LMFDB - Embedded newform 7105.2.a.t.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,1,-4,5,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 11x^{5} + 25x^{4} - 25x^{3} - 16x^{2} + 9x + 4 $ x^8 - x^7 - 10x^6 + 11x^5 + 25x^4 - 25x^3 - 16x^2 + 9x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1015)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.23565$ of defining polynomial
Character	$\chi$	$=$	7105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.23565 q^{2} +1.39789 q^{3} +2.99811 q^{4} -1.00000 q^{5} +3.12519 q^{6} +2.23142 q^{8} -1.04589 q^{9} -2.23565 q^{10} +1.06585 q^{11} +4.19104 q^{12} -4.05345 q^{13} -1.39789 q^{15} -1.00755 q^{16} -5.12519 q^{17} -2.33825 q^{18} -3.92256 q^{19} -2.99811 q^{20} +2.38285 q^{22} +3.09110 q^{23} +3.11929 q^{24} +1.00000 q^{25} -9.06207 q^{26} -5.65573 q^{27} +1.00000 q^{29} -3.12519 q^{30} +2.86680 q^{31} -6.71537 q^{32} +1.48994 q^{33} -11.4581 q^{34} -3.13571 q^{36} +1.88473 q^{37} -8.76946 q^{38} -5.66629 q^{39} -2.23142 q^{40} +7.84255 q^{41} -7.11719 q^{43} +3.19552 q^{44} +1.04589 q^{45} +6.91059 q^{46} +3.80139 q^{47} -1.40845 q^{48} +2.23565 q^{50} -7.16448 q^{51} -12.1527 q^{52} -2.18349 q^{53} -12.6442 q^{54} -1.06585 q^{55} -5.48333 q^{57} +2.23565 q^{58} -11.0310 q^{59} -4.19104 q^{60} -1.84086 q^{61} +6.40916 q^{62} -12.9981 q^{64} +4.05345 q^{65} +3.33097 q^{66} -7.19464 q^{67} -15.3659 q^{68} +4.32102 q^{69} -10.0492 q^{71} -2.33383 q^{72} -6.42384 q^{73} +4.21359 q^{74} +1.39789 q^{75} -11.7603 q^{76} -12.6678 q^{78} +4.78243 q^{79} +1.00755 q^{80} -4.76843 q^{81} +17.5332 q^{82} -1.47525 q^{83} +5.12519 q^{85} -15.9115 q^{86} +1.39789 q^{87} +2.37835 q^{88} -9.88379 q^{89} +2.33825 q^{90} +9.26745 q^{92} +4.00749 q^{93} +8.49856 q^{94} +3.92256 q^{95} -9.38738 q^{96} +5.48280 q^{97} -1.11476 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 4 q^{3} + 5 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{8} + 14 q^{9} - q^{10} + q^{11} + 7 q^{12} + q^{13} + 4 q^{15} + 3 q^{16} - 22 q^{17} + 13 q^{18} - 8 q^{19} - 5 q^{20} - 3 q^{22} + 9 q^{23}+ \cdots - 63 q^{99}+O(q^{100}) $ 8 * q + q^2 - 4 * q^3 + 5 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^8 + 14 * q^9 - q^10 + q^11 + 7 * q^12 + q^13 + 4 * q^15 + 3 * q^16 - 22 * q^17 + 13 * q^18 - 8 * q^19 - 5 * q^20 - 3 * q^22 + 9 * q^23 - 16 * q^24 + 8 * q^25 - 27 * q^26 - q^27 + 8 * q^29 - 6 * q^30 - 3 * q^31 - 26 * q^32 - 14 * q^33 - q^34 - 13 * q^36 + 9 * q^37 + 5 * q^38 + 3 * q^39 + 6 * q^40 - 24 * q^41 + 7 * q^43 + 10 * q^44 - 14 * q^45 - 27 * q^46 - 21 * q^47 + 8 * q^48 + q^50 - 8 * q^51 + 7 * q^52 - 2 * q^53 + 33 * q^54 - q^55 - 10 * q^57 + q^58 - 18 * q^59 - 7 * q^60 - 26 * q^61 - 15 * q^62 + 4 * q^64 - q^65 - 55 * q^66 - 8 * q^67 - 6 * q^68 + 2 * q^69 - 12 * q^71 + 9 * q^72 + q^73 + 14 * q^74 - 4 * q^75 - 15 * q^76 + 8 * q^78 - 14 * q^79 - 3 * q^80 + 40 * q^81 + 42 * q^82 + 10 * q^83 + 22 * q^85 + 19 * q^86 - 4 * q^87 - 29 * q^88 + 4 * q^89 - 13 * q^90 + 47 * q^92 - 24 * q^93 + 30 * q^94 + 8 * q^95 + q^96 + 11 * q^97 - 63 * 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 $

$2$	2.23565	1.58084	0.790420	−	0.612565i	$-0.209862\pi$
$2$	2.23565	1.58084	0.790420	+	0.612565i	$0.209862\pi$
$3$	1.39789	0.807074	0.403537	−	0.914963i	$-0.367781\pi$
$3$	1.39789	0.807074	0.403537	+	0.914963i	$0.367781\pi$
$4$	2.99811	1.49906
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.12519	1.27586
$7$	0	0
$7$	0	0
$8$	2.23142	0.788927
$9$	−1.04589	−0.348631
$10$	−2.23565	−0.706973
$11$	1.06585	0.321365	0.160682	−	0.987006i	$-0.448631\pi$
$11$	1.06585	0.321365	0.160682	+	0.987006i	$0.448631\pi$
$12$	4.19104	1.20985
$13$	−4.05345	−1.12422	−0.562112	−	0.827061i	$-0.690011\pi$
$13$	−4.05345	−1.12422	−0.562112	+	0.827061i	$0.690011\pi$
$14$	0	0
$15$	−1.39789	−0.360935
$16$	−1.00755	−0.251888
$17$	−5.12519	−1.24304	−0.621521	−	0.783397i	$-0.713485\pi$
$17$	−5.12519	−1.24304	−0.621521	+	0.783397i	$0.713485\pi$
$18$	−2.33825	−0.551130
$19$	−3.92256	−0.899898	−0.449949	−	0.893054i	$-0.648558\pi$
$19$	−3.92256	−0.899898	−0.449949	+	0.893054i	$0.648558\pi$
$20$	−2.99811	−0.670398
$21$	0	0
$22$	2.38285	0.508026
$23$	3.09110	0.644538	0.322269	−	0.946648i	$-0.395554\pi$
$23$	3.09110	0.644538	0.322269	+	0.946648i	$0.395554\pi$
$24$	3.11929	0.636723
$25$	1.00000	0.200000
$26$	−9.06207	−1.77722
$27$	−5.65573	−1.08845
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.12519	−0.570580
$31$	2.86680	0.514893	0.257447	−	0.966293i	$-0.417119\pi$
$31$	2.86680	0.514893	0.257447	+	0.966293i	$0.417119\pi$
$32$	−6.71537	−1.18712
$33$	1.48994	0.259365
$34$	−11.4581	−1.96505
$35$	0	0
$36$	−3.13571	−0.522618
$37$	1.88473	0.309848	0.154924	−	0.987926i	$-0.450487\pi$
$37$	1.88473	0.309848	0.154924	+	0.987926i	$0.450487\pi$
$38$	−8.76946	−1.42259
$39$	−5.66629	−0.907332
$40$	−2.23142	−0.352819
$41$	7.84255	1.22480	0.612401	−	0.790548i	$-0.290204\pi$
$41$	7.84255	1.22480	0.612401	+	0.790548i	$0.290204\pi$
$42$	0	0
$43$	−7.11719	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$43$	−7.11719	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$44$	3.19552	0.481743
$45$	1.04589	0.155913
$46$	6.91059	1.01891
$47$	3.80139	0.554490	0.277245	−	0.960799i	$-0.410579\pi$
$47$	3.80139	0.554490	0.277245	+	0.960799i	$0.410579\pi$
$48$	−1.40845	−0.203292
$49$	0	0
$50$	2.23565	0.316168
$51$	−7.16448	−1.00323
$52$	−12.1527	−1.68527
$53$	−2.18349	−0.299925	−0.149963	−	0.988692i	$-0.547915\pi$
$53$	−2.18349	−0.299925	−0.149963	+	0.988692i	$0.547915\pi$
$54$	−12.6442	−1.72066
