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 36.3
Character \(\chi\) \(=\) 85.36
Dual form 85.2.l.a.26.3

$q$-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+(-1.09994 - 1.09994i) q^{2} +(2.77900 - 1.15110i) q^{3} +0.419729i q^{4} +(0.382683 + 0.923880i) q^{5} +(-4.32287 - 1.79059i) q^{6} +(-1.32205 + 3.19170i) q^{7} +(-1.73820 + 1.73820i) q^{8} +(4.27649 - 4.27649i) q^{9} +(0.595282 - 1.43714i) q^{10} +(-3.92026 - 1.62382i) q^{11} +(0.483150 + 1.16643i) q^{12} +0.127392i q^{13} +(4.96485 - 2.05651i) q^{14} +(2.12695 + 2.12695i) q^{15} +4.66329 q^{16} +(-4.11857 - 0.193278i) q^{17} -9.40776 q^{18} +(1.81966 + 1.81966i) q^{19} +(-0.387779 + 0.160623i) q^{20} +10.3916i q^{21} +(2.52594 + 6.09815i) q^{22} +(3.00465 + 1.24457i) q^{23} +(-2.82962 + 6.83130i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(0.140123 - 0.140123i) q^{26} +(3.50841 - 8.47004i) q^{27} +(-1.33965 - 0.554901i) q^{28} +(-1.87644 - 4.53013i) q^{29} -4.67904i q^{30} +(4.95543 - 2.05261i) q^{31} +(-1.65293 - 1.65293i) q^{32} -12.7636 q^{33} +(4.31758 + 4.74277i) q^{34} -3.45467 q^{35} +(1.79497 + 1.79497i) q^{36} +(-1.63681 + 0.677990i) q^{37} -4.00302i q^{38} +(0.146641 + 0.354023i) q^{39} +(-2.27107 - 0.940707i) q^{40} +(3.85069 - 9.29639i) q^{41} +(11.4301 - 11.4301i) q^{42} +(1.79227 - 1.79227i) q^{43} +(0.681566 - 1.64545i) q^{44} +(5.58751 + 2.31442i) q^{45} +(-1.93598 - 4.67388i) q^{46} +4.59479i q^{47} +(12.9593 - 5.36791i) q^{48} +(-3.48941 - 3.48941i) q^{49} +1.55555 q^{50} +(-11.6680 + 4.20377i) q^{51} -0.0534701 q^{52} +(1.15866 + 1.15866i) q^{53} +(-13.1756 + 5.45749i) q^{54} -4.24326i q^{55} +(-3.24984 - 7.84580i) q^{56} +(7.15144 + 2.96222i) q^{57} +(-2.91889 + 7.04683i) q^{58} +(-4.34287 + 4.34287i) q^{59} +(-0.892745 + 0.892745i) q^{60} +(-1.54679 + 3.73428i) q^{61} +(-7.70840 - 3.19293i) q^{62} +(7.99557 + 19.3030i) q^{63} -5.69034i q^{64} +(-0.117695 + 0.0487508i) q^{65} +(14.0392 + 14.0392i) q^{66} -6.88856 q^{67} +(0.0811242 - 1.72868i) q^{68} +9.78255 q^{69} +(3.79993 + 3.79993i) q^{70} +(-6.66802 + 2.76198i) q^{71} +14.8668i q^{72} +(-5.59682 - 13.5119i) q^{73} +(2.54614 + 1.05465i) q^{74} +(-1.15110 + 2.77900i) q^{75} +(-0.763763 + 0.763763i) q^{76} +(10.3655 - 10.3655i) q^{77} +(0.228107 - 0.550699i) q^{78} +(-4.75854 - 1.97105i) q^{79} +(1.78456 + 4.30831i) q^{80} -9.43315i q^{81} +(-14.4610 + 5.98994i) q^{82} +(10.2150 + 10.2150i) q^{83} -4.36164 q^{84} +(-1.39754 - 3.87903i) q^{85} -3.94276 q^{86} +(-10.4293 - 10.4293i) q^{87} +(9.63673 - 