LMFDB - Embedded newform 9800.2.a.cs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.87996$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87996 q^{3} +0.534253 q^{9} +O(q^{10})$	$q-1.87996 q^{3} +0.534253 q^{9} -3.29417 q^{11} +4.19292 q^{13} -1.43737 q^{17} -1.24445 q^{19} +0.272828 q^{23} +4.63551 q^{27} -2.36268 q^{29} +3.72717 q^{31} +6.19292 q^{33} -0.169761 q^{37} -7.88252 q^{39} -11.6630 q^{41} +10.1458 q^{43} +3.12441 q^{47} +2.70220 q^{51} -9.24701 q^{53} +2.33952 q^{57} -9.07107 q^{59} -7.27102 q^{61} +13.1439 q^{67} -0.512907 q^{69} +6.87474 q^{71} -15.2496 q^{73} +14.9510 q^{79} -10.3173 q^{81} +0.167199 q^{83} +4.44175 q^{87} -3.09869 q^{89} -7.00694 q^{93} +6.60894 q^{97} -1.75992 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 6 * q^9	$ 4 q + 2 q^{3} + 6 q^{9} + 2 q^{11} + 10 q^{13} + 6 q^{17} + 4 q^{23} + 14 q^{27} - 2 q^{29} + 12 q^{31} + 18 q^{33} + 14 q^{39} - 12 q^{41} + 8 q^{43} - 2 q^{47} + 2 q^{51} + 4 q^{53} + 8 q^{57} - 8 q^{59} - 20 q^{61} + 8 q^{67} - 24 q^{69} + 4 q^{71} + 16 q^{73} + 22 q^{79} - 20 q^{81} + 36 q^{83} - 18 q^{87} - 40 q^{89} + 32 q^{93} + 26 q^{97} + 12 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^9 + 2 * q^11 + 10 * q^13 + 6 * q^17 + 4 * q^23 + 14 * q^27 - 2 * q^29 + 12 * q^31 + 18 * q^33 + 14 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^47 + 2 * q^51 + 4 * q^53 + 8 * q^57 - 8 * q^59 - 20 * q^61 + 8 * q^67 - 24 * q^69 + 4 * q^71 + 16 * q^73 + 22 * q^79 - 20 * q^81 + 36 * q^83 - 18 * q^87 - 40 * q^89 + 32 * q^93 + 26 * q^97 + 12 * 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
$3$	−1.87996	−1.08540	−0.542698	−	0.839928i	$-0.682597\pi$
$3$	−1.87996	−1.08540	−0.542698	+	0.839928i	$0.682597\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.534253	0.178084
$10$	0	0
$11$	−3.29417	−0.993231	−0.496615	−	0.867971i	$-0.665424\pi$
$11$	−3.29417	−0.993231	−0.496615	+	0.867971i	$0.665424\pi$
$12$	0	0
$13$	4.19292	1.16291	0.581453	−	0.813580i	$-0.302484\pi$
$13$	4.19292	1.16291	0.581453	+	0.813580i	$0.302484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.43737	−0.348614	−0.174307	−	0.984691i	$-0.555768\pi$
$17$	−1.43737	−0.348614	−0.174307	+	0.984691i	$0.555768\pi$
$18$	0	0
$19$	−1.24445	−0.285497	−0.142748	−	0.989759i	$-0.545594\pi$
$19$	−1.24445	−0.285497	−0.142748	+	0.989759i	$0.545594\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.272828	0.0568887	0.0284443	−	0.999595i	$-0.490945\pi$
$23$	0.272828	0.0568887	0.0284443	+	0.999595i	$0.490945\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.63551	0.892104
$28$	0	0
$29$	−2.36268	−0.438739	−0.219369	−	0.975642i	$-0.570400\pi$
$29$	−2.36268	−0.438739	−0.219369	+	0.975642i	$0.570400\pi$
$30$	0	0
$31$	3.72717	0.669420	0.334710	−	0.942321i	$-0.391362\pi$
$31$	3.72717	0.669420	0.334710	+	0.942321i	$0.391362\pi$
$32$	0	0
$33$	6.19292	1.07805
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.169761	−0.0279085	−0.0139543	−	0.999903i	$-0.504442\pi$
$37$	−0.169761	−0.0279085	−0.0139543	+	0.999903i	$0.504442\pi$
$38$	0	0
$39$	−7.88252	−1.26221
$40$	0	0
$41$	−11.6630	−1.82146	−0.910730	−	0.413001i	$-0.864480\pi$
$41$	−11.6630	−1.82146	−0.910730	+	0.413001i	$0.864480\pi$
$42$	0	0
$43$	10.1458	1.54721	0.773607	−	0.633666i	$-0.218451\pi$
$43$	10.1458	1.54721	0.773607	+	0.633666i	$0.218451\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.12441	0.455743	0.227871	−	0.973691i	$-0.426823\pi$
$47$	3.12441	0.455743	0.227871	+	0.973691i	$0.426823\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.70220	0.378384
$52$	0	0
$53$	−9.24701	−1.27018	−0.635088	−	0.772440i	$-0.719036\pi$
$53$	−9.24701	−1.27018	−0.635088	+	0.772440i	$0.719036\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.33952	0.309877
$58$	0	0
$59$	−9.07107	−1.18095	−0.590476	−	0.807055i	$-0.701060\pi$
$59$	−9.07107	−1.18095	−0.590476	+	0.807055i	$0.701060\pi$
$60$	0	0
$61$	−7.27102	−0.930958	−0.465479	−	0.885059i	$-0.654118\pi$
$61$	−7.27102	−0.930958	−0.465479	+	0.885059i	$0.654118\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1439	1.60579	0.802894	−	0.596121i	$-0.203292\pi$
$67$	13.1439	1.60579	0.802894	+	0.596121i	$0.203292\pi$
$68$	0	0
$69$	−0.512907	−0.0617467
$70$	0	0
$71$	6.87474	0.815882	0.407941	−	0.913008i	$-0.366247\pi$
$71$	6.87474	0.815882	0.407941	+	0.913008i	$0.366247\pi$
$72$	0	0
$73$	−15.2496	−1.78483	−0.892414	−	0.451218i	$-0.850990\pi$
