.38997i) q^{47} +(-0.429955 - 0.863466i) q^{48} +(-0.580978 + 1.16676i) q^{49} +(4.19918 - 2.60002i) q^{50} +(-2.23327 - 3.29088i) q^{51} +(4.83159 + 2.99159i) q^{52} +(0.813377 - 0.503622i) q^{53} +(0.451197 + 4.86919i) q^{54} +(-1.10611 + 1.00835i) q^{55} +(-0.220222 + 2.37657i) q^{56} +(0.901028 + 0.557893i) q^{57} +(-5.16989 + 6.84605i) q^{58} +(-0.996875 - 2.00199i) q^{59} +(-0.202637 - 0.125468i) q^{60} +(2.59965 - 5.22080i) q^{61} +(2.14378 - 0.400742i) q^{62} +(2.20174 - 4.42170i) q^{63} +(-0.273663 - 0.961826i) q^{64} +1.40413 q^{65} +(-5.44855 - 2.11078i) q^{66} +(-7.07390 + 6.44871i) q^{67} +(-1.64894 - 3.77902i) q^{68} +(-5.50256 - 5.01625i) q^{69} +(0.262866 + 0.527906i) q^{70} +(8.38498 - 5.19176i) q^{71} +(-0.190955 + 2.06074i) q^{72} +(-2.98202 + 0.557436i) q^{73} +(-0.387146 - 0.777494i) q^{74} +4.76406 q^{75} +(0.811924 + 0.740166i) q^{76} +(8.71293 + 11.5378i) q^{77} +(2.44334 + 4.90688i) q^{78} +(11.4201 - 10.4108i) q^{79} +(-0.182598 - 0.166460i) q^{80} +(0.664953 - 1.33541i) q^{81} +(4.11728 - 8.26862i) q^{82} +(3.87204 - 13.6088i) q^{83} +(-1.38741 + 1.83723i) q^{84} +(-0.837954 - 0.579399i) q^{85} +(-1.16329 - 1.54044i) q^{86} +(-7.71625 + 2.98929i) q^{87} +(-5.15030 - 3.18893i) q^{88} +(-2.00750 - 2.65836i) q^{89} +(0.227933 + 0.457750i) q^{90} +(1.25147 - 13.5055i) q^{91} +(-4.65186 - 6.16006i) q^{92} +(1.96163 + 0.759940i) q^{93} +(-1.24242 - 4.36664i) q^{94} +(0.266842 + 0.0498814i) q^{95} +(0.263973 - 0.927767i) q^{96} +(-2.48361 + 1.53779i) q^{97} +(-1.21539 + 0.470844i) q^{98} +(7.55502 + 10.0045i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q - 14 q^{2} - 2 q^{3} - 14 q^{4} + 28 q^{5} - 2 q^{6} - 4 q^{7} - 14 q^{8} - 24 q^{9} - 6 q^{10} - 18 q^{11} - 2 q^{12} - 12 q^{13} - 4 q^{14} - 22 q^{15} - 14 q^{16} - 16 q^{17} - 24 q^{18} - 12 q^{19}+ \cdots - 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{7}{17}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.739009 + 0.673696i 0.522558 + 0.476375i
\(3\) 0.263973 + 0.927767i 0.152405 + 0.535647i 0.999988 + 0.00498383i \(0.00158641\pi\)
−0.847583 + 0.530663i \(0.821943\pi\)
\(4\) 0.0922684 + 0.995734i 0.0461342 + 0.497867i
\(5\) 0.148902 + 0.197178i 0.0665911 + 0.0881809i 0.830094 − 0.557624i \(-0.188287\pi\)
−0.763503 + 0.645805i \(0.776522\pi\)
\(6\) −0.429955 + 0.863466i −0.175528 + 0.352508i
\(7\) 2.02926 − 1.25646i 0.766988 − 0.474899i −0.0863450 − 0.996265i \(-0.527519\pi\)
0.853333 + 0.521367i \(0.174578\pi\)
\(8\) −0.602635 + 0.798017i −0.213064 + 0.282142i
\(9\) 1.75958 − 1.08949i 0.586527 − 0.363162i
\(10\) −0.0227982 + 0.246031i −0.00720941 + 0.0778019i
\(11\) 0.558927 + 6.03178i 0.168523 + 1.81865i 0.489987 + 0.871730i \(0.337001\pi\)
−0.321464 + 0.946922i \(0.604175\pi\)
\(12\) −0.899453 + 0.348450i −0.259650 + 0.100589i
\(13\) 3.42464 − 4.53495i 0.949823 − 1.25777i −0.0163477 − 0.999866i \(-0.505204\pi\)
0.966171 − 0.257903i \(-0.0830314\pi\)
