LMFDB - Embedded newform 6664.2.a.bc.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 = [12,0,-4,0,8] 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:	$53.2123079070$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 14 x^{10} + 64 x^{9} + 59 x^{8} - 348 x^{7} - 74 x^{6} + 760 x^{5} + 27 x^{4} + \cdots - 14 $ x^12 - 4x^11 - 14x^10 + 64x^9 + 59x^8 - 348x^7 - 74x^6 + 760x^5 + 27x^4 - 616x^3 - 46x^2 + 108*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.01001$ 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)

$f(q)$	$=$	$q-2.01001 q^{3} +1.57380 q^{5} +1.04012 q^{9} -4.02338 q^{11} -5.31968 q^{13} -3.16334 q^{15} +1.00000 q^{17} -2.04717 q^{19} -2.80190 q^{23} -2.52316 q^{25} +3.93936 q^{27} -6.93463 q^{29} -8.06298 q^{31} +8.08701 q^{33} +4.04674 q^{37} +10.6926 q^{39} -4.49040 q^{41} -2.57763 q^{43} +1.63694 q^{45} +1.82856 q^{47} -2.01001 q^{51} +2.98007 q^{53} -6.33198 q^{55} +4.11482 q^{57} +1.54828 q^{59} +11.3423 q^{61} -8.37210 q^{65} -3.28300 q^{67} +5.63183 q^{69} -2.34261 q^{71} +13.4835 q^{73} +5.07157 q^{75} -6.44897 q^{79} -11.0385 q^{81} -8.78103 q^{83} +1.57380 q^{85} +13.9386 q^{87} +17.2060 q^{89} +16.2066 q^{93} -3.22182 q^{95} -0.454764 q^{97} -4.18480 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 8 q^{5} + 8 q^{9} - 4 q^{11} + 12 q^{13} - 8 q^{15} + 12 q^{17} + 8 q^{25} - 16 q^{27} + 8 q^{29} + 4 q^{31} + 16 q^{33} + 12 q^{37} - 16 q^{39} + 24 q^{41} - 4 q^{43} + 28 q^{45} + 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 8 * q^5 + 8 * q^9 - 4 * q^11 + 12 * q^13 - 8 * q^15 + 12 * q^17 + 8 * q^25 - 16 * q^27 + 8 * q^29 + 4 * q^31 + 16 * q^33 + 12 * q^37 - 16 * q^39 + 24 * q^41 - 4 * q^43 + 28 * q^45 + 28 * q^47 - 4 * q^51 - 4 * q^55 + 12 * q^57 + 36 * q^61 + 20 * q^65 + 4 * q^69 + 40 * q^73 + 20 * q^75 - 4 * q^79 - 4 * q^81 + 12 * q^83 + 8 * q^85 + 36 * q^89 - 16 * q^93 - 8 * q^95 + 12 * 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
$3$	−2.01001	−1.16048	−0.580239	−	0.814447i	$-0.697041\pi$
$3$	−2.01001	−1.16048	−0.580239	+	0.814447i	$0.697041\pi$
$4$	0	0
$5$	1.57380	0.703824	0.351912	−	0.936033i	$-0.385532\pi$
$5$	1.57380	0.703824	0.351912	+	0.936033i	$0.385532\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.04012	0.346708
$10$	0	0
$11$	−4.02338	−1.21309	−0.606547	−	0.795048i	$-0.707446\pi$
$11$	−4.02338	−1.21309	−0.606547	+	0.795048i	$0.707446\pi$
$12$	0	0
$13$	−5.31968	−1.47541	−0.737707	−	0.675121i	$-0.764092\pi$
$13$	−5.31968	−1.47541	−0.737707	+	0.675121i	$0.764092\pi$
$14$	0	0
$15$	−3.16334	−0.816771
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.04717	−0.469652	−0.234826	−	0.972037i	$-0.575452\pi$
$19$	−2.04717	−0.469652	−0.234826	+	0.972037i	$0.575452\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.80190	−0.584236	−0.292118	−	0.956382i	$-0.594360\pi$
$23$	−2.80190	−0.584236	−0.292118	+	0.956382i	$0.594360\pi$
$24$	0	0
$25$	−2.52316	−0.504632
$26$	0	0
$27$	3.93936	0.758131
$28$	0	0
$29$	−6.93463	−1.28773	−0.643864	−	0.765140i	$-0.722670\pi$
$29$	−6.93463	−1.28773	−0.643864	+	0.765140i	$0.722670\pi$
$30$	0	0
$31$	−8.06298	−1.44815	−0.724077	−	0.689720i	$-0.757734\pi$
$31$	−8.06298	−1.44815	−0.724077	+	0.689720i	$0.757734\pi$
$32$	0	0
$33$	8.08701	1.40777
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.04674	0.665280	0.332640	−	0.943054i	$-0.392061\pi$
$37$	4.04674	0.665280	0.332640	+	0.943054i	$0.392061\pi$
$38$	0	0
$39$	10.6926	1.71219
$40$	0	0
$41$	−4.49040	−0.701283	−0.350641	−	0.936510i	$-0.614036\pi$
$41$	−4.49040	−0.701283	−0.350641	+	0.936510i	$0.614036\pi$
$42$	0	0
$43$	−2.57763	−0.393084	−0.196542	−	0.980495i	$-0.562971\pi$
$43$	−2.57763	−0.393084	−0.196542	+	0.980495i	$0.562971\pi$
$44$	0	0
$45$	1.63694	0.244021
$46$	0	0
$47$	1.82856	0.266723	0.133361	−	0.991067i	$-0.457423\pi$
$47$	1.82856	0.266723	0.133361	+	0.991067i	$0.457423\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.01001	−0.281457
$52$	0	0
$53$	2.98007	0.409345	0.204672	−	0.978831i	$-0.434387\pi$
$53$	2.98007	0.409345	0.204672	+	0.978831i	$0.434387\pi$
$54$	0	0
$55$	−6.33198	−0.853804
$56$	0	0
$57$	4.11482	0.545021
$58$	0	0
$59$	1.54828	0.201569	0.100784	−	0.994908i	$-0.467865\pi$
$59$	1.54828	0.201569	0.100784	+	0.994908i	$0.467865\pi$
$60$	0	0
$61$	11.3423	1.45224	0.726119	−	0.687569i	$-0.241322\pi$
$61$	11.3423	1.45224	0.726119	+	0.687569i	$0.241322\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.37210	−1.03843
$66$	0	0
$67$	−3.28300	−0.401082	−0.200541	−	0.979685i	$-0.564270\pi$
$67$	−3.28300	−0.401082	−0.200541	+	0.979685i	$0.564270\pi$
$68$	0	0
