LMFDB - Embedded newform 6762.2.a.cj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.23252.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.565882$ of defining polynomial
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.722765 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.722765 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.722765 q^{10} +1.13176 q^{11} -1.00000 q^{12} -1.27724 q^{13} +0.722765 q^{15} +1.00000 q^{16} +3.93685 q^{17} +1.00000 q^{18} -2.00000 q^{19} -0.722765 q^{20} +1.13176 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.47761 q^{25} -1.27724 q^{26} -1.00000 q^{27} -5.76855 q^{29} +0.722765 q^{30} -6.20038 q^{31} +1.00000 q^{32} -1.13176 q^{33} +3.93685 q^{34} +1.00000 q^{36} +6.08232 q^{37} -2.00000 q^{38} +1.27724 q^{39} -0.722765 q^{40} -6.65961 q^{41} +5.52785 q^{43} +1.13176 q^{44} -0.722765 q^{45} -1.00000 q^{46} -3.01371 q^{47} -1.00000 q^{48} -4.47761 q^{50} -3.93685 q^{51} -1.27724 q^{52} +1.68623 q^{53} -1.00000 q^{54} -0.817999 q^{55} +2.00000 q^{57} -5.76855 q^{58} -12.9826 q^{59} +0.722765 q^{60} -12.4913 q^{61} -6.20038 q^{62} +1.00000 q^{64} +0.923140 q^{65} -1.13176 q^{66} +11.5599 q^{67} +3.93685 q^{68} +1.00000 q^{69} +11.8737 q^{71} +1.00000 q^{72} -0.290943 q^{73} +6.08232 q^{74} +4.47761 q^{75} -2.00000 q^{76} +1.27724 q^{78} +0.290943 q^{79} -0.722765 q^{80} +1.00000 q^{81} -6.65961 q^{82} +8.42817 q^{83} -2.84541 q^{85} +5.52785 q^{86} +5.76855 q^{87} +1.13176 q^{88} -0.0631518 q^{89} -0.722765 q^{90} -1.00000 q^{92} +6.20038 q^{93} -3.01371 q^{94} +1.44553 q^{95} -1.00000 q^{96} -0.750180 q^{97} +1.13176 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 3 q^{10} + 2 q^{11} - 4 q^{12} - 5 q^{13} + 3 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 8 q^{19} - 3 q^{20} + 2 q^{22} - 4 q^{23} - 4 q^{24} + 13 q^{25} - 5 q^{26} - 4 q^{27} + 3 q^{29} + 3 q^{30} + 6 q^{31} + 4 q^{32} - 2 q^{33} - 10 q^{34} + 4 q^{36} + q^{37} - 8 q^{38} + 5 q^{39} - 3 q^{40} - q^{41} - q^{43} + 2 q^{44} - 3 q^{45} - 4 q^{46} - 17 q^{47} - 4 q^{48} + 13 q^{50} + 10 q^{51} - 5 q^{52} + 4 q^{53} - 4 q^{54} + 2 q^{55} + 8 q^{57} + 3 q^{58} + 3 q^{60} - 24 q^{61} + 6 q^{62} + 4 q^{64} - 27 q^{65} - 2 q^{66} - 8 q^{67} - 10 q^{68} + 4 q^{69} - 4 q^{71} + 4 q^{72} - 6 q^{73} + q^{74} - 13 q^{75} - 8 q^{76} + 5 q^{78} + 6 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 18 q^{83} - 16 q^{85} - q^{86} - 3 q^{87} + 2 q^{88} - 26 q^{89} - 3 q^{90} - 4 q^{92} - 6 q^{93} - 17 q^{94} + 6 q^{95} - 4 q^{96} - 13 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 3 * q^10 + 2 * q^11 - 4 * q^12 - 5 * q^13 + 3 * q^15 + 4 * q^16 - 10 * q^17 + 4 * q^18 - 8 * q^19 - 3 * q^20 + 2 * q^22 - 4 * q^23 - 4 * q^24 + 13 * q^25 - 5 * q^26 - 4 * q^27 + 3 * q^29 + 3 * q^30 + 6 * q^31 + 4 * q^32 - 2 * q^33 - 10 * q^34 + 4 * q^36 + q^37 - 8 * q^38 + 5 * q^39 - 3 * q^40 - q^41 - q^43 + 2 * q^44 - 3 * q^45 - 4 * q^46 - 17 * q^47 - 4 * q^48 + 13 * q^50 + 10 * q^51 - 5 * q^52 + 4 * q^53 - 4 * q^54 + 2 * q^55 + 8 * q^57 + 3 * q^58 + 3 * q^60 - 24 * q^61 + 6 * q^62 + 4 * q^64 - 27 * q^65 - 2 * q^66 - 8 * q^67 - 10 * q^68 + 4 * q^69 - 4 * q^71 + 4 * q^72 - 6 * q^73 + q^74 - 13 * q^75 - 8 * q^76 + 5 * q^78 + 6 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 18 * q^83 - 16 * q^85 - q^86 - 3 * q^87 + 2 * q^88 - 26 * q^89 - 3 * q^90 - 4 * q^92 - 6 * q^93 - 17 * q^94 + 6 * q^95 - 4 * q^96 - 13 * q^97 + 2 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.722765	−0.323230	−0.161615	−	0.986854i	$-0.551670\pi$
$5$	−0.722765	−0.323230	−0.161615	+	0.986854i	$0.551670\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.722765	−0.228558
$11$	1.13176	0.341240	0.170620	−	0.985337i	$-0.445423\pi$
$11$	1.13176	0.341240	0.170620	+	0.985337i	$0.445423\pi$
$12$	−1.00000	−0.288675
$13$	−1.27724	−0.354241	−0.177121	−	0.984189i	$-0.556678\pi$
$13$	−1.27724	−0.354241	−0.177121	+	0.984189i	$0.556678\pi$
$14$	0	0
$15$	0.722765	0.186617
$16$	1.00000	0.250000
$17$	3.93685	0.954826	0.477413	−	0.878679i	$-0.341575\pi$
$17$	3.93685	0.954826	0.477413	+	0.878679i	$0.341575\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−0.722765	−0.161615
$21$	0	0
$22$	1.13176	0.241293
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−4.47761	−0.895522
$26$	−1.27724	−0.250486
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.76855	−1.07119	−0.535597	−	0.844474i	$-0.679913\pi$
$29$	−5.76855	−1.07119	−0.535597	+	0.844474i	$0.679913\pi$
$30$	0.722765	0.131958
$31$	−6.20038	−1.11362	−0.556810	−	0.830640i	$-0.687975\pi$
$31$	−6.20038	−1.11362	−0.556810	+	0.830640i	$0.687975\pi$
$32$	1.00000	0.176777
$33$	−1.13176	−0.197015
$34$	3.93685	0.675164
$35$	0	0
$36$	1.00000	0.166667
$37$	6.08232	0.999927	0.499964	−	0.866046i	$-0.333347\pi$
$37$	6.08232	0.999927	0.499964	+	0.866046i	$0.333347\pi$
$38$	−2.00000	−0.324443
$39$	1.27724	0.204521
$40$	−0.722765	−0.114279
$41$	−6.65961	−1.04006	−0.520028	−	0.854149i	$-0.674079\pi$
$41$	−6.65961	−1.04006	−0.520028	+	0.854149i	$0.674079\pi$
$42$	0	0
$43$	5.52785	0.842989	0.421495	−	0.906831i	$-0.361506\pi$
$43$	5.52785	0.842989	0.421495	+	0.906831i	$0.361506\pi$
$44$	1.13176	0.170620
$45$	−0.722765	−0.107743
$46$	−1.00000	−0.147442
$47$	−3.01371	−0.439594	−0.219797	−	0.975546i	$-0.570540\pi$
$47$	−3.01371	−0.439594	−0.219797	+	0.975546i	$0.570540\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.47761	−0.633230
$51$	−3.93685	−0.551269
$52$	−1.27724	−0.177121
$53$	1.68623	0.231622	0.115811	−	0.993271i	$-0.463053\pi$
$53$	1.68623	0.231622	0.115811	+	0.993271i	$0.463053\pi$
$54$	−1.00000	−0.136083
$55$	−0.817999	−0.110299
$56$	0	0
