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Properties

Label 85.2.l.a.26.5
Level $85$
Weight $2$
Character 85.26
Analytic conductor $0.679$
Analytic rank $0$
Dimension $24$
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] = [85,2,Mod(26,85)] 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("85.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.l (of order \(8\), degree \(4\), 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: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 26.5
Character \(\chi\) \(=\) 85.26
Dual form 85.2.l.a.36.5

$q$-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+(1.01710 - 1.01710i) q^{2} +(-0.101541 - 0.0420595i) q^{3} -0.0689897i q^{4} +(0.382683 - 0.923880i) q^{5} +(-0.146056 + 0.0604983i) q^{6} +(-0.265997 - 0.642174i) q^{7} +(1.96403 + 1.96403i) q^{8} +(-2.11278 - 2.11278i) q^{9} +(-0.550451 - 1.32891i) q^{10} +(-4.48163 + 1.85635i) q^{11} +(-0.00290167 + 0.00700526i) q^{12} +5.63906i q^{13} +(-0.923703 - 0.382610i) q^{14} +(-0.0777158 + 0.0777158i) q^{15} +4.13322 q^{16} +(1.63113 - 3.78674i) q^{17} -4.29782 q^{18} +(-1.64241 + 1.64241i) q^{19} +(-0.0637382 - 0.0264012i) q^{20} +0.0763945i q^{21} +(-2.67018 + 6.44637i) q^{22} +(4.28390 - 1.77445i) q^{23} +(-0.116823 - 0.282035i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(5.73549 + 5.73549i) q^{26} +(0.251849 + 0.608017i) q^{27} +(-0.0443034 + 0.0183511i) q^{28} +(2.48981 - 6.01093i) q^{29} +0.158090i q^{30} +(-6.12711 - 2.53793i) q^{31} +(0.275837 - 0.275837i) q^{32} +0.533145 q^{33} +(-2.19248 - 5.51052i) q^{34} -0.695085 q^{35} +(-0.145760 + 0.145760i) q^{36} +(0.109595 + 0.0453958i) q^{37} +3.34100i q^{38} +(0.237176 - 0.572593i) q^{39} +(2.56613 - 1.06293i) q^{40} +(-0.412826 - 0.996650i) q^{41} +(0.0777010 + 0.0777010i) q^{42} +(-0.453332 - 0.453332i) q^{43} +(0.128069 + 0.309187i) q^{44} +(-2.76048 + 1.14343i) q^{45} +(2.55237 - 6.16195i) q^{46} +4.93703i q^{47} +(-0.419690 - 0.173841i) q^{48} +(4.60811 - 4.60811i) q^{49} -1.43840 q^{50} +(-0.324894 + 0.315904i) q^{51} +0.389037 q^{52} +(8.47565 - 8.47565i) q^{53} +(0.874571 + 0.362259i) q^{54} +4.85089i q^{55} +(0.738824 - 1.78368i) q^{56} +(0.235850 - 0.0976925i) q^{57} +(-3.58134 - 8.64611i) q^{58} +(7.01329 + 7.01329i) q^{59} +(0.00536159 + 0.00536159i) q^{60} +(0.613413 + 1.48091i) q^{61} +(-8.81322 + 3.65056i) q^{62} +(-0.794779 + 1.91877i) q^{63} +7.70533i q^{64} +(5.20981 + 2.15797i) q^{65} +(0.542263 - 0.542263i) q^{66} +2.99411 q^{67} +(-0.261246 - 0.112531i) q^{68} -0.509622 q^{69} +(-0.706971 + 0.706971i) q^{70} +(-4.33163 - 1.79422i) q^{71} -8.29913i q^{72} +(-2.10442 + 5.08052i) q^{73} +(0.157641 - 0.0652972i) q^{74} +(0.0420595 + 0.101541i) q^{75} +(0.113309 + 0.113309i) q^{76} +(2.38421 + 2.38421i) q^{77} +(-0.341153 - 0.823617i) q^{78} +(-13.7140 + 5.68053i) q^{79} +(1.58171 - 3.81860i) q^{80} +8.89143i q^{81} +(-1.43358 - 0.593808i) q^{82} +(-3.56033 + 3.56033i) q^{83} +0.00527044 q^{84} +(-2.87429 - 2.95609i) q^{85} -0.922169 q^{86} +(-0.505633 + 0.505633i) q^{87} +(-12.4480 - 5.15614i) q^{88} +2.35657i q^{89} +(-1.64470 + 3.97067i) q^{90} +(3.62126 - 1.49997i) q^{91} +(-0.122419 - 0.295545i) q^{92} +(0.515406 + 0.515406i) q^{93} +(5.02145 + 5.02145i) q^{94} +(0.888867 + 2.14591i) q^{95} +(-0.0396102 + 0.0164071i) q^{96} +(-1.03355 + 2.49522i) q^{97} -9.37384i q^{98} +(13.3908 + 5.54664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} + 24 q^{12} - 8 q^{15} - 24 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 32 q^{22} - 16 q^{23} - 8 q^{24} + 16 q^{26} + 24 q^{27} + 48 q^{28} - 8 q^{29} + 16 q^{34} - 32 q^{35}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.01710 − 1.01710i 0.719199 − 0.719199i −0.249242 − 0.968441i \(-0.580181\pi\)
0.968441 + 0.249242i \(0.0801815\pi\)
\(3\) −0.101541 − 0.0420595i −0.0586245 − 0.0242831i 0.353179 − 0.935556i \(-0.385101\pi\)
−0.411803 + 0.911273i \(0.635101\pi\)
\(4\) − 0.0689897i − 0.0344949i
\(5\) 0.382683 − 0.923880i 0.171141 − 0.413171i
\(6\) −0.146056 + 0.0604983i −0.0596271 + 0.0246983i
\(7\) −0.265997 − 0.642174i −0.100538 − 0.242719i 0.865605 − 0.500727i \(-0.166934\pi\)
−0.966143 + 0.258008i \(0.916934\pi\)
\(8\) 1.96403 + 1.96403i 0.694390 + 0.694390i
\(9\) −2.11278 − 2.11278i −0.704260 − 0.704260i
\(10\) −0.550451 − 1.32891i −0.174068 − 0.420237i
\(11\) −4.48163 + 1.85635i −1.35126 + 0.559712i −0.936644 − 0.350283i \(-0.886085\pi\)
−0.414620 + 0.909995i \(0.636085\pi\)
\(12\) −0.00290167 + 0.00700526i −0.000837641 + 0.00202224i
\(13\) 5.63906i 1.56399i 0.623283 + 0.781996i \(0.285798\pi\)
−0.623283 + 0.781996i \(0.714202\pi\)
\(14\) −0.923703 − 0.382610i −0.246870 − 0.102257i
\(15\) −0.0777158 + 0.0777158i −0.0200661 + 0.0200661i
\(16\) 4.13322 1.03330
\(17\) 1.63113 − 3.78674i 0.395606 − 0.918420i
\(18\) −4.29782 −1.01301
\(19\) −1.64241 + 1.64241i −0.376795 + 0.376795i −0.869945 − 0.493150i \(-0.835846\pi\)
0.493150 + 0.869945i \(0.335846\pi\)
\(20\) −0.0637382 − 0.0264012i −0.0142523 − 0.00590349i
\(21\) 0.0763945i 0.0166706i
\(22\) −2.67018 + 6.44637i −0.569283 + 1.37437i
\(23\) 4.28390 − 1.77445i 0.893255 − 0.369998i 0.111632 − 0.993750i \(-0.464392\pi\)
0.781623 + 0.623751i \(0.214392\pi\)
\(24\) −0.116823 − 0.282035i −0.0238464 − 0.0575702i
\(25\) −0.707107 − 0.707107i −0.141421 − 0.141421i
\(26\) 5.73549 + 5.73549i 1.12482 + 1.12482i
\(27\) 0.251849 + 0.608017i 0.0484684 + 0.117013i
\(28\) −0.0443034 + 0.0183511i −0.00837256 + 0.00346803i