$69$	5.63183	0.677992
$70$	0	0
$71$	−2.34261	−0.278017	−0.139008	−	0.990291i	$-0.544392\pi$
$71$	−2.34261	−0.278017	−0.139008	+	0.990291i	$0.544392\pi$
$72$	0	0
$73$	13.4835	1.57813	0.789063	−	0.614312i	$-0.210566\pi$
$73$	13.4835	1.57813	0.789063	+	0.614312i	$0.210566\pi$
$74$	0	0
$75$	5.07157	0.585614
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.44897	−0.725566	−0.362783	−	0.931874i	$-0.618173\pi$
$79$	−6.44897	−0.725566	−0.362783	+	0.931874i	$0.618173\pi$
$80$	0	0
$81$	−11.0385	−1.22650
$82$	0	0
$83$	−8.78103	−0.963843	−0.481921	−	0.876214i	$-0.660061\pi$
$83$	−8.78103	−0.963843	−0.481921	+	0.876214i	$0.660061\pi$
$84$	0	0
$85$	1.57380	0.170702
$86$	0	0
$87$	13.9386	1.49438
$88$	0	0
$89$	17.2060	1.82383	0.911915	−	0.410379i	$-0.134604\pi$
$89$	17.2060	1.82383	0.911915	+	0.410379i	$0.134604\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	16.2066	1.68055
$94$	0	0
$95$	−3.22182	−0.330552
$96$	0	0
$97$	−0.454764	−0.0461743	−0.0230871	−	0.999733i	$-0.507350\pi$
$97$	−0.454764	−0.0461743	−0.0230871	+	0.999733i	$0.507350\pi$
$98$	0	0
$99$	−4.18480	−0.420589
$100$	0	0
$101$	14.5959	1.45234	0.726172	−	0.687513i	$-0.241298\pi$
$101$	14.5959	1.45234	0.726172	+	0.687513i	$0.241298\pi$
$102$	0	0
$103$	8.06798	0.794961	0.397481	−	0.917611i	$-0.369885\pi$
$103$	8.06798	0.794961	0.397481	+	0.917611i	$0.369885\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.05747	−0.392250	−0.196125	−	0.980579i	$-0.562836\pi$
$107$	−4.05747	−0.392250	−0.196125	+	0.980579i	$0.562836\pi$
$108$	0	0
$109$	8.30447	0.795424	0.397712	−	0.917510i	$-0.369804\pi$
$109$	8.30447	0.795424	0.397712	+	0.917510i	$0.369804\pi$
$110$	0	0
$111$	−8.13397	−0.772043
$112$	0	0
$113$	13.3953	1.26012	0.630061	−	0.776546i	$-0.283030\pi$
$113$	13.3953	1.26012	0.630061	+	0.776546i	$0.283030\pi$
$114$	0	0
$115$	−4.40962	−0.411199
$116$	0	0
$117$	−5.53312	−0.511537
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.18755	0.471596
$122$	0	0
$123$	9.02573	0.813822
$124$	0	0
$125$	−11.8399	−1.05900
$126$	0	0
$127$	−8.35298	−0.741207	−0.370603	−	0.928791i	$-0.620849\pi$
$127$	−8.35298	−0.741207	−0.370603	+	0.928791i	$0.620849\pi$
$128$	0	0
$129$	5.18104	0.456165
$130$	0	0
$131$	1.36715	0.119448	0.0597241	−	0.998215i	$-0.480978\pi$
$131$	1.36715	0.119448	0.0597241	+	0.998215i	$0.480978\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.19976	0.533590
$136$	0	0
$137$	−4.64179	−0.396575	−0.198287	−	0.980144i	$-0.563538\pi$
$137$	−4.64179	−0.396575	−0.198287	+	0.980144i	$0.563538\pi$
$138$	0	0
$139$	−15.9522	−1.35305	−0.676523	−	0.736421i	$-0.736514\pi$
$139$	−15.9522	−1.35305	−0.676523	+	0.736421i	$0.736514\pi$
$140$	0	0
$141$	−3.67542	−0.309526
$142$	0	0
$143$	21.4031	1.78982
$144$	0	0
$145$	−10.9137	−0.906334
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22.9439	−1.87964	−0.939818	−	0.341677i	$-0.889005\pi$
$149$	−22.9439	−1.87964	−0.939818	+	0.341677i	$0.889005\pi$
$150$	0	0
$151$	14.6306	1.19062	0.595309	−	0.803497i	$-0.297030\pi$
$151$	14.6306	1.19062	0.595309	+	0.803497i	$0.297030\pi$
$152$	0	0
$153$	1.04012	0.0840889
$154$	0	0
$155$	−12.6895	−1.01924
$156$	0	0
$157$	−1.86434	−0.148791	−0.0743954	−	0.997229i	$-0.523703\pi$
$157$	−1.86434	−0.148791	−0.0743954	+	0.997229i	$0.523703\pi$
$158$	0	0
$159$	−5.98996	−0.475035
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.8505	−1.24151	−0.620753	−	0.784006i	$-0.713173\pi$
$163$	−15.8505	−1.24151	−0.620753	+	0.784006i	$0.713173\pi$
$164$	0	0
$165$	12.7273	0.990820
$166$	0	0
$167$	15.4747	1.19746	0.598732	−	0.800949i	$-0.295671\pi$
$167$	15.4747	1.19746	0.598732	+	0.800949i	$0.295671\pi$
$168$	0	0
$169$	15.2990	1.17685
$170$	0	0
$171$	−2.12930	−0.162832
$172$	0	0
$173$	21.8303	1.65973	0.829863	−	0.557967i	$-0.188418\pi$
$173$	21.8303	1.65973	0.829863	+	0.557967i	$0.188418\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.11205	−0.233916
$178$	0	0
$179$	−18.5978	−1.39007	−0.695033	−	0.718978i	$-0.744610\pi$
$179$	−18.5978	−1.39007	−0.695033	+	0.718978i	$0.744610\pi$
$180$	0	0
$181$	−11.9136	−0.885532	−0.442766	−	0.896637i	$-0.646003\pi$
$181$	−11.9136	−0.885532	−0.442766	+	0.896637i	$0.646003\pi$
$182$	0	0
$183$	−22.7982	−1.68529
$184$	0	0
$185$	6.36875	0.468240
$186$	0	0
$187$	−4.02338	−0.294218
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.630146	0.0455958	0.0227979	−	0.999740i	$-0.492743\pi$
$191$	0.630146	0.0455958	0.0227979	+	0.999740i	$0.492743\pi$
$192$	0	0
$193$	24.7508	1.78160	0.890800	−	0.454395i	$-0.150145\pi$
$193$	24.7508	1.78160	0.890800	+	0.454395i	$0.150145\pi$
$194$	0	0