$55$	−1.06585	−0.143719
$56$	0	0
$57$	−5.48333	−0.726284
$58$	2.23565	0.293555
$59$	−11.0310	−1.43612	−0.718058	−	0.695984i	$-0.754969\pi$
$59$	−11.0310	−1.43612	−0.718058	+	0.695984i	$0.754969\pi$
$60$	−4.19104	−0.541061
$61$	−1.84086	−0.235698	−0.117849	−	0.993032i	$-0.537600\pi$
$61$	−1.84086	−0.235698	−0.117849	+	0.993032i	$0.537600\pi$
$62$	6.40916	0.813964
$63$	0	0
$64$	−12.9981	−1.62476
$65$	4.05345	0.502768
$66$	3.33097	0.410015
$67$	−7.19464	−0.878965	−0.439482	−	0.898251i	$-0.644838\pi$
$67$	−7.19464	−0.878965	−0.439482	+	0.898251i	$0.644838\pi$
$68$	−15.3659	−1.86339
$69$	4.32102	0.520190
$70$	0	0
$71$	−10.0492	−1.19262	−0.596308	−	0.802756i	$-0.703366\pi$
$71$	−10.0492	−1.19262	−0.596308	+	0.802756i	$0.703366\pi$
$72$	−2.33383	−0.275045
$73$	−6.42384	−0.751853	−0.375927	−	0.926649i	$-0.622675\pi$
$73$	−6.42384	−0.751853	−0.375927	+	0.926649i	$0.622675\pi$
$74$	4.21359	0.489820
$75$	1.39789	0.161415
$76$	−11.7603	−1.34900
$77$	0	0
$78$	−12.6678	−1.43435
$79$	4.78243	0.538065	0.269033	−	0.963131i	$-0.413296\pi$
$79$	4.78243	0.538065	0.269033	+	0.963131i	$0.413296\pi$
$80$	1.00755	0.112648
$81$	−4.76843	−0.529825
$82$	17.5332	1.93621
$83$	−1.47525	−0.161929	−0.0809646	−	0.996717i	$-0.525800\pi$
$83$	−1.47525	−0.161929	−0.0809646	+	0.996717i	$0.525800\pi$
$84$	0	0
$85$	5.12519	0.555905
$86$	−15.9115	−1.71578
$87$	1.39789	0.149870
$88$	2.37835	0.253533
$89$	−9.88379	−1.04768	−0.523840	−	0.851817i	$-0.675501\pi$
$89$	−9.88379	−1.04768	−0.523840	+	0.851817i	$0.675501\pi$
$90$	2.33825	0.246473
$91$	0	0
$92$	9.26745	0.966198
$93$	4.00749	0.415557
$94$	8.49856	0.876560
$95$	3.92256	0.402447
$96$	−9.38738	−0.958095
$97$	5.48280	0.556694	0.278347	−	0.960481i	$-0.410214\pi$
$97$	5.48280	0.556694	0.278347	+	0.960481i	$0.410214\pi$
$98$	0	0
$99$	−1.11476	−0.112038
$100$	2.99811	0.299811
$101$	−5.93907	−0.590959	−0.295480	−	0.955349i	$-0.595479\pi$
$101$	−5.93907	−0.590959	−0.295480	+	0.955349i	$0.595479\pi$
$102$	−16.0172	−1.58594
$103$	14.4414	1.42296	0.711478	−	0.702708i	$-0.248026\pi$
$103$	14.4414	1.42296	0.711478	+	0.702708i	$0.248026\pi$
$104$	−9.04495	−0.886930
$105$	0	0
$106$	−4.88151	−0.474134
$107$	−19.0562	−1.84224	−0.921118	−	0.389283i	$-0.872723\pi$
$107$	−19.0562	−1.84224	−0.921118	+	0.389283i	$0.872723\pi$
$108$	−16.9565	−1.63164
$109$	−7.68228	−0.735829	−0.367915	−	0.929860i	$-0.619928\pi$
$109$	−7.68228	−0.735829	−0.367915	+	0.929860i	$0.619928\pi$
$110$	−2.38285	−0.227196
$111$	2.63465	0.250070
$112$	0	0
$113$	−2.55551	−0.240402	−0.120201	−	0.992750i	$-0.538354\pi$
$113$	−2.55551	−0.240402	−0.120201	+	0.992750i	$0.538354\pi$
$114$	−12.2588	−1.14814
$115$	−3.09110	−0.288246
$116$	2.99811	0.278368
$117$	4.23947	0.391939
$118$	−24.6614	−2.27027
$119$	0	0
$120$	−3.11929	−0.284751
$121$	−9.86397	−0.896725
$122$	−4.11552	−0.372601
$123$	10.9631	0.988505
$124$	8.59500	0.771854
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.5616	−1.55834	−0.779169	−	0.626814i	$-0.784359\pi$
$127$	−17.5616	−1.55834	−0.779169	+	0.626814i	$0.784359\pi$
$128$	−15.6284	−1.38137
$129$	−9.94908	−0.875968
$130$	9.06207	0.794796
$131$	14.3597	1.25462	0.627308	−	0.778771i	$-0.284157\pi$
$131$	14.3597	1.25462	0.627308	+	0.778771i	$0.284157\pi$
$132$	4.46700	0.388803
$133$	0	0
$134$	−16.0847	−1.38950
$135$	5.65573	0.486768
$136$	−11.4365	−0.980670
$137$	13.1576	1.12413	0.562065	−	0.827093i	$-0.310007\pi$
$137$	13.1576	1.12413	0.562065	+	0.827093i	$0.310007\pi$
$138$	9.66028	0.822337
$139$	−5.21363	−0.442215	−0.221107	−	0.975249i	$-0.570967\pi$
$139$	−5.21363	−0.442215	−0.221107	+	0.975249i	$0.570967\pi$
$140$	0	0
$141$	5.31394	0.447514
$142$	−22.4663	−1.88533
$143$	−4.32035	−0.361286
$144$	1.05379	0.0878160
$145$	−1.00000	−0.0830455
$146$	−14.3614	−1.18856
$147$	0	0
$148$	5.65063	0.464479
$149$	15.0087	1.22956	0.614780	−	0.788698i	$-0.289245\pi$
$149$	15.0087	1.22956	0.614780	+	0.788698i	$0.289245\pi$
$150$	3.12519	0.255171
$151$	7.10843	0.578476	0.289238	−	0.957257i	$-0.406598\pi$
$151$	7.10843	0.578476	0.289238	+	0.957257i	$0.406598\pi$
$152$	−8.75290	−0.709954
$153$	5.36041	0.433363
$154$	0	0
$155$	−2.86680	−0.230267
$156$	−16.9882	−1.36014
$157$	11.0932	0.885330	0.442665	−	0.896687i	$-0.354033\pi$
$157$	11.0932	0.885330	0.442665	+	0.896687i	$0.354033\pi$
$158$	10.6918	0.850595
$159$	−3.05228	−0.242062
$160$	6.71537	0.530897
$161$	0	0
$162$	−10.6605	−0.837569
$163$	−0.184373	−0.0144412	−0.00722060	−	0.999974i	$-0.502298\pi$
$163$	−0.184373	−0.0144412	−0.00722060	+	0.999974i	$0.502298\pi$
$164$	23.5128	1.83604
$165$	−1.48994	−0.115992
$166$	−3.29813	−0.255984
$167$	18.8370	1.45765	0.728826	−	0.684699i	$-0.240066\pi$
$167$	18.8370	1.45765	0.728826	+	0.684699i	$0.240066\pi$
$168$	0	0
$169$	3.43042	0.263879
$170$	11.4581	0.878798
$171$	4.10258	0.313733
$172$	−21.3381	−1.62702
$173$	15.2581	1.16005	0.580024	−	0.814599i	$-0.303043\pi$
$173$	15.2581	1.16005	0.580024	+	0.814599i	$0.303043\pi$
$174$	3.12519	0.236920
$175$	0	0
$176$	−1.07389	−0.0809479
$177$	−15.4202	−1.15905
$178$	−22.0967	−1.65621
$179$	−0.812914	−0.0607601	−0.0303800	−	0.999538i	$-0.509672\pi$
$179$	−0.812914	−0.0607601	−0.0303800	+	0.999538i	$0.509672\pi$
$180$	3.13571	0.233722
$181$	−0.201642	−0.0149879	−0.00749396	−	0.999972i	$-0.502385\pi$
$181$	−0.201642	−0.0149879	−0.00749396	+	0.999972i	$0.502385\pi$
$182$	0	0
$183$	−2.57333	−0.190226
$184$	6.89754	0.508494
$185$	−1.88473	−0.138568
$186$	8.95932	0.656929
$187$	−5.46267	−0.399470
$188$	11.3970	0.831211
$189$	0	0
$190$	8.76946	0.636204
