Properties

Label 85.2.a.b.1.1
Level $85$
Weight $2$
Character 85.1
Self dual yes
Analytic conductor $0.679$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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(1,85)] 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("85.1"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.678728417181\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 85.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} -0.585786 q^{3} +3.82843 q^{4} -1.00000 q^{5} +1.41421 q^{6} -3.41421 q^{7} -4.41421 q^{8} -2.65685 q^{9} +2.41421 q^{10} -5.41421 q^{11} -2.24264 q^{12} +2.82843 q^{13} +8.24264 q^{14} +0.585786 q^{15} +3.00000 q^{16} -1.00000 q^{17} +6.41421 q^{18} +2.82843 q^{19} -3.82843 q^{20} +2.00000 q^{21} +13.0711 q^{22} -0.585786 q^{23} +2.58579 q^{24} +1.00000 q^{25} -6.82843 q^{26} +3.31371 q^{27} -13.0711 q^{28} +0.828427 q^{29} -1.41421 q^{30} -4.24264 q^{31} +1.58579 q^{32} +3.17157 q^{33} +2.41421 q^{34} +3.41421 q^{35} -10.1716 q^{36} -10.4853 q^{37} -6.82843 q^{38} -1.65685 q^{39} +4.41421 q^{40} +10.4853 q^{41} -4.82843 q^{42} -3.65685 q^{43} -20.7279 q^{44} +2.65685 q^{45} +1.41421 q^{46} +0.828427 q^{47} -1.75736 q^{48} +4.65685 q^{49} -2.41421 q^{50} +0.585786 q^{51} +10.8284 q^{52} +11.6569 q^{53} -8.00000 q^{54} +5.41421 q^{55} +15.0711 q^{56} -1.65685 q^{57} -2.00000 q^{58} -14.8284 q^{59} +2.24264 q^{60} -3.65685 q^{61} +10.2426 q^{62} +9.07107 q^{63} -9.82843 q^{64} -2.82843 q^{65} -7.65685 q^{66} -8.82843 q^{67} -3.82843 q^{68} +0.343146 q^{69} -8.24264 q^{70} +4.24264 q^{71} +11.7279 q^{72} +0.828427 q^{73} +25.3137 q^{74} -0.585786 q^{75} +10.8284 q^{76} +18.4853 q^{77} +4.00000 q^{78} +2.58579 q^{79} -3.00000 q^{80} +6.02944 q^{81} -25.3137 q^{82} -13.3137 q^{83} +7.65685 q^{84} +1.00000 q^{85} +8.82843 q^{86} -0.485281 q^{87} +23.8995 q^{88} -13.6569 q^{89} -6.41421 q^{90} -9.65685 q^{91} -2.24264 q^{92} +2.48528 q^{93} -2.00000 q^{94} -2.82843 q^{95} -0.928932 q^{96} -7.65685 q^{97} -11.2426 q^{98} +14.3848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 2 q^{10} - 8 q^{11} + 4 q^{12} + 8 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 4 q^{23}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 − 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −0.585786 −0.338204 −0.169102 − 0.985599i \(-0.554087\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) 1.41421 0.577350
\(7\) −3.41421 −1.29045 −0.645226 − 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(8\) −4.41421 −1.56066
\(9\) −2.65685 −0.885618
\(10\) 2.41421 0.763441
\(11\) −5.41421 −1.63245 −0.816223 − 0.577736i \(-0.803936\pi\)
−0.816223 + 0.577736i \(0.803936\pi\)
\(12\) −2.24264 −0.647395
\(13\) 2.82843 0.784465 0.392232 − 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 8.24264 2.20294
\(15\) 0.585786 0.151249
\(16\) 3.00000 0.750000
\(17\) −1.00000 −0.242536
\(18\) 6.41421 1.51184
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −3.82843 −0.856062
\(21\) 2.00000 0.436436
\(22\) 13.0711 2.78676
\(23\) −0.585786 −0.122145 −0.0610725 − 0.998133i \(-0.519452\pi\)
−0.0610725 + 0.998133i \(0.519452\pi\)
\(24\) 2.58579 0.527821
\(25\) 1.00000 0.200000
\(26\) −6.82843 −1.33916
\(27\) 3.31371 0.637723
\(28\) −13.0711 −2.47020
\(29\) 0.828427 0.153835 0.0769175 − 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) −1.41421 −0.258199
\(31\) −4.24264 −0.762001 −0.381000 − 0.924575i \(-0.624420\pi\)
−0.381000 + 0.924575i \(0.624420\pi\)
\(32\) 1.58579 0.280330
\(33\) 3.17157 0.552100
\(34\) 2.41421 0.414034
\(35\) 3.41421 0.577107
