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Properties

Label 85.2.a.c.1.1
Level $85$
Weight $2$
Character 85.1
Self dual yes
Analytic conductor $0.679$
Analytic rank $0$
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 = [2,0] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -4.73205 q^{6} -2.73205 q^{7} +1.73205 q^{8} +4.46410 q^{9} -1.73205 q^{10} +4.73205 q^{11} +2.73205 q^{12} -4.00000 q^{13} +4.73205 q^{14} +2.73205 q^{15} -5.00000 q^{16} -1.00000 q^{17} -7.73205 q^{18} -1.46410 q^{19} +1.00000 q^{20} -7.46410 q^{21} -8.19615 q^{22} -8.19615 q^{23} +4.73205 q^{24} +1.00000 q^{25} +6.92820 q^{26} +4.00000 q^{27} -2.73205 q^{28} -3.46410 q^{29} -4.73205 q^{30} +3.26795 q^{31} +5.19615 q^{32} +12.9282 q^{33} +1.73205 q^{34} -2.73205 q^{35} +4.46410 q^{36} -0.535898 q^{37} +2.53590 q^{38} -10.9282 q^{39} +1.73205 q^{40} -3.46410 q^{41} +12.9282 q^{42} -0.535898 q^{43} +4.73205 q^{44} +4.46410 q^{45} +14.1962 q^{46} +12.9282 q^{47} -13.6603 q^{48} +0.464102 q^{49} -1.73205 q^{50} -2.73205 q^{51} -4.00000 q^{52} +6.00000 q^{53} -6.92820 q^{54} +4.73205 q^{55} -4.73205 q^{56} -4.00000 q^{57} +6.00000 q^{58} +2.53590 q^{59} +2.73205 q^{60} -4.92820 q^{61} -5.66025 q^{62} -12.1962 q^{63} +1.00000 q^{64} -4.00000 q^{65} -22.3923 q^{66} -10.0000 q^{67} -1.00000 q^{68} -22.3923 q^{69} +4.73205 q^{70} +11.6603 q^{71} +7.73205 q^{72} +6.39230 q^{73} +0.928203 q^{74} +2.73205 q^{75} -1.46410 q^{76} -12.9282 q^{77} +18.9282 q^{78} +14.5885 q^{79} -5.00000 q^{80} -2.46410 q^{81} +6.00000 q^{82} +8.53590 q^{83} -7.46410 q^{84} -1.00000 q^{85} +0.928203 q^{86} -9.46410 q^{87} +8.19615 q^{88} +4.39230 q^{89} -7.73205 q^{90} +10.9282 q^{91} -8.19615 q^{92} +8.92820 q^{93} -22.3923 q^{94} -1.46410 q^{95} +14.1962 q^{96} -4.92820 q^{97} -0.803848 q^{98} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 8 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 6 q^{22} - 6 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 − 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 2.73205 1.57735 0.788675 − 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −4.73205 −1.93185
\(7\) −2.73205 −1.03262 −0.516309 − 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) 1.73205 0.612372
\(9\) 4.46410 1.48803
\(10\) −1.73205 −0.547723
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 2.73205 0.788675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 4.73205 1.26469
\(15\) 2.73205 0.705412
\(16\) −5.00000 −1.25000
\(17\) −1.00000 −0.242536
\(18\) −7.73205 −1.82246
\(19\) −1.46410 −0.335888 −0.167944 − 0.985797i \(-0.553713\pi\)
−0.167944 + 0.985797i \(0.553713\pi\)
\(20\) 1.00000 0.223607
\(21\) −7.46410 −1.62880
\(22\) −8.19615 −1.74743
\(23\) −8.19615 −1.70902 −0.854508 − 0.519438i \(-0.826141\pi\)
−0.854508 + 0.519438i \(0.826141\pi\)
\(24\) 4.73205 0.965926
\(25\) 1.00000 0.200000
\(26\) 6.92820 1.35873
\(27\) 4.00000 0.769800
\(28\) −2.73205 −0.516309
\(29\) −3.46410 −0.643268 −0.321634 − 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) −4.73205 −0.863950
\(31\) 3.26795 0.586941 0.293471 − 0.955968i \(-0.405190\pi\)
0.293471 + 0.955968i \(0.405190\pi\)
\(32\) 5.19615 0.918559
\(33\) 12.9282 2.25051
\(34\) 1.73205 0.297044
\(35\) −2.73205 −0.461801
\(36\) 4.46410 0.744017
\(37\) −0.535898 −0.0881012 −0.0440506 − 0.999029i \(-0.514026\pi\)
