LMFDB - Embedded newform 6664.2.a.y.1.6

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			1.6
Root			$2.50581$ 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)

$f(q)$	$=$	$q+2.50581 q^{3} -3.79423 q^{5} +3.27908 q^{9} -4.84722 q^{11} +2.44913 q^{13} -9.50763 q^{15} -1.00000 q^{17} +6.99485 q^{19} +4.21329 q^{23} +9.39621 q^{25} +0.699322 q^{27} +3.62181 q^{29} -3.18061 q^{31} -12.1462 q^{33} +4.01047 q^{37} +6.13704 q^{39} -4.63689 q^{41} -11.7265 q^{43} -12.4416 q^{45} -11.3302 q^{47} -2.50581 q^{51} -4.79078 q^{53} +18.3915 q^{55} +17.5278 q^{57} -9.75878 q^{59} +9.55410 q^{61} -9.29256 q^{65} -13.0997 q^{67} +10.5577 q^{69} -0.382777 q^{71} +0.591867 q^{73} +23.5451 q^{75} +8.78161 q^{79} -8.08487 q^{81} +5.09035 q^{83} +3.79423 q^{85} +9.07558 q^{87} -0.425528 q^{89} -7.97000 q^{93} -26.5401 q^{95} -1.02209 q^{97} -15.8944 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.50581	1.44673	0.723365	−	0.690466i	$-0.242594\pi$
$3$	2.50581	1.44673	0.723365	+	0.690466i	$0.242594\pi$
$4$	0	0
$5$	−3.79423	−1.69683	−0.848417	−	0.529329i	$-0.822444\pi$
$5$	−3.79423	−1.69683	−0.848417	+	0.529329i	$0.822444\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.27908	1.09303
$10$	0	0
$11$	−4.84722	−1.46149	−0.730746	−	0.682650i	$-0.760828\pi$
$11$	−4.84722	−1.46149	−0.730746	+	0.682650i	$0.760828\pi$
$12$	0	0
$13$	2.44913	0.679265	0.339633	−	0.940558i	$-0.389697\pi$
$13$	2.44913	0.679265	0.339633	+	0.940558i	$0.389697\pi$
$14$	0	0
$15$	−9.50763	−2.45486
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.99485	1.60473	0.802365	−	0.596834i	$-0.203575\pi$
$19$	6.99485	1.60473	0.802365	+	0.596834i	$0.203575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.21329	0.878532	0.439266	−	0.898357i	$-0.355239\pi$
$23$	4.21329	0.878532	0.439266	+	0.898357i	$0.355239\pi$
$24$	0	0
$25$	9.39621	1.87924
$26$	0	0
$27$	0.699322	0.134585
$28$	0	0
$29$	3.62181	0.672554	0.336277	−	0.941763i	$-0.390832\pi$
$29$	3.62181	0.672554	0.336277	+	0.941763i	$0.390832\pi$
$30$	0	0
$31$	−3.18061	−0.571254	−0.285627	−	0.958341i	$-0.592202\pi$
$31$	−3.18061	−0.571254	−0.285627	+	0.958341i	$0.592202\pi$
$32$	0	0
$33$	−12.1462	−2.11438
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.01047	0.659318	0.329659	−	0.944100i	$-0.393066\pi$
$37$	4.01047	0.659318	0.329659	+	0.944100i	$0.393066\pi$
$38$	0	0
$39$	6.13704	0.982713
$40$	0	0
$41$	−4.63689	−0.724161	−0.362080	−	0.932147i	$-0.617933\pi$
$41$	−4.63689	−0.724161	−0.362080	+	0.932147i	$0.617933\pi$
$42$	0	0
$43$	−11.7265	−1.78828	−0.894139	−	0.447790i	$-0.852211\pi$
$43$	−11.7265	−1.78828	−0.894139	+	0.447790i	$0.852211\pi$
$44$	0	0
$45$	−12.4416	−1.85468
$46$	0	0
$47$	−11.3302	−1.65268	−0.826339	−	0.563173i	$-0.809580\pi$
$47$	−11.3302	−1.65268	−0.826339	+	0.563173i	$0.809580\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.50581	−0.350883
$52$	0	0
$53$	−4.79078	−0.658064	−0.329032	−	0.944319i	$-0.606722\pi$
$53$	−4.79078	−0.658064	−0.329032	+	0.944319i	$0.606722\pi$
$54$	0	0
$55$	18.3915	2.47991
$56$	0	0
$57$	17.5278	2.32161
$58$	0	0
$59$	−9.75878	−1.27048	−0.635242	−	0.772313i	$-0.719100\pi$
$59$	−9.75878	−1.27048	−0.635242	+	0.772313i	$0.719100\pi$
$60$	0	0
$61$	9.55410	1.22328	0.611638	−	0.791137i	$-0.290511\pi$
$61$	9.55410	1.22328	0.611638	+	0.791137i	$0.290511\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.29256	−1.15260
$66$	0	0
$67$	−13.0997	−1.60039	−0.800194	−	0.599741i	$-0.795270\pi$
$67$	−13.0997	−1.60039	−0.800194	+	0.599741i	$0.795270\pi$
$68$	0	0
$69$	10.5577	1.27100
$70$	0	0
$71$	−0.382777	−0.0454273	−0.0227136	−	0.999742i	$-0.507231\pi$
$71$	−0.382777	−0.0454273	−0.0227136	+	0.999742i	$0.507231\pi$
$72$	0	0
$73$	0.591867	0.0692728	0.0346364	−	0.999400i	$-0.488973\pi$
$73$	0.591867	0.0692728	0.0346364	+	0.999400i	$0.488973\pi$
$74$	0	0
$75$	23.5451	2.71876
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.78161	0.988009	0.494004	−	0.869459i	$-0.335533\pi$
$79$	8.78161	0.988009	0.494004	+	0.869459i	$0.335533\pi$
$80$	0	0
$81$	−8.08487	−0.898319
$82$	0	0
$83$	5.09035	0.558738	0.279369	−	0.960184i	$-0.409875\pi$
$83$	5.09035	0.558738	0.279369	+	0.960184i	$0.409875\pi$
$84$	0	0
$85$	3.79423	0.411542
$86$	0	0
$87$	9.07558	0.973004
$88$	0	0
$89$	−0.425528	−0.0451059	−0.0225530	−	0.999746i	$-0.507179\pi$
$89$	−0.425528	−0.0451059	−0.0225530	+	0.999746i	$0.507179\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.97000	−0.826451
