LMFDB - Embedded newform 285.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, 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("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	285.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:	$2.27573645761$
Analytic rank:	$0$
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	$\chi$	$=$	285.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.41421 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.414214 q^{10} +6.24264 q^{11} -1.82843 q^{12} -0.585786 q^{13} -0.585786 q^{14} -1.00000 q^{15} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} -1.00000 q^{19} +1.82843 q^{20} +1.41421 q^{21} -2.58579 q^{22} -3.65685 q^{23} +1.58579 q^{24} +1.00000 q^{25} +0.242641 q^{26} +1.00000 q^{27} -2.58579 q^{28} -1.41421 q^{29} +0.414214 q^{30} -8.82843 q^{31} -4.41421 q^{32} +6.24264 q^{33} -2.82843 q^{34} -1.41421 q^{35} -1.82843 q^{36} -0.585786 q^{37} +0.414214 q^{38} -0.585786 q^{39} -1.58579 q^{40} +8.24264 q^{41} -0.585786 q^{42} +3.75736 q^{43} -11.4142 q^{44} -1.00000 q^{45} +1.51472 q^{46} +3.65685 q^{47} +3.00000 q^{48} -5.00000 q^{49} -0.414214 q^{50} +6.82843 q^{51} +1.07107 q^{52} +8.00000 q^{53} -0.414214 q^{54} -6.24264 q^{55} +2.24264 q^{56} -1.00000 q^{57} +0.585786 q^{58} -4.48528 q^{59} +1.82843 q^{60} -15.3137 q^{61} +3.65685 q^{62} +1.41421 q^{63} -4.17157 q^{64} +0.585786 q^{65} -2.58579 q^{66} +1.65685 q^{67} -12.4853 q^{68} -3.65685 q^{69} +0.585786 q^{70} -5.17157 q^{71} +1.58579 q^{72} +3.65685 q^{73} +0.242641 q^{74} +1.00000 q^{75} +1.82843 q^{76} +8.82843 q^{77} +0.242641 q^{78} -3.00000 q^{80} +1.00000 q^{81} -3.41421 q^{82} +7.17157 q^{83} -2.58579 q^{84} -6.82843 q^{85} -1.55635 q^{86} -1.41421 q^{87} +9.89949 q^{88} -13.8995 q^{89} +0.414214 q^{90} -0.828427 q^{91} +6.68629 q^{92} -8.82843 q^{93} -1.51472 q^{94} +1.00000 q^{95} -4.41421 q^{96} -18.2426 q^{97} +2.07107 q^{98} +6.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{13} - 4 q^{14} - 2 q^{15} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 8 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^13 - 4 * q^14 - 2 * q^15 + 6 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 8 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 8 * q^26 + 2 * q^27 - 8 * q^28 - 2 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 8 * q^41 - 4 * q^42 + 16 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 - 4 * q^47 + 6 * q^48 - 10 * q^49 + 2 * q^50 + 8 * q^51 - 12 * q^52 + 16 * q^53 + 2 * q^54 - 4 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 + 8 * q^59 - 2 * q^60 - 8 * q^61 - 4 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^66 - 8 * q^67 - 8 * q^68 + 4 * q^69 + 4 * q^70 - 16 * q^71 + 6 * q^72 - 4 * q^73 - 8 * q^74 + 2 * q^75 - 2 * q^76 + 12 * q^77 - 8 * q^78 - 6 * q^80 + 2 * q^81 - 4 * q^82 + 20 * q^83 - 8 * q^84 - 8 * q^85 + 28 * q^86 - 8 * q^89 - 2 * q^90 + 4 * q^91 + 36 * q^92 - 12 * q^93 - 20 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 10 * q^98 + 4 * 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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.82843	−0.914214
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.414214	−0.169102
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	1.58579	0.560660
$9$	1.00000	0.333333
$10$	0.414214	0.130986
$11$	6.24264	1.88223	0.941113	−	0.338091i	$-0.109781\pi$
$11$	6.24264	1.88223	0.941113	+	0.338091i	$0.109781\pi$
$12$	−1.82843	−0.527821
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	−0.585786	−0.156558
$15$	−1.00000	−0.258199
$16$	3.00000	0.750000
$17$	6.82843	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$17$	6.82843	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$18$	−0.414214	−0.0976311
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.82843	0.408849
$21$	1.41421	0.308607
$22$	−2.58579	−0.551292
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	1.58579	0.323697
$25$	1.00000	0.200000
$26$	0.242641	0.0475858
$27$	1.00000	0.192450
$28$	−2.58579	−0.488668
$29$	−1.41421	−0.262613	−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421	−0.262613	−0.131306	+	0.991342i	$0.541917\pi$
$30$	0.414214	0.0756247
$31$	−8.82843	−1.58563	−0.792816	−	0.609461i	$-0.791386\pi$
$31$	−8.82843	−1.58563	−0.792816	+	0.609461i	$0.791386\pi$
$32$	−4.41421	−0.780330
$33$	6.24264	1.08670
$34$	−2.82843	−0.485071
$35$	−1.41421	−0.239046
$36$	−1.82843	−0.304738
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	0.414214	0.0671943
$39$	−0.585786	−0.0938009
$40$	−1.58579	−0.250735
$41$	8.24264	1.28728	0.643642	−	0.765327i	$-0.277423\pi$
$41$	8.24264	1.28728	0.643642	+	0.765327i	$0.277423\pi$
$42$	−0.585786	−0.0903888
$43$	3.75736	0.572992	0.286496	−	0.958081i	$-0.407509\pi$
$43$	3.75736	0.572992	0.286496	+	0.958081i	$0.407509\pi$
$44$	−11.4142	−1.72076
$45$	−1.00000	−0.149071
$46$	1.51472	0.223333
$47$	3.65685	0.533407	0.266704	−	0.963779i	$-0.414066\pi$
$47$	3.65685	0.533407	0.266704	+	0.963779i	$0.414066\pi$
$48$	3.00000	0.433013
$49$	−5.00000	−0.714286
$50$	−0.414214	−0.0585786
$51$	6.82843	0.956171
$52$	1.07107	0.148530
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	−0.414214	−0.0563673
$55$	−6.24264	−0.841757
$56$	2.24264	0.299685
$57$	−1.00000	−0.132453
$58$	0.585786	0.0769175
$59$	−4.48528	−0.583934	−0.291967	−	0.956428i	$-0.594310\pi$
$59$	−4.48528	−0.583934	−0.291967	+	0.956428i	$0.594310\pi$
$60$	1.82843	0.236049
$61$	−15.3137	−1.96072	−0.980360	−	0.197218i	$-0.936809\pi$
$61$	−15.3137	−1.96072	−0.980360	+	0.197218i	$0.936809\pi$
$62$	3.65685	0.464421
$63$	1.41421	0.178174
$64$	−4.17157	−0.521447
