LMFDB - Embedded newform 9800.2.a.ci.1.3

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ 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)

$f(q)$	$=$	$q+2.53209 q^{3} +3.41147 q^{9} +O(q^{10})$	$q+2.53209 q^{3} +3.41147 q^{9} -4.22668 q^{11} +5.98545 q^{13} -7.63816 q^{17} +6.94356 q^{19} -1.71688 q^{23} +1.04189 q^{27} +5.18479 q^{29} -0.509800 q^{31} -10.7023 q^{33} +3.98545 q^{37} +15.1557 q^{39} +3.71688 q^{41} +4.17024 q^{43} +1.89393 q^{47} -19.3405 q^{51} +6.39187 q^{53} +17.5817 q^{57} -5.59627 q^{59} -4.31315 q^{61} +13.1925 q^{67} -4.34730 q^{69} +5.08378 q^{71} -5.38919 q^{73} -1.34730 q^{79} -7.59627 q^{81} +8.70233 q^{83} +13.1284 q^{87} +10.8452 q^{89} -1.29086 q^{93} -13.6800 q^{97} -14.4192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3}+O(q^{10}) $ 3 * q + 3 * q^3	$ 3 q + 3 q^{3} - 6 q^{11} - 6 q^{17} + 6 q^{19} + 3 q^{23} + 12 q^{29} - 3 q^{31} - 6 q^{33} - 6 q^{37} + 6 q^{39} + 3 q^{41} - 9 q^{43} + 18 q^{47} - 15 q^{51} + 6 q^{53} + 21 q^{57} - 3 q^{59} + 9 q^{61} + 12 q^{67} - 12 q^{69} + 9 q^{71} - 12 q^{73} - 3 q^{79} - 9 q^{81} + 21 q^{87} + 6 q^{89} + 12 q^{93} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^11 - 6 * q^17 + 6 * q^19 + 3 * q^23 + 12 * q^29 - 3 * q^31 - 6 * q^33 - 6 * q^37 + 6 * q^39 + 3 * q^41 - 9 * q^43 + 18 * q^47 - 15 * q^51 + 6 * q^53 + 21 * q^57 - 3 * q^59 + 9 * q^61 + 12 * q^67 - 12 * q^69 + 9 * q^71 - 12 * q^73 - 3 * q^79 - 9 * q^81 + 21 * q^87 + 6 * q^89 + 12 * q^93 - 21 * q^97 - 9 * 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
$3$	2.53209	1.46190	0.730951	−	0.682430i	$-0.239077\pi$
$3$	2.53209	1.46190	0.730951	+	0.682430i	$0.239077\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.41147	1.13716
$10$	0	0
$11$	−4.22668	−1.27439	−0.637196	−	0.770702i	$-0.719906\pi$
$11$	−4.22668	−1.27439	−0.637196	+	0.770702i	$0.719906\pi$
$12$	0	0
$13$	5.98545	1.66007	0.830033	−	0.557714i	$-0.188322\pi$
$13$	5.98545	1.66007	0.830033	+	0.557714i	$0.188322\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.63816	−1.85252	−0.926262	−	0.376879i	$-0.876997\pi$
$17$	−7.63816	−1.85252	−0.926262	+	0.376879i	$0.876997\pi$
$18$	0	0
$19$	6.94356	1.59296	0.796481	−	0.604663i	$-0.206692\pi$
$19$	6.94356	1.59296	0.796481	+	0.604663i	$0.206692\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.71688	−0.357995	−0.178997	−	0.983850i	$-0.557285\pi$
$23$	−1.71688	−0.357995	−0.178997	+	0.983850i	$0.557285\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.04189	0.200512
$28$	0	0
$29$	5.18479	0.962792	0.481396	−	0.876503i	$-0.340130\pi$
$29$	5.18479	0.962792	0.481396	+	0.876503i	$0.340130\pi$
$30$	0	0
$31$	−0.509800	−0.0915628	−0.0457814	−	0.998951i	$-0.514578\pi$
$31$	−0.509800	−0.0915628	−0.0457814	+	0.998951i	$0.514578\pi$
$32$	0	0
$33$	−10.7023	−1.86304
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.98545	0.655204	0.327602	−	0.944816i	$-0.393759\pi$
$37$	3.98545	0.655204	0.327602	+	0.944816i	$0.393759\pi$
$38$	0	0
$39$	15.1557	2.42685
$40$	0	0
$41$	3.71688	0.580479	0.290240	−	0.956954i	$-0.406265\pi$
$41$	3.71688	0.580479	0.290240	+	0.956954i	$0.406265\pi$
$42$	0	0
$43$	4.17024	0.635956	0.317978	−	0.948098i	$-0.396996\pi$
$43$	4.17024	0.635956	0.317978	+	0.948098i	$0.396996\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.89393	0.276259	0.138129	−	0.990414i	$-0.455891\pi$
$47$	1.89393	0.276259	0.138129	+	0.990414i	$0.455891\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−19.3405	−2.70821
$52$	0	0
$53$	6.39187	0.877991	0.438996	−	0.898489i	$-0.355334\pi$
$53$	6.39187	0.877991	0.438996	+	0.898489i	$0.355334\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17.5817	2.32876
$58$	0	0
$59$	−5.59627	−0.728572	−0.364286	−	0.931287i	$-0.618687\pi$
$59$	−5.59627	−0.728572	−0.364286	+	0.931287i	$0.618687\pi$
$60$	0	0
$61$	−4.31315	−0.552242	−0.276121	−	0.961123i	$-0.589049\pi$
$61$	−4.31315	−0.552242	−0.276121	+	0.961123i	$0.589049\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1925	1.61172	0.805862	−	0.592103i	$-0.201702\pi$
$67$	13.1925	1.61172	0.805862	+	0.592103i	$0.201702\pi$
$68$	0	0
$69$	−4.34730	−0.523353
$70$	0	0
$71$	5.08378	0.603333	0.301667	−	0.953413i	$-0.402457\pi$
$71$	5.08378	0.603333	0.301667	+	0.953413i	$0.402457\pi$
$72$	0	0
$73$	−5.38919	−0.630756	−0.315378	−	0.948966i	$-0.602131\pi$
$73$	−5.38919	−0.630756	−0.315378	+	0.948966i	$0.602131\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.34730	−0.151583	−0.0757913	−	0.997124i	$-0.524148\pi$
