LMFDB - Embedded newform 6664.2.a.y.1.1

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			1.1
Root			$-1.76535$ 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)

$f(q)$	$=$	$q-1.76535 q^{3} -2.46294 q^{5} +0.116446 q^{9} -2.67312 q^{11} +5.98122 q^{13} +4.34793 q^{15} -1.00000 q^{17} -4.36008 q^{19} -5.26403 q^{23} +1.06605 q^{25} +5.09047 q^{27} -2.20043 q^{29} +7.75559 q^{31} +4.71897 q^{33} +6.49652 q^{37} -10.5589 q^{39} +8.67266 q^{41} -1.79605 q^{43} -0.286800 q^{45} -4.70079 q^{47} +1.76535 q^{51} -2.40448 q^{53} +6.58371 q^{55} +7.69705 q^{57} +6.59012 q^{59} -4.86731 q^{61} -14.7314 q^{65} +3.19960 q^{67} +9.29283 q^{69} -10.1793 q^{71} +13.2714 q^{73} -1.88195 q^{75} +16.1856 q^{79} -9.33578 q^{81} -10.4362 q^{83} +2.46294 q^{85} +3.88451 q^{87} +1.12222 q^{89} -13.6913 q^{93} +10.7386 q^{95} +5.03417 q^{97} -0.311275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.76535	−1.01922	−0.509611	−	0.860405i	$-0.670211\pi$
$3$	−1.76535	−1.01922	−0.509611	+	0.860405i	$0.670211\pi$
$4$	0	0
$5$	−2.46294	−1.10146	−0.550729	−	0.834684i	$-0.685650\pi$
$5$	−2.46294	−1.10146	−0.550729	+	0.834684i	$0.685650\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.116446	0.0388155
$10$	0	0
$11$	−2.67312	−0.805975	−0.402987	−	0.915206i	$-0.632028\pi$
$11$	−2.67312	−0.805975	−0.402987	+	0.915206i	$0.632028\pi$
$12$	0	0
$13$	5.98122	1.65889	0.829446	−	0.558586i	$-0.188656\pi$
$13$	5.98122	1.65889	0.829446	+	0.558586i	$0.188656\pi$
$14$	0	0
$15$	4.34793	1.12263
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.36008	−1.00027	−0.500135	−	0.865947i	$-0.666716\pi$
$19$	−4.36008	−1.00027	−0.500135	+	0.865947i	$0.666716\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.26403	−1.09763	−0.548813	−	0.835945i	$-0.684920\pi$
$23$	−5.26403	−1.09763	−0.548813	+	0.835945i	$0.684920\pi$
$24$	0	0
$25$	1.06605	0.213211
$26$	0	0
$27$	5.09047	0.979661
$28$	0	0
$29$	−2.20043	−0.408609	−0.204304	−	0.978907i	$-0.565493\pi$
$29$	−2.20043	−0.408609	−0.204304	+	0.978907i	$0.565493\pi$
$30$	0	0
$31$	7.75559	1.39295	0.696473	−	0.717583i	$-0.254752\pi$
$31$	7.75559	1.39295	0.696473	+	0.717583i	$0.254752\pi$
$32$	0	0
$33$	4.71897	0.821468
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.49652	1.06802	0.534011	−	0.845478i	$-0.320684\pi$
$37$	6.49652	1.06802	0.534011	+	0.845478i	$0.320684\pi$
$38$	0	0
$39$	−10.5589	−1.69078
$40$	0	0
$41$	8.67266	1.35444	0.677221	−	0.735780i	$-0.263184\pi$
$41$	8.67266	1.35444	0.677221	+	0.735780i	$0.263184\pi$
$42$	0	0
$43$	−1.79605	−0.273895	−0.136948	−	0.990578i	$-0.543729\pi$
$43$	−1.79605	−0.273895	−0.136948	+	0.990578i	$0.543729\pi$
$44$	0	0
$45$	−0.286800	−0.0427536
$46$	0	0
$47$	−4.70079	−0.685680	−0.342840	−	0.939394i	$-0.611389\pi$
$47$	−4.70079	−0.685680	−0.342840	+	0.939394i	$0.611389\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.76535	0.247198
$52$	0	0
$53$	−2.40448	−0.330280	−0.165140	−	0.986270i	$-0.552808\pi$
$53$	−2.40448	−0.330280	−0.165140	+	0.986270i	$0.552808\pi$
$54$	0	0
$55$	6.58371	0.887747
$56$	0	0
$57$	7.69705	1.01950
$58$	0	0
$59$	6.59012	0.857961	0.428980	−	0.903314i	$-0.358873\pi$
$59$	6.59012	0.857961	0.428980	+	0.903314i	$0.358873\pi$
$60$	0	0
$61$	−4.86731	−0.623195	−0.311598	−	0.950214i	$-0.600864\pi$
$61$	−4.86731	−0.623195	−0.311598	+	0.950214i	$0.600864\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.7314	−1.82720
$66$	0	0
$67$	3.19960	0.390893	0.195446	−	0.980714i	$-0.437384\pi$
$67$	3.19960	0.390893	0.195446	+	0.980714i	$0.437384\pi$
$68$	0	0
$69$	9.29283	1.11873
$70$	0	0
$71$	−10.1793	−1.20806	−0.604032	−	0.796960i	$-0.706440\pi$
$71$	−10.1793	−1.20806	−0.604032	+	0.796960i	$0.706440\pi$
$72$	0	0
$73$	13.2714	1.55330	0.776651	−	0.629931i	$-0.216917\pi$
$73$	13.2714	1.55330	0.776651	+	0.629931i	$0.216917\pi$
$74$	0	0
$75$	−1.88195	−0.217309
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.1856	1.82102	0.910510	−	0.413486i	$-0.135689\pi$
$79$	16.1856	1.82102	0.910510	+	0.413486i	$0.135689\pi$
$80$	0	0
$81$	−9.33578	−1.03731
$82$	0	0
$83$	−10.4362	−1.14553	−0.572763	−	0.819721i	$-0.694129\pi$
$83$	−10.4362	−1.14553	−0.572763	+	0.819721i	$0.694129\pi$
$84$	0	0
$85$	2.46294	0.267143
$86$	0	0
$87$	3.88451	0.416464
$88$	0	0
$89$	1.12222	0.118955	0.0594775	−	0.998230i	$-0.481057\pi$
$89$	1.12222	0.118955	0.0594775	+	0.998230i	$0.481057\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−13.6913	−1.41972