3.99166i) q^{88} -0.600876i q^{89} +(-3.60019 - 8.69163i) q^{90} +(-0.406598 - 0.168418i) q^{91} +(-0.522381 + 1.26114i) q^{92} +(11.4084 - 11.4084i) q^{93} +(5.05398 - 5.05398i) q^{94} +(-0.984791 + 2.37750i) q^{95} +(-6.49616 - 2.69080i) q^{96} +(-2.93763 - 7.09206i) q^{97} +7.67628i q^{98} +(-23.7092 + 9.82068i) 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 \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.09994 − 1.09994i −0.777774 − 0.777774i 0.201678 − 0.979452i \(-0.435361\pi\)
−0.979452 + 0.201678i \(0.935361\pi\)
\(3\) 2.77900 − 1.15110i 1.60446 − 0.664588i 0.612419 − 0.790533i \(-0.290196\pi\)
0.992037 + 0.125945i \(0.0401964\pi\)
\(4\) 0.419729i 0.209865i
\(5\) 0.382683 + 0.923880i 0.171141 + 0.413171i
\(6\) −4.32287 − 1.79059i −1.76480 − 0.731006i
\(7\) −1.32205 + 3.19170i −0.499687 + 1.20635i 0.449966 + 0.893046i \(0.351436\pi\)
−0.949653 + 0.313305i \(0.898564\pi\)
\(8\) −1.73820 + 1.73820i −0.614547 + 0.614547i
\(9\) 4.27649 − 4.27649i 1.42550 − 1.42550i
\(10\) 0.595282 − 1.43714i 0.188245 − 0.454463i
\(11\) −3.92026 − 1.62382i −1.18200 − 0.489601i −0.296860 − 0.954921i \(-0.595939\pi\)
−0.885143 + 0.465320i \(0.845939\pi\)
\(12\) 0.483150 + 1.16643i 0.139473 + 0.336719i
\(13\) 0.127392i 0.0353322i 0.999844 + 0.0176661i \(0.00562359\pi\)
−0.999844 + 0.0176661i \(0.994376\pi\)
\(14\) 4.96485 − 2.05651i 1.32691 − 0.549625i
\(15\) 2.12695 + 2.12695i 0.549177 + 0.549177i
\(16\) 4.66329 1.16582
\(17\) −4.11857 − 0.193278i −0.998901 − 0.0468767i
\(18\) −9.40776 −2.21743
\(19\) 1.81966 + 1.81966i 0.417458 + 0.417458i 0.884327 − 0.466869i \(-0.154618\pi\)
−0.466869 + 0.884327i \(0.654618\pi\)
\(20\) −0.387779 + 0.160623i −0.0867100 + 0.0359165i
\(21\) 10.3916i 2.26762i
\(22\) 2.52594 + 6.09815i 0.538531 + 1.30013i
\(23\) 3.00465 + 1.24457i 0.626513 + 0.259510i 0.673271 − 0.739396i \(-0.264889\pi\)
−0.0467576 + 0.998906i \(0.514889\pi\)
\(24\) −2.82962 + 6.83130i −0.577593 + 1.39443i
\(25\) −0.707107 + 0.707107i −0.141421 + 0.141421i
\(26\) 0.140123 − 0.140123i 0.0274805 − 0.0274805i
\(27\) 3.50841 − 8.47004i 0.675193 − 1.63006i
\(28\) −1.33965 − 0.554901i −0.253170 − 0.104866i
\(29\) −1.87644 − 4.53013i −0.348446 − 0.841224i −0.996804 − 0.0798877i \(-0.974544\pi\)
0.648358 − 0.761336i \(-0.275456\pi\)
\(30\) − 4.67904i − 0.854272i
\(31\) 4.95543 − 2.05261i 0.890021 − 0.368659i 0.109646 − 0.993971i \(-0.465028\pi\)
0.780375 + 0.625312i \(0.215028\pi\)
\(32\) −1.65293 − 1.65293i −0.292199 − 0.292199i
\(33\) −12.7636 −2.22185
\(34\) 4.31758 + 4.74277i 0.740459 + 0.813378i
