LMFDB - Embedded newform 4704.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, 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("4704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.22833$ of defining polynomial
Character	$\chi$	$=$	4704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.04244 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.04244 q^{5} +1.00000 q^{9} +5.93089 q^{11} -0.888450 q^{13} -1.04244 q^{15} +4.51668 q^{17} +3.47424 q^{19} -1.93089 q^{23} -3.91331 q^{25} +1.00000 q^{27} -8.38755 q^{29} +5.13109 q^{31} +5.93089 q^{33} +5.65685 q^{37} -0.888450 q^{39} +8.39663 q^{41} -0.743541 q^{43} -1.04244 q^{45} -4.21778 q^{47} +4.51668 q^{51} -6.91331 q^{53} -6.18262 q^{55} +3.47424 q^{57} +5.43908 q^{59} +10.9733 q^{61} +0.926159 q^{65} -6.20493 q^{67} -1.93089 q^{69} +3.72596 q^{71} -3.49910 q^{73} -3.91331 q^{75} +12.6053 q^{79} +1.00000 q^{81} -2.94847 q^{83} -4.70838 q^{85} -8.38755 q^{87} -5.00196 q^{89} +5.13109 q^{93} -3.62169 q^{95} +10.1578 q^{97} +5.93089 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{39} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 16 q^{47} + 4 q^{51} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{73} + 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} + 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 4 * q^17 + 8 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 16 * q^47 + 4 * q^51 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 12 * q^69 + 12 * q^71 + 8 * q^73 + 12 * q^75 + 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 + 24 * q^95 + 40 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.04244	−0.466195	−0.233097	−	0.972453i	$-0.574886\pi$
$5$	−1.04244	−0.466195	−0.233097	+	0.972453i	$0.574886\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.93089	1.78823	0.894116	−	0.447836i	$-0.147805\pi$
$11$	5.93089	1.78823	0.894116	+	0.447836i	$0.147805\pi$
$12$	0	0
$13$	−0.888450	−0.246412	−0.123206	−	0.992381i	$-0.539318\pi$
$13$	−0.888450	−0.246412	−0.123206	+	0.992381i	$0.539318\pi$
$14$	0	0
$15$	−1.04244	−0.269158
$16$	0	0
$17$	4.51668	1.09546	0.547728	−	0.836657i	$-0.315493\pi$
$17$	4.51668	1.09546	0.547728	+	0.836657i	$0.315493\pi$
$18$	0	0
$19$	3.47424	0.797045	0.398522	−	0.917159i	$-0.369523\pi$
$19$	3.47424	0.797045	0.398522	+	0.917159i	$0.369523\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.93089	−0.402619	−0.201310	−	0.979528i	$-0.564520\pi$
$23$	−1.93089	−0.402619	−0.201310	+	0.979528i	$0.564520\pi$
$24$	0	0
$25$	−3.91331	−0.782663
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.38755	−1.55753	−0.778764	−	0.627316i	$-0.784153\pi$
$29$	−8.38755	−1.55753	−0.778764	+	0.627316i	$0.784153\pi$
$30$	0	0
$31$	5.13109	0.921571	0.460786	−	0.887511i	$-0.347568\pi$
$31$	5.13109	0.921571	0.460786	+	0.887511i	$0.347568\pi$
$32$	0	0
$33$	5.93089	1.03244
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	−0.888450	−0.142266
$40$	0	0
$41$	8.39663	1.31133	0.655667	−	0.755050i	$-0.272388\pi$
$41$	8.39663	1.31133	0.655667	+	0.755050i	$0.272388\pi$
$42$	0	0
$43$	−0.743541	−0.113389	−0.0566945	−	0.998392i	$-0.518056\pi$
$43$	−0.743541	−0.113389	−0.0566945	+	0.998392i	$0.518056\pi$
$44$	0	0
$45$	−1.04244	−0.155398
$46$	0	0
$47$	−4.21778	−0.615226	−0.307613	−	0.951512i	$-0.599530\pi$
$47$	−4.21778	−0.615226	−0.307613	+	0.951512i	$0.599530\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.51668	0.632462
$52$	0	0
$53$	−6.91331	−0.949617	−0.474808	−	0.880089i	$-0.657483\pi$
$53$	−6.91331	−0.949617	−0.474808	+	0.880089i	$0.657483\pi$
$54$	0	0
$55$	−6.18262	−0.833664
$56$	0	0
$57$	3.47424	0.460174
$58$	0	0
$59$	5.43908	0.708107	0.354054	−	0.935225i	$-0.384803\pi$
$59$	5.43908	0.708107	0.354054	+	0.935225i	$0.384803\pi$
$60$	0	0
$61$	10.9733	1.40499	0.702496	−	0.711688i	$-0.252069\pi$
$61$	10.9733	1.40499	0.702496	+	0.711688i	$0.252069\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.926159	0.114876
$66$	0	0
$67$	−6.20493	−0.758053	−0.379027	−	0.925386i	$-0.623741\pi$
$67$	−6.20493	−0.758053	−0.379027	+	0.925386i	$0.623741\pi$
$68$	0	0
$69$	−1.93089	−0.232452
$70$	0	0
$71$	3.72596	0.442190	0.221095	−	0.975252i	$-0.429037\pi$
$71$	3.72596	0.442190	0.221095	+	0.975252i	$0.429037\pi$
$72$	0	0
$73$	−3.49910	−0.409539	−0.204769	−	0.978810i	$-0.565644\pi$
$73$	−3.49910	−0.409539	−0.204769	+	0.978810i	$0.565644\pi$
$74$	0	0
$75$	−3.91331	−0.451870
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.6053	1.41821	0.709105	−	0.705103i	$-0.249099\pi$