$79$	−1.34730	−0.151583	−0.0757913	+	0.997124i	$0.524148\pi$
$80$	0	0
$81$	−7.59627	−0.844030
$82$	0	0
$83$	8.70233	0.955205	0.477603	−	0.878576i	$-0.341506\pi$
$83$	8.70233	0.955205	0.477603	+	0.878576i	$0.341506\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13.1284	1.40751
$88$	0	0
$89$	10.8452	1.14959	0.574796	−	0.818296i	$-0.305081\pi$
$89$	10.8452	1.14959	0.574796	+	0.818296i	$0.305081\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.29086	−0.133856
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.6800	−1.38900	−0.694499	−	0.719494i	$-0.744374\pi$
$97$	−13.6800	−1.38900	−0.694499	+	0.719494i	$0.744374\pi$
$98$	0	0
$99$	−14.4192	−1.44919
$100$	0	0
$101$	11.1206	1.10654	0.553271	−	0.833001i	$-0.313379\pi$
$101$	11.1206	1.10654	0.553271	+	0.833001i	$0.313379\pi$
$102$	0	0
$103$	19.3259	1.90424	0.952121	−	0.305722i	$-0.0988978\pi$
$103$	19.3259	1.90424	0.952121	+	0.305722i	$0.0988978\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.31820	−0.900824	−0.450412	−	0.892821i	$-0.648723\pi$
$107$	−9.31820	−0.900824	−0.450412	+	0.892821i	$0.648723\pi$
$108$	0	0
$109$	−4.68004	−0.448267	−0.224133	−	0.974558i	$-0.571955\pi$
$109$	−4.68004	−0.448267	−0.224133	+	0.974558i	$0.571955\pi$
$110$	0	0
$111$	10.0915	0.957845
$112$	0	0
$113$	3.57398	0.336212	0.168106	−	0.985769i	$-0.446235\pi$
$113$	3.57398	0.336212	0.168106	+	0.985769i	$0.446235\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	20.4192	1.88776
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.86484	0.624076
$122$	0	0
$123$	9.41147	0.848604
$124$	0	0
$125$	0	0
$126$	0	0
$127$	21.6040	1.91705	0.958523	−	0.285016i	$-0.0919988\pi$
$127$	21.6040	1.91705	0.958523	+	0.285016i	$0.0919988\pi$
$128$	0	0
$129$	10.5594	0.929706
$130$	0	0
$131$	−3.46110	−0.302398	−0.151199	−	0.988503i	$-0.548313\pi$
$131$	−3.46110	−0.302398	−0.151199	+	0.988503i	$0.548313\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.1506	1.12354	0.561768	−	0.827295i	$-0.310121\pi$
$137$	13.1506	1.12354	0.561768	+	0.827295i	$0.310121\pi$
$138$	0	0
$139$	6.46791	0.548601	0.274301	−	0.961644i	$-0.411554\pi$
$139$	6.46791	0.548601	0.274301	+	0.961644i	$0.411554\pi$
$140$	0	0
$141$	4.79561	0.403863
$142$	0	0
$143$	−25.2986	−2.11558
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.57398	−0.374715	−0.187357	−	0.982292i	$-0.559992\pi$
$149$	−4.57398	−0.374715	−0.187357	+	0.982292i	$0.559992\pi$
$150$	0	0
$151$	5.15570	0.419565	0.209782	−	0.977748i	$-0.432724\pi$
$151$	5.15570	0.419565	0.209782	+	0.977748i	$0.432724\pi$
$152$	0	0
$153$	−26.0574	−2.10661
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.6408	−1.08866	−0.544329	−	0.838872i	$-0.683216\pi$
$157$	−13.6408	−1.08866	−0.544329	+	0.838872i	$0.683216\pi$
$158$	0	0
$159$	16.1848	1.28354
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.347296	0.0272023	0.0136012	−	0.999907i	$-0.495670\pi$
$163$	0.347296	0.0272023	0.0136012	+	0.999907i	$0.495670\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.72874	0.133774	0.0668870	−	0.997761i	$-0.478693\pi$
$167$	1.72874	0.133774	0.0668870	+	0.997761i	$0.478693\pi$
$168$	0	0
$169$	22.8256	1.75582
$170$	0	0
$171$	23.6878	1.81145
$172$	0	0
$173$	−13.3327	−1.01367	−0.506835	−	0.862043i	$-0.669185\pi$
$173$	−13.3327	−1.01367	−0.506835	+	0.862043i	$0.669185\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.1702	−1.06510
$178$	0	0
$179$	−15.6013	−1.16610	−0.583049	−	0.812437i	$-0.698140\pi$
$179$	−15.6013	−1.16610	−0.583049	+	0.812437i	$0.698140\pi$
$180$	0	0
$181$	19.6382	1.45969	0.729846	−	0.683611i	$-0.239592\pi$
$181$	19.6382	1.45969	0.729846	+	0.683611i	$0.239592\pi$
$182$	0	0
$183$	−10.9213	−0.807324
$184$	0	0
$185$	0	0
$186$	0	0
$187$	32.2841	2.36084
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.4243	0.826631	0.413315	−	0.910588i	$-0.364371\pi$
$191$	11.4243	0.826631	0.413315	+	0.910588i	$0.364371\pi$
$192$	0	0
$193$	15.8844	1.14339	0.571693	−	0.820467i	$-0.306287\pi$
$193$	15.8844	1.14339	0.571693	+	0.820467i	$0.306287\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.8435	−1.77002	−0.885012	−	0.465567i	$-0.845850\pi$
$197$	−24.8435	−1.77002	−0.885012	+	0.465567i	$0.845850\pi$
$198$	0	0
$199$	0.460170	0.0326206	0.0163103	−	0.999867i	$-0.494808\pi$
$199$	0.460170	0.0326206	0.0163103	+	0.999867i	$0.494808\pi$
$200$	0	0
$201$	33.4047	2.35618
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.85710	−0.407096
$208$	0	0
$209$	−29.3482	−2.03006
$210$	0	0