$73$	−15.2496	−1.78483	−0.892414	+	0.451218i	$0.850990\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.9510	1.68212	0.841061	−	0.540940i	$-0.181931\pi$
$79$	14.9510	1.68212	0.841061	+	0.540940i	$0.181931\pi$
$80$	0	0
$81$	−10.3173	−1.14637
$82$	0	0
$83$	0.167199	0.0183524	0.00917622	−	0.999958i	$-0.497079\pi$
$83$	0.167199	0.0183524	0.00917622	+	0.999958i	$0.497079\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.44175	0.476205
$88$	0	0
$89$	−3.09869	−0.328461	−0.164230	−	0.986422i	$-0.552514\pi$
$89$	−3.09869	−0.328461	−0.164230	+	0.986422i	$0.552514\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.00694	−0.726585
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.60894	0.671037	0.335518	−	0.942034i	$-0.391089\pi$
$97$	6.60894	0.671037	0.335518	+	0.942034i	$0.391089\pi$
$98$	0	0
$99$	−1.75992	−0.176879
$100$	0	0
$101$	−18.8372	−1.87437	−0.937185	−	0.348834i	$-0.886578\pi$
$101$	−18.8372	−1.87437	−0.937185	+	0.348834i	$0.886578\pi$
$102$	0	0
$103$	−1.46756	−0.144603	−0.0723014	−	0.997383i	$-0.523034\pi$
$103$	−1.46756	−0.144603	−0.0723014	+	0.997383i	$0.523034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.5625	−1.31114	−0.655570	−	0.755135i	$-0.727572\pi$
$107$	−13.5625	−1.31114	−0.655570	+	0.755135i	$0.727572\pi$
$108$	0	0
$109$	14.8140	1.41893	0.709463	−	0.704743i	$-0.248938\pi$
$109$	14.8140	1.41893	0.709463	+	0.704743i	$0.248938\pi$
$110$	0	0
$111$	0.319144	0.0302918
$112$	0	0
$113$	13.1903	1.24084	0.620418	−	0.784271i	$-0.286963\pi$
$113$	13.1903	1.24084	0.620418	+	0.784271i	$0.286963\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.24008	0.207095
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.148415	−0.0134923
$122$	0	0
$123$	21.9261	1.97701
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.86118	−0.608831	−0.304416	−	0.952539i	$-0.598461\pi$
$127$	−6.86118	−0.608831	−0.304416	+	0.952539i	$0.598461\pi$
$128$	0	0
$129$	−19.0736	−1.67934
$130$	0	0
$131$	0.345708	0.0302047	0.0151023	−	0.999886i	$-0.495193\pi$
$131$	0.345708	0.0302047	0.0151023	+	0.999886i	$0.495193\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.3137	−0.966595	−0.483298	−	0.875456i	$-0.660561\pi$
$137$	−11.3137	−0.966595	−0.483298	+	0.875456i	$0.660561\pi$
$138$	0	0
$139$	−2.68885	−0.228066	−0.114033	−	0.993477i	$-0.536377\pi$
$139$	−2.68885	−0.228066	−0.114033	+	0.993477i	$0.536377\pi$
$140$	0	0
$141$	−5.87377	−0.494661
$142$	0	0
$143$	−13.8122	−1.15503
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.1340	1.07598	0.537990	−	0.842951i	$-0.319184\pi$
$149$	13.1340	1.07598	0.537990	+	0.842951i	$0.319184\pi$
$150$	0	0
$151$	3.01140	0.245065	0.122532	−	0.992465i	$-0.460898\pi$
$151$	3.01140	0.245065	0.122532	+	0.992465i	$0.460898\pi$
$152$	0	0
$153$	−0.767920	−0.0620826
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.4631	1.55332	0.776662	−	0.629918i	$-0.216911\pi$
$157$	19.4631	1.55332	0.776662	+	0.629918i	$0.216911\pi$
$158$	0	0
$159$	17.3840	1.37864
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.4922	−1.05679	−0.528396	−	0.848998i	$-0.677206\pi$
$163$	−13.4922	−1.05679	−0.528396	+	0.848998i	$0.677206\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.3011	0.951889	0.475944	−	0.879475i	$-0.342106\pi$
$167$	12.3011	0.951889	0.475944	+	0.879475i	$0.342106\pi$
$168$	0	0
$169$	4.58057	0.352351
$170$	0	0
$171$	−0.664852	−0.0508425
$172$	0	0
$173$	2.48975	0.189292	0.0946461	−	0.995511i	$-0.469828\pi$
$173$	2.48975	0.189292	0.0946461	+	0.995511i	$0.469828\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.0533	1.28180
$178$	0	0
$179$	15.1340	1.13117	0.565584	−	0.824690i	$-0.308651\pi$
$179$	15.1340	1.13117	0.565584	+	0.824690i	$0.308651\pi$
$180$	0	0
$181$	−11.0637	−0.822357	−0.411179	−	0.911555i	$-0.634883\pi$
$181$	−11.0637	−0.822357	−0.411179	+	0.911555i	$0.634883\pi$
$182$	0	0
$183$	13.6692	1.01046
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.73495	0.346254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.4517	−1.62455	−0.812274	−	0.583277i	$-0.801770\pi$
$191$	−22.4517	−1.62455	−0.812274	+	0.583277i	$0.801770\pi$
$192$	0	0
$193$	−3.11482	−0.224210	−0.112105	−	0.993696i	$-0.535759\pi$
$193$	−3.11482	−0.224210	−0.112105	+	0.993696i	$0.535759\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.38765	−0.0988659	−0.0494330	−	0.998777i	$-0.515741\pi$
$197$	−1.38765	−0.0988659	−0.0494330	+	0.998777i	$0.515741\pi$
$198$	0	0
$199$	0.413463	0.0293096	0.0146548	−	0.999893i	$-0.495335\pi$
$199$	0.413463	0.0293096	0.0146548	+	0.999893i	$0.495335\pi$
$200$	0	0
