LMFDB - Embedded newform 85.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(1,85)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("85.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	85.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.678728417181$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -4.73205 q^{6} -2.73205 q^{7} +1.73205 q^{8} +4.46410 q^{9} -1.73205 q^{10} +4.73205 q^{11} +2.73205 q^{12} -4.00000 q^{13} +4.73205 q^{14} +2.73205 q^{15} -5.00000 q^{16} -1.00000 q^{17} -7.73205 q^{18} -1.46410 q^{19} +1.00000 q^{20} -7.46410 q^{21} -8.19615 q^{22} -8.19615 q^{23} +4.73205 q^{24} +1.00000 q^{25} +6.92820 q^{26} +4.00000 q^{27} -2.73205 q^{28} -3.46410 q^{29} -4.73205 q^{30} +3.26795 q^{31} +5.19615 q^{32} +12.9282 q^{33} +1.73205 q^{34} -2.73205 q^{35} +4.46410 q^{36} -0.535898 q^{37} +2.53590 q^{38} -10.9282 q^{39} +1.73205 q^{40} -3.46410 q^{41} +12.9282 q^{42} -0.535898 q^{43} +4.73205 q^{44} +4.46410 q^{45} +14.1962 q^{46} +12.9282 q^{47} -13.6603 q^{48} +0.464102 q^{49} -1.73205 q^{50} -2.73205 q^{51} -4.00000 q^{52} +6.00000 q^{53} -6.92820 q^{54} +4.73205 q^{55} -4.73205 q^{56} -4.00000 q^{57} +6.00000 q^{58} +2.53590 q^{59} +2.73205 q^{60} -4.92820 q^{61} -5.66025 q^{62} -12.1962 q^{63} +1.00000 q^{64} -4.00000 q^{65} -22.3923 q^{66} -10.0000 q^{67} -1.00000 q^{68} -22.3923 q^{69} +4.73205 q^{70} +11.6603 q^{71} +7.73205 q^{72} +6.39230 q^{73} +0.928203 q^{74} +2.73205 q^{75} -1.46410 q^{76} -12.9282 q^{77} +18.9282 q^{78} +14.5885 q^{79} -5.00000 q^{80} -2.46410 q^{81} +6.00000 q^{82} +8.53590 q^{83} -7.46410 q^{84} -1.00000 q^{85} +0.928203 q^{86} -9.46410 q^{87} +8.19615 q^{88} +4.39230 q^{89} -7.73205 q^{90} +10.9282 q^{91} -8.19615 q^{92} +8.92820 q^{93} -22.3923 q^{94} -1.46410 q^{95} +14.1962 q^{96} -4.92820 q^{97} -0.803848 q^{98} +21.1244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 8 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 6 q^{22} - 6 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 6 * q^6 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 8 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 - 2 * q^17 - 12 * q^18 + 4 * q^19 + 2 * q^20 - 8 * q^21 - 6 * q^22 - 6 * q^23 + 6 * q^24 + 2 * q^25 + 8 * q^27 - 2 * q^28 - 6 * q^30 + 10 * q^31 + 12 * q^33 - 2 * q^35 + 2 * q^36 - 8 * q^37 + 12 * q^38 - 8 * q^39 + 12 * q^42 - 8 * q^43 + 6 * q^44 + 2 * q^45 + 18 * q^46 + 12 * q^47 - 10 * q^48 - 6 * q^49 - 2 * q^51 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 8 * q^57 + 12 * q^58 + 12 * q^59 + 2 * q^60 + 4 * q^61 + 6 * q^62 - 14 * q^63 + 2 * q^64 - 8 * q^65 - 24 * q^66 - 20 * q^67 - 2 * q^68 - 24 * q^69 + 6 * q^70 + 6 * q^71 + 12 * q^72 - 8 * q^73 - 12 * q^74 + 2 * q^75 + 4 * q^76 - 12 * q^77 + 24 * q^78 - 2 * q^79 - 10 * q^80 + 2 * q^81 + 12 * q^82 + 24 * q^83 - 8 * q^84 - 2 * q^85 - 12 * q^86 - 12 * q^87 + 6 * q^88 - 12 * q^89 - 12 * q^90 + 8 * q^91 - 6 * q^92 + 4 * q^93 - 24 * q^94 + 4 * q^95 + 18 * q^96 + 4 * q^97 - 12 * q^98 + 18 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−4.73205	−1.93185
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	1.73205	0.612372
$9$	4.46410	1.48803
$10$	−1.73205	−0.547723
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	2.73205	0.788675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	4.73205	1.26469
$15$	2.73205	0.705412
$16$	−5.00000	−1.25000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−7.73205	−1.82246
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	1.00000	0.223607
$21$	−7.46410	−1.62880
$22$	−8.19615	−1.74743
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	4.73205	0.965926
$25$	1.00000	0.200000
$26$	6.92820	1.35873
$27$	4.00000	0.769800
$28$	−2.73205	−0.516309
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	−4.73205	−0.863950
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	5.19615	0.918559
$33$	12.9282	2.25051
$34$	1.73205	0.297044
$35$	−2.73205	−0.461801
$36$	4.46410	0.744017
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	2.53590	0.411377
$39$	−10.9282	−1.74991
$40$	1.73205	0.273861
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	12.9282	1.99487
$43$	−0.535898	−0.0817237	−0.0408619	−	0.999165i	$-0.513010\pi$
$43$	−0.535898	−0.0817237	−0.0408619	+	0.999165i	$0.513010\pi$
$44$	4.73205	0.713384
$45$	4.46410	0.665469
$46$	14.1962	2.09311
$47$	12.9282	1.88577	0.942886	−	0.333115i	$-0.108100\pi$
$47$	12.9282	1.88577	0.942886	+	0.333115i	$0.108100\pi$
$48$	−13.6603	−1.97169
$49$	0.464102	0.0663002
$50$	−1.73205	−0.244949
$51$	−2.73205	−0.382564
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−6.92820	−0.942809
$55$	4.73205	0.638070
$56$	−4.73205	−0.632347
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	2.53590	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$60$	2.73205	0.352706
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	−5.66025	−0.718853
$63$	−12.1962	−1.53657
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−22.3923	−2.75630
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−1.00000	−0.121268
$69$	−22.3923	−2.69572
$70$	4.73205	0.565588
$71$	11.6603	1.38382	0.691909	−	0.721985i	$-0.256770\pi$
$71$	11.6603	1.38382	0.691909	+	0.721985i	$0.256770\pi$
$72$	7.73205	0.911231
$73$	6.39230	0.748163	0.374081	−	0.927396i	$-0.377958\pi$
$73$	6.39230	0.748163	0.374081	+	0.927396i	$0.377958\pi$
$74$	0.928203	0.107901
$75$	2.73205	0.315470
$76$	−1.46410	−0.167944
$77$	−12.9282	−1.47331
$78$	18.9282	2.14320
