↑

Properties

Label 85.2.l.a.36.2
Level $85$
Weight $2$
Character 85.36
Analytic conductor $0.679$
Analytic rank $0$
Dimension $24$
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] = [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 36.2
Character \(\chi\) \(=\) 85.36
Dual form 85.2.l.a.26.2

$q$-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+(-1.27691 - 1.27691i) q^{2} +(0.635552 - 0.263254i) q^{3} +1.26102i q^{4} +(-0.382683 - 0.923880i) q^{5} +(-1.14770 - 0.475393i) q^{6} +(1.66158 - 4.01142i) q^{7} +(-0.943613 + 0.943613i) q^{8} +(-1.78670 + 1.78670i) q^{9} +(-0.691061 + 1.66837i) q^{10} +(0.0485041 + 0.0200910i) q^{11} +(0.331970 + 0.801445i) q^{12} -3.02508i q^{13} +(-7.24394 + 3.00054i) q^{14} +(-0.486431 - 0.486431i) q^{15} +4.93187 q^{16} +(3.12202 + 2.69314i) q^{17} +4.56292 q^{18} +(5.52988 + 5.52988i) q^{19} +(1.16503 - 0.482572i) q^{20} -2.98689i q^{21} +(-0.0362810 - 0.0875901i) q^{22} +(0.962654 + 0.398744i) q^{23} +(-0.351305 + 0.848125i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(-3.86277 + 3.86277i) q^{26} +(-1.45495 + 3.51255i) q^{27} +(5.05848 + 2.09529i) q^{28} +(-0.161016 - 0.388726i) q^{29} +1.24226i q^{30} +(-1.27892 + 0.529745i) q^{31} +(-4.41035 - 4.41035i) q^{32} +0.0361159 q^{33} +(-0.547638 - 7.42546i) q^{34} -4.34193 q^{35} +(-2.25306 - 2.25306i) q^{36} +(0.311301 - 0.128945i) q^{37} -14.1224i q^{38} +(-0.796365 - 1.92260i) q^{39} +(1.23289 + 0.510679i) q^{40} +(2.52291 - 6.09084i) q^{41} +(-3.81400 + 3.81400i) q^{42} +(-7.06729 + 7.06729i) q^{43} +(-0.0253352 + 0.0611647i) q^{44} +(2.33443 + 0.966953i) q^{45} +(-0.720064 - 1.73839i) q^{46} +6.13168i q^{47} +(3.13446 - 1.29834i) q^{48} +(-8.38087 - 8.38087i) q^{49} +1.80583 q^{50} +(2.69319 + 0.889748i) q^{51} +3.81469 q^{52} +(-8.52974 - 8.52974i) q^{53} +(6.34307 - 2.62739i) q^{54} -0.0525004i q^{55} +(2.21733 + 5.35312i) q^{56} +(4.97030 + 2.05876i) q^{57} +(-0.290767 + 0.701974i) q^{58} +(3.60468 - 3.60468i) q^{59} +(0.613400 - 0.613400i) q^{60} +(-2.28486 + 5.51614i) q^{61} +(2.30951 + 0.956630i) q^{62} +(4.19844 + 10.1359i) q^{63} +1.39954i q^{64} +(-2.79481 + 1.15765i) q^{65} +(-0.0461170 - 0.0461170i) q^{66} +0.916040 q^{67} +(-3.39611 + 3.93693i) q^{68} +0.716788 q^{69} +(5.54427 + 5.54427i) q^{70} +(3.86169 - 1.59956i) q^{71} -3.37190i q^{72} +(2.06289 + 4.98025i) q^{73} +(-0.562156 - 0.232853i) q^{74} +(-0.263254 + 0.635552i) q^{75} +(-6.97330 + 6.97330i) q^{76} +(0.161187 - 0.161187i) q^{77} +(-1.43810 + 3.47188i) q^{78} +(9.22305 + 3.82031i) q^{79} +(-1.88734 - 4.55645i) q^{80} -4.96488i q^{81} +(-10.9990 + 4.55595i) q^{82} +(-4.61746 - 4.61746i) q^{83} +3.76653 q^{84} +(1.29339 - 3.91499i) q^{85} +18.0487 q^{86} +(-0.204668 - 0.204668i) q^{87} +(-0.0647272 + 0.0268109i) q^{88} +10.2159i q^{89} +(-1.74615 - 4.21559i) q^{90} +(-12.1349 - 5.02642i) q^{91} +(-0.502825 + 1.21393i) q^{92} +(-0.673362 + 0.673362i) q^{93} +(7.82963 - 7.82963i) q^{94} +(2.99275 - 7.22514i) q^{95} +(-3.96405 - 1.64196i) q^{96} +(-7.35663 - 17.7605i) q^{97} +21.4033i q^{98} +(-0.122559 + 0.0507655i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27691 − 1.27691i −0.902915 − 0.902915i 0.0927724 − 0.995687i \(-0.470427\pi\)
−0.995687 + 0.0927724i \(0.970427\pi\)
\(3\) 0.635552 − 0.263254i 0.366936 − 0.151990i −0.191594 − 0.981474i \(-0.561366\pi\)
0.558530 + 0.829484i \(0.311366\pi\)
\(4\) 1.26102i 0.630511i
\(5\) −0.382683 − 0.923880i −0.171141 − 0.413171i
\(6\) −1.14770 − 0.475393i −0.468546 − 0.194078i
\(7\) 1.66158 − 4.01142i 0.628020 − 1.51617i −0.214060 − 0.976821i \(-0.568669\pi\)
0.842080 − 0.539353i \(-0.181331\pi\)
\(8\) −0.943613 + 0.943613i −0.333617 + 0.333617i
\(9\) −1.78670 + 1.78670i −0.595565 + 0.595565i
\(10\) −0.691061 + 1.66837i −0.218533 + 0.527585i
\(11\) 0.0485041 + 0.0200910i 0.0146245 + 0.00605768i 0.389984 − 0.920822i \(-0.372481\pi\)
−0.375359 + 0.926879i \(0.622481\pi\)
\(12\) 0.331970 + 0.801445i 0.0958313 + 0.231357i
\(13\) − 3.02508i − 0.839006i −0.907754 − 0.419503i \(-0.862204\pi\)
0.907754 − 0.419503i \(-0.137796\pi\)
\(14\) −7.24394 + 3.00054i −1.93602 + 0.801928i
\(15\) −0.486431 − 0.486431i −0.125596 − 0.125596i
\(16\) 4.93187 1.23297
\(17\) 3.12202 + 2.69314i 0.757200 + 0.653183i
\(18\) 4.56292 1.07549
\(19\) 5.52988 + 5.52988i 1.26864 + 1.26864i 0.946790 + 0.321851i \(0.104305\pi\)
0.321851 + 0.946790i \(0.395695\pi\)
\(20\) 1.16503 − 0.482572i 0.260509 − 0.107906i
\(21\) − 2.98689i − 0.651792i
\(22\) −0.0362810 − 0.0875901i −0.00773514 − 0.0186743i
\(23\) 0.962654 + 0.398744i 0.200727 + 0.0831439i 0.480782 − 0.876840i \(-0.340353\pi\)
−0.280055 + 0.959984i \(0.590353\pi\)
\(24\) −0.351305 + 0.848125i −0.0717098 + 0.173123i
\(25\) −0.707107 + 0.707107i −0.141421 + 0.141421i
\(26\) −3.86277 + 3.86277i −0.757551 + 0.757551i
\(27\) −1.45495 + 3.51255i −0.280005 + 0.675991i
\(28\) 5.05848 + 2.09529i 0.955964 + 0.395973i
\(29\) −0.161016 − 0.388726i −0.0298999 − 0.0721847i 0.908224 − 0.418484i \(-0.137438\pi\)
−0.938124 + 0.346299i \(0.887438\pi\)
\(30\) 1.24226i 0.226805i
\(31\) −1.27892 + 0.529745i −0.229701 + 0.0951451i −0.494565 − 0.869141i \(-0.664673\pi\)
0.264865 + 0.964286i \(0.414673\pi\)