$94$	0	0
$95$	−26.5401	−2.72296
$96$	0	0
$97$	−1.02209	−0.103778	−0.0518889	−	0.998653i	$-0.516524\pi$
$97$	−1.02209	−0.103778	−0.0518889	+	0.998653i	$0.516524\pi$
$98$	0	0
$99$	−15.8944	−1.59745
$100$	0	0
$101$	2.38574	0.237390	0.118695	−	0.992931i	$-0.462129\pi$
$101$	2.38574	0.237390	0.118695	+	0.992931i	$0.462129\pi$
$102$	0	0
$103$	6.16896	0.607846	0.303923	−	0.952697i	$-0.401703\pi$
$103$	6.16896	0.607846	0.303923	+	0.952697i	$0.401703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.75604	0.653131	0.326565	−	0.945175i	$-0.394109\pi$
$107$	6.75604	0.653131	0.326565	+	0.945175i	$0.394109\pi$
$108$	0	0
$109$	−11.2460	−1.07717	−0.538586	−	0.842571i	$-0.681041\pi$
$109$	−11.2460	−1.07717	−0.538586	+	0.842571i	$0.681041\pi$
$110$	0	0
$111$	10.0495	0.953855
$112$	0	0
$113$	−12.2360	−1.15107	−0.575534	−	0.817778i	$-0.695206\pi$
$113$	−12.2360	−1.15107	−0.575534	+	0.817778i	$0.695206\pi$
$114$	0	0
$115$	−15.9862	−1.49072
$116$	0	0
$117$	8.03088	0.742455
$118$	0	0
$119$	0	0
$120$	0	0
$121$	12.4955	1.13596
$122$	0	0
$123$	−11.6192	−1.04767
$124$	0	0
$125$	−16.6803	−1.49193
$126$	0	0
$127$	0.800731	0.0710534	0.0355267	−	0.999369i	$-0.488689\pi$
$127$	0.800731	0.0710534	0.0355267	+	0.999369i	$0.488689\pi$
$128$	0	0
$129$	−29.3844	−2.58715
$130$	0	0
$131$	−19.1559	−1.67366	−0.836830	−	0.547462i	$-0.815594\pi$
$131$	−19.1559	−1.67366	−0.836830	+	0.547462i	$0.815594\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.65339	−0.228368
$136$	0	0
$137$	−4.53242	−0.387231	−0.193615	−	0.981078i	$-0.562021\pi$
$137$	−4.53242	−0.387231	−0.193615	+	0.981078i	$0.562021\pi$
$138$	0	0
$139$	13.9470	1.18297	0.591483	−	0.806318i	$-0.298543\pi$
$139$	13.9470	1.18297	0.591483	+	0.806318i	$0.298543\pi$
$140$	0	0
$141$	−28.3913	−2.39098
$142$	0	0
$143$	−11.8715	−0.992741
$144$	0	0
$145$	−13.7420	−1.14121
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.5642	−1.52084	−0.760420	−	0.649432i	$-0.775007\pi$
$149$	−18.5642	−1.52084	−0.760420	+	0.649432i	$0.775007\pi$
$150$	0	0
$151$	−20.3810	−1.65858	−0.829292	−	0.558816i	$-0.811256\pi$
$151$	−20.3810	−1.65858	−0.829292	+	0.558816i	$0.811256\pi$
$152$	0	0
$153$	−3.27908	−0.265098
$154$	0	0
$155$	12.0680	0.969323
$156$	0	0
$157$	−0.191776	−0.0153054	−0.00765269	−	0.999971i	$-0.502436\pi$
$157$	−0.191776	−0.0153054	−0.00765269	+	0.999971i	$0.502436\pi$
$158$	0	0
$159$	−12.0048	−0.952040
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.4437	−1.13132	−0.565660	−	0.824638i	$-0.691379\pi$
$163$	−14.4437	−1.13132	−0.565660	+	0.824638i	$0.691379\pi$
$164$	0	0
$165$	46.0856	3.58776
$166$	0	0
$167$	0.601637	0.0465561	0.0232780	−	0.999729i	$-0.492590\pi$
$167$	0.601637	0.0465561	0.0232780	+	0.999729i	$0.492590\pi$
$168$	0	0
$169$	−7.00178	−0.538599
$170$	0	0
$171$	22.9367	1.75401
$172$	0	0
$173$	5.89975	0.448549	0.224275	−	0.974526i	$-0.427999\pi$
$173$	5.89975	0.448549	0.224275	+	0.974526i	$0.427999\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.4536	−1.83805
$178$	0	0
$179$	−8.93532	−0.667857	−0.333929	−	0.942598i	$-0.608374\pi$
$179$	−8.93532	−0.667857	−0.333929	+	0.942598i	$0.608374\pi$
$180$	0	0
$181$	−19.7947	−1.47133	−0.735663	−	0.677347i	$-0.763129\pi$
$181$	−19.7947	−1.47133	−0.735663	+	0.677347i	$0.763129\pi$
$182$	0	0
$183$	23.9407	1.76975
$184$	0	0
$185$	−15.2167	−1.11875
$186$	0	0
$187$	4.84722	0.354464
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.9798	1.44569	0.722845	−	0.691010i	$-0.242834\pi$
$191$	19.9798	1.44569	0.722845	+	0.691010i	$0.242834\pi$
$192$	0	0
$193$	−21.5954	−1.55447	−0.777237	−	0.629208i	$-0.783379\pi$
$193$	−21.5954	−1.55447	−0.777237	+	0.629208i	$0.783379\pi$
$194$	0	0
$195$	−23.2854	−1.66750
$196$	0	0
$197$	19.6358	1.39900	0.699498	−	0.714635i	$-0.253407\pi$
$197$	19.6358	1.39900	0.699498	+	0.714635i	$0.253407\pi$
$198$	0	0
$199$	−20.3568	−1.44306	−0.721529	−	0.692384i	$-0.756560\pi$
$199$	−20.3568	−1.44306	−0.721529	+	0.692384i	$0.756560\pi$
$200$	0	0
$201$	−32.8254	−2.31533
$202$	0	0
$203$	0	0
$204$	0	0
$205$	17.5935	1.22878
$206$	0	0
$207$	13.8157	0.960259
$208$	0	0
$209$	−33.9056	−2.34530
$210$	0	0
$211$	−8.61581	−0.593137	−0.296569	−	0.955012i	$-0.595842\pi$
$211$	−8.61581	−0.593137	−0.296569	+	0.955012i	$0.595842\pi$
$212$	0	0
$213$	−0.959167	−0.0657210
$214$	0	0
$215$	44.4932	3.03441