$211$	0.622674	0.0428667	0.0214333	−	0.999770i	$-0.493177\pi$
$211$	0.622674	0.0428667	0.0214333	+	0.999770i	$0.493177\pi$
$212$	0	0
$213$	12.8726	0.882015
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.6459	−0.922104
$220$	0	0
$221$	−45.7178	−3.07531
$222$	0	0
$223$	−18.7151	−1.25326	−0.626629	−	0.779318i	$-0.715566\pi$
$223$	−18.7151	−1.25326	−0.626629	+	0.779318i	$0.715566\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.3969	1.88477	0.942385	−	0.334530i	$-0.108578\pi$
$227$	28.3969	1.88477	0.942385	+	0.334530i	$0.108578\pi$
$228$	0	0
$229$	−7.39961	−0.488980	−0.244490	−	0.969652i	$-0.578621\pi$
$229$	−7.39961	−0.488980	−0.244490	+	0.969652i	$0.578621\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.10876	0.269174	0.134587	−	0.990902i	$-0.457029\pi$
$233$	4.10876	0.269174	0.134587	+	0.990902i	$0.457029\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.41147	−0.221599
$238$	0	0
$239$	20.5449	1.32894	0.664469	−	0.747316i	$-0.268658\pi$
$239$	20.5449	1.32894	0.664469	+	0.747316i	$0.268658\pi$
$240$	0	0
$241$	1.70140	0.109597	0.0547984	−	0.998497i	$-0.482548\pi$
$241$	1.70140	0.109597	0.0547984	+	0.998497i	$0.482548\pi$
$242$	0	0
$243$	−22.3601	−1.43440
$244$	0	0
$245$	0	0
$246$	0	0
$247$	41.5604	2.64442
$248$	0	0
$249$	22.0351	1.39642
$250$	0	0
$251$	−13.9145	−0.878273	−0.439137	−	0.898420i	$-0.644716\pi$
$251$	−13.9145	−0.878273	−0.439137	+	0.898420i	$0.644716\pi$
$252$	0	0
$253$	7.25671	0.456226
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.8871	1.86431	0.932154	−	0.362062i	$-0.117927\pi$
$257$	29.8871	1.86431	0.932154	+	0.362062i	$0.117927\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	17.6878	1.09485
$262$	0	0
$263$	−19.6955	−1.21448	−0.607239	−	0.794519i	$-0.707723\pi$
$263$	−19.6955	−1.21448	−0.607239	+	0.794519i	$0.707723\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.4611	1.68059
$268$	0	0
$269$	−5.49256	−0.334888	−0.167444	−	0.985882i	$-0.553551\pi$
$269$	−5.49256	−0.334888	−0.167444	+	0.985882i	$0.553551\pi$
$270$	0	0
$271$	13.6527	0.829343	0.414671	−	0.909971i	$-0.363897\pi$
$271$	13.6527	0.829343	0.414671	+	0.909971i	$0.363897\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.6408	−0.699431	−0.349715	−	0.936856i	$-0.613722\pi$
$277$	−11.6408	−0.699431	−0.349715	+	0.936856i	$0.613722\pi$
$278$	0	0
$279$	−1.73917	−0.104121
$280$	0	0
$281$	0.270325	0.0161263	0.00806313	−	0.999967i	$-0.497433\pi$
$281$	0.270325	0.0161263	0.00806313	+	0.999967i	$0.497433\pi$
$282$	0	0
$283$	−5.64084	−0.335313	−0.167657	−	0.985845i	$-0.553620\pi$
$283$	−5.64084	−0.335313	−0.167657	+	0.985845i	$0.553620\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	41.3414	2.43185
$290$	0	0
$291$	−34.6391	−2.03058
$292$	0	0
$293$	−17.5398	−1.02469	−0.512344	−	0.858780i	$-0.671223\pi$
$293$	−17.5398	−1.02469	−0.512344	+	0.858780i	$0.671223\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.40373	−0.255531
$298$	0	0
$299$	−10.2763	−0.594294
$300$	0	0
$301$	0	0
$302$	0	0
$303$	28.1584	1.61766
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.24392	0.242213	0.121107	−	0.992640i	$-0.461356\pi$
$307$	4.24392	0.242213	0.121107	+	0.992640i	$0.461356\pi$
$308$	0	0
$309$	48.9350	2.78381
$310$	0	0
$311$	−6.04458	−0.342757	−0.171378	−	0.985205i	$-0.554822\pi$
$311$	−6.04458	−0.342757	−0.171378	+	0.985205i	$0.554822\pi$
$312$	0	0
$313$	2.69553	0.152360	0.0761801	−	0.997094i	$-0.475728\pi$
$313$	2.69553	0.152360	0.0761801	+	0.997094i	$0.475728\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.5175	−1.71404	−0.857018	−	0.515287i	$-0.827685\pi$
$317$	−30.5175	−1.71404	−0.857018	+	0.515287i	$0.827685\pi$
$318$	0	0
$319$	−21.9145	−1.22697
$320$	0	0
$321$	−23.5945	−1.31692
$322$	0	0
$323$	−53.0360	−2.95100
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.8503	−0.655323
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−23.6209	−1.29832	−0.649162	−	0.760651i	$-0.724880\pi$
$331$	−23.6209	−1.29832	−0.649162	+	0.760651i	$0.724880\pi$
$332$	0	0
$333$	13.5963	0.745071
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.4260	0.567942	0.283971	−	0.958833i	$-0.408348\pi$
$337$	10.4260	0.567942	0.283971	+	0.958833i	$0.408348\pi$
$338$	0	0
$339$	9.04963	0.491508
$340$	0	0
$341$	2.15476	0.116687
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0847	0.595059	0.297529	−	0.954713i	$-0.403837\pi$
$347$	11.0847	0.595059	0.297529	+	0.954713i	$0.403837\pi$
$348$	0	0
$349$	28.6587	1.53406	0.767032	−	0.641609i	$-0.221733\pi$