$195$	16.8280	1.20508
$196$	0	0
$197$	−8.05940	−0.574209	−0.287104	−	0.957899i	$-0.592693\pi$
$197$	−8.05940	−0.574209	−0.287104	+	0.957899i	$0.592693\pi$
$198$	0	0
$199$	9.02266	0.639599	0.319800	−	0.947485i	$-0.396384\pi$
$199$	9.02266	0.639599	0.319800	+	0.947485i	$0.396384\pi$
$200$	0	0
$201$	6.59884	0.465446
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.06698	−0.493579
$206$	0	0
$207$	−2.91432	−0.202559
$208$	0	0
$209$	8.23652	0.569732
$210$	0	0
$211$	13.6053	0.936628	0.468314	−	0.883562i	$-0.344862\pi$
$211$	13.6053	0.936628	0.468314	+	0.883562i	$0.344862\pi$
$212$	0	0
$213$	4.70866	0.322632
$214$	0	0
$215$	−4.05666	−0.276662
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−27.1019	−1.83138
$220$	0	0
$221$	−5.31968	−0.357841
$222$	0	0
$223$	14.5986	0.977597	0.488798	−	0.872397i	$-0.337435\pi$
$223$	14.5986	0.977597	0.488798	+	0.872397i	$0.337435\pi$
$224$	0	0
$225$	−2.62440	−0.174960
$226$	0	0
$227$	−25.2062	−1.67299	−0.836497	−	0.547971i	$-0.815400\pi$
$227$	−25.2062	−1.67299	−0.836497	+	0.547971i	$0.815400\pi$
$228$	0	0
$229$	0.861355	0.0569200	0.0284600	−	0.999595i	$-0.490940\pi$
$229$	0.861355	0.0569200	0.0284600	+	0.999595i	$0.490940\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.70884	−0.439511	−0.219755	−	0.975555i	$-0.570526\pi$
$233$	−6.70884	−0.439511	−0.219755	+	0.975555i	$0.570526\pi$
$234$	0	0
$235$	2.87778	0.187726
$236$	0	0
$237$	12.9625	0.842003
$238$	0	0
$239$	−23.0319	−1.48981	−0.744906	−	0.667169i	$-0.767506\pi$
$239$	−23.0319	−1.48981	−0.744906	+	0.667169i	$0.767506\pi$
$240$	0	0
$241$	−13.5437	−0.872425	−0.436212	−	0.899844i	$-0.643680\pi$
$241$	−13.5437	−0.872425	−0.436212	+	0.899844i	$0.643680\pi$
$242$	0	0
$243$	10.3694	0.665196
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.8903	0.692932
$248$	0	0
$249$	17.6499	1.11852
$250$	0	0
$251$	21.7560	1.37322	0.686612	−	0.727024i	$-0.259097\pi$
$251$	21.7560	1.37322	0.686612	+	0.727024i	$0.259097\pi$
$252$	0	0
$253$	11.2731	0.708733
$254$	0	0
$255$	−3.16334	−0.198096
$256$	0	0
$257$	−3.48461	−0.217364	−0.108682	−	0.994077i	$-0.534663\pi$
$257$	−3.48461	−0.217364	−0.108682	+	0.994077i	$0.534663\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.21287	−0.446465
$262$	0	0
$263$	−7.67545	−0.473288	−0.236644	−	0.971596i	$-0.576048\pi$
$263$	−7.67545	−0.473288	−0.236644	+	0.971596i	$0.576048\pi$
$264$	0	0
$265$	4.69003	0.288106
$266$	0	0
$267$	−34.5841	−2.11651
$268$	0	0
$269$	−0.387946	−0.0236535	−0.0118267	−	0.999930i	$-0.503765\pi$
$269$	−0.387946	−0.0236535	−0.0118267	+	0.999930i	$0.503765\pi$
$270$	0	0
$271$	14.3809	0.873580	0.436790	−	0.899563i	$-0.356115\pi$
$271$	14.3809	0.873580	0.436790	+	0.899563i	$0.356115\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.1516	0.612166
$276$	0	0
$277$	−15.4526	−0.928456	−0.464228	−	0.885716i	$-0.653668\pi$
$277$	−15.4526	−0.928456	−0.464228	+	0.885716i	$0.653668\pi$
$278$	0	0
$279$	−8.38648	−0.502086
$280$	0	0
$281$	21.7055	1.29484	0.647422	−	0.762132i	$-0.275847\pi$
$281$	21.7055	1.29484	0.647422	+	0.762132i	$0.275847\pi$
$282$	0	0
$283$	24.7913	1.47369	0.736844	−	0.676063i	$-0.236315\pi$
$283$	24.7913	1.47369	0.736844	+	0.676063i	$0.236315\pi$
$284$	0	0
$285$	6.47589	0.383598
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0.914078	0.0535842
$292$	0	0
$293$	−19.9643	−1.16633	−0.583163	−	0.812355i	$-0.698185\pi$
$293$	−19.9643	−1.16633	−0.583163	+	0.812355i	$0.698185\pi$
$294$	0	0
$295$	2.43668	0.141869
$296$	0	0
$297$	−15.8495	−0.919684
$298$	0	0
$299$	14.9052	0.861990
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−29.3378	−1.68541
$304$	0	0
$305$	17.8506	1.02212
$306$	0	0
$307$	−11.3409	−0.647262	−0.323631	−	0.946183i	$-0.604904\pi$
$307$	−11.3409	−0.647262	−0.323631	+	0.946183i	$0.604904\pi$
$308$	0	0
$309$	−16.2167	−0.922535
$310$	0	0
$311$	12.3488	0.700235	0.350117	−	0.936706i	$-0.386142\pi$
$311$	12.3488	0.700235	0.350117	+	0.936706i	$0.386142\pi$
$312$	0	0
$313$	−3.88381	−0.219526	−0.109763	−	0.993958i	$-0.535009\pi$
$313$	−3.88381	−0.219526	−0.109763	+	0.993958i	$0.535009\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.23443	0.237829	0.118915	−	0.992904i	$-0.462059\pi$
$317$	4.23443	0.237829	0.118915	+	0.992904i	$0.462059\pi$
$318$	0	0
$319$	27.9006	1.56214
$320$	0	0
$321$	8.15553	0.455198
$322$	0	0
$323$	−2.04717	−0.113907
$324$	0	0
$325$	13.4224	0.744542
$326$	0	0
$327$	−16.6920	−0.923071
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.8740	0.762583	0.381291	−	0.924455i	$-0.375479\pi$
$331$	13.8740	0.762583	0.381291	+	0.924455i	$0.375479\pi$
$332$	0	0
