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Properties

Label 85.2.d.a.16.5
Level $85$
Weight $2$
Character 85.16
Analytic conductor $0.679$
Analytic rank $0$
Dimension $6$
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(16,85)] 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("85.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) 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.d (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 16.5
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 85.16
Dual form 85.2.d.a.16.6

$q$-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)
 
\(f(q)\) \(=\) \(q+1.48119 q^{2} -1.67513i q^{3} +0.193937 q^{4} +1.00000i q^{5} -2.48119i q^{6} +1.28726i q^{7} -2.67513 q^{8} +0.193937 q^{9} +1.48119i q^{10} +0.481194i q^{11} -0.324869i q^{12} -2.15633 q^{13} +1.90668i q^{14} +1.67513 q^{15} -4.35026 q^{16} +(-1.86907 + 3.67513i) q^{17} +0.287258 q^{18} +3.35026 q^{19} +0.193937i q^{20} +2.15633 q^{21} +0.712742i q^{22} -8.24965i q^{23} +4.48119i q^{24} -1.00000 q^{25} -3.19394 q^{26} -5.35026i q^{27} +0.249646i q^{28} -0.649738i q^{29} +2.48119 q^{30} -1.83146i q^{31} -1.09332 q^{32} +0.806063 q^{33} +(-2.76845 + 5.44358i) q^{34} -1.28726 q^{35} +0.0376114 q^{36} -4.31265i q^{37} +4.96239 q^{38} +3.61213i q^{39} -2.67513i q^{40} +11.2750i q^{41} +3.19394 q^{42} +8.15633 q^{43} +0.0933212i q^{44} +0.193937i q^{45} -12.2193i q^{46} -6.54420 q^{47} +7.28726i q^{48} +5.34297 q^{49} -1.48119 q^{50} +(6.15633 + 3.13093i) q^{51} -0.418190 q^{52} -8.57452 q^{53} -7.92478i q^{54} -0.481194 q^{55} -3.44358i q^{56} -5.61213i q^{57} -0.962389i q^{58} +4.96239 q^{59} +0.324869 q^{60} -2.83638i q^{61} -2.71274i q^{62} +0.249646i q^{63} +7.08110 q^{64} -2.15633i q^{65} +1.19394 q^{66} +4.93207 q^{67} +(-0.362481 + 0.712742i) q^{68} -13.8192 q^{69} -1.90668 q^{70} +14.5320i q^{71} -0.518806 q^{72} +13.3503i q^{73} -6.38787i q^{74} +1.67513i q^{75} +0.649738 q^{76} -0.619421 q^{77} +5.35026i q^{78} -9.05571i q^{79} -4.35026i q^{80} -8.38058 q^{81} +16.7005i q^{82} -13.4314 q^{83} +0.418190 q^{84} +(-3.67513 - 1.86907i) q^{85} +12.0811 q^{86} -1.08840 q^{87} -1.28726i q^{88} -16.7816 q^{89} +0.287258i q^{90} -2.77575i q^{91} -1.59991i q^{92} -3.06793 q^{93} -9.69323 q^{94} +3.35026i q^{95} +1.83146i q^{96} +3.66291i q^{97} +7.91397 q^{98} +0.0933212i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} + 2 q^{9} + 8 q^{13} - 6 q^{16} - 2 q^{17} - 10 q^{18} - 8 q^{21} - 6 q^{25} - 20 q^{26} + 4 q^{30} + 6 q^{32} + 4 q^{33} + 6 q^{34} + 4 q^{35} + 22 q^{36} + 8 q^{38} + 20 q^{42}+ \cdots + 86 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.48119 1.04736 0.523681 − 0.851914i \(-0.324558\pi\)
0.523681 + 0.851914i \(0.324558\pi\)
