LMFDB - Embedded newform 6664.2.a.v.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,-3,0,2] 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:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 8x^{5} + 21x^{4} + 25x^{3} - 41x^{2} - 28x + 19 $ x^7 - 3x^6 - 8x^5 + 21x^4 + 25x^3 - 41x^2 - 28x + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.474430$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.474430 q^{3} +1.63521 q^{5} -2.77492 q^{9} -0.241103 q^{11} +1.90325 q^{13} -0.775795 q^{15} +1.00000 q^{17} -5.11412 q^{19} +3.66413 q^{23} -2.32607 q^{25} +2.73980 q^{27} +2.50385 q^{29} -2.41583 q^{31} +0.114387 q^{33} +3.35512 q^{37} -0.902958 q^{39} -9.52792 q^{41} +9.54468 q^{43} -4.53758 q^{45} -8.63945 q^{47} -0.474430 q^{51} -11.6158 q^{53} -0.394255 q^{55} +2.42629 q^{57} -2.07093 q^{59} +10.6760 q^{61} +3.11222 q^{65} -12.5662 q^{67} -1.73837 q^{69} +0.576299 q^{71} +15.5443 q^{73} +1.10356 q^{75} -8.46606 q^{79} +7.02491 q^{81} +1.02923 q^{83} +1.63521 q^{85} -1.18790 q^{87} +11.4269 q^{89} +1.14614 q^{93} -8.36268 q^{95} -3.69602 q^{97} +0.669041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 2 q^{5} + 4 q^{9} - 12 q^{11} + 2 q^{13} - 6 q^{15} + 7 q^{17} - 3 q^{19} - 18 q^{23} + 15 q^{25} - 18 q^{27} + 5 q^{29} - 10 q^{31} + 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} + 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 2 * q^5 + 4 * q^9 - 12 * q^11 + 2 * q^13 - 6 * q^15 + 7 * q^17 - 3 * q^19 - 18 * q^23 + 15 * q^25 - 18 * q^27 + 5 * q^29 - 10 * q^31 + 21 * q^33 + 11 * q^37 - 12 * q^39 - 15 * q^43 + 13 * q^45 + 15 * q^47 - 3 * q^51 + 8 * q^53 - 9 * q^55 - 3 * q^57 + 9 * q^59 - 4 * q^61 - 38 * q^65 - 30 * q^67 - 28 * q^69 - 27 * q^71 - 5 * q^73 - 2 * q^75 + 3 * q^79 - 17 * q^81 - 12 * q^83 + 2 * q^85 + 19 * q^87 + 32 * q^89 - 9 * q^93 - 42 * q^95 - 39 * q^97 - 20 * 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
$3$	−0.474430	−0.273913	−0.136956	−	0.990577i	$-0.543732\pi$
$3$	−0.474430	−0.273913	−0.136956	+	0.990577i	$0.543732\pi$
$4$	0	0
$5$	1.63521	0.731290	0.365645	−	0.930754i	$-0.380848\pi$
$5$	1.63521	0.731290	0.365645	+	0.930754i	$0.380848\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.77492	−0.924972
$10$	0	0
$11$	−0.241103	−0.0726953	−0.0363477	−	0.999339i	$-0.511572\pi$
$11$	−0.241103	−0.0726953	−0.0363477	+	0.999339i	$0.511572\pi$
$12$	0	0
$13$	1.90325	0.527866	0.263933	−	0.964541i	$-0.414980\pi$
$13$	1.90325	0.527866	0.263933	+	0.964541i	$0.414980\pi$
$14$	0	0
$15$	−0.775795	−0.200310
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−5.11412	−1.17326	−0.586630	−	0.809855i	$-0.699546\pi$
$19$	−5.11412	−1.17326	−0.586630	+	0.809855i	$0.699546\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.66413	0.764023	0.382012	−	0.924158i	$-0.375231\pi$
$23$	3.66413	0.764023	0.382012	+	0.924158i	$0.375231\pi$
$24$	0	0
$25$	−2.32607	−0.465215
$26$	0	0
$27$	2.73980	0.527274
$28$	0	0
$29$	2.50385	0.464953	0.232477	−	0.972602i	$-0.425317\pi$
$29$	2.50385	0.464953	0.232477	+	0.972602i	$0.425317\pi$
$30$	0	0
$31$	−2.41583	−0.433896	−0.216948	−	0.976183i	$-0.569610\pi$
$31$	−2.41583	−0.433896	−0.216948	+	0.976183i	$0.569610\pi$
$32$	0	0
$33$	0.114387	0.0199122
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.35512	0.551579	0.275789	−	0.961218i	$-0.411061\pi$
$37$	3.35512	0.551579	0.275789	+	0.961218i	$0.411061\pi$
$38$	0	0
$39$	−0.902958	−0.144589
$40$	0	0
$41$	−9.52792	−1.48801	−0.744006	−	0.668173i	$-0.767076\pi$
$41$	−9.52792	−1.48801	−0.744006	+	0.668173i	$0.767076\pi$
$42$	0	0
$43$	9.54468	1.45555	0.727775	−	0.685816i	$-0.240555\pi$
$43$	9.54468	1.45555	0.727775	+	0.685816i	$0.240555\pi$
$44$	0	0
$45$	−4.53758	−0.676423
$46$	0	0
$47$	−8.63945	−1.26019	−0.630097	−	0.776517i	$-0.716985\pi$
$47$	−8.63945	−1.26019	−0.630097	+	0.776517i	$0.716985\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.474430	−0.0664335
$52$	0	0
$53$	−11.6158	−1.59556	−0.797779	−	0.602949i	$-0.793992\pi$
$53$	−11.6158	−1.59556	−0.797779	+	0.602949i	$0.793992\pi$
$54$	0	0
$55$	−0.394255	−0.0531614
$56$	0	0
$57$	2.42629	0.321370
$58$	0	0
$59$	−2.07093	−0.269612	−0.134806	−	0.990872i	$-0.543041\pi$
$59$	−2.07093	−0.269612	−0.134806	+	0.990872i	$0.543041\pi$
$60$	0	0
$61$	10.6760	1.36693	0.683463	−	0.729985i	$-0.260473\pi$
$61$	10.6760	1.36693	0.683463	+	0.729985i	$0.260473\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.11222	0.386023
$66$	0	0
$67$	−12.5662	−1.53521	−0.767604	−	0.640924i	$-0.778551\pi$
$67$	−12.5662	−1.53521	−0.767604	+	0.640924i	$0.778551\pi$
$68$	0	0
$69$	−1.73837	−0.209275
$70$	0	0
$71$	0.576299	0.0683942	0.0341971	−	0.999415i	$-0.489113\pi$