\(35\) −3.45467 −0.583947
\(36\) 1.79497 + 1.79497i 0.299161 + 0.299161i
\(37\) −1.63681 + 0.677990i −0.269090 + 0.111461i −0.513149 − 0.858300i \(-0.671521\pi\)
0.244059 + 0.969761i \(0.421521\pi\)
\(38\) − 4.00302i − 0.649376i
\(39\) 0.146641 + 0.354023i 0.0234813 + 0.0566890i
\(40\) −2.27107 − 0.940707i −0.359087 − 0.148739i
\(41\) 3.85069 − 9.29639i 0.601377 − 1.45185i −0.270787 − 0.962639i \(-0.587284\pi\)
0.872164 − 0.489214i \(-0.162716\pi\)
\(42\) 11.4301 − 11.4301i 1.76370 − 1.76370i
\(43\) 1.79227 − 1.79227i 0.273318 − 0.273318i −0.557116 − 0.830434i \(-0.688092\pi\)
0.830434 + 0.557116i \(0.188092\pi\)
\(44\) 0.681566 − 1.64545i 0.102750 − 0.248060i
\(45\) 5.58751 + 2.31442i 0.832936 + 0.345013i
\(46\) −1.93598 − 4.67388i −0.285445 − 0.689126i
\(47\) 4.59479i 0.670219i 0.942179 + 0.335109i \(0.108773\pi\)
−0.942179 + 0.335109i \(0.891227\pi\)
\(48\) 12.9593 − 5.36791i 1.87051 − 0.774791i
\(49\) −3.48941 − 3.48941i −0.498487 − 0.498487i
\(50\) 1.55555 0.219988
\(51\) −11.6680 + 4.20377i −1.63385 + 0.588645i
\(52\) −0.0534701 −0.00741498
\(53\) 1.15866 + 1.15866i 0.159155 + 0.159155i 0.782192 − 0.623037i \(-0.214102\pi\)
−0.623037 + 0.782192i \(0.714102\pi\)
\(54\) −13.1756 + 5.45749i −1.79297 + 0.742671i
\(55\) − 4.24326i − 0.572161i
\(56\) −3.24984 − 7.84580i −0.434278 − 1.04844i
\(57\) 7.15144 + 2.96222i 0.947231 + 0.392356i
\(58\) −2.91889 + 7.04683i −0.383269 + 0.925294i
\(59\) −4.34287 + 4.34287i −0.565393 + 0.565393i −0.930834 − 0.365441i \(-0.880918\pi\)
0.365441 + 0.930834i \(0.380918\pi\)
\(60\) −0.892745 + 0.892745i −0.115253 + 0.115253i
\(61\) −1.54679 + 3.73428i −0.198046 + 0.478125i −0.991437 − 0.130587i \(-0.958314\pi\)
0.793391 + 0.608713i \(0.208314\pi\)
\(62\) −7.70840 − 3.19293i −0.978968 − 0.405502i
\(63\) 7.99557 + 19.3030i 1.00735 + 2.43195i
\(64\) − 5.69034i − 0.711292i
\(65\) −0.117695 + 0.0487508i −0.0145983 + 0.00604680i
\(66\) 14.0392 + 14.0392i 1.72810 + 1.72810i
\(67\) −6.88856 −0.841571 −0.420786 − 0.907160i \(-0.638246\pi\)
−0.420786 + 0.907160i \(0.638246\pi\)
\(68\) 0.0811242 − 1.72868i 0.00983776 − 0.209634i
\(69\) 9.78255 1.17768
\(70\) 3.79993 + 3.79993i 0.454178 + 0.454178i
\(71\) −6.66802 + 2.76198i −0.791348 + 0.327787i −0.741485 − 0.670969i \(-0.765878\pi\)
−0.0498626 + 0.998756i \(0.515878\pi\)
\(72\) 14.8668i 1.75207i
\(73\) −5.59682 − 13.5119i −0.655058 − 1.58145i −0.805345 − 0.592807i \(-0.798020\pi\)
0.150287 − 0.988642i \(-0.451980\pi\)
\(74\) 2.54614 + 1.05465i 0.295983 + 0.122600i