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.48311	0.100219
$220$	0	0
$221$	−2.44913	−0.164746
$222$	0	0
$223$	11.1008	0.743366	0.371683	−	0.928360i	$-0.378781\pi$
$223$	11.1008	0.743366	0.371683	+	0.928360i	$0.378781\pi$
$224$	0	0
$225$	30.8109	2.05406
$226$	0	0
$227$	23.8308	1.58171	0.790854	−	0.612005i	$-0.209637\pi$
$227$	23.8308	1.58171	0.790854	+	0.612005i	$0.209637\pi$
$228$	0	0
$229$	−6.53764	−0.432019	−0.216010	−	0.976391i	$-0.569304\pi$
$229$	−6.53764	−0.432019	−0.216010	+	0.976391i	$0.569304\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.7619	−1.36016	−0.680079	−	0.733139i	$-0.738055\pi$
$233$	−20.7619	−1.36016	−0.680079	+	0.733139i	$0.738055\pi$
$234$	0	0
$235$	42.9894	2.80432
$236$	0	0
$237$	22.0050	1.42938
$238$	0	0
$239$	−12.1185	−0.783884	−0.391942	−	0.919990i	$-0.628197\pi$
$239$	−12.1185	−0.783884	−0.391942	+	0.919990i	$0.628197\pi$
$240$	0	0
$241$	18.5851	1.19717	0.598587	−	0.801058i	$-0.295729\pi$
$241$	18.5851	1.19717	0.598587	+	0.801058i	$0.295729\pi$
$242$	0	0
$243$	−22.3571	−1.43421
$244$	0	0
$245$	0	0
$246$	0	0
$247$	17.1313	1.09004
$248$	0	0
$249$	12.7554	0.808344
$250$	0	0
$251$	−21.3552	−1.34793	−0.673966	−	0.738763i	$-0.735410\pi$
$251$	−21.3552	−1.34793	−0.673966	+	0.738763i	$0.735410\pi$
$252$	0	0
$253$	−20.4228	−1.28397
$254$	0	0
$255$	9.50763	0.595391
$256$	0	0
$257$	2.74811	0.171423	0.0857113	−	0.996320i	$-0.472684\pi$
$257$	2.74811	0.171423	0.0857113	+	0.996320i	$0.472684\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	11.8762	0.735120
$262$	0	0
$263$	4.23821	0.261339	0.130670	−	0.991426i	$-0.458287\pi$
$263$	4.23821	0.261339	0.130670	+	0.991426i	$0.458287\pi$
$264$	0	0
$265$	18.1773	1.11662
$266$	0	0
$267$	−1.06629	−0.0652561
$268$	0	0
$269$	2.79189	0.170225	0.0851123	−	0.996371i	$-0.472875\pi$
$269$	2.79189	0.170225	0.0851123	+	0.996371i	$0.472875\pi$
$270$	0	0
$271$	28.2630	1.71686	0.858428	−	0.512934i	$-0.171442\pi$
$271$	28.2630	1.71686	0.858428	+	0.512934i	$0.171442\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−45.5455	−2.74650
$276$	0	0
$277$	−3.01662	−0.181251	−0.0906256	−	0.995885i	$-0.528887\pi$
$277$	−3.01662	−0.181251	−0.0906256	+	0.995885i	$0.528887\pi$
$278$	0	0
$279$	−10.4295	−0.624396
$280$	0	0
$281$	−7.02716	−0.419205	−0.209602	−	0.977787i	$-0.567217\pi$
$281$	−7.02716	−0.419205	−0.209602	+	0.977787i	$0.567217\pi$
$282$	0	0
$283$	8.51900	0.506402	0.253201	−	0.967414i	$-0.418517\pi$
$283$	8.51900	0.506402	0.253201	+	0.967414i	$0.418517\pi$
$284$	0	0
$285$	−66.5045	−3.93938
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.56117	−0.150138
$292$	0	0
$293$	−6.99134	−0.408438	−0.204219	−	0.978925i	$-0.565466\pi$
$293$	−6.99134	−0.408438	−0.204219	+	0.978925i	$0.565466\pi$
$294$	0	0
$295$	37.0271	2.15580
$296$	0	0
$297$	−3.38977	−0.196694
$298$	0	0
$299$	10.3189	0.596756
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.97821	0.343439
$304$	0	0
$305$	−36.2505	−2.07570
$306$	0	0
$307$	−29.8285	−1.70240	−0.851201	−	0.524840i	$-0.824125\pi$
$307$	−29.8285	−1.70240	−0.851201	+	0.524840i	$0.824125\pi$
$308$	0	0
$309$	15.4582	0.879389
$310$	0	0
$311$	−8.76472	−0.497001	−0.248501	−	0.968632i	$-0.579938\pi$
$311$	−8.76472	−0.497001	−0.248501	+	0.968632i	$0.579938\pi$
$312$	0	0
$313$	−7.82420	−0.442250	−0.221125	−	0.975245i	$-0.570973\pi$
$313$	−7.82420	−0.442250	−0.221125	+	0.975245i	$0.570973\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.9683	−0.896869	−0.448435	−	0.893816i	$-0.648018\pi$
$317$	−15.9683	−0.896869	−0.448435	+	0.893816i	$0.648018\pi$
$318$	0	0
$319$	−17.5557	−0.982932
$320$	0	0
$321$	16.9293	0.944903
$322$	0	0
$323$	−6.99485	−0.389204
$324$	0	0
$325$	23.0125	1.27650
$326$	0	0
$327$	−28.1803	−1.55838
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−17.3509	−0.953691	−0.476845	−	0.878987i	$-0.658220\pi$
$331$	−17.3509	−0.953691	−0.476845	+	0.878987i	$0.658220\pi$
$332$	0	0
$333$	13.1507	0.720652
$334$	0	0
$335$	49.7035	2.71559
$336$	0	0
$337$	28.3325	1.54337	0.771685	−	0.636005i	$-0.219414\pi$
$337$	28.3325	1.54337	0.771685	+	0.636005i	$0.219414\pi$
$338$	0	0
$339$	−30.6611	−1.66528
$340$	0	0
$341$	15.4171	0.834883
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−40.0584	−2.15667
$346$	0	0
$347$	11.3760	0.610694	0.305347	−	0.952241i	$-0.401228\pi$
$347$	11.3760	0.610694	0.305347	+	0.952241i	$0.401228\pi$
$348$	0	0
$349$	22.5889	1.20915	0.604577	−	0.796547i	$-0.293342\pi$