$65$	0.585786	0.0726579
$66$	−2.58579	−0.318288
$67$	1.65685	0.202417	0.101208	−	0.994865i	$-0.467729\pi$
$67$	1.65685	0.202417	0.101208	+	0.994865i	$0.467729\pi$
$68$	−12.4853	−1.51406
$69$	−3.65685	−0.440234
$70$	0.585786	0.0700149
$71$	−5.17157	−0.613753	−0.306876	−	0.951749i	$-0.599284\pi$
$71$	−5.17157	−0.613753	−0.306876	+	0.951749i	$0.599284\pi$
$72$	1.58579	0.186887
$73$	3.65685	0.428002	0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685	0.428002	0.214001	+	0.976833i	$0.431350\pi$
$74$	0.242641	0.0282064
$75$	1.00000	0.115470
$76$	1.82843	0.209735
$77$	8.82843	1.00609
$78$	0.242641	0.0274736
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−3.41421	−0.377037
$83$	7.17157	0.787182	0.393591	−	0.919286i	$-0.371233\pi$
$83$	7.17157	0.787182	0.393591	+	0.919286i	$0.371233\pi$
$84$	−2.58579	−0.282132
$85$	−6.82843	−0.740647
$86$	−1.55635	−0.167825
$87$	−1.41421	−0.151620
$88$	9.89949	1.05529
$89$	−13.8995	−1.47334	−0.736672	−	0.676250i	$-0.763604\pi$
$89$	−13.8995	−1.47334	−0.736672	+	0.676250i	$0.763604\pi$
$90$	0.414214	0.0436619
$91$	−0.828427	−0.0868428
$92$	6.68629	0.697094
$93$	−8.82843	−0.915465
$94$	−1.51472	−0.156231
$95$	1.00000	0.102598
$96$	−4.41421	−0.450524
$97$	−18.2426	−1.85226	−0.926130	−	0.377205i	$-0.876885\pi$
$97$	−18.2426	−1.85226	−0.926130	+	0.377205i	$0.876885\pi$
$98$	2.07107	0.209209
$99$	6.24264	0.627409
$100$	−1.82843	−0.182843
$101$	−8.82843	−0.878461	−0.439231	−	0.898374i	$-0.644749\pi$
$101$	−8.82843	−0.878461	−0.439231	+	0.898374i	$0.644749\pi$
$102$	−2.82843	−0.280056
$103$	15.3137	1.50890	0.754452	−	0.656355i	$-0.227903\pi$
$103$	15.3137	1.50890	0.754452	+	0.656355i	$0.227903\pi$
$104$	−0.928932	−0.0910893
$105$	−1.41421	−0.138013
$106$	−3.31371	−0.321856
$107$	−3.31371	−0.320348	−0.160174	−	0.987089i	$-0.551206\pi$
$107$	−3.31371	−0.320348	−0.160174	+	0.987089i	$0.551206\pi$
$108$	−1.82843	−0.175940
$109$	10.4853	1.00431	0.502154	−	0.864778i	$-0.332541\pi$
$109$	10.4853	1.00431	0.502154	+	0.864778i	$0.332541\pi$
$110$	2.58579	0.246545
$111$	−0.585786	−0.0556004
$112$	4.24264	0.400892
$113$	−18.1421	−1.70667	−0.853334	−	0.521364i	$-0.825423\pi$
$113$	−18.1421	−1.70667	−0.853334	+	0.521364i	$0.825423\pi$
$114$	0.414214	0.0387947
$115$	3.65685	0.341003
$116$	2.58579	0.240084
$117$	−0.585786	−0.0541560
$118$	1.85786	0.171030
$119$	9.65685	0.885242
$120$	−1.58579	−0.144762
$121$	27.9706	2.54278
$122$	6.34315	0.574281
$123$	8.24264	0.743214
$124$	16.1421	1.44961
$125$	−1.00000	−0.0894427
$126$	−0.585786	−0.0521860
$127$	−3.31371	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$127$	−3.31371	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$128$	10.5563	0.933058
$129$	3.75736	0.330817
$130$	−0.242641	−0.0212810
$131$	−11.4142	−0.997264	−0.498632	−	0.866814i	$-0.666164\pi$
$131$	−11.4142	−0.997264	−0.498632	+	0.866814i	$0.666164\pi$
$132$	−11.4142	−0.993480
$133$	−1.41421	−0.122628
$134$	−0.686292	−0.0592866
$135$	−1.00000	−0.0860663
$136$	10.8284	0.928530
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	1.51472	0.128941
$139$	14.1421	1.19952	0.599760	−	0.800180i	$-0.295263\pi$
$139$	14.1421	1.19952	0.599760	+	0.800180i	$0.295263\pi$
$140$	2.58579	0.218539
$141$	3.65685	0.307963
$142$	2.14214	0.179764
$143$	−3.65685	−0.305802
$144$	3.00000	0.250000
$145$	1.41421	0.117444
$146$	−1.51472	−0.125359
$147$	−5.00000	−0.412393
$148$	1.07107	0.0880412
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−0.414214	−0.0338204
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	−1.58579	−0.128624
$153$	6.82843	0.552046
$154$	−3.65685	−0.294678
$155$	8.82843	0.709116
$156$	1.07107	0.0857541
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	4.41421	0.348974
$161$	−5.17157	−0.407577
$162$	−0.414214	−0.0325437
$163$	2.10051	0.164524	0.0822621	−	0.996611i	$-0.473786\pi$
$163$	2.10051	0.164524	0.0822621	+	0.996611i	$0.473786\pi$
$164$	−15.0711	−1.17685
$165$	−6.24264	−0.485989
$166$	−2.97056	−0.230560
$167$	5.31371	0.411187	0.205594	−	0.978637i	$-0.434088\pi$
$167$	5.31371	0.411187	0.205594	+	0.978637i	$0.434088\pi$
$168$	2.24264	0.173023
$169$	−12.6569	−0.973604
$170$	2.82843	0.216930
$171$	−1.00000	−0.0764719
$172$	−6.87006	−0.523837
$173$	19.7990	1.50529	0.752645	−	0.658427i	$-0.228778\pi$
$173$	19.7990	1.50529	0.752645	+	0.658427i	$0.228778\pi$
$174$	0.585786	0.0444084
$175$	1.41421	0.106904
$176$	18.7279	1.41167
$177$	−4.48528	−0.337134
$178$	5.75736	0.431532
$179$	−0.485281	−0.0362716	−0.0181358	−	0.999836i	$-0.505773\pi$
$179$	−0.485281	−0.0362716	−0.0181358	+	0.999836i	$0.505773\pi$
$180$	1.82843	0.136283
$181$	−15.1716	−1.12769	−0.563847	−	0.825879i	$-0.690679\pi$
$181$	−15.1716	−1.12769	−0.563847	+	0.825879i	$0.690679\pi$
$182$	0.343146	0.0254357
$183$	−15.3137	−1.13202
$184$	−5.79899	−0.427507
$185$	0.585786	0.0430679
$186$	3.65685	0.268134
$187$	42.6274	3.11723
$188$	−6.68629	−0.487648
$189$	1.41421	0.102869
$190$	−0.414214	−0.0300502
$191$	1.75736	0.127158	0.0635790	−	0.997977i	$-0.479749\pi$
$191$	1.75736	0.127158	0.0635790	+	0.997977i	$0.479749\pi$
$192$	−4.17157	−0.301057
$193$	−9.07107	−0.652950	−0.326475	−	0.945206i	$-0.605861\pi$
$193$	−9.07107	−0.652950	−0.326475	+	0.945206i	$0.605861\pi$
$194$	7.55635	0.542514
$195$	0.585786	0.0419490
$196$	9.14214	0.653010
$197$	1.17157	0.0834711	0.0417356	−	0.999129i	$-0.486711\pi$
$197$	1.17157	0.0834711	0.0417356	+	0.999129i	$0.486711\pi$
$198$	−2.58579	−0.183764
$199$	−10.1421	−0.718957	−0.359478	−	0.933153i	$-0.617045\pi$
$199$	−10.1421	−0.718957	−0.359478	+	0.933153i	$0.617045\pi$
$200$	1.58579	0.112132
$201$	1.65685	0.116865
$202$	3.65685	0.257295
$203$	−2.00000	−0.140372
