LMFDB - Embedded newform 6724.2.a.f.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6724,2,Mod(1,6724)] 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("6724.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6724, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.6914103191$
Analytic rank:	$0$
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} - 16x^{6} + 16x^{5} + 73x^{4} - 58x^{3} - 116x^{2} + 48x + 64 $ x^8 - x^7 - 16x^6 + 16x^5 + 73x^4 - 58x^3 - 116x^2 + 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.95118$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95118 q^{3} -1.45807 q^{5} -1.70016 q^{7} +0.807096 q^{9} -1.73309 q^{11} -0.554924 q^{13} -2.84495 q^{15} -5.62813 q^{17} +1.69717 q^{19} -3.31731 q^{21} +2.98206 q^{23} -2.87404 q^{25} -4.27875 q^{27} +1.85793 q^{29} +4.46183 q^{31} -3.38157 q^{33} +2.47894 q^{35} +7.48456 q^{37} -1.08276 q^{39} +9.93658 q^{43} -1.17680 q^{45} +6.83707 q^{47} -4.10947 q^{49} -10.9815 q^{51} -0.478867 q^{53} +2.52696 q^{55} +3.31149 q^{57} -2.37408 q^{59} -11.2747 q^{61} -1.37219 q^{63} +0.809116 q^{65} +1.94011 q^{67} +5.81853 q^{69} +10.6220 q^{71} +15.6343 q^{73} -5.60777 q^{75} +2.94652 q^{77} +7.46724 q^{79} -10.7699 q^{81} +14.9584 q^{83} +8.20619 q^{85} +3.62514 q^{87} +2.49037 q^{89} +0.943457 q^{91} +8.70583 q^{93} -2.47459 q^{95} -0.505882 q^{97} -1.39877 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} + 3 q^{5} + 9 q^{9} + 8 q^{11} - 10 q^{13} + 5 q^{15} + 7 q^{17} - 3 q^{19} - 5 q^{21} + 6 q^{23} - q^{25} + 5 q^{27} + 17 q^{29} - q^{31} - 22 q^{35} - 8 q^{37} + 12 q^{39} + 18 q^{43} + 8 q^{45}+ \cdots + 56 q^{99}+O(q^{100}) $ 8 * q - q^3 + 3 * q^5 + 9 * q^9 + 8 * q^11 - 10 * q^13 + 5 * q^15 + 7 * q^17 - 3 * q^19 - 5 * q^21 + 6 * q^23 - q^25 + 5 * q^27 + 17 * q^29 - q^31 - 22 * q^35 - 8 * q^37 + 12 * q^39 + 18 * q^43 + 8 * q^45 + 3 * q^47 + 42 * q^49 + 7 * q^51 + 8 * q^53 + 21 * q^55 + 33 * q^57 + 44 * q^59 - 7 * q^61 - 40 * q^63 - 47 * q^65 - 19 * q^67 - 6 * q^69 - 14 * q^71 + 21 * q^73 + 15 * q^75 + 36 * q^77 + 9 * q^79 - 4 * q^81 + 26 * q^83 + 29 * q^85 - 27 * q^87 + 13 * q^89 + 42 * q^91 - 6 * q^93 - 22 * q^95 + 12 * q^97 + 56 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.95118	1.12651	0.563257	−	0.826282i	$-0.309548\pi$
$3$	1.95118	1.12651	0.563257	+	0.826282i	$0.309548\pi$
$4$	0	0
$5$	−1.45807	−0.652067	−0.326034	−	0.945358i	$-0.605712\pi$
$5$	−1.45807	−0.652067	−0.326034	+	0.945358i	$0.605712\pi$
$6$	0	0
$7$	−1.70016	−0.642598	−0.321299	−	0.946978i	$-0.604119\pi$
$7$	−1.70016	−0.642598	−0.321299	+	0.946978i	$0.604119\pi$
$8$	0	0
$9$	0.807096	0.269032
$10$	0	0
$11$	−1.73309	−0.522547	−0.261273	−	0.965265i	$-0.584142\pi$
$11$	−1.73309	−0.522547	−0.261273	+	0.965265i	$0.584142\pi$
$12$	0	0
$13$	−0.554924	−0.153908	−0.0769541	−	0.997035i	$-0.524519\pi$
$13$	−0.554924	−0.153908	−0.0769541	+	0.997035i	$0.524519\pi$
$14$	0	0
$15$	−2.84495	−0.734562
$16$	0	0
$17$	−5.62813	−1.36502	−0.682511	−	0.730875i	$-0.739112\pi$
$17$	−5.62813	−1.36502	−0.682511	+	0.730875i	$0.739112\pi$
$18$	0	0
$19$	1.69717	0.389358	0.194679	−	0.980867i	$-0.437633\pi$
$19$	1.69717	0.389358	0.194679	+	0.980867i	$0.437633\pi$
$20$	0	0
$21$	−3.31731	−0.723895
$22$	0	0
$23$	2.98206	0.621803	0.310901	−	0.950442i	$-0.399369\pi$
$23$	2.98206	0.621803	0.310901	+	0.950442i	$0.399369\pi$
$24$	0	0
$25$	−2.87404	−0.574808
$26$	0	0
$27$	−4.27875	−0.823445
$28$	0	0
$29$	1.85793	0.345008	0.172504	−	0.985009i	$-0.444814\pi$
$29$	1.85793	0.345008	0.172504	+	0.985009i	$0.444814\pi$
$30$	0	0
$31$	4.46183	0.801369	0.400684	−	0.916216i	$-0.368772\pi$
$31$	4.46183	0.801369	0.400684	+	0.916216i	$0.368772\pi$
$32$	0	0
$33$	−3.38157	−0.588656
$34$	0	0
$35$	2.47894	0.419017
$36$	0	0
$37$	7.48456	1.23045	0.615227	−	0.788350i	$-0.289064\pi$
$37$	7.48456	1.23045	0.615227	+	0.788350i	$0.289064\pi$
$38$	0	0
$39$	−1.08276	−0.173380
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	9.93658	1.51531	0.757657	−	0.652653i	$-0.226344\pi$
$43$	9.93658	1.51531	0.757657	+	0.652653i	$0.226344\pi$
$44$	0	0
$45$	−1.17680	−0.175427
$46$	0	0
$47$	6.83707	0.997289	0.498644	−	0.866807i	$-0.333831\pi$
$47$	6.83707	0.997289	0.498644	+	0.866807i	$0.333831\pi$
$48$	0	0
$49$	−4.10947	−0.587067
$50$	0	0
$51$	−10.9815	−1.53772
$52$	0	0
$53$	−0.478867	−0.0657774	−0.0328887	−	0.999459i	$-0.510471\pi$
$53$	−0.478867	−0.0657774	−0.0328887	+	0.999459i	$0.510471\pi$
$54$	0	0
$55$	2.52696	0.340736
$56$	0	0
$57$	3.31149	0.438617
$58$	0	0
$59$	−2.37408	−0.309078	−0.154539	−	0.987987i	$-0.549389\pi$
$59$	−2.37408	−0.309078	−0.154539	+	0.987987i	$0.549389\pi$
$60$	0	0
$61$	−11.2747	−1.44358	−0.721789	−	0.692113i	$-0.756680\pi$
$61$	−11.2747	−1.44358	−0.721789	+	0.692113i	$0.756680\pi$