$79$	14.5885	1.64133	0.820665	−	0.571410i	$-0.193603\pi$
$79$	14.5885	1.64133	0.820665	+	0.571410i	$0.193603\pi$
$80$	−5.00000	−0.559017
$81$	−2.46410	−0.273789
$82$	6.00000	0.662589
$83$	8.53590	0.936937	0.468468	−	0.883480i	$-0.344806\pi$
$83$	8.53590	0.936937	0.468468	+	0.883480i	$0.344806\pi$
$84$	−7.46410	−0.814400
$85$	−1.00000	−0.108465
$86$	0.928203	0.100091
$87$	−9.46410	−1.01466
$88$	8.19615	0.873713
$89$	4.39230	0.465583	0.232792	−	0.972527i	$-0.425214\pi$
$89$	4.39230	0.465583	0.232792	+	0.972527i	$0.425214\pi$
$90$	−7.73205	−0.815030
$91$	10.9282	1.14559
$92$	−8.19615	−0.854508
$93$	8.92820	0.925812
$94$	−22.3923	−2.30959
$95$	−1.46410	−0.150214
$96$	14.1962	1.44889
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	−0.803848	−0.0812009
$99$	21.1244	2.12308
$100$	1.00000	0.100000
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	4.73205	0.468543
$103$	8.92820	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$103$	8.92820	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$104$	−6.92820	−0.679366
$105$	−7.46410	−0.728422
$106$	−10.3923	−1.00939
$107$	17.6603	1.70728	0.853641	−	0.520862i	$-0.174390\pi$
$107$	17.6603	1.70728	0.853641	+	0.520862i	$0.174390\pi$
$108$	4.00000	0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−8.19615	−0.781472
$111$	−1.46410	−0.138966
$112$	13.6603	1.29077
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	6.92820	0.648886
$115$	−8.19615	−0.764295
$116$	−3.46410	−0.321634
$117$	−17.8564	−1.65083
$118$	−4.39230	−0.404344
$119$	2.73205	0.250447
$120$	4.73205	0.431975
$121$	11.3923	1.03566
$122$	8.53590	0.772804
$123$	−9.46410	−0.853349
$124$	3.26795	0.293471
$125$	1.00000	0.0894427
$126$	21.1244	1.88191
$127$	−14.3923	−1.27711	−0.638555	−	0.769576i	$-0.720468\pi$
$127$	−14.3923	−1.27711	−0.638555	+	0.769576i	$0.720468\pi$
$128$	−12.1244	−1.07165
$129$	−1.46410	−0.128907
$130$	6.92820	0.607644
$131$	−2.19615	−0.191879	−0.0959394	−	0.995387i	$-0.530585\pi$
$131$	−2.19615	−0.191879	−0.0959394	+	0.995387i	$0.530585\pi$
$132$	12.9282	1.12526
$133$	4.00000	0.346844
$134$	17.3205	1.49626
$135$	4.00000	0.344265
$136$	−1.73205	−0.148522
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	38.7846	3.30157
$139$	−3.66025	−0.310459	−0.155229	−	0.987878i	$-0.549612\pi$
$139$	−3.66025	−0.310459	−0.155229	+	0.987878i	$0.549612\pi$
$140$	−2.73205	−0.230900
$141$	35.3205	2.97452
$142$	−20.1962	−1.69482
$143$	−18.9282	−1.58286
$144$	−22.3205	−1.86004
$145$	−3.46410	−0.287678
$146$	−11.0718	−0.916308
$147$	1.26795	0.104579
$148$	−0.535898	−0.0440506
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−4.73205	−0.386370
$151$	−1.46410	−0.119147	−0.0595734	−	0.998224i	$-0.518974\pi$
$151$	−1.46410	−0.119147	−0.0595734	+	0.998224i	$0.518974\pi$
$152$	−2.53590	−0.205689
$153$	−4.46410	−0.360901
$154$	22.3923	1.80442
$155$	3.26795	0.262488
$156$	−10.9282	−0.874957
$157$	8.92820	0.712548	0.356274	−	0.934381i	$-0.384047\pi$
$157$	8.92820	0.712548	0.356274	+	0.934381i	$0.384047\pi$
$158$	−25.2679	−2.01021
$159$	16.3923	1.29999
$160$	5.19615	0.410792
$161$	22.3923	1.76476
$162$	4.26795	0.335322
$163$	−0.196152	−0.0153638	−0.00768192	−	0.999970i	$-0.502445\pi$
$163$	−0.196152	−0.0153638	−0.00768192	+	0.999970i	$0.502445\pi$
$164$	−3.46410	−0.270501
$165$	12.9282	1.00646
$166$	−14.7846	−1.14751
$167$	12.5885	0.974124	0.487062	−	0.873367i	$-0.338069\pi$
$167$	12.5885	0.974124	0.487062	+	0.873367i	$0.338069\pi$
$168$	−12.9282	−0.997433
$169$	3.00000	0.230769
$170$	1.73205	0.132842
$171$	−6.53590	−0.499813
$172$	−0.535898	−0.0408619
$173$	3.46410	0.263371	0.131685	−	0.991292i	$-0.457961\pi$
$173$	3.46410	0.263371	0.131685	+	0.991292i	$0.457961\pi$
$174$	16.3923	1.24270
$175$	−2.73205	−0.206524
$176$	−23.6603	−1.78346
$177$	6.92820	0.520756
$178$	−7.60770	−0.570221
$179$	−11.3205	−0.846135	−0.423067	−	0.906098i	$-0.639047\pi$
$179$	−11.3205	−0.846135	−0.423067	+	0.906098i	$0.639047\pi$
$180$	4.46410	0.332734
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	−18.9282	−1.40305
$183$	−13.4641	−0.995295
$184$	−14.1962	−1.04655
$185$	−0.535898	−0.0394000
$186$	−15.4641	−1.13388
$187$	−4.73205	−0.346042
$188$	12.9282	0.942886
$189$	−10.9282	−0.794910
$190$	2.53590	0.183973
$191$	1.85641	0.134325	0.0671624	−	0.997742i	$-0.478605\pi$
$191$	1.85641	0.134325	0.0671624	+	0.997742i	$0.478605\pi$
$192$	2.73205	0.197169
$193$	16.5359	1.19028	0.595140	−	0.803622i	$-0.297097\pi$
$193$	16.5359	1.19028	0.595140	+	0.803622i	$0.297097\pi$
$194$	8.53590	0.612842
$195$	−10.9282	−0.782585
$196$	0.464102	0.0331501
$197$	17.3205	1.23404	0.617018	−	0.786949i	$-0.288341\pi$
$197$	17.3205	1.23404	0.617018	+	0.786949i	$0.288341\pi$
$198$	−36.5885	−2.60023
$199$	10.1962	0.722786	0.361393	−	0.932414i	$-0.382301\pi$
$199$	10.1962	0.722786	0.361393	+	0.932414i	$0.382301\pi$
$200$	1.73205	0.122474
$201$	−27.3205	−1.92704
$202$	16.3923	1.15336
$203$	9.46410	0.664250
$204$	−2.73205	−0.191282
$205$	−3.46410	−0.241943
$206$	−15.4641	−1.07744
$207$	−36.5885	−2.54307
$208$	20.0000	1.38675
$209$	−6.92820	−0.479234
$210$	12.9282	0.892131
$211$	10.1962	0.701932	0.350966	−	0.936388i	$-0.385853\pi$
$211$	10.1962	0.701932	0.350966	+	0.936388i	$0.385853\pi$
$212$	6.00000	0.412082
$213$	31.8564	2.18277
$214$	−30.5885	−2.09098
$215$	−0.535898	−0.0365480
$216$	6.92820	0.471405
$217$	−8.92820	−0.606086
$218$	17.3205	1.17309
$219$	17.4641	1.18011
$220$	4.73205	0.319035
$221$	4.00000	0.269069
$222$	2.53590	0.170198