$94$	0	0
$95$	10.7386	1.10176
$96$	0	0
$97$	5.03417	0.511142	0.255571	−	0.966790i	$-0.417736\pi$
$97$	5.03417	0.511142	0.255571	+	0.966790i	$0.417736\pi$
$98$	0	0
$99$	−0.311275	−0.0312843
$100$	0	0
$101$	−8.43047	−0.838863	−0.419432	−	0.907787i	$-0.637771\pi$
$101$	−8.43047	−0.838863	−0.419432	+	0.907787i	$0.637771\pi$
$102$	0	0
$103$	9.60490	0.946399	0.473200	−	0.880955i	$-0.343099\pi$
$103$	9.60490	0.946399	0.473200	+	0.880955i	$0.343099\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.98481	−0.578573	−0.289287	−	0.957243i	$-0.593418\pi$
$107$	−5.98481	−0.578573	−0.289287	+	0.957243i	$0.593418\pi$
$108$	0	0
$109$	2.60227	0.249252	0.124626	−	0.992204i	$-0.460227\pi$
$109$	2.60227	0.249252	0.124626	+	0.992204i	$0.460227\pi$
$110$	0	0
$111$	−11.4686	−1.08855
$112$	0	0
$113$	2.08626	0.196259	0.0981293	−	0.995174i	$-0.468714\pi$
$113$	2.08626	0.196259	0.0981293	+	0.995174i	$0.468714\pi$
$114$	0	0
$115$	12.9650	1.20899
$116$	0	0
$117$	0.696492	0.0643907
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.85446	−0.350405
$122$	0	0
$123$	−15.3102	−1.38048
$124$	0	0
$125$	9.68906	0.866616
$126$	0	0
$127$	−6.29788	−0.558847	−0.279423	−	0.960168i	$-0.590143\pi$
$127$	−6.29788	−0.558847	−0.279423	+	0.960168i	$0.590143\pi$
$128$	0	0
$129$	3.17065	0.279160
$130$	0	0
$131$	19.5066	1.70430	0.852148	−	0.523301i	$-0.175300\pi$
$131$	19.5066	1.70430	0.852148	+	0.523301i	$0.175300\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.5375	−1.07906
$136$	0	0
$137$	12.1122	1.03482	0.517409	−	0.855738i	$-0.326897\pi$
$137$	12.1122	1.03482	0.517409	+	0.855738i	$0.326897\pi$
$138$	0	0
$139$	−4.52648	−0.383931	−0.191966	−	0.981402i	$-0.561486\pi$
$139$	−4.52648	−0.383931	−0.191966	+	0.981402i	$0.561486\pi$
$140$	0	0
$141$	8.29851	0.698861
$142$	0	0
$143$	−15.9885	−1.33703
$144$	0	0
$145$	5.41951	0.450066
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.5356	1.27273	0.636364	−	0.771389i	$-0.280438\pi$
$149$	15.5356	1.27273	0.636364	+	0.771389i	$0.280438\pi$
$150$	0	0
$151$	−6.08719	−0.495368	−0.247684	−	0.968841i	$-0.579670\pi$
$151$	−6.08719	−0.495368	−0.247684	+	0.968841i	$0.579670\pi$
$152$	0	0
$153$	−0.116446	−0.00941414
$154$	0	0
$155$	−19.1015	−1.53427
$156$	0	0
$157$	−21.8902	−1.74703	−0.873516	−	0.486796i	$-0.838165\pi$
$157$	−21.8902	−1.74703	−0.873516	+	0.486796i	$0.838165\pi$
$158$	0	0
$159$	4.24473	0.336629
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.6364	−1.14641	−0.573204	−	0.819413i	$-0.694300\pi$
$163$	−14.6364	−1.14641	−0.573204	+	0.819413i	$0.694300\pi$
$164$	0	0
$165$	−11.6225	−0.904813
$166$	0	0
$167$	20.1531	1.55950	0.779748	−	0.626094i	$-0.215347\pi$
$167$	20.1531	1.55950	0.779748	+	0.626094i	$0.215347\pi$
$168$	0	0
$169$	22.7750	1.75193
$170$	0	0
$171$	−0.507716	−0.0388260
$172$	0	0
$173$	1.40748	0.107008	0.0535042	−	0.998568i	$-0.482961\pi$
$173$	1.40748	0.107008	0.0535042	+	0.998568i	$0.482961\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.6338	−0.874453
$178$	0	0
$179$	6.48616	0.484798	0.242399	−	0.970177i	$-0.422066\pi$
$179$	6.48616	0.484798	0.242399	+	0.970177i	$0.422066\pi$
$180$	0	0
$181$	22.1415	1.64576	0.822881	−	0.568213i	$-0.192365\pi$
$181$	22.1415	1.64576	0.822881	+	0.568213i	$0.192365\pi$
$182$	0	0
$183$	8.59249	0.635175
$184$	0	0
$185$	−16.0005	−1.17638
$186$	0	0
$187$	2.67312	0.195478
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.5298	−1.41313	−0.706563	−	0.707651i	$-0.749755\pi$
$191$	−19.5298	−1.41313	−0.706563	+	0.707651i	$0.749755\pi$
$192$	0	0
$193$	4.21006	0.303046	0.151523	−	0.988454i	$-0.451582\pi$
$193$	4.21006	0.303046	0.151523	+	0.988454i	$0.451582\pi$
$194$	0	0
$195$	26.0060	1.86233
$196$	0	0
$197$	−22.7221	−1.61888	−0.809441	−	0.587201i	$-0.800230\pi$
$197$	−22.7221	−1.61888	−0.809441	+	0.587201i	$0.800230\pi$
$198$	0	0
$199$	−17.4753	−1.23879	−0.619396	−	0.785078i	$-0.712623\pi$
$199$	−17.4753	−1.23879	−0.619396	+	0.785078i	$0.712623\pi$
$200$	0	0
$201$	−5.64839	−0.398407
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−21.3602	−1.49186
$206$	0	0
$207$	−0.612977	−0.0426049
$208$	0	0
$209$	11.6550	0.806193
$210$	0	0
$211$	−7.23204	−0.497874	−0.248937	−	0.968520i	$-0.580081\pi$
$211$	−7.23204	−0.497874	−0.248937	+	0.968520i	$0.580081\pi$
$212$	0	0
$213$	17.9700	1.23129
$214$	0	0
$215$	4.42356	0.301684
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−23.4287	−1.58316
