LMFDB - Embedded newform 4017.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4017,2,Mod(1,4017)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4017, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4017.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4017 = 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4017.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:	$32.0759064919$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4017.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -3.41421 q^{5} +2.41421 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -3.41421 q^{5} +2.41421 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +8.24264 q^{10} +1.41421 q^{11} -3.82843 q^{12} +1.00000 q^{13} -6.82843 q^{14} +3.41421 q^{15} +3.00000 q^{16} +1.17157 q^{17} -2.41421 q^{18} +5.65685 q^{19} -13.0711 q^{20} -2.82843 q^{21} -3.41421 q^{22} -3.65685 q^{23} +4.41421 q^{24} +6.65685 q^{25} -2.41421 q^{26} -1.00000 q^{27} +10.8284 q^{28} -7.65685 q^{29} -8.24264 q^{30} -7.41421 q^{31} +1.58579 q^{32} -1.41421 q^{33} -2.82843 q^{34} -9.65685 q^{35} +3.82843 q^{36} -3.07107 q^{37} -13.6569 q^{38} -1.00000 q^{39} +15.0711 q^{40} +6.82843 q^{41} +6.82843 q^{42} +2.82843 q^{43} +5.41421 q^{44} -3.41421 q^{45} +8.82843 q^{46} -7.07107 q^{47} -3.00000 q^{48} +1.00000 q^{49} -16.0711 q^{50} -1.17157 q^{51} +3.82843 q^{52} +7.65685 q^{53} +2.41421 q^{54} -4.82843 q^{55} -12.4853 q^{56} -5.65685 q^{57} +18.4853 q^{58} -6.00000 q^{59} +13.0711 q^{60} +4.00000 q^{61} +17.8995 q^{62} +2.82843 q^{63} -9.82843 q^{64} -3.41421 q^{65} +3.41421 q^{66} -7.41421 q^{67} +4.48528 q^{68} +3.65685 q^{69} +23.3137 q^{70} -4.24264 q^{71} -4.41421 q^{72} -16.2426 q^{73} +7.41421 q^{74} -6.65685 q^{75} +21.6569 q^{76} +4.00000 q^{77} +2.41421 q^{78} +10.4853 q^{79} -10.2426 q^{80} +1.00000 q^{81} -16.4853 q^{82} +4.82843 q^{83} -10.8284 q^{84} -4.00000 q^{85} -6.82843 q^{86} +7.65685 q^{87} -6.24264 q^{88} +14.2426 q^{89} +8.24264 q^{90} +2.82843 q^{91} -14.0000 q^{92} +7.41421 q^{93} +17.0711 q^{94} -19.3137 q^{95} -1.58579 q^{96} +11.6569 q^{97} -2.41421 q^{98} +1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{13} - 8 q^{14} + 4 q^{15} + 6 q^{16} + 8 q^{17} - 2 q^{18} - 12 q^{20} - 4 q^{22} + 4 q^{23} + 6 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 16 q^{28} - 4 q^{29} - 8 q^{30} - 12 q^{31} + 6 q^{32} - 8 q^{35} + 2 q^{36} + 8 q^{37} - 16 q^{38} - 2 q^{39} + 16 q^{40} + 8 q^{41} + 8 q^{42} + 8 q^{44} - 4 q^{45} + 12 q^{46} - 6 q^{48} + 2 q^{49} - 18 q^{50} - 8 q^{51} + 2 q^{52} + 4 q^{53} + 2 q^{54} - 4 q^{55} - 8 q^{56} + 20 q^{58} - 12 q^{59} + 12 q^{60} + 8 q^{61} + 16 q^{62} - 14 q^{64} - 4 q^{65} + 4 q^{66} - 12 q^{67} - 8 q^{68} - 4 q^{69} + 24 q^{70} - 6 q^{72} - 24 q^{73} + 12 q^{74} - 2 q^{75} + 32 q^{76} + 8 q^{77} + 2 q^{78} + 4 q^{79} - 12 q^{80} + 2 q^{81} - 16 q^{82} + 4 q^{83} - 16 q^{84} - 8 q^{85} - 8 q^{86} + 4 q^{87} - 4 q^{88} + 20 q^{89} + 8 q^{90} - 28 q^{92} + 12 q^{93} + 20 q^{94} - 16 q^{95} - 6 q^{96} + 12 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 8 * q^10 - 2 * q^12 + 2 * q^13 - 8 * q^14 + 4 * q^15 + 6 * q^16 + 8 * q^17 - 2 * q^18 - 12 * q^20 - 4 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 16 * q^28 - 4 * q^29 - 8 * q^30 - 12 * q^31 + 6 * q^32 - 8 * q^35 + 2 * q^36 + 8 * q^37 - 16 * q^38 - 2 * q^39 + 16 * q^40 + 8 * q^41 + 8 * q^42 + 8 * q^44 - 4 * q^45 + 12 * q^46 - 6 * q^48 + 2 * q^49 - 18 * q^50 - 8 * q^51 + 2 * q^52 + 4 * q^53 + 2 * q^54 - 4 * q^55 - 8 * q^56 + 20 * q^58 - 12 * q^59 + 12 * q^60 + 8 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^65 + 4 * q^66 - 12 * q^67 - 8 * q^68 - 4 * q^69 + 24 * q^70 - 6 * q^72 - 24 * q^73 + 12 * q^74 - 2 * q^75 + 32 * q^76 + 8 * q^77 + 2 * q^78 + 4 * q^79 - 12 * q^80 + 2 * q^81 - 16 * q^82 + 4 * q^83 - 16 * q^84 - 8 * q^85 - 8 * q^86 + 4 * q^87 - 4 * q^88 + 20 * q^89 + 8 * q^90 - 28 * q^92 + 12 * q^93 + 20 * q^94 - 16 * q^95 - 6 * q^96 + 12 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.82843	1.91421
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	2.41421	0.985599
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	8.24264	2.60655
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	−3.82843	−1.10517
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−6.82843	−1.82497
$15$	3.41421	0.881546
$16$	3.00000	0.750000
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	−2.41421	−0.569036
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	−13.0711	−2.92278
$21$	−2.82843	−0.617213
$22$	−3.41421	−0.727913
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	4.41421	0.901048
$25$	6.65685	1.33137
$26$	−2.41421	−0.473466
$27$	−1.00000	−0.192450
$28$	10.8284	2.04638
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	−8.24264	−1.50489
$31$	−7.41421	−1.33163	−0.665816	−	0.746116i	$-0.731916\pi$
$31$	−7.41421	−1.33163	−0.665816	+	0.746116i	$0.731916\pi$
$32$	1.58579	0.280330
$33$	−1.41421	−0.246183
$34$	−2.82843	−0.485071
$35$	−9.65685	−1.63231
$36$	3.82843	0.638071
$37$	−3.07107	−0.504880	−0.252440	−	0.967612i	$-0.581233\pi$
$37$	−3.07107	−0.504880	−0.252440	+	0.967612i	$0.581233\pi$
$38$	−13.6569	−2.21543
$39$	−1.00000	−0.160128
$40$	15.0711	2.38295
$41$	6.82843	1.06642	0.533211	−	0.845983i	$-0.320985\pi$
$41$	6.82843	1.06642	0.533211	+	0.845983i	$0.320985\pi$
$42$	6.82843	1.05365
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	5.41421	0.816223
$45$	−3.41421	−0.508961
$46$	8.82843	1.30168
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	−3.00000	−0.433013
$49$	1.00000	0.142857
$50$	−16.0711	−2.27279
$51$	−1.17157	−0.164053
$52$	3.82843	0.530907
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	2.41421	0.328533
