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Properties

Label 85.4.a.b.1.1
Level $85$
Weight $4$
Character 85.1
Self dual yes
Analytic conductor $5.015$
Analytic rank $1$
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,-5] 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: \(1\)
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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} -5.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} -15.0000 q^{6} -22.0000 q^{7} -21.0000 q^{8} -2.00000 q^{9} -15.0000 q^{10} +60.0000 q^{11} -5.00000 q^{12} -31.0000 q^{13} -66.0000 q^{14} +25.0000 q^{15} -71.0000 q^{16} +17.0000 q^{17} -6.00000 q^{18} -61.0000 q^{19} -5.00000 q^{20} +110.000 q^{21} +180.000 q^{22} -78.0000 q^{23} +105.000 q^{24} +25.0000 q^{25} -93.0000 q^{26} +145.000 q^{27} -22.0000 q^{28} +69.0000 q^{29} +75.0000 q^{30} -31.0000 q^{31} -45.0000 q^{32} -300.000 q^{33} +51.0000 q^{34} +110.000 q^{35} -2.00000 q^{36} +56.0000 q^{37} -183.000 q^{38} +155.000 q^{39} +105.000 q^{40} -6.00000 q^{41} +330.000 q^{42} -538.000 q^{43} +60.0000 q^{44} +10.0000 q^{45} -234.000 q^{46} -465.000 q^{47} +355.000 q^{48} +141.000 q^{49} +75.0000 q^{50} -85.0000 q^{51} -31.0000 q^{52} +723.000 q^{53} +435.000 q^{54} -300.000 q^{55} +462.000 q^{56} +305.000 q^{57} +207.000 q^{58} -753.000 q^{59} +25.0000 q^{60} +35.0000 q^{61} -93.0000 q^{62} +44.0000 q^{63} +433.000 q^{64} +155.000 q^{65} -900.000 q^{66} -322.000 q^{67} +17.0000 q^{68} +390.000 q^{69} +330.000 q^{70} -99.0000 q^{71} +42.0000 q^{72} -1123.00 q^{73} +168.000 q^{74} -125.000 q^{75} -61.0000 q^{76} -1320.00 q^{77} +465.000 q^{78} +488.000 q^{79} +355.000 q^{80} -671.000 q^{81} -18.0000 q^{82} -852.000 q^{83} +110.000 q^{84} -85.0000 q^{85} -1614.00 q^{86} -345.000 q^{87} -1260.00 q^{88} +1215.00 q^{89} +30.0000 q^{90} +682.000 q^{91} -78.0000 q^{92} +155.000 q^{93} -1395.00 q^{94} +305.000 q^{95} +225.000 q^{96} -601.000 q^{97} +423.000 q^{98} -120.000 q^{99} +O(q^{100})\)

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 \(0.322068\pi\)
\(3\) −5.00000 −0.962250 −0.481125 − 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) 1.00000 0.125000
\(5\) −5.00000 −0.447214
\(6\) −15.0000 −1.02062
\(7\) −22.0000 −1.18789 −0.593944 − 0.804506i \(-0.702430\pi\)
−0.593944 + 0.804506i \(0.702430\pi\)
\(8\) −21.0000 −0.928078
\(9\) −2.00000 −0.0740741
\(10\) −15.0000 −0.474342
\(11\) 60.0000 1.64461 0.822304 − 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) −5.00000 −0.120281
\(13\) −31.0000 −0.661373 −0.330687 − 0.943741i \(-0.607280\pi\)
−0.330687 + 0.943741i \(0.607280\pi\)
\(14\) −66.0000 −1.25995
\(15\) 25.0000 0.430331
\(16\) −71.0000 −1.10938
\(17\) 17.0000 0.242536
\(18\) −6.00000 −0.0785674
\(19\) −61.0000 −0.736545 −0.368273 − 0.929718i \(-0.620051\pi\)
−0.368273 + 0.929718i \(0.620051\pi\)
\(20\) −5.00000 −0.0559017
\(21\) 110.000 1.14305
\(22\) 180.000 1.74437
\(23\) −78.0000 −0.707136 −0.353568 − 0.935409i \(-0.615032\pi\)
−0.353568 + 0.935409i \(0.615032\pi\)
\(24\) 105.000 0.893043
\(25\) 25.0000 0.200000
\(26\) −93.0000 −0.701492
\(27\) 145.000 1.03353
\(28\) −22.0000 −0.148486
\(29\) 69.0000 0.441827 0.220913 − 0.975293i \(-0.429096\pi\)
0.220913 + 0.975293i \(0.429096\pi\)
\(30\) 75.0000 0.456435