$333$	4.20911	0.230658
$334$	0	0
$335$	−5.16677	−0.282291
$336$	0	0
$337$	−1.45035	−0.0790054	−0.0395027	−	0.999219i	$-0.512577\pi$
$337$	−1.45035	−0.0790054	−0.0395027	+	0.999219i	$0.512577\pi$
$338$	0	0
$339$	−26.9246	−1.46234
$340$	0	0
$341$	32.4404	1.75675
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.86335	0.477187
$346$	0	0
$347$	17.3969	0.933912	0.466956	−	0.884280i	$-0.345351\pi$
$347$	17.3969	0.933912	0.466956	+	0.884280i	$0.345351\pi$
$348$	0	0
$349$	15.7602	0.843625	0.421812	−	0.906683i	$-0.361394\pi$
$349$	15.7602	0.843625	0.421812	+	0.906683i	$0.361394\pi$
$350$	0	0
$351$	−20.9562	−1.11856
$352$	0	0
$353$	−3.18890	−0.169728	−0.0848640	−	0.996393i	$-0.527046\pi$
$353$	−3.18890	−0.169728	−0.0848640	+	0.996393i	$0.527046\pi$
$354$	0	0
$355$	−3.68680	−0.195675
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.5252	1.18884	0.594418	−	0.804156i	$-0.297382\pi$
$359$	22.5252	1.18884	0.594418	+	0.804156i	$0.297382\pi$
$360$	0	0
$361$	−14.8091	−0.779427
$362$	0	0
$363$	−10.4270	−0.547276
$364$	0	0
$365$	21.2203	1.11072
$366$	0	0
$367$	−24.5361	−1.28078	−0.640388	−	0.768052i	$-0.721226\pi$
$367$	−24.5361	−1.28078	−0.640388	+	0.768052i	$0.721226\pi$
$368$	0	0
$369$	−4.67057	−0.243140
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.72053	−0.244420	−0.122210	−	0.992504i	$-0.538998\pi$
$373$	−4.72053	−0.244420	−0.122210	+	0.992504i	$0.538998\pi$
$374$	0	0
$375$	23.7983	1.22894
$376$	0	0
$377$	36.8900	1.89993
$378$	0	0
$379$	−26.7909	−1.37616	−0.688078	−	0.725637i	$-0.741545\pi$
$379$	−26.7909	−1.37616	−0.688078	+	0.725637i	$0.741545\pi$
$380$	0	0
$381$	16.7895	0.860154
$382$	0	0
$383$	−9.99284	−0.510610	−0.255305	−	0.966861i	$-0.582176\pi$
$383$	−9.99284	−0.510610	−0.255305	+	0.966861i	$0.582176\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.68105	−0.136285
$388$	0	0
$389$	−12.4604	−0.631769	−0.315885	−	0.948798i	$-0.602301\pi$
$389$	−12.4604	−0.631769	−0.315885	+	0.948798i	$0.602301\pi$
$390$	0	0
$391$	−2.80190	−0.141698
$392$	0	0
$393$	−2.74797	−0.138617
$394$	0	0
$395$	−10.1494	−0.510671
$396$	0	0
$397$	8.32627	0.417883	0.208942	−	0.977928i	$-0.432998\pi$
$397$	8.32627	0.417883	0.208942	+	0.977928i	$0.432998\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.46223	0.172895	0.0864476	−	0.996256i	$-0.472448\pi$
$401$	3.46223	0.172895	0.0864476	+	0.996256i	$0.472448\pi$
$402$	0	0
$403$	42.8925	2.13663
$404$	0	0
$405$	−17.3724	−0.863241
$406$	0	0
$407$	−16.2816	−0.807047
$408$	0	0
$409$	22.8498	1.12985	0.564925	−	0.825142i	$-0.308905\pi$
$409$	22.8498	1.12985	0.564925	+	0.825142i	$0.308905\pi$
$410$	0	0
$411$	9.33002	0.460216
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.8196	−0.678375
$416$	0	0
$417$	32.0640	1.57018
$418$	0	0
$419$	−25.2966	−1.23582	−0.617911	−	0.786248i	$-0.712021\pi$
$419$	−25.2966	−1.23582	−0.617911	+	0.786248i	$0.712021\pi$
$420$	0	0
$421$	−22.9085	−1.11649	−0.558246	−	0.829676i	$-0.688525\pi$
$421$	−22.9085	−1.11649	−0.558246	+	0.829676i	$0.688525\pi$
$422$	0	0
$423$	1.90193	0.0924749
$424$	0	0
$425$	−2.52316	−0.122391
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−43.0203	−2.07704
$430$	0	0
$431$	5.95270	0.286731	0.143366	−	0.989670i	$-0.454207\pi$
$431$	5.95270	0.286731	0.143366	+	0.989670i	$0.454207\pi$
$432$	0	0
$433$	23.4270	1.12583	0.562916	−	0.826514i	$-0.309680\pi$
$433$	23.4270	1.12583	0.562916	+	0.826514i	$0.309680\pi$
$434$	0	0
$435$	21.9366	1.05178
$436$	0	0
$437$	5.73595	0.274388
$438$	0	0
$439$	−29.7388	−1.41936	−0.709678	−	0.704526i	$-0.751160\pi$
$439$	−29.7388	−1.41936	−0.709678	+	0.704526i	$0.751160\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.1002	1.38259	0.691296	−	0.722571i	$-0.257040\pi$
$443$	29.1002	1.38259	0.691296	+	0.722571i	$0.257040\pi$
$444$	0	0
$445$	27.0787	1.28365
$446$	0	0
$447$	46.1173	2.18127
$448$	0	0
$449$	28.1998	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$449$	28.1998	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$450$	0	0
$451$	18.0666	0.850721
$452$	0	0
$453$	−29.4075	−1.38168
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.6513	−1.34025	−0.670127	−	0.742247i	$-0.733760\pi$
$457$	−28.6513	−1.34025	−0.670127	+	0.742247i	$0.733760\pi$
$458$	0	0
$459$	3.93936	0.183874
$460$	0	0
$461$	8.61694	0.401331	0.200665	−	0.979660i	$-0.435690\pi$
$461$	8.61694	0.401331	0.200665	+	0.979660i	$0.435690\pi$
$462$	0	0
$463$	5.46431	0.253948	0.126974	−	0.991906i	$-0.459473\pi$
$463$	5.46431	0.253948	0.126974	+	0.991906i	$0.459473\pi$
$464$	0	0
$465$	25.5059	1.18281
$466$	0	0
$467$	6.65358	0.307891	0.153946	−	0.988079i	$-0.450802\pi$