$57$	2.00000	0.264906
$58$	−5.76855	−0.757448
$59$	−12.9826	−1.69020	−0.845098	−	0.534612i	$-0.820458\pi$
$59$	−12.9826	−1.69020	−0.845098	+	0.534612i	$0.820458\pi$
$60$	0.722765	0.0933085
$61$	−12.4913	−1.59935	−0.799675	−	0.600433i	$-0.794995\pi$
$61$	−12.4913	−1.59935	−0.799675	+	0.600433i	$0.794995\pi$
$62$	−6.20038	−0.787449
$63$	0	0
$64$	1.00000	0.125000
$65$	0.923140	0.114501
$66$	−1.13176	−0.139311
$67$	11.5599	1.41227	0.706135	−	0.708077i	$-0.250437\pi$
$67$	11.5599	1.41227	0.706135	+	0.708077i	$0.250437\pi$
$68$	3.93685	0.477413
$69$	1.00000	0.120386
$70$	0	0
$71$	11.8737	1.40915	0.704574	−	0.709630i	$-0.251138\pi$
$71$	11.8737	1.40915	0.704574	+	0.709630i	$0.251138\pi$
$72$	1.00000	0.117851
$73$	−0.290943	−0.0340523	−0.0170262	−	0.999855i	$-0.505420\pi$
$73$	−0.290943	−0.0340523	−0.0170262	+	0.999855i	$0.505420\pi$
$74$	6.08232	0.707055
$75$	4.47761	0.517030
$76$	−2.00000	−0.229416
$77$	0	0
$78$	1.27724	0.144618
$79$	0.290943	0.0327337	0.0163668	−	0.999866i	$-0.494790\pi$
$79$	0.290943	0.0327337	0.0163668	+	0.999866i	$0.494790\pi$
$80$	−0.722765	−0.0808075
$81$	1.00000	0.111111
$82$	−6.65961	−0.735431
$83$	8.42817	0.925112	0.462556	−	0.886590i	$-0.346932\pi$
$83$	8.42817	0.925112	0.462556	+	0.886590i	$0.346932\pi$
$84$	0	0
$85$	−2.84541	−0.308628
$86$	5.52785	0.596083
$87$	5.76855	0.618454
$88$	1.13176	0.120646
$89$	−0.0631518	−0.00669408	−0.00334704	−	0.999994i	$-0.501065\pi$
$89$	−0.0631518	−0.00669408	−0.00334704	+	0.999994i	$0.501065\pi$
$90$	−0.722765	−0.0761861
$91$	0	0
$92$	−1.00000	−0.104257
$93$	6.20038	0.642949
$94$	−3.01371	−0.310840
$95$	1.44553	0.148308
$96$	−1.00000	−0.102062
$97$	−0.750180	−0.0761692	−0.0380846	−	0.999275i	$-0.512126\pi$
$97$	−0.750180	−0.0761692	−0.0380846	+	0.999275i	$0.512126\pi$
$98$	0	0
$99$	1.13176	0.113747
$100$	−4.47761	−0.447761
$101$	16.9195	1.68355	0.841776	−	0.539827i	$-0.181510\pi$
$101$	16.9195	1.68355	0.841776	+	0.539827i	$0.181510\pi$
$102$	−3.93685	−0.389806
$103$	−10.6596	−1.05032	−0.525161	−	0.851003i	$-0.675995\pi$
$103$	−10.6596	−1.05032	−0.525161	+	0.851003i	$0.675995\pi$
$104$	−1.27724	−0.125243
$105$	0	0
$106$	1.68623	0.163782
$107$	−19.5599	−1.89093	−0.945465	−	0.325724i	$-0.894392\pi$
$107$	−19.5599	−1.89093	−0.945465	+	0.325724i	$0.894392\pi$
$108$	−1.00000	−0.0962250
$109$	−4.05490	−0.388389	−0.194195	−	0.980963i	$-0.562209\pi$
$109$	−4.05490	−0.388389	−0.194195	+	0.980963i	$0.562209\pi$
$110$	−0.817999	−0.0779931
$111$	−6.08232	−0.577308
$112$	0	0
$113$	4.65961	0.438339	0.219170	−	0.975687i	$-0.429665\pi$
$113$	4.65961	0.438339	0.219170	+	0.975687i	$0.429665\pi$
$114$	2.00000	0.187317
$115$	0.722765	0.0673981
$116$	−5.76855	−0.535597
$117$	−1.27724	−0.118080
$118$	−12.9826	−1.19515
$119$	0	0
$120$	0.722765	0.0659791
$121$	−9.71911	−0.883555
$122$	−12.4913	−1.13091
$123$	6.65961	0.600477
$124$	−6.20038	−0.556810
$125$	6.85008	0.612690
$126$	0	0
$127$	−17.0878	−1.51630	−0.758148	−	0.652083i	$-0.773895\pi$
$127$	−17.0878	−1.51630	−0.758148	+	0.652083i	$0.773895\pi$
$128$	1.00000	0.0883883
$129$	−5.52785	−0.486700
$130$	0.923140	0.0809648
$131$	−12.4008	−1.08346	−0.541729	−	0.840553i	$-0.682230\pi$
$131$	−12.4008	−1.08346	−0.541729	+	0.840553i	$0.682230\pi$
$132$	−1.13176	−0.0985074
$133$	0	0
$134$	11.5599	0.998626
$135$	0.722765	0.0622057
$136$	3.93685	0.337582
$137$	8.03208	0.686227	0.343114	−	0.939294i	$-0.388518\pi$
$137$	8.03208	0.686227	0.343114	+	0.939294i	$0.388518\pi$
$138$	1.00000	0.0851257
$139$	1.34039	0.113690	0.0568451	−	0.998383i	$-0.481896\pi$
$139$	1.34039	0.113690	0.0568451	+	0.998383i	$0.481896\pi$
$140$	0	0
$141$	3.01371	0.253800
$142$	11.8737	0.996418
$143$	−1.44553	−0.120881
$144$	1.00000	0.0833333
$145$	4.16931	0.346242
$146$	−0.290943	−0.0240786
$147$	0	0
$148$	6.08232	0.499964
$149$	−18.7145	−1.53315	−0.766576	−	0.642153i	$-0.778041\pi$
$149$	−18.7145	−1.53315	−0.766576	+	0.642153i	$0.778041\pi$
$150$	4.47761	0.365595
$151$	−8.10514	−0.659587	−0.329794	−	0.944053i	$-0.606979\pi$
$151$	−8.10514	−0.659587	−0.329794	+	0.944053i	$0.606979\pi$
$152$	−2.00000	−0.162221
$153$	3.93685	0.318275
$154$	0	0
$155$	4.48141	0.359956
$156$	1.27724	0.102261
$157$	−7.04579	−0.562315	−0.281158	−	0.959662i	$-0.590718\pi$
$157$	−7.04579	−0.562315	−0.281158	+	0.959662i	$0.590718\pi$
$158$	0.290943	0.0231462
$159$	−1.68623	−0.133727
$160$	−0.722765	−0.0571396
$161$	0	0
$162$	1.00000	0.0785674
$163$	8.40075	0.657998	0.328999	−	0.944330i	$-0.393289\pi$
$163$	8.40075	0.657998	0.328999	+	0.944330i	$0.393289\pi$
$164$	−6.65961	−0.520028
$165$	0.817999	0.0636811
$166$	8.42817	0.654153
$167$	6.78226	0.524827	0.262414	−	0.964955i	$-0.415482\pi$
$167$	6.78226	0.524827	0.262414	+	0.964955i	$0.415482\pi$
$168$	0	0
$169$	−11.3687	−0.874513
$170$	−2.84541	−0.218233
$171$	−2.00000	−0.152944
$172$	5.52785	0.421495
$173$	−15.0184	−1.14183	−0.570913	−	0.821011i	$-0.693411\pi$
$173$	−15.0184	−1.14183	−0.570913	+	0.821011i	$0.693411\pi$
$174$	5.76855	0.437313
$175$	0	0
$176$	1.13176	0.0853099
$177$	12.9826	0.975835
$178$	−0.0631518	−0.00473343
$179$	−2.03208	−0.151885	−0.0759425	−	0.997112i	$-0.524197\pi$
$179$	−2.03208	−0.151885	−0.0759425	+	0.997112i	$0.524197\pi$
$180$	−0.722765	−0.0538717
$181$	8.82791	0.656173	0.328087	−	0.944648i	$-0.393596\pi$
$181$	8.82791	0.656173	0.328087	+	0.944648i	$0.393596\pi$
$182$	0	0
$183$	12.4913	0.923385
$184$	−1.00000	−0.0737210
$185$	−4.39608	−0.323207
$186$	6.20038	0.454634
$187$	4.45558	0.325824
$188$	−3.01371	−0.219797
$189$	0	0
$190$	1.44553	0.104870
$191$	4.98264	0.360531	0.180266	−	0.983618i	$-0.442304\pi$
$191$	4.98264	0.360531	0.180266	+	0.983618i	$0.442304\pi$
$192$	−1.00000	−0.0721688