$201$	−24.7101	−1.74292
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.145759	0.0101310
$208$	0	0
$209$	4.09944	0.283564
$210$	0	0
$211$	24.6203	1.69493	0.847464	−	0.530853i	$-0.178128\pi$
$211$	24.6203	1.69493	0.847464	+	0.530853i	$0.178128\pi$
$212$	0	0
$213$	−12.9242	−0.885555
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	28.6686	1.93724
$220$	0	0
$221$	−6.02678	−0.405405
$222$	0	0
$223$	−17.9065	−1.19911	−0.599555	−	0.800334i	$-0.704656\pi$
$223$	−17.9065	−1.19911	−0.599555	+	0.800334i	$0.704656\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.5486	−0.832878	−0.416439	−	0.909164i	$-0.636722\pi$
$227$	−12.5486	−0.832878	−0.416439	+	0.909164i	$0.636722\pi$
$228$	0	0
$229$	15.8354	1.04643	0.523215	−	0.852201i	$-0.324732\pi$
$229$	15.8354	1.04643	0.523215	+	0.852201i	$0.324732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.3792	1.53162	0.765811	−	0.643065i	$-0.222338\pi$
$233$	23.3792	1.53162	0.765811	+	0.643065i	$0.222338\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−28.1073	−1.82577
$238$	0	0
$239$	20.9135	1.35278	0.676390	−	0.736544i	$-0.263544\pi$
$239$	20.9135	1.35278	0.676390	+	0.736544i	$0.263544\pi$
$240$	0	0
$241$	3.35746	0.216273	0.108137	−	0.994136i	$-0.465512\pi$
$241$	3.35746	0.216273	0.108137	+	0.994136i	$0.465512\pi$
$242$	0	0
$243$	5.48966	0.352162
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.21789	−0.332006
$248$	0	0
$249$	−0.314327	−0.0199197
$250$	0	0
$251$	−16.5717	−1.04600	−0.522999	−	0.852333i	$-0.675187\pi$
$251$	−16.5717	−1.04600	−0.522999	+	0.852333i	$0.675187\pi$
$252$	0	0
$253$	−0.898744	−0.0565036
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.1414	0.819737	0.409869	−	0.912145i	$-0.365575\pi$
$257$	13.1414	0.819737	0.409869	+	0.912145i	$0.365575\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.26227	−0.0781324
$262$	0	0
$263$	3.75480	0.231531	0.115765	−	0.993277i	$-0.463068\pi$
$263$	3.75480	0.231531	0.115765	+	0.993277i	$0.463068\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.82542	0.356510
$268$	0	0
$269$	−16.4952	−1.00573	−0.502866	−	0.864365i	$-0.667721\pi$
$269$	−16.4952	−1.00573	−0.502866	+	0.864365i	$0.667721\pi$
$270$	0	0
$271$	22.5420	1.36933	0.684665	−	0.728857i	$-0.259948\pi$
$271$	22.5420	1.36933	0.684665	+	0.728857i	$0.259948\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.9982	0.901154	0.450577	−	0.892738i	$-0.351218\pi$
$277$	14.9982	0.901154	0.450577	+	0.892738i	$0.351218\pi$
$278$	0	0
$279$	1.99125	0.119213
$280$	0	0
$281$	−28.0283	−1.67203	−0.836014	−	0.548709i	$-0.815120\pi$
$281$	−28.0283	−1.67203	−0.836014	+	0.548709i	$0.815120\pi$
$282$	0	0
$283$	−26.1715	−1.55573	−0.777866	−	0.628430i	$-0.783698\pi$
$283$	−26.1715	−1.55573	−0.777866	+	0.628430i	$0.783698\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.9340	−0.878468
$290$	0	0
$291$	−12.4246	−0.728340
$292$	0	0
$293$	15.6061	0.911716	0.455858	−	0.890052i	$-0.349332\pi$
$293$	15.6061	0.911716	0.455858	+	0.890052i	$0.349332\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.2702	−0.886065
$298$	0	0
$299$	1.14395	0.0661562
$300$	0	0
$301$	0	0
$302$	0	0
$303$	35.4132	2.03443
$304$	0	0
$305$	0	0
$306$	0	0
$307$	21.3055	1.21597	0.607984	−	0.793949i	$-0.291978\pi$
$307$	21.3055	1.21597	0.607984	+	0.793949i	$0.291978\pi$
$308$	0	0
$309$	2.75895	0.156951
$310$	0	0
$311$	14.8799	0.843760	0.421880	−	0.906652i	$-0.361370\pi$
$311$	14.8799	0.843760	0.421880	+	0.906652i	$0.361370\pi$
$312$	0	0
$313$	2.16273	0.122245	0.0611224	−	0.998130i	$-0.480532\pi$
$313$	2.16273	0.122245	0.0611224	+	0.998130i	$0.480532\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.4595	1.20528	0.602642	−	0.798012i	$-0.294115\pi$
$317$	21.4595	1.20528	0.602642	+	0.798012i	$0.294115\pi$
$318$	0	0
$319$	7.78308	0.435769
$320$	0	0
$321$	25.4970	1.42311
$322$	0	0
$323$	1.78874	0.0995282
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−27.8498	−1.54010
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.6692	−0.641399	−0.320699	−	0.947181i	$-0.603918\pi$
$331$	−11.6692	−0.641399	−0.320699	+	0.947181i	$0.603918\pi$
$332$	0	0
$333$	−0.0906953	−0.00497007
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−17.1403	−0.933693	−0.466846	−	0.884338i	$-0.654610\pi$
$337$	−17.1403	−0.933693	−0.466846	+	0.884338i	$0.654610\pi$
$338$	0	0
$339$	−24.7972	−1.34680
$340$	0	0
$341$	−12.2780	−0.664888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.20070	−0.279188	−0.139594	−	0.990209i	$-0.544580\pi$
