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Properties

Label 85.2.d.a.16.1
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 16.1
Root \(-0.854638 + 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 85.16
Dual form 85.2.d.a.16.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.17009 q^{2} -0.539189i q^{3} +2.70928 q^{4} +1.00000i q^{5} +1.17009i q^{6} -4.87936i q^{7} -1.53919 q^{8} +2.70928 q^{9} -2.17009i q^{10} -3.17009i q^{11} -1.46081i q^{12} +2.63090 q^{13} +10.5886i q^{14} +0.539189 q^{15} -2.07838 q^{16} +(-3.24846 + 2.53919i) q^{17} -5.87936 q^{18} +1.07838 q^{19} +2.70928i q^{20} -2.63090 q^{21} +6.87936i q^{22} +5.21953i q^{23} +0.829914i q^{24} -1.00000 q^{25} -5.70928 q^{26} -3.07838i q^{27} -13.2195i q^{28} -2.92162i q^{29} -1.17009 q^{30} +4.09171i q^{31} +7.58864 q^{32} -1.70928 q^{33} +(7.04945 - 5.51026i) q^{34} +4.87936 q^{35} +7.34017 q^{36} +5.26180i q^{37} -2.34017 q^{38} -1.41855i q^{39} -1.53919i q^{40} -5.60197i q^{41} +5.70928 q^{42} +3.36910 q^{43} -8.58864i q^{44} +2.70928i q^{45} -11.3268i q^{46} -6.78765 q^{47} +1.12064i q^{48} -16.8082 q^{49} +2.17009 q^{50} +(1.36910 + 1.75154i) q^{51} +7.12783 q^{52} +3.75872 q^{53} +6.68035i q^{54} +3.17009 q^{55} +7.51026i q^{56} -0.581449i q^{57} +6.34017i q^{58} -2.34017 q^{59} +1.46081 q^{60} +12.2557i q^{61} -8.87936i q^{62} -13.2195i q^{63} -12.3112 q^{64} +2.63090i q^{65} +3.70928 q^{66} +10.2062 q^{67} +(-8.80098 + 6.87936i) q^{68} +2.81432 q^{69} -10.5886 q^{70} +4.06505i q^{71} -4.17009 q^{72} +11.0784i q^{73} -11.4186i q^{74} +0.539189i q^{75} +2.92162 q^{76} -15.4680 q^{77} +3.07838i q^{78} +6.92881i q^{79} -2.07838i q^{80} +6.46800 q^{81} +12.1568i q^{82} +8.23287 q^{83} -7.12783 q^{84} +(-2.53919 - 3.24846i) q^{85} -7.31124 q^{86} -1.57531 q^{87} +4.87936i q^{88} +7.15449 q^{89} -5.87936i q^{90} -12.8371i q^{91} +14.1412i q^{92} +2.20620 q^{93} +14.7298 q^{94} +1.07838i q^{95} -4.09171i q^{96} -8.18342i q^{97} +36.4752 q^{98} -8.58864i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} + 2 q^{9} + 8 q^{13} - 6 q^{16} - 2 q^{17} - 10 q^{18} - 8 q^{21} - 6 q^{25} - 20 q^{26} + 4 q^{30} + 6 q^{32} + 4 q^{33} + 6 q^{34} + 4 q^{35} + 22 q^{36} + 8 q^{38} + 20 q^{42}+ \cdots + 86 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17009 −1.53448 −0.767241 − 0.641358i \(-0.778371\pi\)
−0.767241 + 0.641358i \(0.778371\pi\)
\(3\) − 0.539189i − 0.311301i −0.987812 − 0.155650i \(-0.950253\pi\)
0.987812 − 0.155650i \(-0.0497473\pi\)
\(4\) 2.70928 1.35464
\(5\) 1.00000i 0.447214i
\(6\) 1.17009i 0.477686i
