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Properties

Label 85.2.l.a.66.4
Level $85$
Weight $2$
Character 85.66
Analytic conductor $0.679$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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(26,85)] 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("85.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.l (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 66.4
Character \(\chi\) \(=\) 85.66
Dual form 85.2.l.a.76.4

$q$-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+(0.680853 + 0.680853i) q^{2} +(-1.01372 - 2.44733i) q^{3} -1.07288i q^{4} +(-0.923880 + 0.382683i) q^{5} +(0.976080 - 2.35647i) q^{6} +(2.85906 + 1.18426i) q^{7} +(2.09218 - 2.09218i) q^{8} +(-2.84049 + 2.84049i) q^{9} +(-0.889577 - 0.368475i) q^{10} +(-2.34612 + 5.66403i) q^{11} +(-2.62569 + 1.08760i) q^{12} -1.16017i q^{13} +(1.14029 + 2.75291i) q^{14} +(1.87311 + 1.87311i) q^{15} +0.703170 q^{16} +(1.25804 - 3.92649i) q^{17} -3.86791 q^{18} +(3.83665 + 3.83665i) q^{19} +(0.410573 + 0.991211i) q^{20} -8.19759i q^{21} +(-5.45373 + 2.25901i) q^{22} +(-1.19300 + 2.88015i) q^{23} +(-7.24113 - 2.99938i) q^{24} +(0.707107 - 0.707107i) q^{25} +(0.789908 - 0.789908i) q^{26} +(2.48908 + 1.03101i) q^{27} +(1.27057 - 3.06743i) q^{28} +(-4.61660 + 1.91226i) q^{29} +2.55062i q^{30} +(-1.42666 - 3.44426i) q^{31} +(-3.70560 - 3.70560i) q^{32} +16.2401 q^{33} +(3.52990 - 1.81682i) q^{34} -3.09463 q^{35} +(3.04750 + 3.04750i) q^{36} +(-0.151817 - 0.366518i) q^{37} +5.22439i q^{38} +(-2.83933 + 1.17609i) q^{39} +(-1.13228 + 2.73356i) q^{40} +(1.57303 + 0.651568i) q^{41} +(5.58135 - 5.58135i) q^{42} +(0.0189720 - 0.0189720i) q^{43} +(6.07683 + 2.51710i) q^{44} +(1.53726 - 3.71128i) q^{45} +(-2.77321 + 1.14870i) q^{46} +5.43715i q^{47} +(-0.712817 - 1.72089i) q^{48} +(1.82202 + 1.82202i) q^{49} +0.962871 q^{50} +(-10.8847 + 0.901513i) q^{51} -1.24473 q^{52} +(-0.244014 - 0.244014i) q^{53} +(0.992732 + 2.39667i) q^{54} -6.13071i q^{55} +(8.45936 - 3.50398i) q^{56} +(5.50028 - 13.2788i) q^{57} +(-4.44519 - 1.84126i) q^{58} +(2.87128 - 2.87128i) q^{59} +(2.00962 - 2.00962i) q^{60} +(-11.4953 - 4.76149i) q^{61} +(1.37369 - 3.31638i) q^{62} +(-11.4850 + 4.75726i) q^{63} -6.45228i q^{64} +(0.443980 + 1.07186i) q^{65} +(11.0571 + 11.0571i) q^{66} -5.62508 q^{67} +(-4.21265 - 1.34973i) q^{68} +8.25804 q^{69} +(-2.10699 - 2.10699i) q^{70} +(4.12510 + 9.95888i) q^{71} +11.8856i q^{72} +(-1.52840 + 0.633083i) q^{73} +(0.146180 - 0.352909i) q^{74} +(-2.44733 - 1.01372i) q^{75} +(4.11627 - 4.11627i) q^{76} +(-13.4154 + 13.4154i) q^{77} +(-2.73391 - 1.13242i) q^{78} +(2.01785 - 