$347$	−5.20070	−0.279188	−0.139594	+	0.990209i	$0.544580\pi$
$348$	0	0
$349$	33.8645	1.81272	0.906362	−	0.422501i	$-0.138848\pi$
$349$	33.8645	1.81272	0.906362	+	0.422501i	$0.138848\pi$
$350$	0	0
$351$	19.4363	1.03743
$352$	0	0
$353$	−23.2451	−1.23721	−0.618606	−	0.785701i	$-0.712302\pi$
$353$	−23.2451	−1.23721	−0.618606	+	0.785701i	$0.712302\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.2843	−1.07056	−0.535281	−	0.844674i	$-0.679794\pi$
$359$	−20.2843	−1.07056	−0.535281	+	0.844674i	$0.679794\pi$
$360$	0	0
$361$	−17.4513	−0.918491
$362$	0	0
$363$	0.279014	0.0146445
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.2738	−0.692887	−0.346443	−	0.938071i	$-0.612611\pi$
$367$	−13.2738	−0.692887	−0.346443	+	0.938071i	$0.612611\pi$
$368$	0	0
$369$	−6.23101	−0.324373
$370$	0	0
$371$	0	0
$372$	0	0
$373$	35.7083	1.84891	0.924453	−	0.381297i	$-0.124522\pi$
$373$	35.7083	1.84891	0.924453	+	0.381297i	$0.124522\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.90652	−0.510212
$378$	0	0
$379$	−12.6878	−0.651728	−0.325864	−	0.945417i	$-0.605655\pi$
$379$	−12.6878	−0.651728	−0.325864	+	0.945417i	$0.605655\pi$
$380$	0	0
$381$	12.8987	0.660823
$382$	0	0
$383$	36.5707	1.86867	0.934337	−	0.356391i	$-0.115993\pi$
$383$	36.5707	1.86867	0.934337	+	0.356391i	$0.115993\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.42040	0.275534
$388$	0	0
$389$	30.1085	1.52656	0.763282	−	0.646066i	$-0.223587\pi$
$389$	30.1085	1.52656	0.763282	+	0.646066i	$0.223587\pi$
$390$	0	0
$391$	−0.392156	−0.0198322
$392$	0	0
$393$	−0.649918	−0.0327840
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.46756	−0.374786	−0.187393	−	0.982285i	$-0.560004\pi$
$397$	−7.46756	−0.374786	−0.187393	+	0.982285i	$0.560004\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.7331	−0.585925	−0.292963	−	0.956124i	$-0.594641\pi$
$401$	−11.7331	−0.585925	−0.292963	+	0.956124i	$0.594641\pi$
$402$	0	0
$403$	15.6277	0.778473
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.559222	0.0277196
$408$	0	0
$409$	−16.8601	−0.833679	−0.416840	−	0.908980i	$-0.636862\pi$
$409$	−16.8601	−0.833679	−0.416840	+	0.908980i	$0.636862\pi$
$410$	0	0
$411$	21.2693	1.04914
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.05494	0.247541
$418$	0	0
$419$	10.3884	0.507507	0.253753	−	0.967269i	$-0.418335\pi$
$419$	10.3884	0.507507	0.253753	+	0.967269i	$0.418335\pi$
$420$	0	0
$421$	13.7758	0.671393	0.335696	−	0.941970i	$-0.391028\pi$
$421$	13.7758	0.671393	0.335696	+	0.941970i	$0.391028\pi$
$422$	0	0
$423$	1.66923	0.0811606
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	25.9664	1.25367
$430$	0	0
$431$	30.4568	1.46705	0.733527	−	0.679661i	$-0.237873\pi$
$431$	30.4568	1.46705	0.733527	+	0.679661i	$0.237873\pi$
$432$	0	0
$433$	25.9182	1.24555	0.622774	−	0.782402i	$-0.286006\pi$
$433$	25.9182	1.24555	0.622774	+	0.782402i	$0.286006\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.339522	−0.0162415
$438$	0	0
$439$	26.9414	1.28584	0.642922	−	0.765931i	$-0.277722\pi$
$439$	26.9414	1.28584	0.642922	+	0.765931i	$0.277722\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.3858	0.778515	0.389257	−	0.921129i	$-0.372732\pi$
$443$	16.3858	0.778515	0.389257	+	0.921129i	$0.372732\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−24.6914	−1.16786
$448$	0	0
$449$	14.2152	0.670858	0.335429	−	0.942065i	$-0.391119\pi$
$449$	14.2152	0.670858	0.335429	+	0.942065i	$0.391119\pi$
$450$	0	0
$451$	38.4201	1.80913
$452$	0	0
$453$	−5.66132	−0.265992
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.5093	0.678716	0.339358	−	0.940657i	$-0.389790\pi$
$457$	14.5093	0.678716	0.339358	+	0.940657i	$0.389790\pi$
$458$	0	0
$459$	−6.66295	−0.311000
$460$	0	0
$461$	17.8933	0.833374	0.416687	−	0.909050i	$-0.363191\pi$
$461$	17.8933	0.833374	0.416687	+	0.909050i	$0.363191\pi$
$462$	0	0
$463$	−15.2657	−0.709457	−0.354729	−	0.934969i	$-0.615427\pi$
$463$	−15.2657	−0.709457	−0.354729	+	0.934969i	$0.615427\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.5492	1.08973	0.544863	−	0.838525i	$-0.316582\pi$
$467$	23.5492	1.08973	0.544863	+	0.838525i	$0.316582\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−36.5898	−1.68597
$472$	0	0
$473$	−33.4219	−1.53674
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.94024	−0.226198
$478$	0	0
$479$	−4.65867	−0.212860	−0.106430	−	0.994320i	$-0.533942\pi$
$479$	−4.65867	−0.212860	−0.106430	+	0.994320i	$0.533942\pi$
$480$	0	0
$481$	−0.711794	−0.0324550