$193$	−7.84161	−0.564452	−0.282226	−	0.959348i	$-0.591073\pi$
$193$	−7.84161	−0.564452	−0.282226	+	0.959348i	$0.591073\pi$
$194$	−0.750180	−0.0538598
$195$	−0.923140	−0.0661075
$196$	0	0
$197$	−15.0878	−1.07496	−0.537480	−	0.843277i	$-0.680624\pi$
$197$	−15.0878	−1.07496	−0.537480	+	0.843277i	$0.680624\pi$
$198$	1.13176	0.0804310
$199$	14.5333	1.03024	0.515119	−	0.857118i	$-0.327748\pi$
$199$	14.5333	1.03024	0.515119	+	0.857118i	$0.327748\pi$
$200$	−4.47761	−0.316615
$201$	−11.5599	−0.815375
$202$	16.9195	1.19045
$203$	0	0
$204$	−3.93685	−0.275635
$205$	4.81333	0.336178
$206$	−10.6596	−0.742690
$207$	−1.00000	−0.0695048
$208$	−1.27724	−0.0885603
$209$	−2.26353	−0.156571
$210$	0	0
$211$	−3.70906	−0.255342	−0.127671	−	0.991817i	$-0.540750\pi$
$211$	−3.70906	−0.255342	−0.127671	+	0.991817i	$0.540750\pi$
$212$	1.68623	0.115811
$213$	−11.8737	−0.813572
$214$	−19.5599	−1.33709
$215$	−3.99533	−0.272479
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−4.05490	−0.274633
$219$	0.290943	0.0196601
$220$	−0.817999	−0.0551495
$221$	−5.02828	−0.338239
$222$	−6.08232	−0.408219
$223$	−5.86379	−0.392668	−0.196334	−	0.980537i	$-0.562904\pi$
$223$	−5.86379	−0.392668	−0.196334	+	0.980537i	$0.562904\pi$
$224$	0	0
$225$	−4.47761	−0.298507
$226$	4.65961	0.309953
$227$	4.13256	0.274287	0.137144	−	0.990551i	$-0.456208\pi$
$227$	4.13256	0.274287	0.137144	+	0.990551i	$0.456208\pi$
$228$	2.00000	0.132453
$229$	−20.4913	−1.35410	−0.677052	−	0.735935i	$-0.736743\pi$
$229$	−20.4913	−1.35410	−0.677052	+	0.735935i	$0.736743\pi$
$230$	0.722765	0.0476577
$231$	0	0
$232$	−5.76855	−0.378724
$233$	−14.8180	−0.970759	−0.485380	−	0.874303i	$-0.661319\pi$
$233$	−14.8180	−0.970759	−0.485380	+	0.874303i	$0.661319\pi$
$234$	−1.27724	−0.0834955
$235$	2.17820	0.142090
$236$	−12.9826	−0.845098
$237$	−0.290943	−0.0188988
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0.722765	0.0466542
$241$	9.80429	0.631550	0.315775	−	0.948834i	$-0.397736\pi$
$241$	9.80429	0.631550	0.315775	+	0.948834i	$0.397736\pi$
$242$	−9.71911	−0.624768
$243$	−1.00000	−0.0641500
$244$	−12.4913	−0.799675
$245$	0	0
$246$	6.65961	0.424601
$247$	2.55447	0.162537
$248$	−6.20038	−0.393724
$249$	−8.42817	−0.534113
$250$	6.85008	0.433237
$251$	−16.1967	−1.02233	−0.511164	−	0.859483i	$-0.670786\pi$
$251$	−16.1967	−1.02233	−0.511164	+	0.859483i	$0.670786\pi$
$252$	0	0
$253$	−1.13176	−0.0711534
$254$	−17.0878	−1.07218
$255$	2.84541	0.178187
$256$	1.00000	0.0625000
$257$	−21.4455	−1.33774	−0.668868	−	0.743381i	$-0.733221\pi$
$257$	−21.4455	−1.33774	−0.668868	+	0.743381i	$0.733221\pi$
$258$	−5.52785	−0.344149
$259$	0	0
$260$	0.923140	0.0572507
$261$	−5.76855	−0.357065
$262$	−12.4008	−0.766121
$263$	11.7686	0.725680	0.362840	−	0.931851i	$-0.381807\pi$
$263$	11.7686	0.725680	0.362840	+	0.931851i	$0.381807\pi$
$264$	−1.13176	−0.0696553
$265$	−1.21875	−0.0748673
$266$	0	0
$267$	0.0631518	0.00386483
$268$	11.5599	0.706135
$269$	−23.8105	−1.45175	−0.725877	−	0.687824i	$-0.758566\pi$
$269$	−23.8105	−1.45175	−0.725877	+	0.687824i	$0.758566\pi$
$270$	0.722765	0.0439860
$271$	−18.4913	−1.12327	−0.561634	−	0.827386i	$-0.689827\pi$
$271$	−18.4913	−1.12327	−0.561634	+	0.827386i	$0.689827\pi$
$272$	3.93685	0.238706
$273$	0	0
$274$	8.03208	0.485236
$275$	−5.06760	−0.305588
$276$	1.00000	0.0601929
$277$	13.1199	0.788296	0.394148	−	0.919047i	$-0.371040\pi$
$277$	13.1199	0.788296	0.394148	+	0.919047i	$0.371040\pi$
$278$	1.34039	0.0803911
$279$	−6.20038	−0.371207
$280$	0	0
$281$	−13.2415	−0.789922	−0.394961	−	0.918698i	$-0.629242\pi$
$281$	−13.2415	−0.789922	−0.394961	+	0.918698i	$0.629242\pi$
$282$	3.01371	0.179464
$283$	−16.9552	−1.00788	−0.503942	−	0.863738i	$-0.668117\pi$
$283$	−16.9552	−1.00788	−0.503942	+	0.863738i	$0.668117\pi$
$284$	11.8737	0.704574
$285$	−1.44553	−0.0856257
$286$	−1.44553	−0.0854759
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.50123	−0.0883074
$290$	4.16931	0.244830
$291$	0.750180	0.0439763
$292$	−0.290943	−0.0170262
$293$	−5.35496	−0.312840	−0.156420	−	0.987691i	$-0.549995\pi$
$293$	−5.35496	−0.312840	−0.156420	+	0.987691i	$0.549995\pi$
$294$	0	0
$295$	9.38339	0.546322
$296$	6.08232	0.353528
$297$	−1.13176	−0.0656716
$298$	−18.7145	−1.08410
$299$	1.27724	0.0738644
$300$	4.47761	0.258515
$301$	0	0
$302$	−8.10514	−0.466398
$303$	−16.9195	−0.971999
$304$	−2.00000	−0.114708
$305$	9.02828	0.516958
$306$	3.93685	0.225055
$307$	33.2782	1.89929	0.949645	−	0.313328i	$-0.101444\pi$
$307$	33.2782	1.89929	0.949645	+	0.313328i	$0.101444\pi$
$308$	0	0
$309$	10.6596	0.606404
$310$	4.48141	0.254527
$311$	12.4639	0.706763	0.353382	−	0.935479i	$-0.385032\pi$
$311$	12.4639	0.706763	0.353382	+	0.935479i	$0.385032\pi$
$312$	1.27724	0.0723092
$313$	−19.0184	−1.07498	−0.537491	−	0.843269i	$-0.680628\pi$
$313$	−19.0184	−1.07498	−0.537491	+	0.843269i	$0.680628\pi$
$314$	−7.04579	−0.397617
$315$	0	0
$316$	0.290943	0.0163668
$317$	−2.23145	−0.125330	−0.0626652	−	0.998035i	$-0.519960\pi$
$317$	−2.23145	−0.125330	−0.0626652	+	0.998035i	$0.519960\pi$
$318$	−1.68623	−0.0945594
$319$	−6.52864	−0.365534
$320$	−0.722765	−0.0404038
$321$	19.5599	1.09173
$322$	0	0
$323$	−7.87370	−0.438104
$324$	1.00000	0.0555556
$325$	5.71896	0.317231
$326$	8.40075	0.465275
$327$	4.05490	0.224237
$328$	−6.65961	−0.367716
$329$	0	0
$330$	0.817999	0.0450294
$331$	9.10894	0.500673	0.250336	−	0.968159i	$-0.419459\pi$
$331$	9.10894	0.500673	0.250336	+	0.968159i	$0.419459\pi$
$332$	8.42817	0.462556
$333$	6.08232	0.333309
$334$	6.78226	0.371109
$335$	−8.35511	−0.456488
$336$	0	0
$337$	18.8563	1.02717	0.513585	−	0.858039i	$-0.328317\pi$
$337$	18.8563	1.02717	0.513585	+	0.858039i	$0.328317\pi$
