LMFDB - Embedded newform 8800.2.a.cj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8800,2,Mod(1,8800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8800 = 2^{5} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$70.2683537787$
Analytic rank:	$0$
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} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 $ x^7 - 8x^5 - 2x^4 + 16x^3 + 5x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.51372$ of defining polynomial
Character	$\chi$	$=$	8800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13376 q^{3} +1.20437 q^{7} +1.55295 q^{9} +1.00000 q^{11} -4.14048 q^{13} -0.621669 q^{17} -4.76215 q^{19} +2.56983 q^{21} +4.66983 q^{23} -3.08766 q^{27} +2.10643 q^{29} +2.08954 q^{31} +2.13376 q^{33} +8.90878 q^{37} -8.83481 q^{39} +9.62524 q^{41} +10.2003 q^{43} -6.90511 q^{47} -5.54950 q^{49} -1.32650 q^{51} -3.32273 q^{53} -10.1613 q^{57} +10.7689 q^{59} +9.40465 q^{61} +1.87032 q^{63} +5.38505 q^{67} +9.96433 q^{69} -8.42895 q^{71} +8.46268 q^{73} +1.20437 q^{77} -6.44641 q^{79} -11.2472 q^{81} +0.675980 q^{83} +4.49462 q^{87} +15.6891 q^{89} -4.98666 q^{91} +4.45860 q^{93} +10.5470 q^{97} +1.55295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{3} + 7 q^{7} + 6 q^{9} + 7 q^{11} + 10 q^{13} - 3 q^{17} + 7 q^{19} + 13 q^{21} + 14 q^{23} + 13 q^{27} - 11 q^{29} - 11 q^{31} + q^{33} + 13 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{43} + 22 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) $ 7 * q + q^3 + 7 * q^7 + 6 * q^9 + 7 * q^11 + 10 * q^13 - 3 * q^17 + 7 * q^19 + 13 * q^21 + 14 * q^23 + 13 * q^27 - 11 * q^29 - 11 * q^31 + q^33 + 13 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^43 + 22 * q^47 + 16 * q^49 - 7 * q^51 - 3 * q^53 - 3 * q^57 + 26 * q^59 - 5 * q^61 + 38 * q^63 + 14 * q^67 + 2 * q^69 + 3 * q^71 + 16 * q^73 + 7 * q^77 - 32 * q^79 - q^81 + 16 * q^83 + 19 * q^87 + 11 * q^89 + 8 * q^91 - 23 * q^93 - 8 * q^97 + 6 * 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
$3$	2.13376	1.23193	0.615965	−	0.787774i	$-0.288766\pi$
$3$	2.13376	1.23193	0.615965	+	0.787774i	$0.288766\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.20437	0.455208	0.227604	−	0.973754i	$-0.426911\pi$
$7$	1.20437	0.455208	0.227604	+	0.973754i	$0.426911\pi$
$8$	0	0
$9$	1.55295	0.517651
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.14048	−1.14836	−0.574181	−	0.818728i	$-0.694680\pi$
$13$	−4.14048	−1.14836	−0.574181	+	0.818728i	$0.694680\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.621669	−0.150777	−0.0753885	−	0.997154i	$-0.524020\pi$
$17$	−0.621669	−0.150777	−0.0753885	+	0.997154i	$0.524020\pi$
$18$	0	0
$19$	−4.76215	−1.09251	−0.546256	−	0.837618i	$-0.683947\pi$
$19$	−4.76215	−1.09251	−0.546256	+	0.837618i	$0.683947\pi$
$20$	0	0
$21$	2.56983	0.560784
$22$	0	0
$23$	4.66983	0.973728	0.486864	−	0.873478i	$-0.338141\pi$
$23$	4.66983	0.973728	0.486864	+	0.873478i	$0.338141\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.08766	−0.594221
$28$	0	0
$29$	2.10643	0.391154	0.195577	−	0.980688i	$-0.437342\pi$
$29$	2.10643	0.391154	0.195577	+	0.980688i	$0.437342\pi$
$30$	0	0
$31$	2.08954	0.375293	0.187647	−	0.982237i	$-0.439914\pi$
$31$	2.08954	0.375293	0.187647	+	0.982237i	$0.439914\pi$
$32$	0	0
$33$	2.13376	0.371441
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.90878	1.46459	0.732297	−	0.680985i	$-0.238448\pi$
$37$	8.90878	1.46459	0.732297	+	0.680985i	$0.238448\pi$
$38$	0	0
$39$	−8.83481	−1.41470
$40$	0	0
$41$	9.62524	1.50321	0.751605	−	0.659614i	$-0.229280\pi$
$41$	9.62524	1.50321	0.751605	+	0.659614i	$0.229280\pi$
$42$	0	0
$43$	10.2003	1.55553	0.777764	−	0.628556i	$-0.216354\pi$
$43$	10.2003	1.55553	0.777764	+	0.628556i	$0.216354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.90511	−1.00721	−0.503607	−	0.863933i	$-0.667994\pi$
$47$	−6.90511	−1.00721	−0.503607	+	0.863933i	$0.667994\pi$
$48$	0	0
$49$	−5.54950	−0.792786
$50$	0	0
$51$	−1.32650	−0.185747
$52$	0	0
$53$	−3.32273	−0.456412	−0.228206	−	0.973613i	$-0.573286\pi$
$53$	−3.32273	−0.456412	−0.228206	+	0.973613i	$0.573286\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−10.1613	−1.34590
$58$	0	0
$59$	10.7689	1.40200	0.700998	−	0.713163i	$-0.252738\pi$
$59$	10.7689	1.40200	0.700998	+	0.713163i	$0.252738\pi$
$60$	0	0
$61$	9.40465	1.20414	0.602071	−	0.798443i	$-0.294342\pi$
$61$	9.40465	1.20414	0.602071	+	0.798443i	$0.294342\pi$
$62$	0	0
$63$	1.87032	0.235639
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.38505	0.657888	0.328944	−	0.944349i	$-0.393307\pi$
$67$	5.38505	0.657888	0.328944	+	0.944349i	$0.393307\pi$
$68$	0	0
$69$	9.96433	1.19956
$70$	0	0