$204$	−12.4853	−0.874145
$205$	−8.24264	−0.575691
$206$	−6.34315	−0.441948
$207$	−3.65685	−0.254169
$208$	−1.75736	−0.121851
$209$	−6.24264	−0.431812
$210$	0.585786	0.0404231
$211$	−15.3137	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$211$	−15.3137	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$212$	−14.6274	−1.00462
$213$	−5.17157	−0.354350
$214$	1.37258	0.0938278
$215$	−3.75736	−0.256250
$216$	1.58579	0.107899
$217$	−12.4853	−0.847556
$218$	−4.34315	−0.294155
$219$	3.65685	0.247107
$220$	11.4142	0.769546
$221$	−4.00000	−0.269069
$222$	0.242641	0.0162850
$223$	−26.6274	−1.78310	−0.891552	−	0.452919i	$-0.850383\pi$
$223$	−26.6274	−1.78310	−0.891552	+	0.452919i	$0.850383\pi$
$224$	−6.24264	−0.417104
$225$	1.00000	0.0666667
$226$	7.51472	0.499872
$227$	−18.9706	−1.25912	−0.629560	−	0.776952i	$-0.716765\pi$
$227$	−18.9706	−1.25912	−0.629560	+	0.776952i	$0.716765\pi$
$228$	1.82843	0.121091
$229$	22.6274	1.49526	0.747631	−	0.664114i	$-0.231191\pi$
$229$	22.6274	1.49526	0.747631	+	0.664114i	$0.231191\pi$
$230$	−1.51472	−0.0998776
$231$	8.82843	0.580868
$232$	−2.24264	−0.147237
$233$	−11.6569	−0.763666	−0.381833	−	0.924231i	$-0.624707\pi$
$233$	−11.6569	−0.763666	−0.381833	+	0.924231i	$0.624707\pi$
$234$	0.242641	0.0158619
$235$	−3.65685	−0.238547
$236$	8.20101	0.533840
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−1.27208	−0.0822839	−0.0411419	−	0.999153i	$-0.513100\pi$
$239$	−1.27208	−0.0822839	−0.0411419	+	0.999153i	$0.513100\pi$
$240$	−3.00000	−0.193649
$241$	−8.34315	−0.537429	−0.268715	−	0.963220i	$-0.586599\pi$
$241$	−8.34315	−0.537429	−0.268715	+	0.963220i	$0.586599\pi$
$242$	−11.5858	−0.744763
$243$	1.00000	0.0641500
$244$	28.0000	1.79252
$245$	5.00000	0.319438
$246$	−3.41421	−0.217682
$247$	0.585786	0.0372727
$248$	−14.0000	−0.889001
$249$	7.17157	0.454480
$250$	0.414214	0.0261972
$251$	−10.2426	−0.646510	−0.323255	−	0.946312i	$-0.604777\pi$
$251$	−10.2426	−0.646510	−0.323255	+	0.946312i	$0.604777\pi$
$252$	−2.58579	−0.162889
$253$	−22.8284	−1.43521
$254$	1.37258	0.0861235
$255$	−6.82843	−0.427613
$256$	3.97056	0.248160
$257$	12.4853	0.778810	0.389405	−	0.921067i	$-0.372681\pi$
$257$	12.4853	0.778810	0.389405	+	0.921067i	$0.372681\pi$
$258$	−1.55635	−0.0968941
$259$	−0.828427	−0.0514760
$260$	−1.07107	−0.0664248
$261$	−1.41421	−0.0875376
$262$	4.72792	0.292092
$263$	27.4558	1.69300	0.846500	−	0.532389i	$-0.178706\pi$
$263$	27.4558	1.69300	0.846500	+	0.532389i	$0.178706\pi$
$264$	9.89949	0.609272
$265$	−8.00000	−0.491436
$266$	0.585786	0.0359169
$267$	−13.8995	−0.850635
$268$	−3.02944	−0.185052
$269$	11.0711	0.675015	0.337507	−	0.941323i	$-0.390416\pi$
$269$	11.0711	0.675015	0.337507	+	0.941323i	$0.390416\pi$
$270$	0.414214	0.0252082
$271$	−5.17157	−0.314151	−0.157075	−	0.987587i	$-0.550207\pi$
$271$	−5.17157	−0.314151	−0.157075	+	0.987587i	$0.550207\pi$
$272$	20.4853	1.24210
$273$	−0.828427	−0.0501387
$274$	−4.14214	−0.250236
$275$	6.24264	0.376445
$276$	6.68629	0.402467
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−5.85786	−0.351331
$279$	−8.82843	−0.528544
$280$	−2.24264	−0.134023
$281$	−17.4142	−1.03884	−0.519422	−	0.854518i	$-0.673853\pi$
$281$	−17.4142	−1.03884	−0.519422	+	0.854518i	$0.673853\pi$
$282$	−1.51472	−0.0902002
$283$	26.3848	1.56841	0.784206	−	0.620500i	$-0.213071\pi$
$283$	26.3848	1.56841	0.784206	+	0.620500i	$0.213071\pi$
$284$	9.45584	0.561101
$285$	1.00000	0.0592349
$286$	1.51472	0.0895672
$287$	11.6569	0.688082
$288$	−4.41421	−0.260110
$289$	29.6274	1.74279
$290$	−0.585786	−0.0343986
$291$	−18.2426	−1.06940
$292$	−6.68629	−0.391286
$293$	−28.4853	−1.66413	−0.832064	−	0.554680i	$-0.812841\pi$
$293$	−28.4853	−1.66413	−0.832064	+	0.554680i	$0.812841\pi$
$294$	2.07107	0.120787
$295$	4.48528	0.261143
$296$	−0.928932	−0.0539931
$297$	6.24264	0.362235
$298$	2.48528	0.143968
$299$	2.14214	0.123883
$300$	−1.82843	−0.105564
$301$	5.31371	0.306277
$302$	−4.34315	−0.249920
$303$	−8.82843	−0.507180
$304$	−3.00000	−0.172062
$305$	15.3137	0.876860
$306$	−2.82843	−0.161690
$307$	26.8284	1.53118	0.765590	−	0.643329i	$-0.222447\pi$
$307$	26.8284	1.53118	0.765590	+	0.643329i	$0.222447\pi$
$308$	−16.1421	−0.919784
$309$	15.3137	0.871166
$310$	−3.65685	−0.207695
$311$	2.24264	0.127168	0.0635842	−	0.997976i	$-0.479747\pi$
$311$	2.24264	0.127168	0.0635842	+	0.997976i	$0.479747\pi$
$312$	−0.928932	−0.0525904
$313$	−33.7990	−1.91043	−0.955216	−	0.295910i	$-0.904377\pi$
$313$	−33.7990	−1.91043	−0.955216	+	0.295910i	$0.904377\pi$
$314$	2.68629	0.151596
$315$	−1.41421	−0.0796819
$316$	0	0
$317$	18.6274	1.04622	0.523110	−	0.852265i	$-0.324772\pi$
$317$	18.6274	1.04622	0.523110	+	0.852265i	$0.324772\pi$
$318$	−3.31371	−0.185824
$319$	−8.82843	−0.494297
$320$	4.17157	0.233198
$321$	−3.31371	−0.184953
$322$	2.14214	0.119377
$323$	−6.82843	−0.379944
$324$	−1.82843	−0.101579
$325$	−0.585786	−0.0324936
$326$	−0.870058	−0.0481880
$327$	10.4853	0.579837
$328$	13.0711	0.721729
$329$	5.17157	0.285118
$330$	2.58579	0.142343
$331$	−0.142136	−0.00781248	−0.00390624	−	0.999992i	$-0.501243\pi$
$331$	−0.142136	−0.00781248	−0.00390624	+	0.999992i	$0.501243\pi$
$332$	−13.1127	−0.719653
$333$	−0.585786	−0.0321009
$334$	−2.20101	−0.120434
$335$	−1.65685	−0.0905236
$336$	4.24264	0.231455
$337$	−27.4142	−1.49335	−0.746674	−	0.665191i	$-0.768350\pi$
$337$	−27.4142	−1.49335	−0.746674	+	0.665191i	$0.768350\pi$
$338$	5.24264	0.285162
$339$	−18.1421	−0.985346
$340$	12.4853	0.677109
$341$	−55.1127	−2.98452
$342$	0.414214	0.0223981
$343$	−16.9706	−0.916324
$344$	5.95837	0.321254
$345$	3.65685	0.196878
$346$	−8.20101	−0.440889