$62$	0	0
$63$	−1.37219	−0.172880
$64$	0	0
$65$	0.809116	0.100359
$66$	0	0
$67$	1.94011	0.237022	0.118511	−	0.992953i	$-0.462188\pi$
$67$	1.94011	0.237022	0.118511	+	0.992953i	$0.462188\pi$
$68$	0	0
$69$	5.81853	0.700469
$70$	0	0
$71$	10.6220	1.26060	0.630299	−	0.776353i	$-0.282932\pi$
$71$	10.6220	1.26060	0.630299	+	0.776353i	$0.282932\pi$
$72$	0	0
$73$	15.6343	1.82985	0.914926	−	0.403621i	$-0.132249\pi$
$73$	15.6343	1.82985	0.914926	+	0.403621i	$0.132249\pi$
$74$	0	0
$75$	−5.60777	−0.647529
$76$	0	0
$77$	2.94652	0.335788
$78$	0	0
$79$	7.46724	0.840130	0.420065	−	0.907494i	$-0.362007\pi$
$79$	7.46724	0.840130	0.420065	+	0.907494i	$0.362007\pi$
$80$	0	0
$81$	−10.7699	−1.19665
$82$	0	0
$83$	14.9584	1.64190	0.820948	−	0.571003i	$-0.193446\pi$
$83$	14.9584	1.64190	0.820948	+	0.571003i	$0.193446\pi$
$84$	0	0
$85$	8.20619	0.890086
$86$	0	0
$87$	3.62514	0.388656
$88$	0	0
$89$	2.49037	0.263979	0.131989	−	0.991251i	$-0.457864\pi$
$89$	2.49037	0.263979	0.131989	+	0.991251i	$0.457864\pi$
$90$	0	0
$91$	0.943457	0.0989012
$92$	0	0
$93$	8.70583	0.902752
$94$	0	0
$95$	−2.47459	−0.253888
$96$	0	0
$97$	−0.505882	−0.0513645	−0.0256823	−	0.999670i	$-0.508176\pi$
$97$	−0.505882	−0.0513645	−0.0256823	+	0.999670i	$0.508176\pi$
$98$	0	0
$99$	−1.39877	−0.140582
$100$	0	0
$101$	3.92338	0.390390	0.195195	−	0.980764i	$-0.437466\pi$
$101$	3.92338	0.390390	0.195195	+	0.980764i	$0.437466\pi$
$102$	0	0
$103$	−12.2534	−1.20737	−0.603683	−	0.797224i	$-0.706301\pi$
$103$	−12.2534	−1.20737	−0.603683	+	0.797224i	$0.706301\pi$
$104$	0	0
$105$	4.83685	0.472028
$106$	0	0
$107$	19.0710	1.84366	0.921832	−	0.387590i	$-0.126692\pi$
$107$	19.0710	1.84366	0.921832	+	0.387590i	$0.126692\pi$
$108$	0	0
$109$	−9.52892	−0.912705	−0.456353	−	0.889799i	$-0.650844\pi$
$109$	−9.52892	−0.912705	−0.456353	+	0.889799i	$0.650844\pi$
$110$	0	0
$111$	14.6037	1.38612
$112$	0	0
$113$	18.1675	1.70905	0.854525	−	0.519409i	$-0.173848\pi$
$113$	18.1675	1.70905	0.854525	+	0.519409i	$0.173848\pi$
$114$	0	0
$115$	−4.34804	−0.405457
$116$	0	0
$117$	−0.447877	−0.0414062
$118$	0	0
$119$	9.56869	0.877161
$120$	0	0
$121$	−7.99639	−0.726945
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.4809	1.02688
$126$	0	0
$127$	−12.2015	−1.08271	−0.541355	−	0.840794i	$-0.682088\pi$
$127$	−12.2015	−1.08271	−0.541355	+	0.840794i	$0.682088\pi$
$128$	0	0
$129$	19.3880	1.70702
$130$	0	0
$131$	−5.84881	−0.511013	−0.255506	−	0.966807i	$-0.582242\pi$
$131$	−5.84881	−0.511013	−0.255506	+	0.966807i	$0.582242\pi$
$132$	0	0
$133$	−2.88546	−0.250201
$134$	0	0
$135$	6.23870	0.536941
$136$	0	0
$137$	8.08891	0.691082	0.345541	−	0.938404i	$-0.387695\pi$
$137$	8.08891	0.691082	0.345541	+	0.938404i	$0.387695\pi$
$138$	0	0
$139$	0.320906	0.0272189	0.0136094	−	0.999907i	$-0.495668\pi$
$139$	0.320906	0.0272189	0.0136094	+	0.999907i	$0.495668\pi$
$140$	0	0
$141$	13.3403	1.12346
$142$	0	0
$143$	0.961734	0.0804242
$144$	0	0
$145$	−2.70898	−0.224968
$146$	0	0
$147$	−8.01831	−0.661339
$148$	0	0
$149$	2.18690	0.179157	0.0895787	−	0.995980i	$-0.471448\pi$
$149$	2.18690	0.179157	0.0895787	+	0.995980i	$0.471448\pi$
$150$	0	0
$151$	−5.50942	−0.448351	−0.224175	−	0.974549i	$-0.571969\pi$
$151$	−5.50942	−0.448351	−0.224175	+	0.974549i	$0.571969\pi$
$152$	0	0
$153$	−4.54244	−0.367235
$154$	0	0
$155$	−6.50565	−0.522546
$156$	0	0
$157$	13.2833	1.06012	0.530062	−	0.847959i	$-0.322169\pi$
$157$	13.2833	1.06012	0.530062	+	0.847959i	$0.322169\pi$
$158$	0	0
$159$	−0.934355	−0.0740991
$160$	0	0
$161$	−5.06997	−0.399569
$162$	0	0
$163$	1.71123	0.134034	0.0670170	−	0.997752i	$-0.478652\pi$
$163$	1.71123	0.134034	0.0670170	+	0.997752i	$0.478652\pi$
$164$	0	0
$165$	4.93055	0.383843
$166$	0	0
$167$	12.5130	0.968283	0.484141	−	0.874990i	$-0.339132\pi$
$167$	12.5130	0.968283	0.484141	+	0.874990i	$0.339132\pi$
$168$	0	0
$169$	−12.6921	−0.976312
$170$	0	0
$171$	1.36978	0.104750
$172$	0	0
$173$	−2.53782	−0.192947	−0.0964734	−	0.995336i	$-0.530756\pi$
$173$	−2.53782	−0.192947	−0.0964734	+	0.995336i	$0.530756\pi$
$174$	0	0
$175$	4.88632	0.369371
$176$	0	0
$177$	−4.63225	−0.348181
$178$	0	0
$179$	13.8049	1.03183	0.515915	−	0.856640i	$-0.327452\pi$
$179$	13.8049	1.03183	0.515915	+	0.856640i	$0.327452\pi$
$180$	0	0
$181$	−17.5949	−1.30782	−0.653909	−	0.756573i	$-0.726872\pi$
$181$	−17.5949	−1.30782	−0.653909	+	0.756573i	$0.726872\pi$
$182$	0	0
$183$	−21.9990	−1.62621
$184$	0	0
$185$	−10.9130	−0.802339
$186$	0	0
$187$	9.75406	0.713288
$188$	0	0
$189$	7.27453	0.529144
$190$	0	0
$191$	−18.6189	−1.34722	−0.673609	−	0.739088i	$-0.735257\pi$
$191$	−18.6189	−1.34722	−0.673609	+	0.739088i	$0.735257\pi$
$192$	0	0
$193$	−0.719152	−0.0517657	−0.0258829	−	0.999665i	$-0.508240\pi$