$79$	12.6053	1.41821	0.709105	+	0.705103i	$0.249099\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.94847	−0.323637	−0.161819	−	0.986821i	$-0.551736\pi$
$83$	−2.94847	−0.323637	−0.161819	+	0.986821i	$0.551736\pi$
$84$	0	0
$85$	−4.70838	−0.510696
$86$	0	0
$87$	−8.38755	−0.899240
$88$	0	0
$89$	−5.00196	−0.530207	−0.265103	−	0.964220i	$-0.585406\pi$
$89$	−5.00196	−0.530207	−0.265103	+	0.964220i	$0.585406\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.13109	0.532069
$94$	0	0
$95$	−3.62169	−0.371578
$96$	0	0
$97$	10.1578	1.03136	0.515682	−	0.856780i	$-0.327539\pi$
$97$	10.1578	1.03136	0.515682	+	0.856780i	$0.327539\pi$
$98$	0	0
$99$	5.93089	0.596077
$100$	0	0
$101$	1.78598	0.177712	0.0888560	−	0.996044i	$-0.471679\pi$
$101$	1.78598	0.177712	0.0888560	+	0.996044i	$0.471679\pi$
$102$	0	0
$103$	−18.5925	−1.83197	−0.915986	−	0.401211i	$-0.868590\pi$
$103$	−18.5925	−1.83197	−0.915986	+	0.401211i	$0.868590\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.23888	0.603135	0.301568	−	0.953445i	$-0.402490\pi$
$107$	6.23888	0.603135	0.301568	+	0.953445i	$0.402490\pi$
$108$	0	0
$109$	−4.74354	−0.454349	−0.227174	−	0.973854i	$-0.572949\pi$
$109$	−4.74354	−0.454349	−0.227174	+	0.973854i	$0.572949\pi$
$110$	0	0
$111$	5.65685	0.536925
$112$	0	0
$113$	−13.3137	−1.25245	−0.626224	−	0.779643i	$-0.715401\pi$
$113$	−13.3137	−1.25245	−0.626224	+	0.779643i	$0.715401\pi$
$114$	0	0
$115$	2.01285	0.187699
$116$	0	0
$117$	−0.888450	−0.0821373
$118$	0	0
$119$	0	0
$120$	0	0
$121$	24.1755	2.19777
$122$	0	0
$123$	8.39663	0.757099
$124$	0	0
$125$	9.29162	0.831068
$126$	0	0
$127$	−0.307985	−0.0273292	−0.0136646	−	0.999907i	$-0.504350\pi$
$127$	−0.307985	−0.0273292	−0.0136646	+	0.999907i	$0.504350\pi$
$128$	0	0
$129$	−0.743541	−0.0654652
$130$	0	0
$131$	9.46139	0.826646	0.413323	−	0.910585i	$-0.364368\pi$
$131$	9.46139	0.826646	0.413323	+	0.910585i	$0.364368\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.04244	−0.0897192
$136$	0	0
$137$	−1.87463	−0.160161	−0.0800803	−	0.996788i	$-0.525518\pi$
$137$	−1.87463	−0.160161	−0.0800803	+	0.996788i	$0.525518\pi$
$138$	0	0
$139$	−7.31371	−0.620341	−0.310170	−	0.950681i	$-0.600386\pi$
$139$	−7.31371	−0.620341	−0.310170	+	0.950681i	$0.600386\pi$
$140$	0	0
$141$	−4.21778	−0.355201
$142$	0	0
$143$	−5.26930	−0.440641
$144$	0	0
$145$	8.74354	0.726112
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.2270	1.82091	0.910454	−	0.413610i	$-0.135732\pi$
$149$	22.2270	1.82091	0.910454	+	0.413610i	$0.135732\pi$
$150$	0	0
$151$	−14.4004	−1.17189	−0.585944	−	0.810352i	$-0.699276\pi$
$151$	−14.4004	−1.17189	−0.585944	+	0.810352i	$0.699276\pi$
$152$	0	0
$153$	4.51668	0.365152
$154$	0	0
$155$	−5.34887	−0.429632
$156$	0	0
$157$	17.9715	1.43428	0.717142	−	0.696927i	$-0.245450\pi$
$157$	17.9715	1.43428	0.717142	+	0.696927i	$0.245450\pi$
$158$	0	0
$159$	−6.91331	−0.548261
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.1755	−1.81524	−0.907622	−	0.419787i	$-0.862105\pi$
$163$	−23.1755	−1.81524	−0.907622	+	0.419787i	$0.862105\pi$
$164$	0	0
$165$	−6.18262	−0.481316
$166$	0	0
$167$	12.9929	1.00542	0.502710	−	0.864455i	$-0.332337\pi$
$167$	12.9929	1.00542	0.502710	+	0.864455i	$0.332337\pi$
$168$	0	0
$169$	−12.2107	−0.939281
$170$	0	0
$171$	3.47424	0.265682
$172$	0	0
$173$	23.1846	1.76269	0.881345	−	0.472472i	$-0.156638\pi$
$173$	23.1846	1.76269	0.881345	+	0.472472i	$0.156638\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.43908	0.408826
$178$	0	0
$179$	24.9366	1.86385	0.931925	−	0.362651i	$-0.118128\pi$
$179$	24.9366	1.86385	0.931925	+	0.362651i	$0.118128\pi$
$180$	0	0
$181$	22.3498	1.66125	0.830625	−	0.556832i	$-0.187983\pi$
$181$	22.3498	1.66125	0.830625	+	0.556832i	$0.187983\pi$
$182$	0	0
$183$	10.9733	0.811172
$184$	0	0
$185$	−5.89695	−0.433552
$186$	0	0
$187$	26.7879	1.95893
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.03967	0.509373	0.254686	−	0.967024i	$-0.418028\pi$
$191$	7.03967	0.509373	0.254686	+	0.967024i	$0.418028\pi$
$192$	0	0
$193$	12.3747	0.890751	0.445375	−	0.895344i	$-0.353070\pi$
$193$	12.3747	0.890751	0.445375	+	0.895344i	$0.353070\pi$
$194$	0	0
$195$	0.926159	0.0663236
$196$	0	0
$197$	13.4262	0.956579	0.478290	−	0.878202i	$-0.341257\pi$
$197$	13.4262	0.956579	0.478290	+	0.878202i	$0.341257\pi$
$198$	0	0
$199$	9.82663	0.696591	0.348296	−	0.937385i	$-0.386761\pi$
$199$	9.82663	0.696591	0.348296	+	0.937385i	$0.386761\pi$
$200$	0	0
$201$	−6.20493	−0.437662
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−8.75301	−0.611337
$206$	0	0
$207$	−1.93089	−0.134206
$208$	0	0
$209$	20.6053	1.42530