$223$	−26.3923	−1.76736	−0.883680	−	0.468092i	$-0.844942\pi$
$223$	−26.3923	−1.76736	−0.883680	+	0.468092i	$0.844942\pi$
$224$	−14.1962	−0.948520
$225$	4.46410	0.297607
$226$	30.0000	1.99557
$227$	22.7321	1.50878	0.754390	−	0.656427i	$-0.227933\pi$
$227$	22.7321	1.50878	0.754390	+	0.656427i	$0.227933\pi$
$228$	−4.00000	−0.264906
$229$	−8.39230	−0.554579	−0.277290	−	0.960786i	$-0.589436\pi$
$229$	−8.39230	−0.554579	−0.277290	+	0.960786i	$0.589436\pi$
$230$	14.1962	0.936067
$231$	−35.3205	−2.32392
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	30.9282	2.02184
$235$	12.9282	0.843343
$236$	2.53590	0.165073
$237$	39.8564	2.58895
$238$	−4.73205	−0.306733
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	−13.6603	−0.881766
$241$	−5.60770	−0.361223	−0.180612	−	0.983554i	$-0.557808\pi$
$241$	−5.60770	−0.361223	−0.180612	+	0.983554i	$0.557808\pi$
$242$	−19.7321	−1.26842
$243$	−18.7321	−1.20166
$244$	−4.92820	−0.315496
$245$	0.464102	0.0296504
$246$	16.3923	1.04514
$247$	5.85641	0.372634
$248$	5.66025	0.359426
$249$	23.3205	1.47788
$250$	−1.73205	−0.109545
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	−12.1962	−0.768285
$253$	−38.7846	−2.43837
$254$	24.9282	1.56413
$255$	−2.73205	−0.171088
$256$	19.0000	1.18750
$257$	−6.92820	−0.432169	−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820	−0.432169	−0.216085	+	0.976375i	$0.569329\pi$
$258$	2.53590	0.157878
$259$	1.46410	0.0909748
$260$	−4.00000	−0.248069
$261$	−15.4641	−0.957204
$262$	3.80385	0.235002
$263$	−1.60770	−0.0991347	−0.0495674	−	0.998771i	$-0.515784\pi$
$263$	−1.60770	−0.0991347	−0.0495674	+	0.998771i	$0.515784\pi$
$264$	22.3923	1.37815
$265$	6.00000	0.368577
$266$	−6.92820	−0.424795
$267$	12.0000	0.734388
$268$	−10.0000	−0.610847
$269$	0.928203	0.0565935	0.0282968	−	0.999600i	$-0.490992\pi$
$269$	0.928203	0.0565935	0.0282968	+	0.999600i	$0.490992\pi$
$270$	−6.92820	−0.421637
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	5.00000	0.303170
$273$	29.8564	1.80699
$274$	0	0
$275$	4.73205	0.285353
$276$	−22.3923	−1.34786
$277$	20.9282	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$277$	20.9282	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$278$	6.33975	0.380233
$279$	14.5885	0.873388
$280$	−4.73205	−0.282794
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	−61.1769	−3.64303
$283$	−5.26795	−0.313147	−0.156574	−	0.987666i	$-0.550045\pi$
$283$	−5.26795	−0.313147	−0.156574	+	0.987666i	$0.550045\pi$
$284$	11.6603	0.691909
$285$	−4.00000	−0.236940
$286$	32.7846	1.93859
$287$	9.46410	0.558648
$288$	23.1962	1.36685
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	−13.4641	−0.789280
$292$	6.39230	0.374081
$293$	−0.928203	−0.0542262	−0.0271131	−	0.999632i	$-0.508631\pi$
$293$	−0.928203	−0.0542262	−0.0271131	+	0.999632i	$0.508631\pi$
$294$	−2.19615	−0.128082
$295$	2.53590	0.147646
$296$	−0.928203	−0.0539507
$297$	18.9282	1.09833
$298$	10.3923	0.602010
$299$	32.7846	1.89598
$300$	2.73205	0.157735
$301$	1.46410	0.0843894
$302$	2.53590	0.145925
$303$	−25.8564	−1.48541
$304$	7.32051	0.419860
$305$	−4.92820	−0.282188
$306$	7.73205	0.442012
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−12.9282	−0.736653
$309$	24.3923	1.38763
$310$	−5.66025	−0.321481
$311$	−16.0526	−0.910257	−0.455129	−	0.890426i	$-0.650407\pi$
$311$	−16.0526	−0.910257	−0.455129	+	0.890426i	$0.650407\pi$
$312$	−18.9282	−1.07160
$313$	−26.3923	−1.49178	−0.745891	−	0.666068i	$-0.767976\pi$
$313$	−26.3923	−1.49178	−0.745891	+	0.666068i	$0.767976\pi$
$314$	−15.4641	−0.872690
$315$	−12.1962	−0.687175
$316$	14.5885	0.820665
$317$	−24.9282	−1.40011	−0.700054	−	0.714090i	$-0.746841\pi$
$317$	−24.9282	−1.40011	−0.700054	+	0.714090i	$0.746841\pi$
$318$	−28.3923	−1.59216
$319$	−16.3923	−0.917793
$320$	1.00000	0.0559017
$321$	48.2487	2.69298
$322$	−38.7846	−2.16138
$323$	1.46410	0.0814648
$324$	−2.46410	−0.136895
$325$	−4.00000	−0.221880
$326$	0.339746	0.0188168
$327$	−27.3205	−1.51083
$328$	−6.00000	−0.331295
$329$	−35.3205	−1.94728
$330$	−22.3923	−1.23266
$331$	−6.53590	−0.359245	−0.179623	−	0.983736i	$-0.557488\pi$
$331$	−6.53590	−0.359245	−0.179623	+	0.983736i	$0.557488\pi$
$332$	8.53590	0.468468
$333$	−2.39230	−0.131097
$334$	−21.8038	−1.19305
$335$	−10.0000	−0.546358
$336$	37.3205	2.03600
$337$	−6.78461	−0.369581	−0.184791	−	0.982778i	$-0.559161\pi$
$337$	−6.78461	−0.369581	−0.184791	+	0.982778i	$0.559161\pi$
$338$	−5.19615	−0.282633
$339$	−47.3205	−2.57010
$340$	−1.00000	−0.0542326
$341$	15.4641	0.837428
$342$	11.3205	0.612143
$343$	17.8564	0.964155
$344$	−0.928203	−0.0500454
$345$	−22.3923	−1.20556
$346$	−6.00000	−0.322562
$347$	3.80385	0.204201	0.102101	−	0.994774i	$-0.467444\pi$
$347$	3.80385	0.204201	0.102101	+	0.994774i	$0.467444\pi$
$348$	−9.46410	−0.507329
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	4.73205	0.252939
$351$	−16.0000	−0.854017
$352$	24.5885	1.31057
$353$	26.7846	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$353$	26.7846	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$354$	−12.0000	−0.637793
$355$	11.6603	0.618862
$356$	4.39230	0.232792
$357$	7.46410	0.395042
$358$	19.6077	1.03630
$359$	−21.4641	−1.13283	−0.566416	−	0.824119i	$-0.691670\pi$
$359$	−21.4641	−1.13283	−0.566416	+	0.824119i	$0.691670\pi$
$360$	7.73205	0.407515
$361$	−16.8564	−0.887179
$362$	4.14359	0.217782
$363$	31.1244	1.63361
$364$	10.9282	0.572793
$365$	6.39230	0.334589
$366$	23.3205	1.21898
$367$	−7.80385	−0.407358	−0.203679	−	0.979038i	$-0.565290\pi$