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.5210	−0.522068	−0.261034	−	0.965330i	$-0.584063\pi$
$487$	−11.5210	−0.522068	−0.261034	+	0.965330i	$0.584063\pi$
$488$	0	0
$489$	25.3648	1.14704
$490$	0	0
$491$	−14.2771	−0.644317	−0.322158	−	0.946686i	$-0.604408\pi$
$491$	−14.2771	−0.644317	−0.322158	+	0.946686i	$0.604408\pi$
$492$	0	0
$493$	3.39605	0.152950
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.9790	1.65541	0.827703	−	0.561167i	$-0.189647\pi$
$499$	36.9790	1.65541	0.827703	+	0.561167i	$0.189647\pi$
$500$	0	0
$501$	−23.1256	−1.03318
$502$	0	0
$503$	−8.08985	−0.360709	−0.180354	−	0.983602i	$-0.557724\pi$
$503$	−8.08985	−0.360709	−0.180354	+	0.983602i	$0.557724\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.61129	−0.382441
$508$	0	0
$509$	28.0806	1.24465	0.622325	−	0.782759i	$-0.286188\pi$
$509$	28.0806	1.24465	0.622325	+	0.782759i	$0.286188\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.76867	−0.254693
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10.2924	−0.452658
$518$	0	0
$519$	−4.68063	−0.205457
$520$	0	0
$521$	12.5810	0.551182	0.275591	−	0.961275i	$-0.411126\pi$
$521$	12.5810	0.551182	0.275591	+	0.961275i	$0.411126\pi$
$522$	0	0
$523$	−1.48453	−0.0649140	−0.0324570	−	0.999473i	$-0.510333\pi$
$523$	−1.48453	−0.0649140	−0.0324570	+	0.999473i	$0.510333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.35733	−0.233369
$528$	0	0
$529$	−22.9256	−0.996764
$530$	0	0
$531$	−4.84624	−0.210309
$532$	0	0
$533$	−48.9022	−2.11819
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−28.4513	−1.22777
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.9135	0.813153	0.406577	−	0.913617i	$-0.366722\pi$
$541$	18.9135	0.813153	0.406577	+	0.913617i	$0.366722\pi$
$542$	0	0
$543$	20.7993	0.892583
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.4403	0.745693	0.372846	−	0.927893i	$-0.378382\pi$
$547$	17.4403	0.745693	0.372846	+	0.927893i	$0.378382\pi$
$548$	0	0
$549$	−3.88456	−0.165789
$550$	0	0
$551$	2.94024	0.125259
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.2106	−0.813981	−0.406990	−	0.913432i	$-0.633422\pi$
$557$	−19.2106	−0.813981	−0.406990	+	0.913432i	$0.633422\pi$
$558$	0	0
$559$	42.5403	1.79926
$560$	0	0
$561$	−8.90152	−0.375823
$562$	0	0
$563$	33.7952	1.42430	0.712150	−	0.702028i	$-0.247722\pi$
$563$	33.7952	1.42430	0.712150	+	0.702028i	$0.247722\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.63635	0.194366	0.0971830	−	0.995267i	$-0.469017\pi$
$569$	4.63635	0.194366	0.0971830	+	0.995267i	$0.469017\pi$
$570$	0	0
$571$	43.8681	1.83582	0.917912	−	0.396785i	$-0.129874\pi$
$571$	43.8681	1.83582	0.917912	+	0.396785i	$0.129874\pi$
$572$	0	0
$573$	42.2083	1.76328
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.27755	0.386230	0.193115	−	0.981176i	$-0.438141\pi$
$577$	9.27755	0.386230	0.193115	+	0.981176i	$0.438141\pi$
$578$	0	0
$579$	5.85574	0.243356
$580$	0	0
$581$	0	0
$582$	0	0
$583$	30.4613	1.26158
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.9971	1.19684	0.598420	−	0.801183i	$-0.295796\pi$
$587$	28.9971	1.19684	0.598420	+	0.801183i	$0.295796\pi$
$588$	0	0
$589$	−4.63829	−0.191117
$590$	0	0
$591$	2.60873	0.107309
$592$	0	0
$593$	−25.5228	−1.04809	−0.524047	−	0.851689i	$-0.675578\pi$
$593$	−25.5228	−1.04809	−0.524047	+	0.851689i	$0.675578\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.777294	−0.0318125
$598$	0	0
$599$	31.0364	1.26811	0.634057	−	0.773287i	$-0.281389\pi$
$599$	31.0364	1.26811	0.634057	+	0.773287i	$0.281389\pi$
$600$	0	0
$601$	−3.56479	−0.145411	−0.0727054	−	0.997353i	$-0.523163\pi$
$601$	−3.56479	−0.145411	−0.0727054	+	0.997353i	$0.523163\pi$
$602$	0	0
$603$	7.02219	0.285966
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6.72620	−0.273008	−0.136504	−	0.990640i	$-0.543587\pi$
$607$	−6.72620	−0.273008	−0.136504	+	0.990640i	$0.543587\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.1004	0.529986
$612$	0	0
$613$	19.7422	0.797381	0.398691	−	0.917085i	$-0.369465\pi$
$613$	19.7422	0.797381	0.398691	+	0.917085i	$0.369465\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.3958	0.660069	0.330035	−	0.943969i	$-0.392940\pi$
$617$	16.3958	0.660069	0.330035	+	0.943969i	$0.392940\pi$
$618$	0	0
$619$	−35.5510	−1.42892	−0.714458	−	0.699678i	$-0.753327\pi$
$619$	−35.5510	−1.42892	−0.714458	+	0.699678i	$0.753327\pi$
$620$	0	0
$621$	1.26470	0.0507506
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−7.70679	−0.307780
$628$	0	0
$629$	0.244010	0.00972930
$630$	0	0