$349$	28.6587	1.53406	0.767032	+	0.641609i	$0.221733\pi$
$350$	0	0
$351$	6.23618	0.332863
$352$	0	0
$353$	−6.58347	−0.350403	−0.175201	−	0.984533i	$-0.556058\pi$
$353$	−6.58347	−0.350403	−0.175201	+	0.984533i	$0.556058\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.77601	−0.0937341	−0.0468670	−	0.998901i	$-0.514924\pi$
$359$	−1.77601	−0.0937341	−0.0468670	+	0.998901i	$0.514924\pi$
$360$	0	0
$361$	29.2131	1.53753
$362$	0	0
$363$	17.3824	0.912338
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.10700	−0.370982	−0.185491	−	0.982646i	$-0.559388\pi$
$367$	−7.10700	−0.370982	−0.185491	+	0.982646i	$0.559388\pi$
$368$	0	0
$369$	12.6800	0.660097
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.0624	1.50480	0.752398	−	0.658709i	$-0.228897\pi$
$373$	29.0624	1.50480	0.752398	+	0.658709i	$0.228897\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.0333	1.59830
$378$	0	0
$379$	−27.9145	−1.43387	−0.716935	−	0.697140i	$-0.754456\pi$
$379$	−27.9145	−1.43387	−0.716935	+	0.697140i	$0.754456\pi$
$380$	0	0
$381$	54.7033	2.80253
$382$	0	0
$383$	−19.6159	−1.00232	−0.501162	−	0.865353i	$-0.667094\pi$
$383$	−19.6159	−1.00232	−0.501162	+	0.865353i	$0.667094\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.2267	0.723183
$388$	0	0
$389$	25.9445	1.31544	0.657719	−	0.753263i	$-0.271521\pi$
$389$	25.9445	1.31544	0.657719	+	0.753263i	$0.271521\pi$
$390$	0	0
$391$	13.1138	0.663194
$392$	0	0
$393$	−8.76382	−0.442076
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.71688	0.136356	0.0681782	−	0.997673i	$-0.478281\pi$
$397$	2.71688	0.136356	0.0681782	+	0.997673i	$0.478281\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.7347	−1.28513	−0.642565	−	0.766231i	$-0.722130\pi$
$401$	−25.7347	−1.28513	−0.642565	+	0.766231i	$0.722130\pi$
$402$	0	0
$403$	−3.05138	−0.152000
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.8452	−0.834987
$408$	0	0
$409$	5.56624	0.275233	0.137616	−	0.990486i	$-0.456056\pi$
$409$	5.56624	0.275233	0.137616	+	0.990486i	$0.456056\pi$
$410$	0	0
$411$	33.2986	1.64250
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.3773	0.802001
$418$	0	0
$419$	28.1516	1.37529	0.687647	−	0.726045i	$-0.258644\pi$
$419$	28.1516	1.37529	0.687647	+	0.726045i	$0.258644\pi$
$420$	0	0
$421$	20.6705	1.00742	0.503710	−	0.863873i	$-0.331968\pi$
$421$	20.6705	1.00742	0.503710	+	0.863873i	$0.331968\pi$
$422$	0	0
$423$	6.46110	0.314150
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−64.0583	−3.09276
$430$	0	0
$431$	19.7442	0.951046	0.475523	−	0.879703i	$-0.342259\pi$
$431$	19.7442	0.951046	0.475523	+	0.879703i	$0.342259\pi$
$432$	0	0
$433$	10.5348	0.506269	0.253135	−	0.967431i	$-0.418538\pi$
$433$	10.5348	0.506269	0.253135	+	0.967431i	$0.418538\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.9213	−0.570272
$438$	0	0
$439$	9.87433	0.471276	0.235638	−	0.971841i	$-0.424282\pi$
$439$	9.87433	0.471276	0.235638	+	0.971841i	$0.424282\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.29355	−0.251504	−0.125752	−	0.992062i	$-0.540134\pi$
$443$	−5.29355	−0.251504	−0.125752	+	0.992062i	$0.540134\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−11.5817	−0.547796
$448$	0	0
$449$	−12.2918	−0.580086	−0.290043	−	0.957014i	$-0.593670\pi$
$449$	−12.2918	−0.580086	−0.290043	+	0.957014i	$0.593670\pi$
$450$	0	0
$451$	−15.7101	−0.739759
$452$	0	0
$453$	13.0547	0.613362
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.5047	−1.05273	−0.526364	−	0.850259i	$-0.676445\pi$
$457$	−22.5047	−1.05273	−0.526364	+	0.850259i	$0.676445\pi$
$458$	0	0
$459$	−7.95811	−0.371453
$460$	0	0
$461$	−4.18479	−0.194905	−0.0974526	−	0.995240i	$-0.531069\pi$
$461$	−4.18479	−0.194905	−0.0974526	+	0.995240i	$0.531069\pi$
$462$	0	0
$463$	−36.1729	−1.68110	−0.840549	−	0.541735i	$-0.817768\pi$
$463$	−36.1729	−1.68110	−0.840549	+	0.541735i	$0.817768\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.58946	−0.166100	−0.0830502	−	0.996545i	$-0.526466\pi$
$467$	−3.58946	−0.166100	−0.0830502	+	0.996545i	$0.526466\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−34.5398	−1.59151
$472$	0	0
$473$	−17.6263	−0.810458
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21.8057	0.998415
$478$	0	0
$479$	−3.50299	−0.160056	−0.0800279	−	0.996793i	$-0.525501\pi$
$479$	−3.50299	−0.160056	−0.0800279	+	0.996793i	$0.525501\pi$
$480$	0	0
$481$	23.8547	1.08768
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−30.1807	−1.36762	−0.683808	−	0.729662i	$-0.739678\pi$