$191$	−0.972823	−0.0703910	−0.0351955	−	0.999380i	$-0.511205\pi$
$191$	−0.972823	−0.0703910	−0.0351955	+	0.999380i	$0.511205\pi$
$192$	−18.1700	−1.31130
$193$	10.4687	0.753554	0.376777	−	0.926304i	$-0.377032\pi$
$193$	10.4687	0.753554	0.376777	+	0.926304i	$0.377032\pi$
$194$	12.2576	0.880044
$195$	5.66629	0.405771
$196$	0	0
$197$	21.5409	1.53473	0.767364	−	0.641211i	$-0.221568\pi$
$197$	21.5409	1.53473	0.767364	+	0.641211i	$0.221568\pi$
$198$	−2.49221	−0.177114
$199$	−12.6462	−0.896464	−0.448232	−	0.893917i	$-0.647946\pi$
$199$	−12.6462	−0.896464	−0.448232	+	0.893917i	$0.647946\pi$
$200$	2.23142	0.157785
$201$	−10.0573	−0.709390
$202$	−13.2777	−0.934212
$203$	0	0
$204$	−21.4799	−1.50389
$205$	−7.84255	−0.547748
$206$	32.2859	2.24947
$207$	−3.23296	−0.224706
$208$	4.08406	0.283178
$209$	−4.18085	−0.289195
$210$	0	0
$211$	20.3492	1.40089	0.700447	−	0.713704i	$-0.252984\pi$
$211$	20.3492	1.40089	0.700447	+	0.713704i	$0.252984\pi$
$212$	−6.54634	−0.449604
$213$	−14.0476	−0.962529
$214$	−42.6030	−2.91228
$215$	7.11719	0.485389
$216$	−12.6203	−0.858704
$217$	0	0
$218$	−17.1749	−1.16323
$219$	−8.97984	−0.606801
$220$	−3.19552	−0.215442
$221$	20.7747	1.39746
$222$	5.89015	0.395321
$223$	29.1542	1.95231	0.976153	−	0.217083i	$-0.0696542\pi$
$223$	29.1542	1.95231	0.976153	+	0.217083i	$0.0696542\pi$
$224$	0	0
$225$	−1.04589	−0.0697262
$226$	−5.71321	−0.380037
$227$	−15.4566	−1.02589	−0.512946	−	0.858421i	$-0.671446\pi$
$227$	−15.4566	−1.02589	−0.512946	+	0.858421i	$0.671446\pi$
$228$	−16.4396	−1.08874
$229$	8.16486	0.539549	0.269775	−	0.962924i	$-0.413051\pi$
$229$	8.16486	0.539549	0.269775	+	0.962924i	$0.413051\pi$
$230$	−6.91059	−0.455671
$231$	0	0
$232$	2.23142	0.146500
$233$	−17.9893	−1.17852	−0.589258	−	0.807945i	$-0.700580\pi$
$233$	−17.9893	−1.17852	−0.589258	+	0.807945i	$0.700580\pi$
$234$	9.47796	0.619594
$235$	−3.80139	−0.247975
$236$	−33.0722	−2.15282
$237$	6.68533	0.434259
$238$	0	0
$239$	−7.24018	−0.468328	−0.234164	−	0.972197i	$-0.575235\pi$
$239$	−7.24018	−0.468328	−0.234164	+	0.972197i	$0.575235\pi$
$240$	1.40845	0.0909151
$241$	0.148391	0.00955872	0.00477936	−	0.999989i	$-0.498479\pi$
$241$	0.148391	0.00955872	0.00477936	+	0.999989i	$0.498479\pi$
$242$	−22.0523	−1.41758
$243$	10.3014	0.660837
$244$	−5.51911	−0.353325
$245$	0	0
$246$	24.5095	1.56267
$247$	15.8999	1.01169
$248$	6.39705	0.406213
$249$	−2.06224	−0.130689
$250$	−2.23565	−0.141395
$251$	20.0758	1.26717	0.633587	−	0.773672i	$-0.281582\pi$
$251$	20.0758	1.26717	0.633587	+	0.773672i	$0.281582\pi$
$252$	0	0
$253$	3.29463	0.207132
$254$	−39.2615	−2.46348
$255$	7.16448	0.448657
$256$	−8.94333	−0.558958
$257$	−19.1469	−1.19435	−0.597174	−	0.802111i	$-0.703710\pi$
$257$	−19.1469	−1.19435	−0.597174	+	0.802111i	$0.703710\pi$
$258$	−22.2426	−1.38476
$259$	0	0
$260$	12.1527	0.753677
$261$	−1.04589	−0.0647392
$262$	32.1033	1.98335
$263$	16.8463	1.03879	0.519394	−	0.854535i	$-0.326158\pi$
$263$	16.8463	1.03879	0.519394	+	0.854535i	$0.326158\pi$
$264$	3.32468	0.204620
$265$	2.18349	0.134131
$266$	0	0
$267$	−13.8165	−0.845556
$268$	−21.5703	−1.31762
$269$	12.5797	0.766998	0.383499	−	0.923541i	$-0.374719\pi$
$269$	12.5797	0.766998	0.383499	+	0.923541i	$0.374719\pi$
$270$	12.6442	0.769502
$271$	−25.2186	−1.53192	−0.765959	−	0.642889i	$-0.777736\pi$
$271$	−25.2186	−1.53192	−0.765959	+	0.642889i	$0.777736\pi$
$272$	5.16390	0.313107
$273$	0	0
$274$	29.4158	1.77707
$275$	1.06585	0.0642729
$276$	12.9549	0.779794
$277$	−1.60938	−0.0966980	−0.0483490	−	0.998831i	$-0.515396\pi$
$277$	−1.60938	−0.0966980	−0.0483490	+	0.998831i	$0.515396\pi$
$278$	−11.6558	−0.699070
$279$	−2.99837	−0.179508
$280$	0	0
$281$	12.2017	0.727892	0.363946	−	0.931420i	$-0.381429\pi$
$281$	12.2017	0.727892	0.363946	+	0.931420i	$0.381429\pi$
$282$	11.8801	0.707449
$283$	1.51089	0.0898128	0.0449064	−	0.998991i	$-0.485701\pi$
$283$	1.51089	0.0898128	0.0449064	+	0.998991i	$0.485701\pi$
$284$	−30.1285	−1.78780
$285$	5.48333	0.324804
$286$	−9.65877	−0.571135
$287$	0	0
$288$	7.02357	0.413868
$289$	9.26762	0.545154
$290$	−2.23565	−0.131282
$291$	7.66437	0.449293
$292$	−19.2594	−1.12707
$293$	−9.66619	−0.564705	−0.282352	−	0.959311i	$-0.591115\pi$
$293$	−9.66619	−0.564705	−0.282352	+	0.959311i	$0.591115\pi$
$294$	0	0
$295$	11.0310	0.642250
$296$	4.20563	0.244447
$297$	−6.02813	−0.349788
$298$	33.5541	1.94374
$299$	−12.5296	−0.724605
$300$	4.19104	0.241970
$301$	0	0
$302$	15.8919	0.914478
$303$	−8.30219	−0.476948
$304$	3.95219	0.226673
$305$	1.84086	0.105408
$306$	11.9840	0.685078
$307$	20.4184	1.16534	0.582670	−	0.812709i	$-0.302008\pi$
$307$	20.4184	1.16534	0.582670	+	0.812709i	$0.302008\pi$
$308$	0	0
$309$	20.1876	1.14843
$310$	−6.40916	−0.364016
$311$	−22.3199	−1.26565	−0.632823	−	0.774297i	$-0.718104\pi$
$311$	−22.3199	−1.26565	−0.632823	+	0.774297i	$0.718104\pi$
$312$	−12.6439	−0.715819
$313$	2.32030	0.131151	0.0655755	−	0.997848i	$-0.479112\pi$
$313$	2.32030	0.131151	0.0655755	+	0.997848i	$0.479112\pi$
$314$	24.8004	1.39957
$315$	0	0
$316$	14.3383	0.806590
$317$	6.55281	0.368042	0.184021	−	0.982922i	$-0.441088\pi$
$317$	6.55281	0.368042	0.184021	+	0.982922i	$0.441088\pi$
$318$	−6.82382	−0.382661
$319$	1.06585	0.0596759
$320$	12.9981	0.726616
$321$	−26.6386	−1.48682
$322$	0	0
$323$	20.1039	1.11861
$324$	−14.2963	−0.794237
$325$	−4.05345	−0.224845
$326$	−0.412192	−0.0228292
$327$	−10.7390	−0.593869
$328$	17.5001	0.966279
$329$	0	0
$330$	−3.33097	−0.183364
$331$	−10.9774	−0.603373	−0.301687	−	0.953407i	$-0.597550\pi$
$331$	−10.9774	−0.603373	−0.301687	+	0.953407i	$0.597550\pi$