$55$	−4.82843	−0.651065
$56$	−12.4853	−1.66842
$57$	−5.65685	−0.749269
$58$	18.4853	2.42724
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	13.0711	1.68747
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	17.8995	2.27324
$63$	2.82843	0.356348
$64$	−9.82843	−1.22855
$65$	−3.41421	−0.423481
$66$	3.41421	0.420261
$67$	−7.41421	−0.905790	−0.452895	−	0.891564i	$-0.649609\pi$
$67$	−7.41421	−0.905790	−0.452895	+	0.891564i	$0.649609\pi$
$68$	4.48528	0.543920
$69$	3.65685	0.440234
$70$	23.3137	2.78652
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	−4.41421	−0.520220
$73$	−16.2426	−1.90106	−0.950529	−	0.310637i	$-0.899458\pi$
$73$	−16.2426	−1.90106	−0.950529	+	0.310637i	$0.899458\pi$
$74$	7.41421	0.861885
$75$	−6.65685	−0.768667
$76$	21.6569	2.48421
$77$	4.00000	0.455842
$78$	2.41421	0.273356
$79$	10.4853	1.17969	0.589843	−	0.807518i	$-0.299190\pi$
$79$	10.4853	1.17969	0.589843	+	0.807518i	$0.299190\pi$
$80$	−10.2426	−1.14516
$81$	1.00000	0.111111
$82$	−16.4853	−1.82049
$83$	4.82843	0.529989	0.264994	−	0.964250i	$-0.414630\pi$
$83$	4.82843	0.529989	0.264994	+	0.964250i	$0.414630\pi$
$84$	−10.8284	−1.18148
$85$	−4.00000	−0.433861
$86$	−6.82843	−0.736328
$87$	7.65685	0.820901
$88$	−6.24264	−0.665468
$89$	14.2426	1.50972	0.754858	−	0.655888i	$-0.227706\pi$
$89$	14.2426	1.50972	0.754858	+	0.655888i	$0.227706\pi$
$90$	8.24264	0.868851
$91$	2.82843	0.296500
$92$	−14.0000	−1.45960
$93$	7.41421	0.768818
$94$	17.0711	1.76075
$95$	−19.3137	−1.98154
$96$	−1.58579	−0.161849
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	−2.41421	−0.243872
$99$	1.41421	0.142134
$100$	25.4853	2.54853
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	2.82843	0.280056
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−4.41421	−0.432849
$105$	9.65685	0.942412
$106$	−18.4853	−1.79545
$107$	13.6569	1.32026	0.660129	−	0.751152i	$-0.270502\pi$
$107$	13.6569	1.32026	0.660129	+	0.751152i	$0.270502\pi$
$108$	−3.82843	−0.368391
$109$	−16.7279	−1.60224	−0.801122	−	0.598501i	$-0.795763\pi$
$109$	−16.7279	−1.60224	−0.801122	+	0.598501i	$0.795763\pi$
$110$	11.6569	1.11144
$111$	3.07107	0.291493
$112$	8.48528	0.801784
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	13.6569	1.27908
$115$	12.4853	1.16426
$116$	−29.3137	−2.72171
$117$	1.00000	0.0924500
$118$	14.4853	1.33348
$119$	3.31371	0.303767
$120$	−15.0711	−1.37579
$121$	−9.00000	−0.818182
$122$	−9.65685	−0.874291
$123$	−6.82843	−0.615699
$124$	−28.3848	−2.54903
$125$	−5.65685	−0.505964
$126$	−6.82843	−0.608325
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	20.5563	1.81694
$129$	−2.82843	−0.249029
$130$	8.24264	0.722927
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	−5.41421	−0.471247
$133$	16.0000	1.38738
$134$	17.8995	1.54628
$135$	3.41421	0.293849
$136$	−5.17157	−0.443459
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	−8.82843	−0.751526
$139$	20.1421	1.70843	0.854217	−	0.519917i	$-0.174037\pi$
$139$	20.1421	1.70843	0.854217	+	0.519917i	$0.174037\pi$
$140$	−36.9706	−3.12458
$141$	7.07107	0.595491
$142$	10.2426	0.859543
$143$	1.41421	0.118262
$144$	3.00000	0.250000
$145$	26.1421	2.17099
$146$	39.2132	3.24531
$147$	−1.00000	−0.0824786
$148$	−11.7574	−0.966449
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	16.0711	1.31220
$151$	2.72792	0.221995	0.110998	−	0.993821i	$-0.464595\pi$
$151$	2.72792	0.221995	0.110998	+	0.993821i	$0.464595\pi$
$152$	−24.9706	−2.02538
$153$	1.17157	0.0947161
$154$	−9.65685	−0.778171
$155$	25.3137	2.03325
$156$	−3.82843	−0.306519
$157$	−16.1421	−1.28828	−0.644141	−	0.764906i	$-0.722785\pi$
$157$	−16.1421	−1.28828	−0.644141	+	0.764906i	$0.722785\pi$
$158$	−25.3137	−2.01385
$159$	−7.65685	−0.607228
$160$	−5.41421	−0.428031
$161$	−10.3431	−0.815154
$162$	−2.41421	−0.189679
$163$	5.65685	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$163$	5.65685	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$164$	26.1421	2.04136
$165$	4.82843	0.375893
$166$	−11.6569	−0.904747
$167$	−24.1421	−1.86817	−0.934087	−	0.357045i	$-0.883784\pi$
$167$	−24.1421	−1.86817	−0.934087	+	0.357045i	$0.883784\pi$
$168$	12.4853	0.963260
$169$	1.00000	0.0769231
$170$	9.65685	0.740647
$171$	5.65685	0.432590
$172$	10.8284	0.825660
$173$	−25.7990	−1.96146	−0.980730	−	0.195366i	$-0.937411\pi$
$173$	−25.7990	−1.96146	−0.980730	+	0.195366i	$0.937411\pi$
$174$	−18.4853	−1.40137
$175$	18.8284	1.42330
$176$	4.24264	0.319801
$177$	6.00000	0.450988
$178$	−34.3848	−2.57725
$179$	13.3137	0.995113	0.497557	−	0.867431i	$-0.334231\pi$
$179$	13.3137	0.995113	0.497557	+	0.867431i	$0.334231\pi$
$180$	−13.0711	−0.974260
$181$	15.6569	1.16376	0.581882	−	0.813273i	$-0.302316\pi$
$181$	15.6569	1.16376	0.581882	+	0.813273i	$0.302316\pi$
$182$	−6.82843	−0.506157
$183$	−4.00000	−0.295689
$184$	16.1421	1.19001
$185$	10.4853	0.770893
$186$	−17.8995	−1.31245
$187$	1.65685	0.121161
$188$	−27.0711	−1.97436
$189$	−2.82843	−0.205738
$190$	46.6274	3.38271
$191$	24.9706	1.80681	0.903403	−	0.428792i	$-0.141061\pi$
$191$	24.9706	1.80681	0.903403	+	0.428792i	$0.141061\pi$
$192$	9.82843	0.709306
$193$	8.72792	0.628250	0.314125	−	0.949382i	$-0.398289\pi$
$193$	8.72792	0.628250	0.314125	+	0.949382i	$0.398289\pi$
$194$	−28.1421	−2.02049
$195$	3.41421	0.244497
$196$	3.82843	0.273459
$197$	7.89949	0.562816	0.281408	−	0.959588i	$-0.409199\pi$
$197$	7.89949	0.562816	0.281408	+	0.959588i	$0.409199\pi$
$198$	−3.41421	−0.242638
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−29.3848	−2.07782
$201$	7.41421	0.522958
$202$	−18.4853	−1.30062
$203$	−21.6569	−1.52001
$204$	−4.48528	−0.314033
$205$	−23.3137	−1.62830
$206$	−2.41421	−0.168206
$207$	−3.65685	−0.254169
$208$	3.00000	0.208013
$209$	8.00000	0.553372