\(32\) −4.41035 − 4.41035i −0.779647 − 0.779647i
\(33\) 0.0361159 0.00628698
\(34\) −0.547638 − 7.42546i −0.0939192 − 1.27346i
\(35\) −4.34193 −0.733920
\(36\) −2.25306 − 2.25306i −0.375510 − 0.375510i
\(37\) 0.311301 − 0.128945i 0.0511775 − 0.0211984i −0.356948 − 0.934124i \(-0.616183\pi\)
0.408125 + 0.912926i \(0.366183\pi\)
\(38\) − 14.1224i − 2.29095i
\(39\) −0.796365 − 1.92260i −0.127520 − 0.307862i
\(40\) 1.23289 + 0.510679i 0.194937 + 0.0807455i
\(41\) 2.52291 − 6.09084i 0.394012 − 0.951230i −0.595044 − 0.803693i \(-0.702865\pi\)
0.989057 − 0.147537i \(-0.0471345\pi\)
\(42\) −3.81400 + 3.81400i −0.588513 + 0.588513i
\(43\) −7.06729 + 7.06729i −1.07775 + 1.07775i −0.0810414 + 0.996711i \(0.525825\pi\)
−0.996711 + 0.0810414i \(0.974175\pi\)
\(44\) −0.0253352 + 0.0611647i −0.00381943 + 0.00922092i
\(45\) 2.33443 + 0.966953i 0.347996 + 0.144145i
\(46\) −0.720064 − 1.73839i −0.106168 − 0.256311i
\(47\) 6.13168i 0.894398i 0.894435 + 0.447199i \(0.147578\pi\)
−0.894435 + 0.447199i \(0.852422\pi\)
\(48\) 3.13446 − 1.29834i 0.452420 − 0.187399i
\(49\) −8.38087 − 8.38087i −1.19727 − 1.19727i
\(50\) 1.80583 0.255383
\(51\) 2.69319 + 0.889748i 0.377122 + 0.124590i
\(52\) 3.81469 0.529002
\(53\) −8.52974 − 8.52974i −1.17165 − 1.17165i −0.981816 − 0.189834i \(-0.939205\pi\)
−0.189834 − 0.981816i \(-0.560795\pi\)
\(54\) 6.34307 − 2.62739i 0.863183 − 0.357542i
\(55\) − 0.0525004i − 0.00707916i
\(56\) 2.21733 + 5.35312i 0.296304 + 0.715340i
\(57\) 4.97030 + 2.05876i 0.658332 + 0.272690i
\(58\) −0.290767 + 0.701974i −0.0381796 + 0.0921737i
\(59\) 3.60468 − 3.60468i 0.469290 − 0.469290i −0.432395 − 0.901684i \(-0.642331\pi\)
0.901684 + 0.432395i \(0.142331\pi\)
\(60\) 0.613400 − 0.613400i 0.0791896 − 0.0791896i
\(61\) −2.28486 + 5.51614i −0.292547 + 0.706270i −1.00000 0.000490243i \(-0.999844\pi\)
0.707453 + 0.706760i \(0.249844\pi\)
\(62\) 2.30951 + 0.956630i 0.293308 + 0.121492i
\(63\) 4.19844 + 10.1359i 0.528954 + 1.27701i
\(64\) 1.39954i 0.174943i
\(65\) −2.79481 + 1.15765i −0.346653 + 0.143588i
\(66\) −0.0461170 − 0.0461170i −0.00567661 − 0.00567661i
\(67\) 0.916040 0.111912 0.0559561 − 0.998433i \(-0.482179\pi\)
0.0559561 + 0.998433i \(0.482179\pi\)
\(68\) −3.39611 + 3.93693i −0.411839 + 0.477423i
\(69\) 0.716788 0.0862912
\(70\) 5.54427 + 5.54427i 0.662667 + 0.662667i
\(71\) 3.86169 − 1.59956i 0.458298 − 0.189833i −0.141577 − 0.989927i \(-0.545217\pi\)
0.599875 + 0.800094i \(0.295217\pi\)
\(72\) − 3.37190i − 0.397382i
\(73\) 2.06289 + 4.98025i 0.241443 + 0.582895i 0.997427 − 0.0716959i \(-0.0228411\pi\)
−0.755984 + 0.654590i \(0.772841\pi\)