$71$	0.576299	0.0683942	0.0341971	+	0.999415i	$0.489113\pi$
$72$	0	0
$73$	15.5443	1.81932	0.909659	−	0.415355i	$-0.136343\pi$
$73$	15.5443	1.81932	0.909659	+	0.415355i	$0.136343\pi$
$74$	0	0
$75$	1.10356	0.127428
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.46606	−0.952507	−0.476253	−	0.879308i	$-0.658005\pi$
$79$	−8.46606	−0.952507	−0.476253	+	0.879308i	$0.658005\pi$
$80$	0	0
$81$	7.02491	0.780545
$82$	0	0
$83$	1.02923	0.112972	0.0564862	−	0.998403i	$-0.482010\pi$
$83$	1.02923	0.112972	0.0564862	+	0.998403i	$0.482010\pi$
$84$	0	0
$85$	1.63521	0.177364
$86$	0	0
$87$	−1.18790	−0.127356
$88$	0	0
$89$	11.4269	1.21125	0.605624	−	0.795751i	$-0.292924\pi$
$89$	11.4269	1.21125	0.605624	+	0.795751i	$0.292924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.14614	0.118850
$94$	0	0
$95$	−8.36268	−0.857993
$96$	0	0
$97$	−3.69602	−0.375274	−0.187637	−	0.982238i	$-0.560083\pi$
$97$	−3.69602	−0.375274	−0.187637	+	0.982238i	$0.560083\pi$
$98$	0	0
$99$	0.669041	0.0672411
$100$	0	0
$101$	8.68120	0.863811	0.431906	−	0.901919i	$-0.357841\pi$
$101$	8.68120	0.863811	0.431906	+	0.901919i	$0.357841\pi$
$102$	0	0
$103$	−14.1120	−1.39050	−0.695248	−	0.718770i	$-0.744706\pi$
$103$	−14.1120	−1.39050	−0.695248	+	0.718770i	$0.744706\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.8529	−1.33921	−0.669605	−	0.742718i	$-0.733537\pi$
$107$	−13.8529	−1.33921	−0.669605	+	0.742718i	$0.733537\pi$
$108$	0	0
$109$	−6.26739	−0.600308	−0.300154	−	0.953891i	$-0.597038\pi$
$109$	−6.26739	−0.600308	−0.300154	+	0.953891i	$0.597038\pi$
$110$	0	0
$111$	−1.59177	−0.151084
$112$	0	0
$113$	−14.5983	−1.37329	−0.686644	−	0.726993i	$-0.740917\pi$
$113$	−14.5983	−1.37329	−0.686644	+	0.726993i	$0.740917\pi$
$114$	0	0
$115$	5.99163	0.558723
$116$	0	0
$117$	−5.28135	−0.488261
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9419	−0.994715
$122$	0	0
$123$	4.52033	0.407585
$124$	0	0
$125$	−11.9797	−1.07150
$126$	0	0
$127$	−14.9323	−1.32502	−0.662512	−	0.749052i	$-0.730510\pi$
$127$	−14.9323	−1.32502	−0.662512	+	0.749052i	$0.730510\pi$
$128$	0	0
$129$	−4.52828	−0.398693
$130$	0	0
$131$	9.23736	0.807072	0.403536	−	0.914964i	$-0.367781\pi$
$131$	9.23736	0.807072	0.403536	+	0.914964i	$0.367781\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.48015	0.385590
$136$	0	0
$137$	−0.201993	−0.0172574	−0.00862872	−	0.999963i	$-0.502747\pi$
$137$	−0.201993	−0.0172574	−0.00862872	+	0.999963i	$0.502747\pi$
$138$	0	0
$139$	−8.80733	−0.747028	−0.373514	−	0.927625i	$-0.621847\pi$
$139$	−8.80733	−0.747028	−0.373514	+	0.927625i	$0.621847\pi$
$140$	0	0
$141$	4.09882	0.345183
$142$	0	0
$143$	−0.458879	−0.0383734
$144$	0	0
$145$	4.09433	0.340016
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.90694	−0.401992	−0.200996	−	0.979592i	$-0.564418\pi$
$149$	−4.90694	−0.401992	−0.200996	+	0.979592i	$0.564418\pi$
$150$	0	0
$151$	−3.03913	−0.247321	−0.123661	−	0.992325i	$-0.539463\pi$
$151$	−3.03913	−0.247321	−0.123661	+	0.992325i	$0.539463\pi$
$152$	0	0
$153$	−2.77492	−0.224339
$154$	0	0
$155$	−3.95040	−0.317304
$156$	0	0
$157$	−19.9083	−1.58886	−0.794429	−	0.607357i	$-0.792230\pi$
$157$	−19.9083	−1.58886	−0.794429	+	0.607357i	$0.792230\pi$
$158$	0	0
$159$	5.51091	0.437044
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.54155	0.434048	0.217024	−	0.976166i	$-0.430365\pi$
$163$	5.54155	0.434048	0.217024	+	0.976166i	$0.430365\pi$
$164$	0	0
$165$	0.187047	0.0145616
$166$	0	0
$167$	3.42008	0.264654	0.132327	−	0.991206i	$-0.457755\pi$
$167$	3.42008	0.264654	0.132327	+	0.991206i	$0.457755\pi$
$168$	0	0
$169$	−9.37765	−0.721358
$170$	0	0
$171$	14.1912	1.08523
$172$	0	0
$173$	11.3536	0.863195	0.431598	−	0.902066i	$-0.357950\pi$
$173$	11.3536	0.863195	0.431598	+	0.902066i	$0.357950\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.982511	0.0738501
$178$	0	0
$179$	14.7667	1.10372	0.551859	−	0.833938i	$-0.313919\pi$
$179$	14.7667	1.10372	0.551859	+	0.833938i	$0.313919\pi$
$180$	0	0
$181$	4.00555	0.297730	0.148865	−	0.988858i	$-0.452438\pi$
$181$	4.00555	0.297730	0.148865	+	0.988858i	$0.452438\pi$
$182$	0	0
$183$	−5.06504	−0.374418
$184$	0	0
$185$	5.48635	0.403364
$186$	0	0
$187$	−0.241103	−0.0176312
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.6248	−1.56471	−0.782357	−	0.622830i	$-0.785983\pi$
$191$	−21.6248	−1.56471	−0.782357	+	0.622830i	$0.785983\pi$
$192$	0	0
$193$	15.5283	1.11775	0.558877	−	0.829251i	$-0.311233\pi$
$193$	15.5283	1.11775	0.558877	+	0.829251i	$0.311233\pi$
$194$	0	0
$195$	−1.47653	−0.105737
$196$	0	0
$197$	0.389479	0.0277492	0.0138746	−	0.999904i	$-0.495583\pi$