$71$	−8.42895	−1.00033	−0.500166	−	0.865929i	$-0.666728\pi$
$71$	−8.42895	−1.00033	−0.500166	+	0.865929i	$0.666728\pi$
$72$	0	0
$73$	8.46268	0.990482	0.495241	−	0.868756i	$-0.335080\pi$
$73$	8.46268	0.990482	0.495241	+	0.868756i	$0.335080\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.20437	0.137250
$78$	0	0
$79$	−6.44641	−0.725278	−0.362639	−	0.931930i	$-0.618124\pi$
$79$	−6.44641	−0.725278	−0.362639	+	0.931930i	$0.618124\pi$
$80$	0	0
$81$	−11.2472	−1.24969
$82$	0	0
$83$	0.675980	0.0741984	0.0370992	−	0.999312i	$-0.488188\pi$
$83$	0.675980	0.0741984	0.0370992	+	0.999312i	$0.488188\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.49462	0.481874
$88$	0	0
$89$	15.6891	1.66304	0.831518	−	0.555497i	$-0.187472\pi$
$89$	15.6891	1.66304	0.831518	+	0.555497i	$0.187472\pi$
$90$	0	0
$91$	−4.98666	−0.522744
$92$	0	0
$93$	4.45860	0.462335
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.5470	1.07088	0.535442	−	0.844572i	$-0.320145\pi$
$97$	10.5470	1.07088	0.535442	+	0.844572i	$0.320145\pi$
$98$	0	0
$99$	1.55295	0.156078
$100$	0	0
$101$	−1.27495	−0.126862	−0.0634311	−	0.997986i	$-0.520204\pi$
$101$	−1.27495	−0.126862	−0.0634311	+	0.997986i	$0.520204\pi$
$102$	0	0
$103$	−3.58766	−0.353503	−0.176751	−	0.984256i	$-0.556559\pi$
$103$	−3.58766	−0.353503	−0.176751	+	0.984256i	$0.556559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.97478	0.480930	0.240465	−	0.970658i	$-0.422700\pi$
$107$	4.97478	0.480930	0.240465	+	0.970658i	$0.422700\pi$
$108$	0	0
$109$	6.63873	0.635875	0.317937	−	0.948112i	$-0.397010\pi$
$109$	6.63873	0.635875	0.317937	+	0.948112i	$0.397010\pi$
$110$	0	0
$111$	19.0092	1.80428
$112$	0	0
$113$	4.16340	0.391660	0.195830	−	0.980638i	$-0.437260\pi$
$113$	4.16340	0.391660	0.195830	+	0.980638i	$0.437260\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.42997	−0.594451
$118$	0	0
$119$	−0.748718	−0.0686348
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.5380	1.85185
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.75874	0.422270	0.211135	−	0.977457i	$-0.432284\pi$
$127$	4.75874	0.422270	0.211135	+	0.977457i	$0.432284\pi$
$128$	0	0
$129$	21.7650	1.91630
$130$	0	0
$131$	10.8864	0.951146	0.475573	−	0.879676i	$-0.342241\pi$
$131$	10.8864	0.951146	0.475573	+	0.879676i	$0.342241\pi$
$132$	0	0
$133$	−5.73537	−0.497320
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0795	−0.861146	−0.430573	−	0.902556i	$-0.641688\pi$
$137$	−10.0795	−0.861146	−0.430573	+	0.902556i	$0.641688\pi$
$138$	0	0
$139$	20.5862	1.74610	0.873048	−	0.487634i	$-0.162140\pi$
$139$	20.5862	1.74610	0.873048	+	0.487634i	$0.162140\pi$
$140$	0	0
$141$	−14.7339	−1.24082
$142$	0	0
$143$	−4.14048	−0.346244
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−11.8413	−0.976656
$148$	0	0
$149$	6.65848	0.545484	0.272742	−	0.962087i	$-0.412069\pi$
$149$	6.65848	0.545484	0.272742	+	0.962087i	$0.412069\pi$
$150$	0	0
$151$	20.8504	1.69678	0.848389	−	0.529373i	$-0.177573\pi$
$151$	20.8504	1.69678	0.848389	+	0.529373i	$0.177573\pi$
$152$	0	0
$153$	−0.965422	−0.0780498
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.8397	−1.26415	−0.632073	−	0.774909i	$-0.717796\pi$
$157$	−15.8397	−1.26415	−0.632073	+	0.774909i	$0.717796\pi$
$158$	0	0
$159$	−7.08992	−0.562267
$160$	0	0
$161$	5.62419	0.443248
$162$	0	0
$163$	1.86398	0.145998	0.0729990	−	0.997332i	$-0.476743\pi$
$163$	1.86398	0.145998	0.0729990	+	0.997332i	$0.476743\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.7054	1.13794	0.568969	−	0.822359i	$-0.307342\pi$
$167$	14.7054	1.13794	0.568969	+	0.822359i	$0.307342\pi$
$168$	0	0
$169$	4.14358	0.318737
$170$	0	0
$171$	−7.39539	−0.565540
$172$	0	0
$173$	−11.9226	−0.906462	−0.453231	−	0.891393i	$-0.649729\pi$
$173$	−11.9226	−0.906462	−0.453231	+	0.891393i	$0.649729\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	22.9784	1.72716
$178$	0	0
$179$	10.9996	0.822151	0.411075	−	0.911601i	$-0.365153\pi$
$179$	10.9996	0.822151	0.411075	+	0.911601i	$0.365153\pi$
$180$	0	0
$181$	−0.0996441	−0.00740649	−0.00370324	−	0.999993i	$-0.501179\pi$
$181$	−0.0996441	−0.00740649	−0.00370324	+	0.999993i	$0.501179\pi$
$182$	0	0
$183$	20.0673	1.48342
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.621669	−0.0454610
$188$	0	0
$189$	−3.71867	−0.270494
$190$	0	0
$191$	−20.5239	−1.48506	−0.742530	−	0.669813i	$-0.766374\pi$
$191$	−20.5239	−1.48506	−0.742530	+	0.669813i	$0.766374\pi$
$192$	0	0
$193$	16.8714	1.21443	0.607214	−	0.794539i	$-0.292287\pi$
$193$	16.8714	1.21443	0.607214	+	0.794539i	$0.292287\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.36235	0.0970631	0.0485316	−	0.998822i	$-0.484546\pi$
