Properties

Label 85.4.a.f.1.1
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 1.1
Root \(3.12489\) 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)
 
\(f(q)\) \(=\) \(q-1.64002 q^{2} -5.37466 q^{3} -5.31032 q^{4} -5.00000 q^{5} +8.81456 q^{6} +2.20103 q^{7} +21.8292 q^{8} +1.88693 q^{9} +8.20012 q^{10} +18.7611 q^{11} +28.5412 q^{12} +62.9220 q^{13} -3.60975 q^{14} +26.8733 q^{15} +6.68211 q^{16} -17.0000 q^{17} -3.09461 q^{18} -47.8179 q^{19} +26.5516 q^{20} -11.8298 q^{21} -30.7687 q^{22} +153.674 q^{23} -117.325 q^{24} +25.0000 q^{25} -103.194 q^{26} +134.974 q^{27} -11.6882 q^{28} -64.4803 q^{29} -44.0728 q^{30} -40.9650 q^{31} -185.593 q^{32} -100.835 q^{33} +27.8804 q^{34} -11.0052 q^{35} -10.0202 q^{36} +32.6681 q^{37} +78.4225 q^{38} -338.184 q^{39} -109.146 q^{40} +159.252 q^{41} +19.4012 q^{42} -111.512 q^{43} -99.6276 q^{44} -9.43465 q^{45} -252.029 q^{46} +614.111 q^{47} -35.9141 q^{48} -338.155 q^{49} -41.0006 q^{50} +91.3692 q^{51} -334.136 q^{52} +308.809 q^{53} -221.361 q^{54} -93.8056 q^{55} +48.0469 q^{56} +257.005 q^{57} +105.749 q^{58} +267.795 q^{59} -142.706 q^{60} +521.030 q^{61} +67.1836 q^{62} +4.15320 q^{63} +250.920 q^{64} -314.610 q^{65} +165.371 q^{66} +118.688 q^{67} +90.2755 q^{68} -825.946 q^{69} +18.0487 q^{70} +1141.21 q^{71} +41.1902 q^{72} -40.7561 q^{73} -53.5765 q^{74} -134.366 q^{75} +253.928 q^{76} +41.2939 q^{77} +554.630 q^{78} +374.144 q^{79} -33.4106 q^{80} -776.387 q^{81} -261.176 q^{82} +826.610 q^{83} +62.8201 q^{84} +85.0000 q^{85} +182.882 q^{86} +346.559 q^{87} +409.541 q^{88} -38.9664 q^{89} +15.4730 q^{90} +138.493 q^{91} -816.060 q^{92} +220.173 q^{93} -1007.16 q^{94} +239.089 q^{95} +997.497 q^{96} -917.727 q^{97} +554.583 q^{98} +35.4009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 9 q^{3} - q^{4} - 15 q^{5} + 33 q^{6} + 34 q^{7} + 39 q^{8} + 60 q^{9} - 15 q^{10} + 52 q^{11} + 17 q^{12} + 19 q^{13} - 2 q^{14} - 45 q^{15} + 59 q^{16} - 51 q^{17} - 153 q^{19} + 5 q^{20}+ \cdots + 2524 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.64002 −0.579836 −0.289918 − 0.957052i \(-0.593628\pi\)
−0.289918 + 0.957052i \(0.593628\pi\)
\(3\) −5.37466 −1.03435 −0.517177 − 0.855879i \(-0.673017\pi\)
−0.517177 + 0.855879i \(0.673017\pi\)
\(4\) −5.31032 −0.663790
\(5\) −5.00000 −0.447214
\(6\) 8.81456 0.599755
\(7\) 2.20103 0.118845 0.0594223 − 0.998233i \(-0.481074\pi\)
0.0594223 + 0.998233i \(0.481074\pi\)
\(8\) 21.8292 0.964725
\(9\) 1.88693 0.0698863
\(10\) 8.20012 0.259310
\(11\) 18.7611 0.514245 0.257122 − 0.966379i \(-0.417226\pi\)
0.257122 + 0.966379i \(0.417226\pi\)
\(12\) 28.5412 0.686594
\(13\) 62.9220 1.34242 0.671209 − 0.741268i \(-0.265775\pi\)
0.671209 + 0.741268i \(0.265775\pi\)
\(14\) −3.60975 −0.0689104
\(15\) 26.8733 0.462577
\(16\) 6.68211 0.104408
\(17\) −17.0000 −0.242536
\(18\) −3.09461 −0.0405226
\(19\) −47.8179 −0.577378 −0.288689 − 0.957423i \(-0.593219\pi\)
−0.288689 + 0.957423i \(0.593219\pi\)
\(20\) 26.5516 0.296856
\(21\) −11.8298 −0.122927
\(22\) −30.7687 −0.298178
\(23\) 153.674 1.39319 0.696593 − 0.717466i \(-0.254698\pi\)
0.696593 + 0.717466i \(0.254698\pi\)
\(24\) −117.325 −0.997867
\(25\) 25.0000 0.200000
\(26\) −103.194 −0.778382
\(27\) 134.974 0.962066
\(28\) −11.6882 −0.0788879
\(29\) −64.4803 −0.412886 −0.206443 − 0.978459i \(-0.566189\pi\)
−0.206443 + 0.978459i \(0.566189\pi\)