$349$	22.5889	1.20915	0.604577	+	0.796547i	$0.293342\pi$
$350$	0	0
$351$	1.71273	0.0914187
$352$	0	0
$353$	−10.7144	−0.570270	−0.285135	−	0.958487i	$-0.592038\pi$
$353$	−10.7144	−0.570270	−0.285135	+	0.958487i	$0.592038\pi$
$354$	0	0
$355$	1.45235	0.0770825
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.3298	−1.81186	−0.905928	−	0.423432i	$-0.860825\pi$
$359$	−34.3298	−1.81186	−0.905928	+	0.423432i	$0.860825\pi$
$360$	0	0
$361$	29.9280	1.57516
$362$	0	0
$363$	31.3114	1.64342
$364$	0	0
$365$	−2.24568	−0.117544
$366$	0	0
$367$	19.1036	0.997198	0.498599	−	0.866833i	$-0.333848\pi$
$367$	19.1036	0.997198	0.498599	+	0.866833i	$0.333848\pi$
$368$	0	0
$369$	−15.2047	−0.791527
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.8981	−0.823171	−0.411585	−	0.911371i	$-0.635025\pi$
$373$	−15.8981	−0.823171	−0.411585	+	0.911371i	$0.635025\pi$
$374$	0	0
$375$	−41.7976	−2.15842
$376$	0	0
$377$	8.87028	0.456843
$378$	0	0
$379$	−15.4844	−0.795382	−0.397691	−	0.917519i	$-0.630188\pi$
$379$	−15.4844	−0.795382	−0.397691	+	0.917519i	$0.630188\pi$
$380$	0	0
$381$	2.00648	0.102795
$382$	0	0
$383$	10.6601	0.544707	0.272354	−	0.962197i	$-0.412198\pi$
$383$	10.6601	0.544707	0.272354	+	0.962197i	$0.412198\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−38.4522	−1.95463
$388$	0	0
$389$	−9.03166	−0.457923	−0.228962	−	0.973435i	$-0.573533\pi$
$389$	−9.03166	−0.457923	−0.228962	+	0.973435i	$0.573533\pi$
$390$	0	0
$391$	−4.21329	−0.213075
$392$	0	0
$393$	−48.0011	−2.42133
$394$	0	0
$395$	−33.3195	−1.67649
$396$	0	0
$397$	7.57689	0.380273	0.190137	−	0.981758i	$-0.439107\pi$
$397$	7.57689	0.380273	0.190137	+	0.981758i	$0.439107\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.03901	0.301574	0.150787	−	0.988566i	$-0.451819\pi$
$401$	6.03901	0.301574	0.150787	+	0.988566i	$0.451819\pi$
$402$	0	0
$403$	−7.78971	−0.388033
$404$	0	0
$405$	30.6759	1.52430
$406$	0	0
$407$	−19.4396	−0.963587
$408$	0	0
$409$	11.3684	0.562133	0.281067	−	0.959688i	$-0.409312\pi$
$409$	11.3684	0.562133	0.281067	+	0.959688i	$0.409312\pi$
$410$	0	0
$411$	−11.3574	−0.560218
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−19.3140	−0.948086
$416$	0	0
$417$	34.9484	1.71143
$418$	0	0
$419$	7.81481	0.381778	0.190889	−	0.981612i	$-0.438863\pi$
$419$	7.81481	0.381778	0.190889	+	0.981612i	$0.438863\pi$
$420$	0	0
$421$	−10.9355	−0.532964	−0.266482	−	0.963840i	$-0.585861\pi$
$421$	−10.9355	−0.532964	−0.266482	+	0.963840i	$0.585861\pi$
$422$	0	0
$423$	−37.1526	−1.80642
$424$	0	0
$425$	−9.39621	−0.455783
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−29.7476	−1.43623
$430$	0	0
$431$	−18.9660	−0.913560	−0.456780	−	0.889580i	$-0.650997\pi$
$431$	−18.9660	−0.913560	−0.456780	+	0.889580i	$0.650997\pi$
$432$	0	0
$433$	2.06964	0.0994607	0.0497304	−	0.998763i	$-0.484164\pi$
$433$	2.06964	0.0994607	0.0497304	+	0.998763i	$0.484164\pi$
$434$	0	0
$435$	−34.4349	−1.65103
$436$	0	0
$437$	29.4714	1.40981
$438$	0	0
$439$	33.8163	1.61396	0.806981	−	0.590577i	$-0.201100\pi$
$439$	33.8163	1.61396	0.806981	+	0.590577i	$0.201100\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.97504	0.426417	0.213208	−	0.977007i	$-0.431609\pi$
$443$	8.97504	0.426417	0.213208	+	0.977007i	$0.431609\pi$
$444$	0	0
$445$	1.61455	0.0765372
$446$	0	0
$447$	−46.5184	−2.20024
$448$	0	0
$449$	−17.8245	−0.841190	−0.420595	−	0.907249i	$-0.638179\pi$
$449$	−17.8245	−0.841190	−0.420595	+	0.907249i	$0.638179\pi$
$450$	0	0
$451$	22.4760	1.05836
$452$	0	0
$453$	−51.0709	−2.39952
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.8033	0.832801	0.416401	−	0.909181i	$-0.363291\pi$
$457$	17.8033	0.832801	0.416401	+	0.909181i	$0.363291\pi$
$458$	0	0
$459$	−0.699322	−0.0326416
$460$	0	0
$461$	1.49574	0.0696634	0.0348317	−	0.999393i	$-0.488910\pi$
$461$	1.49574	0.0696634	0.0348317	+	0.999393i	$0.488910\pi$
$462$	0	0
$463$	−18.5747	−0.863238	−0.431619	−	0.902056i	$-0.642057\pi$
$463$	−18.5747	−0.863238	−0.431619	+	0.902056i	$0.642057\pi$
$464$	0	0
$465$	30.2400	1.40235
$466$	0	0
$467$	−7.53800	−0.348817	−0.174408	−	0.984673i	$-0.555801\pi$
$467$	−7.53800	−0.348817	−0.174408	+	0.984673i	$0.555801\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.480554	−0.0221428
$472$	0	0
$473$	56.8410	2.61355
$474$	0	0
$475$	65.7251	3.01568
$476$	0	0
$477$	−15.7093	−0.719281
$478$	0	0
$479$	34.6707	1.58415	0.792073	−	0.610426i	$-0.209002\pi$