$338$	−11.3687	−0.618374
$339$	−4.65961	−0.253075
$340$	−2.84541	−0.154314
$341$	−7.01736	−0.380011
$342$	−2.00000	−0.108148
$343$	0	0
$344$	5.52785	0.298042
$345$	−0.722765	−0.0389123
$346$	−15.0184	−0.807393
$347$	8.10514	0.435107	0.217553	−	0.976048i	$-0.430192\pi$
$347$	8.10514	0.435107	0.217553	+	0.976048i	$0.430192\pi$
$348$	5.76855	0.309227
$349$	14.4913	0.775703	0.387851	−	0.921722i	$-0.373217\pi$
$349$	14.4913	0.775703	0.387851	+	0.921722i	$0.373217\pi$
$350$	0	0
$351$	1.27724	0.0681738
$352$	1.13176	0.0603232
$353$	−3.60392	−0.191817	−0.0959085	−	0.995390i	$-0.530576\pi$
$353$	−3.60392	−0.191817	−0.0959085	+	0.995390i	$0.530576\pi$
$354$	12.9826	0.690019
$355$	−8.58189	−0.455479
$356$	−0.0631518	−0.00334704
$357$	0	0
$358$	−2.03208	−0.107399
$359$	−30.5333	−1.61149	−0.805743	−	0.592265i	$-0.798234\pi$
$359$	−30.5333	−1.61149	−0.805743	+	0.592265i	$0.798234\pi$
$360$	−0.722765	−0.0380930
$361$	−15.0000	−0.789474
$362$	8.82791	0.463984
$363$	9.71911	0.510121
$364$	0	0
$365$	0.210283	0.0110067
$366$	12.4913	0.652932
$367$	16.7512	0.874405	0.437203	−	0.899363i	$-0.355969\pi$
$367$	16.7512	0.874405	0.437203	+	0.899363i	$0.355969\pi$
$368$	−1.00000	−0.0521286
$369$	−6.65961	−0.346686
$370$	−4.39608	−0.228542
$371$	0	0
$372$	6.20038	0.321475
$373$	13.6862	0.708646	0.354323	−	0.935123i	$-0.384711\pi$
$373$	13.6862	0.708646	0.354323	+	0.935123i	$0.384711\pi$
$374$	4.45558	0.230393
$375$	−6.85008	−0.353737
$376$	−3.01371	−0.155420
$377$	7.36780	0.379461
$378$	0	0
$379$	−7.79138	−0.400216	−0.200108	−	0.979774i	$-0.564129\pi$
$379$	−7.79138	−0.400216	−0.200108	+	0.979774i	$0.564129\pi$
$380$	1.44553	0.0741541
$381$	17.0878	0.875433
$382$	4.98264	0.254934
$383$	−0.462891	−0.0236526	−0.0118263	−	0.999930i	$-0.503765\pi$
$383$	−0.462891	−0.0236526	−0.0118263	+	0.999930i	$0.503765\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−7.84161	−0.399128
$387$	5.52785	0.280996
$388$	−0.750180	−0.0380846
$389$	−24.8061	−1.25772	−0.628860	−	0.777519i	$-0.716478\pi$
$389$	−24.8061	−1.25772	−0.628860	+	0.777519i	$0.716478\pi$
$390$	−0.923140	−0.0467450
$391$	−3.93685	−0.199095
$392$	0	0
$393$	12.4008	0.625535
$394$	−15.0878	−0.760111
$395$	−0.210283	−0.0105805
$396$	1.13176	0.0568733
$397$	−2.03574	−0.102171	−0.0510853	−	0.998694i	$-0.516268\pi$
$397$	−2.03574	−0.102171	−0.0510853	+	0.998694i	$0.516268\pi$
$398$	14.5333	0.728489
$399$	0	0
$400$	−4.47761	−0.223881
$401$	9.11986	0.455424	0.227712	−	0.973729i	$-0.426875\pi$
$401$	9.11986	0.455424	0.227712	+	0.973729i	$0.426875\pi$
$402$	−11.5599	−0.576557
$403$	7.91934	0.394490
$404$	16.9195	0.841776
$405$	−0.722765	−0.0359145
$406$	0	0
$407$	6.88375	0.341215
$408$	−3.93685	−0.194903
$409$	14.9552	0.739488	0.369744	−	0.929134i	$-0.379445\pi$
$409$	14.9552	0.739488	0.369744	+	0.929134i	$0.379445\pi$
$410$	4.81333	0.237714
$411$	−8.03208	−0.396193
$412$	−10.6596	−0.525161
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−6.09158	−0.299024
$416$	−1.27724	−0.0626216
$417$	−1.34039	−0.0656390
$418$	−2.26353	−0.110713
$419$	20.9552	1.02373	0.511865	−	0.859066i	$-0.328955\pi$
$419$	20.9552	1.02373	0.511865	+	0.859066i	$0.328955\pi$
$420$	0	0
$421$	9.32849	0.454643	0.227321	−	0.973820i	$-0.427003\pi$
$421$	9.32849	0.454643	0.227321	+	0.973820i	$0.427003\pi$
$422$	−3.70906	−0.180554
$423$	−3.01371	−0.146531
$424$	1.68623	0.0818908
$425$	−17.6277	−0.855068
$426$	−11.8737	−0.575282
$427$	0	0
$428$	−19.5599	−0.945465
$429$	1.44553	0.0697908
$430$	−3.99533	−0.192672
$431$	13.4502	0.647873	0.323937	−	0.946079i	$-0.394994\pi$
$431$	13.4502	0.647873	0.323937	+	0.946079i	$0.394994\pi$
$432$	−1.00000	−0.0481125
$433$	28.5690	1.37294	0.686470	−	0.727158i	$-0.259159\pi$
$433$	28.5690	1.37294	0.686470	+	0.727158i	$0.259159\pi$
$434$	0	0
$435$	−4.16931	−0.199903
$436$	−4.05490	−0.194195
$437$	2.00000	0.0956730
$438$	0.290943	0.0139018
$439$	0.0631518	0.00301407	0.00150704	−	0.999999i	$-0.499520\pi$
$439$	0.0631518	0.00301407	0.00150704	+	0.999999i	$0.499520\pi$
$440$	−0.817999	−0.0389966
$441$	0	0
$442$	−5.02828	−0.239171
$443$	7.64225	0.363094	0.181547	−	0.983382i	$-0.441890\pi$
$443$	7.64225	0.363094	0.181547	+	0.983382i	$0.441890\pi$
$444$	−6.08232	−0.288654
$445$	0.0456439	0.00216373
$446$	−5.86379	−0.277658
$447$	18.7145	0.885166
$448$	0	0
$449$	−6.52706	−0.308031	−0.154015	−	0.988068i	$-0.549221\pi$
$449$	−6.52706	−0.308031	−0.154015	+	0.988068i	$0.549221\pi$
$450$	−4.47761	−0.211077
$451$	−7.53711	−0.354909
$452$	4.65961	0.219170
$453$	8.10514	0.380813
$454$	4.13256	0.193951
$455$	0	0
$456$	2.00000	0.0936586
$457$	36.4738	1.70617	0.853086	−	0.521770i	$-0.174728\pi$
$457$	36.4738	1.70617	0.853086	+	0.521770i	$0.174728\pi$
$458$	−20.4913	−0.957496
$459$	−3.93685	−0.183756
$460$	0.722765	0.0336991
$461$	23.3550	1.08775	0.543875	−	0.839167i	$-0.316957\pi$
$461$	23.3550	1.08775	0.543875	+	0.839167i	$0.316957\pi$
$462$	0	0
$463$	−35.5159	−1.65057	−0.825283	−	0.564719i	$-0.808984\pi$
$463$	−35.5159	−1.65057	−0.825283	+	0.564719i	$0.808984\pi$
$464$	−5.76855	−0.267798
$465$	−4.48141	−0.207820
$466$	−14.8180	−0.686431
$467$	−40.9341	−1.89420	−0.947101	−	0.320935i	$-0.896003\pi$
$467$	−40.9341	−1.89420	−0.947101	+	0.320935i	$0.896003\pi$
$468$	−1.27724	−0.0590402
$469$	0	0
$470$	2.17820	0.100473
$471$	7.04579	0.324653
$472$	−12.9826	−0.597574
$473$	6.25622	0.287661
$474$	−0.290943	−0.0133635
$475$	8.95522	0.410894
$476$	0	0
$477$	1.68623	0.0772074
$478$	−8.00000	−0.365911
$479$	1.97258	0.0901297	0.0450648	−	0.998984i	$-0.485651\pi$
$479$	1.97258	0.0901297	0.0450648	+	0.998984i	$0.485651\pi$
$480$	0.722765	0.0329895
$481$	−7.76855	−0.354216
$482$	9.80429	0.446573