\(14\) 2.34611 + 0.438565i 0.627026 + 0.117211i
\(15\) −0.143630 + 0.190196i −0.0370850 + 0.0491085i
\(16\) −0.982973 + 0.183750i −0.245743 + 0.0459374i
\(17\) −3.91504 + 1.29322i −0.949538 + 0.313653i
\(18\) 2.03433 + 0.380282i 0.479496 + 0.0896333i
\(19\) 0.811924 − 0.740166i 0.186268 − 0.169806i −0.575190 − 0.818020i \(-0.695072\pi\)
0.761458 + 0.648214i \(0.224484\pi\)
\(20\) −0.182598 + 0.166460i −0.0408302 + 0.0372217i
\(21\) 1.70138 + 1.55101i 0.371271 + 0.338458i
\(22\) −3.65053 + 4.83409i −0.778297 + 1.03063i
\(23\) −6.56300 + 4.06364i −1.36848 + 0.847327i −0.996714 − 0.0810002i \(-0.974189\pi\)
−0.371765 + 0.928327i \(0.621247\pi\)
\(24\) −0.899453 − 0.348450i −0.183600 − 0.0711271i
\(25\) 1.35161 − 4.75041i 0.270322 − 0.950082i
\(26\) 5.58601 − 1.04421i 1.09551 − 0.204786i
\(27\) 3.61379 + 3.29441i 0.695475 + 0.634009i
\(28\) 1.43834 + 1.90467i 0.271821 + 0.359949i
\(29\) 0.791554 + 8.54223i 0.146988 + 1.58625i 0.675522 + 0.737340i \(0.263918\pi\)
−0.528534 + 0.848912i \(0.677258\pi\)
\(30\) −0.234278 + 0.0437941i −0.0427731 + 0.00799568i
\(31\) 1.31429 − 1.74041i 0.236054 − 0.312586i −0.664535 − 0.747257i \(-0.731370\pi\)
0.900590 + 0.434671i \(0.143135\pi\)
\(32\) −0.850217 − 0.526432i −0.150299 − 0.0930609i
\(33\) −5.44855 + 2.11078i −0.948471 + 0.367440i
\(34\) −3.76449 − 1.68185i −0.645605 − 0.288434i
\(35\) 0.549909 + 0.213036i 0.0929515 + 0.0360096i
\(36\) 1.24719 + 1.65155i 0.207865 + 0.275258i
\(37\) −0.809899 − 0.313756i −0.133147 − 0.0515812i 0.293747 − 0.955883i \(-0.405098\pi\)
−0.426893 + 0.904302i \(0.640392\pi\)
\(38\) 1.09867 0.178227
\(39\) 5.11139 + 1.98016i 0.818477 + 0.317080i
\(40\) −0.247085 −0.0390676
\(41\) −2.52782 − 8.88438i −0.394780 − 1.38751i −0.863584 − 0.504205i \(-0.831786\pi\)
0.468804 − 0.883302i \(-0.344685\pi\)
\(42\) 0.212424 + 2.29242i 0.0327777 + 0.353728i
\(43\) −1.89747 − 0.354698i −0.289361 − 0.0540910i 0.0370677 − 0.999313i \(-0.488198\pi\)
−0.326429 + 0.945222i \(0.605845\pi\)
\(44\) −5.95448 + 1.11309i −0.897672 + 0.167804i
\(45\) 0.476829 + 0.184724i 0.0710814 + 0.0275371i
\(46\) −7.58777 − 1.41840i −1.11876 − 0.209132i
\(47\) −3.85994 − 2.38997i −0.563030 − 0.348614i 0.215222 − 0.976565i \(-0.430952\pi\)
−0.778252 + 0.627952i \(0.783894\pi\)
\(48\) −0.429955 − 0.863466i −0.0620586 − 0.124631i
\(49\) −0.580978 + 1.16676i −0.0829968 + 0.166680i
\(50\) 4.19918 − 2.60002i 0.593854 − 0.367699i
\(51\) −2.23327 − 3.29088i −0.312721 − 0.460815i
\(52\) 4.83159 + 2.99159i 0.670021 + 0.414860i
\(53\) 0.813377 − 0.503622i 0.111726 − 0.0691778i −0.469432 − 0.882969i \(-0.655541\pi\)
0.581158 + 0.813791i \(0.302600\pi\)
\(54\) 0.451197 + 4.86919i 0.0614002 + 0.662613i
\(55\) −1.10611 + 1.00835i −0.149148 + 0.135966i
\(56\) −0.220222 + 2.37657i −0.0294284 + 0.317583i
\(57\) 0.901028 + 0.557893i 0.119344 + 0.0738947i
\(58\) −5.16989 + 6.84605i −0.678840 + 0.898930i
