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Properties

Label 85.4.a.c.1.1
Level $85$
Weight $4$
Character 85.1
Self dual yes
Analytic conductor $5.015$
Analytic rank $0$
Dimension $1$
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 = [1,3,10] 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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +10.0000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +30.0000 q^{6} -22.0000 q^{7} -21.0000 q^{8} +73.0000 q^{9} +15.0000 q^{10} -30.0000 q^{11} +10.0000 q^{12} -46.0000 q^{13} -66.0000 q^{14} +50.0000 q^{15} -71.0000 q^{16} +17.0000 q^{17} +219.000 q^{18} +104.000 q^{19} +5.00000 q^{20} -220.000 q^{21} -90.0000 q^{22} +42.0000 q^{23} -210.000 q^{24} +25.0000 q^{25} -138.000 q^{26} +460.000 q^{27} -22.0000 q^{28} -66.0000 q^{29} +150.000 q^{30} +194.000 q^{31} -45.0000 q^{32} -300.000 q^{33} +51.0000 q^{34} -110.000 q^{35} +73.0000 q^{36} +206.000 q^{37} +312.000 q^{38} -460.000 q^{39} -105.000 q^{40} -126.000 q^{41} -660.000 q^{42} -388.000 q^{43} -30.0000 q^{44} +365.000 q^{45} +126.000 q^{46} -540.000 q^{47} -710.000 q^{48} +141.000 q^{49} +75.0000 q^{50} +170.000 q^{51} -46.0000 q^{52} +78.0000 q^{53} +1380.00 q^{54} -150.000 q^{55} +462.000 q^{56} +1040.00 q^{57} -198.000 q^{58} +432.000 q^{59} +50.0000 q^{60} -610.000 q^{61} +582.000 q^{62} -1606.00 q^{63} +433.000 q^{64} -230.000 q^{65} -900.000 q^{66} +848.000 q^{67} +17.0000 q^{68} +420.000 q^{69} -330.000 q^{70} -174.000 q^{71} -1533.00 q^{72} +362.000 q^{73} +618.000 q^{74} +250.000 q^{75} +104.000 q^{76} +660.000 q^{77} -1380.00 q^{78} +398.000 q^{79} -355.000 q^{80} +2629.00 q^{81} -378.000 q^{82} +828.000 q^{83} -220.000 q^{84} +85.0000 q^{85} -1164.00 q^{86} -660.000 q^{87} +630.000 q^{88} +630.000 q^{89} +1095.00 q^{90} +1012.00 q^{91} +42.0000 q^{92} +1940.00 q^{93} -1620.00 q^{94} +520.000 q^{95} -450.000 q^{96} -1486.00 q^{97} +423.000 q^{98} -2190.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 − 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 10.0000 1.92450 0.962250 − 0.272166i \(-0.0877398\pi\)
0.962250 + 0.272166i \(0.0877398\pi\)
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 30.0000 2.04124
\(7\) −22.0000 −1.18789 −0.593944 − 0.804506i \(-0.702430\pi\)
−0.593944 + 0.804506i \(0.702430\pi\)
\(8\) −21.0000 −0.928078
\(9\) 73.0000 2.70370
\(10\) 15.0000 0.474342
\(11\) −30.0000 −0.822304 −0.411152 − 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 10.0000 0.240563
\(13\) −46.0000 −0.981393 −0.490696 − 0.871331i \(-0.663258\pi\)
−0.490696 + 0.871331i \(0.663258\pi\)
\(14\) −66.0000 −1.25995
\(15\) 50.0000 0.860663
\(16\) −71.0000 −1.10938
\(17\) 17.0000 0.242536
\(18\) 219.000 2.86771
\(19\) 104.000 1.25575 0.627875 − 0.778314i \(-0.283925\pi\)
0.627875 + 0.778314i \(0.283925\pi\)
\(20\) 5.00000 0.0559017
\(21\) −220.000 −2.28609
\(22\) −90.0000 −0.872185
\(23\) 42.0000 0.380765 0.190383 − 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) −210.000 −1.78609
\(25\) 25.0000 0.200000
\(26\) −138.000 −1.04092
\(27\) 460.000 3.27878
\(28\) −22.0000 −0.148486
\(29\) −66.0000 −0.422617 −0.211308 − 0.977419i \(-0.567772\pi\)
−0.211308 + 0.977419i \(0.567772\pi\)
\(30\) 150.000 0.912871