$193$	−0.719152	−0.0517657	−0.0258829	+	0.999665i	$0.508240\pi$
$194$	0	0
$195$	1.57873	0.113055
$196$	0	0
$197$	−4.30717	−0.306873	−0.153436	−	0.988159i	$-0.549034\pi$
$197$	−4.30717	−0.306873	−0.153436	+	0.988159i	$0.549034\pi$
$198$	0	0
$199$	13.7649	0.975764	0.487882	−	0.872909i	$-0.337770\pi$
$199$	13.7649	0.975764	0.487882	+	0.872909i	$0.337770\pi$
$200$	0	0
$201$	3.78550	0.267009
$202$	0	0
$203$	−3.15876	−0.221702
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.40681	0.167285
$208$	0	0
$209$	−2.94135	−0.203458
$210$	0	0
$211$	16.3033	1.12237	0.561183	−	0.827692i	$-0.310347\pi$
$211$	16.3033	1.12237	0.561183	+	0.827692i	$0.310347\pi$
$212$	0	0
$213$	20.7254	1.42008
$214$	0	0
$215$	−14.4882	−0.988086
$216$	0	0
$217$	−7.58581	−0.514958
$218$	0	0
$219$	30.5052	2.06135
$220$	0	0
$221$	3.12318	0.210088
$222$	0	0
$223$	22.0340	1.47550	0.737752	−	0.675072i	$-0.235888\pi$
$223$	22.0340	1.47550	0.737752	+	0.675072i	$0.235888\pi$
$224$	0	0
$225$	−2.31963	−0.154642
$226$	0	0
$227$	−17.5387	−1.16408	−0.582042	−	0.813159i	$-0.697746\pi$
$227$	−17.5387	−1.16408	−0.582042	+	0.813159i	$0.697746\pi$
$228$	0	0
$229$	−26.3359	−1.74032	−0.870162	−	0.492767i	$-0.835986\pi$
$229$	−26.3359	−1.74032	−0.870162	+	0.492767i	$0.835986\pi$
$230$	0	0
$231$	5.74919	0.378269
$232$	0	0
$233$	−1.28071	−0.0839019	−0.0419509	−	0.999120i	$-0.513357\pi$
$233$	−1.28071	−0.0839019	−0.0419509	+	0.999120i	$0.513357\pi$
$234$	0	0
$235$	−9.96890	−0.650299
$236$	0	0
$237$	14.5699	0.946418
$238$	0	0
$239$	−15.8350	−1.02428	−0.512140	−	0.858902i	$-0.671147\pi$
$239$	−15.8350	−1.02428	−0.512140	+	0.858902i	$0.671147\pi$
$240$	0	0
$241$	25.8975	1.66820	0.834101	−	0.551611i	$-0.185987\pi$
$241$	25.8975	1.66820	0.834101	+	0.551611i	$0.185987\pi$
$242$	0	0
$243$	−8.17772	−0.524601
$244$	0	0
$245$	5.99188	0.382807
$246$	0	0
$247$	−0.941802	−0.0599254
$248$	0	0
$249$	29.1865	1.84962
$250$	0	0
$251$	−3.85631	−0.243408	−0.121704	−	0.992566i	$-0.538836\pi$
$251$	−3.85631	−0.243408	−0.121704	+	0.992566i	$0.538836\pi$
$252$	0	0
$253$	−5.16819	−0.324921
$254$	0	0
$255$	16.0117	1.00269
$256$	0	0
$257$	17.9355	1.11879	0.559394	−	0.828902i	$-0.311034\pi$
$257$	17.9355	1.11879	0.559394	+	0.828902i	$0.311034\pi$
$258$	0	0
$259$	−12.7249	−0.790688
$260$	0	0
$261$	1.49952	0.0928182
$262$	0	0
$263$	29.2403	1.80304	0.901518	−	0.432742i	$-0.142454\pi$
$263$	29.2403	1.80304	0.901518	+	0.432742i	$0.142454\pi$
$264$	0	0
$265$	0.698220	0.0428913
$266$	0	0
$267$	4.85915	0.297375
$268$	0	0
$269$	23.9903	1.46271	0.731356	−	0.681996i	$-0.238888\pi$
$269$	23.9903	1.46271	0.731356	+	0.681996i	$0.238888\pi$
$270$	0	0
$271$	9.36626	0.568960	0.284480	−	0.958682i	$-0.408179\pi$
$271$	9.36626	0.568960	0.284480	+	0.958682i	$0.408179\pi$
$272$	0	0
$273$	1.84085	0.111413
$274$	0	0
$275$	4.98098	0.300364
$276$	0	0
$277$	17.4413	1.04795	0.523973	−	0.851735i	$-0.324449\pi$
$277$	17.4413	1.04795	0.523973	+	0.851735i	$0.324449\pi$
$278$	0	0
$279$	3.60113	0.215594
$280$	0	0
$281$	−17.5841	−1.04898	−0.524489	−	0.851417i	$-0.675744\pi$
$281$	−17.5841	−1.04898	−0.524489	+	0.851417i	$0.675744\pi$
$282$	0	0
$283$	0.0974883	0.00579508	0.00289754	−	0.999996i	$-0.499078\pi$
$283$	0.0974883	0.00579508	0.00289754	+	0.999996i	$0.499078\pi$
$284$	0	0
$285$	−4.82837	−0.286008
$286$	0	0
$287$	0	0
$288$	0	0
$289$	14.6758	0.863285
$290$	0	0
$291$	−0.987066	−0.0578628
$292$	0	0
$293$	−25.6463	−1.49827	−0.749137	−	0.662416i	$-0.769531\pi$
$293$	−25.6463	−1.49827	−0.749137	+	0.662416i	$0.769531\pi$
$294$	0	0
$295$	3.46156	0.201540
$296$	0	0
$297$	7.41546	0.430289
$298$	0	0
$299$	−1.65482	−0.0957006
$300$	0	0
$301$	−16.8937	−0.973738
$302$	0	0
$303$	7.65520	0.439780
$304$	0	0
$305$	16.4393	0.941310
$306$	0	0
$307$	3.10539	0.177234	0.0886169	−	0.996066i	$-0.471755\pi$
$307$	3.10539	0.177234	0.0886169	+	0.996066i	$0.471755\pi$
$308$	0	0
$309$	−23.9086	−1.36011
$310$	0	0
$311$	−18.5735	−1.05321	−0.526603	−	0.850111i	$-0.676535\pi$
$311$	−18.5735	−1.05321	−0.526603	+	0.850111i	$0.676535\pi$
$312$	0	0
$313$	4.95529	0.280089	0.140045	−	0.990145i	$-0.455275\pi$
$313$	4.95529	0.280089	0.140045	+	0.990145i	$0.455275\pi$
$314$	0	0
$315$	2.00074	0.112729
$316$	0	0
$317$	−10.1159	−0.568167	−0.284084	−	0.958799i	$-0.591689\pi$
$317$	−10.1159	−0.568167	−0.284084	+	0.958799i	$0.591689\pi$
$318$	0	0
$319$	−3.21996	−0.180283
$320$	0	0
$321$	37.2109	2.07691
$322$	0	0
$323$	−9.55190	−0.531482
$324$	0	0
$325$	1.59487	0.0884677
$326$	0	0
$327$	−18.5926	−1.02817
$328$	0	0
$329$	−11.6241	−0.640856
$330$	0	0
$331$	−30.8856	−1.69763	−0.848814	−	0.528692i	$-0.822683\pi$
$331$	−30.8856	−1.69763	−0.848814	+	0.528692i	$0.822683\pi$
$332$	0	0
$333$	6.04076	0.331032
$334$	0	0
$335$	−2.82881	−0.154555
$336$	0	0