$347$	23.4558	1.25918	0.629588	−	0.776929i	$-0.283224\pi$
$347$	23.4558	1.25918	0.629588	+	0.776929i	$0.283224\pi$
$348$	2.58579	0.138613
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	−0.585786	−0.0313116
$351$	−0.585786	−0.0312670
$352$	−27.5563	−1.46876
$353$	−7.65685	−0.407533	−0.203767	−	0.979019i	$-0.565318\pi$
$353$	−7.65685	−0.407533	−0.203767	+	0.979019i	$0.565318\pi$
$354$	1.85786	0.0987444
$355$	5.17157	0.274479
$356$	25.4142	1.34695
$357$	9.65685	0.511095
$358$	0.201010	0.0106237
$359$	−28.8701	−1.52370	−0.761852	−	0.647752i	$-0.775710\pi$
$359$	−28.8701	−1.52370	−0.761852	+	0.647752i	$0.775710\pi$
$360$	−1.58579	−0.0835783
$361$	1.00000	0.0526316
$362$	6.28427	0.330294
$363$	27.9706	1.46807
$364$	1.51472	0.0793928
$365$	−3.65685	−0.191408
$366$	6.34315	0.331562
$367$	22.3848	1.16848	0.584238	−	0.811582i	$-0.301394\pi$
$367$	22.3848	1.16848	0.584238	+	0.811582i	$0.301394\pi$
$368$	−10.9706	−0.571880
$369$	8.24264	0.429095
$370$	−0.242641	−0.0126143
$371$	11.3137	0.587378
$372$	16.1421	0.836931
$373$	3.41421	0.176781	0.0883906	−	0.996086i	$-0.471828\pi$
$373$	3.41421	0.176781	0.0883906	+	0.996086i	$0.471828\pi$
$374$	−17.6569	−0.913014
$375$	−1.00000	−0.0516398
$376$	5.79899	0.299060
$377$	0.828427	0.0426662
$378$	−0.585786	−0.0301296
$379$	8.82843	0.453486	0.226743	−	0.973955i	$-0.427192\pi$
$379$	8.82843	0.453486	0.226743	+	0.973955i	$0.427192\pi$
$380$	−1.82843	−0.0937963
$381$	−3.31371	−0.169766
$382$	−0.727922	−0.0372437
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	10.5563	0.538701
$385$	−8.82843	−0.449938
$386$	3.75736	0.191245
$387$	3.75736	0.190997
$388$	33.3553	1.69336
$389$	−2.97056	−0.150614	−0.0753068	−	0.997160i	$-0.523994\pi$
$389$	−2.97056	−0.150614	−0.0753068	+	0.997160i	$0.523994\pi$
$390$	−0.242641	−0.0122866
$391$	−24.9706	−1.26282
$392$	−7.92893	−0.400472
$393$	−11.4142	−0.575771
$394$	−0.485281	−0.0244481
$395$	0	0
$396$	−11.4142	−0.573586
$397$	−16.6274	−0.834506	−0.417253	−	0.908790i	$-0.637007\pi$
$397$	−16.6274	−0.834506	−0.417253	+	0.908790i	$0.637007\pi$
$398$	4.20101	0.210578
$399$	−1.41421	−0.0707992
$400$	3.00000	0.150000
$401$	−16.2426	−0.811119	−0.405559	−	0.914069i	$-0.632923\pi$
$401$	−16.2426	−0.811119	−0.405559	+	0.914069i	$0.632923\pi$
$402$	−0.686292	−0.0342291
$403$	5.17157	0.257614
$404$	16.1421	0.803101
$405$	−1.00000	−0.0496904
$406$	0.828427	0.0411141
$407$	−3.65685	−0.181264
$408$	10.8284	0.536087
$409$	−7.17157	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$409$	−7.17157	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$410$	3.41421	0.168616
$411$	10.0000	0.493264
$412$	−28.0000	−1.37946
$413$	−6.34315	−0.312126
$414$	1.51472	0.0744444
$415$	−7.17157	−0.352039
$416$	2.58579	0.126779
$417$	14.1421	0.692543
$418$	2.58579	0.126475
$419$	0.585786	0.0286175	0.0143088	−	0.999898i	$-0.495445\pi$
$419$	0.585786	0.0286175	0.0143088	+	0.999898i	$0.495445\pi$
$420$	2.58579	0.126173
$421$	−13.3137	−0.648870	−0.324435	−	0.945908i	$-0.605174\pi$
$421$	−13.3137	−0.648870	−0.324435	+	0.945908i	$0.605174\pi$
$422$	6.34315	0.308780
$423$	3.65685	0.177802
$424$	12.6863	0.616101
$425$	6.82843	0.331227
$426$	2.14214	0.103787
$427$	−21.6569	−1.04805
$428$	6.05887	0.292867
$429$	−3.65685	−0.176555
$430$	1.55635	0.0750538
$431$	31.1127	1.49865	0.749323	−	0.662205i	$-0.230379\pi$
$431$	31.1127	1.49865	0.749323	+	0.662205i	$0.230379\pi$
$432$	3.00000	0.144338
$433$	−25.0711	−1.20484	−0.602419	−	0.798180i	$-0.705796\pi$
$433$	−25.0711	−1.20484	−0.602419	+	0.798180i	$0.705796\pi$
$434$	5.17157	0.248243
$435$	1.41421	0.0678064
$436$	−19.1716	−0.918152
$437$	3.65685	0.174931
$438$	−1.51472	−0.0723761
$439$	−10.3431	−0.493651	−0.246826	−	0.969060i	$-0.579388\pi$
$439$	−10.3431	−0.493651	−0.246826	+	0.969060i	$0.579388\pi$
$440$	−9.89949	−0.471940
$441$	−5.00000	−0.238095
$442$	1.65685	0.0788085
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	1.07107	0.0508306
$445$	13.8995	0.658899
$446$	11.0294	0.522259
$447$	−6.00000	−0.283790
$448$	−5.89949	−0.278725
$449$	26.8701	1.26808	0.634038	−	0.773302i	$-0.281396\pi$
$449$	26.8701	1.26808	0.634038	+	0.773302i	$0.281396\pi$
$450$	−0.414214	−0.0195262
$451$	51.4558	2.42296
$452$	33.1716	1.56026
$453$	10.4853	0.492641
$454$	7.85786	0.368788
$455$	0.828427	0.0388373
$456$	−1.58579	−0.0742613
$457$	27.1716	1.27103	0.635516	−	0.772087i	$-0.280787\pi$
$457$	27.1716	1.27103	0.635516	+	0.772087i	$0.280787\pi$
$458$	−9.37258	−0.437952
$459$	6.82843	0.318724
$460$	−6.68629	−0.311750
$461$	8.34315	0.388579	0.194290	−	0.980944i	$-0.437760\pi$
$461$	8.34315	0.388579	0.194290	+	0.980944i	$0.437760\pi$
$462$	−3.65685	−0.170132
$463$	15.7574	0.732307	0.366153	−	0.930555i	$-0.380675\pi$
$463$	15.7574	0.732307	0.366153	+	0.930555i	$0.380675\pi$
$464$	−4.24264	−0.196960
$465$	8.82843	0.409409
$466$	4.82843	0.223673
$467$	24.3431	1.12647	0.563233	−	0.826298i	$-0.309557\pi$
$467$	24.3431	1.12647	0.563233	+	0.826298i	$0.309557\pi$
$468$	1.07107	0.0495101
$469$	2.34315	0.108196
$470$	1.51472	0.0698688
$471$	−6.48528	−0.298826
$472$	−7.11270	−0.327388
$473$	23.4558	1.07850
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−17.6569	−0.809301
$477$	8.00000	0.366295
$478$	0.526912	0.0241004
$479$	2.92893	0.133826	0.0669132	−	0.997759i	$-0.478685\pi$
$479$	2.92893	0.133826	0.0669132	+	0.997759i	$0.478685\pi$
$480$	4.41421	0.201480
$481$	0.343146	0.0156461
$482$	3.45584	0.157409
$483$	−5.17157	−0.235315
$484$	−51.1421	−2.32464
$485$	18.2426	0.828356
$486$	−0.414214	−0.0187891
$487$	3.51472	0.159267	0.0796336	−	0.996824i	$-0.474625\pi$
$487$	3.51472	0.159267	0.0796336	+	0.996824i	$0.474625\pi$
$488$	−24.2843	−1.09930