$631$	−9.58569	−0.381600	−0.190800	−	0.981629i	$-0.561108\pi$
$631$	−9.58569	−0.381600	−0.190800	+	0.981629i	$0.561108\pi$
$632$	0	0
$633$	−46.2851	−1.83967
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.67285	0.145296
$640$	0	0
$641$	19.5145	0.770778	0.385389	−	0.922754i	$-0.374067\pi$
$641$	19.5145	0.770778	0.385389	+	0.922754i	$0.374067\pi$
$642$	0	0
$643$	−18.9895	−0.748872	−0.374436	−	0.927253i	$-0.622164\pi$
$643$	−18.9895	−0.748872	−0.374436	+	0.927253i	$0.622164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.2570	1.62198	0.810989	−	0.585061i	$-0.198930\pi$
$647$	41.2570	1.62198	0.810989	+	0.585061i	$0.198930\pi$
$648$	0	0
$649$	29.8817	1.17296
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.7495	0.772857	0.386429	−	0.922319i	$-0.373709\pi$
$653$	19.7495	0.772857	0.386429	+	0.922319i	$0.373709\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.14713	−0.317850
$658$	0	0
$659$	10.1830	0.396672	0.198336	−	0.980134i	$-0.436446\pi$
$659$	10.1830	0.396672	0.198336	+	0.980134i	$0.436446\pi$
$660$	0	0
$661$	2.79086	0.108552	0.0542759	−	0.998526i	$-0.482715\pi$
$661$	2.79086	0.108552	0.0542759	+	0.998526i	$0.482715\pi$
$662$	0	0
$663$	11.3301	0.440025
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.644606	−0.0249593
$668$	0	0
$669$	33.6636	1.30151
$670$	0	0
$671$	23.9520	0.924657
$672$	0	0
$673$	−18.2879	−0.704947	−0.352473	−	0.935822i	$-0.614659\pi$
$673$	−18.2879	−0.704947	−0.352473	+	0.935822i	$0.614659\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.0523	0.924404	0.462202	−	0.886775i	$-0.347059\pi$
$677$	24.0523	0.924404	0.462202	+	0.886775i	$0.347059\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	23.5908	0.904002
$682$	0	0
$683$	38.2200	1.46245	0.731224	−	0.682137i	$-0.238949\pi$
$683$	38.2200	1.46245	0.731224	+	0.682137i	$0.238949\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−29.7699	−1.13579
$688$	0	0
$689$	−38.7720	−1.47709
$690$	0	0
$691$	−26.4623	−1.00667	−0.503337	−	0.864090i	$-0.667895\pi$
$691$	−26.4623	−1.00667	−0.503337	+	0.864090i	$0.667895\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.7641	0.634986
$698$	0	0
$699$	−43.9520	−1.66242
$700$	0	0
$701$	−34.5136	−1.30356	−0.651780	−	0.758408i	$-0.725977\pi$
$701$	−34.5136	−1.30356	−0.651780	+	0.758408i	$0.725977\pi$
$702$	0	0
$703$	0.211260	0.00796780
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.40900	0.240695	0.120347	−	0.992732i	$-0.461599\pi$
$709$	6.40900	0.240695	0.120347	+	0.992732i	$0.461599\pi$
$710$	0	0
$711$	7.98763	0.299559
$712$	0	0
$713$	1.01688	0.0380824
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−39.3165	−1.46830
$718$	0	0
$719$	−38.6070	−1.43980	−0.719900	−	0.694078i	$-0.755812\pi$
$719$	−38.6070	−1.43980	−0.719900	+	0.694078i	$0.755812\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.31190	−0.234742
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.5280	−0.835516	−0.417758	−	0.908558i	$-0.637184\pi$
$727$	−22.5280	−0.835516	−0.417758	+	0.908558i	$0.637184\pi$
$728$	0	0
$729$	20.6317	0.764136
$730$	0	0
$731$	−14.5832	−0.539380
$732$	0	0
$733$	−7.50150	−0.277074	−0.138537	−	0.990357i	$-0.544240\pi$
$733$	−7.50150	−0.277074	−0.138537	+	0.990357i	$0.544240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.2985	−1.59492
$738$	0	0
$739$	35.0864	1.29067	0.645336	−	0.763899i	$-0.276717\pi$
$739$	35.0864	1.29067	0.645336	+	0.763899i	$0.276717\pi$
$740$	0	0
$741$	9.80943	0.360358
$742$	0	0
$743$	2.42902	0.0891122	0.0445561	−	0.999007i	$-0.485813\pi$
$743$	2.42902	0.0891122	0.0445561	+	0.999007i	$0.485813\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.0893263	0.00326828
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.9598	0.764833	0.382417	−	0.923990i	$-0.375092\pi$
$751$	20.9598	0.764833	0.382417	+	0.923990i	$0.375092\pi$
$752$	0	0
$753$	31.1542	1.13532
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.9748	1.30753	0.653763	−	0.756699i	$-0.273189\pi$
$757$	35.9748	1.30753	0.653763	+	0.756699i	$0.273189\pi$
$758$	0	0
$759$	1.68960	0.0613288
$760$	0	0
$761$	2.45752	0.0890852	0.0445426	−	0.999007i	$-0.485817\pi$
$761$	2.45752	0.0890852	0.0445426	+	0.999007i	$0.485817\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−38.0343	−1.37334
$768$	0	0
$769$	−21.6994	−0.782501	−0.391250	−	0.920284i	$-0.627957\pi$
$769$	−21.6994	−0.782501	−0.391250	+	0.920284i	$0.627957\pi$
$770$	0	0
$771$	−24.7053	−0.889739
$772$	0	0
$773$	15.6980	0.564618	0.282309	−	0.959324i	$-0.408900\pi$