$197$	1.36235	0.0970631	0.0485316	+	0.998822i	$0.484546\pi$
$198$	0	0
$199$	−20.1085	−1.42546	−0.712728	−	0.701440i	$-0.752541\pi$
$199$	−20.1085	−1.42546	−0.712728	+	0.701440i	$0.752541\pi$
$200$	0	0
$201$	11.4904	0.810472
$202$	0	0
$203$	2.53691	0.178056
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.25203	0.504051
$208$	0	0
$209$	−4.76215	−0.329405
$210$	0	0
$211$	−6.77219	−0.466217	−0.233109	−	0.972451i	$-0.574890\pi$
$211$	−6.77219	−0.466217	−0.233109	+	0.972451i	$0.574890\pi$
$212$	0	0
$213$	−17.9854	−1.23234
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.51658	0.170836
$218$	0	0
$219$	18.0574	1.22020
$220$	0	0
$221$	2.57401	0.173147
$222$	0	0
$223$	−2.11259	−0.141470	−0.0707349	−	0.997495i	$-0.522534\pi$
$223$	−2.11259	−0.141470	−0.0707349	+	0.997495i	$0.522534\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.64802	0.507617	0.253808	−	0.967254i	$-0.418317\pi$
$227$	7.64802	0.507617	0.253808	+	0.967254i	$0.418317\pi$
$228$	0	0
$229$	12.2471	0.809310	0.404655	−	0.914469i	$-0.367392\pi$
$229$	12.2471	0.809310	0.404655	+	0.914469i	$0.367392\pi$
$230$	0	0
$231$	2.56983	0.169083
$232$	0	0
$233$	15.9502	1.04493	0.522467	−	0.852659i	$-0.325012\pi$
$233$	15.9502	1.04493	0.522467	+	0.852659i	$0.325012\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.7551	−0.893492
$238$	0	0
$239$	−28.5217	−1.84491	−0.922456	−	0.386101i	$-0.873822\pi$
$239$	−28.5217	−1.84491	−0.922456	+	0.386101i	$0.873822\pi$
$240$	0	0
$241$	4.51452	0.290806	0.145403	−	0.989373i	$-0.453552\pi$
$241$	4.51452	0.290806	0.145403	+	0.989373i	$0.453552\pi$
$242$	0	0
$243$	−14.7359	−0.945308
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.7176	1.25460
$248$	0	0
$249$	1.44238	0.0914072
$250$	0	0
$251$	−14.1643	−0.894044	−0.447022	−	0.894523i	$-0.647515\pi$
$251$	−14.1643	−0.894044	−0.447022	+	0.894523i	$0.647515\pi$
$252$	0	0
$253$	4.66983	0.293590
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.25007	0.202734	0.101367	−	0.994849i	$-0.467678\pi$
$257$	3.25007	0.202734	0.101367	+	0.994849i	$0.467678\pi$
$258$	0	0
$259$	10.7294	0.666695
$260$	0	0
$261$	3.27118	0.202481
$262$	0	0
$263$	−5.56822	−0.343351	−0.171676	−	0.985154i	$-0.554918\pi$
$263$	−5.56822	−0.343351	−0.171676	+	0.985154i	$0.554918\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	33.4768	2.04874
$268$	0	0
$269$	−14.2454	−0.868557	−0.434279	−	0.900779i	$-0.642997\pi$
$269$	−14.2454	−0.868557	−0.434279	+	0.900779i	$0.642997\pi$
$270$	0	0
$271$	−16.5866	−1.00757	−0.503783	−	0.863830i	$-0.668059\pi$
$271$	−16.5866	−1.00757	−0.503783	+	0.863830i	$0.668059\pi$
$272$	0	0
$273$	−10.6404	−0.643983
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.39484	−0.203976	−0.101988	−	0.994786i	$-0.532520\pi$
$277$	−3.39484	−0.203976	−0.101988	+	0.994786i	$0.532520\pi$
$278$	0	0
$279$	3.24496	0.194271
$280$	0	0
$281$	−17.2362	−1.02823	−0.514113	−	0.857723i	$-0.671879\pi$
$281$	−17.2362	−1.02823	−0.514113	+	0.857723i	$0.671879\pi$
$282$	0	0
$283$	−31.9797	−1.90100	−0.950498	−	0.310732i	$-0.899426\pi$
$283$	−31.9797	−1.90100	−0.950498	+	0.310732i	$0.899426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.5923	0.684273
$288$	0	0
$289$	−16.6135	−0.977266
$290$	0	0
$291$	22.5048	1.31925
$292$	0	0
$293$	−10.1889	−0.595240	−0.297620	−	0.954684i	$-0.596193\pi$
$293$	−10.1889	−0.595240	−0.297620	+	0.954684i	$0.596193\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.08766	−0.179164
$298$	0	0
$299$	−19.3354	−1.11819
$300$	0	0
$301$	12.2849	0.708088
$302$	0	0
$303$	−2.72044	−0.156285
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.3727	1.16273	0.581365	−	0.813643i	$-0.302519\pi$
$307$	20.3727	1.16273	0.581365	+	0.813643i	$0.302519\pi$
$308$	0	0
$309$	−7.65523	−0.435491
$310$	0	0
$311$	−4.63809	−0.263002	−0.131501	−	0.991316i	$-0.541980\pi$
$311$	−4.63809	−0.263002	−0.131501	+	0.991316i	$0.541980\pi$
$312$	0	0
$313$	−25.9376	−1.46608	−0.733040	−	0.680185i	$-0.761899\pi$
$313$	−25.9376	−1.46608	−0.733040	+	0.680185i	$0.761899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.99840	0.393069	0.196535	−	0.980497i	$-0.437031\pi$
$317$	6.99840	0.393069	0.196535	+	0.980497i	$0.437031\pi$
$318$	0	0
$319$	2.10643	0.117937
$320$	0	0
$321$	10.6150	0.592472
$322$	0	0
$323$	2.96048	0.164726
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.1655	0.783353
$328$	0	0
$329$	−8.31628	−0.458491
$330$	0	0
$331$	−6.73753	−0.370328	−0.185164	−	0.982708i	$-0.559282\pi$
$331$	−6.73753	−0.370328	−0.185164	+	0.982708i	$0.559282\pi$
$332$	0	0
$333$	13.8349	0.758148
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.20655	0.174672	0.0873361	−	0.996179i	$-0.472165\pi$