\(30\) −44.0728 −0.268219
\(31\) −40.9650 −0.237340 −0.118670 − 0.992934i \(-0.537863\pi\)
−0.118670 + 0.992934i \(0.537863\pi\)
\(32\) −185.593 −1.02526
\(33\) −100.835 −0.531911
\(34\) 27.8804 0.140631
\(35\) −11.0052 −0.0531489
\(36\) −10.0202 −0.0463898
\(37\) 32.6681 0.145152 0.0725758 − 0.997363i \(-0.476878\pi\)
0.0725758 + 0.997363i \(0.476878\pi\)
\(38\) 78.4225 0.334784
\(39\) −338.184 −1.38853
\(40\) −109.146 −0.431438
\(41\) 159.252 0.606608 0.303304 − 0.952894i \(-0.401910\pi\)
0.303304 + 0.952894i \(0.401910\pi\)
\(42\) 19.4012 0.0712777
\(43\) −111.512 −0.395475 −0.197737 − 0.980255i \(-0.563359\pi\)
−0.197737 + 0.980255i \(0.563359\pi\)
\(44\) −99.6276 −0.341351
\(45\) −9.43465 −0.0312541
\(46\) −252.029 −0.807819
\(47\) 614.111 1.90590 0.952950 − 0.303127i \(-0.0980308\pi\)
0.952950 + 0.303127i \(0.0980308\pi\)
\(48\) −35.9141 −0.107995
\(49\) −338.155 −0.985876
\(50\) −41.0006 −0.115967
\(51\) 91.3692 0.250867
\(52\) −334.136 −0.891084
\(53\) 308.809 0.800344 0.400172 − 0.916440i \(-0.368950\pi\)
0.400172 + 0.916440i \(0.368950\pi\)
\(54\) −221.361 −0.557840
\(55\) −93.8056 −0.229977
\(56\) 48.0469 0.114652
\(57\) 257.005 0.597213
\(58\) 105.749 0.239406
\(59\) 267.795 0.590913 0.295457 − 0.955356i \(-0.404528\pi\)
0.295457 + 0.955356i \(0.404528\pi\)
\(60\) −142.706 −0.307054
\(61\) 521.030 1.09362 0.546812 − 0.837255i \(-0.315841\pi\)
0.546812 + 0.837255i \(0.315841\pi\)
\(62\) 67.1836 0.137618
\(63\) 4.15320 0.00830561
\(64\) 250.920 0.490077
\(65\) −314.610 −0.600347
\(66\) 165.371 0.308421
\(67\) 118.688 0.216418 0.108209 − 0.994128i \(-0.465488\pi\)
0.108209 + 0.994128i \(0.465488\pi\)
\(68\) 90.2755 0.160993
\(69\) −825.946 −1.44105
\(70\) 18.0487 0.0308177
\(71\) 1141.21 1.90756 0.953781 − 0.300503i \(-0.0971546\pi\)
0.953781 + 0.300503i \(0.0971546\pi\)
\(72\) 41.1902 0.0674211
\(73\) −40.7561 −0.0653444 −0.0326722 − 0.999466i \(-0.510402\pi\)
−0.0326722 + 0.999466i \(0.510402\pi\)
\(74\) −53.5765 −0.0841641
\(75\) −134.366 −0.206871
\(76\) 253.928 0.383258
\(77\) 41.2939 0.0611153
\(78\) 554.630 0.805122
\(79\) 374.144 0.532842 0.266421 − 0.963857i \(-0.414159\pi\)
0.266421 + 0.963857i \(0.414159\pi\)
\(80\) −33.4106 −0.0466927
\(81\) −776.387 −1.06500
\(82\) −261.176 −0.351733
\(83\) 826.610 1.09316 0.546580 − 0.837407i \(-0.315930\pi\)
0.546580 + 0.837407i \(0.315930\pi\)
\(84\) 62.8201 0.0815980
\(85\) 85.0000 0.108465
\(86\) 182.882 0.229310
\(87\) 346.559 0.427070
\(88\) 409.541 0.496105
\(89\) −38.9664 −0.0464093 −0.0232046 − 0.999731i \(-0.507387\pi\)
−0.0232046 + 0.999731i \(0.507387\pi\)
\(90\) 15.4730 0.0181222
\(91\) 138.493 0.159539
\(92\) −816.060 −0.924784
\(93\) 220.173 0.245493
\(94\) −1007.16 −1.10511
\(95\) 239.089 0.258211
\(96\) 997.497 1.06049
\(97\) −917.727 −0.960630 −0.480315 − 0.877096i \(-0.659478\pi\)
−0.480315 + 0.877096i \(0.659478\pi\)
\(98\) 554.583 0.571646
\(99\) 35.4009 0.0359387
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.1.1 ✓ 3
3.2 odd 2 765.4.a.k.1.3 3
4.3 odd 2 1360.4.a.p.1.3 3
5.2 odd 4 425.4.b.h.324.2 6
5.3 odd 4 425.4.b.h.324.5 6
5.4 even 2 425.4.a.f.1.3 3
17.16 even 2 1445.4.a.k.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.f.1.1 ✓ 3 1.1 even 1 trivial
425.4.a.f.1.3 3 5.4 even 2
425.4.b.h.324.2 6 5.2 odd 4
425.4.b.h.324.5 6 5.3 odd 4
765.4.a.k.1.3 3 3.2 odd 2
1360.4.a.p.1.3 3 4.3 odd 2
1445.4.a.k.1.1 3 17.16 even 2