$367$	−7.80385	−0.407358	−0.203679	+	0.979038i	$0.565290\pi$
$368$	40.9808	2.13627
$369$	−15.4641	−0.805029
$370$	0.928203	0.0482550
$371$	−16.3923	−0.851046
$372$	8.92820	0.462906
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	8.19615	0.423813
$375$	2.73205	0.141082
$376$	22.3923	1.15479
$377$	13.8564	0.713641
$378$	18.9282	0.973562
$379$	17.8038	0.914522	0.457261	−	0.889332i	$-0.348830\pi$
$379$	17.8038	0.914522	0.457261	+	0.889332i	$0.348830\pi$
$380$	−1.46410	−0.0751068
$381$	−39.3205	−2.01445
$382$	−3.21539	−0.164514
$383$	15.4641	0.790179	0.395089	−	0.918643i	$-0.370714\pi$
$383$	15.4641	0.790179	0.395089	+	0.918643i	$0.370714\pi$
$384$	−33.1244	−1.69037
$385$	−12.9282	−0.658882
$386$	−28.6410	−1.45779
$387$	−2.39230	−0.121608
$388$	−4.92820	−0.250192
$389$	−4.39230	−0.222699	−0.111349	−	0.993781i	$-0.535517\pi$
$389$	−4.39230	−0.222699	−0.111349	+	0.993781i	$0.535517\pi$
$390$	18.9282	0.958467
$391$	8.19615	0.414497
$392$	0.803848	0.0406004
$393$	−6.00000	−0.302660
$394$	−30.0000	−1.51138
$395$	14.5885	0.734025
$396$	21.1244	1.06154
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−17.6603	−0.885229
$399$	10.9282	0.547094
$400$	−5.00000	−0.250000
$401$	12.9282	0.645604	0.322802	−	0.946467i	$-0.395375\pi$
$401$	12.9282	0.645604	0.322802	+	0.946467i	$0.395375\pi$
$402$	47.3205	2.36013
$403$	−13.0718	−0.651153
$404$	−9.46410	−0.470857
$405$	−2.46410	−0.122442
$406$	−16.3923	−0.813536
$407$	−2.53590	−0.125700
$408$	−4.73205	−0.234271
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	8.92820	0.439861
$413$	−6.92820	−0.340915
$414$	63.3731	3.11462
$415$	8.53590	0.419011
$416$	−20.7846	−1.01905
$417$	−10.0000	−0.489702
$418$	12.0000	0.586939
$419$	38.1962	1.86600	0.933002	−	0.359871i	$-0.117179\pi$
$419$	38.1962	1.86600	0.933002	+	0.359871i	$0.117179\pi$
$420$	−7.46410	−0.364211
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	−17.6603	−0.859688
$423$	57.7128	2.80609
$424$	10.3923	0.504695
$425$	−1.00000	−0.0485071
$426$	−55.1769	−2.67333
$427$	13.4641	0.651574
$428$	17.6603	0.853641
$429$	−51.7128	−2.49672
$430$	0.928203	0.0447619
$431$	9.80385	0.472235	0.236117	−	0.971725i	$-0.424125\pi$
$431$	9.80385	0.472235	0.236117	+	0.971725i	$0.424125\pi$
$432$	−20.0000	−0.962250
$433$	−17.8564	−0.858124	−0.429062	−	0.903275i	$-0.641156\pi$
$433$	−17.8564	−0.858124	−0.429062	+	0.903275i	$0.641156\pi$
$434$	15.4641	0.742301
$435$	−9.46410	−0.453769
$436$	−10.0000	−0.478913
$437$	12.0000	0.574038
$438$	−30.2487	−1.44534
$439$	−20.0526	−0.957056	−0.478528	−	0.878072i	$-0.658830\pi$
$439$	−20.0526	−0.957056	−0.478528	+	0.878072i	$0.658830\pi$
$440$	8.19615	0.390736
$441$	2.07180	0.0986570
$442$	−6.92820	−0.329541
$443$	−12.9282	−0.614237	−0.307119	−	0.951671i	$-0.599365\pi$
$443$	−12.9282	−0.614237	−0.307119	+	0.951671i	$0.599365\pi$
$444$	−1.46410	−0.0694832
$445$	4.39230	0.208215
$446$	45.7128	2.16456
$447$	−16.3923	−0.775329
$448$	−2.73205	−0.129077
$449$	−34.3923	−1.62307	−0.811537	−	0.584302i	$-0.801369\pi$
$449$	−34.3923	−1.62307	−0.811537	+	0.584302i	$0.801369\pi$
$450$	−7.73205	−0.364492
$451$	−16.3923	−0.771883
$452$	−17.3205	−0.814688
$453$	−4.00000	−0.187936
$454$	−39.3731	−1.84787
$455$	10.9282	0.512322
$456$	−6.92820	−0.324443
$457$	−36.7846	−1.72071	−0.860356	−	0.509694i	$-0.829759\pi$
$457$	−36.7846	−1.72071	−0.860356	+	0.509694i	$0.829759\pi$
$458$	14.5359	0.679218
$459$	−4.00000	−0.186704
$460$	−8.19615	−0.382148
$461$	−24.9282	−1.16102	−0.580511	−	0.814252i	$-0.697147\pi$
$461$	−24.9282	−1.16102	−0.580511	+	0.814252i	$0.697147\pi$
$462$	61.1769	2.84621
$463$	−23.8564	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$463$	−23.8564	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$464$	17.3205	0.804084
$465$	8.92820	0.414036
$466$	−10.3923	−0.481414
$467$	1.60770	0.0743953	0.0371976	−	0.999308i	$-0.488157\pi$
$467$	1.60770	0.0743953	0.0371976	+	0.999308i	$0.488157\pi$
$468$	−17.8564	−0.825413
$469$	27.3205	1.26154
$470$	−22.3923	−1.03288
$471$	24.3923	1.12394
$472$	4.39230	0.202172
$473$	−2.53590	−0.116601
$474$	−69.0333	−3.17081
$475$	−1.46410	−0.0671776
$476$	2.73205	0.125223
$477$	26.7846	1.22638
$478$	36.0000	1.64660
$479$	−11.6603	−0.532771	−0.266385	−	0.963867i	$-0.585829\pi$
$479$	−11.6603	−0.532771	−0.266385	+	0.963867i	$0.585829\pi$
$480$	14.1962	0.647963
$481$	2.14359	0.0977395
$482$	9.71281	0.442407
$483$	61.1769	2.78365
$484$	11.3923	0.517832
$485$	−4.92820	−0.223778
$486$	32.4449	1.47173
$487$	24.9808	1.13199	0.565993	−	0.824410i	$-0.308493\pi$
$487$	24.9808	1.13199	0.565993	+	0.824410i	$0.308493\pi$
$488$	−8.53590	−0.386402
$489$	−0.535898	−0.0242342
$490$	−0.803848	−0.0363141
$491$	−19.6077	−0.884883	−0.442441	−	0.896797i	$-0.645888\pi$
$491$	−19.6077	−0.884883	−0.442441	+	0.896797i	$0.645888\pi$
$492$	−9.46410	−0.426675
$493$	3.46410	0.156015
$494$	−10.1436	−0.456382
$495$	21.1244	0.949469
$496$	−16.3397	−0.733676
$497$	−31.8564	−1.42896
$498$	−40.3923	−1.81002
$499$	−15.6603	−0.701049	−0.350525	−	0.936554i	$-0.613997\pi$
$499$	−15.6603	−0.701049	−0.350525	+	0.936554i	$0.613997\pi$
$500$	1.00000	0.0447214
$501$	34.3923	1.53653
$502$	−12.0000	−0.535586
$503$	15.1244	0.674362	0.337181	−	0.941440i	$-0.390527\pi$
$503$	15.1244	0.674362	0.337181	+	0.941440i	$0.390527\pi$
$504$	−21.1244	−0.940954
$505$	−9.46410	−0.421147
$506$	67.1769	2.98638
$507$	8.19615	0.364004
$508$	−14.3923	−0.638555
$509$	19.8564	0.880120	0.440060	−	0.897968i	$-0.354957\pi$
$509$	19.8564	0.880120	0.440060	+	0.897968i	$0.354957\pi$