$337$	3.20655	0.174672	0.0873361	+	0.996179i	$0.472165\pi$
$338$	0	0
$339$	8.88371	0.482497
$340$	0	0
$341$	2.08954	0.113155
$342$	0	0
$343$	−15.1142	−0.816090
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.6613	0.679696	0.339848	−	0.940480i	$-0.389624\pi$
$347$	12.6613	0.679696	0.339848	+	0.940480i	$0.389624\pi$
$348$	0	0
$349$	−0.596140	−0.0319106	−0.0159553	−	0.999873i	$-0.505079\pi$
$349$	−0.596140	−0.0319106	−0.0159553	+	0.999873i	$0.505079\pi$
$350$	0	0
$351$	12.7844	0.682381
$352$	0	0
$353$	−21.4134	−1.13972	−0.569860	−	0.821742i	$-0.693003\pi$
$353$	−21.4134	−1.13972	−0.569860	+	0.821742i	$0.693003\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.59759	−0.0845533
$358$	0	0
$359$	17.2379	0.909781	0.454891	−	0.890547i	$-0.349678\pi$
$359$	17.2379	0.909781	0.454891	+	0.890547i	$0.349678\pi$
$360$	0	0
$361$	3.67808	0.193583
$362$	0	0
$363$	2.13376	0.111994
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.2870	1.63317	0.816583	−	0.577228i	$-0.195866\pi$
$367$	31.2870	1.63317	0.816583	+	0.577228i	$0.195866\pi$
$368$	0	0
$369$	14.9475	0.778137
$370$	0	0
$371$	−4.00178	−0.207762
$372$	0	0
$373$	−18.6441	−0.965355	−0.482677	−	0.875798i	$-0.660336\pi$
$373$	−18.6441	−0.965355	−0.482677	+	0.875798i	$0.660336\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.72162	−0.449186
$378$	0	0
$379$	33.4682	1.71915	0.859574	−	0.511011i	$-0.170729\pi$
$379$	33.4682	1.71915	0.859574	+	0.511011i	$0.170729\pi$
$380$	0	0
$381$	10.1540	0.520207
$382$	0	0
$383$	32.4451	1.65787	0.828934	−	0.559347i	$-0.188948\pi$
$383$	32.4451	1.65787	0.828934	+	0.559347i	$0.188948\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.8405	0.805220
$388$	0	0
$389$	−17.2080	−0.872481	−0.436241	−	0.899830i	$-0.643690\pi$
$389$	−17.2080	−0.872481	−0.436241	+	0.899830i	$0.643690\pi$
$390$	0	0
$391$	−2.90309	−0.146816
$392$	0	0
$393$	23.2289	1.17174
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.8359	1.24648	0.623238	−	0.782032i	$-0.285817\pi$
$397$	24.8359	1.24648	0.623238	+	0.782032i	$0.285817\pi$
$398$	0	0
$399$	−12.2379	−0.612663
$400$	0	0
$401$	−11.6749	−0.583017	−0.291509	−	0.956568i	$-0.594157\pi$
$401$	−11.6749	−0.583017	−0.291509	+	0.956568i	$0.594157\pi$
$402$	0	0
$403$	−8.65172	−0.430973
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.90878	0.441592
$408$	0	0
$409$	−32.7457	−1.61917	−0.809584	−	0.587003i	$-0.800308\pi$
$409$	−32.7457	−1.61917	−0.809584	+	0.587003i	$0.800308\pi$
$410$	0	0
$411$	−21.5072	−1.06087
$412$	0	0
$413$	12.9697	0.638199
$414$	0	0
$415$	0	0
$416$	0	0
$417$	43.9260	2.15107
$418$	0	0
$419$	−40.3114	−1.96934	−0.984670	−	0.174429i	$-0.944192\pi$
$419$	−40.3114	−1.96934	−0.984670	+	0.174429i	$0.944192\pi$
$420$	0	0
$421$	−13.2601	−0.646256	−0.323128	−	0.946355i	$-0.604734\pi$
$421$	−13.2601	−0.646256	−0.323128	+	0.946355i	$0.604734\pi$
$422$	0	0
$423$	−10.7233	−0.521385
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.3266	0.548135
$428$	0	0
$429$	−8.83481	−0.426549
$430$	0	0
$431$	−0.602928	−0.0290420	−0.0145210	−	0.999895i	$-0.504622\pi$
$431$	−0.602928	−0.0290420	−0.0145210	+	0.999895i	$0.504622\pi$
$432$	0	0
$433$	35.5966	1.71066	0.855332	−	0.518080i	$-0.173353\pi$
$433$	35.5966	1.71066	0.855332	+	0.518080i	$0.173353\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−22.2384	−1.06381
$438$	0	0
$439$	11.2346	0.536198	0.268099	−	0.963391i	$-0.413605\pi$
$439$	11.2346	0.536198	0.268099	+	0.963391i	$0.413605\pi$
$440$	0	0
$441$	−8.61811	−0.410386
$442$	0	0
$443$	24.5417	1.16601	0.583006	−	0.812468i	$-0.301877\pi$
$443$	24.5417	1.16601	0.583006	+	0.812468i	$0.301877\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.2076	0.671998
$448$	0	0
$449$	−11.7896	−0.556384	−0.278192	−	0.960526i	$-0.589735\pi$
$449$	−11.7896	−0.556384	−0.278192	+	0.960526i	$0.589735\pi$
$450$	0	0
$451$	9.62524	0.453235
$452$	0	0
$453$	44.4898	2.09031
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.9828	−0.981534	−0.490767	−	0.871291i	$-0.663283\pi$
$457$	−20.9828	−0.981534	−0.490767	+	0.871291i	$0.663283\pi$
$458$	0	0
$459$	1.91950	0.0895948
$460$	0	0
$461$	−29.8844	−1.39186	−0.695928	−	0.718111i	$-0.745007\pi$
$461$	−29.8844	−1.39186	−0.695928	+	0.718111i	$0.745007\pi$
$462$	0	0
$463$	−24.4425	−1.13594	−0.567970	−	0.823049i	$-0.692271\pi$
$463$	−24.4425	−1.13594	−0.567970	+	0.823049i	$0.692271\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.13953	−0.145280	−0.0726401	−	0.997358i	$-0.523142\pi$
$467$	−3.13953	−0.145280	−0.0726401	+	0.997358i	$0.523142\pi$
$468$	0	0
$469$	6.48557	0.299476
$470$	0	0
$471$	−33.7982	−1.55734
$472$	0	0
$473$	10.2003	0.469009
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.16004	−0.236262