$483$	0	0
$484$	−9.71911	−0.441778
$485$	0.542203	0.0246202
$486$	−1.00000	−0.0453609
$487$	−36.5701	−1.65715	−0.828574	−	0.559880i	$-0.810847\pi$
$487$	−36.5701	−1.65715	−0.828574	+	0.559880i	$0.810847\pi$
$488$	−12.4913	−0.565455
$489$	−8.40075	−0.379895
$490$	0	0
$491$	−30.1829	−1.36213	−0.681067	−	0.732221i	$-0.738484\pi$
$491$	−30.1829	−1.36213	−0.681067	+	0.732221i	$0.738484\pi$
$492$	6.65961	0.300239
$493$	−22.7099	−1.02280
$494$	2.55447	0.114931
$495$	−0.817999	−0.0367663
$496$	−6.20038	−0.278405
$497$	0	0
$498$	−8.42817	−0.377675
$499$	27.4108	1.22708	0.613538	−	0.789665i	$-0.289746\pi$
$499$	27.4108	1.22708	0.613538	+	0.789665i	$0.289746\pi$
$500$	6.85008	0.306345
$501$	−6.78226	−0.303009
$502$	−16.1967	−0.722895
$503$	21.5097	0.959070	0.479535	−	0.877523i	$-0.340805\pi$
$503$	21.5097	0.959070	0.479535	+	0.877523i	$0.340805\pi$
$504$	0	0
$505$	−12.2288	−0.544175
$506$	−1.13176	−0.0503130
$507$	11.3687	0.504900
$508$	−17.0878	−0.758148
$509$	−8.85532	−0.392505	−0.196253	−	0.980553i	$-0.562877\pi$
$509$	−8.85532	−0.392505	−0.196253	+	0.980553i	$0.562877\pi$
$510$	2.84541	0.125997
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−21.4455	−0.945922
$515$	7.70439	0.339496
$516$	−5.52785	−0.243350
$517$	−3.41081	−0.150007
$518$	0	0
$519$	15.0184	0.659233
$520$	0.923140	0.0404824
$521$	−31.7484	−1.39092	−0.695461	−	0.718563i	$-0.744800\pi$
$521$	−31.7484	−1.39092	−0.695461	+	0.718563i	$0.744800\pi$
$522$	−5.76855	−0.252483
$523$	17.4729	0.764039	0.382019	−	0.924154i	$-0.375229\pi$
$523$	17.4729	0.764039	0.382019	+	0.924154i	$0.375229\pi$
$524$	−12.4008	−0.541729
$525$	0	0
$526$	11.7686	0.513133
$527$	−24.4099	−1.06331
$528$	−1.13176	−0.0492537
$529$	1.00000	0.0434783
$530$	−1.21875	−0.0529391
$531$	−12.9826	−0.563399
$532$	0	0
$533$	8.50589	0.368431
$534$	0.0631518	0.00273285
$535$	14.1372	0.611205
$536$	11.5599	0.499313
$537$	2.03208	0.0876908
$538$	−23.8105	−1.02655
$539$	0	0
$540$	0.722765	0.0311028
$541$	45.5828	1.95976	0.979878	−	0.199598i	$-0.0639637\pi$
$541$	45.5828	1.95976	0.979878	+	0.199598i	$0.0639637\pi$
$542$	−18.4913	−0.794270
$543$	−8.82791	−0.378842
$544$	3.93685	0.168791
$545$	2.93074	0.125539
$546$	0	0
$547$	16.0642	0.686854	0.343427	−	0.939179i	$-0.388412\pi$
$547$	16.0642	0.686854	0.343427	+	0.939179i	$0.388412\pi$
$548$	8.03208	0.343114
$549$	−12.4913	−0.533117
$550$	−5.06760	−0.216083
$551$	11.5371	0.491497
$552$	1.00000	0.0425628
$553$	0	0
$554$	13.1199	0.557410
$555$	4.39608	0.186603
$556$	1.34039	0.0568451
$557$	1.39529	0.0591204	0.0295602	−	0.999563i	$-0.490589\pi$
$557$	1.39529	0.0591204	0.0295602	+	0.999563i	$0.490589\pi$
$558$	−6.20038	−0.262483
$559$	−7.06036	−0.298622
$560$	0	0
$561$	−4.45558	−0.188115
$562$	−13.2415	−0.558559
$563$	39.8799	1.68074	0.840370	−	0.542014i	$-0.182338\pi$
$563$	39.8799	1.68074	0.840370	+	0.542014i	$0.182338\pi$
$564$	3.01371	0.126900
$565$	−3.36780	−0.141685
$566$	−16.9552	−0.712681
$567$	0	0
$568$	11.8737	0.498209
$569$	−30.5792	−1.28195	−0.640974	−	0.767562i	$-0.721470\pi$
$569$	−30.5792	−1.28195	−0.640974	+	0.767562i	$0.721470\pi$
$570$	−1.44553	−0.0605465
$571$	16.6689	0.697571	0.348785	−	0.937203i	$-0.386594\pi$
$571$	16.6689	0.697571	0.348785	+	0.937203i	$0.386594\pi$
$572$	−1.44553	−0.0604406
$573$	−4.98264	−0.208153
$574$	0	0
$575$	4.47761	0.186729
$576$	1.00000	0.0416667
$577$	0.290943	0.0121121	0.00605606	−	0.999982i	$-0.498072\pi$
$577$	0.290943	0.0121121	0.00605606	+	0.999982i	$0.498072\pi$
$578$	−1.50123	−0.0624428
$579$	7.84161	0.325886
$580$	4.16931	0.173121
$581$	0	0
$582$	0.750180	0.0310960
$583$	1.90842	0.0790387
$584$	−0.290943	−0.0120393
$585$	0.923140	0.0381672
$586$	−5.35496	−0.221211
$587$	16.2103	0.669070	0.334535	−	0.942383i	$-0.391421\pi$
$587$	16.2103	0.669070	0.334535	+	0.942383i	$0.391421\pi$
$588$	0	0
$589$	12.4008	0.510964
$590$	9.38339	0.386308
$591$	15.0878	0.620628
$592$	6.08232	0.249982
$593$	16.9615	0.696524	0.348262	−	0.937397i	$-0.386772\pi$
$593$	16.9615	0.696524	0.348262	+	0.937397i	$0.386772\pi$
$594$	−1.13176	−0.0464368
$595$	0	0
$596$	−18.7145	−0.766576
$597$	−14.5333	−0.594809
$598$	1.27724	0.0522300
$599$	−5.31923	−0.217338	−0.108669	−	0.994078i	$-0.534659\pi$
$599$	−5.31923	−0.217338	−0.108669	+	0.994078i	$0.534659\pi$
$600$	4.47761	0.182798
$601$	34.4665	1.40592	0.702959	−	0.711230i	$-0.251862\pi$
$601$	34.4665	1.40592	0.702959	+	0.711230i	$0.251862\pi$
$602$	0	0
$603$	11.5599	0.470757
$604$	−8.10514	−0.329794
$605$	7.02463	0.285592
$606$	−16.9195	−0.687307
$607$	13.2819	0.539096	0.269548	−	0.962987i	$-0.413126\pi$
$607$	13.2819	0.539096	0.269548	+	0.962987i	$0.413126\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	9.02828	0.365544
$611$	3.84921	0.155723
$612$	3.93685	0.159138
$613$	−33.9560	−1.37147	−0.685735	−	0.727851i	$-0.740519\pi$
$613$	−33.9560	−1.37147	−0.685735	+	0.727851i	$0.740519\pi$
$614$	33.2782	1.34300
$615$	−4.81333	−0.194092
$616$	0	0
$617$	−43.9104	−1.76777	−0.883884	−	0.467706i	$-0.845081\pi$
$617$	−43.9104	−1.76777	−0.883884	+	0.467706i	$0.845081\pi$
$618$	10.6596	0.428793
$619$	−9.68323	−0.389202	−0.194601	−	0.980883i	$-0.562341\pi$
$619$	−9.68323	−0.389202	−0.194601	+	0.980883i	$0.562341\pi$
$620$	4.48141	0.179978
$621$	1.00000	0.0401286
$622$	12.4639	0.499757
$623$	0	0
$624$	1.27724	0.0511303
$625$	17.4371	0.697482
$626$	−19.0184	−0.760127
$627$	2.26353	0.0903966
$628$	−7.04579	−0.281158
$629$	23.9452	0.954756
$630$	0	0
$631$	0.336587	0.0133993	0.00669966	−	0.999978i	$-0.497867\pi$
$631$	0.336587	0.0133993	0.00669966	+	0.999978i	$0.497867\pi$
$632$	0.290943	0.0115731
$633$	3.70906	0.147422
$634$	−2.23145	−0.0886220
$635$	12.3504	0.490112
