Properties

Label 85.4.a.f.1.3
Level $85$
Weight $4$
Character 85.1
Self dual yes
Analytic conductor $5.015$
Analytic rank $0$
Dimension $3$
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] = [85,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.363328\) 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)
 
\(f(q)\) \(=\) \(q+4.50466 q^{2} +5.08998 q^{3} +12.2920 q^{4} -5.00000 q^{5} +22.9287 q^{6} -0.616696 q^{7} +19.3340 q^{8} -1.09206 q^{9} -22.5233 q^{10} -8.63535 q^{11} +62.5661 q^{12} +6.44012 q^{13} -2.77801 q^{14} -25.4499 q^{15} -11.2427 q^{16} -17.0000 q^{17} -4.91934 q^{18} +7.96806 q^{19} -61.4600 q^{20} -3.13897 q^{21} -38.8994 q^{22} +66.1209 q^{23} +98.4099 q^{24} +25.0000 q^{25} +29.0106 q^{26} -142.988 q^{27} -7.58043 q^{28} +219.813 q^{29} -114.643 q^{30} -0.608686 q^{31} -205.317 q^{32} -43.9538 q^{33} -76.5793 q^{34} +3.08348 q^{35} -13.4235 q^{36} -216.124 q^{37} +35.8934 q^{38} +32.7801 q^{39} -96.6701 q^{40} +355.508 q^{41} -14.1400 q^{42} +209.947 q^{43} -106.146 q^{44} +5.46028 q^{45} +297.852 q^{46} -324.626 q^{47} -57.2253 q^{48} -342.620 q^{49} +112.617 q^{50} -86.5297 q^{51} +79.1619 q^{52} +189.591 q^{53} -644.114 q^{54} +43.1768 q^{55} -11.9232 q^{56} +40.5573 q^{57} +990.185 q^{58} -257.619 q^{59} -312.830 q^{60} +240.335 q^{61} -2.74192 q^{62} +0.673466 q^{63} -834.942 q^{64} -32.2006 q^{65} -197.997 q^{66} -66.9428 q^{67} -208.964 q^{68} +336.554 q^{69} +13.8900 q^{70} -131.664 q^{71} -21.1138 q^{72} +1172.76 q^{73} -973.565 q^{74} +127.250 q^{75} +97.9434 q^{76} +5.32539 q^{77} +147.663 q^{78} +707.825 q^{79} +56.2136 q^{80} -698.322 q^{81} +1601.44 q^{82} +1007.30 q^{83} -38.5843 q^{84} +85.0000 q^{85} +945.740 q^{86} +1118.85 q^{87} -166.956 q^{88} +1013.01 q^{89} +24.5967 q^{90} -3.97159 q^{91} +812.758 q^{92} -3.09820 q^{93} -1462.33 q^{94} -39.8403 q^{95} -1045.06 q^{96} +973.147 q^{97} -1543.39 q^{98} +9.43029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 9 q^{3} - q^{4} - 15 q^{5} + 33 q^{6} + 34 q^{7} + 39 q^{8} + 60 q^{9} - 15 q^{10} + 52 q^{11} + 17 q^{12} + 19 q^{13} - 2 q^{14} - 45 q^{15} + 59 q^{16} - 51 q^{17} - 153 q^{19} + 5 q^{20}+ \cdots + 2524 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.50466 1.59264 0.796320 − 0.604876i \(-0.206777\pi\)
0.796320 + 0.604876i \(0.206777\pi\)
\(3\) 5.08998 0.979568 0.489784 − 0.871844i \(-0.337076\pi\)
0.489784 + 0.871844i \(0.337076\pi\)
\(4\) 12.2920 1.53650
\(5\) −5.00000 −0.447214
\(6\) 22.9287 1.56010
\(7\) −0.616696 −0.0332984 −0.0166492 − 0.999861i \(-0.505300\pi\)
−0.0166492 + 0.999861i \(0.505300\pi\)
\(8\) 19.3340 0.854451
\(9\) −1.09206 −0.0404465
\(10\) −22.5233 −0.712250
\(11\) −8.63535 −0.236696 −0.118348 − 0.992972i \(-0.537760\pi\)
−0.118348 + 0.992972i \(0.537760\pi\)
\(12\) 62.5661 1.50511
\(13\) 6.44012 0.137397 0.0686987 − 0.997637i \(-0.478115\pi\)
0.0686987 + 0.997637i \(0.478115\pi\)
\(14\) −2.77801 −0.0530324
\(15\) −25.4499 −0.438076
\(16\) −11.2427 −0.175668
\(17\) −17.0000 −0.242536
\(18\) −4.91934 −0.0644167
\(19\) 7.96806 0.0962104 0.0481052 − 0.998842i \(-0.484682\pi\)
0.0481052 + 0.998842i \(0.484682\pi\)
\(20\) −61.4600 −0.687144
\(21\) −3.13897 −0.0326181
\(22\) −38.8994 −0.376972
\(23\) 66.1209 0.599442 0.299721 − 0.954027i \(-0.403106\pi\)
0.299721 + 0.954027i \(0.403106\pi\)
\(24\) 98.4099 0.836993
\(25\) 25.0000 0.200000
\(26\) 29.0106 0.218825
\(27\) −142.988 −1.01919
\(28\) −7.58043 −0.0511631
\(29\) 219.813 1.40753 0.703764 − 0.710434i \(-0.251501\pi\)