$478$	0	0
$479$	12.5709	0.574380	0.287190	−	0.957874i	$-0.407279\pi$
$479$	12.5709	0.574380	0.287190	+	0.957874i	$0.407279\pi$
$480$	0	0
$481$	−36.8866	−1.68189
$482$	0	0
$483$	12.0007	0.546051
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−13.2520	−0.600503	−0.300252	−	0.953860i	$-0.597071\pi$
$487$	−13.2520	−0.600503	−0.300252	+	0.953860i	$0.597071\pi$
$488$	0	0
$489$	3.97729	0.179859
$490$	0	0
$491$	11.3070	0.510278	0.255139	−	0.966904i	$-0.417879\pi$
$491$	11.3070	0.510278	0.255139	+	0.966904i	$0.417879\pi$
$492$	0	0
$493$	−1.30950	−0.0589770
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.1515	−0.455359
$498$	0	0
$499$	−6.06431	−0.271476	−0.135738	−	0.990745i	$-0.543341\pi$
$499$	−6.06431	−0.271476	−0.135738	+	0.990745i	$0.543341\pi$
$500$	0	0
$501$	31.3779	1.40186
$502$	0	0
$503$	39.7943	1.77434	0.887169	−	0.461444i	$-0.152669\pi$
$503$	39.7943	1.77434	0.887169	+	0.461444i	$0.152669\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.84143	0.392662
$508$	0	0
$509$	11.1058	0.492258	0.246129	−	0.969237i	$-0.420841\pi$
$509$	11.1058	0.492258	0.246129	+	0.969237i	$0.420841\pi$
$510$	0	0
$511$	10.1922	0.450875
$512$	0	0
$513$	14.7039	0.649193
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.90511	−0.303686
$518$	0	0
$519$	−25.4401	−1.11670
$520$	0	0
$521$	−8.57831	−0.375822	−0.187911	−	0.982186i	$-0.560172\pi$
$521$	−8.57831	−0.375822	−0.187911	+	0.982186i	$0.560172\pi$
$522$	0	0
$523$	−25.6845	−1.12311	−0.561553	−	0.827441i	$-0.689796\pi$
$523$	−25.6845	−1.12311	−0.561553	+	0.827441i	$0.689796\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.29901	−0.0565856
$528$	0	0
$529$	−1.19266	−0.0518546
$530$	0	0
$531$	16.7236	0.725744
$532$	0	0
$533$	−39.8531	−1.72623
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.4706	1.01283
$538$	0	0
$539$	−5.54950	−0.239034
$540$	0	0
$541$	−5.62955	−0.242033	−0.121017	−	0.992650i	$-0.538615\pi$
$541$	−5.62955	−0.242033	−0.121017	+	0.992650i	$0.538615\pi$
$542$	0	0
$543$	−0.212617	−0.00912427
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.0338	1.88275	0.941374	−	0.337365i	$-0.109536\pi$
$547$	44.0338	1.88275	0.941374	+	0.337365i	$0.109536\pi$
$548$	0	0
$549$	14.6050	0.623325
$550$	0	0
$551$	−10.0311	−0.427340
$552$	0	0
$553$	−7.76384	−0.330152
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.5943	1.72003	0.860017	−	0.510265i	$-0.170453\pi$
$557$	40.5943	1.72003	0.860017	+	0.510265i	$0.170453\pi$
$558$	0	0
$559$	−42.2341	−1.78631
$560$	0	0
$561$	−1.32650	−0.0560047
$562$	0	0
$563$	−6.63333	−0.279562	−0.139781	−	0.990182i	$-0.544640\pi$
$563$	−6.63333	−0.279562	−0.139781	+	0.990182i	$0.544640\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−13.5457	−0.568868
$568$	0	0
$569$	−18.3346	−0.768625	−0.384313	−	0.923203i	$-0.625562\pi$
$569$	−18.3346	−0.768625	−0.384313	+	0.923203i	$0.625562\pi$
$570$	0	0
$571$	9.13047	0.382098	0.191049	−	0.981580i	$-0.438811\pi$
$571$	9.13047	0.382098	0.191049	+	0.981580i	$0.438811\pi$
$572$	0	0
$573$	−43.7932	−1.82949
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−9.93656	−0.413665	−0.206832	−	0.978376i	$-0.566315\pi$
$577$	−9.93656	−0.413665	−0.206832	+	0.978376i	$0.566315\pi$
$578$	0	0
$579$	35.9995	1.49609
$580$	0	0
$581$	0.814127	0.0337757
$582$	0	0
$583$	−3.32273	−0.137613
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.66282	0.0686318	0.0343159	−	0.999411i	$-0.489075\pi$
$587$	1.66282	0.0686318	0.0343159	+	0.999411i	$0.489075\pi$
$588$	0	0
$589$	−9.95073	−0.410013
$590$	0	0
$591$	2.90693	0.119575
$592$	0	0
$593$	22.3866	0.919307	0.459653	−	0.888098i	$-0.347974\pi$
$593$	22.3866	0.919307	0.459653	+	0.888098i	$0.347974\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−42.9069	−1.75606
$598$	0	0
$599$	36.9636	1.51029	0.755147	−	0.655556i	$-0.227566\pi$
$599$	36.9636	1.51029	0.755147	+	0.655556i	$0.227566\pi$
$600$	0	0
$601$	−1.08085	−0.0440887	−0.0220443	−	0.999757i	$-0.507017\pi$
$601$	−1.08085	−0.0440887	−0.0220443	+	0.999757i	$0.507017\pi$
$602$	0	0
$603$	8.36272	0.340556
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.9873	0.648903	0.324451	−	0.945902i	$-0.394820\pi$
$607$	15.9873	0.648903	0.324451	+	0.945902i	$0.394820\pi$
$608$	0	0
$609$	5.41317	0.219353
$610$	0	0
$611$	28.5905	1.15665
$612$	0	0
$613$	−13.6367	−0.550781	−0.275390	−	0.961332i	$-0.588807\pi$
$613$	−13.6367	−0.550781	−0.275390	+	0.961332i	$0.588807\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.9104	−1.64699	−0.823496	−	0.567323i	$-0.807979\pi$
$617$	−40.9104	−1.64699	−0.823496	+	0.567323i	$0.807979\pi$
$618$	0	0
$619$	42.1598	1.69454	0.847272	−	0.531159i	$-0.178243\pi$