$510$	4.73205	0.209539
$511$	−17.4641	−0.772566
$512$	−8.66025	−0.382733
$513$	−5.85641	−0.258567
$514$	12.0000	0.529297
$515$	8.92820	0.393424
$516$	−1.46410	−0.0644535
$517$	61.1769	2.69056
$518$	−2.53590	−0.111421
$519$	9.46410	0.415428
$520$	−6.92820	−0.303822
$521$	4.14359	0.181534	0.0907671	−	0.995872i	$-0.471068\pi$
$521$	4.14359	0.181534	0.0907671	+	0.995872i	$0.471068\pi$
$522$	26.7846	1.17233
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−2.19615	−0.0959394
$525$	−7.46410	−0.325760
$526$	2.78461	0.121415
$527$	−3.26795	−0.142354
$528$	−64.6410	−2.81314
$529$	44.1769	1.92074
$530$	−10.3923	−0.451413
$531$	11.3205	0.491268
$532$	4.00000	0.173422
$533$	13.8564	0.600188
$534$	−20.7846	−0.899438
$535$	17.6603	0.763519
$536$	−17.3205	−0.748132
$537$	−30.9282	−1.33465
$538$	−1.60770	−0.0693127
$539$	2.19615	0.0945950
$540$	4.00000	0.172133
$541$	39.1769	1.68435	0.842174	−	0.539207i	$-0.181276\pi$
$541$	39.1769	1.68435	0.842174	+	0.539207i	$0.181276\pi$
$542$	−5.07180	−0.217852
$543$	−6.53590	−0.280482
$544$	−5.19615	−0.222783
$545$	−10.0000	−0.428353
$546$	−51.7128	−2.21310
$547$	−39.9090	−1.70638	−0.853192	−	0.521597i	$-0.825337\pi$
$547$	−39.9090	−1.70638	−0.853192	+	0.521597i	$0.825337\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	−8.19615	−0.349485
$551$	5.07180	0.216066
$552$	−38.7846	−1.65078
$553$	−39.8564	−1.69487
$554$	−36.2487	−1.54006
$555$	−1.46410	−0.0621477
$556$	−3.66025	−0.155229
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	−25.2679	−1.06968
$559$	2.14359	0.0906643
$560$	13.6603	0.577251
$561$	−12.9282	−0.545829
$562$	22.3923	0.944562
$563$	27.4641	1.15747	0.578737	−	0.815514i	$-0.303546\pi$
$563$	27.4641	1.15747	0.578737	+	0.815514i	$0.303546\pi$
$564$	35.3205	1.48726
$565$	−17.3205	−0.728679
$566$	9.12436	0.383525
$567$	6.73205	0.282720
$568$	20.1962	0.847412
$569$	40.6410	1.70376	0.851880	−	0.523737i	$-0.175463\pi$
$569$	40.6410	1.70376	0.851880	+	0.523737i	$0.175463\pi$
$570$	6.92820	0.290191
$571$	−36.4449	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$571$	−36.4449	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$572$	−18.9282	−0.791428
$573$	5.07180	0.211877
$574$	−16.3923	−0.684202
$575$	−8.19615	−0.341803
$576$	4.46410	0.186004
$577$	−38.6410	−1.60865	−0.804323	−	0.594192i	$-0.797472\pi$
$577$	−38.6410	−1.60865	−0.804323	+	0.594192i	$0.797472\pi$
$578$	−1.73205	−0.0720438
$579$	45.1769	1.87749
$580$	−3.46410	−0.143839
$581$	−23.3205	−0.967498
$582$	23.3205	0.966666
$583$	28.3923	1.17589
$584$	11.0718	0.458154
$585$	−17.8564	−0.738272
$586$	1.60770	0.0664133
$587$	46.3923	1.91482	0.957408	−	0.288740i	$-0.0932362\pi$
$587$	46.3923	1.91482	0.957408	+	0.288740i	$0.0932362\pi$
$588$	1.26795	0.0522893
$589$	−4.78461	−0.197146
$590$	−4.39230	−0.180828
$591$	47.3205	1.94651
$592$	2.67949	0.110126
$593$	19.8564	0.815405	0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564	0.815405	0.407702	+	0.913115i	$0.366330\pi$
$594$	−32.7846	−1.34517
$595$	2.73205	0.112003
$596$	−6.00000	−0.245770
$597$	27.8564	1.14009
$598$	−56.7846	−2.32210
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	4.73205	0.193185
$601$	8.24871	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$601$	8.24871	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$602$	−2.53590	−0.103356
$603$	−44.6410	−1.81792
$604$	−1.46410	−0.0595734
$605$	11.3923	0.463163
$606$	44.7846	1.81925
$607$	−21.6603	−0.879163	−0.439581	−	0.898203i	$-0.644873\pi$
$607$	−21.6603	−0.879163	−0.439581	+	0.898203i	$0.644873\pi$
$608$	−7.60770	−0.308533
$609$	25.8564	1.04775
$610$	8.53590	0.345608
$611$	−51.7128	−2.09208
$612$	−4.46410	−0.180451
$613$	15.8564	0.640434	0.320217	−	0.947344i	$-0.396244\pi$
$613$	15.8564	0.640434	0.320217	+	0.947344i	$0.396244\pi$
$614$	17.3205	0.698999
$615$	−9.46410	−0.381629
$616$	−22.3923	−0.902212
$617$	−27.4641	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$617$	−27.4641	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$618$	−42.2487	−1.69949
$619$	38.5885	1.55100	0.775501	−	0.631347i	$-0.217498\pi$
$619$	38.5885	1.55100	0.775501	+	0.631347i	$0.217498\pi$
$620$	3.26795	0.131244
$621$	−32.7846	−1.31560
$622$	27.8038	1.11483
$623$	−12.0000	−0.480770
$624$	54.6410	2.18739
$625$	1.00000	0.0400000
$626$	45.7128	1.82705
$627$	−18.9282	−0.755920
$628$	8.92820	0.356274
$629$	0.535898	0.0213677
$630$	21.1244	0.841614
$631$	−32.3923	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$631$	−32.3923	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$632$	25.2679	1.00511
$633$	27.8564	1.10719
$634$	43.1769	1.71477
$635$	−14.3923	−0.571141
$636$	16.3923	0.649997
$637$	−1.85641	−0.0735535
$638$	28.3923	1.12406
$639$	52.0526	2.05917
$640$	−12.1244	−0.479257
$641$	31.1769	1.23141	0.615707	−	0.787975i	$-0.288870\pi$
$641$	31.1769	1.23141	0.615707	+	0.787975i	$0.288870\pi$
$642$	−83.5692	−3.29821
$643$	−24.1962	−0.954203	−0.477102	−	0.878848i	$-0.658313\pi$
$643$	−24.1962	−0.954203	−0.477102	+	0.878848i	$0.658313\pi$
$644$	22.3923	0.882380
$645$	−1.46410	−0.0576489
$646$	−2.53590	−0.0997736
$647$	−38.7846	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$647$	−38.7846	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$648$	−4.26795	−0.167661
$649$	12.0000	0.471041
$650$	6.92820	0.271746
$651$	−24.3923	−0.956010
$652$	−0.196152	−0.00768192
$653$	−25.6077	−1.00211	−0.501053	−	0.865416i	$-0.667054\pi$
$653$	−25.6077	−1.00211	−0.501053	+	0.865416i	$0.667054\pi$
