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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [71,2,Mod(2,71)] 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("71.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(71, base_ring=CyclotomicField(70)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.g (of order \(35\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.566937854351\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(5\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label 16.5
Character \(\chi\) \(=\) 71.16
Dual form 71.2.g.a.40.5

$q$-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+(2.13757 + 0.387912i) q^{2} +(-1.78691 + 0.493156i) q^{3} +(2.54627 + 0.955632i) q^{4} +(-0.00692986 + 0.0213279i) q^{5} +(-4.01095 + 0.360992i) q^{6} +(-1.29201 - 2.40095i) q^{7} +(1.34221 + 0.801931i) q^{8} +(0.374497 - 0.223752i) q^{9} +(-0.0230864 + 0.0429018i) q^{10} +(1.10135 - 2.57674i) q^{11} +(-5.02123 - 0.451919i) q^{12} +(2.16765 + 5.07147i) q^{13} +(-1.83040 - 5.63340i) q^{14} +(0.00186505 - 0.0415285i) q^{15} +(-1.53826 - 1.34394i) q^{16} +(-4.52281 + 3.28601i) q^{17} +(0.887311 - 0.333013i) q^{18} +(-0.0803815 - 1.78983i) q^{19} +(-0.0380269 + 0.0476843i) q^{20} +(3.49275 + 3.65313i) q^{21} +(3.35377 - 5.08075i) q^{22} +(6.37911 + 3.07202i) q^{23} +(-2.79388 - 0.771061i) q^{24} +(4.04468 + 2.93863i) q^{25} +(2.66622 + 11.6815i) q^{26} +(3.28425 - 3.43505i) q^{27} +(-0.995377 - 7.34817i) q^{28} +(-3.03565 - 4.59881i) q^{29} +(0.0200961 - 0.0880468i) q^{30} +(-4.13294 + 3.61084i) q^{31} +(-4.71650 - 5.91431i) q^{32} +(-0.697282 + 5.14755i) q^{33} +(-10.9425 + 5.26964i) q^{34} +(0.0601608 - 0.0109176i) q^{35} +(1.16740 - 0.211851i) q^{36} +(5.61345 - 2.70330i) q^{37} +(0.522477 - 3.85708i) q^{38} +(-6.37442 - 7.99326i) q^{39} +(-0.0264048 + 0.0230692i) q^{40} +(0.145743 - 0.638544i) q^{41} +(6.04891 + 9.16370i) q^{42} +(0.361507 + 2.66876i) q^{43} +(5.26676 - 5.50860i) q^{44} +(0.00217694 + 0.00953781i) q^{45} +(12.4441 + 9.04119i) q^{46} +(-6.66098 - 1.83831i) q^{47} +(3.41151 + 1.64290i) q^{48} +(-0.239016 + 0.362093i) q^{49} +(7.50586 + 7.85052i) q^{50} +(6.46133 - 8.10225i) q^{51} +(0.672969 + 14.9848i) q^{52} +(-6.99687 + 2.62597i) q^{53} +(8.35281 - 6.06867i) q^{54} +(0.0473243 + 0.0413460i) q^{55} +(0.191257 - 4.25868i) q^{56} +(1.02630 + 3.15863i) q^{57} +(-4.70499 - 11.0079i) q^{58} +(-5.38067 - 0.484269i) q^{59} +(0.0444349 - 0.103961i) q^{60} +(7.05785 - 13.1157i) q^{61} +(-10.2351 + 6.11521i) q^{62} +(-1.02107 - 0.610061i) q^{63} +(-5.85174 - 10.8744i) q^{64} +(-0.123185 + 0.0110869i) q^{65} +(-3.48729 + 10.7328i) q^{66} +(3.55888 + 1.33567i) q^{67} +(-14.6565 + 4.04494i) q^{68} +(-12.9139 - 2.34352i) q^{69} +0.132833 q^{70} +(-8.03904 + 2.52464i) q^{71} +0.682085 