$619$	42.1598	1.69454	0.847272	+	0.531159i	$0.178243\pi$
$620$	0	0
$621$	−14.4189	−0.578609
$622$	0	0
$623$	18.8954	0.757027
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10.1613	−0.405804
$628$	0	0
$629$	−5.53831	−0.220827
$630$	0	0
$631$	−29.6308	−1.17958	−0.589792	−	0.807555i	$-0.700790\pi$
$631$	−29.6308	−1.17958	−0.589792	+	0.807555i	$0.700790\pi$
$632$	0	0
$633$	−14.4503	−0.574347
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.9776	0.910406
$638$	0	0
$639$	−13.0898	−0.517823
$640$	0	0
$641$	36.4240	1.43866	0.719330	−	0.694668i	$-0.244449\pi$
$641$	36.4240	1.43866	0.719330	+	0.694668i	$0.244449\pi$
$642$	0	0
$643$	26.9899	1.06438	0.532188	−	0.846626i	$-0.321370\pi$
$643$	26.9899	1.06438	0.532188	+	0.846626i	$0.321370\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.6852	1.20636	0.603181	−	0.797605i	$-0.293900\pi$
$647$	30.6852	1.20636	0.603181	+	0.797605i	$0.293900\pi$
$648$	0	0
$649$	10.7689	0.422718
$650$	0	0
$651$	5.36978	0.210458
$652$	0	0
$653$	−21.9274	−0.858085	−0.429042	−	0.903284i	$-0.641149\pi$
$653$	−21.9274	−0.858085	−0.429042	+	0.903284i	$0.641149\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.1421	0.512724
$658$	0	0
$659$	19.1287	0.745148	0.372574	−	0.928003i	$-0.378475\pi$
$659$	19.1287	0.745148	0.372574	+	0.928003i	$0.378475\pi$
$660$	0	0
$661$	−50.9975	−1.98357	−0.991786	−	0.127912i	$-0.959172\pi$
$661$	−50.9975	−1.98357	−0.991786	+	0.127912i	$0.959172\pi$
$662$	0	0
$663$	5.49233	0.213304
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.83667	0.380877
$668$	0	0
$669$	−4.50778	−0.174281
$670$	0	0
$671$	9.40465	0.363062
$672$	0	0
$673$	23.3223	0.899010	0.449505	−	0.893278i	$-0.351600\pi$
$673$	23.3223	0.899010	0.449505	+	0.893278i	$0.351600\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.30482	−0.242314	−0.121157	−	0.992633i	$-0.538660\pi$
$677$	−6.30482	−0.242314	−0.121157	+	0.992633i	$0.538660\pi$
$678$	0	0
$679$	12.7024	0.487475
$680$	0	0
$681$	16.3191	0.625348
$682$	0	0
$683$	36.8716	1.41085	0.705426	−	0.708783i	$-0.250756\pi$
$683$	36.8716	1.41085	0.705426	+	0.708783i	$0.250756\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.1324	0.997013
$688$	0	0
$689$	13.7577	0.524126
$690$	0	0
$691$	35.5795	1.35351	0.676754	−	0.736209i	$-0.263386\pi$
$691$	35.5795	1.35351	0.676754	+	0.736209i	$0.263386\pi$
$692$	0	0
$693$	1.87032	0.0710477
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−5.98372	−0.226649
$698$	0	0
$699$	34.0340	1.28729
$700$	0	0
$701$	12.1069	0.457271	0.228636	−	0.973512i	$-0.426574\pi$
$701$	12.1069	0.457271	0.228636	+	0.973512i	$0.426574\pi$
$702$	0	0
$703$	−42.4249	−1.60009
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.53551	−0.0577486
$708$	0	0
$709$	−31.1844	−1.17115	−0.585577	−	0.810617i	$-0.699132\pi$
$709$	−31.1844	−1.17115	−0.585577	+	0.810617i	$0.699132\pi$
$710$	0	0
$711$	−10.0110	−0.375441
$712$	0	0
$713$	9.75783	0.365433
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−60.8585	−2.27280
$718$	0	0
$719$	−44.1954	−1.64821	−0.824106	−	0.566436i	$-0.808322\pi$
$719$	−44.1954	−1.64821	−0.824106	+	0.566436i	$0.808322\pi$
$720$	0	0
$721$	−4.32086	−0.160917
$722$	0	0
$723$	9.63292	0.358252
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.419917	−0.0155739	−0.00778694	−	0.999970i	$-0.502479\pi$
$727$	−0.419917	−0.0155739	−0.00778694	+	0.999970i	$0.502479\pi$
$728$	0	0
$729$	2.29867	0.0851359
$730$	0	0
$731$	−6.34120	−0.234538
$732$	0	0
$733$	15.6107	0.576595	0.288297	−	0.957541i	$-0.406911\pi$
$733$	15.6107	0.576595	0.288297	+	0.957541i	$0.406911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.38505	0.198361
$738$	0	0
$739$	−12.5531	−0.461773	−0.230887	−	0.972981i	$-0.574163\pi$
$739$	−12.5531	−0.461773	−0.230887	+	0.972981i	$0.574163\pi$
$740$	0	0
$741$	42.0727	1.54558
$742$	0	0
$743$	41.3322	1.51633	0.758166	−	0.652061i	$-0.226096\pi$
$743$	41.3322	1.51633	0.758166	+	0.652061i	$0.226096\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.04976	0.0384088
$748$	0	0
$749$	5.99146	0.218923
$750$	0	0
$751$	−13.6104	−0.496652	−0.248326	−	0.968677i	$-0.579880\pi$
$751$	−13.6104	−0.496652	−0.248326	+	0.968677i	$0.579880\pi$
$752$	0	0
$753$	−30.2233	−1.10140
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.8830	0.577279	0.288640	−	0.957438i	$-0.406797\pi$
$757$	15.8830	0.577279	0.288640	+	0.957438i	$0.406797\pi$
$758$	0	0
$759$	9.96433	0.361682
$760$	0	0
$761$	−0.731594	−0.0265203	−0.0132601	−	0.999912i	$-0.504221\pi$
$761$	−0.731594	−0.0265203	−0.0132601	+	0.999912i	$0.504221\pi$
$762$	0	0
$763$	7.99546	0.289455
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−44.5886	−1.61000
$768$	0	0