$332$	−4.42295	−0.242741
$333$	−1.97123	−0.108023
$334$	42.1129	2.30432
$335$	7.19464	0.393085
$336$	0	0
$337$	24.0404	1.30957	0.654783	−	0.755817i	$-0.272760\pi$
$337$	24.0404	1.30957	0.654783	+	0.755817i	$0.272760\pi$
$338$	7.66921	0.417150
$339$	−3.57233	−0.194022
$340$	15.3659	0.833333
$341$	3.05557	0.165468
$342$	9.17193	0.495961
$343$	0	0
$344$	−15.8815	−0.856271
$345$	−4.32102	−0.232636
$346$	34.1116	1.83385
$347$	26.2453	1.40892	0.704459	−	0.709744i	$-0.251189\pi$
$347$	26.2453	1.40892	0.704459	+	0.709744i	$0.251189\pi$
$348$	4.19104	0.224663
$349$	−20.8284	−1.11492	−0.557460	−	0.830204i	$-0.688224\pi$
$349$	−20.8284	−1.11492	−0.557460	+	0.830204i	$0.688224\pi$
$350$	0	0
$351$	22.9252	1.22366
$352$	−7.15755	−0.381499
$353$	−32.2061	−1.71416	−0.857078	−	0.515187i	$-0.827722\pi$
$353$	−32.2061	−1.71416	−0.857078	+	0.515187i	$0.827722\pi$
$354$	−34.4741	−1.83228
$355$	10.0492	0.533354
$356$	−29.6327	−1.57053
$357$	0	0
$358$	−1.81739	−0.0960519
$359$	12.9643	0.684227	0.342114	−	0.939659i	$-0.388857\pi$
$359$	12.9643	0.684227	0.342114	+	0.939659i	$0.388857\pi$
$360$	2.33383	0.123004
$361$	−3.61349	−0.190184
$362$	−0.450800	−0.0236935
$363$	−13.7888	−0.723723
$364$	0	0
$365$	6.42384	0.336239
$366$	−5.75306	−0.300717
$367$	−15.1148	−0.788985	−0.394493	−	0.918899i	$-0.629080\pi$
$367$	−15.1148	−0.788985	−0.394493	+	0.918899i	$0.629080\pi$
$368$	−3.11444	−0.162351
$369$	−8.20248	−0.427004
$370$	−4.21359	−0.219054
$371$	0	0
$372$	12.0149	0.622943
$373$	2.31068	0.119643	0.0598214	−	0.998209i	$-0.480947\pi$
$373$	2.31068	0.119643	0.0598214	+	0.998209i	$0.480947\pi$
$374$	−12.2126	−0.631498
$375$	−1.39789	−0.0721869
$376$	8.48251	0.437452
$377$	−4.05345	−0.208763
$378$	0	0
$379$	−13.6099	−0.699092	−0.349546	−	0.936919i	$-0.613664\pi$
$379$	−13.6099	−0.699092	−0.349546	+	0.936919i	$0.613664\pi$
$380$	11.7603	0.603290
$381$	−24.5492	−1.25769
$382$	−2.17489	−0.111277
$383$	8.57142	0.437979	0.218990	−	0.975727i	$-0.429724\pi$
$383$	8.57142	0.437979	0.218990	+	0.975727i	$0.429724\pi$
$384$	−21.8468	−1.11487
$385$	0	0
$386$	23.4043	1.19125
$387$	7.44383	0.378391
$388$	16.4380	0.834515
$389$	−22.9465	−1.16343	−0.581717	−	0.813391i	$-0.697619\pi$
$389$	−22.9465	−1.16343	−0.581717	+	0.813391i	$0.697619\pi$
$390$	12.6678	0.641459
$391$	−15.8425	−0.801188
$392$	0	0
$393$	20.0734	1.01257
$394$	48.1579	2.42616
$395$	−4.78243	−0.240630
$396$	−3.34218	−0.167951
$397$	−26.6840	−1.33923	−0.669616	−	0.742708i	$-0.733541\pi$
$397$	−26.6840	−1.33923	−0.669616	+	0.742708i	$0.733541\pi$
$398$	−28.2724	−1.41717
$399$	0	0
$400$	−1.00755	−0.0503776
$401$	−26.5679	−1.32674	−0.663369	−	0.748293i	$-0.730874\pi$
$401$	−26.5679	−1.32674	−0.663369	+	0.748293i	$0.730874\pi$
$402$	−22.4846	−1.12143
$403$	−11.6204	−0.578855
$404$	−17.8060	−0.885881
$405$	4.76843	0.236945
$406$	0	0
$407$	2.00883	0.0995740
$408$	−15.9870	−0.791473
$409$	−12.1541	−0.600982	−0.300491	−	0.953785i	$-0.597150\pi$
$409$	−12.1541	−0.600982	−0.300491	+	0.953785i	$0.597150\pi$
$410$	−17.5332	−0.865902
$411$	18.3929	0.907257
$412$	43.2970	2.13309
$413$	0	0
$414$	−7.22775	−0.355224
$415$	1.47525	0.0724169
$416$	27.2204	1.33459
$417$	−7.28810	−0.356900
$418$	−9.34689	−0.457172
$419$	2.58063	0.126072	0.0630359	−	0.998011i	$-0.479922\pi$
$419$	2.58063	0.126072	0.0630359	+	0.998011i	$0.479922\pi$
$420$	0	0
$421$	16.8779	0.822576	0.411288	−	0.911505i	$-0.365079\pi$
$421$	16.8779	0.822576	0.411288	+	0.911505i	$0.365079\pi$
$422$	45.4935	2.21459
$423$	−3.97585	−0.193312
$424$	−4.87228	−0.236619
$425$	−5.12519	−0.248608
$426$	−31.4056	−1.52160
$427$	0	0
$428$	−57.1327	−2.76162
$429$	−6.03939	−0.291584
$430$	15.9115	0.767322
$431$	−24.7151	−1.19049	−0.595243	−	0.803546i	$-0.702944\pi$
$431$	−24.7151	−1.19049	−0.595243	+	0.803546i	$0.702944\pi$
$432$	5.69844	0.274166
$433$	9.03464	0.434177	0.217089	−	0.976152i	$-0.430344\pi$
$433$	9.03464	0.434177	0.217089	+	0.976152i	$0.430344\pi$
$434$	0	0
$435$	−1.39789	−0.0670239
$436$	−23.0323	−1.10305
$437$	−12.1250	−0.580018
$438$	−20.0757	−0.959256
$439$	−15.0515	−0.718371	−0.359186	−	0.933266i	$-0.616945\pi$
$439$	−15.0515	−0.718371	−0.359186	+	0.933266i	$0.616945\pi$
$440$	−2.37835	−0.113383
$441$	0	0
$442$	46.4449	2.20916
$443$	7.62710	0.362374	0.181187	−	0.983449i	$-0.442006\pi$
$443$	7.62710	0.362374	0.181187	+	0.983449i	$0.442006\pi$
$444$	7.89898	0.374869
$445$	9.88379	0.468537
$446$	65.1783	3.08628
$447$	20.9806	0.992347
$448$	0	0
$449$	−4.22773	−0.199519	−0.0997594	−	0.995012i	$-0.531807\pi$
$449$	−4.22773	−0.199519	−0.0997594	+	0.995012i	$0.531807\pi$
$450$	−2.33825	−0.110226
$451$	8.35895	0.393608
$452$	−7.66169	−0.360376
$453$	9.93683	0.466873
$454$	−34.5555	−1.62177
$455$	0	0
$456$	−12.2356	−0.572985
$457$	−38.3481	−1.79385	−0.896924	−	0.442185i	$-0.854204\pi$
$457$	−38.3481	−1.79385	−0.896924	+	0.442185i	$0.854204\pi$
$458$	18.2537	0.852941
$459$	28.9867	1.35298
$460$	−9.26745	−0.432097
$461$	23.5362	1.09619	0.548095	−	0.836416i	$-0.315353\pi$
$461$	23.5362	1.09619	0.548095	+	0.836416i	$0.315353\pi$
$462$	0	0
$463$	−28.1018	−1.30600	−0.653000	−	0.757358i	$-0.726490\pi$
$463$	−28.1018	−1.30600	−0.653000	+	0.757358i	$0.726490\pi$
$464$	−1.00755	−0.0467744
$465$	−4.00749	−0.185843
$466$	−40.2176	−1.86305
$467$	33.1032	1.53183	0.765917	−	0.642940i	$-0.222285\pi$
$467$	33.1032	1.53183	0.765917	+	0.642940i	$0.222285\pi$
$468$	12.7104	0.587539
$469$	0	0
$470$	−8.49856	−0.392009
$471$	15.5070	0.714527
$472$	−24.6149	−1.13299
$473$	−7.58583	−0.348797
$474$	14.9460	0.686494
$475$	−3.92256	−0.179980
$476$	0	0