4.87153i) q^{79} +(-0.649645 + 0.269092i) q^{80} +4.91441i q^{81} +(0.627376 + 1.51462i) q^{82} +(-8.78638 - 8.78638i) q^{83} -8.79502 q^{84} +(0.340325 + 4.10904i) q^{85} +0.0258342 q^{86} +(9.35987 + 9.35987i) q^{87} +(6.94167 + 16.7587i) q^{88} -3.22930i q^{89} +(3.57348 - 1.48019i) q^{90} +(1.37395 - 3.31701i) q^{91} +(3.09005 + 1.27994i) q^{92} +(-6.98302 + 6.98302i) q^{93} +(-3.70190 + 3.70190i) q^{94} +(-5.01283 - 2.07638i) q^{95} +(-5.31240 + 12.8253i) q^{96} +(11.9502 - 4.94995i) q^{97} +2.48105i q^{98} +(-9.42450 - 22.7528i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} + 24 q^{12} - 8 q^{15} - 24 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 32 q^{22} - 16 q^{23} - 8 q^{24} + 16 q^{26} + 24 q^{27} + 48 q^{28} - 8 q^{29} + 16 q^{34} - 32 q^{35}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.680853 + 0.680853i 0.481435 + 0.481435i 0.905590 − 0.424154i \(-0.139429\pi\)
−0.424154 + 0.905590i \(0.639429\pi\)
\(3\) −1.01372 − 2.44733i −0.585271 − 1.41297i −0.887979 − 0.459885i \(-0.847891\pi\)
0.302708 − 0.953083i \(-0.402109\pi\)
\(4\) − 1.07288i − 0.536440i
\(5\) −0.923880 + 0.382683i −0.413171 + 0.171141i
\(6\) 0.976080 − 2.35647i 0.398483 − 0.962023i
\(7\) 2.85906 + 1.18426i 1.08062 + 0.447609i 0.850728 − 0.525606i \(-0.176161\pi\)
0.229896 + 0.973215i \(0.426161\pi\)
\(8\) 2.09218 − 2.09218i 0.739697 − 0.739697i
\(9\) −2.84049 + 2.84049i −0.946830 + 0.946830i
\(10\) −0.889577 − 0.368475i −0.281309 − 0.116522i
\(11\) −2.34612 + 5.66403i −0.707382 + 1.70777i −0.000939675 1.00000i \(0.500299\pi\)
−0.706442 + 0.707771i \(0.749701\pi\)
\(12\) −2.62569 + 1.08760i −0.757972 + 0.313962i
\(13\) − 1.16017i − 0.321775i −0.986973 − 0.160887i \(-0.948564\pi\)
0.986973 − 0.160887i \(-0.0514356\pi\)
\(14\) 1.14029 + 2.75291i 0.304756 + 0.735746i
\(15\) 1.87311 + 1.87311i 0.483634 + 0.483634i
\(16\) 0.703170 0.175793
\(17\) 1.25804 − 3.92649i 0.305120 − 0.952314i
\(18\) −3.86791 −0.911675
\(19\) 3.83665 + 3.83665i 0.880188 + 0.880188i 0.993553 − 0.113365i \(-0.0361629\pi\)
−0.113365 + 0.993553i \(0.536163\pi\)
\(20\) 0.410573 + 0.991211i 0.0918070 + 0.221642i
\(21\) − 8.19759i − 1.78886i
\(22\) −5.45373 + 2.25901i −1.16274 + 0.481623i
\(23\) −1.19300 + 2.88015i −0.248757 + 0.600552i −0.998099 − 0.0616306i \(-0.980370\pi\)
0.749342 + 0.662183i \(0.230370\pi\)
\(24\) −7.24113 − 2.99938i −1.47809 − 0.612245i
\(25\) 0.707107 − 0.707107i 0.141421 − 0.141421i
\(26\) 0.789908 − 0.789908i 0.154914 − 0.154914i
\(27\) 2.48908 + 1.03101i 0.479025 + 0.198418i
\(28\) 1.27057 − 3.06743i 0.240115 − 0.579690i
\(29\) −4.61660 + 1.91226i −0.857282 + 0.355098i −0.767644 − 0.640877i \(-0.778571\pi\)