\(31\) −31.0000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −45.0000 −0.248592
\(33\) −300.000 −1.58252
\(34\) 51.0000 0.257248
\(35\) 110.000 0.531240
\(36\) −2.00000 −0.00925926
\(37\) 56.0000 0.248820 0.124410 − 0.992231i \(-0.460296\pi\)
0.124410 + 0.992231i \(0.460296\pi\)
\(38\) −183.000 −0.781224
\(39\) 155.000 0.636407
\(40\) 105.000 0.415049
\(41\) −6.00000 −0.0228547 −0.0114273 − 0.999935i \(-0.503638\pi\)
−0.0114273 + 0.999935i \(0.503638\pi\)
\(42\) 330.000 1.21238
\(43\) −538.000 −1.90801 −0.954003 − 0.299798i \(-0.903081\pi\)
−0.954003 + 0.299798i \(0.903081\pi\)
\(44\) 60.0000 0.205576
\(45\) 10.0000 0.0331269
\(46\) −234.000 −0.750031
\(47\) −465.000 −1.44313 −0.721566 − 0.692345i \(-0.756577\pi\)
−0.721566 + 0.692345i \(0.756577\pi\)
\(48\) 355.000 1.06750
\(49\) 141.000 0.411079
\(50\) 75.0000 0.212132
\(51\) −85.0000 −0.233380
\(52\) −31.0000 −0.0826717
\(53\) 723.000 1.87381 0.936903 − 0.349590i \(-0.113679\pi\)
0.936903 + 0.349590i \(0.113679\pi\)
\(54\) 435.000 1.09622
\(55\) −300.000 −0.735491
\(56\) 462.000 1.10245
\(57\) 305.000 0.708741
\(58\) 207.000 0.468628
\(59\) −753.000 −1.66156 −0.830782 − 0.556598i \(-0.812106\pi\)
−0.830782 + 0.556598i \(0.812106\pi\)
\(60\) 25.0000 0.0537914
\(61\) 35.0000 0.0734638 0.0367319 − 0.999325i \(-0.488305\pi\)
0.0367319 + 0.999325i \(0.488305\pi\)
\(62\) −93.0000 −0.190500
\(63\) 44.0000 0.0879917
\(64\) 433.000 0.845703
\(65\) 155.000 0.295775
\(66\) −900.000 −1.67852
\(67\) −322.000 −0.587143 −0.293571 − 0.955937i \(-0.594844\pi\)
−0.293571 + 0.955937i \(0.594844\pi\)
\(68\) 17.0000 0.0303170
\(69\) 390.000 0.680442
\(70\) 330.000 0.563465
\(71\) −99.0000 −0.165481 −0.0827404 − 0.996571i \(-0.526367\pi\)
−0.0827404 + 0.996571i \(0.526367\pi\)
\(72\) 42.0000 0.0687465
\(73\) −1123.00 −1.80051 −0.900255 − 0.435363i \(-0.856620\pi\)
−0.900255 + 0.435363i \(0.856620\pi\)
\(74\) 168.000 0.263914
\(75\) −125.000 −0.192450
\(76\) −61.0000 −0.0920682
\(77\) −1320.00 −1.95361
\(78\) 465.000 0.675011
\(79\) 488.000 0.694991 0.347496 − 0.937682i \(-0.387032\pi\)
0.347496 + 0.937682i \(0.387032\pi\)
\(80\) 355.000 0.496128
\(81\) −671.000 −0.920439
\(82\) −18.0000 −0.0242411
\(83\) −852.000 −1.12674 −0.563368 − 0.826206i \(-0.690495\pi\)
−0.563368 + 0.826206i \(0.690495\pi\)
\(84\) 110.000 0.142881
\(85\) −85.0000 −0.108465
\(86\) −1614.00 −2.02375
\(87\) −345.000 −0.425148
\(88\) −1260.00 −1.52632
\(89\) 1215.00 1.44708 0.723538 − 0.690285i \(-0.242515\pi\)
0.723538 + 0.690285i \(0.242515\pi\)
\(90\) 30.0000 0.0351364
\(91\) 682.000 0.785638
\(92\) −78.0000 −0.0883920
\(93\) 155.000 0.172825
\(94\) −1395.00 −1.53067
\(95\) 305.000 0.329393
\(96\) 225.000 0.239208
\(97\) −601.000 −0.629096 −0.314548 − 0.949242i \(-0.601853\pi\)
−0.314548 + 0.949242i \(0.601853\pi\)
\(98\) 423.000 0.436015
\(99\) −120.000 −0.121823
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.b.1.1 ✓ 1
3.2 odd 2 765.4.a.c.1.1 1
4.3 odd 2 1360.4.a.g.1.1 1
5.2 odd 4 425.4.b.b.324.2 2
5.3 odd 4 425.4.b.b.324.1 2
5.4 even 2 425.4.a.b.1.1 1
17.16 even 2 1445.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.b.1.1 ✓ 1 1.1 even 1 trivial
425.4.a.b.1.1 1 5.4 even 2
425.4.b.b.324.1 2 5.3 odd 4
425.4.b.b.324.2 2 5.2 odd 4
765.4.a.c.1.1 1 3.2 odd 2
1360.4.a.g.1.1 1 4.3 odd 2
1445.4.a.g.1.1 1 17.16 even 2