$467$	6.65358	0.307891	0.153946	+	0.988079i	$0.450802\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.74734	0.172668
$472$	0	0
$473$	10.3708	0.476848
$474$	0	0
$475$	5.16533	0.237002
$476$	0	0
$477$	3.09964	0.141923
$478$	0	0
$479$	20.2838	0.926791	0.463396	−	0.886152i	$-0.346631\pi$
$479$	20.2838	0.926791	0.463396	+	0.886152i	$0.346631\pi$
$480$	0	0
$481$	−21.5274	−0.981564
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.715706	−0.0324985
$486$	0	0
$487$	35.9829	1.63054	0.815270	−	0.579081i	$-0.196589\pi$
$487$	35.9829	1.63054	0.815270	+	0.579081i	$0.196589\pi$
$488$	0	0
$489$	31.8596	1.44074
$490$	0	0
$491$	33.6659	1.51932	0.759660	−	0.650321i	$-0.225365\pi$
$491$	33.6659	1.51932	0.759660	+	0.650321i	$0.225365\pi$
$492$	0	0
$493$	−6.93463	−0.312320
$494$	0	0
$495$	−6.58603	−0.296020
$496$	0	0
$497$	0	0
$498$	0	0
$499$	37.4777	1.67773	0.838865	−	0.544339i	$-0.183220\pi$
$499$	37.4777	1.67773	0.838865	+	0.544339i	$0.183220\pi$
$500$	0	0
$501$	−31.1041	−1.38963
$502$	0	0
$503$	11.0864	0.494316	0.247158	−	0.968975i	$-0.420503\pi$
$503$	11.0864	0.494316	0.247158	+	0.968975i	$0.420503\pi$
$504$	0	0
$505$	22.9709	1.02219
$506$	0	0
$507$	−30.7511	−1.36571
$508$	0	0
$509$	31.3059	1.38761	0.693806	−	0.720162i	$-0.255933\pi$
$509$	31.3059	1.38761	0.693806	+	0.720162i	$0.255933\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.06453	−0.356058
$514$	0	0
$515$	12.6974	0.559512
$516$	0	0
$517$	−7.35699	−0.323560
$518$	0	0
$519$	−43.8790	−1.92608
$520$	0	0
$521$	10.6272	0.465586	0.232793	−	0.972526i	$-0.425214\pi$
$521$	10.6272	0.465586	0.232793	+	0.972526i	$0.425214\pi$
$522$	0	0
$523$	22.8111	0.997461	0.498731	−	0.866757i	$-0.333800\pi$
$523$	22.8111	0.997461	0.498731	+	0.866757i	$0.333800\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.06298	−0.351229
$528$	0	0
$529$	−15.1494	−0.658669
$530$	0	0
$531$	1.61040	0.0698854
$532$	0	0
$533$	23.8875	1.03468
$534$	0	0
$535$	−6.38563	−0.276075
$536$	0	0
$537$	37.3817	1.61314
$538$	0	0
$539$	0	0
$540$	0	0
$541$	36.8312	1.58350	0.791749	−	0.610846i	$-0.209171\pi$
$541$	36.8312	1.58350	0.791749	+	0.610846i	$0.209171\pi$
$542$	0	0
$543$	23.9464	1.02764
$544$	0	0
$545$	13.0696	0.559838
$546$	0	0
$547$	21.7985	0.932039	0.466019	−	0.884775i	$-0.345688\pi$
$547$	21.7985	0.932039	0.466019	+	0.884775i	$0.345688\pi$
$548$	0	0
$549$	11.7974	0.503502
$550$	0	0
$551$	14.1963	0.604785
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.8012	−0.543382
$556$	0	0
$557$	−7.46644	−0.316363	−0.158182	−	0.987410i	$-0.550563\pi$
$557$	−7.46644	−0.316363	−0.158182	+	0.987410i	$0.550563\pi$
$558$	0	0
$559$	13.7121	0.579962
$560$	0	0
$561$	8.08701	0.341434
$562$	0	0
$563$	−0.0308908	−0.00130189	−0.000650946	−	1.00000i	$-0.500207\pi$
$563$	−0.0308908	−0.00130189	−0.000650946		1.00000i	$0.500207\pi$
$564$	0	0
$565$	21.0815	0.886904
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.45536	0.0610121	0.0305060	−	0.999535i	$-0.490288\pi$
$569$	1.45536	0.0610121	0.0305060	+	0.999535i	$0.490288\pi$
$570$	0	0
$571$	20.1584	0.843602	0.421801	−	0.906688i	$-0.361398\pi$
$571$	20.1584	0.843602	0.421801	+	0.906688i	$0.361398\pi$
$572$	0	0
$573$	−1.26660	−0.0529128
$574$	0	0
$575$	7.06964	0.294824
$576$	0	0
$577$	23.4654	0.976876	0.488438	−	0.872599i	$-0.337567\pi$
$577$	23.4654	0.976876	0.488438	+	0.872599i	$0.337567\pi$
$578$	0	0
$579$	−49.7492	−2.06751
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.9900	−0.496573
$584$	0	0
$585$	−8.70801	−0.360032
$586$	0	0
$587$	−18.8264	−0.777050	−0.388525	−	0.921438i	$-0.627015\pi$
$587$	−18.8264	−0.777050	−0.388525	+	0.921438i	$0.627015\pi$
$588$	0	0
$589$	16.5063	0.680128
$590$	0	0
$591$	16.1994	0.666356
$592$	0	0
$593$	30.5147	1.25309	0.626544	−	0.779386i	$-0.284469\pi$
$593$	30.5147	1.25309	0.626544	+	0.779386i	$0.284469\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18.1356	−0.742240
$598$	0	0
$599$	−22.2047	−0.907260	−0.453630	−	0.891190i	$-0.649871\pi$
$599$	−22.2047	−0.907260	−0.453630	+	0.891190i	$0.649871\pi$
$600$	0	0
$601$	−5.59711	−0.228311	−0.114155	−	0.993463i	$-0.536416\pi$
$601$	−5.59711	−0.228311	−0.114155	+	0.993463i	$0.536416\pi$
$602$	0	0
$603$	−3.41472	−0.139058
$604$	0	0
$605$	8.16416	0.331920
$606$	0	0
$607$	25.1268	1.01986	0.509932	−	0.860215i	$-0.329670\pi$
$607$	25.1268	1.01986	0.509932	+	0.860215i	$0.329670\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.72736	−0.393527
$612$	0	0
$613$	−26.5310	−1.07158	−0.535788	−	0.844352i	$-0.679986\pi$
$613$	−26.5310	−1.07158	−0.535788	+	0.844352i	$0.679986\pi$
$614$	0	0