q^{72} +(4.85109 + 0.880343i) q^{73} +(13.0478 - 3.60097i) q^{74} +(-8.67668 - 3.25641i) q^{75} +(1.50575 - 4.63422i) q^{76} +(-7.60960 + 0.684876i) q^{77} +(-10.5251 - 19.5589i) q^{78} +(0.736717 + 0.440168i) q^{79} +(0.0393234 - 0.0234946i) q^{80} +(-4.79481 + 8.91025i) q^{81} +(0.559236 - 1.30840i) q^{82} +(-8.32249 - 0.749038i) q^{83} +(5.40244 + 12.6396i) q^{84} +(-0.0387413 - 0.119234i) q^{85} +(-0.262494 + 5.84489i) q^{86} +(7.69236 + 6.72061i) q^{87} +(3.54461 - 2.57531i) q^{88} +(11.1385 - 4.18034i) q^{89} +(0.000953540 + 0.0212322i) q^{90} +(9.37574 - 11.7568i) q^{91} +(13.3072 + 13.9183i) q^{92} +(5.60448 - 8.49042i) q^{93} +(-13.5252 - 6.51340i) q^{94} +(0.0387304 + 0.0106889i) q^{95} +(11.3446 + 8.24236i) q^{96} +(2.49858 + 10.9470i) q^{97} +(-0.651374 + 0.681284i) q^{98} +(-0.164097 - 1.21141i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 22 q^{2} - 20 q^{3} - 18 q^{4} - 20 q^{5} - 20 q^{6} - 27 q^{7} - 27 q^{8} - 11 q^{9} - 8 q^{10} - 27 q^{11} + 3 q^{12} - 31 q^{13} + 2 q^{14} + 12 q^{15} + 30 q^{16} + 9 q^{17} + 27 q^{18} - 31 q^{19}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{12}{35}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.13757 + 0.387912i 1.51149 + 0.274295i 0.870084 − 0.492904i \(-0.164065\pi\)
0.641409 + 0.767199i \(0.278350\pi\)
\(3\) −1.78691 + 0.493156i −1.03167 + 0.284724i −0.740578 − 0.671971i \(-0.765448\pi\)
−0.291095 + 0.956694i \(0.594020\pi\)
\(4\) 2.54627 + 0.955632i 1.27314 + 0.477816i
\(5\) −0.00692986 + 0.0213279i −0.00309913 + 0.00953813i −0.952594 − 0.304244i \(-0.901596\pi\)
0.949495 + 0.313782i \(0.101596\pi\)
\(6\) −4.01095 + 0.360992i −1.63746 + 0.147374i
\(7\) −1.29201 − 2.40095i −0.488333 − 0.907475i −0.998866 − 0.0476168i \(-0.984837\pi\)
0.510532 − 0.859859i \(-0.329448\pi\)
\(8\) 1.34221 + 0.801931i 0.474541 + 0.283525i
\(9\) 0.374497 − 0.223752i 0.124832 − 0.0745838i
\(10\) −0.0230864 + 0.0429018i −0.00730057 + 0.0135667i
\(11\) 1.10135 − 2.57674i 0.332070 − 0.776917i −0.667410 − 0.744690i \(-0.732597\pi\)
0.999481 − 0.0322270i \(-0.0102599\pi\)
\(12\) −5.02123 − 0.451919i −1.44951 − 0.130458i
\(13\) 2.16765 + 5.07147i 0.601198 + 1.40657i 0.892984 + 0.450088i \(0.148608\pi\)
−0.291786 + 0.956484i \(0.594250\pi\)
\(14\) −1.83040 − 5.63340i −0.489196 − 1.50559i
\(15\) 0.00186505 − 0.0415285i 0.000481554 − 0.0107226i
\(16\) −1.53826 − 1.34394i −0.384566 − 0.335985i
\(17\) −4.52281 + 3.28601i −1.09694 + 0.796975i −0.980558 − 0.196230i \(-0.937130\pi\)
−0.116384 + 0.993204i \(0.537130\pi\)
\(18\) 0.887311 − 0.333013i 0.209141 − 0.0784920i
\(19\) −0.0803815 − 1.78983i −0.0184408 − 0.410616i −0.986889 − 0.161401i \(-0.948399\pi\)
0.968448 − 0.249215i \(-0.0801726\pi\)
\(20\) −0.0380269 + 0.0476843i −0.00850308 + 0.0106625i