$489$	2.10051	0.0949881
$490$	−2.07107	−0.0935613
$491$	−10.2426	−0.462244	−0.231122	−	0.972925i	$-0.574240\pi$
$491$	−10.2426	−0.462244	−0.231122	+	0.972925i	$0.574240\pi$
$492$	−15.0711	−0.679456
$493$	−9.65685	−0.434923
$494$	−0.242641	−0.0109169
$495$	−6.24264	−0.280586
$496$	−26.4853	−1.18922
$497$	−7.31371	−0.328065
$498$	−2.97056	−0.133114
$499$	−10.8284	−0.484747	−0.242373	−	0.970183i	$-0.577926\pi$
$499$	−10.8284	−0.484747	−0.242373	+	0.970183i	$0.577926\pi$
$500$	1.82843	0.0817697
$501$	5.31371	0.237399
$502$	4.24264	0.189358
$503$	−0.828427	−0.0369377	−0.0184689	−	0.999829i	$-0.505879\pi$
$503$	−0.828427	−0.0369377	−0.0184689	+	0.999829i	$0.505879\pi$
$504$	2.24264	0.0998952
$505$	8.82843	0.392860
$506$	9.45584	0.420364
$507$	−12.6569	−0.562111
$508$	6.05887	0.268819
$509$	24.7279	1.09605	0.548023	−	0.836463i	$-0.315381\pi$
$509$	24.7279	1.09605	0.548023	+	0.836463i	$0.315381\pi$
$510$	2.82843	0.125245
$511$	5.17157	0.228777
$512$	−22.7574	−1.00574
$513$	−1.00000	−0.0441511
$514$	−5.17157	−0.228108
$515$	−15.3137	−0.674803
$516$	−6.87006	−0.302437
$517$	22.8284	1.00399
$518$	0.343146	0.0150770
$519$	19.7990	0.869079
$520$	0.928932	0.0407364
$521$	16.2426	0.711603	0.355802	−	0.934562i	$-0.384208\pi$
$521$	16.2426	0.711603	0.355802	+	0.934562i	$0.384208\pi$
$522$	0.585786	0.0256392
$523$	−32.4853	−1.42048	−0.710241	−	0.703959i	$-0.751414\pi$
$523$	−32.4853	−1.42048	−0.710241	+	0.703959i	$0.751414\pi$
$524$	20.8701	0.911713
$525$	1.41421	0.0617213
$526$	−11.3726	−0.495868
$527$	−60.2843	−2.62602
$528$	18.7279	0.815028
$529$	−9.62742	−0.418583
$530$	3.31371	0.143938
$531$	−4.48528	−0.194645
$532$	2.58579	0.112108
$533$	−4.82843	−0.209142
$534$	5.75736	0.249145
$535$	3.31371	0.143264
$536$	2.62742	0.113487
$537$	−0.485281	−0.0209414
$538$	−4.58579	−0.197707
$539$	−31.2132	−1.34445
$540$	1.82843	0.0786830
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	2.14214	0.0920126
$543$	−15.1716	−0.651075
$544$	−30.1421	−1.29233
$545$	−10.4853	−0.449140
$546$	0.343146	0.0146853
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	−18.2843	−0.781065
$549$	−15.3137	−0.653573
$550$	−2.58579	−0.110258
$551$	1.41421	0.0602475
$552$	−5.79899	−0.246821
$553$	0	0
$554$	9.11270	0.387161
$555$	0.585786	0.0248652
$556$	−25.8579	−1.09662
$557$	−2.68629	−0.113822	−0.0569109	−	0.998379i	$-0.518125\pi$
$557$	−2.68629	−0.113822	−0.0569109	+	0.998379i	$0.518125\pi$
$558$	3.65685	0.154807
$559$	−2.20101	−0.0930928
$560$	−4.24264	−0.179284
$561$	42.6274	1.79973
$562$	7.21320	0.304271
$563$	26.2843	1.10775	0.553875	−	0.832600i	$-0.313149\pi$
$563$	26.2843	1.10775	0.553875	+	0.832600i	$0.313149\pi$
$564$	−6.68629	−0.281544
$565$	18.1421	0.763245
$566$	−10.9289	−0.459377
$567$	1.41421	0.0593914
$568$	−8.20101	−0.344107
$569$	16.9289	0.709698	0.354849	−	0.934924i	$-0.384532\pi$
$569$	16.9289	0.709698	0.354849	+	0.934924i	$0.384532\pi$
$570$	−0.414214	−0.0173495
$571$	14.8284	0.620550	0.310275	−	0.950647i	$-0.399579\pi$
$571$	14.8284	0.620550	0.310275	+	0.950647i	$0.399579\pi$
$572$	6.68629	0.279568
$573$	1.75736	0.0734147
$574$	−4.82843	−0.201535
$575$	−3.65685	−0.152501
$576$	−4.17157	−0.173816
$577$	−31.4558	−1.30952	−0.654762	−	0.755835i	$-0.727231\pi$
$577$	−31.4558	−1.30952	−0.654762	+	0.755835i	$0.727231\pi$
$578$	−12.2721	−0.510451
$579$	−9.07107	−0.376981
$580$	−2.58579	−0.107369
$581$	10.1421	0.420767
$582$	7.55635	0.313221
$583$	49.9411	2.06835
$584$	5.79899	0.239964
$585$	0.585786	0.0242193
$586$	11.7990	0.487412
$587$	−23.6569	−0.976423	−0.488211	−	0.872725i	$-0.662351\pi$
$587$	−23.6569	−0.976423	−0.488211	+	0.872725i	$0.662351\pi$
$588$	9.14214	0.377015
$589$	8.82843	0.363769
$590$	−1.85786	−0.0764871
$591$	1.17157	0.0481921
$592$	−1.75736	−0.0722270
$593$	−8.62742	−0.354286	−0.177143	−	0.984185i	$-0.556685\pi$
$593$	−8.62742	−0.354286	−0.177143	+	0.984185i	$0.556685\pi$
$594$	−2.58579	−0.106096
$595$	−9.65685	−0.395892
$596$	10.9706	0.449372
$597$	−10.1421	−0.415090
$598$	−0.887302	−0.0362845
$599$	−36.9706	−1.51058	−0.755288	−	0.655393i	$-0.772503\pi$
$599$	−36.9706	−1.51058	−0.755288	+	0.655393i	$0.772503\pi$
$600$	1.58579	0.0647395
$601$	−20.1421	−0.821615	−0.410807	−	0.911722i	$-0.634753\pi$
$601$	−20.1421	−0.821615	−0.410807	+	0.911722i	$0.634753\pi$
$602$	−2.20101	−0.0897065
$603$	1.65685	0.0674723
$604$	−19.1716	−0.780080
$605$	−27.9706	−1.13717
$606$	3.65685	0.148550
$607$	−0.485281	−0.0196970	−0.00984848	−	0.999952i	$-0.503135\pi$
$607$	−0.485281	−0.0196970	−0.00984848	+	0.999952i	$0.503135\pi$
$608$	4.41421	0.179020
$609$	−2.00000	−0.0810441
$610$	−6.34315	−0.256826
$611$	−2.14214	−0.0866615
$612$	−12.4853	−0.504688
$613$	6.48528	0.261938	0.130969	−	0.991386i	$-0.458191\pi$
$613$	6.48528	0.261938	0.130969	+	0.991386i	$0.458191\pi$
$614$	−11.1127	−0.448472
$615$	−8.24264	−0.332375
$616$	14.0000	0.564076
$617$	11.5147	0.463565	0.231783	−	0.972768i	$-0.425544\pi$
$617$	11.5147	0.463565	0.231783	+	0.972768i	$0.425544\pi$
$618$	−6.34315	−0.255159
$619$	−3.51472	−0.141268	−0.0706342	−	0.997502i	$-0.522502\pi$
$619$	−3.51472	−0.141268	−0.0706342	+	0.997502i	$0.522502\pi$
$620$	−16.1421	−0.648284
$621$	−3.65685	−0.146745
$622$	−0.928932	−0.0372468
$623$	−19.6569	−0.787535
$624$	−1.75736	−0.0703507
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	−6.24264	−0.249307
$628$	11.8579	0.473180
$629$	−4.00000	−0.159490
$630$	0.585786	0.0233383
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−15.3137	−0.608665
$634$	−7.71573	−0.306431
$635$	3.31371	0.131501
$636$	−14.6274	−0.580015
$637$	2.92893	0.116049
$638$	3.65685	0.144776
$639$	−5.17157	−0.204584