$220$	0	0
$221$	−5.98122	−0.402341
$222$	0	0
$223$	−1.93972	−0.129893	−0.0649466	−	0.997889i	$-0.520688\pi$
$223$	−1.93972	−0.129893	−0.0649466	+	0.997889i	$0.520688\pi$
$224$	0	0
$225$	0.124138	0.00827587
$226$	0	0
$227$	2.14748	0.142533	0.0712665	−	0.997457i	$-0.477296\pi$
$227$	2.14748	0.142533	0.0712665	+	0.997457i	$0.477296\pi$
$228$	0	0
$229$	−11.5395	−0.762550	−0.381275	−	0.924462i	$-0.624515\pi$
$229$	−11.5395	−0.762550	−0.381275	+	0.924462i	$0.624515\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.53281	−0.427978	−0.213989	−	0.976836i	$-0.568646\pi$
$233$	−6.53281	−0.427978	−0.213989	+	0.976836i	$0.568646\pi$
$234$	0	0
$235$	11.5777	0.755248
$236$	0	0
$237$	−28.5732	−1.85603
$238$	0	0
$239$	17.9226	1.15932	0.579659	−	0.814859i	$-0.303186\pi$
$239$	17.9226	1.15932	0.579659	+	0.814859i	$0.303186\pi$
$240$	0	0
$241$	18.5519	1.19503	0.597517	−	0.801856i	$-0.296154\pi$
$241$	18.5519	1.19503	0.597517	+	0.801856i	$0.296154\pi$
$242$	0	0
$243$	1.20947	0.0775877
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−26.0786	−1.65934
$248$	0	0
$249$	18.4236	1.16755
$250$	0	0
$251$	−27.7227	−1.74984	−0.874920	−	0.484267i	$-0.839086\pi$
$251$	−27.7227	−1.74984	−0.874920	+	0.484267i	$0.839086\pi$
$252$	0	0
$253$	14.0714	0.884659
$254$	0	0
$255$	−4.34793	−0.272278
$256$	0	0
$257$	−31.4075	−1.95914	−0.979572	−	0.201093i	$-0.935551\pi$
$257$	−31.4075	−1.95914	−0.979572	+	0.201093i	$0.935551\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.256232	−0.0158604
$262$	0	0
$263$	−11.4517	−0.706141	−0.353070	−	0.935597i	$-0.614862\pi$
$263$	−11.4517	−0.706141	−0.353070	+	0.935597i	$0.614862\pi$
$264$	0	0
$265$	5.92207	0.363790
$266$	0	0
$267$	−1.98111	−0.121242
$268$	0	0
$269$	−15.8091	−0.963897	−0.481948	−	0.876200i	$-0.660071\pi$
$269$	−15.8091	−0.963897	−0.481948	+	0.876200i	$0.660071\pi$
$270$	0	0
$271$	−17.2836	−1.04990	−0.524952	−	0.851132i	$-0.675917\pi$
$271$	−17.2836	−1.04990	−0.524952	+	0.851132i	$0.675917\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.84968	−0.171842
$276$	0	0
$277$	−29.8947	−1.79620	−0.898099	−	0.439794i	$-0.855051\pi$
$277$	−29.8947	−1.79620	−0.898099	+	0.439794i	$0.855051\pi$
$278$	0	0
$279$	0.903111	0.0540679
$280$	0	0
$281$	7.52710	0.449029	0.224515	−	0.974471i	$-0.427920\pi$
$281$	7.52710	0.449029	0.224515	+	0.974471i	$0.427920\pi$
$282$	0	0
$283$	0.976369	0.0580391	0.0290196	−	0.999579i	$-0.490761\pi$
$283$	0.976369	0.0580391	0.0290196	+	0.999579i	$0.490761\pi$
$284$	0	0
$285$	−18.9573	−1.12294
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.88705	−0.520968
$292$	0	0
$293$	−18.4896	−1.08017	−0.540087	−	0.841609i	$-0.681609\pi$
$293$	−18.4896	−1.08017	−0.540087	+	0.841609i	$0.681609\pi$
$294$	0	0
$295$	−16.2310	−0.945008
$296$	0	0
$297$	−13.6074	−0.789582
$298$	0	0
$299$	−31.4853	−1.82084
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.8827	0.854989
$304$	0	0
$305$	11.9879	0.686424
$306$	0	0
$307$	−25.4603	−1.45310	−0.726549	−	0.687115i	$-0.758877\pi$
$307$	−25.4603	−1.45310	−0.726549	+	0.687115i	$0.758877\pi$
$308$	0	0
$309$	−16.9560	−0.964592
$310$	0	0
$311$	−10.2438	−0.580872	−0.290436	−	0.956894i	$-0.593800\pi$
$311$	−10.2438	−0.580872	−0.290436	+	0.956894i	$0.593800\pi$
$312$	0	0
$313$	17.2666	0.975966	0.487983	−	0.872853i	$-0.337733\pi$
$313$	17.2666	0.975966	0.487983	+	0.872853i	$0.337733\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1890	0.740769	0.370385	−	0.928878i	$-0.379226\pi$
$317$	13.1890	0.740769	0.370385	+	0.928878i	$0.379226\pi$
$318$	0	0
$319$	5.88199	0.329328
$320$	0	0
$321$	10.5653	0.589695
$322$	0	0
$323$	4.36008	0.242601
$324$	0	0
$325$	6.37630	0.353693
$326$	0	0
$327$	−4.59390	−0.254043
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.7740	−0.702121	−0.351060	−	0.936353i	$-0.614179\pi$
$331$	−12.7740	−0.702121	−0.351060	+	0.936353i	$0.614179\pi$
$332$	0	0
$333$	0.756497	0.0414558
$334$	0	0
$335$	−7.88040	−0.430552
$336$	0	0
$337$	22.8757	1.24612	0.623058	−	0.782176i	$-0.285890\pi$
$337$	22.8757	1.24612	0.623058	+	0.782176i	$0.285890\pi$
$338$	0	0
$339$	−3.68297	−0.200031
$340$	0	0
$341$	−20.7316	−1.12268
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−22.8876	−1.23223
$346$	0	0
$347$	−16.3468	−0.877544	−0.438772	−	0.898598i	$-0.644586\pi$
$347$	−16.3468	−0.877544	−0.438772	+	0.898598i	$0.644586\pi$
$348$	0	0