$773$	15.6980	0.564618	0.282309	+	0.959324i	$0.408900\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.5141	0.520022
$780$	0	0
$781$	−22.6466	−0.810359
$782$	0	0
$783$	−10.9522	−0.391400
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−24.5074	−0.873594	−0.436797	−	0.899560i	$-0.643887\pi$
$787$	−24.5074	−0.873594	−0.436797	+	0.899560i	$0.643887\pi$
$788$	0	0
$789$	−7.05887	−0.251302
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−30.4868	−1.08262
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.73557	0.203164	0.101582	−	0.994827i	$-0.467610\pi$
$797$	5.73557	0.203164	0.101582	+	0.994827i	$0.467610\pi$
$798$	0	0
$799$	−4.49094	−0.158878
$800$	0	0
$801$	−1.65549	−0.0584937
$802$	0	0
$803$	50.2348	1.77275
$804$	0	0
$805$	0	0
$806$	0	0
$807$	31.0104	1.09162
$808$	0	0
$809$	3.48431	0.122502	0.0612510	−	0.998122i	$-0.480491\pi$
$809$	3.48431	0.122502	0.0612510	+	0.998122i	$0.480491\pi$
$810$	0	0
$811$	−25.9953	−0.912819	−0.456409	−	0.889770i	$-0.650865\pi$
$811$	−25.9953	−0.912819	−0.456409	+	0.889770i	$0.650865\pi$
$812$	0	0
$813$	−42.3781	−1.48627
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.6259	−0.441725
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.4988	−1.76242	−0.881210	−	0.472725i	$-0.843271\pi$
$821$	−50.4988	−1.76242	−0.881210	+	0.472725i	$0.843271\pi$
$822$	0	0
$823$	44.8699	1.56407	0.782034	−	0.623236i	$-0.214182\pi$
$823$	44.8699	1.56407	0.782034	+	0.623236i	$0.214182\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.7323	0.547066	0.273533	−	0.961863i	$-0.411808\pi$
$827$	15.7323	0.547066	0.273533	+	0.961863i	$0.411808\pi$
$828$	0	0
$829$	51.7726	1.79814	0.899068	−	0.437808i	$-0.144245\pi$
$829$	51.7726	1.79814	0.899068	+	0.437808i	$0.144245\pi$
$830$	0	0
$831$	−28.1960	−0.978109
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	17.2773	0.597192
$838$	0	0
$839$	2.86118	0.0987788	0.0493894	−	0.998780i	$-0.484272\pi$
$839$	2.86118	0.0987788	0.0493894	+	0.998780i	$0.484272\pi$
$840$	0	0
$841$	−23.4177	−0.807508
$842$	0	0
$843$	52.6921	1.81481
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	49.2014	1.68859
$850$	0	0
$851$	−0.0463156	−0.00158768
$852$	0	0
$853$	45.8645	1.57037	0.785185	−	0.619261i	$-0.212568\pi$
$853$	45.8645	1.57037	0.785185	+	0.619261i	$0.212568\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.1501	−0.790793	−0.395397	−	0.918510i	$-0.629393\pi$
$857$	−23.1501	−0.790793	−0.395397	+	0.918510i	$0.629393\pi$
$858$	0	0
$859$	8.46384	0.288783	0.144391	−	0.989521i	$-0.453878\pi$
$859$	8.46384	0.288783	0.144391	+	0.989521i	$0.453878\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.0923	1.19456	0.597278	−	0.802034i	$-0.296249\pi$
$863$	35.0923	1.19456	0.597278	+	0.802034i	$0.296249\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	28.0753	0.953486
$868$	0	0
$869$	−49.2513	−1.67074
$870$	0	0
$871$	55.1115	1.86738
$872$	0	0
$873$	3.53085	0.119501
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.3372	−0.652971	−0.326486	−	0.945202i	$-0.605864\pi$
$877$	−19.3372	−0.652971	−0.326486	+	0.945202i	$0.605864\pi$
$878$	0	0
$879$	−29.3388	−0.989573
$880$	0	0
$881$	33.0573	1.11373	0.556865	−	0.830603i	$-0.312004\pi$
$881$	33.0573	1.11373	0.556865	+	0.830603i	$0.312004\pi$
$882$	0	0
$883$	28.6238	0.963267	0.481634	−	0.876373i	$-0.340044\pi$
$883$	28.6238	0.963267	0.481634	+	0.876373i	$0.340044\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.1289	0.642285	0.321142	−	0.947031i	$-0.395933\pi$
$887$	19.1289	0.642285	0.321142	+	0.947031i	$0.395933\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	33.9871	1.13861
$892$	0	0
$893$	−3.88818	−0.130113
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.15058	−0.0718057
$898$	0	0
$899$	−8.80611	−0.293700
$900$	0	0
$901$	13.2914	0.442801
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50.9781	1.69270	0.846350	−	0.532627i	$-0.178795\pi$
$907$	50.9781	1.69270	0.846350	+	0.532627i	$0.178795\pi$
$908$	0	0
$909$	−10.0638	−0.333796
$910$	0	0
$911$	−5.52585	−0.183080	−0.0915399	−	0.995801i	$-0.529179\pi$
$911$	−5.52585	−0.183080	−0.0915399	+	0.995801i	$0.529179\pi$
$912$	0	0
$913$	−0.550782	−0.0182282
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−40.6326	−1.34035	−0.670173	−	0.742205i	$-0.733780\pi$
$919$	−40.6326	−1.34035	−0.670173	+	0.742205i	$0.733780\pi$
$920$	0	0
$921$	−40.0535	−1.31981
$922$	0	0
$923$	28.8252	0.948794
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.784047	−0.0257515
$928$	0	0