$654$	47.3205	1.85038
$655$	−2.19615	−0.0858108
$656$	17.3205	0.676252
$657$	28.5359	1.11329
$658$	61.1769	2.38492
$659$	−32.7846	−1.27711	−0.638554	−	0.769577i	$-0.720467\pi$
$659$	−32.7846	−1.27711	−0.638554	+	0.769577i	$0.720467\pi$
$660$	12.9282	0.503230
$661$	−8.14359	−0.316749	−0.158375	−	0.987379i	$-0.550625\pi$
$661$	−8.14359	−0.316749	−0.158375	+	0.987379i	$0.550625\pi$
$662$	11.3205	0.439984
$663$	10.9282	0.424416
$664$	14.7846	0.573754
$665$	4.00000	0.155113
$666$	4.14359	0.160561
$667$	28.3923	1.09935
$668$	12.5885	0.487062
$669$	−72.1051	−2.78774
$670$	17.3205	0.669150
$671$	−23.3205	−0.900278
$672$	−38.7846	−1.49615
$673$	23.4641	0.904475	0.452237	−	0.891898i	$-0.350626\pi$
$673$	23.4641	0.904475	0.452237	+	0.891898i	$0.350626\pi$
$674$	11.7513	0.452643
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	2.78461	0.107021	0.0535106	−	0.998567i	$-0.482959\pi$
$677$	2.78461	0.107021	0.0535106	+	0.998567i	$0.482959\pi$
$678$	81.9615	3.14771
$679$	13.4641	0.516705
$680$	−1.73205	−0.0664211
$681$	62.1051	2.37987
$682$	−26.7846	−1.02564
$683$	−30.8372	−1.17995	−0.589976	−	0.807421i	$-0.700863\pi$
$683$	−30.8372	−1.17995	−0.589976	+	0.807421i	$0.700863\pi$
$684$	−6.53590	−0.249906
$685$	0	0
$686$	−30.9282	−1.18084
$687$	−22.9282	−0.874766
$688$	2.67949	0.102155
$689$	−24.0000	−0.914327
$690$	38.7846	1.47650
$691$	−38.9808	−1.48290	−0.741449	−	0.671009i	$-0.765861\pi$
$691$	−38.9808	−1.48290	−0.741449	+	0.671009i	$0.765861\pi$
$692$	3.46410	0.131685
$693$	−57.7128	−2.19233
$694$	−6.58846	−0.250094
$695$	−3.66025	−0.138841
$696$	−16.3923	−0.621349
$697$	3.46410	0.131212
$698$	−18.6795	−0.707029
$699$	16.3923	0.620014
$700$	−2.73205	−0.103262
$701$	11.3205	0.427570	0.213785	−	0.976881i	$-0.431421\pi$
$701$	11.3205	0.427570	0.213785	+	0.976881i	$0.431421\pi$
$702$	27.7128	1.04595
$703$	0.784610	0.0295921
$704$	4.73205	0.178346
$705$	35.3205	1.33025
$706$	−46.3923	−1.74600
$707$	25.8564	0.972430
$708$	6.92820	0.260378
$709$	4.53590	0.170349	0.0851746	−	0.996366i	$-0.472855\pi$
$709$	4.53590	0.170349	0.0851746	+	0.996366i	$0.472855\pi$
$710$	−20.1962	−0.757948
$711$	65.1244	2.44235
$712$	7.60770	0.285110
$713$	−26.7846	−1.00309
$714$	−12.9282	−0.483826
$715$	−18.9282	−0.707875
$716$	−11.3205	−0.423067
$717$	−56.7846	−2.12066
$718$	37.1769	1.38743
$719$	−5.41154	−0.201816	−0.100908	−	0.994896i	$-0.532175\pi$
$719$	−5.41154	−0.201816	−0.100908	+	0.994896i	$0.532175\pi$
$720$	−22.3205	−0.831836
$721$	−24.3923	−0.908417
$722$	29.1962	1.08657
$723$	−15.3205	−0.569776
$724$	−2.39230	−0.0889093
$725$	−3.46410	−0.128654
$726$	−53.9090	−2.00075
$727$	0.143594	0.00532559	0.00266279	−	0.999996i	$-0.499152\pi$
$727$	0.143594	0.00532559	0.00266279	+	0.999996i	$0.499152\pi$
$728$	18.9282	0.701526
$729$	−43.7846	−1.62165
$730$	−11.0718	−0.409786
$731$	0.535898	0.0198209
$732$	−13.4641	−0.497648
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	13.5167	0.498909
$735$	1.26795	0.0467690
$736$	−42.5885	−1.56983
$737$	−47.3205	−1.74307
$738$	26.7846	0.985955
$739$	−11.6077	−0.426996	−0.213498	−	0.976944i	$-0.568486\pi$
$739$	−11.6077	−0.426996	−0.213498	+	0.976944i	$0.568486\pi$
$740$	−0.535898	−0.0197000
$741$	16.0000	0.587775
$742$	28.3923	1.04231
$743$	−22.7321	−0.833958	−0.416979	−	0.908916i	$-0.636911\pi$
$743$	−22.7321	−0.833958	−0.416979	+	0.908916i	$0.636911\pi$
$744$	15.4641	0.566941
$745$	−6.00000	−0.219823
$746$	−34.6410	−1.26830
$747$	38.1051	1.39419
$748$	−4.73205	−0.173021
$749$	−48.2487	−1.76297
$750$	−4.73205	−0.172790
$751$	43.6603	1.59319	0.796593	−	0.604516i	$-0.206634\pi$
$751$	43.6603	1.59319	0.796593	+	0.604516i	$0.206634\pi$
$752$	−64.6410	−2.35722
$753$	18.9282	0.689782
$754$	−24.0000	−0.874028
$755$	−1.46410	−0.0532841
$756$	−10.9282	−0.397455
$757$	30.6410	1.11367	0.556833	−	0.830624i	$-0.312016\pi$
$757$	30.6410	1.11367	0.556833	+	0.830624i	$0.312016\pi$
$758$	−30.8372	−1.12006
$759$	−105.962	−3.84616
$760$	−2.53590	−0.0919867
$761$	−28.3923	−1.02922	−0.514610	−	0.857424i	$-0.672063\pi$
$761$	−28.3923	−1.02922	−0.514610	+	0.857424i	$0.672063\pi$
$762$	68.1051	2.46719
$763$	27.3205	0.989069
$764$	1.85641	0.0671624
$765$	−4.46410	−0.161400
$766$	−26.7846	−0.967767
$767$	−10.1436	−0.366264
$768$	51.9090	1.87310
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	22.3923	0.806963
$771$	−18.9282	−0.681683
$772$	16.5359	0.595140
$773$	−46.6410	−1.67756	−0.838780	−	0.544470i	$-0.816731\pi$
$773$	−46.6410	−1.67756	−0.838780	+	0.544470i	$0.816731\pi$
$774$	4.14359	0.148938
$775$	3.26795	0.117388
$776$	−8.53590	−0.306421
$777$	4.00000	0.143499
$778$	7.60770	0.272749
$779$	5.07180	0.181716
$780$	−10.9282	−0.391292
$781$	55.1769	1.97439
$782$	−14.1962	−0.507653
$783$	−13.8564	−0.495188
$784$	−2.32051	−0.0828753
$785$	8.92820	0.318661
$786$	10.3923	0.370681
$787$	43.9090	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$787$	43.9090	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$788$	17.3205	0.617018
$789$	−4.39230	−0.156370
$790$	−25.2679	−0.898993
$791$	47.3205	1.68252
$792$	36.5885	1.30011
$793$	19.7128	0.700023
$794$	−24.2487	−0.860555
$795$	16.3923	0.581375
$796$	10.1962	0.361393
$797$	−24.9282	−0.883002	−0.441501	−	0.897261i	$-0.645554\pi$
$797$	−24.9282	−0.883002	−0.441501	+	0.897261i	$0.645554\pi$
$798$	−18.9282	−0.670051
$799$	−12.9282	−0.457367
$800$	5.19615	0.183712
$801$	19.6077	0.692804
$802$	−22.3923	−0.790700
$803$	30.2487	1.06745
$804$	−27.3205	−0.963520
$805$	22.3923	0.789225