$210$	0	0
$211$	−20.1698	−1.38854	−0.694272	−	0.719713i	$-0.744274\pi$
$211$	−20.1698	−1.38854	−0.694272	+	0.719713i	$0.744274\pi$
$212$	0	0
$213$	3.72596	0.255299
$214$	0	0
$215$	0.775099	0.0528613
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.49910	−0.236447
$220$	0	0
$221$	−4.01285	−0.269933
$222$	0	0
$223$	−27.0373	−1.81055	−0.905275	−	0.424826i	$-0.860335\pi$
$223$	−27.0373	−1.81055	−0.905275	+	0.424826i	$0.860335\pi$
$224$	0	0
$225$	−3.91331	−0.260888
$226$	0	0
$227$	8.02231	0.532460	0.266230	−	0.963910i	$-0.414222\pi$
$227$	8.02231	0.532460	0.266230	+	0.963910i	$0.414222\pi$
$228$	0	0
$229$	−12.4920	−0.825493	−0.412747	−	0.910846i	$-0.635430\pi$
$229$	−12.4920	−0.825493	−0.412747	+	0.910846i	$0.635430\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.7013	1.29067	0.645336	−	0.763899i	$-0.276717\pi$
$233$	19.7013	1.29067	0.645336	+	0.763899i	$0.276717\pi$
$234$	0	0
$235$	4.39679	0.286815
$236$	0	0
$237$	12.6053	0.818804
$238$	0	0
$239$	−1.93089	−0.124899	−0.0624496	−	0.998048i	$-0.519891\pi$
$239$	−1.93089	−0.124899	−0.0624496	+	0.998048i	$0.519891\pi$
$240$	0	0
$241$	−15.1026	−0.972846	−0.486423	−	0.873724i	$-0.661699\pi$
$241$	−15.1026	−0.972846	−0.486423	+	0.873724i	$0.661699\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.08669	−0.196401
$248$	0	0
$249$	−2.94847	−0.186852
$250$	0	0
$251$	29.5982	1.86822	0.934111	−	0.356982i	$-0.116194\pi$
$251$	29.5982	1.86822	0.934111	+	0.356982i	$0.116194\pi$
$252$	0	0
$253$	−11.4519	−0.719976
$254$	0	0
$255$	−4.70838	−0.294850
$256$	0	0
$257$	13.8580	0.864440	0.432220	−	0.901768i	$-0.357730\pi$
$257$	13.8580	0.864440	0.432220	+	0.901768i	$0.357730\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.38755	−0.519176
$262$	0	0
$263$	2.78696	0.171851	0.0859255	−	0.996302i	$-0.472615\pi$
$263$	2.78696	0.171851	0.0859255	+	0.996302i	$0.472615\pi$
$264$	0	0
$265$	7.20673	0.442706
$266$	0	0
$267$	−5.00196	−0.306115
$268$	0	0
$269$	−6.44104	−0.392717	−0.196358	−	0.980532i	$-0.562912\pi$
$269$	−6.44104	−0.392717	−0.196358	+	0.980532i	$0.562912\pi$
$270$	0	0
$271$	20.8547	1.26683	0.633415	−	0.773812i	$-0.281652\pi$
$271$	20.8547	1.26683	0.633415	+	0.773812i	$0.281652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−23.2094	−1.39958
$276$	0	0
$277$	−27.0724	−1.62663	−0.813313	−	0.581827i	$-0.802338\pi$
$277$	−27.0724	−1.62663	−0.813313	+	0.581827i	$0.802338\pi$
$278$	0	0
$279$	5.13109	0.307190
$280$	0	0
$281$	6.28450	0.374902	0.187451	−	0.982274i	$-0.439977\pi$
$281$	6.28450	0.374902	0.187451	+	0.982274i	$0.439977\pi$
$282$	0	0
$283$	5.57729	0.331535	0.165768	−	0.986165i	$-0.446990\pi$
$283$	5.57729	0.331535	0.165768	+	0.986165i	$0.446990\pi$
$284$	0	0
$285$	−3.62169	−0.214531
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.40040	0.200023
$290$	0	0
$291$	10.1578	0.595458
$292$	0	0
$293$	−12.6460	−0.738785	−0.369393	−	0.929273i	$-0.620434\pi$
$293$	−12.6460	−0.738785	−0.369393	+	0.929273i	$0.620434\pi$
$294$	0	0
$295$	−5.66993	−0.330116
$296$	0	0
$297$	5.93089	0.344145
$298$	0	0
$299$	1.71550	0.0992101
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.78598	0.102602
$304$	0	0
$305$	−11.4391	−0.655000
$306$	0	0
$307$	−16.2750	−0.928865	−0.464432	−	0.885609i	$-0.653742\pi$
$307$	−16.2750	−0.928865	−0.464432	+	0.885609i	$0.653742\pi$
$308$	0	0
$309$	−18.5925	−1.05769
$310$	0	0
$311$	−28.9929	−1.64404	−0.822018	−	0.569462i	$-0.807152\pi$
$311$	−28.9929	−1.64404	−0.822018	+	0.569462i	$0.807152\pi$
$312$	0	0
$313$	13.5840	0.767812	0.383906	−	0.923372i	$-0.374579\pi$
$313$	13.5840	0.767812	0.383906	+	0.923372i	$0.374579\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.1240	1.35494	0.677469	−	0.735552i	$-0.263077\pi$
$317$	24.1240	1.35494	0.677469	+	0.735552i	$0.263077\pi$
$318$	0	0
$319$	−49.7457	−2.78522
$320$	0	0
$321$	6.23888	0.348220
$322$	0	0
$323$	15.6920	0.873127
$324$	0	0
$325$	3.47678	0.192857
$326$	0	0
$327$	−4.74354	−0.262318
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7.72357	0.424526	0.212263	−	0.977213i	$-0.431917\pi$
$331$	7.72357	0.424526	0.212263	+	0.977213i	$0.431917\pi$
$332$	0	0
$333$	5.65685	0.309994
$334$	0	0
$335$	6.46829	0.353400
$336$	0	0
$337$	1.05153	0.0572803	0.0286401	−	0.999590i	$-0.490882\pi$
$337$	1.05153	0.0572803	0.0286401	+	0.999590i	$0.490882\pi$
$338$	0	0
$339$	−13.3137	−0.723101
$340$	0	0
$341$	30.4320	1.64798
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.01285	0.108368
$346$	0	0
$347$	1.52103	0.0816531	0.0408265	−	0.999166i	$-0.487001\pi$
$347$	1.52103	0.0816531	0.0408265	+	0.999166i	$0.487001\pi$