$479$	34.6707	1.58415	0.792073	+	0.610426i	$0.209002\pi$
$480$	0	0
$481$	9.82215	0.447852
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.87806	0.176094
$486$	0	0
$487$	13.6809	0.619942	0.309971	−	0.950746i	$-0.399681\pi$
$487$	13.6809	0.619942	0.309971	+	0.950746i	$0.399681\pi$
$488$	0	0
$489$	−36.1933	−1.63672
$490$	0	0
$491$	37.3533	1.68573	0.842865	−	0.538124i	$-0.180867\pi$
$491$	37.3533	1.68573	0.842865	+	0.538124i	$0.180867\pi$
$492$	0	0
$493$	−3.62181	−0.163118
$494$	0	0
$495$	60.3072	2.71061
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.68560	0.388821	0.194410	−	0.980920i	$-0.437721\pi$
$499$	8.68560	0.388821	0.194410	+	0.980920i	$0.437721\pi$
$500$	0	0
$501$	1.50759	0.0673541
$502$	0	0
$503$	15.1280	0.674523	0.337261	−	0.941411i	$-0.390499\pi$
$503$	15.1280	0.674523	0.337261	+	0.941411i	$0.390499\pi$
$504$	0	0
$505$	−9.05206	−0.402811
$506$	0	0
$507$	−17.5451	−0.779207
$508$	0	0
$509$	−6.12456	−0.271467	−0.135733	−	0.990745i	$-0.543339\pi$
$509$	−6.12456	−0.271467	−0.135733	+	0.990745i	$0.543339\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.89166	0.215972
$514$	0	0
$515$	−23.4065	−1.03141
$516$	0	0
$517$	54.9199	2.41537
$518$	0	0
$519$	14.7836	0.648930
$520$	0	0
$521$	−11.9757	−0.524666	−0.262333	−	0.964977i	$-0.584492\pi$
$521$	−11.9757	−0.524666	−0.262333	+	0.964977i	$0.584492\pi$
$522$	0	0
$523$	−36.2433	−1.58481	−0.792405	−	0.609995i	$-0.791171\pi$
$523$	−36.2433	−1.58481	−0.792405	+	0.609995i	$0.791171\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.18061	0.138550
$528$	0	0
$529$	−5.24817	−0.228181
$530$	0	0
$531$	−31.9998	−1.38867
$532$	0	0
$533$	−11.3563	−0.491897
$534$	0	0
$535$	−25.6340	−1.10825
$536$	0	0
$537$	−22.3902	−0.966209
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.1816	0.523730	0.261865	−	0.965105i	$-0.415663\pi$
$541$	12.1816	0.523730	0.261865	+	0.965105i	$0.415663\pi$
$542$	0	0
$543$	−49.6017	−2.12861
$544$	0	0
$545$	42.6700	1.82778
$546$	0	0
$547$	−32.7211	−1.39905	−0.699526	−	0.714607i	$-0.746605\pi$
$547$	−32.7211	−1.39905	−0.699526	+	0.714607i	$0.746605\pi$
$548$	0	0
$549$	31.3287	1.33707
$550$	0	0
$551$	25.3341	1.07927
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−38.1301	−1.61853
$556$	0	0
$557$	25.7737	1.09207	0.546033	−	0.837764i	$-0.316137\pi$
$557$	25.7737	1.09207	0.546033	+	0.837764i	$0.316137\pi$
$558$	0	0
$559$	−28.7197	−1.21471
$560$	0	0
$561$	12.1462	0.512813
$562$	0	0
$563$	−37.4518	−1.57840	−0.789202	−	0.614133i	$-0.789506\pi$
$563$	−37.4518	−1.57840	−0.789202	+	0.614133i	$0.789506\pi$
$564$	0	0
$565$	46.4263	1.95317
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.6573	1.78829	0.894143	−	0.447781i	$-0.147786\pi$
$569$	42.6573	1.78829	0.894143	+	0.447781i	$0.147786\pi$
$570$	0	0
$571$	12.4580	0.521349	0.260675	−	0.965427i	$-0.416055\pi$
$571$	12.4580	0.521349	0.260675	+	0.965427i	$0.416055\pi$
$572$	0	0
$573$	50.0657	2.09152
$574$	0	0
$575$	39.5890	1.65098
$576$	0	0
$577$	39.6892	1.65228	0.826141	−	0.563464i	$-0.190532\pi$
$577$	39.6892	1.65228	0.826141	+	0.563464i	$0.190532\pi$
$578$	0	0
$579$	−54.1141	−2.24890
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23.2219	0.961755
$584$	0	0
$585$	−30.4710	−1.25982
$586$	0	0
$587$	−9.17691	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$587$	−9.17691	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$588$	0	0
$589$	−22.2479	−0.916708
$590$	0	0
$591$	49.2037	2.02397
$592$	0	0
$593$	24.8548	1.02066	0.510332	−	0.859978i	$-0.329523\pi$
$593$	24.8548	1.02066	0.510332	+	0.859978i	$0.329523\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−51.0104	−2.08772
$598$	0	0
$599$	−4.45211	−0.181908	−0.0909542	−	0.995855i	$-0.528992\pi$
$599$	−4.45211	−0.181908	−0.0909542	+	0.995855i	$0.528992\pi$
$600$	0	0
$601$	36.3507	1.48277	0.741387	−	0.671077i	$-0.234168\pi$
$601$	36.3507	1.48277	0.741387	+	0.671077i	$0.234168\pi$
$602$	0	0
$603$	−42.9551	−1.74927
$604$	0	0
$605$	−47.4110	−1.92753
$606$	0	0
$607$	−15.4605	−0.627523	−0.313761	−	0.949502i	$-0.601589\pi$
$607$	−15.4605	−0.627523	−0.313761	+	0.949502i	$0.601589\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27.7491	−1.12261
$612$	0	0
$613$	8.95712	0.361775	0.180887	−	0.983504i	$-0.442103\pi$
$613$	8.95712	0.361775	0.180887	+	0.983504i	$0.442103\pi$
$614$	0	0
$615$	44.0858	1.77771
$616$	0	0
$617$	37.4791	1.50885	0.754427	−	0.656384i	$-0.227915\pi$