$210$	−23.3137	−1.60880
$211$	10.8284	0.745460	0.372730	−	0.927940i	$-0.378422\pi$
$211$	10.8284	0.745460	0.372730	+	0.927940i	$0.378422\pi$
$212$	29.3137	2.01327
$213$	4.24264	0.290701
$214$	−32.9706	−2.25382
$215$	−9.65685	−0.658592
$216$	4.41421	0.300349
$217$	−20.9706	−1.42357
$218$	40.3848	2.73520
$219$	16.2426	1.09758
$220$	−18.4853	−1.24628
$221$	1.17157	0.0788085
$222$	−7.41421	−0.497609
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	4.48528	0.299685
$225$	6.65685	0.443790
$226$	24.1421	1.60591
$227$	−16.2426	−1.07806	−0.539031	−	0.842286i	$-0.681209\pi$
$227$	−16.2426	−1.07806	−0.539031	+	0.842286i	$0.681209\pi$
$228$	−21.6569	−1.43426
$229$	−6.68629	−0.441843	−0.220921	−	0.975292i	$-0.570906\pi$
$229$	−6.68629	−0.441843	−0.220921	+	0.975292i	$0.570906\pi$
$230$	−30.1421	−1.98751
$231$	−4.00000	−0.263181
$232$	33.7990	2.21901
$233$	−28.6274	−1.87544	−0.937722	−	0.347386i	$-0.887069\pi$
$233$	−28.6274	−1.87544	−0.937722	+	0.347386i	$0.887069\pi$
$234$	−2.41421	−0.157822
$235$	24.1421	1.57486
$236$	−22.9706	−1.49526
$237$	−10.4853	−0.681092
$238$	−8.00000	−0.518563
$239$	12.3431	0.798412	0.399206	−	0.916861i	$-0.369286\pi$
$239$	12.3431	0.798412	0.399206	+	0.916861i	$0.369286\pi$
$240$	10.2426	0.661160
$241$	−20.7279	−1.33520	−0.667601	−	0.744519i	$-0.732679\pi$
$241$	−20.7279	−1.33520	−0.667601	+	0.744519i	$0.732679\pi$
$242$	21.7279	1.39672
$243$	−1.00000	−0.0641500
$244$	15.3137	0.980360
$245$	−3.41421	−0.218126
$246$	16.4853	1.05106
$247$	5.65685	0.359937
$248$	32.7279	2.07823
$249$	−4.82843	−0.305989
$250$	13.6569	0.863735
$251$	24.4853	1.54550	0.772749	−	0.634712i	$-0.218881\pi$
$251$	24.4853	1.54550	0.772749	+	0.634712i	$0.218881\pi$
$252$	10.8284	0.682127
$253$	−5.17157	−0.325134
$254$	0	0
$255$	4.00000	0.250490
$256$	−29.9706	−1.87316
$257$	30.9706	1.93189	0.965945	−	0.258746i	$-0.0833094\pi$
$257$	30.9706	1.93189	0.965945	+	0.258746i	$0.0833094\pi$
$258$	6.82843	0.425119
$259$	−8.68629	−0.539740
$260$	−13.0711	−0.810633
$261$	−7.65685	−0.473947
$262$	14.4853	0.894904
$263$	−1.17157	−0.0722423	−0.0361211	−	0.999347i	$-0.511500\pi$
$263$	−1.17157	−0.0722423	−0.0361211	+	0.999347i	$0.511500\pi$
$264$	6.24264	0.384208
$265$	−26.1421	−1.60590
$266$	−38.6274	−2.36840
$267$	−14.2426	−0.871635
$268$	−28.3848	−1.73388
$269$	2.82843	0.172452	0.0862261	−	0.996276i	$-0.472519\pi$
$269$	2.82843	0.172452	0.0862261	+	0.996276i	$0.472519\pi$
$270$	−8.24264	−0.501631
$271$	−20.5858	−1.25050	−0.625249	−	0.780426i	$-0.715002\pi$
$271$	−20.5858	−1.25050	−0.625249	+	0.780426i	$0.715002\pi$
$272$	3.51472	0.213111
$273$	−2.82843	−0.171184
$274$	19.3137	1.16678
$275$	9.41421	0.567698
$276$	14.0000	0.842701
$277$	3.17157	0.190561	0.0952807	−	0.995450i	$-0.469625\pi$
$277$	3.17157	0.190561	0.0952807	+	0.995450i	$0.469625\pi$
$278$	−48.6274	−2.91648
$279$	−7.41421	−0.443877
$280$	42.6274	2.54748
$281$	−0.100505	−0.00599563	−0.00299781	−	0.999996i	$-0.500954\pi$
$281$	−0.100505	−0.00599563	−0.00299781	+	0.999996i	$0.500954\pi$
$282$	−17.0711	−1.01657
$283$	−10.1421	−0.602887	−0.301444	−	0.953484i	$-0.597469\pi$
$283$	−10.1421	−0.602887	−0.301444	+	0.953484i	$0.597469\pi$
$284$	−16.2426	−0.963823
$285$	19.3137	1.14405
$286$	−3.41421	−0.201887
$287$	19.3137	1.14005
$288$	1.58579	0.0934434
$289$	−15.6274	−0.919260
$290$	−63.1127	−3.70611
$291$	−11.6569	−0.683337
$292$	−62.1838	−3.63903
$293$	−9.55635	−0.558288	−0.279144	−	0.960249i	$-0.590051\pi$
$293$	−9.55635	−0.558288	−0.279144	+	0.960249i	$0.590051\pi$
$294$	2.41421	0.140800
$295$	20.4853	1.19270
$296$	13.5563	0.787947
$297$	−1.41421	−0.0820610
$298$	9.65685	0.559407
$299$	−3.65685	−0.211481
$300$	−25.4853	−1.47139
$301$	8.00000	0.461112
$302$	−6.58579	−0.378969
$303$	−7.65685	−0.439875
$304$	16.9706	0.973329
$305$	−13.6569	−0.781989
$306$	−2.82843	−0.161690
$307$	−21.0711	−1.20259	−0.601295	−	0.799027i	$-0.705348\pi$
$307$	−21.0711	−1.20259	−0.601295	+	0.799027i	$0.705348\pi$
$308$	15.3137	0.872580
$309$	−1.00000	−0.0568880
$310$	−61.1127	−3.47097
$311$	−22.3431	−1.26696	−0.633482	−	0.773758i	$-0.718375\pi$
$311$	−22.3431	−1.26696	−0.633482	+	0.773758i	$0.718375\pi$
$312$	4.41421	0.249906
$313$	6.68629	0.377932	0.188966	−	0.981984i	$-0.439486\pi$
$313$	6.68629	0.377932	0.188966	+	0.981984i	$0.439486\pi$
$314$	38.9706	2.19924
$315$	−9.65685	−0.544102
$316$	40.1421	2.25817
$317$	−25.4558	−1.42974	−0.714871	−	0.699256i	$-0.753515\pi$
$317$	−25.4558	−1.42974	−0.714871	+	0.699256i	$0.753515\pi$
$318$	18.4853	1.03660
$319$	−10.8284	−0.606276
$320$	33.5563	1.87586
$321$	−13.6569	−0.762251
$322$	24.9706	1.39156
$323$	6.62742	0.368759
$324$	3.82843	0.212690
$325$	6.65685	0.369256
$326$	−13.6569	−0.756383
$327$	16.7279	0.925056
$328$	−30.1421	−1.66432
$329$	−20.0000	−1.10264
$330$	−11.6569	−0.641689
$331$	5.07107	0.278731	0.139366	−	0.990241i	$-0.455494\pi$
$331$	5.07107	0.278731	0.139366	+	0.990241i	$0.455494\pi$
$332$	18.4853	1.01451
$333$	−3.07107	−0.168293
$334$	58.2843	3.18917
$335$	25.3137	1.38304
$336$	−8.48528	−0.462910
$337$	12.6274	0.687859	0.343930	−	0.938995i	$-0.388242\pi$
$337$	12.6274	0.687859	0.343930	+	0.938995i	$0.388242\pi$
$338$	−2.41421	−0.131316
$339$	10.0000	0.543125
$340$	−15.3137	−0.830502
$341$	−10.4853	−0.567810
$342$	−13.6569	−0.738478
$343$	−16.9706	−0.916324
$344$	−12.4853	−0.673161
$345$	−12.4853	−0.672185
$346$	62.2843	3.34842
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	29.3137	1.57138
$349$	30.8701	1.65244	0.826218	−	0.563350i	$-0.190488\pi$
$349$	30.8701	1.65244	0.826218	+	0.563350i	$0.190488\pi$
$350$	−45.4558	−2.42972
$351$	−1.00000	−0.0533761
$352$	2.24264	0.119533
$353$	−14.9289	−0.794587	−0.397293	−	0.917692i	$-0.630050\pi$