$806$	22.6410	0.797496
$807$	2.53590	0.0892679
$808$	−16.3923	−0.576679
$809$	−55.8564	−1.96381	−0.981903	−	0.189383i	$-0.939351\pi$
$809$	−55.8564	−1.96381	−0.981903	+	0.189383i	$0.939351\pi$
$810$	4.26795	0.149960
$811$	44.8372	1.57445	0.787223	−	0.616668i	$-0.211518\pi$
$811$	44.8372	1.57445	0.787223	+	0.616668i	$0.211518\pi$
$812$	9.46410	0.332125
$813$	8.00000	0.280572
$814$	4.39230	0.153950
$815$	−0.196152	−0.00687092
$816$	13.6603	0.478205
$817$	0.784610	0.0274500
$818$	−45.0333	−1.57455
$819$	48.7846	1.70467
$820$	−3.46410	−0.120972
$821$	−24.9282	−0.870000	−0.435000	−	0.900430i	$-0.643252\pi$
$821$	−24.9282	−0.870000	−0.435000	+	0.900430i	$0.643252\pi$
$822$	0	0
$823$	−8.98076	−0.313050	−0.156525	−	0.987674i	$-0.550029\pi$
$823$	−8.98076	−0.313050	−0.156525	+	0.987674i	$0.550029\pi$
$824$	15.4641	0.538718
$825$	12.9282	0.450102
$826$	12.0000	0.417533
$827$	−15.8038	−0.549554	−0.274777	−	0.961508i	$-0.588604\pi$
$827$	−15.8038	−0.549554	−0.274777	+	0.961508i	$0.588604\pi$
$828$	−36.5885	−1.27154
$829$	17.7128	0.615191	0.307596	−	0.951517i	$-0.400476\pi$
$829$	17.7128	0.615191	0.307596	+	0.951517i	$0.400476\pi$
$830$	−14.7846	−0.513181
$831$	57.1769	1.98345
$832$	−4.00000	−0.138675
$833$	−0.464102	−0.0160802
$834$	17.3205	0.599760
$835$	12.5885	0.435642
$836$	−6.92820	−0.239617
$837$	13.0718	0.451827
$838$	−66.1577	−2.28538
$839$	19.2679	0.665203	0.332602	−	0.943067i	$-0.392074\pi$
$839$	19.2679	0.665203	0.332602	+	0.943067i	$0.392074\pi$
$840$	−12.9282	−0.446065
$841$	−17.0000	−0.586207
$842$	−9.46410	−0.326154
$843$	−35.3205	−1.21650
$844$	10.1962	0.350966
$845$	3.00000	0.103203
$846$	−99.9615	−3.43675
$847$	−31.1244	−1.06945
$848$	−30.0000	−1.03020
$849$	−14.3923	−0.493943
$850$	1.73205	0.0594089
$851$	4.39230	0.150566
$852$	31.8564	1.09138
$853$	−23.1769	−0.793562	−0.396781	−	0.917913i	$-0.629873\pi$
$853$	−23.1769	−0.793562	−0.396781	+	0.917913i	$0.629873\pi$
$854$	−23.3205	−0.798011
$855$	−6.53590	−0.223523
$856$	30.5885	1.04549
$857$	31.1769	1.06498	0.532492	−	0.846435i	$-0.321256\pi$
$857$	31.1769	1.06498	0.532492	+	0.846435i	$0.321256\pi$
$858$	89.5692	3.05784
$859$	−25.4641	−0.868824	−0.434412	−	0.900714i	$-0.643044\pi$
$859$	−25.4641	−0.868824	−0.434412	+	0.900714i	$0.643044\pi$
$860$	−0.535898	−0.0182740
$861$	25.8564	0.881184
$862$	−16.9808	−0.578367
$863$	23.0718	0.785373	0.392687	−	0.919672i	$-0.371546\pi$
$863$	23.0718	0.785373	0.392687	+	0.919672i	$0.371546\pi$
$864$	20.7846	0.707107
$865$	3.46410	0.117783
$866$	30.9282	1.05098
$867$	2.73205	0.0927853
$868$	−8.92820	−0.303043
$869$	69.0333	2.34180
$870$	16.3923	0.555751
$871$	40.0000	1.35535
$872$	−17.3205	−0.586546
$873$	−22.0000	−0.744587
$874$	−20.7846	−0.703050
$875$	−2.73205	−0.0923602
$876$	17.4641	0.590057
$877$	−1.21539	−0.0410408	−0.0205204	−	0.999789i	$-0.506532\pi$
$877$	−1.21539	−0.0410408	−0.0205204	+	0.999789i	$0.506532\pi$
$878$	34.7321	1.17215
$879$	−2.53590	−0.0855337
$880$	−23.6603	−0.797587
$881$	−41.3205	−1.39212	−0.696062	−	0.717982i	$-0.745066\pi$
$881$	−41.3205	−1.39212	−0.696062	+	0.717982i	$0.745066\pi$
$882$	−3.58846	−0.120830
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	4.00000	0.134535
$885$	6.92820	0.232889
$886$	22.3923	0.752284
$887$	3.12436	0.104906	0.0524528	−	0.998623i	$-0.483296\pi$
$887$	3.12436	0.104906	0.0524528	+	0.998623i	$0.483296\pi$
$888$	−2.53590	−0.0850992
$889$	39.3205	1.31877
$890$	−7.60770	−0.255011
$891$	−11.6603	−0.390633
$892$	−26.3923	−0.883680
$893$	−18.9282	−0.633408
$894$	28.3923	0.949581
$895$	−11.3205	−0.378403
$896$	33.1244	1.10661
$897$	89.5692	2.99063
$898$	59.5692	1.98785
$899$	−11.3205	−0.377560
$900$	4.46410	0.148803
$901$	−6.00000	−0.199889
$902$	28.3923	0.945360
$903$	4.00000	0.133112
$904$	−30.0000	−0.997785
$905$	−2.39230	−0.0795229
$906$	6.92820	0.230174
$907$	30.7321	1.02044	0.510220	−	0.860044i	$-0.329564\pi$
$907$	30.7321	1.02044	0.510220	+	0.860044i	$0.329564\pi$
$908$	22.7321	0.754390
$909$	−42.2487	−1.40130
$910$	−18.9282	−0.627464
$911$	36.3397	1.20399	0.601995	−	0.798500i	$-0.294373\pi$
$911$	36.3397	1.20399	0.601995	+	0.798500i	$0.294373\pi$
$912$	20.0000	0.662266
$913$	40.3923	1.33679
$914$	63.7128	2.10743
$915$	−13.4641	−0.445109
$916$	−8.39230	−0.277290
$917$	6.00000	0.198137
$918$	6.92820	0.228665
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−14.1962	−0.468033
$921$	−27.3205	−0.900241
$922$	43.1769	1.42196
$923$	−46.6410	−1.53521
$924$	−35.3205	−1.16196
$925$	−0.535898	−0.0176202
$926$	41.3205	1.35788
$927$	39.8564	1.30906
$928$	−18.0000	−0.590879
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	−15.4641	−0.507088
$931$	−0.679492	−0.0222694
$932$	6.00000	0.196537
$933$	−43.8564	−1.43579
$934$	−2.78461	−0.0911152
$935$	−4.73205	−0.154755
$936$	−30.9282	−1.01092
$937$	−32.6410	−1.06634	−0.533168	−	0.846010i	$-0.678999\pi$
$937$	−32.6410	−1.06634	−0.533168	+	0.846010i	$0.678999\pi$
$938$	−47.3205	−1.54507
$939$	−72.1051	−2.35306
$940$	12.9282	0.421671
$941$	48.2487	1.57286	0.786432	−	0.617677i	$-0.211926\pi$
$941$	48.2487	1.57286	0.786432	+	0.617677i	$0.211926\pi$
$942$	−42.2487	−1.37654
$943$	28.3923	0.924581
$944$	−12.6795	−0.412682
$945$	−10.9282	−0.355494
$946$	4.39230	0.142806
$947$	8.19615	0.266339	0.133170	−	0.991093i	$-0.457485\pi$
$947$	8.19615	0.266339	0.133170	+	0.991093i	$0.457485\pi$
$948$	39.8564	1.29448
$949$	−25.5692	−0.830012
$950$	2.53590	0.0822754
$951$	−68.1051	−2.20846
$952$	4.73205	0.153367
$953$	−8.78461	−0.284561	−0.142281	−	0.989826i	$-0.545444\pi$