\(75\) −1.15110 + 2.77900i −0.132918 + 0.320891i
\(76\) −0.763763 + 0.763763i −0.0876096 + 0.0876096i
\(77\) 10.3655 − 10.3655i 1.18126 − 1.18126i
\(78\) 0.228107 − 0.550699i 0.0258280 − 0.0623544i
\(79\) −4.75854 − 1.97105i −0.535378 − 0.221761i 0.0985790 − 0.995129i \(-0.468570\pi\)
−0.633957 + 0.773369i \(0.718570\pi\)
\(80\) 1.78456 + 4.30831i 0.199520 + 0.481684i
\(81\) − 9.43315i − 1.04813i
\(82\) −14.4610 + 5.98994i −1.59695 + 0.661478i
\(83\) 10.2150 + 10.2150i 1.12124 + 1.12124i 0.991555 + 0.129685i \(0.0413967\pi\)
0.129685 + 0.991555i \(0.458603\pi\)
\(84\) −4.36164 −0.475893
\(85\) −1.39754 − 3.87903i −0.151585 − 0.420740i
\(86\) −3.94276 −0.425159
\(87\) −10.4293 − 10.4293i −1.11813 − 1.11813i
\(88\) 9.63673 − 3.99166i 1.02728 − 0.425513i
\(89\) − 0.600876i − 0.0636927i −0.999493 − 0.0318463i \(-0.989861\pi\)
0.999493 − 0.0318463i \(-0.0101387\pi\)
\(90\) −3.60019 − 8.69163i −0.379494 − 0.916179i
\(91\) −0.406598 − 0.168418i −0.0426230 − 0.0176550i
\(92\) −0.522381 + 1.26114i −0.0544620 + 0.131483i
\(93\) 11.4084 − 11.4084i 1.18299 − 1.18299i
\(94\) 5.05398 − 5.05398i 0.521279 − 0.521279i
\(95\) −0.984791 + 2.37750i −0.101037 + 0.243926i
\(96\) −6.49616 − 2.69080i −0.663012 − 0.274628i
\(97\) −2.93763 − 7.09206i −0.298271 − 0.720090i −0.999971 − 0.00761730i \(-0.997575\pi\)
0.701700 − 0.712473i \(-0.252425\pi\)
\(98\) 7.67628i 0.775421i
\(99\) −23.7092 + 9.82068i −2.38287 + 0.987016i
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.36.3 yes 24
3.2 odd 2 765.2.be.b.631.4 24
5.2 odd 4 425.2.n.c.274.4 24
5.3 odd 4 425.2.n.f.274.3 24
5.4 even 2 425.2.m.b.376.4 24
17.3 odd 16 1445.2.a.q.1.4 12
17.5 odd 16 1445.2.d.j.866.18 24
17.9 even 8 inner 85.2.l.a.26.3 ✓ 24
17.12 odd 16 1445.2.d.j.866.17 24
17.14 odd 16 1445.2.a.p.1.4 12
51.26 odd 8 765.2.be.b.451.4 24
85.9 even 8 425.2.m.b.26.4 24
85.14 odd 16 7225.2.a.bs.1.9 12
85.43 odd 8 425.2.n.c.349.4 24
85.54 odd 16 7225.2.a.bq.1.9 12
85.77 odd 8 425.2.n.f.349.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.3 ✓ 24 17.9 even 8 inner
85.2.l.a.36.3 yes 24 1.1 even 1 trivial
425.2.m.b.26.4 24 85.9 even 8
425.2.m.b.376.4 24 5.4 even 2
425.2.n.c.274.4 24 5.2 odd 4
425.2.n.c.349.4 24 85.43 odd 8
425.2.n.f.274.3 24 5.3 odd 4
425.2.n.f.349.3 24 85.77 odd 8
765.2.be.b.451.4 24 51.26 odd 8
765.2.be.b.631.4 24 3.2 odd 2
1445.2.a.p.1.4 12 17.14 odd 16
1445.2.a.q.1.4 12 17.3 odd 16
1445.2.d.j.866.17 24 17.12 odd 16
1445.2.d.j.866.18 24 17.5 odd 16
7225.2.a.bq.1.9 12 85.54 odd 16
7225.2.a.bs.1.9 12 85.14 odd 16