$615$	14.2047	0.572787
$616$	0	0
$617$	1.40092	0.0563988	0.0281994	−	0.999602i	$-0.491023\pi$
$617$	1.40092	0.0563988	0.0281994	+	0.999602i	$0.491023\pi$
$618$	0	0
$619$	−15.7956	−0.634880	−0.317440	−	0.948278i	$-0.602823\pi$
$619$	−15.7956	−0.634880	−0.317440	+	0.948278i	$0.602823\pi$
$620$	0	0
$621$	−11.0377	−0.442927
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.01784	−0.240714
$626$	0	0
$627$	−16.5555	−0.661161
$628$	0	0
$629$	4.04674	0.161354
$630$	0	0
$631$	29.5962	1.17820	0.589102	−	0.808058i	$-0.299481\pi$
$631$	29.5962	1.17820	0.589102	+	0.808058i	$0.299481\pi$
$632$	0	0
$633$	−27.3468	−1.08694
$634$	0	0
$635$	−13.1459	−0.521679
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.43660	−0.0963906
$640$	0	0
$641$	28.1644	1.11243	0.556214	−	0.831039i	$-0.312254\pi$
$641$	28.1644	1.11243	0.556214	+	0.831039i	$0.312254\pi$
$642$	0	0
$643$	−18.8869	−0.744825	−0.372412	−	0.928067i	$-0.621469\pi$
$643$	−18.8869	−0.744825	−0.372412	+	0.928067i	$0.621469\pi$
$644$	0	0
$645$	8.15391	0.321060
$646$	0	0
$647$	−28.9887	−1.13966	−0.569832	−	0.821761i	$-0.692992\pi$
$647$	−28.9887	−1.13966	−0.569832	+	0.821761i	$0.692992\pi$
$648$	0	0
$649$	−6.22931	−0.244522
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.16970	0.280572	0.140286	−	0.990111i	$-0.455198\pi$
$653$	7.16970	0.280572	0.140286	+	0.990111i	$0.455198\pi$
$654$	0	0
$655$	2.15161	0.0840705
$656$	0	0
$657$	14.0245	0.547148
$658$	0	0
$659$	3.12117	0.121583	0.0607917	−	0.998150i	$-0.480637\pi$
$659$	3.12117	0.121583	0.0607917	+	0.998150i	$0.480637\pi$
$660$	0	0
$661$	−26.1370	−1.01661	−0.508306	−	0.861176i	$-0.669728\pi$
$661$	−26.1370	−1.01661	−0.508306	+	0.861176i	$0.669728\pi$
$662$	0	0
$663$	10.6926	0.415266
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.4301	0.752337
$668$	0	0
$669$	−29.3433	−1.13448
$670$	0	0
$671$	−45.6345	−1.76170
$672$	0	0
$673$	−43.8277	−1.68943	−0.844716	−	0.535214i	$-0.820231\pi$
$673$	−43.8277	−1.68943	−0.844716	+	0.535214i	$0.820231\pi$
$674$	0	0
$675$	−9.93965	−0.382577
$676$	0	0
$677$	−51.4149	−1.97603	−0.988017	−	0.154344i	$-0.950674\pi$
$677$	−51.4149	−1.97603	−0.988017	+	0.154344i	$0.950674\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	50.6646	1.94147
$682$	0	0
$683$	30.4069	1.16349	0.581743	−	0.813373i	$-0.302371\pi$
$683$	30.4069	1.16349	0.581743	+	0.813373i	$0.302371\pi$
$684$	0	0
$685$	−7.30524	−0.279119
$686$	0	0
$687$	−1.73133	−0.0660543
$688$	0	0
$689$	−15.8530	−0.603953
$690$	0	0
$691$	−11.7824	−0.448224	−0.224112	−	0.974563i	$-0.571948\pi$
$691$	−11.7824	−0.448224	−0.224112	+	0.974563i	$0.571948\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.1055	−0.952306
$696$	0	0
$697$	−4.49040	−0.170086
$698$	0	0
$699$	13.4848	0.510042
$700$	0	0
$701$	8.76988	0.331234	0.165617	−	0.986190i	$-0.447038\pi$
$701$	8.76988	0.331234	0.165617	+	0.986190i	$0.447038\pi$
$702$	0	0
$703$	−8.28435	−0.312450
$704$	0	0
$705$	−5.78436	−0.217852
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.8056	−0.405812	−0.202906	−	0.979198i	$-0.565039\pi$
$709$	−10.8056	−0.405812	−0.202906	+	0.979198i	$0.565039\pi$
$710$	0	0
$711$	−6.70772	−0.251559
$712$	0	0
$713$	22.5916	0.846063
$714$	0	0
$715$	33.6841	1.25971
$716$	0	0
$717$	46.2943	1.72889
$718$	0	0
$719$	24.6662	0.919894	0.459947	−	0.887946i	$-0.347868\pi$
$719$	24.6662	0.919894	0.459947	+	0.887946i	$0.347868\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	27.2229	1.01243
$724$	0	0
$725$	17.4972	0.649830
$726$	0	0
$727$	−45.8425	−1.70020	−0.850102	−	0.526618i	$-0.823460\pi$
$727$	−45.8425	−1.70020	−0.850102	+	0.526618i	$0.823460\pi$
$728$	0	0
$729$	12.2730	0.454557
$730$	0	0
$731$	−2.57763	−0.0953369
$732$	0	0
$733$	4.65522	0.171944	0.0859722	−	0.996298i	$-0.472600\pi$
$733$	4.65522	0.171944	0.0859722	+	0.996298i	$0.472600\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.2087	0.486550
$738$	0	0
$739$	−30.4547	−1.12030	−0.560148	−	0.828393i	$-0.689256\pi$
$739$	−30.4547	−1.12030	−0.560148	+	0.828393i	$0.689256\pi$
$740$	0	0
$741$	−21.8895	−0.804131
$742$	0	0
$743$	23.3330	0.856003	0.428002	−	0.903778i	$-0.359218\pi$
$743$	23.3330	0.856003	0.428002	+	0.903778i	$0.359218\pi$
$744$	0	0
$745$	−36.1090	−1.32293
$746$	0	0
$747$	−9.13335	−0.334172
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15.4483	0.563716	0.281858	−	0.959456i	$-0.409049\pi$
$751$	15.4483	0.563716	0.281858	+	0.959456i	$0.409049\pi$
$752$	0	0
$753$	−43.7296	−1.59360
$754$	0	0
$755$	23.0255	0.837985
$756$	0	0
$757$	−31.5123	−1.14534	−0.572668	−	0.819788i	$-0.694091\pi$
$757$	−31.5123	−1.14534	−0.572668	+	0.819788i	$0.694091\pi$