$349$	−6.39145	−0.342126	−0.171063	−	0.985260i	$-0.554720\pi$
$349$	−6.39145	−0.342126	−0.171063	+	0.985260i	$0.554720\pi$
$350$	0	0
$351$	30.4472	1.62515
$352$	0	0
$353$	−30.6148	−1.62946	−0.814732	−	0.579838i	$-0.803116\pi$
$353$	−30.6148	−1.62946	−0.814732	+	0.579838i	$0.803116\pi$
$354$	0	0
$355$	25.0710	1.33063
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.0565	−0.530760	−0.265380	−	0.964144i	$-0.585497\pi$
$359$	−10.0565	−0.530760	−0.265380	+	0.964144i	$0.585497\pi$
$360$	0	0
$361$	0.0102913	0.000541650 0
$362$	0	0
$363$	6.80445	0.357141
$364$	0	0
$365$	−32.6867	−1.71090
$366$	0	0
$367$	−17.5048	−0.913743	−0.456871	−	0.889533i	$-0.651030\pi$
$367$	−17.5048	−0.913743	−0.456871	+	0.889533i	$0.651030\pi$
$368$	0	0
$369$	1.00990	0.0525733
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.57413	0.340396	0.170198	−	0.985410i	$-0.445559\pi$
$373$	6.57413	0.340396	0.170198	+	0.985410i	$0.445559\pi$
$374$	0	0
$375$	−17.1045	−0.883275
$376$	0	0
$377$	−13.1612	−0.677838
$378$	0	0
$379$	−27.9951	−1.43801	−0.719006	−	0.695004i	$-0.755402\pi$
$379$	−27.9951	−1.43801	−0.719006	+	0.695004i	$0.755402\pi$
$380$	0	0
$381$	11.1179	0.569589
$382$	0	0
$383$	19.6637	1.00477	0.502385	−	0.864644i	$-0.332456\pi$
$383$	19.6637	1.00477	0.502385	+	0.864644i	$0.332456\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.209144	−0.0106314
$388$	0	0
$389$	0.519642	0.0263469	0.0131734	−	0.999913i	$-0.495807\pi$
$389$	0.519642	0.0263469	0.0131734	+	0.999913i	$0.495807\pi$
$390$	0	0
$391$	5.26403	0.266213
$392$	0	0
$393$	−34.4358	−1.73706
$394$	0	0
$395$	−39.8641	−2.00578
$396$	0	0
$397$	−26.5487	−1.33244	−0.666222	−	0.745754i	$-0.732090\pi$
$397$	−26.5487	−1.33244	−0.666222	+	0.745754i	$0.732090\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.0537	1.55075	0.775375	−	0.631502i	$-0.217561\pi$
$401$	31.0537	1.55075	0.775375	+	0.631502i	$0.217561\pi$
$402$	0	0
$403$	46.3879	2.31075
$404$	0	0
$405$	22.9934	1.14255
$406$	0	0
$407$	−17.3660	−0.860798
$408$	0	0
$409$	25.4302	1.25744	0.628720	−	0.777631i	$-0.283579\pi$
$409$	25.4302	1.25744	0.628720	+	0.777631i	$0.283579\pi$
$410$	0	0
$411$	−21.3823	−1.05471
$412$	0	0
$413$	0	0
$414$	0	0
$415$	25.7038	1.26175
$416$	0	0
$417$	7.99080	0.391311
$418$	0	0
$419$	32.8196	1.60334	0.801672	−	0.597764i	$-0.203944\pi$
$419$	32.8196	1.60334	0.801672	+	0.597764i	$0.203944\pi$
$420$	0	0
$421$	24.7098	1.20428	0.602141	−	0.798390i	$-0.294315\pi$
$421$	24.7098	1.20428	0.602141	+	0.798390i	$0.294315\pi$
$422$	0	0
$423$	−0.547390	−0.0266150
$424$	0	0
$425$	−1.06605	−0.0517112
$426$	0	0
$427$	0	0
$428$	0	0
$429$	28.2252	1.36273
$430$	0	0
$431$	−35.4341	−1.70680	−0.853401	−	0.521255i	$-0.825464\pi$
$431$	−35.4341	−1.70680	−0.853401	+	0.521255i	$0.825464\pi$
$432$	0	0
$433$	29.6822	1.42644	0.713219	−	0.700942i	$-0.247237\pi$
$433$	29.6822	1.42644	0.713219	+	0.700942i	$0.247237\pi$
$434$	0	0
$435$	−9.56731	−0.458717
$436$	0	0
$437$	22.9516	1.09792
$438$	0	0
$439$	−23.5954	−1.12614	−0.563072	−	0.826408i	$-0.690381\pi$
$439$	−23.5954	−1.12614	−0.563072	+	0.826408i	$0.690381\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.81422	−0.418776	−0.209388	−	0.977833i	$-0.567147\pi$
$443$	−8.81422	−0.418776	−0.209388	+	0.977833i	$0.567147\pi$
$444$	0	0
$445$	−2.76396	−0.131024
$446$	0	0
$447$	−27.4257	−1.29719
$448$	0	0
$449$	−10.4607	−0.493670	−0.246835	−	0.969058i	$-0.579391\pi$
$449$	−10.4607	−0.493670	−0.246835	+	0.969058i	$0.579391\pi$
$450$	0	0
$451$	−23.1830	−1.09165
$452$	0	0
$453$	10.7460	0.504891
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.3716	0.812608	0.406304	−	0.913738i	$-0.366817\pi$
$457$	17.3716	0.812608	0.406304	+	0.913738i	$0.366817\pi$
$458$	0	0
$459$	−5.09047	−0.237603
$460$	0	0
$461$	3.49201	0.162639	0.0813196	−	0.996688i	$-0.474087\pi$
$461$	3.49201	0.162639	0.0813196	+	0.996688i	$0.474087\pi$
$462$	0	0
$463$	−16.0554	−0.746158	−0.373079	−	0.927800i	$-0.621698\pi$
$463$	−16.0554	−0.746158	−0.373079	+	0.927800i	$0.621698\pi$
$464$	0	0
$465$	33.7208	1.56376
$466$	0	0
$467$	16.9420	0.783984	0.391992	−	0.919969i	$-0.371786\pi$
$467$	16.9420	0.783984	0.391992	+	0.919969i	$0.371786\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	38.6438	1.78061
$472$	0	0
$473$	4.80105	0.220753
$474$	0	0
$475$	−4.64807	−0.213268
$476$	0	0
$477$	−0.279993	−0.0128200
$478$	0	0
$479$	9.87488	0.451195	0.225597	−	0.974221i	$-0.427567\pi$