$337$	6.54739	0.356659	0.178330	−	0.983971i	$-0.442931\pi$
$337$	6.54739	0.356659	0.178330	+	0.983971i	$0.442931\pi$
$338$	0	0
$339$	35.4480	1.92527
$340$	0	0
$341$	−7.73276	−0.418753
$342$	0	0
$343$	18.8878	1.01985
$344$	0	0
$345$	−8.48381	−0.456753
$346$	0	0
$347$	−3.20923	−0.172281	−0.0861403	−	0.996283i	$-0.527453\pi$
$347$	−3.20923	−0.172281	−0.0861403	+	0.996283i	$0.527453\pi$
$348$	0	0
$349$	33.5968	1.79840	0.899198	−	0.437541i	$-0.144151\pi$
$349$	33.5968	1.79840	0.899198	+	0.437541i	$0.144151\pi$
$350$	0	0
$351$	2.37438	0.126735
$352$	0	0
$353$	15.9300	0.847870	0.423935	−	0.905693i	$-0.360648\pi$
$353$	15.9300	0.847870	0.423935	+	0.905693i	$0.360648\pi$
$354$	0	0
$355$	−15.4876	−0.821994
$356$	0	0
$357$	18.6702	0.988133
$358$	0	0
$359$	−12.3389	−0.651222	−0.325611	−	0.945504i	$-0.605570\pi$
$359$	−12.3389	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$360$	0	0
$361$	−16.1196	−0.848400
$362$	0	0
$363$	−15.6024	−0.818913
$364$	0	0
$365$	−22.7958	−1.19319
$366$	0	0
$367$	12.9183	0.674329	0.337165	−	0.941446i	$-0.390532\pi$
$367$	12.9183	0.674329	0.337165	+	0.941446i	$0.390532\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.814148	0.0422685
$372$	0	0
$373$	−25.4004	−1.31518	−0.657590	−	0.753376i	$-0.728424\pi$
$373$	−25.4004	−1.31518	−0.657590	+	0.753376i	$0.728424\pi$
$374$	0	0
$375$	22.4012	1.15679
$376$	0	0
$377$	−1.03101	−0.0530996
$378$	0	0
$379$	−12.5762	−0.645993	−0.322997	−	0.946400i	$-0.604690\pi$
$379$	−12.5762	−0.645993	−0.322997	+	0.946400i	$0.604690\pi$
$380$	0	0
$381$	−23.8073	−1.21969
$382$	0	0
$383$	16.6036	0.848404	0.424202	−	0.905568i	$-0.360555\pi$
$383$	16.6036	0.848404	0.424202	+	0.905568i	$0.360555\pi$
$384$	0	0
$385$	−4.29623	−0.218956
$386$	0	0
$387$	8.01977	0.407668
$388$	0	0
$389$	−14.5279	−0.736593	−0.368296	−	0.929708i	$-0.620059\pi$
$389$	−14.5279	−0.736593	−0.368296	+	0.929708i	$0.620059\pi$
$390$	0	0
$391$	−16.7834	−0.848775
$392$	0	0
$393$	−11.4121	−0.575663
$394$	0	0
$395$	−10.8877	−0.547821
$396$	0	0
$397$	−31.7497	−1.59347	−0.796736	−	0.604327i	$-0.793442\pi$
$397$	−31.7497	−1.59347	−0.796736	+	0.604327i	$0.793442\pi$
$398$	0	0
$399$	−5.63004	−0.281854
$400$	0	0
$401$	27.4452	1.37055	0.685275	−	0.728284i	$-0.259682\pi$
$401$	27.4452	1.37055	0.685275	+	0.728284i	$0.259682\pi$
$402$	0	0
$403$	−2.47598	−0.123337
$404$	0	0
$405$	15.7032	0.780299
$406$	0	0
$407$	−12.9714	−0.642970
$408$	0	0
$409$	−9.95694	−0.492339	−0.246169	−	0.969227i	$-0.579172\pi$
$409$	−9.95694	−0.492339	−0.246169	+	0.969227i	$0.579172\pi$
$410$	0	0
$411$	15.7829	0.778513
$412$	0	0
$413$	4.03630	0.198613
$414$	0	0
$415$	−21.8103	−1.07063
$416$	0	0
$417$	0.626144	0.0306624
$418$	0	0
$419$	32.7438	1.59964	0.799819	−	0.600241i	$-0.204929\pi$
$419$	32.7438	1.59964	0.799819	+	0.600241i	$0.204929\pi$
$420$	0	0
$421$	18.1208	0.883155	0.441577	−	0.897223i	$-0.354419\pi$
$421$	18.1208	0.883155	0.441577	+	0.897223i	$0.354419\pi$
$422$	0	0
$423$	5.51817	0.268303
$424$	0	0
$425$	16.1755	0.784626
$426$	0	0
$427$	19.1688	0.927641
$428$	0	0
$429$	1.87651	0.0905990
$430$	0	0
$431$	19.1103	0.920511	0.460255	−	0.887787i	$-0.347758\pi$
$431$	19.1103	0.920511	0.460255	+	0.887787i	$0.347758\pi$
$432$	0	0
$433$	7.33396	0.352448	0.176224	−	0.984350i	$-0.443612\pi$
$433$	7.33396	0.352448	0.176224	+	0.984350i	$0.443612\pi$
$434$	0	0
$435$	−5.28570	−0.253430
$436$	0	0
$437$	5.06107	0.242104
$438$	0	0
$439$	40.1433	1.91593	0.957967	−	0.286879i	$-0.0926178\pi$
$439$	40.1433	1.91593	0.957967	+	0.286879i	$0.0926178\pi$
$440$	0	0
$441$	−3.31674	−0.157940
$442$	0	0
$443$	4.16320	0.197800	0.0988998	−	0.995097i	$-0.468468\pi$
$443$	4.16320	0.197800	0.0988998	+	0.995097i	$0.468468\pi$
$444$	0	0
$445$	−3.63112	−0.172132
$446$	0	0
$447$	4.26702	0.201823
$448$	0	0
$449$	22.5856	1.06588	0.532940	−	0.846153i	$-0.321087\pi$
$449$	22.5856	1.06588	0.532940	+	0.846153i	$0.321087\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−10.7499	−0.505073
$454$	0	0
$455$	−1.37562	−0.0644902
$456$	0	0
$457$	−28.4571	−1.33117	−0.665584	−	0.746323i	$-0.731817\pi$
$457$	−28.4571	−1.33117	−0.665584	+	0.746323i	$0.731817\pi$
$458$	0	0
$459$	24.0813	1.12402
$460$	0	0
$461$	−6.28335	−0.292645	−0.146322	−	0.989237i	$-0.546744\pi$
$461$	−6.28335	−0.292645	−0.146322	+	0.989237i	$0.546744\pi$
$462$	0	0
$463$	−12.2714	−0.570299	−0.285150	−	0.958483i	$-0.592043\pi$
$463$	−12.2714	−0.570299	−0.285150	+	0.958483i	$0.592043\pi$
$464$	0	0
$465$	−12.6937	−0.588655
$466$	0	0
$467$	9.73015	0.450258	0.225129	−	0.974329i	$-0.427720\pi$
$467$	9.73015	0.450258	0.225129	+	0.974329i	$0.427720\pi$
$468$	0	0
$469$	−3.29849	−0.152310
$470$	0	0
$471$	25.9181	1.19424
$472$	0	0
$473$	−17.2210	−0.791822
$474$	0	0
$475$	−4.87774	−0.223806
$476$	0	0
$477$	−0.386492	−0.0176962