$617$	37.4791	1.50885	0.754427	+	0.656384i	$0.227915\pi$
$618$	0	0
$619$	26.4780	1.06424	0.532120	−	0.846669i	$-0.321396\pi$
$619$	26.4780	1.06424	0.532120	+	0.846669i	$0.321396\pi$
$620$	0	0
$621$	2.94645	0.118237
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.3078	0.652311
$626$	0	0
$627$	−84.9609	−3.39301
$628$	0	0
$629$	−4.01047	−0.159908
$630$	0	0
$631$	12.7300	0.506774	0.253387	−	0.967365i	$-0.418455\pi$
$631$	12.7300	0.506774	0.253387	+	0.967365i	$0.418455\pi$
$632$	0	0
$633$	−21.5896	−0.858109
$634$	0	0
$635$	−3.03816	−0.120566
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.25516	−0.0496532
$640$	0	0
$641$	−9.84680	−0.388925	−0.194463	−	0.980910i	$-0.562296\pi$
$641$	−9.84680	−0.388925	−0.194463	+	0.980910i	$0.562296\pi$
$642$	0	0
$643$	−19.3523	−0.763182	−0.381591	−	0.924331i	$-0.624624\pi$
$643$	−19.3523	−0.763182	−0.381591	+	0.924331i	$0.624624\pi$
$644$	0	0
$645$	111.491	4.38997
$646$	0	0
$647$	−27.2238	−1.07028	−0.535140	−	0.844764i	$-0.679741\pi$
$647$	−27.2238	−1.07028	−0.535140	+	0.844764i	$0.679741\pi$
$648$	0	0
$649$	47.3029	1.85680
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.3565	1.10968	0.554838	−	0.831958i	$-0.312780\pi$
$653$	28.3565	1.10968	0.554838	+	0.831958i	$0.312780\pi$
$654$	0	0
$655$	72.6821	2.83992
$656$	0	0
$657$	1.94078	0.0757170
$658$	0	0
$659$	28.5156	1.11081	0.555405	−	0.831580i	$-0.312563\pi$
$659$	28.5156	1.11081	0.555405	+	0.831580i	$0.312563\pi$
$660$	0	0
$661$	7.94844	0.309159	0.154579	−	0.987980i	$-0.450598\pi$
$661$	7.94844	0.309159	0.154579	+	0.987980i	$0.450598\pi$
$662$	0	0
$663$	−6.13704	−0.238343
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.2598	0.590860
$668$	0	0
$669$	27.8165	1.07545
$670$	0	0
$671$	−46.3108	−1.78781
$672$	0	0
$673$	43.0255	1.65851	0.829255	−	0.558870i	$-0.188765\pi$
$673$	43.0255	1.65851	0.829255	+	0.558870i	$0.188765\pi$
$674$	0	0
$675$	6.57098	0.252917
$676$	0	0
$677$	8.57456	0.329547	0.164774	−	0.986331i	$-0.447311\pi$
$677$	8.57456	0.329547	0.164774	+	0.986331i	$0.447311\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	59.7155	2.28830
$682$	0	0
$683$	−32.0582	−1.22667	−0.613337	−	0.789821i	$-0.710173\pi$
$683$	−32.0582	−1.22667	−0.613337	+	0.789821i	$0.710173\pi$
$684$	0	0
$685$	17.1971	0.657066
$686$	0	0
$687$	−16.3821	−0.625015
$688$	0	0
$689$	−11.7332	−0.447000
$690$	0	0
$691$	30.4481	1.15830	0.579151	−	0.815220i	$-0.303384\pi$
$691$	30.4481	1.15830	0.579151	+	0.815220i	$0.303384\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−52.9180	−2.00730
$696$	0	0
$697$	4.63689	0.175635
$698$	0	0
$699$	−52.0254	−1.96778
$700$	0	0
$701$	−5.57614	−0.210608	−0.105304	−	0.994440i	$-0.533582\pi$
$701$	−5.57614	−0.210608	−0.105304	+	0.994440i	$0.533582\pi$
$702$	0	0
$703$	28.0527	1.05803
$704$	0	0
$705$	107.723	4.05709
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−35.1910	−1.32163	−0.660814	−	0.750550i	$-0.729789\pi$
$709$	−35.1910	−1.32163	−0.660814	+	0.750550i	$0.729789\pi$
$710$	0	0
$711$	28.7956	1.07992
$712$	0	0
$713$	−13.4008	−0.501865
$714$	0	0
$715$	45.0431	1.68452
$716$	0	0
$717$	−30.3668	−1.13407
$718$	0	0
$719$	6.27050	0.233850	0.116925	−	0.993141i	$-0.462696\pi$
$719$	6.27050	0.233850	0.116925	+	0.993141i	$0.462696\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	46.5708	1.73199
$724$	0	0
$725$	34.0313	1.26389
$726$	0	0
$727$	11.3407	0.420603	0.210301	−	0.977637i	$-0.432556\pi$
$727$	11.3407	0.420603	0.210301	+	0.977637i	$0.432556\pi$
$728$	0	0
$729$	−31.7681	−1.17659
$730$	0	0
$731$	11.7265	0.433721
$732$	0	0
$733$	34.0544	1.25783	0.628915	−	0.777474i	$-0.283500\pi$
$733$	34.0544	1.25783	0.628915	+	0.777474i	$0.283500\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	63.4973	2.33895
$738$	0	0
$739$	−14.3472	−0.527771	−0.263886	−	0.964554i	$-0.585004\pi$
$739$	−14.3472	−0.527771	−0.263886	+	0.964554i	$0.585004\pi$
$740$	0	0
$741$	42.9277	1.57699
$742$	0	0
$743$	26.5838	0.975264	0.487632	−	0.873049i	$-0.337861\pi$
$743$	26.5838	0.975264	0.487632	+	0.873049i	$0.337861\pi$
$744$	0	0
$745$	70.4370	2.58061
$746$	0	0
$747$	16.6917	0.610716
$748$	0	0
$749$	0	0
$750$	0	0
$751$	5.03283	0.183650	0.0918252	−	0.995775i	$-0.470730\pi$
$751$	5.03283	0.183650	0.0918252	+	0.995775i	$0.470730\pi$
$752$	0	0
$753$	−53.5122	−1.95009
$754$	0	0
$755$	77.3303	2.81434
$756$	0	0
$757$	−21.2568	−0.772592	−0.386296	−	0.922375i	$-0.626246\pi$
