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Properties

Label 85.2.a.b.1.2
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 1.2
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)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} -3.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.41421 q^{6} -0.585786 q^{7} -1.58579 q^{8} +8.65685 q^{9} -0.414214 q^{10} -2.58579 q^{11} +6.24264 q^{12} -2.82843 q^{13} -0.242641 q^{14} +3.41421 q^{15} +3.00000 q^{16} -1.00000 q^{17} +3.58579 q^{18} -2.82843 q^{19} +1.82843 q^{20} +2.00000 q^{21} -1.07107 q^{22} -3.41421 q^{23} +5.41421 q^{24} +1.00000 q^{25} -1.17157 q^{26} -19.3137 q^{27} +1.07107 q^{28} -4.82843 q^{29} +1.41421 q^{30} +4.24264 q^{31} +4.41421 q^{32} +8.82843 q^{33} -0.414214 q^{34} +0.585786 q^{35} -15.8284 q^{36} +6.48528 q^{37} -1.17157 q^{38} +9.65685 q^{39} +1.58579 q^{40} -6.48528 q^{41} +0.828427 q^{42} +7.65685 q^{43} +4.72792 q^{44} -8.65685 q^{45} -1.41421 q^{46} -4.82843 q^{47} -10.2426 q^{48} -6.65685 q^{49} +0.414214 q^{50} +3.41421 q^{51} +5.17157 q^{52} +0.343146 q^{53} -8.00000 q^{54} +2.58579 q^{55} +0.928932 q^{56} +9.65685 q^{57} -2.00000 q^{58} -9.17157 q^{59} -6.24264 q^{60} +7.65685 q^{61} +1.75736 q^{62} -5.07107 q^{63} -4.17157 q^{64} +2.82843 q^{65} +3.65685 q^{66} -3.17157 q^{67} +1.82843 q^{68} +11.6569 q^{69} +0.242641 q^{70} -4.24264 q^{71} -13.7279 q^{72} -4.82843 q^{73} +2.68629 q^{74} -3.41421 q^{75} +5.17157 q^{76} +1.51472 q^{77} +4.00000 q^{78} +5.41421 q^{79} -3.00000 q^{80} +39.9706 q^{81} -2.68629 q^{82} +9.31371 q^{83} -3.65685 q^{84} +1.00000 q^{85} +3.17157 q^{86} +16.4853 q^{87} +4.10051 q^{88} -2.34315 q^{89} -3.58579 q^{90} +1.65685 q^{91} +6.24264 q^{92} -14.4853 q^{93} -2.00000 q^{94} +2.82843 q^{95} -15.0711 q^{96} +3.65685 q^{97} -2.75736 q^{98} -22.3848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 2 q^{10} - 8 q^{11} + 4 q^{12} + 8 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 4 q^{23}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 − 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −3.41421 −1.97120 −0.985599 − 0.169102i \(-0.945913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) −1.41421 −0.577350
\(7\) −0.585786 −0.221406 −0.110703 − 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) −1.58579 −0.560660
\(9\) 8.65685 2.88562
\(10\) −0.414214 −0.130986
\(11\) −2.58579 −0.779644 −0.389822 − 0.920890i \(-0.627463\pi\)
−0.389822 + 0.920890i \(0.627463\pi\)
\(12\) 6.24264 1.80210
\(13\) −2.82843 −0.784465 −0.392232 − 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) −0.242641 −0.0648485
\(15\) 3.41421 0.881546
\(16\) 3.00000 0.750000
\(17\) −1.00000 −0.242536
\(18\) 3.58579 0.845178
\(19\) −2.82843 −0.648886 −0.324443 − 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 1.82843 0.408849
\(21\) 2.00000 0.436436
\(22\) −1.07107 −0.228352
\(23\) −3.41421 −0.711913 −0.355956 − 0.934503i \(-0.615845\pi\)
−0.355956 + 0.934503i \(0.615845\pi\)
\(24\) 5.41421 1.10517
\(25\) 1.00000 0.200000
\(26\) −1.17157 −0.229764
\(27\) −19.3137 −3.71692
\(28\) 1.07107 0.202413
\(29\) −4.82843 −0.896616 −0.448308 − 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 1.41421 0.258199
\(31\) 4.24264 0.762001 0.381000 − 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) 4.41421 0.780330
\(33\) 8.82843 1.53683
\(34\) −0.414214 −0.0710370
\(35\) 0.585786 0.0990160
\(36\) −15.8284 −2.63807
\(37\) 6.48528 1.06617 0.533087 − 0.846061i \(-0.321032\pi\)