$636$	−1.68623	−0.0668636
$637$	0	0
$638$	−6.52864	−0.258471
$639$	11.8737	0.469716
$640$	−0.722765	−0.0285698
$641$	−3.45020	−0.136275	−0.0681373	−	0.997676i	$-0.521706\pi$
$641$	−3.45020	−0.136275	−0.0681373	+	0.997676i	$0.521706\pi$
$642$	19.5599	0.771969
$643$	11.6360	0.458879	0.229439	−	0.973323i	$-0.426311\pi$
$643$	11.6360	0.458879	0.229439	+	0.973323i	$0.426311\pi$
$644$	0	0
$645$	3.99533	0.157316
$646$	−7.87370	−0.309786
$647$	−7.68424	−0.302099	−0.151049	−	0.988526i	$-0.548265\pi$
$647$	−7.68424	−0.302099	−0.151049	+	0.988526i	$0.548265\pi$
$648$	1.00000	0.0392837
$649$	−14.6933	−0.576762
$650$	5.71896	0.224316
$651$	0	0
$652$	8.40075	0.328999
$653$	−31.7795	−1.24363	−0.621813	−	0.783165i	$-0.713604\pi$
$653$	−31.7795	−1.24363	−0.621813	+	0.783165i	$0.713604\pi$
$654$	4.05490	0.158559
$655$	8.96282	0.350206
$656$	−6.65961	−0.260014
$657$	−0.290943	−0.0113508
$658$	0	0
$659$	−32.8335	−1.27901	−0.639506	−	0.768786i	$-0.720861\pi$
$659$	−32.8335	−1.27901	−0.639506	+	0.768786i	$0.720861\pi$
$660$	0.817999	0.0318406
$661$	−25.8105	−1.00391	−0.501957	−	0.864893i	$-0.667386\pi$
$661$	−25.8105	−1.00391	−0.501957	+	0.864893i	$0.667386\pi$
$662$	9.10894	0.354029
$663$	5.02828	0.195282
$664$	8.42817	0.327076
$665$	0	0
$666$	6.08232	0.235685
$667$	5.76855	0.223359
$668$	6.78226	0.262414
$669$	5.86379	0.226707
$670$	−8.35511	−0.322786
$671$	−14.1372	−0.545761
$672$	0	0
$673$	−26.9417	−1.03852	−0.519262	−	0.854615i	$-0.673793\pi$
$673$	−26.9417	−1.03852	−0.519262	+	0.854615i	$0.673793\pi$
$674$	18.8563	0.726319
$675$	4.47761	0.172343
$676$	−11.3687	−0.437257
$677$	−1.48127	−0.0569297	−0.0284648	−	0.999595i	$-0.509062\pi$
$677$	−1.48127	−0.0569297	−0.0284648	+	0.999595i	$0.509062\pi$
$678$	−4.65961	−0.178951
$679$	0	0
$680$	−2.84541	−0.109117
$681$	−4.13256	−0.158360
$682$	−7.01736	−0.268709
$683$	−18.1921	−0.696100	−0.348050	−	0.937476i	$-0.613156\pi$
$683$	−18.1921	−0.696100	−0.348050	+	0.937476i	$0.613156\pi$
$684$	−2.00000	−0.0764719
$685$	−5.80530	−0.221809
$686$	0	0
$687$	20.4913	0.781793
$688$	5.52785	0.210747
$689$	−2.15372	−0.0820502
$690$	−0.722765	−0.0275152
$691$	49.6790	1.88988	0.944939	−	0.327246i	$-0.106121\pi$
$691$	49.6790	1.88988	0.944939	+	0.327246i	$0.106121\pi$
$692$	−15.0184	−0.570913
$693$	0	0
$694$	8.10514	0.307667
$695$	−0.968784	−0.0367481
$696$	5.76855	0.218656
$697$	−26.2179	−0.993073
$698$	14.4913	0.548505
$699$	14.8180	0.560468
$700$	0	0
$701$	4.76776	0.180076	0.0900379	−	0.995938i	$-0.471301\pi$
$701$	4.76776	0.180076	0.0900379	+	0.995938i	$0.471301\pi$
$702$	1.27724	0.0482061
$703$	−12.1646	−0.458798
$704$	1.13176	0.0426550
$705$	−2.17820	−0.0820358
$706$	−3.60392	−0.135635
$707$	0	0
$708$	12.9826	0.487917
$709$	8.09588	0.304047	0.152024	−	0.988377i	$-0.451421\pi$
$709$	8.09588	0.304047	0.152024	+	0.988377i	$0.451421\pi$
$710$	−8.58189	−0.322072
$711$	0.290943	0.0109112
$712$	−0.0631518	−0.00236672
$713$	6.20038	0.232206
$714$	0	0
$715$	1.04478	0.0390724
$716$	−2.03208	−0.0759425
$717$	8.00000	0.298765
$718$	−30.5333	−1.13949
$719$	17.6138	0.656885	0.328442	−	0.944524i	$-0.393476\pi$
$719$	17.6138	0.656885	0.328442	+	0.944524i	$0.393476\pi$
$720$	−0.722765	−0.0269358
$721$	0	0
$722$	−15.0000	−0.558242
$723$	−9.80429	−0.364626
$724$	8.82791	0.328087
$725$	25.8293	0.959278
$726$	9.71911	0.360710
$727$	−23.2918	−0.863845	−0.431923	−	0.901911i	$-0.642165\pi$
$727$	−23.2918	−0.863845	−0.431923	+	0.901911i	$0.642165\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.210283	0.00778294
$731$	21.7623	0.804908
$732$	12.4913	0.461692
$733$	−19.9736	−0.737742	−0.368871	−	0.929481i	$-0.620256\pi$
$733$	−19.9736	−0.737742	−0.368871	+	0.929481i	$0.620256\pi$
$734$	16.7512	0.618298
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	13.0831	0.481923
$738$	−6.65961	−0.245144
$739$	11.8737	0.436781	0.218390	−	0.975861i	$-0.429919\pi$
$739$	11.8737	0.436781	0.218390	+	0.975861i	$0.429919\pi$
$740$	−4.39608	−0.161603
$741$	−2.55447	−0.0938408
$742$	0	0
$743$	−21.0283	−0.771453	−0.385726	−	0.922613i	$-0.626049\pi$
$743$	−21.0283	−0.771453	−0.385726	+	0.922613i	$0.626049\pi$
$744$	6.20038	0.227317
$745$	13.5262	0.495561
$746$	13.6862	0.501089
$747$	8.42817	0.308371
$748$	4.45558	0.162912
$749$	0	0
$750$	−6.85008	−0.250130
$751$	4.94430	0.180420	0.0902101	−	0.995923i	$-0.471246\pi$
$751$	4.94430	0.180420	0.0902101	+	0.995923i	$0.471246\pi$
$752$	−3.01371	−0.109899
$753$	16.1967	0.590241
$754$	7.36780	0.268320
$755$	5.85811	0.213198
$756$	0	0
$757$	16.9780	0.617078	0.308539	−	0.951212i	$-0.400160\pi$
$757$	16.9780	0.617078	0.308539	+	0.951212i	$0.400160\pi$
$758$	−7.79138	−0.282996
$759$	1.13176	0.0410804
$760$	1.44553	0.0524348
$761$	13.3999	0.485745	0.242873	−	0.970058i	$-0.421910\pi$
$761$	13.3999	0.485745	0.242873	+	0.970058i	$0.421910\pi$
$762$	17.0878	0.619025
$763$	0	0
$764$	4.98264	0.180266
$765$	−2.84541	−0.102876
$766$	−0.462891	−0.0167249
$767$	16.5819	0.598737
$768$	−1.00000	−0.0360844
$769$	−10.2599	−0.369981	−0.184990	−	0.982740i	$-0.559225\pi$
$769$	−10.2599	−0.369981	−0.184990	+	0.982740i	$0.559225\pi$
$770$	0	0
$771$	21.4455	0.772342
$772$	−7.84161	−0.282226
$773$	−11.7401	−0.422263	−0.211131	−	0.977458i	$-0.567715\pi$
$773$	−11.7401	−0.422263	−0.211131	+	0.977458i	$0.567715\pi$
$774$	5.52785	0.198694
$775$	27.7629	0.997272
$776$	−0.750180	−0.0269299
$777$	0	0
$778$	−24.8061	−0.889342
$779$	13.3192	0.477211
$780$	−0.923140	−0.0330537
$781$	13.4382	0.480857
$782$	−3.93685	−0.140781
$783$	5.76855	0.206151
$784$	0	0
$785$	5.09245	0.181757
$786$	12.4008	0.442320
$787$	−7.25506	−0.258615	−0.129307	−	0.991605i	$-0.541275\pi$