$487$	−30.1807	−1.36762	−0.683808	+	0.729662i	$0.739678\pi$
$488$	0	0
$489$	0.879385	0.0397672
$490$	0	0
$491$	24.2858	1.09600	0.548002	−	0.836477i	$-0.315389\pi$
$491$	24.2858	1.09600	0.548002	+	0.836477i	$0.315389\pi$
$492$	0	0
$493$	−39.6023	−1.78360
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.276311	−0.0123694	−0.00618470	−	0.999981i	$-0.501969\pi$
$499$	−0.276311	−0.0123694	−0.00618470	+	0.999981i	$0.501969\pi$
$500$	0	0
$501$	4.37733	0.195564
$502$	0	0
$503$	0.573978	0.0255924	0.0127962	−	0.999918i	$-0.495927\pi$
$503$	0.573978	0.0255924	0.0127962	+	0.999918i	$0.495927\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	57.7965	2.56683
$508$	0	0
$509$	−37.0506	−1.64224	−0.821119	−	0.570757i	$-0.806650\pi$
$509$	−37.0506	−1.64224	−0.821119	+	0.570757i	$0.806650\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.23442	0.319408
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8.00505	−0.352062
$518$	0	0
$519$	−33.7597	−1.48189
$520$	0	0
$521$	7.30541	0.320056	0.160028	−	0.987112i	$-0.448842\pi$
$521$	7.30541	0.320056	0.160028	+	0.987112i	$0.448842\pi$
$522$	0	0
$523$	32.7297	1.43117	0.715584	−	0.698526i	$-0.246160\pi$
$523$	32.7297	1.43117	0.715584	+	0.698526i	$0.246160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.89393	0.169622
$528$	0	0
$529$	−20.0523	−0.871840
$530$	0	0
$531$	−19.0915	−0.828501
$532$	0	0
$533$	22.2472	0.963634
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−39.5039	−1.70472
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.0729	−0.991979	−0.495990	−	0.868328i	$-0.665195\pi$
$541$	−23.0729	−0.991979	−0.495990	+	0.868328i	$0.665195\pi$
$542$	0	0
$543$	49.7256	2.13393
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.2172	−0.522369	−0.261185	−	0.965289i	$-0.584113\pi$
$547$	−12.2172	−0.522369	−0.261185	+	0.965289i	$0.584113\pi$
$548$	0	0
$549$	−14.7142	−0.627986
$550$	0	0
$551$	36.0009	1.53369
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.0378	−1.06088	−0.530442	−	0.847721i	$-0.677974\pi$
$557$	−25.0378	−1.06088	−0.530442	+	0.847721i	$0.677974\pi$
$558$	0	0
$559$	24.9608	1.05573
$560$	0	0
$561$	81.7461	3.45132
$562$	0	0
$563$	4.96080	0.209073	0.104536	−	0.994521i	$-0.466664\pi$
$563$	4.96080	0.209073	0.104536	+	0.994521i	$0.466664\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.6132	−0.696461	−0.348230	−	0.937409i	$-0.613217\pi$
$569$	−16.6132	−0.696461	−0.348230	+	0.937409i	$0.613217\pi$
$570$	0	0
$571$	6.41921	0.268636	0.134318	−	0.990938i	$-0.457116\pi$
$571$	6.41921	0.268636	0.134318	+	0.990938i	$0.457116\pi$
$572$	0	0
$573$	28.9273	1.20845
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22.9632	−0.955969	−0.477984	−	0.878368i	$-0.658632\pi$
$577$	−22.9632	−0.955969	−0.477984	+	0.878368i	$0.658632\pi$
$578$	0	0
$579$	40.2208	1.67152
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−27.0164	−1.11891
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.00681	0.247927	0.123964	−	0.992287i	$-0.460439\pi$
$587$	6.00681	0.247927	0.123964	+	0.992287i	$0.460439\pi$
$588$	0	0
$589$	−3.53983	−0.145856
$590$	0	0
$591$	−62.9059	−2.58760
$592$	0	0
$593$	−15.6304	−0.641864	−0.320932	−	0.947102i	$-0.603996\pi$
$593$	−15.6304	−0.641864	−0.320932	+	0.947102i	$0.603996\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.16519	0.0476881
$598$	0	0
$599$	−27.8033	−1.13601	−0.568007	−	0.823024i	$-0.692285\pi$
$599$	−27.8033	−1.13601	−0.568007	+	0.823024i	$0.692285\pi$
$600$	0	0
$601$	−3.80604	−0.155251	−0.0776257	−	0.996983i	$-0.524734\pi$
$601$	−3.80604	−0.155251	−0.0776257	+	0.996983i	$0.524734\pi$
$602$	0	0
$603$	45.0060	1.83279
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.63278	0.147450	0.0737250	−	0.997279i	$-0.476511\pi$
$607$	3.63278	0.147450	0.0737250	+	0.997279i	$0.476511\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.3360	0.458607
$612$	0	0
$613$	6.97266	0.281623	0.140812	−	0.990036i	$-0.455029\pi$
$613$	6.97266	0.281623	0.140812	+	0.990036i	$0.455029\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.5996	1.83577	0.917885	−	0.396847i	$-0.129896\pi$
$617$	45.5996	1.83577	0.917885	+	0.396847i	$0.129896\pi$
$618$	0	0
$619$	26.0196	1.04582	0.522908	−	0.852389i	$-0.324847\pi$
$619$	26.0196	1.04582	0.522908	+	0.852389i	$0.324847\pi$
$620$	0	0
$621$	−1.78880	−0.0717821
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−74.3123	−2.96775
$628$	0	0
$629$	−30.4415	−1.21378
$630$	0	0
$631$	−12.5699	−0.500398	−0.250199	−	0.968194i	$-0.580496\pi$