$197$	0.389479	0.0277492	0.0138746	+	0.999904i	$0.495583\pi$
$198$	0	0
$199$	3.06877	0.217539	0.108770	−	0.994067i	$-0.465309\pi$
$199$	3.06877	0.217539	0.108770	+	0.994067i	$0.465309\pi$
$200$	0	0
$201$	5.96180	0.420513
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−15.5802	−1.08817
$206$	0	0
$207$	−10.1676	−0.706700
$208$	0	0
$209$	1.23303	0.0852904
$210$	0	0
$211$	−16.8747	−1.16170	−0.580850	−	0.814010i	$-0.697280\pi$
$211$	−16.8747	−1.16170	−0.580850	+	0.814010i	$0.697280\pi$
$212$	0	0
$213$	−0.273414	−0.0187340
$214$	0	0
$215$	15.6076	1.06443
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.37467	−0.498334
$220$	0	0
$221$	1.90325	0.128026
$222$	0	0
$223$	−4.32590	−0.289683	−0.144842	−	0.989455i	$-0.546267\pi$
$223$	−4.32590	−0.289683	−0.144842	+	0.989455i	$0.546267\pi$
$224$	0	0
$225$	6.45466	0.430311
$226$	0	0
$227$	24.1219	1.60103	0.800514	−	0.599313i	$-0.204560\pi$
$227$	24.1219	1.60103	0.800514	+	0.599313i	$0.204560\pi$
$228$	0	0
$229$	−0.408154	−0.0269716	−0.0134858	−	0.999909i	$-0.504293\pi$
$229$	−0.408154	−0.0269716	−0.0134858	+	0.999909i	$0.504293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.57468	−0.103161	−0.0515804	−	0.998669i	$-0.516426\pi$
$233$	−1.57468	−0.103161	−0.0515804	+	0.998669i	$0.516426\pi$
$234$	0	0
$235$	−14.1274	−0.921567
$236$	0	0
$237$	4.01656	0.260903
$238$	0	0
$239$	−7.31831	−0.473382	−0.236691	−	0.971585i	$-0.576063\pi$
$239$	−7.31831	−0.473382	−0.236691	+	0.971585i	$0.576063\pi$
$240$	0	0
$241$	13.0642	0.841540	0.420770	−	0.907167i	$-0.361760\pi$
$241$	13.0642	0.841540	0.420770	+	0.907167i	$0.361760\pi$
$242$	0	0
$243$	−11.5522	−0.741075
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.73343	−0.619324
$248$	0	0
$249$	−0.488297	−0.0309446
$250$	0	0
$251$	−10.1335	−0.639619	−0.319810	−	0.947482i	$-0.603619\pi$
$251$	−10.1335	−0.639619	−0.319810	+	0.947482i	$0.603619\pi$
$252$	0	0
$253$	−0.883432	−0.0555409
$254$	0	0
$255$	−0.775795	−0.0485822
$256$	0	0
$257$	14.3509	0.895183	0.447591	−	0.894238i	$-0.352282\pi$
$257$	14.3509	0.895183	0.447591	+	0.894238i	$0.352282\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.94797	−0.430069
$262$	0	0
$263$	−17.6427	−1.08789	−0.543947	−	0.839120i	$-0.683071\pi$
$263$	−17.6427	−1.08789	−0.543947	+	0.839120i	$0.683071\pi$
$264$	0	0
$265$	−18.9944	−1.16682
$266$	0	0
$267$	−5.42126	−0.331776
$268$	0	0
$269$	9.00765	0.549206	0.274603	−	0.961558i	$-0.411454\pi$
$269$	9.00765	0.549206	0.274603	+	0.961558i	$0.411454\pi$
$270$	0	0
$271$	−19.5196	−1.18573	−0.592864	−	0.805302i	$-0.702003\pi$
$271$	−19.5196	−1.18573	−0.592864	+	0.805302i	$0.702003\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.560823	0.0338189
$276$	0	0
$277$	26.2675	1.57826	0.789131	−	0.614225i	$-0.210531\pi$
$277$	26.2675	1.57826	0.789131	+	0.614225i	$0.210531\pi$
$278$	0	0
$279$	6.70373	0.401342
$280$	0	0
$281$	−0.117643	−0.00701798	−0.00350899	−	0.999994i	$-0.501117\pi$
$281$	−0.117643	−0.00701798	−0.00350899	+	0.999994i	$0.501117\pi$
$282$	0	0
$283$	−2.86895	−0.170542	−0.0852708	−	0.996358i	$-0.527176\pi$
$283$	−2.86895	−0.170542	−0.0852708	+	0.996358i	$0.527176\pi$
$284$	0	0
$285$	3.96751	0.235015
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.75350	0.102792
$292$	0	0
$293$	−17.8039	−1.04011	−0.520057	−	0.854132i	$-0.674089\pi$
$293$	−17.8039	−1.04011	−0.520057	+	0.854132i	$0.674089\pi$
$294$	0	0
$295$	−3.38641	−0.197164
$296$	0	0
$297$	−0.660573	−0.0383303
$298$	0	0
$299$	6.97374	0.403302
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.11862	−0.236609
$304$	0	0
$305$	17.4576	0.999620
$306$	0	0
$307$	−21.8210	−1.24539	−0.622695	−	0.782465i	$-0.713962\pi$
$307$	−21.8210	−1.24539	−0.622695	+	0.782465i	$0.713962\pi$
$308$	0	0
$309$	6.69516	0.380874
$310$	0	0
$311$	5.94343	0.337021	0.168511	−	0.985700i	$-0.446104\pi$
$311$	5.94343	0.337021	0.168511	+	0.985700i	$0.446104\pi$
$312$	0	0
$313$	−12.8644	−0.727138	−0.363569	−	0.931567i	$-0.618442\pi$
$313$	−12.8644	−0.727138	−0.363569	+	0.931567i	$0.618442\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.31744	−0.186326	−0.0931629	−	0.995651i	$-0.529698\pi$
$317$	−3.31744	−0.186326	−0.0931629	+	0.995651i	$0.529698\pi$
$318$	0	0
$319$	−0.603686	−0.0337999
$320$	0	0
$321$	6.57223	0.366826
$322$	0	0
$323$	−5.11412	−0.284557
$324$	0	0
$325$	−4.42709	−0.245571
$326$	0	0
$327$	2.97344	0.164432
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7.44505	0.409217	0.204608	−	0.978844i	$-0.434408\pi$
$331$	7.44505	0.409217	0.204608	+	0.978844i	$0.434408\pi$
$332$	0	0
$333$	−9.31019	−0.510195
$334$	0	0
$335$	−20.5485	−1.12268
$336$	0	0
$337$	−4.05359	−0.220813	−0.110407	−	0.993887i	$-0.535215\pi$