$769$	−28.1471	−1.01501	−0.507505	−	0.861649i	$-0.669432\pi$
$769$	−28.1471	−1.01501	−0.507505	+	0.861649i	$0.669432\pi$
$770$	0	0
$771$	6.93488	0.249753
$772$	0	0
$773$	27.3629	0.984176	0.492088	−	0.870545i	$-0.336234\pi$
$773$	27.3629	0.984176	0.492088	+	0.870545i	$0.336234\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	22.8941	0.821321
$778$	0	0
$779$	−45.8368	−1.64228
$780$	0	0
$781$	−8.42895	−0.301612
$782$	0	0
$783$	−6.50393	−0.232432
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.3954	0.548788	0.274394	−	0.961617i	$-0.411523\pi$
$787$	15.3954	0.548788	0.274394	+	0.961617i	$0.411523\pi$
$788$	0	0
$789$	−11.8813	−0.422985
$790$	0	0
$791$	5.01426	0.178286
$792$	0	0
$793$	−38.9398	−1.38279
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.3762	1.71357	0.856786	−	0.515673i	$-0.172458\pi$
$797$	48.3762	1.71357	0.856786	+	0.515673i	$0.172458\pi$
$798$	0	0
$799$	4.29269	0.151865
$800$	0	0
$801$	24.3644	0.860872
$802$	0	0
$803$	8.46268	0.298642
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.3963	−1.07000
$808$	0	0
$809$	−34.1856	−1.20190	−0.600950	−	0.799287i	$-0.705211\pi$
$809$	−34.1856	−1.20190	−0.600950	+	0.799287i	$0.705211\pi$
$810$	0	0
$811$	−28.5790	−1.00354	−0.501771	−	0.865000i	$-0.667318\pi$
$811$	−28.5790	−1.00354	−0.501771	+	0.865000i	$0.667318\pi$
$812$	0	0
$813$	−35.3920	−1.24125
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−48.5753	−1.69943
$818$	0	0
$819$	−7.74404	−0.270599
$820$	0	0
$821$	−48.9846	−1.70957	−0.854787	−	0.518978i	$-0.826313\pi$
$821$	−48.9846	−1.70957	−0.854787	+	0.518978i	$0.826313\pi$
$822$	0	0
$823$	−51.9200	−1.80982	−0.904909	−	0.425605i	$-0.860061\pi$
$823$	−51.9200	−1.80982	−0.904909	+	0.425605i	$0.860061\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−55.7187	−1.93753	−0.968765	−	0.247980i	$-0.920233\pi$
$827$	−55.7187	−1.93753	−0.968765	+	0.247980i	$0.920233\pi$
$828$	0	0
$829$	−55.7521	−1.93635	−0.968175	−	0.250274i	$-0.919479\pi$
$829$	−55.7521	−1.93635	−0.968175	+	0.250274i	$0.919479\pi$
$830$	0	0
$831$	−7.24379	−0.251284
$832$	0	0
$833$	3.44995	0.119534
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.45180	−0.223007
$838$	0	0
$839$	19.2438	0.664369	0.332184	−	0.943214i	$-0.392214\pi$
$839$	19.2438	0.664369	0.332184	+	0.943214i	$0.392214\pi$
$840$	0	0
$841$	−24.5630	−0.846999
$842$	0	0
$843$	−36.7780	−1.26670
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.20437	0.0413825
$848$	0	0
$849$	−68.2371	−2.34189
$850$	0	0
$851$	41.6025	1.42612
$852$	0	0
$853$	33.8108	1.15766	0.578830	−	0.815448i	$-0.303509\pi$
$853$	33.8108	1.15766	0.578830	+	0.815448i	$0.303509\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	58.4040	1.99504	0.997522	−	0.0703617i	$-0.0224153\pi$
$857$	58.4040	1.99504	0.997522	+	0.0703617i	$0.0224153\pi$
$858$	0	0
$859$	23.8572	0.813998	0.406999	−	0.913429i	$-0.366575\pi$
$859$	23.8572	0.813998	0.406999	+	0.913429i	$0.366575\pi$
$860$	0	0
$861$	24.7353	0.842976
$862$	0	0
$863$	4.37835	0.149041	0.0745204	−	0.997219i	$-0.476257\pi$
$863$	4.37835	0.149041	0.0745204	+	0.997219i	$0.476257\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−35.4494	−1.20392
$868$	0	0
$869$	−6.44641	−0.218680
$870$	0	0
$871$	−22.2967	−0.755494
$872$	0	0
$873$	16.3790	0.554344
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.8329	1.58143	0.790717	−	0.612182i	$-0.209708\pi$
$877$	46.8329	1.58143	0.790717	+	0.612182i	$0.209708\pi$
$878$	0	0
$879$	−21.7406	−0.733293
$880$	0	0
$881$	2.72179	0.0916993	0.0458496	−	0.998948i	$-0.485400\pi$
$881$	2.72179	0.0916993	0.0458496	+	0.998948i	$0.485400\pi$
$882$	0	0
$883$	−22.8773	−0.769881	−0.384941	−	0.922941i	$-0.625778\pi$
$883$	−22.8773	−0.769881	−0.384941	+	0.922941i	$0.625778\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.0004	−1.24235	−0.621175	−	0.783672i	$-0.713344\pi$
$887$	−37.0004	−1.24235	−0.621175	+	0.783672i	$0.713344\pi$
$888$	0	0
$889$	5.73127	0.192221
$890$	0	0
$891$	−11.2472	−0.376795
$892$	0	0
$893$	32.8832	1.10039
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−41.2571	−1.37753
$898$	0	0
$899$	4.40147	0.146797
$900$	0	0
$901$	2.06564	0.0688164
$902$	0	0
$903$	26.2130	0.872315
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6.01498	−0.199724	−0.0998621	−	0.995001i	$-0.531840\pi$
$907$	−6.01498	−0.199724	−0.0998621	+	0.995001i	$0.531840\pi$
$908$	0	0
$909$	−1.97993	−0.0656703
$910$	0	0
$911$	−9.98170	−0.330708	−0.165354	−	0.986234i	$-0.552877\pi$
$911$	−9.98170	−0.330708	−0.165354	+	0.986234i	$0.552877\pi$
$912$	0	0
$913$	0.675980	0.0223717
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.1112	0.432969
$918$	0	0