\(21\) 3.49275 + 3.65313i 0.762180 + 0.797178i
\(22\) 3.35377 − 5.08075i 0.715027 − 1.08322i
\(23\) 6.37911 + 3.07202i 1.33014 + 0.640559i 0.957772 − 0.287528i \(-0.0928333\pi\)
0.372363 + 0.928087i \(0.378548\pi\)
\(24\) −2.79388 − 0.771061i −0.570298 − 0.157392i
\(25\) 4.04468 + 2.93863i 0.808936 + 0.587726i
\(26\) 2.66622 + 11.6815i 0.522889 + 2.29093i
\(27\) 3.28425 − 3.43505i 0.632053 − 0.661076i
\(28\) −0.995377 − 7.34817i −0.188109 − 1.38867i
\(29\) −3.03565 − 4.59881i −0.563706 − 0.853978i 0.435220 − 0.900324i \(-0.356671\pi\)
−0.998925 + 0.0463464i \(0.985242\pi\)
\(30\) 0.0200961 − 0.0880468i 0.00366903 − 0.0160751i
\(31\) −4.13294 + 3.61084i −0.742297 + 0.648525i −0.943632 − 0.330997i \(-0.892615\pi\)
0.201335 + 0.979523i \(0.435472\pi\)
\(32\) −4.71650 − 5.91431i −0.833768 − 1.04551i
\(33\) −0.697282 + 5.14755i −0.121381 + 0.896073i
\(34\) −10.9425 + 5.26964i −1.87663 + 0.903735i
\(35\) 0.0601608 − 0.0109176i 0.0101690 − 0.00184541i
\(36\) 1.16740 − 0.211851i 0.194566 − 0.0353085i
\(37\) 5.61345 − 2.70330i 0.922846 − 0.444419i 0.0887597 − 0.996053i \(-0.471710\pi\)
0.834086 + 0.551634i \(0.185995\pi\)
\(38\) 0.522477 − 3.85708i 0.0847569 − 0.625701i
\(39\) −6.37442 − 7.99326i −1.02072 − 1.27995i
\(40\) −0.0264048 + 0.0230692i −0.00417497 + 0.00364756i
\(41\) 0.145743 − 0.638544i 0.0227613 − 0.0997238i −0.962271 − 0.272092i \(-0.912285\pi\)
0.985033 + 0.172368i \(0.0551418\pi\)
\(42\) 6.04891 + 9.16370i 0.933367 + 1.41399i
\(43\) 0.361507 + 2.66876i 0.0551294 + 0.406981i 0.997396 + 0.0721181i \(0.0229758\pi\)
−0.942267 + 0.334863i \(0.891310\pi\)
\(44\) 5.26676 − 5.50860i 0.793994 − 0.830453i
\(45\) 0.00217694 + 0.00953781i 0.000324519 + 0.00142181i
\(46\) 12.4441 + 9.04119i 1.83479 + 1.33305i
\(47\) −6.66098 − 1.83831i −0.971604 − 0.268146i −0.256019 − 0.966672i \(-0.582411\pi\)
−0.715585 + 0.698526i \(0.753840\pi\)
\(48\) 3.41151 + 1.64290i 0.492409 + 0.237132i
\(49\) −0.239016 + 0.362093i −0.0341451 + 0.0517276i
\(50\) 7.50586 + 7.85052i 1.06149 + 1.11023i
\(51\) 6.46133 − 8.10225i 0.904767 − 1.13454i
\(52\) 0.672969 + 14.9848i 0.0933240 + 2.07802i
\(53\) −6.99687 + 2.62597i −0.961093 + 0.360704i −0.782105 − 0.623147i \(-0.785854\pi\)
−0.178988 + 0.983851i \(0.557282\pi\)
\(54\) 8.35281 − 6.06867i 1.13667 − 0.825842i
\(55\) 0.0473243 + 0.0413460i 0.00638121 + 0.00557510i
\(56\) 0.191257 − 4.25868i 0.0255578 − 0.569089i
\(57\) 1.02630 + 3.15863i 0.135937 + 0.418371i
\(58\) −4.70499 − 11.0079i −0.617795 − 1.44540i
\(59\) −5.38067 − 0.484269i −0.700503 − 0.0630464i −0.266338 − 0.963880i \(-0.585814\pi\)
−0.434165 + 0.900833i \(0.642957\pi\)
\(60\) 0.0444349 − 0.103961i 0.00573652 − 0.0134213i
\(61\) 7.05785 − 13.1157i 0.903665 − 1.67929i 0.186419 − 0.982470i \(-0.440312\pi\)
0.717246 − 0.696820i \(-0.245402\pi\)