\(36\) −10.1716 −1.69526
\(37\) −10.4853 −1.72377 −0.861885 − 0.507104i \(-0.830716\pi\)
−0.861885 + 0.507104i \(0.830716\pi\)
\(38\) −6.82843 −1.10772
\(39\) −1.65685 −0.265309
\(40\) 4.41421 0.697948
\(41\) 10.4853 1.63753 0.818763 − 0.574132i \(-0.194660\pi\)
0.818763 + 0.574132i \(0.194660\pi\)
\(42\) −4.82843 −0.745042
\(43\) −3.65685 −0.557665 −0.278833 − 0.960340i \(-0.589947\pi\)
−0.278833 + 0.960340i \(0.589947\pi\)
\(44\) −20.7279 −3.12485
\(45\) 2.65685 0.396060
\(46\) 1.41421 0.208514
\(47\) 0.828427 0.120839 0.0604193 − 0.998173i \(-0.480756\pi\)
0.0604193 + 0.998173i \(0.480756\pi\)
\(48\) −1.75736 −0.253653
\(49\) 4.65685 0.665265
\(50\) −2.41421 −0.341421
\(51\) 0.585786 0.0820265
\(52\) 10.8284 1.50163
\(53\) 11.6569 1.60119 0.800596 − 0.599204i \(-0.204516\pi\)
0.800596 + 0.599204i \(0.204516\pi\)
\(54\) −8.00000 −1.08866
\(55\) 5.41421 0.730052
\(56\) 15.0711 2.01396
\(57\) −1.65685 −0.219456
\(58\) −2.00000 −0.262613
\(59\) −14.8284 −1.93050 −0.965248 − 0.261334i \(-0.915838\pi\)
−0.965248 + 0.261334i \(0.915838\pi\)
\(60\) 2.24264 0.289524
\(61\) −3.65685 −0.468212 −0.234106 − 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 10.2426 1.30082
\(63\) 9.07107 1.14285
\(64\) −9.82843 −1.22855
\(65\) −2.82843 −0.350823
\(66\) −7.65685 −0.942494
\(67\) −8.82843 −1.07856 −0.539282 − 0.842125i \(-0.681304\pi\)
−0.539282 + 0.842125i \(0.681304\pi\)
\(68\) −3.82843 −0.464265
\(69\) 0.343146 0.0413099
\(70\) −8.24264 −0.985184
\(71\) 4.24264 0.503509 0.251754 − 0.967791i \(-0.418992\pi\)
0.251754 + 0.967791i \(0.418992\pi\)
\(72\) 11.7279 1.38215
\(73\) 0.828427 0.0969601 0.0484800 − 0.998824i \(-0.484562\pi\)
0.0484800 + 0.998824i \(0.484562\pi\)
\(74\) 25.3137 2.94266
\(75\) −0.585786 −0.0676408
\(76\) 10.8284 1.24211
\(77\) 18.4853 2.10659
\(78\) 4.00000 0.452911
\(79\) 2.58579 0.290924 0.145462 − 0.989364i \(-0.453533\pi\)
0.145462 + 0.989364i \(0.453533\pi\)
\(80\) −3.00000 −0.335410
\(81\) 6.02944 0.669937
\(82\) −25.3137 −2.79543
\(83\) −13.3137 −1.46137 −0.730685 − 0.682715i \(-0.760799\pi\)
−0.730685 + 0.682715i \(0.760799\pi\)
\(84\) 7.65685 0.835431
\(85\) 1.00000 0.108465
\(86\) 8.82843 0.951994
\(87\) −0.485281 −0.0520276
\(88\) 23.8995 2.54769
\(89\) −13.6569 −1.44762 −0.723812 − 0.689997i \(-0.757612\pi\)
−0.723812 + 0.689997i \(0.757612\pi\)
\(90\) −6.41421 −0.676117
\(91\) −9.65685 −1.01231
\(92\) −2.24264 −0.233811
\(93\) 2.48528 0.257712
\(94\) −2.00000 −0.206284
\(95\) −2.82843 −0.290191
\(96\) −0.928932 −0.0948087
\(97\) −7.65685 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) −11.2426 −1.13568
\(99\) 14.3848 1.44572
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.a.b.1.1 ✓ 2
3.2 odd 2 765.2.a.i.1.2 2
4.3 odd 2 1360.2.a.o.1.1 2
5.2 odd 4 425.2.b.e.324.1 4
5.3 odd 4 425.2.b.e.324.4 4
5.4 even 2 425.2.a.f.1.2 2
7.6 odd 2 4165.2.a.q.1.1 2
8.3 odd 2 5440.2.a.ba.1.2 2
8.5 even 2 5440.2.a.bm.1.1 2
15.14 odd 2 3825.2.a.p.1.1 2
17.4 even 4 1445.2.d.f.866.4 4
17.13 even 4 1445.2.d.f.866.3 4
17.16 even 2 1445.2.a.f.1.1 2
20.19 odd 2 6800.2.a.ba.1.2 2
85.84 even 2 7225.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.a.b.1.1 ✓ 2 1.1 even 1 trivial
425.2.a.f.1.2 2 5.4 even 2
425.2.b.e.324.1 4 5.2 odd 4
425.2.b.e.324.4 4 5.3 odd 4
765.2.a.i.1.2 2 3.2 odd 2
1360.2.a.o.1.1 2 4.3 odd 2
1445.2.a.f.1.1 2 17.16 even 2
1445.2.d.f.866.3 4 17.13 even 4
1445.2.d.f.866.4 4 17.4 even 4
3825.2.a.p.1.1 2 15.14 odd 2
4165.2.a.q.1.1 2 7.6 odd 2
5440.2.a.ba.1.2 2 8.3 odd 2
5440.2.a.bm.1.1 2 8.5 even 2
6800.2.a.ba.1.2 2 20.19 odd 2
7225.2.a.o.1.2 2 85.84 even 2