\(7\) − 4.87936i − 1.84423i −0.386921 − 0.922113i \(-0.626462\pi\)
0.386921 − 0.922113i \(-0.373538\pi\)
\(8\) −1.53919 −0.544185
\(9\) 2.70928 0.903092
\(10\) − 2.17009i − 0.686242i
\(11\) − 3.17009i − 0.955817i −0.878410 − 0.477909i \(-0.841395\pi\)
0.878410 − 0.477909i \(-0.158605\pi\)
\(12\) − 1.46081i − 0.421700i
\(13\) 2.63090 0.729680 0.364840 − 0.931070i \(-0.381124\pi\)
0.364840 + 0.931070i \(0.381124\pi\)
\(14\) 10.5886i 2.82993i
\(15\) 0.539189 0.139218
\(16\) −2.07838 −0.519594
\(17\) −3.24846 + 2.53919i −0.787868 + 0.615844i
\(18\) −5.87936 −1.38578
\(19\) 1.07838 0.247397 0.123698 − 0.992320i \(-0.460524\pi\)
0.123698 + 0.992320i \(0.460524\pi\)
\(20\) 2.70928i 0.605812i
\(21\) −2.63090 −0.574109
\(22\) 6.87936i 1.46668i
\(23\) 5.21953i 1.08835i 0.838972 + 0.544174i \(0.183157\pi\)
−0.838972 + 0.544174i \(0.816843\pi\)
\(24\) 0.829914i 0.169405i
\(25\) −1.00000 −0.200000
\(26\) −5.70928 −1.11968
\(27\) − 3.07838i − 0.592434i
\(28\) − 13.2195i − 2.49826i
\(29\) − 2.92162i − 0.542532i −0.962504 − 0.271266i \(-0.912558\pi\)
0.962504 − 0.271266i \(-0.0874422\pi\)
\(30\) −1.17009 −0.213628
\(31\) 4.09171i 0.734893i 0.930045 + 0.367446i \(0.119768\pi\)
−0.930045 + 0.367446i \(0.880232\pi\)
\(32\) 7.58864 1.34149
\(33\) −1.70928 −0.297547
\(34\) 7.04945 − 5.51026i 1.20897 − 0.945002i
\(35\) 4.87936 0.824763
\(36\) 7.34017 1.22336
\(37\) 5.26180i 0.865034i 0.901626 + 0.432517i \(0.142374\pi\)
−0.901626 + 0.432517i \(0.857626\pi\)
\(38\) −2.34017 −0.379626
\(39\) − 1.41855i − 0.227150i
\(40\) − 1.53919i − 0.243367i
\(41\) − 5.60197i − 0.874880i −0.899247 − 0.437440i \(-0.855885\pi\)
0.899247 − 0.437440i \(-0.144115\pi\)
\(42\) 5.70928 0.880960
\(43\) 3.36910 0.513783 0.256892 − 0.966440i \(-0.417302\pi\)
0.256892 + 0.966440i \(0.417302\pi\)
\(44\) − 8.58864i − 1.29479i
\(45\) 2.70928i 0.403875i
\(46\) − 11.3268i − 1.67005i
\(47\) −6.78765 −0.990081 −0.495040 − 0.868870i \(-0.664847\pi\)
−0.495040 + 0.868870i \(0.664847\pi\)
\(48\) 1.12064i 0.161750i
\(49\) −16.8082 −2.40117
\(50\) 2.17009 0.306897
\(51\) 1.36910 + 1.75154i 0.191713 + 0.245264i
\(52\) 7.12783 0.988452
\(53\) 3.75872 0.516300 0.258150 − 0.966105i \(-0.416887\pi\)
0.258150 + 0.966105i \(0.416887\pi\)
\(54\) 6.68035i 0.909080i
\(55\) 3.17009 0.427454
\(56\) 7.51026i 1.00360i
\(57\) − 0.581449i − 0.0770148i
\(58\) 6.34017i 0.832505i
\(59\) −2.34017 −0.304665 −0.152332 − 0.988329i \(-0.548678\pi\)
−0.152332 + 0.988329i \(0.548678\pi\)
\(60\) 1.46081 0.188590
\(61\) 12.2557i 1.56918i 0.620018 + 0.784588i \(0.287125\pi\)
−0.620018 + 0.784588i \(0.712875\pi\)
\(62\) − 8.87936i − 1.12768i