$348$	0	0
$349$	18.1670	0.972457	0.486229	−	0.873832i	$-0.338372\pi$
$349$	18.1670	0.972457	0.486229	+	0.873832i	$0.338372\pi$
$350$	0	0
$351$	−0.888450	−0.0474220
$352$	0	0
$353$	11.2069	0.596483	0.298241	−	0.954490i	$-0.403600\pi$
$353$	11.2069	0.596483	0.298241	+	0.954490i	$0.403600\pi$
$354$	0	0
$355$	−3.88410	−0.206147
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.44381	0.445647	0.222824	−	0.974859i	$-0.428473\pi$
$359$	8.44381	0.445647	0.222824	+	0.974859i	$0.428473\pi$
$360$	0	0
$361$	−6.92968	−0.364720
$362$	0	0
$363$	24.1755	1.26888
$364$	0	0
$365$	3.64761	0.190925
$366$	0	0
$367$	−23.2880	−1.21562	−0.607812	−	0.794081i	$-0.707953\pi$
$367$	−23.2880	−1.21562	−0.607812	+	0.794081i	$0.707953\pi$
$368$	0	0
$369$	8.39663	0.437111
$370$	0	0
$371$	0	0
$372$	0	0
$373$	35.5502	1.84072	0.920360	−	0.391073i	$-0.127896\pi$
$373$	35.5502	1.84072	0.920360	+	0.391073i	$0.127896\pi$
$374$	0	0
$375$	9.29162	0.479817
$376$	0	0
$377$	7.45192	0.383794
$378$	0	0
$379$	−13.4168	−0.689173	−0.344586	−	0.938755i	$-0.611981\pi$
$379$	−13.4168	−0.689173	−0.344586	+	0.938755i	$0.611981\pi$
$380$	0	0
$381$	−0.307985	−0.0157785
$382$	0	0
$383$	−35.6533	−1.82180	−0.910898	−	0.412632i	$-0.864610\pi$
$383$	−35.6533	−1.82180	−0.910898	+	0.412632i	$0.864610\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.743541	−0.0377963
$388$	0	0
$389$	23.3360	1.18318	0.591592	−	0.806238i	$-0.298500\pi$
$389$	23.3360	1.18318	0.591592	+	0.806238i	$0.298500\pi$
$390$	0	0
$391$	−8.72123	−0.441051
$392$	0	0
$393$	9.46139	0.477264
$394$	0	0
$395$	−13.1403	−0.661162
$396$	0	0
$397$	−9.63199	−0.483416	−0.241708	−	0.970349i	$-0.577708\pi$
$397$	−9.63199	−0.483416	−0.241708	+	0.970349i	$0.577708\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.61245	−0.380148	−0.190074	−	0.981770i	$-0.560873\pi$
$401$	−7.61245	−0.380148	−0.190074	+	0.981770i	$0.560873\pi$
$402$	0	0
$403$	−4.55872	−0.227086
$404$	0	0
$405$	−1.04244	−0.0517994
$406$	0	0
$407$	33.5502	1.66302
$408$	0	0
$409$	34.6564	1.71365	0.856825	−	0.515607i	$-0.172434\pi$
$409$	34.6564	1.71365	0.856825	+	0.515607i	$0.172434\pi$
$410$	0	0
$411$	−1.87463	−0.0924688
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.07362	0.150878
$416$	0	0
$417$	−7.31371	−0.358154
$418$	0	0
$419$	−6.53523	−0.319267	−0.159633	−	0.987176i	$-0.551031\pi$
$419$	−6.53523	−0.319267	−0.159633	+	0.987176i	$0.551031\pi$
$420$	0	0
$421$	−10.4778	−0.510655	−0.255327	−	0.966855i	$-0.582183\pi$
$421$	−10.4778	−0.510655	−0.255327	+	0.966855i	$0.582183\pi$
$422$	0	0
$423$	−4.21778	−0.205075
$424$	0	0
$425$	−17.6752	−0.857372
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.26930	−0.254404
$430$	0	0
$431$	−28.9682	−1.39535	−0.697674	−	0.716415i	$-0.745782\pi$
$431$	−28.9682	−1.39535	−0.697674	+	0.716415i	$0.745782\pi$
$432$	0	0
$433$	−7.31116	−0.351352	−0.175676	−	0.984448i	$-0.556211\pi$
$433$	−7.31116	−0.351352	−0.175676	+	0.984448i	$0.556211\pi$
$434$	0	0
$435$	8.74354	0.419221
$436$	0	0
$437$	−6.70838	−0.320905
$438$	0	0
$439$	−12.6863	−0.605484	−0.302742	−	0.953073i	$-0.597902\pi$
$439$	−12.6863	−0.605484	−0.302742	+	0.953073i	$0.597902\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.6136	−0.789335	−0.394668	−	0.918824i	$-0.629140\pi$
$443$	−16.6136	−0.789335	−0.394668	+	0.918824i	$0.629140\pi$
$444$	0	0
$445$	5.21426	0.247180
$446$	0	0
$447$	22.2270	1.05130
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	49.7995	2.34497
$452$	0	0
$453$	−14.4004	−0.676590
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.72357	−0.454850	−0.227425	−	0.973796i	$-0.573031\pi$
$457$	−9.72357	−0.454850	−0.227425	+	0.973796i	$0.573031\pi$
$458$	0	0
$459$	4.51668	0.210821
$460$	0	0
$461$	−17.9451	−0.835787	−0.417894	−	0.908496i	$-0.637232\pi$
$461$	−17.9451	−0.835787	−0.417894	+	0.908496i	$0.637232\pi$
$462$	0	0
$463$	2.37470	0.110362	0.0551809	−	0.998476i	$-0.482426\pi$
$463$	2.37470	0.110362	0.0551809	+	0.998476i	$0.482426\pi$
$464$	0	0
$465$	−5.34887	−0.248048
$466$	0	0
$467$	15.3617	0.710855	0.355428	−	0.934704i	$-0.384335\pi$
$467$	15.3617	0.710855	0.355428	+	0.934704i	$0.384335\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	17.9715	0.828085
$472$	0	0
$473$	−4.40986	−0.202766
$474$	0	0
$475$	−13.5958	−0.623817
$476$	0	0
$477$	−6.91331	−0.316539
$478$	0	0
$479$	−14.0444	−0.641705	−0.320853	−	0.947129i	$-0.603969\pi$
$479$	−14.0444	−0.641705	−0.320853	+	0.947129i	$0.603969\pi$
$480$	0	0
$481$	−5.02583	−0.229158
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.5889	−0.480816