$479$	9.87488	0.451195	0.225597	+	0.974221i	$0.427567\pi$
$480$	0	0
$481$	38.8571	1.77173
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.3988	−0.563002
$486$	0	0
$487$	22.9311	1.03911	0.519553	−	0.854438i	$-0.326098\pi$
$487$	22.9311	1.03911	0.519553	+	0.854438i	$0.326098\pi$
$488$	0	0
$489$	25.8382	1.16845
$490$	0	0
$491$	−10.7775	−0.486380	−0.243190	−	0.969979i	$-0.578194\pi$
$491$	−10.7775	−0.486380	−0.243190	+	0.969979i	$0.578194\pi$
$492$	0	0
$493$	2.20043	0.0991022
$494$	0	0
$495$	0.766650	0.0344583
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6.89689	−0.308747	−0.154374	−	0.988013i	$-0.549336\pi$
$499$	−6.89689	−0.308747	−0.154374	+	0.988013i	$0.549336\pi$
$500$	0	0
$501$	−35.5772	−1.58947
$502$	0	0
$503$	−1.14665	−0.0511265	−0.0255633	−	0.999673i	$-0.508138\pi$
$503$	−1.14665	−0.0511265	−0.0255633	+	0.999673i	$0.508138\pi$
$504$	0	0
$505$	20.7637	0.923973
$506$	0	0
$507$	−40.2058	−1.78560
$508$	0	0
$509$	25.9016	1.14807	0.574033	−	0.818832i	$-0.305378\pi$
$509$	25.9016	1.14807	0.574033	+	0.818832i	$0.305378\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−22.1949	−0.979927
$514$	0	0
$515$	−23.6563	−1.04242
$516$	0	0
$517$	12.5657	0.552641
$518$	0	0
$519$	−2.48468	−0.109065
$520$	0	0
$521$	−27.6760	−1.21251	−0.606253	−	0.795272i	$-0.707328\pi$
$521$	−27.6760	−1.21251	−0.606253	+	0.795272i	$0.707328\pi$
$522$	0	0
$523$	−17.7784	−0.777397	−0.388699	−	0.921365i	$-0.627075\pi$
$523$	−17.7784	−0.777397	−0.388699	+	0.921365i	$0.627075\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.75559	−0.337839
$528$	0	0
$529$	4.71000	0.204783
$530$	0	0
$531$	0.767396	0.0333022
$532$	0	0
$533$	51.8731	2.24687
$534$	0	0
$535$	14.7402	0.637274
$536$	0	0
$537$	−11.4503	−0.494118
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.79587	−0.163197	−0.0815986	−	0.996665i	$-0.526003\pi$
$541$	−3.79587	−0.163197	−0.0815986	+	0.996665i	$0.526003\pi$
$542$	0	0
$543$	−39.0874	−1.67740
$544$	0	0
$545$	−6.40922	−0.274541
$546$	0	0
$547$	−30.2394	−1.29294	−0.646471	−	0.762939i	$-0.723756\pi$
$547$	−30.2394	−1.29294	−0.646471	+	0.762939i	$0.723756\pi$
$548$	0	0
$549$	−0.566781	−0.0241896
$550$	0	0
$551$	9.59403	0.408720
$552$	0	0
$553$	0	0
$554$	0	0
$555$	28.2464	1.19899
$556$	0	0
$557$	29.3195	1.24231	0.621155	−	0.783688i	$-0.286664\pi$
$557$	29.3195	1.24231	0.621155	+	0.783688i	$0.286664\pi$
$558$	0	0
$559$	−10.7426	−0.454363
$560$	0	0
$561$	−4.71897	−0.199235
$562$	0	0
$563$	−34.0341	−1.43437	−0.717184	−	0.696884i	$-0.754569\pi$
$563$	−34.0341	−1.43437	−0.717184	+	0.696884i	$0.754569\pi$
$564$	0	0
$565$	−5.13832	−0.216171
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.3541	−0.643679	−0.321840	−	0.946794i	$-0.604301\pi$
$569$	−15.3541	−0.643679	−0.321840	+	0.946794i	$0.604301\pi$
$570$	0	0
$571$	−21.6957	−0.907936	−0.453968	−	0.891018i	$-0.649992\pi$
$571$	−21.6957	−0.907936	−0.453968	+	0.891018i	$0.649992\pi$
$572$	0	0
$573$	34.4768	1.44029
$574$	0	0
$575$	−5.61173	−0.234025
$576$	0	0
$577$	36.2580	1.50944	0.754720	−	0.656047i	$-0.227773\pi$
$577$	36.2580	1.50944	0.754720	+	0.656047i	$0.227773\pi$
$578$	0	0
$579$	−7.43220	−0.308872
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.42744	0.266197
$584$	0	0
$585$	−1.71542	−0.0709237
$586$	0	0
$587$	1.33460	0.0550850	0.0275425	−	0.999621i	$-0.491232\pi$
$587$	1.33460	0.0550850	0.0275425	+	0.999621i	$0.491232\pi$
$588$	0	0
$589$	−33.8150	−1.39332
$590$	0	0
$591$	40.1124	1.65000
$592$	0	0
$593$	3.11071	0.127741	0.0638707	−	0.997958i	$-0.479655\pi$
$593$	3.11071	0.127741	0.0638707	+	0.997958i	$0.479655\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30.8500	1.26261
$598$	0	0
$599$	0.652838	0.0266742	0.0133371	−	0.999911i	$-0.495755\pi$
$599$	0.652838	0.0266742	0.0133371	+	0.999911i	$0.495755\pi$
$600$	0	0
$601$	−44.7546	−1.82558	−0.912789	−	0.408431i	$-0.866076\pi$
$601$	−44.7546	−1.82558	−0.912789	+	0.408431i	$0.866076\pi$
$602$	0	0
$603$	0.372582	0.0151727
$604$	0	0
$605$	9.49328	0.385957
$606$	0	0
$607$	24.4906	0.994044	0.497022	−	0.867738i	$-0.334427\pi$
$607$	24.4906	0.994044	0.497022	+	0.867738i	$0.334427\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−28.1165	−1.13747
$612$	0	0
$613$	−4.92482	−0.198912	−0.0994559	−	0.995042i	$-0.531710\pi$
$613$	−4.92482	−0.198912	−0.0994559	+	0.995042i	$0.531710\pi$
$614$	0	0
$615$	37.7081	1.52054
$616$	0	0
$617$	−30.6341	−1.23328	−0.616641	−	0.787245i	$-0.711507\pi$