−0.0896380 + 0.995974i \(0.528571\pi\)
\(30\) 2.55062i 0.465677i
\(31\) −1.42666 − 3.44426i −0.256236 − 0.618608i 0.742448 − 0.669904i \(-0.233665\pi\)
−0.998683 + 0.0512962i \(0.983665\pi\)
\(32\) −3.70560 − 3.70560i −0.655064 − 0.655064i
\(33\) 16.2401 2.82703
\(34\) 3.52990 − 1.81682i 0.605373 − 0.311582i
\(35\) −3.09463 −0.523088
\(36\) 3.04750 + 3.04750i 0.507917 + 0.507917i
\(37\) −0.151817 − 0.366518i −0.0249585 − 0.0602551i 0.910909 − 0.412607i \(-0.135382\pi\)
−0.935867 + 0.352352i \(0.885382\pi\)
\(38\) 5.22439i 0.847508i
\(39\) −2.83933 + 1.17609i −0.454657 + 0.188325i
\(40\) −1.13228 + 2.73356i −0.179029 + 0.432214i
\(41\) 1.57303 + 0.651568i 0.245665 + 0.101758i 0.502119 − 0.864799i \(-0.332554\pi\)
−0.256453 + 0.966557i \(0.582554\pi\)
\(42\) 5.58135 − 5.58135i 0.861221 − 0.861221i
\(43\) 0.0189720 − 0.0189720i 0.00289320 − 0.00289320i −0.705659 − 0.708552i \(-0.749349\pi\)
0.708552 + 0.705659i \(0.249349\pi\)
\(44\) 6.07683 + 2.51710i 0.916116 + 0.379468i
\(45\) 1.53726 − 3.71128i 0.229162 − 0.553245i
\(46\) −2.77321 + 1.14870i −0.408888 + 0.169367i
\(47\) 5.43715i 0.793090i 0.918015 + 0.396545i \(0.129791\pi\)
−0.918015 + 0.396545i \(0.870209\pi\)
\(48\) −0.712817 − 1.72089i −0.102886 − 0.248389i
\(49\) 1.82202 + 1.82202i 0.260288 + 0.260288i
\(50\) 0.962871 0.136171
\(51\) −10.8847 + 0.901513i −1.52417 + 0.126237i
\(52\) −1.24473 −0.172613
\(53\) −0.244014 − 0.244014i −0.0335179 − 0.0335179i 0.690149 − 0.723667i \(-0.257545\pi\)
−0.723667 + 0.690149i \(0.757545\pi\)
\(54\) 0.992732 + 2.39667i 0.135094 + 0.326145i
\(55\) − 6.13071i − 0.826664i
\(56\) 8.45936 − 3.50398i 1.13043 − 0.468239i
\(57\) 5.50028 − 13.2788i 0.728530 − 1.75883i
\(58\) −4.44519 − 1.84126i −0.583683 − 0.241769i
\(59\) 2.87128 − 2.87128i 0.373808 − 0.373808i −0.495054 − 0.868862i \(-0.664852\pi\)
0.868862 + 0.495054i \(0.164852\pi\)
\(60\) 2.00962 − 2.00962i 0.259441 − 0.259441i
\(61\) −11.4953 − 4.76149i −1.47182 − 0.609646i −0.504544 − 0.863386i \(-0.668339\pi\)
−0.967273 + 0.253740i \(0.918339\pi\)
\(62\) 1.37369 − 3.31638i 0.174459 − 0.421181i
\(63\) −11.4850 + 4.75726i −1.44698 + 0.599358i
\(64\) − 6.45228i − 0.806534i
\(65\) 0.443980 + 1.07186i 0.0550689 + 0.132948i
\(66\) 11.0571 + 11.0571i 1.36103 + 1.36103i
\(67\) −5.62508 −0.687213 −0.343607 − 0.939114i \(-0.611649\pi\)
−0.343607 + 0.939114i \(0.611649\pi\)
\(68\) −4.21265 − 1.34973i −0.510859 − 0.163678i
\(69\) 8.25804 0.994152
\(70\) −2.10699 − 2.10699i −0.251833 − 0.251833i
\(71\) 4.12510 + 9.95888i 0.489560 + 1.18190i 0.954942 + 0.296792i \(0.0959167\pi\)
−0.465383 + 0.885110i \(0.654083\pi\)
\(72\) 11.8856i 1.40073i