$631$	−12.5699	−0.500398	−0.250199	+	0.968194i	$0.580496\pi$
$632$	0	0
$633$	1.57667	0.0626669
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.3432	0.686086
$640$	0	0
$641$	29.6786	1.17223	0.586117	−	0.810226i	$-0.300656\pi$
$641$	29.6786	1.17223	0.586117	+	0.810226i	$0.300656\pi$
$642$	0	0
$643$	1.56624	0.0617664	0.0308832	−	0.999523i	$-0.490168\pi$
$643$	1.56624	0.0617664	0.0308832	+	0.999523i	$0.490168\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.9290	0.980061	0.490030	−	0.871705i	$-0.336986\pi$
$647$	24.9290	0.980061	0.490030	+	0.871705i	$0.336986\pi$
$648$	0	0
$649$	23.6536	0.928486
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.5262	−1.54678	−0.773390	−	0.633930i	$-0.781441\pi$
$653$	−39.5262	−1.54678	−0.773390	+	0.633930i	$0.781441\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.3851	−0.717270
$658$	0	0
$659$	19.6955	0.767229	0.383614	−	0.923493i	$-0.374679\pi$
$659$	19.6955	0.767229	0.383614	+	0.923493i	$0.374679\pi$
$660$	0	0
$661$	45.3655	1.76451	0.882256	−	0.470770i	$-0.156024\pi$
$661$	45.3655	1.76451	0.882256	+	0.470770i	$0.156024\pi$
$662$	0	0
$663$	−115.762	−4.49581
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.90167	−0.344674
$668$	0	0
$669$	−47.3884	−1.83214
$670$	0	0
$671$	18.2303	0.703773
$672$	0	0
$673$	−42.8726	−1.65262	−0.826308	−	0.563218i	$-0.809563\pi$
$673$	−42.8726	−1.65262	−0.826308	+	0.563218i	$0.809563\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.4219	−0.861744	−0.430872	−	0.902413i	$-0.641794\pi$
$677$	−22.4219	−0.861744	−0.430872	+	0.902413i	$0.641794\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	71.9035	2.75535
$682$	0	0
$683$	−9.45067	−0.361620	−0.180810	−	0.983518i	$-0.557872\pi$
$683$	−9.45067	−0.361620	−0.180810	+	0.983518i	$0.557872\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.7365	−0.714841
$688$	0	0
$689$	38.2583	1.45752
$690$	0	0
$691$	41.3010	1.57116	0.785581	−	0.618758i	$-0.212364\pi$
$691$	41.3010	1.57116	0.785581	+	0.618758i	$0.212364\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−28.3901	−1.07535
$698$	0	0
$699$	10.4037	0.393505
$700$	0	0
$701$	−36.6049	−1.38255	−0.691275	−	0.722592i	$-0.742951\pi$
$701$	−36.6049	−1.38255	−0.691275	+	0.722592i	$0.742951\pi$
$702$	0	0
$703$	27.6732	1.04372
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	21.8476	0.820504	0.410252	−	0.911972i	$-0.365441\pi$
$709$	21.8476	0.820504	0.410252	+	0.911972i	$0.365441\pi$
$710$	0	0
$711$	−4.59627	−0.172373
$712$	0	0
$713$	0.875266	0.0327790
$714$	0	0
$715$	0	0
$716$	0	0
$717$	52.0215	1.94278
$718$	0	0
$719$	5.78073	0.215585	0.107793	−	0.994173i	$-0.465622\pi$
$719$	5.78073	0.215585	0.107793	+	0.994173i	$0.465622\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	4.30810	0.160220
$724$	0	0
$725$	0	0
$726$	0	0
$727$	31.0104	1.15011	0.575057	−	0.818114i	$-0.304980\pi$
$727$	31.0104	1.15011	0.575057	+	0.818114i	$0.304980\pi$
$728$	0	0
$729$	−33.8289	−1.25292
$730$	0	0
$731$	−31.8530	−1.17812
$732$	0	0
$733$	−28.5699	−1.05525	−0.527626	−	0.849477i	$-0.676918\pi$
$733$	−28.5699	−1.05525	−0.527626	+	0.849477i	$0.676918\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−55.7606	−2.05397
$738$	0	0
$739$	30.0452	1.10523	0.552615	−	0.833437i	$-0.313630\pi$
$739$	30.0452	1.10523	0.552615	+	0.833437i	$0.313630\pi$
$740$	0	0
$741$	105.235	3.86589
$742$	0	0
$743$	−10.9153	−0.400443	−0.200222	−	0.979751i	$-0.564166\pi$
$743$	−10.9153	−0.400443	−0.200222	+	0.979751i	$0.564166\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	29.6878	1.08622
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.70914	0.0623674	0.0311837	−	0.999514i	$-0.490072\pi$
$751$	1.70914	0.0623674	0.0311837	+	0.999514i	$0.490072\pi$
$752$	0	0
$753$	−35.2327	−1.28395
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.4475	−1.39740	−0.698699	−	0.715416i	$-0.746237\pi$
$757$	−38.4475	−1.39740	−0.698699	+	0.715416i	$0.746237\pi$
$758$	0	0
$759$	18.3746	0.666957
$760$	0	0
$761$	33.1729	1.20252	0.601259	−	0.799054i	$-0.294666\pi$
$761$	33.1729	1.20252	0.601259	+	0.799054i	$0.294666\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.4962	−1.20948
$768$	0	0
$769$	−17.7124	−0.638727	−0.319363	−	0.947632i	$-0.603469\pi$
$769$	−17.7124	−0.638727	−0.319363	+	0.947632i	$0.603469\pi$
$770$	0	0
$771$	75.6769	2.72544
$772$	0	0
$773$	21.5904	0.776552	0.388276	−	0.921543i	$-0.373071\pi$
$773$	21.5904	0.776552	0.388276	+	0.921543i	$0.373071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.8084	0.924682
$780$	0	0