$758$	0	0
$759$	−22.6590	−0.822468
$760$	0	0
$761$	−14.3047	−0.518547	−0.259273	−	0.965804i	$-0.583483\pi$
$761$	−14.3047	−0.518547	−0.259273	+	0.965804i	$0.583483\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.63694	0.0591838
$766$	0	0
$767$	−8.23635	−0.297397
$768$	0	0
$769$	26.0463	0.939253	0.469626	−	0.882865i	$-0.344389\pi$
$769$	26.0463	0.939253	0.469626	+	0.882865i	$0.344389\pi$
$770$	0	0
$771$	7.00409	0.252246
$772$	0	0
$773$	12.4029	0.446102	0.223051	−	0.974807i	$-0.428398\pi$
$773$	12.4029	0.446102	0.223051	+	0.974807i	$0.428398\pi$
$774$	0	0
$775$	20.3442	0.730785
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.19259	0.329359
$780$	0	0
$781$	9.42521	0.337261
$782$	0	0
$783$	−27.3180	−0.976267
$784$	0	0
$785$	−2.93410	−0.104722
$786$	0	0
$787$	−2.58090	−0.0919991	−0.0459996	−	0.998941i	$-0.514647\pi$
$787$	−2.58090	−0.0919991	−0.0459996	+	0.998941i	$0.514647\pi$
$788$	0	0
$789$	15.4277	0.549240
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−60.3377	−2.14265
$794$	0	0
$795$	−9.42699	−0.334341
$796$	0	0
$797$	20.6778	0.732446	0.366223	−	0.930527i	$-0.380651\pi$
$797$	20.6778	0.732446	0.366223	+	0.930527i	$0.380651\pi$
$798$	0	0
$799$	1.82856	0.0646898
$800$	0	0
$801$	17.8963	0.632336
$802$	0	0
$803$	−54.2493	−1.91441
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.779773	0.0274493
$808$	0	0
$809$	26.1163	0.918199	0.459099	−	0.888385i	$-0.348172\pi$
$809$	26.1163	0.918199	0.459099	+	0.888385i	$0.348172\pi$
$810$	0	0
$811$	−25.7557	−0.904405	−0.452203	−	0.891915i	$-0.649362\pi$
$811$	−25.7557	−0.904405	−0.452203	+	0.891915i	$0.649362\pi$
$812$	0	0
$813$	−28.9058	−1.01377
$814$	0	0
$815$	−24.9455	−0.873801
$816$	0	0
$817$	5.27683	0.184613
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.9583	−1.01065	−0.505325	−	0.862929i	$-0.668628\pi$
$821$	−28.9583	−1.01065	−0.505325	+	0.862929i	$0.668628\pi$
$822$	0	0
$823$	14.6301	0.509974	0.254987	−	0.966945i	$-0.417929\pi$
$823$	14.6301	0.509974	0.254987	+	0.966945i	$0.417929\pi$
$824$	0	0
$825$	−20.4048	−0.710405
$826$	0	0
$827$	−27.8310	−0.967779	−0.483890	−	0.875129i	$-0.660776\pi$
$827$	−27.8310	−0.967779	−0.483890	+	0.875129i	$0.660776\pi$
$828$	0	0
$829$	−29.1377	−1.01199	−0.505997	−	0.862535i	$-0.668875\pi$
$829$	−29.1377	−1.01199	−0.505997	+	0.862535i	$0.668875\pi$
$830$	0	0
$831$	31.0598	1.07745
$832$	0	0
$833$	0	0
$834$	0	0
$835$	24.3540	0.842804
$836$	0	0
$837$	−31.7630	−1.09789
$838$	0	0
$839$	10.4065	0.359271	0.179635	−	0.983733i	$-0.442508\pi$
$839$	10.4065	0.359271	0.179635	+	0.983733i	$0.442508\pi$
$840$	0	0
$841$	19.0891	0.658245
$842$	0	0
$843$	−43.6282	−1.50264
$844$	0	0
$845$	24.0776	0.828293
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−49.8306	−1.71018
$850$	0	0
$851$	−11.3386	−0.388680
$852$	0	0
$853$	8.18851	0.280369	0.140185	−	0.990125i	$-0.455230\pi$
$853$	8.18851	0.280369	0.140185	+	0.990125i	$0.455230\pi$
$854$	0	0
$855$	−3.35109	−0.114605
$856$	0	0
$857$	−31.4898	−1.07567	−0.537835	−	0.843050i	$-0.680758\pi$
$857$	−31.4898	−1.07567	−0.537835	+	0.843050i	$0.680758\pi$
$858$	0	0
$859$	26.8260	0.915291	0.457645	−	0.889135i	$-0.348693\pi$
$859$	26.8260	0.915291	0.457645	+	0.889135i	$0.348693\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−53.4600	−1.81980	−0.909900	−	0.414827i	$-0.863842\pi$
$863$	−53.4600	−1.81980	−0.909900	+	0.414827i	$0.863842\pi$
$864$	0	0
$865$	34.3565	1.16815
$866$	0	0
$867$	−2.01001	−0.0682634
$868$	0	0
$869$	25.9466	0.880180
$870$	0	0
$871$	17.4645	0.591762
$872$	0	0
$873$	−0.473010	−0.0160090
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.8119	0.601464	0.300732	−	0.953709i	$-0.402769\pi$
$877$	17.8119	0.601464	0.300732	+	0.953709i	$0.402769\pi$
$878$	0	0
$879$	40.1284	1.35350
$880$	0	0
$881$	4.98077	0.167806	0.0839032	−	0.996474i	$-0.473261\pi$
$881$	4.98077	0.167806	0.0839032	+	0.996474i	$0.473261\pi$
$882$	0	0
$883$	−28.6744	−0.964971	−0.482486	−	0.875904i	$-0.660266\pi$
$883$	−28.6744	−0.964971	−0.482486	+	0.875904i	$0.660266\pi$
$884$	0	0
$885$	−4.89773	−0.164636
$886$	0	0
$887$	43.3500	1.45555	0.727776	−	0.685815i	$-0.240554\pi$
$887$	43.3500	1.45555	0.727776	+	0.685815i	$0.240554\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	44.4121	1.48786
$892$	0	0
$893$	−3.74337	−0.125267
$894$	0	0
$895$	−29.2692	−0.978361
$896$	0	0
$897$	−29.9595	−1.00032
$898$	0	0
$899$	55.9138	1.86483
$900$	0	0
$901$	2.98007	0.0992806
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.7496	−0.623259
$906$	0	0
$907$	−42.7746	−1.42031	−0.710153	−	0.704047i	$-0.751374\pi$