$486$	0	0
$487$	−30.4004	−1.37757	−0.688787	−	0.724964i	$-0.741856\pi$
$487$	−30.4004	−1.37757	−0.688787	+	0.724964i	$0.741856\pi$
$488$	0	0
$489$	−23.1755	−1.04803
$490$	0	0
$491$	−25.5526	−1.15317	−0.576586	−	0.817036i	$-0.695615\pi$
$491$	−25.5526	−1.15317	−0.576586	+	0.817036i	$0.695615\pi$
$492$	0	0
$493$	−37.8839	−1.70620
$494$	0	0
$495$	−6.18262	−0.277888
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−16.2270	−0.726421	−0.363211	−	0.931707i	$-0.618319\pi$
$499$	−16.2270	−0.726421	−0.363211	+	0.931707i	$0.618319\pi$
$500$	0	0
$501$	12.9929	0.580479
$502$	0	0
$503$	−39.2880	−1.75177	−0.875883	−	0.482523i	$-0.839720\pi$
$503$	−39.2880	−1.75177	−0.875883	+	0.482523i	$0.839720\pi$
$504$	0	0
$505$	−1.86179	−0.0828484
$506$	0	0
$507$	−12.2107	−0.542294
$508$	0	0
$509$	−2.93187	−0.129953	−0.0649763	−	0.997887i	$-0.520697\pi$
$509$	−2.93187	−0.129953	−0.0649763	+	0.997887i	$0.520697\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.47424	0.153391
$514$	0	0
$515$	19.3816	0.854055
$516$	0	0
$517$	−25.0152	−1.10017
$518$	0	0
$519$	23.1846	1.01769
$520$	0	0
$521$	26.3508	1.15445	0.577225	−	0.816585i	$-0.304135\pi$
$521$	26.3508	1.15445	0.577225	+	0.816585i	$0.304135\pi$
$522$	0	0
$523$	40.7048	1.77990	0.889948	−	0.456062i	$-0.150741\pi$
$523$	40.7048	1.77990	0.889948	+	0.456062i	$0.150741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.1755	1.00954
$528$	0	0
$529$	−19.2717	−0.837898
$530$	0	0
$531$	5.43908	0.236036
$532$	0	0
$533$	−7.45999	−0.323128
$534$	0	0
$535$	−6.50367	−0.281178
$536$	0	0
$537$	24.9366	1.07609
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15.3489	−0.659899	−0.329950	−	0.943999i	$-0.607032\pi$
$541$	−15.3489	−0.659899	−0.329950	+	0.943999i	$0.607032\pi$
$542$	0	0
$543$	22.3498	0.959123
$544$	0	0
$545$	4.94487	0.211815
$546$	0	0
$547$	−34.2258	−1.46339	−0.731696	−	0.681631i	$-0.761271\pi$
$547$	−34.2258	−1.46339	−0.731696	+	0.681631i	$0.761271\pi$
$548$	0	0
$549$	10.9733	0.468331
$550$	0	0
$551$	−29.1403	−1.24142
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.89695	−0.250311
$556$	0	0
$557$	−18.6369	−0.789670	−0.394835	−	0.918752i	$-0.629198\pi$
$557$	−18.6369	−0.789670	−0.394835	+	0.918752i	$0.629198\pi$
$558$	0	0
$559$	0.660600	0.0279404
$560$	0	0
$561$	26.7879	1.13099
$562$	0	0
$563$	−2.90047	−0.122240	−0.0611201	−	0.998130i	$-0.519467\pi$
$563$	−2.90047	−0.122240	−0.0611201	+	0.998130i	$0.519467\pi$
$564$	0	0
$565$	13.8788	0.583885
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.0853	−1.13547	−0.567737	−	0.823210i	$-0.692181\pi$
$569$	−27.0853	−1.13547	−0.567737	+	0.823210i	$0.692181\pi$
$570$	0	0
$571$	−3.39467	−0.142063	−0.0710313	−	0.997474i	$-0.522629\pi$
$571$	−3.39467	−0.142063	−0.0710313	+	0.997474i	$0.522629\pi$
$572$	0	0
$573$	7.03967	0.294086
$574$	0	0
$575$	7.55619	0.315115
$576$	0	0
$577$	−22.5545	−0.938958	−0.469479	−	0.882944i	$-0.655558\pi$
$577$	−22.5545	−0.938958	−0.469479	+	0.882944i	$0.655558\pi$
$578$	0	0
$579$	12.3747	0.514275
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−41.0021	−1.69813
$584$	0	0
$585$	0.926159	0.0382920
$586$	0	0
$587$	29.4391	1.21508	0.607540	−	0.794289i	$-0.292156\pi$
$587$	29.4391	1.21508	0.607540	+	0.794289i	$0.292156\pi$
$588$	0	0
$589$	17.8266	0.734533
$590$	0	0
$591$	13.4262	0.552281
$592$	0	0
$593$	−33.9153	−1.39273	−0.696367	−	0.717686i	$-0.745201\pi$
$593$	−33.9153	−1.39273	−0.696367	+	0.717686i	$0.745201\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.82663	0.402177
$598$	0	0
$599$	28.0655	1.14673	0.573363	−	0.819302i	$-0.305639\pi$
$599$	28.0655	1.14673	0.573363	+	0.819302i	$0.305639\pi$
$600$	0	0
$601$	−17.0434	−0.695216	−0.347608	−	0.937640i	$-0.613006\pi$
$601$	−17.0434	−0.695216	−0.347608	+	0.937640i	$0.613006\pi$
$602$	0	0
$603$	−6.20493	−0.252684
$604$	0	0
$605$	−25.2016	−1.02459
$606$	0	0
$607$	36.8639	1.49626	0.748130	−	0.663552i	$-0.230952\pi$
$607$	36.8639	1.49626	0.748130	+	0.663552i	$0.230952\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.74729	0.151599
$612$	0	0
$613$	25.4483	1.02785	0.513924	−	0.857836i	$-0.328191\pi$
$613$	25.4483	1.02785	0.513924	+	0.857836i	$0.328191\pi$
$614$	0	0
$615$	−8.75301	−0.352955
$616$	0	0
$617$	−35.3103	−1.42154	−0.710770	−	0.703424i	$-0.751653\pi$
$617$	−35.3103	−1.42154	−0.710770	+	0.703424i	$0.751653\pi$
$618$	0	0
$619$	40.9599	1.64632	0.823159	−	0.567811i	$-0.192209\pi$
$619$	40.9599	1.64632	0.823159	+	0.567811i	$0.192209\pi$
$620$	0	0
$621$	−1.93089	−0.0774841
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9.88058	0.395223