$353$	−14.9289	−0.794587	−0.397293	+	0.917692i	$0.630050\pi$
$354$	−14.4853	−0.769884
$355$	14.4853	0.768799
$356$	54.5269	2.88992
$357$	−3.31371	−0.175380
$358$	−32.1421	−1.69876
$359$	22.4853	1.18673	0.593364	−	0.804934i	$-0.297800\pi$
$359$	22.4853	1.18673	0.593364	+	0.804934i	$0.297800\pi$
$360$	15.0711	0.794315
$361$	13.0000	0.684211
$362$	−37.7990	−1.98667
$363$	9.00000	0.472377
$364$	10.8284	0.567564
$365$	55.4558	2.90269
$366$	9.65685	0.504772
$367$	−15.1716	−0.791950	−0.395975	−	0.918261i	$-0.629593\pi$
$367$	−15.1716	−0.791950	−0.395975	+	0.918261i	$0.629593\pi$
$368$	−10.9706	−0.571880
$369$	6.82843	0.355474
$370$	−25.3137	−1.31600
$371$	21.6569	1.12437
$372$	28.3848	1.47168
$373$	1.65685	0.0857887	0.0428943	−	0.999080i	$-0.486342\pi$
$373$	1.65685	0.0857887	0.0428943	+	0.999080i	$0.486342\pi$
$374$	−4.00000	−0.206835
$375$	5.65685	0.292119
$376$	31.2132	1.60970
$377$	−7.65685	−0.394348
$378$	6.82843	0.351216
$379$	−4.58579	−0.235556	−0.117778	−	0.993040i	$-0.537577\pi$
$379$	−4.58579	−0.235556	−0.117778	+	0.993040i	$0.537577\pi$
$380$	−73.9411	−3.79310
$381$	0	0
$382$	−60.2843	−3.08441
$383$	−2.58579	−0.132128	−0.0660638	−	0.997815i	$-0.521044\pi$
$383$	−2.58579	−0.132128	−0.0660638	+	0.997815i	$0.521044\pi$
$384$	−20.5563	−1.04901
$385$	−13.6569	−0.696018
$386$	−21.0711	−1.07249
$387$	2.82843	0.143777
$388$	44.6274	2.26561
$389$	10.4853	0.531625	0.265812	−	0.964025i	$-0.414360\pi$
$389$	10.4853	0.531625	0.265812	+	0.964025i	$0.414360\pi$
$390$	−8.24264	−0.417382
$391$	−4.28427	−0.216665
$392$	−4.41421	−0.222951
$393$	6.00000	0.302660
$394$	−19.0711	−0.960787
$395$	−35.7990	−1.80124
$396$	5.41421	0.272074
$397$	−26.8701	−1.34857	−0.674285	−	0.738471i	$-0.735548\pi$
$397$	−26.8701	−1.34857	−0.674285	+	0.738471i	$0.735548\pi$
$398$	9.65685	0.484054
$399$	−16.0000	−0.801002
$400$	19.9706	0.998528
$401$	−25.1716	−1.25701	−0.628504	−	0.777806i	$-0.716332\pi$
$401$	−25.1716	−1.25701	−0.628504	+	0.777806i	$0.716332\pi$
$402$	−17.8995	−0.892746
$403$	−7.41421	−0.369328
$404$	29.3137	1.45841
$405$	−3.41421	−0.169654
$406$	52.2843	2.59482
$407$	−4.34315	−0.215282
$408$	5.17157	0.256031
$409$	15.1716	0.750186	0.375093	−	0.926987i	$-0.377611\pi$
$409$	15.1716	0.750186	0.375093	+	0.926987i	$0.377611\pi$
$410$	56.2843	2.77968
$411$	8.00000	0.394611
$412$	3.82843	0.188613
$413$	−16.9706	−0.835067
$414$	8.82843	0.433894
$415$	−16.4853	−0.809231
$416$	1.58579	0.0777496
$417$	−20.1421	−0.986365
$418$	−19.3137	−0.944664
$419$	−29.6569	−1.44883	−0.724416	−	0.689363i	$-0.757891\pi$
$419$	−29.6569	−1.44883	−0.724416	+	0.689363i	$0.757891\pi$
$420$	36.9706	1.80398
$421$	−10.4853	−0.511021	−0.255511	−	0.966806i	$-0.582244\pi$
$421$	−10.4853	−0.511021	−0.255511	+	0.966806i	$0.582244\pi$
$422$	−26.1421	−1.27258
$423$	−7.07107	−0.343807
$424$	−33.7990	−1.64142
$425$	7.79899	0.378307
$426$	−10.2426	−0.496258
$427$	11.3137	0.547509
$428$	52.2843	2.52726
$429$	−1.41421	−0.0682789
$430$	23.3137	1.12429
$431$	−30.4853	−1.46842	−0.734212	−	0.678920i	$-0.762448\pi$
$431$	−30.4853	−1.46842	−0.734212	+	0.678920i	$0.762448\pi$
$432$	−3.00000	−0.144338
$433$	−3.65685	−0.175737	−0.0878686	−	0.996132i	$-0.528006\pi$
$433$	−3.65685	−0.175737	−0.0878686	+	0.996132i	$0.528006\pi$
$434$	50.6274	2.43019
$435$	−26.1421	−1.25342
$436$	−64.0416	−3.06704
$437$	−20.6863	−0.989560
$438$	−39.2132	−1.87368
$439$	−25.4558	−1.21494	−0.607471	−	0.794342i	$-0.707816\pi$
$439$	−25.4558	−1.21494	−0.607471	+	0.794342i	$0.707816\pi$
$440$	21.3137	1.01609
$441$	1.00000	0.0476190
$442$	−2.82843	−0.134535
$443$	−2.82843	−0.134383	−0.0671913	−	0.997740i	$-0.521404\pi$
$443$	−2.82843	−0.134383	−0.0671913	+	0.997740i	$0.521404\pi$
$444$	11.7574	0.557980
$445$	−48.6274	−2.30516
$446$	0	0
$447$	4.00000	0.189194
$448$	−27.7990	−1.31338
$449$	−12.1005	−0.571058	−0.285529	−	0.958370i	$-0.592169\pi$
$449$	−12.1005	−0.571058	−0.285529	+	0.958370i	$0.592169\pi$
$450$	−16.0711	−0.757597
$451$	9.65685	0.454724
$452$	−38.2843	−1.80074
$453$	−2.72792	−0.128169
$454$	39.2132	1.84037
$455$	−9.65685	−0.452720
$456$	24.9706	1.16935
$457$	−29.4142	−1.37594	−0.687969	−	0.725740i	$-0.741498\pi$
$457$	−29.4142	−1.37594	−0.687969	+	0.725740i	$0.741498\pi$
$458$	16.1421	0.754272
$459$	−1.17157	−0.0546843
$460$	47.7990	2.22864
$461$	−37.9411	−1.76709	−0.883547	−	0.468342i	$-0.844851\pi$
$461$	−37.9411	−1.76709	−0.883547	+	0.468342i	$0.844851\pi$
$462$	9.65685	0.449278
$463$	−18.2426	−0.847807	−0.423904	−	0.905707i	$-0.639341\pi$
$463$	−18.2426	−0.847807	−0.423904	+	0.905707i	$0.639341\pi$
$464$	−22.9706	−1.06638
$465$	−25.3137	−1.17390
$466$	69.1127	3.20158
$467$	12.6863	0.587052	0.293526	−	0.955951i	$-0.405171\pi$
$467$	12.6863	0.587052	0.293526	+	0.955951i	$0.405171\pi$
$468$	3.82843	0.176969
$469$	−20.9706	−0.968331
$470$	−58.2843	−2.68845
$471$	16.1421	0.743790
$472$	26.4853	1.21908
$473$	4.00000	0.183920
$474$	25.3137	1.16270
$475$	37.6569	1.72781
$476$	12.6863	0.581475
$477$	7.65685	0.350583
$478$	−29.7990	−1.36297
$479$	−5.41421	−0.247382	−0.123691	−	0.992321i	$-0.539473\pi$
$479$	−5.41421	−0.247382	−0.123691	+	0.992321i	$0.539473\pi$
$480$	5.41421	0.247124
$481$	−3.07107	−0.140029
$482$	50.0416	2.27933
$483$	10.3431	0.470629
$484$	−34.4558	−1.56617
$485$	−39.7990	−1.80718
$486$	2.41421	0.109511
$487$	−1.55635	−0.0705249	−0.0352625	−	0.999378i	$-0.511227\pi$
$487$	−1.55635	−0.0705249	−0.0352625	+	0.999378i	$0.511227\pi$
$488$	−17.6569	−0.799288
$489$	−5.65685	−0.255812
$490$	8.24264	0.372365
$491$	−29.6569	−1.33840	−0.669198	−	0.743085i	$-0.733362\pi$
$491$	−29.6569	−1.33840	−0.669198	+	0.743085i	$0.733362\pi$
$492$	−26.1421	−1.17858
$493$	−8.97056	−0.404014
$494$	−13.6569	−0.614451
$495$	−4.82843	−0.217022