$617$	−30.6341	−1.23328	−0.616641	+	0.787245i	$0.711507\pi$
$618$	0	0
$619$	−15.3145	−0.615542	−0.307771	−	0.951460i	$-0.599583\pi$
$619$	−15.3145	−0.615542	−0.307771	+	0.951460i	$0.599583\pi$
$620$	0	0
$621$	−26.7964	−1.07530
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.1938	−1.16775
$626$	0	0
$627$	−20.5751	−0.821690
$628$	0	0
$629$	−6.49652	−0.259033
$630$	0	0
$631$	−39.4413	−1.57013	−0.785066	−	0.619412i	$-0.787371\pi$
$631$	−39.4413	−1.57013	−0.785066	+	0.619412i	$0.787371\pi$
$632$	0	0
$633$	12.7671	0.507445
$634$	0	0
$635$	15.5113	0.615546
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.18535	−0.0468916
$640$	0	0
$641$	−32.4131	−1.28024	−0.640119	−	0.768275i	$-0.721115\pi$
$641$	−32.4131	−1.28024	−0.640119	+	0.768275i	$0.721115\pi$
$642$	0	0
$643$	−5.75507	−0.226958	−0.113479	−	0.993540i	$-0.536199\pi$
$643$	−5.75507	−0.226958	−0.113479	+	0.993540i	$0.536199\pi$
$644$	0	0
$645$	−7.80911	−0.307483
$646$	0	0
$647$	15.9816	0.628300	0.314150	−	0.949373i	$-0.398281\pi$
$647$	15.9816	0.628300	0.314150	+	0.949373i	$0.398281\pi$
$648$	0	0
$649$	−17.6162	−0.691495
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.7247	1.12408	0.562042	−	0.827109i	$-0.310016\pi$
$653$	28.7247	1.12408	0.562042	+	0.827109i	$0.310016\pi$
$654$	0	0
$655$	−48.0434	−1.87721
$656$	0	0
$657$	1.54541	0.0602922
$658$	0	0
$659$	−10.1982	−0.397267	−0.198634	−	0.980074i	$-0.563650\pi$
$659$	−10.1982	−0.397267	−0.198634	+	0.980074i	$0.563650\pi$
$660$	0	0
$661$	−17.1278	−0.666194	−0.333097	−	0.942893i	$-0.608094\pi$
$661$	−17.1278	−0.666194	−0.333097	+	0.942893i	$0.608094\pi$
$662$	0	0
$663$	10.5589	0.410075
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.5831	0.448500
$668$	0	0
$669$	3.42428	0.132390
$670$	0	0
$671$	13.0109	0.502280
$672$	0	0
$673$	20.6752	0.796970	0.398485	−	0.917175i	$-0.369536\pi$
$673$	20.6752	0.796970	0.398485	+	0.917175i	$0.369536\pi$
$674$	0	0
$675$	5.42671	0.208874
$676$	0	0
$677$	−36.4902	−1.40243	−0.701216	−	0.712949i	$-0.747359\pi$
$677$	−36.4902	−1.40243	−0.701216	+	0.712949i	$0.747359\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.79104	−0.145273
$682$	0	0
$683$	−2.49361	−0.0954153	−0.0477076	−	0.998861i	$-0.515192\pi$
$683$	−2.49361	−0.0954153	−0.0477076	+	0.998861i	$0.515192\pi$
$684$	0	0
$685$	−29.8316	−1.13981
$686$	0	0
$687$	20.3712	0.777208
$688$	0	0
$689$	−14.3817	−0.547899
$690$	0	0
$691$	−35.8508	−1.36383	−0.681915	−	0.731432i	$-0.738852\pi$
$691$	−35.8508	−1.36383	−0.681915	+	0.731432i	$0.738852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.1484	0.422884
$696$	0	0
$697$	−8.67266	−0.328500
$698$	0	0
$699$	11.5327	0.436206
$700$	0	0
$701$	23.3117	0.880471	0.440235	−	0.897882i	$-0.354895\pi$
$701$	23.3117	0.880471	0.440235	+	0.897882i	$0.354895\pi$
$702$	0	0
$703$	−28.3254	−1.06831
$704$	0	0
$705$	−20.4387	−0.769766
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.2147	0.984514	0.492257	−	0.870450i	$-0.336172\pi$
$709$	26.2147	0.984514	0.492257	+	0.870450i	$0.336172\pi$
$710$	0	0
$711$	1.88475	0.0706838
$712$	0	0
$713$	−40.8257	−1.52893
$714$	0	0
$715$	39.3786	1.47268
$716$	0	0
$717$	−31.6396	−1.18160
$718$	0	0
$719$	45.9485	1.71359	0.856795	−	0.515657i	$-0.172452\pi$
$719$	45.9485	1.71359	0.856795	+	0.515657i	$0.172452\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−32.7506	−1.21801
$724$	0	0
$725$	−2.34577	−0.0871197
$726$	0	0
$727$	−10.3089	−0.382337	−0.191168	−	0.981557i	$-0.561228\pi$
$727$	−10.3089	−0.382337	−0.191168	+	0.981557i	$0.561228\pi$
$728$	0	0
$729$	25.8722	0.958230
$730$	0	0
$731$	1.79605	0.0664294
$732$	0	0
$733$	38.2615	1.41322	0.706611	−	0.707602i	$-0.250223\pi$
$733$	38.2615	1.41322	0.706611	+	0.707602i	$0.250223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.55289	−0.315050
$738$	0	0
$739$	−34.1968	−1.25795	−0.628975	−	0.777425i	$-0.716525\pi$
$739$	−34.1968	−1.25795	−0.628975	+	0.777425i	$0.716525\pi$
$740$	0	0
$741$	46.0378	1.69124
$742$	0	0
$743$	20.6762	0.758537	0.379268	−	0.925287i	$-0.376176\pi$
$743$	20.6762	0.758537	0.379268	+	0.925287i	$0.376176\pi$
$744$	0	0
$745$	−38.2632	−1.40186
$746$	0	0
$747$	−1.21526	−0.0444642
$748$	0	0
$749$	0	0
$750$	0	0
$751$	48.6977	1.77701	0.888503	−	0.458871i	$-0.151746\pi$
$751$	48.6977	1.77701	0.888503	+	0.458871i	$0.151746\pi$
$752$	0	0
$753$	48.9401	1.78348
$754$	0	0
$755$	14.9924	0.545628
$756$	0	0