$907$	−42.7746	−1.42031	−0.710153	+	0.704047i	$0.751374\pi$
$908$	0	0
$909$	15.1815	0.503538
$910$	0	0
$911$	1.37603	0.0455898	0.0227949	−	0.999740i	$-0.492744\pi$
$911$	1.37603	0.0455898	0.0227949	+	0.999740i	$0.492744\pi$
$912$	0	0
$913$	35.3294	1.16923
$914$	0	0
$915$	−35.8797	−1.18615
$916$	0	0
$917$	0	0
$918$	0	0
$919$	53.5456	1.76631	0.883154	−	0.469083i	$-0.155416\pi$
$919$	53.5456	1.76631	0.883154	+	0.469083i	$0.155416\pi$
$920$	0	0
$921$	22.7954	0.751133
$922$	0	0
$923$	12.4620	0.410190
$924$	0	0
$925$	−10.2106	−0.335722
$926$	0	0
$927$	8.39169	0.275619
$928$	0	0
$929$	−6.52983	−0.214237	−0.107118	−	0.994246i	$-0.534162\pi$
$929$	−6.52983	−0.214237	−0.107118	+	0.994246i	$0.534162\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.8211	−0.812607
$934$	0	0
$935$	−6.33198	−0.207078
$936$	0	0
$937$	−7.37739	−0.241009	−0.120504	−	0.992713i	$-0.538451\pi$
$937$	−7.37739	−0.241009	−0.120504	+	0.992713i	$0.538451\pi$
$938$	0	0
$939$	7.80647	0.254755
$940$	0	0
$941$	−44.7809	−1.45982	−0.729908	−	0.683546i	$-0.760437\pi$
$941$	−44.7809	−1.45982	−0.729908	+	0.683546i	$0.760437\pi$
$942$	0	0
$943$	12.5816	0.409714
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.8661	0.580570	0.290285	−	0.956940i	$-0.406250\pi$
$947$	17.8661	0.580570	0.290285	+	0.956940i	$0.406250\pi$
$948$	0	0
$949$	−71.7280	−2.32839
$950$	0	0
$951$	−8.51122	−0.275995
$952$	0	0
$953$	20.7625	0.672562	0.336281	−	0.941762i	$-0.390831\pi$
$953$	20.7625	0.672562	0.336281	+	0.941762i	$0.390831\pi$
$954$	0	0
$955$	0.991722	0.0320914
$956$	0	0
$957$	−56.0804	−1.81282
$958$	0	0
$959$	0	0
$960$	0	0
$961$	34.0116	1.09715
$962$	0	0
$963$	−4.22027	−0.135996
$964$	0	0
$965$	38.9527	1.25393
$966$	0	0
$967$	−27.6601	−0.889488	−0.444744	−	0.895658i	$-0.646705\pi$
$967$	−27.6601	−0.889488	−0.444744	+	0.895658i	$0.646705\pi$
$968$	0	0
$969$	4.11482	0.132187
$970$	0	0
$971$	−14.3533	−0.460619	−0.230310	−	0.973117i	$-0.573974\pi$
$971$	−14.3533	−0.460619	−0.230310	+	0.973117i	$0.573974\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−26.9791	−0.864024
$976$	0	0
$977$	−55.5023	−1.77567	−0.887837	−	0.460157i	$-0.847793\pi$
$977$	−55.5023	−1.77567	−0.887837	+	0.460157i	$0.847793\pi$
$978$	0	0
$979$	−69.2261	−2.21248
$980$	0	0
$981$	8.63767	0.275779
$982$	0	0
$983$	−6.63867	−0.211741	−0.105870	−	0.994380i	$-0.533763\pi$
$983$	−6.63867	−0.211741	−0.105870	+	0.994380i	$0.533763\pi$
$984$	0	0
$985$	−12.6839	−0.404141
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.22224	0.229654
$990$	0	0
$991$	32.4827	1.03185	0.515924	−	0.856634i	$-0.327449\pi$
$991$	32.4827	1.03185	0.515924	+	0.856634i	$0.327449\pi$
$992$	0	0
$993$	−27.8868	−0.884960
$994$	0	0
$995$	14.1998	0.450165
$996$	0	0
$997$	47.8690	1.51603	0.758014	−	0.652239i	$-0.226170\pi$
$997$	47.8690	1.51603	0.758014	+	0.652239i	$0.226170\pi$
$998$	0	0
$999$	15.9416	0.504370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.bc.1.4	✓	12
7.6	odd	2		6664.2.a.bd.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6664.2.a.bc.1.4	✓	12	1.1	even	1	trivial
6664.2.a.bd.1.9	yes	12	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)

\(f(q)\)	\(=\)	\(q-2.01001 q^{3} +1.57380 q^{5} +1.04012 q^{9} -4.02338 q^{11} -5.31968 q^{13} -3.16334 q^{15} +1.00000 q^{17} -2.04717 q^{19} -2.80190 q^{23} -2.52316 q^{25} +3.93936 q^{27} -6.93463 q^{29} -8.06298 q^{31} +8.08701 q^{33} +4.04674 q^{37} +10.6926 q^{39} -4.49040 q^{41} -2.57763 q^{43} +1.63694 q^{45} +1.82856 q^{47} -2.01001 q^{51} +2.98007 q^{53} -6.33198 q^{55} +4.11482 q^{57} +1.54828 q^{59} +11.3423 q^{61} -8.37210 q^{65} -3.28300 q^{67} +5.63183 q^{69} -2.34261 q^{71} +13.4835 q^{73} +5.07157 q^{75} -6.44897 q^{79} -11.0385 q^{81} -8.78103 q^{83} +1.57380 q^{85} +13.9386 q^{87} +17.2060 q^{89} +16.2066 q^{93} -3.22182 q^{95} -0.454764 q^{97} -4.18480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 8 q^{5} + 8 q^{9} - 4 q^{11} + 12 q^{13} - 8 q^{15} + 12 q^{17} + 8 q^{25} - 16 q^{27} + 8 q^{29} + 4 q^{31} + 16 q^{33} + 12 q^{37} - 16 q^{39} + 24 q^{41} - 4 q^{43} + 28 q^{45} + 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 8 * q^5 + 8 * q^9 - 4 * q^11 + 12 * q^13 - 8 * q^15 + 12 * q^17 + 8 * q^25 - 16 * q^27 + 8 * q^29 + 4 * q^31 + 16 * q^33 + 12 * q^37 - 16 * q^39 + 24 * q^41 - 4 * q^43 + 28 * q^45 + 28 * q^47 - 4 * q^51 - 4 * q^55 + 12 * q^57 + 36 * q^61 + 20 * q^65 + 4 * q^69 + 40 * q^73 + 20 * q^75 - 4 * q^79 - 4 * q^81 + 12 * q^83 + 8 * q^85 + 36 * q^89 - 16 * q^93 - 8 * q^95 + 12 * q^97 - 20 * q^99

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