$626$	0	0
$627$	20.6053	0.822898
$628$	0	0
$629$	25.5502	1.01875
$630$	0	0
$631$	−21.9648	−0.874406	−0.437203	−	0.899363i	$-0.644031\pi$
$631$	−21.9648	−0.874406	−0.437203	+	0.899363i	$0.644031\pi$
$632$	0	0
$633$	−20.1698	−0.801676
$634$	0	0
$635$	0.321057	0.0127407
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.72596	0.147397
$640$	0	0
$641$	−31.9520	−1.26203	−0.631014	−	0.775772i	$-0.717361\pi$
$641$	−31.9520	−1.26203	−0.631014	+	0.775772i	$0.717361\pi$
$642$	0	0
$643$	−41.0501	−1.61886	−0.809430	−	0.587217i	$-0.800223\pi$
$643$	−41.0501	−1.61886	−0.809430	+	0.587217i	$0.800223\pi$
$644$	0	0
$645$	0.775099	0.0305195
$646$	0	0
$647$	9.67917	0.380527	0.190264	−	0.981733i	$-0.439066\pi$
$647$	9.67917	0.380527	0.190264	+	0.981733i	$0.439066\pi$
$648$	0	0
$649$	32.2586	1.26626
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.5982	−0.688671	−0.344336	−	0.938847i	$-0.611896\pi$
$653$	−17.5982	−0.688671	−0.344336	+	0.938847i	$0.611896\pi$
$654$	0	0
$655$	−9.86296	−0.385378
$656$	0	0
$657$	−3.49910	−0.136513
$658$	0	0
$659$	4.10427	0.159880	0.0799398	−	0.996800i	$-0.474527\pi$
$659$	4.10427	0.159880	0.0799398	+	0.996800i	$0.474527\pi$
$660$	0	0
$661$	47.0124	1.82857	0.914286	−	0.405070i	$-0.132753\pi$
$661$	47.0124	1.82857	0.914286	+	0.405070i	$0.132753\pi$
$662$	0	0
$663$	−4.01285	−0.155846
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.1955	0.627091
$668$	0	0
$669$	−27.0373	−1.04532
$670$	0	0
$671$	65.0817	2.51245
$672$	0	0
$673$	21.7587	0.838738	0.419369	−	0.907816i	$-0.362251\pi$
$673$	21.7587	0.838738	0.419369	+	0.907816i	$0.362251\pi$
$674$	0	0
$675$	−3.91331	−0.150623
$676$	0	0
$677$	−23.6829	−0.910209	−0.455105	−	0.890438i	$-0.650398\pi$
$677$	−23.6829	−0.910209	−0.455105	+	0.890438i	$0.650398\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.02231	0.307416
$682$	0	0
$683$	32.0655	1.22695	0.613476	−	0.789713i	$-0.289771\pi$
$683$	32.0655	1.22695	0.613476	+	0.789713i	$0.289771\pi$
$684$	0	0
$685$	1.95420	0.0746660
$686$	0	0
$687$	−12.4920	−0.476599
$688$	0	0
$689$	6.14214	0.233997
$690$	0	0
$691$	−14.8782	−0.565992	−0.282996	−	0.959121i	$-0.591328\pi$
$691$	−14.8782	−0.565992	−0.282996	+	0.959121i	$0.591328\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.62412	0.289199
$696$	0	0
$697$	37.9249	1.43651
$698$	0	0
$699$	19.7013	0.745170
$700$	0	0
$701$	31.0150	1.17142	0.585710	−	0.810521i	$-0.300816\pi$
$701$	31.0150	1.17142	0.585710	+	0.810521i	$0.300816\pi$
$702$	0	0
$703$	19.6533	0.741236
$704$	0	0
$705$	4.39679	0.165593
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−51.7105	−1.94203	−0.971014	−	0.239021i	$-0.923173\pi$
$709$	−51.7105	−1.94203	−0.971014	+	0.239021i	$0.923173\pi$
$710$	0	0
$711$	12.6053	0.472737
$712$	0	0
$713$	−9.90759	−0.371042
$714$	0	0
$715$	5.49295	0.205425
$716$	0	0
$717$	−1.93089	−0.0721105
$718$	0	0
$719$	19.7493	0.736523	0.368262	−	0.929722i	$-0.379953\pi$
$719$	19.7493	0.736523	0.368262	+	0.929722i	$0.379953\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−15.1026	−0.561673
$724$	0	0
$725$	32.8231	1.21902
$726$	0	0
$727$	−51.3230	−1.90346	−0.951731	−	0.306932i	$-0.900698\pi$
$727$	−51.3230	−1.90346	−0.951731	+	0.306932i	$0.900698\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.35834	−0.124213
$732$	0	0
$733$	−29.1230	−1.07568	−0.537841	−	0.843046i	$-0.680760\pi$
$733$	−29.1230	−1.07568	−0.537841	+	0.843046i	$0.680760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.8008	−1.35557
$738$	0	0
$739$	9.27855	0.341317	0.170658	−	0.985330i	$-0.445411\pi$
$739$	9.27855	0.341317	0.170658	+	0.985330i	$0.445411\pi$
$740$	0	0
$741$	−3.08669	−0.113392
$742$	0	0
$743$	26.5935	0.975620	0.487810	−	0.872950i	$-0.337796\pi$
$743$	26.5935	0.975620	0.487810	+	0.872950i	$0.337796\pi$
$744$	0	0
$745$	−23.1704	−0.848898
$746$	0	0
$747$	−2.94847	−0.107879
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	0	0
$753$	29.5982	1.07862
$754$	0	0
$755$	15.0116	0.546328
$756$	0	0
$757$	17.8839	0.650001	0.325000	−	0.945714i	$-0.394636\pi$
$757$	17.8839	0.650001	0.325000	+	0.945714i	$0.394636\pi$
$758$	0	0
$759$	−11.4519	−0.415678
$760$	0	0
$761$	−49.5263	−1.79533	−0.897664	−	0.440681i	$-0.854737\pi$
$761$	−49.5263	−1.79533	−0.897664	+	0.440681i	$0.854737\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.70838	−0.170232
$766$	0	0
$767$	−4.83235	−0.174486
$768$	0	0
$769$	−45.0675	−1.62517	−0.812587	−	0.582840i	$-0.801942\pi$
$769$	−45.0675	−1.62517	−0.812587	+	0.582840i	$0.801942\pi$
$770$	0	0
$771$	13.8580	0.499085
$772$	0	0