\(73\) −1.52840 + 0.633083i −0.178886 + 0.0740968i −0.470329 − 0.882491i \(-0.655865\pi\)
0.291443 + 0.956588i \(0.405865\pi\)
\(74\) 0.146180 − 0.352909i 0.0169931 − 0.0410249i
\(75\) −2.44733 − 1.01372i −0.282594 − 0.117054i
\(76\) 4.11627 − 4.11627i 0.472168 − 0.472168i
\(77\) −13.4154 + 13.4154i −1.52883 + 1.52883i
\(78\) −2.73391 − 1.13242i −0.309554 − 0.128222i
\(79\) 2.01785 − 4.87153i 0.227026 − 0.548090i −0.768787 − 0.639505i \(-0.779139\pi\)
0.995813 + 0.0914157i \(0.0291392\pi\)
\(80\) −0.649645 + 0.269092i −0.0726325 + 0.0300854i
\(81\) 4.91441i 0.546045i
\(82\) 0.627376 + 1.51462i 0.0692821 + 0.167262i
\(83\) −8.78638 − 8.78638i −0.964430 − 0.964430i 0.0349583 − 0.999389i \(-0.488870\pi\)
−0.999389 + 0.0349583i \(0.988870\pi\)
\(84\) −8.79502 −0.959616
\(85\) 0.340325 + 4.10904i 0.0369135 + 0.445688i
\(86\) 0.0258342 0.00278577
\(87\) 9.35987 + 9.35987i 1.00348 + 1.00348i
\(88\) 6.94167 + 16.7587i 0.739984 + 1.78648i
\(89\) − 3.22930i − 0.342305i −0.985245 − 0.171152i \(-0.945251\pi\)
0.985245 − 0.171152i \(-0.0547490\pi\)
\(90\) 3.57348 − 1.48019i 0.376678 − 0.156025i
\(91\) 1.37395 − 3.31701i 0.144029 − 0.347717i
\(92\) 3.09005 + 1.27994i 0.322160 + 0.133443i
\(93\) −6.98302 + 6.98302i −0.724106 + 0.724106i
\(94\) −3.70190 + 3.70190i −0.381822 + 0.381822i
\(95\) −5.01283 − 2.07638i −0.514305 − 0.213032i
\(96\) −5.31240 + 12.8253i −0.542195 + 1.30897i
\(97\) 11.9502 − 4.94995i 1.21336 − 0.502592i 0.318070 − 0.948067i \(-0.396965\pi\)
0.895294 + 0.445476i \(0.146965\pi\)
\(98\) 2.48105i 0.250624i
\(99\) −9.42450 − 22.7528i −0.947198 − 2.28674i
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.l.a.66.4 ✓ 24
3.2 odd 2 765.2.be.b.406.3 24
5.2 odd 4 425.2.n.c.49.3 24
5.3 odd 4 425.2.n.f.49.4 24
5.4 even 2 425.2.m.b.151.3 24
17.3 odd 16 1445.2.d.j.866.13 24
17.5 odd 16 1445.2.a.p.1.6 12
17.8 even 8 inner 85.2.l.a.76.4 yes 24
17.12 odd 16 1445.2.a.q.1.6 12
17.14 odd 16 1445.2.d.j.866.14 24
51.8 odd 8 765.2.be.b.586.3 24
85.8 odd 8 425.2.n.c.399.3 24
85.29 odd 16 7225.2.a.bq.1.7 12
85.39 odd 16 7225.2.a.bs.1.7 12
85.42 odd 8 425.2.n.f.399.4 24
85.59 even 8 425.2.m.b.76.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.66.4 ✓ 24 1.1 even 1 trivial
85.2.l.a.76.4 yes 24 17.8 even 8 inner
425.2.m.b.76.3 24 85.59 even 8
425.2.m.b.151.3 24 5.4 even 2
425.2.n.c.49.3 24 5.2 odd 4
425.2.n.c.399.3 24 85.8 odd 8
425.2.n.f.49.4 24 5.3 odd 4
425.2.n.f.399.4 24 85.42 odd 8
765.2.be.b.406.3 24 3.2 odd 2
765.2.be.b.586.3 24 51.8 odd 8
1445.2.a.p.1.6 12 17.5 odd 16
1445.2.a.q.1.6 12 17.12 odd 16
1445.2.d.j.866.13 24 17.3 odd 16
1445.2.d.j.866.14 24 17.14 odd 16
7225.2.a.bq.1.7 12 85.29 odd 16
7225.2.a.bs.1.7 12 85.39 odd 16