$477$	2.28370	0.104563
$478$	−16.1865	−0.740352
$479$	−20.6800	−0.944893	−0.472446	−	0.881359i	$-0.656629\pi$
$479$	−20.6800	−0.944893	−0.472446	+	0.881359i	$0.656629\pi$
$480$	9.38738	0.428473
$481$	−7.63965	−0.348338
$482$	0.331750	0.0151108
$483$	0	0
$484$	−29.5733	−1.34424
$485$	−5.48280	−0.248961
$486$	23.0304	1.04468
$487$	−38.2269	−1.73223	−0.866113	−	0.499848i	$-0.833389\pi$
$487$	−38.2269	−1.73223	−0.866113	+	0.499848i	$0.833389\pi$
$488$	−4.10774	−0.185949
$489$	−0.257734	−0.0116551
$490$	0	0
$491$	−5.53181	−0.249647	−0.124824	−	0.992179i	$-0.539836\pi$
$491$	−5.53181	−0.249647	−0.124824	+	0.992179i	$0.539836\pi$
$492$	32.8685	1.48182
$493$	−5.12519	−0.230827
$494$	35.5465	1.59931
$495$	1.11476	0.0501048
$496$	−2.88845	−0.129695
$497$	0	0
$498$	−4.61043	−0.206598
$499$	20.3931	0.912922	0.456461	−	0.889743i	$-0.349117\pi$
$499$	20.3931	0.912922	0.456461	+	0.889743i	$0.349117\pi$
$500$	−2.99811	−0.134080
$501$	26.3321	1.17643
$502$	44.8824	2.00320
$503$	8.57840	0.382492	0.191246	−	0.981542i	$-0.438747\pi$
$503$	8.57840	0.382492	0.191246	+	0.981542i	$0.438747\pi$
$504$	0	0
$505$	5.93907	0.264285
$506$	7.36563	0.327442
$507$	4.79536	0.212970
$508$	−52.6516	−2.33604
$509$	−22.3317	−0.989833	−0.494917	−	0.868940i	$-0.664801\pi$
$509$	−22.3317	−0.989833	−0.494917	+	0.868940i	$0.664801\pi$
$510$	16.0172	0.709255
$511$	0	0
$512$	11.2626	0.497743
$513$	22.1850	0.979490
$514$	−42.8056	−1.88807
$515$	−14.4414	−0.636365
$516$	−29.8284	−1.31312
$517$	4.05170	0.178193
$518$	0	0
$519$	21.3291	0.936246
$520$	9.04495	0.396647
$521$	−33.3920	−1.46293	−0.731464	−	0.681880i	$-0.761163\pi$
$521$	−33.3920	−1.46293	−0.731464	+	0.681880i	$0.761163\pi$
$522$	−2.33825	−0.102342
$523$	16.6650	0.728711	0.364355	−	0.931260i	$-0.381289\pi$
$523$	16.6650	0.728711	0.364355	+	0.931260i	$0.381289\pi$
$524$	43.0521	1.88074
$525$	0	0
$526$	37.6624	1.64216
$527$	−14.6929	−0.640034
$528$	−1.50119	−0.0653309
$529$	−13.4451	−0.584571
$530$	4.88151	0.212039
$531$	11.5373	0.500675
$532$	0	0
$533$	−31.7894	−1.37695
$534$	−30.8888	−1.33669
$535$	19.0562	0.823873
$536$	−16.0543	−0.693439
$537$	−1.13637	−0.0490379
$538$	28.1238	1.21250
$539$	0	0
$540$	16.9565	0.729692
$541$	1.66228	0.0714668	0.0357334	−	0.999361i	$-0.488623\pi$
$541$	1.66228	0.0714668	0.0357334	+	0.999361i	$0.488623\pi$
$542$	−56.3798	−2.42172
$543$	−0.281874	−0.0120964
$544$	34.4176	1.47564
$545$	7.68228	0.329073
$546$	0	0
$547$	−35.6380	−1.52377	−0.761886	−	0.647711i	$-0.775726\pi$
$547$	−35.6380	−1.52377	−0.761886	+	0.647711i	$0.775726\pi$
$548$	39.4480	1.68513
$549$	1.92535	0.0821718
$550$	2.38285	0.101605
$551$	−3.92256	−0.167107
$552$	9.64203	0.410392
$553$	0	0
$554$	−3.59799	−0.152864
$555$	−2.63465	−0.111835
$556$	−15.6310	−0.662904
$557$	14.8033	0.627238	0.313619	−	0.949549i	$-0.398459\pi$
$557$	14.8033	0.627238	0.313619	+	0.949549i	$0.398459\pi$
$558$	−6.70330	−0.283773
$559$	28.8492	1.22019
$560$	0	0
$561$	−7.63623	−0.322402
$562$	27.2787	1.15068
$563$	35.0508	1.47721	0.738607	−	0.674137i	$-0.235484\pi$
$563$	35.0508	1.47721	0.738607	+	0.674137i	$0.235484\pi$
$564$	15.9318	0.670849
$565$	2.55551	0.107511
$566$	3.37780	0.141980
$567$	0	0
$568$	−22.4239	−0.940886
$569$	−5.69826	−0.238884	−0.119442	−	0.992841i	$-0.538110\pi$
$569$	−5.69826	−0.238884	−0.119442	+	0.992841i	$0.538110\pi$
$570$	12.2588	0.513464
$571$	−37.3826	−1.56441	−0.782206	−	0.623019i	$-0.785906\pi$
$571$	−37.3826	−1.56441	−0.782206	+	0.623019i	$0.785906\pi$
$572$	−12.9529	−0.541587
$573$	−1.35990	−0.0568108
$574$	0	0
$575$	3.09110	0.128908
$576$	13.5946	0.566443
$577$	−39.1382	−1.62935	−0.814673	−	0.579921i	$-0.803083\pi$
$577$	−39.1382	−1.62935	−0.814673	+	0.579921i	$0.803083\pi$
$578$	20.7191	0.861801
$579$	14.6341	0.608174
$580$	−2.99811	−0.124490
$581$	0	0
$582$	17.1348	0.710261
$583$	−2.32726	−0.0963853
$584$	−14.3343	−0.593157
$585$	−4.23947	−0.175281
$586$	−21.6102	−0.892708
$587$	21.9446	0.905750	0.452875	−	0.891574i	$-0.350398\pi$
$587$	21.9446	0.905750	0.452875	+	0.891574i	$0.350398\pi$
$588$	0	0
$589$	−11.2452	−0.463351
$590$	24.6614	1.01530
$591$	30.1119	1.23864
$592$	−1.89896	−0.0780469
$593$	23.0560	0.946796	0.473398	−	0.880849i	$-0.343027\pi$
$593$	23.0560	0.946796	0.473398	+	0.880849i	$0.343027\pi$
$594$	−13.4768	−0.552959
$595$	0	0
$596$	44.9978	1.84318
$597$	−17.6780	−0.723513
$598$	−28.0117	−1.14548
$599$	23.0577	0.942113	0.471057	−	0.882103i	$-0.343873\pi$
$599$	23.0577	0.942113	0.471057	+	0.882103i	$0.343873\pi$
$600$	3.11929	0.127345
$601$	−41.6019	−1.69698	−0.848489	−	0.529214i	$-0.822487\pi$
$601$	−41.6019	−1.69698	−0.848489	+	0.529214i	$0.822487\pi$
$602$	0	0
$603$	7.52483	0.306435
$604$	21.3119	0.867168
$605$	9.86397	0.401028
$606$	−18.5607	−0.753979
$607$	10.8311	0.439622	0.219811	−	0.975542i	$-0.429456\pi$
$607$	10.8311	0.439622	0.219811	+	0.975542i	$0.429456\pi$
$608$	26.3415	1.06829
$609$	0	0
$610$	4.11552	0.166632
$611$	−15.4087	−0.623370
$612$	16.0711	0.649636
$613$	−42.3639	−1.71106	−0.855530	−	0.517753i	$-0.826769\pi$
$613$	−42.3639	−1.71106	−0.855530	+	0.517753i	$0.826769\pi$
$614$	45.6483	1.84222
$615$	−10.9631	−0.442073
$616$	0	0
$617$	−3.36891	−0.135627	−0.0678136	−	0.997698i	$-0.521602\pi$
$617$	−3.36891	−0.135627	−0.0678136	+	0.997698i	$0.521602\pi$
$618$	45.1323	1.81549
$619$	2.04646	0.0822542	0.0411271	−	0.999154i	$-0.486905\pi$
$619$	2.04646	0.0822542	0.0411271	+	0.999154i	$0.486905\pi$
$620$	−8.59500	−0.345183
$621$	−17.4824	−0.701545
$622$	−49.8994	−2.00078
$623$	0	0
$624$	5.70908	0.228546
$625$	1.00000	0.0400000