\(3\) − 1.67513i − 0.967137i −0.875306 − 0.483569i \(-0.839340\pi\)
0.875306 − 0.483569i \(-0.160660\pi\)
\(4\) 0.193937 0.0969683
\(5\) 1.00000i 0.447214i
\(6\) − 2.48119i − 1.01294i
\(7\) 1.28726i 0.486538i 0.969959 + 0.243269i \(0.0782197\pi\)
−0.969959 + 0.243269i \(0.921780\pi\)
\(8\) −2.67513 −0.945802
\(9\) 0.193937 0.0646455
\(10\) 1.48119i 0.468395i
\(11\) 0.481194i 0.145086i 0.997365 + 0.0725428i \(0.0231114\pi\)
−0.997365 + 0.0725428i \(0.976889\pi\)
\(12\) − 0.324869i − 0.0937816i
\(13\) −2.15633 −0.598057 −0.299028 − 0.954244i \(-0.596663\pi\)
−0.299028 + 0.954244i \(0.596663\pi\)
\(14\) 1.90668i 0.509581i
\(15\) 1.67513 0.432517
\(16\) −4.35026 −1.08757
\(17\) −1.86907 + 3.67513i −0.453315 + 0.891350i
\(18\) 0.287258 0.0677073
\(19\) 3.35026 0.768603 0.384301 − 0.923208i \(-0.374442\pi\)
0.384301 + 0.923208i \(0.374442\pi\)
\(20\) 0.193937i 0.0433655i
\(21\) 2.15633 0.470549
\(22\) 0.712742i 0.151957i
\(23\) − 8.24965i − 1.72017i −0.510151 − 0.860085i \(-0.670410\pi\)
0.510151 − 0.860085i \(-0.329590\pi\)
\(24\) 4.48119i 0.914720i
\(25\) −1.00000 −0.200000
\(26\) −3.19394 −0.626382
\(27\) − 5.35026i − 1.02966i
\(28\) 0.249646i 0.0471787i
\(29\) − 0.649738i − 0.120653i −0.998179 − 0.0603267i \(-0.980786\pi\)
0.998179 − 0.0603267i \(-0.0192142\pi\)
\(30\) 2.48119 0.453002
\(31\) − 1.83146i − 0.328939i −0.986382 − 0.164470i \(-0.947409\pi\)
0.986382 − 0.164470i \(-0.0525912\pi\)
\(32\) −1.09332 −0.193274
\(33\) 0.806063 0.140318
\(34\) −2.76845 + 5.44358i −0.474786 + 0.933567i
\(35\) −1.28726 −0.217586
\(36\) 0.0376114 0.00626857
\(37\) − 4.31265i − 0.708995i −0.935057 − 0.354498i \(-0.884652\pi\)
0.935057 − 0.354498i \(-0.115348\pi\)
\(38\) 4.96239 0.805006
\(39\) 3.61213i 0.578403i
\(40\) − 2.67513i − 0.422975i
\(41\) 11.2750i 1.76087i 0.474171 + 0.880433i \(0.342748\pi\)
−0.474171 + 0.880433i \(0.657252\pi\)
\(42\) 3.19394 0.492835
\(43\) 8.15633 1.24383 0.621914 − 0.783086i \(-0.286355\pi\)
0.621914 + 0.783086i \(0.286355\pi\)
\(44\) 0.0933212i 0.0140687i
\(45\) 0.193937i 0.0289104i
\(46\) − 12.2193i − 1.80164i
\(47\) −6.54420 −0.954569 −0.477285 − 0.878749i \(-0.658379\pi\)
−0.477285 + 0.878749i \(0.658379\pi\)
\(48\) 7.28726i 1.05183i
\(49\) 5.34297 0.763281
\(50\) −1.48119 −0.209473
\(51\) 6.15633 + 3.13093i 0.862058 + 0.438418i
\(52\) −0.418190 −0.0579926
\(53\) −8.57452 −1.17780 −0.588900 − 0.808206i \(-0.700439\pi\)
−0.588900 + 0.808206i \(0.700439\pi\)
\(54\) − 7.92478i − 1.07843i
\(55\) −0.481194 −0.0648842
\(56\) − 3.44358i − 0.460168i
\(57\) − 5.61213i − 0.743344i
\(58\) − 0.962389i − 0.126368i
\(59\) 4.96239 0.646048 0.323024 − 0.946391i \(-0.395301\pi\)
0.323024 + 0.946391i \(0.395301\pi\)
\(60\) 0.324869 0.0419404