$337$	−4.05359	−0.220813	−0.110407	+	0.993887i	$0.535215\pi$
$338$	0	0
$339$	6.92586	0.376161
$340$	0	0
$341$	0.582465	0.0315422
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.84261	−0.153041
$346$	0	0
$347$	−35.3082	−1.89544	−0.947722	−	0.319096i	$-0.896621\pi$
$347$	−35.3082	−1.89544	−0.947722	+	0.319096i	$0.896621\pi$
$348$	0	0
$349$	−1.80670	−0.0967104	−0.0483552	−	0.998830i	$-0.515398\pi$
$349$	−1.80670	−0.0967104	−0.0483552	+	0.998830i	$0.515398\pi$
$350$	0	0
$351$	5.21451	0.278330
$352$	0	0
$353$	−0.450656	−0.0239860	−0.0119930	−	0.999928i	$-0.503818\pi$
$353$	−0.450656	−0.0239860	−0.0119930	+	0.999928i	$0.503818\pi$
$354$	0	0
$355$	0.942373	0.0500160
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.2275	1.91201	0.956007	−	0.293344i	$-0.0947681\pi$
$359$	36.2275	1.91201	0.956007	+	0.293344i	$0.0947681\pi$
$360$	0	0
$361$	7.15421	0.376537
$362$	0	0
$363$	5.19116	0.272465
$364$	0	0
$365$	25.4182	1.33045
$366$	0	0
$367$	−19.2616	−1.00545	−0.502724	−	0.864447i	$-0.667669\pi$
$367$	−19.2616	−1.00545	−0.502724	+	0.864447i	$0.667669\pi$
$368$	0	0
$369$	26.4392	1.37637
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.4332	−1.16155	−0.580774	−	0.814065i	$-0.697250\pi$
$373$	−22.4332	−1.16155	−0.580774	+	0.814065i	$0.697250\pi$
$374$	0	0
$375$	5.68353	0.293496
$376$	0	0
$377$	4.76544	0.245433
$378$	0	0
$379$	1.03798	0.0533176	0.0266588	−	0.999645i	$-0.491513\pi$
$379$	1.03798	0.0533176	0.0266588	+	0.999645i	$0.491513\pi$
$380$	0	0
$381$	7.08432	0.362941
$382$	0	0
$383$	−23.1557	−1.18320	−0.591600	−	0.806232i	$-0.701503\pi$
$383$	−23.1557	−1.18320	−0.591600	+	0.806232i	$0.701503\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26.4857	−1.34634
$388$	0	0
$389$	−36.6431	−1.85788	−0.928940	−	0.370231i	$-0.879278\pi$
$389$	−36.6431	−1.85788	−0.928940	+	0.370231i	$0.879278\pi$
$390$	0	0
$391$	3.66413	0.185303
$392$	0	0
$393$	−4.38248	−0.221067
$394$	0	0
$395$	−13.8438	−0.696559
$396$	0	0
$397$	−17.1632	−0.861396	−0.430698	−	0.902496i	$-0.641733\pi$
$397$	−17.1632	−0.861396	−0.430698	+	0.902496i	$0.641733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.95796	0.197651	0.0988256	−	0.995105i	$-0.468491\pi$
$401$	3.95796	0.197651	0.0988256	+	0.995105i	$0.468491\pi$
$402$	0	0
$403$	−4.59793	−0.229039
$404$	0	0
$405$	11.4872	0.570805
$406$	0	0
$407$	−0.808931	−0.0400972
$408$	0	0
$409$	6.06740	0.300014	0.150007	−	0.988685i	$-0.452070\pi$
$409$	6.06740	0.300014	0.150007	+	0.988685i	$0.452070\pi$
$410$	0	0
$411$	0.0958317	0.00472703
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.68301	0.0826157
$416$	0	0
$417$	4.17846	0.204620
$418$	0	0
$419$	5.33057	0.260415	0.130208	−	0.991487i	$-0.458436\pi$
$419$	5.33057	0.260415	0.130208	+	0.991487i	$0.458436\pi$
$420$	0	0
$421$	17.1834	0.837466	0.418733	−	0.908109i	$-0.362474\pi$
$421$	17.1834	0.837466	0.418733	+	0.908109i	$0.362474\pi$
$422$	0	0
$423$	23.9738	1.16564
$424$	0	0
$425$	−2.32607	−0.112831
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.217706	0.0105109
$430$	0	0
$431$	5.62016	0.270713	0.135357	−	0.990797i	$-0.456782\pi$
$431$	5.62016	0.270713	0.135357	+	0.990797i	$0.456782\pi$
$432$	0	0
$433$	−6.15198	−0.295645	−0.147823	−	0.989014i	$-0.547227\pi$
$433$	−6.15198	−0.295645	−0.147823	+	0.989014i	$0.547227\pi$
$434$	0	0
$435$	−1.94247	−0.0931345
$436$	0	0
$437$	−18.7388	−0.896397
$438$	0	0
$439$	38.6667	1.84546	0.922731	−	0.385444i	$-0.125952\pi$
$439$	38.6667	1.84546	0.922731	+	0.385444i	$0.125952\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.8987	−1.65809	−0.829045	−	0.559183i	$-0.811115\pi$
$443$	−34.8987	−1.65809	−0.829045	+	0.559183i	$0.811115\pi$
$444$	0	0
$445$	18.6854	0.885773
$446$	0	0
$447$	2.32800	0.110111
$448$	0	0
$449$	13.1959	0.622752	0.311376	−	0.950287i	$-0.399210\pi$
$449$	13.1959	0.622752	0.311376	+	0.950287i	$0.399210\pi$
$450$	0	0
$451$	2.29721	0.108171
$452$	0	0
$453$	1.44186	0.0677443
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.7706	−1.01839	−0.509193	−	0.860653i	$-0.670056\pi$
$457$	−21.7706	−1.01839	−0.509193	+	0.860653i	$0.670056\pi$
$458$	0	0
$459$	2.73980	0.127883
$460$	0	0
$461$	20.9376	0.975159	0.487579	−	0.873079i	$-0.337880\pi$
$461$	20.9376	0.975159	0.487579	+	0.873079i	$0.337880\pi$
$462$	0	0
$463$	12.4193	0.577175	0.288588	−	0.957453i	$-0.406814\pi$
$463$	12.4193	0.577175	0.288588	+	0.957453i	$0.406814\pi$
$464$	0	0
$465$	1.87419	0.0869136
$466$	0	0
$467$	9.43012	0.436374	0.218187	−	0.975907i	$-0.429986\pi$
$467$	9.43012	0.436374	0.218187	+	0.975907i	$0.429986\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.44512	0.435208
$472$	0	0