$478$	0	0
$479$	−20.1352	−0.920001	−0.460001	−	0.887919i	$-0.652151\pi$
$479$	−20.1352	−0.920001	−0.460001	+	0.887919i	$0.652151\pi$
$480$	0	0
$481$	−4.15336	−0.189377
$482$	0	0
$483$	−9.89241	−0.450120
$484$	0	0
$485$	0.737609	0.0334931
$486$	0	0
$487$	−26.6720	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$487$	−26.6720	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$488$	0	0
$489$	3.33892	0.150991
$490$	0	0
$491$	36.5220	1.64822	0.824108	−	0.566432i	$-0.191677\pi$
$491$	36.5220	1.64822	0.824108	+	0.566432i	$0.191677\pi$
$492$	0	0
$493$	−10.4566	−0.470944
$494$	0	0
$495$	2.03950	0.0916688
$496$	0	0
$497$	−18.0590	−0.810058
$498$	0	0
$499$	25.0318	1.12058	0.560289	−	0.828297i	$-0.310690\pi$
$499$	25.0318	1.12058	0.560289	+	0.828297i	$0.310690\pi$
$500$	0	0
$501$	24.4150	1.09078
$502$	0	0
$503$	−19.2803	−0.859665	−0.429832	−	0.902909i	$-0.641427\pi$
$503$	−19.2803	−0.859665	−0.429832	+	0.902909i	$0.641427\pi$
$504$	0	0
$505$	−5.72054	−0.254561
$506$	0	0
$507$	−24.7645	−1.09983
$508$	0	0
$509$	12.2522	0.543068	0.271534	−	0.962429i	$-0.412469\pi$
$509$	12.2522	0.543068	0.271534	+	0.962429i	$0.412469\pi$
$510$	0	0
$511$	−26.5807	−1.17586
$512$	0	0
$513$	−7.26177	−0.320615
$514$	0	0
$515$	17.8663	0.787284
$516$	0	0
$517$	−11.8493	−0.521130
$518$	0	0
$519$	−4.95174	−0.217357
$520$	0	0
$521$	7.31631	0.320533	0.160267	−	0.987074i	$-0.448765\pi$
$521$	7.31631	0.320533	0.160267	+	0.987074i	$0.448765\pi$
$522$	0	0
$523$	−11.0229	−0.481998	−0.240999	−	0.970525i	$-0.577475\pi$
$523$	−11.0229	−0.481998	−0.240999	+	0.970525i	$0.577475\pi$
$524$	0	0
$525$	9.53408	0.416101
$526$	0	0
$527$	−25.1118	−1.09389
$528$	0	0
$529$	−14.1073	−0.613361
$530$	0	0
$531$	−1.91611	−0.0831520
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−27.8068	−1.20219
$536$	0	0
$537$	26.9359	1.16237
$538$	0	0
$539$	7.12209	0.306770
$540$	0	0
$541$	43.9779	1.89076	0.945379	−	0.325973i	$-0.105692\pi$
$541$	43.9779	1.89076	0.945379	+	0.325973i	$0.105692\pi$
$542$	0	0
$543$	−34.3308	−1.47327
$544$	0	0
$545$	13.8938	0.595145
$546$	0	0
$547$	17.7048	0.757003	0.378501	−	0.925601i	$-0.376440\pi$
$547$	17.7048	0.757003	0.378501	+	0.925601i	$0.376440\pi$
$548$	0	0
$549$	−9.09978	−0.388369
$550$	0	0
$551$	3.15322	0.134332
$552$	0	0
$553$	−12.6955	−0.539866
$554$	0	0
$555$	−21.2932	−0.903845
$556$	0	0
$557$	−20.6032	−0.872986	−0.436493	−	0.899708i	$-0.643780\pi$
$557$	−20.6032	−0.872986	−0.436493	+	0.899708i	$0.643780\pi$
$558$	0	0
$559$	−5.51405	−0.233219
$560$	0	0
$561$	19.0319	0.803528
$562$	0	0
$563$	17.8313	0.751498	0.375749	−	0.926721i	$-0.377385\pi$
$563$	17.8313	0.751498	0.375749	+	0.926721i	$0.377385\pi$
$564$	0	0
$565$	−26.4894	−1.11442
$566$	0	0
$567$	18.3105	0.768968
$568$	0	0
$569$	9.10780	0.381819	0.190909	−	0.981608i	$-0.438856\pi$
$569$	9.10780	0.381819	0.190909	+	0.981608i	$0.438856\pi$
$570$	0	0
$571$	10.3448	0.432916	0.216458	−	0.976292i	$-0.430550\pi$
$571$	10.3448	0.432916	0.216458	+	0.976292i	$0.430550\pi$
$572$	0	0
$573$	−36.3289	−1.51766
$574$	0	0
$575$	−8.57057	−0.357418
$576$	0	0
$577$	2.45889	0.102365	0.0511824	−	0.998689i	$-0.483701\pi$
$577$	2.45889	0.102365	0.0511824	+	0.998689i	$0.483701\pi$
$578$	0	0
$579$	−1.40319	−0.0583148
$580$	0	0
$581$	−25.4316	−1.05508
$582$	0	0
$583$	0.829920	0.0343718
$584$	0	0
$585$	0.653034	0.0269997
$586$	0	0
$587$	−9.21100	−0.380178	−0.190089	−	0.981767i	$-0.560878\pi$
$587$	−9.21100	−0.380178	−0.190089	+	0.981767i	$0.560878\pi$
$588$	0	0
$589$	7.57250	0.312019
$590$	0	0
$591$	−8.40405	−0.345696
$592$	0	0
$593$	18.5719	0.762657	0.381329	−	0.924440i	$-0.375467\pi$
$593$	18.5719	0.762657	0.381329	+	0.924440i	$0.375467\pi$
$594$	0	0
$595$	−13.9518	−0.571968
$596$	0	0
$597$	26.8577	1.09921
$598$	0	0
$599$	2.32941	0.0951772	0.0475886	−	0.998867i	$-0.484846\pi$
$599$	2.32941	0.0951772	0.0475886	+	0.998867i	$0.484846\pi$
$600$	0	0
$601$	−22.3148	−0.910241	−0.455121	−	0.890430i	$-0.650404\pi$
$601$	−22.3148	−0.910241	−0.455121	+	0.890430i	$0.650404\pi$
$602$	0	0
$603$	1.56586	0.0637666
$604$	0	0
$605$	11.6593	0.474017
$606$	0	0
$607$	−33.0334	−1.34078	−0.670392	−	0.742007i	$-0.733874\pi$
$607$	−33.0334	−1.34078	−0.670392	+	0.742007i	$0.733874\pi$
$608$	0	0
$609$	−6.16331	−0.249750
$610$	0	0
$611$	−3.79405	−0.153491
$612$	0	0
$613$	−25.7702	−1.04085	−0.520424	−	0.853908i	$-0.674226\pi$
$613$	−25.7702	−1.04085	−0.520424	+	0.853908i	$0.674226\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.4918	1.26781	0.633906	−	0.773410i	$-0.281451\pi$
$617$	31.4918	1.26781	0.633906	+	0.773410i	$0.281451\pi$
$618$	0	0
$619$	30.3701	1.22068	0.610339	−	0.792140i	$-0.291033\pi$
$619$	30.3701	1.22068	0.610339	+	0.792140i	$0.291033\pi$
$620$	0	0
$621$	−12.7595	−0.512021