$757$	−21.2568	−0.772592	−0.386296	+	0.922375i	$0.626246\pi$
$758$	0	0
$759$	−51.1755	−1.85755
$760$	0	0
$761$	30.6015	1.10930	0.554651	−	0.832083i	$-0.312852\pi$
$761$	30.6015	1.10930	0.554651	+	0.832083i	$0.312852\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	12.4416	0.449827
$766$	0	0
$767$	−23.9005	−0.862996
$768$	0	0
$769$	−2.38477	−0.0859971	−0.0429986	−	0.999075i	$-0.513691\pi$
$769$	−2.38477	−0.0859971	−0.0429986	+	0.999075i	$0.513691\pi$
$770$	0	0
$771$	6.88625	0.248002
$772$	0	0
$773$	15.5402	0.558943	0.279471	−	0.960154i	$-0.409841\pi$
$773$	15.5402	0.558943	0.279471	+	0.960154i	$0.409841\pi$
$774$	0	0
$775$	−29.8857	−1.07353
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.4344	−1.16208
$780$	0	0
$781$	1.85541	0.0663916
$782$	0	0
$783$	2.53282	0.0905155
$784$	0	0
$785$	0.727643	0.0259707
$786$	0	0
$787$	−25.1186	−0.895380	−0.447690	−	0.894189i	$-0.647753\pi$
$787$	−25.1186	−0.895380	−0.447690	+	0.894189i	$0.647753\pi$
$788$	0	0
$789$	10.6201	0.378087
$790$	0	0
$791$	0	0
$792$	0	0
$793$	23.3992	0.830930
$794$	0	0
$795$	45.5489	1.61545
$796$	0	0
$797$	−21.0066	−0.744093	−0.372046	−	0.928214i	$-0.621344\pi$
$797$	−21.0066	−0.744093	−0.372046	+	0.928214i	$0.621344\pi$
$798$	0	0
$799$	11.3302	0.400833
$800$	0	0
$801$	−1.39534	−0.0493020
$802$	0	0
$803$	−2.86891	−0.101242
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.99595	0.246269
$808$	0	0
$809$	−7.40293	−0.260273	−0.130137	−	0.991496i	$-0.541542\pi$
$809$	−7.40293	−0.260273	−0.130137	+	0.991496i	$0.541542\pi$
$810$	0	0
$811$	−36.8579	−1.29426	−0.647128	−	0.762381i	$-0.724030\pi$
$811$	−36.8579	−1.29426	−0.647128	+	0.762381i	$0.724030\pi$
$812$	0	0
$813$	70.8217	2.48383
$814$	0	0
$815$	54.8029	1.91966
$816$	0	0
$817$	−82.0253	−2.86970
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.5521	−1.17098	−0.585489	−	0.810681i	$-0.699097\pi$
$821$	−33.5521	−1.17098	−0.585489	+	0.810681i	$0.699097\pi$
$822$	0	0
$823$	−38.6605	−1.34762	−0.673809	−	0.738905i	$-0.735343\pi$
$823$	−38.6605	−1.34762	−0.673809	+	0.738905i	$0.735343\pi$
$824$	0	0
$825$	−114.128	−3.97344
$826$	0	0
$827$	6.04811	0.210314	0.105157	−	0.994456i	$-0.466466\pi$
$827$	6.04811	0.210314	0.105157	+	0.994456i	$0.466466\pi$
$828$	0	0
$829$	−47.8537	−1.66203	−0.831014	−	0.556252i	$-0.812239\pi$
$829$	−47.8537	−1.66203	−0.831014	+	0.556252i	$0.812239\pi$
$830$	0	0
$831$	−7.55908	−0.262222
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.28275	−0.0789979
$836$	0	0
$837$	−2.22427	−0.0768821
$838$	0	0
$839$	18.7792	0.648329	0.324165	−	0.946001i	$-0.394917\pi$
$839$	18.7792	0.648329	0.324165	+	0.946001i	$0.394917\pi$
$840$	0	0
$841$	−15.8825	−0.547671
$842$	0	0
$843$	−17.6087	−0.606476
$844$	0	0
$845$	26.5664	0.913912
$846$	0	0
$847$	0	0
$848$	0	0
$849$	21.3470	0.732627
$850$	0	0
$851$	16.8973	0.579232
$852$	0	0
$853$	−5.42303	−0.185681	−0.0928404	−	0.995681i	$-0.529595\pi$
$853$	−5.42303	−0.185681	−0.0928404	+	0.995681i	$0.529595\pi$
$854$	0	0
$855$	−87.0272	−2.97627
$856$	0	0
$857$	−19.6060	−0.669728	−0.334864	−	0.942266i	$-0.608690\pi$
$857$	−19.6060	−0.669728	−0.334864	+	0.942266i	$0.608690\pi$
$858$	0	0
$859$	−20.7412	−0.707679	−0.353839	−	0.935306i	$-0.615124\pi$
$859$	−20.7412	−0.707679	−0.353839	+	0.935306i	$0.615124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.7209	−0.875551	−0.437776	−	0.899084i	$-0.644234\pi$
$863$	−25.7209	−0.875551	−0.437776	+	0.899084i	$0.644234\pi$
$864$	0	0
$865$	−22.3850	−0.761113
$866$	0	0
$867$	2.50581	0.0851017
$868$	0	0
$869$	−42.5664	−1.44397
$870$	0	0
$871$	−32.0829	−1.08709
$872$	0	0
$873$	−3.35152	−0.113432
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−57.3001	−1.93489	−0.967443	−	0.253090i	$-0.918553\pi$
$877$	−57.3001	−1.93489	−0.967443	+	0.253090i	$0.918553\pi$
$878$	0	0
$879$	−17.5190	−0.590900
$880$	0	0
$881$	−36.9570	−1.24511	−0.622557	−	0.782575i	$-0.713906\pi$
$881$	−36.9570	−1.24511	−0.622557	+	0.782575i	$0.713906\pi$
$882$	0	0
$883$	−53.0813	−1.78633	−0.893165	−	0.449730i	$-0.851520\pi$
$883$	−53.0813	−1.78633	−0.893165	+	0.449730i	$0.851520\pi$
$884$	0	0
$885$	92.7828	3.11886
$886$	0	0
$887$	−29.5961	−0.993740	−0.496870	−	0.867825i	$-0.665517\pi$
$887$	−29.5961	−0.993740	−0.496870	+	0.867825i	$0.665517\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	39.1892	1.31289
$892$	0	0
$893$	−79.2530	−2.65210
$894$	0	0