\(59\) −0.996875 − 2.00199i −0.129782 − 0.260638i 0.820735 − 0.571308i \(-0.193564\pi\)
−0.950518 + 0.310671i \(0.899446\pi\)
\(60\) −0.202637 − 0.125468i −0.0261604 − 0.0161978i
\(61\) 2.59965 − 5.22080i 0.332851 − 0.668455i −0.664002 − 0.747731i \(-0.731143\pi\)
0.996853 + 0.0792761i \(0.0252609\pi\)
\(62\) 2.14378 − 0.400742i 0.272260 − 0.0508943i
\(63\) 2.20174 − 4.42170i 0.277394 − 0.557082i
\(64\) −0.273663 − 0.961826i −0.0342079 − 0.120228i
\(65\) 1.40413 0.174161
\(66\) −5.44855 − 2.11078i −0.670670 − 0.259819i
\(67\) −7.07390 + 6.44871i −0.864215 + 0.787836i −0.978637 − 0.205596i \(-0.934087\pi\)
0.114422 + 0.993432i \(0.463498\pi\)
\(68\) −1.64894 − 3.77902i −0.199963 − 0.458274i
\(69\) −5.50256 − 5.01625i −0.662430 − 0.603885i
\(70\) 0.262866 + 0.527906i 0.0314185 + 0.0630969i
\(71\) 8.38498 − 5.19176i 0.995114 − 0.616148i 0.0706832 − 0.997499i \(-0.477482\pi\)
0.924431 + 0.381350i \(0.124541\pi\)
\(72\) −0.190955 + 2.06074i −0.0225043 + 0.242860i
\(73\) −2.98202 + 0.557436i −0.349019 + 0.0652430i −0.355340 − 0.934737i \(-0.615635\pi\)
0.00632108 + 0.999980i \(0.497988\pi\)
\(74\) −0.387146 − 0.777494i −0.0450048 − 0.0903818i
\(75\) 4.76406 0.550106
\(76\) 0.811924 + 0.740166i 0.0931340 + 0.0849029i
\(77\) 8.71293 + 11.5378i 0.992930 + 1.31485i
\(78\) 2.44334 + 4.90688i 0.276653 + 0.555595i
\(79\) 11.4201 − 10.4108i 1.28486 − 1.17130i 0.309448 − 0.950916i \(-0.399856\pi\)
0.975412 − 0.220388i \(-0.0707325\pi\)
\(80\) −0.182598 − 0.166460i −0.0204151 − 0.0186108i
\(81\) 0.664953 − 1.33541i 0.0738837 − 0.148378i
\(82\) 4.11728 − 8.26862i 0.454678 − 0.913117i
\(83\) 3.87204 − 13.6088i 0.425011 − 1.49376i −0.393530 − 0.919312i \(-0.628746\pi\)
0.818541 − 0.574448i \(-0.194783\pi\)
\(84\) −1.38741 + 1.83723i −0.151379 + 0.200458i
\(85\) −0.837954 − 0.579399i −0.0908889 − 0.0628446i
\(86\) −1.16329 − 1.54044i −0.125440 − 0.166110i
\(87\) −7.71625 + 2.98929i −0.827269 + 0.320486i
\(88\) −5.15030 − 3.18893i −0.549023 − 0.339941i
\(89\) −2.00750 − 2.65836i −0.212795 − 0.281786i 0.679084 − 0.734061i \(-0.262377\pi\)
−0.891879 + 0.452275i \(0.850613\pi\)
\(90\) 0.227933 + 0.457750i 0.0240262 + 0.0482511i
\(91\) 1.25147 − 13.5055i 0.131190 − 1.41576i
\(92\) −4.65186 − 6.16006i −0.484990 − 0.642230i
\(93\) 1.96163 + 0.759940i 0.203412 + 0.0788021i
\(94\) −1.24242 − 4.36664i −0.128145 − 0.450384i
\(95\) 0.266842 + 0.0498814i 0.0273774 + 0.00511772i
\(96\) 0.263973 − 0.927767i 0.0269416 − 0.0946899i
\(97\) −2.48361 + 1.53779i −0.252173 + 0.156139i −0.646697 − 0.762747i \(-0.723850\pi\)
0.394525 + 0.918885i \(0.370909\pi\)
\(98\) −1.21539 + 0.470844i −0.122773 + 0.0475625i
\(99\) 7.55502 + 10.0045i 0.759308 + 1.00549i
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 578.2.f.b.35.10 ✓ 224
289.256 even 17 inner 578.2.f.b.545.10 yes 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.2.f.b.35.10 ✓ 224 1.1 even 1 trivial
578.2.f.b.545.10 yes 224 289.256 even 17 inner