$640$	−10.5563	−0.417276
$641$	39.3553	1.55444	0.777221	−	0.629227i	$-0.216629\pi$
$641$	39.3553	1.55444	0.777221	+	0.629227i	$0.216629\pi$
$642$	1.37258	0.0541715
$643$	5.61522	0.221443	0.110721	−	0.993851i	$-0.464684\pi$
$643$	5.61522	0.221443	0.110721	+	0.993851i	$0.464684\pi$
$644$	9.45584	0.372612
$645$	−3.75736	−0.147946
$646$	2.82843	0.111283
$647$	−7.85786	−0.308925	−0.154462	−	0.987999i	$-0.549365\pi$
$647$	−7.85786	−0.308925	−0.154462	+	0.987999i	$0.549365\pi$
$648$	1.58579	0.0622956
$649$	−28.0000	−1.09910
$650$	0.242641	0.00951715
$651$	−12.4853	−0.489337
$652$	−3.84062	−0.150410
$653$	37.1716	1.45464	0.727318	−	0.686301i	$-0.240767\pi$
$653$	37.1716	1.45464	0.727318	+	0.686301i	$0.240767\pi$
$654$	−4.34315	−0.169830
$655$	11.4142	0.445990
$656$	24.7279	0.965463
$657$	3.65685	0.142667
$658$	−2.14214	−0.0835091
$659$	26.6274	1.03726	0.518628	−	0.855000i	$-0.326443\pi$
$659$	26.6274	1.03726	0.518628	+	0.855000i	$0.326443\pi$
$660$	11.4142	0.444298
$661$	−43.4558	−1.69024	−0.845118	−	0.534579i	$-0.820470\pi$
$661$	−43.4558	−1.69024	−0.845118	+	0.534579i	$0.820470\pi$
$662$	0.0588745	0.00228822
$663$	−4.00000	−0.155347
$664$	11.3726	0.441342
$665$	1.41421	0.0548408
$666$	0.242641	0.00940214
$667$	5.17157	0.200244
$668$	−9.71573	−0.375913
$669$	−26.6274	−1.02948
$670$	0.686292	0.0265138
$671$	−95.5980	−3.69052
$672$	−6.24264	−0.240815
$673$	32.1838	1.24059	0.620297	−	0.784367i	$-0.287012\pi$
$673$	32.1838	1.24059	0.620297	+	0.784367i	$0.287012\pi$
$674$	11.3553	0.437391
$675$	1.00000	0.0384900
$676$	23.1421	0.890082
$677$	−16.9706	−0.652232	−0.326116	−	0.945330i	$-0.605740\pi$
$677$	−16.9706	−0.652232	−0.326116	+	0.945330i	$0.605740\pi$
$678$	7.51472	0.288601
$679$	−25.7990	−0.990074
$680$	−10.8284	−0.415251
$681$	−18.9706	−0.726954
$682$	22.8284	0.874146
$683$	29.6569	1.13479	0.567394	−	0.823446i	$-0.307952\pi$
$683$	29.6569	1.13479	0.567394	+	0.823446i	$0.307952\pi$
$684$	1.82843	0.0699117
$685$	−10.0000	−0.382080
$686$	7.02944	0.268385
$687$	22.6274	0.863290
$688$	11.2721	0.429744
$689$	−4.68629	−0.178533
$690$	−1.51472	−0.0576644
$691$	41.1716	1.56624	0.783120	−	0.621870i	$-0.213627\pi$
$691$	41.1716	1.56624	0.783120	+	0.621870i	$0.213627\pi$
$692$	−36.2010	−1.37616
$693$	8.82843	0.335364
$694$	−9.71573	−0.368804
$695$	−14.1421	−0.536442
$696$	−2.24264	−0.0850071
$697$	56.2843	2.13192
$698$	−7.45584	−0.282208
$699$	−11.6569	−0.440903
$700$	−2.58579	−0.0977335
$701$	29.3137	1.10716	0.553582	−	0.832795i	$-0.313261\pi$
$701$	29.3137	1.10716	0.553582	+	0.832795i	$0.313261\pi$
$702$	0.242641	0.00915788
$703$	0.585786	0.0220934
$704$	−26.0416	−0.981481
$705$	−3.65685	−0.137725
$706$	3.17157	0.119364
$707$	−12.4853	−0.469557
$708$	8.20101	0.308213
$709$	−4.97056	−0.186673	−0.0933367	−	0.995635i	$-0.529753\pi$
$709$	−4.97056	−0.186673	−0.0933367	+	0.995635i	$0.529753\pi$
$710$	−2.14214	−0.0803929
$711$	0	0
$712$	−22.0416	−0.826045
$713$	32.2843	1.20906
$714$	−4.00000	−0.149696
$715$	3.65685	0.136759
$716$	0.887302	0.0331600
$717$	−1.27208	−0.0475066
$718$	11.9584	0.446282
$719$	13.0711	0.487469	0.243734	−	0.969842i	$-0.421628\pi$
$719$	13.0711	0.487469	0.243734	+	0.969842i	$0.421628\pi$
$720$	−3.00000	−0.111803
$721$	21.6569	0.806543
$722$	−0.414214	−0.0154154
$723$	−8.34315	−0.310285
$724$	27.7401	1.03095
$725$	−1.41421	−0.0525226
$726$	−11.5858	−0.429989
$727$	−23.3553	−0.866202	−0.433101	−	0.901345i	$-0.642581\pi$
$727$	−23.3553	−0.866202	−0.433101	+	0.901345i	$0.642581\pi$
$728$	−1.31371	−0.0486893
$729$	1.00000	0.0370370
$730$	1.51472	0.0560623
$731$	25.6569	0.948953
$732$	28.0000	1.03491
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−9.27208	−0.342239
$735$	5.00000	0.184428
$736$	16.1421	0.595007
$737$	10.3431	0.380995
$738$	−3.41421	−0.125679
$739$	25.6569	0.943803	0.471901	−	0.881651i	$-0.343568\pi$
$739$	25.6569	0.943803	0.471901	+	0.881651i	$0.343568\pi$
$740$	−1.07107	−0.0393732
$741$	0.585786	0.0215194
$742$	−4.68629	−0.172039
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	−14.0000	−0.513265
$745$	6.00000	0.219823
$746$	−1.41421	−0.0517780
$747$	7.17157	0.262394
$748$	−77.9411	−2.84981
$749$	−4.68629	−0.171233
$750$	0.414214	0.0151249
$751$	4.14214	0.151149	0.0755743	−	0.997140i	$-0.475921\pi$
$751$	4.14214	0.151149	0.0755743	+	0.997140i	$0.475921\pi$
$752$	10.9706	0.400055
$753$	−10.2426	−0.373263
$754$	−0.343146	−0.0124966
$755$	−10.4853	−0.381598
$756$	−2.58579	−0.0940441
$757$	52.4264	1.90547	0.952735	−	0.303802i	$-0.0982562\pi$
$757$	52.4264	1.90547	0.952735	+	0.303802i	$0.0982562\pi$
$758$	−3.65685	−0.132823
$759$	−22.8284	−0.828619
$760$	1.58579	0.0575225
$761$	−23.9411	−0.867865	−0.433933	−	0.900945i	$-0.642874\pi$
$761$	−23.9411	−0.867865	−0.433933	+	0.900945i	$0.642874\pi$
$762$	1.37258	0.0497234
$763$	14.8284	0.536825
$764$	−3.21320	−0.116250
$765$	−6.82843	−0.246882
$766$	−11.5980	−0.419052
$767$	2.62742	0.0948705
$768$	3.97056	0.143275
$769$	9.02944	0.325610	0.162805	−	0.986658i	$-0.447946\pi$
$769$	9.02944	0.325610	0.162805	+	0.986658i	$0.447946\pi$
$770$	3.65685	0.131784
$771$	12.4853	0.449646
$772$	16.5858	0.596936
$773$	−14.3431	−0.515887	−0.257944	−	0.966160i	$-0.583045\pi$
$773$	−14.3431	−0.515887	−0.257944	+	0.966160i	$0.583045\pi$
$774$	−1.55635	−0.0559418
$775$	−8.82843	−0.317126
$776$	−28.9289	−1.03849
$777$	−0.828427	−0.0297197
$778$	1.23045	0.0441137
$779$	−8.24264	−0.295323
$780$	−1.07107	−0.0383504
$781$	−32.2843	−1.15522
$782$	10.3431	0.369870
$783$	−1.41421	−0.0505399
$784$	−15.0000	−0.535714
$785$	6.48528	0.231470
$786$	4.72792	0.168639
$787$	−37.1716	−1.32502	−0.662512	−	0.749052i	$-0.730510\pi$