\(29\) 2.48981 − 6.01093i 0.462346 − 1.11620i −0.505086 − 0.863069i \(-0.668539\pi\)
0.967432 − 0.253132i \(-0.0814608\pi\)
\(30\) 0.158090i 0.0288631i
\(31\) −6.12711 − 2.53793i −1.10046 − 0.455826i −0.242819 − 0.970072i \(-0.578072\pi\)
−0.857643 + 0.514246i \(0.828072\pi\)
\(32\) 0.275837 − 0.275837i 0.0487616 − 0.0487616i
\(33\) 0.533145 0.0928087
\(34\) −2.19248 − 5.51052i −0.376008 − 0.945047i
\(35\) −0.695085 −0.117491
\(36\) −0.145760 + 0.145760i −0.0242933 + 0.0242933i
\(37\) 0.109595 + 0.0453958i 0.0180173 + 0.00746302i 0.391674 − 0.920104i \(-0.371896\pi\)
−0.373657 + 0.927567i \(0.621896\pi\)
\(38\) 3.34100i 0.541981i
\(39\) 0.237176 − 0.572593i 0.0379785 − 0.0916883i
\(40\) 2.56613 − 1.06293i 0.405741 − 0.168064i
\(41\) −0.412826 − 0.996650i −0.0644726 − 0.155651i 0.888359 − 0.459149i \(-0.151845\pi\)
−0.952832 + 0.303498i \(0.901845\pi\)
\(42\) 0.0777010 + 0.0777010i 0.0119895 + 0.0119895i
\(43\) −0.453332 − 0.453332i −0.0691325 − 0.0691325i 0.671695 − 0.740828i \(-0.265566\pi\)
−0.740828 + 0.671695i \(0.765566\pi\)
\(44\) 0.128069 + 0.309187i 0.0193072 + 0.0466116i
\(45\) −2.76048 + 1.14343i −0.411508 + 0.170452i
\(46\) 2.55237 − 6.16195i 0.376326 − 0.908531i
\(47\) 4.93703i 0.720139i 0.932925 + 0.360070i \(0.117247\pi\)
−0.932925 + 0.360070i \(0.882753\pi\)
\(48\) −0.419690 − 0.173841i −0.0605770 − 0.0250918i
\(49\) 4.60811 − 4.60811i 0.658302 − 0.658302i
\(50\) −1.43840 −0.203420
\(51\) −0.324894 + 0.315904i −0.0454943 + 0.0442354i
\(52\) 0.389037 0.0539497
\(53\) 8.47565 − 8.47565i 1.16422 − 1.16422i 0.180678 − 0.983542i \(-0.442171\pi\)
0.983542 − 0.180678i \(-0.0578292\pi\)
\(54\) 0.874571 + 0.362259i 0.119014 + 0.0492972i
\(55\) 4.85089i 0.654093i
\(56\) 0.738824 − 1.78368i 0.0987295 − 0.238354i
\(57\) 0.235850 − 0.0976925i 0.0312392 − 0.0129397i
\(58\) −3.58134 − 8.64611i −0.470252 − 1.13529i
\(59\) 7.01329 + 7.01329i 0.913053 + 0.913053i 0.996511 − 0.0834587i \(-0.0265967\pi\)
−0.0834587 + 0.996511i \(0.526597\pi\)
\(60\) 0.00536159 + 0.00536159i 0.000692179 + 0.000692179i
\(61\) 0.613413 + 1.48091i 0.0785394 + 0.189611i 0.958272 − 0.285858i \(-0.0922785\pi\)
−0.879733 + 0.475469i \(0.842278\pi\)
\(62\) −8.81322 + 3.65056i −1.11928 + 0.463621i
\(63\) −0.794779 + 1.91877i −0.100133 + 0.241742i
\(64\) 7.70533i 0.963166i
\(65\) 5.20981 + 2.15797i 0.646197 + 0.267664i
\(66\) 0.542263 − 0.542263i 0.0667479 − 0.0667479i
\(67\) 2.99411 0.365789 0.182894 − 0.983133i \(-0.441453\pi\)
0.182894 + 0.983133i \(0.441453\pi\)
\(68\) −0.261246 − 0.112531i −0.0316808 − 0.0136464i
\(69\) −0.509622 −0.0613513
\(70\) −0.706971 + 0.706971i −0.0844992 + 0.0844992i
\(71\) −4.33163 − 1.79422i −0.514070 − 0.212935i 0.110540 − 0.993872i \(-0.464742\pi\)
−0.624610 + 0.780937i \(0.714742\pi\)
\(72\) − 8.29913i − 0.978062i