$473$	−2.30125	−0.105812
$474$	0	0
$475$	11.8958	0.545817
$476$	0	0
$477$	32.2330	1.47585
$478$	0	0
$479$	0.519314	0.0237281	0.0118640	−	0.999930i	$-0.496223\pi$
$479$	0.519314	0.0237281	0.0118640	+	0.999930i	$0.496223\pi$
$480$	0	0
$481$	6.38563	0.291160
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.04378	−0.274434
$486$	0	0
$487$	−37.5904	−1.70338	−0.851691	−	0.524044i	$-0.824423\pi$
$487$	−37.5904	−1.70338	−0.851691	+	0.524044i	$0.824423\pi$
$488$	0	0
$489$	−2.62908	−0.118891
$490$	0	0
$491$	−27.0798	−1.22210	−0.611048	−	0.791593i	$-0.709252\pi$
$491$	−27.0798	−1.22210	−0.611048	+	0.791593i	$0.709252\pi$
$492$	0	0
$493$	2.50385	0.112768
$494$	0	0
$495$	1.09403	0.0491728
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−28.0722	−1.25668	−0.628342	−	0.777937i	$-0.716266\pi$
$499$	−28.0722	−1.25668	−0.628342	+	0.777937i	$0.716266\pi$
$500$	0	0
$501$	−1.62259	−0.0724920
$502$	0	0
$503$	−33.5454	−1.49572	−0.747858	−	0.663858i	$-0.768918\pi$
$503$	−33.5454	−1.49572	−0.747858	+	0.663858i	$0.768918\pi$
$504$	0	0
$505$	14.1956	0.631697
$506$	0	0
$507$	4.44904	0.197589
$508$	0	0
$509$	11.0980	0.491910	0.245955	−	0.969281i	$-0.420898\pi$
$509$	11.0980	0.491910	0.245955	+	0.969281i	$0.420898\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−14.0116	−0.618629
$514$	0	0
$515$	−23.0761	−1.01686
$516$	0	0
$517$	2.08300	0.0916102
$518$	0	0
$519$	−5.38647	−0.236440
$520$	0	0
$521$	−34.8637	−1.52741	−0.763704	−	0.645567i	$-0.776621\pi$
$521$	−34.8637	−1.52741	−0.763704	+	0.645567i	$0.776621\pi$
$522$	0	0
$523$	23.8737	1.04392	0.521961	−	0.852969i	$-0.325201\pi$
$523$	23.8737	1.04392	0.521961	+	0.852969i	$0.325201\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.41583	−0.105235
$528$	0	0
$529$	−9.57418	−0.416269
$530$	0	0
$531$	5.74665	0.249383
$532$	0	0
$533$	−18.1340	−0.785470
$534$	0	0
$535$	−22.6525	−0.979351
$536$	0	0
$537$	−7.00579	−0.302322
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.4510	0.750276	0.375138	−	0.926969i	$-0.377595\pi$
$541$	17.4510	0.750276	0.375138	+	0.926969i	$0.377595\pi$
$542$	0	0
$543$	−1.90035	−0.0815520
$544$	0	0
$545$	−10.2485	−0.438999
$546$	0	0
$547$	17.0311	0.728198	0.364099	−	0.931360i	$-0.381377\pi$
$547$	17.0311	0.728198	0.364099	+	0.931360i	$0.381377\pi$
$548$	0	0
$549$	−29.6251	−1.26437
$550$	0	0
$551$	−12.8050	−0.545510
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.60289	−0.110487
$556$	0	0
$557$	1.75404	0.0743210	0.0371605	−	0.999309i	$-0.488169\pi$
$557$	1.75404	0.0743210	0.0371605	+	0.999309i	$0.488169\pi$
$558$	0	0
$559$	18.1659	0.768335
$560$	0	0
$561$	0.114387	0.00482941
$562$	0	0
$563$	36.6610	1.54508	0.772539	−	0.634968i	$-0.218987\pi$
$563$	36.6610	1.54508	0.772539	+	0.634968i	$0.218987\pi$
$564$	0	0
$565$	−23.8713	−1.00427
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.7045	1.74835	0.874173	−	0.485615i	$-0.161404\pi$
$569$	41.7045	1.74835	0.874173	+	0.485615i	$0.161404\pi$
$570$	0	0
$571$	−9.07579	−0.379810	−0.189905	−	0.981802i	$-0.560818\pi$
$571$	−9.07579	−0.379810	−0.189905	+	0.981802i	$0.560818\pi$
$572$	0	0
$573$	10.2594	0.428595
$574$	0	0
$575$	−8.52303	−0.355435
$576$	0	0
$577$	45.5424	1.89595	0.947977	−	0.318338i	$-0.103125\pi$
$577$	45.5424	1.89595	0.947977	+	0.318338i	$0.103125\pi$
$578$	0	0
$579$	−7.36711	−0.306167
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.80062	0.115990
$584$	0	0
$585$	−8.63614	−0.357061
$586$	0	0
$587$	−14.6764	−0.605759	−0.302879	−	0.953029i	$-0.597948\pi$
$587$	−14.6764	−0.605759	−0.302879	+	0.953029i	$0.597948\pi$
$588$	0	0
$589$	12.3549	0.509073
$590$	0	0
$591$	−0.184781	−0.00760086
$592$	0	0
$593$	−22.4432	−0.921633	−0.460817	−	0.887495i	$-0.652443\pi$
$593$	−22.4432	−0.921633	−0.460817	+	0.887495i	$0.652443\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.45592	−0.0595868
$598$	0	0
$599$	−19.0834	−0.779728	−0.389864	−	0.920872i	$-0.627478\pi$
$599$	−19.0834	−0.779728	−0.389864	+	0.920872i	$0.627478\pi$
$600$	0	0
$601$	−13.1280	−0.535503	−0.267751	−	0.963488i	$-0.586281\pi$
$601$	−13.1280	−0.535503	−0.267751	+	0.963488i	$0.586281\pi$
$602$	0	0
$603$	34.8702	1.42002
$604$	0	0
$605$	−17.8923	−0.727426
$606$	0	0
$607$	−13.8740	−0.563126	−0.281563	−	0.959543i	$-0.590853\pi$
$607$	−13.8740	−0.563126	−0.281563	+	0.959543i	$0.590853\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.4430	−0.665213
$612$	0	0
$613$	44.1168	1.78186	0.890931	−	0.454138i	$-0.150053\pi$
$613$	44.1168	1.78186	0.890931	+	0.454138i	$0.150053\pi$
$614$	0	0
$615$	7.39172	0.298063
$616$	0	0
$617$	−35.7861	−1.44069	−0.720347	−	0.693614i	$-0.756018\pi$