$787$	−7.25506	−0.258615	−0.129307	+	0.991605i	$0.541275\pi$
$788$	−15.0878	−0.537480
$789$	−11.7686	−0.418972
$790$	−0.210283	−0.00748155
$791$	0	0
$792$	1.13176	0.0402155
$793$	15.9544	0.566556
$794$	−2.03574	−0.0722456
$795$	1.21875	0.0432246
$796$	14.5333	0.515119
$797$	−40.3439	−1.42905	−0.714526	−	0.699609i	$-0.753358\pi$
$797$	−40.3439	−1.42905	−0.714526	+	0.699609i	$0.753358\pi$
$798$	0	0
$799$	−11.8645	−0.419736
$800$	−4.47761	−0.158307
$801$	−0.0631518	−0.00223136
$802$	9.11986	0.322034
$803$	−0.329279	−0.0116200
$804$	−11.5599	−0.407687
$805$	0	0
$806$	7.91934	0.278947
$807$	23.8105	0.838171
$808$	16.9195	0.595225
$809$	6.35511	0.223434	0.111717	−	0.993740i	$-0.464365\pi$
$809$	6.35511	0.223434	0.111717	+	0.993740i	$0.464365\pi$
$810$	−0.722765	−0.0253954
$811$	37.4885	1.31640	0.658200	−	0.752843i	$-0.271318\pi$
$811$	37.4885	1.31640	0.658200	+	0.752843i	$0.271318\pi$
$812$	0	0
$813$	18.4913	0.648519
$814$	6.88375	0.241275
$815$	−6.07177	−0.212685
$816$	−3.93685	−0.137817
$817$	−11.0557	−0.386790
$818$	14.9552	0.522897
$819$	0	0
$820$	4.81333	0.168089
$821$	−7.20942	−0.251610	−0.125805	−	0.992055i	$-0.540151\pi$
$821$	−7.20942	−0.251610	−0.125805	+	0.992055i	$0.540151\pi$
$822$	−8.03208	−0.280151
$823$	24.8973	0.867866	0.433933	−	0.900945i	$-0.357126\pi$
$823$	24.8973	0.867866	0.433933	+	0.900945i	$0.357126\pi$
$824$	−10.6596	−0.371345
$825$	5.06760	0.176431
$826$	0	0
$827$	20.0685	0.697849	0.348924	−	0.937151i	$-0.386547\pi$
$827$	20.0685	0.697849	0.348924	+	0.937151i	$0.386547\pi$
$828$	−1.00000	−0.0347524
$829$	24.5829	0.853799	0.426900	−	0.904299i	$-0.359606\pi$
$829$	24.5829	0.853799	0.426900	+	0.904299i	$0.359606\pi$
$830$	−6.09158	−0.211442
$831$	−13.1199	−0.455123
$832$	−1.27724	−0.0442802
$833$	0	0
$834$	−1.34039	−0.0464138
$835$	−4.90198	−0.169640
$836$	−2.26353	−0.0782857
$837$	6.20038	0.214316
$838$	20.9552	0.723886
$839$	26.6385	0.919661	0.459831	−	0.888007i	$-0.347910\pi$
$839$	26.6385	0.919661	0.459831	+	0.888007i	$0.347910\pi$
$840$	0	0
$841$	4.27622	0.147456
$842$	9.32849	0.321481
$843$	13.2415	0.456061
$844$	−3.70906	−0.127671
$845$	8.21687	0.282669
$846$	−3.01371	−0.103613
$847$	0	0
$848$	1.68623	0.0579055
$849$	16.9552	0.581902
$850$	−17.6277	−0.604624
$851$	−6.08232	−0.208499
$852$	−11.8737	−0.406786
$853$	−2.32201	−0.0795042	−0.0397521	−	0.999210i	$-0.512657\pi$
$853$	−2.32201	−0.0795042	−0.0397521	+	0.999210i	$0.512657\pi$
$854$	0	0
$855$	1.44553	0.0494360
$856$	−19.5599	−0.668545
$857$	−20.9780	−0.716594	−0.358297	−	0.933608i	$-0.616642\pi$
$857$	−20.9780	−0.716594	−0.358297	+	0.933608i	$0.616642\pi$
$858$	1.44553	0.0493495
$859$	−4.75119	−0.162109	−0.0810543	−	0.996710i	$-0.525829\pi$
$859$	−4.75119	−0.162109	−0.0810543	+	0.996710i	$0.525829\pi$
$860$	−3.99533	−0.136240
$861$	0	0
$862$	13.4502	0.458115
$863$	−23.6561	−0.805263	−0.402632	−	0.915362i	$-0.631904\pi$
$863$	−23.6561	−0.805263	−0.402632	+	0.915362i	$0.631904\pi$
$864$	−1.00000	−0.0340207
$865$	10.8547	0.369073
$866$	28.5690	0.970816
$867$	1.50123	0.0509843
$868$	0	0
$869$	0.329279	0.0111700
$870$	−4.16931	−0.141353
$871$	−14.7648	−0.500285
$872$	−4.05490	−0.137316
$873$	−0.750180	−0.0253897
$874$	2.00000	0.0676510
$875$	0	0
$876$	0.290943	0.00983006
$877$	18.6369	0.629322	0.314661	−	0.949204i	$-0.398109\pi$
$877$	18.6369	0.629322	0.314661	+	0.949204i	$0.398109\pi$
$878$	0.0631518	0.00213127
$879$	5.35496	0.180618
$880$	−0.817999	−0.0275747
$881$	33.5655	1.13085	0.565426	−	0.824799i	$-0.308712\pi$
$881$	33.5655	1.13085	0.565426	+	0.824799i	$0.308712\pi$
$882$	0	0
$883$	37.8846	1.27492	0.637459	−	0.770484i	$-0.279985\pi$
$883$	37.8846	1.27492	0.637459	+	0.770484i	$0.279985\pi$
$884$	−5.02828	−0.169119
$885$	−9.38339	−0.315419
$886$	7.64225	0.256746
$887$	−8.29196	−0.278417	−0.139208	−	0.990263i	$-0.544456\pi$
$887$	−8.29196	−0.278417	−0.139208	+	0.990263i	$0.544456\pi$
$888$	−6.08232	−0.204109
$889$	0	0
$890$	0.0456439	0.00152999
$891$	1.13176	0.0379155
$892$	−5.86379	−0.196334
$893$	6.02742	0.201700
$894$	18.7145	0.625907
$895$	1.46872	0.0490938
$896$	0	0
$897$	−1.27724	−0.0426457
$898$	−6.52706	−0.217811
$899$	35.7672	1.19290
$900$	−4.47761	−0.149254
$901$	6.63845	0.221159
$902$	−7.53711	−0.250958
$903$	0	0
$904$	4.65961	0.154976
$905$	−6.38050	−0.212095
$906$	8.10514	0.269275
$907$	−19.4383	−0.645438	−0.322719	−	0.946495i	$-0.604597\pi$
$907$	−19.4383	−0.645438	−0.322719	+	0.946495i	$0.604597\pi$
$908$	4.13256	0.137144
$909$	16.9195	0.561184
$910$	0	0
$911$	−8.42191	−0.279030	−0.139515	−	0.990220i	$-0.544554\pi$
$911$	−8.42191	−0.279030	−0.139515	+	0.990220i	$0.544554\pi$
$912$	2.00000	0.0662266
$913$	9.53870	0.315685
$914$	36.4738	1.20645
$915$	−9.02828	−0.298466
$916$	−20.4913	−0.677052
$917$	0	0
$918$	−3.93685	−0.129935
$919$	−5.86278	−0.193395	−0.0966976	−	0.995314i	$-0.530828\pi$
$919$	−5.86278	−0.193395	−0.0966976	+	0.995314i	$0.530828\pi$
$920$	0.722765	0.0238288
$921$	−33.2782	−1.09656
$922$	23.3550	0.769155
$923$	−15.1655	−0.499179
$924$	0	0
$925$	−27.2343	−0.895457
$926$	−35.5159	−1.16713
$927$	−10.6596	−0.350108
$928$	−5.76855	−0.189362
$929$	30.2880	0.993717	0.496859	−	0.867831i	$-0.334487\pi$
$929$	30.2880	0.993717	0.496859	+	0.867831i	$0.334487\pi$
$930$	−4.48141	−0.146951
$931$	0	0
$932$	−14.8180	−0.485380
$933$	−12.4639	−0.408050
$934$	−40.9341	−1.33940
$935$	−3.22034	−0.105316
$936$	−1.27724	−0.0417477
$937$	−33.6065	−1.09788	−0.548938	−	0.835863i	$-0.684968\pi$
$937$	−33.6065	−1.09788	−0.548938	+	0.835863i	$0.684968\pi$
$938$	0	0
$939$	19.0184	0.620641
$940$	2.17820	0.0710451
$941$	43.0795	1.40435	0.702175	−	0.712004i	$-0.252212\pi$
$941$	43.0795	1.40435	0.702175	+	0.712004i	$0.252212\pi$