\(73\) −2.10442 + 5.08052i −0.246304 + 0.594629i −0.997885 − 0.0650115i \(-0.979292\pi\)
0.751581 + 0.659641i \(0.229292\pi\)
\(74\) 0.157641 − 0.0652972i 0.0183254 − 0.00759065i
\(75\) 0.0420595 + 0.101541i 0.00485661 + 0.0117249i
\(76\) 0.113309 + 0.113309i 0.0129975 + 0.0129975i
\(77\) 2.38421 + 2.38421i 0.271705 + 0.271705i
\(78\) −0.341153 − 0.823617i −0.0386280 − 0.0932563i
\(79\) −13.7140 + 5.68053i −1.54295 + 0.639110i −0.982024 − 0.188757i \(-0.939554\pi\)
−0.560924 + 0.827867i \(0.689554\pi\)
\(80\) 1.58171 − 3.81860i 0.176841 − 0.426932i
\(81\) 8.89143i 0.987937i
\(82\) −1.43358 − 0.593808i −0.158312 − 0.0655752i
\(83\) −3.56033 + 3.56033i −0.390797 + 0.390797i −0.874971 − 0.484175i \(-0.839120\pi\)
0.484175 + 0.874971i \(0.339120\pi\)
\(84\) 0.00527044 0.000575052
\(85\) −2.87429 − 2.95609i −0.311761 − 0.320633i
\(86\) −0.922169 −0.0994400
\(87\) −0.505633 + 0.505633i −0.0542096 + 0.0542096i
\(88\) −12.4480 − 5.15614i −1.32696 − 0.549646i
\(89\) 2.35657i 0.249796i 0.992170 + 0.124898i \(0.0398604\pi\)
−0.992170 + 0.124898i \(0.960140\pi\)
\(90\) −1.64470 + 3.97067i −0.173367 + 0.418545i
\(91\) 3.62126 − 1.49997i 0.379611 − 0.157240i
\(92\) −0.122419 − 0.295545i −0.0127630 − 0.0308127i
\(93\) 0.515406 + 0.515406i 0.0534452 + 0.0534452i
\(94\) 5.02145 + 5.02145i 0.517924 + 0.517924i
\(95\) 0.888867 + 2.14591i 0.0911958 + 0.220166i
\(96\) −0.0396102 + 0.0164071i −0.00404270 + 0.00167454i
\(97\) −1.03355 + 2.49522i −0.104941 + 0.253351i −0.967625 − 0.252394i \(-0.918782\pi\)
0.862683 + 0.505745i \(0.168782\pi\)
\(98\) − 9.37384i − 0.946900i
\(99\) 13.3908 + 5.54664i 1.34582 + 0.557458i
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 85.2.l.a.26.5 ✓ 24
3.2 odd 2 765.2.be.b.451.2 24
5.2 odd 4 425.2.n.f.349.5 24
5.3 odd 4 425.2.n.c.349.2 24
5.4 even 2 425.2.m.b.26.2 24
17.2 even 8 inner 85.2.l.a.36.5 yes 24
17.6 odd 16 1445.2.a.q.1.10 12
17.7 odd 16 1445.2.d.j.866.6 24
17.10 odd 16 1445.2.d.j.866.5 24
17.11 odd 16 1445.2.a.p.1.10 12
51.2 odd 8 765.2.be.b.631.2 24
85.2 odd 8 425.2.n.c.274.2 24
85.19 even 8 425.2.m.b.376.2 24
85.53 odd 8 425.2.n.f.274.5 24
85.74 odd 16 7225.2.a.bq.1.3 12
85.79 odd 16 7225.2.a.bs.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.5 ✓ 24 1.1 even 1 trivial
85.2.l.a.36.5 yes 24 17.2 even 8 inner
425.2.m.b.26.2 24 5.4 even 2
425.2.m.b.376.2 24 85.19 even 8
425.2.n.c.274.2 24 85.2 odd 8
425.2.n.c.349.2 24 5.3 odd 4
425.2.n.f.274.5 24 85.53 odd 8
425.2.n.f.349.5 24 5.2 odd 4
765.2.be.b.451.2 24 3.2 odd 2
765.2.be.b.631.2 24 51.2 odd 8
1445.2.a.p.1.10 12 17.11 odd 16
1445.2.a.q.1.10 12 17.6 odd 16
1445.2.d.j.866.5 24 17.10 odd 16
1445.2.d.j.866.6 24 17.7 odd 16
7225.2.a.bq.1.3 12 85.74 odd 16
7225.2.a.bs.1.3 12 85.79 odd 16