$622$	0	0
$623$	−4.23401	−0.169632
$624$	0	0
$625$	−2.36967	−0.0947869
$626$	0	0
$627$	−5.73911	−0.229198
$628$	0	0
$629$	−42.1241	−1.67960
$630$	0	0
$631$	3.47740	0.138433	0.0692165	−	0.997602i	$-0.477950\pi$
$631$	3.47740	0.138433	0.0692165	+	0.997602i	$0.477950\pi$
$632$	0	0
$633$	31.8106	1.26436
$634$	0	0
$635$	17.7906	0.706000
$636$	0	0
$637$	2.28044	0.0903545
$638$	0	0
$639$	8.57296	0.339141
$640$	0	0
$641$	−48.1780	−1.90292	−0.951458	−	0.307778i	$-0.900415\pi$
$641$	−48.1780	−1.90292	−0.951458	+	0.307778i	$0.900415\pi$
$642$	0	0
$643$	−32.3072	−1.27407	−0.637036	−	0.770834i	$-0.719840\pi$
$643$	−32.3072	−1.27407	−0.637036	+	0.770834i	$0.719840\pi$
$644$	0	0
$645$	−28.2690	−1.11309
$646$	0	0
$647$	−31.0314	−1.21997	−0.609984	−	0.792413i	$-0.708824\pi$
$647$	−31.0314	−1.21997	−0.609984	+	0.792413i	$0.708824\pi$
$648$	0	0
$649$	4.11449	0.161508
$650$	0	0
$651$	−14.8013	−0.580107
$652$	0	0
$653$	−45.8540	−1.79441	−0.897203	−	0.441618i	$-0.854405\pi$
$653$	−45.8540	−1.79441	−0.897203	+	0.441618i	$0.854405\pi$
$654$	0	0
$655$	8.52795	0.333215
$656$	0	0
$657$	12.6184	0.492289
$658$	0	0
$659$	−7.45274	−0.290317	−0.145159	−	0.989408i	$-0.546369\pi$
$659$	−7.45274	−0.290317	−0.145159	+	0.989408i	$0.546369\pi$
$660$	0	0
$661$	3.80325	0.147929	0.0739647	−	0.997261i	$-0.476435\pi$
$661$	3.80325	0.147929	0.0739647	+	0.997261i	$0.476435\pi$
$662$	0	0
$663$	6.09389	0.236667
$664$	0	0
$665$	4.20719	0.163148
$666$	0	0
$667$	5.54045	0.214527
$668$	0	0
$669$	42.9922	1.66217
$670$	0	0
$671$	19.5401	0.754338
$672$	0	0
$673$	−6.93547	−0.267343	−0.133671	−	0.991026i	$-0.542677\pi$
$673$	−6.93547	−0.267343	−0.133671	+	0.991026i	$0.542677\pi$
$674$	0	0
$675$	12.2973	0.473323
$676$	0	0
$677$	17.0626	0.655770	0.327885	−	0.944718i	$-0.393664\pi$
$677$	17.0626	0.655770	0.327885	+	0.944718i	$0.393664\pi$
$678$	0	0
$679$	0.860078	0.0330068
$680$	0	0
$681$	−34.2211	−1.31136
$682$	0	0
$683$	15.0022	0.574044	0.287022	−	0.957924i	$-0.407335\pi$
$683$	15.0022	0.574044	0.287022	+	0.957924i	$0.407335\pi$
$684$	0	0
$685$	−11.7942	−0.450632
$686$	0	0
$687$	−51.3860	−1.96050
$688$	0	0
$689$	0.265735	0.0101237
$690$	0	0
$691$	15.1399	0.575950	0.287975	−	0.957638i	$-0.407018\pi$
$691$	15.1399	0.575950	0.287975	+	0.957638i	$0.407018\pi$
$692$	0	0
$693$	2.37813	0.0903376
$694$	0	0
$695$	−0.467902	−0.0177485
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−2.49889	−0.0945166
$700$	0	0
$701$	−23.3044	−0.880193	−0.440097	−	0.897950i	$-0.645056\pi$
$701$	−23.3044	−0.880193	−0.440097	+	0.897950i	$0.645056\pi$
$702$	0	0
$703$	12.7026	0.479087
$704$	0	0
$705$	−19.4511	−0.732571
$706$	0	0
$707$	−6.67035	−0.250864
$708$	0	0
$709$	29.1253	1.09382	0.546912	−	0.837190i	$-0.315803\pi$
$709$	29.1253	1.09382	0.546912	+	0.837190i	$0.315803\pi$
$710$	0	0
$711$	6.02678	0.226022
$712$	0	0
$713$	13.3055	0.498293
$714$	0	0
$715$	−1.40227	−0.0524420
$716$	0	0
$717$	−30.8969	−1.15387
$718$	0	0
$719$	30.9219	1.15319	0.576596	−	0.817030i	$-0.304381\pi$
$719$	30.9219	1.15319	0.576596	+	0.817030i	$0.304381\pi$
$720$	0	0
$721$	20.8327	0.775852
$722$	0	0
$723$	50.5306	1.87925
$724$	0	0
$725$	−5.33976	−0.198314
$726$	0	0
$727$	40.6202	1.50652	0.753259	−	0.657724i	$-0.228481\pi$
$727$	40.6202	1.50652	0.753259	+	0.657724i	$0.228481\pi$
$728$	0	0
$729$	16.3535	0.605684
$730$	0	0
$731$	−55.9243	−2.06844
$732$	0	0
$733$	16.3370	0.603420	0.301710	−	0.953400i	$-0.402443\pi$
$733$	16.3370	0.603420	0.301710	+	0.953400i	$0.402443\pi$
$734$	0	0
$735$	11.6912	0.431238
$736$	0	0
$737$	−3.36239	−0.123855
$738$	0	0
$739$	−14.9723	−0.550763	−0.275382	−	0.961335i	$-0.588804\pi$
$739$	−14.9723	−0.550763	−0.275382	+	0.961335i	$0.588804\pi$
$740$	0	0
$741$	−1.83762	−0.0675068
$742$	0	0
$743$	−6.43428	−0.236051	−0.118025	−	0.993011i	$-0.537656\pi$
$743$	−6.43428	−0.236051	−0.118025	+	0.993011i	$0.537656\pi$
$744$	0	0
$745$	−3.18864	−0.116823
$746$	0	0
$747$	12.0728	0.441722
$748$	0	0
$749$	−32.4237	−1.18474
$750$	0	0
$751$	−36.8067	−1.34310	−0.671548	−	0.740961i	$-0.734370\pi$
$751$	−36.8067	−1.34310	−0.671548	+	0.740961i	$0.734370\pi$
$752$	0	0
$753$	−7.52436	−0.274203
$754$	0	0
$755$	8.03311	0.292355
$756$	0	0
$757$	−16.7225	−0.607788	−0.303894	−	0.952706i	$-0.598287\pi$
$757$	−16.7225	−0.607788	−0.303894	+	0.952706i	$0.598287\pi$
$758$	0	0
$759$	−10.0841	−0.366028
$760$	0	0
$761$	−23.9071	−0.866632	−0.433316	−	0.901242i	$-0.642657\pi$
$761$	−23.9071	−0.866632	−0.433316	+	0.901242i	$0.642657\pi$
$762$	0	0
$763$	16.2007	0.586503
$764$	0	0
$765$	6.62318	0.239462
$766$	0	0
$767$	1.31743	0.0475697
$768$	0	0
$769$	23.0206	0.830143	0.415071	−	0.909789i	$-0.363757\pi$
$769$	23.0206	0.830143	0.415071	+	0.909789i	$0.363757\pi$
$770$	0	0
$771$	34.9954	1.26033