$617$	−35.7861	−1.44069	−0.720347	+	0.693614i	$0.756018\pi$
$618$	0	0
$619$	−13.6707	−0.549473	−0.274736	−	0.961520i	$-0.588591\pi$
$619$	−13.6707	−0.549473	−0.274736	+	0.961520i	$0.588591\pi$
$620$	0	0
$621$	10.0390	0.402849
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.95901	−0.318361
$626$	0	0
$627$	−0.584987	−0.0233621
$628$	0	0
$629$	3.35512	0.133778
$630$	0	0
$631$	−11.5439	−0.459557	−0.229779	−	0.973243i	$-0.573800\pi$
$631$	−11.5439	−0.459557	−0.229779	+	0.973243i	$0.573800\pi$
$632$	0	0
$633$	8.00586	0.318204
$634$	0	0
$635$	−24.4174	−0.968977
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.59918	−0.0632627
$640$	0	0
$641$	14.1088	0.557263	0.278631	−	0.960398i	$-0.410119\pi$
$641$	14.1088	0.557263	0.278631	+	0.960398i	$0.410119\pi$
$642$	0	0
$643$	31.8841	1.25738	0.628692	−	0.777654i	$-0.283590\pi$
$643$	31.8841	1.25738	0.628692	+	0.777654i	$0.283590\pi$
$644$	0	0
$645$	−7.40472	−0.291560
$646$	0	0
$647$	−2.68038	−0.105377	−0.0526884	−	0.998611i	$-0.516779\pi$
$647$	−2.68038	−0.105377	−0.0526884	+	0.998611i	$0.516779\pi$
$648$	0	0
$649$	0.499307	0.0195995
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.88111	0.151880	0.0759398	−	0.997112i	$-0.475804\pi$
$653$	3.88111	0.151880	0.0759398	+	0.997112i	$0.475804\pi$
$654$	0	0
$655$	15.1051	0.590204
$656$	0	0
$657$	−43.1340	−1.68282
$658$	0	0
$659$	−18.8524	−0.734383	−0.367192	−	0.930145i	$-0.619681\pi$
$659$	−18.8524	−0.734383	−0.367192	+	0.930145i	$0.619681\pi$
$660$	0	0
$661$	−20.4000	−0.793469	−0.396734	−	0.917933i	$-0.629857\pi$
$661$	−20.4000	−0.793469	−0.396734	+	0.917933i	$0.629857\pi$
$662$	0	0
$663$	−0.902958	−0.0350680
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.17442	0.355235
$668$	0	0
$669$	2.05234	0.0793479
$670$	0	0
$671$	−2.57403	−0.0993692
$672$	0	0
$673$	51.4240	1.98225	0.991125	−	0.132931i	$-0.0424389\pi$
$673$	51.4240	1.98225	0.991125	+	0.132931i	$0.0424389\pi$
$674$	0	0
$675$	−6.37297	−0.245296
$676$	0	0
$677$	24.4045	0.937941	0.468971	−	0.883214i	$-0.344625\pi$
$677$	24.4045	0.937941	0.468971	+	0.883214i	$0.344625\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.4442	−0.438542
$682$	0	0
$683$	−23.0029	−0.880182	−0.440091	−	0.897953i	$-0.645054\pi$
$683$	−23.0029	−0.880182	−0.440091	+	0.897953i	$0.645054\pi$
$684$	0	0
$685$	−0.330302	−0.0126202
$686$	0	0
$687$	0.193641	0.00738785
$688$	0	0
$689$	−22.1078	−0.842241
$690$	0	0
$691$	−8.51322	−0.323858	−0.161929	−	0.986802i	$-0.551772\pi$
$691$	−8.51322	−0.323858	−0.161929	+	0.986802i	$0.551772\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.4019	−0.546294
$696$	0	0
$697$	−9.52792	−0.360896
$698$	0	0
$699$	0.747077	0.0282570
$700$	0	0
$701$	−28.3455	−1.07060	−0.535298	−	0.844663i	$-0.679801\pi$
$701$	−28.3455	−1.07060	−0.535298	+	0.844663i	$0.679801\pi$
$702$	0	0
$703$	−17.1585	−0.647145
$704$	0	0
$705$	6.70245	0.252429
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22.2872	0.837012	0.418506	−	0.908214i	$-0.362554\pi$
$709$	22.2872	0.837012	0.418506	+	0.908214i	$0.362554\pi$
$710$	0	0
$711$	23.4926	0.881042
$712$	0	0
$713$	−8.85191	−0.331507
$714$	0	0
$715$	−0.750365	−0.0280621
$716$	0	0
$717$	3.47203	0.129665
$718$	0	0
$719$	−19.1219	−0.713128	−0.356564	−	0.934271i	$-0.616052\pi$
$719$	−19.1219	−0.713128	−0.356564	+	0.934271i	$0.616052\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.19806	−0.230508
$724$	0	0
$725$	−5.82414	−0.216303
$726$	0	0
$727$	−8.01996	−0.297444	−0.148722	−	0.988879i	$-0.547516\pi$
$727$	−8.01996	−0.297444	−0.148722	+	0.988879i	$0.547516\pi$
$728$	0	0
$729$	−15.5940	−0.577555
$730$	0	0
$731$	9.54468	0.353023
$732$	0	0
$733$	36.2759	1.33988	0.669941	−	0.742414i	$-0.266319\pi$
$733$	36.2759	1.33988	0.669941	+	0.742414i	$0.266319\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.02975	0.111602
$738$	0	0
$739$	−36.4999	−1.34267	−0.671336	−	0.741153i	$-0.734279\pi$
$739$	−36.4999	−1.34267	−0.671336	+	0.741153i	$0.734279\pi$
$740$	0	0
$741$	4.61784	0.169640
$742$	0	0
$743$	5.06824	0.185936	0.0929678	−	0.995669i	$-0.470365\pi$
$743$	5.06824	0.185936	0.0929678	+	0.995669i	$0.470365\pi$
$744$	0	0
$745$	−8.02389	−0.293973
$746$	0	0
$747$	−2.85602	−0.104496
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.2136	0.920059	0.460029	−	0.887904i	$-0.347839\pi$
$751$	25.2136	0.920059	0.460029	+	0.887904i	$0.347839\pi$
$752$	0	0
$753$	4.80763	0.175200
$754$	0	0
$755$	−4.96963	−0.180863
$756$	0	0
$757$	−7.28908	−0.264926	−0.132463	−	0.991188i	$-0.542289\pi$
$757$	−7.28908	−0.264926	−0.132463	+	0.991188i	$0.542289\pi$
$758$	0	0
$759$	0.419127	0.0152133
$760$	0	0
$761$	5.31688	0.192737	0.0963683	−	0.995346i	$-0.469277\pi$