\(74\) −0.562156 − 0.232853i −0.0653493 − 0.0270686i
\(75\) −0.263254 + 0.635552i −0.0303980 + 0.0733873i
\(76\) −6.97330 + 6.97330i −0.799892 + 0.799892i
\(77\) 0.161187 − 0.161187i 0.0183690 − 0.0183690i
\(78\) −1.43810 + 3.47188i −0.162833 + 0.393113i
\(79\) 9.22305 + 3.82031i 1.03767 + 0.429819i 0.835477 − 0.549526i \(-0.185192\pi\)
0.202198 + 0.979345i \(0.435192\pi\)
\(80\) −1.88734 − 4.55645i −0.211011 − 0.509427i
\(81\) − 4.96488i − 0.551653i
\(82\) −10.9990 + 4.55595i −1.21464 + 0.503120i
\(83\) −4.61746 − 4.61746i −0.506833 − 0.506833i 0.406720 − 0.913553i \(-0.366672\pi\)
−0.913553 + 0.406720i \(0.866672\pi\)
\(84\) 3.76653 0.410962
\(85\) 1.29339 − 3.91499i 0.140288 − 0.424640i
\(86\) 18.0487 1.94624
\(87\) −0.204668 − 0.204668i −0.0219427 − 0.0219427i
\(88\) −0.0647272 + 0.0268109i −0.00689994 + 0.00285805i
\(89\) 10.2159i 1.08289i 0.840738 + 0.541443i \(0.182122\pi\)
−0.840738 + 0.541443i \(0.817878\pi\)
\(90\) −1.74615 − 4.21559i −0.184061 − 0.444362i
\(91\) −12.1349 − 5.02642i −1.27208 − 0.526912i
\(92\) −0.502825 + 1.21393i −0.0524231 + 0.126561i
\(93\) −0.673362 + 0.673362i −0.0698244 + 0.0698244i
\(94\) 7.82963 − 7.82963i 0.807565 − 0.807565i
\(95\) 2.99275 − 7.22514i 0.307050 − 0.741283i
\(96\) −3.96405 − 1.64196i −0.404579 − 0.167582i
\(97\) −7.35663 − 17.7605i −0.746952 − 1.80330i −0.574916 − 0.818212i \(-0.694965\pi\)
−0.172036 − 0.985091i \(-0.555035\pi\)
\(98\) 21.4033i 2.16206i
\(99\) −0.122559 + 0.0507655i −0.0123176 + 0.00510212i
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.36.2 yes 24
3.2 odd 2 765.2.be.b.631.5 24
5.2 odd 4 425.2.n.c.274.5 24
5.3 odd 4 425.2.n.f.274.2 24
5.4 even 2 425.2.m.b.376.5 24
17.3 odd 16 1445.2.a.p.1.3 12
17.5 odd 16 1445.2.d.j.866.19 24
17.9 even 8 inner 85.2.l.a.26.2 ✓ 24
17.12 odd 16 1445.2.d.j.866.20 24
17.14 odd 16 1445.2.a.q.1.3 12
51.26 odd 8 765.2.be.b.451.5 24
85.9 even 8 425.2.m.b.26.5 24
85.14 odd 16 7225.2.a.bq.1.10 12
85.43 odd 8 425.2.n.c.349.5 24
85.54 odd 16 7225.2.a.bs.1.10 12
85.77 odd 8 425.2.n.f.349.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.2 ✓ 24 17.9 even 8 inner
85.2.l.a.36.2 yes 24 1.1 even 1 trivial
425.2.m.b.26.5 24 85.9 even 8
425.2.m.b.376.5 24 5.4 even 2
425.2.n.c.274.5 24 5.2 odd 4
425.2.n.c.349.5 24 85.43 odd 8
425.2.n.f.274.2 24 5.3 odd 4
425.2.n.f.349.2 24 85.77 odd 8
765.2.be.b.451.5 24 51.26 odd 8
765.2.be.b.631.5 24 3.2 odd 2
1445.2.a.p.1.3 12 17.3 odd 16
1445.2.a.q.1.3 12 17.14 odd 16
1445.2.d.j.866.19 24 17.5 odd 16
1445.2.d.j.866.20 24 17.12 odd 16
7225.2.a.bq.1.10 12 85.14 odd 16
7225.2.a.bs.1.10 12 85.54 odd 16