$953$	−8.78461	−0.284561	−0.142281	+	0.989826i	$0.545444\pi$
$954$	−46.3923	−1.50201
$955$	1.85641	0.0600719
$956$	−20.7846	−0.672222
$957$	−44.7846	−1.44768
$958$	20.1962	0.652508
$959$	0	0
$960$	2.73205	0.0881766
$961$	−20.3205	−0.655500
$962$	−3.71281	−0.119706
$963$	78.8372	2.54049
$964$	−5.60770	−0.180612
$965$	16.5359	0.532309
$966$	−105.962	−3.40926
$967$	39.1769	1.25984	0.629922	−	0.776658i	$-0.283087\pi$
$967$	39.1769	1.25984	0.629922	+	0.776658i	$0.283087\pi$
$968$	19.7321	0.634212
$969$	4.00000	0.128499
$970$	8.53590	0.274071
$971$	54.2487	1.74092	0.870462	−	0.492236i	$-0.163820\pi$
$971$	54.2487	1.74092	0.870462	+	0.492236i	$0.163820\pi$
$972$	−18.7321	−0.600831
$973$	10.0000	0.320585
$974$	−43.2679	−1.38639
$975$	−10.9282	−0.349983
$976$	24.6410	0.788740
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0.928203	0.0296807
$979$	20.7846	0.664279
$980$	0.464102	0.0148252
$981$	−44.6410	−1.42528
$982$	33.9615	1.08376
$983$	58.7321	1.87326	0.936631	−	0.350318i	$-0.113927\pi$
$983$	58.7321	1.87326	0.936631	+	0.350318i	$0.113927\pi$
$984$	−16.3923	−0.522568
$985$	17.3205	0.551877
$986$	−6.00000	−0.191079
$987$	−96.4974	−3.07155
$988$	5.85641	0.186317
$989$	4.39230	0.139667
$990$	−36.5885	−1.16286
$991$	−2.98076	−0.0946870	−0.0473435	−	0.998879i	$-0.515076\pi$
$991$	−2.98076	−0.0946870	−0.0473435	+	0.998879i	$0.515076\pi$
$992$	16.9808	0.539140
$993$	−17.8564	−0.566656
$994$	55.1769	1.75011
$995$	10.1962	0.323240
$996$	23.3205	0.738939
$997$	−2.39230	−0.0757651	−0.0378825	−	0.999282i	$-0.512061\pi$
$997$	−2.39230	−0.0757651	−0.0378825	+	0.999282i	$0.512061\pi$
$998$	27.1244	0.858606
$999$	−2.14359	−0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.a.c.1.1	✓	2
3.2	odd	2		765.2.a.g.1.2		2
4.3	odd	2		1360.2.a.k.1.1		2
5.2	odd	4		425.2.b.d.324.1		4
5.3	odd	4		425.2.b.d.324.4		4
5.4	even	2		425.2.a.e.1.2		2
7.6	odd	2		4165.2.a.t.1.1		2
8.3	odd	2		5440.2.a.bl.1.2		2
8.5	even	2		5440.2.a.bb.1.1		2
15.14	odd	2		3825.2.a.v.1.1		2
17.4	even	4		1445.2.d.e.866.3		4
17.13	even	4		1445.2.d.e.866.4		4
17.16	even	2		1445.2.a.g.1.1		2
20.19	odd	2		6800.2.a.bg.1.2		2
85.84	even	2		7225.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	1.1	even	1	trivial
425.2.a.e.1.2		2	5.4	even	2
425.2.b.d.324.1		4	5.2	odd	4
425.2.b.d.324.4		4	5.3	odd	4
765.2.a.g.1.2		2	3.2	odd	2
1360.2.a.k.1.1		2	4.3	odd	2
1445.2.a.g.1.1		2	17.16	even	2
1445.2.d.e.866.3		4	17.4	even	4
1445.2.d.e.866.4		4	17.13	even	4
3825.2.a.v.1.1		2	15.14	odd	2
4165.2.a.t.1.1		2	7.6	odd	2
5440.2.a.bb.1.1		2	8.5	even	2
5440.2.a.bl.1.2		2	8.3	odd	2
6800.2.a.bg.1.2		2	20.19	odd	2
7225.2.a.l.1.2		2	85.84	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -4.73205 q^{6} -2.73205 q^{7} +1.73205 q^{8} +4.46410 q^{9} -1.73205 q^{10} +4.73205 q^{11} +2.73205 q^{12} -4.00000 q^{13} +4.73205 q^{14} +2.73205 q^{15} -5.00000 q^{16} -1.00000 q^{17} -7.73205 q^{18} -1.46410 q^{19} +1.00000 q^{20} -7.46410 q^{21} -8.19615 q^{22} -8.19615 q^{23} +4.73205 q^{24} +1.00000 q^{25} +6.92820 q^{26} +4.00000 q^{27} -2.73205 q^{28} -3.46410 q^{29} -4.73205 q^{30} +3.26795 q^{31} +5.19615 q^{32} +12.9282 q^{33} +1.73205 q^{34} -2.73205 q^{35} +4.46410 q^{36} -0.535898 q^{37} +2.53590 q^{38} -10.9282 q^{39} +1.73205 q^{40} -3.46410 q^{41} +12.9282 q^{42} -0.535898 q^{43} +4.73205 q^{44} +4.46410 q^{45} +14.1962 q^{46} +12.9282 q^{47} -13.6603 q^{48} +0.464102 q^{49} -1.73205 q^{50} -2.73205 q^{51} -4.00000 q^{52} +6.00000 q^{53} -6.92820 q^{54} +4.73205 q^{55} -4.73205 q^{56} -4.00000 q^{57} +6.00000 q^{58} +2.53590 q^{59} +2.73205 q^{60} -4.92820 q^{61} -5.66025 q^{62} -12.1962 q^{63} +1.00000 q^{64} -4.00000 q^{65} -22.3923 q^{66} -10.0000 q^{67} -1.00000 q^{68} -22.3923 q^{69} +4.73205 q^{70} +11.6603 q^{71} +7.73205 q^{72} +6.39230 q^{73} +0.928203 q^{74} +2.73205 q^{75} -1.46410 q^{76} -12.9282 q^{77} +18.9282 q^{78} +14.5885 q^{79} -5.00000 q^{80} -2.46410 q^{81} +6.00000 q^{82} +8.53590 q^{83} -7.46410 q^{84} -1.00000 q^{85} +0.928203 q^{86} -9.46410 q^{87} +8.19615 q^{88} +4.39230 q^{89} -7.73205 q^{90} +10.9282 q^{91} -8.19615 q^{92} +8.92820 q^{93} -22.3923 q^{94} -1.46410 q^{95} +14.1962 q^{96} -4.92820 q^{97} -0.803848 q^{98} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 8 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 6 q^{22} - 6 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 6 * q^6 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 8 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 - 2 * q^17 - 12 * q^18 + 4 * q^19 + 2 * q^20 - 8 * q^21 - 6 * q^22 - 6 * q^23 + 6 * q^24 + 2 * q^25 + 8 * q^27 - 2 * q^28 - 6 * q^30 + 10 * q^31 + 12 * q^33 - 2 * q^35 + 2 * q^36 - 8 * q^37 + 12 * q^38 - 8 * q^39 + 12 * q^42 - 8 * q^43 + 6 * q^44 + 2 * q^45 + 18 * q^46 + 12 * q^47 - 10 * q^48 - 6 * q^49 - 2 * q^51 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 8 * q^57 + 12 * q^58 + 12 * q^59 + 2 * q^60 + 4 * q^61 + 6 * q^62 - 14 * q^63 + 2 * q^64 - 8 * q^65 - 24 * q^66 - 20 * q^67 - 2 * q^68 - 24 * q^69 + 6 * q^70 + 6 * q^71 + 12 * q^72 - 8 * q^73 - 12 * q^74 + 2 * q^75 + 4 * q^76 - 12 * q^77 + 24 * q^78 - 2 * q^79 - 10 * q^80 + 2 * q^81 + 12 * q^82 + 24 * q^83 - 8 * q^84 - 2 * q^85 - 12 * q^86 - 12 * q^87 + 6 * q^88 - 12 * q^89 - 12 * q^90 + 8 * q^91 - 6 * q^92 + 4 * q^93 - 24 * q^94 + 4 * q^95 + 18 * q^96 + 4 * q^97 - 12 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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