$496$	−22.2426	−0.998724
$497$	−12.0000	−0.538274
$498$	11.6569	0.522356
$499$	−0.585786	−0.0262234	−0.0131117	−	0.999914i	$-0.504174\pi$
$499$	−0.585786	−0.0262234	−0.0131117	+	0.999914i	$0.504174\pi$
$500$	−21.6569	−0.968524
$501$	24.1421	1.07859
$502$	−59.1127	−2.63833
$503$	18.2843	0.815255	0.407628	−	0.913148i	$-0.366356\pi$
$503$	18.2843	0.815255	0.407628	+	0.913148i	$0.366356\pi$
$504$	−12.4853	−0.556139
$505$	−26.1421	−1.16331
$506$	12.4853	0.555038
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	21.4558	0.951014	0.475507	−	0.879712i	$-0.342265\pi$
$509$	21.4558	0.951014	0.475507	+	0.879712i	$0.342265\pi$
$510$	−9.65685	−0.427613
$511$	−45.9411	−2.03232
$512$	31.2426	1.38074
$513$	−5.65685	−0.249756
$514$	−74.7696	−3.29794
$515$	−3.41421	−0.150448
$516$	−10.8284	−0.476695
$517$	−10.0000	−0.439799
$518$	20.9706	0.921394
$519$	25.7990	1.13245
$520$	15.0711	0.660910
$521$	13.7990	0.604545	0.302272	−	0.953222i	$-0.402255\pi$
$521$	13.7990	0.604545	0.302272	+	0.953222i	$0.402255\pi$
$522$	18.4853	0.809079
$523$	−28.1421	−1.23057	−0.615285	−	0.788305i	$-0.710959\pi$
$523$	−28.1421	−1.23057	−0.615285	+	0.788305i	$0.710959\pi$
$524$	−22.9706	−1.00347
$525$	−18.8284	−0.821740
$526$	2.82843	0.123325
$527$	−8.68629	−0.378381
$528$	−4.24264	−0.184637
$529$	−9.62742	−0.418583
$530$	63.1127	2.74144
$531$	−6.00000	−0.260378
$532$	61.2548	2.65573
$533$	6.82843	0.295772
$534$	34.3848	1.48797
$535$	−46.6274	−2.01588
$536$	32.7279	1.41363
$537$	−13.3137	−0.574529
$538$	−6.82843	−0.294394
$539$	1.41421	0.0609145
$540$	13.0711	0.562489
$541$	37.1127	1.59560	0.797800	−	0.602922i	$-0.205997\pi$
$541$	37.1127	1.59560	0.797800	+	0.602922i	$0.205997\pi$
$542$	49.6985	2.13473
$543$	−15.6569	−0.671900
$544$	1.85786	0.0796553
$545$	57.1127	2.44644
$546$	6.82843	0.292230
$547$	−7.17157	−0.306634	−0.153317	−	0.988177i	$-0.548996\pi$
$547$	−7.17157	−0.306634	−0.153317	+	0.988177i	$0.548996\pi$
$548$	−30.6274	−1.30834
$549$	4.00000	0.170716
$550$	−22.7279	−0.969122
$551$	−43.3137	−1.84523
$552$	−16.1421	−0.687055
$553$	29.6569	1.26114
$554$	−7.65685	−0.325309
$555$	−10.4853	−0.445075
$556$	77.1127	3.27031
$557$	−18.9289	−0.802045	−0.401022	−	0.916068i	$-0.631345\pi$
$557$	−18.9289	−0.802045	−0.401022	+	0.916068i	$0.631345\pi$
$558$	17.8995	0.757746
$559$	2.82843	0.119630
$560$	−28.9706	−1.22423
$561$	−1.65685	−0.0699524
$562$	0.242641	0.0102352
$563$	3.31371	0.139656	0.0698281	−	0.997559i	$-0.477755\pi$
$563$	3.31371	0.139656	0.0698281	+	0.997559i	$0.477755\pi$
$564$	27.0711	1.13990
$565$	34.1421	1.43637
$566$	24.4853	1.02919
$567$	2.82843	0.118783
$568$	18.7279	0.785806
$569$	−20.1421	−0.844402	−0.422201	−	0.906502i	$-0.638742\pi$
$569$	−20.1421	−0.844402	−0.422201	+	0.906502i	$0.638742\pi$
$570$	−46.6274	−1.95301
$571$	−10.6274	−0.444744	−0.222372	−	0.974962i	$-0.571380\pi$
$571$	−10.6274	−0.444744	−0.222372	+	0.974962i	$0.571380\pi$
$572$	5.41421	0.226380
$573$	−24.9706	−1.04316
$574$	−46.6274	−1.94619
$575$	−24.3431	−1.01518
$576$	−9.82843	−0.409518
$577$	18.3848	0.765368	0.382684	−	0.923879i	$-0.375000\pi$
$577$	18.3848	0.765368	0.382684	+	0.923879i	$0.375000\pi$
$578$	37.7279	1.56927
$579$	−8.72792	−0.362720
$580$	100.083	4.15573
$581$	13.6569	0.566582
$582$	28.1421	1.16653
$583$	10.8284	0.448468
$584$	71.6985	2.96690
$585$	−3.41421	−0.141160
$586$	23.0711	0.953057
$587$	16.8284	0.694584	0.347292	−	0.937757i	$-0.387101\pi$
$587$	16.8284	0.694584	0.347292	+	0.937757i	$0.387101\pi$
$588$	−3.82843	−0.157882
$589$	−41.9411	−1.72815
$590$	−49.4558	−2.03606
$591$	−7.89949	−0.324942
$592$	−9.21320	−0.378660
$593$	−18.9289	−0.777318	−0.388659	−	0.921382i	$-0.627062\pi$
$593$	−18.9289	−0.777318	−0.388659	+	0.921382i	$0.627062\pi$
$594$	3.41421	0.140087
$595$	−11.3137	−0.463817
$596$	−15.3137	−0.627274
$597$	4.00000	0.163709
$598$	8.82843	0.361021
$599$	−35.1127	−1.43467	−0.717333	−	0.696731i	$-0.754637\pi$
$599$	−35.1127	−1.43467	−0.717333	+	0.696731i	$0.754637\pi$
$600$	29.3848	1.19963
$601$	5.31371	0.216751	0.108375	−	0.994110i	$-0.465435\pi$
$601$	5.31371	0.216751	0.108375	+	0.994110i	$0.465435\pi$
$602$	−19.3137	−0.787168
$603$	−7.41421	−0.301930
$604$	10.4437	0.424946
$605$	30.7279	1.24927
$606$	18.4853	0.750913
$607$	−19.3137	−0.783919	−0.391960	−	0.919982i	$-0.628203\pi$
$607$	−19.3137	−0.783919	−0.391960	+	0.919982i	$0.628203\pi$
$608$	8.97056	0.363804
$609$	21.6569	0.877580
$610$	32.9706	1.33494
$611$	−7.07107	−0.286065
$612$	4.48528	0.181307
$613$	−9.79899	−0.395777	−0.197889	−	0.980224i	$-0.563408\pi$
$613$	−9.79899	−0.395777	−0.197889	+	0.980224i	$0.563408\pi$
$614$	50.8701	2.05295
$615$	23.3137	0.940099
$616$	−17.6569	−0.711415
$617$	−10.0416	−0.404261	−0.202130	−	0.979359i	$-0.564786\pi$
$617$	−10.0416	−0.404261	−0.202130	+	0.979359i	$0.564786\pi$
$618$	2.41421	0.0971139
$619$	−33.1716	−1.33328	−0.666639	−	0.745381i	$-0.732268\pi$
$619$	−33.1716	−1.33328	−0.666639	+	0.745381i	$0.732268\pi$
$620$	96.9117	3.89207
$621$	3.65685	0.146745
$622$	53.9411	2.16284
$623$	40.2843	1.61396
$624$	−3.00000	−0.120096
$625$	−13.9706	−0.558823
$626$	−16.1421	−0.645169
$627$	−8.00000	−0.319489
$628$	−61.7990	−2.46605
$629$	−3.59798	−0.143461
$630$	23.3137	0.928840
$631$	−28.4853	−1.13398	−0.566991	−	0.823724i	$-0.691892\pi$
$631$	−28.4853	−1.13398	−0.566991	+	0.823724i	$0.691892\pi$
$632$	−46.2843	−1.84109
$633$	−10.8284	−0.430391
$634$	61.4558	2.44072
$635$	0	0
$636$	−29.3137	−1.16236
$637$	1.00000	0.0396214
$638$	26.1421	1.03498
$639$	−4.24264	−0.167836
$640$	−70.1838	−2.77426
$641$	−18.6863	−0.738064	−0.369032	−	0.929417i	$-0.620311\pi$
$641$	−18.6863	−0.738064	−0.369032	+	0.929417i	$0.620311\pi$
$642$	32.9706	1.30124
$643$	−17.4558	−0.688391	−0.344196	−	0.938898i	$-0.611848\pi$