0.533087 + 0.846061i \(0.321032\pi\)
\(38\) −1.17157 −0.190054
\(39\) 9.65685 1.54633
\(40\) 1.58579 0.250735
\(41\) −6.48528 −1.01283 −0.506415 − 0.862290i \(-0.669030\pi\)
−0.506415 + 0.862290i \(0.669030\pi\)
\(42\) 0.828427 0.127829
\(43\) 7.65685 1.16766 0.583830 − 0.811876i \(-0.301554\pi\)
0.583830 + 0.811876i \(0.301554\pi\)
\(44\) 4.72792 0.712761
\(45\) −8.65685 −1.29049
\(46\) −1.41421 −0.208514
\(47\) −4.82843 −0.704298 −0.352149 − 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(48\) −10.2426 −1.47840
\(49\) −6.65685 −0.950979
\(50\) 0.414214 0.0585786
\(51\) 3.41421 0.478086
\(52\) 5.17157 0.717168
\(53\) 0.343146 0.0471347 0.0235673 − 0.999722i \(-0.492498\pi\)
0.0235673 + 0.999722i \(0.492498\pi\)
\(54\) −8.00000 −1.08866
\(55\) 2.58579 0.348667
\(56\) 0.928932 0.124134
\(57\) 9.65685 1.27908
\(58\) −2.00000 −0.262613
\(59\) −9.17157 −1.19404 −0.597019 − 0.802227i \(-0.703648\pi\)
−0.597019 + 0.802227i \(0.703648\pi\)
\(60\) −6.24264 −0.805921
\(61\) 7.65685 0.980360 0.490180 − 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 1.75736 0.223185
\(63\) −5.07107 −0.638894
\(64\) −4.17157 −0.521447
\(65\) 2.82843 0.350823
\(66\) 3.65685 0.450128
\(67\) −3.17157 −0.387469 −0.193735 − 0.981054i \(-0.562060\pi\)
−0.193735 + 0.981054i \(0.562060\pi\)
\(68\) 1.82843 0.221729
\(69\) 11.6569 1.40332
\(70\) 0.242641 0.0290011
\(71\) −4.24264 −0.503509 −0.251754 − 0.967791i \(-0.581008\pi\)
−0.251754 + 0.967791i \(0.581008\pi\)
\(72\) −13.7279 −1.61785
\(73\) −4.82843 −0.565125 −0.282562 − 0.959249i \(-0.591184\pi\)
−0.282562 + 0.959249i \(0.591184\pi\)
\(74\) 2.68629 0.312275
\(75\) −3.41421 −0.394239
\(76\) 5.17157 0.593220
\(77\) 1.51472 0.172618
\(78\) 4.00000 0.452911
\(79\) 5.41421 0.609147 0.304573 − 0.952489i \(-0.401486\pi\)
0.304573 + 0.952489i \(0.401486\pi\)
\(80\) −3.00000 −0.335410
\(81\) 39.9706 4.44117
\(82\) −2.68629 −0.296651
\(83\) 9.31371 1.02231 0.511156 − 0.859488i \(-0.329217\pi\)
0.511156 + 0.859488i \(0.329217\pi\)
\(84\) −3.65685 −0.398996
\(85\) 1.00000 0.108465
\(86\) 3.17157 0.341999
\(87\) 16.4853 1.76741
\(88\) 4.10051 0.437115
\(89\) −2.34315 −0.248373 −0.124186 − 0.992259i \(-0.539632\pi\)
−0.124186 + 0.992259i \(0.539632\pi\)
\(90\) −3.58579 −0.377975
\(91\) 1.65685 0.173686
\(92\) 6.24264 0.650840
\(93\) −14.4853 −1.50205
\(94\) −2.00000 −0.206284
\(95\) 2.82843 0.290191
\(96\) −15.0711 −1.53818
\(97\) 3.65685 0.371297 0.185649 − 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) −2.75736 −0.278535
\(99\) −22.3848 −2.24975
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.1.2 ✓ 2
3.2 odd 2 765.2.a.i.1.1 2
4.3 odd 2 1360.2.a.o.1.2 2
5.2 odd 4 425.2.b.e.324.3 4
5.3 odd 4 425.2.b.e.324.2 4
5.4 even 2 425.2.a.f.1.1 2
7.6 odd 2 4165.2.a.q.1.2 2
8.3 odd 2 5440.2.a.ba.1.1 2
8.5 even 2 5440.2.a.bm.1.2 2
15.14 odd 2 3825.2.a.p.1.2 2
17.4 even 4 1445.2.d.f.866.2 4
17.13 even 4 1445.2.d.f.866.1 4
17.16 even 2 1445.2.a.f.1.2 2
20.19 odd 2 6800.2.a.ba.1.1 2
85.84 even 2 7225.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.a.b.1.2 ✓ 2 1.1 even 1 trivial
425.2.a.f.1.1 2 5.4 even 2
425.2.b.e.324.2 4 5.3 odd 4
425.2.b.e.324.3 4 5.2 odd 4
765.2.a.i.1.1 2 3.2 odd 2
1360.2.a.o.1.2 2 4.3 odd 2
1445.2.a.f.1.2 2 17.16 even 2
1445.2.d.f.866.1 4 17.13 even 4
1445.2.d.f.866.2 4 17.4 even 4
3825.2.a.p.1.2 2 15.14 odd 2
4165.2.a.q.1.2 2 7.6 odd 2
5440.2.a.ba.1.1 2 8.3 odd 2
5440.2.a.bm.1.2 2 8.5 even 2
6800.2.a.ba.1.1 2 20.19 odd 2
7225.2.a.o.1.1 2 85.84 even 2