$772$	0	0
$773$	−26.0366	−0.936471	−0.468236	−	0.883604i	$-0.655110\pi$
$773$	−26.0366	−0.936471	−0.468236	+	0.883604i	$0.655110\pi$
$774$	0	0
$775$	−12.8235	−0.460633
$776$	0	0
$777$	−24.8286	−0.890720
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−18.4089	−0.658721
$782$	0	0
$783$	−7.94959	−0.284095
$784$	0	0
$785$	−19.3679	−0.691272
$786$	0	0
$787$	−4.10144	−0.146201	−0.0731003	−	0.997325i	$-0.523289\pi$
$787$	−4.10144	−0.146201	−0.0731003	+	0.997325i	$0.523289\pi$
$788$	0	0
$789$	57.0531	2.03114
$790$	0	0
$791$	−30.8875	−1.09823
$792$	0	0
$793$	6.25661	0.222179
$794$	0	0
$795$	1.36235	0.0483176
$796$	0	0
$797$	−24.9970	−0.885438	−0.442719	−	0.896660i	$-0.645986\pi$
$797$	−24.9970	−0.885438	−0.442719	+	0.896660i	$0.645986\pi$
$798$	0	0
$799$	−38.4799	−1.36132
$800$	0	0
$801$	2.00997	0.0710187
$802$	0	0
$803$	−27.0956	−0.956184
$804$	0	0
$805$	7.39235	0.260546
$806$	0	0
$807$	46.8093	1.64776
$808$	0	0
$809$	−6.11465	−0.214980	−0.107490	−	0.994206i	$-0.534281\pi$
$809$	−6.11465	−0.214980	−0.107490	+	0.994206i	$0.534281\pi$
$810$	0	0
$811$	12.1786	0.427648	0.213824	−	0.976872i	$-0.431408\pi$
$811$	12.1786	0.427648	0.213824	+	0.976872i	$0.431408\pi$
$812$	0	0
$813$	18.2752	0.640941
$814$	0	0
$815$	−2.49509	−0.0873992
$816$	0	0
$817$	16.8641	0.590000
$818$	0	0
$819$	0.761460	0.0266076
$820$	0	0
$821$	29.6501	1.03480	0.517398	−	0.855745i	$-0.326901\pi$
$821$	29.6501	1.03480	0.517398	+	0.855745i	$0.326901\pi$
$822$	0	0
$823$	−31.4073	−1.09479	−0.547395	−	0.836874i	$-0.684380\pi$
$823$	−31.4073	−1.09479	−0.547395	+	0.836874i	$0.684380\pi$
$824$	0	0
$825$	9.71878	0.338364
$826$	0	0
$827$	16.1088	0.560159	0.280080	−	0.959977i	$-0.409639\pi$
$827$	16.1088	0.560159	0.280080	+	0.959977i	$0.409639\pi$
$828$	0	0
$829$	−13.0818	−0.454349	−0.227174	−	0.973854i	$-0.572949\pi$
$829$	−13.0818	−0.454349	−0.227174	+	0.973854i	$0.572949\pi$
$830$	0	0
$831$	34.0311	1.18052
$832$	0	0
$833$	23.1286	0.801360
$834$	0	0
$835$	−18.2447	−0.631385
$836$	0	0
$837$	−19.0910	−0.659883
$838$	0	0
$839$	8.67572	0.299519	0.149760	−	0.988722i	$-0.452150\pi$
$839$	8.67572	0.299519	0.149760	+	0.988722i	$0.452150\pi$
$840$	0	0
$841$	−25.5481	−0.880969
$842$	0	0
$843$	−34.3097	−1.18169
$844$	0	0
$845$	18.5059	0.636621
$846$	0	0
$847$	13.5951	0.467134
$848$	0	0
$849$	0.190217	0.00652823
$850$	0	0
$851$	22.3194	0.765100
$852$	0	0
$853$	−1.57181	−0.0538179	−0.0269089	−	0.999638i	$-0.508566\pi$
$853$	−1.57181	−0.0538179	−0.0269089	+	0.999638i	$0.508566\pi$
$854$	0	0
$855$	−1.99723	−0.0683039
$856$	0	0
$857$	37.8371	1.29249	0.646246	−	0.763129i	$-0.276338\pi$
$857$	37.8371	1.29249	0.646246	+	0.763129i	$0.276338\pi$
$858$	0	0
$859$	−1.61748	−0.0551875	−0.0275938	−	0.999619i	$-0.508784\pi$
$859$	−1.61748	−0.0551875	−0.0275938	+	0.999619i	$0.508784\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.0345	1.39683	0.698416	−	0.715692i	$-0.253889\pi$
$863$	41.0345	1.39683	0.698416	+	0.715692i	$0.253889\pi$
$864$	0	0
$865$	3.70031	0.125814
$866$	0	0
$867$	28.6352	0.972502
$868$	0	0
$869$	−12.9414	−0.439007
$870$	0	0
$871$	−1.07661	−0.0364797
$872$	0	0
$873$	−0.408295	−0.0138187
$874$	0	0
$875$	−19.5193	−0.659872
$876$	0	0
$877$	−17.5169	−0.591503	−0.295751	−	0.955265i	$-0.595570\pi$
$877$	−17.5169	−0.591503	−0.295751	+	0.955265i	$0.595570\pi$
$878$	0	0
$879$	−50.0405	−1.68782
$880$	0	0
$881$	39.0590	1.31593	0.657965	−	0.753049i	$-0.271418\pi$
$881$	39.0590	1.31593	0.657965	+	0.753049i	$0.271418\pi$
$882$	0	0
$883$	18.3243	0.616661	0.308330	−	0.951279i	$-0.400230\pi$
$883$	18.3243	0.616661	0.308330	+	0.951279i	$0.400230\pi$
$884$	0	0
$885$	6.75412	0.227037
$886$	0	0
$887$	13.9531	0.468499	0.234249	−	0.972177i	$-0.424737\pi$
$887$	13.9531	0.468499	0.234249	+	0.972177i	$0.424737\pi$
$888$	0	0
$889$	20.7445	0.695748
$890$	0	0
$891$	18.6652	0.625308
$892$	0	0
$893$	11.6037	0.388302
$894$	0	0
$895$	−20.1285	−0.672822
$896$	0	0
$897$	−3.22884	−0.107808
$898$	0	0
$899$	8.28975	0.276479
$900$	0	0
$901$	2.69513	0.0897876
$902$	0	0
$903$	−32.9627	−1.09693
$904$	0	0
$905$	25.6545	0.852785
$906$	0	0
$907$	3.59595	0.119402	0.0597008	−	0.998216i	$-0.480985\pi$
$907$	3.59595	0.119402	0.0597008	+	0.998216i	$0.480985\pi$
$908$	0	0
$909$	3.16654	0.105028
$910$	0	0
$911$	−48.6712	−1.61255	−0.806275	−	0.591541i	$-0.798520\pi$
$911$	−48.6712	−1.61255	−0.806275	+	0.591541i	$0.798520\pi$
$912$	0	0
$913$	−25.9242	−0.857967
$914$	0	0
$915$	32.0760	1.06040
$916$	0	0
$917$	9.94388	0.328376
$918$	0	0
$919$	−2.65337	−0.0875266	−0.0437633	−	0.999042i	$-0.513935\pi$
$919$	−2.65337	−0.0875266	−0.0437633	+	0.999042i	$0.513935\pi$
$920$	0	0
$921$	6.05916	0.199656
$922$	0	0
$923$	−5.89439	−0.194016
$924$	0	0
$925$	−21.5109	−0.707275
$926$	0	0
$927$	−9.88970	−0.324820