$643$	−17.4558	−0.688391	−0.344196	+	0.938898i	$0.611848\pi$
$644$	−39.5980	−1.56038
$645$	9.65685	0.380238
$646$	−16.0000	−0.629512
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−4.41421	−0.173407
$649$	−8.48528	−0.333076
$650$	−16.0711	−0.630359
$651$	20.9706	0.821901
$652$	21.6569	0.848148
$653$	−16.8284	−0.658547	−0.329274	−	0.944235i	$-0.606804\pi$
$653$	−16.8284	−0.658547	−0.329274	+	0.944235i	$0.606804\pi$
$654$	−40.3848	−1.57917
$655$	20.4853	0.800426
$656$	20.4853	0.799816
$657$	−16.2426	−0.633686
$658$	48.2843	1.88232
$659$	36.6274	1.42680	0.713401	−	0.700756i	$-0.247154\pi$
$659$	36.6274	1.42680	0.713401	+	0.700756i	$0.247154\pi$
$660$	18.4853	0.719539
$661$	8.92893	0.347295	0.173648	−	0.984808i	$-0.444445\pi$
$661$	8.92893	0.347295	0.173648	+	0.984808i	$0.444445\pi$
$662$	−12.2426	−0.475824
$663$	−1.17157	−0.0455001
$664$	−21.3137	−0.827132
$665$	−54.6274	−2.11836
$666$	7.41421	0.287295
$667$	28.0000	1.08416
$668$	−92.4264	−3.57609
$669$	0	0
$670$	−61.1127	−2.36099
$671$	5.65685	0.218380
$672$	−4.48528	−0.173023
$673$	18.6863	0.720304	0.360152	−	0.932894i	$-0.382725\pi$
$673$	18.6863	0.720304	0.360152	+	0.932894i	$0.382725\pi$
$674$	−30.4853	−1.17425
$675$	−6.65685	−0.256222
$676$	3.82843	0.147247
$677$	25.3137	0.972885	0.486442	−	0.873713i	$-0.338294\pi$
$677$	25.3137	0.972885	0.486442	+	0.873713i	$0.338294\pi$
$678$	−24.1421	−0.927173
$679$	32.9706	1.26529
$680$	17.6569	0.677109
$681$	16.2426	0.622419
$682$	25.3137	0.969312
$683$	41.6985	1.59555	0.797774	−	0.602956i	$-0.206011\pi$
$683$	41.6985	1.59555	0.797774	+	0.602956i	$0.206011\pi$
$684$	21.6569	0.828071
$685$	27.3137	1.04360
$686$	40.9706	1.56426
$687$	6.68629	0.255098
$688$	8.48528	0.323498
$689$	7.65685	0.291703
$690$	30.1421	1.14749
$691$	44.8701	1.70694	0.853469	−	0.521144i	$-0.174495\pi$
$691$	44.8701	1.70694	0.853469	+	0.521144i	$0.174495\pi$
$692$	−98.7696	−3.75466
$693$	4.00000	0.151947
$694$	−67.5980	−2.56598
$695$	−68.7696	−2.60858
$696$	−33.7990	−1.28115
$697$	8.00000	0.303022
$698$	−74.5269	−2.82089
$699$	28.6274	1.08279
$700$	72.0833	2.72449
$701$	−27.1127	−1.02403	−0.512016	−	0.858976i	$-0.671101\pi$
$701$	−27.1127	−1.02403	−0.512016	+	0.858976i	$0.671101\pi$
$702$	2.41421	0.0911186
$703$	−17.3726	−0.655219
$704$	−13.8995	−0.523857
$705$	−24.1421	−0.909245
$706$	36.0416	1.35644
$707$	21.6569	0.814490
$708$	22.9706	0.863287
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	−34.9706	−1.31242
$711$	10.4853	0.393229
$712$	−62.8701	−2.35616
$713$	27.1127	1.01538
$714$	8.00000	0.299392
$715$	−4.82843	−0.180573
$716$	50.9706	1.90486
$717$	−12.3431	−0.460963
$718$	−54.2843	−2.02587
$719$	4.20101	0.156671	0.0783356	−	0.996927i	$-0.475039\pi$
$719$	4.20101	0.156671	0.0783356	+	0.996927i	$0.475039\pi$
$720$	−10.2426	−0.381721
$721$	2.82843	0.105336
$722$	−31.3848	−1.16802
$723$	20.7279	0.770880
$724$	59.9411	2.22769
$725$	−50.9706	−1.89300
$726$	−21.7279	−0.806399
$727$	−19.0294	−0.705763	−0.352881	−	0.935668i	$-0.614798\pi$
$727$	−19.0294	−0.705763	−0.352881	+	0.935668i	$0.614798\pi$
$728$	−12.4853	−0.462735
$729$	1.00000	0.0370370
$730$	−133.882	−4.95520
$731$	3.31371	0.122562
$732$	−15.3137	−0.566011
$733$	−10.7868	−0.398419	−0.199210	−	0.979957i	$-0.563837\pi$
$733$	−10.7868	−0.398419	−0.199210	+	0.979957i	$0.563837\pi$
$734$	36.6274	1.35194
$735$	3.41421	0.125935
$736$	−5.79899	−0.213754
$737$	−10.4853	−0.386230
$738$	−16.4853	−0.606832
$739$	−10.6274	−0.390936	−0.195468	−	0.980710i	$-0.562623\pi$
$739$	−10.6274	−0.390936	−0.195468	+	0.980710i	$0.562623\pi$
$740$	40.1421	1.47565
$741$	−5.65685	−0.207810
$742$	−52.2843	−1.91942
$743$	−10.1005	−0.370552	−0.185276	−	0.982687i	$-0.559318\pi$
$743$	−10.1005	−0.370552	−0.185276	+	0.982687i	$0.559318\pi$
$744$	−32.7279	−1.19986
$745$	13.6569	0.500348
$746$	−4.00000	−0.146450
$747$	4.82843	0.176663
$748$	6.34315	0.231928
$749$	38.6274	1.41142
$750$	−13.6569	−0.498678
$751$	8.82843	0.322154	0.161077	−	0.986942i	$-0.448503\pi$
$751$	8.82843	0.322154	0.161077	+	0.986942i	$0.448503\pi$
$752$	−21.2132	−0.773566
$753$	−24.4853	−0.892293
$754$	18.4853	0.673194
$755$	−9.31371	−0.338961
$756$	−10.8284	−0.393826
$757$	−12.3431	−0.448619	−0.224310	−	0.974518i	$-0.572013\pi$
$757$	−12.3431	−0.448619	−0.224310	+	0.974518i	$0.572013\pi$
$758$	11.0711	0.402119
$759$	5.17157	0.187716
$760$	85.2548	3.09252
$761$	−11.4142	−0.413765	−0.206882	−	0.978366i	$-0.566332\pi$
$761$	−11.4142	−0.413765	−0.206882	+	0.978366i	$0.566332\pi$
$762$	0	0
$763$	−47.3137	−1.71287
$764$	95.5980	3.45861
$765$	−4.00000	−0.144620
$766$	6.24264	0.225556
$767$	−6.00000	−0.216647
$768$	29.9706	1.08147
$769$	5.41421	0.195242	0.0976208	−	0.995224i	$-0.468877\pi$
$769$	5.41421	0.195242	0.0976208	+	0.995224i	$0.468877\pi$
$770$	32.9706	1.18818
$771$	−30.9706	−1.11538
$772$	33.4142	1.20260
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	−6.82843	−0.245443
$775$	−49.3553	−1.77290
$776$	−51.4558	−1.84716
$777$	8.68629	0.311619
$778$	−25.3137	−0.907540
$779$	38.6274	1.38397
$780$	13.0711	0.468019
$781$	−6.00000	−0.214697
$782$	10.3431	0.369870
$783$	7.65685	0.273634
$784$	3.00000	0.107143
$785$	55.1127	1.96706
$786$	−14.4853	−0.516673
$787$	−10.3431	−0.368693	−0.184347	−	0.982861i	$-0.559017\pi$
$787$	−10.3431	−0.368693	−0.184347	+	0.982861i	$0.559017\pi$
$788$	30.2426	1.07735
$789$	1.17157	0.0417091
$790$	86.4264	3.07491
$791$	−28.2843	−1.00567
$792$	−6.24264	−0.221823
$793$	4.00000	0.142044
$794$	64.8701	2.30215
$795$	26.1421	0.927166
$796$	−15.3137	−0.542780
$797$	18.6863	0.661902	0.330951	−	0.943648i	$-0.392630\pi$
$797$	18.6863	0.661902	0.330951	+	0.943648i	$0.392630\pi$
$798$	38.6274	1.36740
$799$	−8.28427	−0.293076