$757$	−44.9968	−1.63544	−0.817718	−	0.575620i	$-0.804761\pi$
$757$	−44.9968	−1.63544	−0.817718	+	0.575620i	$0.804761\pi$
$758$	0	0
$759$	−24.8408	−0.901664
$760$	0	0
$761$	−6.75040	−0.244702	−0.122351	−	0.992487i	$-0.539043\pi$
$761$	−6.75040	−0.244702	−0.122351	+	0.992487i	$0.539043\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.286800	0.0103693
$766$	0	0
$767$	39.4170	1.42327
$768$	0	0
$769$	44.1728	1.59291	0.796456	−	0.604697i	$-0.206706\pi$
$769$	44.1728	1.59291	0.796456	+	0.604697i	$0.206706\pi$
$770$	0	0
$771$	55.4451	1.99681
$772$	0	0
$773$	−23.2451	−0.836070	−0.418035	−	0.908431i	$-0.637281\pi$
$773$	−23.2451	−0.836070	−0.418035	+	0.908431i	$0.637281\pi$
$774$	0	0
$775$	8.26787	0.296991
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−37.8135	−1.35481
$780$	0	0
$781$	27.2105	0.973669
$782$	0	0
$783$	−11.2012	−0.400298
$784$	0	0
$785$	53.9142	1.92428
$786$	0	0
$787$	39.0981	1.39370	0.696849	−	0.717218i	$-0.254585\pi$
$787$	39.0981	1.39370	0.696849	+	0.717218i	$0.254585\pi$
$788$	0	0
$789$	20.2162	0.719715
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−29.1125	−1.03381
$794$	0	0
$795$	−10.4545	−0.370783
$796$	0	0
$797$	−41.6163	−1.47412	−0.737062	−	0.675825i	$-0.763788\pi$
$797$	−41.6163	−1.47412	−0.737062	+	0.675825i	$0.763788\pi$
$798$	0	0
$799$	4.70079	0.166302
$800$	0	0
$801$	0.130678	0.00461730
$802$	0	0
$803$	−35.4760	−1.25192
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.9085	0.982426
$808$	0	0
$809$	16.5602	0.582224	0.291112	−	0.956689i	$-0.405975\pi$
$809$	16.5602	0.582224	0.291112	+	0.956689i	$0.405975\pi$
$810$	0	0
$811$	8.27144	0.290450	0.145225	−	0.989399i	$-0.453609\pi$
$811$	8.27144	0.290450	0.145225	+	0.989399i	$0.453609\pi$
$812$	0	0
$813$	30.5115	1.07009
$814$	0	0
$815$	36.0484	1.26272
$816$	0	0
$817$	7.83093	0.273969
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20.8763	−0.728587	−0.364294	−	0.931284i	$-0.618690\pi$
$821$	−20.8763	−0.728587	−0.364294	+	0.931284i	$0.618690\pi$
$822$	0	0
$823$	−39.8696	−1.38977	−0.694883	−	0.719123i	$-0.744544\pi$
$823$	−39.8696	−1.38977	−0.694883	+	0.719123i	$0.744544\pi$
$824$	0	0
$825$	5.03067	0.175146
$826$	0	0
$827$	−3.83809	−0.133463	−0.0667317	−	0.997771i	$-0.521257\pi$
$827$	−3.83809	−0.133463	−0.0667317	+	0.997771i	$0.521257\pi$
$828$	0	0
$829$	4.21208	0.146292	0.0731458	−	0.997321i	$-0.476696\pi$
$829$	4.21208	0.146292	0.0731458	+	0.997321i	$0.476696\pi$
$830$	0	0
$831$	52.7744	1.83073
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−49.6358	−1.71772
$836$	0	0
$837$	39.4796	1.36462
$838$	0	0
$839$	9.33879	0.322411	0.161205	−	0.986921i	$-0.448462\pi$
$839$	9.33879	0.322411	0.161205	+	0.986921i	$0.448462\pi$
$840$	0	0
$841$	−24.1581	−0.833039
$842$	0	0
$843$	−13.2879	−0.457661
$844$	0	0
$845$	−56.0934	−1.92967
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.72363	−0.0591548
$850$	0	0
$851$	−34.1979	−1.17229
$852$	0	0
$853$	−18.3510	−0.628328	−0.314164	−	0.949369i	$-0.601724\pi$
$853$	−18.3510	−0.628328	−0.314164	+	0.949369i	$0.601724\pi$
$854$	0	0
$855$	1.25047	0.0427652
$856$	0	0
$857$	27.5667	0.941662	0.470831	−	0.882223i	$-0.343954\pi$
$857$	27.5667	0.941662	0.470831	+	0.882223i	$0.343954\pi$
$858$	0	0
$859$	−20.7139	−0.706750	−0.353375	−	0.935482i	$-0.614966\pi$
$859$	−20.7139	−0.706750	−0.353375	+	0.935482i	$0.614966\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.4799	−1.00351	−0.501754	−	0.865010i	$-0.667312\pi$
$863$	−29.4799	−1.00351	−0.501754	+	0.865010i	$0.667312\pi$
$864$	0	0
$865$	−3.46652	−0.117865
$866$	0	0
$867$	−1.76535	−0.0599543
$868$	0	0
$869$	−43.2659	−1.46770
$870$	0	0
$871$	19.1375	0.648449
$872$	0	0
$873$	0.586211	0.0198402
$874$	0	0
$875$	0	0
$876$	0	0
$877$	57.1828	1.93092	0.965462	−	0.260542i	$-0.0839013\pi$
$877$	57.1828	1.93092	0.965462	+	0.260542i	$0.0839013\pi$
$878$	0	0
$879$	32.6406	1.10094
$880$	0	0
$881$	49.3021	1.66103	0.830515	−	0.556996i	$-0.188046\pi$
$881$	49.3021	1.66103	0.830515	+	0.556996i	$0.188046\pi$
$882$	0	0
$883$	10.0053	0.336706	0.168353	−	0.985727i	$-0.446155\pi$
$883$	10.0053	0.336706	0.168353	+	0.985727i	$0.446155\pi$
$884$	0	0
$885$	28.6534	0.963174
$886$	0	0
$887$	42.5492	1.42866	0.714331	−	0.699808i	$-0.246731\pi$
$887$	42.5492	1.42866	0.714331	+	0.699808i	$0.246731\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	24.9556	0.836045
$892$	0	0
$893$	20.4958	0.685866
$894$	0	0