$626$	5.18736	0.207329
$627$	−5.84438	−0.233402
$628$	33.2585	1.32716
$629$	−9.65960	−0.385154
$630$	0	0
$631$	37.6451	1.49863	0.749315	−	0.662214i	$-0.230383\pi$
$631$	37.6451	1.49863	0.749315	+	0.662214i	$0.230383\pi$
$632$	10.6716	0.424494
$633$	28.4460	1.13063
$634$	14.6498	0.581816
$635$	17.5616	0.696910
$636$	−9.15109	−0.362864
$637$	0	0
$638$	2.38285	0.0943381
$639$	10.5103	0.415783
$640$	15.6284	0.617766
$641$	47.0240	1.85734	0.928668	−	0.370913i	$-0.120955\pi$
$641$	47.0240	1.85734	0.928668	+	0.370913i	$0.120955\pi$
$642$	−59.5545	−2.35043
$643$	8.74218	0.344758	0.172379	−	0.985031i	$-0.444855\pi$
$643$	8.74218	0.344758	0.172379	+	0.985031i	$0.444855\pi$
$644$	0	0
$645$	9.94908	0.391745
$646$	44.9452	1.76835
$647$	−18.1260	−0.712606	−0.356303	−	0.934370i	$-0.615963\pi$
$647$	−18.1260	−0.712606	−0.356303	+	0.934370i	$0.615963\pi$
$648$	−10.6404	−0.417993
$649$	−11.7574	−0.461517
$650$	−9.06207	−0.355444
$651$	0	0
$652$	−0.552770	−0.0216482
$653$	1.42762	0.0558670	0.0279335	−	0.999610i	$-0.491107\pi$
$653$	1.42762	0.0558670	0.0279335	+	0.999610i	$0.491107\pi$
$654$	−24.0086	−0.938812
$655$	−14.3597	−0.561081
$656$	−7.90178	−0.308513
$657$	6.71865	0.262119
$658$	0	0
$659$	−2.46266	−0.0959316	−0.0479658	−	0.998849i	$-0.515274\pi$
$659$	−2.46266	−0.0959316	−0.0479658	+	0.998849i	$0.515274\pi$
$660$	−4.46700	−0.173878
$661$	12.4726	0.485126	0.242563	−	0.970136i	$-0.422012\pi$
$661$	12.4726	0.485126	0.242563	+	0.970136i	$0.422012\pi$
$662$	−24.5416	−0.953837
$663$	29.0408	1.12785
$664$	−3.29190	−0.127750
$665$	0	0
$666$	−4.40696	−0.170766
$667$	3.09110	0.119688
$668$	56.4755	2.18510
$669$	40.7544	1.57566
$670$	16.0847	0.621405
$671$	−1.96208	−0.0757451
$672$	0	0
$673$	40.5317	1.56238	0.781192	−	0.624291i	$-0.214612\pi$
$673$	40.5317	1.56238	0.781192	+	0.624291i	$0.214612\pi$
$674$	53.7459	2.07022
$675$	−5.65573	−0.217689
$676$	10.2848	0.395569
$677$	−37.1179	−1.42656	−0.713278	−	0.700881i	$-0.752790\pi$
$677$	−37.1179	−1.42656	−0.713278	+	0.700881i	$0.752790\pi$
$678$	−7.98645	−0.306718
$679$	0	0
$680$	11.4365	0.438569
$681$	−21.6067	−0.827971
$682$	6.83117	0.261579
$683$	29.8051	1.14046	0.570230	−	0.821485i	$-0.306854\pi$
$683$	29.8051	1.14046	0.570230	+	0.821485i	$0.306854\pi$
$684$	12.3000	0.470302
$685$	−13.1576	−0.502726
$686$	0	0
$687$	11.4136	0.435456
$688$	7.17094	0.273390
$689$	8.85065	0.337183
$690$	−9.66028	−0.367760
$691$	23.5127	0.894465	0.447232	−	0.894418i	$-0.352410\pi$
$691$	23.5127	0.894465	0.447232	+	0.894418i	$0.352410\pi$
$692$	45.7454	1.73898
$693$	0	0
$694$	58.6751	2.22728
$695$	5.21363	0.197764
$696$	3.11929	0.118236
$697$	−40.1946	−1.52248
$698$	−46.5650	−1.76251
$699$	−25.1471	−0.951150
$700$	0	0
$701$	19.7954	0.747661	0.373830	−	0.927497i	$-0.378044\pi$
$701$	19.7954	0.747661	0.373830	+	0.927497i	$0.378044\pi$
$702$	51.2526	1.93440
$703$	−7.39297	−0.278831
$704$	−13.8540	−0.522141
$705$	−5.31394	−0.200135
$706$	−72.0013	−2.70981
$707$	0	0
$708$	−46.2314	−1.73748
$709$	25.8615	0.971248	0.485624	−	0.874168i	$-0.338592\pi$
$709$	25.8615	0.971248	0.485624	+	0.874168i	$0.338592\pi$
$710$	22.4663	0.843147
$711$	−5.00191	−0.187586
$712$	−22.0549	−0.826543
$713$	8.86157	0.331868
$714$	0	0
$715$	4.32035	0.161572
$716$	−2.43721	−0.0910827
$717$	−10.1210	−0.377976
$718$	28.9835	1.08165
$719$	−35.0624	−1.30761	−0.653804	−	0.756664i	$-0.726828\pi$
$719$	−35.0624	−1.30761	−0.653804	+	0.756664i	$0.726828\pi$
$720$	−1.05379	−0.0392725
$721$	0	0
$722$	−8.07849	−0.300650
$723$	0.207435	0.00771460
$724$	−0.604545	−0.0224677
$725$	1.00000	0.0371391
$726$	−30.8268	−1.14409
$727$	24.6517	0.914282	0.457141	−	0.889394i	$-0.348873\pi$
$727$	24.6517	0.914282	0.457141	+	0.889394i	$0.348873\pi$
$728$	0	0
$729$	28.7056	1.06317
$730$	14.3614	0.531540
$731$	36.4770	1.34915
$732$	−7.71513	−0.285159
$733$	50.0889	1.85007	0.925037	−	0.379876i	$-0.124033\pi$
$733$	50.0889	1.85007	0.925037	+	0.379876i	$0.124033\pi$
$734$	−33.7913	−1.24726
$735$	0	0
$736$	−20.7579	−0.765145
$737$	−7.66837	−0.282468
$738$	−18.3378	−0.675025
$739$	10.2482	0.376988	0.188494	−	0.982074i	$-0.439639\pi$
$739$	10.2482	0.376988	0.188494	+	0.982074i	$0.439639\pi$
$740$	−5.65063	−0.207721
$741$	22.2264	0.816506
$742$	0	0
$743$	−26.1716	−0.960141	−0.480071	−	0.877230i	$-0.659389\pi$
$743$	−26.1716	−0.960141	−0.480071	+	0.877230i	$0.659389\pi$
$744$	8.94240	0.327844
$745$	−15.0087	−0.549876
$746$	5.16587	0.189136
$747$	1.54295	0.0564536
$748$	−16.3777	−0.598827
$749$	0	0
$750$	−3.12519	−0.114116
$751$	16.8899	0.616320	0.308160	−	0.951334i	$-0.400287\pi$
$751$	16.8899	0.616320	0.308160	+	0.951334i	$0.400287\pi$
$752$	−3.83010	−0.139669
$753$	28.0638	1.02270
$754$	−9.06207	−0.330021
$755$	−7.10843	−0.258702
$756$	0	0
$757$	−35.8636	−1.30349	−0.651743	−	0.758440i	$-0.725962\pi$
$757$	−35.8636	−1.30349	−0.651743	+	0.758440i	$0.725962\pi$
$758$	−30.4268	−1.10515
$759$	4.60554	0.167171
$760$	8.75290	0.317501
$761$	22.4997	0.815615	0.407808	−	0.913068i	$-0.366293\pi$
$761$	22.4997	0.815615	0.407808	+	0.913068i	$0.366293\pi$
$762$	−54.8833	−1.98821
$763$	0	0
$764$	−2.91663	−0.105520
$765$	−5.36041	−0.193806
$766$	19.1627	0.692375
$767$	44.7136	1.61451
$768$	−12.5018	−0.451121
$769$	3.47293	0.125237	0.0626185	−	0.998038i	$-0.480055\pi$
$769$	3.47293	0.125237	0.0626185	+	0.998038i	$0.480055\pi$
$770$	0	0
$771$	−26.7653	−0.963928
$772$	31.3864	1.12962
$773$	−43.0058	−1.54681	−0.773406	−	0.633911i	$-0.781449\pi$
$773$	−43.0058	−1.54681	−0.773406	+	0.633911i	$0.781449\pi$
$774$	16.6418	0.598176
$775$	2.86680	0.102979