\(63\) − 13.2195i − 1.66550i
\(64\) −12.3112 −1.53891
\(65\) 2.63090i 0.326323i
\(66\) 3.70928 0.456580
\(67\) 10.2062 1.24689 0.623443 − 0.781869i \(-0.285733\pi\)
0.623443 + 0.781869i \(0.285733\pi\)
\(68\) −8.80098 + 6.87936i −1.06728 + 0.834245i
\(69\) 2.81432 0.338804
\(70\) −10.5886 −1.26558
\(71\) 4.06505i 0.482432i 0.970471 + 0.241216i \(0.0775463\pi\)
−0.970471 + 0.241216i \(0.922454\pi\)
\(72\) −4.17009 −0.491449
\(73\) 11.0784i 1.29663i 0.761374 + 0.648313i \(0.224525\pi\)
−0.761374 + 0.648313i \(0.775475\pi\)
\(74\) − 11.4186i − 1.32738i
\(75\) 0.539189i 0.0622602i
\(76\) 2.92162 0.335133
\(77\) −15.4680 −1.76274
\(78\) 3.07838i 0.348558i
\(79\) 6.92881i 0.779552i 0.920910 + 0.389776i \(0.127448\pi\)
−0.920910 + 0.389776i \(0.872552\pi\)
\(80\) − 2.07838i − 0.232370i
\(81\) 6.46800 0.718667
\(82\) 12.1568i 1.34249i
\(83\) 8.23287 0.903674 0.451837 − 0.892100i \(-0.350769\pi\)
0.451837 + 0.892100i \(0.350769\pi\)
\(84\) −7.12783 −0.777710
\(85\) −2.53919 − 3.24846i −0.275414 − 0.352345i
\(86\) −7.31124 −0.788392
\(87\) −1.57531 −0.168891
\(88\) 4.87936i 0.520142i
\(89\) 7.15449 0.758374 0.379187 − 0.925320i \(-0.376204\pi\)
0.379187 + 0.925320i \(0.376204\pi\)
\(90\) − 5.87936i − 0.619739i
\(91\) − 12.8371i − 1.34569i
\(92\) 14.1412i 1.47432i
\(93\) 2.20620 0.228773
\(94\) 14.7298 1.51926
\(95\) 1.07838i 0.110639i
\(96\) − 4.09171i − 0.417608i
\(97\) − 8.18342i − 0.830900i −0.909616 − 0.415450i \(-0.863624\pi\)
0.909616 − 0.415450i \(-0.136376\pi\)
\(98\) 36.4752 3.68455
\(99\) − 8.58864i − 0.863191i
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.16.1 ✓ 6
3.2 odd 2 765.2.g.b.271.5 6
4.3 odd 2 1360.2.c.f.1121.4 6
5.2 odd 4 425.2.c.b.424.1 6
5.3 odd 4 425.2.c.a.424.6 6
5.4 even 2 425.2.d.c.101.6 6
17.4 even 4 1445.2.a.k.1.3 3
17.13 even 4 1445.2.a.j.1.3 3
17.16 even 2 inner 85.2.d.a.16.2 yes 6
51.50 odd 2 765.2.g.b.271.6 6
68.67 odd 2 1360.2.c.f.1121.3 6
85.4 even 4 7225.2.a.r.1.1 3
85.33 odd 4 425.2.c.b.424.6 6
85.64 even 4 7225.2.a.q.1.1 3
85.67 odd 4 425.2.c.a.424.1 6
85.84 even 2 425.2.d.c.101.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.1 ✓ 6 1.1 even 1 trivial
85.2.d.a.16.2 yes 6 17.16 even 2 inner
425.2.c.a.424.1 6 85.67 odd 4
425.2.c.a.424.6 6 5.3 odd 4
425.2.c.b.424.1 6 5.2 odd 4
425.2.c.b.424.6 6 85.33 odd 4
425.2.d.c.101.5 6 85.84 even 2
425.2.d.c.101.6 6 5.4 even 2
765.2.g.b.271.5 6 3.2 odd 2
765.2.g.b.271.6 6 51.50 odd 2
1360.2.c.f.1121.3 6 68.67 odd 2
1360.2.c.f.1121.4 6 4.3 odd 2
1445.2.a.j.1.3 3 17.13 even 4
1445.2.a.k.1.3 3 17.4 even 4
7225.2.a.q.1.1 3 85.64 even 4
7225.2.a.r.1.1 3 85.4 even 4