$781$	−21.4875	−0.768884
$782$	0	0
$783$	5.40198	0.193051
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−44.1721	−1.57457	−0.787283	−	0.616592i	$-0.788513\pi$
$787$	−44.1721	−1.57457	−0.787283	+	0.616592i	$0.788513\pi$
$788$	0	0
$789$	−49.8708	−1.77545
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−25.8161	−0.916758
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.18210	−0.325247	−0.162623	−	0.986688i	$-0.551996\pi$
$797$	−9.18210	−0.325247	−0.162623	+	0.986688i	$0.551996\pi$
$798$	0	0
$799$	−14.4662	−0.511776
$800$	0	0
$801$	36.9982	1.30727
$802$	0	0
$803$	22.7784	0.803831
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13.9077	−0.489573
$808$	0	0
$809$	15.2686	0.536814	0.268407	−	0.963306i	$-0.413503\pi$
$809$	15.2686	0.536814	0.268407	+	0.963306i	$0.413503\pi$
$810$	0	0
$811$	−11.5202	−0.404530	−0.202265	−	0.979331i	$-0.564830\pi$
$811$	−11.5202	−0.404530	−0.202265	+	0.979331i	$0.564830\pi$
$812$	0	0
$813$	34.5699	1.21242
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.9564	1.01305
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.2449	−1.05555	−0.527776	−	0.849383i	$-0.676974\pi$
$821$	−30.2449	−1.05555	−0.527776	+	0.849383i	$0.676974\pi$
$822$	0	0
$823$	−25.3482	−0.883584	−0.441792	−	0.897118i	$-0.645657\pi$
$823$	−25.3482	−0.883584	−0.441792	+	0.897118i	$0.645657\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.0678	−1.21943	−0.609713	−	0.792622i	$-0.708715\pi$
$827$	−35.0678	−1.21943	−0.609713	+	0.792622i	$0.708715\pi$
$828$	0	0
$829$	36.9736	1.28415	0.642073	−	0.766644i	$-0.278075\pi$
$829$	36.9736	1.28415	0.642073	+	0.766644i	$0.278075\pi$
$830$	0	0
$831$	−29.4757	−1.02250
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.531155	−0.0183594
$838$	0	0
$839$	−27.0506	−0.933889	−0.466945	−	0.884287i	$-0.654645\pi$
$839$	−27.0506	−0.933889	−0.466945	+	0.884287i	$0.654645\pi$
$840$	0	0
$841$	−2.11793	−0.0730319
$842$	0	0
$843$	0.684488	0.0235750
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−14.2831	−0.490195
$850$	0	0
$851$	−6.84255	−0.234560
$852$	0	0
$853$	14.4834	0.495902	0.247951	−	0.968773i	$-0.420243\pi$
$853$	14.4834	0.495902	0.247951	+	0.968773i	$0.420243\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.43107	−0.322159	−0.161080	−	0.986941i	$-0.551498\pi$
$857$	−9.43107	−0.322159	−0.161080	+	0.986941i	$0.551498\pi$
$858$	0	0
$859$	4.76382	0.162540	0.0812698	−	0.996692i	$-0.474102\pi$
$859$	4.76382	0.162540	0.0812698	+	0.996692i	$0.474102\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.5107	−0.732234	−0.366117	−	0.930569i	$-0.619313\pi$
$863$	−21.5107	−0.732234	−0.366117	+	0.930569i	$0.619313\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	104.680	3.55512
$868$	0	0
$869$	5.69459	0.193176
$870$	0	0
$871$	78.9633	2.67557
$872$	0	0
$873$	−46.6691	−1.57951
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.8520	−0.906729	−0.453365	−	0.891325i	$-0.649776\pi$
$877$	−26.8520	−0.906729	−0.453365	+	0.891325i	$0.649776\pi$
$878$	0	0
$879$	−44.4124	−1.49799
$880$	0	0
$881$	−18.5084	−0.623563	−0.311781	−	0.950154i	$-0.600926\pi$
$881$	−18.5084	−0.623563	−0.311781	+	0.950154i	$0.600926\pi$
$882$	0	0
$883$	16.1821	0.544571	0.272286	−	0.962216i	$-0.412220\pi$
$883$	16.1821	0.544571	0.272286	+	0.962216i	$0.412220\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.3087	−1.42059	−0.710294	−	0.703905i	$-0.751438\pi$
$887$	−42.3087	−1.42059	−0.710294	+	0.703905i	$0.751438\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	32.1070	1.07562
$892$	0	0
$893$	13.1506	0.440070
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−26.0205	−0.868800
$898$	0	0
$899$	−2.64321	−0.0881559
$900$	0	0
$901$	−48.8221	−1.62650
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.9864	−1.49375	−0.746874	−	0.664965i	$-0.768446\pi$
$907$	−44.9864	−1.49375	−0.746874	+	0.664965i	$0.768446\pi$
$908$	0	0
$909$	37.9377	1.25831
$910$	0	0
$911$	20.5066	0.679414	0.339707	−	0.940531i	$-0.389672\pi$
$911$	20.5066	0.679414	0.339707	+	0.940531i	$0.389672\pi$
$912$	0	0
$913$	−36.7820	−1.21731
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	18.1848	0.599861	0.299930	−	0.953961i	$-0.403037\pi$
$919$	18.1848	0.599861	0.299930	+	0.953961i	$0.403037\pi$
$920$	0	0
$921$	10.7460	0.354092
$922$	0	0
$923$	30.4287	1.00157
$924$	0	0
$925$	0	0
$926$	0	0
$927$	65.9299	2.16542
$928$	0	0
$929$	13.7145	0.449959	0.224979	−	0.974364i	$-0.427769\pi$