\(62\) −10.2351 + 6.11521i −1.29986 + 0.776632i
\(63\) −1.02107 − 0.610061i −0.128643 − 0.0768605i
\(64\) −5.85174 − 10.8744i −0.731467 − 1.35929i
\(65\) −0.123185 + 0.0110869i −0.0152793 + 0.00137516i
\(66\) −3.48729 + 10.7328i −0.429255 + 1.32111i
\(67\) 3.55888 + 1.33567i 0.434787 + 0.163178i 0.559164 − 0.829057i \(-0.311122\pi\)
−0.124378 + 0.992235i \(0.539693\pi\)
\(68\) −14.6565 + 4.04494i −1.77736 + 0.490521i
\(69\) −12.9139 − 2.34352i −1.55465 − 0.282127i
\(70\) 0.132833 0.0158766
\(71\) −8.03904 + 2.52464i −0.954059 + 0.299620i
\(72\) 0.682085 0.0803845
\(73\) 4.85109 + 0.880343i 0.567777 + 0.103036i 0.454852 − 0.890567i \(-0.349692\pi\)
0.112925 + 0.993603i \(0.463978\pi\)
\(74\) 13.0478 − 3.60097i 1.51678 − 0.418604i
\(75\) −8.67668 − 3.25641i −1.00190 − 0.376018i
\(76\) 1.50575 − 4.63422i 0.172721 − 0.531581i
\(77\) −7.60960 + 0.684876i −0.867194 + 0.0780489i
\(78\) −10.5251 − 19.5589i −1.19173 − 2.21461i
\(79\) 0.736717 + 0.440168i 0.0828871 + 0.0495227i 0.553745 − 0.832686i \(-0.313198\pi\)
−0.470858 + 0.882209i \(0.656055\pi\)
\(80\) 0.0393234 − 0.0234946i 0.00439649 − 0.00262678i
\(81\) −4.79481 + 8.91025i −0.532756 + 0.990027i
\(82\) 0.559236 − 1.30840i 0.0617573 − 0.144488i
\(83\) −8.32249 − 0.749038i −0.913512 − 0.0822176i −0.377099 − 0.926173i \(-0.623078\pi\)
−0.536413 + 0.843956i \(0.680221\pi\)
\(84\) 5.40244 + 12.6396i 0.589454 + 1.37910i
\(85\) −0.0387413 − 0.119234i −0.00420209 − 0.0129327i
\(86\) −0.262494 + 5.84489i −0.0283055 + 0.630271i
\(87\) 7.69236 + 6.72061i 0.824708 + 0.720525i
\(88\) 3.54461 − 2.57531i 0.377857 − 0.274529i
\(89\) 11.1385 − 4.18034i 1.18068 − 0.443115i 0.317618 − 0.948219i \(-0.397117\pi\)
0.863059 + 0.505103i \(0.168546\pi\)
\(90\) 0.000953540 0.0212322i 0.000100512 0.00223807i
\(91\) 9.37574 − 11.7568i 0.982844 − 1.23245i
\(92\) 13.3072 + 13.9183i 1.38737 + 1.45108i
\(93\) 5.60448 − 8.49042i 0.581157 − 0.880415i
\(94\) −13.5252 − 6.51340i −1.39502 − 0.671806i
\(95\) 0.0387304 + 0.0106889i 0.00397366 + 0.00109666i
\(96\) 11.3446 + 8.24236i 1.15786 + 0.841233i
\(97\) 2.49858 + 10.9470i 0.253693 + 1.11150i 0.927862 + 0.372923i \(0.121645\pi\)
−0.674169 + 0.738577i \(0.735498\pi\)
\(98\) −0.651374 + 0.681284i −0.0657987 + 0.0688201i
\(99\) −0.164097 − 1.21141i −0.0164924 − 0.121751i
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 71.2.g.a.16.5 ✓ 120
3.2 odd 2 639.2.v.a.442.1 120
71.18 even 35 5041.2.a.s.1.54 60
71.40 even 35 inner 71.2.g.a.40.5 yes 120
71.53 odd 70 5041.2.a.t.1.54 60
213.182 odd 70 639.2.v.a.253.1 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.16.5 ✓ 120 1.1 even 1 trivial
71.2.g.a.40.5 yes 120 71.40 even 35 inner
639.2.v.a.253.1 120 213.182 odd 70
639.2.v.a.442.1 120 3.2 odd 2
5041.2.a.s.1.54 60 71.18 even 35
5041.2.a.t.1.54 60 71.53 odd 70