$776$	12.2344	0.439191
$777$	0	0
$778$	−51.3003	−1.83920
$779$	−30.7629	−1.10220
$780$	16.9882	0.608273
$781$	−10.7108	−0.383264
$782$	−35.4181	−1.26655
$783$	−5.65573	−0.202119
$784$	0	0
$785$	−11.0932	−0.395932
$786$	44.8770	1.60071
$787$	35.4996	1.26542	0.632712	−	0.774387i	$-0.281942\pi$
$787$	35.4996	1.26542	0.632712	+	0.774387i	$0.281942\pi$
$788$	64.5821	2.30064
$789$	23.5493	0.838379
$790$	−10.6918	−0.380398
$791$	0	0
$792$	−2.48750	−0.0883896
$793$	7.46184	0.264978
$794$	−59.6559	−2.11711
$795$	3.05228	0.108253
$796$	−37.9146	−1.34385
$797$	−46.4556	−1.64554	−0.822771	−	0.568373i	$-0.807573\pi$
$797$	−46.4556	−1.64554	−0.822771	+	0.568373i	$0.807573\pi$
$798$	0	0
$799$	−19.4829	−0.689254
$800$	−6.71537	−0.237424
$801$	10.3374	0.365254
$802$	−59.3964	−2.09736
$803$	−6.84682	−0.241619
$804$	−30.1530	−1.06341
$805$	0	0
$806$	−25.9792	−0.915078
$807$	17.5851	0.619025
$808$	−13.2526	−0.466224
$809$	−44.3202	−1.55821	−0.779107	−	0.626891i	$-0.784327\pi$
$809$	−44.3202	−1.55821	−0.779107	+	0.626891i	$0.784327\pi$
$810$	10.6605	0.374572
$811$	−7.77582	−0.273046	−0.136523	−	0.990637i	$-0.543593\pi$
$811$	−7.77582	−0.273046	−0.136523	+	0.990637i	$0.543593\pi$
$812$	0	0
$813$	−35.2529	−1.23637
$814$	4.49103	0.157411
$815$	0.184373	0.00645830
$816$	7.21858	0.252701
$817$	27.9176	0.976715
$818$	−27.1723	−0.950056
$819$	0	0
$820$	−23.5128	−0.821104
$821$	−1.99847	−0.0697471	−0.0348736	−	0.999392i	$-0.511103\pi$
$821$	−1.99847	−0.0697471	−0.0348736	+	0.999392i	$0.511103\pi$
$822$	41.1201	1.43423
$823$	−1.34858	−0.0470084	−0.0235042	−	0.999724i	$-0.507482\pi$
$823$	−1.34858	−0.0470084	−0.0235042	+	0.999724i	$0.507482\pi$
$824$	32.2249	1.12261
$825$	1.48994	0.0518730
$826$	0	0
$827$	1.03814	0.0360996	0.0180498	−	0.999837i	$-0.494254\pi$
$827$	1.03814	0.0360996	0.0180498	+	0.999837i	$0.494254\pi$
$828$	−9.69277	−0.336847
$829$	−22.9989	−0.798784	−0.399392	−	0.916780i	$-0.630779\pi$
$829$	−22.9989	−0.798784	−0.399392	+	0.916780i	$0.630779\pi$
$830$	3.29813	0.114480
$831$	−2.24974	−0.0780425
$832$	52.6871	1.82660
$833$	0	0
$834$	−16.2936	−0.564202
$835$	−18.8370	−0.651882
$836$	−12.5346	−0.433520
$837$	−16.2139	−0.560433
$838$	5.76937	0.199299
$839$	−33.4822	−1.15593	−0.577967	−	0.816060i	$-0.696154\pi$
$839$	−33.4822	−1.15593	−0.577967	+	0.816060i	$0.696154\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	37.7329	1.30036
$843$	17.0567	0.587463
$844$	61.0090	2.10002
$845$	−3.43042	−0.118010
$846$	−8.88859	−0.305596
$847$	0	0
$848$	2.19998	0.0755475
$849$	2.11206	0.0724856
$850$	−11.4581	−0.393010
$851$	5.82588	0.199709
$852$	−42.1164	−1.44288
$853$	19.1000	0.653971	0.326985	−	0.945029i	$-0.393967\pi$
$853$	19.1000	0.653971	0.326985	+	0.945029i	$0.393967\pi$
$854$	0	0
$855$	−4.10258	−0.140305
$856$	−42.5225	−1.45339
$857$	25.7356	0.879110	0.439555	−	0.898216i	$-0.355136\pi$
$857$	25.7356	0.879110	0.439555	+	0.898216i	$0.355136\pi$
$858$	−13.5019	−0.460948
$859$	−22.7809	−0.777275	−0.388638	−	0.921391i	$-0.627054\pi$
$859$	−22.7809	−0.777275	−0.388638	+	0.921391i	$0.627054\pi$
$860$	21.3381	0.727624
$861$	0	0
$862$	−55.2543	−1.88197
$863$	−16.4420	−0.559694	−0.279847	−	0.960045i	$-0.590284\pi$
$863$	−16.4420	−0.559694	−0.279847	+	0.960045i	$0.590284\pi$
$864$	37.9803	1.29212
$865$	−15.2581	−0.518790
$866$	20.1983	0.686365
$867$	12.9551	0.439980
$868$	0	0
$869$	5.09733	0.172915
$870$	−3.12519	−0.105954
$871$	29.1631	0.988153
$872$	−17.1424	−0.580516
$873$	−5.73442	−0.194081
$874$	−27.1072	−0.916916
$875$	0	0
$876$	−26.9226	−0.909629
$877$	20.1781	0.681367	0.340684	−	0.940178i	$-0.389341\pi$
$877$	20.1781	0.681367	0.340684	+	0.940178i	$0.389341\pi$
$878$	−33.6499	−1.13563
$879$	−13.5123	−0.455759
$880$	1.07389	0.0362010
$881$	5.00231	0.168532	0.0842660	−	0.996443i	$-0.473145\pi$
$881$	5.00231	0.168532	0.0842660	+	0.996443i	$0.473145\pi$
$882$	0	0
$883$	48.4641	1.63095	0.815473	−	0.578795i	$-0.196477\pi$
$883$	48.4641	1.63095	0.815473	+	0.578795i	$0.196477\pi$
$884$	62.2849	2.09487
$885$	15.4202	0.518344
$886$	17.0515	0.572856
$887$	−27.1594	−0.911923	−0.455962	−	0.889999i	$-0.650705\pi$
$887$	−27.1594	−0.911923	−0.455962	+	0.889999i	$0.650705\pi$
$888$	5.87902	0.197287
$889$	0	0
$890$	22.0967	0.740682
$891$	−5.08241	−0.170267
$892$	87.4074	2.92662
$893$	−14.9112	−0.498984
$894$	46.9051	1.56874
$895$	0.812914	0.0271727
$896$	0	0
$897$	−17.5150	−0.584810
$898$	−9.45170	−0.315407
$899$	2.86680	0.0956133
$900$	−3.13571	−0.104524
$901$	11.1908	0.372820
$902$	18.6877	0.622231
$903$	0	0
$904$	−5.70241	−0.189659
$905$	0.201642	0.00670280
$906$	22.2152	0.738052
$907$	−27.7897	−0.922740	−0.461370	−	0.887208i	$-0.652642\pi$
$907$	−27.7897	−0.922740	−0.461370	+	0.887208i	$0.652642\pi$
$908$	−46.3407	−1.53787
$909$	6.21163	0.206027
$910$	0	0
$911$	13.2018	0.437396	0.218698	−	0.975793i	$-0.429819\pi$
$911$	13.2018	0.437396	0.218698	+	0.975793i	$0.429819\pi$
$912$	5.52474	0.182942
$913$	−1.57238	−0.0520383
$914$	−85.7327	−2.83579
$915$	2.57333	0.0850717
$916$	24.4792	0.808814
$917$	0	0
$918$	64.8040	2.13885
$919$	−38.3589	−1.26534	−0.632672	−	0.774420i	$-0.718042\pi$
$919$	−38.3589	−1.26534	−0.632672	+	0.774420i	$0.718042\pi$
$920$	−6.89754	−0.227405
$921$	28.5428	0.940516
$922$	52.6186	1.73290
$923$	40.7337	1.34077
$924$	0	0
$925$	1.88473	0.0619695
$926$	−62.8256	−2.06458
$927$	−15.1042	−0.496087
$928$	−6.71537	−0.220443
$929$	−44.7910	−1.46955	−0.734773	−	0.678313i	$-0.762711\pi$
$929$	−44.7910	−1.46955	−0.734773	+	0.678313i	$0.762711\pi$
$930$	−8.95932	−0.293788
$931$	0	0