$895$	−15.9750	−0.533985
$896$	0	0
$897$	55.5825	1.85585
$898$	0	0
$899$	−17.0656	−0.569170
$900$	0	0
$901$	2.40448	0.0801047
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−54.5330	−1.81274
$906$	0	0
$907$	18.0226	0.598430	0.299215	−	0.954186i	$-0.403275\pi$
$907$	18.0226	0.598430	0.299215	+	0.954186i	$0.403275\pi$
$908$	0	0
$909$	−0.981698	−0.0325609
$910$	0	0
$911$	20.7731	0.688245	0.344122	−	0.938925i	$-0.388176\pi$
$911$	20.7731	0.688245	0.344122	+	0.938925i	$0.388176\pi$
$912$	0	0
$913$	27.8973	0.923265
$914$	0	0
$915$	−21.1627	−0.699619
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.82232	−0.324008	−0.162004	−	0.986790i	$-0.551796\pi$
$919$	−9.82232	−0.324008	−0.162004	+	0.986790i	$0.551796\pi$
$920$	0	0
$921$	44.9463	1.48103
$922$	0	0
$923$	−60.8849	−2.00405
$924$	0	0
$925$	6.92564	0.227713
$926$	0	0
$927$	1.11846	0.0367349
$928$	0	0
$929$	31.0009	1.01711	0.508553	−	0.861030i	$-0.330180\pi$
$929$	31.0009	1.01711	0.508553	+	0.861030i	$0.330180\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.0838	0.592038
$934$	0	0
$935$	−6.58371	−0.215310
$936$	0	0
$937$	−12.9410	−0.422763	−0.211382	−	0.977404i	$-0.567796\pi$
$937$	−12.9410	−0.422763	−0.211382	+	0.977404i	$0.567796\pi$
$938$	0	0
$939$	−30.4815	−0.994727
$940$	0	0
$941$	−41.7589	−1.36130	−0.680651	−	0.732608i	$-0.738303\pi$
$941$	−41.7589	−1.36130	−0.680651	+	0.732608i	$0.738303\pi$
$942$	0	0
$943$	−45.6531	−1.48667
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.4449	−0.371909	−0.185955	−	0.982558i	$-0.559538\pi$
$947$	−11.4449	−0.371909	−0.185955	+	0.982558i	$0.559538\pi$
$948$	0	0
$949$	79.3793	2.57676
$950$	0	0
$951$	−23.2832	−0.755009
$952$	0	0
$953$	3.42466	0.110935	0.0554677	−	0.998460i	$-0.482335\pi$
$953$	3.42466	0.110935	0.0554677	+	0.998460i	$0.482335\pi$
$954$	0	0
$955$	48.1006	1.55650
$956$	0	0
$957$	−10.3838	−0.335659
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29.1492	0.940298
$962$	0	0
$963$	−0.696910	−0.0224576
$964$	0	0
$965$	−10.3691	−0.333793
$966$	0	0
$967$	−25.2293	−0.811319	−0.405659	−	0.914024i	$-0.632958\pi$
$967$	−25.2293	−0.811319	−0.405659	+	0.914024i	$0.632958\pi$
$968$	0	0
$969$	−7.69705	−0.247265
$970$	0	0
$971$	42.9019	1.37679	0.688394	−	0.725337i	$-0.258316\pi$
$971$	42.9019	1.37679	0.688394	+	0.725337i	$0.258316\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−11.2564	−0.360492
$976$	0	0
$977$	−5.92519	−0.189564	−0.0947818	−	0.995498i	$-0.530215\pi$
$977$	−5.92519	−0.189564	−0.0947818	+	0.995498i	$0.530215\pi$
$978$	0	0
$979$	−2.99982	−0.0958747
$980$	0	0
$981$	0.303025	0.00967484
$982$	0	0
$983$	44.0166	1.40391	0.701955	−	0.712221i	$-0.252311\pi$
$983$	44.0166	1.40391	0.701955	+	0.712221i	$0.252311\pi$
$984$	0	0
$985$	55.9631	1.78313
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.45447	0.300634
$990$	0	0
$991$	33.3633	1.05982	0.529911	−	0.848054i	$-0.322226\pi$
$991$	33.3633	1.05982	0.529911	+	0.848054i	$0.322226\pi$
$992$	0	0
$993$	22.5505	0.715617
$994$	0	0
$995$	43.0406	1.36448
$996$	0	0
$997$	−2.62768	−0.0832194	−0.0416097	−	0.999134i	$-0.513249\pi$
$997$	−2.62768	−0.0832194	−0.0416097	+	0.999134i	$0.513249\pi$
$998$	0	0
$999$	33.0703	1.04630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.y.1.1		7
7.3	odd	6		952.2.q.e.681.1	yes	14
7.5	odd	6		952.2.q.e.137.1	✓	14
7.6	odd	2		6664.2.a.v.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.e.137.1	✓	14	7.5	odd	6
952.2.q.e.681.1	yes	14	7.3	odd	6
6664.2.a.v.1.7		7	7.6	odd	2
6664.2.a.y.1.1		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.76535 q^{3} -2.46294 q^{5} +0.116446 q^{9} -2.67312 q^{11} +5.98122 q^{13} +4.34793 q^{15} -1.00000 q^{17} -4.36008 q^{19} -5.26403 q^{23} +1.06605 q^{25} +5.09047 q^{27} -2.20043 q^{29} +7.75559 q^{31} +4.71897 q^{33} +6.49652 q^{37} -10.5589 q^{39} +8.67266 q^{41} -1.79605 q^{43} -0.286800 q^{45} -4.70079 q^{47} +1.76535 q^{51} -2.40448 q^{53} +6.58371 q^{55} +7.69705 q^{57} +6.59012 q^{59} -4.86731 q^{61} -14.7314 q^{65} +3.19960 q^{67} +9.29283 q^{69} -10.1793 q^{71} +13.2714 q^{73} -1.88195 q^{75} +16.1856 q^{79} -9.33578 q^{81} -10.4362 q^{83} +2.46294 q^{85} +3.88451 q^{87} +1.12222 q^{89} -13.6913 q^{93} +10.7386 q^{95} +5.03417 q^{97} -0.311275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 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

Introduction

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