$773$	4.45051	0.160074	0.0800368	−	0.996792i	$-0.474496\pi$
$773$	4.45051	0.160074	0.0800368	+	0.996792i	$0.474496\pi$
$774$	0	0
$775$	−20.0796	−0.721279
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.1719	1.04519
$780$	0	0
$781$	22.0983	0.790739
$782$	0	0
$783$	−8.38755	−0.299747
$784$	0	0
$785$	−18.7343	−0.668656
$786$	0	0
$787$	18.1919	0.648470	0.324235	−	0.945977i	$-0.394893\pi$
$787$	18.1919	0.648470	0.324235	+	0.945977i	$0.394893\pi$
$788$	0	0
$789$	2.78696	0.0992183
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.74926	−0.346207
$794$	0	0
$795$	7.20673	0.255597
$796$	0	0
$797$	−23.8005	−0.843059	−0.421529	−	0.906815i	$-0.638507\pi$
$797$	−23.8005	−0.843059	−0.421529	+	0.906815i	$0.638507\pi$
$798$	0	0
$799$	−19.0504	−0.673953
$800$	0	0
$801$	−5.00196	−0.176736
$802$	0	0
$803$	−20.7528	−0.732350
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.44104	−0.226735
$808$	0	0
$809$	51.8897	1.82435	0.912173	−	0.409805i	$-0.134403\pi$
$809$	51.8897	1.82435	0.912173	+	0.409805i	$0.134403\pi$
$810$	0	0
$811$	−13.6462	−0.479183	−0.239592	−	0.970874i	$-0.577014\pi$
$811$	−13.6462	−0.479183	−0.239592	+	0.970874i	$0.577014\pi$
$812$	0	0
$813$	20.8547	0.731405
$814$	0	0
$815$	24.1591	0.846257
$816$	0	0
$817$	−2.58324	−0.0903761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.69576	0.163883	0.0819416	−	0.996637i	$-0.473888\pi$
$821$	4.69576	0.163883	0.0819416	+	0.996637i	$0.473888\pi$
$822$	0	0
$823$	−15.8045	−0.550912	−0.275456	−	0.961314i	$-0.588829\pi$
$823$	−15.8045	−0.550912	−0.275456	+	0.961314i	$0.588829\pi$
$824$	0	0
$825$	−23.2094	−0.808049
$826$	0	0
$827$	32.8221	1.14134	0.570668	−	0.821181i	$-0.306684\pi$
$827$	32.8221	1.14134	0.570668	+	0.821181i	$0.306684\pi$
$828$	0	0
$829$	3.71688	0.129092	0.0645462	−	0.997915i	$-0.479440\pi$
$829$	3.71688	0.129092	0.0645462	+	0.997915i	$0.479440\pi$
$830$	0	0
$831$	−27.0724	−0.939133
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.5443	−0.468721
$836$	0	0
$837$	5.13109	0.177356
$838$	0	0
$839$	3.16625	0.109311	0.0546556	−	0.998505i	$-0.482594\pi$
$839$	3.16625	0.109311	0.0546556	+	0.998505i	$0.482594\pi$
$840$	0	0
$841$	41.3510	1.42590
$842$	0	0
$843$	6.28450	0.216450
$844$	0	0
$845$	12.7289	0.437888
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.57729	0.191412
$850$	0	0
$851$	−10.9228	−0.374428
$852$	0	0
$853$	8.30404	0.284325	0.142162	−	0.989843i	$-0.454594\pi$
$853$	8.30404	0.284325	0.142162	+	0.989843i	$0.454594\pi$
$854$	0	0
$855$	−3.62169	−0.123859
$856$	0	0
$857$	14.3117	0.488880	0.244440	−	0.969664i	$-0.421396\pi$
$857$	14.3117	0.488880	0.244440	+	0.969664i	$0.421396\pi$
$858$	0	0
$859$	34.1463	1.16506	0.582528	−	0.812811i	$-0.302064\pi$
$859$	34.1463	1.16506	0.582528	+	0.812811i	$0.302064\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.0397	1.32893	0.664463	−	0.747321i	$-0.268660\pi$
$863$	39.0397	1.32893	0.664463	+	0.747321i	$0.268660\pi$
$864$	0	0
$865$	−24.1686	−0.821757
$866$	0	0
$867$	3.40040	0.115483
$868$	0	0
$869$	74.7609	2.53609
$870$	0	0
$871$	5.51277	0.186793
$872$	0	0
$873$	10.1578	0.343788
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.4225	−0.419478	−0.209739	−	0.977757i	$-0.567261\pi$
$877$	−12.4225	−0.419478	−0.209739	+	0.977757i	$0.567261\pi$
$878$	0	0
$879$	−12.6460	−0.426538
$880$	0	0
$881$	−32.4282	−1.09253	−0.546267	−	0.837611i	$-0.683952\pi$
$881$	−32.4282	−1.09253	−0.546267	+	0.837611i	$0.683952\pi$
$882$	0	0
$883$	21.8475	0.735228	0.367614	−	0.929978i	$-0.380175\pi$
$883$	21.8475	0.735228	0.367614	+	0.929978i	$0.380175\pi$
$884$	0	0
$885$	−5.66993	−0.190592
$886$	0	0
$887$	−54.2035	−1.81998	−0.909988	−	0.414634i	$-0.863910\pi$
$887$	−54.2035	−1.81998	−0.909988	+	0.414634i	$0.863910\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.93089	0.198692
$892$	0	0
$893$	−14.6536	−0.490363
$894$	0	0
$895$	−25.9950	−0.868917
$896$	0	0
$897$	1.71550	0.0572790
$898$	0	0
$899$	−43.0373	−1.43537
$900$	0	0
$901$	−31.2252	−1.04026
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.2984	−0.774466
$906$	0	0
$907$	−19.8172	−0.658018	−0.329009	−	0.944327i	$-0.606715\pi$
$907$	−19.8172	−0.658018	−0.329009	+	0.944327i	$0.606715\pi$
$908$	0	0
$909$	1.78598	0.0592374
$910$	0	0
$911$	−32.5583	−1.07870	−0.539352	−	0.842080i	$-0.681331\pi$
$911$	−32.5583	−1.07870	−0.539352	+	0.842080i	$0.681331\pi$
$912$	0	0
$913$	−17.4871	−0.578738
$914$	0	0
$915$	−11.4391	−0.378164
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.95420	−0.0644630	−0.0322315	−	0.999480i	$-0.510261\pi$