$942$	7.04579	0.229564
$943$	6.65961	0.216867
$944$	−12.9826	−0.422549
$945$	0	0
$946$	6.25622	0.203407
$947$	17.9514	0.583343	0.291671	−	0.956519i	$-0.405789\pi$
$947$	17.9514	0.583343	0.291671	+	0.956519i	$0.405789\pi$
$948$	−0.290943	−0.00944939
$949$	0.371603	0.0120627
$950$	8.95522	0.290546
$951$	2.23145	0.0723596
$952$	0	0
$953$	38.7559	1.25543	0.627713	−	0.778445i	$-0.283991\pi$
$953$	38.7559	1.25543	0.627713	+	0.778445i	$0.283991\pi$
$954$	1.68623	0.0545939
$955$	−3.60127	−0.116534
$956$	−8.00000	−0.258738
$957$	6.52864	0.211041
$958$	1.97258	0.0637313
$959$	0	0
$960$	0.722765	0.0233271
$961$	7.44466	0.240150
$962$	−7.76855	−0.250468
$963$	−19.5599	−0.630310
$964$	9.80429	0.315775
$965$	5.66764	0.182448
$966$	0	0
$967$	36.2546	1.16587	0.582935	−	0.812519i	$-0.301904\pi$
$967$	36.2546	1.16587	0.582935	+	0.812519i	$0.301904\pi$
$968$	−9.71911	−0.312384
$969$	7.87370	0.252940
$970$	0.542203	0.0174091
$971$	26.7373	0.858042	0.429021	−	0.903295i	$-0.358859\pi$
$971$	26.7373	0.858042	0.429021	+	0.903295i	$0.358859\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−36.5701	−1.17178
$975$	−5.71896	−0.183153
$976$	−12.4913	−0.399837
$977$	−27.6605	−0.884937	−0.442469	−	0.896784i	$-0.645897\pi$
$977$	−27.6605	−0.884937	−0.442469	+	0.896784i	$0.645897\pi$
$978$	−8.40075	−0.268626
$979$	−0.0714730	−0.00228429
$980$	0	0
$981$	−4.05490	−0.129463
$982$	−30.1829	−0.963174
$983$	−14.0916	−0.449452	−0.224726	−	0.974422i	$-0.572149\pi$
$983$	−14.0916	−0.449452	−0.224726	+	0.974422i	$0.572149\pi$
$984$	6.65961	0.212301
$985$	10.9049	0.347459
$986$	−22.7099	−0.723231
$987$	0	0
$988$	2.55447	0.0812685
$989$	−5.52785	−0.175775
$990$	−0.817999	−0.0259977
$991$	22.7648	0.723146	0.361573	−	0.932344i	$-0.382240\pi$
$991$	22.7648	0.723146	0.361573	+	0.932344i	$0.382240\pi$
$992$	−6.20038	−0.196862
$993$	−9.10894	−0.289064
$994$	0	0
$995$	−10.5042	−0.333004
$996$	−8.42817	−0.267057
$997$	−20.2734	−0.642066	−0.321033	−	0.947068i	$-0.604030\pi$
$997$	−20.2734	−0.642066	−0.321033	+	0.947068i	$0.604030\pi$
$998$	27.4108	0.867674
$999$	−6.08232	−0.192436

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.cj.1.3	✓	4
7.6	odd	2		6762.2.a.cs.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.cj.1.3	✓	4	1.1	even	1	trivial
6762.2.a.cs.1.2	yes	4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.722765 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.722765 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.722765 q^{10} +1.13176 q^{11} -1.00000 q^{12} -1.27724 q^{13} +0.722765 q^{15} +1.00000 q^{16} +3.93685 q^{17} +1.00000 q^{18} -2.00000 q^{19} -0.722765 q^{20} +1.13176 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.47761 q^{25} -1.27724 q^{26} -1.00000 q^{27} -5.76855 q^{29} +0.722765 q^{30} -6.20038 q^{31} +1.00000 q^{32} -1.13176 q^{33} +3.93685 q^{34} +1.00000 q^{36} +6.08232 q^{37} -2.00000 q^{38} +1.27724 q^{39} -0.722765 q^{40} -6.65961 q^{41} +5.52785 q^{43} +1.13176 q^{44} -0.722765 q^{45} -1.00000 q^{46} -3.01371 q^{47} -1.00000 q^{48} -4.47761 q^{50} -3.93685 q^{51} -1.27724 q^{52} +1.68623 q^{53} -1.00000 q^{54} -0.817999 q^{55} +2.00000 q^{57} -5.76855 q^{58} -12.9826 q^{59} +0.722765 q^{60} -12.4913 q^{61} -6.20038 q^{62} +1.00000 q^{64} +0.923140 q^{65} -1.13176 q^{66} +11.5599 q^{67} +3.93685 q^{68} +1.00000 q^{69} +11.8737 q^{71} +1.00000 q^{72} -0.290943 q^{73} +6.08232 q^{74} +4.47761 q^{75} -2.00000 q^{76} +1.27724 q^{78} +0.290943 q^{79} -0.722765 q^{80} +1.00000 q^{81} -6.65961 q^{82} +8.42817 q^{83} -2.84541 q^{85} +5.52785 q^{86} +5.76855 q^{87} +1.13176 q^{88} -0.0631518 q^{89} -0.722765 q^{90} -1.00000 q^{92} +6.20038 q^{93} -3.01371 q^{94} +1.44553 q^{95} -1.00000 q^{96} -0.750180 q^{97} +1.13176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 3 q^{10} + 2 q^{11} - 4 q^{12} - 5 q^{13} + 3 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 8 q^{19} - 3 q^{20} + 2 q^{22} - 4 q^{23} - 4 q^{24} + 13 q^{25} - 5 q^{26} - 4 q^{27} + 3 q^{29} + 3 q^{30} + 6 q^{31} + 4 q^{32} - 2 q^{33} - 10 q^{34} + 4 q^{36} + q^{37} - 8 q^{38} + 5 q^{39} - 3 q^{40} - q^{41} - q^{43} + 2 q^{44} - 3 q^{45} - 4 q^{46} - 17 q^{47} - 4 q^{48} + 13 q^{50} + 10 q^{51} - 5 q^{52} + 4 q^{53} - 4 q^{54} + 2 q^{55} + 8 q^{57} + 3 q^{58} + 3 q^{60} - 24 q^{61} + 6 q^{62} + 4 q^{64} - 27 q^{65} - 2 q^{66} - 8 q^{67} - 10 q^{68} + 4 q^{69} - 4 q^{71} + 4 q^{72} - 6 q^{73} + q^{74} - 13 q^{75} - 8 q^{76} + 5 q^{78} + 6 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 18 q^{83} - 16 q^{85} - q^{86} - 3 q^{87} + 2 q^{88} - 26 q^{89} - 3 q^{90} - 4 q^{92} - 6 q^{93} - 17 q^{94} + 6 q^{95} - 4 q^{96} - 13 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 3 * q^10 + 2 * q^11 - 4 * q^12 - 5 * q^13 + 3 * q^15 + 4 * q^16 - 10 * q^17 + 4 * q^18 - 8 * q^19 - 3 * q^20 + 2 * q^22 - 4 * q^23 - 4 * q^24 + 13 * q^25 - 5 * q^26 - 4 * q^27 + 3 * q^29 + 3 * q^30 + 6 * q^31 + 4 * q^32 - 2 * q^33 - 10 * q^34 + 4 * q^36 + q^37 - 8 * q^38 + 5 * q^39 - 3 * q^40 - q^41 - q^43 + 2 * q^44 - 3 * q^45 - 4 * q^46 - 17 * q^47 - 4 * q^48 + 13 * q^50 + 10 * q^51 - 5 * q^52 + 4 * q^53 - 4 * q^54 + 2 * q^55 + 8 * q^57 + 3 * q^58 + 3 * q^60 - 24 * q^61 + 6 * q^62 + 4 * q^64 - 27 * q^65 - 2 * q^66 - 8 * q^67 - 10 * q^68 + 4 * q^69 - 4 * q^71 + 4 * q^72 - 6 * q^73 + q^74 - 13 * q^75 - 8 * q^76 + 5 * q^78 + 6 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 18 * q^83 - 16 * q^85 - q^86 - 3 * q^87 + 2 * q^88 - 26 * q^89 - 3 * q^90 - 4 * q^92 - 6 * q^93 - 17 * q^94 + 6 * q^95 - 4 * q^96 - 13 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

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