$929$	13.7145	0.449959	0.224979	+	0.974364i	$0.427769\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.3054	−0.501077
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.5098	−1.35607	−0.678033	−	0.735031i	$-0.737167\pi$
$937$	−41.5098	−1.35607	−0.678033	+	0.735031i	$0.737167\pi$
$938$	0	0
$939$	6.82531	0.222736
$940$	0	0
$941$	−50.4407	−1.64432	−0.822160	−	0.569257i	$-0.807231\pi$
$941$	−50.4407	−1.64432	−0.822160	+	0.569257i	$0.807231\pi$
$942$	0	0
$943$	−6.38144	−0.207808
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.7006	1.61505	0.807526	−	0.589832i	$-0.200806\pi$
$947$	49.7006	1.61505	0.807526	+	0.589832i	$0.200806\pi$
$948$	0	0
$949$	−32.2567	−1.04710
$950$	0	0
$951$	−77.2731	−2.50575
$952$	0	0
$953$	−42.2458	−1.36848	−0.684238	−	0.729259i	$-0.739865\pi$
$953$	−42.2458	−1.36848	−0.684238	+	0.729259i	$0.739865\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−55.4894	−1.79372
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.7401	−0.991616
$962$	0	0
$963$	−31.7888	−1.02438
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.4260	−0.367436	−0.183718	−	0.982979i	$-0.558813\pi$
$967$	−11.4260	−0.367436	−0.183718	+	0.982979i	$0.558813\pi$
$968$	0	0
$969$	−134.292	−4.31408
$970$	0	0
$971$	−4.44387	−0.142611	−0.0713053	−	0.997455i	$-0.522716\pi$
$971$	−4.44387	−0.142611	−0.0713053	+	0.997455i	$0.522716\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.9172	0.893149	0.446574	−	0.894746i	$-0.352644\pi$
$977$	27.9172	0.893149	0.446574	+	0.894746i	$0.352644\pi$
$978$	0	0
$979$	−45.8394	−1.46503
$980$	0	0
$981$	−15.9659	−0.509750
$982$	0	0
$983$	−37.6219	−1.19995	−0.599975	−	0.800018i	$-0.704823\pi$
$983$	−37.6219	−1.19995	−0.599975	+	0.800018i	$0.704823\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.15982	−0.227669
$990$	0	0
$991$	−0.527646	−0.0167612	−0.00838061	−	0.999965i	$-0.502668\pi$
$991$	−0.527646	−0.0167612	−0.00838061	+	0.999965i	$0.502668\pi$
$992$	0	0
$993$	−59.8103	−1.89802
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.05550	−0.0334281	−0.0167141	−	0.999860i	$-0.505320\pi$
$997$	−1.05550	−0.0334281	−0.0167141	+	0.999860i	$0.505320\pi$
$998$	0	0
$999$	4.15240	0.131376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.ci.1.3		3
5.4	even	2		9800.2.a.cc.1.1		3
7.2	even	3		1400.2.q.i.1201.1	yes	6
7.4	even	3		1400.2.q.i.401.1	✓	6
7.6	odd	2		9800.2.a.cb.1.1		3
35.2	odd	12		1400.2.bh.h.249.6		12
35.4	even	6		1400.2.q.k.401.3	yes	6
35.9	even	6		1400.2.q.k.1201.3	yes	6
35.18	odd	12		1400.2.bh.h.849.6		12
35.23	odd	12		1400.2.bh.h.249.1		12
35.32	odd	12		1400.2.bh.h.849.1		12
35.34	odd	2		9800.2.a.ch.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.q.i.401.1	✓	6	7.4	even	3
1400.2.q.i.1201.1	yes	6	7.2	even	3
1400.2.q.k.401.3	yes	6	35.4	even	6
1400.2.q.k.1201.3	yes	6	35.9	even	6
1400.2.bh.h.249.1		12	35.23	odd	12
1400.2.bh.h.249.6		12	35.2	odd	12
1400.2.bh.h.849.1		12	35.32	odd	12
1400.2.bh.h.849.6		12	35.18	odd	12
9800.2.a.cb.1.1		3	7.6	odd	2
9800.2.a.cc.1.1		3	5.4	even	2
9800.2.a.ch.1.3		3	35.34	odd	2
9800.2.a.ci.1.3		3	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)

\(f(q)\)	\(=\)	\(q+2.53209 q^{3} +3.41147 q^{9} +O(q^{10})\)	\(q+2.53209 q^{3} +3.41147 q^{9} -4.22668 q^{11} +5.98545 q^{13} -7.63816 q^{17} +6.94356 q^{19} -1.71688 q^{23} +1.04189 q^{27} +5.18479 q^{29} -0.509800 q^{31} -10.7023 q^{33} +3.98545 q^{37} +15.1557 q^{39} +3.71688 q^{41} +4.17024 q^{43} +1.89393 q^{47} -19.3405 q^{51} +6.39187 q^{53} +17.5817 q^{57} -5.59627 q^{59} -4.31315 q^{61} +13.1925 q^{67} -4.34730 q^{69} +5.08378 q^{71} -5.38919 q^{73} -1.34730 q^{79} -7.59627 q^{81} +8.70233 q^{83} +13.1284 q^{87} +10.8452 q^{89} -1.29086 q^{93} -13.6800 q^{97} -14.4192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3}+O(q^{10}) \) 3 * q + 3 * q^3	\( 3 q + 3 q^{3} - 6 q^{11} - 6 q^{17} + 6 q^{19} + 3 q^{23} + 12 q^{29} - 3 q^{31} - 6 q^{33} - 6 q^{37} + 6 q^{39} + 3 q^{41} - 9 q^{43} + 18 q^{47} - 15 q^{51} + 6 q^{53} + 21 q^{57} - 3 q^{59} + 9 q^{61} + 12 q^{67} - 12 q^{69} + 9 q^{71} - 12 q^{73} - 3 q^{79} - 9 q^{81} + 21 q^{87} + 6 q^{89} + 12 q^{93} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^11 - 6 * q^17 + 6 * q^19 + 3 * q^23 + 12 * q^29 - 3 * q^31 - 6 * q^33 - 6 * q^37 + 6 * q^39 + 3 * q^41 - 9 * q^43 + 18 * q^47 - 15 * q^51 + 6 * q^53 + 21 * q^57 - 3 * q^59 + 9 * q^61 + 12 * q^67 - 12 * q^69 + 9 * q^71 - 12 * q^73 - 3 * q^79 - 9 * q^81 + 21 * q^87 + 6 * q^89 + 12 * q^93 - 21 * q^97 - 9 * q^99

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