\(31\) 194.000 1.12398 0.561991 − 0.827143i \(-0.310036\pi\)
0.561991 + 0.827143i \(0.310036\pi\)
\(32\) −45.0000 −0.248592
\(33\) −300.000 −1.58252
\(34\) 51.0000 0.257248
\(35\) −110.000 −0.531240
\(36\) 73.0000 0.337963
\(37\) 206.000 0.915302 0.457651 − 0.889132i \(-0.348691\pi\)
0.457651 + 0.889132i \(0.348691\pi\)
\(38\) 312.000 1.33192
\(39\) −460.000 −1.88869
\(40\) −105.000 −0.415049
\(41\) −126.000 −0.479949 −0.239974 − 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) −660.000 −2.42477
\(43\) −388.000 −1.37603 −0.688017 − 0.725695i \(-0.741518\pi\)
−0.688017 + 0.725695i \(0.741518\pi\)
\(44\) −30.0000 −0.102788
\(45\) 365.000 1.20913
\(46\) 126.000 0.403863
\(47\) −540.000 −1.67590 −0.837948 − 0.545750i \(-0.816245\pi\)
−0.837948 + 0.545750i \(0.816245\pi\)
\(48\) −710.000 −2.13499
\(49\) 141.000 0.411079
\(50\) 75.0000 0.212132
\(51\) 170.000 0.466760
\(52\) −46.0000 −0.122674
\(53\) 78.0000 0.202153 0.101077 − 0.994879i \(-0.467771\pi\)
0.101077 + 0.994879i \(0.467771\pi\)
\(54\) 1380.00 3.47767
\(55\) −150.000 −0.367745
\(56\) 462.000 1.10245
\(57\) 1040.00 2.41669
\(58\) −198.000 −0.448253
\(59\) 432.000 0.953248 0.476624 − 0.879107i \(-0.341860\pi\)
0.476624 + 0.879107i \(0.341860\pi\)
\(60\) 50.0000 0.107583
\(61\) −610.000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 582.000 1.19216
\(63\) −1606.00 −3.21170
\(64\) 433.000 0.845703
\(65\) −230.000 −0.438892
\(66\) −900.000 −1.67852
\(67\) 848.000 1.54626 0.773132 − 0.634245i \(-0.218689\pi\)
0.773132 + 0.634245i \(0.218689\pi\)
\(68\) 17.0000 0.0303170
\(69\) 420.000 0.732783
\(70\) −330.000 −0.563465
\(71\) −174.000 −0.290845 −0.145423 − 0.989370i \(-0.546454\pi\)
−0.145423 + 0.989370i \(0.546454\pi\)
\(72\) −1533.00 −2.50925
\(73\) 362.000 0.580396 0.290198 − 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) 618.000 0.970825
\(75\) 250.000 0.384900
\(76\) 104.000 0.156969
\(77\) 660.000 0.976805
\(78\) −1380.00 −2.00326
\(79\) 398.000 0.566816 0.283408 − 0.958999i \(-0.408535\pi\)
0.283408 + 0.958999i \(0.408535\pi\)
\(80\) −355.000 −0.496128
\(81\) 2629.00 3.60631
\(82\) −378.000 −0.509062
\(83\) 828.000 1.09500 0.547499 − 0.836806i \(-0.315580\pi\)
0.547499 + 0.836806i \(0.315580\pi\)
\(84\) −220.000 −0.285762
\(85\) 85.0000 0.108465
\(86\) −1164.00 −1.45950
\(87\) −660.000 −0.813327
\(88\) 630.000 0.763162
\(89\) 630.000 0.750336 0.375168 − 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(90\) 1095.00 1.28248
\(91\) 1012.00 1.16578
\(92\) 42.0000 0.0475957
\(93\) 1940.00 2.16310
\(94\) −1620.00 −1.77756
\(95\) 520.000 0.561588
\(96\) −450.000 −0.478416
\(97\) −1486.00 −1.55547 −0.777734 − 0.628593i \(-0.783631\pi\)
−0.777734 + 0.628593i \(0.783631\pi\)
\(98\) 423.000 0.436015
\(99\) −2190.00 −2.22327
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.c.1.1 ✓ 1
3.2 odd 2 765.4.a.a.1.1 1
4.3 odd 2 1360.4.a.a.1.1 1
5.2 odd 4 425.4.b.d.324.2 2
5.3 odd 4 425.4.b.d.324.1 2
5.4 even 2 425.4.a.a.1.1 1
17.16 even 2 1445.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.c.1.1 ✓ 1 1.1 even 1 trivial
425.4.a.a.1.1 1 5.4 even 2
425.4.b.d.324.1 2 5.3 odd 4
425.4.b.d.324.2 2 5.2 odd 4
765.4.a.a.1.1 1 3.2 odd 2
1360.4.a.a.1.1 1 4.3 odd 2
1445.4.a.f.1.1 1 17.16 even 2