$919$	−36.3004	−1.19744	−0.598719	−	0.800959i	$-0.704323\pi$
$919$	−36.3004	−1.19744	−0.598719	+	0.800959i	$0.704323\pi$
$920$	0	0
$921$	43.4705	1.43240
$922$	0	0
$923$	34.8999	1.14874
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.57147	−0.182991
$928$	0	0
$929$	25.3480	0.831640	0.415820	−	0.909447i	$-0.363495\pi$
$929$	25.3480	0.831640	0.415820	+	0.909447i	$0.363495\pi$
$930$	0	0
$931$	26.4276	0.866128
$932$	0	0
$933$	−9.89659	−0.324000
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−36.8662	−1.20437	−0.602183	−	0.798358i	$-0.705702\pi$
$937$	−36.8662	−1.20437	−0.602183	+	0.798358i	$0.705702\pi$
$938$	0	0
$939$	−55.3448	−1.80611
$940$	0	0
$941$	10.2385	0.333764	0.166882	−	0.985977i	$-0.446630\pi$
$941$	10.2385	0.333764	0.166882	+	0.985977i	$0.446630\pi$
$942$	0	0
$943$	44.9483	1.46372
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.4386	0.534183	0.267092	−	0.963671i	$-0.413937\pi$
$947$	16.4386	0.534183	0.267092	+	0.963671i	$0.413937\pi$
$948$	0	0
$949$	−35.0396	−1.13743
$950$	0	0
$951$	14.9329	0.484234
$952$	0	0
$953$	−57.7188	−1.86969	−0.934847	−	0.355050i	$-0.884464\pi$
$953$	−57.7188	−1.86969	−0.934847	+	0.355050i	$0.884464\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.49462	0.145290
$958$	0	0
$959$	−12.1394	−0.392000
$960$	0	0
$961$	−26.6338	−0.859155
$962$	0	0
$963$	7.72559	0.248954
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31.9836	−1.02852	−0.514262	−	0.857633i	$-0.671934\pi$
$967$	−31.9836	−1.02852	−0.514262	+	0.857633i	$0.671934\pi$
$968$	0	0
$969$	6.31697	0.202930
$970$	0	0
$971$	47.6080	1.52781	0.763906	−	0.645327i	$-0.223279\pi$
$971$	47.6080	1.52781	0.763906	+	0.645327i	$0.223279\pi$
$972$	0	0
$973$	24.7933	0.794837
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.75975	0.184271	0.0921353	−	0.995746i	$-0.470631\pi$
$977$	5.75975	0.184271	0.0921353	+	0.995746i	$0.470631\pi$
$978$	0	0
$979$	15.6891	0.501425
$980$	0	0
$981$	10.3096	0.329161
$982$	0	0
$983$	31.8921	1.01720	0.508601	−	0.861003i	$-0.330163\pi$
$983$	31.8921	1.01720	0.508601	+	0.861003i	$0.330163\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−17.7450	−0.564829
$988$	0	0
$989$	47.6336	1.51466
$990$	0	0
$991$	−43.6276	−1.38588	−0.692938	−	0.720997i	$-0.743684\pi$
$991$	−43.6276	−1.38588	−0.692938	+	0.720997i	$0.743684\pi$
$992$	0	0
$993$	−14.3763	−0.456218
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.0472	−1.23664	−0.618319	−	0.785927i	$-0.712186\pi$
$997$	−39.0472	−1.23664	−0.618319	+	0.785927i	$0.712186\pi$
$998$	0	0
$999$	−27.5073	−0.870292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8800.2.a.cj.1.6		7
4.3	odd	2		8800.2.a.cc.1.2		7
5.2	odd	4		1760.2.b.f.1409.3	yes	14
5.3	odd	4		1760.2.b.f.1409.12	yes	14
5.4	even	2		8800.2.a.cd.1.2		7
20.3	even	4		1760.2.b.e.1409.3	✓	14
20.7	even	4		1760.2.b.e.1409.12	yes	14
20.19	odd	2		8800.2.a.ci.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1760.2.b.e.1409.3	✓	14	20.3	even	4
1760.2.b.e.1409.12	yes	14	20.7	even	4
1760.2.b.f.1409.3	yes	14	5.2	odd	4
1760.2.b.f.1409.12	yes	14	5.3	odd	4
8800.2.a.cc.1.2		7	4.3	odd	2
8800.2.a.cd.1.2		7	5.4	even	2
8800.2.a.ci.1.6		7	20.19	odd	2
8800.2.a.cj.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13376 q^{3} +1.20437 q^{7} +1.55295 q^{9} +1.00000 q^{11} -4.14048 q^{13} -0.621669 q^{17} -4.76215 q^{19} +2.56983 q^{21} +4.66983 q^{23} -3.08766 q^{27} +2.10643 q^{29} +2.08954 q^{31} +2.13376 q^{33} +8.90878 q^{37} -8.83481 q^{39} +9.62524 q^{41} +10.2003 q^{43} -6.90511 q^{47} -5.54950 q^{49} -1.32650 q^{51} -3.32273 q^{53} -10.1613 q^{57} +10.7689 q^{59} +9.40465 q^{61} +1.87032 q^{63} +5.38505 q^{67} +9.96433 q^{69} -8.42895 q^{71} +8.46268 q^{73} +1.20437 q^{77} -6.44641 q^{79} -11.2472 q^{81} +0.675980 q^{83} +4.49462 q^{87} +15.6891 q^{89} -4.98666 q^{91} +4.45860 q^{93} +10.5470 q^{97} +1.55295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{3} + 7 q^{7} + 6 q^{9} + 7 q^{11} + 10 q^{13} - 3 q^{17} + 7 q^{19} + 13 q^{21} + 14 q^{23} + 13 q^{27} - 11 q^{29} - 11 q^{31} + q^{33} + 13 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{43} + 22 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 7 * q + q^3 + 7 * q^7 + 6 * q^9 + 7 * q^11 + 10 * q^13 - 3 * q^17 + 7 * q^19 + 13 * q^21 + 14 * q^23 + 13 * q^27 - 11 * q^29 - 11 * q^31 + q^33 + 13 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^43 + 22 * q^47 + 16 * q^49 - 7 * q^51 - 3 * q^53 - 3 * q^57 + 26 * q^59 - 5 * q^61 + 38 * q^63 + 14 * q^67 + 2 * q^69 + 3 * q^71 + 16 * q^73 + 7 * q^77 - 32 * q^79 - q^81 + 16 * q^83 + 19 * q^87 + 11 * q^89 + 8 * q^91 - 23 * q^93 - 8 * q^97 + 6 * q^99

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