0.703764 + 0.710434i \(0.251501\pi\)
\(30\) −114.643 −0.697697
\(31\) −0.608686 −0.00352655 −0.00176328 − 0.999998i \(-0.500561\pi\)
−0.00176328 + 0.999998i \(0.500561\pi\)
\(32\) −205.317 −1.13423
\(33\) −43.9538 −0.231860
\(34\) −76.5793 −0.386272
\(35\) 3.08348 0.0148915
\(36\) −13.4235 −0.0621461
\(37\) −216.124 −0.960285 −0.480142 − 0.877191i \(-0.659415\pi\)
−0.480142 + 0.877191i \(0.659415\pi\)
\(38\) 35.8934 0.153229
\(39\) 32.7801 0.134590
\(40\) −96.6701 −0.382122
\(41\) 355.508 1.35417 0.677086 − 0.735904i \(-0.263243\pi\)
0.677086 + 0.735904i \(0.263243\pi\)
\(42\) −14.1400 −0.0519489
\(43\) 209.947 0.744572 0.372286 − 0.928118i \(-0.378574\pi\)
0.372286 + 0.928118i \(0.378574\pi\)
\(44\) −106.146 −0.363684
\(45\) 5.46028 0.0180882
\(46\) 297.852 0.954694
\(47\) −324.626 −1.00748 −0.503740 − 0.863855i \(-0.668043\pi\)
−0.503740 + 0.863855i \(0.668043\pi\)
\(48\) −57.2253 −0.172078
\(49\) −342.620 −0.998891
\(50\) 112.617 0.318528
\(51\) −86.5297 −0.237580
\(52\) 79.1619 0.211111
\(53\) 189.591 0.491364 0.245682 − 0.969351i \(-0.420988\pi\)
0.245682 + 0.969351i \(0.420988\pi\)
\(54\) −644.114 −1.62320
\(55\) 43.1768 0.105854
\(56\) −11.9232 −0.0284519
\(57\) 40.5573 0.0942446
\(58\) 990.185 2.24168
\(59\) −257.619 −0.568459 −0.284230 − 0.958756i \(-0.591738\pi\)
−0.284230 + 0.958756i \(0.591738\pi\)
\(60\) −312.830 −0.673104
\(61\) 240.335 0.504455 0.252227 − 0.967668i \(-0.418837\pi\)
0.252227 + 0.967668i \(0.418837\pi\)
\(62\) −2.74192 −0.00561653
\(63\) 0.673466 0.00134681
\(64\) −834.942 −1.63075
\(65\) −32.2006 −0.0614460
\(66\) −197.997 −0.369269
\(67\) −66.9428 −0.122065 −0.0610326 − 0.998136i \(-0.519439\pi\)
−0.0610326 + 0.998136i \(0.519439\pi\)
\(68\) −208.964 −0.372656
\(69\) 336.554 0.587194
\(70\) 13.8900 0.0237168
\(71\) −131.664 −0.220080 −0.110040 − 0.993927i \(-0.535098\pi\)
−0.110040 + 0.993927i \(0.535098\pi\)
\(72\) −21.1138 −0.0345596
\(73\) 1172.76 1.88029 0.940145 − 0.340774i \(-0.110689\pi\)
0.940145 + 0.340774i \(0.110689\pi\)
\(74\) −973.565 −1.52939
\(75\) 127.250 0.195914
\(76\) 97.9434 0.147827
\(77\) 5.32539 0.00788161
\(78\) 147.663 0.214354
\(79\) 707.825 1.00806 0.504029 − 0.863687i \(-0.331851\pi\)
0.504029 + 0.863687i \(0.331851\pi\)
\(80\) 56.2136 0.0785609
\(81\) −698.322 −0.957918
\(82\) 1601.44 2.15671
\(83\) 1007.30 1.33212 0.666058 − 0.745900i \(-0.267980\pi\)
0.666058 + 0.745900i \(0.267980\pi\)
\(84\) −38.5843 −0.0501177
\(85\) 85.0000 0.108465
\(86\) 945.740 1.18583
\(87\) 1118.85 1.37877
\(88\) −166.956 −0.202245
\(89\) 1013.01 1.20650 0.603249 − 0.797553i \(-0.293872\pi\)
0.603249 + 0.797553i \(0.293872\pi\)
\(90\) 24.5967 0.0288080
\(91\) −3.97159 −0.00457512
\(92\) 812.758 0.921042
\(93\) −3.09820 −0.00345450
\(94\) −1462.33 −1.60455
\(95\) −39.8403 −0.0430266
\(96\) −1045.06 −1.11105
\(97\) 973.147 1.01864 0.509320 − 0.860577i \(-0.329897\pi\)
0.509320 + 0.860577i \(0.329897\pi\)
\(98\) −1543.39 −1.59087
\(99\) 9.43029 0.00957353
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 85.4.a.f.1.3 ✓ 3
3.2 odd 2 765.4.a.k.1.1 3
4.3 odd 2 1360.4.a.p.1.2 3
5.2 odd 4 425.4.b.h.324.6 6
5.3 odd 4 425.4.b.h.324.1 6
5.4 even 2 425.4.a.f.1.1 3
17.16 even 2 1445.4.a.k.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.f.1.3 ✓ 3 1.1 even 1 trivial
425.4.a.f.1.1 3 5.4 even 2
425.4.b.h.324.1 6 5.3 odd 4
425.4.b.h.324.6 6 5.2 odd 4
765.4.a.k.1.1 3 3.2 odd 2
1360.4.a.p.1.2 3 4.3 odd 2
1445.4.a.k.1.3 3 17.16 even 2