$761$	5.31688	0.192737	0.0963683	+	0.995346i	$0.469277\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.53758	−0.164057
$766$	0	0
$767$	−3.94149	−0.142319
$768$	0	0
$769$	−32.0489	−1.15571	−0.577857	−	0.816138i	$-0.696111\pi$
$769$	−32.0489	−1.15571	−0.577857	+	0.816138i	$0.696111\pi$
$770$	0	0
$771$	−6.80849	−0.245202
$772$	0	0
$773$	5.70523	0.205203	0.102601	−	0.994723i	$-0.467283\pi$
$773$	5.70523	0.205203	0.102601	+	0.994723i	$0.467283\pi$
$774$	0	0
$775$	5.61940	0.201855
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48.7269	1.74582
$780$	0	0
$781$	−0.138948	−0.00497193
$782$	0	0
$783$	6.86003	0.245158
$784$	0	0
$785$	−32.5544	−1.16192
$786$	0	0
$787$	39.6366	1.41289	0.706446	−	0.707767i	$-0.250297\pi$
$787$	39.6366	1.41289	0.706446	+	0.707767i	$0.250297\pi$
$788$	0	0
$789$	8.37022	0.297988
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.3191	0.721554
$794$	0	0
$795$	9.01152	0.319606
$796$	0	0
$797$	−30.9698	−1.09701	−0.548503	−	0.836148i	$-0.684802\pi$
$797$	−30.9698	−1.09701	−0.548503	+	0.836148i	$0.684802\pi$
$798$	0	0
$799$	−8.63945	−0.305642
$800$	0	0
$801$	−31.7086	−1.12037
$802$	0	0
$803$	−3.74777	−0.132256
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.27350	−0.150434
$808$	0	0
$809$	−39.9889	−1.40593	−0.702967	−	0.711223i	$-0.748142\pi$
$809$	−39.9889	−1.40593	−0.702967	+	0.711223i	$0.748142\pi$
$810$	0	0
$811$	−8.21437	−0.288446	−0.144223	−	0.989545i	$-0.546068\pi$
$811$	−8.21437	−0.288446	−0.144223	+	0.989545i	$0.546068\pi$
$812$	0	0
$813$	9.26067	0.324786
$814$	0	0
$815$	9.06162	0.317415
$816$	0	0
$817$	−48.8126	−1.70774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−49.3341	−1.72177	−0.860886	−	0.508799i	$-0.830090\pi$
$821$	−49.3341	−1.72177	−0.860886	+	0.508799i	$0.830090\pi$
$822$	0	0
$823$	1.97772	0.0689391	0.0344696	−	0.999406i	$-0.489026\pi$
$823$	1.97772	0.0689391	0.0344696	+	0.999406i	$0.489026\pi$
$824$	0	0
$825$	−0.266072	−0.00926343
$826$	0	0
$827$	−9.58007	−0.333132	−0.166566	−	0.986030i	$-0.553268\pi$
$827$	−9.58007	−0.333132	−0.166566	+	0.986030i	$0.553268\pi$
$828$	0	0
$829$	11.9550	0.415214	0.207607	−	0.978212i	$-0.433432\pi$
$829$	11.9550	0.415214	0.207607	+	0.978212i	$0.433432\pi$
$830$	0	0
$831$	−12.4621	−0.432305
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.59257	0.193539
$836$	0	0
$837$	−6.61889	−0.228782
$838$	0	0
$839$	34.7915	1.20114	0.600569	−	0.799573i	$-0.294941\pi$
$839$	34.7915	1.20114	0.600569	+	0.799573i	$0.294941\pi$
$840$	0	0
$841$	−22.7307	−0.783819
$842$	0	0
$843$	0.0558133	0.00192231
$844$	0	0
$845$	−15.3345	−0.527522
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.36112	0.0467135
$850$	0	0
$851$	12.2936	0.421419
$852$	0	0
$853$	−54.9715	−1.88219	−0.941095	−	0.338143i	$-0.890201\pi$
$853$	−54.9715	−1.88219	−0.941095	+	0.338143i	$0.890201\pi$
$854$	0	0
$855$	23.2057	0.793619
$856$	0	0
$857$	24.5572	0.838858	0.419429	−	0.907788i	$-0.362230\pi$
$857$	24.5572	0.838858	0.419429	+	0.907788i	$0.362230\pi$
$858$	0	0
$859$	33.2337	1.13392	0.566959	−	0.823746i	$-0.308120\pi$
$859$	33.2337	1.13392	0.566959	+	0.823746i	$0.308120\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.37246	0.319042	0.159521	−	0.987195i	$-0.449005\pi$
$863$	9.37246	0.319042	0.159521	+	0.987195i	$0.449005\pi$
$864$	0	0
$865$	18.5655	0.631246
$866$	0	0
$867$	−0.474430	−0.0161125
$868$	0	0
$869$	2.04119	0.0692428
$870$	0	0
$871$	−23.9166	−0.810384
$872$	0	0
$873$	10.2561	0.347117
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0865	1.08348	0.541742	−	0.840545i	$-0.317765\pi$
$877$	32.0865	1.08348	0.541742	+	0.840545i	$0.317765\pi$
$878$	0	0
$879$	8.44670	0.284900
$880$	0	0
$881$	−4.55055	−0.153312	−0.0766559	−	0.997058i	$-0.524424\pi$
$881$	−4.55055	−0.153312	−0.0766559	+	0.997058i	$0.524424\pi$
$882$	0	0
$883$	−20.7818	−0.699362	−0.349681	−	0.936869i	$-0.613710\pi$
$883$	−20.7818	−0.699362	−0.349681	+	0.936869i	$0.613710\pi$
$884$	0	0
$885$	1.60662	0.0540058
$886$	0	0
$887$	−14.9653	−0.502487	−0.251243	−	0.967924i	$-0.580839\pi$
$887$	−14.9653	−0.502487	−0.251243	+	0.967924i	$0.580839\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.69373	−0.0567420
$892$	0	0
$893$	44.1832	1.47853
$894$	0	0
$895$	24.1468	0.807138
$896$	0	0
$897$	−3.30855	−0.110469
$898$	0	0
$899$	−6.04888	−0.201741
$900$	0	0
$901$	−11.6158	−0.386980
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.54993	0.217727
$906$	0	0
$907$	42.0823	1.39732	0.698661	−	0.715453i	$-0.253780\pi$
$907$	42.0823	1.39732	0.698661	+	0.715453i	$0.253780\pi$
$908$	0	0
$909$	−24.0896	−0.799001
$910$	0	0
$911$	4.98375	0.165119	0.0825594	−	0.996586i	$-0.473691\pi$