$919$	−1.95420	−0.0644630	−0.0322315	+	0.999480i	$0.510261\pi$
$920$	0	0
$921$	−16.2750	−0.536280
$922$	0	0
$923$	−3.31033	−0.108961
$924$	0	0
$925$	−22.1370	−0.727861
$926$	0	0
$927$	−18.5925	−0.610657
$928$	0	0
$929$	−23.2641	−0.763272	−0.381636	−	0.924313i	$-0.624639\pi$
$929$	−23.2641	−0.763272	−0.381636	+	0.924313i	$0.624639\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−28.9929	−0.949184
$934$	0	0
$935$	−27.9249	−0.913242
$936$	0	0
$937$	−25.1244	−0.820778	−0.410389	−	0.911911i	$-0.634607\pi$
$937$	−25.1244	−0.820778	−0.410389	+	0.911911i	$0.634607\pi$
$938$	0	0
$939$	13.5840	0.443297
$940$	0	0
$941$	4.67721	0.152473	0.0762363	−	0.997090i	$-0.475710\pi$
$941$	4.67721	0.152473	0.0762363	+	0.997090i	$0.475710\pi$
$942$	0	0
$943$	−16.2130	−0.527968
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.24460	−0.0404441	−0.0202221	−	0.999796i	$-0.506437\pi$
$947$	−1.24460	−0.0404441	−0.0202221	+	0.999796i	$0.506437\pi$
$948$	0	0
$949$	3.10878	0.100915
$950$	0	0
$951$	24.1240	0.782273
$952$	0	0
$953$	−33.0373	−1.07018	−0.535091	−	0.844794i	$-0.679723\pi$
$953$	−33.0373	−1.07018	−0.535091	+	0.844794i	$0.679723\pi$
$954$	0	0
$955$	−7.33845	−0.237467
$956$	0	0
$957$	−49.7457	−1.60805
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4.67190	−0.150707
$962$	0	0
$963$	6.23888	0.201045
$964$	0	0
$965$	−12.8999	−0.415263
$966$	0	0
$967$	13.9799	0.449563	0.224781	−	0.974409i	$-0.427833\pi$
$967$	13.9799	0.449563	0.224781	+	0.974409i	$0.427833\pi$
$968$	0	0
$969$	15.6920	0.504100
$970$	0	0
$971$	5.48708	0.176089	0.0880444	−	0.996117i	$-0.471938\pi$
$971$	5.48708	0.176089	0.0880444	+	0.996117i	$0.471938\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.47678	0.111346
$976$	0	0
$977$	−0.324434	−0.0103796	−0.00518978	−	0.999987i	$-0.501652\pi$
$977$	−0.324434	−0.0103796	−0.00518978	+	0.999987i	$0.501652\pi$
$978$	0	0
$979$	−29.6661	−0.948133
$980$	0	0
$981$	−4.74354	−0.151450
$982$	0	0
$983$	−22.5129	−0.718051	−0.359025	−	0.933328i	$-0.616891\pi$
$983$	−22.5129	−0.718051	−0.359025	+	0.933328i	$0.616891\pi$
$984$	0	0
$985$	−13.9961	−0.445952
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.43570	0.0456526
$990$	0	0
$991$	−43.1649	−1.37118	−0.685588	−	0.727989i	$-0.740455\pi$
$991$	−43.1649	−1.37118	−0.685588	+	0.727989i	$0.740455\pi$
$992$	0	0
$993$	7.72357	0.245100
$994$	0	0
$995$	−10.2437	−0.324747
$996$	0	0
$997$	−24.7743	−0.784609	−0.392305	−	0.919835i	$-0.628322\pi$
$997$	−24.7743	−0.784609	−0.392305	+	0.919835i	$0.628322\pi$
$998$	0	0
$999$	5.65685	0.178975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.a.bz.1.2	yes	4
4.3	odd	2		4704.2.a.bx.1.2	yes	4
7.6	odd	2		4704.2.a.bw.1.3	✓	4
8.3	odd	2		9408.2.a.em.1.3		4
8.5	even	2		9408.2.a.ek.1.3		4
28.27	even	2		4704.2.a.by.1.3	yes	4
56.13	odd	2		9408.2.a.en.1.2		4
56.27	even	2		9408.2.a.el.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bw.1.3	✓	4	7.6	odd	2
4704.2.a.bx.1.2	yes	4	4.3	odd	2
4704.2.a.by.1.3	yes	4	28.27	even	2
4704.2.a.bz.1.2	yes	4	1.1	even	1	trivial
9408.2.a.ek.1.3		4	8.5	even	2
9408.2.a.el.1.2		4	56.27	even	2
9408.2.a.em.1.3		4	8.3	odd	2
9408.2.a.en.1.2		4	56.13	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.04244 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.04244 q^{5} +1.00000 q^{9} +5.93089 q^{11} -0.888450 q^{13} -1.04244 q^{15} +4.51668 q^{17} +3.47424 q^{19} -1.93089 q^{23} -3.91331 q^{25} +1.00000 q^{27} -8.38755 q^{29} +5.13109 q^{31} +5.93089 q^{33} +5.65685 q^{37} -0.888450 q^{39} +8.39663 q^{41} -0.743541 q^{43} -1.04244 q^{45} -4.21778 q^{47} +4.51668 q^{51} -6.91331 q^{53} -6.18262 q^{55} +3.47424 q^{57} +5.43908 q^{59} +10.9733 q^{61} +0.926159 q^{65} -6.20493 q^{67} -1.93089 q^{69} +3.72596 q^{71} -3.49910 q^{73} -3.91331 q^{75} +12.6053 q^{79} +1.00000 q^{81} -2.94847 q^{83} -4.70838 q^{85} -8.38755 q^{87} -5.00196 q^{89} +5.13109 q^{93} -3.62169 q^{95} +10.1578 q^{97} +5.93089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{39} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 16 q^{47} + 4 q^{51} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{73} + 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} + 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 4 * q^17 + 8 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 16 * q^47 + 4 * q^51 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 12 * q^69 + 12 * q^71 + 8 * q^73 + 12 * q^75 + 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 + 24 * q^95 + 40 * q^97 + 4 * q^99

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