$929$	15.5879	0.511423	0.255711	−	0.966753i	$-0.417690\pi$
$929$	15.5879	0.511423	0.255711	+	0.966753i	$0.417690\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−27.9736	−0.915814
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.4794	−0.669034	−0.334517	−	0.942390i	$-0.608573\pi$
$937$	−20.4794	−0.669034	−0.334517	+	0.942390i	$0.608573\pi$
$938$	0	0
$939$	−4.06585	−0.132684
$940$	0	0
$941$	−16.7872	−0.547248	−0.273624	−	0.961837i	$-0.588222\pi$
$941$	−16.7872	−0.547248	−0.273624	+	0.961837i	$0.588222\pi$
$942$	0	0
$943$	−3.18201	−0.103620
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.43928	0.111761	0.0558807	−	0.998437i	$-0.482203\pi$
$947$	3.43928	0.111761	0.0558807	+	0.998437i	$0.482203\pi$
$948$	0	0
$949$	−63.9402	−2.07559
$950$	0	0
$951$	−40.3430	−1.30821
$952$	0	0
$953$	30.8610	0.999686	0.499843	−	0.866116i	$-0.333391\pi$
$953$	30.8610	0.999686	0.499843	+	0.866116i	$0.333391\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−14.6319	−0.472982
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.1082	−0.551877
$962$	0	0
$963$	−7.24582	−0.233493
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34.3195	1.10364	0.551820	−	0.833964i	$-0.313934\pi$
$967$	34.3195	1.10364	0.551820	+	0.833964i	$0.313934\pi$
$968$	0	0
$969$	−3.36276	−0.108027
$970$	0	0
$971$	−31.6203	−1.01475	−0.507373	−	0.861727i	$-0.669383\pi$
$971$	−31.6203	−1.01475	−0.507373	+	0.861727i	$0.669383\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.9454	0.606116	0.303058	−	0.952972i	$-0.401992\pi$
$977$	18.9454	0.606116	0.303058	+	0.952972i	$0.401992\pi$
$978$	0	0
$979$	10.2076	0.326237
$980$	0	0
$981$	7.91443	0.252688
$982$	0	0
$983$	15.2187	0.485402	0.242701	−	0.970101i	$-0.421967\pi$
$983$	15.2187	0.485402	0.242701	+	0.970101i	$0.421967\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.76805	0.0880189
$990$	0	0
$991$	31.3069	0.994496	0.497248	−	0.867608i	$-0.334344\pi$
$991$	31.3069	0.994496	0.497248	+	0.867608i	$0.334344\pi$
$992$	0	0
$993$	21.9377	0.696172
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.1054	1.27015	0.635076	−	0.772450i	$-0.280969\pi$
$997$	40.1054	1.27015	0.635076	+	0.772450i	$0.280969\pi$
$998$	0	0
$999$	−0.786929	−0.0248973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cs.1.1		4
5.4	even	2		1960.2.a.x.1.4	✓	4
7.6	odd	2		9800.2.a.cl.1.4		4
20.19	odd	2		3920.2.a.ce.1.1		4
35.4	even	6		1960.2.q.y.961.1		8
35.9	even	6		1960.2.q.y.361.1		8
35.19	odd	6		1960.2.q.x.361.4		8
35.24	odd	6		1960.2.q.x.961.4		8
35.34	odd	2		1960.2.a.y.1.1	yes	4
140.139	even	2		3920.2.a.cd.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.x.1.4	✓	4	5.4	even	2
1960.2.a.y.1.1	yes	4	35.34	odd	2
1960.2.q.x.361.4		8	35.19	odd	6
1960.2.q.x.961.4		8	35.24	odd	6
1960.2.q.y.361.1		8	35.9	even	6
1960.2.q.y.961.1		8	35.4	even	6
3920.2.a.cd.1.4		4	140.139	even	2
3920.2.a.ce.1.1		4	20.19	odd	2
9800.2.a.cl.1.4		4	7.6	odd	2
9800.2.a.cs.1.1		4	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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.87996 q^{3} +0.534253 q^{9} +O(q^{10})\)	\(q-1.87996 q^{3} +0.534253 q^{9} -3.29417 q^{11} +4.19292 q^{13} -1.43737 q^{17} -1.24445 q^{19} +0.272828 q^{23} +4.63551 q^{27} -2.36268 q^{29} +3.72717 q^{31} +6.19292 q^{33} -0.169761 q^{37} -7.88252 q^{39} -11.6630 q^{41} +10.1458 q^{43} +3.12441 q^{47} +2.70220 q^{51} -9.24701 q^{53} +2.33952 q^{57} -9.07107 q^{59} -7.27102 q^{61} +13.1439 q^{67} -0.512907 q^{69} +6.87474 q^{71} -15.2496 q^{73} +14.9510 q^{79} -10.3173 q^{81} +0.167199 q^{83} +4.44175 q^{87} -3.09869 q^{89} -7.00694 q^{93} +6.60894 q^{97} -1.75992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 6 * q^9	\( 4 q + 2 q^{3} + 6 q^{9} + 2 q^{11} + 10 q^{13} + 6 q^{17} + 4 q^{23} + 14 q^{27} - 2 q^{29} + 12 q^{31} + 18 q^{33} + 14 q^{39} - 12 q^{41} + 8 q^{43} - 2 q^{47} + 2 q^{51} + 4 q^{53} + 8 q^{57} - 8 q^{59} - 20 q^{61} + 8 q^{67} - 24 q^{69} + 4 q^{71} + 16 q^{73} + 22 q^{79} - 20 q^{81} + 36 q^{83} - 18 q^{87} - 40 q^{89} + 32 q^{93} + 26 q^{97} + 12 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^9 + 2 * q^11 + 10 * q^13 + 6 * q^17 + 4 * q^23 + 14 * q^27 - 2 * q^29 + 12 * q^31 + 18 * q^33 + 14 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^47 + 2 * q^51 + 4 * q^53 + 8 * q^57 - 8 * q^59 - 20 * q^61 + 8 * q^67 - 24 * q^69 + 4 * q^71 + 16 * q^73 + 22 * q^79 - 20 * q^81 + 36 * q^83 - 18 * q^87 - 40 * q^89 + 32 * q^93 + 26 * q^97 + 12 * q^99

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