$911$	4.98375	0.165119	0.0825594	+	0.996586i	$0.473691\pi$
$912$	0	0
$913$	−0.248150	−0.00821257
$914$	0	0
$915$	−8.28242	−0.273808
$916$	0	0
$917$	0	0
$918$	0	0
$919$	18.0195	0.594410	0.297205	−	0.954814i	$-0.403946\pi$
$919$	18.0195	0.594410	0.297205	+	0.954814i	$0.403946\pi$
$920$	0	0
$921$	10.3525	0.341128
$922$	0	0
$923$	1.09684	0.0361029
$924$	0	0
$925$	−7.80426	−0.256603
$926$	0	0
$927$	39.1596	1.28617
$928$	0	0
$929$	31.1983	1.02358	0.511791	−	0.859110i	$-0.328982\pi$
$929$	31.1983	1.02358	0.511791	+	0.859110i	$0.328982\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.81974	−0.0923143
$934$	0	0
$935$	−0.394255	−0.0128935
$936$	0	0
$937$	58.5957	1.91424	0.957119	−	0.289695i	$-0.0935540\pi$
$937$	58.5957	1.91424	0.957119	+	0.289695i	$0.0935540\pi$
$938$	0	0
$939$	6.10326	0.199172
$940$	0	0
$941$	33.2705	1.08459	0.542293	−	0.840189i	$-0.317556\pi$
$941$	33.2705	1.08459	0.542293	+	0.840189i	$0.317556\pi$
$942$	0	0
$943$	−34.9115	−1.13687
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.9688	−1.55878	−0.779388	−	0.626542i	$-0.784470\pi$
$947$	−47.9688	−1.55878	−0.779388	+	0.626542i	$0.784470\pi$
$948$	0	0
$949$	29.5846	0.960356
$950$	0	0
$951$	1.57389	0.0510370
$952$	0	0
$953$	−4.51078	−0.146119	−0.0730593	−	0.997328i	$-0.523276\pi$
$953$	−4.51078	−0.146119	−0.0730593	+	0.997328i	$0.523276\pi$
$954$	0	0
$955$	−35.3611	−1.14426
$956$	0	0
$957$	0.286407	0.00925822
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.1638	−0.811734
$962$	0	0
$963$	38.4406	1.23873
$964$	0	0
$965$	25.3921	0.817402
$966$	0	0
$967$	−12.4644	−0.400830	−0.200415	−	0.979711i	$-0.564229\pi$
$967$	−12.4644	−0.400830	−0.200415	+	0.979711i	$0.564229\pi$
$968$	0	0
$969$	2.42629	0.0779438
$970$	0	0
$971$	−1.20656	−0.0387205	−0.0193602	−	0.999813i	$-0.506163\pi$
$971$	−1.20656	−0.0387205	−0.0193602	+	0.999813i	$0.506163\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.10035	0.0672650
$976$	0	0
$977$	−12.0291	−0.384845	−0.192422	−	0.981312i	$-0.561634\pi$
$977$	−12.0291	−0.384845	−0.192422	+	0.981312i	$0.561634\pi$
$978$	0	0
$979$	−2.75506	−0.0880520
$980$	0	0
$981$	17.3915	0.555268
$982$	0	0
$983$	−8.10551	−0.258526	−0.129263	−	0.991610i	$-0.541261\pi$
$983$	−8.10551	−0.258526	−0.129263	+	0.991610i	$0.541261\pi$
$984$	0	0
$985$	0.636882	0.0202927
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.9729	1.11207
$990$	0	0
$991$	16.8504	0.535272	0.267636	−	0.963520i	$-0.413758\pi$
$991$	16.8504	0.535272	0.267636	+	0.963520i	$0.413758\pi$
$992$	0	0
$993$	−3.53216	−0.112090
$994$	0	0
$995$	5.01810	0.159084
$996$	0	0
$997$	−16.1781	−0.512367	−0.256184	−	0.966628i	$-0.582465\pi$
$997$	−16.1781	−0.512367	−0.256184	+	0.966628i	$0.582465\pi$
$998$	0	0
$999$	9.19235	0.290833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.v.1.4		7
7.2	even	3		952.2.q.e.137.4	✓	14
7.4	even	3		952.2.q.e.681.4	yes	14
7.6	odd	2		6664.2.a.y.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.e.137.4	✓	14	7.2	even	3
952.2.q.e.681.4	yes	14	7.4	even	3
6664.2.a.v.1.4		7	1.1	even	1	trivial
6664.2.a.y.1.4		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q-0.474430 q^{3} +1.63521 q^{5} -2.77492 q^{9} -0.241103 q^{11} +1.90325 q^{13} -0.775795 q^{15} +1.00000 q^{17} -5.11412 q^{19} +3.66413 q^{23} -2.32607 q^{25} +2.73980 q^{27} +2.50385 q^{29} -2.41583 q^{31} +0.114387 q^{33} +3.35512 q^{37} -0.902958 q^{39} -9.52792 q^{41} +9.54468 q^{43} -4.53758 q^{45} -8.63945 q^{47} -0.474430 q^{51} -11.6158 q^{53} -0.394255 q^{55} +2.42629 q^{57} -2.07093 q^{59} +10.6760 q^{61} +3.11222 q^{65} -12.5662 q^{67} -1.73837 q^{69} +0.576299 q^{71} +15.5443 q^{73} +1.10356 q^{75} -8.46606 q^{79} +7.02491 q^{81} +1.02923 q^{83} +1.63521 q^{85} -1.18790 q^{87} +11.4269 q^{89} +1.14614 q^{93} -8.36268 q^{95} -3.69602 q^{97} +0.669041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 2 q^{5} + 4 q^{9} - 12 q^{11} + 2 q^{13} - 6 q^{15} + 7 q^{17} - 3 q^{19} - 18 q^{23} + 15 q^{25} - 18 q^{27} + 5 q^{29} - 10 q^{31} + 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} + 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 2 * q^5 + 4 * q^9 - 12 * q^11 + 2 * q^13 - 6 * q^15 + 7 * q^17 - 3 * q^19 - 18 * q^23 + 15 * q^25 - 18 * q^27 + 5 * q^29 - 10 * q^31 + 21 * q^33 + 11 * q^37 - 12 * q^39 - 15 * q^43 + 13 * q^45 + 15 * q^47 - 3 * q^51 + 8 * q^53 - 9 * q^55 - 3 * q^57 + 9 * q^59 - 4 * q^61 - 38 * q^65 - 30 * q^67 - 28 * q^69 - 27 * q^71 - 5 * q^73 - 2 * q^75 + 3 * q^79 - 17 * q^81 - 12 * q^83 + 2 * q^85 + 19 * q^87 + 32 * q^89 - 9 * q^93 - 42 * q^95 - 39 * q^97 - 20 * q^99

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