$787$	−37.1716	−1.32502	−0.662512	+	0.749052i	$0.730510\pi$
$788$	−2.14214	−0.0763104
$789$	27.4558	0.977454
$790$	0	0
$791$	−25.6569	−0.912253
$792$	9.89949	0.351763
$793$	8.97056	0.318554
$794$	6.88730	0.244421
$795$	−8.00000	−0.283731
$796$	18.5442	0.657280
$797$	14.1421	0.500940	0.250470	−	0.968124i	$-0.419415\pi$
$797$	14.1421	0.500940	0.250470	+	0.968124i	$0.419415\pi$
$798$	0.585786	0.0207366
$799$	24.9706	0.883395
$800$	−4.41421	−0.156066
$801$	−13.8995	−0.491115
$802$	6.72792	0.237571
$803$	22.8284	0.805598
$804$	−3.02944	−0.106840
$805$	5.17157	0.182274
$806$	−2.14214	−0.0754535
$807$	11.0711	0.389720
$808$	−14.0000	−0.492518
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0.414214	0.0145540
$811$	31.3137	1.09957	0.549787	−	0.835305i	$-0.314709\pi$
$811$	31.3137	1.09957	0.549787	+	0.835305i	$0.314709\pi$
$812$	3.65685	0.128330
$813$	−5.17157	−0.181375
$814$	1.51472	0.0530909
$815$	−2.10051	−0.0735775
$816$	20.4853	0.717128
$817$	−3.75736	−0.131453
$818$	2.97056	0.103863
$819$	−0.828427	−0.0289476
$820$	15.0711	0.526305
$821$	−2.48528	−0.0867369	−0.0433685	−	0.999059i	$-0.513809\pi$
$821$	−2.48528	−0.0867369	−0.0433685	+	0.999059i	$0.513809\pi$
$822$	−4.14214	−0.144474
$823$	57.0122	1.98732	0.993660	−	0.112426i	$-0.0358622\pi$
$823$	57.0122	1.98732	0.993660	+	0.112426i	$0.0358622\pi$
$824$	24.2843	0.845983
$825$	6.24264	0.217341
$826$	2.62742	0.0914195
$827$	−25.3137	−0.880244	−0.440122	−	0.897938i	$-0.645065\pi$
$827$	−25.3137	−0.880244	−0.440122	+	0.897938i	$0.645065\pi$
$828$	6.68629	0.232365
$829$	29.5147	1.02509	0.512544	−	0.858661i	$-0.328703\pi$
$829$	29.5147	1.02509	0.512544	+	0.858661i	$0.328703\pi$
$830$	2.97056	0.103110
$831$	−22.0000	−0.763172
$832$	2.44365	0.0847183
$833$	−34.1421	−1.18295
$834$	−5.85786	−0.202841
$835$	−5.31371	−0.183888
$836$	11.4142	0.394769
$837$	−8.82843	−0.305155
$838$	−0.242641	−0.00838188
$839$	−38.1421	−1.31681	−0.658406	−	0.752663i	$-0.728769\pi$
$839$	−38.1421	−1.31681	−0.658406	+	0.752663i	$0.728769\pi$
$840$	−2.24264	−0.0773785
$841$	−27.0000	−0.931034
$842$	5.51472	0.190050
$843$	−17.4142	−0.599777
$844$	28.0000	0.963800
$845$	12.6569	0.435409
$846$	−1.51472	−0.0520771
$847$	39.5563	1.35917
$848$	24.0000	0.824163
$849$	26.3848	0.905523
$850$	−2.82843	−0.0970143
$851$	2.14214	0.0734315
$852$	9.45584	0.323952
$853$	−11.1716	−0.382507	−0.191254	−	0.981541i	$-0.561255\pi$
$853$	−11.1716	−0.382507	−0.191254	+	0.981541i	$0.561255\pi$
$854$	8.97056	0.306966
$855$	1.00000	0.0341993
$856$	−5.25483	−0.179607
$857$	35.3137	1.20629	0.603147	−	0.797630i	$-0.293913\pi$
$857$	35.3137	1.20629	0.603147	+	0.797630i	$0.293913\pi$
$858$	1.51472	0.0517116
$859$	17.9411	0.612143	0.306072	−	0.952008i	$-0.400985\pi$
$859$	17.9411	0.612143	0.306072	+	0.952008i	$0.400985\pi$
$860$	6.87006	0.234267
$861$	11.6569	0.397265
$862$	−12.8873	−0.438943
$863$	31.3137	1.06593	0.532966	−	0.846137i	$-0.321078\pi$
$863$	31.3137	1.06593	0.532966	+	0.846137i	$0.321078\pi$
$864$	−4.41421	−0.150175
$865$	−19.7990	−0.673186
$866$	10.3848	0.352889
$867$	29.6274	1.00620
$868$	22.8284	0.774847
$869$	0	0
$870$	−0.585786	−0.0198600
$871$	−0.970563	−0.0328863
$872$	16.6274	0.563075
$873$	−18.2426	−0.617420
$874$	−1.51472	−0.0512361
$875$	−1.41421	−0.0478091
$876$	−6.68629	−0.225909
$877$	−49.0711	−1.65701	−0.828506	−	0.559980i	$-0.810809\pi$
$877$	−49.0711	−1.65701	−0.828506	+	0.559980i	$0.810809\pi$
$878$	4.28427	0.144587
$879$	−28.4853	−0.960785
$880$	−18.7279	−0.631318
$881$	33.7990	1.13872	0.569358	−	0.822089i	$-0.307192\pi$
$881$	33.7990	1.13872	0.569358	+	0.822089i	$0.307192\pi$
$882$	2.07107	0.0697365
$883$	36.0416	1.21290	0.606449	−	0.795123i	$-0.292594\pi$
$883$	36.0416	1.21290	0.606449	+	0.795123i	$0.292594\pi$
$884$	7.31371	0.245987
$885$	4.48528	0.150771
$886$	−7.45584	−0.250484
$887$	−41.9411	−1.40825	−0.704123	−	0.710078i	$-0.748659\pi$
$887$	−41.9411	−1.40825	−0.704123	+	0.710078i	$0.748659\pi$
$888$	−0.928932	−0.0311729
$889$	−4.68629	−0.157173
$890$	−5.75736	−0.192987
$891$	6.24264	0.209136
$892$	48.6863	1.63014
$893$	−3.65685	−0.122372
$894$	2.48528	0.0831202
$895$	0.485281	0.0162212
$896$	14.9289	0.498741
$897$	2.14214	0.0715238
$898$	−11.1299	−0.371411
$899$	12.4853	0.416407
$900$	−1.82843	−0.0609476
$901$	54.6274	1.81990
$902$	−21.3137	−0.709669
$903$	5.31371	0.176829
$904$	−28.7696	−0.956861
$905$	15.1716	0.504320
$906$	−4.34315	−0.144291
$907$	10.1421	0.336764	0.168382	−	0.985722i	$-0.446146\pi$
$907$	10.1421	0.336764	0.168382	+	0.985722i	$0.446146\pi$
$908$	34.6863	1.15111
$909$	−8.82843	−0.292820
$910$	−0.343146	−0.0113752
$911$	31.3137	1.03747	0.518735	−	0.854935i	$-0.326403\pi$
$911$	31.3137	1.03747	0.518735	+	0.854935i	$0.326403\pi$
$912$	−3.00000	−0.0993399
$913$	44.7696	1.48166
$914$	−11.2548	−0.372277
$915$	15.3137	0.506256
$916$	−41.3726	−1.36699
$917$	−16.1421	−0.533060
$918$	−2.82843	−0.0933520
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	5.79899	0.191187
$921$	26.8284	0.884027
$922$	−3.45584	−0.113812
$923$	3.02944	0.0997151
$924$	−16.1421	−0.531037
$925$	−0.585786	−0.0192605
$926$	−6.52691	−0.214488
$927$	15.3137	0.502968
$928$	6.24264	0.204925
$929$	−29.1127	−0.955157	−0.477578	−	0.878589i	$-0.658485\pi$
$929$	−29.1127	−0.955157	−0.477578	+	0.878589i	$0.658485\pi$
$930$	−3.65685	−0.119913
$931$	5.00000	0.163868
$932$	21.3137	0.698154
$933$	2.24264	0.0734208
$934$	−10.0833	−0.329934
$935$	−42.6274	−1.39407
$936$	−0.928932	−0.0303631
$937$	9.79899	0.320119	0.160060	−	0.987107i	$-0.448831\pi$
$937$	9.79899	0.320119	0.160060	+	0.987107i	$0.448831\pi$
$938$	−0.970563	−0.0316900
$939$	−33.7990	−1.10299