$928$	0	0
$929$	59.5977	1.95534	0.977668	−	0.210156i	$-0.0673971\pi$
$929$	59.5977	1.95534	0.977668	+	0.210156i	$0.0673971\pi$
$930$	0	0
$931$	−6.97448	−0.228579
$932$	0	0
$933$	−36.2402	−1.18645
$934$	0	0
$935$	−14.2221	−0.465112
$936$	0	0
$937$	16.7017	0.545621	0.272810	−	0.962068i	$-0.412047\pi$
$937$	16.7017	0.545621	0.272810	+	0.962068i	$0.412047\pi$
$938$	0	0
$939$	9.66865	0.315524
$940$	0	0
$941$	41.4870	1.35244	0.676219	−	0.736701i	$-0.263617\pi$
$941$	41.4870	1.35244	0.676219	+	0.736701i	$0.263617\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−10.6068	−0.345038
$946$	0	0
$947$	−43.6049	−1.41697	−0.708484	−	0.705726i	$-0.750621\pi$
$947$	−43.6049	−1.41697	−0.708484	+	0.705726i	$0.750621\pi$
$948$	0	0
$949$	−8.67583	−0.281629
$950$	0	0
$951$	−19.7380	−0.640048
$952$	0	0
$953$	12.5702	0.407189	0.203594	−	0.979055i	$-0.434738\pi$
$953$	12.5702	0.407189	0.203594	+	0.979055i	$0.434738\pi$
$954$	0	0
$955$	27.1476	0.878477
$956$	0	0
$957$	−6.28271	−0.203091
$958$	0	0
$959$	−13.7524	−0.444088
$960$	0	0
$961$	−11.0921	−0.357808
$962$	0	0
$963$	15.3921	0.496005
$964$	0	0
$965$	1.04857	0.0337547
$966$	0	0
$967$	40.2360	1.29390	0.646951	−	0.762531i	$-0.276044\pi$
$967$	40.2360	1.29390	0.646951	+	0.762531i	$0.276044\pi$
$968$	0	0
$969$	−18.6375	−0.598722
$970$	0	0
$971$	11.5557	0.370841	0.185421	−	0.982659i	$-0.440635\pi$
$971$	11.5557	0.370841	0.185421	+	0.982659i	$0.440635\pi$
$972$	0	0
$973$	−0.545589	−0.0174908
$974$	0	0
$975$	3.11189	0.0996601
$976$	0	0
$977$	−1.99747	−0.0639048	−0.0319524	−	0.999489i	$-0.510172\pi$
$977$	−1.99747	−0.0639048	−0.0319524	+	0.999489i	$0.510172\pi$
$978$	0	0
$979$	−4.31604	−0.137941
$980$	0	0
$981$	−7.69076	−0.245547
$982$	0	0
$983$	26.6664	0.850527	0.425263	−	0.905070i	$-0.360181\pi$
$983$	26.6664	0.850527	0.425263	+	0.905070i	$0.360181\pi$
$984$	0	0
$985$	6.28014	0.200102
$986$	0	0
$987$	−22.6806	−0.721933
$988$	0	0
$989$	29.6315	0.942226
$990$	0	0
$991$	−5.07173	−0.161109	−0.0805543	−	0.996750i	$-0.525669\pi$
$991$	−5.07173	−0.161109	−0.0805543	+	0.996750i	$0.525669\pi$
$992$	0	0
$993$	−60.2634	−1.91240
$994$	0	0
$995$	−20.0701	−0.636264
$996$	0	0
$997$	23.0989	0.731550	0.365775	−	0.930703i	$-0.380804\pi$
$997$	23.0989	0.731550	0.365775	+	0.930703i	$0.380804\pi$
$998$	0	0
$999$	−32.0245	−1.01321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.f.1.7		8
41.16	even	5		164.2.g.a.133.4	yes	16
41.18	even	5		164.2.g.a.37.4	✓	16
41.40	even	2		6724.2.a.g.1.2		8
123.59	odd	10		1476.2.n.f.37.3		16
123.98	odd	10		1476.2.n.f.1117.3		16
164.59	odd	10		656.2.u.g.529.1		16
164.139	odd	10		656.2.u.g.625.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.g.a.37.4	✓	16	41.18	even	5
164.2.g.a.133.4	yes	16	41.16	even	5
656.2.u.g.529.1		16	164.59	odd	10
656.2.u.g.625.1		16	164.139	odd	10
1476.2.n.f.37.3		16	123.59	odd	10
1476.2.n.f.1117.3		16	123.98	odd	10
6724.2.a.f.1.7		8	1.1	even	1	trivial
6724.2.a.g.1.2		8	41.40	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:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q+1.95118 q^{3} -1.45807 q^{5} -1.70016 q^{7} +0.807096 q^{9} -1.73309 q^{11} -0.554924 q^{13} -2.84495 q^{15} -5.62813 q^{17} +1.69717 q^{19} -3.31731 q^{21} +2.98206 q^{23} -2.87404 q^{25} -4.27875 q^{27} +1.85793 q^{29} +4.46183 q^{31} -3.38157 q^{33} +2.47894 q^{35} +7.48456 q^{37} -1.08276 q^{39} +9.93658 q^{43} -1.17680 q^{45} +6.83707 q^{47} -4.10947 q^{49} -10.9815 q^{51} -0.478867 q^{53} +2.52696 q^{55} +3.31149 q^{57} -2.37408 q^{59} -11.2747 q^{61} -1.37219 q^{63} +0.809116 q^{65} +1.94011 q^{67} +5.81853 q^{69} +10.6220 q^{71} +15.6343 q^{73} -5.60777 q^{75} +2.94652 q^{77} +7.46724 q^{79} -10.7699 q^{81} +14.9584 q^{83} +8.20619 q^{85} +3.62514 q^{87} +2.49037 q^{89} +0.943457 q^{91} +8.70583 q^{93} -2.47459 q^{95} -0.505882 q^{97} -1.39877 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} + 3 q^{5} + 9 q^{9} + 8 q^{11} - 10 q^{13} + 5 q^{15} + 7 q^{17} - 3 q^{19} - 5 q^{21} + 6 q^{23} - q^{25} + 5 q^{27} + 17 q^{29} - q^{31} - 22 q^{35} - 8 q^{37} + 12 q^{39} + 18 q^{43} + 8 q^{45}+ \cdots + 56 q^{99}+O(q^{100}) \) 8 * q - q^3 + 3 * q^5 + 9 * q^9 + 8 * q^11 - 10 * q^13 + 5 * q^15 + 7 * q^17 - 3 * q^19 - 5 * q^21 + 6 * q^23 - q^25 + 5 * q^27 + 17 * q^29 - q^31 - 22 * q^35 - 8 * q^37 + 12 * q^39 + 18 * q^43 + 8 * q^45 + 3 * q^47 + 42 * q^49 + 7 * q^51 + 8 * q^53 + 21 * q^55 + 33 * q^57 + 44 * q^59 - 7 * q^61 - 40 * q^63 - 47 * q^65 - 19 * q^67 - 6 * q^69 - 14 * q^71 + 21 * q^73 + 15 * q^75 + 36 * q^77 + 9 * q^79 - 4 * q^81 + 26 * q^83 + 29 * q^85 - 27 * q^87 + 13 * q^89 + 42 * q^91 - 6 * q^93 - 22 * q^95 + 12 * q^97 + 56 * q^99

Introduction

L-functions

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