\(61\) − 2.83638i − 0.363161i −0.983376 − 0.181581i \(-0.941879\pi\)
0.983376 − 0.181581i \(-0.0581213\pi\)
\(62\) − 2.71274i − 0.344519i
\(63\) 0.249646i 0.0314525i
\(64\) 7.08110 0.885138
\(65\) − 2.15633i − 0.267459i
\(66\) 1.19394 0.146963
\(67\) 4.93207 0.602548 0.301274 − 0.953538i \(-0.402588\pi\)
0.301274 + 0.953538i \(0.402588\pi\)
\(68\) −0.362481 + 0.712742i −0.0439572 + 0.0864327i
\(69\) −13.8192 −1.66364
\(70\) −1.90668 −0.227892
\(71\) 14.5320i 1.72463i 0.506373 + 0.862314i \(0.330986\pi\)
−0.506373 + 0.862314i \(0.669014\pi\)
\(72\) −0.518806 −0.0611418
\(73\) 13.3503i 1.56253i 0.624200 + 0.781265i \(0.285425\pi\)
−0.624200 + 0.781265i \(0.714575\pi\)
\(74\) − 6.38787i − 0.742575i
\(75\) 1.67513i 0.193427i
\(76\) 0.649738 0.0745301
\(77\) −0.619421 −0.0705896
\(78\) 5.35026i 0.605798i
\(79\) − 9.05571i − 1.01885i −0.860516 − 0.509423i \(-0.829859\pi\)
0.860516 − 0.509423i \(-0.170141\pi\)
\(80\) − 4.35026i − 0.486374i
\(81\) −8.38058 −0.931175
\(82\) 16.7005i 1.84426i
\(83\) −13.4314 −1.47428 −0.737142 − 0.675738i \(-0.763825\pi\)
−0.737142 + 0.675738i \(0.763825\pi\)
\(84\) 0.418190 0.0456283
\(85\) −3.67513 − 1.86907i −0.398624 − 0.202729i
\(86\) 12.0811 1.30274
\(87\) −1.08840 −0.116688
\(88\) − 1.28726i − 0.137222i
\(89\) −16.7816 −1.77885 −0.889424 − 0.457082i \(-0.848894\pi\)
−0.889424 + 0.457082i \(0.848894\pi\)
\(90\) 0.287258i 0.0302796i
\(91\) − 2.77575i − 0.290977i
\(92\) − 1.59991i − 0.166802i
\(93\) −3.06793 −0.318129
\(94\) −9.69323 −0.999780
\(95\) 3.35026i 0.343730i
\(96\) 1.83146i 0.186922i
\(97\) 3.66291i 0.371912i 0.982558 + 0.185956i \(0.0595383\pi\)
−0.982558 + 0.185956i \(0.940462\pi\)
\(98\) 7.91397 0.799432
\(99\) 0.0933212i 0.00937913i
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.d.a.16.5 ✓ 6
3.2 odd 2 765.2.g.b.271.1 6
4.3 odd 2 1360.2.c.f.1121.5 6
5.2 odd 4 425.2.c.b.424.5 6
5.3 odd 4 425.2.c.a.424.2 6
5.4 even 2 425.2.d.c.101.2 6
17.4 even 4 1445.2.a.k.1.1 3
17.13 even 4 1445.2.a.j.1.1 3
17.16 even 2 inner 85.2.d.a.16.6 yes 6
51.50 odd 2 765.2.g.b.271.2 6
68.67 odd 2 1360.2.c.f.1121.2 6
85.4 even 4 7225.2.a.r.1.3 3
85.33 odd 4 425.2.c.b.424.2 6
85.64 even 4 7225.2.a.q.1.3 3
85.67 odd 4 425.2.c.a.424.5 6
85.84 even 2 425.2.d.c.101.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.5 ✓ 6 1.1 even 1 trivial
85.2.d.a.16.6 yes 6 17.16 even 2 inner
425.2.c.a.424.2 6 5.3 odd 4
425.2.c.a.424.5 6 85.67 odd 4
425.2.c.b.424.2 6 85.33 odd 4
425.2.c.b.424.5 6 5.2 odd 4
425.2.d.c.101.1 6 85.84 even 2
425.2.d.c.101.2 6 5.4 even 2
765.2.g.b.271.1 6 3.2 odd 2
765.2.g.b.271.2 6 51.50 odd 2
1360.2.c.f.1121.2 6 68.67 odd 2
1360.2.c.f.1121.5 6 4.3 odd 2
1445.2.a.j.1.1 3 17.13 even 4
1445.2.a.k.1.1 3 17.4 even 4
7225.2.a.q.1.3 3 85.64 even 4
7225.2.a.r.1.3 3 85.4 even 4