$940$	6.68629	0.218083
$941$	−30.8701	−1.00634	−0.503168	−	0.864189i	$-0.667832\pi$
$941$	−30.8701	−1.00634	−0.503168	+	0.864189i	$0.667832\pi$
$942$	2.68629	0.0875241
$943$	−30.1421	−0.981563
$944$	−13.4558	−0.437950
$945$	−1.41421	−0.0460044
$946$	−9.71573	−0.315886
$947$	−52.8284	−1.71669	−0.858347	−	0.513070i	$-0.828508\pi$
$947$	−52.8284	−1.71669	−0.858347	+	0.513070i	$0.828508\pi$
$948$	0	0
$949$	−2.14214	−0.0695367
$950$	0.414214	0.0134389
$951$	18.6274	0.604035
$952$	15.3137	0.496320
$953$	−2.54416	−0.0824133	−0.0412066	−	0.999151i	$-0.513120\pi$
$953$	−2.54416	−0.0824133	−0.0412066	+	0.999151i	$0.513120\pi$
$954$	−3.31371	−0.107285
$955$	−1.75736	−0.0568668
$956$	2.32590	0.0752250
$957$	−8.82843	−0.285383
$958$	−1.21320	−0.0391968
$959$	14.1421	0.456673
$960$	4.17157	0.134637
$961$	46.9411	1.51423
$962$	−0.142136	−0.00458264
$963$	−3.31371	−0.106783
$964$	15.2548	0.491325
$965$	9.07107	0.292008
$966$	2.14214	0.0689221
$967$	−40.0416	−1.28765	−0.643826	−	0.765172i	$-0.722654\pi$
$967$	−40.0416	−1.28765	−0.643826	+	0.765172i	$0.722654\pi$
$968$	44.3553	1.42563
$969$	−6.82843	−0.219361
$970$	−7.55635	−0.242620
$971$	33.6569	1.08010	0.540050	−	0.841633i	$-0.318405\pi$
$971$	33.6569	1.08010	0.540050	+	0.841633i	$0.318405\pi$
$972$	−1.82843	−0.0586468
$973$	20.0000	0.641171
$974$	−1.45584	−0.0466483
$975$	−0.585786	−0.0187602
$976$	−45.9411	−1.47054
$977$	−24.2843	−0.776923	−0.388461	−	0.921465i	$-0.626993\pi$
$977$	−24.2843	−0.776923	−0.388461	+	0.921465i	$0.626993\pi$
$978$	−0.870058	−0.0278214
$979$	−86.7696	−2.77317
$980$	−9.14214	−0.292035
$981$	10.4853	0.334769
$982$	4.24264	0.135388
$983$	20.6274	0.657912	0.328956	−	0.944345i	$-0.393303\pi$
$983$	20.6274	0.657912	0.328956	+	0.944345i	$0.393303\pi$
$984$	13.0711	0.416690
$985$	−1.17157	−0.0373294
$986$	4.00000	0.127386
$987$	5.17157	0.164613
$988$	−1.07107	−0.0340752
$989$	−13.7401	−0.436910
$990$	2.58579	0.0821817
$991$	13.6569	0.433824	0.216912	−	0.976191i	$-0.430401\pi$
$991$	13.6569	0.433824	0.216912	+	0.976191i	$0.430401\pi$
$992$	38.9706	1.23732
$993$	−0.142136	−0.00451054
$994$	3.02944	0.0960879
$995$	10.1421	0.321527
$996$	−13.1127	−0.415492
$997$	58.0833	1.83952	0.919758	−	0.392487i	$-0.128385\pi$
$997$	58.0833	1.83952	0.919758	+	0.392487i	$0.128385\pi$
$998$	4.48528	0.141979
$999$	−0.585786	−0.0185335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.a.g.1.1	✓	2
3.2	odd	2		855.2.a.d.1.2		2
4.3	odd	2		4560.2.a.bf.1.1		2
5.2	odd	4		1425.2.c.l.799.2		4
5.3	odd	4		1425.2.c.l.799.3		4
5.4	even	2		1425.2.a.k.1.2		2
15.14	odd	2		4275.2.a.y.1.1		2
19.18	odd	2		5415.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.g.1.1	✓	2	1.1	even	1	trivial
855.2.a.d.1.2		2	3.2	odd	2
1425.2.a.k.1.2		2	5.4	even	2
1425.2.c.l.799.2		4	5.2	odd	4
1425.2.c.l.799.3		4	5.3	odd	4
4275.2.a.y.1.1		2	15.14	odd	2
4560.2.a.bf.1.1		2	4.3	odd	2
5415.2.a.n.1.2		2	19.18	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.41421 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.414214 q^{10} +6.24264 q^{11} -1.82843 q^{12} -0.585786 q^{13} -0.585786 q^{14} -1.00000 q^{15} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} -1.00000 q^{19} +1.82843 q^{20} +1.41421 q^{21} -2.58579 q^{22} -3.65685 q^{23} +1.58579 q^{24} +1.00000 q^{25} +0.242641 q^{26} +1.00000 q^{27} -2.58579 q^{28} -1.41421 q^{29} +0.414214 q^{30} -8.82843 q^{31} -4.41421 q^{32} +6.24264 q^{33} -2.82843 q^{34} -1.41421 q^{35} -1.82843 q^{36} -0.585786 q^{37} +0.414214 q^{38} -0.585786 q^{39} -1.58579 q^{40} +8.24264 q^{41} -0.585786 q^{42} +3.75736 q^{43} -11.4142 q^{44} -1.00000 q^{45} +1.51472 q^{46} +3.65685 q^{47} +3.00000 q^{48} -5.00000 q^{49} -0.414214 q^{50} +6.82843 q^{51} +1.07107 q^{52} +8.00000 q^{53} -0.414214 q^{54} -6.24264 q^{55} +2.24264 q^{56} -1.00000 q^{57} +0.585786 q^{58} -4.48528 q^{59} +1.82843 q^{60} -15.3137 q^{61} +3.65685 q^{62} +1.41421 q^{63} -4.17157 q^{64} +0.585786 q^{65} -2.58579 q^{66} +1.65685 q^{67} -12.4853 q^{68} -3.65685 q^{69} +0.585786 q^{70} -5.17157 q^{71} +1.58579 q^{72} +3.65685 q^{73} +0.242641 q^{74} +1.00000 q^{75} +1.82843 q^{76} +8.82843 q^{77} +0.242641 q^{78} -3.00000 q^{80} +1.00000 q^{81} -3.41421 q^{82} +7.17157 q^{83} -2.58579 q^{84} -6.82843 q^{85} -1.55635 q^{86} -1.41421 q^{87} +9.89949 q^{88} -13.8995 q^{89} +0.414214 q^{90} -0.828427 q^{91} +6.68629 q^{92} -8.82843 q^{93} -1.51472 q^{94} +1.00000 q^{95} -4.41421 q^{96} -18.2426 q^{97} +2.07107 q^{98} +6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{13} - 4 q^{14} - 2 q^{15} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 8 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^13 - 4 * q^14 - 2 * q^15 + 6 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 8 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 8 * q^26 + 2 * q^27 - 8 * q^28 - 2 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 8 * q^41 - 4 * q^42 + 16 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 - 4 * q^47 + 6 * q^48 - 10 * q^49 + 2 * q^50 + 8 * q^51 - 12 * q^52 + 16 * q^53 + 2 * q^54 - 4 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 + 8 * q^59 - 2 * q^60 - 8 * q^61 - 4 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^66 - 8 * q^67 - 8 * q^68 + 4 * q^69 + 4 * q^70 - 16 * q^71 + 6 * q^72 - 4 * q^73 - 8 * q^74 + 2 * q^75 - 2 * q^76 + 12 * q^77 - 8 * q^78 - 6 * q^80 + 2 * q^81 - 4 * q^82 + 20 * q^83 - 8 * q^84 - 8 * q^85 + 28 * q^86 - 8 * q^89 - 2 * q^90 + 4 * q^91 + 36 * q^92 - 12 * q^93 - 20 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 10 * q^98 + 4 * q^99

Introduction

L-functions

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