$932$	−53.9338	−1.76666
$933$	−31.2009	−1.02147
$934$	74.0070	2.42158
$935$	5.46267	0.178648
$936$	9.46006	0.309212
$937$	27.1871	0.888165	0.444082	−	0.895986i	$-0.353530\pi$
$937$	27.1871	0.888165	0.444082	+	0.895986i	$0.353530\pi$
$938$	0	0
$939$	3.24353	0.105849
$940$	−11.3970	−0.371729
$941$	49.8902	1.62637	0.813187	−	0.582003i	$-0.197731\pi$
$941$	49.8902	1.62637	0.813187	+	0.582003i	$0.197731\pi$
$942$	34.6683	1.12955
$943$	24.2421	0.789431
$944$	11.1143	0.361740
$945$	0	0
$946$	−16.9592	−0.551392
$947$	29.7809	0.967749	0.483875	−	0.875137i	$-0.339229\pi$
$947$	29.7809	0.967749	0.483875	+	0.875137i	$0.339229\pi$
$948$	20.0434	0.650978
$949$	26.0387	0.845251
$950$	−8.76946	−0.284519
$951$	9.16013	0.297038
$952$	0	0
$953$	−21.5271	−0.697330	−0.348665	−	0.937247i	$-0.613365\pi$
$953$	−21.5271	−0.697330	−0.348665	+	0.937247i	$0.613365\pi$
$954$	5.10554	0.165298
$955$	0.972823	0.0314798
$956$	−21.7069	−0.702050
$957$	1.48994	0.0481629
$958$	−46.2331	−1.49372
$959$	0	0
$960$	18.1700	0.586433
$961$	−22.7814	−0.734885
$962$	−17.0795	−0.550667
$963$	19.9308	0.642261
$964$	0.444893	0.0143291
$965$	−10.4687	−0.337000
$966$	0	0
$967$	21.9037	0.704374	0.352187	−	0.935930i	$-0.385438\pi$
$967$	21.9037	0.704374	0.352187	+	0.935930i	$0.385438\pi$
$968$	−22.0107	−0.707450
$969$	28.1031	0.902802
$970$	−12.2576	−0.393568
$971$	−27.0883	−0.869304	−0.434652	−	0.900598i	$-0.643129\pi$
$971$	−27.0883	−0.869304	−0.434652	+	0.900598i	$0.643129\pi$
$972$	30.8848	0.990632
$973$	0	0
$974$	−85.4618	−2.73837
$975$	−5.66629	−0.181466
$976$	1.85477	0.0593696
$977$	−4.70266	−0.150452	−0.0752258	−	0.997167i	$-0.523968\pi$
$977$	−4.70266	−0.150452	−0.0752258	+	0.997167i	$0.523968\pi$
$978$	−0.576201	−0.0184249
$979$	−10.5346	−0.336687
$980$	0	0
$981$	8.03485	0.256533
$982$	−12.3672	−0.394652
$983$	−36.7362	−1.17170	−0.585851	−	0.810419i	$-0.699240\pi$
$983$	−36.7362	−1.17170	−0.585851	+	0.810419i	$0.699240\pi$
$984$	24.4632	0.779859
$985$	−21.5409	−0.686352
$986$	−11.4581	−0.364901
$987$	0	0
$988$	47.6697	1.51657
$989$	−21.9999	−0.699557
$990$	2.49221	0.0792077
$991$	−60.9399	−1.93582	−0.967910	−	0.251298i	$-0.919143\pi$
$991$	−60.9399	−1.93582	−0.967910	+	0.251298i	$0.919143\pi$
$992$	−19.2517	−0.611241
$993$	−15.3453	−0.486967
$994$	0	0
$995$	12.6462	0.400911
$996$	−6.18281	−0.195910
$997$	13.8562	0.438831	0.219415	−	0.975632i	$-0.429585\pi$
$997$	13.8562	0.438831	0.219415	+	0.975632i	$0.429585\pi$
$998$	45.5918	1.44318
$999$	−10.6595	−0.337252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.t.1.8		8
7.6	odd	2		1015.2.a.l.1.8	✓	8
21.20	even	2		9135.2.a.bh.1.1		8
35.34	odd	2		5075.2.a.ba.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.l.1.8	✓	8	7.6	odd	2
5075.2.a.ba.1.1		8	35.34	odd	2
7105.2.a.t.1.8		8	1.1	even	1	trivial
9135.2.a.bh.1.1		8	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23565 q^{2} +1.39789 q^{3} +2.99811 q^{4} -1.00000 q^{5} +3.12519 q^{6} +2.23142 q^{8} -1.04589 q^{9} -2.23565 q^{10} +1.06585 q^{11} +4.19104 q^{12} -4.05345 q^{13} -1.39789 q^{15} -1.00755 q^{16} -5.12519 q^{17} -2.33825 q^{18} -3.92256 q^{19} -2.99811 q^{20} +2.38285 q^{22} +3.09110 q^{23} +3.11929 q^{24} +1.00000 q^{25} -9.06207 q^{26} -5.65573 q^{27} +1.00000 q^{29} -3.12519 q^{30} +2.86680 q^{31} -6.71537 q^{32} +1.48994 q^{33} -11.4581 q^{34} -3.13571 q^{36} +1.88473 q^{37} -8.76946 q^{38} -5.66629 q^{39} -2.23142 q^{40} +7.84255 q^{41} -7.11719 q^{43} +3.19552 q^{44} +1.04589 q^{45} +6.91059 q^{46} +3.80139 q^{47} -1.40845 q^{48} +2.23565 q^{50} -7.16448 q^{51} -12.1527 q^{52} -2.18349 q^{53} -12.6442 q^{54} -1.06585 q^{55} -5.48333 q^{57} +2.23565 q^{58} -11.0310 q^{59} -4.19104 q^{60} -1.84086 q^{61} +6.40916 q^{62} -12.9981 q^{64} +4.05345 q^{65} +3.33097 q^{66} -7.19464 q^{67} -15.3659 q^{68} +4.32102 q^{69} -10.0492 q^{71} -2.33383 q^{72} -6.42384 q^{73} +4.21359 q^{74} +1.39789 q^{75} -11.7603 q^{76} -12.6678 q^{78} +4.78243 q^{79} +1.00755 q^{80} -4.76843 q^{81} +17.5332 q^{82} -1.47525 q^{83} +5.12519 q^{85} -15.9115 q^{86} +1.39789 q^{87} +2.37835 q^{88} -9.88379 q^{89} +2.33825 q^{90} +9.26745 q^{92} +4.00749 q^{93} +8.49856 q^{94} +3.92256 q^{95} -9.38738 q^{96} +5.48280 q^{97} -1.11476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 4 q^{3} + 5 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{8} + 14 q^{9} - q^{10} + q^{11} + 7 q^{12} + q^{13} + 4 q^{15} + 3 q^{16} - 22 q^{17} + 13 q^{18} - 8 q^{19} - 5 q^{20} - 3 q^{22} + 9 q^{23}+ \cdots - 63 q^{99}+O(q^{100}) \) 8 * q + q^2 - 4 * q^3 + 5 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^8 + 14 * q^9 - q^10 + q^11 + 7 * q^12 + q^13 + 4 * q^15 + 3 * q^16 - 22 * q^17 + 13 * q^18 - 8 * q^19 - 5 * q^20 - 3 * q^22 + 9 * q^23 - 16 * q^24 + 8 * q^25 - 27 * q^26 - q^27 + 8 * q^29 - 6 * q^30 - 3 * q^31 - 26 * q^32 - 14 * q^33 - q^34 - 13 * q^36 + 9 * q^37 + 5 * q^38 + 3 * q^39 + 6 * q^40 - 24 * q^41 + 7 * q^43 + 10 * q^44 - 14 * q^45 - 27 * q^46 - 21 * q^47 + 8 * q^48 + q^50 - 8 * q^51 + 7 * q^52 - 2 * q^53 + 33 * q^54 - q^55 - 10 * q^57 + q^58 - 18 * q^59 - 7 * q^60 - 26 * q^61 - 15 * q^62 + 4 * q^64 - q^65 - 55 * q^66 - 8 * q^67 - 6 * q^68 + 2 * q^69 - 12 * q^71 + 9 * q^72 + q^73 + 14 * q^74 - 4 * q^75 - 15 * q^76 + 8 * q^78 - 14 * q^79 - 3 * q^80 + 40 * q^81 + 42 * q^82 + 10 * q^83 + 22 * q^85 + 19 * q^86 - 4 * q^87 - 29 * q^88 + 4 * q^89 - 13 * q^90 + 47 * q^92 - 24 * q^93 + 30 * q^94 + 8 * q^95 + q^96 + 11 * q^97 - 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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