$895$	33.9027	1.13324
$896$	0	0
$897$	25.8571	0.863345
$898$	0	0
$899$	−11.5196	−0.384199
$900$	0	0
$901$	4.79078	0.159604
$902$	0	0
$903$	0	0
$904$	0	0
$905$	75.1056	2.49660
$906$	0	0
$907$	47.9718	1.59288	0.796439	−	0.604719i	$-0.206714\pi$
$907$	47.9718	1.59288	0.796439	+	0.604719i	$0.206714\pi$
$908$	0	0
$909$	7.82304	0.259474
$910$	0	0
$911$	−18.1999	−0.602990	−0.301495	−	0.953468i	$-0.597486\pi$
$911$	−18.1999	−0.602990	−0.301495	+	0.953468i	$0.597486\pi$
$912$	0	0
$913$	−24.6740	−0.816592
$914$	0	0
$915$	−90.8368	−3.00297
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30.2617	0.998242	0.499121	−	0.866532i	$-0.333656\pi$
$919$	30.2617	0.998242	0.499121	+	0.866532i	$0.333656\pi$
$920$	0	0
$921$	−74.7445	−2.46292
$922$	0	0
$923$	−0.937469	−0.0308572
$924$	0	0
$925$	37.6833	1.23902
$926$	0	0
$927$	20.2285	0.664392
$928$	0	0
$929$	−4.20392	−0.137926	−0.0689631	−	0.997619i	$-0.521969\pi$
$929$	−4.20392	−0.137926	−0.0689631	+	0.997619i	$0.521969\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−21.9627	−0.719027
$934$	0	0
$935$	−18.3915	−0.601466
$936$	0	0
$937$	11.6210	0.379643	0.189822	−	0.981819i	$-0.439209\pi$
$937$	11.6210	0.379643	0.189822	+	0.981819i	$0.439209\pi$
$938$	0	0
$939$	−19.6060	−0.639816
$940$	0	0
$941$	48.1221	1.56874	0.784368	−	0.620295i	$-0.212987\pi$
$941$	48.1221	1.56874	0.784368	+	0.620295i	$0.212987\pi$
$942$	0	0
$943$	−19.5366	−0.636199
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.49362	0.243510	0.121755	−	0.992560i	$-0.461148\pi$
$947$	7.49362	0.243510	0.121755	+	0.992560i	$0.461148\pi$
$948$	0	0
$949$	1.44956	0.0470546
$950$	0	0
$951$	−40.0135	−1.29753
$952$	0	0
$953$	38.4095	1.24421	0.622103	−	0.782935i	$-0.286278\pi$
$953$	38.4095	1.24421	0.622103	+	0.782935i	$0.286278\pi$
$954$	0	0
$955$	−75.8082	−2.45310
$956$	0	0
$957$	−43.9913	−1.42204
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.8837	−0.673669
$962$	0	0
$963$	22.1536	0.713889
$964$	0	0
$965$	81.9382	2.63768
$966$	0	0
$967$	−3.36687	−0.108271	−0.0541357	−	0.998534i	$-0.517240\pi$
$967$	−3.36687	−0.108271	−0.0541357	+	0.998534i	$0.517240\pi$
$968$	0	0
$969$	−17.5278	−0.563073
$970$	0	0
$971$	16.9506	0.543971	0.271985	−	0.962301i	$-0.412320\pi$
$971$	16.9506	0.543971	0.271985	+	0.962301i	$0.412320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	57.6650	1.84676
$976$	0	0
$977$	−4.42459	−0.141555	−0.0707775	−	0.997492i	$-0.522548\pi$
$977$	−4.42459	−0.141555	−0.0707775	+	0.997492i	$0.522548\pi$
$978$	0	0
$979$	2.06263	0.0659219
$980$	0	0
$981$	−36.8766	−1.17738
$982$	0	0
$983$	44.7820	1.42832	0.714162	−	0.699980i	$-0.246808\pi$
$983$	44.7820	1.42832	0.714162	+	0.699980i	$0.246808\pi$
$984$	0	0
$985$	−74.5030	−2.37386
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−49.4072	−1.57106
$990$	0	0
$991$	29.3088	0.931026	0.465513	−	0.885041i	$-0.345870\pi$
$991$	29.3088	0.931026	0.465513	+	0.885041i	$0.345870\pi$
$992$	0	0
$993$	−43.4780	−1.37973
$994$	0	0
$995$	77.2386	2.44863
$996$	0	0
$997$	22.1842	0.702580	0.351290	−	0.936267i	$-0.385743\pi$
$997$	22.1842	0.702580	0.351290	+	0.936267i	$0.385743\pi$
$998$	0	0
$999$	2.80461	0.0887341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.y.1.6		7
7.3	odd	6		952.2.q.e.681.6	yes	14
7.5	odd	6		952.2.q.e.137.6	✓	14
7.6	odd	2		6664.2.a.v.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.e.137.6	✓	14	7.5	odd	6
952.2.q.e.681.6	yes	14	7.3	odd	6
6664.2.a.v.1.2		7	7.6	odd	2
6664.2.a.y.1.6		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)

\(f(q)\)	\(=\)	\(q+2.50581 q^{3} -3.79423 q^{5} +3.27908 q^{9} -4.84722 q^{11} +2.44913 q^{13} -9.50763 q^{15} -1.00000 q^{17} +6.99485 q^{19} +4.21329 q^{23} +9.39621 q^{25} +0.699322 q^{27} +3.62181 q^{29} -3.18061 q^{31} -12.1462 q^{33} +4.01047 q^{37} +6.13704 q^{39} -4.63689 q^{41} -11.7265 q^{43} -12.4416 q^{45} -11.3302 q^{47} -2.50581 q^{51} -4.79078 q^{53} +18.3915 q^{55} +17.5278 q^{57} -9.75878 q^{59} +9.55410 q^{61} -9.29256 q^{65} -13.0997 q^{67} +10.5577 q^{69} -0.382777 q^{71} +0.591867 q^{73} +23.5451 q^{75} +8.78161 q^{79} -8.08487 q^{81} +5.09035 q^{83} +3.79423 q^{85} +9.07558 q^{87} -0.425528 q^{89} -7.97000 q^{93} -26.5401 q^{95} -1.02209 q^{97} -15.8944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 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

L-functions

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