$800$	10.5563	0.373223
$801$	14.2426	0.503239
$802$	60.7696	2.14585
$803$	−22.9706	−0.810614
$804$	28.3848	1.00105
$805$	35.3137	1.24464
$806$	17.8995	0.630483
$807$	−2.82843	−0.0995654
$808$	−33.7990	−1.18904
$809$	7.85786	0.276268	0.138134	−	0.990414i	$-0.455890\pi$
$809$	7.85786	0.276268	0.138134	+	0.990414i	$0.455890\pi$
$810$	8.24264	0.289617
$811$	33.0711	1.16128	0.580641	−	0.814160i	$-0.302802\pi$
$811$	33.0711	1.16128	0.580641	+	0.814160i	$0.302802\pi$
$812$	−82.9117	−2.90963
$813$	20.5858	0.721975
$814$	10.4853	0.367509
$815$	−19.3137	−0.676530
$816$	−3.51472	−0.123040
$817$	16.0000	0.559769
$818$	−36.6274	−1.28065
$819$	2.82843	0.0988332
$820$	−89.2548	−3.11691
$821$	−7.31371	−0.255250	−0.127625	−	0.991822i	$-0.540735\pi$
$821$	−7.31371	−0.255250	−0.127625	+	0.991822i	$0.540735\pi$
$822$	−19.3137	−0.673643
$823$	−44.4853	−1.55066	−0.775330	−	0.631557i	$-0.782416\pi$
$823$	−44.4853	−1.55066	−0.775330	+	0.631557i	$0.782416\pi$
$824$	−4.41421	−0.153776
$825$	−9.41421	−0.327761
$826$	40.9706	1.42555
$827$	32.5269	1.13107	0.565536	−	0.824724i	$-0.308669\pi$
$827$	32.5269	1.13107	0.565536	+	0.824724i	$0.308669\pi$
$828$	−14.0000	−0.486534
$829$	−20.6274	−0.716420	−0.358210	−	0.933641i	$-0.616613\pi$
$829$	−20.6274	−0.716420	−0.358210	+	0.933641i	$0.616613\pi$
$830$	39.7990	1.38144
$831$	−3.17157	−0.110021
$832$	−9.82843	−0.340739
$833$	1.17157	0.0405926
$834$	48.6274	1.68383
$835$	82.4264	2.85248
$836$	30.6274	1.05927
$837$	7.41421	0.256273
$838$	71.5980	2.47331
$839$	11.6569	0.402439	0.201220	−	0.979546i	$-0.435510\pi$
$839$	11.6569	0.402439	0.201220	+	0.979546i	$0.435510\pi$
$840$	−42.6274	−1.47079
$841$	29.6274	1.02164
$842$	25.3137	0.872368
$843$	0.100505	0.00346158
$844$	41.4558	1.42697
$845$	−3.41421	−0.117453
$846$	17.0711	0.586915
$847$	−25.4558	−0.874673
$848$	22.9706	0.788812
$849$	10.1421	0.348077
$850$	−18.8284	−0.645810
$851$	11.2304	0.384975
$852$	16.2426	0.556464
$853$	42.9706	1.47128	0.735642	−	0.677371i	$-0.236881\pi$
$853$	42.9706	1.47128	0.735642	+	0.677371i	$0.236881\pi$
$854$	−27.3137	−0.934656
$855$	−19.3137	−0.660515
$856$	−60.2843	−2.06047
$857$	0.485281	0.0165769	0.00828845	−	0.999966i	$-0.497362\pi$
$857$	0.485281	0.0165769	0.00828845	+	0.999966i	$0.497362\pi$
$858$	3.41421	0.116559
$859$	4.48528	0.153036	0.0765179	−	0.997068i	$-0.475620\pi$
$859$	4.48528	0.153036	0.0765179	+	0.997068i	$0.475620\pi$
$860$	−36.9706	−1.26069
$861$	−19.3137	−0.658209
$862$	73.5980	2.50676
$863$	−47.5563	−1.61884	−0.809418	−	0.587232i	$-0.800218\pi$
$863$	−47.5563	−1.61884	−0.809418	+	0.587232i	$0.800218\pi$
$864$	−1.58579	−0.0539496
$865$	88.0833	2.99492
$866$	8.82843	0.300002
$867$	15.6274	0.530735
$868$	−80.2843	−2.72503
$869$	14.8284	0.503020
$870$	63.1127	2.13972
$871$	−7.41421	−0.251221
$872$	73.8406	2.50056
$873$	11.6569	0.394525
$874$	49.9411	1.68928
$875$	−16.0000	−0.540899
$876$	62.1838	2.10099
$877$	25.2132	0.851389	0.425695	−	0.904867i	$-0.360030\pi$
$877$	25.2132	0.851389	0.425695	+	0.904867i	$0.360030\pi$
$878$	61.4558	2.07403
$879$	9.55635	0.322328
$880$	−14.4853	−0.488299
$881$	34.4853	1.16184	0.580919	−	0.813961i	$-0.302693\pi$
$881$	34.4853	1.16184	0.580919	+	0.813961i	$0.302693\pi$
$882$	−2.41421	−0.0812908
$883$	9.11270	0.306667	0.153333	−	0.988175i	$-0.450999\pi$
$883$	9.11270	0.306667	0.153333	+	0.988175i	$0.450999\pi$
$884$	4.48528	0.150856
$885$	−20.4853	−0.688605
$886$	6.82843	0.229405
$887$	−12.2843	−0.412465	−0.206233	−	0.978503i	$-0.566120\pi$
$887$	−12.2843	−0.412465	−0.206233	+	0.978503i	$0.566120\pi$
$888$	−13.5563	−0.454921
$889$	0	0
$890$	117.397	3.93516
$891$	1.41421	0.0473779
$892$	0	0
$893$	−40.0000	−1.33855
$894$	−9.65685	−0.322974
$895$	−45.4558	−1.51942
$896$	58.1421	1.94239
$897$	3.65685	0.122099
$898$	29.2132	0.974857
$899$	56.7696	1.89337
$900$	25.4853	0.849509
$901$	8.97056	0.298853
$902$	−23.3137	−0.776262
$903$	−8.00000	−0.266223
$904$	44.1421	1.46815
$905$	−53.4558	−1.77693
$906$	6.58579	0.218798
$907$	24.6863	0.819695	0.409847	−	0.912154i	$-0.365582\pi$
$907$	24.6863	0.819695	0.409847	+	0.912154i	$0.365582\pi$
$908$	−62.1838	−2.06364
$909$	7.65685	0.253962
$910$	23.3137	0.772842
$911$	42.6274	1.41231	0.706155	−	0.708058i	$-0.250428\pi$
$911$	42.6274	1.41231	0.706155	+	0.708058i	$0.250428\pi$
$912$	−16.9706	−0.561951
$913$	6.82843	0.225988
$914$	71.0122	2.34887
$915$	13.6569	0.451482
$916$	−25.5980	−0.845781
$917$	−16.9706	−0.560417
$918$	2.82843	0.0933520
$919$	41.1716	1.35812	0.679062	−	0.734081i	$-0.262387\pi$
$919$	41.1716	1.35812	0.679062	+	0.734081i	$0.262387\pi$
$920$	−55.1127	−1.81701
$921$	21.0711	0.694315
$922$	91.5980	3.01662
$923$	−4.24264	−0.139648
$924$	−15.3137	−0.503784
$925$	−20.4437	−0.672183
$926$	44.0416	1.44730
$927$	1.00000	0.0328443
$928$	−12.1421	−0.398585
$929$	40.2843	1.32168	0.660842	−	0.750525i	$-0.270199\pi$
$929$	40.2843	1.32168	0.660842	+	0.750525i	$0.270199\pi$
$930$	61.1127	2.00396
$931$	5.65685	0.185396
$932$	−109.598	−3.59000
$933$	22.3431	0.731482
$934$	−30.6274	−1.00216
$935$	−5.65685	−0.184999
$936$	−4.41421	−0.144283
$937$	7.85786	0.256705	0.128353	−	0.991729i	$-0.459031\pi$
$937$	7.85786	0.256705	0.128353	+	0.991729i	$0.459031\pi$
$938$	50.6274	1.65304
$939$	−6.68629	−0.218199
$940$	92.4264	3.01462
$941$	−0.686292	−0.0223725	−0.0111862	−	0.999937i	$-0.503561\pi$
$941$	−0.686292	−0.0223725	−0.0111862	+	0.999937i	$0.503561\pi$
$942$	−38.9706	−1.26973
$943$	−24.9706	−0.813153
$944$	−18.0000	−0.585850
$945$	9.65685	0.314137
$946$	−9.65685	−0.313971
$947$	−38.3848	−1.24734	−0.623669	−	0.781689i	$-0.714359\pi$
$947$	−38.3848	−1.24734	−0.623669	+	0.781689i	$0.714359\pi$
$948$	−40.1421	−1.30376
$949$	−16.2426	−0.527258
$950$	−90.9117	−2.94956