−0.0440506 + 0.999029i \(0.514026\pi\)
\(38\) 2.53590 0.411377
\(39\) −10.9282 −1.74991
\(40\) 1.73205 0.273861
\(41\) −3.46410 −0.541002 −0.270501 − 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 12.9282 1.99487
\(43\) −0.535898 −0.0817237 −0.0408619 − 0.999165i \(-0.513010\pi\)
−0.0408619 + 0.999165i \(0.513010\pi\)
\(44\) 4.73205 0.713384
\(45\) 4.46410 0.665469
\(46\) 14.1962 2.09311
\(47\) 12.9282 1.88577 0.942886 − 0.333115i \(-0.108100\pi\)
0.942886 + 0.333115i \(0.108100\pi\)
\(48\) −13.6603 −1.97169
\(49\) 0.464102 0.0663002
\(50\) −1.73205 −0.244949
\(51\) −2.73205 −0.382564
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −6.92820 −0.942809
\(55\) 4.73205 0.638070
\(56\) −4.73205 −0.632347
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 2.53590 0.330146 0.165073 − 0.986281i \(-0.447214\pi\)
0.165073 + 0.986281i \(0.447214\pi\)
\(60\) 2.73205 0.352706
\(61\) −4.92820 −0.630992 −0.315496 − 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) −5.66025 −0.718853
\(63\) −12.1962 −1.53657
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −22.3923 −2.75630
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −1.00000 −0.121268
\(69\) −22.3923 −2.69572
\(70\) 4.73205 0.565588
\(71\) 11.6603 1.38382 0.691909 − 0.721985i \(-0.256770\pi\)
0.691909 + 0.721985i \(0.256770\pi\)
\(72\) 7.73205 0.911231
\(73\) 6.39230 0.748163 0.374081 − 0.927396i \(-0.377958\pi\)
0.374081 + 0.927396i \(0.377958\pi\)
\(74\) 0.928203 0.107901
\(75\) 2.73205 0.315470
\(76\) −1.46410 −0.167944
\(77\) −12.9282 −1.47331
\(78\) 18.9282 2.14320
\(79\) 14.5885 1.64133 0.820665 − 0.571410i \(-0.193603\pi\)
0.820665 + 0.571410i \(0.193603\pi\)
\(80\) −5.00000 −0.559017
\(81\) −2.46410 −0.273789
\(82\) 6.00000 0.662589
\(83\) 8.53590 0.936937 0.468468 − 0.883480i \(-0.344806\pi\)
0.468468 + 0.883480i \(0.344806\pi\)
\(84\) −7.46410 −0.814400
\(85\) −1.00000 −0.108465
\(86\) 0.928203 0.100091
\(87\) −9.46410 −1.01466
\(88\) 8.19615 0.873713
\(89\) 4.39230 0.465583 0.232792 − 0.972527i \(-0.425214\pi\)
0.232792 + 0.972527i \(0.425214\pi\)
\(90\) −7.73205 −0.815030
\(91\) 10.9282 1.14559
\(92\) −8.19615 −0.854508
\(93\) 8.92820 0.925812
\(94\) −22.3923 −2.30959
\(95\) −1.46410 −0.150214
\(96\) 14.1962 1.44889
\(97\) −4.92820 −0.500383 −0.250192 − 0.968196i \(-0.580494\pi\)
−0.250192 + 0.968196i \(0.580494\pi\)
\(98\) −0.803848 −0.0812009
\(99\) 21.1244 2.12308
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.c.1.1 ✓ 2
3.2 odd 2 765.2.a.g.1.2 2
4.3 odd 2 1360.2.a.k.1.1 2
5.2 odd 4 425.2.b.d.324.1 4
5.3 odd 4 425.2.b.d.324.4 4
5.4 even 2 425.2.a.e.1.2 2
7.6 odd 2 4165.2.a.t.1.1 2
8.3 odd 2 5440.2.a.bl.1.2 2
8.5 even 2 5440.2.a.bb.1.1 2
15.14 odd 2 3825.2.a.v.1.1 2
17.4 even 4 1445.2.d.e.866.3 4
17.13 even 4 1445.2.d.e.866.4 4
17.16 even 2 1445.2.a.g.1.1 2
20.19 odd 2 6800.2.a.bg.1.2 2
85.84 even 2 7225.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.a.c.1.1 ✓ 2 1.1 even 1 trivial
425.2.a.e.1.2 2 5.4 even 2
425.2.b.d.324.1 4 5.2 odd 4
425.2.b.d.324.4 4 5.3 odd 4
765.2.a.g.1.2 2 3.2 odd 2
1360.2.a.k.1.1 2 4.3 odd 2
1445.2.a.g.1.1 2 17.16 even 2
1445.2.d.e.866.3 4 17.4 even 4
1445.2.d.e.866.4 4 17.13 even 4
3825.2.a.v.1.1 2 15.14 odd 2
4165.2.a.t.1.1 2 7.6 odd 2
5440.2.a.bb.1.1 2 8.5 even 2
5440.2.a.bl.1.2 2 8.3 odd 2
6800.2.a.bg.1.2 2 20.19 odd 2
7225.2.a.l.1.2 2 85.84 even 2