$951$	25.4558	0.825462
$952$	−14.6274	−0.474077
$953$	−1.85786	−0.0601821	−0.0300911	−	0.999547i	$-0.509580\pi$
$953$	−1.85786	−0.0601821	−0.0300911	+	0.999547i	$0.509580\pi$
$954$	−18.4853	−0.598483
$955$	−85.2548	−2.75878
$956$	47.2548	1.52833
$957$	10.8284	0.350033
$958$	13.0711	0.422307
$959$	−22.6274	−0.730677
$960$	−33.5563	−1.08303
$961$	23.9706	0.773244
$962$	7.41421	0.239044
$963$	13.6569	0.440086
$964$	−79.3553	−2.55586
$965$	−29.7990	−0.959263
$966$	−24.9706	−0.803415
$967$	−42.7279	−1.37404	−0.687019	−	0.726640i	$-0.741081\pi$
$967$	−42.7279	−1.37404	−0.687019	+	0.726640i	$0.741081\pi$
$968$	39.7279	1.27690
$969$	−6.62742	−0.212903
$970$	96.0833	3.08505
$971$	24.2843	0.779319	0.389660	−	0.920959i	$-0.372593\pi$
$971$	24.2843	0.779319	0.389660	+	0.920959i	$0.372593\pi$
$972$	−3.82843	−0.122797
$973$	56.9706	1.82639
$974$	3.75736	0.120394
$975$	−6.65685	−0.213190
$976$	12.0000	0.384111
$977$	−42.1421	−1.34825	−0.674123	−	0.738619i	$-0.735478\pi$
$977$	−42.1421	−1.34825	−0.674123	+	0.738619i	$0.735478\pi$
$978$	13.6569	0.436698
$979$	20.1421	0.643745
$980$	−13.0711	−0.417540
$981$	−16.7279	−0.534081
$982$	71.5980	2.28478
$983$	3.45584	0.110224	0.0551122	−	0.998480i	$-0.482448\pi$
$983$	3.45584	0.110224	0.0551122	+	0.998480i	$0.482448\pi$
$984$	30.1421	0.960896
$985$	−26.9706	−0.859354
$986$	21.6569	0.689695
$987$	20.0000	0.636607
$988$	21.6569	0.688996
$989$	−10.3431	−0.328893
$990$	11.6569	0.370479
$991$	8.82843	0.280444	0.140222	−	0.990120i	$-0.455218\pi$
$991$	8.82843	0.280444	0.140222	+	0.990120i	$0.455218\pi$
$992$	−11.7574	−0.373297
$993$	−5.07107	−0.160925
$994$	28.9706	0.918890
$995$	13.6569	0.432951
$996$	−18.4853	−0.585729
$997$	−35.6569	−1.12926	−0.564632	−	0.825343i	$-0.690982\pi$
$997$	−35.6569	−1.12926	−0.564632	+	0.825343i	$0.690982\pi$
$998$	1.41421	0.0447661
$999$	3.07107	0.0971643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4017.2.a.d.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4017.2.a.d.1.1	✓	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4017 = 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4017.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -3.41421 q^{5} +2.41421 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -3.41421 q^{5} +2.41421 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +8.24264 q^{10} +1.41421 q^{11} -3.82843 q^{12} +1.00000 q^{13} -6.82843 q^{14} +3.41421 q^{15} +3.00000 q^{16} +1.17157 q^{17} -2.41421 q^{18} +5.65685 q^{19} -13.0711 q^{20} -2.82843 q^{21} -3.41421 q^{22} -3.65685 q^{23} +4.41421 q^{24} +6.65685 q^{25} -2.41421 q^{26} -1.00000 q^{27} +10.8284 q^{28} -7.65685 q^{29} -8.24264 q^{30} -7.41421 q^{31} +1.58579 q^{32} -1.41421 q^{33} -2.82843 q^{34} -9.65685 q^{35} +3.82843 q^{36} -3.07107 q^{37} -13.6569 q^{38} -1.00000 q^{39} +15.0711 q^{40} +6.82843 q^{41} +6.82843 q^{42} +2.82843 q^{43} +5.41421 q^{44} -3.41421 q^{45} +8.82843 q^{46} -7.07107 q^{47} -3.00000 q^{48} +1.00000 q^{49} -16.0711 q^{50} -1.17157 q^{51} +3.82843 q^{52} +7.65685 q^{53} +2.41421 q^{54} -4.82843 q^{55} -12.4853 q^{56} -5.65685 q^{57} +18.4853 q^{58} -6.00000 q^{59} +13.0711 q^{60} +4.00000 q^{61} +17.8995 q^{62} +2.82843 q^{63} -9.82843 q^{64} -3.41421 q^{65} +3.41421 q^{66} -7.41421 q^{67} +4.48528 q^{68} +3.65685 q^{69} +23.3137 q^{70} -4.24264 q^{71} -4.41421 q^{72} -16.2426 q^{73} +7.41421 q^{74} -6.65685 q^{75} +21.6569 q^{76} +4.00000 q^{77} +2.41421 q^{78} +10.4853 q^{79} -10.2426 q^{80} +1.00000 q^{81} -16.4853 q^{82} +4.82843 q^{83} -10.8284 q^{84} -4.00000 q^{85} -6.82843 q^{86} +7.65685 q^{87} -6.24264 q^{88} +14.2426 q^{89} +8.24264 q^{90} +2.82843 q^{91} -14.0000 q^{92} +7.41421 q^{93} +17.0711 q^{94} -19.3137 q^{95} -1.58579 q^{96} +11.6569 q^{97} -2.41421 q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{13} - 8 q^{14} + 4 q^{15} + 6 q^{16} + 8 q^{17} - 2 q^{18} - 12 q^{20} - 4 q^{22} + 4 q^{23} + 6 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 16 q^{28} - 4 q^{29} - 8 q^{30} - 12 q^{31} + 6 q^{32} - 8 q^{35} + 2 q^{36} + 8 q^{37} - 16 q^{38} - 2 q^{39} + 16 q^{40} + 8 q^{41} + 8 q^{42} + 8 q^{44} - 4 q^{45} + 12 q^{46} - 6 q^{48} + 2 q^{49} - 18 q^{50} - 8 q^{51} + 2 q^{52} + 4 q^{53} + 2 q^{54} - 4 q^{55} - 8 q^{56} + 20 q^{58} - 12 q^{59} + 12 q^{60} + 8 q^{61} + 16 q^{62} - 14 q^{64} - 4 q^{65} + 4 q^{66} - 12 q^{67} - 8 q^{68} - 4 q^{69} + 24 q^{70} - 6 q^{72} - 24 q^{73} + 12 q^{74} - 2 q^{75} + 32 q^{76} + 8 q^{77} + 2 q^{78} + 4 q^{79} - 12 q^{80} + 2 q^{81} - 16 q^{82} + 4 q^{83} - 16 q^{84} - 8 q^{85} - 8 q^{86} + 4 q^{87} - 4 q^{88} + 20 q^{89} + 8 q^{90} - 28 q^{92} + 12 q^{93} + 20 q^{94} - 16 q^{95} - 6 q^{96} + 12 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 8 * q^10 - 2 * q^12 + 2 * q^13 - 8 * q^14 + 4 * q^15 + 6 * q^16 + 8 * q^17 - 2 * q^18 - 12 * q^20 - 4 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 16 * q^28 - 4 * q^29 - 8 * q^30 - 12 * q^31 + 6 * q^32 - 8 * q^35 + 2 * q^36 + 8 * q^37 - 16 * q^38 - 2 * q^39 + 16 * q^40 + 8 * q^41 + 8 * q^42 + 8 * q^44 - 4 * q^45 + 12 * q^46 - 6 * q^48 + 2 * q^49 - 18 * q^50 - 8 * q^51 + 2 * q^52 + 4 * q^53 + 2 * q^54 - 4 * q^55 - 8 * q^56 + 20 * q^58 - 12 * q^59 + 12 * q^60 + 8 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^65 + 4 * q^66 - 12 * q^67 - 8 * q^68 - 4 * q^69 + 24 * q^70 - 6 * q^72 - 24 * q^73 + 12 * q^74 - 2 * q^75 + 32 * q^76 + 8 * q^77 + 2 * q^78 + 4 * q^79 - 12 * q^80 + 2 * q^81 - 16 * q^82 + 4 * q^83 - 16 * q^84 - 8 * q^85 - 8 * q^86 + 4 * q^87 - 4 * q^88 + 20 * q^89 + 8 * q^90 - 28 * q^92 + 12 * q^93 + 20 * q^94 - 16 * q^95 - 6 * q^96 + 12 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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